International Association of Geodesy Symposia Fernando SansO, Series Editor
International Association of Geodesy Symposia Fernando Sansb, Series Editor
Symposium 101: Global and Regional Geodynamics Symposium 102: Global Positioning System: An Overview Symposium 103: Gravity, Gradiometry, and Gravimetry Symposium 104: Sea SurfaceTopography and the Geoid Symposium 105: Earth Rotation and Coordinate Reference Frames Symposium 106: Determination of the Geoid: Present and Future Symposium 107: Kinematic Systems in Geodesy, Surveying, and Remote Sensing Symposium 108: Application of Geodesy to Engineering Symposium 109: Permanent Satellite Tracking Networks for Geodesy and Geodynamics Symposium 11O: From Mars to Greenland: Charting Gravity with Space and Airborne Instruments Symposium 111: Recent Geodetic and Gravimetric Research in Latin America Symposium 112: Geodesy and Physics of the Earth: Geodetic Contributions to Geodynamics Symposium 113: Gravity and Geoid Symposium 114: Geodetic Theory Today Symposium 115: GPS Trends in Precise Terrestrial, Airborne, and Spaceborne Applications Symposium 116: Global Gravity Field and Ist Temporal Variations Symposium 117: Gravity, Geoid and Marine Geodesy Symposium 118: Advances in Positioning and Reference Frames Symposium 119: Geodesy on the Move Symposium 120: Towards an Integrated Global Geodetic Observation System (IGGOS) Symposium 121: Geodesy Beyond 2000: The Challenges of the First Decade Symposium 122: IV Hotine-Marussi Symposium on Mathematical Geodesy Symposium 123: Gravity, Geoid and Geodynamics 2000 Symposium 124: Vertical Reference Systems Symposium 125: Vistas for Geodesy in the New Millennium Symposium 126: Satellite Altimetry for Geodesy, Geophysics and Oceanography Symposium 127: V Hotine Marussi Symposium on Mathematical Geodesy Symposium 128: A Window on the Future of Geodesy Symposium 129: Gravity, Geoid and Space Missions Symposium 130: Dynamic Planet - Monitoring and Understanding a Dynamic Planet with Geodetic and Oceanographic Tools Symposium 131: Geodetic Deformation Monitoring: From Geophysical to Engineering Roles
Paul Tregoning Chris Rizos
(Eds.)
Dynamic Planet Monitoring and Understanding a Dynamic Planet with Geodetic and OceanographicTools lAG Symposium Cairns,Australia 22-26 August, 2005
With 795 Figures
Springer
Volume Editors
Series Editor
Dr. Paul Tregoning
Prof. Fernando Sans6
Research School of Earth Sciences The Australian National University Canberra ACT 0200 Australia
Polytechnic of Milan D.I.I.A.R.- Surveying Section Piazza Leonardo da Vinci, 32 20133 Milan Italy
Dr. Chris Rizos
University of New South Wales School of Surveying and Spatial Information Systems Sydney NSW 2052 Australia
Library of Congress Control Number: 2006936097 ISSN ISBN-10 ISBN-13
0939-9585 3-540-49349-5 Springer Berlin Heidelberg New York 978-3-540-49349-5 Springer Berlin Heidelberg New York
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32/3141/as - 5 4 3 2 1 0
Contents
Part l Joint IAG/IAPSO Papers ..........................................................
1
E K n u d s e n • O. B. A n d e r s e n . R. Forsberg • H. E F 6 h . A. V. Olesen • A. L. Vest D. Solheim • O. D. O m a n g • R. H i p k i n • A. H u n e g n a w . K. H a i n e s . R. B i n g h a m j._E Drecourt • J. A. Johannessen • H. Drange • F. S i e g i s m u n d • F. H e r n a n d e z G. L a r n i c o l . M.-H. R i o . E Schaeffer
Chapter I
Combining Altimetric/Gravimetric and Ocean Model Mean Dynamic Topography Models in the GOClNA Region . . . . . . . . . . . . . . . . . . . . . . .
3
P. K n u d s e n . C. C. Tscherning
Chapter 2
Error Characteristics of Dynamic Topography Models Derived from Altimetry and GOCEGravimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
Z. Z h a n g • Y. L u . H. H s u
Chapter 3
Detecting Ocean Currents from Satellite Altimetry, Satellite Gravity and Ocean Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
F. N. Teferle • R. M. Bingley • A. I. Waugh • A. H. D o d s o n • S. D. P. Williams T. E Baker
Chapter 4
Sea Level in the British Isles: Combining Absolute Gravimetry and Continuous GPS to Infer Vertical Land Movements at Tide Gauges ......... 23
Chapter 5
Snow Accumulation and Snowmelt Monitoring in Greenland and Antarctica .. 31
Chapter 6
H.-P. Plag E s t i m a t i n g Recent Global Sea Level C h a n g e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
Chapter 7
I. v. Sakova • G. Meyers • R. C o l e m a n On t h e L o w - F r e q u e n c y Variability in t h e Indian Ocean
..........................
47
Chapter 8
w. B o s c h . R. Savcenko S a t e l l i t e A l t i m e t r y : Multi-Mission Cross Calibration
.............................
51
S. V. N g h i e m • K. Steffen • G. N e u m a n n • R. Huff
D. N. Arabelos • G. Asteriadis • M. E. Contadakis • D. Papazachariou • S. D. Spatalas
Chapter 9
Assessment of Recent Tidal Models in the Mediterranean Sea . . . . . . . . . . . . . . . . . .
57
Chapter 10
s. M. Barbosa • M. J. Fernandes • M. E. Silva Scale-Based Comparison of Sea Level Observations in the North Atlantic from Satellite Altimetry and Tide Gauges . . . . . . . . . . . . . . . . .
63
M. J. Garc~a. B. P. G 6 m e z . F. Raicich. L. R i c k a r d s . E. B r a d s h a w . H.-P. Plag X. Z h a n g • B. L. Bye. E. Isaksen
Chapter 11
European Sea Level Monitoring: Implementation of ESEASQuality Control ..... 67 R. Dalazoana • S. R. C. de Freitas • J. C. B~iez • R. T. Luz
Chapter 12
Chapter 13
Brazilian Vertical Datum Monitoring Vertical Land Movements and Sea Level Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
M. Tervo • M. Poutanen • H. Koivula Tide Gauge Monitoring Using GPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
Y. C h u . J. Li. W. Jiang. X. Z o u . X. X u . C. Fan
Chapter 14
Determination of Inland Lake Level and Its Variations in China from Satellite Altimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
Vl
Contents M. C. M a r t i n . C. L. Villanoy
Chapter 15
Sea Surface Variability of Upwelling Area Northwest of Luzon, Philippines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
Y. Fukuda • Y. Hiraoka. K. Doi
Chapter 16
Chapter 17
An Experiment of Precise Gravity Measurements on Ice Sheet, Antarctica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
M. Amalvict • P. Willis. K. Shibuya Status of DORIS Stations in Antarctica for Precise G e o d e s y . . . . . . . . . . . . . . . . . . . . . .
94
H. H. A. Schotman • E N. A. M. Visser • L. L. A. Vermeersen
Chapter 18
High-Harmonic Gravity Signatures Related to Post-Glacial Rebound ......... 103
Part II Frontiers in the Analysis of Space Geodetic Measurements ................
Chapter 19
113
C. Urschl • G. Beutler. W. Gurtner • U. Hugentobler • S. Schaer GPS/GLONASS Orbit D e t e r m i n a t i o n Based on Combined Microwave and SLR Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
115
Chapter 20
s. Bergstrand • H.-G. Scherneck • M. Lidberg • J. M. Johansson BIFROST: Noise Properties of GPS Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
123
Chapter 21
w. Bosch Discrete Crossover Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
131
s. SchGn • H. Kutterer
Chapter 22
A Comparative Analysis of Uncertainty Modelling in GPS Data Analysis ...... 137 P. Willis. E G. Lemoine • L. Soudarin
Chapter 23
Looking for Systematic Error in Scale from Terrestrial Reference Frames Derived from DORISData .......................................................
143
A. Nothnagel • J.-H. C h o . A. Roy. R. Haas
Chapter 24
WVR Calibration Applied to European VLBI Observing Sessions ...............
152
Chapter 25
M. Kriigel • D. A n g e r m a n n Frontiers in t h e C o m b i n a t i o n of Space Geodetic Techniques
..................
158
Modifying the Stochastic Model to Mitigate GPS Systematic Errors in Relative Positioning .........................................................
166
D. B. M. Alves • J. F. G. Monico
Chapter 26
E. M. Souza • J. F. G. Monico
Chapter 27
Chapter 28
GPSAmbiguity Resolution and Validation under Multipath Effects: Improvements Using Wavelets .................................................
172
R. F. Leandro • M. C. Santos An Empirical Stochastic Model for GPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
179
R. E Leandro • C. A. U. Silva. M. C. Santos
Chapter 29
Feeding Neural Network Models with GPS Observations: a Challenging Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
186
P. J. Mendes Cerveira • T. Hobiger • R. Weber. H. Schuh
Chapter 30
Spatial Spectral Inversion of the Changing Geometry of the Earth from SOPACGPS Data ..........................................................
194
L. VGlgyesi. L. FGldv~ry. G. Csap6
Chapter 31
Improved Processing Method of UEGN-2002Gravity Network Measurements in Hungary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 32
J. Nastula • B. Kolaczek • R. Weber. H. Schuh • J. Boehm Spectra of Rapid Oscillations of Earth Rotation P a r a m e t e r s
Chapter 33
E. Wei. J. Liu. C. Shi On t h e Establishing Project of Chinese Surveying and Control Network
D e t e r m i n e d during t h e C0NT02 C a m p a i g n
....................................
for Earth-0rbit Satellite and Deep Space Detection
Chapter 34
...........................
T. K. Yeh. C. S. Chen Constructing a System to Monitor t h e Data Quality of GPS Receivers . . . . . . . . .
202
208
215 222
Contents Part III Gravity Field Determination from a Synthesisof Terrestrial, Satellite, Airborne and Altimetry Measurements ...........................
229
E G. L e m o i n e • S. B. Luthcke • D. D. Rowlands • D. S. C h i n n • S. M. Klosko C. M. Cox
Chapter 35
The Use of Mascons to ResolveTime-Variable Gravity from GRACE ............ 231 R. S c h m i d t • F. F l e c h t n e r • R. K 6 n i g . U1. M e y e r . K.-H. N e u m a y e r • Ch. Reigber M. R o t h a c h e r • S. Petrovic • S.-Y. Z h u . A. G i i n t n e r
Chapter 36
GRACE T i m e - V a r i a b l e Gravity A c c u r a c y A s s e s s m e n t
...........................
237
G. S. Vergos • V. N. Grigoriadis • I. N. Tziavos • M. G. Sideris
Chapter 37
Combination of Multi-Satellite Altimetry Data with CHAMP and GRACEEGMsfor Geoid and Sea Surface Topography Determination ..... 244
Chapter 38
A New Methodology to ProcessAirborne Gravimetry Data: Advances and Problems ........................................................
B. A. Alberts • P. D i t m a r • R. Klees
251
N. Kiihtreiber • H. A. A b d - E l m o t a a l
Chapter 39
Ideal Combination of Deflection Components and Gravity Anomalies for Precise Geoid Computation .................................................
259
D. Blitzkow • A. (2. O. (2. de Matos • J. P. Cintra
Chapter 40
SRTM Evaluation in Brazil and Argentina with Emphasis on the Amazon Region .........................................................
266
J. H u a n g • G. Fotopoulos • M. K. C h e n g • M. V 4 r o n n e a u • M. G. Sideris
Chapter 41
On t h e E s t i m a t i o n of t h e R e g i o n a l Geoid Error in C a n a d a
.....................
272
A. L6cher • K. H. Ilk
Chapter 42
A Validation Procedure for Satellite Orbits and Force Function Models Based on a New Balance Equation Approach ..................................
280
L. Z h u . C. Jekeli
Chapter 43
Combining Gravity and Topographic Data for Local Gradient Modelling ...... 288 G. A u s t e n • W. Keller
Chapter 44
Numerical Implementation of the Gravity Space Approach Proof of Concept ................................................................
296
R. Klees • T. W i t t w e r
Chapter 45
Local Gravity Field M o d e l l i n g w i t h M u l t i - P o l e W a v e l e t s . . . . . . . . . . . . . . . . . . . . . . .
303
G. S. Vergos • V. N. Grigoriadis • G. K a l a m p o u k a s • I. N. Tziavos
Chapter 46
Accuracy Assessment of the SRTM90m DTM over Greece and Its Implications to Geoid Modelling .......................................
309
C. H i r t . G. Seeber
Chapter 47
High-Resolution Local Gravity Field Determination at the Sub-Millimeter Level using a Digital Zenith Camera System ............ 316
Chapter 48
A Data-Adaptive Design of a Spherical Basis Function Network for Gravity Field Modelling .....................................................
R. Klees • T. W i t t w e r
322
K. H. I l k . A. Eicker • T. Mayer-Giirr
Chapter 49
Global Gravity Field Recovery by Merging Regional Focusing Patches: an Integrated Approach ........................................................
329
D. N. Arabelos • C. C. T s c h e r n i n g • M. Veicherts
Chapter 50
Chapter 51
External Calibration of GOCESGG Data with Terrestrial Gravity Data: a Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
337
J. P. van L o o n . J. Kusche T o w a r d s a n O p t i m a l C o m b i n a t i o n of S a t e l l i t e Data a n d Prior I n f o r m a t i o n
..........................................................
345
A. J/iggi • G. B e u t l e r . H. B o c k . U. H u g e n t o b l e r
Chapter 52
Kinematic and Highly Reduced Dynamic LEO Orbit Determination for Gravity Field Estimation ....................................................
354
VII
viii
Contents
Chapter 53
E Holota On the Combination of Terrestrial Gravity Data with Satellite Gradiometry and Airborne Gravimetry Treated in Terms of Boundary-Value Problems .....
362
Chapter 54
o. Nesvadba • E Holota • R. Klees A Direct Method and its Numerical interpretation in the Determination of the Earth's Gravity Field from Terrestrial Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
370
Chapter 55
u. Marti Comparison of High Precision Geoid Models in Switzerland . . . . . . . . . . . . . . . . . . .
377
Chapter 56
E Migliaccio. M. Reguzzoni. N. Tselfes GOCE: a Full-Gradient Simulated Solution in the Space-Wise Approach . . . . . . .
383
Chapter 57
Sz. R6zsa. Gy. T6th The Determination of the Effect of Topographic Masses on the Second Derivatives of Gravity Potential Using Various Methods . . . . . . .
Chapter 58
391
s. Bajracharya • M. G. Sideris Density Effects on Rudzki, RTM and Airy-Heiskanen Reductions
in Gravimetric Geoid Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
397
P. Ditmar • X. Liu. R. Klees • R. Tenzer • P. Moore
Chapter 59
Combined Modeling of the Earth's Gravity Field from GRACE and GOCESatellite Observations: a Numerical Study . . . . . . . . . . . . . . . . . . . . . . . . . .
403
R. Tenzer. P. Moore. O. Nesvadba
Chapter 60
Chapter 61
Analytical Solution of Newton's Integral in Terms of Polar Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
410
c. Tocho • (3. Font. M. (3. Sideris A New High-Precision Gravimetric Geoid Model for Argentina ................
416
Gy. T6th. L. VGlgyesi
Chapter 62
Chapter 63
Local Gravity Field Modeling Using Surface Gravity Gradient Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
424
Part IV Earth Processes:Geodynamics,Tides, Crustal Deformation and Temporal Gravity Changes ...............................................
431
M. Amalvict • Y. Rogister. B. Luck. J. Hinderer Absolute Gravity Measurements in the Southern Indian Ocean . . . . . . . . . . . . . . .
433
j. Beavan. L. Wallace. H. Fletcher. A. Douglas
Chapter 64
Slow Slip Events on the Hikurangi Subduction Interface, New Zealand ....... 438
Chapter 65
A Geodetic Measurement of Strain Variation across the Central Southern Alps, New Zealand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
P. H. Denys • M. Denham • C. F. Pearson
445
B. Ducarme • A. P. Venedikov • A. R. de Mesquita • C. A. de Sampaio Franga • D. S. Costa. D. Blitzkow. R. Vieira Diaz. S. R. C. de Freitas
Chapter 66
New Analysis of a 50 Years Tide Gauge Record at CananGia (SP-Brazil) with the VAV Tidal Analysis Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
453
O. Gitlein • L. Timmen
Chapter 67
Chapter 68
Atmospheric Mass Flow Reduction for Terrestrial Absolute Gravimetry in the Fennoscandian Land Uplift Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
461
T. Jahr. (3. Jentzsch • H. Letz. A. Gebauer Tilt Observations around the KTB-Site/Germany: Monitoring and Modelling of Fluid Induced Deformation of the Upper Crust of the Earth ..... 467 W. Jiang. W. Kuang • B. Chao. M. Fang. C. Cox
Chapter 69
Understanding Time-Variable Gravity Due to Core Dynamical Processes with Numerical Geodynamo Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
473
R. Kiamehr
Chapter 70
A New Height Datum for Iran Based on the Combination of the Gravimetric and Geometric Geoid Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
480
Contents R. Klees • E. A. Zapreeva • H. C. Winsemius • H. H. G. Savenije
Chapter 71
Monthly Mean Water Storage Variations by the Combination of GRACE and a Regional Hydrological Model: Application to the Zambezi River ........ 488 A. Koohzare • P. Vanfcek • M. Santos
Chapter 72
The Use of Smooth Piecewise Algebraic Approximation in the Determination of Vertical Crustal Movements in Eastern Canada ...... 496 C. Kroner. T. Jahr. M. Naujoks • A. Weise
Chapter 73
Hydrological Signals in Gravity- Foe or Friend? ...............................
504
Chapter 74
S. M. Kudryavtsev Applications of t h e KSM03 Harmonic D e v e l o p m e n t of t h e Tidal Potential ....
511
Chapter 75
j. Kusche • E. J. O. Schrama • M. J. F. Jansen Continental Hydrology Retrieval from GPS Time Series and GRACE Gravity Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
517
Chapter 76
j. Miiller • M. Neumann-Redlin • F. Jarecki • H. Denker. O. Gitlein Gravity C h a n g e s in Northern Europe As Observed by GRACE . . . . . . . . . . . . . . . . . .
523
Chapter 77
M. Naujoks • T. Jahr. G. Jentzsch • J. H. Kurz. Y. H o f m a n n Investigations a b o u t E a r t h q u a k e Swarm Areas and Processes . . . . . . . . . . . . . . . .
528
K. Nawa. K. Satake • N. Suda. K. Doi. K. Shibuya • T. Sato
Chapter 78
Sea Level and Gravity Variations after the 2004 Sumatra Earthquake Observed at Syowa Station, Antarctica .........................................
536
J. Neumeyer. T. Schmidt • C. Stoeber
Chapter 79
Improved Determination of the Atmospheric Attraction with 3D Air Density Data and Its Reduction on Ground Gravity Measurements .... 541 H.-P. Plag. G. Blewitt • C. Kreemer • W. C. H a m m o n d
Chapter 80
Chapter 81
Solid Earth Deformations Induced by the Sumatra Earthquakes of 2004-2005 ...................................................................
549
I. Prutkin • R. Klees Environmental Effects in Time-Series of Gravity M e a s u r e m e n t s at t h e
Astrometric-Geodetic 0 b s e r v a t o r i u m W e s t e r b o r k (The Netherlands) .. 5 5 7
E. Rangelova • W. van der Wal. M. G. Sideris • P. Wu
Chapter 82
Chapter 83
Numerical Models of the Rates of Change of the Geoid and Orthometric Heights over Canada .........................................
563
S. Rosat Optimal Seismic Source M e c h a n i s m s to Excite t h e Slichter Mode . . . . . . . . . . . . .
571
s. Shimada • T. Kazakami
Chapter 84
Chapter 85
Recent Dynamic Crustal Movements in the Tokai Region, Central Japan, Observed by GPS Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. Sun. S. O k u b o . G. Fu New Theory for Calculating Strains C h a n g e s Caused by
578
Dislocations
in a Spherically Symmetric Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
585
Part V Advances in the Realization of Global and Regional Reference Frames ... 593
Chapter 86
D. A n g e r m a n n • H. Drewes • M. Kriigel • B. Meisel Advances in Terrestrial Reference Frame C o m p u t a t i o n s . . . . . . . . . . . . . . . . . . . . . . .
595
Chapter 87
A. L. Fey The Status and Future of t h e International Celestial Reference Frame . . . . . . . .
603
Chapter 88
R. Ojha. A. L. Fey. D. L. Jauncey. J. E. J. Lovell • K. J. Johnston Is Scintillation t h e Key to a Better Celestial Reference Frame?
610
................
R. Ojha. A. L. Fey. P. Charlot • K. J. Johnston. D. L. Jauncey • J. E. Reynolds A. K. Tzioumis • J. E. J. Lovell • J. F. H. Quick. G. D. Nicolson • S. P. Ellingsen P. M. McCulloch • Y. Koyama
Chapter 89
Improvement and Extension of the International Celestial ReferenceFrame in the Southern Hemisphere ...................................................
616
IX
x
Contents
Chapter 90
J. B e a v a n . G. Blick L i m i t a t i o n s in t h e NZGD2000 D e f o r m a t i o n M o d e l . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
624
A. J o r d a n . P. D e n y s . G. Blick
Chapter 91
Implementing kocalised Deformation Models into a Semi-Dynamic Datum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
631
L. Sanchez
Chapter 92
Definition and Realisation of the SIRGASVertical Reference System within a Globally Unified Height System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
638
R. T. L u z . S. R. C. de Freitas • R. D a l a z o a n a • J. C. B~iez • A. S. Palmeiro
Chapter 93
Tests on Integrating Gravity and Leveling to Realize SIRGASVertical Reference System in Brazil . . . . . . . . . . . . . . . . . . . . . . . . .
646
L. P. S. F o r t e s . S. M. A. C o s t a . M. A. A. L i m a . J. A. Fazan • M. C. Santos
Chapter 94
Accessing the New SIRGAS2000 Reference Frame through a Modernized Brazilian Active Control Network . . . . . . . . . . . . . . . . . . . . . .
653
J. C. B~iez • S. R. C. de Freitas • H. Drewes • R. D a l a z o a n a • R. T. Luz
Chapter 95
Deformations Control for the Chilean Part of the SIRGAS2000 Frame ........ 660 C. C. C h a n g . H. C. H u a n g
Chapter 96
E s t i m a t i o n of H o r i z o n t a l M o v e m e n t F u n c t i o n for G e o d e t i c or M a p p i n g - 0 r i e n t e d M a i n t e n a n c e in t h e T a i w a n Area . . . . . . . . . . . . . . . . . . . . . . . .
665
M. C. Pacino • D. Del Cogliano • G. F o n t . J. M o i r a n o • P. Natalf • E. Laurfa • R. R a m o s • S. M i r a n d a
Chapter 97
Activities Related to the Materialization of a New Vertical System for Argentina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
671
F. G. Nievinski • M. C. Santos
Chapter 98
An Analysis of Errors Introduced by the Use of Transformation Grids .......... 677 Z. A l t a m i m i • X. Collilieux • C. B o u c h e r
Chapter 99
P r e l i m i n a r y Analysis in View of t h e ITRF2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
685
K. Le Bail. M. Feissel-Vernier. J.-J. Valette. W. Z e r h o u n i
Chapter 100 Long Term Consistency of Multi-Technique Terrestrial Reference Frames, a Spectral Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
692
Part Vl
GGOS:Global Geodetic Observing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701 H. Drewes
Chapter 101 Science Rationale of the Global Geodetic Observing System (GGOS) .......... 703 H.-P. Plag
Chapter 102 GGOSand Its User Requirements, Linkage, and Outreach . . . . . . . . . . . . . . . . . . . . . 711 M. P e a r l m a n • Z. A l t a m i m i • N. B e c k . R. F o r s b e r g • W. G u r t n e r S. K e n y o n . D. B e h r e n d • F. G. L e m o i n e • C. M a . C. E. N o l l . E. C. Pavlis Z. M a l k i n • A. W. M o o r e • F. H. W e b b • R. E. N e i l a n • J. C. Hies M. R o t h a c h e r . P. Willis
Chapter 103 GGOSWorking Group on Ground Networks and Communications ............ 719 H.-P. P l a g . G. Beutler • R. Forsberg • C. M a . R. Neilan • M. P e a r l m a n B. R i c h t e r . S. Z e r b i n i
Chapter 104 Linking the Global Geodetic Observing System (GGOS) to the Integrated Geodetic Observing Strategy Partnership (IGOS-P) ......... 727 W. Schlfiter • D. B e h r e n d • E. H i m w i c h • A. N o t h n a g e l • A. Niell. A. W h i t n e y
Chapter 105 IVS High Accurate Products for the Maintenance of the Global Reference Frames As Contribution to GGOS . . . . . . . . . . . . . . . . . . . . . 735 M. P e a r l m a n • C. Noll. W. G u r t n e r • R. N o o m e n
Chapter 106 The International Laser Ranging Service and Its Support for GGOS ........... 741 M. P o u t a n e n . P. K n u d s e n . M. Lilje. T. N o r b e c h . H.-P. Plag H.-G. S c h e r n e c k
Chapter 107
T h e Nordic G e o d e t i c O b s e r v i n g S y s t e m (NGOS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
749
Contents A. Niell. A. W h i t n e y . W. P e t r a c h e n k o • W. Schliiter. N. V a n d e n b e r g . H. Hase Y. K o y a m a • C. M a . H. Schuh • G. Tuccari Chapter 108
VLBI2010: a Vision for F u t u r e G e o d e t i c VLBI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
757
J. I h d e . W. S6hne • W. Schwahn • H. Wilmes • H. W z i o n t e k • T. Kliigel • W. Schliiter
Chapter 109 Combination of Different Geodetic Techniques for Signal Detection a Case Study at Fundamental Station Wettzell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
760
Part VII Systemsand Methods for Airborne Mapping, Geophysics and Hazardsand DisasterMonitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
767
H. K a h m e n . A. E i c h h o r n • M. H a b e r l e r - W e b e r
Chapter 110 A Multi-Scale Monitoring Concept for Landslide Disaster Mitigation .......... 769 H. Kutterer • C. Hesse
Chapter 111
High-Speed Laser Scanning for near Real-Time Monitoring of Structural Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
776
H. N e u n e r
Chapter 112 A Method for Modelling the Non-Stationary Behaviour of Structures in Deformation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
782
H. Z. Abidin • H. A n d r e a s • M. G a m a l • M. A. K u s u m a • M. H e n d r a s t o O. K. S u g a n d a • M. A. P u r b a w i n a t a • F. K i m a t a • I. Meilano
Chapter 113 Volcano Deformation Monitoring in Indonesia: Status, Limitations and Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
790
D. A. Grejner-Brzezinska • C. K. T o t h . S. M o a f i p o o r • E. Paska • N. Csanyi
Chapter 114 Vehicle Classification and Traffic Flow Estimation from Airborne Lidar/CCD Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
799
B. de Saint-Jean. J. V e r d u n . H. D u q u e n n e • J. P. Barriot • S. M e l a c h r o i n o s • J. Cali Fine Analysis of Lever Arm Effects in M o v i n g G r a v i m e t r y . . . . . . . . . . . . . . . . . . . . . .
809
Chapter 115
c. K. T o t h . N. C s a n y i . D. A. Grejner-Brzezinska
Chapter 116
I m p r o v i n g LiDAR-Based S u r f a c e R e c o n s t r u c t i o n Using G r o u n d Control . . . . . . .
817
G. w. R o b e r t s . X. M e n g . C. Brown
Chapter 117 The Use of GPS for Disaster Monitoring of Suspension Bridges . . . . . . . . . . . . . . . .
825
Part VIII AtmosphericStudies Using SpaceGeodeticTechniques . . . . . . . . . . . . . . . . . . . . 835 J. B o e h m • E J. M e n d e s Cerveira • H. Schuh • E Tregoning
Chapter 118 The Impact of Mapping Functions for the Neutral Atmosphere Based on Numerical Weather Models in GPS Data Analysis . . . . . . . . . . . . . . . . . . . .
837
G. Hulley • E. C. Pavlis • V. B. M e n d e s
Chapter 119 Validation of Improved Atmospheric Refraction Models for Satellite Laser Ranging (SLR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
844
T. Tanaka
Chapter 120 Correlation Analyses of Horizontal Gradients of Atmospheric Wet Delay Versus Wind Direction and Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
853
M. A q u i n o • A. D o d s o n • J. Souter • T. M o o r e
Chapter 121
Ionospheric Scintillation Effects on GPS Carrier Phase Positioning Accuracy at Auroral and Sub-Auroral Latitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
859
D. A. G r e j n e r - B r z e z i n s k a • C.-K. H o n g . P. Wielgosz • L. H o t h e m
Chapter 122 The Impact of Severe Ionospheric Conditions on the GPS Hardware in the Southern Polar Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
867
Y. B. Y u a n . D. B. W e n . J. K. O u . X. L. H u o . R. G. Yang. K. F. Z h a n g • R. Grenfel
Chapter 123 Preliminary Research on Imaging the Ionosphere Using CIT and China Permanent GPS Tracking Station Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
876
XI
xII
Contents Part IX Geodesy of the Planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
885
V. D e h a n t • T. Van H o o l s t
Chapter 124 Gravity, Rotation, and Interior of the Terrestrial Planets from Planetary Geodesy: Example of Mars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
887
G. B a l m i n o • J. C. M a r t y . J. D u r o n • O. K a r a t e k i n
Chapter 125 Mars Long Wavelength Gravity Field Time Variations: a New Solution from MGS Tracking Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
895
J. Miiller • J. G. W i l l i a m s . S. G. T u r y s h e v • P. J. Shelus
Chapter 126 Potential Capabilities of Lunar Laser Ranging for Geodesy and Relativity .... 903
Contributors
A b d - E l m o t a a l , H u s s e i n A. . (Chap. 39)
A r a b e l o s , D. N. • (Chap. 9, 50)
Civil Engineering Department, Faculty of Engineering, MiniaUniversity, Minia 61111, Egypt
Department of Geodesy and Surveying, Aristotle University of Thessaloniki, 54124 Thessaloniki, GreeceThessaloniki, Greece
A b i d i n , H. Z. • (Chap.
113)
A s t e r i a d i s , G. . ( C h a p . 9)
Department of Geodetic Engineering, Institute of Technology Bandung, J1. Ganesha 10, Bandung 40132, Indonesia,
[email protected]
Department of Geodesy and Surveying, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece A u s t e n , G. . (Chap. 44)
A l b e r t s , B. A. . (Chap. 38)
Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, PO Box 5058, 2600 GB Delft, TheNetherlands A l t a m i m i , Z. • (Chap. 99, lO3)
Institut Geographique National, ENSG/LAREG,6-8 avenue Blaise Pascal, 77455 Marne-la-Vallee, France
Stuttgart University, Geodetic Institute, Geschwister-SchollStr. 24/D, 70174 Stuttgart, Germany B d e z , ]. C . . (Chap. 12, 93, 95)
Geodetic Sciences Graduation Course (CPGCG), Federal University of Parami (UFPR), Curitiba, Brazil; and Department of Surveying, University of Concepci6n, Chile,
[email protected] B a j r a c h a r y a , S. . (Chap. 58)
Alves, D. B. M. . (Chap. 26)
Department of Cartography, Faculty of Science and Technology, S~o Paulo State University, FCT/UNESP, 305 Roberto Simonsen, Pres. Prudente, S~o Paulo, Brazil
Department of Geomatics Engineering, The University of Calgary, 2500 University Drive N. W., Calgary, Alberta, T2N 1N4, Canada,
[email protected], fax: 403-284-1980 B a k e r , T. E . (Chap. 4)
A m a l v i c t , M . . (Chap. 17, 63)
Institut de Physique du Globe de Strasbourg / t~cole et Observatoire des Sciences de la Terre (UMR 7516 CNRS-ULP), 5 rue Ren~ Descartes, 67000, Strasbourg, France,
[email protected]; and National Institute of Polar Research, 1-9-10 Kaga, Itabashi-ku, 173 8515, Tokyo, Japan A n d e r s e n , Ole B. . (Chap.
B a l m i n o , G. . (Chap.
125)
Centre National d'Etudes Spatiales, 18, Avenue Edouard Belin, 31401 Toulouse Cedex 9, France
1)
Danish National Space Center, Juliane Maries Vej 30, 2100 Kobenhavn O, Denmark A n d r e a s , H. . (Chap. 113)
Department of Geodetic Engineering, Institute of Technology Bandung, J1. Ganesha 10, Bandung 40132, Indonesia Angermann,
Proudman Oceanographic Laboratory, Joseph Proudman Building, 6 Brownlow Street, Liverpool L3 5DA, UK
D e t l e f . (Chap. 25, 86)
Deutsches Geod~itisches Forschungsinstitut (DGFI), Marstallplatz 8, 80539 Munich, Germany,
[email protected]
B a r b o s a , S. M. . (Chap.
10)
Departamento de Matematica Aplicada, Faculdade de Ci~ncias, Universidade do Porto, Rua do Campo Alegre, 687, 4169007 Porto, Portugal B a r r i o t , J. P. . (Chap.
115)
Centre National d'Etudes Spatiales, 18, Avenue Edouard Belin, 31401 Toulouse Cedex 9, France B e a v a n , ]. • (Chap. 64, 90)
GNS Science, PO Box 30368, Lower Hutt, New Zealand A q u i n o , M. . (Chap.
121)
Institute of Engineering Surveying and Space Geodesy - IESSG, The University of Nottingham, University Park, Nottingham, UK, NG7 2RD
Beck, N. . (Chap. 103)
Geodetic Survey Division - Natural Resources Canada, Ottawa, ON K1A OE9, Canada
XIV
Contributors
B e h r e n d , D. • (Chap.
103, 105)
Bye, B e n t e Lilja . (Chap.
11)
NVI, Inc./NASA Goddard Space Flight Center, Code 697, Greenbelt, MD 20771-0001, USA
Geodesi, Norwegian Mapping Authority, 3507 Honefoss, Norway
Bergstrand, Sten . (Chap. 20)
Cali, J. . (Chap.
Onsala Space Observatory, Chalmers University of Technology, 439 92 Onsala, Sweden,
[email protected]
ESGT, 1, Bd. Pythagore, 72000 Le Mans, France
115)
Chang, C. C. . (Chap. 96) Beutler, G. • (Chap. 19, 52, lO4)
Astronomical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland, get-
[email protected]
Department of Information Management, Yuda University, Miaoli 361, Taiwan, ROC Ckao, B. . (Chap. 69)
B i n g h a m , R o r y . (Chap. 1)
University of Reading, Environmental Systems Science Centre, PO Box 238, RG6 6AL Reading, UK
Space Geodesy Laboratory, NASA Goddard Space Flight Center, Greenbelt, MD 20771 Charlot, P. . (Chap. 89)
Bingley, R. M. . (Chap. 4)
Institute of Engineering Surveying and Space Geodesy, University of Nottingham, University Park, Nottingham NG7 2RD, UK Blewitt, G. . (Chap. 80)
Nevada Bureau of Mines and Geology and Seismological Laboratory, University of Nevada, Reno, Mailstop 178, Reno, NV 89557, USA
Observatoire de Bordeaux (OASU) CNRS/UMR 5804, BP89, 33270 Floirac, France Cken, C. S. . (Chap. 34)
Institute of Geomatics and Disaster Prevention Technology, Ching Yun University, No. 229, ]iansing Rd., ]hongli 320, Taiwan, R.O.C.,
[email protected] Cheng, M. K. . (Chap. 41)
Center for Space Research, University of Texas at Austin, 3925 West Braker Ln. #200, Austin, Texas 78759, USA
Blick, G.. (Chap. 90, 91)
Land Information New Zealand, Private Box 5501, Wellington, New Zealand
Chinn, D. S. . (Chap. 35)
B l i t z k o w , D. • (Chap. 40, 66)
SGT Inc., 7701 Greenbelt Road, Greenbelt, Maryland 20770, USA
Escola Polit&nica da Universidade de S~o Paulo, EPUSP-PTR, Code Postal 61548, CEP:05424-970, S~o Paulo, Brazil,
[email protected]; FAX: 55 11 30915716
Cho, l u n g - h o . (Chap. 24)
Bock, H. . (Chap. 52)
Geodetic Institute of the University of Bonn, Nussallee 17, 53115 Bonn, Germany; on leave from Korean Astronomy and Space Science Institute, 61-1, Whaam-Dong, Youseong-Gu, Taejeon, Rep. of Korea 305-348
Astronomical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland B o e h m , ]. • (Chap. 32,
Cku, Y o n g k a i . (Chap.
118)
Institute of Geodesy and Geophysics (IGG), Vienna University of Technology, Gusshausstrasse 27-29, 1040 Vienna, Austria
14)
School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan, 430079, China Cintra, J. P.. (Chap. 4o)
Bosck, W o l f g a n g . (Chap. 8, 21)
Deutsches Geod/itisches Forschungsinstitut (DGFI), Marstallplatz 8, 80539 Mtinchen, Germany
Escola Polit&nica da Universidade de S~o Paulo, EPUSP-PTR, Code Postal 61548, CEP: 05424-970, S~o Paulo, S~o Paulo, Brazil,
[email protected]; FAX: 55 11 30915716
Boucker, C. . (Chap. 99)
C o l e m a n , R. . (Chap. 7)
Conseil g~n~ral des ponts et chauss~es, tour Pascal B, 92055 La D~fense, France
School of Geography and Environmental Studies, University of Tasmania, Private Bag 78, Hobart, Tasmania, Australia, 7001
B r a d s k a w , E l i z a b e t k . (Chap.
11)
British Oceanographic Data Centre, Joseph Proudman Building, 6 Brownlow St., Liverpool, L3 5DA, UK
Collilieux, X. . (Chap. 99)
Institut Geographique National, ENSG/LAREG,6-8 avenue Blaise Pascal, 77455 Marne-la-Vallee, France
B r o w n , Chris. (Chap. 117)
C o n t a d a k i s , M. E. • (Chap. 9)
School of Engineering and Design, Brunel University West London, Uxbridge, Middlesex, UB8 3PH, UK
Department of Geodesy and Surveying, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
Contributors
Costa, D. S. • (Chap. 66)
D e n y s , P. H. . (Chap. 65, 91)
Escola. Polit&nica, Universidade de S~o Paulo, Caixa Postal 61548, 05413-001 Silo Paulo, SP, Brasil
School of Surveying, University of Otago, PO Box 56, Dunedin, New Zealand
Costa, S. M. A. • (Chap. 94)
D i t m a r , P. • (Chap. 38, 59)
Directorate of Geosciences, Brazilian Institute of Geography and Statistics, Av. Brasil 15671, Rio de Janeiro, RJ, Brazil, 21241-051
Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, PO Box 5058, 2600 GB Delft, TheNetherlands
121)
Cox, C. M. . (Chap. 35, 69)
D o d s o n , A. H. • (Chap. 4,
Raytheon at Space Geodesy Laboratory, NASA Goddard Space Flight Center; and Raytheon ITSS, 1616 McCormick Drive, Upper Marlboro, Maryland 20774, USA
Institute of Engineering Surveying and Space Geodesy - IESSG, University of Nottingham, University Park, Nottingham, NG7 2RD, UK
Csanyi, N o r a . (Chap. 114, 116)
Doi, K o i c h i r o . (Chap. 16, 78)
Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State University, 2070 Neil Avenue, Columbus, Ohio 43210
National Institute of Polar Research, Kaga 1-chome, Itabashi-ku, Tokyo 173-8515, Japan D o u g l a s , A. . (Chap. 64)
Csap6, G. . (Chap. 31)
E6tv6s Lor~ind Geophysical Institute of Hungary, 1145 Budapest, Hungary, Kolumbusz u. 17-23
School of Earth Sciences, Victoria University of Wellington, PO Box 600, Wellington, New Zealand Drange, H e l g e . (Chap. 1)
D a l a z o a n a , R . . (Chap. 12, 93, 95)
Geodetic Sciences Graduation Course, Federal University of Parami (UFPR), Curitiba, Brazil,
[email protected]
Nansen Environmental and Remote Sensing Center, Edvard Griegsvei 3a, 5059 Bergen, Norway Drecourt, J e a n - P h i l i p p e . (Chap. 1)
de Freitas, S. R. C. • (Chap. 12, 66, 93, 95)
Geodetic Sciences Graduation Course, Federal University of Parami (UFPR), Curitiba, Brazil,
[email protected]
University of Reading, Environmental Systems Science Centre, PO Box 238, RG6 6AL Reading, UK Drewes, H e r m a n n .
de M a t o s , A. C. O. C. • (Chap. 4o)
Escola Polit4cnica da Universidade de S~o Paulo, EPUSP-PTR, Code Postal 61548, CEP: 05424-970, S~o Paulo, S~o Paulo, Brazil,
[email protected], FAX: 55 11 30915716 de M e s q u i t a , A. R. . (Chap. 66)
(Chap. 86, 95,101)
Deutsches Geod~itisches Forschungsinstitut (DGFI), Marstallplatz 8, 80539 Munich, Germany,
[email protected] D u c a r m e , B. • (Chap. 66)
Chercheur qualifi4 FNRS, Observatoire Royal de Belgique, Av. Circulaire 3, 1180, Bruxelles, Belgique
Instituto Oceanogr~ifico da Universidade de S~o Paulo, SP, Brasil D u q u e n n e , H. . (Chap. de S a m p a i o F r a n f a , C. A. . (Chap. 66)
115)
Instituto Oceanogr~ifico da Universidade de S~o Paulo, SP, Brasil
IGN-LAREG, 6/8 av. Blaise Pascal, Champs sur Marne, 77455 Marne la Vall4e Cedex 2, France
D e h a n t , V.. (Chap. 124)
D u r o n , J. . (Chap. 125)
Royal Observatory of Belgium, Av. Circulaire 3, 1180, Brussels, Belgium;
[email protected]
Observatoire Royal de Belgique, 3, Avenue Circulaire, 1180 Brussels, Belgium; presently at CNES, Toulouse
Del Cogliano, D. . (Chap. 97)
E i c h h o r n , A. . (Chap.
Facultad de Ciencias Astron6micas y Geof~sicas, Universidad Nacional de La Plata, Paseo del Bosque s/n, (1900) La Plata, Argentina,
[email protected]
Institute of Geodesy and Geophysics, Vienna University of Technology, Gusshausstrasse 27-29/E1283, 1040 Vienna, Austria
D e n h a m , M. . (Chap. 65)
Eicker, A. • (Chap. 49)
School of Surveying, University of Otago, PO Box 56 Dunedin, New Zealand
Institute of Theoretical Geodesy, University of Bonn, Nussallee 17, 53115 Bonn, Germany
D e n k e r , H. • (Chap. 76)
Ellingsen, S. P. • (Chap. 89)
Institut ffir Erdmessung, University of Hannover, Schneiderberg 50, 30167 Hannover, Germany
School of Mathematics and Physics, University of Tasmania, Private Bag 37, Hobart, Tasmania 7001, Australia
110)
XV
XVI
Contributors
Fan, C h u n b o . (Chap. 14)
Fu, G u a n g y u . (Chap. 85)
School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan, 430079, China
Earthquake Research Institute, University of Tokyo, Tokyo, Japan
Fang, M. . (Chap. 69)
Fukuda, Y. . (Chap. 16)
Department of Earth and Space Sciences, Massachusetts Institute of Technology, Cambridge, MA 02136
Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan
Fazan, ]. A. . (Chap. 94)
Gamal, M. . (Chap. 113)
Directorate of Geosciences, Brazilian Institute of Geography and Statistics, Av. Brasil 15671,Rio de Janeiro, RJ, Brazil, 21241-051
Department of Geodetic Engineering, Institute of Technology Bandung, J1. Ganesha 10, Bandung 40132, Indonesia
Feissel-Vernier, M. . (Chap.
1OO)
Observatoire de Paris/SYRTE and Institut G~ographique National/LAREG, 8 Av. Blaise Pascal, Champs sur Marne, 77455 Marne la Vall~e Cedex 2, France Fernandes, M. ].. (Chap.
10)
Departamento de Matematica Aplicada, Faculdade de Ci4ncias, Universidade do Porto, Rua do Campo Alegre, 687, 4169007 Porto, Portugal Fey, A. L. . (Chap. 87, 88, 89)
u. s. Naval Observatory, 3450 Massachusetts Avenue, NW, Washington, DC 20392-5420 USA
Garcia, Maria ]esfis. (Chap. 11)
Instituto Espafiol de Oceanografla, Coraz6n de Marla, 8, 28002 Madrid, Spain Gebauer, A. . (Chap. 68)
Institute of Geosciences, Department of Applied Geophysics, University of Jena, Burgweg 11, 07749 Jena, Germany Gitlein, 0 . . (Chap. 67, 76)
Institut ffir Erdmessung, University of Hannover, Schneiderberg 50, 30167 Hannover, Germany G6mez, Begofia Pdrez. (Chap. n )
puertos del Estado, Area de Medio Fisico y Tecnologla de las Infraestructuras, Avda Parten6n, 10, 28042 Madrid, Spain
Flechtner, F.. (Chap. 36)
Department 1, Geodesy and Remote Sensing, GeoForschungsZentrum Potsdam (GFZ), Telegrafenberg A17, 14473 Potsdam, Germany
Grejner-Brzezinska, Dorota A. • (Chap. 114,116, 122)
Fletcher, H. . (Chap. 64)
Satellite Positioning and Inertial Navigation (SPIN) Lab, Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State University, 470 Hitchcock Hall, Columbus, OH 43210-1275, USA
GNS Science, PO Box 30368, Lower Hutt, New Zealand F6h, H e n n i n g P. • (Chap.
1)
Grenfel, R. . (Chap. 123)
Danish National Space Center, Juliane Maries Vej 30, 2100 Kobenhavn O, Denmark
School of Mathematical and Geospatial Sciences, RMIT University, Australia
F6ldvdry, L. . (Chap. 31)
Grigoriadis, V. N. . (Chap. 37, 46)
Department of Geodesy and Surveying, Budapest University of Technology and Economics; Research Group of Physical Geodesy and Geodynamics of the Hungarian Academy of Sciences, 1521 Budapest, Hungary
Department of Geodesy and Surveying, Aristotle University of Thessaloniki, University Box 440, 541 24, Thessaloniki, Greece, fax: +30 231 0995948 Gfintner, A. • (Chap. 36)
Font, G.. (Chap. 61, 97)
Facultad de Ciencias Astron6micas y Geofisicas,Universidad Nacional de La Plata, Paseo del Bosque s/n, (1900) La Plata, Argentina
Department 5, Geoengineering, Section 5.4, Engineering Hydrology, GeoForschungsZentrum Potsdam (GFZ), Telegrafenberg Haus F, 14473 Potsdam, Germany
Forsberg, Ren~. (Chap. 1, 103, 104)
Gurtner, W.. (Chap. 19, lO3, lO6)
Danish National Space Center, Juliane Maries Vej 30, 2100 Kobenhavn O, Denmark,
[email protected]
Astronomical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
Fortes, L. P. S. • (Chap. 94)
Haas, Ri~diger. (Chap. 24)
Directorate of Geosciences, Brazilian Institute of Geography and Statistics, Av. Brasil 15671,Rio de ]aneiro, R], Brazil, 21241-051
Onsala Space Observatory, Chalmers University of Technology, 439 92 Onsala, Sweden
Fotopoulos, G. . (Chap. 41)
Haberler- Weber, M. • (Chap. 11o)
Department of Civil Engineering, University of Toronto, 35 St. George Street, Toronto, ON, M5S1A4, Canada
Institute of Geodesy and Geophysics, Vienna University of Technology, Gusshausstrasse 27-29/E1283, 1040 Vienna, Austria
Contributors
Haines, Keith . (Chap. 1)
Hong, C.-K.. (Chap. 122)
University of Reading, Environmental Systems Science Centre, PO Box 238, RG6 6AL Reading, UK
Satellite Positioning and Inertial Navigation (SPIN) Lab, Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State University, 470 Hitchcock Hall, Columbus, OH 43210-1275, USA
H a m m o n d , W. C. • (Chap. 8o)
Nevada Bureau of Mines and Geology and Seismological Laboratory, University of Nevada, Reno, Mailstop 178, Reno, NV 89557, USA
Hothem, L. . (Chap. 122)
United States Geological Survey, 521 National Center, 12201 Sunrise Valley Dr., Reston, VA 20192 USA
Hase, H. . (Chap. 1o8)
Bundesamt ffir Kartographie und Geod~isie,Observatorio Geod4sico TIGO, Casilla 4036, Correo 3, Concepci6n, Chile Hendrasto, M. . (Chap. 113)
Directorate of Vulcanology and Geological Hazard Mitigation, J1. Diponegoro 57, Bandung, Indonesia
Hsu, Houtse. (Chap. 3)
Institute of Geodesy and Geophysics, Chinese Academy of Sciences, 340 Xudong Street, Wuhan, China, 430077; and Unite Center for Astro-geodynamics Research, CAS, Shanghai, China, 200030 Huang, H. C.. (Chap. 96)
Hernandez, Fabrice. (Chap. 1)
The 401 st Factory, Armaments Bureau, Taichung 402, Taiwan, R.O.C.
Collecte Localisation Satellites, Space Oceanography Division, 8-10 Rue Hermes - Parc Technologique Du Canal, 31526 Ramonville St. Agne, France
Huang, ].. (Chap. 41)
Hesse, Christian. (Chap. 111)
Geodetic Survey Division, CCRS, Natural Resources Canada, 615 Booth Street, Ottawa, Ontario, K1A 0E9, Canada
Geod~itischesInstitut, Universit~itHannover, Nienburger Strasse 1, 30167 Hannover, Germany,
[email protected]
Huff, R. . (Chap. 5)
Himwich, E. . (Chap. lO5)
Cooperative Institute for Research in Environmental Sciences, University of Colorado, Campus Box 216, Boulder, CO 803090216, USA
NVI, Inc./NASA Goddard Space Flight Center, Code 697, Greenbelt, MD 20771-0001, USA
Hugentobler, U. • (Chap. 19, 52)
Hinderer, ].. (Chap. 63)
Astronomical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
Institut de Physique du Globe de Strasbourg/l~cole et Observatoire des Sciences de la Terre, 5 rue Ren4 Descartes, 67000, Strasbourg, France
Hulley, G. . (Chap. 119)
Hipkin, Roger. (Chap. 1)
The University of Edinburgh, School of GeoSciences, West Mains Road, EH9 3JW Edinburgh, UK
Joint Center for Earth Systems Technology (JCET), University of Maryland Baltimore County, Baltimore, MD, USA Hunegnaw, Addisu . (Chap. 1)
Hiraoka, Y. . (Chap. 16)
The University of Edinburgh, School of GeoSciences, West Mains Road, EH9 3JW Edinburgh, UK
Geographical Survey Institute, 1, Kitasato, 305-0811, Japan
Huo, X. L. . (Chap. 123)
Hirt, Christian. (Chap. 47)
Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan 430077, China
Institut ffir Erdmessung, Universit/it Hannover, Schneiderberg 50, 30167 Hannover, Germany,
[email protected]_hannover.de, fax: +495117624006
Ihde, ].. (Chap. lO9)
Hobiger, T.. (Chap. 3o)
Vienna University of Technology, Institute of Geodesy and Geophysics, Research Unit Advanced, Geodesy, Gusshausstrasse 27-29, 1040 Vienna, Austria
Federal Agency for Cartography and Geodesy, Richard-StraussAllee 11, 60598 Frankfurt am Main, Germany Ilk, K. H. . (Chap. 42, 49)
Institute of Theoretical Geodesy, University of Bonn, Nussallee 17, 53115 Bonn, Germany
Hofmann, Y.. (Chap. 77)
ROSEN Technology GmbH, Lingen, Germany
Isaksen, Espen. (Chap. 11)
Geodesi, Norwegian Mapping Authority, 3507 Honefoss, Norway Holota, P.. (Chap. 53, 54)
Research Institute of Geodesy, Topography and Cartography, 25066 Zdiby 98, Praha-wchod, Czech Republic,
[email protected],tel.: +420 323649235,fax: +420 284890056
Ji~ggi, A. . (Chap. 52)
Astronomical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland,
[email protected]
XVII
XVIII
Contributors
]ahr, T.. (Chap. 68, 73, 77)
K a r a t e k i n , 0 . . (Chap. 125)
Institute of Geosciences, Department of Applied Geophysics, University of Jena, Burgweg 11, 07749 Jena, Germany
Observatoire Royal de Belgique, 3, Avenue Circulaire, 1180 Brussels, Belgium
Jansen, M. J. E . (Chap. 75)
K a z a k a m i , T. . (Chap. 84)
DEOS, TU Delft, Kluyverweg 1, PO Box 5058, 2600 GB Delft, The Netherlands
Solid Earth Research Group, National Research Institute for Earth Science and Disaster Prevention (NIED), 3-1 Tennodai, Tsukuba-Shi, Ibaraki-Ken 305-0006 Japan
Jarecki, F.. (Chap. 76)
Institut for Erdmessung, University of Hannover, Schneiderberg 50, 30167 Hannover, Germany
Keller, W. . (Chap. 44)
Stuttgart University, Geodetic Institute, Geschwister-SchollStr. 24/D, 70174 Stuttgart, Germany
]auncey, D. L. • (Chap. 88, 89)
Australia Telescope National Facility, CSIRO, PO Box 76, Epping, NSW 1710, Australia
K e n y o n , S. . (Chap. 103)
National Geospatial-IntelligenceAgency,Arnold, MO 63010-6238, USA
Jekeli, C. . (Chap. 43)
Division of Geodetic Science, School of Earth Sciences, Ohio State University, 125 South Oval Mall, Columbus, OH 43210
K i a m e h r , R. . (Chap. 70)
Department of Infrastructure, Division of Geodesy, Royal Institute of Technology (KTH), 100-44 Stockholm, Sweden
]entzsch, G. . (Chap. 68, 77)
Institute of Geosciences, Department of Applied Geophysics, University of Jena, Burgweg 11, 07749 Jena, Germany
K i m a t a , F. . (Chap. 113)
Research Center for Seismology and Volcanology and Disaster Mitigation (RCSVDM), Nagoya University, Japan
fiang, W. . (Chap. 69)
Joint Center for Earth Systems Technology,University of Maryland at Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21229, USA
Klees, R. • (Chap. 38, 45, 48, 54, 59, 71, 81)
Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology,Kluyverweg 1, PO Box 5058, 2600 GB Delft, TheNetherlands
fiang, W e i p i n g . (Chap. 14)
School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan, 430079, China
Klosko, S. M. . (Chap. 35)
SGT Inc., 7701 Greenbelt Road, Greenbelt, Maryland 20770, USA
] o h a n n e s s e n , J o h n n y A. . (Chap. 1)
Nansen Environmental and Remote Sensing Center, Edvard Griegsvei 3a, 5059 Bergen, Norway
Kliigel, T.. (Chap. lO9)
Federal Agency for Cartography and Geodesy, Sackenrieder Strafle 25, 93444 K6tzting, Germany
]ohansson, Jan M. . (Chap. 20)
Onsala Space Observatory, Chalmers University of Technology, 439 92 Onsala, Sweden
K n u d s e n , Per. (Chap. 1, 2, lO7)
Danish National Space Center, Juliane Maries Vej 30, 2100 Kobenhavn 0, Denmark
Johnston, K. ].. (Chap. 88, 89)
u. s. Naval Observatory, 3450 Massachusetts Avenue, NW, Washington, DC 20392-5420, USA
Koivula, H a n n u . (Chap. 13)
Finnish Geodetic Institute, Geodeetinrinne 2, 02430 Masala, Finland
Jordan, A. . (Chap. 91)
Land Information New Zealand, Private Box 5501, Wellington, New Zealand
Kolaczek, B. . (Chap. 32)
Space Research Centre of the PAS, Bartycka 18a, Warsaw, Poland
K a h m e n , H. . (Chap. 110)
KOnig, R. . (Chap. 36)
Institute of Geodesy and Geophysics, Vienna University of Technology, Gusshausstrasse 27-29/E1283, 1040 Vienna, Austria
Department 1, Geodesy and Remote Sensing, GeoForschungsZentrum Potsdam (GFZ), Telegrafenberg A17, 14473 Potsdam, Germany
K a l a m p o u k a s , G. . (Chap. 46)
Koohzare, A z a d e h . (Chap. 72)
Department of Geodesy and Surveying, Aristotle University of Thessaloniki, University Box 440, 541 24, Thessaloniki, Greece, fax: +30 231 0995948
Department of Geodesy and Geomatics Engineering, University of New Brunswick, PO Box 4400, Fredericton NB, Canada E3B 5A3
Contributors
Koyama, Y.. (Chap. 89,
108)
Leandro, R. F. • (Chap. 28, 29)
Kashima Space Research Center, Communications Research Laboratory, 893-1 Hirai, Kashima, Ibaraki 314-8501, Japan
Department of Geodesy and Geomatics Engineering, University of New Brunswick, PO Box 4400, Fredericton, New Brunswick, Canada, E3B 5A3
Kreemer, C.. (Chap. 80) Nevada Bureau of Mines and Geology and Seismological Laboratory, University of Nevada, Reno, Mailstop 178, Reno, NV 89557, USA
Lemoine, F. G. • (Chap. 23, 35, lO3) NASA Goddard Space Flight Center, Code 697, Greenbelt, Maryland 20771, USA
Kroner, C.. (Chap. 73)
Letz, H. . (Chap. 68)
Institute of Geosciences, Friedrich-Schiller-University Jena, Burgweg 11, 07749 Jena, Germany
Institute of Geosciences, Department of Applied Geophysics, University of Jena, Burgweg 11, 07749 Jena, Germany
Krfigel, M a n u e l a . (Chap. 25, 86) Deutsches Geod~itisches Forschungsinstitut (DGFI), Marstallplatz 8, 80539 Munich, Germany, kruegel@dg?.badw.de
Li, J . . ( C h a p . 14) School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan, 430079, China
Kuang, W. . (Chap. 69) Space Geodesy Laboratory, NASA Goddard Space Flight Center, Greenbelt, MD 20771 Kudryavtsev, S. M. • (Chap. 74) Sternberg Astronomical Institute of Moscow State University, 13 Universitetsky Pr., Moscow, 119992, Russia
Lidberg, M a r t i n . (Chap. 2o) Onsala Space Observatory, Chalmers University of Technology, 439 92 Onsala, Sweden; and Geodesy Division, Lantm~iteriet, 801 82 G~ivle,Sweden Lilje, M i k a e l . (Chap. lO7) National Land Survey of Sweden, 801 82 G~ivle,Sweden
Ki~htreiber, Norbert . (Chap. 39) Institute of Navigation and Satellite Geodesy, TU-Graz, Steyrergasse 30, 8010 Graz, Austria Kurz, ]. H. . (Chap. 77) Institute of Construction Materials, University of Stuttgart, Germany Kusche, ].. (Chap.
51, 75)
DEOS, TU Delft, Kluyverweg 1, PO Box 5058, 2600 GB Delft, The Netherlands Kusuma, M. A. • (Chap.
Department of Geodetic Engineering, Institute of Technology Bandung, J1. Ganesha 10, Bandung 40132, Indonesia
111)
Geodetic Institute, University of Hannover, Nienburger Strasse 1, 30167 Hannover, Germany,
[email protected] Larnicol, Gilles. (Chap.
1)
Collecte Localisation Satellites, Space Oceanography Division, 8-10 Rue Hermes - Parc Technologique Du Canal, 31526 Ramonville St. Agne, France Lauria, E. . (Chap. 97) Instituto Geogr~ifico Militar, Cabildo 391, Buenos Aires, Argentina,
[email protected] Le Bail, K. . (Chap.
Directorate of Geosciences, Brazilian Institute of Geography and Statistics, Av. Brasil 15671, Rio de Janeiro, RJ, Brazil, 21241-051 Liu, ]ingnan . (Chap. 33) President, Wuhan University, Luojia Hill, Wuhan 430072, China; and GPS Engineering Research Center, Wuhan University, 129 Luoyu Road, Wuhan 430079, China Liu, X. . (Chap. 59)
113)
Kutterer, Hansj6rg. (Chap. 22,
Lima, M. A. A. • (Chap. 94)
1OO)
Institut G6ographique National/LAREG and Observatoire de la C6te d'Azur/GEMINI (UMR 6203), 8 Av. Blaise Pascal, Champs sur Marne, 77455 Marne la Vall6e Cedex 2, France
Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands LOcher, A. . (Chap. 42) Institute of Theoretical Geodesy, University of Bonn, Nussallee 17, 53115 Bonn, Germany Lovell, ]. E. ].. (Chap. 88, 89) Australia Telescope National Facility, CSIRO, PO Box 76, Epping, NSW 1710, Australia Lu, Yang. (Chap. 3) Institute of Geodesy and Geophysics, Chinese Academy of Sciences, 340 Xudong Street, Wuhan, China, 430077; and Unite Center for Astro-geodynamics Research, CAS, Shanghai, China, 200030 Luck, B. . (Chap. 63) Institut de Physique du Globe de Strasbourg/t~cole et Observatoire des Sciences de la Terre, 5 rue Ren6 Descartes, 67000, Strasbourg, France
XIX
XX
Contributors
Luthcke, S. B. . (Chap. 35)
Meng, X i a o l i n . (Chap. 117)
NASA Goddard Space Flight Center, Code 697, Greenbelt, Maryland 20771, USA
Institute of Engineering Surveying and Space Geodesy, The University of Nottingham, University Park, Nottingham, NG7 2RD, UK
Luz, R. T. • (Chap. 12, 93, 95)
Department of Geomatics, Geodetic Sciences Graduation Course, Federal University of Paranfi (UFPR), Curitiba, Paranfi, Brazil; and Coordination of Geodesy, Brazilian Institute of Geography and Statistics (IBGE), Brazil,
[email protected]
Meyer, Ul. . (Chap. 36)
Department 1, Geodesy and Remote Sensing, GeoForschungsZentrum Potsdam (GFZ), Telegrafenberg A17, 14473 Potsdam, Germany Meyers, G. . (Chap. 7)
Ma, C.. (Chap. 103, 104, 108)
NASA Goddard Space Flight Center, Greenbelt MD 20771-0001, USA,
[email protected]
CSIRO Marine and Atmospheric Research, GPO Box 1538, Hobart, Tasmania, Australia, 7001 Migliaccio, F.. (Chap. 56)
Malkin, Z. . (Chap. lO3)
Institute of Applied Astronomy, St. Petersburg, 191187, Russia
DIIAR, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
Marti, Urs. (Chap. 55)
Miranda, S. . (Chap. 97)
Federal Office of Topography, Seftigenstrasse 264, 3084 Wabern, Switzerland,
[email protected]
Facultad de Ciencias Exactas, Hsicas y Naturales, Universidad nacional de San Juan San Juan, Argentina,
[email protected]
Martin, M. C.. (Chap. 15)
Marine Science Institute, University of the Philippines, 1101 Diliman, Quezon City, Philippines Mart),, ]. C. . (Chap. 125)
Moafipoor, Shahram . (Chap. 114)
Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State University, 2070 Neil Avenue, Columbus, Ohio 43210
Centre National d'Etudes Spatiales, 18, Avenue Edouard Belin, 31401 Toulouse Cedex 9, France
Moirano, ].. (Chap. 97)
Mayer-Gfirr, T. • (Chap. 49)
Facultad de Ciencias Astron6micas y Geoffsicas, Universidad Nacional de La Plata, Paseo del Bosque s/n, (1900) La Plata, Argentina
Institute of Theoretical Geodesy, University of Bonn, Nussallee 17, 53115 Bonn, Germany McCulloch, P. M. . (Chap. 89)
School of Mathematics and Physics, University of Tasmania, Private Bag 37, Hobart, Tasmania 7001, Australia
Monico, ]. F. G. • (Chap. 26, 27)
Department of Cartography, Faculty of Science and Technology, S~o Paulo State University, FCT/UNESP, 305 Roberto Simonsen, Pres. Prudente, S~o Paulo, Brazil Moore, A. W. . (Chap. lO3)
Meilano, I r w a n . (Chap. 113)
Research Center for Seismology and Volcanology and Disaster Mitigation (RCSVDM), Nagoya University, Japan Meisel, Barbara. (Chap. 86)
Deutsches Geod~itisches Forschungsinstitut (DGFI), Marstallplatz 8, 80539 Munich, Germany
Jet Propulsion Laboratory, California Institute of Technology, Pasadena CA 91109, USA Moore, P. • (Chap. 59, 60)
School of Civil Engineering and Geosciences, University of Newcastle upon Tyne, Newcastle upon Tyne, NE 17RU, UK Moore, T.. (Chap. 121)
Melachroinos, S. . (Chap. 115)
CNES, 18 av. E. Belin, 31401 Toulouse Cedex 9, France
Institute of Engineering Surveying and Space Geodesy - IESSG, The University of Nottingham, University Park, Nottingham, UK, NG7 2RD
Mendes, V. B. . (Chap. 119)
Laboratorio de Tectonofisica e Tectonica Experimental and Departamento de Matematica, Faculdade de Ciencias da Universidade de Lisboa, Lisbon, Portugal
Mfiller, Jiirgen . (Chap. 76,126)
Institut ffir Erdmessung, University of Hannover, Schneiderberg 50, 30167 Hannover, Germany, mueller@ife'uni-hann°ver'de
Mendes Cerveira, P.. ]. • (Chap. 30,118)
Vienna University of Technology, Institute of Geodesy and Geophysics, Research Unit Advanced, Geodesy, Gusshausstrasse 27-29, 1040 Vienna, Austria
Nastula, ].. (Chap. 32)
Space Research Centre of the PAS, Bartycka 18a, Warsaw, Poland
Contributors
Natali, P.. (Chap. 97)
Nievinski, F. G. . (Chap. 98)
Facultad de Ciencias Astron6micas y Geofisicas, Universidad Nacional de La Plata, Paseo del Bosque s/n, (1900) La Plata, Argentina
Department of Geodesy and Geomatics Engineering, University of New Brunswick, PO Box 4400, Fredericton, New Brunswick, Canada, E3B 5A3
Naujoks, M. . (Chap. 73, 77)
Noll, C. F,. . (Chap. lO3, lO6)
Institute of Geosciences, Friedrich-Schiller-University Jena, Burgweg 11, 07749 Jena, Germany
NASA Goddard Space Flight Center, Greenbelt MD 20771-0001, USA
Nawa, K a z u n a r i . (Chap. 78)
N o o m e n , R. . (Chap. 1o6)
Geological Survey of Japan, AIST,AIST Tsukuba Central 7, Higashi 1-1-1, Tsukuba, Ibaraki 305-8567, Japan
Facility of Aerospace Engineering, Delft University of Technology, 2629 HS Delft, The Netherlands
Neilan, R. E. . (Chap. lO3, lO4)
Norbech, Torbjorn. (Chap. lO7)
Jet Propulsion Laboratory, California Institute of Technology, Pasadena CA 91109, USA,
[email protected]
Norwegian Mapping Authority, 3507 Honefoss, Norway Nothnagel, A x e l . (Chap. 24, lO5)
Nesvadba, 0 . . (Chap. 54, 60)
Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, 2629 HS Delft, Kluyverweg 1, The Netherlands; and Research Institute of Geodesy, Topography and Cartography Zdiby 98, Praha v~chod, 25066, The Czech Republic; and Land Survey Office, Pod sldli~t~em 9, Praha 8, 18211, The Czech Republic
Geodetic Institute of the University of Bonn, Nussallee 17, 53115 Bonn, Germany Ojha, R . . (Chap. 88, 89)
Australia Telescope National Facility, CSIRO, PO Box 76, Epping, NSW 1710, Australia Okubo, S h u h e i . (Chap. 85)
N e u m a n n , G. . (Chap. 5)
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, 300-235, Pasadena, CA 91OO7,USA N e u m a n n - R e d l i n , M. . (Chap. 76)
Institut ffir Erdmessung, University of Hannover, Schneiderberg 50, 30167 Hannover, Germany
Earthquake Research Institute, University of Tokyo, Tokyo, Japan Olesen, A r n e V.. (Chap. 1)
Danish National Space Center, Juliane Maries Vej 30, 2100 Kobenhavn 13, Denmark Omang, Ore D. . (Chap. 1)
Neumayer, K.-H. • (Chap. 36)
Norwegian Mapping Authority, Geodetic Institute, Kartverksveien 21, 3504 Honefoss, Norway
Department 1, Geodesy and Remote Sensing, GeoForschungsZentrum Potsdam (GFZ), Telegrafenberg A17, 14473 Potsdam, Germany
Ou, J. K. . (Chap. 123)
Neumeyer, ].. (Chap. 79)
Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan 430077, China; and Graduate School of Chinese Academy of Sciences, Beijing, 100039, China
GeoForschungZentrum Potsdam, Dept. Geodesy and Remote Sensing, Telegrafenberg, 14473 Potsdam, Germany
Pacino, M. C. . (Chap. 97) Neuner, H. . (Chap. 112)
Geodetic Institute, University of Hanover, Nienburger Str. 1, 30167 Hanover, Germany
Facultad de Ciencias Exactas, Ingenierla y Agrimensura, Universidad Nacional de Rosario, Av. Pellegrini 250, (2000) Rosario, Argentina, mpacin°@fceia'unr'edu'ar
Nghiem, S. V.. (Chap. 5)
Palmeiro, A. S. . (Chap. 93)
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, 300-235, Pasadena, CA 91OO7,USA
Geodetic Sciences Graduation Course, Federal University of Parami (UFPR), Curitiba, Brazil, ale-palmeir°@yah°°'c°m'br Papazachariou, D. • (Chap. 9)
Nicolson, G. D. • (Chap. 89)
Hartebeesthoek Radio Astronomy Observatory, PO Box 443, Krugersdorp 1740, South Africa
Department of Geodesy and Surveying, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece Paska, E v a . (Chap. 114)
Niell, A. . (Chap. 105,108)
MIT Haystack Observatory, Off Route 40, Westford, MA 01886-1299, USA
Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State University, 2070 Neil Avenue, Columbus, Ohio 43210, USA
XXI
XXII
Contributors
Pavlis, E. C. • (Chap. lO3,119)
Reigber, Ch. . (Chap. 36)
NASA Goddard Space Flight Center, Greenbelt MD 20771-0001, USA; and Joint Center for Earth Systems Technology (JCET), University of Maryland Baltimore County, Baltimore, MD, USA
Department 1, Geodesy and Remote Sensing, GeoForschungsZentrum Potsdam (GFZ), Telegrafenberg A17, 14473 Potsdam, Germany
P e a r l m a n , M . . (Chap. lO3, lO4, lO6)
R e y n o l d s , ]. E. . (Chap. 89)
Harvard-Smithsonian Center for Astrophysics (CfA), Cambridge, MA 02138, USA, mpearl-
[email protected]
Australia Telescope National Facility, CSIRO, PO Box 76, Epping, NSW 1710, Australia
Pearson, C. E • (Chap. 65)
Richter, B. • (Chap. lO4)
National Geodetic Survey, 2300 S Dirksen Parkway, Springfield IL 62764, USA
Bundesamt ffir Kartographie und Geod~isie,Frankfurt a. M., Germany,
[email protected]
P e t r a c h e n k o , W. . (Chap. lO8)
Rickards, L e s l e y . (Chap.
Geodetic Survey Division, Natural Resources Canada, Dominion Radio Astrophysical Observatory (DRAO), Box 248, Penticton, B. C., V2A 6K3, Canada
British Oceanographic Data Centre, Joseph Proudman Building, 6 Brownlow St., Liverpool, L3 5DA, UK
11)
Ries, J. C. . (Chap. 103) Petrovic, S. . (Chap. 36)
Department 1, Geodesy and Remote Sensing, GeoForschungsZentrum Potsdam (GFZ), Telegrafenberg A17, 14473 Potsdam, Germany Plag, H a n s - P e t e r . (Chap. 6,11, 80, lO2, lO4, lO7)
Nevada Bureau of Mines and Geology and Seismological Laboratory, University of Nevada, Reno, Mailstop 178, Reno, NV 89557, USA,
[email protected] P o u t a n e n , M a r k k u . (Chap. 13, 107)
Center for Space Research, The University of Texas, Austin TX 78712, USA Rio, M a r i e - H e l e n e . (Chap.
1)
Collecte Localisation Satellites, Space Oceanography Division, 8-10 Rue Hermes - Parc Technologique Du Canal, 31526 Ramonville St. Agne, France Roberts, G e t h i n W y n . (Chap. 117)
Institute of Engineering Surveying and Space Geodesy, The University of Nottingham, University Park, Nottingham, NG7 2RD, UK
Finnish Geodetic Institute, Geodeetinrinne 2, 02430 Masala, Finland Rogister, Y.. (Chap. 63) P r u t k i n , I. . (Chap. 81)
Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, 2629 HS Delft, Kluyverweg 1, The Netherlands
Institut de Physique du Globe de Strasbourg/l~cole et Observatoire des Sciences de la Terre, 5 rue Ren4 Descartes, 67000, Strasbourg, France Rosat, S. . (Chap. 83)
P u r b a w i n a t a , M. A. . (Chap. 113)
Directorate of Vulcanology and Geological Hazard Mitigation, J1. Diponegoro 57, Bandung, Indonesia
National Astrogeodynamics Observatory, Mizusawa, Iwate, 023-0861 Japan,
[email protected] R o t h a c h e r , M . . (Chap. 36, lO3)
Quick, J. F. H. . (Chap. 89)
Hartebeesthoek Radio Astronomy Observatory, PO Box 443, Krugersdorp 1740, South Africa
Department 1, Geodesy and Remote Sensing, GeoForschungsZentrum Potsdam (GFZ),TelegrafenbergA17, 14473Potsdam, Germany R o w l a n d s , D. D. • (Chap. 35)
Raicich, Fabio . (Chap. 11)
Consiglio Nazionale delle Ricerche, Istituto di Scienze Marine, Viale Romolo Gessi, 2, 34123, Trieste, Italy
NASA Goddard Space Flight Center, Code 697, Greenbelt, Maryland 20771, USA Roy, A l a n . (Chap. 24)
R a m o s , R. . (Chap. 97)
Instituto Geogr~ificoMilitar, Cabildo 391, Buenos Aires, Argentina
Max-Planck-Institute for Radio Astronomy, Auf dem Hfige169, 53121 Bonn, Germany
R a n g e l o v a , E. . (Chap. 82)
R6zsa, Sz. . (Chap. 57)
Department of Geomatics Engineering, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada, T2N 1N4
Department of Geodesy and Surveying, Budapest University of Technology and Economics, 1521 Budapest, PO Box 91, Hungary,
[email protected]
R e g u z z o n i , M. . (Chap. 56)
Geophysics of the Lithosphere Dept., Italian National Institute of Oceanography and Applied Geophysics (OGS) c/o Politecnico di Milano, Polo Regionale di Como,Via Valleggio,11, 22100 Como, Italy
Saint-Jean, B. d e . (Chap.
115)
IGN-LAREG, 6/8 av. Blaise Pascal, Champs sur Marne, 77455 Marne la Vall4e Cedex 2, France
Contributors
Sakova, L V.. (Chap. 7)
Schrama, E. ]. 0 . . (Chap. 75)
CSIRO Marine and Atmospheric Research, GPO Box 1538, Hobart, Tasmania, Australia, 7001,
[email protected]
DEOS, TU Delft, Kluyverweg 1, PO Box 5058, 2600 GB Delft, The Netherlands
Sanchez, Laura. (Chap. 92)
Schuh, 1-1. • (Chap. 3o, 32, lO8,118)
Deutsches Geod~itisches Forschungsinstitut (DGFI), Marstallplatz 8, 80539 Munich, Germany
Vienna University of Technology, Institute of Geodesy and Geophysics, Research Unit Advanced, Geodesy, Gusshausstrasse 27-29, 1040 Vienna, Austria
Santos, Marcelo C. • (Chap. 28, 29, 72, 94, 98)
Department of Geodesy and Geomatics Engineering, University of New Brunswick, PO Box 4400, Fredericton, New Brunswick, Canada, E3B 5A3
Schwahn, W. . (Chap. lO9)
Federal Agency for Cartography and Geodesy, Richard-StraussAllee 11, 60598 Frankfurt am Main, Germany
Satake, Kenji . (Chap. 78)
Seeber, Gfinter. (Chap. 47)
Geological Survey of Japan, AIST,AIST Tsukuba Central 7, Higashi 1-1-1, Tsukuba, Ibaraki 305-8567, Japan
Institut ffir Erdmessung, Universit~it Hannover, Schneiderberg 50, 30167 Hannover, Germany
Sato, Tadahiro. (Chap. 78)
Shelus, Peter ].. (Chap. 126)
National Astronomical Observatory, Hoshigaoka 2-12, Mizusawa, Iwate 023-0861, Japan
University of Texas at Austin, Center for Space Research, 3925 W. Braker Lane, Austin, TX 78759, USA
Savcenko, R o m a n . (Chap. 8)
Shi, Chuang. (Chap. 33)
Deutsches Geod~itisches Forschungsinstitut (DGFI), Marstallplatz 8, 80539 Mfinchen, Germany
GPS Engineering Research Center, Wuhan University, 129 Luoyu Road, Wuhan 430079, China
Savenije, H. H. G. . (Chap. 71)
Shibuya, K a z u o . (Chap. 17, 78)
Department of Water Management, Delft University of Technology, Stevinweg 1, 2628 CN, Delft, The Netherlands
National Institute of Polar Research, 1-9-10 Kaga, Itabashi-ku, 173 8515, Tokyo, Japan
Schaeffer, Philippe. (Chap.
1)
Collecte Localisation Satellites, Space Oceanography Division, 8-10 Rue Hermes - Parc Technologique Du Canal, 31526 Ramonville St. Agne, France
Shimada, S. . (Chap. 84)
Solid Earth Research Group, National Research Institute for Earth Science and Disaster Prevention (NIED), 3-1 Tennodai, Tsukuba-Shi, Ibaraki-Ken 305-0006 Japan,
[email protected], tel.: +81-29-863-7622, fax: +81-29-854-0629
Schaer, S. . (Chap. 19)
Astronomical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
Sideris, M. G.. (Chap. 37, 41, 58, 61, 82)
Department of Geomatics Engineering, The University of Calgary, 2500 University Drive N. W., Calgary, Alberta, T2N 1N4, Canada
Scherneck, Hans-Georg. (Chap. 2o, lO7)
Onsala Space Observatory, Chalmers University of Technology, 439 92 Onsala, Sweden Schliiter, W. . (Chap.
Siegismund, Frank. (Chap. 1)
Nansen Environmental and Remote Sensing Center, Edvard Griegsvei 3a, 5059 Bergen, Norway
105,108,109)
Bundesamt for Kartographie und Geod~isie,Fundamentalstation Wettzell, Sackenrieder Strasse 25, 93444 K6tzting, Germany
Silva, C. A. U. . (Chap. 29)
Department of Civil construction, Federal Technologic Learning Centre, Bel4m, Para, Brazil
Schmidt, T.. (Chap. 36, 79)
Department 1, Geodesy and Remote Sensing, GeoForschungsZentrum Potsdam (GFZ),TelegrafenbergA17,14473 Potsdam, Germany Sch6n, S. . (Chap.
22)
Engineering Geodesy and Measurement Systems, Graz University of Technology (TUG), Steyrergasse 30, 8010 Graz, Austria Schotman, H. H. A. . (Chap.
18)
Delft Institute of Earth Observation and Space Systems (DEOS), Aerospace Engineering,Delft Universityof Technology,Kluyverweg1, 2629 HS Delft,The Netherlands; and SRON Netherlands Institute for Space Research,Sorbonnelaan 2, 3584 CA, Utrecht, The Netherlands
Silva, M. E. • (Chap. lO)
Departamento de Matematica Aplicada, Faculdade de Ci4ncias, Universidade do Porto, Rua do Campo Alegre, 687, 4169007 Porto, Portugal S6hne, W. . (Chap. 109)
Federal Agency for Cartography and Geodesy, Richard-StraussAllee 11, 60598 Frankfurt am Main, Germany Solheim, Dag. (Chap. 1)
Norwegian Mapping Authority, Geodetic Institute, Kartverksveien 21, 3504 Honefoss, Norway
XXIII
XXIV
Contributors
S o u d a r i n , L. . (Chap. 23)
Tocho, C.. (Chap. 61)
Collecte Localisation Satellite, parc technologique du canal, 31526 Ramonville Saint-Agne, France
Facultad de Ciencias Astrondmicas y Geoflsicas, Paseo del Bosque s/n, 1900 La Plata, Argentina,
[email protected]
Souter, ].. (Chap. 121)
Toth, Charles K. . (Chap. 114,116)
Institute of Engineering Surveying and Space Geodesy- IESSG, The University of Nottingham, University Park, Nottingham, UK, NG7 2RD
Center for Mapping, The Ohio State University, 1216 Kinnear Road, Columbus, Ohio 43212,
[email protected] T6th, Gy. . (Chap. 57, 62)
Souza, E. M. • (Chap. 27)
Department of Cartography, S~o Paulo State University UNESP, Roberto Simonsen, 350, Pres. Prudente, SP, Brazil Spatalas, S. D. . (Chap. 9)
Department of Geodesy and Surveying, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece Steffen, K. . (Chap. 5)
Cooperative Institute for Research in Environmental Sciences, University of Colorado, Campus Box 216, Boulder, CO 80309-0216, USA
Department of Geodesy and Surveying, Budapest University of Technology and Economics, 1521 Budapest, PO Box 91, Hungary; and Research Group of Physical Geodesy and Geodynamics of the Hungarian Academy of Sciences, 1521 Budapest, Hungary, Mtiegyetem rkp. 3 rregoning, P. . (Chap.
118)
Research School of Earth Sciences, The Australian National University, Canberra, ACT,Australia Tscherning, Carl C h r i s t i a n . (Chap. 2, 50)
Department of Geophysics, University of Copenhagen, Juliane Maries Vej 30, 2100 Copenhagen, Denmark
Stoeber, C. . (Chap. 79)
Institute for Geodesy,Technical University Berlin, Germany Suda, N a o k i . (Chap. 78)
Hiroshima University, Kagamiyama 1-3-1, Higashi-Hiroshima, Hiroshima 739-8526, Japan S u g a n d a , O. K. . (Chap. 113)
Directorate of Vulcanology and Geological Hazard Mitigation, J1. Diponegoro 57, Bandung, Indonesia Sun, W e n k e . (Chap. 85)
Tselfes, N. . (Chap. 56)
DIIAR, Politecnico di Milano, Polo Regionale di Como, Via Valleggio, 11, 22100 Como, Italy Tuccari, G. . (Chap. lO8)
Istituto di Radioastronomia/INAF, Contrada Renna, PO Box 141, Noto (SR), 96017, Italy Turyshev, Slava G. • (Chap.
126)
Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109, USA
Earthquake Research Institute, University of Tokyo, Tokyo, Japan,
[email protected]
Tziavos, I. N. • (Chap. 37, 46)
Tanaka, Torao . (Chap. 12o)
Department of Geodesy and Surveying, Aristotle University of Thessaloniki, University Box 440, 541 24, Thessaloniki, Greece, fax: +30 231 0995948
Department of Environmental Science and Technology, Meijo University, 1-501 Shiogamaguchi, Tenpaku-ku, Nagoya 468-8502, Japan Teferle, F. N. . (Chap. 4)
Institute of Engineering Surveying and Space Geodesy, University of Nottingham, University Park, Nottingham NG7 2RD, UK
Tzioumis, A. K. • (Chap. 89)
Australia Telescope National Facility, CSIRO,PO Box 76, Epping, NSW 1710, Australia Urschl, C. • (Chap. 19)
Astronomical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland, claudia'urschl@aiub'unibe'ch
Tenzer, R . . (Chap. 59, 6o)
School of Civil Engineering and Geosciences, University of Newcastle upon Tyne, Newcastle upon Tyne, NE17RU,United Kingdom
Valette, J.-J. . (Chap. 1oo)
Collecte Localisation Satellites (CLS), 8-10 rue Hermbs, 31526 Ramonville-Saint-Agne Cedex, France
Tervo, M a a r i a . (Chap. 13)
van der Wal, W. . (Chap. 82)
Finnish Geodetic Institute, Geodeetinrinne 2, 02430 Masala, Finland
Department of Geomatics Engineering, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada, T2N 1N4
124)
T i m m e n , L. . (Chap. 67)
Van Hoolst, T.. (Chap.
Institut ftir Erdmessung, University of Hannover, Schneiderberg 50, 30167 Hannover, Germany
Royal Observatory of Belgium, Av. Circulaire 3, 1180 Brussels, Belgium,
[email protected]
Contributors
van Loon, ]. P. • (Chap. 51)
V6lgyesi, L . . (Chap. 31, 62)
DEOS, TU Delft, Kluyverweg 1, PO Box 5058, 2600 GB Delft, The Netherlands
Department of Geodesy and Surveying, Budapest University of Technology and Economics; and Research Group of Physical Geodesy and Geodynamics of the Hungarian Academy of Sciences, 1521 Budapest, Hungary, Mfiegyetem rkp. 3
Vandenberg, N. . (Chap. 108)
NVI, Inc./NASA Goddard Space Flight Center, Code 697, Greenbelt, MD 20771-0001, USA
Wallace, L. . (Chap. 64)
GNS Science, PO Box 30368, Lower Hutt, New Zealand Vanicek, P e t r . (Chap. 72)
Department of Geodesy and Geomatics Engineering, University of New Brunswick, PO Box 4400, Fredericton NB, Canada E3B 5A3
Waugh, A. I. . (Chap. 4)
Institute of Engineering Surveying and Space Geodesy, University of Nottingham, University Park, Nottingham NG7 2RD, UK
Veicherts, M. . (Chap. 50)
Webb, F. H. . (Chap. lO3)
Department of Geophysics, University of Copenhagen, Juliane Maries Vej 30, 2100 Copenhagen, Denmark
Jet propulsion Laboratory, California Institute of Technology, Pasadena CA 91109, USA
Venedikov, A. P. • (Chap. 66)
Weber, R . . (Chap. 30, 32)
Geophysical Institute and Central Laboratory on Geodesy, Acad. G. Bonchev Str., Block 3, Sofia 1113
Vienna University of Technology, Institute of Geodesy and Geophysics, Research Unit Advanced, Geodesy, Gusshausstrasse 27-29, 1040 Vienna, Austria
Verdun, ].. (Chap. 115)
IGN-LAREG, 6/8 av. Blaise Pascal, Champs sur Marne, 77455 Marne la Vall~e Cedex 2, France
Wei, E r h u . (Chap. 33)
School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, China
Vergos, G. S. . (Chap. 37, 46) Weise, A. . (Chap. 73)
Department of Geodesy and Surveying, Aristotle University of Thessaloniki, University Box 440, 541 24, Thessaloniki, Greece,
[email protected], fax: +30 231 0995948
Institute of Geosciences, Friedrich-Schiller-University Jena, Burgweg 11, 07749 Jena, Germany; and Society for the Advancement of Geosciences Jena, H61derlinweg 6, 07749 Jena
Vermeersen, L. L. A. • (Chap. 18)
Wen, D. B. . (Chap. 123)
Delft Institute of Earth Observation and Space Systems (DEOS), Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands
Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan 430077, China; and Graduate School of Chinese Academy of Sciences, Beijing, 100039, China W h i t n e y , A. . (Chap. 105,108)
V d r o n n e a u , M. . (Chap. 41)
Geodetic Survey Division, CCRS, Natural Resources Canada, 615 Booth Street, Ottawa, Ontario, K1A 0E9, Canada Vest, A n n e L. . (Chap. 1)
Danish National Space Center, Juliane Maries Vej 30, 2100 Kobenhavn 0, Denmark
MIT Haystack Observatory, Off Route 40, Westford, MA 01886-1299, USA Wielgosz, P. . (Chap. 122)
Satellite Positioning and Inertial Navigation (SPIN) Lab, Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State University, 470 Hitchcock Hall, Columbus, OH 43210-1275, USA
Vieira Diaz, R. . (Chap. 66)
Williams, ] a m e s G. . (Chap. 126)
Instituto de Astronomfa y Geodesia (CSIC-UCM), Facultad de Matem~iticas, Plaza de Ciencias, 3, 28040 Madrid, Spain
Jet propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109, USA Williams, S. D. P. • (Chap. 4)
Villanoy, C. L. . (Chap. 15)
Marine Science Institute, University of the Philippines, 1101 Diliman, Quezon City, Philippines
Proudman Oceanographic Laboratory, Joseph Proudman Building, 6 Brownlow Street, Liverpool L3 5DA, UK Willis, P... (Chap. 17, 23, lO3)
Visser, P. N. A. M. • (Chap. 18)
Delft Institute of Earth Observation and Space Systems (DEOS), Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands
Institut G~ographique National, Direction Technique, 2, avenue Pasteur, BP 68, 94160 Saint-MandG France; and Jet Propulsion Laboratory, California Institute of Technology, MS 238-600, 4800 Oak Grove Drive, Pasadena CA 91109, USA, p ascal'willis@ign'ff
XXV
XXVI
Contributors
W i l m e s , H. . (Chap. lO9)
Z a p r e e v a , E. A. . (Chap. 71)
Federal Agency for Cartography and Geodesy, Richard-Strauss-Allee 11, 60598 Frankfurt am Main, Germany
Delft Institute of Earth Observation and Space Systems (DEOS), Physical and Space Geodesy group, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands
W i n s e m i u s , H. C. • (Chap. 71)
Department of Water Management, Delft University of Technology, Stevinweg 1, 2628 CN Delft, The Netherlands
Z e r b i n i , S. . (Chap. lO4)
Department of Physics, University of Bologna, Bologna, Italy,
[email protected]
W i t t w e r , T.. (Chap. 45, 48)
Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, 2629 HS Delft, Kluyverweg 1, The Netherlands
Z e r h o u n i , W. . (Chap.
1OO)
Centre National des Techniques Spatiales (CNTS) BP 13, Arzew, 31200, Oran, Algeria
Wu, P. . (Chap. 82)
Z h a n g , K. E . (Chap. 123)
Department of Geology and Geophysics, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada, T2N 1N4
School of Mathematical and Geospatial Sciences, RMIT University,Australia
W z i o n t e k , H. • (Chap. lO9)
Z h a n g , X i u h u a . (Chap.
Federal Agency for Cartography and Geodesy, Richard-Strauss-Allee 11, 60598 Frankfurt am Main, Germany
Geodesi, Norwegian Mapping Authority, 3507 Honefoss, Norway
11)
Z h a n g , Z i z h a n . (Chap. 3) X u , X i n y u . (Chap. 14)
School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan, 430079, China
Institute of Geodesy and Geophysics, Chinese Academy of Sciences, 340 Xudong Street, Wuhan, China, 430077; and Graduate School of the Chinese Academy of Sciences, Beijing, China, 100049
Yang, R. G. . (Chap. 123)
Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan 430077, China; and Graduate School of Chinese Academy of Sciences, Beijing, 100039, China
Z h u , L. . (Chap. 43)
Division of Geodetic Science, School of Earth Sciences, Ohio State University, 125 South Oval Mall, Columbus, OH 43210
Yeh, T. K. . (Chap. 34)
Z h u , S.-Y.. . (Chap. 36)
Institute of Geomatics and Disaster Prevention Technology, Ching Yun University, No. 229, Jiansing Rd., Jhongli 320, Taiwan, R. O. C.,
[email protected]
Department 1, Geodesy and Remote Sensing, GeoForschungsZentrum Potsdam (GFZ), Telegrafenberg A17, 14473 Potsdam, Germany
Yuan, Y. B. . (Chap. 123)
Z o u , X i a n c a i . (Chap. 14)
Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan 430077, China
School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan, 430079, China
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Part l Joint IAG/IAPSOPapers Chapter 1
Combining Altimetric/Gravimetric and Ocean Model Mean Dynamic Topography Models in the GOCINA Region
Chapter 2
Error Characteristics of Dynamic Topography Models Derived from Altimetry and GOCE Gravimetry
Chapter 3
Detecting Ocean Currents from Satellite Altimetry, Satellite Gravity and Ocean Data
Chapter 4
Sea Level in the British Isles: Combining Absolute Gravimetry and Continuous GPS to Infer Vertical Land Movements at Tide Gauges
Chapter 5
Snow Accumulation and Snowmelt Monitoring in Greenland and Antarctica
Chapter 6
Estimating Recent Global Sea Level Changes
Chapter 7
On the Low-Frequency Variability in the Indian Ocean
Chapter 8
Satellite Altimetry: Multi-Mission Cross Calibration
Chapter 9
Assessment of Recent Tidal Models in the Mediterranean Sea
Chapter 10
Scale-Based Comparison of Sea Level Observations in the North Atlantic from Satellite Altimetry and Tide Gauges
Chapter 11
European Sea Level Monitoring: Implementation of ESEAS Quality Control
Chapter 12
Brazilian Vertical Datum Monitoring Vertical Land Movements and Sea Level Variations
Chapter 13
Tide Gauge Monitoring Using GPS
Chapter 14
Determination of Inland Lake Level and Its Variations in China from Satellite Altimetry
Chapter 15
Sea Surface Variability of Upwelling Area Northwest of Luzon, Philippines
Chapter 16
An Experiment of Precise Gravity Measurements on Ice Sheet, Antarctica
Chapter 17
Status of DORIS Stations in Antarctica for Precise Geodesy
Chapter 18
High-Harmonic Gravity Signatures Related to Post-Glacial Rebound
Chapter I
Combining altimetric/gravimetric and ocean model mean dynamic topography models in the GOCINA region Per Knudsen, Ole B. Andersen, Rend Forsberg, Henning P. F6h, Arne V. Olesen, Anne L. Vest Danish National Space Center, Juliane Maries Vej 30, 2100 Kobenhavn O, Denmark. Dag Solheim, Ove D. Omang, Norwegian Mapping Authority, Geodetic Institute, Kartverksveien 21, 3504 Honefoss, Norway Roger Hipkin, Addisu Hunegnaw, The University of Edinburgh, School of GeoSciences, West Mains Road, EH9 3JW Edinburgh, UK Keith Haines, Rory Bingham, Jean-Philippe Drecourt, University of Reading, Environmental Systems Science Centre, P.O.Box 238, RG6 6AL Reading, UK Johnny A. Johannessen, Helge Drange, Frank Siegismund, Nansen Environmental and Remote Sensing Center, Edvard Griegsvei 3a, 5059 Bergen, Norway Fabrice Hernandez, Gilles Larnicol, Marie-Helene Rio, and Philippe Schaeffer Collecte Localisation Satellites, Space Oceanography Division, 8-10 Rue Hermes - Parc Technologique Du Canal, 31526 Ramonville St. Agne, France
Abstract. Initially, existing mean dynamic topography (MDT) models were collected and reviewed. The models were corrected for the differences in averaging period using the annual anomalies computed from satellite altimetry. Then a composite MDT was derived as the mean value in each grid node together with a standard deviation to represent its error. A new synthetic MDT was obtained from the new mean sea surface (MSS) KMS04 combined with a regional geoid updated using GRACE gravity and gravimetric data from a recent airborne survey. Compared with the composite MDT the synthetic MDT showed very similar results. Then combination methods were tested for the computation of MDT models from gravity data and MSS data. Both a rigorous and an iterative combination method have been tested in the GOCINA region. At this stage, the iterative combination method with its efficient handling of large data sets covering the whole region appears to give the best solution. Naturally, the errors associated with the MDT can be obtained using the rigorous method only.
1 Background The ocean transport through the straits between Greenland and the UK is known to play an important role in the global circulation as well as on the climate in Northern Europe. Warm Gulf Stream water flows into the Nordic seas and feeds the formation of heavy, cold bottom water that returns back into the Atlantic Ocean. The central quantity associated with the ocean circulation and transport is the mean dynamic topography (MDT), which is the difference between the mean sea surface (MSS) and the geoid. The MDT provides the absolute reference surface for the ocean circulation.
A major goal of the EU project GOCINA (Geoid and Ocean Circulation In the North Atlantic) is to determine an accurate mean dynamic topography model in the region between Greenland and the UK. The improved determination of the mean circulation will advance the understanding of the role of the ocean mass and heat transport in climate change.
4
Knudsenet al. $0°W
40'W
30°W
2O ° W
1O=W
O°
[O°E
2O°L
70~N
7 0~N
• ...:..
~0=N
6 0=N
:~0°N
$0°N
50~W
40~W
3O°W
20~W
1O~W
O~
1O~E
2O°~E
Figure 1. The region of interest in the GOCINA project.
Through the first phases of the GOCINA project the main focus has been on the determinations of the geoid the mean sea surface and the mean dynamic topography using the individual techniques and on the assessment of the models in the region of interest (Figure 1).
2 Description of data Existing mean dynamic topography (MDT) models were collected and reviewed. It was decided to compute a composite MDT model using the available models. The Composite MDT was derived as the mean value in each grid node together with a standard deviation to represent its error. To calculate the best possible synthetic mean dynamic topographies a new MSS (KMS04) has been derived from nine years of multi mission altimetric data (1993-2001). The regional geoid has furthermore being updated using GRACE and gravimetric data from a recent airborne survey. The new synthetic mean dynamic topography model has been computed from this geoid model and the MSS model KMS04. Error estimates associated with the synthetic MDT have been derived from the errors of the geoid and the errors of the mean sea surface model. Compared with the Composite MDT the synthetic MDT derived from the mean sea surface and the geoid showed very similar results (see Figure 2). Details on the data processing and how the individual models have been obtained are described below.
were done in a band from Greenland over Iceland and the Faeroe Islands to Norway and Scotland. A total of 84 airborne flight-hours were flown. The airborne gravity measurements were processed and incorporated with the revised marine gravity data. The gravity data have been cross-over adjusted for survey biases. Two different approaches to bias adjust the marine gravity data has been tested. Mean differences between overlapping surveys, both marine and airborne, were estimated and used as observations in a joint least squares collocation estimation of survey biases. 190 individual surveys were inter-compared and 1531 overlapping areas were identified. Surveys considered to be of superior quality were held fixed in the adjustment; these surveys include airborne surveys and recent marine surveys processed together with a few other sources acknowledged for their reliability. The identification of reliable sources, which should be held fixed in the adjustment. The geoid modelling was based on the adjusted gravity. Some altimetric gravity data from the KMS02 model were patched in areas with larger data voids, i.e. more than 20 km to the nearest marine/airborne data point. A newly released GRACE geopotential model from JPL was used for the longer wavelengths of the gravity/geoid field in the geoid modelling. This model was used up to degree and order 120. The residual geoid was determined with the spherical FFT approach used on gravity data reduced for reference field, restoring gridded terrain-corrected RTM anomalies into Faye anomalies prior to FFT. The FFT routine was applied with Wong-Gore kernel modification to degree 90. The reference geoid then subsequently restored. This final geoid is called NAT04 geoid in the sequel (e.g. Forsberg et al., 2004). For the error assessment a standard deviation was assigned to the gravity data based on the nature (terrestrial, marine, airborne) and the history (collector and processor) of the data. The surface, marine and airborne gravity data was used in a rigorous least-squares collocation error estimate. Because least-squares collocation requires the solution of as many linear equations as the number of data, data were thinned prior to applying the method. The satellite altimetric gravity was omitted in the data set, as these data are not "real" gravity values. 1.2 Altimetry and mean sea surface
1.1
Gravity data and geoid
An airborne survey activity was carried out in June 26-July 18 and Aug 7-9 2003. The measurements
A new global mean sea surface has been derived using the best available dataset for the GOCINA region. In deriving this high resolution MSS grid
Chapter 1
•
Combining Altimetric/Gravimetric and Ocean Model Mean DynamicTopography Models in the GOCINARegion
file for the period 1993-2001 with associated quality indication grid on 2 km or 1/30 ° by 1/60 ° resolution the following scientific achievements were obtained. The MSS model is the only available MSS based on 9 years of data (1993-2001) using T/P as reference. All data have been interpolated using least squares collocation taking into account the varying quality and coverage of the data. Global sea level change over the 1993-2001 was taken into account in the computation. A new method has been derived to account for the inter-annual ocean variability (like the major E1-Nifio event in 199798), as well as sea surface trends and pressure effects on the ocean surface. The MSS is available both with and without correction for the atmospheric pressure correction applied to the altimeter range (inv. barometer correction). Furthermore, annual averages of the sea level with respect to the nine years mean sea surface have been computed, so that mean sea surface from other sources and mean dynamic topographies covering different periods of time may be inter-compared. The MSS model is denoted KMS04 (Andersen et al., personal communication) 1.3 Ocean topography
models
and
mean
dynamic
Existing mean dynamic topography (MDT) models were collected and reviewed. It was found that the best currently available products were those developed by GOCINA project partners and identified as CLSv2 and OCCAM, along with the MDT based on the assimilation results available from the UK Met Office FOAM system. It was decided to compute a composite MDT model using the available models (see Table 1).
computed from satellite altimetry as described above. Also, the high resolution models were smoothed to one by one degree grids. The Composite MDT was derived as the mean value in each grid node. Furthermore, at each node the standard deviation was computed to represent the error of the mean value (Bingham et al., personal communication). The composite MDT and its errors are shown in Figure 2.
3
Combining data sources
The MDT may be obtained simply by subtracting the geoid from the MSS as described in the previous section. If a full coverage of both the gravity data and the altimeter data exist in the region, then this simple approach will probably give a nice result. But, if that is not the case, then more advanced methods may be needed. |n the GOCINA region the coverage of gravity data is not very good in the Northern and the Southwestern parts of the region. For the geoid computation described above altimetric anomalies are inserted into data gaps, hereby, combining the two data sources already at this point. To avoid major errors a so-called draping technique is applied when merging the two data sources. When different data types are combined it is important that it is done in a rigorous way and that the full signal/error content is taken into account. Else fatal inconsistencies between different data types may occur. The MSS consists of the geoid and the MDT as expressed in: h =N+~'+n
(1)
where
MDT
Time period
Resolution
CLS v l
1993-1999
l°xl °
CLS v2
1993-1999
l°xl °
h N
ECCO
1992-2001
l°xl °
4
is the mean sea surface height, is the geoid height, is the mean dynamic topography, and
ECMWF
1993-1995
1.4°x 1.4 °
n
is the measurement noise.
FOAM
May02-May03
1/9°x 1/9 °
O C C A M vl
1993-1995
0.25°x0.25 °
O C C A M v2
1993-1995
0.25°x0.25 °
T a b l e 1: T h e i m p o r t a n t f e a t u r e s o f the M D T s u s e d in this
The geoid height is a quantity associated with the anomalous gravity potential 7". Hence, N can be expressed in terms of a linear functional (or as in this case linearized functional according to Bruns' formula) applied on T (), is the normal gravity):
study
The models were corrected for the differences in averaging period using the annual anomalies
N=LN(T) -
T 2"
(3)
5
6
Knudsen et al.
At this point the important link between altimetry and gravimetry can be made, gravity anomalies are associated with T too. They are expressed as
variances. Then the covariance between T in the points P(%)~) and Q(qD() is expressed as oO
Ag = LAg ( T ) = - c9T_ 2 --T Or r
(4)
K(P, Q)= Z °-TTpi (cosg)
(7)
i=2 The gravity data describes in the previous section are used in the combination solution rather that the estimated geoid. Hereby the structure of the original data source, e.g. data distribution and their individual errors, is maintained and represented in the computations. Furthermore, remaining biases in the individual gravity surveys may be taken into account. For the ship data such a bias is considered. The airborne data are considered to be bias free. In principle, the altimeter data should be used as described in Knudsen (1993). However, since the mean sea surface determination described above include a very large number of data and since those data have been combined using a rigorous method, it was decided to use the KMS04 mean sea surface with its associated errors in the combination solution. MDT information from ocean circulation models may be taken into account if reliable error estimates may be derived. Hence, the composite MDT described in section 1.3 can be used in the combination procedures described in this section.
3.1 Rigorous combination
(5)
An estimate of the a-posteriori error covariance between two estimated quantities, x and x/, is obtained using
Cx'x = Cx'x-Crx (C + D)-' Cx'
TT are degree variances and q~ is the
o-i
spherical distance between P and Q. Hence, eq.(7) only depends on the distance between P and Q and neither on their locations nor on their azimuth (i.e. a homogeneous and isotropic kernel). Expressions associated with geoid heights and gravity anomalies are obtained by applying the respective functionals on K(P,Q), e.g. CNv-=LN(LN(K(P,Q))) (more on collocation by Sans6, 1986, Tscherning, 1986). Then
C N N --
Pi (cos N)
(8)
°-/I Pi (c°s ¢/ )
(9)
Pi (cos ~)
(10)
i=2 oO
- Z
2
i=2 oO
CNAg i=2
The kernel associated with the MDT, eq.(7), is expressed in a similar manner as the gravity fields as
The combination of the mean sea surface heights and the gravity data is done rigorously using the optimal estimation technique called least squares collocation (LSC). Here the results in the following expression x - C~ (C + D)-'y
where
(6)
where cxx/is the a-priori (signal) covariance between x and x/ (see e.g. Moritz, 1980). The elements of the covariance matrices of eq.(56) are calculated according to the mathematical model of the observations. In this case, signals associated with the gravity data and the mean sea surface are considered. The covariance values are obtained using the kernel functions. The kernel associated with the gravity field is derived using eq.(5) and some a-priori
o(3
CG- - Z o-~( Pi (cos ~' )
(11)
i 1
Finally, it is shown how covariance functions associated with the geostrophic ocean surface currents are obtained from the sea surface topography. If accelerations and friction terms are neglected and horizontal pressure gradients in the atmosphere are absent, then the components of the surface currents are obtained from the MDT by
y ~4 u - - - --y -, G v = f R c3~b f R cos ~b 0A
(12)
where f=2mosinq) is the Coriolis force coefficient. Expressions associated with the geostrophic surface currents and the MDT depend on the azimuth between the two points P and Q, apQ as (Knudsen, 1991)
Chapter 1
•
Combining Altimetric/Gravimetric and Ocean Model Mean DynamicTopography Models in the GOCINARegion
2 7
C bl bl m
(- COS~pQCOS£ZQp Cll -
fPfQ
sin apQsin aQp
(13)
Cqq)
2
Y
C VV m
UpUQ
(- sin C~pQsin C~QpCll-
(14)
COS O[pQ COS OCQp Cqq)
-7" cosapQ Ct4"
C.4" -
(15)
fe
Cv( =
2" sin CZpQ C
fP
l
(
(
1
6
)
where z (cos p' C ' ( ( - sin 2 ~ C"(4- ) CM - -~__
Cqq
1
(17) (18)
and C l ( - - ~-k- sin ~ C ' ( (
(19)
The modelling of the covariance function associated with the gravity field is described in Knudsen (1987a). This technique has been applied using empirical covariance values calculated from marine gravity data reduced using a hybrid reference model consisting of the GRACE GGM01 model up to degree 90 and EGM96 from degree 91 to degree 360. As degree variance model a Tscheming/Rapp model (Tscherning & Rapp, 1974) was used. This expression has the advantage that the kernel can be evaluated using a closed expression instead of the infinite sum. The model is GRACE oei
-
-
gi
( k~ k~ ] i+1 cr)~S=b• k3+is k]+i s "s
¢
Re C ( (
O~T T m
0.25. The covariance function associated with gravity anomalies has a variance of (11.8 mgal) 2 and a correlation length (which is the distance where the covariance is 50 % of the variance) of 0.15 °. The corresponding geoid height covariance function has a variance of (0.20 m) 2 and a correlation length of 0.26 ° . A determination of a covariance function model associated with the MDT was carried out using an empirical covariance function was determined and a degree variance model chosen. The degree variance model was constructed using 3 rd degree Butterworth filters combined with an exponential factor. Hence, the spectrum of the MDT is assumed to have a decay similar to the geoid spectrum. Then the model was fitted iteratively to the empirical covariance values as described in Knudsen (1993). This resulted in the model:
i = 2 ..... 90
EGM96 _A_
i : 91 .....
360
where b = 6.3 10 .4 m 2, kl = l, k2 = 90, s = ((R5000.0)2/R2) 2. The variance and correlation length are (0.20 m) 2 and 1.3 ° respectively. The variance and correlation length of the current components are (0.16 m/s) 2 and 0.22 ° respectively. 3.2 I t e r a t i v e c o m b i n a t i o n
As in the geoid determination described in section 1.1 approximation methods based on FFT may be applied as an alternative to the rigorous LSC. A socalled local collocation technique may be used to grid data, so that the FFT technique can be used. This procedure is very efficient compared to LSC and does not require inversion of large equation systems. Hence, geoid computations in large regions with many observations may easily be handled using the FFT method. Stokes integral which relates the gravity anomalies to geoid undulations as (Heiskanen and Moritz, 1967)
(20)
( R2R]i+l i = 361 .....
(i - /) (i - 2)(i+ 4) [--~ )
N = S(Ag)
(22)
are used in the iterative combination method. Two metods may be applied for iterative scheme (Hipkin et al., private communication). One method is based on
where A = 1544850 m4/s 4, RB = R - 6.823 km were found in an adjustment. The error degree variances, ei, associated with the EGM96 model was multiplied by
(21)
'(f,
7
8
Knudsenet al.
where the i'th increment is based on the difference between the MSS and the geoid. In each iteration a geoid is computed using a set of gravity data with fill-in gravity values from MSS-MDT. The geoid is iteratively improved as the synthetic fill-in gravity anomalies computed from the MSS-MDT is iteratively improved. The other method is based on applying Stokes integral on the gravity equivalent of the MDT. That is A ( i - S(S -1 ( h - ( i )
-
AN)
(24)
where the gravity equivalent of the i'th increment which is the difference between the synthetic gravity anomalies and the real data tend to zero iteratively. In both cases the i+l 'th MDT is obtained using
(i+1--(i-t-wA(i-t-we((
c
-- (i )
(25)
where weights are introduced to balance the M D T between a freely iterating procedure and an iteration that is constrained to the composite MDT, e.g. within its errors. Also, the composite MDT may be used as initial MDT. The two procedures have their strengths in cases where the coverage of the gravity data is better or poorer than the altimeter data coverage respectively (Zlotnicki, 1984, Knudsen, 1992).
4
Results
Both the rigorous and the iterative combination methods have been tested in the GOCINA region. in the initial tests results were obtained using gravity data and MSS data only to derive ocean model free MDTs to be assimilated into ocean models. For the computation of a MDT from gravity and MSS data using the rigorous method it was nessesary to devide the region into nine sub-regions to limit the number of observations to overcome the task of solving the equation systems. The subregions were constructed so that they have an overlap of about 1 degree to avoid edge effects when the nine solutions later are to be merged. Furthermore, data were selected with a spacing of about 0.1 by 0.2 degrees for the gravity data and a spacing of 0.2 by 0.4 degrees for the MSS data. The estimated MDT and it error estimates are shown in Figure 3. Compared to the MDTs in Figure 2 this MDT show the same general pattern
though much more details are recovered. Those details are probably spurious in terms of ocean circulation are may be caused by edge effects at coastlines and by unrecovered errors in the data. Furthermore the Southeastern sub-region appeared to be biased by long wavelength signals that are not recovered fully within each of the sub-regions. The errors reflect the actual data distribution and range from 4 to 12 cm. Furthermore, the characteristics of the MDT error covariances were studied. That was done at a few locations within the GOCINA region where error covariances were computed using eq. (6). The results showed that the correlation lengths of the error covariance function all were similar and close to 0.3 degrees. Hence, a expression having a fixed correlation length such as CpQ - epeo cov(q,,po)
g/ = epCQ[l+ ~// I.C 0.18 L
(26)
0.18 a
may be a fairly accurate expression for the MDT error covariances. The result of the iterative combination method was obtained using the second approach, eq. (2425) by Hipkin et al. (private communication). The result has been smoothed at 75 km wavelength. The general pattern is very similar compared with the composite MDT in Figure 2. Parts of the spurious details shown in Figure 3 are also present in Figure 4. At this stage, the iterative combination method with its efficient handling of large data sets covering the whole region appears to give the best solution. However, the errors associated with the MDT can be obtained using the rigorous method only.
5 Perspectives The next tasks of the GOCINA project are associated with the merging of all three data streams. These methods will rely on new techniques of data assimilation. The following experiments will examine the mass and heat exchange across the Greenland-Scotland Ridge, considering the Atlantic inflow, the surface outflow in the East Greenland Current, and the overflows. Also the impact on the current running along the continental shelf from the Bay of Biscay to the northern Norwegian Sea will
Chapter 1
•
Combining Altimetric/Gravimetric and Ocean Model Mean DynamicTopography Models in the GOCINARegion
~1'~" LB.),
f~.." ~"
hU'.%
~11',%"
..~ ' .%"
...-..~,...,.
•g l ' Vl
.~P.I' ~'1.
.,.~1' ~lIw"
[ I~ 1~"
IJ'
111 r.
On
4.0" th
.'~," W
~'~
It)'~h
l'
a Ih'/I-.
40"tY
.W~"
110-~h
10" ~t"
(c
LtFIF:
a:t.~
:Ip'N
•li..~.".'~
~'..,;
,~'5"
,an',,v
,1,i',~,"
., r,"l,,,"
In'~,'l
n"
an,'i.:
ql 0.~
,.1.01
0.02
O.~:a
0.~
0.05
0.~
,.1.~
0.~
0.~
O.IQ
nl
0.414)
Ik4~
4". L't
41.15
(I _.'.'.~
Figure 2. Mean dynamic topography models with error estimates. The Composite MDT model is shown top left and its errors are shown lower left. The synthetic MDT is derived using the mean sea surface and the gravimetric geoid model and shown top right with its estimated errors lower right. Gray scale bar associated with the MDTs is found on the right. be analysed. A best possible ocean circulation experiment will be performed, which will also include sea floor pressure data from GRACE based on methods developed in a separate project. This analysis will give invaluable information on the ocean role in climate. Finally, the GOClNA project will support the GOCE mission in two distinct cases, namely (1) to educate and prepare the community in using GOCE data for oceanography including sea level and climate research as well as operational prediction; and (2) to develop methods for generating regional gravity fields and to use them to generate a best possible regional gravity field and geoid model for the North Atlantic that can be used in validation of the GOCE products.
Acknowledgement GOCINA is a shared cost project (contract EVG1CT-2002-00077) co-funded by the Research DG of the European Commission within the RTD activities of a generic nature of the Environment
and Sustainable Development sub-programme of the 5th Framework Programme. More information on the GOCINA project may be found on the Internet at http://www.gocina.dk
References Andersen, O., P. Knudsen, and R. Trimmer, Improved High Resolution Altimetric Gravity Field Mapping (KMS2002 Global Marine Gravity Field. lAG symposia, Vol. 128, Springer Verlag, ISBN 3-540-24055-1,326-331, 2005. Forsberg, R., A. Olesen, A. Vest, D. Solheim, R. Hipkin, O. Omang, P. Knudsen: Gravity Field Improvements in the North Atlantic Region. Proc. GOCE Workshop, ESA-ESRIN, March 2004. Heiskanen, W. A., and Moritz, H.: Physical Geodesy, W. H. Freeman, San Francisco, 1967. Knudsen, P.: Estimation and Modelling of the Local Empirical Covariance Function using gravity and satellite altimeter data. Bulletin G6od6sique, Vol. 61,145-160, 1987.
9
10
Knudsen
et al.
Knudsen, P.: Determination of local empirical covariance functions from residual terrain reduced altimeter data. Reports of the Dep. of Geodetic Science and Surveying no. 395, The Ohio State University, Columbus, 1988. Knudsen, P.: Simultaneous Estimation of the Gravity Field and Sea Surface Topography From Satellite Altimeter Data by Least Squares Collocation. Geophysical Journal International, Vol. 104, No. 2, 307-317, 1991. Knudsen, P.: Estimation of Sea Surface Topography in the Norwegian Sea Using Gravimetry and Geosat Altimetry. Bulletin Gdoddsique, Vol. 66, No. 1, 27-40, 1992. Knudsen, P.: Integration of Altimetry and Gravimetry by Optimal Estimation Techniques. In: R. Rummel and F. Sans6 (Eds.): Satellite Altimetry in Geodesy and Oceanography, Lecture Notes in Earth Sciences, 50, Springer-Verlag, 453-466, 1993. Knudsen P., R. Forsberg, O. Andersen, D. Solheim, R. Hipkin, K. Haines, J. Johannessen & F. Hernandez. The GOCINA Project- An Overview and Status. Proc. Second International GOCE User Workshop "GOCE, The Geoid and Oceanography", ESA-ESRIN, March 2004, ESA SP-569, June 2004. Moritz, H.: Advanced Physical Geodesy. Herbert Wichmann Verlag, Karlsruhe, 1980. Omang, O., R. Forsberg, G. Strukowski: Comparison of New Geoid models and EIGEN2S in the North Atlantic Region. In: F. Sanso (Ed.): A Window on the Future of Geodesy, Sapporo, Japan, IAG Symposium Series 128, Springer Verlag, pp. 306-309, 2003. Sans6, F.: Statistical Methods in Physical Geodesy. In: Stinkel, H.: Mathematical and Numerical Techniques in Physical Geodesy. Lecture Notes in Earth Sciences, Vol. 7, 49-155, Springer-Verlag, 1986.. Tscherning, C.C.: Functional Methods for Gravity Field Approximation. In: Stinkel, H.: Mathematical and Numerical Techniques in Physical Geodesy. Lecture Notes in Earth Sciences, Vol. 7, 3-47, Springer-Verlag, 1986. Tscherning, C.C., and R.H. Rapp: Closed Covariance Expressions for Gravity Anomalies, Geoid Undulations, and Deflections of the Vertical Implied by Anomaly Degree Variances. Report no. 208, Dept. of Geodetic Science and Surveying, The Ohio State University, Columbus, 1974. Wunsch, C., and V. Zlotnicki: The accuracy of altimetric surfaces. Geophys. J. R. astr. Soc., 78, 795-808, 1984.
Zlotnicki, V.: On the Accuracy of Gravimetric Geoids and the Recovery of Oceanographic Signals from Altimetry. Marine Geodesy, Vol. 8, 129-157, 1984.
-{1,41(I -(I, L~ ~10'
0-05
i),,35
(I. LO ¢, I,~
~,OL'I,
m
~
,,,
Figure 3. Mean dynamic topography model with error estimates from rigorous combination of gravity data and mean sea surface data by least squares collocation.
-di.d
-I}.~
-{I.4
-¢.3
-~_2
-{I. 1
i).¢
0.1
¢.2
I}.3
0.4
¢.:~
Figure 4. Mean dynamic topography model from iterative combination method smoothed at 75 km wavelength (Hipkin et al., private communication).
Chapter 2
Error Characteristics of Dynamic Topography Models Derived from Altimetry and GOCE Gravimetry Per Knudsen Danish National Space Center, Juliane Maries Vej 30, 2100 Kobenhavn O, Denmark. Carl Christian Tscherning University of Copenhagen, Juliane Maries Vej 30, 2100 Kobenhavn O, Denmark.
The impact of the GOCE satellite mission on the recovery of the gravity field is analysed for two simulated cases. In the first case the GOCE Level 2 product is used where the gravity field is approximated by spherical harmonic coefficients up to degree and order 200. In the second case synthetic GOCE Level 1B data are used directly in a gravity field determination using least squares collocation. In case two the full spectrum geoid error was improved from 31 cm to 15 cm and the resolution was doubled. To get reliable errors associated with the mean dynamic topography (MDT) a reliable model for the spectral characteristics of the MDT is needed. Such a model was derived reflecting empirically derived properties such as MDT variance and correlation length. Combining the MDT characteristics with the estimated geoid errors in the spectral domain resulted in a-posteori error estimates. In the two cases the MDT errors were improved from 20 cm to 6 cm and 5 cm respectively. For the geostrophic surface current components the errors were improved from 23 cm/s to 18 cm/s and 16 cm/s. Abstract.
1 Background The GOCE (Gravity and Ocean Circulation Experiment) satellite mission by the European Space Agency is planned to improve the knowledge about the Earth gravity field and to improve the modelling of the ocean circulation. The central quantity associated with the ocean circulation and transport is the mean dynamic topography (MDT), which is the difference between the mean sea surface (MSS) and the geoid. The MDT provides the absolute reference surface for the ocean circulation. In the EU project GOCINA (Geoid and Ocean Circulation In the North Atlantic) the use of GOCE
data in oceanographic modelling is prepared by developing methodology for the determination of accurate mean dynamic topography models in the region between Greenland and the UK (Knudsen et al., 2004). The improved determination of the mean circulation will advance the understanding of the role of the ocean mass and heat transport in climate change. The GOCINA project will support the GOCE mission in a distinct case, namely to educate and prepare the community in using GOCE data for oceanography including sea level and climate research as well as operational prediction. In this study, the impact of GOCE on the mapping of the gravity field is studied. Furthermore, its impact on the estimation of the MDT is analysed.
2
GOCE simulations
The simulations of GOCE impact on the gravity field recovery are done using the full spectra of the signals and the errors. Hence, both commission and omission errors are taken into account when, e.g., a spherical harmonic expansion truncated at a certain harmonic degree and order is considered. This is important because the omission error usually is larger that the commission error. Using spherical harmonic functions, the signal and the error covariances associated with the gravity field between points P and Q may be expressed as a sum of Legendre's polynomials multiplied by degree variances as oo
K(P, Q)= 2
crTTPi (cosp')
(1)
i=2 TT
where cri
are degree variances associated with the
anomalous gravity potential field and ~, is the spherical distance between P and Q. Hence, eq.(1) only depends on the distance between P and Q and
12
P. Knudsen. C. C. Tscherning
not on their locations nor on their azimuth (i.e. K(P,Q) is a homogeneous and isotropic kernel). Expressions associated with geoid heights and gravity anomalies are obtained by applying the respective functionals on K(P,Q), e.g. CuN=LN(LN(K(P,Q))) and become as follows (For more information on collocation, see for example Sans6, 1986, Tscherning, 1986) CNN --
Pi (cos~,)
(2)
0-; 7, Pi (cos ~)
(3)
i=2 oo
i=2 oO
-- Z
i-1 or/TTPi(cosp')
(4)
i=2
where N is the geoid, ), is the normal gravity, and Ag is the gravity anomaly. The determination of the degree variances is essential to obtain reliable and useful signal and error covariance functions. For the gravity field it has been accepted that the degree variances tend to zero somewhat faster than i-3 and that the Tscherning-Rapp model may be used as a reliable model (Tscherning & Rapp, 1974, and Knudsen, 1987). This expression has the advantage that the kernel can be evaluated using a closed expression instead of the infinite sum. When a spherical harmonic expansion of the gravity field up to degree and order N has been used as a reference model and has been subtracted from the gravity field related observations, then the error degree variances, ei, associated with the reference model should enter the expression, eq. (1), up to harmonic degree N, so that the degree variances are expressed as i=2
o.TiT
-
-
A
..... N
(5)
(RS] i-'
7;- u(,-3)(,-+-4) kT)
200 that can fulfil the aim of the satellite mission, which is to model the geoid at a resolution of 100 km with an accuracy of 1-2 cm. Based on mission parameters and extensive simulations it has been demonstrated that GOCE will meet those requirements (e.g. Visser et al., private communication). An important outcome of the simulations is a set of error degree variances that may be included as commission errors in other simulations of the GOCE performance. In this case the simulated error degree variances described above are used as commission errors, i.e. the ai, in eq. (5), in the evaluation of the impact of GOCE on the recovery of the gravity field. The unmodelled part of the gravity field remains unknown and will be considered as the omission error. The omission error was modelled using the Tscherning-Rapp model from degree 201 and up. Hence, the full spectrum error of the GOCE level 2 harmonic expansion as an approximation of the gravity field consists of both the commission and the omission errors as expressed in eq. (5). The degree variances are shown in Figure 1. The error covariance function is shown in Figure 2. it has a variance of (0.31 m) 2 and a correlation length (which is the distance where the covariance is 50 % of the variance) of about 0.3 ° . 2.2 GOCE Collocation Product
As part of the simulations of the GOCE performance alternative methods such as least squares collocation have been tested (Tscherning, 2004). In this case a test was carried out using simulated GOCE Level 1B observations in the GOCINA region in the North Atlantic Ocean. Using both the along track and the vertical second order derivatives, least squares collocation was applied to estimate the geoid in the region using the following expression x = C~r (C + D)-ly
(6)
i = N + I .....
where A = 1544850 m4/s 4, RB = R - 6.823 km were found in an adjustment so that agreement with empirical covariance values calculated from marine gravity data was obtained. This procedure is described in Knudsen (1987). 2.1 GOCE Level 2 Product
The standard Level 2 product coming from GOCE is a spherical harmonic expansion to degree and order
where C and D are covariance matrices associated with the signal and the errors of the observations y. x is the estimated quantity. As covariance function used for determination of the values in the C matrix the Tscherning-Rapp model with the same parameters as in eq.(5) was used from harmonic degree 2 and up. The error covariance matrix D is diagonal and contains the error variance of the observations. Then error covariances were estimated using G , x - c x,~-
d(C+D)' x Cx,
(7)
Chapter2 • Error Characteristics of Dynamic Topography Models Derived from Altimetry and GOCEGravimetry
where c J is the a-priori (signal) covariance between x and x/ (see e.g. Moritz, 1980). The results show that the estimated errors range from 10 cm to 15 cm. Note that these are full spectrum errors demonstrating that the approach will give a significant improvement of the geoid. The estimated error covariances show that the errors are associated with scales shorter that half a degree. Hence, the resolution appears to have been doubled. For the subsequent simulations of the GOCE performance in terms of modelling a MDT a degree variance model was obtained by extending the GOCE part of the previous degree variance model to harmonic degree 360 (also shown in Figure 1). This model give an error covariance function (also shown in Figure 2), which has a variance of (0.15 m) 2 and a correlation length of about 0.1 °. Hence, compared to the standard GOCE Level 2 product, a significant improvement may be obtained by using the GOCE data directly in a determination of the gravity field using least squares collocation.
Modelling the Signal Characteristics of the Mean Dynamic Topography
)
(9)
where b, kl, k2, and s are determined so that the variance and the correlation length agree with empirically derived characteristics. Since geostrophic surface currents are associated with the slope of the MDT, it may be shown how covariance functions associated with the geostrophic surface currents can be obtained. If accelerations and friction terms are neglected and horizontal pressure gradients in the atmosphere are absent, then the components of the surface currents are obtained from the MDT by
u-----, vf R c3~b f R c o s ~ b 02
(10)
where f=2mesino is the Coriolis force coefficient. Covariance functions associated with the geostrophic surface currents depend on the azimuth between the two points P and Q, aeQ (Knudsen, 1991). Hence, the respective current components are not isotropic. They are expressed as 2
Cuu To get reliable results of simulations and tests carried out using least squares methods it is important that both the signal and the error characteristics have been taken into account. In least squares collocation that means that the covariance function models should agree with empirically determined characteristics such as the variance and correlation length. In analysis of errors formally estimated using eq.(7), it is very important that those quantities are reliable. That is also the case when MDT errors are analysed. Hence, a model describing the magnitude and the spectral characteristics of the MDT is needed. A kernel function associated with the MDT, may be expressed in a similar manner as the gravity fields
,+,
o~'~'=b • k~+i 3 k]+i 3 "s
Y
/
(- COS~TpQCOS~Qp Cll -
fpfo
(11)
sin C~pQsin C~Qe C qq) 2 C
VV m
7" fpfQ
(- sinc~pQsin~zQp Cll (12) \
COS 6~'pQ COS 6~'Qp Cqq )
-y COS 6~ pQ Cl(
C,,;- =
(13)
fp
Y Cv( -- - - sin apQ Cl(
f~
(14)
where
as oO
(s) i=l where the degree variances in this expression are associated with the MDT, naturally. The degree variance model was constructed using 3 rd degree Butterworth filters combined with an exponential factor (e.g. Knudsen, 1991). Hence, the spectrum of the MDT is assumed to have properties similar in smoothness and infinite extent to the geoid spectrum. That is
C~ -_ ) 7t (cos ~ C';;- - sin2gtC"4-4 -)
(15)
Cqq - 1 R2 C, ~
(16)
are the longitudinal and the transverse components respectively (see also Tscherning, 1993), and C~4 - - z T sin gt
C'
(4
(17)
With those expressions it is possible to estimate geostrophic surface currents using collocation. They
13
14
P. Knudsen. C. C. Tscherning
also give a very important constraint on the modelling of the degree variance model. In Knudsen (1993) the parameters in the degree variance mode, eq.(9), were fitted iteratively to the empirical covariance values. This resulted in the model where b = 6.3 10 -4 m 2, kl = 1, k2 = 90, s = ((R-5000.O)2/R2) 2. The variance and correlation length are (0.20 m) 2 and 1.3 ° respectively. The variance and correlation length of the current components are (0.16 m/s) 2 and 0.22 ° respectively.
Modelling A-posteori Mean Dynamic Topography Error Characteristics Combining the MDT signal degree variances and the geoid error degree variances it may provide information about the a-posteori errors of an MDT that has been estimated using the geoid and a mean sea surface computed from satellite altimetry. Using least squares to estimate the MDT by degree its error degree variance is expressed as 1
To study the properties of the degree variance model in more detail characteristic parameters associated with the MDT and its associated geostrophic surface current components were derived. This was done for block averages of varying block sizes, because those numbers may be compared with output parameters from ocean circulation models with different grid sizes and resolutions. The MDT signal covariance properties were computed rigorously using the series of Legendre's polynomials to which the smoothing operators associated with the running averages have been applied. Then the following expression may be used oo
C~;- - E / 5 ' 2 (s) cr~" Pi (cos ~,,)
(18)
i 1
where the beta factors are the so-called Pellinen operators that depend on the side length, s, of the cells. The covariance functions associated with the MDT and with the geostrophic surface current components (represented by the Cll, eq. (15)) and averaged in cells of ½, 1, and 2 degree were computed. The resulting variances and correlation lengths are summarized in Table 1. it is important to emphasize that those numbers are statistical expected values representing a region as the GOCINA region. They are not representative for the strong Western boundary currents as the Gulf Stream. Table 1. Standard deviations in cm and cm/s and correlation lengths in degrees of MDT and geostrophic surface current components (u,v) (represented by the Cll) as point values and averaged in cells.
(u,v)
MDT
Points ½o x ½o 1o x 1o 2°x2 °
St.dev. 20 19 18 16
C.length 1.3 ° 1.4 ° 1.6 ° 2.1 °
St.dev. 16 13 10 6
C.length 0.22 ° 0.38 ° 0.56 ° 0.85 °
=
1
1
(19)
where the errors of the mean sea surface have been ignored since they are very small compared to the geoid errors. For both GOCE simulations the a-posteori MDT error degree variances were computed. Subsequently, error covariance functions for the MDT and the surface current components were computed and their variances were found. The results are summarized in Table 2. Table 2. A-posteori errors in cm and cm/s of MDT and geostrophic surface current components (u,v) (represented by the Cll) as point values and averaged in cells as estimated using the two GOCE simulations; to harmonic degree 200 and 360 respectively.
(u,O
MDT
Points ½o x ½o l°xl ° 2°x2 °
200 6 5 3 2
360 5 4 3 2
200 12 8 4 1
360 11 6 3 1
By comparing the numbers in Table 1 with the numbers in Table 2 it is obvious that the GOCE satellite mission will have a large impact on the estimation of the MDT. With the Level 2 product the error of point values is brought down substantially from 20 to 6 cm. The current components are associated with shorter wavelengths and moderately improved from 16 to 12 cm/s. For 1 x 1 degree averages however, the current components is improved from 10 to 4 cm/s. The solution obtained using least squares collocation improved the recovery of the geoid substantially. The impact on the estimation of the MDT is not that pronounced, since most of the signal contents in the MDT have a more long wavelength character. However, the MDT is improved at point values and ½ x ½ degree averages by about 20 %.
Chapter 2 • ErrorCharacteristicsof DynamicTopographyModels Derivedfrom Altimetry and GOCEGravimetry
The improvement has a larger impact of the current components. Compared to the standard deviations of the MDT the current components have been improved twice as much almost.
5
Perspectives
The impact of the GOCE satellite mission on the recovery of the gravity field has been analysed for two simulated cases. In the first case the GOCE Level 2 product is used where the gravity field is approximated by spherical harmonic coefficients up to degree and order 200. In the second case synthetic Level 1B GOCE data are used directly in a gravity field determination using least squares collocation. In case two the full spectrum geoid error was improved from 31 cm to 15 cm and the resolution was doubled. The results are important for the future users of GOCE that need the extra accuracy. Then the impact of the improved geoid on the estimation of the MDT is analysed. To get reliable errors associated with the MDT a reliable model for the spectral characteristics of the MDT is needed. Such a model was derived reflecting empirically derived properties such as MDT variance and correlation length. This model, naturally, is purely empirical and more information needs to be collected to verify the reliability of the model characteristics. Combining the MDT characteristics with the estimated geoid errors in the spectral domain resulted in a-posteori error estimates. In the two cases the MDT errors were improved from 20 cm to 6 cm and 5 cm respectively. For the geostrophic surface current components the errors were improved from 23 cm/s to 18 cm/s and 16 cm/s. Those results depend of the MDT a-priori degree variance model and will not be reliable unless that model is reliable. So the results may change accordingly. However, since this a-priori model actually do reflect empirically derived characteristics, they may not change that much.
Acknowledgement GOCINA is a shared cost project (contract EVG1CT-2002-00077) co-funded by the Research DG of the European Commission within the RTD activities of a generic nature of the Environment and Sustainable Development sub-programme of the 5th Framework Programme.
References
Heiskanen, W. A., and Moritz, H.: Physical Geodesy, W. H. Freeman, San Francisco, 1967. Knudsen, P.: Estimation and Modelling of the Local Empirical Covariance Function using gravity and satellite altimeter data. Bulletin Gdoddsique, Vol. 61,145-160, 1987. Knudsen, P.: Simultaneous Estimation of the Gravity Field and Sea Surface Topography From Satellite Altimeter Data by Least Squares Collocation. Geophysical Journal International, Vol. 104, No. 2, 307-317, 1991. Knudsen, P.: Estimation of Sea Surface Topography in the Norwegian Sea Using Gravimetry and Geosat Altimetry. Bulletin Gdoddsique, Vol. 66, No. 1, 27-40, 1992. Knudsen, P.: Integration of Altimetry and Gravimetry by Optimal Estimation Techniques. In: R. Rummel and F. Sans6 (Eds.): Satellite Altimetry in Geodesy and Oceanography, Lecture Notes in Earth Sciences, 50, Springer-Verlag, 453-466, 1993. Knudsen P., R. Forsberg, O. Andersen, D. Solheim, R. Hipkin, K. Haines, J. Johannessen & F. Hernandez. The GOCINA Project - An Overview and Status. Proc. Second international GOCE User Workshop "GOCE, The Geoid and Oceanography", ESA-ESRIN, March 2004, ESA SP-569, June 2004. Moritz, H.: Advanced Physical Geodesy. Herbert Wichmann Verlag, Karlsruhe, 1980. Sans6, F.: Statistical Methods in Physical Geodesy. In: Stinkel, H.: Mathematical and Numerical Techniques in Physical Geodesy. Lecture Notes in Earth Sciences, Vol. 7, 49-155, Springer-Verlag, 1986. Tscherning, C.C.: Functional Methods for Gravity Field Approximation. In: Stinkel, H.: Mathematical and Numerical Techniques in Physical Geodesy. Lecture Notes in Earth Sciences, Vol. 7, 3-47, Springer-Verlag, 1986. Tscherning, C.C.: Computation of covariances of derivatives of the anomalous gravity potential in a rotated reference frame. Manuscripta Geodaetica, Vol. 18, no. 3, pp. 115-123, 1993. Tscherning, C.C.: Simulation results from combination of GOCE SGG and SST data. 2. Int. GOCE user Workshop, ESRIN, ESA SP-569, March, 2004. Tscheming, C.C., and R.H. Rapp: Closed Covariance Expressions for Gravity Anomalies, Geoid Undulations, and Deflections of the Vertical Implied by Anomaly Degree Variances. Report no.
15
16
P. Knudsen. C. C. Tscherning
208, Dept. of Geodetic Science and Surveying, The Ohio State University, Columbus, 1974. Wunsch, C., and V. Zlotnicki: The accuracy of altimetric surfaces. Geophys. J. R. astr. Soc., 78, 795-808, 1984.
Zlotnicki, V.: On the Accuracy of Gravimetric Geoids and the Recovery of Oceanographic Signals from Altimetry. Marine Geodesy, Vol. 8, 129-157, 1984.
10 -1
i 0 -i
°~E.E'
10 .3
10 .3
>
1 0-5
>
0-5
121
D 10 .7
10 .7
_
GOCE - 200
10-9
10-9
. . . . . . . . . . . . . . . . . . . . 10
100
Harm.
1000
10
Figure 1. Geoid error degree variances associated with.GOCE Level 2 harmonic expansion to degree 200 and to degree 360 simulating the collocation solution
100 Harm. deg.
0.04 .
Coll." v = 15 cm 200+: v = 31 cm
~
.
.
.
.
.
.
~ ' ~
oq
.
.
.
.
.
.
.
0.04
E
L
.
.
.
.
.
t
CII: s = 16 cm/s, cl = 0.22 deg : = , = . deg
~
0.03
8 e-
1000
Figure 3. MDT signal degree variance model shown together with the geoid error degree variances associated with.GOCE Level 2 harmonic expansion to degree 200 and to degree 360 simulating the collocation solution
t
0.08
oOcE_200 -360 I
.....
deg.
°12t ...................
_
....
0.02
g
> 0
>
o
8
-0.04 ' 0
0.5
1.0
1.5
o.ol
2.0 -0.01
Lag [sph. degrees]
0
1
2
3
4
lag (deg)
Figure 2. Geoid error covariance functions associated with.GOCE Level 2 harmonic expansion to degree 200 and to degree 360 simulating the collocation solution
Figure 4. Covariance functions of the MDT and the geostrophic surface current components based of the MDT degree variance model shown in Figure 3.
Chapter 3
Detecting Ocean Currents from Satellite Altimetry, Satellite Gravity and Ocean Data Zizhan Zhang 1'2 Yang L u 1'3 Houtse HSU 1'3 1 Institute of Geodesy and Geophysics, Chinese Academy of Sciences, 340 Xudong Street, Wuhan, China, 430077 2 Graduate School of the Chinese Academy of Sciences, Beijing, China, 100049 3 Unite Center for Astro-geodynamics Research, CAS, Shanghai, China, 200030
Abstract: With the preparation and launch of the high accuracy geodetic missions CHAMP, GRACE and GOCE, the geoid models have been improved greatly. This gives us a good chance to combine a high precision sea surface height with improved geoids to estimate the dynamic ocean topography (DOT) and associated surface currents accurately. Combining the geoid model (EIGEN-CG01 C) from CHAMP and GRACE missions with the timeaveraged sea surface model KMS04, we calculate a new high accurate mean DOT, and then compare it with DOT derived from ocean hydrological data. When taking 3km as the reference depth datum, the correlation coefficient is larger than 0.9 and the discrepancy of mean values is very small between the DOT derived from altimetry/geoid and that from WOA01 salinity and temperature data. The surface geostrophic currents from the altimetry/geoid derived DOT map are very close to those from WOA01 derived DOT map. This independent knowledge of surface currents from the altimetry/geoid derived DOT is used in combination with the salinity and temperature data from NOAA's WOA01 to retrieve the ocean currents as a function of depth. At 2000db, comparisons of the retrieved velocities of the deep geostrophic currents with the WOCE current meter observation show that the spread of velocity deviation is approximately normal distribution, while the spread of velocity deviation is ruleless. The reasons caused these discrepancies need further analysis. Keywords. Ocean currents, satellite gravity, satellite altimetry, WOA01
1 Introduction Ocean currents transport mass and heat between different regions of the Earth, so knowledge of the ocean currents is vitally important for Earth sciences (i.e. climate, atmosphere, hydrology etc.), but global
measurements of them, especially the deep ocean currents, are very difficult. Fortunately, ocean currents are very close to the geostrophic balance on time scales longer than a few days. Therefore, the dynamic ocean topography, the deviation of the stationary sea surface from the marine geoid, is a direct measure of the dynamic pressure at the sea surface that leads to the geostrophic part of the surface current velocity (Dobslaw et al., 2004). Previously published dynamic ocean topography models suffered mainly from the poor quality of the marine geoid when derived solely from satellite tracking data, or were biased when using altimetry measurements for geoid determination (Le Traon et al., 2001). During the last two decades, satellite altimetry has offered an abundance of measurement of the sea surface height (SSH), which results in some high accuracy mean SSH models, such as CLS01 (Hernandez et al., 2001), GSFC00 (Wang, 2001). KMS01, KMS04 (Andersen et al., 2004) etc. With the advent of the new models from the dedicated gravity missions CHAMP and GRACE, the quality of the gravity field models based on satellite tracking data significantly increased (Tapley et al., 2003), especially, the long-wavelength components of gravity field models from GRACE missions (see Figure 1). The goals of this paper are that computing a new mean global DOT by combining the gravity field model EIGEN-CG01C from CHAMP and GRACE missions with the KMS04 mean SSH model from multi-satellite altimetry, revealing the relationship between altimetry/geoid derived DOT and WOA01 hydrological data derived DOT, and retrieving the surface geostrophic currents and deep ocean currents. Additionally, the results of currents are compared to observed velocities from WOCE current meters.
2 Data used Several sources of data are used in this work; they are
18
Z. Zhang. Y. Lu. H. Hsu 50
models (GGM01) actually gives very reliable geostrophic currents in the Arctic region (Andersen, 2004). To perform the comparing analysis, hydrological data from WOA01 annual analysis temperature and salinity fields (Conkright et al., 2002) are used to estimate the oceanography dynamic topography and to retrieve deep ocean currents, and WOCE current meter observation is included.
30
.[ "~
0.5 O.1 0.01 0.001 0
60
120
180
240
300
360
Spheri cal H a r m o n i c D e g r e e
Figure l:Error amplitudes as a function of maximum degree spherical harmonic degree of gravity field models (EGM96, GGM01 S, GGM02S, CG01 C, CG03C) in terms of geoid heights
the gravity field model EIGEN-CG01C, the mean sea surface model KMS04, the WOA01 temperature and salinity data and WOCE current meter observation. The newly obtained global mean gravity field model EIGEN-CG01C is a combination of GRACE mission (376 days out of February to May/July to December 2003 and February to July 2004) and CHAMP mission (860 days out of October 2000 to June 2003) data plus altimetry and gravimetry surface data (F6rste et al. 2005). The altimetry and gravity surface data used in CG01C include geoid undulations over the oceans derived from CLS01 altimetric SSH and ECCO simulated sea surface topography (Stammer et al., 2002), and NIMA gravity anomalies data etc. This model is complete to degree/order 360 in terms of spherical harmonic coefficients and resolves wavelengths of 110 km in the geoid and gravity anomaly fields. Compared to pre-CHAMP/GRACE global high-resolution gravity field models, the accuracy could be improved by one order of magnitude to 4 cm and 0.5 mgal in terms of geoid heights and gravity anomalies, respectively, at a spatial resolution of 400 km wavelength. The overall accuracy of the full model is estimated to be 20 cm and 5 mgal, respectively (Reigber et al., 2004). The mean sea surface KMS04 is derived from a combination of altimetry from a total of 5 different satellites and a total of 8 different satellite missions like the T/P, T/P Tandem Mission, ERS 1 ERM+GM, ERS2 ERM, GEOSAT GM, and GFO-ERM data. New data from the JASON and ENVISAT data are used to validate the mean sea surface models. This model covers the period 1993-2001using T/P as reference (Andersen et al., 2004). The resolution of the mean sea surface is 2 minutes equivalent to 4 km at the Equator. For the first time, the mean DOT derived from KMS04 and GRACE derived geoid
3 Primary theory The geoid is defined as the equipotential surface of the gravitational field that best fits the undisturbed ocean. This ocean will be in a state of equilibrium, subject only to the force of gravity, and free from variations with time (Torge, 2001). The difference the time-averaged sea surface H and the geoid undulation N (both related to the same conventional ellipsoid of revolution) is called the sea surface topography ( : (1)
(=H-N
Mostly, the altimetry-based SSH models have higher spatial resolution than the gravity field models, so a filter is needed to fade out the high frequency signals in SSH models. The motion of a fluid can be generally described by the Navier-Stokes equation combining the effects of pressure gradient, internal friction, gravity gradient and the Coriolis-force originating from the rotation of the Earth. For periods longer than one day, and away from oceanic boundaries (the coasts, the seafloor, the sea surface), the effects of internal friction and the Coriolis-force dominate all other accelerating forces can be neglected. Based on the geostrophic equilibrium and hydrostatic balance assumption, in local Cartesian coordinate system, the surface geostrophic current is o f the form (Apel, 1987, Hwang, 2000, Wahr et al., 2002) ~,
f u
= g__a(
f c~x =
(2)
g0( f
~Y
where Vs and Us are the east and north velocity components of the surface geostrophic currents, respectively; g is the local acceleration of gravity, f = 2f~sin ~b, with f2 being the earth's rotational rate (7.292115x104 rad s-i); ( i s the sea surface dynamic height.
Chapter 3 • Detecting Ocean Currents from Satellite Altimetry, Satellite Gravity and Ocean Data
i
-2.0
2.0
i-
-1.6 1.2
- 0.8
0.8
-0.4
0.4
,5,
- -0.0
--0.0
0 - -0.4
-0.4 -0.8 --1.2
i
--1.6
..,_. m ._1
- -0.8
-1.2
i
-50-
-2.0
0
60
120
180
240
300
- -1.6 - -2.0
m
m
(a)
1.6 - 1.2
0
360
Longitude (deg.)
60
120
180
(b)
240
300
360
Longitude (deg.)
Fig.2 (a) Mean DOT estimated by the K M S 0 4 model minus the CG01C geoid. (b) Mean DOT estimated from W O A 0 1 based on 3000db depth datum.
Let the atmosphere pressure at the sea surface equal to zero, the deep ocean geostrophic currents can be presented in the following form (Pond et al., 1978).
l.pS #
- - + V
z
u
=
g
z
s
----F
f.p
(3)
zi s
, -~y
where Vz and Uz are the east and north velocity components of the geostrophic currents at the z depth level. P is the pressure at z depth level. 4 C o m p a r i s o n of D O T s f r o m altimetry, geoid and f r o m W O A 0 1 The mean DOT (see Fig. 2 (a)) estimated by the KMS04 model minus the CG01C geoid is filtered with 400kin Gauss filter in order to reduce the noise signal and signal wavelength shorter than 400km. Figure 2 (b) is the mean DOT derived from WOA01 annual salinity and temperature data based on the reference depth datum 3000db. Comparing figure 2
0.6
'
I
'
I
'
I
'
.,,_, c
o
0.4
c
0.8
%
o
I
E 4-
0.6
1
I
E ,,,
o (1) o c O 'o-
0.2
~
correlationcoefficent
I _
(a)(b), it can be seen that the altimetry/geoid derived DOT map is very similar with WOA01 derived DOT (mean value is removed). Figure 3 (a)(b) shows the correlation coefficients and deviation between means of altimetry/geoid derived DOT and WOA01 derived DOT based on different reference depth datum in global area and in a test region with bounds of (El70 ° ~ 210 °, N0 ° ~ 40°). From figure 3 (a), one can see that, above 3kin, the correlation coefficients is 0.89 and deviation between the mean values of altimetry/geoid derived DOT and WOA01 derived DOT is o n l y - 0 . 1 c m (altimetry/geoid derived DOT minus WOA01 derived DOT). In the test region, when taking the reference depth shallower than 3km, the correlation coefficients is larger than 0.9 and deviation of mean values of altimetry/geoid derived DOT and WOA01 derived DOT is zero at 3.5kin deep level (see Figure 3 (b)). Both figure 3 (a) and (b) show that the correlation coefficients decrease quickly and deviation mean increase obviously as the reference depth datum becomes deeper than 3km. Several reasons can be used to explain this matter. One is
mOOT AG " mOOT WOA01
0,6
o~ (1) O o ,-
.9 ....,
~
0.4
0
%
E o
0.4
car) t-O~
i 0
(a)
I
1000
i
I
i
I
2000 3000 depth m
i
I
4000
i
0.2 5000
%
% -
o E
0.8
%
%
0.2
o.6 N O o
oo
c
0
0.4 %%%
c (1)
-"-
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-*--, C .o_
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o -0.2 -f.
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_
-0.4
I
0 (b)
%%
m~OrAG-m~orwoAo~ I
'
1000
I
-'
I
2000
-
3000
I
.o .,,_,
_
0.2 o
i
4000
I
0
5000
depth m
Fig. 3 Correlation coefficients and difference of the mean values of DOTs from altimetry/geoid and W O A 0 1 based on different reference depth datums. (a) (N82 ° ~$82 °, 0 ° ~ 360°), mDOT-AGnotes the mean value of DOT derived from altimetry/geoid. (b) (NO ° ~40 °, E170 ° ~ 210 °), mDOT-WOA01notes the mean value of DOT derived from W O A 0 1 salinity & temperature data.
19
20
Z. Zhang. Y. Lu. H. Hsu
I --40
I --20
I --15
--10
--5
I
"
0
5
I 15
10
I 20
I
I" cm,"s 40
-20
-12
,
I
'
i
,
,
-8
-4
0
4
8
12
~ cm,,~
Fig. 4 Surface geostrophic currents from altimetry (KMS04), geoid (CG01C) and WOA01 map on 3000db depth Left-zonal geostrophic currents, positive currents are toward the east. Right-Meridional geostrophic currents, positive currents are toward the north. (a) (top) from DOT derived from altimetry, geoid. (b) (bottom) from WOA01D poor quality or sparsity with ocean salinity and temperature data, as the reference depth deeper than 3km. The other is no enough quality with altimetry data in high latitude areas, near polar area, and geoid. So we consider that the reference depth datum at 3000db is reasonable for oceanography to determine the sea surface dynamic height.
5 Retrieval of surface geostrophic currents Tapley et al. (2003) computed geostrophic currents with an earlier GRACE geoid model. Similar to that study, we compute the currents with this DOT and show the results in Figure 4. From figure 4, it can be seen that both altimetry/geoid derived DOT and WOA01 derived DOT (except near equator N2 ° $2 °) show all major surface geostrophic currents (such as Kuroshio Extension, Gulf Stream, the
..r-I
.~
'
I ~ ' I
'
I
Antarctic Circumpolar Current (ACC)), especially those in the tropics. The locations and magnitudes are very similar to each other. The altimetry/geoid derived DOT maps show better details closer to land due to the depth required for the WOA01 maps. While the ACC from altimetry/geoid derived DOT is stronger than from WOA01 derived DOT, which may be due to sparsity of salt and temperature data or no effective experience density formula in this region. Figure 5 plots the correlation coefficients and the RMS of the difference of zonal components and meridional components from altimetry/geoid derived DOT and that from WOA01 derived DOT based on different depth datums. We can see that the correlation coefficients of two current components from altimetry/geoid derived DOT and WOA01 derived DOT are very larger and the RMS of the
1
' ~t20
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..-
I
'
I
'
I
'
I
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'
correlatio n coefficent
=
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16 o
(D u~
_
¢~ 0.6
correlation c
o o
12 fE
/
o
- - - - - - RMS of the difference
E
or)
.o,. _,e0.4 t,o
0.2
t
0 (a)
ItJ s
i
i 1000
i 2000 3000 depth m
-, 8 rv~
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--4
i
i 4000
t
'-
(19 o
0.8
-
-
. . . .
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16
of the difference
/
¢~ 0.6
/
o o
/
E
-,__....
o
,-,-,----,
0 0
(b)
¢..)
t
s J
,'- 0.2
5000
12"~ 8
_~°"-'¢0.4
0
o
f
1000
2000
3000
4000
co
n,' 4
0 5000
depth m
Fig. 5 Correlation coefficient and RMS of the difference of zonal and meridional surface geostrophic currents from the altimetry/geoid DOT and from the WOA01 DOT maps as a function of depth. (a) Zonal component (b) Meridional component
Chapter 3 • Detecting Ocean Currents from Satellite Altimetry, Satellite Gravity and Ocean Data
. . . . . . .
__-.
__
I
i
..&--l.
-
-
-
I
I
j--; I
,
-12
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-8
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-4
[
,
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--I--
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6
I
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-9
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-6
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_
I
_
,
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3
,
I' cm/s
6
9
Fig. 6 Deep geostrophic currents from altimetry (KMS04), geoid (CG01C) and WOA01 salinity & temperature data Left column-zonal geostrophic currents, positive currents are toward the east. Right column-meridional geostrophic currents, positive currents are toward the north. (a) (top) 1000db (b) (middle) 2000db (c) (bottom) 3000db difference of them is very small when the reference depth datums are shallower 3km than deeper, which indicates the reference depth datum on 3km is optimal. More important, the high correlation indicates that accuracy of the geostrophic currents derived from space measurements has been improved greatly. 6 Retrieval
of deep
geostrophic
currents
Figure 6 shows the deep geostrophic currents retrieved from altimetry/geoid derived DOT and
25
i
i
I
i
I
i _1
i
I
i
I
WOA01 salinity and temperature data at l km, 2km, and 3km depth levels, respectively. From figure 6, we see that deep currents grow weaker with depth, and the changes between them are not obvious, because the velocities of deep currents are very weaker, majority of them no more than 2 cm/s. Figure 7 shows the frequency distribution about the deviation between retrieval velocity from altimetry/ geoid & WOA01 and W O C E current meter observation at 2000db. We collect all W O C E current
25
i
2O
20 -
=~ 15
o 10 5 0
,
-:30 (a)
5 ,
-20
|, ||
-10
0
||,
10
Velocity deviation (cm/sec)
,
20
0 -180
,
30 (b)
-120
-60
0
60
120
180
Direction deviation (deg.)
Fig. 7 Frequency distribution about the deviation between retrieved velocity from altimetry, geoid & WOA01 and the WOCE current meter observations on 2000db. (a) (left) velocity deviation (b) (right) direction deviation
21
22
Z. Zhang. Y. Lu. H. Hsu
meter data at 2000db in global area, and choose the surveying time period longer than one year, from Jan. 1992 to Dec. 1995, to compare with the velocity and direction retrieved from altimetry/geoid and salinity and temperature data. The total number of valid points is 32. We interpolate the altimetry/geoid derived point velocity and direction to the current meter position. The statistic discrepancy about velocity deviation and direction deviation at 2000db are illustrated in Figure7 (a) (b). From figure 7(a), it can be see that the spread of the difference of velocity is approximately normal distribution and the average difference is about 1.4cm/sec. While the spread of the direction deviation at 2000db is ruleless, their differences of direction, beyond 68% of the total number, are larger than 45 ° . It is difficult to explain what caused these discrepancy, because we do not know which, sea surface height errors, geoid errors or uncertainty of salinity and temperature data, has m u c h effect on these discrepancy. Additionally, the current meters provide all components of deep ocean currents, while retrieved currents are only the geostrophic components.
7 Conclusion and further w o r k The correlation is very high and the discrepancy of means is much small between altimetry/geoid derived D O T and WOA01 derived DOT, when WOA01 takes 3km as the reference depth datum. The surface geostrophic currents derived from altimetry and satellite gravimetry recovered earth gravity model (CG01C) are very close to currents from WOA01 based on 3km depth datum, which indicates that accuracy of the geostrophic currents derived from space measurements has been improved greatly. The deep geostrophic currents change unobviously as depth increasing. The retrieved deep ocean currents have discrepancy with W O C E current meter observations. The reasons need to be further analysed.
Acknowledgements The authors are grateful to Dr. Don Chambers and the reviewers of this paper for their helpful recommendations. The data used in this work were obtained from GeoForschungsZentrum Potsdam, NOAA, CSR of Texas University and Ole B. Andersen etc. in Danmark. This work was supported by the National Natural Science Foundation of China (Grant No. 40374007, 40234039), Foundations of Chinese Academy of Science (Grant N o s . KZCX2-SW-T1, KZCX3-SW-132), Foundation of marine 863 (Grant No. 2002AA639280) and the Informatization Construction of Knowledge Innovation Projects of the Chinese Academy of Sciences "Supercomputing Environment Construction and Application" (Grant No. INF 105-SCE).
References Andersen O. B., A. L. Vest, and R Knudsen (2004). KMS04 mean sea surface and inter-annual sea level variability. EGU meeting poster, April 2004, Nice, France Andersen O. B (2004). Sea Level Determination from satellite altimetry-Recent development. Observing and Understanding Sea Level Variations, 36 Apel, J. R (1987). Principles of Ocean Physics, pp. 634, Academic, San Diego Calif. Conkright M. E., R. A. Locarnini and H. E. Garcia et al (2002). WORLD OCEAN ATLAS 2001:Objective Analyses, Data Statistics, and Figures CD-ROM Documentation. National Oceanographic Data Center Internal Report 17 Dobslaw H., R Schwintzer and F. Barthelmes, et al (2004). Geostrophic Ocean Surface Velocities from TOPEX Altimetry, and CHAMP and GRACE Satellite Gravity Models. Scientific Technical Report, GeoForschungsZentrum F6rste C, F. Flechtner and R. Schmidt et al. (2005). A new high-resolution global gravity field model derived from combination of grace and champ mission and altimetry/gravimetry surface gravity data. European Geosciences Union, General Assembly 2005, Vienna, Austria, 24-29, April, 2005 Gruber T and P. Steigenberger (2002). Impact of new Gravity Field Missions for Sea Surface Topography determination, in: I.N. Tziavos (Ed.), Gravity and Geoid, 3rd Meeting of the International Gravity and Geoid Commission (IGGC), Univ. Of Thessaloniki, Greece, 320-325 Hernandez F. and P. Schaeffer (2001). The CLSO1 Mean Sea Surface." A validation with the GSFCO0 surface, in press. CLS Ramonville St Agne, France Hwang C. and Chen S. (2000). Circulation and eddies over the South China Sea derived from TPOEX/Poseidon altimetry. Journal of Geophysical Research, 105(C10): 23943~23965 Le Traon P. Y., P. Schaeffer and S. Guinehut, et al. (2001). Mean Ocean Dynamic Topography from GOCE and Altimetry, Int. GOCE Workshop, Noordwijk Pond, S. and L. Pickard (1978). Introductory Dynamic Oceanography. Pergamon Press. First Edition. Roemmich, D., and C. Wunsch, (1982). On combining satellite altimetry with hydrographic data, Journal Marine Research, 40, 605-619. Reigber C, Schwintzerl P and Stubenvolll R, et al. A high resolution global gravity field model combining CHAMP and GRACE satellite mission and surface gravity data. Joint CHAMP~GRACE Science Meeting, Solid Earth Abstracts, 2004, 17 Tapley B. D, D. P. Chambers and S. Bettadpur, et al. (2003). large-scale ocean circulation from the GRACE GGM01 geoid. Geophysical Research Letter, 30:22, 2163 Torge W (2001). Geodesy, de Gryter, Berlin Wahr J. M., S. R. Jayne and F. O. Bryan, (2002). A method of inferring changes in deep ocean currents from satellite measurements of time-variable gravity. Journal of Geophysical Research, vol. 107, C12:3218 Wang Y. M. (2001). GSFC00 mean sea surface, gravity anomaly, and vertical gravity gradient from satellite altimeter data. Journal of Geophysical Research, vol. 106 (C12): 31167-31174
Chapter 4
Sea Level in the British Isles" Combining Absolute Gravimetry and Continuous GPS to Infer Vertical Land Movements at Tide G a u g e s F.N. Teferle, R.M. Bingley, A.I. Waugh, A.H. Dodson Institute of Engineering Surveying and Space Geodesy University of Nottingham, University Park, Nottingham NG7 2RD, UK. S.D.R Williams, T.F. Baker Proudman Oceanographic Laboratory, Joseph Proudman Building, 6 Brownlow Street, Liverpool L3 5DA, UK.
Abstract.
The current terrestrial reference frame, current global GPS products and current precise GPS processing techniques, limit the determination of accurate, long-term, vertical station velocities from continuous GPS measurements on a global scale. Several authors have reported biases in their vertical station velocities determined from continuous GPS when compared to alternative geodetic methods. It has been argued that until these problems have been resolved, the study of relative land and sea level rates on regional scales is the only way to investigate vertical land movements at tide gauges co-located with continuous GPS. In the UK, we have been operating a network of continuous GPS and absolute gravimetry stations for the purpose of determining vertical land movements at tide gauges for almost ten years. This network consists of ten continuous GPS stations and three absolute gravimetry stations, all of which are either co-located or close to tide gauges. In this paper, we compare vertical land movements obtained from both geodetic methods with estimates of vertical land movements from high quality, independent geological and geophysical evidence, and derive a GPS-specific bias for which the estimates of vertical land movements from all continuous GPS stations are corrected. Based on recently published mean sea level trends by the Permanent Service for Mean Sea Level, we estimate a change in sea level, de-coupled from vertical land movements, for the British Isles.
Keywords. absolute gravimetry, continuous GPS, tide gauge, sea level, British Isles
1
Introduction
Recent studies of 20th century sea level from combined tide gauge and satellite altimetry measurements showed a global averaged rise in sea level of 1.7 to 1.8 mm/yr (Church et al., 2004; Holgate and Woodworth, 2004; White et al., 2005; Church and White, 2006). Furthermore, it was identified that sea level was not rising at the same rate everywhere, but was, e.g. slower in the east Atlantic and North Sea than in the west Atlantic (Church et al., 2004; Holgate and Woodworth, 2004). |n all such studies, evidence from tide gauges is used extensively, which must be de-coupled from the specific vertical land movements (VLM) occurring at each of the tide gauge sites. By using continuous GPS (CGPS) at or close to tide gauges (CGPS@TG) (e.g. Teferle et al., 2002a; Caccamise et al., 2005), estimates of VLM can be obtained from vertical station velocity estimates derived from height time series and used to correct the tide gauge records. However, it is now clear that biases related to, e.g. the terrestrial reference frame (e.g. Herring, 2001; Dong et al., 2002, 2003), the relative modelling of the phase centers at satellite and receiver antennas (e.g. Ge et al., 2005; Schmid et al., 2005) and the neglect of higherorder ionospheric terms (e.g. Fritsche et al., 2005), all affect vertical station velocity estimates in some form or another. These biases currently prevent estimates of VLM for CGPS@TG stations to be obtained with the accuracy required for sea level studies. As it is generally believed that these biases cancel out over small regions, Caccamise et al. (2005) argued that, currently, investigations into VLM and changes in sea level at CGPS@TG stations can only be carried out in a relative sense over regional, and not global, scales.
24
F.N. Teferle • R. M. Bingley. A. I. Waugh • A. H. Dodson • S. D. P. Williams. T. F. Baker
Absolute gravimetry (AG) measurements, however, do not suffer from the above biases, hence, vertical station velocity estimates from this technique provide accurate estimates of VLM, independent of the problems associated with the GPS. Excellent previous results (e.g. Williams et al., 2001; Lambert et al., 2006) confirm the ability of AG to deliver accurate estimates of VLM, and as this paper shows, have enabled an assessment of the vertical station velocity estimates based purely on CGPS and a demonstration of how CGPS and AG may be combined to obtain better estimates of the desired VLM. Since the mid 1990s the Institute of Engineering Surveying and Space Geodesy (IESSG) and the Proudman Oceanographic Laboratory (POL) have been using AG and CGPS to measure the VLM at or close to tide gauges in the UK. Teferle et al. (2002a,b) reported of the establishment of five CGPS@TG stations between 1997 and 1999 and produced the first vertical station velocity estimates for these. Since then, the IESSG and POL have set up five more CGPS@TG stations in the UK. Williams et al. (2001) discussed the establishment of three AG sites close to tide gauges in the UK in 1995 and 1996 and reported of their initial results. This paper presents an update from Teferle et al. (2006) and was prompted by new estimates of VLM and new mean sea level (MSL) trends from the Permanent Service for Mean Sea Level (PSMSL) for the CGPS@TG stations used.
2
CGPS
Measurements
and
Analysis
During the period from 1997 to 2005, the IESSG and POL established CGPS stations at ten tide gauges in the UK, namely Sheerness, Newlyn, Aberdeen, Liverpool, Lowestoft, North Shields, Portsmouth, Lerwick, Stornoway and Dover. All of these CGPS stations were established such that the GPS antennas were sited as close as possible to the tide gauge, i.e. within a few meters of the tide gauge itself. As the CGPS@TG stations Lerwick, Stornoway and Dover have only been established in late 2005, no data from these were used in this study. At Lerwick, data from a CGPS station, initially established in 1998 for the purpose of estimating integrated precipitable water vapour, are available for investigating VLM. This additional CGPS station is located within 5 km of the newly established CGPS@TG station and the GPS antenna is mounted on a survey monument connected to bedrock. Although, this means that this station is not monitoring
the VLM at the tide gauge directly, it is monitoring the underlying geophysical movements in the area (Teferle et al., 2002a). As the observation timespan for the CGPS@TG station Lerwick is still too short to give reasonable results and by assuming that there are no differential VLM between the two CGPS stations within the area, this study only uses data from the CGPS station 5 km from the tide gauge in Lerwick. An earlier study (Sanli and Blewitt, 2001) claimed to detect uplift of the tide gauge in North Shields by analysing episodic and six months of CGPS data collected over a period of only 2.5 years up to early 2000. In contrast, this analysis uses the same data as Sanli and Blewitt (2001) plus data from the period since the installation of the CGPS receiver in 2001. Furthermore, Teferle et al. (2003) showed that GPS data from North Shields are affected by severe radio frequency interference, and that potentially a much longer observation period will be needed in order to form any conclusions on the VLM at that tide gauge with confidence. Therefore, the results for North Shields and Portsmouth, which currently only has an observation timespan of 3.1years, are still considered as preliminary. To complement the eight CGPS stations of Aberdeen, Lerwick, Liverpool, Lowestoft, Newlyn, Portsmouth and Sheerness, the CGPS@TG station Brest in northern France, has also been used in this study, making a total of nine CGPS stations at or close to tide gauges (see Fig 1). The processing for all of the CGPS stations shown in Fig. 1 has been carried out using the GPS Analysis Software (Stewart et al., 2002), which was developed at the IESSG and uses the double-difference observable. The results presented in this study are based on 24-hour, dual frequency GPS data for the period up to November 2004. All CGPS data were processed along with data from IGS stations at Kootwijk, Onsala, Villafranca and Wettzell, and consistent coordinate time series for each of the CGPS stations were obtained in the ITRF2000 (Altamimi et al., 2002). More details of the GPS processing methodology applied can be obtained from Teferle (2003). In order to obtain accurate vertical station velocity estimates and realistic uncertainties from the CGPS results, it is important to understand the timecorrelated (coloured) noise content of height time series, as the often quoted statistical uncertainties, assuming time-uncorrelated (white) noise, can lead to largely optimistic error bounds (e.g. Langbein and Johnson, 1997; Zhang et al., 1997; Mao et al., 1999; Williams et al., 2004). By using Maximum-
Chapter 4 • Sea Level in the British Isles: Combining Absolute Gravimetry and Continuous GPS to Infer Vertical Land Movements at Tide Gauges
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In Fig. 2, and throughout the paper, all uncertainty values quoted are l-or. As can be seen from Fig. 2, all of the CGPS vertical station velocities are positive and their uncertainties are in the range from -+-0.4 to 4-0.8 mm/yr. A comparison of the CGPS vertical station velocities to values previously published is possible for Brest. For Brest, Sella et al. (2002) and Boucher et al. (2004) reported values of -4.2 -+- 3.4 mm/yr and -3.4 -+- 2.3 mm/yr, respectively. In both cases the large uncertainties are indicative of a much shorter observation time span compared to
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Fig. 2:ITRF2000 height time series and vertical station velocity estimates (rate) based on a Maximum-Likelihood estimation (MLE) using a white plus power-law noise model. The figure includes the height time series for Sheerness (SHEE), Lerwick (LERW), Newlyn (NEWL), Aberdeen (ABER), Liverpool (LIVE), Lowestoft (LOWE), Brest (BRST), North Shields (NSTG) and Portsmouth (PMTG). T is the time period used by this study and the vertical dashed lines represent times at which coordinate offsets were estimated.
25
26
F.N. Teferle • R. M. Bingley. A. I. Waugh • A. H. Dodson • S. D. P.Williams. T. F. Baker
this study and the suggested magnitude of subsidence is clearly not supported by the MSL trend for Brest, which is 1.0 + 0.1 mm/yr (PSMSL, 2005). 3
AG
Measurements
and
gauge measurements, which only began in the 1960s and show a fall of MSL from the 1960s to 1990s.
Lerwick 0.2 + 0.2 pgal/yr [-1.1 + 1.1 mm.yr]
Analysis
20
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POL began to make AG measurements near the tide gauges at Newlyn and Aberdeen in 1995 and at Lerwick in 1996 (Williams et al., 2001). These measurements are being made with the POL absolute gravimeter FG5-103, manufactured by Micro-g Solutions Inc., USA. A value of gravity is obtained every 10 s by dropping a test mass in a vacuum and using an Iodine stabilized He-Ne laser interferometer and rubidium atomic clock to obtain distance-time pairs and solve the equations of motion (Niebauer et al., 1995). AG measurements are taken for typically 3 to 4 days every year at each site. The sites were chosen to be on bedrock and FG5-103 is regularly inter-compared with other instruments in Europe and the USA to ensure that it gives consistent results at the 1 to 2 #gal level (Williams et al., 2001). The time series of AG values at Newlyn and Lerwick are of particularly high quality and are used in the present work (see Fig. 3). The uncertainties in the linear trends have been determined by combining an instrumental set-up error with a Gauss-Markov model for the coloured noise (Van Camp et al., 2004). Comparison
of
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Move-
ments
For the British Isles, a number of alternative, high quality and independent evidence of VLM have been published. These data include, estimates based o n : geological information (Shennan and Horton, 2002), denoted as GEO; the negative of the difference between the MSL trend measured by the tide gauge at each site (Woodworth et al., 1999; PSMSL, 2005) and the average sea level rise for Northern Europe of 1.5 mm/yr (Holgate and Woodworth, 2004), denoted as -(MSL-GSL); glacial isostatic adjustment models, e.g. Lambeck and Johnston (1995), Peltier (2001), denoted as GIA(L) and GIA(P) respectively. Table 1 and Fig. 4 show the independent estimates along with the values obtained from CGPS and AG. From Table 1, it is clear that there is generally good agreement between the estimates of VLM based on geology, tide gauges and GIA models, at all sites except Brest and Lerwick. Taking the estimates based on AG into account, these would seem to suggest that the anomaly at Lerwick is in the tide
Table 1" Vertical land movements comparison. All figures shown are in mm/yr. Station CGPS Sheerness 0.2-+-0.4 Lerwick 0.1±0.7 Newlyn 0.7±0.7 Aberdeen 1.4±0.7 Liverpool 1.2±0.7 Lowestoft 0.14-0.6 Brest 1.6+0.8 North Shields 1.2±0.7 Portsmouth 0.5±0.5 Station -(MSL-GSL) Sheerness -0.1 Lerwick 2.3 Newlyn -0.2 Aberdeen 0.6 Liverpool 0.1 Lowestoft - 1.0 Brest 0.5 North Shields -0.4 Portsmouth -0.3 afrom Woodworth et al. (1999)
AG -1.14-1.1 -0.5±0.9
GIA(L) -0.5 - 1.8 -1.0 0.0 -0.3 -0.5 -0.9 0.0 -0.5
GEO -0.7 -11 07 -0 2 -0 6 - 0 2a 02 -0 6 GIA(P) -02 -05 -03 06 04 -04 -03 04 -01
Chapter 4 • Sea Level in the British Isles: Combining Absolute Gravimetry and Continuous GPS to Infer Vertical Land Movements at Tide Gauges
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Fig. 4" Vertical land movement (VLM) estimates for nine CGPS stations close to or at tide gauge sites in the UK and northern France compared to alternative evidence for VLM. l-or uncertainties are shown where available. From Fig. 4, however, it is also clear that the CGPS estimates of VLM are systematically offset from the AG estimates of VLM and from the estimates of VLM from the other independent evidence. It would appear that the AG estimates of VLM are closely aligned to the estimates of VLM from geology and G|A models. Whereas, the CGPS estimates are in the range of 0.7 to 1.8 mm/yr greater than the geology estimates, 0.6 to 2.5 mm/yr greater than the GIA(L) estimates, and 0.4 to 1.9 mm/yr greater than the GIA(P) estimates. Similar offsets of CGPS estimates of VLM from independent evidence have also been reported by Prawirodirdjo and Bock (2004) and MacMillan (2004). From the analysis of a global CGPS network, Prawirodirdjo and Bock (2004) showed average offsets between CGPS and GIA(P) of 1.1 mm/yr for stations in North America, and 1.7 mm/yr for stations in northern Europe. Separately, in a comparison of VLBI and CGPS at 24 globally distributed sites, MacMillan (2004) showed the CGPS estimates
to be on average 1.5 mm/yr greater than the VLBI estimates. In considering the nature of the offsets apparent in our CGPS estimates, it is worth noting that the various sources of independent evidence have their own reference frames, e.g. the GIA(L) and GIA(P) estimates are based on GIA models for the last 10,000 years, and are referred to a Centre of mass of the Solid Earth reference frame (Blewitt, 2003; Dong et al., 2003), and the geology estimates are based on changes in sea level for the last 10,000 years, assuming no net global melting for the last 3,000 to 4,000 years. The ITRF2000 reference frame, to which our CGPS solutions are aligned, has an origin that is defined as the Center of Mass of the Earth Syswm (Blewitt, 2003; Dong et al., 2003), based on Satellite Laser Ranging. However, the CGPS estimates given in this paper are effectively referenced to their own realization of the ITRF2000 reference frame and, hence, their own definition of the centre of mass
27
28
F.N. Teferle • R. M. Bingley. A. I. Waugh • A. H. Dodson • S. D. R Williams. T. F. Baker
3
(CM), which depends mostly on the sub-set of 1GS stations constrained in ITRF2000 and partly on the fact that the IGS final orbit is in its own reference frame that is not exactly in ITRF2000. Hence, the CGPS estimates of VLM could be offset from the truth because the reference frame of the regional network solutions is not identical to ITRF2000, and because the origin of ITRF2000 varies with respect to the true CM (Blewitt, 2003; Dong et al., 2002, 2003).
5
Combining CGPS and AG and Computing a Sea Level Rise
Using weighted least-squares and data for Newlyn and Lerwick it is possible to compute an offset of 1.2+0.4mm/yr between the VLM estimates based on CGPS and those based on AG. This positive offset is consistent with that found in the comparisons between CGPS and the other independent evidence, both in this paper and e.g. Prawirodirdjo and Bock (2004) and MacMillan (2004). Therefore, the combination of CGPS and AG for tide gauges in the UK has been effected by aligning the CGPS estimates of VLM to the AG estimates. The MSL trends and the AG-aligned CGPS estimates of VLM have been used to compute an estimate of sea level rise for the British Isles. Fig. 5 shows the MSL trends compared to the negative of the AG-aligned CGPS estimates of VLM. As stated previously, the Lerwick tide gauge measurements only began in the 1960s and show a fall of mean sea level, which appears to be an anomaly specific to this tide gauge. Considering Sheerness, Newlyn, Aberdeen, Liverpool, Lowestoft, Brest, North Shields and Portsmouth, a sea level rise of 1.3 +0.3 mm/yr is obtained. Furthermore, the exclusion of data for Portsmouth and Lowestoft, the tide gauges with the shortest tide gauge records, does not change this estimate of 1.3 mm/yr.
6
Conclusions
This paper provides details of research that is ongoing in relation to the use of CGPS and AG for measuring vertical land movements (VLM) at tide gauges in the UK. Results, from CGPS time series dating back to 1997 and AG time series dating back to 1995/6, have been used to demonstrate the complementarity of these two techniques, and a series of AG-aligned CGPS estimates of VLM have been computed for nine tide gauges. An initial comparison between these estimates of VLM and changes in MSL observed by the tide gauges, suggests a change in sea level, de-coupled from VLM, as a rise of
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Fig. 5: MSL trends for seven tide gauges compared with the negative of the vertical station velocity estimates (E/S or emergence/submergence rates) based on AG-aligned CGPS. In this figure, SHEE is Sheerness, LERW is Lerwick, NEWL is Newlyn, ABER is Aberdeen, LIVE is Liverpool, LOWE is Lowestoft, BRST is Brest, NSTG is North Shields and PMTG is Portsmouth. Data points on the dashed line would imply a sea level rise of 1.0 mm/yr.
1.3 mm/yr for the British Isles. Clearly, the statistical significance of such results cannot be assured as yet, due to the level of the uncertainties in the CGPS and AG time series. However, these should reduce as the time series are extended into the future. It has recently been argued that studies of VLM using CGPS data and sea level changes at tide gauges are best performed on regional scales and more importantly only in a relative sense. However, the alignment procedure demonstrated in this paper has been shown to have the potential for determining sitespecific VLM at tide gauges, by using a combination of CGPS and AG. Furthermore, the procedure enables multiple CGPS stations to be deployed without the need for simultaneous AG measurements at each site.
Acknowledgements This research was funded by the Department for Environment, Food and Rural Affairs (Defra) R&D Projects FD2301 and FD2319. The CGPS data were taken from BIGF - British Isles GPS archive Facility (www.bigf.ac.uk) funded by the Natural Environment Research Council (NERC). The authors would
Chapter 4 • Sea Level in the British Isles: Combining Absolute Gravimetry and Continuous GPS to Infer Vertical Land Movements at Tide Gauges
also like to thank the two anonymous reviewers for their helpful comments.
References Altamimi, Z., P. Sillard, and C. Boucher (2002), ITRF2000: A new release of the International Terrestrial Reference Frame for Earth science applications. J. Geophys. Res., 107 (B 10), 2214, 10.1029/2001JB000561. Blewitt, G. (2003), Self-consistency in reference frames, geocenter definition, and suface loading of the solid Earth. J. Geophys. Res., 108 (B2), 2103, 10.1029/2002JB002082.
Ge, M., G. Gendt, G. Dick, F.P. Zhang, and C. Reigber (2005), Impact of GPS satellite antenna offsets on scale changes in global network solutions. Geophys. Res. Lett., 32, L06310, 10.1029/2004GL022224. Herring, T.A. (2001), Vertical reference frame for sea level monitoring. EOS Trans. AGU, 82 (47), Fall Meet. Suppl., Abstract G31D-07. Holgate, S. and P.L. Woodworth (2004), Evidence for enhanced coastal sea level rise during the 1990s. Geophys. Res. Lett., 31, L07305, 10.1029/2004GL019626.
Boucher, C., Z. Altamimi, P. Sillard, and M. Feissel-Vernier (2004), The ITRF2000, international Earth Rotation Service (IERS). Technical Note 31, Verlag des Bundesamts ftir Kartographie und Geod/isie, Frankfurt am Main.
Lambeck, K. and P.J. Johnston (1995), Land subsidence and sea-level change: Contributions from the melting of the last great ice sheets and the isostatic adjustment of the Earth. In: Land Subsidence, F.B.J. Barends, F.J.J. Brouwer, and F.H. Schr6der (eds.), 3-18, Balkema, Rotterdam.
Caccamise ||, D.J., M.A. Merrifield, M. Bevis, J. Foster, Y.L. Firing, M.S. Schenewerk, F.W. Taylor, and D.A. Thomas (2005), Sea level rise at Honolulu and Hilo, Hawaii: GPS estimates of differential land motion. Geophys. Res. Lett., 32, L03607,
Lambert, A., N. Courtier, and T.S. James (2006), Long-term monitoring by absolute gravimetry: Tides to postglacial rebound. J. Geodyn., 41,307317, 10.1016/j.jog.2005.08.032.
10.1029/2004GL021380. Church, J.A., N.J. White, R. Coleman, K. Lambeck, and J.X. Mitrovica (2004), Estimates of the Regional Distribution of Sea Level Rise over the 1950-2000 Period. J. Clim., 17, 2609-2625. Church, J.A. and N.J. White (2006), A 20th century acceleration in global sea-level rise. Geophys. Res. Lett., 33, L01602, 10.1029/2005GL024826. Dong, D., P. Fang, Y. Bock, M.K. Cheng, and S. Miyazaki (2002), Anatomy of apparent seasonal variations from GPS derived site position time series. J. Geophys. Res., 107(B4), 10.1029/2001JB000573. Dong, D., T. Yunck, and M. Heflin (2003), Origin of the International Terrestrial Reference Frame. J. Geophys. Res., 108(B4), 2200, 10.1029/2002JB002035. Fritsche, M., R. Dietrich, A. Kn6fel, A. Rfilke, S. Vey, M. Rothacher, and P. Steigenberger (2005), Impact of higher-order ionospheric terms on GPS estimates. Geophys. Res. Lett., 32, L23311,
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Langbein, J. and H.O. Johnson (1997), Correlated errors in geodetic time series: Implications for time-dependent deformation. J. Geophys. Res., 102 (B1), 591-603, 10.1029/96JB02945. MacMillan, D.S. (2004), Rate Difference Between VLBI and GPS Reference Frame Scales. Eos Trans. AGU, 85 (47), Fall Meet. Suppl., Abstract G21B-05. Mao, A., C.G.A. Harrison, and T.H. Dixon (1999), Noise in GPS coordinate time series. J. Geophys. Res., 104 (B2), 2797-2818. Niebauer, T.M., G.S. Sasegawa, J.E. Faller, R. Hilt, and F. Klopping (1995), A new generation of absolute gravimeters. Metrologia, 32, 159-180. Peltier, W.R. (2001), ICE4G (VM2) glacial isostatic adjustment corrections. In: Sea Level Rise History and Consequences. International Geophysics Series, 75, Academic Press, San Diego, 65-96. Penna, N.T. and M.R Stewart (2003), Aliased tidal signatures in continuous GPS height time series. Geophys. Res. Lett., 30(23), B08401, 10.1029/2004JB003390.
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Prawirodirdjo, L. and Y. Bock (2004), Instantaneous global plate motion model from 12 years of continuous GPS observations. J. Geophys. Res., 109 (8), B08405, 10.1029/2003JB002944. PSMSL (2005), Table of MSL secular trends derived from PSMSL RLR data [online]. Liverpool: Permanent Service for Mean Sea Level (PSMSL). Available at: < U R L : h t t p : / / w w w . p o l . a c . uk/psmsl/datainfo/rlr.trends> [Accessed 16 August 2005]. Sanli, D.U. and G. Blewitt (2001), Geocentric sea level trend using GPS and > 100-year tide gauge record on a postglacial rebound nodal line. J. Geophys. Res., 106 (B 1), 713-719, 10.1029/2000JB900348. Schmid, R., M. Rothacher, D. Thaller, and R Steigenberger (2005), Absolute phase center corrections of satellite and receiver antennas. GPS Sol., 9,283293, 10.1007/s 10291-005-0134-x. Sella, G.F., T.H. Dixon, and A. Mao (2002), REVEL: A model for recent plate velocities from space geodesy. J. Geophys. Res., 107 (B4), 10.1029/2000JB000033. Shennan, I. and B. Horton (2002), Holocene landand sea-level changes in Great Britain, J. Quaternary Sci., 17 (5-6), 511-526. Stewart, M.R, G.H. Ffoulkes-Jones, W.Y. Ochieng, P.J. Shardlow, N.T. Penna, and R.M. Bingley (2002), GAS: GPS Analysis Software version 2.4 user manual. IESSG, University of Nottingham, Nottingham, U.K. Teferle, F.N., R.M. Bingley, A.H. Dodson, and T.F. Baker (2002a), Application of the dual-CGPS concept to monitoring vertical land movements at tide gauges. Phys. Chem. Earth, 27, 1401-1406. Teferle, F.N., R.M. Bingley, A.H. Dodson, N.T. Penna, and T.F. Baker (2002b), Using GPS to separate crustal movements and sea level changes at tide gauges in the UK. In: H. Drewes, A.H. Dodson, L.P.S. Fortes, L. Sanchez and P. Sandoval (eds), Vertical Reference Systems. International Association of Geodesy Symposia, 124, SpringerVerlag, Heidelberg Berlin, 264-269. Teferle, F.N. (2003), Strategies for long-term monitoring of tide gauges with GPS. PhD thesis, University of Nottingham, [Available at: < URL : http ://etheses .nottingham. ac. uk>].
Teferle, F.N., R.M. Bingley, A.H. Dodson, P. Apostolidis, and G. Staton (2003), RF Interference and Multipath Effects at Continuous GPS Installations for Long-term Monitoring of Tide Gauges in UK Harbours. In: Proc. 16th Tech. Meeting of the Sat.
Div. of the Inst. of Navigation, ION GPS/GNSS 2003, Portland, Oregon, 9-12 September 2003, pp. 12. Teferle, F.N., R.M. Bingley, S.D.R Williams, T.F. Baker, and A.H. Dodson (2006), Using continuous GPS and absolute gravity to separate vertical land movements and changes in sea level at tide gauges in the UK, Phil. Trans. R. Soc. A, 364, 1841, 10.1098/rsta.2006.1746. Van Camp, M., S.D.R Williams, and O. Francis (2005), Uncertainty of absolute gravity measurements. J. Geophys. Res., 110, B05406, 10.1029/2004JB003497. White, N.J., J.A. Church, and J.M. Gregory (2005), Coastal and global averaged sea level rise for 1950 to 2000. Geophys. Res. Lett., 32, L01601, 10.1029/2004GL021391. Williams, S.D.P. (2003), The effect ofcoloured noise on the uncertainties of rates estimated from geodetic time series. J. Geod., 76 (9-10), 483-494. Williams, S.D.P., T.F. Baker, and G. Jeffries (2001), Absolute gravity measurements at UK tide gauges. Geophys. Res. Lett., 28 (12), 2317-2320, 10.1029/2000GL012438. Williams, S.D.R, Y. Bock, P. Fang, P. Jamason, R.M. Nikolaidis, L. Prawirodirdjo, M. Miller, and D.J. Johnson (2004), Error analysis of continuous GPS position time series. J. Geophys. Res., 109(B3), B03412, 10.1029/2003JB002741. Woodworth, P.L., M.N. Tsimplis, R.A. Flather, and i. Shennan (1999), A review of the trends observed in British Isles mean sea level data measured by tide gauges. Geophys. J. Int. 136, 651-670. Zhang, J., Y. Bock, H.O. Johnson, P. Fang, S.D.P. Williams, J. Genrich, S. Wdowinski, and J. Behr (1997), Southern California Permanent GPS Geodetic Array: Error analysis of daily position estimates and site velocities. J. Geophys. Res., 102(B8), 18035-18055, 10.1029/97JB01380.
Chapter 5
Snow Accumulation and Snowmelt Monitoring in Greenland and Antarctica S. V. Nghiem 1, K. Steffen 2, G. Neumann 1, and R. Huff2 ~Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, 300-235, Pasadena, CA 91007, U.S.A. 2Cooperative Institute for Research in Environmental Sciences, University of Colorado, Campus Box 216, Boulder, CO 80309-0216, U.S.A.
Abstract. Snow deposition, accumulation, and melt on an ice sheet are key components of mass balance. Innovative algorithms using satellite scatterometer data have been developed to monitor snowmelt, ice layer extent, and snow accumulation on Greenland with verifications using in-situ data from the Greenland Climate Network (GC-Net). QuikSCAT/SeaWinds Scatterometer (QSCAT) has collected data over Greenland and Antarctic two times per day since July 1999. QSCAT data show the shortest melt season in 2004, verified by GCNet data at ETH/CU Camp, and detect peculiar snowmelt during wintertime in Greenland in 2005. QSCAT results reveal a record increase in the snow accumulation rate on the Greenland ice sheet including the west flank in January-March 2005 with an estimate of 565 km 3 of total snow accumulation volume. The record snow anomaly is verified by GC-Net snow measurements, showing the largest snow accumulation rate in the first half of 2005 ever recorded in the past decade since the inception of the GC-Net. The QSCAT algorithms developed for Greenland are adapted for Antarctica. QSCAT results show strong melt in 2002 and prolonged melt in 2005 at McMurdo. New extensive ice layers, created by refreezing of melt water in the firn layer, were identified by QSCAT along the Antarctica Walgreen, Bakutis, and Hobbs coasts extending well inland in 2005. Extensive regions of ice layering, evidence of preceding strong melt occurrence, were also found over the Rockefeller Plateau and along the Ross Ice Shelf adjacent to Queen Maud Mountains in 2005.
Keywords. QuikSCAT, ICESat, ice mass balance, snowmelt, snow accumulation, sea level, Greenland, Antarctica
1 Introduction The possibility of future sea level rise necessitates a precise knowledge of the mass balance of the large ice sheets in Greenland and Antarctica. Understanding the ice-sheet melt characteristics is critical to the assessment of ice sheet mass balance and the interpretation of mass balance observations. Because of the positive albedo feedback associated with snow melt and the fact that wet snow absorbs as much as three times more incident solar energy than dry snow (Steffen (1995)), ice sheet melt characteristics play a major role in the energy and mass exchanges at the ice sheet surface. Moreover, surface melt can act to enhance the flow of outlet glaciers through crevasse overdeepening (Robin, (1974), van der Veen (1998)) and is believed to have contributed to the very rapid thinning of a number of outlet glaciers in Eastern Greenland (Krabill et al. (1999)) due to increase of ice velocity (Zwally et al. (2002)). As a result, not only is ice sheet melt directly tied to ablation through surface runoff, but is also indirectly linked to fresh water discharge in forms of ice bergs through the potential increase in ice velocity of tide water glaciers. The northern parts of the Greenland ice sheet and most of Antarctica have low accumulation rates. Consequently, it takes a long time for the temperature signal of the currently warming climate to penetrate into these ice masses; hence, the effects are unlikely to be very large over a century timescale. However, in areas with increased melt water percolation as observed over large regions of the Greenland ice sheet and some low-lying coastal regions of Antarctica, including the Peninsula, the effect of latent heat release is likely to cause a faster response in the thermal regime. In a warming climate, we would expect an increase in vapor flux convergence that may deposit more snow at higher-
32
S.V. N g h i e m • K. Steffen • G. N e u m a n n • R. Huff
altitude regions. These regions are extensive and any change in snow accumulation is crucial to the total mass balance. Furthermore, monitoring the Greenland and Antarctic ice sheet melt characteristics and snow accumulation can facilitate the assessment and interpretation of satellite altimeter data (e.g., ICEsat) in time and in space.
2 Approach 2.1 Snowmelt Algorithm Detection of surface melt at large spatial scales is most effectively accomplished through the use of wide-swath satellite microwave data. There are several approaches to detect and map different snowmelt conditions on ice sheets. Passive microwave data have a clear melt signature due to strong microwave absorption by wet snow during melt onset (M~itzler and Happi (1989)). As such, wet-snow emission approaches the black body behaviour. This change in emission characteristics is detectable by most microwave sensors at frequencies above 10 GHz. (Ulaby et al. (1986)). Passive microwave data provide a long-term record since 1979 for snowmelt assessment (Mote and Anderson (1995), Abdalati and Steffen (1995)). In particular, the cross-polarized gradient ratio (XPGR) algorithm was developed by Abdalati and Steffen (1995) for consistent snowmelt detection. Here, we use an active microwave algorithm for snowmelt mapping based on diurnal change in QSCAT backscatter data (Nghiem et al. (2001), Steffen et al. (2004)). Melt areas are detected based on diurnal change in QSCAT backscatter caused by diurnal differential wetness in melting snow. The advantages of the QSCAT algorithm include: high sensitivity to snowmelt allowing daily delineation of snowmelt and refreezing, applicability to all areas and facies of the ice sheet, and independence from absolute sensor calibration (avoiding inconsistency due to long-term drift in sensor gain factor). Results show that QSCAT can detect early melt and more extensive melt areas. QSCAT melt results correspond to locations where ice surface albedo switches to a low value due to melt, while XPGR results are consistently coincident with melt areas undergoing 10 days to two weeks or longer of melting detected by QSCAT (Steffen et al. (2004)). Thus, for the first time, QSCAT independently
verifies the consistency for which the XPGR algorithm is designed. We use the QSCAT algorithm to obtain updated snowmelt results including 2005.
2.2 Snow Accumulation Algorithm Snow accumulation (SA) rate has been estimated with a C-band synthetic aperture radar in the dry snow zone in Greenland (Munk et al. (2003)). A new method has been developed to measure SA in the percolation zone using QSCAT (Nghiem et al. (2005)), and the results are verified with in-situ data from the GC-Net. The QSCAT algorithm for SA measurement (Nghiem et al. (2005)) relies on the attenuation decreasing QSCAT backscatter as snow accumulates on an ice layer formed during a past melt season. An ice layer consists of large scatterers such as clumps of coalesced ice grains, icicles, ice columns, or ice lenses formed by the percolation of melt water that refreezes in the firn layer. Such scatterers in the ice layer can dominate radar backscatter (Jezek et al. (1993)), which is attenuated by the overlaying SA. GC-Net data are used to determine the attenuation coefficient and QSCAT data measure the backscatter decrease, which is inverted into SA with the GC-Net derived coefficient (Nghiem et al. (2005)). The QSCAT algorithm is applicable in the percolation zone where the last (or previous year's) melt created an ice layer. It is noted that the strong and extensive melt in 2002 (Steffen et al. (2004), Nghiem et al. (2005)) significantly increases the region of validity for the QSCAT algorithm well into the traditional dry-snow zone, defined by Benson (1962), while reducing the validity region of the C-band radar approach. We will use the QSCAT algorithm to derive new SA results over Greenland, and adapt the algorithm to obtain initial SA results around McMurdo in Antarctica.
2.3 Verification Approach We will verify our results using in-situ measurements from networks of climate and meteorological stations. Over Greenland, we utilize data from the GC-Net, established in 1995 to monitor climatological and glaciological parameters at various locations on the ice sheet (Steffen and Box (2001)). GC-Net currently consists of 18
Chapter 5 • Snow Accumulation and Snowmelt Monitoring in Greenland and Antarctica
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descending (des) orbit passes, and (c) air temperature at 2 m above the surface.
automatic weather stations distributed over the entire Greenland ice sheet. The GC-Net objectives are to measure hourly, daily, annual and interannual variability in accumulation rate, surface climatology and surface energy balance parameters at selected locations on the ice sheet, and to monitor nearsurface snow temperatures at the automated weather station (AWS) locations for the assessment of snow densification, accumulation, and metamorphosis. The National Oceanic and Atmospheric Administration (NOAA) National Climatic Data Center (NCDC) maintains the Global Summary Of the Day (GSOD) dataset consisting of meteorological data measured by weather stations in the World Meteorological Organization/Global Telecom-munications System network. Data from NCDC/GSOD include snow depth, precipitation, temperature, humidity, dew point, pressure, and other meteorological data. NCDC/GSOD includes more than 50 stations in Antarctica. To compare satellite and station measurements, we have developed the Special Satellite Station Processing (SSSP) at the Jet Propulsion Laboratory. SSSP uses a mask of stations with a selectable radius up to 60 km around each station, and an effective pointer
algorithm is specifically designed to collect and collocate massive time-series satellite data together with in-situ station data. We apply the SSSP to selected stations from GC-Net and NCDC/GSOD to obtain the results presented in this paper. 3 3.1
Results Greenland
Recent results from QSCAT data revealed the most extensive melt in 2002 when melt areas penetrated well into the traditional dry-snow zone, especially in the northeast region of the Greenland ice sheet (Steffen et al. (2004)). The 2002 melt zone almost connected the west and east coast across the central region at about latitude 75°N (Figure 5 in Nghiem et al. (2005)). The 2002 anomalous melt also created a new significant ice layer in vast areas in the dry snow zone on both the east and west sides of the ice sheet. The QSCAT SA algorithm captures an extreme snowfall in mid-April 2003 when half a meter of snow deposited on the southeast side of the ice sheet in a single day (Nghiem et al. (2005)). Moreover, the semi-annual SA rate from October
33
34
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Fig. 2 Snow accumulation rates obtained by semi-annual linear regression from GC-Net data at the AWS locations: (a) NASA-U at 73.83°N and 49.5°W, (b) Crawford Point 1 at 69.88°N and 46.97°W, and (c) DYE-2 at 66.48°N and 46.28°W. Light-grey bars are for January-June and dark-grey bars are for July-December. QuikSCAT snow accumulation rate is estimated over an area with a radius of 25 km around each station. The left panel shows the map of the GC-Net station locations.
2002 to March 2003 doubled the normal amount in a southeast region, while the SA on the southwest side is less than other years (Nghiem et al. (2005)). This SA anomaly significantly impacts the regional mass balance in the 2002-2003 season. New results from the latest QSCAT data detected peculiar melt in wintertime on the west flank of the ice sheet. The SSSP provides QSCAT satellite backscatter and GC-Net measurement time-series at the ETH/CU Camp AWS (69.57°N, 49.32°W) in 2004-2005 in Figure 1, which extends the result by Steffen et al. (2004) to six years of data since the start of QSCAT. Compared to the past melt season lengths in 2000-2003 (Steffen et al. (2004)), the melt season length in 2004 (26 May to 23 August) of 90 days is the shortest. In 2005, the melt onset occurred on 16 May, followed by a period of partial re-freezing and then a strong melt on 9 June 2005. QSCAT melt results are verified with the AWS temperature data (Figure l c). Moreover, results at ETH/CU AWS capture a peculiar winter melt event in February 2005 (Figure 1), observed for the first time by QSCAT in the last six years. The associated QSCAT diurnal signature on 19 February 2005 (Figure lb) has negative value less than -2 dB indicating a warmer condition in the early morning
(-6:20 am) compared to that in the afternoon (-6:20 pm). This winter melt event was caused by a strong katabatic storm with hourly mean wind speeds exceeding 25 m s-1. The 2005 wintertime melt was also measured by QSCAT and verified by GC-Net temperature data at JAR l, JAR 2, and JAR 3 AWS locations, all located on the west flank of the Greenland ice sheet in a line from the ETH/CU station towards the coast. The west flank underwent a significant anomaly in snow accumulation in 2005. QSCAT results at NASA-U AWS location indicated an SA rate of 1.6 m/year from January to June 2005, which is the same SA rate measured by the in-situ instrument for snow height monitoring at the NASA-U AWS location. In general, SA measured by QSCAT is within 10% of SA measured by GC-Net AWS (Nghiem et al. (2005)). The anomalous SA of 1.6 m/year in January-June 2005 is double the longterm average of 0.8 m/year at NASA-U. This record amount of SA is the highest observed in the full data record at NASA-U over the past decade (Figure 2). At Crawford Point 1, the AWS SA rate of 3.1 m/year for January-June 2005 is the highest record in the decadal dataset, which is significantly larger than the second record of 2.3 m/year in the
Chapter 5
2003
0.25
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•
Snow Accumulation and Snowmelt Monitoring in Greenland and Antarctica
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Fig. 3 Snow accumulation maps from QSCAT snow-accumulation algorithm for the period from 1 January to 31 March in 2003 (left panel), in 2004 (center), and in 2005 (right). The grey scale is for snow depth in meter.
first half of 1999. At DYE-2, the AWS SA rates for January-June 2005 is the largest during the first half of all year since 1996, and among the top four values of all semi-annual SA rates (Figure 2). QSCAT SA rates are close to the AWS values at both Crawford Point 1 and DYE-2, but the QSCAT SA is below the AWS value at Crawford 1 and above the AWS value at DYE-2. Furthermore, another confirmation of the record snow in 2005 is provided by measurements at the NCDC/GSOD Station 042200 (Aasiaat and Egedesminde, 68.7°N, 52.85°W, west coast of Greenland), which show the record snow depth of 0.8 m and the highest value of mean snow depth over winter and spring seasons in 2005 throughout the entire data time-series since 1994. Note that NASA-U, Crawford Point 1, and DYE2 mostly line up from north to south extending about one third of the length of the Greenland ice sheet. However, GC-Net measurements represent localized conditions at each station location. Without extensive spatial data, it is not possible to connect point measurements from the three local locations (NASA-U, Crawford Point 1, and DYE-2)
to extrapolate over a large region with a high confidence level. Here, QSCAT enables the continuous spatial mapping of SA to reveal the enormous region of the SA anomaly in 2005, in particular over the west flank of the Greenland ice sheet. Figure 3 presents SA maps of areas that have more than 0.3 m of SA for a 3-month period from January to March in 2003, 2004, and 2005. These QSCAT maps of SA reveal: (1) the extensive snow accumulation anomaly in 2005, (2) large spatial variability of SA in each year, and (3) large interannual variability in SA during January-March of 2003-2005. These characteristics of SA are also supported by GC-Net results in Figure 2. Results in Figure 3 also show more SA in southeast Greenland that is consistent with the general pattern of snow climatology in Greenland (Bales et al., (2001)). Integrated over the full surface area, QSCAT results yield a January-March SA volume of: 196 km 3 for 2003, 280 km 3 for 2004, and 565 km 3 for 2005. The results are conservative estimates accounting only for areas that have SA larger than 0.3 m (QCAT SA algorithm can be improved in the future for more accurate results over areas with low
35
36
S.V. N g h i e m • K. Steffen • G. N e u m a n n • R. Huff
SA rate). Assuming a bulk density of 330 kg m-3 (Krabill et al. (2004)), the corresponding snow accumulation mass is 6.47x1013 kg, 9.24x1013 kg, and 18.6x 1013 kg in the first quarter of 2003, 2004, and 2005, respectively. Compared to the estimate of total Greenland glacier discharge by calving (Reeh et al. (1999)) of 263 km 3 per year, equivalent to a mass of 5.92x 1013 kg per quarter or a potential sea level rise of 0.16 mm per quarter, the SA in the first quarter of 2005 is more than 3 times larger. 3.2 A n t a r c t i c a
We adapt the QSCAT algorithms to observe snowmelt, ice layer extent, and snow accumulation in Antarctica. Snowmelt is detected when the absolute value of the diurnal backscatter difference is larger than 1 dB and refreezing when the diurnal change decrease to below 1 dB from the melting conditions. At McMurdo, QSCAT data show no detectable snowmelt in the austral summer seasons of 1999-2000 and 2000-2001, and then melting occurred consecutively during the last 4 summer seasons. The strongest melt was in the austral summer of 2001-2002, which preceded the anomalous melt in Greenland in 2002. Air temperature measured at McMurdo (data from NCDC/GSOD Station 896640) was as high as 10.5°C (51 F) on 30 December 2001, corresponding to the strong melt indicated by a sharp drop in QSCAT backscatter. This strong melt created a significant ice layer identified by a large increase in backscatter after the last melt date in January 2002.
on 16 January 2005 (AZ'm=40 days) and is by far the longest one observed by QSCAT since the launch of the satellite mission (Table 2). These melt results are consistent with NCDC/GSOD temperature data measured at McMurdo. The sharp increase in backscatter (-2dB) at McMurdo in January 2005 indicates the formation of a new significant ice layer (Nghiem et al. (2005)) caused by the prolonged melt. In east Antarctica, QSCAT detected the first melt ever recorded at NCDC/GSOD Station 893320 (Elizabeth, 82.62°S, 137.08°W, 549 m in elevation) in January 2005 over the entire six-year record of QSCAT data since July 1999. This melt event was so strong that QSCAT backscatter decreased by more than 13 dB (a decreasing factor of 20 times). This result is verified by data at Station 893320 with a record air temperature of 5.3°C in the 11-year dataset.
hts Coast
/algreen Coast
Queen Maud Mts •
R
er
Table 2. Timing of snowmelt onset and freeze-up at McMurdo (77.85°S, 166.67°E) in Antarctica detected by QSCAT data from July 1999 to June 2005. There was no melt before December 2001 detectable with QSCAT data.
AZ'm
Melt onset
Freeze-up
I 0.5
22 19 01 40
27 06 03 07
18 Jan 25 Jan 04 Jan 16 Jan
Fig. 4 QSCAT map of areal extent (grey areas) of ice layer formed by the 2004-2005 melt season in Antarctica. The grey scale represents the backscatter increase in dB.
days days days days
Dec 2001 Jan 2003 Jan 2004 Dec 2004
2002 2003 2004 2005
In Table 2, the melt season length A'rm(Nghiem et al., (2005)) is defined as the time period between the first and the last melt date, within which multiple melt events can occur. The most recent melt season started on 7 December 2004 and ended
1.0
I 1.5
Figure 4 shows QSCAT ice-layer extent over west Antarctica created by the refreezing of melt water in the firn layer. The ice layer is detected by the strong increase in bi-weekly backscatter signatures before (October) and after (February of the next year) the melt season. QSCAT results show
Chapter 5
extensive melt along the Antarctica Walgreen, Bakutis, and Hobbs coasts extending well inland in 2005 (Figure 4). Extensive melt regions are also detected by QSCAT over the Rockefeller Plateau and along the Ross Ice Shelf adjacent to the Queen Maud Mountains in 2005 (Figure 4). Assuming the same attenuation coefficient derived for Greenland, QSCAT data give a snow accumulation depth of 1.24 m over the McMurdo area during the 325 days between 3 February 2002 and 25 December 2002. This is equivalent to a snow accumulation rate of 1.4 m/year for the period of February-December 2002, which was a heavy snow season after the unusually warm summer. However, the precision of this result is uncertain, and a future field experiment is necessary to calibrate an accurate snow attenuation coefficient for Antarctica snow accumulation inversion and to validate QSCAT SA inversion results. In this regard, QSCAT data can be used to select appropriate locations with different snow accumulation depths for such a field campaign.
• Snow
Accumulation and Snowmelt Monitoring in Greenland and Antarctica
The combination of GC-Net and QSCAT results provides an alternative for independent verification of altimetry results. GC-Net time-series data, consisting of thousands of data points in time, are used to calibrate QSCAT results over a long time period to determine QSCAT accuracy. Since QSCAT has large and frequent coverage in time and space (two times per day over Greenland and Antarctica), calibrated QSCAT results can be compared with altimeter results integrated over the QSCAT pixel in which altimeter data are collected. Since this is a statistical method over a large spatial extent (25 km by 25 km for a QSCAT pixel over large spatial coverage), effects of uncertainties in altimetry data location are minimized while the capability to collocate data in time and space are maximized for the comparison and verification of results. Once validated, altimeter results can provide high-resolution observations at a sub-pixel scale relative to scatterometer measurements. Moreover, a laser altimeter can measure snow height regardless of surface melt conditions when the scatterometer approach is not applicable.
4 Combination with Altimetry 5 Summary and Conclusions Snowmelt and snow accumulation results derived from QSCAT data will facilitate the interpretation of satellite laser altimetry and radar altimetry data. First, melt water formed at the surface may percolate into the snowpack and refreeze to form ice lenses and glands. This densification mechanism can decrease the ice surface height without loss of ice mass. The densification can result in surface lowering, which over a prolonged period of warming will become important for mass balance estimates from satellite altimeters. Frequent QSCAT snow accumulation maps over extensive areas will greatly complement altimeter data that are limited in spatial and temporal coverage. A satellite laser altimeter collects data points along surface ground tracks, and a satellite radar altimeter acquires data with a very narrow beam measuring surface profiles. Ascending and descending orbit passes provide crossover points on the surface where data can be compared for accuracy assessment. Because of the narrow surface profiles, a limited number of crossover points, and uncertainties in data locations, it is difficult to have altimetry data collocated with accurate in-situ or field measurements for result validation in time and in space.
This paper addresses two important components of mass balance using QSCAT data: snowmelt and snow accumulation in Greenland and Antarctica. The innovative scatterometry approach can create a new dataset of snow accumulation to compare with glacier discharge. Such data are continuous and extensive in space and in time, over 6 years so far and extending into a decadal time-series. In Greenland, QSCAT results show the record melt in 2002, prolonged melt in 2003, and peculiar melt in winter 2005. Snow accumulation maps are obtained over Greenland in the first quarter of each year in 2003-2005. The record SA in JanuaryMarch 2005 supplied a massive amount of total snow mass compared to total Greenland glacier discharge by calving over the quarterly period. The record SA results are verified with GC-Net and NCDC/GSOD data. The 2005 snow anomaly highlights the importance of SA in the total estimate of mass balance and sea level change. For major catchment basins of the Greenland ice sheet (Thomas et al. (2001)), QSCAT SA can provide the crucial input to assess mass balance of each basin. In Antarctica, QSCAT detected the strongest melt in the austral summer of 2001-2002 and the longest
37
38
S.V. Nghiem • K. Steffen • G. Neumann • R. Huff
melt season in 2004-2005 at M c M u r d o . Both the strong melt and the long melt created significant ice layer due to the refreezing o f melt water over the area around M c M u r d o . The first map of ice layer extent in Antarctica is derived from Q S C A T data. This map reveals the strong impact o f s n o w m e l t in extensive regions from coastal zones extending well in land. The first initial estimate o f snow a c c u m u l a t i o n is obtained over the area around McMurdo assuming the Greenland snow attenuation coefficients. H o w e v e r , field c a m p a i g n and in-situ m e a s u r e m e n t s are necessary to calibrate and validate Q S C A T results for snow accumulation. To derive results over full annual cycles, each pixel in G r e e n l a n d or Antarctica should be treated with the full time-series Q S C A T data to m o n i t o r melt and re-freezing so that the SA algorithm can be applied to the freezing conditions, over which the backscatter is stable. Effects due to a z i m u t h angle and seasonal temperature should be accurately accounted for improving the performance o f the SA algorithm. M e t h o d s to c o m b i n e satellite altimeter and scatterometer data should be developed to obtain more precise results over extensive coverage. Accurate estimates o f s n o w m e l t and snow a c c u m u l a t i o n will advance the understanding and determination of mass balance and sea level change.
Acknowledgment. The
research carried out at the Jet Propulsion Laboratory, California Institute of T e c h n o l o g y , was supported by the National Aeronautics and Space Administration (NASA). The research carried out at the Cooperative Institute for Research in E n v i r o n m e n t a l Sciences, University of Colorado, was also supported by N A S A .
References
Abdalati, W., and K. Steffen (1995). Passive microwavederived snow melt regions on the Greenland ice sheet, Geophys. Res. Lett., Vol. 22, pp. 787-790. Bales, R. C., J. R. McConnell, E. Mosley-Thompson, and B. Csatho (2001). Accumulation over the Greenland ice sheet from historical and recent records, J. Geophys. Res., Vol. 106, pp. 33813-33825. Benson, C. S. (1962). Stratigraphic Studies in the Snow and Firn on the Greenland Ice Sheet, Res. Report 70, Snow, Ice and Permafrost Research Establishment, Corps Eng., US Army, 1962. Jezek, K. C., M. R. Drinkwater, J. P. Crawford, and R. Kwok (1993). Analysis of synthetic aperture radar data collected over the southwestern Greenland ice sheet, J. Glac., Vol. 39, No. 131, pp. 119-132.
Krabill, W., E. Frederick, S. Manizade, C. Martin, J. Sonntag, R. Swift, R. Thomas, W. Wright, J. Yungel (1999). Rapid thinning of parts of the southern Greenland ice sheet, Science. Vol. 283, pp. 1522-1524. Krabill, W., E. Hanna, P. Huybrechts, W. Abdalati, J. Cappelen, B. Csatho, E. Frederick, S. Manizade, C. Martin, J. Sonntag, R. Swift, R. Thomas, and J. Yungel (2004). Greenland Ice Sheet: increased coastal thinning, Geophys. Res. Lett., Vol. 31, L24402, doi:10.1029/ 2004GL021533. Mfitzler, C. H., and R. Huppi (1989). Review of signature studies for microwave remote sensing of snowpacks, Adv. In Space Res., Vol. 9, pp. 253-265. Mote T. L., and M. Anderson (1995). Variations in snowpack melt on the Greenland ice sheet based on passivemicrowave measurements. J. Glaciol., Vol. 41, pp. 51-60. Munk, J., K. C. Jezek, R. R. Foster, and S. P. Gogineni, An accumulation map for the Greenland dry-snow facies derived from spaceborne radar (2003). J. Geophys. Res., Vol. 108, No. D9, art.4280. Nghiem, S. V., K. Steffen, R. Kwok, and W.Y. Tsai, Detection of snow melt regions on the Greenland ice sheet using diurnal backscatter change (2001). J. Glac., Vol. 47, No. 159, 539-547. Nghiem, S. V., K. Steffen, G. Neumann, an R. Huff, Mapping of ice layer extent and snow accumulation in the percolation zone of the Greenland ice sheet (2005). J. Geophys. Res., Vol. 1 1 0 , F03017, doi: 10.1029/ 2004JF000234. Reeh, N., C. Mayer, H. Miller, H. H. Thomsen, and A. Weidick (1999). Present and past climate control on ~ord glaciations in Greenland: implications for IRD-deposition in the sea, Geophys. Res. Lett., Vol. 26, pp. 1039-1042. Robin, G. de Q., Depth of water-filled crevasses that are closely spaced (1974). J. Glaciol., Vol. 13, pp. 543. Steffen, K. (1995). Surface energy exchange during the onset of melt at the equilibrium line altitude of the Greenland ice sheet, Ann. Glaciol., Vol. 21, pp. 13-18. Steffen, K, and J. E. Box (2001). Surface climatology of the Greenland ice sheet: Greenland climate network 19951999, J. Geophys. Res., Vol. 106, pp. 33065 - 33982. Steffen, K., S. V. Nghiem, R. Huff, and G. Neumann (2004). The melt anomaly of 2002 on the Greenland ice sheet from active and passive microwave satellite observations, Geophys. Res. Lett., Vol. 31, L20402, doi:10.1029/ 2004GL020444. Thomas, R., B. Csatho, C. Davis, C. Kim, W. Krabill, S. Manizade, J. McConnell, and J. Songtag (2001). Mass balance of higher-elevation pasrts of the Greenland ice sheet, J. Geophys. Res., Vol. 126, pp. 33707-33716. Ulaby, F.T, R.K. Moore, and A.K. Fung (1986). Microwave Remote Sensing. Vol. 3, From Theory to Applications, 1120 pp., Artech House Inc., Norwood, MA. van der Veen, C.J., (1998). Fracture mechanics approach to penetration of surface crevasses on glaciers. Cold Regions Sci. Tech., Vol. 27, pp. 31-47. Zwally, H.J.W. Abdalati, T. Herring, K. Larsen, J. Saba, and K. Steffen (2002). Surface melt-induced acceleration of Greenland ice-sheet flow, Science, Vol. 297, pp. 218-222.
Chapter 6
Estimating Recent Global Sea Level Changes H.-P. Plag Nevada Bureau of Mines and Geology and Seismological Laboratory, University of Nevada, Reno, Mailstop 178, Reno, NV 89557, USA, e-mail:
[email protected].
Abstract An empirical model for sea level trends over several decades is set up such that it is consistent with the global pattern of Local Sea Level (LSL) trends observed by the global network of tide gauges. The forcing factors taken into account are steric sea level variations, present-day ice load changes, and post-glacial rebound. Model parameters are determined in a least squares fit of the model to the LSL trends. The model allows the determination of the contribution of each factor to the global average LSL trend. Here we compare the solutions for two different LSL trend sets, namely one determined without and one with taking into account local atmospheric forcing at the tide gauges (denoted here as T1 and T2). From the globally given model, the global average trend over the last 50 years in LSL is found to be of the order of 1.05 + 0.75 mm/yr and 1.20 + 0.70 mm/yr for T1 and T2, respectively. For T1, the contribution of the Antarctic and Greenland ice sheets to the global average are 0.39 + 0.11 mm/yr and 0.10 + 0.05 mm/yr, respectively and for T2 0.31 + 0.16 mm/yr and 0.16 -+- 0.03 mm/yr, respectively. Using T1, the contribution from steric change is clearly identified and found to be at least 0.2 mm/yr with the most likely value being close to 0.35 mm/yr. For T2, there is no correlation between the spatial pattern of the observed LSL trends and the steric sea level trends, and the steric contribution to the global average turns out to be equal to zero. This result indicates a very high correlation between the local atmospheric forcing and the thermosteric sea level changes, which may be the result of a feedback of temperature changes in the upper layer of the ocean into the air pressure and wind field over the ocean. Keywords: global sea level rise, local sea level trends, ice sheets changes
1
Introduction
The mass balance of the global water cycle is of paramount interest for understanding, predicting and mitigating the impact of climate change. Understanding the sea level changes over the last decades and century is a prerequisite for quantifying climaterelated changes in the oceans volume and mass, as
well as for establishing future sea level scenarios. Church et al. (2001) emphasize the considerable uncertainties in the mass balance of the ocean and, in consequence, the global sea level. In particular, the contribution of the large ice sheets to current sea level changes is rather uncertain. The global mass and volume of the ocean are two absolute quantities characterizing the ocean as a reservoir in the global hydrological cycle. Changes in these quantities are directly related to changes in the hydrological cycle and therefore to climate change. Local Sea Level (LSL), which is defined here as the (absolute) vertical distance between the surface of the ocean and the surface of the solid Earth, depends on the distribution of the ocean water in a given topography of the Earth surface. Thus, LSL depends on many different factors, such as the Earth's topography, the (timevariable) geoid, changes of the Earth's rotation, atmospheric circulation, heat and salinity distribution in the ocean, ocean circulation, past and present mass movements in the Earth system, the visco-elastic properties of the Earth's interior, sedimentation, and even anthropogenic subsidence due to groundwater, gas, and oil extraction. At coastal locations, LSL is measured relative to a benchmark on land, which, if properly chosen, follows the vertical motion of the land around the tide gauge, including the ocean bottom below it. Over the last thirty years a number of studies have utilized the unique sea level data set provided by the Permanent Service f o r Mean Sea Level (PSMSL) for the determination of a global sea level rise (see Church et al., 2001, for a review). The global trends estimated in these studies range from + 1 to ÷2.5 mm/yr. This relatively wide range mainly is due to the selection criteria used by the different researchers to select subsets of tide gauges as well as the methodology to determine a global trend. However, the link between LSL changes and changes in the global ocean mass and volume is complex and all forcing factors result in spatially highly variable trends. Taking into account that the global network of tide gauges only samples a small fraction ofthe ocean's surface, any estimate of a global rise not taking into account the spatial variability of the different contribution is bound to be biased. In order to account for the spatial variability of the forcing, Plag (2006) derived a LSL balance equation that accounts for each forcing factor individually.
40
H.-P. Plag
Using an approximate LSL equation which accounts for the contribution due to ocean temperature changes, post-glacial rebound, and the present-day changes in Greenland and Antarctica, he determined a global average rise in LSL of 1.05 + 0.75 mm/yr. Here we extend this approach and study how the local atmospheric forcing affects the estimates of the global average. In the next section, we briefly introduce the LSL equation and discuss the spatial fingerprint of the main forcing factors. In Section 3 we summarize the database used by Plag (2006), which is here complemented with an atmospheric dataset. Then, in Section 4 we consider the effect of the local forcing on the global grid of observed LSL trends, in Section 5, we introduce the regression model and in Section 6 discuss the effect of the local atmospheric forcing on the global estimates by comparing the results of Plag (2006), which do not account for the local atmospheric forcing, to the results obtained here after the local atmospheric forcing has been removed.
3
The database
and methodology
Time series of monthly mean values for LSL are taken from the PSMSL data base (Woodworth & Player, 2003), which contains records from more than 1950 tide gauges, i.e. a major fraction of the global tide gauge data. Those records, for which the history of a local reference can be established are compiled into a subset denoted as Revised Local Reference (RLR) datasets, and these records can be used confidently to determine local LSL trends. Most of the PSMSL records are restricted to the time window of approximately 1950 to 2000. Thus, only in that time interval a spatially sufficient picture of the pattern of LSL trends can be expected. Considering that the steric sea level changes are given for the time window 1950 to 1998 (see below) we choose this time window as a compromise between highest accuracy for the local secular LSL trends and the optimal spatial coverage. For each tide gauge, Plag (2006) determined a secular trend by fitting the model function 2
2
L o c a l Sea Level B a l a n c e
g(t) - a + bt + E Ai sin(wit + 0i)
(2)
i=1
For the discussion of secular trends, we approximate the monthly mean LSL hM(:g, t) at a tide gauge located at a point E on the Earth surface as a sum of several factors, namely
h(:< t) =
t) +
t) + A ( i , t) +
I(i, t) + a(i, t) + T(I, t) + p(1)(t
- to) + V o ( 1 ) ( t - to) +
6v( , t)
where t is time, to an arbitrary time origin, and where we have considered the following contributions to LSL changes: S: steric changes, C: changes in ocean currents, A: changes in atmospheric circulation, I: changes in the mass of large ice sheets, G: changes in the mass of glaciers, T: changes in the terrestrial hydrosphere, P: post-glacial rebound (assuming a timeindependent velocity), V0: tectonic vertical land motion (assuming a time-independent velocity) 6V: nonlinear vertical land motion. (Plag, 2006, see there for a detailed discussion of these factors). The factors that contribute to secular trends with a fingerprint exhibiting large spatial variations on regional to global scales are the post-glacial rebound signal, the steric signal and the present-day contribution from the two large ice sheets in Antarctica and Greenland (Plag, 2006). Moreover, changes in atmospheric circulation are also likely to have regional scales. The database for these factors are discussed in the next Section.
to the series of monthly mean sea levels, where t is time, a is an offset and b the constant secular LSL trend. Ai and ~bi are the amplitude and phase, respectively, of an annual and semiannual constituent, in the fit, the parameters a and b and the amplitudes of the sine and cosine terms of the annual and semi-annual constituents are determined simultaneously. Here we use an alternative equation to determine the LSL trends, i.e. 2
3
g(t) - a + bt + E Ai sin(wit + Oi) + E diai (3) i=1
i=1
where o-i, i = 1, 2, 3, are the relevant components of the atmospheric stress tensor on the sea surface, and di are the respective regression coefficients, which we determine together with the other parameters in the least squares fit to the LSL records. The component of the atmospheric stress tensor perpendicular to the sea surface is the air pressure p. The horizontal components are taken to be proportional to the wind stress components, i.e.
a2 ~ WEV/W~ + w'~ ~
+
(4)
(5)
where WE and WN are the east and north components of the wind vector, respectively. We denote the sets of
Chapter 6 • Estimating Recent Global Sea Level Changes
LSL trends determined with eq. (2) and eq. (3) as T1 and T2, respectively, Monthly mean values of the air pressure and the wind stress components are computed from the ERA40 reanalysis data provided by the European Center for Medium Range Weather Forecast (ECMWF). The ERA40 dataset has a spatial and temporal resolution of 2.5°x2.5 ° and 6 hours, respectively. Monthly means of or2 and era are computed as averages of the six-hourly values of these quantities. With respect to thermosteric sea level changes, post-glacial rebound signal and the fingerprints of the two large ice sheets in Antarctica and Greenland, we use the same data base as Plag (2006). Thermosteric sea level variations are computed from observations of the subsurface temperature field. Currently, two global datasets are available, namely Levitus et al. (2000)and Ishii et al. (2003). The two datasets are, to a large extent, based on the same observations; however, different analysis schemes are used to create the gridded datasets. As pointed out by Plag (2006), the sea level trends derived from these two datasets display considerable differences, with the former having more short wavelength variations and a larger range of local trends. The computed sea level changes depend on the depth interval used for the integration. For the Levitus et al. (2000) dataset, steric sea levels are available for 500 m and 3000 m (denoted as L500 and L3000). For the Ishii et al. (2003) dataset, sea levels are only available for 500 m (denoted here as I500). All grids have a spatial resolution of 1°. For the L500 and L3000 datasets, which are given as annual means, local trends were determined by a least squares fit of a polynomial of degree 1 to the data for each grid point. I500 is given as monthly means, and the model function (2) was used instead. In large parts of the ocean, the thermocline depth is much deeper in the ocean than 500 m. Therefore, L500 and I500 are likely to underestimate the thermosteric sea level variations in these areas. The large uncertainties in the mass changes of the two large ice sheets in Antarctica and Greenland over the last five decades do not allow to use a spatial fingerprint deduced from observations to represent the contribution of these two ice sheets to LSL. Therefore, based on the static elastic sea level equation (Farrell & Clark, 1976), Plag & Jfittner (2001) determined for each ice sheet a fingerprint function for a constant, unit trend over the complete area of the ice sheet. The resuiting Antarctic fingerprint has a distinct zonal component, while the Greenland fingerprint shows more variations with longitude, particularly in the northern hemisphere (Plag, 2006).
The present-day LSL fingerprint of the post-glacial rebound signal (PGS) is fairly well predicted by geophysical models. Plag (2006) used a suite of models (Milne et al., 1999) to study the effect of the uncertainties of the predicted PGS fingerprint in LSL on the global results. Here we also use the same set of models but will not discuss in detail the sensitivity of the global results to the PGS model. The thermosteric contribution is only given on a 1o by 1° grid. Using a regression based on individual LSL trends at tide gauges would require interpolation and in most cases extrapolation of the steric signal from nearby grid points to the exact tide gauge location. Based on a detailed sensitivity study, Plag (2006) chose to create a gridded dataset of LSL trends instead, assigning a weighted average of all available LSL trends to each grid cell with tide gauges. The sensitivity study also indicated that a grid resolution of 2 ° x2 ° was a reasonable compromise between the accuracy of the individual LSL trends assigned to the grids and the spatial resolution required to capture the main features of the spatial fingerprint of e.g. the postglacial signal. This 2 ° grid as defined by the available tide gauges only covers about 3.1% of the global ocean surface. Similar 2 ° grids were created for all available forcing factors by averaging the 1° grids.
4
Effect of the local atmospheric forcing on LSL
At many stations, a large amount (up to 90%) of the variability of monthly LSL is explained by the regression model according to eq. (3), with a large fraction of the model coming from air pressure and to a lesser extent the two wind stress components. In many areas, the regression coefficient for air pressure is close to the equilibrium value of ~ - 1 0 mm/HPa (Fig. 1). However, close to the equator, where air pressure variations are small, much larger values than that are found. The relative importance of the local atmospheric forcing depends mainly on the latitude and the coastal geometry. At stations at latitudes outside the -+-25° band the combined local atmospheric forcing explains most of the intraseasonal variations, while significant interannual to decadal variations are more pronounced in the residuals than in the observations (e.g. Halifax, Cuxhaven, San Francisco, and Sydney). At most stations the main atmospheric contribution is due to air pressure, but at Cuxhaven in the German Bight, the east component of the windstress is by far the dominating contribution. For the stations on the west coasts of the Americas, the E1 Nifios events are visible in sea level. At
41
42
H.-P. Plag 40
40
L~.
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20
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20
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i
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-90
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0
-30
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30
60
90
-90
-60
-30
0
30
60
90
Latitude
Latitude
Fig. 1 • Regression coefficients for the local atmospheric forcing. P" air pressure, E: east component of wind stress, N" north component of wind stress. The horizontal line in the left diagram indicates the equilibrium value of ~ - 1 0 mm/HPa. Station
HALIFAX
Station
40
Station 20
0
-20
0 -40
SANFRANCISCO
60
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0
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1994
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9
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-200 1970
1982
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1982
Year
1994
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1970
1982
1994
2006
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Fig. 2: Regression model for selected tide gauges. For each station, we show O" observations, M: regression model, R: residual, as well as the individual contributions of AP" air pressure, WE: east component of windstress and WN" north component of windstress. All parameters are given in mm. San Francisco, the high sea levels during the largest E1 Nifios are not modeled by the local atmospheric forcing, while at Santa Cruz in the Galapagos Islands, the local forcing captures a large fraction of these high sea levels very well. However, at Santa Cruz, the regression coefficient for air pressure is as large as - 7 2 mm/HPa, which indicates that there air pressure may
be correlated with other factors influencing sea level. At many stations close to the equator, the rather small air pressure variations are dominated by a (likewise) small seasonal signal, which is also present in sea level (e.g. Cochin at the southwest coast of India). Comparing the LSL grid derived from T1 to the grid derived here from the T2 LSL trends, we see
Chapter 6 • Estimating Recent Global Sea Level Changes
0 90
30 i
60
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Fig. 3: Effect of local atmospheric forcing on the secular LSL trend pattern. Upper diagram: LSL trend grid derived by Plag (2006) for T1. Middle: LSL trend grid determined in the present study for T2. Lower Diagram: atmospheric contribution to LSL trends computed as the difference between T2 and T 1. All scales are in mm/yr. In the computation of the grid values, only local LSL trends within 4-12 mm/yr are used and the LSL trends in a grid cell are averaged using the length of the records as weights. Minimum record length is 10 years.
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that the spatial pattern shows some differences particularly in the Pacific (Fig. 3). The effect of the atmospheric forcing on the long-term LSL trends depends on the geographical region. For example, along the west coast of North America, the atmospheric contribution in general is positive, exceeding 2 mm/yr in some areas. For most of the North Atlantic, the con-
-90 360
5
tribution is also positive, while over the western Pacific, large negative contributions are found reaching -5 mm/yr in some areas. For the Mediterranean stations, the atmospheric contribution is negative (particularly in the Adriatic and the eastern Mediterranean, where it reaches values up to -0.8 mm/yr). The results for the Mediterranean are confirmed by Tsimplis et al.
43
44
H.-P. Plag
(2006) on the basis of a hydrodynamical model run.
5
The regression model
The regression model set up by Plag (2006) for the gridded LSL trends is given by K
(6) j=l N
where i is the index of the grid cell, bi is the modeled LSL trend, --I/ij) the LSL trend due to a unit mass change in ice sheet [(J), I£ the number of individual ice sheet fingerprints included in the regression, Pi the predicted PGS, Si the thermosteric LSL trend, and c a mean global LSL trend introduced to collect all unaccounted contributions. The c~j are unknown mean mass trends of the ice sheets, which are determined as a results of the regression analysis. /3 is introduced to account for any scale error in the PGS predictions. With the introduction of /3, we preserve the predicted PGS fingerprint but we allow for adjustments in the amplitude. The same is true for the thermosteric effect, where we have introduce a scale factor ~,. In the following, we have to distinguish between the regression grid, which is defined by the fact that all factors used in a regression are available, and the global (ocean) grid, which is defined as far as possible for the complete ocean surface. Since all models are available globally, the regression grids are determined as the set of grid cells where both the steric sea level trends and the observed LSL trends are given. The global grids are defined by the steric grid used. The regression grids cover typically 2% of the ocean surface, while the steric grids reach up to approximately 95%. Any regression analysis is hampered by the presence of high covariance of the forcing factors. Using T 1, the fingerprints of the Antarctic and Greenland ice sheet are significantly anti-correlated (Table 1). Moreover, the Greenland ice sheet fingerprint is weakly correlated with the present-day PGS fingerprint. For the steric fingerprints, the correlation with the T1 LSL trends are of the order of 0.25 for the L500 and I500 datasets, while the correlation is less than 0.2 for the L3000 dataset (Table 1). Using the T2 trends does not change the cross correlation between LSL and most of the factors (Table 1). However, there is no correlation between the steric sea level heights and the T2 trends, in particular, the cross correlation coefficients for I500 and L500 are 0.00 while for L3000 a non-significant value of 0.04
is found. Taking into account the local atmospheric forcing completely removes the correlation between the LSL and steric trend patterns. This rather surprising results has a profound effect on the results of the regression analysis. It indicates that the steric signal sensed by the tide gauges is mainly due to changes in the upper ocean layer, which appear to be highly correlated with local atmospheric forcing. Thus, a warming of the upper layer and an increase in sea surface temperature will tend to lower the air pressure above the warming area. Both processes will lead to an increase of sea level, which is then fully absorbed in the separate removal of the contribution due to local atmospheric forcing. Nevertheless, in the next section, we will compare selected results to those obtained by Plag (2006) in order to assess the effect of the local atmospheric forcing on the global estimates.
6
Results
In Table 2, results of the regression analysis for a few selected combinations of fingerprints are given for the T1 and T2 LSL trend grids. The actual values of the regression coefficients for T1 are discussed in detail in Plag (2006). Here it is only mentioned that/3 is in the range of 0.4 to 1.5, depending on the PGS model, while -y ranges from approximately 0.4 for L3000 to approximately 1.2 for I500, depending slightly on the PGS model used. For T2,/3 is generally lower, while -y is close to zero for all three steric data sets (see below). For all regressions, the constant c is of the order of 0.5 mm/yr and 0.75 mm/yr for the T1 and T2 LSL trends, respectively. Thus, the unexplained average trend at the tide gauges is between 50 and 80% of the average sea level rise. Moreover, the assumption that c determined for the (small) tide gauge grid can be extrapolated over the whole ocean is uncertain, since most contributions included in eq. (1) are spatially highly variable. Compared to the T 1 results, the T2 results explain a smaller fraction of the spatial pattern in the LSL trends (Table 3). They show a smaller contribution from the Antarctic ice sheet and a larger one from the Greenland ice sheet. The PGS contribution is slightly lower or unchanged, while no contribution comes from the steric forcing. A larger fraction of the LSL trends remains unexplained. For the T1 solutions, the steric signal based on I500, L500 and L3000 contributes significantly to the global LSL pattern. For the L500 dataset, the explained fraction of the variance is the highest and the
Chapter 6 • Estimating Recent Global Sea Level Changes
Table 1: Correlation matrix for the fingerprints of the forcing factors and LSL. Columns are: A, G: fingerprints of the Antarctic, and Greenland ice sheets, respectively, P 1: post-glacial rebound model (ice history is ICE-3G, upper and lower mantle visocities are 1 • 1021 and 2 • 1021 Pas, respectively, and lithosphere thickness is 120 km, see Milne et al. (1999)), I500, L500, L3000: thermosteric fingerprints, and LSL: LSL trend pattern. Matrix is for cross correlation on the regression grid. Upper and lower matrix are for T1 and T2, respectively. For a detailed discussion, see Plag (2006).
A
G
P1
i500
L500
L3000
LSL
A G P1 I500 L500 L3000 LSL
1.000 -0.492 -0.118 -0.026 -0.004 -0.120 -0.097
-0.492 1.000 -0.187 0.100 -0.023 -0.001 -0.074
-0.118 -0.187 1.000 0.014 -0.006 0.023 0.317
-0.026 0.100 0.014 1.000 0.826 0.675 0.248
-0.004 -0.023 -0.006 0.826 1.000 0.832 0.247
-0.120 -0.001 0.023 0.675 0.832 1.000 0.186
-0.097 -0.074 0.317 0.248 0.247 0.186 1.000
A G P1 i500 L500 L3000 LSL
1.000 -0.493 -0.136 -0.030 0.008 -0.102 -0.088
-0.493 1.000 -0.169 0.106 -0.024 0.001 -0.101
-0.136 -0.169 1.000 0.014 -0.007 0.023 0.273
-0.030 0.106 0.014 1.000 0.820 0.676 0.000
0.008 -0.024 -0.007 0.820 1.000 0.827 -0.002
-0.102 0.001 0.023 0.676 0.827 1.000 0.036
-0.088 -0.101 0.273 0.000 -0.002 0.036 1.000
Table 2: Selected results of the regression analysis. The column M (Model) indicates the LSL trend set used and gives the
factors included in the regression function (6). Parameters are as in eq. (6). The other columns are as follows: Emod: mean of modeled RSL trends, (weighted by area); V: fraction of the variance in % explained by the regression model. For each solution, the upper and lower lines give the regression results in mm/yr equivalent contribution to the mean over the (small) regression grid and the (near-global) complete steric grids, respectively (see Plag, 2006, for a more detailed discussion of the global model). P2 is similar to P1 (see Table 1), except for a lower mantle viscosity of 4.75- 1021 Pas. The contributions are given with 95% confidence limits. Bared quantities are spatial averages over the respective grid. M TI" A,G,P2,I500,c
0.61 0.27 T2: A,G,P2,I500,c 0.42 0.19 T 1" A,G,P2,L500,c 0.54 0.24 T2: A,G,P2,L500,c 0.44 0.20 TI: A,G,P2,L3000,c 0.41 0.18 T2: A,G,P2,L3000,c 0.40 0.18
OZA" /-A 4- 0.37 4- 0.16 4- 0.31 4- 0.14 + 0.37 4- 0.16 4- 0.32 4- 0.15 4- 0.37 4- 0.16 4- 0.32 4- 0.14
0.04 0.09 0.07 0.14 0.02 0.05 0.06 0.14 0.02 0.05 0.06 0.13
CtG" /-G 4- 0.03 4- 0.07 4- 0.03 4- 0.06 + 0.02 4- 0.05 4- 0.03 4- 0.07 4- 0.02 4- 0.05 4- 0.03 4- 0.07
-0.09 -0.04 -0.07 -0.04 -0.12 -0.05 -0.09 -0.04 -0.12 -0.05 -0.09 -0.04
regression coefficient is close to one. B a s e d on that, Plag (2006) c o n c l u d e d that the L 5 0 0 is m o s t consistent with the spatial pattern o f the T1 L S L trends. H o w ever, in the o p e n ocean, the L 3 0 0 0 set m a y be better, and for L 3 0 0 0 the global average is 0.3 5 ram/yr. Consequently, the estimates given in Table 2 for the steric contribution are likely to be at the lower end. H o w ever, for the T2 L S L trends, the regression coefficients for all three data sets are very close to zero, leaving no r o o m for a steric contribution. For all regressions, the global average values are lower than the averages obtained for the regression grid, and they are at the very low end of values gen-
ft. /5 4- 0.02 0.08 4- 0.01 0.06 4- 0.02 0.00 4- 0.01 0.00 4- 0.02 0.20 4- 0.01 0.20 4- 0.02 - 0 . 0 2 4- 0.01 - 0 . 0 2 4- 0.02 0.14 0.14 4- 0.01 4- 0.02 0.01 4- 0.01 0.01
444444444444-
7" ~' 0.02 0.02 0.02 0.02 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04
0.46 0.46 0.73 0.73 0.50 0.50 0.80 0.80 0.65 0.65 0.76 0.76
c Emod V 1 10 13.10 4- 0.40 0.83 4- 0.40 1 15 9.08 4- 0.35 1.02 4- 0.35 + 0.40 1 14 13.79 0.94 4- 0.40 1 19 9.79 4- 0.35 1.07 4- 0.35 1 10 12.43 4- 0.40 4- 0.40 0.97 1 15 9.57 4- 0.35 1.04 4- 0.35
Table 3" Comparison of T1 and T2 results. The contributions are to global average.
Factor/parameter Variance explained (%) Antarctica (mm/yr) Greenland (mm/yr) Thermosteric (mm/yr) Unexplained (mm/yr) Global average (mm/yr)
T1 11.3 to 13.8 0.394-0.11 0.104-0.05 0.30-t-0.10 0.354-0.40 0.904-0.75
T2 8.5 to 10.1 0.314-0.13 0.164-0.03 0.004-0.01 0.454-0.35 1.004-0.70
erally reported for the global sea level rise over the last 50 to 100 years. The basic a s s u m p t i o n s for the extrap-
45
46
H.-P. Plag
olation of the models from the regression grid to the global grid are: (1) the regression model is appropriately representing the long spatial wave length in sea level trends, and (2) there are no other open ocean contributions to global sea level rise not sensed by the tide gauges, that would affect the extrapolation of e. Plag (2006) argued that the results for the steric contributions using the L500 and I500 datasets are likely to be minimum estimates, while the global average of the L3000 dataset indicates that the actual steric contribution may be larger by 0.1 to 0.2 mm/yr. Therefore, he considered a global sea level rise value of 1.05 -+- 0.75 mm/yr to be more likely. For the T2 results, it is likely that the steric contribution has been absorbed by the regression of local atmospheric forcing. Therefore, if we add a similar contribution to compensate for a bias of the steric contribution, the global average LSL trend for the T2 grid is of the order of 1.20 -+- 0.70 mm/yr.
7
Conclusions
The comparison of the regression results for the T1 and T2 LSL trends reveals a high correlation between spatial patterns of the LSL trends locally attributed to atmospheric forcing and the thermosteric contribution. This correlation may be due to a feedback from sea surface temperature changes to the regional air pressure and wind fields. The regression results for both LSL trend datasets show that the observed spatial pattern of L S L trends is compatible with melting of both the Greenland and Antarctic ice sheets. In fact, the results assign a high significance to this melting. Based on the regression results for T1, the steric contribution to the global LSL average trend is at least 0.20 + 0.04 mm/yr but more likely to be larger. However, a part of that signal may actually be due to a correlated effect of atmospheric forcing on sea level. The results presented here underline the potential of the fingerprint method to extract useful information from the sea level observations provided by the global network of tide gauges. Potential biases of the regression results and particularly the extrapolation to the global ocean surface are discussed by Plag (2006) and can result from (1) long-term changes in the Greenland and Antarctic ice sheets, (2) errors in the PGS predictions, and (3) unaccounted factors with fingerprints having large spatial variations. Here we have identified an additional error source, which results from the correlation of atmospheric forcing and steric changes. In order to reduce the biases, a more comprehensive
regression model needs to be set up, that takes into account the different forcing factors in one common fit.
Acknowledgements The author would like to thank J. Hunter for a thorough review, J.X.Mitrovica for the provision of the post-glacial rebound models, and Anny Cazenave for the provision of the steric sea levels computed from the Levitus and ishii datasets. The tide gauge data was taken from the database maintained by the Permanent Service f o r Mean Sea Level. Without the long-lasting work of the PSMSL, this study would not have been possible. Part of this work was supported by a NASA grant in the frame of the Interdisciplinary Science Program.
References Church, J. A., Gregory, J. M., Huybrechts, P., Kuhn, M., Lambeck, K., Nhuan, M. T., Qin, D., & Woodworth, P. L., 2001. Changes in sea level, in Climate Change 2001: The Scientific Basis. Contribution of Working Group I to the Third Assessment Report of the Intergovernmental Panel on Climate Change, edited by J. T. Houghton, Y. Ding, D. J. Griggs, M. Noguer, P. J. van der Linden, X. Dai, K. Maskell, & C. A. Johnson, pp. 639-693, Cambridge University Press, Cambridge. Farrell, W. E. & Clark, J. A., 1976. On postglacial sea level, Geophys. J. R. Astron. Soc., 46, 647-667. Ishii, M., Kimoto, M., & Kachi, M., 2003. Historical ocean subsurface temperature analysis with error estimates, Monthly Weather Rev., 131, 51-73. Levitus, S., Stephens, C., Antonov, J., & Boyer, T., 2000. Yearly and year-season upper ocean temperature anomaly field, 1948-1998, Tech. rep., U.S. Gov. Printing Office, Washington, D.C. Milne, G. A., Mitrovica, J. X., & Davis, J. L., 1999. Nearfield hydro-isostasy: the implementation of a revised sea-level equation, Geophys. J. Int., 139, 464482. Plag, H.-P., 2006. Recent relative sea level trends: an attempt to quantify the forcing factors, Phil. Trans. Roy. Soc. London, In press. Plag, H.-P. & Jfittner, H.-U., 2001. Inversion of global tide gauge data for present-day ice load changes, in Proceed. Second Int. Syrup. on Environmental research in the Arctic and Fifth Ny-~lesund Scientific Seminar, edited by T. Yamanouchi, no. Special Issue, No. 54 in Memoirs of the National Institute of Polar Research, pp. 301-317. Tsimplis, M. N., ilvarez-Fanjul, E., Gomis, D., FenoglioMarc, L., & P~rez, B., 2006. Mediterranean sea level trends: Separating the meteorological and steric effects, GRL, In press. Woodworth, P. & Player, R., 2003. The Permanent Service for Mean Sea Level: an update to the 21st century, J. Coastal Research, 19, 287-295.
Chapter 7
On the low-frequency variability in the Indian Ocean I.V. Sakova, G. Meyers CSIRO Marine and Atmospheric Research, GPO Box 1538, Hobart, Tasmania, Australia, 7001 e-mail: Irina.
[email protected] R. Coleman School of Geography & Environmental Studies, University of Tasmania, Private Bag 78, Hobart, Tasmania, Australia, 7001
Abstract. This paper presents the results of an investigation of low-frequency variability in the Indian Ocean (IO) primarily using satellite altimeter sea surface height (SSH) observations and expendable bathy-thermograph (XBT) data. We found that in most regions of the IO the low-frequency part of the SSH spectra (corresponding to signals with periods of a few months or longer) is concentrated in four frequency bands separated by substantial spectral gaps. These bands correspond to periods of approximately 6 months, 12 months, 18-20 months and more than 30 months (hereafter referred to as the 30-month band). For both 18-20-month and 30month bands the spectral density shows a dipolelike pattern with some degree of similarity; however, analysis of the spatial-temporal evolution of these signals suggests that the 18-20-month signal is an internal mode of the IO, while that of the 30month component propagates from the Pacific Ocean, in particular with a much stronger signal during the period from 1998 to 2003. Keywords. Indian Ocean, Indian Ocean Dipole, Satellite Altimetry, spectral analysis 1 Introduction Today more than ever the oceanographic community enjoys the availability of extensive observational data sets including satellite altimetry (Fu and Cazenave 2001; Born 2003). The TOPEX/Poseidon and Jason-1 satellite altimeter data cover a time interval of more than 13 years. Such a long time interval opens new opportunities for analysis of the low-frequency processes in the ocean with characteristic time scales of up to several years. In this paper we present results of an investigation of the low-frequency variability in the IO by applying spectral analysis methods to the satellite altimeter and XBT data sets. The existence of annual and semi-annual processes in the IO is well known; they have been sub-
jects of numerous studies (e.g., Clarke and Liu 1993). In the last decade there has also been a growing interest in studying of the interannual variability in the tropical Indian Ocean (TIO) (Perigaud and Delecluse 1993; Masumoto and Meyers 1998; Feng and Meyers 2003), particularly after the discovery of the Indian Ocean Dipole (IOD) mode (e.g., Saji et al. 1999). The interannual variability of sea surface temperature (SST) is also well studied (e.g., Behera et al. 2000), with only several studies of subsurface variability of the TIO (Tourre and White 1995; Meyers 1996; Murtugudde and Busalacchi 1999; Schiller et al. 2000; Rao et al. 2002; Feng and Meyers 2003). Rao et al. (2002) found that the dominant modes of interannual variability in the IO do not show co-variability between the surface and subsurface. Using empirical orthogonal functions (EOF) for analysis of SSH satellite data for the period 1993-1999, as well as ocean model output, the authors found two dominant modes of the interannual subsurface variability of the TIO. The first mode was found to be governed by the IOD, while the second mode "shows the interesting quasibiennial tendency". In another study, Feng and Meyers (2003) found a 2-year time-scale of the upper-ocean evolution in the TIO near the Java coast that is unique to the IOD (that is, not directly forced by an atmospheric teleconnection from the Pacific). This conclusion was based on the timing of the temperature anomalies, associated with strength of upwelling of Java, when "cold anomaly during 1994 and 1997 [was] followed by warm anomaly during 1995/96 and 1998". In this work, we study the low-frequency variability of the IO by using spectral analysis methods. Spectral analysis enables investigation of spatial and temporal characteristics of the low-frequency processes in the whole IO basin in a systematic way by identifying principal frequencies, calculating spatial distributions of the power spectrum density, and reconstructing the spatial-temporal evolution of the main variability modes.
48
I.V. Sakova. G. Meyers • R. Coleman
2 Data and methods Two data sets were used in this study: gridded (1 degree) SSH weekly-averaged data for the period from October 1992 to August 2004 collected by the ERS/Envisat/TOPEX/Jason- 1.satellites (http://www.j ason. oce anob s. com/html/donn ee s/pro d uits/msla_uk.html) and expendable bathythermograph (XPT) temperature data for the period from January 1989 to December 2002 (http://www.marine. c siro. au/~pigot/REPORT/overv iew.html) (Meyers and Pigot 1999). The spectral analysis of these data sets was conducted by using Discrete Fourier Transform (DFT) method (e.g. Emery and Thomson 1997). The XBT data had irregular time sampling; therefore it was interpolated to a monthly grid. The time series contained 168 monthly entries. For analyzing twodimensional SSH fields, DFT was applied to the time series of SSH at each grid point. The resulting gridded spectra were then either presented as power spectral density maps or used to select a particular frequency band and conduct Inverse Discrete Fourier Transform (IDFT) to investigate the temporal dynamics of the corresponding process. Each time series contained 618 weekly entries.
3 Analysis 3.1 XBT data analysis The broad application of spectral analysis to the SSH data for the whole IO was initially motivated by results of spectral analysis of upwelling offshore of Indonesia using XBT data. Figure l a shows a typical temperature profile near the Sumatra-Java coast obtained by averaging XBT data in a rectangle with opposite comers at 7.0°S, 104.0°E and 8.0°S, 106.0°E. One can see a number of strong upward displacements of the thermocline associated with upwelling that result in the cold water appearing at the sea surface, the most noticeable being during the 1994 and 1997 IOD events, and possibly in 1991. There are also frequent minor displacements that can be easily seen from the shape of the 20°C (D20) isothermal surface shown by the white line. It is well known that the thermocline in this region heaves with a periodicity of two times a year, which is related to the development of the Wyrtki jet (Clarke and Liu 1993), and some of these displacements can be associated with this semi-annual process. It is difficult to make other suggestions about a possible decomposition of the movement of the D20 surface from Figure l a until conducting a DFT of this signal.
1990
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Figure 1. (a) XBT temperature data (°C) at 6-7°S, 104-106°E, where white line - 20°C isotherm; (b) Power spectrum for depth of the 20 ° C isotherm, where psd is power spectral density, and cpm is cycle per month.
Figure lb shows the power spectrum of the depth of the D20 surface. Interestingly, the spectrum contains a number of well-separated maxima. As expected, one can easily identify the semi-annual component. There are also two surprisingly clear and strong low-frequency maxima corresponding to periods of approximately 18.7 months and 34 months. These low-frequency maxima rise far above the "noise" level in the spectrum and do carry much more energy than the 6-month component. Because of the long associated time scales, such processes may be expected to manifest themselves not only near Java coast but at larger spatial scales. This motivated us to conduct a broader spectral analysis of the low-frequency variability in the whole IO.
3.2 SSH data analysis Figure 2 shows power spectra of SSH in a number of different locations in the IO. All these spectra have one common feature: they contain a few rather strong and well separated maxima. Apart from the semi-annual and annual components, there are also two inter-annual variability modes already encountered in the spectrum of D20: those corresponding to periods of approximately 18-20 and more than 30 months. Overall, in different parts of the ocean, the power spectra of SSH may lack one or another of the identified four standalone maxima, but in all cases, all well-separated narrow maxima arising in power spectra do belong to one of these four frequency bands.
Chapter 7 • On the Low-Frequency Variability in the Indian Ocean
I
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suggestive of Rossby waves, while the maximum on the equator in the eastern TIO is suggestive of Kelvin waves. It is well known that the semi-annual signal is generated by wind forcing in the central basin (Birol and Morrow 2001); the remote Kelvin and Rossby responses are then seen in the SSH signal. The figure also shows strong signal along the coast in the Bay of Bengal and in the Arabian Sea close to the Gulf of Aden (Shankar et al. 2002). a
Longitude
6
Period ( m o n t h s )
--
* G 2
20
I v I V V~¢ "--/ ],..-J X.,/XJ
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Figure 2. Spectral analysis of SSH, where psd is normalised power spectral density.
Figure 3 presents statistics (mean, lower quartile and upper quartile values) for the power spectral density in a number of regions in the IO. Compared to Figure 2, the outlines of the spectra are smoother, which points to the possibility that the involved variability modes are localised, reaching maxima at some locations and being insignificant at others. Nevertheless, even with the values of the spectral maxima being smoothed by the spatial averaging, the maxima of 6-, 12-, 18- and 30-month frequency bands can still be clearly seen for most of the regions.
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Figure 4. Spatial spectral density of SSH for (a) 6-month signal; (b) one year signal, (c) 18-20-month signal; and (d) more then 30-month signal.
Period (months)
Figure 3. Statistics, spectral analysis of SSH, where psd is normalised power spectral density.
Following in Figure 4 are the plots of the spatial distribution of the spectral density for 6-, 12-, 18and 30-month bands. Figure 4a shows the spatial distribution of spectral density for the semi-annual signal. It shows strong signals in upwelling regions offshore of Indonesia between approximately 5 N and 8 S and offshore of Africa between approximately 7°N and 5°S. The U-shaped structure in the western TIO with the maxima off the equator is
The spatial distribution of the 1-year signal in Figure 4b has maxima in the Red Sea, Arabian Sea, western part of the Bay of Bengal and south-east of Indonesia. The maximum in the Arabian Sea is obviously caused by strong monsoon winds in this region. The maximum south of the equator is due to wind-forced, annual Rossby waves (Masumoto and Meyers, 1998). Figure 4c shows the spatial distribution of spectral density of the 18-20 month signal. It contains strong signals offshore of Indonesia and in the Bay of Bengal. The two new interesting features are the
49
50
I.V. Sakova. G. tvleyers • R. Coleman
strong maximum in the central IO, in between approximately 5°S and 15°S and 60°E and 95°E and zonally elongated narrow maximum at approximately 23°S and between 70°E and 98°E. The spatial distribution of the 30-month signal in Figure 4d contains two major maxima: offshore of Indonesia and in the central western IO. The location of the latter maximum is different to that for the 18-20-month signal, being wider and more westward. Analysis of the spatial distributions of the power spectral densities can provide important clues to the physics of the corresponding variability modes; however, the power spectrums contain no phase information. To study temporal variability of signals in different frequency bands, we conduct IDFT of the filtered signals and plot series of snapshots of the reconstructed signal in the given frequency band (not shown)
4 Discussion and Summary Using frequency analysis of satellite SSH data, two main frequency bands on the interannual time scale in the IO were identified. These bands correspond to the signals with periods of 18-20 and more than 30 months. While the patterns of spatial density of these signals do have similar dipole-like features, analysis of their temporal evolution (not shown) points at the different physical backgrounds of the corresponding variability modes. It is quite possible that the 18-20-month signal arising in our spectral analysis corresponds to the two-year time scale of variability in the TIO found by Rao et al. (2002) and Feng and Meyers (2003). Concerning the 30-month signal, its temporal dynamics suggests propagation of the signal from the Pacific in 1998-2003. The question of whether the signal in this frequency band correlates with ENSO and/or IOD has to be addressed in the future, including studying of the correlation with surface wind data and spectral analysis of signals over longer time intervals than in the current study. Overall the use of the spectral analysis of the SSH satellite observations allowed us to find strong well separated low-frequency spectral maximums in the spectrums of the signals and to investigate the spatial distribution and temporal dynamics of the corresponding variability modes. We believe that this information gives a strong quantitative basis for subsequent investigations of the underlying complex physical processes in the ocean-atmosphere system.
Acknowledgments The first author is supported by a joint CSIROUTAS PhD scholarship in quantitative marine science (QMS) and a top-up CSIRO PhD stipend (funded from Wealth from Oceans National Research Flagship)
References Behera, S. K., P. S. Salvekar, et al. (2000) Simulation of interannual SST variability in the tropical Indian Ocean. Journal of Climate vol. 13, no. 19: 3487-3499. Birol, F. and R. Morrow (2001) Source of the baroclinic waves in the southeast Indian Ocean. Journal of Geophysical Research 106(C5): 9145-9160. Born, G. H. (2003) Jason-1 calibration/validation. Special issue. Source: Marine Geodesy; 26(3-4): 129-421. Clarke, A. J. and X. Liu (1993) Observations and dynamics of semiannual and annual sea levels near the eastern equatorial Indian Ocean boundary. Journal of Physical Oceanography 23(2): 386-399. Emery, W. J. and R. E. Thomson (1997). Data analysis methods in physical oceanography, Pergamon. Eeng, M. and G. Meyers (2003) Interannual variability in the tropical Indian Ocean: a two-year time-scale of Indian Ocean dipole. Deep-Sea Research Part Ii-Topical Studies in Oceanography 50(12-13): 2263-2284. Fu, L. and A. Cazenave (2001) Satellite altimetry and Earth sciences. A handbook of techniques and applications. International Geophysics Series, 69. Academic Press, San Diego. Masumoto, Y. and G. Meyers (1998) Forced Rossby waves in the southern tropical Indian Ocean. Journal of Geophysical Research-Oceans 103(C 12): 27589-27602. Meyers, G. (1996) Variation of indonesian throughflow and the E1 Nino Southern Oscillation. Journal of Geophysical Research-Oceans 101 (C5): 12255-12263. Meyers, G. and L. Pigot (1999) Analysis of frequently repeated XBT lines in the Indian Ocean. CSIRO Marine Laboratories Report, Hobart, Australia: 43pp. Murtugudde, R. and A. J. Busalacchi (1999) Interannual variability of the dynamics and thermodynamics of the tropical Indian Ocean. Journal of Climate 12(8): 2300-2326. Perigaud, C. and P. Delecluse (1993) interannual sea level variations in the tropical Indian Ocean from Geosat and shallow water simulations. Journal of Physical Oceanography vol.23, no.9: 1916-1934. Rao, S. A., S. K. Behera, et al. (2002) Interannual subsurface variability in the tropical Indian Ocean with a special emphasis on the Indian Ocean Dipole. Deep-Sea Research Part IiTopical Studies in Oceanography 49(7-8): 1549-1572. Saji, N. H., B. N. Goswami, et al. (1999) A dipole mode in the tropical Indian Ocean. Nature 401(6751): 360-363. Schiller, A., J. S. Godfrey, et al. (2000) Interannual dynamics and thermodynamics of the Indo-Pacific oceans. Journal of Physical Oceanography 30(5): 987-1012. Shankar, D., P. N. Vinayachandran, et al. (2002) The monsoon currents in the north Indian Ocean. Progress in Oceanography 52(1): 63-120. Tourre, Y. M. and W. B. White (1995) ENSO signals in global upper-ocean temperature. Journal of Physical Oceanography
Chapter 8
Satellite Altimetry: Multi-Mission Cross Calibration Wolfgang Bosch and Roman Savcenko Deutsches GeoEitisches Forschungsinstitut (DGFI), Marstallplatz 8, 80539 Mtinchen, Germany
A b s t r a c t . Multi-mission satellite altimetry provides
a unique opportunity to perform an innovative, utmost rigorous cross-calibration of all satellites operating simultaneously. Data from TOPEX/Poseidon, ERS-2, GFO, and Jasonl is used to c o m p u t e - in all combinations - nearly simultaneous single and dual satellite crossovers that are unaffected by sea level variability. The total set of crossovers provides a rather dense sampling of the orbits of all satellites and realizes a rigid network with high redundancy to o b t a i n - by a discrete crossover analysis technique - a reliable estimate of the radial error components. The analysis is performed for a sequence of 10 day periods (the cycle 96-457 of TOPEX) with 3 days overlap to neighbouring periods. For all satellites the error estimates for the radial component exhibit significant geographical pattern - even for TOPEX. The error components are also taken to calculate time series of relative range biases and geocentre offsets between all altimeter missions. Satellite altimetry, cross calibration, crossover analysis, least squares
Keywords.
1 Introduction Crossover analysis is a powerful approach to estimate errors and improve observations of satellite altimetry. The orbit error of early altimeter satellites has been essentially improved this way (Schrama 1989). Crossover analysis helps to identify time tag errors and to estimate improved correction models (Chambers 2003). Since the contemporary operation of TOPEX/Poseidon and ERS-1 (later on ERS-2) crossover analysis has become an important tool to cross calibrate altimeter missions with different sampling characteristic (Le Traon & Dibarboure 1999). For several reasons, TOPEX/Poseidon played a dominant role in cross calibration: Due to three independent tracking systems (Laser, GPS and DONS), the first two frequency altimeter sensor (allowing in-situ corrections for the ionosphere) and the mean
orbit height of 1300 km (reducing non-gravitational errors) it was reasonable to consider TOPEX/Poseidon as the most precise altimeter mission. Consequently, TOPEX/Poseidon has been used as reference for the cross calibration of different altimeter missions. Le Traon & Ogor (1998) use crossover differences of ERS and between ERS and TOPEX to adjust the ERS orbits to TOPEX. The TOPEX orbits are not changed. Other altimeter systems (e.g. GFO and ENVISAT) are treated the same way. During the several month tandem phase both, TOPEX and Jasonl observed the same ground track with only 1 minute delay in order to allow an utmost precise cross calibration. For consistency the orbits of Jason l and TOPEX were computed with JGM3. Figure 1 shows, however, systematic differences between TOPEX and Jason l. New, GPS based orbits computed with the GRACE gravity fields confirm significant geographical orbit errors for TOPEX. Thus, there is no longer a justification for the prominent role of TOPEX. In the present paper, the orbits of TOPEX are subject to error estimates and improvements just as all other missions. Crossover differences between all contemporary altimeter missions increasing considerably the redundancy in the crossover network and opens the challenge to identify also systematic errors of the TOPEX mission. In this paper a common analysis of TOPEX, Poseidon (treated independent of TOPEX), ERS-2, GFO, and Jasonl is performed. The extended mission of TOPEX (with shifted ground tracks) is treated separately and is further on identified as T/P-EM.
,
,
I
-30
-20
-10
I
0 mm
I
,
,
10
20
30
Fig. 1. Mean sea surface height differences between TOPEX and Jason l during their tandem phase (TOPEX cycles 344 - 365) reveal systematic errors.
52
W. Bosch. R. Savcenko
2 Discrete Crossover Analysis In the present investigation a discrete crossover analysis (DCA) is performed (Bosch 2006). It does not estimate parameters of any functional error model but solves only for the radial errors at the "observed" crossover locations. To ensure a certain degree of smoothness for adjacent errors a least squares minimization is applied to both, the crossover differences and adjacent differences of the radial errors. The DCA is realized by minimizing mink~=0 I D l x - I ; 1 2
(1)
where the vector d compiles the n observed crossover differences. The vector x keeps the 2n unknown error components which are assumed to be ordered with respect to time. Then, matrix
D--
1
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0
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0
.--
0
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• ..
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:
accounts for the consecutive differences and matrix
n
1
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and the Jasonl data (AVISO & PODACC 2003) were taken from the ftp server of JPL. E R S - 1 0 P R data has not been included, because it is based on different models and software versions which could affect the cross calibration in an unpredictable way. ENVISAT mission data preparation is ongoing. This data will be included in any further analysis. For the computation of sea surface heights the standard corrections have been applied with the following modifications: - for all missions ocean tide corrections have been computed from FES2004 (Letellier et al. 2004)) - for the sea state bias correction for TOPEX, side B the model of Chambers et al. (2003) was used - the pole tide for ERS-2 was added with the same mean pole as used for TOPEX/Poseidon - all sea surface heights were transformed to the TOPEX reference ellipsoid - a common time reference epoch and the TAI time scale was introduced for all missions The orbit data, however, was not changed, although improved orbits are available (e.g. for ERS-2 as provided by the DEOS group). The orbits of TOPEX and Jasonl are based on the same gravity field model, JGM3. Orbits for ERS-2 are based on the precise orbits generated by the "German ERS Processing and Archiving Facility", DPAF. GFO precise orbits are generated by GSFC. No altimeter range bias was a p p l i e d - it will result from the crossover analysis.
m
4 Crossover Computation 0
•••
0
1
0
-1
relates the radial errors to the crossover differences. The linear system (1) has a rank defect of one because the sum of all columns of the coefficient matrix is a null vector• This rank defect is overcome by a single constraint, a linear combination of unknowns x. For n crossover points there are 2n unknown error components to estimate• Thus, the normal equation system is in general huge, but s p a r s e - it is therefore solved by the iterative "Conjugate Gradient Projection" algorithm. More details on the DCA are described in Bosch (2006, this volume).
3 Altimeter Data TOPEX/Poseidon data has been taken from the "Geophysical Data Record" (GDR-M), Version C, distributed by AVISO (1996). The "Ocean Product Record" (OPR) for ERS-2 were delivered by CERSAT. GFO data were kindly provided by NOAA's Laboratory for Satellite Altimetry (NOAA 2002).
Single satellite crossovers are performed as intersections of ascending (A) and descending (D) passes. For satellites with different inclination four types of crossovers are possible: AA, DD, AD and DA. In favour of a large intersection angle only two of these combinations were computed, if both satellites have prograde (or retrograde as ERS-2 and GFO) orbits, DA and AD crossings are favourable. If orbits are different, prograde and retrograde (e.g. TOPEX and ERS-2), only AA and DD crossings were generated. Crossover points were excluded if the absolute value of the crossover difference exceeds 100 cm or if the standard deviation was larger than 10 cm. Finally, crossovers are used only if their time difference is less than three days to ensure that crossover differences are as little as possible affected by sea level variability. Single satellite crossovers with short time difference are concentrated at high latitudes, the distribution of nearly simultaneous dual satellite crossovers is much better and leads to a dense sampling of the orbits of all satellites involved.
Chapter 8 • Satellite Altimetry: Multi-Mission Cross Calibration
5 The A n a l y s i s R e s u l t s
mm2
The crossover analysis was performed for the TOPEX cycles 096-456. Every 9.9156 day period was extended by three days at the beginning and the end such that an overall period of 16 days was analysed by DCA. For this period all single and dual satellite crossovers between all missions and all cycles were computed. A weighting with cosd0 was applied to the crossovers. For two missions there are about 20000 crossovers. With four simultaneously observing altimeters, the number of crossovers may reach 1 0 0 0 0 0 - in spite of limiting the crossover time difference to three days. For every 16 day period the solution was regularized by forcing the sum of all TOPEX error components to zero. The solution was iterated once. Crossovers in the second run were removed, if the absolute value of the residual crossover difference exceeds three times the posteriori unit weight standard deviation a 0. The number of rejected crossovers is always below 1%. Differences between the two errors estimates within the overlapping periods were found to be at most 1 - 2 mm. This justifies to consider further on only those errors that belong to the central cycle period. Concatenating all theses errors a temporal sequence of discrete error components is created for all altimeters. The estimation of empirical autocovariance functions for each mission is straightforward and shown in Figure 2. The overall crossover statistic, illustrated in Figure 3 shows the gain obtained by the DCA. The gain is measured by the ratio of rms values for the crossover differences before and after the analysis. Up to cycle 266 rms values of about are reduced by 1 - 2 cm only.
100
120 140
160
Autocovariance
~K 0
1000
2000
3000
4000
5000
6000
8000
9000
10000
When Jasonl enters the DCA (at TOPEX cycle 344), the gain becomes significant: rms values of about 11 cm are reduced to about 7cm[ A few rather high rms values are due to orbit anomalies of GFO.
6 Relative R a n g e B i a s e s and C e n t r e - o f Origin Shifts It is of particular interest to investigate how the errors are geographically distributed. Systematic error patterns may be explained by inconsistencies in the centre-of-origin implied by the satellite orbit. Non vanishing mean values of the errors are caused by relative range biases between (uncalibrated) altimeter missions, but may also indicate scale differences between different tracking systems used to compute the orbits. To estimate relative range biases and centre-of-origin shifts the model x~ + Vx, -- A r + A x cos ~ sin X + & y cos ~ cos X + & z sin c~ was fitted by least squares to the errors x i of every cycle and for every mission. A weighting with cosqb was applied to account for an increasing number of
180 200 220 240 260 280
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Fig. 3 Crossover rms-values before (black dots) and after (circels) analysis. Up to cycle 266 (with TOPEX and ERS-2 only) the analysis gain is moderate: the rms-values (on average about 9cm) is reduced by 1-2cm. From cycle 344 on (all four missions) the analysis gain is significant: the rms-values of about 11cm is reduced to about 7cm. Large rms-values are due to significant orbit errors. E.g., GFO had a momentum wheel anomaly at cycle 118 (corresponds to TOPEX cycle 409/410).
53
54
W. Bosch. R. Savcenko 1995
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Fig. 4. Range biases relative to TOPEX and T/P-EM (top panel a) and centre-of-origin shifts Ax, Ay, and Az (panel b), c) and d)) for all altimeter systems analysed. The range bias of ERS-2 shows a significant annual variation. Conspicuous is also the ycomponent of the centre-of-origin shifts: Both, TOPEX and ERS-2 show an annual oscillation with a few millimetres amplitude but opposite phasing - even during the period where GFO and Jasonl data is included. The scatter of x- and z-components of the centre-of-origin shifts significantly increases from cycle 266 on, when GFO data enters into the analysis.
crossovers at higher latitude. Figure 4 shows results for the range biases Ar (Figure 4a) and the centre-oforigin offsets Ax, Ay, Az (Figures 4b-d). The range bias for TOPEX (and T/P-EM) is close to zero because in each 16 day analysis periods the sum of all error components for TOPEX (and T/PEM) was forced to vanish. The time series of the ERS-2 range bias exhibits a rather high scatter and shows a clear annual oscillation. At and beyond TOPEX cycle 397 the ERS-2 bias and offsets are af-
fected by a significant loss of data (a failure of the ERS-2 tape recorder). Without these cycles the annual oscillation has 5mm amplitude. For GFO the mean value of the relative range bias is +46 mm, for Jasonl it is +167 mm. The Ay-component of the centre-of-origin shifts is conspicuous: TOPEX, ERS-2, and GFO show an annual oscillation with 3, 4, and 6 mm amplitude respectively. TOPEX and ERS-2 have nearly opposite phasing. Up to now there is no explanation for this
Chapter8 • SatelliteAltimetry:Multi-MissionCrossCalibration
behaviour. Instabilities in the reference frame realization by the different tracking systems are possible but not verified. 7 Geographically
Correlated
The mean orbit error A? is of particular danger because it is not visible in single satellite crossover differences, but it maps one to one into the sea surface heights. Figure 6 shows for all altimeter missions the distribution of the geographically correlated mean errors A7. The error pattern of TOPEX and Jasonl are small (up to ±2 cm), but differ, although the orbit configuration is identical and the same gravity field (JGM3) was used for the orbit computation. This indicates that not only gravity field errors were mapped into the radial component. The mean error patterns for GFO and ERS-2 exhibit a significant higher amplitude and ERS-2 reveals a pronounced track dependency.
Errors
The time series of radial error components allows to assess geographically correlated errors. From Kaula's first order analytic solution of the satellites motion (Rosborough 1986) it is known that the radial orbit errors of ascending and descending passes A r a'~ - A"f + A~
and
A r de'~ - A ~ - A
are composed of a ,,mean" (AT) and a ,,variable" part (AS). Both error components are related to errors of the gravity field harmonics a n d - if k n o w n - can be used to estimate corrections to the gravity field taken to compute the orbits of the altimeter satellites (Bosch et al. 2000). This is, however, far beyond the scope of this paper. For each mission, the error components of ascending and descending tracks were averaged on a 2°×2 ° g r i d - independent from each other. Figure 5 shows - as e x a m p l e s - the mean errors on ascending and descending ground tracks of TOPEX and GFO. For each mission these means were taken as an estimate for Ar "c and AY~'c and then used to compute A7 and A8 according to the inverse relationship
Conclusion
The common DCA of nearly simultaneous single and dual satellite crossover differences was performed for up to four contemporary altimeter systems. The approach creates a strong network with high redundancy and results in a reliable and dense sampling of the radial error components of all satellites. The estimated time series allow to assess the spectral properties of the radial component, captures relative range biases and indicates systematic variations of the centre-of-origin realisations. Most challenging is the capability to assess for all altimeter systems the geographically correlated mean error which is not visible by single satellite crossovers, but maps one to one into the mean sea surface.
A 7 - ( A r a'C + Ard~'c)/2
-0.01
-0.02
-0.03
-0.04
-0.05
Fig. 5. Mean errors [meter]of ascending (upper row) and descending tracks (lower row) for TOPEX (left column) and GFO (right column), averaged on a 2°x2 ° grid. For TOPEX the mean errors remain below +2 cm, for GFO they amount up to +5 cm. For both satellites the geographical pattern indicate systematic errors discoverable only by multi-mission crossover analysis.
55
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W. Bosch. R. Savcenko
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300
Fig. 6. Distribution of geographically correlated (mean) errors [meter] for TOPEX (top left), Jasonl (bottom left), GFO (top right), and ERS-2 (bottom right). The errors of TOPEX and Jasonl (left column) remain below +2 cm. Although TOPEX and Jasonl have identical orbits and the ephemerides for both satellites were based on the JGM3 gravity field, the error pattern differ, in particular in the Atlantic and in the South Indian Ocean. The errors for GFO and ERS-2 (right column) amount up to +5 cm - considerable higher than for TOPEX and Jasonl. The pronounced track dependent pattern of the mean error of ERS-2 can only be explained by the poor ephemeries originally distributed with the OPR-products.
References AVISO (1996): User Handbook- Merged TOPEX/ Poseidon Products (GDR-M), AVI-NT-02-101CN, Ed. 3.0, July 1996 AVISO and PODAAC (2003): User Handbook - IGDR and GDR Jason-1 Products, SMM-MUM5-OP- 13184-CN, Ed. 2.0, April 2003 Bosch W., J. Klokoenik, C. Wagner, J. Kostelecky (2000): Geosat and ERS-1 datum offsets relative to Topex/Poseidon estimated simultaneously with geopotential corrections from multi-satellite crossover altimetry. In: Rummel, R., H. Drewes, W. Bosch, H. Hornik (Eds.): Towards an Integrated Global Geodetic Observing System (IGGOS). lAG Symposia (120) 96-98, Springer, Berlin. Bosch W. (2005): Simultaneous crossover adjustment for contemporary altimeter missions. ESAESTEC, SP 572, ESA/ESTEC Bosch W. (2006): Discrete Crossover Analysis. In: Rizos, Ch. and R Tregoning (Eds.) Dynamic Planet 2005. IAG Symposia, Vol. 13?, Springer, Berlin (this volume) Chambers D.P., S.A. Hayes, J.C. Ries, T.J. Urban (2003): New TOPEX Sea State Bias Models and Their Effect on Global Mean Sea Level. J. Geophys. Res., Vol. 108(C10), 3305-11 Lettellier T., F. Lyard, and F. Lefebre (2004): The new global tidal solution: FES2004. Presented at:
Ocean Surface Topography Science Team Meeting, St. Petersburg, Florida, Nov. 4-6 CERSAT (2001): Altimeter & Microwave Radiometer ERS Products - User Manual, C2-MUT-A01-IF, Ed. 2.3, July 2001 Golub G.E. and C.F. van Loan: Matrix Computation, John Hopkins Press, 1983 Le Traon P.Y., and F. Ogor (1998): ERS 1/2 orbit improvement using T/P: The 2 cm challenge. J. Geophys. Res. 95, 8045-8057 Le Traon RY. and G. Dibarboure (1999): Mesoscale mapping capabilities from multiple altimeter missions. J. Atmos. Ocean Techn., Vol.16, 12081223 Le Traon P.Y., G. Dibarboure and N. Ducet (2001): Use of high resolution model to analyse the mapping capabilities of multiple altimeter missions. J. Atmos. Ocean Techn., Vol 18, NOAA: GFO GDR User Handbook, June 2002 Rosborough G.W. (1986): Satellite Orbit Pertubations due to the Geopotential. Report CSR-86-1, Univ. of Texas at Austin, Centre for Space Research. Schrama E.J.O.(1989): The role of orbit errors in processing of satellite altimeter data. Report 33, Netherlands Geodetic Commission, Publications
Chapter 9
Assessment of recent tidal models in the Mediterranean Sea D. N. Arabelos, G. Asteriadis, M.E. Contadakis, D. Papazachariou, S.D. Spatalas Department of Geodesy and Surveying, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece
Abstract. Ocean tides especially in closed sea areas can deviate considerably from the theoretical values due to unequal water depths and to the fact that the continents impede the movement of water. Satellite altimetry enabled the development of improved tidal models event in closed sea areas, by assimilating altimeter data into hydrodynamic models. The Mediterranean Sea due to its morphology is an ideal test field for the assessment of tidal models, based on this technique. An attempt to assess the recent tidal models TPXO.6, GOT00.2 and NAO.99b was based on (a) an inter-comparison of tidal heights computed from the three models at different time moments on the same 15 t × 15 t grid covering the Mediterranean and (b) on the comparison of the statistics of a crossover analysis of nearly 2.3 years of JASON-1 altimeter data (Cycle 1-86), before and after the tidal correction, using the three tidal models. The inter-comparison in terms of mean value and standard deviation of the differences between the tidal heights on the 15 ~ × 15 ~ grid resulted in mean values up to 3 mm and standard deviations ranging from 16 to 26 mm. However, maximum values of differences exceed several din. On the other hand, the statistics of the crossover analysis showed a 17% decrease of the standard deviation of the JASON-1 crossover differences after the tidal correction. These results show a good agreement between the three tidal models.
Keywords. Ocean tides, satellite altimetry, external assessment
1
Introduction
Ocean tides have been important from theoretical and practical point of view. In the past, tides were measured only by coastal tide gauge stations along continental coastlines and at islands and by bottom pressure recorders at deep-sea sites. The advent of satellite altimetry in the late 1970s offered, for the first
time a way to estimate ocean tides globally. Since the launch of TOPEX/POSEIDON (T/P) in 1992, the study of ocean tides has progressed impressively with the development of models of unprecedented accuracy by numerous authors. In general, the modern global tidal models can be categorized into three groups: hydrodynamic empirical, and assimilation models. Hydrodynamic models are derived by solving the Laplace Tidal Equations (LTE) and using bathymetry data as boundary conditions. Most of hydrodynamic solutions, such as Schwiderski's (1980) and FES94.1 (Le Provost et al., 1994) are undefined in the Mediterranean Sea, due to its bottom morphology and coast complexity (see Fig. 1). Empirical models are derived by extracting ocean tidal signals from satellite altimetry and they describe the total geocentric ocean tides, which include the ocean loading effect. Those models can be used directly in altimetry applications such as ocean tide corrections. Assimilation models are derived by solving the hydrodynamic equations with altimetric and tide gauges data assimilation. Although the amplitude of the tides in the Mediterranean Sea is rather small, the use of the best current tidal model is very essential for many geodetic and geodynamic applications (see, e.g. Arabelos, 2002). The aim of this paper is to provide an assessment of the currently available tidal models in the Mediterranean Sea. These models are described in brief in section 2.1. We tried to assess the models in two ways: by intercomparison and by external assessment. The intercomparison was based on: •
Estimation o f / ~ M S and/~S5' differences between the models.
•
The statistics of the differences of the tidal heights computed from the models at 20 different time moments on a grid covering the Mediterranean Sea.
•
Comparison of the tidal heights computed along 4 JASON-1 tracks in the test area.
58
D.N. Arabelos. G. Asteriadis • M. E. Contadakis • D. Papazachariou. S. D. Spatalas
Table 1. Statistics of the reduced JASON-1 ssh to EGM96.
FES94.1 and several local hydrodynamic models, using 286 cycles of T/P data supplemented in shallow seas and in polar seas (latitudes above 66 °) by 81 cycles of ERS 1 and ERS2 data. The solution consists of 8 constituents (Q1, O1, P1, K1, N2, M2, $2, K2) given on a 0.5 ° × 0.5 ° grid. NAO.99b: (Matsumoto et al., 2000). It uses the hydrodynamic tidal equations derived by Schwiderski (1980) and the tide solution is estimated on a 0.5 ° x 0.5 ° grid by assimilating about 5 years of T/P data (cycles 10-198) into the hydrodynamic model. The solution consists of 16 tidal constituents (Q 1, O 1, P 1, K1, OO 1, M1, J1, N2, M2, $2, K2, 2N2, Mu2, Nu2, L2, T2). This short period model is supplemented by the long period NAO.99L, consisting of 7 constituents (Mtm, Mf, MSf, Mm, MSm, Ssa,
(Unit = m). Number 167780 167618
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Depth Figure 1. Morphology of the Mediterranean basin after
Salech & Pavlis (2002).
The external assessment was based on the statistics of the crossover differences of JASON-1, before and after tidal correction, using the three tidal models. 2
2.1
Data
Tidal Models
The most recent tidal models used in this study are the following: •
•
TPXO.6.2 Current version of the global tidal solution developed by Egbert et al. (1994). This is an assimilation model with a resolution of 0.25 ° x0.25 °, comprising 8 major (Q1, O1, P1, K1, N2, M2, $2, K2) and 2 long period (Mf, Mm) constituents (Egbert & Erofeeva, 2002). Instead of the global model, the regional solution MED released by the same authors was used. This choice was made since, according to authors, the regional inverse solution fits the data significantly better for complex topographic areas such as the Mediterranean Sea. The tides are provided through 4 major diurnal and semidiurnal constituents (O 1, K1, M2, $2) on a 5 ~ x 5 ~ grid. GOT00.2: Update of GOT99 (Ray, 1999). This solution is a long wavelength adjustment of
2.2
Altimeter data
For the external assessment of the tidal models altimeter data from JASON-1 GDRs (AVISO & PODAAC, 2001) were used extracted from cycles 1 to 86 with the exception of cycle 69, covering the Mediterranean Sea. The data were corrected due to ionospheric, wet and dry tropospheric effect, sea state bias, inverse barometric effect and solid earth tide. We avoided performing ocean tide correction using the tidal models provided on the JASON-1 GDR in order to assess the tidal models under consideration. Flagged data with radiometer surface type 1 (land) were removed. In order to eliminate outliers the remaining sea surface heights (ssh) were compared with the EGM96 geopotential model (Lemoine et al. 1998), up to degree 360. The statistics of the differences Ah = J A S O N s s h - E G M 9 6 is shown in Table 1 (row 1). From the amount of 167780 ssh there are only 162 with IZXhl > 2m. These were considered as outliers and consequently were eliminated. The statistics of the comparison of the remaining ssh with EGM96 is shown in Table 1 (row 2). 3
3.1
Accuracy
assessment
Intercomparisons
The internal assessment was based on three different criteria. One of them is the intercomparison of the three models using the R M S deviation of each tidal constituent.
Chapter 9 • Assessment of Recent Tidal Models in the Mediterranean Sea 0o
10 o
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40°
40°
30°
30° 0° f
10 °
1
'
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20 °
'
~
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,
To account for the total effect o f the 4 major constituents we defined the Root S u m of Squares (RSS) (see e.g. Wang, 2004)
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In the comparison the major diurnal and semidiurnal constituents K1, O1, M2, $2 are included. Table 2 illustrates the R M S deviation b e t w e e n each pair o f the three m o d e l s in the M e d i t e r r a n e a n Sea. In Table 2 we can see that the largest R M S difference o f 1.3 cm resulted in M2 constituent b e t w e e n M E D and NAO.99b models. The distribution o f the R M S deviation for each point of a 0.5 / × 0.5 / grid b e t w e e n M E D and GOT00.2 is shown in Fig. 2 for K1 (below) and M2 (upper).
_~.-
:~---~.'..., k'-
RSS
40 °
-
RMS(j)
(2)
~
j-1
30°
i:"",'"' "
A
-
..
,
0°
10 °
20 °
30 °
1
30 °
cm 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 RMS Figure 2. RMS per grid point between MED and GOT00.2 for the constituents K1 (below) and M2 (upper). (Unit=m).
Table 2.
RMS
deviation between the tidal models
(Unit-cm) M2 0.9 1.3 1.3
MED - GOT00.2 MED - NAO.99b GOT00.2 - NAO.99b
$2 0.9 1.2 0.8
K1 0.8 0.7 0.9
O1 0.4 0.4 0.3
To compare two models, the inphase Ck = Hk c o s ( G k ) and the quadrature 5'k - Hk s i n ( G k ) terms o f each constituent k from its amplitude Hk and phase lag G~ are c o m p u t e d for each model, and c o m p a r e d in terms of the R M S deviation of amplitude defined as follows 1
RMS
where j is the index for the 4 tidal constituents stated above. To display the geographical distribution of the difference b e t w e e n the models, the RSSs for the 4 major constituents are c o m p u t e d on a 0.5 ° × 0.5 ° grid for each paired models, according to (2). Then, the obtained RSSs are averaged over the three pairs. The final difference distribution b e t w e e n m o d e l s in terms of RSS for the M e d i t e r r a n e a n Sea is shown in Fig. 3. The comparison b e t w e e n the m o d e l s can also be expressed in another manner, n a m e l y by c o m p u t i n g the statistics of the differences of the tidal heights at different time m o m e n t s . For this reason, tidal heights were c o m p u t e d from the m o d e l s on a 15 / x 15 / grid at a n u m b e r o f different time m o m e n t s selected rand o m l y within the time period from August 2001 to
10°
20°
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{ [C~I
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In (1) N is the total n u m b e r o f locations where the inphase and quadrature terms are c o m p u t e d and the sub indexes 1 and 2 stand for two different models.
1
2
3
4
5
6
7
8
9
10
RSS Figure 3. Difference distribution between models in terms of RSS. (Unit=cm).
59
60
D.N. Arabelos.
G. A s t e r i a d i s • M. E. C o n t a d a k i s • D. P a p a z a c h a r i o u .
0o
10o
20°
S. D. S p a t a l a s
30°
M a y 2005. It was f o u n d that 20 such grids were sufficient to give a stable estimate o f the standard deviation o f the differences for the three pairs o f the models.
40 °
40 ° The n u m b e r o f the differences varies for the three pairs from 7 4 3 2 0 to 84780, d e p e n d i n g on the comm o n points o f the grid w h e r e the c o m p u t a t i o n o f tidal
30°
30° 0o
10o
20 °
30°
height was possible from both m o d e l s o f each pair. The statistics o f these differences is s h o w n in Table 3.
cm Table 3 shows that in terms o f standard deviation the best a g r e e m e n t o c c u r r e d b e t w e e n G O T 0 0 . 2 -
0 1 2 3 4 5 6 7 8 9 10111213141516171819 0°
10°
20 °
30°
N A O . 9 9 b , a l t h o u g h in this case the m a x i m u m value o f differences exceeds 66 cm.
The standard devi-
ation o f the differences b e t w e e n T P X O . 6 . 2 ( M E D ) 40 °
40 °
and G O T 0 0 2 , G O T 0 0 . 2 ( M E D ) and N A O . 9 9 b and T P X O . 6 . 2 ( M E D ) and N A O . 9 9 b are s h o w n in Fig. 4. In this Figure the m a x i m u m values o f standard deviation a p p e a r e d m a i n l y in the closed gulfs o f Gabes, Therrnaic and Venice, as it was expected.
30°
30° 0o
10o
20 °
I n d i v i d u a l tidal heights from the three m o d e l s are
30° cm
135) o f J A S O N - 1 in the test area (see Fig. 5).
0 1 2 3 4 5 6 7 8 9 10 1112 13 14 15 16 0o
10o
20 °
c o m p u t e d along selected passes (No 007, 009, 094,
The tidal heights for the selected tracks are s h o w n in
30°
Fig. 6. F r o m this figure it is s h o w n that tidal heights : ~,.,% . 4 t'.
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a l o n g tracks lying in the o p e n e d sea present differences up to 5 cm, w h i l e for track 94 p a s s i n g t h r o u g h A e g e a n Sea the differences are increased up to 8 cm. This c o u l d be attributed to low a c c u r a c y o f the al-
.
timeter data close to the coasts. •
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.
.
.
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F r o m the variety o f tests c o n d u c t e d to p r o v i d e an
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0
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5 6 7 8 9 10 11 12 13 14 15 STD
Figure 4. Standard deviation of the differences between tidal heights from MED and GOT00.2 (below), from GOT00.2 and NAO.99b (middle) and from MED and NAO.99b (upper). (Unit=cm).
internal assessment o f the tidal m o d e l s resulted an a g r e e m e n t o f t h e m at the level o f several c m in the o p e n e d sea. In closed gulfs the differences are larger, p r e s e n t i n g standard deviations up to 15 cm. The geographical distribution o f the differences b e t w e e n the m o d e l s was c o n f i r m e d from the RSS values. 0o
10o
20°
30°
40°
40°
Table 3. Statistics of the differences of tidal heights computed on a 15' × 15' grid, at 20 different time moments. (Unit = cm).
MED - GOT00.2 MED - NAO.99b GOT00.2 - NAO.99b
Mean 0.0 0.3 0.3
Std 2.3 2.6 1.6
Min -27.8 -29.3 -33.2
Max 31.6 35.5 33.2
30°
30° 0o
Figure
10o
20°
30°
5. JASON-1 individual ground tracks in the Mediterranean. From West to East 007, 009, 094, 135.
Chapter
Table 4. Statistics of the crossover analysis of JASON-1 altimeter data before and after tidal correction. The number of crossovers is 124244. (Unit = cm). Mean -0.86 -0.86 -0.80 -0.86
Uncorrected MED GOT00.2 NAO.99b
3
Std 14.97 12.51 12.41 12.47
Min -70.00 -66.00 -67.00 -68.00
Max 63.00 61.00 58.00 59.00
-
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External assessment
I
. . . . .
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. . . . .
07:22:30
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. . . . .
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'
Summary and conclusion
The statistics of the differences of tidal heights computed from the 3 models on a 15' x 15' grid for 20 different time moments, shows negligible mean differences (up to 3 ram) and standard deviations ranging from 16 to 26 mm. However, minimum and maximum differences between -33 and 36 cm were resulted from this comparison. Maximum values of standard deviation appeared mainly in the closed gulfs (Gabes, Thermaic and Venice), in Gibraltar and in the east coast of the Mediterranean Sea. The geographical distribution of the differences between the models, expressed in terms o f / ~ S S confirms the previous statement.
I . . . . . . . . . .
G O T O 0 . 2 = triangle
7-
"~
Models
A possibility to have an external assessment of the models comes through the comparison of the results of tidal correction applied on heterogeneous data such as the altimeter data. However, altimeter data from T/P and ERS 1/2 are correlated to these models since they have been used for the development of them. For this reason, JASON-1 data have been used instead of T/P and ERS1/2. The assessment was based on the comparison of the statistics of the crossover differences of JASON-1 ssh, before and after tidal correction, using the three models.
4
0'1 . . . . . . . . . . .
+.a
Tidal
[]
2-
-2
o f Recent
In section 2.2 the pre-processing of the altimeter data was described in order to become suitable for this purpose. The statistics of the crossover analysis is shown in Table 4. According to Table 4, the standard deviation of the crossover differences after the tidal correction, using MED, GOT00.2 and NAO.99b was decreased by 16.4%, 17.1% and 16.7% respectively, which represents a considerable amount of the total standard corrections. These results show that the tidal correction from the three models is approximately the same.
10-2
3.2
9 • Assessment
'
07:23:30
UTC
Figure 6. Tidal heights along JASON-1 track 86/007 (a), 86/009 (b), 86/094 (c) and 86/135 (d). Unit = cm.
The comparison in terms of the / ~ M S of the differences of the major diurnal and semi-diurnal constituents resulted in values ranging from 0.4cm for O 1 to 1.3cm for M2. The comparison of the tidal heights along 4 JASON1 tracks computed from the 3 models shows differences varying from 3 to 8 cm. The last case concerns a south going track passing through the islands of the Aegean Sea. The external assessment based on the comparison of the crossover statistics of JASON-1 altimeter data in
61
62
D.N. Arabelos. G. Asteriadis • M. E. Contadakis • D. Papazachariou. S. D. Spatalas
the Mediterranean Sea before any ocean tidal correction and after the tidal correction using the tested models, shows a decrease of the standard deviation of the crossover differences equal to 16.4%, 17.1% and 16.7% for MED, GOT00.2 and NAO.99b respectively. Based on this results, GOT00.2 is slightly better than MED and NAO.99b in the Mediterranean Sea. The intercomparison and the external assessment of the three models in the Mediterranean Sea resulted in a considerable agreement especially in the opened sea. Close to the coasts and in shallow waters, disagreements at the level of dm are showed, obviously related to the known altimetry problems in such areas. Improvement of the models might be achieved by incorporating tidal parameters from the coastal tide gauge stations into the global solutions.
Acknowledgments: Kindly thanks are due to R. Ray (NASA Goddard) for updated information about tidal models.
References Arabelos, D. (2002). Comparison of Earth-tide parameters over a large latitude difference. Geoph. J. Int, i5i, 950-956. AVISO and PODAAC User Handbook (2001). IGDR and GDR Jason Products. Egbert, G. D., A. F. Bennett and M. G. Foreman (1994). TOPEX/POSEIDON tides estimated using a global inverse model. J. Geophys. Res., 99 (C12), 24821-24852. Egbert, G. D. and S. Y. Erofeeva (2002). Efficient inverse modeling ofbarotropic ocean tides, Journal of Atmospheric and Oceanic Technology, 19, N2, 183-204.
Lemoine, F.G., S.C. Kenyon, J.K. Factor, R.G. Trimmer, N.K. Pavlis, D.S. Chinn, C.M. Cox, S.M. Klosko, S.B. Luthcke, M.H. Torrence, Y.M. Wang, R.G. Williamson, E.C. Pavlis, R.H. Rapp, and T.R. Olson (1998). The Development of the Joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) Geopotential Model EGM96, NASA/TP-1998-206861, Goddard Space Flight Center, Greenbelt, MD. Le Provost, C., M. L. Genco, F. H. Lyard, P. Vincent and R Canceil (1994). Tidal spectroscopy of the world ocean tides from a finite element hydrodynamic model. J. Geophys. Res., 99 (C12), 2477724798. Le Provost, C. (2001). Ocean tides. In Satellite Altimetry and earth sciences, Lee-lueng Fu and Army Cazenave (Eds.), International Geophysics Series, Vol. 69, Academic Press, 267-303. Matsumoto, K., T. Takanezawa and M. Ooe (2000). Ocean tide models developed by assimilating TOPEX/POSEIDON altimeter data into hydrodynamic model: A global model and a regional model around Japan. J. of Oceanography, 56, 567-581. Ray, R. D. (1999). A global ocean tide model from Topex/Poseidon altimetry: GOT99.2, NASA Tech. Memo. 209478, Goddard Space Flight Center, Greenbelt, 58 pp. Saleh, J. and N. Pavlis (2002). The development and evaluation of the Global Digital Terrain Model DTM2002. 3 ~ Meeting of the International Gravity and Geoid Commission,I.N. Tziavos (Ed.), 207-212. Schwiderski, E. W. (1980). Ocean tides, Part I: Global ocean tidal equations, Ma~ Geod., 3, 161217, 1980. Wang Y. (2004). Ocean Tide Modeling in the Southern Ocean. Report No. 471, Ohio State University, 2004.
Chapter 10
Scale-based comparison of Sea Level observations in the North Atlantic from Satellite Altimetry and Tide Gauges S. M. Barbosa, M. J. Fernandes, M. E. Silva Departamento de Matematica Aplicada, Faculdade de Ciencias, Universidade do Porto Rua do Campo Alegre, 687, 4169-007 Porto, Portugal
Abstract. A comparative study is carried out for sea level observations in the North Atlantic from tide gauges and satellite altimetry. Monthly tide gauge records from 12 stations in both sides of the North Atlantic from January 1993 to December 2003 and monthly time series of sea level anomalies derived from TOPEX measurements are considered. The degree of association between tide gauge and altimetry observations is analysed for different scales by computing the correlation between the sea level components resulting from a multiresolution analysis based on the maximal overlap discrete wavelet transform. A similar correlation analysis is carried out to assess the relationship between the sea level observations and climate variables: sea surface temperature, precipitation rate and wind speed. The results show that altimetry and tide gauge observations are strongly correlated, as expected, but also that the relation is scale dependent, with covariability driven by the seasonal signal for most stations. For all variables the obtained correlation patterns exhibit significant spatial variability reflecting the diversity of local conditions affecting coastal sea level.
calibration issues. This is also a fundamental step in devising strategies to merge the available observations and take advantage of the distinct spatial and temporal resolutions (Chambers et al. 2002, Church et al. 2004, White et al. 2005). Another issue is the relationship between the sea level and climate variables since the physical processes influencing sea level, and the sea level response itself, can be different at the coast and offshore. This study focuses on the comparison of sea level observations in the North Atlantic from tide gauges and satellite altimetry using a scale-based approach.
2 Data Continuous and high quality tide gauge records covering the TOPEX/Poseidon (T/P) period are required for comparison purposes. Tide gauge stations in the North Atlantic with data available for this period have been selected (Figure 1) and monthly Mean Sea Level (MSL) series obtained from the PSMSL database (Woodworth and Player, 2003). Ocean tide and inverse barometer (IB) corrections are not applied to the tide gauge data.
Keywords. Sea level, satellite altimetry, tide gauges, discrete wavelet transform
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~0°~ , _ , Sea level measurements are available from in-situ instruments (tide gauges) and from space through radar altimeters. The two components of the sea level observing system are complementary, each yielding unique information on sea level change over distinct spatial and temporal scales. Comparisons of tide gauges and satellite altimetry measurements have been used for assessment of the ocean signals retrieved by altimeters (Cheney et al. 1994, Mitchum, 1994), for absolute calibration (Dong et al. 2002, Woodworth et al. 2004) and to monitor the stability of the altimetry system (Mitchum, 1998, 2000). The interest in comparing tide gauge and altimetry measurements is not only related to
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Fig. 1: Studyarea and tide gauges: 1-Newlyn;2-Brest; 3-Vigo; 4-Las Palmas; 5-Portland;6-Newport; 7-C. May; 8-Chesapeake; 9-Wilmington; 10-Charleston; l 1-FortPulaski; 12-VacaKey. TOPEX measurements from 1993 to 2003 are obtained from the AVISO GDR-Ms products (AVISO, 1996). Corrected Surface Heights (SSH) are derived by applying standard instrumental and geophysical corrections, including the wet tropospheric correction (Scharroo et al. 2004), the sea state bias (SSB) (Chambers et al. 2003) and a
64
S.M. Barbosa • M. J. Fernandes • M. E. Silva
residual SSB correction (Berwin, 2003). Ocean tides are removed using the NAO99b model (Matsumoto et al. 2000) since it yields smaller residuals in the coastal regions (Fernandes and Antunes 2003). The pole tide correction and the inverse barometer (IB) correction are not applied. Sea Level Anomalies (SLA) are derived from the GSFC00.1 mean sea surface (Wang, 2001). Along-track measurements for each month are converted onto a 1o regular grid using the algorithm of Smith and Wessel (1990). Monthly time series of precipitation rate (P) and wind speed (WS) are extracted from the 2.5 ° grid NCEP reanalysis dataset while monthly series of sea surface temperature (SST) are obtained from the NOAA 1° grid dataset (Reynolds et al. 2002). For each variable the gridpoint nearest to each tide gauge station is chosen. All time series are standardised by subtracting the mean and dividing by the corresponding standard deviation for the analysed period.
matrix W' is not formed explicitly but obtained using a wavelet filter through a recursive algorithm (Mallat, 1989), defining a multiresolution analysis tl
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Time series from each tide gauge and the nearest altimeter gridpoint (average distance 40 km, maximal distance < 90 km) are analysed. The Pearson correlation coefficient (Table 1) indicates a moderate positive correlation for all sites. Since sea level variability involves a wide range of scales, assessing the correlation as a function of scale is more appealing than using a single number to describe the dependence between tide gauge and altimetry observations. In this study an additive decomposition based on the discrete wavelet transform is carried out for a scale-based comparison. Table 1. Pearson correlation coefficient (r) between tide gauge
and TOPEX observations
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3.1 Wavelet decomposition The wavelet transform is a flexible approach for the description of the local behaviour of a time series. The level J discrete wavelet transform (DWT) for a monthly time series X is defined by an orthonormal transform W=W'X. In practice the transformation
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modified version of the DWT, the maximal overlap discrete wavelet transform, MODWT (Percival and Walden, 2000). A level J=4 MRA based on the MODWT is carried out for each tide gauge and each altimetry series using a Daubechies filter of length 4 (Daubechies, 1988) with periodic conditions. Figure 2 shows the resulting decomposition for Newlyn.
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3.2 Correlation patterns A scale-based description of the degree of association between tide gauge and altimetry observations is obtained by computing the correlation between the MRA components. The resulting correlation patterns (Figure 3) show a strong positive correlation between tide gauge and altimetry observations. The degree of association is however scale dependent. For most stations covariability is mainly driven by the seasonal signals (decomposition level 3). In order to assess the relationship between the sea level and the forcing variables, a similar correlation analysis is carried out for the sea level series from tide gauges (MSL) and from altimetry (SLA) and the climate variables SST, P and WS. The correlation between sea level and sea surface temperature (Figure 3) is dominated on the west coast by the
Chapter 10 • Scale-Based Comparison of Sea Level
seasonal signal while on the east coast higher correlations are found at the largest scales. Very similar correlation patterns are obtained for tide gauge and altimetry observations suggesting a similar response to SST.
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65
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speed has been examined (Figure 4), although the effect of wind on sea level is difficult to establish from scalar wind alone. The correlation is strong for Newlyn, Brest and Vigo and weak at the remaining sites The correlation patterns for altimetry and tide " gauges are similar with differences at the largest scale for most stations. These results show that distinct factors influence the analysed sea level records reflecting local conditions at the coastal sites. At the seasonal scale, the dominant influence is temperature for the stations on the western boundary, and atmospheric and hydrologic effects on the eastern boundary.
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The correlation between sea level and precipitation rate is stronger for Newlyn, Brest and Vigo but the relation between scales is distinct for the three stations: for Newlyn and Vigo the resulting correlation pattern is very similar for tide gauge and altimetry observations, with correlation increasing with scale up to level 3 for Newlyn and level 4 for Vigo and decreasing afterwards; for Brest the correlation is slightly higher for tide gauge measurements, at all scales, and the largest correlations occur at the seasonal scales. The correlation of sea level and precipitation is weaker at Las Palmas and f°r m°st °f the stati°ns °n the west coast; the exception is the southern station, Vaca Key, for which a strong correlation is found at the values.
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66
S . M . Barboss • M . J . Fernandes • M . E . Suva
4 Discussion and Conclusions A comparative study between sea level from tide gauges and satellite altimetry has been carried out from a scale-baserf perspectine . The time series have been decomposed into components associated to specific scales through a multiresolution analysis baserf on the discrete wavelet transform . Choice of the wavelet filter to be used is an issue inherent to any wavelet analysis . The wavelet filter used here has been selected in a trade-off between the nood of a large width filter to avoid sharpness in the components and the resulting increase in the number of coefficients which are affected by boundary conditions . By repeating the analysis using different filters we have concluded that the results are not sensitive to the selected wavelet filter. The degree of association between tide gauge and altimetry records, and between sea level and climate variables has been estimated through the correlation coefficient for the components resulting from the wavelet decomposition . A delicate issue is the significante of the computed correlations . In the case of time series, inference is affected by serial correlation . Nevertheless, useful information can stil) be retrieved from the computed correlations even without significante testing, particularly concerning the scale-baserf correlation patterns . The dependence of correlation across the various scales is more informative than the actual value of the correlation . Changing the degree of smoothing of the time series, for example through filtering, affects the computed correlation nalues but leaves the correlation pattern across scales virtually unchanged, only affecting slightly the first (high frequency) scales . Tide gauge and altimetry observations are, as expected, positively correlated but the relation between sea level as measured by tide gauges and altimetry is scale-dependent . For most stations covariability is drieen by the seasonal signals and is lowest for the first scale (high frequency signals), increasing witti scale . Although the correlation patterns exhibit considerable spatial variability, reflecting local influences on sea level, they also display similar features for nearby stations indicating regional coherency . Here we have described a scale-baserf approach for the comparison of altimetry and tide gauge observations baserf on the discrete wavelet transform . Since each specific method has its advantages and drawbacks, alternativo approaches could be considered in order to strengthen and confirm our results .
Centro de lnvestiga~ o em Ciências Geo-espaciais (CICGE) of the Faculty of Science, University of Porto .
References AVISO (1996) . User handbonk, AVI-NT-02-lal-CN, Ed. 3.0 . Berwin, R . (2003) . Topex/Poseidon Sea Surface Height Anomaly Product. User's Reference Manual, NASA JPL Physical Oceanography DAAL, Pasadena, CA . Chambers, D . P ., C . F . Mehlhaff, T. J . Urban, D. Fujii . and R . S . Nerem (2002). Low frequency variations in global mean sea level :1950-2000. J Geophys Res, 107, doi : 10 .1029/2001JCOa1089 . Chambers, D. P., S . A . Hayes, J . C . Ries and T. J . Urban (2003) . New Topex sea state bias models and their effect on global mean sea level . J Geophys Res,108, pp . 3305-3311 . Cheney, B ., L. Miller, R. Agreen, N . Doyle and J . Lillibridge (1994) . TOPEX/POSEIDON : The 2-cm Solution . J Geophys Res, 99, pp . 24555-24564 . Church, J., N . White, R Coleman, K . Lambek and J . Mitrovica, (2004) . Estimates of the regional distribution of sea level rise over the 1950-2000 period. JClimate, 17, pp . 2609-2625 . Daubechies, 1 . (1988) . Orthonormal bases of compactly supported wavelets . Commun Por Appl Math, 41, pp . 909-996. Dong, X ., P. Moore and R . Bingley (2002). Absolute calibration of the T/P altimeters using UK tide gauges, GPS and precise local geoiddifferences . Mar Geod, 25, pp .189-204 . Fernandes, M . J. and M . A . Antunes (2003). Eight years of satellite radar altimetry in the Northeast Atlantic . In : Proc. 3 Assembleia Luso-Espanhola de Geodesia e Geofisica, Editorial UPV, pp . 226-230 . Mallat, S . 6.1989 . A theory for multiresolution signa) decomposition : the wavelet representation. IEEE T Pattern Anal,11, pp . 674-693 . Matsumoto, K ., T. Takanezawa and 0 . Masatsugu (2000). Ocean tide models developed by assimilating Topex/Poseidon altimeter data into hydrodynamica) model : a global model and a regional model around Japan . JOceanogr, 56, pp . 567-581 . Mitchum G . T . (1994) . Comparison of TOPEX sea Surface heights and tide gauge sea levels . J Geophys Res, 99, pp . 24 541 - 24 553 . Mitchum, G . T. (1998) . Monitoring the stability of satellite altimeters witti tide gauges, JAtmos Ocean Tech, 15, pp . 721-730 . Mitchum, G . T . (2000). An improved calibration of satellite altimetric heights using tide gauge sea levels witti adjustment for land motion, Mar Geod, 23, pp.145-166 Percival, D. and Walden, A . 2000 . Wavelet methods for time series analysis, Cambridge University Press . Reynolds, R. W., N . A . Rayner, T . M. Smith, D . C . Stokes and W . Wang . (2002). An Improved In Situ and Satellite SST Analysis for Climate . JClimate, 15, pp .1609-1625 Scharroo, R., J . L. Lillibridge, W . H . F . Smith and E. J . 0. Schrama (2004) . Cross-calibration and long term monitoring of the microwave radiometers of ERS, TOPEX, GFO, Jason and Envisat, Mar Geod, 27, pp . 279-297 . Smith, W. H . F . and P . Wessel (1990). Gridding witti continuons curvature splines in tension . Geophysics, 55, pp. 293-305 . Wang, Y . M. (2001) . Mean Sea Surface, gravity anomaly, and verfka) gravity gradient from satellite altimeter data . J Geophys Res,106, pp . 31167 -31174. White, N . J ., J . A. Church and J . M . Gregory (2005). Coastal and global averaged sea level rise for 1950 to 2000 . Geophys Res Lett, 32, doi :10 .102912004GL021391 .
Acknowledgements
Woodworth, P. L . and R. Player (2003) . The permanent service for mean sea level : an update to the 2lst century . J Coastal Res, 19, pp . 287-295 .
Statistica) analysis has been carried out witti R and R package waveslim provided by B . Whitcher . This work bas been supported through the
Woodworth, P ., P. Moore, X. K. Dong and R. Bingley (2004) . Absolute Calibration of the Jason-1 Altimeter using UK Tide Ganges . Mar Geod, 27, pp. 95-106 .
Chapter 11
European Sea Level Monitoring: Implementation of ESEAS Quality Control Maria Jesfis Garcia,Instituto Espafiol de Oceanografia, Corazdn de Maria, 8, E-28002 Madrid, Spain Begofia Pdrez Gdmez,Puertos del Estado, Area de Medio Fisico y Tecnologia de las Infraestructuras, Avda Partendn, 10, E-28042 Madrid, Spain Fabio Raicich,Consiglio Nazionale delle Ricerche, Istituto di Scienze Marine, Viale Romolo Gessi, 2, 134123, Trieste, Italy Lesley Rickards, Elizabeth Bradshaw, British Oceanographic Data Centre, Joseph Proudman Building, 6 Brownlow St., Liverpool, L3 5DA, UK Hans-Peter Plag, Nevada Bureau of Mines and Geology and Seismological Laboratory, University of Nevada, Reno, Mailstop 178, NV89557, USA Xiuhua Zhang, Bente Lilja Bye, Espen Isaksen,Geodesi, Norwegian Mapping Authority, 3507 Honefoss, Norway
One of the objectives of the European SeaLevel Service (ESEAS) is to provide access to quality controlled European tide gauge data for research and other uses. The ESEAS quality control is based on a common set of procedures for the quality control of observed sea level data, which to a great extent can be referred to those specified by the Intergovernmental Oceanographic Commission for the Global Sea Level Observing System. Quality control also extends to other information such as documentation of datum information, metadata, exchange format, application classification, and levels of quality control. In relation to tsunamienabled tide gauges, quality control is related to the availability of data in real-time and the use of automatic control and analysis procedures. Abstract.
Keywords. Sea level monitoring, tide gauge, distribution of data, standardization, quality control
1 Introduction The European Sea-Level Service (ESEAS) is an international collaboration of organizations in 23 countries. It has brought
observation and research community in Europe as a major research infrastructure for all aspects related to sea-level, in the fields of climate change research, natural hazards or marine research. One of the main objectives of the ESEAS is to provide access to quality controlled European tide gauge data. Most tide gauge authorities have developed their own methods of quality control and for the ESEAS, a together the formerly scattered sea level
common standardized set of procedures has to be adopted. Based on existing documents such as the IOC manuals (UNESCO, 2002, and the references therein) and ESEAS documents (Plag et al., 2000), these standards were defined within the European Sea Level Service-Research Infrastructure (ESEASRI) project, funded by the European Commission from November 2002 to October 2005. The ESEAS Observing Sites (EOS) are primarily tide gauges providing continuous measurements of the sea level relative to a well monitored benchmark on land (ESEAS Governing Board, 2001). Some of these tide gauges are co-located with geodetic techniques such as continuous Global Positioning System (CGPS), and in some cases episodic absolute gravity measurements are performed. For an EOS committed to the ESEAS, operation has to comply with the accepted standards for each of the applications defined and classified by ESEAS (2004a). At present, there are more than 200 tide gauges in the ESEAS observing network distributed along thousands of kilometres of the European coast. Two levels of quality control depending on the delivery time line are defined, namely L l: quality control for low-latency products, and L2: full quality control and analysis for delayed mode products. L1 is less detailed than the one applied to delayed mode or historical time series, and is tailored for the operational use of sea level data (e.g. storm surges, tsunami warning), where automatic procedures, short sampling intervals and rapid transmission are crucial. For the L2 or delayed mode quality control, the use of additional
68
M.J. Garcia • B. P~rez G6mez • F. Raicich • L. Rickards • E. Bradshaw • H.-P. Plag. X. Zhang • B. Lilja Bye. E. Isaksen
information is mandatory: documentation of datum information, diagrams, maps and other meta-data have to be provided. However, there has been little standardization of the type of information. The ESEAS has agreed on common procedures across the participating organizations. The design of a unique ESEAS web interface is in progress (www.eseas.org). This allows easy access to both the near-real time and historical quality controlled data. The interface provides a single point of access for users, who previously needed to contact individual institutions of different countries in order to get information and data, if available, about sea level. 2
Data
Requirements
From each tide gauge station three basic data sets should ideally be available, namely (1) raw sea level data in digital or graphical form, (2) the calibration data (normally manual data obtained at the site during each calibration), and (3) levelling information. Sometimes ancillary data are also present. Depending on the acquisition system, the actual sea level data are obtained either by digitizing graphical records of analog systems, or from a sensors providing digital output. In the latter case raw sea level data are obtained at a particular sampling interval (at present between 5 and 10 minutes, normally not lower). These raw higher frequency data are processed to L1 quality control, in order to be available for operational purposes. Hourly values are obtained by means of an adequate filter. Additional information (metadata) is needed not only for quality control and archiving, but also for exchanging data or integration of them into a regional or global data set. A wide variety of metadata 'standards' (e.g. FGDC, Dublin Core, ISO19115) are in regular use and metadata recommendations are also available from IOC Manuals and Guides, GE-TADE (IOC/IODE), and various other projects. The ISO standard ISO 19115 for geo-spatially referenced data has recently been ratified, thus the metadata for ESEAS will be compliant with this standard. The ESEAS Quality Control document (ESEAS, 2004b) includes a list of the information that should be stored with the data. The metadata accompanying each data series must be sufficient to ensure that the data are adequately qualified and
may therefore be used with confidence. Such documentation should be stored alongside the data, and where applicable, should describe the site, the data sampling and processing and the instruments used.
3 Quality Control and Quality Assurance As for any measured physical value, the errors that can arise in the sea level data and related parameters can be random or systematic. Examples of the first type are electronic noise, problems in the communications, sensor calibrations, etc. Systematic errors would be, for example, inhomogeneities due to changes in observational practices, or a change of instrumentation. Changes of instrument location can result in sharp discontinuities in the sea level data. Changes in the environment of the station, such as harbour constructions, land movements, etc., can produce trends in the data or changes in the tide parameters. For the L 1 level (low latency), the initial process of quality control consists of performing various checks of the original series and then flagging of suspicious values according to ESEAS quality control codes. Near-real time data applications require the development of automatic procedures that perform the basic quality control also in nearreal time. This automatic procedure consists basically of the following steps applied to the original high frequency raw data: • checks for strange characters • checks of date and time • gap detection • out of range and spikes detection • constant values detection (stability test) • doubtful values An adequate algorithm for detection of spikes has to be used, which leaves real phenomena, like tsunamis and seiches, unflagged. At this stage normally ancillary data or data from nearby stations are not used as the time of processing would become too long and ineffective. The result of the L1 quality control is the original raw data flagged, and a new corrected time series with flags, regular sampling and gaps lower than a certain interval interpolated. As part of the scientific or delayed mode quality control (L2), a more detailed processing of sea level data is performed, applied to longer time series (typically one year) that include not only the steps
Chapter11 • EuropeanSea LevelMonitoring:Implementationof ESEASQualityControl
described for L1, but also the filtering to hourly values, computation of annual harmonic constants, residuals, extremes and means. The examination of the quality of these products is crucial for the detection of problems and malfunction in the tide gauge. The primary quality control of sea level is based on the inspection of both recorded data and meteorological residuals. Raw data are normally registered at time intervals between 1 minute and 1 hour, the most common being 5, 6 and 10 minutes. In regions where seiches occur frequently, or where phenomena such as tsunamis should be detected, sea level data are registered at lminute interval or even less. Apart from keeping higher frequency signals for other purposes, it is always necessary to obtain filtered hourly values for further sea level data processing. The filtering process will eliminate higher frequencies dependent on the frequency cut-off. Pugh (1987) describes useful filters that can be applied to the sea level data at intervals of 5, 10 or 15 minutes to obtain the hourly heights whilst preserving the tidal phenomena. In Godin (1972) there is an extensive discussion on tidal filters. Within the ESEAS the recommendation is to use filters with amplitude response similar to the one of Pugh or Butterworth filters, in order to avoid reduction in tidal ranges and extremes just due to filtering. A study of the importance of higher order harmonics for a particular station may indicate the need for designing a filter, which does not attenuate other high frequencies. A common procedure is to compute the harmonic constants for each year of observed data. Some problems in the data series, such as clock errors, will introduce temporal variation in the normally stable harmonics. An inspection of the variation through the years of the harmonic constants is interesting both for detecting problems and also for having information about changes at the station. For example, changes in the configuration of a harbour can affect the tide parameters. The inspection of meteorological residuals is a very useful tool for the quality control process. All fundamental types of errors that a sea level series can present (e.g. clock malfunction, reference changes, spikes) are easily detected in the residual plot. A shift or drift in the time (caused either by operator error or clock malfunction) can be detected by visualization of the residuals or correlation between observed data and predictive or
neighbourhood station data. The data will then need to be moved according to the lag time (if constant) or interpolated. Correlations can be computed both between data from different stations or sensors and between different parameters at the same station (wind, atmospheric pressure, etc), and this is a valuable tool for detecting problems. Study of the lag correlation between observed and predicted data is used to detect clock malfunctions. Basic statistics from historical data are computed or updated annually; some of these parameters are used for the quality control process, for example: •
upper and lower limits or historical extremes (for range check), • tidal and observed sea level ranges, • extremes, mean and standard deviation of hourly values, meteorological residuals, ranges or mean sea levels, • tables of monthly and annual extremes, • density function for hourly values, tide predictions and residuals, For historical data, the lack of sufficient metadata often complicates the quality control. This may particularly affect the control of datum stability. In this case, some additional checks should be performed to obtain a unique reference. A standard procedure is to work with several sea level series from nearby stations. Different algorithms described in literature can help to detect discontinuities or reference problems in historical data or hourly time series. The Standard Normal Homogeneity Test (SNHT, Alexanderson, 1986) gives the points where an inhomogeneity exists and provides information about the probable break magnitude. An EOF (Empirical Orthogonal Function) analysis over a set of tide gauge records (2 or more) is an available method to estimate the trends and to detect possible errors, apart from other applications. According to Sneyers (1975; 1992), the non-parametric MannKendall test for trends can be used to study the increases and decreases in climatic time series. 4 Software Packages for Sea Quality Control and Processing
Level
Within the ESEAS, the development of a standardized ESEAS sea level data quality control and management software has been proposed. Most
69
70
M.J. Garcia • B. P~rez G6mez • F. Raicich • L. Rickards • E. Bradshaw • H.-P. Plag. X. Zhang • B. Lilja Bye. E. Isaksen
sea level data collecting organizations within ESEAS have developed their own software to validate incoming data in varied formats and media that are specific to their requirements. However, it would benefit the community if there would be a platform independent package available. An on-line tidal analysis facility has been made available on the UK National Tidal and Sea Level Facility website (www.pol. ac.uk/ntslf), and the Israel Oceanographic and Limnological Research (IOLR) Institute has produced a Windows based sea level package. These, together with other software packages currently in use provide the basis of an ESEAS package.
5 ESEAS Data Exchange The ESEAS has agreed on an exchange format, including a set of quality flags. The format is a simple 'spreadsheet' style similar to that in use in many participating organisations and has been defined taking into account international guidelines developed by working groups within the IOC's International Data and Information Exchange Committee. The format includes appropriate ancillary information. Via the website, users can search for data by date, region, country and site. The retrieved data contain information about the quality control of the data, thus allowing each user to decide for what applications the data are suitable for. The basic principles of the web portal are described in Plag (2004). The web portal is designed as a distributed system with the tide gauge data being stored at each ESEAS Operational Centre (EOC). A database is used to store metadata for the tide gauge data as well as details of the EOCs and the EOSs according to a predefined structure. An underlying principle in developing the web portal is that the work required by each EOC should be kept at a minimal level. When a user submits a request for data, the web portal will automatically retrieve the data from the respective EOC, thus ensuring that the data are up-to-date. The standardized quality control described above ensures that the quality of the products from the
web portal is as far as possible independent of the EOC delivering the data.
Acknowledgments The authors would like to thank the many colleagues in the ESEAS and the ESEAS-RI project for the discussions of the ideas reported here. The ESEAS-RI project was supported by the European Commission under the contract number EVR1-CT2002-40025.
References Alexanderson, H. (1986). A Homogeneity Test Applied to Precipitation Data, Journal of Climatology, 6, pp661-675. ESEAS (2004a). Classification of ESEAS Operational Sites for Applications A- D, http://www, eseas,org/eseas-ri/deliverables/d 1.3 ESEAS (2004b). Standards for Quality Control of Tide Gauge Observations, Deliverable D 1.2. http://www, eseas.org/eseas-ri/deliverables/d 1.2 ESEAS Governing Board (2001). European Sea Level Service, Terms of Reference, http://www.eseas.org/websitemap.php Foreman, M.G.G. (1977). Manual for Tidal Heights Analysis and Prediction, Pacific Marine Science Report 77-10, pl00. Godin, G. (1972). The Analysis of Tides, Liverpool University Press, Liverpool, p264. Plag, H.-P., Axe, P., Knudsen, P. Richter, B. and Verstraeten, J.,(2000). European Sea Level Observing Systems (EOSS) Status and Future Developments, European Commission, Cost Action 40, EUR 19682. Plag, H.-P. (2004). The ESEAS Data Portal: Principal Considerations, In Holgate, S. and Aarup, T. (ed.): Workshop on New Technical Developments' in Sea and Land Level Observing Systems, IOC Workshop Report
No.193, ppl08-113. Pugh, D.T. (1987). Tides, Surges and Mean Sea-Level, Chichester, John Wiley and Sons, p472. Sneyers, R. (1975). Sur l'annalyse statistique des sdries d'observations, N° 415 WMO, Ginebra. Sneyers, R. (1992). On the Use of Statistical Analysis for Objective Determination of Climate Change. Meteorol. Zeitschrift, No. 1, pp247-256. UNESCO (2002). Manual on Sea Level Measurement and Interpretation, Volume Ill: Reappraisals and Recommendations as of the Year 2000, IOC Manuals and Guides, No. 14, p55.
Chapter 12
Brazilian Vertical Datum Monitoring- Vertical Land M o v e m e n t s and S e a Level Variations R. Dalazoana 1, S. R. C. de Freitas 1, J. C. Baez 1'2 and R. T. Luz 1'3 Department of Geomatics, Geodetic Sciences Graduation Course Federal University of Paranfi (UFPR), Curitiba, Paranfi, Brazil 2 Department of Surveying, University of Concepci6n, Chile 3 Coordination of Geodesy, Brazilian Institute of Geography and Statistics (IBGE), Brazil Abstract. UFPR's Geodetic Instrumentation Laboratory carried out several geodetic campaigns at Brazilian Vertical Datum during the last nine years. The last campaign performed in the first semester of 2005 had as main goals: calibration of two sea level sensors of the tide gauge station; periodic GPS observations at one SIRGAS station, used to materialize the tide gauge geocentric position; and leveling of the benchmarks in the harbor area. In this work there are presented results obtained with this campaign and some comparisons to past campaigns. The main objective is to check the stability of the GPS station in order to estimate vertical land movements and sea level variations. GPS results indicate a tendency of subsidence of 0.2mm/a with a standard deviation of 0. lmm/a at the station.
Keywords. Vertical land movements; GPS survey; Precise leveling; Sea level variations
about crustal deformations, especially the vertical movements, and how these deformations affect the tide gauge measurements. The establishment of the geocentric position of a point in the vicinities of the tide gauge, linked to the tide gauge by spirit leveling, allows for defining the MSL in a geocentric Geodetic Reference System (GRS). Then, with the tide gauge geocentric position monitoring, absolute sea level variations can be obtained instead of values related to the structure where it is fixed. Geodetic Instrumentation Laboratory from Federal University of Paranfi (UFPR), Brazil, performed during the last nine years geodetic campaigns at Brazilian Vertical Datum (BVD) region. These campaigns included periodical GPS positioning in order to estimate the geocentric position of BVD, and to detect possible vertical movements. Initial results from these campaigns are presented.
2 Tide Gauge Geocentric Position Con1 Introduction In South America, the majority of the vertical Datums were defined between the years 1940 and 1960, through the observation of the Mean Sea Level (MSL) with a time span of at least nine years. So, each vertical Datum is related to a specific epoch and with different periods of sea level observation. As well known the sea level varies in time and space and these variations has been estimated, usually, from tide gauge measurements. The tide gauges are connected by first order leveling to benchmarks (BMs) located on land. The measured water level changes are also relative to land. Then, possible vertical movements of the earth's crust appear as sea level variations in the tide gauge records. Consequently, the study of sea level temporal variations depends on the knowledge
trol SIRGAS project allowed for definition and realization of a 3D reference system in South America with geocentric coordinates consistent with I T R F 2 0 0 0 - International Terrestrial Reference Frame (IBGE, 2002). The two SIRGAS campaigns performed in 1995 and 2000 allowed for velocities estimation for SIRGAS 1995-2000 stations. SIRGAS 2000 campaign also estimated the geocentric position for most tide gauges which define the vertical datum of the leveling networks in each South American country. This campaign included the station IMBI (Figure 1), which materialize the geocentric position of BVD. IMBI station is located at Imbituba harbor, Santa Catarina State, south Brazil, and it is approximately 450m far away from the tide gauge station. The station, built in 1997, consists in
72
R. Dalazoana • S. R. C. de Freitas. J. C. B~ez. R. T. Luz
a pyramidal monument with a bolt that holds the GPS antenna.
Table 1. SIRGAS 2000 Geocentric Coordinates - IMBI station SIRGAS 2000 (ITRF2000 epoch 2000.4) X (m)= 3 714 672.427 q~= -28° 14' 11.8080" Y (m) = -4 221 791.488 X = -48° 39' 21.8825" Z (m)=-2 999 637.883 h = 11.850m
b
.i
.....a
..........
Fig. 1. IMBI station
The connection between the GPS station and the tide gauge is done by spirit leveling. At Imbituba, the tide gauge is located below the harbor's principal pier. In 1997 it was made the first connection between IMBI station and an old BM (3M) on the pier located approximately over the tide gauge (Cordini, 1998). In 2001, it was possible to connect IMBI station to others BMs within the harbor area, including two BMs located close to the tide gauge below the pier, the tide staff and the stilling well. This operation was repeated in 2005 and the observed height differences were adjusted (see section 3). The recommended procedure is to have a dual frequency continuously recording permanent GPS receiver installed at or near the tide gauge (Bevis et al., 2002). But, due to the impossibilities to allocate a permanent receiver at Imbituba, in Brazilian case four GPS campaigns were performed at Imbituba during the last nine years aiming the connection of BVD to a geocentric GRS and to estimate its velocity in order to detect possible vertical motions. First GPS campaign was performed in 1997 during 45 days but only data from GPS days 118, 119, 120, 123 and 124 were recovered. The second campaign was performed in 2000 (GPS days 131 to 140) in the context of SIRGAS Project, Table 1 indicates the resulting coordinates for IMBI station (IBGE, 2003). The third campaign was performed in February 2005 (GPS days 41 to 48) and the fourth in July 2005 (GPS days 189 to 197). The first three campaigns were performed with an Ashtec ZXII receiver and the last one with a Trimble 4000 SSI receiver.
2.1
GPS
Data
Processing
Methodology
The first campaign was reprocessed considering aspects like: improvement of ambiguities resolutions techniques; precise orbit improvement; receiver phase antenna offset determination; satellite offset antenna determination and ocean loading effect modeling. The data was organized in 24 hours period and converted to RINEX format. A priori coordinates and velocities were obtained from SIRGAS 2000 solution. Earth orientation parameters (EOP), precise orbits (sp3) and satellite clock corrections were taken from IGS final products. Antenna phase center offset and variations were obtained at NGS. Antenna heights were related to the antenna reference point (ARP). The campaigns were processed with Bernese GPS Software 5.0 (Hugentobler et al., 2004). IGS stations SANT (Santiago) and BRAZ (Brasilia), and the station PARA (Curitiba) from Brazilian active control network (RBMC) were used as fiducial stations to estimate IMBI position (F i gure 2).
280"
300"
320"
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.
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........,;,._Zt..~ ,~.-;,. ...... ~-,=._..-~."
:-~ .~:~ ....
-60"
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Fig. 2. Fiducial stations
300=
320"
Chapter 12 • Brazilian Vertical Datum Monitoring - Vertical Land Movements and Sea Level Variations
Maximum observation strategy (OBS-MAX) was used to create the baselines. The sampling rate used was 30 sec., with an elevation cut off angle of 5 °. QIF (quasi ionosphere free) strategy was used for Ambiguities Resolution with Niell (Dry and Wet) troposphere modeling (Hugentobler et al., 2004). The Global Ocean Tide M o d e l - GOT00.2 was applied for ocean loading correction. The combination of the daily solutions (NEQ) provided a final solution for each campaign.
2.2 GPS Processing Results Table 2 presents the estimated positions and standard deviations (~s) for IMBI station in each campaign. The processing software gives precisions better than 1.5mm for the coordinates; this is very optimistic, needing more studies in order to verify if these values are realistic. Table 2. Geocentric Coordinates - IMBI Station 1997 Campaign (ITRF2000 X (m)= 3 714 672.4278 c~(mm) = 0.6 Y (m)=-4 221 791.4718 c~(mm) = 0.8 Z (m) = -2 999 637.9143 (mm) = 0.5 2000 Campaign (ITRF2000 X (m)= 3 714 672.4277 c~(mm) = 0.4 Y (m)=-4 221 791.4845 c~(mm) = 0.5 Z (m)=-2 999 637.8821 c~(mm) = 0.3 2005 Campaign (ITRF2000 X (m)= 3 714 672.4333 c~(mm) = 1.4 Y (m) = -4 221 791.5173 c~(mm) = 1.2 Z (m)=-2 999 637.8391 c~(mm) = 0.8 2005 Campaign (ITRF2000 X (m)= 3 714 672.4390 (mm) = 0.4 Y (m)=-4 221 791.5062 c~(mm) = 0.5 Z (m)=-2 999 637.8215 c~(mm) = 0.3
epoch 1997.4) qb=-28 ° 14' 11.809025" X =-48 ° 39' 21.882063" h-- 11.8546m epoch 2000.4) qb=-28 ° 14' 11.807957" )~ =-48 ° 39' 21.882371" h = 11.8477m epoch 2005.1) qb=-28 ° 14' 11.806292" )~ = -48 ° 39' 21.883015" h = 11.8523m epoch 2005.5) qb=-28 ° 14' 11.805857" )~ =-48 ° 39' 21.882588" h = 11.8399m
Estimated and derived velocities from SIRGAS velocity model for IMBI station are indicated on Table 3. There is a small tendency of subsidence at IMBI station. The estimated velocities are similar to
the ones derived from SIRGAS velocity model (Drewes and Heidbach, 2005). The processing software gives precisions better than 0.1mm/a for the estimated velocities. Table 3. Velocities - IMBI station Estimated velocities Vy (m/a) VE (m/a) Vup (m/a) Vx (m/a) Vy (m/a) Vz (m/a)
0.0118 -0.0019 -0.0002 0.0021 -0.0053 0.0105
Velocities derived from SIRGAS velocity model VLAT(m/a) VLONG(m/a) Vx (m/a) Vv (m/a) Vz (m/a)
0.0124 -0.0028 0.0018 -0.0062 0.0109
3 Integrating Temporal Time Series of Sea Level Data Usually, the sea level time series in one station were obtained by different tide gauges. Then, it is necessary to relate all the measurements to a common reference to have a coherent sea level time series. This relationship is done based in a network of BMs periodically leveled and associated to the tide gauge. The necessary information to integrate temporal time series of sea level data are: the sea level values; the tide gauge datum definition; the BMs description and the leveling surveys performed. Regarding the datum definition and the sea level values, BVD was established by the MSL with observations between the years 1949 and 1957. Nowadays the periods with available sea level data are: a) 1949 to 1969 monthly and annual means at Permanent Service for Mean Sea Level (PSMSL) database; b) 1969 to 1987 graphic records not found yet; c) 1987 to 1992 graphic records recovered, which digitizing is being made at UFPR's Geodetic Instrumentation Laboratory; d) 1992 to 1998 tide gauge operation seems to be interrupted; e) since 1998 graphic records being digitized at IBGE's (Brazilian Institute of Geography and Statistics) Coordination of Geodesy and f) since 2001 data from digital sensors are available. The process of graphic records digitizing is not an easy task and it is time consuming; efforts are being made in order to automate this process. Concerning the leveling surveys performed at Imbituba is important to say that the United States Coast and Geodetic Survey (USCGS) performed periodical leveling between 1948 and 1971. IBGE performed leveling surveys in 1946, 1980, 1986,
73
74
R. Dalazoana • S. R. C. de Freitas. J. C. B~ez. R. T. Luz
1995, and annually since 2001. The temporal control of the original BMs implanted at the harbor's area during the period of BVD definition is a great problem due to the destruction of the marks and due to the lack of repeated leveling surveys. From the original BMs (approximately eight), only one (BM 3M) was not destroyed until July 15 th, 2005. This BM was of great importance in the integration between old surveys and newer ones. With the 2005 campaigns it was possible to perform leveling surveys on almost all the actual BMs located within the harbor area. The leveling process includes BMs located below the pier, the tide staff, the stilling well and two new BMs established on July 2005; it was performed in order to make an adjustment of the observed height differences. Height of BM 3M was estimated from USCGS reports and was kept fixed (6.555m). Table 4 gives an overview of the adjusted leveled heights of the existing BMs. Table
4. Adjusted Leveled Heights
BMs UFPR2 3012Z 3010A CBD3A 3010B 4X |MB| PORT3 3012X UFPR1
AdjustedHeight (m) 1.829 2.098 6.160 6.140 9.480 8.642 10.508 6.025 2.008 6.889
StandardDeviation (mm) 0.6 0.6 0.9 0.2 1.4 1.4 1.4 0.9 0.6 0.4
The recovering, digitizing and integration of sea level time series are still under development and would be enhanced. Initial results, presented by Dalazoana et al. (2004), based on the integration of sea level data from PSMSL database and sea level data from the digital sensor, had shown relative increasing of about 2 mm/year in the local MSL. With the evolution of graphic records digitizing and GPS processing results, it will be possible to have an estimative of the absolute sea level trend at BVD.
4 Final Remarks Monitoring tide gauges by GPS positioning can give some information about possible vertical crustal movements allowing the separation between apparent and true sea level changes. Precise geocentric positioning associated with leveling gives the
link of the tide gauge zero point to a geocentric reference system. Initial results had shown relative increasing in the local MSL at Imbituba harbor. Due to the periods with no digital sea level data efforts are been done in order to automate the process of graphic record digitalization. There is a small subsidence at IMBI station indicated by the GPS data. Estimated velocities are very similar to the velocities derived from SIRGAS velocity model. The ideal situation in order to estimate velocities, mainly the vertical component, is to have several years of GPS data. There is a project to install very soon a permanent dual frequency GPS receiver at Imbituba, and the collected data will be very important in future studies.
Acknowledges The authors would like to thank CAPES (Brazilian Public Agency of Foment in Education) and CNPq (National Council of Research) by the financial support through scholarship and support to projects. IBGE for historical leveling and tide gauge data. Companhia Docas de Imbituba and Imunizadora Imbituba for operational support.
References Bevis, M., W. Scherer, and M. Merrifield (2002). Technical Issues and Recommendations Related to the Installation of Continuous GPS Stations at Tide Gauges. Marine Geodesy, v. 25, n. 1-2. Cordini, J. (1998). Estudo dos Aspectos Geodindmicos no Datum da Retie Altimdtrica do SGB. Curitiba. 159 f. Tese (Doutorado em Ciencias Geoddsicas)- Setor de Ciencias da Terra, Universidade Federal do Paranfi. Dalazoana, R., R. T. Luz, S. R. C. de Freitas, and J. C. Baez. (2004). First Studies to Estimate Temporal Variations of the Sea Level at the Brazilian Vertical Datum. In: IA G International Symposium - Gravity, Geoid And Space Missions (GGSM2004). Porto, Portugal, 30th August to 3rd
September 2004. CD-ROM Proceedings. Drewes, H. and O. Heidbach. (2005). Deformation of the South American Crust Estimated from Finite Element and Collocation Methods. IAG Symposia, Springer 2004, vol.
128. Hugentobler, U., R. Dach, and P. Fridez (2004). Bernese GPS Software. Version 5.0, University of Bern. 388p. IBGE. Funda~o Instituto Brasileiro de Geografia e Estatistica. (2002). S I R G A S - Boletim Informativo n. 7.
Access" 4 jun. 2003. IBGE. Fundagfio Instituto Brasileiro de Geografia e Estatistica. (2003). Coordenadas SIRGAS 2000. Access" 4 jun. 2003.
Chapter 13
Tide gauge monitoring using GPS Maaria Tervo, Markku Poutanen, Hannu Koivula Finnish Geodetic Institute, Geodeetinrinne 2, 02430 Masala, Finland
Abstract. In this work we have studied accuracy of GPS for tide gauge stability monitoring as well as possibilities for observing the absolute sea level rise of the Baltic Sea with GPS and tide gauge time series. Our determination give the average sea level rise for the long, up to 120 years, time series 1.9 _+ 1.0 mm/year, and when corrected for the geoid rise, 1.6 _+ 1.0 m m / y e a r . Rates of recent years are even eight times higher. The possibility to detect minor changes in tide gauge benchmark height with GPS may be limited by GPS-related errors, which can be up to several mm even in a 10 km baseline.
Benchmark stability can be monitored also with GPS. GPS offers better temporal coverage, ultimately continuous tracking in real time, and the data collection and analysis can be automated. However, there are periodic and random temporal variations, especially in the vertical component, which may limit the resolving power. These variations have been studied for example in Mao et al. (1999) and Williams et al. (2004). The major component of land movements in Fennoscandia is the postglacial rebound, which has been studied also earlier with GPS (see e.g. Johansson et al., 2002).
K e y w o r d s : Sea level rise, land uplift, Baltic Sea, GPS, tide gauge, stability monitoring
70"
20"
15"
25"
30" 70"
1 Introduction Sea level monitoring is an important part of oceanography and climate investigation. Information of the sea level can be used for forecasts of climate change and marine resources as well as for natural hazard mitigation and improvement of use and protection of coastal areas. The sea level can be observed from the coasts by tide gauges or from space with different satellite borne radars. Tide gauges are spatially limited, while they are of limited number and situated on the coasts, but they provide the longest continuous sea level time series. Satellite data are spatially more evenly distributed but they can be temporally limited and their time series are much shorter. Both data are needed to complete each other. There are several studies combining the methods for shorter and longer time periods (e.g. Church et al. 2004, White et al., 2005, Holgate and Woodworth, 2004). Tide gauges measure the sea level relative to a benchmark. The problem with the tide gauges, besides their spatial limitations, is their vertical movements. Movements of these benchmarks and thus movements of tide gauges and the ground around them have been conventionally observed by levelling. Because levelling is done mostly once a year, or even less frequently, it is not very fast or accurate method if sudden movements happen.
~) Sodankyli /
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Oulu
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Olkiluo~ Hamina
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~0.
Helsinki Hanko . ~
~.
~'0.
Fig. 1. The GPS stations of Finnish permanent GPS network FinnRef are marked with spots and the Finnish tide gauges with triangles.
76
M. Tervo • M. Poutanen • H. Koivula
a)
b)
Earth's surface )
Earth's surface
Sob s
Sea surface
/II\ ~
S
--
Sea surface
bill1 ', ~ _n ~zxS ___Ho ', 8-1. . . . . .
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Ellipsoid
Ellipsoid
Fig. 2. Reference surfaces and heights (a) and their changes (b). Sobs is the observed sea level height, S is the absolute sea level height, H is the orthometric height of a benchmark, h is the height above the ellipsoid, N is the geoidal height. Subscripts 0 and 1 in b) denote epochs 0 and 1. Vertical changes are not in scale.
2 Sea level change
or from ellipsoidal heights using geoid height
We have calculated absolute sea level change rates for the Baltic Sea using time series from Finnish tide gauges and Finnish permanent GPS stations (Figure 1), (Tervo, 2004, Poutanen e t a l . , 2004). We chose six tide gauge - GPS station pairs that were less than 30 km from each other. We also assumed that the Finnish bedrock is stable enough for the GPS land uplift rates to be valid also for the closest tide gauge. By combining these two series we get the absolute sea level rise, i.e. the change relative to the mass centre of the Earth. The GPS data was achieved from the Finnish permanent GPS network, FinnRef, and the tide gauge data was provided by the Finnish Institute of Marine Research (K. Kahma, private communication, 2004). Tide gauges measure the sea level relative to a benchmark; this is the observed sea level height Sobs. The height of the benchmark, and thus the ground in the vicinity of the tide gauge, can be observed either by levelling or with GPS. Levelling gives the orthometric height H and GPS the ellipsoidal height h (Figure 2a). The height of the geoid N is the difference between orthometric and ellipsoidal heights (N = h - H). The absolute sea level height S is the difference between the orthometric height and the observed sea level height. The surfaces change in time (Figure 2b) and changes in their heights can be observed. The deformation of the crust AH between epochs 0 and 1 can be calculated from orthometric heights
z~-H
1 -H
o
(1)
- (h, - h o ) - (N 1 - N O) - a h - AN.
(2)
The observed sea level height is the height between the benchmark and the sea level (3)
Sob s - H - S ,
so the observed sea level change is Sobsl -- Sobs2 =
(H 1-
S 1 ) --
=(H,-Ho)-(S
~o~
= aH
(H 0 - S0)
(4)
, - S o)
- AS.
(5)
The absolute sea level change becomes AS = AH
- ASob s
(6)
and combining this with Eq. (2) gives the equation to be used with ellipsoidal heights AS = Ah - AN - ASob,"
(7)
The observed sea level change contains components of crustal deformation, sea surface topography changes and geoid changes. The crustal deformation can be calculated from GPS observations assuming the long-term stability of the reference frame. This requirement is not yet fully achieved in sub-cm level in the current ITRF realisations, or the frames used in GPS satellite orbit computations. The geoid rise values were calculated using the uplift rates from GPS time series and an empirical relation between the land uplift and geoid rise in Fennoscandia. The relation was derived by Ekman and M/ikinen, (1996) using repeated precise levellings, tide gauge records and gravity data. They obtained the geoid rise to be 6% of the land uplift.
Chapter
Vaasa
o
Ob
o
Tide Gauge
Monitoring
Using
GPS
Table 1. The absolute sea level rise rates of the Finnish tide gauges (mm/yr). 'Long time series' is the longest existing time series for each tide gauge and the ' Short ' covers years 1996-2002. 'With uplift' values are corrected for the land uplift and 'With geoid' also for the geoid rise derived from a model.
GPS
o CO
c_
13 •
Long time series
Short time series
Length (yr)
With Uplift
With ge0id
With uplift
With Ge0id
Helsinki
123
22
1.9
16.3
16.0
Hamina
74
14
1.2
15.5
15.3
Turku
80
26
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I
I
I
1997
I
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1999
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2001
2003
Year
Rauma
69
33
2.9
17.0
16.6
Vaasa
119
16
1.1
16.7
16.2
Oulu
112
05
0.1
14.5
14.1
Vaasa tide gauge 1996 - 2002 LO
To obtain the absolute sea level change rates for the Baltic Sea, we calculated A S using Eq. (7). In Figure 3 one can see examples of the GPS and tide gauge time series used in the calculation. The GPS time series are relative to the Mets~hovi GPS station. Land uplift rates agree well with the previous results, e.g. Mfikinen et al., 2003.
--
E O ¢(.O3
il v
o
,.,?-
"1o LO
"T
i
I
1996
1998
I
I
I
2000
I
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2002
Year
Vaasa
tide gauge
1883
- 2001
o o LO OO
E E
o
o_
¢(.(D
° o
-
-1o o
I
Lo
1880
I
I
1920
I
1960
I
I
2000
Year
Fig. 3. Time series for Vaasa GPS station and for Vaasa tide gauge. Note, that the scale of the GPS time series is 10 cm and the scales of the tide gauge time series are 3 m. Spikes in GPS time series are due to the snow on the antenna radome in winter time.
The tide gauge time series are given for two different time periods, the shorter being years 19962002 and the longer one the longest time series existing for each tide gauge. In Vaasa (63 ° 06' N; 21 ° 34' E, Fig. 3) the tide gauge observations have started in year 1883. The trends for the six tide gauges corrected with land uplift from GPS can be seen in Table 1 (Tervo, 2004). To compute the absolute rates in sea level change, one should use as long tide gauge time series as possible. The average of the long time series gives for the sea level rise 1.9 _ 1.0 m m / y e a r (varying b e t w e e n 0.5 - 3.3 mm/year), and when corrected for the geoid rise, 1.6 _ 1.0 m m / y e a r (varying between 0.1 - 2.9 mm/year). Globally the sea level rise is observed to be 1 - 2 m m / y e a r (Church and Gregory, 2001, Church et al., 2004). There are also regional differences in the sea level rise rates, as Church et al. (2004) present in their study. Our results for the Baltic Sea agree with the previous studies but the scatter is too large to make any further conclusions. One can see the change in observed rates in the short time series. It means that the sea level rise in the Baltic Sea is now different than what it has been before the year 1990. Possible explanations are discussed in (Johansson et al., 2003), but the reason for the change is not yet satisfactory explained.
77
78
M. Tervo • M. Poutanen • H. Koivula
3 Stability and accuracy of the GPS solution
M E T S - M A S B (L1) 0.015
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. 35
40
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Figure 4. Results of the GPS test. Two top rows: Vectors METS-MASB computed with two softwares. Solid line is the actual change in height between METS and MASB. Third figure: Observed height change using only the 3 m vector MASA-MASB. Bottom: Observed drift METS-MASA. There has been no actual change in height between these stations. There is more than 5 mm change during the period of one month.
To simulate the accuracy of a footprint GPS installation to detect minor vertical m o v e m e n t s , we made a test where two identical antennas were 3 m from each other, and a reference station was about 10 k m away. In a footprint technique, the stability of a marker is controlled with a network of reference stations 10 - 15 k m from the site. In our test we m o v e d one test antenna in vertical direction, and the second one was u n m o v e d during the test. The GPS data were processed with two different softwares (Pinnacle and Bernese) to find out the difference in results b e t w e e n different softwares, and to confirm that observed features are not artefacts from the data processing. In both cases a standard troposphere and P C V models were used. W e made a total of 18 sessions, 24 h each, in J a n u a r y - February 2004. The reference station is called M E T S , the fixed test antenna is M A S A and the m o v i n g antenna is M A S B . In Fig. 4. we have three different solutions for M A S B , two c o m p u t e d from M E T S (about 10 km vector) with two different softwares, and one using the 3 m baseline M A S A - M A S B . In the figures, the "ground truth", i.e. the k n o w n vertical shift of M A S B , is shown with a solid line. Height of M A S B was changed 5 - 15 m m during the test. Sudden changes of a few m m in height are hardly visible in some cases in our GPS solution when the 10 k m vector M E T S M A S B is used. However, if we process the data of the 3 m vector, 0.2 m m changes m a y b e c o m e visible. It means that even with the footprint technique sudden sub-cm shifts m a y r e m a i n undetectable in episodic measurements. In continuous m e a s u r e m e n t s the change will b e c o m e visible in long time series, but even in that case, periodic changes m a y degrade the resolving p o w e r (Poutanen et al., 2005). Most notable thing is the observed drift in height M E T S - M A S A . Both antennas were fixed, and we have tried to exclude external reasons for the drift. M E T S is an IGS station, and data are used also for our p e r m a n e n t network analysis. The same pattern in M E T S - M A S A is visible in the results of two independent softwares, thus excluding the software based reason. There are several possible cause for the drift, including e n v i r o n m e n t a l effects (like snow) and crustal loading, as discussed in P o u t a n e n et al., 2005. This data set does not allow us to make any detailed conlusions but longer time series are needed to analyse the temporal variation in this vector.
Chapter 13 • Tide Gauge Monitoring Using GPS
The same drift is visible also in M E T S - M A S B vector. If we c o m p u t e separately M E T S - M A S A and M E T S - M A S B and from these the height difference M A S A - M A S B , the a g r e e m e n t with the ground truth b e c o m e s better than directly f r o m the vector M E T S - M A S B .
4 Conclusion T h e r e are m a n y applications of GPS in tide gauge monitoring. A m o n g the most important ones is the possibility to calculate rates of absolute sea level rise. C o m b i n i n g tide g a u g e s and GPS stations world wide w o u l d give a significant contribution to the sea level monitoring, giving it a well-defined reference frame. W e calculated absolute sea level rates for the Baltic Sea using GPS and tide gauge time series. The rate was found to be b e t w e e n 0.1 - 2.9 m m / y e a r , a v e r a g e being 1.6 m m / y e a r . The results agree with the global rate, though the scatter and uncertainty of the trend are large. R e c e n t rates differ significantly, but these can be a d d r e s s e d to local t e m p o r a l variation and they do not represent the l o n g - t e r m trend. This stresses the i m p o r t a n c e of uninterrupted l o n g - t e r m tide g a u g e time series. W e have s h o w n that GPS m a y be used for controlling the stability of the tide with a s u b - m m accuracy w h e n the baseline is very short. D i s a d v a n t a g e of the short baseline is that the reference antenna m a y m o v e together with the tide g a u g e b e n c h m a r k a n t e n n a if there are local deformations. This is a v o i d e d by locating the reference antenna further a w a y from the tide gauge. In this case the G P S - r e l a t e d errors, which can be up to several m m over a b a s e l i n e of 10 k m limit the accuracy obtained. One should find a suitable c o m b i n a t i o n b e t w e e n the desired accuracy and the distance b e t w e e n the reference antenna and tide gauge.
Acknowledgements This study was partially s u p p o r t e d by Finnish F u n d i n g A g e n c y for T e c h n o l o g y and I n n o v a t i o n T E K E S (decision n u m b e r 40414/04).
References Church, J.A. and J.M. Gregory (Co-ordinating lead authors), 2001, Changes in sea level, In: Climate Change 2001, The Scientific Basis, Contribution of working group I to the third assessment report of the IPCC, Cambridge University Press, p. 639 - 694 Church, J. A., N. J. White, R. Coleman, K. Lambeck, and J. X. Mitrovica, 2004: Estimates of the regional distribution of sea-level rise over the 1950 to 2000 period. Journal of Climate, 17(13), 2609-2625 Ekman M. and J. M~ikinen, 1996, Mean sea suface topography in the Baltic Sea and its transition area to the North Sea: A geodetic solution and comparisons with oceanographic models, Journal of Geophysical Research, 101:11993 - 11999 Holgate S.J. and P.L. Woodworth, 2004, Evidence for enhanced coastal sea level rise during the 1990s, Geophysical Research Letters, 31, L07305 Johansson, J.M., J.L. Davis, H.-G. Scherneck, G.A. Milne, M. Vermeer, J.X. Mitrovica, R.A. Bennett, B. Jonsson, G. Elgered, P. Elosegui, H. Koivula, M. Poutanen, B.O. R6nn~ing and I.I. Shapiro, 2002, Continuous GPS measurements of postglacial adjustment in Fennoscandia, 1. Geodetic results, Journal of Geophysical Research, 107:2157 Johansson, M., K. Kahma and H. Boman, 2003, An improved estimate for the long-term mean sea level on the Finnish coast, Geophysica, 39:51-73 Mao, A., C.G.A. Harrison and T.H. Dixon, 1999, Noise in GPS coordinate time series, Journal of Geophysical Research, 104; 2797 - 2816 M~ikinen, J., H. Koivula, M.Poutanen and V. Saaranen, 2003, Vertical velocities in Finland from permanent GPS networks and repeated precise levelling, Journal of Geodynamics, 38:443-456 Poutanen, M., J. Jokela, M. Ollikainen, H. Koivula, M. Bilker, H. Virtanen, 2005, Scale variation in GPS time series. In: A Window on the Future Geodesy (Ed. F. Sans6). lAG Symposia vol. 128, Springer Verlag, Berlin, pp. 15-20. Poutanen, M., H. Koivula, M. Tervo, K. Kahma, M. Ollikainen and H. Virtanen, 2004, GPS time series and sea level, In: Celebrating a decade of the international GPS s e r v i c e - Proceedings, Meindl, M. (ed.) Astronomical Institute, University of Berne, CD Tervo, M., 2004, Benefits of combining tide gauges and GPS stations, Master's thesis, University of Helsinki, 45 p. White, N.J., J.A. Church and J.M. Gregory, 2005, Coastal and global averaged sea level rise for 1950 to 2000, Geophysical Research Letters, 32:L01601 Williams, S.D.P., Y. Bock, P. Fang P. Jamason, R. M. Nikolaidis, L. Prawirodirdjo, M. Miller and D. J. Johnson 2004, Error analysis of continuous GPS position time series, Journal of Geophysical Research, 109:B03412.
79
Chapter 14
Determination of Inland Lake Level and Variations in China from Satellite Altimetry*
Its
Yonghai Chu, J. Li, Weiping Jiang, Xiancai Zou, Xinyu Xu, Chunbo Fan School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan, 430079, China Email: yh_chu@ 163.com
Abstract. With the rapid development and the widely application of satellite altimetry, it provides an efficient tool and a well-validated technique for real-time and continual monitoring of the inland lake level. However, only few lakes in China are covered with historical altimeters' GDRs. In this study, we investigate the level fluctuations of Poyang and Dongting (Yangtze River area), Qinghai (west China) and Hulun (northeast china) from altimetry and analyze the connections between level changes and the local climate.
Keywords: Satellite Altimetry, Inland Lake, Lake level Variation in China
basin waters from T/P satellite altimetry. Mercier (2002) studied the connections between 12 lakes of Africa and the climate of Indian Ocean using T/P data (1993-1999). Jekeli & Dumrongchai (2003) monitored the vertical datum with satellite altimetry and water-level gauge data on large lakes. Zhang (2002) has studied the Great Lakes level changes with the T/P data (1992-2002). Hwang (2005) used T/P to investigate and analyze the links between lake level variations and ENSO over six lakes in China. In this paper, four lakes level changes are presented using 2 yr. of Envisatl GDRs and 13 yr. of time series of Hulun are shown based on T/P and Jason 1 data.
2 The Chinese Inland Lakes 1 Introduction Altimeters were primarily designed to study the ocean, their ability to investigate marine geoid, gravity anomaly; mean sea surface height (Jiang, 2001) has already been applied since 1970's.This technology is limited inland for the inherent quality of altimeter. However, it has been demonstrated that altimetry has an advantage over other tools in the research of inland waters. Birkett (1994, 1995) evaluated the potentialities of this technology over continental waters and monitored several climatically sensitive lakes level with Geosat and analysed the total error budget of T/P for lake height and obtained 24 lakes level changes including Hulun Lake of China with T/P altimeter data. Morris & Gill (1994) evaluated the T/P altimeter over the Great Lakes based on the previous 40 cycle's data. Ponchaut & Cazenave (1998) investigated the relationships between level fluctuations and climatic environment about four lakes of Africa and three lakes of North America. Birkett (1999) studied the influence of the Indian Ocean climate on the lakes of east Africa. Oliveira (2001) presented the temporal variations of Amazon
There are many lakes in China, but the distributions are very wide and uneven. Moreover, the construction is also varying. Poyang, Dongting, Tai, Hongze and Chao are the five largest fresh lakes which are located at southeast of China, a subsiding region of crust, and regulate the water quantity of Yangtze River. Contrarily, western china, for instance Tibet, is an intense upthrust region. The altitude of lakes there is very high and the type of the lakes is usually closed. The lake supplies depend on the dissolution of ice or snow and rainfall precipitation, such as Qinghai and Bositeng Lake. We selected four lakes as mentioned earlier to study (Fig.l, Tab. 1). T/P and Jasonl data (13 years) are used in Hulun Lake, and Envisat 1 data (2 years) are used in others.
3 Data Analysis We did not employ the same methods of data processing as used in open oceans when using altimetric data to monitor the level changes of inland lakes for satellite altimetry was developed and optimized for open sea.
* This study is funded by the Natural Science Foundation Committee of China under grants 40274004 and 40304001
Chapter
14 • Determination of
Inland Lake Level and Its Variations in China from Satellite Altimetry
Table. 1 Features of 4 studied lakes in China (Hulun: T/P and Jasonl a=6378136.3m, other: Envisatl a=6378137m) Borders Latitude Longitude 28.75-29.75 115.75-116.75 28.50-29.50 112.00-113.00 36.50-37.50 99.50-101.00 48.50-49.50 116.90-117.90
Lake Poyang Dongting Qinghai Hulun
Province
Mean Surface areanr~'2~t,,~..J ellipsoid height (m)
Points Pass
Type
Jiangxi
3960
8.18
144
163,621,980
Open
Hunan
2740
10.76
44
694,879
Open
Qinghai
4635
3149.30
254
094,479,552
Open
Inner Mongolia
2315
533.94
1816
27,36(TP-Jasonl)
Open
3.2 Data Editing and Processing '
'
37.5"
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'
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.
,
.
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,
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To retain the maximum data, the following basic requirements must be considered: 1) data in lake; 2) altitude above reference ellipsoid is valid; 3) valid range. If data quantities are very rich, 4) number of 1Hz ranges points; 5) altimeter echo type or radiometer surface type; 6) correction items valid. This paper only considers items 1)-3).Certainly, abnormal values may exist in the data, artificial elimination is also a very important step to get the final LLH with respect to the reference ellipsoid: LLH=
23.0" ~
28.5" llZ.O"
43.0
,
,
11:~.5"
48.5 113.0"
11 0" 11k5" '
Fig 1 Geographical locations of 4 studied lakes and data distribution (Hulun: T/P and Jasonl, other: Envisat1)
3.1 Distance Corrections
The GDRs (AVISO, ENVISAT) provide many corrections in open sea. Generally, some corrections are not suitable for studying in inland waters, such as troposphere and ionosphere corrections are usually unavailable in the GDRs. Inverted barometer correction is often ignored for the small lake area. The instrumental bias has been applied in 1Hz range data. In this paper, ionosphere, troposphere (dry and wet), solid earth tide, ocean tide, pole tide, sea state biases are used. Among these corrections, the dry troposphere is from model, but the wet troposphere is from radiometer correction. The wet model is used if it is available when the radiometer correction is invalid. The dualfrequency altimeter correction is replaced by DORIS. But it is also used if it is available when the DORIS is invalid.
Ral t
-
-
R - AR
(1)
Where L L H is lake level height, Ra~, is altimeter altitude with respect to the reference ellipsoid, R is the range from altimeter to lake's surface and AR is the range corrections mentioned above. The instantaneous LLH is split into geoid height above the reference ellipsoid and lake surface height above the geoid (Ponchaut, 1998). We average all LLH represent the mean lake level (MLLH). To obtain the monthly average LLH, we also average the LLH month to month and mark symbol as L L H M . Finally, the monthly level variations d L L H are expressed: dLLH
= LLH M - MLLH
(2)
With month number as x-axis and d L L H as y-axis, the graphs of variations plot are shown in Fig2-3.
4 Analysis of Lake Level Variation and Climate 4.1 Poyang Lake Poyang Lake is the largest fresh-water lake that can regulate or control Yangtze River. Its level is affected by five rivers and Yangtze River. The major flood period of five rivers is from Apr to Jun, so the level is influenced by the water discharge of these five rivers, nevertheless, the level is still lower. Although the water quantities of the five rivers
81
82
Y. Chu. J. Li. W. Jiang • X. Zou. X. Xu. C. Fan
Poyang, Dongting and qinghai lake
> -2.5 Time (month) &
Poyang
--
Dongting - - O -
Qinghai ]
Fig. 2 Monthly Mean Level Variation of Poyang, Dongting and Qinghai Lake (2002.11-2005.2)
Hulun Lake
• _°o: v--4
Month Fig. 3 Monthly Mean Level Variation of Hulun Lake (1992.10-2004.11), blue triangle: Jasonl data
decrease from Jul to Sep which is the major flood period of Yangtze River, the level is still steadily rising and keeps high and comes to a head in Jul for the reverse flow of the water. Fig.2 (triangle symbol) shows the seasonal changes of the level.
4.2 Dongting Lake Dongting Lake is located at the middle reaches of Yangtze River. There are four rivers that flow into Dongting from the south and four outfalls which link to Yangtze River from the north. The lake water flows into Yangtze River at Yueyang. Dongting seems to be a natural big reservoir which contains water from the four rivers and gorges or disgorges Yangtze River and regulates flood. So the level feature is affected not only by rainfall, surface runoff and lake conservation, but also by the situations of Yangtze River. The advent of flood season of the four rivers starts from Apr and cause the level to rise (Fig.2 square symbol)). During the raininess season in Yangtze River catchment's area from Jun to Sep, The level of Dongting rises continuously. Generally speaking, the highest water level appears in Jul. When the water intake is less than the water output after Sep, the level declines slowly and arrives at the bottom because it is lowwater season from Sep of previous year to Mar of the ensuing year.
4.3 Qinghai Lake Qinghai Lake, the largest inland lake and the largest saltwater lake in China, belongs to a high and cold, semiarid climate. Although there are more than 40 rivers that flow into Qinghai Lake, the water of its is supplied by surface runoff and precipitations. From Fig.2 (circular symbol), we can say that the level keeps balance. Even so, the level displays a seasonal feature during 2002-2004. In spite of its coldness with its surface freezing in winter, the level rises rapidly from Nov and keeps high level to next Apr for rainfall increasing and evaporation decreasing in spring. However, from Apr to Nov, the amplitude of variation is about 0.5m, less than 0.9m in winter. On the whole, the amount of water in Qinghai keeps balance during 2002-2004. We investigated the environment around Qinghai and found that the policy of reusing farmland for planting grass since 2001 contributes to the rising level due to the conservation of soil and water.
4.4 Hulun Lake Hulun Lake, the fifth largest one in China, located at Inner Mongolia region where the continental climate is very obvious for Great Xingan Mountains
Chapter 14 • Determination of Inland Lake Level and Its Variations in China from Satellite Altimetry
shielding off moist air from ocean in east and the Mongolia Plateau stands at the west. Fig 3 is a 13year time series of level changes from 1992 to 2004 based on T/P and Jason l. From 1992 to 1999, the interannual fluctuations almost kept balance, but it decreased linearly from 2000 to 2004. Flake ice occurs in the last ten-day period of Oct and the total lake surface freezes up at the beginning of Nov, so the level is low in Jan to Mar, and high levels appear from Jul to Sep. In recent years, no regulation and control measures are put into practice, the environment and water supplies around Hulun and its catchment's basin are being destroyed by human activities and natural factors. All these aggravate the decline of level from 2000 to 2004 about 2.5m.
5 Conclusions Water level fluctuations of inland lakes are related to regional to global scale climatic changes. Water level fluctuations reflect variations in evaporation and precipitation over lake area and its catchments area. To study the correlations between level variation and climate, the aridity index, water balance equation or the fundamental equation of lake water balance can be used. The meteorological parameters in these methods include lake area, precipitation, evaporation rates, river runoff, discharge rates, groundwater inflow or outflow. If satellite altimetry data are combined with local hydrologic data, the comparison of lake level derived from altimetry and from the in-situ lake gauge records could be a valid verification method (Hwang, 2005), but unfortunately, no available gauge data can be obtained, the comparison can not be made in the paper. Even so, the results from this study also validate the widely use of satellite altimetry in inland lakes or rivers. If possible, Geosat, ERS1/2, T/P, Jasonl and Envisatl data could be used together to get multi-altimetry and long time series numerical results. Waveform retracking technology also could help improve the accuracy of altimeter range over inland lakes (Chu, 2004, 2005).
References AVISO and PODAAC User Handbook (2003). IGDR and GDR Jason Products SMM-MU-M5-OP-13184-CN (AVISO), Edition 2.0.
AVISO/Altimetry (1996). AVISO user hand book for merged Topex/Poseidon products. AVI-NT-02-101, Edition 3.0. Birkett, C.M., 1 9 9 4 , Radar altimetry: a new conception monitoring lake level changes, EOS, Trans., AGU 75 ( 2 4 ) , 273-275 Birkett, C.M (1995). The Contribution of Topex/Poseidon to the Global Monitoring of Climatically Sensitive lakes, J. Geophys.Res. 100 (C12), 25179- 25204. Birkett, C.M (1998), Contribution of TOPEX NASA radar altimeter to global monitoring of large rivers and wetlands. Water Resour. Res, 34(5), 1223-1239 Birkett, C.M., Murtugudde, R., Allan, T(1999), Indian Ocean climate event brings floods to east Africa's lake and the Sudd Marsh. Geophys. Res. Lett. 26, 1031-1034 Cheinway Hwang, Ming-fong Peng, Jingsheng Ning, Jia Luo, Chung-Hsiung Sui (2005), Lake Level Variation in China from TOPEX/POSEIDON Altimetry: Data Quality Assessment and Links to Precipitation and ENSO, Geophys.J. int, 161, p 1- 11. Chu Yonghai, Li Jiancheng, Zhang Yan, Xu Xinyu, Fan Chunbo, Zou Xiancai (2005), Analysis and Investigation of Waveform Retracking about ENVISAT[J],Journal of Geodesy and Geodynamics, Vol. 25(1). Chu Yonghai, Li Jiancheng, Jiang Weiping Zhang Yan (2005). Monitoring the Hulun Lake Level and Its Fluctuation with Jason-1 Altimetric Data, Journal of Geodesy and Geodynamics, Vol. 25(4), p 11-16. Chu Yonghai(2004). The Theory and Technology of Waveform about Satellite Altimetry, Wuhan University, master dissertation. ENVISAT RA2/MWR Product Handbook (2002), European Space Agency, issue 1.0 Jekeli C. Dumrongchai (2003). On Monitoring a Vertical Datum with Satellite Altimetry and Water-level Gauge Data on Large Lakes, Journal of Geodesy, vol.77, p447453 Mercier, F., Cazenave, A., Maheu, C., (2002). Interannual Lake Level Fluctuations (1993-1999) in Africa from Topex/Poseidon: Connections with Ocean-atmosphere Interactions over the Indian Ocean, Global and Planetary Change, and vol.32:141-163 Morris, CS., Gill, SK. (1994). Evaluation of the TOPEX/POSEIDON Altimeter System over the Great Lakes. J Geophys Res 99(C12):24527-24539 Oliveria Campos, Mercier, F., Maheu, C., Cochnneau, G., Kosuth, P., Blitzkow, D., Cazenave, A., (2001). Temporal Variations of River Basin Waters from Topex-Poseidon Satellite Altimetry. Application to the Amazon basin [J], Earth and Planetary Sciences, v333:633-643 Ponchaut F., Cazenave A (1998). Continental Lake Level Variations from Topex/Poseidon (1993-1996), Comptes Rendus De IAcademie des Sciences Series IIA Earth and Planetary Science, vol.326, 13-20 Weiping Jiang (2001). The Application of Satellite Altimetry in Geodesy, Wuhan University, doctoral dissertation Zhang Ke,(2002). Monitoring Continental Lake Level Variations by using Satellite Altimeter Data, Wuhan University, master dissertation.
83
Chapter 15
Sea Surface Variability of Upwelling Area Northwest of Luzon, Philippines M.C.Martin, C.L. Villanoy Marine Science Institute University of the Philippines, 1101 Diliman, Quezon City, Philippines
Abstract. The upwelling events during winter at the northwest tip of Luzon are among the sea surface features occurring in the South China Sea. The variability of sea surface in the upwelling area was determined using 9-year satellite altimeter data from TOPEX/ Poseidon and ERS 1 & 2. Using Empirical Orthogonal Function (EOF) decomposition the variability of the area was resolved. Two modes were derived representing the sea level anomaly variations of the upwelling area that peak during the monsoon periods. The sea level anomaly associated with the upwelling event coincided with the positive wind stress curl, which has been suggested to be due to the topographic steering of the wind that can be initiated by the northeast wind at the tip of Luzon. ENSO events also appear to modulate the timing of the development and decay of upwelling in the northwest of Luzon. E1 Nifio events initiate early development while La Nifia episodes delay the occurrence of upwelling events in the area. The intensity or strength of E1 Nifio and La Nifia episodes seems to influence the extent/magnitude of the upwelling area. Keywords. Sea level anomaly (SLA), upwelling, sea surface variability, ENSO, South China Sea
1 Introduction Upwelling, the movement of nutrient-rich deep water to the surface, is considered one of the most important processes in the coastal ocean (Tomczak, 1996) because of the role it plays in biological productivity and, consequently, to fisheries. The upwelling area northwest of Luzon, as part of the South China Sea (SCS) (Figure 1), has likewise been subjected to the seasonal variability occurring in the region. The SCS as part of the East monsoon system (Wyrtki, 1961) is dominated by monsoon winds. The northeast monsoon winds prevail from November to March and southwest monsoon winds blow from May to September (Pickard and Emery, 1982).
t on v
l, 100
105
110
115
120
125
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Fig 1. Map of the Philippines. A box highlights the upwelling area northwest of Luzon. The occurrence of upwelling area off Luzon in winter was first documented by Shaw et al (1996). Evidence of this feature was based on in-situ data of temperature, salinity and dissolved oxygen concentration they had gathered during a cruise in 1990. Udarbe-Walker and Villanoy (2001) analyzed the three-dimensional thermal structure and described the extent and timing of upwelling using historical temperature data. In this study, the spatial and temporal variations of upwelling in the area northwest of Luzon were determined using the 9year data set from merged T/P and ERS 1-2 satellite altimeter data. The likely evolution of upwelling as modulated by interannual variations are also presented.
2 Methods The sea level anomaly fields between 11 ° to 22 ° N and 112 ° to 122°E (Figure 1) were extracted from the global sea surface topography maps of the merged TOPEX/POSEIDON (T/P) and ERS-1/2 altimetry data. These 0.25 ° x 0.25 ° gridded SLA maps produced by the CLS Space Oceanography
Chapter 15
Division as part of the Environment and Climate EU AGORA (ENV4-CT9560113) and DUACS (ENV44-T96-0357), were obtained from the ftp://ftp.cls.fr/pub/oceano/AVISO/MSLA site. The 10-day sea level anomaly (SLA) data from October 1992 to September 2001 were grouped on a yearly basis. Time series data with missing values were linearly interpolated and each SLA was Ztransformed to ensure equal weights for each data point. The nine-year sea level anomaly data were then subjected to Empirical Orthogonal Function (EOF) analysis (Emery and Thomson, 1997) to describe the temporal and spatial variability of the upwelling area northwest of Luzon. Further, QUICKSCAT wind data (1999-2002) were examined to determine the possible influence of winds stress curl on upwelling.
•
Sea Surface Variability of Upwelling Area Northwest of Luzon, Philippines MODE A
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Decomposition of sea surface height variability in the upwelling area northwest of Luzon derived two modes that together accounted for 33% of the total variance. Mode A, which explained 2 1 % variance, represents seasonal/annual variation. The spatial and temporal distributions for this mode are shown in Figures 2 and 3. The spatial distribution shows an elongated band of low sea level in the northeastsouthwest orientation located northwest of Luzon that encloses three eddy-like structures within it (Figure 2). The two eddy-like features situated west of Luzon (between 15°-18.5 ° N, 116.5°-119°E) and at the centre of SCS (14.5 °- 16.5°N, 113 °- 115.5°E) are consistent with the two of the three dynamically active areas described in Ho et al (2000a). This likewise conforms to the cyclonic circulation measured using drifters during the experiment conducted by National Taiwan University in winter of 1993-1994 (Ho et al., 2000b). As positive amplitudes indicate low sea level, the positive peaks that were generally observed during December of 1993-1996 and 1998-1999, thus, demonstrated that lowest sea level occurred during this month. The occurrences of highest sea levels, on the other hand, were distributed as the negative peaks in June (1994, 1996, and 2000), July (1993, 1995 and 1999) and May (1997, 1998 and 2001). Periods that corresponded to Mode A positive amplitudes included the months from November to February while the negative amplitude months were from May to September.
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Mode B, representing 12% of the total variance likewise characterized seasonal variation but lagged in phase compared to Mode A (Figure 3). Lag correlation analysis revealed that this lag period was equivalent to 3 months (r2=0.6). Mode B spatial pattern is shown as a core of low sea level centered at 18°N, 119°E (Figure 2). This observation agrees with the upwelling area previously described by Udarbe-Walker and Villanoy (2001). Similarly with Mode A, the core of low sea level occurred during the northeast monsoon while the opposite was observed during the southwest monsoon. The positive peaks (low sea surface levels) generally occurred in March while the negative peaks (high sea surface levels) were frequently observed in September. The monthly sea surface distribution that likely exists in the upwelling area is shown in Figure 4. The core of low sea level was noticeable during the
85
86
M.C. Martin. C. L. Villanoy
northeast monsoon season. With manifestation in November, a peak in January and then lasted until March. High sea level initially appeared in April and then remained until October, which is the transition from the SW to NE monsoon. The piling up of seawater in the upwelling area is expected as a result of the wind-driven southwesterly monsoon. Studies on the seasonal variability of sea surface heights in the SCS derived from TOPEX/POSEIDON altimeter data have provided similar observations (Ho et al. (2000b). Jan 2 / D . / / ~ ~ . . ~ ~ . ~ ' . . ~ ) ) "', / i
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From the monthly wind stress curl averages, an area of positive wind stress curl coincided with the core of low sea level during northeast monsoon months (Figure 5). Conversely, negative wind stress curl coincided with the core of high sea levels during the southwest monsoon period. Positive wind stress curl appears to be formed as the winds turn the comer at the northwest tip of Luzon from the Luzon Strait. This is similar to patterns in wind stress curls observed at the leeward (southwest) part of Taiwan.
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in mm, 0 mm are shown as bold lines. Although it has earlier been suggested that this upwelling is a remotely forced process driven by basin circulation in the SCS and not by local winds (Shaw et al., 1996), the role of positive wind stress curl (Chu et al., 1998; Shaw et al., 1999) in generating upwelling off Luzon may be another potential mechanism (Udarbe-Walker and Villanoy,
2001).
Longitude (°E)
10 .6 N/m 3 Figure 5. Representative of monthly QUIKSCAT wind stress curl average from 1999 to 2002 (source: PO.DAAC, http://podaac.jpl/nasa,gov.
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The effects of the ENSO on sea level distribution were also evident from the time series plots of sea level data (from the center of upwelling area) with the SOI index (Figure 6)(Australia Bureau of Meteorology, 2005), where E1 Nifio and La Nifia events are denoted by negative and positive SOI indices, respectively. Sea level in the center of the upwelling area during years with E1 Nifio events (1992-93, 1994-95 and 1997-98) were lower than the sea levels with attendant La Nifia (1998-1999 and 1999-2000) events. The sea level anomaly distribution of warm-water events and normal year exhibited similar spatial (depicted in Figure 4) and temporal sea level distribution, except that the extent of upwelling was wider and deeper during E1 Nifio years. However, it seems that the initiation and extent of upwelling were also influenced by the intensity of preceding events. For example, upwelling appeared a month earlier i.e., from November to October and persists until April, during periods with moderate (1994-95) and weak (1992-93) E1 Nifio, which were all preceded by strong E1Nifio and non-ENSO events, respectively.
Chapter15 • SeaSurfaceVariabilityof Upwelling AreaNorthwestof Luzon, Philippines -
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In contrast, the occurrence of smaller and shallower upwelling was observed during period with La Nifia events (Figure 6), where the occurrence of a shallower sea level core was observed 1-2 months delayed from its actual development. E1 Nifio events enhanced summer wind-forced upwelling in the southern shelves of Australia (Middleton et al., 2005) and that sea levels in Thevenard and Portland were lowest during summer of 1999 E1 Nifio and highest during the La Nifia winters of 1997 and from 1999 to 2000. The determination of ENSO influence on wind and currents that are associated with the lowering of the sea surface near the coast are of also important considerations.
4 S u m m a r y and C o n c l u s i o n s EOF analysis of merged T/P and ERS 1-2 altimeter data from October 1992 to September 2001 revealed two dynamic features that contributed to the sea surface variability in the upwelling area off Luzon, Philippines. Mode A showed elongated low sea level feature in the northeast-southwest orientation and enclosed three eddylike structures that centered at 15.5°N, 114.5°E; 16°N, 117.5°E and 18.5°N, 119.5°E. Mode B was dominated by a core of low sea level that centered at 19°N, 119°E. Both Modes A and B were found to exhibit seasonal reversals. Lower sea surface was observed during the northeast monsoon and reverses to high sea level during the southwest monsoon, which indicates the influence of the monsoon season on the evolution of upwelling in the northwest of Luzon. This study demonstrated the good agreement of satellite altimetry with observations of mesoscale features using in-situ data. The examination of QUIKSCAT winds revealed an area of positive wind stress curl west of Luzon, which may account for the observed low sea levels and an
upward Ekman pumping during the northeast monsoon months. Sea level distribution likewise showed interannual influences i.e., higher sea level in the upwelling area was noted during La Nifia events and lower sea level was observed during the E1 Nifio period. E1 Nifio events appeared to initiate the early development while La Nifia episodes delayed the occurrence of upwelling
5 Acknowledgment The authors would like to acknowledge the PCAMRD-DOST PACSEA Program for research support, PO.DAAC for the QUICKSCAT data, AVISO for MSLA data, and the International Association for Physical Sciences of the Ocean (IAPSO) and the Office of the Provincial Council of Nueva Ecija for M. Martin's attendance to the Dynamic Planet 2005 Conference.
6 References Australian Bureau of Meteorology, 2005. http://www.bom.gov.au/climate/current/soi2.shtml Chu PC, Chen Y, Lu S (1998) Wind-driven South China Sea deep basin warm-core/cool-core eddies. J Oceanogr 54:347-360. Emery WJ, Thomson RE (1997) Data analysis methods in physical oceanography. Pergamon Press, Great Britain, pp 395 Ho C-R, Kuo N-J, Zheng Q, Soong YS (2000a) Dynamically active areas in the South China Sea detected from TOPEX/ POSEIDON satellite altimeter data. Remote Sens Environ 71:320-321 Ho C-R, Zheng Q, Soong YS, Kuo N-J, Hu J-H (2000b) Seasonal variability of sea surface height in the South China Sea observed with TOPEX/Poseidon altimeter data. J Geophys Res 105, C6:13,981-13,990 Middleton JF, Arthur C, McClathie S, Ward TM, Maltrud M, McClean J, Gill P, Middleton S (2005) ENSO effects and upwelling along Australia's southern shelves: a report. Applied Mathematics Report AM05/01 School of Mathematics, UNSW February 2005 from www.maths.unsw.edu.au/~jffm/ENSO/enso.pd Pickard GL, Emery WJ (1982) Descriptive physical oceanography. 4th ed, Pergamon Press, Oxford, pp 249 Shaw P-T, Chao S-Y, Liu K-K, Pai S-C, Liu CT (1996) Winter upwelling off Luzon in the northeastern South China Sea. J ofGeophys Res 101:16435-16448 Tomczak M (1996) Upwelling dynamics in deep and shallow water from http://www.es.flinders.edu.au/~mattom/Shelf Coast/notes/chapter06.html Udarbe-Walker MJ, Villanoy CL (2001) Structure of potential upwelling in the Philippines. Deep-Sea Res I 48:1499-1518
87
Chapter 16
An Experiment of Precise Gravity Measurements on Ice Sheet, Antarctica Y. Fukuda, Graduate school of Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan. Y. Hiraoka, Geographical Survey Institute, l, Kitasato, 305-081 l, Japan K. Doi, National Institute of Polar Research, Kaga 1-chome, Itabashi-ku, Tokyo 173-8515, Japan
Abstract. Aiming at the future applications of the precise gravity measurements for glaciological studies and the CAL/VAL (Calibration/Validation) purpose of satellite gravity mission data, we conducted several experiments of precise gravity measurements on ice sheet near Syowa Station, Antarctica, during the 45th Japanese Antarctic Research Expedition (JARE-45). We established a test site where a set of 5 by 5 gravity points with a 10-m interval and the other set of 4 by 4 points between the 5 by 5 points are installed, and conducted gravity measurements as well as RTK (Real Time Kinematic) GPS surveys on the gravity points. Since the positions of the gravity points are moving due to the ice sheet flow with the speed of about 5 m/yr at the site, we simulated the gravity measurements on the moving ice sheet by evaluating how well the gravity values of the second set (4 by 4 points) could be estimated from the gravity values of the first sets (5 by 5 points). The result suggests that the measurements with an appropriate data processing enable 10 gGal accuracy even on the moving ice sheet. Keywords. Temporal gravity changes, Ice sheet, Antarctica, Precise gravity measurements, Absolute gravity measurement, Kinematic GPS
1
Introduction
Satellite gravity missions in the 21st Century, especially GRACE (Gravity Recovery And Climate Experiment) mission, are beneficial to multidisciplinary scientific objectives in terms of time varying gravity fields (Tapley et al., 2004). Of particular importance in the polar region is the studies of ice sheet mass
balance (Velicogna, et al., 2005) and/or postglacial rebound related to sea level changes and global water circulation (Fukuda et al., 2002). The data sets will impose a strong constraint on these issues in terms of total mass conservation (Wahr et al., 2000). The Antarctic region is also expected to become a promising CAL/VAL (Calibration/Validation) field (Shum et al, 2001). In general, CAL/VAL of gravity satellite data by means of ground based observations is not an easy task, primarily because of mismatching of accuracy, spatial and temporal resolutions between satellite and in-situ gravity data. In Antarctica, expected gravity signals mainly due to the ice sheet movements are relatively large in amplitude and comparatively simple for modeling. In addition, the detection of mass changes due to regional to local scale ice sheet movements is a challenging study for both satellite gravimetry and in-situ precise gravity measurements. The ice sheet thinning rate of the Shirase Glacier drainage basin in the JARE research area is estimated about 10-20 cm/year (Mac and Naruse, 1978; National Institute of Polar Research, 1989). Although the corresponding gravity signal is relatively large, the detection of the signal still requires very careful processing of gravity measurements. For the precise gravity measurements on ice sheet, Fukuda et al. (2003) proposed a practical configuration which consists of one absolute gravity point and several surrounding relative gravity points. In general, operations on ice sheet were expected to be fraught with several difficulties. Therefore we conducted practical experiments to make clear the logistic problems and to verify the feasibility of the precise gravity measurements on ice sheet during the JARE-45, in 2004-2005.
Chapter 16 • An Experimentof Precise Gravity Measurements on Ice Sheet, Antarctica
In this paper, we first review the basic concept and the configurations of the precise gravity measurements on ice sheet, and then refer to the experiments and their results conduced during the JARE45 period.
2 Basic configurations of the precise gravity measurements on ice sheet Fukuda et al. (2003) proposed the basic idea of the precise gravity measurements on ice sheet. There are several points in conducting the precise gravity measurements. To simplify the problem, we first assume that the ice sheet does not flow (i.e. no horizontal movement) as shown in Fig. 1a. In this case, by repeat gravity measurements, we observe, the gravity changes due to; 1) snow accumulation/ablation, 2) compaction of the ice sheet, 3) uplift/subsidence, and 4) indirect loading effect. Among these, indirect loading effect is negligible in the spatial scale concerned. 1) cause both mass changes and height changes while 2) and 3) cause height changes only. Absolute height changes of the gravity point can be obtained by precise GPS measurements. Thus we can detect the gravity changes due to snow/ice mass changes. Moreover, accurate tie to the geodetic observations at Syowa Station may reveal a major part of the uplift/subsidence due to a postglacial rebound; fresh snow accumulation can be measured by the conventional snowstakes method separately; thus we can obtain compaction rate of the ice sheet or the useful information of the density variations, consequently. Practically, unlike the case of Fig. 1a, the measurement point itself is moving due to the ice sheet flow as illustrated in Fig. lb. Therefore, GPS meas-
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gravimeter
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V
urements are inevitable not only for monitoring the vertical movements but also for determining the precise horizontal position. If we try to conduct repeat gravity measurements at the same geographical position regardless of the ice sheet movement, we may need real time positioning to come back to the same point. However, it is not an easy task practically, nor gravity measurements at single point may not properly represent the gravity changes around the point. A practical method is to measure gravity values as well as GPS positions at several points in a survey area, and calculate the gravity value at a virtual reference point which retains same geographical position regardless of the ice sheet movement. The procedure to calculate the gravity value at the virtual reference point is a kind of spatial interpolation with the consideration of the effects of the local gravity anomalies around the point. Several methods can be employed for the calculation beside the least squares prediction as proposed by Fukuda et al. (2003). A key point is that the measurement points well cover the virtual reference point in consideration of the speed of the ice sheet flow at the site. Another important point related with the gravity measurements on ice sheet is that at least one absolute gravity measurement is indispensable to achieve 10 btGal accuracy of the gravity measurement, because it is very difficult to conduct gravity tie to a gravity base station so frequently and quickly in Antarctica. Therefore a field type absolute gravimeter such as A10 meter of Micro-G solution should be employed to ensure the accuracy of the absolute gravity value. Considering these issues, Fukuda et al. (2003) propose a recommended gravity survey site, which
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Fig. 2. A schematic illustration of a proposed gravity measurement site. The site consists of an absolute gravity measurement point (AG), a GPS point (GPS) and several relative gravity points (+:RG) which are employed to determine the gravity anomalies around the absolute point.
consists of one absolute gravity point and several surrounding relative gravity points, as configured in Fig. 2. In each site, absolute gravity measurements and precise GPS measurements should be conducted at the absolute gravity point; meanwhile, precise relative gravity measurements can be carried out to determine the local gravity anomalies for calculating the value at the virtual reference point, by mean of spring type gravimeters with a kinematic GPS positioning system. GRACE possibly allows us to detect a 10 cm ice sheet mass change in several hundreds km spatial wavelengths, which is an expected case in the Shirase Glacier drainage basin. Assuming an infinite Bouguer plate, the 10 cm ice sheet mass change will cause gravity changes of about 4 gGal after the effects of height changes are corrected. This accuracy is still hard to be attained at single observation site with an interval of one year. Nevertheless, if we can install 20 to 30 survey sites of this kind over several hundred km span with intervals of 10-50 km in the Shirase Glacier drainage basin, repeated measurements with a time span of 2-3 years or longer will reveal the gravity changes related to the drainage basin scale ice mass changes. Although we considered the measurements were probably feasible as a JARE operation in a summer season, we have no experience of this kind of operation. Therefore we carried out several basic experiences as described in the next section.
3 Experiments conducted during JARE-45 3.1
E x p e r i m e n t s in 2004 s u m m e r season
The main target of the experiments is to make clear the logistic problems related to the gravity and GPS measurements and to verify the accuracies of the measurements on ice sheet. As part of the JARE activities, various kinds of observations have been conducted at and around Syowa Station (69.0°S, 39.6°E) which is located on East Ongul Island (Shibuya et al., 2003). The experiments were conducted on ice sheet near Syowa Station. There is a traverse route to Mizuho Station (the Mizuho route), and the experiments were conducted at the S17 point near the coastal line. Figure 3 shows the location map. Figure 4 shows the setting of the gravity points. We arranged two sets of gravity points near the S 17 point; A-set consists 5 by 5 gravity points (A 11 - A55) and B-set 4 by 4 gravity points (B 11 - B44). Intervals of the gravity points are 10 meters and B-set is
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conducted in November 2004. At that time, all the wooden plates were buried under fresh snowfall, thus we only carried out the GPS measurements at the bamboo poles. In February 2005, we had primarily planned to conduct the same experiments as in the 2004 austral summer season again. However we could not do it due to bad weather condition. Because of limited operation time during busy summer season in Antarctica, we finally gave up to conduct the experiments. 4
Results
of the
experiments
Fig. 4. Distribution of the gravity points. 4.1 Results of the gravity measurements
about 5 meters shifted from A-set so as to be included in A-set. Since we did not possess the portable type absolute gravimeter at present, we only measured the gravity differences from the S 17 point for all the gravity points of both A-set and B-set by means of two LaCoste and Romberg gravimeters (G type). The precise positions of the gravity points were determined as follows; we first determined the position of the S 17 point by means of the static positioning with the IGS point of Syowa Station as the reference, and then positions of all the gravity points were determined by mean of RTK measurements with the S 17 position as the reference. We also conducted the rapid-static measurements at selected points to evaluate the accuracies of the RTK measurements. The ice sheet movement near S 17 was expected to be N-W direction with a speed of about 5 m/yr. Therefore we assumed that the positions of B-set represented the A-set positions of one year later, and compared the gravity values of B-set with those estimated from the values of A-set. This test revealed the accuracy of the relative gravity measurements to estimate the gravity value of a virtual reference point. For the gravity point marks, we used wooden plates with a stake which was buried in the snow surface. All the gravity and GPS measurements were conduced directly on the plates. We also stuck up bamboo poles at the comer and the center points of A 11, A 15, A51, A55 and A33 for future marks. The experiments were conducted during January 28 to 30, 2004, and whole the operation took about 1.5 days by two persons.
We first calculated relative gravity differences from the S 17 point at all the gravity points of both A-set and B-set. Because the gravity values differ mainly due to the height differences at the points, we corrected the height effects using the free-air gradient of 0.3086 mGal/meter before comparing the gravity values. After the height corrections, we estimated the gravity values at the points of B-set from surrounding 4 gravity points of A-set, and then the gravity differences between measured values and the estimated ones at the points of B-set were calculated. Figure 5 shows the gravity differences. Fig. 5 clearly shows two jumps of the values at B 11-B 12 and B31-B32. These were steps in the gravimeter readings probably due to the thermal shock to the gravimeter. While more careful measurements should be required to avoid the steps, it is also true that the steps could be estimated if we had conducted some additional measurements. Except the points with large differences by assuming the steps could be corrected,
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A15 Although the main experiments were carried out in January 2004, some additional experiments were
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91
92
Y.Fukada. Y. Hiraoka • K. Doi
the average value of B 12 to B31 points was 0.13 +10.7 pGal. This suggests that the gravity measurements combined with GPS positioning have a potential to reveal the 10 pGal gravity changes even on the moving ice sheet. 4.2
Results of the G P S m e a s u r e m e n t s
As the experiments of the GPS measurements, we conducted 1) the comparison between the rapid static measurements and RTK measurements, and 2) a test of repeatability of the RTK measurements at a point. The results are summarized in Table 1 and 2. Table 1 shows that the RTK measurements are almost same as the rapid static measurements in accuracy. Table 2 also shows that 1cm repeatability is achieved in the RTK measurements. In general, the accuracy of the static GPS measurements at the reference site is considered to be better than 1-cm level. Therefore 1 cm accuracy at the gravity points can be achieved by combining the RTK measurements and the static GPS measurements at the reference site. From the logistic point of views, we also confirmed that there was no problem to conduct the GPS measurements. Table 1. Comparison between RTK and rapid static GPS measurements. AX(m)
AY(m)
AS(m)
A 11
-0.0040
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0.0046
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-0.0083
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0.0065
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-0.0058
0.0144
Table 2. Repeatability of the RTK measurements.
As previously described, we conducted the GPS m e a s u r e m e n t s in N o v e m b e r 2004 at the bamboo poles. Because the positions of the bamboo poles were not exactly same as the wooden plate marks, we could not observe the exact movements of the ice sheet. Nevertheless we observed approximate movement of 1.7 m northward and 3.4 m westward. This movement is consistent with the ice sheet flow at the site. We also observed the height changes of about 40 cm, which indicates the fresh snow coverage. 5
Concluding
remarks
For the studies of the postglacial rebound, present day ice melting and other objectives, not a few absolute gravity measurements have already been carried out at several stations in Antarctica (Amalvict et al., 2005; Fukuda et al., 2005). However the precise gravity measurement on ice sheet is still very challenging. Appropriate combinations of absolute gravity measurements, relative gravity measurements and GPS positioning have the essential importance for the purpose. Although we did not possess a field type absolute gravimeter, we conducted field experiments on ice sheet to verify the concept of the precise gravity measurements. The results are encouraging. We found no problem in GPS measurements and proved that the accuracy of 10 pGal could be achieved by combining relative and absolute gravity measurements. Through the field experiments, we learned several logistic points which should be modified. But we think all these are trivial and the precise gravity measurements on ice sheet are feasible practically. There is no doubt that the satellite gravity missions will bring about a revolution in glaciological studies in Antarctica. However it should be noted that the gravity mission data are essentially potential field data and their interpretation and/or analysis require various in-situ observations. From this point of view as well, in-situ observations become very important to utilize the satellite data. Precise gravity measurements on the ice sheet should bring further possibilities for glaciological studies, especially when combined with satellite gravity mission data.
AX(m)
AY(m)
AS(m)
Ah(m)
0.0012
0.0012
0.0017
0.0036
-0.0037
0.0016
0.0040
0.0134
-0.0024
0.0013
0.0027
0.0216
0.0035
-0.0001
0.0035
0.0029
0.0045
-0.0007
0.0046
-0.0083
0.0012
0.0009
0.0015
0.0027
0.0045
-0.0039
0.0060
-0.0106
Acknowledgments
0.0036
-0.0049
0.0061
-0.0027
-0.0006
-0.0017
0.0018
0.0014
-0.0041
0.0012
0.0043
0.0087
We thank Yukinori Satou and Kazuo Ito for their kind support during the field operations. We also thank all the JARE-45 members led by Prof. H. Kanda for their kind help.
Chapter 16 • An Experiment of Precise Gravity Measurements on Ice Sheet, Antarctica
References Amalvict, M., M~kinen, J., Shibuya, K. and Fukuda Y. (2005): Absolute Gravimetry in Antarctica: Status and Prospects, submitted to J. Geodynamics. Fukuda, Y., Aoki, S. and Doi, K. (2002): Impact of satellite gravity missions on Glaciology and Antarctic Earth sciences, Polar Meteorol. Glaciol., 16, 32-41. Fukuda, Y., Shibuya, K., Doi, K. and Aoki, S. (2003): A Challenge to the Detection of Regional to Local Scale Ice Sheet Movements in Antarctica by the Combination of In-situ Gravity Measurements and Gravity Satellite Data, 3rd Meeting of the International Gravity and Geoid Commission (IGGC), Tziavos(ed.), GG2002, 243-248. Fukuda,Y., Higashi, T., Takemoto, S., Iwano, S., Doi, K., Shibuya, K., Hiraoka, Y., Kimura, I., McQeen, H., Govind, R. (2004) Absolute Gravity Measurements in Australia and Syowa Station, Antarctica, IAG, International Symposium, Gravity, Geoid and Space Mission GGSM2004,C.Jekili, L.Bastos, J.Fernandes (Eds.), IAG Symposia, 129, Springer, 280-285. Mae, S. and Naruse, R. (1978): Possible causes of ice sheet thinning in the Mizuho Plateau, Nature, 273, 291-292.
National Institute of Polar Research (1989): Antarctica: East Queen Maud Land Enderby Land Glaciological Folio, 8 sheets and explanatory text, National Institute of Polar Research, Tokyo, Japan (ISBN 4-906651-00-3 Folio). Shibuya, K., Doi, K. and Aoki, S. (2003): Ten years' progress of Syowa Station, Antarctica, as a global geodesy network site, Polar Geoscience, 16, 29-52. Shum, C.K., Jekeli, C., Keynon, S. and Roman, D. (2001): Validation of GOCE and GOCE/GRACE Data Products, Prelaunch Support and Science Studies, presented at the International GOCE User Workshop, April 23-24th, 2001, ESTEC, Noordwijk, NL, 1-5. Tapley, B., Bettadpur, S., Ries, J., Thompson, P., Watkins, M. (2004): GRACE Measurements of Mass Variability in the Earth System, Science, 305, 23,503-505. Velicogna, I., Wahr, J., Hanna, E and Huybrechts, P (2005): Short term mass variability in Greenland, from GRACE. Geophysical Research Letters, 32, L05501. Wahr, J. D., Wingham, D. and Bentley, C. (2000): A method of combining ICESat and GRACE satellite data to constrain Antarctic mass balance, J. Geophys. Res., 105, 16279-16294.
93
Chapter 17
Status of DORIS Stations in Antarctica for Precise Geodesy M. Amalvict, Institut de Physique du Globe de Strasbourg / l~cole et Observatoire des Sciences de la Terre, 5 rue Ren6 Descartes, 67000, Strasbourg, France, [email protected] National Institute of Polar Research, 1-9-10 Kaga, Itabashi-ku, 173 8515, Tokyo, Japan
R Willis, Institut G6ographique National, Direction Technique, 2, avenue Pasteur, BP 68, 94160 Saint-Mand6, France Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena CA 91109, USA K. Shibuya, National Institute of Polar Research, 1-9-10 Kaga, Itabashi-ku, 173 8515, Tokyo, Japan
Abstract. Polar regions and especially Antarctica are nowadays recognised as exerting a major control upon the global Mean Sea Level (MSL) directly linked to climate changes. Monitoring and understanding the geodynamical behaviour of these regions is then of critical importance. The long-term displacement (or velocity) of reference sites helps constraining the ice sheet evolution prediction models. Several geodetic space techniques, such as GPS, observe displacements of such reference sites. In Antarctica, in addition to numerous GPS stations, four DORIS stations are permanently operating: Belgrano, Rothera, Syowa, Terre Ad61ie. In addition to the permanent DORIS stations, episodic DORIS campaigns took also place at Dome C / Concordia and on Sorsdal and Lambert glaciers. In this paper, we first present general information concerning the stations and the campaigns (exact location, period of measurements, etc). We then discuss the solutions obtained by different analysis centres (when available) for all DORIS stations in the Antarctic region. In particular, we use several ITRFs (from the early ITRF96 to ITRF2000) to see their impact on the derived velocities in Antarctica. An emphasis is given to the investigation and possible explanation of differences observed between each solution. Finally, we compare at these stations, the results of DORIS observations to the solutions from other geodetic techniques (GPS, VLBI) and to the results of repeated absolute gravity measurements (when available). Keywords. Absolute Gravity, Antarctica, DORIS, GPS, Syowa, Terre Ad61ie, VLBI.
1
Introduction
Antarctica is a vast area of about 14 million km 2 of which 98% is covered by an ice cap. The ice volume changes over time; recent studies (Weller, 1998) show that observations in the polar regions are critical to validate climate models which by their very nature incorporate large uncertainties. The present-day thawing of ice is one cause of vertical displacement at the surface of the ice (elastic rebound); horizontal displacements also occur since the thawing is more important at the border of the continent, leading to a flux of ice from inland towards the coast. Because of thawing due to major warming of Antarctica over the past fifty years (Turner et al., 2005), there is an uplift of Mean Sea Level (MSL) due to increasing volume of water (Cazenave and Nerem, 2004). In fact, things are complicated because of the movement of the underlying bedrock. The most important causes of vertical movement of the Antarctic continent is Post Glacial rebound (PGR) and the effect of the present Glacial Isostatic Adjustment (GIA). The PGR is the viscoelastic response to the deglaciation, which occurred 11,000 years ago by the end of the last ice age (Peltier, 1996). The GIA is the elastic response of the surface to the present deglaciation. The vertical displacement of the surface can be monitored either by direct or by indirect ways. The direct way is the straightforward observation of the surface; this is the geometrical observation of the surface through classical geodetic techniques such as levelling or station position using satellite techniques. The indirect way refers to gravity measurements which are dependent on both the
Chapter 17 • Status of DORISStations in Antarctica for Precise Geodesy
redistribution of masses and on the distance of the station from the Earth's centre. When no change in mass redistribution is assumed, any variation in the gravity will lead to a vertical displacement of the site. 2
Geodetic
observations
in Antarctica
2.1 DORIS observations in Antarctica DORIS (Doppler Orbitography by Radiopositioning Integrated on Satellite) provides the position of beacons located all over the world, through the observation of the Doppler shift of a signal received by a constellation of up to 6 satellites orbiting around the Earth (Tavernier et al., 2005). These include: the altimetric satellites TOPEX/POSEIDON (only until November 2004), Jason-1 and ENVISAT and the remote sensing satellites SPOT-2, SPOT-4, SPOT-5 and SPOT-3 (only until November 1996). About 55 ground stations are geographically quite well distributed on the Earth's surface. There are however only four permanent DORIS stations in Antarctica (Figure 1): Terre Ad61ie installed since February 1987, Rothera since November 1991, Syowa since February 1993 and Belgrano that has only been in operation from March 2004 to May 2004 and starting again in summer 2005. In addition to the permanent stations, episodic DORIS campaigns took place at Dome C / Concordia (Vincent et al., 2000) and Sorsdal and Lambert Glaciers (Govind and Valette, 2004).
The DORIS data from the permanent stations are available at the NASA/CDDIS data centre through the International DORIS Service (IDS), see Tavernier et al., 2005. Table 1 shows the data availability of DORIS observations at NASA/CDDIS Data Centre after January 1993, as well as the total number of corresponding IGN/JPL weekly solutions available. As indicated in Table 1, the DORIS system provides long-term continuous observations in Antarctica. As it is an up-link system, no data are recorded on site and the ground beacons do not rely on any human intervention or data communication, as it is the case for GPS. In the case of an hostile environment, such as Antarctica, this is a clear logistical advantage. More specifically, Figure 2 shows the time series of results available at the CDDIS data centre for the IGN/JPL weekly solutions, showing some temporary interruptions for a few stations. However, in general, continuous observations are available during several years for the same beacon. Table 1. Summary of DORIS weekly solutions available from the IGN/JPL Analysis Centre at the NASA/CDDIS data centre (July 2005). Site Terre Adelie Rothera Syowa
0
~
Belgrano
Name ADEA ADEB ROTA ROTB SYOB SYPB BELB
Start Jan 93 Mar 02 Jan 93 Mar 05 Apt93 Apt 99 Mar 04
End Feb 02 Jun 05 Feb 05 Jun 05 Apr 98 Jun 05 May 04
Wks 467 104 574 12 261 319 7
ADEA
o
ADEB ........................................................................................
ROTA
:
N
.
:
:.
....
B
ROTB
¢Xl
SYOB SYPB
D
BELB 1993 180"
Figure 1. DORIS Antarctic stations. Full circles are for permanent stations, open diamonds stand for episodic campaigns.
1995
1997
1999
2001
2003
2005
Figure 2. Availability, at the NASA CDDIS data centre, of data from permanent DORIS Antarctic stations. These data are used for the IGN/JPL weekly solution IGNWD04 (as July 1, 2005)
95
96
M. Amalvict
• P. W i l l i s .
K. S h i b u y a
Several other IDS Analysis Centres (ACs) process these DORIS data and, for each, several types of solutions are available. These include: cumulative solutions - positions and velocities at a reference epoch, derived from the complete data set of DORIS observations (perAC) and available in SINEX format. These solutions can be directly used for geodesy and geophysics (Soudarin et al., 1999; Cr6taux et al., 1998, Willis et al., 2005). These types of DORIS results are available at the cddis ftp-website #1 (see references): time series of station coordinates - weekly or monthly station coordinates in SINEX format (per AC). These solutions can be obtained either in free-network or a loosely constrained solution (to be used in future geodetic combinations) or directly projected and transformed into ITRF2000 (Sillard and Boucher, 2001). These solutions correspond to a more recent way to realize the Terrestrial Reference Frame (TRF) through time series of geodetic results instead DORIS
"
weekly
solutions
- IGN/JPL
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.......
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~. . . . . .
slope =
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Center
DORIS
ign03wd01 R O T A WRMS =
of using simplified linear model of positions and velocities (Altamimi et al., 2005). These types of DORIS results are available at the cddis ftpwebsite #2 (see references) station coordinate differences - station coordinates expressed in ITRF2000 for weekly or monthly solutions in STCD format (STation Coordinate Difference), (Tavernier et al., 2005). This format proposes a more user-friendly presentation (tabulated in XYZ and in North/East/ Vertical results) of time series results for a potentially broader community of users. These DORIS results are available at the cddis ftpwebsite #3 (see references)
b , .....
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I~nO3~lDl.amcL.r'~m.gff
z(x]:~,_bn 3 11 :l':J:4-T " I~Cer'n~'l.llee.rma.811
Figure 3. Doris time series, at Rothera, from IDS Web site. a is the weekly solution and b the monthly solution. Note that for the vertical component, + is up and- is down
Chapter 17 • Status of DORISStations in Antarcticafor PreciseGeodesy
Figure 3 shows an example of plots of time series available at IDS website. It can be noticed (in spite of the poor quality of available pictures) that the precision is improving in the last years. Results from the LEGOS/CLS groups (Figure 3b) are much smoother as they are based on monthly solutions, while results from the IGN/JPL group (Figure 3a) are based on weekly solutions, hence providing more data (this could be an advantage for outlier detection). Both DORIS solutions are in good agreement. These results are regularly updated on Internet: every 3-6 month for LEGOS/CLS and every week for IGN/JPL. It must also be noted that the precision of these DORIS geodetic results strongly depends on the number of available DORIS satellites (Willis et al., 2005), varying between 2 to 5 from 1993 to 2005. Due to ENVISAT and SPOT5, the results have a better precision after March 2002, as they are based on data from 4 or 5 satellites instead of 2 in the early 1993 solutions. We also need to point out here that unfortunately the DORIS/Jason data cannot be used for geodesy due to an extreme sensitivity to radiation affecting the satellite oscillator over the South Atlantic Anomaly (Willis et al., 2004). Finally, satellites with quasi-polar orbits, such as SPOTs and ENVISAT, provide more DORIS data in the polar region and are critical for positioning accuracy.
2.2 Other positioning techniques 2.2.1 GPS The International GPS Service (IGS), see Beutler et al., 1999, includes 7 permanent stations in Antarctica, namely Syowa since 1999, Mawson since 1999, Davis since 1995, Casey since 1995, McMurdo since 1995, O'Higgins since 1995 and Sanae IV since 1999. Out of these, only Syowa is equipped with a DORIS beacon. There are numerous studies based on GPS observations to analyse the horizontal velocities and deformations in Antarctica (James and Ivins, 1998; Larson and Freymueller, 1995; Bouin and Vigny, 2000). Some recent studies include Donelan and Luyendyk (2004) who observe a 12 +4 mm/yr vertical velocity at Matye Byrd Land, Dietrich et al. (2004) who restrain their analysis to the Antarctic Peninsula with 9.5 +5-6 mm/yr at O'Higgins and 2.4 +5-6 mm/yr at Palmer. In addition to the IGS stations, several permanent GPS receivers are stationed in Antarctica. This is the case at Terre Ad61ie since 1998 and at Rothera since 1998. Bouin and Vigny (2000) published the
results of the Terre Ad61ie station (which belongs to a specific geophysical network of the ENS (l~cole Normale Sup6rieure)), for the 1995-1998 period of observation. In addition to these permanent stations, episodic campaigns are also organised, mostly during austral summer. Dietrich et al. (2001) derived horizontal motions from GPS data of 1992 to 1998 campaigns organised by the Scientific Committee on Antarctic Research (SCAR). In addition, specific experiments are also organised during the austral summer season (Tregoning et al., 2000).
2.2.2 VLBI There are 2 VLBI stations in Antarctica: Syowa (11-m antenna) and O'Higgins (9 m antenna) on the Antarctic Peninsula; Syowa is therefore the only station utilising the three co-located techniques. Fukuzaki et al. (2005) analysed VLBI sessions connecting Syowa to 3 other stations in Antarctica, Australia and Africa. VLIBI velocity results were derived for the Syowa station and compared to earlier GPS and DORIS determinations. 2.3 Physical geodesy in Antarctica Opposite to data from space geodesy techniques, absolute gravity measurements are only given episodically. They typically last a few days, averaging results of thousands of drops of a free-falling body in the vacuum chamber of the gravimeter. The measurements should be ideally repeated every couple of years, which is quite a challenge in remote polar regions. The gravity changes (if any) are then interpreted as changes in mass redistribution and/or changes in height of the station itself. The 23 Absolute Gravimetry (AG) measurements in Antarctica, obtained at 12 different stations are listed and analysed by Amalvict et al. (2005). Among them, the only 2 DORIS stations are Terre Ad61ie and Syowa. Syowa is the only DORIS station with repeated AG measurements: 7 measurements between 1992 and 2004. 3
Models
and
predictions
Changes in MSL and ice mass in Antarctica are key indicators of what is referred to 'Global Change'. Numerous predictive models attempt to depict the station displacements and velocities; they are depending on physical parameters, inferred from observations: viscosity, timing of glaciation and deglaciation. The PGR, resulting from the deglaciation started more than 10 000 years ago, involves a long period of
97
98
M. Amalvict • P.Willis. K. Shibuya
time. The rheology of the Earth is then visco-elastic. On the contrary, the GIA, which is the response of the Earth surface to the present-day deglaciation does not involve time, due to its elastic behaviour. As observations result of all these phenomena, the comparison of observations and models can help separating the effect of each phenomenon. Among the models, we can mention Peltier (1996), James and Ivins (1998), Nakada et al. (2000). These models, relying on different glaciation and deglaciation models predict vertical displacements (and sometimes gravity changes). An overview of this problem in Antarctica, is given in Makinen et al. (2005). The value of the ratio dg/dh derived from observations of both changes in gravity g and changes in height h can help in constraining the physical parameters of rebound models (Wahr et al., 1995). 4
Results
and
discussion
We focus on vertical velocities, which is indeed the component involved in PGR and GIA previously described. This is also the component directly derived from the gravity variations. 4.1 DORIS analysis
We have analyzed here the available DORIS station velocities derived by different groups (Table 2). First we have considered the latest ITRF solutions: ITRF96 (Sillard et al., 1998), ITRF97 (Boucher et al., 1998) and ITRF2000 (Altamimi et al., 2002). They all provide positions and velocities based on a global adjustment of several individual DORIS solutions as well as other individual solutions from other techniques (VLBI, SLR and GPS) and geodetic local Table 2. DORIS results analyzed to derive vertical velocities for DORIS stations in Antarctica (June 2005). X = position, V = velocity, week = weekly solution, month = monthly solution.
•
•
Solution
Source
Type
ITRF96 ITRF97 ITRF2000 IGN03D02 IGN04D02 IGN05D02P STCD IGN STCD LCA
IERS IERS IERS IGN/JPL IGN/JPL IGN/JPL IGN/JPL LEGOS/CLS
X/V X/V X/V X/V X/V X/V week month
_
.
_
.
Last data Jul 97 Dec 97 Mar 99 Dec 03 Sep 04 Jun 05 Jun 05 Jan 05
ties properly weighted. Continuous improvement in the adjustment method has been done between these three realizations as well as in the pre-processing (data screening). In the future, a new ITRF2005 should be available soon, combining time series of results instead of cumulative solutions (positions and velocities at a reference epoch) as proposed by Altamimi et al., 2005. We have also considered three recent DORIS solutions from the IGN/JPL DORIS Analysis Center: IGN03D02 (Willis et al., 2005), IGN04D02 (Willis and Heflin, 2004) and a preliminary solution for IGN05D02P, based on the same DORIS analysis strategy but using more DORIS data (Table 1) as well as a refined pre-processing (data screening and also identification of station coordinates discontinuities as described in Willis and Ries, 2005). These three solutions are also cumulative solutions (positions/velocities) but they only take into account DORIS results from the IGN/JPL Analysis Centre as well as DORISDORIS geodetic local ties, with proper constrain, when a new DORIS beacon is installed in close collocation with an older DORIS instrument. Results from other techniques or from other groups are not used in this study. As a test, we have also analyzed two new types of solutions, provided as time series of results expressed in ITRF2000 in the STCD format. We have considered here the latest LEGOS/CLS (LCA) monthly solutions and the IGN/JPL (IGN) weekly solutions as they are regularly updated by these groups and available at NASA/CDDIS through the IDS (Tavernier et al., 2005). These results are available per station (there are potentially several DORIS beacons at the same site, corresponding to successive instrument occupations) and for each solution we derived a weighted slope using the available data on July 2005. Table 3. DORIS vertical velocities estimated for Terre Adelie Source
Station Code
ITRF96 ITRF97 ITRF2000 IGN03D02 IGN04D02 IGN05D02P STCD IGN
ADEA ADEA ADEA ADEA/ADEB ADEA/ADEB ADEA/ADEB ADEA ADEB ADEA ADEB
STCD LCA
V (mm/yr) -5.90 0.93 -0.86 - 1.10 0.47 0.63 -0.01 8.04 1.37 4.43
Sig V (mm/yr) 6.56 4.05 1.39 0.16 0.16 0.15 0.19 0.88 0.16 1.31
Chapter 17 • Status of DORIS Stations in Antarctica for Precise Geodesy
In Table 3, the constant decrease of the formal errors between ITRF96, ITRF97 and ITRF2000 highlights the continuous improvement made by the IERS in the global combination. A similar trend can also be seen in parallel in the three consecutive IGN/JPL solutions. However, the improvement is much smaller because it only corresponds to an increase in the considered DORIS observation data span (the DORIS analysis strategy was exactly the same). The IGN/JPL also provides much smaller formal errors than ITRF solutions. This can come from a different re-weighting of the solutions and also from current systematic errors in the different IERS techniques. Finally, the STCD approach provides less precise results because the DORIS-DORIS local tie information was not used, so all the DORIS results and information (local ties) were not used when more than one DORIS beacon exists at the DORIS site. In Table 4 (Rothera) and Table 5 (Syowa) we observe the same decrease of formal errors than in Table 3. In the case of Rothera, the new beacon ROTB is too recent to provide any valuable information on the
Table 4. DORIS vertical velocities estimated for Rothera Source
Station Code
ITRF96 ITRF97 ITRF2000 IGN03D02 IGN04D02 IGN05D02P
ROTA ROTA ROTA ROTA ROTA ROTA/ROTB ROTA(*) ROTA ROTB ROTA ROTB
STCD IGN STCD LCA
V (mm/yr) -5.40 3.54 1.27 1.95 3.71 1.53 4.78 3.93 -92.07 5.25 -
Sig V (mm/yr) 8.32 5.29 1.95 0.14 0.13 0.56 0.20 0.13 34.25 0.12 -
velocity using the STCD solution. However, it is also possible to use the ROTA and ROTB STCD solutions, as well as the DORIS-DORIS local tie information to provide a longer time series better suited for velocity determination. In this case, the local time precision is assumed to be at 1 mm and does not degrade the precision of this technique. In the case of the Syowa stations, all DORIS estimations show a clear positive and small vertical uplift of the station. In the case of the Belgrano station, as the DORIS results are only based on 7 weeks of data and are currently rather useless for any geodetic or geophysical investigation we present only the preliminary analyses we processed (Table 6). The station does not even appear in the earlier DORIS TRFs. However, more recently, after an interruption of more than a year, a new station (BEMB) has been installed and should provide soon some regular data to the NASA/CDDIS IDS Data Centre. The velocity calculated by August 6, 2005, is -22.2 + 30 mm/yr but the velocity estimation should then rapidly improve as soon as more data become available (a geodetic local tie is provided by IGN/SIMB between BELB and BEMB with a precision of 2 mm). Figure 4 shows the vertical velocities at the 3 stations, according to different models. Figure 5 shows Table 6. DORIS vertical velocities estimated for Belgrano
.
.
Source
Station Code
IGN04D02 IGN05D02P STCD IGN STCD LCA
BELB BELB BELB BELB
_
_
[] [] []
ITRF2000 IGN04D02 IGN05D02
•
ICE4G
V (mm/yr) -75.78 -22.2 -171.43 206.49
VERTICAL VELOCITY 5~
Table 5. DORIS vertical velocities estimated for Syowa Source
Station Code
ITRF96 ITRF97 ITRF2000 IGN03D02 IGN04D02 IGN05D02P STCD IGN
SYOB SYOB SYOB/SYPB SYOB/SYPB SYOB/SYPB SYOB/SYPB SYOB SYPB SYOB SYPB
STCD LCA
V (mm/yr) 2.14 5.89 2.11 1.81 3.61 3.89 4.04 3.50 0.17 6.40
Sig V (mm/yr) 8.58 5.06 1.89 0.25 0.21 0.19 0.49 0.28 0.44 0.26
Sig V (mm/yr) 17.04 30 83.74 187.37
F
4
3
8LU
...................................... ~.................. ...................
1 .......................................................................
i .....
fIN
-3
ILl :::=.
-1
N [~ Terre
Adelie
Rothera
Syowa
Figure 4. Vertical velocities at the 3 Antarctica stations taken for the different solution sets and rebound model.
99
100
M. Amalvict • P. Willis. K. Shibuya
4.4 AG analysis
A ITRF2000 (~) IGN04D02 i IGN05D02 GSRM1.2
HORIZONTAL VELOCITIES Rothera
~o
4.5 5 ...................................... [...................................................................................................................................................................................................... co
v UJ
r7
~=
..................
. i
Syowa
..,-,.
rr
~i
......................................................................................................................................................................................
We follow the analysis of Fukuda et al. (2004) at Syowa station, leading to a change of gravity equal to - 0 . 3 +0.4 ~tGal/yr.
Comparison of results from different techniques
m /,\
0
Let us summarise (Table 7) the tendencies observed at our 3 stations, according to the different techniques. Terre Adelie The value of the GPS vertical velocity at the -10 ........................................................................................................................................................................................................................... Rothera station comes from itrf-ensg website, is rather m~ different from DORIS but the standard deviation -15 -10 -5 0 5 10 15 20 makes it compatible with the DORIS results. It is highly L O N G I T U D E (MM/YR) possible that only a few GPS data were used to derive this GPS velocity (short time series). Future solutions Figure 5. Horizontal velocities in Antarctica (models and should confirm or invalidate our DORIS results. prediction) I-l--
-5
...........................................................................................................................................................................................................................
.d
/,\
the same result for horizontal velocities. We notice again than ITRF2000 leads to slightly different results. The IGN/JPL cumulative solutions based on the same DORIS processing strategy provides closer results. All results differ from ICE-4G models at a few mm/yr, especially in Syowa. Figure 5 shows the agreement between the estimated DORIS horizontal velocities and a plate motion model GSRM 1.2 (Kreemer et al., 2003). All results are in good agreement within a couple of mm/yr. We can notice that the precision is as high for the vertical velocity as for the horizontal one.
Table 7. Comparison of vertical velocities at different DORIS stations. Syowa: GPS and VLBI results from Fukuzaki et al., 2005; DORIS results, IGN04D02 solution. Terre Ad61ie: GPS results from TIGA and MN Bouin, 2005; DORIS results, IGN04D02 solution.
DORIS mm/yr GPS mm/yr VLBI mm/yr
4.2 GPS analysis Fukuzaki et al. (2005) obtain 2.26 +0.33 mm/yr at Syowa for the vertical displacement. The daily JPL GPS results, available on-line provide an estimate with opposite sign 2.53 mm/yr +0.27. These GPS/JPL results are based on a Precise Point Positioning technique (Zumberge et al., 1997). Terre Ad61ie is not an IGS station; the value of the vertical velocity is 0.43 + 1.5 mm/yr according to the TIGA analysis and - 0 . 5 6 +2 according to M.N. Bouin's analysis (personal communication).
4.3 VLBI analysis Fukuzaki et al. (2005) report on VLBI measurements at Syowa from 1999 to the end of 2003. They obtain 4.6 +2.2 mm/yr for the vertical component of velocity.
AG gGal/yr Prediction
Syowa 3.6+0.2
Terre Addlie 0.63 + 0.15
Rothera 1.53 + 0.56
1993/2004 2.3 + 0.3
1993/2004 0.43 + 1.5 -0.56 + 2
1993/2004 -8.5 + 2.0
1999/2003 4.6+ 2.2
N/A
N/A
1999/2003 -0.3 + 0.43
N/A
N/A
1995/2004 - 0
- 0
Table 8. Comparison of horizontal velocities at Syowa station; GPS and VLBI results from Fukuzaki et al., 2005; DORIS results, IGN04D02 solution. Syowa DORIS mm/yr GPS mm/yr VLBI mm/yr
East -6.5 + 0.8
North 3.2 + 0.9
1993/2004 -4.4 + 0.2
1993/2 004 -0.2 + 0.2
1999/2003 -2.5 + 0.6
1999/2003 4.0+0.7
1999/2 003
1999/2 003
Chapter 17 • Status of DORISStations in Antarctica for Precise Geodesy
The agreement between different positioning techniques is fairly good at Syowa; the results are consistent with both AG trend and modelled predictions. At Terre Ad61ie, the 2 GPS solutions can be seen as a nonsignificant displacement, which is in agreement with the modelled prediction. DORIS solution for Terre Adelie is consistent with a very small displacement. Horizontal velocities at Syowa are shown in Table 8. In our opinion, the difference in periods of observations could, partly, explain the differences in results. 5
Conclusions
The present analysis of DORIS data at Antarctic stations shows the sensitivity of the solution to the models (Earth model, data sampling...). Comparison of different positioning techniques at the same station shows a fairly good agreement for vertical velocities (typically 1-2 mm/yr). There is only one station (Syowa) with repeated Absolute Gravimetry measurements, showing good agreement. DORIS proves to be a useful geodetic tool, especially as it provides long-term and continuous measurements in this hostile environment. It is also quite encouraging to see that the most recent results (since 2002) provide far better geodetic precision. It is then important that the number of polar satellites carrying a DORIS receiver remains the same, or even increases, to obtain good geodetic results in the future. In conclusion, we should say that the number of DORIS beacons should be increased in Antarctica. There is now such a possibility as the new satellite receiver allow measurements from shifted frequency transmission from ground station, decreasing possible interferences on-board the satellite and allowing a larger number of DORIS beacons in the same region of the world. We do hope that the coming IPY (International Polar Year) will provide a boost in that direction. Acknowledgments. This study was carried out during the stay of MA at the National Institute for Polar Research (NIPR), Tokyo, Japan under a fellowship from the Japanese Society for the Promotion of Science (JSPS). Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (NASA). Authors thank MN Bouin who provided her GPS results for Terre Ad61ie and K. Doi who drew the map of Antarctic DORIS stations. Authors are thankful to two anonymous reviewers for their helpful comments and constructive suggestions.
References
Altamimi Z., P. Sillard, C. Boucher (2002). ITRF2000, A new release of the International Terrestrial Reference Frame for earth science applications, J Geophys Res, Solid Earth, Vol. 107(B 10), pp. 2214. Altamimi Z., C. Boucher, P. Willis (2005). Terrestrial Reference Frame requirements within GGOS, Journal ofGeodynamics. Vol. 40(4-5), pp. 363-374 Amalvict M., M~ikinen J., Shibuya K. and Fukuda Y., 2005, Absolute Gravimetry in Antarctica: Status and Prospects, Journal of Geodynamics, submitted Beutler G., M. Rothcaher, S. Schaer, T.A. Springer, J. Kouba, R.E. Neilan (1999), The International GPS Service (IGS), an interdisciplinary service in support of Earth sciences, Advances in Space Research, Vol. 23(4), pp. 631-653. Bouin MN, Vigny C (2000), New constraints on Antarctic plate motion and deformation from GPS data, Journal of Geophysical Research, Solid Earth, Vol. 105(B 12), pp. 28279-28293. Boucher C., Z. Altamimi, and P. Sillard (1998), The 1997 International Terrestrial Reference Frame (ITRF97), IERS Techn. Note 27, Paris Observatory. Bouin M.N., 2005, personal communication Cazenave A., Nerem R.S. (2004). Present-day sea level change, Observations and causes, Reviews of Geophysics, Vol. 42(3), RG3001 Cr6taux JF, Soudarin L, Cazenave A, Bouille F (1998) Present-day Tectonic Plate Motions and Crustal Deformations from the DORIS Space System, JGeophys Res, Solid Earth, Vol. 103(B12), pp. 30167-30181 Dietrich R, Dach R, Engelhardt G, Ihde J, Korth W, Kutterer H-J, Lindner K, Mayer M, Menge F, Miller C, Niemeier W, Perlt J, Pohl M, Salbach H, Schenke H-W, Sch6ne T, Seeber G, Veit A, V61ksen C, 2001, ITRF coordinates and plate velocities from repeated GPS campaign in Antarctica--an analysis based on different individual solutions. Journal of Geodesy, Vol. 74(11-12), pp. 756-766 Dietrich R, ROlkeA, Ihde J, Lindner K, Miller H, Niemeier W, Schenke H-W, Seeber G; 2004, Plate kinematics and deformation status of the Antarctic Peninsula based on GPS, Global Planet Change, 42, 313-321 DonnelanA; and Luyendyk B.R, 2004, GPS evidence for a coherent Antarctic plate and for postglacial rebound in Marie Byrd Land, Global and Palnetary Change, 42, 305-311 Feissel-Vernier M., Valette J.J., Soudarin L., Le Bail K. (2005). Report of the 2003 Analysis campaign "Impact of GRACE gravity field models on IDS products", IDS Report.
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Fukuda Y, Higashi T, Takemoto S, Iwano S, Doi K, Shibuya K, Hiraoka Y, Kimura I, McQueen H, Govind R., 2004, Absolute gravity measurements in Australia and Syowa Station, Antarctica. Gravity, Geoid and Space Missions GGSM 2004. IAG International Symposium. Porto, Portugal. August 30 September 3, 2004Series: IAG Symposia, Vol. 129 Jekeli, Christopher; Bastos, Luisa; Fernandes, Joana (Eds.) ftp://cddis.gsfc.nasa.gov/pub/doris/products/ sinex global #1, ftp://cddis.gsfc.nasa.gov/pub/doris/ products/sinex_series #2, ftp://cddis.gsfc.nasa.gov/ pub/doris/products/stcd #3 Fukuzaki Y., Shibuya K., Doi K., Ozawa T., Nothnagel A., Jike T., Iwano S., Jauncey D. L., Nicolson G. D., McCulloch P. M. (2005). Results of the VLBI experiments conducted with Syowa Station, Antarctica, Journal of Geodesy, Vol. 79(6-7), pp. 379-388. Govind et Valette, 2004, The Sordsal and Lambert campaigns: organisational aspects and first results, IDS 2004 Plenary Meeting, http:// lareg, ensg.ign, fr/ID S/events/prog_2004.html, http ://sideshow.jpl.nasa. gov/mbh/series.html, http://ids.cls.fr, http://itrf.ensg.ign.fr/ITRF solutions/ 2000/results/ITRF2000 SCAR.SSC.txt James T.S., E.R. Ivins (1998). Predictions of crustal motions driven by present-day ice sheet evolution and by isostatic memory of the Last Glacial Maximum, Journal of Geophysical Research, Solid Earth, Vol. 103(B3), pp. 4993-5017. Kreemer C., W.E. Holt, A.J. Haines (2003). An integrated global model of present-day plate motions and plate boundary deformation, Geophysical Journal International, Vol. 154(1), pp. 8-34. Larson K, Freymueller J (1995), Relative motions of the Australia, Pacific and Antarctic plates by the Global Positioning System. Geophysical Research Letters, Vol. 22(1), pp. 37-40. Makinen J, Amalvict M, Shibuya K, Fukuda Y (2006) Absolute Gravimetry in Antarctica: Status and Prospects, Journal of Geodynamics, in press Nakada M, Kimura R, Okuno J, Moriwaki K, Miura H, Maemoku H, 2000, Late Pleistocene and Holocene melting history of the Antarctic ice sheet derived from sea-level variations, Marine Geology, Vol. 167, pp. 85-103. Peltier W.R. (1996). Mantle viscosity and ice-age ice sheet topography, Science, Vol. 273(5280), pp. 1359-1364. Sillard P., Z. Altamimi, and C. Boucher(1998). The ITRF96 realization and its associated velocity field, Geophysical Research Letters, Vol. 25(17), pp. 3223-3226. -
Sillard R, and C. Boucher (2001). A review of algebraic constraints in Terrestrial Reference Frame datum definition, Journal of Geodesy, Vol. 75(2-3), pp. 63-73. Soudarin L., J.F. Cr6taux, A. Cazenave (1999). Vertical Crustal Motions from the DORIS space-geodesy system, Geophysical Research Letters, Vol. 26(9), pp. 1207-1210. Tavernier G., H. Fagard, M. Feissel-Vernier, F. Lemoine, C. Noll, J.C. Ries, L. Soudarin, R Willis (2005). The International DORIS Service, IDS, Advances in Space Research, Vol. 36(3), pp. 333-341. Tregoning, P., A. Welsh, H. McQueen and K. Lambeck, 2000, The search for postglacial rebound near the Lambert Glacier, Antarctica Earth, Planets and Space, Vol. 52(11), pp. 1037-1041 Turner J., Colwell S.R., Marshall G.J., Lachlan-Cope T.A., Carleton A.M., Jones RD., Lagun V., Reid P.A., Iagovkina S. (2005). Antarctic climate change during the last 50 years, International Journal of Climatology, Vol. 25(3), pp. 279-294 Vincent C., J.J. Valette, L. Soudarin, J.F. Cretaux, B. Legresy, F. Remy,A. Capra (2000). DORIS campaigns at Dome Concordia, Antarctica in 1993 and 19992000, in Proc. DORIS Day 2000, CNES, France. Wahr J, Han D, Trupin A, (1995) Predictions of vertical uplift caused by changing polar ice volumes on a viscoelastic Earth. Geophysical Research Letters, Vol. 22(8), pp. 977-980. Weller G. (1998). Regional impacts of climate change in the Arctic and Antarctic, Annals of Glaciology, Vol. 27, pp. 543-552 Willis R, M. Heflin (2004). External validation of the GRACE gravity GGM01C gravity field using GPS and DORIS positioning results. Geophysical Research Letters, Vol. 31 (13), L 13616 Willis P., J.C. Ries (2005), Defining a DORIS core network for Jason-1 precise orbit determination based on ITRF2000, Methods and realization, Journal of Geodesy, Vol. 79(6-7), pp. 370-378. Willis R, B. Haines, J.R Berthias, R Sengenes, J.L. Le Mouel (2004). Behavior of the DORIS/Jason oscillator over the South Atlantic Anomaly, Comptes Rendus Geoscience, Vol. 336(9), pp. 839-846. Willis R, C. Boucher, H. Fagard, Z. Altamimi (2005), Geodetic applications of the DORIS system at the French Institut G6ographique National, Comptes Rendus Geoscience, Vol. 337(7), pp. 653-662. Zumberge J.F., M.B. Heflin, D.C. Jefferson, M.M. Watkins, F.H. Webb (1997). Precise point positioning for the efficient and robust analysis of GPS data from large networks, Journal of Geophysical Research, SolidEarth, Vol. 102(B3),pp. 5005-5017.
Chapter 18
High-Harmonic Gravity Signatures Related to Post-G lacial Rebou nd H.H.A. Schotman 12, EN.A.M. Visser 1, L.L.A. Vermeersen ~ 1 Delft Institute of Earth Observation and Space Systems (DEOS), Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg l, 2629 HS Delft, The Netherlands 2 SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA, Utrecht, The Netherlands
Abstract. The earth's shallow layers, up to a depth of about 200 km, can have viscosities that are an order to several orders of magnitude lower than those of surrounding layers. These layers can induce highharmonic (degree and order 50 - 150) gravity anomalies due to the ice and meltwater redistribution in the last glacial cycle. Uncertainties in ice-load histories will induce gravity and geoid anomaly differences in these high harmonics. The GOCE satellite mission is expected to be able to discern differences between various Late-Pleistocene ice-load histories and is also predicted to be sensitive enough to detect the effects of shallow low-viscosity crustal and asthenosphere zones. For example, our earth relaxation models indicate that GOCE should be sensitive to typical differences between ice-load histories up to harmonic degree 140 for a crustal low-viscosity zone and up to harmonic degree 70 for a low-viscosity zone in the asthenosphere. GRACE is mainly sensitive to differences for the latter. We show that for the limiting case of a lateral homogeneous earth, it is possible to constrain properties of crustal low-viscosity layers in the presence of uncertainties in the ice-load history. Keywords. low-viscosity earth layers, post-glacial rebound, satellite gravity
1 Introduction Late-Pleistocene Ice-Age cycles have left observable markings at the earth's surface. Examples include the ongoing post-glacial rebound (PGR) in Fennoscandia and Canada, and secular variations in the earth's rotation. Furthermore, in the geoid and in gravity anomalies the remaining solid-earth deviations from isostasy following the ice and meltwater redistribution from the last glacial cycle are detectable. Magnitudes and spatial patterns of these geoid and gravity anomalies are dependent on two variables: the ice- and waterload distribution history on one side, and material and rheological variables of the solid earth on the other. The earth's shallow layers, up to a depth of
about 200 km, can have viscosities that are an order to several orders of magnitude lower than those of surrounding layers. Beneath oceanic areas the asthenosphere, i.e. the uppermost layer of the mantle, can have such low-viscosity zones (LVZs, see e.g. Pollitz (2003), Stein and Wysession (2003, p. 170)), whereas in continents LVZs can exist in the lower crust (see e.g. Watts and Burov (2003); Ranalli and Murphy (1987)). The layers can superimpose highharmonic (degree and order 50 - 150) contributions (van der Wal et al., 2004) upon the generally lowharmonic (smaller than 50) geoid anomalies resulting from mantle relaxation. Typical amplitudes are a few decimeters underneath and just outside formerly glaciated areas, with scales down to hundred kilometers. Changes in the properties of these crustal LVZs (CLVZs) and asthenospheric LVZs (ALVZs) and uncertainties in the ice-load histories will induce geoid and gravity anomaly differences in these high harmonics with the same order of magnitude and similar resolution. The GOCE satellite mission, to be launched by ESA in February 2007, is predicted to measure the static gravity field with an accuracy of 1 cm in geoid height and 1 regal in gravity anomaly at 100 km resolution (Visser et al., 2002). Such accuracy and resolution opens the possibility to discern differences between various Late-Pleistocene ice-load histories and constrain properties of CLVZs and ALVZs. Several studies on the effect of a CLVZ on PGR observables have been performed in the past years, see e.g. Klemann and Wolf (1999); Di Donato et al. (2000); Kendall et al. (2003); Vermeersen (2003). In this paper we concentrate on gravity anomalies induced by CLVZs and ALVZs. We investigate the sensitivity of the response to a CLVZ and an ALVZ by using two different iceload histories: a modified version of ICE3G (Tushingham and Peltier, 1991), and a more recent model by Kurt Lambeck (see e.g. Lambeck et al. (1998)). We show using spherical harmonic degree amplitudes, that for a CLVZ the differences are above the expected GOCE error level or performance up to spherical harmonic degree 140 and for an ALVZ up to de-
104
H.H.A. Schotman • P. N. A. M. Visser • L. L. A. Vermeersen
gree 70. This performance estimate is based on formal errors as described in Visser et el. (2002). The differences are also above the realized GRACE performance over a 363-day period (GGM02S, Tapley et el. (2005)); up to degree 90 for a CLVZ and degree 60 for an ALVZ. Note that the GRACE performance will improve if a longer measurement period is used, though this will mainly affect the long-wavelength part. This means that GRACE, but especially GOCE, could provide information on the ice-load history in the presence of an LVZ. If we are however interested in constraining properties of an LVZ, uncertainties in the ice-load history are an error source. We show that even in the presence of uncertainties in the ice-load history, it is possible to extract information on LVZs, using spectral signatures. In future studies, we will concentrate on extracting information on LVZs in the presence of other error sources, as errors in modelling time-variable processes (tides, atmospheric and hydrological mass variations, see Hen et el. (2006)), unmodelled crustal and lithospheric mass inhomogeneities due to compositional, thermal and thickness variations, and (shallow) mantle heterogeneities due to for example subduction (Mikhailov et el., 2004) and mantle plumes.
2 Theory
can then be computed by using Brun'sformula (Heiskanen and Moritz, 1967, p. 85), which in our notation reads:
Nnm = R . Cnm
The gravity anomaly is defined as the difference between the gravity acceleration (i.e. the negative of the potential gradient) on the geoid and the normal gravity on the reference surface, and is equal to (Heiskanen and Moritz, 1967, p. 85):
A9-
2.1 Gravity Field from Potential Coefficients If there are no masses above the geoid, or if these have been properly removed, the disturbing potential, i.e. the difference between the actual potential and a normal potential in the same point, outside the geoid can in spherical approximation be expanded as (Heiskanen and Moritz, 1967, p. 35):
T - GM ~ (_~) n+l n R
~ n=O
C~mYnm
(1)
m=O
where G is Newton's gravitational constant, M and R are the mass and radius of the earth, N is the maximum degree of expansion, r is the distance to the center of the earth, Cnm are the fully normalized, dimensionless potential (or Stokes) coefficients of degree n and order m, and Ynm are fully normalized surface spherical harmonics. The geoid is defined as the surface that has the same potential as a reference surface (e.g. the GRS80 ellipsoid). The coefficients of the geoid height, the distance between the geoid and the reference surface,
OT & +
10~
(3)
where "7o = G M / R 2 is the normal gravity at the reference surface. The term 07~Or can be regarded as the free-air correction and is on the surface of the earth equal to:
Or
R
This gives for eq. 3 (Heiskanen and Moritz, 1967, p. 89):
A9 -
In this section, we will concentrate on the definition of the geoid height and gravity anomaly, and how these are computed from potential coefficients, as delivered by GOCE and GRACE, and in our PGR model.
(2)
OT Or
2 RT
(5)
Putting expansion eq. 1 for T, we find for the coefficients of the gravity anomaly:
Agnm = % . ( n - 1)Chin
(6)
From this equation, gravity anomalies can be computed from a set of dimensionless geopotential coefficients Cnm as for example provided by GOCE and GRACE.
2.2 Gravity Field Computation in Post-Glacial Rebound In PGR studies, it is common to compute elastic load Love numbers in the Laplace-transformed domain for an elastic earth. An inverse Laplace transformation then yields viscoelastic Love numbers in the time domain (according to the correspondence principle, see e.g. Peltier (1974)). We use a semi-analytical normalmode relaxation model (Peltier, 1974; Wu and Peltier, 1982; Vermeersen and Sabadini, 1997) to compute the Love numbers in the Laplace domain. The dimensionless potential perturbation coefficients at the undeformed surface, which is in this formalism the reference surface, are equal to (Wahr et el., 1998; Johnston and Lambeck, 1999):
Cnm = 3pL 1 + kn * -L,~m PE 2n + 1
(7)
Chapter 18 where PL and PE are the density of the load and the mean density of the earth respectively, kn is the viscoelastic load Love number for potential perturbation of degree n, the asterisk denotes convolution in time and Lnm are the dimensionless, normalized coefficients of the load thickness. Note that the geoid height, which can again be computed using eq. 2, is now derived from the disturbing potential on the reference surface and not on the geoid, under the assumption that the actual gravity acceleration g at the geoid is equal to the normal gravity at the reference surface 70. To compute the gravity anomaly we proceed as in Mitrovica and Peltier (1989). First we compute the gravity perturbation at the deformed surface under the surface mass load (Longman, 1963), which consists of three terms (Farrell, 1972): - the change in acceleration from moving through the perturbed gravity field, proportional to the radial displacement Love number hn; - the direct attraction of the mass load (the 1-term in eq. 7); - effect of mass redistribution, proportional to kn. We cannot compute the direct attraction due to the surface mass load using expansion eq. 1, as this is only valid outside the masses. Instead we have to use the expansion (Heiskanen and Moritz, 1967, p. 34):
T=GM R
z(r) z n
iv
-R
n=O
C~mY~m
(8)
m=O
Using the free-air correction as an approximation for the gravity change in the unperturbed field, and taking the gradient of eq. 8 for the direct part and of eq. 1 for the internal mass distribution part, we find (Longman, 1963):
3pL --n + (n + 1)k~ - 2h.~ s * Ln~ A gn m -- " / 0 - PE 2n + 1
(9)
In Mitrovica and Peltier (1989) this is called the gravity anomaly at the perturbed surface. To obtain the gravity anomaly (at the geoid), Mitrovica and Peltier (1989) move eq. 9 to the geoid through the unperturbed field to obtain:
3pL --(n + 2) + (n-- 1)k~
~gnm = "70-PE
2n + 1
--
• L,~m (10)
Note that this result can be obtained directly by using the appropriate expansions for eq. 7 (eq. 8 for the direct term and eq. 1 for the mass redistribution term proportional to kn) and using eq. 5, showing the consistency of eq. 6 and eq. 10.
• High-Harmonic
Gravity Signatures Related to
Post-Glacial Rebound
Table 1. Viscosity stratification of our earth models Layer
Depth [km] Viscosity [Pas] CLVZ ALVZ
lithosphere
0-20 20-32 32-80 80-115 115-400 400-670 670-2891 2891-6371
upper mantle
lower mantle core
1 • 10 50 1.10 is 1.10 s° 5 • 10 20 5 10 20 5 . 1 0 21 0 •
1 10 50 •
1. 1018 5- 10 20 5 . 1 0 20 5 . 1 0 21 0
3 Input Parameters 3.1 Earth Stratification The earth model is radially stratified, incompressible, Maxwell viscoelastic and self-gravitating. We choose a model for a crustal low-viscosity zone (CLVZ) with a lower crust starting at a depth of 20 km, with a thickness of 12 km and a viscosity of 1018 Pas. The total lithospheric thickness is 80 km. We have modelled an asthenospheric low-viscosity zone (ALVZ) below a fully elastic lithosphere of 80 km, with a thickness of 35 km and a viscosity of 1018 Pas. The total viscosity stratification of our models is given in Table 1. We take volume-averaged densities and elastic parameters from the earth model PREM (Dziewonski and Anderson, 1981). Note that our current model is laterally homogeneous, which is not very realistic, as it can for example be expected that there are no CLVZs in old and cold lithosphere as for example in Scandinavia. In general CLVZs can be expected in parts of the continental crust with relatively high geothermal heat flux, and ALVZs can be expected more globally, especially below oceanic lithosphere, though with variable thickness.
3.2 Ice- and Sea-Load History We use an ice-load history of Kurt Lambeck and coworkers from ANU, Canberra (see e.g. Lambeck et al. (1998)) as our reference, and a modified version of ICE3G (Tushingham and Peltier, 1991), which we denote I3G, to test the sensitivity of the gravity anomaly perturbations. The modifications to ICE3G are: scaling up of the volume by 20% to an ice-equivalent sea level 1 of about-130 m at last glacial maximum (LGM), filtering to remove holes that arise due to the finite disc definition of ICE3G, and interpolation to 1 The ice-equivalent sea level is equal to the ice mass at a particular time, converted to ocean volume by the density of sea water, divided by the ocean area at that particular time. It is equal to the eustatic sea level change if all the ice mass at that particular time would melt.
105
106
H . H . A . S c h o t m a n • P. N. A. M. Visser • L. L. A. Vermeersen
i ~
i l"
0
ANU "" modified IGE3G I
-40 40 _o
i
~
i
i
•~, -80
.~_-100 -120[ -14~)~)0
100
i
i
i
i
80
60 time (kyr BP)
"-:
i" i,.~
40
i 20
0
F i g . 1. Ice-equivalent sea level for the ANU and modified
ICE3G (I3G) ice-load history calibrated carbon years, as ICE3G is given in uncalibrated carbon years. We added a phase of constant volume from 30 kyrs BP to LGM ( ~ 21 kyr BP) and a linear glaciation phase of 90 kyr starting from 120 kyr BR The latter is also added to the ANU model, which has an ice-equivalent sea level of 130 - 140 m from 30 kyr BP to LGM. In both models, 99% of the ice melted before 6 kyr BR In Figure 1 we have plotted the ice-equivalent sea levels for both histories. In Figure 2 and 3 we have given the ice-load distribution at LGM of the ANU and I3G ice-load histories respectively. The major differences between the models are the larger volume of the Laurentide ice sheet and the smaller (excess) volume over Greenland in the ANU model. The large ice volumes over the Kara Sea and in East-Siberia in the ICE3G model are currently considered to be unrealistic (see e.g. Siegert and Dowdeswell (2004)). We have included the effect of coastline migration and meltwater influx in areas that were once-glaciated and are now below sea level, as described in Mitrovica and Milne (2003), Lambeck et al. (2003) and Schotman and Vermeersen (2005). We have not considered the effect of rotation and have used only one glacial cycle, as the effect of additional glacial cycles on perturbations is small.
0
500 1000 1500 2000 2500 3000 3500 4000 4500 ice height [m]
Fig. 2. ANU ice-load distribution at LGM
560 1600 1500 20'00 2500 3600 3500 40'00 45'00 ice height [m]
Fig. 3. I3G (modified ICE3G) ice-load distribution at LGM perturbation differences, i.e. the difference between perturbations computed with a certain set of earthor ice-model parameters and perturbations computed with our reference model.
4.1 PGR- and LVZ-lnduced Gravity Anomalies 4 Results As we are interested in gravity anomaly perturbations due to an LVZ, we subtract from the results for a model with an LVZ a model without an LVZ. This means that we subtract the results of a model that has the same number of layers as the model with an LVZ, but with a viscosity value for the LVZ that is either very large (i.e. effectively elastic, for a CLVZ) or equal to the upper mantle viscosity (for an ALVZ). Our reference earth model for a CLVZ and an ALVZ is given in Table 1, and our reference ice-model is ANU. To investigate the effect of different properties of the LVZ or different ice-load histories, we use
In Figure 4 we show the predicted gravity anomaly due to PGR, i.e. without an LVZ, using ice-load history ANU. The signal is dominated by negative anomalies in the Hudson Bay area, where mantle material has been pushed towards the bulges, which are the areas with positive anomalies around North-America. The same effect can be seen in Scandinavia. Next we show gravity anomaly perturbations due to a CLVZ, again using ANU (Figure 5). The picture is dominated by small scale perturbations near the edge of the ice-load due to extra mass flow away from glaciated areas during glaciation. As the introduction of a CLVZ increases the relaxation time (Schotman
Chapter 18 • High-Harmonic Gravity Signatures Related to Post-Glacial Rebound
and Vermeersen, 2005) and as the glaciation period is much longer than the deglaciation period, mass has yet to flow back to previously glaciated areas. Perturbation amplitudes are up to 10 mgal which is not small in comparison to the total signal (Figure 4) and the expected performance of GOCE. Due to the larger depth of the ALVZ, Figure 6 shows less spatial detail than Figure 5. Moreover, because an ALVZ shortens the relaxation time (Schotman and Vermeersen, 2005), adjustment to isostasy is faster, leading to negative perturbations in the bulge areas and positive perturbations in formerly glaciated areas.
==%
4.2 Comparison with the Performance of GOCE and GRACE
gravityanomaly[mgal]
Fig. 4. PGR-induced gravity anomaly (no LVZ) To compare the computed gravity anomaly perturbations with GOCE we use degree amplitudes. These are the square roots of the spherical harmonic expansion degree variances of the perturbed field. For gravity anomaly perturbations the degree amplitudes are (compare eq. 6): n
a~-%-(n-1)
EV~mC*nm
(11)
mzO
-'9
-'6
-'3 b 3 6 gravityanomaly[mgal]
9
I
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Fig. 5. CLVZ-induced gravity anomaly perturbation
-'9
:6
-'3
b
3
6
9
lv2
gravityanomaly[mgal]
Fig. 6. ALVZ-induced gravity anomaly perturbation
where the asterisk denotes complex conjungation. In Figure 7 we have plotted the CLVZ- and ALVZinduced perturbation degree amplitudes, the expected performance of GOCE and the realized performance of GRACE (GGM02S). The CLVZ-induced gravity anomaly perturbations are above the GOCE performance up to harmonic degree 140 and above GGM02S up to degree 90. The ALVZ-induced gravity anomaly perturbations have significant amplitude for low degrees; in particular, they are above the GOCE performance up to degree 70 and above GGM02S up to degree 60. This means that GOCE compared to GRACE will probably not add much information on ALVZs, though the ratio of signal to error is more favorable for GOCE, so GOCE is especially predicted to deliver more information on CLVZs. In following, we will concentrate on CLVZs only. One of the largest uncertainties in PGR modeling is the ice-load history. From Figure 8, where we have plotted perturbation differences, we see that GOCE is sensitive to uncertainties in the ice-load history ('I3GANU'); as the difference in perturbations between a model using I3G and our reference model (using ANU) is above the performance of GOCE up to degree 140. In Figure 8 we have plotted the differences between a model with either a thicker CLVZ ('t20t 12') or higher viscosity CLVZ ('v 19-v 18') than our reference model. We see that GOCE is predicted to be sensitive to changes in the properties of the CLVZ up to degree 120-140. Moreover, we see that the curves
107
108
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• P. N. A. M. V i s s e r • L. L. A. Vermeersen
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ence of errors in these models (compare Velicogna and Wahr (2002)). If we assume that all other processes have been removed error-free, then the only uncertainty in this hypothetical case is the ice-load history. In the next section we will show that it is still possible to extract information on the properties of CLVZs from GOCE data in the presence of uncertainties in the ice-load history.
250
harmonic degree
Fig. 8. Differences in CLVZ induced perturbations in gravity anomaly degree amplitudes due to different properties of the CLVZ and different ice-load histories, compared with the performance of GOCE and GGM02S have a specific form, which is related to the different behavior for a thicker CLVZ and a higher viscosity CLVZ. Further on we will deduce from these curves spectral signatures for different properties of the CLVZ. From Figure 8 we conclude that in principle GOCE data will add constraints to estimates of the ice-load history in the presence of a CLVZ (with known parameters) and about properties of a CLVZ (if the iceload history is known). In practice, things are more complicated, because the measured gravity signal consists of large number of contributions, see Section 1. If we consider the gravity field as given by GGM02S, in the spectral range where we expect the largest amplitude (from degree 40 to 90, compare Figure 9), we see no direct relation with the modeled gravity anomalies induced by a CLVZ (Figure 5). Short-scale features are visible, but the amplitudes are much larger than predicted by our PGR model. This means that we have to remove other geophysical signals from the measured gravity field. If models for these geophysical signals are available, the question is if the information on CLVZs is still recoverable in the pres-
4.3
Spectral
Signatures
If we regard the differences between the ANU and I3G ice-load histories as realistic uncertainties in the ice-load history, then it is already clear from Figure 9 that it will be difficult to extract information from GOCE data on the properties of the CLVZ. This can be illustrated more clearly using the degree correlation coefficient, defined as (Mitrovica and Peltier, 1989):
Pn --
~-~nm=° Cnm-Dnm
(12)
with C ~ , D ~ different sets of spherical harmonic coefficients. In Figure 10 we have plotted the degree correlation coefficient between gravity anomaly values computed with our reference model (CLVZ from Table 1 and ice-model ANU) and different test model values. If our test model is the same as our standard model, then the degree correlation coefficient will be equal to one (not plotted). If we use in our test model the I3G iceload history, then the correlation between the reference and test model is very poor (p(Z~gANU~ z~gI3G)). If we know the ice-load history, then we can clearly distinguish between a model with a thicker CLVZ
Chapter 18 • High-Harmonic Gravity Signatures Related to Post-Glacial Rebound
(p(Agtt2, Agt2o), from degree 70) or a higher viscosity CLVZ (p(Agv,8, Agv~9), from degree 100). If we do not know the ice-load history, and correlate for example our reference model with a test model with different parameters of the CLVZ and the I3G iceload history, then the degree correlation coefficient will be poor due to the bad correlation for different ice-load histories (not plotted), and we cannot constrain the properties of the CVLZ. This means that we cannot extract information on the properties of the CLVZ in the presence of uncertainties in the ice-load history. A large part of this uncertainty can be removed by normalizing the degree amplitudes with the degree amplitudes of the ice-load history at LGM. This is because the computed freeair gravity anomaly perturbations are a convolution of the temporal and spatial impulse response of the earth (i.e. the time-dependent Love numbers for a certain CLVZ) and the time- and space-dependent input sequence (i.e. the ice-load history). If we consider the response at a certain time interval and assume the ice-load is constant (which is obviously not the case), then the spatial spectrum (i.e. the degree amplitudes) of the impulse response is equal to the ratio of the spectrum of the output (i.e. the gravity anomalies) and the ice-load. The assumption of constant ice load is approximated by using the ice load at LGM, which is justified by the long period of glaciation compared to deglaciation. In Schotman and Vermeersen (2005) we have shown that the computed normalized degree amplitudes closely resemble the time-dependent Love numbers for a certain CLVZ as a function of harmonic degree. In practice, we do not know the real ice-load history, but we can estimate which ice-load history best fits the gravity anomaly perturbations by correlating the spatial spectrum of the gravity anomaly perturbations with the spectrum of the ice-load history (at LGM). We see from Figure 11 that the correlation is always significantly better for the ice model that generated the anomaly perturbations. Moreover we see that the ANU ice model correlates up to higher degree with the corresponding gravity anomalies than the I3G ice model, mainly because the I3G model has less power in the high harmonics. In Figure 12 we show perturbation degree amplitudes computed with I3G and ANU, normalized by the dimensionless degree amplitudes of the I3G and ANU ice height at LGM, respectively. We can see that the curves are very close, except above degree 70, where the correlation of the perturbations computed from I3G and the ice heights of I3G drop very fast, see Figure 11. If we consider the degree correlations from Figure 11 as a measure for the quality of the normalized degree amplitudes, we can compute a best estimate from the two normalized degree amplitude curves in a weighted least squares (WLSQ) sense, with weights determined by the degree correlations. If we follow the same procedure for different prop-
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and Outlook
We have shown that gravity anomaly perturbations induced by crustal low-viscosity zones (CLVZs) are above the expected GOCE performance up to harmonic degree 140 and above the realized GRACE
109
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H.H.A. Schotman • P. N. A. M. Visser • L. L. A. Vermeersen
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Acknowledgements
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Fig. 13. Spectral signature for different properties of the CLVZ performance (GGM02S) up to degree 90. For an asthenospheric LVZ (ALVZ) the gravity anomaly perturbations are above the expected G O C E performance and G G M 0 2 S up to degree 70 and 60, respectively. G O C E is thus especially useful for detecting CLVZs. It was found that G O C E is also sensitive to changes in theological properties of a CLVZ, which means that in principle G O C E should be able to constrain the theology. G O C E is however also sensitive to uncertainties in the ice-load history, though a large part of this uncertainty can be removed by manipulating the data in the spectral domain to obtain spectral signatures for different CLVZs. Note that this study is valid for the limiting case of a laterally h o m o g e n e o u s earth, which is clearly not realistic everywhere with regard to the presence and properties of CLVZs. In practice, it will be difficult to extract information from satellite gravity data. F r o m a filtered version of G G M 0 2 S we have seen that a large n u m b e r of geophysical signals is present in the gravity field. We therefore need to remove as well as possible all geophysical signals in the frequency range that we are interested in, and use some form of spatio-spectral filtering to isolate the relevant signal (see e.g. Simons and Hager (1997)). In future studies, we will show if
We thank Jerry Mitrovica, an anonymous reviewer and Mark Drinkwater for their constructive comments, Kurt Lambeck and co-workers (ANU, Canberra) for their global ice sheet model and Radboud Koop (SRON, Utrecht) for discussions.
References
Di Donato, G., J.X. Mitrovica, R. Sabadini, and L.L.A. Vermeersen (2000). The influence of a ductile crustal zone on glacial isostatic adjustment; geodetic observables along the U.S. East Coast, Geophys. Res. Lett., 27, pp. 3017-3020. Dziewonski, A.M. and D.L. Anderson (1981). Preliminary Reference Earth Model, Phys. Earth Planet. Inter, 25, pp. 297-356. Farrell, W.E. (1972). Deformation of the earth by surface loads, Rev. Geophys. Space Phys., 10, pp. 761-797. Hart, S.-C., C.K. Shum, E Ditmar, E Visser, C. van Beelen and E.J.O. Schrama (2006) Aliasing effect of high frequency mass variations on GOCE recovery of the earth's gravity field, J. Geodyn., 41, pp. 69-76. Heiskanen, W. and H. Moritz (1967). Physical Geodesy, W.H. Freeman and Co., San Francisco, 364 pp. Johnston, F'. and K. Lambeck (1999). Postglacial rebound and sea level contributions to changes in the geoid and the earth's rotation axis, Geophys. J. Int., 136, pp. 537558. Kendall, R., J.X. Mitrovica and R. Sabadini (2003). Lithospheric thickness inferred from Australian post-glacial sea-level change: The influence of a ductile crustal zone, Geophys. Res. Lett., 30, pp. 1461-1464. Klemann, V. and D. Wolf (1999). Implications of a ductile crustal layer for the deformation caused by the Fennoscandian ice sheet, Geophys. J. Int., 139, pp. 216226. Lambeck, K., C. Smither and P. Johnston (1998). Sea-level change, glacial rebound and mantle viscosity of northern Europe, Geophys. J. Int., 134, pp. 102-144. Lambeck, K., A. Purcell, P. Johnston, M. Nakada, Y. Yokoyama (2003). Water-load definition in the glaciohydro-isostatic sea-level equation, Quat. Sci. Rev., 22, p. 309-318. Longman, I.M. (1963). A Green's function for determining the deformation of the earth under surface mass loads, 2. Computations and numerical results, J. Geophys. Res., 68, pp. 485-496. Mikhailov, V., S. Tikhotsky, M. Diament, I. Panet, V. Ballu (2004). Can tectonic processes be recovered from new gravity satellite data?, Earth Planet. Sci. Lett., 228, 10.1016/j.epsl. 2004.09.035. Mitrovica, J.X. and G.A. Milne (2003). On post-glacial sea level: I. General theory, Geophys. J. Int., 154, pp. 253267.
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Chapter 18 • High-Harmonic Gravity Signatures Related to Post-Glacial Rebound
Mitrovica, J.X. and W.R. Peltier (1989). Pleistocene deglaciation and the global gravity field, J. Geophys. Res., 94, pp. 13,651-13,671. Peltier, W.R. (1974). The impulse response of a Maxwell earth, Rev. Geophys. Space Phys., 12, pp. 649-669. Pollitz, F.F. (2003). Transient rheology of the uppermost mantle beneath the Mojave Desert, California, Earth Planet. Sci. Lett., 215, pp. 89-104. Ranalli, G. and D. Murphy (1987). Rheological stratification of the lithosphere, Tectonophys., 132, pp. 281-295. Schotman, H.H.A. and L.L.A. Vermeersen (2005). Sensitivity of glacial isostatic adjustment models with shallow low-viscosity earth layers to the ice-load history in relation to the performance of GOCE and GRACE, Earth Planet. Sci. Lett., 236, 10.1016/j.epsl.2005.04.008. Siegert, M.J. and J.A. Dowdeswell (2004). Numerical reconstructions of the Eurasian Ice Sheet and climate during the Late Weichselian, Quat. Sci. Rev., 23, pp. 12731283. Simons, M. and B.H. Hager (1997). Localization of the gravity field and the signature of glacial rebound, Nature, 390, pp. 500-504. Stein, S. and M. Wysession (2003). Introduction to Seismology, Earthquakes, and Earth Structure, Blackwell Publishing, Oxford, 498 pp. Tapley, B., J. Ries, S. Bettadpur, D. Chambers, M. Cheng, F. Condi, B. Gunter, Z. Kang, R Nagel, R. Pastor, T. Pekker, S. Poole and F. Wang (2005). GGM02 - An improved earth gravity field model from GRACE, jr. Geodesy, 10.1007/s00190-005-0480-z. Tushingham, A.M. and W.R. Peltier (1991). ICE3G: A new global model of late Pleistocene deglaciation based upon geophysical predications of postglacial relative sea level change, J. Geophys. Res., 96, pp. 4497-4523.
van der Wal, W., H.H.A. Schotman and L.L.A. Vermeersen (2004). Geoid heights due to a crustal low viscosity zone in glacial isostatic adjustment modeling; a sensitivity analysis for GOCE, Geophys. Res. Lett., 31, 10.1029/2003 GL019139. Velicogna, I. and J. Wahr (2002). Postglacial rebound and earth's viscosity structure from GRACE, Jr. Geophys. Res., 107, 10.1029/2001JB001735. Vermeersen, L.L.A. (2003). The potential of GOCE in constraining the structure of the crust and lithosphere from post-glacial rebound, Space Sci. Rev., 108, pp. 105-113. Vermeersen, L.L.A. and R. Sabadini (1997). A new class of stratified visco-elastic models by analytical techniques, Geophys. J. Int., 139, pp. 530-571. Visser, RN.A.M., R. Rummel, G. Balmino, H. Stinkel, J. Johannessen, M. Aguirre, RL. Woodworth, C. Le Provost, C.C. Tscherning and R. Sabadini (2002). The European earth explorer mission GOCE: Impact for the geosciences, In: Ice Sheets, Sea Level and the Dynamic Earth, J.X. Mitrovica and L.L.A. Vermeersen (eds), AGU Geodynamics Series, 29, AGU, Washington DC, pp. 95-107. Wahr, J., M. Molenaar and F. Bryan (1998). Time variability of the earth's gravity field: Hydrological and oceanic effects and their possible detection using GRACE, J. Geophys. Res., 103, pp. 30,205-30,229. Watts, A.B. and E.B. Burov (2003). Lithospheric strength and its relationship to the elastic and seismogenic layer thickness, Earth Planet. Sci. Lett., 213, pp. 113-131. Wu, R and W.R. Peltier (1982). Viscous gravitational relaxation, Geophys. J. R. Astron. Soc., 70, pp. 435-485. Wu, R, H. Wang and H. Schotman (2005) Postglacial induced surface motions, sea-levels and geoid rates on a spherical, self-gravitating, laterally heterogeneous earth, J. Geodyn., 39, pp. 127-142.
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Part II Frontiers in the Analysis of Space Geodetic Measurements Chapter 19
GPS/GLONASS Orbit Determination Based on Combined Microwave and SLR Data Analysis
Chapter 20
BIFROST: Noise Properties of GPS Time Series
Chapter 21
Discrete Crossover Analysis
Chapter 22
A Comparative Analysis of Uncertainty Modelling in GPS Data Analysis
Chapter 23
Looking for Systematic Error in Scale from Terrestrial Reference Frames Derived from DORIS Data
Chapter 24
WVR Calibration Applied to European VLBI Observing Sessions
Chapter 25
Frontiers in the Combination of Space Geodetic Techniques
Chapter 26
Modifying the Stochastic Model to Mitigate GPS Systematic Errors in Relative Positioning
Chapter 27
GPS Ambiguity Resolution and Validation under Multipath Effects: Improvements Using Wavelets
Chapter 28
An Empirical Stochastic Model for Gps
Chapter 29
Feeding Neural Network Models with GPS Observations: a Challenging Task
Chapter 30
Spatial Spectral Inversion of the Changing Geometry of the Earth from SOPAC GPS Data
Chapter 31
Improved Processing Method of UEGN-2002 Gravity Network Measurements in Hungary
Chapter 32
Spectra of Rapid Oscillations of Earth Rotation Parameters Determined during the CONT02 Campaign
Chapter 33
On the Establishing Project of Chinese Surveying and Control Network for Earth-Orbit Satellite and Deep Space Detection
Chapter 34
Constructing a System to Monitor the Data Quality of GPS Receivers
Chapter 19
GPS/GLONASS orbit determination based on combined microwave and SLR data analysis C. Urschl, G. Beutler, W. Gurtner, U. Hugentobler, S. Schaer Astronomical Institute, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland email: [email protected]
Abstract. The combination of space-geodetic techniques is considered as an important tool for improving the accuracy and consistency of the resulting geodetic products. For GNSS satellites, tracking data is regularly collected by both the microwave and the SLR observation technique. In this study, we investigate the impact of combined analysis of microwave and SLR observations on precise orbit determination of GNSS satellites. Combined orbits are generated for the two GPS satellites equipped with Laser retroreflector arrays and for three GLONASS satellites that are currently observed by the ILRS network. The combination is done at the observation level, implying that all parameters common to both techniques are derived from both observation types. Several experimental orbits are determined using different observation weights. As the well-known 5 cm-bias between SLR measurements and GPS microwave orbits is unexplained, SLR range biases as well as satellite retroreflector offsets are estimated in addition to the orbital parameters. The different orbit solutions are then compared in order to determine whether and to which extent the SLR measurements influence a microwave orbit primarily derived from microwave observations. Key words. GNSS orbit determination, Multitechnique combination, GPS, SLR
1 Introduction Different space-geodetic techniques contribute to the generation of the International Earth Rotation and Reference Systems Service (IERS) products. Therefore, the IERS shows an increasing interest in intertechnique combinations, in particular to improve the accuracy and consistency of the IERS products, the International Terrestrial Reference Frame (ITRF), the international Celestial Reference Frame (ICRF), and the Earth Orientation Parameters (EOP). Efforts are underway to develop methods for combining observations and products from individual space-geodetic techniques (Rothacher, 2002; Ray et al., 2004). Sys-
tematic biases between the individual technique solutions are topics of current investigations. Within the scope of an ongoing project "Combined analysis of the major satellite-geodetic observables", this paper presents the estimation of combined GNSS (Global Navigation Satellite System, consisting of GPS and GLONASS) satellite orbits using measurements of two space-geodetic techniques, microwave and SLR measurements. First experiments were carried out by Zhu et. al (1997), who determined two 1-day arc orbits using microwave and SLR tracking data for two days in 1995. Previous studies on validating microwave orbits with SLR observations have shown a constant bias of about 5.5 cm between the range measurements and the GPS microwave orbit (see, e.g., Appleby and Otsubo, 2000; Springer, 2000; Urschl et al., 2005). This bias is not yet understood. We wanted to know whether the bias disappears by including SLR measurements into the orbit determination process, or whether the orbit dynamics is strong enough not to absorb the bias. SLR range biases as well as satellite reflector offsets that might be responsible for the mean biases between SLR and microwave observation are estimated in addition to the orbital parameters. We analyze the resulting orbit to see whether the SLR observations help to improve the combined orbit. The combination strategy is outlined in Section 2. Section 3 describes the data set used. Section 4 introduces three combination experiments and the corresponding experimental orbit solutions. The analysis and comparison of the different combined orbit solutions is done in Section 5. Section 6 derives conclusions from the combination experiments and Section 7 summarizes the results.
2 Combination strategy The weighted least-squares method is used for the parameter estimation process. We apply a combination strategy at the observation level using microwave (double difference phase, ionosphere-free linear combination) and SLR range observations. Technically, the combination is done on the normal equation level by stacking the technique-specific nor-
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C. Urschl • G. Beutler. W. Gurtner • U. Hugentobler. S. Schaer
mal equations. Parameters common to both observation types are derived from the observations of both techniques. These common parameters are orbit parameters, Earth orientation parameters (EOP), geocenter coordinates and coordinates of collocated sites constrained with local ties. As this study focuses on the estimation of common orbit parameters only, we constrain all other parameters to highly accurate apriori values. The geocenter is constrained to the origin of the ITRF2000. Earth orientation parameters are constrained to the weekly EOPs, derived from the CODE (Center for Orbit Determination in Europe) final orbit solution. For the datum definition, the coordinates of the Laser tracking sites and the sites with microwave receivers are constrained to their ITRF2000 estimates. We compute two types of combined orbits, namely 1-day and 3-day arcs, using a development version of the Bernese GPS Software V5.0 (Hugentobler et al., 2005). For 3-day arcs the orbit dynamics is a stronger constraint than for 1-day arcs, affecting the impact of the SLR observations on the resulting orbit. The CODE orbit model is adopted. The six orbital elements, nine dynamical orbit parameters (radiation pressure parameters) and one stochastic pulse at noon and at the day boundaries for the 3-day arcs are estimated.
3 Data set Combined orbits are computed for five GNSS satellites, the two GPS satellites equipped with Laser retroreflectors and three of the GLONASS satellites currently tracked by the ILRS (International Laser Ranging Service) network. Table 1 characterizes the satellites used. The microwave phase measurements from 157 GNSS sites as well as the Laser ranging measurements from 13 SLR sites are used to obtain a time series of 41 days in 2004 (DoY 305-345). Figure 1 shows the global distribution of the used SLR (triangle) and GNSS (circles) sites. Most of the sparsely distributed SLR sites tracking GNSS satellites are located on the northern hemisphere. Only 27 of the 157 sites with microwave receivers track both GPS and GLONASS satellites with the majority (i.e. 130) tracking only GPS satellites.
Table 1. Denotation of the used GNSS satellites Satellite system RINEX ILRS COSPAR GPS G05 35 1993-054A G06 36 1994-016A GLONASS R03 87 2001-053B R22 89 2002-060A R24 84 2000-063B
l
Fig. 1. Geographic location of SLR and GNSS sites; triangle, SLR; circle, GNSS The number of observations available for the parameter estimation process differs greatly for the two techniques and for the two satellite systems, GPS and GLONASS. About 20,000 microwave measurements for each GPS satellite and about 3,000 for each GLONASS satellite are used to generate a combined orbit (1-day arc), considering a sampling rate of 180 seconds. The SLR measurements are so called normal point data, formed by averaging the individual range measurements over 5-min intervals. On a daily average, only 5-20 normal points for each GPS satellite and about 10 to 40 normal points for each GLONASS satellite are available from the selected sites during the considered time interval. Figure 2 shows the greatly varying number of normal points for each satellite over the analyzed time interval.
4 Combination experiments We perform the three following combination experiments.
Experiment I The first experiment is a sort of "quick-look" experiment to study the impact of SLR observations on the combined orbits without attempting to model the 5 cm-bias between SLR and microwave observations. We address the question whether the microwave observation model is able to absorb this bias by, e.g., adapting phase ambiguities or receiver clocks, or whether the orbit dynamics is strong enough not to absorb the bias. The former would result in orbit scaling, while the latter would induce an orbit deformation, if the weight of the SLR observations is increased.
Experiment 2 The second experiment is complementary to the first one. Daily SLR range biases for each station and each satellite are estimated in addition to the orbital parameters. Thus, only the pass-specific SLR informa-
Chapter 19
• GPS/GLONASS Orbit Determination
Based on Combined Microwave and SLR Data Analysis
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tion contributes to the combined orbit. This experiment will show whether there is any influence of the remaining pass-specific SLR contribution on the resulting orbit.
Experiment 3 In the third experiment, we model the mean SLR range biases with two satellite reflector offsets in the radial direction (pointing to the geocenter) of the satellite's body-fixed coordinate system, one for the two GPS satellites and one for the three GLONASS satellites. These offsets absorb the mean bias of all SLR observations to GPS and GLONASS satellites, each over the considered time interval. Compared to the second experiment, the SLR observations to GPS satellites have more influence on the resulting orbit, because the day to day variations of the Laser ranges contribute to the orbit determination. Therefore, the third experiment is the most interesting one to study the impact of additional Laser measurements on the microwave measurements-dominated orbit.
Orbit determination For each experiment several orbits are determined. We characterize the orbits with a scheme of "solution IDs" that is used throughout the paper. Table 2 lists the orbit solution iDs and their characteristics. The table is divided into three parts, corresponding to the three experiments.
Table 2. Listing of orbit solution IDs Orbit ID
SLR weighting
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/31 C1 D1
Additional parameters
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SLR range bias SLR range bias satellite reflector offset satellite reflector offset
The initial character of the solution ID indicates the observation weighting, the digit corresponds to the experiment number. We use four different weighting cases, A , B , C and D. The a priori sigma of the microwave observations (double difference phase, ionosphere-free linear combination) crMw is about 1 cm in all cases, whereas the a priori sigma of the range measurements changes in such a way that the weight of the SLR observations increases with consecutive solution IDs. In case A the a priori sigma of the SLR measurements era is set to infinity, which corresponds to a weight of zero. Thus, the SLR measurements do not contribute to the orbit determination, i.e., orbit A1 is a microwave-only orbit. The a priori sigma cTB corresponds in order of magnitude to the actual measurement accuracy of the range measurements as well as to ~ M w . Both measurement types have approximately the same weight. Cases C and D increase the weight of the range measurements compared to the microwave measurements. In the first experiment, we compute four different orbit solutions with the four different weighting cases, in the second and third experiment only weighting of t y p e / 3 and C are considered.
5 Orbit analysis Let us now analyze the different orbits obtained in the three experiments to study the influence of the additional range measurements on the estimated orbits. When increasing the weight of the range observations we expect the SLR range residuals to show smaller mean values and standard deviations. The residuals indicate whether the small amount of SLR observations is able to "influence" an orbit determined by microwave observations. They do not, however, allow to draw conclusions on the orbit accuracy. In order to assess the "deformation" of the orbits, we compare the orbits directly (in radial, along-track, and out-of plane direction), in addition the parameters of a Helmert transformation will reveal possible translation, rotation and scaling effects in the inertial reference frame.
117
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C. Urschl • G. Beutler. W. Gurtner • U. Hugentobler. S. Schaer
Orbit quality is assessed by the orbit overlap accuracy, i.e., the position differences of consecutive orbital arcs at the day boundaries. If the orbits are well defined, consecutive arcs should fit together well and the overlap difference should be small. 5.1
Discussion
of Experiment
1
For the first experiment, we compute 1-day arcs of the orbit types A 1 , / 3 1 , C1, and D 1 (see Table 2) for each of the 41 days in the considered time interval. Figure 3 shows the obtained SLR range residuals for one of the GPS satellites (G05). Table 3 lists the mean values and standard deviations of the SLR range residuals for all G N S S satellites. For the microwave-only solution A1, the standard deviation of the SLR range residuals is about 2 cm for the GPS satellites and about 5 cm for the G L O N A S S satellites, which corresponds to the microwave orbit accuracy of 1-day arc solutions. The mean residual offset for the GPS satellites is about -5.7 cm, whereas it is very different for the three G L O N A S S satellites with -3 cm, 0 cm and 1 cm. As expected, the SLR range residuals b e c o m e smaller with increasing weight of the SLR observations. For solution/31, significant changes in the SLR range residuals can be noticed already, particularly for the G L O N A S S satellites. The mean value of the range residuals for the G L O N A S S satellites diminishes to zero with increasing weight of the SLR o b -
servations. The mean offset for the GPS satellites, however, remains at 5-6 cm for orbit s o l u t i o n / 3 1 but decreases to 1 m m for the D 1 solution. Next we compare the microwave-only orbit A1 with the combined o r b i t s / 3 1 , C1, and D1. Helmert transformation parameters for each 1-day arc are estimated for the solutions pairs A1 /31, A1 - C1, Table 4. Mean values and standard deviations of the Helmert transformation parameters averaged over 41 days between the 1-day arc orbit solutions A1 - / 3 1 , A1 - C1, and A1 - D1 Satellite
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1.3 1.9 22.9 7.7 12.7
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0.0 -0.1 2.0 0.9 1.2
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Chapter 1 9
• GPS/GLONASS
and A1 - D1. No significant rotations are observable. Table 4 displays the mean values and standard deviations of the translation parameters and the scale difference averaged over the considered 41 days. The mean values of the Helmert parameters of A1 - B1 are zero for the GPS orbits. For GLONASS, variations of up to 4 cm are observed. Both, translation parameters and scale difference, increase with the weight of the SLR observations. For A1 - D1 the translation parameters for GLONASS orbits vary by up to 23 cm about a mean value of up to 5 cm. For GPS the mean value is at -1 cm with a variation of up to 3 cm. The influence of SLR measurements on GLONASS satellites is larger due to the higher number of normal points and the lower number of microwave measurements when compared to GPS satellites. The scale difference is about 0.6 ppb (1.3 cm in radial direction) on the average, but may reach up to about 1 ppb for some orbit solutions. Table 5 gives the standard deviations of the daily orbit differences over the considered 41 days for each GNSS satellite after Helmert transformation. With increasing SLR weight significant orbit differences up to 5 cm for GPS and 40 cm for GLONASS satellites remain in radial, along-track, and out-of-plane directions. Figures 4(a) and 4(b) display the radial orbit overlaps for G05 and for R22. The overlap accuracy of the orbit positions at the day boundaries for solution A1 is about 5 cm for the GPS orbits. This value is much larger for the GLONASS orbits, due to the low number of GLONASS tracking sites. The overlap accuracy in along-track and out-of-plane direction is within the same order of magnitude. We found that the orbit differences at the day boundaries become larger for the combined orbits with increasing weight of the SLR observations. The overlap accuracy of the combined orbit B1 does not change significantly compared to the microwave orbit A1, while it gets worse in cases G'l and D1.
5.2 Discussion of Experiment 2 We estimate daily SLR range biases for each station and satellite when generating the combined orbit in order to model the dominating offsets (solution IDs B2 and C2). As a result, the range measurements lose a lot of their potential influence on the orbit determination, as the "range" is modified. The SLR range residuals derived from the combined orbits B2 and C2 vary around zero with a standard deviation of about 1-2 cm (see Table 6). The mean values of the Helmert parameters between the 1-day arc orbits A1 and B2 are about zero. For GLONASS orbits the translation parameters show variations of about 1 cm. For a larger SLR weight (solution C2) the mean translation parame-
Orbit Determination Based on Combined Microwave and SLR
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(b) R22 Fig. 4. Orbit overlaps in radial direction of the 1-day arc orbit solutions A1,/31, C1, and D1 for the GPS satellite G05 and the GLONASS satellite R22 6. Mean values and standard deviations (cm) of the SLR range residuals derived from the 1-day arc orbit solutions/32 and C2 Table
Satellite G05 G06 R03 R22 R24
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119
120
C. Urschl • G. Beutler. W. Gurtner • U. Hugentobler. S. Schaer
Table 7. Standard deviations of orbit differences of 41 days between the 1-day arc orbit solutions A1 - / 3 2 and A1 C2, after Helmert transformation
Table 8. Estimated satellite reflector offset correction in radial direction (in cm) and formal rms for GPS and GLONASS satellites from solutions/33 and C3.
Satellite
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using larger weights for the range measurements. The standard deviation o f the daily GPS orbit comparisons is below the one-cm-level, see Table 7. But for the G L O N A S S satellites there are orbit differences of up to 5-10 cm, due to the fact that pass-specific patterns in the range m e a s u r e m e n t s are m u c h larger for the G L O N A S S than for the GPS satellites. Figure 5 demonstrates the improvement of the range residual pattern for one pass of the G L O N A S S satellite R24 observed by the SLR site 7090 (Yarragadee) when increasing the weight of the range measurements. The orbit overlap accuracy does not change significantly with increasing SLR weight, neither f o r / 3 2 nor for C 2 (no figure included). 5.3
Discussion
of Experiment
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parameters instead of constraining the reflector offsets to their a priori values. Two satellite-type-specific offset in radial direction are determined for the considered time series, one offset for the GPS satellites and one for the G L O N A S S satellites. Thus, we allow for an additional constant offset for each satellitetype, but not for each satellite and not for each station, as in experiment 2. Table 8 shows the estimated retroreflector offsets with respect to the a priori values for the GPS and G L O N A S S satellites and the 1-day arc s o l u t i o n s / 3 3 , C3. The a priori retroreflectot offsets in radial direction are 0.6584 m for the GPS and 1.5416 m for the G L O N A S S satellites. As expected, the estimated reflector offset correction is about 5 cm for the GPS satellites and close to zero for the G L O N A S S satellites. The values correspond to the mean values of the SLR range residuals with respect to the microwave-only orbits.
Chapter 19 • GPS/GLONASS Orbit D e t e r m i n a t i o n Based on C o m b i n e d M i c r o w a v e and SLR Data Analysis
As for the second experiment, the SLR range residuals derived from the combined orbits/33 and C3 vary around zero but with a slightly larger standard deviation of about 1-3 cm (see Table 9), as expected from the significantly smaller number of estimated parameters. The Helmert parameters for the solution pairs A1 - / 3 3 and A1 - C3 are slightly larger than for the pairs A1 - / 3 2 and A1 - C2 from the second experiment. The mean values of the translation parameters between solution A1 and/33 are zero for the GPS orbits. The GLONASS orbits show mean translations of about 1 cm for A1 - / 3 3 . When increasing the weight of the SLR observations (C3) the mean translations increase up to 3 cm for GLONASS with variations of up to 5 cm. The scale change of the GPS orbits is at the zero level. That of GLONASS orbits varies with 0.5 ppb around zero comparing solutions A1 and C3. Table 10 shows the standard deviations of the daily orbit differences. As for the second set of orbit solutions, only small differences are encountered between the microwave orbit A1 and the combined orbits/33 and C3 for the GPS satellites. The orbit overlaps in all three directions, in radial, along-track, and out-of-plane, are the same for /33 and larger for solution C3, when compared to solution A1. Figures 6(a) and 6(b) show the overlaps in radial direction for G05 and R22. 3-day arcs were analyzed in an analogous manner as 1-day arcs for all three experiments. The second day of the combined 3-day arc was compared to the second day of the microwave 3-day arc. As observations from the neighboring days are included for the orbit determination, the orbit overlaps of the second day are always better defined than the same overlaps of the corresponding 1-day arcs. The overlap accuracy of the combined orbits is, however, not improved compared to the microwave orbit overlap accuracy. The resulting SLR range residuals and orbit parameters for the 3-day arc solutions do not show any significant differences to the analysis results of the 1-day arc solutions. 6
Conclusions
From the first experiment we conclude that the generated combined orbits are sensitive to introducing additional range measurements. Depending on the SLR weighting (with respect to the microwave observations), SLR observations have an impact on a microwave observations-dominated orbit, even if the number of SLR measurements is about three orders of magnitude smaller than the number of microwave measurements. A range observation represents the geometric distance to a satellite and is, therefore, much "stronger" than a microwave phase measurement, which corresponds to a range biased by ambi-
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guity and clock corrections. Table 3 shows that the mean values and standard deviations of the resulting SLR range residuals decrease with increasing SLR weight. Even for solution/31, with the smallest SLR weight, improvements of the standard deviation especially for the GLONASS satellites are observed. The radial bias between microwave- and SLRderived GPS orbits of about 5 cm does not disappear without "damaging" the combined orbit. By "overweighting" the SLR measurements, the bias decreases to zero, but the resulting orbit is deformed rather than scaled. The observed scale change (Table 4) and the change of the along-track component (Table 5) correspond radially to about 1 cm, only. This indicates, that the microwave observations determine the satellite orbit and scale rather well and that the orbit dynamics is strong enough not to absorb the 5 cmbias.
121
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C. Urschl • G. Beutler. W. Gurtner • U. Hugentobler. S. Schaer
In the second experiment we changed the observation model. Daily SLR range biases were estimated in addition to the orbit parameters already set up in experiment 1. These range biases were found to absorb the main range information. When solving for such parameters, the impact of SLR observations on the combined orbits becomes very small. For GLONASS, orbit differences between the microwave orbit A1 and the combined orbits t32 and (72 of up to 10 cm are nonetheless seen (see Table 7) due to the pass-specific residual patterns that are much larger for the GLONASS than for the GPS satellites (e.g., Figure 5). The orbit overlap accuracy does not change significantly with increasing weight of the SLR observations. But the SLR range residuals show clear improvements in the pass-specific patterns (see Fig. 5) when including SLR observations into the orbit determination process. The 5cm-bias for GPS satellites between microwave- and SLR-determined orbits might be attributed to unidentified errors in the assumed location of the retroreflector arrays with respect to the satellite's center of mass (Appleby and Otsubo, 2004). Thus, we adapted the observation model in the third experiment and estimated only two additional parameters, one constant retroreflector offset for all GPS and one for all GLONASS satellites in radial direction. The estimated offset for the GPS satellites correspond with about 5 cm to the mean SLR range bias derived from the microwave-only orbit (Table 8). Thus, the mean of the resulting range residuals decreases to zero. But there are still significant differences of several centimeters in the orbit positions between the microwave-only orbit solution A1 and the combined solutions B3 and 6'3 for the GLONASS satellites (Table 10), showing the impact of SLR observations. The overlap accuracy of the combined orbits does not change significantly when compared to the microwave orbit. 7 Summary
We have shown that from the technical point of view, combined orbit determination analysis of microwave and SLR tracking data is no problem. Additional range measurements have an impact on a combined microwave-SLR orbit, where the impact depends on the observation weights. Significant biases between the two observation techniques do not allow a rigorous combination for GNSS orbit determination without adapting the observation model. The estimation of additional satellite retroreflector offsets absorbs the well-known 5 cm-bias between the microwave orbits and the SLR measurements. The pass-specific patterns of the SLR range residuals improve with increasing weight of the SLR observations.
In this first study, we analyzed combined orbit solutions of only 41 days, not yet allowing to draw final conclusions. There might still be some deficiencies in the orbit modeling. The small number of SLR measurements (in general not equally distributed over the day) represents a challenge when attempting to improve GNSS orbits by using additional SLR data. A follow-up long-term analysis covering several years will allow more precise conclusions on the impact of SLR observations on orbits determined by microwave measurements. We have seen that the impact of SLR on GLONASS orbits is much larger than on GPS orbits due to the small number of GLONASS microwave tracking sites. A similar situation can be expected for the new GALILEO system in its startup phase. Therefore, the use of SLR measurements to the GALILEO satellites should considerably improve the orbits (as compared to pure microwave orbits). 8 Acknowledgment
This work is supported by the Swiss National Science Foundation. Data used in this study was obtained from the ILRS (Pearlman et al., 2002) and the IGS (Beutler et al., 1999). References
Appleby G, Otsubo T (2000) Comparison of SLR measurements and orbits with GLONASS and GPS microwave orbits. In: Proc. of 12th International Workshop on Laser Ranging, Matera, Italy, November 13-17 Appleby G, Otsubo T (2004) Laser ranging as a precise tool to evaluate GNSS orbital solutions. In: Proc. of 14th International Workshop on Laser Ranging, San Fernando, Spain, June 7-11 Beutler G, Rothacher M, Schaer S, Springer TA, Kouba J, and Neilan RE (1999) The International GPS Service (IGS): An interdisciplinary service in support of Earth sciences. Adv Space Res 23(4): 631-635 Hugentobler U, Schaer S, Fridez P (2005) Bernese GPS Software Version 5.0. Druckerei der Universitfit Bern, Switzerland Pearlman MR, Degnan JJ, and Bosworth JM (2002) The International Laser Ranging Service. Adv Space Res 30(2): 135-143 Rothacher M (2002) Combination of space-geodetic techniques. In: IVS 2002 General Meeting Proceedings: 3343, Tsukuba, Japan, February 4-7 Ray J, Dong D, Altamimi Z (2004) IGS reference frames: status and future improvements. GPS Solutions 8: 251266 Springer T (2000) Modeling and validating orbits and clocks using the Global Positioning System. Geodfitisch-geophysikalische Arbeiten in der Schweiz 60, Schweizerische Geodfitische Kommission Urschl C, Gurtner W, Hugentobler U, Schaer S, Beutler G (2005) Validation of GNSS orbits using SLR observations. Adv Space Res 36(3): 412-417 Zhu SY, Reigber C, Kang Z (1997) Apropos Laser tracking to GPS satellites. J Geod 71:423-431
Chapter 20
BIFROST: Noise properties of GPS time series Sten Bergstrand 1., Hans-Georg Scherneck 1, Martin Lidberg 1'2, Jan M. Johansson 1 ~Onsala Space Observatory, Chalmers University of Technology, SE-439 92 Onsala, Sweden *Now with Leica Geosystems. 2Geodesy Division, Lantm~iteriet, SE-801 82 G~ivle, Sweden
Abstract. The BIFROST project uses GPS to observe the intra-continental deformation of Fennoscandia caused predominantly by Glacial Isostatic Adjustment (GIA). The noise in GPS position time series has been proven correlated, so we investigate a fractal model in order to obtain a parameter that can gauge our network stations true velocity uncertainties and utilize an empirical orthogonal function (EOF) to remove the inherent common mode. We employ a Kaiser window to reduce the power spectrum variance and retain independent power estimates based on the window's main lobe width ratio compared to that of a boxcar. As power spectra lack Gaussian distribution properties, we devise a transform that normalizes the power spectrum and subsequently iterate a fractional power law noise model. We find that the spectral indices for the different velocity components in our network are 0.6 for North, 0.5 for East, and 0.7 for the Vertical. As there is no white noise floor in our power spectra to indicate inevitable system noise it is possible that GPS time series should be sampled more frequently then once per day in order to separate between different uncertainty sources.
uncertainties are attributed to incomplete models that very broadly can be classified as being related to the satellites (e.g. clocks, phase centre variations, orbital errors), the propagation media (e.g. ionosphere, troposphere) or the receiver units (e.g. clocks, multipath, elevation cut-off, phase centre variations, monumentation). In addition to the electromagnetic signal perturbations, geophysical phenomena such as tidal forces, loading related to ocean, ground water and atmospheric pressure as well as surface humidity, climate related variations and local geological settings impose further uncertainties on the observed variables. The purpose of this paper is twofold: firstly, to apply an empirical orthogonal function (EOF) to remove a common mode noise from a regional network; secondly to assess an unbiased parameter that can be used to scale the formal uncertainties of GPS derived velocities.
Keywords. GPS, time series analysis, power-law noise, uncertainty
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1 Introduction The use of Global Navigation Satellite Systems (GNSS) in general and the Global Positioning System (GPS) in particular have been proven important for geophysical studies for more than a decade. Nevertheless, it is fair to say that the uncertainties of GPS observations are less than fully understood, much related to the difficulties in separating the different uncertainty sources. The
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124
S. Bergstrand • H.-G. Scherneck. M. Lidberg. J. M. Johansson
In the Baseline Inferences for Fennoscandian Rebound, Sea level and Tectonics (BIFROST) project, we use GPS to assess the current glacial isostatic adjustment (GIA) that pertains to the unloading of the regional Pleistocene ice sheet in Finland and Sweden. The aseismic GIA process yield station velocities superimposed on the Eurasian tectonic plate motion of order 1 mm/yr horizontally and 10 mm/yr vertically, and we aim at a station velocity determination of +0.1 mm/yr, see Fig. 1. Milne et al. (2001, 2004) fitted the GPS velocity observations to a three-layered viscoelastic earth model response to an ice model by Lambeck et al. (1998) and found an optimal model with 120 km lithospheric thickness and upper and lower mantle viscosities of 8x102o Pas and lxl022 Pas, respectively. However, in Milne et al. (2001) the station velocities in the north eastern part systematically showed larger misfits compared to the model. These stations are known to be strongly affected by snow accumulation on the radomes, as first observed in the network by Jaldehag (1996). The analyst's approach to mitigate the effect of snow on radomes has varied somewhat arbitrarily in the past depending on study theme, and to gauge whether the observation uncertainties allow the misfit or if the ice model has to be reassessed, we need an unbiased estimate of the true station velocity uncertainties. Bergstrand et al. (2005) confirmed the inferred rheological variables by Milne (2001) in an independent baseline analysis that required a scaling of formal GPS uncertainties by a factor no more than 1.5 to fit a root-mean square <0.8 mm/yr for the 630 baselines in the network, including the snow affected stations. In this context, 'baselines' means estimating the velocities from individual site-to-site time series. The baseline method is robust as it eliminates a lot of common mode uncertainties in the horizontal direction related to e.g. satellite orbits, mapping functions and systematic errors at reference stations, but yields results that are hard to compare to other techniques such as tide gauges and precise levelling. Milne et al. (2001) and subsequent articles used a GIPSY network solution thoroughly presented by Johansson et al. (2002), and one conclusion by Bergstrand et al. (2005) was that common mode had not significantly affected the single site inferred viscosity parameters. However, we propose EOF as an alternative to baselines to remove the common
mode, partly because it is more sensitive to vertical variations. The scaling of the formal uncertainties mentioned above largely pertains to the fact that the processes that drive the error sources to some extent are autocorrelated. Langbein and Johnson (1997) applied a maximum likelihood estimator (MLE) to electronic distance measuring (EDM) observations and estimated the frequency-power relation with P~(f) = Po
(1)
where jr is the temporal frequency, Po and j~ are normalizing constants, and x is the power law or spectral index (Mandelbrot and Van Ness, 1968). This application to geodetic time series was subsequently adapted to GPS by a number of authors, e.g. Zhang et al. (1997), Mao et al. (1999), Williams et al. (2004) and Beavan (2005). Classic examples of different noise behaviour assign integer values to ~c and include white (K-=0), flicker (r=l), and random walk noise (~=2). Integer Ms are convenient in many ways but are often chosen with little forethought, as there is no physical reason why ~c should be restricted to these values. As pointed out by several authors (e.g. Mandelbrot (1983), Turcotte (1997), Williams (2003), and Williams et al. (2004)), a wide range of observable geophysical phenomena are also represented by non-integer, or fractal ~c. Geophysical phenomena generally have more power at low frequencies, i.e. there is a correlation in time between different observations. Processes in the ranges -I0, blueness to x<0 and whitening to ic approaching 0. In our case, the validity of the power law model is limited at high frequencies by the sampling theorem
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and at low frequencies by the rebound signal that is assumed to be smooth and deterministic. Inevitably, we have to remove a biased signal from the time series before we can analyze the noise, and the lowest frequencies will be affected. With a set of 19 months long time series from a regional network in southern California, Zhang et al. (1997) were unable to distinguish between a pure
Chapter 20 • BIFROST:Noise Properties of GPS Time Series
fractal noise model with index 0.4 and a combination of white and flicker noise. Mao et al. (1999) used three-year time series from a global network and concluded that GPS time series could be estimated with a combination of white and flicker noise but didn't preclude the possibility of fractal white or fractal random walk noise. Williams et al. (2004) investigated a large number of global and regional time series with an average length of 3.4 years and found that the noise in global time series was best characterized with a combination of white and flicker noise, but that regional analysis should solve for spectral index along with the noise amplitudes in the MLE algorithm. The above authors (partially save Zhang et al. (1997)) advocated an a priori assumption that white noise is a significant contributor in GPS daily time series although Beavan (2005, section 5.1) noticed that the white noise component of the PositioNZ network time series often tended towards zero. Williams (2003) and Williams et al. (2004) commented upon the large computational burden involved in the MLE approach, and in the discussion Williams (2003) suggested a simplified route to obtain rate uncertainty estimates. We suggest a new, simple method to estimate the rate uncertainties. In the long run, this will enable us to efficiently gauge the true uncertainties of station velocities and also identify stations with abnormal behaviour. A much simplified route is viable when time series are "long enough" and we investigate whether our time series fulfil this apparently loose
ARJE BORA HASS JOEN JONK KARL KEVO KIRU KIVE KUUS LEKS LOVO MART METS NORR OLKI ONSA OSKA OSTE OULU OVER RIGA ROMU SKEL SODA SUND SVEG TRO1 TUOR
criterion. We investigate the influence of two different GPS processing software and strategies on the estimation of I¢ in the regional BIFROST network, using a previously published data set (Johansson et al., 2002, henceforth referred to as JGR02) and a recently processed solution of data from 1996 to 2004 (Lidberg et al., 2005; henceforth JG05).
2 Data and Processing The data in this study are acquired with the Swedish and Finnish geodetic reference networks SWEPOS and FinnRef and the adjacent IGS stations in Riga and Tromso. The sites have generally been chosen with a free sky view at angles >10 ° and often lower; antenna monuments are, with few exceptions temperature regulated to suppress thermally related vertical variations and mounted on unweathered crystalline bedrock. Data with 30 s sampling interval have been used to obtain daily position estimates and these position time series constitute the data set used for interpretation in this study. The primary reason for reprocessing has been optimal station velocity estimation, and the variables in the different solutions have been chosen independently. Thorough presentations of the networks, processing strategies etc. have been given by Scherneck et al. (2002), Johansson et al. (2002), and Lidberg et al. (2005). Previous publications have applied different editing strategies to the data before interpreting the results; the current analysis is applied to the original data time series. Data availability for the series, significant gaps, etc are shown in Fig. 2.
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We aim to characterize the noise in the different solutions and ran an editing loop that removed outliers larger than 5cy in the time series. This removal is conservative compared to the 3c~ in the edited solution of Johansson et al. (2002), and removed less than 1% from the total data set, compared to e.g. the 1-4% in Nikolaidis (2002) and 4.8% in Beavan (2005). We also note that the stations that are most affected by snow still use close to every observation, in contrast to earlier studies (e.g. Scherneck et al., 2002). When data gaps are present, Langbein and Johnson (1997) suggested adding white noise to their EDM interpolate, and was able to match a value that corresponded to
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3.1 Empirical orthogonal function (EOF) We assume that the noise in general is stationary but with a considerable superimposed seasonal variation. Before the EOF evaluation, we remove a rate and seasonal component with 4 sinusoidal harmonics • from the data through least squares, weighted with the formal uncertainty of each observation. The fit is made to the model:
The common mode reduced G' is then used for the rest of the evaluation.
3.2 Spectral fit Mao et al. (1999) used a boxcar window on the time series autocovariance function, and judged from shown power spectra, so did Zhang et al. (1997), Calais (1999) and Williams et al. (2004). On average our time series are longer, and we apply a 4year Kaiser taper window with shape parameter 37r to the time series' autocovariance functions before evaluating the power spectrum with an ordinary fast Fourier transform (FFT). The Kaiser window yields a power spectrum with less variance compared to that of the boxcar at the expense of spectral resolution (Harris, 1978). In order to get uncorrelated estimates of the frequency powers, we pick independent frequencies co with an interval based on the main lobe width ratio ,o of the Kaiser and implicit full length boxcar windows in the Fourier domain. We then notice from Fig. 4 that our power spectra exhibit little or generally no sign of a white noise floor at high frequencies before we make a ic fit to the spectrum. The starting solution is taken from a
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Chapter 20 • BIFROST:Noise Properties of GPS Time Series
simple linear fit of P versus co in the log-log diagram N
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z¢=0.5 (though JGR02 east appear slightly whiter before the EOF removal). This difference could possibly be attributed to the geometry of the global net. The determination of the spectral index for the Up component isn't affected much by the EOF for JGR02. Strong candidates to the explanation are the large data gaps in the Finnish time series, but further investigation is needed to outrule other possible causes. For JG05 whose time series are more complete than JGR02, the spectral index of the vertical component was reduced from 0.8 to 0.7. However, for the extreme cases of high spectral index the EOF failed to whiten the noise. This indicates that this noise is site specific and not a fundamental part of the system. The obtained degrees of freedom v for the spectral fit of the two solutions were larger than 6 for JGR02 and larger than 8 for JG05. We feel that this is representative of an adequate trade-off between spectral resolution and variance in our power spectra.
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The results of the EOF removal and spectral index estimation are shown in Fig. 5. Except the east components of JGR02 and the vertical of JG05, the EOF doesn't significantly change the spectral index estimate but certainly aids in the accuracy of the determination for the horizontal components' spectral indices. North components are adequately described with z¢=0.6, and East components with
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127
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S. Bergstrand • H.-G. Scherneck. M. Lidberg. J. M. Johansson 5
Discussion
JGR02 and JG05 have an overlapping period from 1996 to 2000, and we don't consider this time span long enough to exclude data outside the period. A direct comparison between the different time series' results in Fig. 5 may therefore result in conclusions of little significance, especially as the majority of Finnish stations lack data in the early years. It is possible that higher order harmonics would improve the fit to the time series and whiten the time series in Fig. 3. However, for stations less affected then KUUS, the risk of fitting nonparsimonious parameters is just as evident. With the high conformity of the network as well as constraints provided by independent observations and model fit to our station velocities, we feel that the chosen parameterization of the linear fit is representative of the factors that are considered important for the network as a whole. Compared to other methods, e.g. baselines (cf. Bergstrand, 2005) the EOF method is a tool to reduce the common mode noise from a set of GPS time series that requires a small amount of computational code. Another asset is that we find more realistic observation uncertainties on individual stations to constrain the GIA earth and ice models. Results are also more easily compared to those of other independent techniques. Langbein and Johnson (1997) used an 1100 sample Hann window on their EDM series and noted that the removal of a secular trend from data removes significant amounts of energy from the low frequency part of the spectra and that ~c estimates therefore are biased low. This could be complemented with a reasoning of leakage between spectral bins; through our subsampling strategy based on the implicit main lobe width relation to that of the full length boxcar, we only use the straight part of the spectrum and are not affected by the trend removal. This approach reduces the redundancy of the spectral fit slightly, but enhances the structure of the spectral power relation and still leaves enough degrees of freedom to fit fairly complicated models if desired. We also sacrifice the very low frequencies that could be resolved with a rectangular window as we consider these biased after the secular motion reduction in equation (3), anyway.
It appears from Fig. 5 that the devised power fit has little influence on the inferred parameters. However, we investigated the crude first order linear regression power fit for the two solutions (Fig. 6) and found offsets to the ideal that are equivalent to the observed shift in Fig. 5. We also notice that for ~c>l.0 in JGR02 and ~c>1.2 in JG05, the crude fit started to diverge but that preliminary tests on the transformed fit for 2500 sample long synthetic data series up to x=2.0 show excellent results. We are thus confident that the obtained results are representative of the time series actual noise content for the chosen model. An advantage of the scheme is that it is reasonably fast. The first crude fits to the power spectra for the whole network are obtained within 5 minutes and with the relaxed iteration the final results are obtained in less than 30 minutes on an ordinary PC without optimizing the underlying MATLAB script. We are thus provided with a tool that makes a comparison between e.g. different outlier removal strategies and their impact on velocity uncertainties feasible. Davis et al. (2003) asked whether the estimate of velocity from GPS is fundamentally limited by one or more error sources and if so, what these error 1.4
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Chapter 20 • BIFROST:Noise Properties of GPS Time Series
sources are. Zhang et al. (1997) could not distinguish between an a priori chosen model with a white noise floor and one where ~c was estimated along with the noise amplitudes. Mao et al. (1999) found a better fit with an a priori chosen model then with estimated ~c, but at the same time failed to fit three out of ten synthetic runs with a white and flicker noise combination (Mao et al., 1999, Table 4). Williams et al. (2004) used a larger data set and also advocated a white noise floor in the power spectra and searched for a dominant error source in the time series. One parameter of the user segment that has received considerable attention in geodetic time series evaluation is the monument stability (Langbein and Johnson (1997), Johnson and Agnew (2000), Williams et al. (2004), Beavan (2005)). Combrinck (2000) gave an introduction to space geodetic monumentation and we adhere to his holistic approach, i.e. that monument stability is more than a strictly geodetic issue. Our stations are essentially invariant to the physical parameters listed by Moore (2004, section 2.3, online) and there is no conspicuous reason to assume dominant monument noise in our network. We are primarily interested in the true station velocity uncertainties and content with parameters that allow us to scale the formal uncertainties of the observations (e.g. as suggested by Williams, 2003). In BIFROST, the (easily identified!) winter time snow coverage of the northern stations' antennas reddens the noise. Should we be able to completely remove the snow e.g. with hot air fans, the noise for these stations would probably whiten and our velocity estimate uncertainties hence be reduced. Although of minor importance for the GIA evaluation, we notice that we are generally unable to observe any white noise floor in the higher end of our power spectra. Beavan (2005, section 5.1) used two different MLE packages and with the Williams et al. (2004) package also noticed a recurring absence of white noise in the time series from the PositioNZ network. This observation could possibly be used for evaluation in areas where shorter term movements are targeted and monument stability in not-well-consolidated sediment might be an impediment to the observations, e.g. the Southern California Integrated Geodetic Network (SCIGN), cf. Williams et al. (2004) and Beavan (2005). Given the complexity of the error sources, we question whether an a priori assumption of white noise
properly addresses the noise process in GPS time series before the contributions from individual sources have been duly separated. As we don't observe any white floor in our power spectra, there is evidence that sampling should be more frequent than once per day in order to get smaller error in parameter uncertainties. Given the complexity of the overall uncertainty, we feel that this is a valid result. The next higher order candidate to address could be a semidiurnal tide related signal resolved with a bihourly time series. However, with daily sampling interval strategies, the questions by Davis et al. (2003) will probably remain unanswered. 6
Conclusion
We investigated two different GPS time series ensembles from the BIFROST network and used an EOF approach to remove the common mode noise. Our time series have reached a length where it is useful to apply a tapering window to the autocovariance functions before we estimate the power spectra using FFT. Power law models appear to accurately represent the noise process in the BIFROST time series. The spectral indices are in the range 0.5--0.7 for the North, East and Up components and determined with excellent precision in the horizontal. We are able to determine the spectral power relationship with a simple least squares analysis in the log-log diagram by using uncorrelated spectral bins, but prefer the devised normalizing transformation since it performs better for a wider range of power spectral indices and yields more accurate results. Winter-time snow coverage of antennas still affects some of our position time series, but we now have an unbiased parameter that can be used to scale the individual stations formal velocity uncertainties and still use more than 99% of the observed data. There is little evidence for a white noise floor in our power spectrum evaluations, and this indicates that a higher sampling frequency than once per day will get smaller errors in parameter uncertainties. This also leads us to the conclusion that a priori assumptions of noise structure may be misguiding in a search for dominant error sources. Acknowledgments. The authors would like to thank Simon Williams for thorough and constructive reviews of this manuscript as well as an earlier version, and John Beavan for the review of the early version. We also appreciate the effort put in by the SWEPOS staff and our colleagues at the Finnish Geodetic Institute for running and maintaining the networks.
129
130
S. Bergstrand • H.-G. Scherneck. M. Lidberg. J. M. Johansson
References Altamimi Z., P. Sillard and C. Boucher (2002). ITRF 2000: A New Release of the International Terrestrial Reference Frame for earth Science Applications. J. Geophys. Res., 107(B10), 2214, doi:10.1029/2001JB000561. Beavan J. (2005). Noise properties of continuous GPS data from concrete pillar geodetic monuments in New Zealand and comparison with data from U.S. deep drilled monuments, J. Geophys. Res., 110, B08410, doi: 10.1029/2005JB003642. Bergstrand S., H.-G. Scherneck, G. A. Milne and J. M. Johansson (2005). Upper mantle viscosity from continuous GPS baselines in Fennoscandia. J. Geodyn., 39, 91--109. Calais E. (1999). Continuous GPS measurements across the western alps, 1996-1998, Geophys. J. Int., 138,221--230. Combrinck L. (2000). Local Surveys of VLBI Telescopes. In: Int. VLBI Service for Geodesy and Astrometry 2000 Gen. Meeting Proc., N. R. Vandenberg and K. D. Bayer (eds.), NASA/CP-2000-209893. Davis J. L., R. A. Bennett, and B. R Wernicke (2003). Assessment of GPS velocity accuracy for the Basin and Range Geodetic Network (BARGEN), Geophys. Res. Lett., 30(7), 1411, doi: 10.1029/2003GL016961. Ekman M. (1996). A consistent map of the postglacial uplift of Fennoscandia, Terra Nova, 8, 158--165. Harris, F. J. (1978). On the use of windows for harmonic analysis with the discrete Fourier transform, Proc. IEEE, vol. 66, 51-83, Jaldehag R.T.K., J. M. Johansson, J. L. Davis, P. E16segui (1996). Geodesy using the Swedish Permanent GPS Network: Effects of snow accumulation on estimates of site positions. Geophys. Res. Lett., 26, 1601-- 1604. Jenkins G. M.and D. G. Watts (1968). Spectral analysis and its applications, Holden-Day, San Francisco. Johansson J. M., J. L. Davis, H. -G. Schemeck, G. A. Milne, M. Vermeer, J. X. Mitrovica, R. A. Bennett, B. Jonsson, G. Elgered, P. E16segui, H. Koivula, M. Poutanen, B. O. ROnn~ing and I. I. Shapiro (2002). Continuous GPS measurements of postglacial adjustment in Fennoscandia: 1. Geodetic results. J. Geophys. Res., 107(B8), 2157, doi: 10.1029/2001JB000400. Johnson H. and D. C. Agnew (2000). Correlated noise in geodetic time series, U.S. Geol. Surv. Final Tech. Rep.,
FTR-1434-HQ-97-GR-03155. Lambeck K., C. Smither and P. Johnston (1998). Sea-level change, glacial rebound and mantle viscosity for northern Europe, Geophys. J. Int., 134(1), doi: 10.1046/j. 1365-
246x.1998.00541.x. Langbein J. O. and H. Johnson (1997). Correlated errors in geodetic time series: Implications for time dependent deformation. J. Geophys. Res., 102, 591--603. Lidberg M., J. M. Johansson, H.-G. Scherneck and J. Davis (2005). An improved and extended GPS derived velocity field of the postglacial adjustment in Fennoscandia, submitted. Mandelbrot B. (1983). The Fractal Geometry of Nature, W.H. Freeman, New York. Mandelbrot B. and J. Van Ness (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev, 1O, 422--439. Milne G. A., J. L. Davis, J. X. Mitrovica, H. -G. Scherneck, J. M. Johansson and M. Vermeer (2001). Space Geodetic constraints on glacial isostatic adjustment in Fennoscandia. Science, 291,2381--2385. Milne G. A., J. X. Mitrovica, H. -G. Schemeck, J. L. Davis, J. M. Johansson, H. Koivula and M. Vermeer (2004). Continuous GPS measurements of postglacial adjustment in Fennoscandia: 2. Modeling results. J. Geophys. Res., 109, B02412, doi:10.1029/2003JB002619. Moore A. W. (2004). IGS Site Guidelines,
http.'//igscb.jpl.nasa.gov/network/guidelines/guidelines.htm 1, cited: 2005-11-01. Nelder J. A. and R. Mead (1965). A simplex Method for Function Minimization, Comput. J., 7, 308--313. Segall P. and J. L. Davis (1997). GPS applications for geodynamics and earthquake studies. Ann. Rev. Earth Planet. Sci., 25, 301--336. Schemeck H.-G., J. M. Johansson, G. Elgered, J. L. Davis, B. Jonsson, G. Hedling, H. Koivula, M. Ollikainen, M. Poutanen, M. Vermeer, J. X. Mitrovica and G. A. Milne (2002). Observing the Three-Dimensional Deformation of Fennoscandia. In: Ice Sheets, Sea Level and the Dynamic Earth. J. X Mitrovica. and B. L. A. Vermeersen (eds.), Geodynamics Series vol. 29, AGU Washington, D.C. Turcotte D. L. (1997). Fractals and Chaos in Geology and Geophysics, 2nd ed., Cambridge University Press, Cambridge. Williams S. D. P. (2003). The effect of coloured noise on the uncertainties of rates estimated from geodetic time series. J. Geodesy, 76, 483--494. Williams S. D. P., Y. Bock, P. Fang, P. Jamason, R. M. Nikolaidis, L. Prawirodirdjo, M. Miller, and D. J. Johnson (2004). Error analysis of continuous GPS position time series. J. Geophys. Res., 109, B03412, doi: 10.1029/2003JB002741.
Chapter 21
Discrete Crossover Analysis Wolfgang Bosch Deutsches Geodfitisches Forschungsinstitut (DGFI), Marstallplatz 8, 80539 Mfinchen, Germany
Abstract. Crossover analysis is performed to improve observations on intersecting profiles. The most prominent application are sea surface heights observed by satellite altimetry. We discuss different versions of a discrete crossover analysis, which do not apply an analytic function of the error model (like polynomials, splines, or Fourier series), but estimate the errors at crossover locations only. In order to achieve for the error a certain degree of smoothness we suggest a least squares approach, minimizing both, crossover differences and consecutive differences. The weighting scheme can be modified to balance the influence of crossovers and consecutive differences. The normal equations are large but sparse and can be solved by iterative algorithms. The approach is flexible and can be applied to global as well as regional crossover analysis and has been demonstrated for multi-mission crossover analysis.
Keywords. crossover differences, least squares, satellite altimetry
1 Introduction Crossover analysis is known as a powerful approach to estimate errors and improve observations along intersecting profiles. It takes advantage of the redundancy given by the two (more or less) independent observations taken at the intersection (the crossover) of the two profiles. Gravity measurements along ship routes can be analysed in this way. The most prominent example, however, are the sea surface height observations of satellite altimetry taken along the intersecting satellite ground tracks. The application of satellite altimetry will be often referred to in order to illustrate the theoretical concept developed below. In general the error of profiled observations is described by an error function modelling the error as a function of time or profile length. Low degree polynomials are used, for example, for short profiles or ground tracks. For extended profiles or sequential ground tracks, as they appear e.g. in satellite
altimetry, observation errors have been modelled by splines or Fourier series. As soon as the parameters of such models are estimated the radial error can be computed by evaluating the error function at any point of the profile. In general, functional error models impose some assumption about the error characteristic. A Fourier series, for example, pre-defines the frequencies that are supposed to explain the dominant error and estimate amplitudes and phases for these frequenciess. For other frequenciess Fourier series are 'blind'. In the present investigation, we do not predefine any functional error model, but consider exclusively the (radial) errors xi and xj for the two observations (the sea surface heights) taken at the crossing of two intersecting profiles (ground tracks). These consideration leads to a discrete form of crossover adjustment subsequently referred to as DCA. The two observations are expected to be identical, their differences reflect radial errors and are subject to a minimization. Minimization of crossover differences alone is the most trivial adjustment procedure - leading, however, as shown below to discontinuous results. In order to achieve a certain degree of smoothness for the radial errors we consider in addition - as a substitution for any analytical model - the consecutive differences between radial errors. The weighted minimization of consecutive differences was first introduced by Cloutier (1983). He used the crossover differences as constraint and did not account for observation errors. We therefore extend the method in order to simultaneously minimize crossover and consecutive differences. It will be shown that the method is flexible and can account for prior knowledge of observation errors and the statistical property of radial errors. First a few remarks are given to crossover differences. Then, the basic concept of discrete error modelling is explained. The notation is used to shortly present Cloutiers method. Section 5 describes the suggested approach to minimize both, consecutive and crossover differences. Finally, numerical aspects of the DCA are discussed.
132
W. Bosch
2 Crossover Differences
X'--(Xl,X2,...,X2n )
The crossing of two intersecting profiles is called a crossover. In satellite altimetry the intersecting profiles are the ground tracks o f the satellite. Ground tracks are the orthogonal projection of the satellite orbit to a mean Earth ellipsoid. At a crossover the height is given twice, once by the observations of the ascending ground track and the second time by the observation along the descending ground track. In general, there is no direct observation at the crossover and a least squares interpolation procedure is applied to obtain 'observations' at the intersection (see Figure 1). Low degree polynomials can be used, for example, to interpolate the heights along the observed profiles. If the crossover location is surrounded by a sufficient number of observation points (e.g. six) and the polynomial degree remains small (e.g. three) there is no risk of oscillating polynomials.
(2)
then the linear relationship between 'observed' crossover differences and radial error components is d = Ax
(3)
with the (n×2n) coefficient matrix 1
0
...
0
1
-1
0 .
"i"
• ..
0
1
0
A --
.
.
-1
0
0 .
(4)
-1
In each row, matrix A has only two non-vanishing elements, namely '+ 1' and '- 1 '. Moreover, every column of A has only a single non-vanishing element. It follows A A ~ = 2I
(5)
where I is the identy matrix. In addition A~A = I - P
(6)
where P is a permutation matrix with a single nonvanishing element in every column and row. Permutation matrices are orthogonal pip = pp1=
I
which implies p-1 = pI Fig. 1 Observations at intersecting profiles are to be interpolated to the crossover location (red)
The difference of the two interpolated 'observations' is called crossover difference. The least squares interpolation is used to associate an error estimate to the crossover differences.
Equation (3) is an underdetermined system, which has no unique solution. If the number of unknowns in x is larger than the number of observations in d a minimum norm solution is given by A
x--A+d with the pseudo inverse A + -- A t ( A A t ) -1
(8)
With equations (7), (8), and (5) it follows
3 Discrete Crossover Analysis (DCA) For a set of n crossover events the observed crossover differences A d k, k = 1 .... n shall be compiled to the vector dt - ( / X d l , / X d 2 , . . . , /Xd~)
(7)
(1)
For every crossover observation we then consider errors for both, the ascending and the descending ground track. Thus, there are 2n radial error components xi(t ), i - 1, ... 2n which shall be sequentially ordered in time such that ti < ts+l. If the error components are compiled in the vector
^ 1 x -- - A ~ d (9) 2 Thus, the minimum norm solution consists in equally distributing crossover differences to the radial error components of the two intersecting ground tracks. In general, the computation of crossover differences is accomplished by error estimates, derived from the least squares interpolation procedure. These error estimates can be used to weight the crossover differences. Assuming uncorrelated crossover differences the weights can be compiled into the diagonal matrix
Chapter 21 • Discrete Crossover Analysis
0
...
above. For the consecutive differences the weights
0
"°"
0
1~ 2
"''
0
•
"'°
i
0
...
o __
w
0
which can be used to make the linear system (3) homogeneous by left multiplying with W d1/2 l/2AAtW
**d
/2
__
2W d
The solution (9) does not change, however, because I1/2)+ (AW
Id
1/2d_
The minimum norm solution of system (3) implies that consecutive error components are adjusted independent of each other such that the time series of the errors may exhibit significant offsets from one error to the next. This property can be hardly accepted, because the actual orbit of a satellite obeys a certain degree of smoothness: the gravity field at satellite height is continuous, an analytic function and the satellite is always moving at some distance to the attracting masses of the Earth. As the discrete crossover analysis abandons any functional error model the smoothness of the errors has to be introduced by some other means. The following section describes a method to ensure smoothness by minimizing the differences between consecutive errors.
4 The method of Cloutier Already in 1983 Cloutier introduced an approach to minimize the variations of consecutive e r r o r s - under the constraint that the crossover equation (1) is fulfilled. The method is summarized with the notation used here - as a foundation for the extensions to be described in section 5. With n crossover events there are 2n error components and 2n-1 differences between consecutive errors. The latter can be expressed by the matrix-vector product D x where 1
0
-1
1
(11)
w~
0
-1
......
0
0
--.
0
are introduced in order to impose smaller variations between error components close in time compared to error components separated by a larger period• Weights can be derived from an empirical auto-covariance function o r - as a first guess - may be set, for example, by the covariance function ~.
w,-
2 C0.5
~-.
w,(At,)-
t2
2
Co.5 + A
where the constant c0. 5 defines the 'correlation length', the time difference where the covariance function is decreased to the value 0.5-wi(0 ). Using these notations, the cost function, introduced by Cloutier, may be written as m i n 21 IDx!-',~v - - rain -1x D ' ~:A,, d=Ax 2
'W
Dx
(12)
As shown in detail in the appendix Cloutiers approach leads to a rank deficient system• A single additional constraint, e.g. a linear combination of the error components k/x - - c is necessary and sufficient to regularize the system• The standard procedure for least squares problems leads - as shown in the a p p e n d i x - to the solution x -- Q-~AZ(AQ-~AZ)-I d
(13)
with Q - T + kk I
(14)
and Wm
-w~
0
.-•
-wi T
m
w~ + w 2
2
-w
0 o
".
.
0
0
--W 2
W2 @1473
".
•
",
•
"•
--WH
0
...
0
-w,
w,
(10)
(15)
is a matrix with 2n-1 rows and 2n columns and x is the vector of error components already introduced
Equation (13) is the solution for the 2n radial errors at the crossover location. Note, left multiplication of (13) with A immediately shows that indeed the crossover equation (3) is fullfiled.
U
m
0
.-.
0
......
0
1
-1
0
0
1
-1
133
134
W. Bosch
5 Minimizing consecutive and crossover differences
f~ -- x~M~WM x _ 2 x M ~ W b + b 1 W b + 2kk~x
Cloutier's method introduces the crossover equations (3) as a constraint. This, however, appears not appropriate and too restrictive: crossover differences are derived from observed quantities which are affected by observation errors. Therefore crossover differences should be treated as observations and introduced in the crossover analysis as quantities that get a residual in order to account for errors and to make the overall system more consistent. Consequently, the following section develops a modification to the method of Cloutier. It essentially consist in minimizing both, consecutive differences and the crossover differences. In contrast to the cost function (12) the following minimum principle shall be applied m i n l M x -- b 2w
(16)
where
with the necessary conditions for a minimum leads to the normal equation system MIWM
k
k ~
0
The coefficient matrix
N2 --
MIWM kI
k 0
is regular and has the inverse N~ ~ -
Qj1- a~ss' OLS !
o
with Q2 - M I W M + k k l the summation vector
1
1]
and the constant is composed of matrix D, equation (10), and matrix A, equation (4). M is now a matrix with 3n-1 rows and 2n columns. The vector
o~ -- ( k ' s ) - ' The product M I W M - D I W D + A/Wd A
bI:l
T + AIWo A
consist of a null-vector of length 2n-l, which may be considered as pseudo-observations of the consecutive differences, followed by the vector d which keeps, compare equation (1), all crossover differences. The weight matrix
w~
0
W-
0
w.
compiles the diagonal weight matrices W v and W d for the consecutive differences and the crossover differences respectively. Correlation between consecutive and crossover differences are not considered. Matrix M has - similar as in Cloutiers approach - a rank defect of one: the sum of all columns results in a null vector such that one of the columns can be represented as a linear combination of the other columns. In order to overcome the rank defect, a single constraint, e.g. k'x -- 0
can be introduced. The Lagrange function
(17)
is composed of the tri-diagonal part T, already defined by equation (15) above and the sparse matrix product A'WjA which may be d e c o m p o s e d - similar to equation (6) - to a diagonal part and a scaled permutation matrix. Elaborating the explicit solution for the 2n radial error components, and considering that s ' M = 0 and M W ' b = A'Wdd one obtains x -- Q 2 1 A I W j d
(18)
with Q2 - T + A I W d A + k k I
(19)
6 Numerical Aspects The system, equations (18) and (19) can become very large, because the size of the normal equation is twice the number of crossovers. In Bosch (2005) the DCA is used to perform a common adjustment
Chapter 21
of all single- and dual-satellite crossovers between 2-4 contemporary altimeter missions. Up to 100000 nearly simultaneous crossover events are considered for a ten-day period with three day overlap to neighbouring periods. Thus the systems to be solved have about 200000 unknowns. A strict solution for systems of this size is difficult, if not impossible. The only way to solve such systems is to apply an iterative procedure such as the conjugate gradient projection (CGP) algorithm as described, for example, by Golub & van Loan (1983). This iterative solution algorithm is applied here - without any preconditioning. Fortunately, the matrix Q of the normal equations (18) to be solved has a rather simple structure. The first component of the matrix Q (see equation (19)) is a weighted Gauss transform of matrix D and has tri-diagonal structure. The second component of Q, the Gauss transform of matrix A, is a sparse matrix with a non-zero diagonal and as many non-zero offdiagonal elements as there are crossovers. If the constraint (17) is taken to fix a single unknown, then the product kk' contributes only to the diagonal. This way the structure of Q can be kept tri-diagonal plus sparse as shown in Figure 2. The CGP algorithm requires to perform repeated products between matrix Q and a vector. The computations can be considerably speed up if the specific structure of Q is taken into account and the product of the tri-diagonal matrix and the remaining sparse matrix is treated separately. We therefore modified the CGP algorithm accordingly with the effect that the iterative solution becomes very fast (systems with 200000 unknowns were solved within a few second computation time on a state-of-the-art PC) and has very low storage requirements.
N.
4000
•
eUe Q
00•
000
•
Appendix According to section 4 Cloutiers approach is based on the cost function,
1] Dx12v
min d=Ax 2
min -1x 'D' W v D X 0=Ax 2
-
(12)
minimizing consecutive differences under the constraint that the crossover differences are fulfilled. The structure of D, equation 10, is as simple as the structure of A, equation 4" in every row there are only two non-vanishing elements, namely +1 and -1. It follows that the product T = D'WvD is a symmetric, tri-didagonal matrix
T
w~
-w~
0
--W 1
W 1 -~ W 2
--W 2
".
0
--W 2
W 2 ---~W 3
".
•
o
...
0
m
0
...
,
-%
0
0 -w~ % (15)
with size 2nx2n. With the Lagrange function -- l x ' T x + A ' ( A x - d) --+ min 2 a vector A with n Lagrange factors is introduced. The two necessary conditions to minimize ~, c~/c~x=0 and c~/c~A=0 lead to the normal equation system [i
Ot]'[A] - [0d]
(20)
This system cannot be solved without further conditions, because it is obvious that T has no full rank: the sum of all rows (columns) is a null-vector and the 2n columns (rows) are not linear independent. The rank of T is, however, 2n-1. Thus, a single additional condition as, for example, k~x--c is necessary and sufficient. The extended Lagrange function is then
•
•00
CrossoverAnalysis
• Discrete
-- -1 x ' T x + A ' ( A x - d) + X o ( k ' x - c) -+ min 2
•
with the additional Lagrange factor )~o and leads to the modified normal equation system
"||, 0@•
•
"ii
Fig. 2 The structure o f the normal equation matrix Q is composed o f a tri-diagonal matrix, and a sparse structure with as many @ d i a g o n a l non-zero elements as there are crossovers.
T
k
Al
x
~ ;0 00 0 ")"° A --
0
(21)
135
136
W. Bosch
The upper left submatrix
A -- - ( A Q - 1 A ' ) - 1 d
NI[ T
(22)
For the remaining solve-for parameters it follows Nix o -- Ao(AQ-~A~)-I d +
is now regular and possesses the inverse
[
such that
o']
X Xo z
with the summation vector s'-[1
X0 -- NI-' (Ao ( A Q - ' A / ) - ' d + Co) Q-1At (AQ-lAl)-I d
1]
1--.
Co
c~s~A~
the constant
Considering again As = 0 and taking into account that N~c0 vanishes it follows )~0 = 0. Finally, we get the solution, already shown in section 4
oL -- (k's) -1 and the matrix.
1..
Q -- T + k k '
(23)
t
I;'] I
x
x o --
0 ,
Co - -
)k 0
References
C
BOSCH, W.: Simultaneous crossover adjustment for
the system (21) is transformed to N~ Ao
A o . xo _ 0
A
To solve the regularized system - first for the Lagrange factors A - we get -1
/
- A o N 1 AoA
-
d-
AoN
-1
1 e,
•
(13)
with Q and T as defined above, by equations (23) and (15) respectively.
With equation (22) and the short hand notations A o --
x -- Q - 1 A t ( A Q - 1 A I ) - I d
•
-1
l
Because As = 0, the coefficient matrix A0N 1 A 0 -1 l -I reduces to A Q A and the product A0N ~ c 0 becomes a null vector• The solution for the Lagrange factors A is therefore
contemporary altimeter mission. ESA Scientific Publications SP 572, ESA ESTEC, 2004 BoscH, W.: Satellite Altimetry: Multi-Mission Cross Calibration. In: Rizos, Ch. et al. (Eds.) Dynamic Planet 2005. l a G Symposia, Vol. 13?, Springer, Berlin (this volume) CLOUTmR, J.R., A Technique for Reducing Low-Frequency, Time-Dependent Errors in network-Type Surveys. J. Geophys. Res., Vol.88 (B 1), 659-663, 1983 GOLUB, G•E. AND C.F. VAN LOAN, Matrix Computation, John Hopkins Press, 1983
Chapter 22
A comparative analysis of uncertainty modelling in G PS data analysis S. Sch6n Engineering Geodesy and Measurement Systems, Graz University of Technology (TUG), Steyrergasse 30, A8010 Graz, Austria H. Kutterer Geodetic Institute, University of Hannover, Nienburger Strasse 1 D-30167 Hannover, Germany
Abstract. A thorough assessment and mathematical treatment of all relevant errors in GPS data processing and analysis are essential for the further use and interpretation of the processing results. In this study two mathematical approaches for the error handling are studied in a comparative way. A probabilistic approach is based on the construction of a fully populated variance-covariance matrix of zero difference phase observations by introducing the uncertainty measures of the respective influence parameters in terms of standard deviations. A deterministic approach interprets these uncertainty measures as error bands and uses formalisms from interval mathematics. Both approaches are applied to a simulated EUREF sub-network. The deterministic approach yields more realistic results, in particular with respect to the dependence of the uncertainty measures on the baseline length.
ters). However, remaining systematic errors may persist. In the following we focus on the uncertainty due to remaining systematic errors in GPS results. Two approaches are studied, a probabilistic and a deterministic one. The common starting point is the wellknown GPS phase observation equation (HofmannWellenhof et al., 2001)
+ri
"}-
ri
or-
v
i
Or- C
(4
'or-
'Jr-
where L~ denotes the metric phase observation, ,o~ the geometric distance between the receiver i and the satellite k. The ionospheric phase advance is denoted by I~, the tropospheric delay by T~, the multipath by M/~, satellite clock errors by 6 ~ , receiver clock errors by 4 , and the velocity of light
Keywords. GPS, systematic errors, correlations, imprecision, interval mathematics
in vacuum by c. The ambiguity parameter is given by N,~ , the wavelength of the carrier wave by 2.
1 Motivation
Eq. (1) is valid for both GPS carrier wave frequencies L 1 and L2 with differences mostly for the ionospheric phase advance and the multipath effects.
In GPS data processing all errors have to be considered which are relevant for the proper interpretation of the obtained results. Various approaches exist for modelling and reducing the arising uncertainty. For the random component advanced weighting schemes can be applied (e.g. Han and Rizos, 1995; Hartinger and Brunner, 1999; Brunner et al. 1999; Wieser, 2002). Fully populated variance covariance matrices can be constructed by means of variance covariance component estimation (Wang et al., 2002; Tiberius and Kenselaar, 2003) or time series analysis (Howind, 2005). For the systematic component different reduction methods are used such as double differencing, linear combination, or model extension (e.g. estimation of tropospheric parame-
Eq. (1) depends on a multitude of parameters like the temperature, air pressure, etc. (influence parameters). As the values of these parameters are imprecise in general, systematic effects remain. In this paper two approaches for the assessment of uncertainty due to these effects are compared and discussed, a probabilistic one and a deterministic one. Both use a forward modelling strategy. The probabilistic one is based on the construction of an additional variance-covariance matrix (vcm) of the zero difference (ZD) phase observations by introducing the influence parameters as random variables (Jfiger and Leinen, 1992; Schwieger, 1996). The deterministic approach interprets these uncertainty measures as error bands and uses formalisms
138
S.Sch6n• H.Kutterer
from interval mathematics (Sch6n 2003, Sch6n and Kutterer 2003, 2005). In the next section both approaches are developed. Then a sub network of 8 EUREF GPS stations is simulated in order to study the behavior of both uncertainty measures. Finally, the obtained results are compared and discussed.
matrices of the partial derivatives are given by F 1 and F 2 . As only the frequency dependent ionospheric parts of these matrices are different this approach leads to high positive correlations between the L1 and L2 ZD observations. The e x t e n d e d v c m of the ZD £ZD is then obtained as
2 Mathematical
£ ~ - £ ~0 + £ v~ .
Concept
2.1 Modelling of influence p a r a m e t e r s
The initial random uncertainty component can be described by the vcm
Mz)z)12zz) l l M r ~-"DD --
,
, 22 M T DD
assuming uncorrelated ZD observations for both frequencies and an elevation depending weighting. In order to describe the systematic component, the impact of the influence parameters on the GPS phase observations is derived by means of a sensitivity analysis. Therefore, Eq. (1) is linearized with respect to the influence parameters. For L1 and L2 phase observations, respectively, this yields a linear relation between differential changes dLk of the phase observations on one side and the partial derivatives contained in the Jacobi matrix F and the differential ds of all considered influence parameters on the other side: dL~ - V ds. (3) The main difference between the probabilistic and deterministic approaches lies in the interpretation of ds and the associated mathematical handling.
- £DD121 £DD.22 with the DD operation described by the matrix M vv. Eq. (6) can be also represented by an initial term and an additional term encountering the uncertainty (covariances) for remaining systematic errors (
-E°z),l 1 --
vcm
/~-" ZD,11
£v
ZD,21
"~
£,
ZD,12 _
FI diag(s)2
F, r
(4)
ZD,22
of the ZD observations describes the uncertainty due to remaining systematic errors. Here, Z' ZD,11 "~'ZD,22
are
the fully populated vcm (in [m2])
,
)
r
-~- ~[] 'DD,11
M DD ( ~-"ZD,22 o -4- '~ ,ZD,22 ) M D Dr
~Z~DD,22 --
"
(7)
0 v -- ~"DD,22 + ~ DD,22
12o~,12 -
12
'o~,12
The ionosphere-free linear combination L3 of the DD L1 and L2 observations can be introduced in a similar way. The associated vcm reads XDD,Ls - MLS12DDM[S
(8)
with M Ls-(
If the differentials of the influence parameters are modelled as random variables, their expectation values are considered as zero. Here, a diagonal vcm is introduced. Hence, the law of variance propagation has to be applied to Eq. (3). When both frequencies L1 and L2 are studied, the fully populated
0
£DD,~ -- MDD £zD,~I + E zD,~ MDD
2.2 Probabilistic a p p r o a c h
\
(6)
e diag [ O-L2
and
MDD~ZO
+
12zv -
£'zD-
Mz)z)12zz),12Mz)z)
DD
MDD]F-"ZD,21MrDD
0
(5)
In order to reduce systematic errors in GPS phase observations double differences (DD) are built as
a,
""
a,
~, / part of L1 DI~ observations
a 2 ""
a2 )
v / part of L2 DI~ observations
and the frequency dependent coefficients f,2 j], a2
.1
al
f2
./~ .if2 .2/] "
Again the vcm can be split into two terms. The second term describes the impact of the probabilistic formulation of additional uncertainty due to remaining systematic errors on the initial n x n vcm
(
,
+ a~X DD,11 nt- 2al
a2~2,DD,12 -~- a ~ £
,
DD,22
)
(9)
_£0
DD,L3 -Jr-~ VDD,L3
Finally, the u × u vcm of the estimated coordinates is obtained as
of the L1 and L2 ZD, respectively, and 12'ZD,12 denotes their covariance matrix. The corresponding
12~ - (ArI2~,LsA) 1
(10)
Chapter 22
assuming a least-squares estimation in the GaussMarkov model (Koch, 1999) with the n x u design matrix A .
•A
Comparative Analysisof Uncertainty Modelling in GPS Data Analysis
tion of Oct 1st 2004. The common reference station of the baselines was POTS which was fixed during the adjustment. The baseline length and ellipsoidal heights of the stations are given in Table 1.
2.3 Deterministic approach o
In the deterministic approach, the uncertainty measures of the influence parameters are interpreted as deterministic error bands and modelled in terms of interval radii s r . These intervals enclose all possi-
o
60 ° ~___~o~ E 1~ E 2o°E
E
.:: NSA (~::-
ble values of the influence parameters. Hence, maximum effects are considered. Methods from interval mathematics (see, e.g., Alefeld and Herzberger, 1983) are applied for the mathematical handling. Here, only the final equation is given. Details on all important derivations can be found in Sch6n (2003) and Sch6n and Kutterer (2003, 2005). The direct application of interval methods to the estimated coordinates leads to the interval radii
2c ~
\
,
Fig. 1: EUREF sub-network with 8 stations
xF
I/A r (E°D,L3
A
°
(I~DD,L3) -1
(11)
Table 1. Overview of baseline lengths and ellipsoidal heights
station
where l e I denotes the element-by-element absolute values of the matrix coefficients. Besides these absolute values which change the matrix structure significantly, there is a formal analogy of Eq. (11) with the measure of external reliability. However, the concepts are different: the minimum detectable biases needed for external reliability are obtained through hypothesis tests, the s r used here are given deterministic values for the forward modelling.
2.4 Relationship between both approaches Both approaches use forward modelling and start with the same linearization. But the interpretation of the uncertainty of the influence parameters is different. In the probabilistic approach (PROB) the total differential ds is associated with the standard deviations % of the influence parameters which leads to a fully populated vcm. In the deterministic approach (DE7) the total differential ds leads to interval radii s r . In order to reflect maximum effects, the relation s r = 3% is set for reasons of comparability in the following.
3 Simulation studies For the comparison of the two approaches a simulation study for an 8 station EUREF sub-network (Fig. 1) was carried out using the satellite distribu-
POTS BOR1 WTZR ONSA GRAZ RIGA GRAS MATE
baseline length [km] with respect to POTS 273.3 360.1 562.9 615.9 870.5 1060.7 1330.6
ellipsoidal height [m] 144.4 124.4 666.0 45.6 538.3 34.7 1.319.3 535.6
Table 2. Overview of the magnitude of the influence parameters
Parameter
sr =
3or
Orbit errors in alongtrack, crosstrack and radial dir. Carrier frequency stability
_+10-12 f0, f0: n o m .
Clock errors
_+10 -12 [S]
Ionosphere -VTEC-representativity -Model constants -Mapping function in terms of the zenith angle Troposphere -Temperature -Pressure -Partial pressure of water vapour -Model constants -Mapping function in terms of the zenith an~le Antenna offsets and variations in a local system Clock stability
+0,1 [m]
frequency
+1% VTEC = +10 -15 [el/m2] +0,5 of last digit +1 ["]
+1 [°C] +1 [mbar] +1 [mbar] +0,5 of last digit +1 ["] +2 [mm] _+5.10-9 f0
139
140
S. Sch6n • H. Kutterer
An 8h session was simulated using a cut-off elevation angle of 15 ° and a sampling rate of 6 min. In addition to the coordinates, tropospheric parameters were estimated with a temporal resolution of 2h using the Niell mapping function. Table 2 shows the applied magnitude of influence parameters. Two scenarios were considered with different vcm of the L3 DD; see Eq. (9). In the scenario S1 only the matrix I:vv,L 3 ° is considered, in the scenario $2
Fig. 3 shows the interval radii obtained by Eq. (11), i.e. the deterministic measure of additional uncertainty due to remaining systematic errors. Obviously, the interval radii increase significantly with the baseline length, especially for the north and height component. Second the interval radii of the height component are greater than those of the horizontal components (up to seven times for short baselines and approximately three times for long baselines). BOR1WTZR ONSAGRAZ
the whole matrix I2DD,L3 is used.
IOt. .N.o.~. h.' ...........
Fig. 2 shows for both scenarios the standard deviations of the north, east and height components of the 7 EUREF stations, respectively. The stations are sorted by increasing baseline length. Three aspects are of interest. First, the standard deviations are rather independent of the baseline length; for numerical values see Table 3. Second, the magnitude of the height component is four to five times the one of the horizontal components. Third, the standard deviation obtained under consideration of additional uncertainty due to remaining systematic errors by the stochastic approach ($2; grey+black) is greater than in the standard case (S1," grey). On average, these changes are very small and certainly below a typically expected magnitude. BOR1WTZR ONSAGRAZ
RIGA
ss°
"°2["Ei"~ ~~ "o
~
~1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
GRAS
MATE
I Iscenar'° 1Z°o L3 "~cenari° 2 ZDD'i3f ~ ~---
~' 10 East.
RIGA
i .... ' ...............
GRAS
' ...........
MATE
'. ..............
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
~. 5I o ............................................................. I-1~ ~ ~ ~ ~ Up
0200
..........
'
400
'
600
800
Baseline length
......
'
1000 [km]
......... '
1200
I 1400
Fig. 3: Interval radii (DET) for the coordinate components of 7 EUREF stations [mm] obtainedfrom Eq (11).
In Fig. 2 and Fig. 3 PROB and DET are depicted with different scales. Their comparison shows that the magnitudes of both uncertainty measures for the horizontal components are similar for short baselines. For the vertical component, the interval radii are more than twice the standard deviation. Due to the strong dependency of the interval radii on the baseline length they dominate the standard deviations for long baselines in all components.
4 Discussion
200
400
600
800
1000 Baseline length [km]
1200
1400
Fig. 2: Comparison of the standard deviations of the coordinate components of 7 EUREF stations for both scenarios [mm]. Grey." PROB for S1, Grey+Black: PROB for $2.
Table 3. Comparison of standard deviations and interval radii for the shortest considered baseline POTS-BOR1 (min. values) and the longest baseline POTS-MATE (max. values) in [mm]
N E U
standard deviation S1($2) min [mm] max[mm] 1.48 (1.49) 1.59 (1.86) 1.12 (1.14) 1.18 (1.30) 7.24 (7.50) 8.21 (8.99)
interval radius S1 min [mm] max [mm] 1.16 12.36 2.17 6.5 14.68 26.15
In order to explain these results the composition of the vcm of the observations is analysed. Here, the consideration of two epochs is sufficient. For the initial ZD observations, four types of correlations are generated which have different impact: (1) Station specific effects (PCV, receiver clocks) lead to strong correlations between observations at one particular station. (2) Satellite specific effects like orbit and satellites clock errors correlate observations to the same satellite over all stations. (3) Tropospheric and ionospheric effects correlate all observations. (4) The additional consideration of an initial, diagonal vcm decorrelates the observations.
Chapter 22 • A Comparative Analysis of Uncertainty Modelling in GPS Data Analysis
Table 4 shows the magnitudes of the contributing matrices in terms of their Frobenius norm. It is obvious and consistent with the analysis of real data that the receiver clock error dominates the uncertainty, cf. fig. 5. In addition, the ZD on L1 and L2 are highly correlated. Table 4. Magnitude of the vcm and contributing parts in log 10 Contribution
total initial vcm orbit satellite clock troposphere ionosphere station specific receiver clock
Frobenius norm of L 1 ZD 1.0166e7 135 4.70e4 0.4 9.89e3 451.1 40.6 1.01 e7
Frobenius norm of L 1 DD 596 527 137 0 118 5 0 0
Frobenius norm of L3 DD 5.4e3 5.4e3 678 0 770 26 0 0
one particular epoch and between successive epochs are obvious. Figures 6 and 7 show the resulting vcm and correlation matrix after double differencing. Since the partial derivations with respect to the receiver and satellite clocks errors are time invariant, their contributions to the vcm are cancelled out, cf. 3 rd column of Table 4. In addition, the time variable parts of, e.g., ionosphere, troposphere and orbit errors are reduced. Hence, the contribution of the initial vcm is increased during the double differencing and finally dominates the resulting vcm. If only the additional vcm is considered (see Eq. (4)), L1 and L2 observations are still highly correlated.
BOR1
•
•
..
,, • , i
L1 epoch1
i
L1 epoch2
L2 epoch1
i L2 epoch2
Fig. 6: Magnitude [logl0] of the elements of the vcm 100
,
BOF~I '~'-..:"-..:'...'-.."- "-.,' epoch1
WTZR ['..,""%"-..."-..,"-., "-.."~
epoch2
Fig. 4: Magnitude [logl0] of the elements of the vcm of L1 ZD
O.SA.'-"-.?"-.X"..."-. GRAZ ;"-..'.. "..%,,,,"-. % %R,GA"-.::"-.."-..'..... . i-"
80
GRAS "" ,'%" ," . . . . ""~'-, MAT E -'%,%, "%, %., BOR1 WTZR ONSA GRAZ RIGA GRAS MATE BOR1 WTZR ONSA GRAZ RIGA GRAS MATE BOR1 WTZR ONSA GRAZ RIGA GRAS MATE
60 40
20
mm
0
",. " , " , ",, " . " . "m "mm " . "~, ~'m
) -20
-40 _
....3", ?-,",~.":"-i"-."-.i ..... I
I
I
L1 epoch1
L1 epoch2
L2 epoch1
Fig. 7: Correlation matrix associated with
epoch1
.~"
%"'-3"~
L2 epoch2
-60
-80 -100
~-"DD'Cf. Eq. (6)
epoch2
Fig. 5: Correlation matrix of L1 ZD associated with
~2ZD,cf.
Eq. (5)
Figures 4 and 5 show the vcm and the corresponding correlation matrix of original L1 ZD observations. The very high correlations at each station at
Finally, the additional application of the L3 ionosphere-free linear combination reduces the magnitudes further; see Fig. 8. This is due to the high correlation between L 1 and L2 DD. The correlation structure of the vcm 12DD,L3 of DD L3 differs only slightly from the one obtained from uncorrelated
141
142
S. Sch6n • H. Kutterer
Z D ( XDD,L3 0 )" Hence, the final standard deviations of the resulting coordinates are close to those obtained without considering remaining systematic errors.
ters. Those studies should be carried out for different baseline lengths, geographic regions, etc. This will also allow to better quantify the magnitudes of uncertainty of the estimated coordinates.
Born % %. "". %'. "% ""- ""-
References
...'-...'-%'-.. "-.. "... ".. ' .= =%==%===% =%= %% =%= =.
-., ,.. .?-.?-.i"..i"-..',?-.."-. _-,-.., "-.. %. "%'%'_--...%..'%.~-,.%...'-%.%,
=%= ==== %% =%= ======% "~=% ==%%==%%==%%==%%== "-.._"%"-. .. -.. %. "; •.% ===.=%=% %% "% ". .. ======== =====~== ==m===
BOR1
_
%=
_
•
% "-% % %% == "%% =% " ~ " . .== %%= . ==%•-..."-...%% "-...'%.. "%."...~ •
%.',...',... "....'.... %..'% I
L3epoch1 L3epoch2 Fig. 8: Correlation matrix associated with ~DD,L3 ' cf. Eq. (9)
5 Conclusions and outlook In this paper, both the probabilistic and deterministic approaches to treat uncertainty caused by remaining systematic errors are compared. For the probabilistic approach two mechanisms are relevant. The uncertainty measure is directly related to and combined with the initial vcm. This initial vcm is increased during the transformation of ZD to L3 DD while the additional vcm is reduced. Considering the intrinsic quadratic error propagation this leads to results (extended vcm) which are very close to those obtained when uncorrelated ZD were used. In general these results are considered to be too optimistic. The deterministic approach provides an adequate and alternative way for the handling of uncertainty caused by remaining systematic errors. Due to its intrinsic linear propagation the uncertainty measures are less optimistic.
The present study indicates the reduced adequacy of a purely probabilistic uncertainty concept. However, the findings have to be verified through the analysis of real GPS data. So future work has to focus on the determination of both, the actual correlation patterns in GPS data as well as the maximum magnitudes (interval radii) of the influence parame-
Alefeld G. and Herzberger J. (1983): Introduction to Interval Computations. Academic Press, New York. Brunner F.K., Hartinger H., and Troyer L. (1999): GPS signal diffraction modelling: the stochastic SIGMA-D model, Journal of Geodesy, 73:259-267. Han S. and Rizos C. (1995) Standardization of the variancecovariance matrix for GPS rapid static positioning. Geomatics Research Australasia 62:37-54 Hartinger H. and Brunner F.K. (1999): Variances of GPS Phase Observations: The SIGMA-e Model. GPS Sol. 2/4:35-43. Hofmann-Wellenhof B., Lichtenegger H., and Collins J. (2001): GPS Theory and Practice, 5 ed, Springer Wien New York. Howind J. (2005): Analyse des stochastischen Modells von Trdgerphasenbeobachtungen. Dissertation. DGK C 584. Mtinchen. J~iger R. and Leinen S. (1992): Spectral Analysis of GPSNetworks and Processing Strategies due to Random and Systematic Errors. In: Defence Mapping Agency and Ohio State University (Eds.) Proceedings of the Sixth International Symposium on Satellite Positioning, Columbus/Ohio (USA), 16-20 March. Vol. (2), p. 530-539. Koch K.-R. (1999): Parameter Estimation and Hypothesis Testing in Linear Models (2d Ed.). Springer, Berlin. Sch6n S. (2003): Analyse und Optimierung geoddtischer Messanordnungen unter besonderer Beriicksichtigung des Intervallansatzes. Dissertation. DGK C 567. Mtinchen. Sch6n S. and Kutterer H. (2003): Imprecision in Geodetic Observations- Case Study GPS Monitoring Network. In: Stiros S. and Pytharouli S. (Eds.): Proceedings of the 1 lth: FIG Inter national Symposium on Deformation Measurements. Geodesy and Geodetic Applications Lab., Patras University. Publication No.2: 471-478. Sch6n S. and Kutterer H. (2005): Realistic uncertainty measures for GPS observations. In: Sanso F. (Ed.): A Window on the Future, Proceedings of the 36 th IAG General Assembly, 23 rd IUGG General Assembly, Sapporo, Japan, 2003, IAG Symposia series, No. 128, 54-59. Schwieger V. (1996): An Approach to Determine Correlations between GPS monitored epochs. Proceedings of the 8th International Symposium on Deformation Measurements, Hong Kong, 17-26 Tiberius C. and Kenselaar F. (2003): Variance Component Estimation and Precise GPS Positioning: Case Study. Journal of Surveying Engineering 129(1 ): 11-18. Wang J., Satirapod C., and Rizos C. (2002): Stochastic assessment of GPS carrier phase measurements for precise static relative positioning, Journal of Geodesy 76:95-104. Wieser A. (2002): Robust and fuzzy techniques for parameter estimation and quality assessment in GPS. Dissertation. Shaker-Verlag, Aachen. Acknowledgement: The first author is recipient of a Feodor Lynen fellowship. He gratefully acknowledges his host Fritz K. Brunner and the Alexander-von-Humboldt-Foundation for their support.
Chapter 23
Looking for systematic error in scale from terrestrial reference frames derived from DORIS data P. Willis Institut G6ographique National, Direction Technique, 2, avenue Pasteur, BP 68, 94160 Saint-Mande, France Jet Propulsion Laboratory, California Institute of Technology, MS 238-600, 4800 Oak Grove Drive, Pasadena CA 91109, USA F.G. Lemoine
Goddard Space Flight Center, Code 697, Greenbelt MD 20771, USA L. Soudarin Collecte Localisation Satellite, parc technologique du canal, 31526 Ramonville Saint-Agne, France
Abstract
1 Introduction
The long-term stability of the scale of Terrestrial Reference Frames is directly linked with station height determination and is critical for several scientific studies, such as global mean sea level rise or ocean circulation with consequences to global warming studies. In recent International Terrestrial Reference Frame (ITRF) solutions, the DORIS technique was not considered able to provide any useful information on scale (derived from VLBI and SLR). We have analyzed three different DORIS time series of coordinates (GSFC, IGN/JPL, LEGOS/CLS) performed independently using different software packages. In the long-term, we show that the DORIS technique, due to its very stable and geographically distributed network, has extremely good stability (<0.1 ppb/yr). In the short-term, the three groups show systematic errors in scale (up to 5 ppb) that could come from their specific analysis strategies. Furthermore, we have investigated on a shorter time period (2004) new results for single-satellite solutions. This analysis is a first step in understanding the systematic errors currently seen in the DORIS-derived scale from different groups.
Several key geophysical studies, such as global change, plate tectonics, or post-glacial rebound cannot be done without a proper geodetic frame. Recently, the International Association of Geodesy (lAG) has started an ambitious project called Global Geodetic Observing System (GGOS) in order to serve as a backbone infrastructure for other geosciences (Rummel et al., 2002). In particular the definition and the maintenance of the Terrestrial Reference Frame (TRF) is a key component of this proposal, (Altamimi et al., 2005). Let us briefly redefine the mathematical relationships relating two TRFs. Equation (1) describes the 14-parameter simplified conformal transformation between those TRFs when the same reference epoch is used for all station coordinates and data sets.
Keywords. DORIS, Terrestrial Reference Frame (TRF), orbit determination
X j = X o + rj + s i X o + R j X o
(1)
j = 20 + ~ + ~jXo + k j X o where X j
(resp. J ( j ) is the vector of station
coordinates (resp. velocities) of the jth data set (in our case, GSFC, IGN/JPL or LEGOS/CLS), and X 0 (resp. J(0) is the vector corresponding to the
144
L.P.Willis.F.6. Lemoine• L.Soudarin
reference data set (ITRF2000, see Altamimi et al., 2002). The transformation parameters and their time derivatives between the jth data set and the reference data set are the following matrices:
Tj (resp. J'j) for translation, R j (resp. /~j) for rotation and the scale factor
Sj (resp.
Sj).
In principle the scale factor is small (up to a few ppb for recent realizations derived from space geodetic techniques). However, a 1 ppb systematic error is equivalent to a constant systematic error of 6.4 mm in station height. A 0.1 ppb systematic error in the scale factor derivative is therefore equivalent to a 0.64 mm/year systematic error in station vertical velocities or any derived geophysical product, such as global sea level rise (Morel and Willis, 2005). These numbers are large relative to the measured rate of sea level change, i.e. 2.7 mm/year from Topex/Poseidon (Cazenave and Nerem, 2004). The DORIS system (Doppler Orbit determination and Radiopositioning Integrated by Satellite) is one of the techniques that contributes to the Global Geodetic Observing System (GGOS) (Rummel et al., 2002; Altamimi et al., 2005). The DORIS contribution of GGOS is organized since 2003 within the International DORIS Service (IDS), (Tavernier et al., 2005). It can be seen on Figure 1 that the DORIS technique relies on a rather dense and geographically well-distributed tracking network, potentially well suited for TRF maintenance (Gerasimenko et al., 2005). In January 2005, 48 DORIS tracking beacons were operating on a permanent basis.
used in this study in Table 1 (data from 2004). All DORIS satellites provided almost continuous DORIS data in 2004, except for TOPEX/Poseidon (T/P), whose DORIS data ended on November 1, 2004. The number of available DORIS satellites is important as the best geodetic results are obtained when using DORIS data from all satellites (Willis et al., 2005a or Tavernier et al., 2005). Table 1. List of satellites carrying a DORIS receiver (2004) Satellite
Launch
SPOT-2 T/P SPOT-4 Jason- 1 ENVISAT SPOT-5
Jan 1990 Aug 1992 July 1998 Dec 2001 Mar 2002 May 2002
Altitude (km) 830 1,330 830 1,330 800 830
In this study, we have considered solutions derived by three different DORIS Analysis Centers (ACs) using three different software packages to investigate possible systematic errors in their realization of the TRF and more specifically of the TRF scale factor. In particular, we would like to assess the accuracy of the derived scale factor and to investigate if some DORIS results could be used in the future to define the scale of the ITRF2004. 2 Terrestrial reference frame scale factor derived from station coordinates 2.1 Description of the c o n s i d e r e d D O R I S w e e k l y T R F solutions
/o
/
.o+
+ + •
• _-
\
i+
-
Fig. 1 DORIS (January 2005).
• Oo .... +,.'
o-.:"
'e,+ • ee • •
•--:+ e"-*e
•
• .
•
.+, ...+
.
00
•
\
\ /
+
0 0 /
permanent tracking network
Since 1990, several remote sensing and altimetric satellites were launched that carry an on-board DORIS receiver. We list the satellites
In this study, we considered solutions from three different ACs, all using the DORIS preprocessed data available at CDDIS: (1) GSC = The Goddard Space Flight Center (GSFC) NASA used the GEODYN software package (version 0407) (Lemoine et al., 1998). (2) IGN = The Institut Geographique National (IGN), France in common with Jet Propulsion Laboratory (JPL) used the Gipsy/Oasis II software package (Webb and Zumberge, 1995; Willis et al., 2005a). (3) LCA = The Laboratoire d'Etudes en
Geophysique (LEGOS)
Localisation
et in
Oceanographie
Spatiale
conjunction with Collecte Satellite (CLS) used the
Chapter 23 •
Looking for Systematic Error in Scale from Terrestrial Reference Frames Derived from DORIS Data
GINS/DYNAMO (version 4.1) software package (Cr6taux et a1.,1998; Soudarin et al., 1999). All groups used their current analysis strategy. No attempt was made to use exactly the same models or processing strategies to minimize possible systematic differences. In particular different gravity fields were used, GGM02C (120x120, including C20-dot, C21-dot and S21dot) for GSFC, GGM01C (120x120) for IGN/JPL and GRIM5-C1 (120x120, truncated at degree 90 for ENVISAT and SPOTs and 75 for TOPEX). The models GGM01C and GGM02C are models derived from the GRACE mission (Tapley et al., 2004), whereas the GRIM5C1 model is a pre-launch model developed in advance of the CHAMP mission (Biancale et al., 2000). In most cases, the use of the recent GRACE-derived gravity field provided enhanced DORIS geodetic results (Willis and Heflin, 2004, Feissel-Vernier et al., 2005). The DORIS data were processed using 7-day arcs for GSFC, 1-day arcs for IGN/JPL and 3.5day arcs for LEGOS/CLS. Atmospheric density models used were MSIS86 for GSFC and DTM94 for IGN/JPL and LEGOS/CLS. Additional drag parameters were estimated every 6-hr (resp. 8-hr) for ENVISAT and SPOT5 (resp. T/P, SPOT2 and SPOT4) by GSFC. Drag parameters were estimated every 6hr for ENVISAT and SPOTs by IGN/JPL and LEGOS/CLS, while estimated every 12-hr for T/P only by LEGOS/CLS, as using longer arcs (3.5-days vs. 30-hours arcs). Center of mass corrections were recomputed by GSFC based on offset and center-of-mass information supplied by the International DORIS Service. The IGN/JPL used the CNES supplied preprocessed data corrections. LEGOS computed its own corrections like GSFC for TOPEX and SPOTs before September 2004 or otherwise used the CNES-supplied data corrections. All ACs estimated tropospheric corrections per satellite pass but IGN/JPL used a more sophisticated approach, adding time-dependant constraints between passes, (Willis et al., 2005a). Atmospheric pressure loading corrections were used by LEGOS/CLS but not by IGN/JPL, potentially leading to small TRF effects, (Tregoning et al., 2005). GSFC does not apply atmospheric loading for the geometric station correction but uses a dynamical correction to the geopotential coefficients to model atmospheric mass variations (Chao and Au, 1991).
GSFC (resp. LEGOS/CLS) used a common a priori data weigh of 0.5 mm/s (resp. 0.4 mm/s) for all DORIS Doppler measurements, while IGN/JPL used 0.4 mm/s for SPOT5 (newer generation) and 0.5 mm/s for all others. GSFC (resp. LEGOS/CLS) preprocessed the DORIS data using a 5-degree (resp. 12-degree) minimum elevation cut-off angle, while IGN/JPL used all available data. Finally, the DORIS data from Jason-1 satellite were not used in this study due to unexpected large sensitivity to radiation leading to erroneous clock accelerations over the South Atlantic Anomaly (SAA), (Willis et al., 2004). Fortunately, Jason- 1 orbit results are less affected, especially when derived using simultaneously GPS, DORIS and Laser data (Luthcke et al., 2003). A correction model is being developed in Toulouse and early results show some improvement in the results (Lemoine and Biancale, in preparation).
2.2 Using multi-satellite solutions As a first step, we have compared weekly station coordinates using all available DORIS data (except Jason-1) from January to December 2004 with a unique reference based on ITRF2000 (Altamimi et al., 2002), on a week-by-week basis and on a AC-by-AC basis using the standard projection and transformation approach, (Sillard and Boucher, 2001). Figure 2 shows that the weekly TRF scale factors are very different from one group to another. The IGN/JPL provides smaller values (typically -3 ppb) while GSFC and LEGOS/CLS provide slightly larger values (typically +5 ppb) but in good agreement for the first 6 months. -e-scale GSFC allsats --=-scale IGN allsats -x--scale LCA allsats 10
o5 O O Ii
0
.......................................................................... - ...................................................................................................................................................
_.o
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o
2004. O0
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2004.50 Year
_
2004.75
2005. O0
Fig. 2 Weekly scale factor determination towards ITRF2000 using multi-satellite SINEX solutions.
145
146
L. R Willis. F. G. Lemoine • L. Soudarin
GSFC (white circle), IGN/JPL (black squares), LEGOS/CLS (crosses). January - December 2004. The IGN/JPL solution also provides a more consistent time series as easily detected by an Allan Variance test over 1 week: 0.168 ppb 2 for IGN/JPL, 0.427 ppb2for LEGOS/CLS and 0.665 ppb 2 for GSFC. This value represents the internal consistency (variance) between consecutive weeks of each individual series. It must be noted that the GSFC and the LEGOS/CLS solutions tend to diverge around September 2004, when LEGOS/CLS changed their strategy to use the phase center corrections directly from the DORIS data files instead of recomputing them. This could be a valuable clue to understand how analysis strategies can lead to systematic errors in scale factor determination. We will test for each software packages if the results obtained using the phase center correction provided by CNES in the data file or recomputing it using the satellite orientation models in space are exactly the same or not for orbit determination and for geodesy. The way the tropospheric correction is treated by the three different groups is also significantly different from one group to the other. This could also explain differences in the derived scale of the Terrestrial Reference Frame. To be more specific, we tried to use different references to do these comparisons, either using directly ITRF2000 (with less points in common as the DORIS tracking network has evolved between the end of 2000 and 2004, see Willis and Ries, 2005), or using differenet internal ACderived solutions based on the 2004 data and transformed into ITRF2000, or even using a unique long-term cumulative DORIS solution (based on more than 10 years of observations and including all 2004 DORIS stations), such as IGN04D02, (Willis et al., 2004). All comparisons led to similar results (differences were less than 0.1 ppb) for each weekly determination of the scale factor. 2.3
Using
single-satellite
solutions
In the second stage of our analysis, we have done the same type of study but using DORIS singlesatellite weekly solutions. These solutions are, of course, noisier (based on lesser data) but these comparisons should help us identify some satellite-related systematic errors. For TOPEX/Poseidon (T/P), GSFC and LEGOS/CLS show a good internal agreement
and also a good agreement with ITRF2000 (see Figure 3). There is again a systematic difference o f - 2 ppb between the IGN/JPL and the two other solutions. No results are available after November 1, 2004 as the DORIS receiver onboard T/P unfortunately stopped functioning after more than 12 years of continuous operation. -e-scale GSFC TOPEX -=-scale IGN TOPEX ~ s c a l e LCA TOPEX 15
,-,'-'10 ........................................................................................................................................................................................................................... 13.
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........................................... i
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/ 2004.40
i...................... 2004.60
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'.
....................................................
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Fig. 3 Weekly scale factor determination towards ITRF2000 using TOPEX/Poseidon SINEX solutions. GSFC (white circles), IGN/JPL (black squares), LEGOS/CLS (crosses). January December 2004. SPOT2 and SPOT4 provide similar results (Figure 4 for SPOT2). On Figure 4, a possible discontinuity may be observed in fall 2004 for the IGN/JPL SPOT2 results. This discontinuity is not observed at all by the two other centers.
-e-scale GSFC SPOT2 --=-scale IGN SPOT2 ~ s c a l e LCA SPOT2
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-10 2004.00
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2004.60
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Fig. 4 Weekly scale factor determination towards ITRF2000 using SPOT2 SINEX solutions. GSFC (white circles), IGN/JPL (black squares), LEGOS/CLS (crosses). January - December 2004.
Chapter 23 • Looking for Systematic Error in Scale from Terrestrial Reference Frames Derived from DORIS Data
Finally, SPOT5, equipped with a newer type of instrument (second generation, see Tavernier et al., 2005) provides more consistent results (Figure 5), especially for the IGN/JPL solution. The GSFC and LEGOS/CLS scale are still very close but far away from the ITRF2000 reference (+7 ppb). Some additional tuning (change in satellite physical model: shape, reflectivity coefficients,...) may be required by the different ACs for processing data from this recent satellite. A large seasonal variation may also be visible in Figure 5 and could come from orbit mismodelling for this sun-synchronous satellite. However a longer time series is really needed to ascertain this point. All components of the radiation pressure modeling could be the source of this effect: (1) macromodel approximations, errors in values of surface properties; (2) failure to account for shadowing effects of one surface by another surface; (3) improper modelling of albedo and thermal emission variations. Lastly, we cannot exclude possible errors due to the solar tides (particularly $2) since they would alias to an annual signal on these sunsynchronous satellites. The argument against the solar tide error is that an annual signal does not appear on SPOT2 which is really quite stable in comparison. -e-scale GSFC SPOT5 15
~.10 CL
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-e-scale GSFC ENVISAT --=-scale IGN ENVISAT ~ s c a l e LCA ENVISAT
Fig. 6 Weekly scale factor determination towards ITRF2000 using ENVISAT SINEX solutions. GSFC (white circles), IGN/JPL (black squares), LEGOS/CLS (crosses). January - December 2004.
-=-scale IGN SPOT5 --~-.-scale LCA SPOT5
O o O
In the second half of 2004, all 3 solutions show an interesting pattern that could be linked to a possible annual signal or long-term oscillation (see also discussion after in §2.4). This could be an indication of remaining orbit errors. This is not surprising for such a large and complex satellite, orbiting at a lower orbit. However, we must also note that the surface/mass ratio of ENVISAT is rather small for this satellite, which is a clear advantage for Precise Orbit Determination. Seasonal variations could also be attributed to mis-modelling in the atmospheric propagation (troposphere or ionosphere). One of the points in the IGN/JPL solution in Figure 6 is also a clear outlier.
2004.40 2004.60 Year
2004.80
2005. O0
Fig. 5 Weekly scale factor determination towards ITRF2000 using SPOT5 SINEX solutions. GSFC (white circles), IGN/JPL (black squares), LEGOS/CLS (crosses). January - December 2004. In the case of ENVISAT, a similar pattern can also be found. The GSFC and the LEGOS/CLS solutions are still very close but far apart from ITRF2000 (around +8 ppb).
In summary, Figures 2 to 6 show that the IGN/JPL solution is always close to a constant value (with an -3 ppb offset from ITRF2000), while the GSFC and the LEGOS/CLS show larger satellite-dependant biases, especially for the newer satellites such as SPOT5 and ENVISAT, for which improved satellite orbits can still be found. In our opinion, this may explain why the IGN/JPL multi-satellite solution showed a better internal consistency (smaller Allan variance) as it presents smaller satellitedependant biases. However, some clear long-term or annual signals are also clearly visible in the results, with an amplitude of 0.5 ppb. They could be attributed to annual variations potentially due to mis-modeled atmospheric effects (ionosphere correction?). Several sources of annual systematic errors can be found in geodetic results (Cretaux et al., 2002;
147
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L.P. Willis. F. 6. Lemoine • L. Soudarin
Meisel et al., 2005). The present analysis, limited to the 2004, is not sufficient to fully answer these questions. If we assume that the DORIS scale factor can be modeled as a constant systematic bias per AC and per satellite, we can derive the following synthetic results (Table 2). Table 2. Terrestrial reference frame scale factor derived from station coordinates. Mean and standard deviation estimated for different Analysis Centers. January - December 2004. Part per billion (ppb) Satellite SPOT2 SPOT4 SPOT5 T/P ENVIS All sats
GSFC 1.5 + 0.7 2.4 + 0.9 6.1+1.3 -0.5 _+ 1.5 7.2 + 3.1 4.0 ___0.7
IGN -2.4 _+ 1.2 -1.9 +1.1 -1.2_+1.2 -2.7 _+ 1.7 -2.5 _+2.4 -2.8 _+0.8
LCA 2.7 + 0.7 3.4 + 0.6 8.1+1.0 0.8 _+ 1.5 8.2 + 2.1 5.2 ___0.8
Table 2 shows that IGN/JPL shows smaller intersatellite biases relative to ITRF2000. However, GSFC and LEGOS/CLS show smaller biases toward ITRF2000 for the older satellites (SPOT2, SPOT4 and TOPEX). 2.4 Testing scale bias
long-term
stability
of
TRF
In parallel, we have combined all IGN/JPL weekly DORIS SINEX solutions available from 1993.0 to 2005.25 to create a refined DORIS cumulative solution (Willis et al., 2005a, Willis and Ries, 2005). DORIS station positions and velocities were all estimated simultaneously, DORIS-DORIS geodetic local ties were also used with proper weighting. This can be considered as an updated solution of the IGN/JPL latest cumulative solution available at NASA/CDDIS data center (1GN04D02, see Willlis et al., 2005a). A direct comparison between this frame and the 1TRF2000, provided the following results for the TRF scale:
scale(2000.O) scale_rate
= -3.12
= -0.075
_+
_+
0.54ppb
(2)
O.046ppb / yr (3)
These results confirm the -3 ppb bias previously observed for the IGN/JPL solutions for all satellites. They also show an extremely good long-term agreement with the ITRF2000 (better
than 0.1 ppb/yr for the formal error and for a potential bias). These numbers confirm on earlier determination of this stability (Willis et al., 2005b). If needed, we could use a time-limited subset of the DORIS data to recalibrate the satellite antenna offsets (center of phase of the antenna center of mass), as currently done by the GPS ACs, to align the DORIS TRF scale on ITRF2000. Using these new empirical values would ensure a long-term consistency with 1TRF2000 at the 1 ppb level for almost 2 decades (from Eq. 3). The LEGOS/CLS group has started this type of activity and provides corrected solutions for the IERS Combination Pilot Project (CPP). As this study shows that the scale biases are more software or analysisspecific than really DORIS technique-specific, we do not propose to adopt such a correction scheme for the time being. On the long term, more investigation is required to better understand the sources of errors in DORIS data processing potentially leading to systematic errors in the TRF scale.
3 Terrestrial reference frame scale factor derived from satellite orbits In a second step, we have compared satellite orbits provided by each AC in 2004, fixing all DORIS station coordinates to their ITRF2000 values. For newer stations, an extension of ITRF2000 was used: DPOD2000 (Willis and Ries, 2005). All satellites orbits were computed individually using only DORIS data. Comparisons were only performed for the ENVISAT satellite as it showed larger discrepancies in earlier tests discussed here. Table 4 shows the consistency between orbits from different solutions. As expected, best results are obtained in the radial component leading to 1-2 cm differences in RMS between all groups. No significant bias could be found in the radial and cross-track component (less than a couple of mm). However, a significant bias could be found for the IGN/JPL group in the along-track component (-10 cm) relative to the 2 other groups. This problem was identified recently and is now fully understood. It was related to an incorrect option in the relativity correction.
Chapter23 • Lookingfor SystematicErrorin Scale from TerrestrialReferenceFramesDerivedfrom
Table 3. ENVISAT orbit comparisons (January - December 2004) between ACs (GSC = GSFC, IGN = IGN/JPL, LCA = LEGOS/CLS). Daily Root-Mean-Squares in the radial, cross-track and along-track components.
LCA-IGN GSC-IGN GSC-LCA
radial 1-2 cm 1-2 cm 1-2 cm
cross 6-8 cm 4-10 cm 4-10 cm
along 11-14 cm 10-12 cm 3-6 cm
It can be seen that even if the 3 ACs have estimated different TRF scales that could differ by almost 10 ppb, the derived orbits do not present a systematic error in the satellite altitude. This result was predictable, looking at earlier simulation results (Morel and Willis, 2005), as the satellite period is directly accessible from the observations and as Kepler's Third Law directly links the satellite orbit period to the semi-major axis of the orbit (and then to the orbit radial bias). In the case of the ENVISAT satellite, orbits can be tested using external sources of information, such as satellite laser ranging data residuals. These tests were performed at GSFC using the G E O D Y N software and are displayed in Table 4. Table 4. ENVISAT Root-Mean-Squares of SLR residuals over 2004. All GSFC IGN/JPL LEGOS/CLS
5.6 cm 9.6 cm 4.9 cm
High elev (>70 deg) 2.7 cm 3.6 cm 3,2 cm
Typically over 50000 laser residuals were tested for ENVISAT in 2004 (56484 for GSFC, 51623 for IGN/JPL and 47693 for LEGOS/CLS). In order to better test the radial component a subcategory was also analyzed, selecting only SLR residuals at high elevation (over 70 degrees). This subset already comprises a lot of data points: 1347 for GSFC, 1248 for IGN/JPL and 1141 for LEGOS/CLS. The number of points is slightly different for each group, because there is an edit criterion imposed at 3-hours at start that affects the three groups in a different manner. Furthermore, some groups did not submit all possible arcs and, usually arcs around maneuvers are not considered at all. It can be seen that GSFC and LEGOS/CLS provide better results in all cases and especially when all residuals are considered (high and low
DORIS Data
elevation together). However, in the case of the IGN/JPL solution, the Laser residual test could be altered by the constant along-track offset previously detected. High elevation Laser residuals would not be affected by a timing error but all other SLR residuals would be. These results also suggest that we should extend this study by comparing single satellite orbits from all groups for all satellites, compare them internally (one against another) and test them using other source of information when available (SLR residuals, altimeter cross-over). Conclusions In order to investigate the stability of the DORIS-derived terrestrial reference frame scale factor, we have analyzed weekly station coordinates and daily orbits obtained by three different groups using three different software and analysis strategies (GSFC, IGN/JPL and LEGOS/CLS). Results show that the TRF scale derived by all groups are significantly affected by satellitedependent biases, even if the IGN/JPL solution seems to be less affected. Typically single satellite TRF solutions can show biases in scale up to almost 10 ppb. However, multi-satellite DORIS solutions show a better agreement with ITRF2000 (typically up to 5 ppb in bias). Seasonal signals are also superimposed in DORIS results with a typical amplitude of 0.5 ppb. However, DORIS provides an excellent long-term stability for scale monitoring (typically 0.05 ppb/yr). Some differences in the analysis strategies were noted and presently it seems that the way the center of phase correction is applied by the ACs could be a possible explanation for differences in results. Preliminary tests on ENVISAT orbit show a 10 cm mis-modeling bias in the along-track component that could also be linked with timing issues. In order to better understand these systematic errors, future tests are needed in which all groups try to use the same analysis strategy. With the recent creation of the International DORIS Service, we hope that future Analysis Centers will join in to perform these tests and discuss these difficult issues. Acknowledgments
Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (NASA).
149
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References
Altamimi, Z., P. Sillard, C. Boucher (2002). ITRF2000, A new release of the International Terrestrial Reference Frame for earth science applications. Journal of Geophysical Research, Solid Earth, vol. 107(B10), 2214. Altamimi Z., C. Boucher, P. Willis (2005). Terrestrial Reference Frame requirements within GGOS, Journal of Geodynamics, Vol.
40(4-5), pp. 363-374 Biancale R., G. Balmino, J.M. Lemoine, et al. (2000). A new global Earth's gravity field model from satellite orbit perturbations, GRIM5-S1, Geophysical Research Letters, Vol. 27(22), pp. 3611-3614. Cazenave A., R.S. Nerem (2004). Present-day sea level change, Observations and causes, Reviews of Geophysics, Vol. 42(3), RG3001. Chao B.F., and A. Au (1991), Temporal variation of the Earth's low-degree field caused by atmospheric mass redistribution1980-1988, Journal of Geophysical Research, Solid Earth, Vol. 96(B4), pp. 6569-6575. Cr6taux J.F., L. Soudarin, A. Cazenave, F. Bouille (1998), Present-day Tectonic Plate Motions and Crustal Deformations from the DORIS Space System. Journal of Geophysical Research, Solid Earth vol. 103(B12), pp. 30167-30181. Cr6taux J.F., L. Soudarin, F.J.M. Davidson, M.C. Gennero, M. Berge-Nguyen, A. Cazenave (2002), Seasonal and interannual geocenter motion from SLR and DORIS measurements, comparison with surface loading data, Journal of Geophysical Research, Solid Earth, Vol. 107(B12), 2374. Feissel-Vernier M., J.J. Valette, K. Le Bail, L. Soudarin (2005). Impact of the GRACE gravity fields models on IDS products,
International DORIS Service Report. Gerasimenko M.D., A.G. Kolomiete, M. Kasahara, J.F. Cr6taux, L. Soudarin (2005). Establishment of the three-dimensional kinematic reference frame by VLBI and DORIS global data, Far Eastern
Mathematical Journal. Lemoine F.G., et al. (1998). The development of the joint NASA/GSFC and the National Imagery and mapping Agency (NIMA) geopotential models, EGM96, in NASA, TP1998-206861, pp. 1-575. Lemoine J.-M., H. Capdeville (submitted). Correction model for the SAA effect on Jason/DORIS data. Journal of Geodesy
Luthcke S.B., N.P. Zelensky, D.D. Rowlands, F.G. Lemoine, T.A. Williams (2003). The 1centimeter orbit, Jason-1 precision orbit determination using GPS, SLR, DORIS, and altimeter data, Marine Geodesy, Vol. 26(3-4), pp. 399-421. Meisel B, D. Angermann, M. Krugel, H. Drewes, M. Gerstl, R. Kelm, H. Muller, W. Seemuller, V. Tesmer (2005). Refined approaches for terrestrial reference frame computations, Advances in Space Research, Vol. 36(3), pp. 350-357. Morel L., and P. Willis (2005). Terrestrial reference frame effects on mean sea level rise determined by TOPEX/Poseidon, Advances in Space Research, Vol. 36(3), pp. 358-368. Rummel R., H. Drewes, G. Beutler (2002). Integrated Global Observing System IGGOS, A candidate IAG Project, In Proc.
International
Association
of
Geodesy,
Vol. 125, pp. 135-143. Sillard P., and C. Boucher (2001). A review of algebraic constraints in Terrestrial Reference Frame datum definition, Journal of Geodesy, Vol. 75(2-3), pp. 63-73. Soudarin L., J.F. Cr6taux, A. Cazenave (1999). Vertical Crustal Motions from the DORIS space-geodesy system, Geophysical Research Letters, Vol. 26(9), pp. 1207-1210. Tapley B.D., S. Bettadpur, M. Watkins, C. Reigber (2004), the Gravity Recovery and Climate Experiment, Mission overview and early results, Geophysical Research Letters, Vol. 31 (9), L09607. Tavernier G., H. Fagard, M. Feissel-Vernier, F. Lemoine, C. Noll, J.C. Ries, L. Soudarin, P. Willis (2005). The International DORIS Service, IDS. Advances in Space Research, Vol. 36(3), pp. 333-341. Tregoning P., T. van Dam (2005). Effects of atmospheric pressure loading and sevenparameter transformations on estimates of geocenter motion and station heights from space geodetic observations, Journal of Geophysical Research, Solid Earth, Vol. 110(B3), B03408. Webb F., and J. Zumberge (Eds.) (1995). An introduction to Gipsy/Oasis II, Report Jet Propulsion Laboratory, Pasadena, USA, JPLM D- 11088. Willis P., and M. Heflin (2004), External validation of the GRACE GGM01C Gravity Field using GPS and DORIS positioning results, Geophysical Research Letters, Vol. 31(13), L13616.
Chapter 23 • Looking for Systematic Error in Scale from Terrestrial Reference Frames Derived from DORIS Data
Willis P., and J.C. Ries (2005), Defining a DORIS core network for Jason-1 precise orbit determination based on ITRF2000, methods and realization. Journal of Geodesy, Vol. 79(6-7), pp. 370-378. Willis P., B. Haines, J.-P. Berthias, P. Sengenes, J.L. Le Mouel (2004), Behavior of the DORIS/Jason oscillator over the South Atlantic Anomaly. Comptes Rendus Geoscience, Vol. 336(9), pp. 839-846.
Willis P., C. Boucher, H. Fagard, Z. Altamimi (2005a). Geodetic applications of the DORIS system at the French Institut Geographique National. Comptes Rendus Geoscience, Vol. 337(7), pp. 653-662. Willis P., Y.E. Bar-Sever, G. Tavernier (2005b), DORIS as a potential part of a Global Geodetic Observing System. Journal of Geodynamics, Vol. 40(4-5), pp. 494-501.
151
Chapter 24
WVR calibration applied to European VLBI observing sessions Axel Nothnagel, Jung-ho Cho 1 Geodetic Institute of the University of Bonn, Nussallee 17, D-53115 Bonn, Germany 1on leave from Korean Astronomy and Space Science Institute, 61-1, Whaam-Dong, Youseong-Gu, Taejeon, Rep. of Korea 305-348 Alan Roy Max-Planck-Institute for Radio Astronomy, Auf dem Hiigel 69, D-53121 Bonn, Germany Riidiger Haas Onsala Space Observatory, Chalmers University of Technology, SE-439 92 Onsala, Sweden
Abstract. From a conceptual point of view water vapour radiometers (WVRs) are ideal instruments for a direct determination of the water vapour content in the atmosphere above a space geodetic observing site. For various reasons the application of WVRs for a direct correction of geodetic Very Long Baseline Interferometry (VLBI) observations for the wet component of atmospheric refraction has not been brought to a stage where it could be employed routinely. The installation of a new type of WVR at the 100 m Effelsberg radio telescope in Germany opens up new possibilities for a direct correction of the VLBI observations using line-ofsight WVR observations. The WVR has been in operation during two geodetic VLBI sessions of a purely European network. In addition, some WVR data at Onsala have been revisited to embark on a project which aims at applying wet delay corrections from WVRs to VLBI sessions of the European geodetic VLBI network for the Effelsberg, Madrid, Onsala and Wettzell sites. In this paper we present the analysis and results of a few examples of Effelsberg and Onsala observing sessions.
Keywords. Water Vapour Radiometers (WVR), atmospheric refraction, Interferometry (VLBI)
Very
Long
Baseline
1 Introduction The rapid variations of water vapour in the atmosphere, both in time and space, belong to the list of factors which limit further progress of
improving the accuracy of space geodetic results based on radio frequency observations. A direct determination of the excess delay caused by water vapour in the atmosphere (wet delay) can be carried out through measurements of the sky brightness temperatures using water vapour radiometers (WVRs) (e.g. Elgered et al. 1982). However, routine operations of WVRs for a direct correction of space geodetic observations like Very Long Baseline Interferometry (VLBI) or GPS have not been achieved due to various unsolved problems. Since early 2003 a new instrument has been in operation at the 100 m radio telescope of the MaxPlanck-Institute for Radio Astronomy at Effelsberg, Germany, which has been constructed with an improved architecture overcoming a number of shortcomings of existing WVRs (Roy et al. 2003). This WVR is installed on the prime focus cabin of the telescope and operates along the line-of-sight taking brightness temperature measurements by sweeping the full frequency band between 18.3 GHz and 26.0 GHz in 25 steps of 900 MHz each. An example spectrum is shown in Fig 1. The radiometer was employed to carry out observations during two geodetic VLBI sessions of the European geodetic VLBI network (Campbell and Nothnagel, 2000) in December 2004 and March 2005. During this period, radiometers were operated at the VLBI telescopes at Madrid Deep Space Complex (Spain), Wettzell Geodetic Fundamental Station (Germany) and Onsala Space Observatory (Sweden). This favourable constellation of four sites equipped with WVR triggered the idea of investigating the current status and future prospects for regular use of WVR data in VLB! data analysis.
Chapter 24 • WVR Calibration Applied to European VLBI Observing Sessions
Here, we put special emphasis on the results of the first Effelsberg observations. In addition, a few older sessions of Onsala are revisited in order to discuss general problems with WVR calibrations on the basis of a case study. Effelsberg WVR Spectrtlnl Measured 2005mar22 17:32:47 UT
J-IT -I_ .',lt) --
observations for instrumental effects. The gain was measured inserting an ambient temperature load and assuming the sky as a cold load. Since the sky is not perfectly cold, its temperature was measured using a tipping-curve. Furthermore, a water vapour saturation correction has been applied to the Effelsberg data applying the linearization method according to Claflin et al. (1978). As a result, the correction of the brightness temperature can be determined with a linear relationship between the brightness temperatures and the air mass values (Elgered 1993).
7(}-
2.2 Delay corrections from brightness temperatures
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Fig. 1 :A sky-temperature spectrum measured with the Effelsberg WVR, showing the broad emission line of water vapour at 22.235 GHz. The water line appears asymmetric and the peak is shifted slightly toward higher frequency due to additional emission from the wing of the oxygen line at 60 GHz. The oxygen emission adds a baseline level that rises with frequency squared.
The final aim of reliably applying WVR corrections to VLBI observations will serve the purpose of eliminating the need for estimating zenith wet delay (ZWD) parameters in the VLBI analysis process. This in turn will lead to the benefit that VLBI observations need not be driven to the lowest possible elevation. Today, low elevation observations have to be scheduled for no other reason than to estimate the wet delay parameter as good as possible. If the need for estimating wet delay parameters disappears, so does the need for low elevation observations, at least for those below 7 ° or 8 °. Ultimately, a reduced number of unknown parameters will stabilize the adjustment process and with it the quality of the results.
2 Effelsberg WVR Data Analysis 2.1 Radiometer calibration Water vapour radiometers measure the sky brightness temperatures along their line-of-sight. The first step in WVR data analysis is, thus, the calibration of the raw brightness temperature
The next step is the inversion process to generate delay corrections for the VLBI observables from brightness temperatures. To separate the continuum of liquid water from the water vapour line emission for the determination of the wet delays, a frequencysquared baseline and a van Vleck-Weisskopf water vapour line profile were fitted to the 25 individual frequency channels of the radiometer output, following Tahmoush and Rogers (2000). In addition, for comparison wet delays were also computed applying the Resch (1983) model adapted for different sets of frequency pairs. Ideally, radiosonde observations should be used to support the determination of the conversion coefficients. However, in the absence of radiosonde information standard model coefficients like the ones developed by Resch for the North American continent may serve as a suitable starting point. The first WVR observations with the new Effelsberg radiometer were made during the Euro74 VLBI session on December 14 and 15, 2004. Unfortunately, a few channels of the WVR malfunctioned during the session which happened to take place just at the beginning of its operation. In order to investigate the implications of missing channels the Resch (1983) model was first applied in a slightly modified version with two healthy channels only. In addition, we also applied the Tahmoush and Rogers (2000) model regardless of the missing channels. Figure 2 depicts the resulting zenith wet delays for Effelsberg during this session for both types of inversions. Here, it becomes obvious how important it is that the full spectrum be sampled and what effect the inversion process has because the resulting wet delays are rather different (see Fig. 2 and 3 bottom parts). Since the WVR did
153
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A.L. Nothnagel. J. Cho. A. Roy. R. Haas
not work to its expectations during the Euro-74 session; further analysis of this data set was abandoned.
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The difference of the Resch and the Tahmoush and Rogers models (Fig. 3 bottom part) show a similar or even larger level of disagreement as in Fig. 2. It is largest where the disagreement between the wet delay estimates and the WVR inversion has a maximum as well. The other features of the WVR inversions in Figure 3 (top part), the apparent positive outliers, originate from clouds due to additional water vapour they contain and perhaps due to imperfect removal of the continuum emission from liquid water droplets in the clouds. This has been verified with a video camera viewing the same line-of-sight as the radiometer. During stability tests the video pictures showed that the passage of clouds through the radiometer beam produced increases, sometimes dramatic, in the spectral baseline level.
20
Time (hour) Fig. 2 Top part: Zenith wet delays derived from WVR at
Effelsberg in the Euro-74 session on Dec. 14 and 15, 2004. Middle part: Median filtered curve of WVR observations and VLBI estimates (both offset by -100 mm for better identification). Bottom part: Differences of raw observations with Tahmoush/Rogers model minus modified Resch model (offset by-200 mm)
The second geodetic VLBI session with participation of Effelsberg on March 22 and 23, 2005 (Euro-75) produced much better WVR results as may be interpreted from the agreement of the wet delay estimates and the WVR inversion (Fig. 3 middle part) however with a noticeable deviation at around 5.00 UT.
3 0 n s a l a WVR Data Analysis At Onsala the WVR operates at two frequencies (20.7 GHz and 31.4 GHz) and the observations are calibrated and inverted to delay corrections using the Johansson algorithm (Johansson et al. 1993). This algorithm is based on empirical coefficients which have been determined with parallel observations by radiosondes. The sky is regularly sampled with a special observing schedule in order to produce a time series of zenith wet delays. In the analysis, a gradient model has been implemented according to Davis et al. (1993). In 1991 the calibration loads for the reference temperatures were changed and in 2001 the WVR underwent a major upgrade.
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In state-of-the-art VLBI data analyses the hydrostatic component of refraction of each individual observation is corrected by applying a hydrostatic zenith delay (e.g. Davis et al. 1985) which is transformed onto the respective elevation using a hydrostatic mapping function (e.g. Niell 2001). The wet component is estimated applying a similar mapping function tailored for the wet path delay. In order to model time-dependent variations either a stochastic process is assumed and a filter adjustment such as the Kalman filter is applied or the variations are considered to be of a less variable type and the model consists of piece-wise linear elements.
Chapter 24 • WVR Calibration Applied to European VLBI Observing Sessions
Instead of estimating the wet component, WVR corrections can be applied. Zenith wet delay corrections have to be mapped to the respective elevation of the observations. If WVR corrections for the wet component are available, it should not be necessary any more to estimate the wet component. Likewise, estimates of the wet component with WVR corrections applied to the VLBI data should be zero within their error margins provided that hydrostatic and ionospheric refraction effects are modelled correctly. So far, a small number of VLBI sessions observed with the European geodetic VLBI network has been processed in this way to provide the basis for a case study. In order to discuss the quality of the WVR corrections, zenith wet delays have been estimated in three different ways employing the Calc/Solve VLBI analysis software with piece-wise linear functions of one-hour segment lengths:
a)
b)
c)
standard solutions have been computed by correcting the hydrostatic component with surface meteorological data and estimating zenith wet delay parameters in the usual way with rate changes at every hour (see Fig. 4, standard). in the second type of solutions we corrected the hydrostatic part in the usual way and also corrected the wet part with WVR wet delays applying the Resch model and still estimated the zenith wet delays as in case a). the third type of solutions differs from case b) only in that the Tahmoush and Rogers model is applied.
5 Results In Figure 4 the zenith wet delay results of the standard solution (a) give a very good indication of the variations which are to be expected during the 24-hour observing period. Curves (b) and (c) depict the residual zenith wet delay estimates if WVR corrections are applied beforehand. Calibrating the VLBI observations with WVR delay corrections and estimating zenith wet delays at the same time in the analysis process is a method used only within this investigation and should normally not be necessary. If the WVR delay corrections describe the wet refraction effect completely and no other systematic effects are absorbed in estimates of the wet delays, the residuals should be zero throughout. It should be emphasised that this assumption is only valid if the estimates of the zenith wet delays truly represent just the water vapour refraction effect and not any additional contamination with a similar signature. In the case of the Effelsberg observations in the Euro-75 session the residual wet delays after WVR calibration are still significant for both inversion models (Fig. 4, lines b and c) at the level of abut 15 20 mm RMS, with no model appearing superior to the other. -
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This type of analysis with its graphical representation in figures 4, 5 and 6 have the advantage that the effects of the least squares estimation with the correlations between all parameters (e.g. also for the clock effects) are correctly transferred to the wet delay estimates. This would not be the case when the observations had not been corrected for the wet delays beforehand. In all sessions analysed here Wettzell was kept fixed at its ITRF2000 coordinates and all other site coordinates were estimated on a session by session basis without constraints.
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155
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A.L. Nothnagel.J. Cho.A. Roy. R. Haas
always have to be positive. There are three possible causes for this phenomenon: the WVR data in its present state over-calibrates the wet component at this time of the observations, or the hydrostatic correction was insufficient, which is less likely, or the wet delay estimates compensate other unmodelled errors. Since these cases occur predominantly at the period of the most rapid changes in the water vapour content it is conceivable that the conversion from line-of-sight data to zenith path delays introduced an additional uncertainty which we will investigate further.
Onsala, Euro 22 2O0
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In addition to the sessions with Effelsberg we would like to discuss two other VLBI observing sessions, Euro-22 on December 28/29, 1994 and Euro-72 on July 13/14, 2004, in which we applied and investigated Onsala WVR delay corrections. They were processed in the same way as in the case of Effelsberg except that here only the Johansson et al. (1993) inversion model was applied. The zenith wet delays estimated after WVR corrections were applied to the VLBI data in the Euro-22 session are predominantly negative (Fig. 5) whilst those of the Euro-72 session are very close to zero (Fig. 6). From the latter fact we deduce that the WVR wet delay corrections very well match the respective estimates, the only exceptions being those data points where the standard wet delay estimates show obvious deviations.
~
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Time (hour) Fig. 5 VLBI-derived ZWD at Onsala for the Euro-22 session employing two different analysis approaches: a) top line: standard solution, b) bottom line: applying WVR corrections with the Johansson et al. (1993)model.
Without trying to further interpret these interim results we now look directly at the baseline repeatability. As a representative example we just look at the baseline Wettzell - Onsala here without using Wettzell WVR data. When applying WVR calibration at Onsala instead of estimating the zenith wet delay there is no noticeable change for the baseline length determined with the Euro-72 session (Fig. 7 and 8). This neutral behaviour of the Euro-72 session can clearly be attributed to the fact that the WVR wet delays match the estimated zenith wet delays as can bee seen in Figure 6 and little change should thus be expected. However, this situation does not imply that this agreement is the desired standard case because we then would not need any WVR observations.
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• WVR
Calibration Applied to EuropeanVLBI Observing Sessions
spectral channels to allow a more refined separation of cloud liquid water and water vapour components. Depending on the ongoing performance tests this technology is well placed to play a leading role for future installations. The brightness temperature results are of a state-of-the-art quality. Currently, the inversion algorithm is being adapted to geodetic requirements of mm-level accuracy and a robust data flow is being developed. Similar activities for data of Wettzell, Madrid and Onsala will prepare the grounds for a better usage of W V R monitoring in VLBI data analysis.
References In the case of the Euro-22 session the baseline length Onsala - Wettzell changes significantly if W V R corrections are applied at Onsala instead of estimating zenith wet delays. This is not unexpected since the application of W V R corrections with simultaneous wet delay estimation had produced a significant non-zero wet delay estimate (Fig. 5). It is not necessary here to look at statistical evidence but from the visual inspection of the baseline length graph alone it may already be concluded that the change has happened to the better.
6 Conclusions This study has shown that those VLBI sessions where the VLBI estimates and the W V R measurements disagree significantly are most promising for further insights into the problem of water vapour refraction effects in geodetic VLBI. The question of why the wet delay estimates are so different from W V R delay corrections in some of the sessions is still open. With four radiometers available at VLBI observatories in Europe there is ample scope for analysis of VLBI data where both ends of a baseline are equipped with WVRs. However, each VLBI session has to be analyzed in detail individually in order to figure out possible reasons for the discrepancies between estimates and measurements. The new water vapour radiometer at Effelsberg provides the lowest thermal noise of any radiometer used so far for geodetic applications. It offers many
Campbell J., A. Nothnagel (2000). European VLBI for Crustal Dynamics. J. of Geodynamics, 30, pp. 321-326 Claflin E.S., S.C. Wu and G.M. Resch (1978). Microwave Radiometer Measurement of Water Vapor Path Delay: Data Reduction Techniques. DSN Progress Report 42-48, pp. 22-30 Davis J., T.A. Herring, I.I. Shapiro, A.E.E. Rogers, G. Elgered (1985). Geodesy by Radio Interferometry: Effects of atmospheric modelling errors on estimated baseline lengths. Radio Science, 20, pp. 1593-1607 Davis J., G. Elgered, A.E. Niell, C. Kuehn (1993). Groundbased measurement of gradients in the "wet" radio refractive index of air. Radio Science, 28, pp. 1003-1018 Elgered G., B.O. R6nn/ing, J.I.H. Askne (1982). Measurements of atmospheric water vapor with microwave radiometry. Radio Science, 17, No. 5, pp. 1258-1264 Elgered, G. (1993). Tropospheric radio path delay from ground-based microwave radiometry, In: Atmospheric remote sensing by microwave radiometry, Chap. 5, Wiley, New-York Johansson, J.M., G. Elgered, J i . Davis (1993). Wet Path Delay Algorithm for Use with Microwave Radiometer Data. Contribution of Space Geodesy to Geodynamics: Technology, Vol. 25, AGU, Washington, D.C., pp. 81-98 Niell, A.E. (2001). Preliminary evaluation of atmospheric mapping functions based on numerical weather models. Phys. Chem. Earth, 26, pp. 475-480 Resch G.M. (1983). Inversion Algorithms for Water Vapor Radiometers Operating at 20.7 and 31.4 GHz. TDA Progress Report 42-76, pp. 12-26 Roy, A i . , U. Teuber and R. Keller (2003). Tropospheric Delay Measurement at Effelsberg with Water-Vapour Radiometry.Proc. 16th Working Meeting on European VLBI for Geodesy and Astrometry, May 9th-10th 2003, Leipzig, Germany, Edited by: V. Thorandt, BKG Leipzig, pp. 53-59 Tahmoush, D.A. and A.E.E. Rogers (2000). Correcting atmospheric path variations in millimetre wavelength very long baseline Interferometry using a scanning water vapor spectrometer. Radio Science, 35-5, pp. 1241-1251
157
Chapter 25
Frontiers in the combination of space geodetic techniques Manuela Kriigel and Detlef Angermann Deutsches GeodStisches Forschungsinstitut (DGFI), Marstallplatz 8, D-80539 Munich, Germany e-mail: kruegel~dgfi.badw.de
Abstract. Co-location sites are one of the key elements in the combination of the reference frames of different space geodetic techniques. They are fundamental for a consistent datum realization of the combined networks. This paper deals with a new strategy for the selection and implementation of local tie information and demonstrates how the ERP as common parameters of the different geodetic space techniques can be used for local tie validation. As one result of the new strategy it is shown, that only a small set of very consistent co-location sites should be used for the combination. Additionally, the capability of a combination using only the lengths of the local tie vectors is investigated. The strengths and weaknesses of this method are characterized. Keywords: Combination methods, space geodetic techniques, local ties, Earth Rotation Parameters
1
Introduction
Rigorous combination of space geodetic techniques has become a more and more expedient method to use the individual strengths of the different techniques for the combined product on the one side and to reduce the influence of their weaknesses on the other side. One iraportant and also complex part in the combination is the connection of the station networks of the different techniques into a unique reference frame. Due to the fact, that no stations are observed by two techniques the combination can only be done by introducing terrestrial measurements performed at so-called co-location sites. The recent global International Terrestrial Reference Frame (ITRF) of the International Earth Rotation and Reference Systems Service (IERS) are realized in this way (Altamimi et al., 2002; Boucher et al., 2004). In 2005 a new realization of the ITRF, namely the ITRF2004, will be com-
puted and published by the IERS. As one of the ITRS Combination Centres, DGFI is involved in this work. Within the TRF realization the selection of terrestrial measurements at co-location sites (so-called local ties) is an important step of work. There are partly large discrepancies between coordinate differences derived from space techniques and the local tie measurements, so that not all available information should be used for the combination (Angermann et al., 2004, Kriigel and Angermann, 2005). Additionally, in most cases no co-variance information is available for the local station networks. Hence a careful selection of suitable local ties is necessary. While previous ITRS realizations only coiltain station coordinates and velocities, now additional Earth Rotation Parameters (ERP) are included. The ERP provide new possibilities for the validation of co-location sites, which are investigated for the co-locations between a GPS and a VLBI network. As common parameters to both techniques the ERP are very suitable to identify rotations between the station networks caused by terrestrial measurements introduced within the combination. While Ray and Altamimi (2005) analysed co-location sites after combining the ERP we used the uncombined ERP series for local tie validation. In addition, simulation studies are performed to demonstrate a combination strategy using only vector lengths instead of local tie vectors in three components (3D).
2
Input data
For the investigations the data of a global GPS and a VLBI network are used. For GPS six years (1999-2004, 302 weeks) of the time series of weekly data are available, which are provided by the IGS for the IERS Combination Pilot Project (Rothacher et al., 2005) in SINEX format. These are combined GPS solutions based on individual solutions of up to 10 IGS analysis centers
Chapter 25 • Frontiersin the Combination of Space GeodeticTechniques
and basically come up to the official IGS time series described by Ferland (2004). In the case of VLBI the D G F I time series of normal equations from 1984 to 2004 is used. T h e y comprehend about 2580 sessions of 24 hours of observing time. The normal equations contain station coordinates and ERP. T h e y are combined to technique specific multi-year solutions using the full variance co-variance information. A detailed description of the combination strategy is given in Meisel et al. (2005). We concentrate on three years (2002-2004) of ERP, since this is sufficient for the purpose of the study and the a m o u n t of p a r a m e t e r s becomes very large using the complete E O P time series. For the GPS and VLBI station networks 26 colocations do exist, which are listed in Table 1. All local ties are provided in the I T R F 2 0 0 0 d a t u m .
A second set of 17 local ties was selected, showing residuals smaller t h a n 2.2 cm in positions and 4.5 m m / a in velocities (Ny Alesund, Onsala, Medicina, Noto, Madrid, Wettzell, Tsukuba, Hartebeesthoek, Algonquin Park, Kokee Park, Westford, N o r t h Liberty, M a u n a
Table 1: Residuals of positions resulting from the 14 p a r a m e t e r H e l m e r t - t r a n s f o r m a t i o n between the VLBI and the GPS networks. Stations with a residual of larger t h a n 8 m m in one c o m p o n e n t are not used for the transformation. T h e y are identified by a *
positions [mm]
3
Selection
of co-location
sites
The analysis of the co-location sites for VLBI and GPS was performed using a 14 p a r a m e t e r H e l m e r t - t r a n s f o r m a t i o n between the GPS and the VLBI network. Therefore, the local tie vector was added to the VLBI station coordinates to obtain identical stations for the transformation. Only stations with residuals smaller t h a n 8 m m in each c o m p o n e n t are used for the transformation. Based on the residuals of the transformation (shown in Table 1 and Table 2) a set of high-quality local ties was identified. T h e residuals show a good agreement (smaller t h a n 8 ram) for 9 stations: Ny Alesund, Onsala, Wettzell, T s u k u b a , Hartebeesthoek, Santiago, Westford, M a u n a Kea and Saint Croix. The residuals of the velocities are smaller t h a n 2.5 m m / a . The thresholds of 8 m m and 2.5 m m / a were found empirically with the precondition t h a t at least a set of 6-7 sites with good ties should be selected. The estimated Helmertt r a n s f o r m a t i o n p a r a m e t e r s differ significant from zero within the 3 sigma significance level. T h e y are not given here, because they are not interpretable, since the technique solutions have only a loose d a t u m . The global distribution of the 9 stations is shown in Fig. 1.
station Ny Alesund Onsala Medicina Noto (NOTO)* Noto (NOT1)* Madrid* Yebes Wettzell Tsukuba Hartrao Algonquin Goldstone* Fairbanks* Kokee* Westford* Fort D avis Pietown* North Liberty* Mauna Kea Fortaleza* Santiago Saint Croix Tidbinbilla* Hobart* Syowa* O'Higgins*
(1) (2) (3) (4) (5)
north -0.2 -0.5 7.1 18.1 17.6 0.1 -4.4 3.9 2.4 1.8 3.0 11.3 13.8 2.1 3.3 5.6 7. 7 0.7 1.1 4.2 -2.7 -0.3 3.3 9.4 11.6 -1.9
east -3.0 -1.4 6.2 9.9 5.6 4.5 -2.3 -1.8 -0.7 -1.2 4.3 3.4 2.8 0.9 -8.3 2.3 10.2 1.4 0.2 3.3 -0.4 -2.2 -5.6 -14.9 -3.4 10.8
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IGN d a t a b a s e ( ftp: / /lar eg. e ns g. i gn. fr / p u b / itrf/ie r s. e cc ) Steinforth et al. 2003 Sarti et al. 2004 Matsuzaka et al. 2004 J o h n s t o n et al. 2004
159
160
M. Kr~igel • D. Angermann
Figure 1" VLBI network (triangles" 9 selected stations with good co-location to GPS)
Table 2" Residuals of velocities resulting from a 14 parameter Helmert-transformation between VLBI and GPS. Stations identified by a * are not used for the transformation (see caption of Tab.l)
Kea, Santiago, Saint Croix, Tidbinbilla, Hobart). The criteria for the definition of the thresholds was to select a number of at least five additional sites to the 9 selected above. Generally, it can be seen that the largest discrepancies occur in the height component (see Table 1). This is consistent with the fact, that in particular for GPS and VLBI the station heights do have a higher uncertainty compared to the horizontal components. In Thaller et al. (2005) comparable results are shown for the 14-day continuous VLBI campaign CONT02. So far only the local tie vectors with seperate components are taken into account. In Fig. 2 the corresponding length of the difference vector ld between the local tie derived from terrestrial measurements and from the space techniques after the Helmert-transformation: ld -- v / d x 2 + @ 2 + dz 2
with:
velocities [mm/a] station Ny Alesund Onsala Medicina Noto (NOTO)* Noto (NOT1)* Madrid* Yebes Wettzell Tsukuba Hartrao Algonquin Goldstone* Fairbanks* Kokee* Westford* Fort Davis Pietown* North Liberty* Mauna Kea Fortaleza* Santiago Saint Croix Tidbinbilla* Hobart* Syowa* O'Higgins*
dx dy -dz -
-
I north 0.5 -0.0 0.0 -0.0 0.2 -1.4 -0.1 -0.0 -0.2 -1.7 -0.4 -0.3 0.1 -0.6 -1.3 0.2 1.2 -0.6 0.9 -2.0 -0.5 -0.1 0.6 1.1 -3.5 2.2
east -0.2 0.0 0.0 -0.3 -0.3 0.1 0.1 0.4 0.5 0.9 -0.7 -0.7 -0.4 -0.1 -0.9 -0.8 0.6 -0.6 -0.8 -1.1 1.7 -1.1 0.2 0.8 1.5 -0.0
up 1.1 -0.6 -0.9 1.6 1.2 0.1 4.5 -0.1 -1.5 -1.7 0.6 0.9 2.0 -1.8 -2.0 0.2 1.5 1.0 -1.0 -1.1 1.5 -2.0 1.8 2.6 -7.1 2.2
(1)
Z2kXspac e -- A Xterr A y s p a c e -- A Y t e r r A Zspace -- A Zterr
is compared to the difference between the vector lengths: dl
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2 2 2 A X s p a c e -+- A Y s p a c e -+- A Z s p a c e
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(2)
The second quantity dl represents the difference of "local ties" containing only information about the distances between the stations. This means, that the datum definition (orientation) in ITRF2000 and the information about the network geometry are removed from the local tie. It is clear, that the difference between the vector lengths is smaller in all cases, but for some stations it is extremly smaller, e.g. for Hobart. Possible reasons could be: • For most co-locations the horizontal extension of the local network is much larger than the vertical. Hence a discrepancy between space techniques and terrestrial measurements in the height component has a much smaller effect on the distance between the stations as on the difference vector. For example, for a local network with a horizontal extension of 100 m, a discrepancy in the
Chapter 25
•
Frontiers in the Combination of Space Geodetic Techniques
Table 3: Solution types 20~15-~ ~10-
i
,I, i i . I o_LLi I l i L L I Ii ..ILl • differenceofvectors
• differenceofdistances
Figure 2: Differences between space geodetic solutions (VLBI and GPS) and local tie measurements. Compared are the length of the difference vector and the difference of the vector lengths. Only stations with a length of the difference vectot smaller than 2.2cm are displayed.
height component of 10 cm affects the vector length by only 0.04 mm. • Secondly, a possible rotation of the local network with respect to the space technique solutions is eliminated because no datum information is included in the tie information any more. Thus, it can be concluded that the use of distances between stations instead of local ties in three components minimizes the problems in height component and eliminates errors in the orientation. However, the information content, which can be used to connect the networks is reduced compared to the conventional 3D case and more co-location sites are necessary to ensure stable combination results.
4
Analysis of co-locations using ERP
In the section above sets of 9 and 17 local ties are identified on the basis of the results of a Helmerttransformation. The ERP provide additionally information to validate local ties. Based on the same GPS and VLBI input data we analysed and compared these sets of local tie information with respect to the following two criteria: • The consistency of the combined reference frame should be maximized. • Secondly, the deformation of the networks due to the combination should be minimized.
solution discription type SING stand-alone VLBI and GPS solution DIST vector lengths of all 26 VLBI-GPS local ties 1YD3 3D local ties of 17 sites 9D3 9 good 3D local ties (Ny Alesund, Onsala, Wettzell, Tsukuba, Hartrao, Westford, Santiago, Mauna Kea, St. Croix)
To investigate the first criteria the station networks are combined but not the EOP. The offsets between xpole and ypole values from VLBI and GPS are a measure of the achieved consistency (unless there are systematics in the ERP estimation of one of the techniques). To quarttify the deformation (second criteria) of the combined network caused by the local ties, the RMS of the transformation between stand-alone technique solutions and the combined solution was estimated. Different solution types (see Table 3) were iraplemented and analysed with respect to the two criteria. The combinations are performed introducing the local tie information as pseudo observations with a priori standard deviations of 0.1 ram, 0.3 ram, 1.0 mm and 3.0 ram. In the case of standalone technique solutions the datum for stations and velocities was defined for GPS by no-netrotation (NNR) conditions w.r.t ITRF2000 and not-net-rotation and not-net-translation (NNT) conditions w.r.t. ITRF2000 for VLBI. For all combined solutions NNR conditions for station positions and velocities were applied using only GPS stations. The weighting factors between both techniques within the combination are 1.0. The results are displayed in Fig. 3, which include the offset in the pole values (upper part) and the RMS values estimated within the transformation (lower part). The RMS values are only displayed for the transformations of the VLBI part of the combined solutions to the stand-alone VLBI solution, because due to a much larger number and the better global distribution of stations the GPS network is much less deformed.
161
162
M. Kr~igel • D. Angermann
The most stable results can be achieved using a set of only a few good and well distributed ties. The RMS values for the solution type 9 D 3 are nearly independent from the chosen a priori standard deviation. This proofs the high consistency of the selected ties. The offset in the pole values are also smaller than those of the other solution types. 150 100 so 0 -50
-1 O0 -150
E -ff ~5
~0 rr
E
E
E
E
~
~
~
E
~
~
E
E
a priori standard deviation of local ties
• •
x-pole offset (VLBI-GPS) RMS of positions
• •
y-pole offset (VLBI-GPS) RMS of velocities
Figure 3: Accuracy evaluation of different solution types for the combination of VLBI and GPS. Top figure: pole offsets. The standard deviations of the pole offsets are lower than 5 #as. Lower figure: RMS values of Helmert-transformation. They are estimated including all stations and all coordinate components. Note, that for better perceptibility the RMS values for the velocities are multiplied with a factor 10. If 17 instead of 9 local ties (solution type: 17D3) are used, the RMS values and thus the deformation of the network increases significantly with decreasing a priori standard deviations. The pole offsets increase as well. This indicates that the quality of the local tie information is obviously more important than the quantity of co-location sites. A combination introducing only the lengths of the local tie vectors (solution type: D I S T ) gives RMS values and pole offsets in the same order of magnitude as the combinations using local ties in three components. It demonstrates that this solution type works in principle. If smaller a pri-
ori standard deviations are used, the RMS values increase, whereas the pole offsets are getting smaller. It is obvious that the connection of the VLBI and GPS networks is too weak for an a priori standard deviation of the length of local tie vectors of 3 ram. It has to be noted, that the implementation of vector lengths requires a set of well distributed co-locations and local networks with different relative positions of the GPS and the VLBI stations. Assuming, for example, that all co-locations are orientated identically, i.e., the VLBI stations are located in the west direction of the GPS stations, the use of length information only, will lead to singularities in the translation of the VLBI network. Fortunately, the relative positions of GPS and VLBI stations at the 26 co-location sites are very different. The results in Fig. 3 have shown, that there are large variations in pole offsets between GPS and VLBI between different solution types. This demonstrates the high dependency of the (realized) datum from the selected type of datum definition. If both technique solutions are solved separately, the d a t u m was introduced by NNR conditions w.r.t. ITRF2000 using a set of good and well distributed stations. The offsets in xand y-pole between both solutions a r e - 9 0 and 143 # a s respectively. Whereas, after the combination introducing 9 very good ties with an a priori standard deviation of 0.1 m m and defining the datum by NNR conditions using only GPS stations, the offsets become 4 and 35 # a s as shown in Fig. 4. Thus, it can be recognized, that offsets of 100 to 200 # a s between GPS and VLBI, as they are often seen in comparisons of both techniques, may result from inconsistencies of the underlying reference frames. 5
Simulation
studies
As shown in the previous section, it is generally possible to connect the station networks of the different techniques by introducing the lengths of the local tie vectors. At present, 26 co-location sites for GPS and VLBI with local tie information are available. It has been demonstrated that a combination based on the lengths of these 26 local tie vectors is not an approximate alternative compared to the introduction of 3D local tie information, because the information used for the connection is comparatively weak. In the following we investigate the situation for
Chapter 25 • Frontiers in the Combination of Space Geodetic Techniques L
1.0
J
GPS I
0.5 E
Z o.o 0 13_ x -0.5
GPS II VLBI
-1.0 2002
2003
2004
2005
Years
Figure 5: Simulated station distribution at colocation sites
xpole
RMScps RMSvLBI Offset
0.06 m a s 0.18 m a s 0.004 m a s
1.0
0.5
j 0.0 0 13_ >-0.5
-1.0 2002
r
1
2003
2004
2005
Years
ypole RMSGps RMSVLBI Offset
0.05 m a s 0.17 m a s 0.035 m a s
Figure 4- GPS (grey) and VLBI (black) x- and ypole time series w.r.t IERS-C04 for solution type 9 d 3 (or= 0.1ram). For RMS computation the offset to IERS-C04 is removed.
tation. To get a stand-alone VLBI solution it was solved by adding N N R and N N T conditions w.r.t ITRF2000. The simulated GPS network was created by setting up exactly the same dat u m information as for the VLBI solution. For the combination studies the GPS normal equation with full d a t u m information and the d a t u m free VLBI normal equation were used. The resulting transformation parameters of a 14 p a r a m e t e r Helmert-transformation between the combined solution and the stand-alone VLBI solution are a measure of the consistency of the combined frame. For these studies no discrepancies between the local tie information and the networks of the space techniques do exist and no deformation of the network occur within the combination. The local tie information was introduced with a priori s t a n d a r d deviations ranging from 1 to 100 mm. The transformation parameters of the different solution types are compared in Fig. 7.
a VLBI and a GPS network, assuming t h a t all VLBI stations are co-located to GPS. For this purpose a GPS network was simulated which contains 2 GPS stations per VLBI site. The network was created in such a way, t h a t the GPS stations form a rectangular triangle together with the VLBI station, as shown in Fig. 5. The VLBI network is a real network based on the d a t a described above. Figure 6 shows the network configuration. The 17 stations identified by a triangle are stations with a good co-location to GPS in reality. One solution type was performed connecting the terrestrial networks using only the length of local tie vectors. For comparison, solutions with local tie information in three components are solved. Altogether 6 different solution types (see Table 4) are compared.
Figure 6: Global VLBI network used for simulation studies. Stations signed with a triangle have good co-locations to GPS in reality.
The VLBI normal equation was free of dat u m information concerning translation and ro-
Using an a priori s t a n d a r d deviation of 1 m m for the local tie information the transformation
163
164
M. KriJgel • D. Angermann I
Table 4: Solution types for simulation studies solution type all ties 17 ties all dist 17 dist all dist x2
17 dist x2
description 3D local ties at all 56 VLBI sites 3D local ties at 17 VLBI sites the length of one local ties vector at all 56 VLBI sites the length of one local tie vector at 17 VLBI sites the lengths of two local ties vectors at all 56 VLBI sites the lengths of two local tie vectors at 17 VLBI sites
o
......
I
¢¢=¢CC"" go
~;o
a priori standard deviation of local ties/lengths [mm] 0
1
L.
-10
-20
-30 '0 a pdod standard deviation of local tie~engths [mm]
100
0
-10
-20
-30
parameters become zero for all solution types. That means, that the combined network is really consistent in these cases. By increasing the a priori standard deviations the connection of the VLBI and GPS network is getting weaker and weaker. Thereby, solution types with a small change of the transformation parameters are more stable than those with a higher change.
i 50 a pdod standard deviation of local ties/lengths [ram]
1 O0
I
"'
"= == ~
I
~
-10
-~
"zz.
-20
~
~
.
-30
50
1 O0
a pdon standard deviation of local ties/lengths [mm]
The investigations show, that for the current global VLBI network a combination based on only vector lengths of local ties at all VLBI sites provides a larger consistency of the combined network, than the use of 17 3D local ties. This can be explained by the amount and the global distribution of information used for the connection. If two vector lengths to GPS stations are introduced per co-location site, as expected, a further significant stabilization is achieved. Additionally, the elimination of possible orientation errors of the ties and the reduction of discrepancies in the height component argue for an application of vector lengths. The usage of free norreal equations of local networks (without any information about the orientation) instead of local ties in ITRF2000 datum also eliminates possible errors in orientation, however, the impact of the discrepancies in the height component does still exist. It can be eliminated by introducing only the north and east component of the tie. But at present only for a few sites (e.g. Ny Alesund) free normal equations of the local measurements are available.
I
-10
~
~
~
~
-20
-30
5'0
1~o
a priori standard deviation of local ties/lengths [mm]
-10
-20
-30
s'o
loo
a pdori standard deviation of local ties/lengths [mm]
all ties (#168) all dist x2 (#112) all dlst (#56)
-
-- 17 ties (#51) -- 17 dist ](2 (#34) 17 dlst (#17)
Figure 7: Transformation parameters (translation and rotation) of the combined solution w.r.t. the VLBI stand-alone technique solution for different combination strategies and different a priori standard deviations of the local ties or vector lengths. The number in paranthesis gives the sum of used local tie information.
Chapter 25 • Frontiers in the Combination of Space Geodetic Techniques
6
Conclusions and discussion
In this paper new strategies for the selection and implementation of local ties based on time series of space geodetic solutions containing station positions and E a r t h Rotation P a r a m e t e r s (ERP) have been discussed. Main criteria for the selection of co-location sites and local ties were a high consistency in the d a t u m realisation of the combined networks (i.e., a small offset between the E R P estimates of different techniques) and a minimal deformation of the networks due to the combination. The investigations have shown t h a t the most stable results arc obtained if only a subset of high-quality co-locations is used instead of all available co-locations. The results obtained from the different solution types for the combination of VLBI and GPS indicate, t h a t the offsets between the E R P estimates of both techniques are sensitive w.r.t, the d a t u m definition. By using only 9 co-locations the offset becomes almost zero. However, the sets of 9 co-locations between GPS and VLBI, which were identified here are only valid for this study. Using other input data other constellations of co-location sites may be selected. An alternative m e t h o d for local tie implementation was addressed, which uses only the length of local tie vectors instead of 3D components. A clear advantage is, t h a t the influence of possible errors in the height component and in the orientation of the local networks is significantly reduced or even elimated. Simulation studies for the connection of GPS and VLBI networks have shown, t h a t this approach provides good and stable results, if the co-location sites are well distributed. Additionally, it was shown, t h a t the combination gets more stable if two GPS antennas per site are installed. Besides this, such a constellation would enable the permanent mutual monitoring of the GPS stations to observe the effects of station events (e.g. an antenna change) with millimeter accuracy. Acknowledgements. A significant part of this work is supported by the G e r m a n Bundesministerium fiir Bildung und Forschung within the Geotechnologien project (03F0425C).
References Altamimi Z., Sillard P., Boucher C. (2002) ITRF2000: A new release of the International Terrestrial Reference Frame for earth science applications. J Geophys Res 107 (B7), 2214, doi: 10.1029/2001JB000561.
Angermann D., Drewes H., Krfigel M., Meisel B., Gerstl M., Kelm R., Miiller H., Seemiiller W., Tesmer V. (2004) ITRS Combination Center at DGFI: A terrestrial reference frame realization 2003. Deutsche Geod/itische Kommission, Reihe B, Heft Nr. 313. Boucher C., Altamimi Z., Sillard P., Feissel-Vernier M.(2004) The ITRF2000, IERS ITRS Centre, IERS Technical Note No.31, Verlag des Bundesamtes fib Karthographie und Geodgsie, Frankfurt am Main Ferland, R., Reference Frame Working Group technical report, in I G S 2001-2002 Technical Reports, Jet Propulsion Labratory Publication, Pasadena, California, 2004 Johnston, G., Dawson J. (2004) The 2002 Mount Pleasant (Hobart) radio telescope local tie survey, Geosience Australia, Record 2004/21, Canberra, Australia Kr/igel M., Angermann D. (2005) Analysis of local ties from multi-year solutions of different techniques. Proc. of the IERS Workshop on site co-location, Richter B, Dick W, Schwegmann W (Eds), IERS Technical Note No. 33, 32 37, Bundesamt fSr Kartographie und GeodS~sie, Frankfurt am Main. Matsuzaka S., Masaki Y., Tsuji H., Takashima K., Tsumtsumi T., Ishimoto Y., Machida M., Wada H., Kurihara S. (2004) V L B I co-location results in Japan In: Vandenberg N. and Bayer K.: International VLBI Service for Geodesy and Astrometry 2004 General Meeting Proceedings, NASA/CP2004-212255:138-142 Meisel B., D. Angermann, M. Kriigel, H. Drewes, M. Gerstl, R. Kelm, H. Miiller, W. Seemiiller, V. riles_ mer (2005) Refined apporaches for terrestrial reference frame computations Adv. Space Res 33(6) : 350-357, DOI: 10.1016/j.asr.2005.04.057 Ray J., Altamimi Z., (2005) Evaluation of co-location ties relating the V L B I and the GPS frames. Journal of Geodesy 79(4-5):189-195 doi: 10.1007/s00190005-0456-z
Rothacher M., Dill. R, Thaller D. (2005) I E R S Analysis Coordination. Observation of the Earth System from Space, Rummel, Reigber, Rothacher, BGdecker, Schreiber, Flury (Eds), Springer Verlag, in press. Sarti P., Sillard P., Vittuary L. (2004) Surveying colocated space-geodetic technique instruments for TRF computation, Journal of Geophysical Research, doi: 10.1007/s00190-004-0387-0 Steinforth, C., R. Haas, M. Lidberg, A. Nothnagel (2003) Stability of Space Geodetic Reference Points at Ny-Alesund and their Excentricity Vectors. In: W. Schwegmann and V. Thorandt: Proceedings of the 16th Working Meeting on European VLBI for Geodesy and Astrometry, Bundesamt ffir Kartographie und Geod~sie, Leipzig, Germany.
Thaller D., Dill R. Kriigel M., Steigenberger P., Rothacher M., Tesmer V. (2005) C O N T 0 2 Analysis and Combination of Long E O P Time Series. Observation of the Earth System from Space, Rummel, Reigber, Rothacher, BGdecker, Schreiber, Flury (Eds), Springer Verlag, in press.
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Chapter 26
Modifying the Stochastic Model to Mitigate GPS Systematic Errors in Relative Positioning D.B.M. Alves, J.F.G. Monico Department of Cartography, Faculty of Science and Technology Silo Paulo State University, FCT/UNESP, 305 Roberto Simonsen, Pres. Prudente, Silo Paulo, Brazil
Abstract.
The GPS observables are subject to several errors. Among them, the systematic ones have great impact, because they degrade the accuracy of the accomplished positioning. These errors are those related, mainly, to GPS satellites orbits, multipath and atmospheric effects. Lately, a method has been suggested to mitigate these errors: the semiparametric model and the penalised least squares technique (PLS). In this method, the errors are modeled as functions varying smoothly in time. It is like to change the stochastic model, in which the errors functions are incorporated, the results obtained are similar to those in which the functional model is changed. As a result, the ambiguities and the station coordinates are estimated with better reliability and accuracy than the conventional least square method (CLS). In general, the solution requires a shorter data interval, minimizing costs. The method performance was analyzed in two experiments, using data from single frequency receivers. The first one was accomplished with a short baseline, where the main error was the multipath. In the second experiment, a baseline of 102 km was used. In this case, the predominant errors were due to the ionosphere and troposphere refraction. In the first experiment, using 5 minutes of data collection, the largest coordinates discrepancies in relation to the ground truth reached 1.6 cm and 3.3 cm in h coordinate for PLS and the CLS, respectively, in the second one, also using 5 minutes of data, the discrepancies were 27 cm in h for the PLS and 175 cm in h for the CLS. In these tests, it was also possible to verify a considerable improvement in the ambiguities resolution using the PLS in relation to the CLS, with a reduced data collection time interval.
Keywords. Functional and Stochastic Penalised least squares, Systematic errors
models,
1 Introduction The GPS is a satellite-based radio navigation system providing precise three-dimensional position, navigation, and time information to suitably equipped users. The system is continuously available to a user anywhere in the world at any time, and is independent of the meteorological conditions (Seeber (2003)). But the GPS observables, like all other observables involved in the measurement processes, are subject to random, outliers and systematic errors. The random errors are inevitable, being, therefore, considered an inherent property of the observations. Outliers should be eliminated through the quality control process. A procedure extensively used in the navigation field is denominated Detection, Identification and Adaptation (DIA) (Teunissen (1998b)). Systematic errors can be modeled or eliminated by appropriate observation techniques. These errors can have a significant effect on GPS observables. So, this is a critical problem for high precision GPS positioning applications. In medium and long baselines, the major systematic error sources are the ionosphere and troposphere refraction and the GPS satellites orbit errors. But, for short baselines, the multipath is more relevant (Alves (2004b)). Recently, the semiparametric model and the penalised least squares technique have been proposed to mitigate these systematic errors, using single frequency receiver data (Jia (2000); Jia et al. (2001)). This method uses a natural cubic spline, whose smoothness is determined by a smoothing parameter, computed by using the generalized cross validation to model the errors as a function which varies smoothly in time. The systematic errors functions, ambiguities and station coordinates are estimated simultaneously. It will be shown that it is equivalent to change the stochastic model, in which
Chapter 26 • Modifying the Stochastic Model to Mitigate GPS Systematic Errors in Relative Positioning
the error functions (functional model) are incorporated in the stochastic model. As a result, the ambiguities and the station coordinates are estimated with better reliability and accuracy than the conventional least square method. Besides, the solution may require a shorter data collection interval, minimizing costs (Alves (2004a)). In this paper, the penalised least squares and the semiparametric model were used to mitigate the systematic errors in GPS relative positioning, using single frequency receiver data. Two experiments were accomplished. In first one, with a short baseline (-2 km), the multipath was the main error source. But, in the other one, with medium baseline lengths (-102 km), the atmosphere refraction and orbital errors were predominant. The theoretical revision, results and analyses are presented in this paper. 2 Systematic
Errors
Systematic errors on GPS signals can reduce significantly the precision and reliability of GPS relative positioning. However, for short baselines, errors can be reduced by double differencing, as for example, receiver and satellite clocks, satellites orbits and atmosphere refraction. But multipath is not reduced. Multipath is a phenomenon in which a signal arrives to the receiver antenna for multiple ways, due the reflection (Braasch (1991)). This effect is different for each station and depends of the antenna receiver localization. It is caused mainly due signal reflections in surfaces close to the receiver, such as constructions, cars, trees, hills, etc. Secondary effects are caused by reflections in the own satellite and during the signal propagation (Hofmann-Wellenhof (1997)). Multipath introduces significant errors in code and carrier measurements. So, several techniques have been developed to mitigate it. These techniques include the use of special antennas, several antennas arrangement, antenna localization strategies, software techniques, etc (Souza and Monico (2004)). But, these techniques are, in general, very difficult or expensive. In medium and, principally, in long baselines, the atmosphere conditions are poorly correlated. So, the effects caused in the rover and base stations are different. Therefore, the double differences don't
mitigate all errors caused by atmosphere effects and satellites orbit. In relation to atmosphere effects, the errors caused by troposphere and ionosphere can be compared. But the variability of ionosphere errors is larger than the troposphere, and it is also more difficult to model. The ionosphere errors vary from few meters to dozens of meters, while in the troposphere the zenith errors are usually between two and three meters (Klobuchar (1996)). The propagation delay in the ionosphere depends on the electron content along the signal path and on the frequency used. So, the GPS users can use double frequency receivers to take advantage of this property. But single frequency users don't have this possibility (Klobuchar (1996)). Besides, the ionospheric effect also varies with geographic localization and time (Seeber (2003)). The contribution of the ionosphere to the differential positioning error budget is estimated to be 1 to 2 parts per million (ppm). However, gradients up to 10 (15) ppm in the auroral region and up to 40 ppm in the equatorial region have already been reported (Fortes (2002)). The tropospheric delay is independent of the frequency. Usually, the wet and hidrostatic components express the troposphere influence in GPS measurement. The wet component depends on the distribution of the water vapor in the atmosphere, and is harder to model. But, it comprises just 10% of the total tropospheric refraction. The hidrostatic component is precisely described by familiar models (Seeber (2003)). The contribution of the troposphere to the differential positioning error budget varies typically from 0.2 to 0.4 ppm, after applying a model. However, before applying a model, the contribution of the wet component varies from 1 to 4 ppm, which depends strongly on the satellite elevation angle (Fortes (2002)). Discrepancies between the ephemerides available to the user and the actual orbit are propagated into the determined positions of the user antenna (Seeber (2003)). Information about the GPS satellites orbits can be obtained by broadcast or precise ephemerides. The broadcast ephemerides are transmitted by GPS satellites in the navigation files, and your accuracy is about 2 m (IGS, (2005)). The precise ephemerides are provided by International GNSS Service (IGS), and your accuracy is about 10 cm in real time (IGS, (2005)).
167
168
D.B.M.Alves.J. F.G.Monico 3 Semiparametric Model and Penalised Least Squares Technique In a semiparametric model the computed variables are divided into two parts: the parametric part and the nonparametric one. Usually, the parametric part is of interest to the users. In the GPS case, the parametric part can be the site coordinates and the carrier phase ambiguities. The nonparametric part can represent a combination of any error functions that vary smoothly with time (Jia et al. (2001)). A vector semiparametric model can be expressed as:
y, = A , x + M , g ( 6 ) + g ,
i=1,2 ..... n,
where E -1 is the weigh matrix to the observations, czj is the smoothing parameter (Alves (2004b)) and g~(t) is the second-order derivative of the jth function with respect to time. Equation (3) defines the penalised quadratic form. The first part of equation (3) is the least squares residual quadratic form, and the second one is the roughness penalty term. The second part represents the roughness of the functions. The roughness penalty term can be expressed as (Fessler (1991)):
(1)
where Yi • ~ m are the observations (carrier phase or pseudorange double differences); A i e [Rm~ is the design matrix; x • [P-,P is the parameters vector including the carrier phase ambiguities and site coordinates; Mi • J R "Xq is the incidence matrix (Green and Silverman ( 1 9 9 4 ) ) ; g ( t ~ ) • l R q are the systematic error functions; ti is the time index;ci •JR m are the random errors; n is the number of epochs; m is the number of observations per epoch; q is the number of error functions; and p is the number of estimated parameters. Equation (1) contains m*n observations and q*n+p unknowns. So, two cases can be considered for this situation. First, if the number of unknowns is larger than the number of the observations. In this case, equation (1) can not be solved by using the traditional least squares technique. Second, even if the number of the unknowns is less than the number of the observations, equation (1) can not provide a stable solution when the traditional least squares technique is used, due the number of unknowns to be larger than the usual, because the error functions (g(ti)) also have to be computed. in order to obtain a reliable resolution, an additional constraint must be added. This constraint is called roughness penalty term, giving as:
(4)
j=l
where Q and R are the matrices related in Fessler (1991) and Alves (2004b); Iq is an q xq identity matrix; D(c~) = diag(a~l, ..., %); @denotes the Kronecker product. So, substituting equation (4) into equation (3), and minimizing in relation to x and g, the following equations are obtained:
(5)
®z -') ®
® ,,, Xe
(e ®
(6)
®
where In is an n x n identity A = (A 1 A,)r, y = ( y , .., y,)r, ~-, , '"''
=
'"
matrix, ~_ ,
----'''----
n'
"',gl ,g2,'",g2 , " ' , g , , ' " ,
Equations (5) and (6) can be solved using a direct method (Green e Silverman (1994)), substituting equation (6) into equation (5). So, after accomplishing some mathematical manipulations, the final result is given as: (7)
a Ign(/~)2d[.
(2)
Thus, the penalised least squares technique is used. So, the function that will be minimized is:
(y,- A,x- u,g(,.))~Z;'
(y, - A,x - U,g(,,
))+
i-1
~ c ~ / I(g;f(t))2dt-min, j 1
where As = SA is the design matrix smoothed, y~. = Sy is the observaction vector smoothed, and S is the smoothing matrix, givin as:
(3)
(s)
Chapter 26 • Modifying the StochasticModel to Mitigate GPS SystematicErrorsin Relative Positioning
The equation (7) can be written by:
x = [AT ((In ® Z-1)- (In ® Z-1)S )A 1
(9)
It can be seen in equation (9) that the original weight matrix ( I n ® Z -1 ) was changed by another weight matrix ((In ® Z -1)- (I n ® Z-')S)o This happens because the functional model was augmented with extra parameters. An equivalent result could be obtained if the stochastic model was changed instead. This equivalence introduces great flexibility into estimation algorithms, with a wide variety of geodetic applications (Blewitt (1998)).
4 Experiments
and
Analysis
In order to test the penalised least square technique and the semiparametric model, data from two baselines, UEPP-TAKI (~2 km) and UEPP-QUIN (~ 102 km), were used. The data were colleted on 13 and 20 September 2003 to the first baseline, and on 16 and 17 July 2003 to the other one. It was used a Trimble 4600 LS and an Asthech ZXII receivers in the rover stations, which are of single and double frequency respectively. But just data collected with the single frequency receiver were processed. The double frequency receiver data was used to compute the considered "ground truth" coordinates. In all experiments, UEPP station was used as the base station. UEPP is a permanent station from the Brazilian Continuous GPS Network (Rede Brasileira de Monitoramento Continuo - RBMC), located in Silo Paulo state, Brazil (Fortes (1997)). It collects data continuously with a Trimble 4000 SSI (double frequency) receiver. Researchers from the Faculty of Science and Technology (FCT/UNESP) in Brazil have been developing a GPS data processing software, called GPSeq (Machado and Monico (2002)). It was implemented with the conventional least squares method for single frequency data, where phase and pseudoranges double difference are the basic observables. The penalised least squares technique and the semiparametric model were implemented in the GPSeq software. So, in this paper, the results obtained by penalised least squares (PLS) were compared with the conventional least squares (CLS). In all experiments the data were collected with sampling rate of 15 seconds and a cut off elevation
angle of 15 °. During the data processing, the original sampling rate was maintained. Broadcast ephemerides were used to compute the satellite positions and no tropospheric and ionospheric models were used. For solving the ambiguities, the Least Squares Ambiguity Decorrelation Adjustment (LAMBDA) method was used. This method first decorrelates the ambiguities and next computes integer least-squares estimates for the ambiguities in a highly efficient way (Teunissen (1998a)). The ambiguity resolution validation was performed by using the ratio test (Teunissen (1998a)). In this test it is computed the ratio of the second best norm of the residual by the first one. The threshold value of the ratio test was 1.5, accordingly to Jia et al. (2001). Besides, the satellite that had the large elevation angle was chosen as base satellite. The first experiment had as main goal to verify if the penalised least squares can mitigate the multipath effects on GPS signals. So, a short baseline was used, UEPP-TAKI, where the atmosphere effects and the orbital errors are reduced by double differencing. Therefore, initially, the data were collected near a reflector surface (a cart), that was 6 m far from the receivers and 1.3 m from the ground. Its dimensions were 13 m and 2.5 m of length and width respectively. Later on, the cart was removed and another data collection was accomplished in the same place and considering the same sideral time interval. This second data collection was realized just for computing the "ground truth". For testing the proposed method, four sub-sets of data were processed; two sub-sets of 5 min (M1 and M2 sessions) and two of 10 min (M3 and M4 sessions), all with 8 satellites. The resultant coordinates discrepancies (AE2 + AN 2 + Ah 2)~ in relation to the ground truth coordinates, between PLS and CLS are shown in Figure 1.
0,035 0,03 0,025 "t~ 0,02 ~. 0,015
~,
-=
[] CLS 131
O,Ol
----
o 0,005
;~
0. M1
M2
M3
Hh
[] PLS
M4
Sessions
Fig. 1. Resultant coordinates discrepancies in relation to the ground truth for 5 and 10 min of data processing in the UEPP-TAKI baseline
169
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D.B.M. Alves. J. F. G. Monico
From Figure 1 one can observe that the coordinates discrepancies estimated from the PLS are better than those from the CLS. The largest coordinate discrepancies were in h component, whose values are shown in table 1.
resultant coordinates discrepancies in relation to the ground truth coordinates, between PLS and CLS methods.
2 1,5
Table 1. Largest coordinates discrepancies for the UEPPTAKI baseline Sub-Set Method Session Discrepancies(mm) M1 11 CLS 5 min M2 33 M1 9.3 PLS M2 16 M3 18 CLS M4 12 10rain M3 12 PLS M4 7.7
._B Q
[] CLS
I!
0,5
7
0 Q1
Q2
[] P L S
, Q3
Q4
Sessions
Fig. 3. Resultant coordinates discrepancies in relation to the ground truth for 5 and 10 min of data processing in the UEPP-QUIN baseline (~ 102 km)
The largest coordinates discrepancies are shown in table 2. Table 1 shows that the largest coordinates discrepancies are also related with CLS method. In relation to the ambiguity resolution, Figure 2 shows the values for the ratio test. It is possible to verify that, in all cases, the values obtained for the PLS are larger than that of the CLS. This indicates that the PLS solution may be more reliable than CLS.
Table 2. Largest coordinates discrepancies for the UEPPQUIN baseline Sub-Set 5 rain
10min
18
~o
~.
Method Session Discrepancies(cm) CLS Q1 175 in h Q2 166 in h PLS Q1 20 in h Q2 27 inh CLS Q3 150 in E Q4 92 in h PLS Q3 15 in E Q4 14 in E
[] C L S 9
[] PLS
3 0 M1
M2
M3
M4
Sessions
Fig. 2. Ratio test for 5 and 10 min of data processing in the UEPP-TAKI baseline
The other experiment had the goal of verifying if the PLS could be used to mitigate the atmospheric effects and the orbital errors for a medium baseline. So, the UEPP-QU1N (~102 km) baseline was used. In this baseline four sub-sets of data, two with 5 min (Q 1 and Q2) and two with 10 minutes (Q3 and Q4), were considered. Besides, these data were collected at about 2 pm (local time), when the ionospheric effects are more relevant. Figure 3 shows the
In these experiment, the PLS coordinates quality is also better than the CLS. This indicates the efficiency that the proposed method may provide for mitigating systematic errors. In relation to the ambiguity reliability, Figure 4 shows the ratio test values.
2,5 2 I-
1,5
0,5 0,
I I Q2
Q3
II II Q4
Sessions
Fig. 4. Ratio test for 5 and 10 min of data processing in the UEPP-QUIN baseline
Chapter 26 • Modifying the StochasticModel to Mitigate GPS SystematicErrorsin Relative Positioning
From Figure 4 it can be verified that, again, the values of the ratio test are larger in the PLS method than in CLS. This indicates, as stated before, that the PLS solution may be more reliable than CLS. However, due to the long baseline length together to the short occupation time, it may imply some wrong ambiguities. 5 Conclusions
In this paper, the semiparametric model and the penalised least squares technique were used to mitigate systematic errors on GPS signals. The fundamental of the method was presented. It was shown that changing the functional model is equivalent to change the stochastic model instead. In order to test the advantages of the method, two experiments were realized. While in the first experiment multipath was the main error source, in the other one, the atmosphere refraction and orbital errors were predominant. Results showed that in all experiments the coordinates discrepancies obtained by penalised least squares were always smaller than those obtained with conventional least squares. Additionally, the ratio test values were always larger in the penalised least squares method than in the conventional one. This indicates that the ambiguity solution by this method may be more reliable. These preliminary results have shown that the proposed method may be quite efficient in mitigating systematic errors. Therefore, more experiments must be realized in the future to effectivey proof this statement. They are been carried out now to better qunatify the results Acknowledgments
The authors would like to thank FAPESP for the financial support (01/11858-9 process) for the first autor and to CNPq and Fundunesp for funding the participation in the Dynamic Planet 2005 conference. References
Alves, D. B. M. (2004a). M~todo dos Minimos Quadrados corn Penalidades: Aplicaq~o no posicionamento relativo GPS. 133f. Dissertagfio (Mestrado em Ci~ncias
Cartogrfificas) - Faculdade de Ci6ncias e Tecnologia, Universidade Estadual Paulista, Presidente Prudente. Alves, D. B. M. (2004b). Using Cubic Splines to Mitigate Systematic Errors in GPS Relative Positioning. In: Proc. of ION GNSS 2004, Long Beach, California. Blewitt, G. (1998). GPS Data Processing Metodology. In: Teunissen, P. J. G.; Kleusberg, A. GPS for Geodesy. Berlin: Springer Verlage, pp.231-270. Braasch, M. S. (1991). A Signal Model for GPS, Navigation. vol. 37, no. 4, pp. 363-377. Fessler, J. A. (1999). Nonparametric Fixed-Interval Smoothing With Vector Splines. In: Proc. IEEE Transactions on Signal Processin, pp. 852-859. Fortes, L. P. S. (1997). OperacionalizaqCto da Rede Brasileira de Monitoramento Continuo do Sistema GPS (RBMC). 152f. Disserta~fio (Mestrado em Ci6ncias em Sistemas e Computa~o)- Instituto Militar de Engenharia (IME), Rio de Janeiro. Fortes, L. P. S. (2002). Optimising the Use of GPS MultiReference Stations for Kinematic Positioning. 2002. 355f. Thesis (PhD)- University of Calgary, Calgary. Green, P. J. and B. W. Silverman (1994). Nonparametric Regression and Generalized Linear Models': a roughness penalty approach. 1.ed. London: Chapman & Hall. Hofmann-Wellenhof, B. et al. (1997). GPS Theory and Practice. Wien: Spring-Verlage. 326p. Jia, M. (2000). Mitigation of Systematic Errors of GPS Positioning Using Vector Semiparametric Models. In: Proc. oflON GPS 2000, Salt Lake City, pp. 1938-1947. Jia et al. (2001). Mitigation of Ionospheric Errors by Penalised Least Squares Technique for High Precision Medium Distance GPS Positioning. In: Proc. of KIS 2001, Banff, Canada. Klobuchar, J. (1996). A. Ionospheric Effects on GPS. In: Parkinson, B. W. and J. J. Spilker. Global Positioning System: Theory and Applications. Cambridge: American institute of Aeronautics and Astronautics, pp.485-515. Machado, W. C. and J. F. G. Monico (2002). Utiliza~o do software GPSeq na solu~fio rfipida das ambigfiidades GPS no posicionamento relativo cinemfitico de bases curtas. In: Pesquisa em Geocidncias, Porto Alegre, pp.89-99. Seeber, G. (2003). Satellite Geodesy: Foundations, Methods, and Applications. Berlin, New York: Walter de Gruyter. Souza, E. M. and J. F. G. Monico (2004). Wavelet Shirinkage: High frequency multipath reduction from GPS relative positioning. GPS Solutions, vol. 8, no. 3, pp. 152159. Teunissen, P. J. G. (1998a). GPS Carrier Phase Ambiguity Fixing Concepts. In: Teunissen, P. J. G.; Kleusberg, A. GPS for Geodesy. Berlin: Springer Verlage, pp.271-318. Teunissen, P. J. G. (1998b). Quality Control and GPS. In: Teunissen, P. J. G.; Kleusberg, A. GPS for Geodesy. Berlin: Springer Verlage, 1998b, pp.271-318.
171
Chapter 27
GPS Ambiguity Resolution and Validation Under
Multipath Effects: Improvements using Wavelets E.M. Souza, J.F.G. Monico Department of Cartography S~o Paulo State University- UNESP, Roberto Simonsen, 350, Pres. Prudente, SP, Brazil
Abstract. Integer carrier phase ambiguity resolution is the key to rapid and high-precision global navigation satellite system (GNSS) positioning and navigation. As important as the integer ambiguity estimation, it is the validation of the solution, because, even when one uses an optimal, or close to optimal, integer ambiguity estimator, unacceptable integer solution can still be obtained. This can happen, for example, when the data are degraded by multipath effects, which affect the real-valued float ambiguity solution, conducting to an incorrect integer (fixed) ambiguity solution. Thus, it is important to use a statistic test that has a correct theoretical and probabilistic base, which has became possible by using the Ratio Test Integer Aperture (RTIA) estimator. The properties and underlying concept of this statistic test are shortly described. An experiment was performed using data with and without multipath. Reflector objects were placed surrounding the receiver antenna aiming to cause multipath. A method based on multiresolution analysis by wavelet transform is used to reduce the multipath of the GPS double difference (DDs) observations. So, the objective of this paper is to compare the ambiguity resolution and validation using data from these two situations: data with multipath and with multipath reduced by wavelets. Additionally, the accuracy of the estimated coordinates is also assessed by comparing with the "ground truth" coordinates, which were estimated using data without multipath effects. The success and fail probabilities of the RTIA were, in general, coherent and showed the efficiency and the reliability of this statistic test. After multipath mitigation, ambiguity resolution becomes more reliable and the coordinates more precise.
Keywords. Ambiguity, multipath, wavelets, Ratio Test Integer Aperture (RTIA)
1 Introduction Rapid and high precision relative positioning using GNSS requires the use of the very precise carrier phase measurements. However, the carrier phases are ambiguous by an unknown number of cycles. The double difference (DD) ambiguities are known to be integer-valued, and this knowledge has been exploited for the development of integer ambiguity resolution algorithms (Verhagen 2005). The estimation process consists of three steps. Firstly, a least-squares adjustment is applied in order to obtain the float solution. All unknown parameters are estimated as real-valued. In the second step, the integer constraints on the ambiguities are considered. Thus, the float ambiguities will be mapped to integer values. In relation to the map, different choices are possible. The simplest is to round the float ambiguities to the nearest integer values or to conditionally round so that the correlation between the ambiguities is taken into account. The optimal choice is to use the integer least-squares estimator (LAMBDA Method), which maximizes the probability of correct integer estimation. After fixing the ambiguities to their integer values, the last step is to adjust the remaining unknown parameters considering their correlation with the ambiguities (Verhagen 2005). However, as important as the integer ambiguity estimation, it is their validation, because, even when one uses an optimal, or close to optimal, integer ambiguity estimator, one can still come up with an unacceptable integer solution. This can happen for example when the data are degraded by multipath, which can affect the real-valued float ambiguity solution, and it can lead to an incorrect integer (fixed) ambiguity solution. Thus, it is important to use a statistic test that has a firm theoretical and probabilistic footing, which is possible by using the Ratio Test Integer Aperture (RTIA) estimator (Teunissen and Verhagen 2004).
Chapter 27 • GPS Ambiguity Resolution and Validation under Multipath Effects: Improvements Using Wavelets
In order to analyze the performance of ambiguity resolution and validation, an experiment was carried out using data with and without multipath effects. Reflector objects were placed surrounding the receiver antenna aiming to cause multipath. A method based on Multi-Resolution Analysis (MRA) by wavelet transform is used to reduce multipath of the GPS DD observations. So, the objective of this paper is to compare the ambiguity resolution and validation using data from these two situations: data with multipath and with multipath reduced by wavelets. Furthermore, the accuracy of the coordinates is analyzed by comparing with the "ground truth" coordinates estimated using data not affected by multipath.
Teunissen (1998a) introduced a new class of ambiguity estimators, and it is called the class of |nteger Aperture (IA) estimators. It is defined by dropping the condition that no gaps are allowed between the pull-in regions, so that all pull-in regions should not cover the complete space R""
U f~ -f~'
where f2 c R"
(3a)
ZE Z n
s,,,(a
s
,(n ) - e . v . . z
f~ -z+f~0,
z"...
z
Vze Z n
where ~ c R ~ is called the aperture space. Thus, the integer aperture estimator (Teunissen and Verhagen 2004) is given by
(3b) (3c) m
N
2 Integer Estimation and Validation In this section a brief introduction related to integer estimation and validation is presented based on the works developed at the Delft University of Technology. In the integer estimation, different real-valued ambiguity vectors N are mapped to the same integer vector. So, a subset &oR ~ can be assigned to each integer vector z~ Z" (Teunissen 1998a):
&={xe R"lz=S(x)},ze z".
(1)
The subset S~ has all real-valued float ambiguity vectors that will be transformed to the same integer vector z, and it is called the pull-in region of z. An integer estimator N is said to be admissible if its pull-in regions S~ satisfies (Teunissen 1998a)
Us Z- R ~
(2a)
zE Z n
Int(S )c~ Int(S Z) - •, Vu, z • Z n , u :/: z
S Z - z + S O, V z • Z "
(2b) (2c)
where Int denotes the interior of the subset. The first condition means that all real-valued vectors will be transformed to an integer vector. For the second condition, the float solution is mapped to just one integer vector. The last condition is related to the translational invariance. An admissible integer estimator is the Integer Least-Squares (ILS) estimator, which is used in the LAMBDA (Least squares AMBiquity Decorrelation Adjustment) (Tiberius and de Jonge 1995; Teunissen 1998b).
Therefore, for the last equation, when ~r~ f~ the ambiguity will be fixed using one of the admissible integer estimators, otherwise the float solution should be maintained. This means i)
Ne ~N
success" correct integer estimation;
ii) N ~ ~ \ ~ N fail: incorrect integer estimation. Consequently, the corresponding success (Ps) and fail (Pj) probabilities or rates are given by (4a)
P~ - P 0 V - N ) - ff:~ (x)dx
(4b) (4c)
where f ~ ( x )
is the probability density function of
_ £r_ ~ and f~ (x) of the float ambiguities. However, for a user it is important that the fail rate is below a certain limit. The approach of integer aperture estimation allows choosing a threshold for the fail rate, and then determines the size of the aperture pull-in regions such that the fail rate will be equal to or below this threshold. So, applying this approach means that the ambiguity estimate is validated using an appropriate criterion (Teunissen and Verhagen 2004). Thus, an IA estimator is the RTIA, which is the inverse of the popular Ratio test: Accept N if: ~ - ~, 2Q~ ~ _ ~ 2
<s,,
o
(5)
173
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E.M. Souza. J. F. 6. Moni¢o
where the critical value is denoted as ¢t and Q~ denotes the vc-matrix of N . The acceptance region or aperture space is given by
~ - x e R"
x - 2, 2Q,,~ ~ fil X__ ~ 2 2Q; , 0
(6)
a , b ~ R,
a :/= O
(7)
where a represents the dilatation parameter and b the translation parameter (Daubechies 1992). The wavelet transform of the DD signal f is given by (\ vf , ~t"'h/. The signal can be reconstructed from: /
with 2, and 22 the best and the second best ILS estimator of x. Thus, the acceptance region is qualified and quantified by means of equation (6). Furthermore, an exact and overall probabilistic evaluation of the ambiguity estimation and validation is available. in Teunissen and Verhagen (2004) simulations were carried out which indicated that the Ratio Test gives a close to optimal performance. However, this is only valid when the fixed fail rate approach is used, but not if the classical approach of using a fixed critical value is applied. More details are presented in the experiment description and results.
3 GPS DD Signal Denoising There are several methods for denoising GPS signal. A recent one is based on a unitary transform Time Frequency Denoising Operator (TFDO) that involves fractional Fourier transform for multipath reduction (Jarrot et al 2005). This method showed to be very promising but the accuracy of the results was not analyzed. Another one uses a Reweighed Least Squares for COrrelated observations (RLSCO) (Wieser and Brunner 2002). To take multipath, diffraction and noise into account, a fuzzy variance model was used. It combines the signal quality provided by GPS receiver together with the quality derived from the parameter estimation process (Wieser 2001). Other denoising methods involve wavelets of any form and have provided very good results (Xia 2001; Satirapod et al 2001). In this work the denoising is performed by MRA and wavelet shrinkage.
3.1 Wavelet Method (WM) Wavelets are building block functions localized in time or space. They are obtained from a single function g4t), called the mother wavelet, by translations and dilations, given by:
f = ~1f ~ f ~ ( f ' g t " ' b )gt.,b dadb a '
(8)
where the constant C~, depends only on N and is given by (Daubechies 1992)"
C¢ =2r~+i ~ " ( ~ ) 2 d ~ < o° .
(9)
--co
A way of facilitating the discretization and allowing obtaining fast algorithms becomes possible with an appropriate choice of the mother wavelet. We used Symmlets with 8 coefficients. Souza and Monico (2004) showed that this mother wavelet is better than other Symmlets and Daubechies wavelets. To perform the decomposition, the MRA was used, where a p p r o x i m a t i o n s and d e t a i l s are words frequently used. The approximations are the high scale, low frequency components of the signal. The details are the low scale, high frequency components. In the filtering process, the original DD signal passes through two complementary filters and emerges as two signals. The approximation is then itself split into a second level of approximation and detail, and the process is repeated (Mallat 1998). The DD signal is broken down into several lower resolution components (wavelet coefficients). After the decomposition, the wavelet shrinkage is performed to remove the components relative to high frequency multipath and noise using a threshold. There are many ways of applying a threshold. To deal with the time variant GPS DD signal, an automatic threshold for the wavelet coefficients is necessary to obtain an efficient multipath reduction. For this purpose, the hard threshold function and the universal threshold are applied to the d coefficients. In this step, it is important to compute the noise level for each o f the = m e d i a n l [[[_d ~ :0 < k < n/21/O 6745 / J [__ ' n DD observatfons in the first decomposition level (finest scale) (Donoho and Johnstone 1994). More [
,
Chapter 27
• GPS Ambiguity Resolution
details and results of this process are given in Souza and Monico (2004). It is important to note that when these methods are applied in the DD observations, not all multipath effects are mitigated, because the low one cannot be separated from the other DD signal components. Just the high frequency multipath from long delays was attenuated. But even in this case, significant improvements can be obtained (Souza 2004).
and Validation under Multipath Effects: Improvements Using Wavelets
resolution at the last epoch and the success (P,) and fail (Pu) probabilities of RTIA were computed to verify the correctness of the ambiguities. The wavelet method (WM) was also applied to the same situation in order to compare the RTIA results before and after multipath mitigation. In Figure 1 are shown the best and the worst DD carrier phase residuals for the processing using data affected by multipath from one of the three days (September 17th) experiments.
4 Experiments Description Aiming to verify the RTIA performance, an experiment was conducted such that the probability of successful ambiguity resolution was degraded. Data were collected on September 16 to 21, 2003, at Pres. Prudente, Brazil. UEPP permanent station from the Brazilian Continuous GPS Network (RBMC) was used as reference station. This experiment was performed using data contaminated by multipath, which affect the realvalued float ambiguity solution, and it can lead to an incorrect integer (fixed) ambiguity solution. Thus, reflector objects were placed surrounding an Astech ZXII receiver antenna to cause reflection of the GPS signals. Later on the experiment was repeated but without the reflector objects. This allows computing the "ground truth" coordinates and to assess the actual level of multipath in each of the experiments. In each configuration, data were collected for three consecutive days to verify the multipath repeatability in the DD residuals and to make sure that the errors were due to multipath. As the baseline length was of about 1 km, the atmospheric effects are assumed to be insignificant after the differentiation process. Therefore, DD residuals may exhibit, mainly, multipath errors.
5 Experiments Results and Analysis To verify the RTIA efficiency and reliability for ambiguity validation under multipath effects, 97 processings were accomplished using only the L1 GPS carrier. The GPSeq software, which is under development at UNESP, was used (Machado and Monico 2002). Firstly, the three initial epochs were processed. Later on the 4 initials and so forth until all 100 epochs were processed. In each processing, the LAMBDA method (Tiberius and de Jonge 1995; Teunissen 1998b) was used for ambiguity
0,08 0,06 0,04 0,02
t,k,
-0,02 -0,04
DD23-18
. . . .
' LJ"
0
DD23-21
-0,06 -0,08 0
10
20
30
40
50
60
Epochsof 15 s
70
80
90
100
Fig. 1 Best and worst carrier phase DD residuals set. From Figure 1 it is possible to observe that after the 25 th epoch the multipath errors increased. The RTIA values and its Ps and Pf were computed before and after applying the WM to mitigate multipath. Ps and Pf are shown in figures 2 and 3, respectively. For implementation, the approach described by Teunissen and Verhagen (2004) and Verhagen (2005) was followed. To determine the aperture parameter for a given fail rate and consequently the 'size' of the aperture pull-in regions ~z such that the float ambiguity solution lies on the boundary of ~ , we proceeded as follows. First, the float and fixed solution were computed, and then it was determined what would be the fail rate if the float solution would fall on the boundary of f~'~. In this process
'/'/' = )Q -- IV1 o~ 2 / )Q -- N2 20~ w a s
computed and the fail rate was found using 500000 simulations. If this fail rate was larger than 0.025 (user-defined), ~ ' was larger than allowed and the float solution lied outside the aperture space corresponding to Pf- 0.025. The fixed solution was accepted if Pf was lower than 0.025, otherwise it was rejected and the float solution used.
175
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E.M. 5ouza. J. F. 6. Monico
1
1
0,8
0,5
0,6 0,4
¢
Ps.
•
Ps. Wav
N Float
0
N Fixed
............. . ~
N Way -0,5
0,2 0
II
.
.
D
.
10
~ _ ~ ~ _ _ ~ _
.
20
30
40 50 60 70 Epochs of 15 s
80
',
90
-1 0
100
10 20
30
40
50
60
70
80
90 100
Epochs of 15 s
Fig. 2 Success probabilities before (Ps) and after WM (Ps Wav).
Fig. 5 ComponentN discrepancies
1 0,8 1
0,6
,
0,4
-
Pf. m._wav
U Float
0,5 --~
Iio
0,2 o
0
10
. . . . . 30 40 50 60 70 80 Epochs of 15 s
20
F i x Fail lO~Rate: 0 . 0 2 5
90
U Fked U Way
-0,5 -1 0
Fig. 3 Fail probabilities before (Pf) and after WM (PfWav)
10
20
30
40
50
60
70
80
90
100
Epochs of 15 s
Fig. 6 Component U discrepancies It can be seen in Figure 2 that after the 30 th epoch, the Ps became very low and PU(Figure 3) was larger than 0.025, indicating the fixed ambiguity solution should be rejected. This is coherent with the data, because after the 30 th epoch, there was a larger incidence of multipath (Figure 1). Once the WM was applied and multipath reduced, the fail probability decreased significantly. It improved up to 98% and after the 90 th epoch, the RTIA indicated that the fixed ambiguity could be accepted (Figure 3). It shows how the RTIA estimator is sensitive to multipath and the effectiveness of reducing multipath using wavelets The user is, in general, more interested in the impact of the ambiguity solution (float or fixed) on the quality and reliability of the coordinates. In order to compare the accuracy of the coordinates components E,N,U, the discrepancies in relation to the "ground truth" coordinates were computed for each of the 97 processings (Figures 4, 5 and 6). It is
important to emphasize that the "ground truth" coordinates were estimated using data collected in the absence of the reflector objects. The large discrepancies related to the float solution, from the beginning to around epoch 50 (Figures 4, 5 and 6) are due to the fact that the float ambiguities require some time to converge (see Figure 7). The best and the worst standard deviations of the float DD ambiguities are shown in Figure 7.
21,81,61,41,2~. 1"-" 0,80,60,40,20-0,2 0
--e-- DD23-18 DD23-21
10
20
30
40
50
60
70
80
90
100
Epochs of 15 s
Fig. 7 Best and Worst standard deviations of the ambiguities
1 E Float
0,5 I 0
n n
-1 0
10
20
30
40
50
60
70
Epochs of 15 s
Fig. 4 Component E discrepancies
80
90 100
E Fixed
One can see in Figure 7 that the precision of the float solution stabilized around the 5 0 th epoch. This is in agreement with figures 4, 5 and 6, where the coordinate components using the float solution presented smaller discrepancies than those using the fixed solution without the WM.
Chapter 27 • GPS Ambiguity Resolutionand Validation under Multipath Effects:Improvements UsingWavelets
The root mean squares (RMS) of these discrepancies are shown in Figure 8. For the float solution, only results from epoch 50 forward were considered. The coordinate discrepancies after multipath mitigation using the W M were the smallest ones for the three components: E, N and U. This can also be seen from figures 4, 5, 6 and 8. It reached 75%, 80% and 75% of improvement in the RMS of E, N and U components respectively in relation to the coordinates obtained using the fixed solution without the WM. Therefore, one can conclude that multipath was the main error affecting the ambiguity resolution and that the W M reduced significantly this error.
0,10
reliable and the fail probabilities decreased significantly (Figure 2). The accuracy of the coordinates after applying the W M also improved up to 80% (Figure 8). Although the experimental verification is still very limited, the results indicated that the W M is very promising to mitigate multipath and the RTIA pointed it out. A more complete data set is being analyzed and the results will be confronted with other multipath mitigation procedures.
Acknowledgments
The authors thank FAPESP for the financial support to the first author (03/12770-3) and Sandra Verhagen for the contributions.
0,08
0,08
~" 0,06
o,o ,ill)
o,oo, Illllllllll~)))m
---li'06)~
"ll l
llllllllll!i ,,,,
References .Float .
,xe0
.way
E
Fig. 8 RMS of the coordinate component discrepancies
Furthermore, after about the 50 th epoch, the coordinates from the float solution were more accurate than those from the fixed solution without mitigation of multipath. This is due to the convergence of the ambiguities (Figure 7) together with the multipath effects. This result has quite good coherence with the RTIA estimator, which indicated that the fixed solution could not be accepted from epoch 50 to 100.
6 Conclusions
The integer ambiguity estimation and validation, as well as a preliminary probabilistic evaluation of the solution with the RTIA were analyzed. The sensitivity of this test to multipath was verified in the experiments. When the multipath significantly affects the data, the RTIA indicated the fixed solution should be rejected and the results were, in general, better if the float solution were used. Furthermore, after multipath mitigation using the WM, the ambiguity solution became more
Daubechies I (1992) Ten Lectures on Wavelets. Vol. 61 Regional Conference, SIAM, Philadelphia, PA. Donoho DL, Johnstone IM (1994) Ideal Spatial Adaptation by Wavelet Shrinkage. Biometrika, Vol 81, p.425-455. Jarrot A et al (2005) Multi-Component Signal Denoising Using Unitary Time-Frequency Transforms, In: Eusipco 2005, Antalya, Turkey, 4-8 September. Machado WC, Monico JFG (2002) Utiliza?~o do software GPSeq na Solu?~o r/Ipida das ambigfiidades GPS. Pesquisas em GeociYncias, v. 29 (2), pp 89-99. Mallat S (1998) A wavelet tour of signal processing. United States of America: Academic Press, 577 p. Satirapod C et al (2001) An approach to GPS analysis incorporating wavelet decomposition. Artificial Satellites, 36, pp 27-35. Souza EM (2004) Multipath Reduction from GPS Double Differences using Wavelets: How far can we go? In: ION GNSS, 17, Long Beach, CA, pp 2563-2571. Souza EM, Monico JFG (2004) Wavelet Shrinkage: High frequency multipath reduction from GPS relative positioning. GPS Solutions. Vol 8, No. 3, pp 152-159. Tiberius CCJM, De Jonge P (1996) The LAMBDA method for integer ambiguity estimation: implementation e aspects.
Delft Geodetic Centre (LGR). Teunissen PJG (1998a) On the integer normal distribution of the GPS ambiguities. Artificial Satellites, 33(2), pp 49-64. Teunissen PJG (1998b) GPS Carrier Phase Ambiguity fixing concepts. In: Teunissen, P.J.G.; Kleusberg, A. GPS for Geodesy. 2 ed. Berlin: Springer, pp 319-388. Teunissen PJG (1999) An optimality property of the integer least-squares estimator. Journal of Geodesy, 73:587-593. Teunissen PJG (2003b) Towards a unified theory of GNSS ambiguity resolution. Journal of GPS, 2(1), pp 1-12. Teunissen PJG, Verhagen S (2004) On the foundation of the popular ratio test for GNSS ambiguity resolution. In: ION GNSS, 17. Long Beach, CA, pp 2529-2540.
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E.M. Souza. J. F. G. Monico
Tiberius CCJM, De Jonge P (1995) Fast Positioning using LAMBDA Method. In: International Symposium on Differential Satellite Navigation Systems, 4, Bergen, 30. Verhagen S (2005) The GNSS integer ambiguities: estimation and validation. PhD Thesis, Delft University of Technology. Xia L (2001) Approach for multipath reduction using wavelet algorithm. In: ION GPS, Salt Lake City, UT, pp
2134-43. Wieser A (2001) Robust and fuzzy techniques for parameter estimation and quality assessment in GPS. Ph.D. Thesis. Surveying Engineering, Graz University of Technology. Wieser A, Brunner FK (2002) Short Static GPS Sessions: Robust Estimation Results. GPS Solutions, Vol. 5, No. 3, pp. 70-79.
Chapter 28
An Empirical Stochastic Model for GPS R.F. Leandro, M.C. Santos Department of Geodesy and Geomatics Engineering, University of New Brunswick, P.O. Box 4400, Fredericton, New Brunswick, Canada, E3B 5A3 Abstract. In addition to functional models, stochastic modeling of observations plays an important role in GPS data processing. The stochastic model has influence over several issues of coordinates determination with GPS, such as the covariance matrix of the observations (which leads to weighting scheme) and the estimated covariance matrix of the parameters. In this paper we present an empirical stochastic approach to create observation covariance matrices for GPS. Our approach aims to a more realistic and complete information about the stochastic behavior of GPS observations and an improvement in quality and quality control of estimated coordinates. In the empirical approach the correlation functions and variances are computed using the observed data, instead. Therefore it is not necessary any a-priori assumption linking observables variances and correlations to elevation angle or time lags, usually given by formal models. The first step of this approach is a functional reduction of the observables, which is made according to the functional model used in the late adjustment. The objective of this first step is leading the observation time series to stationarity. An auto-regressive model is then adjusted, with the determination of its parameters and order. Parameters are estimated using least squares adjustment and the order determined by an assessment of residuals. Once all parameters of the stochastic model were determined empirically, it is used to create the observation covariance matrix to be used in the functional model. In the present case study, GPS baselines were processed. Analyses were made in terms of obtained coordinates and their estimated covariance matrices, aiming at an improvement in GPS quality control. Improvements of at least 11% in precision and accuracy were found when this empirical stochastic approach was used. Future research aims to the enhancement of the method until it is general enough to be used in any case of GPS data processing. Keywords. Empirical, Stochastic, GPS.
1 Introduction The covariance matrix of observations plays a fundamental role in GPS data adjustment. Usually the method of least squares is used to compute the coordinates of a given receiver. In the adjustment the covariance matrix of observations drives how each one of the observations will contribute for the update of the parameters. This matrix includes not just variances, but also a relation between all observations, the covariances. A usual approach to estimate the covariance matrix of observations is to set a weight for all observations and then, in case of baseline, propagate it using the double difference operator. It is not assumed any correlation between observations made in two different epochs, the so called autocorrelations. The weights can be set based on different types of information, such as elevation angle of the satellite or signal-to-noise ratio. Sometimes even the identity matrix (equally weighted observations) is used. In this work, we are presenting an empirical approach to build the covariance matrix of observations. The main objective of such technique is populating the covariance matrix with realistic information, estimated by means of a stochastic analysis of the raw data. Trying to have as much as possible information inside the covariance matrix is a way to carry into the adjustment model a realistic picture of the quality and behavior of the observations. Eventually, the coordinates will be adjusted using this matrix. The ultimate goal of this work is to improve the quality of GPS data processing, as well as quality of the estimated precision for coordinates. In order to analyze the data assuming a stochastic behavior a few requirements need to be satisfied, such as stationarity. Because of that, the GPS data need to undergo some modifications before it can be used. In our Empirical Stochastic model (which will be called herein with ESto model) all analysis is made based on raw data, without any external information, before the adjustment. In next section a background explanation about stochastic processes and their analysis will be made. Next the treatment given to GPS data in order to
180
R. E Leandro. M. C. Santos
make its analysis as a stochastic process possible will be shown. A case study was carried out and its results are shown, as well as conclusions and recommendations for future work.
oo
(5)
S 2 -- E [ ( z t _ ~)2 ]-- ~(z-- ~ - ) 2 p ( z ) d z . -oo
The variance of a process can be estimated with the following equation:
2 Stochastic Processes S
A time series is a series of observations made through time. An important feature of a series is that, usually, the observations made at subsequent epochs are dependent on each other. The analysis of a time series is based on this dependency. A stochastic process is a statistical phenomena which occurs through time according to probability laws, [Box et. al., 1994]. A time series can be considered as a realization of a stochastic process. There is a particular process, called a stationary process, which has a particular statistic equilibrium state. A stochastic process is called widely stationary when its properties remain unaffected when the time origin is changed. This means that the joint probability function of a process with n observations zq,z& ..... zt,, observed at time instants t 1 , t 2 ..... t n
is
the
observations
same
as
a process
with
n
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2
1
(6)
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n t=l
The stationarity of a process implies that the joint probability function V ( z t l , z t 2 ) is the same for any instants tl and t2, and for the same time interval between them. This means that we can estimate the joint probability function of a process for different time intervals k . The covariance between the values z t and Z t + k , separated by a time interval k , which is constant for every t for a stationary process is called the auto-covariance function, and can be defined by: Z)(Zt+k - Z ) ] .
Zk : c o v ( z t , z t + k ) : E [ ( z t -
(7)
It is possible to estimate the auto-correlation function of a process for a given lag k with the following equation:
instants t 1 + k , t 2 + k ..... t n + k for every integer k . If a process is stationary, we can say that: Pk tO(zt) = tO(Zt+k) 'V' k >_ O,t >_ O ,
where p ( z t )
(1)
is the probability density function, and
because of that, we can conclude that the mean of a stationary process is constant: oo
Pt = E(zt)
=
~z.,o(z)dz
,
(2)
-oo
and therefore: /dr =/~,+k ,
(3)
for every integer k. The mean of a process can be estimated with the equation: _
1
n
z = --. 2 z, . n t=l
And the variance of a process can be defined as:
(4)
~/E[(z t _ Z)2].E[(zt+k _ Z)2]
t:l
s2
"
Based on this equations and the assumption behind them, the variances for GPS observables are computed in the ESto model, which will be explained in next session.
3 ESto Model In ESto model, the previously described approaches are used to estimate the correlations between different observations (made at different time and/or from different satellite pairs), as well as the variances for each one of the observed satellites. The input of the model is the raw data, and the output is the stochastic parameters. As stated in the previous section, a stochastic process needs to have certain characteristics in order to be used for estimation of variances and covariances. Since the objective of this work is to analyze GPS data as a stochastic process, it is necessary to modify the original time series of observations to satisfy the assumptions made in stochastic analysis. The main assumption that needs to be satisfied is the stationarity of the process. It is
Chapter 28 • An Empirical Stochastic Model for GPS
clear that GPS observations are not stationary, because they are related to the distances between satellite and receivers (plus errors and biases) and always vary with time. Figure 1 shows an example of double differenced pseudoranges through time.
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lO
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Fig. 1 Double differenced pseudoranges time series.
These observations could not be used directly in a stochastic analysis, and therefore need to be modified. In order to get a stationary series from the measurements, they are reduced by using the approximate geometric distances between receiver and satellite antennas, according to:
AVPre d = AVPob s - A V p ,
-2 / 0
10
20
30 40 50 "l]me [30 seconds interval]
60
0
10
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30 40 50 "l]me [30 seconds interval]
60
%
;0
2'o
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where AVPobs is the double differenced observed pseudorange, AVp is the double differenced geometric distance between receiver and satellite antennas and AVPred is the double differenced reduced pseudorange. This simple reduction can be used in this case because, as it is going to be mentioned later, we are dealing with short length baselines, where the most of errors and biases are supposedly eliminated. Figure 2 shows the double differenced C/A pseudoranges before and after the reduction made in the ESto model, for each one of the satellites shown above. In Figure 2 below, the light lines represent the original series, and the dark lines represent the reduced series. After this step the reduced series are used in the stochastic analysis. This part of the ESto model is based on the approaches mentioned in the previous section.
3o
4'o
5'o
6o
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Fig. 2 Reduction of DD pseudoranges, in ESto model.
As an illustration of the stochastic part of the model, Figure 3 shows the auto-correlation function estimated with ESto model and also computed based on the adjustment residuals, for satellites with different elevation angles (65, 31 and 17 degrees). The observables are C/A pseudoranges, observed for a short baseline (5 km) during half hour. The sampling interval s 30 seconds, thus one epoch lag represents 30 seconds in time.
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ESto can provide a good approximation of the residuals auto-correlation and therefore for the observations. It can be also noticed how the correlations get smaller as the elevation angle gets lower, with the autocorrelation function getting closer to zero with smaller lags. Another issue is the variance which is assigned to each satellite. In ESto model, as said previously, it is also computed by analyzing the raw data. Figure 4 shows the pseudorange standard deviations estimated by ESto for different elevation angles. .-. 1.2
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Fig. 4 Estimated standard deviation for different elevation angles.
The standard deviation varies with the elevation angle with values between around 0.2 m and 1 m. In Figure 5, it is shown the agreement between the estimated standard deviations and the residuals standard deviation after the adjustment.
estimated
with
ESto
and
In Figure 5, the dots represent the residuals of each of the observations, during a half hour observation session with 30 seconds of sampling interval (7 satellites observed). The dashed lines represent the one sigma standard deviations estimated with ESto. For each satellite, the color of the lines and dots is the same. Table 1 shows the estimated values for the standard deviations. Table 1. Comparison of residuals tandard deviation with ESto estimated standard deviations.
Satellite- elev. angle 1 (blue) - 65 ° 2 (green) - 17 ° 3 (red) - 52 ° 4 (cyan) - 44 ° 5 (magenta) - 45 ° 6 (black)- 31 o
Standard dev. [m] 0.15 0.84 0.19 0.30 0.25 0.48
ESto model has been developed in order to allow a estimation of a fully populated observation covariance matrices for GPS. It also potentially allows a more complete analysis of the raw data before the adjustment, such as outliers detection. This type of pre-analysis is not explored in this work, however it is another potential contribution brought by the use of our empirical approach. In terms of validation, ESto has been compared with other techniques to build covariance matrices. The first one, called here Formal DD, the simplest of the ones explored in this work, is a matrix generated by propagating a identity matrix with the double difference operator, thus all satellites have the same weight. The second approach uses the elevation angle as a parameter to estimate the standard deviation of the observations (here this approach is being called Elevation based). After the standard
Chapter 28 • An Empirical Stochastic Model for GPS
deviations are estimated, they are propagated using the double difference operator as well. Figure 6 shows a representation of covariance matrices estimated using these three schemes.
i
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20
30 Formal
40
50
60
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20
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short baseline using C/A pseudoranges. The choice of such data set is justified by the elimination of several errors in the measurements, such as clock errors and atmospheric refraction, which would need a special treatment in the stochastic analysis. It was used 24 hours of data, which were processed in half hour batch adjustments along the day. It was done for the baseline, UNB 1-FRDN with approximately 5 km. It were used the C/A pseudoranges as observables. Figure 7 shows the results of the adjustment. In the plots below, zero is the published coordinate of FRDN station.
"',~..,
.J ' "" ",..'
E
. .. :..,:,.: . . . . . .
60 10
20
30
0.2
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Fig 6 Representation of covariance estimated using different schemes.
0.4 0.3
,, " , "-"' 40 50 60 ~
0.1
"
0
g
matrices
0
5
10
15
20
25
30
35
25
30
35
40
45
50
45
50
1
E
In the figure above, different colors represent different values within the matrices. Each one of them has its own scale, and they were generated from the same data set. In the case of Formal DD, it can be noticed that the elements on the diagonal have always the same color, what means the same variance. The green squares around the diagonal are the correlation between different satellite pairs within the same epoch. This correlation exists with this pattern due to the use of a common reference satellite. In the second case (Elev. based), the main difference from the previous in the different weights for different satellites (this can be noticed by the different colors in the diagonal). The blue squares are due to the same as the green ones in the previous. The dark blue means zero in both of them. When the ESto model is used, it can be noticed that a large amount of information is placed into the covariance matrix. These values were derived from raw data, and represent the correlation of each observation with respect to each other. The next step to validate the ESto model should be an analysis with real data processing, explored in next session.
0.5 .c o~ -r 0 ~me[0.5
In order to investigate the efficiency of this empirical stochastic approach with respect to the conventional techniques, it was used to process a
40
hoursessions]
I~
F o r m a l DD
•
Elev.
>F
ESto
based
Fig. 7. Results for the adjustment of the baseline UNB1FRDN.
Although all results are fairly good, when ESto was used, the mean bias was quite lower than other solutions. However the standard deviation didn't show a good improvement. The use of ESto shows some errors quite larger than their mean values, which can be affecting standard deviation values. Table 2 shows the results summary for this data processing. Table 2. Comparison of results. Formal Mean error
(m) RMS (m)
4 Data Processing
20
0
Latitude Longitude Height Latitude Longitude Height
DD 0.14 -0.08 0.52 0.24 0.14 0.60
Elev. based
ESto
0.13 -0.08 0.50 0.23 0.15 0.57
0.07 -0.04 0.15 0.16 0.16 0.39
Tables 3 and 4 show the absolute and relative improvements achieved by the use of ESto with respect to the other models.
183
184
R.F. Leandro. M. C. Santos
show the error bars for the three models (latitude, longitude and height, respectively) at 1 sigma level.
Table 3. Absolute improvements when using ESto. w.r.t, Formal DD 0.07 0.04 0.37 0.08 -0.02 0.21
Latitude Longitude Height Latitude Longitude Height
Mean error (m)
RMS (m)
w.r.t. Elev. based 0.06 0.04 0.35 0.07 -0.01 0.18
1
0.5
Latitude Longitude Height Latitude Longitude Height
Mean error RMS
.
.
•
Table 4. Relative improvements when using ESto. w.r.t, Formal DD 50% 50% 71% 33% 14% 35%
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
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.
.
.
.
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.
.
.
.
.
.
.
l
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i
1
w.r.t. Elev. based 46% 50% 70% 30% 7% 46%
ESto model provided improvements ranging from 4cm to 37cm in bias and -2cm to 21cm in RMS. The greatest improvements were achieved for height determination. In terms of relative results, the improvements are very good, ranging from 46% to 71% for mean bias and -7% to 46% for RMS. These results show that the use of the empirical approach can definitively bring advantages in terms of quality of the estimated coordinates. Figure 8 shows the solutions in the horizontal position for the three models.
Fig.
0
5
10
15
20 "rime [0.5
25 hour
30
35
40
45
sessions]
Estimated standard deviations for latitude at 1 sigma
9.
level.
-1
0
5
10
15
20 "lime [0.5
25 hour
30
35
40
45
sessions]
Fig. 10. Estimated standard deviations for longitude at 1 sigma level.
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30
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40
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Fig. 11. Estimated standard deviations for height at 1 sigma level.
i
I
I
-
5
i
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%
0 -0.5
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]I
I
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It can be noticed that in the plot above the coordinates estimated with ESto are generally grouped closer to (0,0), than the solutions provided with the other techniques. Further analysis in terms of estimated coordinate variances was also carried out. Figures 9, 10 and 11
Clearly when ESto model is used, the estimated standard deviations are more realistic. Table 5 shows the percentage of the times when the real error was less than the estimated standard deviation (1 sigma) with the three weighting schemes: Table 5. Comparison of estimated standard deviations (1 sigma) and real errors.
Latitude Longitude Height
Formal DD 30 % 46 % 11%
Elev. based 28 % 39 % 11%
ESto 85 % 78 % 65 %
Chapter28 • An EmpiricalStochasticModelfor GPS
According to Table 5, the use of the empirical approach provides a better agreement between the estimated variances and the real errors, what means that in this case, the covariance matrix of the parameters computed in the adjustment is more trustful. The agreement for the two other models was quite lower than the supposed probability at 1 sigma level (68 %), while with ESto the agreement was higher than this value, except for height, when the agreement was of 65 %, which is still several times closer than for the other models (both 11%). 5
Conclusions
and
Future
Work
It was shown that the empirical model is capable of providing a good stochastic modeling of observables without any external source of information, before any adjustment. The implementation of the model is computationally efficient, and can be easily implemented for GPS data processing. Although its use makes the process slower, it is not a great computational cost when compared with the gain in terms of the adjustment results. The greater time is due the computation of auto-correlation and crosscorrelation functions that involve several combinations in order to have almost fully populated covariance matrices. A baseline was processed using three different weighting schemes for observations. In general, when ESto was used a good improvement in the bias was obtained. The mean biases were at least 46 % smaller. The RMS values did not improve as well as the biases, with one case of degradation (-8 % for longitude), but still had relatively god improvements with values up to 46 %. The greater improvements were achieved for height determination, with improvements in the order of 70% in mean bias for both comparisons. In terms of standard deviation, the solutions obtained with ESto showed small or no improvements. There are still some outliers in the solution when ESto is used, what means that there
is room for improvement in this sense. An attenuation of those outliers would help to make the standard deviations lower. In terms of estimated standard deviation, ESto provided much more realistic estimates, with agreements (compared against the real errors) ranging between 65 % and 85 %. For the other two stochastic models the estimates had a lower agreement with the real errors, with values ranging from 1 1 % to 46 %. The model certainly brings improvement to the solution, and future investigation is needed to investigate the source of solution outliers and improvements in the stochastic analysis for longer baselines. Further research to apply ESto to point positioning and carrier phase measurements is also a desirable future step in the development and validation of the model. 6
References
Box, G.E.P.; Jenkins, G.M.; Reinsel, G.C. (1994). Time Series Analysis- Forecasting and Control. Prentice-Hall International, London. Collins, J. P. and R. B. Langley (1999). Possible weighting schemes for GPS carrier phase observations in the presence of multipath. Final contract report for the U.S. Army Corps of engineers Topographic Engineering Center, No DAAH04-96-C-0086 / TCN 98151, March, 33pp. E1-Rabbany, A. (1994). The effect of physical correlations on the ambiguity resolution and accuracy estimation in GPS differential positioning. Ph.D. Thesis - University of New Brunswick, Department of Geodesy and Geomatics Engineering, Fredericton, NB. Hofmann-Wellenhof, B., H. Lichteneger and J. Collins (2001). Global Positioning System: Theory and Practice. Springer-Verlag Wien New York. Kim, D and R. Langley (2001). Estimation of the Stochastic Model for Long-Baseline Kinematic GPS Applications. ION National Technical Meeting, January 22-24, 2001, Long Beach-CA. Langley, R. B. (1997). GPS observation noise. GPS World, Vol. 8, No. 6, April, pp 40-45. Vanicek, P. and E. Krakiwsky (1982). Geodesy: The Concepts. North-Holland Publishing Company, Amsterdan - New York - Oxford.
185
Chapter 29
Feeding Neural Network Models Observations: A Challenging Task
with
GPS
R.F. Leandro Department of Geodesy and Geomatics Engineering, University of New Brunswick, P.O. Box 4400, Fredericton, New Brunswick, Canada, E3B 5A3 C.A.U. Silva Department of Civil construction, Federal Technologic Learning Centre, Belam, Para, Brazil M.C. Santos Department of Geodesy and Geomatics Engineering, University of New Brunswick, P.O. Box 4400, Fredericton, New Brunswick, Canada, E3B 5A3 Abstract. Much has been done in terms of functional and stochastic modelling of observations in space geodesy, aiming at the development of adequate adjustment models. One of the techniques, which has been the focus of more attention in the last years, is the Neural Network model. Although not trivial to be used, this kind of model provides an extreme adaptation capability, which can be an issue of fundamental importance for certain applications. In this paper we discuss the use of GPS observations in Neural Networks models, providing a brief description how a neural model works and what are its restrictions, as well as how to treat the GPS observations in order to satisfy them. A Neural Network is an information processing system formed by a big number of simple processing elements, called artificial neurons. Typically the input values must be normalized, with typical range [0,1], or alternatively [-1,1]. After processed, the signal can be transformed back to its original origin and amplitude. When dealing with GPS observations, namely ranges and range rates, the absolute numerical values are usually pretty large (e.g. order of 20 millions of meters for ranges) coupled with precisions in the order of mm for carrier-phase and meter for pseudoranges. The observations need to be modified to avoid degrading their precision during the normalization, in order to make the application of neural models suitable for GPS data. In this work methods to make the use of GPS data possible in neural models are discussed and showed with real examples. The analysis is made for both pseudoranges and carrier-phases. It is demonstrated that with the adequate treatment the use of those observables can be made without degradation of precision.
Keywords. Neural Networks, GPS.
1 Introduction Modelling plays a fundamental role in Geodesy. Signal processing, physical phenomena functional modelling, interpolation, forecasting, stochastic modelling are a few examples of the applications that require modelling in Geodesy. In most cases adjustments are used, and in this case, the most used technique is the least squares procedure. Filters are also widely used, such as Kalman filter, which involves some of the least squares technique concepts. In the ninety's a new technique appeared to be useful in Geodesy, called Neural Networks. Primarily developed for computing applications, such as pattern recognition, neural networks have been adapted to be used in several fields of science, including Geodesy. Those adaptations are needed because usually the situations and problems encountered in computer science are sometimes very different than in other fields. Geodesy was not an exception in this case, and because of that, the Neural Network analyst for geodesy needs to have a good knowledge in neural data processing in order to be able to use this technique successfully. Adaptation of the geodetic data may be needed in some cases to make the data useful in a neural model. This is the case explored in this paper, where the problem of using GPS data in neural networks is shown. Compatibility between GPS and the neural model has to be made possible by means of some modification in the original GPS data. This innovating synergy has made necessary the development of novel techniques in terms of GPS data handling.
Chapter 29 • Feeding Neural Network Models with GPS Observations: a Challenging Task
2 Neural N e t w o r k M o d e l s A Neural Network is an information processing system formed by a big number of simple processing elements, called artificial neurons, or simply neurons. The first artificial neuron model was presented by Rosenblatt (1958), who called it perceptron.
It is possible to introduce a functional link into the network as an additional layer of neurons, called a hidden layer. This layer can be composed of one or more neurons. The input signal of the hidden layer neurons is generated by the output signal of the input layer. The output signal of the hidden layer is used to generate the input signal to the output layer. It is also possible to introduce not just one, but several hidden layers into the model. Neural Network Multllayer Perceptron
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Figure 2 shows a scheme of a neural network with one hidden layer. In this example, x(1), x(2) and x(3) are the input parameters and y(t) is the output parameter. Each element, excepting the biases, is a neuron. Each of these neurons is a processing element that works according to equation (1). The synaptic links (the lines in the draw) connect the different layers, carrying the output signal of a previous one to generate the input signal of the next one. Each synaptic link of the network has a corresponding synaptic weight that is applied to the flowing signal that is going through it. Another issue of a neural network model is the number of neurons of each layer. This number is fixed to the input and output layers, in function of the input and output parameters. For the hidden layers this number is arbitrary. The model resulting from adding hidden layers between the input and output layers is called Multilayer Perceptron (MLP). The MLP is not the only type of neural network model, but is one of the most popular ones, due to its high adaptation capability and its applicability to a wide group of different applications. It is necessary not just to know which model will be used, but also all its characteristics, such as the number of hidden layers, the number of neurons in each hidden layer, the activation function of each
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layer, etc. There are others more specific characteristics that will not be discussed here. Once we have a model defined, it is necessary to train the neural network with real data. Such data is composed by a set of known input and output parameters. The training process is not more than an adjustment of the synaptic weights to the data set. This adjustment attempts to decrease the residuals of the output of the network. The residuals are the difference between the computed output and the known output. Based on these residuals it is performed an update of the synaptic weights. Due to the complexity of neural networks, the adjustment can not be done with a direct computation, so the so called training algorithms, which are a type of iterative adjustment of the synaptic weights, are used. One of these algorithms is the backpropagation training algorithm, which is composed by two steps. The first one is the feedforward, when the Network is fed with a set of inputs that are propagated through the links, from the input layer to the output layer. After that the output value is compared with the known output (from the same set) and the residuals are computed. The second step is the feed-backward. In this step the network is fed with the errors of the previous step, which are propagated through the network from the output layer to the input layer (backwards). During the feed-backward step the synaptic weights are adjusted, using the partial derivatives of each activation function with respect to each synaptic weight. In this step, an additional parameter is also used, which is the learning rate. The learning rate controls how much the weights are going to be updated, given the derived correction. These two steps are carried out several times, for all training sets (which are also called training patterns), and for several epochs (one epoch is the cycle where all patterns are used once in the training process). It is made several times to each parameter up to the residuals converge to a desired threshold value. After the training process we have a Neural Network Model with adjusted synaptic weights according to the training parameters. Figure 3 shows a plot of a training process. In the example above, the goal of the training process was to achieve a value for the summation of the squared residuals (or errors) of 0.001 (the straight line in the plot). This number is dimensionless because of the normalization process that all data has to go through before it can be used
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Fig. 3 Neural network training process in the neural model. In the case of this figure, the goal was achieved in 222 epochs, when the network got a value less then the imposed goal. Usually the initial value of the residuals is high, because the weights of the network have to be initialized with arbitrary or randomic values. Neural networks can be used in three distinct ways: as intelligence simulation systems, as real time adaptation processors or as data analysis systems. One of the advantages of neural networks models is their high flexibility, allowing solving problems for dynamic and non linear systems. The generalization capability allows estimations of values not used in the training process. In this case, neural models can be successfully used for approximation, interpolation, extrapolation, time prediction. Their complex nature allows the use of hyper surfaces to represent the phenomena modeled, and the training process makes them self-adaptative. These characteristics make the neural networks an attractive solution, since they are theoretically capable of approximate any surface (and therefore modeling any phenomena) without a high knowledge of the phenomena modeled. In order to successfully conceive a neural network model, some steps are necessary. One of the steps is the definition of the optimal architecture of the network, for the specific application it is going to be used. The architecture consists of the definition of the number of layers, and the number of neurons in each layer. Together with this step, there are also other parameters to be established, such as the learning rate, type of activation functions and
Chapter29 • FeedingNeural NetworkModelswith GPSObservations:a ChallengingTask
normalization interval. All of them play an important role in the training process, and an important role in the generalization capability of the neural model. Seeking the optimal set of parameters and architecture for a neural network can be a hard work. Although some authors define some rules to setup a model, there is not a defined procedure to follow in order to build a model. In the testing procedure (which is performed to test the efficiency of candidate models for certain application) depending on the complexity of the architectures the training procedure can be time consuming, even for fast computers. The training time of a neural network can vary from a few seconds to several days. Even though it is necessary to spend some time to set an optimal configuration, the results of the neural network estimations usually pay off the effort made. 3
Neural
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Neural networks showed to be an attractive alternative in terms of modeling for several areas, and it was not different for geodesy. Since early nineties several authors have published papers reporting the use of neural models in geodesy, for different applications. These applications include navigation (e.g. Dumville and Tsakiri (1994), Chansarkar (1999), Vickery and King (2002)), geoid approximation (e.g. Kuhar et al (2001)), weather parameters (atmosphere and space) interpolation and forecasting (e.g. Xenos and Stergiou (2002), Leandro (2004), Leandro and Santos (2004)), prediction of Earth orientation parameters (e.g. Schuh et. al. (2002)), design of networks (e.g. Chang et. al. (1996)) and others. Usually the utilization of neural models is attempted in order to solve problems where deterministic and statistic tools still requires improvements, or where there isn't a very good knowledge about the modeled phenomena. A potential application is also the dynamic filters where the real time adaptation capability of neural networks fits pretty well. This is the case for example of the navigation applications, when the neural model has to "learn" the pattern of the movement, playing the role of a predictive filter. The models designed for approximation and interpolation usually target phenomena for which the modeling with linear models is complicated, or the deterministic functions don't fit very well on the approximated surface. As said in the previous
section, a neural network can theoretically approximate any surface, depending only on whether an optimal configuration (architecture and other parameters of the neural model) is found or not for the specific case. So, in theory, with the optimal configuration, neural network models are capable of perfect fitting for these cases. The difficulties to achieve it are the determination of the optimal configuration, and the size of the data set available to train the network. In terms of forecasting, neural networks have also a good advantage, given by their adaptation capability. While when using deterministic models a beforehand, complete knowledge of the modeled effect is necessary, in case of neural networks the model can adapt itself to the behavior of the time series, in order to be able of forecast future occurrences. Also, when it's considered that the series has a stochastic component, there is need of using a stochastic model coupled with the deterministic model. In case of neural networks there is no such need, because a single neural model can be used for either or both components. Leandro and Santos (2004) have showed this kind of application, using neural networks for space weather parameters forecasting. Another example that involves time series is the case of GPS cycle slip detection and correction, where neural models can be used to predict values for GPS carrier phase measurements. Far from being a concurrent to the actual modeling techniques, such as least squares, neural networks can play an important role as an additional key for complex problems. Neural models differ from least squares adjustments in several aspects, such as: ¢" ¢" ¢"
¢" ¢" ¢" ¢"
Synaptic weights in neural models play the role of parameters in least squares; The architecture of a neural model plays the role of the functional model in least squares; Patterns used in the training process can play the role of observations and/or constraints in a least squares adjustment, depending on the training RMS goal; Test patterns play the role of control points; Errors as called in neural processing are similar to observation residuals in adjustments; The training process plays the role of the parameters adjustment; The learning rate plays the role of an observation weight (it controls the update of the parameters/synaptic weights);
189
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R.F. Leandro • C. A. U. Silva. M. C. Santos
¢"
An epoch in neural training process plays the role of an iteration in an adjustment; ¢" In least squares adjustment, the functional model has to be built to represent the phenomena. In neural networks, the architecture is built in order to be able of adapting itself to the phenomena. Even thought there are several differences between them, in some cases neural networks and least squares can be along side. This is the case of using neural networks as the transition part of Kalman filtering. For example one can take advantage of the well known functional model for positioning to build the update adjustment of a navigation filter, together with a neural network playing a role of a self learning navigation estimator as the transition. Neural networks, when used properly, are a very attractive tool that can make a difference enhancing actual techniques with a high adaptation capability. Its potential applications are countless, being restricted only by the geodesist's imagination to link a neural model to a geodesy problem. As the proposed subject of this work, some of those techniques are discussed in next section.
4 Feeding Neural Networks with GPS observations When we talk about using GPS data in neural networks as a relevant subject, the first question that arises is why should the data need special treatment to be useful. Actually, there are some reasons to that and the first one (maybe the more important one) is the fact that in neural processing, all values should be set between 0 and -1, or between -1 and 1. This implies that values of the order of tens of thousands of kilometers (in the case of pseudoranges) or up to hundreds of thousands of cycles (in the case of carrier phase) must be fit into small intervals, and the model still needs to be capable of providing estimations with precision compatible with the original data. Other aspect to be considered is the compatibility between training patterns from different observations and the compatibility between training and testing patterns. The generalization capacity of the model depends on this coherency when transforming the original data set. The value which is most used in neural networks for performance analysis is the MSE (Mean Squared Error). The MSE can be computed with the equation:
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Chapter 2 9 • Feeding
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The disadvantage of using this technique comes from the added slips themselves. For long time series, several slips have to be used, and for each subsequent series between slips, the errors of the estimations get accumulated when building a slip free estimated series. Figure 7b shows how small the differences between L 1 and L2 are when the data is normalized. The neural model should be sensible to these variations. Implicitly, by minimizing the normalization factor these small differences are more easily detected when processed in a neural network.
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As can be seen in the figure above, there is a difference of reference between the two frequencies, given by the different numbers assigned for the initial cycle ambiguity, and a difference in rate, given by the different wavelengths. Because of these differences, it would be hard to relate L 1 and L2, as wanted. This is a typical problem of coherence between patterns generated from different observables. Since the initial value of the phase counter is arbitrary, mistreating this difference can lead to high normalization factors. A second implication is that each receiver-satellite pair would have a different value for this difference. To solve this problem the meter was used as common unit for both frequencies. Also the counters were synchronized, assigning the same initial phase count for both (in this case zero) as shown in Figure 6.
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After the above modifications, the problem to relate the two frequencies is solved. However the values for normalization factors would be still high, because of the wide range of carrier phase values, and there are a few techniques that can be used.
Another technique that can be used is the time differencing of the observations. This approach reduces the range in which the phase counters vary. However, similarly to the previous case, the errors in the prediction get accumulated through time when rebuilding a slip free time series. This effect should be compensated by the much smaller value of the normalization factor. Depending on the type of series and generalization capability of the network it can be a good option. However, usually it does not perform very well for long time series. Figure 8 shows an example of the application of this technique.
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Other techniques than the ones shown in this work can also be used. One example of this is the functional reduction used by Leandro et. al. (2005) to modify pseudorange time series in order to feed stochastic models. This reduction consists of removing the mathematical distance and other systematic effects from the observations. In terms of neural processing this approach can be useful, however when in some cases it can bring some difficulties. For example for the sake of the coherence between patterns, coordinates of different receivers should be well known. In case one or more receivers have unknown position the technique could introduce biases in the reduced series, what can complicate the training process. This is a reason why this technique was not used in this specific case. However it can be useful in other situations (e.g. in applications dealing with only one receiver). In terms of the data processing where all those techniques were applied, the best results were found when the time differencing technique was used for carrier phases and the inverse truncation was used for pseudoranges. Although these techniques performed better in this case, doesn't mean they are better than the other ones, but that they were more appropriate for the situation explored. In other cases, the other shown techniques might be more adequate. 5
Conclusions
Neural network models require data fitted into normalized intervals, what require modification in the original data set. In the process of modifying the data several techniques can be used. Some of the
Chapter 29 • Feeding Neural Network Models with GPS Observations: a Challenging Task
techniques that can be useful for using neural models in geodesy, as well as their advantages and disadvantages, were show in this paper. Definitively the choice of the adequate procedure when preparing the data to be used by neural models can produce large variations in quality of the estimations when brought back to their original range. The techniques shown in this work are not all existent alternatives and other ones maybe even better than those might exist. The variety of solutions for this kind of problem can be as large as the possibilities w h e n using neural model. However, an important contribution of this work is showing the importance of certain problems w h e n dealing with data from geodetic measurements (such as GPS), as well as some useful solutions. If they are not applicable for a specific case, alternatives can arise from the concepts explored in them. It is impossible to point which one of the shown techniques is the best one, because in each case a different solution can be the optimal. It was the case shown in this work, where the technique used for pseudoranges was different from the one used for carrier phase. Although the data used in this work were GPS measurements, the procedures can be potentially applied to any type of measurement with same characteristics.
6 Acknowledgements This search was partially funded by NSERC.
7 References Chang, Y. M.; Chen, C. H.; Chen, C. S. (1996). Optimal Observation Design of Surveying Network using Artificial Neural Network. Geomatics Research Australasia, No.64, June, 1996, pp. 1-16. Chansarkar, M. (1999). GPS Navigation using Neural Networks. 12th International Technical Meeting of the
Satellite Division of the Institute of Navigation, September 14-17, 1999, Nashville Convention Center, Nashville, Tennessee. Dumville, M. and Tsakiri, M. (1994). An Adaptive Filter for Land Navigation Using Neural Computing. 7th International Technical Meeting of The Satellite Division of The Institute of Navigation, September 20-23, 1994, Salt Palace Convention Center- Salt Lake City, UT. Haykin, S. (1999). Neural Networks- A Comprehensive Foundation. Prentice H a l l - Upper Saddle River, New Jersey. Kuhar, M.; Stopar, B.; Turk, G.; Ambrozic, T. (2001). The use of artificial neural network in geoid surface approximation. Allgemeine Vermessungs-Nachrichten, Vol. 108, No. 1,2001, pp. 22-27. Leandro, R. F. (2004). A New Technique to TEC Regional Modeling using a Neural Network. ION GNSS 2004, September, 2004, Long Beach, California. Leandro, R. F. and Santos, M. C. (2004). Comparison between autoregressive model and neural network for forecasting space environment parameters. Bollettino di Geodesia e Scienze Affini, Vol.63, No.3, 2004, pp. 197212. Maia, T.C.B., Silva C.A.U., Leandro R.F., Segantine P.Ci., Romero R.A.F. (2002). Predi~fio da Contagem de Ciclos da Portadora GPS Utilizando uma Modelagem Conexionista T e m p o r a l - FIR MLP. XVI Brazilian Symposium on Neural Networks. Porto de Galinhas, Recife, Brazil. Schuh, H.; Ulrich, M.; Egger, D.; Mueller, J.; Schwegmann, W. (2002). Prediction of Earth orientation parameters by artificial neural networks. Journal of Geodesy, Vol.76, No.5, 2002, pp. 247-258. Vickery, J. L. and King, L. R. (2002). Use of Neural Networks and Expert Systems for Rapid Differential GPS Navigation. ION GPS 2002, September 24-27, 2002, Oregon Convention Center, Portland, Oregon. Xenos, T. D. and Stergiou, D. C. (2002). One day before foF2 neural network based prediction models: A performance comparison between ordinary, fuzzy and recurrent neural networks. Acta Geodaetica et Geophysica Hungarica, Vol.37, No.2-3, 2002, pp. 293-296.
193
Chapter 30
Spatial spectral inversion of the changing geometry of the Earth from SOPAC GPS data P.J. Mendes Cerveira, T. Hobiger, R. Weber, H. Schuh Vienna University of Technology, Institute of Geodesy and Geophysics, Research Unit Advanced Geodesy, Gusshausstrasse 27-29, A-1040 Vienna, Austria
Abstract. Estimates of station displacements, derived from global GPS networks, contain valuable information about surface deformations. For this reason, we performed a temporal surface spherical harmonics (SH) expansion (up to degree-ll) solely for vertical station variations. Our goal was to estimate stable height displacement coefficients, which in turn can be converted into equivalent height coefficients of a column of seawater. New in our approach is a new data set, plus a new method of creating constraints to deal with the truncation and spatial aliasing problem. To achieve our goal, the data of 675 stations were obtained from the Scripps Orbit and Permanent Array Center (SOPAC). To overcome aliasing effects, high-frequency spatial local variations, and high correlations between the estimated harmonic coefficients, we performed an iterative 2D-interpolation approach, based on input from low-degree spherical harmonic coefficients, with a spatial resolution of 10 deg for longitude and latitude. Considering the assumption that the radius of a spherically symmetric Earth model is not supposed to change because of Earth's constant mass, we tested two options: one with and one without setting degree-0 to zero. One half of the amplitude of degree-0 coefficient will be absorbed by the zonal degree-2 coefficient, if the former is not estimated. This study agrees with recent results on higher load coefficients published in Kusche and Schrama (2005).
Keywords. Reference frame, geocenter, surface loading, deformation
1
Introduction
The Earth's crust is deformed by a multitude of sources, e.g., by gravitational forces of celestial bodies, by variations of the centrifugal potential, by processes inside the Earth or by surface mass exchanges within the atmosphere, (van Dam et
al. 1994; Petrov and Boy 2004), the oceans and continental water storages. Geodetic observations are either corrected by applying theoretical deformation models, or through semi-empirical corrections. In this paper, we focus on the Earth's crust deformation exclusively for the vertical component, derived from site coordinate time series created by a SOPAC refined model (Nikolaidis 2002). We choose the SOPAC data, because the combined International GNSS Service (IGS) data show evident seasonal irregularities and aliasing effects (Mendes Cerveira et al. 2005). A software bug, in the solid Earth tides corrections, has been detected at the Analysis Center (AC) of the Center for Orbit Determination in Europe (CODE) in mid 2004. The contribution of CODE to the combined IGS coordinate time series is not negligible, but re-processing of the combined IGS solution has not been performed so far. We can expect that SOPAC data is not influenced by the previously mentioned bug.
2
Background
Blewitt et al. (2001) presented a new global mode of Earth deformation using five years of Global Positioning System (GPS) data, acquired by 66 stations of the IGS network. Free network solutions in the center of mass (CM) frame were analyzed to produce site coordinate time series in the center of figure (CF) frame. In that paper, a maximum downward deformation of 3 mm was found, pointing close to the North Pole during February to March, and to the South Pole during August to September. Wu et al. (2002) pointed out that the CF approximation for the center of network (CN) in the inversion for load coefficients introduces nonnegligible errors. Moreover, they showed that it is difficult to retrieve all significant harmonics when using a sparse and uneven GPS network
Chapter 30 • Spatial Spectral Inversion of the Changing Geometry of the Earth from SOPACGPS Data !
lacking of coverage in the polar areas, oceans and southern hemisphere. In addition, they stated that if load-induced deformation components of higher degrees are not estimated, the inverted results for the degree-1 load and geocenter motion are contaminated. Blewitt et al. (2003) used the published empirical seasonal model of degree-1 deformation, described in Blewitt et al. (2001), to weigh sea level in static equilibrium with surface mass redistribution. Recently, Kusche and Schrama (2005) used a different method to deal with the problem of lacking coverage, where pseudodata is replaced by an analytical augmentation of the least squares cost functional, i.e., using an ocean load variability constraint. In that paper, they find similar patterns of annual variations of continental mass redistribution from ICS and Gravity Recovery and Climate Experiment (GRACE) data, as well as from a global hydrology model. Nevertheless, they also point out systematic differences between these three data sets.
3
Theory
( 2 / t @ 1), f i e U i n m __
3psh~
fnSinm
(1)
!
where h~ are the height LLNs, which are taken here in the center of figure frame for the Cutenberg-Bullen GB-A Earth model, kg p z - 5 5 1 4 ~ w the mean density of the Earth, kg ps=1025 ~-~ the load density of seawater, and ~,~ the cosine and 2~,~ the sine components for degree-n and order-re. The conversion factors f~, which transform geometric height displacement SH into equivalent height of surface load SH coefficients, are tabulated in Table 1.
4
1: L o a d L o v e N u m b e r s h~ in t h e c e n t e r
of figure frame (CF) for a Gutenberg-Bullen GBA Earth model, and the conversion factors f~ up to degree 11. !
Degree n 0 1 2 3 4 5 6 7 8 9 10 11
h~ 0 -0.269 -1.001 -1.052 -1.053 -1.088 -1.147 -1.224 -1.291 -1.366 -1.433 -1.508
f~ 0 -20.0000 -8.9286 -11.9048 -15.3846 -18.1818 -20.4082 -21.7391 -23.8095 -25.0000 -26.3158 -27.0270
annual fluctuations for stations that have at least two years and one year of data, respectively, and which are updated on a regular basis. Input data to their model are the daily processed site positions calculated at SOPAC using the GAMIT and GLOBK software packages (King and Bock
2oo ).
Following the Load Love Numbers (LLNs) formalism, Farrell (1972), and Blewitt and Clarke (2003), a very simple correspondence between spectral height displacement coefficients Sinm and the surface load coefficients Tinm, given in units of [m], is achieved by the following equation: Zinm
Table
Approach
The SOPAC model takes into account offsets (e.g. co-seismic) and linear velocities. Additionally, SOPAC analysts estimate annual and semi-
Not any change and no additional transformations were applied to the SOPAC data in this study. Annual and semi-annual amplitudes and their phases were taken as provided by SOPAC. These were used to derive synthetic station height coordinate time series, as input for the SH expansion, described from subsection (5.2) onwards. Besides, we also took a few SOPAC coordinate time series, but only for the purpose of investigating temporal aliasing effects, described in subsection (5.1). For extensive details concerning tile pre-processing and processing of the GPS data at SOPAC, we refer to the thesis of Nikolaidis (2002), that can be downloaded fl~oln the SOPAC homepage. We thus choose a global network (see Fig. 1) of 609 GPS stations among 675 (downloaded from SOPAC on 7th of August 2005) fulfilling the requirements that for that specific epoch • the standard deviations of annual amplitudes in the horizontal components (north and east) are smaller than 1 ram, and in the vertical component smaller than 2.5 m m (see Fig. 2), and • the standard deviations of semi-annual amplitudes in the horizontal components are
195
196
P.J. Mendes Cerveira • T. Hobiger • R. Weber. H. Schuh
80 60
•
4o
•
~&
•"°
Ci'~
0-
~'
"0
,~ w
20
°
o• •
-40
h
• •
2• •
-60
-80 L 0
50
1 O0
150 200 longitude [deg]
250
300
350
Fig. 1 Station distribution of 609 GPS SOPAC sites. 50 c-
._o
40 0
30 .Q
E ¢.G20 o r'-
~10 GY
2 4 standard deviation [mm]
6
Fig. 2 Standard deviation of the annual amplitudes of the time series of vertical components of 675 GPS sites. smaller than 0.75 ram, and in tile vertical component smaller than 1.75 ram. In order to overcome spatial aliasing effects mentioned in Wu et al. (2002, 2003), we performed an iterative 2D-interpolation approach with a spatial resolution of 10 deg for longitude and latitude. This iterative approach consists of three main steps: firstly, we estimate degree1 spectral coefficients from the irregular sampled data, secondly, we perform a gridded 2Dinterpolation with a spatial resolution of 10 degrees, based upon the degree-1 coefficients, which were estimated previously, and thirdly, we add tile irregular sampled vertical components to this 2D-grid, and estimate degree-2 coefficients. The iteration is carried out till degree-11. These 2D-
interpolation pseudodata form our constraints. By this method, the 609 GPS height coordinate time series (in the ITRF2000) were inverted into spherical harmonics series up to degree and order 11. The pseudo-observations obtained from the 2D-interpolation are weighted by one, whereas the synthetic real SOPAC data are weighted by two. A comparison of these estimated coeffcients versus the coeffcients derived from an expansion till degree and order seven, shows maximum differences of 0.2 ram. Moreover, a comparison of this degree-11 expansion versus the point-bypoint SOPAC data, does not show any abnormal aliasing effects. A spatial resolution of 10 degrees spacing is a judicious choice, as the role of the pseudodata should only convey tile stabilization of the normal equation system. With respect to a 10 degree spacing, a five degree spacing experiment shows apparently the same pattern in the variation of coefficients. However, there is a maximum increase of 0.2 mm in the spectrum of degree-l, as well as a maximum decrease of 0.3 mm in the degree-2 and degree-5 spectrum. Other spectrum degrees are not influenced by more than 0.1 ram. In this study, we only estimated the height displacement SH coefficients Hi~,~, because the horizontal displacement components show a random spatial pattern. 5
5.1
Results
Temporal aliasing effects
Penna and Stewart (2003) describe that temporal aliasing effects are mainly a function of the design of the GPS constellation, the length of the processing session and the quality of tidal models applied in the GPS data processing procedure. They give evidence that this kind of aliasing effects might result in incorrect conclusions drawn from GPS coordinate time series, regarding the presence of seasonal crustal motions. An heuristic spectral analysis of SOPAC height time series using a software routine provided by Mautz (2001) revealed significant energy around 13.66 days (see Fig. 3 and Tab. 2), which is exactly the period predicted from the GPS sidereal orbit aliasing effect for both M2 and O1 tidal constituents. However, this period could also be attributed to the lunar fortnightly MI tidal constituent. In Nikolaidis (2002) we find that ocean loading and pole tide were only modeled after November 26, 2000. However, the 13.66-days period also shows up for sites far from the coast.
Chapter 30 • Spatial Spectral Inversion of the Changing Geometry of the Earth from SOPACGPS Data
].5
0.8
'
/ 0.6 ~
E
5
0.4
1
'
'
NN~
~,-HIO0 l- ° -dHR201
ssS ~ ~" m
0.2
(U
"o
"-..\
.1
Q.
E < 0.5
-0.2
1
-0.4 -0.6
,., ,,,,,.,....,..,,,,., ....,,,
5
10
15
20
Period [days]
Fig. 3 Fourier spectrum (window is from 2 to 20 days) of height time series for station BAHR located at Manama, Bahrain.
This brings us to the conclusion t h a t this period cannot uniquely be a t t r i b u t e d to unmodeled ocean loading. Table 2: Result of the heuristic spectral analysis (Mautz 2001) derived from the height time series for station BAHR located at Manama, Bahrain. Eight frequencies were estimated. The phase shift is referenced to epoch June 6th, 1996.
Period [days] 588.09 569.73 360.67 200.59 116.49 87.14 69.94 13.65
5.2
A m p l i t u d e [mm] 5.15 5.94 7.30 1.39 1.43 1.34 1.34 1.30
P h a s e [0..2pi] 3.93 6.21 1.34 1.90 3.47 2.59 2.26 4.18
Degree-O investigation
Degree-0 represents an apparent scale variation in the GPS SOPAC coordinate time series. Lavallee and Blewitt (2002), Blewitt et al. (2003), and Gross et al. (2004) start the expansion of the height function H~,~ with degree-i, stating t h a t the existence of a degree-0 would imply an average change of the E a r t h ' s radius and a degree-0 load different from zero. In theory, e.g., in a spherical non-rotating elastic and isotropic (SNREI) E a r t h model, such a radial variation is not permitted, with the assumption t h a t the total surface mass (i.e. oceans and atmosphere) is constant because of conservation of
-0.8 2
4
6 8 month of year
10
Fig. 4 Temporal variation of Hloo, and temporal difference of dH12o = H12owithoutH12owith, by fixing degree-0 to zero (with) and not fixing it (without). mass. But, on a real Earth, nothing prevents the E a r t h ' s radius from shrinking or expanding. Here, the meaning of H100 differs completely from the meaning of the coefficient derived from the gravitational potential. In the latter case, the degree-0 coefficient is effectively a function of the E a r t h ' s mass. Geometry (volume=V) and m a t t e r ( m a s s = m ) are related by density p = rn/V. If the volume changes, the density may change propertionally, without affecting the mass. We investigated two options: one with and one without setting degree-0 to zero, first to evaluate the pattern of H100 (see Fig. 4), and second to test its effect on the degree-2 order-0 coefficient H120 (see Fig. 4). From Fig. 4 we may derive t h a t the E a r t h ' s radius apparently seems to be expanding from February to March (+0.8 ram) and shrinking from September to October ( - 0 . 8 ram), wrt. to ITRF2000. However, the degree-0 coefficients are possibly biased due to Helmert transformations applied to the SOPAC d a t a set. Hence, degree-0 is subject to frame-related errors. Systematic errors of the orbit models may be removed by applying 7- or 14-parameter transfermations, but the latter also remove the apparent scale due to loading signals, t h a t are aliased by the uneven and sparse GPS network. Such transformations lead to frame errors and may bias the load signals. From inspecting Fig. 4 it appears t h a t coefficient H120 absorbs approximately one half of tile amplitude of coefficient H100, if the latter is not estimated.
197
198
P.J. Mendes Cerveira • T. Hobiger • R. Weber. H. Schuh
5.3
Degree-1 investigation
Degree-1 conveys a real (seasonal) motion of the solid Earth's (:enter of mass. As pointed out by Wu et al. (2002) degree-1 deformation may be considered as equivalent to geocenter motion. In the same paper they argue that degree-1 is insufficient to represent surface loading deformations: the SH expansion should be done at least to degree and order six. Farrell (1972) and subsequently Blewitt et al. (2003) derive that degree1 deformation (from a surface displacement field in poloidal functions) is not a pure translation, because only three coefficients are independent out of six due to the no-net translation condition. However, we interpret the degree-1 height displacement coefficients as a translation. Fig. 5 shows the temporal variation of degree-1 coefficients. Before the 7-parameter transformation is applied to the station coordinates, the degree-1 gravity coefficients are set to zero for GPS orbital integration. Thus the fiducial-free solution is in the center of mass of the Earth's system (CM) frame. But the CM link is pretty weak, which is reflected by the large uncertainties in such solutions. After the 7-parameter transformation, the coordinate origin is then fixed to the ITRF2000 origin, which is realized by constant and linear coordinates of the ITRF2000 coordinates. At the seasonal scale, since no motion is allowed for the ITRF2000 stations, the ITRF2000 origin is closer to the center of network (CN) frame than to CM. Having used coordinates in the ITRF2000 (considered as a CF frame), we deduce from Fig. 5 that a maximal positive z-shift of 1.2 m m takes place from September to October, an x-shift of 1.8 m m in August, and an y-shift of 2.5 m m in July. This y-shift points towards the Asian continent where large positive vertical seasonal deformations occur at that epoch.
5.4
Degree-2 investigation
Degree-2 coefficients are related to the Earth's inertia tensor (postulating a constant density) and hence to changes in Earth rotation. Changes in Hi20 can be referred to variations in lengthof-day (LOD), changes in H121 and H221 to polar motion excitation. Thus, these GPS-derived height displacement coefficients allow a comparison with observed Earth rotation changes, after removing tidal and motion effects, e.g., winds and currents. Gross et al. (2004) investigate
2.5[ 2i1.51
~ /= /
...... Hl10 - * -Hlll --*--H211[
~l _~.'--~,
###oi# I
0.5
!
~
.....
0 %... -0.5
-1
"'"'"I ........ ,."""
-1.5 -2 2
4
6 8 month of year
10
12
Fig. 5 Temporal variation of Hllo, Hlll and H2~, with 1~ formal errorbars.
degree-2 mass loads from GPS. They state that low-degree SH coefficients determine the surface density (mass load) that is acting to change the Earth's shape. But, on the other hand, if a load variation is assumed to be known, then Load Love Numbers may be computed to test Earth's mechanical properties (Blewitt et al. 2001; Blewitt and Clarke 2003). Fig. 6 shows tile temporal variation of three degree-2 coefficients in the vertical displacement component. The flattening of the Earth, described by H120, is maximal in September (2.5 ram) and should theoretically slow down the angular speed of Earth's rotation axis during that month. Generally, the variation of coefficient H122 represents an asymmetry of the equatorial axis with respect to the rotational axis or in other words an ellipticity of the equator, and the variation of coefficient H222 indicates that tile principal axes of the vertical deformation oscillate around the conventional X (Greenwich meridian) and Ydirections. Fig. 7 presents the angular variation a of the principal axes of the vertical deformation, wrt. to the Greenwich meridian, estimated by: O~-
1.ArcTan(H222I
~
H122
(2)
A rapid angular variation c~ is noticed from February to April.
6
Discussion
Despite the low magnitude of lateral Love numbers the relative information of horizontal displacements is usually assumed to be significant,
l'~,
Chapter 30 • Spatial Spectral Inversion of the Changing Geometry of the Earth from SOPAC GPS Data
11i
......--',,/,
g0.5[ i
t
. ,.....,...... I,,, .X'"
]]
""...~...
~i "'x ....... "
-o.
\
]
~
1
.5 T'
g'-1.5
o.'
'""I,.'
-1.5
2
4
6 8 month of year
10
12
00
2
4
6
I
I
8
10
degree
F i g . 6 Temporal variation of vertical component degree-2 coefficients H~o, H121, and H221, with l a formal errorbars.
Fig. 8 Power spectrum of height displacement SH sine and cosine coefficients (the twelve curves correspond to the twelve months of one year).
80 60
2
...... • _ , . . . ~
"'*" H130
01
1
0 ,-o, 40
"0
ii
~E0
01 20 i
0
-1
-2
-20 2
4
6 8 month of year
10
12
-3
."
IE....... 3["
4
6
month of year
8
1.
12
F i g . 7 Angular variation ct of the principal axes of the vertical deformation wrt. to the Greenwich meridian derived from H~22 coefficients.
Fig. 9 Zonal coefficients containing large power are: H130, H14o and H~50.
as horizontal variances are typically ten times smaller t h a n vertical variances for global referenced station coordinates. However, due to t h e r a n d o m spatial distribution, we believe t h a t so far the lateral component only adds noise. This is a topic of investigation for the future. In artificial cases, i.e when mass redistribution compensates the effect of height displacement, it is possible to obtain a non-zero¢ LLN ! h 0 (Varga 1983). However, as the LLN h 0 does practically not influence the results of load calculations, most authors simply neglect it. Fig. 8 shows the monthly power spectrum of the height displacement SH coefficients/-/[/,.~ over one year. A first insight reveals t h a t degree three, five and six also contain a large con-
tribution to the complete vertical displacement field. Fig. 9 shows the zonal coefficients of degree three, four and five divulging large amplitudes. The zonal coefficient /-/150 has a positive m a x i m u m in April (2.1 mm). The load moment vector, as defined by equation (58) in Blewitt and Clarke (2003), is simply obtained by multiplying the degree-1/-/i1,~ coefficients by E a r t h ' s mass! M, and dividing it by t h e load love number h 1. Kusche and Schrama (2005) find a similar load moment vector as Blewitt et al. (2001) over the period 1999.5-2001.0. Our transformed degree-1 coefficients (reflecting t h e load moment vector) fit close to Figure 2 of Kusche and Schrama (2005) after the year 2002. Tab. 3 shows the discrepancy of our degree-1
199
200
P.J. Mendes Cerveira • T. Hobiger • R. Weber. H. Schuh
coefficients with the ones published in Blewitt and Clarke (2003). In our study, the annual amplitude of the sectorial H.ill coefficients is larger than that of the zonal coefficient Hl10. A probable cause for this difference could originate from the different epochs under consideration. Blewitt and Clarke (2003) used IGS data from 1996.0 to 2001.0. The diminution in amplitude of the variation in the z-component of the load moment vector from 2002 upwards is obvious on Figure 3 of Kusche and Schrama (2005).
I
I
"'"'.. ""-. "-. \
81| 6~.| 41
E
E
,,
/ "-.
\
\/
\:
i
H211 OS /-/11o
/-/111 /-/211
AA 2.97 ± 0.12 0.90 ± 0.15 1.30 ± 0.13 AA 1.22 1.61 2.22
AP 236 ± 2 266 ± 9 165 ± 6 AP 279.56 171.77 156.61
SA 0.67 ± 0.12 0.31 ± 0.15 0.27 ± 0.12 SA 0.37 0.70 0.57
SP 2 7 ± 10 249 ± 26 121 ± 25 SP 74.38 95.25 44.16
Fig. 10 shows four low-degree load harmonic 4rrlnormalized coefficients: T120, T122, T221 and T144. The load spectral coefficients were calculated by using the conversion factors f~ provided in Tab. 1. A comparison with Figure 3 of Kusche and Schrama (2008) shows excellent agreement between their and our coefficients. This statement is in general also valid for the projected annual and semi-annual gravity changes from Satellite Laser Ranging (SLR) analysis to the loading mass variations, also included in Figure 3 of Kusche and Schrama (2005). However, the higher load harmonic coefficients in our study as well as the ones of Kusche and Schrama (2005) show less power than those obtained from SLR analyses. This good agreement to Kusche and Schrama (2005) suggests that either the "seasonal irregularities" (mentioned in section 1) in the combined IGS data are not that relevant for our study, or that similar irregularities are also present in the SOPAC data. We tend to accept the first explanation, as the IGS combined data results from
/
/ 2
j / / ,,
Z----.
/
// I
I
2
4
~ ......
,,,"
I
6 8 month of year
I
10
Fig. 10 Low-degree load harmonic 47cnormalized coefficients T120, T122, T221 and T144, to be compared with Figure 3 of Kusche and Schrama (2005).
noted by BaC) and our study (OS). AA=Annual Amplitude [mini, SA=Semi-annual Amplitude [mm], AP=Annual Phase [deg], SP=Semiannual Phase [deg], using the same conventions as Blewitt and Clarke (2003).
BaC Hllo Hlll
.-"
:
44
\
-10
T a b l e 3: C o m p a r i s o n of height f u n c t i o n spect r a l degree-1 coefficients Hil.~ f r o m G P S inversion b e t w e e n B l e w i t t a n d C l a r k e (2003) (de-
.._
;~ , \
4
,."
\ 4 /" 44, ,"
\
\,"
-o
--4 -T120 - - - T122 , , " ' . . . . T221 ,'" '-" 'T144 ," .-
\ ',
s~.
-, -2
I
," ""4 ," ~ 4 4 ,' / \'4 : s / \4
~e/
2V~ -
I
, - ,.
eight different Analysis Centers, thus minimizing the impact and contribution of the CODE solution in global studies. 7
Conclusion
Inversion of the changing geometry of Earth's surface is a valuable source of information for many scientific fields. A geometric and physical interpretation of the surface spherical harmonics Hi~,~, from degree-0 to degree-2, provides details on the Earth's properties. We discussed degree-0 extensively, pointing to the still remaining unresolved issues of that coefficient. Our iterative 2D-interpolation approach is another possibility to obtain stable spectral coefficients from a sparse and uneven GPS network with lacking coverage in many areas. Given the present global distribution of GPS stations, an SH expansion up to degree and order 11 is absolutely justified, if we apply constraints via our iterative 2D-interpolation approach for a 10 degree spatial resolution grid. This 2D-interpolation is integrally based on SH coefficients, estimated from lower degrees of the irregular sampled data. We showed that conservation of the Earth's mass influences the H120 coefficient significantly in the order of 0.5 mm. Recent results on higher load coefficients (Kusche and Schrama 2005) have been confirmed independently by our approach, using SOPAC data. Finally, it must be emphasized that our results are only valid under the assumption that other error sources with annual periods, which
Chapter 30 • Spatial Spectral Inversion of the Changing Geometry of the Earth from SOPACGPS Data
might cause a p p a r e n t geophysical deformations, e.g. deficiencies in the used m a p p i n g functions for high latitudes ( B o e h m and Schuh 2004), have been completely removed prior to our analysis. In this context, very recent studies ( B o e h m et al. 2005) show t h a t the i m p a c t of the new Vienna M a p p i n g Functions 1 (VMF1) reduces seasonal signals in station height time series significantly over m a n y regions of the globe. However, these new m a p p i n g functions are not yet routinely used at SOPAC. Acknowledgments
We thank the GPS analysts at SOPAC who make space geodetic data publicly available. We are aspecially grateful to Xiaoping Wu for providing a constructive review of this paper. The first author wishes to express his gratitude to Peter Varga, Olivier Francis and Tonie van Dam for scientific support. A significant part of this work is funded by the Bourse Formation Recherche BFR03,/029 and CEDIES awarded by the Ministhre de la Culture, de l'Enseignement Sup6rieur et de la Recherche of the Grand-Duchy of Luxembourg. The first author acknowledges the financial support awarded by the Bourse Fondation Mathieu, University of Luxembourg. The I~TF_/K package was kindly provided by Gilles Celli and Patricia Codran.
References
Blewitt, G., D. Lavallee, P. Clarke and K. Nurutdinov (2001). A new global mode of Earth deformation: Seasonal cycle detected, Science, Vol. 294, No. 5550, pp. 2342-2345. Blewitt, G. (2003). Self-consistency in reference frames, geocenter definition, and surface loading of the solid Earth, Geophys. Res. Lett., 108 (B2), 2103, doi: 10.1029/2002JB002082. Blewitt, G., and P. Clarke (2003). Inversion of Earth's changing shape to weigh sea level in static equilibrium with surface mass redistribution, J. Geophys. Res., 108 (B6), 2311, doi: 10.1029 / 2002 JB 002290. Boehm, J., and H. Schuh (2004). Vienna Mapping Functions in VLBI analyses, Gcophys. Res. Lett., 31, L01603, doi:10.1029/2003GL018984. Boehm, J., P.J. Mendes Cerveira, H. Schuh and P. Tregoning (2005). The impact of mapping runetions for the neutral atmosphere based on numerical weather models in GPS data analysis, IA G Scientific Assembly 2005, Springer Verlag, in press.
Farrell, W.E. (1972). Deformation of the Earth by surface loads, I~ev. Geophys. J. Int., 10, pp. 761797. Gross, R. S., G. Blewitt, P. J. Clarke and D. Lavallee (2004). Degree-2 harmonics of the Earth's mass load estimated from GPS and Earth rotation data, Geophys. Res. Lett., 31, L07601, doi: 10.1029 / 2004G L019589. King, R.W., and Y. Bock (2005). Documentation for the GAMIT GPS processing software Release 10.2, Mass. Inst. of Technol., Cambridge, MA. Kusche, J., and E.J.O. Schrama (2005). Surface mass redistribution inversion from global GPS deformation and GRACE gravity data, J. Geophys. Res., 110, B09409, doi:10.1029/2004JB003556. Lavallee, D., and G. Blewitt (2002). Degree-one Earth deformation from very long baseline interferometry, Geophys. Res. Lctt., Vol 29(20), doi:10.1029/2002GL015883. Mautz, R. (2001). Zur Loesung nichtlinearer Ausgleichungsprobleme bei tier Bestimmung von Frequenzen in Zeitreihen, Deutsche Geodaetische Komrnission, Reihe C, Nr. 532, Muenchen. Mendes Cerveira, P.J., R. Heinkelmann, J. Boehm, R. Weber and H. Schuh (2005). Contributions of GPS and VLBI in understanding station motions, Journal of Geodynarnics, in press. Nikolaidis, R. (2002). Observation of Geodetic and Seismic Deformation with the Global Positioning System, Ph.D. Thesis, University of California, San Diego. http://sop ac. ucsd. edu/input/processing/pubs / nikoThesis.pdf. Parma, N.T., and M.P. Stewart (2003). Aliased tidal signatures in continuous GPS height time series, Geophys. Rcs. Lett., 30(23), 2184, doi: 10.1029 / 2003G L018828. Petrov, L., and J.-P. Boy (2004). Study of the atmospheric pressure loading signal in VLBI observations, J. Geophys. Res., Vo1.109, No.B03405, 10.1029 / 2003JB002500. van Dam, T.M., G. Blewitt and M.B. Heflin (1994). Atmospheric Pressure Loading Effects on GPS Coordinate Determinations, J. Geophys. Res., Vol.99, pp. 23939 23950. Varga, P. (1983). Potential Free Love Numbers, Manuscripta Geodetica, Vol.8, pp. 85 92. Wu, X., D.F. Argus, M.B. Heflin, E.R. Ivins and F.H. Webb (2002). Site distribution and aliasing effects in the inversion for load coefficients and geocenter motion from GPS data, Geophys. Res. Left., 29 (24), 2210, doi:10.1029/2002GL016324. Wu, X., M.B. Heflin, E.R. Ivins, D.F. Argus and F.H. Webb (2003). Large-scale global surface mass variations inferred from GPS measurements of load-induced deformation, Gcophys. Res. Lett., 30 (14), 1742, doi:10.1029/2003GL017546.
201
Chapter 31
Improved processing method of UEGN-2002 gravity network measurements in Hungary L. V61gyesi, L. F61dvfiry Department of Geodesy and Surveying, Budapest University of Technology and Economics; Research Group of Physical Geodesy and Geodynamics of the Hungarian Academy of Sciences, H- 1521 Budapest, Hungary. G. Csap6 E6tv6s Lorfind Geophysical Institute of Hungary, H-1145 Budapest, Hungary, Kolumbusz u. 17-23.
Abstract. A new method and software has been
made and tested in the Hungarian part of the UEGN-2002 network. Making a suitable base gravity network and providing proper data for UEGN2002 has required some experimental measurements and many important investigations. In case of precise gravity measurements the determination of the vertical gradient's real value is necessary. Using the real or normal value of vertical gradients may give 6-10 gGal differences of height reductions depending on the reference height of the instruments. Taking into account the periodical errors of LCR gravimeter's reading device is very important; neglecting the periodical errors may give 25-30 gGal errors of Ag between measured points depending on the instrument (in case of our instruments LCR 963 and 1919 it was found to be below 2 gGal). We have found that the accuracy of the parameter estimation increases with fewer periods estimated. We suggest to always using a full parameter set for the estimation of the periodical correction. Based on our investigations the reliability of the MGH-2000's adjusted data is significantly better, than the reliability of the European network's one (probably because of the different reliabilities of the different European countries' gravity data). According to our plans, after the final adjustment of UEGN-2002 we are going to readjust the Hungarian MGH-2000 taking into account the adjusted g values of UEGN-2002. Keywords. Gravimeters, gravity measurements, gravity networks, vertical gradient, periodical errors, corrections of measurements, adjustment
1 Introduction The International Union of Geodesy and Geophysics (IUGG) has long been planning to set up a unified scale and datum gravimetric network which could be applicable in the whole continent of Europe. Its conditions have been established when several countries have got absolute gravimeters,
providing unified scale in accordance with the current accuracy specifications. At the same time the need for increasing the accuracy of global geodetic reference systems, and for solving several geodynamic and geotectonic problems, have brought about the realisation of this objective as a daily routine. Among others this purpose was served by the establishment of Unified European Gravity Net (UEGN-93) by 11 countries (Boedecker, 1993). Later further countries such as Hungary have been joined to UEGN and completed some necessary works (making new absolute points and performing common measurements on the neighbouring countries' network) for the joining. Unfortunately significant inhomogeneities of the UEGN-93 point distribution can be seen, investigating the networks of the different UEGN countries. As far as we know there is no country which has transformed its own network to the UEGN datum up to now. However this is an important problem and according to our plans after the final adjustment of UEGN-2002 we are going to readjust the Hungarian MGH-2000 taking into account the adjusted g values of UEGN2002 referring to Hungary as constraints of a fixed network.
2 The former Hungarian works
gravity net-
We provide an overview of processing and adjustment methods that have been applied for the gravimetric network in Hungary from the beginning of the fifties up to now. In former times to make use of complex equations was not feasible according to the computational capacity of the time, so many factors have been neglected, e.g. instrument drift. For the first time, the Hungarian gravimetry network has been determined in the 1950s (referred as MGH-50). Both the processing and the adjustment have been done manually; solving this problem was a definitely time consuming procedure. In the eighties due to the development of computers, processing of data with much larger set of unknowns be-
Chapter31 • ImprovedProcessingMethodof UEGN-2002GravityNetworkMeasurementsin Hungary came executable. The registration of the observations, the processing method and also the adjustment became much more easily feasible, providing more space for optimization of the solution, e.g. the parameterisation of the processing sequence could be done in different manners (a priori and a posteriori as well), or tests could be performed for an optimal adjustment method. Previously the memory limits of computers allowed solving for some hundred unknowns in a short duration. Nevertheless, in order to be able to handle more unknowns in a more flexible way and to consider more effects than before (e.g. changes in water table, periodical errors of the data registration), the development of an up-to-date software became necessary. A new method and software has been made and tested in the Hungarian part of the UEGN-2002 network and the accuracy of the adjusted gravity network has been improved. Hungary's first gravity network (MGH-50) covering the entire territory of the country was established by the Lorfind E6tv6s Geophysical Institute (ELGI) during the early 1950s. The measurements were carried out by a Heiland GSC-3 astatic gravimeter. Both the processing and the adjustment of these measurements have been carried out manually in 1954. This network was containing 16 first order and 493 second order points. Description of measurements, processing's and adjustment's method and the results can be found in (Renner and Szilfird 1959). Constraints of the second order network adjustment were the adjusted g values of the first order network points. MGH-50 gravity network was adjusted in the Potsdam Gravity System. It is interesting to mention that a correction of magnetic azimuth was applied here for the first time; this type of correction was applied nowhere else before. At the same time the height correction was not applied here. According to our estimations considering that all measurements were made on a special tripod, omitting the height corrections may cause about 515 gGal errors of adjusted g values. As a result of industrial and infrastructural developments during the 1960-70s, most of the base points established mainly along national roads were beginning to deteriorate or simply became unsuitable for their original purposes. This was the main reason why a new gravity network had been established during the 1980-88s. One part of network measurements was performed in international cooperation (Csapd et al, 1994), and 8 Sharpe, 4 Worden and 1 LCR-G gravimeters were applied in the measurements. Before the adjustment process a lot of different investigations were performed (Csap6, Sfirhidai 1990a) and the results were applied to our
newest gravity network. The network has been adjusted as a fixed one by LSQ method, this was the common adjustment of the first and the second order network. The constraints were the g values of the 5 absolute points measured by the GABL absolute gravimeter. A lot of different adjustment version was investigated, but each version of adjustment used the measured Ag values between points as an independent measurement data (Csap6, S/trhidai 1990b). The error of unit weight of the adjusted network is ;10 = +16 ~tGal, the errors of adjusted values are +2-9 gGal for the 408 points (1 gGal = 10 8ms 2).
3 N e c e s s i t y of d e t e r m i n a t i o n Hungarian gravity network
of a n e w
The establishment of the newest gravity network MGH-2000 in Hungary and the necessity of a new processing of the measurements are essential due to several reasons (Csap6, V61gyesi, 2001). First of all, several new absolute measurements have been performed in the country. Moreover the points of Hungarian part of UEGN-2002 are taken out from the points of the new MGH-2000 network demanding an increased need in accuracy. Making a suitable base gravity network and providing proper data for UEGN-2002 has required some experimental measurements and many important investigations. In what follows these investigations and results are presented.
4 I n f l u e n c e of local v e r t i c a l g r a d i e n t s on the v a l u e s of Ag b e t w e e n p o i n t s Generally only the normal value of vertical gradient (0.3086 mGal/m) is used for height reduction of gravity measurements instead of the real value. During last years we have determined the real value of vertical gradients at different points and found a difference 20-25% between the real and the normal values (Csap6, V61gyesi, 2004). In Table 1 values of measured vertical gradients can be seen as examples in some points in Hungary. In this table (,o and 2 denote ellipsoidal coordinates, H is the height of the point and VG is the vertical gradient. In case of precise gravity measurements (e.g. measurements on a calibration base line or measurements on special polygons for investigations of local variations of gravity) the use of observed vertical gradient's real value is necessary. Using the observed or normal value of vertical gradients may give 6-10 gGal differences of height reductions depending on the reference height of an instrument
203
204
L. V61gyesi • L. F61dv~ry. G. Csap6
above the benchmark (e.g. the reference height LCR gravimeters are about 50-250 mm). So it obvious that instrument height should be chosen small as it is possible and should be the same different points. Table 1. Measured vertical gradient in ~tGal/m (1 btGal = 10 -s ms -2)
Point
I
(o
(VG) values 2
106/Ercsi/
47-14-58
92.0/Madocsa/ 2142/Tfiborhegy/ 82/Budapest UEGN/ 821 /Mfityfishegy/ 103/Tolna/ 2143/Hfirmashatfirh./ 107/Buda6rs/
46-41-1918-57-40 47-33-01 19-00-31 47-32-00 19-01-00 47-32-00 19-00-57 46-25-20 18-47-32 47-33-23 19-00-10 47-26-58 18-59-14
18-53-57
of is as at
in Hungary
H I VG 124
94 412 202 201 100 463 126
-309.3 -255.2 -354.7 -251.9 -262.5 -310.7 -386.0 -308.2
5 Investigation of periodical errors of LCR gravimeter's reading device Periodical errors of LCR gravimeter's reading device are discussed in many publications, e.g. Becker (1984), Lederer (2004). Neglecting the periodical errors can give 25-30 ~tGal errors of Ag between measured points. Determination of periodical errors is possible by measurements in special calibration lines or in laboratories (Chao, Baker 1984). Lederer (2004) has determined the periodical errors of several LaCoste gravimeters, including our two instruments LCR 963 and LCR 1919. His analysis is based on observations at two gravimetric baselines in Pecny-Chocerady (Czech) and in Modra-Piesok (Slovakia) in 2002. The range of these calibration lines is about 40 mGal. The Pecny-Chocerady baseline contains 16 points, the intervals varies between 0.2 + 16.2 mGal. The Modra-Piesok baseline contains 31 points with uniform intervals of about 4 mGal. After the adjustment of UEGN-2002 we are planning to re-process and re-adjust the measurements of MGH-2000, therefore taking into account the periodical errors of gravimeters is important in our new software. So the analysis has been reperformed for LCR 963 and LCR 1919 gravimeters based on the earlier observations in the calibration line of Pecn~-Chocerady. The reason of the reanalysis is that the classical sine-cosine and the spectral representations of the Fourier series used in Lederer (2004) were found to be inconsistent. Probably the phase has been provided incorrectly. The periodical error was defined as a discrete Fourier-series, with the discrete periods being related to the radius of the gears inside the instrument.
The range of the observations with LCR 963 varies between 4378 and 4425 mGal, and with LCR 1919 between 4439 and 4482 mGal, therefore no longer periods of the instruments could be determined. Periods which can be determined are 1.00, 3.67, 7.33 and 36.67 mGal (or equivalently CU - counter unit) in case of LCR 963, and 1.00, 3.94, 7.88 and 35.47 reGal (CU) in case of LCR 1919. The equation of the periodical errors is x ( t ~ ) - Ao + ~
Ck cos
t~-To
k=l
where ti is a dial reading, x is the variation of gravity after the adjustment, A0 is the bias of the observations, P is the periods, K is the number of the periods, C and S are the Fourier coefficients, To is a phase shift, which can be chosen arbitrarily. In spectral representation, it reads x(t~ ) - Ao + ~K ( Ak sin 12Jr (t~ - To ) + k =1
~
1/ (/)i
'
where A i - ~/C 2 + S 2 and (,oi - arctan(Ci/S i) . Parameters of the periodical errors have been determined considering 4 different periods (1.00, 3.67, 7.33 and 36.67 CU for LCR 963, and 1.00, 3.94, 7.88 and 35.47 CU for LCR 1919). The results are shown in Table 2 and 3 for LCR 963 and Table 4 and 5 for LCR 1919. Dimensions of P and To are CU (Counter Unit), that of ~0 is decimal degree, and all the other quantities are in gGal. Comparing the parameters to Lederer (2004), significant differences can be detected. Unfortunately no certain information of Lederer's adjustment is available. However, there are still some known differences of the parameterisation, e.g. Lederer has chosen To below the minimum of the observations, t, while in our case it was chosen to be the mean of them. Due to the difference in choice of To, comparison only in the non-phaserelated variables can be sought for, e.g. A or A0 but not in C, S and ~0. The a posteriori standard deviations in these tables are denoted by m, and are in the same order as the signals themselves. Comparing the parameters in Table 2 with Table 3, and Table 4 with Table 5, it can be seen that the accuracy of the parameter estimation increases with less periods estimated. In general, it is clear that the parameters vary notably by the choice of the estimated periods. We have investigated how large variations of the amplitude
Chapter 31 • Improved Processing Method of UEGN-2002 Gravity Network Measurements in Hungary
can be found due to the choice of the periods. All possible combinations of these 4 periods have been adjusted; it provided 8 estimations for every period. The accuracy estimate of the amplitudes is provided in Table 6.
Table 2. Estimated parameters of periodical correction for LCR 963 considering 4 periods.
P
To
Ao
mAo
A
1.00 3.67 7.33 36.67
4401.5625 4401.5625 4401.5625 4401.5625
23.15 23.15 23.15 23.15
1.25 1.25 1.25 1.25
1.47 0.65 1.97 0.96
mA
To
1.00 4401.5625 3.67 4401.5625 7.33 4401.5625
Ao
mAo
A
1.44 142.98 1.74 86.75 1.84 7.06 2.30 102.61
mA
According to our investigations the gravity difference AgAB between points A and B can be expressed by the formula
AgAB -- M[(CB~ B --CAg A) q- ~)gH q- ~ p if- ~ tide q- ~ s k if- ~ drift]
~9
Table 3. Estimated parameters of periodical correction for LCR 963 considering 3 periods.
P
6 Corrections of gravity measurements
q)
2 2 . 5 8 0 . 7 2 1.48 1.32 137.40 2 2 . 5 8 0.72 0.53 1.41 54.30 2 2 . 5 8 0 . 7 2 1.61 1.67 3.46
Table 4. Estimated parameters of periodical correction for
LCR 1919 considering 4 periods. P
To
Ao
mAo
A
mA
~o
1.00 3.94 7.88 35.47
4466.0160 4466.0160 4466.0160 4466.0160
-5.52 -5.52 -5.52 -5.52
0.47 0.47 0.47 0.47
0.74 0.35 0.18 0.31
0.54 0.60 0.62 0.67
259.95 224.31 317.99 136.28
where M is the scale factor of a gravimeter referring to the actual year determined on the national calibration line, CA and cB are the mean calibration factor or the value of the scale function referring to the scale readings ~ A and ~ B, 6gH is the height reduction, dgp is the barometric reduction, dgadeis the tidal correction, dg,k is the correction of periodical errors, and 6gd~/~ is the drift correction. Height reduction dg H can be computed as the product of the reference height of the gravimeter's sensing mass and the vertical gradient, so mmaccuracy of reference height is necessary. In case of computation of barometric reduction ~gp taldng into account of DIN standard No. 5450 is necessary: Pn = 1013.25 exp.(1013.25 x 10 -6 H ) m b a r so the barometric reduction: c~gp - ( p - pn )r .
Table 5. Estimated parameters of periodical correction for
LCR 1919 considering 3 periods. P
To
1.00 4466.0160 3.94 4466.0160 7.88 4466.0160
Ao
mAo
A
mA
q)
- 5 . 6 8 0.28 0.81 0 . 5 0 257.61 - 5 . 6 8 0.28 0.43 0 . 5 6 218.99 - 5 . 6 8 0.28 0 . 0 7 0 . 5 8 295.74
According to experiences of Boedecker and Richter (1984) to a similar climate we use the same empirical value of r they employed, r = 0.3 gGal/mbar. Tidal correction c7~g,de can be computed by 6g,d ~ - ~
Table 6. Accuracy estimate of the amplitude A due to the
choice of the periods. P [cu] 1.00
3.67 7.33 36.67
LCR 963 A &G~l] 1.43 +/- 0.05
0.90 +/- 0.33 1.79 +/- 0.23 0.71 +/- 0 . 3 7
P [CUJ 1.00
3.94 7.88 35.47
LCR 1919 A &Goi] 0.78 +/- 0.05
0.43 +/- 0.08 0.48 +/- 0.32 0.35 +/- 0.09
The statistics show that no relevant differences at period 1.00 CU can be found, but the others can differ notably. According to the results, we suggest to always use a full parameter set for estimation of the periodical correction.
~A~ cos(~o~ + co~t+ K~), i
where i is the index of tidal wave, A is the theoretical amplitude, (p is the theoretical phase, co is the angular velocity, 6 is the deformation coefficient, ~c is the phase delay and t is the universal time (UT). Altogether 237 different tidal waves have been considered in our computations. A value of the deformation coefficient 6 = 1.16 have been used (Lassovszky, 1956). This procedure of tidal computation has been chosen because a lot of partners in our former international cooperation were using this type of computation too (Holub, 1988). Drift correction 6gdr ~
is computed by the so-
called slope method (Csap6, Sfirhidai 1990a).
205
206
L. V61gyesi • L. F61dv~ry. G. Csap6
7 Adjustment of gravity network MGH2000 The MGH-2000 network consists of 22 absolute points (7 points can be found in the neighbouring countries from these) and 442 further base points as it can be seen on Fig. 1. 5544 connections were measured by six LCR-G gravimeters between these points, partly in international cooperation. The accuracy of absolute points varies between 2+4 gGal. Two types of adjustment of these measurements have been performed, as a fixed network (i.e. the absolute points are fixed) and as a free network (i.e. values of absolute points can change during the adjustment). First the MGH-2000 network has been adjusted as a fixed network by Least Squares method. The constraints of the network were the g values of the 22 absolute points. Performing the adjustment the Danish iteration procedure was applied. To decrease the effect of relatively large errors in the adjustment the weights of measured data having been greater errors should be decreased, but before the adjustment the errors are not known. This contradiction can be solved by iteration. In the first step (i= 1) all observed data has equal weight (pil = 1). In the further steps the weight of the ith measurement will be: py-
where j is the actual iteration step and v j _ 1 is the residual from the previous step. The coefficient a k is appropriate when p = 0.25 for the erroneous measurement (Soha, 1986). The threshold of errors can be taken as the function of the errors of unit weight, then: ak
31v~
-
where vk - 3¢t0
if
Vmax > 3/10
vk - 2¢t0
if
2/z 0 < Vmax < 3/10
V~ --/tO
if
//0
2~o •
< Vmax <
The erroneous measurements will get less weight by each subsequent iteration step. The iteration should be repeated until the error of unit weight is decreasing in a considerable way. In the adjustment of MGH-2000 three steps of iteration proved to be sufficient. The error of unit weight of the adjusted network is +14 gGal, the average error of the adjusted values is +7 gGal. In the second case the MGH-2000 network has been adjusted as a free network. In this case the g value of Budapest absolute station has been used for reference level. Differences of adjusted g values are about +20 pGal coming from the two types of adjustment.
2
l + a k Vj_l I
I
I
PLESIVEC , --
I
•
I
,.01 BRATISLA
KAISEREICkl
E
~111 HURBANOVO
2(30000-t
,k'l
•
i? v
•
UEGN points
A Absolute points z~ Absolute points 149
•
I.-I1. order base I
500000
600000
700000
Fig. 1. The structure of MGH-2000
800000
'
900000
'
Chapter 31 • Improved Processing Method of UEGN-2002 Gravity Network Measurements in Hungary
8 Comparison the results of different adjustment's data After the simultaneous adjustment of the Hungarian, Czech and Slovakian gravity network, comparison of the three adjusted results was possible. Each country considered the common gravity points and the partner's measurements for their own adjustment. The Czech and Slovakian adjustment have been performed as a free network's adjustment (Charamza, Trfiger 1971). We compared our adjusted values to the Czech and Slovakian results on those points which were included in the adjustment all of three networks. The maximum deviation we have obtained on identical points was about 3-8 gGal. The results can be regarded as excellent taken into consideration the differences of the three networks (different gravimeters, database, and adjustment). Similarly, two further comparisons were made. We compared the Hungarian, Austrian and U E G N ' 9 4 gravity datum (Csap6 et all, 1993) based on 8 common points (i.e. points which are included in both networks). The Hungarian datum proved to be higher than the Austrian by 18 gGal. We compared the gravity values obtained for common points in the adjustment of UEGN '94 and MGH-2000. We could do this because five Hungarian points which were part of the Austrian-Hungarian interconnecting measurements in 1992-93 were already included in the adjustment of UEGN '94. Based on the five points the Hungarian datum is higher by 14 gGal than the UEGN '94 one (see Table 7).
Table 7. Comparison of UEGN '94 and MGH-2000 data Point
Gravity [mGal]
UEGN'94 Fert6d Hegyesh. K6szeg Sopron V61csej
980824,222+ 8,0 980844,449+12,0 980784,705+15,0 980808,350+14,0 980802,189+14,0
]
MGH-2000 980824,234 + 4,9 980844,460 + 7,0 980784,713 + 5,0 980808,375 + 5,4 980802,203 + 4,1
DW: [gGal] 12 11 8 25 14
It can be seen from Table 7 that the reliability of the MGH-2000's adjusted data is significantly better than the reliability of European network's one (because of the different reliabilities of the different European countries' gravity data). According to our plans after the final adjustment of UEGN-2002 we are going to readjust the Hun-
garian MGH-2000 taking into account the adjusted g values of UEGN-2002 referring to Hungary as constraints of a fixed network.
Acknowledgements Our investigations are supported by the National Scientific Research Fund OTKA T-037929.
References Becker M (1984): Analyse von hochpr~izisen Schweremessungen. DGK, Reihe C, Nr 294 (Dissertationen), pp. 37-50. Mfinchen. Boedecker G, Richter B (1984): Das Schweregrundnetz 1976 der Bundesrepublik Deutschland. Tell II. DGK, Reihe B, Nr 271, pp. 30-31. Mfinchen. Boedecker G (1993): Ein einheitliches Schweregrundnetz ffir Europa. Zeits.f.Verm.wesen 8/9, pp. 422-428. Chao D, Baker E M (1985): A study of the optimization problem for calibrating a LaCoste and Romberg ,,G" gravity meter to determine circular errors. OSU, Dep. of Geod. Sci. and Surveying, 1985. No 364, Columbus. Charamza F, Trfiger L (1971): Bearbeitung der Messungsergebnisse auf den Schwerepolygonen. Geofysikalni Sbornik. XIX, pp. 109-119. Praha. Csap6 G, Sfirhidai A (1990a): The new gravity basic network. Geod. 6s Kart. 42, 2, pp. 181-190. (In Hungarian) Csap6 G, Sfirhidai A (1990b): The adjustment of the new Hungarian gravimetric network (MGH-80). Geoddzia ds Kartogrfifia, 42, 3, pp. 181-190. (In Hungarian) Csap6 G, Maurers B, Ruess D, Szatmfiri G (1993): Interconnecting gravity measurements between the Austrian and the Hungarian network. Geoph.Trans. 38, 4, pp. 251-259. CsapdG, Szatmfiri G, Klobusiak M, Kovacik J, Olejnik S. Trage L (1994): Unified Gravity Network of the Czech Republic, Slovakia and Hungary. Springer, IAG Symposia, 113, pp.72-81 Csap6 G, V61gyesi L (2001): Hungary's New Gravity Base Net (MGH-2000) and It's Connection to the UEGN. (IAG Symposia Vo1.125) Springer, pp.72-77. Csap6 G, V61gyesi L (2004): New measurements for the determination of local vertical gradients. Reports on Geodesy, No.2 (69), pp. 303-308, Warsaw Holub J (1988): Tidal Observations with Gravity Meter Gs 15 at Station Pecnst Travaux Geoph., XXXIV, 4. Praha Lassovszky K (1956): Determination of the Earth's deformation parameter from gravity measurements. Geod6zia 6s Kartogrfifia, V., 1., pp.: 18-26. (In Hungarian) Lederer M (2004): Periodical errors of gravimeters LaCoste & Romberg. Land Survey Office, Dep. of Gravimetry, Prague. (Report) Renner J, Szilfird J (1959): Gravity Network of Hungary. Acta Techn. Academiae Scientamm Hungaricae, XXIII., 4, pp. 365-395. Soha G (1986): A robust adjustment with weights depending from measurement corrections. Geoddzia 6s Kartogrfifia, 38, 4, pp. 267-271. (In Hungarian)
207
Chapter 32
Spectra of rapid oscillations of Earth Rotation Parameters determined during the CONT02 Campaign J. N a s t u l a , B. K o l a c z e k , S p a c e R e s e a r c h C e n t r e o f the P A S , B a r t y c k a 18a, W a r s a w , P o l a n d R. W e b e r , H. Schuh, J. B o e h m Institute o f G e o d e s y a n d G e o p h y s i c s ( I G G ) , U n i v e r s i t y o f T e c h n o l o g y , Vienna, Austria Abstract. Over the two weeks of the CONT02 campaign in October 2002 the Earth rotation parameters (ERP) were determined with a resolution of one hour by VLBI and also by GPS. Analyses of these two very precise polar motion and UT1-UTC series reveal oscillations with periods of 8 hours and 6 hours. Their detection independently by two accurate space geodetic techniques is a first evidence for a real sub-semidiumal variation of the ERP. On the other hand the oscillations might still be an artefact stemming from similar methods of data sampling or from un-recovered diurnal drifts in the time series. Additionally, polar motion excitation functions were computed from the VLBI and GPS data series and analyzed in order to estimate the power necessary to excite rapid polar motion. The Atmospheric Angular Momentum (AAM) and Oceanic Angular Momentum (OAM) data series with high time resolution were used in the analyses, too. Time variable spectra of rapid polar motion oscillations and their excitation functions were computed by the Fourier Transform Band Pass Filter, showing time variations of the amplitudes of these oscillations. Keywords. Sub-diurnal Polar Motion, Atmospheric and Oceanic Excitation, Geodetic Space Techniques 1. Introduction Modem space geodetic techniques such as the VLBI and GPS allow to determine the Earth rotation parameters (ERP) with high accuracy and time resolution and to compare the observations with atmospheric and oceanic excitation for periods as short as a few days (Schmitz-Ht~bsch and Schuh, 2003). Time series with two hour or one hour resolution even allow to study the ultra-rapid oscillations of polar motion (xpol, ypol) and UT1-UTC with subdaily periods which were already detected in former investigations dealing with GPS and VLBI data from 1997 onwards (Kolaczek et al., 2000; Weber et al., 2002).
0.02
FTBPF X i -Y, 199
_
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_
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0.00 '
-15
'
-10
'
-5
I
0
'
I
5
'
I
10
'
I
15
hours
Fig. 1 FTBPF spectra of rapid oscillations of GPS polar motion derived from CODE ERP-series (Weber et a1.,2002). While the polar motion spectra clearly demonstrate the existence of distinct peaks both in the retrograde as well as prograde 6 hours and 8 hours band (e.g. derived from GPS time series covering the period 1997-1999, see Fig.l) the discussion about the origin of these oscillations is still ongoing. Although showing up in the series of both techniques the former are most likely artifacts (e.g. a daily drift in the ERP-estimates due to deficiencies in the orbit modelling would cause such a pattern of higher order harmonic terms at 12, 8, 6, 4 hours). On the other hand the retrograde 8 hours oscillation detected also in the GPS data set of the period 1999-2001 and in the Atmospheric Angular Momentum (AAM) variations of the year 1995 (Weber et al., 2002) may be a real oscillation of polar motion. To continue and intensify the former investigations we now make use of high resolution VLBI results available over the restricted period of the CONT02 campaign as well as the Oceanic Angular Momentum data series with temporal solution of one hour recently established for the period 1993-2002.2. 2. Analyses and Results The two weeks CONT02 campaign observed in October 2002 by the International VLBI Service for Geodesy and Astrometry (IVS) delivered dense ERP series with a time resolution of one hour. Simultaneous GPS observations were analysed by the CODE (Center for Orbit Determination in Europe, Univ. of Berne, Switzerland) Analysis Center. The following data series were treated in this paper:
Chapter 32 • Spectra of Rapid Oscillations of Earth Rotation Parameters Determined during the CONT02 Campaign
VLBI ERP data for the time interval of the CONT02 campaign, Oct. 16-31, 2002 (MJD 52563.0-52578.0); resolution one hour, determined at the Institute of Geodesy and Geophysics, University of Technology, Vienna, Austria using the OCCAM6.0 software (Titov et al., 2001). Two different solutions were carried out, one with the nutation fixed to model MHB2000 (Mathews et al., 2002), the second one with nutation parameters estimated in the least-squares fit. Median formal errors for the hourly ERP are +/-0.09 mas for xpol, +/-0.08 mas for ypol and +/-0.004 msec for UT1-UTC for the first solution and about 20-30% bigger for the second solution due to the high correlation between nutation parameters and the hourly ERP (see Tesmer et al. 2001). The accuracy is usually larger by a factor 2 to 2.5.
GPS ERP solution of CODE for the extended time interval Sept. 27 until Nov. 6, 2002 (MJD 52544.0-52584.0); resolution one hour, interpolated from 2 hours solution (same formal errors as reported above). In principal satellite techniques do not provide UT1UTC but length-of-day (LOD). The investigated GPS series was obtained by integrating LOD-estimates. More details about the CODE GPS solutions can be obtained from Hefty et al. (2000) and Rothacher et al. (2001). Oceanic diurnal and semi-diurnal variations provided by Ray (Ray et al., 1994, see IERS Conventions 2003) were a priori subtracted from the mentioned ERP observations. The one hour resolution of the GPS and VLBI data allows - according to the Nyquist limit - detections of oscillations with periods longer than two hours. Spectra of these polar motion and UT1-UTC data computed by the Fourier Transform Band Pass Filter (FTBPF) (Kosek, 1995) clearly show similarities (Figs. 2, 3). In particular oscillations with a period of 8 hours appear in the results of both techniques besides the well-known remaining semi-diurnal oscillations. The weak 6 hours oscillation is visible in Figure 2, too.
GPS ERP solution of CODE for the time interval of the CONT02 campaign, Oct. 16-31, 2002 (MJD 52563.0-52578.0); resolution one hour, interpolated from 2 hours solution. The formal errors are +/-0.02mas for the pole coordinates and +/-0.002 msec for UT1-UTC; the accuracy can be assumed as about five times bigger.
CONT02- 52563.0 - 52578.0 MJD
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GPS; Ray model removed mOrn VLBI Nutation model fixed; Ray model removed VLBI Nutation model estimated" Ray model removed
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period (hours) Fig. 2 FTBPF spectra of polar motion computed from GPS and VLBI during CONT02.
209
210
J. Nastula • B. Kolaczek • R. Weber. H. Schuh. J. B o e h m
4,5 -
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Fig. 3 FTBPF spectra of UT1-UTC computed from GPS and VLBI (nutation model estimated) during CONT02. The oscillations in the VLBI spectra of polar motion and UT1-UTC are more energetic than those in the GPS spectra (probably due to the higher noise level) except the 12 hours oscillation of UT1-UTC. There is a high correlation between the nutation parameters and the hourly ERP if all are estimated at the same time in VLBI analysis (Tesmer et al., 2001). This is the reason why in one of the solutions we fixed nutation to the best values available (Fig. 2) In the prograde part of the GPS spectra of polar motion the 8 hours oscillation is very weak (Fig. 2). The detection of rapid oscillations with a period of about 8 hours by two different techniques suggests that they may be real oscillations, e.g. the third harmonic of the diurnal ERP variations. On the other hand these oscillations might still be an artefact stemming from similar methods of data sampling or from un-recovered diurnal drifts in the time series. The oscillation with a period of about 9.5 hours in polar motion could be the effect of time variations of the 8 hours oscillation, created by the spectral analysis. Next, the spectra of excitation functions of polar motion were computed in order to estimate the power necessary to excite rapid variations of polar motion. These spectra were obtained in frequency domain through multiplication of the polar motion spectra by the transfer function (Brzezifiski, 1992, 1994, Brzezifiski et al., 2001). The equations below describing the transfer of polar motion to excitation of polar motion comprises of two parts which are the transfer functions of the pressure term Tp and wind term Tw:
p ( a ) - 7" ( a ) x ~ + L ( a ) x " (a)
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)
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) O"f
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where p(G) is the polar motion spectrum, yP,z ~ are spectra of the atmospheric excitation function of polar motion for pressure and wind, respectively, cr denotes the angular frequency and ~f= - ~ (1+1/430) is the observed value of the Free Core Nutation (FCN) angular frequency of resonance, o-c ~ ~/433 days is the Chandler wobble frequency, ~=7292115× 10-]]rad/sec is the mean angular velocity of the Earth's rotation, and ap=9.2x 10 -2, aw=5.5x 10 .4 are dimensionless constants.
--O-- Transfer function for the pressure ~O~ Transfer function for the wind ---I--- Transfer function for the wind= pressure
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Fig. 4 Transfer function of polar motion to excitation function of polar motion.
Figure 4 shows diagrams of the wind and pressure components of the transfer functions. It is obvious that the transfer function results in a considerable intensification of the rapid part of the excitation functions spectra in relation to the spectra of polar motion. In this paper we considered only the pressure influence on polar motion and assumed that the wind part is equal zero, because the AAM high time resolution (3 hours) series currently available includes only pressure data. Besides it is known that the pressure term is much more important that the wind term in polar motion excitation. Thus the amplification caused by the transfer function is a maximum case. Figure 5 shows the FTBPF spectra of polar motion excitation functions computed from GPS and VLBI data of the CONT02 campaign using the transfer function of the pressure term (Brzezifiski, 1992, 1994). This allows the comparison with the atmospheric and oceanic excitation functions of polar motion. Unfortunately a lack of dense sets of AAM and Oceanic Angular Momentum (OAM) for the period of the CONT02 Campaign prevents direct comparisons.
Chapter 32 • Spectra of Rapid Oscillations of Earth Rotation Parameters Determined during the CONT02 Campaign
However, the oscillation with a period of 8 hours was also detected in the series of available AAM functions with a resolution of 3 hours (Weber et al., 2002) (Fig. 6). Those results were obtained from the pressure term of excitation function of polar motion computed from a special analysis of the U.S. NASA GEOS Data Assimilation System for the year 1995.
100 GPS Excitation - computedbytransferfunction
90
VLBI Excitation computed by transfer funct on
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Recently, a dense set of OAM with a resolution of one hour was computed by Ponte and Ali (2002) for the time interval 1993.0-2002.2 (see also Brzezifiski et al., 2004). The ocean model is driven by surface wind and pressure fields from the NCEP-NCAR reanalysis (Kalnay et al., 1996) available solely at 6 hour intervals. Figure 6 (lower part) shows rapid spectra of the OAM containing both components, bottom pressure and currents, for the period 19932002.2. The spectrum is dominated by the semidiurnal oscillation however small peaks corresponding to an oscillation of about 8 hours are again visible.
I
8-16-14-12-10-8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 period (hours)
Fig. 5 FTBPF spectra of rapid oscillations of geodetic polar motion excitation from GPS and VLBI (nutation model estimated) covering the CONT02 time span.
1.00-
E
FTBPF Chil +i Chi2, AAM, 1995 -
1.00
-
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-5
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It is necessary to notice that the amplitudes of excitation functions (AAM and OAM) for the 8 hours oscillation are about one hundred times smaller than the geodetic excitation functions. It means that the energy of the known geophysical excitation is much too small. This needs further investigations of the transfer function and of the modelling of geophysical fluids with one hour resolution.
0.00
I
ho0urs 5
10
15
Finally, to compare time variations of the rapid oscillations, time variable Fourier Transform Band Pass Filter (FTBPF) spectra of polar motion variations and their excitation functions were computed for the ERP time series during CONT02. These variations of amplitudes, which are usually larger in the case of VLBI data are shown in Figures 7 and 8.
FTBPF Chil + i Chi2, OAM, 1993-2002,2
2
Additionally, time variable FTBPF spectra of hourly polar motion based on GPS observations in the o I ' I ' I ' I ' I ' I ' longer time interval 2002.737-2002.844 (Fig. 9) were 10 15 -15 -10 -5 hOrsou 5 computed. The variations obtained here differ slightly from the results of the shorter CONT02 data Fig. 6 FTBPF spectra of rapid oscillations of AAM (Weber series (Fig. 7). Nevertheless the only remaining et al., 2002) and OAM (Brzezifiski et al., 2004) polar motion major feature below the semi-diurnal band is again excitation functions, the retrograde 8 hours oscillation. 0
211
212
J. Nastula • B. Kolaczek. R. Weber. H. Schuh. J. Boehm
FT BPF time variable spectra, x -iy, CONT 2002 VLB! 0~1
~
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Fig. 7 FTBPF time variable spectra of x-iy from GPS and VLBI (nutation model estimated) data computed over CONT02 for the time interval 2002.8-2002.83; 52566.7305-52577.6523 MJD (units mas).
FT BPF time variable spectra, excitation functions, CONT 2002 VLBI ]~.
,/
•
.,
# ----._ ---------_
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~20 ;~0
. '
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Fig. 8 FTBPF time variable spectra of polar motion excitation from GPS and VLBI (nutation model estimated) computed over CONT02 for the time interval 2002.8-2002.83; 52566.7305-52576.2695 MJD (units mas).
Chapter 32 • Spectra of Rapid Oscillations of Earth Rotation Parameters Determined during the CONT02 Campaign
FT BPF time variable spectra, x-iy, GP8
FT BPF time variable spectra, exitation function, GPS 1
Fig. 9 FTBPF time variable spectra of x-iy and excitation function of polar motion computed from GPS data for the time interval 2002.737-2002.844; 52543.7227-52582.7812 MJD (units mas).
3. Conclusions Acknowledgements A common retrograde oscillation with a period of 8 hours could be detected in the spectra of both, the VLBI and GPS series of polar motion. This 8 hours oscillation is also apparent in the VLBI UT1-UTC (respectively integrated GPS LOD) series. The spectra have similar character but the variations in the VLBI spectra are more distinct. Time variable spectra of VLBI and GPS of rapid oscillations of both polar motion and excitation functions have similar character but the variations in VLBI are stronger.
The authors are grateful to the international services that provided the GPS and the VLBI data used for our study, i.e. to the IVS and the IGS. The research reported here was supported by the Polish State Committee for Scientific Research through project No 0001/T12/2004/26, and by the WTZ cooperation 24/2003/Poland-Austria/of the Austrian Exchange Service (OeAD) on the "Determination and analysis of Earth rotational motion, atmospheric and oceanic angular momentum ".
References Retrograde oscillations with periods of 8 hours and weak oscillations of 6 hours are present in the spectra of the atmospheric excitation function. The oceanic excitation function of polar motion shows very weak oscillations in the 8 hours band. Nevertheless, the amplitudes of the AAM and OAM excitation functions in the investigated spectral band are very small compared to the corresponding geodetic excitation function. This needs further investigations concerning the transfer function and the modelling of geophysical fluids with high time resolution as well as further investigations concerning systematic effects of the methods of ERP determination.
Brzezifiski, A. (1992). Polar motion excitation by variations of the effective angular momentum function: considerations concerning the deconvolution problem. Manuscripta Geodaetica, Vol. 17, pp. 3-20. Brzezifiski, A. (1994). Polar Motion excitation by Variations of the Effective Angular Momentum Function: II extended model, Manuscripta Geodaetica, Vol. 19, (3), pp. 157-171. Brzezifiski, A., Ch. Bizouard, and S.D. Petrov (2001). Influence of the atmosphere on Earth rotation: what new can be learned from recent atmospheric angular
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momentum esimates? Surveys in Geophysics, Vol.23, pp. 33-69. Brzezifiski, A., R.M. Ponte, and A.H. Ali (2004). Nontidal oceanic excitation of nutation and diurnal/semidiurnal polar motion revisited. JGR, Vol. 109, Bl1407, doi: 10.1029/2004JB003054. IERS, Conventions (2003). IERS Technical Note, No. 32, D.D. McCarthy and Gdrard Petit (eds.), Verlag des Bundesamts fiJr Kartographie und Geod~isie Frankfurt am Main, 2004. Hefty, J., M. Rothacher, T. Springer, R. Weber, and G. Beutler (2000). Analysis of the First Year of Earth Rotation Parameters with a Sub-Daily Resolution gained at the CODE Processing Center of the 1GS Journal of Geodesy, Vol. 74, No. 6, pp. 479-487. Kalnay, E., et al. (1996). The NMC/NCAR 40-year reanalysis project, Bull. Am. Meteorol. Soc., 77 (3), pp. 437-471. Kotaczek, B., W. Kosek, and H. Schuh (2000). Short Period Oscillations of Earth Rotation. "Polar Motion: Historical and Scientific Problems", IAU Colloquium
178. Astronomical Society of the Pacific Conference Series, VoÂ.208, pp. 533-544. Kosek, W. (1995). Time Variable Band Pass Filter Spectra of Real and Complex-Valued Polar Motion Series, Artificial Satellites, Planetary Geodesy, 30, Warsaw, Poland, pp. 27-43. Mathews, P.M., T.A. Herring, and B.A. Buffett (2002). Modeling of nutation-precession: New nutation series for nonrigid Earth, and insights into the Earth's Interior, J. Geophys. Res., 107, B4, 10.1029/2001JB000390.
Ponte, R.M., and A.H. Ali (2002). Rapid ocean signals in polar motion and length of day, Geophys. Res. Lett., 29(15), doi: 10.1029/2002GL015312. Ray, R.D., D.J. Steinberg, B.F. Chao, and D.E. Cartwright (1994). Diurnal and Semidiurnal Variations in the Earth's Rotation Rate induced by Oceanic Tides, Science, 264, pp. 830-832. Rothacher, M., G. Beutler, R. Weber, and J. Hefty (2001). High-Frequency Earth Rotation Variations From Three Years of Global Positioning System Data, JGR, 106, B7, pp. 13711-13738. Schmitz-Ht~bsch, H., and H. Schuh (2003). Seasonal and Short-Period Fluctuations of Earth Rotation Investigated by Wavelet Analysis. Springer-Verlag,
"Geodesy, The Challenge of the Third Millenium"; E.W. Grafarend, F.W. Krumm, V.S. Schwarze (eds), pp. 125-134. Titov, O., V. Tesmer, and J. Boehm (2001). OCCAM Version 5.0 Software User Guide, AUSLIG Technical Report 7. Tesmer, V., H. Kutterer, B. Richter, and H. Schuh (2001). Reassessment of Highly Resolved EOP Determined with VLBI. In: D. Behrend and A. Rius (Eds.):
Proceedings of the 15th Working Meeting on European VLBI for Geodesy and Astrometry, Institut d'Estudis Espacials de Catalunya, Consejo Superior de Investigaciones Cientificas, Barcelona, Spain. Weber, R., J. Nastula, B. Kotaczek, and D.A. Salstein (2002). Analysis of Rapid Variations of Polar Motion Determined by GPS. Springer-Verlag, "Vistas for Geodesy in the New Millenium", J. Adam, K.R Schwarz (eds); Vol 125, pp. 445-450.
Chapter 33
On the Establishing Project of Chinese Surveying and Control Network for Earth-orbit Satellite and Deep Space Detection Erhu WEI School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079,China Jingnan LIU President, Wuhan University, Luojia Hill, Wuhan 430072, China GPS Engineering Research Center, Wuhan University, 129 Luoyu Road,Wuhan 430079,China Chuang SHI GPS Engineering Research Center, Wuhan University, 129 Luoyu Road,Wuhan 430079,China Abstract: The relationship between the surveying
and control network(CSN) for earth-orbit satellite and spatial geodesy, and between the CSN for deep space celestial bodies and detectors and deep space detection are briefly summarized, and so are the basic technical needs of the DSN. Then, the techniques, the constituents and the distributing of Chinese satellite CSN (CSCSN) and other radio observing establishments in China are introduced. Lastly, with the primary CSCSN and other observing establishments, some projects for China to rebuild a more perfect CSCSN, and to establish a DSN are analyzed and stated. Key words= Satellite surveying and control Network; DSN; Microwave Radio Radar; LLR; DSCCs; Radio observation; Space-based Station
space detection project (see Liang B et al,2003).The basic establishments and technological demands of DSN have been introduced (see Liu J N e t al,2004). China has set up a network named Chinese satellite controlling and surveying network (CSCSN) for earth-orbiting satellite and spacecraft. Chinese academy of science (CAS) has also set up many radio observation establishments for radio astronomy. So it will be very important and valuable to organically combine all these establishments both of CSCSN and CAS, and to rebuild them to a more perfect CSCSN and a Chinese DSN. In this way, it will avoid to repeat construction, to save money and to improve the efficiency.
2
Chinese Space Controlling Surveying Establishments
and
1 Introduction
2.1 The CSCSN
To realize the project of spatial geodesy by geodetic satellites, it is needed to establish a satellite controlling and surveying network(SCSN) to track, to survey , to remotely control and to communicate with earth-orbit satellite. To realize the project of deep space detection, a deep space controlling and surveying network(DSN) is needed to survey and track the celestial bodies in deep space ,and to navigate, to remotely control and to communicate with the deep space detectors. Therefore, to setup a SCSN and DSN are the key task in China's spatial geodesy project and deep
From 1960s, China has set up a controlling and surveying network for near-earth satellite, named CSCSN which began to operate with 'Dongfanghong-l' (the first satellite of China) launched in April 24, 1970 . When China's Shenzhou- 5 spaceflight which has carried a people was launched successfully, the CSCSN has become the controlling and surveying network of Chinese spaceflight. In the CSCSN , the diameter of each antenna's curved face is less than 20m, therefore the network can only provide controlling and surveying service with high precise for the space detectors
Funded by the national '973 Project' of China (No. 2006CB701301) and the basic research of geomatics and geodesy of the key laboratory of geospace environment & geodesy ministry of education, China (No. 03-04-10)
216
E.Wei. J. Liu • C. Shi
within the distance of 3.6× 104km from the earth, such as rockets, lower earth-orbit satellites , sun-synchronous satellites , geosynchronous satellites, airships carrying people and so on. 1 ) The construction of the CSCSN The CSCSN (see Guangming Daily 2003) is mainly composed of controlling and surveying stations and controlling and surveying centers. Chinese satellite controlling and surveying stations mainly include the fixed stations on the land and the mobile stations on the sea. The fixed stations on the land are divided into two parts. One part is in China and the other part is in other countries over the world. The fixed stations in China are respectively located in regions of Xiamen, Qingdao, Weinan(also named Xi'an station) , Jiuquan , Hetian, Neimenggu, Hetian and Kashi. They have almost covered with the whole China. The fixed stations in other countries are respectively the Karachi station in Pakistan , Malindi station in Kenya and the Namibia station in Namibia. The mobile stations on the sea include mainly four "Yuanwang" ships to control and survey the spaceflight. The different track missions, the different locations of the ships. As shown in Figure.1 Chinese Satellite Control Network, in the track of the flight of "Shenzhou-5", the "Yuanwang-1" ship is located on the Pacific ocean to the southeast Japanese ocean, and the "Yuanwang-2" ship is located on Pacific ocean to the eastern New Zealand, and the "Yuanwang-3" ship is located on the Atlantic ocean to the western South Africa, and the "Yuanwang-4" ship is located on the India Ocean to the western Australia.
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and Jiuquan(Dongfeng command center), which send operating instructions , navigation instructions , attempering instructions and so on to the controlling and surveying stations, and to the satellites or airships. 2) The radio frequencies for control and survey At present, CSCSN surveys and controls the flying targets in the ultra short wave, C wave band and S wave band. 3 ) The controlling and surveying equipment The main controlling and surveying equipments of each controlling and surveying station include(see Zhao Y F et al, 2002): (~The communicating system of satellite, which makes sure that the message can be transmitted between each controlling and surveying station and center. (~Single impulse radar, which is used for catching and tracking the spaceflights. @The microwave uniform system for control and survey, which is the main controlling and surveying equipment working on the principle of responsion radar. Spaceflights receive the microwave signal sent out by the microwave uniform system and later send out the answering signal. The transmission of digital signal of remote surveying, remote controlling and communication is completely executed on it. In this way, it has formed a bidirectional expedite message loop among the command center, the controlling and surveying ships and the spaceflights. 4) The observation techniques adapted by the controlling and surveying stations: • Radio interferometry: to survey the angle of the spaceflights by comparing the received phases. • Dual frequencies Doppler: to survey the speed of the spaceflights. • Microwave radar surveying: to survey the distance from the ground station to the spaceflights and get the images of the spaceflights.
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CAS has respectively set up two radio telescopes with the diameter of 25 m at Shanghai astronomical observatory in 1987 and at Urumchi astronomical observatory in 1993. The two radio telescopes have formed the fixed VLBI observation stations of China. And the mobile VLBI station developed by Xi'an surveying and mapping institute and Shanghai astronomical observatory is parked in
Chapter 33 • On the Establishing Project of Chinese Surveying and Control Network for Earth-Orbit Satellite and Deep Space Detection
Kunming, which has formed the VLBI observation network of China with the two fixed VLBI observation stations above. In 1984, the Beijing astronomical observatory has set up a metric wave synthetical aperture composed of 28 radio antennas with each a diameter of 9m. And the purple mountain observatory has installed a millimeter wave radio telescope with a diameter of 13.7m in Delingha of Qinghai province in 1990. With the coming of the 21 century, the development of radio astronomy requires bigger antennas to receive radio signals. And the international radio astronomy group has advised to set up a huge telescope(square kilometer area, SKA ) with a receiving area of lkm z. To realize the SKA plan, Chinese astronomers have tabled a proposal to use the karst depressions in Pingtang and Puding counties of Guizhou province, and to establish the telescope arrays of about 30 individual unit telescopes with each roughly 200m in diameter to implement the lkm ~ receiving area. A Five hundred meter Aperture Spherical Telescope (FAST) proposed to be built is the forerunner project of the SKA plan.
communicate with the satellites or airships. Therefore, four mobile surveying and controlling stations have been added on the sea in the CSCSN. But, the conditions of earth rotation, water waving, ship swing, antenna shaking, target moving and the spacecrafts with high moving speed have seriously affected the precision of positioning the ships and the directional precision of tracking the satellites or spaceflights in real time. In order to increase the precision of observation and direction, it will obviously be a better way to add more fixed surveying and controlling stations to replace the mobile ones on the sea. /--:;-,7 ,..... ~.... ~.~-,:~-:~;~:!-;~,. I' [ .......
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-
~Built
3.1 The reconstruction of the CSCSN The earth is approximately a elliptical sphere with a major radius of about 6378140m and a flat rate of about 1/298.257. Therefore one circle flying voyage around the earth of a spaceflight about 300 km above the earth is about 40000 km (the perimeter of elliptical sphere) on the ground. Because of the influence of the earth's curvature and the barrier on the ground , a surveying and control station can only track a spaceflight with about 3500 km voyage on the ground, so the distance between the surveying and control stations should be less than 7000 km. As shown in Figure.1 Chinese Satellite Control Network, at present , the fixed stations of CSCSN are mainly located in the Asian countries such as china and Pakistan, and in the African countries such as Kenya(Malindi station) and Namibia. And from Figure.1 we can get acknowledged that the fixed stations have not covered with the American Continent and the CSCSN only has the Namibia fixed station in the southern hemisphere, so the CSCSN can't continuously track , survey and
,,
. . . . . . ~" ":" ~
THE
WORLD
Antactie-~
station
~
":'i "
" ......
Australia , S ' ' ~
:
I~i.
3
Kashi ~-Hetiern L ,
'
~
.=
P l a n n e d station
Fig.2 Plan of Chinese
Satellite Control
Fig.3 Space Visibility
of Chinese
Network
I Satellite Control
Network
As shown in (Fig.2 Plan of Chinese Satellite Control Network and Fig.3 Space Visibility of Chinese Satellite Control Network), in order to set up the surveying and controlling stations on land to replace the ships, it is needed for China to set up a South America station in south America which is about 7000 km away from the Namibia station, and in Europe it is needed to set up a North Europe station which is about 7000 km away both from the
217
218
E.Wei. J. Liu • C. Shi
Namibia station and Hashi, and in north America it is needed to set up a North America station which is about 7000 km away both from the South America station and North Europe station, and it is needed to set up a Australian station in Australia or New Zealand, which is about 7000 km away both from the Xiamen station and the Namibia station. In the Antarctica, it is needed to set up a Antarctica station which is about 7000 km away both from the Australian station and South America station. In this way, the CSCSN has become a new network which cover with the whole world and which is able to continuously track and observe the satellites and spacecrafts. When China cannot set up any fixed stations in some places for political reasons or diplomatic reasons, the spaceflight surveying and controlling ships can be berthed on the high seas near the planned fixed stations to ensure that the spaceflights can continuously be tracked. Of course, the limit of the stability of the ships will affect the precision and efficiency of survey. If the situation is allowable, we can increase the precision and efficiency by applying the surveying and controlling stations established in the countries or regions near our planned places.
3.2 Establishing the DSCCs At present, the diameters of the radio antennas in the CSCSN are all equal to or less than 20m, which can only observe the near-earth satellites and spacecrafts no more than 3.6 ×104km from the earth. So our country need to add some radio antennas with different diameters , such as 25m , 35m and about 70m, which will form a group continuously tracking, surveying and communicating with the earth-orbit satellites and spaceflights and detectors of the solar system. This group is called a deep space communication complex (DSCC). The celestial bodies in deep space are a great distance away from the earth. The moon which is the nearest celestial body of the earth is averagely 384400 km away from the earth. Because of the earth rotation and the relative movement between the earth and the celestial bodies, and between the spacecrafts and the earth, a station on the ground can only track a spacecraft for a voyage of 19800km once(the ground distance of the spacecraft flying around the earth is 40000 km). So, to be able to continuously observe the deep space celestial
bodies and spaceflights, at least three DSCCs covering with the earth are needed in Chinese DSN, which provides some regions where the received signals can overlap when the detector is flying from one DSCC to another one, and where the interferometry can be carried out. ~k
~
V-
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
~
Longitude(Degrees)
Fig.4 Plan of Chinese DSCCs When the DSN tracks and observes the detectors or the celestial bodies, it is mainly the rotation of the earth that interrupts the continuous observation of a DSCC. So at least three DSCCs are needed which are respectively about 120°apart away along the longitude. To make sure that each DSCC can possess a continuous observation time of 8 to 14 hours, the DSCCs should be located near the equator. The first DSCC can be selected round the Xiamen station in the CSCSN, and the second DSCC can be selected round the Namibia station of the CSCSN, which is located in Africa mainland of the south hemisphere. And the third DSCC can be selected round the North America station of the CSCSN. Selecting stations of the Chinese DSN in this way can ensure the station interval is about 120°along the longitude and cover with the southern and northern Hemisphere. With each of the DSN station as a center, a circle with 19000 kilometer observation diameter is drawn on each station. And each circle expresses the observation area. Projected on the plane, the stations and circles are shown in Figure 4, where the observation areas have almost covered with the earth. The area A and the area B in the Figure 4 represent the uncovered regions. The ground distance between every two neighboring DSCCs is about 12000 km. In the overlap regions of the received signals, the resolution of the observation angle can reach 1.29mas (for radio wave of 7.5cm wavelength in the S wave band). To establish the Chinese D S N , it is very important to investigate the locations, category and function of the observation equipments of the CAS and of the CSCSN. And then the 25m, 35m, and 75m radio antennas should be constructed
Chapter33 • Onthe EstablishingProjectof ChineseSurveyingand ControlNetworkfor Earth-OrbitSatelliteand DeepSpaceDetection
additionally to form a domestic DSCC, avoiding the repetition of construction. On the Namibia station and North America station, on the one hand, it is needed to construct the antennas of different diameters to form a DSCC. On the other hand , the surveying and controlling equipments established by other countries can be used as the parts of the DSCC in the Chinese DSN, by which the charge will be saved and the structure of the DSCCs is strengthened and the tracking covering area on the earth is enlarged. Such controlling equipments established by other countries are as the European VLBI Network(EVN), the U.S. Very Long Baseline Array (VLBA) and DSN of NASA, the networks in the USSR and in Japan, networks in the Southern Hemisphere(e.g., the Australia Telescope (AT) and Australian VLB1 antennas), networks in Asia (e.g., India) and in Canada. 3.3
The d e e p s p a c e s y s t e m of C h i n a
radar
observation
At present, the CSCSN has adopted the microwave radar system to observe the distance between the ground station and the lower earth-orbit satellites or spaceflights and image them. But the distance between the earth and the celestial bodies in the deep space such as the planets and the distance between the earth and detectors are very far. The echo received by the radar is very faint. Adopting synthetic aperture radar formed with multi-radar receiving to proceed interferometry will add the synthetical antenna's diameter, increase the energy of the receiving signal and the observing resolution and the imaging quality of radio objects. B u t , the effective receiving region of the echo by radar on the ground is in a area of several hundred kilometers. The further distance between the center and the echo, the more faint the received signal. Thus the length of the interferential baseline on the ground and the synthetic aperture are limited, therefore , using the higher frequency of radar wave is more feasible to increase the observing resolution of the deep space celestial bodies and imaging quality. The electromagnetic wave that can traverse the atmosphere and arrive at the ground is divided into several frequency bands (see Li Z H et al, 1998). At present, the international deep space radar system has used the X band which is 3-4 times higher than the S band used by the CSCSN, and the
resolution of radar increases the same times. The international deep space radar system has already received the echoes from some planets(Mercury, Venus, Mars, Jupiter and the satellite of Saturn , moon , asteroid , comet and so on of solar system ). Some scientific research institutions in China have begun to approve the X bands microwave radar to be able used in altimetry, and they are researching the collectivity design of the altimetry microwave radar (see Xu M Q et al, 2003). So we have the basis to increase the frequency of radar system to X band in the CSCSN. The power of the radar wave transmitted by the radar subsystem should be increased above 500kw, and the radar subsystem need to convert the alternating current into the powerful X band microwave transmitted to the deep space target. And it is also needed to develop the technology of Low Noise Amplifiers to improve the quality of the received echoes. Adopting the antenna with 70m diameter can realize to receive the radar signal by a single antenna, and to associate the antenna with 70m diameter with the other surveying and controlling stations with antennas of 35m and 25m diameter to form a synthetic aperture radar and to improve the resolution and imaging quality. According to the request of the precision of the surface manufacture of the radio telescope antenna, which is rms< ~/20 ()~ is the wavelength of the observed radio). So, when the wavelength of the X band is 3.75cm, the precision of the manufacture of the radio antenna's surface should be rms<0.19cm. 3.4 Building the lunar laser ranging s y s t e m of C h i n a
From America's starting up the Apollo moon project in 1961, three laser reflectors which are respectively named Apollo 11, Apollol4 and Apollo l5 have been installed on the moon. And the former Soviet Union has also installed two reflectors named L1 and L2 by the unpiloted spacecrafts. All of these reflectors have provided the targets for lunar laser ranging(LLR) . At present, the international LLR stations on the ground have adopted the laser wave with very narrow beam and with 0.1ns lasting time of the pulse, and the precision of LLR can reach 0.03m. At present, China has set up several satellite laser
219
220
E. Wei. J. Liu. C. Shi
ranging (SLR)stations which is necessary to be developed to LLR stations which is useful for surveying the distance between the ground station of China and lunar laser reflectors with high precision, and which will have important meaning for lunar detection and deep space detection. @ Adopting the technology of LLR can determine the lunar moving orbit with high precision and can also improve the lunar calendar which is the necessary condition to precisely design the flying orbit of the lunar detectors. @Using the laser observing stations with the known coordinates, we can resect the coordinates of each reflector and precisely survey the lunar rotation and the lunar librations, which can provide good conditions to research lunar dynamics, and to set up lunar coordinate system and to connect the lunar coordinate system with earth coordinate system, and which has the important meaning for China to digitize the lunar, to realize the lunar detection and to construct the lunar testing base and SO o n .
3.5 Setting up the space based CSCSN In CSCSN, because of the diameter limit of the earth, the resolution of angle is restricted for the observation with VLBI. Because of the influence of aerosphere, the frequence of radio observation will be within the band of the atmospheric window. Therefore, it is needed for China to launch an earth-orbit satellite with a radio telescope which can be associated with the radio telescopes on the ground to carry out interferometry. The ground telescopes and the space telescopes have formed the space VLBI and have developed the space based surveying and controlling network. In this way, the length of interferential baseline of the space VLBI break through the restriction of earth diameter, and also the resolution of the observation angle break through the limit of the ground VLBI. The time delays and time delay rates of the same radio signals received by the space VLBI antenna and the ground VLBI antenna are concerned with three reference systems(see I. Fejes et al, 1998) , the first of which is the radio celestial Conventional Inertial reference System(CIS) defined by the radio source, the second of which is the dynamic inertial reference system defined by the satellite motion, and the third of which is the Conventional Terrestrial reference System(CTS) defined by the ground stations. However, the observations of other spatial geodesy technologies(SLR, DORIS, GPS
or satellite altimetry) are concerned only with CTS and the dynamics reference system without concerning with the CIS. Thus the space VLBI is the unique technology tying the three reference systems together. The signal received at the antenna on the satellite is virtually unaffected by the atmosphere due to the high altitude of the satellite, so the observation band will not be limited by the atmospheric window. In the future, the radio satellite of China can form the pure space VLBI system with other radio satellites launched by other countries of the world, which can proceed interferometry in much wider band of radio frequency. Therefore, the space based surveying and controlling network will found the terminal condition to fulfill the precision request of the deep space observation, to obtain the electromagnetic wave message of the entire band and to found up the reference frame for deep space positioning. This will bring a revolutionary influence and infinite development potential for the detecting technology of deep space and for deep space geodesy. Until Now, it is only Japan who has launched the space VLBI antenna(VSOP), the main mission of which is for astrophysics. But its physical characters and orbit characters are not much applicable to geodynamics which require more regular figure of the satellite, less area/mass ratio and longer life time, and the precision of determining the orbit of the satellite should reach the same level (better than several centimeter) as the observation precision. B u t , the figure of the VSOP are quite irregular, and its area/mass ratio is much big. And the estimated life time of VSOP is only 3 ~ 5 a . The precision of determining the orbit of VSOP can hardly reach the level of 10cm. And the VSOP are also not convenient to frequently change the observation objects, but this is necessary for astrogeodynamics. All of these are very important for China to establish the space based surveying and controlling network.
4 conclusions 1 ) It is needed to additionally establish ground surveying and controlling stations in CSCSN and to research their distribution for the CSCSN to precisely and continuously track the satellites in different near-earth orbits. This paper puts forward some suggestions and plans for this. 2) It is needed to rebuild the CSCSN to the DSN
Chapter 33 • On the Establishing Project of Chinese Surveying and Control Network for Earth-Orbit Satellite and Deep Space Detection
and also to construct DSCCs to synthetically serve for tracking, surveying, navigation and communication with various detectors . 3) Developing Chinese LLR will set up the base to precisely determine the lunar orbit, to establish and precisely survey the control network on the lunar surface, to study the lunar dynamics and lunar rotation and to carry out selenodesy. 4) Constructing the deep space radar system of China will set up the base for carrying out the initiative observation of the near-earth planets to offset the incapacity of the ground laser survey which cannot survey the planets, and for getting the radar images of the planets and analyzing their material distribution. And it is also very important for planet ranging, determining their orbits and positions. And it will set up the base for designing the orbits of planet detectors, for planet geodesy and for the deep space geodesy. 5 ) Setting up the space based surveying and controlling network can directly connect the CTS, the dynamic inertial system and the CIS which are concerned with the deep space detectors and the deep space geodesy. And the space based surveying and controlling network can avoid the system error by adopting different technologies to indirectly connect the reference systems. And the precision of surveying, connecting and maintaining the reference systems is improved.
References
Liang B, Wang W,Wang C E(2003). The pilot assumptionfor China to develop the Lunar vehicle in the future. Spacechina,2003(1)(in Chinese). http ://www.space.cetin.net.cn/docs/ht0301/ht0301 sktc01 .ht m Liu J N, Wei E H, Huang J S(2004).Deep space network and its applications. Geomatics and Information Science of Wuhan University,29(7),pp.565-569 Guangming Daily(2003). Yu Zhijian--the chief designer of the surveying, control and communication system in the Chinese spaceflight engineering of caring people. http ://www.china.org.cn/chinese/zhuanti/425509.htm. Zhao Y F(2002). The technical development of satellite surveying and control network. Journal of spacecraft TT&C technology, 21(3),pp. 1-4 Li Z H, Xu D B, Dong Y Y(1998). The theoretical basis of spatial geodesy. Wuhan: publishing company of Wuhan university of surveying and mapping. Xu M Q, Duan F, Li X(2003). The argumentation of the PD system for altimetry radar in X band. China Radar, 2003(1),pp. 13-20 I. Fejes, I. Almar, J.Adam, Sz. Mihaly. Space-VLBI: A new Technique for Unification of Reference Frames.Earth Rotation and Coordinate Reference Frames. International Association of Geodesy Symposia 105,Springer-Verlag 1989,pp.158-165
221
Chapter 34
Constructing a System to Monitor the Data Quality of GPS Receivers T. K. Yeh and C. S. Chen Institute of Geomatics and Disaster Prevention Technology, Ching Yun University No. 229, Jiansing Rd., Jhongli 320, Taiwan, R.O.C., [email protected]
Abstract.
In Taiwan, there are more than one
1 Preface
stations maintained by
GPS tracking stations are multifunctional base
Ministry of the Interior (MOI), Academia Sinica,
stations that receive GPS signals 24 hours a day,
Central Weather Bureau and Central Geological
and serve as the basis for the national coordinate
Survey. In the future, they may take place of the
system. In addition to functioning as the superior
GPS controlling points after given a legal status.
controlling point when the satellite controlling
Therefore, the data quality of the tracking stations
points perform the surveys. Besides, GPS tracking
has become more and more important. This paper
stations can also provide delicate, detailed, and
addresses the feasibility of establishing a system for
correct GPS data for the use of every sector.
hundred
GPS tracking
monitoring GPS receivers.
In order to establish a complete, centralized, and
In this study, many data quality indexes were
delicate essential monitoring system, the Ministry
adapted and the relationships of the indexes and the
of the Interior planed on systematically setting up
positioning precision were found. The frequency
eight GPS tracking stations around the Taiwan area.
stability of GPS receiver is the most important
Among them, in year 1993, four of stations were
index, the cycle slip is the second index, and the
established
multipath is the third index. According to the results,
Ken-Ding
the auto-analytical system of GPS data quality was
King-Men (KMNM). While in year 1994 the other
established and the MOI's tracking stations were
four were set up in Ma-Zu (MZUM), Pei-Kan
monitored. When the receiver get some problems or
(PKGM),
there is a change of the station's environment, we
(TNSM) (MOI, 1998). After ten years of operation,
hope to find and resolve the problems earlier to
the monitored data received have been widely
ensure the high data quality of the tracking stations.
applied to fields such as engineering
Keywords. GPS, Tracking Station, Data Quality,
cadastral
Monitoring System
geodynamics. The data has become the basis for
in
Yan-Ming
(KDNM),
Tai-Ma-Li
survey,
Mountain
Feng-Lin
(TMAM),
crust
(YMSM),
(FLNM),
and
and
Tong-Sa
measurement,
survey, and
essential monitoring surveys as well. Therefore, the
Chapter 34 • Constructing a System to Monitor the Data Quality of GPS Receivers
quality of the data received at GPS tracking stations
the DOY 203 in 2003 as the basis, there was
is gaining more and more importance.
nothing wrong with the data received before the
In addition, the number of GPS tracking stations
first 35 days. However, while the operation of the
established by Ministry of the Interior, Academica
equipment was normal, there was a decrease in the
Sinica, Central Weather Bureau, Central Geological
number of the received data, an increase in the
Survey, and other academic research institutions has
number of cycle slips, let alone soaring noises in the
outnumbered one hundred. If there be the ordination
data received in the past one month. After careful
of a legal status in the future, the GPS tracking
inspection,
stations may well replace the function of existing
signals in the neighborhood were the culprit. The
first-rank satellite controlling points. In other words,
situation was solved only after negotiations on the
future users will not need to set up GPS at known
adjustment of frequencies.
points when implementing GPS static relative
similar to frequencies that convey
?
?
,
,
,
,
500
,?
,
,
35
30
25
,?
,
oo,
20
15
,
450 20000
positioning. They will only have to download from
~* *
400
*
050 15000
000
* •
***%. * 250
the Internet the information on GPS tracking
200
10000
150 100
5000
stations around the surveyed area. After data
50 o 45
reprocessing, they will be able to acquire the
I 40
I 35
I 3o
I 25
day~ before
coordinates of unknown points with accuracy. In
2.5
?
I 20
I 15
y2004,
I lO
I 5
0 o
45
days b e f o r e
?
2,5
*%
1,5
examination on known points that used to be carried
provider (ex: Ministry of the Interior). Therefore, a monitoring system needed to be established in order to monitor the quality of the
10
5
0
d203
@
$
i,
*
0.5
@
0.5
0
out before GPS surveys will be done by the data
y2004,
2
this way, the need for human and material resources on surveys will be greatly reduced, and the
40
d203
0 45
40
35
da~s
30
25
before
20
15
~2004,
10 d203
5
45
40
35
days
30
25
before
20
15
y2004,
10
5
0
d203
Fig. 1 Data Quality Chart of TWTF at IGS (IGS, 2003) (2) Case no. 2: the Ken-Ting station of Ministry of the Interior
data received at GPS tracking stations. We hope to
Fig. 2 is the chart of data quality monitor of
identify problems and deal with things that go
Ken-Ting and Yan-Ming Mountain stations of
wrong as soon as possible when the equipment
Ministry of the Interior. These two stations adopt
breaks down or when there are changes in the GPS
receivers
tracking station environment, so as to maintain the
purchased at the same time. Therefore, the data
quality of the data received at each tracking station.
quality of the two stations should be similar. Yet we
The followings are two life examples explaining the
can find out that the cycle slips at the Ken-Ting
importance and benefit of this system:
station are obviously larger. After inspection, the
(1) Case no. 1: the TWTF station at IGS
result came from a wrong parameter setup. After
Fig. 1 is the chart of data quality monitor of TWTF at IGS. We can see in the chart that, taking
of the
same brand and they were
adjustments, the number of cycle slips was reduced from 100 to 20 out of every 1000 observations.
223
224
T.K. Yeh. C. S. Chen
can also be regarded as the data of the noises of
i~NM I~alty. mmdbe~ olnbserca~lis . . . . . . .
2SOLOi ,
(.'~ck ulllPs~I BBO,%llsnl-alkmn
!
~,'~ " ~ ,
the carrier phase observation. The unit of the
100 i ' ' " "g'~ ' ~ ' ~ " " ~ + " " ~ " ' + ' - ' " ~ 1t11~,
•
oi
o
(4) RMS MP2: The multipath effect on L2, which "
"+*"
Owyo~ Y~lir (20(13)
'[ o.1.,+!
can also be regarded as the data of the noises of
DIy ¢1"YeW"4;~l~.J
R~IS .MPI I L l .~lllhlpll~)
o.. .i. . . + ,...,+"
noises is meter.
m,I
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o.~, . .,.++ ...
*"'
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0.4p
'.'+ .:.. ::-.2: '~
+
,*..'.,.:':...',,'*.:"...*'!
o~I...'/.
the carrier phase observation. The unit of the noises is meter.
"
°"" °'~I 0"
70
80 90 100 Day o+ Year ( ~ |
110
i i'
0.1 0
120
70
~
9(I 100 Dayaf Yell' 42gQ31
t10
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mp2-
(.)~ll i~lli'~ I | I I i n r I I l i l l l i
mmo!2mmP'"~"'~"~+'"L+"""'~++:'di ~"
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'~°1~
~['" ++l •
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."
+
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YMSM Ilil~. I I I I l I e ! d d l I l r r I i I l I
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'
130
[
P2 -
/ / / 2a ~ c~ - I
+
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)
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I
In the above two formulas: P = Pseudo range observation
ol
:
I Oily ¢4"Yell+124~1:~j K.~,IS M P 2 I L l M u l d F a l k )
•
,o,°l
:21 °.
i
o'
= Carrier phase observation
OIly ¢2 YP.-Br(20(1~) R,~IS M P I i l l .~tulnpMh)
0.31,, -~ + . +
!
°+;
:i
°
:i
, +"
'
I
~o
.o
+o
mo
.o
Day o+ Year (~116)
~o
N
N
N
7N
~
Day of Year 42(KI$I
+,+l
t
7+o
Fig. 2 Data Quality Chart of Ken-Ting and Yan-Ming Mountain Stations of Ministry of the Interior
fl -
Frequency of L 1 - 1 5 7 5 . 4 2 M H z
f2 -
Frequency of L 2 - 1227.60MHz
The Matlab programming language has made all the calculating works automatic while all the data quality indexes are made into statistical charts
2 Monitor
the Data
Quality
We use Teqc software to make initial judgments on data quality in the first phase of this monitoring system. The indexes of data quality adapted are as follows: (1) Daily number of observations: a total number of monitored data. (2) Cycle slips x 1000 / observations: as the title
automatically as well. The charts are updated daily on the Internet; when users download data through the Internet, the can also inquire results the calculation of data quality at any GPS tracking stations of Ministry of the Interior at the same time. The URL of the inquiry website is as follow: http ://www. gp s.moi, gov.tw/satellite/j sp/C ontent9.j s 12.
implies, this index refers to "the number of
Upon choosing the inquired date and year, one
cycle slips" multiplies 1000 and then divided
shall pick an inquired station (here we shall take the
by "the number of observations". In other
DOY 300-365 in year 2003 at Feng-Lin station for
words, it means the number of cycle slips out
instance). Then, click on "start inquiry", and the
of every 1000 observations.
statistics of data quality will be obtained, as shown
(3) RMS MPI: The multipath effect on L1, which
in Fig. 3. If desiring to examine ready-made
Chapter34 • Constructinga Systemto Monitor the Data Qualityof GPS Receivers
statistical charts, click on "examine" under a Teqc
with the conclusion made at the 13th General
picture, and a chart of data quality will appear, as
Conference on Weights and Measures in 1967: The
shown in Fig. 4.
duration of 9192631770 periods of the radiation corresponds to the transition between the two hyperfine levels of the ground state of the cesium
i~l~ lINt H I - ."..-'.'.~-~:,:,':'~c, I~:
Ill:
ik,:n.
I~I.:~M:
~"M,,~,-"t"7~G.m=)
133 atom (ISO, 1995). Therefore, in this research, the TWTF maintained
Q|
by National
:)~v5.~2':l'0J;~
25":'7J;7
i1~II-Itt" " i~t]~[tl, "I~TNI[[R. I'VJ."~..~:[II:.
21_~{ 2f"l'? )i .~." ,"IZI'~
"l l~b'7,'"1,9~_,'7 :~ PI'.'I '~ 5~'.~, "~'~'~i i .',,~,ii O.MJ. 14~Y;
"]~..~)1~o ~..~;.~" ~.::,~.~ .¥;~I~,
ILl tlli
Laboratory is chosen as the referenced station.
I~]~..~U~.
~-~'.)~'
i ~
'~L,N,'~
i.'lf~? J -q ~:~ :)~47 i3"170 ";)G~7 "lO2,',~ i 0~'~.~, 1 L~'~,
tllt tlt ltli
The GPS receiver adopted by this station is Ashtech
:~1.~_~ 2"[t'(d 21.:'.~] 21 t.%
tLt tUi
Z-XII3T (as shown in Fig. 5). This receiver uses a
)t~ )1)5 3"~ ~?t "K'~
~.~.b3".(;'.~'l
F.~'.'#
~.h_~"_l ;)1 ~.5~)".'102 "-.~.0".i 0) ~'IX'II.'I-('K
I~rl~" [~'¢tl-' " i'~iS~ I(|')~
~
~lX'l~'l...O'i. •
I.,-.~0"~
.'~[(0.'.I ~ ~¢.k~3"'1i.~" ~.,',i~"_f,,> ~ ? : i ..)
I~X~.It; i~i~tl~ t~.~'~ " i'h~:llll'~. IC~,t," i~.~.'IIl~'" t(i:,~ ~.~.tt~[!1,
J~. I 3~.~" "3'.3 3~z
111 i§ ' "~ff.tllN:
:Y,~
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27.t~4;~ 27.:)3'~7 '.~'.','t;~l ~.":~3
-I~ : k :_..'_~_2 r ", .~1 rllt O. .m.i BJ~K ]
HP5071A cesium atomic clock (as shown in Fig. 6)
Fig. 3 Inquiry Result of Statistics of Data Quality
and a ASH701945C_M
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.
.
.
.
.
i
15000 ~ 2t),. 1~O3[
.
.
.
.
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.
,
a day, all year round. Bernese 4.2 software developed by Astronomical
I •
1¢i
Institute, University
O~y of Year t,-"O03:,
calculation.
Oey ol ~¢~r l~¢J3t
ll~ISi .MPI | L ! .~lullllp~lh) •
i I. "
0 5'
*'
fi i i
¢'~'.'f"'~'"
'
=
! ".'.." , -'..",-'.'."4",'""i
I O: 32~
is adopted
for
In order to obtain more accurate
33~ 3113 35O Of ~'Lmllr1.2011r'~)
360
information on receiver frequency, in addition to implementing the examination of cycle clips and data smoothing on the phase data by using the
0.5i O
310
of Berne,
R M $ ,MP2 [ L I MuNllp=lh)
1.5-
1
chokering antenna (as
shown in Fig. 7); it also receives GPS data 24 hours
FLNM 25001:
Standard of Time and Frequency
i 310
320
330
340
350
2~0
Olt.~' o-r ~'~Nkl" 1~0031
Fig. 4 Inquiry Result of Charts of Data Quality
"Rnxsmt" program before the calculation. In the process of using "Gpsest" program to calculate an unknown value, we also use the "Resrms" program
3 Monitor
the
Receiver
Clock
Error
to constantly erase the worse periods and satellites
The second phase is to analyze the stability of the frequency in the receiver. The receiver clock error is divided frequency
into
two
categories:
stability.
As
time
offset
and
for
calculation,
by making use of output residuals. In this way, we will be able to acquire more accurate time offset and frequency stability of the receiver (Beutler et al., 2001).
undifferenced GPS phase data is utilized to infer the receiver clock error. Nevertheless,
a referenced
frequency source has to be made while processing the data (that is, assuming the time offset of the frequency source as zero). The definition of the international time frequency standard still accords
Fig. 5 GPS Receiver Used by TWTF Station
225
226
T.K. Yeh. C. S. Chen
In the formulas" x k • T h e drift of every time measurement m
Yk" The average value of every two neighboring time measurement z'" Sampling interval
N - The number of time measurement Under the Linux system, the construction of BPE (Bemese Processing Engine) automatic calculation has been done. The Matlab programming language is adopted to transform the calculative results of the
Fig. 6 Cesium Atomic Clock Used by TWTF Station
time offset and frequency stability into statistical charts.
The
charts are updated daily in the
monitoring computers of the Ministry of the Interior. The charts of calculative results are shown as in Fig. 8 (taking the DOY 300-365 in year 2003 at Feng-Lin for instance). z 10
4.
2,
,
÷
i. 0-"
-2-
Fig. 7 Chokering Antenna Used by TWTF Station
-4.
Day
ol~ Y c e r
i .'~1~:3~
However, since the frequency of the atomic clock is constantly drift, it is hard to calculate the average value accurately;
even the estimated standard
25-
• .
2b-
.
+
..
÷ ."
.
,
...
.
.
.
.. 15.
.
.
0
deviation would diverge. Therefore, the Modified +
Allan Deviation defined by Allan is adopted here to
:[~i~faf g e l l f
estimate the offset and stability of the frequencies
Fig. 8 Calculative Result of Receiver Clock Error
I."O03)
(Lesage and Ayi, 1984). The calculative method of Modified Allan Deviation is to calculate the estimated
standard
deviation
after
the
double-differencing of the data. The formulas are as follows:
-
The last step is to analyze and calculate the results of the positioning coordinates acquired at each station. First, via the Internet, we would
x~ - (xk+ 1 + x k) / 2 y-;
4 Monitor the GPS Coordinates
-
download data received at 12 IGS tracking stations
/
that are geologically stable and receive better
MODcry( r )
-
i
1
2 ( N - 3 m + 1)
-
k=l
quality data. The coordinates of the stations are fixed and these stations are respectively as follows:
Chapter 34 • Constructing a System to M o n i t o r the Data Quality of
GPS Receivers
(1) Australia : ALIC station; DARW station
Ministry of the Interior, presently, the construction
(2) China : BJFS station; LHAS station; WUHN
of
station
the
BPE
automatic
calculation
has
been
completed under a Linux system. Likewise, the
(3) The United States : GUAM station
Matlab programming language is adopted here to
(4) Singapore : NTUS station
transform the result of the daily coordinates into
(5) The Philippines : PIMO station
statistical
(6) Taiwan : TCMS station; TNML station
automatically on the webpage of Ministry of the
(7) Japan : TSKB station; USUD station
Interior, and the URL is: http://www.gps.moi.gov.tw
The distribution of these IGS tracking stations
charts.
The
charts
are
updated
/satellite/j sp/Content9.j sp.
around Taiwan is as shown in Fig. 9. Next, via the
Upon choosing the inquired tracking station (here
Internet, we would acquire precise ephemeris at
taking Feng-Lin station as an instance), click on
IGS, and earth rotation parameter at AIUB. In this
"coordinate
way, with the 8 GPS tracking stations of Ministry of
coordinate charts of that station will be acquired, as
the Interior, there would be 20 stations in total for
shown in Fig. 10.
calculations. When calculating, the coordinates of
~. _
•
charts"
will
be
obtained
by
fixing
.--
the
coordinates of the 12 IGS tracking stations for data
.... ~ x
.
.,.
.
...
.
.
.
i'
:'
!
. " z -.. . . . .
.......
-,:
[].:~.~;; .... .., _ _ .... :_1~]I~" ......... • -;.:,=.' guao
-
.- • , [ ]
I ._
ulab
i : ~ i
~
. .--"i
..
. [] kh'~j
[] ........
'
;.."
~..,... . ,.......l ,, ~,_=
•
:'
.
.
.
.
.
.
.
.
.
.
. .
.
.,
,.:'
.s~.ii.~
I []
-,~ F.
,,=... .... >. , =...1 . . ,-_
•,
•
:
,'
'\ \.
..,:.,
'
, ,.,
•
.........
: .... u,,.,.~,,. ' " ~ , - ~ - " . , . . 4-,' . . . .
/; y
:
....
...~ :--~,
"
E,.
"--'~
_,:.:.,
.; ' . ~ - ~ - ~ . ,
-
," ,-~ .'~."*,
+."."
_ ,',,
~..
ix~Y
l~t--J =.
Fig. 10 Inquired Result of Three-dimensional Coordinates
~';
~ . o .,0.,~o ~""
\. -I
~....-...
i:: .:'-" ;' ....
..
-=
.i-l]~l
SeO.Ul
..,._
L~)
'
,.
....~'1. -.ma_= " • • -. u ..,-= T. -: -.i"7 -.-.~ ' ,,<.... ,-r. !~,.,,.,-r., . S h a l ~ g h a i
~.'~
.....
• :...
, ...-.;"':. ~ ! i = l ) i ~ "
'¢1ii~-,,
b,'j'g,s
[]
:
~ .
/
.,
' u
xian
I
I
- . . . ~ . .
. ....
'
a~.i
s&~=-l,
I-"-I(-I'I-.I:lJd'N)
.- - "
-Beijing.
.'
',~ .........
\, -
[] .... /",-~.o,.;F~
[] . . . . . ,.. . . . . . .
..
.\'..
[]
,
.:.
I~i,ilrullr, '
~...::., • "~.... ~
,,,,.,~
=,.: :,,.
....:.... ,,~:~-~ ,~,,-,,,-,,;':. . . . . . . . . . . . [] i '-, ~," i,~.
'"
,..
.
.:....... ....
processing. ,,
three-dimensional
r.,i,...~,..,ri,-,~ I,t I LN~
~,'
.,..
Interior
the
} ......."........' ~''"'"'"""~'-"""~',""'"r' <.
the 8 GPS tracking stations belong to Ministry of the
and
.~. 5 Conclusions
'
,~' ,£
,,." '=."-,,
...<"
~ .
";:.
Through the monitoring procedures
~,,,<==,:.
of data
quality, receiver clock error, and GPS coordinates, '
',-_:-_-~--:;"i--
"\.r:
:
::
. . . .
..t.
with the analyses of the influence of data quality on
.... """
the accuracy of GPS positioning, the user or system operator will be able to discern whether the Fig. 9 Distribution
of the IGS Tracking
Taiwan Area (SOPAC,
Stations Around
the
2005)
receivers at the GPS stations function well quickly and accurately with simple clear charts. When the
As for calculations on the three-dimensional
equipment breaks down or there are changes in the
coordinates at the 8 GPS tracking stations belong to
tracking station's environment, problems can be
227
228
T.K. Yeh. C. S. Chen
discovered and resolved as son as possible in order
IGS,
2003.
International
GNSS
Service,
to ensure the data quality received at each tracking
http ://igscb.j pl.nasa, gov/network/site/twtf.html.
station. In this way, the accuracy and reliability of
ISO, 1995. Guide to the Expression of Uncertainty
GPS data will be ensured, and the user will feel
in Measurement, International Organization for
much more confident with themselves. In the future,
Standardization, 2nd ed., ISBN 92-67-1018809.
the need for human and material resources for GPS
Lesage, R, T. Ayi,
1984. Characterization of
surveys will be greatly reduced and further uplift
Frequency Stability: Analysis of the Modified
the competence of the nation.
Allan Variance and Properties of Its Estimate, IEEE
Trans.
Instrum.
Meas.,
IM-33(4),
pp.332-336.
References
MOI, 1998. Final Report on the Measurement Plan Beutler G., E. Brockmann, R. Dach, R Fridez, W.
of Basic Control Points in Taiwan-Fuchien Area
Gurtner, U. Hugentobler, J. Johnson, L. Mervart,
Where GPS are Applied, Ministry of the Interior.
M. Rothacher, S. Schaer, T. Springer, R. Weber,
(in Chinese)
2001.
Bernese
GPS
Software
Version 4.2,
Astronomical Institute, University of Berne.
SOPAC, 2005. Scripps Orbit and Permanent array Center, http://sopac.ucsd.edu/maps/.
This page intentionally blank
Part III Gravity Field Determination from a Synthesis of Terrestrial, Satellite, Airborne and Altimetry Measurements Chapter 35 Chapter 36 Chapter 37 Chapter 38 Chapter 39 Chapter 40 Chapter 41 Chapter 42 Chapter 43 Chapter 44 Chapter 45 Chapter 46 Chapter 47 Chapter 48 Chapter 49 Chapter 50 Chapter 51 Chapter 52 Chapter 53 Chapter 54 Chapter 55 Chapter 56 Chapter 57 Chapter 58 Chapter 59 Chapter 60 Chapter 61 Chapter 62
The Use of Mascons to Resolve Time-Variable Gravity from GRACE GRACE Time-Variable Gravity Accuracy Assessment Combination of Multi-Satellite Altimetry Data with CHAMP and GRACE Egms for Geoid and Sea Surface Topography Determination A New Methodology to Process Airborne Gravimetry Data: Advances and Problems Ideal Combination of Deflection Components and Gravity Anomalies for Precise Geoid Computation SRTM Evaluation in Brazil and Argentina with Emphasis on the Amazon Region On the Estimation of the Regional Geoid Error in Canada A Validation Procedure for Satellite Orbits and Force Function Models Based on a New Balance Equation Approach Combining Gravity and Topographic Data for Local Gradient Modelling Numerical Implementation of the Gravity Space Approach - Proof of Concept Local Gravity Field Modelling with Multi-Pole Wavelets Accuracy Assessment of the SRTM 90m DTM over Greece and Its Implications to Geoid Modelling High-Resolution Local Gravity Field Determination at the Sub-Millimeter Level using a Digital Zenith Camera System A Data-Adaptive Design of a Spherical Basis Function Network for Gravity Field Modelling Global Gravity Field Recovery by Merging Regional Focusing Patches: an Integrated Approach External Calibration of GOCE SGG Data with Terrestrial Gravity Data: a Simulation Study Towards an Optimal Combination of Satellite Data and Prior Information Kinematic and Highly Reduced Dynamic LEO Orbit Determination for Gravity Field Estimation On the Combination of Terrestrial Gravity Data with Satellite Gradiometry and Airborne Gravimetry Treated in Terms of Boundary-Value Problems A Direct Method and Its Numerical Interpretation in the Determination of the Earth's Gravity Field from Terrestrial Data Comparison of High Precision Geoid Models in Switzerland GOCE: a Full-Gradient Simulated Solution in the Space-Wise Approach The Determination of the Effect of Topographic Masses on the Second Derivatives of Gravity Potential Using Various Methods Density Effects on Rudzki, RTM and Airy-Heiskanen Reductions in Gravimetric Geoid Determination Combined Modeling of the Earth's Gravity Field from GRACE and GOCE Satellite Observations: a Numerical Study Analytical Solution of Newton's Integral in Terms of Polar Spherical Coordinates A New High-Precision Gravimetric Geoid Model for Argentina Local Gravity Field Modeling Using Surface Gravity Gradient Measurements
Chapter 35
The use of mascons to resolve time-variable gravity from GRACE F. G. Lemoine, S. B. Luthcke, D. D. Rowlands NASA Goddard Space Flight Center, Code 697, Greenbelt, Maryland 2077 l, USA D.S. Chinn, S. M. Klosko SGT Inc., 7701 Greenbelt Road, Greenbelt, Maryland 20770, USA C. M. Cox Raytheon ITSS 1616 McCormick Drive, Upper Marlboro, Maryland 20774, USA
Abstract. We have analyzed intersatellite data from the GRACE mission (Gravity Recovery and Climate Experiment) to resolve timevariable gravity using a local mascon approach. The spherical harmonic solutions have resolved the signal from surface hydrology over land areas at spatial scales of 750 to 1000 km over one month intervals (Wahr et al., 2004; Tapley et al., 2004a). In our local approach, we solve explicitly for the mass of water in surface blocks using only the intersatellite data collected as the GRACE spacecraft overfly the region of interest. The local representation of gravity minimizes leakage of errors from other areas due to aliasing or mismodelling. We review solutions for mascons from July 2003 to August 2004 over three regions: the A m a z o n , the Indian subcontinent, and the continental United States. We solve for mass change at 10-day intervals using 4°x 4 ° blocks using a temporal and spatial constraint. The mascon solutions clearly resolve the annual signal over all three regions and show evidence of shorter period variations in parts of the continental U.S. K e y w o r d s . GRACE, time-variable gravity, mascons, continental hydrology. 1 Introduction The prime objective of the GRACE mission is to resolve mass flux on the Earth with the highest possible spatial and temporal resolution permitted by the data. The challenges include a non-repeating ground track from month to month, the fact that the intersatellite observations cannot completely define the orbit of the two GRACE satellites, removal of non-conservative forces from the signal, and spatial and temporal aliasing of geophysical signals. One approach has been to generate monthly global solutions in
spherical harmonics (cf. Wahr et al., 2004; Tapley et al., 2004a). Unfortunately, aliasing and modelling errors cause north-south striping in the maps of the temporal field differences and require smoothing on scales of 1000 or more km. Ray et al. (2003) showed that errors in ocean tidal constituents leak into the land areas when solutions are made using spherical harmonics. Han et al. (2004), and Thompson et al. (2004) characterize the aliasing due to atmosphere, tidal, or oceanic mismodelling on the G R A C E monthly solutions. Errors in one dynamical model in one part of the globe can corrupt the time-variable gravity recovery elsewhere, for example over land areas. Thus, we were motivated to attempt a local mass flux recovery using mass concentrations (mascons~), rather than using global basis functions such as spherical harmonics. Mascons can be localized in both time and space, and hence, shortwavelength errors (e.g. due to ocean tides) should not leak into land areas. In simulations, with intersatellite tracking data, it can be shown that the residuals from a simulated mass anomaly lie on top of the mass anomaly.
2 Experiment design Chao et al. (1987) show how to compute a set of differential coefficients corresponding to a mass distribution in a small additional surface layer spread over the entire Earth: AAlm(t)- (1 + k[)n~a(t)
(2t + 1)M /'~,.(~)d~
(1)
The term 'mascon' was first used to describe mass concentrations on the Moon, detected from tracking of the Lunar Orbiters by Mueller and Sjogren (1968).
232
F.G. Lemoine.
S. B. L u t h c k e
• D. D. R o w l a n d s
• D. S. C h i n n
• S. M . K l o s k o • C. M . C o x
where 1 and m are the harmonic degree and order, kl" is the Earth's load Love number of degree l R is the mean Earth radius and M is the mass of the Earth, £2 is a representation of surface area, and Ylm is the spherical harmonic of degree and order 1 and m corresponding to the potential coefficient Arm and o(t) is the mass of the layer over a unit of surface area at the epoch t.
for a Stokes coefficient at degree 1 and order m is just the coefficient from the set of differential coefficients at the same degree and order. We estimate the scale factor for each 4°x4 ° block within a region of interest. We introduce spatial and temporal 'neighbour constraints' to help stabilize the mascon solution. For each region, mascon blocks are constrained equal to each using an exponential weighting function in time and space, i.e.
exp[2 d,j
t,j,]l
D Although equation (1) was derived for a global surface mass function (ty(0 in equation (1)), it is clear that it can be adapted to any subset of the Earth, for example a 4°x4 ° block of interest. The set of differential coefficients that corresponds to the addition of 1 cm of water in a particular 4°x4 ° block can be found by defining the surface mass function in equation (1) to be a step function such that ty(t) = 10 kg/m 2 inside the 4°x4 ° block and ty(t) = 0 kg/m 2 elsewhere. We find that generating a set of differential coefficients complete through degree 60 is sufficient to describe the mass step function for a 4°x4 ° block. The operation of finding a set of differential coefficients corresponding to any depth of water is entirely linear. For example, the step function for 2 cm of water inside the same 4°x4 ° block is exactly twice as large as for 1 cm and the same can be said for each of the resulting coefficients in the differential set. The linear relationship described just above enables a scale factor parameterization for mascons. The additional mass in a 4°x4 ° block (as compared to the mean background gravity field) can be represented as a scale factor on the set of differential coefficients corresponding to 1 cm of water in the block. The process of estimating a scale factor for a set of differential coefficients is not much different than for the well known process of estimating individual global Stokes coefficients. In either case (individual coefficient estimation or scale factor estimation), the partial derivative of the GRACE intersatellite measurement with respect to the parameter of interest must be computed. The partial for the scale factor parameter is just a linear combination of partials for individual Stokes coefficients. The multiplier of the partial
(2)
T
where T and D are the correlation time and distance, dij is the distance between the pair of blocks i and j and ti~ is the difference in the time tags of the blocks, in units of ten days. This procedure follows closely the application of neighbour constraints on empirical accelerations in reduced-dynamic orbit solutions (Luthcke et al., 2003). The introduction of these neighbour constraints allows us to resolve time-variable gravity from the GRACE data on a sub-monthly basis. In our solutions we applied a correlation time of 10 days and a correlation distance of 250 km. The number of mascons estimated in each tenday period was 195 for the Amazon; 81 for India; and 119 for the continental U.S. Because of data gaps, solutions were obtained in two segments: July - December 2003; and February - J u l y 2004. The typical ground track coverage in a 10-day period for the three regions is depicted in Figures 1, 2, and 3. 280 ° i
12 ° 8o
288 ° i
296 °
304 °
i ,
I
i
312 ° i
320 ° i
328 ° i
..'
"
4o 0o
_4 ° _8 ° _12 ° _16 ° _20 ° _24 ° _28 ° .32 °
~
. "
Fig. 1 GRACE ground tracks over the mascon solution region centered on the Amazon for July 2130, 2003.
Chapter 3 5 68 °
72 °
76 °
80 °
84 °
88 °
92 °
32 °
28 °
24 °
20 °
16 °
12 °
Fig. 2 GRACE ground tracks from the mascon solution region over the Indian subcontinent for July 21-30, 2003.
248
°
256
°
264
°
272
°
280 °
The Use of Mascons to ResolveTime-Variable G r a v i t y
from
GRACE
data. Thus, our mascon solutions are based solely on the KBRR data.
96 °
36 °
240 °
•
288 °
296 °
48°
40 ° 32 °
Fig. 3 GRACE ground tracks from the mascon solution region over the continental U.S. for July 2130, 2003.
3 Reduction of the G R A C E data We processed the GRACE intersatellite K-Band range-rate (KBRR) data in 1-day arcs. We started with the precise orbits provided by the G R A C E project, which are produced from analysis of the GPS data. These orbits have an accuracy of a few cm, as verified by independent satellite laser ranging (SLR) data (Bertiger et al., 2002). We adjust only three components of the baseline vector between the two co-orbiting spacecraft since the KBRR data alone cannot support the free adjustment of the state vectors for both GRACE satellites. The components of the transformed state vectors that we do adjust are: (1) the pitch of the GRACE intersatellite baseline; (2) the magnitude of the GRACE intersatellite baseline velocity; and (3) the pitch of the GRACE intersatellite baseline velocity (Rowlands et al., 2002). Our procedure 'refines' the a p r i o r i orbits, and we avoid the computational burden of processing the GPS
The force modelling includes the GGM02C GRACE gravity model (Tapley et al., 2005). The ocean tides are modelled according to G O T 0 0 . 2 with 36 separate constituents, including M2 to 70x70, and 21 other major constituents to 50x50. The treatment of earth tides is consistent with IERS2003 conventions. The nonconservative forces are modelled by the G R A C E precise accelerometry data. We perform a daily calibration on these data. The atmospheric gravity is forward modelled (cf. Chao and Au, 1991) using potential coefficients to 50x50 at six-hour intervals, derived from NCEP pressure grids (Petrov, 2005). With respect to the preliminary mascon solutions for the Amazon, described in Rowlands et al. (2005), the improvements include the use of GGM02C (instead of GGM01C (Tapley et al., 2004b), an improved atmospheric gravity series (to 50x50 rather than 20x20 in spherical harmonics), as well as refinements to the procedures for accelerometer calibration. The new second-generation mean gravity fields from G R A C E ( G G M 0 2 C , E I G E N - C G 0 1 C ) are responsible for a substantial reduction in the GRACE data fits from an average of 0.55 ~t/sec to about 0.36 g/sec. Other improvements, including the improved atmospheric gravity series (cf. from Petrov, 2005), the expansion of the tidal modelling (cf. M2 to 70x70) reduce our fits to the GRACE KBRR data to just under 0.3 ~/sec.
4 M a s c o n results 4.1 Description of the peak-to-peak water mass variations in the study regions In Figure 4, we depict the peak-to-peak changes in the estimate of water storage in the mascon 4°x4 ° blocks for the Amazon. We can examine the peak-to-peak changes in each mascon block, for each of the three study regions. The total annual changes, block by block, range from 8 to 58 cm, over the Amazon study region. In these solutions, the peak change in the hydrological signal occurs south of the Amazon river in the block centered on 7 ° S, 306°E. This feature is observed in the global spherical harmonic solutions (Tapley et al., 2004b).
233
234
F.G. Lemoine. S. B. Luthcke • D. D. Rowlands • D. S. Chinn. S. M. Klosko. C. M. Cox
i 8
i 12
i 16
280':'
i 20
i 24
288'
i 28
i 32
296'
i 36
i 40
304"
i 44
312'
i 48
i 52
320'
Over the US study region, we can perform a similar analysis. The peak-to-peak changes range from about 6 cm over the Great Basin to 18-20 cm o v e r L a b r a d o r and Q u e b e c , on the northeastern limits of the study region. Broadly speaking this is consistent with the semi-arid nature of the Great Basin region, and the heavy snow load that occurs in those regions of Canada.
I i 60
i 56 328'
_16 ~
-16':
_24 ~
-24':
_32 ~
-32':'
4.2 Examples
280 ~
Fig.
288'
296 ~
304 °
312 ~
320 ~
328'
Peak-to-peak changes in continental water storage from July 2003 to July 2004 in each of the mascon blocks over the Amazon region. The units are cm, and the change ranges from 8 to 58 cm. 4
The peak-to-peak changes in the estimate of water storage are depicted in Figure 5, for the India region. The changes range from 7 to 51 cm, with the peak not occurring over the Indian subcontinent, but on the eastern boundaries of the m a s c o n region, over South East Asia. Specifically it is the 4°x 4 ° block centered on 24°N, 100°E that has the largest amplitude signal. With the mascon solutions over the Indian subcontinent, we resolve the differences in the change of annual water storage between southern India (peak signal 10-12 cm), northern India along the Ganges (30-40 cm).
i 8
i 12
i 16
i 20
i 24
i 28
i 32
i 36
--
i 40
i 44
i 48
p 52
40 ~
40 ~
36 ~
36 ~
32 ~
32 ~
of mascon
time
series
The individual mascon estimates in each block can be examined as a distinct time series. We depict three examples in Figures 6-8, where we show the time series of water mass estimates with r e s p e c t to the base g r a v i t y m o d e l , GGM02C. A time series over the A m a z o n is depicted in Figure 6, and over southeastern Venezuela in Figure 7. The block centers are 8 ° apart in latitude, and yet show peaks occurring at different times of the year, as expected from the b e h a v i o u r of the hydrological basins of the A m a z o n and the Orinoco. The first block is centered at 3°S, 298 E °, just to the west of Manaus. According to the mascon solutions, the peak occurs in the ten-day solution of April 1120, 2004. The m i n i m u m occurs in the ten-day solution of Oct. 21-30, 2003. The second block is centered over southeastern Venezuela at 5°N, 298°E in a region that drains into the Orinoco river. The m i n i m u m occurs in the solution of February 11-20, 2004. The maximum occurs in a broad peak (either July 2003 or June 2004), indicating the phase of m a x i m u m amplitude in this region is distinct from that in the previous example. 40.0
........!..........I........Antazon
30.0
............................A n n u a l f i t t e d a m p l i t u d e : R M S of fit: 3.1 cm.
B l o c k '111:
I
(~os-,os: 2,6oE-300oE).
24.0 cm.
'
'
'
I
'
'
'
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii iiii~iiiiiiiiiiiiill ............ iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii~
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!!
. . . . . . . . . . . . . . . . . . . . . . . . . . .
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i
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!
Fitted Values
|iiiiiiiiiiiiiiiiiiiiii
12: -3o.o , 2003.4
6 8 ':'
72'
7 6 ~'
80'
84':'
88'
92 °
96'
,
, i , 2003.6
,
, i , 2003.8
,
,
i , 2004
,
, i , 2004.2
,
, i , 2004.4
,
, 2004.6
100"
Fig. 5 Peak-to-peak changes in continental water
Fig. 6 Time series of recovered water mass (wrt.
storage from July 2003 to July 2004 in each of the mascon blocks over the India study region. The units are cm, and the change ranges from 7 to 51 cm.
GGM02C) for mascon block in the Amazon. This block is centered at 3°S, 298 E °, just to the west of Manaus. The total annual variation is 51 cm.
Chapter 35 • The Use of Mascons to Resolve Time-Variable Gravity from GRACE
30.0
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-iiiiiiiiiiiiiiiiiiiiiiiii:::iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiill ~ N ~ . ~6~300°E). ....................................... i................................. A n n u a l f i t t e d a m p l l t u c l e : 19.7 cm. ~;2 ......................R i S o f fit: 3.6 cm. -:::::::::::::::::: : 222.2 ° (
ii::iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiill ~=....................................... = ; ~ -
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U.S. In each case, we resolve the annual cycle in each region. Qualitatively, we can say that the trends in the solutions make sense climatologically. They correlate well with other solutions, in terms of the amplitude of the signals, and the locations of the minima and maxima.
~::::::::::i::::::::::~i::::::::::j::?~:;:=======================:~!~:::~====================================================================
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Fig. 7 Time series of recovered water mass (wrt. GGM02C) for mascon block in southeastern Venezuela. This block is centered at 5°N, 298 E. The total annual variation is 48 cm. An example of a mascon solution over the US is shown in Figure 8, for a block centered on 36°N, 268°E that includes the confluence of the Ohio and Mississippi rivers. The total annual change is 13 cm, however we notice variation with respect to the annual signal. The minimum occurs in the solution for Nov. 1-10, 2003, and the maximum does not occur six months later, but rather in the solution of February 1-10, 2004. The water storage remains at or near that high level through late April 2004, before beginning to decrease. This could be a manifestation of late Winter and early Spring precipitation and runoff, however careful correlations with hydrological models will need to be made to see if this assertion can be substantiated.
The formal uncertainties of the mascons vary between 0.2 to 0.3 cm over the period of the year for all the study regions. Of course, these uncertainties do not take into account other factors, which may contribute to the error. These issues include errors in ocean tide modelling, the effect of m i s m o d e l l i n g of the atmospheric gravity signal due to data errors or uncertainties in the input pressure grid (Han et al., 2004; Thompson et al., 2004). One upper bound on the error would be to consider the residuals with respect to a harmonic fit of an annual signal in each mascon block. As can be seen in Figures 68, these residuals with respect to the harmonic fits are 1-3 cm. This approach will naturally ascribe legitimate shorter period variations as error, but nonetheless give us an upper bound on the error inherent in the mascons. So between the formal uncertainties in these solutions and the residuals from the harmonic fits, we think a legitimate error estimate would typically be on the order of 0.75 cm for each mascon block.
5 Summary and Discussion
Another significant issue is the impact of the constraint on the solution. We experimented with the application of the constraint (equation 2) at different weights, or scale factors in the least squares solution. We attempted solutions with a scale factor of unity, (1/100), and (1/1000). The solutions using the scale factor of unity and 1/100 tended to overconstrain the mascon estimates, smearing out the maps and damping the amplitude of the peaks. The optimum scale factor which we chose for the results presented here was (1/1000). This means the contraint matrix has a weight of 0.001 with respect to a weight of unity for the KBRR data. We find that lighter constraints (e.g. 1/10,000) produce mascon solutions that are too noisy. It is possible that the weighting of the constraint might need to be optimized for different mascon regions.
In this paper, we have presented the results from our mascon solutions over three study regions" the Amazon, India, and the continental
Future work will consider comparisons with hydrology models and expansion of the mascon solutions to other regions.
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B l o c k 44 ( C a i r o , III): :::::::i:::::ii ...................................... ii ...................................... (34°N-38°N: 270°E -274°E). ......... A n n u a l f i t t e d a m p l i t u d e : 4.2 c m . :::::::::::::i ...................................... ii .......... R M $ o f f t 1 7 c m .......................... !....................................... ; ...................................... ::::: P h a s e o f f i t t e d m a x : 95.7 ° (2004,27).11111111111111111111 ~ ::::::::::::::::::i:::::::::::::::::::..........
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Fig. 8 Time series of recovered water mass (wrt. GGM02C) for mascon block in North America. This block is centered at 36°N, 268 E. The total annual variation is 13 cm.
235
236
F.G. Lemoine. S. B. Luthcke • D. D. Rowlands • D. S. Chinn. S. M. Klosko. C. M. Cox
Acknowledgments We a c k n o w l e d g e the G R A C E p r o j e c t for providing the L e v e l l B data, which we retrieved from JPL/PODAAC. We acknowledge Gerard Kruizinga (NASA/JPL) for aid in sorting out data and spacecraft anomalies over the period of the study. John J. McCarthy (SGT Inc.) prepared the constraint matrices used in the m a s c o n solutions.
References Bertiger, W., Y. Bar-Sever, S. Bettadpur, S. Desai, C. Dunn, B. Haines, G. Kruizinga, D. Kuang, S. Nandi, L. Romans, M. Watkins, and S. Wu (2002), GRACE: Millimeters and microns in orbit, Proceedings of ION GPS 2002, Portland, Oregon, September 24-27, 2002, pp. 2022-2029. Chao, B.F, W.P. O'Connor, A.T.C. Chang, D.K. Hall, and J. L. Foster, (1987), Snow load effects on the Earth's rotation and gravitational field, 1979-1985, Journal of Geophysical Research, 92(B9), pp. 9415-9422. Chao, B.F, and A. Au, (1991), Temporal variation of the Earth's low-degree field caused by atmospheric mass distribution- 1980-1988, Journal of Geophysical Research, 96 (B4), pp. 6569-6575. Han, S-C., C. Jekeli, and C.K. Shum, (2004), Timevariable aliasing effects of ocean tides, atmosphere, and continental water mass on monthly mean GRACE gravity field, Journal of Geophysical Research, 109, doi: 10.1029/2003JB002501. Luthcke, S.B., N.P. Zelensky, D.D. Rowlands, F.G. Lemoine, and T.A. Williams (2003), The 1-cm orbit: Jason-1 precision orbit determination using GPS, SLR, DORIS, and altimeter data, Marine Geodesy, 26(3-4), pp. 399-421. Mueller, P.M. and W.L. Sjogren (1968), Mascons-lunar
mass concentrations, Science, 161, pp. 680. Petrov, L. (2005), Personal communication, NASA Goddard Space Flight Center, Greenbelt, Maryland, USA. Ray R.D., D.D. Rowlands, and G. Egbert (2003), Tide models in a new era of satellite altimetry, Space Science Reviews, 108, pp. 271-282. Rowlands, D.D., R.D. Ray, D.S. Chinn and F.G. Lemoine, (2002), Short-arc analysis of intersatellite tracking data in a mapping mission, Journal of Geodesy, 76(6-7), pp. 307-316. Rowlands, D.D., S.B. Luthcke, S.M. Klosko, F.G. Lemoine, D.S. Chinn, J.J. McCarthy, C.M. Cox, and O. A. Andersen (2005), Resolving mass flux at high spatial and temporal resolution using GRACE intersatellite measurements, Geophysical Research Letters, 32, L04310, doi: 10.1029/2004GL021908. Tapley, B.D., S. Bettadpur, J. Ries, P. Thompson, and M. Watkins, (2004a), GRACE measurements of mass variability in the Earth system, Science 305(5683), pp. 503-505. Tapley, B.D., S. Bettadpur, M. Watkins, C. Reigber (2004b), the Gravity Recovery and Climate Experiment, Mission overview and early results, Geophysical Research Letters, 31(9), L09607, doi: 10.1029/2004GL021908. Tapley, B.D., J. Ries, S. Bettadpur, D. Chambers, M. Cheng, F. Condi, B. Gunter, Z. Kang, P. Nagel, R. Pastor, T. Pekker, S. Poole, and F. Wang (2005), GGM02 - An improved Earth gravity field model from GRACE, Journal of Geodesy, 79(8), pp. 467478. Thompson, P.F., S.V. Bettadpur, and B.D. Tapley, (2004), Impact of short period, non-tidal mass variability on GRACE gravity estimates, Geophysical Research. Letters 31, L06619, doi: 10.1029,2003GL019285 Wahr, J., S. Swenson, V. Zlotnicki, and I. Velicogna, (2004), Time-variable gravity from GRACE: First results, Geophysical Research Letters, 31(11), L 1150110, doi: 10.1029/2004GL019779.
Chapter 36
GRACE Time-Variable Gravity Accuracy Assessment R. Schmidt, F. Flechtner, R. K6nig, U1. Meyer, K-H. Neumayer, Ch. Reigber, M. Rothacher, S. Petrovic, S-Y. Zhu Department 1 Geodesy and Remote Sensing GeoForschungsZentrum Potsdam (GFZ), Telegrafenberg A17, 14473 Potsdam, Germany A. Gtintner Department 5 Geoengineering, Section 5.4 Engineering Hydrology GeoForschungsZentrum Potsdam (GFZ), Telegrafenberg Haus F, 14473 Potsdam, Germany Abstract. This study aims at the calibration of the formal variance covariance matrices of monthly GRACE-only gravity models that are known to give too optimistic accuracy measures for derived gravity functionals respectively surface mass anomalies. Based on 16 monthly solutions generated at GFZ Potsdam, a simple degree-dependent scaling of given variance-covariance matrices is considered, using difference degree amplitudes of monthly GRACE-only gravity field models that are separated by 12 months. It turns out that for the recent GFZ models the mean accuracy level of the monthly fields is about 20 - 40 times lower than the GRACE baseline accuracy. An observed scatter of the accuracy of the monthly models can be attributed to the amount of data used to generate the monthly model, but a more significant contribution arises from variations in the ground track coverage from month to month. Error propagation based on variance-covariance matrices calibrated with respect to 20 times the GRACE baseline, indicates that on global average the level of the model uncertainty is well reproduced when compared to residual surface mass variability in the space domain. Comparisons between time series of basin averages of surface mass variations from GRACE and state-of-the-art hydrology models seem to confirm this calibration factor for river basins with large annual mass variations. For basins with small variations the results suggest that a calibration with respect to 40 times the baseline may be a more representative value, however. As a general result it is observed that the formal respectively calibrated variance-covariance matrices of the monthly gravity models do not represent additional spurious gravity signals of significant amplitude caused by aliasing effects.
Keywords. GRACE, time-variable gravity, accuracy assessment, spherical harmonics.
1 Introduction Recent results based on monthly GRACE-only (Gravity Recovery And Climate Experiment, Tapley and Reigber (2001)) gravity models generated by the various groups of the GRACE Science Data System (SDS) have demonstrated the mission's sensitivity to time-variable gravity signals caused by geophysically and climatologically driven processes. In particular seasonal mass redistributions in the continental water cycle are traceable in GRACE data (see e.g. Wahr et al. (2004), Han et al. (2005) or Schmidt et al. (2006)), indicating that global hydrology modeling will be a first major application of GRACE-derived surface mass variability. Other applications are expected to follow as the quality of the future monthly solutions will improve due to expected advances e.g. in background modeling and refined processing strategies and methods. In any application, besides the GRACE-derived spatio-temporal distribution of changing surface mass, the accuracy of the inferred mass variability will be needed. In principle these quantities can be determined by rigorous error propagation from the given variance-covariance matrices that are obtained simultaneously with the solved spherical harmonic coefficients recovered by the leastsquares adjustment. However, the GRACE formal uncertainties are obviously too optimistic, i.e. both the error amplitude and the spatial distribution of the GRACE errors are not well represented by the given variance-covariance matrices. On the one hand, this originates from the L2-norm adjustment process which is based on the assumption that all systematic influences have already been eliminated prior to the adjustment. This requirement cannot be met, because some of such influences are unknown and not represented in the applied physical background models. Additionally, it is well known that in general the variance-covariance matrices never represent the real accuracy, but merely an inner consistency of the system (precision). Thus an appropriate calibration of the formal GRACE error information is needed.
238
R. S c h m i d t • F. F l e c h t n e r • R. K 6 n i g • U. L. M e y e r . K-H. N e u m a y e r • Ch. R e i g b e r • M. R o t h a c h e r • S. P e t r o v i c • S-Y. Z h u . A . G i J n t n e r
In this contribution we investigate the possibility of calibrating the formal variance-covariance matrices by means of GRACE-internal comparisons in the spectral and the space domain. In order to achieve this, a simple degree-dependent scaling of given variance-covariance matrices is investigated. Section 2 briefly describes the input data of the study, i.e. the period of 16 monthly GRACE-only gravity models period generated at GFZ Potsdam, highlighting the too optimistic level of formal errors. The approach to derive degree-dependent scaling factors is covered in Sect. 3, including an alternative approach to be used for comparison. Section 4 compiles results of the two calibration approaches. In Sect. 5 the calibrated variancecovariance matrices are applied for the accuracy assessment of surface mass variability in selected river basins and compared to output from state-ofthe-art hydrology models. The final Sect. 6 contains conclusions.
2 Monthly GRACE-only Gravity Models The basis of the study are 16 monthly GRACE-only gravity models in the period February 2003 to July 2004 generated at GFZ Potsdam. June 2003 and January 2004 are omitted due to larger gaps in the GRACE science data. The monthly models are given in terms of spherical harmonics and are derived from GRACE observations (GPS pseudo range and carrier phase, accelerometer, star camera and the K-band range-rate observations) using the dynamic method as described in Reigber et al. (2005). In the least-squares adjustment process the spherical harmonics are obtained together with the variance-covariance matrices. These are labeled as formal variance-covariance matrices or formal errors in the remainder of this paper. Figure 1 depicts the signal degree and the error degree amplitudes of the 16 monthly solutions in terms of geoid heights. The plotted error degree amplitudes are computed from the formal variances of the spherical harmonic coefficients. It can be seen that the error degree amplitudes vary from month to month. One possible explanation is the variable amount of data-days going into the individual solutions, which varies from 18 days for the 03/2003 field to 26 days for the 03/2004 field. However, the large difference in formal errors between the most accurate month (03/2004, 26 days of data) and the least accurate month (07/2004, 24 days of data) obviously cannot be explained by the amount of data. This distinction is evidently caused by the uneven ground track coverage in July 2004 (and also for June 2004, the second least accurate monthly model) when GRACE was approaching an exact 4d-repeat orbit in September 2004. In March
2003 and the other months the ground track has been much more uniform. This result indicates that the ground track coverage has quite a significant impact on the accuracy of monthly GRACE-only gravity models. Another impact that is known to contribute to the uncertainty of monthly GRACE-only gravity models are aliasing effects from errors in the background models of time-variable gravity signals such as short-term mass variations in the atmosphere and the oceans, ocean tides and other. These cause the known stripping patterns in spatial distributions of gravity functionals computed from monthly GRACE-only models (see e.g. Han et al. 2004). However, such contributions are not represented by the formal errors of the models. This is illustrated in Fig. 1 for the example of oceanic short-term mass variations. 101
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Fig. 1 Degree and error degree amplitudes of 16 monthly GRACE-only gravity models generated at GFZ Potsdam. The dashed lines represent the error degree amplitudes from the formal errors of spherical harmonic coefficients of the 16 monthly fields. For two state-of-the-art models describing oceanic short-term mass variations the difference degree amplitudes are plotted in Fig.1. It can be seen that the resulting curve for the model difference amplitudes is significantly larger than the formal errors up to degrees 15 - 20. Assuming that the existing model differences are a realistic measure for the uncertainty in the used background model, the comparison with the formal errors amplitudes reveals the underestimation of the true error level for this contribution. Similar results can be obtained for other time-variable signals such as atmospheric mass variations or ocean tides (not shown). Since these contributions add up to the errors in the shortterm oceanic mass variations the actual error level may even be larger. In this way the formal errors must be regarded as too optimistic, at least in the long wavelength part of the solved gravity spectrum.
Chapter 36 • GRACE Time-Variable Gravity Accuracy Assessment
3 Calibration Approaches Since no global data sets of comparable strength and homogeneity exist, a calibration based on GRACE-internal comparisons is considered. It has however to be noted that these comparisons are less internal than the variance-covariance matrices, which refer to internal consistency of single monthly solutions. The approach is to use differences between monthly GRACE-only solutions that are separated by 12 months assuming that the model differences are a representative measure for the true model error. The idea is to cancel out the dominant hydrological signal present in the monthly models, provided that the hydrological signal patterns are equivalent from year to year. The final goal is to derive degree-dependent scaling factors that can be applied degree-wise to the formal variance-covariance matrices. To this end the model differences are transformed into the spectral domain by computing degree difference amplitudes to be compared to the formal error degree amplitudes. In detail the following steps are carried out: From the recovered monthly solutions difference amplitudes are computed for the months February to May in 2003 and 2004. The difference July 2003 - 2004 is not considered because of the degraded ground track coverage in 07/2004. For the resulting difference amplitudes the average (arithmetic mean per degree) is calculated and compared to the GRACE baseline accuracy. It turns out that the averaged difference amplitudes are about 40 times the GRACE baseline (see Fig. 2(a)), which is consistent with a value found by Wahr et al. (2004) based on monthly GRACE-only models generated by the Center of Space Research (CSR), Austin/Texas. Using the 40 times the GRACE baseline accuracy as calibrated mean accuracy of the monthly solutions the scatter of the monthly solutions is added to this curve by applying degree-dependent scaling factors of the formal errors relative to the mean formal error of the monthly solutions. Instead of the 16 fields only 14 monthly models respectively their formal errors are used. The models for June and July 2004 are excluded because of the degraded ground track coverage during these months. The result for the calibrated error degree amplitudes per month is shown in Fig.2 (b). Comparing these calibrated, monthly error degree amplitudes degree-wise to their formal error degree amplitudes gives the degree-dependent scaling factors which are then applied to the
formal variance-covariance to obtain the calibrated variance-covariance matrices. Finally, the calibrated variance-covariance matrices are used for error propagation of gridded gravity functionals respectively surface mass anomalies in the space domain.
-
For comparison, an alternative approach in the space domain is considered. In this second approach, the true error of the monthly GRACE-only models is to be represented by the residual gravity respectively surface mass variability of time series of global grids from GRACE, where in each grid point a constant, a linear trend and a sinusoidal signal with annual frequency are removed. The resulting residual grids can then be compared to the gridded errors of gravity respectively surface mass based on error propagation from the calibrated variance-covariance matrices derived from the 1 st approach.
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239
240
R. Schmidt. F. Flechtner. R. K6nig • U. L. Meyer. K-H. Neumayer. Ch. Reigber. M. Rothacher. S. Petrovic • S-Y. Zhu. A.G~intner
To this end, global grids ( l ° x l °) of surface mass variability in terms of the thickness of an equivalent mass of water for the monthly models with respect to their mean are computed. Only the m o n t h l y gravity models of 2003 and their mean is used in the following. The reason for this limitation is to cover almost exactly an annum period on the one hand and to exclude the degraded fields June and July 2004 on the other. The grids are filtered using the Gaussian averaging filter for different radii of 1500, 1000, 750 and 500 km, respectively. In each grid c o m p a r t m e n t a bias and linear trend plus an annual sine is fitted to the time series of data points and removed. Finally, the R M S of the residuals per grid point is taken as the uncertainty of the monthly solutions.
Figures 3 (a) and 4 (a) show the spatial distribution obtained from error propagation using the resulting calibrated variance-covariance matrix for
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4 Comparison of Approaches Table 1 lists the global weighted RMS (cosine latitude weighing) of the global grids of the errors of surface mass variability from error propagation based on the mean calibrated error level scaled to 40 times the baseline and for residual R M S variability from approach 2. The surface mass variability is expressed in terms of the thickness of an equivalent mass of water. Table 1. Global weighted RMS of global grids of errors of
surface mass variability from error propagation based on the mean calibrated error level scaled to 20/40 times the GRACE baseline and for residual RMS variability from approach 2. Surface mass in terms of the thickness of an equivalent mass of water. Unit is centimeter. Mean formal error level scaled to
[cm] 0.0
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Fig. 3 (a) Spatial distribution of errors of surface mass variability (thickness of equivalent mass of water, unit cm) from error propagation using the complete, calibrated variance-covariance matrix for month 08/2003 (mean level scaled to 20 times the GRACE baseline). Gaussian averages for a filter radius of 750 km. (b) Spatial distribution of absolute values of residual surface mass variability for 08/2003 after removal of a best-fitting constant plus a linear trend plus an annual sinusoid per grid point. Same units and Gaussian filtering applied as in Fig. 3 (a).
Spherical harmonic degree, half-wavelength X/2 in [km]
GRACE baseline X 40 X 20
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The comparison shows that the global error level obtained from the calibrated variance-covariance matrix is significantly larger than the level from approach 2 over all spatial wavelengths. This may indicate that a global scaling of the mean error level of the monthly solutions by a value of 40 is too conservative. Systematically reducing this global scaling factor and re-computing the error propagation, a value of 20 times the G R A C E baseline is obtained for the mean calibrated error level of the m o n t h l y models which eventually gives a better agreement b e t w e e n the two approaches (cf. Tab. 1). Consequently calibrated variance-covariances relative to this level are determined.
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Fig. 4 (a) and (b) as in Fig. 3 (a), (b) but Gaussian averages for a filter radius of 500 kin.
Chapter 36 • GRACE Time-Variable Gravity
August 2003 as an example. The full matrix including the covariances has been used for error propagation and Gaussian averages of 750 km (Fig. 3 (a)) and 500 km (Fig. 4 (a)) radius are computed. The displayed maps show meridional oriented patterns that are typical for the propagated errors of the monthly models: largest errors are obtained in the equator region, smallest errors are given towards the poles. This kind of distribution obviously correlates with the ground track coverage, where the sampling along the G R A C E orbit increases in direction of the poles due to the high inclination of the orbital plane (i ~- 89.5 o). Towards the equator larger unsampled areas occur, because of the evolution of the orbit in longitudinal direction. The spatial distribution of the residual variability from approach 2 is shown in Fig. 3 (b) and 4 (b). Again meridional oriented patterns are visible, h o w e v e r the given features differ significantly in space and amplitude from the maps of error propagation (cf. Fig. 3 (a) and 4 (a)). Over continents such deviations can be explained to some extent from non-reduced hydrological signals, e.g. the variability in the Niger region in Africa. However, other signals, in particular of the oceans, cannot be explained by non-reduced hydrology, but represent obviously aliasing effects from various sources like deficiencies in the applied background models for time-variable gravity, but also possible inefficiencies in the data processing like e.g. an imperfect orbit parametrization. Table 2 confirms that on the global scale, represented by the weighted RMS, the two approaches give comparable results. However, spurious signals of large amplitude (cf. m a x i m u m values in Tab.2) caused by aliasing are not covered by the calibrated variance-covariance matrices. Table 2. Statistics of grids from Fig. 3 (a) - Fig. 4 (b). Units cm.
Grid Fig. 3 (a) Fig. 3 (b) Fig. 4 (a)
Min. 0.6 0 0.6
Max. 1.7 7.8 3.3
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Fig. 5 (a) Time series of basin averages of surface mass for the Ganges river derived from GRACE and the two hydrology models WGHM (D611 et al., 2003) and LaD (Milly and Shmakin, 2002) using the method of Swenson and Wahr (2002). The acceptable satellite model error is set to 2 cm, the model variances are calibrated relative to 40 times the GRACE baseline accuracy. Lines represent the annual-varying part derived from the individual time series. (b) same as (a), but model variances calibrated with respect to 20 times the GRACE baseline accuracy. Lena (2461 000 km 2) 80 :
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5 Accuracy Assessment of Basin Averages of Surface Mass Variability As an additional test to assess the value of the calibrated variance-covariance matrices time series of surface mass variability for selected river basin, are derived from the 16 monthly G R A C E - o n l y gravity models and monthly output from two state-of-theart hydrology models. The method described in Swenson and W a h r (2002) is applied to derive the basin averages, that are constrained by the given satellite model errors respectively a user-defined value for the acceptable satellite model error (see Swenson and W a h r (2002) for details). In this study
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Fig. 6 (a) and (b) as in Fig.5 (a) and (b) but for the Lena/Ob river basin.
241
242
R. Schmidt. F. Flechtner. R. K~nig • U. L. Meyer. K-H. Neumayer. Ch. Reigber. M. Rothacher. S. Petrovic • S-Y. Zhu. A.G~intner
a value of 2 cm water column is taken as the constraint for the satellite model error, using the monthly calibrated errors for the two versions of 40 times and 20 times the G R A C E baseline from the previous section. Figures 5 and 6 depict the resulting time series of the basin averages from G R A C E and the two hydrology models for the Ganges (Fig. 5) and the Lena/Ob river (Fig.6). Cases (a) and (b) show the variations using 40 times (a) and 20 times (b) the G R A C E baseline for calibrating the average monthly accuracy. For the Ganges the deviations between G R A C E and the hydrology models are well within the derived error bounds for both versions of calibrated variances. However, for case (a), i.e. supposing the mean accuracy level of the monthly solutions at 40 times the G R A C E baseline, the signal amplitude is much smaller than for case (b). This is a feature of the applied filter method that causes a stronger smoothing in case of a larger uncertainty of the gravity coefficients as it is the case for version (a). However, in view of the good coherence between the time series from G R A C E and hydrology models for this basin it is difficult to set a preference on one of the two scaling levels. For the Lena/Ob river (cf. Fig. 6 (a) and (b)) the amplitude and phase differ significantly between G R A C E and hydrology models. Some GRACEderived averages look rather as outliers which are presumably due to spurious gravity signals shown in the previous section. The feature is of course more pronounced when using version (b), i.e. 20 times the G R A C E baseline for calibrating the mean accuracy of the monthly solutions. Thus version (a). i.e. 40 times the baseline may be more representative for the true mean model errors than version (b), at least in this region. 6
Conclusions
This study aimed at the calibration of variancecovariance matrices of monthly GRACE-only gravity models by means of degree-dependent scaling factors derived in the spectral domain from GRACE-only model difference that are separated by 12 months. Based on such model differences an average accuracy level of the monthly GRACEonly gravity models of about 20 - 4 0 times the G R A C E baseline accuracy is inferred. Error propagation based on two extreme versions of calibrated variance-covariance matrices of the monthly gravity models (i.e. 20 and 40 times the G R A C E baseline) indicates that a value of 20 times the baseline may be representative, when compared to global averages of residual surface mass vari-
ability from G R A C E in the space domain. Comparisons between time series of basin averages of surface mass variations from G R A C E and state-ofthe-art hydrology models confirm the plausibility of this value, however only for regions with large mass variations. In regions with small variations a value of 40 times the G R A C E baseline may be a better estimate of the mean accuracy level of the monthly solutions. In general it turns out that any version of G R A C E gravity model errors calibrated by means of degree-dependent scaling of the formal variancecovariance matrices considered here does not represent the significant systematic gravity model errors that are caused by aliasing effects. It seems that the obtained formal variance-covariance matrices mainly reflect the sampling geometry along the G R A C E orbit proportional to the amount of data. The former has a strong impact on the model accuracy in months with a particularly uneven ground coverage (e.g. months with repeat orbit patterns). It has been argued that degree-dependent scaling may have little physical meaning and that the selected GRACE baseline accuracy taken as reference may itself be arbitrary. Regarding this latter aspect it is true that the level of the anticipated accuracy from G R A C E largely depends on items such as orbit characteristics (inclination and altitude), the expected instruments performance (e.g. accelerometers and the K-band link) and the degree of detail of the simulations. However, the general shape of the G R A C E baseline degree error amplitudes is assumed to be reasonable as it is verified by results of comprehensive numerical simulations (e.g. Kim (2000)) and in this way may serve as a plausible reference. With respect to the physical meaning, a degree and order dependent scaling could be more flexible allowing for individual scale factors for the single elements of the variance-covariance matrices. On the other hand, such a procedure could be less robust and might lead to an almost perfect adaptation of the variance-covariance matrices to the results of the selected calibration approach. However, such results depend on the chosen comparison and may be arbitrary as well, i.e. cannot represent all aspects of the physical truth. This general problem is illustrated by the differences between the two approaches considered here. In this way the proposed method has to be considered as a first, rather heuristic procedure giving preliminary results. Further investigations are needed to derive more comprehensive estimates of the monthly GRACE-only gravity model accuracy.
Chapter 36 • GRACE Time-Variable Gravity Accuracy Assessment
Acknowledgements. The German Ministry of Education and Research (BMBF) supports the GRACE project within the GEOTECHNOLOGIEN geoscientific R+D programme under grant 03F0326A. We are also grateful to two anonymous reviewers for the helpful comments and suggestions.
References DGll, P., F. Kasper, B. Lehner (2003). A global hydrological model for deriving water availability indicators: model tuning and validation. J. Hydrol., 270, pp. 105-134. Hart, S.C., C.K. Shum, C. Jekeli (2004). Time-variable aliasing effects of ocean tides, atmosphere, and continental water mass on monthly mean GRACE fields. J. Geophys. Res., 109, B04403, doi: 10.1029/2003JP002501. Hart, S.C., C.K. Shum, C. Jekeli, D. Alsdorf (2005). Improved estimation of terrestrial water storage changes from GRACE. Geophys. Res. Lett., 32, doi: 10.1029 / 2005GL022382. Kim, J. (2000). Simulation Study of A Low-Low Satelliteto-Satellite Tracking Mission., Dissertation, The University of Texas at Austin. Milly, P.C.D., and A.B. Shmakin (2002). Global modeling of land water and energy balances: Part I. The Land Dyam-
ics (LAD) Model. J. Hydrometeor. 3 (3), pp. 283-299. Tapley, B.D., and Ch. Reigber (2001). The GRACE mission: Status and future plans. EOS Trans A GU, 82 (47), Fall Meet. Suppl., G41 C-02. Reigber, Ch., R. Schmidt, F. Flechtner, R. KGnig, U1. Meyer, K-.H. Neumayer, P. Schwintzer, S.-Y. Zhu (2005). An Earth gravity field model complete to degree and order 150 from GRACE: EIGEN-GRACE02S. J. Geodynamics, 39, 1-10, doi: 10.1016/j.jog.2004.07.001. Schmidt, R., P. Schwintzer, F. Flechtner, Ch. Reigber, A. Gtintner, P. DGll, G. Ramillien, A. Cazenave, S. Petrovic, H. Jochmann, J. Wtinsch (2006). GRACE observations of changes in continental water storage. Global and Planetary Change, 50, pp. 112-126, doi: 10.1016/j.glopacha.2004. 11.018 Swenson, S., and J. Wahr (2002). Methods for interring regional surface-mass anomalies from Gravity Recovery and Climate Experiment (GRACE) measurements of time-variable gravity. J. Geophys. Res., Vol. 107, No. B9, doi: 10.1029/2001 JB000576. Wahr, J., S. Swenson, V. Zlotnicki, I. Velicogna (2004). Timevariable gravity from GRACE: First results. Geophys. Res. Lett., 31, L11501, doi: 10.1029/2004GL019779.
243
Chapter 37
Combination of multi-satellite altimetry data with CHAMP and GRACE EGMs for geoid and sea surface topography determination G.S. Vergos[], V.N. Grigoriadis, I.N. Tziavos Department of Geodesy and Surveying, Aristotle University of Thessaloniki, University Box 440, 541 24, Thessaloniki, Greece, Fax: +30 231 0995948, E-mail: [email protected]. M.G. Sideris Department of Geomatics Engineering, University of Calgary,
Abstract. Since the launch of the first altimetric missions a wealth of data for the sea surface has become available and utilized for geoid and sea surface topography modeling. The data from the gravity field dedicated satellite missions of CHAMP and GRACE provide a unique opportunity for combination studies with satellite altimetric observations. This study focuses on the combination of data from GEOSAT, ERS1/2, Topex/Poseidon, JASON-1 and ENVISAT with Earth Gravity Models (EGMs) generated from CHAMP and GRACE data to study the mean sea surface (MSS)/marine geoid in the Mediterranean Sea. Various combination methods, i.e., weighted least squares and least squares collocation are investigated and conclusions on the most appropriate combination strategy are drawn. Then, a remove-compute-restore scheme is followed to estimate the MSS model. Comparisons with other MSS models referenced to EGM96 and CHAMP/GRACE EGMs are performed in terms of the geoid height values at various control points. Finally, a sea surface topography model for the eastern part of the Mediterranean Sea is determined by a combination of the altimetric geoid and the CHAMP/GRACE EGM. The latter is validated against a sea surface topography model derived from altimetric data, in-situ oceanographic observations and an ocean general circulation model. K e y w o r d s . CHAMP/GRACE EGMs, ENVISAT and JASON-1 validation, mean sea surface, sea surface topography.
1
Introduction
During the last twenty five years altimeters on-board satellites have offered a tremendous amount of highaccuracy measurements of the instantaneous height of the sea surface above a reference ellipsoid known as sea surface heights (SSHs). With the advent of technology new missions emerged offering always a more accurate picture of the ocean surface and continuing the missions of previous satellites. The latter, i.e., the continuity of one satellite mission from another is of very highimportance, since it offers a long time series of exactly
repeating measurements of the sea surface. Such satellites are ERS 1, ERS2 and ENVISAT and TOPEX/Poseidon (T/P) with JASON-1. JASON-1 and ENVISAT are the latest on orbit satellites (December 2001 and March 2002, respectively) and are both set on exact repeat missions (ERM). This long series of altimetric observations has been widely used for studies on the determination of MSS models (Andersen and Knudsen 1998; Cazenave et al. 1996), global and regional geoid models (Andritsanos et al. 2001; Lemoine et al. 1998; Vergos et al. 2005) as well as on the recovery of gravity anomalies from altimetric measurements (Andersen and Knudsen 1998; Hwang et al. 1998; Tziavos et al. 1998). The main advantage of altimetric SSHs over shipborne gravity data can be viewed in terms of their high precision and resolution, homogeneity and global coverage. One of the main aims of the present study was the validation of the data acquired so far from JASON-1 and ENVISAT with respect to their accuracy compared to the latest Mean Sea Surface (MSS) models. Both satellites are supposed to continue the missions of their predecessors. Therefore, their accuracy should be at least comparable to that acquired from ERS1/2 and T/P. Another goal was the combination of multi-satellite altimetry data for the determination of a high-accuracy and high-resolution MSS model for the eastern part of the Mediterranean Sea. Both ERM and Geodetic Mission (GM) data have been used to achieve maximum resolution in the computed field. The MSS model was based on data from GEOSAT (ERM and GM), ERS1 (ERM and GM), ERS2, and T/P, while it was decided that the new mission's data would be used only if they provided accurate results during their validation. The estimated MSS models were validated against the two latest KMS MSS models and a local one derived during an earlier study. The altimetric MSS model to be computed in the frame of the present study actually coincides with the marine geoid, since they only deviate by the quasistationary sea surface topography (QSST) term, which is unknown for the Mediterranean Sea due to the absence of local models and the fact that global ones are inappropriate for closed sea areas. Therefore, the terms MSS and marine geoid models are considered the same for the present study, of course under the aforemen-
Chapter 37 • Combination of Multi-Satellite Altimetry Data with CHAMP and GRACE EGMs for Geoid and Sea Surface
tioned condition, i.e., the absence of a sea surface topography model. The QSST in the area under study was estimated from a combination of the computed MSS/marine geoid model and a gravimetric geoid model computed during an earlier study. The resulting QSST was validated against a recent oceanographic QSST model for the Mediterranean Sea, which was based on altimetric data, in-situ oceanographic observations and a parallel ocean circulation model.
2
The validation was based on comparisons between the available SSHs from each satellite with highaccuracy (+4-6 cm) and high-resolution ( l ' x l ' ) altimetric and gravimetric geoid models for the area under study (Vergos et al. 2005). Additionally, a cycle-bycycle analysis of the satellite records has been performed to conclude on their precision. Finally, stacked JASON-1 and stacked and crossover adjusted ENVISAT datasets have been constructed and compared with the available geoid models. This would allow the removal of (a) sea surface variability effects and (b) orbital errors from the GDRs and would make the comparisons more representative. It was not necessary to crossover adjust the JASON-1 data since this has already been done by AVISO. In all cases the comparisons were performed as S S H - N ~, where N ~ is the predicted geoid height at the sub-satellite point using least squares collocation for the interpolation. The computed differences were minimized using a 3ra order polynomial model for bias and tilt fit according to the formula
E N V I S A T and JASON-1 data validation
For the validation of the ENVISAT and JASON-1 SSHs the available Geophysical Data Records (GDRs) from their launch until May 2005 have been collected. The data became available by ESA/CNES (ESA 2004) and CNES/AVISO (AVISO 2003), respectively. The ENVISAT SSHs span from September 4, 2003 to March 16, 2005, corresponding to cycles 15-34; JASON-1 SSHs span from January 15, 2002 to January 15, 2005. It should be mentioned that due to some problems detected in the ENVISAT CDs distributed by the responsible agency (lack of satellite cycle and track numbers) it was not possible to process cycles 1-14 of the satellite since the new records became available only in early August 2005 when the MSS and QSST models have already been developed. The area under study is bounded between 33 ° < q~ < 38 ° and 20 ° < X < 30 ° and the total number of SSH records was 26072 and 31109 from ENVISAT and JASON-l, respectively (see Figure 1). ~.
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tude of the sub-satellite points respectively. The selection of a 3rd order polynomial model for the minimization of the differences between altimetric SSHs and geoid heights was based on a validation performed by testing various parametric models. In this test, 1st, 2 nd, and 3ra order polynomial models as well as the well-known four- and five-parameter similarity transformation models have been employed. From the results acquired, it was concluded that the 3rd order polynomial model provided the best fit, i.e., managed to give the smallest standard deviation (std) for the differences between the heights compared. Table 1 presents the statistics of the differences between JASON-1 and the gravimetric geoid model for some of its cycles as well as for the entire dataset. The last two rows of the table give the differences of the stacked data before and after (italics) the bias and tilt fit. From that table it is evident that even after stacking the JASON-1 data and minimizing its differences with the geoid model, we end up to acr of +20 cm. This is far worse than the +14 cm that T/P provided when compared in the same area and with the same geoid model (Vergos et al. 2005). Even when the stacked JASON-1 data was compared to the altimetric geoid model, the (std) of the differences after the bias and tilt fit reached the +15.5 cm level compared to +8 cm for T/P. The aforementioned results combined with the fact that JASON- 1 data are interrupted far away from the coastline
245
246
G.S. Vergos. V. N. Grigoriadis. I. N. Tziavos • M. G. Sideris Table 1. Statistics of differences between JASON-1 and the gravimetric geoid model. Unit: [m]. cycles
max
1 2 4 5 45 90 1 - 111 stacked 1 - 111 stacked 1 - 111
0.530 0.615 0.426 0.317 0.449 0.773 1.184 0.175 0.578
min
-0.452 -0.542 - 1.380 -0.633 -0.850 -0.346 -3.536 -1.049 -0.519
mean
std
-0.009 -0.012 -0.057 -0.084 -0.222 0.165 0.041 -0.339 0.000
-+0.167 _+0.281 _+0.248 _+0.254 +0.245 +0.324 _+0.316 +0.254 -+0.202
(see Fig. 2 for the T/P case), forced us to decide not to use them in the MSS/geoid model determination. The latter may be correlated with the problems encountered with the radiometer on-board the satellite. On the other hand, when the ENVISAT SSHs were compared with the gravimetric and altimetric geoid models, the results acquired were very encouraging (see Table 2), since the differences with the former after stacking and crossover adjustment (second last row) were at the +22 cm and dropped to less than +13 cm after the minimization procedure (last row). When compared to the altimetric geoid model, the ENVISAT SSHs presented a difference of only +12.5 (+9 cm after the fit); therefore they were considered as very satisfactory and were used for the subsequent MSS model determination.
ber of 299662 observations of the sea surface were available for the determination of the MSS model (see Fig. 2). In Fig. 2 the geodetic mission data from ERS1 and GEOSAT are denoted by the very dense dots covering the entire area. All data were provided by AVISO (1997) except from the GEOSAT SSHs which were provided by NOAA (1997). In all cases the geophysical and instrumental corrections proposed by the respective agency were implemented to construct corrected SSHs. For the determination of the MSS/geoid model all data had to be consistent, i.e., no biases between them should exist. The ERS 1 and ERS2 data have been reference to T/P by AVISO, since their orbits were recomputed based on that of T/P. The GEOSAT-GM data were processed during an earlier study (Andritsanos et al. 2001) by estimating and removing their bias and tilt w.r.t. T/P. Therefore, only the ENVISAT SSHs had to be referenced to T/P, so their bias (~14 cm) and tilt w.r.t, the latter were estimated and removed. In this way a homogeneous dataset has been constructed for use in MSS/marine geoid determination.
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3
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Estimation
After the validation of the ENVISAT and JASON-1 data and the conclusion that the latter will not be used, a database of all available altimetric observations for the area under study was created to determine the MSS/geoid model. The altimetric SSHs came from: (i) the GEOSAT-GM mission (25402 SSHs), (ii) the ERS 1-ERM mission phases c and g (34323 SSHs), (iii) the ERS1-GM mission (14901 SSHs), (iv) the ERS2ERM mission (30991 SSHs), (v) nine years of the T/P mission (136864 SSHs), and (vi) the already used ENVISAT mission (26072 SSHs). Therefore, a total num-
i// Fig. 2" Distribution of ERS 1-ERM, ERS2, ENVISAT, ERS 1-GM, GEOSAT-GM and T/P data.
Consequently, a remove-compute-restore method has been followed to determine the final MSS model, i.e., the altimetric SSHs were referenced to an earth geopotential model, gridded and then the GM contribution was restored to construct the final MSS. During the remove step and in order to assess the improvement that the latest CHAMP and GRACE EGMs offer, two models have been employed, namely the traditional EGM96 (Lemoine et al. 1998) and EIGEN-CG03C (F6rste et al. 2005). The latter is the latest combination model by GFZ using CHAMP and GRACE data and is complete to degree and order 360. Table 3 summarizes the statistics of the SSHs before and after the reduction to the EGMs. From that table it is clear that EGM96 provides a (marginally) better reduction of the data by about 2 cm in terms of the std and 6 cm in terms of the range compared to EIGEN-CD03c. On the other hand the latter
Chapter 37 • Combinationof Multi-Satellite Altimetry Data with CHAMPand GRACEEGMsfor Geoid and Sea SurfaceTopographyDetermination
reduces the mean by 9 cm more than EGM96. Therefore, no clear conclusion can be drawn for their performance other than that the models are comparable.
Table 3. Statistics SSHs before and after the reduction to EGM96 and EIGEN-CG03c. Unit: [m]. max
SSHs
40.401 SSUsred EGM96 1.461 S SHsred EIGEN-CG03c 1.627
min
-0.358 -2.367 -2.264
mean
14.541 -0.266 -0.175
std
+8.665 +0.340 +0.358
To grid the data and generate the reduced MSS/marine geoid mesh at l ' x l ' resolution three methods have been identified, i.e., conventional least squares, splines in tension and least squares collocation. From the analysis performed and the comparisons with global MSS models it was concluded that the LSC solution provided superior results by about +7-11 cm (in terms of the std of the differences) compared to the other methods, something in line with an earlier study (Tziavos et al. 2004), where these algorithms were also tested. Due to the limited space available, only the results from LSC will be reported herein. Table 4 presents the gridded reduced MSS heights as well as the final MSS models referenced to EGM96 and EIGEN-CG03c. The EGM96 MSS model is also depicted in Fig. 3.
of high-importance in terms of the absolute MSS/marine geoid error. In a next step, the computed differences were minimized using Eq. 1 to remove any bias and tilts between the KMS and the compute MSS models. This resulted in smaller std values at the level of +14.2, + 15.3, + 10.7, and + 11.4 cm for the differences between KMS01, KMS04 and the EGM96 and EIGEN-CG03c MSS models, respectively. Therefore, even after the minimization of the differences the MSS referenced to EGM96 outperformed the EIGEN-CG03c model even by a slight margin. In Tziavos et al. (2004) MSS models for the same area were developed and also compared to the KMS models giving an overall best std at the +17 cm (after bias and tilt fit). So, it can be concluded that the MSS models developed in this study are about 5 cm more accurate than the previous ones. Finally, the estimated MSS models agree better by almost +9 cm with KMS04 compared to KMS01, which gives evidence that the latest KMS MSS model is indeed an improved version of its predecessor. The differences between KMS04 and the referenced to EGM96 MSS model (see Fig. 4) are almost zero in marine areas, while they reach their minimum and maximum values close to the coastline, where both models suffer due to the inherent problems of satellite altimetry in such areas.
T a b l e 5. Statistics of the differences between the KMS and estimated MSS models. Unit: [m]. max
max
M S sred EGM96 MsSred EIGEN-CG03c M S S EGM96
MSSEI~EN-cC°3c
0.940 1.060 40.139 39.851
min
-1.720 -1.660 0.596 0.714
mean
-0.370 -0.280 19.828 19.828
std
+0.350 +0.360 +10.839 +10.839
The validation of the estimated MSS models was performed through comparisons with the latest KMS (Danish Survey and Cadastre) MSSs, namely KMS01 (Andersen and Knudsen 1998) and KMS04 (Andersen et al. 2003). Table 5 summarizes the results of the comparisons for both MSS models developed, while Fig. 4 depicts the differences between KMS04 and the referenced to EGM96 MSS model. From that table it becomes once again evident that the two EGMs give almost the same results, but EGM96 outperforms EIGENCG03c by +1 cm in terms of the std and 50 cm in terms of the range, even though KMS04 is referenced to GGM01C ( C H A M P - G R A C E combination EGM). This is an indication that EGM96 can be regarded as a dominant geopotential model and is still not outperformed by the new EGMs. Of course this is true for the present study (relative accuracy), the data used, the area under study and may not repeat in other regions. Furthermore, EIGEN-CG03c gives a much smaller cumulative geoid error (30 cm compared to 42 cm for EGM96), which is
4
min
KMS01-MSSEl~ENcG°3c KMS04 - MSSEGM96 KMS04-MSSEloENcG°3c
1.194 1.363 1.199 1.179
Sea Surface
Topography
K M S 0 1 - M S S wGM96
Table 4. Statistics of the reduced MSS heights and the final MSS models referenced to EGM96 and EIGEN-CG03c. Unit: [m].
mean
-0.959 -1.109 -0.339 -0.919
0.207 0.208 0.127 0.126
std
+0.194 +0.201 +0.117 4-0.122
Estimation
For the determination of the quasi-stationary (QSST) model the estimated MSS model referenced to EGM96 was combined with the gravimetric geoid available for the area under study. The latter was estimated from airborne (Olesen et al. 2003), shipborne and land gravity data (Vergos et al. 2005). To derive a first QSST model, the differences between the MSS and the gravimetric geoid were formed as QSST = MSS air - N grav
(2)
where the gravity anomalies used to determine the gravimetric geoid are free-air reduced, i.e., reduced from the sea surface to the geoid, and the MSS heights refer to the sea surface. The statistical characteristics of this preliminary QSST are given in Table 6, from which it is evident that the QSST estimated presents some unreasonably large variations in the area (3.5 m) and reaches a maximum of almost 2 m. Therefore it is clear that blunders are present in the estimated field. Finally, noisy features are evident, thus low-pass filtering (LPF) was needed in order to reduce these effects.
247
248
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max
QSST QSST (after 3cy) QSST (after 3~ and LPF) MDT04 - QSST
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4
(3)
std
-1.448 -0.125 +0.318 -0.953 -0.131 +0.287 -0.510 0 . 0 1 4 +0.238 -0.265 0 . 0 0 0 4-0.072
For the detection and removal of blunders, a simple 3cy test was performed, i.e., points with a QSST value larger than 3 times the a of the preliminary field were removed. The statistics of the QSST model after this test are given in Table 6 as well. To low-pass filter the preliminary QSST model, a collocation-type of filter (Wiener filtering) was used, assuming the presence of white noise in the QSST field. Furthermore, it was assumed that Kaula's rule for the decay of the geoid power spectrum holds, i.e., that the geoid height power spectral density decays like k -4 where k is the radial wavenumber. These resulted in the flowing filtering function
where co is the radial frequency, ( t ) = 4 U 2 -]-V 2 , and coc the cut-off frequency. To filter the wanted field, the desired cut-off frequency needs to be selected. The latter relates to the final resolution of the filtered field and the reduction of the noise in the data. Thus, a trade-off is necessary, since higher resolution means more noise will pass the filter, while higher noise reduction means lower resolution of the final model. A high resolution is vital in the determination of regional to local QSST models, since if a high value cannot be achieved then a so-derived local model has little to offer compared to a global solution. It can be clearly seen, that the disadvantage of Wiener filtering is that the selection of the cut-off frequency is based on the spectral characteristics of the field only, while its spatial characteristics are not taken into account. Furthermore, the selection of the cut-off fie-
Chapter 37 • Combination of Multi-Satellite Altimetry Data with CHAMPand GRACEEGMsfor Geoid and Sea Surface Topography Determination
quency is based on solely objective criteria. Thus, a trial and error process, based on maximum noise reduction with minimum signal loss, is needed to determine the desired cut-off frequency. Various cut-off frequencies have been tested corresponding to wavelengths of 5, 10, 20, 40, 60, 100, 110 and 120 km and finally a wavelength of 100 km (about 1o or harmonic degree 180) was selected since it offered the minimum signal loss with maximum noise reduction. Wavelengths shorter than 100 km left too much noise in the field, while those larger than 100 km were reducing not only the noise but some spatial characteristics of the QSST as well. If a longer wavelength was selected, then, and if the area was significantly larger (e.g. the entire Mediterranean Sea) it would have been possible to identify larger-scale QSST features and distinguish them from smaller ones. The problem in this case is that for the rest of the Mediterranean Sea only few ship tracks with gravimetric observations are available, therefore, a gravimetric geoid model cannot be determined at least at such high resolution (1'). The answer in such cases for geoid modeling is the combination of shipborne gravity data with satellite altimetry, but such a combination model cannot be used for QSST modeling (at least in the present context) due to the high correlation with the MSS model. The final QSST field after filtering is shown in Fig. 5 (top), while the statistics are given in Table 6 (last row). From the aforementioned figure it can be seen that the noise present in the preliminary model is reduced significantly, while blunders cannot be identified. For validation purposes the estimated QSST model was compared with a Mean Dynamic Topography (MDT04) model estimated for the entire Mediterranean Sea from an analysis of satellite altimetry and oceanographic data (Rio 2004). The latter was given as a grid of mean QSST values of 3.75'x3.75' resolution in both latitude and longitude. The statistics of the differences between the MDT and the estimated QSST models are given in Table 6 (last row). From the comparison it can be concluded that the two models agree very well to each other (std at the +7 cm level only). The maximum and minimum values of the differences are found close to land areas only, where both models are inadequate. This comparison gives evidence that the estimated QSST model is in good agreement with existing regional oceanographic MDT models. Furthermore, it is a welcoming fact, which supports the appropriateness of the proposed methodology for the determination of a geodetic QSST model. 5
Conclusions
A first validation of the ENVISAT and JASON-1 data in the eastern Mediterranean Sea has been performed, from which it was found that the former provide accurate results comparable to the other altimetric missions, while the latter are of lower accuracy compared to T/P
and present extensive gaps. The latter can be attributed to the radiometric correction problems in the JASON-1 data. The MSS models developed present very good agreement with the corresponding KMS01 and KMS04 ones, with their smallest difference being at the + 11 cm level. Compared to earlier results achieved by the authors, the newly developed MSS is of higher resolution (1' comparing to 5') and accuracy (+ 11 cm comparing to +17 cm) and presents an improved version. Furthermore, it can be concluded that at least in the present stage EGM96 is still comparable to the EIGEN/GRACE type of EGMS, but of course not in terms of the cumulative geoid error. Finally, the estimated QSST model provided very encouraging when compared to an oceanographic MDT model, with its differences only at the +7 cm. This is a tremendous improvement, since it can be used for local/regional geoid and gravity field modeling in the area, due to the inappropriateness of global MDT models in closed sea areas.
Acknowledgement
This research was funded from the Greek Secretariat for Research and Technology in the frame of the 3rd Community Support Program (Opp. Supp. Progr. 2000 - 2006), Measure 4.3, Action 4.3.6, Sub-Action 4.3.6.1 (International Scientific and Technological Co-operation with non-EU countries), bilateral co-operation between Greece and Canada.
References
Andritsanos VD, Vergos GS, Tziavos IN, Pavlis EC and Mertikas SP. (2001) A High Resolution Geoid for the Establishment of the Gavdos Multi-Satellite Calibration Site. In: Sideris MG (ed) Proc of International Association of Geodesy Symposia "Gravity Geoid and Geodynamics 2000", Vol. 123. Springer Verlag Berlin Heidelberg, pp 347-354. Andersen OB, Knudsen P (1998) Global gravity field from ERS1 and Geosat geodetic mission altimetry. J Geophys Res 103(C4): 8129-8137. Andersen OB, Vest AL, Knudsen P (2003) Altimetric Mean Sea Surfaces and inter-annual ocean variability. 2003 JASON-1 Sciency Working Team Meeting, Aries, France. AVISO (1998) AVISO User Handbook- Corrected Sea Surface Heights (CORSSHs), AVI-NT-011-311-CN, Ed 3.1. AVISO (2003) AVISO & PoDaac User Handbook-IGDR & GDR Jason-1 Products, SMM-MU-M5-OP-13184-CN, Ed. 2.0. Cazenave A, Schaeffer P, Berge M, Brosier C, Dominh K, Genero MC (1996) High-resolution mean sea surface computed with altimeter data of ERS 1 (geodetic mission) and TOPEX/POSEION. Geophys J Int 125: 696-704. ESA (2004) ENVISAT RA2/MWR Handbook, Issue 1.2. F6rste C, Flechtner C, Schmidt R, Meyer R, Stubenvoll R, Barthelmes F, K6nig R, Neumayer KH, Rothacher M, Reigber Ch, Biancale R, Bruinsma S, Lemoine J.-M, Raimondo JC (2005) A New High Resolution Global Gravity Field Model Derived From Combination of GRACE and CHAMP Mission and Altimetry/Gravimetry Surface Gravity Data. EGU General Assembly 2005, Vienna, Austria, April 24-29.
249
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G.S. Vergos. V. N. Grigoriadis. I. N. Tziavos • M. G. Sideris
I
I
I
I
I 0.4
0,6
,
Im
.°'3 .o2 .~, 0'o o11 o12 0'. . . . . . . Fig. 5: The final (top) QSST model and its comparison with the oceanographic model.
Hwang C, Kao EC, Parsons B (1998) Global derivation of marine gravity anomalies from Seasat, Geosat, ERS1 and TOPEX/POSEIDON altimeter data. Geophys J Int 134: 449459. Lemoine FG, Kenyon SC, Factor JK, Trimmer RG, Pavlis NK, Chinn DS, Cox C, Klosko SM, Luthcke SB, Torrence MH, Wang YM, Williamson RG, Pavlis EC, Rapp RH, Olson TR (1998) The development of the join NASA GSFC and NIMA geopotential model EGM96, NASA Technical Paper, 1998 206861. National Oceanographic and Atmospheric Administration NOAA (1997) The GEOSAT-GM Altimeter JGM-3 GDRs. Olesen AV, Tziavos IN, Forsberg R (2003) New Airborne Gravity Data Around Crete - First results from the CAATER Campaign. In: Tziavos IN (ed) Proc of the 3rd Meeting of the Gravity and Geoid Commission "Gravity and Geoid 2002", pp 4044. Rio M.-H. (2004) A Mean Dynamic Topography of the Mediter-
ranean Sea Estimated from the Combined use of Altimetry, InSitu Measurements and a General Circulation Model. Geoph Res Let Vol. 6, 03626. Tziavos IN, Sideris MG, Forsberg R (1998) Combined satellite altimetry and shipborne gravimetry data processing. Mar Geod 21: 299-317. Tziavos IN, Vergos GS, Kotzev V, Pashova L (2004) Mean Sea Level and Sea Level Variation Studies in the Black Sea and the Aegean. Presented at the Gravity Geoid and Space Missions 2004 (GGSM2004) conference, August 30 - September 3, Porto, Portugal (accepted for publication to the conference proceedings). Vergos GS, Tziavos IN, Andritsanos VD (2005) On the Determination of Marine Geoid Models by Least Squares Collocation and Spectral Methods Using Heterogeneous Data. In: Sans6 F (ed) Proc of International Association of Geodesy Symposia "A Window on the Future of Geodesy", Vol. 128. SpringerVerlag Berlin Heidelberg, pp 332-337.
Chapter 38
A new methodology to process airborne gravimetry data: advances and problems B.A. Alberts, R Ditmar and R. Klees Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, PO box 5058, 2600 GB Delft, The Netherlands.
Abstract. In the framework of the development of a new methodology for the processing of airborne gravity data, we discuss the problem of edge effects and the treatment of long-wavelength errors in the local geoid solution. The new methodology combines several pre-processing steps, such as filtering and cross-over adjustment, with the parameter estimation. The base functions used in the analytical representation of the disturbing potential are the fundamental solutions of Laplace's equation in Cartesian coordinates. This implicitly assumes periodicity in the gravity data, which does not hold in practice. The limitation to the local area introduces highly oscillating distortions in the adjusted gravity disturbances, which are mainly located along the boundary of the area. These distortions cause long-wavelength errors in the geoid over the whole area. We investigate five approaches to reduce these effects, using a simulated data set. Among these methods, least-squares prediction used to extend the data set gives the best gravity field solution in terms of gravity disturbances. However, the solution still suffers from long-wavelength geoid errors, which partially reflects the non-uniqueness of local geoid determination from airborne gravity data. Therefore, two alternative methods are investigated, which aim to solve this problem. We show that it is possible to slightly reduce the long-wavelength errors, in particular at the center of the area.
Keywords: Airborne gravimetry, regional gravity field determination, edge effects
1
Introduction
For many applications in gravity field modeling, the resolution of satellite-only models derived from current and future satellite missions will not be sufficient. The most suitable technique to determine the short-wavelength information is airborne gravimetry because it can provide gravity observations in a fast
and efficient way. In Alberts et al. (2005) a new methodology is proposed for the processing of airborne gravity measurements, aimed at the computation of gravity field functionals, e.g. geoid heights or gravity anomalies. It is based on a spectral representation of the gravity field. The gravitational potential is parameterized as a linear combination of harmonic functions, which are fundamental solutions of Laplace's equation in Cartesian coordinates. The parameters of this representation are estimated using least-squares techniques. The methodology uses a frequency-dependent data weighting strategy, similar to the one developed by Klees and Ditmar (2004) for the processing of CHAMR GRACE and GOCE data. The base functions used in the representation of the gravity potential are periodic in the horizontal directions. As a consequence, the gravity signal is assumed to be periodic as well, which does not hold in practice. Inequality at the opposite boundaries of the computation area result in strong oscillations that propagate inside the area. Furthermore, when we compute geoid heights the results get distorted by long-wavelength errors, which affect the whole computation area. An additional factor that increases these low-frequency errors is that the gravity signal outside the computation area is neglected. In the first part of this paper five methods that may be used to reduce edge effects are discussed and compared in terms of gravity disturbance and disturbing potential errors. First, the computation area may be extended at flight level, by computing gravity disturbance values using either a taper technique or leastsquares prediction. Especially the latter method reduces edge effects significantly, but may not be applicable for the processing of data contaminated by colored noise. Second, the set of base functions can be extended in such a way that the requirement of periodicity is evaded. The last part of the paper deals with the reduction of the low-frequency errors, that distort the computed geoid heights. In order to solve this problem we corn-
252
B.A. Alberts
• P. D i t m a r
• R.
Klees
pare two approaches, both of which make use of prior long-wavelength information added to the functional model.
2
Representation of the gravity field
2.1 Model description For the representation of the disturbing potential we use a linear combination of harmonic functions, that are the fundamental solutions of Laplace equation in Cartesian coordinates (Alberts et al., 2005): L
with C O S ~2~zx ~
~
1> __ 0
sin 2,~lllx 1 < 0
COS
(Y) --
2rcmy du
~
sin 2~l,~ly dv ,
Ax)TQy-l(y-
//Z >
0
fr~ <
0
(4)
ATQy-ly.
(5)
Gibbs phenomenon
The series of Eq. (1) converges to the true signal at points of continuity and to the average of the two limits at points of discontinuity. As a result, near points of discontinuity, a "ringing" known as the Gibbs phenomenon can occur (see e.g. Bracewell, 2000). Notice that different function values at the opposite borders are also considered discontinuities. The Gibbs effect is illustrated in figure 1 for a straight line.
and
.........
:..........
10 . -
Ax),
which is equivalent to solving the system of normal equations
2.2
rn=--M
(1)
~z(x) -
¢~(x) = ( y -
(ATQy-IA):k-
2U
I=-L
where y is the n x 1 observation vector containing the gravity disturbances Tz, e is the vector of residuals with zero expectation ET{e} = 0, Qy is the covariance matrix of the observations and x is the r × 1 vector of unknown coefficients. The optimal model of the gravity field can be obtained by minimizing the quadratic functional
,..........
, ..........
, ..........
, ..........
, ........
.............. ::.......... i.......... ::.......... : :
:
:
80'
90'
+
_
where dx and d v determine the area size. The relation between linear functionals of the disturbing potential can easily be computed from Eq. (1). For gravity disturbances it is Zz
--
Oz L
- Z
Z
C,~ ~,(x)g)~(y)"7,~e -'rz"~,
(2)
where a small difference between the z-direction and the radial direction is neglected, i.e. OT/Oz OT/Or. Because for I--m--O, "71~ is equal to zero, the coefficient Coo cannot be determined. It means that the constant part of the gravity disturbances Tz is not modeled. Therefore, we add an extra term ecz to Eq. (1), where ec represents the constant part of the observations. The functional model of gravity field recovery from airborne gravimetry observations may be written as a standard Gauss-Markov model D{e}-Qy,
(3)
10'
20'
30'
40'
50' x
60'
70'
1:~
- 00 M
l=--Lm=--M
y-Ax+e;
-100
I
10
20
30
40
50 x
60
70
80
diffe . . . . .
90
;0
t
100
Figure 1. The Gibbs effect caused by discontinuities at the edges in ID, illustrated for maximum degree L -- 10. In case of regional or local gravity field modeling similar discontinuities occur at the edges of the area. The results for a test with simulated gravity data generated from GPM98b (Wenzel, 1998) are shown in figures 2 and 3 for gravity disturbance errors and disturbing potential errors, respectively. For this test 10201 gravity disturbances were computed at flight level (h= 4 km) for an area of 400 x 400 km, on a grid with a spacing of 5 km. The frequency content is limited to spherical harmonic degrees 360 to 1800. In the course of data processing the coefficients Cl,~ were estimated up to L = 32, M = 32, from which
Chapter 38 • a New Methodology to Process Airborne Gravimetry Data: Advances and Problems
disturbing potential values and gravity disturbances were computed at ground level (h= 0 km). Figure 2 clearly shows the large errors at the edges and the propagation of the errors inside the area. When we compute potential errors, the figure shows that these edge effects results in long-wavelength errors that affect the whole area.
continuity is now half of the gravity value at the former edge. Therefore, zero-padding is only beneficial if the gravity values at the two edges have an opposite sign. Note that when the area is extended, dx and dy become larger and L and M have to be changed accordingly.
2. Extension using a cosine taper A different way to make the signal periodic, but without keeping the discontinuity at the edge, is to use a cosine taper. Values ~0 outside the computation area are computed as
.hL .
-
1
71-d
1
# - #~ (~ co~ - ~ + ~),
Figure 2. Gravity disturbance errors at ground level. 0.05 0.04
(6)
where ge is the original gravity disturbance value at the edge, D defines how far the area is extended and d is the distance of the point where the value is computed to the nearest point at the edge. The transition between the observed signal and the computed values may, however, not be very smooth. Oscillations can therefore still occur near the former edges.
0.03 0.02
~].:
0,
"
0.01 0.00 -0.01 -0.02
,,,-" -0.03 -0.04 -0.05
Figure 3. Geoid height errors at ground level.
3 Reduction of edge effects We investigated several approaches to reduce the edge effects. The performance of each approach is assessed using the same data set as above. For comparison we also computed a solution with leastsquares collocation as described in Alberts and Klees (2004), using the attenuated planar logarithmic covariance model of Forsberg (1987).
3.1
Methods
1. Extension by zeroes The most simple approach to reduce edge effects is to extend the computation area with zero-observations, i.e. zero-padding, which is a common procedure in discrete Fourier techniques. This way the signal becomes periodic, but the discontinuity at the (former) edge of the area remains, resulting again in high oscillations. The average of the two limits at the dis-
3. Extension using least-squares prediction The third approach to reduce edge effects is to estimate gravity disturbances outside the computation area from the observed signal using least-squares (LS) prediction. This way the transition of the observed signal to the values outside the computation area is smooth and due to the nature of LS prediction the signal gradually approaches zero for distances larger than the correlation length. However, when processing observations contaminated by colored noise, the noise will propagate into the predicted values, which may cause large errors. Furthermore, the resulting error-covariance matrix that is used in the frequency-dependent weighting scheme will no longer be Toeplitz and the numerical costs to obtain the optimal solution may become too large. 4. Modification of the base functions Alternatively to the representation of the disturbing potential in Eq. (1), the potential may also be represented as a series of cosine functions only, with a repeat interval of 2dx and 2d v along the x-axis and y-axis, respectively. Then, the representation of the disturbing potential becomes 2L
2M
7rlx
7Tmy
o -jg x /:0 m:O
(7)
253
254
B.A.
Alberts
• P. Ditmar
• R. Klees
with
~r,~ -
i~k........, ......... ~.......... ~.......... ~.......... ~.......... ~.......... ~....... t ~o
~V/(Z/dx) ~ + (.~/4) ~
Although the requirement of periodicity is now evaded, the solution will only be free of edge effects when the derivative of the signal is zero at the boundary of the computation area. Similarly, only the sine functions can be used in the expansion. Then, the signal must be zero at the boundary to prevent edge effects.
L+I
A,I--? I
I=--L-1
m=-M-1
'%
where ~ , ( z ) , ~m (Y) are the same as defined in Eq. (1), except for 1 - + ( L + 1) and m - -+-(M + 1)"
COS~~x ~(z)-
l--L+1
s i n -jT, ~ 1-- - L - 1 roy
cos ~-7, m g ~ (Y)
Try
sin -dT' m
M -M
t 1 - 1
and the expression for 7z~ becomes
i
,
2
2 ,-L
-2F/<
7l-
_~_(2m
<
L,
m < 2F/
,Z-+(L+~), -M<m<M
i
2
2 ,-L
<1
~-
i
,
2
<
L,
+ ( ~ + 1)
2 ,Z-+(L+~), ~ - + ( : U + I )
The same test as shown in figure l, the estimation of straight line, was also done using the additional base functions. The result in figure 4 clearly shows a large improvement (the errors are smaller by a factor 1000), which means that the additional base functions are well suitable for signal representation.
::..........
::..........
::.........
i .........
gg~pu,ed II i ........
io
,'0
~'o
io
a'o
~'o
;o
~'o
~'o
,;o
x
0.01,
1 , o.e.....~] o oo~ ........ ~......... ~......... ~......... ,.......... ,.......... ,.......... ,......... , ......... !..... ~1 o ..
.::..
..::....
i....
i
I'0
20'
30'
40'
...:
":
"
:
'
" ........
t
-0.005
1
0
50' x
60'
70'
80'
90'
;0
Figure 4. The estimation of a straight line using the additional base functions in 1D, illustrated for maximum degree L -- 10.
3.2
(8)
::..........
_
~ 0 ~0
5. Additional base functions The last approach we investigate in this study is to include additional base functions that are not periodic on the computation domain:
..i .........
Results
The error statistics of the results obtained with noisefree simulated data sets are given in tables 1 and 2 for the computation of gravity disturbances and geoid heights at ground level, respectively. The first row gives the statistics for the results of the inversion using the original base functions, as was shown in figures 2 and 3. In this case no reduction method was used. The second row gives the error statistics of the LS collocation and shows the accuracy that may be expected for this data set. For the first three methods the area was extended by 50 k m on all sides, resulting in an area size dz, dy of 500 km. To obtain the same resolution, the coefficients Cl,~ were estimated up to L = 40, M = 40. The statistics are given for the original area at ground level. Table 1 shows that with zeropadding and with the cosine taper, the errors can even be larger than without a reduction method applied. This is because there are still discontinuities at the former edges. Nevertheless, the resulting RMS geoid height errors are somewhat smaller, indicating that the long-wavelength errors do not distort the computation area as much as when no reduction method is used. A m o n g the methods that use an extension of the area, the best results were obtained when using LS-prediction. The solution is almost completely free of edge effects, due to the smooth transition at the (former) edge of the computation area. The correlation length for this data set was about 20 km, resulting in predicted values that are nearly zero at the edge of the extended area. As such, the signal is periodic and without discontinuities. We believe that the area should at least be extended by the correlation length, to obtain a solution without oscillations.
Chapter 38
• a New
The other two methods, cosine functions and additional base functions, show an improvement in terms of gravity disturbance errors, but the results are worse for the computation of the disturbing potential at ground level. For the additional base functions the disturbing potential gets completely distorted by very large long-wavelength errors, which is shown in the first plot of figure 6. However, figure 4 shows that these errors are not caused by the non-periodicity of the signal. In the next section two approaches will be discussed to reduce these long-wavelength errors.
Table 1. Statistics of gravity disturbance errors [mGal] at ground level. Method Inversion LSC 1. Zero-padding 2. Cosine taper 3. LS prediction 4. Cos functions 5. Add. functions
Min -10.24 -1.88 -17.03 -6.85 - 1.23 -6.68 -4.45
Max Mean 1 5 . 9 4 0.078 1.69 0.005 21.63 0.015 1 8 . 6 3 0.033 1.95 0.006 7.73 0.055 2.91 -0.085
RMS
1.12 0.15 1.34 0.88 0.11 0.57 0.32
Table 2. Statistics of geoid height errors [m] at ground level. Method Inversion LSC 1. Zero-padding 2. Cosine taper 3. LS prediction 4. Cos functions 5. Add. functions
4
Min -0.119 -0.072 -0.130 -0.066 -0.085 -0.103 -2.218
Max 0.115 0.040 0.158 0.092 0.036 0.074 2.240
Mean 0.003 -0.005 0.003 0.003 0.003 0.003 0.390
RMS
0.017 0.015 0.014 0.013 0.011 0.024 0.600
Combination with prior information
Long-wavelength errors are mainly caused by the propagation of edge effects, the absence of observations outside the computation area and imperfections in the reference GPM when using the remove-restore technique. Furthermore, they may be the result of the intrinsic non-uniqueness of the inversion problem, that is, we try to solve a boundary value problem without having data at all the boundaries. To remove long-wavelength effects, we discuss two approaches, both of which make use of prior information added to the functional model. This prior information is that the disturbing potential at low spatial frequencies is zero.
Methodology to ProcessAirborne Gravimetry Data: Advancesand Problems
4.1
Methods
Addition of pseudo observations To obtain a solution without low-frequency errors, we add pseudo-observations T = 0 to the functional model. The model of Eq. (3) then becomes Y Yp
EA B
Xp-}-
e ep
e
0
[opl
(9)
where yp are the pseudo observations T = 0, ep is the vector of residuals with expectation E { e p } = 0, and B is the design matrix which describes the relation between the disturbing potential and the coefficients (Eq. (1)). The entries of the covariance matrix Qp are computed using the covariance function N1
Z
n=2
N2 n=N1 +1
where f is the spherical distance, c~ are the error degree variances of the geopotential model that was subtracted from the gravity signal up to degree N1 and o-~ are signal degree variances which may be computed using Kaula's rule. The optimal solution is found by solving the following system of normal equations: (ATQy-IA+BTQp-IB)fcp--ATQy-ly.
(11)
Addition of fixed constraints A different approach to reduce long-wavelength errors is to add integral constraints to the functional model. Because we only want to compute the highfrequency part of the disturbing potential, we add the constraints
/ f T(P)Yz,~,(P) d~ - O,
(12)
where Yz,,~,(P) are the spherical harmonics up to the minimum degree (or wavelength) we want to solve for and the integration domain f~ is the computation area. In discretized form this becomes N
E T(Pi)Yz,,~,(Pi) - 0.
(13)
i=1
Inserting the representation of the disturbing potential T in Cartesian coordinates, given by Eq. (1), this
255
256
B.h. hlberts • R Ditmar. R. Klees
results in
found by minimizing the Lagrange function N
• (x, .,k) -- ( y - A x ) T Q y - l ( y - A x ) - 2 . , k ( c - K x ) ,
Z c~ ~ ~(x~)~(~)~ -~z, 1,m
(16)
i=1
x Yz',~'(Pi)
-- O.
(14)
yielding the normal equations
These constraints can be added to the functional model of Eq. (3). The constrained Gauss-Markov model, with fixed constraints is given as (see e.g. Koch, 1999)
Ey7 c
K
xf+
Eo] 0
;
D{e}-Qy,
[NKT][,,] [ATQy'yK 0 c
The constrained parameter vector is obtained by eliminating the Lagrange multiplier ~ from Eq. (17), resulting in
(15)
where c is the vector of zero-observations related to the constraint equations, and K is the design matrix relating the coefficients to be estimated to the constraint observations. The optimal solution 2f is
0
300 - - ~ /
100
~
200
300
~ ~ , ~ ' ~ J ~
400
J { - 300
0
100
300 ~
~
(17)
2f -- 2 + N - 1 K T ( K N - 1 K T ) - I ( c -
K2)
(18)
The solution can thus be computed as an update to the unconstrained solution 2.
200
300
400
0
~l- 300
100
200
300
400
300 -
300
100 -
100
_oo 100
"~--~~ - 100 S
o~ ~ , ~ f o -0.10-0.08-0.06-0.04-0.02
- 100
o , ~ ~
0.00 0.02 0.04 0.06 0.08 0 . 1 0
-0.10-0.08-0.06-0.04-0.02
o o~,_
0.00 0.02 0.04 0.06 0.08 0 . 1 0
,~~~~!
-0.10-0.08-0.06-0.04-0.02
o
0.00 0.02 0.04 0.06 0.08 0.10
Figure 5. Geoid height errors for the computations with prior information using the original set of base functions (Eq. 1). From left to right: no addition of prior information, addition of pseudo observations, addition of a fixed constraint.
0
100
200
300
400
400
0
1O0
300
100
0
200
400/p'~\-~~~,~,~
300
~~
400
,oo
0.5
1.0
1.5 2.0 2.5
1O0 -
4
200 0
300
~
Q
100
-0.10-0.08-0.06-0.04-0.02
0 0.00 0.02 0.04 0.06 0.08 0.10
400
0
1oo
300
0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0
0 400- ! ~
°II IfUI °i -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5
1.0
1.5 2.0 2.5
Figure 6. Geoid height errors for the computations with prior information using the additional base functions (Eq. 8). From left to right: no addition of prior information, addition of pseudo observations, addition of a fixed constraint.
Chapter 38 • a New Methodology to Process Airborne Gravimetry Data: Advances and Problems
4.2
Numerical experiment
show a much smaller improvement. Again the addition of more constraints may improve the solution.
Both approaches were tested for the original set of base functions (Eq. 1) and for the additional base functions (Eq. 8). Especially for the latter approach the long-wavelength errors were very large. The resuits are shown in figures 5 and 6. For the test with pseudo observations, the entries of the covariance matrix Qp (Eq. 10) were computed using Kaula's rule. Because the frequency content of the data is limited to the interval from spherical harmonic degree N~ = 361 up to degree N2 = 1800, the error degree variances c,~ are set to zero. For the test with fixed constraints we added the constraint
fI
T(P)
df~ -
O,
(19)
at z = 0. This means that we exclude the constant part of the potential at z = 0, instead of removing a bias from the computed disturbing potential at ground level h = 0.
Table 3. Statistics of geoid height errors [m] at ground level for the original set of base functions. Method no prior information pseudo observations fixed constraint
Min -0.119 -0.117 -0.107
Max 0.115 0.112 0.124
Mean 0.003 0.003 0.003
RMS 0.017 0.016 0.015
Table 4. Statistics of geoid height errors [m] at ground level for the additional base functions. Method no prior information pseudo observations fixed constraint
Min -2.218 -0.208 -2.326
Max 2.240 0.212 2.117
Mean 0.390 0.002 0.004
RMS 0.600 0.030 0.514
From figure 5 and table 3 it can be seen that both approaches slightly improve the solution for the original set of base functions. The improvement can especially be seen in the middle of the computation area. However, long-wavelength errors are still visible, which means that more constraints should be added. For the additional base functions (figure 6 and table 4) the best results are obtained when we add pseudo observations. Because we used Kaula's rule for the computation of the covariance function, the pseudo observations may get too much weight resulting in a solution, which is too smooth. Instead, a more realistic power spectrum could be used. The results for the computation with one fixed constraint
5
Summary and conclusions
In a previous study on the development of a new methodology for the processing of airborne gravimetry data we faced the problem of strong oscillations near the edge of the computation area. Furthermore, when computing disturbing potential values at ground level, the results were distorted by lowfrequency errors. Several approaches to reduce edge effects were investigated. The results from a numerical study with simulated high-frequency data showed that edge effects can be reduced by extending the gravity disturbance signal at flight level outside the computation area. This way the signal becomes periodic. The best results were obtained when LS-prediction is used for the computation of gravity disturbances outside the computation domain. As an alternative, non-periodic base functions can be used to reduce edge effects, although the results for the computation of the disturbing potential computed at ground level did not improve. For the reduction of long-wavelength errors we compared two approaches, both of which make use of prior information to the functional model. Numerical tests with both methods showed that the solution can be improved, but more constraints may be needed. In this research we only considered simulations with noise-free data. To assess the flexibility of the methods discussed in the paper, more tests with data corrupted by white noise and colored noise should be performed. Further research will focus on these topics as well as the processing of real airborne gravity data. Besides the processing of data contaminated by colored noise, other essential processing steps such as the computation of terrain corrections will then play an important role.
References Alberts BA, Klees R (2004) A comparison of methods for the inversion of airborne gravity data. J Geodesy 78: 55-65 Alberts BA, Klees R, Ditmar P (2005) A new strategy for processing airborne gravity data. In: Jekeli C, Bastos L, Fernandes J (Eds.) Proc. IAG International Symposium on Gravity, Geoid and Space Missions 2004 (GGSM2004), Porto, Portugal Bracewell RN (2000) The Fourier transform and its applications. 3rd ed., McGraw-Hill, Boston, USA
257
258
B.A. Alberts • P. Ditmar • R. Klees
Forsberg, R (1987), A new covariance model for inertial gravimetry and gradiometry. Journal of Geophysical Research, Vol. 92, No. B2, pp. 1305-1310 Klees R, Ditmar P (2004), How to handle colored noise in large least squares problems in the presence of data gaps? Proc. V Hotine-Marussi Symposium on Mathematical Geodesy, Matera, Italy Koch KR (1999), Parameter estimation and hypothesis
testing in linear models. 2nd ed., Springer, Berlin, Germany Wenzel G (1998) Ultra high degree geopotential models GPM98A, B and C to degree 1800. Proc Joint Meeting International Gravity Commission and International Geoid Commission, 7-12 September 1998, Trieste, Italy
Chapter 39
Ideal Combination of Deflection Components and Gravity Anomalies for Precise Geoid
Computation Norbert Kiihtreiber Institute of Navigation and Satellite Geodesy, TU-Graz, Steyrergasse 30, A-8010 Graz, Austria Hussein A. Abd-Elmotaal Civil Engineering Department, Faculty of Engineering, Minia University, Minia 61111, Egypt Abstract. Recently, a high-precision and economic system for the real-time measurement of astronomical deflections of the vertical has been available (Miiller et al., 2005). Based on the use of a modern digital zenith camera a measurement precision in the order of 0.1 - 0.3 arcseconds can be achieved. Using this efficient and precise technique, a huge amount of deflection measurements can be done in mountainous and desert areas, which usually suffer from a lack of gravity field information (especially gravity anomalies). The combination of the deflections of the vertical with existing gravity networks in order to compute a gravimetric geoid needs two steps. First of all gravity anomalies have to be computed from the deflections of the vertical measured. Second of all an integration of the computed gravity anomalies and the available gravity networks has to be done. This paper analyzes the gravity anomalies computed from the deflection components and the resulting gravimetric geoid solution. The case study is performed for an area in the Austrian Alps where two sets of gravity field data (716 deflections of the vertical and 5796 gravity anomalies) are available. Keywords. zenith camera, gravimetric geoid, deflections of the vertical, data combination
1
Introduction
As shown by Ktihtreiber (2003), precise geoid determination should preferably be done by combining all available signals of the earth's gravity field. In the ideal situation (like Austria) a homogenous data set of gravity anomalies and deflection components exists. Additional informations used for gravity field modelling are a global geopotential earth model (e.g. EGM96) and a detailed elevation model. Another
reason that Austria may be considered an excellent test area for gravity field investigations is its varying topography (high mountains in the Central Alps and lowlands in the Vienna Basin). In many cases homogeneous gravity field data sets are not available. It is more likely that large areas with no data exist. In order to fill these data gaps (e.g., in the gravity anomaly data), an economical and fast method, like the use of an automated zenith camera (MOiler et al., 2005) is needed. With a zenith camera deflections of the vertical can be observed efficiently with a reasonable good accuracy of 0.1 - 0.3 arcseconds. Still the challenge of the data combination for the determination of the geoid remains. This article is an attempt to answer the critical question, which data technique is the best for combining gravity anomalies and deflections of the vertical to compute a gravimetric geoid.
2 2.1
Data Gravity data
The gravity data base at the Institute of Navigation and Satellite Geodesy in Graz was established by Kraiger and Kfihtreiber. The database includes Austrian data and data from the neighboring countries. All gravity measurements are stored in a uniform system for position, height and gravity. Regarding position, the local Austrian coordinate system based on the Bessel ellipsoid was chosen. The height system used is the Austrian orthometric height system based on the tide gauge of Triest. All gravity measurements refer to the IGSN71 system. For the following computations only a subset of the aforementioned gravity measurements is used and transformed to WGS84. Figure 1 shows the se-
260
N. Ktihtreiber• H. A. Abd-Elmotaal
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Two different digital elevation models are available for this investigation: a coarse model with a resolution of 90" × 150" in the latitude and the longitude directions and a fine model with a resolution of 11.25" x 18.75" in the latitude and the longitude directions. The fine DEM covers the window: 44•75°N _< q~ _< 50.25°N and 7•75°E _< )v _< 19.25°E. The coarse model covers the window: 40°N _< ~ _< 5 2 ° N and 5°E _< )v _< 22 ° E. Details on the used elevation models are given in Graf (1996). 2.4
lected 5796 gravity data points which are uniformly distributed. Gravity measurements were selected with a mean distance of 6 km in the area: 46.20 ° N < q~ _< 49.21°N and 9.25°E _< )~ _< 17•25 °E. Additional data with a mean distance of 12 km were added around this inner rectangle up to the maximum limits of 45.7 ° N _< (~ _< 49.7 ° N and 8.5 ° E _< )v _< 18.2 ° E.
2.2
GPS benchmarks
Figure 3 shows the distribution of the available GPS benchmarks with known orthometric heights in Austria. It shows that the number of stations in the eastern part of Austria is bigger, but more or less, a good distribution of the GPS benchmarks in Austria can be stated•
D e f l e c t i o n s of the v e r t i c a l •
For Austria a dataset of 716 homogeneously distributed deflections of the vertical exists. For the following investigation 169 deflections of the vertical were selected for the window: 46.9°N _< ~) _< 48.6°N and 13.9°E < )~ < 15.6°E. The deflections
0
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13
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17
Figure 2. Selected deflections of the vertical.
of the vertical (see Fig. 2) basically refer to the Bessel ellipsoid and to the datum of the Military Geographic Institute. They were transformed to the WGS84 system.
The geoid computation was based on the removerestore technique. This implies that the effect of the global geopotential model and the effect of the topography and its compensation was removed from the gravity anomalies used for computing the geoid. Mathematically this can be expressed as follows: Ag -
Agobs -- Agh -- AgGM ,
(1)
Chapter
39
• Ideal
Combination
of
Table 1. Gravity reduction using the standard density value of 2.67 g/cm ~ and the adapted geopotential model EGM96. Statistics are based on 5796 points• mgal
Agobs Agobs -- Aga~
Ag
min --154.1 --204.3 --72.0
max 187.2 224.0 85.4
mean 9.8 -- 1.1 0.6
std.dev. -t-42.2 -+-47.6 -t-23.6
where Agobs stands for the observed free-air anomalies, Agh is the effect of the topography and its compensation on the gravity anomalies, and Ag~M is the effect of the global geopotential earth model on the gravity anomalies. Thus the computed geoid height N can be expressed by: N -- NGM + NAg + Nh,
(2)
where NGM gives the contribution of the global geopotential earth model, Nag gives the contribution of the reduced gravity anomalies, and Nh gives the contribution of the topography and its compensation (the indirect effect). The adapted EGM96 (Abd-Elmotaal and Kfihtreiber, 2001) was used to compute the contribution of the global geopotential earth model. The contribution of the topography and its compensation was computed using the TC-program (Forsberg, 1984). Table 1 shows the statistics for the reduction of the gravity anomalies.
Deflection Components and Gravity Anomalies for Precise Geoid Computation
3.3
Gravimetric geoid solution
The dataset of gravity anomalies is homogeneous and uniform and serves as a basis for computing a gravimetric geoid which is used as reference for comparisons in this study. The computation was done using the 1D-FFT technique based on a 3 / × 3 / grid of gravity anomalies created by the Kriging interpolation technique.
Combining gravity anomalies and deflections of the vertical 4.1
Test scenario
In order to study the combination of gravity anomalies and deflections of the vertical the following test scenario was created. First of all a gravity anomaly dataset with an artificial rectangular gap with the limits 47.05°N _< $ _< 48.45° N and 14.05°E _< ~ _< 15.45°E was generated. This gravity dataset was combined with a
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where y is the normal gravity, R is the mean earth radius and S(~) stands for the Stokes function• Equation (3) can be rewritten as convolution integral and thus be evaluated by FFT technique (Strang van Hees, 1990, p. 236). The computation was done using the 1D-FFT technique introduced by Haagmans et al. (1993) with the help of the program FFTGEOID by Li and Sideris (Sideris and Li, 1993).
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The contribution of the reduced gravity anomalies to the geoid NAg(in the following only called N) can be calculated from the gridded gravity anomalies by using the well known Stokes' integral (Heiskanen and Moritz, 1967, p. 94)
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Figure 4. Combined data set. Gravity anomalies shown as dots, deflections of the vertical shown as triangles.
dataset of 169 deflections of the vertical shown in Figure 2. The deflections were chosen for a rectangle of 46.9° N _< ¢ _< 48.6°N and 13.9° E _< )~ _< 15.6° E which is a little bit bigger than the gap in the gravity anomalies dataset in order to avoid edge effects• Figure 4 shows the combined dataset. Using this combined dataset a prediction of geoid heights (31 x 31 grid) by collocation was performed• The following subsection gives details on the used covariance function model.
261
262
N. K~ihtreiber • H. A. Abd-Elmotaal
4.2
Covariance function
The well-known Tscherning-Rapp covariance function model was used for all following LSC applications. The global covariance function of the gravity anomalies Cg(P, Q) given by Tscherning and Rapp (1974, p. 29) is written as 10 oo
n -- 1
Cg (P, Q) - a ~_~ (n - 2)(n + B) n=3
sn+ZPn(cOs~)
where Pn (cos ~) denotes the Legendre polynomial of degree n, ~ is the spherical distance between P and Q and A, B and s are the model parameters. A closed expression for (4) is available in (ibid., p. 45). The local covariance function of gravity anomalies C(P, Q) given by Tscherning-Rapp can be defined as oo
n -- 1
sn+2p
C(P, Q) - a ~_~ (n - 2)(n + B)
n
(cos~/). (5)
NN+ I
Modelling the covariance function in practice means fitting the empirically determined covariance function (through its three essential parameters, the variance Co, the correlation length ~ and the variance of the horizontal gradient Go) to the covariance function model. Hence the four parameters A, B, NN and s are to be determined through this fitting procedure. A simple fitting of the empirical covariance function was done using COVAXN-Subroutine (Tscherning, 1976). With a fixed value B = 24, the following Tscherning-Rapp covariance function model parameters were determined on the basis of the gravity anomalies data set: s = 0.997065, A = 746.002 mgal 2 and NN = 76. The parameters were used for the following LSC solutions.
4.3
12
14
16
(4)
Combined geoid solution
Figure 5 shows the difference between the combined geoid (combination of gravity anomalies and deflections of the vertical) and geoid heights derived from the GPS/levelling benchmarks. A high-order polynomial structure of the difference is apparent. After scaling the combined geoid solution by a fourth order polynomial function a comparison with the gravimetric geoid solution (reference geoid computed in Section 3.3) can be done. The result is shown in Figure 6.
Figure 5. Difference of combined geoid solution and the GPS/levelling points. The difference is given in m. The contour interval is 0.1 m
Big differences near the border of Austria are apparent. The reason for this circumstance is, that the gravimetric geoid solution (computed by 1D-FFT) and the combined geoid solution (computed by LSC) are scaled by different fourth order polynomial functions and only GPS/levelling benchmarks inside Austria are used for scaling. Therefore the differences increase at the border and outside. In addition differ-
/
48.5-
46.510
12
4
16
Figure 6. Difference of the scaled gravimetric geoid solution and the scaled astrogravimetric geoid solution. The differences are given in cm.
ences exist in the area of the gap in the gravity data. Most of the differences are less than 4 cm. Only in few parts the error is bigger. No edge effect is visible. In order to check whether a correlation of the differences with the topography exists, the differences are plotted together with the elevation model. Figure (7) reveals that in some parts a correlation with the topography exists.
Chapter 39 • Ideal Combination of Deflection Components and Gravity Anomalies for Precise Geoid Computation
ii1
° 0
tion 2.1. The gravity anomalies range from - 4 2 . 3 to 67.5 mgal with a standard deviation of 23.0 mgal
Height
Using LSC, gravity anomalies may also be predicted from deflections of the vertical. The difficulty within this procedure is that deflections of the vertical mainly contain short wavelengths while gravity anomalies mainly contain medium wavelengths. Thus the gravity anomalies predicted from deflections of the vertical will miss medium wavelengths. This makes a combination of predicted and measured gravity anomalies difficult.
13 2000
15oo
II
000
-5oo m
l_~ %.0
"~1~'5"
,~.o -
1;. . . . . .
Figure 7. Checking the correlation between the differences in the geoid solution and the topography
Prediction of gravity anomalies from deflections of the vertical Another possible way of combining the above mentioned datasets of deflections of the vertical and gravity anomalies is to introduce an intermediate step. The gap in the gravity anomaly data may be filled by gravity anomalies predicted from deflections of the vertical. Figure 8 shows isostatic anomalies for
The dilemma can be solved if deflections of the vertical (in areas with no gravity data) are combined with gravity anomalies of the surrounding areas. This means in terms of our test scenario that gravity anomalies for the "gap" region are predicted with the data set described in Section 4.1. The result of this prediction is shown on the right hand side of Figure 8. Comparing the two plots in Figure 8 reveals that the main features of the gravity anomalies are identical. Also the statistics for the 35 x 35 interpolated grid values are nearly equal showing a range from - 4 4 . 2 to 67.2 mgal with a standard deviation of 23.4 mgal for the gravity anomalies predicted from deflections of the vertical. The quality of the interpolation of gravity anomalies from deflections of the vertical becomes evident, if the difference to the measured gravity anomalies is computed (see Figure 9). Nearly all differences are less than 6 mgal and even more, 80% of the differences are less than 3 mgal. Again a correlation of the differences with topography can be seen for some parts.
-40
-20
0
20
40
60
mgal
Figure 8. Gravity anomalies. Left: gravity anomalies predicted from measured gravity anomalies. Right: gravity anomalies predicted from deflections of the vertical. The values are given in mgal and the contour interval is 10 mgal.
the area of the "gap" computed by different methods. The left plot of Figure 8 is based on 35 × 35 grid values which were interpolated by the Kriging technique using the full data set described in sec-
Geoid solution by combining measured gravity anomalies and predicted gravity anomalies A grid of gravity anomalies was computed by Kriging interpolation technique using measured and predicted gravity anomalies. The geoid was computed using 1D-FFT technique. The comparison of this geoid with the geoid given by GPS/levelling points shows that the difference has a similar high order polynomial structure as the difference shown in Figure 5. The scaling of the geoid solution to the GPS/levelling points was done using a fourth order polynomial. Finally the scaled astrogravimetric
263
264
N. K~ihtreiber. H. A. Abd-Elmotaal
48.4-
48.2-
I
i
48-
47.8-
47.6-
47.4-
47.2-
. . . .
__.~1
Figure 9. Difference between measured gravity anomalies and gravity anomalies predicted from deflections of the vertical. The values are given in mgal and the contour interval is 3 mgal.
the computation of the geoid can be done in several successful ways. One possibility is the combination of the gravity anomalies and deflections of the vertical using least squares collocation. For large areas the computation of a gravimetric geoid by the FFT technique is more efficient. In that case gravity anomalies inside the gap can be predicted from deflections of the vertical. Both procedures give reasonable good results. The differences of the geoid heights compared to a reference solution are of the same order for both cases. They are small and exceed 4 cm only in few parts. The differences to the reference geoid are smoother for the case of using predicted gravity anomalies.
References geoid was compared to the reference geoid (gravimetric geoid computed in section 3.3). The comparison is shown in Figure 10. Nearly all the differences are less than 4 cm. The result is even better than that of the first combination method shown in Figure 5.
48.5-
x.....w~Y
46.510
12
-8
4
-4
4
16
8
Abd-Elmotaal, H. and Ktihtreiber, N. (2001) Precise Geoid Computation Employing Adapted Reference Field, Seismic Moho Information and Variable Density Anomaly, Presented at the Scientific Assembly of the International Association of Geodesy IAG 2001, Budapest, Hungary, September 2-8, 2001. Forsberg, R. (1984) A Study of Terrain Reductions, Density Anomalies and Geophysical Inversion Methods in Gravity Field Modelling, Department of Geodetic Science, Ohio State University, Columbus, Ohio, 355.
cm
Figure 10. Difference between the scaled gravimetric geoid solution based on measured gravity anomalies and the scaled gravimetric geoid solution based on measured gravity anomalies combined with gravity anomalies predicted from deflections of the vertical. The differences are given in cm.
7
Abd-Elmotaal, H. (1998) An Alternative Capable Technique for the Evaluation of Geopotential from Spherical Harmonic Expansions, Bollettino di Geodesia e Scienze Affini, 57, 25-38.
Conclusion
Incomplete gravity field information given by gravity anomaly networks with gaps can be complemented using deflections of the vertical for this gaps. The combination of the two data sets which is needed for
Graf, J. (1996) Das digitale Gel/~ndemodell ffir Geoidberechnungen und Schwerereduktionen in Osterreich, Proceedings of the 7th International Meeting on Alpine Gravimetry, Vienna, Osterreichische Beitr~ge zu Meteorologie und Geophysik, 14, 121-136. Haagmans, R., de Min, E. and van Gelderen, M. (1993) Fast Evaluation of Convolution Integrals on the Sphere Using 1D FFT, and a Comparison with Existing Methods for Stokes' Integral, Manuscripta Geodaetica, 18, 227-241. Heiskanen, W.A. and Moritz, H. (1967) Physical Geodesy, Freeman, San Francisco. Ktihtreiber, N. (2003) High Precision Geoid Determination of Austria Using Heterogeneous Data,
Chapter 39 • Ideal Combination of Deflection Components and Gravity Anomalies for Precise Geoid Computation
Proceedings of the 3rd Meeting of the International Gravity and Geoid Commission, Gravity and Geoid 2002, 144-149. Mtiller, A. Btirki, B. and Kahle, H.-G (2005), First Results from new High-precision Measurements of Deflections of the Vertical in Switzerland, International Association of Geodesy Symposia, 129 Gravity, Geoid and Space Missions, 143-148. Rapp, R.H. (1982) A Fortran Program for the Computation of Gravirnetric Quantities From High Degree Spherical Harmonic Expansions, Department of Geodetic Science, Ohio State University, Columbus, Ohio, 334. Sideris, M.G. and Li, Y.C. (1993) Gravity Field Convolutions Without Windowing and Edge-Effects, Bulletin Geodesique, 67, 107-118. Strang van Hees, G. (1990) Stokes Formula Using Fast Fourier Techniques, Manuscripta Geodaetica, 15, 235-239.
Tscherning, C.C. and Rapp, R.H. (1974) Closed Covariance Expressions for Gravity Anomalies, Geoid Undulations, and Deflections of the Vertical Implied by Anomaly Degree Variance Models, Department of Geodetic Science, The Ohio State University, Columbus, Ohio, 208. Tscherning, C.C. (1976) Implementation of AlgolProcedures for Covariance Computation on the RC 4000-Computer, The Danish Geodetic Institute, 12. Tscherning, C.C., Rapp, R.H. and Goad, C. (1983) A Comparison of Methods for Computing Gravimetric Quantities From High Degree Spherical Harmonic Expansions, Manuscripta geodaetica, 8, 249-272. Tscherning, C.C., Forsberg, R. and Knudsen, P. (1992) The GRAVSOFT Package for Geoid Determination, Presented at the 1st Continental Workshop on the European Geoid, Prague, May, 1992.
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Chapter 40
SRTM evaluation in Brazil and Argentina with emphasis on the Amazon region D. Blitzkow, A.C.O.C. de Matos, J.P. Cintra Escola Polit6cnica da Universidade de Silo Paulo, EPUSP-PTR, Code Postal 61548, CEP:05424-970, Silo Paulo, Silo Paulo, Brazil E-mail: [email protected], [email protected], [email protected]; FAX: 55 11 30915716
Abstract. An extensive process of validation of the
data derived from SRTM (Shuttle Recovery Topography Mission) was carried out in different parts of South America using heights available from several sources (Matos, 2005). This paper addresses the attention to Argentina, Brazil and Amazon area. Over Brazil a total of 62,030 BMs (Bench Marks) and 13,723 over Argentina, derived from spirit leveling, were used for a direct comparison. SRTM height was estimated in the BMs from a 3" grid using a bilinear interpolation. In 85% of the BMs in Brazil as well as in Argentina the differences remained in the interval of 0 to 20m, consistent with the absolute error of SRTM that is 16 m. In a total of 2,525 and 1,050 points in the two countries respectively, the differences were greater than 50m. In the Amazon area SRTM was compared with 7 DTM global models. The analyses have been carried out in blocks of 1° x 1°. The attention was addressed to areas with different characteristics like: basin, rivers, umbriferous forest and anthropic areas. The results show different features in the mentioned areas. Several profiles have been drawn in blocks of 1° x 1° passing through the highest point of the block where the features can be seen. Keywords: SRTM, Radar altimetry, height, DTM
1. Global Models in South America Amazon area is a challenge for many scientists interested in discovering the very many unknowns and secrets concealed by the strong and immense forest. In order to address the attention of the DTM to this area the radar mission SRTM and 7 other DTM global models were analysed: ETOPO5, TERRAINBASE, JGP95E, GTOPO30, GLOBE, ETOPO2, and DTM2002. ETOPO5 is a model constructed by National Oceanic and Atmospheric Administration/National Geophysical Data Centre (NOAA, 1988). It is derived from terrestrial and
oceanic data in a grid of 5'. In South America the model used data digitised from maps in the scale of 1:1,000,000. T E R R A I N B A S E , developed by NOAA/NDC too, is a model in a grid of 5' and the difference with respect to the previous one is the quantity and the quality of the data used. In Brazil, in the Cerrado area (Savana), it used data provided by International Centre for Tropical Agriculture and a 3' DTM in Chile and Peru, acquired from Cornell University (NOAA, 1995). JGP95E was intended to solve the problem with the requirement for a 5' DTM when the EGM96 was computed (Lemoine et al. 1998a; Lemoine et al. 1998b; Arabelos, 2000). GTOPO30 (Global Topographic Data), constructed by EROS Data Centre (EDC), involves the terrestrial part of the Earth and it is a grid of 30". The minimum and maximum elevations are: 407m and 8,752m respectively. The grid was created using the Australian National University Digital Elevation Modeling (ANUDEM) developed by Hutchinson (1989). The data source were DTED (Digital Terrain Elevation Data), the Digital Chart of the World (DCW), maps printed by AMS (Army Map Service), see U.S. Geological Survey (1997). G L O B E (Global Land One-kilometer Base Elevation), version 1.0, c o n s t r u c t e d by NOAA/NGDC, has a grid space of 30" with default value o f - 5 0 0 for the oceans. The data source for South America were DCW, AMS maps in the scale of 1:1,000,000 digitised by Geographical Survey Institute (GSI), IBGE maps in Amazon in the scale 1:1,000,000, maps of the Defence Ministry in Peru in the scale of 1:1,000,000 and several DTEDs of different versions (Hasting and Dunbar, 1999). ETOPO2 has a grid space of 2' and was constructed by NOAA (2001). The continental part of this model used GLOBE regridded for 2' using bicubic interpolation. DTM2002 is a model with a resolution of 30", constructed by Raytheon ITSS Corporation. It combines data from GLOBE and from ACE (Altimeter Corrected Elevation), see Saleh and Pavlis (2002). ACE, from Earth and
Chapter 40 • SRTM Evaluation in Brazil and Argentina with Emphasis on the Amazon Region
Planetary Remote Sensing Laboratory, University of Montfort, UK, is a global model derived from altimetry data (Johnson et al., 2001). These models have the grid position, latitude and longitude, referred to World Geodetic System 84 (WGS84). The grid values represent height in meters above Mean Sea Level. Finally, SRTM, the most important and updated height information in the world, is a joint project undertaken by NASA (National Aeronautic and Space Administration), DLR (Deutschen Zentrum for Luft und Raumfahrt) and ASI (Agenzia Spatiale Italiana) carried out with a special radar system on board of Endeavour, placed at orbit on February 1 l th, 2000 and returned to the Earth on the 22nd. Data have been collected on a global basis, from 60 ° N to 56 ° S. The result is a DTE (Digital Terrain Elevation) with a resolution of 3". It represents the best global terrain elevation model now available, with accuracies in different regions of the world subjected to evaluation, but in general the prescribed error is of + 16m. It is expected that the error in the position is on the order of 15m. The heights are in meters referred to the geoid implied by EGM96/WGS84 and the grid is referred to WGS84 ellipsoid (Lemoine et al. 1998a; Hensley et al., 2001; JPL, 2004).
2. Evaluation of global models using BMs The classical levelling network in Brazil and Argentina was used for the validation of the global models, including SRTM. The coordinates of the BMs were derived from maps on 1:50,000 and 1:100,000 scales, so that the accuracy of the position is compatible with the maps. Nowadays the GPS is being used for that purpose. The coordinates of the BMs may have inconsistencies with respect to SRTM grid because they have been derived from topographic maps in the scales of 1:50,000 and 1:100,000, very often with the impossibility to identify the correct position of the BM. An error of 50m in horizontal position can be expected. The precision of the terrestrial leveling network involved is about 10 cm, a worthless value in comparison with the other errors. On the other hand, a method of interpolation can introduce an error of a meter.
In order to estimate the height of the BM from the grid, bilinear interpolation was used for SRTM and bicubic for the others. The SRTM present many points with no information and by this reason, the bilinear interpolation calculates the height of the BM in more points than the bicubic (Matos, 2005), a method that require a complete neighborhood. The horizontal coordinates of the BM in Brazil were originally in SAD69 and they were transformed to WGS84; in Argentina the coordinates of the BM are truncated to minutes and the Campo Inchauspe reference was maintained. There are available 62,030 BMs in Brazil and 13,723 in Argentina (Figures l a and 2a). The comparison of the height was restricted to intervals of 10m up to 50m. The reason for the value of 10m is due to the error of S R T M , - 1 6 m , with 90% of confidence level, which means 1.5 c~, so that G 10m. On the other hand, 50m is the accepted error for the maps used. Tables 1 and 2 show that 85% of the interpolated values from SRTM, in Brazil and in Argentina, are consistent with the BMs heights. Figures lb and 2b present the distribution of the 2,525 BMs (Brazil) and 1,050 BMs (Argentina) with differences greater than 50m with respect to SRTM. In the Argentina case it is visible the concentration of the values in the Andes, where the strong irregularity becomes a difficulty for the representation of the topography. The other models of the Tables 1 and 2 present more inconsistency with BMs heights; these can be seen by the percentages where the differences were greater than 50 m.
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Fig. 1 Distribution of the BMs over Brazil.
-4~"
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267
268
D. Blitzkow • A. C. O. C. de Matos. J. P. Cintra
Table 1. Percentages of the BMs comparison over Brazil. Models
SRTM (3") DTM2002 GLOBE GTOPO30 ETOPO2 ETOPO5
N. of points
A < 10
10 < A < 20
20 < A < 30
compared 61,860 62,030 60,791 60,818 62,030 62,030
% 70.57 20.90 20.42 17.49 17.33 10.65
% 14.94 15.99 16.00 14.75 14.15 10.51
% 5.72 11.98 12.35 12.02 11.78 9.01
30
40
A >50 %
2.95 9.69 9.48 9.93 9.57 7.95
1.74 7.51 7.53 7.82 7.83 6.97
4.08 33.93 34.23 37.99 39.35 54.92
(A = BM height - interpolated value, in meters)
Table 2. Percentages of the BMs comparison over Argentina.
Models
N. of points
A < 10
compared
%
10
20
30
40
A>50 %
9.33 12.96
4.13 6.44
2.15 3.91
1.42 2.57
7.66 12.18
23.53 23.94 24.70 17.69
7.88 8.70 9.50 10.75
3.96 4.01 4.56 6.90
2.41 2.45 2.95 5.16
11.35 11.54 14.92 33.02
SRTM (3") 13,703 75.31 DTM2002 13,723 61.95 GLOBE 13,679 50.87 GTOPO30 13,678 49.36 ETOPO2 13,723 43.36 ETOPO5 13,723 26.47 (A = BM height - interpolated value, in meters)
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3. Amazon region An important region to be studied is Amazon due to the existence of many areas of dense forest (it complicates the mapping with R A D A R as well as with aerophotogrametry) and of immense areas covered by rivers (Matos, 2005). This paper shows several profiles that crosses the maximum height in blocks with different characteristics like dense
forest, plains and great extension rivers. These profiles show the behavior of the eight mentioned models (Figure 3). Figure 4 shows the longitudinal profiles in the block N02W063 with the different models; it is situated in the Y a n o m a m i Indigenous Reserve, where the vegetation is umbriferous umbrageous forest. The models of 30" present vertical offsets with respect to SRTM and ETOPO2 has a behavior much smoother than the other models previously mentioned. The profiles of the different models referred to block N 0 0 W 0 7 0 are shown at Figure 5. It corresponds to Pari Cachoeira indigenous reserve. SRTM shows a height spike equal to 432 m (%= 0,1083333 °, ) ~ = - 6 9 , 4 8 5 8 3 3 3 ° ) , which does not exist in the other models. On the other hand, there is a topographic difference between the profiles. The vegetation in this block also belongs to umbriferous umbrageous forest. The longitudinal profiles for the different models at block S03W063, Jail National Park (Figure 6), show that the area is flat, with a type of vegetation equal to the two previous regions. The profiles of the models of 30" and of 2' practically coincide and are shifted vertically by 30 meters with regard to SRTM. A similar vertical offset also occurs in the Indigenous Reserve T.I. of Javari's Valley (Figure
Chapter 40 • SRTM Evaluation in Brazil and Argentina with Emphasis on the Amazon Region
7) regarding the models DTM2002 and SRTM with respect to the other mentioned models; in this block the vegetation is umbriferous bulbiferous forest. The other models present very different results when compared each one with the others. Figures 8 and 9 present profiles of plains close to each other; the first is situated between Branco and Negro Rivers and the second is localized between the last one and Solim6es River, therefore, regions situated in opposite margins of the Negro River. Figure 8 shows that the models of 30" and of 2' are similar. SRTM indicates a profile with lower heights with respect to the mentioned ones, except in the left hand side of the block. The vegetation is prairie and umbriferous forest. SRTM and DTM2002 profiles (Figure 9) present a similar topographic behavior, with a vertical offset around 30 m; the vegetation is umbriferous umbrageous forest. The environmental protected area Mordaga's Cave is important due to the Balbina Dam inside of
SRTM
....
DTM2002
......
GLOBE
.......
GTOPO30
.....
it. Figure 10 presents the latitudinal profiles over this dam. The models of 30" are coincident to each other and show a soft decay of the height in the dam if compared to SRTM. At the beginning and at the end of the profile of this block the vegetation is umbriferous umbrageous forest. Finally, Figures 11, 12 and 13 show profiles intercepting the Negro, Solim6es and Tapajds Rivers, respectively. The profiles of the models of 30" over Tapajds river present behavior similar to SRTM, but only in this case. The vegetation in the proximity of Solim6es and Negro rivers is also umbriferous umbrageous forest and the profiles show vertical offsets around 30 meters. In Figure 13, at the beginning of the profile, the vegetation is umbriferous umbrageous forest and next to the Tapajds River is anthropic forest. The considerably different behavior that some of the global models exhibit can be explained in most cases by the different grid size and the consequent interpolation.
......
ETOPO2
.....
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.......
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Fig. 3 Symbols that represent the profiles in the eight DTM.
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Fig. 7 Longitudinal profiles - block S06W072.
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269
270
D. Blit zkow • A. C. O. C. de M a t o s . J. P. Cintra
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Fig. 12 Latitudinal profiles - block S04W064.
Fig. 8 Longitudinal profiles - block SO1W063.
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279.0 ]
4. Conclusions
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-1.0
In general, when compared with the BMs heights, SRTM shows differences compatible with the precision specified for the model. In Tables 1 and 2, r e g a r d i n g SRTM, the differences greater than 50 m are due whether to the uncertainty in BM coordinates or the quality of the model. The position of the BM has to be improved. Most BMs in the Amazon region, although in a reduced number and with a located distribution, present a satisfactory behavior. In the considered region profiles (Figures 4 to 13) show that the models with 5' spacing are not representative of the t o p o g r a p h y . The a n a l y s i s also s h o w s that GTOPO30 and GLOBE coincide to each other in many of these blocks, the same occurring with T E R R A I N B A S E and JGP95E. The reason is that the two pairs of models use the same information source. The R A D A R signal theoretically reflects on the water. Therefore, the implied model should provide values of the height referred to that surface. Nevertheless, due to the SRTM vertical accuracy, the height over the rivers oscillates around a mean value. In the regions of umbriferous forest SRTM
Chapter 40 • SRTM Evaluation in Brazil and Argentina with Emphasis on the Amazon Region
30" profiles, show vertical offset around 30 meters. This fact suggests that radar wave is reflecting on the canopy and not on the ground.
5. Acknowledgements W e a c k n o w l e d g e to the B r a z i l i a n Institute o f G e o g r a p h y and Statistics ( I B G E ) , the M i l i t a r y Geographic Institute (IGM) of Argentina and Maria Cristina Pacino of the Rosfirio National University, for m a k i n g a v a i l a b l e the i n f o r m a t i o n o f the levelling network.
References Arabelos, D. (2000). Intercomparisons of the global DTMs ETOPO5, TerrainBase and JGP95E. Physics and Chemistry of the Earth Part A, 25(1), pp. 89-93. Hasting, D.A., and P.K. Dunbar (1999). Global Land Onekilometer Base Elevation (GLOBE) Digital Elevation Model, Documentation, Volume 1.0. Key to Geophysical Records Documentation (KGRD) 34. National Oceanic and Atmospheric Administration, National Geophysical Data Center, 325 Broadway, Boulder, Colorado 80303, U.S.A. Hensley, S., R. Munjy, P. Rosen (2001). Interferometric synthetic aperture radar. In: Maune, D. F. (Ed.). Digital elevation model techonoligies applications." the D E M users manual. Bethesda, Maryland: ASPRS (The Imaging & Geospatial Information Society), cap. 6, pp. 142-206. Hutchinson, M.F. (1989). A new procedure for gridding elevation and stream line data with automatic removal of spurious pits. Journal of Hydrology, 106, pp. 211-232. JPL (2004). SRTM - The Mission to Map the World. Jet Propulsion Laboratory, California Inst. of Techn., http://www2.jpl.nasa.gov/srtm/index.html.
Johnson, C.P., P.A.M. Berry, and R.D. Hilton (2001). Report on A CE g e n e r a t i o n , Leicester, UK, http:// www.cse.dmu.ac.uk/geomatics/ace/ACE_report.pdf. Lemoine, F.G., N.K. Pavlis, S.C. Kenyon, R.H. Rapp, E.C. Pavlis, and B.F. Chao (1998a). New high-resolution modle developed for Earth' gravitational field, EOS, Transactions, AGU, 79, 9, March 3, No 113, 117-118, 1998. Lemoine, F.G., S.C. Kenyon, J.K. Factor, R.G. Trimmer, N.K. Pavlis, D.S. Chinn, C.M. Cox, S.M. Klosko, S.B. Luthcke, M.H. Torrence, Y.M. Wang, R.G. Williamson, E.C. Pavlis, R.H. Rapp and T.R. Olson (1998b). The development of the joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) geopotential model E G M 9 6 , NASA/TP-1998-206861. National Aeronautics and Space Administration, Maryland, USA. Matos, A.C.O.C. (2005). lmplementa¢~to de modelos digitais de terreno para aplica¢Oes na Orea de GeodOsia e Geofisica na AmOrica do Sul. PhD thesis - Escola Politdcnica, Universidade de Silo Paulo, Silo Paulo, 355 p. NOAA (1988). Data Announcement 88-MGG-02, Digital relief of the Surface of the Earth. NOAA, National Geophysical Data Center, Boulder, CO. http ://www.ngdc.noaa. gov/mgg/global/etopo5.html. NOAA (1995). TerrainBase Global Digital Terrain Model, Version 1.0, NOAA, National Geophysical Data Center, Boulder, CO. http://www.ngdc.noaa.gov/seg/fliers/se1104.shtml. NOAA (2001). 2-minute Gridded Global Relief Data (ETOP02). NOAA, National Geophysical Data Center, Boulder,CO. http ://www.ngdc.noaa. gov/mgg/fliers/01mgg04.html. Saley, J., and N.K. Pavlis (2002). The development and evaluation of the global digital terrain model DTM2002, 3rd Meeting of the International Gravity and Geoid Commission, Thessaloniki, Greece. U.S. Geological Survey (1997). GTOP030 Global 30 Arc Second Elevation Data Set, at http:/!edcwww.cr.usgs.gov/landdaac/gtopo30/gtopo30.ht ml.
271
Chapter 41
On the Estimation of the Regional Geoid Error in Canada J. Huang (l~, G. Fotopoulos (2), M. K. Cheng (3), M. Vdronneau (1), M. G. Sideris (4~ (1) Geodetic Survey Division, CCRS, Natural Resources Canada, 615 Booth Street, Ottawa, Ontario, K1A 0E9, Canada (2) Dept. of Civil Engineering, University of Toronto, 35 St. George Street, Toronto, ON, M5S 1A4, Canada (3) Center for Space Research, University of Texas at Austin, 3925 West Braker Ln. #200, Austin, Texas 78759, USA (4) Dept. of Geomatics Engineering, 2500 University Drive N.W. Calgary, Alberta T2N 1N4 Canada The geoid errors for the Canadian Gravimetric Geoid 2005 (CGG05) model are estimated from the error information of the satellite and terrestrial gravity data. Calibration is conducted through the application of variance component estimation (VCE) with GPS-leveling data and their associated covariance matrices. Preliminary results suggest that the error of the geoid heights is generally smaller than 6 cm in Canada, with a range from 6 cm to 31 cm for the Western Cordillera area. Overall, the average error of the CGG05 model is estimated at 5.5 cm.
(CGG05) model (includes the satellite gravity model, terrestrial gravity data, gravity reduction methods and the far-zone contribution error), and (ii) to develop an adequate method for the estimation of the gravimetric geoid error from these error sources. In particular, the iterative almost unbiased estimation (IAUE) scheme is implemented to validate/calibrate the geoid error using existing GPS-leveling data across Canada.
Keywords. Geoid, gravity, error estimation
The CGG05 model was computed using the degreebanded Stokes integral, which is described in Huang and Vdronneau (2005). In order to illustrate how errors propagate into the geoid model, the formula for the geoid determination is simplified as
Abstract.
1 Introduction With the increased use of GPS-based positioning, the demand for directly converting ellipsoidal heights to heights referred to a regional vertical datum with sufficient accuracy is also increasing. For Canada, the use of a gravimetric geoid as the national height reference surface is currently under study. The recent revolutionary development of space gravimetry (i.e., CHAMP, GRACE and the upcoming GOCE mission) offers the opportunity to pursue such a dramatic change in the local vertical datum. However, before a vertical datum based on a gravimetric geoid is adopted, as opposed to one based on conventional leveling networks, it is important to conduct a reliable assessment of the systematic and stochastic errors of the geoid model. This is a challenging task as there is limited or poor information regarding the quality of the terrestrial gravity data, difficulty in quantifying the gravity reduction and interpolation errors, as well as only approximate estimates of the errors associated with the satellite models. The purpose of this paper is twofold, namely (i) to provide a detailed examination of all error sources of the Canadian Gravimetric Geoid 2005
2 Methodology 2.1 Estimation of Gravimetric Geoid Error
N - N S C + R iSDB(V)(Agm_AgSC)d.O,+Fy (1) 4~7 ~40' where the first term on the right hand side of eq. (1) represents the geoid components below spherical harmonic degree L+ 1 from a satellite model (SG), R is the mean radius of the Earth, 7 is the normal gravity on the reference ellipsoid, and Ag denotes the gravity anomalies. The degree-banded Stokes kernel can be expressed as mTG 2 n + l S DB(~1/) -- Z Pn (COS~)
n--L+1 n - 1
It is used as a band-pass filter to compute the geoid components from degree L+I to mT6. The upper limit mTG is dependent on the terrestrial gravity (TG) data spacing. The Stokes integration is performed within a spherical cap limited to a spherical angular distance of 6 arc-degrees. This implies that the band-pass filtering is incomplete and renders aliasing geoid errors that account for an RMS of approximately 2 cm over Canada.
Chapter 41
Finally, FN is the far-zone contribution outside the Stokes integration. It can be evaluated from a combined global spherical harmonic gravity (GGM) model up to its maximum degree (preferably larger than degree 200):
R ~ DB CG 7_. Qn gn 27 n=L+l
FN
•
on the Estimation of the Regional Geoid Error in Canada
terrestrial gravity data in Canada, which inherently contains a combination of errors originating from several sources, including gravity measurements, height measurements at the gravity points, topographic reduction, interpolation of gravity values, digital elevation models (DEM) and actual topographical density distribution. 2.2 Calibration of the Geoid Error
where QDB(V0) -- fo SDB(V)Pn (COSV) sin v d v The geoid error is primarily comprised of errors from the satellite-only gravity model, the combined global gravity models, and the terrestrial gravity data. The satellite gravity signal usually dominates the low-degree part of the geoid components in a combined model while the terrestrial gravity data complete the GGM for higher degrees and orders (Sideris and Schwarz, 1987). For regional geoid determination, the lower limit, L, must be selected according to the quality of the satellite data. Empirical tests show that it should not exceed 30 for GGMs prior to the CHAMP/GRACE missions, if a decimeter-level accurate geoid is sought. A simplified expression for the geoid error is given by: V N = VSG + VTG + VCG
where _ V SG
R (2 ~
n=2
n- 1
Geoid heights can also be determined at co-located GPS and leveling stations, which provides an independent external means to validate and calibrate the gravimetric geoid model and its precision. The discrepancies between the GPS/levelingderived and gravimetric geoid heights can (predominantly) be attributed to a combination of systematic and random errors in the ellipsoidal heights (h), the orthometric heights (H), and the gravimetric geoid heights (N), as discussed in Kotsakis and Sideris (1999). The following general linear functional model was used for the combined (multi-data) least-squares adjustment of the heterogeneous height data: Ax + Bv + w - 0, E{v}- 0
(6)
B-[I
(7)
where
(2)
+ QDB n
V--
-I VH
-I VSG
w-h-H-N R ~SDB (v)eTGAO , VTG = 4rc7 R
mcG ,--. DB CG
- 2_t~ v c~ = - 2"7
~
-I VTG
-I] VCG
(8)
(9)
(4)
(5)
n=L+l
Vso, vm, and Vco represent the geoid errors from the satellite model, the terrestrial gravity data, and the combined model, respectively. Given the covariance (CV) matrices for each of these three types of errors, the geoid standard deviation (std) can be estimated based on eqs. (3) to (5) via error propagation. In our case, the geoid std may be evaluated only approximately because the CV matrices for the satellite and combined harmonic gravity models are approximate. Furthermore, only approximate error values are available for the
The deterministic term, Ax, introduced in eq. (6) represents the parametric model used to approximately model the systematic errors inherent in and among all three types of heights. The selection procedure for the type of model and assessing its validity has been discussed extensively in Fotopoulos (2003). For this particular case a simple four-parameter model was found to be sufficient and therefore incorporated for all calculations. Individual variance components are estimated using the adjustment model in eq. (6) to (9) and the a-priori CV matrices for each of the data types (see Fotopoulos, 2003 for the detailed procedure). This procedure was followed in this study in order to achieve a more realistic estimate of the geoid model
273
274
J. Huang • G. Fotopoulos • M. K. Cheng • M. Vl:ronneau • M. G. Sideris
error that incorporates five individual variance components for the ellipsoidal heights and the orthometric heights at the GPS-leveling benchmarks,
denoted
by
crh2
and
2 crH,
respectively. The geoid height errors are separated for the satellite gravity model, terrestrial gravity data, and the combined gravity model, denoted by craG, cr2G and O'~G, respectively.
The CGG05 geoid model is validated using 430 colocated GPS-leveling stations with a distribution as depicted in Figure 1. The computed h-H-N residuals for the 430 stations plotted as a function of longitude and latitude are shown in Figure 2. The overall standard deviation of these residuals is 10.2 cm. A negative mean value of-40 cm indicates that the zero-height point of the leveling network is approximately 40 cm lower than the CGG05 geoid model.
3 Gravity and GPS-leveling data .
The lower degrees (2 to 90) of the GRACE-based GGM02C model (Tapley et al., 2005) are used for the determination of the long wavelength components of the geoid while the higher degrees (91 to 200) determine the far-zone contribution of the Stokes integration. EGM96 (Lemoine et al., 1998) is used to extend the GGM02C up to degree and order 360. The local residual terrestrial gravity data, i.e., ground, airborne, shipboard (including satellite altimetry-derived), are used to compute the geoid components above degree 90. These terrestrial data are the same as those used for the CGG2000 model (V6ronneau, 2002). The latest model, CGG05, is a high-resolution geoid model for North America with a geographical spacing of 2 minutes of arc along latitudes and longitudes. Its reference ellipsoid is GRSS0 and the reference frame is ITRF (no specific realization). Canadian GPS surveys after year 1994 were used in a least-squares adjustment to compute the ellipsoidal heights with respect to the GRS80 reference ellipsoid and their associated variances/covariances (Craymer and Lapelle, 2004; pers. comm.). The reference frame is ITRF97. The geopotential numbers are determined from a minimally constrained least-squares adjustment (via Helmert-blocking) of the geodetic leveling observations after year 1981. The single fixed station is a benchmark located along the StLawrence River in Rimouski, Qudbec, which is the same constraint used for the North American Vertical Datum of 1988 (NAVD88). Gravity values are interpolated at each benchmark from local measurements and converted to mean values along the plumbline (from geoid to topography) by correcting for the variable terrain. The variances and covariance within each Helmert block are implemented for the calibration of the geoid error, while the correlation between neighboring Helmert blocks have been omitted at this stage.
•
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Figure 2. Height residuals (h-H-N) versus longitude and latitude
The observed north-south and east-west slopes evident in Figure 2 result in a standard deviation of about 8 cm and are caused by systematic errors attributed to all three types of heights. In general, the uncertainty in the location of the ITRF97 reference frame geocenter can generate systematic errors of a few centimeters in the ellipsoidal heights (Altamimi et al., 2002), while the GRACE data may introduce systematic errors of less than a few
Chapter 41 • on the Estimation of the Regional Geoid Error in Canada
centimeters in the low-degree components of the gravimetric geoid heights. However, the leveling data are most likely the major source of systematic errors that accumulate over the 6000 km separation between the east and west coasts. Current knowledge about the mathematical and physical characteristics of the systematic errors in the leveling data is limited and therefore it is difficult to accurately model and correct for these discrepancies.
4 Geoid Error from the GGM02C Model The error model of the GGM02C model propagates into the low-degree components and far-zone contribution of the Canadian geoid model. The error information of the GGM02C model is provided in terms of error coefficients obtained from the diagonal elements of the covariance matrix. Figures 3a and 3b show the low-degree geoid error estimates from the diagonal-only terms and from the fully-populated covariance matrix of the GGM02C model, respectively. In this case, the use of diagonal-only elements does not provide sufficient information for the estimation of the geoid errors.
0.1
Io. I ~ "W
O'
¢~'W
~'0"E
t-'O' E
t-
0.0
0.3
I~'W
t.O 1.5 ~) F,om da~gor~l CV rn~,~ (~'W
0~
~O'E
2.0
2.5
The far-zone contribution error was found to be closely approximated by a constant of 1.6 cm according to the diagonal CV matrix and 0.7 cm to 1.8 cm if the fully-populated form of the CV matrix is utilized. Again, it was determined that the fullypopulated covariance matrix is needed to evaluate the far-zone contribution error. Initial covariance matrices of the low-degree and far-zone contribution components of the geoid heights at the 430 GPS-leveling stations have been estimated from the CV matrix of the harmonic coefficients of GGM02C.
5 Geoid Error from the Terrestrial Data The terrestrial gravity error is comprised of measurement, datum, data reduction and interpolation errors. A fully populated covariance matrix for this data is not available; however, the standard deviation at each gravity station can be estimated from the measurement and elevation standard deviations. By neglecting the covariance between any two gravity stations, and the datum and interpolation errors, the initial geoid error standard deviations can be estimated via simple error propagation of Stokes integration (Li and Sideris, 1994). Figure 4 depicts the computed geoid error standard deviations based on the terrestrial gravity data, which provides an average error of 1.5 cm across Canada and a maximum of 15.7 cm in the western region. These values are most likely too optimistic due to the obvious omissions mentioned previously. However, since this is the best information currently available, these values were used to construct the initial CV matrix for the terrestrial gravity component at the 430 GPSlevelling points.
L20'E
6 Estimation of Variance Components
o.!
Io, I~'W
0.0
0.3
O'
¢~'W
1.0
e",O"E 1.5
L"O' I-] 2.0
b~,From E,,IIIC..,V,'n~lTi~
Figure 3. GGM02C geoid error estimates for degrees 2 to 90 based on (a) diagonal-only CV matrix and (b) fully populated CV matrix
2.5
Given the initial covariance matrices corresponding to the h, H, Ns~, NTG and Ncc data, the geoid errors estimated from the GGM02C model and from the terrestrial data can be verified and calibrated at the GPS-leveling stations as per the procedure described in section 2.2. Figure 5 shows the mean covariances with respect to the spherical distance computed from the initial CV matrices for the (a) ellipsoidal heights, (b) orthometric heights, (c) satellite and (d) terrestrial geoid errors at the 430 GPS-leveling stations, respectively.
275
276
J. Huang • G. Fotopoulos • M. K. Cheng • M. V[~ronneau • M. G. Sideris
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2
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Figure 4. Regional geoid error propagated from estimated errors of the terrestrial gravity data through the degreebanded Stokes integration a)h
b)H
300 250 200
3500 : .............................................................................
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..............................................................................
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.............................................................................
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............................................................................. 300
.............................................................................
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50
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10 20 30 40 Spherical Distance (degree)
50
Figure 5. Initial mean covariance (in mm 2) via spherical distance for (a) ellipsoidal heights, (b) orthometric heights, (c) satellite geoid heights, and (d) residual terrestrial geoid heights
Chapter 41
The ellipsoidal heights are wealdy correlated across the GPS-leveling network with a range in standard deviations from 0.2 cm to 7.6 cm and an average value of 1.3 cm. In general, these values obtained directly from the results of the post-processing software tend to be on the optimistic side. The orthometric heights are strongly spatially correlated within each of the Helmert blocks. The corresponding standard deviations increase as the distance with respect to the 'fixed' station in Rimouski increases. These values reach a maximum of approximately 9 cm with a mean standard deviation of 5.4 cm. The low-degree geoid components from the GRACE model values are correlated because they are evaluated from the same set of spherical harmonic coefficients. Standard deviations for this data vary from 0.5 cm to 1.4 cm with a mean of 0.9 cm. The covariance matrix for NcG is similar to that of Nso (Figure 5c) and therefore not shown. The residual geoid components evaluated from the degree-banded Stokes integral exhibit a high correlation only when common data have been used between any two computational points. In this study, the integration cap radius is 6 arc-degrees, which indicates a correlation for any two computational points located within a spherical angular distance of less than 6 arc-degrees. The standard deviations of the residual components of the geoid model are evaluated from the errors of the gravity anomaly data and range from 0.3 cm to 8.8 cm with a mean value of 1.4 cm. Using the iterative almost unbiased variance component estimation scheme (Horn and Horn, 1975; Fotopoulos, 2003) and the a-priori CV matrices described above, five individual variance components were estimated. These values are tabulated in Table 1 for two scenarios, namely (i) diagonal-only CV matrices and (ii) fully-populated CV matrices (where available). The sensitivity of the estimated variance factors to the a-priori covariance information is evident from the differences between the estimated variance factors in each scenario. As expected, if only the diagonal information of the matrices is used as an approximation, the computed variance components are (in general) low for data where correlation is evident (e.g., orthometric heights). The estimates in Table 1 suggest that the a-priori CV matrices corresponding to the ellipsoidal, orthometric and geoid heights are too optimistic,
•
on the
Estimation of
the Regional Geoid Error in Canada
with final estimated variance components suggesting a re-scaling of the a-priori CV matrices of more than 3. The result for the far-zone contribution is less conclusive. In all cases, the number of iterations remained constant at approximately 70. Table 1. Estimated variance factors using fullypopulated and diagonal a-priori covariance matrices (n is the number of iterations)
CV
"2 (3"h
@2 H
(~2 SO
(~2 TO
C~co
"2
n
Diagonal
2.24
0.03
7.71
2.69
6.00
72
Full
9.09
5 . 8 5 3.19
3.61
0.01
69
7 Total Geoid Error The total geoid error is finally computed from the scaled CV matrices (after variance component estimation) of the three estimated components corresponding to SG, TG and CG. Assuming that the variance factors in Table 1 are applicable for non-GPS/leveling points (albeit a bold assumption), the total calibrated geoid error for CGG05 is illustrated in Figure 6 on a 2' x 2' grid. This assumption will be further tested using additional data that was not implemented in this study. The calibrated geoid error ranges from 1 cm to 32 cm, with a mean error of 5.5 cm across the entire Canadian landmass. The mountainous areas of western Canada (up to Alaska) exhibit the largest errors due to sparse gravity anomaly error information (i.e., 2' in the mountains is insufficient). The geoid error in central Canada is generally smaller than 6 cm, with the hilly regions in eastern Canada showing slightly larger geoid errors. In particular, the geoid errors in the Fox Basin and Ungava Bay are significantly larger than those of the surrounding regions, due to the lack of terrestrial gravity data.
8 Discussion of Future Work The progress made in this study represents a significant step forward to achieving realistic error estimates for the Canadian gravimetric geoid model. However, it should be stated that the total geoid errors shown in Figure 6 are preliminary and refinements are ongoing. In particular, major improvements are expected on three fronts, namely (i) the inclusion of additional GPS-leveling data for a regional calibration based on the geographical
277
278
J. Huang • G. Fotopoulos • M. K. Cheng • M. V[~ronneau • M. G. S i d e r i s
"~m,r • .,m..-, "'m'm'mM'~l''m
÷
,_
4+
L ¥ " ,m
"r
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tie
I
I
0
2
1 0 0 ~"~ly'
~*vvr
8/()" ~ r
'"
•
E
4
6
8
gll
[0
Figure 6. Regional geoid error map for Canada estimated using GPS-leveling data
heterogeneity of the data, (ii) the incorporation of the correlation between the terrestrial gravity data and (iii) the verification of the reliability of the estimated variance components, through an external validation process. Further tests will also be conducted to re-evaluate the suitability of the deterministic term introduced in the functional model for the systematic errors (i.e., type of parametric model).
Acknowledgements The authors thank Dr. Mike Craymer and Dr. Joe Henton for their constructive comments. Appreciation is also extended to Dr. Robert Tenzer and an anonymous reviewer for their critical review.
References Altamimi Z, P Sillard, and C Boucher (2002) ITRF2000: A new release of the International terrestrial Reference Frame for earth science applications, J. Geophys. Res. 107(B10), 2214, doi: 10.1029/2001 JB00056
Fotopoulos G (2003) An analysis on the optimal combination of geoid, orthometric and ellipsoidal height data. PhD Thesis, University of Calgary, Dept. of Geomatics Engineering, No. 20185, Calgary, Canada Horn SD and RA Horn (1975) Comparison of estimators of heteroscedastic variances in linear models. Journal of the American Statistical Association, 70( 132):872-879 Huang J and M V6ronneau (2005) Applications of downward continuation in gravimetric geoid modeling - case studies in Western Canada, Journal of Geodesy, 79:135-145 Kotsakis C and MG Sideris (1999) On the adjustment of combined GPS/leveling/geoid networks. Journal of Geodesy, 73:412-421 Lemoine FG, SC Kenyon, JK Factor, RG Trimmer, NK Pavlis, DS Chinn, CM Cox, SM Klosko, SB Luthcke, MH Torrence, YM Wang, RG Williamson, EC Pavlis, RH Rapp, TR Olson (1998) The development of the joint NASA GSFC and NIMA geopotential model EGM96. NASA/TP1998-206861 GSFC, Greenbelt, MD
Chapter 41 • on the Estimation of the Regional Geoid Error in Canada
Li Y and MG Sideris (1994) Minimization and estimation of geoid undulation errors. Bulletin GOodOsique, 68(4):201-219 Sideris MG and KP Schwarz (1987) Improvement of medium and short wavelength features of geopotential solutions by local gravity data, Bolletino di Geodesia e Scienze Affini, 3:207-221 Tapley B, J Ries, S Bettadpur, D Chambers, M
Cheng, F Condi, B Gunter, Z Kang, P Nagel, R Pastor, T Pekker, S Poole, and F Wang (2005) GGM02 - An Improved Earth Gravity Field Model from GRACE, Journal of Geodesy, 79:467-478 Vdronneau M. (2002) Canadian gravimetric geoid model of 2000 (CGG2000). Report of the Geodetic Survey Division, Natural Resources Canada, Ottawa, ON (available by request)
279
Chapter 42
A Validation Procedure for Satellite Orbits and Force Function Models Based on a New Balance
Equation Approach A. L6cher, K.H. Ilk Institute of Theoretical Geodesy, University of Bonn, Nussallee 17, D-53115 Bonn, Germany
Abstract. Since the availability of CHAMP and GRACE data, the energy approach has become an important tool for the recovery of the gravity field based on continuously observed satellite orbits. Up to now, only the total energy of the satellite's threedimensional motion has been considered which is known as the Jacobi integral if formulated in a constantly rotating Earth-fixed reference frame. Beside this, additional energy integrals can be found for the components of the satellite's motion and various combinations hereof, starting from the three scalar components of Newton' s equation of motion. Furthermore, integrals of motion based on the linear momentum and the angular momentum can be formulated which show even better mathematical characteristics than the Jacobi integral for the determination of the gravity field. Therefore, this new approach seems to be appropriate to validate the consistency of gravity field models and precisely observed satellite orbits and to improve, subsequently, these gravity field models. The advantages and critical aspects of this approach are investigated in this paper. First results with real data were presented using kinematic CHAMP orbits.
Keywords. CHAMP, GRACE, integrals of motion, energy integral, Jacobi integral, balance equations, gravity field recovery, validation
1 Introduction The continuously observed orbits of low flying satellites (LEO - Low Earth O r b i t e r ) b y Global Navigation Satellite Systems (GNSS) as GPS or GLONASS and in future Galileo suggest a paradigm shift of gravity field determination techniques. Instead of the analysis of accumulated orbit perturbations of artificial satellites caused by the inhomogeneous structure of the gravity field the local in-situ analysis techniques gain more and
more importance. These new gravity field determination concepts can be divided in three groups. All of them require densely tracked orbits of the low flying satellites with high accuracy by a GNSS (CHAMP type satellites) or precisely measured inter-satellite functionals between two satellites (GRACE-type twin satellites). A first possibility is based on the formulation of Newton's equation of (relative) motion as integral equation of Volterra or Fredholm type. A second approach uses the equation of motion directly and a third intermediate technique exploits the balance equations of classical theoretical mechanics. The latter technique has been applied yet by using the energy balance principle in form of the so-called Jacobi integral. For an overview of modem techniques of gravity field determination with artificial satellites cf. Schneider (2002). Besides the use of the energy integral approach for gravity field recovery tasks there is also another area of application which has been demonstrated by Ilk and L6cher (2003) and L6cher and Ilk (2005): Because of the balance principle of the energy integral it represents a sort of "absolute" criterion for the proof of consistency of observed orbit and the dynamical model of the satellite's motion along the orbit. If the various energy constituents do not sum up to a constant then either the orbit is incorrect or the force function models are wrong or imperfect. The size of the constant is only of secondary importance, while the structure of the deviations from the constant may give hints to specific force function or orbit determination deficiencies. This is based on the fact that the energy exchange relations caused by the various force function components show typical properties which can be used to separate the different sources of inconsistency, especially those who show specific patterns of deviations in the space and the time domain. It is obvious that validation and gravity field improvement cannot be separated rigorously because any systematic inconsistency of observations and reference model can be used as well
Chapter 42
• A
Validation Procedure for Satellite Orbits and Force Function Models Based on a New Balance Equation Approach
to improve the parameters of the force function models. In this respect the proposal made in the following can be used in a first step as validation procedure to proof the consistency between observations and reference force function model and in a second step to improve those parameters of the reference model which have obviously led to the inconsistencies. This paper addresses alternative integrals of motion for gravity field improvement and validation tasks in comparison to the use of the Jacobi integral. Obviously it has been overlooked in the past that there are numerous alternative balance equations which can be used for the gravity field recovery as well. It will be shown that these alternative balance equations seem to have partly much better properties for validation and gravity field improvements than the frequently used Jacobi integral. In Sect. 1, the integrals of motion are reviewed, then some alternative integrals of motion are presented and the dependencies between the various balance equations are presented. In Sect. 2 selected balance equations such as the balance equation of linear and angular momentum, the Jacobi integral and the coordinate related energy integrals are discussed in the time domain. Sect. 4 includes a discussion of these balance equations in the space domain. In Sect. 5 the important question is investigated, in which way insufficiently modeled surface force constituents acting on the satellites are mapped into the gravity field parameters. Sect. 6 concludes this paper.
time interval[t0,t ] . A similar relation for the angular momentum can be derived. Vector multiplication of Newton's equation of motion with R , MRxR-RxK =0, (2.3) and integration over the time interval [to,t ] results in, t
L - [ R x K dt - L o •
(2.4)
to
This relation connects the angular momentum at time t, L = MR x R , with the angular momentum at time to, L0, and the torque, integrated within the time interval [t0,t ] . The energy balance equation has been derived following a similar procedure: It follows by scalar multiplication with R , MR.R-R.K = 0, (2.5) and after integration over the time interval [t0,t],
_1M R ~ 2
t
(2.6)
fR. K dt - E. d [0
This relation represents the energy integral along the satellite's orbit. Based on this equation the wellknown Jacobi integral can be derived easily if K are conservative forces and referred to a constantly rotating Earth fixed reference system. We realize that the general procedure to derive balance equations based on Newton's equation of motion consists in formulating the basic relation as follows,
f(M,R,R,
ft)-g(M,R,R,K)-O,
(2.7)
and integrating over the time interval [t0,t], t
F(M,R,R)-Ig(M,R,R,K)dt-C.
2 Integrals of motion 2.1 Some classical balance equations We start from the well-known Newton's equation of motion, which was derived from the balance equation of linear momentum by considering a constant mass M:
MR-K =0. (2.1) The quantity R is the acceleration of the satellite and K the force function acting on the satellite. Integration over a time interval [t0,t ] starting from an initial time t o results in t
P - JK dt - P0,
(2.8)
to
(2.2)
The left hand side represents the "kinetic" term, the right hand side the force function integral. In the next section we will derive various alternative energy balance equations following this generalized procedure.
2.2 Alternative energy balance relations We can derive "energy balance equations in the coordinate directions" if we begin with MR i - K i = 0 , for i e {x,y,z}
(2.9)
and multiply these equations with
to
which connects the linear momentum at a time l, P = M R , with the initial linear momentum at time 10, P0, and the integrated force function within the
MRik i - Ki/~ i = 0.
(2.10)
By integration over the time interval [t0,t ] we receive three energy balance relations of the form:
281
282
A.L6cher• K.H. Ilk t .
.
.
0E 0R = P0.
.
(2.17)
to
These formulae represent the energy integrals of the three-dimensional motion of the satellite in the coordinate directions e~, i E {x,y,z} . Another three energy balance relations can be derived if we start from Mki - K~ = 0,
(2.12)
for i E {x, y.,z}, and multiply these equations crosswise with R/ and /~/ for i, j ~ {x, y,z} resulting in the equations
Mk, k~ - K~k/ - o
(2.13)
Mkyk~ -Kyk~ - O,
(2.14)
and respectively. The integrals over the time interval [t0,t ] for all combinations of the sums of(2.13) and (2.14) represent three "energy balance equations in the coordinate surfaces",
The sum of the three energy constants in the coordinate directions (2.11) corresponds to the three-dimensional energy constant in Eq. (2.6), on the one hand, ~x + ~ + ~ - E , (2.1 s) while the derivatives of the energy components Eii with respect to the velocity components R i correspond to the respective components of the linear momentum, on the other hand, c~Eii a/~ = ~ " (2.19) The latter components of the linear momentum can be derived as well by differentiation of the "energy constants in the coordinate surfaces (ei,%)" with respect to the velocities /~/, ~E c~/~ = P" (2.20) .1
M[~[~/ - i(Ki[~/ + Kj[~i )dt - E!j .
(2.15)
tO
These balance equations are the energy integrals of the projections of the three-dimensional motion of the satellite onto the coordinate surfaces (ei, % ). Finally, a "balance equation of the momentum volume" can be derived in a similar way as before, resulting in
Mkk x
y
k
z
-
t
t0
(2.16) it should be pointed out that these 'projected' energies can be formulated in any reference flame, most easily in an inertial or an Earth-fixed reference flame, but also in a body-fixed reference flame. Details of formulating balance equations and integrals of motion related to the gravity field determination are treated by Schneider (2005), such as sensitivity aspects of balance equations, etc. 2.3 The d e p e n d e n c i e s between the balance equations
Except the balance equation for the angular momentum, the balance equations derived in the last sections are not independent. The differentiation of the energy integral (2.6) with respect to the velocity R results in the balance equation for the linear momentum (2.2)
Finally, the "energy constants in the coordinate surfaces (e~,% ) " can be derived by differentiating the "energy constant of the momentum volume" with respect to the velocities / ~ , c~E!Jk = P .
(2.21)
Despite these dependencies the various balance equations show specific characteristics if they are applied for validation and gravity field determination tasks. This will be demonstrated in the next section. Because of lack of space only the balance equations of the linear and angular momentum, the Jacobi integral and the "energy balance equations in the coordinate directions" are treated in the following. 3 Analysis
in t h e t i m e d o m a i n
The balance equations can be evaluated along the orbits. The graphs in these cases reflect the consistency of the force function and the orbit in the time domain. The computation steps consist essentially in a differentiation of the observed orbit on the one hand and in an integration of the force functions on the other hand (Fig. 1). We apply the procedure as shown in Fig. 1 to a half-revolution arc of CHAMP crossing the Himalaya region starting from the polar area in the North and ending in Antarctica in the South. The orbit of CHAMP has been derived either by integration based on the gravity field model EI-
Chapter 42 • A Validation Procedure for Satellite Orbits and Force Function Models Based on a New Balance Equation Approach
GEN-GRACE01S (Reigber et al. 2003) and considered error-free or determined by a precise kinematic orbit determination procedure (Svehla and Rothacher, 2003). The former case will be labeled in the following as "simulated", the latter one as "kinematic". The gravity field model used in the balance equations is EGM96 (Lemoine et al. 1998). Because of the different gravity field models used in the balance equations on the one hand (EGM96) and as a basis for the orbit determination on the other hand (EIGEN-GRACE01S), inconsistencies of gravity field model and orbit must occur in the balance equations as deviations from the specific constants.
coincides with the reality only approximately can be recognized. The graphs on top of Fig. 2 and Fig. 3 and labeled by "simulated" are based on different integrals of motion but should reflect essentially the same sort of inconsistencies. The same should be true for the bottom graphs and labeled "kinematic". It is interesting to note that the different integrals of motion show different sensitivities for the inconsistency features. Also the different coordinates of the specific integrals of motion show different characteristics.
0.002
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transformation
into inertial system
Earth orientation parameters
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integrand of the force function integral quadrature of the integrand
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Fig. 2: Deviation of the linear momentum from a constant along the half-revolution arc in m / s ; top" simulated, bottom: kinematic.
quadrature ith polynomials I
motionintegral
Fig. 1" Computation scheme for evaluating the balance equations along the orbits. The graphs in Fig. 2 and Fig. 3 reflect the inconsistencies of the force function and the ephemeris of a half-revolution arc as deviations from the respective constants of the balance equation for the linear momentum and the energy balance in x,y,z, respectively. The top graphs show the "simulated" case, the bottom graphs are derived by using a "kinematic" orbit. Both graphs show similar but slightly different features. The top graphs in Fig. 2 and Fig. 3 reflect only the inconsistencies caused by the different gravity field models EGM96 and EIGENGRACE01S. The bottom graphs contain additional noise originating from the noisy observations and additional forces acting on the satellite which are not considered in the integrals of motion. Therefore, the graphs labeled by "simulated" and "kinematic" for the specific integrals of motion show slightly different inconsistencies; only the most pronounced features caused by the fact that EGM96
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4 Analysis in the space domain If the inconsistencies of force function and orbit in the time domain are referred at every time to its subsatellite point then we get an impression of the inconsistencies in the space domain. The Jacobi integral directly reflects these inconsistencies in the
283
284
A. LScher. K. H. Ilk
space domain in terms of the gravity potential. This is not the case for the alternative integrals of motion as derived in Sect. 2. In this case the inconsistencies have to be expressed in terms of gravity potential inconsistencies at satellite altitude along the satellite orbits. These inconsistencies can be modeled by spherical harmonics. The corresponding potential coefficients can be considered as corrections to the parameters of the inconsistent gravity field model as well. Again, it becomes clear that the validation procedure corresponds to a global gravity field improvement procedure.
ity field. As reference field we select again EGM96. If we identify the real gravity field with EIGENGRACE01S then the graphics can be compared with Fig. 4. The inconsistencies are plotted along a kinematic 30-days orbit of CHAMP. Fig. 5 shows the inconsistencies of the linear and angular momentum, of the Jacobi integral and of the energy balances in x , y , z - all of them represented by spherical harmonic expansions of the gravity potential. The residual patterns in the time domain and in the space domain can be used to discriminate different causes for the inconsistencies.
Fig. 4" Potential differences of gravity field models EGM96 minus EIGEN-GRACE01S along the satellite orbit in m 2 / s 2 .
Fig. 5: Deviations of the integrals of motion from constants transformed to gravity potential inconsistencies along the satellite orbit in m 2 / s 2 . -150
-100
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50
-150
-100
-50
0
50
100
150
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linear momentum
-150
-100
-50
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angular momentum Fig. 4 shows the potential differences of the gravity field model EGM96 and the gravity field model EIGEN-GRACE01S along the satellite orbit. Fig. 5 illustrates the inconsistencies as seen in the various balance equations as a consequence of the differences between a reference model and the real grav-
-150
-100
-50
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100
150
-50
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50
100
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Jacobi integral
- 150
- 1O0
energy balance in x,y,z A careful look at Fig. 5 shows interesting differences in the inconsistency patterns of the various balance equations. We recognize the stripe pattern as well as the rough features in case of the Jacobi integral compared to, e.g., the linear and the angular momentum.
Chapter 42 • A Validation Procedure for Satellite Orbits and Force Function Models Based on a New Balance Equation Approach
The triangle plots of the formal errors of the coefficients describing the potential inconsistencies in Fig. 5 are shown in Fig. 6. In case of the Jacobi integral the stripe pattern in the space domain (Fig. 5) correlates clearly with the formal errors of the spherical harmonic coefficients. These errors increase with increasing degrees and orders. The situation is completely different in case of the linear momentum and also in case of the angular momentum.
But also the formal errors of the coefficients of the inconsistencies of the energy balances in the coordinates x,y,z show a completely different behaviour than the formal errors of the inconsistencies based on the Jacobi integral. This is remarkable in so far as the Jacobi integral represents nothing else as the sum of the balance equations in the coordinates x,y,z. Again, the Jacobi integral seems to be the worst choice for gravity field validation and gravity improvement tasks. order
order
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Fig. 6: Formal errors of the spherical harmonic coefficients, corresponding to the inconsistencies of the integrals of motion. 4E-008 -
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_ _ [ _ _ [ _ _ [
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:
:
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20
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60
Fig. 7: Degree RMS of the formal errors of spherical harmonic coefficients.
0
20
40
Fig. 8" Difference degree RMS with respect to EIGEN-GRACE01S ("true error").
60
285
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A. LScher. K. H. Ilk
The graphs in Fig. 7 and Fig. 8 show the true and the formal error degree variances- and confirm the results discussed before. The formal but also the true error degree variances for the inconsistencies based on the Jacobi integral are the worst of all integrals of motion considered here. The best result can be achieved with the angular momentum balance, but also the energy balances in the coordinates x,y,z show a remarkable good result. Again, this is interesting in so far as the sum of all three coordinates corresponds to the Jacobi integral.
100
used. The model of the surface forces has been based on the measured accelerations of CHAMP, filtered and approximated by spline functions. Then the surface force model has been falsified by an amount of 10% of the total surface forces. The question is whether the surface forces are shifted into the gravity field parameter corrections and which part of these forces can be recognized after the recovery procedure when the integrals of motion are inspected in the time domain.
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5 Energy shifts: a simulation
example
An interesting question for the validation of force function models and precisely observed orbits is the question how time dependent orbit effects, e.g., insufficiently modeled or measured surface forces are shifted into the gravity field parameters during the gravity field recovery procedure. A simulation experiment shall demonstrate the consequences of not correctly modeled dissipative force function constituents for the gravity field recovery. Surface forces cause a transfer from kinetic energy to heat energy. If the total of heat will be assimilated into the gravity field parameters then it is very difficult to detect these effects by the integrals of motion. The simulated reality is approximated by the gravity field model EIGEN-GRACE01S and a surface force model. For the gravity field recovery the reference gravity field model EGM96 has been
Fig. 9: Gravity field inconsistency shown as potential defects along the satellite orbit in m 2 / s 2 for the various balance equations.
The Fig. 9 shows the gravity field inconsistencies as total potential effects in m 2 / s 2 . The "observations" used in the gravity recovery process are the deviations of the different balance equations from the respective constants. Then the complete spectrum of coefficients of a spherical harmonic expansion complete up to degree n=60 have been recovered for all four cases. The differences between the recovered potential coefficients and the true coefficients reflect the gravity field inconsistencies. It is remarkable that all integrals of motion show different gravity field recovery properties and the procedure based on the Jacobi integral is by far the worst one.
Chapter 42 • A Validation Procedure for Satellite Orbits and Force Function Models Based on a New Balance Equation Approach
Only a minor part of the dissipative surface forces has been transferred to the gravity potential coefficients; the main part can be detected if the integrals of motion are determined with the improved gravity field model. Fig. 10 shows the effects which are caused by these dissipative forces in terms of total energy. In real applications this would be a clear hint that still model misconceptions exist in the mathematical-physical model of the gravity field recovery procedure. 0.0
~.-.,%%~
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temporal and local disturbances which have to be investigated in the future.
Acknowledgement.
We appreciate the anonymous reviewer and J. Kusche for their helpful comments and suggestions. We are grateful to Prof. Dr. M. Schneider for his continuous partnership in discussing various problems related to this research. Our special thanks go to D. Svehla and M. Rothacher for placing the kinematic orbits of CHAMP at our disposal. The support of BMBF (Bundesministerium f'tir Bildung und Forschung) and DFG (Deutsche Forschungs-Gemeinschaft) of the GEOTECHNOLOGIEN programme is gratefully acknowledged. References
-40.0 -
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-60 o
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Fig. 10: Energy contribution shifted from 10% of the surface forces into the motion balance constants illustrated in the time domain. 6
Conclusions
In this paper alternative integrals of motion have been presented which show interesting properties both for gravity field recovery as well as for validation tasks. It is surprising that the Jacobi integral, exclusively used as balance equation approach within the last years for gravity field recovery task in case of densely and precisely observed low satellite orbits, is by far the worst choice under various other possibilities. The balance equations of the linear momentum, of the angular momentum or of the coordinate energy balances show much better properties for gravity field recovery as well as for validation of the consistency of force functions and observed orbits. There are alternative balance equations mentioned in Sect. 2 but additional ones which should be investigated in detail. It seems that the proposal of using modified integrals of motion for gravity field recovery and validation tasks is very encouraging. Nevertheless there are some important open questions as the detailed separation of
Ilk KH, L6cher A (2003) The Use of Energy Balance Relations for Validation of Gravity Field Models and Orbit Determination Results, F. Sans6 (ed.) A Window on the Future of Geodesy, IUGG General Assembly 2003, Sapporo, Japan, International Association of Geodesy Symposia, Vol. 128, pp. 494-499, Springer Lemoine FG, Kenyon SC, Factor JK, Trimmer RG, Pavlis NK, Chinn DS, Cox CM, Klosko SM, Luthcke SB, Torrence MH, Wang YM, Williamson RG, Pavlis EC, Rapp RH, Olson TR (1998) The development of the joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) geopotential model EGM96, NASA/TP- 1998-206861, Goddard Space Flight Center, Greenbelt, MD L6cher A, Ilk KH (2005) Energy Balance Relations for Validation of Gravity Field Models and Orbit Determinations Applied to the Results of the CHAMP Mission, in C. Reigber, H. Lfihr, P. Schwintzer, J. Wickert (Eds.): Earth Observation with CHAMP, Results from Three Years in Orbit, pp. 53-58, Springer Mayer-Giirr T, Ilk KH, Eicker A, Feuchtinger M (2005) ITG-CHAMP01: A CHAMP Gravity Field Model from Short Kinematic Arcs of a One-Year Observation Period, Journal of Geodesy (2005) 78:462-480 Reigber C, Schmidt R, Flechtner F, K6nig R, Meyer U, Neumayer KH, Schwintzer P, Zhu SY (2003) First EIGEN Gravity Field Model based on GRACE Mission Data Only, in preparation for GRL Schneider M (2002) Zur Methodik der Gravitationsfeldbestimmung mit Erdsatelliten, Schriftenreihe IAPG/ FESG, 15, Institut ffir Astronomische und Physikalische Geodfisie, Mtinchen, 3-934205-14-3 Schneider M (2005) Beitr/ige zur Gravitationsfeldbestimmung mit Erdsatelliten, Schriftenreihe IAPG/ FESG, 21, Institut ftir Astronomische und Physikalische Geod/isie, Mfinchen, 3-934205-20-8 Svehla D, Rothacher M (2003) Kinematic and reduced dynamic precise orbit determination of low-Earth orbiters, Adv. Geosciences 1:47-56
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Chapter 43
Combining Gravity and Topographic Data for Local Gradient Modeling L. Zhu, C. Jekeli Division of Geodetic Science School of Earth Sciences Ohio State University, 125 South Oval Mall, Columbus, OH
Abstract. The local modeling of gravity gradients
supports gradiometry systems by predicting observables whose residuals then yield new geophysical and geodetic information in the survey area. Gradients are usually modeled using local digital terrain elevation data (DTED). We supplement this with available (lower-resolution) gravity anomaly data for the longer-wavelength features and analyze the total model in terms of its spectral content over a local region such as the San Andreas Fault. New spherical models of transforming gravity anomalies to all components of the gradient tensor using Green's functions are developed and consistently combined with forward models of terrain elevation from very dense Shuttle Radar Topography Mission (SRTM) measurements. The modeling is applied to the case of airborne gradiometry at about 400 m over moderately rough terrain and yields an upper bound on the power spectral density to be expected in this case. Such gradient modeling should contribute to the design of appropriate filters in the processing of airborne gradiometric data. Keywords. gravity gradients, Stokes integral
1
DEM
modeling,
Introduction
The importance of gravity gradient modeling becomes more significant as instrumentation, particularly in airborne applications, becomes more accurate. A good gradient model derived from existing data sources aids in the pre-processing of airborne gradient data, as well as in detecting and interpreting residual density anomalies sought out by an airborne gradiometric survey. Gravity gradients can be modeled theoretically either from surface gravity or topographic data, or a combination of both. Some early studies in obtaining gradients from gravity data include those
43210
of Agarwal and Lal (1972) and Gunn (1975). More recent methods relied on Fourier Transform relationships among various derivatives of the geopotential field (e.g., Mickus and Hinojosa, 2001). Most modeling of gravity gradients, however, is based on topographic data since the gradients reflect primarily the near structures of the Earth's mass. A recent review of the various associated methods was given by (Jekeli and Zhu, 2005); see also references therein. Fortuitously, topographic data usually have much higher resolution than gravity data and thus topographically-derived gradient models inherit the resolution more nearly appropriate to this type of gravitational quantity. Nevertheless, depending on their resolution, gravity data grids also provide information on gradients at corresponding wavelengths. In fact, they more accurately reflect lateral density variation that may be missing in the usual topographically-derived model that is usually based on a constant mass density. We develop a unified approach to modeling gradients at aircraft altitude from a combination of surface gravity and terrain elevation data, and consider the power spectral densities at different altitudes. The objectives of the analysis are 1) to verify the significant attenuation of the gradient signal with altitude, regardless of the high resolution of the data; 2) to study the dependence of the model at aircraft altitude on typical density variability in the topographic masses; and 3) to develop a validation and calibration approach for airborne gradiometry.
2 2.1
Theory Basic Definitions
Before introducing our approach to modeling the gravity gradient, let us review some basic definitions. Let W be the gravity potential; then the gravity vector is given by
Chapter 4 3
rOW" Ox~ g-
g2
- VW -
(l)
3x2
g3
3W
v Ox3 where we have introduced a local Cartesian coordinate system, (x~,x2,x 3 ), with axes pointing north, east and down. The gravity gradient tensor is given by the second derivatives of the gravity potential:
F-
VV~Ig - Vg ~ -
where,
3gj/OXj
/ 3g 1
3g 2
3g 3
Ox,
3x,
3x,
3g~
3g 2
3g 3
~X2
OX2
ON2
3g~ Ox3
3g 2
3g 3
3x3
3x3
(2)
is called the inline gradient, and
Ogi/Oxj the cross gradient. The gradient disturbance is the difference between the total
•
CombiningGravity and TopographicData for Local Gradient Modeling
where GM is Newton's gravitational constant times Earth's total mass, R is a mean Earth radius, Cnm are the harmonic coefficients, and Y,,,n are spherical harmonic functions. Stokes's integral solves (in spherical approximation) the boundary-value problem for the disturbing potential (Heiskanen and Moritz, 1967), and thus,
R I I A (O,/~, ) 02 1-'j~ Ag - ~ g ' OxjOxk S (r, gt)dcr cy
(4)
where cr represents the unit sphere, Ag is the gravity anomaly on the geoid, S is the generalized Stokes function, and g is the spherical distance between integration and evaluation points. Finally, we apply Newton's density integral in terms of local Cartesian coordinates to model the gradients due to local topographic masses:
a ~ ~v~ dv, E/~ - Cp ax/Ox, Ix- x
(5)
gravity gradient and a normal gradient, as implied, e.g., by the standard Geodetic Reference System of 1980 (GRS80). In the following, we use the same symbol, /-'j~, to denote a component of the gradient disturbance tensor.
Here, the volume element is d v - dxldx'2dx; , and the density, p , is assumed constant. The second-order local derivatives in terms of the spherical derivatives are given by 02 1 0 1 02 _ _ z ____-71-____ OXl2 r Or r 2 O02
2.2
32 m cot0 3 1 3 1 32 -Jr-m m -Jr-_ _ 0X22 r 2 0 0 r Or r 2 sin 2 0 022
Individual Models
Three types of models can be used to represent the gradient disturbances: a global spherical harmonic model derived from an existing set of geopotential coefficients; the solution to a boundary value problem in which gravity anomalies constitute boundary values; and Newton's density integral. In each case, we start with the corresponding expression for the disturbing potential and simply obtain second-order derivatives in a local coordinate system. For the global spherical harmonic model, e.g., EGM96 (Lemoine et al., 1998), given in spherical coordinates, ( 0 , , t , r ) , being co-latitude, longitude, and radial distance, we have:
O2
O2
OX32
Or e
32
cot0
OXlOX2 O2
3x~Ox3 02
sin 0 02
1 O r 2 00 1
OX2OX3
2.3
r 2
O
1 r 2
02
sin 8 003A
1 32 t
r {)O~r 0
r 2 sin 00,;L
1
32
r sin 00,2cOr
(6)
The Combined Model
The basic principle that consistently combines these various representations of the gradients is the --~-'~2 32 ( ( R / n + l / remove/restore technique, often used to combine C k EGM96 G M Cnm --Ynm (0,,,~) models of the disturbing potential from different R n=2 m=-n {)XJ~)Xk sources. We use Stokes's integral as the basic model and include the global model and the effect of (3)
289
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L. Zhu. C. Jekeli
local terrain by first removing these from the gravity anomaly. In order to keep the residual gravity anomalies small, the terrain effect is "removed" in the form of the Helmert condensation, approximated by the usual terrain correction (resulting in the Faye anomaly). However, here we must be careful to note that the corresponding restored gradient is not simply the gradient due to the residual terrain, but that due to the restored Helmert condensation. The final formula for the gradient is given by
which the gradients were computed (elevation of 1100 m and 1290 m, respectively), about 400 m above the mean along-track terrain in each case.
t, .:,7 .a'."
I ;Ih', I'11,, li~lpa
VI.41 O3
q)
,'.~.;i I
"{D
Ti:41
"11", ,:. nil I p.',"
u',, "4 I 4-, ='J:41
1--'jk -- C - k IEGM9 6
4~:4n
+4~
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--
Agterrain -- AgEGM96 )
~T
OZS (r,~) dc r
.~'a i
ax~axk
15 ".'
_-1"4 i .'a'," - I .°11 ~lKr
-
- I '11 .i., .~i'¢
~,
ilr,:il-
longitude [deg]
-I-/-"jk Iterrain (7) ~.% i.,,'- 'II "
.1,.l r.~l
where
-1,.|
AgEGM96 =
GM R2
Z36°2
-(n-1)CnmYnm ( O,/~)
(8)
III
n=2 m = - n
Agterrain - -c
• It,,'~r,'q~" "O
(9)
and c is the usual terrain correction. /-'jk Jte,~ain is the difference of gradient effects due to the residual terrain and the Helmert condensation layer on the geoid:
/--'jk [terrain--1--'jk[res.terr.--1--'jk[layer
3
11,
'-'-' ~Cr)
(10)
Examples and Discussion
3.1 Data We chose an area in California over the San Andreas Fault, being the test site for a recent airborne gradiometric survey. Digital terrain data were available from USGS in the form of 30 m gridded values, derived from the Shuttle Radar Topography Mission (SRTM) conducted by N G A and NASA. A total of 286 free-air gravity anomalies obtained from NGA and approximately evenly scattered in this 22.55<15' area was interpolated onto a l'xl' grid. Figure 1 shows the Digital Elevation Model (DEM) (mean elevation = 706 m) for this area, as well as the gravity anomalies, and the two test profiles on
- l.:i
{
_ .~:,
m
- I.:l -13
;.= '1 " Ill • - I ~:l 4~.',.41 •
- I ~., "1~" =11
-
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-I .'1: _~)"l:r
longitude [deg] Fig. 1 30 m resolution DEM (top), and 15<1' resolution gravity anomaly (bottom), with test profiles indicated.
3.2 Modeled Gradients Models for the gravitational gradient disturbances were constructed by consistently combining boundary-value problem solutions (Stokes integral and spherical harmonic expansions) with Newton's density integral (equation (7)); see Figure 2. The gravity gradient implied by EGM96 is almost constant along these 10-km profiles, since the resolution of EGM96 is several times larger than the profile length. Figure 3 gives the corresponding spectral content of the modeled gradients, in terms of power spectral densities (psd's), for both profiles shown in Figure 1. Also included are the psd's for the modeled gradients on corresponding profiles much lower at 140 m above the mean terrain. Clearly, the upward continuation strongly filters the
Chapter 43
high frequencies, as expected. Qualitatively, one could say that most of the signal energy at the higher profiles occurs at frequencies below 2 cy/km. To determine if the modeled gradients at 400 m above the mean terrain lack high frequency content because of the constant density assumption, we synthesized a terrain with a plausible variation in the density (as deduced from the inversion of gravity data in the region), as shown in Figure 4(left). Figure 4(right) verifies that the density assumption does not significantly determine the gradient power at high frequencies. We also considered a gradient model derived solely from residual gravity anomaly data on a very dense grid, again with the objective of determining if gradients could have greater power at high frequencies than suggested by the terrain-derived model. In this case, we used another similar area over southern
CombiningGravity and TopographicData for Local Gradient Modeling
•
California where high-resolution (200m) data are available. A 32 km test profile with 35 m sampling interval was designed at altitude 400 m above the mean elevation along the profile. Figures 5 and 6, respectively, show the modeled gradients and their psd's from the residual gravity using Stokes's integral. We see from these examples that even from high-resolution gravity data, the spectral power of the modeled gradients is concentrated at frequencies below 2 cy/km. Further results (not shown) in this case also verify that the field attenuation associated with the upward continuation is largely responsible for the lack of high-frequency power. It is interesting, however, to note the increase in gradient magnitude due to gravity data as the resolution of the latter increases (compare Figures 2 and 6).
50
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i
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Ui ~
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i
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Fig. 2 Modeled gradients (64 m computational resolution) along profile 1.
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Fig. 3 Along-track psd's of the modeled gradients; profiles 1 and 2; different heights above mean elevation.
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Fig. 4 Plausible density variation of topographic elements (left); power spectral densities of F33 gradient derived from constant- and variable-density topographic elements (right; dotted and solid lines, respectively).
Chapter 43 • Combining Gravity and Topographic Data for Local Gradient Modeling .14 I:1:1
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50
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Fig. 6 Gradients from high resolution residual gravity data.
293
294
L. Z h u . C. Jekeli
10 7 E
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0s
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2 4 6 0 2 4 6 Frequency [cy/km] Frequency [cy/km] Fig. 7 Spectra of gradients along test profile derived from high-resolution gravity data.
Conclusions
We have developed a consistent approach to modeling gravity gradients from multiple sources, including a global model, regional gravity anomaly data, and local terrain data. Stokes's integral serves as the basis for a remove/restore procedure. High-resolution data, typically topographic data (but also high-resolution gravity data) contribute the most to the total gradient signals. Clearly, as it is well known, the upward continuation acts as a strong low-pass filter. High-frequency variations in the density model do not affect the high-frequency spectrum of the gradients at altitude. Knowing the cut-off of the upward continuation filter of the signal, airborne gradiometer systems can be better tuned with appropriate processing filters to remove high frequency errors due to aircraft dynamics and other systematic sources. For example, airborne gradiometer observations over this test area (400 m above the mean terrain) with spectral content significantly greater than indicated by the model at
8
frequencies higher than 2 cy/km could be suspected as being erroneous. The appropriate filter to use in airborne gradiometry, of course, would also take the instrument noise into account, but the models considered here provide at least an upper bound on the type of signal to be expected.
Acknowledgments This work was supported with funding through contracts with the National GeospatialIntelligence Agency, contract nos. NMA401-02-1-2005 and HM1582-05-1-2009.
References Agarwal, B. N. P. and Lal, T. (1972): A generalized method of computing second derivative of gravity field. Geophysical Prospecting, 20, 385-394. Gunn, P.J. (1975): Linear transformations of gravity and magnetic fields. Geophysical Prospecting, 23, 300-312.
Chapter 4 3
Heiskanen, W.A. and H. Moritz (1967): Physical Geodesy. W.H. Freeman and Co., San Francisco. Jekeli, C. and L. Zhu (2005): Comparison of methods to model the gravitational gradients from topographic data bases. Geophysical Journal International, in press. Lemoine, F.G., Kenyon S.C., Factor J.K., Trimmer R.G., Pavlis N.K., Chinn D.S., Cox C.M., Klosko S.M., Luthcke S.B., Torrence M.H., Wang Y.M., Williamson, R.G., Pavlis E.C., Rapp R.H., Olson T.R. (1998): The development of
•
CombiningGravity and TopographicData for Local Gradient Modeling
the joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) geopotential model EGM96, NASA Technical Paper NASA/TP-1998-206861, Goddard Space Flight Center, Greenbelt. Mickus, K.L. and J.H. Hinojosa (2001): The complete gravity gradient tensor derived from the vertical component of gravity: a Fourier transform technique. Journal of Applied Geophysics, 46, 159-174.
295
Chapter44
Numerical implementation of the gravity space approach - proof of concept G. Austen, W. Keller Stuttgart University, Geodetic Institute, Geschwister-Scholl-Str. 24/D, 70174 Stuttgart, Germany
Keywords. gravity space, geodetic boundary value problem, telluroid
Abstract
unknown surface OG to the unit sphere or. We are looking for a function V : R 3 \ G ~ R and a diffeomorphism ~ with the following properties aVtx)
=
0,
V aa The classical geodetic boundary value problem is a linear free boundary value problem, which implies considerable mathematical difficulties for the investigation of its existence and uniqueness properties. In 1977 F. Sans?) found a breakthrough by transforming the problem into the gravity space, using Legendre transformation. Nevertheless, the transformed problem still suffers from a singularity at the origin. W. Keller proposed in 1987 a modified contact transformation, which provides a boundary value problem free of singularities. Despite its conceptual advantages the gravity space problem was not yet implemented numerically. The paper aims at a study of this numerical implementation in the global case. It gives indication that gravity field determination can also successfully be carried out in the gravity space.
1
Gravity space theory
vv
oc
xca
~\G
f og
(4)
g o ~,
(5)
with (7 being the closure of G. From its mathematical structure (3)-(5) is a socalled free boundary value problem with a high complexity both in the theory of existence and uniqueness as well as in the numerical implementation. In 1977 F. Sansb (Sansb, F. 1977) found a brilliant approach to transform the problem (3)(5) into a boundary value problem with a fixed boundary by introducing new coordinates (6)
:= V V t x )
and a new potential ~, the so-called adjoint potential by ~(~) = x ~ V V ( x ) -
V(x),
x = x(~).
~
-
({TV{~)- ~) {=g
f:cr
--+
R
(1)
g:cr
-+
IRa .
(2)
o
(s)
f.
(9)
Equations (8) and (9) constitute a nonlinear boundary value problem with a Monge-Amp6re type of differential equation and a linear boundary condition. Unfortunately the asymptotic behaviour of the gravitational potential v~
1
,
Ix
-~
x
leads to the asymptotic behaviour The concatenations (f o 9~)(q):= f ( g ( q ) ) and (g o g ) ( q ) : = g(g(q)) map the data given at the
(7)
In the new coordinates ( the adjoint potential ~) solves the following boundary value problem t~(v~) ~ -(t~v~)
The classical geodetic boundary value problem can be formulated as follows: Let G c R 3 be a bounded, open, simply connected domain. Its boundary OG is supposed to be diffeomorphic to the unit sphere or. The diffeomorphism is denoted by g : OG ~ o-. On cr two functions f, g are defined
(3)
(~o)
Chapter44 • NumericalImplementationof the GravitySpaceApproach- Proofof Concept
of the adjoint potential. Equation (11) describes the so-called singularity of the gravity space approach at the origin: The adjoint potential is not differentiable at the origin but it has to solve a differential equation of second order at exactly that place, since (8),(9) is an interior problem. The singularity reduces the mathematical beauty of the gravity space approach and has its origin in the fact that the transformation (6) maps the point at infinity to the origin. Since the concept of differentiability has no meaning at the point at infinity, it is not a surprise that in the new gravity coordinates the differentiability is lost at the origin. The singularity can only be avoided by a similar gravity space transformation, which leaves the point at infinity as a fixed point.
2
Regular
gravity
In (Keller, W. 1987) a gravity space transformation was given, which does not generate a singularity. Similar to equations (6) to (9) the modified transformation is described by OV P~ " - Ox~ '
i = 1,2,3
(12)
& : = - ( G M ) ~ / 2P,ii = 1,2,3 ilpll~/~, , := p i x i - V
-GM (
-
ai~,Vkz
:-
and
&G )
C~ik'Ykj = 5ij
9~
(13) (14)
T h e o r e m 1 Let the coefficients flij~ be defined by
=
R e m a r k : In the geometry space two versions of the Molodensky problem are considered: The so-called vectorial Molodensky problem, where the complete gravity vector g = VV is used as boundary data and the so-called scalar Molodensky problem, where only the modulus g = IIVVII enters the problem as boundary data. From the observational point of view the modulus 9 is much easier to access than the complete gravity vector g. Hence, in the geometry space the scalar Molodensky problem is the prevailing form of dealing with geodetic boundary value problems. The gravity space approach introduced by F. Sansb is an example of a so-called contacttransformation. Contact transformations were first used by F. Klein (Klein, F. 1891) and are based on the idea to consider the solutionmanifold of a differential equation as the envelope of its tangential spaces and to transform the equation for the manifold into an equivalent equation for its tangential spaces. The latter then is hopefully easier to treat than the original one. Since the quantity, which describes the tangential plane of a manifold V is the gradient VV the complete gradient and not only its modulus has to be known in order to apply a contact-transformation. Therefore a contacttransformation (or in other words a gravity space transformation) for the scalar Molodensky problem cannot be found, despite the fact that the scalar formulation is much closer to reality.
space
transformation
(16)
(97 i rn
(17)
®k
and let • denote the following m a t r i x :
a2~
Then
AV=0
~=~
1
d~t+(t~+~-[t~+l
2)
-o
(19)
and
w
- f ~(--~
-~)
-f (20)
holds.
Equations (19) and (20) again establish a nonlinear boundary value problem with a fixed boundary. But this time for tile adjoint potential ~/: the asymptotic relation
W~
1
~,
~1-~o~
(2,)
holds, which means that tile modified gravity space approach is free from any singularity.
297
298
6. Austen• W. Keller 3
Linearised
problem
Nevertheless, the singularity-free boundary value problem (19), (20) is still vastly complicated. Since with the normal potential u :-
GM
(22)
Ilxll
we have a first approximation of the actual gravitational potential U ~ V, we can expect that the adjoint normal potential ~0, given by :=-2--
GM
(23)
I111
is a first approximation of the actual adjoint potential ~. The linearisation of (19), (20) at ~0 leads to the following boundary value problem for T := ~/: -- ~0: Theorem
~- := ~ -
2 The adjoint disturbing potential ~0 solves the following boundary value
problem
-([
w
+
- Vloo
- Ulo
- av
(25)
where E is the gravimetric telluroid defined by the following equation: g = VV(x)=
x •
oc,
•
(26)
Basically, the gravimetric telluroid E is the image of the Earth's surface OG under the mapping (13). In (Keller, W. 1987)it is shown, that for a spherical normal potential U this deftnition of the boundary surface E is equivalent to the implicit definition by (26). It is remarkable that the linearised problem (24), (25) is mathematically of the same structure as the linearised Molodensky problem. Only potential and gravity have changed their places: In the linearised Molodensky problem the boundary surface is defined by a p o t e n t i a l relation V(x) = U(~); in the linearised gravity space approach tile boundary surface is defined by a g r a v i t y relation VV(x) = VU(~). On the other hand, in the linearised Molodensky problem the boundary values are g r a v i t y anomalies, while in the linearised gravity space approach the boundary values are p o t e n t i a l disturbances. Due to Remark:
the identical mathematical structure all algorithms and procedures developed for tile Moledensky problem can be carried over to the linearised gravity space approach. The consideration of the linearised gravity space approach instead of the usual Molodensky problem has two conceptual advantages and two disadvantages: 1. The boundary data 6V in the gravity space approach are smoother than the boundary data Ag of the Molodensky problem. This is conceptual understandable since both field quantitles are measured at tile Earth's surface but the potential is smoother than its gradient. The fact that in the gravity space approach the incasured potential values are mapped to the gravimetric telluroid does not alter their smoothness. Besides this, for a global simulation study this greater smoothness of potential data will be numerically proved. 2. The potential data 6V are related to spiritleveling and are therefore available with a higher density than the gravity measurement related gravity anomalies Ag of the Molodensky problem. Even more: On the oceans (besides the small influence of dynamical topography) the potential V is constant. Hence, the boundary values on the oceans are continuously available. 3. The only disadvantage of the gravity-space formulation is that the necessary gravity vector g = VV is not available but only its modulus 9 -- ]]VVII. This results ill a horizontal uncertainty of the telluroid E smaller than 1500 m or, expressed as a relative deviation, of 2 . 1 0 -4. On tile other hand, the boundary surface E will never actually been used as computation surface. Instead of that all data given on E will be harmonically continued to a sphere. The deviation between the telluroid E and the sphere S enters the harmonic continuation formula only as a correction parameter and has not to be very precise. This means, a deviation error of 2.10 -4 is tolerable for the purpose of harmonic continuation. 4. The normal potential U approximates the actual potential V only with an accuracy of 10 -3. This leads to the unfavourable situation that the telluroid undulations and the potential anomalies 6V get rather large. This means that for the continuation of the boundary values 5V from the telluroid E to the computation surface much more care than in the usual Molodensky case has to be taken. The numerical studies will show that a continuation with a sufficient accuracy is
Chapter 44 • Numerical Implementation of the Gravity Space Approach - Proof of Concept
possible, but on a computationally high price. A linearisation at an ellipsoidal normal potential would very much simplify the continuation process. But at the time being it is not clear whether or not a new linearisation point would preserve the Molodensky-type structure of the linearised problem. Despite those conceptual advantages and the strong similarity to the well known Molodensky problem the gravity space approach was never implemented numerically. With the global sireulation study presented in the following the feasibility of the numerical implementation will be shown.
4
Proof of concept
For the global study the TUG87 model (Wieser, M. 1987) up to degree and order 180 was used as a model for the Earth's topography. As a model for the gravitational field of the Earth the GPM98B model (Wenzel, H.-G. 1998) up to degree and order 720 was applied. On a ~ , j . AA grid, which was designed to meet the requirements of the subsequent Gaussian integration step, the gravitational vector
gij
:=
~TvGPM98b(~gi,j" A,k, hTUG87(qDi,j" /k)~)) (27)
on the Earth' surface was computed. Solving the equations g~j = V U ( ~ j ) (28)
The boundary surface E is shown in Figure 1 by plotting the deviations of the gravimetric telluroid wrt. to a best-fitting ellipsoid of revolution E r . The differences between E and E z (Figure 1) vary f r o m - 1 km to 5 kin, which reflects both the spatial variation of the potential V and the variation of the topographic heights h, though the prevailing influence on E is clearly the topography. The corresponding parameters of E r , i.e. semi-major axis a s and flattening f r , were determined by least-squares methods from rz~j -- a~(1 - fz sin 2 ~i), where rx~j is known from
rr~-
~j
,
~j~E
(30)
and represents the geocentric distance of the gravimetric telluroid points. The following parameters were estimated: a s = 6372993m and e~ - 2 f z - f~ - 0.001599. This means the boundary surface in gravity space features a significant lower flattening than in geometry space (cf. e ~ c M - 0.006694), which leads, on the one hand, to separations of the gravimetric telluroid and the physical Earth surface of up to 10 kin, as a result of using a spherical normal potential as linearisation point. On the other hand a reduced flattening can be of benefit for the continuation process of the boundary data from the gravimetric telluroid to the computational sphere, as will be discussed in the next paragraph.
by Newton's method for (ij, the gravimetric telluroid E was determined point-wise. [m] 5250 1000 700 400 200 0 -200 -400 -650 -850 -1050
Figure 1. Gravimetric telluroid
(29)
299
300
G. Austen • W. Keller
Next, p o t e n t i a l d i s t u r b a n c e s (~V = V[oa- U[oa were c o m p u t e d at the E a r t h ' s surface cOG. U n d e r the m a p p i n g of (13) t h e y represent the b o u n d a r y values on E in gravity space. G r a v i t y anomalies on a reference ellipsoid were d e t e r m i n e d for comparison. Figures 2 and 3 display the Fourier spectra of b o t h d a t a types, confirming t h a t potential d i s t u r b a n c e s are indeed m u c h s m o o t h e r t h a n gravity anomalies. G r a v i t y anomalies on the E a r t h ' s surface would even be m u c h rougher.
2d FFT of potential
T a b l e 1. Geoid accuracies after tion.
..........
i ................................... ~.................. i .................................... ' ..................
0 . 1 5 ......... i ................. i ................. !.................. i .................. ~................. i ..................
disturbances
.............~...............i................ii................i...............................i........................
......i.........~........~........~........i........,........i........ ............i...............i................~................!...............................~.........................
o.1 .......... ~.................. :'................. ~.................. ~.................. ~................. ~..................
0.05 .......
o-
~ ................. ~................. ~.................. i .................. ! ................. !..................
~
............i...............i................',................~...............~................~.........................
.................
[l/rad]
[llrad]
F i g u r e 2. Fourier spectrum of potential disturbances on the gravimetric telluroid
2d 0.12
.........................
" ................. ~ .................
i ..................
FFT
of
gravity anomalies
! ........................................................................................................
~...............................
~........................
0.1-
0.05 .......
~................. "................. T
................ ~.................. ~.................. ~.............................................
0.05 ..........i.................~................. ..-................. i .................. ~.................. ::..................
0.04 ..........
i................ i................ i ............... i.......................................
........... ii ............... !................ i................ i ............... i........................................
...........i ...............i................i................i...............................~................~....... i .................
i .................
~ .................
[l/rad]
! ..................
~..................
2 nd
itera-
no. of
RMS
mean
std
max
rain
iteration
[cm]
[cm]
[cm]
[cm]
[cm]
1
3.3
1.9
+2.8
50.7
-42.9
2
1.5
0.8
±1.2
36.7
-41.9
6Vn-FIIs
-- 6rnls
-]-- c
T E c E
......i.........i........~ '........i................~........~........ i ................. ~...................................................... ~................. ~..................
and
In a following step the p o t e n t i a l anomalies on the gravimetric telluroid were iteratively u p w a r d continued to an enclosing Brillouin sphere (R = aEGM = 6378136.3m) using collocation in a remove-compute-restore mode
............................................................. i ................................................................................................................... i......................................................... O.Z5 . . . . . . .
1 st
~..................
[l/rad]
F i g u r e 3. Fourier spectrum of gravity anomalies on a reference ellipsoid (a -- 6378136.3m, e 2 = 0.006694)
E-1
(6V
--
6Vn)lE
(31) In e q u a t i o n (31) S denotes the Brillouin sphere, C r r the auto covariance m a t r i x of the ~ V ~Vn values on the gravimetric telluroid E and C s z the cross-covariance between the ~ V (~Vn values on the gravimetric telluroid E and on the Brillouin sphere S. T h e iteration was s t a r t e d with 5Vo c o m p u t e d from the E I G E N G R A C E 0 2 S model, a satellite-only spherical harmonic model complete up to degree and order 150. T h e auto- and cross-covariances were derived from the E G M 9 6 model above degree and order 150. Two iterations were carried out. T h e deviations between the continued p o t e n t i a l d i s t u r b a n c e s after each iteration step and the exact values comp u t e d from G P M 9 8 B are displayed in Figures 4 and 5. After the second iteration the continuation errors are below 0.3m2s -2 in t e r m s of potential values or 3 cm in t e r m s of resulting geoid u n d u l a t i o n errors. Errors above are the e x t r e m e values! T h e vast m a j o r i t y of errors is two orders of m a g n i t u d e smaller, giving a root m e a n square (RMS) value of 1.5 cm (compare Table 1). Keeping in m i n d t h a t the c o n t i n u a t i o n was p e r f o r m e d over a distance of up to 10 km the o b t a i n e d accuracy is stunning. W i t h a c o m b i n a t i o n of F F T a n d Gaussian q u a d r a t u r e the spherical h a r m o n i c coefficients v~,~ of the continued p o t e n t i a l d i s t u r b a n c e s ~V were c o m p u t e d (2~
n
Z Z n=2 m=--n
(32)
Chapter 44 • Numerical I m p l e m e n t a t i o n of the Gravity Space Approach - Proof of Concept
[m2/s 2] 5.0 1.0 0.5 0.3 0.1 0.0 -0.1 -0.3 -0.5 -1.0 -4.0
F i g u r e 4. Upward continuation error after first iteration [m2/s 2] 5.00 0.30
60 °
-
.
.
.
.
.
.
.
.
60"
0°
0.10 0.20
0.05 0 o ~,
0o
0.00
~,~.~
-0.05
-30 °
303°
-0.10 -0.20 -0.30 -4.00
F i g u r e 5. Upward continuation error after second iteration This is done by representing the definition of the spherical h a r m o n i c coefficients v~.~ as an i t e r a t e d integral
Vnm
1 f 6v. Y~dS 47cR2 Js
4~_R~, ×
P~(cos 0)
sin rn,k
/o
~v(o,~)
sin OR2dAdO.
(33)
T h e inner integral is evaluated by the t r a p e z i a n q u a d r a t u r e rule
A closer look to the e q u a t i o n (34) shows t h a t the quantities an.~ (0), bn.~ (0) are exactly the discrete Fourier coefficients of 6V(O,A). This m e a n s instead of using the t r a p e z i a n rule these quantities can be c o m p u t e d more efficiently by F F T . T h e r e m a i n i n g outer integral 1 vn.~ = 47cR2
/0
P ~.~ (cos 0)
{
a~.~ (0) bn.~ (0)
}
sin 0R 2dO
(35)
can be t r a n s f o r m e d to
Vnm = 47C
1
P~(~)
b,~(~rcco~)
d~ (36)
and evaluated by a G a u s s i a n q u a d r a t u r e formula N/2
a~.~ (0) b~m (0)
_ 7r
2:r
--ff E-ff-V(O'i--N )
cos rn~ 7 s i n m i 727r
i=0
(34)
v~.~ -- 47c E
j=0
wjP~.~(xj)
anm(arccosxj) bn~(arccosxj)
(37)
301
302
6.
Austen • W.
Keller
with wj and xj being the weights and the nodes of the Gaussian quadrature formula respectively. The (pj = a r c s i n z j , i . AA) grid was designed in such a way that (besides rounding errors) the spherical harmonic analysis is exact up to degree and order 383. (The maximum degree of exact spherical harmonic analysis depends upon the largest available set of nodes and weights for the Gaussian quadrature, which is, if taken from Strout/Secrest, 383). Therefore, the only error contained in the spherical harmonic coefficients v~,~ stems from the downward continuation error and reaches a RMS value of 1.5 cm for the geoid undulations. Since outside the topographic masses
and
1 o% 2 ~l 0 ~ - - T - - 6 V
(39)
holds, the coefficients 7-~ of the spherical harmonic expansion of the adjoint disturbing potential
- Z
Z 77~
(40) - - Tt
are related to the v~,~ in the simple way Trim =
2 n-1
- - V n m
(41)
and with these two relations the adjoint disturbing potential T is known up to an accuracy which corresponds to a RMS value of 1.5 cm for the resulting geoid undulation commission error.
5
Conclusions
For error free simulated data the gravity space approach was tested in the global case at first. Despite the fact that all numerical procedures were implemented rather straightforwardly, the resulting accuracy is about 1.5cm RMS in terms of geoid undulation errors. Further improvements of the numerics can still reduce this value considerably.
References Keller W. (1987): On the treatment of the gcode-
tic boundary value problem by regular contact transformations. Gerl. Beitr. z. Geophysik, 96(4), pp. ls6-196 Klein F. (1891): Considdrations comparatives sur los recherches gdomdtriques modernes (suite). Annales scientifiques de I't~.N.S. 3e s6rie, tome 8, pp. 173-199 Sansb F. (1977): The Geodetic Boundary Value Problem in Gravity Space. Mere. Akad. Naz. Lincei, 14, pp. 39-97 Stroud A. H. and Secrest D. (1966): Caussian Quadrature Formulas. Englewood Cliffs, N J: Prentice-Hall Wenzel H. G. (1998): Ultra-high degree geopotcntial models GPM98A, B and C to degree 1800. Proceedings of the joint meeting of the International Gravity Commission and the International Geoid Commission, Sept. 7-12, Trieste Wieser M. (1987): The global digital terrain model TUG87. Internal report on set-up, origin and characteristics, Institute of Mathematical Geodesy, Technical University, Graz
Chapter 45
Local gravity field modelling with multi-pole wavelets R. Klees and T. Wittwer Delft Institute of Earth Observation and Space Systems (DEOS) Delft University of Technology, 2629 HS Delft, Kluyverweg 1, The Netherlands
Abstract. We propose an alternative approach to the application of spherical wavelet functions for local gravity field modelling from terrestrial, airborne and/or space-borne data. Instead of estimating the finest scale coefficients and recover the detail information by band-pass filtering, we propose a stripping algorithm, which starts at the coarsest scale and directly estimates the wavelet coefficients at the various scales by least-squares. The geographical location and the bandwidth of the spherical wavelet functions are fixed level-by-level using a data-adaptive strategy, which takes the data distribution, the signal variation, and the data noise into account. This reduces the number of spherical wavelet functions significantly and avoids overfitting. The excellent performance of the method in terms of accuracy and numerical efficiency is demonstrated using multi-pole wavelets and a simulated, heterogeneous, scattered airborne gravity data set. K e y w o r d s . Local gravity field modelling, multipole wavelets, spherical basis functions, thresholding, data-adaptive network design.
1
Introduction
The new gravity field missions CHAMP, GRACE, and GOCE have triggered a number of activities devoted to local gravity field modelling using overflight data. The major goal of these developments is two-fold. Firstly, to improve the spatial resolution in local regions with a better signal-to-noise ratio than on a global scale. Secondly, to directly combine overflight data with high-quality terrestrial and/or airborne gravity data set, which allows for higher spatial resolutions over selected areas than satellite data can provide. Wavelet functions are of special interest for local gravity field modelling due to their space- and frequency localizing properties, and the possibility of a multi-resolution analysis.
Wavelet functions, which, when restricted to the surface of a sphere, only depend on the spherical angle between the centre of the function and the computation point, have several favourable properties for local gravity field modelling: (i) they are rotated zonal functions and, therefore, are directly related to spherical harmonics, which are used to represent the long-wavelength part of the gravity field; (ii) a variety of different wavelet functions can be selected by simply changing the definition of the mother wavelet without any need to change the algorithms. We refer to this class of wavelet functions as spherical wavelets throughout the paper. Compared to standard approaches of gravity field modelling using spherical wavelets (e.g. Schmidt et al. 2004, Freeden et al. 1998), we propose two major alternatives: Firstly, we do not compute the wavelet coefficients from the solution at the finest scale using filter operations. Instead, we directly estimate the wavelet coefficients using a stripping algorithm, which starts at the coarsest scale. The next higher scale models the residual signal, i.e. the signal left in the data after the solution at the next coarser level has been subtracted from the data. Secondly, commonly used multi-resolution techniques locate the centres of the spherical wavelets according to some subdivision scheme. Among the most popular choices are a triangulation of an icosahedron, a homogeneous point distribution on the sphere or an equal angular grid. Both hierarchical and non-hierarchical grids are used. We propose a data-driven strategy to design an optimal network of spherical wavelet centres starting with an arbitrary template network of basis functions. This strategy, originally developed by Klees and Wittwer (2005) for single-scale spherical basis functions, determines simultaneously the geographical location and the optimal bandwidth of the spherical wavelets as function of the data distribution, the data noise, and the signal variation in the area. It is also an alternative to wavelet shrinkage techniques.
304
R. Klees • T. Wittwer
It will be shown that this approach yields excellent results in terms of approximation quality and numerical efficiency. In particular, overfitting is avoided, and no regularization is needed for terrestrial and airborne data sets. The outline of the paper is the following: in section 2 we briefly introduce the multi-pole wavelets and the representation of the disturbing potential. Section 3 is devoted to our strategy to directly estimate the wavelet coefficients starting with the coarsest scale. In section 4 we show how the geographical position and the bandwidth of the multi-pole wavelets are chosen. In section 5 the test set-up is described. Section 6 contains the results of the simulations and a discussion of the accuracy and efficiency of the pursued approach. In section 7 we summarize the most iraportant conclusions from this study.
2
Representation of the local gravity field with multi-pole wavelets
It is a standard procedure in local gravity field modelling to subtract an existing, global geopotential model and also a topographic model from the data prior to parameter estimation. This remove step reduces the signal power at the low and the high frequency part of the spectrum and facilitates the estimation of the residual disturbing potential from data distributed over a local area. In this paper, we are only interested in the determination of the residual disturbing potential from the reduced observations within a local area. The data distribution is supposed to be heterogeneous and may also show gaps. In this situation it is advantageous to represent the residual disturbing potential, T~e~, as a linear combination of spherical basis functions (SBFs) at various levels ranging from J,~in to J,~ax: J~ax
T~e~(x) -- E
E /3J,n q2j(x, yj,~),
(1)
where/~j,~ are the coefficients of the wavelet expansion. We consider a class of spherical basis functions, which has the following structure:
Y) -
2/+1
(_~1) l+l
Z l=L+l
~)j,1 -- (aj l) m e - a j 1
(3)
where m is the order of the multi-pole wavelet and aj is a level-dependent scale parameter. The latter is related to the centre of the multi-pole wavelet by y-
(4)
i ~ e - a j y,
see (Holschneider et al. 2003). Figure 1 shows a typical spectrum of the multi-pole wavelets for order m = 3 and various levels. We observe that for a fixed order m, the scale parameter aj determines the shape and position of the spectrum of the wavelet. The larger aj, the deeper the basis function is located below the surface and the more the spectrum is concentrated at lower frequencies. With decreasing aj, the spectrum covers higher frequencies. Correspondingly, the centre of the basis function moves towards the surface, which is equivalent to a better space localization.
6,~~=0.012
~
a7=0.006
Nj
j = J ~ i ~ n=l
o~
of the SBF to the surface of a sphere is a function of the spherical angle between the location of the basis function y and the computation point x. The sampling properties of the SBF are determined by the coefficients ~j,1. They depend among others on the level j and the location of the center of the SBF y. We use the multi-pole wavelets introduced in (Holschneider et al. 2003):
(2)
x ~) - l-y[' Y and R is the radius of where 2 - K[' a reference sphere. In particular, the restriction
500 1000 spho.rJp.~lh~rmnnie,deor~
1500
Figure 1: Spectrum of the multi-pole wavelets for order m = 3 and various levels j. Note that the maximum of the spectrum is located at m/aj. The smaller the scale parameter aj, the higher the maximum harmonic degree that is essentially covered by the multi-pole wavelet. The parameter aj is approximately equal to the depths of the multi-pole wavelet below the surface of the unit sphere.
Chapter 4 5
Estimation of the multi-pole wavelet coefficients Suppose we are given Nob~ observations {li • i -1 . . . Nob~}, which are linear functionals of the residual disturbing potential: -
&....
Nj
Z Z
j=Jmin
yj,.)+
i -
1...Nob~.
n=l
We assume that the number of observations exceeds the number of wavelet coefficients to be estimated. The errors {e~ : i = 1 . . . Noa.~} are normally distributed as A/'(0, C), where C is the noise covariance matrix. Suppose, 1 is the vector of observations, A is the design matrix, and x is the vector of unknown multipole-wavelet coefficients. At a first glance, one may think of computing the solution of the linear model (5) that minimizes the quadratic functional (I)(x) -- ( 1 - A x ) T C - I ( 1 -
Ax) - axTx,
(6)
where c~ _> 0 is the regularization parameter. However, this is not useful, due to the significant overlap of the spectrum of the multi-pole wavelets at various levels (see Figure 1). Instead, we propose the following procedure. We start with the coarsest level j - J,~i~ and estimate the wavelet coefficients at that level, {/3&,,~,~ " n - 1 . . . Nj,,,~}, by minimization of the quadratic functional, Eq. (6). The residuals of the least-squares adjustment are taken as observations to estimate the wavelet coefficients at level J,~i~ + 1. This procedure is continued until the coefficients of the m a x i m u m level J , ~ x have been estimated. Note that the noise covariance matrix of the residuals at level j is the noise covariance matrix of the observations at level j + 1. In the test case mentioned here, the initial covariance matrix C was used for all levels. This introduces small errors in the estimated coefficients, which were negligible. If the data-adaptive algorithm of section 4 is applied, the choice of J,~i~ is up to the user, and J , ~ x is fixed automatically. The coefficients of level J,~i~ represent the solution at the coarsest scale. The coefficients of level j represent information, which is not
•
Local Gravity Field Modelling with Multi-Pole Wavelets
included in the coefficients of the levels J,~i~J~,i~ + j - 1. Since with increasing level tile number of basis functions increases and the bandwidth decreases, the coefficients of higher levels represent short-scale information in the gravity field. Correspondingly, higher levels represent detail information not included in the coarser levels. Therefore, it is justified to call Eq. (1) a multi-scale representation.
4
Choice of the multi-pole wavelet centres and bandwidths
To estimate the wavelet coefficients, we first have to fix the centres and the bandwidths of the multi-pole wavelets. Both are important design criteria to get a good local gravity field model. For a fixed order m, the bandwidth of the multipole wavelet is solely determined by the scale parameter a. Moreover, the scale parameter fixes the depth of the multi-pole wavelet according to Eq. (4). Therefore, fixing the centre and the bandwidth of a multi-pole wavelet means that 3 parameters have to be determined: (i) the location on the unit sphere (2 parameters) and (ii) the scale parameter. Klees and Wittwer (2005) have proposed a data-adaptive network design (DAND) strategy to select the centres and tile bandwidths of radial basis functions automatically using information about data distribution, data quality, and signal variation. We generalize this approach to the multi-scale situation. To keep the derivations concise, we only address the aspects related to the multi-scale situation. Aspects, identical with the single-scale version are skipped; the reader is referred to (Klees and Wittwer, 2005). We start with a template network of points on the unit sphere and compute the region of influence (ROI) of a network point, which is a spherical cap with radius A 2
(1 -
Xo
/
(7)
centered at the network point. A is the size of the area projected onto tile unit sphere. Each level knows its own template network and ROIradius f r o 1 . In the simulations, section 5, we use equal-angular template networks. The generation starts with a square on the unit sphere, bounded by meridians and parallels, and covering the data area. Then, the square is successively subdivided in 4j-1 smaller squares, where j denotes the level. In that way we obtain a
305
306
R. K l e e s • T.
Wittwer
template network of points on the unit sphere for each level j = J , ~ i n . . . Jmax. The template network of level j has 4j-1 points. When the template networks have been generated, we proceed with the coarsest level j = d,~in. We select a candidate parameter p , compute the bandwidth cry = p - ¢ R o I , j and the scale parameter aj, select the centres {yj,~ : n = 1 . . . Nj} (cf. Klees and Wittwer, 2005), and estimate the coefficients ( f l j , n : n = 1 . . . N j } by least-squares. This procedure is repeated until the m a x i m u m level J,~ax is reached. Thereafter, we compute the Generalized Cross Validation functional for the chosen parameter p. The value Pg~v t h a t rainimizes the GCV functional is the optimal p. In this way, we obtain the centres of the multi-pole wavelets in 3D-space for each level j. The solution {flj,n, j = d,~i~ . . . g m a x , n = 1 . . . N j } obtained with the value Pgcv is the optimal solution. If regularization is applied (i.e. for a non-zero c~ in Eq. (6)), a second iteration loop inside the GCV loop is required. The optimal regularization parameter is found using Variance Component Estimation techniques (see Koch and Kusche, 2002).
258"
36 °
• ~
o
34"
34"
32 °
32"
30"
30 °
258 °
260"
Ao
-~o
262 °
o
, mGal
do
~o
Figure 2" Gravity disturbances over the test area. The RMS signal variation is 6.2 regal, the range is - 3 0 to 45 regal. 258
°
±:'i~':"-:-=::i-'.~
260
°
• "i..: .......• .}:: _ ' : ' : i .
262
°
i .-2~.v. "; L:5"i'a.':
--:-i:i-i::~i:.i-L-=-:f: ::. i -- -:4. .-:..-: -.: :.~i :-::-.--:ii:.-:.-::i~ 36 °
Simulations
To investigate the performance of the multi-pole wavelet approach pursued in sections 3 and 4, we did extensive numerical test with simulated terrestrial and airborne gravity data sets. The results to be presented refer to an airborne gravity data set. The area is located in the US, North to the Gulf of Mexico. The size is about 6 x 6 degrees. The gravity disturbances at a flight altitude of 2 km range from - 3 0 reGal to 45 reGal; the RMS signal is about 6.2 regal. The frequency content of the data is limited to degrees 121 to 1800. Therefore, the parameter L in Eq. (2) is set equal to 120. Figure 3 shows the gravity signal over the test area. We generated 5453 gravity disturbances, h e t e r o g e n e o u s l y d i s t r i b u t e d above the area at an altitude of 2 kin, see Figure 3. The data are corrupted with white noise with a standard deviation of 1.5 reGal. All computations are done with multi-pole wavelets of order m = 3. The lowest level is J,~in = 5.
6
262"
36"
ae°
5
260"
Results and discussion
We have computed the level 5-7 multi-scale solution using the approach described in the sections
34 ° i :i -i-.iiiL i iK i::::_.:-:__:_.-:i;2:=-
34 °
32 °
30 °
30 ° 258 °
260 °
262 °
Figure 3: Data distribution over the test area. The data set consists of 5453 observations. Note the heterogeneous distribution with significant gaps in certain areas. 3 and 4. The template networks comprise 256 (level 5), 1024 (level 6), and 4096 (level 7) centres. The DAND procedure outlined in section 4 selects 244 (level 5), 599 (level 6), and 467 centres, if
Chapter 45 • Local Gravity Field Modelling with Multi-Pole Wavelets
level
~ROI
bandwidth
depth
[degree I 0.2128 0.1064 0.0532
[km]
[kml
28.39 14.19 7.10
83.56 46.40 27.73
Table 1: Spherical radius of the region of influence ~Rof, bandwidth, and depth of tile multipole wavelets for levels 5-7 as selected by the multi-scale DAND procedure (cf. section 4). The optimal parameter Pgcv is 1.2.
a threshold of 3o = 4.5 reGal is used. This is a reduction of about 75%. Compared with a level-7 single-scale solution, the reduction is still significant: 1672 basis functions vs. 1319 basis functions (i.e. about 21%). Table 6 shows for the multi-pole wavelets of level 5-7 the results of the multi-scale DAND procedure. Figure 4 shows the estimated residual disturbing potential at ground level for the levels 5, 6, and 7. The level-5 multi-scale solution represents the solution at the coarsest scale. The solution at level 6 represents detail information, which is not included in the solution at level 5. The solution at level 7 represents detail information that is not included in the sum of the solutions at level 5 and 6. The DAND procedure places multi-pole wavelets only in areas with sufficiently high residual signals. Figure 5 shows a histogram of the true errors at ground level in terms of potential values. The RMS error is 0.27 m2/s ~. The low number of basis functions compared to the number of observations is also an indicator of the numerical efficiency of the pursued multiscale approach. The number of basis functions is 76% less than the number of observations. A Least-Squares Collocation solution with tile covariance function of Forsberg (1987) gives comparable results.
7
Conclusions
The following conclusions are drawn from the conducted numerical experiment. They are confirmed by other experiments with simulated terrestrial and airborne gravimetry data.
The pursued multi-scale approach is wellsuited for regional gravity field modelling from terrestrial and airborne gravity data. It is easy to implement and the numerical complexity is low. The quality of the obtained gravity field model is comparable with Least-Squares Collocation, although the number of basis functions is a factor 4-5 less than the number of observations. The multi-scale version of the DAND algorithm leads to a significant reduction of the number of basis functions at each level. The efficiency gain is maximum for heterogeneous data distributions. The performance of the approach for the regional inversion of satellite data has to be investigated. A particular problem to be addressed is the effect of colored observation noise on the performance of the DAND algorithm.
References Forsberg R (1987) A new covariance model for inertial gravimetry and gradiometry. J Geophys Res 92 (B2), 1305-310. Freeden W, Gervens T, Schreiner M (1998) Constructive approximation on the sphere. Clarendon Press, Oxford. Holschneider M, Chambodut A, Mandea M (2003) From global to regional analysis of the magnetic field on the sphere using wavelet frames. Physics of the Earth and Planetary Interiors 135, 107-124. Klees R, Wittwer T (2005) A data adaptive design of a spherical basis function network for gravity field modelling. Proceedings Dynamic Planet 2005, Cairns, Australia. Koch KR, Kusche J (2002) Regularization of geopotential determination from satellite data by variance components. Journal of Geodesy 76, 259-268. Schmidt M, Fabert O, Shum CK, Hart SC (2004) Gravity field determination using multiresolution techniques. Proceedings Second International GOCE User Workshop, ESA-ESRIN, Frascati, Italy.
307
308
R. Klees • T. Wittwer 258 36"
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Figure 4: Multi-scale solution of the residual disturbing potential at ground level. From left to right: level 5, 6, and 7. The number of multi-pole wavelets per level is 244 (level 5), 599 (level 6), and 467 (level 7). The total number of basis functions is 1310, which is about 24% of the number of observations. The single-scale solution at level 7 needs 1672 basis functions. 15Q0
,
,
O0 1000
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Error [m2/s 2 ] Figure 5: Multi-scale level 5-7 solution. Histogram of the errors at ground level in terms of potential values. The RMS error is 0.27 m2/s 2.
Chapter 46
Accuracy assessment of the SRTM 90m DTM over Greece and its implications to geoid modelling G.S. Vergos[], V.N. Grigoriadis, G. Kalampoukas, I.N. Tziavos Department of Geodesy and Surveying, Aristotle University of Thessaloniki, University Box 440, 541 24, Thessaloniki, Greece, Fax: +30 231 0995948, E-mail: [email protected].
Abstract. With the realization of the Shuttle Radar Topographic Mission (SRTM) and the distribution of the 3 arcsec (90 m) data over Europe, a high-resolution digital terrain model (DTM) became available for Greece. Until today, high-resolution DTMs for Greece were generated by the Hellenic Military Geographic Service (HMGS) only and present variable resolutions with the finest one being set to 100 m. These DTMs were usually determined by digitizing topographic maps and are thus of variable and sometimes unknown accuracy. When a high-resolution, e.g., 0.5 - 1 arcmin geoid is needed, it is absolutely necessary to employ a very high resolution DTM to compute the terrain effects to gravity and the indirect effect to the geoid. If this information is not available and a coarser DTM is used, then the topographic effects computed are aliased, due to the insufficient resolution of the topographic data used. The scope of this work is twofold. First, a validation and accuracy assessment of the SRTM 90 m DTM over Greece is performed through comparisons with existing global models, like GLOBE and GTOPO30, as well as with the Greek 450 m DTMs delivered by HMGS. Whenever a misrepresentation of the topography is identified in the SRTM data, it is "corrected" using the local 100 m DTM. This processes resulted in an improved SRTM DTM called SRTMG, which was then used to determine terrain and RTM effects to gravity field quantities. Then, all available DTMs were used to compute terrain effects on both gravity anomalies and geoid heights at variable spatial resolutions. From the results acquired, the performance of the SRTMG model with respect to geoid modeling was assessed and conclusions on the effect of the DTM resolution were drawn. K e y w o r d s . SRTM, terrain effects, gravity field, geoid modelling.
1
Introduction
Digital Terrain Models (DTMs) play a crucial role in gravity field related studies, since they provide the highfrequency content of the gravity field spectrum. This is due to the high correlation of the short wavelength gravimetric features with the topography. According to Schwarz (1984) about 2% and 34% of the geoid height and gravity anomaly spectrum, respectively, are contained in the high frequencies (harmonic degrees 3 6 0 36000), where terrain effects play a significant role.
Furthermore, modem-day gravity field and geoid approximation is based heavily on the well-known remove-compute-restore (rcr) procedure, during which the terrain data are used to smooth gravity and geoid height observations to aid field gridding, transformations and predictions and avoid aliasing effects (Forsberg and Tscherning 1981; Forsberg 1985; Forsberg and Solheim 1988; Tziavos et al 1988, 1992; Vergos et al. 2005). With the advent and continuous launch of new altimetric and gravity field satellite missions and the collection and availability of new and higher in resolution data, it has become apparent that high-quality and highresolution DTMs should be available for geoid and gravity field approximation. In several countries around the world high-resolution local DTMs are not available due to confidentiality reasons, since they are most commonly generated by the respective geodetic/cartographic military agencies. Furthermore, the DTMs available are usually not homogeneous, since they are derived by a (simple in most cases) merging of available height data. On the other hand, in 2000 the Shuttle Radar Topographic Mission (SRTM) was launched on-board space shuttle Endeavour and collected a wealth of data of the Earth's topography in global scale and with homogeneous coverage. This resulted in the release of a global 3" (roughly 90 m) SRTM DTM by NASA and the National GeospatialIntelligence Agency (NGA). Thus, it was obvious that such a global DTM would offer a great aid in local, regional and global gravity field and geoid determinations, since it could be used to fill-in gaps and densify local and regional/continental DTMs. The first main goal of the present contribution is the validation of the SRTM 90 m DTM over Greece through comparisons with a national DTM generated by the Hellenic Military Geographic Service (HMGS). Their differences are analyzed and a new corrected SRTM DTM called SRTMG05 is generated for the area under study. The second main objective is the evaluation of the generated SRTMG05 DTM, against the national DTM and other global models, for gravity field and geoid determination. This is achieved by estimating the contribution of all models to gravity anomalies and geoid heights through a number of terrain reduction techniques.
2
Digital Terrain Models and Area
For the evaluation of the SRTM DTM over Greece a national DTM for the area under study and the GLOBE
310
G.S. Vergos.V. N. Grigoriadis• G. Kalampoukas.I. N. Tziavos
and ETOPO2 global DTMs were used. The SRTM mission took place in Feb 11-20, 2000 on-board the space shuttle Endeavour. Its main instrumentation was a spaceborne imaging radar modified with a mast like the one used in the International Space station and an additional antenna, so that a 60 m long interferometer could be formed. The SRTM data coverage ranges between 60 ° north to 54 ° south and covers about 80% of the Earth's total landmasses. Bamler (1999) and Farr and Kobrick (2000) should be consulted for more information on the SRTM mission and data. The data used in the present study come from the released "research grade" SRTM 90 m dataset, which means that they were unedited so could contain blunders and voids (gaps). Furthermore, no special processing of the data has been done so that in many cases the measured heights represent what was captured from the radar and not the real elevation. The latter, also known as roof effect, is especially evident over dense forests and populated areas with high trees and high buildings, respectively. The SRTM data for the area under study, bounded between 39 ° < q) < 40.5 ° and 21 ° < )v < 22.5 °, were downloaded from the corresponding US Geological Survey (USGS) ftp site (USGS 2005) and consisted of a total number of 3243601 elevations. Their statistics are presented in Table 1 while Fig. 1 depicts the SRTM 90 m DTM. The total number of undefined elevations (black dots in Fig. 1) in the area was 43199 representing roughly a 1.4% of the total dataset. They were mainly located over river basins and sea areas as well as over the Pindos range stretching from the north-west to the south-west corner of the area. The SRTM data are referenced to the EGM96 global geopotential model, the horizontal datum is WGS84 and their accuracy is at the 16 m level. The local D T M obtained from HMGS, being identified with the same name herein, had a 15" (-450 m) resolution and was generated from the digitization of 1"50,000 topographic maps (HMGS 2005). This is the standard set of heights available in Greece for surveying and engineering applications, it covers the entire country and the heights provided have a formal vertical accuracy of 20 m. A denser 100 m resolution D T M is also available from H M G S but its status is declared as confidential and is not available to the public. In the first part of this study, the 15" HMGS D T M will serve as the ground truth data set against which SRTM will be compared in order to develop a corrected SRTM DTM. Then, the corrected SRTM DTM will be used as reference in the investigation of the D T M effects on the gravity field and the geoid. The statistics of H M G S are presented in Table 1 as well. Moreover, the G L O B E and ETOPO2 DTMs were considered as well to investigate the performance of other, than SRTM, global models in the area under study. GLOBE (GLOBE 2005) is a 30" global D T M generated from a mosaic of vector and raster data sources. Its horizontal datum is WGS84, it refers to the mean sea level and has a formal accuracy for Greece at the 30 m level. Finally, the ETOPO2 D T M (ETOPO
2005) is a global model of 2' (about 3.7 km) resolution generated by assimilating a number of other DTMs and digital depth models (DDM). The models used in the computation of ETOPO2 were GLOBE, ETOPO5, DBDB5, D B D V and the Sandwell and Smith DDM. Using these DTMs and DDMs, ETOPO2 was constructed by regridding them to 2 arcmin resolution by bicubic spline interpolation. Its horizontal datum is WGS84 also and it refers to the mean sea level. The statistics of both GLOBE and ETOPO2 are also listed in Table 1. Gaps in the DTMs over marine areas were replaced by zeros.
Table
1. Statistics of the DTMs and their differences. Unit: [m]. DTMs
max
SRTM HMGS SRTMG05 GLOBE ETOPO2 SRTM-HMGS SRTMG05-HMGS SRTM-SRTMG05
min
2884.00 -23.00 2734.41 -5.68 2884.00 0.00 2710.00 1.00 2552.00 -95.00 653.54 -407.59 5 9 8 . 2 6 -406.94 33.35 -60.54
mean
704.20 716.32 707.16 703.75 690.09 -11.87 -11.73 0.00
std
_+455.04 _+456.33 _+457.77 _+457.41 _+460.28 +70.43 +70.13 +0.94
The first step of the present work was the validation of the SRTM D T M against H M G S and its correction in places where voids in the data existed. Table 1 presents the statistics of the differences between SRTM and HMGS, which have a mean value o f - 1 1 m only and a standard deviation (std) of ___70 m. Taking into account that the topography in the area under study varies significantly, it can be concluded that SRTM provides very good results. The absence of a significant bias between the two models can be attributed to the fact that no roof effect is present in the data, at least in the area under study. Therefore, the only processing done to construct a "corrected" SRTM dataset was to fill-in existing voids with heights predicted from HMGS using spline interpolation. This resulted in the so-called SRTMG05 (SRTM Greece 2005) DTM with the statistical characteristics presented in Table 1. SRTMG05 presents a smaller range of differences with HMGS compared to SRTM by about 60 m. For both models, the larger differences with HMGS are located over the Pindos mountain range and the smallest ones over the plain of Thessaly (central and central-east part of the area). Finally, some large differences can be found approximately at q0=39.5 ° and )v=21.5 ° where the Valia Calda national park, a densely forest-covered area, is located. Nevertheless, the comparisons against the national DTM give significant evidence that the SRTM dataset gives a realistic picture of the topography of the area under study. DTM oid
effects
on the gravity
field and ge-
To investigate the performance of the SRTM D T M and its implications to gravity field and geoid modelling, all
Chapter 46 • AccuracyAssessment of the SRTM90m DTM over Greece and Its Implications to Geoid Modelling
21" 00'
Fig. 1:
21" 30'
22" Off
22" 30'
The original (left) SRTM and the corrected (right) SRTMG05 90 m DTMs. Gaps in data are shown as black dots.
available DTMs (SRTMG05, HMGS, GLOBE and ETOPO2) have been used to estimate various types of topographic corrections on both gravity anomalies and geoid heights. Furthermore, aliasing effects on terrain corrections, i.e., the loss of detail when using coarser in resolution DTMs was studied. This was achieved by constructing lower resolution SRTMG05 DTMs at resolutions of 15", 30", 1', 2' and 5'. The topographic effects on gravity and the geoid computed were (a) full topographic effects, i.e., the combined effect of the Bouguer and terrain corrections, (b) terrain correction (TC) effects, (c) residual terrain model effects and (d) isostatic effects using the Airy model. Furthermore, indirect effects on the geoid have been computed estimating all three terms. The effects from all models were estimated and then compared on a l ' x l ' grid for the area under study, which corresponds to cases that a geoid and/or gravity field model of that resolution is needed. Such a high resolution l'x 1' geoid model is clearly within reach nowadays in the presence of new gravity-field related data. Due to the limited space available no formulations are given since the evaluation of topographical effects is well documented. A very detailed analysis can be found in Forsberg (1984), Heiskanen and Moritz (1967) and Tziavos (1992). The indirect effect on the geoid is explicitly described in Wichiencharoen (1982). Tables 2, 3, 4 and 5 present the statistics of the estimated full topographic effects, terrain corrections, RTM and isostatic effects on gravity from the available DTMs, respectively. From these tables it is evident that the performance of SRTMG05 is directly comparable to that of the HMGS DTM. Their difference in the computed full topographic effect is at the +6.5 mGal level in
terms of the std and ranges between -33 to +36 mGal. This is a very encouraging result, since it shows clearly that the SRTM DTM is indeed accurate and does not introduce any errors when used in gravity field and geoid determination. The same conclusions hold for the computation of the other topographic effects on gravity, since the std of the differences between SRTMG05 and HMGS is +3.3 mGal in the terrain corrections and +6.6 mGal for the RTM and isostatic effects. On the other hand, the differences almost double in magnitude when comparing the topographic effects computed from GLOBE with those derived from either SRTMG05 or HMGS. For example the std of the differences between the TC effects on gravity computed from SRTM and GLOBE are at the +6.5 mGal and reach the +13 mGal on the rest of the effects computed. Moreover, the range of the differences increases from about 60 mGal to 120 mGal. This is evidence that indeed SRTMG05 manages to depict more detail of the topography in the area under study, while GLOBE's
Table
2. Full topographic effects on gravity. Unit: [mGal]. DTMs
max
min
mean
SRTM3" SRTM15" SRTM30" SRTMI' SRTM2' SRTM5' HMGS GLOBE ETOPO2'
244.47 248.29 246.42 250.87 238.54 209.52 257.96 252.79 249.30
- 1.86 -1.56 -2.07 -1.69 -1.16 -9.55 - 1.37 -3.57 -3.05
72.88 72.57 71.63 74.64 75.07 75.88 75.37 73.64 73.78
std
+46.93 4-46.56 4-45.99 4-45.56 +45.30 4-44.37 +46.96 4-46.97 +47.39
311
312
G.S. Vergos. V. N. Grigoriadis • G. Kalampoukas. I. N. Tziavos Table
3. Terrain corrections on gravity. Unit: [mGal]. DTMs
SRTM3" SRTM15" SRTM30" SRTMI' SRTM2' SRTM5' HMGS GLOBE ETOPO2'
Table
min
mean
47.00 52.54 52.71 56.84 68.89 125.61 50.56 70.87 134.24
0.04 0.03 0.08 0.02 0.01 -0.02 0.01 0.02 0.03
6.58 7.14 8.64 8.09 6.36 9.37 5.47 8.24 12.49
std
+5.92 +6.38 +7.55 +7.92 +8.17 +13.41 +5.91 +9.26 +16.04
4. Residual terrain model effects on gravity. Unit: [mGal]. DTMs
SRTM3" SRTM15" SRTM30" SRTMI' SRTM2' SRTM5' HMGS GLOBE ETOPO2'
Table
max
max
183.03 183.51 180.91 185.15 174.35 141.38 191.79 183.57 190.82
min
-91.38 -91.41 -88.63 -88.77 -83.48 -82.10 -84.68 -101.90 -87.52
mean
-3.93 -4.26 -5.16 -2.27 -1.76 -1.28 -2.81 -3.42 -1.39
std
+37.76 +37.39 +36.77 +38.04 +36.57 +33.08 +37.51 +38.31 +38.94
5. Isostatic effects on gravity. Unit: [mGal]. DTMs
SRTM3" SRTM15" SRTM30" SRTMI' SRTM2' SRTM5' HMGS GLOBE ETOPO2'
max
215.27 217.41 214.70 218.81 207.26 175.09 225.95 221.73 217.49
rain
-35.88 -34.54 -36.28 -34.57 -36.17 -35.61 -34.67 -33.60 -33.20
mean
37.63 37.23 36.21 39.01 39.09 38.36 39.42 38.59 38.31
std
+43.62 +43.25 +42.66 +44.21 +43.12 +40.40 +43.64 +43.73 +44.13
resolution is inadequate compared to the high-resolution D T M s available. The results achieved from E T O P O 2 are disappointing, since the differences between its topographic effects and the ones computed by either SRTMG05 or H M G S are at the +25 mGal in terms o f the std and reach 230 mGal in terms of the range. This is a clear indication that in the presence of the SRTMG05 elevation data, D T M s of the E T O P O 2 class should not be used for geoid and gravity field modeling anymore. Comparing the topographic effects from the SRTMG05 DTMs at the generated resolutions (15", 30", 1', 2' and 5') to those from the original 3" D T M some aliasing effects are clear. The m a x i m u m and std values in the TC effects increase gradually as moving from the dense to the coarser resolutions. In the TC effects on gravity anomalies, the std of the differences between the 3" and the rest o f the SRTMG05 models increases from +2.2 mGal for the 15", to +3.7, +3.8, +6 and +10.7 mGal for the 30", 1', 2' and 5' models. For
the R T M effects the corresponding std of the differences is at the +3.2, +4.7, +7.5, +12 and +19 mGal level. The same trends hold for the rest o f the topographic effects as well. Therefore it can be concluded that aliasing occurs when using coarser resolution D T M s for gravity field and geoid modeling. Fig. 2 depicts the TC and R T M effects on gravity as computed from the 3" SRTMG05 model. Fig. 3 (left) presents the differences between the R T M effects computed from SRTMG05 and HMGS. The same topographic effects have been computed for geoid heights as well, considering the case when the restore step is reached in the rcr procedure and the effects of the topography previously removed from the gravity data have to be restored to the estimated residual geoid heights. Tables 6 and 7 present the TC and R T M effects on geoid heights computed from the available DTMs. Once again the 3" SRTMG05 model agrees very well with H M G S with the std of the differences being at the +3.5, +2.7, +2.1 and +1.6 cm level for the computed full topographic, isostatic, TC and R T M effect, respectively. The corresponding range o f the differences is at the 20, 17, 16 and 12 cm level showing once again the very good agreement between the two models. The differences between SRTMG05 and G L O B E are again slightly larger and reach the +5.9 cm in terms of the std and the 40 cm in terms of the range for the computed R T M effects on geoid heights. For E T O P O 2 the differences with SRTMG05 have a std at the +9.4, +8.5, +8.9 and +9.2 cm level for the computed full topographic, isostatic, TC and R T M effect, respectively. Fig. 3 (right) depicts the differences in the R T M effects on geoid heights between the 3" SRTMG05 and HMGS. Table
6. TC effects on geoid heights. Unit: [m]. DTMs
SRTM3" SRTM15" SRTM30" SRTMl' SRTM2' SRTM5' HMGS GLOBE ETOPO2'
Table
max
rain
0.723 0.792 0.930 0.878 0.907 1.146 0.627 0.932 1.357
0.257 0.285 0.344 0.124 0.259 0.382 0.206 0.300 0.467
mean
0.484 0.525 0.635 0.624 0.458 0.677 0.403 0.607 0.919
std
+0.101 +0.114 +0.129 +0.127 +0.132 +0.099 +0.090 +0.126 +0.177
7. RTM effects on geoid heights. Unit: [m]. DTMs
SRTM3" SRTM15" SRTM30" SRTMI' SRTM2' SRTM5' HMGS GLOBE ETOPO2'
max
1.184 l.179 1.174 1.158 1.102 0.972 1.154 1.011 1.273
rain
-0.638 -0.639 -0.640 -0.643 -0.647 -0.802 -0.634 -0.655 -0.685
mean
0.060 0.058 0.055 0.049 0.038 0.027 0.055 0.010 0.071
std
+0.340 +0.340 +0.340 +0.340 +0.339 +0.335 +0.337 +0.336 +0.386
Chapter 46 • Accuracy Assessment of the SRTM 90m DTM over Greece and Its Implications to Geoid Modelling 21" 00' ~'~0
~o' °
o
>
a9 ° ~ '
rib 21" oo'
21" 30'
22" 00'
Fig. 2: TC (left) and RTM (right) effects on gravity anomalies from SRTMG05 (C.I. 5 mGal and 20 mGal respectively). 3' 40°~
39" 00'
39" OC
Fig. 3: Differences of TC effects on gravity (left) and of RTM effects on the geoid (right) between SRTMG05 and HMGS. Investigating the aliasing effects on the geoid from the use of coarser resolution DTMs, the same conclusions were reached. The differences between the TC effects on geoid heights from the original 3" SRTMG05 model and the 15", 30", 1', 2' and 5' DTMs was at the +1, +2.6, +10.1, +10.8 and +12 cm level, respectively. Therefore it can be concluded that an error of that
amount is introduced in geoid determination when coarser resolution DTMs are used. The final computation performed was to estimate the indirect effect (the first three terms of the expansion) on the geoid from all available DTMs. Table 8 presents the statistics of the results acquired. Once again, SRTMG05
313
314
G.S. Vergos. V. N. Grigoriadis • G. Kalampoukas. I. N. Tziavos
agrees very well with HMGS since their difference is again at the few cm level. The G L O B E model behaves much better than ETOPO2, even though the latter resulted from just a re-gridding of the former. So, the bad performance of ETOPO2 cannot be attributed to its coarser resolution alone. Aliasing is evident in the computed indirect effects on the geoid which are more pronounced when reaching the 5' resolution. A noticing fact is the very large std of the computed indirect effect from the 5' SRTMG05 model (+85 cm) when it is only +3 cm for the original 3" DTM. This is a very good example of the error introduced in geoid determination when a low-resolution DTM is used.
Table
8. Indirect effects on geoid heights. Unit: [m]. DTMs
SRTM3" SRTM15" SRTM30" SRTM 1' SRTM2' SRTM5' HMGS GLOBE ETOPO2'
4
max
0.121 0.227 0.146 0.115 0.435 0.934 0.207 0.156 0.811
min
-0.185 -0.220 -0.218 -0.286 -0.710 -4.010 -0.218 -0.219 -0.782
mean
-0.009 -0.012 -0.029 -0.078 -0.243 -1.417 -0.012 -0.029 -0.239
std
4-0.033 4-0.038 4-0.036 +0.053 4-0.151 ±0.848 +0.037 +0.036 +0.161
Conclusions
The SRTM 90 m DTM was evaluated over Greece through comparisons with a national DTM and the GLOBE and ETOPO2 global models. A corrected SRTM DTM called SRTMG05 was constructed by filling-in voids in the original dataset with interpolated values from the HMGS DTM. From the results acquired it can be concluded that the SRTM DTM is very accurate, at least in the area under study, since the topographic effects on both gravity anomalies and geoid heights are very close, if not identical, to those estimated using the national model. The std of the differences of the computed topographic effects between SRTMG05 and HMGS are at the +3.3 - +6.6 mGal and the +1.6 - +3.5 cm level for gravity anomalies and geoid heights, respectively. These results are comparable to those acquired in Germany by Denker (2004) and in Switzerland by Marti (2004), proving that the 3" SRTM DTM is indeed a valuable model and can be used for gravity field and geoid determination even at national scales at the absence of higher-resolution national models. From the results acquired for G L O B E and ETOPO2 it can be concluded that the former is indeed a good model, at least for its time, and provided a very useful set of elevation data for global geoid determinations. But, in view of the SRTM data sets, it is of little use, since it introduces an error of +6.5 - + 13 mGal and +3.5 - +5.9 cm to gravity field and geoid determination, re-
spectively. The corresponding results for ETOPO2 are far more disappointing, since the error is at the +14 +25 mGal and +8.5 - +9.5 cm. Finally, from the study on the aliasing effects introduced in gravity field and geoid determination by using coarser resolution DTMs, it can be concluded that a DTM with at least a 15" resolution should be used. In this case the error introduced in geoid heights does not exceed +1 cm. If coarser resolution is used, then errors up to +12 cm can be introduced. The use of the 15" resolution SRTMG05 model introduced an error of +2.5 mGal in gravity anomalies, which reached the +10 mGal for the 5' model. Therefore, the use of a DTM with resolution lower than 15" is prohibitive, if a highaccuracy geoid determination is needed. The next goal is to extent the present study in a larger part of the country to validate and investigate the SRTM performance nationwide. Acknowledgement
This research was funded from (a) the Greek Secretariat for Research and Technology in the frame of (a) the 3rd Community Support Program (Opp. Supp. Progr. 2000 - 2006), Measure 4.3, Action 4.3.6, Sub-Action 4.3.6.1 (international Scientific and Technological Co-operation with non-EU countries), bilateral cooperation between Greece and Canada and (b) the Ministry of Education under the O.P. Education II program <
Bamler R (1999) The SRTM Mission: A World-Wide 30 m Resolution DEM from SAR Interferometry in 11 Days. In: Fritsch D and Spiller R (eds), Photogrametric Week 99, Wichmann Verlag Heidelberg: 145-154. Denker H (2004) Evaluation of SRTM3 and GTOPO30 Terrain Data in Germany. Presented at the Gravity Geoid and Space Missions 2004 (GGSM2004) conference, August 30 - September 3, Porto, Portugal. ETOPO (2005) 2-Minute Gridded Global Relief Data (ETOPO2), http://www.ngdc.noaa.gov/mgg/fliers/0 lmgg04.html. Farr TG, Kobrick M (2000) Shuttle Radar Topography Mission produces a wealth of data. LOS Trans Amer Geophys U 81: 583-585. Forsberg R (1984) A study of terrain corrections, density anomalies and geophysical inversion methods in gravity field modelling. Rep no 355, Dept of Geod Sci and Surv, The Ohio State University, Columbus, Ohio. Eorsberg R (1985) Gravity field terrain effect computation by EFT. Bull G6od 59:342-360. Forsberg R, Solheim D (1988) Perfomance of EFT methods in local gravity field modeling. In Chapman conference on Progress in the Determination of the Earth's Gravity Field. Bahia Mar Hotel, Fort Lauderdale, Florida, September 13-16, 100103. Forsberg R, Tscherning CC (1981) The Use of Height Data in Gravity Field Approximation by Collocation. J Geophys Res 86(B9): 7843-7854.
Chapter 46 • AccuracyAssessment of the SRTM 90m DTM over Greece and Its Implications to Geoid Modelling
GLOBE (2005) The Global Land One-km Base Elevation (GLOBE) Project- A 30-arc-second (1-km) gridded, qualitycontrolled global Digital Elevation Model (DEM), http://www.ngdc, noaa.gov/mgg/topo/globe.html. Heiskanen WA, Moritz H (1967) Physical Geodesy. WH Freeman, San Francisco. HMGS (2005) Hellenic Military Geographic Service Digital Terrain Model (personal communication). Marti U (2004) Comparison of SRTM data with national DTMs of Switzerland. Presented at the Gravity Geoid and Space Missions 2004 (GGSM2004) conference, August 30 September 3, Porto, Portugal. Schwarz K-P (1984) Data Types and their Spectral Properties. In: Schwarz K-P (ed), Proc of the Int Summer School on Local Gravity Field Approximation, Beijing, China, 1-66. Tziavos IN (1992) Alternative numerical techniques for the efficient computation of terrain corrections and geoid undulations. Presented at the 1~t Continental Workshop on the Geoid in Europe - "Towards a Precise Pan European Reference Geoid for the Nineties", Prague, May 11-14. Tziavos IN, Sideris MG, Forsberg R, Schwarz KP (1988) The el-
fect of the terrain on airborne gravity and gradiometry. J of Geophys Res 93(B8): 9173-9186. Tziavos IN, Sideris MG, Forsberg R, Schwarz KP (1992) A study of the contribution of various gravimetric data types on the estimation of gravity field parameters in the mountains. J of Geophys Res 97(B6): 8843-8852. USGS (2005) United States Geological Survey, ftp ://edcsgs9.cr.usgs.gov/pub/data/srtm. Vergos GS, Tziavos IN, Andritsanos VD (2005) On the Determination of Marine Geoid Models by Least Squares Collocation and Spectral Methods Using Heterogeneous Data. In: Sansd F (ed) Proc of International Association of Geodesy Symposia "A Window on the Future of Geodesy", Vol. 128. Springer Verlag Berlin Heidelberg, 332-337. Wessel P, Smith WHF (1998) New improved version of Generic Mapping Tools released. EOS Trans Amer Geophys U 79(47): 579. Wichiencharoen C (1982) The indirect effects on the computation of geoids undulations. Rep no 336, Dept of Geod Sci and Surv, The Ohio State University, Columbus, Ohio.
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Chapter 47
High-Resolution Local Gravity Field Determination at the Sub-Millimeter Level using a Digital Zenith Camera System Christian Hirt and Giinter Seeber Institut ftir Erdmessung, Universit~it Hannover, Schneiderberg 50, 30167 Hannover, Germany E-mail: [email protected] Fax: +49 511 762 4006 Abstract. During the last years the observation of vertical deflections experienced a revival due to the development of state-of-the-art Digital Zenith Camera Systems in Zurich and Hanover. Other than analogue instruments of geodetic astronomy, the new digital observation systems provide vertical deflection data very fast and highly-accurate. One main application for these instruments is the precise gravity field determination in local areas applying the classical method of astronomical leveling. This paper presents preliminary results of an ongoing local gravity field survey carried out in a test area near Hannover in Northern Germany. Here, a subterranean salt deposit influences the fine structure of the gravity field. At the Earth's surface a profile was established with densely arranged stations (50 m spacing). At 131 stations high-precision vertical deflection data has been collected using the Hanover Digital Zenith Camera System TZK2-D. Based on extensive instrumental calibration and the highly redundant data acquisition at each station, an unprecedented accuracy level of about 0t.I08 is reached for the deflection data. The local equipotential profile is directly obtained through integration of vertical deflections in the course of the profile. Error propagation shows that the variations of the local gravity field are determined 7-f~m at an accuracy level of 0.1 ~ required for example in engineering projects (linear accelerators). As a conclusion the study demonstrates that Digital Zenith Cameras are ideal instruments for accessing the submillimeter accuracy domain for gravity field studies in local areas.
Keywords. Digital Zenith Camera System, vertical deflection, astronomical leveling, gravity field fine structure, least squares collocation
1
Introduction
Over the past five years, considerable advances have been made in the field of geodetic astronomy by the development of Digital Zenith Camera Systems (DZCS) in Zurich and Hannover. Due to the application of digital imaging and a high degree of au-
tomation, this new type of instrument provide the deflection of the vertical very fast at an extremely low noise level. Using a DZCS, the complete observation procedure of determining vertical deflections at a single station lasts 20-30 min including instrumental set up, time for 50 single observations and data processing. If compared to techniques from the analogue era of geodetic astronomy, today vertical deflection data can be provided with utmost efficiency and accuracy. These benefits led to an intensive use of DZCSs in local and regional gravity field determinations in the recent time. At some 100 stations vertical deflections have been observed, e.g. in Switzerland 2003 for regional gravity field modeling (MOiler et al. 2004; Brockmann et al. 2004), in Northern Germany (Hirt 2004; Hirt et al. 2004), in further campaigns in Bavaria and Portugal 2004, Switzerland and Greece 2005 (currently yet unpublished). A substantial field of application for DZCSs is the highly-precise determination of the gravity field at local scales. Applying the traditional method of astronomical leveling, information on the local gravity field can be easily derived. The spatial resolution and the accuracy of the gravity field profile are a function of station spacing which can be adapted to the problem to be solved. Shortening the station spacing leads to a nearly unlimited spatial resolution and opens the possibility to access the sub-millimeter accuracy level over distances of a few kilometer. An initial study showing the potential of astronomical leveling carried out with a DZCS has been already presented by Hirt and Reese (2004). While the general characteristics of the Earth's gravity field are well-known for the most part of its spectrum, there is currently only less knowledge available regarding very high-frequent fine structures with wavelengths in the kilometer-range or even smaller. This is the point where this contribution enters into: The aim of this paper is to present preliminary results of an ongoing astrogeodetic highresolution local gravity field survey carried out above a salt-dome in Northern Germany. The derived data set consists of 131 vertical deflection stations with a close spacing of 50 m thus allowing to obtain empi-
Chapter 47 •
High-Resolution Local Gravity Field Determination at the Sub-Millimeter Level using a Digital Zenith Camera System
rical information on the high-frequent portion of the local gravity field. The chosen test area near Hannover is completely even and situated at a height of approximately 40 m above mean sea level. The astrogeodetic gravity field profile is referred to this height as it is directly computed from the deflection data. The impact of the curvature of the plumb line is assumed to be rather small and neglected in the presented data analysis. The derived gravity field profile located in observation height is subsequently called "equipotential profile" with the corresponding variable A N . It directly corresponds to the shape a free water surface follows. Strictly spoken neither the local geoid nor the quasigeoid is obtained although the equipotential profile derived approximates the corresponding geoid profile very well due to the small elevation of the area. General results regarding the fine structure are not depending on this proceeding. It should be emphasized that the results presented have preliminary character as the gravity field survey is still ongoing.
2
The Digital Zenith Camera System TZK2-D
The Digital Zenith Camera System TZK2-D developed and operated by the Institut ftir Erdmessung, University of Hannover is used as sensor for this local gravity field study. In this paper it is only briefly presented since its characteristics have already been published (cf. Hirt 2004, Hirt and Btirki 2002). The TZK2-D consists of two major components. Firstly, a digital image sensor (CCD) is applied for the astrogeodetic determination of the plumb line (~, A). For the data processing the highly-accurate star catalogue UCAC (Zacharias et al. 2004) serves as celestial reference. The fully-automated data processing from image data to vertical deflections is done with the Hannover astrogeodetic processing system AURIGA (cf. Hirt 2004). Secondly, a GPS-receiver is used for precise timing and positioning (ellipsoidal coordinates ~, )~). Vertical deflections (~, r]) are obtained by combining both components: = ~,-~
~ = ( A - 2,)cos~.
(1)
In early 2005, the TZK2-D was undergoing a thorough reworking by installing a set of motors in order to automatically perform necessary motions during the instrumental set up (leveling and focusing) and observation procedure (camera rotation). These technical enhancements resulted in a complete automation of observation so that the determination
Figure 1. The Hannover Zenith Camera System TZK2-D in action above the salt dome of vertical deflections now is a "one-mouse-clickapplication". A recently developed and tested observation strategy eliminates the influence of instrumental zerooffset variations on the deflection data. This improvement has increased the accuracy level of the deflection data from 0'/10 - 0'/15 (derived in Hirt et al. 2004) significantly in the unprecedented range of 0'.'05 - 0'! 101 .
3
Project Area
As a test area the salt dome "Bokeloh", forming a part of the salt structure "Steinhuder-Meer-Linie" near Hannover in Northern Germany, has been selected. The salt dome creates an extended subterranean geophysical anomaly as its density contrasts from the surrounding sediment layers. Due to the geometrical extensions of the salt dome (width of approximately 1 km, length of 12 km, depth of 3 km) and complexly folded neighbouring sediment layers the test area is ideal for studying the local gravity field fine structure. Small disturbances of the local gravity field may be expected. Since the salt dome "Bokeloh" is presently exploited by the German min1The accuracy obtained in practice is essentially a function of the number of single observations and the presence of refraction anomalies during observation. An accuracyof about 0'.'05 is usually reached on the basis of about 100 or more repeated observations.
317
318
C. Hirt. G. Seeber (a) 11 52.4
salt dome ,,Bokeloh"
a~ .-,=_-o "" 52.39
cannel ,,Mittellandkana~180
~ ; ' o
J l "
10.5
230 220
~ " -
~o ~ o - -
10
200
9.5
19o E
1
1
i
t
9.4
9.42
9.44
9.46
longitude • [deg]
mate location of the salt dome. Note the different scales of the longitude- and latitude axis. ing company "K+S GmbH" the geometrical position of the salt structure is well known on the basis of exploration data (cf. Sessler and Hollander 2002). However, precise three-dimensional information on the mass and density distribution in the neighbourhood of the salt dome is not available. In order to sample the local gravity field with highresolution, a profile with a length of 6.5 km consisting of densely arranged stations with a spacing of 50 m has been set up at the Earth's surface. Due to aligning the profile line to the course of the "Mittellandkanal" (a cannel in East-West-direction) a relatively smooth profile course was achieved (Figure 2).
Astrogeodetic Observations
Within thirteen clear nights in spring 2005 a total of 150 observation series have been carried out at 131 stations using the TZK2-D. Because a single observation series of the deflection of the vertical consists of 50 repeated observations, the derived astrogeodetic (~, r/) data set is basing on a total of approximately 6700 single solutions 2. According to the efficiency aspects mentioned in section 1, in average two stations have been occupied per hour. Depending on the length of night, within most nights deflection data has been collected at 10-17 different stations. A first accuracy estimation is obtained from the residuals of double measurements in different nights which is in the order of 0/./08 for ~ and q. These figures underline the utmost efficiency and accuracy of modern astrogeodetic observation systems like the TZK2-D.
Data Processing and Analysis
As a first step of data processing the double observations carried out at identical stations have been averaged. As a second step the geometrical course of the profile has been slightly smoothed in order to sup2A single observation of the deflection of the vertical is composed of a pair of digital images. Since a single zenithal image contains 20-30 stars in average, a total of more than 300.000 processed directions to stars contribute to the (~, q) data set (!).
"••'• ... :•"'•'•"'•,•'','.....•.
8.5
".•••° ° • ••°••,,,°%••°•,
8 7.5
Figure 2. Course of the astrogeodetic profile and approxi-
5
•°••
9
100 110 120 130 1 0
9.38
4
'••'•,.°,,'•••,••°°•o,.• ,••-,•;•,,. ,,•",,';•
7 100 (b)
lio
1;.o
4
1;o
station
1;o
2~o 2~o
3.5 3
,••••••°'~°,.,"'••••°°*•° °°°•°•Z••°••••••,,•°•°•
2.5 =.--,
'~
2
••°•
1.5
•°• ,•°o•"~'"% •°•°•°.• °;°•--°°•°°°
1
"*°°°°°° • •°,.,°•°• ,j•••°, °
0.5 0
0
10'00 20'00 3600
40'00 ~0'00 60'00
distance [m]
Figure 3. Vertical deflections (~, r/) in the course of the astrogeodetic geoid profile crossing the salt dome "Bokeloh" press small peak-structures in the equipotential profile.
5.1
Analysis of the Deflection Data
The (~, q)- data set obtained is depicted in Figure 3 as a function of distance. In the profile's course a variation of about 2//(~) and 3'J5 (q) is visible. Both components show local minima located approximately in the middle of the profile (station range 160-180) where the main part of the salt dome is located. The very low noise level of the data sets is clearly visible in both figures. Applying the method of least squares collocation for separating signal from noise (cf. section 5.2), the noise level of the deflection data is found to be about 0//08 (~) and 0//07 (q)3. As such it corresponds to the initially made accuracy estimation yielded by double observations (cf. section 4).
5.2 Computation and Analysis of the Equipotential Profile Applying the well-known classical formulae of astronomical leveling, the equipotential profile A N is 3However, the noise of the ~-component is obviously not normally-distributed as seen by locally strong correlated parts (e.g. stations 150-160 or stations 210-220). This is most likely due to the presence of small, very local refraction anomalies which may occur perpendicular to the profile over the cannel's water surface.
Chapter 47 • High-Resolution Local Gravity Field Determination at the Sub-Millimeter Level using a Digital Zenith Camera System
obtained. Using the average of the vertical deflections at every pair of adjacent stations Pi and Pi+l, the vertical deflection ci ~ + ~+ 1 r/~ + q~+ 1 ci = cos c~ + sin c~ 2 2
(2)
i=230
~
ei-si.
4
(3)
120
140
160 station
•-
180
200
220
,o,~,.'~°%o'~,o,~ ,°. ",.,o"o,.,., °o%,.,
3.5
is obtained being the tilt of the equipotential profile in azimuth c~ of the connecting line between stations Pi and Pi+l (according to Torge 2001, pp 294-300). The equipotential profile A N (= height difference between Pi and P~) is obtained by integrating single height increments ci • si from the beginning (station no. 100) to the end (station no. 231) of the profile:
AN---
O0 4.5
°,,.
:..o%..,.,°°°"°:°""°" "%.,,,.,~.,,,," =° %° ,.,°~
3
,. °°° °
2.5
°'%°.o,,,°g°
2 1.5 1
0
1000
2000
3000 4000 distance [m]
5000
6000
Figure 4. Vertical deflection component c in the course of the astrogeodetic geoid profile. Averaging of the deflections from neighboured stations yields to a smoother profile course if compared with { and r/(Fig. 3).
i=100 0 •" ~ , ~
Figure 4 shows the course of the vertical deflection component e. The salt dome's signal of about ltt is clearly visible in the range of stations 160-180. Contrary to the relatively smooth course from station 130 to 231 the range between station 100 and 130 reveals correlated noise which is certainly not coming from the Earth's gravity field 4. The result of the computation of the equipotential profile is depicted in Figure 5. The height of the equipotential profile changes by approximately 10 cm over a distance of 6.5 km. In order to extract information on the fine structure contained in the A N data the method of least squares collocation (LSC) can be applied which is described in detail by Moritz (1980). The general form of LSC observation equation reads as (Torge 2001, p. 303): 1 -
Ax
+ s + n.
(4)
For the analysis of astrogeodetic profile measurements, 1 is the observation vector comprising the gravity field quantities A N l i in the course of the profile: lT - [ A N 1 1 , / k N 1 2 , . . . , / k N l n ] (5) and x is the parameter vector with profile distances from 0 m to about 6500 m. The product A x expresses the deterministic part, e.g. a straight trend line as first approximation. The signal vector s and the noise vector n are random parts of the observation vector 1. Whereas the signal vector s includes correlated (quasi-deterministic) parts, the latter usually represents observation errors, namely noise. The 4The noise reflects variations of instrumental zero offsets which are not adequately eliminated during the observation procedure at the very beginning of the campaign.
140
160
180
200
220
station
-0.02 •~ -0.04 E z -0.06 -0.08
"-,,,,,,,
-0.1 -0.12
0
1000
2000
3000 4000 distance [m]
5000
6000
Figure 5. Equipotential profile A N stochastic properties are described by the signal covariance matrix C and the noise covariance matrix D. For the computation of x, s and n the reader is referred to Moritz (1980) or Torge (2001). In LSC, two parameters essentially control the separation of signal s and noise n: At first, the ratio between a priori standard deviations o-s and crn for signal and noise is necessary. Secondly, the covariance function cov (1,1') containing information on the autocorrelation between neighboured stations P and P' is required for the determination of the signal covariance matrix C. As the LSC results are relatively insensible to the chosen covariance function, an exponentially decreasing correlation function can be used with a correlation length of about 1 km. Instead of an estimated signal-to-noise-ratio of about 10 a smaller value of 1 has been used for collocation. This has the advantage that interesting fine structures with small amplitudes are separated from the main signal of the equipotential profile. Following that way, the information contained in 1 with amplitudes of a few 0.1 mm or less is included in the noise vector n and becomes visible. Figure 6 (a) shows the filtered equipotential profile. It is equivalent to the difference between the ob-
319
320
C. Hirt. G. Seeber (a) 3
"..°° •.. ~2 •, I13
140
160
180
200
220
station
.,,,,',,.,,,,. .....
"*,
•=- 1 o t-,..., ._
120
."
"*°•
0
•.
"*',.,.,,,.,•,,,.
.
~-1 ~'-2 E
**°,..,
"°'%.....,~...°°'*
Z-3
(b)
3 =
2
=
1
vE
0
'~'•'•°°°•°
°•,°°°°••°"°"'*" ,,,,,....... ,, •,~,°°•,••° o. %•%
Z•
"•'*"*'..o,.,.,°
E '-"-2 Z
°• .*" •°°...,...,......"
<1_ 3
(c) 0 . 3 , . = 0.2 ._~ or) 0.1 -
5 119
°
•
•
-
(1)
•
0
.
• 1 " " ; . . . " " " " " %;
+ -0.1 ._~
• °•-: .,.o.
o -0.2
.
° .......
%*
"**..,,,,%,.,,..
t t I (2) ( 3 ) ( 3 )
"-."
. . . . . -..
"•-
-%.,.
ZE
"~
0
10'00
20'00
30'00 40'00 distance [m]
50'00
60'00
Figure 6. Equipotential profile. In subfigure (a) the filtered profile is depicted (= 1 - A x = s + n). The signal part s is displayed in subfigure (b). Residual signal parts (fine structure) and observation errors are contained in the noise vector n illustrated in subfigure (c). servation vector 1 and the deterministic part A x . The latter corresponds to the long-wave regional part of the gravity field. Subtracting the noise vector n, the signal vector s is obtained (Figure 6 (b)). A wavelike structure forms the dominant part of the graph with an amplitude of about 2 m m and an estimated length of 5 km. As such it reflects the local mass distribution. The noise vector n as residual profile is shown in Figure 6 (c) as the most interesting depiction. This vector contains measurement noise and highly correlated patterns which basically may come from atmospheric refraction anomalies and inhomogeneous subterranean mass distribution. The visible features may be explained as follows: (1) The presence of atmospheric refraction anomalies leads to physical correlation of neighboured stations if measured during the same night. Mean amplitudes of this effect come usually close to some 0'/01 (estimation based on results of repeated observations in Hannover 2005). However, extreme weather situations may cause refraction anomalies estimated to be in the order of 0'!1. Consequently, for
the analysis of the residual profile refraction anomalies are not negligible. For example the correlated point patterns occurring in the station range 135-140 (cf. arrow in Figure 6) are believed to come from refraction anomalies. (2) Within the station range 160-210 various fine structure features with different characteristics are visible. The most striking fine structure is visible within the station range 170-185 (estimated amplitude 0.1 mm, half wavelength 800 m). A first comparison with detailed geological data (courtesy K+S GmbH) reveals a strong local correlation with subterranean sediment layers. As such this structure most likely reflects the very local mass distribution. (3) Even finer structures are visible from station 185 to 195 (amplitude 0.05 mm, half wavelength 400 m) and 195 to 210 (0.02 ram, 350 m). However, it is not clear where these structures come from. Currently it can neither be excluded nor confirmed whether these patterns are also mirroring very local mass anomalies. As a future task further repeated astrogeodetic measurements will be carried out in order to find out more about these interesting structures. On the basis of Figures 6 (a) - (c) it can be concluded that the sediments located at both sides of the salt dome have different densities and are not homogeneously distributed.
5.3
Accuracy
of the Equipotential
Profile
The accuracy O-Am of the equipotential profile is a function of the observation accuracy o-~, station spacing ds and number of stations r~. According to the derived formulae described in Hirt and Reese (2004) the accuracy O-Am in [mm] is given by O-Am -- V/r~ -- 1 • 4.8. ds • o-~
(6)
where cr~ is introduced in ['] and ds in [km]. Applying equation 6 the standard deviation orAN is found for an observation accuracy to be about 0.09 ~ of about 0'/08 (cf. Table 1). Even a lower observation accuracy of 0'/1 still yields an accuracy beff~ ff/~ ing in the order of 0.1 kv/-U~. The sub-millimeter level is also clearly achieved for profile lengths up to 10 km. These figures underline that astronomical leveling can provide local gravity field information on an unprecedented accuracy level. Anyway, remaining systematic errors due to refraction are not yet taken into account. Therefore adequate consideration of refractivity (e.g. modeling or reduction by observation techniques) remains as future task.
Chapter 47 • High-Resolution Local Gravity Field Determination at the Sub-Millimeter Level using a Digital Zenith Camera System
cr~ ['1 0.05 0.07 0.08 0.10 r~
50 0.012 0.017 0.019 0.024 2
Total length of profile [m] 600 l O O O 6500 0.042 0.054 0.139 0.059 0.076 0.194 0.067 0.087 0.222 0.084 0.109 0.278 13 21 131
10000 0.172 0.241 0.275 0.344 201
T a b l e 1. A c c u r a c y o-zxN as a function of profile length and observation accuracy o-~ for a station spacing of 50 m.
6
Conclusion and Outlook
In this paper preliminary results of a high-resolution local gravity field survey have been presented using the Digital Zenith Camera System TZK2-D for observation of vertical deflections. The sub-millimeter accuracy level of about 0.1 ~f / ~ TFb has been reached for the equipotential profile due to the unsurpassed low noise level of the deflection data of about 0~.I08 and the very dense station spacing of 50 m. Analysis of the gravity field profile using least squares collocation indicates the existence of very short-wavelength fine structure with amplitudes of some 0.02 mm and wavelengths in the order of 1 km. Having highly-resolved data opens the possibility to a better understanding of the finest components of the gravity field. Besides geodesy, geology and geophysics could benefit from new astrogeodetic data sets like the presented one as data inversion can provide precise information on the density distribution of the subterranean masses. Furthermore, even highest accuracy requirements in engineering projects can benefit from astronomical leveling using digital instrumentations. E.g., the alignment of new linear accelerators require exceptionally precise information on the geometry of the gravity field at an level of about 0.1 mm per 600 m (cf. Becker et al. 2002; Schl6sser and Hetty 2002). As seen from Table 1, astronomical leveling basically meets this requirement. The presented study shows that some work remains to be done in the future. In general, attention has to be laid on the appropriate consideration of refraction anomalies. With respect to the validation of geoid and quasigeoid profiles, the role of the curvature of the plumb line respectively normal line should be discussed in detail. In particular, selected stations of the equipotential profile (e.g. station range 160-210) should be observed repeatedly using the system TZK2-D. This new data set to be obtained under completely different atmospheric conditions could help to answer the question whether the fine structure features with amplitudes of a few 0.01 mm
which are visible in the data (section 5.2) come from the Earth's gravity field or from the refraction field.
7
Acknowledgement
The application of the Digital Zenith Camera System for gravity field determination is supported by the German National Research Foundation DFG. The authors are grateful to the students Ilka Rehr, Niels Hartmann, Eiko Mtinstedt and Rene Gudat for their unrestless and engaged support of the astrogeodetic measurements. Dr. Holl~inder (K+S GmbH) is acknowledged for providing detailed geological data.
References Becker, F., Coosemans, W. und Jones, M. (2002). Consequences of Perturbations o/ the Gravity Field on HLS Measurements. Proc. of 7th Int. Workshop on Accelerator Alignment (IWAA): 327-342, SPring-8, Japan. Brockmann, E., Becker, M., Btirki, B., Gurtner, W., Haefele, R, Hirt, C., Marti, U., Mtiller, A., Richard, P., Schlatter, A., Schneider, D., and Wiget, A. (2004). Realization of a Swiss Combined Geodetic Network (CH-CGN). EUREF'04 Symposium of the IAG Commission 1 - Reference Frames, Subcommission 1-3a Europe (EUREF), Bratislava, Slovakia. Hirt, C. (2004). Entwicklung und Erprobung eines digitalen Zenitkamerasystems fiir die hochpriizise Lotabweichungsbestimmung. Wissenschaftliche Arbeiten der Fachrichtung Geod~isie und Geoinformatik an der Universit~it Hannover Nr. 253. Hirt, C. and Btirki, B. (2002). The Digital Zenith Camera A New High-Precision and Economic Astrogeodetic Observation System for Real-Time Measurement of Deflections of the Vertical. Proc. of the 3rd Meeting of the International Gravity and Geoid Commission of the International Association of Geodesy, Thessaloniki, Greece (ed. i. Tziavos): 161-166. Hirt, C. and Reese, B. (2004). High-Precision Astrogeodetic Determination of a Local Geoid Profile Using the Digital Zenith Camera System TZK2-D. Electronic Proc. IAG GGSM2004 Meeting in Porto, Portugal. Published also in: CHGeoid 2003, Report 03-33 A (ed. U. Marti et al), Bundesamt ftir Landestopographie (swisstopo), Wabern, Schweiz. Hirt, C., Reese, B., and Enslin, H. (2004). On the Accuracy of Vertical Deflection Measurements Using the High-Precision Digital Zenith Camera System TZK2-D. GGSM 2004 IAG International Symposium Porto, Portugal (ed. C. Jekeli et al.), Springer, Heidelberg: 197-201. Mtiller, A., Btirki, B., Kahle, H.-G., Hirt, C., and Marti, U. (2004). First Results .fEom New High-precision Measurements of' Deflections of the Vertical in Switzerland. GGSM 2004 IAG International Symposium Porto, Portugal (ed. C. Jekeli et al.), Springer, Heidelberg: 143-148. Moritz, H. (1980). Advanced Physical Geodesy. Wichmann Karlsruhe. Schl6sser, M. and Hetty, A. (2002). High Precision Survey and Alignement of Large Linear Colliders - Vertical Alignement. Proc. of 7th Int. Workshop on Accelerator Alignment (IWAA): 343-355, SPring-8, Japan. Sessler, W. and Holl~inder, R. (2002). Das Kaliwerk Sigmundshall der K+ S Aktiengesellschaft. Ver6ffentlichungen der Akademie der Geowissenschaften zu Hannover 20: 7-76. Torge, W. (2001). Geodesy, Third Edition. W. de Gruyter, Berlin, New York. Zacharias, N., Urban, S. E., Zacharias, M. i., Wycoff, G. L., Hall, D. M., Monet, D. G., and Rafferty, T. J. (2004). The Second US Naval Observatory CCD Astrograph Catalog (UCAC2). The Astronomical Journal 127: 3043-3059.
321
Chapter 48
A data-adaptive design of a spherical basis function network for gravity field modelling R. Klees and T. Wittwer Delft Institute of Earth Observation and Space Systems (DEOS) Delft University of Technology, 2629 HS Delft, Kluyverweg 1, The Netherlands
Abstract. A data-adaptive strategy for the design of spherical basis function (SBF) networks is presented. The strategy comprises a datadriven selection of the SBFs prior to parameter estimation and an automatic selection of the bandwidth of the SBFs. The selection procedure takes the data distribution and the data noise into account. The strategy can be applied to all interpolation and approximation problems that use SBFs. It is particularly suited for the local inversion of magnetic and gravity data acquired by terrestrial, airborne or space-borne sensors. It is shown that for inhomogeneous and scattered data distribution, the method requires significantly less basis functions than commonly used strategies, which range from template networks of SBFs (e.g. from hierarchical subdivision schemes or equal angular grids) to SBFs located below each data point or below a subset of the data points. Moreover, the system of normal equations is much better conditioned, which often make regularization superfluous. Finally, it is shown that the approximation errors are smaller than when using standard SBF networks. K e y w o r d s . Spherical basis functions, network design, gravity field modelling, interpolation and approximation.
1
Introduction
Spherical basis functions (SBFs) have found a wide range of applications in interpolation and approximation of data given on a sphere. They become more and more popular in magnetic and gravity field modelling from terrestrial, airborne and space-borne data. The performance of the SBFs depends critically on the design of the SBF network. Among the approaches commonly used are subdivision schemes, e.g. based on a triangulation of an icosahedron (e.g. Eicker et al. 2004), homogeneous point distribution, derived from the theory of numerical integration of spherical
functions (e.g. Schmidt et al. 2004) or equal angular grids. In gravity field modelling from terrestrial and airborne data, the SBFs are usually located below the data points or below a subset of the data points (e.g. Marchenko et al. 2001). The depths of the SBFs is often fixed a-priori, e.g. as function of the density of the data points (e.g. Heikkinen 1981). More advanced strategies involve Generalized Cross Validation techniques or adaptation to the local signal covariance function (Marchenko et al. 2001). Commonly used choices of SBF networks may have a number of drawbacks, which reduce the quality of the solution and/or the numerical efficiency. Often, these standard networks of SBFs contain too many SBFs. However, using all the coefficients may involve serious overfitting with the result that the estimator is extremely noisy. Moreover, the norreal equations may become very ill-conditioned. For instance, a non-homogeneous distribution of data points may not allow to estimate all coefficients. Moreover, if the bandwidth of the SBF is not adapted to the data density, the normal matrix may be extremely ill-conditioned, and regularization will become indispensable. The objective of this paper is to present a strategy, which automatically designs a network of SBFs. The design involves the choice of the location of the SBFs in 3D-space and the choice of the bandwidth of the SBFs. The design is data-driven, i.e. it depends on the data distribution, the data density, and the data noise. Information about the noise level can be replaced by some user-specific threshold value, which limits the quality of the fit to the data to some prescribed value. The approach can be used without any limitation for interpolation and approximation of spherical data (e.g. topographic, bathymetric, magnetic, gravity) using SBFs. It is particularly suited for the local inversion of grayity or magnetic data. The example used in this paper is taken fl'om airborne gravimetry. More specific, we demonstrate the performance of the
Chapter 48
•A
Data-Adaptive Design of a Spherical
data-adaptive network strategy for the local inversion of airborne gravimetry data into gravity anomalies and disturbing potential values at ground level. The application of the strategy to multi-scale modelling using spherical wavelets is straightforward and is addressed in (Klees and Wittwer, 2005). In section 2 we introduce the SBFs and the representation of the disturbing potential in terms of SBFs. In section 3 the data-adaptive network design strategy is presented. It is shown how the network is designed starting with an arbitrary template network. Tile design is driven by the data distribution, the data density, and the data noise level. The optimal bandwidth of the SBFs is determined using Generalized Cross Validation. In section 4 the test set-up is described. Section 5 contains the results of the simulations and a discussion of the accuracy and efficiency of the pursued approach.
2
Local gravity field modelling and SBFs
The disturbing potential T is represented as the sum of a long-wavelength part TL and a residual part T,~¢s: =
(1)
+
The long-wavelength part is usually represented in terms of spherical harmonics up to a maximum degree L. The primary task of this term is to model the long wavelength part of the spectrum not covered by the local area. The residual part is written as a linear combination of N spherical basis functions (SBFs) ~ ( z , y): N
We consider a class of spherical basis functions, which has the following structure: (XD
•
(3)
/=L+I
where 2 - ~x, ~ ) - ~Y, and /~ is the radius of a reference sphere. In particular, the restriction of the SBF to the surface of a sphere is a function of the spherical angle between the location of the basis function y and the computation point z. The sampling properties of the SBF are determined by the coefficients ~z. They depend on
Basis
Function Network for Gravity Field Modelling
the location of the center of the SBF y. Sometimes, they also depend on some additional parameters. SBFs are quite flexible w.r.t, the data density and distribution, and are directly related to the spherical harmonic representation of the long-wavelength part. This makes the representation Eq. (1) very suitable for local geopotential modelling. Assuming that Nobs linear functionals li LiT are observed, we obtain the system of observation equations N -
> x,
-
(4)
~z=l
which can be solved by minimization of the quadratic functional e(x) = (1- Ax)TC-~(1- Ax)-
axTx,
(5)
where 1 is the vector of reduced observations, A is the design matrix, x is the vector of unknown SBF coefficients, and a is tile regularization parameter.
3
The data-adaptive (DAN D) strategy
network
design
The performance of a SBF network critically depends upon the chosen centres of the SBFs, or, in other words, upon the design of the sampling network. In multi-scale modelling, various subdivision schemes are used, generating hierarchical or non-hierarchical grids of points. Examples are the well-known geographical grids, subdivisions of the faces of icosahedrons projected on the sphere, and homogeneous point distributions on the sphere like the Reuter grids (e.g. Freeden et al. 1998). Alternatively, the basis functions are located below the data points at some depth, a strategy, which is often used in pointmass modelling (e.g. Heikkinen 1981, Marchenko et al. 2001). However, placing the basis functions below a subset of the data points is clearly an unsatisfactory method for building a sampling network. The resulting network often performs poorly or has a large size. On the other hand, simply using a sampling network of centres is also not the method of choice; it may cause overfit or instabilities. Finally, the choice of the centres of the SBF network cannot be considered independently of the choice of the bandwidth of the SBFs. Both
323
324
R. Klees • T. Wittwer
aspects are inter-related and have to be addressed jointly to get a high-performance network. In the following we will propose a dataadaptive network design (DAND) strategy, which selects the SBFs and their bandwidths automatically from the data. The network design starts with a template network, generated by any subdivision scheme. As first step, we remove a basis function if there are less than q _> 1 data points within its region of influence (ROI). The latter is defined as a spherical cap of radius ~Rof, which is only a function of the point density of the template network. The spherical radius of the ROI is determined according to A
2~(1
-
~os~o~)
-
Xov~'
(s)
where A is the size of the area under investiagation projected onto the unit sphere, and Novs is the number of data points in that area. If at least q data points are within the region of influence, the SBF is selected and the q nearest data points are removed from the index list of data points. Then, the next SBF is considered. This process is continued until no data points are left in the index list. As second step, the bandwidth of the SBF is selected (for the definition of the bandwidth, see (Narcowich and Ward 1996)). For the multi-pole wavelets, to be introduced in section 4, the bandwidth is a non-linear function of the distance of the multi-poles from the origin of the co-ordinate system, i.e. from the depths below the surface. The deeper the multi-poles are located, the larger the bandwidth is. In the DAND algorithm, the bandwidth is adapted optimally to the data and the sampling network density. For that reason, we write the bandwidth as
and determine the optimal factor p using Generalized Cross Validation techniques (cf. Golub et al. 1979, Klees and Wittwer, 2005). The first two steps of the data-adaptive network design guarantee that more SBFs are located in data-rich areas, while data-poor areas are represented by fewer SBFs. Li (1996) uses the concept of a region of influence and a proper selection of the bandwidth for the global modelling of temperature fields from a sparse set
of points heterogeneously distributed over the sphere. However, the number of SBFs in a certain area should not be driven by the number of data points, but by the signal variation in that area. If too many SBFs are placed in an area with smooth signal, they mostly model noise. Therefore, in the third step of the data-adaptive network strategy, we look at the signal at the data points. If there are only data points within the ROI that have a signal smaller than some threshold, the SBF is removed from the network. The threshold may be a function of the expected noise variance at the data points, provided this information is available. Otherwise, the user may specify its own threshold value. The third step guarantees that mostly signal above the threshold is modelled, whereas signal and noise below this level are not modelled; this reduces the risk of overfitting. Then, the overall estimation procedure looks as follows: The estimation process starts with the choice of the template network. Next, we compute the ROI of the template network, Eq. (6). Then, we select a candidate parameter p and compute the bandwidth cr of the SBFs, according to Eq. (7). From the bandwidth, we determine the depths of the SBFs by a Newton iteration scheme. The horizontal positions of the SBFs are given by the template network. Next, the SBFs are selected using the ROI criterion. Thereafter, the potential coefficients are estimated by minimizing the quadratic functional, Eq. (5). Thereafter, the GCV functional is computed. This procedure is repeated until the parameter p has been found that minimizes the GCV functional. The corresponding solution represents the residual disturbing potential. If regularization is applied (i.e. for a non-zero c~ in Eq. (5)), a second iteration loop inside the GCV loop is required. The optimal regularization parameter is found using Variance Component Estimation techniques (see Koch and Kusche 2002). The DAND algorithm has been generalized to multi-scale problems. For more details, the reader is referred to (Klees and Wittwer, 2005). 4
Simulations
Our approach to local gravity field modelling is independent of the choice of the SBFs. The re-
Chapter 48 • A D a t a - A d a p t i v e Design of a Spherical Basis Function N e t w o r k for Gravity Field Modelling
sults, to be presented later, have been obtained with the so-called multi-pole wavelets introduced in (Holschneider et al. 2003)"
258 °
260 °
262 °
36"
36"
¢(~, ~) l+1 l----L+1
P~(~)
4rrR 2
(8)
• ~,
m is the order of the multi-pole wavelet. The parameter a determines the distance of the multipole wavelet w.r.t, the origin of the spherical coordinate system according to y __ /~ ~ - - a !),
l) - -
Y I'Y
°
34"
34"
32"
32"
(9)
For a fixed m, the scale parameter a determines the shape and position of the spectrum of the wavelet. The larger a, the deeper the basis function is located below the surface, and the more the spectrum is concentrated at lower frequencies. With decreasing a, the spectrum covers higher frequencies. Correspondingly, the centre of the basis function moves towards the surface, which is equivalent with a better space localization. We did extensive numerical test with simulated terrestrial and airborne gravity data. The results to be presented later refer to an airborne gravity data set. The area is located in the US, North to the Gulf of Mexico. The size is about 6 x 6 degrees. The gravity disturbances at a flight altitude of 2 km range from - 3 0 regal to 45 regal; the RMS signal is about 6.2 regal. The frequency content of the data is limited to degrees 121 to 1800. Therefore, the parameter L in Eq. (8) is set equal to 120. Figure 2 shows the gravity signal over the test area. We generated 5453 gravity disturbances, heterogeneously distributed at an altitude of 2 km above the area, see Figure 2. The data have been corrupted with white noise with a standard deviation of 1.5 regal. Multi-pole wavelets of order m = 3 have been used in all simulations. In section 5, the results for two different template networks are presented: (i) an equal angular grid at level 7 with 4096 points and (ii) the 5453 data points themselves. Once the SBF coefficients have been estimated, gravity disturbances and disturbing potential values are computed at flight level and at ground level. The estimation at flight level has been done at the data points and at a set of control points. The latter are located in between the data points.
30"
30" 258"
260"
262"
, mGal
Ao
-io
10
o
~o
Figure 1: Gravity disturbances over the test area. The RMS signal variation is 6.2 regal, the range is - 3 0 to 45 regal. 258 °
260 °
262 °
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258 °
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5
Results and discussion
Five different scenarios have been investigated" • Scenario 1: An equal angular grid is used as template network; the DAND algorithm is applied with a threshold equal to zero; no regularization.
325
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R. Klees • I. Wittwer
Scenario 2: The SBFs are placed on an equal angular grid at level Y, which consists of 4096 points. This point density is sufficient to resolve the spectral content in the data. Regularization is applied. • Scenario 3: An equal angular grid is used as template network; the DAND algorithm is applied with a 3~-threshold, where cr = 1.5 r e g a l is the data noise level; no regularization. • Scenario 4: The data points are used as ternplate network; the DAND algorithm is applied with a 3or-threshold, where cr = 1.5 r e g a l is the data noise level; no regularization. • Scenario 5: A SBF is placed below every data point; regularization is applied. Figure 3 shows the RMS error in the gravity disturbances computed at the data points and at the control point at flight altitude for the various scenarios. First of all, we see that the RMS error at the control points is the smallest if the DAND algorithm is applied (scenarios 3 and 4). The RMS error at the data points, however, is larger for scenarios 3 and 4 compared with scenarios 1,2, and 5, which do not use the DAND algorithm. This can be explained by the presence of 'overfitting' for scenarios 1, 2, and 5. Therefore, one important conclusion is that the DAND algorithm avoids or at least reduces 'overfitting'. A second important conclusion can be drawn from a comparison of the results for scenario 3 and 4. They differ w.r.t, the template network used in the DAND algorithm. In scenario 3, an equal angular grid is used as a ternplate network, whereas scenario 4 uses the data points as a template network. Obviously, if the DAND algorithm is applied, the choice of the template network does not matter, i.e. almost the same results are obtained with various ternplate networks. T h a t is exactly what one would expect from a good algorithm. Remarkable is that the DAND algorithm reduces the number of SBFs significantly. Scenarios 3 and 4 use only 1672 and 1584 SBFs, respectively, whereas 5453 SBFs are needed when placing a SBF below every data point, although the latter provides a significantly larger RMS error at the control points. Moreover, if the DAND algorithm is used, no regularization is necessary. If, however, a SBF is placed below every data point, we found in all
2,5 2 o~
L9 E
1 0,5 0 1
2
3
4
5
Figure 3: Gravity disturbance errors at flight altitude at the data points and the control points for various scenarios. 1: equal angular ternplate network, DAND applied, no regularization, zero threshold; 2 : 4 0 9 6 SBFs on an equal angular grid, regularization applied; 3: equal angular template network, DAND applied, no regularization, 3or-threshold; 4: data points as ternplate network, DAND applied, no regularization, 3a-threshold; 5: one SBF below every data point, regularization applied. The data noise is 1or = 1.5 mGal.
simulations that no physically meaningful solution is obtained without regularization. Figure 4 shows the RMS error in the residual disturbing potential computed at a set of control points at ground level. The conclusions drawn from the results shown in Figure 3 are fully supported. In particular, the DAND algorithm provides the best solution with the smallest number of SBFs. A geometrical explanation for getting suboptimal results if a SBF is placed below every data point is given by the factors p, which determine the depths of the SBFs (cf. Eq. (7)). In that case, we obtain an optimal value p = 2.4. This is much smaller than the value of p obtained when DAND is applied (p = 3.4 for an equal angular grid as template network and p = 3.0 if the data points act as template network). A smaller value of p means that the SBFs are placed more shallow. This can give small residuals, but a worse fit between the data points and at the ground level. Figure 5 shows the performance of the DAND algorithm for various threshold values varying between 0 (i.e. no threshold) to 4or. The RMS error is the smallest for a threshold value of 1or, which corresponds to the data noise level. As one can expect, the RMS error increases with increasing threshold, whereas the number of SBFs to be used decreases. Important for practical ap-
Chapter 48
• A
Data-Adaptive Design of a Spherical Basis Function Network for Gravity Field Modelling
0,4
0,3
0,35 0,3 ' ~ 0 2, 5 % 0,2 E ''0,15 0,1 0,05
0
0,25 ~~ ¢/)
0,2
¢,1
0,15 0,1 1
2
3
4
5
0,05
0 0
Figure 4: Disturbing potential errors at ground level for various scenarios. 1: equal angular ternplate network, DAND applied, no regularization, zero threshold; 2:4096 SBFs on an equal angular grid, regularization applied; 3: equal angular template network, DAND applied, no regularization, 3or-threshold; 4: data points as ternplate network, DAND applied, no regularization, 3or-threshold; 5: one SBF below every data point, regularization applied. The data noise is l a = 1.5 mGal. plications is that the RMS error increases quite moderately, whereas the number of SBFs to be used decreases significantly. For instance, if a threshold of 1or is used, 2876 SBFs are selected by the DAND algorithm. A threshold of 3a reduces the number of SBFs to 1672, a reduction by about 42 %. The RMS error, however, increases at the same time from 0.21 m 2/s 2 to 0.23 m2/s 2, i.e. by only about 10%.
6
Conclusions
The following conclusions are drawn from the conducted numerical experiments. They are confirmed by other experiments with simulated terrestrial and airborne gravimetry data and for other choices of radial basis functions. • The DAND algorithm reduces the number of SBFs significantly. The ratio of the number of SBFs to the number of observations is typically about 30%. This is much less than for Least Squares Collocation or classical point mass modelling. • The DAND algorithm can make regularization superfluous. In all experiments with terrestrial and airborne data, regularization was not necessary, contrary to the use of point masses below the data points. • When DAND is applied, the location of the
1G
2a
3c
4G
Figure 5: Disturbing potential errors at ground level for various thresholds: 1or = 1.5 regal corresponds to the standard deviation of the data noise. DAND algorithm is applied with an equal angular template network. No regularization is used. The number of SBFs used is (from left to right) 3368, 2876, 2231, 1672, and 1207.
SBFs does not matter. That is, any template network can be used including the set of data points. The quality of the solution when the DAND algorithm is applied is superior to the solution without DAND algorithm. The DAND algorithm is a generic tool and has widespread applications in interpolation and approximation of data on the sphere. Experiments with simulated satellite data and real terrestrial, airborne and satellite data are going on and will be reported elsewhere.
References Eicker A, Mayer-Guerr T, Ilk KH (2004) Global gravity field solutions from GRACE SST data and regional refinements by GOCE SGG observations. Proceedings IAG International Symposium Gravity, Geoid and Space Missions (GGSM2004), Porto, Portugal. Freeden W, Gervens T, Schreiner M (1998) Constructive approximation on the sphere. Clarendon Press, Oxford. Golub GH, Heath M, Wahba G (1979) Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics 21, 215-223. Heikkinen M (1981) Solving the shape of the Earth by using digital density models. Finnish
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Geodetic Institute, Report 81:2, Helsinki, Finland. Holsehneider M, Chambodut A, Mandea M (2003) From global to regional analysis of the magnetic field on the sphere using wavelet frames. Physics of the Earth and Planetary Interiors 135, 107-124. Klees R, Wittwer T (2005) Local gravity field modelling with multi-pole wavelets. Proceedings Dynamic Planet 2005, Cairns, Australia. Koch KR, Kusche J (2002) Regularization of geopotential determination from satellite data by variance components. Journal of Geodesy 76, 259-268. Li TH (1996) Multiscale representation and analysis of spherical data by spherical wavelets.
SIAM J. Sci. Comput. 21,924-953. Marchenko AN, Barthelmes F, Meyer U, Schwintzer P (2001) Regional geoid determination: an application to airborne gravity data in the Skagerrak. Scientific Technical Report 01/07, GeoForschungsZentrum, Potsdam, Germany. Nareowich F J, Ward, JD (1996) Nonstationary wavelets on the m-sphere for scattered data. Appl. Comp. Harm. Anal. 3, 324-336. Schmidt M, Fabert O, Shum CK, Han SC (2004) Gravity field determination using multiresolution techniques. Proceedings Second International GOCE User Workshop, ESAESRIN, Fraseati, Italy.
Chapter 49
Global gravity field recovery by merging regional focusing patches: an integrated approach K.H. Ilk, A. Eicker, T. Mayer-Gfirr Institute of Theoretical Geodesy, University of Bonn, Nussallee 17, D-53115 Bonn, Germany
Abstract. Usually the gravity potential is modelled
by a spherical harmonic expansion. Simulation tests and real-data investigations based on POD (precise orbit determination) and SST (satellite-to-satellite tracking) data demonstrated that the heterogeneity of the gravity field cannot be properly taken into account by base functions with global support. It is preferable to model the gravity field only up to a moderate safely determinable spherical harmonic degree without any regularization to cover the long and medium wavelengths characteristics; the specific detailed features tailored to the individual gravity field characteristics in areas of rough gravity field signal can be modelled additionally by space localizing base functions. In a final step, a spherical harmonic expansion up to a maximum degree, only limited by the most detailed structures of the gravity field, can be derived based on a Gauss-Legendre-Quadrature procedure. This last step can be performed without stability problems and without losing the regional details of the gravity field. The proposed integrated gravity field recovery approach integrates consistently a regional gravity field zoom-in into a global gravity field solution. The technique has been applied to the determination of gravity field models based on SST data of GRACE. Keywords. CHAMP, GRACE, global gravity field recovery, regional gravity field recovery
1 Introduction As a result of the dedicated space-borne gravity field missions CHAMP (Challenging Minisatellite P a y l o a d - Reigber et al. 1999) in orbit since 2000 and especially GRACE (Gravity Recovery And Climate Experiment- Tapley et al. 2004), in orbit since 2002, a breakthrough in accuracy and resolution of gravity field models has been
achieved. Subsequent solutions by using the observations collected over a period of time of, e.g., one month, enable the derivation of time dependencies of the gravity field parameters. The innovative character of these missions lies in the continuous and precise observations of the low flying satellites by the Global Positioning System (GPS) and the highly precise line-of-sight range and range-rate Kband measurements between the twin-satellites of the GRACE mission. In addition, the surface forces acting on these satellites are measured and can be considered properly during the recovery procedure. As a result of this mission, the presently best combination static model, EIGEN-CG03C, has been derived from 376 days of GRACE observations and three years of CHAMP (F6rste et al. 2005) as well as monthly snap-shots of the gravity field, showing clearly temporal variations of the gravity field closely correlated to the hydrological water cycle. Another comparable precise GRACE gravity field model is GGM02C, based on the analysis of 363 days of GRACE in-flight data (Tapley et al., 2005). it is represented by a spherical harmonic expansion up to degree 200 and constrained with terrestrial gravity information. The gravity field recovery based on the new types of satellite gravity observations poses a challenge in many respects. One of the key problems seems to be related to the representation of the gravity field by an appropriate set of base functions and of the associated gravity field parameters. The established and approved way is to model the gravity field by spherical harmonics up to a certain maximum degree. This degree is limited by the significance of the gravity field signal in the observations and its ratio to the observation noise. Because of the inhomogeneous gravity field of the Earth the signal content varies in the space domain. The gravity field in regions with rough gravity field features should be modelled up to a higher degree than within regions with smooth gravity field features. But the gravity
330
K.H.Ilk. A. Eicker.T.Mayer-GOrr field cannot be resolved globally up to this degree because corresponding gravity signals in the SSTobservations do not exist over the predominant parts of the world. Therefore a spherical harmonic gravity field model up to this maximum degree results in instabilities; the missing high-resolution signal in the observations of the predominant part of the world acts similar as the polar gaps. The application of the frequently applied Tikhonov regularization leads to a mean damping of the global gravity field features with the consequence that the high frequent gravity field signal in the observations is lost again in some geographical regions. An alternative integrated approach is to determine a global gravity field solution with high long and medium wavelength accuracy and improve this global solution in regions with characteristic gravity field features by an adapted regional recovery procedure. The global solution is parameterized by spherical harmonic coefficients up to a moderate and safely determinable degree and the regional solutions are represented by space localizing base functions, e.g., by spherical splines. This procedure provides several advantages. The regional approach allows exploiting the individual signal content in the observations and a tailored regularization for regions with different gravity field characteristics. In addition, the resolution of the gravity field determination can be chosen for each region individually according to the spectral content of the signal in the specific region. Furthermore, the regional approach has the advantage of dealing with regions with different data coverage more easily, if no data at all is available the regional refinement can be skipped. For regions with sparse data coverage a coarser parameterization can be selected• Several regional solutions with global coverage can be merged by means of numerical quadrature methods to obtain a global solution, in principle, up to an arbitrary degree, only limited by the signal content of the gravity observations. Due to the regionally adapted strategy this method provides better results than calculating a spherical harmonic solution by recovering the potential coefficients directly.
the success of this integrated approach in case of selected examples related to the GRACE mission. Sect. 4 concludes this article with a summary and some perspectives for a future work. 2 Computation
procedure
2.1 Setup of the mathematical models If precise intersatellite functionals as line-of-sight ranges or range-rate measurements are available, as in case of the GRACE mission, the mathematical model can be based on Newton's equation of relative motion, i;,2(t) = g(t; r~2,r~,/'1,/'2;x). (1) This differential equation can be formulated as a boundary value problem, r,2 (t) - (1- r) r,2,A + rr,2,8 -
-T2 I K(r'r')g(t;r12'rl'rl'r2;x)dr" z-'=0
with the integral kernel,
(3) the normalized time variable, r -
t-t A
with T - t~ - tA, t E [tA,t~ ], T as well as the boundary values
r,~,~ := r,~(t~),
r,~,~ := r,~(t~),
t~ < t~.
(4)
(5)
Differentiation with respect to the time results in the relative velocity between both satellites, •
1
) dK(:,:') g(r"
- T ~':0
dr
(6) • d r .' ,rl2,r 1, /'l,/'2,x)
The mathematical model for range observations follows by projecting the relative vector to the lineof-sight connection in combination with Eq. (2), r~2(r) = e,2 (r). r12(r). (7) Analogously, the mathematical model for range-rate measurements in combination with Eq. (6) reads as follows,
~,~(~) = e,~ (~). i~,~(~). Sect. 2 of this article reviews various aspects of the computation procedure for processing the observations of the new gravity satellite missions. It is our concern to point out the importance of combining a regional gravity field zoom-in with a global gravity field recovery to assure the consistency of both views of the gravity field. Sect. 3 demonstrates
(2)
(s)
In both equations, e~2 is the unit vector in the lineof-sight direction of both twin-satellites. This vector is lonown with high accuracy, assuming that the satellite positions are measured with an accuracy of a few cm and taking into account the distance of approximately 200km between the two satellites. The
Chapter 49 • Global Gravity Field Recovery by Merging Regional Focusing Patches:an Integrated Approach
specific force function for the relative motion according to Eq. (1) can be separated into various parts, g(t;r12,r1,i'1,i'2; x) = gd (/;rl,r2,i'l,i'2) +
The anomalous potential T(r, Ax) reads for a global
-~-V ~(12)E (t; 1"12 , r 1 ; x 0 ) + V' T(12)E (t; 1"12 , r l ; A x ) ,
gravity field recovery,
(9)
The quantity gj is the disturbance part, which represents the non-conservative disturbing forces and V V~12)e is the reference part, modeled by the tidal potential of the Earth (E) acting on the satellites 1 and 2, V~(12)E (/~;rl2,rl ; Xo) -- V (V(F, + r , 2 ) - V ( r , ) ) ,
(10)
and represents the long-wavelength gravity field features. The anomalous part, VT(,2)E (t;r,2,r,; Ax) - V ( r ( r , + r,2)- r ( r , ) ) ,
(11)
Q~ (0,2) = p m(COSO)COSm2,
(13)
S .... (,9,2)- P['(cos,9)sinm2.
F
n=2 m=0
(]4)
with the corrections Acre,As" • Ax to the reference potential coefficients cn,,,, s,, • x 0 . In case of a regional recovery the anomalous potential T(r) is modeled by parameters of space localizing base functions, I
models the high frequent refinements, parameterized either by corrections Ax to the global gravity field parameters x 0 or by parameters Ax of a linear approximation with space localizing base functions, modeling the regional gravity field refinements. Details can be found in Mayer-Gfirr et al. (2005a). In case of the analysis of observation sets of months to years, the observation equations are formulated in space domain by dividing the total orbit in short pieces of arcs with a length of approximately 30 minutes. The length of the arcs is not critical at all and can be adapted to the uniformity of the data flow. Because of the fact, that a bias for each of the three components of the accelerometer measurements along a short arc will be determined, the arc length should not be too small to get a safe redundancy and not too long to avoid accumulated not modelled disturbances. When the normal equations of all arcs are merged, for every short arc a variance factor has to be determined by an iterative computation procedure, to take the (possibly) varying accuracy of the short arcs into account (cf. Koch and Kusche (2003) for the variance component estimation, and Mayer-Giirr et al. (2005a) for the iterative computation procedure).
2.2 Gravity field representation The reference potential can be formulated in the usual way as follows,
i=l
with the unknown field parameters a i arranged in a column matrix Ax := (ai, i = 1,...,I) r and the base functions, N...... f R -~n+l
The coefficients k are the degree variances of the (difference) gravity field spectrum to be determined, m=0
with the fully normalized potential coefficients A~ .... A~,,. Re is the mean equator radius of the Earth, r the distance of a field point from the geocentre and P (r,%,) are the Legendre polynomials depending on the spherical distance between a field point P and the nodal points Qi of the set of base functions located at the nodal points i. The maximum degree NMaX in Eq. (16) should correspond to the envisaged maximum resolution expected for the regional recovery. With the definition in (16) the base functions ¢~(r,ro, ) can be interpreted as isotropic and homogeneous harmonic spline functions (Freeden et al., 1998). The nodal points are defined on a grid generated by a uniform subdivision of an icosahedron of twenty equal-area spherical triangles. In this way the global pattern of spline nodal points Qi shows approximately uniform nodal point distances. Details can be found e.g. in Eicker et al. (2005).
2.3 Combination of normal equations F
n 0m=0
(12) with the surface spherical harmonics,
For the analysis of CHAMP or GRACE observations not only the gravity field parameters have to be estimated, but also arc-related parameters as for ex-
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K.H. Ilk. A. Eicker. T. Mayer-G~irr
ample the two boundary position vectors of each arc and additional bias parameters to take into account residual surface force effects. These parameters sum up to thousands of additional unknowns for an analysis period of one month in case of short arcs with a mean arc length of approximately 30 minutes. To reduce the size of the normal equation matrices, the arc-related parameters are eliminated before the arcs are merged to the complete system of normal equations. Every short arc builds a (reduced) partial normal equation. To combine the normal equation matrices for the short arcs, separate variance factors for each arc have to be determined, to consider the variable precision of the range and range-rate observations. Furthermore, due to the fact that the gravity field recovery process is improperly posed, an additional regularization factor and a regularization matrix can be introduced into the gravity field recovery procedure. For details of the iterative combination scheme combined with a variance component estimation and the computation of the regularization factor refer to Mayer-Gfirr et al. (2005a).
2.4 Merging of regional refinement patches For many applications it is useful to derive a global gravity field model by spherical harmonics without losing the details of a regional zoom-in. This can be performed by a direct stable computation step. In our approach the coefficients of the spherical harmonic expansion are calculated by means of the Gauss-Legendre-Quadrature (cf. e.g. Stroud and Secrest, 1966). This method is also referred to as Neumann's method, as described in Sneeuw (1994) among different other quadrature methods,
s
t
- GM4Jr ~ TkP'' (cos 0~) ~ sin(m2~
w~,
(~8)
with the area weights,
T~ is the gravitational potential at the K nodes of the numerical quadrature, P~ are associated Legendre functions and P¢+I are the first derivatives of the Legendre polynomials of degree N + 1, where N is the maximum degree to be determined. This method requires the data points to be located at a specific grid, called Gauss-Legendre-Grid. From the regional spline solutions the gravitational potential can be calculated at the nodes of this grid
without loss of accuracy. It has equiangular spacing along circles of latitude; along the meridian the nodes are located at the zeros of the Legendre polynomials of degree N + 1. This quadrature method based on this specific grid has the advantage of maintaining the orthogonality of the Legendre functions despite the discretization procedure, which allows an exact calculation of the potential coefficients. The resolution of the grid has to be adapted to the envisaged maximum degree of the spherical harmonic expansion and should be limited only by the signal content of the gravity signals in the observables.
Results of regional gravity field refinements The complete recovery procedure consists of three steps which can be applied independently as well: • Least squares global gravity field recovery based on a spherical harmonic expansion up to a moderate degree without any regularization to provide a basis for further refinements, • Least squares regional refinements of the gravity field by spherical splines as space localizing base functions, adapted to the specific gravity field features with respect to nodal point distribution and base function characteristics, if possible covering the globe, • Determination of a global gravity field model by merging the regional refinement solutions and deriving potential coefficients by a numerical quadrature technique without loosing the regional details. We want to point out that a combination of data sets of different origin is simply possible, either based on satellite, airborne or terrestrial gravity information. This can be done, in principle, at every step of the 3step procedure. The global gravity field determination of the first step is performed independently from the subsequent regional refinement computations and no correlations between the parameters have been taken into account yet. The following recovery results refer to the K-band range-rate measurements of the GRACE twin satellite mission for the months September 2003 and July 2003, respectively. The observations are corrected for the tides caused by the Sun, the Moon and the planets. The ephemeris are taken from the JPL405 data set. Effects originating from the deformation of the Earth caused by these tides are modelled following the IERS 2003 conventions. Ocean tides are
Chapter49
• Global Gravity Field Recovery by Merging Regional Focusing Patches: an Integrated
computed from the FES2004 model. Effects of high frequency atmosphere and ocean mass redistributions are removed prior to the processing by the GFZ AOD de-aliasing products. The 30-days-orbit has been split into 1500 short arcs of approximately 30 minutes arc length. For each arc the coordinates of the boundary vectors have been determined as well as an accelerometer bias (Mayer-Gfirr et al., 2flflSh~_
-50 -40 -30 -20 -10
0
10
20
30
40
Approach
has been applied. The arc-related parameters are eliminated before merging the normal equations for each short arc to the total system of normal equations as outlined in Sect. 2.3. Fig. 1 shows the differences of this monthly solution and the EIGENCG03C, here taken up to a maximal degree n=140, representing a slightly higher resolution as our onemonth solution. Fig. 2 demonstrates that a maximum degree of n=140 results only in a slight improvement from 40,7cm to 37,9cm in terms of RMS. This means that a higher resolution of the global model based on spherical harmonics cannot significantly improve the result.
50
[cm]
RMS: 40.70 cm
avg: 29.69cm
Max: 392.05cm
Fig. 1: Differences of direct monthly solution ITGGRACE-2003-09-d (n-120) minus EIGEN-CG03C (n=140).
-50 -40 -30 -20 -10
RMS: 37.90 cm
0
10
[crn] avg: 30.03 cm
20
30
40
50
Max: 196.88 cm
Fig. 2: Differences of direct monthly solution ITGGRACE-2003-09-d (n = 140) minus EIGEN-CG03C (n=140).
In the first step, a global spherical harmonic solution up to degree n=120 beginning from degree n=2 has been determined directly for the month September 2003 from the GRACE range-rate measurements, in the following designated as gravity field model ITG-GRACE-2003-09-d ("d" means direct spherical harmonic solution). The mathematical model (8) with (6) based on a pure spherical harmonic gravity field representation according to (9) and (10) with the spherical harmonic model (12)
Fig. 3: Regional refinement patches covering the complete surface of the Earth with additional frame around a recovery area. For the regional refinement solutions the same mathematical model as used for the global solution and formulated in (8) with (6) has been applied except for the gravity field representation. Based upon the global spherical harmonic solution up to degree n-120 the additional gravity field refinements are represented according to (9) with (11) represented by spherical spline functions according to (15). The number of nodal points I as well as the maximum degree NMax of (16) has been chosen identical for all patches in the present case. To avoid geographical truncation effects at the region boundaries, gravity field parameters defined in an additional strip of 10 ° width around the specific regions have to be taken into account. The regional patches are shown in Fig. 3. This is of course not an ideal adaptation to the different gravity field characteristics. Here the pole regions north of 60 ° and south of-60 ° are separate patches. The nodal points are located at a regular grid with a mean distance between the nodal points of approximately 140kin. A global gravity field model represented by spherical harmonics can be calculated again by a proper application of the Gauss - Legendre - Quadrature method based on the global coverage with regional refinements.
333
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K.H. Ilk. A. Eicker. T. Mayer-GiJrr
E r~
8 .~
0.1
> tL',(j~o~.'~-~.,ru~e,.c~.an m~.-...l'e'J ~'
,~,,,~W'~"
"
0.01
o.ool
60 °
70 °
80 °
90 °
110 o
120 ° 90 °
,
;
-1;0-80-E~0-4~0-2'0
100 o 2'0
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6'0
80
I
100 o
110 o
120 °
100-100-80-60-40-20
140 °
150 °
I
0 20 [cm]
[cm]
RMS: 56,07 cm Avg: 39,42 cm Max: 355,78 cm
130 °
. . . . . . . . . . 40
6-i
60
80
RMS: 4,62 cm Avg: 3,60 crn Max: 19,27 cm r
I°
90 ° .
100 o .
-10
-8
.
.
-6
110 o .
-4
.
-2
120 ° .
.
.
.
0 2 [cm]
4
6
8
I 10
Fig. 4: Regional refinements of adjacent patches with differences (based on the direct global solution ITG-GRACE-2003-09-d, n=120). Fig. 4 demonstrates the excellent matching of two regional refinements with a common overlapping recovery strip. It should be pointed out that both residual fields are computed independently with individually determined regularization parameters. The bottom graph at the left-hand side has been plotted with the same scale as the two regional refinements at the top of Fig. 4, while the graph at the righthand shows the same overlapping region with a magnified scale.
d~-ecl~ 0
100
RMS: 56,58 crn Avg: 41,41 cm Max: 317,50 cm
ii
0.0682 & 0 0001 10
20
:243
40
50
h~"rr~n~: ~ 60
70
80
n-=140,:~ ~,,~i.,,lanz,i,-,,on.', 90
100
110
t20
t30
t40
spherical harmomc degcee
Fig. 6: Difference degree variances of monthly solutions (September 2003) versus EIGEN-CG03C. Fig. 5 shows the differences of the merged monthly solution ITG-GRACE-2003-09 minus EIGEN-CG03C, both models up to a spherical harmonic degree of n=140. The comparison with Fig. 2 demonstrates a significant improvement, especially in the higher degrees from n=120 upwards. Fig. 6 confirms this result by the corresponding difference degree variances related to EIGEN-CG03C. It should be pointed out that the slightly better results of the direct solution in the long and medium wavelength part of the gravity field spectrum are very small and maybe not really significant. The resolution of the regional refinements corresponds to a spherical harmonic degree of approximately 140. In the following we will compare our merged September 2003 solution to various models by restricting the upper degree to n=120. The statistics of the geoid height differences between the merged monthly solution ITG-GRACE-2003-09 and the GFZ model CG03C are: RMS=15,52cm, avg=12,01cm and Max=74,82. The corresponding values for the differences with respect to the CSR model GGM02C are slightly smaller and read: RMS=13,50cm, avg=10,59cm and Max=72,95.
E
a~ ,~-
0.1
C
'~ -50
-40
-30
-20
-10
0
10
20
30
40
50
0.001
._
[ore] RMS: 23.83 cm
avg: 18.28 cm
Max: 135.49 cm
Fig. 5: Differences of merged monthly solution ITG-GRACE-2003-09 (n=140) minus EIGENCG03C (n = 140).
I~
10
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40 50 ~ 70 813 9'0 ~1~n¢4~1 h B r r n o ~ i ¢ ~%=~ree
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Fig. 7: Difference degree variances of monthly solutions (September 2003) versus EIGEN-CG03C.
Chapter49 • GlobalGravityField Recoveryby Merging RegionalFocusing Patches: an Integrated Approach
The difference degree variances of our merged solution ITG-GRACE-2003-09 together with the corresponding GFZ and CSR solutions for September 2003 are displayed with respect to the GFZ model EIGEN-CG03C in Fig. 7 and with respect to the CSR gravity field model GGM02C in Fig. 8. It is remarkable that our monthly model has a significant better coincidence with the superior models EIGEN-CG03C and CSR-GGM02C from degree n-70 upwards than the corresponding GFZ and CSR monthly models. In the long and medium frequencies the situation is slightly different; here the GFZ and the CSR monthly models show a better coincidence with the corresponding combined models of these organisations, albeit the differences are very small if the logarithmic scaling is properly taken into account.
i •
ences of the corresponding merged monthly solutions GFZ-GRACE-2003-09 minus GFZ-GRACE2003-07 provided by the GFZ Potsdam and Fig. 11 the differences of these monthly solutions by CSR Austin. We detect a remarkable good fit in the pole regions for our models in contrast to the GFZ- and C S R - solutions. But also the RMS, the average and maximum values are significantly better for our solutions than the GFZ and CSR solutions. The large deviations along the +/-60°-latitudes cannot be explained yet, but they occur also in the other solutions.
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30
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50 i~} 70 80 90 sphe~cal hann~ic degree
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110
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Fig. 10: Differences of monthly solutions GFZGRACE-2003-09 minus GFZ-GRACE-2003-07.
Fig. 8: Difference degree variances of various monthly solutions (September 2003) versus CSRGGM02C.
-50
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RMS: 28.14 cm
-50
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-10
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avg: ~1r[~7 cm
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30
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Fig. 11: Differences of monthly solutions CSRGRACE-2003-09 minus CSR-GRACE-2003-07.
Max: 98.54 cm
Fig. 9: Differences of monthly solutions ITGGRACE-2003-09 minus ITG-GRACE-2003-07. The inner consistency of the monthly solutions can be checked, especially in the high-frequent spectral range, if two monthly solutions are compared. Fig. 9 shows the differences of the merged monthly solutions ITG-GRACE-2003-09 minus ITG -GRACE-2003-07. Fi~. 10 displays the differ-
4 Conclusions
The gravity field determination approach by merging regional recovery patches integrates a regional zoom-in with a global gravity field recovery. The achievement of this proposal has been tested based on GRACE low-low satellite-to-satellite data. All tests demonstrated the advantage of our approach
335
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K.H. Ilk. A. Eicker. T. Mayer-G~irr
compared to deriving a global gravity field solution in terms of spherical harmonics directly. The reasons for the superior quality of our procedure are versatile: The tailored calculation of a regularization parameter for each region allows a tailored filtering according to the individual gravity field features. The modelling of these structures can be performed very effectively by tailored space localizing base functions as e.g. spherical splines as applied in our approach, but others as spherical wavelets can be applied as well. To utilize the manifold advantages of spherical harmonics as base functions, our approach allows the calculation of the potential coefficients as well with the advantage that the quadrature procedure does not limit the resolution to an upper degree. A consequence is that no stability problems occur and no regional details of the gravity field are lost. Further advantages are the following: the method is modest in terms of computation costs, as the complete problem is split up into much smaller problems. This procedure enables the computation of a global gravity field model up to an arbitrary resolution on a single PC. It is very simple to include additional types of gravity field information in the computation scheme; combination of terrestrial date causes no problem and the weighting of this information can be determined within the frame of the variance component estimation step. Further improvements are expected with respect to the refinement of the regularization strategy to enable smoother transition zones between the zoom-in-regions and by tailoring the zoom-in areas more accurately to the characteristics of the gravity field in the specific regions. Furthermore, a more precise selection of the base functions and the nodal point distribution adapted to the roughness of the gravity field, possibly combined with a multiresolution strategy, seems to be possible. Additional points of future research are a careful investigation of the aliasing effects originating from the patching of several regional solutions and a homogenisation of the regional solutions to avoid long wavelength errors. A rigorous variance-covariance computation of the final result based on the three computation steps as outlined in Sect. 3 is one additional aspect of a forthcoming research.
Acknowledgement.
We appreciate the anonymous reviewers for their helpful comments and suggestions. The support of BMBF (Bundesministerium fiir Bildung und Forschung) and DFG (Deutsche
Forschungs-Gemeinschaft) of the GEOTECHNOLOGIEN programme is gratefully acknowledged.
References Eicker A, Mayer-Gfirr T, Ilk, KH (2005) Global Gravity Field Solutions Based on a Simulation Scenario of GRACE SST Data and Regional Refinements by GOCE SGG Observations, in C. Jekeli, et al.: Gravity, Geoid and Space Missions- GGSM2004, Porto, Portugal, lAG International Symposium, International Association of Geodesy Symposia, Vol. 129, pp. 6671, Springer F6rste C, Flechtner F, Schmidt R, Meyer U, Stubenvoll R, Barthelmes F, Neumayer KH, Rothacher M, Reigber C, Biancale R, Bruinsma S, Lemoine JM, Raimondo JC (2005) A New High Resolution Global Gravity Field Model Derived From Combination of GRACE and CHAMP Mission and Altimetry/Gravimetry Surface Gravity Data, Poster presented at EGU General Assembly 2005, Vienna, Austria, 24-29, April 2005 Freeden W, Gervens T, Schreiner M (1998) Constructive Approximatiuon on the Sphere, Oxford University Press, Oxford, Ilk KH, Mayer-Gfirr T, Eicker A, Feuchtinger M, (2004) The Regional Refinement of Global Gravity Field Models from Kinematical Orbits, New Satellite Mission Results for the Geopotential Fields und Their Variations, Proceedings Joint CHAMP/GRACE Science Meeting, GFZ Potsdam, July 6-8 Koch KR, Kusche J (2003) Regularization of geopotential determination from satellite data by variance components, Journal of Geodesy 76(5):259-268 Mayer-Gfirr T, Ilk KH, Eicker A, Feuchtinger M (2005a) ITG-CHAMP01: A CHAMP Gravity Field Model from Short Kinematical Arcs of a One-Year Observation Period, Journal of Geodesy (2005) 78: 462-480, Springer-Verlag Mayer-Gfirr T, Eicker A, Ilk KH (2005b) Gravity field recovery from GRACE-SST data of short arcs, in: R. Rummel, et al. Observation of the Earth System from Space (in preparation), Springer Reigber C, Schwintzer P, Lfihr H (1999) The CHAMP geopotential mission, Boll. Geof. Teor. Appl., 40:285289 Sneeuw N (1994) Global Spherical Harmonic Analysis by Least Squares and Numerical Quadrature Methods in Historical Perspective, Geophys. J. Int. 118:707-716 Stroud AH, Secrest D (1966) Gaussian Quadrature Formulas, Prentice-Hall, Englewood Cliffs, N.J. Tapley BD, Bettadpur S, Watkins M, Reigber Ch (2004) The gravity recovery and climate experiment: mission overview and early results., Geophys Res Lett 31, L09607: doil 0.1029/2004GL019920 Tapley BD, Ries J, Bettadpur S, Chambers D, Cheng M, Condi F, Gunter B, Kang Z, Nagel P, Pastor R, Pekker T, Poole S, Wang F Ch (2005) GGM02- An improved Earth gravity field model from GRACE, Journal of Geodesy (2005) 79(8): 467-478, Springer-Verlag
Chapter 50
External calibration of GOCE SGG data with terrestrial gravity data: A simulation study D. N. Arabelos Department of Geodesy and Surveying, Aristotle University of Thessaloniki, GR-54 124 Thessaloniki, Greece C.C. Tscherning, M. Veicherts Department of Geophysics, University of Copenhagen, Juliane Maries Vej 30, DK-2100 Copenhagen, Denmark
Abstract.
Terrestrial gravity anomalies selected from three extended continental regions having a smooth gravity field were used in order to determine the appropriate size of the area for gravity data collection as well as the required data-sampling for calibration of the GOCE satellite gravity gradient (SGG) data. Using Least Square Collocation (LSC), prediction of gravity gradient components was carried out at points on a realistic orbit. Based on the mean error estimation it was shown that up to 80% of the signal of the gravity gradient components, as it is expressed through the covariance function of the terrestrial gravity data, can be recovered in the case of an optimal size of the collection area and of the optimum resolution of the data. These optimal conditions e.g. for the Australian gravity field, correspond to an 10 ° × 12 ° area extend and a 5 f data-sampling. It was also numerically demonstrated that it is possible to calibrate the GOCE SGG data for systematic errors such as bias and tilt.
Keywords. Satellite Gravity gradiometer data, External calibration, Systematic error parameters
1
Introduction
In the last decade numerous investigations were published, concerning the calibration of GOCE satellite gravity gradiometer data, (e.g. Bouman et al., 2004; Wolf & Denker, 2005). The use of Least Squares Collocation (LSC) as a element of the space-wise approach methods has also been discussed in a number of papers (e.g. Tscherning, 2005). The aim of this work was to determine the size of required areas with terrestrial gravity data, as well as the required resolution and accuracy of the gravity data needed for calibration when LSC is used. The aim is to detect possible systematic errors in the GOCE SSG data. The "simple" LSC method was used for the tests concerning the size of the area and the resolution of the
data, while the parametric LSC was used for the tests concerning the detection of systematic errors. The terrestrial gravity data sets used in this study are described in details in section 2. In the first part of this study errors of gravity gradients were computed at points on the realistic orbit of the lAG SC7 simulated data set with the gradients in a reference frame aligned with the velocity vector and the z-axis which lie in the plane formed by this vector and the position vector. In the last part of the study points on a similar orbit were used, but with the gradients given in a more realistic instrument reference frame provided by ESA (R. Floberghagen, private communication 2004). The precision of the calibration will be directly proportional to the gravity field standard deviation for example expressed as the standard deviation of gravity anomalies from which the contribution of a reference field have been subtracted. The areas studied here are therefore areas with a very smooth gravity field. We have used EGM96 to degree 360 (Lemoine et al., 1998) for the reduction of gravity anomalies, in order to smooth as far as possible the gravity anomalies used in all test areas. Then, using the reduced gravity anomalies we have predicted gradient values at points on tracks crossing the areas, further on called control points. Since real GOCE data are not yet available we have used the error estimates given by collocation (eq. 66 in Moritz, 1978), instead of the statistics of the differences between predicted and control values. More specifically, we relate the mean collocation error, depending on the choice of the covariance function, with the formal standard deviation of the gravity gradient components, depending also on the covariance function used, in order to draw conclusions about the optimal size of the area and the resolution of the data needed for the calibration. The mean collocation error is computed as the mean error of the collocation error estimates over all predicted
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D.N. Arabelos • C. C. Tscherning • M. Veicherts
points. Note, however, that the error in the middle of the area typically is 90% of the mean error.
free-air gravity field is shown in Fig. 1. The corresponding statistics is shown in Table 1.
It will be numerically shown in the next, that in all test areas, using LSC and terrestrial gravity anomalies, up to 80% of the formal standard deviation of the gravity gradient signal can be recovered, in the case of a high data accuracy, size of the area of collection of terrestrial gravity anomalies and of the datasampling. Furthermore, it will be also numerically shown that in this way it is possible to calibrate the G O C E SGG data for systematic errors such as bias and tilt. Here we have used that the expected accuracy of 1 s sampled data in the measurement bandwith will have an error equal to or above 7 mE.
2. Surface gravity anomalies from Australia (further on called region B)
In the computations it was attempted to keep the data-sampling and the size of the terrestrial data collections areas constant for the corresponding experiments from area to area.
2
Gravity data used
For reasons discussed in earlier work (e.g. Arabelos & Tscherning, 1998), for the requirements of the calibration, the terrestrial data have to be collected from areas with possible smooth free air gravity anomaly field. A further reason for this is to avoid topographic reductions to smooth the gravity field, due to errors that could be introduced to gravity anomalies from erroneous altitudes and density hypotheses.
The 2004 edition of the Australian National Gravity Database contains over 1,200,000 point data values in the area bounded by - 4 8 ° _< q0 _< - 8 °, 108 ° _< )~ _< 162 °. This data was made available by Geoscience Australia. The data set covering the continental Australia and the surrounding ocean (1,117,054 point values) was reduced to E G M 9 6 up to degree 360 was removed from the gravity anomalies. The statistics of the original and reduced free-air gravity anomalies is shown in Table 1. As it is shown in Fig. 2 the gravity field is very smooth in the central Australia and consequently, appropriate for the calibration requirements. For this reason, point gravity anomalies were selected from the area bounded by - 3 2 ° _< q0 _< - 2 0 °, 124 ° _< )~ _< 144 ° .
236 °
240 °
244 °
248 °
252 °
68 °
68 °
66 °
66 °
Another requirement was to collect data from extended regions in different geographic latitudes due to the dependence of the distribution of the G O C E data on the latitude.
64 °
64 °
For all these reasons, data from the Canadian plains, Australia and Scandinavia were used.
62 °
62 °
60 °
60 °
58 °
58 °
56 °
56 °
The terrestrial gravity data have errors which we consider random. However, we are aware that systematic errors are present in the data, especially due to height datum problems. Assuming an error of 1 m in the height datum, the correlated noise of the gravity anomalies equals to 0.3 mgal. The calibration procedure may very well take such error correlations into account, if they are known.
1. Terrestrial free-air gravity anomalies from the Canadian plains (further on called region A) This data set was already described in (Arabelos & Tscherning, 1998). In the present paper the reduced values, i.e. the free-air gravity anomalies after the removal of the contribution of the geopotential model E G M 9 6 up to degree 360, were used within the area bounded by 56 ° _< q0 _< 68o,236 ° _< )~ _< 254 °. The
236 °
240 °
244 °
248 °
252 °
~
mGal
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Gravity Figure 1o The free-air anomaly-EGM96 gravity fi eld in the Canadian plains (simplified)
Chapter50 • ExternalCalibrationof GOCESGGData with TerrestrialGravityData:a SimulationStudy
Table 1. Statistics of the free-air gravity anomalies used in the regions A, B and C. Unit is mGal Region A, 14,177 point values Observations EGM96 Difference Mean -10.768 -10.678 -0.090 Standard Dev. 22.419 17.497 13.418 Max. Value 133.000 51.044 114.168 Minimum value -81.100 -72.210 -124.857 Region B, 1,117,054 point values Mean 4.901 5.158 -0.258 Standard Dev. 24.504 22.504 12.102 Max. Value 248.602 94.223 219.875 Minimum value -211.327 -104.930 -194.732 Region C, 62,126 terrestrial values Mean -8.466 -8.118 -0.349 Standard Dev. 18.285 16.229 8.789 Max. Value 71.740 36.066 76.805 Minimum value -84.021 -77.351 -47.250 Region C, 4,778 air-borne values Mean -18.443 -19.735 1.292 Standard Dev. 19.834 18.209 10.024 Max. Value 29.660 21.235 35.085 Minimum value -80.540 -68.107 -31.675
F r o m Table 1 it is s h o w n that the r e d u c e d to E G M 9 6 free-air gravity a n o m a l i e s present very similar statistical characteristics. This is m o r e evident f r o m the shape of the c o r r e s p o n d i n g covariance functions (see Fig. 4). For the reasons discussed in section 1 the formal standard deviation of the control values used in the n u m e r i c a l e x p e r i m e n t s of section 3 is s h o w n in Table 2 for the three test regions. This formal standard deviation for each area is based on the c o r r e s p o n d i n g covariance function of Fig. 4. 112°
Try
Txz Tyy Tyz Tzz
Region A 0.0042 0.0049 0.0050 0.0042 0.0050 0.0072
Region B 0.0035 0.0041 0.0042 0.0035 0.0042 0.0060
Region C 0.0047 0.0054 0.0056 0.0047 0.0056 0.0079
3. Gravity anomalies from Scandinavian (further on called region C) Terrestrial as well as air-borne gravity a n o m a l y data sets were m a d e available by R. Forsberg, R K n u d sen and G. Strykovski (private c o m m u n i c a t i o n ) . T h e terrestrial data set (62,126) cover the area 53.99 ° _< q0 _< 64 °, 11.97 ° _< ~, _< 30.02 °. After the r e d u c t i o n to E G M 9 6 we obtain present the statistics given in Table 1. T h e air-borne data set cover the area 54.58 ° < q0 _< 60.12 °, 12.01 ° _< )v _< 26.86 °. T h e statistics of the r e d u c e d to E G M 9 6 air-borne gravity a n o m a l i e s (4778 point values) is s h o w n in Table 1. In the collocation e x p e r i m e n t s both data sets were used jointly with c o m m o n accuracy equal to 2 m G a l . T h e free-air gravity field r e d u c e d to E G M 9 6 is shown in Fig. 3.
128°
136°
144°
152°
160°
_12°
_12°
_18°
_18°
_24°
_24°
_30°
_30°
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_36°
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Table 2. Signal standard deviation of data at the control points used in the numerical experiments. (E)
Txx
120°
120°
128°
136°
144°
152°
160° mGal
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Gravity
Figure 2. The free-air anomaly-EGM96 gravity field in Australia (simplifi ed)
3
Numerical Experiments
3.1 T e s t s c o n c e r n i n g t h e r e q u i r e d s i z e of t h e a r e a a n d t h e r e s o l u t i o n of t h e d a t a T h e e x p e r i m e n t s c o n c e r n r e c o v e r y of 5 s sampling noise-free simulated G O C E data p r o v i d e d by IAG, along 1 m o n t h realistic orbit (250 km), using terrestrial gravity anomalies. For the d e t e r m i n a t i o n of the required size of the area for terrestrial data collection in all cases 10 arcmin d a t a - s a m p l i n g was used and prediction e x p e r i m e n t s were carried out in four areas with different size. For the d e t e r m i n a t i o n of the required data-sampling, the prediction e x p e r i m e n t s were carried using data with 5, 7.5, 10, 15 and 20 arcmin, sampling in areas with constant size. In all c o m p u t a t i o n s the G R A V S O F T p r o g r a m s (Tscherning et al., 1992) E M P C O V , C O V F I T and G E O C O L were used.
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D.N. Arabelos • C. C. Tscherning • M. Veicherts
15 ° I
mm
mm
20 ° mm
mm
25 ° mm
mm
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ror estimation concerning all gradient c o m p o n e n t s is shown in Fig. 5. This i m p r o v e m e n t is more significant in the case of Tzz (39%). The mean error of 0.0027 EU correspond to a 37% of the formal standard deviation of Tzz (see Table 2 Region A), resulting from the covariance function of free-air gravity anomalies in this region. This could be interpreted as the ability of the m e t h o d to recover the 63% of the signal, in the case of real S G G data.
(b) Experiments for the determination of datasampling Concerning the determination of the required datasampling, experiments were carried out with terres55°
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(a) Experiments for the determination of the size of the area for terrestrial data collection The experiments were carried out using terrestrial gravity data with constant 10 arcmin sampling. Accuracy equal to 1 m G a l was adopted for these data. Prediction experiments were carried out collecting terrestrial data from four areas with size 5 ° x 6 ° 6 ° 8°,8 ° × 10 ° and 10 ° × 12 ° respectively. The first of them correspond to size of the control points collection area. The results of these numerical experiments in terms of the mean collocation error estimation, are shown in Fig. 5(a). It is well known that the assessment of the prediction results in collocation may be based not only on the statistics of the differences between observed (control) and predicted quantities, but also on the collocation error estimation of the prediction. Since we do not yet have any real G O C E data used the formal error estimates. With the increase of collection area from 5 ° z 6 ° to 10 ° × 12 ° a continuous i m p r o v e m e n t of the mean er-
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Chapter 50 • External Calibration of GOCESGG Data with Terrestrial Gravity Data: a Simulation Study trial data collected f r o m the same 6 ° x 8 ° area (58 ° _<
q~ _< 64 °, - 1 1 9 ° _< )~ _< - 1 1 1 o) but with different resolution (5, 7.5, 10, 15 and 20 arcmin). The results of these experiments are shown in Fig. 5(b)
2~< 144 o.
The simulated G O C E data were collected from the area - 3 0 . 5 0 ° _< q~ _< - 2 5 . 5 0 °, 127 ° _< )~ _< 133 ° (245 values). The terrestrial data were assumed to be accurate to 1 mGal.
(a) Experiments concerning the size of required area with terrestrial gravity data
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The experiments were carried out using terrestrial gravity data with constant 10 arcmin sampling. Prediction experiments were carried out collecting terrestrial data from four areas with size 5 ° × 6 °, 6 ° × 8°,8 ° × 10 ° and 10 ° × 12 ° respectively. The first of them correspond to size of the control points collection area. The results of the prediction in terms of the mean collocation error estimation are shown in Fig. 6(a).
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I
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5
10
15
20
Data-sampling (arcmin)
Figure g. Region A" Mean collocation error estimation for different sizes of the area for terrestrial data collection (a), mean collocation error for different data-sampling (b). Txx = square, Txv = circle, Txz = triangle, Tyy = inverted trinagle, Tyz -- hexagon, T,z -- diamont.
From Fig. 5(b) it is shown that for the same size of the test area, the mean collocation error estimation was decreased with increasing resolution of the data, for all the gravity gradient components. The decrease is more evident in the case of Tzz because it drops from 0.0050 to 0.0032 EU when the resolution was increased from 20 to 5 arcmin. In terms of percentage of the formal standard deviation of T~z this correspond to a change from 69% to 44%.
Region B Experiments were carried out with constant 10 arcmin. data-sampling. The empirical covariance function shown (together with its analytical fitting) in Fig. 4(b) was computed from the 10 arcmin gravity data covering the area - 3 2 ° _< q0 _< - 2 0 °, 124 ° _<
In the region B the formal standard deviation of Tzz is 0.0060 EU when the covariance function is computed from terrestrial gravity anomalies in the area - 3 2 ° _< q~ _< - 2 0 °, 124 ° _< )~ _< 144 °, while is increased to 0.0089 EU when the covariance function is computed from terrestrial data in the area - 3 0 . 5 0 ° _ < q ~ _ < - 2 5 . 5 0 ° ,127 °_<)~_< 133 °. From this point of view, the mean collocation error estimation should be considered as a measure of the part of the real signal that could be recovered, if of course, the covariance function used reflects the statistical characteristics of the real signal. From Fig. 6(a) it is shown that e.g. in the case of T~, the mean collocation error is rapidly changing from 54% to 31% of the formal standard deviation of Tzz, when the size of the collection of terrestrial data was increased from 5 ° × 6 ° to 10 ° × 12 °. This means that with an area extent of 10 ° × 12 ° about 65% of the signal of the real data could be recovered, which is a value acceptable for the calibration requirements.
(b) Experiments concerning the data-sampling Terrestrial gravity anomalies with data-sampling 5, 7.5, 10, 15 and 20 arcmin were collected from the area bounded by - 3 2 ° _< q~ _< - 2 0 °, 124 ° _< )~ _< 144 °. The prediction results in terms of mean collocation error are shown in Fig. 6(b). From Fig. 6 it is shown that the mean error of estimation for the same size of the area of collection of terrestrial gravity anomalies was decreased when the density of the data was increased, for all the gravity gradient components. In terms of percentage of the formal standard deviation of the various components this decrease is more significant in the case of T~z ,
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Table 3.
Results of experiments in the region (B) with data-sampling 20', showing that further increase of the size of data collection area has no signifi cant effect. (E)
Comp.
Area extent
Control points
Mean coll. err.
T~z
6 ° × 8°
436
0.0043
T~z Tzz
10 ° × 12 ° 12 ° × 20 °
982 2163
0.0041 0.0041
Table 4. Improvement of the mean error in region (B) when terrestrial gravity anomalies with 101,7.5 t and 5 t sampling are used in the same (10 ° × 12 °) area. (E) Comp. T~z Tzz Tzz
Data-sampl. 101 7.51 51
Control points 3879 6834 13476
Mean coll. err. 0.0018 0.0014 0.0012
since it drops from 71% to 30% as the data-sampling changes from 20 to 5 arcmin. As it is shown in another e x p e r i m e n t (see Table 3), a further increase of the size of the area in the case of 20' data-sampling, has no significant effect on the prediction results e.g. of T~. C o m b i n i n g the results shown in Fig. 6 (a) and (b) it is r e a s o n a b l e to expect that using data in the larger area ( - 3 3 ° _< q0 _< - 2 3 °, 124 ° _< )~ _< 136 °) with the m o r e dense data-sampling (5 arcmin) we will get the better results in terms of the m e a n collocation error estimation. This was verified with a relevant experim e n t (see Table 4)
Region C The empirical covariance function shown (together with its analytical fitting) in Fig. 4(c) was computed from gravity data covering the area 54 ° _< q0 _< 64 °, 12 ° _< ~ _< 30 °. Simulated G O C E data were collected from the area b o u n d e d by 57 ° _< q0 _< 62 °, 21 ° _< ~ _< 27 ° (250 values) in order to be used as control data. For these data accuracy equal to 0.001 EU was adopted.
0.003
(a) Experiments concerning the size of required area with terrestrial gravity data
0.002
I
I
-5Ox 6° 6°x 8°
I
8°xlO°
lO°xl]Y~
Size of collection area
I
I
I
I
0.004 o "~ 0.003
The experiments were carried out using terrestrial gravity data with constant 10 arcmin sampling. Accuracy equal to 2 m G a l was adopted for these data. Prediction experiments were carried out collecting terrestrial data from four areas with size 5 ° x 6 °, 7 ° x 80,9 ° x 10 ° and 10 ° x 12 °, respectively. The first of t h e m c o r r e s p o n d to size of the control points collection area. The results of these numerical experiments are shown in Fig. 7(a) These results are very similar to the previous ones in Figures 5 and 6(a). Increasing the collection area from 5 ° x 6 ° to 10 ° x 12 °, the m e a n error estimation, e.g. in the case of Tzz was decreased up to 51% of its original value. In terms of p e r c e n t a g e on the formal standard deviation of Tzz the m e a n estimation error drops from 53.8% to 27.5%.
(b) Experiments concerning the requirements in data-sampling
0.002 I
I
I
I
5
10
15
20
Data-sampling (arcmin)
Figure 6. Region B: Mean collocation error estimation for different sizes of the area for terrestrial data collection (a), mean collocation error for different data-sampling (b). Txx = square, Txy = circle, Txz = triangle, Tyy = inverted trinagle, Ty: = hexagon, ~z = diamont.
For this p u r p o s e terrestrial gravity anomalies with data-sampling 5, 7.5, 10, 15 and 20 arcmin were collected from the area b o u n d e d by 56 ° _< q) _< 63 °, 28 ° _< ~, _< 28 °. The prediction results are shown in Fig. 7(b). Also these results are similar to the results of corresponding experiments in the regions A and B.
Chapter 50 • External Calibration of GOCESGG Data with Terrestrial Gravity Data: a Simulation Study
Table 5. GOCE data used for the estimation of systematic parameters
to the following equation (Moritz, 1978)
S - ScocE + A X + o. Region A B C
No of tracks 40 41 42
Area extent 56 ° _< q)_< 6 6 ° , - 1 2 2 ° _< L_< - 1 1 2 ° - 3 3 ° _< q0 _< - 2 3 °, 124 ° _< )k, _< 136 ° 54 ° _< q0_< 64 ° , 18 ° _< ~_< 30 °
(a)
0.003 O
0.002 I
I
I
7Ox 8 °
I
9Ox10 o
lOOx12 o
Size of collection area
0.005
I
I
I
The results in terms of the mean collocation error estimation of the systematic parameters in the 3 test regions are shown in Table 6. Note that the m e a n error of the estimated bias parameters in all cases are below 8 mE.
0.004 O
0.003
0.002
The estimation of the parameters was carried out trackwise. For each region the tracks which lie within the borders shown in Table 5 were taken into account. The "data" were regarded as having alongtrack correlated errors with error-covariance functions derived from the expected noise power density spectrum. We are hereby accounting for the fact that G O C E data will be band-limited.
I
O
O o
In (1) S represent an (anomalous) gravity gradient c o m p o n e n t , SGOCE stand for G O C E simulated (anomalous) c o m p o n e n t , and A = 1 w h e n X represent a bias, A = d t when X represent a tilt and A -the u n - r e d u c e d gradient c o m p o n e n t w h e n 1 - X is a scale factor. Finally, (y stand for a r a n d o m error. Because bias and scale factor are very strongly correlated, it does not make sense to estimate both simultaneously. Therefore, the estimation of scale factors was carried out separately from the estimation of the bias and tilt.
~,~0.004
5Ox 6 °
(1)
4
I
5
I
I
I
10
15
20
Data-sampling (arcmin) Figure 7. Region C: Mean collocation error estimation for different sizes of the area for terrestrial data collection (a), mean collocation error for different data-sampling (b). Txx = square, Txy = circle, Txz = triangle, Tyy = inverted trinagle, Tyz = hexagon, Tz,' = diamont.
3.2 Tests concerning the detection of systematic errors As it was stated in section 1, the parametric L S C was used for the detection of systematic errors. In order to estimate bias, tilt and scale factor parameters using collocation the possibility for the determination of scale factors was added in the G R A V S O F T program G E O C O L . The systematic parameters were estimated relative to E G M 9 6 "observations" according
Conclusion
E x t e n d e d prediction experiments were carried out using terrestrial gravity anomalies from the Canadian plains, from Australia and from Scandinavian, in order to determine the appropriate size of the area for gravity data collection as well as the required datasampling for calibration of the G O C E S G G data. The experiments concerning the size requirements showed that in all regions the m e a n error of estimation was continuously decreasing up to 35% (in average) w h e n the size of the area was continuously increased from 5 ° × 6 ° to 10 ° × 12 °. This 10 ° × 12 ° size is considered satisfactory since according to the m e a n error of estimation a 70% of the signal of the real data can be recovered by the m e t h o d used. In the same way, the results of the experiments for the determination of the required resolution of the terrestrial gravity data showed that for constant size 6 ° × 8 ° of the collection area, w h e n the data-sampling was c h a n g e d from 20 t to 5 t, the m e a n error of estimation dropped up to 50% (in average) of its original value.
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D.N. Arabelos • C. C. Tscherning • M. Veicherts
Table 6. Mean collocation error estimation of the systematic parameters in the 3 test regions
Component Trx Try
Tx~ Tyy Tw ~ Trx
Txy Tx~ ~,y ~,~ ~
Txx Try
Tx~ Try Tyz ~z
Region A Bias Tilt EU EU/s 0.006467 0.000250 0.007859 0.000300 0.007794 0.000300 0.007795 0.000298 0.007850 0.000299 0.007975 0.000303 Region B 0.004273 0.000138 0.005299 0.000169 0.005152 0.000166 0.005234 0.000167 0.005219 0.000167 0.005239 0.000167 Region C 0.002914 0.000143 0.003883 0.000178 0.003439 0.000170 0.003717 0.000175 0.003637 0.000173 0.003511 0.000170
Scale fac. 0.000003 0.000496 0.000874 0.000004 0.000159 0.000002 0.000003 0.000182 0.000563 0.000003 0.000034 0.000002 0.000003 0.000353 0.000400 0.000004 0.000081 0.000002
In this case, the mean error estimation of Tzz correspond to 30% of its formal standard deviation. More sensitive in both cases (size of collection area as well as data-sampling) is T~.~. It was expected, since it has the largest (signal) standard deviation comparing to the other components of the gravity gradients. However, the combination of 5 t data-sampling in an 10 ° × 12 ° area results in a mean error estimation that drops further up to 20% of the formal standard deviation of Tzz. Hence, by selecting the optimal area extend and data sampling we are much below the expected noise level in the measurement band-with. Finally, using parametric LSC it was shown that the estimation of systematic errors such as bias and tilt in the SGG data is possible using a combination of terrestrial and satellite data. Here again the error of the estimated biases are at the level or below the error in the measurement band-with. The estimation of tilts (drifts) is however above the error if one considers that it takes between 100 and 200 s for GOCE to cross one of the areas. The scale factors of the diagonal elements are also at the error level, considering that the absolute value of the gravity gradients does not exceed 3000 E. The determination of systematic parameters bias,
tilt and scale factor was carried out using least squares collocation. For this purpose the possibility to determine a scale factor was implemented in the GRAVSOFT program GEOCOL. However, an off-line estimation of scale-factors, which has been used for CHAMP accelerometer data (Howe et al., 2003) might be a good alternative. A c k n o w l e d g m e n t : This is a contribution to the ESA funded GOCE HPF development project.
References Arabelos, D. and C.C. Tscherning (1998). Calibration of satellite gradiometer data aided by ground gravity data, Journal of Geodesy, 72, 617-625. Bouman, J., R. Koop, C.C.Tscherning and RVisser (2004). Calibration of GOCE SGG Data Using High-Low SST, Terrestrial Gravity data, and Global Gravity Field Models, Journal of Geodesy, 78, no. 1- 2. 124-137. Howe, E., L.Stenseng and C.C.Tscherning (2003). Analysis of one month of CHAMP state vector and accelerometer data for the recovery of the gravity potential, Advances in Geosciences, 1, 1-
4. Lemoine, F.G., S.C. Kenyon, J.K. Factor, R.G. Trimmer, N.K. Pavlis, D.S. Chinn, C.M. Cox, S.M. Klosko, S.B. Luthcke, M.H. Torrence, Y.M. Wang, R.G. Williamson, E.C. Pavlis, R.H. Rapp, and T.R. Olson (1998). The Development of the Joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) Geopotential Model EGM96, NASA/TP-1998-206861, Goddard Space Flight Center, Greenbelt, MD. Moritz, H. (1978). Least-squares Collocation, Review of Geophysics and Space Physics, 16, no 3, 421-430. Tscherning, C.C., Testing frame transformation, gridding and filtering of GOCE gradiometer data by Least-Squares Collocation using simulated data. lAG Symposia, Vol. 128,277-282, 2005. Tscherning, C.C., R. Forsberg and R Knudsen (1992). The GRAVSOFT package for geoid determination. Proc. 1. Continental Workshop on the Geoid in Europe, Prague, May 1992, 327-334, Research Institute of Geodesy, Topography and Cartography, Prague. Wolf, K. I., H. Denker. Upward Continuation of Ground Data for GOCE Calibration/Validation Purposes. lAG Symposia, Vol. 129, 60-65, 2005.
Chapter 51
Towards an optimal combination of satellite data and
prior information J.R van Loon, J. Kusche DEOS, TU Delft, Kluyverweg 1, PO box 5058, 2600 GB Delft, The Netherlands
Abstract. With the CHAMP and GRACE satellite gravity missions and the upcoming GOCE mission, millions of gravity-related observations are being released to the geodetic community. In order to provide an optimal gravity model in a statistical sense, it is common practice to combine satellite-only normal equations with prior information derived from terrestrial data or from previous satellite missions in the form of an existing gravity model. The weighting could be derived from formal error estimates, but more often these are adjusted based on heuristics like inspection of the residuals or of subset solutions. In recent years, rigorous approaches based on variance component estimation techniques have been developed and enjoy increasing popularity. At the same time, these techniques aim to provide more realistic error assessments of the combination solutions.
The problem that all these approaches face is that typically systematic inconsistencies exist between the prior information and the information obtained from different observation systems, which may dominate over the random errors. In this contribution, we investigate different methods that can deal with inconsistencies: model augmentation to accommodate for a vector of deterministic inconsistencies, and (partial) downweighting in a variance-component estimation procedure. These methods are tested thoroughly and applied to combine CHAMP data and normal equations with the EGM96 model. Keywords. combination techniques, inconsistencies, CHAMR gravity models, variance component estimation
1
Introduction
It is common practice in gravity field modelling to combine satellite data sets with other data sets, e.g. from different missions, satellite gravity data from different periods (e.g. individual arcs) within one mission, a combination with terrestrial or altimetric data or a combination with an a priori model. The
problem of optimal combining the different data sets is of increasing relevance for data analysis from the present satellite missions (e.g. CHAMR GRACE) and the upcoming GOCE mission. Another, quite new, application is the optimal estimation of hydrological surface mass changes from a combination of GRACE temporal gravity fields and GPS time series measuring the geometrical loading of the Earth [Kusche et al., submitted]. One should however be careful when combining different gravity data sources. Inconsistencies can occur in both functional and stochastic model. Examples of modelled inconsistencies in the functional model are datum inconsistencies [Fotopoulos, 2005], systematic biases [Kotsakis, 2005] and geographically correlated errors. However, rarely the existence of the inconsistencies is tested before estimating them. In [Schaffrin, 1987] and [Schaffrin and Iz, 2001 ], equations are derived for the hypothesis testing of partial inconsistencies in the functional model, which we will review in the light of satellite gravity modelling. Augmentation of the stochastic model has been a subject for many years, e.g. [Rao, 1971, 1973] and [Koch, 1986]. In [Lerch et al., 1991], [Schwintzer et al., 1997] and [Mayer-Gtirr et al., 2005] stochastic models were adjusted in satellite gravity field modelling based on variance component estimation or similar techniques. We have derived equations for the MINQUE method and Maximum Likelihood method, in case of disjunctive observation groups with multiple (co)variance components to be estimated.
2
Least-squares combination
Assuming we have one batch of satellite gravity data and we want to estimate a set of spherical harmonic coefficients, the linear(ized) functional model reads
X/~--y+e
(1)
346
J.P. van Loon. J. Kusche
and the stochastic model is E(e)-0
;
The increase in the quadratic form
D(e)-E
(2)
with n x u design matrix X, u x 1 vector of unknown parameters/3, n x 1 vector of observations y, and n x 1 vector of stochastic observation errors e. The least-squares estimate of the vector of unknowns, i.e. /3, can be obtained by solving the system of normal equations NO) - b
a --(y--X/~)T~ -l(y-x~))
-- (y--X/~)T~--ly, (10)
with/3 from (eq. (3), can be a good test statistic for this test (see e.g. [Schaffrin, 1987]). The quadratic form, after the addition of prior information, reads aLS -
(y--X~Ls)T~--ly+(ap "-~LS) TNpIO P^ (11)
(3) The increase 5 ~ L S : ~ L S -- ~ is distributed as
with N = xTE-1X
;
H 0 : 6f~LS "" x2(n)
b = xTE-ly.
;
HA : a~LS "" x2(n, OLS). (12)
The addition of prior information/~p will be rather straightforward if one assumes E(~p)
- E(~)
-
E(tg).
(4)
(~-~LS
can
be computed as [Schaffrin, 1987]
6 ~ L S -- ( ~ p --
a ) T [ N -1 -+- N p 1 ] - l ( ~ p -- ~)
The functional model then reads X
y
and the non-centrality parameter 0LS is defined as
e
OLS -- ( E ( ~ p - ~ ) ) T [ N - l +NP1] -1 ( E ( ~ p - ~ ) ) . where I is an u x u identity matrix, ~ p is the u x 1 a priori solution vector, and e p is the u x 1 vector of stochastic errors of the prior information. Assuming no correlations between the satellite data and the prior information, the stochastic model can be written as E ( [ eep ]
ep
)-0
;
0
(6)
01 Np ]
(7)
where N p is the u x u normal matrix of the prior information. As well-known, the least-squares solution using the combined functional and stochastic model, i.e./3LS, can be obtained when solving the following system of normal equations: (N + N P ) ~ L S -- b + Np/~p
(8)
In practice, the satellite-only model can appear tailored, i.e. it does not match the prior model. One way to look at the problem is to question the unbiasedness of the prior model. Two hypotheses will be tested against each other:
- E(/9)
;
HA
# E(a) (9)
(14) Rejection of this test can be caused by either an incorrect functional model or an incorrect stochastic model.
3 Augmentation of the functional model Rejection of the functional model of (eq. (5)), via rejection of the test (eq. (12)) can have many reasons, e.g. systematic errors in the satellite observations, residual transformations in the underlying cartesian (ITRF) coordinate systems, geographically correlated errors caused by mismodelling in geophysical corrections. Any of these effects can in principle be modelled by a simple augmentation of the functional model, i.e. the addition of extra inconsistency parameters O to account for these partial inconsistencies.
3.1
Combination with prior information
The functional model, after the addition of the extra inconsistency parameters, can be modelled as
[Xl xo] [y] [el In1
0
0
0 In2 0
--
¢)P,1
/~P,2
+
ep,1
ep,2
(15)
Chapter 51 • Towardsan Optimal Combinationof Satellite Data and Prior Information
where 5 is a u0 x 1 vector of inconsistency parameters and X0 the n x u0 design matrix, connecting the inconsistency parameters to the satellite data. Alternatively, it would be possible to assign 5 to the prior information, but this creates just an equivalent model and will therefore not be discussed further. The vector of unknown parameters/3 has been split into two parts,/~1 and ¢/2, to make an easy adaption in case any inconsistencies in the unknown parameters have to be modelled directly as coefficient biases, e.g. a misfit for certain spherical harmonic degrees between the satellite-only solution and the prior information. If this is the case, the matrix X0 will equal X , . The stochastic properties of the satellite data and the prior information are assumed to remain unaltered, i.e.
E(
and
D(
[e] ep,1 ep,2
[e] ep,1 ep,2
The least-squares solution of the inconsistency vector 6AF follows then from the back-substitution step
3.2 Hypothesis testing for the inconsistency vector
Naturally, by adding an increasing number of inconsistency parameters, we can improve the fit of the model. However, any augmentation of the functional model should be tested first, before implementing it. Therefore, two hypotheses are tested against each other: H0"E(aAF) --0
(16)
) -- 0
0 )-
Np,11 Np,12 Np,21 Np,22
0
-1
(17)
The vector of unknown spherical harmonic coefficients/3 and the vector of inconsistency parameters can be jointly estimated by solving the normal equations
[Sll+S11s12+s12Sl0] N21 -Jr-Np,21 N22 + Np,22 N2o Nol No2 Noo
;
HA'E(aAF)
7kO.
(21)
When accepting the H0-hypothesis, no inconsistency vector should be estimated and the functional model of (eq. (5)) should be adopted as the functional model. Rejecting the H0-hypothesis results in accepting the existence of the inconsistency vector 6 and therefore adopting the functional model of (eq. (15)). To this end, the two hypotheses can be tested by analyzing the difference
(22)
5~AF = ~LS -- ~AF, where ~AF -- (Y - [ X I X 2 J ~ ) A F
^
(20)
N00aAF -- bo -[NolNo2]/3AF.
--
XOaAF)T~--lY
=
(23)
AF The difference ( ~ A F is distributed as
b2 -+- Np,21/~p,1 Jr- Np,22/~p,2 bo
(18) where " A F " means augmentation of the functional model and
Nij
-
xT~-~--1Xj
;
bi - x T E - I y .
Partial elimination of 6AF reduces the normal equations to [N + N p -
Nlo N2o
Nlo N2o
(24)
and can be re-written as (cf. [Schaffrin and Iz, 2001 ]) AT 5f~AF - (~AF - ~LS)T( b + NP/~F) + 5AFbo ,,T ,,T = -6AF[NolNog,][N + N p ] - l ( b + NpC)p) + 6AFbo
--/)AF
Noo -[NolNo2][N + Np] -1
Nlo N2o
5AF.
(25) The non-centrality parameter OAF is defined as
N o 1 [NolNo2]]/~AF =
b + NeC)p -
Ho : ( ~ A F "" X2(U0) HA : ( ~ A F "" )(.2(Uo,OAF)
Noolbo
OAF_~T[Noo_[NolNo2][N+Np]_I
(19)
[NlON2o I ~
(26)
347
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J.P. van Loon. J. Kusche
E(anF).
with 5 -
3.3
3.4 Hypothesis testing for the partial combination
Partial combination
Up till now, it was assumed that prior information is provided on the full vector of unknown parameters. In this section, we will only use parts of the prior information. If an inconsistency vector is modelled, regularity of the normal equations is not guaranteed anymore. Those prior observations that enable an estimation of the inconsistency vector are put in ]~P,1, the redundant observations for estimating the inconsistencies are put in ¢)p,3. The functional model (eq. (15)) changes to
[XlX X3Xol E,] e]+ [ Inl 0 0
0
0 I,~a
0
--
0
~3
/~P,1
~P,3
eY,3
PC
(27) with the stochastic model
E(
[e l [eli
ep,1 ) -- 0 ep,3
D(
ep,1 )-ep,3
(28)
o
o ]
0 [Np1111 [Np1113 . 0 [Npl131 [Np1]3 3
(29)
The added prior information will be tested for unbiasedness. Therefore, two hypotheses will be tested against each other:
Ho" H A " E(J~p,3) ¢
The test statistic, (~PC, is the difference between the residual square sums of the situation, with the addition of the redundant observations/~P,a and the situation, without the addition of these redundant observations. The test statistic is x2-distributed with u3 being the degrees of freedom.
4 Augmentation of the stochastic model In the former section, questioning of the prior information resulted in the augmentation of the functional model, either by estimating inconsistencies, or by using only a part of the prior information. However, it appears very often that the given prior information is unbiased, but the stochastic model is incorrect. In this section, augmentation of the stochastic model will be considered.
4.1 The stochastic model is changed by removing rows and columns of the VC-matrix N p 1 of the prior information. Introducing
[Zllz13]_ [ N llll ENlll3l Z31 Z33
--1
(30)
[NP1331 [Np1333
the normal equations for the partial combination become ^
Nll + Zll N12 N13 + Z13 N10 N21
N22 N23
~,
/33
No1
ao
bl
Representation of the stochastic model
We assume the stochastic model of the new data to be internally consistent, i.e. we will only estimate a scale factor for the variance-covariance matrix. However, we will investigate a splitting of the variance-covariance matrix of the prior information into a linear combination of cofactor matrices, with the (co)variance components as the linear coefficients. This can be motivated by the fact that a priori gravity models may be derived from many different types of gravity information and that the assigned VC-matrix may have undergone different calibration procedures. The stochastic model therefore reads
N20
N31 + Z31 N32 N33 + Z33 N30 No2 No3
(32)
S(j~3)
Noo
PC
+ Zll~p,1 -~- Zl3aP,3
D(
"1- I4q0 o 1
ep )
2
O
o%Qi
(33)
i=1
b2
-- b3 -~- Z31~)P,1 -~- Z33]~P,3 bo
(31)
where Q0 is the variance-covariance matrix of the satellite data, which will be scaled by the variance
Chapter 51 • Towards an Optimal Combination of Satellite Data and Prior Information
component o-g and Qi are the r cofactor matrices of the prior information, with ai2 the corresponding (co)variance components.
4.2
ML VCE
N [NllNI N10] N21
)
(35)
A formula to estimate the (co)variance components of the prior information is derived in the appendix AT
--2
epNp(ri QiNp6p
tr(NpQi)--tr(NAIF
[NpQiNp
o
O] o )
(36) Monte Carlo simulations can be used to reduce the computational burden [Kusche, 2003].
4.3
--1
-
MINQUE VCE
Another approach to estimate the variance components is the Minimum Norm Quadratic Unbiased Estimator (MINQUE), see e.g. [Rao, 1971, 1973]. MINQUE iterations converge faster, but the method requires evaluation of more complicated expression. The equations for the special case with disjunctive observation group and multiple variance components to be tested within an observation group, are derived in the appendix. For this specific functional (eq. (15)) and stochastic (eq. (33)) model, the variance components can be retrieved by solving the equation system S 0 0 ~01 " ' " ~ 0 r
~_2
•0
S10 Sll
a l2
Ul
-.. Slr
(37)
S~0 &l
&~
2tr(NAF
1
NAF
o )
[NpQiNpQjNpO] 0
0
0 NAF
)
0
l
0 )
and
ui -- eTNpQiNpei
(34)
with
--2 0-i
o
uo - ~r0 4eoTqoleo
tr(Nn~l~I )
N22 N2o No1 No2 N0o
NpQiNp 0 ]
Sij = tr(QiNpQjNp)
6°TQ°16° n-
N-
SO 0 -- 0-0 4 (n--2tr(N AIF~f)+tr(N AIFSINAiF~q))
S0i - Si0 - #o2tr(NA~IQNA~
The scale factor (or variance component) of the variance-covariance matrix of the new satellite data, assuming the functional model (eq. (15)), i.e. with the estimation of an inconsistency vector, can be estimated by the Maximum Likelihood Variance Component Estimation (MLVCE) for disjunctive observation groups [Koch, 1986]: 6-~) -
with
^'2
(7r
Ltr
for/= 1...r.
Here N is defined according to (eq. (35)) and 6-g is the estimated variance component of crg of the previous iteration. Again, Monte Carlo simulations can be used to speed up the computations [Van Loon and Kusche, 2005].
5 Case study: Static gravity field from combination of CHAMP data with EGM96 A common problem in satellite gravity modelling is to combine new satellite data (e.g. from CHAMP, GRACE and GOCE) with existing SH models in an optimal way. In this case study, CHAMP satellite data will be combined with the a priori gravity model EGM 96 [Lemoine et al., 1998].
5.1
Test setup
A detailed description of the CHAMP data that we use here, can be found in [Van Loon and Kusche, 2005]. There, we have computed a global gravity field model, up to degree and order 75, using 299 days of CHAMP pseudo-observations. These observations were derived from kinematic orbit data [Svehla and Rothacher, 2003] and accelerometer and quaternion data provided by GFZ, using the energy balance approach (see e.g. [Jekeli, 1999] and [Visser et al., 2003]).
5.2 Combination with the original EGM96 stochastic model The functional model of (eq. (5)) and the stochastic model of (eq. (7)) are used to combine the EGM96 mode, complete to degree and order 75, with the CHAMP-only model. This combined solution showed latitude weighted rms geoid differences
349
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J.P. van Loon. J. Kusche
with EIGEN-GRACE02S of 7.86 cm (L=50). However, the test for the addition of the prior information, i.e. (eq. 12), was rejected as 5f~LS turned out to be 14080.70, well above the critical value t~(n = 5772, a = 0.01) = 5525.00. A test has therefore been performed to look for inconsistencies in the spherical harmonic coefficients of either the CHAMP model or EGM96. After setting X0 equal to X1, equations (19) and (20) change to
estimated with the spherical harmonic coefficients. This improved the solution in the lower degrees significantly, as can be seen in Fig. 2. 10-1
Z 1°-2 !!!!!!!!!!!!!!!::!!!!!!!!!!!!!!!!i!!!!!!!!!!!!!!!!::!!!!!!!!!!!!!!!!i!!!!!!!!!!!!!!!!::!!!!:::::••
8
............
; ................
. . . . . . . . . . .
:. . . . . . . . . . . .
! ............
~,,
; ................
.......
^
0
0
[o N22-N21N N12]] Np~p
+
I
1~ . . . . . . . . . . . . .
:. . . . .
._ -o
g
0
10 -a
b2 - N21N]-11bl
(38) 10-4 0
and (39)
N l l ~ A F -- b l - [N11N12]/~AF
An inconsistency vector was estimated for every degree 2 to 75. The test-statistic per degree, together with the critical value (oz = 0.01), can be seen in Fig.1.
5
10
15
Degree
20
25
1 30
Figure 2" Geoid difference with EIGEN-GRACE02S per degree. The solid line represents the situation without the estimation of an inconsistency vector for degree 2 and 4. The dashed line represents the situation with such an estimation. The cumulative geoid difference with EIGENGRACE02S improved to 7.75 cm. Partial combinations of EGM96 with the CHAMP-model were tested, but either the test was rejected or the solution deteriorated considerably.
0
.o_ 1500 ._
5.3
:i: .:
Augmentation
of the EGM96
stochastic
model I- lOOO
!! :9
i
.:i
9
g&~_, °o
10
........... 20
-~. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 40 50
Degree
60
70
80
Figure 1" Test for any inconsistency vectors between CHAMP and EGM96, per degree. The dash-dotted line represents the critical values (a -- 0.01) per degree. According to this test (Fig. 1), one should estimate inconsistency vectors for every degree, which is practically not feasible. However, it is clearly visible, that degrees 2 and 4 are particularly bad (something that many groups working with these data had observed, e.g. [Reubelt et al., submitted]) and may cause, due to correlations, large test statistics for the other degrees. Another possible explanation is the use of an over-optimistic stochastic model. Consequently, an inconsistency vector for degrees 2 and 4 is jointly
We assume the stochastic properties of the satellite data to be internally consistent and to be externally inconsistent, i.e. we only estimate a scaling factor for the variance-covariance matrix of the CHAMP-only model, as it is constructed using one data type and the data has already been re-weighted using ML VCE [Van Loon and Kusche, 2005]. The EGM96-model, however, is constructed using a large number of data sets of different types, e.g. satellite data, terrestrial data, marine data. Weighting of the input data was performed by trial and error and independent solutions calibrations [Lerch et al., 1991 ]. The variancecovariance matrix is dense up to degree and order 70 and diagonal for higher degrees. We therefore decided to split the variance-covariance matrix into several cofactor matrices, which will be factorised by (co)variance components. The obvious division would be to use one cofactor matrix for every original data set. As this information is not available, we decided to split the variance-covariance matrix using
Chapter 51 • Towards an Optimal Combination of Satellite Data and Prior Information
an eigenvalue decomposition:
r
£
T
T
kC@
i--1 k E(I)i
i--1
(40) where ~ is the full class of eigenvalues and ~i are sub-classes of ¢~. The eigenvalues are plotted in Fig. 3. We have chosen to split the eigenvalues into four different groups. The construction of the cofactor matrices showed that group IV is almost identical to the variance-covariance matrix of the degrees 71 to 75, which is diagonal. Such a statement cannot be made for the other groups. However, as e.g. the eigenvectors with the lowest eigenvalues are put in group I, this group mainly contains of the lowest degrees. lO -1;'
............... ! ................ i ................ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ...................................................................................................
! ................
i ................
!...............
lO -18
information (EGM96) were tested for inconsistencies. The hypotheses testing clearly showed inconsistencies between the spherical harmonic coefficients of degree 2 an 4, something we expected. Augmenting the normal equations with corresponding bias parameters improved the combined solution considerably in the lower degrees. Augmentation of the stochastic model showed an even greater improvement. The EGM96 VC-matrix was modified using a combination of eigenvalue decomposition and ML VCE. The augmentation of the functional and stochastic model decreased the rms geoid difference to EIGEN-GRACE02S from 7.86 cm to 6.96 cm, a considerable improvement.
Acknowledgements We are grateful to GFZ Potsdam for providing CHAMP ACC data. Thanks go also to IAPG, TU Munich, for providing CHAMP kinematic orbits. J.v.L. acknowledges financial support by the GO-2 program (SRON EO-03/057).
lO -1~
References 1 0 -20
c~ ~ lO -2~
10 -22
................................................................................................... ............... : ................ :. . . . . . . . . . . . . . . . .
: ................
,. . . . . . . . . . . . . . . . .
:...............
...............
: ................
,. . . . . . . . . . . . . . . . .
:...............
: ................
~ ................
:...............
: ................
; ................
:...............
: ................
:. . . . . . . . . . . . . . . . .
. . . . ................................................................................................... ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ...................................................................................................
.
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ............... 10 -23
: ................
~ ................
............... : ................ ; ................ ................................................................................................... ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ................................................................................................... ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ...............
10-240
: ................
; ................
: ................
; ................
:...............
t
t
t
I
t
1000
2000
3000
4000
5000
6000
# Eigenvalue
Figure 3" Eigenvalues of EGM96, in ascending order. The eigenvalues are divided into four groups. A Monte-Carlo variant of (eq. (34)) and (eq. (36)) is used to estimate the variance components of the CHAMP-only model and the EGM96 model. The variance component of the CHAMP-only model was estimated to be 1.23, the variance components of the EGM96-model are 16.82, 3.20, 5.85 and 4.181 respectively. Hence, the standard deviations of the group I (see Fig. 3), i.e. with the lowest eigenvalues, have to be increased by a factor 4.1. The newly estimated model, using the adjusted EGM96 VC matrix, has a rms geoid difference with EIGEN-GRACE02S of only 6.96 cm (L=50). For all partial combinations, the hypothesis test (eq. (32)) was rejected.
6
Discussion
The satellite-only (CHAMP) solution and the prior
Crocetto N, Gatti M, Russo P (2000) Simplified formulae for the BIQUE estimation of variance components in disjunctive observation groups, Journal of Geodesy 74, pp. 447-457 Fotopoulos G (2005) Calibration of geoid error models via a combined adjustment of ellipsoidal, orthometric and gravimetric geoid height data, Journal of Geodesy 79, pp. 111-123 Jekeli C (1999) The determination of gravitational potential differences from satellite-to-satellite tracking, Cel. Mech. Dyn. Astr. 75, pp. 85-100 Koch K-R (1986) Maximum Likelihood estimate of variance components, Bull Geod 60, pp. 329-338 Kotsakis (2005) A type of biased estimators for linear models with uniformly biased data, Journal of Geodesy 79, pp. 341-350 Kusche J ( 2 0 0 3 ) A Monte-Carlo technique for weight estimation in satellite geodesy., J Geodesy 76, pp. 641-652 Kusche J, Schrama EJO, Jansen MJF Continental hydrology retrieval from GPS time series and GRACE gravity solutions, Proceedings Joint Assembly of IAG, IAPSO and IABO (Dynamic Planet), session G3, submitted Lemoine FG, Kenyon SC, Factor JK, Trimmer RG, Pavlis NK, Chinn DS, Cox CM, Klosko SM, Luthcke SB, Torrence MH, Wang YM, Williamson RG, Pavlis EC, Rapp RH, Olsen TR
351
352
J.P. van Loon• J. Kusche
(1998) The development of the joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) geopotential model EGM96. NASA/TP1998-206861, NASA-GSFC, Greenbelt MD Lerch FH, Marsh JG, Klosko SM, Patel GB, Chinn DS, Pavlis EC, Wagner CA (1991) An improved error assessment for the GEM-T1 gravitational model, J. Geophys. Res. 96, pp. 20023-20040 Mayer-Gtirr T, Ilk KH, Eicker A, Feuchtinger M (2005) ITG-CHAMPOI : a CHAMP gravity field model from short kinematic arcs over a one-year observation period, Journal of Geodesy 78, pp. 462-480 Rao CR (1971) Estimation of variance and covariance components- The MINQUE theory, J. of Multivariate Anal. 1, pp. 257-275 Rao CR (1973) Linear statistical inference and its" applications, John Wiley and Sons, New York Reubelt T, G6tzelmann M, Grafarend EW A new CHAMP gravitational field model based on the GIS acceleration approach and two years of kinematic CHAMP data, submitted to Reigber C, Tapley B (eds.) New satellite mission results for the geopotential fields" and their variations, proceedings of the Joint CHMAP/GRACE Science Meeting, 6th-8th July 2004, GFZ Potsdam, EGU Schaffrin B (1987) Less sensitive tests by introducing stochastic linear hypotheses, Proc. Second International Tampere Conference in Statistics, Univ. of Tampere, pp 647-664 Schaffrin B and H.B. Iz (2001) Integrating heterogeneous data sets with partial inconsistencies, in Siderius M. (ed.), Gravity, Geoid and Geodynamics 2000, Springer, Berlin Heidelberg, pp. 49-54 Schwintzer P, Reigber Ch, Bode A, Kang Z, Zhu SY, Massmann F-H, Raimondo JC, Biancale R, Balmino G, Lemoine JM, Moynot B, Marty JC, Barlier F, Boudon Y (1997) Long-wavelength global gravity field models: GRIM4-S4, GRIM4C4, Journal of Geodesy 71, pp. 189-208 Svehla D, Rothacher M (2003) Kinematic and reduced-dynamic precise orbit determination of low earth orbiters, Adv, Geosciences 1, pp. 4756 Van Loon JP, Kusche J (2005) Stochastic model validation of satellite gravity data: a test with CHAMP pseudo-observations, in Jekeli C, Bastos L and Fernandes J (ed.) Gravity, Geoid and Space Missions, IAG symposia, vol. 129, Springer Visser R Sneeuw N, Gerlach C (2003) Energy integral method for gravity field determination from satellite orbit coordinates, Journal of Geodesy 77, pp. 207-216
Appendix- Estimation of multiple variance components in disjunctive observation groups We assume the general linear Gauss-Markov model X/Y=y+e
;
E(e)=0
(A-l)
The variance-covariance matrix of the observations y will now be written as a linear combination of (co)variance components and cofactor matrices: p 2
D(e) -- N -- E
°-i Qi
(A-2)
i--1
where cri2 are the (co)variance components and Qi are the n x n cofactor matrices. The (co)variance components can be estimated by the Minimum Norm Quadratic Unbiased Estimator (MINQUE), see e.g. [Rao, 1971, 1973]. The MINQUE can be obtained by solving the equations S& = u
(A-3)
with -
.
Sij = t r ( R Q i R Q j ) R
-
E-~P
(A-4)
P = In - - x ( x T ~ - ] - - I x ) - - I x T ~ ] -1 Uj = y T R Q j R y The VC-matrix of the observations will now be split into disjunctive observations groups:
X] - -
E1 0 . . . 0 0 E2... 0 . • • • . O
.
. O
P -- Z
. ...
JiX~iJ~
(A-S)
i=1 ~']p
with J i the n x ni matrix, defined as
Ji-
[O...In~...O]
T
(A-6)
The VC-matrix of each disjunctive observation group can be written as a linear combination of (co)variance components and cofactor matrices: -
ql
O
. o o
O
k=l q2
0 E
__
~ ~kC~k
.
.
0
.
(A-7)
k=l
qp
0
0
... ~ k=l
2 Cpk %k
Chapter 51 • Towards an Optimal Combination of Satellite Data and Prior Information
Re-writing this matrix gives
P qu 0.uk Quk
(A-S)
the MINQUE for disjunctive observation groups with multiple (co)variance components can be estimated by
u = l k=l
80" -- u
where
(A-13)
with
Q~k - J ~ C ~ k J ~
(A-9)
Inserting (eq. (A-5)) and (eq. (A-9)) into (eq. (A-3)) results in S& -- u
(A-10)
Sij u=v - t r ( E v i C v k E v l C v l ) - 2 t r (N-1XvT }--]v 1Cv k y]vl Cvl Y]vl Xv ) v "-'v ,-,vk ~-]vI X v N - I x Tv- 1~v C v l ~ v l Xv )
+tr(N-lXTv-lo Sij uCv =
t r ( N - 1 X T E u l C u k E u 1XuN -1 XvT ~"]v i c ~ l ~-]v 1Xv ) with (7" -- (O-21 " --0.12q10.21 " --0.22q2 "''0.pl"''0.pqp) 2 2
Uj -- e^T v E v-,CvlE;-~av. (A-14)
P
J~Er-lJY]J~C~kJ~
Sij - t r ( p T [ E
We use the general simplified procedure of [Crocetto et al., 2000] to derive the equations of the maximum likelihood estimator of the (co)variance components from the equations of MINQUE:
P - P T [ E JsE~ 1JT]JvC~,JT) 8----1
P
P rzl
,,2
0.vI --
"[E JsE-1jy]Py s=l
(A-11) 2 the jth Note that 0.~k2 is the ith element of 0. and 0.vt element. Using the properties jTjj j~cji J? P JJ J iTP T Y
-O for all i ¢ j - I~ - JTJJ - r C x i- xT -ei
(A-12)
" T E v l - 20.vl C v l E v l ~ v ev tr(Y]vl Cvl ) - t r ( N - l X T ~ v l Cvi Y]vl Xv ) (A-15)
This equation is similar to the one derived by [Crocetto et al., 2000] using a different approach. With only one variance component for a certain observation group, the equation simplifies to the well-known equation [Koch, 1986] ^2 0.i -
AT
--1
ei Qi ei
_
ni - t r ( N - 1 N i )
(A- 16)
353
Chapter 52
Kinematic and highly reduced-dynamic LEO orbit determination for gravity field estimation A. J~iggi, G. Beutler, H. Bock, U. Hugentobler Astronomical Institute, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland email: adrian.jaeggi @aiub.unibe.ch
Abstract. Kinematic positions of low Earth orbiting satellites equipped with spaceborne GPS receivers are widely used as input for subsequent gravity field estimation procedures. Positions relying on reduceddynamic orbit determination, however, are often considered as inappropriate for this task, because they depend to some extent on the gravity field model underlying the orbit estimation. We review the principles of reduced-dynamic orbit determination and give the mathematical background for a very efficient estimation scheme of reduced-dynamic satellite trajectories using least-squares methods. Simulated as well as real data from the CHAMP GPS receiver are used to show the equivalence of kinematic and reduceddynamic orbits in the kinematic limit and to present a highly reduced-dynamic orbit determination scheme as an alternative to kinematic point positioning.
CHAMP models by means of the energy integral method. Gerlach et al. (2003) reported that gravity field models derived from reduced-dynamic orbits (RD-orbits) are biased towards the a priori gravity field model used for the preceding orbit estimation. This article focuses on both reduced-dynamic and kinematic precise orbit determination (POD) for the purpose of gravity field estimation. We analyze so-called highly RD- (HRD-) and maximum RDorbits (MRD-orbits) as alternatives to kinematic orbits. Simulated and real GPS data of the CHAMP receiver are used to investigate the properties of such orbits and to assess their value for a subsequent gravity field estimation, where the main issue consists of clarifying dependencies of orbital positions and velocities on the a priori gravity field models used.
2 LEO orbit determination Key words. Low Earth orbiter, reduced-dynamic orbit determination, kinematic orbit determination
1 Introduction A new era in using data from spaceborne GPS receivers on board low Earth orbiters (LEOs) for gravity field determination was opened in the framework of the CHAMP mission (Reigber et al., 2002). The combined analysis of high-low GPS satellite-tosatellite tracking data and STAR accelerometer data (Touboul et al., 1999) enabled the derivation of a whole series of high quality global gravity field models with unprecedented accuracy (see, e.g., Reigber et al., 2003). Due to the heavy demands posed on computational resources in the case of classical numerical integration techniques, alternative methods have been developed and established as well, e.g., relying on satellite positions used as pseudo-observables (see, e.g., Visser et al., 2003). Gerlach et al. (2003) used kinematic CHAMP positions, which were precedingly derived by Svehla and Rothacher (2003), as pseudo-observations together with accelerometer data and showed that gravity field models can be estimated with a quality comparable to the official
This section briefly introduces kinematic, dynamic and reduced-dynamic orbit modelling techniques applied to LEOs equipped with on board GPS receivers. The main focus lies on a brief review of a novel approach proposed by Beutler et al. (2006) for a very efficient computation of any type of RD-orbits. RDorbit modelling techniques are of particular interest because they contain the two other above mentioned techniques as special (asymptotic) cases.
2.1 Kinematic orbit determination The geometric strength and the high density of GPS observations allows for a purely geometrical approach to determine LEO positions at the observation epochs by precise point positioning (Svehla and Rothacher, 2004). The ephemeris are represented by a time series of three kinematic coordinates per kinematic epoch, which are determined in a standard least-squares adjustment process of GPS observations together with all other relevant parameters without using any information on LEO dynamics.
2.2 Dynamic orbit determination The equation of motion of an Earth orbiting satellite including all perturbations reads in the inertial frame
Chapter 5 2
•
Kinematic and Highly-Reduced Dynamic LEO Orbit Determination for Gravity Field Estimation
as
r -- - G M T f f + f l ( t , r , ~ , q l , . . . , q d ) with
a
set
of
initial
conditions
(1)
r(k)(to)
=
:p(k) (_EI~ ... '/~76 ; t o ) , k -- 0, 1 where E l , . . . , 1776 a r e
the six Keplerian elements pertaining to epoch to. ql,..., qd denote d additional dynamical parameters considered as unknowns, which describe the perturbing acceleration acting on the satellite. Let us assume that an a priori orbit to(t) is available, e.g., from a GPS code solution. Dynamic orbit determination may then be set up as an orbit improvement process, i.e., the actual orbit r(t) is expressed as a truncated Taylor series with respect to the unknown orbit parameters Pi about the a priori orbit, which is represented by the parameter values Pio" ~(t) - ~o(t) +
~
Oro(t) . Api op~
orbital change may be expressed only by a change in the Keplerian elements (see, e.g., J~iggi et al., 2004a). Therefore, the partial derivative of the a priori orbit with respect to a pulse Vij at time ti in direction j, subsequently denoted as zvij, may be expressed as a linear combination of the partial derivatives with respect to the initial conditions E l , ..., E6:
0
;t
6
z ~ , (t) -
E
9~j,k ' zE~ (t) ; t >_ t~ "
(3)
k=l
Independent of the number of pseudo-stochastic pulses set up, only the six variational equations referring to the initial conditions have to be integrated numerically to subsequently compute all partial derivatives with respect to the pulses as simple linear combinations with constant coefficients.
(2)
where Api " Pi - Pio denote the n " 6 + d corrections to be estimated. Numerical integration techniques must usually be applied to solve the so-called variational equations (see, e.g., Beutler, 2004) to obtain the partial derivatives of the a priori orbit to(t) with respect to the parameters Pi, which allow the solution for the corrections Api in a standard leastsquares adjustment process of GPS observations together with all other relevant parameters. Eventually, the improved orbit may be computed according to Eq. (2). As the obtained orbit is a particular solution of the equation of motion, the trajectory fully depends on the dynamical model defined by Eq. (1).
2.3 Reduced-dynamic orbit determination
2.3.2 Piecewise constant accelerations Piecewise constant accelerations are attractive for RD LEO POD, as well, because a large number of accelerations can be set up efficiently, as well. J~iggi et al. (2004a) showed that the partial derivative of the a priori orbit with respect to an acceleration ai acting in the subinterval ti-1 <_ t < ti in direction j may be written as a linear combination with time-dependent coefficients of the partial derivatives with respect to the initial conditions E l , ..., E6:
0 6 E ~ij,k(t)"ZEk(t)
;t
6
Y~. / 3 i j , k ( t i ) ' z . ~ ( t ) ; t _> ti k=l
We use pseudo-stochastic orbit modelling techniques (J~iggi et al., 2006) as a realization for RD-POD (Wu et al., 1991), which makes use of both the geometric strength of the GPS observations and the fact that satellite trajectories are particular solutions of an equation of motion. The attribute "pseudo" is used to distinguish our approach from methods considering the satellite motion as a stochastic process, whereas the attribute "stochastic" refers to the introduction of additional parameters to the deterministic equation of motion, which may have a priori known statistical properties. In this article we make use of two types of additional parameters, namely instantaneous velocity changes (pulses) and piecewise constant accelerations.
2.3.1 Instantaneous velocity changes Pulses are attractive for RD LEO POD mainly because a large number of pulses can be set up efficiently. This is due to the fact that a pulse-induced
(4) or, alternatively, as a linear combination with constant coefficients of the same partial derivatives and one additional partial derivative with respect to a constant acceleration pointing in the same direction and acting over the entire orbital arc. Therefore, all partial derivatives with respect to the accelerations may be constructed from a very limited set of numerically integrated partial derivatives.
2.3.3 Normal equation system We give a short overview of the structure of the resulting normal equation system for the standard leastsquares adjustment process of GPS observations. For the sake of simplicity, we consider only the six orbital elements and the pulses in three orthogonal directions at times ti, i = 1, ..., n - 1 as parameters. For a more detailed derivation, also considering different parameter types like piecewise constant accelerations and
355
356
h. J~iggi • G. Beutler • H. Bock. U. Hugentobler
additional parameter types like carrier phase ambiguities, we refer to Beutler et al. (2006). The pulse-epochs divide the orbital arc into n subintervals. Let us write all no~ observation equations of the subinterval Ii = [ti, ti+l ) in a convenient matrix notation: i
Ai . AE
+ Ai . ~
B ~ . Ziv,~ - ZicPi - pi ,
the structure of Eq. (6) and found that for a large variety of applications the solution vector and the associated full variance-covariance information may be computed with sufficient efficiency. However, when striving to the kinematic limit, i.e., pseudo-stochastic parameters set up at a rate close or equal to the observation sampling rate, the procedures become inefficient due to the unavoidably large normal equation matrix, which has to be inverted.
m=l
(5) where A i is the first design matrix with no~ lines and six columns, A E the column array containing the six increments of the initial osculating elements, B ilk,j] = /~ij,k the matrix with six lines and three columns containing the coefficients of Eq. (3), a v i the column array containing the three pulses at time ti, A O i the column array containing the no~ terms "observed-computed", and Pi the column array containing the no~ residuals. Note that all pulses set up before the subinterval [i remain active and contribute to the observation equations of subinterval Ii as predicted by the linear combination of Eq. (3). Therefore, the last subinterval eventually contains the contributions due to all pulses of the orbital arc, provided that the initial conditions are still referring to the beginning of the orbital arc. To study the structure of the resulting normal equation matrix, it is instructive to use the contributions N i " A T P i A i per subinterval to the normal equation matrix of dynamic POD, i.e., POD without pulses. Obviously, these contributions form the complete normal equation matrix of dynamic POD as n--1 N " ~-~i=0 A ~ P i A i , but they are also the building blocks of the complete, symmetric normal equation matrix in the presence of pulses, which reads as /
n--1
N
~
n--1
NiB1
"'"
. Bf
Z
Bf
y~
n--1 T
.
\
Bn_
- AEi_I
NiBn-1
1 i=n-1
(8)
+ Bi" z~vi.
It is instructive to apply the transformation given in Eq. (8) each time after having processed all observations of one subinterval. The resulting normal equation matrix then reads as" / ¢
0
N
n--2
- ~ NiB1 i=0 0
....
~
'~
NiB~-I
i=0 0
• BTENiBI"'" i=0
B TZNiB~-I i=0
n--2
"
• BT-1 E
NiBn-1 J
(9) NiBn-i
i=n--1
i=1
AEi
i=0
n--1
...
where the term in parentheses denotes the column array containing the six orbital elements pertaining to epoch to, but characterizing the trajectory within this particular subinterval. This set of elements, subsequently denoted as z l E i , is simply related to the set of elements of the previous subinterval by:
k "
NiBn-1
i=n--1
NiB1
Rearranging all observation equations (Eq. (5)) of the subinterval Ii shows that the orbit may be represented within this subinterval by only six Keplerian elements"
"~
~
i=1 n--1
2.3.4 Transformation of Keplerian elements
J
(6) Equation (6) illustrates that the normal equation matrix (and also the corresponding right hand side of the normal equation system) has a simple structure, but grows monotonically after having processed all observations of one subinterval. Note in particular that it is not possible to pre-eliminate the pulses at any observation epoch which is indicated by the upper summation limit. Beutler et al. (2006) made full use of
The solution vector obtained from the transformed normal equation system contains the same pulses as the untransformed system, but the set of elements A E n _ I referring to the last subinterval (instead of A E 0 referring to the first subinterval). A comparison with the untransformed normal equation matrix (compare Eq. (6)) reveals the benefit of the applied transformation because it is now possible to preeliminate the pulses after each subinterval as the upper summation limits in Eq. (9) indicate. Beutler et al. (2006) made full use of the structure of Eq. (9) and proposed a very efficient pre-elimination and backsubstitution scheme for different types of pseudostochastic parameters, which allows it to efficiently realize the kinematic limit with RD-orbits.
Chapter52 • Kinematicand Highly-ReducedDynamicLEOOrbitDeterminationfor GravityFieldEstimation l0
,
,
+
u
,
+
,
++ +
,
IAPG MUB
+
8
x
.~~
' {x
+ 6
'~
•
5
"~
1 min. 3 rain. . -.
o -5
0
i
i
i
100
150
200
i
i
i
i
250
300
350
400
-10 0
i
i
i
i
5
10
15
20
15
20
(DOY)
th)
Fig. 1. Daily (1-dim.) RMS of differences for AIUB kinematic and IAPG kinematic orbits w.r.t, conventional RDorbits for days 071/2002 to 070/2003.
5
o
IIHIll )
IIIll
I
i1TIIII
i
-s
3 Processing
of real C H A M P
data
-10 0
5
10 (h)
The GPS final orbits and the 30s high-rate satellite clock corrections from the CODE analysis center were used together with attitude data from the star tracker on board of CHAMP provided by GFZ (GeoForschungsZentrum Potsdam) and the gravity field model EIGEN-2 (Reigber et al., 2003) to conventionally process undifferenced CHAMP GPS phase tracking data covering a one year time period from day 071/2002 to 070/2003. For a subset of tracking data, covering GPS weeks 1173-1176, 10 s GPS satellite clock corrections were generated in order to perform tests with several kinds of HRD orbit parametrization. All computations were performed with a development version of the Bernese GPS Software (Hugentobler et al., 2001). 3.1 Results of k i n e m a t i c P O D
Figure 1 shows daily (1-dim.) RMS values of orbital differences derived from our kinematic orbits (AIUB) with respect to conventional RD-orbits with pseudo-stochastic parameters set up every six minutes. As a reference, the differences emerging from the kinematic orbits (IAPG) computed by Svehla and Rothacher (2003) are displayed as well. The two curves show, on the one hand, that both sets of kinematic orbits are of similar quality, but, on the other hand, they reveal for both solutions a considerable number of poorly determined trajectories, mainly due to data quality issues. As a consequence of the very low degree of freedom per epoch, kinematic positions react very sensitively to the density and quality of GPS observations. This makes a robust preprocessing a greater challenge than for conventional RD-POD. 3.2 R D - P O D at the k i n e m a t i c limit
The estimation scheme presented in section 2.3.4 makes it possible to efficiently approach the kinematic limit with RD-orbits. For one particular day Fig. 2 (top) puts the cross-track differences for the kinematic orbit together with the differences emerg-
Fig. 2. Cross-track differences of the kinematic and different HRD-orbits w.r.t, a conventional RD-orbit (top) and differences (radial/cross-track shifted by 5 mm) between the kinematic and the MRD-orbit (bottom) for day 198/2002. ing from two HRD-orbits, i.e., orbits which are represented by six initial conditions and three unconstrained pulses set up every one and every three minutes, respectively. We see that the HRD-orbits approach the kinematic orbit when the number of pulses increases. Because the differences for the MRD-orbit would completely overlap with the differences for the kinematic orbit, Fig. 2 (bottom) displays the differences between both orbits separately and confirms their equivalence to the numerical precision provided by the SP3 orbit file format, apart from a few exceptions which are discussed in the following paragraph.
3.2.1 Properties of MRD-orbits The least-squares adjustment process for the estimation of a MRD-orbit, either based on pulses or accelerations, results in a regular normal equation system like in the case of kinematic POD. It is instructive to have a closer look at MRD-orbits in order to emphasize possible benefits of HRD-orbits. For the sake of simplicity we confine ourselves to discuss the results achieved with pulses, because MRD-orbits based on accelerations do not provide more insight in this respect. For MRD-orbits, three unconstrained pulses are set up at all hobs observation epochs, except for the very first and the very last one. Therefore, together with the six initial conditions, a total number of 3-hobs orbit parameters are estimated, which is obviously the same number of unknowns as in the case of a kinematic orbit. Provided that at least four GPS observations are available for every observation epoch, all epoch parameters can be determined for both approaches, i.e., three pulses and three kinematic coordinates, respectively, and a receiver clock correction. Both orbit ephemeris are equivalent at the observation epochs as illustrated in Fig. 3 (top).
357
358
A. J~iggi • G. Beutler • H. Bock. U. Hugentobler
Table 1. Overall RMS of velocity differences w.r.t, conventional RD-orbits (GPS weeks 1173-1176). Solution o
~
,
............. ,:~ ....""-~:-
Fig. 3. MRD-orbit (solid line) based on pulses in comparison with the true orbit (dotted line) (top) and impact of an increased sampling rate (bottom). The trajectory of the MRD-orbit is defined inbetween the observation epochs, as well. The positions at the left and at the right boundary epochs provide the necessary six conditions to define a trajectory between these two points, which solves the equation of motion (Eq. (1)). Note that the positions at the observation epochs are completely independent of the force field, but the trajectory in-between and in particular the orbital velocity is given by the apriori force model. It is thus not possible to derive more independent information concerning the force field from a MRD-orbit than from a kinematic orbit. The same statement holds if accelerations are set up with the highest possible resolution, where the trajectory in-between is allowed to have large excursions ("slalom"-orbit). If there are less than four GPS observations available at a certain epoch, it is not possible to estimate all three pulses. Figure 2 (bottom) includes such epochs where the filtering due to the dynamic orbit model starts to affect the MRD-trajectory, which is responsible for a few larger differences. A closer inspection shows that this effect is limited to the neighboring epochs of intervals with few observations. For a few days, however, we also found deviations lasting longer, a case to be investigated further.
3.2.2 Properties of HRD-orbits Section 3.2.1 showed that MRD-orbits may be considered as equivalent to kinematic orbits. In order to make use of filtering effects associated with RDorbits, it is thus necessary to set up pseudo-stochastic parameters at subintervals of length Tp longer than the observation sampling interval Ts. Because Tp determines, in essence, the achievable resolution for a subsequent gravity field recovery procedure, Ts should be rather decreased than Tp increased. Figure 3 (bottom) illustrates this for pulses in comparison to the case of maximum resolution (top). The trajectory is filtered with the force model because Ts < Tp, which leads to a trajectory with a reduced scatter. As a drawback, however, the results show a dependency on the force model as illustrated in Fig. 3 (bottom), even if the orbital positions are evaluated only at the pulse epochs. A simulation study in section 4 is used
kinematic 30s pulses 30s acc. 60s pulses 60s acc.
30s GPS-sampling (mm/s) 0.19 (0.15) (9.26) 0.14 0.17
10s GPS-sampling (mm/s) 0.23 0.24 0.30 0.15 0.15
to establish the relationship between the additional force field dependency and the reduction of noise. 3.3
Results
of HRD-POD
Commonly used gravity field recovery procedures do not only use the orbital positions as pseudoobservables. The energy integral method, e.g., requires instantaneous orbital velocities to compute disturbing potential values along the orbit (FSldvfiry et al., 2004). Table 1 gives an impression of the scatter of four weeks of C H A M P orbital velocities obtained for different solutions with respect to conventional RD-orbit velocities. The solutions in parentheses (e.g., the "slalom"-orbit, see section 3.2.1) have no value for gravity field recovery and are listed just for completeness. In general, we recognize that pulse-based solutions show a smaller RMS than acceleration-based solutions for high resolutions, but would encounter an opposite behavior if the resolution was further decreased. We have to keep in mind, however, that a small noise for pulse-based solutions does not necessarily indicate a better qualification for gravity field recovery, e.g., due to dependencies on the a priori gravity field model. The results listed in Table 1 are in good agreement with the expectations from the simulation study following in section 4, with two exceptions. First, the 10 s kinematic velocities are slightly noisier than the 30s based velocities, even if the same 7-point Newton-Gregory interpolation was applied to the kinematic positions (F61dvfiry et al., 2004). This might indicate a problem with the 10 s GPS satellite clock corrections, in particular because an almost identical RMS of 0.20 mm/s results, if the velocities are evaluated every 30 s only. Secondly, simulated data predict only a slightly larger RMS for 30 s acceleration-based velocities with 10 s sampling than for the 10 s kinematic velocities. This prediction is not confirmed by the observed RMS of 0.30mm/s and needs to be investigated further.
4 Simulation study A 24 h dynamic C H A M P orbit in a true gravity field, defined by the gravity field model EIGEN-2 up to degree and order 120, served as the true orbit to sim-
Chapter52 • Kinematicand Highly-ReducedDynamic LEO OrbitDeterminationfor GravityFieldEstimation 0.3
0.3
a priori signal ( R M S = 2.5 m m ) noise signal ( R M S = 2.4 mm)
L'
0.25
I
0.25 0.2
0.2
:[
30s ki . . . .
tic ( R M S = 2.9 m m ) .......... 10s kinematic ( R M S = 2.6 m m ) noise signal ( R M S = 1.6 mm)
[i
0.15 0.1 E
0.05 0
i 50
100
150
200
250
300
50
100
150
period (s) ,~
0.3
a priori signal ( R M S = 0.6 m m ) - noise signal ( R M S = 2.4 mm)
0.25
200
250
300
period (s)
,~
0.2
,
, a priori signal (~VIS = 0.2 m m ) noise signal ( R M S = 1.6 mm)
0.25 0.2
0.15
0.15 ..
0.1 0.05
o.1 0.05
0
o
50
100
150
200
250
300
period (s)
'
50
J
.
100
.
.
.
.
.
.
.
'
-
150
' -
200
•
- '
-
250
300
period (s)
Fig. 4. Amplitude spectra due to noise and due to force model errors for HRD-orbit positions based on pulses (top) and accelerations (bottom) with T~ - 30 s and Tp - 60 s.
Fig. 5. Noise spectra (top) for 10 s and 30 s kinematic orbits and HRD-orbit based on accelerations with Ts = 10 s and Tp = 30 s and spectrum due to force model errors (bottom).
ulate undifferenced GPS phase observations, which were generated either free of noise or, alternatively, with a white noise of 1 mm RMS error. A different a priori gravity field model was then used to reconstruct the true orbit with different POD strategies. Eventually, Fourier analysis techniques were used to study the differences of the estimated orbital positions and velocities with respect to the true values in the frequency domain. A rather poor a priori gravity field, realized by the EIGEN-2 model truncated at degree and order 20, was used to reconstruct the true orbits. In addition, the low degree and order spherical harmonic coefficients (_< 20) were slightly modified according to the formal error estimates provided by the EIGEN-2 model.
range. It is in particular possible for RD-orbits based on pulses and larger values of Tp that the signal spectrum exceeds the noise spectrum also for few periods larger than 2 - T v. As expected, accelerations show a slightly better noise reduction and a greatly reduced dependency on the a priori gravity field model over the entire frequency range, because the unexplained gravity field signal is not only absorbed at certain discrete epochs. Figure 5 shows analogue spectra for the more interesting case with T8 = 10 s and Tp = 30 s based on accelerations. Figure 5 (bottom) indicates a negligible influence of the a priori gravity field signal in comparison to the noise level, at least over the considered one-day time interval (see section 4.3). Figure 5 (top) also shows the noise spectra of 10 s and 30 s kinematic orbits. Taking into account that the level of the last mentioned spectrum is only apparently higher by x/3 because of its lower sampling, we see that the HRD-orbit exhibits a better noise characteristic than both kinematic orbits in the highest frequency range and still around the region of interest at 2 • Tp = 60 s.
4.1 Fourier analysis of orbital positions Figure 4 shows amplitude spectra of orbital position differences over one day emerging from HRD-orbits based on pulses (top) and accelerations (bottom), respectively, with Ts = 30 s and Tp = 60 s. The amplitude spectra denoted as 'a priori signal' characterize the residual impact of the a priori gravity field model in a noise-free simulation. The amplitude spectra denoted as 'noise signal' characterize the impact of the 1 mm GPS phase observation noise in absence of any gravity field model errors. Fig. 4 confirms, in essence, that orbit difference signals with periods T < 2 • T v (indicated by a vertical line in all spectra) are dominated by the a priori gravity field model as the a priori gravity field induced signal amplitudes exceed the apparent amplitudes caused by the observation noise, which is strongly reduced in the highest frequency range due to the RD-filtering. The effect illustrated in Fig. 3 (bottom), however, explains that the impact of the a priori gravity field model is not only restricted to periods T < 2- Tp as one might expect from an ideal filter, but also leaks into the lower frequency
4.2 Fourier analysis of orbital velocities As mentioned in section 3.3, gravity field recovery procedures like the energy integral method also require instantaneous orbital velocities as input data. This is the motivation to perform an analogue analysis for velocities as it was done in section 4.1 for positions. Note that orbital velocities are by-products of all types of HRD-orbits, which may be obtained by taking the time derivative of Eq. (2). For kinematic orbits, however, only approximate procedures can be applied to the kinematic positions like a 7-point Newton-Gregory interpolation proposed by F61dv~ry et al. (2004). Figure 6 shows amplitude spectra of orbital velocity differences over one day emerging from HRD-
359
360
A. J~iggi • G. Beutler • H. Bock. U. Hugentobler
= ~
0.012 0.01 0.008 0.006 0.004 0.002
~.
50
~
0.012
a priori'signal(RMS = 0.044mnffs) noise signal(RMS = 0.077mm/s)
0.012 ' 0.01 0.008 0.006 0.004 0.002 0 ........~_ ' ..-~_ .?. 50
100
'
150 period (s)
200
250
f
!
I 0.0080"01
=
0.006
300
I I
:!~ i'! '": ~' ',.i ":i', '.[ I
0 0.012 0.01
a priori signal (RMS = 0.012 mln/s) noise' signal (RMS = 0.093 mm/s)
.!i 30s kinematic (,~I~IS = 0.090 mm/s) .......... :Ji !~i:[:~= ,i=[~, ,.""'"i"m =: [=~i[.n°ise : If" msignal ~. / (RMS' s ',',::i )!',!!I. = 0'"93
50
100
'
.' :.i
o.oo8 0.004 0.006 ~ ...... 100
0.002 150 period (s)
200
250
0
300
6. Amplitude spectra due to noise and due to force model errors for HRD-orbit velocities based on pulses (top) and accelerations (bottom) with T~ = 10 s and Tp = 30 s. Fig.
orbits based on pulses (top) and accelerations (bottom), respectively, with T8 = 10s and Tp = 30s, i.e., spectra corresponding to the situation shown in Fig. 5 (bottom). Figure 6 (bottom) indicates for the acceleration-based velocities a negligible influence of the a priori gravity field signal in comparison with the noise level, at least over the considered one-day time interval (see section 4.3). The pulse-based velocities (top) show a similar picture if they are computed at the pulse epochs as the mean values of the left- and fight-hand side limits of the discontinuous velocity vectors. Note that the stronger dependency on the a priori gravity field model favors accelerationbased velocities for the task of gravity field recovery. Comparing both noise spectra in Fig. 6 implies, on the other hand, that highly-resolved pulse-based solutions exhibit a more favorable noise reduction for the highest frequency range than the corresponding acceleration-based solutions, which was already observed in Table 1 for real data. Figure 7 shows the amplitude spectra from Fig. 6 (bottom) amended by spectra of kinematic velocities, which were established by the Newton-Gregory interpolation from noise-free and noisy 30s kinematic positions, respectively. Taking the apparently higher level of the last mentioned spectrum into account, we see similar noise for kinematic velocities and velocities from the acceleration based HRDorbit, except for the highest frequency range. There, the performance of the kinematic velocities is better due to a more efficient smoothing of high frequency signals by the relatively long interpolation-intervals. Application of 60 s piecewise constant accelerations would lead to a similar effect for the HRD-solution. Note that comparable results would be obtained for the kinematic velocities if the same 7-point NewtonGregory interpolation (30 s spacing between the positions used for interpolation) was applied to 10 s kine-
150 period (s)
.~
250
300
~ ~, i~ !::
I
,[ii~lll~i, ~iii~ ~ii
........... 50
200
a priori' signal (RMS = 0.012 min)s) 30s ki . . . . tic (I~M~ = 0.029 mm/s) ..........
ii i!Iii i
',
lt~!~[ ~ ~ ~ !! :i
100
150 period (s)
200
~ 250
300
Fig. 7. Noise spectrum analogue to Fig. 6 (bottom) and total spectrum for 30 s kinematic velocities (top) and corresponding model error spectra (bottom). matic positions, but much worse results if the spacing of the used interpolation points was changed to 10 s. Figure 7 (bottom) gives the corresponding noisefree spectra separately and shows that the NewtonGregory interpolation introduces comparably large interpolation errors when compared to the HRDspectrum, although the absolute errors do not exceed the level of 0.1 mm/s. The discrete spectral lines are not restricted to the period range shown in Fig. 7 (bottom), but continue to lower frequencies down to the orbital frequency. Figure 7 (top) shows that some of these lines even exceed the noise level of the total spectrum of the considered one day data set. 4.3
Impact
of data
accumulation
All gravity field recovery procedures make use of the accumulation of individual (daily) solutions in order to reduce random errors for a most reliable estimation of gravity field coefficients. Figure 8 illustrates for the most trivial error model how the HRD signal and noise spectra from Fig. 6 (bottom) would look if data had been accumulated over 400 days. It is simply assumed that all systematic errors are not reduced, which would be the (worst) case for a daily repeat orbit, whereas random errors are reduced by the square root of the number of accumulated solutions. Figure 8 indicates that for long data sets the systematic errors would become more important than the random errors under the above mentioned assumptions, because the impact of the rather poor a priori force field (truncated at degree and order 20: right vertical line) starts to exceed the noise level in the lower frequency range. It remains to be seen whether such biases towards the a priori model would actually occur in real gravity recovery experiments. It is just as well possible that small systematic errors are reduced in a combination like random errors due to a permanently changing orbit geometry.
Chapter 5 2
•
Kinematic and Highly-Reduced Dynamic LEO Orbit Determination for Gravity Field Estimation
Steady-State Ocean Circulation Mission. ESA SP-1233
0.0006
(1)
0.0005 0.0004 0.0003 0.0002
0.0001 0 0
50
100
150
200
250
300
period (s)
Fig. 8. Amplitude spectra from Fig. 6 (bottom). The noise spectrum is downscaled to simulate the worst case effect of data accumulation (see text). To get an impression of the impact of the a priori model for real data, we repeated the processing described in section 3.3 with the gravity field model EGM96 (Lemoine et al., 1997) instead of EIGEN2 and compared the corresponding orbital velocities with each other. For the most interesting solution with Ts - 10 s and Tp = 30 s we found an overall RMS of velocity differences of 0.026 mm/s for pulses and 0.016 mm/s for accelerations due to the changed force model, which is comparable to the simulated results.
5 Conclusions We presented a very efficient method to compute RDorbits based on pseudo-stochastic parameters with resolutions Tp >_ T8 and showed that MRD-orbits are, in essence, equivalent to kinematic orbits. The pre-processing of GPS data was found to be the most important aspect when generating the one-year data set of kinematic CHAMP positions. This problem is of course not removed when generating MRD-orbits. An extensive simulation study showed that not only kinematic orbits but HRD-orbits, as well, could be interesting as input data for gravity field recovery, in particular for the upcoming GOCE mission (ESA, 1999) which is expected to provide 1 s GPS data. The influence of interpolation errors on kinematic velocities was found to be larger than the influence of the a priori gravity field model on HRD velocities, even if a very poor a priori model was used. First experiences gained with four weeks of real CHAMP GPS data confirmed, in essence, the expectations from the simulation study. However, the side-issue of generating most reliable 10 s GPS satellite clock corrections and the issue of a larger velocity noise level than expected from simulations for the acceleration-based HRD solutions need to be further studied.
References Beutler G (2004) Methods of celestial mechanics. Springer, Berlin Heidelberg New York Beutler G, J~iggi A, Hugentobler U, Mervart L (2006) Efficient orbit modelling using pseudo-stochastic parameters. J Geod 80:353-372 European Space Agency ESA (1999) The Four Candidate Earth Explorer Core Missions: Gravity Field and
F61dv~iry L, Svehla D, Gerlach C, Wermuth M, Gruber T, Rummel R, Rothacher M, Frommknecht B, Peters T, Steigenberger P (2004) Gravity model TUM-2Sp based on the energy balance approach and kinematic CHAMP orbits. In: Reigber C, Liahr H, Schwintzer P, Wickert J (Eds) Earth observation with CHAMP, results from three years in orbit. Springer Verlag, Berlin Heidelberg New York, pp 13-18 Gerlach C, F61dv~iry L, Svehla D, Gruber T, Wermuth M, Sneeuw N, Frommknecht B, Oberndorfer H, Peters T, Rothacher M, Rummel R, Steigenberger P (2003) A CHAMP-only gravity field model from kinematic orbits using the energy integral. Geophys Res Lett 30(20) Hugentobler U, Schaer S, Fridez P (2001) Bernese GPS Software Version 4.2. Documentation, Astronomical Institute University of Berne J~iggi A, Beutler G, Hugentobler U (2004a) Efficient stochastic orbit modeling techniques using least squares estimators. In: Sanso F (Ed) The proceedings of the international association of geodesy: a window on the future of geodesy. Springer, Berlin Heidelberg New York, pp 175-180 J~iggi A, Hugentobler U, Beutler G (2006) Pseudostochastic orbit modelling techniques for low Earth orbiters. J Geod 80:47-60 Lemoine FG, Smith DE, Kunz L, Smith R, Pavlis EC, Pavlis NK, Klosko SM, Chinn DS, Torrence MH, Williamson RG, Cox CM, Rachlin KE, Wang YM, Kenyon SC, Salman R, Trimmer R, Rapp RH, Nerem RS (1997) The development of the NASA GSFC and NIMA Joint Geopotential Model. In: Segawa J, Fujimoto H, Okubo S (Eds) lAG Symposia: Gravity, Geoid and Marine Geodesy. Springer-Verlag, pp 461-469 Reigber C, Balmino G, Schwintzer P, Biancale R, Bode A, Lemoine JM, Koenig R, Loyer S, Neumayer H, Marty JC, Barthelmes F, Perosanz F, Zhu SY (2002) A high quality global gravity field model from CHAMP GPS tracking data and accelerometry (EIGEN-1S). Geophys Res Lett 29(14) Reigber C, Schwintzer P, Neumayer KH, Barthelmes F, K6nig R, FSrste C, Balmino G, Biancale R, Lemoine JM, Loyer S, Bruinsma S, Perosanz F, Fayard T (2003) The CHAMP-only Earth Gravity Field Model EIGEN2. Adv Space Res 31(8) 1883-1888 Svehla D, Rothacher M (2003) Kinematic and reduceddynamic precise orbit determination of CHAMP satellite over one year using zero-differences, presented at EGS-AGU-EUG Joint Assembly, Nice, France Svehla D, Rothacher M (2004) Kinematic precise orbit determination for gravity field determination. In: Sanso F (Ed) The proceedings of the international association of geodesy: a window on the future of geodesy. Springer, Berlin Heidelberg New York, pp 181-188 Touboul P, Willemenot E, Foulon B, Josselin V (1999) Accelerometers for CHAMP, GRACE and GOCE space missions: synergy and evolution. Boll Geof Teor Appl 40 321-327 Visser PNAM, Sneeuw N, Gerlach C (2003) Energy integral method for gravity field determination from satellite orbit coordinates. J Geod 77:207-216 Wu SC, Yunck TP, Thornton CL (1991) Reduced-dynamic technique for precise orbit determination of low Earth satellites. J Guid, Control Dyn 14(1): 24-30
361
Chapter 53
On the Combination of Terrestrial Gravity Data with
Satellite Gradiometry and Airborne Gravimetry Treated in Terms of Boundary-Value Problems P. Holota Research Institute of Geodesy, Topography and Cartography, 25066 Zdiby 98, Praha-vychod, Czech Republic e-mail: [email protected], Tel." +420 323649235, Fax: +420 284890056
Abstract. The unprecedented progress in satellite, but also airborne and terrestrial measurements is associated with new possibilities for refined studies on Earth gravity field. At the same time, however, these advances open a number of new problems of theoretical nature. The purpose of this paper is to study the synthesis of satellite, airborne and terrestrial measurements and to show that it brings impulses for the formulation of boundary-value problems which together with some optimization concepts may offer a reasonable way for the combination of the mentioned data. In a sense the approach offers a generalization or modification of the problems, their importance is well-know in physical geodesy. Mathematical properties and the solution of these problems are discussed. The apparatus of spherical harmonics is applied and it illustrates the synthesis in a spectral domain. The optimization procedure enables to treat the fact that the problems are overdetermined by nature.
Keywords. Earth's gravity field, geodetic boundary-value problems, overdetermined problems, optimization
1 Introduction In gravity field modeling a challenging and also frequently discussed problem is the combination of heterogeneous gravity data. In parallel also more data become available than necessary. Hence certain kinds of overdetermined problems have to be solved. The purpose of this paper is to extend theoretical studies performed in Holota and Kern (2005) concerning the combination of terrestrial and airborne gravity data and, in view of the GOCE mission and the so-called space-wise approach, also the combination of terrestrial gravity and (satellite) gradiometry data. At the same time values of some parameters were changed. In the sequel 12 means a layer bounded by two surfaces. With some simplification we can even
suppose that 12 is bounded by two spheres of radius R i and Re, respectively, assuming that Ri < Re .
As usual we will refer our considerations to Euclidean three-dimensional Euclidean space R 3 with rectangular Cartesian coordinates xi, i = 1, 2, 3 and the origin at the center of gravity of the Earth. Then x = ( x I , x 2 , x 3 )
is a general point in R 3
In the sequel we will also use the spherical coordinates r (radius vector), q) (geocentric latitude) and 2(geocentric
longitude),
which
are related to
x 1, x 2 , x 3 by the equations x 1 = r cos (,ocos 2 ,
x 2 = r cos (p cos 2
x 3 = r sin q)
(1) (2)
We will assume that the Earth is a rigid body and that the system of coordinates rotates together with the Earth with a known constant angular velocity co around the x 3-axis.
2 Mixed boundary problem for terrestrial and airborne gravimetry We start our considerations with the problem to find iF such that in
AT-O
aT ~T c3r
--
_(2 Ag
(3) for
r
-
R i
(4)
R i
c3T --=-6g Or
for
r-R e
where A is Laplace's operator, A g gravity anomaly and 6 g
(5) is the usual
is the gravity disturbance.
Note, however, that for T as above we do not explicitly use the term "disturbing potential", since T does not necessarily coincide with this notion (which is common in geodesy) if defined by Eqs. (3)- (5), see below.
Chapter 53 • On the Combination of Terrestrial Gravity Data with Satellite Gradiometry and Airborne Gravimetry Treated in Terms of Boundary-Value Problems
In case that T really represents the disturbing potential in the sense as in geodesy, then for physical reasons T is regular at infinity, which means that asymptotically where
T(r, (,o,2) = O(r -a) for r ~ oo,
(_9(.) is the usual Landau symbol (i.e.
(17 - 1)T~/) -(17 4- 2)q"Tj e) = (, + 1)qn+'rj
')
-
, rj
(14)
R i Ag n
=
(15)
in contrast to Eqs. (7) and (8). Its determinant is D, = (17+ 1)(17+ 2) q2n+l _
n(17- 1)
(16)
IT(r, (p,2) ] < const./r for r --~ oo ). In this case one In case that D, ~ 0 one obtains
usually writes
T(,i) --[Ri17Ag~-Re(17+2)q~Ggn]/D~ T- Z ~ ,=0 \ /
T~ (~p,Z)
(6)
where T~ are the respective Laplace surface spherical harmonics. If by chance a solution of the problem above exists in the form of Eq. (6) then in view of the orthogonality of spherical harmonics (n - 1) T,, =
Ri Ag,
(7)
(17 + 1) q'+aT, = Re 6g, where
(8)
Ag, and 6gn are Laplace's surface spherical Ag and 6 g , i.e.,
harmonics in the expansions for
(17)
and T,(e) - - [ R / ( n + l ) q
n+,zag,,-Re(n
- 1)Ggn] /D~ (1 8)
However, for some 17 the determinant D n may be rather small or even zero. This illustrates the following figure.
100 -
i ~f~
50
oooo<J~o¢o
=
.999'37
I 30
40
Xk
0, ¢¢~ IL 50
oo
Ag(q~, 2) - Z Ag, (q), 2)
I "~60
(9)
n=0 oo
ag(e,
-
a g , (e,
(10)
-100
I
c~~l~l
q = 0.94099
n=0
-150
while Ri
q =-Re
(11)
Ag 1 = 0, as
a condition and from Eqs. (7) and (8) it results that 17-1 =
and selected values of R e :
R e = R i + 4 k m , i.e., q = 0.99937 ; R e = Ri + 250 k m , i.e.,
Hence from Eq. (7) we necessarily have
Ag,
Fig. 1. D n for R i = 6378km
(n+l)q
n+2
6g,
(12)
q = 0.96228 ; R e = R~ + 400 k m , i.e., q = 0.94099.
From Eq. (16) one can also deduce that D, = (2n + 1)[2- c(n + 1)(n + 2)]
(19)
Thus it is clear that the problem under consideration is an overdetermined problem, when the solution is supposed to be regular at infinity. [A similar conclusion for a non-spherical analogue of our problem can be deduced from classical potential theory as represented e.g. by Kellogg (1953, Chap. IX and XI).] On the other hand recall, that .(2 is a bounded domain and that formally in this case T = T(r, q~,2)
where c - 1 - q
is represented by
The negative root has no meaningful interpretation in our considerations, so that we can confine ourselves just to
T - ~ -°°2/ ~ ~=0 \
/ n+lT(i)((tg,~)_+ - ° ° ( ~ e l n
~
/
T(,e)(q),A)(13)
,=0
see also Holota (1995) and Grafarend and Sans6 (1984). Thus inserting into Eqs. (2) and (3), we obtain for any individual n (i.e. for the individual harmonics) the following system
and only terms linear in c where
kept. Thus Dn = 0 implies a quadratic equation
(17+1)(17+2)-2/c=0
(20)
and (taking n for a rational number) its two roots are t71, 2
--
1,5+ x/2,25 + 2 ( 1 / c - 1 )
171 - - 1,5 + x/2,25 + 2 ( 1 / ~ - 1)
(21)
(22)
In general n a is not an integer, so that rigorously, D n ~ 0 for n = 0,1,2,...,c~, but it can be very small for n close to n 1 .
363
364
P. Holota
vimetry data seems to be less complicated. We can consider the following problem
Note also that Dn - 0 means that q
2~+1 =
n(n-1)
(23)
(n + 1)(n + 2)
Hence, clearly, q - 0 for n - 0 or 1, which represents the case when R e --~ oc. On the other hand
c~T
Thus, considering our system for
2
(28)
Ag
for
r-
(29)
Ri
c~2T
Example. For q - 6378 k m / 6382 km - 0.99937 we
n - 5 5 the system of Eqs. (14) and (15) is nearly singular and in fact it can be solved only in the case that the equations are linearly dependent. Clearly, the solution is then given not uniquely.
_Q
--+--T-c3r R i
q --~ 1 as n --~ oc, see also Holota and Kern (2005).
have n 1 - 54, 86. This, however, means that e.g. for
in
AT-O
c~r 2
=G
for
r-R e
(30)
Here the input from satellite gradiometry is symbolized by oo
G(~o, 2) - ~ G~ (~o, Z)
(3 i)
n=0
and T~(e)
where again G, are the respective Laplace surface
and assuming that q - 0 , 9 9 9 3 7 and n - 55, we can
spherical harmonics. Hence, in view of the orthogonality of the spherical harmonics we can deduce for any individual n the following system
T (i)
easily verify that the ratio between the elements in the first and the second column of the matrix of the system is very close to 1, in particular n-1 (n + 1)q "+1 = 0,998
and
(n +2)q ~ n = 1,001 (24)
respectively. This means that the same ratio (close to 1 ) between the right hand sides of the system, i.e., R edg55 = R i A g 5 5
or
6g55 = q A g 5 5
(25)
actually represents the condition for the solvability of the system. In this connection recall that Eq. (25) means 2 n + l conditions between the individual harmonic components which form 6g55 and Ags5 and note that for n = 55 this actually is 111 conditions that have to be met by the respective scalar coefficients [cf. with Eq. (12) which, however, has to hold for all n ]. If the condition, i.e. Eq. (25), is met we obtain (e.g. from the first equation) that T5(~) - ( 57 / 54 ) q55 T5(~) + (1/54) R i Ag55
(26)
In consequence, using the possibility of a free choice, we naturally put Ts(~) - 0 and thus obtain T¢{) - (1 / 54) Ri Ag55
(27)
Finally, recall again that the dimension of the nullspace of the boundary problem under consideration depends on the particular value of the parameter q (on the thickness of the layer .(2 ), see also Sect. 8.
( n - 1)T(i) - (n + 2)q nT/~(e) - R i Ag~
(32)
(n + 1)(n + 2)qn+lT (i) + n ( n - 1)T(e) - R e2 G n
(33)
Its determinant is Dgn - n ( n - 1)2 + (n + 1)(n + 2)2q 2n+1
(34)
and it is clear that it always differs from zero. Thus T (i) - [Rin(n - 1)Ag~ + R e2 (n + 2)q ~ G~ ] / D g (35)
and T (e) - - [ R i (n + 1)(n + 2)q
n+l
Ag n (36)
- R e2(n
1)G,]//D g
4 Optimization in H 2 In both the cases we found a solution in the form of Eq. (13) which represents a harmonic function in the layer .(2. The problem, however, is that in general the continuation of T for r > R e is not a regular function at infinity. This can be considered a consequence of a contamination of the input data by some measurement errors and thus they do not exactly represent the same function. Indeed, the data given on the sphere of radius R i are enough to determine (under the respective solvability conditions) a harmonic function in the whole
}
3 Mixed problem for gravimetry and gradiometry
domain U2ext = { x
In comparison with the case above the combination of satellite gravity gradiometry with terrestrial gra-
nature of excess data and in general (when some measurement errors are present) give rise to the
• R 3", r > R i
and thus also in
,(2 c -Oext • Therefore, the data for r = R e have the
Chapter 53 • On the Combination of Terrestrial Gravity Data with Satellite Gradiometry and Airborne Gravimetry Treated in Terms of Boundary-Value Problems
("internal") terms (r / R e)" T~e) that are not regular
which equals zero at the point f ,
at infinity. The problem calls for a regularization that at the same time can reduce errors. In the sequel we will continue the discussion already started in Holota and Kern (2005). By nature the problem may be ranged under overdetermined problem which in general have been already treated in literature, see e.g. Sacerdote and Sans6 (1985) and Rummel et al. (1989). Here we approach it in a slightly different way, through analytical regularization. In solving the overdetermined problem as above, we will look for a harmonic function f which is
Kern (2005). In calculus of variations the identity (39) represents Euler's necessary condition for the functional @ to have a minimum at the point f
regular at infinity and meets some optimization criteria. We start with the minimization of the following functional
function. Then
@(f) - I( f - T ) 2 dx (37) 12 where dx =dxldx2dx 3 is the volume element in Cartesian coordinates. In particular, we will suppose that H 2 (X2ext) is the space of harmonic func-
The integral identity given by Eq. (39) is a natural starting point for a numerical solution. First, however, we put for the indices n = 0,1, 2,...,oc and m = - n , - n + 1,...,-1, 0,1,...,n - 1,n cosm2 form_>0 Pf/Iml(sin ('°) L sin m 2 for m < 0
t
where
vf/m
1
I f g7
dx
(38)
1 2 6"xt
and we will look for a function f • H 2 (-C2ext) that minimizes the functional @. Recall that the inner product above induces the norm ]l f l] = ( f , f)1/2 and that, roughly speaking,
H 2 (.(2e~t) is a space of harmonic functions, which are defined on .(2e~t and square integrable under the weight r -2 . Note also that the regularity at infinity of functions from H2(_C2~t) is actually implied in the definition of the space H 2 (.(2ext). The functional @ is a quadratic functional and one can show that it attains its minimum in H2(X2ext). This was already discussed in Holota and Kern (2005) in detail. The respective reasoning follows the proof based on the theory of non-linear functional, as e.g. in Necas and Hlavacek (1991). On the contrary suppose that at a point f • H 2 (.C2ext) the functional 1"2 has its minimum. In this case we can deduce that necessarily
Pnlml
(40)
is the usual (associated) Legendre
\-~/
Y~m((p, 2)
(41)
are the solid spherical harmonics and it is known that in general O0
f-
tions endowed with inner product
( f ' g) -
cf. Holota and
1Tl = f/
Z Z ff/mVf/m f/=Om=- f/
(42)
where ff/m are scalar coefficients. In consequence, using the orthogonality of spherical harmonics, Eq. (39) transforms into the following system for the coefficients ff/m :
ff/m IV2rn d x - ITvnm dX D D
(43)
Here
IVf/L d x 12
R3 ( 1 - - q 2 f / - ' ) IYn2((/9,/~)do" (44) 2n-1 o-
and do- is the surface element of the unit sphere o-. As to the integral on the right hand side of Eq. (43), we first recall that in general m--f~
T~ i) - T,(i)((,°,2) - ~
"f/m'~(i)yf/m((,O,2)
(45)
/T/=--f/ In
r(,
- r(,
----f/
- Z
.,m
Lm(e.Z)
(46)
/T/------f/
while ~"/f///n .~(i) and u.~(e) are the respective coefficients. f///n Thus we obtain
I fv dx - I Tv dx 12
(39)
12
for all v • H 2 (..C2ext ) . R e m a r k 1. To see it one has to compute the functional derivative of @ (Gfiteaux' differential of @ )
I Tvnm dx - "f/m ._,(i) I vf/m 2 dx + .(2 .(2 2 %m
(47)
2 cr
365
366
P Holota
and subsequently ~ ) + a nanm (e) fnm - a (nm
which holds for all v (2n-1)(1-q2) n-2 an = q 2n-1 ) 2(1-q
(49)
Hence, inserting into Eq . (42), we get
[T(i) + anT(e)
i
J(grad T, grad v) dx (53)
(48)
where
f -
=
J(grad f, grad v) dx
E H2 1)
(Qext ) . It represents the
respective Euler's necessary conditien in analogy to Eq . (39) . Interpreting Eq. (53) in terras of our function basis, we can again expres the function f by means of Eq . (42), but new for the coefficients fnm we obtain the following system
(50)
]
n=0~ r ~
fnm
J
grad vnm
$ (grad T, grad vnm ) dx (54)
In particular ene can show that ao = (1+ q) / 2q and that lira an = 0
n ->
as
oo .
Further values of an
which is an analogue to Eq . (43) . Here
are in Fig . 2 . J(grad T, grad v nm ) dx = an,
grad vnm 2 dx
(55)
1,2
and in consequente a nm (l)
mm
0,8
(56)
Thus we finally get
0,6
~ n+1
T(i) n
c
0,4
f
fR n=0 r
(57)
0,2
which for large lira a n 0
25
50
Fig. 2 . Values of n and a 360
a~
75
100
125
150
(q as in Fig . 1) . In addition for higher
q = 0 .99937
some further values of a n
= 0 .9919, x 720 = 0 .9672 , and
a1440
= 0
n
n
as
course, for i (e)
= 0
approaches Eq . (50), since oo , as we already know . Of (Eidl compatibility of data for
r = R . and r = Re ) both the formulas coincide .
are :
6 Optimization in
= 0 .8763 .
H21 ~ -
Traces
Considering the fact that functions belonging to
5 Optimization in Let on
H21) (sext) SZext
(f ,
Hz1 ~
Hz1) (0ext)
be the space of harmonie functions
= J
(grad f , grad g) dx
boundary of Q, we can also look for a function f u HZ> > (Qext )
which is equipped with inner product
g) 1
have precisely defined traces on the
(51)
off>=
that minimines the functional
j(f_T) 2 dS
(58)
where ( ., .) is the stalar product of two vettors in
As above ene can show, the functional 0 attains its
R 3 . We will look for a function
minimum in
f u H21) (Qext )
Hz1) (Qext)
and that new
that minimines the functional $ fv dS = $TvdS (f) =
grad ( f - T ) 2 dx
(52)
(59)
a~
As in the last sectien we can deduce that the func-
valid for all v
tional
conditien for 0 to have a minimum at the point f
attains its minimum in
H21) (Qext )
and
E H2 1) (Qext)
is the respective Euler's
that in this case the function f is defined by the
(i .e . for the function f) . Hence for the coefficients
following integral identity
fnm in Eq . (42) we new have the following system
Chapter 53
• On the
Combination of Terrestrial Gravity Data with Satellite Gradiometry and Airborne Gravimetry Treated in Terms of Boundary-Value Problems
(60)
fnm I Vn., 2 6t8--fTvnmdx
a.c2
co(f)
-
( f -- r )2
I
an
#.O
I Vnm 2 dS -- f Vnm 2 dS+ f Vnm 2 dS
- I w c ~ ( f - r ) 2 dS an ~.Q
where
~
r =R i
r =R e
(61)
: R 2 ( l + q n-l) Iy£2dcr
as -
On (69) + 2
I w grad
(f
- T)
2dx
{2
Our aim is now to find a function w such that
Aw-1
cr
and
Ow
in .(2
and
w-0
on c~.(2
(70)
(i.e. for r = R i and r = R e ). As is well-known we can split w into two parts, i.e., w = v + z , where z
f Tvnm d x _ Unrn ._,(i) f Vnm 2 6[8
(62)
+ Unm-(e)(Ri + Re)Riqn 1 8 2 doO"
is a particular solution of the Poisson equation Az = 1 and v is a harmonic function in .(-2 such that v = - z on c%O. For the particular solution we can take, e.g.,
In consequence
z = r 2/6 fn., - Unto .~(i) + r-BnCtnm _(e) ,
fin -
1 + n-1 q q n-1 l+q
(63)
and
and see that, indeed,
Hence v has to be found as a solution of the following problem
Av-O -
[ T(i) +
~--o2
tinT. {e) ]
(64)
J
One can show that /3o - 1
and that lim f i n - 0 as
n ---> oo, which qualitatively is close to what we get for the functional co.
v--R v--R
n=0 k
we will consider the following functional {P(f) - I/9 ( f - T )2 dx
in
.(2
2/6 2
e/6
(71)
for
r-R i
(72)
for
r-R e
(73)
For this aim recall that in general
7 A note on a m u t u a l tie In this section we try to throw more light upon the tie among c o ( f ) , 7-'(f) and O ( f ) . For this reason
Az = 1 in £2.
J
v, ((o,2) + Z n=0
((,o,2) (74)
Using now the orthogonality of spherical harmonics, we easily deduce that Eqs. (72) and (73) yield
v; i) + v~e) - - R 2 / 6 (65)
(75)
and
(R i/Re)v~ i) + v~e) - - R e2 /6
(76)
where /9 is a positive function (weight function). In addition we will suppose that there exists a function w such that t 9 - Aw in .(2. Thus
~ ( f ) - I ( f - T ) 2 A w dx
(66)
[2
and it follows from Greens' identity that
{~(f) - I ( f - T )2
_f w a
c?X2
(67) a(/-T)2
c~n
dS+
IwA( f
-
since f
v~i) - (R 2 + ReR i ) / 6
(77)
v~e) - - (R 2 + ReR i + R2 ) / 6
(78)
w = w ( ~ ) = v ( ~ ) + ~(~)
1
T) 2 dx
=-6[(Re+Ri)
ReRi r 2 2 r + - R e - R e R i - R[]
(79)
[2
Inspecting this function, we immediately see that w = 0 on c%O. Moreover, for r ~ (Ri,Re) it is not
Moreover, on the right hand side
A(f -T) 2 - 2 grad(f-T)
Eqs. (75) and (76) we have
and consequently
dS
On
012
-
Oral4'
while for n > 0 they give v(~i) - v(/) - O. Thus from
2
and T are harmonic functions. Hence
(68)
extremely difficult to show that w(r) is a negative function which attains its m i n i m u m Wmin for
367
368
P. Holota
r -/'min so
that
-
3%/(Re+ Ri)ReRi / 2
"
(R e + Ri) / 2
(80)
Wmi n -- W ( r m i n ) " -- ( R 2 + R ? ) / 8 .
Summing up, for w as above we have
ence of this problem if compared with the usual Stokes' problem that has a tie to the boundary condition for T , we consider on the sphere of radius R~, see Eq. (4). One can also say that for the combination of A g
~ ( f ) - c0(f) -
~ ( f - r )2 c3w dS c~n (?X2
and 6 g the solution of our auxiliary problem in the (81)
+ 2 1 w g r a d ( f - T ) 2d x In addition, c~w
c~w . Or
.
On for r
-
R i
(R e - R i)(2Ri + R e) . R e - R i . . . . 6R i 2
(82)
and
aw = a__w__w= (R e - R i ) ( R i + 2Re) _. -RRei c3n c3r 6R e 2
(83)
we are faced by a problem which is manageable more easy way, see Holota and Kern (2005). In case also the dimension of the null-space of auxiliary problem in 12 equals one for all finite
for r - R e . Hence, approximately, cO(f) - Re - R / O ( f ) 2 + 2 [ w] grad ( f - T ) 2 dx
(84)
12 where
w-w(r)
domain .(2 may be used as an aid (if combined with an optimization concept), but in this case it is better to confine the combination to higher harmonic components of the disturbing potential only. This, in fact, causes not a big problem, since a restriction like this is natural for the use of terrestrial and airborne gravity data. Nevertheless the problem needs further investigation. Note. In case that the combination concerns gravity disturbances given for r = Ri and r = R e
is given by Eq. (79). Eq. (84)
shows that there is no perfect tie between 0~, 7~ and O . Nevertheless, we can see from this equation how the values of T on the boundary c~12 and the smoothness of T in the layer 12 are represented in the functional ¢o. R e m a r k 2. From Eq. (84), if also Wmin is taken into consideration, one can deduce that
in a this the q.
As regards the problem in Section 3, we see in contrast that the use of the solution of an auxiliary boundary problem in the domain 12 offers a way for combining terrestrial gravity data and satellite gradiometry which has relatively favorable mathematical properties. It seems, that this is associated with the fact that derivatives of different orders appear in the boundary conditions on the lower and upper part of the boundary, respectively, i.e. for r = R i and r = R e . Hence for the problem studied in Section 3 the optimization in terms of the functional 0~ leads to
2
R~ - R i O ( f ) < q ) ( f ) + R~ + R2i T ( f ) 2 4
(85) n=0
( R "~"+1E
~-1Agn (86)
i.e., a special case of an important inequality which appears in a trace theorem that is well-know in functional analysis.
+ A~(e)
Re Gn (n + 1)(n + 2)
with
8 Concluding remarks In order to apply our optimization to the problems discussed in Sections 2 and 3 we still have to return to Laplace's surface spherical harmonics T~(i) and
A(O _ n 2 - 1 E n ( n - 1 ) D,g n+i
2)qn+l 1
(87)
and A(e) _ (n + 1)(n + 2)
_
T,(e) as they were derived in this two cases The problem in Section 2 (the combination of terrestrial gravity data with airborne gravity data) has somewhat more difficult nature. It is a singular problem, as we have already mentioned. The dimension of its null-space varies with the thickness of the domain (layer) 12 and may attain considerable values. This actually causes an essential differ-
-%(n+
D~
F-lL(n+2)qn+%(n_l)j
(88)
Note that for n = 1 1 A~( i ) n-1
c~1--0.33 3q
Recall also that in Eq. (86) R i A g n / ( n - l )
(89) repre-
sents the typical structure of a spherical harmonic component as it appears it the solution of Stokes'
Chapter 53 • On the Combination of Terrestrial Gravity Data with Satellite Gradiometry and Airborne Gravimetry Treated in Terms of Boundary-Value Problems
problem for the exterior of a sphere of radius R i 2 while R e G, / (n + 1)(n + 2) is a harmonic component that typically appears in the solution of the gradiometric problem for the exterior of a sphere of radius R e .
1,8 1,6 ", ~ ~
1,4 1 ,2
A (e) n
I k :
1
~
, ~
-
(i)
-
Ar~
~"'~'-/ j ,/ " ~ --~
0,8
0,6 0,4 0,2
~,q
= 0.96228
q = 0.94099
-
n
-0,2 -0,4
As regards the functional O Eqs. (86) - (89) may be used again, but in this case a , - f i n for all n . The respective figure for the coefficients A,(i) and A~(e) shows a faster attenuation of the influence of gradiometric data with increasing n again. Qualitatively, however, it does not differ too much from Fig. 3. For this reason we skip its presentation here. The illustrations also show that the optimization applied offers a natural (and analytical) concept for weighting the effect of the input data on the final solution. Possibly, the concept may also be taken for an alternative view of the balance between the terrestrial and gradiometry data which usually is solved on the basis of stochastic considerations.
Acknowledgements. The work on this paper was supported 0
25
50
75
100
125
150
Fig. 3. Coefficients A F/(i) and A /7(e) for the functional q) q = 0.96228 and q = 0.94099, i.e. for Re = R i+ 250km and Re = R i + 400kin
by the Grant Agency of the Czech Republic through Grant No. 205/04/1423 and partly by the Ministry of Education, Youth and Sports of the Czech Republic through Projects No. LC506. All this support is gratefully acknowledged. Thanks are also due to two anonymous reviewers for their constructive criticism and comments.
The values of the coefficients A~ i) and A~e) can be seen from Fig. 3. Note that the meaning of these coefficients slightly differs from their interpretation in Holota and Kern (2005). The other two optimization concepts associated with the functionals T and O yield similar results in principle, but in comparison with ¢o both of them attenuate the influence of gradiometric data with increasing n more quickly. In particular, for the functional T the optimized solution is given by Eqs. (86) - (89) again, but with a~ = 0 for all n . The respective values of the coefficients A~i) and A~(e) are depicted in Fig 4. 1,8 1,6 1,4 1,2 ~ A ,11
o,8 0,6 0,4 0,2 0
(e) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
-
A(i)
__
n
~. ~ ' " ' -
..... 0.96228
,~
" ~, ~ = l
rl
,-- q = 0 . 9 4 0
-0,2 -0,4
~
m~====~ =======,
I 0
25
50
75
100
125
150
Fig. 4. Coefficients A~i) and A~e) for q as in Fig. 3, but for the functional 7", i.e. for a n = 0 for all n.
References
Grafarend E and Sans6 F (1984) The multibody space-time geodetic boundary value problem and the Honkasalo term. Geophys. J.R. astr. Soc. 255-275 Holota P (1995) Boundary and initial value problems in airborne gravimetry. In: Proc. IAG Symp. on Airborne Gravity Field Determination (IAG Symp. G4), IUGG XXI General Assembly, Boulder, Colorado, USA, July 214, 1995, (Conveners: Schwarz K-P, Brozena JM, Hein GW). Special report No. 60010 of the Dept. of Geomatics Engineering at The University of Calgary, Calgary, 67-71 Holota P and Kern M (2005) A study on two-boundary problems in airborne gravimetry and satellite gradiometry. In: Jekeli C, Bastos L and Fernandes J (Eds.) Gravity, Geoid and Space Missions. GGSM 2004, IAG Intl. Symposium, Porto, Portugal, August 30 - September 3, 2004. Intl. Association of Geodesy Symposia, Vol. 129, Springer, Berlin-Heidelberg-New York, 2005, 173-178. Kellogg OD (1953) Foundations of potential theory. Dover Publications, Inc., New York Necas J and Hlavacek I (1981) Mathematical theory of elastic and elasto-plastic bodies: An introduction. Elsevier Sci. Publ. Company, Amterdam-Oxford-New York Rektorys K (1974) Variacni metody v inzenyrskych problemech a v problemech matematicke fyziky. SNTL Publishers of Techn. Literature, Prague 1974; also in English: Variational methods. Reidel Co., Dordrecht-Boston, 1977 Rummel R, Teunissen P and Van Gelderen M (1989) Uniquely and over-determined geodetic boundary value problem by least squares. Bull. Gdoddsique, Vol. 63, 1-33 Sacerdote F and Sans6 F (1985) Overdetermined boundary value problems in physical geodesy. Manuscripta Geodaetica, Vol. 10, No. 3, 195-207
369
Chapter 54
A Direct Method and its Numerical Interpretation in the Determination of the Earth's Gravity Field from Terrestrial Data O. Nesvadba ~'~'3, P. Holota ~ and R. Klees ~ Delft Institute of Earth Observation and Space Systems (DEOS) Delft University of Technology, 2629 HS Delft, Kluyverweg 1, The Netherlands 2 Research Institute of Geodesy, Topography and Cartography Zdiby 98, Praha vj~chod, 250 66, The Czech Republic 3 Land Survey Office. Pod sfdligt6m 9, Praha 8, 182 11, The Czech Republic
Abstract. In the determination of the gravity field of the Earth the present accuracy requiremerits represent an important driving impulse for the use of direct and numerical methods in the solution of boundary value problems for partial differential equations. The aim of this paper is to discuss the numerical solution of the linear gravimetric boundary value problem. The approach used follows the principles of Galerkin's approximations. It is formulated directly for the surface of the Earth as the boundary of the domain of harmonicity. Thus, the accuracy of the gravity field model is only limited by the accuracy of gravimetric data and the data coverage, and by the capability of the computer hardware. The approach offers a certain freedom in the choice of a function basis suitable for representing the gravity potential of the Earth. Extensive numerical simulations have been done using simulated gravity data derived from the EGM96. Solutions for spherical and more general boundaries have been computed. The solutions also show the oblique derivative effect, usually neglected in geodesy. Moreover, different function bases (such as point masses, reproducing kernels, and Poisson multi-pole wavelets) were used to represent the disturbing potential. The computed global gravity field models are compared with the EGM96 input data in terms of potential values and gravity disturbances at points on the boundary surface.
Keywords. Gravity field modelling, linear gravimetric boundary value problem, Galerkin's method, oblique derivative effect.
1
Introduction
In gravity field studies a linear gravimetric boundary value problem governs the relation between the gravity disturbance and the disturbing potential. As known, the disturbing potential meets Laplace's equation outside the Earth. (We assume that effects of extraterrestrial masses are removed by corrections.) The problem can then be solved uniquely. A uniqueness proof based on variational method can be found in Holota (1997). In addition, we will interpret the solution as a limit of a sequence of Neumann problems, see Holota (2000). This sequence can be understood as a successive rectification of an oblique derivative in the respective boundary condition, where in each iteration step the boundary data are modified. The process avoids any simplifications and leads to an exact solution of the linear gravimetric boundary value problem. The concept of the weak formulation makes it possible to solve the problem directly by means of Galerkin's method. For this purpose spherical basis functions (SBFs) are used to represent the disturbing potential. The mentioned numerical solution enables to solve the problem also for general boundary representing the surface of the Earth. Moreover, oblique derivative effects can be discussed.
2
Problem formulation
Weighted Sobolev space W~'2(fl). Following Holota (1997), in this paper we work with the weighted Sobolev's space Wl'2(t2) that is
Chapter 54 • A Direct Method and Its Numerical Interpretation in the Determination of the Earth's 6rarity Field from Terrestrial Data
equipped with the inner product given by (u, v ) s -
(gvadu, g r a d v ) d x
~-~ d x +
apply Green's identity and show that, indeed, in the classical sense Eq. (6) yields (1)
Here ft is an unbounded domain in R 3 with Lipschitz' boundary 0f~ that represents the E a r t h surface.
Neumann's problem. Loosely speaking, to solve the classical Neumann problem for Laplace's equation means to find a function u such that Au
(gradu, n)
=
0
- f
i~
(2)
on 0f~
Au
(gradu, n + a × n}
n+a×n
= f
A1 (u, v) = (f , V)L2(aa)
(4)
0
- f
in ft
(8)
on c9~2 (9)
This can be found in Holota (1997). In particular, considering the usual formulation of the L G B V P as in Koch and Pope (1972), Bjerhammar and Svensson (1983) or Grafarend (1989) and taking into account (n, n + a x n} - 1 we can show that
(3)
where n is the unit vector of the outer normal of Oft. In this paper, we will use a concept of a weak solution. A function u • Wl'2(ft) is a weak solution of the Neumann problem for Laplace's equation, if the identity
--
--
grad U on0f~ (10) {n, grad U} grad U @ on Oft (11) ( n, grad U}
where U is the normal gravity potential, 9 is the measured gravity and 5g - g grad U is the gravity disturbance on the boundary 0f~ (i.e. on the surface of the Earth). In this case the function u represents the disturbing potential as known in geodesy. In addition,
holds for all v • Wl'2(f~) and
As (u, v) - ~ (gradu, gradv} dx
(5)
A tie between classical and weak formulation may be shown by means of a reasoning based on Green's identity. Finally, note that the unique solvability of the problem in WS'2(f~) follows from the Lax-Milgram lemma. For details see Ne(:as (1967, Chap. 1, § 3.1) and Rektorys (1977, Part IV), where, however, the problem is investigated for a bounded domain. Linear
gravimetric
boundary
value
problem.
(i.e. u is regular at the infinity) which follows automatically from the properties of the weighted space Wl'2(f~). Note that O is the so-called Landau symbol.
Sequence of Neumann problems. Our aim is to solve the identity (6) by means of successive approximations. For this reason we will consider approximations u,~ E Wl'2(f~) defined by the following sequence of problems
A1 (u,~+l, v) - (f , V)L2(Oa) + A2(u,~, v)
(13)
The linear gravimetric boundary value problem (in the sequel only LGBVP) is an oblique derivative problem. In analogy to Neumann's problem above, a function u • Wl'9(f~) is a weak solution of the L G B V P if
for all v C Wl'2(ft). One can show the sequence of u,~ has a limit u C Wl'2(ft) and it yields the solution of the LGBVP, see Holota (2000). In case u,~ is sufficiently smooth, we can apply Green's identity to the bilinear form A2 and rewrite Eq. (13) as follows
A(u, v) - As (u, v) - A2(u, v) - (f , V)L~(a~) (6)
A1 (u,~+l, v) - (f,~, v)c2(On)
(14)
f,~ - f -
(15)
holds for all v • WS'e(f~). Here, in addition to As
A2(u, v)
-
./(){gradv, a × gradu} dx
+
/~ v ( c u r l a , gradu} dx
(7)
ai • L°~(~2)and x (curla)~ • L°°(ft). For sufficiently smooth functions from Wl'2(ft) we can
where
(a × n, gradu,~}
see Holota (2000). The initial iteration step of the computing process approximates the L G B V P by Neumann's problem. Next iterations then rectify the direction of the derivative in the boundary condition by modifying the right-hand side in Eq. (14).
371
372
O. Nesvadba • P. Holota. R. Klees
3
Practical realization
Numerical approximation. Galerkin's method can be directly applied to the solution of our weakly formulated problems. For any finite-dimensional subspace W~ c Wl'~(ft), l i m ~ o ~ W~ = Wl'2(ft) (n denotes the dimension of W~) we can form the respective Galerkin system of linear equations, see e.g. Rektorys (1977). This means u C W,~, i.e. /It
~(~) - } ~ ~ ( ~ )
case as vertices of some level of the sphere triangulation mentioned above. Thus we will deal with: (i) Reciprocal Distance (RD) oo
Ix -
/=0
Yil
(~9)
which is well-known in geodesy. Note that here Ibi is the angle between the position vectors x and Yi, while Yi c R (~2 U 0f~).
(16)
(ii) Reproducing Kernel (RK)
i=l
vi(x)
and for the basis functions vi, vj C V ~ we have
\
zi
oo 2 l + 1
4~
]
Z
+-----T 1 ~P~(~°~)~
(20)
/=0
i=1
where j = 1 , . . . , n . Moreover, it can be shown that Galerkin's matrix
Aij = Al(vi, vj)
i, j = 1 , . . . , n
where zi = /~2/l~llY~l, y~ ~ ~, we us~ th~ r~producing kernel of a Hilbert space of all harInonic functions froln IvVI'2(~) which is equipped with an equivalent inner product given by A~, see Holota (2004). This function can be written in a closed form
(~s)
1
(2zi
~(~) - ~ is symmetric and positive definite. Therefore, for the solution of Galerkin's system the Conjugate Gradients Method with a diagonal preconditioning can be used, see Ditmar and Klees (2002).
Integration over the boundary. In Galerkin's system, see Eq. (17), also surface integrals have to be computed. To this aim a triangulation of the faces of the icosahedron is used to generate a subdivision of the boundary surface, see Fig. 1. A Romberg integration method is used, which exploits the hierarchical subdivision of the boundary surface. Romberg's method gives a more precise approximation of the integral and also an estimate of the integration error.
Figure 1: Icosahedral refinement on the sphere. Zero, first and second subdivision levels.
Choice of the basis. In this work we use some special cases of SBFs located at points y~ of a parking grid. The parking grid points are spread over the sphere with radius lYil = const, in our
Li+zi-cos~)
- ~
~ - co~
(21)
2 Thanks to the w h e r e L ~ - V/1 2 z ~ c o s ~ + z i. reproducing property, this basis function provides very simple setup of Galerkin's matrix for a spherical boundary with radius/~.
(iii)
Poisson Wavelets (PW)
Z ( 2 l + 1)
~n(co~)
(22)
/=0
wh~r~ ~ = R/I~I. This choice gives us an advantage of spectral decomposition of the solution with respect to parameters ai (scale factor) and si (order of multipole), see Holschneider (2003).
4
I
~
Numerical example - spherical boundary
Source data. For the boundary we take a sphere of radius /~ = 6371km. Although simple, this boundary enables to show tile effect of the oblique derivative, since g r a d U does not coincide with the normal to the boundary. Note explicitly that in this work U is defined by the parameters given in Moritz (1980). The gravimetric data on the boundary are simulated by means of EGM96 published by Lemoine at al. (1997), see Fig. 2. This simulation also enables to check the quality of the solution.
Chapter 54 • A Direct Method and Its Numerical Interpretation in the Determination of the Earth's Gravity Field from Terrestrial Data
•
,,,
300mGal 4e
60.
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30.
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....
%'0
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[,).~ [...\ .,,a~.:-,.: ~'" ......"'"i~0mGa'. .'ixk.
" ,.:.
[.,
,.~....
, '
-120
6 longitude
[deg]
6'0
120
-60
180
, 60-
,.
180 ~130 GPU
~ ~ .
;"
~-2sGPu "
.,,,,,:~,ll2oGPu
0-
For the RD basis we put [fli[ = 0.9856R (i.e. the points fli are in the depth of ca 92 kin), for the RK basis ly~l = 1.0146R (i.e. the points fl~ are in the height ca 93 kin), while the fl~ points themselves are vertices of the icosahedral refinement of the sixth level on the sphere. This produces 40962 points of the parking grid, therefore the obtained solution belongs to the W4096~, approximation subspace. The setup of Galerkin's matrix is easy for all the basis functions. Indeed, in view of the harmonicity and smoothness of the basis functions, the use of Green's identity yields
Moreover, for our basis functions and the spherical boundary cgf~ we are able to find an analytical expression for the elements of Galerkin's matrix. The matrix itself is constant during the whole computation process. Right hand side of Eq. (17) have to be computed at every iteration step by numerical surface integration described above. Starting with a trivial disturbing potential u0 (i.e. u0 = 0), the iteration process can be terminated after seven iteration steps. Actually, the obtained solution u = u7 in Fig. 3 represents the disturbing potential related to EGM96. Thus, it can be compared with the exact solution (i.e. the disturbing potential derived directly from the EGM96 and GRSS0 gravity field), for instance at the points of the boundary surface 0f~, see Fig. 3. For the IRK basis we can see the maximal difference of 67 GPU, mean value equal to 0.05 GPU and rms of 3.35 GPU, where GPU is GeoPotential Unit,
~-..~
120
30-
Solution for V~40962 approximation subspace.
(23)
60
[deg]
""~,;,:'; .:;;
Figure 2: Gravity disturbance 59 ( E G M 9 6 GRSS0) on the boundary. It simulates the input data [1 r e g a l - 10-5ms-9].
Al(vi,vj) = - ( v j , (gradvi,n))L2(O~)
0 longitude
-60-
,
........
-90
I
-180
-120
' "t
'
, " " '
-60
.
0 longitude
. . . . . . . . .
60
.
120
/=,0o 0 /11 /ml._.2SGPu ' "q~---3o
180
GPU
[deg]
Figure 3: An obtained solution u (after seven iteration steps) for RK basis (top) and differences between the obtained solution and the exact solution (bottom) depicted on the boundary [1 G P U - lm2s-2].
1 GPU - l m 2 s -2 The detected zero order harmonic term -0.0583 km 3s -2 corresponds well to 5cM - -0.0585km3s -2 for E G M 9 6 - GRSS0. The results for the RD and P W basis are similar to the RK basis. 9O
60 ~i~ll
.~ 300
_
-180
-120
-60
0 longitude
60
120
180
[deg]
Figure 4" Oblique derivative effect (as a difference between the first and the final iteration) in the case of W40962 RK basis and the spherical boundary [1 G P U - 1 m2s-2].
373
374
O. Nesvadba • P. Holota. R. Klees
Oblique derivative effect. From the iteration process it is evident that the effect of the oblique derivative cannot be neglected. When taking the oblique derivative boundary condition into account, we can see the positive impact on the LGBVP solution. This is evident from the successive iteration steps which considerably suppress low frequency error components that appear in the first iteration (Neumann's problem approximates the LGBVP) and in some cases have an amplitude of about 0.7 GPU, see Fig. 4.
..:; '.~>..~.:,.," ..-~.. ..b~
<
.-,.. •-...
,,
.."..~
r
,
-&...
j'-..
~:.,
,.., ~2
,t
~-~ ..... .
'~...'-':.-,"~t~... . :};
• ,
- 180
-60
-120
..~,
120
0 longitude [deg]
180
Numerical example - non-spherical boundary
Figure 6: Gravity disturbance @ on the E T O P O 5 boundary. It simulates the input data [1 regal - 10-Sms-2].
In practice we do not observe gravimetric data on the sphere but o n the physical surface of the Earth. The following example shows the numerical computation based on the E T O P O 5 ( h t t p ://www. ngdc. noaa. gov/mgg/g~oba~/) terrain data. This boundary is much closer to the real physical surface of the Earth.
to the boundary reach ca 40 deg, see Fig. 5. The gravimetric data are again simulated by means of the EGM96, but now for the E T O P O 5 boundary, see Fig. 6.
Source data. The boundary surface is more complicated now. Maximal tilts of the normal
,,~,.:~
_ *_~"qiillllr .:-,.,p~',J~pr
~.
f
.....
,,-.
"--
IIl~o00m
6(>
3O-
i
-30-
1.0599s7 (i.~. G753 kin).
ly,~l =
.••
0-
Solution for W2562 approximation subspace. In contrast to the spherical case treated before, the construction of the Galerkin matrix has to be done by numerical integration, see Fig. 7. Due to a considerable extent of computational work needed, only the W256~, approximation subspace has been used. For the RD basis we take ly~l- 0 . 9 4 3 5 / 7 (i.e. 6011kin), for the R K basis
l~?kl ~ I !-~ ':, •
,t :£~....,,. 160
480
640
800
960
1120 1280 1440 1600 1760 1920 2080 2 2 4 0 2 4 0 0 2 5 6 0
2400
~#*
,,
320
.
22402080= ~ _ um
_ -180
-120
-60
0 longitude [deg]
90'
. 60 . . . .
..... ~ ~ i " .::.,..~ ~ / , i t
....~,:,,~;:~ . . .
.
. • " ~"
......
.
..,
~#-
.
", . . . . . ~.~ .... , ,q,o.
0"
"-<.. ..,~,:.
~
,,~..:~...,.;~
~,.~...;.-,.;~:.!,,,. .
"
:. ". .... ~,-*~",~ -' d - "..:: ... . . . ..,~.,.~..:'~' ~ . .~ """'.!.,'" -,¢.,j "~ .{:.... .:"" '
1920-
-lOdeg
1760-
-gd~
1600-
-30.
!
.
..,--
,.
i
:!7.
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1280-
- 5 dell
960-
- 4 deg
800-
-3 dell
640-
- 2 dell
480-
"-"
-60.
"
'~
"
-
''
;,~
~
~
'
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1440-
-,deQ-'d"
1120-
-:::, O'
i
/:
l
:,, ..
~
l~\ , . .
!
:t
~,' =
:
"
/
~
/
~
.
~
~
~
-
,
--..,.I
~
-
..... "
i
~
.
-
':
i
.
I
-
-'.''-='' I ~-'~-°°' ~:
"
:
~.o= ---,~..
-
320-
-9-C180
i~:;;$!i;: ii ::==~:':-(~0 '~--~'~:~-~:;~'~6 :--:'' ...... -120
~:
6"0
120
-'°~
- 0 dell
longitude [deg]
Figure 5" E T O P O 5 surface data (top) and the inclination of the boundary normal with respect to the grad U direction (bottom).
160-
i
. . ...................... . . . . .
.
"
-.
_.................................. - . . . . . . . . . . . . . . . . . . . . . --. . . . . . . . . . . . . ..
..
~.,,=-o=i..
.....
- . .1. E. . - O ~
".
Figure 7: The example of the structure of Galerkin's matrix for the RK basis of the W25G2 and the E T O P O 5 boundary.
Chapter
901
60
.
.
......
Method
":
and
Its Numerical
.
:.";'.'.
" '
".".'.'.".
" : ;'.'.'-'-
:.:.::.v.,.j.v,..;-.,~¢~:
'.
.......
IOOGPU
. . . . . .
..
i
.
;
". . :. : ;. : :.'.'.'.'.'.'.'~t.'..'.-.'. . . . .. . ..............
.'"...'..'~: . . . . . . .
...
: : . :. :..~. .':.',. " - : . ' . ' . ' . ' . .,, . . . , ; , ~ . . . . ... . . . . .
,~ oti;'~-;i;??i.:::::::::::::::i::~i-: ::::::::.:iii'.))Z.)X.X.X.:::::::::: ::::!::;.:: :':';~-;-bi:"-.~..':]l I-oGPu "-
of the
Earth's
Gravity
Field
from
Terrestrial
120
180
Data
6
p 50G U
~.........!:!:!:i:``..::~:i.~:!:.::~:..:~.:.~...:.~..:!:!:~:i:::i:i:!:.:`f :.}..:..:. ...
Determination
90
..111
?'..:.-....~
•" . . v . . ' - . : . : . : ....... . . . . . . . .......... . . . . . . 30 " ."M" ' I ' . ' - ,""' . ; ' " '"- .' ". .- .' : ' ' ' " : ' " - "":' ": - " -"' . . . . . . . . ".''~" . '. ........ . ; : ' " : ' - ' :". ~ ' " ~' -:~ - : ' ; : " : ''~2 " "" " i " ". " 2 : - ' " "" I...'.,i!iii~.'.'.--~-;;.'..'.-'-:.:.;:~:...'.
in the
." .:. ". i..'..:..'-.
'. " : " ." . ' . ' . ' . "
..',.~... ~. :,-~-. :~,-" ::: :.:;
". " . : . v . - . ' . : :
Interpretation
1~-150 GPU
"..~ .:".. ::( ".. i .".':'.'.
: :.".'.'.".
.'.
• A Direct
.
..'..:-.~, i ." .:. 1.-; .... ."-~.'."
54
~
=========================== :,::::-"."'".'.Z')?7'i:::':i::'~.:!:i:!:iii:i:iil:!'"'"-"'."~£"
' . : "2.'.i . :,".::*i ' . : ' . i ...,: "'.'~i'-'.." : ' . i ' . : ":-"
4
• " "
"
" " "
,o "
"
o
" "
:
longitude
" " " [deg]
,o "
"
" "
:
" " '~'
-9o -180 IE
" ;":
:"
:"""
'
'
:
-120
-60
0
longitude
90
150 GPU
: " " v " ' II !|
[deg]
90
°
,O
g ~::~!~:~;i~!!!!:!~!:~!:!~!!?!i!~:~)~:~:~:~N(.::i't"::II ;~?~;i!!!I-o~,u!:!~!~?~:i;i F'-'---'.";~2' ~a~Ii~!~:~::::::::..:::~~:~:.~.~~:.i~::::~:~:~:::::::::~:~ -'-".-'-'.'.v I~~°aPu -
============================ ~:::--.-'.:'.':v))i'?7'/:-!::.'t~.'.!:i:i:iii:i:i!!
-,0]-.:..L-.-.:: ..-...-...-.:.4i... .~. :.-:-.;.L..:. i: -."2t". i : ... 2.:-:l~-,00GPu . ". ".." : ".. i...,: ".'..' . ! . "." ..'°: . . i . . "." .~.
4:
; 0 - ;0:
,, : ',0
-60
Figure 8: Differences between the obtained solution for the R K basis in W25G2 and the exact solution on the boundary. The case of the E T O P O 5 b o u n d a r y (top) and the spherical b o u n d a r y (bottom). Also the parking grid points of the W2562 are depicted. Simultaneously, for comparison purposes we solve the problem in the same a p p r o x i m a t i o n subspace also for the spherical boundary. The iteration process is slightly affected by the E T O P O 5 b o u n d a r y irregularity now, but c o m p a r e d to eight iteration for the spherical boundary, twelve iterations is enough to obtain an acceptable solution. We have c o m p a r e d the obtained results with the exact disturbing potential on the boundary, see Fig. 8. For the E T O P O 5 b o u n d a r y and RK basis we have maximal difference of I ? 9 G P U , m e a n value equal to - 2 . 3 0 G P U and rms of 1 4 . 8 G P U . For the spherical b o u n d a r y the m a x i m a l difference is 204 GPU, m e a n value equal to 0.04 G P U and rms of 14.5 GPU. The results for the RD basis are similar, but in the case of the E T O P O 5 boundary it seems to be s o m e w h a t affected by the oscillation of the coefficients (rms of 14.9 GPU).
Oblique derivative effect. In the case of the E T O P O 5 b o u n d a r y the oblique derivative effect
0
longitude
longitude [deg]
60 [deg]
Figure 9: Oblique derivative effect (as a difference between the first and the final iteration) for the R K basis. The effect for the E T O P O 5 b o u n d a r y (top) and the effect for the spherical b o u n d a r y (bottom). is clearly visible in the continental and especially m o u n t a i n o u s areas (e.g. - 2 . S G P U in Nepal), see Fig. 9. Again, it is evident t h a t the effect noticeably improves the accuracy of the solution. Oblique derivative effects for the spherical b o u n d a r y are similar, i.e. they are similar for the RD and R K basis and also for the W2~G2 and W40962 a p p r o x i m a t i o n space. In general we can say t h a t the oblique derivative effect mostly depends on the g e o m e t r y of the b o u n d a r y and has a r a t h e r complex behaviour.
6
Conclusions
The weak formulation has an essential tie to the numerical approach and provides tools for dealing with the E a r t h gravity field in a fairly general way. (i) It was experimentally proved t h a t tile solution based on the sequence of N e u m a n n ' s problem converges and t h a t the procedure is numerically stable also for a more general b o u n d a r y like
375
376
O. Nesvadba • P. Holota. R. Klees
the E T O P O 5 E a r t h ' s topography.
References
(ii) The numerical implementation of our approach offers a solution whose accuracy is liraited, but only by tile capability of the hardware used. Already now it makes it possible to solve tile problem under consideration with an accuracy and resolution which are comparable or even better t h a n that for the current geopotential models (e.g. EGM96).
Bjerhammar A and Svensson L (1983) Oil the geodetic boundary-value problem for a fixed boundary surface A satellite approach. Bull. G6od. 57, 382-393. Ditmar P and Klees R (2002) A method to compute the Earth's gravity field from S G G / S S T data to be acquieted by the G O C E satellite. DUP Science.
(iii) The oblique derivative effect should not be neglected in the solution of the LGBVP. If neglected, the solution may be contaminated by an error exceeding 2.5 G P U in some cases. (iv) The RK function seems to be more suitable than the RD function. The developed numerical procedure offers a big freedom in the use of various function bases, not confined just to RK, P W and RD functions. Therefore, future research will be aimed to an optimal choice of basis functions and their parameters, including the multiresolution approach.
Acknowledgements
Much of this paper was writ-
ten during the visit of the first author at the Delft University of Technology, Institute of Earth Observation and Space System (DEOS), supported by the Erasmus/Socrates Program. This support and discussion with the host, Prof. R. Klees, are gratefully acknowledged. The work of the second author on this paper was supported by the Grant Agency of the Czech Republic through Grant No. 205/04/1423. Computing facilities (SGI Altix 3700 system) were kindly provided by Stichting Nationale Computerfaciliteiten (NCF), grant SG-027. Thanks are also due to two anonymous reviewers for their comments.
Grafarend E W (1989) The geoid and the gravimetric boundary value problem. Report no. 18 from the Dept. of Geodesy. The Royal Inst. Of Technology, Stockholm. Holota P (1997) Coerciveness of the linear gravimetric boundary-value problem and a geometrical interpretation. Journal of Geodesy 71, 640-651. Holota P (2000) Direct methods in physical geodesy. In: Schwarz KP (ed.): Geodesy Beyond 2 0 0 0 - The Challenges of the First Decade. IAG Symposia Vol. 121. Springer, Berlin etc., 163-170. Holota P (2004) Some topics related to the solution of boundary value problems in geodesy. In: Sanso F (ed.): 5th Hotine-Marussi Symposium on Mathematical Geodesy. IAG Symposia Vol. 127. Springer, Berlin etc., 189-200. Holschneider M, C h a m b o d u t A and Mandea M (2003) From global to regional analysis of the magnetic field on the sphere using wavelet frames. Physics of the Earth and Planetary Interiors 135, 107-124. Koch KR and Pope AJ (1972) Uniqueness and existence for the geodetic boundary-value problem using the known surface of the Earth. Bull. G6od. 106, 467476. Lemoine FG et al. (1997) The development of the NASA GSFC and NIMA joint geopotential model. In: Segawa J, Fujimoto H and Okubo S(ed.): Gravity, Geoid and Marine Geodesy. IAG Symposia Vol. 117. 461-469. Moritz H (1980) Geodetic Reference System 1980. Bull. GSod. 54, The Geodesist's Handbook, 395-405. Ne~as J (1967) Les m~thodes directes en th~orie des 5quations elliptiques. Academia, Praha. Rektorys K (1977) Variational problems. drecht.
Reidel, Dor-
Chapter 55
Comparison of High Precision Geoid Models in Switzerland Urs Marti Federal Office of Topography, Seftigenstrasse 264, CH-3084 Wabern, Switzerland E-mail: [email protected]
The recently released national geoid model CHGeo2004 of Switzerland was determined by combining gravity, vertical deflections and GPS/leveling. Its accuracy is in the order of 2-3 cm as could be verified by comparison with independent data. In addition to the standard models (topography and global geopotential model), a simple 3D density model of the Earth's crust has been introduced for the reduction of the observations. The method for the calculation was basically least squares collocation with a slight modification in the way that the parameters of the covariance function have been chosen to minimize the resulting residuals between the astrogeodetic, the gravimetric and the GPS/leveling geoid model. Abstract.
Although CHGeo2004 is the result of a combination of different data sets, individual pure astrogeodetic, gravimetric and GPS/leveling solutions have been calculated as well. This gave us the opportunity to investigate the systematic discrepancies. With a pure astrogeodetic solution, we obtained a global accuracy of about 6 cm with some larger systematic differences to the combined solution of up to 20 cm. The deflections of the vertical are the most sensitive data set with respect to the variation of the covariance model. Gravity and GPS/leveling are less sensitive to changing the covariance function. Nevertheless, especially GPS/leveling shows rather large differences to the solutions of the other data sets in some areas. The official geoid model that has been released to the surveyor community is strongly based on GPS/leveling, since their principal use of the geoid model is GPS height determination, which should be in agreement with leveling.
Keywords: Local geoid Switzerland, combined geoid
determination,
1 Introduction The main goal of the calculation of a new geoid model for Switzerland was to set up a consistent height system where the orthometric heights out of leveling (and gravity) are compatible with the heights out of GPS and the geoid model. Nowadays, almost all height determinations with lower accuracy demands are performed by GPS measurements and a geoid model giving the same results as leveling is a principal request of the surveyors. This request implies, that GPS/leveling data had to be introduced into the geoid calculation with a rather high weighting.
2 The data set for C H G e o 2 0 0 4 In the last few years there have been efforts to cover the country with very precise GPS/leveling measurements. So, many GPS stations have been connected to the first order leveling network and many leveling benchmarks have been observed by GPS in sessions of at least 24 hours. This gave us for most of the stations an accuracy of better than 1 cm. Presently we can use about 200 GPS/leveling measurements for the geoid computation. Another improvement of the data set for the new geoid computation was the densification of the already rather dense network of astrogeodetic stations. In regions with known problems in the existing geoid model CHGeo98, additional deflections of the vertical have been determined with the digital zenith cameras of the ETH Zurich (Mfiller et al. 2004) and of the TU Hanover (Hirt et al. 2004). In just 1 month about 60 stations could be observed with accuracy in the order of 0.1 arc seconds. For older astrogeodetic observations with analog zenith cameras or by classical methods, the accuracy is in the order of 0.3 to 1.0 arc seconds. On the contrary to the geoid calculation of CHGeo98, where gravity was only used to model the difference between geoid and quasigeoid (downward continuation), gravity data has been introduced directly as observations in the
378
Marti
U.
determination of CHGeo2004. All the available gravity data was preprocessed to bring them to a common datum and to eliminate gross errors. These more than 30 000 gravity values have been gridded with a resolution of 5 km. This gridded gravity data set has been introduced into the geoid determination with an accuracy of 0.5 mgal. This gives us the data set shown in fig. 1 with about 2200 gridded gravity values, about 700 deflections of the vertical and about 200 GPS/leveling observations. Another set of 270 'artificial' GPS/leveling observations has been introduced in the regions in neighboring countries where we had no or only few data. There, we introduced the height anomalies of the European quasigeoid model EGG97 (Denker and Torge, 1997) directly as observations just to avoid the drifting away of our solution in these areas. The weighting of these observations was chosen in a way that they reproduce exactly the values of EGG97 on the introduced grid points themselves but also that they have no influence on the solution inside Switzerland. This can be reached by improving artificially the distance between these stations and the 'real' observations in the least squares collocation approach. o
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As further models we introduced a rather rough density model of the Earth's crust into the calculation. This model mainly includes a Moho model, a model of the Ivrea body (a large, high density mass anomaly in Northern Italy and Southern Switzerland causing gravity anomalies of up to 160 mgal), sediments of rivers, water masses of lakes and the ice thickness of large glaciers. Since a part of the effect of most of these models is already included in the global model, only the remaining part may be used for the reduction. In our calculations, we used the difference between the influence at station height and the influence at sea level for the reduction. This reproduces exactly the differences between normal heights and orthometric heights as we use them in our national leveling network LHN95 (Marti and Schlatter, 2001). Therefore, it also does not matter if we introduce normal heights or orthometric heights for the GPS/leveling observations in the geoid determination. If we apply the correct reduction, we obtain exactly the same results for the geoid and the quasigeoid, if we also apply a small correction to the GPS/leveling data and the deflections of the vertical in the form of the product of the residual gravity and station height.
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Fig. 1: Dataset for the determination of CHGeo2004
3 Mass models and reduction of data The reduction of the observations has been performed in a first approach with the standard method by using a global geopotential model (EGM96) and a digital terrain model (DTM). As digital terrain model we used our national 25 meter DTM and outside of its area we used SRTM3 data. The full resolution of the DTM is only used in an area of about 150 meters around each data point. Further away we use re-sampled models with a resolution of 50, 500 and 10000 meters.
The residual field of all observations has been interpolated by least squares collocation with the 3rd order Markov model as the covariance function. Several tests by varying the parameters of the model and by changing the weights of the individual data sets have been performed, which mainly showed that the most sensitive data set with respect to the variation of the model parameters are the deflections of the vertical, whereas gravity and GPS/leveling are not so much affected by changing the model. The definitive parameters have been chosen in a way that the residuals on GPS/leveling become minimal for a pure astrogeodetic / gravimetric solution, which is a slight modification of the concept of least squares collocation. For the official solution of CHGeo2004, a combination of all three data sets (GPS/leveling, gravity and deflections of the vertical) has been used, but GPS/leveling data got a very high weighting, so that these observations got practically no residuals. This is the solution that guarantees the consistency in the height system and therefore, is preferred by the surveyor community but certainly it hides some problems of the GPS/leveling data set. The restore from the co-geoid to the geoid and to the quasigeoid is simply done by adding the
Chapter 55 • Comparison of High PrecisionGeoid Models in Switzerland
formerly removed effects of the mass models and the global model. Figure 2 shows the calculated geoid model CHGeo2004, whereas figures 3 and 4 show this step in the case of the geoid and in case of the quasigeoid, where we see the contribution of each part. The differences to the former geoid model CHGeo98 (fig. 5) are mainly caused by the strong weighting of GPS/leveling in CHGeo2004 in some areas and - outside of Switzerland - by the use of data in areas which were not covered in the calculation of CHGeo98. The recently measured deflections of the vertical have only a minor influence and the introduction of gravity data had only an important effect in remote alpine areas Fig. 2: Swiss geoid model CHGeo2004 which are not covered by astrogeodetic measurements or GPS/leveling. Topography (H=0)
Density models (diff. up - down)
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Fig. 4: Restore step of the quasigeoid
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One test was to compute a geoid model out of GPS/leveling data alone (fig. 7). This solution shows a rather good agreement with the combined solution in areas where GPS/leveling data is available. In regions without sufficient data coverage, the differences become very large and reach up to 60 cm in the southern areas, mainly in the region of the Ivrea body. Another test with only half of all available GPS/leveling points showed almost the same result. The differences to the solution with all GPS/leveling were everywhere smaller than 2 cm. Fig. 5:CHGeo2004 minus CHGeo98 The quasigeoid has basically been calculated for compatibility reasons with European projects such as UELN, EVRS (Ihde et al. 2001) or the European Gravity and Geoid Project EGGP (Denker et al. 2004). The differences between geoid and quasigeoid are in general smaller than 10 cm but they canreach amounts of up to 60 cm in mountainous regions. These differences are shown in fig. 6.
Fig. 6: Quasigeoid minus geoid
5 Comparison of solutions The official solution of CHGeo2004 is a combined solution of GPS/leveling, gravity and deflections of the vertical with a very high weighting of the GPS/leveling observations. This gives the surveyors a geoid model they can use for height determination with GPS. Besides of this official model, several other solutions have been computed. The variations include the parameters of the covariance model and mainly the weighting of the observations. This chapter and the next figures present a selection of these calculations and the differences of the results to the official combined solution.
The pure astrogeodetic solution (fig.8) is very sensitive to the parameters of the covariance function. Depending on these parameters we get a variation of the result in the order of 10 cm. In the east of the country we get differences of more than 20 cm compared to the combined solution. This indicates a discrepancy between the deflections of the vertical and GPS/leveling in this region. An explanation for this discrepancy could not be found yet. This area is well covered by astrogeodetic data and by GPS/leveling points. Also the Austrian leveling data are in agreement with our leveling results. But as we see in fig. 8, in most parts of the country the differences to the combined solution are smaller than 5 cm. The mean GPS/leveling residuals of the astrogeodetic solution are about 6 cm. The pure gravimetric solution (fig. 9) also shows a very good agreement with the combined solution in most parts of the country - even better than the astrogeodetic solution. Only at the borders towards Italy there are larger differences of up to 20 cm. In these areas the gravimetric data coverage is rather poor and especially the effect of the Ivrea body is clearly visible. In the east of the country the gravimetric solution shows about the same tendency as the astrogeodetic solution. The gravimetric solution is rather insensitive to changing the parameters of the covariance model. The combined solution of astrogeodetic and gravimetric observations without GPS/leveling (fig. 10) is very close to the official model except for the region in the east, where GPS/leveling does not fit to the other observation types and in some remote areas at the borders where we have GPS/leveling stations. This solution is also very close to the pure astrogeodetic solution except in some areas of a very good coverage with gravity data and poor coverage with deflections of the vertical. Fig. 11 displays the difference between the gravimetric and the astrogeodetic solution. The characteristics show once again the already mentioned problems in the east and the difficulties
Chapter 55 • Comparison of High Precision Geoid Models in Switzerland
in the region of the Ivrea body and in the Southern Alps where the gravity coverage is rather poor. As a summary, it can be said that for more than 90% of the country all the tested solutions are in agreement in the order of 2-3 cm. This is especially the case in the flatter areas in the north and west but also in most regions of the central Alps. The main problems of the geoid determination in Switzerland are found in the mountainous areas in the east and in the south towards the borders with Italy. One special case is the Ivrea zone, where we have larger differences of more than 20 cm between the different solutions. Future local gravity field investigations in Switzerland will have to concentrate on the explanation of the discrepancies between GPS/leveling on the one side and of gravity and deflections of the vertical on the other side in some areas where we still have unexplained effects of
more than 20 cm. A further main problem is the rather rough modeling of the Ivrea body, which disturbs heavily the geoid determination in the southern Switzerland and in northern Italy.
Fig. 9:CHGeo2004 minus gravimetric solution
Fig. 7:CHGeo2004 minus GPS/leveling solution
Fig. 10:CHGeo2004 minus combined gravimetric / astrogeodetic solution
Fig. 8:CHGeo2004 minus astrogeodetic solution
Fig. 11: Gravimetric minus astrogeodetic solution
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Conclusions
The geoid and quasigeoid model CHGeo2004 was calculated by the combination of all available data sets of gravity, deflections of the vertical and GPS/leveling. For the reduction of the observations and for the downward continuation we used the geopotential model EGM96, a digital terrain model with a resolution of 25 meters and a simple density model of the Earth's crust. The method of computation was least squares collocation. The parameters of the covariance function have been chosen to minimize the GPS/leveling residuals in a pure astrogeodetic/gravimetric solution. The accuracy of the calculated model is in the order of 2-3 cm in most parts of the country as it could be verified in comparisons with independent data sets. In some mountainous regions the accuracy is in the order of 5 cm or even 10 cm. The GPS/leveling data was introduced into the adjustment with a very strong weighting. This leads to a geoid model whose general form is fixed by these GPS/leveling measurements. Only in regions without leveling lines gravimetric and astrogeodetic measurements have a major influence. In the original plan it was foreseen to calculate a geoid model out of gravity and vertical deflections only. In this case, the GPS/leveling residuals would have been separated to the individual data sets in an additional adjustment as described in Marti (2001). But this lead to a solution which was very near to the combined solution which we finally chose. Therefore we rejected this more complicated method. The comparison of different individual solutions of pure GPS/leveling, astrogeodetic and gravimetric geoid models showed that in most parts of the country they are in a very good agreement of better than 3 cm. Only in some remote mountainous regions in the south we get larger discrepancies of more than 20 cm. The main reasons for this are usually local weaknesses of the data sets. But there are also other reasons such as the insufficient modeling of the Ivrea body. In the east of the country there also exists a discrepancy between GPS/leveling and the other data sets which could not be explained sufficiently yet. This area will be a
major focus for gravity field investigations in the near future. The calculation of the Swiss Geoid model CHGeo2004 was part of setting up a new consistent national height system that will be used from now on as the reference for all national geodetic work. The geoid model has been released to the surveyors and it is integrated in most geodetic GPS receivers and software sold in Switzerland. With this geoid model, it is possible to obtain orthometric and normal heights that are compatible with leveling better than 1 cm. Unfortunately, cadastral surveying will not change to this new modern height system and we will continue to have to deal with local height transformations (Marti, 2002). References
Denker H., W. Torge (1997): The European gravimetric quasigeoid EGG97. In: Forsberg, Feissel, Dietrich (eds.): Geodesy on the Move, IAG Symposia vol. 119. Denker H., J.-P. Barriot, R. Barzaghi, R. Forsberg, J. Ihde, A. Kenyeres, U. Marti, I.N. Tziavos (2004): Status of the European Gravity and Geoid Project EGGP. In: Jekeli, Bastos, Fernandes (eds.): Gravity, Geoid and Space Missions GGSM 2004, IAG Symposia vol. 129. Hirt Ch., B. Reese, H. Enzlin (2004): On the accuracy of Vertical Deflection Measurements Using the HighPrecision Digital Zenith Camera System TZK2-D. In: Jekeli, Bastos, Femandes (eds.): Gravity, Geoid and Space Missions GGSM 2004, IAG Symposia vol. 129. ihde J., W. Augath, M. Sacher (2001): The Vertical Reference System for Europe. In: Drewes, Dodson, Fortes, Sanches, Sandoval (eds.): Vertical Reference Systems. IAG Symposia vol. 124. Marti U., A. Schlatter (2001): The new height system in Switzerland. In: Drewes, Dodson, Fortes, Sanches, Sandoval (eds.): Vertical Reference Systems. IAG Symposia vol. 124. Marti U. (2001): The way to a Consistent National Height System for Switzerland. In: Adam, Schwarz (eds): Vistas for Geodesy in the New Millennium. IAG Symposia vol. 125. Marti U. (2002): Modeling of Differences of Height Systems in Switzerland. In: Tziavos (ed.): Gravity and Geoid 2002. Proceedings of the 3rd Meeting of the International Gravity and Geoid Commission, Thessaloniki, Greece. Mfiller A., B. Bfirki, H.-G. Kahle (2004): First results from new High-precision Measurements of Deflections of the Vertical in Switzerland. In: Jekeli, Bastos, Fernandes (eds.): Gravity, Geoid and Space Missions GGSM 2004, lAG Symposia vol. 129.
Chapter 56
GOCE: a full-gradient simulated solution in the
space-wise approach F. Migliaccio (1), M. Reguzzoni (2), N. Tselfes (3) (1~DIIAR - Politecnico di Milano - Piazza Leonardo da Vinci, 32 - 20133 Milano - Italy (2~Geophysics of the Lithosphere Dept. - Italian National Institute of Oceanography and Applied Geophysics (OGS) c/o Politecnico di Milano, Polo Regionale di Como - Via Valleggio, 11 - 22100 Como - Italy (3~DIIAR - Politecnico di Milano, Polo Regionale di Como - Via Valleggio, 11 - 22100 Como - Italy GOCE will be the first satellite gradiometric mission with the purpose of estimating the stationary gravitational field to a high accuracy and spatial resolution. The on-board gradiometer will provide measurements of the second derivatives of the gravitational potential along the instrumental axes, which are used together with the satellite tracking data to retrieve the spherical harmonic coefficients of the geopotential model. To this aim, a possible strategy is the so-called space-wise approach, basically exploiting the spatial correlation between data. This requires a pre-processing procedure consisting in filtering the observations in time and then producing grids of second derivatives and potential on a boundary sphere at mean satellite altitude. In the past, this approach was applied only to simulated second radial derivatives. A timeseries of the anomalous potential along the orbit was also used in order to improve the estimate at low frequencies. In this work we aim at integrating the information coming from all the three diagonal components, as measured in the gradiometer reference frame. In particular the performance of the space-wise approach has been tested on the basis of realistic end-to-end simulated data, showing that the proposed method is able to estimate the spherical harmonic coefficients up to degree and order 200. The results have to be evaluated also taking into account the time-length of the available data, namely one month, which is critical for any space-wise approach. Abstract.
Keywords. GOCE mission, satellite gradiometry, space-wise approach.
1 The GOCE mission GOCE (Gravity field and steady-state Ocean Circulation Explorer) is a satellite gradiometry mission designed by ESA (European Space Agency),
the launch being scheduled for November 2006 (ESA, 1999). The main goal of the mission is to determine the stationary part of the gravity field to a high degree of accuracy and spatial resolution; in particular the requirements of the mission are to estimate a global model represented by a spherical harmonic expansion with a commission error of 1 + 2 cm (geoid undulation) and 1 + 2 mgal (gravity anomalies), up to degree and order 200. The main instrument on board the satellite will be a triaxial gradiometer composed by three pairs of electrostatic accelerometers assembled according to orthogonal axes, allowing for the measure of the second derivatives of the potential along the satellite orbit (the so-called gravitational gradients). Another information on the gravity field can be achieved from the tracking of the satellite orbit by means of the GPS receiver, combined with the measure of the non-gravitational forces obtained by the gradiometer common mode accelerations. These observations are complemented by each other: in fact the gravitational gradients mainly provide information on the high degrees of the spherical harmonic expansion (i.e. the high frequency details of the field), while the satellite orbit tracking allows to estimate with better accuracy the low degrees (i.e. the behaviour of the field at low frequencies). In order to analyze the data coming from the GOCE mission, a Consortium has been founded, named EGG-C (European GOCE Gravity Consortium), which comprises Universities and Research Centres in Europe having a long experience in the gravity field estimation by satellite data (Balmino, 2001; Pummel et al., 2004). Three different approaches are under development for the estimation of the gravity field from GOCE observations: the direct approach (Abrikosov and Schwintzer, 2004; Bruinsma et al., 2004), the time-wise approach (Pail and Plank, 2002; Pail et al., 2005) and the space-wise approach (Rummel et al., 1993; Migliaccio et al., 2004a).
384
F.Migliaccio• M. Reguzzoni• N.Tselfes 2 The space-wise
approach
The space-wise approach basically consists in estimating the global model of the gravity field by solving a boundary value problem on a reference sphere having radius equal to the mean satellite altitude. In other words, the spatial correlation of the field is exploited by gridding the observations on a spherical mesh and afterwards applying numerically efficient algorithms of integration or collocation to estimate the coefficients of the spherical harmonic expansion. In more detail, the space-wise approach consists of the steps described below (see also Figure 1). 2.1 Potential e s t i m a t e along orbit
At the first step, the Satellite-to-Satellite Tracking (SST) data are used and the potential is directly estimated along the orbit, via the energy conservation method (Jekeli, 1999; Visser et. al., 2003). 2.2 W i e n e r filter
Afterwards, the gravitational potential data along the orbit and the gradiometric observations (in particular the diagonal components of the gravitational tensor, which are measured with higher accuracy) are transformed to the frequency domain and filtered by a Wiener filter (Papoulis, 1984) along the orbit. Basically, for a certain frequency f, one can write:
W(f)- S, (f)[S,, ( f )
+ S~
(f)]-I
(1)
where W is the Wiener filter, S~ and S V are the signal and noise spectra respectively. Note that in the N-dimensional case, S~ and Sv are NxN matrices, including also cross-spectra. According to this filtering, the spectrum of the estimation error S e can be computed as
S~(f)-S,(f)-S,(f)[S,(f)+ S~(f)]-~S,(f)
(2)
and the corresponding covariance function can be derived by inverse Fourier transform. Applying such a filter (which means working in a time-wise fashion, exploiting the time correlation of data) is to be considered as a data pre-processing in
the space-wise approach: such an operation is necessary because of the strongly coloured noise affecting gradiometric observations (Albertella et al., 2004). The Wiener filter allows to obtain time series with a much lower level of noise with respect to the initial observations; besides, for each observed quantity it allows to compute a proper covariance function of the estimation error along the orbit, to be used in the subsequent data interpolation on the spherical grid. The latter result is particularly important, considering that an accurate prior information on the covariance function of the observation noise is not available. Finally, it has to be remarked that the Gradiometer Reference Frame (GRF) (x,y,z being the instrumental axes) does not coincide with the Local Orbital Reference Frame (LORF) (~: along-track; rl: crosstrack; r: radial), because the FEEP (Field Emission Electric Propulsion) attitude control system is no longer part of the satellite design. Depending on this misalignment, the observed gravitational gradients cannot be considered a time stationary signal and the Wiener filter should be applied to the corresponding quantities in the LORF system. However, a direct rotation cannot be applied to the gravitational tensor, because the large error in the non-diagonal components of the tensor would be propagated to all the other components, increasing their noise level of two or three orders of magnitude. For this reason, a preliminary estimate of the diagonal components is performed in LORF, to be used in the Wiener filter, by ignoring the effects on the rotation due to the non-diagonal terms; these effects will be iteratively corrected (Migliaccio et al., 2004b).
2.3 Data interpolation on a spherical grid
The filtered data, properly transformed back to the time domain, are interpolated on a spherical grid with radius equal to the mean satellite altitude. The potential values along the orbit and the gravitational gradients are treated separately; the latter are predicted in a local East-North-Radial reference frame (e,n,r). The gridding is performed by a least squares collocation algorithm applied to local patches of data (Tscherning, 2005). In this way it is possible to "homogenize" observations which are close in space but far away in time, filtering out long period systematic effects.
Chapter56 2.4 Harmonic analysis on the sphere
W c (f) = I-
Energy i conservation
3.1 The end-to-end data set The test data set has been provided by ESA. It consists of observations spanning the duration of one month, at a sampling rate of 1 second. The gravity gradients in the GRF are based on the EGM96 model (Lemoine et al., 1998), up to degree and order 360, and have been contaminated with heavily coloured noise. The satellite orbit, including positions and velocities, was simulated without noise, again using the EGM96 model up to degree and order 50. The misalignment between GRF and LORF (in other words, the angular attitude of the satellite) is described by means of quaternions.
hi
Final model
along orbit
Legend
q Wiener filter
complementary j Wiener filter
(3)
3 The s i m u l a t i o n
There are two reasons which explain the need for the computation of synthesized values of the observations (potential data and gravitational gradients) along the orbit, starting from the previously estimated coefficients: first, one has to recover the effects of the terms which were neglected in the rotation between the GRF and the LORF (due to the ignored non-diagonal components of the tensor); secondly, one has to
Data
W(f)
where I is the identity matrix. After taking into account both corrections (which result in a significant decrease of the estimation error along the orbit) the gridding and harmonic analysis procedures must be iterated until convergence. Based on numerical tests, the iterative scheme, reported in Figure 1, is known to converge rapidly (Migliaccio et al., 2004a).
2.5 Data synthesis along orbit
SGG data
in the Space-WiseApproach
recover the signal lost in the initial filtering procedure by means of a complementary Wiener filter W c ( f ) , defined as
Using the gridded values, it is possible to apply a harmonic analysis operator to estimate the coefficients of the spherical harmonic expansion of the gravitational potential. Two solutions are available: - a numerically efficient collocation algorithm, known as Fast Spherical Collocation (FSC) (Sans6 and Tscherning, 2003), which exploits the statistical prior knowledge of the gravitational field in terms of degree variances (i.e. in terms of the covariance function of the field); - an integration method (INT) (Migliaccio and Sans6, 1989), based on the orthogonality property of the spherical harmonic functions. In the flame of EGG-C, the FSC algorithm is considered the baseline solution for the space-wise approach, while integration is used as check solution.
SST data
• GOCE:a Full-Gradient Simulated Solution
Hamlonic analysis
LORF/GRF correction
I
! iI ! I ......
Data gridding I
SST = Satellite to Satellite Tracking
j g,.,=
SGG = Satellite Gravity-Gradiometry FFT = Fast Fourier Transform LORF- Local Orbital Reference Frame GRF - Gradiometer Reference Frame
Fig. 1 Scheme of the space-wise approach to GOCE data analysis.
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Among simulated data, also differential and common mode accelerations between pairs of accelerometers constituting the gradiometer are provided; in particular, common mode accelerations (when properly calibrated) give a direct measure of non-gravitational forces acting on the satellite and can be used in the analysis of orbit tracking data. To conclude the description of the data used in this experiment, two remarks are in order. First of all, only the diagonal components T××, Tyy and Tzz of the gravitational tensor were used, since the other measurements are affected by a much higher noise, especially T×y and Tyz. Secondly, the effect of the coefficients up to degree 24 was subtracted both from the potential and from the gravitational gradients. 3.2 Results
of the simulation
Regarding the first step of the space-wise approach, in this experiment the energy conservation
algorithm was not used to estimate the potential along the orbit. Instead, these data were directly synthesized starting from EGM96 data to degree and order 360, consistently with the gradiometric data. A white noise with standard deviation of 0.3 m2/s 2 (corresponding to an orbital error of about 3 cm) was added to the time series of the potential data. Regarding the filtering procedure inside the spacewise approach, in Figure 2 the Power Spectral Densities (PSD) of the signals to be treated and the corresponding simulated noise are shown. The presence of low-frequency peaks in the spectra of the gradiometric data noise clearly reveals that it has non stationary characteristics, thus making the subsequent filtering procedure non optimal. Besides, the signal and noise are not completely uncorrelated. For all the above reasons, it was decided not to apply a unique Wiener filter to the data, i.e. a unique multidimensional filter based on the full covariance structure among the gradients and the potential.
10 2
10 2
10 o
10 0
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~
10-2
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10-2
10 .4
~
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10 -6
10 -6
I
10.4
10 -2
10-81
10 -8
10 -6
i I I I
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10-2
10 0
10 -6
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10 4
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I I
~
10.4 . . . . . . . . .
10 0
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~ ~
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~-
I I I
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I
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10.4
10 0
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10 -4
[Hz] Fig. 2 Power Spectral Densities (PSD) of the simulated signals (in black) and noise (in gray).
10 -2
[Hz]
10 0
Chapter56 • GOCE:a Full-GradientSimulated Solution in the Space-WiseApproach
Only the second radial derivatives Trr and the potential T were jointly filtered, in order to exploit the complementary spectral characteristics of the two signals: in fact, the data of the potential T especially carry information on the low frequencies (where the signal to noise ratio is high), while Trr is less noisy at high frequencies. The other two diagonal components of the gravitational tensor were instead filtered separately, for each one using a one-dimensional Wiener filter. The final result of the filtering procedure was a noticeable reduction of the error along the orbit, as shown in Table 1. As an example, Figure 3 shows the empirical error covariance function of Trr (computed from the residuals after the filtering with respect to the true signal coming from EGM96) and the corresponding predicted covariance (Eq. 2).
Table 1. The root mean square (r.m.s.) error of the data along the orbit, before and after the Wiener Filtering (WF).
r.m.s.
before WF after WF
T~ [mE]
Tnn [mE]
Trr [mE]
T [m2/s2]
269.2 4.369
246.8 24.428
548.0 9.968
0.299 0.088
100
80
00X 40
I I I
I I
Since the second derivative Tnn in the cross-track direction has a much higher estimation error than Trr and T~ (see again Table 1), it was decided not to consider it in the gridding procedure; in fact, its use would not bear any additional information. The other two components Trr and T~ were jointly treated in order to produce a spherical grid of values of Trr and Tnn (n = north). The potential T was separately interpolated on the same spherical grid. Note that point values were produced and polar caps were left empty. It has to be remarked that in order to reduce the computational time, all filtered data were under-sampled with a rate of 5 seconds (which is not critical, due to the strong time correlation of both the signal and the noise). Table 2 shows the global root mean square errors for the different grids which were produced; the prediction of the error standard deviations for the second radial derivatives Trr by using the collocation algorithm is reported in Figure 4. It can be easily seen that the predicted errors are about one order of magnitude smaller than the real one. In other words, the error estimates provided by collocation can be considered quite optimistic and need to be better calibrated; one possible solution could be to use locally adapted signal covariance functions (Arabelos and Tscherning, 2003). In any case, the errors on the grid are smaller than the ones along the orbit, especially at high latitudes where the observations are denser (except polar caps). Once the data had been gridded, it was possible to estimate the coefficients of the spherical harmonic expansion either by using Fast Spherical Collocation (FSC) or an integration algorithm (INT).
1
\
.....
. . . . .
4-
. . . . .
/
I
20
1.2
40
1.1
I
I I
0
20'00
7""--'I I
40'00
6000
8000
60
1
"=
80
0.9
_~
1oo
0.8
120
0.7
140
0.6
160
0.5
10000
Is] Fig. 3 Empirical (in black) and predicted (in gray) covariance function of the Trr estimation error.
Coming to the gridding, this step was implemented by applying collocation to a moving window of 10°x 10 ° (2 ° overlapping). The final grid had a resolution of 0.72 ° x 0.72 °, which divides the latitude interval covered by data into an almost integer number of parts (there are polar gaps of about 6.5°).
6
-150-100-50
0 50 longitude
100 150
[mE]
Fig. 4 Predicted error standard deviations of the second radial derivatives Trr, computed cell by cell by collocation.
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Table 2. The root mean square (r.m.s.) error of the data estimated on the spherical grid.
r.m.s.
Tnn [mE]
Trr [mE]
T [m2/s2]
gridding
3.04
8.41
0.0675
In the case of integration, only the T and Tr~ grids were used, obtaining two different estimates of the harmonic coefficients which were afterwards combined using weights inversely proportional to the error degree variances. On the contrary, the FSC algorithm allows to use all the available data, included the Tun grid, in a unique step. The improvement due to the use of Tnn is shown in Figure 5.
effects of the two corrections are shown, either separately or jointly applied to the different observations. It can be seen that the T~ component mainly benefits from the rotation correction between GRF and LORF, while the complementary Wiener filter (mainly affecting the low frequencies) has a larger effect on the other two components of the tensor (see signal to noise ratio at low frequencies in Figure 2).
Table 3. The root mean square (r.m.s.) error of the data along the orbit, after the rotation correction between LORF and GRF (ROT) and after the application of the complementary Wiener Filter (wFc). The two effects are evaluated either individually or jointly. r.m.s.
10 -14
i I
I
EGM96 II ~._ I 1 0 -16
~
. . . .
50
Trr [mE]
T [m2/s2]
after WF
4.369
24.428
9.968
0.088
+ROT
2.050
24.132
9.680
0.088
+WF c
4.106
11.330
4.571
0.054
+ROT +WF c
1.334
10.447
3.686
0.054
lI FSC2 I I 4-
. . . . .
-1
. . . . .
10 .20
0
Tnn [mE]
FSC1
~ 10 -18
T~ [mE]
100 150 [ degree ]
200
250
Fig. 5 Error degree variances of the spherical harmonic
coefficients estimated at the first iteration by Fast Spherical Collocation using T and Trr only (FSC1) and using T, Trr and Tnn (FSC2). The reference model is EGM96.
Both methods were applied in this experiment, giving rise to very similar results as displayed in Figure 6 in terms of empirical error degree variances, i.e. the power of the differences between predicted and true coefficients, degree by degree. The coefficients estimated by FSC were used in the subsequent iterations, since FSC is the baseline solution in the space-wise approach. After obtaining these first coefficients estimates, it was possible to compute both the rotation correction between GRF and LORF and the correction based on the complementary Wiener filter. In Table 3 the
At this point it was possible to perform again the gridding and harmonic analysis steps, obtaining a new estimate of the potential coefficients (see Figure 7). The maximum resolution of degree 200 was reached by both harmonic analysis methods, however it has to be remarked that FSC has a better performance at low degrees, while integration works better at high degrees (see also Migliaccio et al., 2004c). This consideration points towards the study of a combined solution. The whole procedure was iteratively repeated until convergence. The errors along orbit for the different observed quantities at each iteration are reported in Table 4; the improvement was considerable at the first iteration, while it became negligible already at the second step, thus showing that the method reached convergence (in general, two or three iterations are sufficient). Regarding the accuracy of the estimated model, the total error (except polar caps) in terms of gravity anomalies is equal to 5 mgal up to degree 200 and shows higher values in the areas where the gravity field is less smooth (like in the Andes or Himalayas). Although this result does not yet satisfy the mission requirements, one has to consider that the
Chapter 56 • GOCE: a Full-Gradient Simulated Solution in the Space-Wise Approach
simulated data only cover a time span of one month and that the observations have been under-sampled by a factor 5 in the gridding step. As a consequence, a remarkable improvement can be expected both in terms of accuracy and spatial resolution when data become available covering all the 12 months of the mission life time.
4
10 -14 I
10 -16 . . . . . .
I
I I
I
o
,~
-~
I I
I I
-_
2;0
1;o
250
[ degree ] Fig. 6 Error degree variances of the spherical harmonic coefficients estimated at the first iteration by Fast Spherical Collocation (FSC) and by integration (INT). The reference model is EGM96.
10 "14
I I
I
,-'~l
EGM96 -i ~ , , ~ i
10 -16 . . . . . .
-
I I
I
I
i I
i I i
i
- - - l -I . . . . .
-~I . . . .
i
i
I
I
~
7 I i
I,f
INTl.
I.
j" r
i i
------
~
Ir
~"
_firI FSC
[mE]
T [mZ/s2]
iteration 0 iteration 1 iteration 2
4.369 1.334 1.323
24.428 10.447 10.406
9.968 3.686 3.618
0.088 0.054 0.054
Conclusions
and
future
Trr
work
-
~
- -i-
7,A~-,-r"
,~l
--i--
i i
i i
i
i
i
'
'
'
150
200
I I
I I I
50
100
--
Nevertheless, some aspects of the approach were not completely tested (for example, the optimal dimension of the moving window in the gridding step) due to the short time span covered by the simulated data. Regarding the future work, the first action to be taken will be the introduction in the space-wise processing chain of the energy conservation algorithm for the estimate of the potential, also exploiting the information of the common mode accelerations. Besides, in the next computations also the second derivative Txz will be used, being, among the three non-diagonal components of the gravitational tensor, the one measured with highest accuracy.
I. --I
i 0"if
•
10 .20 0
t ~
Tqq [mE]
The simulation was valuable to assess the methodology and the software implementing it, and gave the opportunity to identify ways to obtain improvements.
50
~a
T~ [mE]
The solution based on the simulation presented here confirms that the space-wise approach to the GOCE data analysis is able to estimate a global model of the gravity field in terms of a spherical harmonics expansion up to degree 200.
FSC
10 .20
r.m.s.
I
EGM96
10 -18 _ _ _
10 -18
Table 4. The root mean square (r.m.s.) error of the data along the orbit, after each iteration of the space-wise scheme.
I I I
I I I
250
Finally, particular care will have to be devoted to the gridding procedure, for example by applying an a-priori statistical homogenization of the observed field (Migliaccio et al., 2004d) and by locally calibrating the covariance functions to be used by the collocation algorithm.
[ degree ]
Fig. 7 Error degree variances of the spherical harmonic coefficients estimated at the second iteration by Fast Spherical Collocation (FSC) and by integration (INT). The reference model is EGM96.
One open problem concerns the study of the possibilities offered in the frame of the space-wise approach by a combined solution coming from the results of the Fast Spherical Collocation and of the harmonic analysis by integration.
389
390
F. Migliaccio • M. Reguzzoni • N. Tselfes
Acknowledgement This work was performed under ESA contract No. 18308/04/NL/NM (GOCE High-level Processing Facility). The authors wish to thank Prof. Sans6 for the fruitful discussions while carrying out the experiment. Many thanks are also due to Prof. Tscherning and Dr. Veicherts for providing the results of data gridding and Fast Spherical Collocation.
References Abrikosov, O., and P. Schwintzer (2004). Recovery of the Earth's gravity field from GOCE satellite gravity gradiometry: a case study. In: Proc. of 2 nd International GOCE User Workshop. Frascati, italy, March 8-10 2004. Albertella, A., F. Migliaccio, M. Reguzzoni and F. Sans6 (2004). Wiener filters and collocation in satellite gradiometry, in: International Association of Geodesy Symposia, '5thHotine-Marussi Symposium on Mathematical Geodesy', F. Sans6 (ed), vol. 127, Springer-Verlag, Berlin, pp. 32-38. Arabelos, D., and C.C. Tscherning (2003). Globally covering a-priori regional gravity covariance models. Advances in Geosciences, 1, pp. 143-147. Balmino, G. (2001). The European GOCE Gravity Consortium (EGG-C). In: Proc. of 1~'~International GOCE Workshop. ESA-ESTEC, Noordwijk, The Netherlands, April 23-24 2001. Bruinsma, S., J.C. Marty and G. Balmino (1997). Numerical simulation of the gravity field recovery from GOCE mission data. in: Proc. of 2 nd International GOCE User Workshop. Frascati, Italy, March 8-10 2004. ESA (1999). Gravity Field and Steady-State Ocean Circulation Mission. ESA SP-1233 (1). ESA Publication Division, c/o ESTEC, Noordwijk, The Netherlands. Jekeli, C. (1999). The determination of gravitational potential differences from satellite-to-satellite tracking. Celestial Mechanics and Dynamical Astronomy, 75, pp. 85-101. Lemoine, F.G., S.C. Kenyon, J.K. Factor, R.G. Trimmer, N.K. Pavlis, D.S. Chinn, C.M. Cox, S.M. Klosko, S.B. Luthcke, M.H. Torrence, Y.M. Wang, R.G. Williamson, E.C. Pavlis, R.H. Rapp and T.R. Olson (1998). The development of the joint NASA GSFC and NIMA geopotential model EGM96. NASA/TP-1998-206861, Goddard Space Flight Center, Greenbelt, Maryland. Migliaccio, F., and F. Sans6 (1989). Data processing for the Aristoteles mission. In: Proc. of the ltalian Workshop on the European Solid-Earth Mission Aristoteles. Trevi, Italy, May 30-31 1989, pp. 91-123.
Migliaccio, F., M. Reguzzoni, and F. Sans6 (2004a). Space-wise approach to satellite gravity field determination in the presence of coloured noise. Journal of Geodesy, 78, pp. 304-313. Migliaccio, F., M. Reguzzoni, F. Sans6 and P. Zatelli (2004b). GOCE: dealing with large attitude variations in the conceptual structure of the space-wise approach. In: Proc. of 2 nd International GOCE User Workshop. Frascati, Italy, March 8-10 2004. Migliaccio, F., M. Reguzzoni, F. Sans6 and C.C. Tscherning (2004c). An enhanced space-wise simulation for GOCE data reduction. In: Proc. of 2"d International GOCE User Workshop. Frascati, Italy, March 8-10 2004. Migliaccio, F., M. Reguzzoni, F. Sans6 and C.C. Tscherning (2004d). The performance of the space-wise approach to GOCE data analysis, when statistical homogenization is applied. Newton's Bulletin, 2, pp. 60-65. Pail, R., and G. Plank (2002). Assessment of three numerical solution strategies for gravity field recovery from GOCE satellite gravity gradiometry implemented on a parallel platform. Journal of Geodesy, 76, pp. 462-474. Pail, R., W.D. Schuh and M. Wermuth (2005). GOCE Gravity Field Processing. In: International Association of Geodesy Symposia, 'Gravity, Geoid and Space Missions', C. Jekeli, L. Bastos and J. Fernandes (eds), vol. 129, Springer-Verlag, Berlin, pp. 36-41. Papoulis, A. (1984). Signal analysis. McGraw Hill, New York. Rummel, R., T. Gruber and R. Koop (2004). High Level Processing Facility for GOCE: Products and Processing Strategy. In: Proc. of 2 nd International GOCE User Workshop. Frascati, Italy, March 8-10 2004. Rummel, R., M. van Gelderen, R. Koop, E. Schrama, F. Sans6, M. Brovelli, F. Migliaccio and F. Sacerdote (1993). Spherical harmonic analysis of satellite gradiometry. Netherlands Geodetic Commission Publications on Geodesy, New Series, N. 39. Sans6, F., and C.C. Tscherning (2003). Fast Spherical Collocation: theory and examples. Journal of Geodesy, 77, pp. 101-112. Tscherning, C.C. (2005). Testing frame transformation, gridding and filtering of GOCE gradiometer data by Least-Squares Collocation using simulated data. In: International Association of Geodesy Symposia, 'A window on the Future of Geodesy', F. Sans6 (ed), vol. 128, Springer-Verlag, Berlin, pp. 277-282. Visser, P.N.A.M., N. Sneeuw and C. Gerlach (2003). Energy integral method for gravity field determination from satellite orbit coordinates. Journal of Geodesy, 77, pp. 207-216.
Chapter 57
The determination of the effect of topographic masses on the second derivatives of gravity
potential using various methods
Sz. Rdzsa, Gy. T6th Department of Geodesy and Surveying, Budapest University of Technology and Economics, H-1521 Budapest, P.O. Box 91, Hungary, e-mail: [email protected]
Abstract In the space gradiometry the determination of the effect of topographic masses is crucial for the validation and downward continuation of the gravitational signal. This paper focuses on the determination of the effect of topographic masses on the second derivatives of the potential. During the investigation two methods are compared to each other. The first method is the direct numeric integration using planar approximation and mass prism topographic model, while the second one is the application of tesseroids. Both of the techniques are investigated over Europe. The application of the tesseroids seem to have many advantages over the first method. The effect of the topography is computed using the tesseroids for the whole globe, too. In order to do this, the ETOPO5 digital elevation model has been used. The results show that the effect of topography is significant on the altitude of the LEOs, reaching the level of 10 E6tv6s in all gravity gradients.
In the last years many papers have been published, which deal with topographic effects on the second derivatives of Earth's gravity potential (e.g. W i l d - Heck, 2004, Heck- Wild, 2005, CsapdPapp, 2000, R d z s a - Tdth, 2005, etc.) however most of the papers focus on the second radial derivatives of the potential only. In this paper we investigate the effect of spherical and planar approximation using two different methods. Moreover the topographic effects on all of the gravity gradients are computed.
2 Methods
2.1 Gravity gradients of the rectangular prism The equations for determining various parameters of the gravity field induced by the right rectangular prisms are given in (Nagy et al, 2000). P ram,
X1
1 Introduction Due to the emerging of satellite gravity gradiometry in the last decade the determination of the topographic effects on the second derivatives of Earth's gravity potential becomes important. While the sufficient accuracy can be achieved with planar approximation in case of torsion balance measurements, the application of spherical or even ellipsoidal approximation seems to be necessary for satellite gradiometry.
X2
iiiiiiiiiii:: ;;i...... i:~:i .... i .....
i. z .'.
!. 1 ]
. . . . . .
;
.."
:
-._" .
j" .
.
.
.
-
s" . . . . . . . . . . . . . .
2
Fig. 1. The mass prism and the applied coordinate system for planar approximation
X
392
Sz. R6zsa.Gy.Toth
In case of planar approximation, the gravity gradients can be computed using the following formulae: Y2
Vxx -
G
-tan
-1 Y~ X1x2
(1) Yl
Vxy - G ln(z + r)X2xl yIY2
(2)
Zlz~ Vxz - G ln(y
+
v
(3)
r) x2 Y2 Xl Yl Z1
Vyy -
G
yr _tan_ 1 zx
X2 Y2
z2
(4)
~ ~1~1 ~2 -
- tan -1 x y
x2
r
r - / x 2 -k-y2 + z2 .
is
Yl
(7)
Vxz = - 3 / 3Gm (e r "ez)(e~ .ez)
(9)
[3(e
-el) 2 - 1
]
(10)
Gm
/3 (e ~ -e;)(e~ .e;)
(11)
z2
z1
the
The
]
(8)
Vy~ - - 3 (6)
Xl
1-6
Y2
z
-1
°el ~ e @ ° e 1 ~
(5)
~i
Eq
xx Gm Vxy _ 3 ~ ( e
z3
Xl Yl Z1
In
=~m[3 ( e ~ . e ~ )
Vyy-Gm
c ln( +r)
Vzz-C
the tesseroid potential and its derivatives up to second order. These formulae, however, are not yet published. If we introduce as a first approximation to the gravity gradients of the tesseroid the point mass m placed at its mid-point Q= (A '= (21+22)/2, (/9'= ((,o1+(,o2)/2, r ' = (r1+r2)/2), the six elements of the symmetric gravity gradient tensor in the local x=East, y=North, z=Up system are
Euclidean definition
of
distance further
Vzz-13Gm [3 (er .e t )2-1]
(12)
Here G denotes the gravitational constant and l, is the distance between P and Q; e~, %, er are unit vectors of the moving frame attached at point P (Fig. 2).
variables and the applied coordinate system can be seen on Fig. 1. er
2.1 Gravity gradients of the tesseroid Newton's integral can be evaluated by discretised numerical integration dividing the topographic masses into spherical volume elements (Kuhn and Seitz, 2004). In a spherical approximation, this is a natural way to express the topographic masses that are given by a DEM with respect to a geographical coordinate system. The spherical volume element, illustrated in Fig. 2, is bounded by two cones corresponding to the spherical latitudes ¢1 and ¢2, two meridians, defined by the spherical longitudes A1 and 22 and two concentric spheres defined by the radii rl and r2. For such a geometrical body, the term "tesseroid" was introduced by Anderson (1976). Seitz et al (2003) proposed a method for an efficient calculation of
/
/
I=r-r'
r
e'
/
r,
'~~5 ~
_~
¢ /
m = p r'2cos(p'A~'/~'
-',
Fig. 2. Geometry of the spherical tesseroid placed at its mid-point at Q
Chapter 57 • The Determination of the Effect of Topographic Masses on the Second Derivatives of Gravity Potential Using Various Methods
1 0
1
10.2 o o)
¢ 104
=>
. . . . . . . (.,::1)
1 0
.4
. . . . . . . . . . . . . . .
.,. . . . . . . . . . . . . . .
.., . . . .
' oi i i i i i i i i i i i l lO °
i
0
i i
1000 8000 27000 number of subdivisions
64000
Fig. 3. Dependence of the percent relative error of gravity gradients Vzz and Vyyon the number of subdivisions of the tesseroid. A tesseroid of size 5' x5'x2 km was placed at height 0, whereas the computation height was 250 km.
2 1 0
: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : . . . . . . . . . . . . |. . . . . . . . . . . ~ . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |. . . . . . . . . . . 4 . . . . . . . . . . . + . . . . . . . . . . . ~ . . . . . . . . . . .
.
. . . . . . . . . . . .
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d . . . . . . . . . . .
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. . . . . . . . . . .
~
. . . . . . . . . . .
tesseroid. The first-order series expansion corresponds to point mass approximation. Terms of second order cancel due to symmetry (if the Taylor point is chosen in the centre of the tesseroid). Therefore third-order terms represent corrections to the point mass approximation (1)-(6). The validity of the point mass approximation was investigated by computing gravity gradients at 250 km height of a tesseroid of size 5'x5'x2 km placed at latitude 45 ° on the surface of the mean Earth sphere (radius R = 6371 km). The reference gravity gradients were computed for the comparison by subdividing the tesseroid into n 3 mass elements (point masses). Figure 3 shows the dependence of the computed gravity gradients on the number of subdivisions of the tesseroid. It seems that relative error of the point approximation is at the level of 0.05%, so it can be acceptable for gravity gradient computation at the altitude of the forthcoming GOCE satellite mission. At lower altitudes or for higher accuracy requirements, however, the thirdorder approximation may be necessary, as it can be seen from Figure 4., that relative error of the point mass approximation reaches the level of 1% at altitude of about 60 km.
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiill ii 101
o
._> ~
~ 10
0
c--
3 Computations
u L.. c5
-1
10 . . . . . . . . . . . .
|. . . . . . . . . . .
4 . . . . . . . . . . .
+ . . . . . . . . . . .
)- . . . . . . . . . . .
. . . . . . . . . . . .
, ,. . . . . . . . . . .
, ~ . . . . . . . . . . .
, ° . . . . . . . . . . .
, ~. . . . . . . . . . . .
10 -2 0
1
50
100
1
1
150
200
250
height (kin) Fig. 4 Relative approximation errors of gravity gradients Vzz and Vyy from point mass modeling of a tesseroid of size 5 ' x 5 ' x 2 k m as function of the altitude of the computation point. The reference values of gravity gradients were approximated by 403 = 64000 subdivisions of the tesseroid.
The formulae (7)-(12) are written in a coordinate-flee manner, since they contain only scalar products and other quantities that are invariant to translations and rotations of the coordinate system. This is what was expected for, since they are elements of a second-rank tensor. Also, the validity of the Laplace equation Vxx + Vyy + Vzz can immediately be verified from (1)-(6). In order to evaluate the potential of a tesseroid, Seitz et al. (2003) proposed to expand the integrand into Taylor series around the mid-point Q of the
Two test areas have been chosen for the investigations. The first one covers Central Europe (5<X<25, 40>q0>60), while the second one covers the whole globe. In order to be able to compute the effect of topography on the vertical gravity gradients, the ETOPO5 elevation model has been used. ETOPO5 has the advantage that it contains elevation as well as bathymetric information, which is important for global computations. In Section 2 it was demonstrated that the effect of the tesseroids can be approximated by a simple mass point in the center of the tesseroid with sufficient accuracy. Therefore this strategy has been used to compute the effect of topography on the various gravity gradients in case of spherical approximation. This solution led to much less computational time than using planar approximation and the prism model. All computations have been carried out using the altitude of the GOCE orbit (250 kin). Therefore all the results presented in this paper refer to this altitude.
393
394
Sz. R6zsa. Gy. Toth
To decrease the edge effect in the first study area (Europe), the elevation model has been used for a larger area ((0<)~<30, 30>q0>70)). After computing the total topographic effect on all of the gravity gradients for the aforementioned area, the results of the spherical mass point and planar mass prism approximation has been compared to each other. The statistics of this comparison can be found in Table 1. The topographic effect on the second radial derivatives of the Earth's potential using the spherical and planar approximation can be found in Fig. 5 and 6 respectively, while the difference of the two data sets can be seen in Fig. 7. The results show that the planar approximation using the mass prism model is in good agreement with the spherical approximation. However it must be noted that the numeric evaluation of the mass point model in spherical approximation is much more effective than the numeric evaluation of the mass prism formulae. After the comparison of the two methods, the spherical approximation and the tesseroid model have been chosen to compute the effect of global topography on the second derivatives of the Earth's potential. In order to decrease the computational time the resolution of the ETOPO5 has been decreased to 1 degree. The statistical properties of the total topographic effects on the gravity gradients can be found in Table 2. The statistical properties show that the total topographic effect is not negligible for any of the gravity gradients. The effect reaches even the level of 5-10 E6tv6s. The topographic effects can be seen on Fig. 8.
'~'~.. ',.
,...-".,
"7,'-~. ; ~ "'. "
-
, 2
.,,
I
1
0
-1 -2 -3
,~
;~"
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. " : "o " , ' , : '
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.;
% /
•
•
,'~'-
d.--x
"
:
.,.
"o'V,
~ . --. "
,
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60[~j ..._~~.~p~,.;_. .. ; ,,\, ":.~ ,~.~_~.,... _
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2. The statistical properties of the total topographic effect on the gravity gradients. Table
Min -90
Max 33
Mean 0
Std. dev 28
55
20
Table 1. The statistics of the differences of the
topographic effect on Vzz using spherical and planar -60
approximation(units are in mE))
-80 -100
Min
Max
Mean
Std. dev.
Vxx Vxy Vxz Vyy Vyz Vzz
-7.75 -4.14 -7.11 -5.40 -8.68 -7.26
4.94 2.80 10.30 5.26 7.49 7.84
0.48 0.00 0.00 0.67 -0.19 -1.15
1.53 0.65 1.56 1.60 1.85 2.75
Fig. 7. The difference of the topographic effects on Vzz using planar and spherical approximation (units are in mE)
Chapter
57 • The Determination
of the Effect of Topographic
Masses on the Second Derivatives
of Gravity Potential
Using Various Methods
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Sz.R6zsa.Gy.Toth 4 Conclusions
and Outlook
The results of the investigations showed that the effect of the tesseroid can be approximated by a simple mass point placed in the center of the tesseroid. This approximation cause a relative error of less than 0,05 percent at the altitude of the GOCE orbit. Moreover the computations using the spherical approximation and mass point topographic model are much faster than the planar approximation and mass prism model. The reason for this is that the formulae (1)-(6) have to be evaluated 8 times to compute the effect generated by one prism. The comparison of the two methods gave nearly identical results for the area of Central Europe. The two solutions agreed on the level of 28 mE for the second radial derivate in terms of standard deviation. Summarizing the results, we could state that for larger study areas the mass-point approximation provides better results, while for local investigations the planar mass prism model can be suitable for computing the effect of topography on the gravity gradients. The results showed that the effect of topography can not be neglected in GOCE gravity gradient observations. The effects reached the level of 5-10 E6tv6s for all of the gravity gradients. However it must be noted that all of the results contain the effect of the whole topography without isostatic compensation. Other aspects of the topographic effect should also be investigated in the future. For instance the density information on the topographic masses can be easily incorporated in both of the applied methods.
Acknowledgements Our investigations are supported by the National Scientific Research Fund (OTKA T-046718) and the Jfinos Bolyai Research Grant of the Hungarian Academy of Sciences.
References Anderson EG (1976): The effect of topography on solutions of Stokes's problem. UNISURV Report S 14, University of New South Wales, Kensington, Australia, pp. 252. Csap6, G - Papp, G (2000): Measuring and Modelling of the vertical gradient of gravity- Hungarian examples (in Hungarian, Geomatikai K6zlemdnyek III. pp. 109123.) Heck, B., Wild, F. (2005) Topographic-Isostatic Reductions in Satellite Gravity Gradiometry Based on a Generalized Condensation Model. Proceedings of the IUGG 23 rd General Assembly, lAG Symposia Vol 128. pp 294-300. Kuhn, M. Seitz, K. (2004) Comparison of Newton's Integral in the Space and Frequency Domains. Sapporo Proceedings, Sapporo. lAG Symposia Vol XXX, Springer Verlag pp. 386-391. Nagy, D., Papp, G., Benedek, J. (2000): The gravitational potential and its derivatives for the prism. Journal of Geodesy Vol 74. No. 7-8., pp. 552-560. Rdzsa, Sz., T6th, Gy. (2005) Prediction of Vertical Gravity Gradients Using Gravity and Elevation Data. Proceedings of the IUGG 23 rd General Assembly, lAG Symposia Vol 128. pp 344-350. Seitz, K. Wild, F. Heck, B. (2003) Efficient calculation of the tesseroid potential and its derivatives up to second order. Poster presented at the XXXth General Assembly of the IUGG. Sapporo, Japan. Wild, F., Heck, B. (2004) Effects of Topographic Isostatic Masses in Satellite Gravity Gradiometry. Proceedings of the Second International GOCE Workshop, ESAESR1N, Frascati/Italy, March 8-10, 2004.
Chapter 58
Density Effects on Rudzki, RTM and AiryHeiskanen Reductions in Gravimetric Geoid Determination S. Bajracharya, M.G. Sideris Department of Geomatics Engineering The University of Calgary, 2500 University Drive N.W., Calgary, Alberta, T2N 1N4, Canada E-mail: [email protected]; fax: 403-284-1980 Abstract. This paper investigates the density effects on geoid determination using Rudzki inversion gravimetric reduction scheme, residual terrain model (RTM) and Airy-Heiskanen topographic-isostatic (AH) method. The direct topographical density effect (DTDE) on gravity and geoid for Rudzki reduction is studied. The DTDE on gravity and geoid and the density effects on RTM restored effect and Bouguer correction term are studied for RTM reduction method. The DTDE on gravity and geoid and the indirect density effect ODE) on geoid for AH scheme is studied. A rugged area in the Canadian Rockies bounded by latitude between 49°N and 54°N and longitude between 236°E and 246°E is selected to carry out numerical tests. The lateral density information, which is available in the form of digital density model (DDM) of 30" grid resolution of the test area, is used for this study along with the same resolution of digital terrain model (DTM). The total density effects on geoid for Rudzki inversion scheme, RTM reduction method and AH scheme are as much as 8 cm, 15 cm, and 26 cm respectively. Our results indicate that the actual density information is important for precise geoid determination with centimtre accuracy. Keywords. Rudzki, RTM, AH, variable density,
terrain effects 1 Introduction Real density information is required for every gravimetric terrain reduction method, which is chosen to remove all topographical masses above the geoid as required by Stokes's formula for geoid determination. The gravimetric reduction method can be either the removal of topographical masses alone (Bouguer method), or with compensated masses (Airy-Heiskanen method, Pratt-Hayford method, etc.) or with condensed masses (Helmert's first or second condensation method) or with inverted masses
(Rudzki method) or with reference masses (RTM method). The unavailability of actual density information in the past restricted the geodetic community to use variable density in gravimetric reduction methods for geoid determination. The challenge for centimeter geoid determination in recent years has forced geodetic community to study density effects (among other aspects) on reduction methods if the actual density information is available. The recent studies on density effects (Tziavos et al., 1996; Huang et al., 2000; Tziavos and Featherstone, 2000; Bajracharya et al., 2002) are based on Helmert's second method of condensation. Ktihtrelber (1998) and Kuhn (2003) have studied the density effects on RTM and topographic-isostatic models, respectively. The main purpose of this paper is to investigate the density effects on Rudzki reduction. This reduction method does not require the computation of indirect effects on geoid (which is regarded as a major advantage compared to other methods), nor does the indirect density effects. Thus the total density effect on geoid for this reduction scheme is merely due to DTDE on gravity. This paper also studies the density effects using RTM and Airy-Heiskanen topographic isostatic methods in the test area of Canadian Rockies. The density effects on gravity and geoid, on restored terrain effect, and on Bouguer correction term (required to convert the quasigeoid to geoid) are studied for RTM method. The DTDE on gravity and geoid and indirect effect on geoid are studied for AH scheme. The three dimensional density model is a must to strictly represent the real density distribution of crustal masses. However, only two dimensional digital density models are available in some parts of the world. The use of two dimensional density information assumes that its vertical variation is constant.
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S. Bajracharya • M. G. Sideris
2 Computational formulas
IDE. The first term, DTDE, in equation (2) can be formulated as
The total density effects on geoid using any gravimetric reduction method can be expressed as"
ANDTDE - 4rc7~s~f(AgF-AgT(c°n) - AgGM)S(~)d~ 4W
ANTDE = NCon - NDen
(1)
R
R
R
ff(AgF - AgT(den) - AgGM)S(w)do - 4rc7 R
off(AgT(con)- AgT(den))S(v)des= ---4rc7c~SAgDTDE I S(V)des where ANTDE, represents the total density effect on geoid. Ncon and NDen are total geoid using constant and variable density, respectively. The following formula can be obtained after introducing three components of total geoid (the residual geoid, the indirect effect on geoid, and the long wavelength part of geoid) using constant and variable density in equation (1): ANTDE = (NAg(con) - NAg(den)) + (Nind(con) - Nind(den)) = ANDTDE + ANIDE
(2) where NAg(con) and NAg(den)are residual geoid using constant and variable density. Nind(con) and Nind(den) are the indirect effects on geoid using constant and real density. The reference gravity field is computed from the EGM96 geopotential model (which is nominally based on free air anomalies) in this test for all reduction methods both using constant and variable density information. The long wavelength component of the geoid, thus, is cancelled out in equation (2) for this study. However, in theory, either a geopotential model corresponding to each mass reduction scheme should be used, or the corresponding correction for each reduction scheme should be applied to the coefficients (Stinkel, 1985; Kuhn, 2003; Bajracharya and Sideris, 2004). The coefficients of geopotential model computed from topo-isostatic anomalies using constant and variable density would be different, and thus, there would be a third term in equation (2) which represents the density effect on the long wavelength part of the geoid. The total density effect, in this paper, represents all effects except the one on long wavelength part of the geoid.
ANDTDE and ANIDE represent the direct topographical density effect on geoid (DTDE) and indirect density effect (IDE) on geoid, respectively. Thus, the total density effect for every gravimetric reduction scheme contains two components of density effect, the DTDE and the
(3) where AgF is the free-air anomaly, AgT(con) (AgT(den)) is the direct topographical effect on gravity in each reduction method using constant density (variable density), and AgGM is the reference gravity anomaly from a geopotential model. R is the mean Earth's radius, 7 is the normal gravity, and S(V) is Stokes' function. The commonly used normal gravity gradient of 0.3086 mGal/m is used in the computation of free-air anomalies AgF. The direct topographical density effect on geoid according to equation (3) is equal to the negative of the geoid effect obtained when the DTDE on gravity is applied to Stokes's formula. The direct topographical effect on gravity in equation (3), AgT, for Rudzki, RTM and AH gravimetric reduction scheme can be formulated as: Ag T - A - A(inv,Ref,Comp)
(4)
where A is the attraction of all topographic masses above the geoid. A(Inv, Ref,Comp) represents the attraction of inverted topographical masses, the reference topographic masses, and the compensated masses for the Rudzki, RTM, and AH reduction schemes, respectively. The details of the computation of two components on the right hand side of Equation (4) for different mass reduction schemes are given in Bajracharya and Sideris (2004). The RTM reduction, in principle, gives the quasigeoid (Forsberg, 1984). Equations (1), (2) and (3) are valid for RTM method as well replacing a notation of geoid, N, with that of quasigeoid, ~. The second term, ANIDE, in equation (2) is replaced by the topographical density effect on restored RTM, A~ RTDE. The separation between quasigeoid to geoid is required for the computation of geoid using RTM method and is given in Heiskanen and Moritz (1967). The total density effect on geoid using RTM gravimetric scheme can be written as ANTDE = A~DTDE + A~RTDE +SNDcor r
(5)
Chapter 58 • Density Effects on Rudzki, RTM and Airy-Heiskanen Reductions in Gravimetric Geoid Determination
where ~SNDcorr represents the density effect on Bouguer correction. Stokes's integral formula with the rigorous spherical kernel by the onedimensional fast Fourier transform algorithm is used in this study (Haagmans et al., 1993). The details for the computation of the indirect effect on geoid in equation (2) for AH scheme can be found in Bajracharya and Sideris (2004). 3 Numerical tests A rugged area in Canadian Rockies bounded by latitude between 49°N and 54°N and longitude between 236°E and 246°E is chosen to study the density effects using Rudzki, RTM, and AiryHeiskanen reduction methods. There are 9477 gravity measurements used for this test. The constant density of topographical masses is assumed to be 2.67 g/cm 3. The grid resolution of two-dimensional DDM available for this study is 30". A 30" grid spacing of DTM and DDM is used. Figure 1 shows the topography of the test area. The maximum and mean elevations of test area are 3937 m and 1396 m with a standard deviation of 543 m. Figure 2 shows the topographical density model of the test area. It has a large contrast between maximum and minimum values of 2.98 g/cm 3 and 2.49 g/cm 3, respectively. --4000 -3500 3000 2500 2000 1500 1000 500 16
Longitude
0
(o)
Fig 1. Digital terrain model in the test area (m) ~A
2.9
using Rudzki, RTM, and AH methods. A radius of 300 km is used around the computation point to compute the gravitational attraction of the topography, the attraction of the inverted masses, and the attraction of compensated masses. The height of the smooth reference surface, href, is computed with the resolution of 100 km. The normal crust thickness for the AH model is assumed equal to 30 km. The density of upper mantle is assumed equal to 3.27 g/cm 3 for this test. The long wavelength part of the gravity field is computed from the EGM96 geopotential model (Lemoine et al., 1998) complete to degree and order 360. The density information is incorporated in all steps of the geoid computational process for Rudzki, RTM, and AH reduction schemes. The direct topographical density effect on geoid is the only density effect, which should be taken into account in Rudzki geoid computational process, whereas the RTM method requires the computation of density effects on direct topographical effect, restored terrain effect, and the Bouguer correction term (if geoid needs to be determined using RTM method). Similarly, AH scheme requires the computation of density effects on direct topographical effect on gravity and indirect effect on geoid. The topographical density effect on gravity for all reduction methods are given in table 1. The range (around 20 mGal) of direct topographical density effect for Rudzki reduction scheme is nearly equal to that of Helmert's second method of condensation (see Huang et al., 2000; Bajracharya et al., 2002). The range of DTDE is greater for RTM and AH reduction methods. However, the mean value of this effect for all reduction schemes did not exceed over half a milligal. This DTDE on gravity for every reduction method is correlated with the topography and topographical density as shown in figures 3, 4 and 5. cA
2.8
2.7
2.6
2.5 .6
Longitude
(°)
Fig 2. Digital density model in the test area (g/cm3) The program TC developed by Forsberg (1984) is modified to adapt the digital density model
~6
Longitude
(°)
Fig. 3 DTDE on gravity for Rudzki scheme (mGal)
399
400
S. Bajracharya • M. G. Sideris
Table 2 Density effect on geoid (m)
54
53
i
5
i
51
50
-10
Scheme Rudzki RTM
-15 49 -124
-122
-120
-118 (°)
-116
-114
Longitude
Fig. 4 DTDE on gravity for RTM scheme (mGal)
AH
~a
Effectson geoid Total density effect Density effect on restored RTM Density effect on Bouguer term Total density effect Indirect effect on geoid Total density effect
Max
Min
Mean
STD
0.08
-0.05
0.02
0.02
0.04
-0.06
0.05
0.02
0.04
-0.02
0.00
0.01
0.15
-0.09
0.04
0.04
0.46
-0.50
-0.07
0.19
0.26
-0.04
0.12
0.06
,6 Longitude
(°)
Fig.5 DTDE on gravity for AH scheme (mGal) 0.06 0.04
Table 1. The DTDE on gravity (mGal) and on geoid undulation (m) Reduction scheme Rudzki
Effect
Max
Min
Mean
STD
gravity
11.9
-5.5
-0.04
1.1
~
.02
-0.02 -0.04 4
geoid
0.08
-0.05
-0.02
0.02
RTM
gravity
10.1
-16.3
-0.4
2.0
geoid
0.14
-0.10
0.04
0.04
AH
gravity
21.3
-9.6
-0.5
3.3
geoid
0.64
-0.42
0.23
0.27
Longitude
(°)
Fig. 6 DTDE on geoid for Rudzki scheme (m)
0.1
.05
The D T D E on geoid, given in table 1, for all reduction m e t h o d s suggests that the density effect should be taken into account for the centimeter geoid determination using every reduction scheme. It affects as m u c h as 64 cm, 14 c m and 8 c m for AH, R T M , and Rudzki reduction schemes respectively and is correlated with both the topographical density and t o p o g r a p h y o f Canadian Rockies as s h o w n in figures 6, 7, and 8. The other density effects on gravimetric quantities of RTM geoid computational process, n a m e l y restored R T M terrain effect and B o u g u e r correction, given in table 2, also exhibit the importance o f incorporating realistic density information in
4
Longitude
(°)
Fig. 7 DTDE on geoid for RTM scheme (m)
40
Longitude
(o)
Fig. 8 DTDE on geoid for AH scheme (m)
I
-0.05
Chapter 58 • Density Effects on Rudzki, RTM and Airy-Heiskanen Reductions in Gravimetric Geoid Determination 54 0 . 0 3
0 . 0 2
5 3
0 . 0 1
52
~_~
0
- 0 . 0 1
- 0 . 0 2
- 0 . 0 3
5 0
- 0 . 0 4
- 0 . 0 5
4 9 - 1 2 4
- 1 2 2
- 1 2 0
- 1 1 8
Longitude
- 1 1 6
- 1 1 4
(o)
Fig. 9 Topographical density effect on restored RTM (m)
geoid determination with centimeter accuracy since they can alter the geoid as much as 6 cm and 4 cm respectively. These two quantities, which are also correlated both with the topography and topographical density (shown in figures 9 and 11), do not seem to change the
ao
DTDE is most prominent one compared to other density effects in RTM geoid determination. The indirect density effect on geoid for AH scheme (figure 10) can alter the geoid as much as 50 cm. However, the DTDE on geoid and indirect density effect on geoid for this reduction scheme cancel each other and it brings the maximum total density effect on geoid down to 26 cm. The total density effect on geoid is lowest for Rudzki reduction scheme (maximum value of 8 cm) and highest for AH ( the maximum value of 26 cm). The GPS leveling data set is used to evaluate the precision of Rudzki, RTM, and AH gravimetric geoid solutions using constant and variable density. The 258 GPS benchmarks were available for this test; their distribution is given in figure 12. The statistics of the differences between Rudzki, RTM, and AH gravimetric geoid solutions using constant and variable density with the GPS- levelling geoid after the fit is shown in table 3. Table 3 The statistics of difference between gravimetric geoid solutions using constant and variable density with
GPS-levelling geoid (after fit) (m) L o n g i t u d e
(o)
Fig. l0 Indirect effect on AH geoid (m)
Geoid Model Rudzki
RTM
Rudzki
Density
Max
Min -0.82
Mea n 0.00
Constant
0.56
Variable
STD 0.20
0.58
-0.79
0.00
0.20
Constant
1.15
-0.52
0.00
0.25
Variable
1.17
-0.52
0.00
0.25
Constant
1.39
-0.86
0.00
0.36
Variable
1.39
-0.86
0.00
0.35
.6 L o n g i t u d e
(o)
Fig. 11 Density effect on Bouguer correction term o
o
°
o
°
°
o
°
o
°
.
o
.
.
.
.
o
o
o
.
°
°
o
°
o .
°
o
°
o o
°
o
o Oo
o
o ° o
o
° o ° .
o
°
o
°
°
°
°
o o °
o
°
°
o
o
o
o
o
.
.
°
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.
o
.
°
°
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°
~.
o °
-
°
°
o
°
°
o °
2 4 0 L o n g i t u d e
.
o
-
.
°
o
o
-
~.
2 3 8
o o
o o
°.
°
o
° :
.
o
o
o
o
o . "
o
o
°
° o o
"
o °
o
o
4 2 3 6
o
°
°
o
o
~.~51
50
°
o
° ~__~52
*O°o
o
....
2 4 2
°
.
° .
.
..,.
2 4 4
2 4 6
(°)
Fig. 12 Distribution of GPS leveling points in the test area
statistics of total density effect (table 2) from those of DTDE for RTM reduction scheme. The
There is no difference in statistics using constant and variable density after fit with GPS-levelling geoid for every reduction scheme. However as stated earlier, the maximum total density effect can range from 8 cm to 26 cm depending on the reduction method chosen to treat topographical masses. The better statistics of Rudzki, RTM, and AH geoid solutions using real density information are not observed. There can be several factors playing role on geoid solutions using actual density information; (i) the accuracy of digital density model used (ii) the assumption of constant density in vertical direction using two dimensional DDM (iii) the use of low resolution of DTM as also pointed out by Kahtrelber (1998) (iv) the accuracy of GPS
401
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S. Bajracharya • M. G. Sideris
measurements (v) the use of constant density in Orthometric height determination, which along with GPS ellipsoidal height, is used to validate gravimetric geoid solutions.
Natural Sciences and Council of Canada.
4 Conclusion
Bajracharya S, Kotsakis C, Sideris MG (2002) Aliasing effects on terrain correction computation, international Geoid Service, Bulletin N. 12, April, 2002.
This paper showed the significance of incorporating actual density information (if available) not only for Helmert's second method of condensation as usually discussed in most geodetic literature, but also for other gravimetric reduction schemes in the context of precise geoid determination. The density information should be applied in all steps of geoid computational process required for every reduction scheme. Not only is the direct topographical density effect on geoid using any reduction scheme significant to be considered for centimeter geoid determination, but also other gravimetric quantities (for example, density effect on RTM restored part, Bouguer correction in RTM method and indirect effect on geoid for AH scheme) are important to be considered. The total density effect on geoid for Rudzki reduction scheme is the lowest which reaches as maximum as 8 cm and that for AH scheme is the highest, which goes up to 26 cm in our test. These significant density effects suggest that the density (if available) should be applied for geoid computation using any gravimetric reduction method. The density effects on all gravimetric quantities for each reduction scheme are correlated with both the topography and topographical density of the test area. The overall improvement on each gravimetric geoid solution used in this study, when compared with GPS leveling geoid solution after fit, is not seen using actual density information for the test area. The factors (which can affect density effects), like the use of high resolution DDM, the use of levelling data (in which the actual density information is incorporated in Orthometric height determination), the information on the change of density in vertical direction, should be considered in future research of geoid determination with density information especially in rugged areas.
Acknowledgements: This research has been supported by grants from GEOIDE NCE and the
Engineering
Research
References
Bajracharya S, Sideris MG (2004) The Rudzki inversion gravimetric reduction scheme in geoid determination. Journal of Geodesy 78: 272-282. Forsberg R (1984) A study of terrain reductions, density anomalies and geophysical inversion methods in gravity field modeling. Report No. 355, Dept. of Geodetic Science and Surveying, The Ohio State University, Colombus, Ohio. Haagmans R, De Min E, Van Gelderen M (1993) Fast evaluation of convolution integrals on the sphere using 1D FFT and a comparison with existing methods for Stokes' integral. Manuscripta Geodaetica, 18:227-241. Heiskanen WA, Moritz H (1967) Physical geodesy. W. H. Freeman and Company. San Fransisco. Huang J, Vanicek P, Pagiatakis SD, Brink W (2000) Effect of topographical density on geoid in the Canadian Rocky Mountains. Journal of geodesy 74: 805-815. Kfihtrelber N (1998) Precise geoid determination using a density variation model. Physics and chemistry of the Earth, Vol. 23, No 1, p. 59 -63 Kuhn M (2003) Geoid determination with density hypotheses from isostatic models and geological information. Journal of Geodesy 77: 50-65. Lemoine FG, Kenyon SC, Factor JK, Trimmer RG, Pavlis NK, Chinn DS, Cox CM, Klosko SM, Luthcke SB, Torrence MH, Wang YM, Williamson RG, Pavlis EC, Rapp RH, Olson TR (1998) The development of the joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) geopotential model EGM96. Nasa/TP- 1998-206861, NASA, Maryland Sfinkel H (1985) An isostatic Earth model. Rep 367. Department of Geodetic Science and Surveying. The Ohio State University, Columbus Tziavos IN, Featherstone WE (2000) First results of using digital density data in gravimetric geoid computation in Australia. lAG Symposia, Vo1.123 Sideris (ed.), GGG2000, ©Springer Verlag Berlin Heidelberg (2001) p. 335-340 Tziavos IN, Sideris MG, Sunkel H (1996) The effect of surface density variation on terrain modeling - A case study in Austria. Proceedings, EGS Society General Assembly. The Hague, The Netherlands, May, 1996. Report of the Finnish Geodetic Institute.
Chapter 59
Combined modeling of the Earth's Gravity Field from GRACE and GOCE Satellite Observations: a Numerical Study R Ditmar, X. Liu, R. Klees Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology Kluyverweg 1, 2629 HS Delft, The Netherlands R. Tenzer, R Moore School of Civil Engineering and Geosciences, University of Newcastle upon Tyne Newcastle upon Tyne, NE1 7RU, United Kingdom
Abstract. A numerical study has been conducted in order to estimate how a gravity field model obtained from GOCE data can be improved in the range of low degrees by addition of GRACE data. The GRACE data are simulated as inter-satellite accelerations. Different types of noise in the inter-satellite accelerations are considered, including white noise. The gravity field model is represented as a series of spherical harmonics; the Stokes coefficients are computed by a least-squares adjustment. It is shown that the incorporation of GRACE data may improve a GOCE-based gravity field model up to degree 120 or even higher depending on the type of noise in the inter-satellite accelerations. Moreover, the joint model at lower degrees may show a significantly higher quality that either a stand-alone GOCE-based or a stand-alone GRACE-based model. It is important, however, that proper covariance matrices of the involved data sets are used in the joint data processing. Keywords. Earth's gravity field; GOCE; GRACE; gravity gradiometry; inter-satellite accelerations.
1
Introduction
The Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) will be launched in 2006 in order to deliver information about the Earth's gravity field with an unprecedented combination of accuracy and resolution (ESA, 1999). According to the estimations performed, the geoid height accuracy will reach the 1-cm level at the spatial scale of 200 km (or 100 km in terms of half-wavelengths) (Stinkel, 2000; Bouman, 2000; Sneeuw, 2002; Ditmar et al., 2003). The necessary data will be collected by a gravity gradiometer, which is being purposely developed for the GOCE mission (Cesare, 2002). This in-
strument, however, will be characterized by a limited accuracy at low frequencies (below 5 mHz). Therefore, a gravity field model obtained solely on the basis of Satellite Gravity Gradiometry (SGG) data will show an unacceptable error level at low spherical harmonic degrees (approximately, below degree 30). As a consequence, GOCE SGG data have to be processed jointly with other information that can improve the gravity field model of the lowest degrees. Until now, it was routinely assumed that GOCE SGG data are to be processed jointly with GOCE HighLow Satellite-to-Satellite Tracking (HL-SST) data, which will be obtained by processing the observations from the on-board GPS receiver (Stinkel, 2000). In the meantime, a success of another satellite mission - GRACE (Tapley et al., 2004, 2005; Reigber et al., 2005b) - has triggered intensive discussions on a possibility to combine instead GOCE SGG and GRACE data. For example, Eicker et al. (2004) have considered how a solution based on GRACE data can be refined by GOCE SGG observations at selected geographical regions. In this paper, we estimate the accuracy of a global gravity field model produced by the joint data processing and compare the results with the solution produced from GOCE SGG data only.
2 2.1
Theory GOCE gravity gradiometry data
The GOCE gradiometer will provide information about the second derivatives of the gravitational potential V: V x x
°~v Vxy 0--2~,
o°x~avY , ... V z z
a2v where X-, Y-, and Z-axis are defined in the OZ 2 , gradiometer reference frame (roughly speaking, the X-axis is along-track, the Y-axis is cross-track, and the Z-axis is radial). It is important to notice that the diagonal elements of the matrix of second deriva-
404
P. D i t m a r • X. L i u . R. Klees • R. Tenzer • P. M o o r e
tives will be provided with a higher (or even much higher) accuracy than the off-diagonal elements (see e.g. Mtiller, 2003). In the paper, we assume that the Earth's gravitation is represented by means of spherical harmonics. This means, in particular, that the explicit expression for the gravitational potential is as follows (Moritz, 1980b):
V(r, o, A)
1
GM -
Z
R
/=0
O"~H'~"~('~, o, A),
m=--l
(,)
where r, 0, A are the spherical coordinates; G M is the the geocentric gravitational constant; /~ is the semi-major axis of a reference ellipsoid; L , ~ x is the maximum spherical harmonic degree of a particular model" Cz,~ are the fully normalized Stokes coefficients; and/Jl,~ (0, A) are the fully normalized solid spherical harmonics: Hz,~ (r, 0, A ) (- R )
z + l -p l . ~ ( c o s
-
0) ×
r
cos m A sin m A
(2) if if
m _> 0 m < 0
with Pl.~ (cos 0) the fully normalized associated Legendre functions. Obviously, the relationship between the Stokes coefficients and the gravitational potential is linear. Consequently, the dependence of the second derivatives of the gravitational potential on the Stokes coefficients is linear, too. Then, the relationship between the set of observed gravity gradients Ysgg c a n be related to the set of Stokes coefficients s in terms of linear algebra as follows: Ysgg -- Asgg
s,
(3)
where A~gg is the design matrix related to the SGG data set.
Thomas, 1999):
a(~) _ f i _ Av2-(Av.e)2, P
where A v is the velocity difference between the satellites (in an inertial frame), and e is the unit vector that defines the line-of-sight direction. Thus, the set of Inter-Satellite Accelerations (ISA) can be directly derived from the KBR data, provided that the correction expressed by the second term in Eq. 4 can be done with a sufficiently high accuracy. Furthermore, an inter-satellite acceleration is equal to: a (~8) - ( g r a d ( A V ) . e ) ,
(5)
where A V is the difference between gravitational potentials at the satellite locations. Thus, the GRACE inter-satellite accelerations can be related to the gravitational potential gradients linearly. Consequently, the vector Yisa composed by the observed intersatellite accelerations can be linearly related to the Stokes coefficients. This relationship can be written in terms of a matrix-to-vector multiplication as follows: Y i s a = A i s a s, (6) where Aisa is the design matrix related to the intersatellite accelerations. 2.3
Joint data p r o c e s s i n g
It is advisable to represent the vector of unknown Stokes coefficients s in the form s -- s (°) + x,
(7)
where the vector s (°) corresponds to a reference gravity field, and x is the vector of corrections to be estimated. The reference model is used to compute a set of reference SGG data j-s g(o) g and a set of reference _ (o) inter-satellite accelerations Yisa" j~(o) sgg _ Asggs(O)
2.2
(4)
'
y (isa O ) _ A i s a s (°)
(8)
G R A C E data
The relevant GRACE data are provided by the KBand Ranging (KBR) system. In essence, the KBR data are (biased) ranges between the twin satellites that comprise the GRACE mission. Temporal variations of the inter-satellite ranges are recorded with an accuracy at the level of 10 # m (Reigber et al., 2005a). The second derivative of the range/9 with respect to time is related to the inter-satellite acceleration a (~8) (the projection of the satellite acceleration difference onto the line-of-sight) as follows (e.g.
The optimal estimation of the vector x can be obtained by means of the least-squares adjustment: -- N - 1 A T C d ~ d ,
(9)
where the matrix A and the vector d are the joint design matrix and the joint data vector, respectively: Asgg
A-
Aisa
,
d-
Ysgg - ,ysgg
.
y!O)
,
Y i s a --
lsa
(~o)
Chapter 59 • Combined Modeling of the Earth'sGravity Field from GRACEand GOCESatellite Observations:a Numerical Study
Cd is the joint data covariance matrix:
10000 5000
C d --
0
Cisa
2000
(11 )
1000 500
with Csgg and Cisa the covariance matrices of SGG and ISA data sets, respectively; and N is the combined normal matrix: N -- A T C d l A
+ C ~ o) 1
where ct is the Tikhonov regularization parameter, which provides the necessary scaling. Technically, a numerically efficient least-squares adjustment of SGG and ISA data can be performed with the algorithm proposed by Ditmar and Klees (2002), which is based on the preconditioned conjugate gradient method. The preconditioner related to the ISA data can be built in the same way as the preconditioner for the XX-component of the gravity gradients. In other words, the twin GRACE satellites are considered as a large gradiometer.
3.1
Numerical study SGG
data
The simulated set of SGG data is similar to the "long data set" described by Ditmar et al. (2003); it is based on a realistic 6-month GOCE orbit. We have computed along this orbit the diagonal gravity gradient components with the 1-s sampling interval on the basis of the EIGEN-CG03C gravity field model (F6rste et al., 2005) truncated at degree 300. The simulated data have been artificially contaminated by realistic colored noise (Stinkel, 2000); the corresponding square-root Power Spectral Density (SQRT-PSD) is shown in Fig. 1. 3.2
200 100
03 I3_ "~ rr 0 CO
50 20 10 5
(12)
with C~(o) the covariance matrix of the reference gravity field model. Since the true covariance matrix of the reference model is frequently unknown, it is common to replace it by a Tikhonov regularization matrix R: --1 Cs(o~ - a R , (13)
3
L•v
ISA data
To simulate ISA data, we used the real 123-day orbits of GRACE twin satellites (Jul. 31 to Nov. 30, 2002) distributed by the Jet Propulsion Laboratory. Furthermore, the EIGEN-CG03C model truncated at degree 300 was exploited in the simulations. The simulated data were contaminated by an artificially generated noise. In the first instance, noise was white (i.e. frequency independent). A reasonable noise standard deviation was determined as follows. First of all, an
0.0001
0.;01
0.01
0 1
F r e q u e n c y (Hz)
Fig 1. SQRT-PSD of noise added to the simulated SGG data. Dotted, dashed, and solid line corresponds to the XX-, YY-, and ZZ-component, respectively.
auxiliary set of ISA data was simulated on the basis of the EIGEN-CG03C model truncated at degree 100. Then, this set was contaminated by white noise with the unit standard deviation. On the basis of this data set, a model of the Earth's gravity field up to degree 100 was computed; the GRS80 gravity field was used as a reference (Moritz, 1980a); the regularization was switched off. The errors in the computed model were represented in terms of geoid heights per degree and compared with those of the EIGENCG03C model. Since an up- or down-scaling of data noise leads to the same up- or down-scaling of errors in the computed gravity field model, the comparison of the gravity field models allowed us to determine the appropriate scaling factor to be applied to the simulated data noise (which coincided with the standard deviation to be used because the original standard deviation was set equal to one). As a test, the gravity field recovery was repeated with scaled noise. As expected, the errors of the computed model in terms of geoid heights per degree (the solid red curve in Fig. 2) match very well the errors of the EIGENCG03C model (cf. Fig. 5 in F6rste et al., 2005). In both cases, in particular, the errors reach the level of 1 m m at degree 65. Notice that simulated noise must be lower than real noise because the appropriate accuracy of the gravity field model is reached on the basis of a relatively short data set (123 days vs a 376day GRACE data set exploited in the compilation of the EIGEN-CG03C model). On the other hand, the GRACE mission still delivers data and, therefore, by the time of the GOCE mission it will result in more accurate gravity field models than EIGEN-CG03C. Thus, our simulation can be considered as a worstcase scenario. It is important to notice that the appropriate scaling factors could not be determined on the basis of
405
406
P. Ditmar • X. Liu. R. Klees • R. Tenzer • P. Moore
3.3 Gravity field modeling
Geoid height errors (cumulative and per degree) I-
Fig 2. Accuracy of gravity field models produced from the auxiliary sets of ISA data in the presence of properly scaled noise. Three noise realizations are considered: white noise; differentiated white noise (alternative noise II); and twice differentiated white noise (alternative noise I). The black lines depict geoid height errors per degree; the grey lines correspond to the cumulative geoid height errors.
First of all, a gravity field model was produced on the basis of the SGG data only. The normal (GRS80) gravity field was specified as the reference one. According to the suggestions of Ditmar et al. (2003), the first-order Tikhonov regularization was applied; the regularization parameter was set equal to 1000 x 1012 = 1015. The errors of the computed model (i.e. the differences w.r.t, the EIGEN-CG03C model truncated at degree 300) are shown in terms of geoid heights in Fig. 3. The low-degree errors caused by strong noise in SGG data at low frequencies are obvious. Furthermore, one can clearly see large errors at the poles, which are due to the absence of data in polar areas (the inclination of the GOCE orbit is 96.65°). Finally, one can notice a slightly increased magnitude of errors at the equator (say, in the latitude range f r o m - 3 0 ° to 30°). The latter effect can be explained by the fact that the density of satellite tracks at the equator is lower than at high latitudes.
the original ISA data set (produced on the basis of the EIGEN-CG03C model truncated at degree 300). This is because an attempt to recover a gravity field up to degree 300 failed in the absence of regularization due to the fact that the normal matrix is illposed. On the other hand, an attempt to reduce the maximum degree caused additional errors due to the aliasing phenomenon.
After that, the set of SGG data and the set of ISA data with white noise were jointly processed. The regularization was applied as before. This can be justified by the fact that the regularization is needed to stabilize only very high degrees (say, above 200), which are not influenced by GRACE data. The joint processing resulted in a significantly more accurate gravity field model (Fig. 4): both the low-degree errors and the errors at the polar areas are not visible anymore.
0.01
,4:
',Jl"
no'(,, "; 03
$ 0.001
,.:," "u4'''~?'
0.0001 - - 4 -
.'~~'"
--~ i~, ~:,~., ~--j~" -: ="v ~ " 1e-05
0
10
20
30
Alternative noise, Alternative noise II White noise
40
50 60 Degree
70
80
90
100
In practice, the ISA data are produced from the KBR data by a double numerical differentiation. Therefore, the assumption that noise in KBR data can be considered as white may not be realistic. For this reason, we simulated two alternative noise realizations: (i) by a double numerical differentiation of a white noise realization and (ii) by a single numerical differentiation of a white noise realization. The first case can be interpreted as an assumption that noise is white in terms of ranges (provided that the contribution of the correction term in Eq. (4) to the error budget is negligible). Correspondingly, the second case can be interpreted as the assumption that noise is white in terms of range-rates. The appropriate scaling factors in both cases were determined as described above. The errors of gravity field models obtained from these two data sets are shown in Fig. 2 as blue and green lines. Obviously, the chosen standard deviation of noise in both cases is very reasonable. Remarkably, the results of stand-alone ISA data processing are not very sensitive to the type of noise.
Finally, a gravity field model was produced by a joint processing of SGG data and ISA data contaminated by the alternative noise realizations as discussed in section 3.2. It is important to notice that in all the cases the ISA covariance matrix in gravity field modeling was set proportional to the identity 2 a was esti2 I. The variance crib matrix: Cisa - o-isa mated numerically as the total variance of a particular noise realization. All the four computed gravity field models are compared in terms of geoid height errors per degree in Fig. 5. Remarkably, incorporation of the ISA data improves the quality of the gravity field (in comparison to the stand-alone processing of SGG data) up to a surprisingly high degree: between 120 and 180, depending on the type of noise in ISA data. The maximum improvement is reached when the ISA data are contaminated by a white noise realization. This is not surprising because the accuracy of a model at high degrees is primarily dependent on the level of data noise at high frequencies, and this level is increased after a numerical differentiation of noise. Large er-
Chapter 59 • Combined Modeling of the Earth's Gravity Field from GRACE and GOCE Satellite Observations: a Numerical Study
150°W 120°W
90°W
60°W
30°W
0°
30°E
60°E
-
90°E
120°E
n
150°E
m
60°N I
60°N • ,•"
30°N
.
J
. . . .
" ..
,.
.":L '-i: O
).
: ~
..
.,,-~ •
.,,
.. ,
,.- .
"~..~!... :..
•
,-L
........ . .
• -.'-
0
•
• 30°S
~.
.- -..: d."
..:""
~-..-,..:
•
. •. •
•
..
":
\~
.-
"...
• ....
,..,.. , I.: ~'' •' ..... "";::"" : . " = ii-
-|
.." • ""
..
"'"
..
_"
• .
30os '
m m
•
.
..
•
m"
..
. . . "-"
•• -
..
•
60°S
-2.5
• •
•
60°S
..
150°W 120°W
90°W
60°W
30°W
0°
30°E
60°E
90°E
1200E 150°E
-0.5
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
-0.4
0
.,..........
. • •
,.
m
.... : -'
l
'
30°N I -|.
.-.
I r~ .~-"::.: "':"' "
.
,.
"..~--...-- -:..'...,.%-" " ; " " . ";4:•
0.5
2.5
m e t e r s
Fig 3. Geoid height errors of the gravity field model produced solely from the SGG data set.
rors at low degrees (below 10) in case of alternative noise realizations can be explained by an inadequate covariance matrix Cisa used in the joint data processing. As a result, the ISA data at low frequencies are down-weighted and overwhelmed by inaccurate SGG data. Finally, it is remarkable that the combined solution at intermediate degrees may be orders of magnitude more accurate than a stand-alone solution obtained from either data set. For example, the SGG-based solution shows at degree 100 the accuracy of 3 mm, and the ISA-based solutions are characterized at this degree by an accuracy of 5 m m to 2 cm (see Fig. 2). At the same time, the accuracy of the joint solution in case of white noise in ISA data reaches the 0.1-mm level at degree 100. We explain this phenomenon by the fact that the GRACE observations are highly anisotropic: they are not sensitive to sectorial features of the gravity field. Incorporation of SGG data, which show less anisotropy, improves the estimation of near-sectorial harmonics, and this results in a significant overall improvement of a gravity field model. Another way to analyze the accuracy of gravity field models is to compute cumulative geoid height
errors. Unfortunately, results of the standard procedure - the formal summation of degree variances would be somewhat misleading in our case. Firstly, large errors below degree 10 would dominate the cumulative errors up to a very high degree. Secondly, both errors per degree and the cumulative errors characterize the average accuracy on a sphere, including the polar areas. Thus, these errors in a GOCE-based model gets over-estimated if the polar areas are of no interest. For these reasons, we computed the cumulative geoid height errors in an alternative way. We produced a set of truncated gravity field models using different truncation degrees; the coefficients below degree 10 were truncated too. Each of these models was compared in the latitudinal range f r o m 80 ° to 80 ° with the consistently truncated true model. The rms geoid height difference between these models just gave us one point for the cumulative geoid height errors plot (Fig. 6). One can see that this plot confirms the conclusions made above. In particular, the maximum degree where ISA data still may contribute to a GOCE-based solution is in the range 120 - 180.
407
408
P. Ditmar • X. Liu. R. Klees • R. Tenzer • P. Moore
150°W 120°W
90°W
60°W
30°W
0°
30°E
60°E
90°E
120°E "-~---
60°N
30°N
0
•
30°S
150°E .-2--
~
-
60°N
"
30°N
0
30°S
60°S
-2.5
60°S
150°W 120°W
90°W
60°W
30°W
0°
30°E
60°E
90°E
120°E
150°E
-0.5
-0.3
-0.2
-0.1
0
O.1
0.2
0.3
0.4
0.5
-0.4
2.5
meters Fig 4. Geoid height errors of the gravity field model produced by the joint processing of the SGG and ISA data sets.
Geoid height errors per degree 0.1
0.01
0.1
0.001
0.01
1 e-04
0.001
."4[e].1
0.0001
1 e-06 50
100
150 Degree
200
250
300
Fig 5. Stand-alone processing of SGG data and joint SGG+ISA processing: accuracy of gravity field models in terms of geoid heights per degree.
4
1 e-05
0
50
100
150 Degree
200
250
300
Fig 6. Stand-alone processing of SGG data and joint SGG+ISA processing: accuracy of gravity field models in terms of cumulative geoid height starting from degree 10; the polar areas are excluded.
Discussion and conclusions
The quality of a GOCE-based gravity field model can be improved by an incorporation of GRACE data up to degree 120 or even higher. Furthermore, the accu-
racy of a joint model at lower degrees may be significantly better (possibly, as much as an order of magnitude better) than the accuracy of either stand-
Chapter 59 • Combined Modeling of the Earth's Gravity Field from GRACEand GOCESatellite Observations: a Numerical Study
alone gravity field model. Thus, we believe that the incorporation of G R A C E data is an excellent way to improve a G O C E - b a s e d gravity field model at low degrees. A pre-requisite of a successful joint data processing is building accurate stochastic models of data sets involved, otherwise a highly accurate information from one data set may be over-weighted by a less accurate information from another data set.
5
Acknowledgments
Computations were done on the SGI Altix 3700 super-computer in the framework of the grant SG027, which was provided by "Stichting Nationale Computerfaciliteiten" (NCF). The G O C E orbit used in the numerical study was computed by E. Schrama from TU Delft.
References J. Bouman. Quality Assessment of Satellite-based Global Gravity Field Models, Ph.D Thesis. Delft University of Technology, 2000. S. Cesare. Performance requirements and budgets for the gradiometric mission. Issue 2 GO-TN-AI-O027, Preliminary Design Review. Alenia, Turin, 2002. P. Ditmar and R. Klees. A Method to Compute the Earth's Gravity Field from SGG / S S T data to be Acquired by the GOCE Satellite. Delft University Press, 2002. P. Ditmar, J. Kusche, and R. Klees. Computation of spherical harmonic coefficients from gravity gradiometry data to be acquired by the GOCE satellite: regularization issues. Journal of Geodesy, 77:465-477, 2003. A. Eicker, T. Mayer-Gtirr, and K. H. Ilk. Global gravity field solutions from GRACE SST data and regional refinements by GOCE SGG observations. In C. Jekeli, L. Bastos, and J. Femandes, editors, Gravity, Geoid, and Space Missions. GGSM 2004. lAG International Symposium, Porto, Portugal, August 30- September 3, 2004. International Association of Geodesy Symposia, volume 129, pages 66-71. Springer, Berlin, 2004.
Resolution Global Gravity Field Model Derived From Combination of GRACE and CHAMP Mission and Altimetry/Gravimetry Surface Gravity Data. Poster presented at EGU General Assembly 2005, Vienna, Austria, 24-29, April 2005. Available as http://www.gfzpotsdam.de/pb 1/op/grace/results/grav/g004_EGU05-A04561.pdf, 2005. H. Moritz. Geodetic reference system 1980. Bull. Gdod, 54:395-405, 1980a. H. Moritz. Advanced Physical Geodesy. Herbert Wichmann Verlag Karlsruhe, 1980b. J.
Mtiller. GOCE gradients in various reference frames and their accuracies (available as http://www.copernicus.org/EGU/adgeo/2003/1/adg1-33.pdf). Advances in Geosciences, 1:33-38, 2003.
C. Reigber, R. Schmidt, F. Flechtner, R. K6nig, U. Meyer, K.-H. Neumayer, R Schwintzer, and S. Y. Zhu. An earth gravity field model complete to degree and order 150 from GRACE: EIGEN-GRACE02S. Journal of Geodynamics, 39:1-10, 2005a. C. Reigber, R Schwintzer, R. Stubenvoll, R. Schmidt, F. Flechtner, U. Meyer, R. K6nig, H. Neumayer, C. F6rste, F. Barthelmes, S. Y. Zhu, G. Balmino, R. B iancale, J.-M. Lemoine, H. Meixner, and J. C. Raimondo. A high resolution global gravity field model combining CHAMP and GRACE satellite mission and surface gravity data: EIGEN-CG01C. Journal of Geodesy (accepted), 2005b. N. J. Sneeuw. Validation of fast pre-mission error analysis of the GOCE gradiometry mission by a full gravity field recovery simulation. Journal of Geodynamics, 33:4352, 2002. H. Stinkel, editor. From EOtvOs to reGal. Final report, ESA/ESTEC Contract 13392/98/NL/GD. European Space Agency, Noordwijk, 2000. B. Tapley, J. Ries, S. Bettadpur, D. Chambers, M. Cheng, F. Condi, B. Gunter, Z. Kang, R Nagel, R. Pastor, S. Poole, and F. Wang. GGM02 - an improved Earth gravity field model from GRACE. Journal of Geodesy (in press), 2005.
ESA. Gravity field and steady-state ocean circulation missions. Reports for mission selection. The four candidate Earth explorer core missions, SP-1233(1). European Space Agency, Noordwijk, 1999.
B. D. Tapley, S. Bettadpur, M. Watkins, and C. Reigber. The gravity recovery and climate experiment: Mission overview and early results. Geophysical Research Letters, 31, 2004. L09607, doi: 10.1029/2004GL019920.
C. F6rste, F. Flechtner, R. Schmidt, U. Meyer, R. Stubenvoll, F. Barthelmes, R. K6nig, K. H. Neumayer, M. Rothacher, C. Reigber, R. Biancale, S. Bruinsma, J.-M. Lemoine, and J. C. Raimondo. A New High
J. B. Thomas. An analysis of gravity-field estimation based on intersatellite Dual-l-Way biased ranging (JPL Publication 98-15). Jet Propulsion Laboratory. Pasedena, California, 1999.
409
Chapter 60
Analytical Solution of Newton's Integral Terms of Polar Spherical Coordinates
in
R. Tenzer, P. Moore School of Civil Engineering and Geosciences, Newcastle University, Newcastle upon Tyne, United Kingdom, NE 1 7RU O. Nesvadba Department of Advanced Geodesy, Czech Technical University in Prague, Faculty of Civil Engineering, Thfikurova 7, Praha 6 - Dejvice, Czech Republic 166 29 Abstract. In association with precise modelling of
the Earth's gravity field, analytical integration can be used as an alternative to numerical integration, particularly for the intermediate neighbourhood of the computation point. Accordingly, closed analytical formulae for the gravitational potential and attraction are derived after expressing Newton's integral in terms of polar spherical coordinates. As the elemental volume for the integration element is defined by finite changes of the polar spherical coordinates, the actual mass density distribution is discretized so that each integration element is represented by a constant value of density. Keywords. Gravity, Newton's integral, potential, tesseroid
1 Introduction Various numerical and analytical methods have been proposed for evaluating Newton's integral. For example, a simple form such as the right rectangular parallelepiped (prism) can be used for the integration element in gravity field modelling. A closed analytical expression for the potential of a prism was derived by Bessel (1813) while potentialrelated formulae for a prism were also considered by Zach (1811), Mollweide (1813), Everest (1830), Mader (1951), and recently by Nagy et al. (2000). The analytical solution for potential and gravitational attraction of a spherical cap can be found in Ktihtreiber et al. (1989). In addition, Pick (1984) summarized expressions for the gravitational attraction of various geometrical bodies. In Anderson (1976), Martinec (1998) and Kuhn (2000), a spherical prism (tesseroid) was used for an evaluation of the potential and its radial derivative. Moreover, analytical solution for the radial component of the integration domain of tesseroid was adopted (from Bronstein and
Semendjajew (1977) in Martinec (1998) and an efficient method of numerical integration proposed for the surface integration with respect to spherical coordinates. In this study, an analytical solution of Newton's integral is developed in terms of polar spherical coordinates and subsequently utilised to develop an analytical solution for gravity. As consequences, errors within discretized numerical integration are eliminated. It is worthwhile to mention that the closed analytical solution of the Newton integral in terms of geocentric spherical coordinates does not exist (elliptical integral).
2 Analytical Solution for Potential To derive an analytical solution of Newton's integral consider the geocentric spherical coordinate system (~b, 2 , r ), where ~b and 2 are latitude and longitude respectively (- rr / 2 ___~b< rr / 2; 0 _
Chapter 60 • Analytical Solution of Newton's Integral in Terms of Polar Spherical Coordinates
ct
where G is Newton's gravitational constant and g the Euclidean spatial distance. To find an analytical solution of Newton's integral in Eq. (2), classical integration theory (Fubini's theorem, see e.g. Fubine 1958) for triple integral is applied. Thereby
I
~u2
r~
aV(r,a)=Go f IS=dr'd 17 ~=~1r'=r{ 5
=Gp
SF dr' ]~2~tlI~2tyl
r'=r(
=GoF,,,,r, I~~' [~2 ~,, [~2 ,~. The analytical functions F and F Fig. 1 Geocentric spherical and polar spherical coordinates.
(3) in Eq. (3) are
given by L = O~~-1 (]"'~/"]/'t) jr't2 sing ( r . r ' A g .
Let A g , Ac~ and Ar' denote finite changes of the polar spherical coordinates g , a (see Fig. 2) and the geocentric radius r' of the integration point. The volume of the integral element is then given by A~9 = r'2Ar ' s i n g A g A a . Furthermore, assume the mass density p is constant within the element.
],,,! F v -a--g(r,g,r') F
0),
(r~O).
(4)
(5) of
Consequently, the primitive function F~,w, Newton's integral is found to be
[1 F,w,r,- ~-~F
e3(F, ~/J,Pt)-[- -~i
,)cos ( ,_rcos )
2
+ - - c o s y sin2 gt in 12g(r,~u, r')+ 2 r ' - 2r cos ~u 2 (r ~ 0).
~tlt t (,, x)
1
(6)
A proof of the analytical solution of Newton's integral, given by the partial differentiation of the functions F and F,r, with respect to g and r' respectively, is given in Appendix I. The integration of Newton's kernel with respect to the azimuth a in Eq. (4) is trivial.
Fig. 2 Surface component of the integration element in polar spherical coordinates (template).
The gravitational potential, OT, generated by the mass element Am-pA~9 is given by (e.g. MacMillan 1930)
Newton's integral for a potential, and the components of the gravitational attraction vector (discussed in the next section), are singular when the integration point coincides with the computation point. However, the singularity is weak and thus removable, cf. e.g. (Kellogg 1929, p. 151).
3 Analytical Solution for Gravity a2 gt2 r~ I I g-l(r'l//'r')r'2 dr'singdgdc~, a=al ~'=VIr'=r(
O-V(r,F~)=G9 I
(2)
By analogy with Eq. (3), the meridian go, primevertical gx and radial gr components of the vector g of gravitational attraction, i.e.
411
412
R. Tenzer • P. Moore. O. Nesvadba
(7) can be derived. Considering the gravitational attraction of tesseroid, the following relations are written
r
respect to the geocentric spherical coordinates ¢ and 2 in Eqs. (8-11) are easily seen to be 8~ - - = --COS 6g ,
c ~ _--cosCsina,
80
82
1 ( S F , , 8c~
=GPTk. c~°~:' 80
(12)
12Iii
a~, a¢
(8)
c~
sin a
8¢
tan
(~ ~ 0 ) ,
'=sin2.lsin
c°t" /
c9-~ r cos ¢ =Gp
1
82
(SF,
rcos¢
,
8a ,,r + 8 a 82
tan(,;L'-A)
(2'-2>
0>.
Finally, the partial derivatives of the primitive function F r, with respect to c~, ~ and r can be
.... 8~u q'
expressed as: cq
(9)
(r-~O;r~-r'Ag/~=O),
8F ,_ F , c~c~ c~
(13)
~:~z[g(r,w,r')r'sinw 8~ c~r
c~r (10)
Utilising Eqs. (8-10), the gravitational attraction generated by tesseroid takes the following form
(cosg/
+
2g~r, p', r
,)rr's i n ~ ( r ' - r
cosg/)
- ~(r'~'r'~)(r'- 2r c o s ~ ) s i n ~ 2
2
- r - - s i n ~/0--3 COS2 ~]) 2
xln2g(r,g,r')+2r'-2rcosgl + r3 cos~sin3 ~ ( r ' + g ( r , w , r ' ) I 1 2g(r,g,r') g(r,g,r')+r'-rcosg
1
1
c30~(r, f2
+ 2-'7-~ r cos ¢ 8F~, =
GO
1
c3A
2
8F ,
c~r
r
c~a
(r ¢ r ' A g/ ¢ 0), 8a
8F ,
8~
c~¢
c~
c~¢
(SF ~, c?c~
+ ~2co~2~( - I g~
~
C~F< v,r' = ~ r r c o s g / s i n 2g/ 8r L
xln2~(r, gt, r')+ar'-2rcos~l
+--
+
(ll) With reference to Eq. (1), the partial derivatives of the polar spherical coordinates ~ and ~z with
e(~,~,~'t(4~_>,~ 6r
5
-2rr'cos~-3r 2 cos 2 ~) +
(c°s~, 2g~r, ¢/, r
')(~, cos ~ ~'
(14)
Chapter 60 • Analytical Solution of Newton's Integral in Terms of Polar Spherical Coordinates
- r r2 cosp,- 1/2 cosg/+rr r)
number of the integration elements such as topography or atmosphere. Consequently, the nonlinear (higher order) terms of the Taylor series for analytical upward and downward continuation can be obtained Vn ~ (2,+o¢)'c~ n V/Or".
1 r 2 COS{Z/sin2p'
2
x
Ir
~(r,p',r')
rcos/]
~(~, ~ , , / ) + ~' - ~ cos~,
(r ~ 0; r ~ r ' A ~ ~ 0).
(15)
4 Applications Once the analytical solution of Newton's integral for a potential is found analytical expressions for all the representative quantities of the Earth's gravity field can be derived. In accordance with the derivation of the components of gravitational attraction vector in the previous section these quantities can be obtained by differentiation of the analytical expression for a potential with respect to the geocentric spherical coordinates. Specifically, the second order partial derivatives of a potential with respect to the geocentric spherical coordinates yield the analytical expressions for components of the Marussi gravity gradient tensor. On neglecting the deflection of the plumbline from the geocentric radial direction, cos(-g(r, f2),r°)~ 1, where r ° denotes the unit vector in the geocentric radial direction, the linear vertical gravity gradient 0g/c~h can be approximated by the linear change of gravity with respect to the geocentric radial direction c~g/c~r. Moreover, utilising the spherical approximation (i.e. g ~-c~V/c~r ) the linear gravity gradient becomes
0 g(r, f2) ~ 0 g(r, f2______~) ~ _ 02 V(r, f2) . 0h c~r c~r2
where g' and g~' are the mean values of the gravity components generated by topography and atmosphere, and g~ the corresponding quantity generated by the masses within the geoid. The mean gravity 7' generated by topography in Eq. (18) is defined as a potential difference of values referred to the geoid (r ,f2) and the physical surface of the Earth (r,,f~) multiplied by the reciprocal value of the orthometric height H °, i.e. 1
z
f,. (~)+,o(~) t (r, f2) dr
vt(%~,~)--vt(l/'t,~) (19)
H°(a)
(16)
The analytical solution for the linear gravity gradient can then be found as the negative of the second radial derivative of the analytical function F r, in Eq. (6). Thereby
) ~2 F(x,ug,r' (nO.)l 0 2 V ( r, f2.___G Zp(AO.) , 5r ~ . ' c~r 2
Analytical integration can also be implemented in association with a rigorous definition of the orthometric heights. In such cases, a precise determination of the mean value ~ of gravity along the plumbline within the topography is required. In particular, the analytical solution can be used for efficient evaluation of the topographic effect on the mean gravity. On decomposition of the Earth's gravity field into its natural components generated by the atmosphere, topography and geoid, the mean gravity ~ becomes (Tenzer et al. 2005)
With reference to Eq. (19), the mean gravity component generated by topography g' can then be evaluated according to the analytical expression for a potential in Eq. (6), namely
gt(~'-~)--~ H
Zp( AO" ) Fct,%r,(Ao.)lr=rg(D ) . / /
(17)
(20)
where the summation is carried out over all mass elements Am - p(AOi ) A0 i of the volume element
where the function ~ ,Fr']
0 - ~ AOi, i E (1, I}, where I identifies the total
computation point at the geoidal surface r(f~)
i
is evaluated for the
413
414
R.Tenzer• P.Moore. O. Nesvadba
while IF,v,r'] c* c
c,(~)
is evaluated at the Earth's surface
To utilize the analytical formulae in terms of polar spherical coordinates for a practical use, digital terrain and density models usually discretized in a regular grid of geographical coordinates have to be transformed for each computation point to the local polar spherical coordinate system.
5
Summary
and Conclusions
Appendix I: Proof of the analytical solution for Newton's integral: The partial derivative of the analytical function F in Eq. (5) with respect to ~ gives ~a F=, ~ I r' a g(r, ~ , r') = ~ I r' 2rr' sin ') r
0~
r 2~(r,~,r
=c~ ~ ' ( r , p ' , r ' ) r '2 s i n ~ / - F
.
0.1)
Further, the partial derivative of the analytical function F r, in Eq. (6) with respect to r' yields
--=o~ cqr'
If +Co.
p',r
')
-
(c'-ccos ,)
2
((r,p',r')
+ r 2 coswsin2
'e(c. +-2
c°s '
r
2
cosg/
')cos p',r
~ / g ( r , w , r ' ) + r'_- rcos~, 1
I r~
2 g(r,g, r') ]/,¢
Application of Fubini's theorem to Newton integral in Eq. (2), where the spatial domain is expressed in terms of polar spherical coordinates, leads to the analytical solution of Eq. (3). Subsequently, analytical expressions for the gravitational attraction vector components are found in Eqs. (810). Use of analytical formulae for Newton's integral for evaluation of the gravitational potential and attraction eliminates approximation and hence errors within discretized numerical integration although discretization errors for the mass density distribution and orthometric heights remain. However, since a functional model for the actual mass density distribution is usually unknown, discretization errors through the adoption of a constant value of density within the integration element are common to both analytical and numerical solutions. The inaccuracy of the Earth's gravity field quantities computed by analytical Newtonian integration is then mostly caused by errors in the existing density and terrain models.
Op"
cos
,)
r 2 + rr2 --2rr r cos
)1 (I.2)
F
References
Anderson E.G. (1976). The effect of topography on solutions of Stokes' problem. Unisurv S-14, Rep School of Surveying, University of New South Wales, Kensington, Australia Bessel F.W. (1813). Auszug aus einem Schreiben des Herrn Prof. Bessel. Zach's Monatliche Correspondenz zur BefSrderung der Erd- und Himmelskunde, XXVII, pp. 8085. Bronstein I.N., and K.A. Semendjajew K.A. (1977). Taschenbuch der Mathematik. B.G. Teubner, Leipzig. Everest G. (1830). An account of the measurement of the arc of the meridian between the parallels o./'18 ° 3" and 24 ° 7". The Royal Society, Printed by J.L. Cox, London. Fubine G. (1958). Sugli integrali multipli. Opere scetle, Vol.
2, Crelnonese, pp. 243-249. Heiskanen W. H., and H. Moritz (1967). Physical geodesy. W.H. Freeman and Co., San Francisco. Kellogg O.D. (1929). Foundations of potential theory. Springer. Berlin. Kuhn M. (2000). Geoidbestimmung unter Verwendung verschiedener Dichtehypothesen. Reihe C, Heft Nr. 520, Deutsche Geod~itische Kommission, MiJnchen. KiJhtreiber N., G. Kraiger, and B. Meurers (1989). Pilotstudie ft~r eine neue Bouguer-Karte von Osterreich. In: Osterreichische Beitrdge zu Meteorologie und Geophysik. Heft 2, Wien. MacMillan W.D. (1930). The theory of the potential. Dover, New York. Mader K. (1951). Das Newtonsche Raumpotential prismatischer K6rper und seine Ableitungen bis zur dritten Ordnung. Sonderhefi 11 der Osterreichischen Zeitschrift fiir Vermessungswesen. Osterreichischer Verein fiJr Vermessungswesen, Wien. Martinec Z. (1998). Boundary value problems for gravimetric determination of a precise geoid. Lecture notes in earth sciences, Vol. 73, Springer. Mollweide K.B. (1813). Aufl6sung einiger die Anziehing von Linien Fl~chen und K6pern betreffenden Aufgaben unter denen auch die in der Monatl Corresp Bd XXIV. S. 522. vorgelegte s i c h findet. Zach's Monatliche Correspondenz zur BefSrderung Himmelskunde, Bd XXVII, pp. 26-38.
der
Erd-
und
Chapter 60 • Analytical Solution of Newton's Integral in Terms of Polar Spherical Coordinates
Nagy D., G. Papp, and J. Benedek (2000). The gravitational potential and its derivatives for the prism. Journal of Geodesy, Vol. 74, 7/8, pp. 552-560. Pick M. (1984). The gravitational effect of bodies with variable density. Studia Geophysica et Geodaetica, Vol. 28, Academy of Science of the Czech Republic, Geophysical institute Prague, pp. 381-392.
Tenzer R., P. Vani6ek, Santos M., W.E. Featherstone, and M. Kuhn (2005). The rigorous determination of orthometric heights. Journal of Geodesy, Vol. 79(3), pp. 82-92. Zach "s Monatliche Correspondenz zur Befi4rderung der Erdund Himmelskunde, (1811) November, Bd XXVII: p. 522.
415
Chapter 61
A new high-precision gravimetric geoid model for
Argentina
C. Tocho and G. Font Facultad de Ciencias Astrondmicas y Geofisicas, Paseo del Bosque s/n, 1900 La Plata, Argentina. E-mail: cto cho @ fc aglp. unlp. edu. ar M.G. Sideris Department of Geomatics Engineering, University of Calgary, 2500 University Drive N.W., Calgary, Alberta, T2N 1N4, Canada.
Abstract. This paper presents a new gravimetric geoid model for Argentina. The methodology applied for its computation as well as the most recent evaluated data are discussed. Argentina is bounded to the west by the highest mountain range in America, the Andes, so different topographic reduction methods are crucial in practical gravimetric geoid determination. To the east, Argentina borders on the Atlantic Ocean so the combination of shipborne gravity data and altimetry derived free-air gravity anomalies was also taken into account. The remove-compute-restore technique was adopted for the geoid determination and the contribution of the local data to the geoid was computed by FFT. As an external evaluation, the gravimetric geoid was compared with geoid undulations of 539 GPS/levelling points available for the country. The importance of this new gravimetric geoid lies in the fact that it will be the official gravimetric geoid for the country, and thus it will be of interest to the entire scientific community utilizing geospatial data.
Keywords. gravimetric geoid, Argentina, FFT, aliasing, topography
1 Introduction This paper describes the determination process of a high-accuracy and high-resolution gravimetric geoid model for Argentina. The theoretical background related to the estimation of the gravimetric geoid modeling will be briefly outlined together with the description of the data available in the area under study. Finally, the numerical results and accuracies obtained are briefly presented.
2 Area under study and data availability The area under study ranges from 20°S to 55°S in latitude and 53°W (307°E) to 76°W (284 ° E) in longitude. Part of this area is offshore, so the
KMS02 2'x2' altimetry derived free-air gravity anomaly field (Andersen et al., 2003) has been used to fill in information in the Atlantic and Pacific Oceans with the purpose to improve the quality and accuracy of the geoid. KMS02 is the newest compilation of a global altimetry-derived marine free-air gravity field at the National Danish Survey and Cadastre The distribution of the gravity data is depicted in Figure 1. The gravity data were referred to the Geodetic Reference System 1980 (GRS80) (Moritz, 2000). The gravity values were based on the International Gravity Standardization Net 1971 (IGSN71). To derive the long-wavelength information of the gravity field we used and compared the results of two global gravity field models: EGM96 and EIGEN CG01C. Two gravimetric geoid solutions will be presented in this paper; one solution was referenced to the EGM96 global geopotential model (Lemoine et al., 1998) complete to degree and order 360 and the other solution was referenced to the combined global gravity model EIGEN_CG01C (Reigber et al., 2005). The high-resolution global gravity field model EIGEN_CG01 C, complete to degree and order 360, was generated using CHAMP and GRACE satellite gravity data combined with 0.5 x 0.5 degree surface data (gravimetry and altimetry). The topographic data used to compute terrain corrections and indirect effects on the geoid by the second method of Helmert's condensation were those of the GTOPO30 DEM model, with an original grid spacing of approximately 1 km by 1 kin; however, these quantities were computed from the GTOPO30 with a grid spacing 4 km x 4 km due to the numerical instabilities encountered as the inclination of the topography increased. A total of 539 GPS ellipsoidal heights on levelling benchmarks were available in the computation area and they were used to derive the control geoid undulations for comparison with the gravimetric geoidal heights.
Chapter 61 • A New High-Precision Gravimetric Geoid Model for Argentina 75"W
70"W
65"W
60"W
where AgFA are the free-air gravity anomalies,
55'W
AgGM are the reference 25"S
25"S
30"S
30"S
35"S
35"S
411"$
40"8
GM-derived gravity
anomalies, Ag T is the topographical effect on gravity and 8g is the indirect effect on gravity, which was neglected in this study for being very small. The direct topographical effect on gravity due to Helmert's second method of condensation is equal to the negative of the terrain corrections (Bajracharya and Sideris, 2005). Equation (1) can be re-written as: Ag - Ag F A
-
Ag G M
(2)
+ C
The final gravimetric geoid is obtained in the restore step as: 4.5"S
45"S
N - N(AgFA -AgGM) + N(topo) + NGM + Nin d (3) where N(AgFA
50"S
50"S
-
AgGM) represents
geoid calculated using
(Ag F A
-
the Ag G M
residual )
in the
Stokes's formula and N(topo) is the geoid part due to the terrain corrections computed by Stokes's integral, N GM is the geoid contribution due to the 55"S
55"S
75"W
70"W
65"W
60"W
55"W
Figure 1: Distributionof gravity data in the area under study
3 Computational methodology 3.1 Gravimetric geoid modeling The gravimetric geoid determination was based on terrestrial, shipborne and satellite altimetry-derived gravity anomalies, which were used to fill in the sparse shipborne data in the Atlantic and Pacific Oceans, offshore Argentina. The gravimetric geoid was computed using the remove-compute-restore technique, employing Stokes's formula for the prediction of residual geoid heights. Before the prediction of the residual geoid, the flee-air gravity anomalies have to be reduced by a geopotential model (GM) during the remove step. Furthermore, the effect of the topography has to be taken into account through a topographic reduction. In this study, the Helmert's second method of condensation was used to account for the terrain effects. The reduced gravity anomalies can be expressed as: Ag - Ag FA - Ag GM - Ag T + 8g
(1)
reference GM, and Nind i s the indirect effect on the geoid due to Helmert's second method of condensation. Different approximations to Stokes's kernel function were investigated to compute residual geoid undulations, all in the spectral domain. Finally, it was decided to employ the 1D-FFT spherical Stokes convolution (Haagmans et al., 1992) N(q)p,)~p) = q0M RAq)A)~ Fll{ ~F{Sgtp}F{Ag(q)p,)~p)COSq)p}} 4~7 q0p=¢l
(4) where F1 and F{ 1 represent the one-dimensional Fourier transform operator and its inverse, which are performed in the longitudinal direction, while the summation is carried out in the latitudinal direction, and A¢ and A)~ are the grid spacings in latitude and longitude. The FFTGEOID 1D-FFT software, developed by Yecai Li and Michael Sideris at the University of Calgary (Sideris and Li, 1993), was used. The indirect effect of Helmert's reduction on the geoid up the second order is given, in planar approximation, following Wichiencharoen (1982) as:
417
418
C. Tocho • G. Font. M. G. Sideris
rcG9 Gp ['['h3 - h 3 Nin d - - - - h 2jj 13 dxdy (5) y ~-y E where 1 is the planar distance between computation and running point; G is the gravitational constant and 9 is the density. The evaluation of the geoid topographic indirect effect was evaluated via 2DFFT (Sideris and Li, 1993).
During the past years, many studies were carried out in order to reduce the effect of aliasing in gravity and geoid heights, using terrain digital data, for Helmert's second method of condensation (Featherstone and Kirby, 2000; Bajracharya, 2003; Bajracharya and Sideris, 2005). The free-air gravity anomalies on land were gridded in the following way: 1) Simple Bouguer anomalies (AgB) were computed at each of the gravity observations by: -
2rcGph
Table 1" Statistics of the free-air gravity anomalies (5'x 5'). Unit: [mGal] Land & KMS02
min
Ag FA
3.2 Gravity gridding
Ag B - Ag F A
At this point, we had mean free-air gravity anomalies in a 5' x 5' grid to be used in the computation of the geoid. Table 1 shows the statistics of the reconstructed free-air gravity anomalies.
(6)
2) The simple Bouguer anomalies were interpolated at the grid nodes that define the DEM, in this case 2 arc-minute by 2 arc-minute, yielding a grid of simple grid Bouguer anomalies denoted by (Ag B •
-250.50
max
mean
400.75
9.13
The
Two gravimetric geoid models were determined using the computational procedure described in section 3. They basically differ in the global gravity model used to model the long wavelength part of the geoid. One solution, namely ARG05_egm96, was computed using EGM96 (Lemoine et al., 1998) and the other solution; namely ARG05_eigen_cg01c, was calculated using the combined gravity field model EIGEN_CG01C (Reigber et. al., 2005) as reference field. The geoid heights were referred to the GRS80 ellipsoid. Table 2 presents the statistics of the two gravimetric geoid solutions (ARG05_egm96 and ARG05_eigen_cg01c) for the area under study, which are also depicted in Figures 2 and 3, respectively.
(m) 50 46 42 38 34 3o 26 22 18 14 10 6
(7)
reconstructed free-air gravity grid anomaly grid (AgFA contained values on both land and at sea, so the values at sea were eliminated and filled in with KMS02 free-air gravity anomalies. The area was bounded between latitudes 20 ° S to 55 ° S (1051 rows) and longitudes 284 ° E to 307°E (691 columns). The grid has 706241 points from which 407313 were computed as described above and 318928 at sea were KMS02 gravity anomalies
resultant
4) The free-air gravity anomalies in each cell were averaged into a 5' x5' coarser grid.
45.65
4 Gravimetric geoid model development
3) Free-air anomalies were reconstructed at each point of the grid where the simple Bouguer anomalies were interpolated by adding the Bouguer plate reduction. The Bouguer plate was computed using the height of DEM in each cell and the same topographic density used to compute the simple Bouguer anomalies. The reconstructed free-air anomalies were on a grid of 2 arc-minute by 2 arcminute and they were computed by: (AgFA)grid _(Ag B )grid + (2rtGph DEM )grid
a
z.
-z.
-6
-75
-70
-65 -60 Longitude
-55
Figure 2: ARG05_ egm96 geoid
Chapter 61 • A
New High-Precision Gravimetric Geoid Model for Argentina
the largest discrepancies between the models are located. 5.2 Comparisons at GPS benchmarks 48 44 40 36 32 28 24 20 16 12 8 4 O -4
-3 .1__.
-4
-?5
-?0
-65 -60 Longitude
-55
Figure 3:ARG05_eigen_cg0 lc geoid 5 V a l i d a t i o n of the estimated geoid models 5.1 Comparisons between geoid models Four geoid models were used for comparison in order to investigate the accuracy of the two new geoid models. These models are: EGM96, EIGEN_CG01 C, ARG05_egm96 and ARG05_eigen_cg01 c. Table 2 presents the statistics of the four gravimetric geoid solutions and their differences for the area under study. Table 2: Statistics for various geoid models and their differences. Unit: [m]. Geoid model
min
max
mean
cr
EGM96 EIGEN CG01C
-4.74
4 7 . 6 7 16.68 8.82
-4.43
4 6 . 2 8 16.70 8.75
EGM96-EIGEN CG01C
-5.84
6.64
-0.79
ARG05_egm96
-4.12
48.53
17.47 8.94
ARG05_ eigen_cg01c
-3.36
4 8 . 5 7 17.47 8.95
ARG05_egm96 ARG05_ eigen_cg0lc
-1.53
1.00
1.11
- 0 . 0 0 0.25
The comparison between the two gravimetric geoid models shows that the main differences are located along the Andes, especially in the south part of the country called Patagonia. These differences are correlated with the differences between the geoid undulations from EIGEN CG01C and EGM96. Figure 1 shows the sparse gravity measurements located in southern Argentina, where, as expected,
The accuracy of the computed models was assessed through comparisons with interpolated values of the gravimetric
geoid
GPS/levelling
( N p RAv)
points
at
( N p ps).
a network The
of
computed
differences between GPS/levelling and each geoid solution were minimized using the four-parameter transformation model of the form N GPs - N pRAy - b 0 + b 1cosq0cos;~ + b 2 cosq0sinX -b3sinq0+ o i (8) where the parameters b0, bl, b2 and b3 were calculated using the least squares technique. The four-parameter transformation model given in equation (8) absorbs all the systematic differences between the gravimetric geoid and the OPS/levelling data as well as all possible long wavelength errors and biases of the geoid. A total of 539 GPS/levelling points with the outliers removed were used as external control for the quality of the gravimetric geoid solutions. These GPS/levelling points belong to eight GPS/levelling networks, which are located in different topographies. The distribution of GPS/levelling points in Argentina is shown in Figure 4. 5.2.1 Absolute differences between gravimetric geoid models and the GPS/levelling geoid The statistics of the absolute differences before and after the fit between the GPS/levelling derived-geoid and the estimated gravimetric geoid solutions for the entire Argentina are given in Table 3. The values in bold are the results after fit. From Table 3 it can be seen that the overall agreement between the gravimetric geoid based on EGM96 and the one based on EIGEN CG01C present near the same external accuracy, which is at the 0.41-0.42 m level, before fit and at the 0.32-0.33 m level after fit. This result suggests that the accuracy and resolution of the gravity data have to still be improved in Argentina. Table 3 also shows that the global model EIGEN CG01C describes better than EGM96 the long-wavelength structure of the gravity field. After fit, the EGM96 alone fits the GPS/levelling derived geoid with a standard deviation of 54 cm while the EIGEN CG01C alone fits with a standard deviation of 36 cm. Before the fit, EIGEN CG01C alone reduces by approximately 50% the standard
419
420
C. Tocho • G. Font. M. G. Sideris
Table 3: Geoid height difference between various geoid models and GPS/levelling-derived geoid model (All of Argentina). Unit: [m].
deviation of the differences compared to EGM96 alone. 75"W
70"W
65'W
60'W
55'W
Geoid model
N EGM96_N aPs 25 IS
25"S
/
?
NEIGEN_CG01C
San~Fea
•
J
•
30"S
30"S
li Q o,,,
?
35"S
-.J
't
; f ~,,
NARG05_egm96_N GPS
r,
. °.-o,'~-
NARG05_eigen._cg01cNaPS
35'S
.',.,. ".'~
--..-::-"
Aires
40"S
•( 45'S
45"S
,J 50"S
50'S
del Fue 75"W
-70"W
~ 65'W
55"S 60'W
max
mean
cr
-2.43
1.89
0.15
0.80
-2.18
2.00
0.00
0.54
-1.37
1.60
0.48
0.40
-1.74
1.21
0.00
0.36
-0.63
2.69
1.38
0.41
-1.74
0.99
0.00
0.32
-0.83
3.02
1.43
0.42
-1.93
1.15
0.00
0.33
A regional analysis of the absolute differences between the gravimetric geoids was carried out for each GPS/levelling network. The standard deviations of the absolute differences (after fit) between the gravimetric geoids and the GPS/levelling-derived geoid at each regional GPS/levelling network can be seen in Figure 5. In Buenos Aires, the agreement level is approximately 8 cm in terms of the standard deviation for both gravimetric geoid solutions computed. In Santa Fe, the best agreement was achieved with ARG05_eigen_cg01 c (20 cm). These two networks are located in flat areas with good gravity data coverage, density and quality.
40"S
I -I >~3
"o ,Chubut
55"S
_ NaPS
rain
55'W
Figure 4: Distribution of GPS benchmarks in Argentina
0.
0.
&
0.
.~ 0. >
,~ 0.
~0. +.~ ¢/3
0. 0. i
All of Argentina
i
Buenos Aires
i
Chubut
i
Mendoza
1
Neuquan
u
i
POSGAR
SantaF6
i
Tierradel Fuego
Uruguay
Ii egm96 D A R G 0 5 _ e g m 9 6 D eigen_cg01 c D A R G 0 5 _ e i g e n _ c g 0 1 c F i g u r e 5: standard deviation of the absolute differences (after fit) between the gravimetrically geoids and the GPS/levelling-derived geoid
Chapter 61 • A New High-Precision Gravimetric Geoid Model for Argentina
Mendoza and Neuqudn are GPS/levelling networks located in the rough areas in Western Argentina. In the Neuqudn area, both global models have similar standard deviation agreement with the GPS/levelling data (44 cm), but in Mendoza, the global gravity field EIGEN_CG01C is superior by 7 cm compared to EGM96. This result is reflected in the results of the corresponding gravimetric geoids. In Neuqudn, even though both global models present similar behaviour, ARG05_egm96 is better than ARG05_eigen_cg01 c by 2 cm. The Chubut GPS/levelling network shows a very different behaviour with respect to the global models. The reason could be that the ellipsoid heights are not referred to the same datum of the other GPS points The Tierra del Fuego network is located in the Southern part of Argentina; both gravimetric geoid solutions have the same level of agreement (15 cm) but these results are slightly worse than the ones obtain with the global models alone. In the Uruguay network, the differences in the standard deviation between the gravimetric solution ARG05_eigen_cg0 lc and ARG05_egm96 solution is around 4 cm.
5.2.2 Relative differences between gravimetric geoid models and the GPS/levelling geoid To evaluate the relative accuracy of the best four geoid models with respect to the GPS/levellingderived geoid, relative geoid heights differences (ANGRAV-ANGps) were formed for all the baselines
and plotted as a function of the baseline length (spherical distance in km) in parts per million (ppm) The relative differences in ppm were formed after all outliers were removed. Figure 6 and Figure 7 show the relative differences across the entire Argentina before and after fit, respectively. The two global gravity field models have the same relative accuracies up to baselines lengths of 15 km, ranging from 8.5 ppm to 1.6 ppm. For larger baseline lengths ranging from 15 to 125 km, we can see an improvement in the long wavelength structure of the EIGEN_CG01C global model compared to the EGM96. For baseline lengths larger than 125 km to near 500 km, both models show similar relative accuracies. For 500 km to 1200 km, we can observe again an improvement of the EIGEN_CG01C global model compared to the EGM96, tending to 0 ppm for lengths over 1800 km. The two new geoid models (ARG05_egm96 and ARG05_eigen_cg01 c) present for the entire country, similar behavior for all baseline lengths, except for baselines between 15 to 115 km where EIGEN_CG01C is slightly better than EGM96 and for baselines 115 km to 700 km where EGM96 performs slightly better than EIGEN_CG01C. Comparing Figure 6 and Figure 7, we can appreciate that there is a significant improvement in the relative agreement after the fit, especially for distances greater than 225 km where both gravimetric geoid models perform better than the global geopotential models. This demonstrated the importance of using local gravity data to improve the relative accuracy.
EGM96 - • - EIGEN CG01C - -, - ARG05_egm96 - • - ARG05_eigen_cg01c
0.
0.
0.
0.
0 125
250
375
500
625
750
875 1000 1125 1250 1375 1500 1625 1750 1875 2000 distance (km)
Figure 6: Relative accuracy between geoids models and GPS/levelling-derived geoid across Argentina (before fit)
421
422
C. Tocho • G. Font. M. G. 5ideris
0.8
EGM96 - •-
EIGEN CG01C
- -~ - A R G 0 5 _ e g m 9 6 - 4 - ARG05_eigen_cg01 c
0.6
0.4
0.2
0 125
250
375
500
625
750
875
1000
1125
1250
1375
1500
1625
1750
1875
2000
distance (km)
Figure 7:
R e l a t i v e a c c u r a c y b e t w e e n g e o i d s m o d e l s a n d G P S / l e v e l l i n g - d e r i v e d g e o i d a c r o s s A r g e n t i n a ( a f t e r fit)
6 Conclusions and future plans Two new gravimetric geoid models for Argentina were developed at the Department of Geomatics Engineering during the stay of the first author at the University of Calgary. The area covered by both solutions is from 20°S to 55°S in latitude and 53°W (307°E) to 76°W (284 ° E) in longitude with a grid spacing of 5'. The computation of the two new gravimetric Argentinean geoids models, namely ARG05_egm96 and ARG05_eigen_cg01c, was based on the classical remove-compute-restore technique using the most accurate current gravity database for Argentina. The comparison of both geoid solutions with the GPS/levelling data show that the absolute agreement with respect to the GPS/levelling-derived undulations (after the systematic datum differences were removed) is near 32 cm in terms of standard deviation for. ARG05_egm96 and 33 cm for ARG05_eigen_cg01 c. A regional analysis was carried out and the statistics show that the absolute agreement level of the differences between the gravimetric solutions and the GPS/levelling-derived geoid for each network are different in flat areas like the Buenos Aires province and rugged areas like the Andes. The lack of gravity data and the roughness of the topography are similar in the areas where the GPS networks are located, so it is necessary to investigate the accuracy of the GPS/levelling-derived geoid
heights especially in rough areas where the accuracy of the levelling heights is much poorer The best overall agreement with the GPS/levelling data is achieved for the whole Argentina by the ARG05_egm96 gravimetric geoid, with a standard deviation of 0.32 m. We can conclude that with the improvement of gravity data coverage, quality and density mainly in the Andes, it will be possible to improve the accuracy of the geoid to meet the requirements needed nowadays for modern geodetic, oceanographic and geophysics applications. The densification of gravity data in the Andes can be carried out with modern measurement techniques like airborne gravimetry. As digital elevations models play an important role in the removecompute-restore technique, the SRTM3 (JPL, 2004) model with a resolution of 3" x 3" will be evaluated in Argentina. Also, the spectrum of the geoid from various gravity field signals will be investigated in future work in order to be used in the optimal combination of the geopotential model and local gravity data. Finally, a numerical solution for the altimetry-gravimetry boundary value problem (AGBVP) will be evaluated in order to combine different types of gravity data along the coastline. Also the effect on geoid modeling of applying smoothing conditions along the coastline to remove data discontinuities has to be investigated as proposed by Grebenitcharsky (2004).
Chapter 61 • A New High-Precision Gravimetric Geoid Model for Argentina
Acknowledgments The visits of the first author to the Department pf Geomatics Engineering of the University of Calgary have been supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). and GEOIDE NCE grants of the third author. References
Andersen OB, Knudsen P, Trimmer R (2003): Improved high-resolution altimetric gravity field mapping (KMS02 Global marine gravity field), international Association of Geodesy Symposia, vol. 128, Sans6 F (Ed.), Springer. Proceedings of the Symposium 128: A window on the future of Geodesy, Sapporo, Japan, June 30-July 11, 2003, pp. 326-331. Bajracharya S (2003): Terrain effects on geoid determination. UCGE Reports, Number 20181, The University of Calgary. Bajracharya S and Sideris MG (2005): Terrain aliasing effects on gravimetric geoid determination, Geodesy and Cartography vol. 54, no. 1, pp. 3-16. Featherstone WE and Kirby JF (2000): The reduction of aliasing in gravity anomalies and geoid heights using digital terrain data, Geophysics Journal International, no. 141, pp. 204-212. Grebenitcharsky R (2004): Numerical solutions to altimetry gravimetry Boundary Value Problem in coastal region. UCGE Reports, Number 20195, The University of Calgary. GTOPO30 (1996): http ://edc daac.us gs. gov/gtopo 30/gtopo 30.html.
Haagmans R, de Min E and van Gelderen M (1993): Fast evaluation of convolution integrals on the sphere using 1D FFT and a comparison with existing methods for Stokes's integral, Manuscripta Geodaetica, vol. 18, pp. 227-241. JPL (2004): SRTM-The mission to map the World, Jet Propulsion Laboratory, California Institute of technology. http ://www2 .jp 1.nasa. g ov./srtm/in de x.html. Lemoine FG, Kenyon SC, Factim JK, Trimmer RG, Pavlis NK, Chinn DS, Cox CM, Klosko SM, Luthcke SB, Torrence MH, Wang YM, Williamson RG, Pavlis EC, Rapp H and Olson TR (1998): The development of the joint NASA, GSFC and the National Imagery and Mapping Agency (NIMA) Geopotential Model EGM96, NASA, Technical Publication- 1998-206861, July, 1998. Sideris MG. and Li Y (1993): Gravity field convolutions without windowing and edge effects. Bulletin Gdoddsique vol. 67, no. 2. Moritz H. (2000): Geodetic Reference System 1980, Journal of Geodesy, vo|. 74, pp. 128-162. Reigber CH , Schwintzer P , Stubenvoll R, Schmidt R, Flechtner F, Meyer U , K6nig R, Neumayer H, F6rste Ch, Barthelmes F, Zhu SY, Balmino G , Biancale R, Lemoine J , Meixner H, Raimondo JC (2004): A High Resolution Global Gravity Field Model Combining CHAMP and GRACE Satellite Mission and Surface Gravity Data: EIGEN-CG01C accepted by Journal of Geodesy and abstract from Joint CHAMP/GRACE Science Meeting, GFZ, July 5-7, 2004 (page 16, no. 24 in Solid Earth. Wichiencharoen C (1982): The indirect effects on the computation of geoid undulations, Report of the Department of Geodetic Science and Surveying no. 336, The Ohio State University, Columbus, Ohio.
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Chapter 62
Local gravity field modeling using surface gravity gradient measurements Gy. T6th, L. VOlgyesi Department of Geodesy and Surveying, Budapest University of Technology and Economics; Research Group of Physical Geodesy and Geodynamics of the Hungarian Academy of Sciences, H-1521 Budapest, Hungary, Mtiegyetem rkp. 3.
Abstract. Almost 100,000 surface gravity gradient measurements exist in Hungary over an area of about 45 000 km 2. These measurements are a very useful source to study the short wavelength features of the local gravity field, especially below 30 km wavelength. Our aim is to use these existing gravity gradient data in gravity field modeling together with gravity anomalies. Therefore we predicted gravity anomalies from horizontal gravity gradients using the method of least-squares collocation. The crosscovariance function of gravity gradients and gravity anomalies was estimated over the area and a suitable covariance model was estimated for the prediction. The full covariance matrix would require about 15 GB storage, however, the storage requirement can be reduced to about 300 MB by inspecting the structure of the cross-covariance function. Using sparse linear solvers the computation proved to be manageable, and the prediction of gravity anomalies for the whole area was performed. The results were evaluated at those sites where Ag values were known from measurements in the computational area. Keywords: horizontal gradients of gravity, gravity gradient measurements, least-squares collocation, sparse matrix solvers
1 Introduction In view of the increasing accuracy demands of local gravity field determination in the GPS era, it seems advantageous to combine all available measurements to the gravity field for the purpose. Since our knowledge of the local gravity field is based mainly on gravity measurements, the combination of other kind of gravity field parameters (e.g. observations on the direction of the gravity vector or its horizontal gradient) with gravity measurements is preferable. Several authors developed methodologies to combine horizontal gravity gradient and gravity meas-
measurements for local gravity field determination. Vassiliou, for example, showed how to process and downward continue airborne gradiometer data (Vassiliou, 1986). Hein discussed many ways of dealing with gravity gradient measurements that are available in Germany in the Upper Rhine Valley (Hein, 1981). He processed altogether 21616 such measurements in view of local gravity field determination. Another method, the so-called gradient kriging with terrestrial gravity gradients was proposed and used by Menz and Knospe (2002) for local gravity field determination. The problem is particularly interesting to us since in Hungary we have almost 100,000 surface gravity gradient measurements. Our aim is therefore to combine these measurements with gravity data as well as other data in view of a new geoid solution (V61gyesi et al, 2004). This is the reason why the least-squares collocation method was chosen, since it is well known that within this method it is relatively easy to process different kind of gravity field parameters in a theoretically sound framework. The main problem of least-squares collocation is that it is computationally demanding as it requires the solution of a linear system of which the number of unknowns equal to the number of measurements. Therefore several authors proposed compactly supported covariance functions that lead to sparse matrix techniques to reduce the computational burden (Sans6 and Schuh, 1987). First, we briefly review the necessary details of the least-squares collocation method. Next within the framework of an application example (involving an area of about 45 000 km 2) the chosen method is investigated. Finally the results are discussed and several conclusions are drawn.
2 Optimum estimation of gravity anomalies The well-known method of least-squares collocation (Moritz, 1980) is proven to be suitable in grav-
Chapter62 • Local Gravity Field Modeling Using Surface Gravity Gradient Measurements ity field modeling, since it allows estimating any gravity field parameter from measurements of other gravity field parameters. The prediction s at any point is obtained through the following linear system
C~ -
I C xz,xz
Cxz'Yz 1 +
C,,,, + Cn~ - Cyz,xz
C yz, yz
Cnn ,
(4)
C,~ - [Cx~,Ag Cyz,Ag ]
(1)
the blocks each contain covariance functions (2) and (3), evaluated at the distance d(i, k) and azimuth
where ( is the measurement vector, Css and Cnn denote signal and noise covariance matrices, respec-
a(i, k) of measurement and/or prediction points Pi, Pk and Qi, Qk, respectively. The solution of Eq (1)
S -- C s ~ ( C s s --1-C n n ) - l g ,
tively, and Csc is the cross-covariance matrix of measured and predicted quantities. In our case now we would like to predict gravity anomalies, i.e. s = Ag(Qk) at points Qk, (k = 1 . . . . Kmax) from measured horizontal gravity gradients
Vx: Vy: at points Pi, (i = 1. . . . Nmax) i.e. ( = [Vxz(Pi) Vyz(Pi)]. The necessary isotropic covariances can be written as functions of distance d and azimuth a (counted anticlockwise from East towards North) between any pair of points in the local x = East, y = North
system
Cvyz, Vyz(d, a), CAg, Vyz(d, a).
as
Cvxz, Vxz(d, a), C/xg,wrz(d, a) and
is an optimum estimation in the least-squares sense to gravity anomalies Ag. An important restriction on the choice of the covariance function C(Ag,Ag), besides its positive definiteness, is the existence of its six derivatives defined in Eqs. (2) and (3). A simple analytical covariance function model, which behaves well under repeated differentiation and is physically possible, is the two-parameter Gaussian covariance function
C ( d ) - A e -Bd2 .
follows:
Cwrz, Vyz(d, a),
(5)
The covariance functions (2) and (3) can be obtained immediately as
The auto- and cross-covariance functions of horizontal
gradients Cvxz, Vxz(d, a), Cvxz, Vyz(d, a) are obtained by dif-
gravity
Cvyz, Vyz(d, a),
ferentiation from auto-covariance function of grav-
_ O__CC= 2ABd e -Bd2 cos oy Ox OC -Bd: - -- = 2ABd e
sin a'
~y
ity anomalies CAg, Ag(d) = C(Ag, Ag)
02C
---
= ABe-Bd2[1-
0x 2
2Bd 2
-(1 + 2Bd2)cos 2o(]
~2
Cvxz, Vxz ( d, ce) - - - - 7 C ( Ag , Ag ) 3x ~2
Cvxz,Vyz (d, ce) -
Ox~)yC(Ag, Ag) .
(2)
~2 Cgyz,gy z (d, ce) ----~-C(Ag,Ag) Oy Also
the
cross-covariances
CAg,
O2--~C= - A B e -Bd2 (1 + 2Bd 2)sin 2or OxOy O2C --= A B e - B d 2 [ 1 - 2 B d 2 + (1 + 2Bd2)cos 2a'] 0y 2 -
Vxz(d, a)
(6)
and
Czxg, Vyz(d, a)can be written similarly
All of the above covariance functions are azimuthdependent (non-isotropic). However, it is possible to introduce the isotropic functions
OC m__
= 2ABd e-Bd
2
Od O C Ag,Vxz (d, o~) - -~x C(Ag,Ag )
O:C = 2AB(1 (3)
c Ag, vy~ (d, cO - ~ c (Ag,/,,g) ely
The linear system (1) is composed of six covariance matrix blocks
G~,Ag, Cy~,A~"
Cxz, xz, Cyz,yz, Cxz,yz, Cyz,yz,
,
(7a,b)
2B dZ ) e -Bd2
0d 2 which are useful to estimate the parameters A and B in (5) from measured horizontal gravity gradients (Tscherning, 1976). The isotropic covariance functions (7) are illustrated for the parameters A = 6.5 mGal 2 and the parameter B, implicitly defined through the correlation length do - x/ln 2 / B - 6 km
425
426
Gy. T6th.
L. V 6 1 g y e s i
in Fig. 1. The correlation length do is by definition the distance where the covariance is half of the variance C(0) (i.e. C(do) - 0.5C(0)). 30 . . . 25 ._ %
, vd)
20
a compactly supported covariance function (Gneiting, 2002). Our second choice is to keep the covariance model (5) with infinite support, but neglect the covariances beyond a certain distance dmax. Through either of these achievements the covariance matrix will be sparse and thus efficient sparse matrix techniques can be used.
III
I
I
•
c~ 10 E
40000
.
.
I.. :
.
.
I 1
.,- ...... t'
=
.
.
.
I
-, .
.
I
.
.
.I
.
::: : I - . , - . . . . . . t'
-,
.
¢.J t"r-
0
o
• j,~.
....
o
30000. .....
_~
:"
!'_,,'", r'"'.
•
-10 -15 O
5
i
i
10
15
,
,::
"I'...~
• ,:
i
'./'~'~'": ......
,.
;:.
.':'
,
._,~
,
~
"
'
....
" .
:'
.....
!'_,,°',
:z
.-
r~-,
.~, .,.. :'.
'.,~''"':
,/7
"," r . . . ~
,.
,';
,
' ....
,
._,-t
,
"~ '
"
" .
--
20000
20
distance [km]
Fig. 1 Example of isotropic Gaussian cross-covariance function (7a) of gravity anomalies and horizontal gravity gradients C(Vd, Ag) as well as isotropic auto-covariance function (7b) of horizontal gravity gradients C(Vd, Vd). The parameters of the covariance function for this example are A = 6.5 mGal 2, do = 6 km. The two parameters necessary to define a Gaussian covariance function can be estimated from the empirical isotropic auto-covariance function (Tb) of
,oooo1
I
!;:. ..... ,~.~-!
20000
30000
,. ~
40000
I
I
I
I
I
I
I
-~ i : . : 4 / .
40000-
'4
IL
30000-
.....
iil,
........ ..
3 An
10000-
Our example application of the collocation equation (1) is the estimation of gravity anomalies from the surface gravity gradient dataset of Hungary (V61gyesi et al., 2004). The dataset contains 44 818 gravity gradients and cover an area of about 45 000 km2 (See statistical parameters in Table 1). The covariance matrix thus has slightly more than 2 billion elements and that would require 15 GB capacity to store the full matrix in double precision. To make our problem numerically tractable, we have two choices. First, we could use instead of (5)
,...
• ~ !';'!,:_,
I:~ ~:5
20000-
example
,. ;
Fig. 2 Nonzero pattern of the sparse C~ matrix (4) before preordering. The number of nonzero elements is 4 049 268 or 0.2%.
125 E 2, whereas the correlation length dg is about 0.7-1 km. Hence the parameters of the Gaussian covariance function (5) are approximately A - 0.83 mGal 2 and d o - 1-2 km. application
i;. ...... ~..'
10000
horizontal gravity gradients Vxz, Vyz. Since from (7b) the variance is 2AB, and the correlation length dg is connected to do according to the formula dg = 0.532 do, one can take these two parameters for the estimation of A and B. The actual gravity gradient data in Hungary, which were reduced to the normal and topographic effects, show an average variance
,... :-!-;..,_,
..
~. ~
..~"" • ,.;-#
,,
"'~
.~ ', .~ i
10000
,_ i
20000
i
30000
,,. i
40000
Fig. 3 Nonzero pattern of the Ce~ matrix (4) after approximate minimum degree (AMD) preordering
Our example computations were based on the second choice. All auto- and cross-covariances were truncated in the same way at dmax. If the parameters of the Gaussian covariance function are A - 0.5
Chapter 62 • Local Gravity Field Modeling Using Surface Gravity Gradient Measurements
mGal 2 and do - 1.5 km, the covariance drops below 5% of its m a x i m u m value at dmax = 4 km. Beyond this maximum distance all covariances were considered to be zero. Truncating covariances like this will tend to remove from the estimated Ag field any long-range (wavelength dmax ) systematic patterns present in the gradiometric data. This way the number of nonzero elements in the covariance matrix has been reduced to 4 049 268. In efficient compressed column format (Davies, 2005) with the necessary bookkeeping information the matrix can be stored in 46 MB. It was found that about 300 MB in-core memory was consumed during the assembly and solution stages of the problem, which is entirely acceptable even on a standard PC. Table 1 Statistical parameters of horizontal gravity gradients used in the calculations. All units are E (1 E6tv6s = 10.9 s-2) min
Vxz Vvz
max
mean
std
-82.80
98.90
-0.45
11.17
-173.90
225.40
0.92
11.79
For numerical tests we have developed Fortan 90 code and interfaced it with the C language LDL library (Davies, 2005). The preordering of the matrix for efficient factorization was performed through calls to the A M D library (Amestroy et al., 2004). It can be seen that the original covariance matrix (Fig. 2) after the approximate minimum degree (AMD) preordering step has a number of large nonzero blocks (Fig. 3), and this permutation prevents fill-in during the next step, the Cholesky factorization of the matrix.
The solution of the sparse linear system (1) provided us gravity anomalies at 22 409 points. The measurements were considered to be uncorrelated and with uniform noise variance. After several tests runs the noise standard deviation of horizontal gravity gradients was chosen to be +13.5 E. With this value the variance of predicted gravity anomalies was in agreement with the variance of the chosen covariance model. Moreover, this noise variance level is in agreement with the actual errors of _+1015 reGal found by Hein (1981) from his collocation experiments with horizontal gravity gradients in the Upper Rhine Valley in Germany. The histogram of predicted gravity anomalies can be seen on Fig. 4. Although there are several extremely big values (up to 400 reGal!), these are restricted only to a small area and more than 99% of the predictions fall within the _+25 reGal range. It was interesting to us to make comparisons of these results with gridded l ' x l . 5 ' free-air gravity anomalies. These anomalies were reduced to the effect of the EGM96 geopotential model. To get comparison also with the high frequency part of gravity anomalies, low-pass filtered anomalies with a Gaussian filter of length 15 km were removed. We found that the agreement seems better with high-pass filtered gravity anomalies (Fig. 5) than with the original ones. This was expected, since our previous experiences have shown that gravity gradients are more sensitive to local features of the gravity field than gravity anomalies. gravity anomalies
10000
, t_]~8~
760
63 1000 63
7 0
780
predicted
"'/11
o
E
790
high-pass filtered
100
e--
/ I -400
-300
-200
-100
0
100
200
300
400
gravity anomaly [mGal]
Fig. 4 Histogram of gravity anomalies predicted from horizontal gravity gradients at 22409 points with 150 bins. Notice the logarithmic scale on the vertical axis. Only less than 1% of the predictions are above _+25reGal.
760
770
780
790
760
770
780
790
x [kin]
Fig. 5 Comparison of gravity anomalies, high-pass filtered gravity anomalies and predictions over a selected 30x30 km2 nearly flat area. Contour interval is 1 reGal.
427
428
Gy. T6th. L. VSIgyesi
Fig. 5 also shows that predicted gravity anomalies in this region have less power than the actual gravity field. Truncating covariances would partially account for the loss of signal in these areas. On the other hand the chosen covariance model may be inappropriate for this almost flat area - especially the correlation length is too small. On the other hand if the area has non-flat topography, the predicted anomalies have considerably more power than reference gravity anomalies (Fig. 6). This raises the problem of non-stationarity of the gravity gradient signal, the variance of which is very strongly correlated with the topography of the area. The histogram of average point distances (Fig. 7) reflects the difference between flat and non-flat areas as well. This is a problem for least-squares collocation, since homogeneity and isotropy are essential assumptions of the method (Kearsley, 1977). Other methods like kriging may be interesting in this respect, which do not require stationarity assumption on the signal, only stationarity of signal increments, i.e. the intrinsical stationarity (Gneiting et al., 2000). We have also the possibility to smooth the gravity field by removing additional topographical effects from the gravity gradient signal or to make the predictions separately for flat and nonflat areas.
4 Conclusions and recommendations In the present study it was shown how efficient sparse matrix techniques can be used in local gravity field modeling with horizontal gravity gradients. The example computation with Hungarian gravity gradient data has suggested that non-stationarity of gravity gradients makes it difficult to achieve a uniformly good prediction in areas of different topography. On the other hand gravity anomalies predicted from horizontal gradients may show significant details at short wavelengths of the gravity field which are not necessarily present in gravity anomalies.
3000
non-flat areas
2500 t-
5
{D.
Y
2000
0
flat areas
15oo
E e--
1000 500
gravity anomalies 1
780
785
790
795
800
high-pass filtered
.1/ o i/ 780
785
790
795
contour interval: 20 mGal
800
780
x [km]
785
790
795
3 4 5 point distance (km))
6
7
Fig. 7 Histogram of average point distances of gravity gradient observations. It can be observed that fiat and non-fiat areas have different average point distances
contour int.: 5 mGal
predicted
2
800
contour interval: 5 mGal
Fig. 6 Comparison of gravity anomalies, high-pass filtered gravity anomalies and predictions over a selected 20x20 km2 non-fiat area. Contour interval is 5 mGal for the upper and right subfigures and 20 mGal for left subfigure. Notice the very high variance of predicted gravity anomalies from the horizontal gradients
Hence we propose to use gravity gradients together with topography and gravity measurements to yield a better model of the local gravity field than from gravity measurements alone. Our results have shown, however, that the topography of the area has a strong impact on the gravity gradient signal and it must be considered carefully. Further tests should also be done with other covariance models and especially compactly supported covariance functions. Non-stafionarity of gravity gradients can be a problem in combined modeling of the gravity field and it deserves further attention and research.
Ac knowledgements Our investigations are supported by the National Scientific Research Fund (OTKA T-037929 and
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Chapter 62 • Local Gravity Field Modeling Using Surface Gravity Gradient Measurements
T - 0 4 6 7 1 8 ) . T h e careful and thoughtful r e v i e w o f our p a p e r b y M. V e r m e e r is also g r e a t l y appreciated.
References Amestroy, P.R. Davis, T.A. Duff, I.S. (1996) An approximate minimum degree ordering algorithm. SlAM J. Matrix Anal. Applic., Vol. 17(4), pp. 886-905. Amestroy, P.R. Davis, T.A. Duff, I.S. (2004) Algorithm 837: AMD, an approximate minimum degree ordering algorithm. ACM Trans. Math. Softw., Vol. 30(3), pp. 381-388. Davies, T.A. (2005) Algorithm 8xx: a concise sparse Cholesky factorization package. Dept. of Computer and Information Sci. and Eng., Univ. of Florida, Gainesville, USA. (http ://w ww. cise. ufl. edu/-da vies). Gneiting, T. Sasvfiri Z. Schlather, M. (2000) Analogies and Correspondences Between Variograms and Covariance Functions. NRCSE Technical Report No. 056. Gneiting, T. (2002) Compactly Supported Correlation Functions. Journal of Multivariate Analysis, Vol. 83, pp. 493508. Hein, G (1981) Untersuchungen zur terrestrischen Schweregradiometrie. VerOffentlichungen der Deutschen Geodiitischen Kommission bei der Bayerischen Akademie der Wissenschaften Reihe C, Heft Nr. 264, Mtinchen.
Kearsley, W. (1977) Non-stationary Estimation in Gravity Prediction Problems. OSU Report No. 256, The Ohio State University, Dept. of Geod. Sci, Columbus, Ohio. Menz, J, Knospe, S (2002). Lokale Bestimmung des Geoids aus terrestrischen Gradiometermessungen unter Nutzung der geostatistischen Integration, Differentiation und Verkntipfung. Zeitschriftf~r Vermessungswesen, Vol 127. No 5. pp. 321-342. Moritz, H. (1980) Advanced Physical Geodesy. Herbert Wichmann Verlag Karlsruhe & Abacus Press, Tunbridge Wells Kent. Sans6, F. Schuh, W.-D. (1987) Finite covariance functions, Bull. Gdodesique, Vol. 61, pp. 331-347. Tscherning, C.C. (1976) Covariance expressions for second and lower order derivatives of the anomalous potential. OSU Report No. 225, The Ohio State University, Dept. of Geod. Sci, Columbus, Ohio. Vassiliou, A. A. (1986) Numerical Techniques for Processing Airborne Gradiometer Data. UCSE Report No. 20017, The University of Calgary, Calgary, Alberta, Canada. V61gyesi L, T6th Gy, Csap6 G (2004): Determination of gravity anomalies from torsion balance measurements. Gravity, Geoid and Space Missions GGSM 2004. IAG International Symposium Porto, Portugal. Jekeli C, Bastos L, Fernandes J. (Eds.) Springer Verlag Berlin, Heidelberg, New York; Series: IAG Symposia, Vol. 129. (in press)
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Part IV Earth Processes:Geodynamics, Tides, Crustal Deformation and Temporal GravityChanges Chapter 63 Chapter 64 Chapter 65 Chapter 66 Chapter 67 Chapter 68 Chapter 69 Chapter 70 Chapter 71 Chapter 72 Chapter 73 Chapter 74 Chapter 75 Chapter 76 Chapter 77 Chapter 78 Chapter 79 Chapter 80 Chapter 81 Chapter 82 Chapter 83 Chapter 84 Chapter 85
Absolute Gravity Measurements in the Southern Indian Ocean Slow Slip Events on the Hikurangi Subduction Interface, New Zealand A Geodetic Measurement of Strain Variation across the Central Southern Alps, New Zealand New Analysis of a 50 Years Tide Gauge Record at Canan~ia (SP-Brazil) with the VAV Tidal Analysis Program Atmospheric Mass Flow Reduction for Terrestrial Absolute Gravimetry in the Fennoscandian Land Uplift Network Tilt Observations around the KTB-Site/Germany: Monitoring and Modelling of Fluid Induced Deformation of the Upper Crust of the Earth Understanding Time-Variable Gravity Due to Core Dynamical Processes with Numerical Geodynamo Modelling A New Height Datum for Iran Based on the Combination of the Gravimetric and Geometric Geoid Models Monthly Mean Water Storage Variations by the Combination of GRACE and a Regional Hydrological Model: Application to the Zambezi River The Use of Smooth Piecewise Algebraic Approximation in the Determination of Vertical Crustal Movements in Eastern Canada Hydrological Signals in Gravity- Foe or Friend? Applications of the KSM03 Harmonic Development of the Tidal Potential Continental Hydrology Retrieval from GPS Time Series and GRACE Gravity Solutions Gravity Changes in Northern Europe As Observed by GRACE Investigations about Earthquake Swarm Areas and Processes Sea Level and Gravity Variations after the 2004 Sumatra Earthquake Observed at Syowa Station, Antarctica Improved Determination of the Atmospheric Attraction with 3D Air Density Data and Its Reduction on Ground Gravity Measurements Solid Earth Deformations Induced by the Sumatra Earthquakes of 2004-2005 Environmental Effects in Time-Series of Gravity Measurements at the Astrometric-Geodetic Observatorium Westerbork (The Netherlands) Numerical Models of the Rates of Change of the Geoid and Orthometric Heights over Canada Optimal Seismic Source Mechanisms to Excite the Slichter Mode Recent Dynamic Crustal Movements in the Tokai Region, Central Japan, Observed by GPS Measurements New Theory for Calculating Strains Changes Caused by Dislocations in a Spherically Symmetric Earth
Chapter 63
Absolute Gravity Measurements in the Southern Indian Ocean M. Amalvict, [email protected] Institut de Physique du Globe de Strasbourg / Ecole et Observatoire des Sciences de la Terre (UMR 7516 CNRS-ULP), 5 rue Rend Descartes, 67000, Strasbourg, France National Institute of Polar Research, 1-9-10 Kaga, Itabashi-ku, 173 8515, Tokyo, Japan Y. Rogister, B. Luck, J. Hinderer Institut de Physique du Globe de Strasbourg / I~cole et Observatoire des Sciences de la Terre, 5 rue Rend Descartes, 67000, Strasbourg, France
Abstract. In March - April 2005, Absolute Gravity (AG) has been measured by the Strasbourg gravimetry team using gravimeter FG5#206 at 4 French scientific stations in the Indian Ocean. Crozet, Kerguelen and Amsterdam are isolated Islands located approximately 3000 km south of La Reunion Island. Four times a year, the multi-purpose ship Marion-Dufresne II connects them to La Reunion. Measurements were taken at Crozet from 8 to 10 March, Kerguelen from 14 to 18 March, and Amsterdam from 20 to 24 March. Moreover AG was measured at the Volcanic Observatory of the Piton de la Fournaise, La Reunion Island, from 31 March to 4 April. The 2005 AG measurements were the first ever made at Amsterdam Island. In contrast, AG measurements were previously done at Crozet in 2003, Kerguelen in 2001 and 2003, and finally at La Reunion in 2001 and 2003. Here, we report on the first AG measurement at Amsterdam, present the 2005 AG values at each station and compare them to previous ones. Then we pay special attention to the comparison of the gravity changes with available vertical velocities derived from precise positioning techniques (GPS, DORIS) at Kerguelen and La Reunion. The uncertainty on AG variations is too high for inferring any change and stability seems quite likely. DORIS solutions show a small uplifte (a few mm/yr) at both locations.
Keywords. Absolute Gravity, DORIS, GPS, Kerguelen, Crozet, Amsterdam, La Reunion, indian Ocean
1
Introduction
The southern Indian Ocean covers a large area with only a few emerged islands, which are mainly vol-
canic. The Kerguelen, Crozet and Amsterdam Islands host permanent scientific bases. Due to this sparseness of lands, the knowledge of AG at these few points is important. It provides data for the worldwide coverage of AG values, for the global gravity field of the Earth, and for the Mean Sea Level (MSL). A 5-year program of the Gravimetric Observatory of Strasbourg, supported by the French Polar institute, is devoted to the measurement of AG at these scientific bases. The last campaign of the program took place in March-April 2005 with measurements at each of the bases; in addition, a measurement was taken at La Reunion Island, which is the departure place of the ship.
Absolute gravity measurements in Indian Ocean in 2005 AG measurements were performed during the logistic cruise of the multi-purpose ship MarionDufresne II, from 3 to 30 March 2005. The MarionDufresne II departed from La Reunion at the end of February 2005 and came back at the beginning of April, after a counter clockwise trip, calling chronologically at Crozet, Kerguelen and Amsterdam islands. The duration of the measurements at each station was determined by the duration of the logistic stop. AG was measured with the ballistic absolute gravimeter FG5#206 of the Strasbourg Gravimetric Observatory. The instrument was unloaded, set up and loaded back on board at each station. Raw data were processed using the version 3.0806 of the "g" software from Micro-g Compaw. All the raw data were reduced in a similar way for solid Earth tides (ETGTAB), ocean loading (Schwiderski model for the Kerguelen data, Shwiderski
434
M. Amalvict. Y. Rogister • B. Luck. J. Hinderer
(1980) and FES model for the Amsterdam, Crozet and La Rdunion data), atmospheric pressure (admittance factor-0.3 gGal/hPa for the difference between observed and nominal pressures), polar motion (position of the pole from IERS website) and vertical gravity gradient measured at each station. Whenever measurements had been previously made at a given station, we used the same parameters (tide models, vertical gravity gradient, and admittance for atmospheric pressure) as previously to reduce the data. The procedure for the AG observations was the same at each station, and was the same in 2001 and 2005. Moreover, the technical operator was the same for all the measurements.
b. A G m e a s u r e m e n t s at K e r g u e l e n - P o r t - a u x Frangais
Measurements took place from March 14 to March 18 at the new reference station established in 2003 in the sacristy of the Port-aux-Frangais (PAF) church. There are 79 hourly sets of 160 drops, every 20 seconds. c.
A G m e a s u r e m e n t s at A m s t e r d a m - M a r t i n de Vivi/~s
The station, in a closed garage, has been chosen in 2001; a measurement started in 2003, which had to be interrupted because the ship had to leave for rescue mission at Kerguelen. There are 43 hourly sets of 100 drops, every 10 seconds, from March 22 to March 24. d. A G m e a s u r e m e n t s at L a R 6 u n i o n I s l a n d Volcanologic Observatory
F i g u r e 1. Geographical zone of measurements a. A G m e a s u r e m e n t s at C r o z e t - A l f r e d F a u r e
Measurements took place from March 8 to March 10, at the point in the shooting hall, which has been chosen in 2001 and established as a reference station in 2003. There are 52 hourly sets of 100 drops, every 10 seconds.
Measurements took place from March 31 to April 3 at the new reference station established in 2003 in one of the garage slots at the Volcanological Observatory of the Piton de la Fournaise volcano. There are 75 hourly sets of 100 drops, every 10 seconds. Table 1 shows the value of gravity at each station; the value is given at the ground level, after all reductions. Table 1 provides altogether the coordinates of the stations, the nominal pressure calculated according to IAGBN standards (Boedecker, 1988), the vertical gradient of gravity at the reference point, the number of measured drops and the date of measurements.
T a b l e 1 - Station parameters and gravity value (2005), at ground level. Station
Latitude Longitude Height (m)
Nominal pressure
Number of drops
Date
g value
(2005)
(~Gal)
Crozet Alfred Faure
46.430 ° S 51.861 o E 140
996.5441
-3.23
5176
8-10 March
980 964 482.69 ± 3.31
Kerguelen Port-aux-Frangais Sacristy
49.35 ° S 70.2 ° E 17
1011.21
-3.41
9224
14-18 March
981 059 354.49 ± 3.22
Amsterdam Martin de Vivi~s
37.68 ° S 77.53 ° E 35
1009.05
-3.24
3581
22-24 March
980 092 602.88 ± 3.83
R6union Volcanological Observatory
21.21 o S
840.38
-3.40
7060
31 March -3 April
978 638 097.67 ± 5.04
55.56 ° E 1550
(hPa)
Vertical gradient of gravity
(gGal/cm)
Chapter 63 • Absolute Gravity Measurements in the Southern Indian Ocean
Previous AG measurements Previous campaigns and results have been reported by Amalvict et al., 2001 and Amalvict et al., 2003.
AG measurements have been taken 3 times at both Kerguelen and La Reunion islands. Table 2 shows the value of gravity at each site and we detail below the different measurements.
T a b l e 2 - Time series of gravity values (in btGal) at Kerguelen and La Reunion, at ground level.
* Amalvict et al., 2001, ** Amalvict et al., 2003. 2001"
2003**
981 061 041.1 + 5.0
981 061 039.40 + 4.75
Kerguelen sacristy
---
981 059 349.93 + 3.76
981 059 354.49 + 3.22
La Reunion Le Port
978 919 406.7 + 5.3
978 919 424.25 + 4.36
---
---
978 638 114.34 + 2.82
978 638 097.67 + 5.04
Kerguelen B 1
2005
...........................................................................................................................................................................................................................
La Reunion Volcanological Observatory
a.
Kerguelen
AG was measured for the first time in 2001; the station was on a pillar in the shelter B1. This shelter is to be removed in the future. Therefore, in 2003, a new reference station has been established in the sacristy of the church about 100 m from B1. In 2003, AG was also measured in B 1. Taking into account the 2003 AG tie, we refer all the values to the sacristy point. The AG horizontal gradient between the 2 locations measured in 2003 is g(B1)-g(sacristy) = 1 689.47 + 6.06 btGal. We use this value to reduce to the "sacristy" the value measuremed in 2001 at the B~ site. Thus g-gr is 351.63 + 7.85 luGal in 2001, 349.93 + 3.76 gGal in 2003 and 354.49 + 3.22 gGal in 2005, where g~ is 981 059 000.00 gGal. Let us notice that the standard deviation is large in 2001, not because of the measurement itself, but because of addition of variances in the transfer of the value from B1 to the sacristy, that is to say [(4.75)2+(3.76)2+(5.0)2] 1/2= 7.85 btGal.
Thanks to the 2003 AG tie all the values are referred to the Volcanological Observatory site. The AG horizontal gradient between the 2 locations measured in 2003 is g(Meteo)-g(Obs) = 281 309.91 + 5.19 laGal. We use this value to reduce to the Observatory site the "MeteoFrance" measurement in 2001. Thus g-gr is 96.79 + 7.42 btGal in 2001, 114.34 + 2.82gGal in 2003 and 097.67 + 5.041aGal in 2005, where gr is 978 638 000.00 luGal. Let us note again that the standard deviation is large in 2001 because of addition of variances in the transfer of the value, that is to say [(4.36)2+(2.82)2+(5.3)2] ~/2 = 7.42 laGal. ¢. A G a n a l y s i s
We can now summarise the previous discussion on AG values in Table 3 and Figures 2 and 3. T a b l e 3 - Gravity values (g-gr) at Kerguelen and La Reunion, at ground level, in gGal. gr = 981 059 000.00 gGal at Kerguelen, and g~ = 978 638 000.00 gGal at La Reunion.
b. L a R 6 u n i o n
AG was measured for the first time in 2001; the station was in the building of MeteoFrance in Le Port. Unfortunately, it is built on an embankment and a new reference station had to be established in 2003 at the Volcanological Observatory, on the top of the volcano Piton de la Fournaise. AG has also been measured at the MeteoFrance station in 2003.
LaReunion Vol. Obs. Ker
Sacristy
2001
2003
2005
096.79 + 7.42
114.34 + 2.82
097.67 + 5.04
351.63 +7.85
349.93 +3.76
354.49+ 3.22
Fig. 2 shows the time series of AG values at Kerguelen (referred to the sacristy). A linear fit of
435
436
M. Amalvict. Y. Rogister • B. Luck. J. Hinderer
the data, weighted by the error bars leads to a slope of + 1.34 + 1.79 laGal/yr. Fig. 3 shows the time series of AG values at La R6union (referred to the Volcanological Observatory). A linear fit of the data, weighted by the error bars leads to a slope o f - 2.03 + 2.13 pGal/yr. This value is obviously meaningless. In other words, there is no significant linear trend at any station which is in favour of a stability of the sites. The 2003 high value of gravity at the Piton de la Fournaise could be related to volcanic activity (an eruption indeed occurred in May 2003), cf Section 4.
Kerguelen - Sacristy
365 360 355 (9
350
g 345
340 335 2000
.
. 2001
.
.
. 2003
2002
. 2004
2005
2006
Comparison of height changes and gravity variations The measurements of absolute gravity at a given station are episodic, contrary to the techniques of precise positioning such as GPS or DORIS system. It is therefore very important to operate measurements at sites equipped with different techniques. Co-located sites are now the required standard to monitor (vertical) displacements. a. DORIS analysis
DORIS (Doppler Orbitography by Radiopositioning Integrated on Satellite) observations started in 1987 at Amsterdam, Kerguelen and La R6union and at the end of December 2003 at Crozet. The present DORIS beacon is the third one at Amsterdam (AMSTB) and Kerguelen (KERB) and the second one (REUB) at La R6union. Cr6taux et al. (1998) give the first results for DORIS observations. Weekly or monthly solutions can be found on Internet at ftp ://cddis.gsfc.nasa.gov/pub/doris/products/sinexseries. DORIS values of the vertical displacement given in Table 4 come from this website.
YEAR
b. GPS analysis Figure 2 Time series of absolute gravity values at Kerguelen, at ground level and referred to the sacristy.
140
La Raunion Volc Obs
130 120 o~
The stations that we are interested in are also equipped with GPS (Global Positioning System) receivers. Kerguelen GPS receiver (KERG) is installed since November 1994 and belongs to IGS (International GPS Service). Bouin and Vigny (2000) gave the first GPS analyses in that part of the world. The GPS station at La R6union belongs to OGS for a few months. The daily JPL results are available at the website: http://wwwgpsg.mit.edu/-tah/MIT_IGS_AAC/index2.html, from where we got the solutions we present below.
(9 110 v
c. Comparison of results from different techniques
lOO
~o'0~
'
~0'0~
'
~0'0~
'
~0'0,
'
~0'0~
YEAR
Figure 3 Time series of absolute gravity values at La R6union, at ground level and referred to the Volcanologic Observatory.
The values of vertical displacement observed by DORIS and GPS are shown in Table 4, together with the AG changes. Note that this comparison is not presented for Crozet because there are only two AG measurements and no GPS data for this site. The DORIS observations are from 2001 at Kerguelen and 1999 at La R6union. The GPS solutions presented for Kerguelen are from two different
Chapter 63 • Absolute Gravity Measurements in the Southern Indian Ocean
analysis centres (jpl and cod) but for the same period of observation (from the end of 1996 to mid2001). GPS solutions are very different from one analysis centre to an other one. We have chosen the longest time series available at the website. Table 4 - . Gravity change vs height change at Kerguelen and La Reunion. The 2 GPS solutions are provided by 2 different analysis centres.
DORIS (mm/yr) GPS (mm/yr) AG (btGal/yr)
Kerguelen 4.6 + 0.3 + 2.3 + 0.2< < + 7.3 + 0.3 + 1.34 + 1.79
La Rdunion 2.1 + 0.2 not available 2.03 + 2.13
At Kerguelen, both DORIS and GPS observations suggest a few mm/yr uplift. AG measurements suggest stability. La Reunion Island is a "hot-spot" intraplate volcano. Its total height is 7500 m of which 3000 emerge; its basal diameter on the oceanic floor is 240 kin. The Observatory has been established in 1979 at 15 km from the summit of the volcano and geophysical networks are deployed since then. (http://ovp.iniv-reunion.fr). The analysis of these geodetic and geophysical observations is under investigations by several teams and not yet published. Eruptions are quite frequent, the last ones are: October 2000, March, June and November 2001, January and November 2002, May 2003, February and October 2005. Note that the 2003 eruption occurred only slightly after our AG measurements. There were no GPS solution available for this study at La Reunion island. DORIS data show a small uplift, in agreement with the decreasing gravity, but the uncertainty on the AG trend does not allow to draw any final conclusion.
5
Conclusions
AG measurements have been successfully conducted at Amsterdam for the first time and AG measurements have been successfully repeated at Crozet, Kerguelen and La Reunion, in March-April 2005. Repeated measurements at Kerguelen and La Reunion lead to the following remarks: the gravity variations at Kerguelen and at La Reunion could suggest stability of both sites, nevertheless comparison of AG changes and vertical velocity derived from positioning observations is far from being straightforward, because of the insufficient number of AG measurements. All these AG measurements
need to be remade to have a better estimate of any gravity trend that could be compared to height changes. Moreover GPS solutions are very different from one analysis centre to an other one. In addition, let us note that the knowledge of AG values is useful for both relative marine gravimetry and gravity network at the Volcanological Observatory.
Acknowledgments. This study was carried out during the stay of MA at the National institute for Polar Research (NIPR), Tokyo, Japan under a fellowship from the Japanese Society for the Promotion of Science (JSPS). Authors thank all the people from IPEV, TAAF, and Marion-Dufresne II for their support and help. The authors thank the two reviewers for their constructive comments which helped improving the manuscript.
References
Amalvict M., Hinderer J., Boy J.P. and Luck B., 2001, Gravity at Kerguelen (indian Ocean): Absolute Gravity Measurements and Tidal Analysis from a relative gravimeter data, lAG General Scientific Assembly, Budapest, September 2001, proceedings on CD-Rom Amalvict M., Bouin M-N., Hinderer J. and Luck B., 2003, Results of the first absolute gravity measurements at Crozet Island, and repetition at Kerguelen and La Reunion islands, 9 th Symposium on Antarctic Earth Sciences, Potsdam, Germany, September 2003. Boedecker, G. 1988. International Absolute Gravity Basestation Network (IAGBN). Absolute gravity observations data processing standards & station documentation. BGI Bull. Inf. 63, 5157. Bouin M.N. and Vigny C., 2000, New constraints on Antarctic plate motion deformation and deformation from GPS data, Journal of Geophysical Research Solid Earth, Vol. 105(B12), pp. 28279-28293. Cretaux J.F., L. Soudarin, A. Cazenave, F. Bouille (1998). Present-day Tectonic Plate Motions and Crustal Deformations from the DORIS Space System, Journal of Geophysical Research, Solid Earth, Vol. 103(B 12), pp. 3016730181 Schwiderski E.W., 1980, On charting global ocean tides, Rev. Geophys. Space Phys., 18, 1,243268
437
Chapter 64
Slow Slip Events on the Hikurangi Subduction Interface, New Zealand J. Beavan, L. Wallace, H. Fletcher GNS Science, P O Box 30368, Lower Hutt, New Zealand A. Douglas School of Earth Sciences, Victoria University of Wellington, P O Box 600, Wellington, New Zealand Abstract. In common with other regions where continuously-recording Global Positioning System (CGPS) networks have been established above subduction zones, several aseismic deformation episodes have been observed in New Zealand since 2002. We interpret these episodes to result from slow slip on the subduction interface, though with the current density of CGPS stations the details of most events recorded to date are not well resolved. We have observed events with accompanying surface displacements ranging from 5-30 mm magnitude, and lasting several days to more than a year. Modelling suggests that the events are occurring near the down-dip end of the locked seismogenic part of the subduction zone, in the transition zone between the interseismically coupled and creeping portions of the interface. Comparison of event sizes, inter-event deformation rates, and long-term deformation rates suggest a repeat time of 2-3 years for an October 2002 event recorded near G isborne in the northern Hikurangi margin. A similar-sized event recorded in November 2004 supports this estimate. Two longer-duration slow slip events beneath the central and southern North Island may have triggered a series of small to moderate earthquakes over the past two years. There is preliminary indication of seismic tremor associated with the Gisborne events, as has been observed in Japan and western North America, but more work is needed to confirm this. The best-documented slow-slip event in the North Island occurred beneath the Manawatu region, and lasted from early 2004 until June 2005. The event caused displacements at up to seven CGPS sites, and probably resulted from up to 300 m m of slip on the subduction interface.
Keywords. GPS, aseismic slip, slow earthquakes
1 Introduction Transient fault slip episodes, occurring over much longer time periods (days, months) than earthquakes, have been recorded with continuouslyrecording Global Positioning System (CGPS) instruments located at several subduction margins on the Pacific Rim (e.g., Dragert et al. (2001); Ozawa et al. (2001, 2003); Larson et al. (2004)). The episodes are detected at the Earth's surface as non-linear motion of CGPS sites that is often rapid compared to normal tectonic plate motions. The physics of the deformation mechanisms underlying these so-called slow slip events, or slow earthquakes, is not yet well understood. The events may trigger other types of deformation, such as actual earthquakes, and may make a significant contribution to moment release in subduction zones. Quantifying their size, location and frequency is therefore a key task in characterizing seismic hazard for subduction zones. Subduction of the Pacific Plate occurs beneath the North Island of New Zealand, and at least five distinct slow slip events have been observed at CGPS sites in the North Island over the past three years (e.g., Beavan et al. (2003); Douglas et al. (2005)). The CGPS sites have been established as part of the PositioNZ (www.linz.govt.nz/positionz) and GeoNet (www.geonet.org.nz) networks. The slow slip events have occurred in at least three different locations on the subduction interface (Figure 1). Some have lasted only a week, while others have continued for more than a year. Some events have caused small (~5 mm) displacements at the surface, while others have caused CGPS sites to move more than 30 mm. In all cases, the events seem to occur near the down-dip end of the well coupled, or seismogenic, part of the subduction interface, consistent with observations from Cascadia and Japan (e.g., Dragert et al. (2001); Ozawa et al. (2003)). We find that the recurrence intervals of some events may be two to three years,
Chapter 64 • Slow Slip Events on the Hikurangi Subduction Interface, New Zealand
and that other events may take five years or longer to recur. The diverse characteristics of slow slip events observed thus far at the Hikurangi margin highlights the importance of this location as a natural laboratory for understanding aseismic deformation events at subduction margins.
GEOLOGICAL
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2 Tectonic setting and interseismic coupling distribution The North Island of New Zealand lies in the b o u n d a r y zone between the obliquely converging Pacific and Australian plates. North Island active tectonics is dominated by westward subduction of the Pacific Plate beneath the eastern North Island at the Hikurangi T r o u g h (Figure 2). Clockwise rotation of the eastern North Island forearc leads to back-arc rifting in the Taupo Volcanic Z o n e (TVZ), while strike-slip faulting in the eastern North Island dextral fault belt ( N I D F B ) occurs due to the oblique convergence b e t w e e n the Pacific and Australian plates. Wallace et al. (2004) used campaign GPS data to estimate forearc rotation and interseismic coupling on the Hikurangi subduction interface. The coupled zone is wider beneath the southern North Island, and it narrows and shallows towards the north (Figures 1 and 3). Much of the North Island overlies the "transition zone" b e t w e e n the coupled and creeping portions of the subduction interface (see also Reyners (1998)).
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439
440
J. Beavan • L. Wallace. H. Fletcher. A. Douglas
3 Analysis techniques and time series We process the CGPS data to give daily position estimates using Bernese version 4.2 and 5 software (Beutler et al. (2001); Hugentobler et al. (2004)). We use fixed IGS final orbits then place the daily coordinate results in a global reference frame using a least-squares fit to the ITRF2000 coordinates of a set of regional IGS stations. For the New Zealand sites, we then remove outliers and apply regional filtering to reduce remaining common-mode noise in the daily position time series, in an iterative process (Zhang et al. (1997); Beavan (2005)).
4 Gisborne events
indication that some of the Hastings events follow a few months later than Gisborne events, suggesting a possible along-strike migration with time similar to that observed by Dragert et al. (2001) in Cascadia and Ozawa et al. (2003) on the Boso Peninsula. The deformation at HAST in late 2004 and early 2005 may result largely from the Manawatu event discussed below. (a) E" 150 E E
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In October 2002, a rapid (compared to normal plate motion) surface deformation event of 20-30 mm magnitude was observed over a 10 day period on two CGPS instruments near Gisborne (GIS 1 and GISB; Figure 4). The event occurred in the preliminary stages of CGPS network development above the northern Hikurangi subduction zone so was not well recorded spatially. Forward modelling by Douglas et al. (2005) indicates that the event was probably due to about 180 mm of aseismic slip on the subduction interface just offshore of the Gisborne region. The event was fairly shallow, at ~10-14 km depth, but is consistent with slip occurring near the deeper end of the well-coupled part of the plate interface in this region. By balancing the magnitude of slip and the long-term plate motion, Douglas et al. (2005) predict that events of similar magnitude could recur every 2-3 years. A similar event was, in fact, recorded in November 2004, within the predicted interval. Slow slip events in Japan and Cascadia, have been associated with a seismic noise signal, or "subduction zone tremor" (Obara (2002); Rogers and Dragert (2003); Obara et al. (2004)). Douglas (2005) has made a preliminary attempt to identify such signals in regional broadband seismic data for the 2002 and 2004 Gisborne events. There does seem to be an indication of these signals, and further work is planned.
5 Hastings events Several small displacement events have been observed at a CGPS site at Hastings (HAST) since its installation in late 2002 (Figure 5). These events are generally not observed at many CGPS sites because of poor spatial sampling. There is an
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Fig. $ Smoothed time series showing several deformation events detected at sites HAST and RIPA (Figure 1). Linear fits have been subtracted to make the velocities between events approximately zero. The 2003 event lags the similar Gisborne event by ~2 months. The early 2005 deformation is well modelled as part of the Manawatu slow slip event.
Chapter 64 • Slow Slip Events on the Hikurangi Subduction Interface, New Zealand
6 Kapiti Coast event A CGPS site at PAEK on the Kapiti Coast (Figure 6) moved steadily westward at 25 mm/yr (relative to the Australian Plate) from 2000 to about May 2003, when it suddenly slowed to only 15 mm/yr westward. At the same time PAEK began relative uplift at about 10 mm/yr. During the same interval, WGTN continued to show fairly steady westward motion at 30 mm/yr and no clear change in vertical motion. The changes at PAEK lasted about a year then the site resumed approximately its previous motion.
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The signal can be explained by ~500 mm of slip occurring slowly on a deep part of the subduction interface between about 30 km and 40 km depth. In order to produce the observed 10 mm/yr change in horizontal motion at PAEK, while producing little change in motion at WGTN, the area of slip must be fairly small; in particular the southeastern and southwestern edges (see Figure 9) are fairly well constrained. Also, the inferred slip direction is o b l i q u e - it is approximately parallel to the plate motion direction (west-southwest) rather than being approximately down dip of the subducted plate (northwest). For oblique slip to occur on the deeper part of the subduction interface is reasonable, since the proposed slip event is occurring to the west of most of the upper-plate strike-slip faults. Several swarms of earthquakes up to ML ~5 have been felt in the region from mid 2003 to early 2005. The earthquakes are located up-dip of the supposed slow slip event (Figure 7). They are predominantly normal faulting events and their hypocentres, estimated by double differencing, line up on two planes dipping approximately southeast within, but near the top of, the subducting plate (M. Reyners, pets. comm., 2004, 2005). Coulomb stress changes calculated for the postulated slow slip event are in the correct sense to favour this style of faulting (R. Robinson, pets. comm., 2004). A plausible explanation for these observations is that the stress changes produced by the slow slip event triggered incremental slip on pre-existing faults in the crust of the subducted plate.
7 Manawatu regional event Up to seven CGPS sites in the central North Island experienced displacements (a few to 30 mm) during an eighteen month period from early 2004 until June 2005 (Figure 8). The sites with the largest movements were TAKP, WANG and DNVK, which straddle the Manawatu r e g i o n hence our name for the event. Additionally, TAKP and DNVK underwent at least 20 mm of uplift during this period. In Figure 8 we plot smooth curves as well as the original data. These are smoothing spline fits to the data, with the smoothing parameters chosen by trial and error. We have also subtracted a linear signal from the time series so the inter-event velocities are approximately zero.
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To examine its time evolution, we split the Manawatu event into three parts (Figure 8). The first sub-event covers the period from early 2004 to January 2005. The displacements at TAKP, W A N G and D N V K are less marked during this period compared to January-June 2005, but they are still significant. The second sub-event occurs from January-March 2005, and the third from MarchJune 2005. We separate the second and third subevents by the change in direction of motion at TAKP, which is displaced southwards from January-March, and northwards from March-June. We use software of McCaffrey (1995) to invert the observed displacements during each sub-event for slip on an array of dislocations in an elastic halfspace (Okada (1985)). W e assume that the slip occurs on the subduction interface defined from seismicity data by Ansell and Bannister (1996), and constrain the slip directions to be parallel to the direction of long-term relative motion on the interface estimated by Wallace et al. (2004). In the inversion, we solve for the amount of slip at nodes on the fault (Figure 9). Slip on the fault patches between the nodes is estimated by bi-linear interpolation.
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Fig. 9 Slip distribution on the subduction interface over the total duration of the 2004-05 Manawatu slow slip event. Black lines show the transition from relatively strong to relatively weak coupling in the southern North Island and Gisborne regions, from Figure 3. Black rectangles (solid where well constrained, dashed otherwise) show inferred locations of slow slip in the Gisborne and Kapiti events. All events lie near or below the coupling transition.
C h a p t e r 64 • S l o w Slip Events on t h e H i k u r a n g i S u b d u c t i o n I n t e r f a c e , N e w Z e a l a n d
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The data are best fit by slip of up to 300 mm on the subduction interface, over an area of approximately 100 km by 100 km (Figure 9). The distribution of slip in the event follows the transition zone between the interseismically coupled and creeping portions of the interface estimated from campaign GPS (Figure 3). During early 2004 to January 2005, most of the slip occurred further down-dip on the subduction interface, relative to the later stages of the event. From January-June 2005, the event propagated up dip and then slightly southwards along strike. We show only the total slip distribution from all three sub-events in Figure 9, with the match between observed and predicted displacements in Figure 10. If this aseismic slip had occurred rapidly in an earthquake, it would have resulted in a Mw ~7.0 event. If the estimated displacement of up to 300 mm is correct, the expected recurrence interval for an aseismic slip event of this type is ~7.5 years, given that the relative plate motion accommodated on this part of the interface is 40 mm/yr (Wallace et al. (2004)). However this calculation assumes there is 100% coupling on the interface in the periods between aseismic slip events, and that these events release 100% of plate motion accumulated between the events. If less than 100% of the plate boundary strain is accumulated between events (and 100% of accumulated strain is released during events), then
All the slow slip events observed thus far in the North Island occur on the portion of the subduction interface that is in transition between aseismic creep and interseismic coupling. This is shown in Figure 9, where we plot the estimated slip regions for the Gisborne and Kapiti events in addition to the slip distribution of the Manawatu event. This is consistent with observations of slow slip events in Japan and Cascadia (e.g., Dragert et al. (2001); Obara et al. (2004)), which are also inferred to occur in this transition zone. The Manawatu and Kapiti slow slip events occurred during a time period of frequent small to moderate earthquakes in the lower North Island. It is possible that this seismicity was somehow triggered by the slow slip events, as argued above for the Kapiti case. Robinson (1987) suggested that a change in the character and rate of earthquake activity documented in the Wellington region in mid 1981 was due to a possible episode of slow slip on the subduction interface. Moreover, Robinson (2003) suggested that the Weber earthquake sequence in 1990 (close to the location of the Manawatu slow slip event) was triggered by a possible slow slip event in that region. We may speculate that the 2004-05 Manawatu slow slip event is a repeat of the slow slip event that may have triggered the Weber earthquakes. Such speculation implies that large slow slip events in the southern North Island may occur roughly every 1015 years. Slow slip events at the Hikurangi margin are highly variable in duration, frequency of occurrence, and magnitude of slip. Events near Gisborne occur in as little as 7-10 days, and appear to be the result of at least 200 mm of slip on the interface, recurring every 2-3 years. We have observed smaller magnitude events near Hastings on a regular basis, at least once yearly. These events last for up to a month. The Kapiti and Manawatu events are the slowest, lasting for more than a year, and have occurred near the border of the largest well-coupled portion of the interface. These events also occur at deeper depths (20-40 km) compared to
443
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J. Beavan • L. Wallace. H. Fletcher. A. Douglas
the Gisborne and Hastings events. The Kapiti and M a n a w a t u events are also likely to take longer to recur in comparison to Gisborne and Hastings events, which seem to recur every 1 to 3 years. There seems to be an indication in our data that shallower events occur faster than deeper ones. W e speculate that this could represent part of a steady progression from stick-slip behaviour (earthquakes) at shallow depths through to genuinely steady deformation at depths below the deepest slow slip events. A rich variety of aseismic deformation events has been observed during just the first few years of G e o N e t continuous GPS network development. This emphasises the promise of G e o N e t for contributing to our understanding of plate boundary kinematics and dynamics in New Zealand. I m p r o v e d monitoring, detection and analysis of future aseismic events is important for assessing the seismic hazard posed by the subduction zone, as well as providing greater knowledge and understanding of the fundamental physical processes that control the occurrence of slip (aseismic and seismic) on subduction interfaces.
Acknowledgements The CGPS sites are operated by the G e o N e t project at G N S Science, with funding from the N e w Zealand Earthquake C o m m i s s i o n (EQC) and Land Information N e w Zealand. Our thanks to Martin Reyners, Susan Ellis and Paul Tregoning for their reviews of the manuscript. This research was supported by the New Zealand Foundation for Research, Science and T e c h n o l o g y , and by EQC. G N S Contribution 3427.
References Ansell, J., and S. Bannister (1996), Shallow morphology of the subducted Pacific plate along the Hikurangi margin, New Zealand., Phys. Earth Planet. Int., 93, 3-20. Beavan, J. (2005), Noise Properties of Continuous GPS Data From Concrete-Pillar Geodetic Monuments in New Zealand, and Comparison With Data From U.S. Deep Drilled Braced Monuments, J. Geophys. Res., l10(B8), B08410, doi: 10.1029/2005JB003642. Beavan, J., L. Wallace, and T. Hurst (2003), Observation of an aseismic deformation episode above the northern Hikurangi subduction zone, New Zealand, EGS Geophys. Res. Abstr., 5, 04839. Beutler, G., et al. (2001), Bernese GPS software version 4.2, Astronomical Institute, University of Bern, Bern, Switzerland.
Douglas, A. (2005), A Geodetic Investigation of Slow Slip on the Hikurangi Subduction Zone Beneath Raukumara Peninsula, New Zealand, MSc thesis, Victoria University of Wellington. Douglas, A., J. Beavan, L. Wallace, and J. Townend (2005), Slow slip on the northern Hikurangi subduction interface, New Zealand, Geophys. Res. Lett., 32(16), L16305, doi: 10.1029/2005GL023607. Dragert, H., K. Wang, and T. James (2001), A silent slip event on the deeper Cascadia subduction interface, Science, 292, 1525-1528. Hugentobler, U., et al. (2004), Bernese GPS software version 5.0, Astronomical Institute, University of Bern, Bern, Switzerland. Larson, K., A. Lowry, V. Kostoglodov, W. Hutton, O. Sanchez, K. Hudnut, and G. Suarez (2004), Crustal deformation measurements in Guerrero, Mexico, J. Geophys. Res., 109(B4), B04409. McCaffrey, R. (1995), DEF-NODE users guide, Rensselaer Polytechnic Institute, Troy, New York. Obara, K. (2002), Nonvolcanic deep tremor associated with subduction in southwest Japan, Science, 296, 1679-1681. Obara, K., H. Hirose, F. Yamamizu, and K. Kasahara (2004), Episodic slow slip events accompanied by non-volcanic tremors in southwest Japan subduction zone, Geophys. Res. Lett., 31, L23602. Okada, Y. (1985), Surface deformation due to shear and tensile faults in a half-space, Bull. Seism. Soc. Am., 75, 1135-1154. Ozawa, S., M. Murakami, and T. Tada (2001), Timedependent inversion study of the slow thrust event in the Nankai Trough subduction zone, southwestern Japan, J. Geophys. Res., 106(B 1), 787-802. Ozawa, S., S. Miyazaki, Y. Hatanaka, T. Imakiire, M. Kaidzu, and M. Murakami (2003), Characteristic silent earthquakes in the eastern part of the Boso Peninsula, Central Japan, Geophys. Res. Lett., 30(6), 1283, doi: 10.1029/2002GL016665. Reyners, M. (1998), Plate coupling and the hazard of large subduction thrust earthquakes at the Hikurangi subduction zone, New Zealand, N. Z. J. Geol. Geophys., 41,343--354. Robinson, R. (1987), Temporal variations in coda duration of local earthquakes in the Wellington region, New Zealand, Pure Appl. Geophys., 125(4), 579-596. Robinson, R. (2003), Fault interactions and subduction tectonics: a re-examination of the Weber, New Zealand, earthquake sequence of 1990, Geophys. J. Int., 154, 745753. Rogers, G., and H. Dragert (2003), Episodic tremor and slip on the Cascadia subduction zone: the chatter of silent slip, Science, 300, 1942-1943. Wallace, L. M., J. Beavan, R. McCaffrey, and D. Darby (2004), Subduction Zone Coupling and Tectonic Block Rotations in the North Island, New Zealand, J. Geophys. Res., 109:B12, B12406, doi:10.1029/2004JB003241. Zhang, J., Y. Bock, H. Johnson, P. Fang, S. Williams, J. Genrich, S. Wdowinski, and J. Behr (1997), Southern California Permanent GPS Geodetic Array: Error analysis of daily position estimates and site velocities, J. Geophys. Res., 102, 18,035-18,055.
Chapter 65
A geodetic measurement of strain variation across the Central Southern Alps, New Zealand P.H. Denys, M. Denham School of Surveying, University of Otago, PO Box 56 Dunedin, New Zealand C.F. Pearson National Geodetic Survey, 2300 S Dirksen Parkway, Springfield IL 62764, USA Abstract. An analysis of both continuous and campaign GPS data has been used to determine the degree of strain rate variability that is occurring, due to tectonic deformation, from two GPS networks that cross the Alpine Fault and the Main divide in the central Southern Alps, New Zealand. The campaign GPS data was observed during four regional campaigns covering an eight year period between 1994 and 2002, while a network of permanently tracking receivers has collected data over a 5.5 year period. Both data sets enable the estimation of high precision horizontal velocities that measure the velocity variation and hence crustal deformation that is occurring across the region. The permanently tracking receivers clearly show significant localised site displacements that appear to be caused or induced by seasonal changes in the environmental conditions. Such site dependant localised displacements exceed 1cm at several sites and demonstrates the highly dynamic nature of the Southern Alps. In general, the long term strain rates determined from the permanently tracking network and the regional strain rates are in agreement, but there does appear to be evidence of variation in strain on a regional scale. In particular, there is evidence for a zone of extension even though the Alpine Fault is an oblique thrust fault caused by convergence across the plate boundary zone. Keywords. Campaign GPS, Continuous Strain, Southern Alps Strain Variation.
the region. We compare our localised strain rates with those determined largely from average rates estimated from GPS campaigns since 1994. Our current knowledge of the recurrence of an earthquake on the Alpine Fault is based on palaeoseismic evidence (including forest age and tree ring chronologies) derived using data spanning time scales from 100's to 1000's of years. To verify the applicability of this long-term evidence to current processes, we are endeavouring to understand the present-day deformation of the Alpine fault zone and the processes responsible for them through the use of near-real time continuous measurements rather than through discrete geodetic campaigns nominally at intervals of 2-3+ years. Recent geodetic work in New Zealand provides a reliable measure of the strain rates over most of the South Island, but at regional scales this will average the deformation signal. Variations in both the spatial uniformity and temporal persistence of the signal may exist. Although the region is relatively remote, it is likely to produce a major earthquake with a rupture length longer than other great earthquakes and would significantly affect central South island locations on both sides of the Main Divide (Yetton et al. 1998). Knowledge of the strain rate and possible variations is necessary if we want to improve our understanding of earthquake recurrence in this region.
GPS,
1 Introduction This is a geodetic experiment, using an established Continuous GPS (CGPS) network that crosses the Alpine Fault in the central Southern Alps, to test whether constant strain build-up is occurring over time, and, to compare this single transect with the (long term) average strain accumulation derived from campaign geodetic measurements observed in
2 Methodology Geodetic measurements of (discrete) station positions derived from GPS data provide important constraints on the kinematics of the Earth's crust. Over time, from either epoch style campaigns or Continuous GPS (CGPS) data, it is possible to determine highly precise station velocities. Combining stations together to form a network enables a regional velocity field to be determined. A change in the velocity field, i.e. velocity gradient, implies a regional deformation field. Conversely if the veloc-
446
P.H. Denys• M. Denham.C. F.Pearson
ity field is uniform, then the region is translating without deformation. Strain rates are calculated from the gradient of the velocity. Our aim is to compute deformation parameters from the observed GPS data collected from a small regional network. We process and analysis the GPS data in a three step process: 1. Geodetic Parameters: The GPS data is processed on a daily basis to provide loosely constrained estimates of the station coordinates. 2. Network Combination and Analysis: The daily solutions (loosely constrained) are combined to form a single network solution (station coordinates and velocities). Minimal constraints are imposed on coordinates and velocities to define a uniform reference frame. 3. Deformation Parameters: Different combinations of strain or strain rate parameters can then be computed based on selected stations. In addition to the primary parameters (geodetic and deformation parameters), the process allows for the estimation or modelling of auxiliary parameters that account for time-dependent displacements (e.g. seasonal effects) or changes in instrumentation (e.g. antenna changes).
sonal displacements simultaneously. Crustal deformation parameters such as strain or strain rates can also be obtained (e.g. Feigl et al. 1993; Bock et al. 1997; Hager et al. 1999; Shen et al. 2000; Shen et al. 2001) and seasonal displacements (e.g. Soudarin et al. 1999; Mangiarotti et al. 2001; van Dam et al. 2001; Dong et al. 2002; Zhang et al. 2002). Once any outlier observations have been removed, appropriate covariance matrix scaling is applied. Together with site coordinate and velocity parameters, we estimate the magnitude and phase of annual and semi-annual terms (total of up to 6 parameters per site) and, where appropriate, offsets in the time series (e.g. antenna changes, coseismic displacements). The periodic terms contained in the station time series are fitted using the sine function with Aper,od,~ - A sin{co(t- t o )+ ~bo]
where the two estimated parameters are A - amplitude and ~ - initial phase. For the annual term c o - 2 ~ c / y e a r ; the semi-annual term co- 4 ~ c / y e a r ; to = initial epoch (e.g. 2000.0) and t = time in years. 2.3 Strain rate e s t i m a t e s
2.1 G e o d e t i c P a r a m e t e r E s t i m a t i o n
The GPS data were processed with the Bernese Post Processing Software Version 4.2 (Hugentobler et al. 2001) using the Bernese Processing Engine (BPE). A daily set of station coordinates is estimated from simultaneously observed data over one day (24 hours) using the carrier phase double difference observable forming the ionosphere-free linear combination (see Denys et al. 2005). For each day a final set of geodetic parameters (station coordinates and variance-covariance matrix) is estimated. This solution is generated as a loosely constrained set of coordinates (10cm) based on a subset of international GPS Service (IGS) stations in and close to New Zealand. 2.2 N e t w o r k C o m b i n a t i o n and A n a l y s i s
Two common methods of strain parameterisation using geodetic data are the eigenvalue strain and engineering shear strain (Feigl et al. 1990). The eigenvalue parameterisation, £1, £2,0, eb, of a twodimensional velocity field includes strain rate eigenvalues: maximum extension strain rate, g'l, and maximum contractional strain rate, £2, in the directions of the principal axes of extension (el) and compression (e2).
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positive with ob~ > £2. The angle, 0, is the direction of the principal axes of compression (e2) from the north axis, measured clockwise. An arbitrary rigid body rotation rate is defined by, d), to account for a non-symmetric velocity gradient, but is generally not well defined from geodetic observations Engineering sheer parameterisation, y~, Y2, £, eb,
The Quasi-Observation Combination Analysis (QOCA) software package was used to combine the geodetic measurements as loosely constrained coordinate and velocity solutions (as quasi observations) to obtain crustal deformation information (Dong et al. 1998). QOCA solves for site positions, velocities, network parameters, and will also parameterise coseismic and/or postseismic deformation and sea-
includes the engineering parameters, Yl, and, Y2, that are used to describe the angular shear strain rate components. These values can be given a geophysical interpretation in terms of either simple shear or pure shear (see Feigl et al. 1990). The rate of areal dilation is defined as the sum of the two m
strain rate components, £ - £ i + £2, and represents
Chapter 65 • A Geodetic Measurement of Strain Variation across the Central Southern Alps, New Zealand
the net increase (positive) or decrease (negative) in the area over which the strain rate is computed. The rotation, &, is the same as defined for the eigenvalue parameterisation. Two derived quantities include the maximum engineering shear rate, F, and principal direction of contraction, /7, with = ~/}72 + 772
/3 = tan -1 -}72
})l The principal direction of contraction, /3, is related to the direction of the maximum right lateral shear, 0, with f l = 0 + 4 5 °.
3 The CSI and SAGENZ Networks The Central South Island (CSI) network is part of the Institute of Geological and Nuclear Sciences (GNS) national deformation programme. It is a regional network that extends between the west and east coasts of the South Island and roughly from the Waitaki to Rakai rivers. It was first observed as a network of 47 stations in 1994, extended to 115 sites in 1996 and included some 119 sites in 2002 (excluding CGPS sites). The first strain analysis, using the 1994-1998 data, was reported in (Beavan et al. 1999) The 14 station SAGENZ network was established in February 2000 with the primary objective of measuring the tectonic uplift of the Southern
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Alps. The transect extends 70km across the Southern Alps with 12 of the 14 sites concentrated within a 30km distance crossing the Main Divide. These receivers are operated in two modes: the first are sites that are permanent installations (CGPS); the second mode utilizes three receivers that are regularly moved between six sites (semi-CGPS). The semi-CGPS mode of operation allows us to increase the spatial resolution of the project at the expense of not having a continuous data record. The CGPS stations (Figure 1, black arrows) include four sites (QUAR, KARA, CNCL and NETT) that have been operating since the start of the project. WAKA was upgraded to operate continuously (September 2002). A sixth site, HORN, was established in November 2002. The remaining 6 semiCGPS sites (Figure 1, grey arrows), are located between the Alpine fault and the main divide.
3.1 GPS Equipment and Data Operating at each site is a dual frequency GPS receiver to measure and log GPS data collected through a geodetic dual frequency antenna. The data is downloaded from the CGPS sites each hour using a radio telemetry link. If the data link cannot be made, the data are stored in the receiver until a connection can be made. Each receiver can store 10 Mbytes of data, which corresponds to about 10-12 days of GPS observations, depending upon the sky visibility at a particular site. The semi-CGPS sites are similar to the CGPS sites except that they do not have radio telemetry equipment, have two solar panels, three batteries and no metrological sensor. Also, having less equipment allows for the site to be transported easily between semi-CGPS sites via helicopter.
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447
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P.H. Denys • M. Denham. C. F. Pearson
The availability and continuity of the observed data is shown in Figure 2. The seven sites at the top of Figure 2 are the CGPS sites and provide a continuous data stream with only a few data outages. These outages are generally caused by equipment failures, (e.g. receiver, antenna, cables), but also power failures. Environmental conditions such as short daylight hours in winter (compounded by high topography and low sun elevations) and snow accumulation over a period of time do not allow the batteries to be charged adequately and therefore they slowly drain.
3.2 Antenna Offsets and Seasonal Terms Continuous GPS networks provide precise time series of station positions. The achieved precision is depends inherently on the continuous use of the same equipment on a solid monument. When equipment fails, or more particularly, when a GPS antenna is changed, a jump in the time series is often observed. This is usually attributed to different antenna characteristics that are modelled inadequately using current (relative) antenna phase centre variation models. Provided there is sufficient data before and after the antenna change, the discontinuity can be modelled as an offset. In the vertical component the offset can be over 10mm but the horizontal offsets are generally less than 1-3mm. Antenna changes have occurred on the SAGENZ sites but the horizontal offsets are not significant. More details can be found in (Beavan et al. 2004; Denys et al. 2005). As pointed out by (Dong et al. 2003), the current ITRF accounts for secular motion, but does not account for seasonal variations. Seasonal terms have been modelled at all sites with average amplitudes of 5mm east and 3 mm north. Several sites have maximum seasonal terms with amplitudes close to 10mm and one site, NETT, at over 20mm. Further analysis is given in both (Beavan et al. 2004; Denys et al. 2005).
3.3 Horizontal Velocities The geodetic parameters estimated in the combined network solution includes the station velocities, position offsets caused by antenna changes and periodic terms that account for seasonal changes in the station positions. The horizontal velocities in an east coast fixed reference frame (MQZG fixed) are tabulated in Table 1. The stations are listed approximately west to east and so the gradual decrease in the velocity components is clearly seen.
Table 1. Station velocities are with respect to the nominal (global) IGS station velocities. Relative velocities (right hand side) are computed holding MQZG fixed (East coast fixed). One sigma standard errors. Site
OIJAR LEOC KARA WAKA VEXA CNCL MAKA PILK REDD MCKE NETT HORN MTJO MQZG
East
{~East
North
(mm/yr) 26.6 23.0 23.0 23.3 18.4 18.0 15.8 19.8 15.0 17.3 7.5 11.0 6.2 0.0
(mm/;yr) 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
(mm/yr) 10.9 11.5 10.4 11.8 6.8 6.8 5.9 6.7 5.5 6.4 0.8 2.4 0.7 0.0
¢YNo~th (mm/~cr) 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3
This change in velocity (or velocity gradient) implies strain acculmulation in the Southern Alps. Most of the deformation is occouring in the 70km wide zone between MTJO and QUAR. Over this distance, the magnitude change in velocity (and hence rate of strain) is 22mm/yr or nearly 80% of the plate motion across the South Island. As shown in Figure 1, the motion with respect to the east coast of the South Island has a predominatly strike-slip component parallel with the Alpine fault. The NETT velocity appears to be inconsistent with the other sites in the SAGENZ network. Although the NETT data extends over nearly 5.5 years of data, the site is likely to be affected by the poor data quality and significant seasonal motion. The velocity at MCKE has only been determined from data spanning an interval of one year.
4 Long term strain rates 4.1 Network strain rates A comparison of the long term strain rates can be made based on the CSI and SAGENZ networks. This compares the CSI campaign data spanning 8 years (1994-2002) with the SAGENZ data from the CGPS sites spanning 5.5 years. Two triangles, with sites from both networks that are within reasonable proximity of each other have been used. Listed in Table 2 are the sites used from each network and the approximate distance between the campaign sites (CSI) and the CGPS sites (SAGENZ). Three sub-networks are considered. Subnetwork 1 (SN1) is a small triangle with interconnecting sides of 9-16km, straddling the Alpine
Chapter 65
• A
Geodetic Measurement of Strain Variation across the Central Southern Alps, New Zealand
degree, due to overall satellite configuration and receiver/antenna technology improvements. Sub-network 2: This sub-network involves the CSI SN2 sites (Table 3 and Figure 3) and the SAGENZ SN2 sites (see footnote 4 in Table 3 and Figure 4). The extension and contraction strain rates and the principal axis of contraction determined from both networks are the same within the statistical uncertainties. Sub-network 3: The final sub-network estimates an average strain rate over the respective regions. The CSI SN3 includes the four sites used in CSI SN1 and SN2 and the sites surrounding, but mainly to the immediate north, of the SAGENZ network (Table 3 and Figure 3). The estimated extension (0.22+0.02 ppm/yr) and contraction (-0.15+0.01 ppm/yr) strain rates are depicted by the upper strain rate figure (grey) in Figure 3. When compared to the values given in Beavan et al. (1999), both strain rate values based on the CSI SN3 data spanning 8 years decrease (Table 3 and the lower strain rate figure (grey) in Figure 3). The contraction strain rate is statistically different (at the 1% significance level). A comparison of the CS| SN3 and SAGENZ SN3 data based on the 12 SAGENZ sites, including both CGPS and semi-CGPS sites (but excluding MTJO), has also been made. The estimated extension and contraction strain rates are 0.282+0.001 ppm/yr and -0.263+0.001 ppm/yr respectively (grey strain rate figure in Figure 4). Compared to the CSI SN3 network spanning 8 years, this is an increase in
Table 2. Inter-site distances between adjacent sites in the CSI and SAGENZ networks SAGENZ
CSI
Distance
QUAR
6718 (JI Karangarua SD)
1.95km
KARA
6720 (JM Karangarua)
400m
CNCL HORN
6732 (Conical Hill) 6700 (M Mt Hodgkinson)
2m 1.2km
fault. Sub-network 2 (SN2) extends over a larger region from the Alpine fault to just east of the main divide, a distance of 30km. Sub-network 3 (SN3) computes an average strain rate using both the semi-CGPS and CGPS sites and a selection of the CSI sites. Table 3 lists the estimated strain rate values and the eigenvalue strain rates for each subnetwork are given in Figure 3 and Figure 4. Sub-network 1: This sub-network involves the CSI SN1 sites (Table 3 and Figure 3) and the SAGENZ SN1 sites (Table 3 and Figure 4). The extension strain rate for both networks is similar (0.213+0.001 ppm/yr), but the contraction strain rate for the SAGENZ (-0.274+0.002ppm/yr) is 60% larger than that of the CSI network. The orientation of the principal axis of contraction of both networks agree within the statistical uncertainty. The estimated precision, from the CGPS data is improved by approximately three times, (data spans 5.5 years compared to 8 years for the campaign data). This improvement in precision is, for the most part, due to using permanent (non-changing) equipment on solid monuments but also, to a lesser
Table 3. Maximum shear strain, eigenvalue strain and areal dilation strain rates for the CSI and SAGENZ networks. The eigenvalue strain parameters, ~l and 032 are in the directions of the principal axes (eigenvectors). By convention, extension is positive, ~l > ~2, and 0 is measured clockwise from north to the principal axis of the contraction eigenvalue. M a x i m u m Shear Strain Rate
Eigenvalue Strain Rates
Areal Dilation
(prad / yr)
(ppm / yr)
(ppm / yr)
]~2
O"1=2
Sub-network 1 0.415 0.168 0.488 0.002 Sub-network 2 0.505 0.123 0.574 0.002 Sub-network 3 0.88 0.11 0.371 0.026 0.546 0.001
Maximum Direction of Contraction
Network
~1
O'21
~2
era2
~1 + ~2
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0.249 0.213
0.094 0.001
-0.166 -0.274
0.135 0.002
0.083 -0.061
0.164 0.002
104.4 100.0
12.0 0.2
CSI SN11 SAGENZ SN12
0.369 0.420
0.106 0.002
-0.136 -0.154
0.054 0.001
0.233 0.266
0.118 0.002
116.8 121.7
6.9 0.1
CSI SN23 SAGENZ SN24
0.39 0.219 0.282
0.09 0.022 0.001
-0.50 -0.151 -0.263
0.06 0.012 0.001
-0.11 0.068 0.019
0.11 0.025 0.001
110.1 111.6 108.0
2.0 0.1
CSI SN35 CSI SN36 SAGENZ SN37
CS! SN1 sites: 6718, 6720, 6732; 2 SAGENZ SN1 sites: QUAR CNCL KARA; 3 CSI SN2 sites: 6716, 6732, 6700; 4 SAGENZ SN2 sites: QUAR CNCL HORN; 5 C SI SN3 sites 6702, 6715,6718, 6719, 6720, 6732, 6733, 6735, 6736, 6706, 6737, 6714, 6734, from the Beavan et al. (1999) analysis; 6 C SI SN3 sites 6702, 6715, 6718, 6719, 6720, 6732, 6733, 6735, 6736, 6706, 6737, 6714, 6734; 7 SAGENZ SN3 sites: QUAR KARA CNCL WAKA NETT HORN VEXA PILK REDD MAKA LEOC MCKE
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both the extension and contraction strain rates of approximately 30% and 70% respectively. Although the compression rates do not agree within the statistical uncertainty, care must be taken with any comparison as the CSI and SAGENZ regions over which the strain is estimated are not identical. The orientations of the maximum contraction do not change substantially. 4.2 Network Areal Dilation
Except for the SAGENZ SN1 and the regional CSI SN3, all the areal dilation estimates are positive. This indicates that there is an extensional regime with I~,l> Io~2l i.e. ~, + o~2 > 0, and hence an increase in area (area creation). Since the Southern Alps are caused by convergence across the plate boundary zone and as the Alpine Fault is an oblique thrust fault, it could be expected that there is an overall contraction. However, elastic dislocation theory does predict a zone of extension above the hanging wall of a thrust fault during periods of strain accumulations between earthquakes. Although the areal dilation determined by the CSI SN1 data indicates extension, the errors are considerably larger and the dilation is only marginally significant. In contrast, the precision of the areal
dilation determined from the SAGENZ SN1 data is considerably better. This is an interesting result since the Alpine fault is predominantly strike-slip with a significant contribution of reverse slip. Combined, sub-networks 2 and 3 extend across the most tectonically active part of the central Southern Alps. It is worth noting that (Walcott 1998, see Figure 22) documented the existence of normal faults east of the Alpine Fault. 4.3 SAGENZ Sub-network strain rates
Subdividing the whole SAGENZ network into smaller triangles provides further detail on strain rate variation in the region. Using the SAGENZ sites (semi-CGPS and CGPS), a network of 13 triangles was created from 10 sites (excluding NETT and MCKE). The eigenvalue strain rates are plotted in (Figure 5). For the western most triangle, the high contraction rate is most probably due to the north-south elongated triangle as the east-west spatial extent is inadequate. The contraction strain rate is likely to be poorly determined. There is evidence for an along strike variation in the geodetic strain rate. Current tectonic models of the central Southern Alps assume the Alpine fault trends at 55 ° with possibly a second parallel antithetic structure at the same strike, but opposite dip, located some hundred kilometres to the SE. The expected gradient of the strain rate is along a line
Chapter 65 • A Geodetic Measurement of Strain Variation across the Central Southern Alps, New Zealand ,
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perpendicular to both structures (i.e., in the SE-NW direction), while position along the plate boundary zone (NE-SW direction) should not affect the strain rate (significantly). This pattern of greater extension to the north of the profile is seen in the triangle KARA-WAKA-VEXA, even though it is closer to the Alpine fault. This suggests that there is a localised area undergoing extension in the region of the network bounded by WAKA, VEXA, PILK and HORN. A shear strain pattern without any significant dilation or areal contraction is typical near an oblique reverse predominantly strike-slip fault such as the Alpine fault. The network does suggest that areal dilation (contraction or extension) is occurring and that there is regional variability (Figure 5). Comparing the strain rates for the northern regions with triangles at comparable distances from the fault on the southern side of the transect, suggests a pattern of extension along a NW-SE axis dominates along the northern side of the transect.
5 Summary This project has investigated contemporary strain variation using the SAGENZ network of semiCGPS and CGPS that crosses the Southern Alps between Karangarua and Mt Cook village. We have compared our strain rate estimates with those from a previous study and also investigated the
strain variation in the same network. Previous geodetic work, typically using data from several GPS campaigns, has provided a picture of the geodetic strain spanning a period of years at the regional scale. Using the SAGENZ data, the strain parameters were determined for several sub networks involving subsets of both the semi-CGPS and CGPS stations. In addition, the strain rates from the (localised) SAGENZ network were compared to those derived from the CSI regional campaign network that broadly covers the whole region between the east and west coasts of the South Island. Finally, a detailed investigation of the strain variability within the region covered by the SAGENZ network was considered. Consistent with the velocity estimates, a comparison of the CSI and SAGENZ strain parameters shows that they are in general agreement (within the statistical uncertainties of the data). The SAGENZ strain rate values are approximately an order of magnitude more precise. Using only the SAGENZ CGPS data, two sub networks involving 3 stations in each were compared with nearby sites from the CSI network. Although the CSI and SAGENZ strain rates and the orientation of the strain rates agree, with a denser network and the better precision of the SAGENZ data, there appears to be evidence of a zone of extension or increase in area. A wider scale regional strain rate estimate using a subset of the CSI sites and all the SAGENZ sites was also made. Again the SAGENZ strain rates broadly agree with the CSI data, but there are differences that are statistically significant. In particular the SAGENZ contraction strain rate (-0.263+0.001 ppm/yr) is nearly double the rate determined from 8 years of CSI data. This is perhaps not surprising given that the sub networks being compared are not identical (the CSI sites cover a greater region), but does demonstrate that there is regional variability at smaller scales. A shear strain pattern without any significant dilatation or areal contraction is typical around an oblique reverse predominantly strike-slip fault such as the Alpine fault. Comparing the strain rates for the northern regions with triangles at comparable distances from the fault on the southern side of the network suggests a pattern of extension along a NW-SE axis dominates along the northern side. The reason for this is still under investigation. Possible explanations are along strike variations of deformation style or subtle strain variations being caused by seasonally induced processes.
451
452
P.H. Denys • M. Denham. C. F. Pearson
Acknowledgements.
This research was funded by a grant from the Earthquake Commission (New Zealand) (EQC Project 01/457). Funding for the SAGENZ project was provided by Otago University (OU) (ORG MFWB03, ORG MFWB01), Foundation for Research, Science and Technology (FRST) (contracts C05X0203, C05X0402), and the National Science Foundation (NSF) (EAR9903183). GPS data collection has been carried out by field teams including the institutions involved in the SAGENZ project (OU, GNS, MIT, UNAVCO, University of Colorado at Boulder) as well as the many different organisations involved in the collection of the CSI data. The Department of Conservation has been instrumental in granting of access and helicopter landing permits and there are several land owners from whom we have sought permission for access to their land. CGPS data has been provided by the GeoNet (funded by EQC) and the IGS data and product archives. Several figures were plotted using the public domain GMT software (Wessel and Smith 1995). Matlab Version 7.01 (R14) was used for data analysis and figures.
References Beavan, J., D. Matheson, P. Denys, M. Denham, T. Herring, B. Hager and P. Molnar (2004). A Vertical Deformation Profile across the Southern Alps, New Zealand, from 3.5 Years of Continuous GPS Data. In T. v. Dam and O. Francis, (Eds) Proceedings of the workshop: The state of GPS vertical positioning precision: Separation of earth processes by space geodesy, Luxembourg, Cahiers du Centre Europ0en de G0odynamique et de SOismologie. 23: 111-123. Beavan, J., M. Moore, C. Pearson, M. Henderson, B. Parsons, S. Bourne, P. England, D. Walcott, G. Blick, D. Darby and K. Hodgkinson (1999). Crustal Deformation During 1994-1998 Due to Oblique Continental Collision in the Central Southern Alps, New Zealand, and Implications for Seismic Potential of the Alpine Fault. Journal of Geophysical Research-Solid Earth 104(B 11): 2523325255. Bock, Y., S. Wdowinski, P. Fang, J. Zhang, S. Williams, H. Johnson, J. Behr, J. Genrich, J. Dean, M. Vandomselaar, D. Agnew, F. Wyatt, K. Stark, B. Oral, K. Hudnut, R. King, T. Herring, S. Dinardo, W. Young, D. Jackson and W. Gurtner (1997). Southern California Permanent GPS Geodetic Array - Continuous Measurements of Regional Crustal Deformation between the 1992 Landers and 1994 Northridge Earthquakes. Journal of Geophysical Research-Solid Earth 102(B8): 18013-18033. Denys, P. H., C. F. Pearson and M. Denham (2005). Strain Accumulation across the Central Southern Alps, New Zealand - a Geodetic Experiment to Characterise the Accumulation of Strain. Dunedin, New Zealand, Otago University: 58. Dong, D., P. Fang, Y. Bock, M. K. Cheng and S. Miyazaki (2002). Anatomy of Apparent Seasonal Variations from GPS-Derived Site Position Time Series - Art. No. 2075. Journal of Geophysical Research-Solid Earth 107(B4): 2075-2075. Dong, D., T. A. Herring and R. W. King (1998). Estimating Regional Deformation from a Combination of Space and
Terrestrial Geodetic Data. Journal of Geodesy 72(4): 200214. Dong, D., T. Yunck and M. Heflin (2003). Origin ofthe International Terrestrial Reference Frame. Journal of Geophysical Research-Solid Earth 108(B4). Feigl, K. L., D. C. Agnew, Y. Bock, D. Dong, A. Donnellan, B. H. Hager, T. A. Herring, D. D. Jackson, T. H. Jordan, R. W. King, S. Larsen, K. M. Larson, M. H. Murray, Z. K. Shen and F. H. Webb (1993). Space Geodetic Measurement of Crustal Deformation in Central and Southern California, 1984-1992. Journal of Geophysical ResearchSolid Earth 98(B12): 21677-21712. Feigl, K. L., R. W. King and T. H. Jordan (1990). Geodetic Measurements of Tectonic Deformation in the Santa Maria Fold and Thrust Belt, California. Journal of Geophysical Research 95(B3): 2679-2699. Hager, B. H., G. A. Lyzenga, A. Donnellan and D. Dong (1999). Reconciling Rapid Strain Accumulation with Deep Seismogenic Fault Planes in the Ventura Basin, California. Journal of Geophysical Research-Solid Earth 104(B11): 25207-25219. Hugentobler, U., S. Schaer and P. Fridez, Eds. (2001). Bernese GPS Software Version 4.2. Berne, Astronomical Institute, University of Berne. Mangiarotti, S., A. Cazenave, L. Soudarin and J.-F. Crdtaux (2001). Annual Vertical Crustal Motions Predicted from Surface Mass Redistribution and Observed by Space Geodesy. Journal of Geophysical Research 106(B3): 42774291. Shen, Z. K., M. Wang, Y. X. Li, D. D. Jackson, A. Yin, D. N. Dong and P. Fang (2001). Crustal Deformation Along the Altyn Tagh Fault System, Western China, from GPS. Journal of Geophysical Research-Solid Earth 106(B 12): 30607-30621. Shen, Z. K., C. K. Zhao, A. Yin, Y. X. Li, D. D. Jackson, P. Fang and D. N. Dong (2000). Contemporary Crustal Deformation in East Asia Constrained by Global Positioning System Measurements. Journal of Geophysical ResearchSolid Earth 105(B3): 5721-5734. Soudarin, L., J.-F. Cr6taux and A. Cazenave (1999). Vertical Crustal Motions from the Doris Space-Geodesy System. Geophysical Research Letters 26(9): 1207-1210. van Dam, T. M., J. Wahr, P. C. D. Milly, A. B. Shmakin, G. Blewitt, D. Lavallde and K. M. Larson (2001). Crustal Displacements Due to Continental Water Loading. Geophysical Research Letters 28(4): 651-654. Walcott, R. i. (1998). Modes of Oblique Compression: Late Cenozoic Tectonics of the South island of New Zealand. Reviews of Geophysics 36(1): 1-26. Wessel, P. and W. H. F. Smith (1995). New Version of the Generic Mapping Tools Released. EOS Trans. AGU 76: 329. Yetton, M. D., A. Wells and N. J. Traylen (1998). Probablility and Consequences of the Next Alpine Fault Earthquake, New Zealand. Wellington, New Zealand, New Zealand Earthquake Commission, Wellington, New Zealand. EQC Research Foundation Report 95/193: p 161. Zhang, F. P., D. Dong, Z. Y. Cheng, M. K. Cheng and C. Huang (2002). Seasonal Vertical Crustal Motions in China Detected by GPS. Chinese Science Bulletin 47(21): 1772-1779.
Chapter 66
New analysis of a 50 years tide gauge record at Canan6ia (SP-Brazil) with the VAV tidal analysis program B. Ducarme Chercheur Qualifi6 FNRS, Observatoire Royal de Belgique, Av. Circulaire 3, B- 1180, Bruxelles, Belgique. A.P. Venedikov Geophysical Institute & Central Laboratory on Geodesy, Acad. G. Bonchev Str., Block 3, Sofia 1113 A.R. de Mesquita, C.A. de Sampaio Fran~a Instituto Oceanogrfifico da Universidade de Silo Paulo, SP, Brasil. D. S. Costa, D. Blitzkow Escola. Politdcnica, Universidade de Silo Paulo, Caixa Postal 61548, 05413-001 Silo Paulo, SP, Brasil. R. Vieira Diaz Instituto de Astronomfa y Geodesia (CSIC-UCM). Facultad de Matemfiticas. Plaza de Ciencias, 3. 28040. Madrid, Spain. S.R.C. de Freitas CPGCG - Universidade Federal do Paranfi, Caixa Postal 19001, 81531-990 Curitiba, PR, Brasil.
Abstract. A homogeneous high quality tide gauge hourly record covering the period 1954-2004 was obtained at Canan6ia (SP-Brazil). A previous analysis of a 36 years data set has shown many interesting features, especially very long period signals. This re-analysis benefits from a 40% longer time series and the powerful program VAV for tidal data processing is used to determine the parameters of the tidal constituents derived from the tidal potential, including the long period tidal waves, and of the shallow water and radiation tides. Long period terms are determined from the tidal residues by a semi-automatic research algorithm. The variation of the mean sea level is estimated after subtraction of the ocean tides and estimation of the long period terms. The mean sea level rate of change is estimated to 0.5666 ___ 0.0070 cm/year. Special attention is given to the determination of the ocean pole tide as well as the 11 years term directly related to the solar activity.
Keywords. Ocean tides, mean sea level, ocean pole tide, radiation tides, Solar activity
1 Introduction The paper presents some of the results from the application of the tidal program VAV (Venedikov et al., 2001, 2003, 2005) on the ocean tide (OT) data from the Canan6ia tide gauge (SP- Brazil)
(q0 = 25 ° 01.0' S, )~ = 47 ° 55.5' W). This series of data covers 50 years in the time interval 26.02.1954 - 31.12.2004 and contains 444410 hourly ordinates. A previous analysis of a 36 years data set (Mesquita et al., 1995, 1996) has shown many interesting features, especially very long period signals. The reanalysis benefits from the longer time series and a powerful tidal analysis program. Tide gauge, station and data characteristics can be found in (Mesquita et al., 1983, 1995). The present paper is mainly focusing on the investigation of the mean sea level (MSL), including its secular variation, and the detection of various low frequency components.
2 The VAV tidal analysis program VAV is originally designed for the processing of Earth tide (ET) data. Now, it has been supplied by specific options (Ducarme et al., 2006a), corresponding to the OT characteristics and problems (Godin, 1972; Munk and Cartwright, 1966). VAV has various options for tidal data processing. The main one for the OT is the determination of the amplitudes and the phases of all 1200 tides in the development of Tamura (1987), including the long period (LP) tides. For large series of data it can also study the time variations of the tidal parameters, as shown by Figure 1 for the tidal wave M2.
454
B. Ducarme • A. P. Venedikov • A. R. de Mesquita • C. A. de Sampaio Fran(;a • D. S. Costa. D. Blitzkow • R. Vieira Diaz. S. R. C. de
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I
tm
Figure 4 represents the whole set of tidal residues or apparent sea level variations d(t) (18570 daily values).
250 225 200 175 150 125 100
-
"
Fig. 3. Sample of the observed hourly tidal data and the residues (thick line), which represents the sea level, free of the tidal signals. The drift values d(t)are determined every 24 hours (black dots).
..
~ I
'
I L I
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Time in days
o
Frequency in cpd (cycles/day) Fig. 2. Amplitudes of quarter-diurnal shallow water tides in Canan6ia, the grey line representing the 99.99% confidence level.
VAV can also determine an arbitrary number of shallow water and radiation tides in the frequency bands from 1 till 12 cpd. The millimetre accuracy can be achieved (Figure 2). Here we shall pay special attention to the investigation of the tidal residues d(t)obtained by subtraction of the tidal signal. These residues should represent the sea level, free of the known tidal signals. However they can still include LP radiation tides with unknown periods. The investigation is made with the help of our computer program POLAR for regression and spectral analysis, initially created for the study of the gravity effect of the polar motion. Figure 3 shows a sample of the sea level, determined by VAV. It is obtained after the reduction of all estimated tides, including the LP and the shallow water tides.
Fig. 4. The total curve of the sea level variations d(t) (one value per day).
3 General model of the sea level variations The general model used for
d(t) is
M
d(t) - L(t) + Z W(coj,t)
(1)
j=l
The first term L(t) is a polynomial of the time t, which we consider to represent the mean sea level (MSL). Each term
W(o3j,t) = ely cos ~0jt +13j sin~0jt
(2)
is a sinusoidal function of the time, representing a wave with frequency COy and unknown amplitude and phase. The frequencies, as well as their number M, are also a priori unknown quantities.
Chapter 66 • New Analysis of a 50 Years Tide Gauge Record at Canan~ia (SP-Brazil) with the VAV Tidal Analysis Program
After many numerical experiments, combined with some theoretical considerations, we have accepted the hypothesis that the MSL L(t) should be treated as a linear function, i.e.
L(t) = a o + alt
(3)
than the previous AIC, i.e. AIC(WMin)< AiC(mm). This is accepted as a confirmation that one more wave W(m,,+l,t) with frequency o),,+1 = Wvi n really exist in d(t), so that the expression (4) has to be replaced by m+l
with unknown regression coefficients a 0 and a 1 . In such a way we accept that all non-linear components of d(t) can be represented through the periodic terms W(ogj,t), 1 < j < M . Once the model (3) is fixed, it remains to find the frequencies my of the waves W(ogj,t), i.e. we have to solve the problem of finding the hidden periodicities in the sea level d(t).
4 Search of hidden periodicities
m
(4)
j=l Through the application of the least squares we get easily the estimates of all unknowns a0, al, a j, [~j (1 < j < M ) , as well as the corresponding value of AIC, denoted AIC(m m) . Next we add a new wave W(w,t) with frequency w , which does not coincide with any of the existing frequencies c0j, i.e. the expression (4) is replaced by m
d(t) = L(t) + Z W ( m j , t ) + W(w,t ) -+
-+
~ac(%+, )
(6)
j=l
The process is reiterated in the same way, by looking for a new frequency o9,,+2. In the case of two close frequencies we can apply a 2D research algorithm for a minimum AIC. Instead of the model (5) with one new wave, we can add two waves with variable frequencies w ' & w " , i.e. instead of (5) we use m
Our solution of the problem uses the Akaike Information Criterion AIC (Sakamoto et al., 1986), based on the principle of the maximum likelihood. The practical use of AIC is based on the following procedure. Let AIC(A) and AIC(B) be the values of AIC, obtained by the least square solutions using two different models A and B. If it happens AIC(A) < AIC(B), then we may give our preference to the model A. At a given stage we know m frequencies, i.e. we have reached to the concrete expression of our model (1)
d(t) - L(t) + Z W ( m j , t ) --> AIC(%,)
d(t) - L(O + ~ w ( % , t )
d(t) = L(t) + Z W ( m j ' t ) + W( w', t ) + W( w", t ) -->AI C('v~/,v()
Now w ' & w " are allowed to cover two neighboring frequency intervals. For every couple w ' & w" we get the least square solution of (7), accompanied by a corresponding AIC value. In such a way we get AIC = AIC(w',w") as a function two frequencies. The values of w ' & w" at which AIC has a minimum can be accepted as two new frequencies c0m+l and O)m+2 As shown by Table 1 and Table 2 we have found in total M = 16 frequencies. We shall demonstrate the final stage of finding the first two of them, related with the Chandler period. Final stage means that we shall use first in (5), then in (7) m = M - 2 = 14 frequencies, namely those with numbers 3 to 16, while the two first frequencies will be found through frequency variations. Figure 5 is obtained by the application of the model (5) where the variable frequency w is moving between 0.80 + 0.95 cpy. Graphics like this one can be considered as an AIC spectrum.
AIC(w) (5)
j=l In the expression (5) the new frequency w is variable within a selected frequency interval. For every value of w we get the least squares solution of (5), accompanied by a corresponding value of AIC. In such a way we obtain AIC as a function of w , i.e. AIC = A I C ( w ) . Let for a given value w = WMi. AIC(wMi o) be the minimum of the function AIC(w) in the frequency interval and, still, AIC(WMin) be lower
(7)
j=!
>
160000 159990 159980 159970 159960
159950 159940 159930
Frequency in cpy (cycles/year) Fig.5. The values of AIC = AIC(w) as function of the frequency w .
455
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B. Ducarme • A. P. Venedikov • A. R. de Mesquita • C. A. de Sampaio Fran~ja • D. S. Costa. D. Blitzkow • R. Vieira Diaz. S. R. C. de Freitas
The absolute minimum of AIC in this case is at WMin = 0.84402 cpy, i.e. at period 432.75 days. We have got also a relative minimum at frequency 0.87993 cpy, i.e. at period 415.09 days.
deeper difference is that the usual spectral analysis does not take in consideration the dependence between the frequencies found, which can be important when some of them are very close. On the contrary, our spectral analysis takes into account the effect of the frequencies already found, in this case of the m = M - 2 = 1 4 frequencies. The procedure can be repeated as many times as necessary in order to clean up completely such kind of dependences. Table 2. Summary results for very low frequency components.
%_
"
10 11 12 13 14 15 16
",7
Fig. 6. A I C = A I C ( w ' , w " ) with a minimum at w' = 0.84180 cpy and w" = 0.88102 cpy.
Due to the existence of two minima at very close frequencies we had to apply the model (7). In such a way we get the picture presented by Figure 6, through which we get the final values of the first 2 components in Table 1. The existence of two close periods at 433.6 and 414.3 day around the Chandler period of 430 day is due to the amplitude modulation of the signal. Table 1. Summary of the periodicities found by the automatic research procedure for the Chandler term, the annual one and its harmonics
Chandler Annual 1 cpy Annual at 2 cpy Annual at 3 cpy
1 2 3 4 5 6 7 8 9
Freq. cpy 0.8418 0.8810 0.9973 1.0295 1.9881 2.0154 2.9189 2.9555 3.0298
Period Amplit. Days cm 433.6 1.522 414.3 1.271 366.0 5.111 354.5 1.384 183.6 1.241 1 8 1 . 1 0.497 1 2 5 . 0 0.945 1 2 3 . 5 0.637 1 2 0 . 5 0.912
MSD cm _+0.188 _+0.188 _+0.191 _+0.192 _+0.193 _+0.193 _+0.189 _+0.189 _+0.189
The usual spectral analysis deals with the amplitude spectrum, i.e. the amplitude, represented as a function of the frequency, looking for its maximum. Here we deal with AIC as a function of the frequency, looking for the minimum of AIC. A
Freq. cpy
Period Years
Amplit. cm
MSD Cm
0.0413 0.0931 0.1479 0.1951 0.2601 0.3159 0.3795
24.22 10.74 6.76 5.13 3.84 3.17 2.64
3.39 1.48 @.97 0.87 1.39 1.61 1.36
+0.21 +0.19 +0.19 +0.19 +0.19 +0.19 +0.19
As shown in Table 2 we have identified 7 very long periods. The component with the second largest period (10.74 years) corresponds to the known solar cycle with a period close to 11 years. Harmonics of this term are also present: 5.13 year (order 2), 3.84 and 3.17 (order 3) and 2.64 (order 4). The periodicity of 6.76 year, already found by Mesquita et al. (1995), is not easy to interpret, but could be the fourth harmonic of the longest period. The total contribution to the sea level variations at Cananria of periodicities longer than two year is plotted in Figure 7. The peak to peak amplitude (double amplitude) reaches 15 cm.
8 4 -4 -8
Time in days Fig. 7. Total contribution to sea level variations of the modeled periodicities with period larger than two years.
Chapter 66 • New Analysisof a 50 YearsTide Gauge Recordat Canan~ia (SP-Brazil)with the VAVTidal AnalysisProgram 5 Evolution
of the
mean
sea
level
If M denotes the total number of the frequencies, the MSL L(t), as it was defined by (3), is found through the least square solution of the model M
d(t) - a o + ajt + Z W ( o i , t )
(8)
159770 159760 < 159750 159740 159730 --= 159720 159710 159700
'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1
j=l
The use of the linear function a o +alt as representing L(t) requires some explanations. It appeared that when L(t) is represented by a polynomial of power higher than 1, this polynomial interferes with the low frequency components, especially with the 24.2 year wave. Namely, we start getting too high coefficients of the polynomial with too high amplitudes of the periodic constituents. Due to this, we have accepted that the non-linearity in the development of d(t) is entirely represented by the periodic constituents, while a reasonable representation of the MSL is by a polynomial L(t) of power 1. A problem in this simple model of L(t) is whether there are not some discontinuities. Namely, whether there are not one or several time points TDisc in which is an offset in L(t), i.e. some changes in the constant a 0 or changes in the general behavior, i.e. changes in the coefficient aj. Testing the data in this sense is generally necessary for Earth tides data, since the gravimeters and the clinometers are suffering from offsets due to instrumental problems. It is rather unlikely to happen with tide gauges data, except during long interruptions of the records or inside interpolated sections. Our conception is that in any case we should suppose possible discontinuities and check the data for their existence. Let us check whether in a point T is a discontinuity. Then (8) should be used with different models of L(t) before and after the point T , namely
L ( t ) - a ' o +a~t for t < T ] "
L ( t ) - a o + a~'t for t >
T~ --->AIC(T)
(9)
We can let vary the point T in a time interval. For every value of T we can apply the least squares and thus obtain the AIC = AIC(T) as a function of T. If AIC(T) at T=TMi n has a minimum and AIC(TMin) is lower than AIC for the model (8), then we have to accept that TMin is a point of discontinuity, i.e. TDisc = TMin .
Time point T in days Fig. 8. AIC as a function of a supposed point T of discontinuity with TMin - 6349 days (16.07.1971).
This is the case of Figure 8, showing a TDisc = 6 3 4 9 days. The result from the analysis with this TDisc is shown in Figure 9. A significant offset (jump) of 4.00 cm _ 0.68 cm has been found. However, the difference between the slopes of L(t) before and after the discontinuity appeared to be not significant: 0.511 +0.006 cm/year against 0.574+0.018 cm/year. Other points of discontinuities have not been found. 240 220 200 180 160 140 120
Time in days Fig. 9. The data d(t) and the MSL L(t) (white line) with a discontinuity in TDisc - 6 3 4 9 days. On the basis of this result the data d(t) have been corrected for the offset but, due to the lack of significant differences of the slopes before and after the jump, we applied later on the model (8) without discontinuity. The final result about L(t) are: a 0 - 171.279 + 0.094 at the epoch 31.01.1979 a~ - 0.5666 + 0.0070 cm/year The slope is slightly higher than the 0.405 cm/year reported by Mesquita et al, 1995 or Harari et al, 2004.
457
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B. Ducarme • A. P. Venedikov • A. R. de Mesquita • C. A. de Sampaio Fran(ja • D. S. Costa. D. Blitzkow • R. Vieira Diaz. S. R. C. de Freitas
6 Modeling of the ocean pole tide The existence of Chandler frequencies indicates that the astro-geophysical phenomenon known as "ocean pole tide", generated by the polar motion, is effectively present in our data. The polar motion changes the position of the axis of rotation inside the Earth. At a given location on the Earth it produces a change of latitude measured by astronomical methods and a change of gravity, associated with the corresponding change of centrifugal force, measured by the superconducting gravimeters (Ducarme & al., 2006b). Moreover the associated P~ potential is able to excite a tidal deformation of the sea surface. In section 4 the study of the ocean pole tide was made in the frequency domain. It is certainly interesting to study the direct effect of the polar motion in the time domain. At a point of coordinates ((,0,)v) of the ocean surface, the equilibrium ocean pole tide can be written
Here we use only M - 2 of the frequencies, the Chandler frequencies in Table 1 being excluded. As we know already all M - 2 frequencies, the only problem to estimate 6p is the unknown At, which participates non-linearly in this model. The problem has been solved by applying again a Bayesian approach, i.e. through the least square solution of (12) for a set of different values of At. In this case, since we are mostly interested in the values of 6p we have chosen as a criterion the estimated MSD of this unknown. As shown by Figure 10 we get this MSD as a function of At. A broad minimum exists between 8 and 24 days, but we can accept, as a most reliable value of the time lag, the value of At at which we get the minimum of MSD i.e. At = 21 days. It means that the effect of the polar motion has 24 degrees of phase lag. 0.4430 -
i,
0.4425 0.4420 0.4415
.
0.4410 2/2
p(t) -- 72~'-'1
/2g
(~o)
•{x(t) COS)V+ y(t) sin )v} sin 2q~
0.4405
Time lag At in days, At > 0 : retardation where 72 - 1 + k 2 - h 2 , h 2 and k 2 being the Love numbers for radial deformation and change of the potential respectively, x(t) and y(t) are the coordinates of the pole at time t, fl is the angular velocity of the Earth, r is the radius of the Earth and g is the acceleration of gravity. The equilibrium pole tide has been computed for the daily coordinates of the pole, available from IERS since January 1, 1962, i.e. nearly 3000 days later than the start of the Canandia data. In analogy with the gravity effect of the polar motion (Ducarme & al., 2006b) we have accepted that the effect of the polar motion on our data A d(t) will be Ad(t) - 8pp(t-At)
Fig. 10. MSD of 6p as a function of the time lag At in the model (12), with a minimum at At - 21 days Finally, by applying A t = 2 1 , we have got through the solution of (12) for the ocean pole tide an amplification factor 6p - 2.19 + 0.44. 2.0 1.0 0.0 -1.0 -2.0
.
Time in days, 1.01.1962 - 31.12.2004
(11)
Fig. 11. Estimated ocean pole tide. where 6p is an unknown regression coefficient, called also amplitude factor, and At is a time lag, positive for retardation. This expression is included in our model in the following way M-2
d(t) - ao +a¢ +
~-~ff(%.,t)+apAt-m) j=l
Figure 11 displays the estimated ocean pole tide at Canandia, showing the effect of the secular drift of the pole position.. Using the model (12) the MSL variation becomes a 1 0.547 + 0.014 cm/year, with a lower precision than in the previous section. One of the reasons is that here a shorter series of data is used. -
(12)
Chapter 66 • New Analysis of a 50 Years Tide Gauge Record at Canan~ia (SP-Brazil) with the VAV Tidal Analysis Program
7 Modeling of the direct effect of the solar activity Figure 12 shows the daily series of the sunspot index, measured by the number of Wolf W(t), taken from the SIDC (Solar Index Data Center) (Vanderlinden et al, 2005). The series nearly coincides with the Canandia ocean data. The phenomenon has obviously a period, very close to the period of our component 11 in Table 2. Following the same idea as in the previous section we have attempted to study the direct effect of W(t) on our data. The model now used is
The computed effect of the solar activity on the data is shown by Figure 13. It reaches a peak to peak amplitude of 4cm. The estimated slope of L(t) obtained in this way is a 1 = 0 . 5 7 4 + 0 . 0 1 0 . Here again, as in the case of the polar data, the new value of a 1 does not differ significantly from a 1 in section 5 and, in the same time, it has a lower precision. _
-1
-
-
~-2 -3
-
M-1
d(t) - ao + alt + Z W ( m / ,t) +6wW(t-kt)
(13)
-4
'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1
j=l
where 6 w is an amplifying coefficient, At is a time lag, positive for retardation and W(t) represents the solar activity. The number of waves is reduced by 1, because the component 11 is excluded.
Time in days, 26.02.1954 - 9.09.2003 Fig. 13. Estimated effect of the solar activity on the ocean data.
8 Conclusions 350 300 250 200 150 "~ 100
Time in days, 26.02.1954 - 9.09.2003 Fig. 12. Solar activity (sunspot index,, measured by the number of Wolf) and a least squares filtered variant (white curve): cut off 0.001cpd, half length 1,000 days. We have used W(t) in the following variants: (i) the raw data, (ii) filtered data by a least square low pass filtering with characteristics cut off frequency 0.003cpd, half length 300 and (iii) filtered with cut off frequency 0.00 l cpd, half length 1000 (the white curve in Figure 12). In all cases, using the scheme illustrated in (Figure 10), we have got a zero time lag, i.e. At - 0. With At - 0 in (13) we have got an estimate of 6 w with highest signal-to-noise ratio by using the smoothed data in variant (iii), namely 6w - -1.88 _+0.28 cm/(100 units of W(t)). It is thus in opposition of phase with the excitation.
An automatic procedure is used to determine hidden long period frequencies from the daily residues also called non-tidal components. The amplitude modulation of the annual radiation tide and its harmonic can be represented by a frequency splitting. The situation is similar for the Chandler period, associated with the ocean pole tide. Two periodicities dominate the very low frequency spectrum at 24.2 year and 10.7 year. The second one is clearly associated with the Solar cycle. The ocean pole tide was modeled using the equilibrium tide. The results show an amplification factor close to 2 with a time lag between 10 and 24 days. In a similar way we tried to associate the 10.7 year period with the daily sunspot number. We found a perfect anti-correlation, without any time lag. For what concerns the MSL variations, obtained after subtraction of the ocean tide signal and all the long period harmonic terms (pole tide and radiation tides), we found a mean linear drift rate of 0.5666 ___0.0070 cm/year. This high rate is probably due to ground subsidence. GPS observations at the same site seem to confirm this fact (Blitzkow and Costa, personal communication). The necessity of a careful determination of the very long period harmonics for the estimation of MSL has to be underlined.
459
460
B. Ducarme • A. P. Venedikov • A. R. de Mesquita • C. A. de Sampaio Franga • D. S. Costa. D. Blitzkow • R. Vieira Diaz. S. R. C. de Freitas
Acknowledgements The work of A.P. Venedikov on this paper has been supported by the Royal Observatory of Belgium and the International Center for Earth Tides in Brussels and by the project REN2001-2271/RIES of the Institute of Astronomy and Geodesy in Madrid.
References
Ducarme B., Venedikov A.P., Arnoso J., Vieira (2005a). Analysis and prediction of ocean tides by the computer program VAV. Journal of Geodynamics, 41, 119-227. Ducarme, B., Venedikov, A.P., Arnoso, J., , Chen X.D., Sun H.P., Vieira, R. (2005b). Global analysis of the GGP superconducting gravimeters network for the estimation of the pole tide gravimetric amplitude factor. Journal of Geodynamics, 41,344-344. Godin G., 1972. The analysis of tides. Liverpool University Press, 263 pp. Harari J., Fran~a C. A. S., Camargo R. (2004). Variabilidade de longo termo de componentes de mards e do nivel medio do mar na costa b ras il e ira.h ttp ://www. ma re s. io. usp. b r/aa g n/aa g n 8/ressi/ressimgf html. Mesquita A.R., Harari J. (1983). Tides and tide gauges of Canandia and Ubatuba-Brazil. Relat. Int. Inst. Oceanogr. Univ. Sao Paulo, 11, 1-14
Mesquita A.R., Harari J., Fran~a C.A.S. (1995). Interannual variability of tides and sea level at Canandia, Brazil, from 1955 to 1990. Publfao. Esp. Inst. Oceanogr., Sao Paulo, 11-20. Mesquita A.R., Harari J., Fran~a C.A.S. (1996). Changes in the South Atlantic: Decadal and Interdecadal Scales. An. Acad. Bras. Ci, 88 (Sup P1), 105-115. Munk, W. H., Cartwright, D. F. (1966). Tidal spectroscopy and prediction. Philosophical Transactions of the Royal Society of London A259 (1105) 533-581 Sakamoto, Y., Ishiguro, M., Kitagawa, G. (1986). Akaike information criterion statistics, D. Reidel Publishing Company, Tokyo, 290 pp. Tamura, Y., 1987. A harmonic development of the tide-generating potential. Bulletin informations des MarOes Terrestres, 99, 6813-6855. Vanderlinden R.A.M. and SIDC team (2005). On line catalogue of the sunspot index, http://sidc.oma.be/html/sunspot.html Venedikov, A.P., Arnoso, J., Vieira, R. (2001). Program VAV/2000 for tidal analysis of unevenly spaced data with irregular drift and colored noise. Journal of the Geodetic Society of Japan, 47 (1), 281-286. Venedikov, A.P., Arnoso, J., Vieira, R. (2003). VAV: a program for tidal data processing. Computers & Geosciences, 29, 487-502. Venedikov, A.P., Arnoso, J., Vieira, R. (2005). New version of the program VAV for tidal data processing. Computers & Geosciences, 31, 667-669
Chapter 67
Atmospheric Mass Flow Reduction for Terrestrial Absolute Gravimetry in the Fennoscandian Land Uplift Network O. Gitlein, L. Timmen Institut fiir Erdmessung, University of Hannover, Schneiderberg 50, D-30167 Hannover, Germany
Abstract. Temporal variations of the atmospheric density distribution induce changes in the gravitational air mass attraction at a specific observation site. Additionally, the load of the atmospheric masses deforms the Earth's crust and the sea surface. Variations in the local gravity acceleration and atmospheric pressure are known to be correlated with an admittance of about -3 nms -2 per hPa as an average factor, which is in accordance with the IAG Resolution No. 9, 1983. A more accurate correlation factor for a gravity station is varying with time and depends on the total global mass distribution of the atmosphere. For the absolute gravimetric observations of the Fennoscandian land uplift, the atmospheric attraction effect of the local zone has been calculated with 3D atmospheric data describing different pressure levels up to a height of 50 km. To model the regional and global attraction and deformation components with Green's functions method, 2D surface atmospheric data have been used. The improved atmospheric effects have been computed for the position-dependent absolute gravity observations in Fennoscandia performed by the Institut fiir Erdmessung (liE) in 2003. The objective is to ensure an air mass reduction within +3 nms -2 accuracy. For the 2003 campaigns, the use of atmospheric actual data has improved the reductions by about 9 rims-2 (max. 14 rims-2).
Keywords. Atmospheric attraction and deformation, Green's functions, 3D atmospheric data, air pressure reductions, absolute gravimetry
1
Introduction
Fennoscandia is a key study region for the research of glacial isostatic adjustment, and it offers an opportunity for testing the GRACE results. In the centre of the Fennoscandian land uplift area, a temporal geoid variation of 3 mm is expected (Ekman and M/ikinen, 1996) over a period of five
years (life time of GRACE). With terrestrial absolute gravimetry, the corresponding gravity change of about 100 rims -2 can be observed with an accuracy of +10 to 20 nms -2 for a 5-year period, cf. Van Camp et al. (2005). Since 2003, annual absolute gravity measurements with different FG5 absolute gravimeters are being performed. Fig. 1 shows the station distribution of the Fennoscandian land uplift network and the stations occupied by liE in 2003. The FG5 design and features are described in detail by Niebauer et al. (1995). The gravimeter is a sensor measuring gravity variations of different sources. The absolute gravity measurements have to be reduced by the effects of the solid Earth tides and ocean tides and polar motion. Also the atmospheric effects have to be modelled and removed. To ensure the high accuracy of the FG5 absolute gravimeter results, the total reduction uncertainty should be as small as possible. Especially in research applications, a total uncertainty of +10 rims -2 or even better is striven for. In this study, the effects of real global atmospheric deviations from a standard atmosphere are investigated. In general, the atmospheric effect is just considered with a correlation factor of-3 rims -2 per hPa, whereby the local air pressure measurement at the gravimetry site is applied. In the following, this common procedure is also called the "-3 nms -2 per hPa" rule. The improvements of the global atmospheric modelling compared to the commonly applied rule are pointed out. The calculation procedure of the atmospheric mass attraction and load is described in section 2. In section 2.2 the calculation of the atmospheric effects of local, regional, and global zones using 2D atmospheric surface data is presented. The data are convolved with Green's functions derived by Merriam (1992). Section 2.3 describes the calculation method of the direct Newtonian attraction effect of the local zone using 3 D atmospheric data in different pressure levels. The results are discussed in section 3.
462
O. G i t l e i n • L. T i m m e n
xl0 ~ 0
+;,oo
11.
~=-2 .G
+65°
~-3 z O
-4
-5
...............................................................
10-4 ÷60°
%
10.3
10.2
10-1
10o
101
10=
...............................................................
700
6O0 .~, 500 n
~ 400 +55
~ 300 .G
+
+ o 15
+20°
+ ° 25
+
1. Fennoscanian land uplift network. Absolute gravity stations occupied by IfE in 2003 are indicated by big dots.
Fig.
~
200
100
o -100
2 2.1
Calculation of Atmospheric Mass Attraction and Load
Fig. 2. The atmospheric load gravity Green's functions for the Newtonian attraction term GN(~) (top) and the elastic deformation term GE(~) (bottom), (Merriam, 1992).
Atmospheric Data
From the European Centre for Medium-Range Weather Forecasts (ECMWF), global 2D and 3D data are available. The data used in this study were provided by the University of Cologne in cooperation with the Deutsches Klimarechenzentrum (DKRZ, German Computing Centre for Climateand Earth System Research). They are logged every 6 hours on a 1.125°× 1.125 ° grid for 2D data and 0.75°×0.75 ° grid in 21 pressure levels (1000 to 1 hPa) for 3D data. The data sets provided for this study contain the parameters barometric pressure p, geopotential V, temperature T and relative humidity r.
2.2
~ ............................................................... 10-4 10-3 10 -2 10 -1 10 o 101 10= Sphedcel Distance ¥ in [=J
Atmospheric Attraction and Deformation from 2D Surface Data
Variations in the Earth's air pressure affect the absolute gravimetric measurements due to the direct Newtonian attraction of the air masses and due to the elastic deformation of the Earth's surface (loading effect). Farrell (1972) derived gravity Green's functions for the elastic part for a point mass load on the Earth's surface. The air mass distribution in a column of up to 60 km height follows in good approximation a mathematical approach based on surface data such as temperature and pressure. This allows to model the Newtonian contribution of the column. Merriam (1992) presented column load gravity Green's func-
tions in a tabulated form. They are plotted in Fig. 2. The Newtonian attraction of an air column of area dA and density p at a spherical distance ¢ can be computed by Zmax
g(¢) _ _ /
Gp(Z)r2sin a dAdz,
(1)
0
(Merriam, 1992; Sun, 1995; Boy et al., 2002). G is the gravitational constant, z is the vertical height in the column, r is the vector distance between the volume of the mass (dAdz) and the gravity station, and a denotes the angle between the attraction direction and local horizon at the station. The density, air pressure and temperature are related by the ideal gas law and depend on height z. Applying cosine law, rearranging and normalization of the formula (1) for attraction and deformation effects, the gravity effect is obtained using the Newtonian GN(¢) and elastic deformation GE(¢) Green's functions:
g(¢)_GN(¢)+GE(¢) 104fig[tad]
A 27T[1__COS(1O)] (P--PN) • (2)
The Green's functions are insensitive to the detailed structure of the model atmosphere. To obtain the pressure p at station height H above the sea level, the input data (global pressure at sea level P0 and
Chapter 67 • Atmospheric Mass Flow Reduction for Terrestrial Absolute Gravimetry in the Fennoscandian Land Uplift Network
z i.................. i........................ inA;m'°pressure sphericdaialevels 21 ................ ........ii:':::::':"! ........................................... "~ ii (1000 to 1 hPa) ...........
.. .=.---~.....,.. - - ~ -../] .i;~~"I--, ..~~. ' , , .-"', •
S i l l ....................................o ~ ' ~ .......... II i
:z:
i
'e
,,.
i
'o
'o
i ,,-".
,'.
......................;'-~~_o~.. =~_, lib
....-
........... .......... ........... ................................
'
dy
. .~....-"
..'""
y..'-
~Absolute ..............."'-of ~ Gravity i ............~ ~' J Station i ......... J '~
,~
Y
/7 ................................. i: . . . .
X ~"
~=1 °
/
Fig. 3. Division of the global pressure data into three grid zones (local, regional, global) around the gravity station.
Fig. 4. Atmospheric mass volume in cartesian coordinates.
the temperature T at surface) have to be transformed (Liljequist and Cehac, 1984):
putation of the local attraction effect from 3D atmospheric data considers more realisticly the real density variations within the atmosphere. Only the local zone is considered here (cf. Fig. 3). The local attraction effect induces by far the greater contribution to the total atmospheric effect. Investigation with 3D atmospheric data for the local attraction can also be found in Neumeyer et al. (2004) and Simon (2003). Atmospheric Newtonian attraction component for a small air mass volume dz dy dz in cartesian coordinates (Fig. 4) is determined by:
H g~
P-p0exp
~T,
(3)
where g,J - 9.80665 ms-2 is the mean gravity at the geoid and RL -- 287.05 J k g - ~ K - ~ is the gas constant for dry air. The normal air pressure P x at gravity station height must be substracted, to get the pressure and the corresponding gravity difference with respect to the standard atmosphere. As reference, the U.S. Standard Atmosphere, 1976 is applied, see also IGC (1988). Because the Green's functions are varying with the spherical distance ~ the globe is divided into three zones, which contribute to gravity in different proportions. The global pressure data have been subdivided into grid zones around the gravity station and interpolated as follows (see Fig. 3): • Local zone with I) _< 1° and 0.01°x0.01 °, • Regional zone with I) < 10 ° and 0.2°x0.2 ° and • Global zone with ~ _< 180 ° and 1.125°x 1.125 °.
///
9Q -- --Gpdiff
x
y
(:c2 + y2 + z2)3/2'
(4)
z
with fldi f f
--
fl -
(5)
fiN
as the difference between the density p calculated from 3D data and the density p x of the U.S. Standard Atmosphere, 1976. It is considered as the mean value of the mass volume. The density p may be derived from the input data temperature T and relative humidity r at the pressure levels (Liljequist and Cehac, 1984): p-
0.378e
Summing up all small effects g(~) of the grid points, the total gravity effect for each zone is obtained.
P
2.3
with vapour pressure e, which can be computed from the relative humidity:
Atmospheric Attraction Local Zone with 3D Data
from
the
Atmospheric pressure variations affect surface gravity by the Newtonian attraction of the atmospheric masses. Just considering the changes of the surface atmospheric data is not sufficient as the higher layers with their deviating air densities contribute significantly to the total Newtonian attraction. The corn-
t~L T[oK] '
=
r
(6)
E,
(7)
100 whereby E is the saturation vapour pressure (Bolton, 1980): ( E - - 6.112exp
17"67 T[°c] )
T[oc] + 243.5
(8) "
463
464
O. 6itlein • L. Timmen
Table 1. Contributions (atmospheric gravity effects) from the local, regional and global zones. For the local zones the results from 2D and 3D data are compared (A" attraction effect, D: deformation effect). Differences resp. improvements to the "-3 rims-2 per hPa" rule effects ("-3 rule") are shown in the right two columns.
[runs -2 ] Cop/Bud Cop/Vest Helsingor Tebstrup Onsala Borfis Metsfihovi Vaasa-AA Vaasa-AB Skelleftefi Arjeplog Kramfors 0stersund Trondheim ~lesund Trysil Honefoss mean rms + abs. max.
3D local A - 15 - 15 - 15 -2 - 17 -6 8 51 2 45 -8 -32 -28 1 12 44 9 2.0 23.9
Effects with Green's functions local regional A D A D -15 1 -2 2 -13 1 -3 0 -14 1 -2 0 0 0 -3 -2 -14 1 -1 3 -3 0 -2 1 8 -1 0 -1 51 -4 0 -9 3 0 0 -1 47 -3 -1 -9 -3 0 -3 -2 -29 2 -2 4 -24 2 -2 4 4 0 -1 -4 16
-1
-1
51 14 4.7 24.4
-4 -1 -0.3 1.7
-1 3 -1.2 1.9
47
-4
-9 1 -1.6 4.4 10
V (1 - k cos 2 ~ ) g ~
,
-4
-4 -4 -4 -1 -3 -1 -3
-2 -1 -3 -4 -2 -2 -2
-2
-2
-3 -1 -1.9 2.1 10
(9)
where k = 0.002637. The gravity attraction is determined by summing up the results for each small mass volume in z, y and z direction. The calculation method for the changes in the atmospheric mass attraction in a rectangular cartesian coordinate system is given in e.g.: Jung (1961), N a g y (1966, 2000), Sun (1995).
Improved Atmospheric Effects for Fennoscandia 2003 The atmospheric effects have been calculated for 17 absolute gravity stations in Fennoscandia occupied by IfE in June, August, and September 2003 with the absolute gravimeter FG5-220. Averagely, each station has been occupied over a measuring period of about 30 hours with about 3000 free-fall experiments. The atmospheric effects have been computed
D -1 -1 -2 -1 -1 -1
-6
1 -4 -2.9 3.3
The coordinates of the input data (interpolated to 0.1° × 0.1° grid, dz = 500 m) have to be transformed to cartesian coordinates with the origin at the gravity station. The height H at a specific latitude 6 can be calculated from the geopotential (Liljequist and Cehac, 1984): H =
from 2D global A -3 -3 -5 -3 -4 -3
Total effect with local 3D 2D -17 -17 -20 -18 -22 -22 -ll -9 -19 -16 -12 -9 -4 -3 32 32 -5 -3 25 27 -17 -12 -33 -29 -28 -24 -9 -6 2 6 28 36 8 14 -5.9
-3.2
19.7 33
19.3 36
"-3 rule" -14 -15 -13 -2 -13 1 5 36 1 36 -4 -24 -20 5 11 36 8 2.1 18.6 36
Diff. to total effect 3D 2D -3 -2 -5 -3 -10 -9 -9 -7 -5 -2 -13 -10 -9 -9 -5 -5 -6 -5 -11 -9 -14 -9 -8 -5 -8 -4 -14 -11 -9 -5 -7 0 0 5 -8.0 -5.2 8.8 6.5 14 ll
exactly for the measuring time. Interpolation is needed due to the 6 h spacing of the data. Table 1 shows the computation results of the attraction (A) and deformation (D) components for the local, regional and global zones from 2D data for the stations in chronological order. For the local zones, also the results from 3D data are given. The zones are divided as described in section 2.2. The cumulative influences from the atmospheric data (with local 3D and local 2D) and the effects computed with the " - 3 rims -2 per hPa" rule are also shown in Fig. 5. The contributions of the different zones are depicted in Fig. 6. For the local Newtonian attraction effects a max. value of 51 n m s -2, and a rms of + 2 4 n m s -2 have been obtained. The local deformation effects from the local zones are small (rms = + 1 . 7 n m s -2) compared to the attraction and have the opposite sign. The regional and global effects contribute each with up to 10 rims -2. The global zone has a significant signal and should not be neglected for precise gravity observations. The contribution of the regional zone is sitedependent and may include oceanic areas, involving an inverse barometer correction which cannot be properly performed, cf. van D a m and Wahr (1987), Merriam (1992). In this study the globe is divided
Chapter 67 • Atmospheric Mass Flow Reduction for Terrestrial Absolute Gravimetry in the Fennoscandian Land Uplift Network
CBud
-"
i- ,
i
_
3D
.....C V e s ..........i............................................... .... I Hel
t
'-~
CBud
2D -3 rule
_
C,Ves,,,,! .......~...
.......................-..-. I
....
--I
Hel
!
319 local 2 D local regional global
Teb Ons •,
r
Ons
i
B,OF......................... ~ ....................~ Mets
.,iBor .................i............................~
...................................................
==
Mets i
- V - A A ..................... ~...................................................
V-AB
;
=~D
'
....Kra ....... ,
~
~==
Ost
i
J
El I...........................i.................................................................. ........
i
-
-
i
•T r o ..................................~...............
Ale
......................................................... " ................................................................................
--
,
i
•'Try ............................ i ................................................... ~
-40
..............................................................................................
V-AB ] .................... . ....... ]
Ost
Hen
i
....V-AA,,,, ,,,,!.................... ~
;.......... i ......
• S k e l ..................................................................................
Arje
.................................................................................................................
!
,
ii
~t _ _ J
i
i
i
0
20
-20
Ale
I ......
• Try
.................... ~..................... ~
Hen
i
40
-40
i -20
.............................................................................. ...........................
=:U =
i
0
20
40
60
[ n m / s 2]
[ n m / s 2]
Fig. 5 . Total effects calculated from 3D and 2D, only 2D data and with the " - 3 nms -2 per hPa" rule (columns 9 to 11, Table 1).
Fig. 6. Attraction and deformation contributions of different zones, determined from 2D and 3D atmospheric data (derived from columns 2 to 8 of Table 1).
into the three zones for all station locations in the same way, independent of the oceanic vicinity. A non-inverted barometer hypothesis for the oceanic and the Baltic Sea response has been chosen. This simplified assumption considers the still existing uncertainties in modelling the actual response of the Baltic Sea and the oceans. Atmospheric load effects at GPS sites (BIFROST stations) in Fennoscandia have been discussed and analysed in detail by Scherneck et al. (2003). Virtanen (2004) applied the non-inverse barometric response at the Baltic Sea and successfully compared the modelled sea loading with observations of the superconducting gravimeter at Metsfihovi (Finland). The inverted barometer response along Geosat altimeter tracks in the open ocean has been investigated by van Dam and Wahr (1993). Boy et al. (2002) compared loading results with non-inverted and with inverted barometer assumptions for several superconducting gravimeters of the Global Geodynamics Project (GGP) network. For the absolute gravity determinations in Fennoscandia 2003 (this paper), an uncertainty of some nms -2 in the reductions of the atmospheric mass attraction and load can not be excluded. Compared with the originally applied " - 3 nms -2 per hPa" rule for the absolute gravimetry campaign
in 2003, the largest improvement has been achieved with the 3D atmospheric data set. The differences to the " - 3 nms -2 per hPa" rule effects reach up to 14 nms -2 (column 12, Table 1) and vary with arms of +9 nms -2. The difference to the total effect from 2D data only vary with arms of +7 nms -2. Using atmospheric data distributed over different height levels (3D) changes the effects of the local zone in the range of several nms -2 compared to 2D calculations (max. 7 nms -2, rms = +3 nms-2). The greater differences have been found for the last seven stations occupied in September, when the wheater become more unstable, cf. Figs. 5 and 6. Only the 3D data imply the actual height dependent density information from real atmospheric observations. These reduction improvements should be considered for precise absolute gravity determinations. 4
Summary
In this study, the Newtonian attraction and elastic deformation effects induced by atmospheric mass flow have been computed for the absolute gravity observations performed by IfE in the Fennoscandian land uplift area in 2003. The effects have been computed globally using surface 2D data convolved with Green's functions, whereby the globe has
465
466
O. Gitlein • L. T i m m e n
• for high accurate absolute gravity applications, a special support with most advanced models about surface deformations due to mass loading (atmosphere, oceans and seas, continental hydrosphere) would be useful.
Ekman, M. and Mfikinen, J. (1996). Recent postglacial rebound, gravity change and mantle flow in Fennoscandia. Geophysical Journal International, 126:229-234. Farrell, W. (1972). Deformation of the Earth by Surface Loads. Reviews of Geophysics and Space Physics, 10(3):761-797. IGC (1988). International Absolute Gravity Basestation Network (IAGBN) Absolute Gravity Observations Data Processing Standards & Station Documentation Ont. Grav. Com.- WGII: World Gravity Standards). Bulletin d'Information, Bur. Grav. Int., 63:51-57. Jung, K. (1961 ). Schwerkrafiverfahren in der angewandten Geophysik. Geest & Portig K.-G., Leipzig. Liljequist, G. and Cehac, K. (1984). Allgemeine Meteorologie. Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig. Merriam, J. (1992). Atmospheric pressure and gravity. Geophysical Journal International, 109:488-500. Nagy, D. (1966). The gravitational attraction of a right rectangular prism. Geophysics, XXXI(2):362-371. Nagy, D. (2000). The gravitational potential and its derivatives for the prism. Journal of Geodesy, 74:552-560. Neumeyer, J., Hagedoorn, J., Leitloff, J., and Schmidt, T. (2004). Gravity reduction with three-dimensional atmospheric pressure data for precise ground gravity measurements. Journal of Geodynamics, 38:437-450. Niebauer, T. M., Sasagawa, G. S., Faller, J. E., Hilt, R., and Klopping, F. (1995). A new generation of absolute gravimeters. Metrologia, 32:159-180. Scherneck, H.-G., Johansson, J. M., Koivula, H., van Dam, T., and Davis, J. L. (2003). Vertical crustal motion observed in the BIFROST project. Journal of Geodynamics, 35:425-441.
Acknowledgements Within the R&D-Programme GEOTECHNOLOGIEN this study is funded by the German Ministry of Education and Research (BMBF) and the German Research Foundation (DFG), Grant Mu 1141/3-1 and 3-2. We like to thank the University of Cologne for providing ECMWF (European Centre for Medium-Range Weather Forecasts) data in cooperation with the Deutsches Klimarechenzentrum (DKRZ, German Computing Centre .for Climate- and Earth System Research). Steven S. Pietrobon is acknowledged for providing a software for the U.S. Standard Atmosphere, 1976.
Simon, D. (2003). Modelling of the gravimetric effects induced by vertical air mass shifts. Mitteilungen des Bundesamtes ftir Kartographie und Geodfisie, Frankfurt am Main. Sun, H.-P. (1995). Static Deformation and Gravity Changes at the Earth's Surface due to the Atmospherical Pressure. Dissertation, Catholic University of Louvain, Belgium. Van Camp, M., Williams, S. D. P., and Francis, O. (2005). Uncertainty of absolute gravity measurements. Journal of Geophysical Research, 110, B05406, doi: 10.1029/2004JB003497.
been devided into local, regional and global zones. Additionaly, the attraction effect of the local zone has been deduced using 3D data, including the information of the density distribution up to a height of 50 km. The total effects have been compared with the effects, computed with the common "-3 rims -2 per hPa" rule. For the absolute gravimetry purpose, the total atmospheric signal has been considered relating to a reference standard atmosphere. For the 2003 absolute gravimetry observations in Fennoscandia • the use of atmospheric data improves the reductions by 9 rims -2 in the average (rms value, max. change 14 nms -2) compared with the reductions by the "3 rims -2 per hPa" rule; • the global effect contributes with 5 rims -2 (max. 10 rims -2 ) and should not be neglected for precise gravity determinations; • local differences between 2D and 3D effects are in the range of several rims -2 (max. 7 rims -2 ) and should be considered to keep the reduction errors as small as possible; • for applications in geodynamics, the global calculation of the atmospheric reductions with Green's functions with 2D surface data should be performed at least;
References Bolton, D. (1980). The Computation of Equivalent Potential Temperature. Monthly Weather Review, 108:10461053. Boy, J.-P., Gegout, P., and Hinderer, J. (2002). Reduction of surface gravity data from global atmospheric pressure loading. Geophysical Journal International, 149:534545.
van Dam, T. M. and Wahr, J. (1993). The atmospheric load response of the ocean determined using Geosat altimeter data. Geophysical Journal International, 113:1-16. van Dam, T. M. and Wahr, J. M. (1987). Displacements of the Earth's Surface Due to Atmospheric Loading: Effects on Gravity and Baseline Measurements. Journal of Geophysical Research, 92(B2): 1281-1286. Virtanen, H. (2004). Loading effects in Mets/ihovi from the atmosphere and the Baltic Sea. Journal of Geodynamics, 38:407-422.
Chapter 68
Tilt Observations Around the KTB-Site / Germany: Monitoring and Modelling of Fluid Induced Deformation of the Upper Crust of the Earth T. Jahr, G. Jentzsch, H. Letz, A. Gebauer Institute of Geosciences, Department of Applied Geophysics, University of Jena, Burgweg 11, D-07749 Jena, Germany Abstract. At the German Continental Deep Drilling site (KTB) the pilot borehole of a depth of 4000 m was used to inject water with a medium rate of 180 litres/minute over a period of one year. To monitor the expected surface deformation five borehole tiltmeters of the ASKANIA type Gbpl0 were installed in the surrounding area of the KTB location in mid 2003. The deformation was detected at kilometre scale, together with the observations of induced seismicity in the area. We observed elastic as well as inelastic responses: changes of the rheologic properties due to pore pressure increase caused changes in the tidal parameters. First we quantified the expected additional drift for different injection scenarios at each tiltmeter site by numerical modelling. It could be demonstrated that for long term injection phases of up to four months a maximum tilt effect of about 40 nrad is modelled, which should be detectable. We expected changes of the drift curve, slow variations correlating with the injection rate as well as with changes of the rate. First results of the monitoring are presented: they reveal a slight increase of the tidal parameters (main tidal constituent O1 and M2, north-south component) and drifts associated with the long-term injection. Comprehensive numerical modelling using the Finite Element software ABAQUS is in preparation.
Keywords. Surface tilt measurements, fluid injection, KTB drill hole, ASKANIA tiltmeter, tidal parameters, upper Earth crust
1 Introduction A tiltmeter array, consisting of five high resolution borehole tiltmeters of the ASKANIA type Gbp 10 was installed in the surrounding area of the KTB location in mid 2003 (Fig. 1). There, an injection experiment of water started in June, 2004, to be completed in May 2005. The injection had an average rate of 180 litres/minute into the KTB pilot
borehole (4000 meters deep). The aim of the research project was to observe the induced deformation of the upper crust at kilometre scale. Together with the observations of tilt changes, induced seismicity in the area was monitored by a local seismograph network. Numerical modelling revealed that a constant injection of about 180 1/min would cause a deformation of about 40 nrad within 4 months.
"Fig. 1 The area around the KTB-site: The five tiltmeter stations are connected via WLAN (white lines) to transmit the data online. Data transmission from Mittelberg is routed via Stockau. Depths of boreholes are given in meters. The depths of the boreholes were chosen as such that the tiltmeters could be installed in hard bed rock well below the ground water table. Since it is necessary to eliminate locally induced interference, (e.g. arising from groundwater variations) a second borehole was drilled at all stations to observe ground water changes, especially pore pressure
468
T. Jahr. G. Jentzsch • H. Letz. A. Gebauer
changes. All groundwater monitoring boreholes were fitted with tube packers (Jahr et al., 2005a). In this way the instantaneous changes of pore pressure could be monitored to avoid damping and time delay due to water exchange between the rock and the well.
Thus, the quality of the raw tiltmeter data and the remotely performed instrument calibration can be checked every day. In Fig. 3 the data are plotted. We also installed a meteorological station at KTB to monitor air pressure and precipitation.
2 I n s t r u m e n t s and Data
The locations of the tiltmeter stations are centred around KTB at distances of 1.6 to 3.2 km. The depths of the instruments range between 24.5 and 45.5 m depending on the thickness of the sediment cover, which usually is less than 20m. The tiltmeters are equipped with a 3-D geophone set in order to complement the local seismic network (Fig. 2). The groundwater level is monitored using Van Essen's DIVER sensor with a sample rate of 5 minutes. Tilt, groundwater level and seismic data are downloaded via a wireless LAN (WLAN) to KTB and further via Internet to the GeoForschungsZentrum (GFZ)-Potsdam and the Institute of Geosciences, Jena. The project is carried out in close cooperation with seismologists of the GFZ-Potsdam who operate the seismic network and the WLAN. EW - component (in msec x 10)
2000 ,ooo°
-1000 -2000 -3000
1~
Fig. 2. Top of the ASKANIA Borehole Tiltmeter GBP10 (instrument resolution < 0.2 msec) fitted with a 3-component geophone. To measure the azimuth of the tiltmeter against true North by optical methods in the borehole, two sets of Light Emitting Diodes are mounted above the geophones.
NS - c o m p o n e n t (in msec x 10)
Berg
_---.--" . ~
"
f~~-'~
"~'~**~~
~
10000
L
8000 6000 4000 2000 0
Piillersreuth
-1000 -2000 -3000 4000
_.____.._f ~ --v
~'~'~.pii~-"-~ ~'--~--.___ /~ "*'"~'w"""~ '''w
-2000 -4000 -6000
~--'*'" ~ ~'~ I ° Stockau
4000 0 -4000 -8000
F
10000 4 . . . . . . . . . 5000 ~ . ~ _ f ~ 0 -5000 Eiglasdorf
6000 4000 ~ ~ . _ 2000
~--
o
-2000
~,~'~
4000 3000
~
_
0 -10000
~
-20000 Mi~elberg
~
2000 ~ 1000
"
'
~
-
'
~
~ ' ~ ~
'I'I'I'I'I'I'I'I'I'I'I'I'I'I'I'I'I' Nov-03
4000
~
Jan-04 Mar-04 May-04
Jul-04
~ ' ~ ~ ~ .
'I'I'I'I'
Sep-04 Nov-04 Jan-05 Mar-05 May-05
Jul-05
2000 0 -2000
Sep-05
Fig. 3. After pre-processing the time series of the tiltmeter observations the drift behaviour can be visualised. The data is displayed in milliseconds of arc times 10. The maximum tidal amplitudes are about 100 nrad (about 20 msec). Start of injection (June 04) and completion (May 05) is indicated. Data gaps at Eiglasdorf and Berg were caused by malfunctioning data loggers. In Stockau the tiltmeter failed in July 2004. Mittelberg shows the lowest drift rates, while in Ptillersreuth non-linear parts were caused by heavy precipitation.
Chapter 68 • Tilt Observations around the KTB-Site / Germany: Monitoring and Modelling of Fluid Induced Deformation of the Upper Crust of the Earth
The tilt data presented in Fig. 3 consist of long and uninterrupted sections. The comparison of the tidal amplitudes reveals the different long-term drifts at each station. There are also different reactions to precipitation and local ground water changes. Heavy rainfall usually results in strong deviations with a relaxation of several days.
3 First Results The Data are analysed for tidal parameters and for the long-term drift. The correction of the data for local meteorological effects (precipitation, air pressure) and ground water is in progress.
are quite similar, and the station Stockau as well, but with an additional strong (linear?) drift. The variations seen at Pallersreuth may be caused by local hydrological variations.
Hodograms Since the tiltmeters used are vertical pendulums with two degrees of freedom we can plot the movement of the tip of the pendulum over ground. The resulting curves contain the tidal movements, especially the 14-days beating period of M2 and $2, and the long-term drift. The stronger the drift the smaller the tidal oscillations in the figure, serving as a reference.
Tidal analyses The tidal analyses should provide 3,-values (the measured amplitudes of the tidal waves divided by the theoretical amplitudes) to be in the order of 0.7, but they vary quite a bit over the whole array, although the distances are less than 5 km (Tab. 1). There is also no correlation between the values for the tidal waves O1 and M2. We assume that local geological inhomogeneities and topography are the reason. The error bars are also quite big, which points to temporal variations of the values. On the basis of successive 2-monthly intervals (shifted by one month) this temporal behaviour is analysed: In Fig. 4 the parameter function for O1 (North-South) is given as a sample. Each point stands for the value for each interval. As can be seen, the y-values start to increase slightly at the beginning of the injection. Although the error bars are too big to take this result as significant for the individual stations, the common behaviour for all stations justifies this interpretation of an increasing value for tidal parameters (which is observed for M2 as well, but again only for the North-South-direction).
In Fig. 6 the hodograms for all five stations are given. The first figure top left shows the main geological strikes and boundaries. Two important times are also noted: The start of the injection and the time when the main borehole became artesian. It can be clearly seen that the drift directions correlate with the geological strikes; and the changes in drift correlate with the times mentioned. Also the variations of the injection rate seem to be visible, but this has still to be verified.
Table 1. Tidal analyses for O 1 and M2: calculated are the 7values for NS and EW components of the tiltmeter array at the KTB using the whole recording interval.
O1
Tiltmeter station
NS
EW
NS
EW
BER
0.696 +.138
0.653 +.055
0.758 +.028
0.790 +.018
EIK
0.595 +.085
0.707 +.166
0.656 +.ll7
0.739 +.113
STO
0.227 +.193
0.646 +.135
0.593 +.040
0.856 +.033
MIT
0.545 +.027
0.666 +.020
0.737 +.014
0.793 +.010
PUE
0.556 +.050
0.474 +.110
0.504 +.022
0.547 +.044
Long-term drifts' We can also present first results of the separation of the long-term drift. Although these plots are still preliminary because local effects have to be analysed and corrected, we can see common trends. In Fig. 5 these drifts towards the injection centre are given for all stations except Eiglasdorf, where we suspect strong local effects have masked the expected injection signal and have to be separated first. The curves for stations Berg and Mittelberg
M2
469
470
T.Jahr. G.Jentzsch • H. Letz.A. Gebauer
Parameter Variation KTB array north-south, 01
0,8
m
0.6 t._
E m
0.4
© Eiglasdorf I--1 Stockau Mittelberg / Berg • POllersreuth
0.2
begin I
Nov-03
I
Jan-04
I
of injection
I
I
Mar-04 May-04
Jul-04
I
I
Sep-04
Nov-04
I
Jan-05
I
I
Mar-05 May-05
Jul-05
Fig. 4. Parameter variation of O 1, north-south component: Since the beginning of the injection the parameter increased by about 1.2% per month as indicated by error weighted linear regression.
200 ~
,
~
Berg
Ollersreuth
-200 G) u~
E e-
C~
-400
-600 --
-800 --
startof i
I
i
I
i
MainH
injection I
i
I
i
I
o i
~ I
i
I
i
I
Nov-03 Jan-04 Mar-04 May-04 Jul-04 Sep-04 Oct-04 Dec-04 Feb-05 Fig. 5. Long-term drift of four stations: Shown is the drift in radial direction (as seen from the KTB) in msec (1 msec ~ 5 nrad). The data of the station in Eiglasdorf is superimposed by a strong additional drift and thus needs further treatment before being included here. A negative trend correlates with a movement of the bottom of the borehole in direction to the KTB and vice versa.
Chapter68 • Tilt Observationsaroundthe KTB-Site/ Germany:Monitoring and Modellingof Fluid InducedDeformationof the UpperCrustof the Earth
5~ begin of injection (9 June 2004)
Mittelberg
3000
KTB-HB artesian 36°-39°
~"
170
(11 00
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2000
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c
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i
i
i
1000
i
2000
3000
4000
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I
2000
4000
Stockau
I
-8000 -4000
I
I
0
4000
8000
POIlersreuth
2000
8000
I
start
o
6000
\ y ....
E
4000
-2000
/
O
, O
c
2000 0
-2000
j
~J
start
I
\ \
I
I
I
J
2000 4000 6000 8000 1000012000 east-west (msec)
s
-4000
-6000
I
-6000
I
I
-4000 -2000 0 east-west (msec)
I
2000
Fig. 6. Hodograms: Movement of the tip of the tiltmeter at all stations; note the amplitude of the fortnightly beating period of the tides (M2 and $2 constituents) pointing to the amplitude ratios of drift and tides.
471
472
T. Jahr. G. Jentzsch • H. Letz. A. Gebauer 4
Discussion
We found three different injection induced effects: Variations of tidal parameters, long-term drifts, and direct elastic tilts. These signals now need numerical interpretation. The tiltmeters we used at KTB are the most sensitive instruments available. The high resolution is accompanied by high sensitivity to many external sources, the first being the problem of adequate coupling to the rock. Thus, reliable measurements can only be achieved by carefully drilling the borehole and cementing the casing. In previous projects we observed tilt caused by the load of an artificial lake (Jentzsch and Kog, 1997). Our work in Finland (Weise et al., 1999) provided valuable information not only about the regional representation of the monitored tilt but also about the effects of ground water (pore pressure) changes even in granite (see KiJmpel et al., 1988). Westerhaus and Zschau (2001) reported that no earthquake precursors could be observed because of influences of the local geology and the coupling of the instrument casing. Tilt measurements are very sensitive to meteorological influences like air pressure and precipitation as well as ground water changes. Therefore, these disturbances have to be corrected prior to interpretation (Braitenberg and Zadro, 2001; Ishii et al., 2001). The injection tests at KTB were accompanied by induced seismicity (Zoback and Harjes, 1997). Here, we have a combination of surface deformation and seismicity which was already discussed by Jentzsch et al. (2001). The interpretation with regard to physical properties of the rock and fluid induced processes (Fujimori et al., 2001) will be assessed by numerical modelling. We have already modelled the expected deformation applying the program POEL (by R. Wang), but for detailed modelling of the geologic structure we plan to use the Finite Element software ABAQUS. The first model is already being developed (Jahr et al., 2005b). 5
Acknowledgements
We thank the KTB-Operational Support Group for their help. Special thanks go to G. Asch and J. K u m m e r o w (GFZ) for the organisation of the data collection and transfer. R.J. Wang provided the modelling program POEL which helped to estimate the expected deformation. The local landowners
permitted to drill the boreholes on their properties and accepted much discomfort caused by the drilling work. This is gratefully acknowledged. We thank the G e r m a n R e s e a r c h F o u n d a t i o n (DFG) for financial support, grant no. JA-542/1 11,6. For helpful corrections and comments we thank Dr. L. Wallace and an anonymous reviewer. References
Baitenberg, C., and M. Zadro, (2001). Time Series Modelling of the Hydrologic Signal in Geodetic Measurements. Proc. 14th International Symposium on Earth Tides, Special Issue of the Journal of the Geodetic Soc. of Japan, Vol.
47/1, pp. 95-100. Fujimori, K., H. Ishii, A. Mukai, S. Nakao, S. Matsumoto, and Y. Hirata, (2001). Strain and tilt changes measured during a water injection experiment at the Nojima Fault zone, Japan. The lsland Arc, 10, pp. 228-234. Ishii, H., G. Jentzsch, S. Graupner, S. Nakao, M. Ramatschi, and A. Weise, (2001). Observatory Nokogiriyama / Japan: Comparison of different tiltmeters. Proc. 14~h International Symposium on Earth Tides, Special Issue of the Journal of the Geodetic Soc. of Japan, 47/1,155-160.
Jahr, T., G. Jentzsch, H. Letz, H., and M. Sauter (2005a) Fluid injection and surface deformation at the KTB location: Modelling of expected tilt effects. Geofluids 5, pp. 20-27. Jahr, T., Letz, H. & Jentzsch, G., (2005b) The ASKANIA borehole tiltmeter array at the KTB location / Germany.Journ. of Geodynamics (accepted). Jentzsch, G., and S. KoB, (1997). Interpretation of longperiod tilt records at Blfi Sjo, Southern Norway, with respect to the variations of the lake level. Phys. Chem. Earth, 22, pp. 25-31. Jentzsch, G., P. Malischewsky, M. Zadro, C. Braitenberg, A. Latynina, E. Bojarsky, T. Verbytzkyy, A. Tikhomirov, and A. Kurskeev, (2001). Relations between different geodynamic parameters and seismicity in areas of high and low seismic hazards. Proc. 14 th International Symposium on Earth Tides, Special Issue of the Journal of the Geodetic Soc. of Japan, 47/1, 82 - 87.
Kfimpel, H.-J., J.A. Peters, and D.R. Bower, (1988). Nontidal tilt and water table variations in a seismically active region in Quebec, Canada, Tectonophysics, 152, pp. 253-265. Weise, A., G. Jentzsch, A. Kiviniemi, and J. KfifiriNnen, (1999). Comparison of long-period tilt measurements: Results from the two clinometric stations Metsfihovi and Lohja, Finland. J. of Geodynarnics, 27, pp. 237-257. Westerhaus, M., and J. Zschau, (2001). No clear evidence for temporal variations of tidal tilt prior to the 1999 Izmit and DOzce earthquakes in NW-Anatolia. Proc. 14th International Symposium on Earth Tides, Special Issue of the Journal of the Geodetic Soc. of Japan, 47/1,448- 455.
Zoback, M.D., and H.-P. Harjes, (1997). Injection-induced earthquakes and crustal stress at 9 km depth at the KTB deep drilling site, Germany. J. Geophys. Res., 102 (B8), pp. 18477-18492.
Chapter 69
Understanding Time-variable Gravity due to Core Dynamical Processes with Numerical Geodynamo Modeling W. Jiang Joint Center for Earth Systems Technology, University of Maryland at Baltimore County 1000 Hilltop Circle, Baltimore, MD 21229, USA W. Kuang, B. Chao Space Geodesy Laboratory, NASA Goddard Space Flight Center, Greenbelt, MD 20771 M. Fang Dept. Earth and Space Sciences, Massachusetts Institute of Technology, Cambridge, MA 02136 C. Cox Raytheon at Space Geodesy Laboratory, NASA Goddard Space Flight Center
Abstract. On decadal time scales, there are three major physical processes in the Earth's outer core that contribute to gravity variations: (i) the mass redistribution due to advection in the outer core, (ii) the mantle deformation in response to (i), and (iii) the (core) pressure loading on the core-mantle boundary. Except the last one, they cannot be evaluated from surface observations. In this paper we use MoSST core dynamics model and PREM model to understand the gravity anomalies from the three processes. Our numerical results show that, the gravity anomalies are comparable in magnitude, though that from the process (i) is in general the strongest. The gravity anomalies from the first two processes tend to offset each other ("mantle shielding"). Consequently the pressure loading effect contributes more to axisymmetric part of the net gravity variation, while the net effect from the first two processes is more important to non axisymmetric components.
Keywords.
Gravity anomalies, core convection, numerical model.
1 Introduction For more than two decades, space geodetic techniques have observed quite accurately the synoptic or low-degree temporal variations of the Earth's gravity field (AGU 1993; Cox and Chao 2002). These minute changes (relative to the mean gravity of the Earth) arise from various geophysical
processes on the surface and in the deep interior of the Earth that give rise to mass redistribution over space and time (e.g., Chao et al., 2000). While the major effort in studying the timevariable gravity (TVG) has been pertaining to the processes that happen in the surface "geophysical fluids" (e.g. atmosphere, hydrosphere and cryosphere) and the post-glacial rebound in the mantle, only limited attempts have been made on possible contributions from physical processes in the Earth's fluid outer core, in particular those on decadal time scales. Fang et al (1996), Dumburry and Bloxham (2004) attempted to understand the TVG from non-hydrostatic pressure acting on the core-mantle boundary (CMB). Greff et al (2004) discussed the gravity variation and surface deformation due to core flow. Kuang et al (2004) used their numerical geodynamo solutions to investigate gravity variation due to advection of core fluid density anomalies. However, those studies on the gravity variations from core are incomplete in many aspects. The incompleteness can be identified simply by examining possible geophysical processes in the core that contribute to the decadal TVG. There are at least three major geophysical processes for our attention. The first is the mass redistribution inside the Earth's fluid outer core due to convection (Kuang et al 2004). We denote by 6q~p the corresponding gravity anomaly. From surface geomagnetic observations, and likely dynamic balances in the outer core, one could estimate that 6q~p can be observed at the surface of the Earth (Kuang et a12004). The second process is the loading on the CMB by the non-hydrostatic pressure in the core. This
474
W. Jiang • W. Kuang • B. Chao. M. Fang. C. Cox
loading results in mantle deformation, and then gravity changes over time and space (Fang et al 1996, Dumburry and Bloxham 2004). We denote by 601, the gravity anomaly from this CMB pressure loading. The third process is the response of the mantle to gravity changes inside the fluid outer core 6Op. This process is similar to the solid tides resulting from external gravity field changes (e.g. orbital motion of the moon). We denote by 6q)l the corresponding gravity anomaly. Thus, the results from previous studies, even if justified by themselves, are at best partial as far as a complete comprehension on the net, core-induced TVG & b - cT~p +cT~j +cT~p.
(1)
Surface geomagnetic observations can shed light only on 6q5l,, even though such information is incomplete. The observations could provide the geomagnetic field itself and its secular variation (the time derivative). With these data, the "frozenflux" and the "tangential geostrophy" assumptions, one could obtain part of the pressure p on the CMB (e.g. Jault et al 1988), and therefore 6 ~ . However, there is no surface observation on the density anomalies inside the core, and thus we are unable to determine 6~ and 6~,. Therefore, we shall use numerical model to understand the properties of the three gravity anomalies, in particular to find possible relationships among them. The model used in our study is the MoSST core dynamics model. For the details of the model, we refer the reader to Kuang and Bloxham (1999), Kuang and Chao (2002). Though it targets a (mathematically) simplified Earth's core, the numerical model is self-consistent; and all physical variables describing the outer core are determined dynamically. Therefore we hope that the dynamically consistent dynamo model solutions could shed light on the properties of the three gravity anomalies, and thus their net contribution to the TVG observable at the Earth's surface. This paper is organized as follows: the mathematical formulations are described in Section 2. In Section 3, numerical results with selected parameters are presented. Discussion on possible geophysical applications is given in Section 4.
2 Mathematical Model The numerical model used for this study is the MoSST core dynamics model in NASA Goddard Space Flight Center. For the details of the model,
we refer the readers to Kuang and Bloxham (1999), and Kuang and Chao (2003). In this paper, we simply summarize part of the model relevant to our study. In the MoSST core dynamics model, the outer core is approximated to leading order as a spherical shell. The mean radii of the inner core boundary (ICB) and the CMB are the same as those of the Earth inferred from seismic studies. The coordinate used in the model is defined relative to the "mean", spherically symmetric model Earth: it is fixed to the mantle, and centered at the geometric center (also the mass center). In this coordinate system (or reference frame), the mass center of the model Earth will change due to core convection, which results in mass redistribution in the core, and the mantle deformation discussed in this paper. However, the change of the mass center depends also on the inner core deformation due to loadings on the ICB. However, we do not discuss this phenomena in this paper. All physical quantities are scaled as follows: the length scale r ~ is the mean radius of the CMB; the time scale r is the magnetic diffusive time 2
r - rcmh lrl,
where q is the magnetic diffusivity of the core fluid; the magnetic field scale B0is given by B 0 - ~J2P0f2/ar/ , where P0 is the mean density of the core fluid, f2is the mean rotation rate of the Earth, and /1 is the magnetic permeability in the core (almost identical to that of the free space). In the model, the density anomaly kp is assumed arising from thermal effects
Ap -- --6g T (r -- T O ), /90
where c~r is the thermal expansion coefficient of the core. The temperature perturbation is scaled by r , , . h T (the ambient temperature gradient at the ICB). With these scaling rules, the momentum balance in the fluid core is given by the following nondimensional equation, Vp - J x B - 2 x v - R,h O r o
- - + ~ x V
)
(2) + EV2v,
where B is the magnetic field, a ( - V x B ) is the current density, v is the velocity, m ( - V x v) is the
Chapter 69 • Understanding Time-Variable Gravity Due to Core Dynamical Processes with Numerical Geodynamo Modelling
vorticity, ® is the density perturbation, p is the modified pressure, ~ is the unit vector along the z-axis (the mean rotation axis of the Earth), r is the position vector, R,, is the Rayleigh number, R ° is the magnetic Rossby number and E is the Ekman number. The equation (2) is not in the traditional form because we intend to use it to solve the pressure p. Taking the divergence of Equation (2) and applying the continuity condition V. v = 0, one obtains, V219 - V .
(J x B - k x v + R,hOr
(3)
-RoO~X v ) . This equation is solved together with the boundary conditions
-
f.[JxB-2xv-R,hOr
m
m
h/ '
-
(7)
where h~m are the spherical coefficients of the terms on the right hand side of (4). The pressure p in our model is non-dimensional, scaled by p0 = 2pX2~7. Therefore the dimensional pressure used in Fang et al (2005) is given by p & . Furthermore, the dimensional gravity potential is scaled by ~0-4aGP0ro,2,b (Kuang et al, 2004). Applying the relevant scales to the formulation in Fang et al (2005), we can obtain the following nondimensional gravity anomaly:
6oo
8 --pOr
Pt
-
Or
(4)
- -dP0~< _ _, 2tk;+ 1 p'~(" -~)~" (o, 0,) + c.c. (8) L
- Z~7~(o,d+c.c
- RoO~ x v + EV2v~
O<_m<_l
at the ICB and the CMB. It should be pointed out here that in deriving the boundary conditions (4), we assume that the ICB and the CMB are impermeable spherical surfaces. The boundary conditions (4) will be different if the ICB and/or the CMB are non-spherical. Also, the density anomaly in the core can be related to the non-dimensional quantities via (Kuang et al, 2004)
Ap
= _flR,hO
'
/9o and f l = 1 0 -~ in the Earth's core. see clearly that Rayleigh determines the magnitude of anomaly. All physical variables above spherical harmonic expansions model. In particular,
where k'~ are the load Love numbers, and ? =
(9)
In the above equation, g is the gravitational acceleration at the C M B , p E is the mean density of the entire Earth, r ~ is the mean radius of the CMB, and G is the gravitational constant. In the Earth's core, ~ =1.4. The gravity anomaly from advection of density anomaly in the core can be also written as
From this one can number directly the core density are expressed in in the MoSST
3g 4 7cGp E r.mb
L
, - ~_, ¢"r, m, , (o, ~o)+ c.c.
(~ o)
O<_m
The gravity anomaly from the mantle deformation due to the changes (10) can be expressed as L
~,
L
p -
Y~p;"(nt)Y,"(o,e)+
c.c.
(s)
-= Z
where y/" are the spherical harmonic functions, and (r,O,(,o) defines the spherical coordinate system. Combined (3) and (5), we have d2
m
2 d 1(1+1) r" m _ _ _ m_ r d - r pl r2 Pl ~,l '
+---
(6)
where f/" are the spectral coefficients of the terms on the right hand side of Equation (3). The boundary conditions (4) can be written as
.~,,-," ~""'-
(o, ~,)+ c.c.
O<m
O<m
d r 2 Pl
- Z
(11)
.,.,~"",''~(o, ., ~)+ c.c.
O<_m<_l
where k~ are the body Love numbers of the mantle. In our model, the mantle is approximated as a spherical shell; its gravity is therefore zero in the outer core. In the real Earth, the mantle is nearly spherical shell, and its gravity in the core is of second order effect compared to the mean gravity at the surface of the Earth. Thus we do not consider in our paper the mass loading on the CMB because it is negligible compared to the pressure loading.
475
476
W. Jiang • W. Kuang • B. Chao. M. Fang. C. Cox
To evaluate those numbers, we assume a simple PREM model for the mantle, and the values of the numbers are shown in the table below: Table 1. The load and body Love numbers for the first 6 degrees with the PREM model. Degree 1
2 3 4 5
Load Love Number k' 0.22776 0.31735 0.10924 0.04381 0.01990 0.00976
Body Love Number kt 0.41739 -0.68271 -0.34480 -0.19583 -0.13024 -0.09750
~bt,,,
Our objective is to understand the properties of the three gravity anomalies given by (8), (10) and (11). In particular we are interested in the properties of individual spherical harmonic coefficients. It should be pointed out that the effect of the loading on the CMB is opposite to that on the surface of the Earth. Therefore the Love numbers used in our calculation are opposite in sign to those used for surface loading. For the definitions of the Love numbers, we refer the reader to Heiskanen and Moritz (1968).
3 Numerical Result In our numerical tests, we choose the following parameter values: R o
= E = 1.25 x
(12)
1 0 -6
These values are much larger than those appropriate for the Earth's core: R = 10 -9, and E = 10 -15. Therefore we should be careful in explaining numerical results. However, numerical results can still provide important, dynamically consistent information on the properties of, and the relations among the three gravity anomalies defined in (8), (10) and (11). Since the Rayleigh number Rth in the Earth's core is not well known, and since it directly affects the magnitude of the density anomaly in the core, we select two different values Rth = 15000
and
Rth = 25000,
oscillations possess a frequency co~ oc ~ / R ° . Thus, the Rossby number in (12) could potentially result in one and a half order increase in the typical time scales of the numerical TVG. However, due to computational constraint, we shall not discuss in detail the actual changes in time scales of the core gravity anomaly. Note that the Stokes coefficients customarily used in geodesy and the spherical harmonic coefficients in our model can be converted into each other via
(13)
aiming at understanding how the numerical results vary with R,h. On the other hand, the Rossby number R ° directly affects the time scales of the numerical TVG. Braginsky (1967) derived that the torsional
1 { ......
~
(14)
where i is the imaginary number. Our main effort is to understand the relative importance of the individual gravity anomalies in determining the net gravity anomaly (1). But this is not a trivial question. Because we are only interested in time variation part of the gravity anomaly, therefore, given a specified time domain, we focus only on the "linearly de-trended" spherical harmonic coefficients ~b~m:
l
+ b;,)
~-t are the coefficients of the best where a l//7 and hm linear fit to ~b~' in the given time domain. Unless otherwise specified, the coefficients discussed in the rest of the paper are all filtered via (15). In Figure 1 we plot the spherical harmonic coefficients of gravity anomalies of degrees 1 < 3. From the figure we can first observe that the three gravity anomalies are comparable in magnitude. However, because the body Love numbers (see Table 1) are all negative (except for the degree 1 = 1), c~q)~ offsets partially the gravity anomaly c~q) from the mass redistribution in the core. We also observe from Figure 1 that the gravity anomaly c~O from the CMB pressure loading contributes the most to the equatorially and axially symmetric (m = 0 and even l) part of the net gravity anomaly c~q). This result is very interesting, and suggests that torsional oscillations in the core may be investigated from TVG. The pressure p on the CMB is largely determined by the Coriolis force [the second term on the right hand side of (2)]. Therefore its axisymmetric components are then largely determined by the axisymmetric zonal flow. On the other hand, torsional oscillations are axisymmetric, invariant along the z-axis, and vary on decadal time scales (Braginsky, 1967). Their time variation does not directly depend on the buoyancy force (and thus
Chapter 69 • Understanding Time-Variable Gravity Due to Core Dynamical Processes with Numerical Geodynamo Modelling
density anomalies in the core). Therefore, one expects that, on decadal time scales, torsional oscillations contribute largely to the axisymmetric, time-varying pressure on the CMB, and thus to the axisymmetric TVG.
The non-axisymmetric part of g(I) is very different. From Figure 1 we find that, the gravity anomaly c~q)p+c~q)d arising from the mass redistribution in the core contributes the most to the component qi~ (see the second row of Figure 1).
Figure 1 The spherical harmonic coefficients O~ r pIm (dashed lines), ~b~m (dash-dotted lines), ~bp zm (dotted lines) of the gravity anomalies from the numerical solutions. The solid lines are the coefficients for the net gravity anomaly. The horizontal axis is the (non-dimensional) time used in numerical simulation. The vertical axis are the harmonic coefficients ~lm / / 8 .
But its contributions to the rest of the components are comparable to c~q)p. Figure 2 Similar to Figure 1, but coefficients c 21 and s 21 for R,h-15000. 1
0 C4
the
1
C2
S2
0.01
0.01
i¢s~
.
0 C2
are
0
,
,
~ "
0 %
.
. , v !
,
"
~
~,,
.
2
1
~Om01
0.2
0
~,,'fq 0.2
"
,;
0.4
0.6 t
0
_.
-0.2
0.2
0.4
0.6
t
-0.4
0.8
0.2
0.4
0.6
t
1 C1
0.8
1 S1 4
.
.
.
.
2
0 i
oi~
o'.4
oi~
t
i
o'.~
-
o
o=
1
.
o4
o.~ :
:'~:.,
o~" ;1
~
o~
t
2 C2
o
2 S2
1
,,:~,,-,
;,., ~,,.
o.~ ~
,~a?., ,l'i~i:.i, ~ ~;,
, " ; --' '~,'~,
i',
,,
.
.
,, ,~
,, :
.
o
,.
.
~,,.<.i. , ,
.
.
, ~,
.
~
.,
",,, ::,,,
.
!...j -0.
o;~
o.~
o'.~
t
t
C31
S31
0.4
0.4
/ ~,
0.2 o
.
o.I
"-:~~'
",~',..," "~:'
':'
-o
o;4
t
o;.
o;8
t 3
3
C3 0.4
S3 .
:!.
0.4
.
,,
]
.
,,
:1
0.2
:1
; "
~
o ' ": "":'r::':2:/i:~:,~,: '~,,, '~ ~ .,,:"+""' o :"~",":~:' ",, ' "::~'::f"~i" ',' ".~ : ~ ,; '.:' },.,~ -o.,
o
"
'
":~
"
o'.4
i
o16 t
:~
o'.8
'
-o.,
o 4;
ii
0:2
0.'4
o'.6 t
o'.8
0.8
-0.01 0
0.2
0.4
0.6
0.8
t
The equatorially anti-symmetric components (l + m - odd) of the gravity anomalies are much smaller than the equatorially symmetric components (1 + m = even). To demonstrate this, we plot in Figure 2 the coefficients for the component qt~. One can observe clearly that their variations are at least an order of magnitude smaller than those shown in Figure 1. This property is also consistent with the numerical dynamo solutions. In numerical solutions, the poloidal magnetic field is dominantly dipolar, and the toroidal field is dominantly equatorially symmetric (Kuang and Bloxham, 1999). Since the Coriolis force, the Lorenz force [the first term on the right hand side of (2)] and the buoyancy force [the third term on the right hand side of (2)] are comparable in the bulk of the fluid outer core, the density perturbations and pressure field are dominantly equatorially symmetric, so are the gravity anomalies. As we discussed before, the Rayleigh number R,h used in our system (2) is not well known for the Earth's core. To investigate possible influences of the Rayleigh number R,h on the gravity anomalies, we analyze the solutions for R , , - 25000. The results are shown in Figure 3. From the figure we can observe similar properties of the gravity anomalies. For example, &I)p still contributes the most to the axisymmetric part (m = 0) of the net gravity anomaly from the core, while g@p+cS@d contributes the most to the component ~b~. They are comparable to all other equatorially symmetric components. This test suggests that our findings do not vary qualitatively with moderate changes in the Rayleigh number R,~.
477
478
W. J i a n g • W. K u a n g • B. C h a o . M. F a n g . C. Cox
Figure 3 Similar to Figure 1, but are the spherical harmonic coefficients for R,h - 25000. c:
co 1
\,, \
,
/
\.,, 0.2
0.4
0.6
0.2
0.8
0.4
0.6
t
0.8
t
ci
sl
lO
lO
o '-
.................. ,
-5
-
................ ..'L~.....
",', ,'
-1
0.2
0.4
0.6
t
0.8
t
c:
s: ,:-,
i...... "-.": j ~ I
0.2
: / : ~."~..;', . "
"
0.4
0.6
0.8
""
0.2
~
0:4
t
0.6
0:8
1
t
c:
s:
=
2
1
.
o
-
-'.:...l~" "
1
"
,=.-, :,
_
_ 0.2
0.4
o, /
,,,
0.2 "
,, : :
0.6
0.8
'c:
0.4
0.6
0.8
~:
/0, ~
o
0.2
0.2 o
v
-0.6 0 t
02
04
06
08
t
However, we should point out that from Figure 3 one can find that the magnitudes of the gravity anomalies increase significantly, and that the contribution of 8~p +8~d is more significant to the net gravity anomaly.
4 Conclusion In this paper we used our MoSST core dynamics model to understand the TVG arising from mass redistribution in the core, the deformation in
response to the core mass-redistribution, and pressure loading on the CMB. Our main findings from numerical model solutions include that, the TVG is dominantly equatorially symmetric. In other words, if the gravity anomalies are expressed as spherical harmonic expansions, the spectral coefficients with l + m = even are much stronger than those for l + m = odd. The three gravity anomalies (8), (10) and (11) under investigation are comparable in magnitude, but differ in time variation. The gravity anomaly 8~d from the mantle body tides offsets the anomaly S~p from the mass redistribution inside the core. This cancellation results in the complications on the relative importance of the individual contributions to the net gravity anomaly (1): the pressure loading effect 8 ~ contributes the most to the axisymmetric part of the gravity anomaly (m = 0). On the other hand, the effect 8~p from the mass redistribution in the core contributes the most to the l = m = 1 mode. To the rest of the components, all three are equally important. The above properties on the three gravity anomalies (8), (10) and (11) do not change significantly with the Rayleigh number R,h. We observe similar phenomena when R,h increases from 15000 to 25000. However, we find that the magnitudes of the gravity anomalies increase with the Rayleigh number; and that the combined effect 8~p+8~dis more significant to the net gravity anomaly 8~ from the core. The relative importance of the individual effects could also provide certain clues on the core flow from the gravity field variation. For example, the relative importance of the pressure loading effect on the axisymmetric part of the gravity anomaly indicates that this part of gravity anomaly can be used to probe torsional oscillations in the core. On the other hand, the importance of the mass redistribution in the core implies that the nonaxisymmetric part of the gravity anomaly can be used to cross-examine the convective flow in the core, and may provide valuable insight to the flow deep inside the fluid core. Because the parameters (12) used in numerical simulation differ by orders of magnitude from those appropriate for the Earth's core, there always exist questions on how to apply numerical results to the real Earth. In particular, the main questions to our study of gravity anomalies from the core include the magnitude and the time scales of the gravity anomalies. The former may depend more on the
Chapter 69 • UnderstandingTime-Variable Gravity Due to Core Dynamical Processeswith Numerical GeodynamoModelling
leading order force balance in the core, while the latter depends on the magnetic Rossby number Ro, which, for example, directly determines the period of the torsional oscillations. Currently one may not be able to find the final answers before systematic studies with numerical parameters approaching significantly to those of the Earth's core. However, we may still find many properties discovered here applicable to the Earth's core, because it is likely that the magnetostrophic balance is established in the bulk of the fluid core: ~xv ~-Vp+JxB+
Rth®r.
(16)
Such leading order force balance is also established in our numerical dynamo solutions (Kuang, 1999). But further studies are necessary to fully understand the issues. It should be pointed out here that the our results on likely correlations between the axisymmetric gravity anomaly and torsional oscillations in the core, and between the non-axisymmetric part of the gravity anomaly and the mainstream core flow, could be useful for future combined geodetic and geomagnetic studies on the dynamical processes in the Earth's fluid core.
Acknowledgement
This work is supported by NASA Solid Earth and Natural Hazard (SENH) program.
References
AGU, Contributions of Space Geodesy to Geodynamics: Technology (eds. D. E. Smith and D. L. Turcotte), Geodynamics Series 25, AGU, Washington, D.C., 1993 Braginsky, S. I. (1967), Magnetic waves in the Earth's core, Geomagnetism and Aeronomy, 7, pp. 851-859.
Chao, B. F., V. Dehant, R. S. Gross, R. D. Ray, D. A. Salstein, M. M. Watkins, and C. R. Wilson (2000), Space geodesy monitors mass transports in global geophysical fluids, EOS, Trans. Amer. Geophys. Union, 81, pp. 247-250. Chao, B. F. (2005), On inversion for mass distribution from global (time-variable) gravity field, Journal of Geodynamics, 39, pp. 223-230. Cox, C. M., and B. F. Chao (2002), Detection of a large-scale mass redistribution in the terrestrial system since 1998, Science, 297, pp. 831-833. Dumburry, M. and J. Bloxham (2004), Variation in the Earth's gravitational field caused by torsional oscillations in the core. Geophysical Journal International, 159, pp. 417-434. Fang, M., B. H. Hager and T. A. Herring (1996), Surface deformation caused by pressure changes in the fluid core. Geophysical Research Letters, 23, p.p. 1493-1496. Fang, M., B. H. Hager, W. Kuang and B. F. Chao (2005), On the Core Tides, submitted to Journal
of Geophysical Research. Greff, M., Pals and J.-L. LeMouel (2004), Surface gravitational field and topography changes induced by the Earth's fluid core motions, Journal of Geodesy, 78, pp. 386-392. Heiskanen W. A. and H. Moritz, 1968. Physical Geodesy, Freeman & Company. Jault, D., C. Gire and J.-L. LeMou 1 (1988), Westward drift, core motions and exchanges of angular momentum between core and mantle, Nature, 333, pp. 353-356. Kuang, W. (1999), Force balances and convective state in the Earth's core, Physics of Earth and Planetary Interiors, 116, pp. 65-79. Kuang, W. and J. Bloxham (1999), Numerical Modeling of Magnetohydrodynamic Convection in a Rapid Rotating Spherical Shell: Weak and Strong Field Dynamo Action. Journal of Computational Physics 153, pp. 51-81. Kuang, W. and B. F. Chao (2003), Geodynamo modeling and core-mantle interaction. In: The core-mantle boundary region (eds. Dehandt, Creager, Karato and Zatman) Geodynamics Series, 31, AGU, pp. 193-212. Kuang, W., B. F. Chao, C. Cox and M. Fang (2004), Time-variable Gravity Signal From Core Mass Flow Based on Geodynamo Simulation, submitted to Geophysical Journal International.
479
Chapter 70
A NEW HEIGHT DATUM FOR IRAN BASED ON THE COMBINATION OF THE GRAVIMETRIC AND GEOMETRIC GEOID MODELS R. Kiamehr Department of infrastructure, Division of Geodesy, Royal Institute of Technology (KTH), SE 100-44 Stockholm, Sweden
A new geoid model for Iran (IRG04) is computed based on the method of least squares modification of Stokes formula based on the most recent gravity anomaly database, SRTM high resolution Digital Elevation Model (DEM) and GRACE GGM02 Global Geopotential Model. In order to define a new height datum for Iran, we attempt to combine this high resolution gravimetric geoid model with GPS/levelling data by using a corrective surface idea. The corrective surface is constructed based on 224 GPS/levelling points and then evaluated with 35 independent points. Different interpolation techniques were tested for the creation of the corrective surface; among them the Kriging method gives the minimum RMS and noise level versus the suggesting that GPS/levelling data. The RMS of fitting the new combined geoid model versus GPS/levelling data is 0.09 m, it is near 4 times better accuracy compared with the original gravimetric geoid model. Comparing the later model, the new surface should be more convenient and useful in definition of the new height datum, specifically in engineering and GPS/levelling projects. Abstract.
Keywords. Least squares modification, geoid, SRTM, GRACE, GPS/levelling, Corrective surface, Kriging.
north and west or central deserts of Iran, is very tedious and time consuming. Nowadays the combined use of GPS and geoid heights presents an alternative potential to the classic geometric levelling. For many applications, conventional levelling using spirit level is being replaced by height determination using GPS. The GPS technique has been used for levelling projects, e.g., to monitor local subsidence due to water or natural gas removal, earth crustal movements and to control heights across water bodies in connection with bridge construction. Coordinate determination from GPS measurements uses the known positions of satellites and the measured distances between satellites and the unknown points. In principle, this methodology applies to both point and relative (differential) positioning. The heights directly derived from GPS measurements are geodetic heights referring to the ellipsoid defined by WGS84. The combined use of GPS, levelling, and geoid information has been a key procedure for various geodetic applications. Although these three types of height information are considerably different in terms of physical meaning, reference surface definition/realization, observational methods, accuracy, etc., they should fulfil the simple geometrical relationship:
h~H+N,
(1)
1 Introduction The determination of orthometric heights by traditional spirit levelling techniques is known to be a difficult task. This is especially evident in big countries like Iran, where the establishment of a levelling network covering all parts of the country would be impractical from the financial point of view. Moreover, levelling over areas with rough terrain, like the Alborz and Zagros mountains in the
However, in practice equation (1) is never satisfied due to random noise in the values of h, H and N, datum inconsistencies and other possible systematic distortions in the three height data sets (e.g. longwavelength systematic errors in N, distortions in the vertical datum due to an over constrained adjustment of the levelling network, deviation between gravimetric geoid and reference surface of the levelling datum, etc.), various geodynamic-
Chapter 70
rlc,arl Surf'a:,
_. Fig.
~
l
~
r
J
• A
New Height Datum for Iran
Rp.inN ~ v
~,
i, B:od:fi= H=i.qht
flit
hZ _
Blip=pid "'--"-----
1. Principal of GPS/levelling (Fotopoulos, 2005).
effects (tectonic, land subsidence, mean sea level rise) and theoretical approximations in the computation of either H or N (e.g. omitted density variation of topography in the geoid solution, negligence of the Sea Surface Topography (SST) at the tide gauges, error-free assumption for the tide gauge observations and etc.). The statistical behaviour and modelling of the misclosures of equation (1), computed in a network of levelled GPS benchmarks, have been the subjects of many recent studies (e.g. Kotsakis et al. 2001a and 2001b, Kotsakis and Sideris 1999, Fotopoulos 2005). Using the corrective surface idea is one of the most interesting and practical subject in this area. The development of corrector surfaces can be used for optimal height transformation between geoid and levelling datum surfaces for reducing the low-wavelength gravimetric geoid errors and also for general gravimetric geoid refinement. The gravimetric geoid determined using Stokes's integral is deficient in the zero-degree term because the exact values of the product of the mass of the Earth and the Universal gravitational constant and the potential of the geoid remain unknown (e.g. Kirby and Featherstone, 1997; Smith, 1998). Therefore, it is necessary to apply a bias to such a geoid model to account for this so-called zerodegree term. Heiskanen and Moritz (1967) recommend using geometrical geoid information to control this term, which is now available from the absolute ellipsoidal and orthometric heights described earlier. On the other hand, monitoring, testing, and/or improving of already existed vertical datums can be investigated by using the corrective surface idea. In view of the many mentioned different uses, the main purpose of this paper is to present a modelling scheme that can be employed for creation a corrective surface for improving the
Based
on the Combination of the Gravimetric and Geometric Geoid M o d e l s
accuracy of the gravimetric geoid model in GPS/levelling applications. However, the primary problem of transforming GPS derived geodetic heights into orthometric heights is the determination of reliable and precise geoidal undulations, which involves a combination of satellite and terrestrial gravity measurements or other data. This paper is mainly constructed in two different parts. In the first part we explain briefly the procedure and data that were constructing the new hybrid precise geoid model for Iran (Kiamehr 2005) based on the least squares modification of Stokes' (LSMS) formula (SjGberg 1984, 1991, 2003b and 2003c). Then we explain the procedure for creating of the corrective surface and combination of the gravimetric and geometric geoid models. Then the accuracy of the combined geoid model (as a new height datum for Iran) is evaluated and compared with the original gravimetric geoid model.
2 Iranian precise gravimetric geoid model based on the KTH approach The new Iranian gravimetric geoid model (IRG04) (Kiamehr, submitted); is computed based on the KTH approach (SjGberg 2003b) with additive correction terms. The surface gravity anomalies and GGMs are used to determine approximate geoidal heights ( N ) and all necessary corrections are added directly to N (see Eq. 2). In other approaches, these corrections are usually computed one by one in separate corrections in steps, such that in the first step the surface gravity anomalies are corrected by removing the effects of topographic and atmospheric external masses (or reducing them inside the geoid) as direct effects, and then, after applying Stokes's integral, their effects are restored (indirect effects). In addition, the gravity anomalies in Stokes's formula must refer to the geoid, so that a reduction of the observed gravity from the Earth's surface to the geoid is necessary; that is called downward continuation (DWC). In the KTH approach, all these separate effects are replaced by a total topographic effect (SjGberg 2001). The computational procedure for estimating the geoid height N in the KTH approach can be summarized by the following formula: N
f ( - N- + 5Nr°P°comb+ 5NL)wc + 5N~omb + 5 N , Topo
where cYNcomb
(2)
is the combined topographic correction and it includes the sum of direct and
481
482
R. Kiamehr
indirect topographical effects on the geoidal heights, 6NDw c is the correction for the downward
..-..,.~:~-'-.:.,.x<... ' 40-T,.,~ ~t~-~-~:~>~~~~
continuation effect (Sj6berg 2003b), cyNaomb is the combined atmospheric correction (Sj6berg 2001)
.
~
35-
,
" ,,., .
and it includes the sum of direct and indirect
~
~
" i!~ ~!"], '-){
atmospherical effects, and g N e is the ellipsoidal correction for the spherical approximation of the geoid in Stokes's formula to ellipsoidal surface of reference. (For more details about these correction terms we refer to the references). The approximate geoid height ( N ) computed based on the general modification model for Stokes' formula by defining two sets of arbitrary modification parameters (sn andb, ) is as follows:
<_~..,.:....::.'..!....,. -.: -~ . .....:.. "
I~ a0
i " ~: . 2S
5'0
5'5
6'0
6's
Fig. 2. Terrestrial gravity anomalies data in the target area.
M
,t ~ E G M N~ - ~C jf fj S L (~)A,~dcr + c~-" b/_xg,, , 2a" o-0 n=2
where
c = R / ( 2 ~,), b - Q C + s n
(3)
, QL n is the
truncation coefficients and can be calculated by =
sk% where
n=2
Q~ = }s (~)P
2 (cos ~z) sin ~zd~z and
e,k are
the
functions of limiting radius of the integration cap that can be computed by some recursive algorithms and, R is the mean Earth radius, ~ is the geocentric angle, A~ is the gravity anomaly, d~ is an infinitesimal surface element of the unit sphere ~, and ~/is normal gravity on the reference ellipsoid. The modified Stokes's function is expressed as SL( ~ ) -
S (~/)-£ n=2
2n +___~1skPk (COS ~ ) , 2
(4)
where S ( W is the Stokes's function, P cos(~z)is the Legendre polynomial of degree n, and s n are the least squares modification parameters (for more details see, Sj6berg 1991). The upper limit L (in Eq. 3) is arbitrary and generally not equal to M. For the generation of the gravity database a total number of 26125 point and mean gravity data were collected from different data sources (Kiamehr 2005a). Various databases including Bureau Gravimetriqe International (BGI), National Cartographic Centre of Iran, original 0.5o× 0.5 ° surface free-air gravity anomaly data
(that was used in the modelling of the EGM96) and satellite altimetry data (Sandwell at al., 1997) were used in this database. (For more details, see Kiamehr, ibid). Figure 2 shows the distribution of the gravity data and the final gravity anomaly grid file (free from outliers). The final geoid model was computed based on free-air gravity anomalies in 80" × 90" grid size, GGM02S global geopotential model and Shuttle Radar Topography Mission (SRTM) based on 100 m digital elevation model (Kiamehr and Sj6berg, submitted). The full potential of the GGM02S (http ://www.csr.utexas.edu/grace/gravity/) model with maximum degree and order of 110 (Kiamehr and Sj6berg 2005) was also used in determining the least squares modification parameters. The integral cap size 3 degrees and the pre-estimated accuracy for gravity data CrAg=10 mGal (See Kiamehr 2005a for more details) were used in estimating of the modification parameters. Figure 3 shows the 3D contour map of the IRG04 geoid model. For more information and details about IRG04 gravimetric geoid model see Kiamehr (submitted). For the evaluation of the new gravimetric and combined geoid models, 260 GPS/levelling data points where used. From them, 35 points belongs to the precise 1st order Iranian GPS and levelling network, and the rest belongs to 2 nd order networks. From the result of the latest network adjustment (Nilforoushan 2003, personal communication), the mean standard deviation of the geodetic heights ( crh ) was estimated to approximately 0.2 m.
Chapter 70
• A
New Height Datum for Iran Based on the Combination of the Gravimetric and Geometric Geoid Models
40
-
I
I
.
I
I
- - 2S - - 24 _- -_ 2.D --'e, __ --'2 F. 4
--3 ---4
---z -2.3 -24
i ---
i
++
0
-32
~-... ~;'-.
./~<~
,,i~>"-,..._....
2='
I
50 E.E. 60 Fig. 4. Location of the 260 GPS/levelling points [including 4E.
35 precise points (circles)]. Fig. 3. The new Iranian gravimetric geoid model (IRG04). Contour interval is 2 m. Unit: metre.
3 GPS/levelling data The orthometric heights ( H ) are believed to be accurate to 0.7 m in the absolute level o f accuracy, because of neglecting the effect of the sea surface topography, presence of different systematic errors in observations and uncertainty about definition and establishment of the height reference system used in the adjustment of the network (Hamesh, 1991). The relative accuracy of the orthometric heights for the 1st order levelling network is quite well and estimated near the 3 ppm (Hamesh, ibid). The minimum, maximum and average distances between these points are 52, 115 and 80 km, respectively. Figure 4 shows the location of the selected GPS/levelling points and selected traverses on the topographic map of Iran.
4 Construction of the corrective surface 4.1 Fitting models The computation of a corrective surface for the gravimetric geoid models usually starts with removing the trend ( a~ x , see Eq. 5 below) and tilt between gravimetric and GPS/levelling geoid models. An external evaluation for the quality of a gravimetric geoid model can be performed by comparing their interpolated values (N) at a network of GPS benchmarks with the corresponding GPS/levelling-derived geoid heights
( N GPS ). Such a comparison is traditionally based on the following model" AN-
NiCPS - N
- h -H
- N i - a r x + c;,(5)
where x is a n x 1 vector of unknown parameters, aiis a n x l
vector of known coefficients, and
gi denotes a residual random noise term. r The parametric model a i x is supposed to describe the systematic errors and datum inconsistencies inherent in the different height data sets. Its type varies in form and complexity depending on a number of factors. In practice, the various wavelength errors in the gravimetric solution may be approximated by different kinds of functions in order to fit the quasigeoid to a set of GPS levelling points through an integrated least squares (LS) adjustment. Several models can be used ranging from a simple linear regression to more complicated seven parameter similarity transformation model (Kotsakis et al. 2001 b). According to our numerical results for these three models, the seven parameter model gives the best fitting with minimum standard error deviation in gravimetric geoid model, so in this paper we show just the results of fitting using the seven parameter model. a i = (cos fo cos ~ cos ~o,sin 2, sin fo cos fo sin fo cos ~ / W cos r,o sin ¢, sin ~ / W sin ~rp, / W 1)~
and x
(x 1
X2
ooo
X6
X 7 )T
(6)
483
484
R. Kiamehr
where
(/9 and
,~are
the horizontal geodetic
coordinates of the network or baseline points and W
__
( 1 - e 2sin 2(p~
)1/2
(7)
where e is the first eccentricity of the reference ellipsoid. Then we obtain the matrix system of observation equations Ax
- ~ -
6,
(8)
where A is the design matrix composed of one row T
a i for each observation ANi. The least squares
adjustment to this equation, utilizing the sum of squares of the residuals 3i, becomes
yielding the residuals
(lo)
Table 1. Results of fitting the corrective surfaces (Based on different interpolation techniques) versus independent 35 GPS/levelling points. Unit: In. RMS" Root Mean Square. Griding method/ Stat Minimum Maximum Mean RMS
Inverse Distance -0.607 0.742 -0.058 0.212
Griding method/ Stat Minimum Maximum Mean RMS
Polynomial Regression (Simple) -0.815 0.757 -0.078 0.272
Griding method/ Stat Minimum Maximum Mean RMS
Local Local Local Polynomial Polynomial Polynomial Degree =1 Degree =2 Degree =3 -0.627 -0.796 -0.607 0.732 0.751 0.744 -0.079 -0.061 0.007 0.255 0.209 0.195
Griding method/ Stat Minimum
The standard deviation of the adjusted values for the residuals ~ are traditionally taken as the external indication of the geoid model absolute accuracy, it is important to mention here that the final residual values are not the exact errors of the gravimetric geoid models (or height datum), because they can include also some part of errors from GPS and levelling observations. The RMS of fitting between the IRG04 geoid model and the 35 most precise GPS/levelling data evaluated before and after 7 parameter fitting reach 0.36 and 0.29 m, respectively. (See Table 2, for more details)
4.2. Corrective surface From the 260 available GPS/levelling points (see Fig. 4), 224 points were used for creation of the corrective surface, and 35 accurate points (these points belongs to the first degree national GPS and levelling network) were used for validation purpose. We should create a continuous surface from the discrete GPS/levelling data by using some interpolation (prediction) techniques, the choice of which is crucial for the result. Some studies from recent years emphasising this fact can be mentioned, like Kotsakis and Sideris (1999) and Lee and Mezera (2000). A comprehensive -
Maximum Mean
RMS
Modified Shepard Minimum Method Curvature -0.572 -2.553 0.587 1.275 0.002 -0.190 0.096 I 0.621 Polynomial Regression (Bilinear) -0.821 0.691 -0.109 0.261
Natural Neighbour -0.498 0.523 0.026 0.098
Polynomial Polynomial Regression Regression (Quadratic) (cubic) -0.855 -0.618 0.675 0.517 -0.149 -0.027 0.271 0.1808
Kriging
Radial Basis Function
Cubic Spline
-0.474 0.433 0.002 0.0885
-0.49 0.480 -0.007 0.0889
-0.476 0.435 0.003 0.089
-compilation of different methods can be found in Watson (1992) and Burrough and McDonnell (1998). Different interpolation techniques were tested for creating a corrective surface, by using the SURFER software (Golden Software, Inc.). Table 1 presents the statistics of fitting between the combined surfaces, by using different interpolation methods, versus 35 GPS/levelling data. Among them more and less, the Minimum curvature, Natural Neighbour, Cubic @line and Kriging give the minimum and same RIMS values, but at the same time the Kriging method also gives the lowest noise level (The difference between maximum and minimum values). Thus, we choosed the Kriging method for creating the corrective surface in this study. Kriging (Matheron 1963) is a geostatistical approach to interpolate data based on spatial variance and has proven useful and popular in many fields in geodesy as well. A thorough textbook on kriging in general is Stein (1999), on applied geostatistics is Isaaks and Srivastava (1989) and on kriging in GIS in particular is Burrough and
Chapter 70 • A New Height Datum for Iran Based on the Combination of the Gravimetric and Geometric Geoid Models
McDonnell (1998). The reason that we choose Kriging method instead of least squares collocation (Moritz 1973) is just the availability of software and that Kriging has become an extremely important interpolation tool in geostatistics and, as such, has been given a lot of attention from scientists and software producers. Both kriging, and least squares collocation are generalised estimation methods combining the behaviour of a systematic part and two random parts. A difference between Kriging and least squares collocation lies just in the treatment of the covariance structure of the random field, where in Kriging semi-variograms are used (Blais 1982, p. 327) to estimate weights and in least squares collocation covariance functions (Moritz 1973, p. 26). Except this difference, the two methods are essentially equivalent. In this study, we used a quadratic trend and a linear semi-variogram, which is considered to be the best for the research area, in particular a method called universal Kriging (Mfirtensson 2001). Table 2 and Figures 5 and 6 show the result of this comparison. By applying the corrective surface to a gravimetric geoid model, the RMS of fits between the new combined geoid model (IRG04C) and the 35 independent precise GPS/levelling data reduce to 0.09 m, which shows a significant improvement (four times) compared with the original gravimetric geoid model (see Table 2). It apparently the effect of different systematic errors in the gravimetric geoid model were significantly reduced by using the corrective surface model. However, a warning should be attached to corrective surface fitting, because it models also any errors present in the GPS and levelling data, thus sometimes giving over-optimistic error estimates. In the practical view, in order to get more accurate and trustable results in levelling by GPS and corrective geoid surface, the following recommendations should be useful (Zilkoski 1997):
- The best way to use GPS for height determination is to set up a network including a minimum of four benchmarks, and constraining these values in network adjustment. It is usually best to perform a minimally constrained adjustment then to add additional constraints to detect possible problems with the fixed control. • Use relative geoid height values instead of single value. • Try to keep project areas within a 20 km radius of control points. • Process GPS data with Precise Ephemeris.
Table 2. Validation of the IRG04 and new combined geoid
model versus 35 GPS/levelling data. Unit: m. Model IRG04 geoid gravimetric model IRG04 after 7 parameter fitting approach New combined model (IRG04C)
40
,:/
L
35
0.223 -0.652 0.362
-0.518
0.924 0.000
0.288
-0.474
0.433 0.000
0.088
.
^
Q.I._;,
Max
-1.284
":
I
L
Min
I
\
Mean
RMS
I
-
f
~)
~
3
-_~
.3.7 "3.4
30
~3
-'.4 -'.7
i-
-,
00
i
Figure 5. The difference between IRG04 and IRG04C
combined geoid model over Iran. Contour interval 0.3 m. Unit: m.
• • • •
•
Use dual frequency GPS receivers. GPS occupation times of less than 1 hour Observe when Vertical Dilution of Precision (VDOP) is less than 5. Use fixed tripods poles and identical geodetic quality antennas with ground plane. Process with a minimum elevation mask of 15 degrees.
6. Conclusion The uses of a corrective surface to IRG04 gravimetric geoid model has been shown to significantly improve the determination of heights from GPS in Iran. Such an approach can account for any differences between the gravimetric geoid and the vertical datum in any area. Using the corrective surface approach in Iran improved the RMS of fitting the GPS/levelling data significantly. The RMS of fitting between gravimetric geoid model (before 7 parameter fitting model) and
485
486
R. Kiamehr
45
50
55
GO
Figure 6. The new combined geoid model over Iran (IRG04C). Contour interval 2 m. Unit: m. GPS/levelling reduced from 36 cm to 9 cm after using the corrective surface model (four times). However, it should be noted that the approach can give too optimistic estimates o f accuracy as it also absorbs any errors committed during the GPS or levelling surveys. For any future works it is r e c o m m e n d e d to use much denser, well distributed and high quality GPS/levelling points in the creation o f the corrective surface (especially in mountainous areas).
Acknowledgments.
The author is grateful to
Prof. Lars Sj6berg for his constructive discussions and review of this paper.
References
Anonym (1999). Golden Software, Surfer 8, User's Guide: Contouring and 3D surface mapping for scientist and engineers, Colorado, USA Ardalan R. and E. Grafarend (2004). High Resolution geoid computation without applying Stokes's formula case study: High resolution geoid of Iran, Journal of Geodesy, 78, 138- 156. Blais J.A.R. (1982). Synthesis of Kriging Estimation Methods. Manuscripta Geodaetica 7: pp 325-352 Burrough PA, McDonnell RA 1998: Principles of Geographical Information Systems. Oxford University Press Inc, New York Fotopoulos G. (2005). Calibration of geoid error models via a combined adjustment of ellipsoidal, orthometric and gravimetric geoid height data, Journal of Geodesy, Vol 79(1-3), 111- 123 GRACE (2004). GRACE Gravity Model GGM02. University of Texas at Austin, Center for Space Research. http ://www.csr.utexas.edu/grace /gravity/(2004)
Heiskanen W.A. and H. Moritz H (1967). Physical Geodesy. W H Freeman and Co., New York, London and San Francisco Isaaks E.H. and R.M. Srivastava (1989). An Introduction to Applied Geostatistics. Oxford University Press, Inc., New York Kiamehr R. (2005a). Qualification and refinement of the Iranian gravity database, NCC Geomatics 84 Conferences, Tehran, Iran. Kiamehr, R., 2006, A strategy for determining the regional geoid in developing countries by combining limited ground data with satellite-based global geopotential and topographical models: A case study of Iran, J. Geodesy, 79(10,11): 602-612 Kiamehr R. and L.E. Sj6berg (2005). The qualities of Iranian gravimetric geoid models versus recent gravity field missions, J. Studia Geophysica et Geodaetica, Vol 49, 289-304. Kiamehr R. and L.E. Sj6berg (2005). Effect of the SRTM global DEM in the determination of a high-resolution geoid model of Iran, Jr. Geodesy, 79(9):540-551. Kirby J.F. and W.E. Featherstone (1997). A study of zeroand first-degree terms in geopotential models over Australia. Geomatics Research Australasia 66, 93-108. Kotsakis C. and M.G. Sideris (1999). On the adjustment of combined GPS/levelling/geoid networks. J. Geodesy, Vol 73(8), 412-421. Kotsakis C., G. Fotopoulos and M.G. Sideris (2001a). A Study on the Effects of Data Accuracy and Datum Inconsistencies on Relative GPS Levelling. In Proceedings of the International Association of Geodesy Symposium on Vertical Reference Systems, Cartagena, Colombia, February 20-23, IAG Symposia, vol. 124, pp. 113-118. Kotsakis C., G. Fotopoulos and M.G. Sideris (2001b). Optimal fitting of gravimetric geoid undulations to GPS/levelling data using an extended similarity transformation model. Presented at the 27th Annual Meeting joint with the 58th Eastern Snow Conference of the Canadian Geophysical Union, Ottawa, Canada, May 14-17. Lee J.T., D.F. Mezera (2000). Concerns Related to GPSderived Geoid Determination. Survey Review 35: pp 379397 Martensson S.G. (2001). Height Determination by GPS - A Practical Experiment in Central Sweden. Report 1054, Division of Geodesy, Royal Institute of Technology, Stockholm Matheron G. (1963). Principles of Geostatistics. Economic Geology, 58, 1246-1266. Molodenskii M.S., V.F. Eremeer and M.I. Yurkina (1962). Methods for Study of the External Gravitational Field and Figure of the Earth. Transl. from Russian (1960), Jerusalem, Israel Program for Scientific Translation. Moritz H. (1973). Least-Squares Collocation. Deutsche Geodatische Kommission. Reihe A,Heft Nr. 75 Sj6berg LE 1984: Least squaress modification of Stokes' and Vening Meinesz' formulas by accounting for truncation and potential coefficient errors. Manuscr Geod: 9, 209-229 Sj6berg L.E. (1991). Refined least squares modification of Stokes' formula. Manuscr Geod: 16:367-375 Sj6berg L.E. (2000). On the topographic effects by the Stokes-Helmert method of geoid and quasi-geoid determinations. Journal of Geodesy, 74(2): 255-268
Chapter 70 • A New Height Datum for Iran Based on the Combination of the Gravimetric and Geometric Geoid Models
Sj6berg L.E. (2001). Topographic and atmospheric corrections of gravimetric geoid determination with special emphasis on the effects of degree zero and one. Journal of Geodesy, 75:283-290 Sj6berg L.E. (2003a). A solution to the downward continuation effect on the geoid determined by Stokes' formula. Journal of Geodesy, 77:94-100 Sj6berg L.E. (2003b). Improving modified Stokes' formula by GOCE data. Boll Geod Sci Aff61 (3): 215-225 Sj6berg L.E. (2003c). A computational scheme to model the geoid by the modified Stokes formula without gravity reductions. Journal of Geodesy, 74, 255-268. Sj6berg L.E. (2003d). A general model of modifying Stokes' formula and its least squares solution. J Geod 77:459-464 Sj6berg L.E. (2003e). The ellipsoidal corrections to topographic geoid effects. J Geod 77:804-808
Sj6berg LE and Nahavandchi H 2000: The atmospheric geoid effects in Stokes' formula. Geophys. J. Int. 140: 95-100 Smith D.A. (1998). There is no such thing as "the" EGM96 geoid: subtle points on the use of a global geopotential model. International Geoid Service Bulletin 8, 17-28. Stein M.L. (1999). Interpolation of spatial data: some theory for kriging. Springer-Verlag, New York Watson D.F. (1992). Contouring- A Guide to the Analysis and Display of Spatial Data. Pergamon Press, Oxford Zilkoski D., J.D. Onofrio and S. Erakes (1997). Guidelines for Establishing GPS-Derived Ellipsoid Heights (Standards: 2 cm and 5 cm), Version 4.3, NOAA Technical Memorandum NOS NGS-58, National Geodetic Survey Information Center, Silver Spring, MD 20910.
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Chapter 71
Monthly mean water storage variations by the combination of GRACE and a regional hydrological model: application to the Zambezi River. R. Klees, E.A. Zapreeva, Delft Institute of Earth Observation and Space Systems (DEOS), Physical and Space G e o d e s y group, Delft University of Technology, K l u y v e r w e g 1, 2629 HS, Delft, The Netherlands H.C. Winsemius, H.H.G. Savenije, D e p a r t m e n t of Water M a n a g e m e n t , Delft University of Technology, Stevinweg 1, 2628 CN, Delft, The Netherlands Abstract. The lack of hydrological data and the low quality of global hydrological models' output caused by incapability of proper calibration is the main problem when using hydrological models to study the closure of the water balance at river basin scales. Monthly GRACE gravity field solutions can be used to infer water storage variations at river basin scale. However, the estimates are erroneous due to measurement errors, aliasing effects, and limited spatial resolution, which makes it difficult to verify and improve the hydrological models. Currently, a combination of GRACE gravity data and regional hydrological models appears to be an integral part of future investigations for the estimation of the monthly mean water storage variations over river basins as large as 105 - 106 km 2. This paper presents the results of the analysis of 17 monthly GRACE gravity models and of a comparison with the output of the LEW regional hydrological model (Lumped Elementary Watershed) for the estimation of monthly water storage variations over the upper Zambezi River catchment (Southern Africa). The comparison of GRACE and LEW shows significant differences in amplitude, which cannot be attributed unambiguously to an improper calibration of the LEW model, but may also be attributed to the contribution of storage variations in surrounding areas, errors in the GRACE monthly gravity solutions, and low-pass filter errors. The contribution of the surrounding areas to the GRACE monthly mean water storage variations over the upper Zambezi has a maximum amplitude of about 20 mm/month. Applying this correction to the GRACE estimates reduces the maximum amplitudes of the GRACE monthly mean water storage variations to about 20 - 30 mm/month. At the same time the amplitude of the unfiltered water storage
variation of the upper Zambezi from LEW is about 60 80 mm/month. When applying the same filter to the LEW output as used to smooth the GRACE monthly estimates, the LEW amplitudes reduce dramatically to about 6 - 8 mm/month. This is significantly smaller than the 20 - 30 mm/month obtained from GRACE. The total effect of mass variations in atmosphere and oceans on the GRACE monthly mean water storage variations vary between - 1 0 and +18 mm/month. Ocean effects are of the same order of magnitude as atmospheric effects. Therefore, it is necessary to quantify uncertainties in the de-aliasing procedure currently applied to GRACE data prior to the estimation of gravity field parameters.
Keywords. GRACE, time-varying gravity field, continental hydrology, regional hydrological model LEW, water storage variations.
1
Introduction
Since April 2002 GRACE provides monthly gravity field models, reflecting time variations in the Earth's gravity field. The dominant contributors are mass transport processes in the oceans, the atmosphere, and continental water storage variations. The capability of GRACE to measure continental water storage allows the monitoring of monthly water storage variations at river basin scale, provided that the size of the area is well above the spatial resolution of GRACE and the amplitudes of the water storage variations are well above the noise level of GRACE monthly estimates. Estimating water storage variations at river basin scale has been addressed re-
Chapter 71 • Monthly Mean Water Storage Variations by the Combination of GRACEand a Regional Hydrological Model: Application to the Zambezi River
cently in several publications, e.g. Wahr et al. (1998), Rodell and Famiglietti (2001), Swenson and Wahr (2002), Swenson et al. (2003), and Seo and Wilson (2005). In this paper, we focus on the Zambezi fiver basin, which is one of the most important river basins on the African continent (see figure 1). The goal is to compare monthly mean water storage variations of the upper Zambezi sub-catchment inferred from GRACE with the output of a state-of-the-art regional hydrological model. Moreover, we want to quantify the influence of atmospheric and oceanic mass transport processes on the estimated upper Zambezi water storage variations. Finally, we want to estimate the water storage variations in the surrounding areas of the upper Zambezi using the regional hydrological model mentioned before. These estimates are used to quantify the effect of water storage variations in those areas on the GRACE monthly mean water storage variations of the upper Zambezi sub-catchment.
the LEW regional hydrological model and compare it with the GRACE estimates. Moreover, we quantify the effect of mass transport in the atmosphere and oceans on the GRACE estimates of the monthly mean water storage variations over the upper Zambezi. Finally, water storage variations over the surroundings of the upper Zambezi are analyzed. Section 5 contains the summary and the main conclusions.
2
Methodology
The currently adopted approach to analyze GRACE gravity data has been suggested by Swenson and Wahr (2002, 2003). This approach is based on the representation of changes in the gravitational potential from one 30-day period to another, AV(O, I, r), in terms of spherical harmonics. When restricted to a mean Earth sphere with radius _R, the changes in gravitational potential can be written as Av(o,
?,) -
aM
;
Clm
/--0 m----1
(1)
c~
co
o
~
co
=(Av) with C~m the 4~--normalized spherical harmonic coefficients representing the difference of two monthly gravity fields, Yt~(0, A) the 47r-normalized surface spherical harmonics, G Newton's gravitational constant, and M the mean mass of the Earth. Changes in the surface mass distribution, like continental water storage variations, generate corresponding changes in the gravitational potential. They can also be expanded in spherical harmonics. If
~(AS)
F i g u r e 1: The upper Zambezi river sub-catchment.
The outline of the paper is the following: in section 2, we briefly explain the data analysis methodology. Basically, we follow the procedure proposed by Swenson and Wahr (2002). Alternatively, we also use a space domain approach, which is based on numerical integration instead of spherical harmonic analysis and synthesis, in order to quantify the effect of the Gibbs phenomenon on the monthly mean water storage variations. In section 3, the regional hydrological model LEW is described. This model allows the computation of the monthly mean water storage and water storage variations for the upper Zambezi sub-catchment. This model is also used to quantify the effect of water storage variations in surrounding areas on the estimated monthly mean water storage variations over the upper Zambezi subcatchment. In section 4, we present the output of
lm
are the coefficients of this expansion, the rela-
=(AV)
tion between them and the coefficients elm of the induced change in the gravitational potential is (e.g. Wahr et al. 1998, Swenson and Wahr 2002)
~(AS) _ KI=(AV) lm
(2)
elm
with
RpE (21 + 1) :
3.p
(3)
(1 +
kl are the load Love numbers, Pw is the density of water, and pm is the average density of the solid Earth. For studying water storage variations at river basin scale, the primary quantity of interest to hydrologists is the monthly storage variation averaged over the river basin. If B is the scaled characteristic function of the fiver basin, i.e. 4~
B(O , I)
-
~ b ~
0
inside the river basin outside
(4)
489
490
R. Klees • E. A. Zapreeva • H. C. Winsemius • H. H. G. Savenije
where O-basi n is the area of the river basin, the storage variation averaged over the river basin (or simply the mean storage variation) is A&
_ ~
_(A&)
cl~
,
Now, it is straightforward to express the error in the estimated mean storage variation as the difference A & - AS~"
(5) l>l~,~ax ,rn
with
+
_(:x&)
Clr n
--
=(:xs) =(B) _ Kl=(axv)=(B)
Clr n
Clr n
Clr n
,_
(6)
Clrn ,
+
where cz~ =(B) are the coefficients of the expansion of the (scaled) characteristic function /3, Eq. (4), into spherical harmonics" oc
/3(0, A ) - ~
1
~
c,,~=(s)~ ( 0 ,
A).
(7)
/=0 m=--I
_(~) Current GRACE monthly solutions {cz~ } are limited to a maximum degree l ~ a x , which varies from month to month between 70 and 120. This corresponds to a spatial resolution in terms of full wavelengths between 550 and 350 km. Moreover, the
=(Lx~)
GRACE estimates cl~
are erroneous, i.e. they dif-
_(Av) fer from the exact coefficients cl, ~ . The noise is very strong at high frequencies. To reduce the highfrequencyAcomponent of the noise, a spatial averaging of A V is applied, which yields a filtered (i.e. smooth) version A V. From A V , a filtered (smooth) estimate A S of surface mass variations can be computed. In principle, any low-pass filter that reduces sufficiently well the noise in the GRACE estimates, can be used. If e=(w) l , ~ are the properly scaled coefficients of the filter function, the estimated low-pass filtered coefficients of A V and A S are
_(:x~) -- e=(:x9) =(w) , lm elm
elm
=(axe)_
elm
K1
cl,,~
,
(9)
with -- Clm
Clrn
Clm
lrn
lrn
~lm=(/3)k Clm[-(AV)
-
Clm-(A~')/")
.
(11)
We call the first term the omission error, the second term the low-pass filter error, and the third term the c o m m i s s i o n error. Note that this definition of the individual error terms differ from the ones of Swenson et al. (2003), w h o - among others - introduce the socalled leakage error. Commonly, a leakage error is attributed to mass variations outside the area of interest, which leak into the estimated mean storage variation if the low-pass filter W is applied. According to our definition, a low -pass filter error will always be introduced independently whether there are mass variations outside the area of interest or not, simply because of the application of a low-pass filter. However, the low-pass filter error will be larger if there are mass variations outside the area. In particular, mass variations outside but close to the area of interest may cause serious errors in the estimated mean water storage variation. Instead of using expansions in spherical harmonics, the analysis may also be done in the space domain using numerical integration techniques. We have implemented both the spectral and the spacedomain approach.
Regional hydrological model LEW
(8)
l <_l,,. . . . . m
,
K, l<_lmax , m
3
=(axe) .
elm
Now, the GRACE estimate of the storage variation averaged over the river basin can be defined as --
~;:)
"
Often, a Gaussian filter function is used; then, the coefficients el, =(w) ~ are zero for all m # 0. Swenson and Wahr (2002) proposed a filter function that minimizes the contribution of some error sources while constraining others to some pre-scribed value.
It is well known that global hydrological models show large uncertainties in terms of water storage estimates at river basin scales. That is why in this study a regional hydrological model has been used. So far, two regional models have been developed for the Zambezi river basin, STREAM and LEW (see Aerts and Bouwer, 2003; Winsemius et al., 2005). The LEW hydrological model is chosen for the combination with the GRACE model as it has some advantages: 1. The limited computational effort of LEW allows the search for a confident model structure through Monte-Carlo simulations. 2. The LEW model structure better represents
Chapter 71 • Monthly MeanWater StorageVariations bythe Combination of GRACEand a RegionalHydrologicalModel:Application to the ZambeziRiver
heterogeneities in the hydrology of the upper Zambezi: soil moisture capacity is described by a distribution function and the effective surface of the watersheds is divided in an upland and a wetland surface that are interconnected, but behave quite differently. 3. In the STREAM approach, runoff is immediately routed toward the watershed's outlet, while in reality it is stored in floodplain areas where it is a subject to percolation and evaporation. This is considered to be a large shortcoming of the STREAM approach compared to LEW, especially in larger catchments with long residence time. To determine a correct model structure, different concepts have been tested on a small tributary of the Kabompo river using the GLUE approach (see Beven and Binley, 1992) as a tool to test identifiability of parameters. That is, we can have a quite high confidence in the chosen structure. The L E W structure is a conceptual model consisting of three storages (unsaturated, saturated and wetland) that all have a linear outflow behavior. The storage compartments are separated by thresholds.
4
Numerical
4.1
results
Estimation
of
mean
water
storage
variations
Firstly, the L E W regional hydrological model is used to estimate the monthly mean (in time) water storage S c z w averaged (in space) over the upper Zambezi and its change per month A S r z w . The delineated watersheds used for the computation are shown in figure 2; the monthly mean water storage variations (i.e. the change per month of the mean water storage) are shown in figure 3 for the period April 2002 until June 2004. To quantify the water storage variations in the neighborhood of the upper Zambezi catchment, watersheds in the surroundings of the upper Zambezi are delineated and modelled (see figure 4). The L E W model is used to estimate the monthly mean water storage variations per month ASs. The results are shown in figure 5.
The model is forced and calibrated first using in situ rainfall data and discharge time series between 1960-1972 as they cover very well all regions of the Zambezi river basin. It reaches a Nash & Sutcliffe efficiency (Nash and Sutcliffe, 1970) of 0.88 for Lukulu gauging station and 0.8 for Victoria Falls. To force the L E W model during the period of GRACE measurements, rainfall products from the Famine Early Warning System (FEWS) (Herman et al. 1997) have been used. Monthly averaged pixel values of FEWS are compared with in situ data of rainfall stations that lie within the pixel. A very good agreement is found. To quantify the water storage variations over the surroundings of the upper Zambezi catchment, we also use the L E W regional hydrological model. The information about the water balance this model provides can be used to get at least an idea about the order of magnitude of water storage variations in the surroundings, although some components of the water balance, particularly evaporation and storage variation, are difficult to model at the required temporal and spatial scales and difficult to verify (see Beven and Freer, 2001).
Figure 2: Delineated watersheds used for the computation of the water balance of the upper Zambezi. The size of the area is about 0.5 • 106 km2. The region covers parts of Zambia, Angola and Namibia. to
0
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Figure 3" Monthly mean water storage variations averaged over the upper Zambezi shown in figure 2 in [mm/month] inferred from the LEW regional hydrological model (the sampling rate is 1 month).
491
492
R. Klees • E. A. Zapreeva • H. C. Winsemius • H. H. G. Savenije
Figure 4 : Delineated watersheds for the estimation of the water storage over the surrounding areas of the upper Zambezi. The total size of the watersheds is about 2.6 • 106 km2, i.e. much larger than the upper Zambezi sub-catchment.
This output of the LEW model is used to assess the contribution of the surrounding areas to the GRACE estimates of the storage variations averaged over the upper Zambezi sub-catchment.
phase is not changed when one removes these coefficients from the analysis or when they are replaced by the corresponding values from the analysis of EOP and SLR data. The estimated amplitudes of the computed storage variations strongly depend on the correlation length of the spatial filter. The larger the correlation length, the smaller the estimated amplitude of the monthly mean water storage variations (cf. figure 6). For a Gaussian filter, we find that the amplitude changes with about 15-20% if the correlation length changes with 500 km. By visual inspection, we finally choose a filter with correlation length 1000 km. If a smaller correlation length is chosen, we observe strong ground-track patterns in the estimated storage variations. There is no sound methodology of how to choose the optimal correlation length. An alternative to the approach we followed is to determine the correlation length such that on a global scale the amplitudes of GRACE estimates fits in a least-squares sense the output of a global hydrological model. This results in a slightly lower correlation length of about 800 km. Unfortunately, whether this is a better choice has to be left open.
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17 monthly GRACE gravity models, covering the period from April 2002 until June 2004, are used to estimate the mean water storage variations over the upper Zambezi sub-catchment. The spherical harmonic coefficients of degree 0 and 1 and the degree 2 and order 0 coefficient have always been removed from the solution. Chen et al. (2005) have shown that these coefficients do not significantly contribute to the amplitude of the monthly water storage variation averaged over the Zambezi fiver basin. They also showed that the
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o~o1~o~ o~o~o,
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Figure 6: GRACE monthly water storage variations averaged over the upper Zambezi sub-catchment in [ram/month]. A Gauss filter with various correlation lengths has been used: 600 km (dashed), 1000 km (solid), 1500 km (dashdot). AvailableGRACE solutions are shown by squares; values in between have been obtained by spline interpolation.
GRACE data are not continues (see figure 6, squares). There are about I to 3 gaps per year. Therefore the gaps are filled by applying the cubic spline interpolation to the GRACE solutions in terms of water storage. The obtained analytic function for storage is then used to compute GRACE estimates of the monthly mean water storage variation ASGRACE.
Chapter
71
• Monthly
Water Storage Variations
Mean
Combination of GRACE and a Regional Hydrological Model: Application
by the
The results are shown in figure 7.
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procedure as for the computation of the G R A C E estimates is applied using the same Gaussian filter (correlation length 1000 km).
8O
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to the
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-40
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i" .................................
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Figure 7: Monthly mean water storage variations in [ram/month] over the upper Zambezi sub-catchment. Dotted line: GRACE estimates if a Gauss filter with correlation length 1000 km is used. Dashed-dotted line: the unfiltered monthly mean water storage variations of the LEW hydrological model. Dashed line: the monthly mean water storage variations of LEW if a Gauss filter with correlation length 1000 km is used. The sampling rate of LEW is 1 month; available GRACE solutions are shown by squares.
The seasonal variation is clearly visible and in agreement with the output of the L E W regional hydrological model. The amplitudes, however, differ significantly. The amplitudes of the unfiltered output of the L E W model are much larger than the G R A C E amplitudes. Vice versa, if the output of the L E W model is filtered using the same filter as applied to GRACE, the L E W amplitudes are much smaller. In both cases, the differences can reach values of 3 0 - 4 0 mrrdmonth. There are several potential contributors to these differences: (i) the L E W regional hydrological model is not calibrated properly; (ii) the G R A C E estimates are not correct; (iii) significant contribution of storage variations in neighbored areas (see section 4.2); (iv) residual contributions of atmospheric and ocean mass variations (see section 4.3).
4.2 Quantification of the hydrological contribution of the surrounding areas To quantify the hydrological contribution of the surrounding areas to the G R A C E estimates of the monthly mean water storage variations over the upper Zambezi sub-catchment, the L E W regional hydrological model is used. The output of the model is expanded into spherical harmonics. Then the same
0~J(~1/01
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Figure 8: Monthly mean water storage variations in [mm/month] over the upper Zambezi sub-catchment. Dashed line: the contribution of water storage variations over surrounding areas to GRACE monthly mean water storage variations of the upper Zambezi. Solid line: GRACE estimates corrected for the influence of the surrounding areas. Dashed-dotted line: unfiltered monthly mean water storage variations from the LEW model. Dotted line: filtered monthly mean water storage variations from the LEW model. Available GRACE data are indicated by squares. The sampling rate of the LEW model output is 1 month.
Finally, the water storage variation averaged over the upper Zambezi sub-cathment is estimated. The estimates are shown in figure 8. The influence of the surrounding areas on the monthly G R A C E estimates has amplitudes up to 20 mm/month, which is quite significant. W h e n correcting G R A C E estimates for this influence, the differences between the filtered L E W m o d e l estimates and the G R A C E estimates b e c o m e smaller. Nevertheless, the amplitudes still differ significantly: 6 - 8 mrrdmonth from L E W compared with about 20 - 30 m m from GRACE. One possible explanation for this difference is the limited size of the study area (0.5-106 k m 2) in combination with the need to filter G R A C E monthly models due to highfrequency noise. This results in a significant loss of energy for the L E W model estimates. Therefore, we compared G R A C E estimates with (i) the L E W m o d e l and (ii) the N O A A / C P C global hydrological model for the area covering the upper Zambezi subcatchment a n d the surrounding areas. The size of this area is about 3 . 106 k m 2, i.e. six times larger than the area of the upper Zambezi sub-catchment. The results are shown in figure 9. Apparently, the fit between the G R A C E estimates and the output of the regional L E W model is much better than for the upper
494
R. K l e e s
• E. A . Z a p r e e v a
• H. C. Winsemius
• H. H. G.
Savenije
Zambezi sub-catchment (figure 8). They also agree well with the global NOAA/CPC model, although the latter gives slightly higher amplitudes compared with GRACE and LEW.
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F i g u r e 10: Total contribution of atmosphere (dashed) and oceans (dashdot) to the GRACE monthly mean storage variations over the upper Zambezi when assuming that GRACE measurements are not corrected for atmosphere and oceans. GRACE monthly mean storage variations are shown for comparison (solid line). Available GRACE solutions are indicated by squares.
04107101
F i g u r e 9: Mean monthly water storage variations over the upper Zambezi sub-catchment and the surrounding area after smoothing with a Gauss filter with 1000 km correlation length. The area is about 3- 106 km 2. Black squares: available filtered GRACE solutions. Solid line: interpolated filtered GRACE estimates. Dashed line: filtered NOAA/CPC model. Dash-dotted line: filtered L E W model.
4.3 Quantification of the total contribution of atmosphere and oceans Monthly mass variations in the atmosphere and the oceans provided by GFZ are used to quantify the total contribution of the atmosphere and oceans to the GRACE monthly mean water storage variations over the upper Zambezi (i.e. assuming that no corrections for atmosphere and oceans are applied to the GRACE measurements). The same standard averaging procedure as described above is used. The results are shown in figure 10.
The contribution of the atmosphere varies between - 2 0 mm/month and +18 mm/month. The contribution of the oceans varies between - 1 0 mm/month and +18 mm/month. The effects are quite significant. Therefore, it is necessary to investigate the amplitude of the residual atmospheric and oceanic effects for the upper Zambezi subcatchment. This is an on-going activity, and the results will be reported elsewhere.
5
S u m m a r y a n d conclusions
The LEW hydrological model has been applied for the first time to the upper Zambezi sub-catchment. The storage variations consist of changes in the unsaturated zone, the saturated zone, the wetlands and the channel storage. LEW regional hydrological model estimates and GRACE estimates of the monthly water storage variations over the upper Zambezi differ significantly in terms of amplitude. Maximum differences are about 20 mm/month. The contribution of the hydrological signal over surrounding areas to the GRACE monthly mean storage variations over the upper Zambezi has amplitudes up to 20 mm/month. Therefore, when storage variations over the upper Zambezi sub-catchment are studied, it is necessary, to accurately model the contribution of the surrounding areas. Filtered LEW estimates of the monthly storage variations over the upper Zambezi sub-catchment are a factor of 10 smaller than the unfiltered estimates. This loss of energy is due to the fact that the area of the upper Zambezi sub-catchment is relatively small (about 0.5- 106 km 2) compared with the total surface of the Earth. GRACE monthly mean storage variations after correction for the contribution of the surrounding areas have maximum amplitudes of about 30 mm/month. This is much smaller than the amplitudes of the unfiltered LEW model output (maximum amplitudes of 80 mm/month, but significantly larger
Chapter 71 • Monthly Mean Water Storage Variations by the Combination of GRACEand a Regional HydrologicalModel:Application to the Zambezi River
than the amplitudes of the filtered LEW model output (maximum amplitudes of about 8 mm/month). There is no definite explanation for the discrepancies between GRACE estimates and (filtered) LEW model output. The limited size of the upper Zambezi sub-catchment and the need to smooth GRACE estimates due to high-frequency errors may be the most significant contributor. This is supported by the results shown in figure 9. The contribution of the atmosphere and oceans to the GRACE monthly mean storage variations over the upper Zambezi can reach up to 20 mm/month if GRACE measurements are not corrected for. These large amplitudes motivate further investigations into the residual atmospheric and oceanic contribution to the monthly GRACE estimates of mean water storage variations over the upper Zambezi. The contribution of the errors caused by the application of spherical harmonic expansion has also been investigated. The same methodology (see paragraph 2) is used to compute monthly mean water storage variation averaged over the upper Zambezi by using the Gauss-Legendre method of numerical integration as alternative to spherical harmonic expansions. The contribution of the Gibbs phenomenon when using spherical harmonics is found to be below 2 %. Therefore, this artefact cannot be the source of the discrepancy in the amplitudes between GRACE and LEW model. The estimated amplitudes of the monthly mean water storage variations as inferred from GRACE strongly depend on the correlation length of the filter function, which is used to reduce the influence of GRACE errors. The proper choice needs further investigations. The quantification of the uncertainties in GRACE estimates of the monthly mean water storage variations as well as in the hydrological model output is an important subject for future studies and a pre-requisite for a combined solution using hydrological data and models and GRACE data.
Acknowledgments
The project is supported by the Dutch Organization for Scientific Research (NWO) and the Water Research Center Delft (WRCD). The support is gratefully acknowledged. References
Aerts J and Bouwer L (2003) STREAM manual. Amsterdam. The Netherlands
Beven KJ and Binley AM (1992) The future of distributed models: model calibration and uncertainty prediction. Hydrol. Proc. 6:279-298 Beven KJ and Freer J (2001 ) Equifinality, data assimilation and uncertainty estimation in mechanistic modeling of complex environmental systems using the GLUE methodology. Journal of Hydrology 249:11-29 Chen JL, Rodell Matt, Wilson CR, and Famigletti JS (2005) Low degree spherical harmonic influences on Gravity Recovery and Climate Experiment (GRACE) water storage estimates. Geophys. Res. Let.32, L14405 Herman A, Kumar VB Arkin PA and Kousky JV (1997) Objectively determined 10-day African rainfall estimates created for famine early warning systems. Int. J. Remote sensing. 18(10): 21472159 Jekeli C (1981) Alternative methods to smooth the Earth's Gravity field. Report No. 327, Department of Civil and Environmental Engineering and Geodetic Science,The Ohio Sate University, Columbus, Ohio Nash JE and Sutcliffe JV (1970) River flow forecasting through conceptual models, Part 1: A discussion of principles. J. of Hydrol. 10:282-290 Seo KW and Wilson CR (2005) Simulated estimation of hydrological loads from GRACE. Journal of Geodesy 78:442-456 Swenson S and Wahr J (2002) Methods of inferring regional surface mass anomalies from Gravity Recovery and Climate Experiment (GRACE) measurements of time-variable gravity. J. of Geop. Res. 107(B9), 2193:3.1-3.13 Swenson S, Wahr J and Milly P (2003) Estimated accuracies of regional water storage variations inferred from the Gravity Recovery and Climate Experiment (GRACE). Water Resources Research 39(8), 1223:11.1-11.13 Wahr J, Molenaar M and Bryan F (1998) Time variability of the Earth's gravity field: Hydrological and oceanic effects and their possible detection using GRACE. J. ofGeop. Res. 103(30): 205-229 Winsemius HC, Savenije HHG, Gerrits AMJ, Zapreeva EA and Klees R (submitted, 2005) Comparison of two model approaches in the Zambezi river basin with regard to model reliability and identifiability. Hydrology and Earth System Sciences Discussions Xie P and Arkin PA (1997) Global Precipitation: A 17-Year Monthly Analysis Based on Gauge Observations, Satellite Estimates, and Numerical Model Outputs. Bull. Amer. Meteor. Soc. 78:2539-2558
495
Chapter 72
The use of smooth piecewise algebraic approximation in the determination of vertical crustal m o v e m e n t s in Eastern C a n a d a Azadeh Koohzare, Petr Vanfeek, Marcelo Santos Department of Geodesy and Geomatics Engineering, University of New Brunswick, P.O.Box 4400. Fredericton NB., Canada E3B 5A3
Abstract. The objective of this study is to compile
Keywords. Least Square Approximation, crustal
a physically meaningful map of vertical crustal movements (VCM) for Eastern Canada. Average vertical velocities over the past century are determined by repeated precise levelling and monthly mean sea level observations from 17 tide gauges. The spatial vertical velocities may be mathematically expressed in any number of ways.
movements, Adjustment
In this study, the uplift rate is calculated using smooth piecewise algebraic polynomial approximation. The mathematical model of the approximation for the geodetic data is given. We show how a vertical velocity surface is approximated using piecewise algebraic polynomials and what conditions should be satisfied to guarantee the smoothness of the surface. First, we divide Eastern Canada into zones. The vertical movement is represented by a different polynomial surface in each zone. The polynomials are joined together at nodal points along the border of adjacent zones in such a way that a certain degree of smoothness (differentiability) of the resulting function is guaranteed. This study shows that piecewise polynomial surfaces can represent the available data in a unified map. The pattern of a northwest to southeast gradient of crustal movements is consistent with the existing Glacial Isostatic Adjustment (GIA) models. Present-day radial displacement predictions due to postglacial rebound over North America computed using VM2 Earth model and ICE-4G adopted ice history show a zero line (hinge line) very similar to ours along the St. Lawrence River. The main advantage of the presented technique is its capability of accommodating in one model, different kinds of information when the re-levelled segments are scattered not only in time but also in space. Piecewise approximations make it easier to get the physically meaningful details of the map, without increasing the degree of polynomials.
geodynamics,
Glacial
Isostatic
1 Introduction It has been recognized for several decades that the determination of a Vertical Crustal Motion model is of importance in geosciences. In geophysics, for example, it is of primary interest in the study of the theology of the mantle and lithosphere which is crucial in understanding geodynamical processes. In geodesy, they are important in the definition of vertical datum which is in turn, required in many application areas such as navigation, mapping, and environmental studies. The first VCM model in Canada was compiled by Vanf6ek and Christodulides (1974) using scattered geodetic relevelled segments and the first study which covered the whole of Canada was carried out by Vanf6ek and Nagy (1981) using precise re-leveled segments and tide gauge records. The country was divided into regions and polynomial surfaces of order 2, 3 and 4 were calculated by the method of least squares for each region to obtain representations of the vertical movements. A considerably larger database has been gathered since then, and this, together with additional insight into the nature of the data, led to the recompilation of the map of vertical crustal movement of Canada by Carrera et al. (1994) in which a vertical polynomial was fitted to the data. In order to infer a physically meaningful VCM, it is necessary to combine the geodetic and geophysical data, theories, methodologies and techniques that are somehow linked together. Hence, finding the best approach to reconcile geodetic data with geological phenomena is required. In North America, the most significant geophysical process that has an evident effect on the shape of the viscoelastic earth is postglacial rebound
Chapter 72 • The Use of Smooth Piecewise Algebraic Approximation in the Determination of Vertical Crustal Movements in Eastern Canada
or Glacial Isostatic Adjustment (GIA). During the last major glaciation event, immense masses of ice accumulated over regions of North America, causing subsidence of the Earth's crust in these ice covered regions, and uplift in peripheral regions. When this ice has melted during the last 20,000 years, the viscoelastic rebounding of the crust in the ice covered regions started and has been ongoing since (Peltier, 1996). In this paper, some ideas are exploited in an effort to infer a more physically meaningful VCM model for Canada. Using smooth piecewise approximation method, the velocity surfaces are computed in pieces, and then they are tied together to guarantee their continuity across the zone boundaries.
2 Sea Level data and re-levelled segments The data used in this study are of two kinds: sealevel records and relevelled segments of the firstorder levelling network. A number of 17 permanent tide gauge stations with long enough records were selected in the area of interest (Figure 1). The subset of 17 sites was then selected to include all stations for which continuous records of at least 10 years duration are available. In the studies of vertical crustal motion, tide gauge records with longer time span are considered more reliable. Sea 6500W I
5000N
/
level records with duration of less than 50 years may not be taken as representative for the secular trends sought, if they are studied individually. However, when they are treated in pairs, the secular variations can be accurately estimated. There is a well documented feature of tide gauge records: their striking similarity when they are obtained at two close-by locations. (Vanf6ek and Carrera, 1993). This spatial coherence is caused by common atmospheric and oceanic noise. Clearly, a large portion of these variations disappears when the records are differenced. This behaviour offers an alternative way of treating sea level trends in closeby tide gauges. In this study, a straight--forward trend analysis was carried out on monthly mean values for all stations. Then, it was decided to use the differencing technique to treat the sea level records. The regional correlation matrices and correlation coefficient confidence interval is used to select the optimum pairing of sites, i.e., a tree diagram for optimum differencing, that gives the most precise and accurate velocity differences to be used in the modelling. Figure 1 demonstrates the optimum pairing of tide gauges in Eastern Canada. A total of 14168 relevelled segments from Maritimes and southern Quebec were chosen for this study. They were observed during the period between 1909 and 2002. The distribution of data is shown in Figure 1.
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i',
/ .~
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-
~Pairing • 7000W
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Fig 1: Data distribution used in computations. The optimum tree diagram of tide-gauges for differencing is shown by red lines.
497
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A. Koohzare • P. Vani?ek. M. Santos
3 Mathematical Model
m+z) . Here, q, represents the maximum number of the nodal points in each border.
In order to predict the spatial vertical velocities, or uplift rates, a vertical velocity surface should be fitted to the sea level linear trends and levelling height difference differences data reviewed in the previous section. Therefore, the main concern is to provide an approximation to a function V(x,y) from geodetic data. The assumptions underlying this approach are that the uplift rates are linear in time and that they vary smoothly with location. The velocity surface is first obtained in the form of
In order to piece the polynomials together, the following conditions should be satisfied:
V(x, y) - £ co x~y j
,
(1)
G
(Xm,k'
(3.a)
av,,(x, y) &
8Vm(x,y)
i,j=O
In general, if we divide the area of study into m zones and the degree of all the algebraic polynomials is n, the resulting function is a polynomial function of degree n with m zones. A given polynomial in the m-th zone looks as follows:
Vm(X,y)-- ~C~j,,.(X--X,.,~)~(y-- ym,~)j ,
m 8Vm+l (X, y) X=Xm ,k - y= ym,k (iX
_
X=Xm,k y= ym,k
ovm+,(x,y)
_
X=Xm,k - y=ym,k
where (x,y) is the location of the points in an arbitrary selected local horizontal coordinate system, n is the degree of polynomials, and c,j are the sought coefficients. Here, the algebraic functions are the simplest functions to deal with numerically and are adequate when the solution is confined to the regions where sufficient data exits; the poor behaviour appears only when the solution is used in an extrapolation mode (Vanf6ek and Nagy, 1981). The procedure of fitting a surface to the geodetic data involves the use of both the point rates and the gradients simultaneously, together with their proper weights. The point rates are determined from some of the tide gauge data which were selected to be used in the point velocity mode, and the gradients come from relevelled segments and tide gauge pairs. To get the details needed for the map to be meaningful, the order of the velocity surface would have to be too high to be numerically manageable. A practical way to avoid this is to divide the area of study into zones, and seek the velocity surface piecewise.
(2)
i,j=0
where V,,, is the algebraic least squares velocity surface for zone m, fitted to the desired data (x,y). The pair (x,,,k ,y,,,k) for k=l,2 . . . . . q represents the position of each node (P ....k) located in the predefined border between two zones (zones m and
Vk- 1,2..... q
G+I (X ....k' Y ....k )
Ym,k ) --
X=Xm ,k y=ym,~
Vk = 1,2..... q (3.b)
8~Vm(x,y)
~2Vm+l
c~x 2
X=Xm,k - y=ym,k
a 2~/~m(X, y) 2
X=Xm,k - y=ym ,/,
(x,
y)
c~x 2
X=Xm ~k y=ym,k
aaVm+l (x, y) ~17 v/, 2
X=Xm ,k y= ym ,1,
Vk = 1,2..... q (3.c)
Conditions (3.a) make sure that the piecewise polynomial fits to the nodal points (P,,,1, P ....2. . . . . Pm,k=q). These conditions imply that the function is continuous everywhere in the region. Conditions (3.b) and (3.c) ensure that the polynomials are continuous in slope and curvature respectively throughout the region spanned by the points (x,y). Assuming the velocity to be constant in time, the difference of the two levelled height differences divided by the time span between the two levellings gives the velocity difference between the two levelling segment's ends. These 'observations' are used to compute the coefficients by means of leastsquares method. The main mathematical model is equation (2) while all the conditions under (3) show the existence of constraints on the main model. To find the least square solutions, equations (2) and (3) can be simplified in a general form:
f ( ¢ , l ) - O,
(4.a)
f c (¢) - 0.
(4.b)
Chapter 72 • The Use of Smooth Piecewise Algebraic Approximation in the Determination of Vertical Crustal Movements in Eastern Canada
Here, 1 is the vector of observations and c is the vector of unknown coefficients. It will be assumed that it is possible to solve for c, using only the main model (4.a). The auxiliary modelfc consists of some constraint functions that enforce the conditions which should be guaranteed. The above models are next linearized to yield:
N - (Ar (BCrBr u-
Ar (BCrBr
)-I A)-' )-t w
(9) (10) (11)
6(1) - - N -l u .
Equation (11) represents the solution from the main A6 + Br + w-O,
D6+
(5)
w e -0.
In equations (5), r is the vector of expected residuals. Matrices A and D are the Jacobian matrices of transformation from parameter space to the two model spaces, valid for a small neighborhood of c(0,. Matrix B is the Jacobian matrix of transformation from observation space to the main model space. It is observed that equations (5) are merely the differentional form of the original non-linear mathematical model equations (4.a) and (4.b) and describe the relations of quantities in the neighborhoods of c,o,, the point of expansion in the parameter space, and w~0,, the misclosure vector, where,
d - c - c (°~,
I • + 2 k r ( A d + B r + w) + 2 k r (DO + wc ),
(7) where Cr~C/ is the covariance matrix of the observations. Here, there are two sets of Lagrange correlates: k, kc, reflecting the fact that two models are present. The minimum with respect to r is found by the Lagrange approach (Vanf6ek and Krakiwsky, 1986) as
(DN-IDr)~
(w~ + Dg(1) ),
(8) where
C~ - N - l - N - 1 D r
(DN-1Dr
)-I DN-1.
(12)
The appropriate degree of the velocity surface is determined by testing the estimated accuracy, or the 'a posteriori standard deviation'. This is computed from ~Tc[I~
d-02 = - - ,
(13)
where t; is the vector of least square residuals and v denotes the number of degree of freedom.
The variation function for finding the least-squares solution is written as,
- (j(l) _ N - I D r
The next task is to obtain the covariance matrix of the parameters. It is given by Vanf6ek and Krakiwsky (1986) as:
(6)
w(o~ - f(l~o~,c(o~).
¢ -- r T C r
model f alone, and the corrective term 6 - d (') in equation (8), arises from the enforcement of the constraints.
Due to the geophysical diversity in Eastern Canada, for example, different geological characteristics and different rate of seismicity, Eastern Canada was divided into two zones: the Maritimes zone, and the zone containing the southern part of St. Lawrence River (Figure 2). The border of these two zones is dictated by the actual data distribution and the present knowledge of the geodynamics of the area. For example, the estuary of the St. Lawrence River is an area where 50 to 100 earthquakes are detected yearly. The region, known as the Lower St. Lawrence Seismic Zone, was originally defined by spatial clustering of magnitude (M) <5 earthquakes (Basham et al., 1982 from M. Lamontagne et al., 2003). This information was used to select the zone boundaries. The vertical movement was then represented by a different polynomial surface in each zone. The polynomials were joined together at the nodal points along the zone border in such a way that the desired degree of smoothness (differentiability) of the resulting function was guaranteed.
499
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A. Koohzare • P.Vani?ek. M. Santos 6500W I
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?
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/,///,. ../
/ / ,.,.....,"1"
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~
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~
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/
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ISouthSt.
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I 6500W
Fig 2" The polygonal subdivision used to compute the partial solutions describing the trends of VCM.
4 Results Several tests were made to determine the appropriate degree of the velocity surface to be computed. Table 1 shows the a posteriori variance factors for the degrees 2, 3 and 4. All degrees of the polynomials yielded the a posteriori variance factors between 8.1-8.5. The value n=3 was finally selected as the highest degree compatible with data distribution.
Tablel. The a posteriori variance factors of polynomial surfaces of degree 2, 3 and 4.
Degree of polynomials
Degree 2
Degree 3
Degree 4
a posterior variance factor
8.4
8.1
8.3
The map of vertical crustal movements in Eastern Canada produced by smooth piecewise algebraic polynomials is shown in Figure 3. The standard deviation for the area of interest is typically 1.4 mm/a. The solution is evidently much generalized. This is due to the sparseness of data which imposes the use of smooth functions. The map depicts clearly the zero line of the postglacial rebound. The zero line follows the St. Lawrence River. Present-day radial displacement predictions of postglacial rebound over North America computed using VM2 Earth model and ICE-5G adopted ice history (Figure 4, Peltier 2004) show a zero line very similar to ours along St. Lawrence River.
The general Northwest Southeast trend of vertical crustal movements is consistent with the predictions of Glacial Isostatic Adjustment models of Mitrovica et al. (1994), Peltier(1994), Wu(2002) and Peltier(2004).
Chapter 72 • The Use of Smooth Piecewise Algebraic Approximation in the Determination of Vertical Crustal Movements in Eastern Canada 6500W I
50 0 0 N I
60 0 0 W i
5000N 7"0 0 0 W -4500N
Velocity(mm/yrl
I2-a I = I~1 1-2
-6000W
I ~ 1 o-1 -1 -o I~1-1 I -2--1 I-2 I -a--2
450 0 N
i
i
7000W
6500W
Fig 3:.Pattern of vertical crustal movements in Eastern Canada. Contours are in millimetre/year
ICE5G vl.2
Rate of C h a n g e of Radial Disp ( m m / y r )
it 1
[iiil 1
1
LI ~11
Fig 4. ICE-5G (VM2) prediction of the present-day rate of vertical movement (rate of radial displacement) of Earth' s crust over the Eastern North American continent. (Adopted from Peltier, 2004)
501
502
A. Koohzare • P. Vani?ek. M. Santos
With respect to the individual features, caution is required when interpreting the map. The technique used here was designed to model only linear vertical movements, i.e., movements with velocities constant in time. It is unlikely that all parts of Eastern Canada are undergoing such a steady vertical movements. The subsidence in Maritimes predominantly in Nova Scotia and eastern New Brunswick is due to postglacial rebound. This area lies immediately outside of the region that was covered by the Laurentide Ice Sheet at the last glacial maximum (see Peltier, 1994 for maps of surface ice cover from LGM to present). As the Laurentian ice started to decay, leading to the postglacial rebound of the crust in the once ice covered region, the forebulge began to collapse to accommodate the uplift in the central region (Peltier, 1996). The map of VCM in this area reflects this phenomenon and is also compatible with the recent map of gravity changes (See Pagiatakis, 2003 for the map of gravity changes). The pattern shown in the north eastern margin of the former Laurentide ice sheet (the border of which has been postulated to have been parallel to St.Lawrence river) is complicated due to the probable fragmentation of the crust in this zone. The map seems to justify the concentration of seismicity in Lower St. Lawrence Zone (See Lamontagne et al., 2003 for the definition of Lower St.Lawrence Siesmic Zone), which opens new doors into the study of geodynamics of this complex area. The earlier reported uplift of the northern New Brunswick and the subsidence of the south St. Lawrence River (Carrera, G. and P. Vanfeek 1994) are here more sharply defined.
define different surfaces for geophysically different areas and still maintain the continuity and smoothness throughout the region of interest. However, the computed value of 8.1 for the a posteriori variance factor indicates the probability of the existence of some shorter wavelength features that could not be modelled by a surface of such a low degree. Increasing the number of intervals (zones) in the area of computation, might be a solution for representing shorter wavelength features of VCM which would be the next step in our studies.
Acknowledgement We acknowledge financial support provided by GEOIDE (GEOmatics for Informed DEcisions) Network of Centres of Excellence of Canada and CIDA (Canadian International Development Agency).
References Carrera, G. and P. Vanicek. (1994). Compilation of a new map of recent vertical crustal movements in Canada. The 8-th International Symposium on Recent Crustal Movements, Kobe, Japan, December 6-11, 1993. Lamontagne, M., P. Keating and S. Perreault. (2003). Seismotectonic characteristics of the Lower St.Lawrence Seismic Zone, Quebec: insights from geology, magnetic, gravity and seismic. Can. J. Earth Sci. 40: 317-336.
Mitrovica, J.X., J. L. Davis, and I. I. Shapiro (1994). A spectral formalism for computing three-dimensional deformations due to surface loads: 2. Present-day glacial isostatic adjustment, J. Geophys. Res., 99(B4), 70757101.
5 Conclusions The technique of smooth piecewise polynomial approximation is capable of accommodating in one model, different kinds of information when the relevelled segments are scattered not only in time but also in space. Piecewise approximations make it easier to get the details of the map to be physically meaningful, without increasing the degree of polynomials. The general pattern of the map reflects the main geophysical phenomenon in the region, postglacial rebound. The local pattern of the map gives more details of the South St. Lawrence River, compared to the previous maps. This is mainly due to the improvements in the methodology that enables us to
Pagiatakis. S. D. (2003). Historical relative gravity observations and the time rate of change of gravity due to postglacial rebound and other tectonic movements in Canada. Journal of geophysical research, vol. 108 no. B9, 2406, di: 1029/2001 JB001676.
Peltier, W.R. (2004). Global Glacial isostasy and the surface of the ice-age earth: the ICE-5G (VM2) model and GRACE. Annu. Rev. Earth Planet. Sci. 2004. 32:11149.
Peltier, W.R. (1996). Global sea level rise and glacial isostatic adjustment: An analysis of data from the east coast of North America, geophysical research letters, 23, 717-720. Peltier, W. R. (1994). Ice age paleotopography, Science, 265, 195-201, 1994.
Vanf6ek, P. and D. Christodulids (1974). A method for evaluation of vertical crustal movements from scatted geodetic revellings. Can. J. Earth Sci. 11, pp. 605-610. Vanf6ek, P. and G. Carrera (1993). Treatment of sea-level records in modelling linear vertical crustal motion. Proceeding of the CRCM '93, December 6-11,305-309.
Chapter 72 • The Use of Smooth Piecewise Algebraic Approximation in the Determination of Vertical Crustal Movements in Eastern Canada
Vanfeek, P. and D. Nagy (1981). On the compilation of the map of contemporary vertical crustal movements in Canada. Tectonophysics, 71, 75-86. Vanf6ek, P., and E. Krakiwsky, (1986). Geodesy: The Concepts, 2 nd ed. North-Holland, Amsterdam,
Netherlands, 1986. Wu, P. (2002). Effects of mantle flow law stress exponent on postglacial induced surface motion and gravity in Laurentia, Geophys. J. Int., 148,676-686
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Chapter 73
Hydrological signals in gravity- foe or friend? C. Kroner 1, T. Jahr 1, M. Naujoks 1, A. Weise ~'2 'Institute of Geosciences Friedrich-Schiller-University Jena, Burgweg 11, D-07749 Jena, Germany 2Society for the Advancement of Geosciences Jena H61derlinweg 6, D-07749 Jena
Abstract. Although hydrological effects on gravity are known nearly as long as the influence of barometric pressure, they are not as well understood as the latter. The improvement of gravity data quality during the last years adds weight to the importance of understanding the hydrological influence. Moxa observatory is one station at which studies regarding hydrological effects are carried out. From soil moisture, water level and meteorological observations the effects of different hydrological contributors including snow can be modelled and compared to the gravity residuals of the superconducting gravimeter (SG). The total peak-to-peak amplitude amounts to 35 nm/s 2. Contributions from the various areas around the observatory partly compensate due to the hilly morphology. The comparison between residuals and computed total hydrological effect yields a good agreement, but also shows that not all hydrological influences have been taken into account. A significant additional hydrological influence is due to the hill flank near the SG. Besides the possibility of giving an additional constraint to water balance computations, gravity observations might become of interest to hydrologists studying interflow processes. A local gravity network was established around Moxa observatory in order to find out whether slow, e.g. seasonrelated hydrological changes or large-scale fluctuations as caused by snow melt can be detected by repeated gravity measurements. From the six campaigns carried out so far a trend becomes visible: Wet conditions lead to a decrease in the gravity differences between observation points at the foot of the hill and on the upper part of the hill flank. Dry conditions result in increased gravity differences. The changes in the differences which are in the range of several ten nm/s 2 can be explained by variations in the amount of water stored in the hill flank. Keywords. Superconducting gravimetry, repeated gravity measurements, hydrological effects
1 Introduction The existence of hydrologically-induced effects on gravity is well known for years (e.g. Bonatz (1967), Lambert & Beaumont (1977), Elstner & Kautzleben (1982), Elstner (1987), M~ikinen & Tattari (1988), Peter et al. (1995), Bower & Courtier (1998), Imanishi (2000), Kroner (2001), Harnisch & Harnisch (2002), Francis et al. (2004), Abe et al. (2006)). Despite this the influence of hydrological variations on gravity is nowhere near as well understood as the effect of barometric pressure, which is due to the fact that we usually deal with several hydrological sources and with station-specific hydrological settings making generalizations difficult. Also often not all necessary hydrological parameters are monitored. In the last years the endeavours have been intensified to gain a better understanding of local hydrological changes influencing gravity, as for many geodynamic studies this influence needs to be removed from the data (e.g. Crossley et al. (1998), Amalvict et al. (2004), Llubes et al. (2004), Imanishi et al. (2006), Kroner & Jahr (2005)). The spectral range affected is broadband and ranges from a few hours up to years. Depending on the periods the signals can reach a magnitude of some nm/s 2 up to a few ten nm/s 2. One station at which hydrological effects are studied in detail is Moxa Observatory, Germany. During the last years sensors for water table and soil moisture monitoring have been installed in the near area. Additionally, experiments have been carried out in order to improve the knowledge of hydrological influences at the station and to develop reliable reduction algorithms for the data of the superconducting gravimeter (SG) CD034 recording at Moxa observatory (Kroner & Jahr (2005)). From discussions with hydrologists and hydrogeologists it became clear that for them an additional constraint on their modelling e.g. provided by gravity even on a local scale would be very helpful, not only for water balance computations, but also
Chapter73 • HydrologicalSignalsin Gravity- Foeor Friend? for studies related to interflow. Moxa is suited for this kind of study as it is not only in a hilly area, but is directly located at the foot of a hill flank in which lateral water movement along the transition zone between weathering layer and bedrock very likely occurs. Deploying a superconducting gravimeter, gravity variations of some nm/s 2 can be resolved in the time domain for the period of up to a few days and a few tenths nm/s 2 in the frequency domain. The noise level of superconducting gravimeters is about one to two orders of magnitude below the signals to be expected from hydrology (Kroner et al. (2005)). As typically no superconducting gravimeter will be available and this instrument can only be used for a stationary monitoring, the question arose whether repeated measurements with field gravimeters might be deployed to significantly detect gravity changes in the order of some 10 nm/s z related to the slow component of hydrological mass shifts. The order of magnitude of seasonal hydrological effects was derived from the observations with the SG. For the repeated precise gravity measurements a local network consisting of twelve observation points was established, six of the points are located on a W-E running profile which is perpendicular to the main topographic structure and includes the hill flank into which the observatory is built.
2 Hydrological effects at the SG site Hydrological effects on gravity can reach up to some 10 nm/s 2 on both, short- (up to a few days) and long-term scale at Moxa Observatory. Fig. 1 gives an example for a strong event related to snow melt in March 2005. Within two days, gravity increased by 35 nm/s 2 which was caused by the melting snow and the resulting water moving in the hill flank in direction of the observatory. From the meteorological and hydrological observations the gravity effects of soil moisture and water table variations as well as the effect of snow were modelled. Using elevation data the area up to a distance of 700 m from the SG was divided into cells of 5 x 5 m 2. For each rectangular prism the effect of soil moisture variations and additionally for the valley bottom the effect due to water level changes was calculated (Nagy (1966)). For soil moisture fluctuations a layer of 1 m thickness divided into 10 cm thick layers was considered using soil moisture data from three different sites (valley bottom in front of the observatory, forest near top of
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1800 1600 1400
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' I ' I ' 05/03/14 05/03/1605/03/1805/03/2005/03/22
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Fig. 1 Gravity residuals (black) and water level changes (grey) in gap around gravimeter pillar during snow melt, hourly values, 05/03/14-/03/22. the hill flank, and intermediate area) and two different depths (0.3 and 1 m). Between the two depths a linear change in soil moisture was assumed. For the computation of the gravity effect of water level changes a porosity of 25% was assumed representing a mixture of gravel and loam. The lower boundary of the water-saturated zone was set to a depth of 2 m below surface as found by resistivity measurements. For simplification the water level variations obtained at the observation site directly in front of the observatory went into the calculation. The effect of water level variations in the direct vicinity of the gravimeter was computed applying the water level data from the site and the reduction algorithm derived from an injection experiment (Kroner & Jahr (2005)). The effect of snow is computed by accumulating precipitation during periods with temperatures below 0°C. For temperatures above 0°C a simple snow melt model based on degree-days above a reference temperature is used (Chow (1964), Bower & Courtier (1998)). In Fig. 2 the modelled effects of the different hydrological contributions, their total effect as well as unreduced gravity and gravity residuals reduced for hydrological effects are summarized for a period of seven months (04/12/15-05/07/25). The effect of barometric pressure variations on gravity was reduced by applying a frequency-dependent admittance function (Crossley et al. (1995)). For comparison in Fig. 2 the local barometric pressure change during the time under consideration is included emphasizing that the remaining signal in the gravity residuals is not pressure-related. The residual pressure effect in gravity is in the range of up to some nm/s 2, mostly for singular events, and is in general about one order of magnitude below the hydrological signals. The total hydrological effect amounts to 35 nm/s 2 with soil moisture and water table effect partly compensating due to the hilly area. The hydrological reduction leads to a visible
505
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C. Kroner. T. Jahr. M. Naujoks • A. Weise 20 m
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Fig. 2 Gravity effect of different hydrological contributions, unreduced gravity, summarized hydrological effect and gravity residuals reduced for hydrological influence; for comparison barometric pressure and accumulated precipitation (linear trend subtracted) are shown, hourly values, 04/12/15-05/07/25.
decrease in the peak-to-peak amplitude of the gravity residuals as well as in the general fluctuations. Nonetheless, a significant hydrological effect in the range of ten nrrds 2 remains in the gravity data which can be shown by comparing the residuals with e.g. accumulated precipitation also shown in Fig. 2. Between reduced gravity residuals and precipitation an anti-correlation is visible which is due to the fact that most of the hydrological mass changes occur above gravimeter level. Part of the remaining hydrological signals in gravity can be explained by simplifications made in the present estimates. These relate mainly to the assumption of a uniform change in the water table level, a uniform depth of the aquifer and in parts not densely enough distributed topographic data (distance: 10 to 20 m) used as a basis for the modelling in the direct vicinity of the observatory. The total uncertainty due to these simplifications amounts to a few nrrds 2 which is about one order of magnitude below the hydrological effect still present in the gravity residuals. Thus the remaining signal in the gravity residuals surmounts the uncertainties in the modelling. Not taken into account yet is the change of water mass stored in
the hill flank. From irrigation experiments it is known there is water movement through clefts from the top to the foot of the hill (Kroner & Jahr (2005)). Most of the water level changes below the SG are due to water migrating downwards inside the hill.
3 Repeated
gravity
measurements
For the repeated gravity measurements a network consisting of twelve observation points with six on a WE-profile is available (Fig. 3). For the observation points either existing concrete covered sites were used or concrete pillars built. Each point is marked and the gravimeter position indicated. One observation point is next to the SG, so the measurements can be compared and the SG data can be used as a reference. The stability of the observation sites is checked with repeated levellings. So far no significant height changes could be found. For the repeated gravity measurements four to five LaCoste & Romberg field gravimeters (4 G, 1 D) of high quality are deployed. The calibration of
Chapter 73 • Hydrological Signals in Gravity - Foe or Friend?
E.T,~
I45 m
300 m Fig. 3 Location of the observation points on the profile of the local gravity network. all instruments (including periodic calibration terms) has been checked on the calibration line in Hanover within the time frame of the measurement campaigns. Between November 04 and July 05 six campaigns have been carried out. The dates of the campaigns were chosen according to hydrological conditions. Of special interest is the campaign of mid-March 05 which took place during snow melt. Late winter in 2005 there was an unusual great snow depth of 0.5 m. At the time of the measurements in November 04 not all hydrological sensors had been installed making the modelling of hydrological effects more difficult. Mostly measurements are not made on all points of the network during each campaign, but are concentrated on the WEprofile for which the biggest hydrological effects are to be expected. ---÷ dry a) o~ 2
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Between the six points of the profile nine differences are observed. Each gravity difference is measured at least five times with three readings at minimum with each gravimeter at each site. In total about 180 observed differences enter the least squares adjustment. Fig. 4 gives an example of the hydrological variations near observation point MB which is located at the valley bottom. Soil moisture in the surface layer varies by 8%. In 1 m depth the soil moisture variation is about 4% peak-to-peak. The water table fluctuates by 0.8 m. The variations in both, soil moisture and water table steeply increase within a few hours and gradually decrease within a period of several days up to some weeks. In the following we focus on the change in the gravity differences between the observation points MB, SG, and ET. The observation point MB is located in front of the observatory, SG next to the superconducting gravimeter directly at the foot of the hill flank, and ET on the upper part of the hill flank (Fig. 3). The coordinates and elevations are summarized in Tab. 1. The height differences between the sites at the valley bottom and the one at the upper hill flank are 24 and 25 m, resp. Next to all three sites hydrological sensors are installed. Fig. 5a gives the results for the six gravity campaigns, of which the first four were carried out during winter/early spring time. The measurements
E400 200
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05/04/10
05/06/10
I 05/08/09
Fig. 4 (a) Soil moisture variations in two different depths and (b) water level fluctuations (sensor depth: 2.4 m below surface) near observation point MB, hourly values, 04/12/0905/07/25; the vertical arrows indicate gravity measurement campaigns, the horizontal arrows the change in the hydrological conditions; the first gravity campaign is missing here, as the hydrological sensors were installed at a later date.
Tab. 1 Coordinates and elevations of the observation points discussed. observation point MB SG ET
easting
northing
elevation [ m]
4472842.3 4472867.2 4472893.8
5612205.7 5612210.9 5612213.0
454.1 455.3 479.5
507
508
C. Kroner. T. Jahr. M. Naujoks • A. Weise
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dry
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05/07/14
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Fig. 6 Gravity effect of the hill flank for observation points MB, SG, and ET, increase of 1% in pore volume assumed.
MB-ET
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Fig. 5 (a) Observed gravity differences between points MB, SG, and ET. The error bars show standard deviations ranging from 10 to 20 nm/sZ; (b) estimates for differences due to soil moisture and water table variations (time scale not linear).
of each campaign were separately processed using the data of all gravimeters deployed. Then the differences between the gravity values of the three observation points were calculated. The total variation of the gravity differences with time is in the order of several 10 nm/s 2 with standard deviations between 10 and 20 nm/s 2. Between all three points changes occur, generally more pronounced for the differences containing the observation point at the upper part of the hill flank. The maximum peak-topeak variation obtained so far amounts to 130 nm/s 2 for the difference SG-ET. From the comparison of the change in the gravity differences with the general hydrological conditions a trend emerges for an increase in the differences for increasing dryness in the area and vice versa, most striking is the change in the difference between snow melt in March and the dry period in
July. As described in section 2 the gravity effect due to soil moisture and water table variations was calculated for the observation points MB and ET, too. Fig. 5b shows the gravity differences resulting from these computations. The maximum variation is in the range of a 50 nm/s 2, which is partly only about one sixth of the observed gravity effects. At times the computed curves for differences including observation point ET even show opposite trends. This means variations in soil moisture and water level for the different observation points cannot explain a major part of the observed differences. The only hydrological effect not accounted for yet stems from the hill flank. An additional indication for an important influence of this part of the observatory surroundings can be found in the fact that the biggest changes in the gravity differences occur between the observation points closest to the hill flank with SG at its bottom and ET at the upper hill flank. To get an idea of the order of magnitude a hydrological effect from the hill flank could have, an estimate has been made in which a general change of 1% in the water-filled pore volume was assumed (Fig. 6). According to this an increase in the amount of water stored leads to a gravity increase for observation point ET by about 40 nm/s 2, to a decrease in the range of 10 nm/s 2 for the site SG, and has only a minimum effect for point MB. From this follows e.g. a drop in the gravity difference between observation point SG and ET as observed for increasingly wet conditions and vice versa. Besides the polarity of the changes in the differences, the order of magnitude of the variations could be explained by different amounts of water mass stored in the hill flank.
Chapter 73 • Hydrological Signals
4 Conclusions
1400
--
1200
--
in Gravity- Foe or Friend? --10
_
--8
_
The studies at the Geodynamic Observatory Moxa show that different hydrological contributions can be identified and separated in gravity observations with a superconducting gravimeter. Of special interest are changes in the residuals of the SG in the order of magnitude of 10 nm/s 2 caused by hydrological variations occurring in the hill flank as these observations might help hydrologists in their study of interflow processes. These studies will likewise be of use for the removal of this influence from the data which is necessary for analyses of small geodynamic signals. For an improved reduction of effects due to soil moisture variations and snow detailed topographic information of the direct observatory surroundings will be incorporated in the modelling. Regarding the influence of water table changes the various gauge measurements at the valley bottom will be analyzed and compared. Apart from this the aquifer depth along the valley bottom will be determined by resistivity measurements. Presently this information is only for the immediate area in front of the observatory available. From this an improvement in the reduction of water table-related effects on gravity can be achieved. The results of the repeated gravity measurements are promising. Indications for a correlation between gravity changes and hydrological condition have been found. In the next step the observed variations in the gravity differences need to be put in contrast to a detailed modelling of hydrological changes and their gravity effect. In order to enlarge the data base more repeated gravity measurement campaigns will be carried out. In order to gain a better understanding of the hydrological variations in the hill flank hydrologists should be involved. For a more sophisticated computation of hydrological changes in the hill flank we can rely on results of irrigation experiments already carried out. Additionally, precipitation data and water level measurements in the gap around the SG pillar can be used to derive indications about water movements inside the hill. The gauge data clearly show a time lag with regard to rain events (Fig. 7) or snow melt which results from the time the water needs to migrate through clefts from the upper part of the hill to the foot of the hill. As can be seen from Figure 7, not all rain events produce water movement towards the foot of the hill which is also an important information for a modelling. Also the
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Fig.7 Water level in gap around SG pillar (black) and precipitation (grey) data during a dry period, hourly values, 05/06/30-/07/14. water flow continues until a few days after a rain event.
Acknowledgements
The extensive field work would not have been possible without the help of a number of persons. We thank Raphael Dlugosch, Anne Hegewald, Katrin Heinzig, Holger Hartmann, Gerhard Jentzsch, Britta Merten, Stefanie Zeumann, and Albrecht Ziegert for their help. The work of Matthias Meininger and Wernfrid Ktihnel in establishing and maintaining the hydrological observation network is gratefully acknowledged. We are indebted to the Institut for Geodesy und Geoinformationtechnology (TU Berlin), the Leibnitz Institute for Applied Geosciences (GGA), and the Federal Agency for Geodesy and Cartography (BKG) for lending us their instruments. We thank the two anonymous reviewers for their constructive and helpful comments. This research is supported by the German Research Foundation (DFG) by grant KR 1906/6-1.
References
Abe, M., Takemoto, S., Fukuda, Y., Higashi, T., Imanishi, Y., Iwano, S., Ogasawara, S., Kobayashi, S., Dwipa, S., and Kusuma, D.S. (2006). Hydrological effects on the superconducting gravimeter observation at Bandung. J. Geodynamics, 41(1-3): 288-295, doi: 10.1016/j.jog.2005.08.030 Amalvict, M., Hinderer, J., M~ikinen, J., Rosat, S., and Rogister, Y. (2004). Long-term and seasonal gravity changes at the Strasbourg station and their relation to crustal deformation and hydrology. J. Geodynamics, 38, no. 3-5, pp. 343-354. doi: 10.1016/j.jog.2004.07.010. Bonatz, M. (1967). Der Gravitationseinflu6 der Bodenfeuchte. ZfV 92, pp. 135-139. Bower, D.R. and Courtier N. (1998). Precipitation effects on gravity measurements at the Canadian absolute gravity site. Phys. Earth Planet. Int. 106, pp. 353-369.
509
510
C. Kroner. T. Jahr. M. Naujoks • A. Weise
Chow, H.T. (1964). Handbook of applied hydrology. McGraw-Hill, pp. 10-35. Crossley, D.J., Jensen, O.G., and Hinderer, J. (1995). Effective barometric admittance and gravity residuals. Phys. Earth Planet. Int. 90, pp. 355-358. Crossley, D., Xu, S., and van Dam, T. (1998). Analysis of superconducting gravimeter data from Table Mountain, Colorado. Geophys. J. Int. 135, pp. 835-844. Elstner, C. (1987). On common tendencies in repeated absolute and relative gravity measurements in the central part of the G.D.R. Gerlands Beitr. Geoph. 96, pp. 197-205. Elstner, C. and Kautzleben, K. (1982). Results of annual gravity measurements along a W-E profile inside the GDR for the period 1970-1980. Proc. General Meeting of the lAG, Tokyo, pp. 341-348. Francis, O., Van Camp, M., van Dam, T., Warnant, T., and Hendrickx, M. (2004). Indication of the uplift of the Ardenne in long term gravity variations at Membach (Belgium). Geophys. J. Int. 158, pp. 346-352. doi 10.1111/j. 1365-246X2004.02310.x Harnisch, M. and Harnisch, G. (2002). Seasonal variations of hydrological influences on gravity measurements at Wettzell. Bull. d'Inf Mardes Terr. 137, pp. 10849-10861. |manishi, Y. (2000). Present status of SG T011 at Matsushiro, Japan. Cahier du Centre Europden de Gdodynamique et de Sdismologie 17, pp. 97-102. Imanishi, Y. Kokubo, K., and Tatehata, H. (2006). Effect of underground water on gravity observation at Matsushiro, Japan. J. Geodynamics, 41(103): 221-226, doi: 10.1016/j.jog.2005.08.31
Kroner, C. (2001). Hydrological effects on gravity data of the Geodynamic Observatory Moxa. J. Geod. Soc. Japan 47, pp. 353-358. Kroner, C. and Jahr, T. (2005). Hydrological experiments around the superconducting gravimeter at Moxa Observatory. J. Geodynamics 41(1-3): 268-275, doi: 10.1016/j.jog.2005.08.012 Kroner, C., Dierks, O., Neumeyer, J., and Wilmes, H. (2005). Analysis of observations with dual sensor superconducting gravimeters. Phys. Earth. Planet. Int., in press, doi: 10.1016/j.pepi.2005.07.002 Lambert, A. and Beaumont, C. (1977). Nano variations in gravity due to seasonal groundwater movements implications for the detection of tectonic movements. J. Geophys. Res. 82, pp. 297-305. Llubes, M., Florsch, N., Hinderer, J., Longuevergne, L., and Amalvict, M. (2004). Local hydrology, the Global Geodynamics Project and CHAMP/GRACE perspective: some case studies. J. Geodvnamics 38, no, 3-5, pp. 355-374. doi: 10.1016/j.jog.2004.07.015. M~ikinen, J. and Tattari, S. (1988). Soil moisture and groundwater: Two sources of gravity variations. Bull. d'Inf Mardes Terr. 62, pp. 103-110. Nagy, D. (1966). The gravitational attraction of a right rectangular prism. Geophysics 31 (2), pp. 363-371. Peter, G., Klopping, F.J., and Berstis, K.A. (1995). Observing and modeling gravity changes caused by soil moisture and groundwater table variations with superconducting gravimeters in Richmond, Florida, U.S.A. Cahier du Centre Europden de Gdodvnamique et de Sdismologie 11, pp. 147-159.
Chapter 74
A p p l i c a t i o n s o f the K S M 0 3 H a r m o n i c D e v e l o p m e n t o f the T i d a l P o t e n t i a l S. M. Kudryavtsev Sternberg Astronomical Institute of Moscow State University, 13 Universitetsky Pr., Moscow, 119992, Russia
Abstract. The KSM03 harmonic development of the Earth tide-generating potential [TGP] (Kudryavtsev 2004) includes 26,753 terms of amplitudes down to the level of 1× 10.8 m2/s2 and is made in a reference frame defined by the true geoequator of date with an origin at the point - that being the projection of the mean equinox of date. An advantage of that choice for the reference frame is the TGP series have TDB time argument and do not include a much less stable UT1 time argument necessary to eventually calculate the TGP values in the Terrestrial reference frame [TRF]. It makes the series of KSM03 development valid over a longterm interval of time (1000-3000), and the relevant accuracy of the gravity tides calculation is estimated to be at the sub-nGal level. For consistence with the available Earth tide analysis programs (like ETERNA) the coefficients of the KSM03 series have been re-calculated in the TRF and transformed into the HW95 (Hartmann and Wentzel 1995) normalization and format (in total 28806 waves). The KSM03 harmonic development of the Earth TGP represented in the standard HW95 format is available at http://lnfml, sai.msu.ru/neb/ksm/tgp/ksm03.dat. It can be directly used by nutation theories and in precise calculations of tidal effects observed in the TRF. On the base of the KSM03 series we obtained compact analytical representation of variations of the geopotential coefficients caused by the solid Earth tides for the case of frequency-independent Love numbers.
Key words: tidal potential, KSM03 harmonic development, HW95 format, tidal variations of geopotental
1
were done by Cartwright and Tayler (1971), Cartwright and Edden (1973), Bfillesfeld (1985), Xi (1987, 1989), and Tamura (1987, 1995). The latest harmonic developments of the TGP are HW95 by Hartmann and Wenzel (1994, 1995), RATGP95 by Roosbeek (1996), and KSM03 by Kudryavtsev (2004). This article presents our recent results on transformation of the original KSM03 coefficients into the standard HW95 format and application of the KSM03 for obtaining compact analytical series for variations of the geopotential coefficients caused by the solid Earth tides. 2
Formulation of T G P expansion p r o b l e m
The classical representation of the Earth TGP generated by external attracting bodies (the Moon, Sun, planets) at an arbitrary point P on the Earth's surface at epoch t is oo
•
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(t)
where Vis the value of the TGP a t P ;
r is the
geocentric distance to P; p j , 0 are, respectively, the gravitational parameter and geocentric distance to the fh body; E'j is the angle between P and the jth body as seen from the Earth center; P,, is the Legendre polynomial of degree n. The expression (1) is expanded in our study as oo
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oo
/7
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[C,m (t)cosmO (A) (t) + S/7m(t)sinmO (A) (t)] (2) where
Introduction
Doodson (1921) first performed an accurate representation of the Earth tide generating potential (TGP) by harmonic series. Subsequent expansions
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-fi31(sindj(t))sina (A) (t) . (4)
a(A)(t) dj(t) are, respectively, the instantaneous j right ascension and declination of the fh body referred to the true geoequator of epoch t with an origin point A - that being the projection of the mean equinox of date (see Fig.l); o(A)(t) is the local sidereal time at P reckoned from the same point A - so that it is related to the Earth fixed east longitude (from Greenwich) ~ of P simply as (5)
( G M S T is Greenwich Mean Sidereal Time); (p' is the geocentric latitude of the point P;Pnm is the normalized associated Legendre function related to the unnormalized one (P~m) as m
Pnm = NnmPnm where
(6)
Nnm - /~m (2/7 -Jr-1)(/7-
(10)
J
is the mean Earth equatorial radius;
0 (A) (t) - 3, + GMST
(9)
J
Fig. 1 Spherical coordinates used when developing the TGP
m)!
(n+m)! and 6 m _ ~ l i f m - O
~2 if m , O . The classical expression for the TGP (1) has to be completed by some additional terms reflecting the main effect of Earth's flattening (Wilhelm 1983; Dahlen 1993; Hartmann and Wenzel 1995; Roosbeek 1996) which can be re-written as follows
J2 is the normalized value for the dynamical formfactor of the Earth (J2 = J2 / N20 )The coefficients C,m (t) , Snm (t) contain complete information about instantaneous positions of the attracting bodies at every epoch t at which one calculates the TGP value V(t)+V~q(t). Angles tzj "~'(A)(t) and C~j(t) in expressions (3), (4), (8)-(10) are reckoned along and from the true geoequator of the epoch t, so the relevant values for the coefficients C,m (t), S,m (t) fully take into account the effects of both precision and nutation in obliquity. Because of the choice for the origin point A (see Fig. 1) nutation in longitude is not included to the values for C~m (t), S,m (t). Otherwise, one would have to repeatedly take the same effect into account in (5), i.e. substitute there Greenwich True Sidereal Time for Greenwich Mean Sidereal Time, what is more complicated. (When expanding (1) to (2) one gets just differences o(A) (t) -- . j between the local sidereal time at P and the right ascension of every perturbing body.) Having obtained harmonic expansions for Cnm(t), Snm(t ) one can further calculate the timedependent values of the TGP at an arbitrary point P(r, cp', A) on the Earth's surface by using relations (2), (5) and (7). The tidal acceleration along the Earth radius (or <>) is obtained as the radial derivative of the TGP
g(t) =- 8(V(t) + r j' (t)) = ~ n ~ V m(t) + 1 V~ (t)" (11) C~F
,~=2/" m=O
F
Chapter 74 • Applications of the KSM03 Harmonic Development of the Tidal Potential
K S M 0 3 H a r m o n i c D e v e l o p m e n t of T G P and its C o m p a r i s o n to H W 9 5 and RATGP95
xMz
S~a
20.
In (Kudryavtsev 2004) the coefficients Cnm(t),
Snm(t )
are
x
(
expanded to finite 2"d-order Poisson
series of the following form
e-
x $2
,
,
5
10
~
15
,
,
20
25
w~
.
30
35
.
.
.
40
45
5,
-20
?: -40
c t + A k2 c f. 2 ]cos cok(t) + C(S),m : ZN ~A;o + A kl m
k=l
<
[A2o + A~klt -I- A'k2f'2]sincok(t)}
X HW95-KSM03 o RATGP95-KSM03
(12) Frequency [ deglhour ]
where Ak~0 , Ak~l , ' " ,
Aks2
are constant coefficients,
and arguments cok (t) are 4th-degree polynomials of
Fig. 2 Differences between amplitudes of major waves in HW95 / RATG95 and KSM03
time (Ok ( t ) - Vkt + Vkzt
2
3
+ Vk3t + Vk4t
4
.
(13)
Such a development was made with help of a new spectral method of harmonic expansion of an arbitrary tabulated function of the Sum/Moon/planets coordinates to Poisson series (Kudryavtsev 2004). Tables of numerical values of the coefficients C,m, S,m were calculated according to (3), (4), (8)-(10) at every six hours within the time interval of two thousand years, 1000-3000. The latest JPL long-term ephemerides DE/LE-406 (Standish 1998a) were employed as a source of the Moon, Sun and planets coordinates. When calculating the coefficients we used values for the planetary gravitational parameters from m Standish (1998a) and values for J2 and R e from the IERS Conventions (McCarthy and Petit 2003). (The value for the latter constant which further has to be used in (2) and (7) along with expansions of the coefficients is 6378136.3 m.)
We made a comparison of KSM03 series to the previous developments of the Earth TGP: HW95 (Hartmann and Wenzel 1995) and RATGP95 (Roosbeek 1996). As our analysis shows, the main differences in amplitudes of major waves in HW95, RATGP95 and KSM03 (see Fig. 2) arise from their either aliasing with or separating from some other small waves of very close frequencies (differing by +2ps where psis the mean longitude of solar perigee). However, Dehant and Bretagnon (1998) strongly recommend combining such close terms in the Earth TGP developments and prove this procedure just leads to a polynomial form of the combined terms' amplitudes. If to merge terms of frequencies differing by +2ps in HW95,
S,m in the final series is 10.8 m2/s2, and maximum
RATGP95 and KSM03, then amplitudes of the combined major terms become almost identical in all the three developments. Dehant (1997) also states the aliasing or separating terms differing in frequency by ~bs does not change the result of
degree n is equal to 6.
calculation of the Earth tides.
The minimum amplitude of the coefficients C,m,
The total number of terms included to the KSM03 development is equal to 26,753 [what is the total sum of all N in expansions (12) made for every coefficient]. The complete set of coefficients of KSM03 development of the Earth TGP can be found at http ://lnfm 1.sai.msu.ru/neb/ksm/tgp/coeff.zip. (The description of the data format is done in file http://lnfml.sai.msu.ru/neb/ksm/tgp/readme.pdf.).
In time domain the accuracy of KSM03 expansion of the Earth TGP has been checked by computation of the gravity tide values (11) at a midlatitude station. As the latter we choose Black Forest Observatory (BFO) Schiltach: r = 6366836.9 m, ~o = 48.3306 ° N , A = 8.3300 ° E at which Hartmann and Wenzel (1995) and Roosbeek (1996) also computed the tidal gravity by using their expansions of the TGP.
513
514
S.M. Kudryavtsev
[
T
~
O .¢.~
r.m.s, tlsl~luals m i x . tea Idusls
i .
41
i
i
| |, I/
im lgli'jr- J'Y~) (l.n
==*J
,
n G i ] in
Numl~r
1 ~
21~
~,.,~.~n~cxw,~ea~u ~ .....
15131:.~-
,
I
.--Llrll~"~='.'u'~'-~=lo(~, [] ~ Ni'lriD~ XS~aD, { 0 . ~ =G=I i,, 1SO0-22W)
e l ' t e r m s t a k e n f r o m ICSM0$ eOclpmnslon e l ' t h e E m 4 h TOP
Fig. 3 Residuals between gravity tides calculatedby KSM03 expansion of the TGP and those obtained from DE/LE-405 ephemeris (at BFO station over time interval 1600-2200) First, we calculated the total tidal gravity at that station by means of strict expressions (2)-(4) and (7)-(11) where the Moon, Sun and planets spherical coordinates were computed using the most precise JPL ephemeris DE/LE-405 (Standish 1998a). The gravity tides at BFO were calculated at every hour within the whole time span covered by that ephemeris, 1600-2200. Then we calculated the gravity tides at the same point and at the same set of epochs by using KSM03 expansion of the TGP and compared the results with the exact values. The maximal deviation between the two sets of data at any epoch within the whole time span of six hundred years length does not exceed 0.39 nGal (1 nGal = 10-11 m/s2). The corresponding r.m.s. difference between the two sets of data over the same interval is less than 0.025 nGal.
Fig.3 shows how the accuracy of calculation of the gravity tide at BFO depends on the number of terms taken from KSM03 series. It is also interesting to estimate how much KSM03 series can be truncated in order to ensure the accuracy of both RATGP95 and HW95 solution (approximately, because KSM03 series have a different format). Thus, the maximum residual 1.23 nGal (the accuracy of HW95 which includes 12935 terms) is reached when taking 12770 terms from KSM03, and maximum residual 5 nGal (obtained in RATGP95, 6499 terms) is ensured by some 5800 terms from KSM03. Let us remember, that residuals in KSM03 solution are estimated over 1600-2200, while those in HW95 and RATGP95 are done over 1850-2150 and 1987-1993, respectively.
4 Transformation of the K S M 0 3 series to the standard H W 9 5 format Unlike the previous expansions of the Earth TGP, wave frequencies of the original KSM03 series are done not in an Earth-fixed reference frame but in a celestial reference frame defined by the true geoequator of date and the projection of the mean equinox of date as the origin of right ascensions (Fig.l). This format for the series explicitly separates two different time arguments to be used in calculating the wave frequencies in an Earth-fixed frame. One of them is Teph, the time argument for JPL planetary/lunar ephemerides (Standish 1998b), which is close to TDB. It is used when calculating the components of the wave frequencies caused by the Moon, Sun and planets motion, and is exactly the time argument in KSM03 series for coefficients Cnm(t), Snm(t) in (3), (4), (8-10). The second time argument is UT1, which is used for calculating another component of the final frequencies- the sidereal time (or Earth rotation) rate. In our expansion of the TGP the latter time argument is used in (5) only (GMST is a function of UT1). The UT1 time scale is much less stable than Teph, and UT1 time can hardly be predicted for long to sufficient a c c u r a c y - so it is an advantage that coefficients Cnm(t), Snm(t) are both calculated and expanded without any use of UT1 time argument. Such an approach makes the KSM03 series valid over a long-term interval of time (1000-3000) and helps to increase the development accuracy. When calculating the TGP and gravity tide at a certain epoch the relevant value for UT1 can be taken from, e.g. IERS publications (or determined from observations) and used in (5) for calculating just terms mO (A) in (2) and (7). However, for practical applications it is valuable to obtain development of the TGP in the Earth-fixed TRF. Therefore, the coefficients of the KSM03 series have been transformed into the standard HW95 (Hartmann and Wentzel 1995) normalization and format. (For that the Earth rotation was assumed to be uniform, like it is done in all previous expansion of the Earth TGP.) The KSM03 development represented in the HW95 format includes 28806 terms and is available at http://lnfm 1.s ai.msu.ru/neb/ksm/tgp/ksm03, dat. The KSM03 series given in that standard format can be directly used by many present Earth tide analysis programs (like ETERNA) for precise calculations of tidal effects observed in the TRF as well as in development of modem theories of the Earth nutation.
Chapter 74 • Applications of the KSM03 Harmonic Development of the Tidal Potential
5 Harmonic series for variations of gravity coefficients due to solid Earth tides The IERS Conventions (McCarthy and Petit 2003) describe the main effect of solid Earth tides on the geopotential through variations in normalized gravity coefficients zXC~,~ A-~~S~ of degree n and
AC~mS~ zXC,,, cos(m × GMST)+ zXS,,, sin(m × GMST)
(15) A 7,2 - As m cos(m × CMXr)- AC m sin(m x GMST) (16) where
order m as follows AC~-t'~"AF~'~ - 2n+lk~ m
3
~,, (sin~j)e -~m;~.~
/jj Re
(14) R E,/d E are the
where k~m are Love numbers;
AC .... - Re (Rek ....C~m + imk.mS.m ) We
(17)
AS.... - RE (Rek ....S .... -imk,mC~m) /Je
(18)
Earth's equatorial radius and gravitational parameter; ~j, rJ, ~ / and A/• are respectively the
and Rekn,1, I m k , , a r e respectively the real and
gravitational parameter, geocentric distance, latitude and longitude (from Greenwich) of the Moon (/'=2) and Sun (/'=3) at epoch; P,,, are the
the case of anelastic Earth). We calculated harmonic series for coefficients AC,,m and ASnn' according to (17)-(18). The series
normalized associated Legendre functions. [Eq. (14) can also be used for calculating the effect of planets (/'=4, 5,...) on variations of gravity coefficients.] At the first step of calculation of the variations A~-~~ A~ ~~ Love numbers k~m are assumed to be
for every coefficient has a form
nm
~
imaginary parts of the complex value for knm (in
No
AC(S)n,, - ~-] [A;0 cos co, (t)+ A'~T0sin co, (t)]+ k=l N1
nlfl
Z [A;,t cos o k (t) + A;,t sin col,(t)] (19)
frequency independent. The second step corrects for the deviations of k2m of several of the constituent tides from the constant nominal values for k2m assumed at the first step. The latter corrections by the IERS Conventions (McCarthy and Petit 2003) are given in analytical form. However, there is still no accurate analytical representation of the major part of the variations, i.e. those obtained according to (14) at the first step. At present corrections (14) can only by accurately calculated with use of numerical ephemerides of the Moon and planets (like DE/LE-405). However, compact analytical series for variations AC.... s~, A-s~,S .... will reduce the general amount of numerical data necessary in calculations (modern numerical ephemerides of the Moon and planets are of several hundred megabytes), and can be directly used, e.g. in analytical theories of Earth satellite motion (Kudryavtsev 2002). Thus, we have developed so far missing analytical series for variations of the gravity coefficients AC ~~ A~-~~ nm
~
nm
caused by solid Earth tides (for the case of frequency independent Love numbers) by using KSM03 expansion of the Earth TGP. By comparing (3)-(4) and (14) one can obtain the following direct relation between the original coefficients C,m, S ~ in KSM03 development of the Earth TGP and variations AC-S~ AS-S~ rim
~
nm
k=l
where cok(t) are defined by (13). The latest IERS Conventions (McCarthy and Petit 2003) recommend taking into account the relevant variations of all gravity coefficients of degree 2 and 3 as well as of degree 4 and order 0, 1, and 2 which increase the cutoff level of 10-13. Table 1 gives the number of terms in the series (19) for all considered coefficients. Every number is the sum of terms of constant amplitude (No) and terms of linear amplitude (N~) which exceed the chosen cutoff level over the interval of time 10003000. It proved to be that no terms of quadratic amplitude increase the cutoff level of 10 -13 over
1000-3000. Table 1. The number of terms in harmonic series for coefficients AC..... / AS.....
n/m"
0
1
2
2
182/-
164/200
192 / 176
3
19/-
20/20
24 / 24
4
13/-
13 / 16
12 / l0
3 20/21
The obtained series are precise, compact and valid over two thousand years.
515
516
S.M. Kudryavtsev
Acknowledgement The work was supported in part by grant 05-0216436 from the Russian Foundation for Basic Research.
References Bfillesfeld FJ (1985) Ein Beitrag zur harmonischen Darstellung des gezeitenerzeugenden Potentials. Deutsche Geodfitische Kommission, Reihe C, Heft 314, Mtinchen. Cartwright DE, Tayler RJ (1971) New computation of the tide generating potential. Geophys J R Astron Soc 23: 4574 Cartwright DE, Edden AC (1973) Corrected tables of tidal harmonics. Geophys J R Astron Soc 33:253-264 Dahlen FA (1993) Effect of the Earth's ellipticity on the lunar tidal potential. Geophys J Int 113:250-251 Dehant V (1997) Report of the WG on theoretical tidal model. Bull Info Mardes Terrestres 127:9716-9728 Dehant V, Bretagnon P (1998) About the usage of tidal Arguments. Bull Info Mardes Terrestres 129:9946-9952 Doodson AT (1921) The harmonic development of the tide generating potential. Proc R Soc Lond A 100:305-329 Hartmann T, Wenzel H-G (1994) The harmonic development of the Earth tide generating potential due to the direct effect of the planets. Geophys Res Lett 21:1991-1993 Hartmann T, Wenzel H-G (1995) The HW95 tidal potential catalogue. Geophys Res Lett 22:3553-3556
Kudryavtsev SM (2002) Precision analytical calculation of geodynamical effects on satellite motion. Celest Mech Dyn Astron 82:301-316 Kudryavtsev SM (2004) Improved Harmonic Development of the Earth Tide Generating Potential. J Geodesy 77: 829838 McCarthy DD, Petit G (eds.) (2003) IERS Conventions (2003), Verlag des Bundesamts f'tir Kartografie und Geod/isie, Frankfurt am Main Roosbeek F (1996) RATGP95: a harmonic development of the tide-generating potential using an analytical method. Geophys J Int 126:197-204 Standish EM (1998a) JPL Planetary and Lunar Ephemerides DE405/LE405. JPL IOM 312.F-98-048, Pasadena Standish EM (1998b) Time scales in the JPL and CfA Ephemerides. Astron Astrophys 336:381-384 Tamura Y (1987) A harmonic development of the tidegenerating potential. Bull Info Mardes Terrestres 99: 6813-6855 Tamura Y (1995) Additonal terms to the tidal harmonic tables. In: Proceedings of the 12th int Symp on Earth Tides, Science Press, Beijing/NewYork, pp 345-350 Wilhelm H (1983) Earth's flattening on the tidal forcing field. Geophysics J Int 52:131-135 Xi QW (1987) A new complete development of the tidegenerating potential for the epoch J2000.0. Bull Info Mardes Terrestres 99:6766-6812 Xi QW (1989) The precision of the development of the tidal generating potential and some explanatory notes. Bull Info Mardes Terrestres 105:7396-7404
Chapter 75
Continental hydrology retrieval from GPS time series and GRACE gravity solutions J. Kusche, E.J.O. Schrama, M.J.E Jansen DEOS, TU Delft, Kluyverweg 1, PO box 5058, 2600 GB Delft, The Netherlands
Abstract. Monitoring of continental hydrology by measuring gravity variations is one of the primary aims of the Gravity Recovery and Climate Experiment (GRACE) mission. We proposed to combine the GRACE measurements with GPS time series provided by the IGS to strengthen the long wavelengths of the hydrology solution. This strategy is necessary because GRACE monthly gravity solutions not yet match the targeted baseline accuracies at the lower spherical harmonic degrees. The method of GPS inversion has been proposed by Blewitt et al (2001), and sensitivity studies have shown that GPS could contribute up to 60% to degrees 2-4, and up to 30% for selected higher degrees in a combination. New in our approach is a) the introduction of a physically motivated regularization method that guarantees stable GPS-inversions up to higher degrees, while minimizing spatial aliasing, and b) the combination with GRACE in a weighted least-squares sense. We find geocenter estimates which are consistent with previous studies. We estimate a variation in the Earth's flattening term which closely resembles independent estimates from SLR. Finally, we provide some lowdegree combined GPS+GRACE mass redistribution grids.
Keywords.
Surface mass redistribution, GPS, GRACE, inversion, hydrology
We have used the International GPS Service combination solutions for this purpose. This is particularly interesting as long as GRACE monthly gravity solutions not yet match the targeted baseline accuracy at the lower degrees. Fig. 1, modified from Blewitt and Clarke (2003), is an illustration of the dynamic model behind the methods of GPS and GRACE inversion for surface loads. Load Love number (LLN) theory relates load distribution, geoid change and Earth deformation. In this approach the estimated degree-1 surface mass redistribution, obtained from 3D network deformation by consistent choice of degree1 load Love numbers, consequently yields geocenter estimates. It is important to note that these estimates are not derived within the GPS orbit determination process, but rather follow from the conservation of momentum in the Earth-load system. In theory, a purely geometrical technique like VLBI would be capable to provide geocenter estimates through a loading inversion (assuming there would be a sufficient number of homogeneously distributed observation sites).
Other: Earth Tides
I~~' ' I Groundw....
1
~~~~~~~~
....
d )~,
Satellits Dynamics
I\ Li.....M......... i "X~~
Introduction
One of the main objectives of the GRACE mission is to monitor hydrological mass redistribution through its gravitational effect (Tapley et al, 2004). Monthly gravity fields have been released to the public, covering a time span from April 2002 onwards. It has been proposed by Blewitt et al (2001), Wu et al (2003), and others, that at larger spatial scales surface load changes may be derived independently by geometrically measuring the elastic response of the Earth.
Other: Tectonics, PGR
....
Fig.1. The dynamic model used in this study. In principle, gravity field changes sensed by GRACE and surface loading observed by GPS networks include the combined direct and indirect effect of all mass redistributions within the Earth and its atmo-
518
J. Kusche. E. J. O. Schrama • M. J. F. Jansen
spheric and fluid envelope. It is well-known that one cannot uniquely solve for 3D density distributions from gravity data; however, the majority of the mass transports important on time scales from daily to interannual occur at or near the Earth's surface. Under this hypothesis, gravity changes on these time scales can be uniquely inverted into mass redistribution within a spherical shell at the surface. Because the atmospheric contribution to the surface density change can be reasonably modelled using atmospheric pressure data, GRACE gravity and GPS displacements allow to detect changes in the Earth's hydrological storage systems. One is forced to constrain solved-for mass configurations either by low-degree truncation, by spatial averaging (Swenson and Wahr, 2002), or by regularization operations employing mathematically or physically motivated constraints. This is because the errors in GRACE or GPS-derived spherical harmonic coefficients are not "white" over the spectral domain but increase with higher resolution, because load Love numbers quickly lose their power and so the spectral sensitivity decreases, and, in the case of GPS inversion, the Earth's coverage with observations is far from homogeneous. Combining satellite gravity and geometrical displacements in a joint inversion, proposed by Kusche and Schrama (2005), is expected to relieve these constraints and improve the reliability of estimates. In this paper, we combined GRACE time-variable gravity fields with global weekly GPS time series obtained by the International GPS Service (IGS) in a weighted leastsquares sense, for the period August 2002 - July 2004. We find that geocenter estimates from this combination fit quite well to earlier estimates, including our own in Kusche and Schrama (2005) obtained using a regularized GPS inversion. Moreover, changes in the Earth's flattening from the combination fit remarkably well to those from SLR (Cheng and Tapley, 2004).
2
Dynamical Model
As usual, we parameterize the surface mass density change Act, i.e. the anomaly with respect to a longterm reference average, in spherical harmonics
polynomials, and ACz%, AS'z% are the spherical harmonic coefficients of the surface density anomaly. Using Farrell's (1972) loading theory, they can be related to the spherical harmonic coefficients of geoid change (AC~,~, AS~,~), height deformation (ACzh~, AS{~,~), and lateral deformation (ACl~ , A S z,~J, ~ through the load Love numbers/q, hi, ll AC~,,~ - 3pw (1 +/el) ACl~ ~ p~ 2l + 1
(2)
AClhm _ 3pw
p~ 2l + 1 AQ''~ ~'
(3)
3p~ It p~ 21+ 1 ACz''~" %
(4)
ACl~_
hi
Degree-I Love numbers deserve special attention. Starting with hi, ll values from Farrell (1972), referring to the center of mass of the solid Earth (CE), Blewitt (2003) derived hi, 11 for the center of figure (CF) frame as follows l~ - - ~1( h ~cE _ l~ e ) - 0 . 1 3 4
(5)
hi - ~2 ( h ~ E _ l e e ) - - 0 . 2 6 9 .
(6)
We implemented these numbers, which implies that our estimates for AC]I , AS]I , AC]0 and the geocenter refer to the motion of the CF origin as defined in Blewitt (2003) with respect to the center of mass of the Earth-load system. This follows from the conservation of momentum m in the Earth-load system. Geocenter estimates relate to the surface mass change then as follows:
1 ~mx-
1 47ca4p~ AC~I M 3
(7)
1 1 47ca4p~ ASI~ Y - ---~my - M 3
(8)
Z-
(9)
X-
-~ m
M1 4 7ca 4 P ACid° "
In the method of GPS inversion, 3D displacement vectors are used to infer for the load distribution. With height Ah, East Ax and North Ay components, the relations to be employed are
Ah=a oc
(10) 1
/=0 m=O a oc
l
Ax oc
/=0m=0
Here a is Earth's mean radius, pw seawater density, Plm are the fully normalized associated Legendre
(11)
sin 0
×~
1
~
m ( - A C l ~ sin mA + AS12 cos m)~)
/=lm=0
x Pl.~(cos 0)
Chapter 75 • Continental Hydrology Retrieval from GPS Time Series and GRACEGravity Solutions
Ay = - a c~
×~
(12) 1
~
( A C , ~ cos mA + A S , ~ sin mA)
/ = 1 rn=O
0
with observations collected in y, stochastic residuals e, and variance covariance matrix C. Note this interpretation requires that spatial non-stochastic aliasing effects are small; meaning that the truncation degree L must be sufficiently high. 3.2
GRACE observes changes in the Stokes coefficients of the Earth's gravity field, typically averaged over one month. This results in observation equations
Using GPS data from I sites with the dense variance covariance matrix CGps, we have Yaps
A t as
p~
21 + 1 AC~,~(z-)dz-~
(13)
the integral being replaced by a sum over four subsequent weeks in our analysis. 3
Inversion
In what follows, our inversion strategy will be briefly described. Previous studies (Wu et al, 2002, Blewitt and Clarke, 2003, Kusche and Schrama, 2005) have shown that tailoring the inversion strategy is a serious issue, and hinted that diverging results in the literature may be due to different parameterization, truncation, or regularization approaches. 3.1
Parameterization
In a least-squares model, after truncation of the spherical harmonic series eq. (1) at degree L, the unknowns of the inverse problem described in the previous chapter may be collected in the finite-sized vectors X 1 - - (X, Y, Z) T and x2 = ( A C ~ 0 , . . . , A S ~ L ) T. Augmenting this with a set of nuisance parameters 6, we finally have to estimate
x
-
-
(') xl
.
(14)
X2
We have run extensive tests with residual nuisance Helmert parameters (Kusche and Schrama, 2005). Although the weekly IGS combination solutions that we use in our analysis are all in the same (ITRF) frame, we routinely estimate residual translations and rotations below the l m m level (they are separable from degree-I deformation and the associated geocenter motion because the height and lateral degree-1 LLN's eq. (5) are distinctly different). Using GPS or GRACE observations, or both, a Gauss-Markov model (Koch 1999) can be formulated y+e-Ax,
E{e}-0,
E{ee'}-C
(15)
GPS inversion
-- (Ahi, Axi, Aye) T
i - 1...I
(16)
and the design matrix A follows from eq. (10-12). The least-squares cost function if0--
Yops-Ax
2
(17)
will be used exclusively to generate intermediate solutions for outlier removal purposes. The reason is that we have few data for the oceanic areas, rendering the problem into an ill-posed one especially for higher truncation degrees. Instead, we minimize J~--
yaes-Ax
2c_1 GPS
-+-ct[ (Ao-)2dcd. Jo
(18)
where (9 is the ocean function. This is, mass redistribution for ocean regions is penalized, which is physically reasonable having in mind that due to the inverse barometric effect and the fact that tides are removed in the data products we use, these will be significantly smaller than over land. The factor ct can be found from oceanographic or from GRACE-derived models, see Kusche and Schrama (2005). An alternative workable method to derive mass redistributions at higher degrees appears to apply the (truncated) singular value decomposition technique to the problem (X. Wu, pers. comm.). Constraining with respect to time evolution can additionally be implemented, when imposing time evolution as a Markov process with time constants derived from geophysical models, and formulating a Kalman filter scheme. 3.3
Joint G R A C E - GPS inversion
For those periods covered by the GRACE data products, we can implement a joint inversion scheme. We typically combine one month GRACE fields with four subsequent weeks of GPS data. Atmospheric and oceanographic background fields removed in the GRACE processing are first added back to the GRACE fields before combination with GPS. At resolution L, GRACE provides us with YGRACE = ( A C ~ 0 , . . . , AS~L) T, CoaAcr,, and one can minimize the cost function /2~
(19)
519
520
J. Kusche • E. J. O. Schrama • M. J. F. Jansen
l YGPS -- A x 2c o- - 1. s + p n YGaACE - Bxn 2C ~ A C E where matrix B derives from eq. (13), with zero rows for 6 and X l. Two cases are considered: With # = 1 one formulates a true weighted least-squares problem, whereas for # --+ oc the GRACE solution uniquely determines the surface mass distribution from degree 2 til degree L. This last case is equivalent to first removing GRACE-derived load estimates from the GPS data, and then estimating geocenter motion and nuisance parameters from the reduced GPS-data. 4
Results
We have analysed GPS IGS combination solutions and GRACE monthly gravity field models as shown in Fig. 2. 150
.. oa
•
140
,i"
"
"
"-
"• , : "o •
135
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Fig. 4 then provides in the center column some lowdegree (1 = 2 . . . 7) combined GRACE+GPS surface mass fields. For comparison, in the left column we show GRACE-derived surface mass fields, in the fight column those from Milly and Shmakin's (2002) global hydrologic Land Dynamics model. For the GRACE fields, here the atmospheric and oceanographic background fields were not reinstituted, for the combination fields they were subtracted again after the combination with GPS. Both should represent mainly continental hydrology for the continental areas. Larger differences between GRACE and the GPS+GRACE combination occur for the higher latitudes - this is because predominantly the GRACE AC~0 is smoothed by the combination with GPS. But the general picture appears consistent, and the combination field fits the hydrologic model better for example for Siberia and also for South America in August 2003, but not in general.
Chapter
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this study. However, experimentally we have build a dense GRACE variance covariance matrix for August 2003 by assuming that the primary GRACE observations can be well represented through potential differences between the GRACE A and B spacecraft positions every 5 seconds. We have then taken the low-degree part of this matrix and scaled it coefficient-wise to match the sigma's provided by the GRACE project. This alters the combination results by 10-20%, which appears not too much.
521
522
.l. Kusche• E..l.O. Schrama• M..l.F. Jansen 5 Robustness motion
against
non-geophysical
It has been suspected that geophysical station displacents may be masked by other periodic effects: multipath, snow coverage, other site-specific or agencyor antenna-specific effects. Indeed it appears plausible that these non-geophysical effects as well as e.g. aliasing due to tidal mismodelling are partly responsible for the difference of GPS-derived mass distribution fields and GRACE-derived ones. On the other hand, it is however not likely that all potential distortions pass uncontrolled into GPS inversion or GPS+GRACE combination estimates. First, apparent 3D displacements due to non-geophysical effects are not necessarily consistent with the loading theory in mathematical terms they are not within the range of the forward operator described in section 2 of this p a p e r - which means there is a resonable chance that they get caught in an outlier rejection procedure, regardless the spherical harmonic resolution. Second, if they are consistent with the loading theory, they act like local geophysical loading effects. Then, dependend on the network density and homogeneity, it is not necessarily the case that they will distort lowdegree spherical harmonic estimates - of course for isolated stations it may happen. Fig. 5 gives a typical diagnostic statistics of the leverage factor percentage of the GPS data we used: A high factor, close to one, means that a corrupted displacement is not likely do be detected in the outlier rejection procedure. Fortunately, there is only a small fraction of stations where this situation is given. -
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Conclusions
We have implemented a scheme for the joint inversion of global 3D GPS time series data and GRACE gravity fields into low-degree surface mass redistribution fields. We arrive at reasonable geocenter esti-
mates through an elastic loading theory. The choice of the cost function is an issue of investigation, but if either regularized or combined with GRACE, GPS inversion results are largely consistent. A J2 time seties, derived within the same inversion, fits quite well independent results from SLR/CHAME Finally, we provide low-degree mass redistribution fields from the GPS+GRACE combination and compare them with the original GRACE fields and with a hydrologic model. Acknowledgements
We acknowledge the use of IGS GPS station coordinate SINEX files made publicly available through ftp. Thanks go also to the GRACE project for using their level 2 products, and to Minkang Cheng for providing his J2 time series. References
Blewitt G, Lavallee D, Clarke P, Nurutdinov K (2001). A new global mode of Earth deformation: seasonal cycle detected. Science 294:2342-2345 Blewitt G (2003). Self-consistency in reference frames, geocenter definition, and surface loading of the solid Earth. JGR 108(B2), doi:10.1029/2002JB002082 Blewitt G, Clarke P (2003). Inversion of Earth's changing shape to weigh sea level in static equilibrium with surface mass redistribution. JGR 108(B6), doi: 10.1029/2002JB 002290 Cheng MK, Tapley BD (2004). Variations in the Earth's oblateness during the past 28 years, JGR 109, B09402, doi: 10,1029/2004JB 003028 Farrell W (1972). Deformation of the Earth by surface loads Rev Geophys Space Phys 10(3):761-797 Koch KR (1999). Parameter estimation and hypothesis testing in linear models. Springer, Berlin Kusche J, Schrama EJO (2005) Surface mass redistribution inversion from global GPS deformation and GRACE gravity data, JGR 110 (B9), doi: 10.1029/2004JB003556 Milly PCD, Shmakin AB (2002) Global modeling of land water and energy balances, Part h The Land Dynamics (LAD) Model, J. Hydrometerol., 3:283-299. Swenson S, Wahr J (2002). Methods for inferring regional surface-mass anomalies from Gravity Recovery and Climate Experiment (GRACE) measurements of time-variable gravity. JGR 107(B9), doi: 10.1029/2001JB 000576 Tapley B, Bettadpur S, Ries JC, Thompson PF, Watkin MM (2004) GRACE Measurements of mass variability in the Earth system, Science, 305:503-505 Wu X, Argus DF, Heflin MB, Ivins ER, Webb FH (2002). Site distribution and aliasing effects in the inversion for load coefficients and geocenter motion from GPS data. GRL 29(24), doi: 10.1029/2002GL016324 Wu X, Heflin MB, Ivins ER, Argus DF, Webb FH (2003). Large-scale global surface mass variations inferred from GPS measurements of load-induced deformation. GRL 30(14), doi: 10.1029/2003GL017546
Chapter 76
Gravity Changes in Northern Europe as Observed by GRACE J. Mfiller, M. Neumann-Redlin, F. Jarecki, H. Denker, O. Gitlein Institut ffir Erdmessung, University of Hannover, Schneiderberg 50, 30167 Hannover, Germany [email protected]
Abstract. During the mission duration of GRACE (about five years), a temporal geoid change of about 3 mm is expected in the centre of the Fennoscandian land uplift area (the Bothnian Bay), corresponding to a gravity change of about 100 nm/s 2. The uplift effect reaches up to 11 mm/year at the Earth's surface. It can be observed geometrically by GPS, and gravitationally in space by GRACE as well as on the ground by absolute gravimetry. In this study, we determine gravity changes in Northern Europe from the monthly GRACE solutions as provided by CSR, University of Texas, and compare them with recent terrestrial gravity measurements carried out at some selected sites in the region. Each of the gravity data sets is affected by various geophysical processes such as atmospheric, oceanic and hydrological effects which conceal the uplift signal and therefore have to be removed. In this respect, also the processing and filtering of the various data sets is considered.
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Keywords. Postglacial rebound, GRACE, secular and periodic gravity changes, absolute gravimetry ,Pots
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1
Motivation
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In Fennoscandia, the Earth's crust is rising continuously as a result of deglaciation since the last glacial maximum. The spatial extension is about 2000 km in diameter (e.g. Ekman 1996); see also Fig. 1 for the approximate shape and location. To monitor and investigate the postglacial rebound, various geodetic data have been collected since 1892 (mareographs, levellings, relative gravity measurements since 1966). According to Ekman (1996), these observations reveal a maximum orthometric height change of 10.2 mm/a over the Bothnian Bay and show symmetry around the maximum. The height change in the centre is associated with a maximum gravity variation of 20 nm/s 2 per year. Based on these numbers, a geoid change of 0.6 mm/a has been derived for the central area (e.g. Ekman and M/ikinen 1996).
0
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8
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12
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uplift rates (observed)
Fig. 1. The land uplift as determined by BIFROST, 19932000, the uplift rates are given in mm/a (see Schemeck et al. 2003). The BIFROST project (Baseline Inferences for Rebound Observations, Sea Level, and Tectonics), based on GPS techniques and geophysical modelling has delivered a maximum ellipsoidal height change of more than 11 mm/a (see Fig. 1, cf. Milne et al. 2001, Johansson et al. 2002, Scherneck et al. 2003), but the location of the centre and the geometrical structure of the uplift process differs from the previous model. The analysis of GRACE data offers a further possibility to investigate the uplift effect. During the mission duration of GRACE (about five years), a temporal geoid change of 3 mm is expected in the
524
J. M~iller • M. N e u m a n n - R e d l i n • F. Jarecki • H. D e n k e r . O. Gitlein
centre of the Fennoscandian land uplift area, corresponding to a gravity change of about 100 nm/s 2. As the geoid derived from GRACE data can be determined on a monthly basis with an accuracy of about 1 mm at a spatial resolution of 800 km (see also Tapley et al. 2004), the land uplift causes a measurable signal in the observations (NRC 1997, Wahr and Velicogna 2002). The problem of separating this effect from other time-variable gravity changes is a big challenge for the satellite as well as for the terrestrial approaches.
2
The GRACE Project
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GRACE is a joint project between the National Aeronautics and Space Administration (NASA) and the Deutsches Zentrum ffir Luft- und Raumfahrt (DLR). The primary objective of the GRACE mission is to provide global models of the Earth's gravity field with unprecedented accuracy for a period of up to five years. The mission allows to compute mean gravity field models as well as temporal gravity changes on a monthly basis. For example, observations of seasonal variations in the Earth's gravity field place important constraints on models of global mass variability and temporal exchange among the land, ocean, and atmosphere (cf. GRACE 2005). Besides mean global gravity field models from GRACE, for example EIGEN-GRACE02S (e.g. Reigber et al. 2005), 32 monthly solutions are available from CSR, University of Texas, so far. From these monthly solutions, gravity changes such as the post-glacial rebound (PGR) or hydrological variations (e.g. Schmidt et al. 2005) can be investigated. In this context, the spatial resolution of the satellite results has to be considered. Up to now, only temporal variations at the very long wavelengths can be determined significantly from the the monthly GRACE solutions, as for shorter wavelenghts (spherical harmonic degree I _> 15) the errors exceed the signals. The atmospheric and oceanic influence as well as the tidal effects are reduced during the standard GRACE processing by applying corresponding global models. Thus the residual monthly GRACE solutions are mainly affected by hydrological and other contributions.
3
Temporal GRACE
Gravity Variations
from
For our computations, a sequence of 32 monthly gravity field solutions, provided by the Center of Space Research (CSR) at the University of Texas,
Fig. 2. Secular gravity changes over Northern Europe derived from 32 monthly GRACE solutions between 2002 and 2005 (units: nm/s2/a) based on a Gaussian filter with radius 800 km
from April 2002 to March 2005 have been used to determine secular and periodic variations of the gravity field in the area of Fennoscandia. Monthly solutions are released in the form of spherical harmonic geopotential coefficients up to degree and order 120 with corresponding "best guess" calibrated error files (GRACE 2005). It has to be mentioned, that the monthly solutions do not strictly agree with the calendar months and that the sequence contains data gaps from May to July 2002 and in June 2003. From the monthly GRACE solutions, spherical harmonic expansions up to degree and order 50 (without C~o) were computed, followed by a smoothing with a Gaussian filter with a radius of 800 km. We have tested different Gaussian filter radii (1500 km, 1000 km, 500 km), where 800 km seemed to be appropriate considering the present accuracy of the spherical harmonic models and the number and time span of the monthly solutions. (72o was excluded because of its anomalously large variations and error estimates. For each monthly solution gravity values have been computed in a 2°x2 ° grid. In each grid point the data have been used to determine secular (23) and periodic (amplitudes Ci and Di of typical periods wi) gravity variations between April 2002 and March 2005: dg(~,A,t) -
i=2
A + B A t + ~~/=1 C/cos (wiAt) + D i sin (wiAt).
(1)
At is the time difference relative to January 2002. Index i= 1 indicates the semi-annual and i=2 the annual period.
Chapter 76
Fig. 2 shows the secular gravity changes as determined from GRACE data between April 2002 and March 2005. The bright colour indicates a gravity increase corresponding to an increase in the geoid height. The location and magnitude differs from the expected Fennoscandian uplift effect as, e.g., shown in Fig. 1. Clearly, non-PGR signals (e.g. from hydrology) are still present in the input data and have to be investigated further. For a better understanding of the signal, comparisons with hydrological models and absolute gravimetry are performed (see below). 80 18 75
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•
Gravity Changes in Northern Europe As Observed by GRACE
Eq. (1) directly to the spherical harmonic coefficients accompanied by the Gauss filtering; this approach led to identical results. We also performed tests excluding all degree-2 spherical harmonics (not only C2o), because they were determined less accurate by GRACE. The results, however, did not change very much for Fennoscandia. A first attempt of reducing the hydrological contribution in the GRACE data was performed by applying the WaterGAP Hydrological Model (WGHM) (D611 et al. 2003). Monthly solutions have been made available in the form of spherical harmonic coefficients. The WGHM data was processed in the same way as the GRACE data. The secular gravity changes reduced for the effect of the hydrological model are shown in Fig. 4. The hydrological data seems to improve the separation of the land uplift signal, but the location and size is still not in agreement with the expected PGR signal. The reason for this behaviour is not understood at the moment and has to be investigated in upcoming studies.
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50
Fig. 3. Annual (top) and semi-annual (bottom) amplitudes in nm/s~. Results from the analysis of the periodic gravity changes are shown in Fig. 3. In Fennoscandia, annual gravity changes of up to 13 nm/s 2 can be detected in the GRACE data (Fig. 3, top), which peaks in winter. In the Bothnian Bay, there is also a small semi-annual signal with an amplitude of about 4 nm/s 2 (Fig. 3, bottom). However, the semi-annual signals do not seem to be reliable when considering a global analysis. To check our analysis procedure, additional computations were done by fitting
Fig. 5 shows GRACE calculations done for selected sites as well as first results (quadratic points with dashed trend-line) derived from terrestrial absolute gravity measurements carried out during that period. For Onsala the terrestrial and satellite data agree quite well. The terrestrial results for Vaasa indicate a larger gravity increase than GRACE. However, in this connection one has to consider also the accuracy of the absolute gravity values (~ 20-30 nm/s2), the relatively short time span, and that the reduction of the gravity data is still to be improved (e.g., to get rid of the local effects, especially the hydrological signals). Nevertheless, over the entire
525
526
J. M(iller • M. Neumann-Redlin • F. Jarecki • H. Denker. O. Gitlein
mission duration of GRACE, the accuracy of absolute gravimetry will allow the detection of the land uplift signals that can then be used as ground truth for GRACE (e.g., Mfiller et al. 2005). Wilmes et al. (2005) confirmed this in presenting results of absolute gravity measurements at some selected sites in that region covering a time span of 10 to 12 years. In contrast to GRACE, the absolute gravity measurements are affected much more by local effects like groundwater table variations, etc. (see also Mfiller et al. 2005); a more detailed discussion will be presented in a future publication. The GRACE based results obtained in Fennoscandia show differences in magnitude and location of the expected post-glacial rebound signal, which indicates the presence of additional effects not considered so far. They may be caused, e.g., by errors in the GRACE data, mis-modelling in the reductions, contributions from local and global hydrology (see Fig. 5) as well as further regional effects like the so-called surges in the Baltic Sea. More accurate GRACE data over a longer time span and more terrestrial data are required to determine the origin of the gravity changes in Fennoscandia. In addition, the reductions and side-effects have to be considered in more detail.
4
Summary
and
Outlook
Temporal gravity variations were determined in Northern Europe based upon the monthly gravity field solutions from GRACE. We used two different computation approaches and applied different Gaussian filter radii. The GRACE data clearly show temporal gravity variations in Scandinavia. These variations have periodic and secular signatures. The secular variations are not in good agreement with other PGR studies, requiring the continuation of our investigations. First results show, that hydrological effects can be reduced partly by considering corresponding hydrological models. However, the separation of the individual signal parts is still a challenging task, where different reductions, models and auxiliary measurements have to be applied to the terrestrial and the satellite data respectively. Here the reduction of local signals is most challenging. Also the processing and filtering of the various data sets have to be considered more extensively. In the future, better GRACE results over a longer time span and more terrestrial absolute gravity data are expected, which may help to improve the estimation of the land uplift signal. We would like to thank the GRACE science team for overall support, CSR, University of Texas, for providing the monthly GRACE solutions, and GFZ Potsdam as well as Prof. D611 and her group for making the WGHM data available. The research was financially supported by the Geotechnologien-Projekt of the German Federal Ministry of Education and Research (Bundesminsterium ffir Bildung und Forschung BMBF) and the German Research Foundation (Deutsche Forschungsgemeinschaft DFG). This is publication no. GEOTECH-198 of the R&D-Programme GEOTECHNOLOGIEN. Acknowledgments.
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m o n t h s (April 2002 - March 2005)
D611, E, Kaspar, F., Lehner, B. (2003): A global hydrological model for deriving water avaiability indicators: model tuning and validation. Journal of Hydrology 270, 105-134.
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Fig. 5. Gravity changes for Vaasa (Finland) and Onsala (Sweden)
Ekman, M. (1996): A consistent map of the postglacial uplift of Fennoscandia. Terra Nova 8, 158-165. Ekman, M. (1996b): Extreme annual means in the Baltic Sea level during 200 years. Small Publications in Historical Geophysics, No. 1-12, Summer Institute for Historical Geophysics,/~land Islands, 1995-2003. Ekman, M. and M/ikinen, J. (1996): Recent postglacial rebound, gravity change and mantle flow in Fennoscandia. Geophyical Journal International 126, 229-234. GRACE (2005), web pages: http://www.csr.utexas.edu/grace/ or http ://op.gfz-potsdam, de/grace/.
Chapter 76 • Gravity Changes in Northern Europe As Observed by GRACE
Johansson, J.M., Davis, J.L., Scherneck, H.-G., Milne, G.A., Vermeer, M., Mitrovica, J.X., Bennett, R.A., Jonsson, B., Elgered, G., Elsegui, R, Koivula, H., Poutanen, M., R6nnfing, B.O., Shapiro, I.I. (2002): Continuous GPS measurements of postglacial adjustment in Fennoscandia, 1. geodetic results. J. Geophys. Res. 107, B8, ETG 3, 1-27. Milne, G.A., Davis, J.L., Mitrovica, J.X., Scherneck, H.G., Johannson, J.M., Vermeer, M., Koivula, H. (2001): Space-geodetic constraints on glacial isostatic adjustment in Fennoscandia. Science 291, 23,2381-2385. Mtiller, J., Timmen, L., Gitlein, O., Denker, H. (2005): Gravity Changes in the Fennoscandian Uplift Area to be Observed by GRACE and Absolute Gravimetry. In Gravity, Geoid and Space Missions. C. Jekeli, L. Bastos, J. Fernandes (eds.), Springer, lAG Symposia 129, 304-309. NRC (1997): Satellite gravity and the geosphere. National Research Council, National Academic Press, Washington. Reigber, Ch., Schmidt, R., Flechtner, F., K6nig, R., Meyer, U., Neumayer, K.-H., Schwintzer, P., Zhu, S.-Y. (2005): An earth gravity model complete to degree and order 150
from GRACE: EIGEN-GRACE02S, Journal of Geodynamics, 39(1), 1-10. Scherneck, H.-G., Johannson, J.M., Koivula, H., van Dam, T., Davis, J.L. (2003): Vertical crustal motion observed in the BIFROST project. Journal of Geodynamics 35,425-441. Schmidt, R., Schwintzer, R, Flechtner, F., Reigber, Ch., Gfintner, A., D611, R, Ramilien, G, Cazenave, A., Petrovic, S., Jochmann, H., Wtinsch, J. (2005): GRACE observations of changes in continental water storage, Journal of Global and Planetary Change, in print. Tapley, B., Bettadpur, S., Ries, J., Thompson, R, Watkins, M. (2004): GRACE measurements of mass variability in the Earth System. Science 305,503-505. Wahr, J. and Velicogna, I. (2002): What might GRACE contribute to studies of postglacial rebound? Space Sciences Series oflSSI 18, Kluwer Academic Publishers, Dordrecht. Wilmes, H., Falk, R., Roland, E., Lothhammer, A., Reinhold, A., Richter, B., Plag, H.-R, Mfikinen, J. (2005): Long-term gravity variations in Scandinavia from repeated absolute gravity measurements in the period 1991 to 2003. Proceedings (CD publ.) of IAG Symposium GGSM2004, Porto.
527
Chapter 77
Investigations about earthquake swarm areas and processes M. Naujoks 1, T. Jahr l, G. Jentzsch 1, J.H. Kurz 2 and Y. Hofmann 3 1) Institute of Geosciences, Friedrich-Schiller-University Jena, Germany 2) Institute of Construction Materials, University of Stuttgart, Germany 3) ROSEN Technology GmbH, Lingen, Germany Abstract. Earthquake swarms are observed worldwide, especially in connection with fluid movement and volcanism. Two regions are compared by numerical investigations using the finite element method: The Vogtland/NW-Bohemia area situated at the border between Germany and the Czech Republic and the Magadi region in the Kenya Rift. For the Vogtland area a high-precision three-dimensional gravity model was developed. That modelling shows an interaction between geometries of geological structures and geodynamic processes and yields strong indications that a magmatic system at the crust mantle boundary is much more probable than an upwelling mantle as a source of the earthquake swarms. The geodynamic models for the two regions under investigation take into account the regional stress field and thermal stresses as well as creep and plasticity with a porous elastic rheology. The investigations are focussed on the interaction between pore pressure variations, temperature changes, fluid movements, stress accumulation and deformations. It is suspected that these processes play an essential role in the generation of earthquake swarms. An essential result of the modelling is that the existence of the regional stress field alone neither explains the occurrence of the earthquake swarms in the Vogtland area nor in the Magadi area. Temperature changes and periodic pore pressure variations in the earth's crust are most important for the geodynamic processes, although they are weighed differently in each focal area.
Keywords. Earthquake swarms, gravimetric modelling, geodynamic modelling, finite element method
1 Introduction An earthquake swarm is a temporary and regional sequence of earthquakes with several thousend events within a period lasting from hours to days. It usually occurs in an area of just a few square kilometers without any prevailing single event (Mogi, 1963; Neunh6fer & Gtith, 1988; Yamashita, 1998; Hem-
mann et al., 2003). Earthquake swarms are worldwide observed phenomena. As there are indications from observations in different earthquake swarm areas, they obviously are associated with fluid movement in the earth's crust, recent volcanism and/or particular geological structures. The occurrence of earthquake swarms is often linked to active volcanism (Fig. 1). However, earthquake swarms also occur intracontinental and at continental rifts. For this study, the Vogtland/NW-Bohemia intracontinental and the Magadi/Kenya-Rift earthquake swarm areas are chosen. Both areas have a high geodynamic and hydrodynamic activity. Furthermore they are also geologically, geodetically and geophysically well investigated. Hence, adequate boundary conditions for realistic modelling are available. Using both, geodynamic and gravimetric modelling, the earthquake swarm areas are compared regarding the physical mechanisms, which lead to the phenomenom of earthquake swarms.
2 Investigation areas In the intraplate region Vogtland/NW-Bohemia, situated at the border between Germany and the Czech Republic (Fig. 2), the periodic occurrence of earthquake swarms is a characteristic phenomenon (Wirth et al., 2000). For the recurrence of stronger earthquake swarms with magnitudes ME > 3 Grfinthal (1989) found an average period of 69 to 74 ± 10 to 12 years. Weaker swarms with magnitudes ME < 3 appear about triennial (Wirth et al., 2000). Due to the fact that the seismicity is dominated by earthquake swarms, a number of scientific investigations on the earthquake swarm phenomenon were carried out in the region over the last decades. The area of interest includes two marginal fault zones: The Eger Rift and the Marifinzk6 Lfizn6 fault zone which intersect near the focal area of the earthquake swarms (Kurz et al., 2004). Earthquake swarm occurrence is only one of the various manifestations of recent geodynamic activity in the Vogtland/NW-
Chapter 77 • Investigations about Earthquake Swarm Areas and Processes
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The Kenya Rift is an active intercontinental divergent plate boundary (Schliiter, 1997). Ever since geological and geophysical studies were first carried out in the Kenya Rift, its part in Southern Kenya at the Tanzanian border around Lake Magadi (Fig. 3) has been known as a region of high geodynamic
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529
530
M. Naujoks • T. Jahr. G. Jentzsch • J. H. Kurz. Y. Hofmann
activity expressed by recent volcanism, geothermal activity and a high rate of seismicity (Atmaoui & Hollnack, 2003; Baker, 1958). The area around Lake Magadi is the section of the Kenya Rift seismically most active. Since 1994, as a part of the Kenya Rift International Seismic Project (KRISP), not only seismic and gravimetric investigations were carried out in the rift but also a seismic network was established to record the local seismicity (Prodehl, 1997). Many local micro earthquakes and some earthquake swarms were recorded. Underneath the northern end of Lake Magadi the occurence of earthquake swarms in shallow depth may be caused by fluid movement or magma intrusions (Ibs-von Seht et al., 2001).
3 Gravimetric modelling Based on the Bouguer anomaly map computed by Hofmann et al. (2000) a high precision three-dimensional (3-D) density model of the Vogtland/NW-Bohemia region (Hofmann et al., 2003) is developed. Fig. 4 shows a SW-NE section of this model. The 3-D gravimetric model indicates detailed geometries of the geological settings, especially the earthquake hypocentres correlate well with the structural
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boundaries. The most important geological units effecting the gravity field in that section are the Eibenstock granite and the Marifinzk6 Lfizn6 fault separating the Fichtelgebirge granite in the west from phyllites in the east. Regarding the periodic occurrence of earthquake swarms the existence of an upwelling mantle or a magmatic body at the crust mantle boundary is investigated. This is realised by an additional model body dealing with two different densities at the crust mantle boundary. The first case describes an upwelling mantle by a density of 3370 kg/m 3. In the second case a nephelinitic magma system with a density of 2850 kg/m 3 is assumed. The comparison of the modelled gravity of both cases with the measured one shows that obviously an upwelling mantle cannot explain the observed gravity (Hofmann et al., 2003). Therefore a magmatic system as postulated by Weinlich et al. (1999) is supported for the Vogtland/NW-Bohemia region.
4 Geodynamic modelling The geodynamic models for the Vogtland/NW-Bohemia and the Magadi/Kenya Rift area are constructed in consideration of the particular geological, tectonical and seismological circumstances (Fig. 2 & 3). According to the results of both deep seismic profiles and gravity measurements/modelling (Behr et al., 1994; Hofmann et al., 2003; Simiyu & Keller, 2001), both models consist of four layers (Fig. 5 & 6, left) reaching from the surface to the crust mantle boundary of the particular region including the upper, middle und lower crust (Kurz et al., 2003; Naujoks et al., 2004). To ensure realistic modelling, lateral variations in the material properties are likewise included. Fig. 5 (left) also shows the distribution of two different materials included in the Vogtland/NW-Bohemia model. The black boxes mark the material of granitic composition, the rest of the model has metamorphic material properties. In the Magadi/Kenya Rift area the upper 2.5 km are an interbedded stratification of sediments and volcanic rocks. Therefore, the top layer of the model was constructed with averaged material properties of that rocks. The other layers are of metamorphic composition. Tab. 1 summarises the material properties. In the Magadi/Kenya Rift area two special features of the crust, described by Ibs-von Seht et al. (2001), are included into the model: First, an upwelling of the brittle-ductile transition (dark grey in Fig. 6, left) with material properties of the underlying ductile crust. Second, a body in the earth-
Chapter 77
....
• Investigations
about Earthquake Swarm Areas and Processes
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Figure 6: The Magadi/Kenya Rift model: Construction (left) and discretisation with included geological faults (right). quake swarm hypocenter region underneath the northern end of Lake Magadi with increased porosity and permeability (light grey in Fig. 6, left). This region in the crossing area of the passing fault zones is interpreted as strongly jointed and highly fractured (Ibs-von Seht et al., 2001). Therefore, the density of that area in comparison to the surrounding rocks is reduced by two percent. Geological faults are included in the models as weakness zones (Fig. 5 & 6, right). They differ in their physical values from the surrounding rocks. The Youngs's and bulk modulus are reduced by ten percent (Kurz et al., 2003) and the permeability in that faults is six orders of magnitude higher than that of the surrounding rocks, as found by Matthfii & Roberts (1997). In the models a porous elastic rheology as well as Mohr Coulomb plasticity in the upper crust and Newtonian creep in the middle and lower crust is realised. Therefore the brittle-ductile transition between upper and middle crust is characterised by a change in the anelastic material properties.
The question which physical processes cause the earthquake swarms in the Vogtland and Magadi areas is analysed in different steps with varying boundary conditions. At the bottom side the models are always fixed, the surface is free to move in all directions. The surrounding crust at the vertical sides of the models is simulated by elastic element foundation, which approximates the pressure of the surrounding crust to the next main geological units. The first step is to gain the long term influence of the regional stress field acting in the particular region on the model's stress and strain fields over a time period of 10,000 years. Therefore a stress field based on the results of Brudy et al. (1997), Wirth et al. (2000), Ibs-von Seht et al. (2001) and Atmaoui & Hollnack (2003) is applied to the models. This leads to values in layer 1 of about 80 MPa in the NW-SE direction and 40 MPa perpendicular for the Vogtland/NW-Bohemia and to 50 MPa in the NNE-SSW direction and 40 MPa perpendicular for the Magadi/Kenya Rift region respectively. In layer 2 363 MPa (NW-SE) and 183 MPa
531
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M. Naujoks • T. Jahr. G. Jentzsch • J. H. Kurz. Y. Hofmann
Table 1: Parameters for the sedimentary (s), granitic (g) and metamorphic (m) rock in the Vogtland/NW-Bohemia and Magadi/Kenya Rift area after Angenheister (1982); Meissner et al. (1987); Carmichael (1989); Behr et al. (1994); Birt et al. (1997); ibs-von Seht et al. (2001); Simiyu & Keller (2001); (p density, E Young's modulus, v Poisson ratio, ~: bulk modulus, ~y cohesion yield stress (layer 1 and 2 only), 1"1viscosity (layer 3 and 4 only)). p E layer [1~r3] [GPa] Vogtland/NW-Bohemia 1m 2620 71.88 2m 2780 89.47 3m 2880 102.64 4m 3050 129.12 lg 2630 72.32 2g 2700 87.68 3g 2820 81.34 4g 3050 129.12 Magadi/Kenya Rift 1s 2550 41.90 2m 2750 91.39 3m 2850 102.09 4m 3000 130.27
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•y [MPa] 11 [Pa.s]
0.252 0.228 0.234 0.229 0.255 0.239 0.233 0.229
48.35 54.87 64.37 79.41 49.14 55.94 50.89 79.41
370 370 2.5.1022 4.0.1022 510 510 2.5.1022 4.0.1022
0.264 0.247 0.240 0.231
29.53 60.30 65.54 80.67
250 370 2.5.1022 4.0.1022
(perpendicular) for the Vogtland/NW-Bohemia and 360 MPa (NNE-SSW) and 285 MPa (perpendicular) for the Magadi/Kenya Rift region are applied. For the middle and lower crust it is only known, that the differential stress (~1- (Y3 drops to zero when passing the brittle ductile zone (Brudy et al., 1997). Therefore we assume that for the middle and lower crust all three main stress components are equal and take the value of the vertical component which is the lithostatic stress. In the next steps the influence of short term periodic pore pressure variations with an amplitude of 50 MPa and a period of 10 years alone and combined with temperature changes of 0.5 K (Safanda & 0ermfik, 2000) is analysed over a time period of 100 years. The period of the pore pressure variations is an average value of the periodic occurrence of the earthquake swarms (Ibs-von Seht et al., 2001; Neunh6fer & Gtith, 1988). In the Vogtland/NW-Bohemia region the pore pressure and temperature variations are applied to the bottom of the model, because there are indications for a magmatic system (Vrfina & St6drfi, 1997). This could also explain the strong CO2degassing at the surface (Weinlich et al., 1999). Due to hot CO2 springs at the surface (Schltiter, 1997) as well as seismological investigations a magmatic system is also assumed in the upper crust in the Magadi/Kenya Rift area underneath the northern end of
lake Magadi (Ibs-von Seht et al., 2001). There the pore pressure und temperature variations are applied to the model. 5 Results
In the Vogtland/NW-Bohemia area earthquake swarms occur in depths between 7 and 15 km (Wirth et al., 2000), in the Magadi/Kenya Rift area in depths from 0 to 10 km (Ibs-von Seht et al., 2001). The aim is to analyse the influence of the local geology and the regional stress field over a time period of 10,000 years alone and in combination with periodic pore pressure changes and linear temperature changes in a time period of 100 years. The shear stresses ~512, accumulated under these boundary conditions are given in Fig. 7 for the Vogtland/NW-Bohemia region and in Fig. 8 for the Magadi/Kenya Rift region respectively. The other shear stress components (Y13 and c~23 show a similar behavier as (Y12 in the region of maximum stress as well as in the amplitude of the values. Fig. 7 (top) shows the influence of the regional stress field and the local geology in the Marifinzk6 Lfizn~ fault zone of the Vogtland after 10,000 years. Obvious is a small gradient at the intersection of the two fault zones. In the Magadi depression (Fig. 8, top) the regional stress field and the local geology lead to shear stresses which are accumulated at the intersection of the fault zones in a depth from 4 to 13 km as well. Both the amplitude and the gradients are higher compared to the Vogtland region. Fig. 7 (centre) shows an increasing shear stress in the Marifinzk6 Lfizn6 fault zone (Vogtland) under periodic pore pressure variations with an amplitude of 50 MPa and a period of 10 years over a time period of 100 years. Here the gradient in shear stress increases in the crossing region especially in a depth from 12 to 16 km which corresponds to the brittle-ductile transition zone. In the Magadi depression (Fig. 8, centre) the shear stress also increases with a maximum at a depth of about 8 km. Combining these periodic pore pressure changes with linear temperature changes of 0.5 K within 100 years the shear stress increases by the factor of 5 in the Vogtland (Fig. 7, bottom) and the factor 3 in the Magadi area (Fig. 8, bottom). Compared to the calculations, in which only the regional stress field and periodic pore pressure variations are acting, the gradients are higher and the maximum shear stresses are clustered in a smaller region. In the Vogtland as well as in the Magadi region the shear stress accumulations occur where the earthquake swarms take place in reality.
Chapter 77 • Investigations about Earthquake Swarm Areas and Processes
6 Conclusion An essential result of the modelling is that the regional stress field alone can neither explain the occurrence of earthquake swarms in the Vogtland nor in the Magadi region. Indeed, in the Magadi regionin contrast to the Vogtland region- local shear stress
accumulations appear over a long lasting time period caused by the regional stress field. But the computed stress values are too low to cause earthquake swarms. This is a clear indication that in both regions other physical processes must act additionally. As there are obvious indications from observations of fluid movement and temperature anomalies in the earth's crust,
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Figure 8. Magadi/Kenya Rift: Shear stress (Y12 in the Magadi depression under different boundary conditions; top: Regional stress field only; center: 50 MPa pore pressure variation; bottom: 50 MPa pore pressure and 0.5 K temperature variation.
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M. Naujoks • T. Jahr. G. Jentzsch • J. H. Kurz. Y. Hofmann
Table 2" Comparison and evaluation of the results of the modelling in the regions Vogtland/NW-Bohemia and Magadi/Kenya Rift as well as an estimation of the earthquake swarm areas worldwide. structures, rheologies and processes
Vogtland/NW-Bohemia
Magadi/Kenya Rift
Earthquake swarm areas worldwide
local geology and brittle-ductile transition
important for locating the shear stress gradient essential influence, particulary in crossing areas only weak shear stress accumulations in the middle crust crucial; applied to the bottom of the model crucial; intensifies geodynamic effects significantly essential; fluid movement in a southward direction
controls amongst others the local stress distribution in the focal area essential influence, likewise in crossing areas only little influence for long running model loading crucial; applied to the lower border of the upper crust crucial; intensifies geodynamic effects significantly essential; fluid circulation in the model
important for locating the geodynamic processes play obviously a very essential role in the local stress distribution little influence, but possibly important in tectonically very active regions the crucial factor for local shear stress accumulations process-controlling factor assumedly essential for all earthquake swarm areas essential; process-controlling factor in earthquake swarm areas
fault zones
regional stress field pore pressure variations e.g. by fluid movement or degassing magma temperature changes e.g. by magma systems or mobile hot fluids fluids
which can occur in the particular region in very different ways, in the next step periodic pore pressure variations and linear temperature changes are applied to the models to determine the influence of these processes on the stress and strain fields. Linear pore pressure changes do not lead to a significant change in the calculated shear stresses. But a strong effect of periodic pore pressure variations, e. g. caused by degassing of magma intrusions or fluid movements in the earth's crust, can be proved. Both in the Vogtland and in the Magadi area they lead to a concentration and local increase of shear stresses in that section of the model where the earthquake swarms occur in reality (|bs-von Seht et al., 2001; Wirth et al., 2000). The computed shear stresses are higher in the Vogtland than in the Magadi area. The consideration of temperature changes, probably caused by magma intrusions, showed that increasing temperatures are responsible for a significant rise in the occurring shear stresses. A 0.5 K temperature rise causes an amplification of the occurring shear stress by the factor of 5 in the Vogtland and 3 in the Magadi area. Probably because of the different geological settings the Vogtland model reacts more sensitively to temperature changes than the Magadi model. The influence of the regional stress field is superposed by short term and stronger processes, the pore pressure and temperature variations. They cause shear stress accumulations where the earthquake swarms occur in reality. The gravimetric modelling
in the Vogtland region shows that a magmatic system as postulated by Weinlich et al. (1999) seems to be the source of the CO2-rich mineral springs in the region, which may cause pore pressure variations and so may be the crucial factor for local stress accumulations leading to the earthquake swarms. Hence, periodic pore pressure variations may be a local trigger mechanism of the occurrence of earthquake swarms. These results are valid for both the Magadi area as well as the Vogtland area, even though they are weighed differently in each focal area. They are consistent with observations made in different earthquake swarm areas. Modelling is an progressive process. Many models were computed and the geometry, the material properties and the boundary conditions were varied. As a result an evaluation of the influence of parameters on the results of the modelling is possible. And therefore it is possible to determine which parameters in which dimension may have an effect to the occurrence of earthquake swarms in regions worldwide with different geological setting (Tab. 2).
7 Acknowledgements The authors wish to thank the German Research Foundation (DFG) for their funding of this research. Constructive reviews of the manuscript by Bert Vermeersen and an anonymous reviewer are gratefully acknowledged.
Chapter 77 • Investigations about Earthquake Swarm Areas and Processes
References Angenheister, G. (Ed.) (1982). Physical Properties of Rocks, Vol. V l a & l b of Landolt-B6rnstein, Springer, Berlin. Atmaoui, N. and D. Hollnack (2003). Neotectonics and extension direction of the southern Kenya Rift, Lake Magadi area. Tectonophysics, 364, pp. 71-83. Baker, B.H. (1958). Geology of the Magadi Area, Kenya Geol. Surv. Rept., 42. Behr, H.-J., H.-J. Dfirbaum and P. Bankwitz (1994). Crustal structure of the Saxothuringian Zone: Results of the deep seismic profile MVE-90 (East), Z geol. Wiss., 22 (6), pp. 647-769. Benoit, J. and S. McNutt (1996). Global volcanic earthquake swarm database 1979-1989, U.S. Geological Survey, Open-file Report, pp. 69-96. Birt, C. S., Maguire, P. K. H., Khan, M. A. et al. (1997). The influence of pre-existing structures on the evolution of the southern Kenya Rift Valley- evidence from seismic and gravity studies, Tectonophysics, 278, pp. 211242. Brudy, M., Zoback, M., Fuchs, K., Rummel, F. & Baumgfirtner, J., (1997). Estimation of the complete stress tensor to 8 km depth in the KTB scientific drill holes: Implications for crustal strength, J. Geophys. Res., 102 (8), pp. 18453-18475. Carmichael, R. (1989). Practical Handbook of Physical Properties of Rocks and Minerals, CRC Press, Boca Raton, USA. Grfinthal, G. (1989) About the history of the seismic activity in the focal region Vogtland/Western Bohemia. In: Monitoring and Analysis of the earthquake swarm 1985/86 in the region Vogtland/Westem Bohemia (P. Bormann, Hg.), ZIPE Ver6ffentlichung Akademie der Wissenschafien der DDR, Potsdam, 110, pp. 30-34. Hemmann, A., T. Meier, G. Jentzsch and A. Ziegert (2003). Similarity of waveforms and relative relocalisation of the earthquake swarm 1997/1998 near Werdau, J. Geodyn., 35, pp. 191-208. Hofmann, Y., T. Jahr and G. Jentzsch (2003). Threedimensional gravimetric modelling to detect the deep structure of the region Vogtland/NW-Bohemia, J. Geodyn., 35, pp. 209-220. Hofmann, Y., T. Jahr, G. Jentzsch, P. Bankwitz and K. Bram (2000). The gravity field of the Vogtland and NW Bohemia: Presentation of a new project, Studia geoph. et geod., 44 (4), pp. 608-610. Ibs-von Seht, M., S. Blumenstein, R. Wagner, D. Hollnack and J. Wohlenberg (2001). Seismicity, seismotectonics and crustal structure of the southern Kenya-Rift--new data from the Lake Magadi area, Geophys. J. Int., 146, pp. 439-453. Kurz, J.H., T. Jahr and G. Jentzsch (2003). Geodynamic modelling of the recent stress and strain field in
the Vogtland swarm earthquake area using the finiteelement-method, J. Geodyn., 35, pp. 247-258. Kurz, J.H., T. Jahr and G. Jentzsch (2004). Earthquake swarm examples and a look at the generation mechanism of the Vogtland/Western Bohemia earthquake swarms, Phys. Earth Planet. Int., 142 (1), pp. 75-88. Matthfii, S. & G. Roberts (1997) Fluid Flow and Transport in Rocks: Mechanisms and Effects, Chapter 16, pp. 263295, Chapman & Hall, London. Meissner, R., Wever, T. and E.R. Fltih, (1987). The Moho in Europe - Implications for crustal development, Anhales Geophysicae 5B (4), pp. 357-364. Mogi, K. (1963). Some discussions on aftershocks, foreshocks and earthquake swarms - The fracture of a semiinfinite body caused by an inner stress origin and its relation to the earthquake phenomena, Bull. Earthquake Res. Inst. Tokyo Univ., 41 (3), pp. 615-658. Naujoks, M., T. Jahr, G. Jentzsch and J.H. Kurz (2004). Den Schwarmerdbeben auf der Spur: Vergleichende geodynamische Modellierungen zu Kenia-Rift und Vogtland, Proc. of the 16th German-speaking ABAQUS User Conference, K6nigswinter, Germany, Sept. 20-21. Neunh6fer, H. and B. Tittel (1981). Mikrobeben in der DDR, Z. geol. Wiss., 9 (11), pp. 1285-1289. Neunh6fer, H. and D. Gtith (1988). Mikrobeben seit 1962 im Vogtland, Z. geol. Wiss., 16 (2), pp. 135-146. Prodehl, C. (1997). The KRISP 94 lithospheric investigation of southern Kenya - the experiments and their main results in Structure and Dynamic Processes in the Lithosphere of the Afro-Arabian Rift system, Tectonophysics, 278, pp. 121-147. Safanda, J. and V. Cermfik (2000). Subsurface temperature changes due to the crustal magmatic activity - numerical simulation, Studia geoph, et geod., 44, pp. 327-335. Schltiter, T. (1997). Geology of East Africa. Contributions to the regional geology of the earth, Vol. 27, Gebrfider Bomtrfiger, Berlin, Stuttgart. Simiyu, S.M. and R. Keller (2001). An integrated geophysical analysis of the upper crust of the southern Kenya rift, Geophys. J. Int., 147, pp. 543-561. Vrfina, S. and V. St6drfi (1997). Geological model of western Bohemia related to the KTB borehole in Germany, J. Geol. Sci., Vol. 47, Prague. Weinlich, F. H., K. Brfiuer, H. Kfimpf et al. (1999). An active subcontinental mantle volatile system in the eastern Eger rift, Central Europe: Gas flux, isotopic (He, C and N) and compositional fingerprints, Geochem. et Cosmochem. Acta, 63 (21), pp. 3653-3671. Wirth, W., T. Plenefisch, K. Klinge, K. Stammler and D. Seidl (2000). Focal mechanisms and stress field in the region Vogtland/Westem Bohemia, Studia geoph, et geod., 44 (2), pp. 126-141. Yamashita, T. (1998). Simulation of seismicity due to fluid migration in a fault zone, Geophys. J. Int., 132, pp. 674686.
535
Chapter 78
Sea level and gravity variations after the 2004
Sumatra Earthquake observed at Syowa Station, Antarctica Kazunari Nawa, Kenji Satake Geological Survey of Japan, AIST, AIST Tsukuba Central 7, Higashi 1-1-1, Tsukuba, Ibaraki 305-8567, Japan Naoki Suda Hiroshima University, Kagamiyama 1-3-1, Higashi-Hiroshima, Hiroshima 739-8526, Japan Koichiro Doi, Kazuo Shibuya National Institute of Polar Research, Kaga 1-9-10, Itabashi, Tokyo 173-8515, Japan Tadahiro S ato National Astronomical Observatory, Hoshigaoka 2-12, Mizusawa, Iwate 023-0861, Japan
Abstract. The Indian Ocean Tsunami reached Syowa Station, Antarctica, in approximately 12.5 hours after the 2004 Sumatra-Andaman Earthquake. We have analyzed the tsunami records of the tide gauge, including the superconducting gravimeter (SG) at the station. The synthetic tsunami and the induced gravity variations were calculated in order to compare with observations. It was found that the gravity effects of the tsunami exhibited an amplitude of microGal (10 .8 m/s2); obtained from the Syowa SG. Furthermore, the effects of the tsunami on the Earth's free oscillation records of the SG were subtracted by applying a transfer function method, using the tide gauge records as input. The improvement of S/N at frequencies of 0.3 mHz is remarkable. Keywords. Tsunami, tide gauge, superconducting gravimeter, synthetic waveform, free oscillations, noise reduction.
1 Introduction The Sumatra-Andaman earthquake on the 26 th of December 2004 generated a massive tsunami (Fig.l) in the Indian Ocean and excited the Earth's free oscillations. The tsunami propagated over the
Indian Ocean and also reached the coast of Antarctica (e.g. Dumont D'Urville, see Merrifield et al. (2005)). The tide gauge at the Syowa Station, Antarctica (69S 39.6E), detected the tsunami (the amplitude was sub-meter) approximately 12.5 hours after the occurrence of the earthquake. The Earth's free oscillations, excited by the Sumatra-Andaman earthquake, were also observed using a superconducting gravimeter (SG) at the station.
.90 °
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Fig. 1 Water height distribution of simulated Indian Ocean tsunami 12.5 hours after the 2004 Sumatra-Andaman earthquake. Red color means that the water height is higher than normal while blue means lower. Initial water height distribution is calculated by assuming a fault of 600 km in length and 150 km in width (strike 340 deg., dip 8 deg., rake 90 deg. and slip of 5 m). Syowa Station (SYO: black circle) is located at 69S, 39.6E.
Chapter 78 • Sea Level and Gravity Variations after the 2004 Sumatra Earthquake Observed at Syowa Station, Antarctica
At Syowa Station, sea level variations at frequencies between 0.2 and 2.5 mHz, in the seismic normal mode band, were detected simultaneously from on-ice GPS (Aoki et al., 2000) and SG records on calm days (Nawa et al., 2003). Since the Indian Ocean tsunami causes vibrations in the sea around the station, the gravity effects of the tsunami should be detectable using the SG. In such a case the SG could be employed as a tsunami gauge as long as the SG is installed near the coast, as at the Syowa Station. Compared to signals that originate the solid Earth, however, the signals of the gravity effects of the tsunami are on the scale of "noise". In this study we analyze sea level and gravity variations after the Sumatra-Andaman earthquake and compare observations with synthetic waves. We then try to remove the effects of the tsunami from the Earth's free oscillation spectra by applying a transfer function method.
bottom mount (CT-type), replaced the old SG in 2003, which was of a regular size and employed a top mount (TT-type). In figure 3 the original sea level and gravity variation data are presented. The sampling rates of the tide gauge and SG are 30 sec and 1 sec, respectively. One can clearly see in Figure 3 how the tsunami and seismic waves modify the tide waves. The induced Tsunami arrived 12 hours and 40 minutes after the start of the Sumatra-Andaman earthquake. In Fig. 3(b), although the amplitude becomes small, gravity variations caused by the Earth's free oscillations persist while the variation in the sea level continues, Fig. 3(a). Hence, in the time domain, it is difficult to resolve the tsunami effects from the SG records.
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2 Observation The Syowa Station is on East Ongul Island in the coastal zone of the southern Indian Ocean (Fig.2). Tide gauge (ocean bottom pressure gauge) observations, maintained by the Japan Coast Guard, are on-going at the Nishi-no-ura Cove. The SG observations have continued since 1993. However, a new type of SG, compact in size and employing a
min.
Fig. 3 Time series of (a) sea level and (b) gravity variations, caused by the 2004 Sumatra-Andaman earthquake. The data is presented as a function of time, where t=0 is the beginning of the earthquake.
The spectrum of sea level variations after the arrival of the tsunami, obtained from tide gauge, is presented in figure 4 where the data exhibits a maximum power at approximately 0.3 mHz. The tsunami observed at the Syowa Station exhibits a large power in the low frequency seismic mode band (see Fig.7). Although the power is large, the frequencies of these peaks (including around 0.3 mHz) in this frequency band coincide with that of signals on calm days detected simultaneously from sea level variation records by on-ice GPS and gravity variation records by the SG (Nawa et a1.,2003).
537
538
K.Nawa.K.Satake• N.Suda.K.Doi. K.Shibuya.T.Sato the synthetic sea level variation. The high frequency component (> about 1 mHz) of the observation reflects influences of local topography that our modeled bathymetry cannot represent. We have only confirmed that the tsunami waveform of the ocean bottom pressure gauge installed at Lutzow-Holm Bay (Doi et al., this meeting) is comparable to the synthetic tsunami of that site.
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Fig. 4 A spectrum of the Indian Ocean tsunami after the 2004 Sumatra-Andaman earthquake, obtained from 48 hours of tide gauge data. Vertical broken lines indicate frequencies of sea level variations observed by on-ice GPS and SG, with high coherency at the Syowa Station on calm days in 1998 (Nawa et al., 2003). Gravity effects of the tsunami are overlapped in the Earth's free oscillations frequency band. However, the tsunami exhibits a power at frequencies lower than that of the lowest frequency modes, 0S2 (0.3 mHz). In order to extract the tsunami effects and reduce the free oscillation signals, gravity variations are bandpass filtered at frequencies between 0.1 and 0.2 mHz (Fig.6a). This will be compared with a synthetic gravity variation in the next section.
3 Modeling of tsunami and comparison with observations In order to calculate the gravity effects of the tsunami we first need to calculate the tsunami waveforms of the global ocean. Initial water height distribution based on seafloor deformation is computed using Okada's (1985) formulas from an assumed fault model (see caption of Fig. 1). The tsunami source was estimated from the analysis of tide gauge records (Lay et al., 2005). Tsunami waveforms are computed by assuming linear long-waves using the finite-difference method. The grid size is defined as 10 min of the arc. The bathymetry grid was made from a global digital topography dataset (ETOPO2). The details of the tsunami numerical computation are described in Satake (2002). Synthetic sea level variations at the Syowa Station are presented in figure 5b. The high-cut filtered tide gauge record (Fig. 5a) is very similar to
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Fig. 5 A comparison of (a) observed and (b) synthetic tsunami sea level variations. Black and gray lines in (a) show bandpass (0.1 - 1 mHz) filtered data and residual data that only removed tidal components, respectively. The time axis starts at the origin time of the earthquake.
Second, we compute the gravity variation at the Syowa Station induced by the tsunami. We compute the gravity effects, attraction and loading (elastic) effects, assuming the 1066A earth model with a modified version of GOTIC (a program for Global Oceanic Tidal Correction by Sato and Hanada, (1984)) for synthetic global water height distributions (Fig.l) every 5 minutes. The computed gravity variation induced by the tsunami is very similar to the filtered SG record (Fig.6) with peak to peak maximum amplitudes at approximately 1400 rain. 1.5 microGal for observation and 1.7 microGal for synthetic, respectively. Furthermore, the Root-mean-square amplitides from 1200 to 2600 rain. are 0.40 microGal from the observation and 0.45 microGal from the synthetic, respectively. The "noise" level of the SG in this frequency band is found to be less than 0.1 microGal.
Chapter 78 • Sea Level and Gravity Variations after the 2004 Sumatra Earthquake Observed at Syowa Station, Antarctica
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Fig. 6 A comparison of (a) observed gravity variation and (b) synthetic gravity effects, of the tsunami. Black and gray lines in (a) are the bandpass (0.1 - 0.2 mHz) filtered data and residual data (with tidal components and atmospheric effects removed), respectively. Right scale is for bandpass filtered data (black) and left scale is for residual data (gray). Black and gray lines in (b) are bandpass filtered synthetic data at the same frequencies as observation and original synthetic data, respectively. The data is presented as a function of time, where t=0 is the beginning of the earthquake.
4 R e d u c t i o n of s e a level v a r i a t i o n effects for seismic normal mode observation We apply the reduction of the sea level variation to the SG Sumatra-Andaman earthquake record using the transfer function presented by Nawa et al. (2003), with the tide gauge records as input. The dominant frequencies of the tsunami spectrum are the same as those of the spectrum of a calm day, as described previously. We also confmned that the response fac-tor (1.6 microGal/m) between the synthetic sea level and gravity variations computed in section 3 is comparable to the findings of Nawa et al. (2003). As a result, the sea level variation effects from low-frequency free oscillations were reduced (Fig. 7). From the data given in figure 7, it is clear that a significant improvement in the S/N, at frequencies of 0.3 mHz near the 0S2 mode, has been gained (Fig.7). Two peaks (0.291, 0.330 mHz), observed on both sides of the 0S2 m o d e (Fig.7), are clearly
Fig. 7 An example of the reduction sea level variation, in order to resolve the Earth's free oscillations excited by the 2004 Sumatra-Andaman earthquake. Black and gray lines show spectrum after reduction and that before reduction of the sea level variation by the tsunami, respectively. The power spectral densities (PSDs) are averages of three PSDs. Each PSD is calculated from 3 day long and 1 day overlapped data. Vertical dashed lines indicate the frequencies of the fundamental spheroidal modes. The vertical line of the lowest frequency indicates 0S2 mode.
reduced in amplitude by applying the correction for the effect of sea level variations. The S/N of the 0S2 mode increases from 1.4 to 2.9. It is important to note that the frequencies of the two peaks are similar to the observed splitting of the 0S2 mode (e.g. Park et al., 2005). The difference, however, between the frequencies is larger than the difference between the lowest and the largest singlet frequencies of 0S2 (Rosat et al., 2005). Therefore, we recognize that the two peaks are not due to the Earth's normal mode. However, it may be advantageous to estimate the tsunami effect at the mid latitude stations, especially close to the sea coast, in order to increase the analysis accuracy at the sites discussed thus far, and to characterize the effect of the tsunami on the Earth's normal modes. In 1998, we employed a transfer function obtained from sea level variations observed using GPS and gravity variations observed using the TT-type (before 2003) SG. In order to confirm the stability of the transfer function and to increase the accuracy of the reduction the transfer function should be recalculated using recent data of sea level variations of the tide gauge and gravity variations of the new CT-type SG, installed in 2003.
539
540
K. Nawa. K. Satake • N. Suda. K. Doi. K. Shibuya. T. Sato
5 Concluding remarks The tide gauge at the S y o w a Station, Antarctica, collected data of the sub-meter Indian Ocean tsunami triggered by the 2004 S u m a t r a - A n d a m a n earthquake. A superconducting gravimeter at the station also detected the tsunami, which exhibited an amplitude of less than 2 m i c r o G a l (10 .8 m/s2), peak to peak, at frequencies of 0.1 - 0.2 mHz. The results give clear indication that the SG could be e m p l o y e d as a tsunami gauge. With respect to the solid Earth observations, the tsunami and/or seiche are on the scale of "noise". Continuous observation of sea level variations around the S y o w a Station is important for the "noise" reduction of the S G record.
A c k n o w l e d g m e n t s . W e would like to thank the Japan Coast Guard for obtaining and distributing tide gauge data at S y o w a Station, Antarctica.
References Aoki, S., T. Ozawa, K. Doi and K. Shibuya (2000). GPS observation of the sea level variation in Lutzow-Holm Bay, Antarctica. Geophys. Res. Lett., 27, 2285-2288. Doi, K. et al. (2005). Presented at Dynamic Planet 2005, session GP03, August 22-26, Cairns, Australia. Lay, T., H. Kanamori, C. J. Ammon, M. Nettles, S. N. Ward, R. C. Aster, S. L. Beck, S. L. Bilek, M. R. Brudzinski, R.
Butler, H. R. DeShon, G. Ekstorom, K. Satake, and S. Sipkin (2005). The Great Sumatra-Andaman Earthquake of 26 December 2004, Science, 308, 1127-1133. Merrifield, M. A., Y. L. Firing, T. Aarup, W. Agricole, G. Brundrit, D. Chang-Seng, R. Farre, B. Kilonsley, W. Knight, L. Kong, C. Magori, R Manurung, C. McCreery, W. Mitchell, S. Pillay, F. Schindele, E Shillington, L. Testut, E. M. S. Wijeratne, E Caldwell, J. Jardin, S. Nakahara, E -Y. Porter, and N. Turetsky (2005). Tide gauge observations of the Indian Ocean tsunami, December 26, 2004. Gephys. Res. Lett., 32, L09603. Nawa, K., N. Suda, S. Aoki, K. Shibuya, T. Sato and Y. Fukao (2003). Sea level variation in seismic normal mode band observed with on-ice GPS and on-land SG at Syowa Station, Antarctica. Geophys. Res. Lett., 30, 1402. Okada, Y. (1985). Surface deformation due to shear and tensile faults in a half-space. Bull. Seism. Soc. Am., 75, 1135-1154. Park, J., T. A. Song, J. Tromp, E. Okal, S. Stein, G. Roult, E. Clevede, G. Laske, H. Kanamori, R Davis, J. Berger, C. Braitenberg, M. V. Camp, X. Lei, H. Sun, H. Xu and S. Rosat (2005). Earth's free oscillations excited by the 26 December 2004 Sumatra-Andaman earthquake. Science, 308, 1139-1144. Rosat, S., T. Sato, Y. Imanishi, J. Hinderer, Y. Tamura, H. McQueen and M. Ohashi (2005). High-resolution analysis of the gravest seismic nomal modes after the 2004 Mw = 9 Sumatra earthquake using superconducting gravimeter data. Geophys. Res. Lett., 32, L13304. Satake, K. (2002). Tsunamis. In: International Handbook of Earthquake and Engineering Seismology, 81A, edited by W. H. K. Lee, H. Kanamori, E C. Jennings and C. Kisslinger, 437-451. Sato, T. and Hanada, H. (1984). A program for the computation of oceanic tidal loading effects 'GOTIC'. Publ. Int. Latitu. Mizusawa, 18, 63-82.
Chapter 79
Improved determination of the atmospheric attraction with 3D air density data and its reduction on ground gravity measurements J. Neumeyer, T. Schmidt GeoForschungZentrum Potsdam, Dept. Geodesy and Remote Sensing, Telegrafenberg, 14473 Potsdam, Germany C. Stoeber Institute for Geodesy, Technical University Berlin, Germany Abstract. Ground gravity measurements based on a test mass (relative and absolute gravimeters) are influenced by mass redistribution within the atmosphere which induces gravity variations (atmospheric pressure effect) in ~gal range (about 15 ~gal for the Sutherland superconducting gravimeter (SG) station). These variations are disturbing signals in gravity data and they must be reduced very carefully for detecting weak gravity signals. The atmospheric pressure effect consists of a deformation and a Newtonian attraction term. The deformation term can be modelled well with twodimensional (2D) surface atmospheric pressure data e.g. with the Green's function method. For precise modelling of the Newtonian attraction term threedimensional (3D) data are required. From European Centre For Middle Weather Forecasts (ECMWF) 3D atmospheric pressure, humidity and temperature data are available. These data are used for modelling of the Newtonian attraction term. Two 3D models for the attraction term have been tested based on mass point attraction and gravity potential of the air masses. The results show a surface pressure independent (SPI) part of gravity variations induced by mass redistributions within the atmosphere in the order of some ~gal. In the past, different methods have been developed for modelling of the atmospheric pressure effect. These methods use 2D atmospheric pressure data measured at the Earth's surface and a standard model for the height dependency of the atmospheric pressure. With these models the SPI part couldn't be detected. For different SG sites and one absolute gravimeter location the 3D attraction models have been applied and the SPI part was calculated. Its influence is shown on tidal parameter computation for long periodic tidal waves, SG measured polar motion, comparison of SG with GRACE and hydrology model derived gravity variations. Furthermore it is shown how absolute gravity measurements are
affected by the SPI part. The omission of the SPI part correction in the gravity data leads to a misinterpretation of about 2 pgal. It can be shown that the application of the SPI part increases the precision of the atmospheric pressure reduction on gravity data. Keywords. Atmospheric pressure correction, gravimetry, 3D attraction models, superconducting gravimeter, absolute gravimeter, surface pressure independent gravity variations (SPI).
1 Introduction The redistribution of the air masses induces temporal Earth gravity field variations (atmospheric pressure effect) up to about 20 ~gal. These variations are disturbing signals in the gravity recordings and they must be removed as far as possible for detection of weak gravity effects. In the past different methods have been developed for modelling of the atmospheric pressure effect which generally fall into two categories: empirical and physical approaches. These methods use local or two-dimensional (2D) atmospheric pressure data measured at the Earth's surface and a standard height-dependent air density distribution. The empirical methods, Warburton and Goodkind (1977), Crossley et al. (1995), Neumeyer (1995) use the local atmospheric pressure for determining the single and complex admittance based on regression and cross-spectral analysis. The physical approaches (Merriam 1992, Sun 1995, Kroner 1997, Boy et al. 1998, Kroner and Jentzsch 1998, Neumeyer et al. 1998, Vauterin 1998) based on atmospheric models determine the attraction and deformation terms according to Green' s function (Farrell 1972). The atmospheric pressure effect is composed of the attraction and elastic deformation terms. The deformation term can be well modelled with 2D surface atmospheric pressure data, for instance with
542
Neumeyer J.. T. Schmidt • C. Stoeber
the Green's function method. For modelling of the attraction term, three-dimensional (3D) data are required in order to consider the real density distribution within the atmosphere. Preliminary studies have been done by Hagedoorn et al, (2000). A first approach with data of radio sounding launchings was developed by Simon (2002). From European Centre for Medium-Range Weather Forecasts (ECMWF) 3D atmospheric pressure, humidity and temperature data are available. The ECMWF data are characterised by a spacing of AdO= 0.5 ° and A)~ = 0.5 °, 60 height levels up to about 60 km and an interval of 6 hours. These data have been used for modelling of the atmospheric attraction term by Neumeyer et al. (2004). Two attraction models have been developed based on point mass attraction of the air segments and gravity potential of the air masses. According to Etling (2002), the air density is calculated as a function of height from 6 hourly ECMWF data at coordinates ~b= 32.5 ° and A = 21 ° for a time span of 31 days (July 2003). Fig. 1 shows the vertical changes in density at different heights. The density changes at the same height can reach up to about 0.1 kg/m 3.
mass redistribution within the atmosphere. For calculation the atmosphere is divided into spherical air segments sV. These air segments are assumed as point masses at segment centre positions. The air density p of the air segments is calculated from ECMWF atmospheric pressure, humidity and temperature data. Equation 1 describes the vertical component of the gravitational acceleration gaz~ which acts on the test mass of the gravimeter caused by mass redistributions within the atmosphere.
A,,~,. =
/~'s - rt [sin ~%s sin ~9m(cos A~s cos A. + sin % sin A~) + cos ~9~scos ~9m] 3
(RG,s 2 + rt2 - 2Rvsr~ [sin~%s sin 3,, (coS AGs cosA, + sinAas sinAn) + cos~%s cos~gm])7
7"= Gravitational constant, p - density of the air segment, s V - spherical air volume segment, R~s = Earth radius at the gravimeter station, r - R~s + height of the spherical air segment sV, ORs - colatitude of the gravimeter site, )~Rs- longitude of the gravimeter site, 0 and 2 - coordinates of s V .
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2 3D-models for the attraction term 2.1 Point mass model
The model (Neumeyer et al. 2004), based on the law of gravitation, calculates the gravitational acceleration at the gravimeter test mass caused by
The second attraction model has been derived from the gravitational potential of the air masses. For mathematical reasons a coordinate system is introduced in such a way that the north pole of this system coincides with the gravimeter site (GS). The gravitational potential of the air masses is calculated with respect to the gravimeter site By partial derivation of the gravitational potential of air masses and summation over all spherical air segments we obtain the vertical component of the gravitational acceleration ga;r(Rcs, 0, 0) (Eq. 2) caused by mass redistributions within the atmosphere. More details can be found in Neumeyer et al. (2004). For calculating of g a i r ( R a s , O , O ) a coordinate transformation of the air density data into the gravimeter global coordinate system (X, Y, Z) is necessary for the reason of the virtual gravimeter site location at the Z-axis (pole) of this system. The coordinate transformation is performed by two rotations according to Torge (2003). Because of the wide spacing (0.5 °) the ECMWF data must be interpolated, which is accomplished by a two-dimensional (0 and 2) bi-linear interpolation. The amount of interpolated points of 10 x 10 can be regarded as adequate.
Chapter 79 • Improved Determination of the Atmospheric Attraction with 3D Air Density Data and Its Reduction on Ground Gravity Measurements
gair(RGs, O, O) -- - - y Z t91 .... (/~1 --/El-l) 1 Bmn l .... 6R2s '
(2)
-4r2_1 + R~s - 2rm_,RGscos,9- , •(2r2_,- R~s + 2R~srm_lCOS~9,,_1+ 3R~s cos 2,9,,_,
-4r2m + R~s-ZrmRGsC°SO. 1 "(2rZ-R~s + 2Rcsrm c°sO. 1 + 3R~s c°s2'9,, 1)
-4r2_, + R~s-Zrm_,RGsCOSO. • (2r2_, - P~s +
2RGsrm_lCOSO n +
3R~s cos20.)
+4rm2 + R~s - 2rmRvscos#.-(2rm2 - R~s + 2Rvsrmcos ~9,,+ 3R~s cos 2~9.)
Bm.n= -lrl[rml-RGscosO n l+4r21+R~s-2rmlRGsCOSOn II6RGsCOSOn 1sin2tgn 1 +ln[r m - RGSCOS~
1-t- 4 4 -t- R~s-
2rmI~s cos,9.116P~s cos,9., sin2 tgn 1
+ln[rm 1- R~s cos0. + 4 4 1 + R2s- 2r iR~s cos,9 16R~s cos,9 sin2 ~9
-l.[r m- RGsc o s a + 4 4 + RZs - 2rmRvscosO. ]6R3s cos a sin2 ~9.
7gravitational constant, p air density as function of the spherical coordinates (r, O, 2 ) , Rcs = Earth radius at gravimeter's site.
Green's function model using 3D-ECMWF atmospheric pressure, humidity and temperature data.
4.1 Attraction term The attraction term was calculated with the GFZ program 3DAP based on point or potential model, Stoeber (2005). Both models deliver the same results within small error bars of about +0.1%. The application of the 3D attraction model shows a seasonal surface atmospheric pressure independent part (SPI) of the attraction term that is caused by the movement of the air masses without changes of the surface atmospheric pressure. In the summer season the air masses move up and the atmospheric attraction term decreases whereas in winter the air masses move down, causing a larger attraction term. Fig. 2 illustrates this fact.
3. Model for the deformation term 2
This model is based on the calculation of gravity changes caused by a point load on the Earth's surface using the appropriate Green's functions, The Green's functions for atmospheric loading have been calculated and tabulated by Merriam (1992) and Sun (1995). They used a column load of the COSPAR standard atmosphere for mid-latitudes and calculated the atmospheric pressure admittance coefficients as functions of the angular distance between the footprint centre of the column load and the gravimeter site. For calculation of the deformation term the Green's function delivers appropriate results by using 2D surface atmospheric pressure data according to equation 2, Sun (1995).
la
a horizontal plane Earth s u r f a c e
Fig. 2 Mass relocation within the atmosphere
g (~/)._ GE(w)r dr2 105~ 2~c[l_cos(lO)] ~gal/hPa
(3)
With G E ( q / ) r - tabulated temperature corrected Green's function for the elastic deformation term, = angular distance from gravimeter to the air column and d.(2 - footprint element of the air column.
Calculation of the atmospheric pressure induced gravity variations For different superconducting gravimeter sites atmospheric attraction term was calculated with point mass and potential model whereas deformation term has been determined with
the the the the
The gravity effect caused by mass relocation is different above and to the side of the gravimeter. When we assume a mass relocation from point 1 to point 2 above the gravimeter, the attraction term becomes smaller (gair_2 < gait_l). When we assume a mass relocation from point l a to point 2a to the side of the gravimeter the vertical component of the attraction term, which acts on the gravimeter test mass, becomes larger (g~r_2a > g~r_la). This effect is height dependent. With local or 2D data we can not detect this effect. For demonstration of this effect the attraction term (Sg~r_attr) is calculated for different air columns above and beside the gravimeter with ECMWF data from January to December 2001 for station Moxa. The results are displayed in Fig. 3.
543
544
Neumeyer J.. T. Schmidt • C. S t o e b e r
,..,
6
4 ~ Jt~ ~1~ I
A.q,~0.;°,'O o.~.oi~,
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t.<-,~ o 31[~ l,ll Nllll ~1 Iltlilt~l~&l~._~gt~llll •~,'
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I j
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j
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rl,I~l 1 I~I1'1,,~r,l,w,.i
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j
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j
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1
i
i
l
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'~"1~''' ....
llJlr''lI"I''l
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-5~ Sutherland / South Africa
I
.~o "
~"
"
"
~5"
I
"
I ~5'o"
I ~5'o"
~"
~Su
1.'o" "/.~o"
Fig. 5 3D attraction term (grey) at Vienna and Sutherland stations with fitted annual wave (black) The Vienna station (d0= 48.249 °, )~ = 16.358 ° h = 192.4 m) in the northern hemisphere has a larger attraction in winter and smaller in summer. An opposite trend is seen at Sutherland station (q~ = 32.381 o, )~ = 20.811 o, h = 1791 m) in the southern hemisphere.
4.2 Surface term (SPI)
l !l
/
Vienna[Auslria
1. °
j
The attraction term induced by a small air column ( ~ = 0 ° to 0.1 o) around the gravimeter has a strong seasonal behaviour (grey curve). The gravity effect is smaller in summer and larger in winter. For a "hollow cone" like column beside the gravimeter ( ~ = 0.1 ° to 0.2 °) the effect is reversed (black curve), in summer larger than in winter. This continues with decreasing amplitude as, for example, for u / = 0.4 ° to 0.5 ° (Fig. 3, dark grey c u r v e ) .
j
I
10--
Fig. 3 3D attraction term (Agair_attr_cy) above A~=0 ° to 0.1 ° (grey) and beside A~=0.1°to 0.2 ° (black), A~=0.4 ° to 0.5 ° (dark grey) the gravimeter
-12
J
tL!;
j
.~,
5
Ill
pressure
independent
attraction
Different density distributions in the atmosphere at the same surface pressure cause different gravity attractions. From the 3D attraction term (3Dattr) calculated for SG station Metsahovi/Finland (dO = 60.217 °, )~ = 24.396 °, h = 56 m) only the gravity values were selected that have the same surface pressure, in this example (1001 + 0.25) hPa (Fig. 6).
oo,-~',o~-~'~o@' 2-
Fig. 4 3D attraction term (Agair_attr) for ~t=0.1° (grey) and ~t=l.5 ° (black)
1.5- " =~'
i::"
° ~- ':tl I
N o w we take the cumulative air masses around the gravimeter and summarise the previous curves (Fig. 4). For ~ = 0.1 o we see the strong seasonal effect. The difference between summer and winter seasons becomes smaller with increasing ~ (e.g. for ~ = 0.5°). This continues up to acquisition of the entire amount at ~ = 1.5 °. For a larger angular distance, > 1.5 °, the effect is small and not considered. Therefore, the following calculations were carried out with ~ = 1.5°. For comparison of the different seasonal trends in the northern and southern hemisphere an annual wave is fitted to the attraction term calculated with E C M W F data form January 2001 to December 2003 (Fig. 5).
".l., •
'
o
o5-
o-
•
. e
.
"" ~'.-,J
~'*'.
,.
•
J • ..t
"::
iil
, :
~"
.,
•
.
" .
...:•
. .:
.
"
..
• 1
'
-0.5 -
~.~ ,~.~ ,~.~ ...~ ^.~q. ,~.~q.,~.~m...~q. _~ ,~.~ ,\~ ~..~ ^~ Fig. 6 Surface pressure independent part (SPI) of the attraction term at surface pressure of (1001 +0.25) hPa (Metsahovi/Finland)
This signal is the surface pressure independent part SPI of the attraction term for the selected surface pressure. It includes both the seasonal behaviour and gravity changes at shorter periods.
Chapter 79 • Improved Determination of the Atmospheric Attraction with 3D Air Density Data and Its Reduction on Ground Gravity Measurements 1-
0.5-
I
i
I
J
,J
-0.5--
-1-
1t'[1
'"~
'
' '
-1.5 -
Fig. 7 Surface pressure independent part (Metsahovi/Finland) as the difference of sadm and 3Dattr atmospheric pressure corrections.
The total SPI part was separated by subtraction of the single admittance attraction term (single admittance coefficient -4.31agal/hPa) from the 3D attraction term calculated with 3 D - E C M W F data. The result is shown in Fig. 7 for the SG station Metsahovi. The total effect is about 2 lagal.
gravity variations of different sources, the atmospheric pressure induced gravity variations were removed. Two reduction methods were applied: a) 3Dmodel for the attraction term using 3 D - E C M W F data in combination with the Green's function model for the deformation term using 2 D - E C M W F data. The results are summarised to 6g apsD; b) single admittance method (sadm) using local atmospheric pressure data. It calculates the total atmospheric pressure effect 6g_ap~adm.
Table 1. Tidal analysis of long periodic waves Tidal wave ME Ampl.
4.3 Deformation term With the GFZ program 2DAP based on equation (3) the deformation term has been calculated using 2D E C M W F surface atmospheric pressure data applied to a data grid of about 10 ° around the gravimeter. The influence of larger angular distances can be neglected.
5 Applications The importance of the SPI part correction on ground gravity measurements has been investigated by analysing different gravity effects. With reduction of the SPI part the gravity signal becomes smaller in summer and larger in winter compared to the reduction based on local or 2D atmospheric pressure data. The total effect can reach up to 2 ggal.
5.1 Long-periodic tidal waves In a pre-processing procedure, spikes larger than 0.2 ggal and steps that do not have their origin in atmosphere or groundwater level induced gravity variations like instrumental (liquid helium transfers or lightening strikes) and other perturbations such as earthquakes are carefully removed from the raw superconducting gravimeter recordings at stations Moxa/Germany (~) = 50.645 °, )~ = 11.616 °, h = 455 m), Metsahovi/Finland and Sutherland/South Africa. Then, the data are low-pass filtered with a zero phase shift filter (corner period 300 sec) and reduced to 1 hour sampling rate. From these preprocessed gravity data (6graw), which include
MO
SU
theor 8WD 8osadm 8o3D ~:osadm ~:°3D Ampl. theor 8oWD 8°sadm 8°3D ~csadm ~:o3D Ampl. theor 8°WD 8osadm 8°3D ~:osadm Ko3D
SA 33.91
SSA 37.62
Mm 42.72
Mf 80.87
1.161 1.408 1.544 -9.04 -9.05 21.31
1.161 1.276 1.286 0.89 -0.32 23.64
1.161 1.177 1.186 0.47 -0.55 26.84
1.161 1.159 1.161 -0.10 -0.05 50.81
1.165 1.074 1.398 -9.19 -9.91 3.89
1.165 1.143 1.162 -0.22 0.41 4.31
1.165 1.174 1.174 -0.09 0.70 5.45
1.165 1.161 1.160 -0.35 -0.15 9.27
1.115 0.752 1.6439 -12.434 -12.80
1.115 0.995 0.894 4.06 14.92
1.073 1.063 1.126 2.94 2.96
1.115 1.142 1.134 0.66 0.06
The attraction term was calculated with the program 3DAP and the deformation term with 2DAP. The single admittance coefficient between the local atmospheric pressure and the gravity variations was determined by regression analysis. Both reductions (6g_aps D and 6g_aP,adm) were compared for analysing the longperiodic tidal waves using the E T E R N A analysis program, Wenzel 1996. Table 1 shows the theoretical amplitude, the WahrDehant-model amplitude factor 8, Dehant (1987), and the analysed tidal parameters amplitude factor 8 and phase v: for the longperiodic tidal waves SA, SSA, M m and Mf. For the annual tidal wave SA the amplitude factor 8 is larger with 3D atmospheric pressure correction compared to single admittance (local atmospheric pressure) correction, whereas the semi-annual waves SSA and M m or Mf show nearly no differences.
545
546
NeumeyerJ.. T.Schmidt.C.Stoeber 5.2 Polar motion Polar motion causes changes in centrifugal acceleration, which can be measured with the SG. In order to separate polar motion from the preprocessed gravity data dgr~~ the following quantities were subtracted (Eq. 4): -
-
-
-
Analysed Earth tides (ET) calculated with the ETERNA program based on Wahr-Dehantmodel Ocean loading calculated with the GFZ program OCLO, Dierks (2004), based on Francies and Mazzega (1990). OCLO calculates the gravity variations induced by the ocean loading (dgo) in the time domain using data from the ocean tidal model FES2002 (Lefevre et al. 2002, Le Provost et al. 2002). Local groundwater level effect ( 6 g ~ ) based on a single admittance coefficient between local ground water level changes and gravity. Instrumental drift (dr) based on polar motion measured by SG and calculated from IERS data (PT). It is simulated by a first-order polynomial d r ( t ) = a o + alt and the drift parameters a 0 and
For comparison, the gravity effect of the polar motion was calculated for the Sutherland station with IERS (International Earth Rotation Service) polar motion data xp(t) and yp(t) (Bulletin B) (black curve) according to Torge (1989). The 3D attraction reduction 6g_ap3 D fits better to the IERS polar motion than the single admittance reduction 6g_ap ~d~.
5.3 Comparison of hydrology models variations
SG, G R A C E and derived gravity
A further consideration of the SPI part is shown by comparison of SG measurements with GRACE satellite (GFZ Potsdam solutions) and the global
"0
c
0 SG._3D = SG._sadm ~ . - ~ - - ~ GRACE (GF-Z)Imax=10 ~ VVGHMmodel Imax=10
6--
a~ are determined by a linear fit of d gsopo~ -
and PT. Atmospheric pressure 6g_ap~,~ respectively 6gso eo,(t)
=
effect
dg_ap~
6gr~(t)- ET(t)-6g_ap(t)
_
and
(4)
- 6 go~(t) - 6 gg~z(t) - dr(t)
The polar motion due to equation (4) measured at Sutherland station is shown in Fig. 8. The two different atmospheric pressure reductions have been applied, in dark grey 6g_ap~D and in grey 6g_aps,~ ~.
6-
-4--
Fig. 9 Comparison of superconducting gravimeter (SG_3D and SG_sadm), GRACE and hydrological model (WGHM) derived gravity variations hydrological model WGHM, D611 et al. (2003), derived gravity variations (Fig. 9). Details about GRACE and hydrology model data processing can be found in Neumeyer et al (2005). The SG gravity variations are calculated due to equation (5). 6gso(t ) = 6graw(t ) -- E T ( t ) - P T ( t ) - 6 g _ a p ( t )
(5)
-6go, U) - 6 gg~,(t) - dr(t)
-6-
Fig. 8 Polar motion (Sutherland/South Africa) with 6g apsn (dark grey) and 6g apsadm (grey) atmospheric pressure reduction in comparison with IERS polar motion (black)
The two different atmospheric pressure reductions 6g_ap3 ~ (SG_3D) and 6g_apsad m (SG_sadm) have been applied (Fig. 9). The SG gravity variations (SG_3D) with 6g_ap3 ~ atmospheric pressure reduction are closer to the hydrological model and also closer to GRACE derived gravity variations at Metsahovi station compared to the single admittance reduction with cYg_apsad~ (SG_sadm). We clearly see the difference of both gravity signals of about +1 lagal between winter and summer season.
Chapter79 • Improved Determination of the AtmosphericAttractionwith 3D Air Density Data and Its Reductionon GroundGravityMeasurements 5.4 A b s o l u t e gravity m e a s u r e m e n t s
Another example shows the influence of the SPI part on absolute gravity measurements. At station Pecny/Czechia gravity measurements have been carried out frequently with the absolute gravimeter FG5 from August 2001 to August 2003. Both atmospheric pressure correction methods (6g_ap~ D and 6g aP~adm)have been applied on these gravity measurements. Fig. 10 shows the results, single admittance correction with local atmospheric pressure (FG5_sadm) and 3D reduction with ECMWF data (FG5_3D). In winter season the li ~ ~ FG5_sadm 7 o FG5._3D I 6--
FG5_3D_Hy~
-4-
N1-
11"
~ 0 ~
~.
'~
-1-
Fig. 10 3D (FG5_3D) and sadm (FG5_sadm) atmospheric pressure reduction on absolute gravity measurements (Pecny/Czechia) and subtraction of WGHM model derived gravity variations (FG5_3D_Hy)
single admittance method subtracts to less compared to the 3D reduction. The difference is in total about 2Mgal. For interpretation of the absolute gravity measurements this fact must be considered. When we apply the gravity correction according to the hydrological model WGHM (Neumeyer et al. 2005), the oscillation of the signal is reduced to about _+2 Mgal. This is close to the accuracy of the absolute gravimeter.
6 Conclusions
The mass redistribution within the atmosphere induces a gravitational attraction on the test mass of the gravimeter which is independent from surface atmospheric pressure. The gravity recordings contain this attraction (SPI) that has a magnitude of about 2 Mgal. It can only be detected with 3D atmospheric pressure, humidity and temperature data. These data are globally available at European Centre for Middle Weather Forecasts (ECMWF).
The SPI part influences the metering precision of longperiodic gravity variations (Earth tides, polar motion, hydrology signal measured with e.g. SG's) and the absolute gravity measurements. Disregarding the SPI part can introduce a reduction error of up to a few Mgal. The quality of the attraction term computation depends first of all on the quality of the 3D atmospheric pressure data. Unfortunately the spacing (0.5 °) and the interval (6 hours) of the present 3D-ECMWF data is inadequate for precise calculation. A spacing of about 0.1 o and an interval of 1 hour or shorter is required. Consequently, higher precision can be achieved at the moment only by interpolation. For determination of the attraction term a data grid of 5 ° around the gravimeter with 60 height levels (up to 60 km) is sufficient. The deformation term should be determined with a data grid of about 10 ° around the gravimeter. The deformation effect of the air masses redistribution can be determined adequately with 2D surface pressure data using the Green's function method With the present quality of the 3D atmospheric pressure data seasonal gravity atmospheric pressure corrections on gravity data can be performed more precisely compared to corrections with local or 2D data. For the future, a further improvement is anticipated using pressure data with higher resolution in space and at shorter time intervals. For improving the attraction program 3DAP the topography around the station should be considered in more detail.
Acknowledgments
We like to thank Jan Kostelecky Department of Advanced Geodesy, Faculty of Civil Engineering CTU Prague, Czechia; Corinna Kroner Institute of Geosciences Friedrich Schiller University of Jena; Germany; Bruno Meurers Institute of Meteorology and Geophysics University of Vienna, Austria; Heiki Virtanen Finnish Geodetic Institute Masala; Finland; Herbert Wilmes Federal Agency for Cartography and Geodesy (BKG), Germany for providing the gravity data.
References
Boy J.-P., Hinderer J., Gegout P. (1998) The effect of atmospheric loading on gravity. Proc. of the 13t" Int. Symp. on Earth Tides, Brussels, 1997, Eds.: B. Ducarme and P. Paquet, 439-440.
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Bulletin B, http://www.iers.org/iers/products/eop/ monthly.html Crossley D. J., Jensen O. G., Hinderer J. (1995) Effective barometric admittance and gravity residuals. Phys. Earth Planet. Int., 90, 221-241. Dierks, O., 2004. Ocean loading program OCLO. Internal report GFZ Potsdam. D611 P, Kaspar F, Lehner B (2003) A global hydrological model for deriving water availability indicators: model tuning and validation. J. Hydrol. 270:105-134. Etling D. (2002) Theoretische Meteorologie: Eine Einftihrung, Springer Verlag Farrell W. E. (1972) Deformation of the Earth by surface loads. Rev. Geophys. Space Phys., 10, 761-797. Francis, O., Mazzega, P., (1990). Global charts of ocean tide loading effects. J. Geophys. Res., 95, 11411-11424 Hagedoom J., Wolf D., Neumeyer J. (2000) Modellierung von atmosph~irischen Einfltissen auf hochgenaue Schweremessungen mit Hilfe elastischer Erdmodelle. Scient. Technical Report GFZ Potsdam STR 00/15. Kroner C. (1997) Reduktion von Luftdruckeffekten in zeitabh~ingigen Schwerebeobachtungen, Dissertation, Technische Universit~it Clausthal Kroner C., Jentzsch G. (1998) Methods of air pressure reduction tested on Potsdam station. Marees Terrestres Bulletin d'Informationes, Bruxelles, 127, 9834-9842. Lefevre F., Lyard F. H., Le Provost C., Schrama E. J. O., 2002. FES99: a global tide finite element solution assimilating tide gauge and altimetric information. J. Atmos. Oceanic Technol., Vol. 19, 1345-1356. Le Provost C, Lyard F, Lefevre F, Roblou L (2002) FES 2002 - A new version of the FES tidal solution series. Abstract Volume Jason-1 Science Working Team Meeting, Biarritz, France Merriam J. B. (1992) Atmospheric pressure and gravity. Geophys. J. Int., 109, 488-500. Neumeyer J. (1995) Frequency dependent atmospheric pressure correction on gravity variations by means of cross spectral analysis. Marees Ten'estres Bulletin d'Informationes, Bruxelles, 122, 9212-9220.
Neumeyer J., Barthelmes F., Wolf D. (1998) Atmospheric Pressure Correction for Gravity Data Using Different Methods. Proc. of the 13 th Int. Symp. on Earth Tides, Brussels, 1997, Eds.: B. Ducarme and P. Paquet, 431-438. Neumeyer J.,. Barthelmes F., Dierks O., Flechtner F., Harnisch M., Harnisch G.,. Hinderer J.,. Imanishi Y., Kroner C.,. Meurers B., Petrovic S., Reigber Ch., Schmidt R.,. Schwintzer P., Sun H.-P., Virtanen H. (2005) Combination of temporal gravity variations resulting from Superconducting Gravimeter recordings, GRACE satellite observations and global hydrology models. J. Geodes., (accepted). Neumeyer, J., Hagedom, J., Leitloff, J., Schmidt, T.,(2004) Gravity reduction with three-dimensional atmospheric pressure data for precise ground gravity measurements. J.Geodyn. 38,437-450.. Sun H.-P. (1995) Static deformation and gravity changes at the Earth's surface due to the atmospheric pressure. Observatoire Royal des Belgique, Serie Geophysique Hors-Serie, Bruxelles. Simon D. (2002) Modelling of the field of gravity variations induced by seasonal air mass warming 1990-2000. Marees Ten'estres Bulletin d'Informationes, Bruxelles, 136, 1082110836. Stoeber C. (2005) Modellierung und Analyse des Einflusses der 3D Luftdruckkorrektur in Supraleitgravimeter Registrierungen auf die langperiodischen Gezeitenparameter. Diplomarbeit, Institut for Geod~isie und Geoinformationstechnik Technische Universit~it Berlin (unpublished) Torge W. (2003) Geodesy, Walter de Gmyter, Berlin New York Vauterin P. (1998) The correction of the pressure effect for the Superconducting Gravimeter in Membach (Belgium). Proc. of the 13 th Int. Symp. on Earth Tides, Brussels, 1997, Eds.: B. Ducarme and P. Paquet, 447-454 Warburton R. J., Goodkind J. M. (1977) The influence of barometric - pressure variations on gravity. Geophys. J. R. Astr. Soc., 48, 281-292. Wenzel H.-G. (1995) Tidal data processing on a PC. In Hsu H.T. (ed), Proceedings of the 12~hInternational Symposium on Earth Tides, August 4 - 7, 1993, Science Press Beijing China, 235-244.
Chapter 80
Solid Earth Deformations Induced by the Sumatra Earthquakes of 2004-2005: GPS Detection of Co-Seismic Displacements and Tsunami-Induced Loading H.-P. Plag, G. Blewitt, C. Kreemer, W.C. Hammond Nevada Bureau of Mines and Geology and Seismological Laboratory, University of Nevada, Reno, Mailstop 178, Reno, NV 89557, USA.
Abstract The two great Sumatra earthquakes of December 26, 2004, and March 28, 2005, are associated with a number of geodynamic processes affecting Earth's shape, gravity field and rotation on a wide range of spatial and temporal scales, including co-seismic strong motion, static co-seismic displacements, surface loading due to the tsunami, normal modes, and transient postseismic deformations. The December 26, 2004, earthquake is the first one with Mw > 9.0 observed with space-geodetic techniques. Thus, new insight into the earthquake processes and associated phenomena can be expected from studies of space-geodetic, and particularly GPS, observations. Here we focus on the static co-seismic offsets and the tsunami loading signal. The global patterns of the static co-seismic offsets induced by the earthquakes are determined from time series of daily displacements resulting from an analysis of the GPS data from selected stations of the global IGS network. Our estimates are compared to similar estimates computed in other studies. The differences between the independent estimates are mostly within the uncertainties of the offsets (2 to 4 mm). The global pattern of the computed offsets is in agreement with the spatial fingerprint predicted by a reasonable rupture model. Initial model predictions of the tsunami-induced loading signal show that the peak amplitudes are of the order of 10 to 20 mm. If detectable in GPS time series with low latency, the loading signal could be utilized in a tsunami early warning system. Keywords: earthquakes, coseismic displacements, tsunami loading.
1
Introduction
Great earthquakes like the Sumatra events of 26 December 2004 (Mw = 9.2, denoted as "event A") and 28 March 2005 (Mw = 8.7, denoted as "event B") are associated with a number of geodynamic phenomena on a wide range of spatial and temporal scales including co-seismic strong motion and static displacements, free oscillations of the solid Earth, and, if located in an oceanic region, tsunamis. The redistri-
bution of water mass in the ocean associated with the tsunami induces transient perturbations of the Earth's surface and gravity field. Moreover, post-seismic deformations can continue for months and years after such great earthquakes. Event A was the first earthquake of Mw > 9.0 to be observed with space-geodetic techniques, in particular the global network of tracking stations for the Global Positioning System (GPS), that is coordinated by the International GNSS Service (IGS). The expected new insight from studies of the co- and post-seismic displacements into the rupture process of the earthquake and the associated phenomena has stimulated a number of GPS-based studies (e.g. Khan & Gudmundson, 2005; Banerjee et al., 2005; Vigny et al., 2005; Kreemer et al., 2005). GPS observations potentially provide information on the co-seismic displacements, including the static co-seismic offsets, post-seismic non-linear displacements, and tsunami induced loading signals. In order to determine the static offset with high accuracy, time series of daily or sub-daily coordinate estimates are required, and these time series also contain the postseismic signal. For co-seismic motion and tsunamiinduced loading, time series with high temporal resolution down to 30 sec or better are needed. The static co-seismic offsets are valuable constraints for models of the rupture processes. Moreover, the magnitudes of earthquakes determined from initial broadband estimates tend to be too low for large earthquakes (Menke, 2005), compromising early warning efforts (Kerr, 2005). Therefore, if available in near-real time, GPS estimates of the static offsets could help to improve the initial magnitude estimates for large earthquakes. Tsunamis travel the ocean as barotropic waves and thus induce a loading signal. If detectable by GPS with low latency, these signals could be integrated in an early warning system. Here, we first consider the static co-seismic offsets determined from GPS observations and, by comparing our own estimates to those computed by others, assess the accuracy to which these offsets are determined. Then we will briefly comment on the post-
550
H.-P. Plag. G. Blewitt • C. Kreemer. W. C. Hammond
Table 1° Selected GPS stations used in this study and their nominal distances (D) from the rupture zone. Not all farfield stations are shown. No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Stat. samp ntus coco bako iisc mald hyde kunm dgar lhas pimo karr wuhn tnml seyl darw pert kit3 reun alic bahr guam lael tskb mali tow2 mbar hrao
D km
Long. o
Lat. o
498 1064 1722 1757 2142 2304 2340 2605 2688 2973 3208 3654 3711 3744 4394 4440 4502 4870 5003 5220 5313 5677 5953 5956 6047 6281 7072 7841
98.72 103.68 96.83 106.85 77.57 73.53 78.55 102.80 72.37 91.10 121.08 117.10 114.36 120.99 55.48 131.13 115.89 66.89 55.57 133.89 50.61 144.87 146.99 140.09 40.19 147.06 30.74 27.69
3.61 1.35 -12.19 -6.49 13.02 4.19 17.42 25.03 -7.27 29.66 14.64 -20.98 30.53 24.80 -4.67 - 12.84 -31.80 39.14 -21.21 -23.67 26.21 13.59 -6.67 36.11 -3.00 -19.27 -0.60 -25.89
seismic displacements observed at regional stations after the two events. The surface loading caused by the tsunami is studied on the basis of a model prediction of the sea surface heights variations caused by the tsunami. We use the well validated Green's function approach to estimate the solid Earth deformations induced by the tsunami on the basis of modeled sea surface displacements. Finally, we will consider the necessary improvements to the observing system and the analysis strategy in order to fully utilize the potential of GPS and other Global Navigation Satellite Systems for studies of large earthquakes and early warning systems.
2
Observed co-seismic displacements
surface
In order to determine the static co-seismic offset for the two events, GPS data from a total of 39 stations within 7,600 km of the rupture zone (Table 1) were processed for the interval 1 January 2000 to 21 May 2005 using the GIPSY-OASIS II software package from the Jet Propulsion Laboratory (JPL). Daily station coordinates were estimated using the precise point posi-
tioning method (Zumberge et al., 1997) with ambiguity resolution applied successfully across the entire network by automatic selection of the ionospheric- or pseudorange-widelane method (Blewitt, 1989). Satellite orbit and clock parameters, and daily coordinate transformation parameters into ITRF2000 were obtained from JPL. Ionosphere-free combinations of carrier phase and pseudorange were processed every 5 minutes. Estimated parameters included a tropospheric zenith bias and two gradient parameters estimated as random-walk processes, and station clocks estimated as a white-noise process. For all stations, secular velocities where estimated for the interval 1 January 2000 to 25 December 2004. These velocities are assumed to represent the interseismic tectonic motions prior to event A, and they are used to detrend the data. Moreover, using the stations more than 4000 km away from the epicenter as reference stations, all daily solutions were then transformed by a 7-parameter Helmert transformation onto the constant velocity solution (denoted as "spatial filtering"). In Fig. 1, the detrended time series for four stations close to the rupture zones are shown for the interval 1 January 2004 to 21 May 2005. Particularly the sites SAMP and NTUS show large offsets at the time of the two events and significant post-seismic deformations in the months following each of the events. For sites further away, the offsets appear to be hidden in the long-period variations present in the time series. As illustrated by these examples, the determination of the static co-seismic offsets is hampered by the presence of noise and other signals in the time series. In particular, it is difficult to estimate reliable errors. An early analysis of the time series of SAMP and NTUS for event A was presented by Khan & Gudmundson (2005). They used 5 day averages of the north and east displacements to estimate simultaneously a linear trend and the offsets at the time of event A. Their static offsets are (given in mm for east, north) ( - 1 4 5 . 2 + 3.2, - 1 2 . 1 + 1.8) and ( - 2 2 . 0 + 2.0, 6.1 -+- 1.6) for SAMP and NTUS, respectively. Banerjee et al. (2005) analysed the data of 41 farfield and regional stations using the GAMIT/GLOBK software. Static offsets for event A were then determined by differencing the mean positions in five days before and after the event A. They claim to see a coherent surface motion for distances up to 4500 km from the epicenter. Vigny et al. (2005) computed static co-seismic offsets from time series of 79 regional and global CGPS sites. These time series were determined using the GIPSY-OASIS ii software in precise point positioning mode. The daily solutions for 14 days before and after
Chapter80 • Solid Earth DeformationsInducedby the SumatraEarthquakesof 2004-2005 80
80
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Fig. 1: Detrended displacement time series for permanent GPS sites close to the rupture zone. For each station, the vertical (RA), north (LA) and east (LO) component of the displacement vector are shown. Lower left diagrams are for SAME upper left for NTUS, lower right for IISC, and upper right for HYDE. Vertical lines indicate the times of the two events.
event A were combined into campaign-like solutions, which were then projected onto ITRF. The offsets were computed as the differences of the 14-day averages after and prior to event A. Kreemer et al. (2005) using the same time series as in the present study (see above), computed the coseismic displacements by differencing the average coordinates from 28 days after and before the event. While increasing the span of days before and after the event reduces the noise in the computed average, there is a trade-off with bias from post-seismic displacement, at least for the two stations in the near field of the event (SAMP and NTUS). Therefore, for these two stations, Kreemer et al. (2005) estimated a logarithmic function describing the post-seismic displacements simultaneously with the static offsets. The alternative approach chosen here is to model the time series x by the function NH
Nc
x(t) - a + bt + E ~iH(ti) + E Aj sin(wit + Cj) i=1
j=l
(1) where t is time, a is a constant, b a constant rate, and l l is a Heaviside function, with ai giving the displacements associate with event i. We have chosen NH = 2 with the times tl and t2 coinciding with the events A and B, respectively. The harmonic constituents are used to represent a seasonal cycle in the GPS time
series, and we have chosen to include an annual and semi-annual constituent (i.e. N c = 2). In the fit, we solve for a, b, Oil, OL2 and the amplitudes of the cosine and sine terms of the harmonic constituents. The data prior to approximately December 2002 was found to be of lower quality. Therefore, the data interval for the fit ofEq. 1 to the displacement time series was constrained to 1 January 2003 to 21 May 2005. Examples of the resulting models and the residuals for the stations included in Fig. 1 illustrate that the model function is appropriate for most stations not having significant post-seismic motion (Fig. 2). Particularly for SAMP, the large post-seismic displacement after event B biases the estimated offset for this event if the postseismic displacement is not modeled properly. For stations with significant post-seismic deformations, the approach by Kreemer et al. (2005) is appropriate. The resulting offset parameters are listed in Table 1. The co-seismic offsets estimated by Banerjee et al. (2005), Vigny et al. (2005) and Kreemer et al. (2005) are also given for comparison. For most stations, the estimates differ by several mm and the error bars do not overlap for all stations. This indicates that the determination of the offsets depends on the GPS processing strategy and the model used to approximate the time series prior and after the offsets. However, there is no clear systematic difference between the four sets (Fig. 3).
551
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Fig. 2" Determination of co-seismic static offsets using eq. (1). Diagrams marked with O show the observations and the model function, while those marked/ig show the residuals, that is the difference between the observations and the fitted model function.
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Fig. 3: Comparison of the computed offsets. For each station, the deviation from the mean of all estimates are shown for the east (lower) and north offsets. Triangles: Banerjee et al. (2005), inverted triangles: Vigny et al. (2005), squares: Kreemer et al. (2005), diamonds: this study. Kreemer et al. (2005) computed predicted static offsets using a layered elastic spherical Earth model (Pollitz, 1996) and a rupture model that best fits their static offsets computed from the GPS series. The predicted displacement field is a dipole with the main motion on both sides of the fracture being directed towards the rupture zone (Fig. 4). Surprisingly large displacements of the order of 5 mm are predicted for areas
as far away as South Africa and South America. The offsets computed in the various studies are in general agreement with the predictions of the model derived by Kreemer et al. (2005) (Fig. 5). Despite the fact that the model is a best fit to the Kreemer et al. (2005) offsets, no systematic differences are detected between the predictions and any of the four sets of estimates. Deviations are generally of the order of a few mm. The estimates for SAMP and NTUS computed by Khan & Gudmundson (2005) fall well into the range covered by the other four estimates. The largest differences are found for SAME where the east offset derived here is biased by the unaccounted post-seismic deformations. For the stations being further than 4000 km away from the rupture zone, the offsets derived here are generally smaller than those determined by Kreemer et al. (2005). Since these offsets should be close to zero, this may be taken as an indication that using a model function provides slightly better estimates. However, in order to prove this, a more detailed comparison of the observed spatial fingerprint to the predicted fingerprint would be required. It is emphasized here that Kreemer et al. (2005) used identical time series for the determination of the static offsets as the present study but a different ap-
Chapter 80 • Solid Earth Deformations Induced by the Sumatra Earthquakes of 2004-2005
T a b l e 2" Comparison o f static co-seismic offsets determined b y four independent analyses.
Stat. samp ntus COCO
bako iisc mald hyde kunm dgar lhas pimo karr wuhn tnml seyl darw pert kit3 reun alic bahr guam lael tskb mali tow2 mbar hrao
Banerjee et al., 2005 (~E (~N OE ON mm mm mm mm -135.0 -14.8 6.0 2.2 -13.8 2.4 3.0 1.6 3.7 1.1 3.4 1.8 0.9 -3.7 3.6 1.8 14.9 -1.4 2.7 1.5 9.9 -2.8 11.5 1.0 -10.8 -1.8 -3.8 -9.2 -1.6 -2.5 1.8 1.6
-2.7 -8.5 2.2 -4.2 -4.2 1.5 -4.5 -2.1 0.3 -0.6 1.0 -1.5
-1.5
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2.6 3.5 3.8 5.2 4.0 3.8 3.5 3.4 7.7 4.5 1.4 1.8
6.6
1.5 1.8 2.1 2.6 2.1 2.1 2.0 2.0 4.3 2.3 1.3 1.1
Vigny et al., (~E (~N mm mm -142.8 -12.7 -19.4 6.4 2.2 5.0 -1.7 1.0 11.7 -0.1
2005 O'E mm 3.6 2.6 2.5 4.1 2.7
O'N mm 2.0 1.3 2.4 1.9 2.4
7.0 -6.4 5.7 1.7 -7.1 0.9 -2.0
-0.8 -6.9 5.3 -1.3 -1.3 1.1 -2.9
2.5 3.9 3.0 2.5 5.4 4.4 3.5
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Kreemer et al., 2005 (~E (~N O'E mm mm mm -139.0 -9.0 1.1 -22.0 8.0 0.6 1.3 4.9 0.6 2.3 3.7 1.3 11.7 -1.1 0.6 9.9 4.0 1.1 5.8 -1.4 0.6 -6.2 -5.0 1.1 5.4 6.0 0.6 -1.1 -1.3 0.8 -7.9 0.7 0.6 -1.9 2.3 0.6 -3.9 -1.9 0.6 -5.5 -1.8 0.6 -3.5 -3.1 0.8 -3.6 0.8 0.6 1.0 1.6 -1.9 0.1 -4.2 1.9 -2.6 2.0 1.6 1.4 1.5
1.1 3.7 1.1 2.5 2.3 -2.0 -1.8 0.7 -0.7 0.5 -0.4
O'N mm 0.6 0.4 0.4 0.6 0.4 0.6 0.4 0.6 0.4 1.1 0.6 0.4 0.6 0.4 0.4 0.4
0.6 1.3 0.4 0.4 0.6 0.8 0.4 0.8 0.6 0.6 0.6
0.6 0.6 0.4 0.4 0.4 0.6 0.6 0.6 0.4 0.4 0.6
~E mm -160.0 -24.6 -1.5 -0.7 11.7 9.0 6.8 -8.4 6.7 -1.6 1.0 -1.3 -6.4 -10.2 -8.8 -3.4 -0.8 -1.4 -1.4 -0.6 -0.9 -1.9 -0.5 -4.2 0.1 0.6 0.1 0.5
This study (~N O'E mm mm -15.0 0.6 7.7 0.6 1.6 0.6 2.7 0.6 -2.7 0.6 1.9 1.0 -1.4 0.6 -5.5 0.6 6.1 1.1 0.2 0.8 3.9 0.4 -1.2 0.6 -0.9 0.6 -0.1 0.6 -3.8 0.6 2.0 1.0 -1.4 0.6 -1.1 0.7 0.5 0.9 -1.6 0.6 1.3 0.6 5.1 0.7 1.0 1.1 2.7 0.6 -0.7 0.8 0.3 0.6 -0.1 1.2 1.9 0.7
O'N mm 0.6 0.6 0.6 0.6 0.6 1.0 0.6 0.6 1.1 0.6 0.4 0.6 0.6 0.6 0.6 1.0 0.6 0.7 0.9 0.6 0.6 0.7 1.1 0.6 0.8 0.6 1.2 0.7
0
-60 300 330
0
30
60
90 120 150 180 210 240 270
Fig. 4- Predicted co-seismic displacement field. The predictions are for a layered elastic spherical Earth model and a rupture model that best fits the offsets computed b y K r e e m e r et al. (2005). p r o a c h , w h i l e V i g n y et al. ( 2 0 0 5 ) u s e d t h e s a m e s o f t ware package for the GPS analysis but a different ap-
and the other estimates. T h e c o m p a r i s o n o f t h e f o u r sets o f e s t i m a t e s r e v e a l s
p r o a c h to r e a l i z e t h e r e f e r e n c e f r a m e a n d to c o m p u t e
t h a t (1) t h e e r r o r e s t i m a t e s o f K r e e m e r
t h e o f f s e t s . B a n e r j e e et al. ( 2 0 0 5 ) , o n t h e o t h e r h a n d ,
o u r l e a s t s q u a r e s e r r o r s a p p e a r to b e o v e r - o p t i m i s t i c ,
et al. ( 2 0 0 5 ) a n d
used time series obtained independently with a differ-
w h i l e t h e e r r o r e s t i m a t e s o f V i g n y et al. ( 2 0 0 5 ) a n d
ent s o f t w a r e p a c k a g e . N e v e r t h e l e s s , no s y s t e m a t i c dif-
a l s o B a n e r j e e et al. ( 2 0 0 5 ) m i g h t b e p e s s i m i s t i c ; (2)
f e r e n c e s a r e f o u n d b e t w e e n t h e B a n e r j e e et al. ( 2 0 0 5 )
the actual u n c e r t a i n t i e s o f the steps are m o r e o f the
553
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H.-P. Plag. G. Blewitt. C. Kreemer. W. C. Hammond
number of observed time series without distorting the predicted spatial pattern. Eq. (2), in fact, can be used directly in the search for the best-fit model. Static offsets detected with statistical significance for event B are constrained to relatively few stations (Fig. 6). It is interesting to note that the offset for SAMP is too large to be explained by a dislocation model.
A
14
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6
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Fig. 5: Comparison of the observed and predicted offsets for
event A. Symbols are the same as in Fig. 3. ,0J
. . . . .
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3
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. ~ ° & ~-
ii 101
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=
~0 ~, 9', ;, ,00 ,;, ,;0 ..... 0 ,2, Fig. 6: Co-seismic displacements for Event B. The grey arrow for SAMP is scaled down by a factor of four. The offsets for SAMP and NTUS are determined with the inclusion of the post-seismic deformation term. 2'6,
7;
;
order of 2 to 4 mm; and (3) only the offsets for stations experiencing displacements larger than a few mm, i.e. if they are in the areas on both sides of the rupture zone, can be determined with statistical significance. We expect that using the spatial finger-print as predicted by models (see Fig. 4 above) in a multi-station regression for the static offsets, will allow a much better determination of the observed displacement field. For that, we suggest using the modified model function Np
Nc
~(t) - a + bt + E 7i~H(ti) + E Aj sin(wit + Cj) i=l
j=l
(2) to represent the vector of the horizontal components, w h e r e / ~ is the horizontal displacement vector predicted by a rupture model. While the c~i in eq. (1) are the displacements determined independently for each time series, the regression coefficients ~/i are global quantities scaling the model predictions to fit a large
Tsunami
loading
signal
A tsunami travels barotropically through the ocean and causes movements of the ocean water mass comparable to ocean tides. Tsunami similarly load and deform the solid Earth. The displacements of the Earth's surface and the changes in the Earth's gravity field induced by surface loads can be computed using the theory of Farrell (1972). Predictions of the perturbations are computed through a convolution of the surface load expressed through the surface pressure and mass density field with the appropriate Green's function. The initial computations carried out here to estimate the order of magnitude of the induced displacements uses a static Green's function, i.e. the load Love numbers required to compute the Green's functions (see Farrell, 1972, for the details) are calculated neglecting the acceleration term in the field equations for the displacements. Considering that the longest elastic eigenmodes for a non-rotating Earth are of the order of 53 minutes (e.g. Lapwood & Usami, 1981), this static approximation is appropriate for loading with periods of several hours or longer. Tsunamis waves, however, have periods of 30 minutes and less, and the static solution can only give a first order estimate of the amplitude, while arrival times of the loading signal will be strongly biased. Using dynamic Green's function requires not only a convolution in space but also a convolution over the complete history of the tsunami. Thus, for a more accurate modeling of the amplitudes and particularly the temporal variation of the loading signal, a far more complex computation needs to be implemented. For the tsunami caused by Event A, the ocean bottom pressure variations are computed from the sea surface height anomalies predicted by the MOST model of the NOAA Tsunami Research Center (see Titov et al., 2005, and the reference therein). The sea surface heights are given with a mean spatial resolution of 0.3°in longitude and ~ 0.1°in latitude and a temporal resolution of 5 minutes. For the initial loading cornputation, a spatial resolution of 2.5 °was used, with the resolution being increased to 0.25 °in coastal areas and
Chapter80
3O
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Earth Deformations Induced by the Sumatra Earthquakesof 2 0 0 4 - 2 0 0 5
120 .
30
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close to the observer point. Shortly after the earthquake the peak deviation in ocean bottom pressure reaches values of over 200 hPa in the vicinity of the earthquake region (Fig. 7). After two hours, the peak values are still close to 200 hPa with all land areas around the Indian ocean experiencing vertical displacements of 5 to 12 mm (Fig. 7). After nearly a day, the energy of the tsunami has been distributed through the global ocean, with peak bottom pressure values nearly reaching 10 hPa (Fig. 8). The resulting vertical displacements of the Earth's surface are of the order of +1 ram. Peak signals of the order of 20 mm were found in vertical displacements. These peak signals lend to the prospect of detecting tsunami loading using GPS. In particular, the deformational signal with spatial wavelength of the order of 102 km should be detectable. However, the processing of GPS observations with high temporal resolution of better than five minutes will have to be improved in order to be able to detect such signals with low latency. In particular, sidereal filtering (Choi et al., 2004) and appropriate regional filtering (Wdowinski et al., 1997, e.g.) will have to be utilized. Peak signals of the order of 20 mm were found in vertical displacements. These peak signals lend to the prospect of detecting tsunami loading using GPS. In particular, the deformational signal with spatial wavelength of the order of 100 km might be detectable if improvements can be made to the quality of current high rate GPS estimates of the vertical. To as-
o
-
.
-30
-30
-60
7ooo 14ooo21;oo
Fig. 7: Ocean bottom pressure and induced vertical displacements computed from predicted sea surface heights. Upper diagrams: 30 minutes after the earthquake, lower diagrams: 2 hours after the earthquake. Left: ocean bottom pressure in Pa; right: vertical displacement of the solid Earth's surface in mm.
30
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Fig. 8: Same as Figure 7 but for 23 hours after the earthquake.
sess the potential of GPS to detect tsunami loading, we estimated station positions for a 10-station network as white noise every 30 seconds while simultaneously estimating satellite and station clocks as white noise, for the 7 days before and after the earthquake. We then applied a sidereal filter to calibrate for multipath (Nikolaidis et al., 2001), and then averaged the 30second vertical estimates into 5-minute normal points. The resulting time series show a scatter of the order of 10 ram, however systematic variations can be seen that sometimes exceed 20 mm during a 24-hour period. Moreover, a tsunami might happen to occur on a day of poor weather, when passing weather fronts can cause significant systematic error (Gregorius & Blewitt, 1999). Clearly, much research and development is needed to improve high rate GPS vertical positioning, for example, by the application of more effective methods of sidereal filtering (Choi et al., 2004) and spatial filtering (Wdowinski et al., 1997), which might be adapted for high rate network solutions.
4
Conclusions
Event A (26 December 2005) caused the entire Earth surface to deform at a geodetically significant level (> 0.1 ram), with important implications for terrestrial reference frame maintenance. Co-seismic displacements > 2 mm were clearly observed at GPS stations as far away as 4000 km from the rupture zone. The comparison of the co-seismic offsets determined in four different studies shows that the determination of these offsets depends on the GPS analysis as well as the assumptions for the subsequent analysis of the
555
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H.-P. Plag. G. Blewitt. C. Kreemer. W. C. Hammond
displacement time series. Most of the differences are attributed to noise in the displacements time series. Based on the comparison, the uncertainties in the offsets are estimated to be 2 to 4 mm. Using the predicted spatial fingerprint of the displacements in the estimation of the observed offsets is expected to improve the estimates of the displacement field. However, a main limitation for the determination of the displacement fields of large earthquakes arises from the number of available permanent GPS sites. At a minimum, attempts should be made to densify the global tracking network in regions with potentially large or great earthquakes to a spatial resolution of 500 km to 1000 km. Combined with models, the stations could be used to determine the effect of these great earthquakes on the global reference frame and to account for it. The tsunami caused by event A led to transient redistributions of oceanic water mass comparable to those associated with ocean tides. Similar to ocean tidal loading, the tsunami induced displacements of the Earth's surface and gravity changes, with the vertical displacements reaching up to 20 mm. With a sensitive observing system based on e.g. kinematic GPS, these deformations in principle could be sensed in real-time and integrated into a warning system. However, in order to extract this signal in near-real time, other effects such as the apparent displacements caused by variations in multi-path associated with orbital repletions will have to be reduced sufficiently.
Acknowledgment The authors are grateful to the IGS, IERS and JPL for providing the GPS data, the reference frame, and GIPSY OASIS II software as well as precise estimates of satellite orbits, clocks, and reference frame transformations, respectively. The sea level height data for the tsunami was made available by Vasily Titov of the NOAA Tsunami Research Center. Paul Denys and an anonymous reviewer provided valuable comments. The work at UNR was funded by grants from NASA Solid Earth and NASA Interdisciplinary Science.
References Banerjee, R, Pollitz, F. F., & Bfirgmann, R., 2005. The size and duration of the Sumatra-Andaman earthquake from far-field static offsets, Science, 308, 1769-1772. Blewitt, G., 1989. Carrier phase ambiguity resolution for the Global Positioning System applied to geodetic baselines up to 2000 km, d. Geophys. Res., 94(B8), 10187-10283.
Choi, K., Bilich, A., Larson, K. M., & Axelrad, R, 2004. Modified sidereal filtering: Implications for high-rate GPS positioning, Geophys. Res. Lett., 31, L22608, doi:l 0.1029/2004GL021621. Farrell, W. E., 1972. Deformation of the Earth by surface loads., Rev. Geophys. Space Phys., 10, 761-797. Gregorius, T. L. H. & Blewitt, G., 1999. Modeling weather fronts to improve GPS heights: A new tool for GPS meteorology?, J. Geophys. Res., 104, 15,261-15,279. Kerr, R., 2005. Failure to gauge the quake crippled the warning effort, Science, 307, 201. Khan, S. A. & Gudmundson, O., 2005. GPS analysis of the Sumatra-Andaman earthquake, EOS, Trans. Am. Geophys. Union, 86, 89-94. Kreemer, C., Blewitt, G., Hammond, W. C., & Plag, H.-R, 2005. Global deformations from the great 2004 Sumatra-Andaman earthquake observed by GPS: implications for rupture process and global reference frame, Earth Planets Space, In press. Lapwood, E. & Usami, T., 1981. Free Oscillations of the Earth, Cambridge University Press, Cambridge. Menke, W., 2005. A strategy to rapidly determine the magnitude of great earthquakes, EOS, Trans. Am. Geophys. Union, 86, 185,189. Nikolaidis, R. M., Bock, Y., de Jonge, R J., Agnew, D. C., & Van Domselaar, M., 2001. Seismic wave observations with the Global Positioning System, J. Geophys. Res., 106, 21,897-21,916. Pollitz, F. F., 1996. Coseismic deformation from earthquake faulting on a layered spherical Earth, Geophys. J. Int., 125, 1-14. Titov, V., Rabinovich, A. B., Mofjeld, H. O., Thomson, R. E., & Gonzfilez, F. I., 2005. The global reach of the 26 December 2004 Sumatra tsunami, Science, 309, 2045-2048. Vigny, C., Simons, W. J. F., Abu, S., Bamphenyu, R., Satirapod, C., Choosakul, N., Subarya, C., Socquet, A., Omar, K., Abidin, H. Z., & Ambrosius, B. A. C., 2005. Insight into the 2004 SumatraAndaman earthquake from GPS measurements in southeast Asia, Nature, 436, 201-206. Wdowinski, S., Bock, Y., Zhang, J., & Fang, R, 1997. Southern California Permanent GPS geodetic array: spatial filtering of daily positions for estimating cosesimic and postseismic displacements induced by the 1992 Landers earthquake, J. Geophys. Res., 102, 18,057-18,070. Zumberge, J. F., Heflin, M. B., Jefferson, D. C., & Watkins, M. M., 1997. Precise point positioning for the efficient and robust analysis of GPS data from large networks, J. Geophys. Res., 102, 5005-5017.
Chapter 81
Environmental effects in time-series of gravity measurements at the Astrometric-Geodetic Observatorium YYest-
erbork (The Netherlands)
I. Prutkin and R. Klees Delft Institute of Earth Observation and Space Systems (DEOS) Delft University of Technology, 2629 HS Delft, Kluyverweg 1, The Netherlands
Abstract. The effect of air pressure variations, soil moisture variations and groundwater level variations on time-series of gravity measurements acquired with the tidal gravimeter ET-15 at the Astrometric-Geodetic Observatorium Westerbork (WAGO), The Netherlands, has been investigated. The gravity measurements have been corrected for gravimeter drift, Earth body tide, and ocean loading effects. The residual signal has been smoothed using a variational smoothing algorithm. An agro-hydrological model provided the change in soil moisture in the vicinity of the gravity bunker over the measurement period. A precise finite element model of the gravity bunker and the surrounding layers was developed to compute the impact of changes in soil moisture content and ground water level variations on gravity. The combined effect of soil moisture variations and groundwater level variations varies between-7 #Gal a n d - 3 #Gal. It is dominated by soil moisture variations in the layers above the gravimeter between-7 #Gal a n d - 5 . 5 #Gal. Soil moisture variations below the gravimeter and groundwater level variations contribute between 0 #Gal and 2.5 #Gal. The analysis of two years of gravity and local air pressure variations show high correlation factors above 80% for periods of one day and shorter, whereas periods between one day and half a week are correlated between 50% and 80%. Over periods shorter than half a week, an admittance factor of-0.37 # G a l / m b a r has been obtained. The correlation for periods longer than half a week is very low, i.e. these periods should not be used to determine an admittance factor from local air pressure data. Rainfall events have a significant influence on gravity measurements at the WAGO site. A proper modeling requires measurements of precipitation, evaporation and run-off. This is the subject of flmlre studies.
Keywords.
Environmental effects on gravity,
admittance factor, soil moisture, groundwater level variations.
1
Introduction
In 1998, the Westerbork Astrometric-Geodetic Observatory (WAGO) became operational. WAGO is the successor of the Observatory for Space Geodesy at Kootwijk (KOSG). It has facilities for satellite laser ranging, GPS tracking, VLBI, and high-accurate relative and absolute gravity measurements corresponding to international standards. The gravimetry platform is a concrete cube of 3 x 3 x 3 m 3, in order to obtain the highest stability. Moreover, it is protected against the nuisance of moving VLBI telescopes in the vicinity of the station by the construction of surrounding barrages. Over the platform, a climatecontrolled bunker is built wherein absolute and relative gravimeters can operate. A weather station controls pressure, humidity, temperature, speed and direction of wind. Since 2002 permanent gravity measurements are performed with the LaCoste-Romberg ET15 tidal gravimeter and several Scintrex CG3M gravimeters. Moreover, absolute gravity is measured yearly with an FG-5 gravimeter. Since 2004 hydrological information (soil moisture, ground water level, and precipitation) is being gathered. The fluctuation of the groundwater level in the aquifers due to precipitation was measured using several piezometers with divers. In the shaft of the bunker, the height of the free water level was measured, too. At the Royal Dutch Meteorological Institute (KNMI) stations Hoogeveen and Eelde, which are located within a radius of 20 km around the bunker, evaporation, radiation and temperature measurements were acquired. These data are used as input to an agro-hydrological model, which quantifies soil
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moisture variations at the WAGO site. Soil toolsture variations, groundwater level variations and local atmospheric pressure variations are used to analyze their effect on time series of gravity measurements at the WAGO site. In this study, we will discuss the results of these activities. The main focus is on the quantification of the hydrological constituents soil moisture and groundwater level variations on gravity observed with the ET-15 gravimeter. Moreover, the relation between local air pressure data and gravity data at the Westerbork station is investigated. Finally, the effect of ocean-tide loading on the Westerbork station is quantified using different ocean-tide loading models. The outline of the paper is the following: in section 2 we review the data pre-processing strategy applied. The main focus is on the effect of gravimeter drift and the amplitude and quality of ocean tide loading corrections. The latter is important because the WAGO site is only 50 km away from the coastline. After subtraction of the contributions of tides and drift, we applied some smoothing to reduce residual effects caused by the imperfectness of the models. The smoothing is done using a new variational smoothing algorithm, see section 2. The effect of soil moisture variations, groundwater level variations, and atmospheric pressure variations on gravity is addressed in sections 3 and 4. In section 3, the results of the hydrological modeling are presented. In particular, we quantify the contribution of soil moisture and ground water level variations to the time series of gravity measurements. Some other examples of the estimation of hydrological effects on gravity data can be found in e.g. [5] and [7]. Section 4 is devoted to the admittance between local atmospheric pressure variations and gravity variations. The frequency dependent admittance is well known and investigated by different authors (see, e.g. [2] and {9]). It will be shown that significant correlations between the two time series can only be obtained for certain frequency bands. Almost no correlation between air pressure and gravity variations is found for periods above half a week, whereas correlations for shorter periods vary between 50% and 90%. Periods of anti-correlation can be attributed to strong rainfall events. Section 5 contains the main conclusions of this study.
2
Pre-processing of gravity data
The body Earth tides of the Westerbork station have been calculated by means of the package ETERNA, version 3.30 ([11]). To estimate the effect of ocean tide loading, three programs have been compared: LOAD89 [4], O L F G / O L M P P [10] and a new program CARGA [1]. CARGA and O L F G / O L M P P use the ocean tide model FES99, whereas LOAD89 uses the model FES95.2. These ocean tide roodels are quite similar. CARGA uses the most sophisticated approach and its accuracy for the Westerbork station is better than 1% [1]. Therefore, this model is used as a reference. The differences between O L F G / O L M P P and LOAD89 w.r.t, the CARGA model are 11% (0.24 #Gal) and 16% (0.34 #Gal), respectively. Ocean tide
loading (CARGA)
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Figure 1: Gravity ocean loading at the WAGO site calculated with various programs. Top panel: output of CARGA [ 1 ] . Bottom panel: difference between CARGA and O L F G / O L M P P [10].
Among several approaches to model the gravimeter drift, the best results have been obtained using Chebyshev polynomials. For the standard time-series of approximately 40 days, we found a Chebyshev polynomial of the 5th order to give the best results. The order of the polynomial was changed successively until the best fit in terms of amplitude and frequency character of residual gravity and expected gravity signal of local atmospheric pressure variations has been found. Of course, usage of such a drift model means the elimination of the lowest frequencies in the time series of gravity measurements. This is not critical for the analysis of at-
Chapter 81
• Environmental Effects
in Time-Series of Gravity Measurements at the Astrometric-Geodetic Observatorium Westerbork (The Netherlands)
mospheric pressure, soil moisture and groundwater level variations, because they contain almost no power at these frequencies. The same holds for the very high frequencies, which mostly represent noise and un-modeled signal. Therefore, these frequencies have been removed using a variational smoothing algorithm. Assume that u0 is a function to be smoothed (e.g. the residual gravity data). The main idea is to find the smoothest possible function u among all functions, equally distant from the given function u0. The mathematical formulation leads to two conditions, which have to be met simultaneously: I /z In
2 ~zO L L2
3
Evaluation of the hydrological signal
The time-series of gravity measurements have been acquired at the Westerbork station with the tidal gravimeter ET-15. The gravimeter platform in the bunker (Fig. 3) is located 75 cm below ground level, which has significant consequences for the effect of soil moisture variations on gravity. The overall contribution of soil moisture
__ (~2
i rain
(1)
Eq. (1) are discretized, which gives the following variational problem: S
-
u °
_
~//Gravitimete Centre line of gravltl Z'mete r 75 crn b e l o w surface
Problem (2) is solved iteratively by means of Lagrangian multipliers. The effect of this algorithm on the time series of gravity measurements and on the signal power spectral density are shown in Fig. 2. Effect o f
i 200
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Figure 3: Cross section of the gravity bunker at the WAGO site. Notice the location of the gravimeter, which is 75 cm below ground level.
variational smoothing algorithm
,
0
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Figure 2" The effect of variational smoothing in the time domain (top panel) and in the frequency domain (bottom panel).
variations to the gravity signal is a combination of three factors: (i) soil moisture variations in the layers above the gravimeter level; an increasing soil moisture in this layer reduces the observed gravity; (ii) soil moisture variations in the layers below the gravimeter level; an increasing soil moisture in these layers will increase the observed gravity; (iii) the direct gravitational effect of groundwater level variations. Note that there is a relation between (ii) and (iii) in the sense that an increased groundwater level reduces the volume of the layers below the gravimeter level. Special emphasis has been made on a proper modeling of the constituents (i)-(iii). To model soil moisture variations in the layers above the gravimeter level, a precise finite element model of the bunker and the surrounding subsurface layers has been developed. The whole volume was divided into a number of small prisms (see
559
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I. Prutkin • R.
Klees
Fig. 5). The gravitational effect of soil moisture variations over each prism was computed using Gauss-Legendre cubature formulas. The total gravitational effect of soil moisture variations in these layers was obtained by summing up the contribution of all prims. To calculate the Newton integral with 4 correct digits, the area or 350 m around the gravimeter has been taken into account. The information about the variation of groundwater level and soil moisture between the driest and the wettest days of the summer 2004 has been provided by Wageningen University [3]. The information is taken as the output of an agro-hydrological model driven by measurements of precipitation, evaporation, radiation, temperature, and groundwater levels in the aquifers around the Westerbork station. The layer between the ground surface and the centre line of gravimeter is the most influenced by precipitation (see Fig. 4, from [3]). Due to increase of soil moisture, which amounts to 0.3 cma/cm a, this layer of thickness 75 cm generates quite considerable additional gravitational signal. Because the layer is located above the gravimeter (see Fig. 3), an increased soil moisture content gives a negative gravitational signal. For summer 2004, this signal varies between - 5 . 5 and - 7 #Gal. Maximum variation of soil moisture (%) 0 0
5
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,
,
,
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Figure 4: Distribution of soil moisture variation with depth at the WAGO site during summer 2004 [3]. The contribution of soil moisture variations in the layer below the gravimeter level and of groundwater level variations to the gravity signal has been approximated conventionally using the Bouguer plate approximation. This is sufficiently accurate for modeling the soil moisture
variation in this layer, because the amplitude is very small. The Bouguer plate approximation is also sufficiently accurate to model the groundwater level variations at the WAGO site. The maximum groundwater level at WAGO is 70 cm below the gravimeter level; the lowest groundwater level is 290 cm below the gravimeter level (Fig. 4). The positive effect of groundwater level variations and soil moisture variations in the subsurface layer between gravimeter level and groundwater level is less than 2.5 #Gal. Thus, the combined seasonal effect of soil moisture and groundwater level variation at the WAGO site is in the range [-7,-3] #Gal. This is significantly above the noise level of the ET-15 gravimeter, which is about 0.5 #Gal.
3 2.5 2 1.5 1 0.5 0
0
~ ~
14
Figure 5: Finite-element model of the bunker to compute the gravity signal of soil moisture variations above the gravimeter level.
4
Correlation between air pressure and gravity variations
We used a time series of about 2 years to analyze the relation between local air pressure variations and gravity variations. Our study has confirmed, that the correlation between air pressure and gravity variations strongly depends on the frequency band. For periods of one day and less, we observe a high correlation of 8 0 - 9 5 % . This is in agreement with known results (e.g. [8]). One typical record is shown in Fig. 6. The correlation between gravity and air pressure for periods longer than one day is more difficult to access. To compute an admittance factor, we analyzed the whole two-year time-series of gravity and air pressure observation. For all frequencies below 1 cycle per day, we found an averaged value of-0.37 #Gal/mbar, which is not too far from the value-0.356 #Gal/mbar, suggested by
Chapter 81 • Environmental Effects in Time-Series of Gravity Measurements at the Astrometric-Geodetic Observatorium Westerbork (The Netherlands) lO
r
"r
5
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/ ~;,t~
A .,:
/i IV" ~.
i
o
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t
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Figure 6: Gravity variations and gravitational effect of air pressure variations for periods less than one day. The correlation is 81°-/0. The gravity variations are corrected for Earth body tides, ocean loading tides and gravimeter drift prior to correlation analysis.
Merriam [6] as a local admittance on the base of theoretical considerations. However, the correlation between gravity variations and air pressure variations is very low for frequencies below 1 cycle per day. For this reason, we split the frequency band in two parts: a high-frequency part covering periods from one day till half a week, and a low-frequency part, covering periods longer than half a week. For the high-frequency part we found reasonable correlation coefficients between gravity and air pressure variations; typical correlation coefficients vary between 54% and 71%. For the low frequency part, however, we could not find any significant correlation. Therefore, we only determined an admittance over the frequencies between 1 cycle per day and 0.3 cycles per day. Again, we found a value of-0.37 # G a l / m b a r for this frequency band. Un-modeled rainfall events have a strong influence on the correlation between gravity and atmospheric pressure variations. This situation is displayed in Fig. 7. After a long relatively dry period the hydrological effect on gravity is minimal. During a strong rainfall event, air pressure drops down, and gravity goes up. In the high frequency part of gravity variations, the same maximum could be easily observed. But the low frequency part of the gravity signal has at this moment not a maximum, but a minimum. The moment of 'anti-correlation' can be clearly related to a strong rainfall event. It is quite in agreement
"f 15
J
Figure 7: Gravity and air pressure variations for different frequency bands during a strong rainfall event. Top panel: periods between one day and half a week; mid panel: periods longer than half a week; bottom panel: rainfall events.
with the considerations in the previous section, that after a strong rainfall event, the total gravirational effect of the increased soil moisture and of the uplift of groundwater level should be negative. This result has been noticed many times when comparing time series of gravity and atmospheric pressure variations at the WAGO site.
5
Summary and conclusions
A first analysis of the gravity signal of atmospheric pressure variations, groundwater level variations and soil moisture variations at the Westerbork Astrometric-Geodetic Observatory (WAGO) has been done. The following conclusions are drawn: i. The ocean tide loading effect at the WAGO site is ±2 #Gal. The differences between various ocean tide loading algorithms can reach values op to 0.5 #Gal, which is comparable with the noise level of the ET-15 gravimeter. 2. The drift of the ET-15 gravimeter has to be modeled properly prior to studying environmental effects on gravity. Best results have been obtained using Chebyshev polynomials. The degree of the polynomial has to be
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chosen carefully and is mainly a function of the length of the time series. A methodology for the automatic selection of the optimal polynomial degree is still missing. 3. The total effect of soil moisture and groundwater level variation is negative at the WAGO site, due to the particular location of the platform below ground level. The effect varies between [-7,-3] #Gal, and is significantly above the noise level of the ET-15 gravimeter. 4. Strong rainfall events show-up as a negative minimum in the low-frequency part of the spectrum of gravity variations. This is in agreement with the modeling of the total gravitational effect of soil moisture and ground water variations at the WAGO site. So far, only two hydrological constituents have been analyzed at the WAGO site. Recently, the station has been equipped with several rainfall meters and a lysimeter. They will provide more detailed information about precipitation, evaporation, and run-off. Moreover, the network of piezometers will be extended and the reading will be automized. The results of the analysis of these data will be the subject of a forthcoming paper.
Acknowledl~ments Prof. Trevor Baker from P r o u d m a n Oceanographic Laboratory in Liverpool, United Kingdora, has provided us with the ET-15 gravimeter. This support is gratefully acknowledged.
References [1] Bos, M.S. and Baker, T.F. (2005). An estimate of the errors in gravity ocean tide loading computations. Y. Geod., 79:50-63.
[2] Crossley, D. J., Jensen, O. G., Hinderer, J.
[3]
[4]
[5]
[6]
(1995). Effective barometric admittance and gravity residuals. Phys. Earth Planet. Int., 90:221-241. de Jong, B. and Ros, G. (2004). The effect of water storage changes on gravity near Westerbork. MSc thesis, Hydrology and Quantitative Water Management Group, Wageningen University, The Netherlands. Francis, O. and Mazzega, P. (1990). Global Charts of Ocean Tide Loading Effects. J. Geophys. I~es., 95(C7):11,411-11,424. Kroner, C. (2001). Hydrological effects on gravity data of the Geodynamic Observatory Moxa. J. Geod. Soc. Japan, 47(1):353-358. Merriam, J.B. (1992). Atmospheric pressure and gravity. Geophys. J. Int., 109:488-500.
[7] Meurers, B., Van Camp, M., Petermans, T., Verbeeck, K., Vanneste, K. (2005). Investigation of local atmospheric and hydrological gravity signals in Superconducting Gravimeter time series. Geophysical Research Abstracts, 7:07463. [8] Mukai, A., Higashi, T., Takemoto, S., Naito, I. and Nakagawa, I. (1995). Atmospheric effects on gravity observations within the diurnal band. J. Geod. Soc. Japan, 41:365-378. [9] Neumeyer, J. (1995). Frequency dependent atinospheric pressure correction on gravity variations by means of cross spectral analysis. Bulletin d'Information Mardes Terrestres, 122:92129220. [10] Scherneck, H.-G. (1991). A parametrized solid Earth tide mode and ocean loading effects for global geodetic base-line measurements. Geophys. J. Int., 106(3):677-694. [11] Wenzel, H.-G. (1996). The Nanogal Software: Earth tide data processing package ETERNA 3.30. Bulletin d'Information Mattes Terrestrcs, 124:9425-9439.
Chapter 82
Numerical models of the rates of change of the geoid and orthometric heights over Canada E. Rangelova, W. van der Wal, M.G. Sideris, Department of Geomatics Engineering, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada, T2N 1N4 P. Wu, Department of Geology and Geophysics, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada, T2N 1N4
Abstract. For the purpose of modernization of the vertical datum in Canada, rates of change of geoid and orthometric height are computed independently of postglacial rebound simulations. The designed procedure is based on the general least-squares collocation approach and uses as input rates of change of absolute gravity and absolute vertical motion velocities. The procedure is applied in a stepwise manner in order to account for new available data. The predicted geoid rate has a peak value of 1.3-1.4 mm/yr over Hudson Bay and agrees within the computational accuracy, i.e. _+0.1 mm/yr, with the postglacial rebound simulated rate. The predicted rate of change of orthometric height has the general pattern of the vertical motion of the rebounding crust, but also shows some local effects mainly inherited from the rates of gravity data, e.g., the uplift predicted over Southern Alberta as a result of the strong negative gravity rate signal and the subsidence in the Mackenzie River delta. It is found that the achieved prediction accuracy of _+2.4 mm/yr can be increased by using more accurate GPS velocities and extending their coverage to the north as well as incorporating precise relevelled data. Also, improvement is expected from using vertical velocities from a combination of tide gauge records and satellite altimetry sea surface heights. Keywords. Postglacial rebound, rate of change of gravity, vertical crustal motion, rate of change of geoid, vertical datum, inverse multiquadrics.
1 Introduction Traditionally, the definition of the vertical datum is based on the concept of the geoid as a particular equipotential surface of the gravity field that coincides with the mean sea level (MSL). MSL is computed by averaging records of sea level variations in order to define a conventional "zero" (origin at the geoid) for orthometric heights determined and propagated via precise levelling. In
this way, a vertical control network, through which the vertical datum is accessible, is established. The vertical datum is subject to large systematic errors and distortions. They are one of the main factors that contribute to discrepancies between orthometric, geoid, and GPS (geodetic) heights (H, N, and h, respectively) at benchmarks of the vertical control network (see, e.g., Fotopoulos 2003). The discrepancies are computed via a simple relationship given by Heiskanen and Moritz (1967, Eq.4.58, p. 176) On the other hand, with the new geopotential models (provided by CHAMP and GRACE) of increased accuracy at long wavelengths, and expected considerable improvement in accuracy of medium and short wavelengths by GOCE, the computation of a cm-level geoid will become realizable in the very near future (Tscherning et al. 2000). Thus, the geoid becomes an attractive alternative to the traditional vertical datum in those countries, such as Canada, where large territories with harsh environmental conditions are not covered by vertical control networks (V6ronneau 2001). Large scale and magnitude secular vertical crustal motion and temporal changes in the geopotential can also contribute to the discrepancies between orthometric, geoid, and GPS heights, especially if they refer to different epochs. In Canada, the temporal effects are mainly due to the prominent postglacial rebound (PGR), a viscoelastic response of the Earth to the melting of the Laurentide ice-sheet about 18,000-20,000 years ago. Although not as large as the distortions in the levelling network, these temporal effects must be quantified if the vertical datum is to be updated or re-established using the geoid as a reference surface for orthometric heights. In the context of the temporal variations, the absolute vertical crustal motion measured by GPS, 1~, and the rate of change of orthometric height, Iit, differ by the rate of change of geoid, lq, i.e., 1~ - Iq + I2I.
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E. Rangelova • W. van der Wal. M. G. Sideris • P. Wu
In Canada, the availability of absolute rates of change of gravity, g, and measured GPS vertical velocity, t~, allows studies of the spatial variations of the temporal changes of gravity field and vertical crustal motion over the whole country to be conducted independently of postglacial rebound simulations. Compiled maps of vertical crustal motion in Canada based on relevelled data are documented in a series of papers starting with Vanf~ek and Christodulidis (1974), Vanf~ek and Nagy (1981), and later Carrera et al. (1991). Relevelled segments of the first order network contain information about vertical motion, but they are not considered at this stage of the study. The objective of this paper is to investigate a procedure based on the general least-squares collocation (LSC) approach for modelling rates of change of geoid and orthometric heights using solely the available gravity rate data and GPS vertical velocities. However, the procedure can also accommodate repeated levelling data as well as rates of vertical motion from tide gauges and radar satellite altimetry data. In addition to PGR, assumed to be the only factor that contributes to the general trend in data, the procedure is designed to also account for gravity and height rates of change over Canada due to subsidence and erosion described by Pagiatakis and Salib (2003). In the next section, the computational algorithm is explained. That is followed by a description of the numerical tests and a comparison with postglacial rebound simulations. Finally, future improvements in the proposed methodology are suggested.
2 Methodology 2.1
Stepwise Least-Squares Collocation Procedure
The two-dimensional space representation of velocity surfaces derived from scattered relevelling data, traditionally uses a bivariate polynomial series (see, e.g., Vanfeek and Christodulidis 1974). This pure functional approach, though offering a simple mathematical representation of the velocity surfaces, has disadvantages such as numerical instabilities due to the failure of the polynomials to reproduce accurately a surface in areas with lack of data and a risk to over-parameterize the surface in the presence of a limited number of data. In this paper, a general least-squares collocation approach is adopted. It was used by Hein and Kistermann (1981) to separate the local motion of a non-tectonic origin (which takes on a stochastic
description) from the regional trend due to tectonics (described analytically). In the case of isostatic adjustment of the crust, the LSC approach has been followed, e.g. by Danielsen (2001), to model the rates of the vertical uplift in Fennoscandia. A polynomial surface of order 4 has been used to describe the trend in vertical crustal motion data. In this paper, in contrast, inverse multiquadrics (MQs) of Hardy (1990) are used. The MQ analysis is based on a linear combination of (different shape) hyperboloids (basis functions). The basis functions can be located at data points or at arbitrary nodes. In this respect, MQs are able to interpolate more accurately a velocity surface from scattered data than polynomials are (see Holdahl and Hardy 1979, Hardy 1990). Another important difference from the studies of PGR using LSC approach is that the output of the procedure consists of the rates of change of geoid in addition to the rates of change of orthometric heights. This requires considering an additional step in the procedure, i.e., converting the estimated rate in the data into the rate of the geoid change. For example, after the trend of the rate of vertical motion is estimated in the LSC procedure, it can be transformed into the rate of geoid change via a mass flow model (see Eq.7). The observation equation of the general leastsquares collocation model is given by:
1= A X + t + n
(1)
where 1 is the data vector, t is the signal vector component of the data, n is the observation noise, and A is the coefficient matrix of the unknown trend parameters, X. With the a priori covariance matrices of the signal and noise, Ctt = coy(t, t) and Cnn = cov(n,n), respectively, the least-squares collocation solution is given by the well known formulas (Moritz, 1980, p.144) as follows: - (ATC-'A) -jATC-jl,
~ - CstC-' ( 1 - AX) (2a)
where C =Ctt +Cnn ; and s is the signal to be predicted. The error covariance matrices are given by: E x x - ( A T C - 1 A ) -1,
E s s - C s s - C s t C - 1 C t s (2b)
In case a new data set is available and the covariance matrices between the old and new data sets are known, the solution can be derived in a stepwise manner. The estimated parameters, X , and the predicted signal, ~, are corrected as follows"
Chapter 82 • Numerical Models of the Rates of Change of the Geoid and Orthometric Heights over Canada
corr -- ~ nt- 8 X ,
Scorr -- ~ + 8 S .
(3)
Eqs.(3) correspond to the formulas (19-20) and (1927) in Moritz (1980, pp. 144-156). The corrections 8X and 8s stand for the improvement of the initial estimates due to the new data included. Analogously, the error covariances at the first step are corrected using the formulas (20-23) and (2024) given in Moritz (1980, pp. 144-156).
2.1.1
Modelling the Trend Components
The inverse MQs are used to model the trend component of the data. The inverse MQ basis function is defined as follows:
lp = O ( d p j ) [ O ( d ~ j
) f cD(dij )]-10(d ij )f 1.
(6)
O(dij ) will correspond to the coefficient matrix A in Eqs.2a and b if the least-squares collocation approach is followed. The trend in the data, attributed entirely to PGR, is converted into a trend surface of the rate of geoid change using a mass flow model. It is based on the assumption that the crustal uplift is accompanied by an inflow of masses (with density 9) from the upper mantle (see Sj6berg 1982). According to this model, the geoid rate at a point i depends on the differential m a s s pfljd(yj
at a point j located on the internal
sphere with radius r as follows: l~li = 1 Gpla jdoj
(I)(dj) - (dj2 + A2 ) -1/2 ,
where
(4)
d j is the Euclidean distance between the
points (x, y) and (x j, y j ), and A is a parameter that controls the shape of the basis function. A can be varied, so that the basis functions range from a cone (A=0) to flat sheet-like surfaces in order to interpolate accurately the particular data. In the interpolation mode, the basis functions are centered at nodes coinciding with the data points. The MQ equation is then given (in matrix notation) by: ocO(dij ) = 1
(5a)
where dij is the distance between the locations of the i th data point a n d jth basis function; 0c is a vector of
the unknown coefficients to be determined from the known components 1i of the data vector in the linear system of Eq.5a. Values at the grid points p are interpolated as follows:
Y /
where Lij -- ~R2 -k- r 2 - 2 r R c o s l l / i j
is
the
spatial
distance between points i and j, R is the mean Earth radius, and y is the mean gravity.
2.1.2
Modelling the Stochastic Components
The absolute vertical motion (measured by GPS) itself does not contain information about changes in the geopotential due to the mass redistribution below the Earth's crust. Only combined with measurements of gravity changes does it suffice to derive the changes in the geopotential. Since the rate of change of geopotential, w, is given by the following integral relation (see, e.g., Heck 1984): 2y @ - S t ( g +--~I2I),
R St(. ) - - ~ g ~ ( . )S(gt)d•,
(8)
(y
the two components of w can be evaluated as: I - St(g),
lp =O(dpj )O(dij )-11.
(7)
Lij
x~2 - --~ St(tit) •
(9)
(5b) Following Heck (1984),
In least-squares approximation the number of nodes and their locations can be varied and the selection is usually based on accuracy tolerance. A forward selection algorithm based on orthogonal least-squares (see, e.g., Chen et al. 1991) is designed to select the nodes that can explain certain amount of data power. The unexplained power defines the accuracy tolerance. The least-squares prediction of the v e c t o r lp is as follows:
( K i j ) k m = COV{(XV i ) k , ( X ~ q j ) m },
k,m = 1,2
(10)
defines the covariance kernel as covariance between @k and v~m , k , m = 1,2 at any two points i and j. Note that the points i and j are both located on the sphere with radius R that represents the geoid, while, in Eq.7, j was used to designate the differential mass on the interior sphere.
565
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E. Rangelova • W. van der Wal. M. G. Sideris • P. Wu
The covariance kernels can be described by the Gaussian model as follows: 2 Kk m _ Ak m ~km
d 2
exp(-
),
k , m = 1,2
-q [ Rate of geopot, change / via modified kernel Input ~g,G~ -gravity / ~I~ Stokes integral 1= {l~,Oh _vertical vel./ wl = St(g) W2 : (23'/R)St(fl)
rate|
!
(11) 0.06 < cy~ < 0.88 ~tGal/yr
2%km
c~fi > 1 m m / y r
Here d denotes the distance, A k m ' C 2 m / 2 k , m = 1,2 stand for variances, and the parameter "c is related to the correlation length. The auto- and cross-covariances between the rates of change of gravity, orthometric, and geoid heights can be formed using the inverse of the relationships in Eq.9:
I2I - R--~-St-l(0e2).
Stepwise LSC : - estimate the trend - predict the signal in data
(12)
A flowchart of the applied computational algorithm is shown in Figure 1. The input data consist of the absolute rates of change of gravity and GPS vertical velocities, and the output is in terms of rates of change of geoid and orthometric height. Since the rate of geopotential depends on the latter (see Eq.9), the computations are iterated after the absolute vertical velocities, 1~, are replaced by estimated I2I. The evaluation of the Stokes integral (Eqs.8 and 9), via fast Fourier transform (FFT) requires gridding of noisy data by an appropriate method. Among all investigated methods (continuous curvature splines in tension, (inverse)MQs and thinplate splines), only inverse MQs have been able to produce at the same time an accurately interpolated and smooth surface from the irregularly distributed noisy gravity rates and GPS vertical velocities. In order to derive the stochastic information, the long wavelengths below a certain degree are filtered out via applying a two-dimensional (2D) FFT with a spheroidal Stokes kernel (see, e.g., Vanfeek and Featherstone 1998, Eq.23, p. 687) in the evaluation of Eq.9. The components of the rate of geopotential, Wl and w z, are interpolated at the scattered data point locations, and the empirical covariances are computed by averaging products of any two values in the predefined bins with respect to the distance. Then, the parameters Akm and 1;km of the covariance models in E q . l l are fitted to the empirical covariance functions by least-squares adjustment with constraints on the variances
variance factor: accept the model Yes I Compute the trend and signal components of the rates of change of geoid and orthometric heights
The reader can consult Heck (1984) for a list of analytical auto- and cross-covariance functions.
Iterative Computational Algorithm
Corrections for a new
data set
Z 2 - test on a posteriori
23'
2.1.3
Fitting analytical kernels - Gaussian model for the rates of geopotential change due to g and h
!
Correct the model
- st-l(wl),
1~is replaced by ~I
Output N, I:I,o s, on
Figure 1 The iterative computational algorithm based on stepwise least-squares collocation.
2 Akm'l;km
[ 2 k , m = 1,2 to be equal to the empirically estimated variances. A Z2 test on the a posteriori variance factor at a 95% confidence level is set as a criterion for accepting or rejecting tested models. The algorithm can test automatically a number of models defined by combinations of adopted functional descriptions for the data trend component and the covariance matrices computed from the stochastic information derived with different degree (e.g., 10, 15, or 20) spheroidal kernels.
2.2
Postglacial Rebound Modeling
The postglacial rebound simulation consists of two parts. The first is the response of the Earth to a surface load; the Peltier-Wu normal mode theory is used to simulate this (see Wu and Peltier 1982). The input is a spherical, radially symmetric Earth with 6 layers that represent the major discontinuities found in the Earth. The stress-strain behaviour is modeled as Maxwell rheology, which assumes a linear relation between stress and strain. Density and rigidity for each of the six layers are obtained by volume averaging values from the widely used Preliminary Reference Earth Model (PREM) (Dziewonski and Andersen 1981). Viscosity values are selected close to the VM2 model of Peltier
Chapter 82 • Numerical Models of the Rates of Change of the Geoid and Orthometric Heights over Canada
(2004), see Table 1. The output of this part of the simulation are visco-elastic Love numbers, which represent the response of the Earth to a unit load in the spectral domain. The second part of the simulation describes the ice-ocean-earth interaction during growth and melt of ice sheets in an ice age cycle. The process is described by the sea-level equation, which in its basic form is (Farrell and Clark 1976): S(% ~, t ) = C(% ~, t){G(q~, ~, t ) - R(q), ~,, t)}. (13) ~'bU
G and R are the perturbations of the geoid and solid surface, respectively, induced by postglacial rebound. Both have a convolution form of the surface load with the respective Greens functions that contain the visco-eastic Love numbers. The function C, by definition, takes the values 1 for ocean and 0 for land. It is time dependent to account for ice which occupies ocean area (e. g., Hudson Bay) during part of the glaciation. Apart from this so-called marine-based ice, the coastline is taken to be equal to the present-day coastline. Also, the rotational feedback mechanism is not considered in the simulation because it leads to a degree 2 order 1 signal with geoid rate in North America less than 0.1 mm/year (Peltier 1999, Figure 13). Since Eq.13 contains the sea level both on the left side and on the right side (as part of the load that causes change in G and R), the equation is an integral equation solved by the method of Mitrovica and Peltier (1991). With sea level computed and ice level known, the present day geoid perturbation can be computed. The ice model used is the smoothed ICE-3G of Tushingham and Peltier (1991), which provides a good fit to observed sea level data and is independent of the data used in this study. The growth of the ice sheets is simulated by reversing the ICE-3G history, taking 7,000 years instead of 1,000 years for each step, as in Milne et al. (1999). The simulated rate of the geoid change is plotted in Figure 2. The maximum rate of 1.3 - 1.4 mm/yr is located over Hudson Bay; it gradually decreases to 0.6 - 0.8 mm/yr in the Great Lakes area.
Table 1. Stratification for the Earth model used in the postglacial rebound simulation, with viscosities selected to
be close to the VM2 model (Peltier 2004). Layer Depth, [km] Visc., [Pa.s]
Lith. UM1 UM2 LM1 0 115 400 670 ~
4.10 20 4"10 20 2"10 21
LM2
Core
1171 4"10 21
2891 0
. 0.1
.
. 0.3
.
ZIU
. 0.5
0.7
0.9
I 1.1
I 1.3
I mm/year 1.5
Rate of Geoid C h a n g e Figure 2 Rate of change of geoid from a PGR simulation (ICE-3G), in mm/yr.
;ites
, 0.1
, 0.3
, 0.5
LDU , 0.7
Z/U , 0.9
I I .I
I 1.3
I mm/yr 1.5
Rate of Geoid G h a n g e (trend) Figure 3 Trend of the geoid rate from a mass flow model, in mm/yr.
3. Numerical Tests and Discussion of Results The first data set used in the numerical tests consists of the historical time rates of change of absolute gravity from the readjustment of the primary Canadian Gravity Standardization Network (Pagiatakis and Salib 2003). The second data set is the GPS vertical velocities provided by GSD/Natural Resources Canada. Although data from two networks with a different time span of measurements are combined, it is assumed that both data represent the same temporal changes. This would be true for the secular changes associated with postglacial rebound; for local effects this might be disputable. The LSC procedure can be designed such that corrections for the trend and covariance functions are computed from gravity rates and GPS velocities at collocated absolute gravity/GPS sites.
567
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E. R a n g e l o v a • W. v a n d e r W a l . M. G. Sideris • P. W u
Model b
Model a
;ites
;ites
, 0.1
, 0.3
LbU , 0.7
, 0.5
/bU
V_./U I 0.9
i I .I
, 1.3
I mm/yr 1.5
. 0.1
.
. 0.3
Z/U I 0.9
. 0.5
0.7
I 1.1
I 1.3
I mm/yr 1.5
Rate of Geoid Change
Rate of Geoid Change
ites
ites
. -8
.
. -6
. -4
-'' . . -2
260" . 0
2
4
270 ° I 6
I 8
~,~v I 10
I 12
I 14
I mm/yr 16
, -8
, -6
, -4
-,-,u , -2
, 0
260" , 2
, 4
270" I 6
I 8
~,~v I 10
I 12
I 14
I mm/yr 16
Rate of Change of Orthometric Height
Rate of Change of Orthometric Height
ites
ites
LDU Z/U . . . . . . . . . I I I I l i I I mm/yr 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 5.2 5.6 6.0 6.4 6.8 7.2
LbU Z/u . . . . . . . . . I I I I I I I mm/yr 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 5.2 5.6 6.0 6.4 6.8 7.2
St. Dev. of Rate of Change of Orthometric Height
St. Dev. of Rate of Change of Orthometric Height
Figure 4 Rate of change of geoid and orthometric height predicted by LSC and accuracy of rate of change of orthometric height, in mrrdyr.
Chapter 82 • Numerical Models of the Rates of Change of the Geoid and Orthometric Heights over Canada
While the rates of change of gravity are well distributed over the territory of Canada, almost all GPS sites are located south of the 60 degree parallel. Since long wavelengths of postglacial rebound can be modelled by the gravity rates alone, the procedure is modified so that, at the first step, only gravity information is considered. To derive the vertical velocities, a value of-0.16 gGal/mm for the gravity-to-height ratio has been computed using Eq.18 in Fang and Hager (2001, p. 541). This value is characteristic for PGR, so the vertical motion and changes in geopotential will be entirely interpreted in terms of the latter. The rates of change of geoid and orthometric height and the standard deviation are plotted in Figure 4, Model (a); see also Tables 2 and 3. To predict the signal, the wavelengths below degree 15, which contain 62 percent of the total data power, have been filtered out (see section 2.1.3). To model the data trend component, 13 nodes that can explain the filtered data power have been selected as significant. At the second step, the Model (a) estimates are corrected by adding the GPS velocities to obtain the combined solution, i.e. Model (b) in Figure 4. The comparison between the postglacial rebound geoid rate (Figure 2) and the trend component of the LSCestimated geoid rate of Model (b) in Figure 3 reveals a similar pattern with a peak value over Hudson Bay, an identical surface gradient in most areas, and agreement at the level of the standard error of the predicted rate, i.e. _0.1 mm/yr. In Western Canada, where other processes govern gravity and height changes as well as in the areas with lack of data, the difference increases to 0.2-0.3 mm/yr. It is worth repeating that the PGR model is completely independent of the data used in this study. This result verifies the ability of the inverse MQs to approximate correctly the general changes in the velocity surface using the absolute rates of change of gravity and GPS velocities and also the correctness of the assumed mass inflow model. The use of accurate vertical velocities is critical for prediction of the rate of change of orthometric height. It can be seen from the comparison between Models (a) and (b) and Table 2 that GPS velocities only slightly decrease the magnitude of the geoid rate and the prediction error. In contrast, the peak value for the rates of orthometric height in the eastern part of Hudson Bay decreases by 2-3 mm/yr, while accuracy is improved by 1.3 mm/yr. Note that the extreme prediction errors are encountered in the areas with lack of data (see Figure 4). The minimum error, associated with the sites where both gravity and GPS data are available, is below 1 mm/yr.
Table 2. Standard deviation of the predicted rate of change of geoid, in mrrdyr. mean _+0.13 _+0.10
Model (a) Model (b)
min _+0.11 _+0.08
max _+0.14 _+0.12
Table 3. Standard deviation of the predicted rate of change of orthometric height, in mm/yr. mean ___3.7 ___2.4
Model (a) Model (b)
min ___2.1 _+0.8
max __.7.2 _+5.2
14 12 o10 c-~
. . . j~ jl' o,~}, ii~,,,&~.
.
~-
LSC geoid rate ' LSC orth. heigth rate ..... PGR geoid rate _ - . - PGR v ert. displ, r_ate_
--,
~8 ~6 o
~4 .c_ 2 0
5
10
Figure 5 Power
15
20
25
30
35
angular degree
40
45
50
spectra of rates of change of geoid and orthometric height.
Finally, the power spectra of the LSC-estimated and PGR-simulated rates are compared in Figure 5. Both spectra of the PGR and LSC-predicted rates of change of geoid have a peak at degree 5. The slight difference (less than 1% of the total power) in the band from degrees 13 to 25 can be due to mismodelling effects, but it also reflects the signal component in the geoid rate (being between-0.2 to 0.2 mm/yr across the studied territory) which may include the contribution from local processes. Significant decrease in the low degrees and a shift of the power towards higher degrees is observed for LSC-estimated rate of orthometric height. The major contribution comes from the dome-like structure of rates that is due to still insufficient data density mainly in north, but some local patterns, e.g. in Western Canada contribute as well.
4 Conclusions The rates of change of geoid and orthometric heights can be predicted in a stepwise least-squares collocation procedure at the level of accuracy of 0.1 and 2.4 mm/yr, respectively. The good agreement
569
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E. Rangelova • W. van der Wal. tvl. G. Sideris • P. Wu
of the postglacial r e b o u n d simulation and L S C p r e d i c t e d geoid rates (within the given data density and the c h o s e n Earth and ice m o d e l p a r a m e t e rs) confirms that the d e s i g n e d p r o c e d u r e is able to predict the m a i n g e o d y n a m i c signatures over C a n a d a using solely absolute rates of c h a n g e of gravity and GPS vertical velocities. T h e s e results are subject to further i m p r o v e m e n t s before being useful for the m o d e r n i z a t i o n of the vertical d a t u m in Canada. For instance, the accuracy of the p r e d i c t ed rate of c h a n g e of o r t h o m e t r i c height, m o s t l y in Southern Canada, will increase after precise relevelling data are included. U s i n g additional information in terms of vertical velocities derived f r o m water tide gauges and radar satellite altimetry data in the Great L a k e area as well as using c o l l o c a t e d absolute g r a v i t y / G P S sites to i m p r o v e the trend c o m p o n e n t and signal c o v a r i a n c e functions are next possible steps.
Acknowledgements The authors gratefully acknowledge Dr. L.L.A. Vermeersen for kindly providing normal mode codes and GSD, Natural Resources, Canada for providing GPS data, as well as the funding received by the GEOIDE NCE. The authors are also very grateful to reviewers for their constructive criticism and suggestions. Most of the figures are potted by The Generic Mapping Tools (Wessel and Smith 1998).
References Carrera, G., P. Vanf6ek and M.R. Craymer (1991). The compilation of a map of recent vertical crustal movements in Canada, Open file 50SS.23,244-7-4257, Dep. of Energy, Mines and Resour., Ottawa, Canada. Chert S., C.F.N. Cowan and P.M. Grant (1991). Orthogonal Least Squares Learning Algorithm for Radial Basis Function Networks. IEEE Transactions on Neural Networks, Vol. 2, No. 2, pp. 3 0 2 - 309. Danielsen, J.S. (2001). A land uplift map of Fennoscandia. Survey Review, Vol.36, pp. 282-291. Dziewonski, A.M. and D.L. Anderson (1981). Preliminary reference Earth model. Phys. Earth Planet. Inter., Vol. 25, pp. 279-356. Fang, M. and B.H. Hager (2001). Vertical deformation and absolute gravity. Geophys. J. Int., Vo1.146, pp. 539-548. Farrell, W.E. and J.A. Clark (1976). On postglacial sea level. Geophys. J. R. Astr. Soc., Vol. 46, pp. 647-667. Fotopoulos, G. (2003). An Analysis on the Optimal Combination of Geoid, Orthometric and Ellipsoidal Height Data. Doctoral thesis. URL: http://www, geomatic s. uc al gary. c a/links/GradThes es. html. Hardy, R.L. (1990). Theory and applications of the multiquadric-biharmonic method: 20 years of discovery 1968-1988. Computers Math. Applic., Vol.19 (8/9), pp. 163-208. Heck, B. (1984). Zur Bestimmung vertikaler rezenter Erdkrustenbewegungen und zeitlicher )~nderungen des Schwerefeldes aus wiederholten Schweremessungen und
Nivellements. Deutsche geodi:itische commission, Reihe C, Nr.302, Mtinchen 1984. Hein, G.W. and R. Kistermann (1981). Mathematical foundation of non-tectonic effects in geodetic recent crustal movement models. Tectonophysics, Vol. 71, pp. 315-334. Heiskanen, H. and H. Moritz (1967). Physical Geodesy. Freeman, San Francisco. Holdahl, S. and R.L. Hardy (1979). Solvability and multiquadric analysis as applied to investigations of vertical crustal movements. Tectonophysics, Vol. 52, pp. 139-155. Milne, G.A., J.X. Mitrovica, J.L. Davis (1999). Near-field hydro-isostasy: the implementation of a revised sea-level equation. Geophysical Journal International, Vol. 139, pp. 464-482. Mitrovica, J.X. and W.R. Peltier (1991). On Postglacial Geoid Subsidence over the Equatorial Oceans. J. Geophys. Res., Vol. 96, No B 12, pp. 20053-20071. Moritz, H., (1980). Advanced Physical Geodesy. H.Wichmann, Karlsruhe, Germany. Pagiatakis, S.D. and P. Salib (2003). Historical relative gravity observations and the time rate of change of gravity due to postglacial rebound and other tectonic movements in Canada. J. Geophys. Res., Vol. 108, No B9, 2406, doi:10.1029/2001JB001676, 2003. Peltier, W.R. (1999). Global sea level rise and glacial isostatic adjustment. Global and Planetary Change, Vol. 20, pp. 93-123. Peltier, W.R. (2004). Global Glacial Isostasy and the Surface of the Ice-Age Earth: The ICE-5G (VM2) Model and GRACE. Annu Rev. Earth Palnet. Sci., Vol. 32, pp. 111. Sj6berg, L.E. (1982). Studies on the land uplift and its implications on the geoid in Fennoscandia. University of Uppsala, Institute of Geophysics, Department of Geodesy, Report No 14. Tscherning, C.C., D. Arabelos and G. Strykowski (2000). The 1-cm geoid after GOCE. In: International Association of Geodesy Symposia, Vol. 123 Sideris (ed.): Gravity, Geoid, and Geodynamics 2000, Springer- Verlag Berlin Heidelberg 2001, pp. 267-270. Thushingam, A.M. and W.R. Peltier (1991). ICE-3G: a new global model of late Pleistocene deglaciation based upon geophysical predictions of post-glacial relative sea-level change. J. Geophys. Res., B 96, pp. 4497-4523. Vanf6ek, P. and D. Christodulidis (1974). A method for the evaluation of the vertical crustal movement from scattered geodetic relevelling. Canadian Journal of Earth Sciences, Vol. 11, No 5, pp. 605-610. Vanf6ek, P. and D. Nagy (1981). On the compilation of the map of contemporary vertical crustal movements in Canada. Tectonophysics, 71, pp. 75- 86. Vanf6ek, P, and W.E. Featherstone (1998). Performance of three types of Stokes's kernel in the combined solution for the geoid. Journal of Geodesy, 72, pp. 684-697. V6ronneau, M. (2001). The Canadian vertical datum: A new perspective for the year 2005 and beyond, Presented at lAG Symposium 124 Vertical Reference Systems, Cartagena, Columbia, Feb. 20-23,2001. Wessel, P., and W.H.F. Smith (1998). New, improved version of Generic Mapping Tools released, EOS Trans. Amer. Geophys. U., Vol. 79 (47), pp. 579. Wu, P. and W.R. Peltier (1982). Viscous gravitational relaxation. Geophys. J. R. Astron. Soc., Vol. 70, pp. 435486.
Chapter 83
Optimal Seismic Source Mechanisms to Excite the Slichter Mode S. Rosat National Astrogeodynamics Observatory, Mizusawa, Iwate, 023-0861 Japan [email protected]
Abstract. The recent M w = 9.3 SumatraAndaman earthquake on December 2 6 th 2004 has strongly excited the low-frequency seismic modes of the Earth. In particular the large amplitude of the fundamental radial mode 0S0, which consists of a roughly uniform compression and dilatation of the whole Earth, and of the seismic core modes, like 3S2, means that much energy was radiated toward the core. Also the degree one mode 2S~, which corresponds to a translation of the core in the mantle, was clearly observed for the first time without any stacking process on seismometers and on gravimeters. 2S, is the first harmonic of the degree one mode ,S,, the so-called Slichter triplet. In fact, the Slichter mode is not an elastic spheroidal mode like 2S, since its main feedback force is the Archimedean force. The Slichter mode corresponds to a translation of the solid inner core inside the fluid outer core. The theory concerning this mode is still poorly constrained and no convincing detection has been suggested yet. This paper presents theoretical predictions of the amplitude of the Slichter mode after the 2004 Sumatra-Andaman earthquake as well as after some largest events in the past decades. The computation is based on the usual normal mode theory for a spherically symmetric non-rotating Earth model. The source parameters are finally investigated to find the optimal seismic mechanism to excite the translational motion of the inner core. Keywords.
Normal modes, gravimeter, Slichter modes
superconducting
1 Introduction The development of a global Superconducting Gravimeters network under the framework of the Global Geodynamics Project (Crossley et al., 1999)
has given new opportunities of studies. These instruments are very stable and the recent generations are less noisy than the spring gravimeters and even less noisy than the longperiod seismometers for frequencies below 1 mHz (Van Camp, 1999; Widmer-Schnidrig, 2003; Rosat et al., 2003-2004). The large Sumatra-Andaman earthquake that occurred on December 26, 2004 with a moment magnitude larger than 9 (M w = 9.3 by Stein and Okal, 2005; M w = 9.15 by Park et al., 2005), has strongly excited the low-frequency seismic modes; in particular the fundamental radial mode 0S0, associated with changes in the Earth's circumference, has been largely excited. Although the source was shallow, the earthquake was strong enough to excite the core sensitive seismic modes. Therefore much energy was radiated toward the core. The harmonic degree one seismic mode 2S, that can be seen as a translational oscillation of the core has been observed for the first time directly on individual records of seismometers and gravimeters (e.g. Rosat et al., 2005a). In the normal mode theory, the seismic mode 2S1 is the first harmonic of the sub-seismic mode ,S,, the so-called Slichter mode (Slichter, 1961). The Slichter mode corresponds to a translation of the solid inner core inside the fluid outer core. The predicted period of this mode is 5.42 h for the seismological reference Earth's model PREM (Dziewonski and Anderson, 1981). The latest theoretical developments predict that the period of the Slichter mode is between 4 h (Rieutord, 2002) and 6 h (Rogister, 2003). Since the first claim by Smylie (1992) of a possible detection of the Slichter mode, many unfruitful attempts have been performed to detect this mode in superconducting gravimeter records (Hinderer et al., 1995; Jensen et al., 1995; Rosat et al., 2003; 2005b). The observation of this free oscillation of the inner core is fundamental: since the main restoring
572
S. Rosat
forces are the Archimedean forces, the period of the Slichter mode is directly linked to the density jump at the inner core boundary (ICB). The PREM model predicts a density jump at the ICB of 600 kg/m 3. However two recent papers have reached diverging conclusions whether the actual density jump is larger or smaller than for PREM model. The first paper, by Masters and Gubbins (2003), is based on the analysis of the free oscillations of the Earth and proposes that the density jump at the inner core boundary could be as large as 800 kg/m 3. While the second paper by Koper and Pyle (2004) suggests, from the analysis of amplitude ratios of PKiKP/PcP reflected waves that the density jump at the ICB could be as small as 300 kg/m 3. It is in fact difficult to constrain the density jump from these two techniques as, in the case of the seismic modes, the resolution is weak and in the case of the seismic waves, the phases used have very low signal-tonoise ratios. Rosat et al. (2005b) have developed a detection tool based on the splitting, by rotation and ellipticity of the Earth, of the Slichter mode into a triplet of frequencies in their search of the surface gravity effect of the Slichter mode in superconducting gravimeter (SG) data. They computed the splitting for different PREM-like Earth's models having density jumps at the ICB ranging from 300 to 900 kg/m 3. However no convincing evidence for a probable Slichter triplet has been found. The objective of this paper is to investigate a possible excitation of the Slichter mode by the 2004 Sumatra earthquake and by the past largest events like the 1960 Chile and to estimate the best seismic source mechanism to excite ,S,. In the next section we present the formulations to compute synthetic seismograms in a spherically symmetric nonrotating Earth. In section 3, a comparison of our results to previous computations is presented and then, in a last part, we discuss the seismic source mechanism to excite the Slichter mode.
2 Amplitude excitation computation in a spherically symmetric Earth 2.1 Moment tensor response In the following, we use the notation U, V and P for the radial eigenfunctions which respectively correspond to the radial displacement, tangential displacement and the Eulerian potential perturbation. The response of the Earth to a moment
tensor source has been explained in details by Dahlen and Tromp (1998). We only recall here the main equations we have used to compute the amplitude excitation of the harmonic degree one Slichter mode to a moment tensor source M situated at x = (r, t3, qb) in the spherical coordinate convention used here. The receiver is located at x = (r, 0, qb). The angular epicentral distance ® between the source and the receiver is given by cos ® = cos 0 cos 0~ + sin 0 sin 0~ cos (qb- qb) and we note • the azimuth to the receiver measured counter clockwise from due south at the source. The excitation amplitude A x at the receiver x is expressed by: A×(x) =
(2/+llD(r,O, OO)a(o,o0),(l) 4zc
where D is the displacement operator that in the case of the response of a gravimeter in a nonrotating Earth reduces to the radial displacement eigenfunction U at the receiver position x. The real scalar function A is defined for degree harmonic 1=1 modes by: A ( O , ~ ) = A 0 c o s O + A 1 sinOcos~+BlsinOsin • (2) where A o, A, and B, have the following expressions:
A0 -
/
+ (Moo +
' v,-
' kv,
,
The equations (3) result from the contraction of the moment tensor with the deformation tensor. The components M 0 are the elements of the moment tensor M given a s :
M-- MrO MOO MOO The dot on the radial eigenfunctions U and V denotes the derivative with respect to the radius vector r and the subscript s means that it is evaluated at the source x S. k (1(1+1)) 1/2 = x/2. The radial eigenfunctions U, V and P have been computed as in Crossley (1975) and are plotted in Figure 1. These radial functions are close to zero near the Earth's surface, yet they contribute directly (and to the square) to the excitation amplitude through the displacement operator D = U in =
Chapter 83 • Optimal Seismic Source Mechanisms to Excite the Slichter Mode
equation (1) and through the deformation tensor (derivative of the radial functions) implicitly present in equations (3). Therefore, near the Earth's surface, the excitation amplitude is the square of a very small quantity. Dis ~lacements
Ufree -
2 co-2g a-' U and
Upo t -
(1+ 1) co-2a-' P,
where a is the mean Earth's radius, co the pulsation of the angular order 1 mode (in our case/=1) and g is the acceleration of gravity. The relative magnitudes of the inertial, free-air and potentialperturbation for 1S1 are compared in Table 1 with the values obtained by Dahlen and Tromp (1998). The values differ only in the case of the potentialperturbation contribution because of differences in the radial eigenfunction P.
CMB
CMB
We can now compute the excitation amplitude of the Slichter mode after some major earthquakes.
.
.
.
.
ICB
Table 1. Relative magnitudes of the inertial, free-air and potential-perturbation contributions to the accelerometer response for the Slichter mode 1S1. The tabulated values are for the PREM model.
Fig. 1 Displacement and potential eigenfunctions of the Slichter mode.
U
and V
are respectively the radial
displacement and tangential displacement and P (right hand side plot) is the perturbation of the gravitational potential.
Dahlen and
Tromp (1998) This paper
U/U*
UereJU*
Upo/U*
0.032
0.960
0.008
0.032
0.967
2.2e-5
ICB refers to the inner core boundary and CMB to the coremantle boundary. From equation (2), the excitation amplitude of the Slichter mode can be estimated. This computation takes into account the self-gravitation but not the gravitational effects on the instrument itself, in addition to the gravitational effect on the housing (Dahlen and Tromp, 1998, pp. 143-144), that are quite significant for some of the low-frequency spheroidal oscillations, like ,S,.
2.2 Instrumental gravitational effect A gravimeter as well as a seismometer on the Earth's surface responds to changes in the Earth's gravitational field in addition to the acceleration of the instrument housing. In order to account for the free-air change in gravity due to the radial displacement of the instrument Ufree and for the perturbation P in the gravitational potential due to the redistribution of the Earth's mass Upot, the radial displacement eigenfunction U must be replaced by U :~ -- U "/r" Ufree "~- Upo t. The free-air effect Ufree dominates the vertical response of the Slichter mode 1S, so we must incorporate these self-gravitational corrections in the excitation amplitude A x. Ufree and Upot are defined by:
3 Computation results after past major earthquakes We consider the 1960 Chilean, the 1964 Alaska, the 1994 Bolivian, the 2001 Peruvian and the 2004 Sumatra earthquakes. The source mechanisms used and the predicted excitation amplitudes of ,S, for the spherically symmetric PREM model at 10 SG sites well-distributed on the globe (see figure 2) are summarized in Table 2. The last row shows the results on the whole Earth's surface by Crossley (1992) based on the method described in Crossley (1988). To estimate his results at each SG site, the value in Table 2 must be multiplied by the degree one spherical harmonic function at the station coordinates, namely by the sine of the latitude in a spherical model. His results predict an amplitude excitation of the Slichter mode of the same order than our computation. SGs are presently the most sensitive instruments in the Slichter mode frequency range. However their detection threshold is of the order of 1 nGal (10 -12 g) and the noise level of the best sites is a few nGal in the sub-seismic frequency band (Rosat et al., 2004). So in order to be able to detect the Slichter mode in SG records, its amplitude should reach at least 1 nGal at the Earth's surface. From
573
574
S. Rosat
T a b l e 2, o n l y the 1960 C h i l e a n e a r t h q u a k e c o u l d
Chilean
h a v e s u f f i c i e n t l y e x c i t e d 1S1 for it to b e d e t e c t e d b y
Kanamori and Cipar,
S G on the E a r t h ' s surface, if w e c o n s i d e r the t w o
source.
I
l:~¢'cc'~,on l,lr,,w..l of ~
~
Ny-Alesund
l
L ± ~ I
Cant ley
_ 7 i ~ ~ I
[:~t¢ctI¢io~1 ofn~ l % Boulder
iil°,
lil
~
I~,.,:
.......
(foreshock
..
y
Iil
Sutherlnnd
~--
I
~ ~
~..,~.. ! 1
~ I
=2
°
shock,
l:~l~le£tlle~l on of
~,~o,~o o" ~. ~i'~ro~ ~80"
t-
main
Strasb°ur9
•
•
i
.._t,
. . :_o ~
and
1974) in o n e m a i n s e i s m i c
-~'-~i i I Karnioki3
,
iii ,
events
N
0o,u, ;--
Sy0~0 , I H~ o_ I :
~
ii
i
= II
~
~
~
:~
Fig. 2 Excitation amplitude of the Slichter mode at 10 SG sites for the seven seismic events considered in Table 2. From left to right, at each site, the events considered are the 1960 Chile 1, Chile 2, Chile 1+2, 1964 Alaska, 1994 Bolivia, 2001 Peru and 2004 Sumatra earthquakes. The horizontal line corresponds to the 1 nanoGal detection threshold. Table 2. Excitation amplitude of the Slichter mode at various SG sites after the major past earthquakes for PREM model. Event Date Moment (N.m) Mw Depth (km) Dip (o) Strike (°) Slip (o) Reference Site (latitude in degrees N)
Chile l 1960 2.7 1023
Chile2 1960 3.5 1023
9.5 25 10 170 80 Kanamori
Chile 1+2 1960 6.2 1023
Alaska 1964 7.5 1 0 22
Bolivia 1994 2.6 1021
9.2 8.2 8.4 9.6 9.8 50 38 640 30 50 10 10 20 10 18 170 170 114 302 310 80 80 270 -60 63 Harvard CMT* and Cipar (1974) Kanamori (1970) Surface gravity effect in nGal (= 10-2 nm/s2) for PREM model
Sumatra 2004 1.1 1023 9.31 28 8 329 110
Boulder (40.13)
0.263
0.342
0.605
0.117
0.005
0.004
0.253
Canberra (-35.32) Cantley (45.58)
0.399 0.006
0.519 0.008
0.918 0.014
0.175 0.160
0.004 0.005
0.006 0.0007
0.049 0.198
Kamioka (36.43) Ny-Alesund (78.93)
0.503 0.019
0.654 0.024
1.157 0.043
0.047 0.196
0.005 0.007
0.002 0.007
0.177 0.179
Strasbourg (48.62) Sutherland (-32.38)
0.353 0.656
0.459 0.853
0.811 1.509
0.193 0.037
0.006 0.004
0.010 0.006
0.042 0.286
Syowa (-69.01) Tigo-Concepcion (-36.84)
0.350 0.243
0.455 0.315
0.804 0.558
0.153 0.074
0.007
0.001
0.277
Bandung (-6.90)
0.173
0.225
0.398
0.068
0.005 0.0005
0.007 0.004
0.093 0.072
Crossley (1992) (whole surface) 0.724 0.835 1.52 0.336 I 0"022 Stein and Okal (2005) 2 personal communication * The Harvard Centroid Moment Tensor is available at: http://www.seismology.harvard.edu/CMTsearch.html 1
Peru 2001 4.7 1021
_
Chapter 83
•
Optimal Seismic Source Mechanismsto
Excite the Slichter Mode
Table 3. Moment magnitude needed to excite the Slichter mode at the nanoGal level in surface gravity effect for different ideal source mechanisms. The source is situated at a depth of 500 km on the equator at the longitude 90 ° and the receiver is located at the longitude 40°E also on the equator. Type
Moment Tensor
/°iJ /i°)
Explosion
1
Vertical dip-slip
0
1
Pure compensated linear vector dipole
"eyeball" or "fried-egg"
Crossley (1988) came to the conclusion that even under very favourable conditions (i.e. an M w - 9.3 earthquake), it is not possible to exceed a 1 nGal signal at the Earth's surface. We have confirmed it by our computation after the M w - 9.3 Sumatra earthquake. The maximum excitation amplitude expected is 0.3 nGal at the ideal receiver position from the source, which is 47 ° N and 210 ° E.
4 Discussion Intuitively we can think that to excite the free oscillation of the inner core, we need a very deep source with a fault mechanism that enables to radiate as much energy as possible toward the core. This type of focal mechanism would be a vertical dip-slip source. We have computed the moment magnitude required to excite the S lichter mode at the nanoGal level in surface gravity effect, for different focal mechanisms. The results are summarized in table 3. A vertical strike-slip mechanism should not happen and can not excite the S lichter mode. So it is not considered here. The best natural focal mechanism to excite ~S~ is therefore a vertical dip-
dip 10 ° slip 80 °
dip 8° slip 110 °
Fig. 3 Focal mechanisms of the 1960 Chilean earthquake (left) and of the 2004 Sumatra event (right).
/i0 / 0
~/~
0
Mw 9.6
0 ~
45 °-dip thrust
Focal mechanism
I
)
)
9.7
9.8
1
/ ° i/
9.8
r-2 0 0" 0 1 0 q6 0 0 1
9.6
1
~ -2
'/6 , 0 1
/'7"_
slip source as we need a smaller energetic earthquake than other mechanisms to excite 1S1 at the nanoGal level. However compared to a 45°-dip thrust for instance, the difference in magnitude is only of the order of one tenth of a magnitude unit. A pure compensated linear vector dipole, indicative of simultaneous vertical extension and horizontal compression, is not a usual focal mechanism. It has been observed in shallow earthquakes of moderate size (M>5) beneath volcanoes (Nettles and Ekstr6m, 1998). An "eyeball" mechanism would give the largest amplitude excitation of 1S1. However such non-double-couple source mechanisms are unusual and usually associated with shallow earthquakes induced by volcanism or geothermic, so they can unlikely induce an Mw = 9.6 earthquake. Both 1960 Chile and 2004 Sumatra earthquakes were dip-slip sources. If we consider their focal mechanisms represented in figure 3, we can notice that the Chile fault plane was closer to a pure vertical dip-slip focal mechanism than the Sumatra earthquake. If we consider the 1960 Chile source parameters with different moment magnitudes, we obtain the upper graph of figure 4. It represents the estimated excitation amplitude of ~$1 as a function of the magnitude Mw. With a mechanism similar to the one of 1960 Chile event, the nanoGal level is reached with a moment magnitude of about 9.7. The combination of Chile 1 and 2 events had a moment magnitude of 9.8.
575
576
S. Rosat
Now, if we consider the source parameters of the Chile event but we modify the source depth, we obtain the lower graph of figure 4. We have also computed the excitation amplitude of 1S1 as a function of the source depth with the source parameters of the 2004 Sumatra event (dot-dashed line of lower graph in figure 4). With source parameters similar to the Chile 1+2 event and with a moment magnitude of 9.8, the nanoGal level in surface gravity is reached even for shallow sources. In the case of the Sumatra earthquake, even with a source in the lower mantle, the excitation amplitude of the Slichter mode does not reach the nanoGal signal amplitude at the surface. Some "core-quakes" would be needed, which is impossible.
10 -1
i 10_2 ......
12.G._al.d._e
10_3
__=1o-4 E
F
0- 5
10 .6
10 .7 6
f
/ ~ 6.5
~ 7
~ 7.5
~ 8 i
~ 8.5
~ 9
i
10
w
ICB
CMB
1¢
7
~ 9.5
i
..... 2004 M = 9.3 Sumatra w
102
~
--
1960Mw - 9.8 Chile 1+2
10°
© x
< 10-4 mantle
10-6 0
inner core
outer core
i
i
i
i
i
i
1000
2000
3000
4000
5000
6000
From figure 4 we can clearly see that the excitation amplitude depends much more on the magnitude of the earthquake than of its source depth. The effect of the source depth is represented by 1/rs in equations (3), while the seismic moment directly scales the excitation amplitude. The weak influence of the source depth is mainly due to the radial eigenfunctions, which are almost constant with radius in the mantle. In fact they are even close to zero (figure 1).
5 Conclusion The excitation of the translational motion of the inner core by a seismic source is difficult to detect via its surface gravity effect even with the superconducting gravimeters, which are presently the most appropriate instruments. A 1960 Chile-like earthquake with a vertical dip-slip mechanism and huge energy release (Mw > 9.7) is the only way to excite the Slichter mode with amplitude that can reach and even overpass the nanoGal level in surface gravity effect, if we consider the usual normal mode theory in a spherically symmetric PREM Earth's model. The introduction of a more realistic Earth's model that accounts for the rotation, ellipticity and lateral heterogeneities will be necessary to complete this study; however we can a priori think that it will not increase the amplitude excitation of the Slichter mode. Smith (1976) has demonstrated that the surface gravity effect of the Slichter triplet is strongly dependent on the stability or not of the density stratification of the fluid outer core. This point will also have to be further studied. A seismic excitation of the Slichter mode is not the best way to have it detectable at the Earth's surface. Therefore we must consider other possible sources of excitation, like turbulent flows in the liquid outer core at the core-mantle boundary or an atmospheric and oceanic (individually or coupled) excitation. No studies have been done yet about a possible excitation of the Slichter mode by one of these phenomena, so the debate is still open.
Source depth (kin)
Acknowledgements Fig. 4 Influence on the Slichter m o d e ' s surface excitation amplitude of (upper plot) the magnitude of an earthquake with focal mechanism similar to the one of the 1960 Chilean (Chile 1+2) event and of (lower plot) the source depth of an earthquake with focal mechanism respectively similar to the 2004 Sumatra event (dot-dashed line) and to the Chilean (Chile 1+2) event (solid line).
1960
The author is grateful to David Crossley for his code to compute the eigenfunctions of the Slichter mode and to Luis Rivera and Jacques Hinderer for their useful comments on this paper. This study is granted by the Japan Society for the Promotion of Science.
Chapter 83 • Optimal Seismic Source Mechanisms to Excite the Slichter Mode
References Crossley, D.J. (1975). The free-oscillation equations at the centre of the Earth. Geophys. J. R. Astron. Soc., 41, pp. 153-163. Crossley, D.J. (1988). The excitation of core modes by earthquakes. In: D.E. Smylie and R. Hide (Editors), Structure and Dynamics of Earth's Deep Interior. Geophys. Monogr. Am. Geophys. Union, 46(1), pp. 41-50. Crossley, D.J. (1992). Eigensolutions and seismic excitation of the Slichter mode triplet for a fully rotating Earth model, EOS, 73, p. 60. Crossley, D., Hinderer, J., Casula, G., Francis, O., Hsu, H.T., Imanishi, Y., Jentzsch, G., K~i~iri~iinen, J., Merriam, J., Meurers, B., Neumeyer, J., Richter, B., Shibuya, K., Sato, T. and T. Van Dam. (1999). Network of superconducting gravimeters benefits a number of disciplines, EOS, 80, 11, pp. 121/125-126. Dahlen, F.A. and J. Tromp (1998). Theoretical Global Seismology, Princeton: Princeton Univ. Press., Princeton, NJ, 1025 pp. Dziewonski, A. M. and D. L. Anderson (1981). Preliminary reference Earth model (PREM), Phys. Earth Planet. Int., 25, pp. 297-356. Hinderer, J., Crossley, D. and O. Jensen. (1995). A search for the Slichter triplet in superconducting gravimeter data, Phys. Earth Planet. Int., 90, pp. 183-195. Jensen, O.G., Hinderer, J. and D.J. Crossley. (1995). Noise limitations in the core-mode band of superconducting gravimeter data, Phys. Earth Planet. Int., 90, pp. 169-181. Kanamori, H. (1970). The Alaska earthquake of 1964: Radiation of long-period surface waves and source mechanism. J. Geophys. Res., 75, pp. 5029-5040. Kanamori, H. and J. J. Cipar (1974). Focal process of the great Chilean earthquake May 22, 1960. Phys. Earth Planet. Int., 9, pp. 128-136. Koper, Keith D. and L. Moira Pyle (2004). Observations of PKiKP/PcP amplitude ratios and implications for Earth structure at the boundaries of the liquid core. J. Geophys. Res., 109(B3), B03301 10.1029/2003JB002750. Masters, G. and D. Gubbins (2003). On the resolution of the density within the Earth, Phys. Eart Planet. Int., 140, pp. 159-167. Nettles, M. and G. Ekstr6m (1998). Faulting mechanism of anomalous earthquakes near Bfirdarbunga Volcano, Iceland, J. Geophys. Res., 103 (B8), pp. 17,973-17,984.
Park, J., Song, T-R A., Tromp, J., Okal, E., Stein, S., Roult, G., Clevede, E., Laske, G., Kanamori, H., Davis, P., Berger, J., Braitenberg, C., Van Camp, M., Lei, X., Sun, H. and S. Rosat (2005). Earth's Free Oscillations Excited by the 26 December 2004 Sumatra-Andaman Earthquake. Science, 308, pp. 1139-1144. Rieutord, M. (2002). Slichter modes of the Earth revisited. Phys. Earth Planet. Int., 131, pp. 269-278. Rogister Y. (2003). Splitting of seismic free oscillations and of the Slichter triplet using the normal mode theory of a rotating, ellipsoidal Earth. Phys. Earth Planet. Int., 140, pp. 169-182. Rosat, S., Hinderer, J., Crossley, D. and L. Rivera (2003). The search for the Slichter mode: Comparison of noise levels of superconducting gravimeters and investigation of a stacking method. Phys. Earth Planet. Int., 140, pp. 183-202. Rosat, S., Hinderer, J., Crossley, D. and J.-P. Boy (2004). Performance of superconducting gravimeters from longperiod seismology to tides. J. of Geodynamics, 38, 3-5, pp. 461-476. Rosat, S., Sato, T., Imanishi, Y., Hinderer, J., Tamura, Y., McQueen, H. and M. Ohashi (2005a). High resolution analysis of the gravest seismic normal modes after the 2004 Mw=9 Sumatra earthquake using superconducting gravimeter data, Geophys. Res. Lett., 32, L13304, doi: 10.1029/2005GL023128. Rosat, S., Rogister, Y., Crossley, D. and J. Hinderer, J. (2005b). A search for the Slichter Triplet with Superconducting Gravimeters: Impact of the Density Jump at the Inner Core Boundary. J. of Geodynamics, accepted August 30, 2005. Slichter, L. B. (1961). The fundamental free mode of the Earth's inner core. Proc. Nat. Acad. Sci., 47 (2), pp. 186-190. Smith, M.L. (1976). Translational Inner Core Oscillations of a Rotating, Slightly Elliptical Earth. J. Geophys. Res., 81 (17), pp. 3055-3065. Smylie, D.E. (1992). The Inner Core Translational Triplet and the Density Near Earth's Center, Science, 255, pp. 1678-1682. Stein S. and E. Okal (2005). Speed and size of the Sumatra earthquake. Nature, 434, p. 581. Van Camp, M. (1999). Measuring seismic normal modes with the GWR C021 superconducting gravimeter. Phys. Earth Planet. Int., 116, pp. 81-92. Widmer-Schnidrig, R. (2003). What can superconducting gravimeters contribute to normal mode seismology?. Bull. Seismol. Soc. Am., 93 (3), pp. 1370-1380.
577
Chapter 84
Recent dynamic crustal movements in the Tokai Region, Central Japan, observed by GPS Measurements S. Shimada and T. Kazakami Solid Earth Research Group, National Research Institute for Earth Science and Disaster Prevention (NIED), 3-1 Tennodai, Tsukuba-Shi, Ibaraki-Ken 305-0006 Japan e-mail: [email protected]; Tel.: +81-29-863-7622; Fax: +81-29-854-0629
Abstract. We present the recent dynamic crustal movements in the Tokai Region, Central Japan, observed by Global Positioning System (GPS) measurements. Since October 2000 the abnormal crustal movements began to move southeastward in the area inland Tokai Region, the reverse motion to the secular crustal movements, suggesting the slow slip between the subducting Philippine Sea slab and the subducted southwest Japan (SWJ). The abnormal crustal movements were fast and wide during October 2000 and December 2001, and slow and narrow during January and December 2002, and again fast and wide during January 2003 and April 2004. Since around June 2004, the abnormal crustal movements became again slow and narrow, and then in September 2004 a M7.4 earthquake occurred at around 200 km southern southwest of the Tokai Region near the Nankai Trough. Co-seismic motions are southward 10 - 30 mm displacement in the Tokai Region. Since the occurrence of the earthquake the southward motion began in the southeast end of SWJ including the Tokai Region. On the other hand the abnormal crustal movements are still ongoing to be slow and narrow.
network (Fig. 1.2) routinely, until midst of 2000, secular crustal movements consistent with the tectonic setting in the region progressed. GEONET is the Japanese nation-wide GPS permanent network established and managed by Geographical Survey Institute (GSI) (Miyazaki et al. (1998)). Abnormal crustal movement began around October 2000, after the wide-area crustal motion occurred associated with the earthquake swarm and dyke intrusion near Kozu Island about 150 km southeast of the Tokai Region during the period of July and August 2000. The slow event indicates the reverse motion in the Tokai Region and caused by the aseismic slip of
i
36 ° •
~ /. / /
'-~%---~ ~"
I
'
I,," .{-...,l
"_ I
Keywords: Global Positioning System., Crustal movement, Slow slip, Plate subducting zone, Tokai Region
1. I n t r o d u c t i o n
In the Tokai Region, Central Japan, the Philippine Sea Plate (PHS) is subducting along the Suruga Trough beneath SWJ (Fig. 1.1), and usually subducted Tokai Region is compressed and deformed north-westward. Since 1996 when the National Research Institute for Earth Science and Disaster Prevention (NIED) began the analysis of the part of the Tokai Region of the GEONET
~' •
~~,/~
[I J 1,36"
PHS 138"
_22
Fig. 1.1. Tectonic Setting around the Tokai Region. The Izu Peninsula, the northern tip of the Philippine Sea Plate (PHS) is colliding to the main part of Honshu. PHS is subducting along the Suruga trough and Nankai Trough beneath the southwest Japan (SWJ), and along the Sagami trough beneath the northeast Japan (NEJ). The 2004 South coast of Honshu earthquake occurred near the Nankai Trough in the PHS slab.
Chapter 84
•
Recent DynamicCrustal Movements in the Tokai Region,Central Japan,Observed by GPS Measurements
I ÷ USUD
36"
i....
* USUD IGS SITE . N IED SITi;
I
. GEONET SITE
35-5" I
i
=, ,,, " ,,, ,, •
II i I
,"
. . . . ,
•
,,.I
-.
.
.,
L.
,,
u:>
.=-" .. •
,-:
.
.
•
.
• -
~
-', ,
34.5"~
L-
,
POINT
__..', PAc'~tc oc~N 137,5"
138"
• •
138.5"
I SEP 1996 - JUN 2000
~USUD
.s
~." ~.
-, 139"
.,
,.,
\
13c)..5"
Fig. 1.2. GPS sites monitored in NIED with the tectonic setting of the area. The dashed line is the focal region of the hypothesized Tokai earthquake, which is thought to occur between SWJ and subducting PHS slab. The Usuda (USUD) IGS site indicates solid diamond, the NIED GPS sites solid squares, and the GEONET sites solid circles.
SWJ against the Philippine Sea slab (Ozawa et al., 2002). On this paper we will introduce the time evolution of the abnormal crustal movements, which seems to be still ongoing. From the observation of the borehole tiltmeter, the slow event also occurred around 1988 although the period is only about two years and the magnitude of the slow event is smaller than the ongoing event (Yamamoto et al., 2005). On September 2004, a M7.4 large earthquake occurred at about 200 km southern southwest of the Tokai Region near the Nankai trough where PHS is subducting to SWJ. The earthquake affects considerably on the crustal movements in the Tokai Region. This paper also mentions on the co-seismic displacement and after slip of the earthquake. 2. A n a l y z i n g
r e f e r e n c e site
".~ -~ /
" /
,'_/._
I
• "/-',,;.*-'~/ "'~ • / /.
~J._W, ~ ,~7> ~g/~
L--...... ,ok~~ -~--~..~ , ./ / :HAMANA OMAEZAKt • !
Q
GLOBK GPS analyzing software (King and Bock, 2004; Herring 2004). Some of the GEONET sites, mainly western part of Tokai Region, are excluded during September 1996 and December 1999, some of the GEONET sites are abandoned during the period, and some of the GEONET and NIED sites established in the midst of the period.
data
Fig. 1.2 shows the analyzing network in Tokai Region, the Izu Peninsula, and the north-west of the Izu Peninsula, where GEONET network sites and NIED network sites are introduced, with the Usuda IGS site (USUD). The north-west of the Izu Peninsula belongs to northeast Japan (NEJ). About 55 GEONET sites and 5 NIED GPS permanent sites exist in the analyzing network by NIED. We analyzed the network data with twenty IGS sites in and around Japan as fiducial site during September 1996 to July 2005. We use GAMIT/
y
0
m.r
Fig. 3.1. The velocity field during September 1996 and June
2000 as background secular motion. Circles indicates 90% confidential ellipses.
3.
Background
ongoing
slow
secular
motion
and
event
Fig. 3.1 shows usual secular motion as the velocity field during September 1996 and June 2000. In the Tokai Region, the velocity is largest at the Omaezaki point locating the southeast end of SWJ, and the large velocity along the coast facing the Suruga Trough and along the area facing the Pacific Ocean, decreasing inland Tokai Region. The mean velocity field of the ongoing abnormal crustal movements is shown in Fig. 3.2 for the period of October 2000 and July 2005 in the inland area of SWJ. Excluding the area in and around Izu Peninsula, the fastest velocity can be seen in the area northeast of Lake Hamana, just covering some part of the boundary of the focal area of the Tokai earthquake. The area of abnormal crustal movements extended to the southern coast region facing the Pacific Ocean and the west coast of the Suruga Bay. The western extension was not obvious in the term of observed velocity. According to Ozawa et al. (2002), the abnormal crustal movements are extend-
579
580
S. Shimada• T. Kazakami
O C T 2000 - JUL 2005
• U S U D reference site
JAN 2002 - DEC 2002
,
• U S U D reference site
,
L_ -.--=--,'Y I
:'
'20 mm/yr
Fig. 3.2. The velocity field during October 2000 and July 2005, showing the velocity field of the whole period of the slow event ongoing. Circles indicate 90% confidential ellipses.
ed 50 km more to the west in SWJ.
4. T i m e e v o l u t i o n
of the slow event
By estimating the velocity fields using shorter time span, it is possible to study the episodic feature of the slow event. The time evolution of the velocity field is seen in Figures 4.1 to 4.3. During October 2000 and December 2001, the area affected by the slow event was very wide, extending to the areas
O C T 2000 - DEC 2001
• U S U D reference site
Fig. 4.2. The velocity field during January and December 2002. Circles indicate 90% confidential ellipses.
facing the Pacific Ocean in the south and facing the coast of the Suruga Bay to the east (Fig. 4.1). The affected area must continue to the ocean bottom of the Pacific Ocean in the south. The velocity was also very fast in general. During January and December 2002 the area affected by the slow event shrank only to the inland Tokai Region, and the velocity became slower (Fig. 4.2). During the period of February and May 2003, GEONET system was upgraded by replacing the antenna, the antenna radome, and the receiver at each site to minimize the multi-path effects and to improve the phase center variation correction. HowJUN 2 0 0 3 - A U G 2004
~ U S U D reference site
t I
"~
Fig. 4.1. The velocity field during October 2000 and December 2001. Circles indicate 90% confidential ellipses.
i,.,.~
'''~er
.
'~
Fig. 4.3. The velocity field during June 2003 and August 2004. Circles indicate 90% confidential ellipses.
Chapter84
• Recent Dynamic
CrustalMovementsin the Tokai Region, Central Japan, Observed by GPS Measurements
ever, the upgrade caused offsets in the site coordinate solutions in our GPS analysis. GSI also has changed the elevation cut-off angle of the GEONET receiver from 15 ° to 5 ° to improve the vertical repeatability. This change caused significant offset both in horizontal and vertical components of the site coordinate solutions. It is very difficult to obtain the accurate offset estimates due to such a major changes in the observation setup. Therefore we omitted the data during January and May 2003, Fig. 4.3 shows the annual velocity field during June 2003 and August 2004, revealing that the areas affected by the slow event again became very wide and the velocity was also very fast, almost same as those in the period of Fig. 4.1.
5. Co-seismic and after slip displacements of South coast of Honshu earthquake On September 5, 2004, a M7.4 earthquake occurred at about 200 km southern southwest of the Tokai Region. The earthquake occurred in the PHS slab near the Nankai Trough where PHS is subducting under SWJ. The main shock mechanism is almost pure strike slip approximately perpendicular to the trough axis. The foreshock and the largest after shock are nearly parallel to the trough axis and the
• =_.::-.-
i [
"jd,.~ •,
~ .... :~:I' ,~,
, ..
.... .
•
; ..
,~.~,'~.. . :
...~,~
....- .~-...,:..'
j
l
]20ram Fig. 5.2. Co-seismic displacement at the September 4, 2004, South coast of Honshu earthquake.
mechanisms of the earthquakes are almost pure dip slip in the PHS slab. Fig. 5.1 shows the epicenter of large shocks, rupture displacement contour of large shocks, distribution of the epicenter of the earthquakes during September 5 and 8, and the asperity between SWJ and the PHS slab. Mostly caused by the main shock of the above earthquakes, large co-seismic step arose in the Tokai Region. The amount of step is 1 0 - 30 mm. Fig, 5.2 shows the co-seismic displacement at each site in the network. Fig. 5.3 shows the annual velocity field during
•
i:~?: :~: SEP 12 2004 - JUL 2005
.~.j~
.
.
.
.
~ -.~.:,• •
135
~ U S U D reference site
==:. ~.
•
s
i ~ ~
,~;:
t36
...
[37
138
t39
Fig. 5.1. September 5, 2004, South coast of Honshu earthquake. Stars are the epicenters of large earthquakes. Dots are the epicenters of the earthquakes occurred during September 5 and 8. Black contours are the rupture displacements of large earthquakes. The grey contours are the rupture displacements of the M8.3 Tonankai earthquake in 1944 and the white shades are the asperity between the PHS slab and SWJ. Grey shade and grey contours are the bathymetry (after Yamanaka, 2004).
:
'20 m m / y r
Fig. 5.3. The velocity field during September 12, 2004 and July 2005. Circles indicate 90% confidential ellipses.
581
582
S. Shimada. T Kazakami
TAT-USUD B A S E L I N E V E C T O R
USUD-TSKB BASELINE VECTOR N-S C O M P O N E N T
4
0
COMPONENT 40 .E-W ...........
~
N-S C O M P O N E N T
40..
,
.
,
. . . . .
,
.
.
•
-40 ~. 2000
U-D C O M P O N E N T 60
,
,
,
,
40 2 0
,
.
,
•
,
.
,
.
,
60
,
' 2002
.
'
.
•
,
,
i
. . . . . . . 2000 2002
:
f
~
%00
2006
.
,
.
,
'
.
t
• 2006
-.40/ 2000
,
.
•
I 2002
2004
2006
BASELINE LENGTH
,
•
40
,
,
,
,
,
.....
¢
" 0 ' "
.40f
' ' ' -40 . . . . . . . . . 2006 2000 2002 2004
.
. .,..i,~,~..
2 0 1
2004
. . . .
,,~i. ~oI
._..20~
20
' 2004
. . . . .
~ot
20 ~
~u--
.
,
20
U-D C O M P O N E N T
BASELINE LENGTH 40
'
E-W C O M P O N E N T
40..
t
°
-2_420!0
26o2 2oo4
2oo6
2002
6
2oo4 2006
Fig. 5.4. Time series of the USUD-TSKB baseline during 2000 and 2005. The data are the weekly values estimated from the daily values using Kalman filtering.
Fig. 6.1. Time series of the TAT-USUD baseline during 2001 and 2005. The data are the weekly values estimated from the daily values using Kalman filtering.
September 12, 2004, and July 2005. Although the error ellipses are large, the slow event in the Tokai Region became slow, especially the area around the site 3097, the central area of the former slow slip, and the center of the largest velocity moves to northeast of the former center, to the area around the 0819 site. The fast velocity sites are also seen some sites northeast inland of the Tokai Region, one of them the 3078 site. The sites on the Izu Peninsula and NEJ seem to move northward. Actually the USUD reference site moved southward, affected by the after slip of the South coast of Honshu earthquake. To confirm the southward motion of the USUD sites, we plot the time series of the Tsukuba (TSKB) - USUD baseline vector. The TSKB site is the IGS site in NEJ. Fig. 5.4 shows the time series, eliminating the co-seismic step at the South coast of Honshu earthquake. From the occurrence of the South coast of Honshu earthquake, showing at the epoch of the arrow in the figure, southward motion of the USUD site began, and terminated in December 2004. Actually for the period of June 2003 and August 2004, the N-S component velocity of the USUD-TSKB baseline is 0.07 +0.01 (+ means one sigma), although the velocity is -13.94 + 0.61 for the period of September 12 and the end of December 2004.
background secular motion in the Tokai Region (Fig. 3.1), and those two points still indicates westward motions even in the slow slip period (Figs. 4.1, 4.2, 4.3, and 5.3), which suggests that the slow slip event does not reach to the upper-most end of the PHS slab along the Suruga Trough near the Omaezaki Point. To examine the slow slip phenomena during January and May 2003 when GEONET sites were not available because of the system revision, we examine the time series of the baseline between the TAT NIED site indicated in Fig. 5.3 and the USUD IGS sites, both are free from the noise of the GEONET maintenance. Fig. 6.1 shows the time series of the TAT- USUD
3097-USUD BASELINE VECTOR N-S C O M P O N E N T 40
..401 2000
,of
.
,
i
,
.
,
i 2002
.
.
.
~.2o~r 01
.
,
.
,
i
,
_~(ll
.
.
.
"~
.
"
~ ~ ~ "
~-2o 2000
The area near the Omaezaki Point, especially two GEONET sites in the southeast end of the Tokai Region, indicates the largest westward motion in the
,
,
i 2004
k
E-W C O M P O N E N T
,
.
i
,
40
.401 2000
2006
,
U-D C O M P O N E N T 60
E
6. Discussion
.
,
I
I
2002
,
I
40
"~ ,
I
2004
.
i
,
,
i 2002
.
,
,
i
,
,
, 2004
I
,
.
,
, / 2006
BASELINE LENGTH
.
,
,
.
,
,
,
~22,
/
,
,
I
I
|
.401
,
o I
I
I
,
I
I
I
I
0
Fig. 6.2. Time series of the 3097-USUD baseline during 2000 and 2005. The data are the weekly values estimated from the daily values using Kalman filtering.
Chapter
84
• Recent
Dynamic
Crustal
Movements
0819-USUD BASELINE V E C T O R N-S
C O M P O N E N T
E-W
C O M P O N E N T
,!ooo 60
U-D C O M P O N E N T . , . , . . . . .
+o
.
40
,+o!, +ooo BASELINE . , . ,
.
LENGTH . ,
,
,
.
,.
-6£)
~
2000
,
~ , 2002
1 '
V
~
I
,
2004
2006
-40 2000
........ 1 2002
2004
2006
Fig. 6.3. Time series of the 0819-USUD baseline during 2000 and 2005. The data are the weekly values estimated from the daily values using Kalman filtering.
baseline, eliminating the co-seismic step at the South coast of Honshu earthquake. For the period of January and December 2002, the E-W component velocity of the baseline is 1.81 + 0.26 (+ means one sigma), although the velocity is 8.36 + 0.04 for the period of January 2003 and June 2004. Thus the E-W component indicates that the slow event became faster significantly at the beginning of 2003. We re-evaluate the start time of the slow-down of the abnormal crustal movements at midst of 2004 concerning the relation of the occurrence of the South coast of Honshu earthquake and the change of the velocity field in the SWJ. Fig. 5.3 indicates the sites whose time series were examined. Fig. 6.2 shows the time series of the 3097 - USUD baseline,
3078-USUD BASELINE V E C T O R 40
.
N-S ,
.
C O M P O N E N T , . , . ,
,
.
40
20
~
-20 ,
2000
-201"
C O M P O N E N T , . , . , ,
I
Eo g
•
.
L
-20
•..40
g+ o
E-W , .
20
"fro g
60
•
.
2002
2004
U-D C O M P O N E N T , . , . , . ,
-:;L' ' "
,
.
-40 21000
40
2002
•
BASELINE , • , .
"
" "
"
"~
, 2006
2004
.
.
LENGTH . .
,,
,
,?. •
2006
2006
2000
2002
2004
Tokai
Region,
Central
Japan,
Observed
by
GPS
Measurements
elimiating the co-seismic step at the South coast of Honshu earthquake. The arrow indicates the epoch of the change of the trend. The E-W component velocity of the baseline for the period of April 2003 and March 2004 is 11.17 +0.21 (+means one sigma), although the velocity is 2.29 + 0.23 for the period of April 2004 and March 2005. It seems that the change of the trend occurred around April 2004, obviously before the occurrence of the South coast of Honshu Earthquake, judging from the data of the E-W component. Fig. 6.3 shows the time series of the 0819-USUD baseline, eliminating the co-seismic step at the South coast of Honshu earthquake. The arrow indicates the epoch of the change of the trend, judging mainly from the E-W component. The E-W component velocity of the baseline for the period of July 2003 and June 2004 is 12.00 + 0.26 (+ means one sigma), although the velocity is 1.77 + 0.37 for the period of July 2004 and June 2005. The E-W component seems to indicate some gap in the midst of the year and then the velocity changes around July 2004, also before the occurrence of the South coast of Honshu earthquake, although the change delayed three or fore months after that of 3097 site. Fig. 6.4 shows the time series of the 3078-USUD baseline, eliminating the co-seismic step at the South coast of Honshu earthquake. The arrow indicates the epoch of the change of velocity judging from the N-S and E-W components. The N-S and E-W components velocities of the baseline for the period of August 24 2003 and September 4 2004 are 0.11 + 0.22 and-1.67 + 0.22 (+ means one sigma), although the velocities are -0.51 + 0.35 and 3.02 + 0.34 for the period of September 12 2004 and September 17 2005. The epoch seems to coincide with occurrence of the South coast of Honshu earthquake. Thus the epochs of the velocity changes in 2004 are earlier at the sites near Hamana Lake and later at the sites NE Tokai Region, suggesting the migration of the place of the slow slip at the plate boundary.
7. Conclusion
•
t
"
+o4' '
in the
,, t 2006
Fig. 6.4. Time series of the 3078-USUD baseline during 2000 and 2005. The data are the weekly values estimated from the daily values using Kalman filtering.
The recent dynamic crustal movements in the Tokai Region, Central Japan, are examined, observed by GPS measurements. The abnormal crustal movements began to move southeastward, the reverse motion to the secular crustal movement in the Tokai Region, since October 2000, indicating the slow slip between the subducting PHS slab and the subducted SWJ.
583
584
S. Shimada • T. Kazakami
The abnormal crustal movements were fast and wide during October 2000 and December 2001, and slow and narrow during January and December 2002, and again fast and wide during January 2003 and May 2004. Since around April 2004, the abnormal crustal movements became again slow and narrow in the area northeast of Lake Hamana, and then in September 2004 a M7.4 earthquake occurred around 200km southern southwest of the Tokai Region near the Nankai Trough. Co-seismic motions are southward 1 0 - 30 mm displacement in the Tokai Region. During the occurrence of the earthquake and December 2004 the after slip southward motion occurred in the southeast end of SWJ including the Tokai Region. On the other hand the abnormal crustal movements are still ongoing to be slow and narrow.
References
Herring, T. A., (2004). GLOBK global Kalman filter VLBI and GPS analysis program. Mass. Inst. of Technol., Cambridge, Massachesetts. King, R. W., and Bock, Y., (2004). Documentation for the GAMIT GPS analysis software. Mass. Inst. of Technol., Cambridge, Massachusetts. Miyazaki, S., Hatanaka, Y., Sagiya, T., and Tada, T. (1998). The nationwide GPS array as an Earth observation system, Bull. Geogr. Surv. Inst., 44, pp. 11-22. Ozawa, S., Murakami, M., Kaidzu, M., Tada, T., Sagiya, T., Hatanaka, Y., Yarai, H., and Nishimura, T., (2002). Detection and monitoring of ongoing aseislnic slip in the Tokai Region, Central Japan. Science, 298, pp. 1009-1012. Yamamoto, E., Matsumura, S., and Ohkubo, T., (2005). Repeatedly occurring slow rebound of the continental plate at the Kanto-Tokai area of Japan detected by continuous tilt observation. Earth Planetary Space, 57, pp. 917-923. Yamanaka, Y., (2004). http://www.eri.u-tokyo.ac.jp/sanchu/ Seismo Note/2004/EIC153e.html.
Chapter 85
New Theory for Calculating Strains Changes Caused by Dislocations in a Spherically Symmetric Earth Wenke SUN, Shuhei OKUBO and Guangyu FU Earthquake Research Institute, University of Tokyo, Tokyo, Japan sunw@ eri.u-tokyo, ac.j p
Abstract. A new theory is presented for calculating co-seismic strain caused by four independent types of seismic source in a spherically symmetric, non-rotating, perfectly elastic, and isotropic (SNREI) Earth model. Expressions are derived by introducing strain Green's functions. A proper combination of these expressions is useful to calculate co-seismic strain components resulting from an arbitrary seismic source at any position in the Earth. Numerical computations are performed for four independent sources at a depth of 32 km inside the 1066A Earth model. Results in the near field agree well with that calculated for a half-space Earth model. A case study is performed and Earth model effects are investigated. Furthermore, the effects of spherical curvature and the stratified structure of the Earth in computing co-seismic strain changes are also investigated using the present dislocation theory and Okada's (1985) formulation. Curvature effects are small for shallow seismic events, but they are larger for greater source depths. Stratified effects are very large for any depth and epicentral distance, reaching a discrepancy greater than 30% almost everywhere. Key Words: Co-seismic Deformation, change, Dislocation, Earthquake.
1.
Strain
Introduction
Advances in modern geodetic techniques such as GPS and InSAR allow better detection of co-seismic deformations such as displacement, gravity change, and strain. Such geophysical geodetic information is useful for studying seismic mechanisms, Earth structure, and even earthquake forecasting. A quasi-static dislocation theory is necessary to properly apply the observed geophysical phenomena to interpret or invert the seismic parameters. To study co-seismic deformation in a half-space Earth model, numerous studies have been undertaken by many scientists; among them are Steketee (1958), Maruyama (1964), and Okada (1985). They presented analytical expressions for
calculating surface displacement, tilt, and strain resulting from various dislocations buried in a semi-infinite (half-space) medium. However, the validity of these theories is strictly limited to a near field because Earth's curvature and radial heterogeneity are ignored. As modern geodetic techniques can detect and observe far field crustal deformation, even a global co-seismic deformation, a dislocation theory for a more realistic Earth model is demanded to interpret far-field deformation. Efforts to develop formations for such an Earth model were advanced through numerous studies (e.g., Ben-Menahem and Singh, 1968; Ben-Menahem and Israel, 1970; Smylie and Mansinha, 1971). Such studies have revealed that Earth's curvature effects are negligible for shallow events, whereas vertical layering may have considerable effects on deformation fields. However, Sun and Okubo's (2002) recent study comparing discrepancies between a half-space and a homogeneous sphere (accounting for self-gravitation) and between a homogeneous sphere and a stratified sphere indicates that both curvature and vertical layering have marked effects on co-seismic deformation. Stratified sphere models, such as the 1066A model (Gilbert and Dziewonski, 1975) or the PREM model (Dziewonski and Anderson, 1981), are the most realistic: they reflect both sphericity and stratified structure of the Earth. For such an Earth model, Pollitz (1992) solved the problem of regional displacement and strain fields induced by dislocation in a viscoelastic, non-gravitational model. Sun and Okubo (1993, 1998) and Sun et al. (1996) presented methods to calculate co-seismic displacements and gravity changes in spherically symmetric Earth models introducing dislocation Love numbers and Green's functions. Okubo (1993) proposed a reciprocity theorem for connecting solutions of dislocation and tidal, shear, and load deformations. That study found that deformation on the Earth's surface caused by dislocations at source radius r = ~ .
are expressible simply by a linear
combination of tide, shear, and load deformations at
586
W.
Sun. S. Okubo. G. Fu
Theoretically, a co-seismic strain tensor can be obtained numerically by taking spatial derivatives of the co-seismic displacement results from first principles. However, it would be a very tedious work for repeat computations. On the other hand, since co-seismic deformations very rapidly as distance increases, the numerical methods may cause a considerable error. Therefore, it calls a convenient solution in a straightforward manner. This paper presents a new theory for calculating the co-seismic strain for a realistic (e.g., SNREI: Spherically Symmetric, Non-rotating, purely Elastic and Isotropic) Earth model. The theory is given by a set of expressions by strain Green's functions multiplied by an epicentral distance related factor for four independent seismic sources. These formulations can be used to calculate any strain components caused by any kind of dislocation at an arbitrary position in the Earth. 2.
Basic E q u a t i o n s of Elastic R e s p o n s e s of a Sphere to a Point Dislocation
We start with a dislocation model (Fig. 1), which is defined at radial distance r s on an infinitesimal fault dS by slip vector v, normal n, slip angle ~, and dip angle 6 in the coordinate system ( e l , e 2 , e 3); unit vectors e 1 and e 2 are taken in the equatorial plane in the directions of longitude ~ = 0 and 7r/2, respectively, and e 3 along polar axis r. movement (dislocation) of the two fault defined as (U/2)-(-U/2) =U. Note that for opening, the slip vector and the normal equal: v=n.
Relative sides is a tensile become
form of three components along with spherical coordinates ( e r, e o, e e ) as (Sun et al., 1996): u(a,O,(p)= ~
y
m
n~ij
m
I(~R. ( O, (19) + Y 3,m S n (0,(19)
i,j
.~,t,n,ijTm (0, (/9)] vin j ++l,m
UdS a
(1)
2
where
R~(o, ~)- ey2(o,~) I 0 S~(O, q~) - e o -0--~+ e~
1 0] sin 0 Oq~
Y~(O, (p)
e~
r2(o, ~)
E '
T~ (0,~o)- e o sin0 c~(p
(2)
Y~ (0, ~o) = P~ (cos 0)e i~e P [ (cos 0) is the associated Legendre function and a is the Earth radius; the superscript s represents spheroidal deformation and t stands for toroidal deformation. The y-variables ,,n,is :~,m(a) and y t,n,ij
~,m (a)are obtainable by solving the linearized
first order equations of equilibrium, stress-strain relation, and Poisson's equation for excited deformation (Saito, 1967; Takeuchi and Saito, 1972). Because i=1, 2, 3 and j = l , 2, 3, the combination of i and j is nine, i.e., total solutions of all y~ should be nine. However, because of the symmetry and intrinsic symmetry within fault geometry of source functions S ' " , the number of independent solutions of :~,m(a)""'u and :~,~,,""'iJ(a) is four. Consequently, if any four independent solutions are obtained, other solutions among the nine are also obtained easily. In this study, we choose ( Y~im n 12 _ n,32 n 22 n 33 ' Y~,m , Yk;m , Y~im ) (and the
Fig. 1. The top row shows three Earth models: l e f t homogeneous half-space; m i d d l e - homogeneous sphere; r i g h t - heterogeneous sphere. The bottom represents four seismic sources. If dislocation occurs in a spherical Earth, such as in a homogenous sphere or a SNREI Earth (spherically symmetric, non-rotating, perfectly elastic and isotropic, Dahlen 1968), excited displacement u (a, 0, ~o) (radial distance, co-latitude and longitude) are describable in the
corresponding toroidal solution) for four independent solutions. They are excited respectively by a vertical strike-slip, a vertical dip-slip, a horizontal opening along a vertical fault, and a vertical opening along a horizontal fault. Once the displacement components ui(a,O,~o ) in (1) are obtained (Sun et al., 1996), the co-seismic strain tensor can be derived and numerically calculated, i.e., the work of this study. 0
E x p r e s s i o n s of Co-seismic Strain for the F o u r I n d e p e n d e n t Sources
According to conventional theory of elasticity, the components of the co-seismic strain are expressible
Chapter 85 • New Theory for Calculating Strains Changes Caused by Dislocations in a Spherically Symmetric Earth
in spherical coordinates as (Takeuchi and Saito, 1972) the following.
e~, = -C~U - r
(3)
8r
1 8u o
1
eoo = - - +-u r r c~O r 1
eoe
rsin0
e oo = e~
e"°
c~o
r
r
1
1
--+-uo
r 80
cot 0 + - u ~ r
u o cot 0 + - 8u o + - -
c~u o
(6)
8r
8u o 1 . . . uo c~r r
18u,. - - r c30
+
(22)
c o s (/5~32O (a,
0)
e0022'0(1:7/,O, 0/9) -- eoo~22'°(a, O) - 0
(23)
%" (a, o, ~o) - %~"(a, o) - o
(24)
ek z'`ij (a, 0) are the strain Green's functions from"
err-
Yl,2 (6/)
a(2 + 2//) n=2
(25)
- n(n + 1)y3~'~2 (a))/~,2 (cos O)
1
---ue
r sin 0 c~(p .
1
(5)
r sin 0 8 ( , o
8u,.
=
CO32o (a, 0, (,o) -
Therein, the epicentral distance-related variations 1
1
(21)
(4)
c~u o
1 c~u~o
12 (a, 0, (,o) - cos 2(,~;2o (a, 0) CO o
(7)
r
,,32 2,~ £(2yn,32 = ~ ,,, err a(,,~ + 2/./) n=l
(8)
(a)
(26)
1\ n,32
- n(n + UY3,1 (a))P2(cosO) = a(A + 2/a) ,=0
err
Yl,0 (a)
(27)
ix n,22
+ rt(n + l)Y3, 0 ( a ) > , (cos0) Using the displacement components defined in the section above, the expressions of the strain components are derived in the following. The last two components e ro and e,.o vanish on the free Earth surface. Only the remaining four components are considered hereafter. Inserting the displacement components in (1) into (3)-(8), taking into account the responses to the four independent sources, the above four strain components are reducible into the following 16 components, expressed as an epicentral distance-related variation (called Green's function) multiplied by a (,o-related factor, yields:
,2 (a, 0, O)) - sin 2(,~I. 2 (a 0) C/./,
(9)
32(a, O, (,o) - sin Oer~ ,,32(a, O) Crr
(10)
"22,0 (a O) err22,0 (a, O, (/9) -- err ,
(11)
~" (a, o) err" (~ o, ~o) - err
(12)
12 (a, O, (,o) - sin 20~; 2 (a, O) eoo
(13)
32 (a , 0, O) - sin ~eoo "`32(a, O) eoo
(14)
22,0 (a, O, 0/9) -- "22,0 (a O) Coo Coo ,
(15)
"(a Coo
0, e) - Coo ~" (a, o)
(16)
e~012(a, O, (p) - sin 2q~% 0"`12(a, O)
(17)
e~32 (a,
(18)
,
,
O, (,0) - sin
"`32(a, 0) ee~
eoe22'0(a, O, 0/9) -- C00"22'0(a, O)
(19)
eoo33(a, 0, q)) - eoo~33(a, 0)
(20)
"`33 -
/~
err
£(
n33 (a) y,,~
a(/~ + 2/a)~0
(28)
. . . . " ( a ) ~ (cos O) +n(n+UY3,o d 2 P,) (COS 0) Coo
--
--
--
Y3,'2
(a)
dO 2
a n=2
-- yl~)12P f (cos {9) - ~zyl,t,n,12 2 (a)
I 1 dPf(c°sO) dO
(29)
c°s-O 2
• sin 0
I]
sin2 0 P~ (cos O)
"32 - - 2~_.[_yn,32 eoo 3,, (a) d2 C 1(cosO)
a
1
dO 2
-- yln] 32 ( a ) P 1 (cos 0) - yl]l '32 (a)
dP~(cosO) dO COO =
cosO 1 sin 2 0 P~ (cos O)
.y3n '22 (15/) a n=o L
sin 0
(30)
J
/Dn (COS 0)
dO2
(31)
+ yln~22 (a)P n (cos 0)] "33 _ 1~-]~
COO
~,33
d2p~(c°sO)
-~__o~Y~,o (~) + yl~O"(a)P~ (co~ 0)]
dO ~
(32)
587
588
W. Sun. S. Okubo. 6. Fu
"`12 __ Z eel, a ~:2 -
{ yn,12 ( 3,2 (a) 4P] (cosO) sin 0 sin 0
m=2. The above formulas give solutions for the case of m=0; solutions of the m=2 case can be derived from e,l,2 (a, 0, (p) because the following relation holds.
cos 0 dp2 (cos 0) 1 "'~ dO - Y~,2 (a)
(33)
t,n,12
0:1,2 (a)
• p ] ( c o s o) +
t,n,22
Vj - 1,2"s.1:,+2
sin 0
c°s°, OPn2 1 t
•
-- COS 0
cot
-
dO
"32 __ Z eel, a ,--1
{ yn,32
3,1 (a) sin 0
( | Pn (COS0) sin 0
dpl (COS 0) )
n 32
dO
- y~'] (a)
(34) 0
• C' (cos o) + -
[
- cot Of" (cos O)
dO
"`22,0 1 £ I c o t 0 y n , 2 2 ~,0 ( a ) e~°~° = -a n=0
]}
dP. (cos 0) dO
(35)
+ y/,~)22(a)p n(COS 0)] "`33 -_1 ~0Icot0yn,333,0(a) dPn(cosO)
e~
a :
dO
a n,12(a) I "`12 2 ~ ~Y3,2 e°e--- ~ a :2 sin0
(36)
dP n2 (COS0)
dO
+ cot Op2 (cosO)]
-'ky;:2'12Icot O(a) dP2 (cos O) dO I 4P] (cos O) sin 20
"`32 2 eoe-~ a ,, 1
(37)
d2p2(cosO)]} dO ~
3,1 (a) sin 0
Numerical Earth
sin 0
d n (COS 0)
dO
+ cot OP2(cos O)) + y[:i''32 (a)/cot 0 alP1(cos 0)
(38)
dO P2(coS0)sin20 _ d2p2(cosO))}_dO 5
22 (a , 0, (/9) is The case of e~l
--
(39)
__ • t n 12
-t-lSj',+2
Expressions (9)-(38) illustrate the main results of this study. They are useful to calculate strain components excited by four types of sources buried in a spherically symmetric Earth model. In combination, these components allow calculation of a strain field that is excited by an arbitrary seismic source.
(cos O)
y~:i'32(a) •
n,122 V j - 1 , - . . 6 " s ~ ( 22 - -T",sji_+
special because
the source function contains two parts with m=0 and
Results
for
a Homogeneous
Model
To bolster the validity of the theory described above, a numerical test is made by considering two Earth models: a half-space model and a homogeneous sphere. Both Earth models have identical media parameters, which are equivalent to those of the top layer of the 1066A Earth model (Gilbert and Dziewonski, 1975). The numerical calculations are performed for the two models and the results are compared. The above co-seismic strain theory can be considered valid if the results agree well in the near field. For this purpose, Okada's theory (1986) is used for the half-space Earth model; the current theory is used for the homogeneous model. The Green's functions in (25)-(38) show that the important computations are the y-solutions and Legendre functions and their derivatives. The y-solutions
,, "'/j e~,m(a)
and
,, t,,,/j (a) :~,,,~
have already
been discussed in previous papers (e.g., Sun and Okubo (1993) and Sun et al. (1996). The Legendre functions and their derivatives can be calculated using recurrence formulas. Green's functions are then obtainable through series summations. The computation for the near field is relatively difficult in comparison to that for the far field because of the slow convergence of the series of the former. Therefore, some skills are required to accelerate computation (see Sun and Okubo, 1993). On the other hand, because co-seismic deformations in near field dominate over those of the far field, they are useful for comparison with the result of a half-space model. Therefore, the following discussions are limited in the near field. Assuming four independent point sources at a depth of 32 km in the homogenous Earth model, the
Chapter 85 • New Theory for Calculating Strains Changes Caused by Dislocations in a Spherically Symmetric Earth
y2;~(a)
y-solutions
and
y~',"2y(a)
are
~,,mm 11
,.
Computations are performed under the condition of UdS/a2=l, and the Earth radius a=6371 km is taken for all the Green's functions. This assumption requires that, for a practical computation of a strain component, the parameters UdS and a should use the same kilometer unit so that the final strain components become dimensionless. Once these Green's functions ekl"O(a,O) are calculated, the co-seismic strain components in Eqs. (9)-(24) are easily computed. Figure 2 depicts the co-seismic strain components err12 (a, 0, (p) caused by the vertical strike-slip source at a depth of 32 km in the Earth model. The horizontal axis indicates co-latitude and longitude to 2°; the vertical axis represents the strain magnitude with the unit of km -1. As expected, the strain component err12 ( a , 0 , (/9) for the vertical strike-slip source appears as a quadratic distribution pattern. For comparison, the corresponding strain components caused by the four sources in a half-space Earth model are also calculated using Okada's formulation (Okada, 1985), and plotted in Fig. 3. Comparison of Fig. 2 and Fig. 3 illustrates that the two results agree well in both distribution pattern and magnitude. However, discrepancies in magnitude exist for some components (details to be seen in Figs. 5 and 6). These discrepancies are inferred to result from sphericity and vertical inhomogeneous effects.
) s,
W
lllm •
m
m •
m
L ;; .i
.ram
"
l[
ii
l
m" m
|m "i . m m m~
Fig. 3. Co-seismic strain caused by the vertical strike-slip source at a depth of 32 km in the half-space Earth model, as calculated by Okada' formulation (Okada, 1985).
0
Numerical Model
Results
for a SNREI
Earth
Next, we test the above co-seismic strain theory by performing a numerical calculation. Assuming the four independent point sources at a depth of 32 km in the 1066A Earth model (Gilbert and Dziewonski, 1975), we first calculate the strain Green's functions
by the vertical strike-slip source at a depth of 32 km in the 1066A Earth model. Results for other strain ij (a, O, q~) , e~i; (a, O, q~) , and components eoo
,i
\
nun "
Once these Green's functions e~1 ~/; (a,0) are calculated, the co-seismic strain components in Eqs. (9)-(24) are easily computable. Figure 4 depicts the co-seismic strain components err12(a, 0, ~p) caused
i¢
4,,
I¢
IN
m
^/; (a, 0) in Eqs. (25)-(38). Again, the computations are performed under the condition of UdS/a2=l; Earth radius a=6371 km is assumed for all the Green's functions.
Ai
12
)Ni
first
computed. Then the strain Green's functions e~1~i;(a,0) are obtainable using Eqs. (25)-(38).
m $ i-,¢.
m
Fig. 2. Co-seismic strain components err,2 (a, 0, ~) caused by the vertical strike-slip source at a depth of 32 km in a homogeneous Earth model.
589
e~o(a,O,~o ) are also computed, but the plots are omitted here. They show a similar distribution pattern to the component e~r(a,O,~o ) , but with different magnitude.
590
W. Sun. S. Okubo • G. Fu
,mm
m(l
Ill m, _-m
! I .
11
k II
•
..i in m
.l."
l m
•
I I
%
•m
,m_',
",m, •
m m n ~
Fig. 4. Co-seismic strain components elr~(a, 0, q~) caused by the vertical strike-slip source at a depth of 32 km in the 1066A Earth model. 0
Case S t u d y - Application to 1994 Far Off Sanriku Earthquake (M7.5)
In this section, the above co-seismic strain theory is applied to compute the strain changes caused by the 1994 Far Off Sanriku earthquake, which occurred on 28 December 1994, about 180 km east of Hachido, Aomori prefecture, Japan. The fault plane of the earthquake comprises two rectangles with parameters listed in Table 1. This earthquake caused crustal deformations in a large area: displacement detected by GPS, and strain changes observed by extensometers installed at observation points HSK and FDA. The following computations consider only those strain changes to which the above theory applies. Table 1. Source parameters for the main shock of the 1994 Far Off Sanriku earthquake. (Faculty of Science, Tohoku University, 1994) Parameter Fault # 1 Fault #2 3.4 x 1027 1.7 x 1 0 27 M0 dyne x cm dyne x cm Mw 78 Strike angle N 184°E N184°E Dip angle 8° 35 ° Slip angle 70 ° 90 ° LengthxWidth 60x 100 km 50x60 km Dislocation 1.57 m 1.57 m Latitude 40.24°N 40.55°N Longitude 144.04°N 142.85 °N Depth 10 km 24 km The point source theory cannot be used directly because a large error is expected to occur as a result of the large geometrical fault size relative to
the epicentral distance. To surmount this obstacle, a numerical integration of the point source over the fault plane could be done, or a segment-summation scheme could be used, as pointed out by Fu and Sun (2004). The latter is adopted in our practical computation. First, a homogenous half-space model and a homogenous sphere model are used to calculate the strain changes caused by this earthquake. The former is calculated using Okada's (1985) formulation, whereas the latter is computed using the present theory. Results are given in Table 2 (second and third rows). Both results are nearly identical and basically coincide with the observed results. This fact indicates that the current theory is valid and correct. Table 2. Observed and calculated strain changes at the points HSK and FDA Dilatation N87°E N177°E at FDA at HSK at HSK 0.77x 10-6 0.77x 10-6 -0.66 x 10-6 Observed 0.20xlO -6 0.10xl04 -0.78x10 -6 Okada 0.20x 10-6 0.11xl04 -0.80x 10-6 HomoSph O.16xlO -6 0.11xl04 -0.82 x 10-6 1066A Then, we perform a computation for a spherically symmetric Earth model, 1066A (Gilbert and Dziewonsk, 1975). Results in Table 2 (fourth row) show that the volume change is 20% different from that of the homogenous models. Although the results calculated by the above theory fundamentally agree well with observed strain changes, some discrepancies remain for two reasons: 1) the fault model is inaccurate, and 2) the topography effect. The second reason is considered as dominant because this earthquake occurred in the deep Pacific trench and this theory ignores complicated terrain effects. This effect will be addressed in a future study.
7.
Effects of the Earth's Spherical Curvature and Radial Heterogeneity
Two Earth models are used to study effects of sphericity: a homogeneous half-space and a homogeneous sphere. Numerical calculations are made for the four independent seismic sources mentioned previously: two shear strikes and two tensile openings. Media parameters used in both models are equal to those of the top layer of the 1066A Earth model. A comparison of the near field co-seismic 12(a, 0, (,o) strain changes of component err calculated for the 1066A Earth model and the
Chapter 85 • Hew Theory for Calculating
homogeneous half-space caused by a vertical strike-slip source at a 32 km depth show that the two Earth models are almost identical: it is difficult to identify their differences. This similarity indicates that the effect of sphericity is very small for a shallow seismic source. However, as depth increases, their discrepancy becomes large. This fact is illustrated in Fig. 5, which shows a comparison of the co-seismic strain changes of the component err12(a , 0,(,o) for source depths of 3, 10, 100 and 300 km, respectively. We conclude that the effect of spherical curvature (proportional) increases as source depth increases. Because the homogeneous half-space model is non-gravitating, whereas the homogeneous sphere is self-gravitating, it is notable that the difference obtained by comparison in this section includes both the sphericity effect and the self-gravitation effect. Independent comparisons should be made to distinguish the two effects: a comparison between a homogeneous half-space and a homogeneous sphere with non-gravitation, and a comparison between two homogeneous spheres with and without self-gravitation. However, to assume that the self-gravitating effect is less than the spherical effect (to be confirmed) and to simplify discussion, we do not discuss them separately, and refer to them as effect of sphericity for convenience.
Strains
Changes Caused by Dislocations in a Spherically Symmetric Earth
Numerical calculations are made using the same parameters as those in the previous section. 32(a, 0, (,o) Figure 6 shows the strain components ekt (from top to bottom: k l = r r , 0 0 , ~o~o, 0~o ) calculated for the dip-slip source at 32-km depth. The blue curve shows results calculated for the 1066A Earth model; the red line shows homogeneous sphere results. The figure shows clearly that discrepancies between the two models are greater than 30% almost everywhere, including the epicenter. Discrepancies caused by the stratified structure are much larger than those of sphericity. Hence, we infer that the half-space dislocation theory might create an error of 30%. To investigate the effect of stratified structure, we further consider the 1066A Earth model with a new top layer of 11 km with parameters equivalent to those at 30 km. The corresponding calculated numerical results are plotted in Fig. 6 using a green color line. A difference appears for t h e er32 component (and also for all strain components of other sources) from comparison of the homogeneous and 1066A Earth models. It again confirms the importance of the Earth structure in computing co-seismic strain changes. "ll~l .-gO, • I~.l.
• I.'l"
..,. L"
J.
m £ I
"iI
£ •
'11
£ ".
'11
£ ."
I1
£ I
I
L I
.i.I
L ."
.l'l
i. ~.
.Iq
i. "
.i I
i. ~
I
='F
r ~
I
.1. L. ."gO, •
[ I ~'.IL • I r~z HiT •
i..
•
~,
• L
~.
• ~:,:
I, ..~
I I'1", ;4"C,.
g
I:'.Cl
L'"
I: I "
L-."
I a'J
L J
I ."
L II
3-1:1
L :l
f.
:~,.
.
.
.
.
.
.
.
i',.
~
~ . .
I.
I1~. I" I ' -
r,
r
"
."El', • ,:-'.,.
, .,.
;
, .,.
.I "lgl',
~I
r.
C'l
1".~
c".
1"~
c ,.
Li-.¢-
."Hi L - - . I r
l"d
r -
..n
(34.
r ~1. 1...£
... I
"l
]
i Hi
L..
/
I
II 'd
i
I
l
II
L'rIJLr
Fig. 5. Comparison of co-seismic strain components 12
err (a, 0, (,o) calculated for the 1066A Earth model and the homogeneous half-space caused by a vertical strike-slip source at depths of 3, 10, 100 and 300 km, respectively. The blue curve shows results calculated for the 1066A Earth model; the red line shows homogeneous sphere results. To study the effect of stratified structure, we consider the homogeneous sphere and the heterogeneous sphere (1066A model) and compare results calculated from the two models.
Fig. 6. Comparison of co-seismic strain changes 32 ( a , O , ~o) calculated for dip-slip source at 32-kin ekt depth in the homogeneous sphere, 1066A Earth model and with its new top layer. Top to bottom panels show respective results for k l = r r , OO , fafa , O~o . The blue curve shows results calculated for the 1066A Earth model; the red line shows homogeneous sphere results. 8.
Summary
This research presents a new theory for calculating co-seismic strain change for a SNREI Earth model. Four independent point sources are assumed to be
591
592
W.Sun. S. Okubo. G. Fu
located at the polar axis. Corresponding expressions for calculating co-seismic strain components are derived by introducing strain Green's functions. A proper combination of these expressions is useful to calculate co-seismic strain components caused by an arbitrary seismic source at any geographical position in the Earth. Numerical computations are performed for calculating surface strain components in the near field caused by four independent sources at a depth of 32 km inside the 1066A Earth model. Results agree well with those calculated for a half-space Earth model. Results confirm the validity of the theory presented herein. This theory is also applicable to compute co-seismic strain effects caused by the 1994 Far Off Sanriku earthquake. Results indicate that the new theory can approximately predict the observed results. Effects attributable to the Earth models themselves were also investigated. Furthermore, the effects of spherical curvature and the stratified structure of the Earth in computing co-seismic strain changes are also investigated using the present dislocation theory and Okada's (1985) formulation. Curvature effects are small for shallow seismic events, but they are (fractionally) larger for greater source depths. Stratified effects are very large for any depth and epicentral distance, reaching a discrepancy greater than 30% almost everywhere Acknowledgements. This research was supported
financially by JSPS Grant-in-Aid for Scientific Research (C16540377) and "Applications of Precise Satellite Positioning for Monitoring the Earth's Environment". Helpful comments by J. Beavan and G. Blewitt are highly appreciated. References
Ben-Menahem, A., and M. Israel (1970), Effects of major seismic events on the rotation of the Earth, Geophys. J. R. Astron. Soc., 19, 367-393. Ben-Menahem, A., and S. J. Singh. (1968), Eigenvector expansions of Green's dyads with applications to geophysical theory, Geophys. J. R. Astron. Soc., 16, 417.452. Dahlen, F. A. (1968) The normal modes of a rotating, elliptical Earth, Geophys. J. R. Astron. Soc., 16, 329-367. Dziewonski, A. M., and D. L. Anderson (1981), Preliminary Reference Earth Model, Phys. Earth Planet. Inter., 25, 297-356. Faculty of Science, Tohoku University (1994), Faulting Process of the 1994 Far East Off Sanriku Earthquake inferred from GPS observation, The Report of CCER The
coordinating Committee for Earthquake Prediction, Japan, Vol. 54, 97-98. Fu, G., and W. Sun (2004), Effects of Spatial Distribution of Fault Slip on Calculating Co-seismic Displacements- Case Studies of the Chi-Chi Earthquake (m=7.6) and the Kunlun Earthquake (m=7.8), Geophys. Res. Lett., Vol. 31, L21601, doi: 10.1029/2004GL020841. Gilbert F., and A. M. Dziewonski (1975), An application of normal mode theory to the retrieval of structural parameters and source mechanisms from seismic spectra, Phil. Trans. R. Soc. London A, 278, 187-269. Maruyama, T. (1964), Statical elastic dislocations in an infinite and semi-infinite medium, Bull. Earthquake Res. Inst. Univ. Tokyo, 42, 289-368. Okada, Y. (1985), Surface deformation due to shear and tensile faults in a half-space, Bull. Seism. Soc. Am., 75, 1135-1154. Okubo, S. (1993), Reciprocity theorem to compute the static deformation due to a point dislocation buried in a spherically symmetric Earth, Geophys. J. Int., 115, 921-928. Pollitz, F. F. (1992), Postseismic relaxation theory on the spherical Earth, Bull. Seismol. Soc. Am., 82, 422-453. Saito, M. (1967), Excitation of free oscillations and surface waves by a point source in a vertically heterogeneous Earth, J. Geophys. Res., 72, 3689-3699. Smylie, D. S., and Mansinha L. (1971), The elasticity theory of dislocation in real Earth models and changes in the rotation of the Earth, Geophys. J. R. Astron. Soc., 23, 329-354. Steketee, J. A. (1958), On Volterra's dislocations in a semi-infinite elastic medium, Can. J. Phys., 36, 192-205. Sun, W., and S. Okubo (1993), Surface potential and gravity changes due to internal dislocations in a spherical E a r t h - I. Theory for a point dislocation, Geophys. J. Int., 114, 569-592. Sun, W., and S. Okubo (1998), Surface potential and gravity changes due to internal dislocations in a spherical E a r t h - II. Application to a finite fault, Geophys. J. Int., 132, 79-88. Sun, W., Okubo S., and P. Vanicek (1996), Global displacement caused by dislocations in a realistic Earth model, J. Geophys. Res., 101, 8561-8577. Sun, W., and S. Okubo (2002), Effects of the Earth's Spherical Curvature and Radial Heterogeneity in Dislocation Studies - For a Point Dislocation, Geophys. R.L., V. 29, No. 12, 46 (1-4). Takeuchi, H., and M. Saito (1972), Seismic surface waves, Methods Comput. Phys., 11, 217-295.
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Part V Advances in the Realization of Global and Regional Reference Frames Chapter 86
Advances in Terrestrial Reference Frame Computations
Chapter 87
The Status and Future of the International Celestial Reference Frame
Chapter 88
Is Scintillation the Key to a Better Celestial Reference Frame?
Chapter 89
Improvement and Extension of the International Celestial Reference Frame in the Southern Hemisphere
Chapter 90
Limitations in the NZGD2000 Deformation Model
Chapter 91
Implementing Localised Deformation Models into a Semi-Dynamic Datum
Chapter 92
Definition and Realisation of the SIRGAS Vertical Reference System within a Globally Unified Height System
Chapter 93
Tests on Integrating Gravity and Leveling to Realize SIRGAS Vertical Reference System in Brazil
Chapter 94
Accessing the New SIRGAS2000 Reference Frame through a Modernized Brazilian Active Control Network
Chapter 95
Deformations Control for the Chilean Part of the SIRGAS 2000 Frame
Chapter 96
Estimation of Horizontal Movement Function for Geodeticor Mapping-Oriented Maintenance in the Taiwan Area
Chapter 97
Activities Related to the Materialization of a New Vertical System for Argentina
Chapter 98
An Analysis of Errors Introduced by the Use of Transformation Grids
Chapter 99
Preliminary Analysis in View of the ITRF2005
Chapter 100
Long Term Consistency of Multi-Technique Terrestrial Reference Frames, a Spectral Approach
Chapter 86
Advances in terrestrial reference frame computations Detlef Angerrnann, Hermann Drewes, Manuela Kriigel, Barbara Meisel Deutsches Geod5tisches Forschungsinstitut (DGFI), Marstallplatz 8, D-80539 Munich, Germany e-mail: [email protected]
Abstract. In its function as an ITRS Combination Center, DGFI has developed refined methods for the terrestrial reference frame computation. The advanced approach is based on the combination of epoch normal equations (weekly/ daily data sets) containing station positions and Earth orientation parameters (EOP) obtained from different geodetic space techniques such as VLBI, SLR, GPS and DORIS. This refined approach allows to account for nonlinear effects (e.g., periodic signals and discontinuities) in station positions and to ensure consistency between T R F and EOP. The ITRF2004 submissions were used as input for a refined realization of the terrestrial reference frame. This paper presents the combination methodology and the current status of the ITRF2004 computations at DGFI. Key words: ITRF computation, EOP, VLBI, SLR, GPS, DORIS
1
Introduction
Today, space geodetic techniques allow to determine geodetic parameters (e.g., station positions, Earth rotation) with an accuracy up to the millimeter level. However, this high accuracy is not fully reflected in current realizations of the terrestrial reference frame. The reasons are manyfold and reach from remaining biases between different observation techniques to deficiencies in the combination methodology. With the high accuracy of the space geodetic techniques time-variable effects of station positions and datum parameters (e.g., T R F origin) become detectable, that are not considered in recent T R F realizations based on multi-year solutions with station positions and velocities. The most recent realization of the International Terrestrial Reference Frame, the ITRF2000, has been performed in 2000 based on solutions of the space geodetic techniques available at that time (Altamimi et al., 2002; Boucher et al., 2004). Since then almost five years of addi-
tional data have become available, new sites have joined the global network, the processing strategies and models have been improved and some station positions and velocities are no longer valid because of events (e.g., earthquakes, equipmerit changes, etc.). Furthermore, until now, the major products of the International Earth Rotation and Reference Systems Service (IERS), namely the ITRF, the International Celestial Reference Frame (ICRF), and the EOP are computed (combined) separately by different IERS Product Centers. Consequently, the IERS products are not consistent. The results of the IERS Analysis Campaign to align EOP to I T R F 2 0 0 0 / I C R F reveal that significant biases between EOP series exist (e.g., Dill and Rothacher, 2003). This means that there are clear deficiencies in the present IERS product generation, which have to be overcome by a rigorous combination of station positions, EOP (and quasar positions). Towards this aim, the IERS Combination Pilot Project (CPP) has been initiated in 2004 (as a follow-on project of the SINEX Combination Campaign) to develop suitable methods for a rigorous combination of the IERS products, and to prepare the product generation on a weekly basis. The general scope and the objectives of the CPP are presented in Rothacher et al. (2005). Taking into account the deficiencies of current realizations of the terrestrial reference frame a Call for long time series of "weekly" SINEX files for ITRF2004 and a supplementation of the CPP was released in December 2004. The ITRF2004 is based on the combination of time series of station positions and EOP. Weekly or (daily VLBI) contributions will allow better monitoring of nonlinear motions and other kinds of discontinuities in the time series. The ITRS Combination Centers, namely DGFI, IGN (Institut Geographique National, Paris), and NRCan (National Resources Canada, Ottawa), coordinated by the ITRS Product Center (IGN), are in charge for the generation of the ITRF2004 solution.
596
D. A n g e r m a n n • H. Drewes • M. KriJgel • B. Meisel
2
ITRS Combination Center at DGFI
DGFI serves as an ITRS Combination Center (ITRS-CC) within the IERS. Various software programmes of the DGFI Orbit and Geodetic Parameter Estimation Software (DOGS) are implemented. The methodology for the terrestrial reference frame computations at DGFI is based on the combination of unconstrained normal equations. The processing flow and the major comportents of the ITRS-CC are shown in Fig. 1. A detailed description of the combination methodology is provided in Angermann et al. (2004). DGFI has computed a terrestrial reference frame realization 2003 based on multi-year solutions with station positions and velocities (see Angermann et al., 2004; Angermann et al., 2005; Drewes et al., 2005). The contributing space geodetic techniques are Very Long Baseline Interferometry (VLBI), Satellite Laser Ranging (SLR), the Global Positioning System (GPS) and the Doppler Orbitography and Radiopositioning Integrated by Satellite (DORIS). The T R F computations provided valuable results to assess the current accuracy of the terrestrial reference frame, to identify remaining deficiencies and to enhance the combination methodology. The refined approach is based on the com-
Auxilliary Data
I
• StationInformation • Localties
IndividualSolutions/ NEQ's (VLBI, SLR, GPS, DORIS) .
.
.
.
.
.
.
.
.
bIN~-Z,t-orma[
SNXCHK CheckSINEX Format
IndividualFormats (e.g. DOGS,OCCAM)
/
DOGS-AS Analysis of solutions
SNX2DOGS ConvertionSINEX--~--DOGSFormat Generationof free NEQ's
Individual NEQ's (unconstrained) DOGS-OV OutputVisualisation I P,otso~are I
DOGS-C8 Combination& Solutionof NEQ's IPrep ...... ing of individualNEO's I
[ Analysisof ti. . . . . ies ]
I
Intra-technique I Comparison/ Combination I Inter-technique I Comparison/ Combination
I Helmert-T.... formation I
l
Final
CombinedSolution
)
Figure 1" T R F combination procedure.
1
bination of epoch (weekly/daily) SINEX files of the different space geodetic techniques containing station positions and EOP. The advantages are that nonlinear effects in site motions (e.g., caused by earthquakes or instrumentation changes) are detectable and that secondly consistency between the terrestrial reference frame and the EOP can be achieved. The results of an exemplary T R F realization obtained from an accumulated five years solution (1999-2004) are encouraging (Meisel et al., 2005). The ongoing activities at DGFI focus on the processing for the official IERS product ITRF2004.
3
ITRF2004 input data
According to the specifications of the IERS Call for long time series of epoch SINEX files for ITRF2004 and a supplementation for the IERS Combination Pilot Project the input data have to follow the SINEX Version 2.0 format standard and should comply with one of the following constraints categories: (1) Free normal equations; (2) Loosely constrained solutions; (3) Solutions with removable constraints; (4) Solutions with T R F minimum/inner constraints. The temporal resolution of the SINEX files is one week (Sunday to Saturday) for DORIS, GPS and SLR, and one 24 hour session (17 hr to 17 hr) in the case of VLBI. The parameters to be included should comprise site positions, a set of EOP for each day (pole offsets and rates, LOD as well as UT1 for VLBI). If SINEX files contain the variancecovariance matrix, all constraints applied to the solution should be given in the a priori variancecovariance matrix. Solicited solutions are of three types: (A) Official single-technique combined solutions from the Technique Centers. (B) If solutions of type A are not available or for specific reasons, individual Analysis Center solutions are solicited under the coordination of the corresponding Technique Center. (C) Solutions that result from a combination of various techniques at the observation level may be submitted as well. Table 1 summarizes the major characteristics of the ITRF2004 submissions as available by August 2005. In the case of GPS, SLR and VLBI official single-technique combined solutions were submitted by the Techniques' Combination Centers, namely NRCan for the International GNSS
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• Advances in Terrestrial Reference
FrameComputations
Table 1: Summary of ITRF2004 submissions. Techn.
Service AC
Data
Time period
Parameters
Constraints
GPS
IGS NRCan
weekly solutions
1996 2004 from June 1999 from March 1999
Station positions EOP (pole rates, LOD) geocenter
NNT: 0.1 mm NNR: 0.3 mm NNS: 0.02 ppb
VLBI
IVS GIUB
daily sessions free NEQs
1984
Station positions EOP (pole, UT1 + rates)
none
SLR
ILRS ASI
weekly solutions
1993 - 2004
Station positions EOP (pole + LOD)
1m 1m
DORIS
IGN
weekly sol.
1993 2004
loose
INA
weekly sol.
10/92
LCA
weekly sol.
1993
Station positons EOP (pole, UT1 + rates) Station positions EOP (pole, UT1 + rates) Station positions EOP (pole)
2004
06/2004 2004
Service (IGS), the Geodetic Institute of the University Bonn (GIUB) for the International VLBI Service for Geodesy and Astrometry (IVS), and the Agenzia Spaziale Italiana (ASI) for the International Laser Ranging Service (ILRS). Until now, no combined DORIS solution is available by the IDS. Three solutions of individual DORIS Analysis Centers (IGN; INA: Institute of Astrononly, Russian Academy of Sciences; LCA: LEGOS/CLS, France) were submitted as ITRF2004 contributions. In addition to the SINEX solutions the Technique Centers also provided a list with information about discontinuities (e.g., equipment changes, earthquakes) in station positions, which are used as input by the ITRS Combination Centers. The ITRF2004 submissions were analysed by the ITRS Combination Centers and intratechnique combinations have been performed to validate their combination procedures and to give feedback to the Technique Centers and contributing Analysis Centers. The current status of the ITRF2004 submissions is that the ILRS and IVS will provide updated SINEX files until Oct./Nov. 2005, which will then serve as official input for the ITRF2004 computation. In the case of GPS the input data are almost fihal, however an updated version will be submitted as well. Hopefully, also the IDS will provide combined DORIS solutions as official ITRF2004 submission.
4
none
loose loose
Combination methodology
The combination methodology for the ITRF2004 computation applied at DGFI comprises the fob lowing major steps: • Analysis of ITRF2004 submissions as input data and generation of normal equations • Analysis of time series and combination pertechnique (intra-technique combination) • Comparison and combination of different techniques (inter-technique combination) • Final combined solution
Analysis of input data and generation of normal equations: In a first step the ITRF2004 submissions were analysed concerning various aspects, such as the SINEX format compatibility, the suitability for a rigorous combination of station positions and EOP, and the a priori constraints that were applied by the Technique Centers or individual Analysis Centers. If possible, the reported constraints were removed and free normal equations were generated. The resulting SLR normal equations contain information to realize the scale and the T R F origin; they have a rank defect of three w.r.t, the rotations. The VLBI submissions in the form of normal equations could be used directly (without inversion and reduction of constraints). They
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have a rank defect of six (three translations and three rotations) and contain information to realize the scale. Since the weekly GPS and DORIS solutions are defined in an '~arbitrary '~ reference frame~ it was necessary to reduce the datum information by setting up respective Hehnerttransformation parameters. I n t r a - t e c h n i q u e c o m b i n a t i o n : The epoch normal equations were accumulated for each technique separately to compute multi-year solutions with station positions, velocities and EOP. In a first step we include all stations available except those without sufficient data to estimate velocities (e.g.~ less than 2.5 years} and we use the information on discontinuities that is provided by the Technique Centers. We compute epoch solutions by applying datum constraints and inverting the matrices and align them to the multi-year reference solution by a seven parameter similarity transformation to investigate the resulting time series (station positions and transformation parameters} w.r.t, discontinuities or other nonlinear effects (e.g.~ periodic signals and postseismic deformations). To incorporate these effects we change the parameterization for the accumulation of the epoch data (e.g.~ by setting up new positions and velocities where necessary). Postseismic deformation may be parameterized by piecewice linear functions. To account for annual signals it is possible to introduce amplitudes and phases as additional parameters. Finally the intra-technique solutions are obtained by applying minimum datum constraints. Inter-technique combination: Input for it are the accumulated intra-technique normal equations for VLBI~ SLR~ GPS and DORIS. The parameters comprise station positions, velocities and daily EOP. Concerning the combination of EOP of the different space techniques it has to be considered~ that the VLBI estimates are referred to the midpoint of a daily VLBI session (from 17 hr to 17 hr)~ whereas the EOP values of the other techniques are referred to 12 h. Thus the VLBI EOP estimates have to be transformed to the reference epochs of the other techniques. A key issue within the inter-technique combination is the implementation of local tie information. For this purpose the EOP are essential to validate the local tie selection and to stabilize the inter-technique combination as additional "global ties". Other issues include the equating of station velocities of co-located instruments and the weighting between different tech-
niques. The intra-technique normal equations are added by applying the weighting factors. Final c o m b i n e d solution: The combined inter-technique normal equations are completed by pseudo-observations for the selected local ties and for equating station velocities at co-location sites. To generate the final combined solution~ we add datum conditions and invert the resulting normal equation system. The geodetic datum is realized by no-net-rotation (NNR) conditions~ minimizing the common rotation of the ITRF2004 solution w.r.t, its approximate values for the orientation at the reference epoch 2000.0~ and minimizing the horizontal velocity field over the whole Earth for the time evolution of orientation. For the realization of the kinematic reference frame it is necessary to compute a present-day model representing the entire motions of the Earth surface~ such as the Actual Plate Klnematik and Deformation Mode]~ APKIM2002 (Drewes and Meise], 2003). The origin (translation and their rates) is realized by SLR, and the scale and its rate by SLR and VLBI. The final ITRF2004 solution will comprise station position~ velocities and daily EOP estimates as primary results. In addition epoch position residuals and geocenter coordinates will be obtained from the time series combination. Status
of
ITRF2004
computations
at
DGFI
The ITRF2004 submissions were analysed at DGFI and first intra-technique solutions were computed on the basis of the weekly input data. As the ITRF2004 submissions are not final at the moment~ the major focus of the computations is (1) to identify remaining deficiencies; (2) to provide feedback to the Technique Centers and to the contributing Analysis Centers; (3) to validate and enhance the combination procedure; (4) to perform first comparisons among the ITRS Combination Centers. The current status is that the intra-technique combination for GPS can be considered as almost final (see sect. 5.1)~ and comparisons among the three ITRS Combination Centers were performed (see sect. 5.2). The intra-technique combination for SLR, VLBI and DORIS are preliminary as new ITRF2004 input will be submitted by the corresponding Technique Centers. Concerning the inter-technique combination specific investigations were performed and re-
Chapter86 • Advancesin TerrestrialReferenceFrameComputations
fined methods were developed (see chapter 6), which can directly by implemented for the computation of the final ITRF2004 solution. 5.1
G P S intra-technique combination
82.72
,
,
,
,
+'
i
i
500
1000
1500
82.7 82.68 E" .,..,
82.66
•~ I
82.64
82.62
In a first step the weekly SINEX solution files of the IGS were analysed and normal equations in DOGS-format were generated. As the resulting GPS normal equations are not singular, we reduced the a priori datum information by setting up seven Helmert-transformation parameters for each week within the accumulation to avoid possible network deformations caused by the a priori datum. We used the information about discontinuities that is provided by the IGS for the accumulation of the weekly normal equations. Specific information related to the GPS intratechnique combination is provided below:
82.6 82.58 1000
• So far the velocities are not set equal yet for different solution numbers at the same station; this procedure will be done within the inter-technique combination. The accumulated GPS normal equation has 14 degrees of freedom as it should be since the datum information was reduced by setting up Helmert-transformation parameters.
• Finally, minimum constraints were applied to generate the final combined GPS solution. The estimated parameters comprise station positions, velocities and daily EOP. As an example for the GPS intra-technique computation Fig. 2 shows the time series of weekly station positions along with the position and velocity estimations of the accumulated multi-year solution for the GPS station HOFN, Iceland. A discontinuity in the station heights of about 5 cm was caused by an antenna and receiver change, which led to two different solutions on this station.
0
Time lid2000]
Figure 2: Position time series and velocity estimates for the height component of GPS station HOFN, Iceland.
Table 2: Vertical station velocity estimates of different solutions at GPS station HOFN, Iceland.
• A few additional discontinuities were introduced, which were obtained from the time series analysis. • All sites with a time span with less than 2.5 years were excluded (since no reliable station velocities can be estimated). In addition solutions of a specific solution number for a station with less than 26 weeks were not used.
500
Velocities
Epochs JD2000
Estimates [mm/yr]
Solution 1 Solution2
-786.5- 619.7 627.5- 1825.5
4.5 + 1.4 8 . 7 + 1.1
Difference
4.2 ± 1.8
As shown in Table 2 the velocity estimates of these two solutions before and after the event differ by 4.2 mm/yr. An important issue (also for many other GPS stations) is the question, whether the estimated velocities for different solutions on a station should be equated or not. For the application of statistical tests it is important that the standard deviations are realistic, which requires sophisticated weigthing methods. 5.2
Comparison of G P S solutions I T R S C o m b i n a t i o n Centers
among
The new IERS structure with three ITRS Combination Centers (IGN, NRCan and DGFI) provides an optimal basis for the accuracy evaluation of the terrestrial reference frame computations. For first comparisons among the ITRS Combination Centers results of the GPS intratechnique combination were used. It has to be considered that different strategies were applied (e.g., at DGFI and NRCan velocities of different solutions for a station were not equated, IGN equated most of the velocities). Thus we used for the comparisons a subset of about 65 IGS reference frame stations without discontinuities to estimate RMS differences for station positions and velocities between the different GPS solutions. As shown in Table 3 the results of the
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D. Angermann • H. Drewes • M. Kriigel • B. Meisel
Table 3: RMS differences for station positions and velocities. RMS position differences [mm]
ITRS CC
DGFI IGN DGFI NRCan IGN NRCan
RMS velocity differences [mm/yr]
A
H
A
H
0.70
0.62
1.89
0.34
0.32
0.56
0.89
0.84
1.22
0.39
0.38
0.46
0.91
0.83
1.56
0.38
0.42
0.54
Table 4: Comparison of EOP of the combined GPS solutions computed at IGN and DGFI. Parameter
Offset
RMS differences (Offset removed)
xpole ypole xpole rate ypole rate
9.7 3.2 0.0 0.0
2.8 4.1 1.6 1.6
#as #as #as/yr #as/yr
#as #as #as/yr #as/yr
three ITRS Combination Centers are in a good agreement. Furthermore, the daily EOP estimates of the combined GPS solutions computed at DGFI and IGN are compared. The discrepancies between both solutions are very small (see Table 4). 6
Inter-technique
combination
studies
As the final ITRF2004 submissions are not fully available at present, we used the currently available input data for exemplary studies as well as for the development and validation of advanced methods for the inter-technique combination. 6.1
Selection
o f local ties
New strategies for the selection and implementation of local ties were investigated on the basis of time series of space geodetic solutions containing station positions and EOP. As shown for example in Angermann et al. (2004) and Kriigel and Angermann (2005), the selection of suitable local ties for the inter-technique combination is difficult because the number and the spatial distri-
bution of "high quality" co-location sites is not optimal. As common parameters of the different space geodetic techniques the station velocities at co-location sites and the EOP represent additional ties to integrate the technique-specific networks into a unique datum. Furthermore the EOP estimates provide valuable information for the local tie validation. In a first step the space geodetic solutions are compared with the local tie measurements. Since GPS is the dominant technique regarding the number and spatial distribution of colocations with the other techniques, we consider GPS as the reference for this specific study. We used the local ties to refer the VLBI, SLR and DORIS solutions to the GPS markers of the colocation sites. Then we performed a 14 parameter Helmert-transformation between the GPS solution and the "transformed" solutions of the other techniques. Table 5 shows as an exampie the position residuals of a transformation between VLBI and GPS for 17 co-location sites. The 3-D discrepancies for these stations are below 2 cm in positions and below 4.5 m m / y r in velocities. The upper part represents 9 good colocations with an agreement better than 1 cm and 2.5 m m / y r in positions and velocities. In a second step the quality of the selected co-location sites is analysed using EOP and applying the two following criteria: (1) The consistency of the combined frame should be maximized. If we combine the station networks (without EOP), the offsets between the xpole and ypole estimates of VLBI and GPS are a measure for the consistency of the combined solution. (2) The deformation of the networks due to the combination should be minimized. The RMS of the transformation between a singletechnique and the combined solution represents the network deformation caused by the selected local ties. The combinations were performed by introducing the local tie in%rmation as pseudo observations with a priori standard deviations of 0.1 ram, 0.3 ram, 1 mm and 3 ram. The datum of the combined solutions was realized by applying a NNR condition. Two test combinations of VLBI and GPS networks were performed: (a) using all 17 co-locations displayed in Table 5; (b) using only the 9 "high-quality" ties.
Chapter86 • Advancesin TerrestrialReferenceFrameComputations
Table 5: Residuals in positions of the 14 parameter Helmert-transformation between VLBI and GPS (the upper part represents the 9 "highquality" co-locations). VLBI-GPS station Ny Alesund Onsala Wettzell Tsukuba Hartrao Westford Mauna Kea Santiago Saint Croix Medicina Noto (NOT1) Madrid Yebes Algonquin Kokee North Liberty Tidbinbilla Hobart (1)
(2) (3) (4) (5)
north [mm] -0.2 -0.5 3.9 2.4 1.8 3.3 1.1 -2.7 -0.3 7.1 17.6 0.1 -4.4 3.0 2.1 0.7 3.3 9.4
east up [ m m ] [mm] -3.0 -1.4 -1.8 -0.7 -1.2 -8.3 0.2 -0.4 -2.2 6.2 5.6 4.5 -2.3 4.3 0.9 1.4 -5.6 -14.9
2.4 -6.8 -2.4 2.5 4.1 -1.0 -0.0 5.3 4.5 -2.9 10.3 -16.2 1.6 -4.8 -15.2 14.2 14.8 12.1
~
.-.
-.
.
o _.e
-1oo -200
Source
0
1
2
3
a [mm]
(2) (1) (1) (4) (1) (1) (1) (1) (1) (3) (1) (1) (3) (1) (1) (1) (5) (5)
IGN database (ftp://lareg.ensg.ign.fr/pub/itrf/iers.ecc) Steinforth et al. 2003 Sarti et al. 2004 Matsuzaka et al. 2004 Johnston et al. 2004
The results of both test combinations according to the criteria mentioned above are shown in Fig. 3 and 4. The offset between the VLBI and GPS pole estimates is much smaller if only 9 ties are used. Also the RMS residuals between the combined solution and the single VLBI solution indicate that the network is less deformed if 9 good ties are used instead of 17 ties. These investigations suggest, that the most stable results are obtained if only a subset of high-quality colocations is used instead of all available local ties. 6.2
100
Figure 3: Offset in ypole between GPS and VLBI solutions (17 ties: dashed line, 9 ties" solid line).
4 ""
=--
.,.....,
n,
0
o
~
~
3
a [mrn]
Figure 4: RMS differences of station positions between the single VLBI solution and the combined GPS and VLBI solution (17 ties: dashed line, 9 ties: solid line).
proper implementation of local tie information and the handling of remaining biases between techniques. The time series solutions provide valuable information to validate the results of the variance component estimation. As an example the repeatabilities of the weekly (daily) estimated positions using a subset of good reference stations for each space technique are shown in Table 6. Note that these results are preliminary as the official ITRF2004 submissions are not available yet for all techniques. Furthermore it has to be considered that the RMS position differences are influenced by uncertainties of the transformation of the epoch solutions, which is in particular a problem in the case of VLBI and SLR. For the computation of weighting factors the different time resolution of VLBI compared to the other techniques has also to be taken into account.
W e i g h t i n g of different t e c h n i q u e s
The weighting of the heterogeneous input data from the different space geodetic observation methods may be performed by a variance component estimation. This method has been iraplemented at DGFI (Kelm, 2003), and detailed studies for the feasibility of this method are carried out under the specific conditions of the intertechnique combination. Problematic issues are a
7
Conclusions
The new approach for the terrestrial reference frame computation based on time series combination of station positions and EOP has major advantages compared to past T R F realizations. The advanced methodology allows to account for nonlinear effects in site motions and ensures consistency between the terrestrial reference frame
601
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D. A n g e r m a n n • H. Drewes • M. Kr~igel • B. Meisel
Table 6: Repeatability of station position estimates for different space techniques. Note, that in the case of VLBI daily sessions were used, whereas the other technique solutions are weekly. Techn. GPS VLBI SLR DORIS
TC/AC IGS IVS ILRS IGN/JPL
North
East
Up
[cm]
[cm] [cm]
0.22 0.53 1.2 2.7
0.26 0.52 1.2 3.7
0.64 1.5 1.3 2.5
and the EOP. First results obtained from a comparison of the GPS intra-technique solutions among the ITRS Combination Centers are very promising. Advanced methods for the intertechnique combination have been developed and will be applied for ITRF2004 computation at DGFI, as soon as the official ITRF2004 submissions become available. The ITRF2004, which will be computed by the three ITRS Combination Centers, namely IGN, NRCan and DGFI, will guarantee a new level of accuracy and consistency for the terrestrial reference frame and the EOP. A c k n o w l e d g e m e n t s . The work is supported by the German Bundesministerium fiir Bildung und Forschung within the G E O T E C H N O L O GIEN project, Grant (IERS)03F0336C.
References Altamimi Z, Sillard P, Boucher C (2002) ITI~F2000: A new release of the International Terrestrial Reference Frame for earth science applications, J. Geophys. Res. 107 (B7), 2214, doi: 10.1029/2001JB000561. Angermann D, Drewes H, Kr/igel M, Meisel B, Gerstl M, Kelm R, Mfiller H, Seemiiller W, Tesmer V (2004) I T R S Combination Center at DGFI: A terrestrial reference frame realization 2003, Deutsche Geod£tische Kommission, Reihe B, Heft Nr. 313. Angermann D, Drewes H, Gerstl M, Kelm R, Kriigel M, Meisel B (2005) I T R F Combination - Status and Recommendations for the Future, Sanso F (Ed): A Window on the Future of Geodesy, IAG Symposia, Vol. 128, , 3-8, Springer. Boucher C, Altamimi Z, Sillard P, Feissel-Vernier M (2004), The ITI~F2000, IERS ITRS Centre, IERS
Technical Note No.31, Verlag des Bundesamtes fSr Karthographie und Geod£sie, Frankfurt am Main. Dill R, Rothacher M (2003) I E R S analysis campaign to align E O P ' s to ITt~F2OOO/ICI~F, GEOTECHNOLOGIEN Science Report No. 3, 36 39, Koordinierungsbiiro Geotechnologien, Potsdam. Drewes H, Meisel B (2003) A n actual plate motion and deformation model as a kinematic terrestrial reference system, GEOTECHNOLOGIEN Science Report No. 3, 40 43, Koordinierungsbiiro Geotechnologien, Potsdam. Drewes H, Angermann D, Gerstl M, Kr/igel M, Meisel B, Seetnfiller W (2005) Analysis and I~efined Cornputations of the International Terrestrial Reference Frame, Observation of the Earth System from Space, Rummel, Reigber, Rothacher, BSdecker, Schreiber, Flury (Eds), Springer Verlag, in press. Johnston G, Dawson J (2004) The 2002 Mount Pleasant (Hobart) radio telescope local tie survey, Geosience Australia, Record 2004/21, Canberra, Australia. Kelm R (2003) Rank defect analysis and variance component estimation for inter-technique combination, Proceedings of the IERS Workshop on Combination Research and Global Geophysical Fluids, B. Richter, W. Schwegmann, W.R. Dick (eds), IERS Technical Note 30, 112-114, Bundesamt f/ir Kartographie und Geod£sie, Frankfurt a. M. Kriigel M and Angermann D (2005) Frontiers in the combination of space geodetic techniques, this volUIIle.
Matsuzaka S, Masaki Y, Tsuji H, Takashima K, Tsumtsumi T, Ishimoto Y, Machida M, Wada H, Kurihara S (2004) V L B I co-location results in Japan In: Vandenberg N. and Bayer K.: International VLBI Service for Geodesy and Astrometry 2004 Gerneral Meeting Proceedings, NASA/CP-2004-212255: 138142 Meisel B, Angermann D, Kriigel M, Drewes H, Gerstl M, Kelm R, Miiller H, Seemiiller W, Tesmer V (2005) Refined approaches for terrestrial reference frame computations, Adv Space Res, Elsevier, in press. Rothacher M, Dill R, Thaller D (2005) I E R S Analysis Coordination, Observation of the Earth Systern from Space, Rummel, Reigber, Rothacher, BSdecker, Schreiber, Flury (Eds), Springer Verlag, in press. Sarti P, Sillard P, Vittuary L (2004), Surveying colocated space-geodetic technique instruments for TRF computation, Journal of Geophysical Research, doi: 10.1007/s00190-004-0387-0 Steinforth C, Haas R, Lidberg M, Nothnagel A: Stability of Space Geodetic Reference Points at Ny-Alesund and their Excentricity Vectors. In: W. Schwegmann and V. Thorandt: Proceedings of the 16th Working Meeting on European VLBI for Geodesy and Astrometry, Bundesamt f/ir Kartographie und Geod/isie, Leipzig, Germany, 2003
Chapter 87
The Status and Future of the International Celestial Reference Frame A.L. Fey U.S. Naval Observatory 3450 Massachusetts Avenue, NW, Washington, DC 20392-5420 USA
Abstract. At the XXIII General Assembly of the International Astronomical Union (IAU) held on 20 August 1997 in Kyoto, Japan, the International Celestial Reference Frame (ICRF) was adopted as the fundamental celestial reference frame. The ICRF is currently defined by the radio positions of 212 extragalactic objects. Since its inception there have been two extensions to the ICRF. Current efforts toward ICRF maintenance include both analysis and observational improvements and an effort is now underway to define a second realization of the ICRF by the year 2009.
Keywords. Astrometry, catalogs, quasars, international celestial reference frame (ICRF), reference systems, interferometry
1
Introduction
For millenia, celestial reference frames have been used for measuring the passage of time, for nayigation, and for studying the dynamics of the solar system. Using optical telescopes, reference frames with roughly 0.1 arcsecond accuracy were produced. With the advent of the technique of Very Long Baseline Interferometry (VLBI), rapid improvements in positional accuracy became possible, reaching the milliarcsecond level in the late 1980s and the sub-milliarcsecond level by the mid-1990s. Stellar reference frames are time-dependent because stars exhibit detectable motions (i.e., parallax and proper motion) the knowledge of which is imprecise. Extragalactic radio sources, on the other hand, are assumed to be very distant and thus should exhibit little or no detectable motions. A reference frame defined by the positions of extragalactic radio sources may be said to be a quasi-inertial fl'ame (i.e., a frame nonrotating with respect to an inertial frame) with little or no time dependency.
24 h
0h
Figure 1: Distribution of the 212 ICRF Defining sources shown on an Aitoff equal-area projection of the celestial sphere.
An almost three order of magnitude improvement in the establishment of an inertial reference frame has been made possible by the advent of the technique of VLBI coupled with the stability of the positions of strong extragalactic radio sources such as quasars and radio galaxies.
2
T h e ICRF
The International Celestial Reference Frame (ICRF) [12] is currently defined by the radio positions of 212 extragalactic objects. The positions on the sky of the 212 ICRF "defining" sources are shown in Figure 1. The radio positions are based upon a general solution for all applicable dual frequency 2.3 GHz and 8.4 GHz VLBI data available through the middle of 1995 (dual-frequency observations allow for an accurate calibration of the frequency-dependent propagation delay introduced by the Earth's ionosphere). The ICRF "defining" sources set the direction of the ICRF axes and were chosen based on their observing histories and the stability and accuracy of their position estimates. The positional accuracy of the ICRF sources is estimated to be better than about 1 mas in both coordinates. In addition to the defining sources, positions for 294 less observed "candidate" sources along with 102 less suitable "other"
604
A.L.
Fey
sources were also given by [12] to densify the fraIne. The I C R F has replaced the FK5 optical catalog as the fundamental celestial reference frame as of 1998 J a n u a r y 1. As a consequence, the definitions of the axes of the celestial reference systern are no longer related to the equator or the ecliptic but have been superseded by the defining coordinates of the ICRF. The I C R F has accomplished its p r i m a r y goal of providing an accurate and stable frame conceptually independent of the motion of the E a r t h in space. However, from its inception, it has been known t h a t the I C R F has a less t h a n desirable density of sources with a less t h a n uniform distribution of sources on the sky (see Figure 1). This situation makes it difficult to assess and control any local deformations in the ICRF. Systematic effects should be greatly reduced in frames derived from extragalactic radio sources but mis-modeling of b o t h the troposphere and source structure will introduce systematic errors in the reported positions. The deficit and nonuniform distribution of I C R F sources also precludes the use of the I C R F as a catalog of phase reference calibrators serving as fiducial points to determine the relative positions of nearby weaker objects (eg. radio stars, pulsars, etc.). The angular separation between the calibrator and target sources should be a few degrees at most in such observations.
2.1
ICRF-Ext.1
The p r i m a r y objectives of extending the I C R F were to provide positions for extragalactic radio sources observed since the definition of the I C R F and to refine the positions of candidate and other sources using additional observations. A secondary objective was to monitor sources to ascertain whether they continue to be suitable for use in the ICRF. The d a t a added in I C R F Ext.1 [7] spanned December 1994 t h r o u g h April 1999 and were obtained from both geodetic and astrometric observing programs. Improved positions and errors for the candidate and other sources were estimated and reflect the changes in the d a t a set and the analysis. Coordinates in the frame of the I C R F were estimated for a total of 59 new sources. Positions and errors of the I C R F defining sources remained unchanged.
2.2
ICRF-Ext.2
Again, the p r i m a r y objective of extending the I C R F was to provide positions for extragalactic radio sources observed since the definition of the I C R F and to refine the positions of candidate and other sources using additional observations. The d a t a added to the I C R F in I C R F Ext.2 [7] spanned May 1999 t h r o u g h May 2002 and were obtained fl'om both geodetic and astrometric observing programs. Improved positions and errors for the candidate and other sources were estim a t e d and reflect the changes in the d a t a set and the analysis. Coordinates in the frame of the I C R F were estimated for a total of 50 new sources. The positions and errors for the defining sources were unchanged from the ICRF.
2.3
A Brief History of the ICRF
• 1988: The IAU sets up working groups to establish a new reference frame. • 1991: The IAU establishes the theoretical basis for a new reference frame. • 1994: The IAU defines the ensemble of fiducial points for a new reference frame as extragalactic objects. • 1995: A sub-group of the IAU Working Group on Reference Frames is tasked to construct a new reference frame based on VLBI observations of quasars. • 1997: The IAU establishes the International Celestial Reference System (ICRS) and adopts the ICRF. • 1998: On J a n u a r y 1st, the I C R F replaces the FK5 optical catalog as the fundamental celestial reference frame. • 1999: The first extension of the ICRF, I C R F Ext.1, is completed adding 59 new radio sources with I C R F coordinates. • 2001: The ICRS P r o d u c t Center is formed and tasked with overseeing the maintenance of the ICRF. • 2002: The second extension of the ICRF, I C R F Ext.2, is completed adding 50 new radio sources with I C R F coordinates.
Chapter 87 • The Status and Future of the International Celestial Reference Frame 9O°
3
ICRF
Maintenance
The IAU has charged the International E a r t h Rotation and Reference Systems Service (IERS) with the maintenance of the ICRF. Maintenance activities are run jointly by the IERS ICRS Product Center and the International VLBI Service (IVS). The IERS ICRS Product Center is directly responsible for the maintenance of the ICRS and ICRF. The Center is run jointly by the Observatoire de Paris and the U.S. Naval Observatory (USNO). More information can also be obtained from the Product Center Web page at http://www, iers. org/iers/pc/icrs/. The IVS is an international collaboration of organizations which operate or support VLBI. The IVS provides a service which supports geodetic and astrometric work on reference systems, E a r t h science research, and operational activities. Many of the observing programs for maintenance of the I C R F are coordinated by the IVS. More information about the IVS can be obtained from the IVS Web page at http://ivscc.gsfc.nasa.gov/.
3.1
Observing Programs
The following observing programs are among those that contribute astrometric d a t a for maintenance of the ICRF. • I V S I C R F E x p e r i m e n t s : These 24 hr duration VLBI experiments, coordinated by the IVS, concentrate primarily on observation of southern hemisphere I C R F sources for monitoring and to increase the sky density of I C R F defining sources. • V L B A R D V E x p e r i m e n t s : These 24 hr duration Very Long Baseline Array (VLBA) R&D with VLBA (RDV) experiments are part of a collaborative program of geodetic and astrometric research between the USNO, Goddard Space Flight Center (GSFC) and the National Radio Astronomy Observatory (NRAO). VLBA RDV experiments concentrate primarily on observation of northern hemisphere I C R F sources (~ > - 3 0 °). Intrinsic source structure information is also obtained from these experiments.
• IVS G e o d e t i c / A s t r o m e t r i c Experiments: These 24 h~ duration VLBI exper-
24h
Oh
-gO o
Figure 2: Distribution of 74 new southern hemisphere sources with milliarcsecond accurate astrometric VLBI positions which are now available for inclusion in the next realization of the ICRF.
iments, coordinated by the IVS, concentrate primarily on observation of sources for geodetic purposes and for E a r t h Orientation P a r a m e t e r estimation but are also useful for astrometric purposes.
• V L B A Calibrator Surveys: These 24 hr duration VLBA experiments are part of a joint N R A O / G S F C program to expand both the list of high quality geodetic sources and the list of phase reference calibrators for imaging.
• E V N Experiments: These 24 hr duration VLBI experiments are part of a Bordeaux Observatory [2] program to expand the list of I C R F defining sources in the northern hemisphere using the European VLBI Network (EVN).
• L B A : These 24 hr duration VLBI experiments are part of a joint USNO and Australia Telescope National Facility (ATNF) program [6, 8] to expand the list of I C R F sources in the southern hemisphere using the ATNF Long Baseline Array (LBA). Intrinsic source structure information is also obtained from these experiments. The positions on the sky of 74 new southern hemisphere sources with measured astrometric VLBI positions from this program are shown in Figure 2. These sources are now available for inclusion in the next radio realization of the ICRF.
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Astrometric Analysis Improvements
4.1
Modeling Capabilities
The largest contributors to inaccuracy and instability in VLBI astrometric results are the propagation media and source structure. While the charged particle media propagation effects have been effectively calibrated using two observing frequencies, the modeling of the troposphere has improved in discrete steps associated with development of new troposphere mapping algorithms [14, 15] and modeling of asymmetry and variability [13]. Current research is directed toward the use of global weather data in computing the mapping function through ray tracing or a proxy. Source structure and changes in source structure put a floor on the stability of source positions. Modeling such effects has been tested on massive scales by [16]. Results of this analysis show that the effects of intrinsic source structure on astrometric position estimation amounts to a significant fraction of the systematic error budget, thus confirming that source structure does affect VLBI analysis even though it is not currently the dominant error. The geometric distribution of terrestrial and celestial references in many of the early VLBI sessions was not good enough to ensure proper de-correlation of the source coordinates errors from errors in the terrestrial frame. Therefore in the analysis strategy that was selected for the ICRF, the station coordinates were estimated as arc-parameters (i.e., arc-parameters depend only on the data from an individual observing session and are estimated separately for each epoch of observation). The drawback of this strategy is that the link between the celestial frame and the terrestrial frame is lost. As shown by [3], this is no longer the case and it should be possible to include the station positions and velocities as global parameters (i.e. parameters dependent upon the data from all observing sessions) in the analysis without contamination of the celestial reference frame. Consequently, the contribution of VLBI to the unification of reference frames would be strengthened.
4.1.1
Source Suitability Based on Intrinsic Structure
Source structure and changes in source structure put a floor on the stability of estimated source positions using VLBI astrometric data. An investigation of the astrometric suitability of
a large sample of ICRle sources (about two-thirds of the total number of ICRE sources) by [4, 5] found correlations between the observed radio structure and the astrometric position accuracy and stability of the sources. These correlations indicated that the more extended sources have larger position uncertainties and are less positionally stable than the more compact sources. The analysis of [4, 5] also revealed that, despite the stringency of the initial selection criteria, about one-third of the ICRle defining sources are somewhat spatially-extended and thus may not be appropriate for defining the celestial frame with the highest level of accuracy. This suggests that revision of source categories would be m a n d a t o r y upon realization of a new ICRF.
4.1.2
Source Position Stability Time Series Analysis
Based on
In order to address the question of finding an improved method of selecting stable and potential defining sources in a subsequent ICRF realization, [3] used position time series, computed in parallel with the ICRF-Ext.2 analysis, to derive a global source stability index based on the repeatability of the source positions from epoch to epoch. This analysis showed that over long time intervals the ICRF defining sources are not as positionally stable as could be hoped. The analysis also identified a set of sources that could potentially be better and which could be used to improve the next realization of the ICRF. Initial analysis shows an improvement in the stability of the direction of the ICRF axes that is better by about a factor of four or five over that of the current 212 ICRF defining sources. This ability to better define the reference frame was confirmed by [1].
4.2 4.2.1
Observational Improvements Observing Strategy
The source stability analysis of [3] found that only about half of the available sources had sufficient data for a time series analysis. The lack of data for stability analysis is most pronounced in the southern hemisphere. The non-uniformity between the northern and southern sky is due primarily to the fact that astrometric/geodetic observations have historically concentrated on northern hemisphere sources. The organization of global geodetic/astrometric observing is now closely coordinated by the IVS and special era-
Chapter87 • The Statusand Futureof the InternationalCelestialReferenceFrame
phasis is being placed on observations of identified stable and potentially stable sources. Consequently, it should be possible to fill the gaps in temporal and spatial coverage in the next few years.
4.2.2
Data Selection
Due to sensitivity limitations of the early VLBI systems, early sessions were limited to observing only 12-15 of the strongest radio sources in a single session. Improvements in data acquisition technology and the consequent increases in sensitivity allowed the observing profile to evolve over time to the point where 2 0 - 40 sources could be routinely observed in a typical session. In more recent years (i.e., after 1995), the number of sources observed in a session has risen to about 4 0 - 50. The stability of source positions derived from yearly sub-sets of the d a t a has been shown to improve significantly around 1990 [10], largely as a result of changes in observing strategy and observing networks (the radio telescopes used). Consequently, a considerable improvement can be expected in any new realization of the I C R F by simply using only the data acquired since 1990.
4.2.3
Data Acquisition Technology
Technical innovations incorporated into newer VLBI recording systems now allow the use of both wider spanned bandwidths and higher recorded data rates. The consequent increased sensitivity will allow observations of weaker sources not previously accessible. As these systems are more widely deployed, much higher sensitivity observations will become the norm, increasing the number of observations and increasing the number of observable radio sources. Together with specific efforts by the IVS this should provide increased observations on stable and potentially stable sources.
5
Extending the ICRF to Higher Radio Frequencies
VLBA observations to extend the I C R F to radio frequencies of 24 GHz and 43 GHz began in 2002 May. The use of this frequency pair was motivated by the National Aeronautics and Space Administration (NASA) decision to move future spacecraft telemetry from the current 8.4 GHz to 32 GHz and the availability of 24 GHz and 43 GHz receivers on the VLBA. One of the
goals of these observations was to study whether the sources were more compact at 24 GHz and 43 GHz in order to improve the astrometric accuracy at these frequencies. Initial imaging results [9] show that sources are indeed more spatially compact at these higher frequencies than those currently used for the ICRF. The initial reference frame derived from these data [11] shows agreement with the I C R F to roughly the 0.3 mas level with zonal errors dominating the differences. The accuracy of a celestial reference frame defined at these higher observing frequencies has the potential of exceeding that of the current ICRF. 6
Space-based
Optical
Astrornetric
Satellites
In the coming decades, there will be significant advances in the area of space based optical astrometry. Planned missions such as the NASA SIM PlaneQuest and the European Space Agency Gaia will achieve positional accuracies well beyond that presently obtained by any ground based radio interferometric measuremerits. SIM PlaneQuest will be a space based optical interferometer that will be able to determine the positions of stars with a precision approaching the microarcsecond (#as) level. SIM PlaneQuest is planned as a pointed mission with a limited number of target objects and limited sensitivity. The SIM astrometric grid, which will consist almost entirely of stellar sources (and of order 100 extragalactic objects to remove the global rotation of the stellar frame), will ultimately be more accurate than the current I C R F but will not be quasi-inertial as the stars that will be observed are nearby objects in comparison to the quasars that make up the radio frame. Gala on the other hand is planned as a survey mission and will make observations of order 109 objects and, with a limiting magnitude of rn~ ~ 20, will be able to observe almost all known extragalactic reference sources. Because of the large number of extragalactic objects accessible by Gala, the astrometric grid defined by Gala can be constructed in such a way as to be quasi inertial. If the projected accuracies for Gaia are realized, the Gala astrometric grid will be serious competition for the radio realization of the I C R F and must prompt reevaluation of the spectral regime at which the I C R F is defined.
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Discussion
Modeling and analysis capabilities have advanced significantly since the ICRF was defined, particularly for handling the troposphere and source structure. Refinements in modeling the motion of the stations, especially in various loading effects, should permit the unification of analysis for the celestial reference frame, the terrestrial reference frame and E a r t h orientation parameters, which were separated in the ICRF analysis. The institution of the IVS has significantly improved the organization and coordination of global geodetic/astrometric VLBI observations. As a consequence of changes in observing strategy and networks over the lifetime of astrometric,/geodetic VLBI, the stability of source positions derived from individual year data sets has improved significantly since about 1990. A considerable improvement in the ICRF can be expected by using only the data obtained after 1990. Special emphasis by the IVS, especially in the southern hemisphere, is now being placed on observations of the stable and potentially stable sources identified as possible defining sources for a next realization of the ICRF. Technical innovations now allow the use of both wider spanned bandwidths and higher recorded data rates. The consequent increase in sensitivity allows observation of weaker sources and hence increases the available pool of new sources.
A trend toward observing at higher radio frequencies where the radio sources are intrinsically more compact is anticipated. Further, the detected source components at these higher frequencies are expected to remain closer to the central engine of the active galactic nuclei and thus be more astrometrically stable in the long term. Within the next 10-15 years, optical astrometric satellites will present serious competition to the radio based ICRF. Reevaluation of the spectral regime at which the ICRF is defined will then be necessary.
8
Summary
The ICRF forms the stable reference points from which the orientation of the E a r t h in three dimensional space can be measured. The technique of VLBI provides the unique tie to the inertial
reference frame realized by the ICRF. The ICRF is used for navigation and positioning applications on the E a r t h and in space and will continue to be used as fiducial reference points for astrophysical observations. With continued applicable VLBI observations and improvements in analysis, a better realization of the ICRF is now possible and an even better realization is feasible in the foreseeable future. Under the direction of the ICRS Product Center, which is responsible for the maintenance of the ICRF, planning for a second radio realization of the ICRF is currently underway with a projected completion date concurrent with the 2009 IAU General Assembly.
References [1] Arias, E. F. and Bouquillon, S. (2004), Representation of the International Celestial Reference System (ICRS) by different sets of selected radio sources, Astronomy and Astrophysics, 422, 1105 [2] Charlot, P. et al. (2000), A Proposed Astrotnetric Observing Program for Densifying the ICRF in the Northern Hemisphere, in International VLBI Service for Geodesy and Astrornctry 2000 General Meeting Proceedings, edited by Nancy R. Vandcnbcrg and Karcn D. Bayer, NASA/CP-2000209893, p. 168 [3] Feissel-Vernier, M. (2003), Selecting stable extragalactic compact radio sources from the permanent astrogeodetic VLBI program, Astronomy and Astrophysics, 403, 105 [4] Fey, A. L., and Charlot, P. (1997), VLBA Observations of Radio Reference Frame Sources. II. Astrometric Suitability Based on Observed Structure, Astrophysical Journal Supplement Series, 111, 95 [5] Fey, A. L., and Charlot, P. (2000), VLBA Observations of Radio Reference Frame Sources. III. Astrometric Suitability of an Additional 225 Sources, Astrophysical Yournal Supplement Series, 128, 17 [6] Fey, A. L., Ojha, R., Jauncey, D. L., Johnston, K. J.,Reynolds, J. E., Lovell, J. E. J., Tzioumis, A. K., Quick, J. F. H, Nicolson, G. D., Ellingsen, S. P., McCulloch, P. M., and Koyama, Y. (2004), Accurate Astromctry of 22 Southern Hemisphere Radio Sources, Astronomical Journal, 127, 1791 [7] Fey, A. L., Ma, C., Arias, E. F., Charlot, P. Feissel-Vernier, M., Gontier, A.-M., Jacobs, C. S., Li, J., and MacMillan, D. S. (2004), The Second
Chapter 87 • The Status and Future of the International Celestial Reference Frame
Extension of the International Celestial Reference Frame: ICRF-EXT.2, Astronomical Journal, 127, 3587 [8] Fey, A. L., Ojha, R., Reynolds, J. E., Ellingsen, S. P., McCulloch, P. M., Jauncey, D. L., and Johnston, K. J. (2004), Astrometry of 25 Southern Hemisphere Radio Sources from a VLBI ShortBaseline Survey, Astronomical Journal, 128, 2593 [9] Fey, A. L. and Boboltz, D. A., Charlot, P., Fomalout, E. B., Lanyi, G. E., Zhang, L. D. and the K-Q VLBI Survey Collaboration (2005), Extending the ICRF to Higher Radio Frequencies- First Imaging Results, in Future Directions in High Resolution Astronomy: The lOth Anniversary of the VLBA, ASP Conference Proceedings, Vol. 340. Edited by J. Romney and M. Reid. San Francisco: Astronomical Society of the Pacific, 2005., p.514 [10] Gontier, A . - M . , Le Bail, K., Feissel, M., Eubanks, T. M. (2001), Stability of the extragalactic VLBI reference frame, Astronomy and Astrophysics, 375, 661 [11] Jacobs, C. S. et al. (2005), Extending the ICRF to Higher Radio Frequencies: Astrometry, in Future Directions in High Resolution Astronomy: The 10th Anniversary of the VLBA, ASP Conference Proceedings, Vol. 340. Edited by J. Romney and M. Reid. San Francisco: Astronomical Society
of the Pacific, 2005., p.523 [12] Ma, C., Arias, E. F., Eubanks, T. M., Fey, A. L., Gontier, A. M., Jacobs, C. S., Sovers, O. J., Archinal, B. A., and Charlot, P. (1998), The International Celestial Reference Frame as Realized by Very Long Baseline Interferometry, Astronomical Journal, 116, 516 [13] MacMillan, D. S., and Ma, C. (1997), Atmospheric gradients and the VLBI terrestrial and celestial reference frames, Geophys. Res. Left., 24, 453 [14] Niell, A. E. (1996), Global mapping functions for the atmosphere delay at radio wavelengths, Journal of Geophysical Research, 101, 3227 [15] Niell, A. E. and Tang, J. (2002), Gradient Mapping Functions for VLBI and GPS, in International VLBI Service for Geodesy and Astrometry 2002 General Meeting Proceedings, edited by Nancy R. Vandenberg and Karen D. Baver, NASA/CP-2002-210002, p. 215 [16] Sovers, O. J., Charlot, P., Fey, A. L., and Gordon, D. G. (2002), Structure Corrections in Modeling VLBI Delays for RDV Data, in International VLBI Service for Geodesy and Astrometry 2002 General Meeting Proceedings, edited by Nancy R. Vandenberg and Karen D. Bayer, NASA/CP2002-210002, p. 243
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Chapter 88
Is Scintillation the Key to a Better Celestial Reference Frame? R. Ojha Australia Telescope National Facility, CSIRO PO Box 76, Epping, NSW 1710, Australia A.L. Fey U.S. Naval Observatory 3450 Massachusetts Avenue, NW, Washington, DC 20392-5420 USA D.L. Jauncey, J.E.J. Lovell Australia Telescope National Facility, CSIRO PO Box 76, Epping, NSW 1710, Australia K.J. Johnston U.S. Naval Observatory 3450 Massachusetts Avenue, NW, Washington, DC 20392-5420 USA
Abstract. Extragalactic radio sources, such as those that define the International Celestial Reference Frame (ICRF), display a variety of structure down to milliarcsecond scales. Further they are all variable on scales of years to weeks. This departure from the point source approximation introduces error in the observable variables (delay and rate). The effect of source structure on position can be as large as tens of milliarcseconds (e.g. [4]). Also, as the structure of these sources varies with time, it is important to image them at several epochs in order to define a time dependent source model. Multi-epoch observations, using the Very Long Baseline Array (VLBA) to image northern hemisphere sources and the Australian Long Baseline Array (LBA) to image southern hemisphere sources have been progressing successfully for a few years. However, the need for such extensive monitoring programs could be dramatically reduced if a population of extragalactic radio sources that have little or no milliarcsecond scale structure was available. Very Long Baseline Interferometry (VLBI) observations of extragalactic radio sources which exhibit interstellar scintillation suggest that such sources are among the most compact in the sky. In particular, the most variable weak sources,
might be the most point-like and, thus, some of the best candidates for densification of the ICRF and consequent improvement in its accuracy. Further, the advent of the MarkV VLBI recording system, with its enhanced sensitivity, will make use of weaker sources easier. We present the evidence for and discuss the viability of this idea which has the potential to revolutionize future upgrades of the ICRF.
Keywords. Astrometry, catalogs, quasars, international celestial reference frame (ICRF), reference systems, interferometry, scintillation, interstellar scintillation (ISS), intraday variability (IDV), interstellar medium (ISM)
Introduction" The International Celestial Reference Frame Since the late 1970s, Very Long Baseline Interferometry (VLBI) observations at two frequencies (8.4 and 2.3 GHz) have been used to determine the positions of compact radio sources with unprecedented accuracy. Submilliarcsecond accurate radio positions of a group of 212 selected strong extragalactic radio sources, currently define the fundamental celestial reference frame, the International Celestial Reference
Chapter 88 • Is Scintillation the Key to a Better Celestial Reference Frame?
Frame ( I C R F ) [ 1 6 ] . These "defining" sources were chosen for the stability and accuracy of their position estimates as demonstrated by their observing histories over two decades. Partly due to the difficulty in finding suitable objects, the ICRF suffers from a lack of defining sources particularly in the southern hemisphere where VLBI resources are more scarce. Thus any phenomenon that might be a marker of astrometric suitability of extragalactic radio sources would be a valuable tool in the densification of the ICRF and consequent improvement in its accuracy and accessibility. Among the many benefits flowing from a more accurate ICRF would be an improvement in the measurements of VLBI baselines resulting in better station positions and ultimately a more accurate International Terrestrial Reference Frame (ITRF).
Im-
practice, extragalactic radio sources display flux variation and a wealth of morphologies that often vary with time. This has made it necessary to conduct multi-epoch VLBI observations to develop time dependent source models of ICRF sources and evaluate their continuing suitability as reference frame sources. Programs to monitor sources in the northern hemisphere using the Very Long Baseline Array [5] and southern hemisphere using the Long Baseline Array (LBA) [19], [20] are ongoing and represent a major investment of astronomical facilities and skilled manpower. This problem of extended structure is exacerbated by the fact that the relatively stronger sources that the current ICRF is based on are even more likely to be highly variable due to their morphological nature. Thus, the development of techniques to locate point-like sources is crucial to densification and improvement of the the accuracy and accessibility of the ICRF.
For an extragalactic radio source to be suitable for inclusion in the ICRF it must be strong at erawavelengths where the observations are made, should have little or no structure at milliarcsecond resolution (ideally a "point" source), and should have a stable position on the sky.
One possible way forward is to observe at higher frequencies where it has been demonstrated [8] that extragalactic radio sources have systematically less flux in extended components as compared to the current standard 8.4 GHz band. An exploratory observing program at 22 GHz and 43 GHz is in progress [9].
The ICRF is currently limited by a number of factors. One is a deficit of defining sources which is particularly acute in the southern hemisphere. This is mostly a reflection of the relative scarcity of VLBI observing resources in the south. This shortcoming is being addressed with a major program to identify and measure astrometric positions for possible new ICRF sources
Here we wish to present another possible answer to the problem of locating the most compact sources for reference frame use. Our unrelated investigation of the properties of scintillating Intraday Variable (IDV) sources suggests that the key to a better ICRF might lie with these sources. So what are IDV sources and what is scintillation?
2
Current Limitations and Future provement
[6], [71, Another constraint is the limited number of sources that are strong enough for VLBI observations at era-wavelengths. Current ICRF sources have fluxes of the order of one Jansky at these frequencies. The advent of the MarkV VLBI recording system brings an increase in sensitivity that should allow sources above 100 m J y to be used, greatly expanding the pool of possible candidates. However, probably the most difficult condition for a potential ICRF source to satisfy is the "point-source approximation". This assumption of the absence of significant source structure is made in astrometric analysis and its violation has been shown (e.g. [4]) to affect the source position by as much as tens of milliarcseconds. In
3
IDV and ISS
Intraday Variable (IDV) sources are fiatspectrum extragalactic sources which show amplitude variations at centimeter wavelengths. Variation timescales are typically less than a day (Figure 1) with the most egregious IDV sources showing several quasi-periodic variations (with extremes of up to 50%) over a 12-hour period [13], [2], [1]. While some sources appear to be "persistent" scintillators, others are "episodic" with scintillation behaviour appearing and disappearing at different epochs in an essentially unpredictable manner. Interstellar Scintillation (ISS) is the rapid variability in the light curves of radio sources arising from radio wave propagation through turbu-
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Figure 1" Light curves of IDV/ISS sources with tiinescales ranging from a few hours to approximately one day (from [15]). lent electron density fluctuations in the interstellar medium. While many questions remain, it is now well established that ISS is the principal cause of the intra-day variability (IDV) seen in many compact, flat-spectrum radio sources at cm-wavelengths. Time difference of minutes have been discovered in the variability pattern arrival times at widely spaced telescopes for the three most rapidly variable sources [10], [3], [11]. The motion of the irregularities in the interstellar medium (ISM) can cause the the patterns to arrive at different times at two widely spaced radio telescopes, something that would not happen if the variations were intrinsic. In addition, an annual cycle in the characteristic time scale of the variability has also been found in a number of sources e.g. [12]. For such observations, ISS is the only plausible explanation.
Milliarcsecond Scale Structure of Scintillating Sources 4.1
MASIV motivation
In order to facilitate the study of scintillation phenomena it was necessary to expand the number of known IDV sources from the few then known. The 5 GHz VLA Micro-Arcsecond Scintillation-Induced Variability (MASIV) survey [15] is a large variability survey of the north-
ern sky that aimed to construct a sample of scintillating extragalactic sources with which to examine both the microarcsecond structure and the parent population of these sources, as well as to probe the interstellar medium (ISM) responsible for the scintillation. It has discovered ~ 150 new scintillating sources over the northern sky. An immediate result from MASIV is that the fraction of scintillating sources increases strongly with decreasing flux density. This may be because the microarcsecond scintillating components are simply brightness temperature limited, as might be expected from an inverse Compton limited brightness temperature. Alternatively, it may be the result of sampling a different source population amongst the weaker sources, in the sense that the weaker sources are more "core" dominated, or rather, less milliarcsecond "jet" dominated. Further, the presence of ISS implies the presence of microarcsecond structures and measurement of ISS characteristics can provide access to information at far higher resolution than is achievable with conventional VLBI, including space VLBI [17].
4.2
VLBA Observations and Results
We have been using the National Radio Astronomy Observatory's, Very Long Baseline Array (VLBA) to conduct a series of experiments to image the millarcsecond scale morphology of compact radio sources in order to study the phenomenon of ISS starting with the above issues raised by the MASIV survey. We have completed VLBA snapshot imaging of 75 MASIV AGN. For details of sample selection, the experiment and the images see [21]. The results indeed show that most of the scintillating sources we observed with the VLBA have an extremely compact, core-dominated morphology (e.g. see Figure 2) which has important implications for interpretations of the scintillation seen in these sources. This result first suggested that these scintillating sources are also potential candidates for inclusion in the International Celestial Reference Frame. Further, we have compared the milliarcsecond scale structure of the MASIV scintillating and non-scintillating sources using the above data supplemented by published a n d / o r publicly available 8.4 GHz VLBA images [22]. Using multiple measures, we found that all scintillating sources, both low and high flux density samples, are significantly more core dominated than n o n -
Chapter88 • Is Scintillationthe Keyto a BetterCelestialReferenceFrame? I map.
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Figure 2" Typical VLBA image of a scintillating source (from [21]). Coutours are in v/2 steps starting at 3or
scintillating sources. Further, the overall source size of the scintillating sources is significantly smaller than that of non-scintillators. There does not appear to be any significant difference between the milliarcsecond-scale morphologies of low and high flux density scintillators. These results demonstrate that it is the core of the radio source that is scintillating. Surprisingly, we also found 25~ (19 of 75) of the low flux density scintillators were unresolved, with a dynamic range in excess of several hundred, at 8.4 GHz on all VLBA baselines. This is remarkable, since none of the high flux density MASIV sources, scintillators or non-scintillators are unresolved and only ~ 8% of sources in the NRAO 15 GHz VLBA survey [14] are unresolved. This result underscores the potential for scintillators to provide the highest quality candidate ICRF sources.
the sky, such sources have the potential to provide a much denser grid of accurately positioned sources leading to a significant improvement in the ICRF. Surveys similar to the MASIV survey could be a quick and efficient method of locating large numbers of scintillators which can then be followed up with VLBI observations to test for astrometric suitability. Further, the absence of structure will greatly reduce the need for multiepoch VLBI observations to ensure the continuing suitability of existing ICRF sources. The major caveat is that the positional stability of these compact scintillating sources will need to be established. There is the possibility that the flux density variation resulting from scintillation of the micro-arcsecond component could produce position shifts due to refraction in the interstellar medium. However, such elfects should be limited to the scattered component sizes (a few micro-arcseconds). Scintillation from any "Christmas tree" effects (when small components brighten and dim within the larger, mostly invisible, structure) could also cause position shifts. These and other possible factors impacting positional stability will need to be studied before we can conclude that scintillating sources fulfill the promise of providing a pool of good reference frame sources. 6
Conclusions
Flat spectrum, extragalactic radio sources that exhibit rapid variability in their light curves appear to be the most compact sources in the sky. As such, they raise the exciting possibility of providing a large and easily identifiable pool of candidates for inclusion in future extensions of the ICRF. As the number density of scintillating sources increases with decreasing flux density, the coincidental advent of MarkV VLBI recording systems, with their greater sensitivity, is fortuitous and will make use of these weaker sources viable. Provided their positional stability can be established, scintillating sources could play a crucial role in the establishment, expansion and upkeep of the next generation astrometry and geodesy reference frames. References
5
Implications
for
Improvement
of
the
ICRF
As sources that exhibit scintillation have proved to be amongst the most compact sources in
[I] Bignall, H. E., Jauncey, D. L., Lovell, J. E. J., Tzioumis, A. K., Kedziora-Chudzcer, Macquart, J-P., Tingay, S. J., Rayner, D. P., Clay, R. W. (2003). Rapid Variability and An-
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nual Cycles in the Characteristic Timescale of the Scintillating Source PKS 1257-326 The Astrophysical Journal 585, pp. 653-664 [21 Dennett-Thorpe, & J., de Bruyn, G. (2000). The Discovery of a Microarcsecond Quasar: J1819+3845. The Astrophysical Journal 529, pp. L65-L68 [3] Dennett-Thorpe, & J., de Bruyn, G. (2002). Interstellar scintillation as the origin of the rapid radio variability of the quasar J1819+3845. Nature 415, pp. 57-60 [4] Fey, A. L. & Charlot, P. (1997). VLBA Observations of Radio Reference Frame Sources. II. Astrometric Suitability Based on Observed Structure. The Astrophysical Journal Supplement Series, 111, pp. 95-142 [5] Fey, A. L. & Charlot, P. (2000). VLBA Observations of Radio Reference Frame Sources. III. Astrometric Suitability of an Additional 225 Sources. The Astrophysical Journal Supplement Series 128, pp. 17-83 [61 Fey, A. L., Ojha, R., Jauncey, D. L., Johnston, K. J.,Reynolds, J. E., Lovell, J. E. J., Tzioumis, A. K., Quick, J. F. H, Nicolson, G. D., Ellingsen, S. P., McCulloch, P. M., & Koyama, Y. (2004a). Accurate Astrometry of 22 Southern Hemisphere Radio Sources. The Astronomical Journal 127, pp. 1791-1795 [71 Fey, A. L., Ojha, R., Reynolds, J. E., Ellingsen, S. P., McCulloch, P. M., Jauncey, D. L., & Johnston, K. J. (2004b). Astrometry of 25 Southern Hemisphere Radio Sources from a VLBI Short-Baseline Survey. The Astronomical Journal 128, pp. 2593-2598 [8] Fey, A. L., Boboltz, D. A., Chariot, P., Fomalont, E. B., Lanyi, G. E., Zhang, L. D. 85 The K-Q VLBI Survey Collaboration (2005). Extending the ICRF to Higher Radio Frequencies: First Imaging Results. Future Directions in High Resolution Astronomy: The lOth Anniversary of the VLBA, ASP Conference Series 340, pp. 514 [91 Jacobs, C. S., Lanyi, G. E., Naudet, C. J., Sovers, O. J., Zhang, L. D., Charlot, P., Gordon, D., Ma, C. & The KQ VLBI Survey Collaboration (2005). Extending the ICRF to Higher Radio Frequencies: Astrometry at 24
and 43 GHz. Future Directions in High Resolution Astronomy: The l Oth Anniversary of the VLBA, ASP Conference Series 340, pp. 523 [101 Jauncey, D. L. et al., (2000) The Origin of Intra-Day Variability. In: Astrophysical Phenomena Revealed by Space VLBL Proceedings of the VSOP Symposium. Institute of Space and Astronautical Science, Sagamihara, Kanagawa, Japan, January 19-21, H. Hirahayashi, P. G. Edwards, and D. W. Murphy (eds.), Published by the Institute of Space and Astronautical Science, pp. 147-150 [111 Jauncey, D. L., Bignall, H. E., Lovell, J. E. J., Kedziora-Chudczer, L., Tzioumis, A. K., Macquart, J-P., & Rickett, B. J. (2003). Interstellar Scintillation and Radio Intra-Day Variability. In: Radio Astronomy at the Fringe, ASP Conference Proceedings. Zensus, A. J., Ros, E., & Cohen, M. H. (eds.) 300, 199 [12] Jauncey, D. L. & Macquart, J.-P. (2001). Intra-day variability and the interstellar medium towards 0917+624. Astronomy ~ Astrophysics 370, pp. L9-L12 [13] Kedziora-Chudczer, L., Jauncey, D. L., Wieringa, M. H., Walker, M. A., Nicolson, G. D., Reynolds, J. E., Tzioumis, A. K. (1997). PKS 0405-385: The Smallest Radio Quasar? The Astrophysical Journal 490, pp. Lg-L12 [14] Kellerman, K. I. et al. (2004). SubMilliarcsecond Imaging of Quasars and Active Galactic Nuclei. III. Kinematics of Parsecscale Radio Jets. The Astrophysical Journal 609, pp. 539-563 [15] Lovell, J.E.J., Jauncey, D. L., Bignall, H. E., Kedziora-Chudczer, L., Macquart, J . , P., Rickett, B. J., & Tzioumis, A. K. (2003). First Results from MASIV: The Microarcsecond Scintillation-induced Variability Survey. The Astronomical Journal 126, pp. 1699-1706 [16] Ma, C., Arias, E. F., Eubanks, T. M., Fey, A. L., Gontier, A.-M., Jacobs, C. S., Sovers, O. J., Archinal, B. A., & Charlot, P. (1998). The International Celestial Reference Frame as Realized by Very Long Baseline Interferometry. The Astronomical Journal, 116, pp. 516-546
Chapter 88 • Is Scintillation the Key to a Better Celestial Reference Frame?
[17] Macquart, J. -P., & Jauncey, D. L. (2002). Microarcsecond Radio Imaging Using the Earth-Orbit Synthesis. The Astrophysical Journal 572, pp. 786-795 [18] Ojha, R. et al. (2006). hnprovement and Extension of the International Celestial Reference Frame in the Southern Hemisphere. These Proceedings [19] Ojha, R., et. al (2004a). VLBI Observations of Southern Hemisphere ICRF Sources. I. The Astronomical Journal 127, pp. 3609-3621 [20] Ojha, R., et. al (2004b). VLBI Observations of Southern Hemisphere ICRF Sources. II. As-
trometric Suitability Based on Intrinsic Structure. The Astronomical Journal 130, pp. 25292540 [21] Ojha, R., Fey, A. L., Lovell, J. E. J., Jauncey, D. L. & Johnston, K. J. (2004c). VLBA Snapshot Imaging Survey of Scintillating Sources. The Astronomical Journal 128, pp. 1570-1587 [22] Ojha, R., Fey, A. L., Jauncey, D. L., Lovell, J. E. J. & Johnston, K. J. (2004d). Milliarcsecond Structure of Microarcsecond Sources: Comparison of Scintillating and Nonscintillating Extragalactic Radio Sources. The Astrophysical Journal 614, pp. 607-614
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Chapter 89
Improvement and Extension of the International Celestial Reference Frame in the Southern Hemisphere R. Ojha Australia Telescope National Facility, CSIRO PO Box 76, Epping, NSW 1710, Australia A.L. Fey U.S. Naval Observatory 3450 Massachusetts Avenue, NW, Washington, DC 20392-5420 USA P. Charlot Observatoire de Bordeaux (OASU)- CNRS/UMR 5804 BP 89, 33270 Floirac, France K.J. Johnston U.S. Naval Observatory 3450 Massachusetts Avenue, NW, Washington, DC 20392-5420 USA D.L. Jauncey, J. E. Reynolds, A. K. Tzioumis, J.E.J. Lovell Australia Telescope National Facility, CSIRO PO Box 76, Epping, NSW 1710, Australia J.F.H. Quick, G.D. Nicolson Hartebeesthoek Radio Astronomy Observatory P. O. Box 443, Krugersdorp 1740, South Africa S.P. Ellingsen, P.M. McCulloch School of Mathematics & Physics, University of Tasmania Private Bag 37, Hobart, Tasmania 7001, Australia Y. Koyama Kashima Space Research Center, Communications Research Laboratory 893-1 Hirai, Kashima, Ibaraki 314-8501, Japan
Abstract.
The U.S. Naval Observatory (USNO) and the Australia Telescope National Facility (ATNF) are collaborating in a continuing VLBI research program in Southern Hemisphere ICRF source imaging and astrometry using USNO, ATNF and ATNF-accessible facilities. These observations are aimed specifically toward improvement of the International Celestial Reference Frame (ICRF) in the Southern Hemisphere. We present the contributions of this program to the densification of the ICRF
in the Southern Hemisphere and the imaging of all southern hemisphere ICRF sources for calculation of a "structure index". The impact of both these contributions to the improvement of the ICRF in the Southern Hemisphere will be discussed.
Keywords. Astrometry, catalogs, quasars, international celestial reference frame (ICRF), reference systems, interferometry
Chapter 89 • Improvement and Extension of the International Celestial Reference Frame in the Southern Hemisphere
1
Introduction
Since the 1980's Very Long Baseline Interferometry (VLBI) observations at two frequencies, 8.4 (X-band) and 2.3 GHz (S-band) have been used to locate the positions of compact radio sources with unprecedented accuracy. Observations of selected strong compact extragalactic radio sources, using this now mature technique, have been used to define and maintain a radio reference frame with sub-milliarcsecond precision. This International Celestial Reference Frame (ICRF)[14] was adopted by the XXIII IAU General Assembly in 1997 as the fundamental celestial reference frame. Currently, the ICRF is defined by the radio positions of 212 extragalactic sources chosen for their positional stability and accuracy ascertained from VLBI observations over two decades. Figure 1 shows the distribution of these sources on the sky. [14] also include the positions for 294 less observed "candidate" sources and 102 less suitable "other" sources used to densify the frame. In this paper we describe an ongoing program which seeks to improve and extend the ICRF in the Southern Hemisphere. We explain the limitations of the ICRF in the Southern Hemisphere which motivate our observations and set the goals of our research program. After a description of the observations, we report on the progress of both the astrometric and imaging components of this project. We state a few of the many benefits, spread over many different fields, that will result from this program. Not the least of these, from a geodetic perspective, will be reduced errors in VLBI baseline lengths, hence VLBI station positions, leading to improvements in the International Terrestrial Reference Frame (ITRF) and better determination of the Earth Orientation Parameters e.g. see [4], [5].
2
Limitations of the ICRF in the Southern Hemisphere
Though, in principle, two fixed points can define the reference frame [12], in practice numerous fixed points are needed to define the reference frame globally as the definition of the axes of the frame improves with the number of points used to define them. Also more reference points are necessary to allow access to the reference frame anywhere on the celestial sphere. Even a cursory examination of Figure 1 reveals an acute deficit of defining ICRF sources in the South-
ern Hemisphere. In fact less than 30% of the 212 defining sources are located in the Southern Hemisphere. Since the establishment of the fundamental Southern Hemisphere reference frame for the ICRF, [21], [20], [11], there have been very few additions. The scarcity of Southern Hemisphere sources relative to northern hemisphere sources simply reflects the greater availability of observing resources in the northern hemisphere and the resulting concentration of astrometric and geodetic observations on predominantly northern hemisphere sources. Thus, there is a need to locate suitable new Southern Hemisphere radio sources for inclusion in future versions of the ICRF and to conduct astrometric VLBI in order to determine their positions precisely. Even at the high resolution of VLBI, the extragalactic sources used to define the ICRF exhibit variable flux density and frequently display structure that can vary with time. As astrometric analysis commonly makes a "point-source" approximation, any departure therefrom introduces error in the observable quantities (group delay and group rate). Demonstration of the effect of source structure on VLBI astrometric positions [2], [6], led to the establishment of ongoing multi-epoch VLBI observations of northern hemisphere ICRF sources using the Very Long Baseline Array (VLBA) in order to determine a time-dependent source model [7]. There is clearly a need for corresponding observations in the southern hemisphere.
3
Program Goals
The Australia Telescope National Facility (ATNF) and the United States Naval Observatory (USNO) have an ongoing research program that seeks to address both of the above related needs which have precision astrometry as their central driver. The goals of the program are to increase the sky density of Southern Hemisphere ICRF sources and to image all existing and any new Southern Hemisphere ICRF sources to estimate the effect of their structure on their positional accuracy. Such observations will also provide a tie between the hemispheres through an overlap of common sources. A denser and more uniform distribution of ICRF sources in the Southern Hemisphere will allow exploration of systematic errors in the ICRF e.g deformations caused by tropospheric effects [15]. Fur-
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24 h
NSW and the 22-meter near Coonabarabran, NSW all of which are operated by the ATNF. The remaining two telescopes of the LBA are the 26-meter at Hobart, Tasmania and the 30meter at Ceduna, South Australia both of which are operated by the University of Tasmania. In addition to the LBA this program has used the 26-meter at Hartebeesthoek, South Africa, the 70-meter Deep Space Network antenna at Tidbinbilla, ACT, the 34-meter at Kashima, Japan and the 20-meter at Kokee Park, Hawaii. All VLBI data recorded using the $2 VLBI recording system [1] were correlated at the LBA Correlator [22] located in Marsfield, NSW. All VLBI data recorded in the MkIII/MkIV format were correlated at the Washington Correlator [13] located in Washington DC.
0h
5 5.1
-9O °
Figure 2: Distribution of 74 new Southern Hemisphere sources with milliarcsecond accurate astrometric VLBI positions which are now available for inclusion in the next realization of the ICRF. ther, the improved accuracy of the ICRF will lead to greater stability of VLBI station networks and to an improvement of e.g. the International Terrestrial Reference Frame (ITRF) and the Earth Orientation Parameters. Finally, such a grid will also enable greater use of the phase calibration technique in VLBI observations where strong calibrators serve as fiducial sources to determine positions of weaker nearby objects. Such observations are possible only if the separation between calibrator and target source is within a few degrees at most.
4
Observations
As detailed below, different observations in this program have used different subsets of USNO, ATNF and ATNF-accessible facilities. The telescopes used include those that form the Australian Long Baseline Array (LBA). The LBA includes the six 22-meter telescopes at Narrabri, NSW (these comprise the Australia Telescope Compact Array and usually five were used as a combined "tied" array), the 64-meter at Parkes,
Astrometry Identification of new sources
In order to identify potential new extragalactic radio sources for astrometric observation and eventual inclusion in the ICRF we conducted short VLBI scans at 8.4 GHz on extragalactic sources that had been detected on shorter baselines, typically the 6 km baseline of the Australia Telescope Compact Array (ATCA). Since no imaging was intended a single baseline was adequate for this type of observation and subsets of the five LBA telescopes, Tidbinbilla, Hartebeesthoek and Kokee Park were used. These observations were recorded in $2 mode and correlated at the LBA Correlator in Marsfield, NSW. Sources that were detected on trans-oceanic baselines are suitable candidates for the ICRF. Others that were detected only on intraAustralian baselines increase the number of phase reference sources with accurate positions in the Southern Hemisphere but can also be used to increase the sky density of Southern Hemisphere ICRF sources, although with reduced accuracy. This search for suitable candidates is ongoing.
5.2
Astrometry Observations
As candidate sources were identified, a series of dedicated astrometric Mark III VLBI observations, each of 24-hour duration, were commenced and are continuing. A duration of about 24-hours is needed to recover and separate parameters for nutation and polar motion. Individual experiments have used combinations of
Chapter 8 9
•
Improvement and Extension of the International Celestial Reference Frame in the Southern Hemisphere
the Parkes, Hartebeesthoek, Tidbinbilla and Hobart telescopes and all these data were correlated at the Washington Correlator. All observations were made in a bandwidth-synthesis mode at standard frequencies of 2.3 and 8.4 GHz using standard Mark III compatible VLBI systems that have been used for geodesy and astrometry since about mid-1979 [3]. For details of each observation and of the analysis of the astrometric data please see [8], [9], [10].
5.3
Astrometry Results
So far, milliarcsecond-accurate radio positions of 74 new southern hemisphere extragalactic radio sources have been determined in this program. All of these new sources are located south of declination - 3 0 ° , the part of the sky that is beyond the reach of northern hemisphere telescopes and precisely where new ICRF sources are most needed. Since the ICRF was originally defined, this is by far the largest group of new milliarcsecond accurate astrometric positions for sources in this declination range. The positions on the sky of these 74 new Southern Hemisphere sources with measured astrometric VLBI positions are shown in Figure 2. These sources are now available for inclusion in the next realization of the ICRF. The paucity of compact radio sources with well determined astrometric positions is also a constraint on the study of faint astrophysical objects that can only be observed using the technique of phase-referencing. This technique requires the use of a strong calibrator source that is no more than a few degrees away from the target object. This list of 74 new objects will significantly enhance the probability of finding a suitable nearby phase calibrator for observations of objects south of declination - 3 0 ° .
6 6.1
Imaging Imaging Observations
VLBI imaging observations of ICRF sources were made at 8.4 GHz in order to determine their morphology at milliarcsecond resolution and evaluate their continued suitability for reference frame use on the basis of their intrinsic structure. The effect of intrinsic structure on bandwidth synthesis VLBI observations was quantified by the calculation of a "structure index", defined below. These imaging observations were made using the five telescopes of the LBA in combination
with the Hartebeesthoek, Kashima and Kokee Park telescopes. As there is little mutual visibility between South Africa and Japan/Hawaii an observing strategy was developed wherein each observing epoch consisted of a 24 hour session with the LBA and Hartebeesthoek followed by another 24 hour session with the LBA, Kokee Park and Kashima. The data from the two 24 hour sessions is calibrated separately and then combined to yield the best possible uv-coverage (effectively the synthesized telescope aperture, it indicates where the array samples the Fourier transform of the source image). The typical angular resolution of our synthesized beam was 1.5 by 0.7 milliarcsecond in size, with the higher resolution in the east-west direction. Each target source was observed in multiple short scans (for better uv-coverage) with a total on-source time of 60 to 75 minutes. The data were recorded in $2 format and correlated at the LBA Correlator. For details of each observation, the calibration and the imaging proceduces see [16], [18]. High fidelity VLBI imaging requires information at as many different spatial frequencies as possible which in turn requires proper distribution of telescope locations and observations long enough for the rotation of the earth to allow "aperture synthesis". Further, the use of similar antennas and receiver/backends greatly ameliorates the challenges of calibrating VLBI data. Due to the ad-hoc nature of the array used in these observations, the individual telescopes are not optimally located for generation of uniform uv-coverage needed to make high fidelity VLBI images. In particular, the large gap in location between the Australian and overseas telescopes results in a "hole" in the uv-coverage (Figure 3) which is the most serious constraint for this imaging program. Also the heterogenous nature of antennas and antenna systems used in these observations makes calibration of the data a highly non-trivial exercise. However, comparision of the images of sources that are mutually visible to this array and the VLBA show them to be consistent. Further, all sources for which we have images at more than one epoch show consistent structure from epoch to epoch. Thus we are confident that the images obtained in this observing program are reliable, robust and repeatable.
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Imaging Results
Imaging observations of all existing Southern Hemisphere ICRF sources have been made at least once. This imaging program will continue to image both the new ICRF sources (see Section 5) and to obtain second epoch images of all sources concentrating on those that show extended structure and thus most need their suitability as ICRF sources to be monitored and evaluated. Final images of 110 of these sources (some with images at multiple epochs) are presented in [16] and [18]. Reduction and imaging of the remaining sources is in progress. A preliminary morphological classification of the imaged sources was made by inspection of the images and of Gaussian models fitted to the self-calibrated visibility data using the Caltech Difmap package. Barely 40% of these sources exhibit a compact structure with a single fitted component. A more robust way to quantify the effect of intrinsic source structure on calculations of bandwidth synthesis delay is to define a source "structure index", following [6], according to the median value of the structure delay corrections calculated for all projected VLBI baselines that could possibly be observed with Earth-based VLBI. About 35% of the sources in our imaged sample have a structure index of I or 2, indicative of compact or very compact structures as exemplified by Figure 4 and Figure 5. The remaining sources have a structure index of 3 or 4 indicating the presence of more extended emission structures (see Figure 6 and Figure 7). Pointing out that such structures are more likely to affect the observed VLBI synthesis delays, [6] recommended that they be avoided in astrometric and geodetic VLBI experiments requiring the highest accuracy. This stipulation applies particularly to sources with a structure index of 4 which form approximately 35% of this sample. This high percentage of sources with extended structure underscores the importance of this imaging program in maintaining and improving the accuracy of the ICRF. Apart from its stated goal of providing morphological information necessary to evaluate the continuing suitability of ICRF sources, this imaging program forms the most extensive VLBI imaging program of Southern Hemisphere extragalactic radio sources undertaken. As such, it provides an unique opportunity for two types
Chapter 89 • Improvement and Extension of the International Celestial Reference Frame in the Southern Hemisphere
of study. First, it furnishes a large (over 200 sources including the newly added ones) sample of sources for study of a wide range of astrophysical phenomena associated with extragalactic sources in a manner analogous to the PearsonReadhead [19] and follow up surveys in the northern hemisphere. The sample includes subgroups of great physical interest e.g. objects with energy spectra extending up to gamma-ray energies (the so called E G R E T sources) and sources with X-ray jets (radio jets which also emit in the X-ray) detected by the Chandra space observatory. It also contains morphologically defined sub-groups such as BLLacertae objects (blazars with weak emission lines) and quasars which are suitable for study of unification schemes that invoke geometric orientation effects and the presence of an obscuring torus to account for the variety of morphologies exhibited by extragalactic radio sources. Second, the survey includes a number of sources that are individually very interesting and have remained relatively unstudied only because they are too far south for northern hemisphere telescopes to observe. Just one example is the gigahertz peaked spectrum (GPS) galaxy PKS 1934-638 (Figure 7) where we may have detected proper motion over the longest time baseline for any such object [17].
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Map center: Map peak: 0 . 1 7 3 d y / b e a m Contours: 0 . 0 0 2 2 6 J y / b e a m x (-1 1 2 4 8 16 32 64 ) Beam FWHM: 0.881 x 0.727 (mas) at - 5 8 . 4 °
Figure 6: Example of an ICRF source with structure index of 3. A significant fraction of the total flux is located in a component some distance away from the core. Such structures are more likely to affect observed VLBI synthesis delays. Clean
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structure, are in progress. Images and "structure index" calculations for 110 of these sources are complete. About 35% of these sources have "structure index" indicative of compact or very compact structure. Thus ahnost twothirds of these sources may only be suitable for high-precision astrometry or geodesy after making correction for structure. This imaging program furnishes the largest set of VLBI images of extragalactic radio sources in the Southern Hemisphere and makes available unique data for the study of both individually interesting sources and large samples of interesting types of sources that have remained unstudied due to their southerly location. Fundamentally, this program of imaging and addition of new sources strengthens the ICRF, which is the fundamental celestial reference frame, by allowing the choice of better and additional sources as defining sources in the next realization of the ICRF.
References [1] Cannon, W. H., Baer, D., Feil, G., Feir, B., Newby, P., Novikov, A., Dewdney, P. E., Carlson, B. R., Petrachenko, W. T., Popelar, J., Mathieu, P., Wietfeldt, R. D. (1997). The $2 VLBI system. Vistas in Astronomy 41, pp. 297-302 [2] Charlot, P. (1990). Radio-source structure in astrometric and geodetic very long baseline interferometry. The Astronomical Journal 99, pp. 1309-1326
9 f o o
E
[3] Clark, T. A. et al. (1985). Precision Geodesy Using the Mark-III Very Long Baseline Interferometer System. IEEE Trans. Geosci. Remote Sens. 23, pp. 438-449
g o c
o
cl
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[4] Dehant, V., Feissel-Vernier, M., de Viron, O., Ma, C., Yseboodt, M., Bizouard, C. (2003). Journal of Geophysical Research (Solid Earth) 108, pp. ETG 13-1
a o
210
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Map center: Map peak: 0 . 3 7 8 d y / b e a m Contours: 0 . 0 1 3 5 J y / b e a m x (-1 1 2 4 8 16 ) Beam FWHM: 3.78 x 2.9 (mas) at 70.5 °
Figure 7: Example of an ICRF source with structure index of 4. Such double structures are best avoided in those astrometric and geodetic experiments requiring high accuracy.
[5] Feissel-Vernier, M., Ma, C., Gontier, A.-M., Barache, C. (2005). Sidereal orientation of the Earth and stability of the VLBI celestial reference frame. Astronomy and Astrophysics 438, pp.1141-1148 [6] Fey, A. L. & Charlot, P. (1997). VLBA Observations of Radio Reference Frame Sources. II. Astrometric Suitability Based on Observed
Chapter 89 • Improvement and Extension of the International Celestial Reference Frame in the Southern Hemisphere
Structure. The Astrophysical Journal Supplement Series, 111, pp. 95-142
ometry. The Astronomical Journal, 116, pp. 516-546
[7] Fey, A. L., & Charlot, P. (2000). VLBA Observations of Radio Reference Frame Sources. III. Astrometric Suitability of an Additional 225 Sources. The Astrophysical Journal Supplement Series 128, pp. 17-83
[15] MacMillan, D. S. & Ma, C. (1997). Atmospheric gradients and the VLBI terrestrial and celestial reference frames. Geophysical Research Letters 24, pp. 453-456
[8] Fey, A. L., Ojha, R., Jauncey, D. L., Johnston, K. J., Reynolds, J. E., Lovell, J. E. J., Tzioumis, A. K., Quick, J. F. H, Nicolson, G. D., Ellingsen, S. P., McCulloch, P. M., & Koyama, Y. (2004a). Accurate Astrometry of 22 Southern Hemisphere Radio Sources. The Astronomical Journal 127, pp. 1791-1795 [9] Fey, A. L., Ojha, R., Reynolds, J. E., Ellingsen, S. P., McCulloch, P. M., Jauncey, D. L., & Johnston, K. J. (2004b). Astrotnetry of 25 Southern Hemisphere Radio Sources from a VLBI Short-Baseline Survey. The Astronomical Journal 128, pp. 2593-2598 [10] Fey et al. (2006). In Preparation [11] Johnston, K. J. et al. (1995). A Radio Reference Frame. The Astronomical Journal 110, pp. 880-915 [12] Johnston, K. J. & de Vegt, C. (1999). Reference Frames in Astronomy. Annual Reviews of Astronomy and Astrophysics 37, pp. 97-125 [13] Kingham, K. A. (2003). Washington Correlator. In: International VLBI Service for Geodesy and Astrometry 2002 Annual Report N. R. Vandenberg & K. D. Bayer (eds) (NASA Tech. Pap. 211619)(Greenbelt, MD:GSFC), pp. 201-202 [14] Ma, C., Arias, E. F., Eubanks, T. M., Fey, A. L., Gontier, A . - M . , Jacobs, C. S., Sovers, O. J., Archinal, B. A., & Charlot, P. (1998). The International Celestial Reference Frame as Realized by Very Long Baseline Interfer-
[16] Ojha et al. (2004a). VLBI Observations of Southern Hemisphere ICRF Sources. I. The Astronomical Journal 127, pp. 3609-3621 [17] Ojha, R., Fey, A. L., Johnston, K. J., Jauncey, D. L., Tzioumis, A. K., Reynolds, J. E. (2004b). VLBI Observations of the Gigahertz-Peaked Spectrum Galaxy PKS 1934-638. The Astronomical Journal 127, pp. 1977-1981 [18] Ojha et al. (2004). VLBI Observations of Southern Hemisphere ICRF Sources. II. Astrometric Suitability Based on Intrinsic Structure. The Astronomical Journal 130, pp. 25292540 [19] Pearson, T. J., & Readhead, A. C. S. (1988). The milliarcsecond structure of a complete sample of radio sources. II - First-epoch maps at 5 GHz. The Astrophysical Journal 328, pp. 114-142 [20] Reynolds, J. E., Jauncey, D. L., Russell, J. L., King, E. A., McCulloch, P. M., Fey, A. L., Johnston, K. J. (1994). A radio optical reference frame. 7: Additional source positions from a Southern hemisphere short baseline survey. The Astronomical Journal 108, pp. 725-730 [21] Russell, J. L. et al. (1994). A radio/optical reference frame. 5: Additional source positions in the inid-latitude southern hemisphere. The Astronomical Journal 107, pp. 379-384 [22] Wilson, W. E., Roberts, P. P., Davis, E. R. (1995). The ATNF Long Baseline Array Correlator. In: Proceedings of ~th A P T Workshop Sydney, King, E. A. (ed) pp. 16-20
623
Chapter 90
Limitations in the NZGD2000 Deformation Model J. Beavan GNS Science, PO Box 30368, Lower Hutt, New Zealand G. Blick Land Information New Zealand, Private Box 5501, Wellington, New Zealand
Abstract. The New Zealand Geodetic Datum 2000 (NZGD2000) is a semi-dynamic datum, in that coordinates are fixed to their values at 1 January 2000 and velocities from a horizontal deformation model are used to transform the coordinates of data collected before or after that date. The deformation model was calculated from GPS campaign data collected between 1991 and 1998, and was aligned with ITRF96. We examine the performance of this model in 2005 from two points of view: (1) how different are the ITRF2000 velocities from ITRF96, and (2) for new stations, and older stations where additional data have been collected, how well do the newly estimated velocities match those in the deformation model (after the ITRF96-ITRF2000 transformation)? We have calculated ITRF2000 velocities at points throughout New Zealand. The ITRF96 and ITRF2000 velocities differ by 4.8 mm/yr at azimuth -101 ° in the southwest of the country, and by 5.4 mm/yr at azimuth -111 ° in the northeast. The differences between newly-calculated ITRF2000 site velocities and velocities from the deformation model transformed to ITRF2000 range between zero and about 4 mm/yr. Velocities of some continuous GPS stations installed in the past few years differ by more than this (in two cases by >7 mm/yr), in part because the new velocities cannot be estimated reliably from relatively short spans of data, and in part because the velocities at some sites are not linear. Significant vertical velocities up to a few mm/yr are estimated for continuous GPS stations that have been established for at least four years. These comparisons suggest that an upgrade to the deformation model should be considered. Alignment of the deformation model with ITRF2000 (or its successors) will have benefits in combining newly collected data with existing data, as the ITRF96 to ITRF2000 transformation step will no longer be needed.
Keywords. Geodetic datum, deformation model, crustal deformation, New Zealand
1 Introduction In 1998 Land Information New Zealand (LINZ) implemented a new geocentric datum for New Zealand, New Zealand Geodetic Datum 2000 (NZGD2000) with a reference epoch of 1 January 2000 (2000.0). NZGD2000 is realised in terms of ITRF96 and uses the GRS80 ellipsoid; (see Grant et al., 1999; Blick et al., 2003; Office of the SurveyorGeneral, 2003a). A major conceptual departure from the definition of the previous national datum (New Zealand Geodetic Datum 1949) and other international datums is that NZGD2000 accommodates the effects of crustal deformation. This is achieved by applying a deformation model when generating new coordinates, enabling them to be transformed from one epoch to another following a method similar to that described by Snay (1999). For most users, it has the appearance of a static datum. The accuracy criteria aimed at for NZGD2000 are that a mark's coordinate accuracy relative to adjacent marks of the next highest order shall not exceed 0.05 m horizontally and 0.15 m vertically (Office of the Surveyor-General, 2003b). The deformation model must be of sufficient accuracy to enable these accuracy requirements to continue to be achieved over time. Where the computed positions of marks using the deformation model differ from the surveyed positions by greater than these limits, consideration will need to be given to refining the deformation model. The deformation model must be able to reflect the true deformation field with adequate accuracy and resolution. A model should include both the long term deformation trends and, potentially, discrete events such as earthquakes, where the model definition could include surface fault ruptures. This would depend on the extent to which fault movement should be reflected by the deformation field, and the extent to which it should be represented by changing the coordinates of survey marks. The deformation model used in NZGD2000 (Figure 1) is now seven years old and it is time to
Chapter90 • Limitationsin the NZGD2000DeformationModel
consider if the accuracy requirements of the datum are still being met. This paper examines the performance of the current model in 2005 from two points of view: (1) how different are the ITRF2000 velocities from ITRF96, and (2) for new stations, and older stations where additional data have been collected, how well do the newly estimated velocities match those in the deformation model (after the ITRF96-ITRF2000 transformation)?
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coordinates and velocities for all other stations in the solution. A 3-parameter (3 orthogonal rotations) transformation was derived by comparing the horizontal velocities of these 29 points in the deformation model with their estimated ITRF96 values. This transformation was used to convert the Australia-fixed deformation model into ITRF96, and this is the model used by LINZ for the NZGD2000 datum. The surveys used to determine the deformation model are now on average nearly 10 years old. As time passes, errors in the determination of the velocities used in the deformation model lead to increasing errors in the calculated position of marks in terms of the reference epoch of 2000.0. In effect, the spatial accuracy of the datum is steadily degrading. Also, the datum and the effectiveness of the deformation model may be degraded by localised and temporally non-linear deformation, for example earthquakes and recently observed "slow earthquakes" (e.g., Douglas et al., 2005; Beavan et al., 2006). In addition, the current deformation model is aligned with ITRF96, and NZGD2000 will therefore drift from future and more accurate realisations of the ITRF. These limitations are shown schematically in Figure 2.
400 Surveys from this period used
Fig 1. NZGD2000 deformation model, with horizontal velocities relative to the Australian Plate plotted at the nodes of the model. Velocities of points away from the nodes are provided by bicubic spline interpolation. 2 Limitations Deformation
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Coordinates and velocities for 29 primary l storder network stations were generated from 4 to 6 repeat GPS surveys made between 1993 and 1998 (Office of the Surveyor-General, 2000). Data from these and other repeat surveys observed between 1991 and 1998 were used to generate a horizontal deformation model relative to the Australian tectonic plate (Figure 1), assuming constant site velocities (Beavan and Haines, 2001). The velocities at the 29 primary points were also calculated relative to ITRF96 by including their data in global (Morgan and Pearse, 1999) and regional (Beavan, 1998) GPS analyses. In these analyses, a number of global or regional GPS stations are constrained to their ITRF96 positions and velocities, allowing the estimation of ITRF96
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We can easily convert the deformation model used in NZGD2000 from ITRF96 to ITRF2000 (or another realisation of the ITRF). We simply need to repeat the global or regional GPS analysis of the 29 1st order stations using exactly the same data that went into the original analysis, but with the positions and velocities of the global or regional
625
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J.Beavan.G.Blick reference stations constrained to ITRF2000 rather than ITRF96. Figure 3 shows the differences between the ITRF96 and ITRF2000 positions of the 29 1st order stations between epochs 2000.0 and 2010.0. The differences have grown by -50 mm over this 10 year period, because of the -5 mm/yr velocity difference between ITRF96 and ITRF2000 in the New Zealand region. Our estimate of this velocity difference is 4.8 mm/yr at azimuth -101 ° in the southwest of the country, rising to 5.4 mm/yr at azimuth -111 ° in the northeast (westward velocities are faster in ITRF2000 than in ITRF96). Because the differences are quite uniform across the country, the absolute position accuracy of the datum is compromised; however, the relative accuracy between marks is maintained.
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While the change from ITRF96 to ITRF2000 velocities can be managed by a standard 7parameter transformation, the difference o f - 5 mm/yr is quite significant; after 10 years it exceeds the NZGD2000 horizontal accuracy criterion cited earlier. We expect that the changes from ITRF2000 to any future realisation of ITRF will be much smaller than this, so that it will be worthwhile aligning any update of NZGD2000 to ITRF2000 (or whatever ITRF realisation is current at the time of the update), rather than retaining ITRF96.
4 Deformation Model Distortions There are two types of potential distortion in the velocity model itself, even assuming that all points move linearly in time (i.e., with constant velocity). The first is that the velocity estimated at an individual site from 1992-1998 data is likely to be improved by the inclusion of additional (later) data collected from the same site (i.e., more time). The second is that the interpolation of the velocity field between those points where the velocity was actually measured is likely to be improved by the inclusion of velocities from additional spatially distributed points (i.e., more sites). At the time of computation of the NZGD2000 deformation model 371 points were used. There are significant regions where the data were geographically relatively sparse, and the interpolation relied heavily on the minimum strain-rate constraint and the plateboundary velocity conditions applied at the margins of the model. More than 800 points are now available where velocity estimates can be made and which could be included in a recalculation of the velocity model. These new data have largely been collected in GPS surveys carried out for scientific purposes associated with plate tectonic and earthquake hazard research. We can test the two types of distortion by comparing the NZGD2000 velocity model with (1) observed velocities at points used in the velocity model where additional data have been collected since 1998, and (2) observed velocities at points not used in the velocity model where at least two wellseparated epochs of data have been collected since 1998. In these comparisons, we use a version of the NZGD2000 velocity model that has been transformed from ITRF96 to ITRF2000 as explained in Section 3. We compare this model with observed velocities that we estimate in ITRF2000 by including a set of regional IGS stations in the GPS data analysis and aligning the daily solutions with the ITRF2000 coordinates of these stations. This means we are testing the velocity model itself, with minimal effect from differences in the datum on which it is based. There are a large number of points that fulfill one or other of the above criteria, but for this study we consider a limited number of points with rather stronger criteria. In the first category (points updated from the NZGD2000 deformation model), we use the 20 1st order points where additional data have been collected since 1998 (Table 1). These were the points that had the best history of occupation in NZGD2000, with at least four epochs
Chapter 90 • Limitations in the NZGD2000 Deformation Model
o f observation between 1993 and 1998. In general either one or two additional epochs o f data have been obtained since 1998. In the second category (new points since 1998) we take velocities from 24 continuous GPS stations with at least 2 years o f available data. W e use these, rather than campaign stations, because the extra data from continuous stations are likely to provide more accurate velocity estimates than we could obtain from the two (or at most three) available campaign occupations.
5 Comparison
at u p d a t e d p o i n t s
Figure 4 and Table 1 show the velocity differences and estimated uncertainties for the first category: points w h o s e velocities have been updated since 1998 by the collection o f additional data. Since the new velocities use the 1992-1998 data as well as the new data, the uncertainties between the two estimates are not independent. The uncertainties we plot are from the deformation model (which are larger than those from individual site velocity estimates). Only a few o f the differences (3 out o f 40; shown in bold type in Table 1) fall significantly outside 3 standard deviations. This suggests that the individual site velocites used in the construction o f N Z G D 2 0 0 0 were quite reliable, at least for the frequentlymeasured 1 st order sites.
Table 1. Tabulated velocity differences and 1a uncertainties shown in Figure 4. Site
(~(Ve)
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1004 1103 1153 1181 1231 1259 1273 1305 1314 1367 1501 5508 5509 6731 A31C A33D AUCK B03W OUSD WGTN
-0.50 -0.08 0.18 0.41 3.59 -0.07 -0.24 -1.94 0.64 1.53 1.14 -0.67 -0.26 0.82 -0.64 0.34 2.27 0.09 1.40 2.30
-1.16 -1.41 -0.87 0.46 1.74 -0.66 0.98 2.54 2.06 0.48 0.88 -2.69 0.41 0.38 -2.47 -0.17 -0.02 -1.12 0.72 1.56
1.03 0.81 0.91 0.90 0.91 0.83 0.95 0.97 0.87 0.86 0.92 0.60 1.12 0.90 1.12 0.76 0.74 1.17 0.60 0.79
0.88 0.71 0.72 0.74 0.75 0.65 0.76 0.78 0.69 0.69 0.71 0.52 0.93 0.75 0.94 0.61 0.54 0.99 0.52 0.66
7.9 11.0 10.9 10.0 11.0 10.0 11.0 11.0 11.0 11.0 11.0 10.9 11.0 7.9 5.9 10.0 12.3 7.9 10.3 8.3
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l~J uncertainties are from the deformation model. East-north correlations are -0. Velocities and uncertainties are in mm/yr. Differences greater than 3c~ are shown in bold type. aTotal duration of data series in years. bNumber of additional epochs of data at campaign sites.
6 Comparison
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Figure 5 and Table 2 show the velocity differences and estimated uncertainties for the second category o f points, those newly observed since the NZGD2000 calculations. In this case the uncertainties in the model and those in the site velocity estimates are independent, so we combine them by summing variances. A substantially larger fraction o f the differences (7 out o f 48; shown in bold type in Table 2) fall significantly outside 3 standard deviations in this case. Also, the absolute values o f the differences are considerably larger, e x c e e d i n g - 5 m m / y r at five stations and 7 m m / y r at two o f these. This indicates either that the velocity model is not performing particularly well, or that the new velocity estimates are inaccurate in some way. Differences o f this size indicate that distortions in the current velocity model may, in some regions, exceed the N Z G D 2 0 0 0 50 m m horizontal accuracy criterion in less than 10 years.
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Vertical velocities were also computed for continuous sites with a >4 year data span. The computed velocities all fall b e t w e e n - 1 . 6 m m / y r and +4.3 mm/yr. The largest rates are within the Southern Alps where a more rigorous analysis by Beavan et al. (2004) has shown sites to be rising at 3-5 mm/yr relative to sites on the east coast of the South Island. The definition of vertical rates is subject to some fundamental questions (e.g., Blewitt, 2003), and there are probably regional slopes within the ITRF vertical motion field which affect our estimated velocities. The vertical rates are everywhere small enough that they do not need to be considered in N Z G D 2 0 0 0 , given the vertical accuracy criterion detailed above.
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Table 2. Tabulated velocity differences shown in Figure 5. Site 6(vo) 6(Vn) o(v°) O(Vn) Years a CNCL -0.15 2.14 0.99 0.86 5.1 CORM -0.38 0.20 1.49 1.81 2.2 DNVK 2.12 3.21 1.93 1.85 2.7 DUNT 1.35 0.72 0.54 0.51 5.6 GISB -1.68 0.45 1.14 1.01 3.0 GRAC 2.54 0.31 0.82 0.72 6.5 HAMT -0.73 -0.78 1.20 1.11 2.2 HAST 4.70 0.80 1.40 1.50 2.8 HIKB -7.24 2.63 2.12 2.44 2.1 HOKI 0.95 1.27 1.05 0.92 6.7 KARA -0.96 1.90 0.95 0.81 5.1 MAST 3.77 -0.17 1.23 1.21 2.5 MQZG -0.32 -0.26 0.85 0.76 5.2 MTJO -0.17 -1.08 0.75 0.74 4.6 NETT 4.33 -0.21 1.05 0.95 5.1 NPLY -0.14 -2.06 0.98 0.93 2.3 PAEK 5.00 -0.39 0.91 0.79 5.2 QUAIl 0.80 1.54 0.96 0.84 5.1 TAKL 2.16 0.08 0.85 0.78 3.7 TAUP 5.27 2.29 1.08 1.01 3.3 TRNG 2.78 1.59 1.36 1.15 2.4 WANG 7.09 -1.05 1.81 1.92 2.2 WGTT 2.63 1.50 0.79 0.70 5.1 WHNG 1.66 -0.87 1.02 0.83 2.2 Uncertainties are combined from deformation model and site velocity estimates. Other details as in Table 1.
8 Discussion It is interesting to explore the reasons for the large horizontal velocity differences between the deformation model and several of the new sites (Table 2). A majority of the large differences are at sites where we have observed non-linear site velocities since the establishment of continuous GPS stations (e.g., HAST, W A N G , D N V K and PAEK). We believe this non-linear deformation is caused by slip episodes lasting from days to months on the deeper part of the subduction interface where the Pacific Plate descends beneath the North Island (Douglas et al., 2005; Beavan et al., 2006). These events are often known as slow slip events or slow earthquakes; the adjective "slow" refers to their rate of slip compared to the several km per second slip rate in normal earthquakes. Another site with a large velocity discrepancy is HIKB. Though this has shown a fairly steady velocity since the continuous station was installed, it is in a region where slow earthquakes appear to be common. It is possible that the data from this region used in the construction of the N Z G D 2 0 0 0 deformation model had been affected by slow earthquakes that were unrecognised in the campaign GPS data available at the time. The widespread occurrence of slow slip events, at least in the North Island, has implications for the N Z G D 2 0 0 0 deformation model. So far, the largest event we have observed (in 3 years) has had a magnitude of - 3 0 m m at the Earth's surface. This is within the N Z G D 2 0 0 0 horizontal accuracy specification, so it is possible that such events can be ignored at the N Z G D 2 0 0 0 accuracy level. However, to achieve
Chapter 90 • Limitations in the NZGD2000 Deformation Model
this it is important that the velocities in the N Z G D 2 0 0 0 deformation model are estimated using long enough spans of data that an average velocity is obtained. In the Gisborne region we have evidence that such events recur as often as twoyearly, implying that it should be easy to obtain an average velocity here; the small residual at GISB in Figure 5 indicates we may have achieved this. Though we have noted that steady vertical velocities may be neglected in respect of the N Z G D 2 0 0 0 deformation model, it is still important to estimate these velocities accurately when constructing the datum. This is particularly the case when positions need to be extrapolated to the reference time of the datum. An example is the 2000.0 reference epoch of N Z G D 2 0 0 0 , for which the data defining the datum were collected between 1992 and 1998. We know of at least one case where the vertical velocity estimated from 1992-98 data was significantly in error. This meant that the height coordinate for that point extrapolated to 2000.0 was in error (though not by more than the N Z G D 2 0 0 0 vertical criterion). To mitigate this problem it is desirable to estimate velocities using long data spans, and to set the reference epoch within the available data span.
9 Conclusions Our tests indicate that the ITRF96 datum drifts at about 5 mm/yr relative to ITRF2000 in the N e w Zealand region and the N Z G D 2 0 0 0 deformation model probably has errors >5 mm/yr in some regions, implying that the N Z G D 2 0 0 0 horizontal accuracy criterion will be exceeded within the next few years. Some N Z G D 2 0 0 0 stations have larger than desirable vertical position errors because of poor vertical velocity estimates and extrapolation to the 2000.0 reference epoch. There are now >800 sites in N e w Zealand with GPS velocity estimates, whereas there were <400 when the N Z G D 2 0 0 0 deformation model was constructed in 1998. For these reasons it is desirable to construct a new model based on ITRF2000 (or the most current ITRF realisation) within the next few years. Given the NZGD2000 vertical accuracy requirements, it is not necessary to include a vertical deformation model in the N Z G D 2 0 0 0 or successor datums (though it is necessary to use vertical velocities w h e n constructing the datum). Non-linear site velocities, particularly those related to slow slip events on the North Island subduction zone, have been observed at continuous
GPS stations. These events may not be a problem at the specified 50 m m horizontal accuracy level of N Z G D 2 0 0 0 , provided their future amplitudes do not substantially exceed the 30 m m observed to date (from 3 years of data).
Acknowledgements Thank you to Laura Wallace, Matt Amos, Hilary Fletcher and Mike Craymer for suggestions that improved this paper. JB was supported by the N e w Zealand Foundation for Research, Science and Technology. GNS Contribution 3410.
References Beavan, R. J. (1998). Revised horizontal velocity model for the New Zealand geodetic datum, IGNS Client Report 43865B for Land Information NZ, 46 pp., November 1998. Beavan, J., and J. Haines (2001). Contemporary horizontal velocity and strain-rate fields of the Pacific-Australian plate boundary zone through New Zealand, J. Geophys. Res., 106:B1, 741-770. Beavan, J., D. Matheson, P. Denys, M. Denham, T. Herring, B. Hager, and P. Molnar (2004). A vertical deformation profile across the Southern Alps, New Zealand, from 3.5 years of continuous GPS data, Cahiers du Centre Europden de Gdodynamique et de Sdismologie, Vol. 23, 111-123, Luxembourg. Beavan, J., L. Wallace, and H. Fletcher (2007). Aseismic slip events on the Hikurangi subduction interface, New Zealand, this volume, Chapter 64, pp 438-444 Blewitt, G. (2003). Self-consistency in reference frames, geocenter definition, and surface loading of the solid earth, J. Geophys. Res., 108:B2, doi:10.1029/2002JB002082. Blick, G. H., C. Crook, D. Grant, and J. Beavan (2003). Implementation of a Semi-Dynamic Datum for New Zealand. Proceedings of the International Association of Geodesy General Assembly, Sapporo, Japan, A Window on the Future of Geodesy, pp. 38-43. Douglas, A., J. Beavan, L. Wallace, and J. Townend (2005). Slow slip on the northern Hikurangi subduction interface, New Zealand, Geophys. Res. Lett., 32, L16305, doi: 10.1029/2005 GL022395. Grant, D. B., G. H. Blick, M. B. Pearse, R. J. Beavan, and P. J. Morgan (1999). The development and implementation of New Zealand Geodetic Datum 2000, Presented at IUGG 99 General Assembly, Birmingham, UK, July 1999. Morgan, P., and M. B. Pearse (1999). A first-order network for New Zealand, Report UNISURV S-56, School of Geomatic Engineering, Univ. NSW, Sydney, NSW 2052, Australia, 134 pp., December 1999. Office of the Surveyor-General (2000). Realisation of the New Zealand Geodetic Datum 2000, OSG Tech. Report 5, Land Information New Zealand. (Can be found on-line at www.linz.govt.nz)
629
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J. Beavan. G. Blick
Office of the Surveyor-General (2003a). Implementation of the NZGD2000 Velocity Model, OSG Tech. Report 20, Land Information New Zealand. (Can be found on-line at www.linz.govt.nz) Office of the Surveyor-General (2003b). Accuracy Standards for Geodetic Surveys, OSG Standard 1, vl.1,
Land Information New Zealand. (Can be found on-line at www.linz.govt.nz) Shay, R. A. (1999). Using the HTDP software to transform spatial coordinates across time and between reference frames, Surv. and Land lnfo. Systems, Vol. 59(1), No. 1, pp. 15-25.
Chapter 91
Implementing Localised Deformation Models into a Semi-Dynamic Datum A. Jordan ~, P. Denys 2, G. Blick 1 1. Land Information New Zealand, Private Box 5501, Wellington, New Zealand 2. School of Surveying, University of Otago, Box 56, Dunedin, New Zealand
Abstract. The New Zealand Geodetic Datum 2000 (NZGD2000) has an associated deformation model that allows horizontal coordinates to be reduced to a reference epoch of 2000.0 and is therefore a semidynamic datum. Currently, the deformation model does not account for either discrete or continuous horizontal deformation that can be associated with localised events such as earthquakes and landslides. Such events will distort the geodetic network by as little as a few millimetres a year, or as much as several metres in a matter of seconds. Using the August 2003 M 7.2 Fiordland earthquake as a case study, we demonstrate the implementation of a Localised Deformation Model (LDM) into the NZ National deformation model. LDMs employ triangulated interpolation techniques that predict surface displacement. In the Fiordland earthquake, eleven existing monitoring sites were resurveyed immediately following the earthquake and movements of up to 17 cm were observed. To model the associated coseismic deformation, a LDM was generated from surface displacements predicted by a dislocation model of the earthquake. The LDM covered an area of approximately 56,000 km 2. The implementation of a LDM into the NZGD2000 deformation model was successfully demonstrated.
Keywords. Geodetic Datum, earthquakes, deformation model, deformation, New Zealand
1. Introduction New Zealand is prone to land deformation due to the steady motion between the Australian and Pacific tectonic plates as well as the episodic release of accumulated energy on the plate interface, known as earthquakes. The current geodetic datum for New Zealand is the New Zealand Geodetic Datum 2000 (NZGD2000). NZGD2000 was released on 25 August 1999 (Pearse 2000) replacing New Zealand Geodetic Datum 1949 (NZGD1949) as New
Zealand's primary reference datum. NZGD1949 was a static datum and as a result, distortions in the geodetic network of up to 5 m had been measured due to tectonic motion, earthquakes and survey errors (Bevin and Hall 1995). NZGD2000 was realised as a semi-dynamic datum with a geocentric origin that is aligned with the International Terrestrial Reference Frame 1996 (ITRF96) at a reference epoch of 2000.0 and adopts the Geodetic Reference System 1980 (GRS80) ellipsoid. NZGD2000 is considered a semi-dynamic datum because it has an associated national horizontal deformation model that is used to account for tectonic motion. The National Deformation Model (NDM) associated with NZGD2000 accounts for the horizontal deformation resulting from the steady motion between the Australian and Pacific tectonic plates. Ideally, the datum and NDM would meet two requirements; (1) model national and localised deformation to a sufficient degree to allow old and new observations to be used together; and (2) supply coordinates that are "in terms" with the current positions of marks in the ground. However, the current NDM does not account for the horizontal deformation associated with localised events such as earthquakes and landslides. A significant event will create distortions in the geodetic network that will downgrade its integrity and accuracy in the affected area. Currently, there are no strategies or procedures in place to update the geodetic network after such events. This research addresses this problem through an approach that involves the use of Localised Deformation Models (LDM) that can be integrated into the NDM to account for horizontal movement associated with localised deformation events. Through modelling areas of significant land deformation, NZGD2000 can be updated to maintain and realign the geodetic network in order to retain/regain the integrity of the network by reflecting the true positions of geodetic stations on the Earth's surface.
632
A. J o r d a n . P. Denys • G. Blick
2. The National Velocity Model and the NDM The terms "National Deformation Model" (NDM) and "National Velocity Model" (NVM) in existing documentation have been readily interchangeable. However, when giving reference to the NVM model in association with NZGD2000, LINZ (2003a) clarified that the term NVM is not strictly correct. LINZ (2003a) states that the principal purpose of the model is to predict position, not to predict velocity, and that it is better termed a deformation model. For the purposes of this paper, however, both terms will be used, with each representing separate models. NVM will be the term used to represent the velocity model developed by the Institute of Geological and Nuclear Sciences (GNS) for Land Information New Zealand (LINZ), GNS NVM version 2.1 (see Beavan and Haines 2001). The term NDM will represent a model consisting of a number of deformation models, including the NVM. This distinction is covered further in Section 4. The NVM is currently the only deformation model included in the NDM. The NVM is represented as a complex curvilinear grid (Figure 1). To reduce the complexity and increase the efficiency of its evaluation at any given point, the NVM is implemented as a regular grid. The NVM is calculated on a 0.2 x 0.2 degree grid between longitude E 165 ° to E 180 ° and latitude S 48 ° to S 32 ° (Crook 2003). The velocity of a given point is evaluated by interpolation of the velocities at the four points that form the cell in which the point lies. From herein, the term NVM will represent this grid.
3. Dealing with Localised Deformation in the NDM It is possible that during an earthquake marks could move metres relative to one another. Due to such deformation the 'authoritative' coordinates of the affected marks will not reflect the true location in the ground. The deformation model that is currently associated with NZGD2000 cannot replicate or account for earthquakes, major landslides, or other localised deformation events. This is because the NVM predicts a constant velocity for any given point and is spatially coarse (LINZ 2003a). Therefore, after deformation events, the two requirements of the datum and NDM, stated early, are no longer met. LINZ (2003a) discusses and recommends the options for updating the NDM to account for
localised deformation. It was recommended to define a separate grid (regular or irregular) to represent a LDM that is applied as a 'patch' on the NVM. Therefore, the NDM would include the NVM and potentially one (or more) LDMs. A LDM is defined over both spatial and temporal extents for which the localised deformation event exists.
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At a minimum, a LDM needs to be of sufficient accuracy to maintain relative cadastral accuracy standards in New Zealand. The relative horizontal accuracy standard for cadastral traverse marks (Sixth Order) is 0.02 m + 100 ppm (LINZ 2003b). If these standards can be met, then cadastral surveys can still be carried out.
4. National Concept
Deformation
Model-
The
A NDM is made up of one or more layers of deformation models (Figure 2) and it makes no distinction between a NVM and a LDM; these terms describe what is being modelled and how it is represented. Each layer (deformation model) is represented by a 'sequence' of one or more deformation 'components'. Figure 3 illustrates the structure of the NDM and the relationship between deformation-sequences and deformationcomponents. Essentially, by allowing a deformation event to be represented as a sequence of deformation-components, variations in the deformation through time can be easily managed.
Chapter91 For example, the earthquake cycle predicts further deformation after the main earthquake, i.e. postseismic deformation. Therefore, a LDM for an earthquake might include a model of an initial displacement (co-seismic deformation), with models of the post-seismic deformation added periodically after the earthquake. In this case, the deformation-sequence would consist of a deformation-component representing the co-seismic deformation, plus one or more components representing post-seismic deformation.
• Implementing
LocalisedDeformationModelsintoa Semi-Dynamic
represented by a triangulated grid (LINZ 2003a) (Figure 4 (a)).
(a)
with triangles and nodes explicitly defined. (b) An illustration of a rectangular grid (regular), with grid and nodes explicitly defined.
North
5. Case S t u d y Earthquake
LDM
East Figure 2. An illustration of how the NDM is made of layers with the NVM being the base layer, supplemented by a number of LDM layers.
M) (NVM)/
A u g u s t 2003 Fiordland
Fiordland is situated in the south-west corner of the South Island of New Zealand. On the night of 21/22 August 2003 a magnitude 7.2 (M 7.2) earthquake hit the area. The earthquake was located near Secretary Island. The fault surface is an area about 35 km long by 20 km wide, and is located on the dipping interface between the subducting Australian plate and the over-riding Pacific plate at a depth of about 13 km (Beavan and Wallace 2004; Reyners et al. 2003).
~GNS NVM v2. 166"
(LDM)/
(b)
Figure 4. (a) An illustration of a triangulated grid (irregular),
NVM
NI]
Datum
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168"
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¢
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"" 2/'-0"~, ,,oo, (x~Post-seismic de~/J Figure 3. An illustration of how the NDM consists of one or more deformation-sequences, which in turn consist of one or more deformation-components.
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o 166"
The N V M is represented on a rectangular (regular) grid (Figure 4 (b)). However, due to the complexity of the deformation associated with local deformation events an LDM would need to be
z~
4o-f;
r 167"
!) 1GS"
Figure 5. Plot of the 11 resurveyed campaign stations in the
Fiordland area. The star indicates the location of the earthquake below the surface. The solid black line indicates the location of the ruptured fault projected to the surface.
633
634
A. Jordan. P. Denys • G. Blick
Although there was little LINZ geodetic control in the area, a deformation network had been established in 2001 by Otago University (OU) and GNS for scientific purposes (Beavan and Wallace 2004). In the week following the earthquake, GNS re-observed parts of this network and calculated coseismic displacements. This included 11 stations within the earthquake's deformation zone (Figure
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5). 5.1. Methods There are three steps in the process used to calculate station displacements in this case study: Step 1: Project February 2001 coordinates to immediately before the earthquake at epoch 2003.638 (2003') (see Figure 6).
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I Jan 2000 Feb 2001
Step 2: Calculate the difference between the projected August 2003 coordinates and the observed August 2003 coordinates. AE
x
x
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x-(
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2003
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[]2003 [N]2o03N]200,+[]Nvt'-'°l AE OUSD
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X
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The variance-covariance values associated with the displacement calculated at each station were calculated using Equation 5. Note that the error associated with the NVM is assumed zero for this study. This is the current assumption made when LINZ generates NZGD2000 coordinates. However, Beavan and Blick (2005) demonstrate that this is not a correct assumption. Future studies should apply an appropriate error weighting to the NVM values.
I Aug 2003
Figure 6. An illustration of projecting the February 2001 coordinates to immediately before the August 2003 earthquake. The solid line represents the motion described by the NVM. The dashed line represents the true motion. The projected ordinate is represented by the light cross.
5.2. Displacements The data processing results are tabulated in Table 1 and shown in Figure 7. The largest movement was calculated at DF4Q as -0.162 m and 0.076 m east and north respectively. The smallest movement was calculated at B047 as -0.015 m and 0.009 m east and north respectively. The results show that stations DF4Q, DF4R and DF4U have the greatest movement in the WNW direction. They also show that while most stations move in a WNW direction, the three most south western stations (AODW, B047, and DF4L) move in a SW direction. Significant vertical movement was also observed, with a maximum of 13 cm of subsidence at DF4Q. However, only the horizontal movement is of interest in this study. The displacements calculated here agreed well with those calculated by Beavan and Wallace (2004). Using their displacements, they generated a dislocation model for the earthquake (GNS Model 205). From the dislocation model, a grid of 14,000 surface displacement nodes at 2km spacing (280 km (east-west) and 200 km (north-south)) was generated.
Chapter 91 • Implementing Localised Deformation Models into a Semi-Dynamic Datum
Table 1. Tabulated differences in meters shown in Figure 7.
Site
dE
dN
l~dE
(~dN
1002
-0.084
0.025
0.004
0.003
1004
-0.048
0.027
0.001
0.002
AODW
-0.008
-0.063
0.003
0.003
A1TH
-0.058
0.005
0.003
0.003
B047
DF4K
-0.015 -0.043
-0.009 0.048
0.003 0.003
0.003 0.003
DF4L
-0.027
-0.042
0.003
0.003
DF4M
-0.083
-0.006
0.004
0.003
DF4Q
-0.162
0.076
0.004
0.003
DF4R
-0.148
0.034
0.004
0.003
DF4U
-0.130
0.089
0.003
0.003
5.3. L o c a l i s e d D e f o r m a t i o n M o d e l
An optimised triangulated grid (irregular) was generated from the 14,000 displacement nodes (Figure 8). The grid covers a 56,000 km 2 area, and the perimeter nodes have zero deformation. This grid forms the LDM for the Fiordland earthquake. Evaluating this grid using linear interpolation, illustrates the surface deformation as a result of the earthquake, as predicted by the dislocation model (Figure 9). •
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Figure 7. Fiordland station displacements relative
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,,
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•
The 95% error ellipses for calculated displacements are shown.
/ ~ ~
-45"
.
~
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.
0 06 0 04 0 02 0 O0
Figure 9. Contour and directional plot of the Fiordland LDM. The solid black line indicates the position of the fault projected to the surface. Contours are spaced at 0.01 m intervals. The scale bar indicates the amount of deformation represented by each grey tone.
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46"
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Figure 8. Optimised triangulated grid derived from GNS model 205.
Comparing the displacement as evaluated by the LDM with that calculated at the 11 stations listed in Table 1 and shown in Figure 10, results in RMS values of 8mm and 12mm east and north respectively. This is within the constant 2cm
635
636
A.Jordan.P.Denys• G. Blick
portion of the 6th order accuracy standard previously stated and is therefore considered as an accurate representation of the observed displacement. However, it does not necessarily represent the accuracy of the model between stations. 5.4. A n e w N a t i o n a l D e f o r m a t i o n M o d e l
A new NDM was compiled using the NVM and the Fiordland earthquake LDM. When the NDM is evaluated for epochs after 22 nd August 2003, the displacements as predicted by the LDM will be applied to all affected coordinates, as well as the NVM predictions. Figure 11 illustrates the evolution of surface movement. 166"
.
167" .
.
.
168"
169'
months after the earthquake, however the deformation was considered too minor to materially impact on the datum. If any post-seismic deformation is detected in future surveys, an updated LDM could be produced. This paper has demonstrated a method for producing an LDM for an earthquake scenario, where the displacements are applied instantaneously at a specified epoch. However, this may not always be the case with other forms of localised deformation, such as slow earthquakes and creeping landslides. For such events, a velocity rather than a displacement maybe more appropriate. Also, not all localised deformation events will justify the cost of producing an LDM and implementing it into the datum. Jordan (2005) discusses these issues, including comprehensive case studies on the August 2003 Fiordland earthquake and a landslide scenario.
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Figure 11. Evolution of the NDM in the Fiordland Area
relative to OUSD. 6. D i s c u s s i o n
It is important to note that using a dislocation model based on only l l repeat observations to produce a LDM for an affected surface area (land) of approximately 28,000 klTl2 is unlikely to be sufficient to accurately model the deformation for the entire area. The spatial extents are probably sufficient, however the density of the network is likely to be insufficient. It also means that there is no redundancy as all observations will be required to produce the model. Therefore, there is no way to independently assess the accuracy of the model produced. However, this dataset is probably typical of the datasets that will be available after such events in rural situations, and serves well as a case study. In an urban situation, it is likely that the current LINZ geodetic network will provide a sufficient density of marks for resurvey after a significant earthquake event. It is also important to note that the LDM produced here does not account for any ongoing post-seismic deformation. A survey was carried 5
A new NDM structure has been defined, allowing for multiple deformation models to be compiled together, as recommended by LINZ (2003a). The NDM can now be made up of the NVM and one or more LDMs. Beavan and Wallace (2004) modelled the August 2003 Fiordland using a dislocation model (GNS Model 205). An optimised grid of deformation nodes, generated by the dislocation model, has been defined. An LDM has been produced using the optimised grid. The current NVM and the August 2003 Fiordland earthquake LDM have been successfully compiled into a new NDM, which can be implemented into NZGD2000.
8. R e f e r e n c e s
Beavan, J. and Blick, G. (2005). Limitations in the NZGD2000 deformation model. This issue. Beavan, J. and Haines, J. (2001). Contemporary horizontal velocity and strain rate fields of the Pacific-Australian plate boundary zone through New Zealand. Journal of Geophysical Research, 106, B 1, 741-770. Beavan, J. and Wallace, L. M. (2004). The effect of earthquakes on the New Zealand Geodetic System: A scenario for a major Wellington Fault earthquake, and results from the August 2003 Fiordland earthquake. Report. Institute of Geological & Nuclear Sciences Ltd, pp. 49. Bevin, A. J. and Hall, J. (1995). The review and development of a modern geodetic datum. New Zealand Survey Quarterly, 1, 14-18.
Chapter 91 • Implementing Localised Deformation Models into a Semi-Dynamic Datum
Crook, C. N. (2003). Landonline Adjustment Datablade Design. Land Information New Zealand. Land Information New Zealand (2003a). Implementation of a Deformation Model for NZGD2000: OSG Technical Report 20 Land Information New Zealand. Land Information New Zealand (2003b). Accuracy Standards for Geodetic Surveys Land Information New Zealand, Wellington, pp. 26. Jordan, A. M. (2005). Implementing Localised Deformation
Models in a Semi-Dynamic Datum. Masters Thesis, School of Surveying, University of Otago, New Zealand. In press. Pearse, M. B. (2000). Realisation of the New Zealand Geodetic Datum 2000: OSG Technical Report 5 Land Information New Zealand, pp. 18. Reyners, M., and others (2003). The Mw 7.2 Fiordland earthquake of August 21, 2003: Background and preliminary results. Bulletin of New Zealand Society of Earthquake Engineering, 36, 4, 233-248.
637
Chapter 92
Definition and Realisation of the SIRGAS Vertical
Reference System within a Globally Unified Height System Laura Sanchez DGFI, Marstallplatz 8, D-80539 Munich, Germany. Abstract. The new SIRGAS vertical reference system
is based on the determination of an equipotential reference surface 14/o within a global definition, i.e. optimally fitting the worldwide mean sea surface. The corresponding 147ovalue (mean geopotential value over the total ocean surface) is empirically estimated using different combinations of global gravity models (EGM96, TEG4, GGM02S, EIGEN-CG03C) and mean sea surface models (CLS01, KMS04, GFSC00.1 and a series of annual models from 1993 to 2001 derived at DGFI from T/P). The results show the 1470 dependence on the GGM's degree n, on the latitudinal extension, and on time. The recommended Wo value (62 636 853,4 m:s 4) is derived from EIGEN-CG03C (n = 120) and referred to the epoch 2000.0. It differs from previous computations by 3 m2s4 (e.g. Bursa et al. 2002, Bursa et al. 2004). A preliminary realisation of this new reference level is accomplished by transforming the existing South American classical height datums (defined individually at different tide gauges) through the combination of GNSS positioning, high resolution (quasi)geoid models and physical heights derived from spirit levelling and terrestrial gravity data. Keywords. Global reference level Wo, unified vertical reference system, world height system, height datum unification.
1. Introduction The new vertical reference system for South America is based on two components: a geometrical and a physical one (Drewes et al. 2002). The geometrical component corresponds to ellipsoidal heights referred to the SIRGAS datum, i.e. the GRS80 ellipsoid (SIRGAS 1997). This component has already been accomplished by adopting and using the SIRGAS system as the official geodetic reference frame in the South American countries. The realisation of the physical component (normal heights and quasigeoid) implies, among others, the computation of geopotential numbers in a continental adjustment, the determination of a unified quasigeoid model for the
region, and the transformation of the existing classical height systems to the new proposed system. The readjustment of the national vertical networks in terms of geopotential numbers allows to remove the existing internal inconsistencies due to: a) omitting the Earth's gravity field effects in previous computations, and b) the over constraining of tide gauges used as reference levels by forcing more than one tide gauge to zero height. The computation of geopotential numbers is a standard procedure (e.g. Torge 2001, Heiskanen and Moritz 1967), it does not represent any theoretical problems, nor does it demand any additional measurements other than the potential differences derived from levelling and gravity data. Care is however required, when undertaking a minimum constraint solution. In this case, just one geopotential value 141oshall be used as reference level for the entire continental network. Under the classical definition of height systems, the reference level Wo for South America could be the equipotential surface of the Earth's gravity field passing through a selected tide gauge mark. This selection could solve the required unification of the existing height systems in the region, but it would still be a local (or individual) system; i.e. its associate normal heights (or geopotential numbers) would be consistent in South America only, the incompatibilities with other height systems around the world would persist. It would not permit the precise combination of the geometrical reference system (ellipsoidal heights h) with the physical one (geoid N and orthometric heights H °, or quasigeoid ( a n d normal heights/-F) in global representations. Other possibilities to define the South American 147o could be to determine the average of the potential values at the reference tide gauges, or at all available tide gauges in the region (not only those of reference), or along the coast line. However, each of these possible solutions still incurs the previously mentioned problems. A globally defined reference level W0 can be estimated following the Gauss-Listing definition: 14/oshould be the geopotential averaged over the (ideal) ocean surface in a totally undisturbed state. That is, Wo
Chapter 92 • Definition and Realisation of the SIRGASVertical ReferenceSystem within a Globally Unified Height System
should correspond to the equipotential surface to which the averaged sea surface topography (SSTop) is zero when sampled globally over all oceanic areas (i. e. the geoid (Mather 1978)). Under this condition, the physical heights derived from [H°=h-N] or [/~=h-~, being N and ~" also globally defined, would be compatible with those obtained from levelling and gravity data. Since geoid and quasigeoid are identical in oceanic regions, it is not necessary to make distinctions between these surfaces to determine/4/0. Using these concepts, the first section of the present report is centred on the empirical determination of Wo using different combinations of mean sea surface models (MSS) and global gravity models (GGM). Then, after recommending a global reference value Wo, the transformation of the existing height systems in South America to the new reference level is discussed in the concluding section of this paper. 2. E m p i r i c a l
determination
of
Wo
According to the Gauss-Listing definition, the Wo value should satisfy the condition: I(W-Wo)dSo -min
[1]
So
where So represents the global ocean surface. The present state o f the art allows observing the sea surface directly by means of satellite altimetry missions, that data analysis of which leads to accurate MSS models. These models, in combination with GGMs, facilitate the empirical evaluation of Eq. 1 in a discrete manner. The potential value W at every point describing the mean sea surface is given by (e.g. Torge 2001, Heiskanen and Moritz 1967): W = 1 (o2r2cos2(900_ O) + 2 GM 1+ F
£ (Qm cosm2 + S......sin mA)P,,,,,(cosO) n =
[2]
m = 0
where (r, 0,2) are the spherical coordinates of the evaluation point, GM is the geocentric gravitational constant, co is the angular velocity of the Earth's rotation, C,,m, S,,,,, are the GGM's spherical harmonic coefficients and P~m are the fully normalized polynoms of Legendre. Eq. 2 is evaluated by introducing (r, 0,2) for all the open ocean surface (MSS heights) in 1° x 1° block values. The continental and the coastal areas are excluded. Then, Wo is determined by averaging the individual Wblock values using cos(p as a weight. The selected models are: MSS: CLS01 (Hernandez and Schaeffer 2001 a), KMS04 (Andersen et al. 2004),
and GSFCO0.1 (Koblinsky et al. 1999). GGM: EGM96 (Lemoine et al. 1998), TEG4 (Tapley et al. 2001), GGM02 (Tapley et al. 2005), and EIGENCG03C (F6rste et al. 2005). The GGMs are reduced to epoch 2000.0 using the provided rates for the lower degree/order coefficients (C20, C21, $21 or C20, C30, C40) in each model. GGMs in a tide-free system are also transformed into the zero tide system to make them consistent with the MSS models. Although W0 does not change from one to another tide system, its determination must be based on a MSS model and a GGM in the same tide system. To avoid the W0 dependence on seasonal sea ice cycle and on polar glaciations/melting effects, a north-south delimited zone must be adopted. The US National Snow and Ice Data Center (NSIDC; http://nsidc.org) shows how the ice concentration on sea water varies from winter to summer in the polar regions. The approximate seasonal extremes estimated from remote sensing between 1978 and 2002 indicate that in winter a 100% ice concentration on water covers the caps defined by 0~> 60°N (in the Arctic), and 0~<60°S (in the Antarctic). In summer, this ice concentration retrocedes to ~ 7 2 ° N in the North Pole and almost to the coast line in the south. In order to observe the variation of Wo as a function of the extension of the computation zone, it was calculated by varying in 1o the latitudinal limits from q~= 82 ° N/S to ~o= 50 ° N/S (figure 1). Results show Wo decreases if the computation zone is reduced. The mean Wo variation between q~= 82°N/S and ~p= 66 ° N/S is-0,05 m2s-2 only. This apparent stability is due to the little surface change in the polar regions. At the middle latitudes (from ~ = 65 ° N/S to ~o= 50 ° N/S) the Wo mean rate is -0,13 m2s-2. That means, if the computation zone is reduced by one degree in latitude (really two degrees, one in northern hemisphere and the other in the southern hemisphere), the Wo surface would be 1 cm higher. According to the geographical coverage of the T/P mission (reference data for modelling the used mean sea surfaces applied in this study), the Wo computation zone could be delimited by 0~= 66° N/S; however, keeping in mind the water ice content presented by NSIDC, it would be better to exclude those zones where the seasonal sea ice cycle is too dominant. This study therefore adopts a zone delimited by ~o= 60°N/S for all computations. This decision is also supported by the fact that the large discrepancies between the MSS models applied occur at high latitudes. Hernandez and Schaeffer (2001b) found a systematic difference o f - 3 c m between CLS01 and GSFC00.1 models, the largest disagreement (+ 5 cm) is north/south of (,o= 60 ° N / ( p = 60 ° S. The divergence
639
640
L.Sanchez
between KMS04 and CLS01 models (Andersen et al. 2004) is at the same level (+ 5 cm), and the extreme
values appear also at high latitudes.
Wo [mZs z]
' s3.s
-
~ _:
S3.01
_
-
......
...~ ,._
. . . . . . . . . . . . . . . . . . .
~
~2.s s2.o
.~
I ]
I
.
I
,
J I
.
............
[
j
.
.
.
.....
:
.
.
.
.
[
l "N'~'~j~[ ,atitu.a limits [.°] -
"
-"-'~'~'~-
I
Fig. 1 W0 dependence on latitudinal extension (GGM: EGM96, n=360; MSS model: CLS01, l°x 1°), the value 62 636 800 m2s should be added. wo [~s -2]
iI I
ss.s .
54.5
.
.
.
.
.
i, I
J L.
.
~.~
I
-
i
' ,,-t-
r",,,,I
¢~
~
~
4o
~
co
I ~
~
I
,
~ 1--1
r-,,,l 1-1
I'
~
....
I
.
s3.o
1
I
_l .....
"t
.., .
.
.
.
.
I
, ¢-~ ~
~ .,--i
i.t-i 1-1
.
',.D ,,-,
~ ~
'
i
] t .
.
.
I
I
I
130 .,,..,4
,0", ,.-,
I
i .
.
i t .
.
.
.
.
i
. O ¢',,,I
~ r--.l
,
,r"q ~
on ¢.,,,i
~ P..,i
[ [
I
[
~ I'M
,,,,.O ¢.,,,i
.
.
.
.
I
'
i
i
_;
!
!
', ~ r--.,i
tzO ~
0"l ¢-,,I-
I O ~
~ .r~
I
I
l
¢-4 ¢,"1.
t'~ P-'l
~ r',-'l
r,..t3 0'3
,,.,',0 r,1
Fig. 2 Wo dependence on the GGM's harmonic degree n (GGM: EGM96, MSS model" CLS0 l, 1o x 1o, (p = 60 ° N/S), the value 62 636 800 mZs-2 should be added.
w,
,=,
.c:,
m
c~
c:, B l a c k
size
r,~ ~.
Fig. 3 Wo dependence on the MSS's spatial resolution (GGM: EGM96, n=360, MSS model" CLS01, (p = 60 ° N/S), the value 62 636 800 mZs-2 should be added.
To represent the variation of Wo as a function of n, it is calculated using the same GGM, but changing the retained harmonic degree between n=lO and n=360 (figure 2). The dependence of Wo on the lower degree/order expansions is evident, from n=lO to n=20 Wo changes by -1,46 mZs-2, from n=20 to n=30 it varies -0,52 mZs-2. In general, when the retained harmonic degree n grows, the difference between the corresponding Wo values decreases. Nevertheless, up to n =120 the variation of Wo is smaller than 0,001 mZs-2. This demonstrates the dependence of Wo on the harmonics n>120 is negligible. Following this conclusion, Wo can be calculated with a GGM derived exclusively from satellite data, e. g. the new gravity missions CHAMP, GRACE and GOCE. They provide the highest accuracy in the lower frequency band of the Earth's gravity field. In the same context, it is preferable to exclude terrestrial gravity data (n>120) of determining Wo, because their anomalies refer to the level surfaces passing through the reference tide gauges rather than to a unique global surface; i.e. they are affected by the influences of vertical datum inconsistencies. Figure 3 shows the variation of Wo, if the MSS block size is changed. The largest deviation (0,1 mZs-2)
occurs between block sizes of 30' and 45', while Wo is very similar when using the remaining cell sizes. In this way, one can say that 1° x 1° is a representative cell size, and it will be taken as the standard in the following computations. The Wo estimates presented above are based on the models EGM96 and CLS01. In order to verify the reliability of these values, 141ois also computed using other GGMs and MSS models. In these computations GM corresponds to the value of each GGM and co to 7 292 115 x 10 -11 tad s -1 (IAG SC3 Rep. 1995). The GGMs computed up to n = 360 (EGM96, EIGENCG03C) are also truncated by n = 200 and n = 120 to compare the Wo values with those derived from lower degree models (TEG4, GGM02S). Table 1 summarizes the results. The largest Wo variations (1,26 m2s-2) are due to the extension of the computation area ((,o = 60 ° N/S or ~o = 80 ° N/S). However, over the same latitudinal range the Wo values are consistent. The Wo variation from one GGM to another (at the same degree and latitude coverage) is less than 0,02 m2s-2. The discrepancies between the Wo values derived from different MSS models are greater by including data from high latitudes O-=80°N/S than by middle latitudes
Chapter 92 • Definition and Realisation of the SIRGAS Vertical Reference System within a Globally Unified Height System
(p=60°N/S. It is assumed these differences are a consequence of the diverse models applied to analyse the altimetric data in each MSS model, and also the
inter-annual ocean variability averaged over distinct periods of time, which does not permit to define a specified reference epoch for the MSS heights.
Table 1.14Io values derived from different GGMs and MSS models [in m2s-2].
I
MSS
n
EIGEN-CG03C
EGM96
TEG4
GGM02S
120
62 636 853,35
62 636 853,37
62 636 853,38
62 6368 53,36
CLS01
200
53,35
53,37
53,37
q~
[N/S]
60/60 60/60
360
53,35
53,36
60/60
KMS04
360
53,24
53,26
60/60
GSFC00.1
360
53,58
53,59
60/60
120
62 636 854,61
62 636 854,62
62 636 854,65
200
54,61
54,62
54,64
360
54,61
54,61
82/80
KMS04
360
54,46
54,45
82/82
GSFC00.1
360
54,93
54,93
80/80
CLS01
62 636 854,61
82/80 82/80
WO [mzs 'z]
"--....
53.50
"
"
.
" " " " "
53.35
" I
"~
if' ~
m ~I
i
~
4
i
h
53.30
~ . . . .
53_25
1993
EGM96 [rb=3601 m 1994
.
. T E G 4 In=200] m 1995
-
- GGM02S [n=120] 1.996
CG03[n=1201 1.997
'
year 1.998
1999
2000
2001
Fig. 4 Annual Wo values derived from different GGMs and yearly MSS models from T/P 1-365 cycles (1 ° x 1°, (p = 60°N/S), the value 62 636 800 m2s -2 should be added.
To identify the variation of Wo as a function of time, annual MSS models are computed using the T/P altimetric data (cycles 1-365) available in the MGDR's Version C AVISO altimetry project (AVISO 1996). The FES2004 global tidal solution (Lettelier et al. 2004) and the sea state bias models of Chambers et al. (2003) were applied to the MGDR-C data. The yearly Wo values are obtained combining each annual MSS model (latitude limits (p = 60 ° N/S) with the GGMs reduced to the same epoch, i. e. MSS heights at 1997.0 are combined with GGM coefficients at 1997.0 to estimate Wo at 1997.0. Although the Wo changes as a function of the reference epoch of the GGM coefficients is insignificant (~0,002 mZs-2 from 1986 to 2000), they are reduced to the corresponding annual epochs (1993...2001) to be consistent with the MSS models. The results are shown in figure 4.
The variation of 14/o is highly correlated with the variation of MSS. The largest annual changes happened in 1996/1997 (-0,06 m 2s-2), 1997/1998 (-0,05 mZs-2) and 2000/2001 (-0,05 mZs-2). The total Wo variation between 1993 and 2001 is-0,20 mZs-2. The absolute level differences between the used GGMs are assumed insignificant, since they are lower than 5 mm. According to the above presented Wo values, one may conclude: Wo is almost independent of the GGM, slightly dependent on the MSS model and strongly dependent on the latitudinal extension. Its variation with time (until now) is almost negligible, but the sea surface is constantly changing and after some years this dependence could reach significant values. Therefore, it is necessary to define a reference epoch of 141oas a convention. This convention should also include, among other adoptions, the 14/o annual velocity, the latitudinal extension of the computation
641
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L.Sanchez
area and the spatial resolution of the MSS model to be used. From the computations presented in this study, the value Wo = 62 636 853,4 m2s-2 is recommended. It is derived using the following specifications: _
_
[3]
being W~the actual gravity potential of a point referred to the height system i.
_
Extension:
q) = 60 ° N ... q0= 60 ° S
Resolution:
1° x 1°
MSS model:
T/P derived MSS heights at the epoch 2000.0
GGM:
EIGEN-CG03C, n - 120, reference epoch 2000.0
Constants:
o~, - w 0 - ~
P
W,
,~WJ-,,,,
W~
GM = 398 600,4415 x 10 9 m3s -2
dWj,,,,
co = 7 292 115 x 10-11 rad s-1 Procedure:
Eqs. 1 and 2, averaging Wi values with weight equal to cosq)
The proposed Wo value differs from previous computations by ~3 m2s-2 (Bursa et al. 1997, Bursa et al. 1998, Bursa et al. 1999, Bursa et al. 2002, Bursa et al. 2004). The reasons for this disagreement are not yet clear. It is necessary to make a detailed comparison between the different methodologies applied and the used GGMs and MSS models. In fact, the first computations of the presented work were carried out using the models EGM96 and CLS01 only. Since the obtained Wo value varies from those proposed by Bursa et al., it was decided to evaluate the reliability of our results by using two additional independent software suites (Smith 1998, Rapp 1982, Pavlis 1996), other MSS models (KMS04, GSFC00.1) and different GGMs (TEG4, GGM02S, EIGEN-CG03C). All the combinations yield to the here proposed Wo value. In consequence, it will be used as reference level in the next section.
3. Unification of the South American height datums into a global vertical system The relationship between the classical height datums and the proposed global vertical system is represented in figure 5. The existing physical heights (HN, I4°) and the local (quasi)geoid models (N, Q refer to the corresponding surfaces passing through the vertical datum WA or We, respectively. They disagree among each other by dWAB and with respect to the global reference level Wo by dWA or dWB, respectively. (e.g. Rummel and Teunissen 1988, Pavlis 1991, Rapp and Balasubramania 1992, Rapp 1994, Heck 2004). In general, dW~ can be written as:
B ~'T. ,.
;
.
.
,..
Fig.
5 Relationship between classical height datums and a global vertical reference system
In units of meters, Eq. 3 corresponds to: [4]
l~Hi = ~ Yi
where ~ is the normal gravity of the evaluation point on the Earth's surface. Having already introduced a normal gravity field (associated, of course, to a reference ellipsoid), Eq. 4 can be expressed, in terms of normal heights H N, as:
~Hi - h - H i - (
[5]
where the ellipsoidal heights h and the height anomalies ( m u s t be associated to the same reference ellipsoid. Similarly, the discrepancies between two vertical datums/,j would be: 5/-/u ~ = Y
5Wj N N =Hj -H i Y
[6]
Eqs. 4, 5, and 6 indicate that the unification of classical height datums into a global system is feasible by combining adequately high resolution gravity field models, precise geometrical coordinates, and physical heights derived from levelling and terrestrial gravity data. According to this, the proposed methodology is: a)
Determination of 6W~rc (dH/c) at the main tide gauges (including reference gauge site) in each country (or datum zone). It requires GNSS positioning, spirit levelling and a high resolution quasigeoid at the tide gauge marks.
Chapter92 • Definition and Realisation of the SIRGASVertical Reference System within a Globally Unified Height System
b)
Determination of ~W~ssJ~p (6H ss~°p) at the marine areas surrounding the tide gauge. It requires the estimation of the SSTop at each tide gauge included in a).
c)
Determination of c~WiR~ ( 6 H F ) at the vertical reference frame stations, i. e. a set of reference points (well-defined and reproducible benchmarks) should be provided with accurate geometrical coordinates (referred to the ITRS), physical heights and a high resolution quasigeoid model.
d)
Determination
of
6~; (6H!j) by
connecting
precisely neighbouring levelling networks. e)
Combination of the different solutions given by a), b), c), and at) by least squares adjustment.
This process should be iterated until the required reliability (1 ram-level) is achieved. It implies also the adoption of a reference epoch (to) and the determination of changes to h,/4 N, (, and SSTop over time, i.e.:
074 ssr'') (t/, ) - h ss~,,, (t k ) _ ((t/, ) oT-Ii
- h
- I4i
(~Hij(t~) = H j ( t ~ ) -
[71 -
)
H i (tk)
with: h(t
) - h(to) +
8h
- to)
81-1
n(t~) - H(t0) + - ~ ( t ~ -to) a( ((t~) - ((to)
+ -~(t,,
-
[8]
to)
In the particular case of South America, the new vertical reference flame was established in 2000 through a 10 day GPS campaign (Luz et al 2002). It includes 180 stations, among them the main tide gauges of each country, levelling points at the international borders, and the SIRGAS reference flame determined in 1995. The geometrical component corresponds to SIRGAS2000, i.e. ITRS realized by ITRF2000, epoch 2000.4. The physical component is being improved by controlling the national first order levelling networks, by checking the existing terrestrial gravity data and by calculating geopotential numbers as input data for the normal height determination at the reference stations. Once each country had readjusted its levelling networks, 3"Wir~, 3"WiR~ and cyW° can be determined and the continental adjustment of the geopotential numbers
will be feasible. To complement these results, the refined terrestrial gravity data shall be combined with global gravity models derived from CHAMP, GRACE, and GOCE to obtain a unified high resolution quasigeoid model within a global vertical level. The time dependence of heights (h, H N) and SSTop is also being estimated by analysing tide gauge records, satellite altimetry data and continuous GNSS positioning. As a first approximation, the discrepancies 6 H ~ from the existing South American height datums with respect to the proposed vertical system Wo are estimated by means of Eq. 4 or Eq. 5 (the results are identical); the actual gravity potential W~ at each tide gauge is computed using Eq. 2 at [(p, 2, h-/-F] and the EIGEN-CG03C model, n = 360. Since the geometrical coordinates refer to the SIRGAS datum (i.e. GRS80), ~ (in Eq. 4) is derived from the GRS80 ellipsoid, and ((in Eq. 5) takes into account the degree zero terms arising from the different G M values and the difference between Uo and Wo (Eq. 2-182, Heiskanen and Moritz 1967). All height coordinates are in a tide-free system. Figure 6 illustrates the corresponding 6/4/C values at the reference (main) tide gauges. Chile has three different height datums, because the tide gauges located in the south (Chile I and Chile II) cannot be connected by spirit levelling with each other or with the central gauge (Chile III). Those countries that included more than one tide gauge in the reference frame have been denoted with (A). For instance, if Colombia takes into account only the reference tide gauge, c~H~~ would be +52 cm, while by including the other two, it would be -6 cm. The same occurs with Chile III and Venezuela. These large variations of ~H/c as a function of the included tide gauges show the low reliability of linking a height system to another by just one point or few points. It is necessary to include a well-distributed network with very high accuracy on h, H x and (.. Table 2 compares the cyHire" and (~H~fsr°p derived using the EIGEN-CG03C model and the high resolution quasigeoid GeoCol2004 (GGM + high quality terrestrial gravity data, Sanchez 2003) at the Colombian tide gauges. The 6//rc values derived from the high resolution local quasigeoid are (of course) more consistent with each other than those obtained from the lower resolution GGM. One identifies a bias of--22 cm, which can be assumed as an absolute level difference between the local height system and the global one. ;HiSSr°P shows a very similar behaviour but, in this
643
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L.Sanchez
vertical level 14/o and combined with those of the neighbouring countries. The results presented here should be understood as preliminary. They shall be refined by determining 6H Re which are not computable at this current stage, "-'J Col '
case, the systematic difference reaches -26 cm. The inconsistency (4 cm) between 6H i~C and 6H iss>,' should be solved by including 6~;/-/ie~ and performing the corresponding adjustment (Eq. 8). Once a unique 6Hco l (6Wow) for Colombia is obtained, the existing
since the normal heights still require processing.
geopotential numbers can be linked to the global Ecuador 82 cm
Chile I 19 cm
Colombia (A) 6 cm "1"
i±
IIl -5 cm
Chile
Chile II - 17
cm
Venezuela (A) -13 cm
Chil -
II
Venezuela -19 cm
(A)B
cm
Argentina 6 cm T
H0
i Brazil -20 cm
la
-35 cm
Colombia -52 cm Fig. 6 Discrepancies between the individual classical height datums in South America and the proposed global vertical reference system.
Table 2. Level differences between the Colombian height datum and the global vertical system derived from a high resolution quasigeoid model and the model EIGEN-CG03C. GeoCol2004 Name
EIGEN-CG03 C
( ~ H iTG
~ iSST°P
av£
( ~ H SSTop i
[cm]
[cm]
[cm]
[cm]
TG !
- 18
-22
56
43
Ref. TG
-25
-30
-52
-41
TG ii
-23
-20
15
29
-22_+3
-26_+6
6_+53
10+49
Average II
4 Final c o m m e n t s The empirical determination of I47obecomes feasible since accurate derived satellite altimetry MSS models and precise GGMs are available. Nevertheless, as in any reference system, 147oshould be based on some adopted conventions, which guarantee its reliability and repeatability. It is needed, among others, to define a reference epoch (common to both, GGM and MSS model), the spatial resolution and the latitudinal extension of the MSS model, and to use a GGM derived from satellite gravity data only. This study concludes with a value of Wo = 62 636 853,4 m2s-2 recommended as a global reference level. It differs b y - 3 mZs-2 from other computations, but the empirical evaluation of the gravity potential using
different combinations of four GGMs (EGM96, TEG4, GGM02S and EIGEN-CG03C) and four MSS models (CLS01, KMS01, GSFC00.1 and one derived from T/P with yearly representations) proves the reliability of our results. The unification of the classical height systems should be done in a global flame. If they refer to a selected tide gauge mark or to the average of several, its related heights would be confined to be valid only in a determined region and they may not be combined with the geometrical reference system (ellipsoidal heights). The transformation of the height datums into the new global vertical reference system should be done through a set of accurate reference flame stations with precise geometrical coordinates (GNSS positioning and satellite altimetry in sea regions), high resolution (quasi)geoid models (GGM + refined terrestrial gravity data), and physical heights derived from spirit levelling and gravity data.
References Andersen, O. B., A. L. Vest, P. Knudsen, (2004). KMS04 mean sea surface model and inter-annual sea level variability. Poster presented at EGU Gen. Ass. 2005, Vienna, Austria, 24-29, April. AVISO (1996). A VISO user handbook. Merged Topex/Poseidon products (GDR-Ms). CLS/ CNES, AVI-NT-02-101-CN. 3 d Ed., July. Bursa, M., K. Radej, Z. Sima, S. True, V. Vatrt, (1997). Determination of the geopotential scale factor from
Chapter 92 • Definition and Realisation of the SIRGASVertical Reference System within a Globally Unified Height System
Topex/Poseidon satellite altimetry. Studia geoph, et geod. 41: 203-215. Bursa, M., J. Kouba, K. Radej, S. True, V. Vatrt, M. Vojtiskova, (1998). Mean Earth's equipotential surface from Topex/Poseidon altimetry. Studia geoph, et geod. 42: 456-466. Bursa, M., J. Kouba, M. Kulnar, A. Miiller, K. Radej, S. True, V. Vatrt, M. Vojtiskova, (1999). Geoidal geopotential and world height system. Studia geoph, et geod. 43: 327-337. Bursa, M., S. Kenyon, J. Kouba, K. Radej, V. Vatrt, M. Vojtiskova, J. Simek, (2002). World height system specified by geopotential at tide gauge stations. IAG Symposia, 124:291-296. Springer. Bursa, M., S. Kenyon, J. Kouba, Z. Sima, V. Vatrt, M. Vojtiskova, (2004). A global vertical reference frame based on four regional vertical datums. Studia geoph, et geod. 48: 493-502. Chambers, D., S. A. Hayes, J. C. Pies, T. J. Urban, (2003). New Topex sea state bias models and their effect on global mean sea level. J. Geophys. Res. 108 (C10), 3305,10.1029/2003JC001839. Drewes, H., L. Sanchez, D. Blitzkow, S. de Freitas, (2002): Scientific foundations of the SIRGAS vertical reference system. IAG Symposia 124:297-301. Springer. FGrste, C., F. Flechtner, R. Schmidt, U. Meyer, R. Stubenvoll, F. Barthelmes, R. KGnig, K.H. Neuanayer, M. Rothacher, Ch. Reigber, R. Biancale, S. Bruinsrna, J.-M. Lemoine, J.C. Raimondo, (2005) A New High Resolution Global Gravity Field Model Derived From Combination of GRACE and CHAMP Mission and Altimetry/Gravimetry Surface Gravity Data. Poster presented at EGU General Assembly 2005, Vienna, Austria, 24-29, April. Heck, B. (2004). Problems in the definition of vertical reference frames. IAG Symposia 127:164-174. Heiskanen W. And H. Mofitz (1967). Physical Geodesy. W. H. Freeman and company. San Francisco. Hernandez, F., Ph. Schaeffer (200 l a). MSS CLSO1 http:// www'cls'fr/html/°cean°/pr°jects/mss/cls 01_en.html Hernandez, F., Ph. Schaeffer (2001b). The CLSO1 mean sea su~ace." a validation with the GFSCO0.1 surface. Available at http://www.cls.fr/html/oceano/projects/ mss/cls 01 en.html IAG SC3 Pep. (1995). IA G SC3 final report, Travaux de L 'Association Internationale de Gdod&ie, 30:370 - 384 Koblinsky et al. (1999). NASA Ocean Altimeter Pathfinder Project, Report 1." Data processing handbook, NASA/TM- 1998 -208605, April. Lemoine, F., S. Kenyon, J. Factor, R. Trimmer, N. Pavlis, D. Chinn C. Cox, S. Kloslo, S. Luthcke, M. Torrence, Y. Wang, R. Williamson, E. Pavlis, R. Rapp, T. Olson. (1998). The Development of the Joint NASA GSFC and the National Imagery and Mapping Agency (NIMA)
Geopotential Model EGM96, NASA, Goddard Space Flight Center, Greenbelt. Lettelier, T., F. Lyard, F. Lefebre, (2004). The new global tidal solution." FES2004. Presented at: Ocean Surface Topography Science Team Meeting. St. Petersburg, Florida. Nov. 4-6. Luz, R. T., L. P. S. Fortes, M. Hoyer, H. Drewes, (2002): The vertical reference frame for the Americas - the SIRGAS 2000 GPS campaign, lAG Symposia 124: 301-305, Springer. Mather, R. S. (1978). The role of the geoid in fourdimensional geodesy. Marine Geodesy, 1:217-252. Pavlis, N. (1991). Estimation of geopotential differences over intercontinental locations' using satellite and terrestrial measurements. The Ohio State University, Department of Geodetic Science and Surveying. Report No. 409. Pavlis, N. (1996). Modification of program f477 (Rapp 1982) Rapp, R. (1982). A FORTRAN program for the computation of gravimetn'c quantities from high degree spherical harmonic expansions. Rep. No. 344. Dept of Geodetic Science and Surveying, The Ohio State University, Columbus Ohio. Rapp, P.; N. Balasubramania. (1992). A conceptual formulation of a world height system. The Ohio State University, Department of Geodetic Science and Surveying. Report No. 421. Rapp, R., (1994). Separation between reference surfaces of selected vertical datums. Bull. GGod. 69:26-31. Rummel, R.; P. Teunissen. (1988). Height datum definition, height datum connection and the role of the geodetic boundary value problem. Bull. GGod. 62: 477498. S/mchez, L. (2003): Bestimmung der HShenreferenzfliiche fiir Kolumbien. Diplomarbeit. TU Dresden. SIRGAS (1997): Final Report Working Groups I and IISIRGAS RelatGrio Final Grupos de Trabalho 1 e 1L Insfituto Brasileiro de Geografia e Estatistica, Rio de Janeiro. Smith (1998). Program geopot97, v. 0.4c. http://www. ngs.noaa.gov/GEOID/RESEARCH_SOFTWARE/rese arch soflware.html. Tapley M. Kiln, S. Poole, M. Cheng, D. Chambers, J. Ries, (2001). The TEG-4 Gravity field model. AGU Fall 2001. Abstract G51A-0236 Tapley J., Ries, S. Bettadpur, D. Chambers, M. Cheng, F. Condi, B. Gunter, Z. Kang, P.Nagel, R. Pastor, T. Pekker, S.Poole, F. Wang,. (2005). GGM02." An improved Earth gravity field model from GRACE. Journalof Geodesy, doi 10.1007/s00190-005-0480-z. Torge (2001). Geodesy. 3rd. Edition. De Gruyter. Berlin, New York.
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Chapter 93
Tests on Integrating Gravity and Leveling to Realize SIRGAS Vertical Reference System in Brazil R. T. Luz 1'2, S. R. C. de Freitas 1, R. Dalazoana ~, J. C. Baez ~'3, A. S. Palmeiro ~ Geodetic Sciences Graduation Course, Federal University of Paranfi (UFPR), Curitiba, Brazil 2 Coordination of Geodesy, Brazilian Institute of Geography and Statistics (IBGE), Rio de Janeiro, Brazil 3 Department of Geomatics, University of Concepci6n, Chile ( robtluz, sfreitas, regiane ) @ ufpr.br, jbaez @ udec.cl, ale_palmeiro @ yahoo.com.br Abstract. The integration of gravity data within the Brazilian leveling network is very difficult due to the historical dissociation between leveling and gravity surveys. This research was started, in the context of SIRGAS Project, to resolve this problem by evaluating different strategies and procedures with the aim of establishing some kind of gravity coverage over vertical reference stations. Data employed in this article comes from one of the few areas where the points of the Brazilian fundamental leveling network are entirely covered with gravity surveys and, in addition, connect three permanent GPS stations, two of which belong to the SIRGAS 2000 Reference Network. This will allow for several analyses regarding the Brazilian realization of SIRGAS Vertical System, including: an evaluation of the effects of adopting different types of heights; and the investigation of strategies and procedures to solve the absence of gravity values over benchmarks. Values of geopotential differences were computed and adjusted, for a 2300 km network consisting of six loops with perimeters ranging from 136 km to 690 km. Dynamic, Helmert, Normal and Normal-Orthometric reductions were generated and compared. The status of the development of a GIS designed for the tasks involved in geopotential numbers computation to be applied in the studies related to the SIRGAS Vertical Reference System is also presented.
mation to leveling lines is an underlying cause for this problem. Primary vertical surveys conducted by IBGE from the 1940's to the 1980's spread to cover almost the entire country (except for the Amazon Region) (Luz et al., 2002b). IBGE also conducted gravity surveys, which were initially concentrated on horizontal datum region and, since the 1980's, on filling in the gravity gaps. Gravity surveys over new leveling lines were only recently established as a routine procedure at IBGE (Blitzkow et al., 2002). Other institutions also carried out gravity densification, but these efforts were not sufficient to configure an adequate coverage of the leveling lines. An example of this situation is presented in Figure 1.
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Keywords. Heights, Geopotential numbers, SIRGAS.
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1 Introduction The present status of the Brazilian Geodetic System (SGB) does not allow for obtaining precise values of physical heights using modem space geodesy techniques. The inability to integrate gravity infor-
0
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Fig. 1. Map (upper) and profile (lowed of the distribution of vertical stations (RRNN, black crosses) of the leveling loop including Brazilian Vertical "Imbituba" Datum, and gravity stations (EEGG, gray circles) in the same area (see Figure 2).
Chapter 93 • Tests on Integrating Gravity and Leveling to Realize SIRGASVertical Reference System in Brazil
0
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Consequently, heights of the SGB Vertical Network have a certain conceptual weakness due to the absence of gravity information and, therefore, cannot be called orthometric. The normal-orthometric reduction is used since 1993, but it deals only with the effects of theoretical variations of gravity derived from differences in latitude. There are similar difficulties with regard to the height systems of other South American countries (Drewes et al., 2002). The effects of this situation include, e. g., inconsistencies in the geopotential models covering the continent, difficulties for the integration of results from other space techniques, like satellite altimetry, and problems in connecting vertical datums
around the world (e. g., Hemfindez et al., 2002; Ihde and Augath, 2002). To deal with these kinds of problems in the South American context, the Working Group on Vertical Datum (WG-III) of the SIRGAS Project ("Geocentric Reference System for the Americas") was created in 1997. The main goal of WG-III is to improve and unify the vertical reference for the entire continent through the establishment of a network of stations with geometric and physical heights (Drewes et al., 2002). Ellipsoidal heights were achieved after the SIRGAS 2000 GPS Campaign (Luz et al., 2002a), whose stations form the SIRGAS Reference Network (Figure 2). For the
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R.T. Luz. S. R. C. de Freitas • R. Dalazoana. J. C. B~ez. A. S. Palmeiro
physical component of the SIRGAS Vertical Reference System, WG-III recommended the use of normal heights based upon geopotential numbers obtained from leveling lines with gravity, connecting those stations. Detailed analysis of the joint distribution of Brazilian leveling and gravity stations led to the conclusion that almost all leveling paths have at least some gravity points, even thought all leveling stations were not occupied by gravity surveys (Figure 1). Despite being far from ideal, this configuration allows for the interpolation of gravity values, as recommended by the SIRGAS WG-IlI. The following sections describe the procedures and preliminary results from the integration of gravity data to the leveling lines of a test area, whose observed height differences were reduced aiming the computation of dynamic, normal and Helmert heights.
The tests described in next section used the following concepts (e. g., Freitas and Blitzkow, 1999, Drewes et al., 2002). Geopotential (number) difference (AC) between successive leveled points are computed with the respective observed height difference (AH°b') and mean observed gravity (gOb,) : g oh, AHOb,
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3 Integrating Gravity to Leveling
2 Types of Heights
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(4)
where garbitris a constant, arbitrarily chosen value. For the test area, the value 978 170 mGal was used, and additional analyses were performed using the value 978 270 mGal.
As previously stated, the leveling lines of the SGB do not commonly have adequate gravity coverage. Consequently, the computation of geopotential numbers is difficult, except for those few areas where recent leveling stations were occupied by homogeneous gravity surveys. One of these subnetworks, shown in Figure 2 ("Nortesul"), was chosen as a test area for verifying conditions, procedures and impacts regarding the implementation of the WG-III resolutions and recommendations. Besides providing good gravity integration, these leveling lines connect three continuous monitoring GPS stations, two of them (Braz and Bomj) belonging to the SIRGAS Vertical Reference Network. This initial effort considered only gravity surveys performed by IBGE, to avoid problems regarding quality heterogeneity, data access and documentation. Parallel activities are under development to collect the large amounts of data from other institutions. A Fortran program was developed to analyze leveling networks, helping to criticize connections of lines observed in different dates and to integrate gravity data. The program identifies network nodes, filters out the open lines, computes geopotential differences and respective gravity reductions (dynamic, Helmert, and normal) to the observed height differences. Future improvements include the analysis and implementation of some kind of gravity interpolation. The test network will also be im-
Chapter 93 • Tests on Integrating Gravity and Leveling to Realize SIRGASVertical Reference System in Brazil
portant as a standard to be used in the validation of procedures and in the definition of corresponding minimum gravity spacing values. Initial runnings of the program helped to identify some problems regarding the stability of 20 of the stations and the lack of gravity for 7 other stations, which were removed from the tests. Figure 3 and Table 1 represent the resulting network used in the tests, with 861 stations, forming six loops with a total length of about 2300 km.
With the network defined, the computation o f reductions was performed in two steps. First, geopotential and respective dynamic height differences were computed, considering a reference gravity value of 978 170 mGal. Figure 4 presents the results for loop 3. To investigate the effects of using different reference gravity v a l u e s , the reduction was computed using also the value 978 2 7 0 m G a l , which is approximately symmetrical to the other value in terms o f observed gravity in the region.
Table 1. Loop closures (mm) for the test network Loop 1 2 3 4 5 6
Perimeter (km) 231.46 198.94 592.22 670.34 690.35 136.59
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Fig. 4. Leveled heights, observed gravity, gravity anomalies and dynamic reductions (also in Fig. 6) for loop 3. Position of the leveling stations identified in the "Height" graph can be observed in Fig. 3.
649
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R.T. Luz. S. R. C. de Freitas • R. Dalazoana. J. C. B~ez. A. S. Palmeiro 0 0
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Dynamic height differences (garbitr = 978 170 reGal) were adjusted according the common parametric model, referring to an approximate height value for one of the central nodes of the network. Figure 5 shows the standard deviations of the adjusted dynamic heights. After the adjustment of dynamic height differences, normal and Helmert reductions were computed. Figure 6 shows these results. Interesting direct and inverse correlations to topogr aphy can be observed. The direct correlation of the first set of dynamic reductions is clearly dependant upon the gravity reference value used. Using the second value reverses the correlation. So, for a larger network, it can be expected that the reductions will also become larger, as stated by many authors (e. g., Torge, 2001, p. 251). Helmert and normal reduction values show larger variations than those for dynamic reductions. This is probably caused by the geographical configuration of loop 3, i. e., incorporating the effects of latitude variations. This can be observed in the normalorthometric reduction.
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Chapter 93 • Tests on Integrating Gravity and Leveling to Realize SIRGASVertical Reference System in Brazil
4 Next Activities A Geographical Information System (GIS) dedicated to the preparation of leveling lines for adjustment including integration of gravity data is being developed. This GIS will also perform the interpolation of gravity, where needed, and the analysis of the effects of the future adoption of SIRGAS heights. The system should also aid in the study and integration of results from the continuous monitoring GPS stations with regard to possible vertical crustal movements. Considering the existence of three such GPS stations in the test area and the existence of older leveling lines in the same area, some interesting studies could be carried out employing the proposed GIS. The linking of the Santana Vertical Datum (Figure 7) to the main Imbituba Datum is another important study associated to the realization of the SIRGAS Vertical System in Brazil. These studies can provide interesting aspects for the integration of various types of geodetic and oceanographic tools, such as the data from the Geodetic Tide Gauge Network (RMPG), observations from satellite altimetry and results from a local hydrodynamic circulation model ( e. g., Heck and Rummel, 1990; Sans6 and Usai, 1995).
n
RMPG data are also being used to link satellite altimetry results in the Imbituba region to the data used in the definition of the Imbituba Datum, stored at PSMSL (Dalazoana et al., 2005). The integration of present RMPG results (starting in 2001) to the old PSMSL data (1949-1969) is difficult, but the recent recovering of records for 1985 to 1988 promises to be an important way to aid in that integration.
5 Final Remarks The identification of a region where leveling and gravity surveys are already integrated to the SIRGAS Vertical Reference Network allowed for beginning implementation and analysis of procedures for computing geopotential numbers and physical heights. Data corresponding to 2300 km of leveling were selected, forming a network with 861 stations and six loops, whose perimeters and closures range from 136 km to 690 km and from 0.1 mm(km) 1/2 to 3.1 mm(km)l/2, respectively. Values of geopotential differences were computed and adjusted in a preliminary way due to the arbitrary reference value chosen, as there is yet no connection of this test network to any reference surface. A Fortran program that performs not only the integration of gravity to leveling, but also identifies and builds the network, was developed. Dynamic, Helmert, normal and normalorthometric reductions, based on the adjusted geopotential differences, were computed and compared. This comparison could not highlight any pattern that would be suitable to guide the choice for any kind of reduction. An enlargement of the test network would be needed to verify and implement procedures for gravity interpolation. A first analysis, without any interpolation, based only on the results already achieved, will be performed by the increase of gravity spacing. Quantifying differences from respective solutions will allow a preliminary evaluation of the required gravity distribution over leveling stations.
Acknowledgements
Fig. 7. Marajo Island area, where the connection of the local datum for the small state network (Santana Datum) to the main Imbituba Datum will be studied.
The authors wish to acknowledge the Brazilian agencies for research and education (CNPq and CAPES) for financial support ; IBGE, for the data made available and for the study license for R. T. Luz ; B. Heck (GIK/Uni-Karlsruhe, Germany), for
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kind s u g g e s t i o n s and c o m m e n t s ; H. Drewes. W. B o s c h an d W. Seemfiller ( D G F I , Mfinchen, G e r m a n y ) , for fruitful d i s c u s s i o n s ; and I A G , for the grant s u p p o r t i n g R. T. L u z a t t e n d a n c e at the D y n a m i c P l a n e t 2005 Joint A s s e m b l y .
References Blitzkow D, Cancoro de Matos AC, Lobianco MCB (2002) Data Collecting and Processing for Quasi-Geoid Determination in Brazil. In: Drewes H et al. (eds) Vertical Reference Systems (lAG Symposia, vol. 124). Springer, Berlin, pp. 148-151. Dalazoana R, Baez JC, Luz RT, Freitas SRC (2005) Brazilian Vertical Datum Monitoring - Vertical Land Movements and Sea Level Variations. In: Program & Abstract Book, Dynamic Planet 2005, IAG-IAPSO-IABO Joint Assembly, Cairns, p. 257. Drewes H, Sfinchez L, Blitzkow D, Freitas S (2002) Scientific Foundations of the SIRGAS Vertical Reference System. In: Drewes H et al. (eds) Vertical Reference Systems (IAG Symposia, vol. 124). Springer, Berlin, pp. 297-301. Freitas SRC, Blitzkow D (1999) Altitudes e Geopotencial. In: Sans6 F et al. (eds) Bulletin No. 9, International Geoid Service (Special Issue for South America). IgeS, Milano, pp. 47-61. Freitas SRC, Medina AS, Lima SRS (2002a) Associated Problems to Link South American Vertical Networks and Possible Approaches to Face Them. In: Drewes H et al. (eds) Vertical Reference Systems (IAG Symposia, vol. 124). Springer, Berlin, pp. 318-323. Freitas SRC, Schwab SHS, Marone E, Pires AO, Dalazoana R (2002b) Local Effects in the Brazilian Vertical Datum. In: Adam J, Schwarz KP (eds) Vistas for Geodesy in the New Millennium (IAG Symposia, vol. 125). Springer, Berlin, pp. 102-107. Heck B (2004) Problems in the Definition of Vertical Reference Frames. In: Sans6 F (ed) V Hotine-Marussi Symposium on Mathematical Geodesy (IAG Symposia, vol. 127). Springer, Berlin, pp. 164-173.
Heck B, Rummel R (1990) Strategies for solving the vertical datum problem using terrestrial and satellite geodetic data. In: Sfinkel H, Baker T (eds) Sea Surface Topography and the Geoid (IAG Symposia, v. 104). Springer, New York, pp. 116-128. Heiskanen WA, Moritz H (1967) Physical Geodesy. Freeman, San Francisco, 364 pp. Hernfindez JN, B1 itzkow D, Luz R, Sfinchez L, Sandoval P, Drewes H (2002) Connection of the Vertical Control Networks of Venezuela, Brazil and Colombia. In: Drewes H et al. (eds) Vertical Reference Systems (IAG Symposia, vol. 124). Springer, Berlin, pp. 324-327. Hwang C, Hs iao YS (2003) Orthometric corrections from leveling, gravity, density and elevation data: a case study in Taiwan. Journal of Geodesy 77:279-291. Ihde J, Augath W (2002) The European Vertical Reference System (EVRS), Its Relation to a World Height System and to the |TRS. In: Adam J, Schwarz KP (eds) Vistas for Geodesy in the New Millennium (lAG Symposia, vol. 125). Springer, Berlin, pp. 78-83. Luz RT, Fortes LPS, Hoyer M, Drewes H (2002a) The Vertical Reference Frame for the Americas - the SIRGAS 2000 GPS Campaign. In: Drewes H et al. (eds) Vertical Reference Systems (|AG Symposia, vol. 124). Springer, Berlin, pp. 302-305. Luz RT, Guimarfies VM, Rodrigues AC, Correia JD (2002b) Brazilian First Order Levelling Network. In: Drewes H et al. (eds) Vertical Reference Systems (IAG Symposia, vol. 124). Springer, Berlin, pp. 20-22. Sans6 F , Usai S (1995) Height datum and local geodetic datums in the theory of geodetic boundary value problems. Allgemeine Vermessungs-Nachrichten 102:343 355. Tenzer R, Vanicek P, Santo s M, Featherstone WE, Kuhn M (2005) The rigorous determination of orthometric heights. Journal of Geodesy 79: 82-92. Torge W (2001) Geodesy. 3rd edn., Walter de Gmyter, Berlin, 416 pp. Vanicek P, Krakiwsky EJ (1986) Geodesy." the Concepts. 2nd edn., Elsevier, Amsterdam, 697 pp.
Chapter 94
Accessing the New SIRGAS2000 Reference Frame through a Modernized Brazilian Active Control Network L.P.S. Fortes, S.M.A. Costa, M.A.A. Lima, J.A. Fazan Directorate of Geosciences Brazilian Institute of Geography and Statistics, Av. Brasil 15671, Rio de Janeiro, RJ, Brazil, 21241-051 M.C. Santos University of New Brunswick, Fredericton, Canada
Abstract. Since the beginning of its establishment, in December of 1996, the Brazilian Network for Continuous Monitoring of GPS - RBMC has been playing the role as the fundamental geodetic frame in the country, providing users with a direct connection to the Brazilian Geodetic System- SGB. This role has become more relevant with the adoption of the new geodetic system, SIRGAS2000, as of February 25, 2005. In this paper, the current RBMC status is presented, as well as the expansion and modernization plans for its structure, functionality and services to be provided to users. RBMC currently works in postmission mode, where users are able to freely download from the |nternet data collected by each of its 19 stations 24 hours after the observations are collected. The modernization plans specify, in a first step, the network expansion with six additional stations in the Amazon region, including the reactivation of Manaus station, and the connection of all stations to the lnternet, to support real time transfer of 1 Hz data to the control center, in Rio de Janeiro. When available at the control center, the data will support WADGPS (Wide Area Differential GPS) corrections to be transmitted, in real time, to users in Brazil and surrounding areas. This new service is under development based on a cooperation signed at the end of 2004 with the University of New Brunswick, supported by the Canadian International Development Agency and the Brazilian Cooperation Agency. It is estimated that users will be able to achieve a horizontal accuracy around 0.5 m (1-~) in static and kinematic positioning. The expected accuracy for dual frequency receiver users is even better. The availability of the WADGPS service- at no costwill allow users to tie to the new SIRGAS2000 system in a more rapid and transparent way in positioning and navigation applications, it should be emphasized that support to post-mission static
positioning will continue to be provided to users interested in higher accuracy levels.
Keywords. RBMC, real time, SIRGAS2000
1 Introduction The Brazilian Network for Continuous Monitoring of GPS - RBMC (Fortes et al., 1998; IBGE, 2005a) is an active geodetic network which constitutes the main geodetic framework of the country, providing users with the possibility of precise linking to the Brazilian Geodetic System- SGB. This role is even more relevant at the current moment, when the new geodetic system SIRGAS2000 (Drewes et al., 2005) has been officially adopted in Brazil as of February 25, 2005 (IBGE, 2005b), considering that this new system is mainly realized in the country throughout the RBMC stations. In this paper, the current RBMC status is presented, as well as the expansion and modernization plans for its structure, functionality and services provided to users.
2 Current RBMC Status The Brazilian Institute of Geography and Statistics IBGE started to establish RBMC at the end of 1996, when the Curitiba/PR and Presidente Prudente/SP stations were installed with the support of the National Fund for the Environment- FNMA and of the Politechnic School of the University of Silo Paulo -EPUSP. Nowadays, the network is composed of 19 continuous operating GPS stations (Fig. 1), distributed across the national territory, being automatically monitored and remotely controlled by the control center localized in Rio de Janeiro. Among these 19 stations, the ones in Brasilia and Fortaleza are part of the International GNSS Service - IGS global network (IGS, 2005a), whereas the remaining ones compose the IGS
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L.P.S. Fortes. S. M. A. Costa. M. A. A. Lima. J. A. Fazan • M. C. Santos
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Fig. 1 RBMC stations in operation, in test, and being established
densification network in South America and surroundings, whose data is processed on a weekly basis by the IGS Regional Network Associate Analysis Center for the continent- IGS RNAAC SIR (Seemueller and Drewes, 2004). These characteristics include the Brazilian geodetic reference framework in the global structures in a consistent way, which guarantees its continuous monitoring and update. Each RBMC station is equipped with a dual frequency GPS receiver and a chokering antenna. At the end of each 24 hour observing session, the collected data are automatically transferred to a local computer. Few minutes later, the computer server at the control center downloads data from the local computer to Rio de Janeiro, through a dial-up connection or using the Internet. After being transferred, data is checked and made freely
available at the site http://www.ibge.gov.br/home/ geociencias/geodesia/rbmc/rbmc.shtm in general within 24 hours after the observation date. During its almost ten years of operation, the network has been largely used by the national and international communities, as demonstrated by many projects and published papers carried out based on the RBMC. Currently around 3500 daily observation files are downloaded each month (Fig. 2). This high demand is related to the increasing use of GPS for positioning applications in general. Among these applications, the following can be listed: • •
Support to GPS relative positioning in general; Topographic and cadastral systematic mapping;
Chapter 94 •
Accessingthe New SIRGAS2000 Reference Frame through a Modernized Brazilian Active Control Network DOWNLOADS RBMC
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Fig. 2 Number of RBMC daily observation files downloaded from the Internet each month from December 2004 to April 2005
• • • • • • • •
Area delimitation (political-administrative, environmental, etc.); Land use (land reform, precise agriculture, etc.); Navigation (currently in post-mission mode); Ionosphere modeling; Support to climatology and meteorology; Fleet control and management; Integration into global geodetic networks; Regional and global Geodynamics.
3 RBMC Structure Expansion and Modernization Plans As it can be seen in Fig. 1, the RBMC interstation distances vary from over 200 km in the Southeast, to more than 1000 km, in the Amazon region, where due to regional characteristics the network is sparser. Increasing the density of network points becomes a necessity. In its current configuration, RBMC supports GPS positioning applications using long baselines, requiring longer observation sessions to generate adequate results, due to the spatial decorrelation of the positioning residual errors, especially those caused by the ionosphere. A denser RBMC will provide users with stations closer to their area of interest,
allowing them to achieve their desired accuracy faster. Five stations are expected to be installed in the Amazon, including the reactivation of the Manaus station (Fig. 1). The installation of these stations, as well as of the Beldm and Macapfi stations, is result of cooperation with SIVAM project (System for the Vigilance of the Amazon), in order to improve the coverage of the network in the region. It must be emphasized the understandings being established with the National Institute for Land R e f o r m - 1NCRA, towards integrating their Network of Community GPS Base S t a t i o n s RIBaC to RBMC. RIBaC was established to support land surveys directly or indirectly carried out by INCRA. It is currently composed of 31 continuously operating GPS stations distributed in the Brazilian territory. RIBaC stations are equipped with single frequency receivers, which significantly restricts the coverage area of each station, especially in Brazil, where error gradients caused by the ionosphere on GPS signals can easily reach values around tens of parts per million, (Fortes, 2002). The RIBaC stations must also have their coordinates tied to SGB through RBMC. Based on the afore mentioned understandings, around 30 to 40 last generation
655
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L.P.S. Fortes. S. M. A. Costa. M. A. A. Lima. J. A. Fazan • M. C. Santos
dual frequency GPS receivers are expected to be purchased in order to replace the majority of receivers currently used in both RBMC and RIBaC. This receiver replacement has the following objectives:
Fig. 3 presents the intended configuration for the resulting network, with the stations to be modernized being shown. The selection of these stations satisfied the following priorities: •
• •
•
To equip RIBaC stations with dual frequency receivers; To equip RBMC and RIBaC stations with receivers with good GPS signals tracking performance, especially with respect to L2, as Brazil is located under the Equatorial Anomaly where occurrence of scintillations is very common (Fortes, 2002); To equip both networks with receivers capable of real time operation, at 1 Hz, directly connected to the Internet, without the need of a local computer, in order to support real time applications described in next section.
In order to achieve the above objectives, the new receivers have to satisfy the following specifications" •
•
• • •
•
•
•
• • •
At least 12 L1 and 12 L2 channels to track carrier phase and C/A (L1) e P (El e L2) codes; Low carrier phase and code noises (few decimeters for codes and < 0.01 cycle for carrier phase); Chokering or equivalent antenna for multipath mitigation; Observation rates up to 1 Hz; L2-tracking technique with good performance under high ionospheric activities (e.g., semicodeless or equivalent); IP network port for connecting the receiver to LAN/Internet with no local computer interface; Possibility of remote controlling the receiver and real time transferring of observations through the Internet; Possibility of storing observations on the receiver memory at the same time as transferring them through the Intemet to the network Control Center; Enough memory to store 30 days of observations at 1 Hz; External oscillator port; Possibility of L2C signals tracking or upgradeability to that.
• •
Availability of local connection to the Internet, with good stability and quality (i.e., broad band); Existence of long time dual frequency data series collected at the station; Existence of stable monuments.
4 Plans for Modernization of RBMC's Functionality and Services to Users Until today, RBMC has provided support to applications that rely on post-processing data, mostly in relative mode. A modernization of RBMC's functionality is being proposed in order to increase the range of applications, most notably those which require real-time information. Those applications include navigation, either air, maritime or terrestrial. The new functionality planned for a modernized RBMC involves:
•
• • •
Real-time transmission of the data collected by each station to the Control Center in Rio de Janeiro; Reduction of the current 15-second observation interval to 1 second. Real-time computation of WADGPS-type corrections at the Control Center. Real-time availability of the corrections to users, at no via the Internet or satellite link.
The WADGPS corrections are those for the satellite and clock orbits and for the delay provoked by signal propagation through the ionosphere and troposphere. Since the RBMC stations have highly accurate coordinates the data collected can be used to quantify the actual errors and to predict the corrections to be transmitted to users.
This new service is being developed in cooperation with the University of New Brunswick under the National Geospatial Framework Project (PIGN), a technology transfer project sponsored by the Canadian International Development Agency (CIDA) with the support of the Brazilian Cooperation Agency (ABC). The PIGN (PIGN, 2005) has a main object to
Chapter 94 • Accessing the New SIRGAS2000 Reference Frame through a Modernized Brazilian Active Control Network
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Fig. 3 Active geodetic network resulting from the integration of RBMC and RIBaC. The stations represented as triangles will have real time capability, whereas the remaining ones will continue to work in post-mission mode
collaborate and assist in Brazilian efforts towards the adoption of a geocentric coordinate system (SIRGAS2000) compatible with modern satellite positioning technology. Project activities include technical issues, study on the impacts resulting from the adoption of the new system and communication with user community. The modernization of the RBMC corresponds to PIGN Demonstration Project#7. This Demo Project aims at providing the background for the implementation of a modern reference structure that facilitates the connection to the Brazilian geodetic system by users. Since the corrections will be implicitly attached to SIRGAS2000, their application by the users will result in SIRGAS2000 coordinates. Users will be directly attached to SIRGAS2000 in their positioning and
navigation applications. The participation of the Geodetic Survey Division of Natural Resources Canada is being discussed considering the expertise this institution holds from the development of the Canada-Wide DGPS ServiceCDGPS (CDGPS, 2005a). It is expected that users will be capable of performing (real-time) static and kinematic positioning at the 1 m 95% confidence level (0.5 m DRMS). For dual-frequency users, these figures drop to 0.3 m at 95% confidence level (less than 0.2 DRMS) (CDGPS, 2005b; Rho et al., 2005). The real-time functionality of RBMC will allow an even closer collaboration with the IGS
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L.P.S.Fortes. S. M. A. Costa. M. A. A. Lima. J. A. Fazan • M. C. Santos
within the IGS Real Time Working Group (IGS, 2005b). 5
Conclusions
The development of global navigation satellite system (GNSS) positioning technology has set up new standards to national geodetic infrastructure via active networks such as RBMC. RBMC has been the fundamental geodetic infrastructure in Brazil since its inception in 1996, providing accurate connection to the Brazilian Geodetic System, in post-processing mode. In an attempt to follow the technological evolution, IBGE is proposing the expansion and modernization of the geodetic infrastructure, functionality and services provided to users. With these purposes in mind, the densification of the network in the Amazon region is being planned with the establishment of five new stations. In addition, an on-going understanding with INCRA may result in the purchase of 30 to 40 new generation dualfrequency GPS receivers, to be used in RBMC and also in the network maintained by INCRA in support of land reform activities. The resulting infrastructure will provide capabilities for realtime services, by means of computation of WADGPS-type corrections and consequent transmission of these corrections to users in Brazil, and possibly in neighboring regions. The modernization of the RBMC is being carried out under the National Geospatial Framework Project, supported by CIDA.
The application of the WADGPS corrections will allow users to be attached to SIRGAS2000 in a direct and clear way in positioning and navigation applications. It is believed that users will be capable of real-time static and kinematic positioning with a 2D accuracy of 0.5 m (DRMS) or better, depending on the type of receiver used. The current post-processing service will still be offered, allowing users to reach the highest accuracy possible.
CONDERCompanhia de Desenvolvimento Urbano do Estado da Bahia EPUSP - Escola Polit6cnica da Universidade de Silo Paulo DSG - Diretoria do Servigo Geogrfifico do Ex6rcito, 4 a DE, Manaus F N M A - Fundo Nacional do Meio Ambiente I M E - Instituto Militar de Engenharia I N P E - instituto Nacional de Pesquisas Espaciai, Cuiabfi e Euzdbio Marinha do Brasil - Capitania dos Portos, Bom Jesus da Lapa Pr6 G u a i b a - Fundo Prd-Guaiba, Governo do Estado do Rio Grande do Sul SIVAM/SIPAM- Sistema de Vigilfincia/Proteg~o da Amaz6nia UFPE- Universidade Federal de Pernambuco UFPR- Universidade Federal do Paranfi UFRGS - Universidade Federal do Rio Grande do Sul UFSM- Universidade Federal de Santa Maria U F V - Universidade Federal de Vigosa UNESP - Universidade Estadual Paulista, Campus de Presidente Prudente U R C A - Fundag~o Universidade Regional do Cariri
7
In M e m o r i a m
This paper is dedicated to the memory of Eng. Kfitia Duarte Pereira. Kfitia was responsible for the operation of the RBMC since its establishment. Kfitia passed away, prematurely, in April 2005.
References
CDGPS (2005a). The Real-Time Canada-Wide DGPS Service. http ://www.cdgps.com/. CDGPS (2005b). CDGPS Features. http://www.cdgps. com/e/features.htm.
The RBMC network is a reality with the support of the following institutions:
Drewes, H.; K. Kaniuth; C. V61ksen; S.M.A. Costa; L.P.S. Fortes (2005). Results of the SIRGAS Campaign 2000 and Coordinates Variations with Respect to the 1995 South American Geocentric Reference Frame. In A Window on the Future of Geodesy (ed) F. Sans6, International Association of Geodesy Symposia, Vol. 128, pp. 32-37.
CEFET/UNEDI - Centro Federal de Educagfio Tecnoldgica, imperatriz C E M I G - Companhia Energdtica de Minas Gerais
Fortes, L.P.S. (2002). Optimising the Use of GPS MultiReference Stations for Kinematic Positioning. PhD Thesis, UCGE Report Number 20158, The University of Calgary, Calgary, 323 p.
6 Acknowledgments
Chapter 94 • Accessing the New SIRGAS2000 Reference Frame through a Modernized Brazilian Active Control Network
Fortes, L.P.S.; R.T. Luz; K.D. Pereira; S.M.A. Costa; D. Blitzkow (1998). The Brazilian Network for Continuous Monitoring of GPS (RBMC): Operation and Products. In Advances in Positioning and Reference Frames (ed) F.K. Brunner, International Association of Geodesy Symposia, Vol. 118, pp. 73-78. IBGE (2005a). Rede Brasileira de Monitoramento Continuo. http ://www.ibge.gov.br/home/geociencias/ geodesia/rbmc/rbmc, shtm. IBGE (2005b). Projeto Mudanga do Referencial Geod6sico. http ://www.ibge.gov.br/home/geociencias/ noticia_sirgas.shtm. IGS (2005a). Network. html.
International GNSS Service Tracking http ://igscb.jpl.nasa.gov/network/netindex.
IGS (2005b). IGS Real Time Working Group. http ://igscb .jpl.nas a. gov/proj ects/rtwg/index.htm 1. INCRA (2005). Rede INCRA de Bases Comunitfirias do GPS. http ://www.incra.gov.br/_htm/serveinf/_htm/_asp/ estacoes_dcn/default.asp. PIGN (2005). National Geospatial Framework Project. http ://www.pign.org/. Rho, H.; R. Langley; A. Kassam (2003). The Canada-Wide Differential GPS Service: Initial Performance, in Proceedings of the 16th International Technical Meeting of the Satellite Division of the Institute of Navigation ION GPS/GNSS 2003. Portland, Oregon, pp. 425-436. Seemueller, W.; H. Drewes (2004). Annual Report 2002 of IGS RNAAC SIR. In IGS 2001-2002 Technical Reports (eds) K. Gowey, R. Neilan e A. Moore, IGS Central Bureau, Pasadena, California, pp. 129-131.
659
Chapter 95
Deformations Control for the Chilean Part of the S IRGAS 2000 Frame J. C.
B f i e z 1'2 ,
S. R. C. de Freitas ~, H. Drewes 3, R. Dalazoana ~R. T.
L u z 1'4
1 Geodetic Sciences Graduation Course (CPGCG), Federal University of Paranfi (UFPR), Curitiba, Brazil. 2 Department of Surveying, Universidad de Concepcidn (UdeC), Chile. 3 Deutsches Geod~itisches Forschungsinstitut, Manchen, Germany. 4 Brazilian Institute of Geography and Statistics (IBGE), Coordination of Geodesy, Brazil. With the adoption of the Geocentric Reference System for the Americas (SIRGAS), the determination of temporal variations and deformations in its frame are a necessary task. These tasks are fundamental in order to maintain the consistency of the frame with respect to the definition of SIRGAS, especially in the Chilean deformation area, where the variations are larger compared with the rest of the plate due to the convergence between the South American and Nazca plates. 21 permanent GPS stations was used.14 of them are regional stations of the International GNSS Service (IGS), and 7 permanent GPS stations are from different regional networks. 10 of these 21 stations are located in the Chilean Andes were used to study the kinematic effects in this area. Some of them are also SIRGAS stations. Five years of GPS observations from 2000 to 2004 were selected in order to have a sufficient time period with enough coverage for the considered stations. Daily solutions were generated with the Bernese GPS Software 5.0 using IGS precise ephemeris, clocks and ERP. The normal equations were then accumulated and solved in a final combination where the coordinates and velocities of selected stations are weighted for the datum definition compatible with ITRF2000. The estimated velocities of the north Chilean Andes regions are -2630 + 2mm/yr, decreasing up to -16-25 + 2mm/yr and for the south Chilean Andes the estimated velocities are-9-12 + 2ram/yr. For the stable part of the continent the estimated velocities are -8-12 + 2ram/yr. The final results are discussed and compared with the NUVEL-1A and SIRGAS velocity models (SIRGAS-VM).
ties to observe and measure the traditional triangulation network for the entire continent, and to define a unique Geodetic Reference System (GRS). Space geodetic techniques like GNSS are suitable to develop a GRS and its realization. The SIRGAS project is the most important effort to develop an unique GRS, and it was realized in two extensive campaigns during 1995 and 2000, the later one performing the SIRGAS 2000 realization (IBGE, 2002). One of the necessary tasks is the observation of kinematic effects in the frame due to the continental drift which is affecting the position of the stations. One special case is the Andes region which is affected by the subduction area of the Nazca under the South American plate, and produces large movements and deformations of the continental crust. Several plate models have been derived based on evidences from geophysical studies; one of them is the NUVEL-1A (DeMets etal., 1994), which explains the kinematic effects for the past million of years, and is mostly sufficient for the stable part of the crust. Geodetic models were developed to explain the recent kinematic effects, like the Actual Plate Kinematic Model (APKIM 2002) (Drewes, 2003). Nevertheless, the geophysical and geodetic models are not sufficient to completely explain the kinematic effects in the deformation areas. Continuous GPS observations from 21 stations, 14 of them of the International GNSS Service (IGS), are used in selected periods to estimate velocities for the Andes region, and to study the kinematic effects and its correlation with the deformations of the Chilean part of SIRGAS 2000 frame.
Keywords. effects.
2 Chilean part of SIRGAS 2000 frame
Abstract.
GRS, velocities estimation, kinematic
1 Introduction Almost one century after the first realizations of geodetic reference networks in South America, it was necessary to improve them, due to the difficul-
After the SIRGAS 2000 realization, the lnstituto Geogrfifico Militar (IGM) of Chile, responsible for the realization and maintenance of the Chilean Geodetic Reference Frame (GRF), decided to increase the number of 20 original stations included in the SIRGAS 2000 realization (IBGE, 2002), by more than 300 stations, including 10 permanent
Chapter 95 • Deformations Control for the Chilean Part of the SIRGAS 2000 Frame
GPS stations. With this frame it was possible to adopt the SIRGAS2000 as the official Geodetic Reference System (GRS), and its frame was realized for all stations. Figure 1 shows the stations in the Chilean SIRGAS realization (small circles) and the 10 permanent GPS stations (triangles). The IGS and permanently observing GPS stations are serving as fiducials for the final coordinate and velocity estimation of the whole Chilean Network. Considering the deformations of the Andes regions, the IGM of Chile also decided to adopt the SIRGAS 2000 using a mean epoch of 2002.0 which differs from the 2000.4 adopted for the SIRGAS 2000 final solution. The network densification is based on the original 20 SIRGAS stations. In that way, the error propagation is small, because some of the stations used during the campaign are monitored, and its velocities are well known. But, the remaining station, velocities must be estimated from a model with uncertainties, because in deformation areas simple geophysical models fail and geodetic models do not represent the true velocities.
can Geodynamic Activities Project (SAGA). To stabilize the geometry, some other stations were included: 14 IGS stations, 4 Argentinean Geodetic Position stations (POSGAR), and 2 Brazilian Network of Continuous Monitoring (RBMC) stations. Data of ten days were selected from the epochs 2000.4, 2000.9, 2001.4, 2001.8, 2002.4, 2002.7, 2003.3, 2003.8 and 2004.2. Data files contain 24 hours of continuous observations with a sampling interval of 30 s. The stations used are shown in Figure 2. IGS products (precise ephemeris, EOPs, and clocks) were used to process the daily solutions. Carrier phase double differences were formed using the ionospheric-delay free observable. The troposphere was modeled using a combination of Saastamoinen zenith path delay and Niell mapping function. A tropospheric parameter was estimated every two hours. Daily ionospheric maps were used during the ambiguities fixing solutions.
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3 Data collection and processing The network used in this experiment was originally formed by 8 stations from the permanent GPS network of Chile and 2 stations from the South Ameri-
Fig. 2. Permanent GPS stations included in the processing. The GPS data processing was carried out with Bernese GPS Software (Hugentobler et al., 2004). Baselines were created using the maximum observations strategy. During the data processing the observations were referred to the antenna phase centre using the National Geodetic Survey (NGS) GPS antenna calibrations and variations. Daily solutions were generated in a free network and saved in normal equation files.
661
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J.C. B&ez. S. R. C. de Freitas • H. Drewes • R. Dalazoana • R. T. Luz
For the final solution, all the normal equations were combined in a least squares adjustment, including the ITRF2000 coordinates of five IGS stations (SANT, LPGS, RIOG, PARA, CORD) and velocities from the Regional Network Associate Analysis Centre for SIRGAS (RNAAC-SIR), and its precisions to estimate coordinates and velocities compatible with the IRTF2000 frame. Table 1 summarizes the stations velocities and its standard deviations.
stations coordinates of the individual daily solutions (Xi), and the modeled coordinates (X0) epoch (to) and velocities V At=(ti-t0) (1)
4 Velocity comparison
As shown in Figure 3, the estimated velocities (grey vectors), are consistent with those of the NNR-NUVEL-1A model (black vectors) for the stable part of the South American plate (stations: RIOG, PARC, PARA, UEPP, LPGS, VBCA, IGM0, RWSN).
Considering all lithospheric plate motions, a kinematic reference frame must be introduced for its temporal variations (e.g., Larson et al., 1997). The geophysical plate models like NNR-NUVEL-1A provide quit good velocities for the stable part of the continent, but inconsistent, however, in the deformation areas. The estimated solutions (gray vectors) and NNR-NUVEL-1A (black vectors) are compared in Figure 3.
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Table 1. Estimated Velocities. Station
North
East
(mm/yr)+o
(mm/yr) +o
ANTC
7.7+0.5
15.9+0.5
ANTO
15.9+0.9
30.0+ 1.1
CFAG
8.7+1.3
6.4+1.6
CONZ
15.8+1.0
25.7+1.0
COPO
16.6+ 1.1
20.1+0.9
CORD
9.1+0.7
1.0+0.9
COYQ
8.7+0.5
-3.9+0.6
IGM0
8.2+ 1.7
-1.2+0.7
IQQE
10.1+1.3
25.4+1.6
LPGS
9.6+0.6
-1.8+0.5
PARA
8.5+0.6
-4.6+0.7
PARC
11.2+0.5
3.6+0.7
PMON
11.7+0.8
0.0+0.6
RIOG
11.0+0.3
4.0+0.3
RWSN
9.5+0.9
-2.4+0.5 21.1 +0.8
SANT
13.8+0.5
TUCU
6.6+0.5
1.6+0.6
UEPP
10.1+0.5
-3.4+0.4
UNSA
10.9+1.2
5.7+1.6
VALP
20.6+ 1.1
24.3+ 1.2
VBCA
8.9+0.6
-0.8+0.8
Table 1 present the estimated velocities and their standard deviations (~). The latter ones were computed from the differences between the estimated
,/,
..
,' I ~r'/R~""Ae°"° (, .t' ~-i.
", , ,,
; ., t ,i" ;'
"~,
;
J..,,/
1
VBCA ~
~
-
- ........ '
.....
-40"
-40
.
~.
¢.,.-
~ 1
•- , . . _
~,~ ....
280"
cm,,a NNR-NUVEI.-1A
~.
300"
Fig. 3. Comparison between estimated and NNR-NUVEL1A velocities.
The SIRGAS-VM model (Drewes and Heidbach, 2005) was developed using data from SIRGAS campaign (1995 and 2000), and point velocities from different geodynamic projects. The SIRGAS executive recommended using this model for the South American plate. The estimated solutions for the stable part of the plate (UEPP, PARA, IGM0, LPGS, VBCA, RWSN) are in agreement with the SIRGAS-VM
Chapter95 • DeformationsControl for the Chilean Part of the SIRGAS2000 Frame
model with RMS value of N=2.8mm and E=I.0 mm. For the Andes regions the differences are larger with RMS N=3.8mm and E=3.9 mm, and the variations in three stations located in the northern part (IQQE, ANTO, COPO) are rather large, due to the seismic activities. For the central part (SANT, VALP), the velocities are in complete agreement with the SIRGAS model. For the southern part (PARC, RIOG), the comparison with the SIRGASVM model was not possible due to the limitation of the model, which is developed to -43 ° of latitude only. Figure 4 shows the SIRGAS-VM model and Table 2 summarizes the velocity differences between the two models, (SIRGAS-VM and NNRNUVEL-1A), and the estimated velocities in this research paper. The station AREQ was not used in this investigation, because two earthquakes happened in 2001 affecting the position of the station (Kaniuth et al., 2002). More data are necessary to estimate reliable velocity of this station. The same is valid for the station IQQE (earthquake June 2005), therefore data from this period were not included in this research. The comparison of estimated solution and the SIRGAS-VM model is shown in Figure 5.
280"
30{)"
Table 2. Tabulated differences between velocities of NNRNUVEL-1A and SIRGAS-VM model, and that from this study.
Station
Differences Esti.-NUVEL-1A North East (mm/yr) (mm/yr)
Differences Esti.-SIRGAS-VM North East /mm/yr) /mm/yr)
ANTC
-1.6
16.0
-4.4
ANTO
6.4
32.5
-1.3
-2.7 8.7
CFAG
-1.0
7.9
-3.9
-2.2
CONZ
6.7
25.7
-5.2
-5.3
COPO
7.1
22.1
-0.6
-1.2
CORD
-1.0
2.9
-2.5
0.2
COYQ
-0.6
-5.6
-4.2
-5.5
IGM0
-2.5
1.0
-3.4
-0.3
IQQE
0.6
28.4
-6.5
4.9
LPGS
-1.1
0.5
-2.0
-0.3
PARA
-2.8
-0.5
-3.8
-1.2
PARC
1.8
-0.2
PMON
2.5
-0.9
-3.7
-2.3 -1.1
RIOG
1.2
0.7
RWSN
-0.6
-2.5
-2.6
SANT
4.4
22.1
-1.1
1.2
TUCU
-3.5
4.1
-5.3
-1.8
UEPP
-1.1
0.7
-2.5
-0.3
UNSA
0.9
8.4
-1.7
-0.6
VALP
12.3
25.2
1.0
-0.0
VBCA
-1.4
0.3
-2.2
-2.1
4.1
14.8
3.5
3.2
,~, ---,.,-,- ..,- --~ •, ~ _204
rms
.-~,,
-20'
,~ill
5 Final remarks
/
x
t
-40"
r
-40"
_Jii
& IGS stations POSGAR stations • RBMC stations , SAGA stations
~&f.{-f"T '--... _,
•;
•~ 2
<..- .-::
c m,'yr
~.J :-
%"L
.
280'
%
., .'.,.,..
30,0'
Fig. 4. SIRGAS-VM model (adapted from Drewes and
Heidbach, 2004).
The adoption of the SIRGAS 2000 realization in Chile, together with the installation of a permanent GPS network, allow the monitoring of kinematic effects in the Andean deformation areas which is an important task to maintain the reference frame. This paper presents results of the estimation of velocities from 10 continuous GPS stations in the Chilean Andes region and 11 stations on the stable part of the South American plate. The estimated velocities in this research show larger differences for the stations located in the Andes regions (IQQE, ANTO, COPO, ANTC, CONZ, PMON, COYQ) with respect to the SIRGAS velocity model. The reason is that observation data was not available in time to include them in the SIRGAS model. The velocities for the stations located in the stable part of the plate (PARA, UEPP, LPGS, IGM0, VBCA, RWSN, CORD, TUCU, UNSA) are -2-3 mm/yr smaller than in the SIRGAS model. The estimated velocities of SANT and VALP are in
663
664
J.C. B&ez. S. R. C. de Freitas • H. Drewes • R. Dalazoana • R. T. Luz
agreement with respect to the SIRGAS estimations with differences not larger than ~1 mm/yr. The estimated velocities of this investigation, especially for the Andes region, demonstrate larger differences with respect to the SIRGAS-VM, and therefore they can be used to upgrade them. 280"
300"
',,~",,~'., ......
-
.
t _~.
,'
: ~'~ .... ]-..__2__'~.
-20"
f
-20'
I
Acknowledgements The authors acknowledge the Brazilian agencies for research and education (CNPq and CAPES), and the German "Deutscher Akademischer Austausch Dienst" (DAAD) for financial support, the Instituto Geogrfifico Militar of Chile and the Geoforschungszentrum (GFZ) Potsdam for the data made available; and the TIGO station from Concepci6n.
References
L~-,t
st
~TUCU.
~
PA.~e~ll
vA!,,~,r0m
"
~ p ~ - ....
[, " ~
VBC/IMz~ . . . . . . . . . . ;"/
-40'
-40"
.~
fl~-~ ~
L,~ ~L. .'~
:~ a.=,
"-
280"
A IGS s t a t i o n s
" OY(~
POSGAR
/-'~
$ .-" - ..'~ ..,
*~.
stations
• RBMC $lallon~ • SAGA slallons ~ 1 cm,'a
"4M~IO~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
300 +
Fig. 5. Differences between the SIRGAS-VM and the estimated velocity of this study.
Drewes H., Heidbach O. (2005). Deformation of the South American Crust Estimated from Finite Element and Collocation Methods. IA G Symposia, Vol. 128, 544-549. Drewes, H, Meisel, B, (2003). An Actual Plate Motion and Deformation Model as a Kinematic Terrestrial Reference System, Geotechnologien Science Report, no 3. DeMets C, Gordon RG, Argus DF, Stein S (1994) Effect of recent revisions to the geomagnetic reversal time scale on estimates of current plate motions. Geophys. Res. Lett. 21(20): 2191-2194. Hugentobler, U., R. Dach, and P. Fridez (2004). Bernese GPS Software. Version 5.0, University of Bern. IBGE. Instituto Brasileiro de Geografia e Estatistica (2002). Sistema de Refer~ncia Geoc~ntrico para a Am6rica do Sul - SIRGAS. Boletim lnformativo No. 7. Rio de Janeiro, l llp. Located at . Kaniuth K., Mtiller H., Seemtiller W. (2002) Displacement of the space geodetic observatory Arequipa due to recent earthquakes. Zeitsch. Fiir Verm. 127 (4): 238-243. Larson KM, Freymuller JT, Philipsen S (1997) Global plate velocities from global positioning system. J. Geophy. Res., 102, (B5): 9961-9981.
Chapter 96
Estimation of Horizontal M o v e m e n t Function for Geodetic- or M a p p i n g - O r i e n t e d Maintenance in the Taiwan Area C. C. Chang Department of Information Management, Yuda University, Miaoli 361, Taiwan, ROC H. C. Huang The 401 st Factory, Armaments Bureau, Taichung 402, Taiwan, ROC
Abstract. The concept of dynamic datum, in which the coordinates of sites change as a function of time, was relied on to estimate the regional land movement functions for any geodetic or mapping applications. The test area was selected for a deformation active island, Taiwan, and categorized into several regions depending on the similar pattern of the horizontal land movement. The velocity field computed for the Taiwan area was based on the coordinates of first-order control stations measured in 1997 and 2000, both using high precision GPS. The regional movement functions derived from the velocity information can then be estimated with several mathematical models, e.g. averaging, weighted mean and surface fitting. Furthermore, those regional functions were practically applied to estimate and examine the horizontal coordinate movement in the test regions. It is further expected to utilize those regional functions to assess the coordinate qualities of the control stations over a period of time, in order to determine an applicable duration for the geodetic maintenance. It is realized from the internal errors that the horizontal movement velocity obtained using the regional functions based on a second-degree surface fitting model are tested to be better than 0.4 cm/yr. When these regional functions were tested with another independent set of GPS-based coordinates measured in 2002, the external estimation errors of those regional functions were assessed to be approximately 1.7 cm/yr in all test regions. It is, hence, suggested that the coordinate accuracy of the first-order GPS control stations in the Taiwan area can only be maintained for no more than 4 years. In addition, the revision of 1/1,000 digital maps is estimated to be probably required for every 10 years when regional land movement, instead of the topographic changes, is taken into account.
Keywords. Velocity regional function
field,
land
movement,
1 Introduction For the establishment of the geodetic networks, the Global Positioning System (GPS) is particularly important for its great performances on the working scale from local, regional, to global. In addition, the GPS receivers are relatively easy to operate so that the geodetic observations using GPS can be effectively implemented. Most importantly, millimeter to centimeter level positioning accuracy has been widely demonstrated over the baseline lengths from tens to thousands of kilometers. With this space geodetic technique, it has been proved to be feasible to establish a regional or continental scale of reference system by GPS observations. It is generally believed that once a GPS-based reference system is established in a crustal deformation or tectonic plate motion area, such as Taiwan, a procedure linked to its regular maintenance must be implemented. Its purpose is to ensure the position accuracy of GPS control station that might be degraded by any natural hazards or effects, e.g. earthquakes. If regular maintenance is not properly carried out for those GPS sites, the geocentric reference system based on this fundamental GPS network will be distorted or may even be no longer reliable for any geodetic or mapping applications. Additionally, if the operation of geodetic maintenance is frequently made for those GPS sites, it would jeopardize the public administration works for land planning and management, as the coordinates of the control stations are changed too often. Therefore, the land movement in the area of interest is investigated, in order to determine a suitable frequency for
666
C.C. Chang. H. C. Huang
maintaining the fundamental GPS network to ensure accurate positions. The geocentric reference system of Taiwan Geodetic Datum 1997 (TWD97), whose three-dimensional coordinates were established using GPS and connected to sixty-four IGS (International GNSS Service) sites and referred to the ITRF94 (International Terrestrial Reference Frame 1994), has acted as the geodetic datum in the Taiwan area since 1997 (Chang and Tseng (1999)). In addition, one other data set measured at the TWD97 first-order control stations was provided in 2000, as a part of government re-construction work following the 1999 Taiwan earthquake (Mw=7.6). It is possible to realize the time evolution of this reference system and to study its maintenance period through the estimation of the regional land movement functions for the Taiwan area.
2 Regional Movement Function A dynamic mode of geodetic datum is believed to be capable of describing the time evolution of land movement for an active deforming area (Blick et al. (2003)). Based on this concept, it can offer users the spatial and temporal information for three-dimensional coordinates with time variants for the geodetic control stations. For its definition, three components are required to represent such a dynamic datum: a set of geodetic observations for the network of sites; a rigorous adjustment of the geodetic observations; and a procedure for modeling a time varying function of the site coordinates (Tregoning and Jackson (1999)). To investigate this four-dimensional type of datum, the velocity field calculated from the geodetic observations must be accurately determined. In such cases, it may be more appropriate to adopt discrete site velocities estimated from GPS measurements made at well-distributed control stations. It is also noted that when any regional movement function is available to well represent the trend of horizontal movement within an area, it can allow users to compute site displacement at any epoch by simply inputting the site's geographic location and time span. The information provided by the regional movement function would be helpful to easily judge how long the coordinate accuracy can be maintained and what duration the geodetic measurements need to be re-observed. Since a variety of mathematical algorithms has
been suggested by some surveying applications (Wielgosz, et al. (2003))(Yanalak and Baykal (2001)), four basic models, namely the averaging, weighted mean, first-degree and second-degree polynomial surface fitting, are applied and tested for their effectiveness. For those models, the most simplified form can be based on the algorithm of averaging, i.e. ~=~ +~++C n
(1)
where V is an average value of velocity field in the region; Vi is the site velocity at the i-th data point; and n is the total number of data point in the region. It is noted that the main advantage of this model is easy to be estimated and utilized, if a common pattern of the land movement is existed in the investigated area. However, it will be a location-independent value for all sites within the region. The model of weighted mean can also be designed, based on the velocities known at the nearby data points, to obtain the site velocity at the computed point by means of an interpolation with the distance weight, i.e.
£ P iVi V- s
=
i- 1
P; i
(2)
1
where Vs is the estimated velocity at the interpolation point; Vi is the velocity of the i-th data point; n is the total number of data point within the region; and Pi is the weight which can be simply given by Pi-1/di with the distance (di)between the data point and the interpolation point. This model, also named as distance weighting, has the advantages: (1) A regional pattern of the land movement can be presented since a limited range of data points has been applied to estimate the velocity information for the region; (2) The number of the data point in the velocity database can be flexibly adopted for the computations. Moreover, it is able to apply a surface fitting model to establish a velocity-based regional function to estimate land movement in the area. Two basic types of surface fitting model are proposed for the study. (1) First-degree Polynomial (Plane Fitting) It is able to utilize a plane type of model to approximate the land movement for a small or flat area. It can be assumed that a relationship, as
Chapter 96 • Estimation of Horizontal Movement Function for Geodetic- or Mapping-Oriented Maintenance in the Taiwan Area
listed below, is existed between the velocity component (Vi) and the plane coordinate of data point (xi, Yi), where a0, al, a2 are model parameters: Vi = a o +
(3)
alxi + a 2 Y i
(2) Second-degree Polynomial (Surface Fitting) A curve type of model, based on a second-degree polynomial, is also available to estimate the land movement at expected site when those data points' velocities are well fitted. As listed below, six parameters (a0, al...as) need to be defined by: 2 Vi -
a o +
trends might be too variable for a uniform model to be determined. For this reason this study attempts to simplify the estimation of the horizontal displacements using regional functions.
2
a l x i + a 2 Y i -I- a 3 x i + a 4 Y i + a s x i Y i
lWr.~97 ;'J(,5.'",.hire,:5tm'0ns:.
,
f
,
.:?r-g:n,~.[]v.',,1easrred Quake.a~i,~r.',.[e.:.~,d..'o:l at 1997i~,~ a[2,~,:4,~
Re-surveyed a: %~",x~
I
(4)
Re~.'.,,nal\!,,vement .Func]:~,~nEstlmal.lcn
3 E s t i m a t i o n of H o r i z o n t a l M o v e m e n t 3.1 Zoning Design The data used to test models is mainly composed of the high accuracy 3-D coordinates measured by the government at a total of 105 first-order GPS control stations well distributed in the Taiwan area. The network adjusted coordinates were obtained from three independent GPS campaigns at epochs of 1997.0, 2000.4 and 2 0 0 2 . 0 . The coordinate differences in horizontal components, based on the comparisons between any two consecutive data sets, were then applied to either estimate the function parameters or test the estimation errors for the horizontal displacement (see Fig. 1). Due to the fact that a part of the control network was found to have significant displacement caused by the damaging earthquake that occurred in central Taiwan in September 1999 (Chang (2000)), this area was not included in the analysis. A total of five zones were categorized by the same geological regions and the similar trends of horizontal displacement (see Fig. 2). The first set of station displacement obtained for the period 1997.0 to 2000.4 at the selected sites are shown in Fig. 3. It can be clearly seen in Fig. 3 that most of sites in Northern Taiwan, i.e. Zone I and I I , shifted in a southeastern direction, whereas the sites to the South and Southeast appeared to have a more complicated displacement pattern. It does, however, indicate a similar trend of displacement stowards the southeast and southwest in Zone III and IV, respectively. It is also possible to see northwestern displacements in Zone V, although it is inconsistent in rate as the sites are located within a plate collision area. It can be also seen that if site velocities are individually investigated, horizontal displacement
L
~t,'rnal and E.',:icmal Emir E.~1~malion
•
Apl:,l:cab]¢Du:a::on Estinr~tl.:n
Fig. 1 Data Sets Applied and Analyzed
11~1'
~;~
I~1
~;v.~.
"a.J '"
I
I
..
E~
~4
/-3
22
119
~21~
121
~'i'
Fig. 2 Zoning Design for Five Test Regions (Excluding Earthquake Area in Central Taiwan)
667
668
C.C. Chang. H. C. Huang
Table 1. Internal Errors of the Regional Functions (unit: cm/yr) 2~
I
"
..
;--J
76
Zone
Averaging Weighted Mean
Plane Fitting
Surface Fitting
0.24 0.35 0.24 0.44 0.59 0.56 1.21 0.53 0.79 0.47 0.35 0.44 0.44 1.06 1.03 0.97 1.09 1.00
0.24 0.26 0.24 0.38 0.29 0.44 0.88 0.29 0.68 0.06 0.24 0.09 0.18 0.65 0.65 0.35 0.35 0.41
..~.. :
~s •
"~,--_# '~.-~ ~ r ~ .~<~~.
24
2s
24
Ill
IV -"
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23"
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23"
~'~ 22"
..,,,,,~,~,,,, ,..~-~
•
ll~/.lf-
119"
E
V
.,L.,
22'
E N L E N L E N L E N L
N L E Avg N L
0.53 0.41 0.47 0.68 1.38 1.00 2.50 0.65 1.09 0.62 2.09 0.68 1.47 2.26 2.56 1.15 1.35 1.15
0.35 0.38 0.35 0.56 1.44 0.94 2.15 0.59 1.03 0.59 1.68 0.68 1.53 2.24 2.53 1.03 1.26 1.09
r.
12Q
12~"
122"
Fig. 3 Horizontal Displacement (1997.0-2000.4) 3.2
Estimation
of Model Errors
Based on the data, the horizontal land displacements over a time span of 3.40 years, as shown in Fig. 3, can be used to estimate the site velocities for every data point. The four models were then tested using the data from each zone, i.e. a total of 21, 9, 20, 9 and 7 points in Zone I-V respectively. The internal errors of the regional functions established were further assessed by comparing data points' measured and estimated velocities. The RMS errors of the regional function estimates are listed in Table 1, in which the E, N and L stand for the horizontal component in easting, northing and x/E2+ N2 respectively. It can be found in Table 1 that the averaging, weighted mean and first-degree surface fitting models are not satisfactory to provide the best velocity estimates due to their larger RMS errors. On the whole, the surface fitting model using the second-degree polynomial shows the smallest error of the displacement estimate in all test regions, in which an average of 0.4 cm/yr is obtained. However, it is also seen in Zone Ill that a larger error of around 0.9 cm/yr still occurred on the E-component. ,
Furthermore, to check with the external errors of the regional functions, one other independent GPS data set at epoch of 2002.0 was tested. The site velocities were computed using two sets of coordinate over a time span of 5.0 years, i.e. from 1997.0 to 2002.0. The site velocities were then also derived using the four regional functions, and compared with those measured to determine estimation errors. As those measured velocities are based on a period of 5 years, an extension of 1.6 years from the fitting function data, this set of external errors, as shown in Table 2, indicates the capability of using the regional functions to represent the land displacement to be occurred. It can be seen in Table 2 that the second-degree polynomials once again provide the best fitted velocities for the points within the tests regions, although the differences between the model estimates are less significant. In addition, it is found that the external errors are generally larger than those of the internal, e.g. the smallest error of the function estimate is increased from 0.4 cm/yr to 1.7 cm/yr, due to the time span extension. For the increasing errors more significantly occurred in Zone II, it might show an irregular movement trend happened in this area.
Chapter
96
• Estimation
of
Horizontal Movement Function for Geodetic- or Mapping-Oriented Maintenance in the Taiwan Area
Table 2. External Errors of the Regional Functions (unit: cm/yr)
Zone E N L E N L E III N L E IV N L E V N L E Avg N L
Averaging Weighted Mean 0.96 0.86 1.14 2.00 2.50 2.84 3.14 1.48 1.52 1.42 2.58 1.58 2.38 2.66 3.22 1.98 2.02 2.06
0.96 0.86 1.14 1.92 2.52 2.76 2.72 1.46 1.62 1.30 2.14 1.62 2.52 2.58 3.32 1.88 1.92 2.10
Plane Fitting
Surface Fitting
1.02 0.86 1.20 1.80 2.16 2.56 1.74 1.36 1.56 1.34 1.30 1.60 1.54 0.90 1.74 1.50 1.32 1.74
1.02 0.80 1.20 1.82 2.06 2.58 1.36 1.26 1.50 1.26 1.26 1.64 1.44 0.72 1.50 1.38 1.22 1.68
4 years in the test regions are also derived using the best fitting model and shown in Fig. 5. Using information provided by Fig. 4 and Fig. 5, the two boundaries of the applicable duration can be estimated and listed in Table 3, along with the annual velocity, for the test regions in Taiwan. =,,Ill . I ~
_-.
L4
,-
~,
12
,
~
~
"t=lllt ~dd="
_1.1
o []
g 2, @
I
2
3
4
5
Fig. 4 Movement Allowances for the Control Stations
4 Applicable Durations " 0 ~.~.'d I • ,~,'~!
4.1 Datum Maintenance
Based on the regional functions fitted to express the movement velocities in the test regions, the estimation of an applicable duration for the re-survey of the geodetic control stations has become possible. Using the elements of the variance-covariance matrix constructed by the network adjustment, the original coordinate precision, i.e. 1~, in TWD97 can be realized to be 0.3 cm and 0.6 cm for the latitudinal and longitudinal components, respectively. When a 95% confidence circle describing the horizontal accuracy of points was introduced, its radius can serve as a boundary value to judge if any significant site movements occurred (Geomatics Canada (1996)). It has been calculated for the first-order GPS control stations in TWD97 that a tolerance value of 3.4 cm in horizontal component was requested to meet the original point accuracy. Moreover, due to the existence of the estimation errors in the regional functions, the uncertainties have to be taken into account. When the best fitting model was applied to estimate the applicable duration for the first-order control stations, the movement allowances were designed to be between 3.4 cm + 0.41 cm/yr and 3.4 cm + 1.68 cm/yr for the inclusion of two types of estimation errors (see Fig. 4). The regional movement accumulated over
' []
.~'-~" :5 [] .~..i," " "
"q.'l • ' '~-" '-
~¢,
:
~,
.
;
II
Ill
,'.
".÷"
- 0"~
Fig. 5 Regional Movement Accumulated for Four Years Table 3. Applicable Durations First-order GPS Control Stations
Zone I II III IV V
Velocity (cm/yr) 3.5 4.6 2.7 2.8 3.2
Estimated
for
Duration (yr) 2 1-2 2-4 2-4 2-3
It can be generally seen in Table 3 that the first-order horizontal coordinate accuracy can be confirmed for less than 4 years in Taiwan. This is,
669
670
C.C. Chang. H.C. Huang
of course, highly related to the significant site movement with an average velocity of more than 3 cm/yr. It is, hence, suggested to set up a maintenance procedure to ensure high accuracy of geodetic applications in this area
4.2 Map Revision Similarly, the regional function based on the second-degree surface fitting model can also be applied to estimate the duration for map revision. According to the nation's mapping standard, any plane error over 0.2 mm shown on the maps is not allowed for all scales of mapping. In other words, a regional movement of 20 cm was realized to be the error allowance for digital mapping on a scale of 1/1,000. If the internal and external estimation errors were also counted, the allowances between 20 cm + 0.41 cm/yr and 20 cm + 1.68 cm/yr can be set up. Based on such information, the applicable durations for 1/1,000 digital maps are estimated and listed in Table 4 for the test regions in Taiwan.
Table 4. Applicable 1/1,000 Digital Maps Zone I II II! IV V
Durations
Velocity (cm/yr) 3.5 4.6 2.7 2.8 3.2
Estimated
for
Duration (yr) 7-11 5-7 9-27 9-24 8-14
When the most detailed topographic maps currently utilized in Taiwan were estimated for their applicable duration, it is known from Table 4 that the map revision cycles vary widely from 5 years in Zone II to probably 27 years in Zone III. If a lower estimation error, i.e. 20 cm + 0.41 cm/yr, is adopted, a reduced cycle of approximately 10 years is still necessary. However, it must be emphasized that such long cycle of map revision is only based on the horizontal land movement. When any topographic or geomorphic changes are included, the map revision cycle must be accordingly shortened to carry on the cartographic work.
5 Conclusions and Suggestions Two sets of high precision GPS coordinate data measured in 1997.00 and 2000.40 were adopted to
compute the site movement velocity for a part of the first-order control stations in Taiwan. The regional functions were then fitted using four mathematical models. Based on the precision indicators, the second-degree surface fitting model provided the best velocity estimates with an average of 3.4 cm/yr and the variations from 2.7 to 4.6 cm/yr in five categorized zones. Moreover, the lowest estimation error of the regional function was tested to be around 0.4 cm/yr or 1.7 cm/yr, depending on an internal (1997.0-2000.4) or external (1997.0-2002.0) data set assessed. The regional functions were further applied to estimate the applicable durations specifically for the first-order control stations and 1/1,000 digital maps in Taiwan. Under a specific level of accuracy standard and estimation uncertainty, it is suggested that a suitable maintenance period of 4 years for the control stations and 10 years for the 1/1,000 digital maps is required, both based on the use of the regional movement functions estimated. References Blick, G.H., C. Crook, D. Grant, and J. Beavan (2003). Implementation of a Semi-Dynamic Datum for New Zealand. Proceedings of the IA G General Assembly, Sapporo, Japan, pp.38-43. Chang, C.C., and C.L. Tseng (1999). A Geocentric Reference System in Taiwan. Survey Review, Vol. 35, No. 273, pp. 195-203. Chang, C.C. (2000). Estimates of Horizontal Displacement Associated with the 1999 Taiwan Earthquake. Survey Review, Vol. 35, No. 278, pp. 563-568. Geomatics Canada (1996). Accuracy Standards for Position. Version 1.0, Geodetic Survey Division, Geomatics Canada. Tregoning, P., and R. Jackson (1999). The Need for Dynamic Datums. Geomatics Research Australasis, 71, pp. 87-102. Wielgosz, P., D. Grejner-Brzezinska, and I. Kashani (2003). Regional Ionosphere Mapping with Kriging and Multiquadric Methods. Journal of Global Positioning Systems, Vol. 2, No. 1, pp. 48-55. Yanalak, M., and O. Baykal (2001). Transformation of Ellipsoid Heights to Local Leveling Heights. Journal of Surveying Engineering, pp. 90-103.
Chapter 97
Activities Related to the Materialization of a New
Vertical System for Argentina M. C. Pacino
Facultad de Ciencias Exactas, Ingenieria y Agrimensura Universidad Nacional de Rosario, Av. Pellegrini 250, (2000) Rosario, Argentina [email protected] D. Del Cogliano, G. Font, J. Moirano, P. Natali Facultad de Ciencias Astrondmicas y Geofisicas Universidad Nacional de La Plata, Paseo del Bosque s/n, (1900) La Plata, Argentina [email protected]
E. Lauria, R.Ramos Instituto Geogrfifico Militar Cabildo 391, Buenos Aires, Argentina [email protected] S. Miranda Facultad de Ciencias Exactas, Fisicas y Naturales Universidad nacional de San Juan San Juan, Argentina [email protected]
Abstract. The Geodesy Subcommitee of the National Commitee of |UGG created the working group "Geopotential Origin" in December 2000 in order to coordinate the national activities, establish and set down a new Vertical Reference System and to interact with Group III of SIRGAS (Geocentric Reference System for the Americas) Project. For that, all the activities were organised in four items. This contribution describes the main results obtained in each of them as well as future tasks for the near future: • Tide Gauges: The origin of the Argentine Vertical Reference System, defined by means of tide gauge records, is affected by the sea surface topography at the gauge locations. A project to observe and model the sea level variations using both, tide gauges and satellite altimetry, is being carried out since 1998. Four permanently observing GPS stations have already been installed close to gauges along the Argentine Atlantic coast and integrated in the International GPS Tide Gauge (TIGA) Monitoring Project. • Geopotential Numbers: All first order levelling network unadjusted height differences were migrated to digital media and merged with the benchmark positions and gravity values. The resulting database consists of 16,000 level
differences distributed along about 3,170 lines. Gravity values are missing for 20 percent of the benchmarks. The data set is currently being checked for consistency. • Linking of Altimetric Networks from Neighbour Countries: During 2002 the first link between the altimetric networks of Chile and Argentina was done. The result was a difference of 22 cm, the higher value corresponding to Argentine levelling lines. The border point, where the levelling lines of both countries are connected, is Puesto Monte Aymond, near Estrecho de Magallanes in the south extreme of South America. The corresponding activities in order to make two or three new comparisons along the 5,000 km long borderline between these countries are being coordinated. Recently, the determination of a natural water surface (Fagnano Lake) was used like an equipotential surface to connect the altimetric networks of Chile and Argentina where the classical spirit levelling is not accessible due to the lack of roads. • Compensation of Height and Gravity Networks: Adjustment of the national gravity and altimetric networks and their linking with the nets of neighbouring countries are in progress, including both new absolute and relative gravity
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determinations, which are necessary for the adjustment and optimisation processes. Keywords. Tide Gauges, Geopotential Numbers, Altimetric Networks.
1 Introduction The Vertical Reference System for Argentina was realized through a short series of tide gauge observations collected in Mar del Plata in the year 1924. During the 1940s, the reference mark on the tide gauge was connected by high precision geodetic levelling to a highly more stable mark in Tandil, located approximately 200 km away from the coastline. This point remains today as the origin of the national height system. This reference frame was extended to the whole country by high precision geodetic levelling. The first order network was completed in 2001 by Instituto Geogrdfico Militar (IGM). It consists of roughly 16,000 points distributed along several tens of thousands of kilometres of high precision geodetic levelling lines. Since 1997, SIRGAS Working Group III, vertical datum, has striven towards the establishment of a unified vertical reference frame for the American Continent (Laurfa et al., 2002). This involves the revision and unification of both the national vertical reference realisations and their densifications. In connection with the latter, GTIII has recommended the participating countries to compute the geopotencial numbers corresponding to their high precision levelling networks whenever measured gravity information is available. As this is the case for Argentina, the Geodesy Subcommitee of the National Commitee of IUGG has been working to produce a consistent set of geopotencial numbers for the national first order levelling network.
Argentina coastline. A subgroup among these stations was selected for GPS monitoring. Several selection criteria were applied, having been considered some key factors such as: existence of a long history of measurement series, continuity of operation, their location, and the possibility of installing a permanent GPS station near the tide gauge. Based on this criteria Mar del Plata, Puerto Belgrano and Puerto Madryn tide gauges were selected. In 1998 a permanent GPS station was installed in Bahia Blanca. Since then, episodic GPS campaigns at the tide gauges in Mar del Plata and Puerto Belgrano were carried out. At the end of 1999 a second permanent station at Rawson and a new GPS station at the tide gauge of Puerto Madryn were introduced. Finally in 2002, a third permanent station was installed close to Mar del Plata tide gauge. Figure 1 shows the distribution of the tide gauges, new permanent stations and also the IGS stations involved in the processing.
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2 Tide Gauges The Facultad de Ciencias Astrondmicas y Geofisicas (FCAG) and the Deutsches Geodfitisches Forschungsisntitut (DGFI) are engaged in a project called "Sistema de Referencia Vertical en Argentina por Mare6grafos y Altimetria Satelital" (SIRVEMAS) since December 1998. The main aim of this work is to contribute to the South American Vertical Reference System realisation; in particular, to a reference frame consistent with modern space geodetic techniques.
2.1 Permanent Stations and SlRVEMAS Campaigns A group of twelve tide gauges maintained by the Servicio de Hidrografia Naval (SHN) lie along the
Fig. 1: SIRVEMAS Stations
Until now nine episodic campaigns were carried out, each of the measurement campaigns consisted of a minimum of seven consecutive days of simultaneous 24-hour GPS observations.
2.2 GPS Data Processing All data sets were uniformly processed with the Bernese GPS Software V4.2 (Hugentobler et al., 2001). The main characteristics of the data processing can be summarised as follows:
Chapter 97 • Activities Related to the Materialization of a New Vertical System for Argentina
• Final IGS products for orbits, Earth orientation parametres and phase centre variations were used; • The L1 and L2 phase ambiguities were resolved using the QIF strategy. No ionospheric model was introduced at any stage of the processing; • Ocean tide loading corrections following the FES 95.5 model were applied to each station; • An a priori tropospheric delay was computed using the Saastamoinen model (Saastamoinen, 1973) and the dry mapping function by Niell (Niell, 1996), both available as standard options in the Bernese software version 4.2. In addition corrections to the zenith delay were estimated in the adjustment every two hours (Kaniuth, 1998); • The elevation cut off angle in the daily adjustment was set to 10 °, and no elevation depending weighting was applied in order to fully exploit the low elevation data; • In many cases a baseline per baseline processing involving a careful residual analysis and introduction of new ambiguities was necessary. After the pre-processing phase, the daily solutions were held free, that is, only low weights were applied to the fiducial points. After that, for each campaign, an almost free solution of the entire network was obtained, for which ephemeris defined the reference frame.
Table 1: Summary of height velocities [mm/yr]. Station MPLA BELG MPMPY
ITRF97 +0,9 _+0,1 +1,1 _+0,1 -0,2 _+ 0,2
ITRF2000 -1,8 _+ 0,1 -1,9 _+ 0,1 -2,8 _+ 0,2
SSC(DGFI)04P01 -1,3 _+ 0,1 -1,6 _+ 0,1 -2,3 _+ 0,2
The table comprises the estimates and their standard deviations. The velocities estimated using the solutions ITRF97 are different compared with the other two sets. These differences might come from the fact that ITRF97 has less data than the other two solutions. From these results we can not claim which solution is the most reliable, but just show the sensitivity of the results to variations in the reference frame realisation.
3- Geopotential numbers The complete national levelling database is by now in digital format. It consists of 370 levelling lines composed by 16,320 benchmarks, including 225 nodes (Fig. 2).
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2.3 Results The purpose is to obtain precise coordinates for the tide gauge benchmark related to the International Terrestrial Reference System (ITRS). The introduction of the reference frame through the fiducial stations with their weights is a critical issue in order to obtain a reliable set of velocities. The reference frame was introduced by applying suitable weights to the fiducials. In this case BRAZ, LPGS, SANT and RIOG were selected as fiducial stations. These IGS stations were considered to be the most reliable in the region as they are processed by several global and regional IGS analysis centres. We estimated vertical velocities for the SIRVEMAS network using the following solutions:
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• the ITRF realisations 1997 and 2000 of the International Earth Rotation Service (IERS); • the regional GPS solution generated by DGFI, SSC(DGFI)04P01.
The distances between adjacent benchmarks range from 3km to 9 km. The level difference precision depends on the distance between adjacent benchmarks as 3mm times square root of the levelled distance in km.
The vertical velocities resulting from the uniform processing of nine episodic campaigns spanning five years is summarized in Table 1.
Almost every benchmark in the network has geocentric coordinates. Their accuracy however varies from centimeters for the most recent ones
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M.C. Pacino • D. Del Cogliano • G. Font. J. Moirano • P. Natali. E. Lauria • R. Ramos. S. Miranda
determined by GPS, to a few thousand meters in the case of those coming from topographic maps, an usual procedure in older days. Approximately 84% of the benchmarks in the network have measured gravity values. One of the problems to solve prior to the geopotential number computation is to fill the gravity "holes" properly. The analysis of the gravity data indicates that there are 1,200 gravity "holes" on the lines. Among them, 75% are only one isolated missing value and 50 cases consist of large gravity holes including a few cases of complete lines with no gravity observation at all. Most of the gravity values in the network were originally referred to the Potsdam frame but today they have been converted to IGSN71 through the application of a shift o f - 1 4 . 9 3 mGal to the measured values. This conversion formula has been tested on more than 800 points that have measurements on both systems, being the mean difference 0.2 mGal -/+ 0.3 mGal. Apart from the reported measuring methodology and instrumentation, this fact leads us in principle to assume an accuracy for the gravity measurements of at least 0.5 mGal. The nodes of the network show a mean occupation of 3.2 times. Indeed, 76% was occupied more than once. Up to now, both levelling and gravity networks datasets provided by IGM, were analyzed and checked in order to: • compute all closures within the levelling network in terms of raw or measured height differences; • compute geopotential numbers along all levelling lines; • compute orthometric and normal heights along all levelling lines.
heights and compared with the geopotential model EGM96 (Perdomo et al., 2001). Recently, • geopotential numbers, • normal heights, and • quasi-geoid heights were evaluated for the above-mentioned net points. These results show comparisons between geoid undulations and quasi-geoid heights in the GPS/Benchmarks points. (Font and Perdomo, 2004).
4- Linking of Altimetric Networks Several activities related to altimetric networks of different countries in South America started from the IAG meeting that took place in Cartagena (Colombia), where Argentina and Chile decided to compare their respective altimetric nets in different points along their border line, extended for about 5,000 km in the Cordillera de los Andes. Field works to achieve the first comparison were done during 2002. The meeting took place at Monte Aymont, a border step next to Estrecho de Magallanes, in the southernmost continental zone of both countries (Fig. 3). A difference of 0,22 + 0,025 metres was found. I
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On the other hand, a precise GPS geodetic network has been established in Buenos Aires province with benchmarks belonging to the IGM first order Argentinean levelling network. The geodetic network has been related to the reference flames POSGAR 94 and POSGAR 98 (epoch 1995.4). The latter is the materialization of the ITRF in Argentine (Moirano, 2000). In these conditions a height transformation model was developed. This model, named FCAG98 (Perdomo and Del Cogliano, 1999) constitutes an approximate representation of the geoid in Buenos Aires and has been used to transform ellipsoidal heights into IGM
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Some considerations have to be taken into account: The altimetric network of Argentina is referred to a point located 3,000 km north of Monte Aymond, while the altimetric network of Chile is referred to the tide gauge in Punta Arenas, 180 km far from the comparison point. Furthermore, the result is the same order of the differences between the Argentine network and several tide gauges (D'Onofrio et al., 1999). Another comparison was made using the levelling line between Monte Aymond and Rio Gallegos
Chapter 97 • Activities Related to the Materialization of a New Vertical System for Argentina
(Argentina). In this case, the levelling line is referred to a tide gauge located 80 km from the border point. The difference decreased to 0,06 + 0,025 metres ,,51 . . .-., ~
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Recently a new geoid model for Tierra del Fuego (Fig.4) was performed by means of the GPS levelling and the level surface of the Fagnano Lake (Del Cogliano et al, 2005). The lake extents 105 km wide in E-W direction and connects Argentina with Chile. There, the classical spirit levelling is not accessible due to the lack of roads. The determination of the lake level was based on measurements of a GPS buoy and permanent operation of three pressure tide gauges during two years. The integration of the MLL (mean lake level) information contributed notably to the representation of the local geoid. The next task is to link the Chilean altimetric network to a benchmark close to the lake in order to compare both, Argentine and Chile altimetric networks, in the southernmost zone of the American continent. Continuing with the tasks related to the neighbouring countries altimetric networks link, driven by the SIRGAS Project, and aiming at the determination of the Continental vertical Datum between the years 2003 and 2004, the Argentine IGM measured four levelling lines (Paso Tromen, Paso Icalma, Paso Samor~ and Paso Futaleuf{t) closer to the Republic of Chile. The Chilean IGM has showed the same result with regard to its levelling line at Paso Huemules. This year both IGMs are considering finishing the link between their respective networks at "Paso
Caracoles" (International Tunnel at "Paso Cristo Redentor" for the Argentine Republic).
5- Compensation of Height and Gravity networks. Compilation of the original levelling data has been concluded. Meanwhile the recovery of gravity primitive records is in progress. With regard to the network compensation, different robust methodologies are being tried in order to adjust data that have been measured using different instruments for nearly a century. In this sense, Miranda et al. (2004) have settled a local experimental gravity network in San Juan (Argentina). The net was designed to form a chain of triangles in order to hold redundant measurements to the adjustment by the robust method of Becker (1990). The network inner adjust is better than 7 ~tGal. Besides, in scientific cooperation with Brazilian and Uruguayan institutions, several research projects have been proposed, one of them just launched, with the view to set up new absolute gravity stations in Argentina. These stations will be used as fix points in the gravity network adjustment process. On the other hand, a special care will be attempted to get measurements on existing sites where other geodetic observations are available (continuous GPS for instance), so that a repetition network of precise reference geodetic stations can be set up. These sites include five existing absolute gravity stations established in Argentina in 1988 and 1991, which will contribute to better characterise possible ground deformations and gravity temporal changes. Furthermore, strategies for linking the national gravity network with the nets of Brazil and Uruguay are being designed.
5. Conclusions and further work. The results on tide gauges presented in this paper are based on six episodic GPS campaigns spanning over the last three years. The results obtained show that the methodology used in the processing of the GPS data to determine vertical crustal movements is correct. However, the velocities estimated are simply first approximations and a longer series of observation is required. A careful analysis in the realisation of the reference frame is still necessary. The next step is to analyse the tide gauge records for the gauges involved in this project. There are several remaining tasks on geopotential numbers calculation. Firstly, a convenient way to interpolate the gravity values where they are missing will be selected. Secondly, geopotential number differences between benchmarks will be computed. Misclosures of polygons within the
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network will then be analysed as a consistency check for the measurements. Next, a consistent set of geopotential values will be derived through an adjustment on the nodes. Finally, a set of heights will be derived following the adoption of a vertical reference frame and a height definition. These will be selected in accordance with the SIRGAS recommendations in order to ensure the maximum compatibility of the national height systems in the America Continent. Once the link between Argentina and Chile is established, there are views to running similar projects with the rest of the neighbouring countries. References
Becker, M. (1990). Adjustment of Microgravimetric Measurements for Detecting Local and Regional Vertical Displacements. In: Gravity, Gradiometry, and Gravimetry, Symposium N ° 103, R. Rummel and R. Hipkin (Ed.), Edimburgh, Scotia, pp: 149-160. D'Onofrio, E., M. Fiore, F. Mayer, R. Perdomol R. Ramos, (1999). La referencia vertical. In: Contribuciones a la Geodesia en la Argentina de fines del siglo XX. UNR Editora, pp: 101-130. Del Cogliano, D., R. Dietrich, A. Richter, R. Perdomo, J. L. Hormaechea, G. Liebsch, M. Fritsche. 2006. Regional geoid determination in Tierra del Fuego including GPS levelling. Geologica Acta, Special Issue. Accepted. Font, G . , R. Perdomo, 2004. Altura del geoide y cuasigeoide en la Red GPS de la Pcia. de Buenos Aires. XXII Reunidn Cientifica de la AAGG. Buenos Aires. Accepted. Font, G., J. Moirano, R. Ramos, 2004. Anfilisis de la Red altimdtrica y gravimdtrica del pais para la obtencidn de las respectivas cotas geopotenciales. XXII Reunidn Cientffica de la AAGG. Buenos Aires. Accepted. Hugentobler, U. S. Schaer and P. Friedez (eds), 2001. Bernese GPS Software Version 4.2, Astronomical Institute-University of Berne. Kaniuth K., D. Kleuren and H. Tremel, (1998). Sensitivity of GPS height estimates to tropospheric delay modelling, AVN No. 6. Laufia, E., F. Galbfin, C. Brunini, C, G. Font, R. Rodriguez, M.C. Pacino (2002). The vertical
reference system of the Argentine Republic. in: Drewes, H., A. Dodson, L.P. Fortes, L. Sfinchez, P. Sandoval (Eds.): Vertical Reference Systems. IAG Symposia, Vol. 124, pp 11-15, Springer, Germany. Lemoine F. G., Kenyon S. C., Factor J. K., Trimmer R. G., Pavlis N. K., Chinn D. S., Cox C. M., Klosko S. M., Luthcke S. B., Torrence M. H., Wang Y. M., Williamson R. G., Pavlis E. C., Rapp R. H., and Olson T. R. 1998. The Development of Joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) Geopotential Model EGM96. NASA/TP-1998206861. Moirano J., 2000. Materializaci6n del Sistema de Referencia Terrestre Internacional en Argentina mediante observaciones GPS. Ph D Thesis. Universidad Nacional La Plata (FCAG). Miranda S., A. Herrada, J. Sisterna, (2004). Redes de Gravedad/Nivelacidn. Disefio, Medicidn, Cfilculo y Compensacidn de una Red Experimental Local. TOPICOS DE GEOCIENCIAS. Un Volumen de Estudios Sismoldgicos, Geoddsicos y Geoldgicos en Homenaje al Ing. Fernando Sdptimo Volponi. Editorial EFU, Universidad Nacional de San Juan, Argentina. 334 pp. Niell A., (1996). Global mapping functions for the atmospheric delay at radio wavelengths. Journ. Geophys. Res. (101) 3227-3246. Perdomo R. and Del Cogliano D., 1999. The Geoid in Buenos Aires region. International Geoid Service. Bulletin N. 9. Special Issue for South America. Perdomo R. and Del Cogliano D., 1999. The Geoid in Buenos Aires region. International Geoid Service. Bulletin N. 9. Special Issue for South America. Perdomo R., Del Cogliano D., Di Croche N., Neuman K., 2001. Advances in the calculation of a height transformation model in Buenos Aires Province. International Symposium on Vertical Reference Systems. Cartagena, Colombia. Vo1.124 p.75. Saastamoinen J., (1973). Contribution to the theory of atmospheric refraction. Part II, Refraction corrections in satellite geodesy. Bull. Gdod. (107) 13-34.
Chapter 98
An Analysis of Errors Introduced by the Use of Transformation Grids F.G. Nievinski, M.C. Santos Department of Geodesy and Geomatics Engineering University of New Brunswick, P.O. Box 4400, Fredericton, New Brunswick, Canada, E3B 5A3
Abstract. In this paper we analyze the errors introduced by the use of transformation grids. A transformation grid is an intermediate step during the transformation of coordinates of points attached to distinct geodetic reference frames. The use of transformation grids simplifies the transformation when compared to the reference transformation that the grid represents. Transformation grids have become a standard way of making transformation distortion models available for end-users. They are widely accepted by the GIS industry, being already supported by a host of commercial and free programs. They are adopted in countries like Canada and Australia, and are currently being considered for adoption in Brazil. The work described in this paper was conducted by addressing a number of questions in the following sequence: (i) "Is there an upper bound in the error introduced by a transformation grid?"; (ii) "What is the coarsest spacing between nodes for a transformation grid to introduce only negligible errors?"; and, (iii) "How does the error introduced by a transformation grid vary spatially?" To answer these questions we transformed a set of random test points twice, once using a transformation grid and once using the reference transformation that the grid represents. Then, we analyzed the difference between the two results. We show that: (i) yes, there is an upper bound in the error introduced by the grid; (ii) the coarsest spacing can be found by plotting error versus distance to nearest grid node; and, (iii) the error varies spatially partially in proportion to the norm of the second derivative of the reference transformation.
Keywords. Geodetic reference systems and frames, distortion modeling, transformation grids.
1 Introduction Geodetic reference frames have intrinsic distortions due to the positioning techniques used for their materialization. The ones materialized by classical
terrestrial techniques have their distortions due to the surveying techniques employed in the past. Modern, satellite-based ones have considerably less distortions. When one establishes relationships between reference frames, distortions should be taken into account. The modelling of distortions becomes even more important when relating coordinates of points between an "old" reference frame (materialized by classical techniques) and a "modern" reference frame. The distortions in the former should be modelled and taken into account. The modelling of the distortions in the materialization of a reference frame provides better transformation results, but it makes it harder for non-expert users to apply them, especially if those users are using third-party software tools over which they have little control. That is because, as there is no commonly used form for transformations involving distortion modelling, popular software tools usually do not support them. Transformation grids, sometimes also called "gridshift" files, represent the field of shifts (the total shift, not the distortion only) in coordinates between two reference frames. It plays the role of a facilitating step in the transformation involving distortion modelling. The surface representing the shift field does not need to be generated many times, but only once. The grid will correspond to this surface. Usually the procedure to use the grid is simpler than the reference transformation that was used to generate it. That is the reason why the grid makes it easier for users to apply the transformation. Also the procedure to use the grid is standardized, which makes it easier for software developers to support it in their programs. The aim of this work is to analyze the errors introduced by the use of grids in the transformation between geodetic reference frames. The work has been conducted by addressing a number of questions, in the following sequence: (i) "Is there an upper bound in the error introduced by a transformation grid?"; (ii) "What is the coarsest spacing between nodes for a transformation grid to introduce only negligible errors?"; and, (iii) "How
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does the error introduced by a transformation grid vary spatially?" This work was developed to support the transition from a classic non-geocentric reference frame to a modern geocentric reference frame in Brazil. We hope that by sharing our experience interested people or agencies facing a similar transition can benefit, in the same way we have benefited from work conducted to support similar transitions elsewhere in the world. In Section 2 we present a background discussion about transformation of coordinates between geodetic reference frames in general, with a specific focus on transformation grids. In Section 3 we describe the data and techniques used in our analysis. Section 4 contains a discussion on the results obtained. In the final section we summarize the findings of this work. 2 Background 2.1 T r a n s f o r m a t i o n b e t w e e n g e o d e t i c reference frames
A similarity transformation is sufficient to transform points between defined reference systems (not frames'). A tri-dimensional similarity transformation is fully described by seven parameters, as follows: one translation in each of the three Cartesian directions, one rotation around each one these directions, and a uniform scale factor. Although the scale factor is not necessary to transform one reference system onto another (Vanf ek and Steeves, 1996), it is needed to transform points between reference systems. As reference systems are idealized abstractions, we have access to them only through their materialization, called reference frames. Geodetic reference frames have intrinsic distortions due to the surveying techniques employed in their materialization. Distortions may, for instance, increase the farther away the point is from the origin, in the case of classical reference frames (IBGE, 1996). As we transform coordinates of points between two reference frames the distortions in one of them can be taken into account (assuming that the distortions in the newer, e.g., satellite-based one, are negligible compared to the distortions in
1We are following the IERS usage of the terms reference systems and reference frames (see, e.g., McCarthy and Petit, chapter 4).
the older, classical one). The modelling of distortions is important so that they do not propagate through the transformation from the old to the new reference frame, and are added back when transforming from the new to the old one, in order to guarantee a one-to-one correspondence between points in them. Underlying the use of the similarity transformation there is the assumption that the distortions are non-existent or negligible (Collier, 2002). With the greater accuracy achieved by the use of geodetic space techniques (e.g., GPS), that assumption is no longer valid, particularly when the frames are of different nature, e.g., a classical and a space-based one. In this case, the similarity transformation alone is not enough. We have to either augment it by introducing a component to model the distortions, or choose a different transformation model. There are many models of transformation between geodetic reference frames with distortions modelling. Collier et al. (1998) provides an overview on the problem. Junkins and Erickson (1996), Oliveira et al. (1998), and Costa et al. (1999) report additional investigations. In the general case, the transformation between coordinates can be treated as an interpolation problem (Wolberg, 1999), and then the various general spatial interpolation models (e.g., inverse distance weighting, splines, Kriging, etc.) can be applied. We have not found in the literature a commonly used model for distortion modelling.
2.1 T r a n s f o r m a t i o n grids
As mentioned in the previous section, there is no one common model for the transformation between reference frames with distortion modelling. That represents a challenge for non-expert users who use a variety of CAD, GIS, and Image Processing software tools provided by third parties. These tools usually offer a limited set of predefined transformation models. The inclusion of a different model is beyond the regular use of the tools, usually requiring modifications at the source code level. One way to overcome this challenge is to introduce an intermediate form in the transformation, whose format and usage are standardized. That intermediate form is the so-called transformation grid. No matter what transformation was employed to generate it, the transformation grid can
Chapter 98 • An Analysis of Errors Introduced by the Use of Transformation Grids
be used always in the same way. By "standardized" we mean that there is a specification describing it. Therefore, if we generate a grid complying with the specification, automatically our grid will be supported by the tools developed earlier that already comply with it. The transformation grid is an array of shifts. At each node of a two-dimensional regular array the shifts in the two horizontal coordinates are given. These shifts should be added to the "old" coordinates in order to get the "new" ones. The positions of the nodes are described in the old reference frame. The procedure to transform coordinates using a transformation grid is as follows: (1) find in the grid the four nodes nearest to the point to transform; (2) apply a bi-linear interpolation to the shifts at those four nodes; (3) sum the interpolated shifts to the old coordinates of the point to transform. The result is coordinates of the same point in the new reference frame. By an iterative procedure the same grid can be used to do the inverse transformation, i.e., from new to old coordinates. The grid is generated by evaluating the reference transformation procedure at each of its nodes, and then computing the difference between the transformed (i.e., new) and the old coordinates. Once generated, the grid needs to be formatted in a standard way (see, e.g., Mitchell & Collier, 2000). It can be coded first in free text and later converted to a binary format. Transformation grids were adopted in Canada, the USA, and Australia. This approach is independent of a specific software tool. Therefore, the tools developed to support the use of transformation grids in one country could be used in a different country. Nowadays there is a host of commercial and free programs for both the end-user and the programmer that support transformation grids 2.
3 Data and methods
Figure 1. Each control point has coordinates in two different Brazilian realizations of the South American Datum of 1969 (SAD69): the original one, and its 1996 realization (SAD69/96). We will be considering these two realizations as two different reference frames. For each control point the shift in coordinates between the two reference frames was computed by simply taking the difference between them. The shifts in latitude are almost 10 times larger than the shifts in longitude, making the shifts in longitude hardly noticeable when plotted together. For a better visualization, we show in Figure 2 the normalized shifts 3.
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2 For a list of free programs, please contact the first author; for commercial ones see ICSM (2005) and GSD (2005).
3 From now on, whenever we show the shifts in latitude and in longitude together, we will be showing normalized shifts.
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3,2 Reference transformation
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To analyze the errors introduced by the grid we need a reference transformation to be used as a benchmark for the comparisons to follow. After obtaining that reference transformation we assume it as "true" or, conversely, that it does a "perfect" job modelling the observed shifts, which are then abandoned. This is valid because we are not interested in the errors introduced by the reference transformation itself, only in the errors introduced by the grid alone. To allow us to answer the third question posed in this paper (section 4.3), we needed a transformation that would yield results with high spatial variability. Therefore we discarded simple well-behaved models such as global polynomial surfaces of low degree. Other than that, the choice was arbitrary. The chosen reference transformation is as follows. We interpreted the shifts in latitude and longitude as two separate two-dimensional scalar fields, varying over the horizontal space. Then, we used the trianglebased bi-cubic interpolation (Fortune, 1997) to interpolate the shifts at the desired points. A sample of the reference transformation's result is shown in Figure 3. Figure 4 shows the so-called Delaunay triangulation of the control points, an intermediate result required by the chosen transformation.
We generated an array of regularly spaced nodes (spaced 13 ° 30' in latitude and longitude) enclosed by the convex hull of the control points, as shown in Figure 5. Then, we evaluated the reference transformation at each grid node. The results of this evaluation are shown in Figure 6.
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To evaluate the performance of the grid we overlaid on it 10,000 points at random positions, as shown in Figure 7. The number of random points was chosen arbitrarily. We transformed the coordinates of each random point using (i) the reference transformation, and (ii) the grid. The difference between (i) and (ii) represents the error introduced by the grid.
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• An
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As we sample the grid error with more and more test points, we find that there is an upper bound in the error curves. In our case, we noticed that after 5,000 test points the error stops increasing, meaning that (i) that sample is representative of the grid error, and (ii) the error is no greater than 3.5 x 10 .6 degrees or 0.0126". The upper bound on the error curves depends on a balance between grid spacing and spatial variability of the reference transformation results. The next two questions below address the problem of "tuning" a grid so that it introduces only negligible errors, in an efficient manner. 1/] 6
Error in longitude for each random point
3.5 Use of the transformation grid We have used a program conforming to the transformation grids specification (the Canadian NTv2) to verify that the grids we were generating followed the specified text format. The grid is to be used for interpolating bi-linearly (Press et al, 1992) the shifts given at the grid nodes. We did so with a Matlab implementation (function interp2) of that interpolator.
4 Results and discussion 4
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cumulative error (thin, stepped curve), and mean cumulative error (thick, continuous line) versus number of test points.
Obviously the grid may introduce an error with respect to the reference transformation. That is so because the grid transformation model (which is made of the grid itself a n d the bi-linear interpolator to be applied on it) m a y not be able to capture all the variability described by the reference transformation. We would like to assess how bad this error can get. To answer this question we analyzed the cumulative mean and m a x i m u m errors of the r a n d o m test points as we increased the n u m b e r of points (see Figure 8). In that figure, each dot represents an individual test point; the thin, stepped curve is the m a x i m u m error; the thick continuous line is the mean error. These error curves are cumulative, meaning that each value along them is calculated from the test points to the left of it.
4.2 What is the coarsest spacing between nodes for a transformation grid to introduce only negligible errors?
4 Due to lack of space, in the following we have figures only for longitude. The corresponding figures for latitude show curves with similar behaviour and values 10 times larger.
First of all it is required that the ones generating the grid define what error would be negligible in their application. This value might be based, e.g., on the error already introduced by the reference transformation itself. W e will describe an example later in this section. The question will help us to tune the grid so that the inevitable errors introduced by it do not affect the application we have in mind. Here we will assume that there is only one uniformly spaced grid covering the area of interest. In the next question we will be interested in the case of reducing the grid spacing locally (instead of globally, as we do here). To answer the question we have obtained, for each test point, the "distance ''5 to its nearest grid
5 The "distance" is actually the Euclidean norm of the difference in geodetic coordinates between a given point and
681
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F.G. Nievinski • M. C. Santos
node. We sorted the points based on that distance. Then we analyzed how the error increases as that distance increases (see Figure 9). Intuitively, the closer a point is to a grid node (i.e., the smaller that distance), the better the grid model represents the actual shift at that point or, conversely, the smaller the error is at that point. lO .0
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Figure 9 can be used to find the coarsest spacing between nodes for a transformation grid to introduce only negligible errors. To do so, first, we specify the maximum acceptable error; second, we find the corresponding maximum distance to a grid node, using the error curve in Figure 9; third, we regenerate the grid using that distance times 2 (see Figure 10) as the spacing between its nodes.
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its nearest node. It is not, e.g., the ellipsoidal distance between the two.
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4.3 How does the error introduced by a transformation grid vary spatially? In the previous question we assumed that we would be using only one, uniformly spaced grid. But transformation grids allow the use of sub-grids, i.e., denser grids covering a subset of the main grid. This property is important when a national grid is
Chapter 98 • An Analysis of Errors Introduced by the Use of Transformation Grids
densified by a state or provincial grid. It is useful to be able to predict the maximum error spatially because the areas with high maximum errors would be strong candidates for sub-gridding. To help us tackle the posed question we have used a denser 6 set of regularly spaced test data. We did so to improve visualization. The denser point set depicts the patterns in more detail, and the fact that it is regularly spaced allows us to plot the set as an image, which is a lot faster than to plot each individual point. We expected to find a spatial portrayal of the behaviour shown in Figure 9, i.e., error increasing as a function of distance to nearest node. But what we found was the intriguing pattern shown in Figure 13. Maxima seem to concentrate near edges and vertices of the Delaunay triangulation.
case, the piece-wise maximum error depends strongly on the absolute value of the second derivative of the reference function. We could not find or develop an expression describing a similar dependence in the 2dimensional case. Then we went to investigate empirically whether a similar behaviour is observed. To do so, we calculated numerically the second gradient of shift in each coordinate, by means of the central difference numerical derivative (Press et al., 1992) in each direction. The norm of that gradient is shown in Figure 14.
Figure 14 Norm of the second gradient of shift in longitude (red- large values, blue- small values), versus latitude and longitude, with edges of triangulation overlaid.
Figure 13 Error (red- large values, blue- small values) in longitude versus latitude and longitude, with edges of triangulation overlaid. In the search for an explanation for that behavior, it was brought to our attention the existence of the following formal error bound for the 1-dimensional linear interpolation (Wikipedia, 2005):
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6 The test points were 10 times denser than the transformation grid nodes.
We see roughly the main peaks at the same positions (see, e.g., SE-, NE-, and NW-corners). Despite that, the overall matching is weak, as attested by the correlation coefficients: 0.4 and 0.3 for shifts in latitude and in longitude, respectively. The main difference between the error field and the second gradient norm field is that the latter is better defined, with sharper variations, while the former seems like a locally-averaged version of the latter. To verify the interpretation above, we defined grid cells, which are rectangular areas delimited by four nodes of the original grid (the grid on which the errors are b a s e d - see Figures 5 and 10). Then, we computed the maximum values of error and norm of second gradient p e r g r i d cell. At this time the correlation is stronger: 0.76 and 0.75 for shifts in latitude and longitude, respectively (see scatter plot in Figure 15). Therefore, in this case, the norm of the second gradient of shifts predicts partially the error per grid cell.
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As stated at the beginning of this section, in the process of developing and tuning transformation grids, the areas with high errors would be strong candidates for sub-gridding. We should recognize that for this purpose it might be more practical to simply obtain and analyze the error field directly (Figure 13) instead of predicting it only partially. 5
Conclusions
In this paper we analyzed the error introduced by the use of transformation grids. We have shown that: (i) there is an upper bound in the error introduced by the grid; (ii) the coarsest spacing can be found by plotting the error versus distance to nearest grid node; and, (iii) the m a x i m u m error vary spatially partially in proportion to the norm of the second gradient of the shifts. We believe these conclusions and, more importantly, the analyses presented in this paper, might be useful to individuals and agencies considering, developing, or tuning transformation grids to support the transition from a classic to a modern reference frame.
Acknowledgements
Funds for this study have been provided by the Canadian International Development Agency (CIDA) in support of the Brazilian National Geospatial Framework Project ().
References
Australian Intergovernmental Committee on Surveying and Mapping (ICSM). Geocentric Datum of A u s t r a l i a Software. Available at: . Last access on 08/May/2005. Brazilian Institute of Geography and Statistics (IBGE) (1996). Adjustment of the planimetric network, Brazilian Geodetic System (in Portuguese). Canada Geodetic Survey Division (GSD). Commercial Software with NTv2 feature. Available at: . Last access on 08/May/2005. Collier P.A., Argeseanu V., Leahy F. (1998). "Distortion Modelling and the transition to GDA94", The Australian Surveyor, Vol. 43, No. 1, March 1998. Collier, P.A. (2002). "Development of Australia's National GDA94 Transformation Grids". Consultant' s Report to the Intergovernmental Committee Surveying and Mapping. February, 2002. Costa, S. M. A., M. C. Santos and C. Gemael (1999). "The integration of Brazilian geodetic system into terrestrial reference systems." Abstracts, XXII General Assembly, International Union of Geodesy and Geophysics, Birmingham, England, 19-24 July, p. A.415. Fortune, S. (1997). Voronoi diagrams and Delaunay triangulations, Handbook of Discrete and Computational Geometry, J.E. Goodman, J. O'Rourke, eds., pp. 377-388, CRC Press, New York. Junkins, D.R. and Erickson, C. Version 2 of the National Transformation Between NAD-27 and NAD-83 and Its Importance for GPS Positioning in Canada. Geodetic Survey Division Geomatics. Canada, 1996. Oliveira, L.C., J.F.G. Monico, M.C. Santos and D. Blitzkow (1998). "Some considerations related to the new realization of the SAD-69 in Brazil." Advances in Positioning and Reference Frames, F. K. Brunner (Ed.), International Association of Geodesy Symposia, Vol. 118, Springer, Berlin, pp. 205-210. McCarthy, D.D. and G. Petit. (2004). IERS Conventions 2003 (IERS Technical Note 32) Frankfurt am Main: Verlag des Bundesamts fiir Kartographie und Geoddsie. 127 pp., paperback, in print. Mitchell, D.J., Collier, P.A. (2000). GDAit Software Documentation Version 2.0. Available at: . Last access on: 07/Sep/2004. Press W.H., S.A. Teukolsky, W.T. Vetterling and B.P. Flannery (1992). Numerical Recipes in C - The Art of Scientific Computing. Second Edition, Cambridge University Press. Vaniek, P., R.R Steeves (1996) "Transformation of coordinates between two horizontal geodetic datums". Journal of Geodesy, Vol, 70, No, 11, pp. 740-745. "Linear interpolation." Wikipedia: The Free Encyclopedia. 2005. Available at . Last access on 01/Aug/2005. Wolberg, G. (1990) Digital Image Warping, IEEE Computer Society Press, Los Alamitos, CA.
Chapter 99
Preliminary Analysis in view of the ITRF2005 Z. Altamimi, X. Collilieux Institut Geographique National, ENSG/LAREG, 6-8 avenue Blaise Pascal, 77455 Marne-la-Vallee, France C. Boucher Conseil gdndral des ponts et chaussdes, tour Pascal B, 92055 La Ddfense, France
Abstract. Unlike the previous versions of the International Terrestrial Reference Frame (ITRF), the ITRF2005 will be constructed with input data under the form of time series of station positions and Earth Orientation Parameters (EOP' s). This paper presents some preliminary results of the analysis of the time series submitted to the ITRF2005, focusing on the frame parameters and in particular the origin and the scale, as well as the EOP alignment to the combined frame. Quality assessment of the preliminary combinations is addressed in view of the ITRF2005 official solution.
Keywords. Reference System, Reference Frame, Earth Rotation, Combination, Space Geodesy, ITRF
site instability, seasonal loading effects, etc. rigorously and consistently including EOPs in the combination and ensuring their alignment to the combined frame. The results presented in this paper are intended to be preliminary and should not be taken as final or official products of the ITRF2005. Most importantly, some individual solutions analyzed here will not enter the official ITRF2005 combination. They are analyzed in order to assess the current accuracy of some frame parameter, in particular the scale and origin. Moreover, the ITRF2005 analysis will be performed by the 3 ITRF combination centers Natural Resources Canada (NRCan), and Deutsches Geod~idtisches Forschungsinstitut (DGFI) and IGN. The official final ITRF2005 will be delivered by the ITRF Product Center hosted by IGN.
1 Introduction 2 ITRF2005 Input Data Contrary to previous ITRF versions, the ITRF2005 will integrate time series of station positions and daily Earth Orientation Parameters (EOP's). The time series solutions are now provided in a weekly basis by the Services of the International Association of Geodesy (lAG) of satellite techniques (IGS: International GNSS Service, ILRS: International Laser Ranging Service, IDS: International DORIS Service) and in a daily (VLBI session-wise) basis by the International VLBI Service (IVS). Reasons for which it was decided to use time series of station positions and EOP as input to ITRF2005 include: monitoring of non-linear station motions and all kinds of discontinuities in the time series: Earthquake related ruptures,
As input data to the ITRF2005, it was decided to consider official combined time series of solutions provided by the IAG services, known as Technique Centers (TC) by the IERS. These official TC's solutions result obviously from a combination of the corresponding individual analysis centers solutions. Official time series of solutions were submitted to the ITRF2005 by the IGS, IVS and ILRS. At the time of writing, the IVS and ILRS solutions still need some refinement as the quality of their solutions needs to be improved. For DORIS, official weekly combined solutions do not exist yet so that individual solutions are provided by 3 analysis centers. Table 1 summarizes the submitted solution to ITRF2005 and individual additional solutions analyzed in this paper.
686
Z. Altamimi. X. Collilieux • C. Boucher
Table 1. ITRF2005 Submitted and analyzed solutions TC
AC*
Time span
Type of constraints/solution
Comment
IVS
1984-2005
Normal Equation
Official submission but still need refinement at the time of writing
IVS - GSFC
1984-2005
Normal Equation & var-covar
Analyzed in this paper
IVS - GEOS
1984-2005
Loose; var-covar
Analyzed in this paper
IVS - DGFI
1984-2005
Normal Equation
Analyzed in this paper
ILRS
1992-2005
Loose; var-covar
Official submission but still need refinement at the time of writing
ILRS - ASI
1992-2005
Loose; var-covar
Analyzed in this paper
ILRS - DGFI
1992-2005
Loose; var-covar
Analyzed in this paper
ILRS - GEOS
1992-2005
Loose; var-covar
Analyzed in this paper
ILRS - GFZ
1992-2005
Loose; var-covar
Analyzed in this paper
ILRS - JCET
1992-2005
Loose; var-covar
Analyzed in this paper
ILRS - NSGF
1992-2005
Loose; var-covar
Analyzed in this paper
IGS
1996-2005
Minimal/Inner; var-covar
Official submission to ITRF2005
IDS - IGN/JPL
1993-2005
Loose; var-covar
Official submission to ITRF2005
IDS
INA
1993-2005
Loose ; var-covar
Official submission to ITRF2005
IDS
LCA
1993-2005
Loose ; var-covar
Official submission to ITRF2005
* TC: Technique Center, AC: Analysis Center, GSFC: Goddard Space Flight Center, NASA, USA, GEOS: Geosciences Australia, DGFI: Deutsches Geod~idtisches Forschungsinstitut, Germany, ASI: Agencia Spaziale Italian, GFZ: GeoForschungsZentrum Potsdam, JCET: Joint Center for Earth System Technology, at GSFC, NSGF: NERC Space Geodesy Facility (NSGF), formely RGO Satellite Laser Ranging Group, U K , IGN: Institut Gdographique National, France, JPL: Jet Propulsion Laboratory, INA: INstitute of AStronomy Russian Academy of Sciences, LCA: Laboratoire d'Etudes en Geophysique et Oceanographie Spatiale (LEGOS) in cooperation with Collecte Localisation par Satellite (CLS), France. I I R F 2 0 0 5 Derivation
3 Analysis Strategy "W1
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The strategy adopted for the ITRF2005 generation consists of the following steps, illustrated in Figure 1.:
~LRI I
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•
• • •
• •
Remove original constraint (if any) and apply minimum constraints equally to all solutions Use as they are minimally constrained solutions Form per-technique combinations (TRF + EOP) Identify and reject/de-weight outliers and properly handle discontinuity using piece-wise approach Combine if necessary all solutions of a given technique into a unique solution Combine per-technique combination adding local ties at co-location sites
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Fig l. I T R F 2 0 0 5 Derivation
3.1 Stacking of Times Series CATREF Software is used to combine (rigorously stacking) the per technique time series of station positions and EOP using the following two sets of equations, (Altamimi and Boucher, 2003):
Chapter99 • PreliminaryAnalysisin Viewof the ITRF2005 -
+ (t i - to)2
. +
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+
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.) R3 k
4 Discussion results
(2) Jc~
=
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-
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time
derivatives
2f,
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~9~
and
LOD.
is the conversion factor
unit, as the EOPs considered here are provided at daily basis. Note that the link between EOP and the TRF is ensured by the 3 rotation angles and their time derivatives.
3.2 D a t u m D e f i n i t i o n When stacking the daily/weekly solutions using equations (1) and (2) the constructed normal equation system is singular and has 14 degrees of freedom. This rank deficiency corresponds to the number of parameters which are necessary for the datum definition of the combined Terrestrial Reference Frame (TRF). The latter is defined through minimum constraint approach using the equation: -
0
(3)
some
preliminary
In addition to the combinations (rigorously stacking) of the individual time series (per technique and or per analysis centers), a first tentative global combination was also performed for the purpose of this paper. We present here some results, focusing on the frame parameters, namely the origin and the scale. In addition to the analysis of the time series, we provide some results on apparent seasonal variations and a quality evaluation of the individual solutions. Preliminary results of Earth Rotation Parameters included in our combinations are also included here.
'
from universal to sidereal time. Considering LOD--A 0 --7-, dc;r A 0 is equal to one day in time
a
of
4.1 Origin results
and
Scale
preliminary
It is expected that the origin of the ITRF2005 will be defined by the ILRS solution and the scale by the average of IVS and ILRS solutions. In order to illustrate the origin and the scale behavior over time, Figure 2 displays the three translation components as well as the scale of the 6 AC's composing the ILRS combined time series. Figure 3 shows the scale behavior of the IVS preliminary time series as well as 3 other IVS AC solutions. All results show here are with respect to ITRF2000. From these results we see, in average, a good origin and scale consistency between the individual analyzed series and the ITRF2000. Meanwhile the IVS scale variation exhibits significant seasonal variation which might be attributed, at least partly, to thermal deformations of the VLBI radio telescopes. For completeness, Figure 4 presents the time variations of the origin and the scale of the two analyzed DORIS solutions.
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4.2 Seasonal Variations GS F(_', GEOS
IVS
The authors plan to make available to users the residual time series of the station positions as results from the individual combination/stacking. In order to illustrate the type of seasonal variations as seen for some individual stations, Figure 5 displays the residual time series of some GPS/IGS stations as indicated. We note that this type of seasonal variations is more pronounced and clearly identified in the IGS GPS station time series, compared to the other techniques and that this effect is most significant in the vertical component. In order to illustrate that effect, Figure 6 shows the annual amplitude and phase of the vertical components of most pertinent IGS sites.
(mm)
Scale
Fig. 3. Time variation of IVS AC's scale in mm.
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Chapter 99 • Preliminary Analysis in View of the ITRF2005
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Fig 6. Annual Vertical Amplitude and phase of some IGS stations. The arrow defines the amplitude and its orientation (eastward) defines the phase., modeled by A.sin(2 ~ t - to) + qk) 1.0 0.5
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4.3 E a r t h O r i e n t a t i o n
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2005
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2000
2005
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, 1995
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(mas) as result from the global combination test.
Parameters
As results from the preliminary global combination, similar to what will be the ITRF2005, Figure 7 displays a zoom of _+ 1 milli-arcseconds (mas) of polar motion residuals. This combination test includes the following solutions: VLBI/GSFC, SLR/ILRS preliminary solution, GPS/IGS official solution as well as IGN DORIS solution.
The approximate average of the Weighted Root Mean Square Error (WRMS) on polar motion is around (in gas ) 50 for GPS, between 150 and 200 for VLBI and SLR and around 1000 for DORIS. Moreover, a comparison of the resulting polar motion series with the IERS C04 series was performed and the differences are plotted in Figure 8. The mean of these differences indicates a significant bias of about 100 gas in the Y component between our EOP series (expressed in ITRF2000) and the IERS C04.
689
690
Z. A l t a m i m i . X. Collilieux • C. Boucher
4.4 Quality evaluation
conclusions from this preliminary analysis are the following:
As an internal quality indicator of the analyzed times series, we utilise the WRMS of the individual series. Figure 9 displays the weekly (daily for VLBI) WRMS per solution for the horizontal and vertical component. The rough average of these values is summarized in Table 2.
•
• Table 2. Rough average of internal precision per solution, based on computed weekly (daily for VLBI) WRMS resulting from the stacking of the individual time series. Solution
2-D WRMS (mm) 7 6 10 20
VLBI GSFC GPS IGS SLR ILRS DORIS IGN
UP-WRMS (mm) 3 3 10 20
•
•
5 Conclusions • Preliminary analyses are presented in this paper as preparation for the ITRF2005. The major
the use of time series as input to the ITRF2005 combination definitely allows monitoring the stations behavior and consequently improve their estimated linear velocities. the origins of the SLR solutions analyzed are, in average, consistent with the ITRF2000. Their scales as well as the scale of VLBI solutions are also consistent with the ITRF2000. However some significant VLBI scale seasonal variation is apparent. significant seasonal variations are detected for GPS/IGS station time series and most significant in the vertical component. the quality of the analyzed time series is encouraging, promising an improved ITRF2005 solutions compared to previous ITRF versions more work and analysis are still to be done before the official release of the ITRF2005.
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1997
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99 • Preliminary Analysis in View of the ITRF2005
Chapter
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Fig. 9. Weekly (daily VLBI) WRMS
Acknowledgments The ITRF activities are funded by the Institut G6ographique National, France and partly by the Group de Recherche de G6od6sie Spatiale. The IERS ITRF Product Center is indebted to the Analysis Centers and Technique Services of the 4 techniques for their contribution to the ITRF2005.
References Altamimi, Z., P. Sillard, C. Boucher (2002). ITRF2000, A new release of the International
Terrestrial Reference Frame for earth science applications. Journal of Geophysical Research, Solid Earth, vol. 107(B 10), 2214. Altamimi Z, Boucher C (2003) Multi-technique combination of time series of station positions and Earth orientation parameters, in Proceedings of the IERS Workshop on Combination Research and Global Geophysical Fluids, IERS Technical Note No. 30, Richter B, Schwegmann W, Dick W (eds.), Verlag des Bundesamts ffir Kartographie und Geodfisie, Frankfurt am Main, Germany, 102-106
691
Chapter 100
Long term consistency of multi-technique terrestrial reference frames, a spectral approach K. Le Bail Institut Gdographique National/LAREG and Observatoire de la C6te d'Azur/GEMINI (UMR 6203) 8 Av. Blaise Pascal, Champs sur Marne, 77455 Marne la Vallde Cedex 2, France M. Feissel-Vernier Observatoire de Paris/SYRTE and institut Gdographique National/LAREG 8 Av. Blaise Pascal, Champs sur Marne, 77455 Marne la Vallde Cedex 2, France J.-J. Valette Collecte Localisation Satellites (CLS) 8-10 rue Hennas, 31526 Ramonville-Saint-Agne Cedex, France
W. Zerhouni Centre National des Techniques Spatiales (CNTS) BP 13, Arzew - 31200, Oran, Algeria Abstract. Analysing time series of space-geodetic station coordinates, we show that VLBI, SLR and DORIS station motions have a white noise error spectrum, while the majority of GPS station motion have a flicker noise error spectrum. In the latter case, discontinuities in the series do not account for this spectrum. Atmospheric loading has a white noise spectrum, at a much lower level than geodetic measurement errors in the long term. The series of transformation parameters derived for the GPS colocation sub-networks with VLBI, SLR and DORIS have very close spectral characteristics, reaching the 1 mm stability level at one-year interval in a white noise context.
Keywords. Space Station stability.
geodesy.
Reference
frames.
1 Introduction With the advent of the combination of time series of station coordinates to construct ITRF2004 (Altamimi et al. 2005), a key element in the long term stability of a multi-technique Terrestrial Reference Frame (TRF) is the control of the time consistency of the motion measured by the various techniques for colocated stations. Time series of station coordinates are available from global space geodetic programs operated with GPS, SLR, VLBI and DORIS. Good quality series go back to the early 1990's. The major signature in time series of station coordinates is usually modelled as a tri-dimensional linear drift. The horizontal component is mostly related to tectonic plate motion, while the vertical
component is assumed to reflect local uplift or subsidence. The remaining component may be interpreted as noise related to local geophysical phenomena, instrumentation, or to the analysis strategies and modelling. The hypothesis of linear motion is also a key one in most uses of space geodetic positioning. For this reason, our study is focusing on the non-linear content of the time series of coordinates; i.e. residuals relative to a linear motion model for the station. The derivation of spectral characteristics of the time behaviour of the station coordinates from the four ITRF techniques is described in Section 2. In Sections 3 and 4 the possible influence of two types of perturbing phenomena on station stability is investigated. These are 1) confirmed or suspected coordinate discontinuities in position or velocity, and 2) atmospheric loading. The long term consistency of the combined reference frame depends directly on the stability of the tie between the VLBI, SLR, GPS and DORIS networks. In Section 5, based on the stability analysis and associated quality criteria, a detailed analysis of the time behaviour of the GPS-colocated sub-networks is proposed.
2 Spectral characteristics of non linear signals in time series of station coordinates The data analysed are series of station coordinates derived from the VLBI, SLR, DORIS and GPS techniques. They are typically provided as series of geocentric Cartesian coordinates or, equivalently, series of offsets in local directions to the East, North
Chapter 100 • Long Term Consistency of Multi-Technique Terrestrial Reference Frames, a Spectral Approach
and Up (ENU), in some defined terrestrial reference frame. GPS and DORIS series are available at oneweek intervals. The average time distribution of the SLR and VLBI series is comparable but somewhat irregular, due to observing conditions
The time series analysed are listed in Table 1. The VLBI series was derived by Ma (2005) using the CALC-SOLVE software package; the SLR series was derived by Coulot and Bdrio (2005) using GINS-DYNAMO; the GPS series was derived by R. Ferland (IGS 2005a) as the combination of the IGS analysis centres series; the IGN DORIS series was derived by Willis (2005) using GIPSY-OASIS; the LCA DORIS series was derived by Soudarin and Crdtaux (2005) using GINS-DYNAMO. The IGN series was used as the DORIS series for figures 4, 5, 7, and table 3; the LCA series was used as the DORIS series in figures 1, 4, 6, and table 3.
Table 1. The sets of station coordinates analysed. The numbers of stations that (a) passed the time density threshold (see text), and (b) that are sparsely observed are given. Techn. VLBI SLR DORIS DORIS GPS
Solution name gsfc_2004b slr oca05 ignwd05 lcawdl 2 igs_2004
Data span 90.0-04.3 93.0-05.0 93.0-05.3 93.0-05.1 95.0-05.4
Sites # 34 20 59 59 175
Stations (a) (b) . 24 13 18 3 44 38 47 36 137 52
The data analysis is based on the time series of residuals with respect to linear motion for each station individually. Station time series are selected on the basis of time span (at least four years), missing weeks (less than 30%), and data gaps (shorter than 200 days). Data with postfit residuals larger than three times the standard deviation are edited. Table 1 gives the numbers of sites and stations available from each technique, considering separately the stations whose time series of coordinates did or did not pass the above threshold. The analysis approach is described in detail by Le Bail (2004) and Le Bail and Feissel-Vernier (2005). The three-dimensional geodetic station position signal is submitted to Principal Component Analysis (PCA) in the time domain. Then the analysis is made in both the local reference frame and the 3D Principal Components (PCTs) derived from PCA.
The PCTs are the projections of the initial series on the space generated by a set of eigenvectors which are defined so that the leading component describe the temporally coherent pattern that maximises its variance. The spectral behaviour of the series is characterised by the change of the Allan variance of the time series of coordinates as a function of the sampling time (Allan 1987). This statistics allows one to identify white noise (spectral density S independent of frequency j), flicker noise (S proportional to f-l), and random walk (S proportional to f-2). In a white noise context, the variance of residual motion is lessened as the data span is extended. In a flicker noise context, the errors are not diminished with the expansion of the data span. A convenient and rigorous way to relate the Allan variance of a signal to its error spectrum is the interpretation of the Allan graph, which gives the changes of the Allan variance for increasing values of the sampling time "c. In logarithmic scales, slopes -1, 0 and + 1 correspond to white noise, flicker noise and random walk, respectively. Note that the signature of a linear slope in the signal in the Allan graph is a +2 slope. Here, the upstream correction for a linear motion makes it little probable to find a random walk spectrum in the residuals. The presence of a cyclic variation is recognised by the superimposition of a dip when "c is equal to the cycle period and a bump at about 1/2 the cycle period, with a size that depends on the signal/noise ratio of the amplitude of the cyclic component. To avoid biasing due to this effect, the Allan graph slopes are computed on time series corrected for their annual term. The following statistical parameters are derived. - Stability for a one-year sampling time of the first Principal Component in the Time domain, measured by the Allan standard deviation, and its projection in the local ENU frame. - Spectral law followed by the same parameters, measured by the slope of the Allan graph.
Figure 1 shows the values of these two indicators obtained for the coordinate time series (a) of Table 1 (dense series). The two-dimensional graphs show for each station the one-year Allan standard deviation as a function of the slope of its Allan graph. The quantity considered is the first principal component in the time domain (PCT), which explains in general
693
694
K. Le Bail. M. Feissel-Vernier. J.-J. Valette • W. Zerhouni
over 80% of the non-linear, non-seasonal signal. To relate this component to the stations local conditions, the projections of the PCT characteristics in the local ENU frame are also shown. A remarkable, and not unexpected, feature is the occurrence of largest values of the Allan standard deviation in the Up direction for GPS and VLBI, and for a part of the SLR series. The case of DORIS is different, with relatively homogeneous levels in all three directions and some noisier values in the East direction.
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for only a part of the GPS stations, as illustrated by figure 2, that shows the geographical distribution of stations whose non linear signal principal Component in the Time domain has white noise and those with flicker noise spectra. Only about 1/3 of the GPS time series have white noise in the principal component in the time domain, as well as in the three directions in the local frame. This result is compatible with those of Mao et al. (1999), Williams (2003), Williams et al. (2004), who found that the GPS variance law of the station signal is a mix of white and flicker noise.
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It should be noted that, whereas the GPS series are the combination of many centres series, the VLBI and SLR ones come from only one Center, and the presence of outliers that would no longer exist in a combined solution cannot be ruled out. Note also that the DORIS data analysed cover the whole 1993-2005 period, while the data become more precise after 2002.5, when five satellites became available for the first time. The latest segments of the time series are still too short to be analysed separately. The VLBI, SLR, and DORIS time series are found to have a white noise spectrum, while it is the case
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Fig. 2 Stability spectrum of time series of GPS station coordinates (1995-2005). Up: white noise (59 stations). Bottom: Flicker noise (125 stations). Stars and triangles show the location of well observed stations and sparsely observed ones, respectively. Note that post-seismic data at Arequipa and Fairbanks are ignored.
Our analysis approach is quite different from that of the above mentioned studies. The latter is based on the modelling of the signal spectral law of each station as a mix of white and flicker noise with proportions that are the unknowns to be determined. Here we first isolate by Principal Component Analysis the principal signal component in time and then we determine its spectral law. In effect, while the first PCT generally explains over 80% of the variance of the non-linear, non-seasonal signal and has flicker noise for the majority of GPS time series,
Chapter 100
• LongTerm
Consistency of Multi-Technique Terrestrial Reference Frames,
the ratio of white noise to flicker noise is close to one in the other two PCTs.
3 The influence of discontinuities on the GPS noise spectral law
White noise measurement spectrum in the long term is an implicit hypothesis of the concept of reference frame maintenance through repeated observation by permanent space-geodetic networks. The diagnostics illustrated by figure 1 confirm its non validity for a part of the GPS stations. We study here a noise source that could be invoked to explain the observed flicker measurement spectrum is investigated, namely the presence of discontinuities in the time series of coordinates associated to nearby earthquakes, at the millimetre to centimetre level. As a consequence of the precision of GPS positioning, time series of coordinates are known to exhibit minor but visible changes due to regional tectonic events such as identified earthquakes, or to other unnoticed transient events. Outliers may also be present, as in any measurement series.
To manage the discontinuities of the first type, the IGS maintains a file giving lists of confirmed and probable discontinuities in the station time series (IGS 2005a). Both position and velocity discontinuities are considered. In addition, the combined IGS series was submitted to detailed screening and a number of minor defects were corrected (Ferland 2005). The expected influence of such events on the Allan variance depends on the relative size of the discontinuity. Small values should impact the response of the test in the short term, and large values the long term.
The measurement spectra of three categories of stations is evaluated, considering on the one hand the uncorrected version of the IGS combined time series of coordinates over 1995-2005 (IGS 2005b) and on the other the corrected one (IGS 2005c). The categories correspond to the stations with confirmed or probable discontinuities, and those with no discontinuity detected, respectively. The frequencies of white and flicker noise spectra derived by the Allan variance method for the uncorrected and corrected data sets are illustrated in
a SpectralApproach
figure 3. The proportion of stations with white measurement noise are given in table 2 for the various categories. Before corrections, the three categories have similar percentages of white noise stations (12%-14%). The largest improvement after data corrections, from 12% to 40% white noise stations, is found for the stations with no discontinuities proposed, showing the importance of careful data screening.
Confirmeddisoont. 30 20 10 0
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30%t Wlite~ 20 10 0 , , -1 0
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Fig. 3 Measurement error spectra in connection with discontinuities and their correction in the coordinate time series over 1999-2005: histograms of the Allan graph slopes for stations with confirmed, probable, or no detected discontinuities. Bottom: discontinuities not corrected; up: discontinuities and other detected defects corrected. The white and flicker noise zones are shown.
Table 2. Percentage of GPS stations with white noise in the first PCT according to uncorrected/corrected position discontinuities (1999-2005). Station status wrt discontinuities None detected Confirmed Probable
Uncorrected Corrected series series 12% 40% 12% 28% 14% 28%
For the stations where discontinuities are detected, the improvement after correction is more modest. These stations might be more affected by nearby seismicity than expected. However, the flicker noise spectrum is still there in 60% of the non disturbed stations. Other causes of flicker measurement spectrum are worth investigating.
695
696
K. Le Bail. M. Feissel-Vernier. J.-J. Valette • W. Zerhouni
4 Atmospheric loading
The crust to which the stations are attached is submitted, among other effects, to deformation under the load of variable atmospheric mass. The IERS established a Special Bureau of Loading (SBL) under their GGFC (Global Geophysical Fluids Center), in charge of providing valid estimates of surface mass loading effects for the purpose of correcting geodetic time series. 3-D deformations due to variations in atmospheric pressure are provided at six-hour intervals for all IERS sites (SBL 2005). These deformations are dominated by effects with periods two weeks and one year. The effects are larger at mid-latitudes and may reach 20 mm peak-to-peak (Van Dam et al. 2002). These short- and medium-term effects should play no role in the long term stability of the series of stations coordinates. Their influence in the longer term may be questioned. To test the possible contribution of the atmospheric loading to the long term stability of the stations, we selected the following two sites with observations by the four IERS techniques (Figure 4). -
-
Hartebeesthoek in South Africa (latitude: 26 ° S), with stations for VLBI (7232; brought to 49-day intervals, 1990.1-2004.7), SLR (7501; brought to 10-day intervals, 2000.6-2005.0), DORIS (HBKA, HBLA, HBKB; weekly, 1993.0-2005.2; IGN series), and GPS (HARB, weekly, 2000.9-2005.4 - the station plotted on the bottom graph of figure 2). A time series of atmospheric load deformation at seven-day intervals was derived from the 1993.0-2004.3 data available at the SBL. Krasnoyarsk in Siberia (latitude: 56°N), with stations for DORIS (KRAB; weekly over 1999.4-2005.1, LCA series), and GPS (KSTU, weekly over 1997.7-2004.7). A time series of atmospheric load deformation at seven-day intervals was derived from the 1997.7-2004.3 data available at the SBL.
The geodetic spectra at Hartebeesthoek are consistent with the general characters derived in section 2, with white noise for VLBI, SLR, and DORIS, and flicker noise for GPS. With the exception of VLBI, the noise level converges towards 2-3 mm for the 1.2-year sampling time. The atmospheric loading level is about three times lower than the GPS signal at 7-day sampling time, then for
longer sampling times it becomes negligible with respect to the geodetic noise, as a result of its white noise spectrum. The atmosphere loading deformation spectrum starts a flicker noise behaviour after about one-year sampling times, at a level one order of magnitude lower than the geodetic noise.
Hartebeesthoek
Krasnoyarsk
3
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O
-1
-2
1
2
3
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2
3
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Fig. 4 Allan variance of principal component in the time domain of atmosphere loading deformation series (solid line), and geodetic series. GPS: blue diamonds; DORIS: green stars; SLR: red triangles; VLBI: purple circles. All series were de-seasonalised prior to the analysis. Sampling times ranging from 7 days to 4 years for the longest series.
Table 3. Determining the signal spectra by the slope of the Allan variance graphs (figure 4): white noise (Wh) for slopes close to -1" flicker noise (F1) for slopes close to 0. Technique Atmosph. VLBI SLR DORIS GPS
Hartebeesthoek Slope Spectr. -0.8 + 0.1 Wh
Krasnoyarsk Slope Spectr. -1.0 ± 0.1 Wh
-0.8 + 0.1 Wh -0.6 + 0.1 Wh -0.7 + 0.1 Wh -0.1 +0.1 F1
-0.6 + 0. 1 Wh 0.1 +0. 1 F1
The atmosphere loading signal at Krasnoyarsk has also a white noise spectrum and a start of flicker noise spectrum after one-year sampling times, at a higher level than Hartebeesthoek, so that it matches the GPS signal at 7-day sampling time. Like at Hartebeesthoek the GPS signal remains at the 2-3 mm level in the long term, due to flicker noise spectrum, and the atmosphere loading level is nearly one order of magnitude below. The DORIS signature is that of white noise.
Chapter 100 • Long Term Consistency of Multi-Technique Terrestrial Reference Frames, a Spectral Approach
The examples of time behaviour of the geodetic and atmosphere loading signals in Hartebeesthoek and Krasnoyarsk show that it is unlikely that the atmospheric loading influence the long term stability of the terrestrial reference frame derived from series of station coordinates. In addition, they show that in the long term one could expect similar performance of the four techniques, if the GPS flicker noise spectrum stays as is.
5. The stability of the colocation sub-networks
90" 60"
We could extract from the series (a) of table 1, i.e. series longer than four years with no data gap longer than 200 days, a set of colocated time series of coordinates for a number of stations (see table 4). The presence of flicker noise in a number of GPS station coordinates leads to special consideration of their role in the maintenance of the ITRF internal consistency through the stability of the multitechnique colocation sub-networks. Figure 5, similar to figure 2, shows the GPS colocation subnetworks with VLBI, SLR and DORIS stations with at least four-year data span for the white noise and flicker noise GPS stations, respectively.
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The maintenance of the internal consistency of a multi-techniques terrestrial reference frame such as ITRF2004 is in part dependent on the stability of the inter-technique tie through the colocated stations. Having determined the spectral characteristics of the long term behaviour of measurement errors in the four IERS techniques, we now investigate the time behaviour of the colocation sub-networks by means of time series of their Helmert transformation parameters.
Po P°D PL
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-90" 225"
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Fig. 5 GPS colocation sub-networks, based on the 19962005 data, considering station time series longer than three years in all techniques, for white noise GPS stations (Up) and flicker noise GPS stations (Bottom). The symbols R, L, D, P stand for VLBI, SLR, DORIS, and GPS stations, respectively.
combination over colocation sub-networks. These are a stability index to select the stable stations to be considered in the combination, and a scale factor for the data uncertainties for weighting the data from the different technique in a homogeneous way.
5.1 Stability indices and data weighting Based on the joint PCA-Allan variance method, we define two statistical criteria to qualify the noise in time series of coordinates in view of their
Table 4. Number of colocation sites over 1996-2005, considering station time series longer than three years in all four techniques. Technique VLBI SLR DORIS
SLR 7
DORIS 9 7
GPS 26 19 31
The station stability index is defined as a linear combination of the Allan standard deviation for a one-year sampling time, which represents the noise level, and the slope of the Allan graph, which represents the stability expectation. The construction of the stability index is based on three pairs of partial indices, one for each of the three PCTs, as follows. The first three partial indices are based on the Allan standard deviation for a one-year sampling time (Asdi, i - 1,3) of the time series of residuals to linear motion in their time eigenspace. Each ASdi is the ratio of the ith Allan standard deviation to a reference value czi, i -
697
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K. Le Bail. M. Feissel-Vernier. J.-J. Valette • W. Zerhouni
1,3: 2.5, 1.5, and 0.5, respectively. To give an example a 5 mm Allan standard deviation for PCT1 will produce a partial index A S d l equal to 2. The reference values are empirically chosen to allow a common definition of the stability index for the four global geodetic techniques, VLBI, SLR, GPS, and DORIS. The second set of partial indices, (Aratei, i = 1,3) are derived from the slopes of the Allan graphs for the series of continuous residuals in the PCT coordinate system. They are equal to the value of the slope +1, i.e. 0 in the case of white noise, and 1 in the case of flicker noise. The decimal values of the estimated slopes are used. Values of the slope lower than-1 are set to this value. The three pairs of partial indices are combined
A scale factor of the uncertainties associated with the data is defined as the ratio of a 3D Allan standard deviation of the time series at the one-year sampling time with the average cumulated 3D formal uncertainties of the series. Larger scale factors indicate larger underestimation of the uncertainties associated with the data. The ranges of scale factors are 1-5 for VLBI, 0.5-3 for SLR, 1-4 for DORIS, and 0.5-5 for GPS, with outlying large values for all techniques. In a white noise context the scale factor is independent of the sampling time. In a flicker noise context, it is not. The GPS scale factors relative to a one-week sampling time are all found smaller than one.
5.2 Time consistency of colocated sub-networks
as
1
~ ( A S d i + Aratei)xPc i
1 0 0 i=1
Where (Pci, i = 1,3) are the percentages of variance explained by PCT1, PCT2 and PCT3, respectively (by definition, the sum of the three percentages is equal to 100). With the relative weights chosen for the combination of these two components, the major contribution comes from the one-year Allan standard deviation, the stability index being degraded only in the case of flicker noise. Most stability indices for stations with dense series longer than four years range from 1 (most stable) through about 4 (least stable), with a few anomalous larger values. Figure 6 summarises the spectral situation of the three GPS-colocated sub-networks and the respective stability of the stations.
For each of the three colocation sub-networks, a series of weekly transformation parameters is computed by weighted least squares. Only stations with a stability index lower than 5 are considered. The weekly observations equations are weighted according to the quadratic combination of the data uncertainties multiplied by the corresponding station scaling factors introduced in section 5.1. The data spans selected for the experiment were taken in the most recent years to take into account the improvement in the colocation sub-networks configuration. Data were kept starting with 2000.5 for VLBI, 2001.5 for SLR, and 2002.5 for DORIS.
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Fig. 6 Measurement error spectrum (top) and stability (bottom) of coordinates time series of GPS and colocated space-geodetic stations: VLBI, SLR, DORIS.
Fig. 7 Allan variance graph of weekly series of transformation parameters (Translations Tx, Ty, Tz, Rotations Rx, Ry, Rz, and Scale D) for colocated subnetworks: VLBI-GPS (2000.5-2004.2, red triangles), SLR-GPS (2001.5-2005.0, black stars), and DORIS-GPS (2002.5-2005.2, blue open circles).
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Chapter 100 • Long Term Consistency of Multi-Technique Terrestrial Reference Frames, a Spectral Approach
The spectral characteristics of the time series of the Helmert parameters are determined by their Allan variance. The Allan variance graphs are plotted in figure 7. A remarkable feature of the Allan variance graph is the common level white noise signature obtained for the three colocated techniques, including the supposedly less precise DORIS technique. The noise of the colocation time series for the one-year sampling time is at the millimetre level for all seven transformation parameters, with a few exceptions, e.g. Rx for VLBI-GPS or Rz for DORIS-GPS.
6. Summary The measurement errors in series of VLBI, SLR and DORIS stations coordinates have a white noise spectrum. A majority of GPS stations have a flicker noise spectrum. The correction of discontinuities connected to nearby seisms improves only slightly the ratio of white noise GPS stations. The data editing in the time series using the strength of multiple combined series such as the IGS one succeeds in bringing a part of the stations to a white noise spectrum. The contribution of the atmosphere loading to long term instabilities is found to be negligible. The GPS-colocated sub-networks have a white noise spectrum, reaching the 1 mm level at one year for VLBI, SLR, as well as for DORIS. This estimation concerns only random errors. It is insensitive to systematic errors such as apparent technique-dependent local velocities and other difficulties attached to the unified multi-technique terrestrial frame ambition. These conclusions need indeed to be confirmed using intra-technique combined series for all techniques. Considering spectral aspects in time series of station coordinates sheds new light on the problems encountered. Such an approach could be fruitful in many contexts: analysis centres dealing with individual stations, analysis coordinators dealing with individual analysis centres within one spacegeodetic technique, or global combiners dealing with all networks and all techniques.
Acknowledgements. We thank Chopo Ma for providing to us time series of VLBI station coordinates, and David Coulot for providing time series of SLR station coordinates. We are most
thankful to Rdmi F erland (NRCan) for discussing in detail some of our results and helping us to get better acquainted with the GPS time series of station coordinates. Thanks to Xavier Collilieux (IGN/LAREG) for helping us in the use of the SBL time series of atmosphere loading.
References Allan, D.W. (1987),"Time and frequency characterisation, estimation, and prediction of precision clocks and oscillators", IEEE Trans UFFC, vol 34, n. 6 Altamimi, Z. et al. (2005). Status of the ITRF2004. This volume. Coulot D., P. Bdrio (2005). Private communication. Ferland, R. (2005). Private communication. IGS (2005a). ftp://macs.geod.nrcan.gc.ca/pub/requests/sinex/discontinuities IGS (2005b). ftp://macs.geod.nrcan.gc.ca/pub/requests/sinex/coord r/' station'.igs.utm IGS (2005c). ftp://macs.geod.nrcan.gc.ca/pub/requests/sinex/coord r/' station'.igs.utm Le Bail, K. (2004). Etude statistique de la stabilitO des stations de gOodOsie spatiale. Application ~ DORIS. PhD Thesis dissertation. Observatoire de Paris. Dec. 2004. Le Bail, K., M. Feissel-Vemier (2005). Estimating the noise in space-geodetic positioning. The case of DORIS. J. of Geodesy (submitted) Ma, C. (2005). Private communication. Mao, A, C.G.A. Harrison, T.H. Dixon (1999). Noise in GPS coordinate times series. JGRB, Vol. 104, pp. 27972816. Soudarin, L.; J. F. Crdtaux, J. J. Valette, A. Cazenave (2002). 9-year Monitoring of The Doris Sites 3-d Motions. EGS XXVII GA, Nice, 21-26 April 2002, abstract #6186. Special Bureau of Loading (2005). http://www.sbl.statkart.no/products/research/ Van Dam, T., H.-P. Plag,, O. Francis, P. Gegout (2002). GGFC Special Bureau for Loading: Current Status and Plans. http://www.sbl.statkart.no/archive/pos_papeLfinal.pdf Williams, S.D.P. (2003). The effect of coloured noise on the uncertainties of rates estimated from geodetic time series, J. of Geodesy, Vol. 76, pp. 483-494. Williams, S.D.P., Y. Bock, P. Fang, P. Jamason, R. Nikolaidis, L. Prawirodirdjo, M. Miller, D. Johnson (2004). Error analysis of continuous GPS position time series. JGRB Vol. 109. Willis, P., C. Boucher, H. Fagard, Z. Altamimi (2005), Geodetic applications of the DORIS system at the French Institut Gdographique National, C.R. Geoscience, Vol. 337, 653-662
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Part Vl GGOS:Global Geodetic Observing System Chapter 101
Science Rationale of the Global Geodetic Observing System (GGOS)
Chapter 102
GGOS and Its User Requirements, Linkage, and Outreach
Chapter 103
GGOS Working Group on Ground Networks and Communications
Chapter 104
Linking the Global Geodetic Observing System (GGOS) to the Integrated Geodetic Observing Strategy Partnership (IGOS-P)
Chapter 105
IVS High Accurate Products for the Maintenance of the Global Reference Frames As Contribution to GGOS
Chapter 106
The International Laser Ranging Service and Its Support for GGOS
Chapter 107
The Nordic Geodetic Observing System (NGOS)
Chapter 108
VLBI2010: a Vision for Future Geodetic VLBI
Chapter 109
Combination of Different Geodetic Techniques for Signal Detection a Case Study at Fundamental Station Wettzell -
Chapter 101
Science Rationale of the Global Geodetic Observing System (GGOS) Hermann Drewes Deutsches Geod/itisches Forschungsinstitut (DGFI), Marstallplatz 8, D-80539 Mfinchen, Germany E-mail: [email protected]
Abstract. Geodesy is the science of measuring and mapping the Earth including the associated variations with time. The Global Geodetic Observing System (GGOS) as the "flagship" of the International Association of Geodesy (lAG) aims at the coordination and integration of the corresponding observations in order to generate a uniform set of geodetic parameters for monitoring the phenomena and processes in the System Earth. Coordination means to bring the different geodetic observing techniques and analysis methods together for a consistent performance. The same standards, conventions, models and parameters have to be used in the reduction of observations and in the data analysis for producing compatible results in identical reference systems. Integration implies the inclusion of all relevant information for parameter estimation. The elements of the System Earth are transmitting a variety of signals observable by geodetic techniques. They all have to be considered in a combined analysis. This comprises in particular the integration of geometric and gravimetric data, and the common estimation of all the necessary parameters representing the solid Earth, the hydrosphere (including oceans, ice-caps, continental water), and the atmosphere. Geometric parameters describe primarily the shape of the Earth and its deformation; the gravimetric parameters are functions of the Earth's mass distribution. Considering these facts, the scientific rationale of GGOS is focussing on the central theme "Global deformation and mass exchange in the System Earth". Keywords. Global Geodetic Observing System, System Earth, geodetic space techniques, global change, geodynamics
1 Introduction The International Association of Geodesy (IAG) established a new structure at the XXIII General Assembly of the International Union of Geodesy and
Geophysics (IUGG) in Sapporo, Japan, in July 2003 (Beutler et al., 2004). A fundamental component of this structure are the lAG Projects, which shall be of broad scope and of highest interest and importance for the entire field of geodesy. They shall serve as the flagships of the Association for a long time period (decade or longer). The Global Geodetic Observing System (GGOS) proposed by Rummel et al. (2002) was established in Sapporo 2003 as the first and up to date only IAG Project (Reigber and Drewes, 2004). The initial definition phase from end of 2003 until mid of 2005 served as a period to review the original terms of reference and to set up the GGOS science rationale, objectives, and structure (Drewes and Reigber, 2004). We summarize here the science rationale and recommend scientific objectives and the required structure for its performance. 2 The Vision of G G O S GGOS shall be a system which integrates different geodetic observing techniques, models, and approaches in order to achieve a better understanding of the geodynamic and global change processes as a basis for all Earth science research. Integration in this sense does not only consist of a co-location of different instruments in a geographic site in order to use the same infrastructure and to derive similar mathematical parameters for physical resemblances, although this definitely is an important topic. In the frame of GGOS, integration also means to bring different types of observation together for a common parameter estimation in a mathematical collocation approach using compatible methods of observation modelling and data processing. The result of the integration leads to a consistent set of parameters describing the Earth System by using identical physical models and being compatible with related parameters, e.g., provided by other disciplines. The Earth System is seen, from the geodetic point of view, as a whole including its solid, fluid and gaseous components (figure 1).
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The corresponding geodetic parameters originate from the fundamental fields of geodesy - geometry and kinematics of the Earth's surface, - orientation and rotation of the Earth, - Earth gravity field and its variation with time. Due to quite different observation and analysis methods, geometry may be split into point positioning and (land and sea) surface representation. Orientation and rotation is seen in space (precession, nutation, UT1) and on the body of the Earth (polar motion). The gravity field includes in particular the geoid and the gravity anomalies. GGOS shall ensure the consistency of all the parameters estimated from geodetic observations to describe the characteristics of these fundamental fields. The analysis of the time dependent parameters shall lead to a reliable description of the processes in the System Earth. Solar System (Magnetism, Tides) ~ / / ~ o
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Fig. 2. Interactions of the elements of the System Earth between each other and with the surrounding space
The physical processes in the System Earth are characterized by the permanent interaction of the system components (elements) between each other and with the system's environment, i.e., the EarthMoon system, the solar system, the galactic system, and the universe as a whole (figure 2). The signals transmitted from these processes are measured by geodetic observation systems. They carry information of the mass distribution, deformation and mass transport of the system components. GGOS shall integrate all the observations and all the elements into one unique mathematical and physical model. 3 The Scientific Rationale of GGOS
Based on the above general vision, the central theme of GGOS is defined as "Global deformation and mass exchange processes in the System Earth". The master product within this theme encompasses the global and regional patterns of tectonic deformation, i.e., plate tectonics, inter-plate and intraplate deformations, as well as the quantification of effects due to the mass transfer between the solid Earth, the atmosphere, and the hydrosphere (including continents, oceans and ice-caps). It includes the stationary parts of the geometry and gravity field. 3.1 Signals of Earth System Processes
The dynamic processes of global deformation and mass transport within the solid, fluid and gaseous components of the System Earth and its interaction with the surrounding space affect the geodetic observations and the derived parameters. Processes in the solid Earth include convection, subduction, earthquakes, volcano eruptions, and isostatic adjustment; fluid and gaseous processes include precipitation, evaporation, ice ablation, atmosphere and water currents, water storage and runoff. These processes cause geodetically measurable effects such as point motions, sea surface changes, polar motion, length of day and gravity variations. Some examples are given in Table 1. It is important to note that the effects of the processes are not directed to each geodetic parameter group individually but the influence enters likewise into various fields of geodesy; and only a partial signal is detected by a single geodetic observation, namely that part for which the specific observation is sensitive. Consequently we must not look at one parameter in an isolated way to detect a physical effect but we have to analyse jointly all the affected geodetic parameters. This requires, of course, an identical parameterisation of the physical properties and processes in all different observation equations.
Chapter 101
1. Examples of processes in the System Earth affecting different geodetic parameters by various actions Table
Process
acts as
and affects
Core/mantle convection
- plate driving force - mass displacement - angular momentum - ground water storage - moment of inertia - water flow off - loading force - pressure - angular momentum
--~ point position gravity field --~ Earth rotation gravity field Earth rotation sea surface point position Earth surface Earth rotation
Precipitation
Atmospheric and oceanic currents
•
Science Rationale of the Global Geodetic Observing System (GGOS)
Table 2 gives some examples where signals from the surrounding space are affected by different Earth System elements. Comparing it with Table 1 we see that the same elements may affect both, the geodetic signals and the geodetic parameters derived from the observations. For example, the atmosphere causes refraction affecting the electromagnetic (radio and optical) wave propagation and loading forces affecting the derived point positions. It is clear that these effects have to be modelled simultaneously. 2. Examples of effects of System Earth elements on signals of the system environment
Table 3.2 Signals
of the Earth
System
Environment Signal
Some signals of the Earth System environment are used as primary geodetic observations. A classical signal in geodesy is the light of the fixed stars and the sun, which served over millennia as the basis for astronomic positioning (latitude, longitude and azimuth determination in geodetic astronomy). In combination with terrestrial triangulation and trilateration it is also applied to local and regional gravity field determination (astro-geodetic levelling). Today, optical geodetic astronomy is largely replaced by radio interferometry (Very Long Baseline Interferometry, VLBI) which receives its signals from extragalactic quasi-stellar astronomic radio sources (quasars). In both cases the observed signals (light or microwaves, respectively) pass the Earth's atmosphere and are affected by refraction. The solar system (Sun and planets) and the Moon act mainly through gravitational forces on the Earth System causing tides and variations in Earth orientation and rotation (nutation). Bodies of the solar system affect both types of geodetic observations: gravimetric parameters directly by the tidal potential, geometric parameters indirectly by the deformation of the Earth and by the gravitational effect on satellite orbits. The interactions of the Earth System elements and the signals of the surrounding space are from the geodetic point of view primarily "disturbances" at the observations. They are in general simply reduced. For Earth System research, however, they provide important information which can be used for a comprehensive system modelling. The physical models derived (e.g., models of the ionosphere, troposphere, solid Earth rheology) provide a much better basis for data analysis to geodesy than the present-day auxiliary quantities (e.g., total electron content, tropospheric time delay, gravimetric amplitude factor). The physical modelling is thus of mutual benefit for geodesy and Earth System research.
with effects of
causes
Quasars' radio waves - atmosphere Fixed stars' light - atmosphere Sun' s gravitation - oceans - rock rheology Moon' s gravitation - oceans - rock rheology
3.3
Geodetic
System
Earth
Parameters Elements
- radio refraction - optical refraction - nutation, tides - Earth tides - nutation, tides - Earth tides
Affected
by
Different
and Processes
In the previous chapters we showed how the Earth System elements act on the geodetic observations and parameters. If we look in the opposite direction, extracting from Tables 1 and 2 the dependence of the geodetic results from the various system elements, we see that a variety of effects is integrated in all the geodetic products (Table 3). There is no individual influence to be studied in a particular geodetic parameter, it is always the sum of effects. All geodetic parameters reflect the integrated influences of Earth System elements and processes. 3. Examples of geodetic parameters affected by different elements and processes of the System Earth
Table
Parameter
is a f f e c t e d b y
o f p r o c e s s e s in
Point position
-
plate motion - loading effects
Earth surface
- deformation - water flow-off - air pressure - winds, pressure - ocean currents - deformation - geodynamics - ground water - deformation
- solid geosphere - ocean, hydro-/ atmosphere - solid geosphere - hydrosphere - atmosphere - atmosphere - hydrosphere - solid geosphere - geosphere - hydrosphere - solid geosphere
Earth rotation
Gravity field
705
706
H. Drewes
The arguments provided indicate that only a comprehensive combined analysis of all the relevant geodetic observations in a parameter estimation procedure including all System Earth components can lead to reliable results. If we neglect one type of observation we may not be able to catch the corresponding effect of a particular system component. If we neglect one system element in the parameter estimation process we may misinterpret its impact on the observations and attribute it to parameters of other system components. 3.4 The Challenge for GGOS The challenge for GGOS is thus to fully exploit all the information provided by the different geometric and gravimetric observations in a consistent integration of the measured signals in order to estimate all the affected parameters by comprehensive inversion approaches. Geodesy has proven in the past that it is - in principle - capable of observing both, the deformation and the mass transport processes. . .N. . .
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Fig. 3. Observedand modelled solid Earth deformations in South America: plate motions, earthquakes, inter-plate and intra-plate deformations (Drewes and Heidbach, 2005) Space geodetic positioning methods (Very Long Baseline Interferometry, VLBI, Satellite Laser Ranging, SLR, Global Navigation Satellite Systems, GNSS) have demonstrated the extremely high
accuracy in monitoring surface deformations due to tectonic and isostatic adjustment processes (e.g., Peltier, 2001). Present-day rigid plate velocities are nowadays determined from long-term point positioning with an accuracy of better than one millimetre per year. Intra-plate and plate boundary deformations including co-seismic and post-seismic earthquake slip vectors can be detected and modelled with a few millimetres accuracy. An example is shown in Figure 3 (Drewes and Heidbach, 2005). Differential interferometric synthetic aperture radar (D-INSAR) is capable of providing the deformation of a complete area of the solid Earth's surface, e.g., to detect the crustal deformations after an earthquake. Normally, these observations are combined with GPS measurements as a reference (e.g., Reigber et al., 1997). The high precision allows an interpretation of the deformation processes. Significant sea level variations are seen in the tide gauge records (Holgate and Woodworth, 2004). The (nearly) complete global sea surface variations are precisely monitored by satellite altimetry (e.g., Cazenave and Nerem, 2004). There are, however, important challenges left for the future improvement of these excellent geometric results. If we really want to monitor Earth System deformation processes in near real time, e.g., to contribute to a better understanding of the mechanism that causes earthquakes, we have to achieve permanently an accuracy of significantly better than a millimetre because eventual precursor deformations to be detected are very small. Furthermore, we have to reduce those effects from the observed point motions which are not caused by tectonic processes but by episodic or periodic influences (man-made constructions, loading, seasonal variations,...). Terrestrial gravimetric observations provide precise information about mass displacements, which may be attributed to deformation processes (e.g., Pagiatakis and Salib, 2003). Vertical point motions can be modelled in particular. However, the separation of the point movement in the gravity field and the stationary gravity change is problematic. The latter one may be produced, e.g., by nearby mass displacements like groundwater variations or by engineering constructions. A comparison with geometric (positioning) results is often done, however mostly not in a common adjustment, but by independent procedures which are not equivalent to a joint analysis (e.g., Richter et al., 2004). Modem space gravity missions are capable of sensing the global mass transport. Mass displacements generate signals of gravity field changes. The displacement of water masses at the Earth's surface is nowadays detectable by analysing the observa-
Chapter 101
tions of the Gravity Recovery And Climate Experiment (GRACE, e.g., Andersen et al., 2005, Ramillien et al., 2005, Visser and Schrama, 2005). Apart from the gravity effect, modem satellite missions may also be used for sensing the atmosphere, in particular its water vapour content (e.g., Beyerle et al., 2004, 2005). All the observations mentioned for monitoring the deformations and the mass transport in the System Earth are usually processed and analysed separately of each other, at present. There is no combined analysis which ensures the consistency of estimated geometric and gravimetric deformations; the mass balance of the water cycle between ice sheets, oceans, continents and atmosphere is not checked. The most important challenge of GGOS is therefore to combine all the geometric and gravimetric methods in one common procedure. This has not necessarily to be done by a unique approach or software. It can be achieved by distributed processing and a final combination, as it is done today in the geometric and gravimetric community. It has only to be ensured that the used constants, conventions, models, and parameters are identical and that the approaches are consistent.
4 Integration of Heterogeneous Geodetic Observations by Combination 4.1 Status of Combination The combination of homogeneous observation data from different techniques is nowadays done as a standard procedure in all fields of geodesy. For point geometry and kinematics, the space geodetic methods GPS, SLR, VLBI, DORIS are processed by individual analysis centres or by the techniques' services (IGS, ILRS, IVS, IDS) and combined for point positioning and velocity determination, e.g., to derive the International Terrestrial Reference Frame (ITRF, Altamimi et al., 2002) by Combination Centres (Angermann et al., 2005). The same geometric observations are independently combined for estimating the Earth orientation and rotation parameters (Gambis, 2004). Surface geometry and its variation over the oceans is obtained by combining different satellite altimetry mission data and by cross calibration (Bosch 2003). Over land, the D-INSAR observations are normally combined with GPS and other observations (e.g., Delouis et al. 2000). The global Earth gravity field is determined, in general, by combining space-borne, air-borne, shipborne and terrestrial measurements (Lemoine et al.,
• Science
Rationale of the Global Geodetic Observing System (GGOS)
1998, Kern et al., 2003). Regional densifications are obtained by combining global space geodetic observations with regional terrestrial measurements (Schmidt et al. 2005). The principal shortcomings in the present combination approaches have to be seen in the fact that individual analysts or groups do not always use identical constants, models and conventions, e.g., the Geodetic Reference System 1980 (GRS80) or the IERS Conventions. This problem has been discussed in detail by Groten (2001, 2004). Different, not fully compatible sets of parameters are estimated, which may be correlated with each other and transfer effects of one parameter to another (e.g., loading effects and vertical velocities in point positioning). Even the international reference frames are not always used consistently (e.g., ITRF, IGSN71), and the state of scientific methods is not the same in all the software packages. The main imperfection, however, has to be seen in the fact that geometry, orientation and gravity are only rarely combined in routine geodetic product generation. Even the terrestrial and celestial reference frames (ITRF and ICRF) and the connecting Earth orientation parameters (EOP) are not yet consistently combined in the latest realizations. There are, however, activities to overcome this problem (e.g., Rothacher, 2003, Rothacher et al., 2003). The classical combination of geometric and gravimetric data is the determination of physical heights from spirit levelling. This procedure is generalized by the modem space techniques to include GPS for height determination (e.g., Denker et al. 2000). Gravity, sea level and Earth rotation parameters are also combined frequently (e.g., Gross, 2001), but not at the observation level including all common observations and parameters, but at the level of independently estimated results. The same approach is performed for the combination of modem satellite gravity missions with geometric (GPS) observations (e.g., Kusche and Schrama, 2005).
4.2 Requirements for Future Integration The future integration of geodetic observations in GGOS has to start from the original geometric and gravimetric measurements. Consistent models for data reduction and compatible parameterisations have to be applied for a rigorous estimation of parameters related to the same reference frame using identical conventions and comprising all the relevant System Earth elements. The procedure has not necessarily to be done in a complete common adjustment, but the observation equations may be
707
708
H.
Drewes
computed in a distributed data processing and summed up to normal equations, which are then accumulated and solved in an integrated (collocation) procedure. Physical constraints (conservation of mass, angular momentum, energy) have to be included for an overall verification. In deformation studies the balance of forces and displacements (stress and strain) may be verified by gravimetric and geometric measurements in combination with geophysical models. No geodetic model should be released without representing both types of observations and parameters in accordance with the models. The new satellite gravity missions play an important role in this context. For mass transport investigations, the modem space geodetic methods are capable of observing the complete hydrological cycle by integrating satellite altimetry for monitoring variations of the ocean and ice surfaces, GPS sounding for the water vapour in the atmosphere, and satellite gravity missions for detecting water storage and flow-off in the continents (figure 4).
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ship has to be known precisely by local surveys. All the relevant information of the observations has to be well documented and provided as metadata. Observations can only be combined consistently if the processing methodologies are compatible (e.g., adjustment and filter techniques, respectively, treat the parameters differently and reduce the noise differently). Solutions to be combined must not internally be deformed, e.g., by too many constraints; they must not fix the geodetic datum before combining them. One of the greatest problems in combining heterogeneous observations is its weighting. Variance-covariance estimation has to be done in a sophisticated way. A common data processing can only be done using consistent constants, conventions and models. Unfortunately, this condition is not always met today. The use of fundamental constants (e.g., GM, c, c0) refers frequently to different time systems (Groten, 2004). The permanent deformation of the Earth by Sun and Moon (permanent tides) is treated differently in geometric and gravimetric applications. Geophysical models (e.g., PREM with 2600 kg/m 3 crustal mass density) are not respected uniformly. This has to be considered in future. Geodetic results are in general provided with respect to (arbitrary) reference systems. They may be defined by heterogeneous parameters and realized by different reference frames (Table 4). It has to be ensured that the definitions and realizations are compatible for geometric and gravimetric quantities. The consistency of hierarchic reference frames has to be guaranteed, too (e.g., ICRF, EOP, ITRF).
~ ~ _ ; . . ~
Geodeticmonitoring of the hydrological cycle
Table 4. Examples of possible inconsistencies between geometric and gravimetric reference frames Definition
The rigorous combination of geodetic observations in an integrated approach thus requires consistent observations (in stations, missions), consistent processing methodologies using consistent constants, conventions, models in - consistent reference frames in order to estimate - consistent parameters for generation of - consistent products for consistent applications. The basic observations in modem geodesy are time measurements (signal travel times, epoch timing). All observations have to use the same time system (e.g., geocentric time, TCG) and the same speed of light (c), reduced in the same way due to the effects of the medium of propagation (atmosphere). They have to refer to a well-defined reference point, or their geometric / gravimetric relation-
Origin Orientation Scale Deformation
Geometric
Xo, Yo, Zo Xp, Yp, DUT 1 c AX, AY
Gravimetric
Clo,
Cll,
Sll
C21,$21,C20,C22
GM
h,k
Physical effects may be estimated as parameters or reduced from the observations (if not needed). They should be reduced only, if the error of reduction is negligible compared to the accuracy of the measurement. If they are reduced, identical models have to be used. An example are the loading effects which are partly reduced, partly included in the models (e.g., ocean tides), and partly estimated as parameters in the data processing (e.g., atmospheric loading). Some effects may be shifted from one parameter to another (e.g., a constant part in all point
Chapter 101
velocities may be shifted to Earth rotation). Nonparameterised effects may affect other parameters (e.g., in other techniques or other physical effects). Non-modelled seasonal effects may be used as an example for this danger. If the coordinates of satellite tracking stations are only transformed to the observation epoch by linear velocities (as usual), existing seasonal variations may enter into the estimated orbit ephemeris. From the orbits they may affect the sea surface or gravity field determination. Figure 5 shows the gravity variations derived from GRACE and the height variations of the IGS station Brasilia. Both show annual periods with a peak in the months of March/April which may be due to the same physical effects or induced by the parameterisation. As they are not processed jointly, a unique interpretation is not possible. It is indispensable that similar relevant parameters are modelled identically in all steps of the data processing. I
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Fig. 5. Gravity field variation in Brazil from GRACE and height variations at IGS station Brasilia (triangle in the map)
The effects of the Earth System processes to be monitored are in general so small, that highest precision and reliability is required. The targeted relative accuracy and consistency of GGOS products is of the order of 10-9 or better. This is a challenge for the consistency of geometric and gravimetric parameters. A classical product of combined geometry and gravity are physical heights. At present, physical heights cannot be derived directly from the services' results, because geometry and gravimetry use different models, e.g., for reducing the permanent
•
Science Rationale of the Global Geodetic Observing System (GGOS)
tides: H ~ h - N. A unified global vertical reference system is a major objective of GGOS. Combined geometric and gravimetric products are required in many applications (e.g., ESA 2003). Astronautics needs consistent Earth gravity field and orientation models for launch. Flight navigation requires consistent GNSS positioning, gravity (inertia) and geoid (barometric height control during landing). The GPS levelling in surveying requires consistent position and high resolution geoid information. Some engineering constructions (e.g., electron accelerators) require consistent threedimensional positions and height anomalies (geoid heights). GGOS, as the flagship of geodetic science, has to ensure the consistency and reliability of the products for all those applications. Summary
and
Conclusion
There is one unique Earth System. Various signals are transmitted from its system elements and from the dynamic processes between themselves and in interaction with the system's environment. Geodesy is measuring these signals with different sensors (geometric, gravimetric) in different domains (geophysical properties, location, time, spectra, ...). The principal scientific challenge for GGOS is to integrate all the measured signals again to exhaust its full information. The inversion of the complete set of signals into model parameters of the System Earth has to be done in an integrated, but not necessarily centralized, processing using identical constants, conventions and models, related to a unique reference frame, and including all the relevant system components and processes. It shall provide a consistent and reliable image of the system characteristics and dynamics. To achieve the scientific objectives, GGOS needs a structure that allows the coordination of the geodetic activities within IAG, namely the coordination of stations and networks; geodesy-related satellite missions; data and information systems; - conventions, models, analyses; - generated products. These coordination activities have to be performed in very close cooperation with the IAG Services and Commissions. They have the necessary infrastructure and experience. Therefore, the Services and Commissions must be seen as the backbones of GGOS. Geodesy has a responsibility for modem society. It is capable of observing signals from geodynamic and global change processes in a quantitative way.
709
710
H. Drewes
The scientific foundations o f this capability have to be a d v a n c e d in a sophisticated w a y in order to facilitate the d e v e l o p m e n t o f m e t h o d s and p r o c e d u r e s to p r e v e n t d a m a g e due to disasters from natural hazards g e n e r a t e d b y different e l e m e n t s o f the Earth S y s t e m to the extent possible.
References Altamimi, Z., P. Sillard, C. Boucher (2002): ITRF2000: A new release of the International Terrestrial Reference Frame for Earth science applications. J. Geophys. Res. (107) B 10, ETG 2, 1-19, doi: 10.1029/2001JB000561. Andersen, O.B., J. Hinderer, F.G. Lemoine (2005): Seasonal gravity field variations from GRACE and hydrological models. Springer, IAG Symposia, 129, 316-321. Angermann, D., H. Drewes, M. Gerstl, R. Kelm, M. Krtigel, B. Meisel (2005): ITRF combination - status and recommendations for the future. Springer, lAG Symp., 128, 3-8. Beutler, G., H. Drewes, A. Verdun (2004): The new structure of the International Association of Geodesy (IAG) viewed from the perspective of history. J. Geodesy (77) 566-575. Beyerle, G., J. Wickert, T. Schmidt, Ch. Reigber (2004): Atmospheric sounding by global navigation satellite system radio occultation: An analysis of the negative refractivity bias using CHAMP observations. J. Geophys. Res. (109) DO! 01106. Beyerle, G., T. Schmidt, G. Michalak, S. Heise, J. Wickert, C. Reigber (2005): GPS radio occultation with GRACE: Atmospheric profiling utilizing the zero difference technique. Geophys. Res. Lett. (32) L13806, 1-5, doi: 10.1029/2005GL023109.
Bosch, W. (2003): Geodetic application of satellite altimetry. Springer, IAG Symposia, 126, 3-22. Cazenave, A., R.S. Nerem (2004): Present-day sea level change: Observations and causes. Rev. Geophys. (42) RG3001, 1-20. Delouis, B., P. Lundgren, J. Salichon, D. Giardini (2000): Joint inversion of InSAR and teleseismic data for the slip history of the 1999 Izmit (Turkey) earthquake. Geophys. Res. Lett. (27) 3389-3392. Denker, H., W. Torge, G. Wenzel, J. Ihde, U. Schirmer (2000): Investigation of different methods for the combination of gravity and GPS levelling data. Springer, lAG Symposia, 121, 137-142. Drewes, H., O. Heidbach (2005): Deformation of the South American crust estimated from finite element and collocation methods. Springer, lAG Symposia, 128, 544-549. Drewes, H., Ch. Reigber (2004): The IAG project "Integrated Global Geodetic Observing System" (IGGOS) - Setup of the initial phase. IVS 2004 General Meeting Proceedings NASA/CP-2004-212255, 32-37. European Space Agency (2003): ESA's Gravity Mission GOCE. ESA Publication BR-209, Noordwijk. Gambis, D. (2004): Monitoring Earth orientation using space-geodetic techniques: state-of-the-art and prospective. J. Geodesy (78) 295-303. Gross, R. (2001): Gravity, oceanic angular momentum, and the Earth's rotation. Springer, IAG Symp., 123, 153-158. Groten, E. (2001): A discussion of fundamental constants in view of geodetic reference systems. Springer, IAG Symposia, 123, 21-27.
Groten, E. (2004): Fundamental parameters and current (2004) best estimates of the parameters of common relevance to astronomy. J. Geodesy (77) 724-731. Holgate, S.J., P.L. Woodworth (2004): Evidence for enhanced coastal sea level rise during the 1990s. Geophys. Res. Lett. (31) L07305, 1-4. Kern,M., K.P. Schwarz, N. Sneeuw (2003): A study on the combination of satellite, airborne, and terrestrial gravity data. J. Geodesy (77) 217-225. Kusche, J., E.J.O. Schrama (2005): Mass redistribution from global GPS time series and GRACE gravity fields: inversion issues. Springer, lAG Symposia, 129, 322-327. Lemoine, F.G., S.C. Kenyon, J.K. Factor, R.G. Trimmer, N.K. Pavlis, D.S. Chinn, C.M. Cox, S.M. Klosko, S.B. Luthcke, M.H. Torrence, Y.M. Wang, R.G. Willianson, E.C. Pavlis, R.H. Rapp, T.R. Olson (1998): The develpoment of the joint NASA GSFC and NIMA Geopotential Model EGM96. NASA GSFC TP-1998-206861. Pagiatakis, S.D., Ph. Salib: Historical relative gravity observations and the time rate change of gravity due to postglacial rebound and other tectonic movements in Canada. J. Geophys. Res. (108) B9, ETG 1, 1-16, doi: 10.1029/2001JB001676, 2003. Peltier, R. (2001): Earth physics and global glacial isostasy: from paleo-geodesy to space-geodesy. Springer, lAG Symposia, 123, 7-12. Ramillien, G., A. Cazenave, Ch. Reigber, R. Schmidt, P. Schwintzer (2005): Recovery of global time-variations of surface water mass by GRACE geoid inversion. Springer, lAG Symposia, 129, 310-315. Reigber, Ch., Y. Xia, G.W. Michel, J. Klotz, D. Angermann (1997): The Antofagasta 1995 earthquake: Crustal deformation pattern as observed by GPS and D-INSAR. Proc. 3rd ERS Symp. on Space at the service of our environment (ESA SP-414) 507-513. Reigber, Ch., H. Drewes (2004): lAG Project: Integrated Global Geodetic Observing System (IGGOS). In: O.B. Anderson (Ed.), Geodesist's Handbook 2004, J. Geodesy (77) 673-676. Richter, B., S. Zerbini, F. Matonti, D. Simon (2004): Longterm crustal deformation monitored by gravity and space techniques at Medicina, Italy and Wettzell, Germany. J. Geodynamics (38) 281-292. Rothacher, M. (2003): Towards a rigorous combination of space geodetic techniques. IERS Techn. Note 30, 7-18. Rothacher, M., J. Campbell, A. Nothnagel, H. Drewes, D. Angermann, D. Griinreich, B. Richter, Ch. Reigber, S.Y. Zhu (2003): Integration of space techniques and establishment of a user center in the framework of the International Earth Rotation and Reference Systems Service (IERS). Geotechnologien Science Report No. 3, 137-141. Rummel, R., H. Drewes, G. Beutler (2002): Integrated Global Geodetic Observing System (IGGOS): A candidate lAG project. Springer, IAG Symposia, 125, 609-614. Schmidt, M., J. Kusche, J.P. van Loon, C.K. Shum, S.-C. Han, O. Fabert (2005): Multi-resolution representation of a regional geoid from satellite and terrestrial gravity data. Springer, lAG Symposia, 129, 167-172. Visser, P.N.A.M., E.J.O. Schrama (2005): Space-borne gravimetry: determination of the time variable gravity field. Springer, lAG Symposia, 129, 6-11.
Chapter 102
GGOS and it User Requirements, Linkage and Outreach H.-P. Plag, Nevada Bureau of Mines and Geology and Seismological Laboratory, University of Nevada, Reno, Mailstop 178, Reno, NV 89557, USA, email: [email protected].
Abstract
and increasing international transport demand uniform geodetic reference frames as a basis. In fact, such a The Global Geodetic Observing System (GGOS) is go- reference frame, sufficiently accurate, homogeneous in ing to be the interface between the IAG services and a space and time, and easy to access and use, is a prewide range of external users ranging from the Global requisite for an effective support of many of the existEarth Observation System of Systems (GEOSS) over ing and emerging applications. Despite a general unthe European Global Monitoring for Environment awareness of the public of geodesy and its services, the and Security (GMES) Programme and the Themes current economic benefit of the geodetic infrastructure of the Integrated Global Observing Strategy Partner- and the future potential is astonishingly large: a reship (IGOS-P) to practical application in monitoring cent study of the requirements for the national geodetic of infrastructure and the control of processes. Taking infrastructure in Canada (Williams et al., 2005) conthe user requirements (UR) compiled by the different cluded that the Canadian Spatial Reference System IGOS-P Themes and GEOSS as a starting point, a de- (CSRS) contributes directly 60 to 90 billion dollars antailed set of URs for the GGOS products is derived. nually (7 to 10%) to Canada's Gross Domestic ProdThe URs are specified in terms of spatial and temporal uct and several economic fields depend heavily on the resolution, accuracy, and latency. The analysis of the CSRS. The study also diagnosed that geodesy is in a URs reveals that GGOS is facing a rather demanding revolution based on the ability to determine highly acset of URs ranging from high accuracy observations curate and reliable point coordinates in a global referof the Earth's surface displacements in near-real time ence frame 'ad hoc', emphasizing the importance of to long-term observations of Earth system parameters the International Terrestrial Reference Frame (ITRF). such as gravity, Earth's rotation and sea level over sevGeodesy also is fundamental in the frame of Earth eral decades. observation as the provider of both the global reference In order to integrate GGOS into on-going pro- frame and observations of the fundamental geodetic grams such as GEOSS and IGOS-P and to enable quantities, that is the Earth's geometry, its gravity field users to fully exploit the products and services offered and its rotation (see Plag et al., 2006, for a more deby GGOS, outreach activities are a central issue for tails). Recently, this has been widely acknowledged GGOS, including educational measures. A suggested in the process of setting up the Global Earth ObserIGOS-P Theme centered around mass transports in the vation System of Systems (GEOSS, see GEO, 2005b). Earth system and the associated dynamics has consid- The geodetic quantities and their temporal variations erable potential to achieve the integration of GGOS are observed with space-geodetic techniques using a into the main global Earth observation programs and combination of space-borne and air-borne sensors and to enable the full implementation of GGOS. in-situ networks. After three decades and an increase in accuracy of more than three orders of magnitude Keywords:Global Geodetic Observing System, User (Chao, 2003), the space-geodetic techniques are apt requirements', Global Earth Observation System of to observe the integrated mass transport in the Earth Systems, Integrated Global Observing Strategy system and particularly the global water cycle, as well as the dynamics of the system and the kinematics of the surface with unprecedented accuracy. The geode1 Introduction tic observations thus provide a truly global monitoring Geodesy is in a transition of methods brought about of mass movements as well as the associated Earth sysby the advent of space-geodetic techniques and tem dynamics. the combination of these with rapidly improving The last few years have seen a rapid programmatic communication technologies and capacities. The development in Earth observations on global scale. development in computer technology, data commu- Following up the recommendations of the Johannesnication, satellite based positioning and navigation burg conference, the first Earth Observation Summit creates many new opportunities and increases the (EOS-I) was held in Washington, DC, in July 2003 and demand for geographical information technologies. initiated an unprecedented global effort towards coorCooperation between countries, globalisation, dination of global Earth observation. Through its dec-
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laration (see Annex 1 in GEO, 2005b), EOS-I established the ad hoc Group of Earth Observation (GEO) with the task to draft a 10-year Implementation Plan for the GEOSS. Subsequently, this ad hoc GEO met six times, and, supported by several Subgroups, the requested plan was drafted (GEO, 2005a) together with a reference document containing many details of the vision for GEOSS (GEO, 2005b). This implementation Plan was adopted by EOS-|II in February 2005 in Brussels. The work of GEO was guided by the Framework document adopted by the EOS-II, held in Tokyo in April 2004 (see Annex 2 in GEO, 2005b). This Framework document identifies nine major societal benefit areas of Earth observations (Box 1). The International Association of Geodesy (lAG) over the last decade has set up a number of successful technique-specific services that provide valuable observations and products to scientific and increasingly non-scientific users (for a brief overview, see Plag et al., 2006). The internationally coordinated geodetic observations carried out by the lAG Services result in a global terrestrial reference frame, which is determined and monitored on the basis of observations provided continuously by the geodetic station networks. This well-defined, long-term stable, and highly accurate reference frame is the basis for all precise positioning on and near the Earth's surface. It is the indispensable foundation for all sustainable Earth observations, in situ, as well as air-borne and space-borne. Recognizing the need for a consistent treatment of the geodetic observations across technique-specific boundaries in a Earth system approach and also the need for a unique interface for users to the geodetic products, the lAG in 2003 established the Global Geodetic Observing System (GGOS, see next section). The presence is dominated by the first steps towards an implementation of GEOSS. lAG is involved in this process in order to ensure that GGOS is developed consistently with the needs and progress of GEOSS for a maximum mutual benefit. The development of the scientific basis and the implementation of GGOS needs to be guided by a well defined and consistent set of user requirements (UR). The level to which these URs are met by the available observing system, including the generation of higher level products, is an important indicator for the assessment of global, regional and national geodetic infrastructure, in several countries, assessment studies have been carried out (e.g. Williams et al., 2005; Plag, 2006). However, a thorough specification of the global geodetic infrastructure that would meet a wide range of user requirements is still missing. A first step towards such a specification is to set up specific URs in
terms of accuracy, resolution, latency, and availability for the geodetic observations and products. in the next Section, the role of GGOS as an interface to global geodesy will be discussed briefly. In Section 3, the URs particularly from Earth Observation and scientific applications will be considered. The linkage of GGOS to its main user groups is sketched in Section 4, and in Section 5 recommendations related to the implementation of GGOS are given.
2
G G O S : the e x t e r n a l i n t e r f a c e to users
GGOS is envisaged as the unique interface for a wide range of external users. A main task for GGOS will be to ensure that this interface is fully interoperable with other systems contributing to GEOSS. Considering that IGOS-P will be a major driver in the development of the scientific basis for Earth observation systems, GGOS is intended to contribute to the IGOS-P Themes (see Plag et al., 2006). Internally, GGOS will facilitate a fully consistent data processing, quality control and modeling. GGOS also has the task to advocate standardization of products for a better service for the users. GGOS is envisaged to be an umbrella for the existing services. Like GEOSS, GGOS has the important task to identify gaps and deficiencies and to facilitate the necessary steps to close the gaps and remove the deficiencies. GGOS also has the task to identify the external users and their requirements, as well as to promote new products and the transition from data to information, as required by the users. The structure of GGOS will have to reflect this set of tasks very clearly.
3
The
users
and
their
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ments For Earth observations in general, the awareness of the benefit of the observations and knowledge of the requirements decreases with distance to the observation system (G. Foley, 2004, personal communication). Scientists involved in studies of the Earth system are mostly fully aware of their needs with respect to Earth observation. However, the end users involved in decision making normally show little awareness of the requirements for Earth observations. For GGOS and the geodetic contribution to Earth observation, this lack of explicit knowledge of the requirements is even more pronounced. In fact, geodesy faces specific challenges with respect to the UR. On
Chapter 1 0 2
the one hand, users are often not aware of their needs with respect to geodetic observations and products. They are often not aware of the fact that they are using tools that would not be possible or less practical without geodesy providing crucial input. On the other hand, IAG services and GGOS evolve in a mainly scientific environment (affiliation to IUGG and ICSU) without clear links to an increasing group of nonscientific users. The ITRF is the basis not only for scientific applications but also for Earth observation and, increasingly, other applications, though often not recognized by the users. With respect to reference frames, the user groups and URs are fairly well known internally in IAG and Earth science, but far less externally in the wider range of societal applications. For Earth system observations, user groups are less known, and URs in the frame of GEOSS are unclear, particularly for long-term observations. The Subgroup User Requirements and Outreach of the ad hoc GEO discussed the URs for Earth observations, and a number of UR studies were stimulated, mainly on national level. Generally, the goal of these studies was to identify the extent to which Earth observations are required for societal applications. For example, an extensive inquiry carried out in Canada revealed a clear need for Earth observations across a broad range of societal areas, but the results were complex and hard to interpret in terms of quantitative requirements (B6chard, 2005, personal communication). Access to highly accurate geodetic positions is fundamental for many scientific and non-scientific applications. This is equivalent to requiring access to an unique, technique-independent reference frame decontaminated for short-term fluctuations due to global Earth system processes. Providing instantaneous and ad hoc access to highly accurate positions in such a unique, global, long-term stable reference frame would considerably ease present applications and support many new applications, particularly if combined with the rapidly developing communication tools and geo databases. Global Navigation Satellite Systems (GNSS) techniques are, in principle, able to provide such positions relative to a unique, global reference frame ad hoc, that is without simultaneous measurements at local reference points. However, only the integration of the space-geodetic techniques into a consistent system monitoring the Earth surface kinematics, rotational perturbations and gravity field changes will eventually enable the realization of the reference frame as well as the determination of the surface velocity field with sufficient accuracy and long-term stability required for the utilization of the full potential of ad hoc positioning,
•
GGOSand Its User Requirements, Linkage, and Outreach
GEO advocates an strongly user-driven approach to the implementation of GEOSS (see GEO, 2005b), which may serve as a guideline for GGOS. GEO recommends to establish and maintain a distinct and common UR database. The database will be oriented on the CEOS/WMO database of URs and observation system capabilities, and it will be ensured that the database provides a mechanism for the analysis of gap. GEO will use the WMO approach of Rolling Review of Requirements (RRR) as a basis. The study of the URs for geodetic observations and products has to address three key areas, namely • Earth observation for sustainable development, which includes a global component that allows the derivation of information on all spatial scales from global to local and from short time scales of warnings for extreme events and disasters to long-term predictions;
• scientific applications that study the Earth system on all spatial and temporal scales; and • non-scientific applications including surveying on land and in the ocean, mapping of the Earth's surface, steering of processes, monitoring of infrastructure and environmental parameters, and navigation.
Here we will focus mainly on the requirements resulting from Earth observation and scientific applications. The former can be derived on the basis of the requirements of the nine benefit areas identified by GEO (see Box 1) as well as the requirements discussed in the frame of the IGOS-P Themes (see Nag et al., 2006). The latter are linked to key scientific problems currently studied in the wide range of Earth sciences. A discussion of the URs from non-scientific applications can be found in e.g. Plag (2006). Currently, there is a broad political and scientific consensus that comprehensive monitoring of the Earth system is a crucial prerequisite for sustainable development. The monitoring techniques need to be developed within the research community and transformed into operational activities. The necessary properties of a sustainable monitoring include long-term stability, operational mode, homogeneity in time, multiparameter sites, global coverage and participation, and integrated observation and data sets. The GEO Reference Document (GEO, 2005b) provides for each of the nine benefit areas an overview of the requirements in terms of observable and status of the observational capacity. Many of these requirements include or depend on quantities provided by GGOS (see Table 1 in Nag et al., 2006). The next step is to convert the information provided in the
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- Disaster: reducing loss of life and property from natural and human-made disasters Health: understanding environmental factors affecting human health and well being -Energy resources: improvingmanagementof energy resources understanding, assessing, predicting, mitigating, and adopting to climate variability and change - Water: improvingwater resource managementthrough better understanding of the water cycle - Weather: improvingweather information, forecasting, and warning Ecosystems: improvingthe managementand protection of terrestrial, coastal, and marine ecosystems supporting sustainable agriculture and combating desertification Biodiversity: understanding, monitoring and conserving biodiversity -
-Climate:
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-Agriculture: -
Box 1: The nine societal benefit areas identified by EOS-II (see Appendix 2 in GEO, 2005b). GEOSS Reference document into URs in terms of accuracy, latency and resolution. A number of the requirements are being discussed in the frame of the IGOS-P Themes. Currently, IGOS-P has several approved themes and others are in the planning or proposal stage (see Plag et al., 2006). These themes address different components of the Earth system, different processes, or different societal issues. In general, all these themes will address space-borne or air-borne observations that require highly accurate positioning of the sensors. Thus, the themes are linked to the global geodetic networks through their requirements for access to an accurate and stable reference frame. The relevant themes potentially benefiting to a considerable extent from observations of the fundamental geodetic quantities are those addressing dynamics and mass transport in the Earth system or being affected by these processes (see Plag et al., 2006). Here we emphasize only two of the themes, namely the Geohazards and the Global Water Cycle Themes. Geohazards such as earthquakes, volcanic eruptions, landslides, subsidence, and precarious rocks are intimately connected to displacements and deformations of the Earth's surface (Marsh & the Geohazards Theme Team, 2004). Thus, key observing techniques for an anticipated integrated solid Earth observing system complementing the existing systems such as GOOS, GCOS and GTOS would have to be the geodetic techniques capable of observing surface displacements on local to global scales at the highest possible accuracy. The existing global and regional geodetic networks, in fact, could provide the basis for such a Solid Earth Observing System (SEOS). For the purpose of the Geohazards Theme, SEOS would have to rely on a strong global component providing the stable reference frame as well as the technologies and products required to get easy and reliable access to this frame. This is one of the prime objectives of GGOS. The required frame and associated products are today partly available through the IERS and the other lAG services, particularly the IGS. However, the part of SEOS dedicated to geohazards would also have to
be flexible in spatial and temporal resolution, as well as readiness on demand. In many parts of the world, dedicated ground-based networks are needed, some of them on temporary basis (e.g. at certain volcanoes, in areas with unstable slopes and large rocks), which could be established under the guidance of GGOS in coordination with the Geohazards Theme. Of particular importance for monitoring surface deformations with high spatial resolution is lnterferometric Synthetic Aperture Radar (InSAR). The establishment of an international InSAR service has been suggested as an important step towards coordination, capacity building, and the transition to operational applications. A link between InSAR and the global terrestrial reference frame through integration of GNSS and InSAR is a step that would greatly improve the applicability of InSAR for monitoring of surface displacements. A major process moving mass throughout the Earth system is the global water cycle. Changes in the distribution of water stored on land, in the ocean and in the atmosphere affect geodetic observations related to the time-variable gravity field, shape and rotation of the Earth. At time scales of months to a decade, loading of the solid Earth by fluids dominates non-secular variations in the geodetic observables. Space geodetic observations on surface mass variability are inherently strong at the regional to global scale and thus provide a unique tool to complement traditional in-situ measurements of terrestrial water storage. The Global Geophysical Fluid Center (GGFC) of the IERS was established in acknowledgment of the interactions between the solid Earth and its fluid envelop and the necessity to understand these interactions in order to improve the interpretation of the geodetic observations. Thus, the satellite gravity field mission, already provide new insight into the motion and storage of water in the different components of the Earth system. InSAR is increasingly applied to monitor surface displacements induced by changes in groundwater levels. The GNSS are increasingly used to extract information on atmospheric water contents from regional
Chapter102 networks. The Water Cycle Theme report (Lawford & the Water Theme Team, 2004) acknowledges that a strategy for a water cycle observing system will have to rely to a large extent on contributions from geodetic techniques, and in particular, from GGOS. Moreover, geodetic techniques such as in-situ gravity measurements and lnSAR will also have to be integrated, particularly for the management of water resources. The report Living on a Restless Planet (Solomon & The Solid Earth Science Working Group, 2002) gives an excellent overview of the many scientific problems that need to be solved in order to better understand the Earth system processes that affect human well-being. The understanding of these processes and their interactions is a prerequisite for sustainable development. With respect to the Earth system processes, a number of scientific questions and problems can be identified, that are associated with mass transport and thus with changes in the gravity field and displacements of the solid Earth's surface. The certainly incomplete list given in Box 2 is indicative of the scientific problems that benefit directly from geodetic observations. Based on the current state of the art in the different areas, it is possible to derive specific requirements in terms of accuracy, spatial and temporal resolution, and long-term reproducibility for the geodetic quantities that would allow for an improvement in our scientific understanding of the problems. Table 1 compiles some of these requirements given at a high level. The list of relevant quantities includes but is not limited to 3-d displacements and velocities of the Earth's surface, strain rates, the static geoid, temporal variations of the Earth's gravity field, motion of the geocenter with respect to the origin of the reference frame, and perturbations of the Earth's rotation. For most scientific applications requiring knowledge of the Earth surface kinematics, we have identified the accuracy requirement to be of the order of 1 mm/yr or better. Similarly, using ad hoc positioning for the determination of coordinates in a national reference frame also requires knowledge of the velocity field with an accuracy of 1 mm/yr in all three components. Monitoring of infrastructure and hazardous areas have the same requirement on the accuracy of the velocity field. Moreover, the positioning of sensors (e.g. in airborne gravimetry, hydrographic survey, satellite altimetry) has similar accuracy requirements. The static geoid is relevant for studies of ocean circulation (mean dynamic sea surface topography, MDT), as well as the mass distribution in the interior of the Earth (see e.g. Ilk et al., 2005). In both cases, the accuracy requirement are better than 1 cm,
• GGOSand Its User Requirements, Linkage, and Outreach
and ocean circulation studies require spatial resolution down to small scales of a few kilometers. Similarly, for the full utilization of satellite altimetry a geoid accurcay of 1 cm for wavelength down to a few tens of km is required (Drinkwater et al., 2003), translating into an accuracy of 10 .9 or better, in order to monitor the mass movements in the Earth system and particular the global water cycle, accuracy requirements are on the order of 10 mm of equivalent water column for spatial wave length of 500 km, which translates into 0.2 mm in geoid height and 0.3 #Gal for gravity. Temporal resolution is of the order of 1 month. Temporal variation in the Earth's gravity field appears to be the most important parameter for monitoring the mass transport in the Earth system on global to regional scales, which is particularly relevant for the global water cycle, including variations in ocean circulation, water storage on land, sea level changes, ice load changes, etc. (see e.g. |lk et al., 2005). The currently accepted accuracy requirements are better than 10 - 9 with a tendency to 10 - 1 0 (e.g. for ice load changes and ocean circulation changes), and spatial resolution of a few hundred kilometers. Mass transport in the global water cycle shows strong variations at sub-seasonal scales, and temporal resolution of several weeks are considered as a reasonable requirement. Two prominent application of sea level observations are the study of climate change, where the volume and mass changes of the ocean are of importance for studies of the global water cycle, and impact studies, where scenarios for future local sea levels are required. Both applications have demanding requirements with respect to the relation between reference frame origin and the Center o f Mass (CM) of the Earth system. For global sea level studies, an accuracy of 0.1 mm/yr is required for the tie of the frame orgin to the CM, while local studies of sea level often require an accuracy of 0.5 mm/yr for vertical land motion. A rather crucial limitation for sea level studies and, more general, applications in both Earth observations and scientific studies originates from a significant uncertainty in the relation between ITRF and the CM. The ITRS is defined to have its origin in the CM. However, due to the particular sensitivity of the different techniques with respect to geocenter variations, there is a time-dependent difference between the origin of ITRF and the CM. The effect of this unknown differential movement, which is estimated to be of the order of 2 mm/yr (see Fig. 1), on observations of surface displacements, sea level, and comparative studies between local gravity changes and vertical displacements is severe, as it demonstrated in Fig. 1. For the example shown there, the effect on global mean sea level
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Convection: Are the anomalies in seismic velocities detected by seismic tomography in the Earth's mantle due to chemical anomalies or temperature anomalies? This is equivalent to the question whether convection is throughout the whole mantle or layered. Plate tectonics: the location of and the processes at plate boundaries still pose several questions. Likewise, the extent of deformation zones is uncertain in many regions of the Earth surface. Ice sheets/glaciers and sea level: there are large uncertainties with respect to the ice load history, in particular, for Antarctica. The present-day changes in ice sheets are still not known, even to the sign. Consequently, their contribution to sea level changes are highly uncertain. The global global ocean volume and mass changes are not well constrained. Post-glacial rebound: the appropriate rheology of the Earth mantle and its dependency on time scales is not well understood. Ocean circulation: improved monitoring is required in order to separate steric and non-steric components, and to determine absolute circulation. Hydrological cycle: better quantification of the fluxes between the different reservoirs is required. How large are groundwater movements? What are the variations in continental water storage? Seasonal variations: What is the contribution from the terrestrial hydrosphere? For the cryosphere: what is the seasonal mass balance? For sea level: what part of the seasonal variations is steric and what non-steric? Atmospheric circulation: reconstruction of past wind fields on the basis of Earth rotation? Past and present air pressure field? Tides: validation of ocean tide models. Seismic waves and free oscillations: structure and mechanical parameter of the solid Earth?
Box 2: Selected scientific p r o b l e m s requiring geodetic observations. T a b l e 1: U R s for selected scientific applications. S.R. stands for spatial resolution, T.R. for Temporal resolution, Fr. stands for F r a m e , where we distinguish L: local frames, N: national frames, G: global frame. R. stands for Reproducibility and gives the time w i n d o w over w h i c h the parameters are expected to be reproducible with the stated accuracy. M o r e detailed r e q u i r e m e n t s for the geoid and gravity field can be found in e.g. D r i n k w a t e r et al. (2003). N o t e that an accuracy o f 10 - 9 for the geoid translates into an accuracy o f 1 cm. Application Mantle convection and plate tectonics Postglacial rebound
Climate change, including present changes in ice sheets and sea level
Ocean circulation Hydrological cycle Seasonal variations
Atmospheric circulation Earth tides
Surface loading Seismotectonics Volcanoes Earthquakes, tsunamis
Parameter 3-d velocities static geoid secular strain rate 3-d velocities
Accuracy < 1 mm/yr < 10 - 9 10-15 S-- 1 < 1 mm/yr
S.R. n/a n/a
Ft. G G G G
R. several decades and longer
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n/a 102 km n/a 2 to 10 • 102 km 10 .). km < 102 km < 102 km 200 km
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Trend (mm/yr) Fig. 1: Effect of differential motion between the origins of ITRF and IGP-00 on local vertical motion. The vertical motion is given for a differential velocity between the origin of ITRF2000 and IGS-P00, that is a reference frame determined by GPS alone, which is estimated by Kierulf & Plag (2006) as ~7 = (-1.5,-2.2, -2.1) mm/yr. The mean vertical motion over the complete surface of the ocean is 0.4 mm/yr.
estimates would be as large as 0.4 mm/yr, which is far above the uncertainties discussed in the IPCC assessments. Requirements on Earth rotation result mainly from scientific applications. For these applications, an increasing accuracy of the observations normally leads to new applications.
4
Linkage and Outreach activities
Besides maintaining the links to the scientific organizations in IUGG, the implementation of GGOS in agreement with the URs requires firm links to the Earth observation environment, and in particular GEO, the IGOS-E the relevant UN agencies and other international programs. The links to GEO are well established, (a) through IAG being a participating organization in GEO, with the representative coming from GGOS. Moreover, IAG has representatives in all relevant GEO Working Groups, and these representatives are recruited from appropriate GGOS Working Groups. The relationship between GGOS and IGOS-P is currently being progressed towards a full membership of GGOS in IGOS-P (see Plag et al., 2006, for more details). It is expected that this process eventually will lead to the establishment of a 'Earth System Dynamics Theme', which will have a major contribution from GGOS. Moreover, the links to the existing IGOS-P Themes deserve considerable attention, too.
• GGOSand Its User Requirements, Linkage, and Outreach
However, GGOS will have to maintain on a high level outreach activities in order to increase the awareness in the societies of the importance of the geodetic contribution to science, Earth observation and many non-scientific applications. In particular, links to and formal acknowledgments through relevant UN agencies have to be established. It can be expected that both the increased awareness of the importance of the geodetic contribution, which has already been achieved in the process of GEO, as well as the formal acknowledgment through GEO, IGOS-E and eventually UN agencies, will help to secure the funding for the operational activities in GGOS. The process of establishing a comprehensive UR database and converting these in quantitative system specifications is an important tool for the identification of gaps and deficiencies in the current system. An example is the uncertainty in the relation between reference frame origin and geocenter, which is mainly caused by the low number of SLR stations, their insufficient geographical distribution, and a lack of loworbit SLR targets. The channels through GEO can be used to raise the awareness for this crucial gap in the geodetic observing system, and eventually also the funds to close this gap. In its outreach activities, GGOS also needs to educate the users both with respect to their requirements and the benefits from the geodetic observations and products. Knowledge of the geodetic standards and conventions is important for using the products and therefore, it is crucial to support the dissemination. The implementation of GGOS most likely will have to be based on regional implementation, similar to the structures of GOOS, GCOS and GLOSS. Such components are already developing, e.g. the European Combined Geodetic Network (ECGN) and the Nordic Geodetic Observing System (NGOS), and the links to the regional implementations need to be well maintained and formalized. GGOS also will have to be open for new developments in observational techniques. An example is InSAR, which receives considerable attention from GEO, while GGOS has not fully recognized the potential of InSAR, particularly in combination with GNSS. The increasing need for early warning systems for natural and man-made hazards in order to prevent or mitigate disasters requires also the integration of geodetic observations and products in these systems. Here, too, GGOS outreach activities are needed in order to avoid unnecessary duplication of efforts.
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Conclusions and Recommendations There is an emerging consensus that geodetic techniques are indispensable for Earth observation systems. GGOS coordinates an impressive number of operational global networks for monitoring displacements, gravity variations and Earth's rotation variations. With the ITRF, geodesy provides the metrological basis for all Earth observations. Moreover, GGOS provides observations related to the dynamics of the Earth, relevant in particular (but not only) for geohazards and the understanding of the global water cycle. Nevertheless, many users are (still) not fully aware of the potential of geodetic observations. The URs originating from the nine benefit areas of GEO and from open scientific questions are demanding and not all can be met with the currently available geodetic observing system. GGOS therefore should use the URs to systematically identify the gaps in the observing systems and the deficiencies in products with the goal to improve the system so that the URs are eventually met. In order to promote the awareness of the geodetic contribution outside of the traditional scientific environment, it is recommended that GGOS establishes strong links to IGOS-P and the IGOS-P Themes. One element in these links could be a lAG Commission or GGOS WG on geodesy and geohazards, which would not only provide a formal interface to the IGOS-P Geohazards Theme but also to international programs such as the International Strategy for Disaster Reduction (ISDR) of the UN. Acknowledging the importance of InSAR as a technique for observations of surface deformations with high spatial resolution, it is recommended that GGOS initiates or facilitates the establishment of an InSAR service as one component in GGOS. GGOS also should promote work towards combined analysis techniques for GNSS and InSAR observations.
References Chao, B. F., 2003. Geodesy is not just for static measurements anymore, EOS, Trans. Am. Geophys. Union, 84(16), 145-150. Drinkwater, M. R., Floberghagen, R., Haagmans, R., Muzi, D., & Popescu, A., 2003. GOCE; ESA's first Earth Explorer Core mission, in Earth Gravity Fieldfrom Space -from Sensor to Earth Sciences, edited by G. B. Beutler, M. R. Drinkwater, R. Rummel, & R. von Steiger, vol. 18 of Space Science Series of ISSI, pp. 419-432, Kluwer Academic Publishers, Dordrecht.
GEO, 2005. The Global Earth Observing System of Systems (GEOSS) - 10-Year Implementation Plan, Distributed at and endorsed by the EOS-III Meeting, Brussels, 16 February 2005. Document prepared by the Ad hoc Group on Earth Observations (GEO) Impementation Plan Task Team IPTT. Avaliable at http://earthobservations.org. GEO, 2005. Global Earth Observing System of Systems GEOSS - 10-Year Implementation Plan Reference Document- Draft, Tech. Rep. GEO 204/ESA SP 1284, ESA Publication Division, ESTEC, PO Box 299, 2200 AG Noordwijk, The Netherlands, Final Draft Document 204, prepared by the Ad hoc Group on Earth Observations (GEO) Impementation Plan Task Team IPTT. Available at http ://earthobservations.org. Ilk, K. H., Flury, J., Pummel, R., Schwintzer, P., Bosch, W., Haas, C., Schr6ter, J., Stammer, D., Zahel, W., Miller, H., Dietrich, R., Huybrechts, P., Schmeling, H., Wolf, D., G6tze, H. J., Riegger, J., Bardossy, A., Gfinter, A., & Gruber, T., 2005. Mass transport and mass distribution in the earth system, Tech. rep., GOCE-Projectbfiro Deutschland, Technische Universitfit Mfinchen, GeoForschungsZentrum Potsdam. Kierulf, H. P. & Plag, H.-P., 2006. Precise point positioning requires consistent global products, in EUREF Publication No. 14, edited by J. A. Torres & H. Hornik, vol. BKG 35 of Mitteilungen des Bundesamtes flit Kartografie und Geodiisie, pp. 111-120, Bundesamtes ffir Kartografie und Geodfisie, Frankfurt am Main. Lawford, R. & the Water Theme Team, 2004. A Global Water Cycle Theme for the IGOS Partnership, Tech. rep., IGOS integrated Global Observing Strategy, Report of the Global Water Cycle Theme Team, April 2004, available at http://www.igospartners.org. Marsh, S. & the Geohazards Theme Team, 2004. Geohazards Theme Report, Tech. rep., IGOS Integrated Global Observing Strategy, Available at http ://www.igospartners.org. Plag, H.-P., 2006. National geodetic infrastructure: status today and future requirements, Project report, Nevada Bureau of Mines and Geology, University of Nevada, Reno, Project report, in review. Plag, H.-P., Beutler, G., Forsberg, R., Ma, C., Neilan, R., Pearlman, M., Richter, B., & Zerbini, S., 2006. Linking the Global Geodetic Observing System (GGOS) to the Integrated Global Observing Strategy Partnership (IGOS-P) through the Theme 'Earth System Dynamics', in Proccedings of the IAG Meeting Cairns, 2005, this proceedings. Solomon, S. C. & The Solid Earth Science Working Group, 2002. Living on a restless planet, NASA, Jet Propulsion Laboratory, Pasadena, California, also available at http ://solidearth.jpl.nasa.gov. Williams, D., Rank, D., Kijek, R., Cole, S., & Pagiatakis, S., 2005. National geodetic infrastructure requirements study, Study report NRCan 03 - 0628, Natural Resources Canada, Final unpublished Report, prepared by BearingPoint; available on request from NRC.
Chapter 103
GGOS Working Group on Ground Networks and Communications M. Pearlman
Harvard-Smithsonian Center for Astrophysics (CfA), Cambridge, MA 02138, USA Z. Altamimi Institut G6ographique National, 77455 Marne-la-Vallee, France N. Beck Geodetic Survey Division- Natural Resources Canada, Ottawa, ON K1A OE9, Canada R. Forsberg Danish National Space Center, DK-2100 Copenhagen, Denmark W. Gurtner Astronomical Institute University of Bern, Bern, CH-3012, Switzerland S. Kenyon National Geospatial-Intelligence Agency, Arnold, MO 63010-6238, USA D. Behrend, F.G. Lemoine, C. Ma, C.E. Noll, E.C. Pavlis NASA Goddard Space Flight Center, Greenbelt MD 20771-0001, USA Z. Malkin Institute of Applied Astronomy, St. Petersburg, 191187, Russia A.W. Moore, F.H. Webb, R.E. Neilan Jet Propulsion Laboratory, California Institute of Technology, Pasadena CA 91109, USA J.C. Ries Center for Space Research, The University of Texas, Austin TX 78712, USA M. Rothacher GeoForschnungsZentrum Potsdam, Potsdam, D- 14473, Germany P. Willis Institut Gdographique National, 94160 Saint-Mande, France Jet Propulsion Laboratory, California Institute of Technology, Pasadena CA 91109, USA Properly designed and structured ground-based geodetic networks materialize the reference systems to support sub-mm global change measurements over space, time and evolving technologies. Over this past year, the Ground Abstract.
Networks and Communications Working Group (GN&C WG) has been organized under the Global Geodetic Observing System (GGOS) to work with the IAG measurement services (the IGS, ILRS, IVS, IDS and IGFS) to develop a
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strategy for building, integrating, and maintaining the fundamental network of instruments and supporting infrastructure in a sustainable way to satisfy the long-term (10-20 year) requirements identified by the GGOS Science Council. Activities of this Working Group include the investigation of the status quo and the development of a plan for full network integration to support improvements in terrestrial reference frame establishment and maintenance, Earth orientation and gravity field monitoring, precision orbit determination, and other geodetic and gravimetric applications required for the long-term observation of global change. This integration process includes the development of a network of fundamental stations with as many co-located techniques as possible, with precisely determined intersystem vectors. This network would exploit the strengths of each technique and minimize the weaknesses where possible.
Keywords. GGOS, GEOSS, IAG, GPS, SLR, VLBI, DORIS, Gravity, Tides, Geoid 1 Introduction The Ground Networks and Communications Working Group (GN&C WG) of the Global Geodetic Observing System (GGOS) is charged with developing a strategy to design, integrate, and maintain the fundamental space geodetic network. In this report, we review the significance of geodetic networks and the GGOS project. We also summarize the present state of, as well as future improvements to, and requirements on space geodetic networks, services, and products. The approach of the GN&C WG and preliminary conclusions follow. 1.1 Significance of the Terrestrial Reference Frame Space geodesy provides precise position, velocity and gravity on Earth, with resolution from local to global scales. The terrestrial reference system defines the terrestrial reference frame (TRF) in which positions, velocities, and gravity are reported. The reference surface for height reckoning, the geoid, is defined through the adopted gravity model, which is referenced to the TRF. The TRF is therefore a space geodesy product that links each of these observable quantities to other geophysical parameters on Earth. Its position, orientation and evolution in space and time are the basis through which we connect and compare such measurements over space, time, and evolving technologies. It is the means by which
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we verify that observed temporal changes are geophysical signals rather than artifacts of the measurement system. It provides the foundation for much of the space-based and ground-based observations in Earth science and global change, including remote monitoring of sea level, sea surface and ice surface topography, crustal deformation, temporal gravity variations, atmospheric circulation, and direct measurement of solid Earth dynamics. A precise TRF is also essential for interplanetary navigation, astronomy and astrodynamics. The realization of the TRF for its most demanding applications requires a mix of technologies, strategies and models. 1.2 The Role of GGOS In early 2004 under its new organization, the International Association of Geodesy (IAG) established the GGOS project (www.ggos.org) to coordinate geodetic research in support of scientific applications and disciplines (Rummel, 2002). GGOS is intended to integrate different geodetic techniques, models and approaches to provide better consistency, long-term reliability, and understanding of geodetic, geodynamic, and global change processes. Through the IAG's measurement services (IGS 1 , ILRS 2 , IVS 3 , IDS 4 , IGFS 5, and future IAS 6), GGOS will ensure the robustness of the three aspects of geodesy: geometry and kinematics, Earth orientation, and static and time-varying gravity field. It will identify geodetic products and establish requirements on accuracy, time resolution, and consistency. The project will work to coordinate an integrated global geodetic network and implement compatible standards, models, and parameters. A fundamental aspect of GGOS is the establishment of a global network of stations with colocated techniques, to provide the strongest reference frames. GGOS will provide the scientific and infrastructural basis for all global change research and provide an interface to geodesy for the scientific community and to society in general. GGOS will strive to ensure the stability and ready access to the geometric and gravimetric reference frames by establishing uninterrupted time series of state-of-the-art global observations. 1 International GNSS Service, formerly the International GPS Service 2 International Laser Ranging Service 3 International VLBI Service for Geodesy and Astrometry 4 International DORIS Service 5 International Gravity Field Service 6 International Altimeter Service
Chapter103 • GGOSWorkingGroupon GroundNetworksand Communications 1.3 Role of the G r o u n d Networks and Communications Working Group
The ground network of GGOS is fundamental since all GGOS data and products emanate from this infrastructure. The Charter of the Ground Networks and Communications Working Group (GN&C) within GGOS is to develop a strategy to design, integrate and maintain the fundamental geodetic network of instruments and supporting infrastructure in a sustainable way to satisfy the long-term (10-20 years) requirements identified by the GGOS Science Council. At the base of GGOS are the sensors and observatories situated around the world providing the timely, precise and fundamental data essential for creating the GGOS products. Primary emphasis must be on sustaining the infrastructure needed to maintain evolving global reference frames while at the same time ensuring support to the scientific applications' requirements. Opportunities to better integrate or colocate with the infrastructure and communications networks of the many other Earth Observation disciplines now organizing under the Global Earth Observation System of Systems (GEOSS) should be sought and taken into account (Group on Earth Observations, 2005). Recognizing that the infrastructure and operations collectively contributing to the Services of the IAG are possible solely due to the voluntary contributions of the globally distributed collaborating agencies and their interest in maximized system performance and sustainable long term efficient operations, the Working Group is made up of representatives of the measurement services plus other entities (e.g., IERS, ITRF and data center and analysis communities) that are critical to guiding the activities of the Working Group. 2 Global Geodetic Network Infrastructure
All infrastructure, and resulting analysis and products of GGOS and its constituent services are made possible through the goodwill voluntary contributions of national agencies and institutions and are coordinated by the IAG governance mechanisms. The ground network of GGOS includes all the sites that have instruments of the lAG measurement services either permanently in place or regularly occupied by portable instruments. Some sites have more than one space geodesy technique co-located, and knowledge of the precise vectors between such co-located instruments (known as "local ties") is essential to full and accurate use of these co-locations.
Analysis centers use the ground networks' data for various purposes including positioning, Earth orientation parameters (EOP), the TRF, and the gravity field. The ground stations of the satellite techniques provide data for precise orbit determination (POD). The individual sites' reference points of the contributing space geodesy networks are the fiducial points of the TRF. 2.1 lAG M e a s u r e m e n t Services
Each service coordinates its own network, including field stations and supporting infrastructure. Here we will review the current status of each measurement service. 2.1.1 IGS
The foundation of the International GNSS Service (IGS, formerly the International GPS Service) is a global network of more than 350 permanent, continuously operating, geodetic-quality GPS and GPS/GLONASS sites. The station data are archived at three global data centers and six regional data centers. Ten analysis centers regularly process the data and contribute products to the analysis center coordinator, who produces the official IGS combined orbit and clock products. Timescale, ionospheric, tropospheric, and reference frame products are analogously formed by specialized coordinators for each. More than 200 institutes and organizations in more than 80 countries contribute voluntarily to the IGS, a service begun in 1990. The IGS intends to integrate future GNSS signals (such as Galileo) into its activities, as demonstrated by the successful integration of GLONASS. (Beutler et al., 1999). 2.1.2 lENS The International Laser Ranging Service (ILRS), created in 1998, currently tracks 28 retroreflectorequipped satellites for geodynamics, remote sensing (altimeter, SAR, etc.), gravity field determination, general relativity, verification of GNSS orbits, and engineering tests (Pearlman et al., 2002). Satellite altitudes range from a few hundreds of kilometers to GPS altitude (20K kilometers) and the Moon. The network includes forty laser ranging stations, two of which routinely range to four targets on the Moon. Satellites are added and deleted from the ILRS tracking roster as new programs are initiated and old programs are completed. The collected data are archived and disseminated via two centers, and several analysis centers voluntarily and routinely deliver products for TRF, EOP, POD, and gravity modeling and development.
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2.1.3 IVS The International VLBI Service for Geodesy and Astrometry (IVS) was established in 1999 and currently consists of 74 permanent components: coordinating center, operation centers, network stations, correlators, analysis centers, and technology development centers. The IVS observing network includes about 30 regularly-observing IVS stations and 20-30 collaborating stations participating in selected IVS programs on an irregular basis (Behrend and Baver, 2005). 24-hour sessions twice per week as well as other less frequent sessions are used to determine the complete set of EOP (polar motion, celestial pole coordinates, UT1-UTC), station coordinates and velocities, and the positions of the radio sources. Daily 1-hour single baseline sessions are used to monitor Universal Time (UT1) with low latency (Schlueter et al., 2002). 2.1.4
IDS
The International DORIS Service (IDS) was created in 2003 (Tavernier et al., 2005). The current ground tracking network is composed of 55 stations allowing an almost continuous tracking of the current five satellites (SPOT-2,-3 and-4 used for remote sensing applications, Jason-1 and Envisat used for satellite altimetry). The main applications of the DORIS system are precise orbit determination, geodesy and geophysics (Willis et al., 2005). Using improved gravity Earth models derived from the GRACE mission (Tapley et al., 2004), DORIS weekly station positions can now be regularly obtained at the 10 mm level (Willis et al., 2005). DORIS data are available at the two IDS Data Center since 1990 (SPOT-2). The French space agency (CNES) has the leading role in the IDS. 2.1.5 IGFS The International Gravity Field Service (IGFS) was created in 2003 to provide coordination and standardization for gravity field modeling. It supports the lAG scientific and outreach goals and therefore GGOS, through activities such as collecting data for fundamental gravity field observation networks (e.g., a global absolute reference network, co-located with satellite stations and other geodetic observation techniques), data collection and release of marine, surface and airborne gravity data for improved global model development (e.g., EGM96 (Lemoine et al., 1998)), and advocating consistent standards for gravity field models across the IAG services. Establishing new methodology and science applications, particularly in the integration and validation
of data from a variety of sources, is another focus of the service. The IGFS is composed of a variety of primary service entities: Bureau Gravim6trique International (BGI), International Geoid Service (IGeS), International Center for Earth Tides (ICET), and International Center for Global Earth Models (ICGEM), with the National GeospatialIntelligence Agency (NGA) participating as an IGFS Technical Center. 2.2 Communications
Most of the data collected from network instruments are transmitted to data centers and processing or analysis centers primarily via Internet or satellite communications. Due to the terabyte-volume of data, VLBI data are currently shipped on recorded media, but transmission of data via high-speed fiber is a future goal. Gravity data are currently exchanged via Internet or massive storage media on an "as needed" basis. Control and coordination information is also routinely and primarily sent via Internet. Communications costs are borne by the operating agencies, which in remote areas is often at considerable expense, although most sites are ideally situated in areas with easy access to Internet. The GN&C WG will improve the efficiency through coordinated implementation of modem methods and additional sharing of communications facilities and infrastructure. 3 Synergy
of the
Observing
Tech-
niques
At the dawn of space age, the individual national classical systems started slowly to be replaced by initially crude global equivalents (e.g., the SAO Standard Earth models), and later on, by more sophisticated "World Geodetic Systems" (e.g., the US DoD-developed WGS60, 66, 72, and WGS84). As space techniques proliferated, it soon became apparent that the optimal approach would be to make use of all available systems, and to share the burden of the development through international coordination and cooperation. 3.1 The
Terrestrial
Reference
Frame
The dramatic improvement of space geodesy techniques in the eighties, thanks to NASA's Crustal Dynamics Project and Europe's WEGENER Project, has dramatically increased the accuracy of TRF determination (Smith and Turcotte, 1993). However, none of the space geodesy techniques alone is able to provide all the necessary parameters for the TRF datum definition (origin, scale, and orientation). While satel-
Chapter 103 • GGOSWorkingGroupon GroundNetworksand Communications
lite techniques are sensitive to Earth's center of mass, VLBI is not. The scale is dependent on the modeling of some physical parameters, and the absolute TRF orientation (unobservable by any technique) is arbitrary or conventionally defined through specific constraints. The utility of multitechnique combinations is therefore recognized for the TRF implementation, and in particular for accurate datum realization. Since the creation of the International Earth Rotation and Reference Systems Service (IERS), the current implementation of the International Terrestrial Reference Frame (ITRF) has been based on suitably weighted multi-technique combination, incorporating individual TRF solutions derived from space geodesy techniques as well as local ties of co-location sites. The IERS has recently initiated a new effort to improve the quality of ties at existing co-location sites, crucial for ITRF development (Richter et al., 2005). The particular strengths of each observing method can compensate for weaknesses in others. SLR defines the ITRF2000 geocentric origin, which is stable to a few mm/decade, and SLR and VLBI define the absolute scale to around 0.5 ppb/decade (equivalent to a shift of approximately 3 mm in station heights) (Altamimi et al., 2002). Measurement of geocenter motion is under refinement by the analysis centers of all satellite techniques. The density of the IGS network provides easy and rigorous TRF access world-wide, using precise IGS products and facilitates the implementation of the rotational time evolution of the TRF in order to satisfy the No-Net-Rotation condition over tectonic motions of Earth's crust. DORIS contributes a geographically welldistributed network, the long-term permanency of its stations, and its early decision to co-locate with other tracking systems. We recognize that we will need to consider non-linear motions in future reference frame solutions. A first step towards this goal is the use of time series analysis rather than just position and velocity products. 3.2 E a r t h O r i e n t a t i o n P a r a m e t e r s
Earth orientation parameters measure the orientation of Earth with respect to inertial space (which is required for satellite orbit determination and spacecraft navigation) and to the TRF, which is a precondition for long-term monitoring. Polar motion and UT1 track changes in angular momentum in the fluid and solid components of the Earth system driven by phenomena like weather patterns, ocean tides and circulation, post-glacial rebound and great earthquakes. The celestial pole position, on the other hand, is dependent on the deep structure of Earth. Only VLBI measures celestial pole position and UT1, and VLBI also
defines the ICRF (International Celestial Reference Frame) (Ma et al., 1998), whose fiducial objects (mostly quasars) have no detectable physical motion across the sky because of their great distance. The two-decade VLBI data set contributes a long time series of polar motion, UT1 and celestial pole position. Satellite techniques (GPS, SLR and DORIS) measure polar motion and length of day relative to the orbital planes of the satellites tracked. In practice, recent polar motion time series are derived from GPS with a high degree of automation, and predictions of UT1 rely on GPS length of day and atmospheric excitation functions. 3.3 G r a v i t y , G e o i d , a n d V e r t i c a l D a t u m
Gravity is important to many scientific and engineering disciplines, as well as to society in general. It describes how the ~vertical" direction changes from one location to another, and similarly, it defines at each point the equipotential surface; therefore, it describes the direction that "water flows". Global scale models of terrestrial gravity and geoid (Lemoine et al., 1998) are now routinely delivered on a monthly basis by missions like GRACE, with a resolution of 200 km or so, and high accuracy (Tapley et al., 2004). The addition of surface gravity observations can extend the resolution of these models down to tens of kilometers in areas of dense networks. Worldwide databases of absolute and relative gravity, airborne and marine gravity are collected and maintained by IGFS. Astronomically-driven temporal variations of gravity (Earth, ocean and atmospheric tides) are also a product of this and other IAG services. The combination of all this information is crucial in precisely determining instantaneous position on Earth or in orbit, the direction of the vertical and the height of any point on or around Earth, and the computation of precise orbits for near-Earth as well as interplanetary spacecraft. Similarly, the vertical datum is the common reference for science, engineering, mapping and navigation problems. Achieving a globally consistent vertical datum of very high accuracy has been a prime geodetic problem for decades, and only recently (thanks to missions like CHAMP and GRACE) is a successful result in reach. Strengthening and maintaining a close link between the "geometric" and "gravimetric" reference frames is of paramount importance to the goals of GGOS. 3.4 P r e c i s e O r b i t D e t e r m i n a t i o n
Precise orbit determination is one of the principal applications of the satellite techniques (GPS, SLR, DORIS), and has direct application to many
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different scientific disciplines such as ocean topography mapping, measurement of sea level change, determination of ice sheet height change, precise geo-referencing of imaging and remote sensing data, and measurement of site deformation using synthetic aperture radar (SAR) or GPS. The techniques have evolved from meter-level orbit determination of satellites such as LAGEOS in the early 1980's to cm-level today. The computation of precise orbits allows these satellite tracking data to be used for gravity field determination (both static and time-variable) and the estimation of other geophysical parameters such as post glacial rebound, ocean tidal parameters, precise coordinates of tracking sites, or the measurement of geocenter motion. Precise orbit determination, which requires precise UT1 and gravity models, underpins the analysis that in parallel has resulted in improved station coordinate estimation, and thereby improved realizations of the TRF (e.g., ITRF2000); There is close synergy between POD and TRF realization. The density of data available from GPS (and in the future from other GNSS including Galileo) allows the estimation of reduceddynamic or kinematic orbits with radial accuracy of a few cm even on low-altitude satellites such as CHAMP and GRACE. Only a few satellites carry multiple tracking systems, but space-based co-location is invaluable. The detailed intercomparison of orbits computed independently from SLR, DORIS, and GPS data confirms that Jason1 orbits have a one-cm radial accuracy (Luthcke et al., 2003). These techniques are complementary; the precise but intermittent SLR tracking of altimeter satellites, such as Envisat or TOPEX/Poseidon, is complemented by the dense tracking available from the DORIS network. SLR tracking of the GPS, GLONASS or future Galileo satellites is and will be vital to calibrating GNSS satellite biases and assuring the realization of a high quality TRF.
4 Future Requirements The measurement requirements for GGOS will be set by the GGOS Project Board with guidance from the Science Council (Rummel, 2002). Until these requirements are formally specified, we judge the practical useful target for the TRF and space geodetic measurement accuracy to be roughly a factor of 5 to 15 below today's levels. Given that the TRF and global geodesy are now accurate to the order of 1 cm (or 5-15 mm for different quantities) and 2 mm/yr, we foresee near-term utility in global measurements with absolute accuracies at or below 1 mm and 0.2
mm/yr. Corresponding levels of improvement are required for Earth orientation and gravity.
5 Evolution of the Techniques Each of the GGOS Services techniques envisions technological and operational advances that will enhance measurement capability. Some advances are currently being implemented while others are in the process of design or development. In addition, each technique-related service is seeking to improve not only data quality and precision, but also reliability of data and product delivery, performance, continuity, station stability, data latency (which in the case of GNSS includes realtime), and data handling techniques and modeling. While making these improvements, contributors seek operational efficiencies in order to minimize costs.
5.1 GNSS Geodetic GNSS has already evolved from GPSonly operations to inclusion of GLONASS, and upgrades to next-generation receivers will allow full benefit from modernized GPS signal structures, Galileo signals, and GLONASS signals. Studies leading to improved handling of calibration issues such as local signal effects (e.g., multipath) and antenna phase patterns are underway, as are initiatives to fill remaining network gaps, particularly in the southern hemisphere. Elsewhere, station density is less problematic and the focus has shifted to consolidation of supplementary instrumentation such as strain meters and meteorological sensors.
5.2 Laser Ranging Newly designed and implemented laser ranging systems operate semi-autonomously and autonomously at kilohertz frequencies, providing faster satellites acquisition, improved data yield, and extended range capability, at substantially reduced cost. Improved control systems permit much more efficient pass interleaving and new higher resolution event-timers deliver picosecond timing. The higher resolution will make twowavelength operation for atmospheric refraction delay recovery more practical and applicable for model validation. The current laser ranging network suffers from weak geographic distribution, particularly in Africa and the southern hemisphere. The comprehensive fundamental network should include additional co-located sites to fill in this gap. Improved satellite retroreflector array designs will reduce uncertainties in center-of-mass corrections, and optical transponders currently under development offer opportunities for extraterrestrial measurements.
Chapter 103
5.3 VLBI The VLBI component of the future fundamental network will be the next-generation system now undergoing conceptual development. Critical elements include fast slewing; high efficiency 1012 m diameter antennas; ultra wide bandwidth front ends with continuous radio frequency (RF) coverage; digitized back ends with selectable frequency segments covering a substantial portion of the RF bandwidth; data rate improvements by a factor of 2-16; a mixture of disk-based recording and high speed network data transfer, near real time correlation among networks of processors, and rapid automated generation of products. Better geographic distribution, especially in the southern hemisphere, is required.
5.4 DORIS The DORIS tracking network is being modernized using third-generation antennae and improvements to beacon monumentation (Tavernier et al., 2005). Efforts are underway to expand the network to fill in gaps in existing coverage. DORIS beacons are also being deployed to support altimeter calibration, co-location with other geodetic techniques, or specific short-term experiments. A specific IDS working group is selecting sites and occupations for such campaigns, using additional DORIS beacons provided by CNES to the IDS.
5.5 Gravity Gravity observations are most sensitive to height changes; they therefore provide an obvious way to define and control the vertical datum. A uniformly-distributed network of regularly crosscalibrated absolute gravimeters supported by a well-designed relative measurement network that will be repeatedly observed at regular intervals, and a sub-network of continuously operating superconducting tidal gravimeters are expected in a fundamental network of co-located techniques. These permanent networks should be augmented with targeted airborne and ship campaigns to collect data over large areas that are devoid of gravimetric observations. A well-distributed global data set of surface data is necessary to calibrate and validate products of the recent (CHAMP and GRACE) and upcoming (GOCE) high-accuracy and-resolution missions. The organization of a gravity field service is underway and the integration of its activities should emerge shortly.
• GGOS Working Group on
Ground Networks and Communications
6 Approaches to Network Design The final design of the GGOS network must take into consideration all of the applications including the geometric and gravimetric reference frames, EOP, POD, geophysics, oceanography, etc. We will first consider the TRF, since its accuracy influences all other GGOS products. Early steps in the process are: 1. Define the critical contributions that each technique provides to the TRF, POD, EOP, etc. 2. Characterize the improvements that could be anticipated over the next ten years with each technique. 3. Examine the effect in the TRF and Earth orientation resulting from the loss of a significant part of the current network or observation program. 4. Using simulation techniques, quantify the improvement in the TRF, Earth orientation and other key products as stations are added and station capability (co-location, data quantity and quality) is improved. We will also explore the benefit of adding new SLR targets.
7 Sustaining the Ground Over the Long Term
Network
The measurement techniques services have each maintained their own networks and supporting infrastructure, routinely producing data, but suffer from severe budget constraints of the voluntarily contributing agencies that prevent appropriate maintenance and development of physical and computational assets. This degradation of the observing network capability coincides with the deployment of high-value science investigations and missions, such as sea level studies from ocean and ice-sheet altimetry missions, eroding their scientific return and limiting their ability to meet the mission goals. Many of the elements of the current networks are funded from year to year and depend upon specific activities. Stations are often financed for capital and maintenance and operations costs through research budgets, which may not constitute a long-term commitment. Sudden changes in funding as priorities and organizations change have resulted in devastating impacts on station and network performance. On the other hand, missions and long-term projects have assumed that the networks will be in place at no cost to them, fully functioning when their requirements need fulfillment. GGOS will be proactive in helping to persuade funding sources that the networks are interdependent infrastructure that needs long
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term, stable support. The GGOS community must secure long-term commitments from sponsoring and contributing agencies for its evolution and operations in order to support its users with highquality products. Since the present networks must support current as well as future requirements, the GGOS network must evolve without interruption of data and data products. In view of the difficulties in securing long-lasting and stable financial support by the interested parties, new financial models for the networks must be developed. This Working Group will work with the Strategy and Funding Working Group to develop an approach.
8 Summary A permanent geodetic network of complementary yet interdependent space geodetic techniques is critical for geodetic and geophysical applications and underpins the Global Earth Observation System of Systems. Thanks to the generous and voluntary contributions of many national agencies and institutions around the world, the IAG has been able to coordinate global collaborations for geodetic technique based services from which all benefit. There is a strong need for coordination of the planning, funding and operation of future integrated geodetic networks to maximize performance in meeting evolving requirements while taking into account the need for sustainable infrastructure and efficient operations. The GGOS Ground Networks & Communications Working Group has initiated studies, which will guide the services in infrastructure planning for optimal benefit to Earth science and associated engineering and societal concerns.
Acknowledgments The authors would like to acknowledge the support of IAG services (IGS, ILRS, IVS, IDS, IGFS, and IERS) and their participating organizations. Part of this work was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.
References Altamimi, Z., P. Sillard, C. Boucher (2002). ITRF2000, A new release of the International Terrestrial Reference Frame for earth science applications. Journal of Geophysical Research, Solid Earth, vol. 107(B10), 2214.
Behrend D., K.D. Baver (Eds) (2005). International VLBI Service for Geodesy and Astrometry 2004 Annual Report, NASA/TP-2005-212772, 2005. Beutler G., M. Rothacher, S. Schaer, T.A. Springer, J. Kouba, R.E. Neilan (1999). The International GPS Service (IGS), An interdisciplinary service in support of Earth Sciences, Advances in Space Research, Vol. 23(4), pp. 631-653. Group on Earth Observations (2005) Global Earth Observation System of Systems GEOSS 10-Year Implementation Plan Reference Document - GEO 1000R, 209 pages. Lemoine, F.G., S.C. Kenyon, J. K. Factor, R. G. Trimmer, N.K. Pavlis, D.S. Chinn, C.M. Cox, S.M. Klosko, S.B. Luthcke, M.H. Torrence, Y.M. Wang, R.G. Williamson, Pavlis, E.C., R.H. Rapp, and T.R. Olson, (1998). The Development of the Joint NASA GSFC and the National imagery and Mapping Agency (NIMA) Geopotential Model EGM96, NASA/TP-1998-206861, Goddard Space Flight Center, Greenbelt, Maryland, 575 pages. Luthcke S.B., N.P. Zelensky, D.D. Rowlands, F.G. Lemoine, T.A. Williams (2003). The 1-centimeter orbit, Jason-1 precision orbit determination using GPS, SLR, DORIS, and altimeter data, Marine Geodesy, Vol. 26(3-4), pp. 399-421. Ma C., F. Arias, T.M. Eubanks, A.L. Fey, A.M. Gontier, C.S. Jacobs, O.J. Sovers, B.A. Archinal, P. Charlot (1998). The International Celestial Reference Frame as realized by Very Long Baseline interferometry, Astronomical Journal, Vol. 116(1), pp. 516-546. Pearlman, M.R., J.J. Degnan, J.M. Bosworth (2002). The International Laser Ranging Service, Advances in Space Research, Vol. 30(2), pp. 135-143. Richter, B., W. Schwegmann, W. Dick (eds.) (2005), Proceedings of the IERS Workshop on site co-location. Matera, Italy, 23 - 24 October 2003 (IERS Technical Note ; 33). Rummel R., H. Drewes, G. Beutler (2002). integrated Global Observing System IGGOS, A candidate IAG Project, In Proc. International Association of Geodesy, Vol. 125, pp. 135-143. Schlueter W., E. Himwich, A. Nothnagel, N. Vandenberg, A. Whitney (2002). IVS and its important role in the maintenance of the global reference systems, Advances in Space Research, Vol. 30(2), pp. 145-150. Smith D., D. Turcotte (eds.) (1993), Contributions of Space Geodesy to Geodynamics: Crustal Dynamics (23), Earth Dynamics (24) and Technology (25), AGU Geodynamics Series. Tapley B.D., S. Bettadpur, M. Watkins, C. Reigber (2004), the Gravity Recovery and Climate Experiment, Mission overview and early results, Geophysical Research Letters, Vol. 31 (9), L09607. Tavemier G., H. Fagard, M. Feissel-Vernier, F. Lemoine, C. Noll, J.C. Ries, L. Soudarin, P. Willis (2005). The International DORIS Service, IDS. Advances in Space Research, Vol. 36(3), pp. 333-341. Willis P., C. Boucher, H. Fagard, Z. Altamimi (2005). Geodetic applications of the DORIS system at the French Institut Geographique National. Comptes Rendus Geoscience, Vol. 337(7), pp. 653-662.
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Linking the Global Geodetic Observing System (GGOS) to the Integrated Global Observing Strategy Partnership (IGOS-P) through the Theme 'Earth System Dynamics' H.-P. Nag, Nevada Bureau of Mines and Geology and Seismological Laboratory, University of Nevada, Reno, Mailstop 178, Reno, NV 89557, USA, email: [email protected]. G. Beutler, Astronomical Institute, University of Bern, Bern, Switzerland, email: [email protected] R. Forsberg, Danish National Space Center, Copenhagen, Denmark, email: [email protected] C. Ma, Goddard Space Flight Center, Greenbelt, MD, USA, email: [email protected] R. Neilan, Jet Propulsion Laboratory, Pasadena, CA, USA, email: [email protected] M. Pearlman, Harvard-Smithonian Center for Astrophysics, Cambridge, MA, USA, email: mpearlman @c fa. harvard, edu B. Richter, Bundesamt fiir Kartographie und Geodfisie, Frankfurt a.M., Germany, email: [email protected] S. Zerbini, Department of Physics, University of Bologna, Bologna, Italy, email: [email protected]
Abstract When setting up GGOS as a project, the lAG Executive Committee asked the GGOS Steering Committee to establish a relationship with IGOS-R IGOSP addresses a number of problems and components of Earth observing systems in the frame of specific Themes. The IGOS-P Theme process will also be an important mechanism for the development of the components of the Global Earth Observation System of Systems (GEOSS). Many of the burning questions related to the water cycle, the climate, global change, and geohazards cannot be solved without sufficient knowledge of mass transports throughout the Earth system and the associated dynamics. All these processes affect the three fundamental geodetic quantities, namely the Earth's figure (geometry), its gravity field and its rotation. Thus, GGOS is an unique contribution to Earth observation in its capability to provide detailed information on the dynamics of the solid Earth and its fluid envelop on all relevant spatial and temporal scales, Focusing on the observing system for the mass transports within the Earth system the suggested 'Earth System Dynamics' Theme has the goal to develop the science basis for and to facilitate the implementation of GGOS. The Theme will define the role of GGOS, the underlying strategy and its interface to the other components of GEOSS. The interaction of GGOS with the other IGOS-P Themes will facilitate the full exploitation of the geodetic contribution by all other global observing systems. The Theme will ensure that GGOS meets the user requirements both from the IGOS-P Themes and the nine societal benefit areas identified by the Earth Observation Summit Ii.
Keywords: Global Geodetic Observing System, Integrated Global Observing Strategy, Earth Observation, Earth System Dynamics
1
Introduction
The need for information on the current state of the Earth System and its processes is today greater than ever before. The increasingly visible effects of a growing population and economic development on the environment has raised the public and political awareness of the significance of the changes in the Earth environment. The ability to detect and understand the various processes in the Earth system, including those causing climate change is fundamental in the quest for sustainable development and in order to reduce uncertainties, assess impacts, and predict changes. Long-term monitoring of the Earth system provides the indispensable data base for these studies and tasks. Over the last decade, the recognition of the fundamental role of Earth observation in managing the planet in a sustainable way has led to a number of initiatives and programs aiming at a better coordination, better coverage in terms of observed parameters, domains and spatial and temporal scales, more user-orientation and less duplication, which recently culminated in the plan fora Global Earth Observation System of Systems (GEOSS). In the frame of the International Association of Geodesy (IAG), the Global Geodetic Observing System (GGOS) is under implementation and GGOS will contribute to GEOSS. When setting up GGOS as a project, the IAG Executive Committee asked the GGOS Steering Committee to establish also a relationship with the Integrated Global Observing Strategy Partnership (IGOS-P). IGOS-P addresses a num-
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ber of problems and components of Earth observation systems in the frame of specific Themes. The IGOSP Theme process is also expected to be an important mechanism for the development of the components of the GEOSS. In order to ensure that the implementation of GGOS is in agreement with the Integrated Global Observing Strategy (IGOS), and that GGOS optimally serves the needs of GEOSS and the different IGOS-P Themes, GGOS has applied for membership in IGOSP. Moreover, GGOS has suggested a new 'Earth System Dynamics' Theme, focusing on the observation system for mass transport and dynamics of the Earth system in the frame of a 'whole system' approach. In the following, we first give a brief overview of geodesy's contribution to Earth observations, and then in Section 3 describe briefly the background, goals, and current status of GGOS. Then, in Section 4 we provide background information on IGOS and IGOSP and summarize in Section 5 the currently established and proposed Themes with focus on those with direct links to geodesy. The development towards a GGOS membership in IGOS-P and an 'Earth System Dynamics' Theme is reported in Section 6.
2
Geodesy's Contribution Earth Observation
to
The Earth is a dynamic system. Dynamic processes in the Earth's interior and the associated mass displacements lead to plate tectonics, volcanism, and earthquakes. Mass movements in the atmosphere, terrestrial hydrosphere, oceans and cryosphere associated with weather, climate and global change are caused by dynamic processes in the atmosphere and oceans. To study these processes is important for the understanding of atmospheric and oceanic circulation, transport of water in the hydrological cycle and the mass balance in the terrestrial and oceanic hydrosphere, as well as volcanic processes, generation of earthquakes, glacial isostatic processes, mantle convection, and processes in the core. Many of the burning questions related to these processes cannot be solved without sufficient knowledge of mass transports throughout the Earth system and the associated dynamics. Mass transport and changes in the dynamics all affect the three fundamental geodetic quantities, that is, the Earth's figure (geometry), its gravity field and its rotation. These geodetic quantities and their temporal variations are observed with space-geodetic techniques using a combination of space-borne and air-borne sensors and in-situ networks (Figure 1). These observa-
Geometry, kinematics GNSS, altimetry, InSAR, mobile SLR, remote sensing, leveling, tide gauges
Earth Rotation VLB1, LLR, SLR, GNSS, DORIS Classical: astronomy F'uture: terrestrial gyroscopes
Gravitational field
Orbit analysis lli-lo & lo-lo SST Satellite gradiometry Ship/air-borne gravimclry absolute gravimetry gravity recordings
Fig. 1: Contribution of geodesy to Earth observation and other societal applications. Modified from Rummel (2000).
tions allow the detection of mass movements in the Earth's subsystems with unprecedented accuracy and with high temporal resolution, thus linking the subsystems together and providing a truly global monitoring of mass movements and the associated Earth system dynamics. Observations of the displacements of the Earth's surface furnish records of the movements and deformations associated with atmosphere and ocean dynamics as well as earthquakes, volcanoes, tsunamis, natural and man-made subsidence, landslides, and other potential hazards. Geodesy monitors the variations of the Earth rotation as an indicator of all angular momentum exchange inside, on or above the Earth, as well as of the interaction between the Earth and the Sun and Moon. Geodesy explores the Earth's gravity field, both the stationary field and the time variable field due to changes of mass distribution in the Earth system as a whole including the solid Earth, liquid core, atmosphere, oceans, hydrosphere, and cryosphere. Moreover, geodesy monitors the atmosphere, oceans and cryosphere with space geodetic remote sensing techniques. The internationally coordinated geodetic observations result in a global terrestrial reference frame, which is determined and monitored on the basis of observations provided continuously by the networks of geodetic stations. This well-defined, long-term stable, highly accurate and easily accessible reference frame is the basis for all precise positioning on and near the Earth's surface. It is the indispensable foundation for all sustainable Earth observations, in situ, as well as air-borne and space-borne. In summary, geodesy provides a unique frame for the monitoring, understanding and prognosis of the Earth system as a whole. Modem space-geodetic
Chapter 104 • Linking the Global Geodetic Observing System (GGOS) to the Integrated Geodetic Observing Strategy Partnership (IGOS-P)
techniques are inherently strong on global to regional scales and they constitute an important complement to traditional in situ observation systems. Thus, GGOS is an unique contribution to Earth observation in its capability to provide detailed information on the dynamics of the solid Earth and its fluid envelop on all relevant spatial and temporal scales.
3
The G G O S
Over the last decade, IAG has established a system of services, which provides a number of products to a wide range of scientific and non-scientific users. The organizational development within international space geodesy has been inspired by the success of the International GNSS Service (IGS), which was established by IAG in 1994. Since then, the IGS has facilitated the creation of a global network of GPS tracking stations which today consists of more than 300 stations. These stations provide observations on an hourly or daily basis to data centers, from where the data are freely available. A number of IGS Analysis Centers (AC) determine satellite orbits and clocks as well as Earth Rotation Parameters (ERP) on a routine basis with a variety of latency and accuracy (Ray et al., 2004), which are widely used in scientific and, increasingly, also nonscientific applications. The success of the IGS stimulated the establishment of other technique-specific space-geodetic services by IAG, such as the International VLBI Service (IVS), the International Laser Ranging Service (ILRS), and the International DORIS Service (IDS). These services provide continuously observations from their ground-based tracking networks, which are also used to determine station displacements, deformations of the solid Earth, geocenter motion, and ERPs. Both, observations and products, are made available to a wide range of users, though mainly in scientific fields. The products of the technique-specific services are the basis on which the International Earth Rotation and Reference Systems Service (IERS) determines and monitors the International Terrestrial Reference Frame (ITRF) as the most accurate frame realization of the International Terrestrial Reference System (ITRS). For that purpose, a number of ITRF ACs submit single and multi-technique solutions, which are then combined to provide the so-called regularized station coordinates and secular velocities (McCarthy & Petit, 2003) of a given ITRF version. The latest version is ITRF2000, which is considered to be the most accurate realization of the ITRS so far (Altamimi et al., 2002).
It is based on a network of more than 400 stations, many of them co-located by two or more techniques. The IERS includes also the Global Geophysical Fluid Center (GGFC), which was established in 1998. The GGFC and the associated seven Special Bureaus (SB) for Atmosphere, Oceans, Tides, Hydrology, Mantle, Core, Gravity/Geocenter, and Loading have the responsibility of supporting, facilitating, and providing services to the worldwide research community, in areas related to the variations in earth rotation, gravitational field and geocenter that are caused by mass transport in the geophysical fluids. The GGFC provides a general frame for research related to the further development of the products delivered by the IERS and the lAG services. Following the example of the IGS and the other space-geodetic services (IVS, ILRS, IDS), very recently services related to the gravitational field have been initiated. In particular, the International Gravity Field Service (IGFS) takes the responsibility for all aspects of the Earth's gravitational field. In addition, the establishment of an International Altimetry Service (IAS) is contemplated. Such a service was pointed out as missing and an evident complement of the IAG system of services. The increasingly important role of Interferometric Synthetic Aperture Radar (InSAR) for many applications, ranging from the monitoring of small scale surface displacements to changes in the biosphere, has brought up the idea of setting up an international InSAR service (LaBrecque, 2004, personal communication). InSAR observations are relevant to geodetic applications and therefore should be studied and developed under the umbrella of IAG. The organizational model of the IAG services has also been applied to interdisciplinary fields, e.g. to the European Sea Level Service (ESEAS), which integrates geodetic and hydrographic techniques into a sea level observing system for the European coasts (Plag, 2002). These services have established considerable observing infrastructure, comprising global groundbased networks of observing sites, data and analysis centers, and web sites giving access to the products. They utilize signals and data from operational satellite systems and dedicated satellite missions. Organizationally, most of these geodetic services are based on the 'best effort' principle and depend on the contributions of globally distributed institutes. in order to establish a coherent geodetic observing systems and thus to meet the user requirements in a consistent and efficient way, the structure of IAG Services, which has been build up over the last decade, is currently complemented by the GGOS. GGOS was established through the decision of the IAG Execu-
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Fig. 3" The envisaged role of GGOS in Earth observation. tive Committee at the 23-rd IUGG General Assembly, 2003 in Sapporo, Japan. This decision was supported through an IUGG Resolution of the same assembly. The Executive Committee of IAG at its meetings in August 2005 in Cairns, Australia, decided to transform the current GGOS project into a permanent observing system. The scientific basis for GGOS is summarized in Rummel (2000) and Rummel et al. (2005). The GGOS
"aims at maintaining the stability of and providing the ready access to the existing time series of geometric and gravimetric reference frames by ensuring the generation of uninterrupted time series of state-of-theart global observations related to the three pillars of geodesy" (Beutler et al., 2003). The accuracy level targeted by GGOS for the three fundamental geode-
tic quantities (and their mutual consistency level) is 10 -9 or better. At this level of accuracy, a big variety of mechanical interactions between the different Earth system components are relevant and need to be treated consistently. In this respect, modem geodesy requires a system approach to the dynamics of the Earth and involves expertise from all Earth sciences in the analysis and interpretation of the geodetic observations. GGOS aims at integrating the different levels of the geodetic observing systems from ground-based stations as level (1), over Low Earth Observing satellites as well as gravity and altimetry missions as level (2), and navigation satellite systems as level (3), to quasars as level (4) into one coherent observation system (Figure 2) and to analyze and interpret the observations in a consistent Earth system frame. The way to achieve this goal is long and will require considerable developments, both in observational capabilities and physical modeling, including theoretical developments. In particular, the transition from a mainly research-based and science-driven system to an operational, user-driven system will deserve special attention. From an organizational point of view, GGOS is particularly needed to create a unique interface between GEOSS and other users on the one side and the IAG Services on the other side (Figure 3). Externally, GGOS is envisaged as the unique interface between the observing systems maintained by IAG, which will provide geodetic observations and products to GEOSS, the IGOS-P themes, and other users outside of IAG. Internally, GGOS will facilitate steps towards fully consistent data processing, which will improve the quality and accuracy of the products made available internally and externally. Moreover, GGOS will advocate standardization of the products and ensure that the interface giving access to products is fully interoperable with the other systems contributing to GEOSS. The establishment of GGOS is an appropriate response to the emerging broad range of scientific and practical requirements with respect to geodesy. The Framework document resulting from the Earth Observation Summit II (EOS-II), which formed the basis for the 10-year Implementation Plan for the Global Earth Observing System of Systems (GEOSS) (GEO, 2005a) and the associated Reference Document (GEO, 2005b), identifies nine societal benefit areas for Earth Observations (see Appendix 2 in GEO, 2005b). For each of these areas the Reference Document provides an overview of the requirements in terms of observables and an assessment of the status of the observational capacity. Extracting the quantities potentially provided by geodesy (Table 1)demonstrates that geodesy will be a major contributor to GEOSS. More-
Chapter 104
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Linkingthe Global Geodetic ObservingSystem (GGOS)to the Integrated Geodetic Observing Strategy Partnership (IGOS-P)
Table 1: Requirements for geodetic observables for the nine benefit areas. The status is indicated with the follow classes: 0: ok; 1: marginally acceptable accuracy and resolution; 2: could be ok within two years; 3: could be available in six years; 4: still in research. Observable quantity Deformation monitoring, 3-D, over broad areas Subsidence maps Strain and creep monitoring, specific features or structures Gravity, magnetic, electric fields - all scales Gravity and magnetic field anomaly data Groundwater level and pore pressure Tides, coastal water levels Sea level Glacier and ice caps Snow cover Moisture content of atmosphere/water vapor Extreme weather and climate event forecasts Precipitation and soil moisture
Status 3 3 2 3 2/3 4-1 1 2-1 2 2 2 3 3-1
over, a geodetic reference frame, which is not explicitly mentioned in any of these requirements, is indispensable for GEOSS to reach its goals.
4
IGOS and IGOS-P
"The Integrated Global Observing Strategy (IGOS) seeks to provide a comprehensive framework to harmonize the common interests of the major space-based and in-situ systems for global observation of the Earth. It is being developed as an over-arching strategy for conducting observations relating to climate and atmosphere, oceans and coasts, the land surface and the Earth's interior. IGOS strives to build upon the strategies of existing international global observing programs, and upon current achievements'. It seeks to improve observing capacity and deliver observations in a cost-effective and timely fashion." (cite from http://www.igospartners.org). IGOS is a strategic planning process, providing a structure that helps determine observation gaps and identify the resources to fill observation needs. IGOS is intended to cover all forms of data collection concerning the physical, chemical, biological and human environment including the associated impacts. Being user driven, it is expected that the results will increase scientific understanding and guide early warning, policy-setting and decision-making for sustainable development and environmental protection. The IGOS-P was established in June 1998 by the 13 founding Partners for the definition, development and implementation of the IGOS. IGOS-P brings together a number of international bodies concerned with the ob-
servational component of global environmental issues, both from a research and a long-term operational programme perspective, with the principal objectives to address how well user requirements are being met by the existing mix of observations, and how they could be met in the future through better integration and optimization of remote sensing (especially space-based) and in-situ systems. IGOS-P serves as guidance to those responsible for defining and implementing individual observing systems (while the implementation of the Strategy, i.e. the establishment and maintenance of the components of an integrated global observing system, lies with those governments and organizations that have made relevant commitments). IGOS-P has adopted an incremental "Themes" approach to aid the development of the strategy based on perceived priorities.
The IGOS-P Themes and Their Links to Geodesy The goal of IGOS-P is a small number of Themes with strong linkages to critical social issues. The process of Theme selection is based on an assessment of the relevant scientific and operational priorities for overcoming deficiencies in information, as well as analysis of the state of development of relevant existing and planned observing systems. In general, all IGOS-P Themes address space-borne or air-borne observations that require highly accurate positioning of the sensors, and thus are linked to the global geodetic networks through their requirements for access to an accurate and stable reference frame. In the following list of existing and planned Themes, we summarize the objectives (see http://www.igospartners.org for more details) and emphasize links to geodetic observations (besides the requirement to have access to a reference frame) where appropriate: Atmospheric Chemistry Theme: ensure the longterm continuity and spatial comprehensiveness of the monitoring of atmospheric composition and to integrate ground-based and space-borne measurements using models and assimilation tools. Carbon Observations Theme: enhance the scientific understanding of the global carbon cycle, provide for advanced Earth system observation capabilities, and deliver an improved knowledge base for better policy-making. Geohazards Theme: integrate disparate, multidisciplinary, applied research into global, operational systems, and through this improve the provision of
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timely, reliable and cost-effective information to those responsible for managing geohazards. Plate tectonics, pre-, co- and post-seismic strain, processes associated with volcanoes, early warning for tsunamis, subsidence, precarious rocks, landslides, and local and regional predictions of sea level rise are examples of topics that link this theme to geodetic observations. Ocean Theme: develop a strategy for an observing system for the oceans that serves the research and operational oceanographic communities and a wide range of scientific and non-scientific users. Ocean circulation, sea level rise, isostasy, and dynamic sea surface topography are linked to the three geodetic quantities, both for the monitoring and studies of the ocean's variability as well as model validation. Water Cycle Theme: provide a framework for guiding decisions regarding the maintenance and enhancement of water cycle observations that support monitoring of climate, water management and water resource development, provision of initial conditions for numerical weather forecasts and climate predictions, and research related to the water cycle. The geodetic observations provide a unique tool to monitor the global to local scale movements of water through the Earth system and the Theme is strongly linked to geodesy. Coast Observation Theme: coordinate and strengthen present and future coastal observational capabilities, both in situ and space-borne as a basis for a better understanding of the changes in the coastal zone and a service to the decision-making process (under development). Ocean circulation, sea level, and vertical land motion are relevant parameters influencing the dynamic processes in the coastal zone and linking this Theme to geodesy. Coral Reef Sub-Theme: develop a strategy for the observation system of this particular ecosystem taking into account the unique characteristics of coral reefs requiring special observation techniques. Cryosphere Theme: create a framework for improved coordination of cryospheric observations and the generation of data and information needed for both operational services and research (proposal in preparation). Ice mass balance, glacial isostasy, and induced sea level variations all are important parameters, that are directly observed by the geodetic observation techniques. Land Theme: provide a global strategy for a land observations system focusing on globally needed observations for topics such as land cover and land use, human settlement and population, managed and natural ecosystems, soils, biogeochemical cycles, and el-
evation changes (under development). Changes in the elevation are directly observed by geodetic techniques.
6
Steps towards a 'Earth System Dynamics' Theme
In order to foster the implementation of GGOS and to further detail the science basis for GGOS, as well as to strengthen its linkage to existing and new Earth observing systems, such as GEOSS, the IAG has taken a first step to propose a specific IGOS-P Theme addressing the dynamics of the Earth system from a focus on mass movements. The proposal was presented and discussed at the 11-th Meeting of IGOS-P in June 2004 in Rome. In its response, the IGOS-P members requested that IAG/GGOS should study the IGOS Process Paper as well as existing IGOS Theme reports to identify complementary or competing elements and to demonstrate that the idea o f launching a Dynamic Earth Theme could be realized without repetition or overlap. In response to IGOS-P, a document was prepared (Plag et al., 2005) and submitted to IGOS-P for consideration at the 12-th meeting in May 2005 in Geneva. In that document, the following action plan was recommended: (1) GGOS becomes a member of IGOS-P as a new Global Observing System; (2) GGOS establishes links to the existing IGOS-P Themes, allowing GGOS to influence the development of the different theme-specific strategies and to determine the way in which GGOS can best serve the observing systems implemented under these strategies; (3) GGOS together with relevant members of IGOSP develops the 'Earth System Dynamics' Theme further and prepares a proposal for consideration by the IGOS-P members. This action plan was accepted by IGOS-P at the 12-th meeting. Currently, a formal membership proposal is in preparation for the next IGOS-P meeting in November 2005. The relevance of the contribution of GGOS to Earth observations in general and to most of the IGOS-P Themes in particular is obvious, and so is the relevance of the user-oriented issues addressed by a potential 'Earth System Dynamics'Theme. The fundamental contribution of GGOS is widely acknowledged. However, a major concern regarding the establishment of an 'Earth System Dynamics' Theme is the potentially considerable overlap between the suggested new theme and the existing or planned ones. A detailed survey of the requirements of the other themes, both
Chapter104 • Linking the Global Geodetic Observing System (GGOS)to the Integrated Geodetic Observing Strategy Partnership (IGOS-P) existing, proposed, or in planning stage, and the nine societal benefit areas identified by EOS-II shows that these themes and benefit areas would greatly benefit from GGOS. Parts of their observing systems would overlap with GGOS and the 'Earth System Dynamics' Theme would help to ensure full exploitation of synergies. GGOS can be considered as the metrological basis for Earth sciences. Only a holistic, geodesy-driven approach to the entire Earth system, and its dynamics will ensure that geodesy will contribute in an optimum way to Earth sciences in the wider sense. One might think that geodesy's contribution is already available to and trivial to understand for the Earth sciences and observation communities. This statement is false. In order to understand the necessity to establish GGOS and the necessity for the establishment of the 'Earth System Dynamics' Theme it is important to summarize the current status and the planned development of GGOS. Thanks to continuous monitoring and analyzing activities using the space geodetic techniques VLBI, SLR/LLR, GNSS of the relevant lAG services IVS, ILRS, IGS, respectively, and thanks to the IERS, which analyzes the products of the technique-specific services, we have today a m e a n terrestrial and a celestial reference frame available, which are both accurate on the (or close to the) 10-9-level. This essentially implies (sub-)cm and (sub-)0.1 mas accuracies, respectively. Thanks to the established reference frames it is "easily" possible (the IGS, IVS, ILRS, IERS would probably not agree with the attribute "easily") to derive the Earth rotation parameters (precession, nutation, length of day, UT1-UTC, polar motion) on the same accuracy (and time resolution) level. The two reference frames and the time series of Earth rotation parameters can and should now be made mutually consistent on the 10-9-level. The mentioned accuracies and consistency only can be claimed in a mean sense for the terrestrial system (mean site coordinates and "velocities"). Thanks to more than thirty years of SLR/LLR and mainly thanks to the dedicated gravity missions CHAMP, GRACE, GOCE, we are approaching the 10-9-level in gravity (with a spatial resolution of 50100 km, half wavelength) within the next few years, as well. This accuracy implies that a global geoid (equipotential surface near sea level) with a 1-cm accuracy will become available. The accuracy-claim, on the other hand, is based (among other) on the availability of the above mentioned geometrical reference frames (needed to determine the orbits of the low-
orbiting spacecrafts). Unfortunately, the accuracy is needed not in the "mean", but in the "instantaneous" sense: The short-period variations of station coordinates and gravity field (including ocean and atmosphere contributions) would need to be known or estimated (in principle) in one and the same parameter estimation process. Without going into details one can therefore conclude that the generation and maintenance of the geometrical and gravitational reference frames on an accuracy- and mutual consistency-level of 10 -9 with a temporal resolution down to the sub-daily domain will be the challenge for GGOS. It will also be the central issue of the proposed "Earth System Dynamics" Theme within IGOS-P. The suggested 'Earth System Dynamics' Theme has the objective to establish the overall strategy, requirements, and background for a global observing system that consistently monitors the mass movements and dynamics of the Earth system at an accuracy level required in particular by the relevant IGOS-P Themes and the nine societal benefit areas identified by the EOS-II. Utilizing the full potential of the geodetic observations will not be possible without taking a comprehensive system approach considering all mechanical interactions between the different system components. The need for such a theme is mutual both from the IAG/GGOS side as well as from the user side, in particular the other IGOS-P Themes and the nine societal benefit areas guiding GEOSS. The Theme will provide the science basis for the implementation of GGOS and ensure that GGOS can be fully integrated in the frame of IGOS. The Theme takes into account the fundamental difference between GGOS and other observing systems in that GGOS requires an Earth system approach for its full development. The 'Earth System Dynamics' Theme has the task to ensure that the integrated observing system for the dynamic Earth system focusing on mass movements and dynamics is built in a cost-effective way to serve most of the existing themes and GEOSS in two major areas: (1) provision of a stable and accurate global reference frame as well as tools to access this frame anywhere on the globe including air- and space-borne sensors, and (2) provision of long-term observations of the time-variable shape, gravity field and rotation of the Earth, which are related to mass transport and dynamics, both for research in the frame of the other themes as well as applications in the areas addressed by these themes.
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Conclusions
A membership of GGOS in the IGOS Partnership and its active participation in the relevant IGOS-P Themes would, on the one hand, raise the awareness for the needs of these Themes inside the lAG and GGOS in terms of observations provided by GGOS and in terms of the reference frame. On the other hand, GGOS would provide to these Themes expert knowledge concerning the potential and the limitations of the geodetic observing systems. Most of the existing and planned Themes require access to a stable and highly accurate reference frame that will allow us to monitor small changes in physical quantities. Most Themes need information related to mass transport, displacements of the Earth's surface, and variations in the dynamics. GGOS provides both, the basis for the reference frame and observations related to mass transport, surface displacements and Earth system dynamics. GGOS is thus relevant and in many cases indispensable for the observational systems considered in the themes. A piecewise integration of the geodetic techniques into the existing themes would hinder the development of the integrated GGOS and thus of the global facility required by these Themes. Without the suggested 'Earth System Dynamics' Theme (or a similar approach outside the IGOS-P frame), the development in the different Themes might easily result in competing, theme-specific implementations of parts of the geodetic observing system, and the neglect of other parts, that do not directly serve a specific theme but are crucial for the overall performance of GGOS. The establishment of an 'Earth System Dynamics' Theme as the frame for the implementation of GGOS would take into account the fundamental difference between GGOS and other observing systems in that GGOS requires an Earth system approach for its full development. While most of the existing Themes focus on sub-systems (like the Ocean Theme) or specific problem areas (like the Geohazards Theme), the 'Earth System Dynamics' Theme would be truly a whole system theme, comparable in this whole-system aspect only to the Global Water Cycle Theme.
References Altamimi, Z., Sillard, P., & Boucher, C., 2002. A new release of the International Terrestrial Reference Frame for earth science applications, J. Geophys. Res., 107, 2214, doi: 10.1029/2001JB000561. Beutler, G., Drewes, H., Reigber, C., & Rummel, R.,
2003. Proposal to Establish the Integrated Global Geodetic Observing System (IGGOS) as IAG's First Project, distributed to IVS and IGS mailing lists on 21 August 2003. GEO, 2005. The Global Earth Observing System of Systems (GEOSS) - 10-Year Implementation Plan, Distributed at and endorsed by the EOS-III Meeting, Brussels, 16 February 2005. Document prepared by the Ad hoc Group on Earth Observations (GEO) Impementation Plan Task Team IPTT. Available at http ://e arthob servati on s. org. GEO, 2005. Global Earth Observing System of Systems GEOSS - 10-Year Implementation Plan Reference Document- Draft, Tech. Rep. GEO 204/ESA SP 1284, ESA Publication Division, ESTEC, PO Box 299, 2200 AG Noordwijk, The Netherlands, Final Draft Document 204, prepared by the Ad hoc Group on Earth Observations (GEO) Impementation Plan Task Team IPTT. Available at http ://e arthob servations, org. McCarthy, D. D. & Petit, G., 2003. IERS Conventions 2003, IERS Technical Note 32, International Earth Rotation Service, Also available at http ://www.i ers. org. Plag, H.-R, 2002. European Sea Level Service (ESEAS): Status and plans, in Proceedings of the 14-th General Meeting of the Nordic Geodetic Commission, Espoo, Finland, 1-5 October 2002, edited by M. Poutanen & H. Suurmfiki, pp. 80-88, Nordiska Kommissionen f6r Geodesi. Plag, H.-P., Beutler, G., Forsberg, R., Ma, C., Neilan, R., Pearlman, M., Richter, B., & Zerbini, S., 2005. The Global Geodetic Observing System (GGOS): observing the dynamics of the Earth system, Available at http://www.unep.org/dewa/igos. Ray, J., Dong, D., & Altamimi, Z., 2004. IGS reference frames: Status and future improvements, GPS Solutions, pp. DOI: 10.1007/s 10291-004-0110-x. Rummel, R., 2000. Global Integrated Geodetic and Geodynamic Observing System (GIGGOS), in Towards an Integrated Global Geodetic Observing System, edited by R. Rummel, H. Drewes, W. Bosch, & H. Hornik, vol. 120 of International Association of Geodesy Symposia, pp. 253-260, Springer, Berlin. Rummel, R., Rothacher, M., & Beutler, G., 2005. Global Geodetic Observing System (GGOS): Science rationale, J. Geodynamics, 40, 357-362.
Chapter 105
IVS High Accurate Products for the Maintenance of the Global R e f e r e n c e F r a m e s a s C o n t r i b u t i o n to
GGOS W. Schlater Bundesamt for Kartographie und Geod~isie, Fundamentalstation Wettzell, Sackenrieder Strasse 25, D-93444 K6tzting, Germany D. Behrend, E. Himwich NVI, Inc./NASA Goddard Space Flight Center, Code 697, Greenbelt, MD 20771-0001, USA
A. Nothnagel Geodetic Institute of the University of Bonn, Nussallee 17, D-53115 Bonn, Germany A. Niell, A. Whitney MIT Haystack Observatory, Off Route 40, Westford, MA 01886-1299, USA
Abstract. VLBI provides highly accurate and unique products for the realization and maintenance of the celestial and terrestrial reference frames, ICRF and ITRF, as well as for the Earth Orientation Parameters. In 2001 the products were reviewed with respect to obtaining highest accuracy and to improving the observing sessions in order to make best use of the resources which are made available by the IVS member institutions. Since 2002 improved observing sessions were coordinated by the IVS aiming at fast turn-around products for EOP and better products for TRF and CRF. The number of observations increased by more than 30% from 2002 to 2004. New products, e. g. the troposphere zenith path delay, are now being generated. Concerns about the aging technology, which has been used for the past three decades, and about radio interference problems that decrease the number of useable observations, led to the establishment of IVS Working Group 3. The working group was asked to examine current and future requirements for geodetic VLBI, including all components from antennas to analysis, and to create recommendations for a new generation of VLBI systems. The results are summarized in a vision paper, VLBI2010 (Niell et al., 2005), which is recommended for coordination of new developments in VLBI and for plans to invest in new components by member institutions. This paper reviews IVS activities of recent years and gives a perspective about new developments for meeting future requirements, which are set up by GGOS.
Keywords. International VLBI Service, global reference frame, VLBI, IVS, GGOS
1 Introduction High-precision global reference frames are needed for precise positioning in geodesy, for navigation on the Earth and in space, for Earth and space research, and for applications in precision surveying. This is particularly true if satellite-based navigation systems such as GPS and GLONASS (or the future GALILEO) are employed. An inertial reference frame that is fixed in space is needed for orbit determination of space vehicles, of the moon and the planets, and for the description of the positions of stars and extragalactic sources. An Earth-fixed reference frame, i.e. a frame rotating with the Earth, is required for point positioning on the Earth. These two reference frames are connected through the dynamic motions of the Earth in space, including rotation, polar motion, nutation, and precession. The inertial frame is defined by quasars- radio sources which are billions of light years a w a y - and realized through the description of their positions through a pair of coordinates, known as right ascension (a) and declination (~5) in the defined Celestial Reference Frame (CRF). The Earth-fixed frame is defined by a set of stations that are located on the Earth's surface and are globally distributed. The stations' positions are described through coordinates (latitude, longitude, and height, or
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W. SchRiter • D. Behrend • E. H i m w i c h • A. N o t h n a g e l • A. N i e l l . A.
Whitney
geocentric x, y, and z) at a given epoch (e.g. 2000.0), and their change with time is given by their velocity (dx/dt, dy/dt, dz/dt) in a defined Terrestrial Reference Frame (TRF). For the transformation from CRF to TRF, and vice versa, the dynamic orientation of the Earth in space must be known. A set of five parameters, named Earth Orientation Parameters (EOP), describe the orientation of the Earth. In combination with precession and nutation models, two parameters (d~, de) fix the instantaneous rotation axis in the CRF, two parameters (Xp, yp) fix the instantaneous axis with respect to TRF, and one parameter describes the speed of rotation (DUT1). DUT1 is the difference of the Universal Time, UT1, based on the angle of rotation of the Earth at any given instant, and the Universal Time Coordinated, UTC, which is generated by stable atomic clocks. Due to the continuous changes in the Earth system (atmosphere, ocean, biosphere, lithosphere, mantle, and core), and in particular due to the variation of the mass distribution within the Earth system, all EOP parameters are subject to continuous changes. Since no models currently exist to describe these changes with sufficient precision, continuous observations are needed to accurately measure the EOP parameters. Very Long Baseline Interferometry (VLBI) plays a fundamental role in the process of the realization and maintenance of the global reference frames and in the determination of the EOP: Only VLBI allows the observation of quasars, which realize the CRF. • Only VLBI can provide the complete set of EOP and is unique for the determination of DUT1. • VLBI provides most precisely the length of intercontinental baselines, which strongly support the realization and maintenance of TRF. •
The International Association of Geodesy (IAG) is currently establishing the project GGOS (Drewes and Reigber, 2004), which aims to realize a global reference frame with a precision of 10-9, consistent for decades, which demands a strong support by national and international agencies coordinated through the International Services of the IAG. The International VLBI Service for Geodesy and Astrometry (IVS), as one of the lAG services, is tasked to coordinate VLBI components operated and provided by its member organizations and to generate the required EOP parameters precisely and in a timely manner. In 2001, a review of the existing products and observing programs was carried out by IVS Working Group 2 (Schuh et al.,
2002), which clearly defined the necessary IVS products and prescribed an IVS observing program optimized to make best use of the available resources to create these products. Many hardware components for VLBI (such as radiotelescopes) supporting IVS were constructed three to four decades ago; some are badly worn and inefficient in operation and in urgent need of an upgrade or replacement. The increase of radio interference, particularly at S-Band (2.0-2.4 GHz), caused by modern wireless technology (cell phones and satellite radio, among others) severely degrades the quality of many current observations and will force a move to higher frequencies. Additionally, the increasing demand for higher-quality global reference frames requires better global coverage. All these factors led the IVS to create the report called 'VLBI2010' which specifies guidelines for the future evolution of VLBI, for the refinement of its components, and for the replacement of old components and the development of new components (Niell et al., 2005).
2 VLBI Products and Goals In 2001 the IVS Working Group 2 (WG2) reviewed the IVS products and the corresponding observing programs in terms of meeting the service requirements. The WG2 report proposed a development of the products and the observing programs up to the year 2005. Even if, due to the limited resources, the proposed goals cannot be achieved in the expected period, the general view remains valid for years to come. The IVS products are summarized in Table 2.1. The demands on a next generation VLBI system need to be based on the requirements imposed on the products IVS has to provide. With respect to accuracy and latency, the upcoming lAG project GGOS will drive the future demands. The categories summarized in Table 2.1 have different requirements and conditions in terms of operation and timeliness. Some of the products, such as the EOP or TRF (partly), have to be provided operationally and in near-real time. Other products, in particular scientific products for CRF and TRF (partly), necessitate individual studies. Here the accuracy is of primary interest, whereas timeliness is not. The various requirements will be reflected in the specifications for the |VS components and in the demands for the observing programs and the data handling. Combinations of VLBI, GNSS (Global Navigation Satellite Systems), and SLR products will significantly improve the overall accuracy of all products. Systematic errors will be detected if time series of comparable precision are made avail-
Chapter 105 • IVS High Accurate Products for the Maintenance of the Global Reference Frames As Contribution to GGOS
able by all techniques. For the combination of all techniques to be useful, a reasonably timely delivery of products will be required. By making use of the technical potential of the Internet, e-VLBI will be realized in general and timely solutions will become more economical.
Table 2.1. IVS products and the demands on a next generation VLBI system
Cate- Product Accuracy Freq Reso- Timegory of sol lution liness CRF a,6 0.25 mas yearly 1 mo a,6-TS 0.5 mas 1/mo 1 mo 1 mo sou struc 1/mo 1 mo 3 mo flux dens 7 d/w 1 hr NRT TRF x,y,z-TS 2...5 mm 7 d/w 1 d 1d episo event 2...5 mm 7 d/w <1 d NRT annual soln yearly 1 mo coordinates 1...2 mm velocities 0.1...0.3mm/y EOP DUT1 7d/w 10min NRT 5 gs d% de 25...50 gas 7 d/w 1 d NRT Xp, yp 25...50 gas 7d/w 10min NRT dxp/dt 8... 10/aas/d 7d/w 10min dyp/dt GDP solid Earth 0.1% ly ly lmo tides ocean load 1% ly ly lmo atmosload 10% ly ly lmo PP trop param NRT zen delay 1...2 mm 7 d/w 10 min gradients 0.3...0.5 mm 7 d/w 2 h iono map 0.5 TECU 7 d/w 1 h NRT light deft 0.1% 1y session 1 mo param all GDP, geodynamical parameters; PP, physical parameters; NRT, near-real time; TS, time series; episo, episodic
3 Current Status of Geodetic VLBI Coordinated by IVS Figure 3.1 shows the distribution of the IVS components. As of 2004, IVS consists of • 30 Network Stations, acquiring VLBI data ; • 3 Operations Centers, coordinating the activities of a network of Network Stations; • 6 Correlators, processing the acquired data; • 6 Data Centers, distributing the products to users, providing storage, and archiving functions; • 21 Analysis Centers, analyzing the data, processing the results and products;
• 7 Technology Development Centers, developing new VLBI technology; •
1 Coordinating Center, coordinating the daily and long term activities.
In total, there are 74 Permanent Components, representing 37 institutions in 17 countries w i t h - 2 5 0 Associate Members. IVS coordinates the activities of all VLBI components for geodetic and astrometric use based on the proposals made by the institutions in reply to a call for participation. The contributions are dependent on the individual possibilities of the institutions, meaning that each institution provides as much as their resources allow.
3.1
Network
Stations
From Figure 3.1 it is apparent that most of the components are located on the northern hemisphere. The distribution of components is clearly not distributed homogeneously. The non-homogeneity is enhanced further by the fact that the network stations contribute differently to the amount of observations. Some stations carry a high load of observations and are included in most of the observing sessions, while other stations can only contribute to dedicated campaigns. Stations like Wettzell and Kokee Park are regularly involved in the entire observing program. Other stations, such as O'Higgins or Syowa, contribute only campaign-wise, due to their very remote location. The number of technical failures is increasing due to worn-out instrumentation or due to RFI (radio frequency interference). The various data recording techniques, such as Mark IV and Mark 5 (developed at Haystack Observatory for NASA), K4 and K5 (developed at NICT, Kashima, Japan), and $2 (developed in Canada), lead to additional constraints for the combination of network stations in common observing sessions, because the correlators are dedicated to one recording system only. The development of the VLBI Standard Interface (VSI) will help to overcome such a limitation in the near future. Most of the antennas are not primarily designed for geodetic and astrometric VLBI. Deformations of the telescope structures will lead to variations of the reference point, which is usually assumed as an invariant point of the telescope. Systematic errors due to such deformations will have to be considered. The transportation of the data to the correlator is one of the major reasons for the latency of the products. Access to high speed data links for data
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transfer is highly welcome. Nevertheless, high speed Internet access is currently not available at many stations due to the last mile problem and the high costs. Time and frequency information is provided by H-Masers for precise frequency generation and by employing GPS time transfer receivers to compare the local timescale to a global time scale, such as UTC(GPS). The current stability of the masers supports the accuracy of the observable to 20 to 30 ps; the comparison via GPS is good to approximately 100 ns.
tion resources, dynamical scheduling offers the promise of great improvement. Quasar positions are provided by the ICRF. The positions are available only for S- and X-band observations. Observations employing other frequency bands than S/X will require the determination of quasar positions in the respective frequency band, including long-term monitoring to ensure the required stability.
3.2 O p e r a t i o n s Centers
Most of the observing sessions are correlated with the Mk4 correlators at the US Naval Observatory, the Max-Planck-Institut ftir Radioastronomie, and Haystack Observatory. Some experiments are correlated with the K4 and K5 correlators in Kashima and Tsukuba, Japan, and some with the $2 correlator in Penticton, Canada. A real step forward was made by the development of the disk based recording systems Mark 5 and K5. The VLBI Standard Interface (VSI) will help to overcome the dif-
A master schedule is prepared for each calendar year by the IVS Coordinating Center. There are three Operations Centers, which coordinate the network stations for dedicated observing programs by preparing the detailed observation schedules. The schedule is set up before the session starts. As a means of gaining more flexibility in the event of a station failure, and for making best use of the sta-
3.3 Correlators
Chapter 105
• IVS
High Accurate Products for the Maintenance of the Global Reference Frames As Contribution to GGOS
ferent formats and recording philosophies. The Mk4 correlators have the capability to correlate up to 16 stations, but due to the limited available number of recording systems and station units, the capacity is limited to 8 or 9 stations for one correlator run. Since access to high speed data links has not been realized, the latency from observation to product delivery is dominated by the shipment of data carriers. First experiences are being gathered with a software correlator at the Geographical Survey Institute in Tsukuba, Japan, for the baseline WettzellTsukuba. Software correlators are likely to become important for providing products in real time.
international floor to contribute strongly to global and to regional projects. Close to real time products, time series derived in post-processing mode to achieve highest quality, and products acquired from special observing campaigns are provided, as summarized in the Table 4.1.
Table 4.1. General future demands to IVS products
Products
mode of ge- product neration availability
Processing
overall requirements
EOP
CONTOBS, TS w / h i g h est resolution
POST for best accuracy
COMB
TRF
CONTOBS for episodic events a campaign-
POST for best accuracy POST
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3.4 Data and Analysis Centers There are 21 Analysis Centers but only five or six Analysis Centers are handling the regular data flow, employing software programs such as CALC/SOLVE, OCCAM and SteelBreeze. Many of the Analysis Centers are doing special investigations and research. The results derived from the same data set obtained by different Analysis Centers show biases on the order of the internal precision. Studies to reduce the differences are under way. The results of the various Analysis Centers are combined to obtain the IVS solutions. To improve the product reliability more Analysis Centers and more software packages need to be involved in the procedure, in particular supporting the internal IVS combination routines. The routine analysis process requires more automation for near-real time provision of products.
3.5 Technology Developing Centers Technology developments for IVS are mainly carried out by the Haystack Observatory and by NICT in collaboration with related groups. The development of digital data recorders, the developments in e-VLBI, and the progress in the VLBI Standard Interface should be noted as significant steps forward in the last two years. These developments will play a key role in the evolution of the IVS.
4 Future Demands on IVS Products, and Required Steps of Improvement Future demands on IVS products are placed by the requirements set up by users of highly precise global reference frames and by the scientific community as well. The IAG has established the project GGOS, which will result in the provision of a precise global reference system, consistent for decades. Several groups expressed their willingness on the
CRF
close to RT for operational maintenance close to RT for episodic events
wise b by-product NRT POST COMB of EOP, TRF and CRF campaigns SV on demand POST PP, physical parameters; SV, space vehicles; CONTOBS, continuous observation over a short time interval; TS, time series; RT, real time; NRT, near-real time; POST, postprocessing; COMB, combination with other techniques a In combination with the observing sessions, time series for precise position monitoring. b In combination with source monitoring observing sessions. PP
The first attempt to optimize the overall resources resulted from the WG2 report. Nevertheless, to meet the upcoming service requirements and to guarantee the products provision for the maintenance of global reference frames, much more effort is required. Urgent steps of improvement are: • to overcome the unbalanced network configuration, • to increase the observing capabilities, • to reduce technical failures of old components (antennas), • to avoid frequency interferences, • to obtain compatibility in technology, in particular in data recording, • to develop dynamical scheduling to make best use of observation resources, • to speed up the data transmission to the correlators, • to reduce the latency between observations and product provision,
739
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W. SchKiter • D. Behrend • E. Himwich • A. Nothnagel • A. Niell. A. Whitney
• to reduce the systematic errors of the instrumentation, • to reduce systematic errors caused by the analytical and numerical models, • to increase the automation in the data handling process from the correlator to the final analysis, and • to support the combination with the other techniques. The observable in VLBI is the time delay "c, which is currently acquired with an accuracy of approximately 30 ps. To increase the overall accuracy in VLBI towards the demands of GGOS, which is approximately mm-accuracy in station position, the time delay "c has to be derived with an accuracy of 4 ps. Such an increase in accuracy may require time and frequency stability that is almost one order of magnitude more precise than is possible with the systems currently employed. Because of the correlations among the station height, the atmospheric refraction, and the time scale, new and better ways of correcting for the clocks and atmospheres will be needed. For example, the inclusion of water vapour radiometers will have also to be considered. As currently the precise positions of the quasars have been obtained only through S- and Xband observations, the catalogue needs to be extended to the other frequency bands that will be observed in the future. Moreover, variations in source positions and in the strengths of the sources need long term, continuous monitoring. The capac-
ity to link dedicated satellites to the CRF by means of VLBI observations should be considered. Such features will become important for observing very remote space missions or for connecting navigation satellites to the CRF. For planning the future of the VLB! system, the long term continuity of VLB! must be taken into consideration. The changes that are needed must be accomplished through wellorganized transition plans, and the realization will need approval of all contributing institutions.
References
Drewes, H., C. Reigber (2004) The IAG Project "Integrated Global Geodetic Observing System (IGGOS)" Setup of the initial Phase, in International VLBI Service for Geodesy and Astrometry 2004 General Meeting Proceedings,
edited by N. Vandenberg and K. Bayer, NASA/CP-2004212255, pp. 186-190. (http://ivscc.gsfc.nasa.gov/publications/gm2004/drewes) Niell, A., A. Whitney, W. Petrachenko, W. Schlfiter, N. Vandenberg, H. Hase, Y. Koyama, C. Ma, H. Schuh, G. Tuccari (2005) VLBI2010: Current and Future Requirements for Geodetic VLBI Systems, IVS WG3 Report, http://ivscc.gsfc.nasa.gov/about/wg/wg3. Schuh, H, P. Charlot, C. Klatt, H. Hase, E. Himwich, K. Kingham, C. Ma, Z. Malkin, A. Niell, A. Nothnagel, W. Schltiter, K. Takashima, N. Vandenberg (2002) IVS Working Group 2 for Product Specification and Observing Programs, in International VLBI Service jor Geodesy and Astrometry 2001 Annual Report, edited by N. Vandenberg and K. Baver, NASA/TP-2002-210001, pp. 13-45. (http://ivscc.gsfc.nasa.gov/about/wg/wg2).
Chapter 106
The International Laser Ranging Service and Its
Support for GGOS M. Pearlman
Harvard-Smithsonian Center for Astrophysics (CfA), Cambridge, MA 02138, USA C. Noll NASA Goddard Space Flight Center, Greenbelt, MD 2077 l, USA W. Gurtner Astronomical Institute, University of Bern, Bern, Switzerland R. Noomen Facility of Aerospace Engineering, Delft University of Technology, 2629 HS Delft, The Netherlands Abstract. The International Laser Ranging Service (ILRS) was established in September 1998 as a service within the lAG to support programs in geodetic, geophysical, and lunar research activities and to provide data products to the International Earth Rotation and Reference Systems Service (IERS) in support of its prime objectives. The ILRS develops the standards and specifications necessary for product consistency and the priorities and tracking strategies required to maximize network efficiency. This network consists of more than forty SLR stations, routinely tracking nearly thirty retroreflectorequipped satellites and the Moon in support of user needs. The Service collects, merges, analyzes, archives and distributes satellite and lunar laser ranging data to satisfy a variety of scientific, engineering, and operational needs and encourages the application of new technologies to enhance the quality, quantity, and cost effectiveness of its data products. The ILRS works with the global network to improve station performance, new satellite missions in the design and building of retroreflector targets to maximize data quality and quantity, and science programs to optimize scientific data yield. The ILRS Central Bureau maintains a comprehensive web site (http://ilrs.gsfc.nasa.gov) as the primary vehicle for the distribution of information within the ILRS community. During the last few years, the ILRS has addressed very important challenges: (1) Data from the network stations are now submitted hourly and made available immediately through the data centers, (2) Tracking on low orbit satellites has been
significantly improved through the sub-daily issuance of predictions, drag functions, and the realtime exchange of time biases, (3) Analysis products are now submitted in SINEX format for compatibility with the other space geodesy techniques, (4) The Analysis Working Group is now generating an operational station position and Earth Orientation Parameter (EOP) product, and (5) SLR has significantly increased its participation in the International Terrestrial Reference Frame (ITRF) activity. Keywords. Space geodesy, satellite laser ranging (SLR), lunar laser ranging (LLR), International Laser Ranging Service (ILRS), terrestrial reference frame, GGOS.
1 Role of Laser Ranging within GGOS In early 2004, under its new reorganization, the International Association of Geodesy (lAG) established the Global Geodetic Observatory System (GGOS) project to coordinate geodetic research in support of scientific and applications disciplines. GGOS is intended to integrate different geodetic techniques, models and approaches to provide better consistency, long-term reliability, and understanding of geodetic, geodynamic, and global change processes. Through the IAG's measurement services (1GS, IVS, ILRS, IDS, and IGFS), GGOS will work to ensure the robustness of the three aspects of geodesy: geometry and kinematics, Earth orientation and rotation, and static and time varying
742
M. Pearlman• C. Noll.W. Gurtner.R. Noomen
gravity field. GGOS will identify geodetic products and establish requirements on accuracy, time resolution, and consistency. The project will work to coordinate an integrated global geodetic network and implement compatible standards, models, and parameters. A fundamental aspect of GGOS is the establishment of a global network of stations with collocated techniques, working together to provide the strongest reference system. The ILRS will be one of the service participants in GGOS, bringing its unique strengths to the geodetic complex. 1.1 T h e G l o b a l T e r r e s t r i a l R e f e r e n c e F r a m e
The terrestrial reference system is the basis through which we connect and compare measurements over space, time, and evolving technologies. It is the means by which we know that measured change over time is real and not corrupted with instabilities in our measurement technique. One of the best-known scientific realizations of a global terrestrial reference system is the International Terrestrial Reference Frame (ITRF), updated every few years by the International Earth Rotation and Reference Systems Service (IERS). It is based on contributions from the four different space geodetic techniques, consisting of solutions for the positions and velocities of all participating tracking stations in an Earth-fixed geocentric coordinate system. The most important contributions of the laser ranging technique to the reference frame are the fixing of its origin (defined with respect to the center-of-mass of the Earth, including oceans and atmosphere) and its scale (defined by the speed of light, realized mainly through the measurement of the time of propagation, i.e., the ranges to satellites). Origin and scale are crucial elements, not only for "classical" referencing purposes (i.e., crustal deformation studies), but also as providers of an absolute reference for investigations on sea level change, ice budget, etc. ILRS contributions come either as multiyear solutions based on ranges to the geodynamic satellites LAGEOS-1 a n d - 2 or, within the current IERS Combination Pilot Project, as time-series of weekly solutions. The latter are combinations of the individual solutions from several ILRS Analysis Centers to generate one "best" technique-specific submission for the IERS. 1.2 Earth in S p a c e
The connection between the terrestrial and the celestial reference systems is given by the current
position of the Earth's body with respect to its current axis of rotation (polar motion), its orientation in space (UT1) or the change of its orientation (angular velocity, length-of-day), and the orientation of the axis of rotation in space (precession and nutation). One of the official ILRS products is the timeseries of polar motion and length-of-day, submitted weekly to the IERS for combination with the timeseries generated by the other space geodetic techniques. 1.3 M o d e l s of Earth's G r a v i t y Field
Until recently global gravity field models were based on a combination of ground-based satellite tracking and surface measurements, with SLR playing a very major role. New, dedicated gravity satellite missions including new types of observations like satellite-to-satellite tracking, in-orbit observation of derivatives of the gravity field, and oceansurface altimetry have provided much higher spatial and temporal resolution of the gravity field. However, lower-degree terms of this field and their secular changes (e.g., of the dynamic flattening of the Earth) are best observed by laser ranging. Also in view of the long time-history of the solutions, SLR provides extremely valuable information on (changes in) the overall mass distribution of the Earth. In this way, SLR provides extremely valuable information on (changes in) the overall mass distribution of the Earth. 1.4 P r e c i s e Orbit D e t e r m i n a t i o n and Verification
SLR provides routine precise orbit determination for some missions and verification and calibration of precise orbits determined with other tracking techniques such as GPS or DORIS for others. The high accuracy and unambiguous nature of SLR data makes it an independent source of quality control and calibration for other tracking techniques. In particular, SLR has been used in all of the recent ocean and ice topography missions to support altimeter measurements and for a number of special engineering activities (e.g., altimeter calibration). SLR, with its totally passive spaceborne reflectors, also acts as a backup for active tracking techniques. It has saved satellite missions (ERS-1, GFO- 1, TOPEX/Poseidon, and Meteor-3M) after the failure of the primary tracking system. The ILRS continues to encourage new missions with high precision orbit requirements to include retroreflectors as a fail-safe backup tracking system,
Chapter 106 • The International Laser Ranging Service and Its Support for GGOS
to improve or strengthen overall orbit precision, and to provide important intercomparison and calibration data with onboard microwave navigation systems. 1.5
Lunar
Laser
synergy. The majority of ILRS stations have a collocated GPS receiver that adheres to International GPS Service (IGS) standards.
I!
Ranging
I_ IERSRep. iI Coord.
The two ILRS stations currently capable of routinely tracking the four lunar targets have a long history of providing LLR data: the McDonald Observatory in Texas has been in operation since the Apollo 12 mission (since 1985 with the current system), whereas the Grasse Observatory in France started lunar laser ranging in 1987. Several stations have demonstrated lunar capability while others have tracked the Moon for some periods of time (e.g., Maui, Hawaii, 1984-1990). A number of stations (e.g., Matera, Italy, and Mount Stromlo, Australia) are planning to include lunar tracking in their future activities. Applications in gravitational physics include: testing of the Equivalence Principle; (limits for) time-variation of the gravity constant G; and the assessment of the geodetic precession. Applications in lunar science include the determination or the improvement of lunar ephemerides and rotation; dissipation-caused (negative) acceleration; and an assessment of the interior and the Lunar Love numbers. More details about current SLR/LLR contributions to science can be found e.g., in the proceedings from the Science Session of the 2002 International Workshop on Laser Ranging (Noomen et al, (2003)). 2
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The ILRS is organized into the following components: Tracking Stations and Subnetworks, Operations Centers, Global Data Centers, Analysis and Associate Analysis Centers, a Central Bureau, Working Groups, and a Governing Board (see Figure 1 and Pearlman et al (2002) and Gurtner et al (2005)). Stations in the ILRS Tracking Network range to the approved constellation of artificial satellites and the Moon and transmit their data in near-real time to the ILRS Data Centers. The full network currenfly consists of about forty SLR stations as shown in Figure 2. The ILRS has given strong encouragement to the development of Fundamental Reference Stations, where a combination of several space geodetic techniques including SLR, VLBI, GPS, DORIS, and absolute gravimetry are collocated to strengthen reference system constraints and system
ILRS Operations Centers collect and merge the data from the tracking sites, provide initial quality checks on these incoming data, reformat and compress the data if necessary, and relay the data to an ILRS Data Center. Two Global Data Centers archive all the ranging data and auxiliary data (e.g., station log files and satellite orbit predictions), make the data available to the ILRS Analysis Centers and external users of the data, and act as distribution centers for the primary ILRS products. The Analysis and Associate Analysis Centers routinely generate the official ILRS products (station coordinates and derivatives at one-week intervals, EOP at one-day intervals) as well as special products, such as satellite predictions, time-bias information, precise orbits for special-purpose satellites, or scientific data products of a mission-
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M. Pearlman • C. Noll. W. Gurtner. R. Noomen
specific nature. Lunar Analysis Centers process ranging data to the targets on the Moon and produce lunar-specific data products. There are currently about thirty ILRS Analysis and Associate Centers. The Central Bureau (CB) is responsible for the daily coordination and management of the ILRS. The CB maintains the ILRS web site (http://ilrs.gsfc.nasa.gov), a source for all SLR- and LLR-related information, details about the organization and operation of ILRS, and also an entry point to the data and products stored at the Data Centers. The CB maintains the ILRS documentation, organizes meetings and workshops, and issues service reports. The ILRS has four standing Working Groups that provide the expertise necessary to make technical decisions, to plan programmatic courses of action and are responsible for reviewing and approving the content of technical and scientific databases maintained by the Central Bureau. They are: • Missions Working Group reviews requests for laser tracking and recommends the relative tracking priority with respect to other approved satellites. • Data Formats and Procedures Working Group develops and maintains formats and standardized procedures. • Networks and Engineering Working Group facilitates the generation, collection and distribution of data to the user community in a timely and efficient manner, and examines new technologies to improve network performance. • Analysis Working Group (AWG) provides feedback to the network regarding data quality, maintains standards for methods of analysis, provides quality control on the analysis products, and organizes and coordinates activities for the official ILRS analysis products. An ad hoc Working Group on Signal Processing studies the effects introduced by the satellite retroreflector arrays on ranging accuracy. The ILRS Governing Board (GB) is responsible for the general direction of the service and defines official ILRS policy and products, determines satellite-tracking priorities, and develops standards and procedures. The sixteen-member body interacts with other services and organizations and is selected from ILRS associates representing all components of the service. 3
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Satellites
Since its inception, SLR has tracked more than fifty satellites with retroreflectors. Currently, the ILRS
tracks 28 satellites for geodynamics, remote sensing (altimeter, SAR, etc.), gravity field determination, general relativity, verification of global navigation systems satellite orbits, and engineering tests. Altitudes range from a few hundreds of kilometers to GPS altitude (20,000 km). Two stations, Grasse, France and McDonald, USA, routinely range to four targets on the Moon. Satellites are added and deleted from the ILRS tracking roster as new programs are approved by the GB and old programs are completed. The ILRS assigns satellite tracking priorities in an attempt to maximize data yield on the full satellite complex while at the same time placing greatest emphasis on the most immediate data needs. Nominally, tracking priorities decrease with increasing orbital altitude and increasing orbital inclination (at a given altitude). Priorities of some satellites are then increased to intensify support for active missions (such as altimetry), special campaigns, and post-launch intensive tracking campaigns. Finally, daily priority adjustments are made based on actual data yield over the previous ten days. New missions scheduled for ILRS tracking support over the next year include ALOS and OICETS, and the first two engineering versions of the Galileo satellites (GSTB-V2A and GSTB-V2B). It is anticipated that the full thirty satellite Galileo complex, to be launched between 2007 and 2008, will require, at least intermittent, S LR tracking. The ILRS supports space engineering studies on some rather unique missions. The Russian Reflector satellite included retroreflectors over its nearly 1.5 meter length (Figure 3). Differences in the laser return time-of-arrival were used to interpret the orientation and dynamics of the satellite (Figure 4). Another mission, the Naval Research Laboratory's Tether Physics and Survivability satellite, (TIPS) with retroreflector arrays on two satellites separated by a four-kilometer tether was tracked by SLR to study tether dynamics in space. 4
Network
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Laser ranging stations use short pulse lasers to measure the distance to passive targets on satellites and the Moon. The measured range is the roundtrip travel time corrected for optical refraction, spacecraft center-of-mass, and relativistic effects. The prime data product from the ILRS stations are normal points, which are full-rate ranging data averaged over time intervals ranging from fifteen seconds to five minutes depending upon the satellite altitude. Absolute accuracy is typically better than a
Chapter 106 • The International Laser Ranging Service and Its
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Since the inception of the ILRS through 2003, the data yield of the network continued to improve (Figure 5) as stations implemented more automated procedures and new satellites are added to the tracking roster. In 2004, the NASA network suffered some severe budget reductions resulting in the loss of two stations and decreased operations in the others. Efforts are underway now to resurrect the closed stations. Most of the current laser-tracking stations range ten times per second during part or all of the satellite pass, with many stations interleaving passes from different satellites (see Figure 6). An example
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745
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M. Pearlman• C.Noll.W.Gurtner.R.Noomen 5 Data Products From the beginning, the ILRS has put the generation of official analysis products high on its agenda. The Analysis Working Group has initiated several pilot projects, to address questions on the official products, to define standards, to reach consensus on product definition and, ultimately, to arrive at a reliable and high-quality operational product.
of older data focuses on the 1992-2003 timeframe (i.e., the time period since the launch of LAGEOS2), but SLR measurements on LAGEOS-1 obtained in the years prior to 1992 will also be processed for input into the subsequent ITRF solutions.
Table 1. Scatter of Helmert parameters w.r.t. ITRF2000 of successive weekly ILRS solutions for 2004 (geocenter in mm, scale in ppb).
5.1 Pilot Project " P o s i t i o n i n g + E O P " Individual One Pilot Project focused on the computation of the best-possible ILRS product for station coordinates and Earth Orientation Parameters (EOPs). Various scenarios were defined and tested for establishing the proper satellite mix, means of representing the results, computational strategies, etc. This resulted in the scheme of processing that is currently operational. At this moment, five different analysis groups (ASI/Italy, DGFI/Germany, GFZ/Germany, JCET/USA and NSGF/UK) deliver weekly solutions on LAGEOS-1 and -2 for global station coordinates and EOPs on Tuesday of each week. These solutions are merged into a combination solution by ASI and DGFI. Based on the contributions that the analysis groups made to the Pilot Project and an evaluation of the quality of their resuits, the ILRS named ASI as its official Combination Center, with first responsibility for the generation of combination solutions for external customers such as the IERS. DGFI was named the official Backup ILRS Combination Center with the same product generation schedule. Several other institutions such as BKG/Germany, Geosciences Australia, and CSR/USA are also well on their way in the development of their contributions to the official ILRS product. As an illustration of the quality of the individual solutions as well as that of the official combination product (the quality of the primary and the backup solutions is effectively identical), Table 1 gives a summary of the scatter of weekly solutions for geocenter components and global scale. It is clearly visible that the combination solutions give the best result, and that these products offer the best that ILRS can provide. The combination solutions are used for a variety of purposes: the IERS Combination Pilot Project, the IERS/NOAA Bulletin A, etc. At the request of IERS, the ILRS AWG has started a back-processing of older SLR data in a similar fashion to serve as input for a successor to ITRF2000 and possible other applications. At this moment, the processing
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5.2 Pilot Project " B e n c h m a r k i n g " A Benchmarking Pilot Project has been established to provide internal quality checks and quality control over the analysis process. Initially, this was used to scrutinize individual elements of the SLR observations, measurement corrections and parameter solutions. Having reached a fully operational status, the Benchmarking Pilot Project is now being used to assess the quality of new candidate contributors to the ILRS combination products and to identify possible errors. Other pilot projects have also been developed to assess candidate products and analysis techniques.
5.3 New Products In addition to the Station Coordinate and EOP Product, the AWG is studying the production of other official data products including satellite ephemerides and geocenter time history.
6 Modeling The precision of the travel-time measurement is now of the order of several tens of picoseconds, corresponding to a few millimeters in distance. Ranging accuracy is limited mainly by errors in modeling for refraction propagation and the extrapolation from the reflectors to the satellite centerof-mass.
6.1 Refraction Since the launch of LAGEOS-I in 1976, laser ranging stations have used the Marini and Murray model (Marini and Murray, (1973)) for atmospheric
Chapter 106 • The
propagation correction. The model works well at higher elevations, but degrades substantially below 20 degrees. A new model now available (Mendes and Pavlis (2004)) provides improved refraction correction at lower elevations, a region now of greater importance as we try to expand SLR orbital coverage.
6.2 Retroreflector Array Early retroreflector designs, even those of the LAGEOS era, relied on multi-cornercube returns to maximize return signal strength. Satellite center-ofmass correction is highly dependent upon return signal strength and detector configuration, neither of which was well considered in the correction models. Errors could be as much as a cm. Models are now being implemented that use ground systems characteristics to improve the modeled correction to the level of a mm.
7 Advances Underway A number of advances are currently being implemented that will substantially improve data productivity and quality, while at the same time reduce operational costs. Many of these advances that are now working in one or two stations are envisioned as general characteristics of the future SLR network.
InternationalLaserRangingServiceandIts Support forGGOS
7.3 New, More Powerful Stations A number of new systems with large meter-sized telescopes and state-of-the-art optical and mechanical performance are now operational. This helps to give the tracking network a mix of capabilities to better match the range of targets that now appear on the ILRS roster. The Matera station in southern Italy and the remotely controlled Tanegashima station in Japan both use powerful lasers and large optics to achieve single-shot range precision of a few millimeters.
7.4 Two-Color Ranging Several groups are using two-wavelength ranging which provides a promising technique for developing better models for the refraction delay imposed by the atmosphere. Two-color ranging at 423 nm and 846 nm has been underway at Concepci6n for the past two years. The Zimmerwald station has recently begun routine operations at the same wavelengths. Other stations (including GSFC and Matera) have also demonstrated dual-wavelength capability, some of them with superior accuracy using streak cameras. As this technique matures, it is anticipated that the data will help in the improvement of the refraction models, and hence also the ILRS science products.
7.1 Automation Stations are implementing increased automation to reduce personnel costs and facilitate data throughput. The new Mount Stromlo station was designed from the beginning with around-the-clock fully automated, unmanned operations. The Zimmerwald laser station operates autonomously for periods of several hours each day. The fully automated NASA SLR2000 is currently under development.
7.2 KHz Ranging With higher-repetition-rate lasers, faster event timers, and better control software, SLR systems are now able to significantly improve the ranging signal-to-noise conditions. The new Graz SLR station was the first to successfully operate a 2 kHz laser system, increasing the full-rate data volume by up to two orders of magnitude. The SLR2000 prototype is also being developed with this capability, as are upgrades at several other stations.
7.5 Improved Satellite Retroreflector Array Design Early retroreflectors were designed to provide multi-cornercube returns to maximize return signal strength. Even with LAGEOS, the return signal is smeared over several centimeters, making measurements highly dependent upon signal strength and ground system properties. Efforts are underway to parameterize the ground stations and standardize models. Retroreflector array designs are also improving. Most of the recently launched satellites are using standardized arrays with restricted cornercube view. The spherical satellite GFZ-1, Westpac, and LARETS experimented with special reflector geometries to limit access to a very few (or single) cubes. Using another approach, the Russian Space Agency has provided the ILRS with a space borne test Luneburg Sphere that gives the same array correction for a wide variety of aspect angles (MeyerArendt, (1995)).
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M. Pearlman • C. Noll. W. Gurtner. R. Noomen
7.6 Transponders Optical transponders for extraterrestrial ranging are currently under early development by several research teams. An optical transponder is a combination of a laser-ranging receiver and a separate laser pulse transmitter. As opposed to two-way ranging with retroreflector targets, one-way ranging with a transponder would offer the exciting opportunity of ranging to Mars, planetary moons or orbiters, and deep space missions (Degnan, (2002)). These transponders will also help to connect the terrestrial reference frame with reference systems used for planetary missions.
8 Conclusions Laser ranging has proven to be a fundamental component of the space-geodetic complex, offering a straightforward, conceptually simple and highly accurate observable. It provides essential contributions to geosciences, space sciences and fundamental physics. It will play an important role in the GGOS project. Current and future challenges lie in the improvement of the accuracy, reliability and availability of the data and in the long-term support of the network. Many of the technological building blocks for the next generation of laser ranging have already been demonstrated. Their comprehensive implementation will bring dramatic improvements to the capability of the technique.
Acknowledgements The authors would like to acknowledge the support of the International Laser Ranging Service and its contributing organizations.
References Degnan, J. J. (2002). Asynchronous laser transponders for precise interplanetary ranging and time transfer. Geodynamics, Special Issue on Laser Altimetry, Vol. 34. pp. 551-594. Gurtner, W., R. Noomen, M.R. Pearlman (2005). The International Laser Ranging Service: Current Status and Future Developments. Advances in Space Research, In Press, Available online 1 February 2005. Marini, J.W. and C.W. Murray (1973). Correction of Laser Range Tracking Data for Atmospheric Refraction at Elevations Above 10 Degrees. GSFC Report X-591-73-351. Mendes, V.B. and E.C. Pavlis (2004). High-accuracy zenith delay prediction at optical wavelengths. Geophysical Research Letters, Vol. 31, No. 14, L14602.
Meyer-Arendt, J.R. (1995). Introduction to Classical and Modern Optics (4 th ed.), Prentice Hall, Englewood Cliffs, NJ 02632. Noomen, R., S. Klosko, C. Noll, and M. Pearlman (eds.) (2003). Proceedings from the 13th International Laser Ranging Workshop, Washington D.C., October 7-11, 2002. NASA/CP-2003-212248. Pearlman, M.R., J.J. Degnan, and J.M. Bosworth (2002). The International Laser Ranging Service. Advances in Space Research, Vol. 30, No. 2. pp. 135-143.
Chapter 107
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The Nordic Geodetic Observing System (NGOS) Markku Poutanen Finnish Geodetic Institute, Geodeetinrinne 2, 02430 Masala, Finland Per Knudsen Danish National Space Center, Juliane Maries Vej 30, 2100 Copenhagen, Denmark Mikael Lilje National Land Survey of Sweden, SE-801 82 G~ivle, Sweden Torbj 0rn N0rbech Norwegian Mapping Authority, 3507 Honefoss, Norway Hans-Peter Plag University of Nevada, 1664 North Virginia Street, Reno, NV 89557-0042, USA Hans-Georg Scherneck Onsala Space Observatory, SE-439 92 Onsala, Sweden Abstract. The Nordic Geodetic Observing System (NGOS) integrates fundamental geodetic techniques for the long-term observation of Earth system parameters. The Nordic Geodetic Commission (NKG) established a Task Force with the mission to prepare a document providing the definition and draft for the implementation of the NGOS. The regional system aligns with international efforts such as the Global Earth Observation System of Systems (GEOSS) and Global Observing Systems and adheres to the Integrated Global Observing Strategy (IGOS). Recent developments in the structure and function of the International Association of Geodesy cumulated in the establishment of the Global Geodetic Observing System, GGOS. NGOS is planned as a regional implementation and densification of the GGOS. The NGOS is proposed as a system that will serve a wide range of scientific and practical applications. For the Nordic countries, a main focus will be on crustal motion, dynamics of glaciated areas and sea level. NGOS aims to provide geodetic observations for the Nordic area that are of sufficient quantity and quality to serve most of the needs of global Earth observation as well as practical and scientific applications in the region. In particular, NGOS will contribute to the GGOS and other IAG Services; European activities such as EUREF, ECGN, EUVN, and ESEAS; provide the reference frames for the Nordic countries, as well as contribute to the global ones; support scientific projects related to the geodynamics of the Nordic area and provide ground-truth for satellite missions. In this paper we describe the plans and current status of the NGOS.
Keywords: geodetic observing system, geodetic observations, reference frames
1 Introduction The three pillars of geodesy according to Bruns as stated in the 19 th century are geometrical geodesy (constructing the terrestrial polyhedron), astronomical geodesy (tracing the motion of the polyhedron in inertial space) and gravimetry (derive gravity potential or height differences between the points of the polyhedron). This concept is still valid, although the resolving power of the primary parameters has moved to the parts per billion level, and many more processes can be studied at this level. The space-based technologies allow us now to determine positions in a globally coherent and highly accurate reference frame. Key variables of the Earth system such as the movements of the tectonic plates, land movement, Earth rotation, changes in the gravity field, and sea level changes are now observable in a globally consistent reference frame. There is an increasing demand for accurate geodetic observations for many scientific and non-scientific applications. However, the accuracy level achieved reveals inconsistencies between the global reference frame and the regional and national frames established for practical use. The regional system aligns with international efforts such as the Global Observing Systems and adheres to the Integrated Global Observing Strategy (IGOS, see http://www.eohandbock.com/igosp/and the documents available there). However, Plag (2000) pointed out that none of these observing
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M. Poutanen • P. Knudsen • M. Lilje • T. N~rbech. H.-P. Plag. H.-G. Scherneck
systems includes a geodetic component or is directly connected to the extensive global geodetic observing networks established over the last decade. Thus, until very recently, the fundamental role of geodesy as the backbone for all Earth observation was not formally acknowledged in the frame of global Earth observations. Over the last five years, the European Commission and the European Space Agency have jointly proposed a programme for Global Monitoring for Environment and Security (GMES). The needs of GMES in terms of observational infrastructure as well as data management and information system can be expected to be a major driver for the development of observation networks and applications over the next decade. During the last two years, the establishment of the Global Earth Observation System of Systems (GEOSS, see GEO, 2005a, b) has dramatically changed the landscape of global Earth observation. IAG was actively involved in the development of the GEOSS Implementation plan, and the fundamental role of geodesy as the provider of the global reference frame and important observations of the Earth's shape, gravity field and rotation is widely acknowledged (see GEO, 2005b). The new structure of the International Association of Geodesy (IAG) beyond 2000 defines projects as an entity of coordinated long-term activities (Beutler et al., 2000a, b). At the XXIII General Assembly of the IUGG in Sapporo, Japan, July 2003, the Integrated Global Geodetic Observing System (IGGOS, see Rummel et al., 2001, Beutler et al, 2003; later on GGOS, Global Geodetic Observing System) has been established as such a project. GGOS will contribute to large international observation and science programs. Large parts of the physical observing network of the GGOS are already in place through the efforts of the national mapping authorities and other institutions involved in operational monitoring. These networks are contributing e.g. to the International Celestial and Terrestrial Reference Frames (ICRF and ITRF). In parallel and in response to the international developments, geodesists in the Nordic countries have worked towards the definition of an integrated geodetic observing system in the Nordic countries. The Nordic Geodetic Commission (NKG) has recognized the necessity for contributions to global geodetic networks as well as regional homogenisation and standardization and promoted these in the Nordic region. Responding to the international development in Earth observation, since 1998 NKG working groups have discussed the
rational and objectives as well as the potential applications of a Nordic Geodetic Observing System (NGOS). In 2002, the NKG presidium set up a Task Force with the mission to prepare a document providing the definition and draft for the implementation of the NGOS. The authors of this paper were nominated to prepare the plan and the present document. The full document of the NGOS Task Force can be found in the web pages of the NKG, http://www.nkg.fi (Poutanen et al., 2005b, c).
2 Application of geodetic observations A global geodetic observing system provides the infrastructure and observations to determine and maintain an accurate and stable global terrestrial reference frame, and delivers observations of the changes in the geometry and rotation of the solid Earth as well as changes in the Earth's gravity field. In order to detect slow changes in the Earth system, the long-term stability of the reference frame is crucial. The geodetic observations from permanent networks are fundamental for the definition and maintenance of these reference frames. Gravity measuring satellites CHAMP, GRACE and GOCE will provide us an information on the Earth's gravity field which eventually will enable a precise realization of a global vertical datum but also allows us to monitor the gravity changes. Changes in the Earth's geometry, gravity field and rotation are caused by mass movements and dynamical processes in the Earth system. Consequently, observations of these quantities provide a means to monitor the dynamics of the Earth system and associated mass movements, such as fluxes in the hydrological cycle including ocean circulation, ground water storage, terrestrial surface flows, sea level changes and ice changes.
IGOS
IAG ,_qGOS Service
ECGN EUREF
G3OS GLOSS
ESEAS NGOS
GMES
Figure 1. Relations of NGOS to other activities. (ECGN = European Combined Geodetic Network; ESEAS = European Sea-Level Service; EUREF = Reference frame subcommission for Europe; G3OS = Global Observing Systems; GGOS = Global Geodetic Observing System; GLOSS = Global Sea Level Observing System; GMES = Global Monitoring for Environment and Security; IAG = International Association of Geodesy; IGOS = Integrated Global Observing Strategy; NGOS = Nordic Geodetic Observing System)
Chapter107 • The NordicGeodeticObservingSystem(NGOS) 2.1 Positioning, c h a n g e s in positions, and displacements
In maintaining the reference frames and monitoring the Earth system, GPS augmented with the products of the IGS (International GNSS Service) has served both as the primary measuring device as well as a tool to position sensors with unprecedented accuracy. Ten years of experience have shown that accuracies are of the order of better than 1 cm in daily or sub-daily positions and better than 1 mm/yr in secular stability. Today, the International Terrestrial Reference Frame (ITRF) is the most accurate global frame available. On the global scale, monitoring the surface kinematics through the ITRF network and particularly its densest component, the IGS network, has contributed significantly to improvements of the global plate motion model. Additional improvements can be expected mainly through better coverage of the Earth surface with continuous GPS (CGPS) sites and also other spacegeodetic stations. Accuracy requirements are of the order of 1 mm/yr or better. For the vertical velocities no general global model exists. However, for postglacial rebound as one of the major causes in Northern Europe and Canada, geophysical models provide predictions with accuracy on the 2-3 mm/yr level. Most of the Fennoscandian uplift, however, can be reconstructed better than 1 mm/yr due to repeated precise levelling since last 100 years, and current CGPS networks. In the B IFROST project (see e.g. Scherneck et al., 1998, Milne et al., 2001, Johansson et al., 2002), both the vertical and horizontal post-glacial rebound signal could be detected in long (> five years) CGPS time series of a regional network. Significant differences of model predictions due to differences in the Earth and ice models used to compute the present-day kinematics are of the order of 1 mm/yr and < 0.5 mm/yr in the vertical and horizontal components, respectively. On seasonal and subseasonal time scales, the changes in the shape of the Earth as observed by the global and regional geodetic networks have been used to extract the signal due to surface loading (e.g. van Dam and Herring (1994), van Dam et al. (1994), and van Dam et al. (2001)). In order to contribute to a validation of different loading models, this sets a requirement for the precision of daily coordinates (e.g. Poutanen et al., 2005a). Increasingly, GNSS is used to monitor the motion and stability of large infrastructure such as oil platforms, reservoir dams and bridges. User
requirements for monitoring of such infrastructure are of the order of less than 1 cm for sub-daily to daily positions available with a latency of a few days, 2-3 mm on monthly to seasonal time scales, and 1 mm/yr for long-term stability (Plag, 2004). A key limitation for monitoring infrastructure at points far away from a stable reference point is the temporal stability of the global reference frame. Many applications require an accuracy of 5 cm over a time span of up to 50 years, which is equivalent to a long-term stability of 1 mm/yr. 2.2 Earth rotation
The NGOS as a regional effort offers a high-quality capability contributing to global monitoring of earth rotation, primarily with VLBI. The aim is submillimetre stability and calls for corresponding stability monitoring. The methods are, besides collocation of techniques, regional footprinting in e.g. GPS networks. This is an essential requirement in order to achieve the necessary consistency of reference frames. Earth rotation monitoring is carried out in sparse networks. The stability of the participating stations plays a key role. Future demands of site stability will probably range below 0.1 mm (horizontal standard deviation). Emphasis is therefore given to a few well-collocated stations that can be followed up with local footprint studies under long time. The three stations Ny Alesund, Metsfihovi and Onsala continue to be important international resources. Collocated GNSS stations will help to improve the short-term resolution of EOP time series, but VLBI is the only technique available that is capable to link the terrestrial reference frame to the celestial frame on a routineous basis. Earth orientation parameters are key parameters for global reference frames. Need for continuous monitoring and rapid solutions is likely to increase somewhat. Continuous monitoring is already implemented on a global scale; however, increasing need for higher accuracy will probably call for a somewhat larger number of stations in simultaneous networks. Climate-related processes, especially the E1 Nifio phenomenon in the Pacific, leave discernible fingerprints in earth rotation variations. This area has high potential for further discoveries. With the increasing length of observational time series and settling of uncertainties at still better levels more subtle changes in the atmospheric-oceanic angular momentum budget and effects of atmospheric and oceanic torques on nutation and polar motion can probably be observed.
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M. Poutanen • P. Knudsen • M. Lilje • T. N~rbech. H.-P. Plag. H.-G. Scherneck 2.3 Gravity
a n d its c h a n g e s
National gravity networks have been maintained with relative gravimetric measurements. There have been only a limited number of absolute gravity points. In recent years, possibilities to make absolute measurements have been improved. We are moving toward the situation where the primary reference network is maintained by absolute measurements and the sites are common with those of other techniques, e.g. permanent GPS stations. Measuring the secular change of gravity in a gravimetric network of permanent stations over long periods of time affords a unique method to monitor large-scale mass movements. Of particular interest in our area is the phenomenon of postglacial rebound. Combining gravity change with surface displacement promises to help discern ice load related effects from earth structure. In order to resolve models of mass redistribution in glacial isostatic rebound models, gravity change needs to be determined at 1 to 2 nm/s2/yr reproducability. Measurements with modern instruments like superconducting gravimeters and ancillary monitoring of environmental parameters like ground water, rain, snow, lake and sea levels constitute an ongoing research effort, partly in its own right and partly in direct support of the absolute gravity network. Consistency is required at the 0.1 nm/s 2 level, which still is a challenge at seasonal to inter-annual periods. Superconducting gravimeters are very sensitive instruments to monitor gravity changes. However, they have no absolute scale of their own, and due to (unknown) drift, need to control with regular absolute measurements at the same site. In the Nordic area, one needs to ensure that the existing superconducting gravimeter (currently only one) has an adequate good tie to the frame measured by absolute gravimeters. Satellite gravity missions such as the ongoing Gravity Recovery and Climate Experiment (GRACE) and the upcoming Gravity Field and Steady-State Ocean Circulation Mission GOCE, maps the Earth gravity field with high accuracy almost globally. For calibration and validation of the data a-priori information about the gravity field is needed at least in some regions. In those regions, information about the spatial characteristics of the gravity field can be compiled using terrestrial measurements combined with airborne and shipborne surveys. Absolute gravity is essential as fundamental information to define the level as well as to calibrate and validate the gravity relative gravity data.
2.4 Geoid
The geoid is an equipotential surface associated with the Earth gravity field. The geoid as an equilibrium surface defines the reference for physical processes such as mass movements. With the increased use of the Global Navigation Satellite System (GNSS) in height determination, accurate information about the geoid has become essential. Satellite positioning is carried out with the respect to a global reference frame and the height is referring to the ellipsoid. To convert ellipsoidal heights to orthometric heights a geoid model is needed. In ocean science the differences between the ocean surface and the geoid (called the dynamic topography) is important for studies of the dynamics of the ocean, its currents and heat transport. An accurate geoid is required to study the changes in the currents and the heat transport and their role in climate. On the surface of the solid earth the movements of water and ice naturally depend on the gravity. In the Earth's interior movements of mass are caused by the variations in the gravity potential field combined with the visco-elastic properties of the earth materials. The postglacial rebound in the Nordic region, where the crust in the central part of the region moves up by 1 cm per year, is such a phenomenon. As surveying on land by levelling etc. is carried out with respect to the geoid, positioning and navigation by gyros in Inertial Navigation Systems (INS) refer to the geoid, or more correctly to the gravimetric equipotential surface, the geoid defines. Use of data of the gravity satellites CHAMP, GRACE and GOCE will eventually produce a global geoid model, the accuracy of which is better than couple of centimetres. This enables creation of vertical datums anywhere which are globally consistent within the geoid model accuracy. 2.5 Combination
of o b s e r v a t i o n s
Space-based segments can remotely observe several key quantities (global ocean surface, ice surface, gravity field at long- and medium-range spherical harmonic degree), while ground-based observations are sensitive to the situation at a very local scale. In the case of a few global quantities, local observations will hardly become obsolete. GNSS is well suited for measuring the positions of single points in a global reference system. To obtain high accuracies (better than one centimetre) observations need to be acquired continuously on a
Chapter 107 • The Nordic Geodetic Observing System (NGOS)
permanent basis. Other space techniques, especially VLBI and SLR, give positions with a higher accuracy. Classical techniques such as trilateration and levelling can be used for measuring precise relative positions, which after repeated measurements can give local displacements and deformations. Precise levelling is used to determine accurate changes in height to recover local land uplift and subsidence. Land uplift and subsidence relative to mean sea level may be measured using tide gauges. The three techniques, however, measure vertical displacements in three different systems, i.e. relative to the ellipsoid, relative to the geoid, and relative to the mean sea level.
A well-established, accurate and stable reference is essential for the determination of the changes described above. The realisation and maintenance of the terrestrial reference frame is carried out by means of a global or regional cluster of fiducial points with precise positions relative to some external reference associated with them. The reference frame for gravity is established and maintained using a global or regional cluster of absolute gravity measurements combined with relative measurements.
The Earth's rotation is studied using coordinate time series from space techniques as VLBI, SLR and GNSS. Accurate gravity variations from superconducting gravimeters are also used. The determination of earth rotation parameters needs a global coverage of stations.
NGOS aims to provide geodetic observations for the Nordic area that are of sufficient quantity and quality to serve most of the needs of global Earth observation as well as practical and scientific applications in the region. Taking into account the specific phenomena of the region, NGOS will have a particular focus on geodynamics. Particular focus will be the long-term stability in the observing system, homogeneity in time, and a sufficient capability to perform its tasks also in the future. The geographic extent of NGOS is currently defined as the Nordic countries, including Greenland and Svalbard (Fig. 2). It is recommended that the geographical region is extended to include also the area of Baltic States. This covers the area of the ice-covered part of the Northern Europe during the last ice age, and therefore the common geophysical interest.
The determination of the gravity field including the geoid is based on measurements of the accelerations obtained by absolute and spring gravimeters. Spring gravimeters are used for the spatial densification of the gravity network. To study the details of the dynamics of the earth the gravity needs to be measured by superconducting gravimeters. Glacial isostatic adjustment (GIA) is an illustrative example for a geodynamic problem where coordinated multi-component geodetic observations are essential. The dynamics of the GIA problem is constituted by a planet that has a specific rheology, guiding the response to surface loading. Today we are beginning to retrieve fully 3-D surface motion at 0.1 (0.05) mm/yr resolution, sea level change below 0.2 mm/yr, while gravity changes are monitored at a gGal/yr level with absolute or relative gravimeters. For mitigating this shortcoming the Nordic absolute gravimetry plan has been launched and several institutes have already increased their level of activity. The major obstacle for data combinations across all time scales is that the components of the systems have different sampling schemes. For a foreseeable future it is unrealistic to expect permanent absolute gravimeter installations on much tighter than annual campaign schemes. To overcome the drawback the preferred method would be to start and install recording gravimeters (remotely controlled, emphasis on low instrumental drift) at some of the absolute gravity sites.
3 Components of the Nordic Geodetic Observing System
Figure 2. The geographical area covered by the NGOS and proposed NGOS stations.
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M. Poutanen • P. Knudsen • M. Lilje • T. Norbech. H.-P. Plag. H.-G. Scherneck
Table
1. Summary of techniques considered in NGOS.
Technique
Objective Point positioning relative to VLBI space Point positioning relative to SLR many satellites Point positioning relative to a GNSS satellite system Point positioning relative to DORIS satellites Height differences of points Levelling relative to the geoid Height of points relative to sea Tide gauges level Absolute gravimetric Absolute gravimeters accelerations Superconducting Relative gravimetric gravimeters accelerations Relative gravimetric Spring gravimeters accelerations
Accuracy 0.001 ppb 0.1 mas < 1 cm (range) 1-2 cm E: 1-2 cm *) C: 1-2mm 1-5 cm < 1 mm/km1/2 E: 10 cm C: 1 cm 2-3 btGal 0.1 btGal (< 1 nGal periods) 2-3 gGal
Component(s) Surface displacement; Earth rotation; Reference frame Surface displacement; Earth rotation; Reference frame Surface displacement; Reference frame Surface displacement; Reference frame Surface displacement; Reference frame Surface displacement; Reference frame Surface displacement; Earth rotation; Gravity; Reference frame Surface displacement; Earth rotation; Gravity; Reference frame Gravity; Reference frame
"~E means episodical and C continuous measurements'
Only the ground components of the geodetic observation techniques and infrastructures are considered in NGOS. Densifications, e.g. in special target areas such as glaciers, tectonically active structures, or near tide gauges, can be accomplished using remote sensing techniques, such as space and airborne radar and laser systems. Sea level can be monitored using satellite altimetry. The large scales of the gravity field and its changes are studied using observations of the motions of satellites by SLR and GPS or by dedicated gravity satellite missions. NGOS is planned to be a regional implementation of GGOS. Hereby, NGOS is related to the international geodetic cooperation work in the lAG and its associated services. Furthermore, the activities should be coordinated with other international services outside IAG such as PSMSL, and with some European services such as ESEAS. Concerning spatial variations of gravity and its temporal changes, we propose concentration on the use of modern, absolute gravimeters as expressed in NGOS/Absolute Gravimetry plan (Scherneck e t al., 2002) and "Draft plan for absolute gravity campaigns in the Fennoscandian land uplift area" by M~ikinen (2003). The absolute gravity measurements in the Nordic area is made as a cooperation of Danish National Space Center, The Finnish Geodetic Institute, Norwegian Mapping Authority, National Land Survey of Sweden, Onsala Space Observatory (Sweden), Norwegian University of Life Sciences in Aas, University of Hannover (Germany), and Federal Agency for Cartography and Geodesy (BKG, Germany). We also emphasize the importance of geoid determination. By nature, it is not limited to the
NGOS stations or NGOS plan. In the future, the new gravity satellite missions, especially GRACE and GOCE will give their contribution also to the Nordic geoid models.
4 Current situation in the Nordic area NKG has tried to act as a platform for sharing the knowledge concerning construction of various geodetic networks and in co-operation of geodetic campaigns. However, at the end it has always been the responsibility of an individual country to implement the work in practice. The Nordic countries have historically been building up their geodetic networks quite independently. There has been only a limited amount of co-operation between the countries or techniques. In each country there are networks of permanent GNSS stations that partly are operated by the national geodetic authorities. The co-ordination concerning e.g. location, construction, facilities and products was not optimal when the stations were built. This means e.g. that the stations have different monumentations, have different types of equipment, produce slightly different products and possibly are not optimally spread over the Nordic Area. However, the basic observables are the same at all stations, thus allowing e.g. the common Nordic computation of the EPN block, or the collaboration in the projects like B IFROST. GNSS data of some Nordic stations are used in IGS, but a more wide selection of Nordic GNSS stations belong to the EPN, the European Permanent GPS Network, coordinated by EUREF.
Chapter 107
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Figure 3. NGOS plan. Absolute gravity points (triangles), Nordic permanent GPS network (upside down triangles) and tide gauges (circles). All absolute gravity points are occupied with a GNSS instrument. The Nordic countries have adopted national ETRS89-realisations as their national reference systems. NKG is responsible for one of the EPN analysis centres and the responsible organisation is currently the National Land Survey of Sweden. NKG has urged the countries for Nordic cooperation concerning the Nordic height systems. The joint Nordic adjustment of the levelling networks is headed towards the common vertical datum in the area. NGOS is in line with these initiatives towards Nordic co-operation concerning the national geodetic networks and geodetic stations. The Finnish Geodetic Institute has been active in absolute gravimetry and has been performing measurements on stations over the Nordic Area for many years. Groups from USA and Germany have also made absolute gravity measurements in the Nordic Area. The current NGOS AG plan clearly demonstrates the current co-operation in this field. Since 2003, also the Norwegian University of Life Sciences in Aas has an absolute gravimeter which is a substantial increase in the resources. Most of the measurements in Finland and Sweden have been
• The Nordic Geodetic
ObservingSystem(NGOS)
performed at stations with other geodetic techniques, such as permanent GNSS stations and/or tide gauges. There is a superconducting gravimeter at Metsfihovi and Ny-Alesund. These two stations and additionally Onsala, are also equipped with a geodetic VLBI. Mets~ihovi has additionally collocated other two space geodetic equipment, namely the SLR and DORIS. Since 1978 Metsfihovi has participated the SLR programmes, and is currently the Northernmost SLR station. Stations in geodetic VLBI and SLR contribute to the work of IVS and ILRS which are the IAG services. Products of the NGOS, data access and archives are to be defined. Geodetic data are spread between the various countries and organisations and are also in various formats. Very little has been done in standardisation of data archives as well as assure the access to data. Different policies in the countries concerning accessibility to data due to pricing policies have complicated this issue. The characteristics of the NGOS, and NKG behind it, do not imply a common data archive covering all data and products of the NGOS. The role of NGOS data policy should be co-operative, utilising current infrastructure also in data policy and archiving. There already exist working examples of products and data archiving, including the Nordic permanent GPS network, levelling data bank and gravity data. At least in the first step, a NGOS database is needed to list the existing data, how to access to it, and possibly some auxiliary information concerning the use of data and products. It is responsibility of individual institutions to maintain the databases, possibly making a common agreement on data exchange and use, and also the data delivering policy is a national decision.
5 Conclusion We have described the general principles of the NGOS. The final selection of the sites depends on the resources available. We will begin with the sites proposed for the ECGN, added with the sites for absolute gravimetry. One should emphasize the multi-technique approach, and put the priority according to that. During the discussions to be done in the near future we are able to fix the core set of the NGOS stations. The work of the Task Force will continue, because the Presidium of the NKG gave a new task to find out the existing infrastructures and data archives, prepare a status report and recommendations for the practical implementation of the NGOS. We also propose that the work on the
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M. Poutanen • P. Knudsen • M. Lilje • T. N~rbech. H.-P. Plag. H.-G. Scherneck
N G O S on the practical level will be continued in the N K G G e o d y n a m i c s W o r k i n g Group.
Acknowledgements The N G O S report is a c o m b i n e d effort of a large n u m b e r of people. In addition to the authors of this paper, m a n y other p e o p l e have contributed the work, including R. Forsberg, B.R. Pettersen, D. Solheim, J. M~ikinen, M. V e r m e e r and several other Nordic colleagues not m e n t i o n e d here. W e like to express our sincere thanks to those who h a v e h e l p e d us to p r e p a re the N G O S d o c u m e n t .
References Beutler, G., Drewes, H., and Rummel, R., 2000a. Reflections on a new structure for IAG beyond 2000-conclusions from the IAG Section II Symposium in Munich, in Schwarz, K.P. (ed.): Geodesy Beyond 2 0 0 0 - The Challenges of the First Decade, pp. 430-437, IAGSymposia 121, Springer. Beutler, G., Drewes, H., Reigber, Ch., and Rummel, R., 2000b. Space techniques and their coordination within IAG and in future, in R. Rummel, H. Drewes, W. Bosch, H. Hornik (eds.)" Towards an Integrated Global Geodetic Observing System (IGGOS), pp. 22-32, IAG-Symposia 120, Springer. Beutler G., Drewes H., Reigber C., Rummel, R. (2003): Proposal to establish the Integrated Global Geodetic Observing System (IGGOS) as IAG's First Project. http ://www.~fv.ku.dk/-i a~/ig~os_prop_iune_03.htm. GEO, 2005a. The Global Earth Observing System of Systems (GEOSS) - 10-Year Implementation Plan. ESA Publication Division, ESTEC, Noordwijk, The Netherlands. Available at http://earthobservations.org/. GEO, 2005b. Global Earth Observing System of Systems GEOSS 10-Year Implementation Plan Reference Document. ESA Publication Division, Noordwijk, The Netherlands. No. GEO 1000R/ESA SP 1284, February 2005. Available at http://earthobservations.org/. Johansson, J. M., Davis, J. L., Scherneck, H.-G., Milne, G. A., Vermeer, M., Mitrovica, J. X., Bennett, R. A., Jonsson, B., Elgered, G., E10segui, P., Koivula, H., Poutanen, M., ROnn~ing, B. O. and Shapiro, I. I., 2002: Continuous GPS measurements of postglacial adjustment in Fennoscandia 1. Geodetic results, J. Geophys. Res., 1t)7, DOI 10.1029/2001JB000400. M~ikinen, J., 2003. Draft plan for absolute gravity campaigns in the Fennoscandian land uplift area. Meeting of the NKG Working Group of Geodynamics, Copenhagen, April 29, 2003, 11 p. Available at http://www.oso.chalmers.se/-hgs/NKGWG/Docs/AbsGravCamps03_draft_J M.pdf Milne, G. A., Davis, J. L., Mitrovica, J. X., Scherneck, H.-G., Johansson, J. M., Vermeer, M., Koivula, H., 2001" Spacegeodetic constraints on glacial isostatic adjustment in Fennoscandia. Science, 291, 2381-2385. -
Plag, H.-P., 2000. Integration of geodetic techniques into a global Earth monitoring systems and its implication for Earth system sciences, in R. Rummel, H. Drewes, W. Bosch, H. Hornik (eds.): Towards' an Integrated Global Geodetic Observing System (IGGOS), pp. 84-90, IAG Symposia 120, Springer. Plag, H.-P., 2002.: European Sea Level Service (ESEAS): Status and Plans, in Poutanen, M. and Suurmfiki, H. (eds.): Proceedings of the 14 ~h General Meeting of the Nordic Geodetic Commission, Espoo, Finland, 1-5 October 2002, pp 80-88, Finnish Geodetic Institute. Plag, H.-P., 2004: The IGGOS as the backbone for global observing and local monitoring: a user driven perspective. In Rummel, R., Drewes, H., Bosch, W. and Hornik, H. (eds.): International Association of Geodesy Symposia, Springer, Berlin, in press. Poutanen M., J. Jokela, M. Ollikainen, H. Koivula, M. Bilker, H. Virtanen, 2005a: Scale variation of GPS time series. In F. Sans6 (Ed.) A Window on the Future of Geodesy. IAG General Assembly in Sapporo, Japan 2003. pp. 15-12. lAG Symposia 128, Springer Verlag. Poutanen M., P. Knudsen, M. Lilje, T. N0rbech, H.- P. Plag, H.-G. Scherneck, 2005b. NGOS. Report of the Nordic Geodetic Commission Task Force. http://www.nkg.fi /nggos.html. 30 pages. Poutanen M., P. Knudsen, M. Lilje, T. N0rbech, H.- P. Plag, H.-G. Scherneck, 2005c. N G O S - The Nordic Geodetic Observing System. Accepted for the Nordic Journal of Surveying and Real Estate Research. 20 pages. Rummel, R., Drewes, H., and Beutler, G., 2001. Integrated Global Geodetic Observing System (IGGOS): A candidate IAG project. In Vistas for Geodesy in the New Millennium: lAG 2001 Scientific Assembly, Budapest, Hungary, September 2-7, 2001 (Ed. J. Adam, K.-P. Schwarz). pp. 609-614. IAG Symposia 125. Springer. Scherneck, H.-G., Johansson, J. M., Mitrovica, J. M., Davis, J. L., 1998: The BIFROST project: GPS determined 3-D displacements in F ennoscandia from 800 days of continous observations in the SWEPOS network. Tectonophysics, 294, 305-321. Scherneck, H.-G., Vermeer, M., Forsberg, R., Schmidt, K.E., M~ikinen, J., Ollikainen, M., Poutanen, M., Ruotsalainen, H., Virtanen, H., V61ksen, Ch., Plag, H.-P., Lidberg, M., and Olsson, A., 2003. The Nordic Geodetic and Geodynamic Observing System (NGGOS): An NKG plan for the contribution from an absolute gravimetry network (NGGOS/AG), http://www.oso.chalmers.se/simhgs /NKGWG/Docs/Abs Grav Plan.pdf van Dam, T. M. and Herring, T. A., 1994: Detection of atmospheric pressure loading using very long baseline interferometry measurements. J. Geophys. Res., 99, 45054517. van Dam, T. M., Blewitt, G. and Heflin, M. B., 1994: Atmospheric pressure loading effects on Global Positioning System coordinate determinations. J. Geophys. Res., 99, 23939-23950. van Dam, T. M., Wahr, J., Milly, P. C. D., Shmakin, A. B., Blewitt, G., Lavalee, D. and Larson, K. M., 2001: Crustal displacements due to continental water loading, Geophys. Res. Let., 28, 651-654.
Chapter 108
VLBI2010: A Vision for Future Geodetic VLBI A. Niell, A. Whitney MIT Haystack Observatory, Off Route 40, Westford, MA 01886-1299, USA
W. Petrachenko Geodetic Survey Division, Natural Resources Canada Dominion Radio Astrophysical Observatory (DRAO), Box 248, Penticton, B.C., V2A 6K3, Canada W. Schlfiter Bundesamt fiir Kartographie und Geodfisie, Fundamentalstation Wettzell, Sackenrieder Strasse 25, D-93444 K6tzting, Germany N. Vandenberg NVI, Inc./NASA Goddard Space Flight Center, Code 697, Greenbelt, MD 20771-0001, USA H. Hase Bundesamt fiir Kartographie und Geodfisie, Observatorio Geod6sico TIGO, Casilla 4036, Correo 3, Concepcidn, Chile Y. Koyama Kashima Space Research Center, NICT, 893-1 Hirai, Kashima, Ibaraki 314-8501, Japan C. Ma NASA Goddard Space Flight Center, Code 697, Greenbelt, MD 20771-0001, USA
H. Schuh Institute of Geodesy and Geophysics, Vienna University of Technology, Gusshausstrasse 27-29/128-1, A- 1040 Vienna, Austria G. Tuccari Istituto di Radioastronomia/INAF, Contrada Renna, P.O. Box 141, Noto (SR), 96017, Italy Abstract. This article summarizes the results of IVS Worldng Group 3 'VLBI2010', which was charged with creating a vision for a new geodetic VLBI instrument that will meet requirements for the coming decades. This comes at a time when problems with aging antennas, a deteriorating radio frequency environment due to interference, obsolete electronics, and high operating costs are making it difficult to achieve the required level of performance. Fortunately, recent advances in antenna manufacture, digital electronics, and data transmission technology are enabling the development of systems and modes of operation unimaginable only a few years ago, along with much reduced costs. A set of criteria to be met by a future geodetic VLBI system was established based on recommendations in reports compiled by IVS, GGOS, and NASA. These criteria are: 1 mm measurement accuracy on
global baselines, continuous measurements for time series of station positions and Earth orientation parameters, and turnaround time to initial geodetic results of less than 24 hrs. Additionally, it is also vital to continue the measurements for the determination of UT1, nutation, and the celestial reference frame. A number of performance-enhancement strategies were studied, including reducing random and systematic components of the delay observable, increasing the number of antenna sites and improving their global distribution, reducing susceptibility to external radio frequency interference (RFI), increasing the number of observations per unit time, and developing new observing strategies. Recommendations for a new observing system include small (10-12m) antennas covering a broad frequency range (-1-14GHz), high
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A. Niell • A. Whitney. W. Petrachenko • W. SchKiter • N. Vandenberg • H. Hase. Y. Koyama • C. Ma. H. Schuh • G. Tuccari
data rates (up to 8 or 16 Gbps/station), new correlator systems, and automated data analysis. This goes along with further improvements of the VLBI data analysis software and analysis strategies. Specific steps are recommended in order to develop, deploy, and bring the new system into operation, including system studies and simulations in parallel with development and prototyping of new antenna and data-collection systems.
Keywords. Very Long Baseline Interferometry, IVS, lAG, GGOS 1
• 1 mm measurement accuracy on global baselines, • continuous measurements for time series of station positions and Earth orientation parameters, • turnaround time to initial geodetic results of less than 24 hrs. While the new requirements are significant challenges, it is vital to continue the measurements for which VLBI is the unique space geodetic technique:
Motivation
Geodetic VLBI stands at the brink of a new era. Such societally relevant issues as climate change and natural hazards are placing ever increasing demands on performance. This comes at a time when problems with aging antennas, a deteriorating RFI environment, obsolete electronics, and high operating costs are making current levels of accuracy, reliability, and timeliness difficult to sustain. Attaining modern requirements for significantly greater accuracy, continuous data flow, and shortened times to product delivery challenge the continuing progress made by geodetic VLBI over the past 30 years. Fortunately, recent advances in antenna manufacture, digital electronics, and data transmission technology are enabling modes of operation unimaginable only a few years ago. Furthermore, the capital investment and reduced operating costs associated with the new technology make complete renewal of present infrastructure appear cost effective. A new instrument that will meet requirements for decades to come can now be envisioned. 2
Group Report (SESWG; http://solidearth.jpl.nasa.gov/ seswg.html). These criteria are:
Future
requirements
for VLBI
IVS Working Group 3 (WG3; Niell et al., 2005) was asked to examine current and future requirements for geodetic VLBI, including all components from antennas to analysis, and to create recommendations for a new generation of VLBI systems. To constrain these recommendations, a new set of criteria by which to measure the next generation geodetic VLBI system was established based on the recommendations for future IVS products detailed in the IVS Working Group 2 Report (Schuh et al., 2001), on the requirements of the Global Geodetic Observing System (GGOS; http://www.ggos.org) project of the International Association of Geodesy (IAG), and on the science driven geodetic goals outlined in the NASA Solid Earth Science Working
• UT1 and nutation, • the celestial reference frame (CRF). UT1 and the CRF are currently defined by VLBI, and there is no alternative for the foreseeable future. It is recognized that achieving long term accuracy at the level of 1 mm or better is a daunting task. Of equal importance VLBI, together with satellite laser ranging (SLR), must continue to provide the scale of the Terrestrial Reference Frame. 3
Strategies
achieve
the
and
recommendations
to
goals
From the outset WG3 sought approaches for the design of the new system that would enable the following performance enhancing strategies: • Reduce the random component of the delayobservable error, i.e., the per-observation measurement error, the stochastic properties of the clocks, and the unmodeled variation in the atmosphere, • Reduce systematic errors, such as thermal effects on electronics or structure, loading errors, and radio source structure effects, • Increase the number of antennas and improve their geographic distribution, with strong emphasis on collocation with the other techniques, • Reduce susceptibility to external radio-frequency interference, • Increase observation density, i.e. the number of observations per unit time, • Develop new observing strategies. All of the above considerations, along with the need for low cost of construction and operation, required a complete examination of all aspects of geodetic VLBI, including equipment, processes, and observational strategies. The results of this examination have led WG3 to make the following recommendations:
Chapter 108 • VLBI2010:a Vision for Future GeodeticVLBI
• Design a new observing system based on small antennas. The new system will be automated and operate unattended and will be based on small (10-12 m diameter), fast-moving, mechanically reliable antennas that can be replicated economically. The observing should be done over a broad, continuous frequency range, perhaps 1-14 GHz, which includes both the current S-band and X-band frequencies for backwards compatibility, but allows much more agility to avoid RFI and more bandwidth to significantly improve delay measurement precision. At the same time, the best of the existing large antennas will be updated for compatibility with the new small-antenna system; this will allow them to co-observe with the small-antenna systems to preserve continuity with the historical record, as well as to improve measurements contributing to the CRF, which has been defined primarily by observations with the large antennas. • Transfer data with a combination of high-speed networks and high data-rate disk systems. Data recording rates and transmission rates are rapidly increasing courtesy of vast investments by the computer and communications industries. Examine the possibilities for new correlator systems to handle the anticipated higher data rates, including correlation based on commodity PC platforms, possibly widely distributed. Automate and streamline the complete dataanalysis pipeline, enabling rapid turnaround and consistent TRF, CRF, and EOP solutions. Because the new systems should be fully backwards compatible with the existing systems, the transition from the old to the new systems can be gradual and deliberate, maintaining important continuity of geodetic results and measurements series while dramatically upgrading the quality, precision, and timeliness of new observations. Furthermore, with more of the new VLBI systems co-located with the suite of complementary space-geodetic techniques, the space-geodetic program as a whole will be greatly strengthened.
4
Next
Steps
The above recommendations describe a system that can begin to become reality very soon. The IVS WG3 report (Niell et al., 2005) identifies specific steps that need to be taken next in order to develop, deploy, and bring the system into operation. The next steps include two broad categories of efforts:
• System studies and simulations: error budget development, decisions on observing frequencies, optimal distribution of new sites, including those to be co-located with the other techniques, number of antennas per site, new observing strategies, and a transition plan. • Development projects and prototyping: small antenna system, feed and receiver, cost and schedule, higher data rate system, correlator development, backend development, and data management and analysis software. Almost all of the recommended next steps can be done in parallel, and WG3 hopes that various IVS components will find the resources to support one or several of these studies and development projects. Results of these studies and projects should be well communicated within the community and coordinated by IVS so that common goals for the new vision are recognized and met. It is important for IVS to make a strong recommendation that some of the resources dedicated today to routine product generation and technology development be directed to address the studies and projects recommended in this report. These studies must move forward so that a detailed plan can be generated, including defensible costs and schedules. Building on the efforts of WG3, the results of these studies and projects will provide the final element required for IVS members to move forward with requests for augmented funding to implement the new vision. We believe that this vision will renew the interest of current funding resources and inspire new interest from universities, industry, and government, based on the exciting possibilities for a more accurate and data-rich geodetic VLBI system.
Acknowledgement. We thank Dirk Behrend for assembling and formatting this contribution and the associated poster.
References
Niell, A., A. Whitney, W. Petrachenko, W. Schlfiter, N. Vandenberg, H. Hase, Y. Koyama, C. Ma, H. Schuh, G. Tuccari (2005) VLBI2010: Current and Future Requirements for Geodetic VLBI Systems, IVS WG3 Report, http://ivscc.gsfc.nasa.gov/about/wg/wg3. Schuh, H, P. Charlot, C. Klatt, H. Hase, E. Himwich, K. Kingham, C. Ma, Z. Malkin, A. Niell, A. Nothnagel, W. Schliiter, K. Takashima, N. Vandenberg (2002) IVS Working Group 2 for Product Specification and Observing Programs, in International VLBI Service for Geodesy and Astrometry 2001 Annual Report, edited by N. Vandenberg and K. Baver, NASA/TP-2002-210001, pp. 13-45. (http://ivscc.gsfc.nasa.gov/about/wg/wg2).
759
Chapter 109
Combination of different geodetic techniques for signal d e t e c t i o n - a case study at Fundamental Station Wettzell on the occasion of the SumatraAndaman earthquake (Dec 26, 2004) J. Ihde, W. S6hne, W. Schwahn, H. Wilmes, H. Wziontek Federal Agency for Cartography and Geodesy, Richard-Strauss-Allee 11, D-60598 Frankfurt am Main, Germany T. Kliigel, W. Schltiter Federal Agency for Cartography and Geodesy, Sackenrieder StraBe 25, D-93444 K6tzting, Germany
Abstract. At the Fundamental Station Wettzell a
wide range of geodetic measurement systems has been installed: an array of six different GNSS receivers, superconducting gravimeter, ring laser, 20 m radio telescope for VLBI and a Satellite Laser Ranging System. The precise ring laser "G", operated in an underground laboratory, is able to directly measure polar motion. Signals down to 10.8 of the Earth's rotation rate can be detected continuously. Due to the high relative resolution of 10 -11 and the small drift of the instrument the superconducting gravimeter is able to detect e.g. the seismic induced eigenmodes of the Earth. Additionally, seismometers and tilt meters are also operated at the station. The Sumatra-Andaman earthquake on December 26, 2004, 0:58 UTC was one of the largest earthquakes ever and the first one of this magnitude (> 9.0) since continuous earthquake monitoring. It will be described how the different sensors at Fundamental Station Wettzell record the signals induced by the seismic waves. While deformation due to an earthquake in terms of elastic rebound could only be detected in local or regional distances to the centre of the earthquake by means of GPS, the horizontal deformation due to seismic surface waves was large enough to be analysed with highrate GPS data even 9000 km apart from the centre. The rotational component of the seismic waves is clearly recorded by the ring laser "G" which is the only type of instrument being able to measure this quantity. The results of the different sensors will be compared, the benefit of each sensor referred to seismic waves will be summarized. Conclusions for further investigations with respect to the Global
Geodetic Observing System (GGOS) will be discussed. Keywords. Seismic waves, highrate GPS, superconducting gravimeter, ring laser, Sumatra earthquake
1 Introduction The Fundamental Station Wettzell (Fig. 1) of Bundesamt ftir Kartographie und Geodfisie (BKG) is jointly operated with Forschungseinrichtung Satellitengeodfisie (FESG) of Technical University Munich. It is located in the Bavarian Forest. The Fundamental Station Wettzell is one of a few stations worldwide possessing main geodetic observing systems which are complemented by a number of other sensors and instrumentation: • Very Long Baseline Interferometry (VLBI): a 2 0 m radio telescope specially designed for VLBI • Laser Ranging: Wettzell Laser Ranging System (WLRS) designed for Satellite Laser Ranging (SLR) as well as Lunar Laser Ranging (LLR) • Global Navigation Satellite System (GNSS): Different GPS and GPS+GLONASS receiver/antenna pairs mounted on a concrete survey tower, about 7.5 m over ground, observing permanently. The stations are included in the Network of the International GNSS Service (IGS), the EUREF Permanent Network (EPN) and the German Reference Network (GREF) • Superconducting gravimeter: a GWR SG-29 with a relative resolution of 10-11
Chapter 109 • Combination of Different Geodetic Techniques for Signal Detection - a Case Study at Fundamental Station Wettzell
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• Ring Laser: a 4 m x 4 m ring laser for monitoring Earth rotation, installed in a special subterranean laboratory • Seismometer: standard b r o a d b a n d s e is m o m e te r STS-2 installed in a seismic vault; additionally Lennartz seismometers installed in the ring laser laboratory • Tiltmeter: 6 tilt meters installed in the ring laser laboratory • Time and frequency: three cesium frequency standards, three h y d r o g e n masers and two GPS time receivers
• Meteorological sensors: sensors for air temperature, pressure, humidity, precipitation, wi n d speed and direction, g r o u n d moisture; g r o u n d water level; water v a p o u r radiometer. The location o f the main c o m p o n e n t s is s h o w n in Figure 2.
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Fig. 3 Principle scheme of surface wave propagation caused by the Sumatra-Andaman earthquake to Germany The white circles indicate the wave front, the ray path is shown in black.
761
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J. Ihde. W. S6hne. W. Schwahn • H. Wilmes • H. Wziontek • T. KKigel. W. SchKiter
2 Seismic waves The ground deformation following the SumatraAndaman earthquake of Dec 26, 2004, 0:58:50 UTC (second-of-day 3530) was large enough to be detected at far distances to the epicentre. The main types of seismic waves are P, S, Love and Rayleigh waves. Since P and S waves are body waves they will arrive first at far distance locations. Surface waves like the Love waves caused by the SumatraAndaman earthquake are expected to arrive in Germany from the East direction (Fig. 3) with a travel time of approx. 2000 seconds. Therefore the Love wave will reach Germany approx. 5530 seconds after 00:00:00 UTC, Dec 26. The sensitivity of the various sensors to the main seismic waves induced by the Sumatra-Andaman earthquake and detectable at Fundamental Station Wettzell is expected to be different. Due to their spatial design the seismometers should be able to detect all four main seismic waves. GPS with its higher noise level mutually
will only detect the horizontal displacements caused by the surface waves while the ring laser only measures rotations around its sensitive axis with high precision. This unique property led to the development of the Geosensor, a new instrument for seismology, Schreiber et al. (2003). 3 GPS With GREF and the network of the Satellite Positioning Service (SAPOS ®) operated by the Surveying Authorities of the German Lfinder a dense network of GNSS permanent stations is available in Germany. While for most applications a data sampling rate of 30 seconds is sufficient the data are internally stored with a sampling rate of 1 Hz. The derivation of positioning time series was carried out using the Bernese GNSS analysis software version 5.0. The software has the option to output
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4 Time series of the North-South component of GREF site WTZT (12.9°/49.1 °) with S A P O S ® site Kaiserslautern (7.8 ° / 49.4°) as reference station (baseline length 373 km) for three consecutive days, differences against the individual mean coordinate. The curves for days-of-year 359 and 360 were shifted by 472 and 236 seconds, resp., to take the sidereal day into account. Beside the good overall agreement the signatures as a result of the Love wave can clearly be recognized.
Fig.
Chapter 109 • Combination of Different Geodetic Techniques for Signal Detection - a Case Study at Fundamental Station Wettzell
kinematic coordinates at every observation epoch after fixing the ambiguities for the whole session. From the unfiltered kinematic positioning results for a West-East orientated baseline (Fig. 4) it can be seen that there are common signatures of the corresponding time series reflecting the common behaviour as a result of, e.g. satellite constellation or multi-path. The East-West component (not shown) in general has the smallest noise level (rms + 5-7 mm) whereas the rms values for the North-South component are in the range of + 7-10 mm. For the vertical component the noise level is higher with + 13-20 mm. A distinct signal can be detected in the North-South component for day-of-year 361 starting approximately at second 5650 followed by two extrema at seconds 5720 and 5780 with an amplitude of 20-30 mm. Results for other baselines in Germany can be found in Soehne et al. (2005).
4 Superconducting Gravimeter The superconducting gravimeter GWR SG-29 at Fundamental Station Wettzell records the gravitational and inertial acceleration in the local vertical direction with two identical sensors (sensor 1 and 2). Main use of the instrument is the observation and investigation of Earth gravity tides and other long-period and non-periodic signals (Richter et al. (2004)). For this purpose the instrument is equipped with an analogue low-pass filter (approx. 44 sec phase lag). The recorded gravity data (sampling rate 5 s) allow the interpretation of observations in the frequency band 0 to 8 mHz.
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The gravimeter includes an active verticality control with two cryogenic tilt meters separated at an angle of 120 deg and operated in a feedback loop using thermally controlled levellers. During ~ 85 minutes after arrival of the first seismic signals of the Sumatra-Andaman earthquake, the gravity record shows a systematic deviation which is obviously correlated with a perturbation of the tilt compensation system. Whereas an interpretation of the gravity records during the first two hours after the earthquake does not seem reasonable due to the signal deviation and the applied low-pass filtering, an analysis of the following data series clearly reveals spheroidal modes of the excited Earth free oscillations and their decay. The spheroidal modes can be detected very clearly and e.g. the 0S0 mode can be observed until June 2005. Figure 5 shows the amplitude spectrum in the frequency range up to 2 mHz for the time period immediately after the earthquake (Dec 28 - Dec 31, 2004). Figure 6 demonstrates the decay of the radial mode 0S0 until June 2005 which was excited again by the seismic event on March 28, 2005 (Northern Sumatra, magnitude 8.4).
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Ring lasers are rotation sensors. The 4 m x 4 m ring laser at Fundamental Station Wettzell (Fig. 7) is designed for measuring variations in Earth rotation with high precision in near real time. Rotational components of passing seismic waves are recorded as well. The horizontal installation makes the instrument sensitive for rotations around the vertical axis, e.g. Love waves. Regarding a plane shear wave, the rotation rate is in phase with acceleration
763
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J. Ihde. W. S6hne. W. Schwahn • H. Wilmes • H. Wziontek • T. KliJgel. W. SchliJter
having an amplitude of a/2c (a: acceleration, c: phase velocity), Igel et al. (2005). A number of tilt meters, measuring horizontal accelerations, are located on top of the ring laser body to monitor the orientation of the instrument in different directions. A Lennartz seismometer type LE3D/20s is co-located at the same place. Beside this a STS-2 broadband seismometer of the German Regional Seismic Network is operated in a seismic vault in a distance of 250 m. The time series are numerically differentiated to obtain accelerations and integrated to obtain displacements (Figs. 9 and
and 5780 s. This signal is present in records of other sensors, too (Fig. 4, Fig. 9). It is not visible in the East-West component (not presented here).
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Chapter 109 • Combination of Different Geodetic Techniques for Signal Detection - a Case Study at Fundamental Station Wettzell
6 Comparisons
6.2 Vertical accelerations
6.1 Horizontal displacements
The monitored accelerations from the superconducting gravimeter and the seismometer show almost no correlation (Fig. 10). The low pass filter implemented in the SG-29 damps the high frequency parts and causes a frequency dependent phase lag. Therefore, the upper curve for SG-29 is mainly a result of this filter. For the seismometer the values are calculated by numerical derivation of the original vertical velocity output. Although both signals represent vertical accelerations, discrepancies appear caused by the low sampling rate and the applied filtering of the superconducting gravimeter. It is recommended for future investigations to use a low pass filter with significantly smaller time constant and an increased sampling rate of 1 second.
Time series of horizontal displacements are available from GPS and from the seismometer. For the seismometer the values were derived by integration from the velocity output. Since the GPS solution is derived by differential positioning the seismometer results are also differentiated (Fig. 9). The station BFO is used for this, nearly in the same direction and distance to Wettzell (see map in Fig. 4). The second peak shows a small time offset. This is the result of the arrival of the surface waves at the reference stations which are not on the same longitude.
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Fundamental Stations play an important role with respect to a global observing system like GGOS since they provide the instrumentation and the observations for the determination of the fundamental geodetic observables. Beyond this, the co-location of geodetic and geophysical sensors at the Fundamental Station Wettzell gives the unique possibility to study the sensitivity of each sensor to small signals induced by the Sumatra-Andaman earthquake. All of the five different sensor types considered within this paper were able to detect those signals. With respect to the principle of construction of the sensor and/or the noise level of the results the sensors were differently sensitive to the individual signals: - GPS is able to directly give a quantification of the displacement. As a pure geometric technique GPS may help verifying or even calibrating displacements derived by seismometer data. For detecting small seismic signals with GPS improved solutions with additional processing, e.g. sidereal filtering and spatial stacking, are necessary, Bock et al. (2004). - The superconducting gravimeter primarily reveals spheroidal modes of the excited free oscillations of the Earth and their decay. - The ring laser measures rotations around its sensitive axis (the vertical in our case). This new quantity in seismology has never been recorded before with such a precision and gives access to the complete description of the deformation field.
765
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J. Ihde. W. S6hne. W. Schwahn • H. Wilmes • H. Wziontek • T. KKigel. W. SchKiter
- Seismometers measure inertial ground velocities in each direction. Low noise and high resolution allow the derivation of accelerations and displacements by differentiation and integration, resp. Tilt meters record tilts with respect to the plumb line as well as horizontal accelerations and are thus sensitive to horizontal and vertical components of seismic waves. -
References
Bock, Y., L. Prawirodirdjo, T.I. Melbourne (2004). Detection of arbitrarily large dynamic ground motions with a dense high-rate GPS network. Geophys. Res. Let., 31, L06604. Igel, H., U. Schreiber, A. Flaws, B. Schuberth, A. Velikosetsev, A. Cochard (2005). Rotational motions induced by the M8.1 Tokachi-Oki earth-
quake, September 25, 2003. Geophys. Res. Let., 32, L08309. Richter, B., S. Zerbini, F. Matoni, D. Simon (2004). Long-term crustal deformation monitored by gravity and space techniques at Medicina, Italy and Wettzell, Germany. Journal of Geodynamics, 38 (2004) p. 281-292. Schreiber, U., A. Velikosetsev, H. Igel, A. Cochard, A. Flaws, W. Drewitz, F. Miiller (2003). The GEOSENSOR: a new instrument for seismology. Geotechnologien science report, no. 3, 148-151. Soehne, W., W. Schwahn, J. Ihde (2005). Earth surface deformation in Germany following the Sumatra Dec 26, 2004 earthquake using 1 Hz GPS data. Report on the Symposium of the IAG Subcommission for Europe (EUREF), Vienna, 01-03 June 2005, Mitteilungen des Bundesamtes fiir Kartographie und Geodiisie, (in press)
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Part VII Systems and Methods for Airborne Mapping, Geophysicsand Hazardsand DisasterMonitoring Chapter 110
A Multi-Scale Monitoring Concept for Landslide Disaster Mitigation
Chapter 111
High-Speed Laser Scanning for near Real-Time Monitoring of Structural Deformations
Chapter 112
A Method for Modelling the Non-Stationary Behaviour of Structures in Deformation Analysis
Chapter 113
Volcano Deformation Monitoring in Indonesia: Status, Limitations and Prospects
Chapter 114
Vehicle Classification and Traffic Flow Estimation from Airborne Lidar/CCD Data
Chapter 115
Fine Analysis of Lever Arm Effects in Moving Gravimetry
Chapter 116
Improving LiDAR-Based Surface Reconstruction Using Ground Control
Chapter 117
The Use of GPS for Disaster Monitoring of Suspension Bridges
Chapter 110
A Multi-Scale Monitoring Concept for Landslide Disaster Mitigation H. Kahmen, A. Eichhorn, M. Haberler-Weber Institute of Geodesy and Geophysics, Vienna University of Technology, Gusshausstrasse 27-29/E1283, 1040 Vienna, Austria Abstract. There has already been much research on how the kinematic and geodynamic behaviour of landslides can be predicted. However, until now, there has been no decisive breakthrough for a monitoring and evaluation system combined with an alert system. Therefore an interdisciplinary international project OASYS (Integrated Optimization of Landslide Alert Systems) was commenced to progress research in this area. The members of the project, supported by the European Union, believe a multidisciplinary integration of different methods has potential for substantial progress in natural hazards management. This project proposes a new method consisting of three different steps:
1. Detection of boundary lines of potential landslides based on large scale information. 2. Detection of "taking-off-domains" and permanent local scale monitoring of these regions with high sensitivity geotechnical measurement methods. 3. Knowledge-based derivation of real time information regarding actual risks to support alert systems. Following the methodology of the OASYS project, a mathematical method for the detection of takingoff-domains and the concept of a knowledge-based alert system is described in this paper. Keywords. Landslides, Risk Assessment, Alert System, Fuzzy Techniques, Kalman Filtering Techniques. 1 Existing Technologies introduction
- an
There has already been a wide range of research work undertaken on landslides (see, e.g., AlmeidaTeixeira et al., 1991). Most of this work had a bias towards one discipline, such as remote sensing or geology.
Currently the investigation of landslides and unstable slopes is based on two groups of information sources: 1.1 Information sources on a regional scale
- Data on historical landslides: The relevant data should include information of when and where landslides occurred in the critical areas. - General conditions of the areas: Map data of the general conditions, including digital elevation models, geology, tectonics, geomorphology, vegetation, climate, and land use, may be indicators for active landslides (Cui, 1999). - Remote sensing data: Further sources for potential landslide risk can be differential satellite image analysis, time series of airborne sensor data (photogrammetry, laser scanners, radar systems, geophysical measuring devices) and terrestrial mapping. 1.2 Information sources on a local scale
- Geodetic: GPS, precise levelling, tacheometers (measurement robots), multisensor remote sensing techniques (using optical- and radarsensors). - Geophysical: geoelectric and electromagnetic field measurements. - Geodynamical: borehole tiltmeters, extensometers, hydrostatic tiltmeters. - Seismological: sensors for microseismic activity, seismic reflexion measurements. -Hydrological: sensors for groundwater level variations, water level variations, groundwater stream variations. - Meterological: sensors for temperature, air pressure, precipitation. The information of the first source enables detection of the site of the landslide, while those of the second source are used to describe the mechanism of the process(es). Normally, however, only some of the parameters (e.g. deformation vectors, water pore pressure) are used for the investigations and the measurement points are widely spaced across the
770
H. Kahmen. A. Eichhorn• M. Haberler-Weber
landslide domain. The deformation measurements are frequently based only on one measurement method (e.g. GPS, tacheometry). As these measurement methods are relatively costly, usually only a limited number of observation points are observed and the measurement systems are not operated continuously. This methodology has made it possible to follow the evolution of the landslides precisely but yet it is difficult to predict the evolution. By a better integration of these information sources a more reliable prediction should be possible in the future. 2
Integrated
results will be fields of vectors describing the displacements and velocities (Theilen-Willige, 1998). An advanced and generalised deformation analysis algorithm in addition based on geometrical as well as topographical, geological, hydrological and meteorological information has to be developed in order to improve the detection of taking-offdomains, see e.g. Haberler, 2003; Zhang et al., 2001.
Optimization
An advanced model is now under investigation based on large scale monitoring as a first step, regional monitoring as a second step, culminating in a multi-component knowledge-based alert system. 1 st step: Detection of potential landslides (large scale monitoring). To get information about the long-term geodynamical processes a large scale evaluation has to be performed. This includes the historical data, and any information describing the general conditions of the area, as outlined in section 1.1, and remote sensing data, such as aerial photographs, optical and radar images from satellites (Fig. 1).
Fig. 1 Large scale monitoring with airborne and satellite remote sensing.
More specific remote sensing techniques (e.g. InSAR), differential GPS (using phase observations) and tacheometric measurements shall be used to obtain additive information about the deformation process (Fig. 2). The measurements will be performed only three or four times a year, and the
Fig. 2 Geodetic landslide monitoring using GPS and total stations.
2 "0 step: High precision permanent measurements in the taking-off-domains. High precision borehole tiltmeters, extensometers, hydrostatic levelling and further relative measurement systems shall be used in the area of the taking-off-domains to obtain online information about the geodynamical process, see Savvaidis et al., 2001. This multi-sensor system will be running continuously and can therefore support a real time alert system (Fig. 3).
Fig. 3 Continuous local scle measurements and the information transfer.
3 ra step: Impact and risk assessment; development of strategies for knowledge-based alert systems. Risk assessment comprises the analysis of the empirical data and the development of an alert system. The analysis of empirical data includes the
Chapter 110
detection and interpretation of velocity fields in order to define zones of increased deformation. The final analysis will be supported by expert systems, using methods of cluster analysis, neural networks, fuzzy techniques and others (Haberler, 2003). The process of risk analysis can be divided into hazard analysis and vulnerability analysis. Hazard analysis is the review of the potential hazardous processes. Scenarios for the evolution of a landslide area of interest have to be set up, including an estimation of the probability of these scenarios. Different scenarios will affect different areas and therefore have a different impact on people and property. The assessment of the impact of different hazard processes on the affected population and its property is called vulnerability analysis. The integration of hazard and vulnerability analysis will lead to an estimation of the actual risk situation of the affected population. The risk management measures will depend heavily on the specific conditions and will include landuse planning, technical measures (e.g. build drainage systems), biological measures (e.g. afforestation) and temporary measures (e. g. evacuation). The integrated workflow for landslide hazard management is depicted in Fig. 4. Two research areas concerning the integrated work flow are described below. A
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• A Multi-Scale
Monitoring Concept for Landslide Disaster Mitigation
following step geotechnical sensors can be installed across these boundaries to obtain high precision monitoring data of the critical areas as an input for the knowledge-based system in the risk assessment. In the last few years modern techniques like fuzzy systems, neural networks and knowledge-based systems have started to be used also in geodesy (see, e.g., Heine, 1999; Wieser, 2002). One advantage of these methods is that they can reproduce the human way of thinking, so that problem solving is done in a rather intuitive way. Here, fuzzy techniques are used for finding a modern method for the automated detection of consistent block deformation. The classical deformation analysis results in a set of displacement vectors for the observed points in the landslide area. The task is now to find groups of points with a similar pattern of movement so that the boundaries between these blocks can be identified. Two different types of parameters are used here to do the block separation. Firstly, geodetic influence factors are determined to assess the state of deformation within this block. But for an automated block detection these indicators are not sufficient. Hence a second group of parameters is used. The human way of assessing a graph of displacement vectors is imitated by finding displacement vectors with similar direction and length, which are called "visual" influence factors here. Fuzzy systems, which are the suitable tools for imitating the human way of thinking, are used for the analysis of both groups of indicators mentioned above. A deeper insight into these parameters is given in the next sections.
classification of landslides
,,,
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I
assessment
I
o _1
3.1 Geodetic indicators
planning
V
of monitoring concepts for alert systems
measures
Fig. 4 Integrated workflow for integrated landslide hazard management.
An over-determined affine coordinate transformation is used to assess the movement of the observed points between two subsequent epochs of measurements. This means that the coordinates x - (x, y ) r of the points of epoch n are mapped
3 Detection of taking-off-domains based on a Fuzzy System
onto the coordinates points of epoch n+ 1"
x'-(x',y') v
of the same
x'-F.x+t One task within OASYS is that from classical geodetic monitoring measurements, the boundaries between the stable and the unstable, or between unstable areas, moving with different velocities in different directions have to be found, so that in the
(1)
where: x - (x, y)~ ... coordinates of epoch n
oordin.tes of epoch deformation)
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771
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H. Kahmen. A. Eichhorn • M. Haberler-Weber
t
-
F -
(t,, ty )r ...translational parameters c3x' c3x' -~x @,
c3y . matrix of deformation @,..
ax
ay
The indicators used as input parameters in the fuzzy system can be determined by the results of the sequence of affine transformations. A group of points moving in the same direction (assuming that they are lying on one common block) is characterized by, e.g., a small standard deviation of unit weight So within an over-determined affine coordinate transformation. On the other hand, if points of different blocks are considered simultaneously the indicating parameters are significantly larger. In the next step, the following strain parameters are derived from the affine transformation parameters: the infinitesimal strain components exx, eyy (rate of change of length per unit length in the direction of the x-axis and y-axis respectively), exy (= eyx, rate of shear strain) and the derived rotation angle co (see Haberler, 2005). Since these strain parameters are dependent on the coordinate system it is better to transform them into the principal strain axes system, represented by the strain ellipse (Tissot indicatrix). The elements of the strain ellipse (the semi-axes el, e2 and the orientation 0 of the maximum strain rate), which fully describe the state of deformation, are calculated from the strain parameters according to the geodetic point error ellipse (for further explanations see, e.g., Welsch et al., 2000). The strain ellipse parameters are the basic geodetic indicators for the block detection.
vectors with similar directions can belong to one common block. As a second indicator, two or more vectors are said to be similar if the lengths of the displacement vectors are similar. The combined analysis of the direction and the length of the vectors gives a clear distinction if points show the same pattern of movement. In addition, the property of "neighbourhood" is determined from a Delaunay triangulation. The example in Fig. 5 shows the modelling of the input variable "direction" in the fuzzy system. If the directions of several vectors under investigation are within a range of approximately 20 gon, they are assessed as "similar" by the fuzzy system. The greater the difference in azimuth, the smaller the property of "similarity" will be according to this human thinking.
not similar, neg
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Fig. 5 Example of the modelling of one input variable for the fuzzy system.
3.3 Structure of the algorithm 3.2 Visual indicators Another group of indicators for the block detection is based on the human way of thinking, represented by fuzzy systems. The human mind is able to determine blocks of a similar pattern of movement simply by looking at the graph of displacement vectors. Reproducing this intuitive pattern recognition in the fuzzy systems, the similarities of the length and of the direction of the vectors and the property of neighbourhood are basic indicators. According to human thinking it seems to be clear that only
The block detection algorithm was implemented in Matlab ®. With the displacement vectors given by the deformation analysis, the algorithm starts by finding all possible blocks consisting of four neighbouring points (see Fig. 6). A minimum of four points per block is necessary due to the over-determined calculation of the strain analysis. The fuzzy system selects the 'best' set and in an iterative process the best fitting neighbouring points are added to the block until the following fuzzy systems 3 and 4 determine that the block is complete, i.e. that no neighbouring points with a similar pattern of
Chapter 110 • A Multi-Scale Monitoring Concept for Landslide Disaster Mitigation
movement exist. Then the algorithm starts again finding four neighbouring points out of the remaining points.
of stabilizing forces (= shear strength) to destabilizing forces (= shear stress) (Wittke and Erichsen, 2002):
FS= Find all combinations of 4 neighbouring points
_.,
Soil Shear Strength Equilibrium Shear Stress
FS >> 1 (stable)
(1)
ys ~1 (failure)
Fuzzy System 1' choose best block /
Fuzzy System 2 choose I . next best point FFuzzy System 3 assess block properties
yes
oq
Remove last point, terminate block
Fig. 6 Structure of the developed algorithm.
Using this method the task of block detection as a subsequent step following the geodetic deformation analysis can be automated for the first time. Several examples have shown the applicability of this method for local and regional scale landslide areas.
and must be calculated in different parts of the slope. It is very sensitive to external influences (i.e. rain, and other loads) and internal structural changes (Crozier, 1986). The reliability of the allocation can be enhanced by combination of additional quantitative and qualitative expert knowledge. The basic idea for the development of the alert system is the combination of a classical data-based system analysis with a knowledge-based system analysis. The data-based part of this knowledgebased alert system is responsible for prediction and statistical evaluation of the slope's motion using calibrated deformation transfer functions (i.e. finite element models, VOLTERRA-Series or neural networks). The integration of additional sources of hybrid (expert) knowledge and the establishment of automated decision processes is realized by the knowledge-based part.
4.1 Architecture of the alert system 4 Concept of a knowledge-based alert system Impact and risk assessment in the (possible) landslide area primarily requires the definition and reliable separation of different kinematic / geomechanical conditions of the slope. In OASYS five decision levels are defined to evaluate the current stability status, and to take adequate measures for instrumentation, monitoring and alerting (TU Braunschweig et al., 2004): Normal operation ~ Low Margin Operation Warning ~ Emergency ~ Post Mortem To provide suitable indicators for allocation of the different levels is one major goal of the analysis of the landslide process and the task of an alert system. A typical numerical value is represented by the factor of safety FS which is defined as the ratio
In Fig. 7 the architecture of the alert system is shown. In the configuration phase the knowledgebased decision is made, to determine which deformation models and observation designs are suitable for the quantification of the present state of the landslide. The knowledge base must include, e.g., available instrumentation and measurement results from preliminary investigations, possible measuring rates, economic restrictions and accessible additional expert knowledge (geophysics, soil mechanics, etc.). The selected deformation models can be descriptive or causative, static or dynamic, respectively parametric or non-parametric. The identification of the deformation process is realized using geodetic and geotechnical measurements. This calibration step is a precondition for the close-to-reality prediction of the progress of the landslide.
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Fig. 7 Architecture of the knowledge based alert system.
The calculated predictions of the slide's motion are used as one main input for the knowledge acquisition component of the expert system. The system has to decide whether the predicted progress will conform with reality or not (verification step). Considering additional hybrid knowledge, the verification is not only based on incoming measurements (classical procedure) but also on expert prognoses. It enables the evaluation of the predictions at an early stage (preverification). If the predictions are identified as not conforming with reality a feedback to the system's configuration is induced for modification / new adaptation of the deformation models (e.g. static to dynamic). In addition this may be an efficient indicator for a significant change in the slide's motion. The predictions are used as input for the first dec&ion step in order to decide whether to change the alert level (decision level) or not. Additional quantitative or qualitative expert knowledge (excluding the predictions) is used to define the
second dec&ion step. The final dec&ion is made combining the results from the first and the second decision steps. In the case of a change of the alert level the verification component of the alert system will indicate if an additional modification (and identification) of the deformation models is required. In this concept the integration of suitable deformation models into the data-based system analysis, and its calibration to deformation predictors, must be the first principal task.
4.2 Main task" parametric identification of structural landslide models Structural models of the landslide, i.e. FLAC-2D/3D (Kampfer, 2005), SLOPE/W (Geoslope International, 2005) or PFC2D/3D (HcItaska Company, 2005), offer the most comprehensive possibilities for the analysis and prediction of critical states of the slope. Representing its inner structure, the stress distribution indicators such as the FS (1) can be calculated. Nevertheless one main problem is the model adaptation to reality. A suitable tool for the adaptation of structural models (= parametric identification) adaptive KALMAY-filtering techniques (see Fig. 8, Gelb, 1974) are suggested. In contrast to common trial and error-methods KALMAY-filtering offers the optimal combination of theoretical model and empirical measurements. In Fig. 8 the basic principle for the identification of a structural model is shown. Observed influence quantities u (i.e. changes in water regime, mechanical loads, etc.) at time t~ are used as input to calculate the prediction x of the geometrical and structural state of the slope at time tk+l. Together with geodetic and geotechnical measurements L and the linearized model (represented by the transition matrix T, coefficient matrix of correcting B and disturbing variables S) the least-squares estimation x is calculated at tk+l for model update. The recursive adaptation algorithm is finished when predictions and measurements finally show no more significant deviations. This is statistically proven by testing the filter innovations d.
Chapter 110 • A Multi-Scale Monitoring Conceptfor Landslide Disaster Mitigation
Structural model of landslide
Xk+1
Structural parameters
Kinematic State
Xp,~
x,
Prediction
Geometrical and geotechnical observations
Observed influences
Ilk
t k => tk+~
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11
]~
Tk+l,~; Bk+ 1,k ; Sk+l,k Linearized model
~?k+l ~?p,k,l Update of model
Fig. 8 Parametric identification of a landslide model with adaptive KALMAN-filtering.
The statistical evaluation of the KALMAN-filter's innovation is an efficient tool for detecting contradictions between the predictions and observations. The innovations are sensitive to changes in the slope's structure and motion, and can be used as an additional indicator for the change of alert levels in the data-based component of the alert system.
References Almeida-Teixeira M.E. et al. (1991). Natural hazards and engineering geology: Prevention and control of landslides and other mass movements. Proceedings of the European School of Climatology and Natural Hazards course held in Lisbon from 28 March to 5 April 1990. Crozier M.J. (1986). Landslides- Causes, Consequences and Environment. Croom Helm, London. Cui Z.Q. (1999). Geology and geomechanics of the Three Gorges Projects. Bureau of Investigations and survey, Changjiang Water Resources Commission, MWR. Gelb A. (1974). Applied Optimal Estimation. The M.I.T. Press, Cambridge London. Geoslope International (2005). www.geo-slope.com, last access 07/2005. Haberler M. (2003). A fuzzy system for the assessment of landslide monitoring data. Osterreichische Zeitschrifl ftir Vermessung und Geoinformation, Heft 1/2003, 91. Jahrgang, 92-98. Haberler M. (2005). Einsatz von Fuzzy Methoden zur Detektion konsistenter Punktbewegungen. Geowissenschaflliche Mitteilungen, Schriftenreihe der Studienrichtung Vermessungswesen und Geoinformation der Technischen Universitfit Wien, Nr. 73. HcItaska Company (2005). www.itascacg.com/pfc.html, last access 07/2005.
Heine K. (1999). Beschreibung von Deformationsprozessen durch Volterra- und Fuzzy-Modelle sowie Neuronale Netze, Deutsche Geod~itische Kommission, Reihe C, Heft Nr. 516, Verlag der Bayerischen Akademie der Wissenschaften, Mtinchen. KampferG. (2005). Zusammenfassung der Ergebnisse der geotechnischen Modellierung der RWE Testb6schung. Report of the project OASYS. Savvaidis P., K. Lakakis, A. Zeimpekis (2001). Monitoring ground displacements at a national highway project: The case of "Egnatia Odes" in Greece. Proceedings of the IAG Workshop on Monitoring of Constructions and Local Geodynamic Process, Wuhan, China, 2001. Theilen-Willige B. (1998). Seismic hazard localization based on lineament analysis of ERS- and SIR-C- radar-data of the Lake Constance Area and on field check. Proceedings of the European Conference on Synthetic Aperture Radar. 25-27 May 1998 in Friedrichshafen, VDE-Verlag, Berlin, 447-550. TU Braunschweig, TU Vienna, Geodata GmbH (2004). Instrumentation, Data Analysis and Decision Support for Landslide Alarm Systems. Report of the project OASYS. Welsch W., O. Heuneke, H. Kuhlmann (2000). Auswertung geod~itischer lJberwachungsmessungen. Wichmann Verlag, Heidelberg. Wieser A. (2002). Robust and fuzzy techniques for parameter estimation and quality assessment in GPS, Shaker Verlag, Aachen. Wittke W., C. Erichsen (2002). Stability of Rock Slopes. Geotechnical Engineering Handbook, Ernst & Sohn. Zhang Z.L., Huang Q.Y., Chmelina K. (2001). Research on geological landslide problems related to the Three Gorges Project. Proceedings of the IAG Workshop on Monitoring of Constructions and Local Geodynamic Process, Wuhan, China, 2001.
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Chapter 111
High Speed Laser Scanning for Near Real-Time Monitoring of Structural Deformations Hansj6rg Kutterer, Christian Hesse Geodfitisches Institut, Universit/it Hannover, Nienburger Strasse 1, D-30167 Hannover, Germany E-Mail: [kutterer,hesse] @gih.uni-hannover.de
Abstract. Several tasks in structural deformation
monitoring require real-time or near real-time data acquisition methods. In one case, a lock gate is being deformed within a time period of 14 minutes due to the change of the water level inside the lock. In order to accomplish a comprehensive monitoring of the gate's deformations, the high speed laser scanner Z÷F Imager 5003 has been used. The main advantage of this procedure is that object deformations and the corresponding variances can be derived without reflectors and with data acquisition rates of up to 500,000 points per second. The gate was scanned completely at different stages as well as in horizontal profiles at high speed. The analysis of the observed deformations was carried out by using methods from statistics such as Principle Component Analysis, which is well suited for the analysis of the interrelations of multiple data series. For this purpose, the object data was segmented into equally spaced blocks, which were treated as realizations of a time series. In combination with derived statistical measures, it was possible to determine the regions of the object which were mainly affected by deformations. Keywords. Laser scanning, deformation analysis, structural deformation monitoring, lock gate, near real-time, principle component analysis, statistics
1 Introduction Terrestrial laser scanners are new and promising tools for monitoring deformations of artificial and natural objects. At present, there are several types of scanners which can meet different requirements. For example time-of-flight-based laser scanners offer long ranges and can acquire data from widespread objects. Phase-shift scanners offer much higher data rates and are thus the preferred scanner for kinematic tasks or the monitoring of objects in motion. Note that the particular measurement task,
the environmental conditions and other factors such as measurement speed, maximum range, object dimensions and accuracy have to be considered when choosing the most appropriate scanner (Schulz and Ingensand, 2004; B6hler et al., 2003) In any case, three-dimensional (3d) Cartesian coordinates defined in a scanner system are derived from spherical coordinates. The number of scanned points typically numbers up to several million or more (3d point clouds). Hence, the objects are observed quasi-continuously in space and depending on the speed of scanning - with a (very) high temporal resolution. The goal of this study was the derivation of twoand three-dimensional deformation patterns due to varying external forces, such as water pressure, with high temporal and spatial resolution. Here, the changing deformations of a lock gate are of interest when the water level falls or rises. Consequently, the potential of kinematic terrestrial laser scanning for geodetic object monitoring will be assessed. For this study the Leica HDS 4500 scanner was used as it is very convenient for kinematic close range applications. The original equipment manufacturer (OEM) of this scanner is Zoller + Fr6hlich (Z+F), hence it is also known as the Z+F Imager 5003. At present it is one of the fastest scanners on the market with data acquisition rate reaching 500,000 points per second. It has a field of view of 360 ° horizontal and 310 ° vertical with a maximum unique measurement range of 53.5 m. The accuracy of the distance measurement is denoted by the manufacturer as being >3 mm at 25 m distance. A more detailed overview of the technical data of the scanner can be obtained from Table 1. Besides its use in 3d mode, it is possible to observe in 2d mode (single horizontal or vertical profiles) by disabling the low speed motor that is responsible for a rotation around the vertical axis. Then a
Chapter 111
• High-Speed
frequency of 33 Hz can be reached with the noise reduction parameter set to default noise, or a reduced frequency of 12 Hz for the low noise mode. In addition to this it is possible to stop both rotating mirrors in order to realize a one-dimensional distance measurement with up to 500 kHz. This option was not used for this study. Table 1. Technical specifications ofLeica HDS 4500 Parameter Field of view Min/Max range Data acqusition rate
Range accuracy @ 25 m Spot size @ 25 m Min spot spacing @ 25 m Weight
Performance 360 ° (Hz); 310 ° (V) 0.1 m / 5 3 . 5 m Points/s: 500.000; Profiles/s: 33 (default)or 12 (low noise) 9 mm (20% reflectivity); 3 mm (100% reflectivity) 8.5 mm 4.4 m m (Hz); 7.8 mm (V) 13 kg
The paper is organized as follows. In the subsequent section time series from 3d and 2d kinematic terrestrial laser scans observed in order to describe the deformations of a lock gate are presented and discussed. For the analysis, methods from multivariate statistics were used: the derivation of an empirical variance-covariance matrix and its Principal Component Analysis.
2. Lock gate Uelzen I 2.1 Object description
Laser Scanning for near Real-Time Monitoring of Structural Deformations
The lock Uelzen I is located between Hamburg and Hannover in northern Germany. The lock (Fig. 1) was built at the Elbe side channel which connects two major domestic shipping routes in Germany: the river Elbe and the Midland channel. The lock was built to transcend a difference in water level of about 23 m. It has an overall length of 185 m and therefore can accommodate one Euro Class II pusher unit, which has a defined length of 183.6 m. In filled state the lock keeps a water volume of 5 4 , 0 0 0 m 3. When bringing ships downwards, 60% of this water can be reused for filling because of three basins beside the lock. The Geodetic Institute of the University of Hannover carried out five major deformation measurement campaigns at this site within the last 20 years to monitor the deformations induced by the periodic change of the water level inside the lock. Since the completion of the lock its opposing side walls moved continuously away from each other with a magnitude of nearly 15 cm. Because of these deformations two additional campaigns took place in April 2004 (Hesse and Stramm, 2005) and July 2005 at which the deformations of the steel lift gate (Fig. 2) were to be determined. The vertically moving gate has a width of 12 m, a height of 11 m and a thickness of approximately 1.2 m. The water level inside the lock changes from 42 m (low level) to 65 m (high level) to bring ships upwards or downwards. Hence, a water column of 23 m resides behind the gate. The water level has a sink rate of 2.15 m/min and a climb rate of 1.94 m/min. These parameters suggest that a very high temporal resolution of the scanner is needed for measuring deformations at discrete epochs.
Fig. 2" Lock gate Uelzen (width: 12 m, height: 7m above outside water level) from an outside point of view.
2.2 3D scans
Fig. 1; Lock gate Uelzen (width:
12 m, height: 7m above outside water level)from an outside point of view.
In order to obtain a temporal sequence of the deformation patterns of the lock gate between the two extreme water levels (maximum and minimum) the gate was scanned repeatedly with the Leica HDS 4500 in July 2005. The fastest scanning mode
777
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H. Kutterer • C.
Hesse
("preview") was used so a reduced point spacing has to be taken into account. This procedure yielded 32 single scans in total with each one consisting of about 8500 points. Each scan took about 21 s. Fig. 3 shows an example of a 3d point cloud of such a scan.
In the next step the matrix elements (i.e. the associated time series of median values) were mapped row by row into new row vectors r (k) starting with the lowest vector N}k). Formally this reads:
r(k) =[l~}k)~k)~k)~k)l
(3)
These vectors were then rearranged as rows of the matrix:
Fr O)
R-lr(: 2) -Ira,,, m,,2 1~1,3 "'" 1~4,9
m4,10]-(4)
/r °t
Fig. 3: 3D point cloud of the lock gate
where n denotes the number of epochs and which comprises the time series of all grid elements. The vectors Ni,j contain the time series of median
For all further analysis the point clouds were referred to a regular vertical grid which covers the lock gate more or less completely. Therefore, grid elements (classes) with the size 1 m x 1 m were chosen, which are defined column by column starting at the lower left comer (1,1) and ending at the upper right comer (4,10). Each point of the cloud was uniquely assigned to a grid element. The coordinate components orthogonal to the lock gate plane were considered as deformation values. For each grid element and each scan epoch the median value was calculated. This reduced the data noise significantly since each grid element contained more than 200 single points. The noise level of the original data before the analysis was about + 2cm. For further references about laser scanning of a lock gate see Kopacik and Wunderlich (2004).
values of each grid element. The result of this procedure is shown in Fig. 4. In addition, the respective median values are given in Fig. 5 for a 50 cm × 50 cm grid in the lock gate plane for four selected epochs. Actually, the temporal evolution along the axis 'Epochs' shows the release of the deformation since the initial state reflects the maximum deformation. Some properties are of interest. First, there is a clear linear deformation of about 14 to 16 mm combined with a slight trend (translation) for each grid element from Epoch 1 to Epoch 29. It is caused by the reduction of water load on the surface of the lock gate. Second, beginning with Epochs 27 and 28 but mainly between Epoch 29 and Epoch 30, there is a sudden translation of about 16 to 18 mm which is caused by a movement of the whole gate away from the sealings into its regular position. Finally, the actual noise is on average below 1 mm and the number of irregularities (which might be potential outliers) is zero.
r~
Uelzen from outside," grey value coded by intensity. The light vertical lines are caused by water on the surface.
By this way a temporal sequence (time series) of 32 matrices of the size 4 x 10 such as"
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m41(k im4,1 m4,2 ... m4,10 (k) n (k) = m3 = / m3'1 m3'2 m3'1° m2 /m2,1 m2,2 m2,10 ml [_ml,1 ml,2 ml,lO with
the respective mi,j, i = 1,..., 4, j = 1,..., 10
(1)
median values as elements was
obtained; the index k denotes the respective epoch. Afterwards the matrix elements were referred to the initial deformation state M (1) (maximum water level) by subtracting the values of the first epoch. This yielded: ~(k) = M(k) _ M(1). (2)
Epochs [ ]
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0
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Time series of the deformation of the grid elements representing the lock gate plane aligned row by row from the left to the right (matrix R according to Eq. (4)).
Chapter 111 Epoch 10
•
High-Speed Laser Scanning for near Real-Time Monitoring of Structural Deformations
Epoch 20
Std.-Deviation vs. Principle Component Analysis
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0.005
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Fig. 5: Temporal slices of the deformation of the grid elements represented the lock gate plane.
For a more refined interpretation a Principal Component Analysis based on the empirical variance-covariance matrix (vcm) of the column vectors ~ i j of the grid elements was computed. The components of this vcm are the empirical covariances of each pair of the time series vectors ~ i j . Hence, the empirical variances are its main diagonal elements.
Fig. 6 shows the eigenvalue spectrum of this vcm. There is one dominant eigenvalue which represents 94 % of the sum of all eigenvalues (trace of the vcm). Looking at the associated principal component (Fig. 6), is it obvious that the principal component (dominant eigenvector) explains most of the actual variations. There is also a clear null space of the matrix which has not been understood yet. It will be analyzed in detail in a further study.
The angle on the right edge of Fig. 7 is of some interest; it is also visible in Fig. 4. This right hand part is connected with the uppermost row of the grid. Obviously the deformations in the centre element and at the edges differ in their magnitude. This could be due to the steel bar construction on the inside of the lock gate.
2.3 2D scans
In a second scenario a large number of horizontal profiles were scanned in May 2004 with the Z+F Imager 5003. Observation of profiles only allows a significantly higher temporal resolution than of complete surfaces. Nearly 7600 epochs were scanned, with each consisting of about 7300 single points. Hence, the corresponding frequency was about 12 profiles/s. The profile points were associated with 22 equidistant classes with a width of 50 cm. As in the 3d case the noise level of the original data was reduced by calculating median values for each class. A further smoothing was gained by a moving median filter in the time domain, i.e. medians for each class using a window length of 151 epochs.
Eigenvalues of Empirical VCM
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The following figures show the resulting time series. In Fig. 8 all epochs are represented. As in Fig. 4 there are clear translation and deformation patterns until a sudden change during the final quarter of the epochs. The first three quarters are shown in Fig. 9. The properties are here similar to the ones in the 3d case. The final peak at the centre of the profiles has not been explained yet. Note the similar effect in Fig. 4 which was already referred to in Section 2.2.
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a number of nearly 7600 epochs and equidistant classes along the profile (width." 50 cm)
variance-covariance matrix of the time series of the profile classes in semilogarithmic scaling. Std.-Deviation vs. Principle Component Analysis
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4. Discussion and conclusions
Fig. 9: Same time series as shown in Fig. 8 but without the
values of the sudden translation
Another possibility for smoothing the denoised point data, and obtaining deformation values at constantly spaced grid points, is using spline interpolation (Schfifer et al., 2004). As in the 3d case, an empirical covariance matrix was derived for the time series of the 22 profile classes. Again an eigenvalue decomposition was computed. The obtained eigenvalue spectrum is shown in Fig. 10. One eigenvalue is dominant. It represents about 89 % of the sum of all eigenvalues (trace of the matrix). In contrast to the 3d case all eigenvalues are positive, so there is no null space. The deformation between the centre of the profile and its edges is given in Fig. 11. The empirical standard deviations reflect the variability which was already visible in Fig. 9 and Fig. 10. The amount of deformation on the left side of the lock gate is significantly smaller than for the right side. The
Kinematic terrestrial laser scanning is a new and promising method for the fast observation of surfaces and profiles of artificical and natural objects of local scale. Thus, it is a basis for a refined deformation analysis. The experiences gained and the results derived for the lock gate Uelzen I show clearly that today 3d scans are repeatable within time spans which are significantly below one minute. The quality of the results is high. With standard statistical methods it is already possible to obtain a sub-cm precision. In the case of 2d scans (profiles) even higher frequencies for the repetition of scans are possible. In the example described here a frequency of 12 profiles/s was achieved. The statistical processing of the data showed that the noise could be reduced significantly through averaging of observations. Hence, there is a good possibility to control the noise level by the respective spatial and temporal resolution. Without doubt it is possible to apply more sophisticated statistical methods such as estimation, prediction and filtering, Fourier or wavelet transforms, etc. These topics were not the subject of this study but will be investigated in future.
Chapter 111 • High-Speed Laser Scanning for near Real-Time Monitoring of Structural Deformations
Successful tests with other structures, such as the pylon of a wind energy plant (Hesse et al., 2005), indicate further the great potential of terrestrial laser scanning for geodetic monitoring and deformation analysis - in particular in the kinematic mode. Future work has to consider aspects of the instrumentation such as systematic effects and calibration, of the observation configuration such as the merging of scans, the integration with other (complementary) observation techniques, and efficient and effective statistical analysis.
References B6hler, W., Bordas, V., Marbs, A. (2003). Investigating Laser Scanner Accuracy. The International Archives of Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXIV, Part 5/C15, 696-701. Hesse, C., Stramm, i4. (2004). Deformation Measurements with Laser Scanners- Possibilities and Challenges. International Symposium on Modem Technologies, Education and
Professional Practice in Geodesy and Related Fields, Sofia, Nov. 04-05 2004, pp. 228-240. Kopacik, A., Wunderlich, T. (2004). Usage of Laser Scanning Systems at Hydro-technical Structures. In: Proceedings of the FIG Working Week- The Olympic Spirit in Surveying, Athens, May 22-27 2004, http://www.fig.net/pub/athens/papers/ts23/TS23 4 Kopacik Wunderlich.pdf. Schfifer, T., Weber, T., Kyrinovic, P., Zamecnikova, M. (2004). Deformation Mesaurement using Terrestrial Laser Scanning at the Hydropower Station of Gabcikovo. INGEO 2004 and FIG Regional Central and Eastern European Conference on Engineering Surveying, Bratislava, Slovakia, Nov. 11-13 2004,
http ://www.fig.net/pub/bratislava/papers/ts_O2/ts_O2_schaef er etal.pdf. Schulz, T., Ingensand, H. (2004). Influencing Variables, Precision and Accuracy of Terrestrial Laser Scanners. INGEO 2004 and FIG Regional Central and Eastem European Conference on Engineering Surveying, Bratislava, Slovakia, Nov. 11-13 2004,
http://www.fig.net/pub/bratislava/papers/ts_O2/ts_O2_schulz _ingensand.pdf.
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Chapter 112
A method for modelling the non-stationary behaviour of structures in deformation analysis H. Neuner Geodetic Institute, University of Hanover, Nienburger Str. 1, 30167 Hanover, Germany
Abstract. The standard description and modelling of the normal behaviour of structures under influence of factors like temperature or tides is done by means of correlation functions and power spectra that assume the condition of stationarity of the observed process. As a consequence of influencing parameters with highly irregular patterns, aging or damage to the structure the measured reaction of the structure may have non-stationary characteristics. Hence these functions give only a coarse description of the object's reaction and more detailed analysis techniques are needed. For these cases a method of data analysis based on the discrete wavelet decomposition and variance homogeneity testing of the wavelet coefficients is applied in this paper. After identifying the frequencies with high energy in a spectral analysis, the signal is decomposed on a scale base in order to analyse the different stationary and non-stationary components separately. The wavelet coefficients of each scale are separated into intervals with homogeneous variance by performing a test based on the cumulative sum of squares. The amplitude and phase shift of the detected frequencies are calculated for each interval in an adjustment step. Finally, using the modelled data the signal is reconstructed by wavelet synthesis. Because the wavelet transform is done without loss of information, an objective evaluation of the modelled data is possible. Continuously recorded tilt measurements at the tower of a wind energy plant were analysed both in the classical manner and with the proposed method, in order to test the latter's performance. It is shown that the new method performs well and lowers the standard deviation of the residuals by 30%, while the standard procedure does not remove the high energy components. Keywords.
Discrete Wavelet Variance homogeneity test.
Transformation,
1 Introduction One of the main goals of structural deformation analysis is to characterise the behaviour of the monitored object by means of a dynamic deformation model. In the standard approach, the causality between influencing factors acting on the object and the deformation signals they are generating is expressed by a linear filtering model. The gain and phase of the transfer function expresses the amplification and delay of the input and thus reflects the object's physical properties. To estimate these two components, correlation or spectral functions are used. This method provides good results, as shown in previous engineering projects (Kuhlmann, 1996; Neuner et al., 2004), if the analysed deformations are slow in relation to the recording rates, and the time series are at least stationary up to order two. That means, all time points must have the same mean and variance, and the covariance must depend only on the interval between the time points (Priestley, 1981). This property is not always fulfilled by the recorded time series because of blunders or effects like non-stationary influences on the structure (e.g. wind, traffic), measuring of various direction-dependent components of the deformation signal, changing of calibration parameters due to automatic calibration, or high-speed recordings with modem sensors. These effects may cause various types of non-stationarity like jumps, spikes or change of variability. To properly account for nonstationarity, other data analysis methods are required. The present paper deals with a method of analysing time series that are periodic, but encounter changes of variance in time. For this kind of signal, the dominant frequencies can be detected by a Fourier Transformation, and the corresponding amplitude and phase can be estimated separately in an adjustment model of the kind: m
X k -X
qt_ Z A i
i .COS(2rcfitk)+B i .sin(2rcfitk)
(1.1)
Chapter112 • A Methodfor Modellingthe Non-StationaryBehaviourof Structuresin DeformationAnalysis
For a correct functional model, one needs to detect the time points at which the statistical properties change, such that the coefficients A~ and B~ are estimated only for data with homogeneous statistical moments. This can be done by using a statistical goodness-of-fit test. The sensitivity of such tests is low if signals with different power and statistical properties overlap. A way to improve the sensitivity is the a priori filtering of the recorded signal in order to separate its different periodic components, test these separately for variance homogeneity, and estimate the corresponding amplitude and phase individually. Once this step is completed, one needs to find an inverting filter operation that should put the signal components together, to regain a unique modelled signal. The Wavelet Transform is most appropriate for accomplishing this task, because of its orthogonality property. The remainder of the paper is organised as follows. The second and third section contain the theoretical background of the proposed analysis procedure. The results obtained by applying this procedure for modelling the oscillations of the tower of a wind energy plant are presented in the fourth section and compared to the ones obtained by the "overall" estimation. Concluding remarks and possible future work are discussed in the last section. 2 The Wavelet Transform time series analysis
method
for
The Wavelet Transform offers the possibility to extract and study local characteristics of the signal at subsequent resolution levels. This can be done in different ways according to the purpose of the data decomposition (B/ini, 2002). The Discrete Wavelet Transform (DWT) is appropriate if further processing of the transformed data is needed, and therefore this paper is focused on this particular analysis method. The DWT is formulated in terms of an orthogonal filter bank and consists of passing the n-valued low frequency signal component u, separated in a previous step j, through a quadrature mirror filter pair, and decimating the components for reasons of energy conservation by retaining every other value:
Uj+I, n -- Z hk-2n " U J, k k Vj+l,n -- Z gk-2n " Uj,k" k
The high-pass filter g is referred to as the wavelet filter and the output signal components v, form the wavelet coefficients. Similarly, the low-pass filter h is referred to as the scaling filter and the output signal components u, form the scaling coefficients. In terms of the wavelet theory, every decomposition level j is referred to by a scale sj=2j-1. To interpret the physical meaning of signal components in the scale sj, the transfer function of the applied cascade of scaling and wavelet filters, denoted by capital letters of the corresponding filters, can be derived from the relations (Percival et al., 2002): j-2
Gj (f) - G(2 j-'. f). ]--[ H(2 k. f) k=0 j-1
Hi(f) - UH{21' •f).
(2.2)
k=0
The first relation is the transfer function of a band-pass filter with nominal passband 1 / 2 j + l - - 1/2j, while the second one represents the transfer function of a low-pass filter with nominal passband 0 - 1/2j+l. Fig. 1 illustrates the gain functions of the 4 th order Daubechies scaling and wavelet filters at the third decomposition level. Because these filters are not ideal, signal components with frequencies in the transition band, as well as those introduced by leakage effects, can occur in the spectrum of the wavelet coefficients. Beside these effects, the wavelet coefficients, generated at each level of the DWT, are mainly signal components with frequencies located in the band-pass of the equivalent filter sequence. Due to downsampling, the respective frequencies will appear in the frequency spectrum of the wavelet coefficients as aliases. Accounting for these facts, it is
31....
2.5
,i
=2[
-g
li
'~//
/ii, 02
nominal passband of thewavelet-filter
0.2
0.3
0.4
o15
frequency(Hz) (2.1)
Fig. 1 Gain function of the third level Daubechies scaling (dashed line) and wavelet-filter (continuous line)
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possible to relate dominant frequencies in the spectrum of the wavelet coefficients to frequencies contained in the original series and to derive their corresponding magnitudes. In order to perform the DWT it is necessary to know the coefficients of the filters to be used. Daubechies extended the mathematical constraints imposed on filters of orthogonal perfect reconstruction filter banks with regularity conditions, and obtained a filter family appropriate for decomposing and analysing the signal components according to the goals of the wavelet transform (BriM, 2002). The main properties of the filters are determined by the order of the root f = 0.5 of the equation H(f)=0. By increasing the order of this root the transition bands of the filters represented in Fig. 1 become steeper and thus the frequency localisation improves. On the other hand, the filter length also increases and more coefficients become affected by boundary conditions in the convolution operation. These opposite effects make it necessary to select the filter's structure according to the goals of the data analysis. The DWT performed in this analysis aims to separate the signal components of different frequencies. To do so, good localisation properties in the frequency domain are required. Due to the sufficient spacing between the peaks of dominant frequencies, a Daubechies filter of the fourth order seemed to be a good choice. Because of its orthogonality, the DWT is an energy preserving transform. Thus, without concern about the stationarity properties, the original time series and the resulting wavelet and scaling coefficients have the same information content. This can be expressed by the relation:
j-1
where V and U stand for the series of wavelet and scaling coefficients at scale 2j-1 and J for the maximal decomposition level. Regarding the energy of a stationary process in terms of variability, it can be claimed that the DWT decomposes its variance on a scale-by-scale basis (Percival et al., 2002). This is a useful property that permits the separate analysis of the contributions of each scale to the sample variance. For the kind of non-stationary processes treated in this paper, the property (2.3) makes it possible to follow the buildup of the variance in each scale, and hence to study the change of variance on a scale-by-scale basis.
The time points of variance change in each scale can be identified by using a goodness-of-fit statistical test, like the one presented in the following section.
3 Testing for variance homogeneity To test the variance of time series for homogeneity, several procedures are available, but some of them aim to detect a single variance change, or refer to particular models. A more general approach comes from Inclan and Tiao (1994), who investigated the detection of multiple variance changes of a sequence of uncorrelated normally-distributed random variables ak, with mean 0 and variance c~, k = 1..N, based on their cumulative sum of squares Ck -- ~ i k l a i2 • Because Ck is a positive, monotonically increasing function with a slowly varying slope, a sharp detection of a variance change point is difficult. Hence the test value is based on the normalised and centred cumulative sum of squares, defined as: (3 1)
Dk = C k _ k
CN
N'
In the case of variance homogeneity, Dk oscillates around the horizontal zero axis and changes its slope drastically (even in sign) at the change points of the variance. The search for variance change points is done sequentially, in points k* where ]Dk*] attains its maximum. To evaluate the significance of Dk, a distribution relation of the measure defined in (3.1) has to be found. If Dk, exceeds a certain percentage quantile, than a change of variance occurs and the identified point k* is a first evaluation of the change point. For a certain k, the F-statistic could be used to test the equality between the variances c~02and c~ 2 of the samples ai i=l..k and aj j=k+l..N respectively. Expressing the ratio of the two variances in terms of the measure Dk, the testing of rio: C~o2 = c~12leads to:
k.(N-k) (1--FNk.k., c~) / P Dk-~--N k+(N ,~-~----z--~--~--,,1 k)-FNk ~ =l-c~ (3.2) To correctly estimate the quantile and the degrees of freedom, the exact location of k must be known, while looking for max IDkl requires the determination of the location of the change point. The asymptotic distribution of Dk can be determined by using convergence properties of probability measures presented in Billingsley (1999). Under the assumption that the sequence of random variables ak have ho-
Chapter 112 • A Method for Modelling the Non-Stationary Behaviour of Structures in Deformation Analysis
mogeneous variances c~, the random variables ~k build as ~k -- ak2 -- ~a2 with E(~k) -- 0 and Var (~k) -2~S~, can be used to form a function XN(t) of the type:
Table 1. Quantiles of the probability distribution of the Brownian bridge function W°
1 LNtJ ( N . t - k N . t J ) (3.3) XN(t)- ,]-2(s~x[-N ~ ~ + x/~rs~x/-N '~N-tJ+l According to Donsker's theorem (Billingsley, 1999, p. 90) this function converges weakly to the Wiener probability measure W, used to describe the path of a particle in Brownian motion. Based on the Wiener measure, a Brownian bridge function may be defined as W( = W t - t W l . Its sample version is obtained by introducing (3.3): 1 (k k.~i;+ W°(t) = XN(t)-t'X'(t) = x/-2~x/~ i ~ i --N- i l
@
(N.t-~N.tJ) ~V/~ 2~-N " ~N.t~+l
(3.4)
Reversing the prior change of variables, one gets the normalised and centred cumulative sum of squares in the first term:
N2/ / W°(t)=,4/2~77/~ .~a k
~ a 2k k ~a,2, N i1 i=l
-t-
(N.t-LN.tJ)
N i~a2k (N.t-[N.tJ) %/-20"~ %/-N-" Dk @ -~/-2-~a~--~N/r--N - " ~N.tJ+l
(3.5)
For large sample numbers the second summand vanishes, and a concise relation between the Brownian bridge and the cumulative sum of squares follows:
W'~ (t) - ~-~ • Dk
(3.6)
The distribution relation of the Brownian bridge function is introduced in Billingsley (1999): P(maxlW([ < b) - 1 + 2 ~ (_1) k -2k2b2 k=l e
(3.7)
The quantiles were determined numerically with a precision of 10 .6 and are presented for the most common confidence levels in Table 1. For the data analysis presented in the next section, a confidence level of 95% was chosen. The method presented so far offers the possibility to examine a sample of zero-mean uncorrelated
normally-distributed random variables for variance homogeneity. The test value based on their cumulative sum of squares fulfills under the null hypothesis of variance equality, the distributional relation (3.7). If the test value exceeds the quantile corresponding to a prior fixed confidence level, then k* corresponding to max(Dk) is marked as a possible change point. To check for further variance change points the samples are divided into subseries [al..ak,] and [ak*+l..aN], and the procedure is applied again for each of them. This process is iterated until the test value falls below the quantile for every subset of variables [atl..at2]. Thus, a set of variance change point candidates is obtained. To avoid overestimation, the algorithm includes a further step, in which the number and location of the change points is refined. Therefore the candidates are included in ascending order in a vector c, whose first and last elements are 0 and N respectively. The change point detection method is applied successively on each interval c[i- 1]+l..c[i+ 1], aiming to detect c[i]. If a new change point location occurs, it will be added to the vector c and considered in a following iteration step. Similarly if the change point is not confirmed it will be removed from c. This step is iterated until the number of points remains unchanged and their location varies below a certain level. For the data analysis in the following section a maximum modification of two positions was allowed. Because the convergence of this step has not been proven mathematically, it is advisable to set a maximum number of iterations. Before using this method for testing the homogeneity of the variance of the wavelet coefficients in each scale, one needs to make sure that the wavelet coefficients match the stochastic properties imposed on the sample variables ak. While the expectation value of the wavelet coefficients is always zero due to the structure of the high-pass filter, the condition on stochastic independency requires further attention. For long- and short-memory processes the decorrelating effect of the wavelet transform was pointed out by Whitcher (1998). As presented earlier, this effect does not hold in the case of periodic signals. A way to account for existing correlations is
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H. Neuner
to replace in the test value (3.6) the sample number N by the equivalent degrees of freedom Nedf. If the samples of the analysed time series are normallydistributed the squares of the wavelet coefficients will follow a )(,2 distribution and the equivalent degrees of freedom can be calculated by (Priestley, ]98]):
Nedf =
o¢
(3.8)
2 c2 (k) k=-oe vv
where Cvv denotes the autocovariance function and (~v2 the variance of the wavelet coefficients unaffected by the boundary conditions. With this modification of the test value (3.6), the test procedure based on the cumulative sum of squares can be used to analyse in each scale the build-up of the variance and hence to set the intervals on which separate estimation of the parameters in the model (1.1) is necessary.
4 Modelling the non-stationary behaviour of the pylon of a wind energy turbine The wind energy sector owes its continuous development to wind turbine plants with ever larger dimensions, and to the improved technique of acquiring and transforming the mechanic energy into electric power. The increased financial investment necessary for the use of modem technology, coupled with the placement of this structures in relatively inaccessible locations, makes the monitoring of their stability particularly relevant. The static properties of the wind energy turbine distinguish themselves from those of other structures by their ability to resist very high short-term loads, such as those generated by strong wind squalls. This is why the monitoring activity aims more at the detection of material fatigue than at describing the effect of an unusual event. This can be done by continuously measuring the oscillations of the turbine's components at established epochs, and by assessing the change of dominant frequencies and their corresponding amplitude. Hence, the precise estimation of the oscillation's amplitudes is one of the most important tasks of monitoring wind energy turbines. A project performed at the Geodetic Institute Hanover had the main goal to point out possible contributions of engineering geodesy to the monitoring of a wind energy turbine. The studied object
was a wind power plant of the type Tacke 1.5s with variable rotation speed and nominal output of 1.5 MW. To achieve an optimal power level at each wind condition the nacelle rotates following the wind's direction and the blade's pitch can be varied. At high wind speeds the variable pitch prevents the build-up of extreme loadings and throttles, maintaining the rotation speed at a maximal constant velocity of 21 rpm. The project was focused on the determination of the behaviour of the tower, which consists of a 5 m high concrete base and a steel pylon of 77 m. The pylon comprises 3 segments with heights of 25.9 m, 25.9 m and 25.0 m, joined by circular flange connections. A platform is situated inside the pylon at each connection level between the segments. The data were collected over a period of ten days. Due to the variation of the wind speed it was possible to analyse the oscillating behaviour for rotation speeds between 0 and 20.1 rpm. The monitoring was performed with GPS, which was mounted on the top of the nacelle, and with inclinometers, set up on the three platforms and on the base. Only the inclinometer measurements will be treated in this paper because they have the highest sampling rate. However effects similar to those presented in this paper have been detected also from the GPS data. The tilts were measured in one direction only, in order to achieve maximal sampling rates. The directions of the various sensors are roughly, but not exactly, the same. On the first and third platform Rotlevel inclinometers were installed with a maximal recording rate of 1 Hz. At the second level, a Schaevitz inclinometer was used, with a recording rate that was restricted by the attached A/D converter at 6.1 Hz. Because these data have the highest sampling rate, the presentation of the modelling results refers mainly to them.
The analysed time series contains 14671 samples, which correspond to a period of 40.08 minutes. During this period the turbine had an almost constant power output of 110 kW and a rotor velocity of 12 rpm. The spectral analysis was performed only on measurements recorded over periods with nearly constant wind and functioning conditions in order to give a description of the structure's behaviour for distinct operation states. This analysed period is one of the longest in the project which meets these requirements. The dominant frequencies in the spectrum are visible in Fig. 2.
Chapter 112 • A Method for Modelling the Non-Stationary Behaviour of Structures in Deformation Analysis
0.01
0.01
0.009
0.009
0.008
/
0.007 g
0.008
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'
,
i
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~.
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'.
......
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ii
.... "~.?,
i,.
_
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Fig. 2 Amplitude spectra of the time series.
Fig. 3 Amplitude spectra of the residuals of the overall model.
The leftmost peak labelled l p corresponds to the rotation velocity of 0.2 Hz. f0 and fl denote the first and second eigenfrequency of the pylon respectively. The frequency 3p is attributable to the blade frequency and hence is three times l p. The other higher dominant frequencies are harmonics of the blade frequency. Some aspects can be noticed from the spectrum. The first one is the high amplitude resulting from the overlapping of the second eigenfrequency of the pylon with the fourth harmonic of the blade frequency. More important for the estimation of the amplitudes is the energy spreading over neighbouring frequencies in the region corresponding to the second harmonics of the blade frequency. For these only the frequency with the maximal amplitude in the group was introduced in the adjustment model. In addition to these seven frequencies, further peaks occur at other frequencies (e.g. 0.42 Hz, 1.52 Hz or 1.92 Hz) for which no physical interpretation could be given, nor repeatability established. Hence, they were treated as local effects and were not further considered in the adjustment model. If the amplitudes and phases of the seven dominant frequencies are estimated in the adjustment model (1.1) using the data of the entire time series, one expects white noise in the residual series. The residuals spectra are shown in Fig. 3. For comparison purposes the ordinate axes are scaled equally. The diminishing of the magnitude of the peaks is clearly observable, but the leftover energy visible primarily in the eigenfrequencies indicates that further improvement of the model can still be attained. The insufficient model accuracy could come either from small variations of the frequencies, or from changes of the corresponding amplitudes in time.
To account for variation of the frequencies one can introduce further neighbouring frequencies in the model. This is justified by the small variations of the wind speed only in the case of the eigenfrequencies. For rotation-induced frequencies this action could cause an overfitting of the model, because during the analysed time period the rotation velocity and pitch angle remain unchanged and thus new introduced frequencies cannot be justified from a physical point of view. Variations of the amplitudes could be caused by the rotation of the nacelle and by the changing of the wind velocity. They express themselves in a change of the variance. A useful method to detect variance change points is the iterative cumulative sum of squares algorithm presented in the third section. If the test is applied directly on the data of the original time series, the test statistic does not indicate any variance change. But this procedure is rather insensitive because the larger variance on some frequencies might cover effects occurring on frequencies with lesser variance. Additionally, the nonstationary effect cannot be attributed to a certain frequency, which makes it hard to interpret. To overcome these disadvantages the signal was decomposed by a Discrete Wavelet Transform using a Daubechies filter of the fourth order. After the transformation the signal components corresponding to the scale's pass-band were obtained. The distribution of the seven dominant frequencies over the scales is presented, together with the nominal passbands in Table 2. To project all dominant frequencies onto the corresponding wavelet scales, four decomposition levels were necessary. Some of the frequencies, like f0 and 9p, can appear in more than one scale due to the FIR nature of the filters.
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H. Neuner
2. Distribution of the dominant frequency components over scales Table
Scale Nominal frequency band (Hz) Contained frequencies
20 2] 1.530,763.05 1,53 15p, 9p, 6p, 9p 12p + fl
22 0,380,76 3p, f0
0.015
alias of fO at the fourth level
23 0,190,38 f0 +lp
/
0.01
fO at the third level 0.005
The amplitudes corresponding to these frequencies were estimated in both scales on the intervals of homogeneous variance. Their separate processing improves the reliability of the estimates, though it is an increased computational burden. In the spectra of the wavelet coefficients the peaks do not appear at the same frequencies as in the original time series, but on aliases with respect to the lower end of the nominal pass-band. This effect occurs due to the downsampling made at each level of the decomposition. In the adjustment model (1.1) based on the wavelet coefficients the alias frequencies were used instead of the original ones. But not all frequencies occurring in a scale's spectra must be interpreted as aliases. Frequencies in the transition band of the filters below the nominal lower end appear as real ones. This situation occurred for f0 and is illustrated in Fig. 4. In order to follow and understand the characteristics exposed by the spectra of the wavelet coefficients of a certain scale, not only the ideal pass-band but also the gain of the applied filter sequence must be derived and analysed carefully. The variance homogeneity test was applied to each of the four series of wavelet coefficients. At each scale the test indicated several points of variance change. The maximum number of change points was 14 and occurred in the second scale. This can be explained by irregularities on the 6p frequency that already showed up in Figures 2 and 3. Because this frequency contributes little to the total energy budget, the following presentations focuses on effects related to the first eigenfrequency. In each scale containing f0, seven variance change points were identified. Thus, the idea that effects on this frequency cause non-stationarity seems justified. Fig. 5 illustrates the test value (3.6) based on the centred cumulative sum of squares of the wavelet coefficients of the third and fourth level. Vertical dashed lines indicate the variance change points. To compare visually the respective homogeneity intervals, the graphic of the fourth de-
0.4 0 Frequency (Hz)
0.2
0.1
0.2
Fig. 4 Peaks corresponding to f0 at the third (left) and at the fourth (right) decomposition level.
composition level is scaled by two. As can be noticed, only the last three intervals are equal, clearly indicating non-stationary effects on f0. At the other intervals the additional effects on l p and 3p lead to different variance homogeneity domains. This result confirms the justification to decompose the signal into frequency components. It also shows, however, that an even finer segmentation should be aimed at in order to gain good insight on the effects associated with each frequency. One way to accomplish this task is to use the Wavelet Packet Transform, which further decomposes each series of wavelet coefficients contained in a scale. The amplitudes of the f0 oscillation estimated on intervals of homogeneous variance are presented in Table 3. The amplitudes in brackets are obtained on the corresponding intervals of the third level. The good agreement between them confirms the non-
4
,
,
500
1000
4
%
I
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I '
looo
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2000
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Fig. 5 Test value (3.6) for the wavelets coefficients of the third (top) and fourth (bottom) scale.
Chapter
112 • A
Method for Modelling the Non-Stationary Behaviour of Structures in Deformation Analysis 0.01
Table 3. Amplitudes of the f0 oscillation estimated from wavelet coefficients of the fourth (third) level.
0.009 0.008
Interval 1 Amplitude 40 (cc)
2 22
3 54
4 5 192 49
6 7 8 25 66 44 (31) (63) (44)
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0.003
stationary character of the f0 wave. With the estimated coefficients it is now possible to set up a model of the form (1.1), which describes the recorded tilts on each interval of homogeneous variance. Substituting the wavelet coefficients by the calculated data, a modelled "overall" signal is constructed in a wavelet synthesis step. Due to the orthogonality property of the wavelet transform, the deviation from the recorded data is coming exclusively from the model. This allows its objective evaluation by analysing the residuals. The spectrum of the residuals is shown in Fig. 6. Compared to the energy budget remaining after modelling the entire time series, an improvement can be observed especially for the eigenfrequencies. The improvement is expressed also in the standard deviation of the residuals, which decrease by 30% if they are treated as uncorrelated and by 59% if still existing correlations are accounted for.
5 Conclusions and outlook The estimation of the amplitudes and phases of dominant deformation frequencies based on the entire time series is biased, if the recorded signal contains changes of variability. This paper presents a method to account for variance changes which extends the standard modelling approach. The variance change points are detected with a goodness-offit test based on the centred cumulative sum of squares. The sensitivity of this test improves if it's applied on individual frequency components of the signal. To obtain these, the use of the D W T is proposed, because it decomposes the energy on a scaleby-scale basis without loss of information. The performance of the method was discussed on a tilt series recorded at a wind energy turbine, which is mainly non-stationary due to effects occurring on the eigenfrequencies. This work will be continued with the analysis of further measurement series containing sudden changes of other influence factors. An improvement of the method is expected by using the Wavelet
0.002 0.001
00
0.5
1
1.5
2
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2.5
3
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Packet Transform which further decomposes the wavelet coefficients in each scale, and hence makes an assignment of the variance change points to a single frequency and allows a better interpretation of the effects.
References
B/ini, W. (2002). Wavelets. Oldenbourg Verlag, Mfinchen. Billingsley, P. (1999). Convergence of Probability Measures. Second Edition. John Wiley & Sons, Inc., New York. Inclan, C., Tiao G. C. (1994). Use of Cumulative Sums of Squares for Retrospective Detection of Changes of Variance. Journal of the American Statistical Association, Vol. 89, No.427, pp. 913-923. Kuhlmann, H. (1996). A Contribution to the Monitoring of Bridges with continuously recorded measurements. (Ein Beitrag zur (2]berwachung von Briickenbauwerken mit kontinuierlich registrierten Messungen) Ph.D. Thesis, University of Hanover. Mallat, S. (2001). A Wavelet Tour Of Signal Processing. Second Edition. Academic Press, San Diego. Neuner, H., Ch. Hesse, R. Heer (2004). Combination of geodetic sensors for the monitoring of quay walls. (Kombination verschiedener geodfitischer Sensoren zu Uberwachung von Kaimauern). In: Kinematic measurements (Kinematische Messmethoden). Schwieger, V., K. Foppe (eds.) Schriftenreihe DVW, Vol. 45. Wigner, Augsburg, pp. 185 212. Percival, B.D., A.T. Walden (2002). Wavelet Methods for Time Series Analysis. Cambridge University Press. Priestley, M.B. (1981). Spectral Analysis And Time Series. Volume 1: Univariate Series. Academic Press Inc. (London) Ltd. Whitcher, B.J. (1998). Assessing Nonstationary Time Series Using Wavelets. Ph.D. Thesis, University of Washington.
Acknowledgement: The author thanks the German Research Foundation (DFG) for supporting the participation at the conference.
789
Chapter 113
Volcano Deformation Monitoring in Indonesia:
Status, Limitations and Prospects H.Z. Abidin, H. Andreas, M. Gamal, M.A. Kusuma Department of Geodetic Engineering, Institute of Technology Bandung, J1. Ganesha 10, Bandung 40132, Indonesia, E-mail : [email protected] M. Hendrasto, O. K. Suganda, M.A. Purbawinata Directorate of Vulcanology and Geological Hazard Mitigation, J1. Diponegoro 57, Bandung, Indonesia F. Kimata, Irwan Meilano Research Center for Seismology and Volcanology and Disaster Mitigation (RCSVDM), Nagoya University, Japan
Abstract. Indonesia has 129 active volcanoes.
With a population of approximately 200 million, and the fact that the most populated island in Indonesia (i.e. Java) has the largest number of active volcanoes, then it is obvious that the Indonesian people live under the very real threat of volcanic eruptions. Monitoring volcanic activities can be done by using several methods, of which one of them is the deformation method. In Indonesia volcano deformation has been monitored using Tiltmeter observations, EDM measurements, Levelling Surveys, and Repeated or Continuous GPS Survey techniques. This paper will describe and discuss the implementation status of the GPS survey method for deformation monitoring of Indonesian volcanoes, along with their limitations and prospects. The obtained results and experiences will be presented to illustrate the discussions. Keywords. Volcano deformation Indonesia, GPS survey
monitoring,
1 Introduction Indonesia has 129 active volcanoes and 271 eruption points as a consequence of interactions and collisions among several continental plates. With a population of around 200 million, and the fact that the most populated island in Indonesia (i.e. Java) has the largest number of active volcanoes, then it is obvious that the Indonesian people live under the very real threat of volcanic eruptions. According to (Katili & Siswowidjojo, 1994), around 10% of Indonesians live in areas endangered by volcanic eruptions, and several million of them live in the danger zones. This fact alone suggests that in Indonesia the monitoring of volcano activity should be performed not only
routinely, but should also be done as reliably as possible. In relation to the deformation of a volcano, it is already well known that explosive eruptions are usually preceeded by relatively large inflation of its body [Scarpa and Gasparini, 1996, Scarpa and Tilling, 1996]. In the case of a volcano that has been 'quiet' for sometimes, the deformation of its body is one of the reliable indicators of its reawakening phase. Moreover, according to Van der Laat (1996) and Dvorak & Dzurisin (1997), the deformation of a volcano body, represented by the point displacement vectors and their velocity vectors, could provide information on the characteristics and dynamics of the magma chamber. Monitoring the deformation of the volcano itself can be done using several methods. In Indonesia, volcano deformation has been monitored using Tiltmeter observations, EDM measurements, Levelling Surveys, and Repeated or Continuous GPS Survey techniques. This paper will describe and discuss the implementation status of the GPS survey method for deformation monitoring of the Indonesian volcanoes.
2 Deformation Monitoring of Indonesian Volcanoes Monitoring volcanoes in Indonesia is continuously and routinely conducted by the Directorate of Vulcanology and Geological Hazard Mitigation of Indonesia. Up to now, the seismic and visual methods are the most used methods for monitoring Indonesian volcanoes. The deformation method is also widely used. The Department of Geodetic Engineering, Institute of Technology Bandung (ITB), in cooperation with the Directorate of Volcanology
Chapter 113 • Volcano Deformation Monitoring in Indonesia: Status, Limitations and Prospects
and Geological Hazard Mitigation, has conducted several GPS surveys in order to study the deformation of several volcanoes in Indonesia. 105-"
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Processing of all GPS surveys data is done using BERNESE 4.2 scientific software (Beutler et al., 2001). Processing is done in radial mode from a certain reference station. The reference station is assumed to be stable for the deformation study, and its coordinates are computed in relation to Indonesian IGS stations in Bakosurtanal, Cibinong, Bogor. For all computations, the precise ephemeris is used, and residual tropospheric and residual ionospheric biases are estimated. All cycle ambiguities are successfully resolved and the final position solution is obtained using the narrow-lane signal [Hofmann-Wellenhof et al., 1994].
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1998c, 1998d, 2002, 2004, 2005). In the following pages some aspects that have been learned are presented and discussed. 3.1 D e s i g n i n g Network
the Volcano
Monitoring
Establishment of the GPS network for monitoring the volcano deformation consists of two main activities, i.e. choosing the locations for the points to be monitored, and establishing the monuments representing those points on the ground. In principle, the monitoring network should be designed in such a way so that the network geometry is good and reliable, the GPS surveys can be carried out in an efficient and effective manner, and the deformation signal and its characteristics can be revealed with high quality (low uncertainty). In designing the GPS monitoring network for volcano deformation monitoring and choosing the
791
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H.Z. Abidin • H. Andreas • M. Gamal • M. A. Kusuma • M. Hendrasto • O. K. Suganda • M. A. P u r b a w i n a t a • F. K i m a t a . I.
location of its monitored points, there are several factors that should be taken into account, namely: geological and geophysical structure and characteristics of the volcano and its surrounding area, history of volcanic activities, GPS-related observing requirements, location of the existing GPS points in the area, the existing sites of other monitoring techniques being implemented/used (e.g. EDM, levelling, tiltmeter, etc.), site accessibility, site stability, and pillar safety. However, rough topography and dense vegetation often has hampered the establishment of good quality monitoring networks. In such cases, geometrical optimization can not always be done and therefore the observation strategy has to be optimized, and combination with other monitoring techniques has to be undertaken.
3.2 Monumentation After the locations for GPS points have been decided then the next activity is to set up the monuments at those points. Since the volcano deformation is studied by observing the movement of the monuments, the monumentation has to be carefully established such that the erected monuments are only sensitive to the deformation signal of the volcano body, and not to localized movements caused by other phenomena. In order to fulfill that condition each monument should have a robust construction, placed on a stable and safe location, and have a strong and deep foundation, if possible reaching the bedrock beneath it. If in the selected location large and deep rooted rock is available at the surface, then rather than erecting the monument, the stone could be drilled and the GPS point established there. In erecting GPS monuments in a volcano environment, the main problem is generally related to the transportation of materials such as stone, sand, cement and water to the monumentation sites. For relatively difficult sites, such as on the summit and flank of the volcano, monumentation construction would require much local labours and could be quite time consuming. For example in the case of the Guntur volcano, erecting two monuments, i.e. on the summit and the flank, required about 20 local labourers and one working day. In addition several days are required for concrete drying and strengthening of the monument. Therefore the monumentation process should be done at least several days before the first survey is conducted, and, if possible, it should not be done in rainy season.
Meilano
3.3 Planning and Executing GPS Surveys in a Volcano Environment In planning and executing GPS surveys in a volcano environment there are several factors that should be taken into account, namely the timing and length of the GPS campaign, time interval between surveys, the length of an observation session, the number of teams and their members, movement of the observing teams between sites, communication method among the teams, and the availability of power supply, logistics and accommodation, local labours, and supporting tools and equipments (e.g. tents, flashlights, sleeping bags, etc.). In performing a GPS survey in the unfriendly environment of a volcano, the survey should be preferably done when the weather conditions are relatively good and the activity of the volcano poses no danger for the survey team. However it should be noted that the weather in a volcanic area is sometimes quite unpredictable. Therefore in carrying out the survey in a volcanic area, the survey personnel should always be well equipped to overcome the worst possible weather conditions. In monitoring volcano deformation, the time interval between two consecutive GPS surveys has to be synchronized with the activity level of that volcano. Determining the most optimal time interval between two surveys, however, is not an easy task. Besides the fact that the character of a volcano is difficult to precisely predict, the rate of volcano deformation is also not always linear. The results provided by other monitoring techniques, e.g. seismic, EDM, levelling, tiltmeter, should also be taken into account in deciding the optimal time interval between GPS surveys. For each GPS survey, the campaign length should be appropriate for the required accuracy of coordinates, the number of points in the network, the number of available GPS receivers, the travel times required for team movement between points, and also the available budget.
3.4 Some Issues in Data Processing The quality of the coordinates of the monitored points derived by GPS survey, which in turn would dictate the achievable quality of the deformation information, would strongly depend on the data processing technique and strategy being implemented. The higher the quality being sought, the more stringent the strategy and technique that should be used.
Chapter 113 • Volcano Deformation Monitoring in indonesia: Status, Limitations and Prospects
For data processing of GPS surveys, there are some issues that are worthwhile elaborating, such as commercial vs. scientific software, single- vs. dual-frequency solutions, static vs. kinematic solutions, broadcast vs. precise ephemeris solutions, radial vs. network solutions, the effects of tropospheric bias, and the effects of multipath.
network is relatively small (i.e. a few kms in size), or the volcano deformation signal to be detected is of the level of several cm or more.
3.5 Single vs. Dual Frequency Solutions The obtained monitoring results for several volcanoes shows that in general the differences between single- and dual-frequency solutions for epoch-by-epoch solutions are of the order of a few cm. However, whenever a new satellite is rising and its related ambiguity could not be resolved correctly, the differences could go up to 1-2 dm, as illustrated by an example shown in Figure 2 between epochs 300 and 340.
Based on the authors' experiences, it can be concluded that for data processing of GPS surveys intended for volcano deformation monitoring, which requires a relatively high accuracy, then it is advisable to use a scientific software package rather than the commercial software provided by GPS instrument manufacturers. Commercial GPS software might be used when the monitoring 25 E
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quite significant in the case of GPS monitoring in a volcano area, where it can be expected that the altitude of stations have relatively large variations between each other and the meteorological conditions will be different from station to station, as depicted in Figure 3. Moreover in the case of an eruption during the survey, the eruption column will also contribute another tropospheric bias effect.
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delay on the point displacements could reach up to a few cm. In the case of vertical displacements, the effects usually become larger with an increase in station altitude, as illustrated by an example shown in Figure 4. These results are from GPS
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3.7 Multipath in a Volcanic Environment GPS observations in volcanic environment are also prone to multipath, as illustrated in Figure 5. The GPS signals can be reflected either by cliffs, caldera rim, surface of crater lake, volcano flank, or other reflective objects in the area.
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Chapter 113 • Volcano Deformation Monitoring in Indonesia: Status, Limitations and Prospects
3.8 G P S - d e r i v e d Results
Figure 6 shows an example of the horizontal and vertical displacements of GPS points in Bromo volcano, which is derived from two GPS surveys conducted in February 2001 and June 2002. These displacements indicate the inflation of Bromo volcano during that observed period.
In studying the characteristics of volcano deformation, there are several important variables that can be provided by GPS, namely: horizontal and vertical displacements, horizontal distance changes, temporal height variations and pressure source characteristic. F
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Horizontal distance changes derived from the corresponding GPS coordinates are also the important variables to be examined in the case of volcano deformation. In comparison with EDM measurement, GPS survey is more flexible, since
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Fig.7 GPS-derived horizontal dista~e cha~es (Bromo aM Papar~ya~ volcanoes). Figure 7 shows the distance changes from Bromo volcano (left Figure) and Papandayan volcano (right Figure). In this case, Bromo represents the normal active period and for Papandayan the observed period contain the eruption events of November 2002. The results shown in these Figure suggest that the 2002 Papandayan eruption had introduced the inflation of area around the
crater in the order of 1 to 2 dm. The area closer to the crater experienced the largest displacement. The fact that the changes were observed on June 2003, which is about 6 months after the eruption period, shows the viscoelastic nature of the area around Papandayan crater. If its data processed in kinematic mode, GPS survey can also reveal the inflation process before
795
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H.Z. Abidin • H. Andreas • M. Gamal • M. A. Kusuma • M. Hendrasto • O. K. Suganda • M. A. Purbawinata • F. K i m a t a . I.
the eruption. The following Figure 8 show the temporal height variation of GPS stations prior to November 2 0 th Papandayan erup-tion which indicate the inflation of the stations. The results
Meilano
suggested that the avai-lability of GPS continuous moni-toring system can be very useful in anticipating the coming of large eruption event of a volcano.
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3.9 Interpretation of the Results In order to gain a correct picture of volcano deformation, the interpretation of displacement vectors derived by GPS should be done by simultaneously considering the geological and geophysical characteristics, and structures of the volcano and its surrounding area. In addition, the data and information provided by other monitoring techniques, e.g. seismic, EDM, levelling, tiltmeter, etc., should be taken into account and somehow should be correlated with information provided by GPS. In this regard there are several issues that are interesting to be elaborated, such as : • how to derive the physical meaning of the computed GPS vector displacements? • how to determine the characteristics of the magma chamber from the surface vector displacements? • how to correlate the GPS-derived information with information from other monitoring methods? The other challenge is to determine the magma chamber characteristics, i.e. its depth, size, shape, and supply rate, from the ground displacement
vectors of the points located on the body of volcano and its surrounding area. In this case, there are several things that have to be defined, namely the model to be used for transforming the displacement vectors into the characteristics of the magma chamber, the parameters of which can properly describe that model, and the estimation mechanism for computing those parameters. Pressure source modelling of volcano deformation actually can be based on several models (Mogi, 1958; Okada, 1985; Trasatti, 2003). It should be noted in this case that the Mogi model is by far the most widely used, and the simplest model to fit ground deformations in volcanic areas. The model assumes that the Earth's crust consists of an elastic half-space, the source of deformation is small and spherical as a point-like-source with radial expansion, and it exerts hydrostatic pressure on the surrounding rocks. Whilst none of those assumptions strictly apply, many volcanoes show deformation patterns close to those predicted for the Mogi theoretical model (McGuire et al., 1995). Figure 9 show an example of the pressure location of Bromo, Guntur and Kelut volcanoes based on Mogi model, as estimated using GPS-derived displacement vectors.
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Closing
Volcano Deformation Monitoring in Indonesia: Status, Limitations and Prospects
reliable method for studying and monitoring volcano deformation. The method is capable of detecting a deformation signal that has a relatively small magnitude, of the order of a few cm, or even several mm (although achieving this level of accuracy is not an easy task). In this case the use of dual-frequency geodetic-type GPS receivers is mandatory, along with good survey planning, careful observation strategy, and stringent data processing methodology using scientific GPS data processing software. Considering its relatively high accuracy, all-time weather-independent operational capability, wide spatial coverage, and its user friendliness, the use of repeated GPS surveys for volcano deformation monitoring is highly recommended.
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Based on the results obtained from deformation monitoring of several volcanoes in Indonesia, it can be concluded that the GPS survey method is a
Besides the technical aspects in using repeated GPS surveys there are some non-technical issues which have to also be addressed. Based on the authors' experience gained from conducting surveys on some Indonesian volcanoes, operational issues such as team movement, availability of sufficient power supply and local labourers, preparation of logistics and accommodation for survey personnel, and communication mechanisms among survey teams, are a real challenge. The unfriendly and harsh environment of a volcano also should be taken into consideration when selecting the survey team members. Although GPS surveys can provide accurate ground displacement vectors, in order to gain better and more detail information on the volcano deformation characteristics, the GPS survey method should be integrated wherever possible with other monitoring techniques such as EDM, levelling, tiltmeter measurements and InSAR (Interferometric Synthetic Aperture Radar) (Massonnet and Feigl, 1998). By increasing the available data and information, more reliable deformation and pressure source modelling can be obtained. References
Abidin,H.Z., I. Meilano, O.K. Suganda, M.A. Kusuma, D. Muhardi, O. Yolanda, B. Setyadji, R.Sukhyar, J. Kahar, T. Tanaka (1998a). Monitoring the deformation of Guntur Volcano using repeated GPS survey method. Procs XXI International Congress of FIG, Commission 5 : Positioning and Measurements, Brighton, U.K., 19-25 July, 153-169. Abidin,H.Z., O.K. Suganda, I. Meilano, M.A. Kusuma, B. Setyadji, R. Sukhyar, J. Kahar, T. Ta-naka, C. Rizos (1998b). Deformation monitoring of Indonesian volcanoes using repeated GPS survey method: Status and plan. Proceedings of Symposium on Japan-Indonesia IDNDR
797
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Project Volcanology, Tectonics, Flood, and Sediment Hazards, Bandung, 21-23 September, 3 9 - 50. Abidin,H.Z. (1998c). Monitoring the deformation of volcanoes in Indonesia using repeated GPS survey method: status, results and future plan. Proceedings of the International Workshop on Advances in GPS Deformation Monitoring, ISBN: 1-86308-073-2, Curtin University, Perth, Australia, 24-25 September, Paper no. 18. Abidin,H.Z., I. Meilano, O.K. Suganda, M.A. Kusuma, O. Yolanda, D. Muhardi, B. Setyadji, R. Sukhyar, J. Kahar, T. Tanaka (1998d). Some aspects of volcano deformation monitoring using repeated GPS surveys. Proceedings of Spatial Information Science, Technology and Its Applications, Wuhan, China, 13-16 December, 92- 109. Abidin, H.Z., M. Hendrasto, H. Andreas, M. Gamal, F. Kimata, O.K. Suganda, M.A. Kusuma, M.A. Purbawinata (2002). "Studying the deformation of Bromo Volcano by using GPS and EDM surveys. Program and abstracts of International Symposium on Geod-esy in Kanazawa and 98th Meeting of the Geodetic Society of Japan, Kanazawa, 28-30 October, Paper I10, 19-20. Abidin, H.Z., H. Andreas, M. Gamal, M. Hendrasto, Ony K. Suganda, M.A. Purbawinata, Irwan Meilano, and F. Kimata (2004). "The deformation of Bromo Volcano as detected by GPS surveys method". Journal of Global Positioning Systems, Vol.3, No. 1-2, 16-24. Abidin, H.Z., H. Andreas, M. Gamal, O.K. Suganda, I. Meilano, M. Hendrasto, M.A. Kusuma, D. Darmawan, M.A. Purbawinata, A.D. Wirakusumah, F. Kimata (2005). Ground deformation of Papandayan Volcano before, during and after eruption 2002 as detected by GPS surveys. GPS Solutions. Accepted for publication Beutler, G., H. Bock, E. Brockmann, R. Dach, P. Fridez, W. Gurmer, U. Hugentobler, D. Ineichen, J. Johnson, M. Meindl, L. Mervant, M. Rothacher, S. Schaer, T. Springer, R. Weber (2001). Bernese GPS software version 4.2. U. Hugentobler, S. Schaer, P. Fridez (Eds.), Astronomical institute, University of Berne, Switzerland, 515 pp.
Dvorak, J.J. and D. Dzurisin (1997). Volcano geodesy : The search for magma reservoirs and the formation of eruptive vents. Review of Geophysics, Vol. 35, No. 3,343- 384 DVMBG (Direktorat Vulkanologi dan Mitigasi Bencana Geologi) (2003). Situs internet, alamat : http://www.vsi.edsm.go.id, September 2005. Hofmann-Wellenhof, B., H. Lichtenegger, and J. Collins (1994). Global Positioning System, Theory and Practice. Third, Revised Edition, Springer Verlag, Wien. Massonnet, D. and K i . Feigl (1998). "Radar interferometry and its application to changes in the earth's surface. Reviews of Geophysics, Vol. 36, No. 4, 441-500. McGuire B.. C.R.J. Kilburn. and J. Murray (Eds) (1995) Monitoring Active Volcanoes. UCL Press Limited. London. 421 pp. Mogi, K. (1958). "Relations between the eruptions of various volcanoes and the deformation of the ground surfaces around them". Bulletin of Earthquake Research Institute Uniiversity of Tokyo, Vol. 36, 99-134. Okada, Y. (1985). "Surface deformation due to shear and tensile faults in a half-space". Bulletin Seismological Society of America, Vol. 75, 1135-1154. Scarpa, R. & P. Gasparini (1996). "A review of volcano geophysics and volcano-monitoring methods. In "Monitoring and Mitigation of Volcano Hazards" by R. Scarpa & R.I. Tilling (Eds.), Springer Verlag, Berlin, 3-22. Scarpa. R. & R.I. Tilling (Eds.) (1996). Monitoring and Mitigation of Volcano Hazards. Springer Verlag. Berlin. 841pp. Trasatti, E., C. Giunchi, M. Bonafede (2003). ,,Effects of topography and rheological layering on ground deformation in volcanic regions" Journal of Volcanology and Geothermal Research, Vol. 122, .89-110. Van der Laat, R. (1996). "Ground-deformation methods and results. In "Monitoring and Mitigation of Volcano Hazards" by R. Scarpa and R.I. Tilling (Eds.), Springer Verlag, Berlin, 147 - 168.
Chapter 114
Vehicle Classification and Traffic Flow Estimation from Airborne LiDAR/CCD Data Dorota A. Grejner-Brzezinska 1, Charles K. Toth 2, Shahram Moafipoor 1, Eva Paska 1, Nora Csanyi 1 ~Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State University, 2070 Nell Avenue, Columbus, Ohio 43210 2Center for Mapping, The Ohio State University, 1216 Kinnear Road, Columbus, Ohio 43212 Abstract. This paper provides a review of a 3-year
research program on the feasibility of using airborne LiDAR (Light Detection and Ranging) and imagery collected simultaneously over transportation corridors for estimation of traffic flow parameters such as: (1) vehicle counts, (2) vehicle classification, (3) velocity per vehicle category, and (4) intersection movement patterns. This work is conducted by The National Consortium for Remote Sensing in Transportation-Flows (NCRST-F), led by The Ohio State University, supported by the U.S. Department of Transportation and the National Aeronautics and Space Administration (NASA). The major focus is on improving the efficiency of transportation systems by integration of remotely sensed data with traditional ground data to monitor and manage traffic flows. The two primary research activities presented in our earlier publications were the accuracy and reliability of the sensor geolocation/georegistration and the image processing and object modelling. This paper is a summary overview of the NCRST-F research accomplishments over that past three years. The emphasis is on the demonstration of the operational/processing capabilities, while the algorithmic developments and implementation were published earlier in several papers cited here. Recent enhancements in spatial and temporal resolution of LiDAR data can now allow for effective detection and tracking of moving objects. Thus, in this paper, a special focus is on the LiDARbased traffic flow extraction techniques. The methodology of extracting vehicle information together with the road surface modelling supported by precisely georeferenced LiDAR data augmented by LiDAR intensity information is discussed. We demonstrate that algorithms developed under the NCRST-F program are capable of fast and robust identification of the vehicles by their shape (especially the vertical profiles), proving LiDAR's ability to preserve the geometry of a moving object better as compared to conventional imaging, where the projection from 2D to 3D can significantly distort shape in the image domain. We demonstrate that if
LiDAR data of sufficient spatial density are available, vehicle extraction and their classification can be effectively performed simultaneously with efficient and automated road surface extraction and modelling. It is shown, however, that for better accuracy and reliability, a fusion of LiDAR with frame image data is desirable. The examples of the vehicle extraction and the road surface modelling are based on two high-density (2-4 points/m 2) LiDAR datasets collected on February 19, 2004 over the downtown Toronto area, and on December 2, 2004 over the U.S. Route 40 in the Madison County, OH calibration range, with the Optech ALTM 30/70 LiDAR system. High-accuracy georegistration of the LiDAR data was provided by the Applanix POS AV system (http://www.applanix.com/). Keywords. Traffic flows, LiDAR, remote sensing, data fusion
1 Introduction Transportation represents a major pillar of any developed economy. Road traffic, measured in various parameters of passenger and cargo volume, has been steadily increasing worldwide for several decades. For example, the total vehiclemiles travelled in a year, the most frequently quoted traffic parameter that includes travelled distances for all vehicle categories, has almost doubled in the US in the last 15 years (US Bureau of Transportation Statistics, http://www.bts.gov). Moreover, the continuous increase of traffic is much faster than the population growth in the US, which reached about 20% during the same 15-year period. Furthermore, the paved road network grew only a modest 10% for the same period; thus the increasing traffic on the practically stagnating road infrastructure results in more severe traffic congestion. Federal and local government transportation management services monitor and control the
800
D.A. Grejner-Brzezinska• C. K. Toth. S. Moafipoor. E. Paska• N. Csanyi
traffic over the urban road network and the nation's highway system. These agencies collect data for both long-term planning and real-time traffic control. Real-time information is usually gathered from many sources, such as electronic sensors in the pavement (loop detectors), road tubes, ramp meter sensors, and video and digital cameras, which are sent to the traffic management centre at various times. Most of this information is only recorded; a small part of it is analyzed in real-time and used for immediate traffic control and decision making. Commonly, the density and flow of traffic are the two main parameters for describing the traffic stream. Namely, the density is the number of vehicles occupying a road lane per unit length at a given time, while traffic flow represents the amount of vehicles travelling over a road segment in a given time period. With the increasing number of vehicles entering the existing transportation network annually, the effectiveness of traffic management is becoming more crucial, simply because new road construction does not keep up with the growth of the traffic volume. The key to better traffic management, however, is access to better and more complete data and, of course, the capability for immediate processing of these data to provide a real-time response. Hence the interest in the new sensors that can provide large volumes of data in (near) real-time is steadily growing. Currently, loop detectors produce the largest amount of traffic data (Burns and Wendt, 2003). Although, this sensor technology is well established, the installation and maintenance are not simple, and the associated cost could be also high. Recently, more and more imaging-based systems have been introduced. These pole- or bridgemounted cameras have shown significant performance improvements and, for example, they are used to calibrate loop-detectors. The trend of switching toward imaging technologies is expected to increase. The routine use of airborne and/or spaceborne remote sensing technology is the expected next step in traffic monitoring and management, as it can provide data with large spatial extent and varying temporal resolutions for almost any area of interest, as compared to the location-based sensing. Clearly, the distinctive characteristics of using remote sensing is that it can be deployed virtually anytime and anywhere - a definite advantage over the local sensors. This paper provides a summary overview of the results of a multi-year feasibility study on using modern imaging sensors in airborne platforms to derive traffic flow parameters. While our earlier publications were focused on the development and performance analysis of the GPS/IMU-based
(Inertial Measurement Unit) sensor georegistration, calibration and modelling (see, for example, Grejner-Brzezinska et al., 1998; GrejnerBrzezinska and Toth, 2002), the most recent publications addressed the issue of image and LiDAR data processing, road and vehicle information extraction and classification, and traffic flow estimation (see, for example, Toth et al., 2003a and b; Ramprakash, 2003; GrejnerBrzezinska and Toth, 2003a and b; Toth and Grejner-Brzezinska 2004a and b; Toth et al., 2004; Paska and Toth, 2004 and 2005). 2
Airborne
Flow
Imaging
Sensors
for Traffic
Extraction
Airborne mapping and surveying witnessed phenomenal changes in the past five years. For example, introduction of GPS/IMU-based direct georeferencing represents a fundamentally different approach to image sensor orientation, which when combined with all-digital imaging sensors has led to a paradigm shift in mapping processes. The new all-digital sensors include the large-format digital mapping cameras and the new active imaging sensors, such as LiDAR (Light Detection and Ranging) and InSAR (Interferometric Synthetic Aperture Radar). These technological developments related to airborne remote sensing resulted for the first time in a totally digital system design, which, in turn, provided the foundation for high-level automation of data extraction and processing, and thus created the potential for real-time (or near real-time) operations. GPS/IMU is the backbone of any real-time imaging system. Naturally the quality of image georegistration is crucial, and is a function of various factors, such as the accuracy, continuity and reliability of the GPS/IMU solution, the accuracy of sensor calibration, the mission parameters, the accuracy of the LiDAR range measurement, image processing accuracy, etc. Table 1 presents examples of the achievable accuracy of a ground feature, as a function of the accuracy of geolocation and sensor intercalibration (boresight), and flight altitude, while Table 2 illustrates the LiDAR point accuracy on the ground as a function of geolocation and boresight calibration accuracies at a flight altitude of 600m, that is typical of the transportationoriented LiDAR applications. In order to extract vehicles and to estimate traffic flows, the most obvious choice from the new totally digital sensor suite was the use of CCD-based optical frame imagery, as discussed in
Chapter 114 • Vehicle Classification and Traffic Flow Estimation from Airborne Lidar/CCD Data
(Toth et al., 2003). Digital cameras, in general, provide good traffic data for visual interpretation, but the automated vehicle extraction and processing represents a difficult task, as brightness fluctuations, shadows, varying sun angle, camera pose or image orientation, etc., can significantly impact the performance of an automated vehicle extraction system. Table 1. Ground feature accuracy as a function of geolocation and boresight calibration accuracies and flight altitude for a 4k by 4k sensor with a focal length of 55 mm (assumed 9 btm error), and assumed error in image measurement of 5 gm.
Navigation errors
Position [cm] (3" Attitudc
H = 300 m
H = 600 m
H = 1000 m
10
20
10
20
30
c~ [arc sec]
10
20
30
RMS X [m]
0.063
0.116
0.211 0.245
[arc sec] Boresight misalignment
5
R M S Y [m]
0.073
0.132
RMS Z [m]
0.306
0.578
1.136
RMS RMS RMS RMS RIMS
[m] [m] [m] [m] [m]
0.089 0.103 0.407 0.145 0.162
0.143 0.167 0.686 0.207 0.229
0.256 0.283 1.211 0.321 0.350
R M S Z [m]
0.600
0.900
1.443
X Y Z X Y
To eliminate most of the problems related to the arbitrary camera position in object space, and thus lessen the central projection effect on object shape, we developed a technique that is based on image orthorectification. The method was successfully tested on a helicopter platform in cooperation with the University of Arizona. The two important prerequisites of the method are the high-precision sensor orientation, which is provided by a direct GPS/IMU-based navigation solution of the platform motion, and the availability of good quality terrain/surface data (DEM, Digital Elevation Model) to support the orthorectification process, in these tests, the OSU AIMS T M hardware and software system was used (see, for example, GrejnerBrzezinska et al., 1998; and Toth and GrejnerBrzezinska, 1999). Figure 1 shows two images overlaid on each other that were taken from a helicopter hovering above a busy intersection next to the University of Arizona campus; the images were acquired at a 5-6 sec interval. Due to the orthorectification, the two images are almost identical (however, imperfections in the orthoprocess could introduce some errors), with the differences related only to the objects that moved between the two exposures. Thus, if objects that are expected to move, such as vehicles, are coloured red in one image and blue in another one,
and they will show in red (here, in black in the b/w image) and blue (here, in white in the b/w image) in the orthoimagery, unless they remained still, in which case they show in grey. Most importantly, the vehicle's shape in the horizontal footprint is preserved at true object scale in the orthorectified imagery. Table 2. LiDAR point accuracy on the ground as a function of geolocation and boresight calibration accuracies with sensor parameters of range measurement accuracy of 2 cm, beam divergence of 0.2 mrad, and the scan angle [3 error = (1/1000) x[3. H
[m]
cr.'o7
°-ro¢,,c
°ro~od,o
[m]
[. . . . . .
[. . . . . .
l
0.05
10
10
0.05
20
20
0.10
10
10
0.10
20
20
0.20
10
l0
0.20
20
20
0.20
30
30
600
1
,/~
RMxS
RMyS
[deg]
[m]
[m]
RMzS [m]
0 5 10 0 5 10 0 5 10 0 5 10 0 5 10 0 5 10 0 5 10
0.073 0.090 0.128 0.102 0.115 0.147 0.114 0.125 0.155 0.134 0.144 0.170 0.207 0.214 0.232 0.219 0.225 0.243 0.238 0.243 0.260
0.073 0.074 0.074 0.102 0.103 0.104 0.114 0.114 0.114 0.134 0.134 0.135 0.207 0.207 0.207 0.219 0.220 0.220 0.238 0.238 0.239
0.051 0.051 0.055 0.051 0.052 0.056 0.100 0.102 0.102 0.100 0.101 0.103 0.200 0.200 0.201 0.200 0.200 0.202 0.200 0.201 0.202
Thus, for overlapping images, the detection of moving vehicles (as well as any moving targets) can be accomplished by a simple image subtraction, as shown in Figure 1, while detection of non-moving vehicles is a much more complex task. Both processes can be supported by the available road geometry data, such as road centreline or edge lines. Test images acquired from a helicopter and fixed-winged aircraft were used to monitor traffic flow over the road segments, and to determine turning volumes at intersections. For example, Figure 2 shows the traffic flow dynamics at the intersection for a 2-minute interval. Results showed good performance for extracting moving vehicles (Grejner-Brzezinska et al, 2004; Paska and Toth, 2004; Paska and Toth, 2005). Vehicle tracking, however, still needs more research, as the implemented solution produced unreliable results, which is partially due to the slow image acquisition rate (-0.2 images/s) and/or lack of adequate overlap (Toth and Grejner-Brzezinska, 2004).
801
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D.A. Grejner-Brzezinska • C. K. Toth. S. Moafipoor. E. Paska • N. Csanyi
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t
-
"
-
[
•
•
o
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Figure 1. Detecting moving objects in the orthoimage domain. Balance of inbound and outbound traffic flow 15 10
extraction use of LiDAR, to more sophisticated feature extraction, such as building or vehicle extraction (Vosselman and Dijkman, 2001). It is fair to expect that transportation and other agencies will be deploying LiDAR systems over transportation corridors at an increasing rate in the future - mainly to support infrastructure mapping to create accurate surface information of highways and areas around highways. Primarily for engineering purposes, the road surface must be determined at sub-decimetre level accuracy. In general, the vehicles on the road represent obstructions to the LiDAR pulses sent to reflect off the pavement. Therefore, a substantial amount of processing must be devoted to "removing the vehicle signals." Rather than removing and discarding the signals, however, they can be turned into traffic flow information. In this way, LiDAR surveys devoted to surface extraction will soon be able to provide a valuable by-product with little or no additional effort, as large amounts of data are collected during corridor surveys over highways.
5
i ° >
Input LiDAR data
-5
-*,,,
-10
Intensity information
-15 -20
--,--East +
West
North
•
South I
Road Area Filtered I
Figure 2. In-bound and out-bound traffic flow of the selected intersection.
3 Vehicle Extraction from LiDAR LiDAR (or airborne laser scanning) systems became dominant players in high-precision spatial data acquisition in the late 1990's. Installed in aircraft and helicopters, these active sensor systems can deliver surface data at decimetre-level vertical accuracy in an almost totally automated way. In fact, this new technology has quickly established itself as the main source of surface information in commercial mapping. Despite the initially high price, these systems have made remarkable market penetration. LiDAR has a unique feature as it preserves shape in 3D (see a truck in Figure 4), and thus can support object extraction in a very robust manner. Recent technical and algorithmic advances have further improved the capabilities of this remote sensing technology. In particular, intensity data became available, usually on all four multiple returns, and the laser repetition rate has reached 100kHz. These developments provide an unprecedented point density on the ground, which, in turn, helps to accelerate the process of moving from simple surface
Road Definition Data- CAD/GIS
Road Edge Extraction Road Median Extraction Road Surface Modelling I Vehicle Extraction
Vehicle Parameterization
I
Vehicle Modelling and Parameter Analysis
Transformation to Feature Space
Classification in Feature Space
Velocity Estimates Based on Vehicle Category Data
Vehicle Data Augmented by Vehicle Category
Computation of Traffic Flow Data
Figure 3. LiDAR data processing architecture for road and vehicle extraction.
Chapter 114
• Vehicle Classification
In the NCRST-F program it was shown for the first time that civilian vehicles could be extracted from LiDAR data with good accuracy (Toth et. al., 2003c). Based on these encouraging initial results, additional algorithmic research was carried out to further the development of the concept that finally resulted in a prototype implementation. Figure 3 illustrates the architecture of the totally automated data processing, including four major phases: road boundary detection, vehicle extraction, vehicle classification, velocity estimates and flow data derivation. The road area extraction is based on a combination of range and intensity data. The processing of the range data includes a local analysis of surface data, in particular the examination of the road crossprofiles. The intensity data are first segmented, followed by the boundary fitting. Both processes are supported by additional information, such as centreline data available from CAD or GIS databases. Once the road surface areas have been approximated using the elevation and intensity (if available) data, a final consistency check using object space constraints should take place to determine and delineate the road. At this point, the road direction and width are approximately known, and the objective is to determine the edge lines of the road. As the changes in the road geometry are limited in the road direction, a similarity analysis is performed over smaller road segments with the length comparable to the road width. Figure 4 shows automatically extracted road surface boundaries of a divided highway (U.S. Route 40 in Madison County, Ohio). -
-
-
Figure 4. Results of road boundary extraction. Once the road boundaries are available, the vehicles can be extracted by a simple thresholding, since the segments of the road between the edge lines are approximated by a plane. To follow the changes
and Traffic Flow
Estimation from Airborne Lidar/CCD Data
in the road surface orientation, the thresholding scheme should be adaptive, which guarantees that candidate points representing a vehicle will have true perpendicular height values with respect to the actual road s u r f a c e - in this way the very same vehicle description is obtained regardless whether the road is horizontal or of steep grade. Figure 5 shows point clusters extracted as vehicle candidates. Note that besides the vehicles there are other extracted objects that are definitely not vehicles, such as vegetation or guardrails on the side, and thus should be removed during the subsequent processing. 0 )
Roadboundary Cross profiles 0 Ascendedobjects
Figure ft. Initial vehicle extraction. The point cloud of a vehicle can contain a varying number of points, mainly depending on the laser point density and the relative speed between the vehicle and the LiDAR sensor. The effect of the latter is more important and requires a parameterization of points that can, at least, partially reduce the effect of vehicle shortening and elongation due to the scanning-type of data acquisition, see Figure 10. The selection of parameters has a major impact on the classification process, in particular how reliably the different vehicle groups can be separated. The basic model was formed from six parameters, including the length and the width of the vehicle and four height values, representing an average height over four equal segments of the vehicle (further details on parameterizations can be found in Toth et a., 2003a; Toth and Grejner-Brzezinska, 2004a). To reduce the parameter space dimensionality and to eliminate the parameter correlation, Principal Component Analysis (PCA) was used. Using a 72vehicle training dataset, a PCA was formed. Using three major vehicle categories, cars, MUVs (Multipurpose Utility Vehicles) and trucks, the vehicles were represented in the classification domain defined by the two largest eigenvalues. As
803
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D.A. Grejner-Brzezinska • C. K. Toth. S. Moafipoor. E. Paska • N. Csanyi
seen in Figure 6, the groups (three vehicle and the classification mented by ruled-based
4~4[
vehicles are clustered in six categories in two directions) space can be reliably segboundaries.
parameters, is available, and thus can be used as assigned average length for the three vehicle categories.
Trucks Right side
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2.5
t-
@
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Left side (vehicles and LiDAR sensor move in the same direcUon)
+ +
% +
+
yassen~le
o
(a) 05 ~ 0
'~ 5
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20
25
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Figure 6. Vehicle distribution in the classification space using a 6-parameter vehicle model. -. .-=--~--;~--~.~ ~-..~:~--~.::g~;.~--`.?.--;-~--:-.~:~.----~`-~.-~`-~7----:-:~;.~--.--. -----~-~-~--22.\ -~~ ~. ~ ~ ~
The estimation of vehicle velocity is possible based on the LiDAR-sensed vehicle length. Due to the continuous scanning, the vehicles appear shorter or longer depending on their relative motion direction to the LiDAR motion. Equation (1) describes the relation between the actual size, s, and the LiDAR-measured size, m, of the vehicles moving in both directions: Vveh _-- m - s VLiDAR sin(0) m
V~eh = ~ s - m V~iDAe m
along
(1)
sin(0) against
where v~,~ is the velocity of the LiDAR platform, v~h is the vehicle velocity and 0 is the intersection angle of the LiDAR scan line and vehicle direction (usually small enough so that it can be ignored). The problem with this estimation is that the dimension parameters could have significant errors. Firstly, the LiDAR footprint has a non-negligible size. Then the LiDAR scanlines are separated by an even larger distance, which results in setting a lower limit for the accuracy of the length estimation. Secondly, the actual vehicle size is unknown; only the broad vehicle categories are determined in the classification phase. Therefore, only a size distribution is available for the computation. According to a study by Ramprakash (2003), the percentage of the three vehicle category market share in the USA, with the corresponding length and height
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.
.
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.
.
.
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.
(b) Figure 7. LiDAR target and the GPSVanT M moving on the road, as seen by the digital camera (a) and by the LiDAR (b). The Madison County, Ohio, test flight was dedicated to a performance validation and therefore included a "moving" target, the OSU Center for Mapping GPSVan TM (He et al., 1994), a vehicle equipped with high performance GPS/IMU hardware. This vehicle, shown in Figure 7a-b, was constantly moving in the test area and was mapped by both sensors in several flights over the same area. Using the accurate navigation data of the GPSVan TM, the performance of the velocity estimation was checked. Table 3 shows the results from this test flight, including the vehicle identification and classification performance, as well as the accuracy of the derived velocities. The table also lists the execution time on a mediumgrade PC. For vehicle classification, a 3-parameter vehicle model, approximating vehicle volume, was tested, in which case only the largest eigenvalue was used for classification. The Toronto, Ontario, test area included a busy highway corridor in the downtown area. Figure 8a
Chapter 114
•
Vehicle Classification and Traffic Flow
s h o w s the e l e v a t e d r o a d s t r u c t u r e e x t r a c t e d f r o m the L i D A R data. A s the r o a d s u r f a c e is n e a r h o r i z o n t a l , the v e h i c l e s are e a s i l y i d e n t i f i e d b a s e d o n their e l e v a t i o n . F i g u r e 8b s h o w s the r o a d b o u n d a r i e s o v e r l a i d o n the L i D A R data, a n d the e x t r a c t e d vehicles are grey-scale-coded based on the
Estimation
from Airborne Lidar/CCD Data
c l a s s i f i c a t i o n results. In this test t h e r e w a s no r e f e r e n c e for the v e l o c i t y v a l i d a t i o n , a n d t h u s o n l y statistical " w o r s t c a s e " a s s u m p t i o n s w e r e u s e d in the e v a l u a t i o n p r o c e s s . T h e traffic f l o w p a r a m e t e r s c a l c u l a t e d for e a c h lane, as w e l l as totals for b o t h d i r e c t i o n s , are listed in T a b l e 4.
(a) (b) Figure 8. LiDAR data over a busy freeway in the Toronto downtown area: (a) road surface and objects extracted, and (b) extracted and classified vehicles.
Table 3. Overall performance results for the Madison County, OH, tests. Strip number Str Str Str Str Str Str Str Str Str
5 12 9 11 13 14 8 4 7
Largest eigenvalue (PCA) [%] 100 99 99 100 100 100 96 100 99
Vehicle identification [%] 92+ 86+ 90+ 90+ 91+ 91+ 85+ 86+ 89+
Vehicle classification [%] 89+ 95+ 90+ 94+ 95+ 92+ 95+ 93+ 88+
True velocity [km/h] 92.6 85.6 71.9 80.0 84.2 64.8 85.0 89.2
Difference Velocity [kin/h] 21.7 5.3 15.6 24.3 -
Time [minutes] < < < < < < < < <
20 20 20 20 20 20 20 20 20
Table 4. Overall performance results for the Toronto, Ontario, tests; P- passenger car, M- multipurpose utility vehicle, T-truck.
Flight
Velocity ± error [km/h]
Vehicle
Density ± error [Vehicle/h]
Flow ± error [km/h]
i N
Left
,___.
Right
Z
70
P
M
T
P
M
T
Bridgel
19
1
0
3
2
0
Bridge2
22
2
2
4
6
1
Bridge3
18
1
1
5
1
0
Bridge4
13
0
1
6
1
0
117.02 9.1 114.9 27.8 120.7 23.8 74.5 12.7
123.9 13.8 123.9 6.6 77.7 11.7 90.9 25.6
208 0.2 158 0.2 153 0.2 219 0.2
59 0.2 79 0.1 63 0.4 174 0.3
24629 3595 17729 2596 18315 2124 19607 3285
7312 600 9790 375 5116 557 14029 1299
805
806
D.A. Grejner-Brzezinska• C. K.Toth. S. Moafipoor. E. Paska• N. Csanyi 4
Fusion
of LiDAR
with
Imagery
As discussed in the previous sections, both sensors are capable of providing vehicle counts and velocity estimates, but the quality varies. Since the sensors' limitations and strengths are complementary, their fusion may lead to a better traffic flow estimation. Therefore, the next step in our research is to combine the LiDAR outstanding vehicle extraction performance with the excellent velocity estimation of the optical imagery. To overcome the errors in the true vehicle length estimation in the LiDAR data due to generalization or possible misclassifications, the actual length of the vehicle must be determined from other sensory data, such as imagery collected simultaneously with the LiDAR data. Although a single image does not provide the absolute size information, the image may preserve the relative object size information, such as the width/height ratio of a vehicle. Although an extra effort, such as using an adequate matching technique, is required to identify the identical vehicles in the two datasets, the combination of the two datasets could eventually lead to an improved velocity estimation of the moving vehicles (Paska and Toth, 2005). Figure 9 shows the vehicles extracted from LiDAR data, as they are overlaid on an orthoimage formed from simultaneously acquired imagery. LiDAR vehicle points are represented in green and red, corresponding to the motion along or against the flying direction, respectively. For referencing, some static objects, such as one point on the centreline and points on the guardrail, are also marked in the figure. This figure illustrates: (1) the elongated (when vehicles are moving along the flight direction) and shortened (when vehicles are moving against the flight direction) lengths of the moving objects, as sensed by the LiDAR, and (2) the relationship between the corresponding vehicles on the imagery and in the LiDAR data. The matches of the corresponding vehicles in the two datasets are highlighted by rectangles with identical colours. Due to the different nature of the two data acquisition techniques, the continuous scanning mode of the LiDAR sensor and instantaneous capturing of the frame imagery, the locations as well as the shapes of the corresponding vehicles differ in the two datasets. The white triangle in Figure 9 shows the approximate location of the LiDAR beam when the image was taken. The vehicle velocity estimates, shown in Table 5, illustrate that the larger error was introduced by the incorrect vehicle length. The GPSVan T M has a true length of 5.5m, but it falls into the vehicle category with a class length value of 4.7m. This length could
be effectively decreased by the vehicle length estimation from the LiDAR-measured width by using the image measured width/length ratio. The statistics, shown for the cases when the vehicle and the aircraft traveled in the same direction (shaded area in Table 5), clearly indicate that the accuracy of the true length-based velocity estimation can be achieved with the combined LiDAR and image solution. The opposite direction case (white cells in Table 5) offers only a small improvement, with the estimated bias and variance of 2.39 and 1.73 (not shown in the table). However, it is still important, as it helps to obtain a better overall accuracy in velocity when the average velocity of a group of vehicles is computed. See (Paska and Toth, 2005) for further discussion on the error characteristics of the LiDAR-based length and velocity estimation 5 Summary
and Conclusions
An overview of the research results performed within the NCRST-F, focused on using airborne remote sensing based on state-of-the-art LiDAR and digital camera systems, was presented. Earlier publications, referenced in this paper, provided details of the algorithmic approach and the performance assessment of the geolocation and imaging components of the system. The research work performed over the three-year project duration proved that airborne remote sensing is capable of providing valuable traffic flow data that can effectively support traffic monitoring and management. In particular, LiDAR has proven to be a good source of vehicle extraction and for course classification, while digital imagery supports accurate velocity estimation. Our experiences with using LiDAR for obtaining traffic flow data show encouraging results. The conceptual framework and its prototype implementation proved that high-point density LiDAR can effectively support traffic monitoring and management by delivering a variety of traffic flow data. The proposed system represents an addon capability to the existing airborne LiDAR mapping of the road infrastructure. In particular, this technique extracts the vehicles as a byproduct during the process of road surface extraction and modeling. The recent introduction of reflectance information is expected to further improve the road extraction process, while the vehicle classification seems to be unaffected by the availability of intensity data. Naturally, the success of the information extraction with high geolocation accuracy is a function of the quality of the sensor orientation and inter-calibration. These topics were discussed in the references cited here.
Chapter 114 • Vehicle Classification and Traffic Flow Estimation from Airborne Lidar/CCD Data
In summary, the methods discussed here may seem complex, intellectually challenging and computer-intensive. Experiences with several larger datasets, however, showed a good performance of the OSU-developed batch process. Except for the non-negligible execution time, the results were consistently very good, clearly indicating that the method is ready for deployment. Therefore, the authors believe that the real value of the approach presented here is to demonstrate that survey/mapping data can be successfully used for applications different from the original purpose. In fact, the introduction of the technique into daily operations is mainly an institutional decision, namely, it is a question of cooperation between the mapping/surveying and traffic management departments. Moreover, this discussion, in a broader sense, addresses the problem of mapping moving objects, which is an emerging field of the geospatial information science. Obviously, transportation, and in particular traffic management, needs this data, but rapid/emergency mapping also demands this type of geospatial data acquisition and processing. Our investigations provide an insight into the difficulty of mapping moving objects and clearly indicate that only multisensory systems can adequately solve the problem of collecting high spatial and temporal resolution geospatial data in a preferable highly redundant manner.
Acknowledgements This research was partially supported by the N C R S T - F and by the Ohio Department of Transportation.
References Burns, S. G . - Wendt, J. J., 2003, Inductive Loop Detector Vehicle Signature Analysis, ITS Institute Advanced Transportation Technologies Seminar Series Chowdhury, M. A . - Sadek, A., 2003. Fundamentals of intelligent transportation system planning, Artech House Publishers. Csanyi, N. - Toth, C. - Grejner-Brzezinska, D. - Ray, J., 2005. Improving LiDAR Data Accuracy Using LiDARSpecific Ground Targets, ASPRS Annual Conference, Baltimore, MD, March 7-11, CD-ROM. Grejner-Brzezinska D. A., Da, R., Toth C., (1998): GPS Error Modeling and OTF Ambiguity Resolution for HighAccuracy GPS/INS Integrated System, Journal of Geodesy, 72(11), pp. 628-638. Grejner-Brzezinska, D. A. - Toth, C., 2002. Modern Remote Sensing Techniques Supporting Traffic Flow Estimates, Proceedings, ION GPS, September 24-27, CD ROM, pp. 2423-2433. Grejner-Brzezinska, D. A . - Toth, C., 2003a. Deriving Vehicle Topology and Attribute Information over
Transportation Corridors from LiDAR Data, Proceedings, ION 59th Annual Meeting, June 23-25, Albuquerque, New Mexico, CD ROM, pp. 404-410. Grejner-Brzezinska, D. A . - Toth, C., 2003b. Airborne Remote Sensing: Redefining a Paradigm of Traffic Flow Monitoring, Proceedings, ION GPS/GNSS, Sept 9-12, Portland, Oregon, CD ROM. He, G . P . - Novak, K . - Tang, W., 1994. The Accuracy of Features Positioned with the GPSVan, Syrup. ISPRS Comm. II Symposium, Vol. 30, Part 2, pp. 480-486. McCord, M. - Goel, P. - Jiang, Z. - Coifman, B. - Yang, Y. - Merry, C., 2002. Improving AADT and VDT Estimation With High-Resolution Satellite Imagery, Pecora 15/Land Satellite Information IV/ISPRS Commission I/FIEOS 2002 Conference Proceedings. Merry, C . J . - McCord, M . R . - Goel, P.K., 1999. Satellite data use for traffic monitoring, Proceedings of the 14th William T. Pecora Memorial Remote Sensing Symposium (CD-ROM), 6-10 December, Denver, Colorado, pp. 183190. Paska, E . - Toth, C., 2004. A Performance Analysis on Vehicle Detection from Remotely Sensed Imagery, Proceedings of the ASPRS Annual Conference, Denver, May 23-28, CD-ROM. Paska, E . - Toth, C., 2005. Vehicle Velocity Estimation from Airborne Imagery and LiDAR, ASPRS Annual Conference, Baltimore, Maryland, CD ROM. Pline, J.L., 1992. Transportation and Traffic Engineering H a n d b o o k , 4 th Edition, Prentice Hall. Ramprakash, V.L., 2003. Detection and estimation of Vehicular Movement on Highways using a LiDAR sensor, MS.C. Thesis, The Ohio State University. Toth C. K. and Grejner-Brzezinska D. A., (1999): Modern All-Digital Airborne Data Acquisition Systems; presented at 47 th Photogrammetric Week, Stuttgart, Germany, September 20-24. Toth, C. - Grejner-Brzezinska, D. - Lovas, T., 2003a. Traffic Flow Estimates from LiDAR Data, Proceedings, ASPRS Annual Conference, May 5-9, pp. 203-212, CD ROM. Toth, C. - Grejner-Brzezinska, D. A. - Merry, C., 2003b. Supporting Traffic Flow Management with HighDefinition Imagery, ISPRS Workshop on High Resolution Mapping from Space 2003, Hanover, Germany, October 68, CD ROM. Toth, C. - Grejner-Brzezinska, D., 2004a. Vehicle Classification from LiDAR Data to Support Traffic Flow Estimates, Proc. of 3rd International Symposium on Mobile Mapping Technology, Kunming, China, March 29-31, 2004, CD-ROM. Toth, C. - Grejner-Brzezinska, D. A., 2004b. Traffic Management With State-Of-The-Art Airborne Imaging Sensors, International Archives of Photogrammetry and Remote Sensing, Vol. XXXIV, Part B2, pp. 897-904, 2000pp. 848-843. Toth C . - Brzezinska, D . - Moafipoor, S., 2004. Precise Vehicle Topology and Road Surface Modeling Derived from Airborne LiDAR Data, The 60th Annual Meeting of ION 2004, Dayton, OH, June 7-9, 2004. CD-ROM. Toth C. and Grejner-Brzezinska D.: Traffic Flow Estimation From Airborne Imaging Sensors: A Performance Analysis, Proc. of Joint ISPRS Workshop on High Resolution Earth Imaging and Geospatial Information, Hannover, Germany, May 17-20, 2005, CD ROM.
807
808
D.A. Grejner-Brzezinska • C. K. Toth. S. Moafipoor. E. Paska • N. Csanyi
(a)
(b)
(c)
Figure 9. Vehicles extracted from the LiDAR data and overlaid on the orthoimage; (a) match o f corresponding vehicles in the two datasets is marked with identical colors. Also shown are vehicle elongation (b) and vehicle shortening (c). Table 5. Velocity estimation performance for various sensor settings for LiDAR-only and for combined LiDAR and image data; vehicle and LiDAR move in the opposite directions (white cells), vehicle and LiDAR move in the same directions (shaded cells). Vehicle velocity computed from different vehicle length measurements [m/s]
Im
=" -~ "~-~ [ =
Using vehicle class length (4.7 m)
Derived from LiDAR width using image ratio
True vehicle length (5.55 m)
Velocity
Difference
Velocity
Difference
25.71
-3.90
21.22
0.59
19.75
2.06
21.81
27.93
-4.16
23.18
0.59
23.85
-0.08
23.77
20.14
-0.18
17.51
2.45
33.31
-13.35
19.96
11
26.79
-4.58
21.48
0.73
21.94
0.27
22.21
13
26.16
-2.77
22.45
0.94
21.71
1.68
23.39
14
14.64
3.37
22.83
-4.82
26.85
-8.84
18.01
10
5.22
-5.15
1.33
- 1.26
2.85
-2.78
0.07
4
20.75
-5.28
15.55
-0.08
15.15
0.32
15.47
8
11.28
7.87
20.18
- 1.03
23.11
-3.96
19.15
12
Mean
Std
of the absolute differences when vehicle and LiDAR move in the same direction (shaded cells)
Velocity
GPS reference velocity [m/s]
Difference
4.14
0.58
0.88
0.93
0.32
0.91
Chapter 115
Fine analysis of lever arm effects in moving gravimetry B. de Saint-Jean, J. Verdun, H. Duquenne IGN-LAREG, 6/8 av. Blaise Pascal,Champs sur Marne, 77455 Marne la Vallde Cedex 2, France J.P. Barriot, S. Melachroinos CNES, 18 av. E. Belin, 31401 Toulouse cedex 9, France J. Cali ESGT, 1, Bd. Pythagore, 72000 Le Mans, France
Abstract. A
moving inertial gravimetric system is now being developed, consisting of three high precision accelerometers measuring accelerations along three non-parallel axes. The system has been designed to make high resolution gravity measurements on motor vehicles, ships or aircraft at mGal level of precision. Position, velocity and attitude of the platform, needed for computing acceleration corrections, are provided by a 4antenna GPS system rigidly mounted on the platform with a sampling rate of 1 or 2 Hz. Because acceleration and GPS measurements are not made at the same point, the signals derived from the accelerometers have to be corrected for lever arm effect. To this end, we derived the complete relationship connecting accelerometer and GPS measurements. The correction for the lever arm effect requires the computation of seven acceleration terms. The amplitude of the lever arm effect depends not only on the lever arm length, but also on the platform attitude. Our own findings based on both simulations and real measurements suggest that some of the seven lever arm terms have to be considered to reach a few mGals precision for gravity. We present a reliable method to assess the precision that can be achieved for lever arm effect determination given the precision of attitude and lever arm measurements. Keywords. Vector gravimetry, lever arm, accuracy analysis.
1 Introduction The recent development of both kinematic GPS systems and small size, high precision accelerometers makes it possible to design portable gravimetry systems suitable for geodetic and geophysical applications. As an illustration, airborne
gravimetry is particularly well-suited for the quick recovery of the Earth's gravity field over both inaccessible continental and oceanic regions (e.g. mountainous regions, deserts, deep forests such as the Amazonian forest, margins, etc) with a spatial resolution from 5 to 100 km, thus filling the gap between ground-based and spaceborne gravimetry techniques (Klingel6 et al., 1997; Verdun et al., 2003). When compared to classical shipborne or airborne scalar gravimetry systems, consisting of a stabilized vertical accelerometer (LaCoste et al., 1982; Valliant, 1991), the systems integrating GPS and accelerometers are significantly lighter and cheaper, and allow the determination of the entire gravity field that is, the three spatial components of the gravity vector (vector gravimetry). z~ z~ Aec.e!erometer ~ I(Up)Po~n
z~
/ / x .IF/~/Q~..m= " - X--
~
y
/
Y~
~ : center o f Inertial Frame [IF] O : center o f the Earth fixed Frame [EF] A : center o f the NavigationFrame [NF] center o f the Body (vehieule) Frame [BF] M : position o f one aecelerometer
Fig. 1 Definition of the reference frames used in the calculations. The Inertial Frame IF centred at point f2 is the reference frame where Newton's second law of motion can be applied. The Earth fixed Frame EF, centred at the Earth's centre of mass O, rotates relative to the IF frame with rotation v e c t o r (OEF/I/ . The Navigation Frame NF, centred at the vehicle's centre of mass A, moves with the vehicle and its three a x e s A X N F , A YNF, AZNF are respectively parallel to the local east, north and up directions. The Body Frame BF centred at the same point A, is fixed relative to the vehicle. The relative position of BF and NF reference frames defines the attitude of the vehicle, e.g. its pitch, roll and yaw. Point M, fixed in the BF frame, corresponds to the location of one accelerometer. Vector A M is called lever arm.
810
B. de Saint-Jean. J. Verdun. H. Duquenne. J. P. Barriot • S. Melachroinos. J. Cali
The determination directly on Newton's respect to an Inertial (Fig. 1) the total given by:
dZf2M dt 2
of the gravity field is based second law of motion. With Frame (IF), centred at point acceleration of a point M is
=G(M)-Fa(M),
(1)
~IF
where __...
• •
G(M) is the gravitational acceleration at point M, and F a (M) is the opposite of the restoring force by
mounted on the platform (Fig. 3), with a sampling rate of 1 or 2 Hz. The positions of each antenna are simultaneously determined, and then combined in order to derive the sensor platform attitude. By so doing, acceleration and position measurements are not located at the same point. Rather, acceleration is measured at point M, whereas the GPS system provides the positions of the master antenna of the GPS system located at point A, different from point M (Fig. 1). This lever arm gives rise to unavoidable additional acceleration terms which have to be taken into account, as correcting terms for accelerometer measurements.
unit of mass exerted by the accelerometers at point M, i.e. the acceleration actually sensed. In other words, the gravitational acceleration at point M can be simply calculated from the difference between the total acceleration, which can be derived from the GPS solution, e.g. the linear acceleration can be kinematically determined by differentiating GPS-derived positions twice, and the restoring acceleration measured by the accelerometers at the same point.
Fig 3 View of the 4 antenna GPS system used for positioning. Each GPS antenna is mounted at the end of a 0.5 m long arm. The 4 antenna can therefore be used to form 6 independent baselines which can be processed together to derive the
position, velocity and attitude of the vehicle. The system can be fixed on the case containing the accelerometer triad (fig. 2) so as to minimize the lever arm.
Fig 2 View of the triad of accelerometers. The accelerometers are mounted face to face to three plane faces of a regular pyramid. This permits their sensitive axes to be orthogonal to the pyramid faces. By so doing, the accelerometer axes can be made exactly orthogonal. Each accelerometer is connected to five wires carrying the supply voltage, and returning the two signals provided by the acceleration sensor and the internal temperature sensor, which are then digitized by an AD converter.
Our own system consists of three accelerometers (type QA3000, Q-Flex from Honeywell) forming an orthogonal trihedron (Fig. 2), measuring the acceleration components with a sampling rate up to 250 Hz. The positioning parameters of the platform, i.e. its position, velocity, and attitude, are provided by an independent 4-antenna GPS system rigidly
Our approach in this paper is first to derive the expression for the lever arm acceleration as a function of the platform's movement parameters and estimate its order of magnitude. Then, given the precision of GPS-derived movement parameters, we estimate the precision, which can be achieved for the calculation of the lever arm acceleration.
2 Lever Arm Acceleration For calculating the total acceleration in equation (1), as a function of movement parameters, vector E2M has to be broken up into three terms involving the Inertial Frame (IF), the Earth Frame (EF) and the Navigation Frame (NF) (Fig. 1), as follows:
f~M = f~O + ~
+AM,
(2)
Chapter 115 • Fine Analysis of Lever Arm Effects in Moving Gravimetry
The vector A M , named the lever arm, accounts for the distance separating point A (centre of the Navigation Frame given by the master GPS antenna) and point M where the acceleration is sensed. Equation (2) is then differentiated twice, taking care of relative rotations of BF, NF, E F and IF reference frames with respect to each other (Jekeli, 2001; Melachroinos, 2004): •
is the gravity vector at point M, and C(M,A)
-
{TERM I]
aft /ff /,,rf ,
EF/IF: rotation of the Earth Frame relative to the Inertial Frame
•
•
d O A
dt
NF/EF: rotation of the Navigation Frame relative to the Earth-fixed Frame
[TERM 4] + dco~l:/Nf: A AM dt / BF
BF/NF: rotation of the Body (platform) Frame relative to the Navigation Frame
[TERM 5] +
d (ONF / EF
-
dt
A AM / NF
The resulting equation contains 11 acceleration
[TERM6]+COL.,./H:A(COL.,./H. A A M )
terms, 8 of which depend on the lever arm A M (the symbol A is used to denote the cross product, see appendix for derivation):
. A AM [TERM 7] + COBF/xF A COB~/NF
d2~M df 2
z ~IF
.
[TERM2]+(COx~/E~ + 2coE~/~) A
/ EF
A AM / BF
d cox,,,/ L,,/ [TERM 5] + - /~ A M dt / NI: [TERM 6] + o)E,/~F A ( COEF/~~ A AA/])
(
[TERM8] + CON~/E~/~ COiv~/E~ A A M
)
[TEEM 91 + 2COEF/I;A ((J')BF/NFA A J ~ )
,(
, A AM [TERM 10 ] + 2CONF/ E,~ A COBF/NF
) B
(3)
By substituting the total acceleration given by equation (3) into equation (1), and regrouping term 3 with term 6, we obtain the equation giving the acceleration sensed by the accelerometers at point M:
F'(M)- g(M)-C(M,A), where
,
(6)
is the GPS-derived correction acceleration to be applied to the accelerometer measurements.
dOA dt
dt
)
[TERM 10] + 2coi,vF/ eF A COVF/NFA A
d dOA [TERM 11 -~- aft / f~ /~,F
[TERM 4] + dcov"/ivF
,(
(M) [TERM 11] + 2coee/~e(A cox~/e~' A AM)
I-I
/ EF
(4)
The first two terms correspond to the inertial accelerations induced by the general movement of the platform. Their vertical components are well known in airborne scalar gravimetry as the vertical and E6tv6s accelerations respectively (Harlan, 1968). The seven remaining terms (terms 2 to 11) come specifically from the lever arm, and four of them (terms 4, 7, 9, 10) depend explicitly on the attitude of the platform through the rotation vector of the Body Frame with respect to the Navigation Frame. The term 4 in equation (6) is known as the "lever arm correction". Using the open-source, symbolic computation software Maxima (Rand, 2005), we calculated its three components in the Navigation Frame as functions of the attitude angles ~ (yaw), 0 (pitch), ~b (roll) and their first and second derivatives. If the lever arm is longitudinal (i.e. parallel to the axis of the platform), the vertical component of the lever arm correction is surprisingly simple: V4,up - ( g cos 0 - ~ ¢ cos 2 0 ) A M ,
(7)
For an aircraft flying at 100 m/s, affected by a phugoid motion with amplitude of 0.005 radian in pitch and period of 15 seconds, this component can be as large as 100 mGal per metre of lever arm. To
811
812
B. de Saint-Jean. J. Verdun. H. Duquenne. J. P. Barriot • S. Melachroinos. J. Cali
compute this correction the attitude angles should be measured with an unrealistic precision, as we shall see in section 4. Fortunately, equation (7) shows that its mean value is probably zero and hence the residual errors affecting this term can be eliminated by filtering. The terms 5 to 11 in equation (6) are not generally accounted for. They are assumed to be negligible, or their mean value to be zero. The computation of these terms as functions of the attitude parameters and their derivatives shows that this is not always true. Let us for instance examine the term 7 (T 7 ), which is a centrifugal acceleration, the gyration radius of which is the lever arm. Applying the formula of the vector triple product, we find:
T7 --(AM'coB~/xF)OOBF/xe
-(co.,,/x.
a constant altitude. Besides negligible terms (amplitude below 1 mGal), term 4 and, to a lesser extent, term 7 in equation (6), have proved to be the major lever arm terms. It should be noticed that apart from the lever arm, these two terms depend only on the attitude variations. This suggests that when flight conditions are normal, the major disturbing lever arm acceleration is due to the aircraft attitude variations. As a consequence, terms 4 and 7 have to be very carefully estimated in order to reach a few mGals precision for airborne gravity surveys. This model can not be used when the aircraft trajectory is too far from a straight line.
Table 1 Upward component amplitudes of terms in equation (6) for an aircraft flying to the East with the attitude defined by equations (7). The lever arm is 1 metre long and is oriented in order to have the same component in each direction of the Body Frame (BF):
[1~,]-3(right side);1~,f 3( front);1/ ~r3(up)1
"co.:,./x,..) A M
•
(8)
For an aircraft flying at 100 m/s, affected by a phugoid motion with amplitude of 0.05 radian and period of 40 seconds, both the terms on the right side of equation (8) can reach 10 mGal, and their signs are constant. Hence the term 7 (or its vertical component) cannot be eliminated by filtering, but only by a favourable configuration (horizontal lever arm) or by explicit computation.
3 Lever Arm Effect Amplitude In order to identify the major terms coming from the lever arm effect, a simulation was carried out for an airborne gravimetry scenario. The aircraft movement parameters have been set using position and attitude measurements collected during several airborne gravimetry surveys (Boedecker et al., 1994; Bruton, 2000; Duquenne et al., 2003). The equations giving the aircraft attitude that come closest to explaining the observations are the following
yaw: ~(t) = 0.050(1 + b(t))cos(2zct / 40) pitch : O(t) = 0.005(1 + b(t)) cos(2zct / 15) , (9) roll : ~b(t)= 0.050(1 + b(t))cos(2~t / 40 + Jr / 4) where g 0, ~b are the yaw, pitch and roll angles in radians, respectively, and b is a Gaussian noise of zero mean and a standard deviation at 1/15 of the attitude angle amplitude, imitating atmospheric disturbances. in Table 1, the vertical component amplitudes of all the terms of equation (6) are given for an aircraft flying to the East with a speed of about 300 km/h at
Mean
Standard Min Max Deviation mGal mGal mGal mGal Term 1 0.000 0.000 0.000 0.000 Term 2 -1313.813 0 . 0 0 0 -1313.813 -1313.813 Term 4 1.289 106.671 -278.807 299.683 Term 5 0.000 0.000 0.000 0.000 Term 7 -0.801 1.462 -6.847 1.949 Term 8 0.000 0.000 0.000 0.000 Term 9 0.000 0.041 -0.108 0.099 Term 10 0.000 0.007 -0.019 0.018 Term 11 0.000 0.000 0.000 0.000
4 Precision of Lever Arm Effect Determination 4.1
Theory
Our aim is now to estimate the precision that can be achieved in determining the correcting terms of equation (6). We focus on terms 4 and 7 which have been identified as being the major terms, although our method can be carried out for any other term. Both terms 4 and 7 depend on 12 parameters, consisting of the attitude angles O(t), ~(t), ~(t), respectively, their first and second derivatives t)(t), ¢,(t), ¢~(t), ~/(t), ~(t), ~(t), respectively, and also the three components Be, Be, Bs of the vector AM. It should be noted that only lever arm components do not depend on time. Denoting term 4 and term 7 by T4 and T7 , respectively, we have" ;,
i,
,i-
4,7
Chapter 115 • Fine Analysis of Lever Arm Effects in Moving Gravimetry
with
similar multipath effects. However, this spatial correlation is diminished by considering baselines, which result from GPS position differences, rather than point-wise GPS data. This suggests that the spatial correlations between attitude angles are weaker than those affecting the GPS data used to determine these angles.
U = r] (o(t), o(t), O(t), ~(t),~(t),V,(t), O(t),~(t),~(t), BI ,B2,B3)
i = 4,7,
j = 1,2,3
Let Cm(t ) be the covariance matrix of the 12 component vector
at time t. Then, according to the law of variancecovariance propagation, the covariance matrices ____+
____+
Cr4 and Cr7 of the vectors T4 and T7, respectively,
may be expressed as:
where
CT4 (t) -- P~4Cm (t)P~ ,
(1 O)
Cr~ (t) - nr~ C,, ( t ) n / ,
(11)
P~,i=4,7
are
the
3×12
propagation
matrices of term 4 and term 7, given by:
~7~i 1
P~- ~
~77i 2
~/3
ax
ox
X - O,O,O,~/,~,i/),(),(b,~',B,,B2,B
The atmospheric disturbing effects on GPS signals vary very slowly compared with the GPS data sampling rate. It is likely that the GPS data, and thus the attitude angles, are also highly correlated in time. Hence, the time correlation of attitude angles is reduced by numerical differentiation, which suggests that the attitude angle derivatives are much less time-correlated than the attitude angles themselves. Accordingly, the decrease of time and space correlations between the attitude angles and their first and second derivatives gives rise to an increase of their individual variances. In order to overestimate those variances, we postulate that there is no correlation in space and in time between the attitude angles, and their first and second derivatives. Under this assumption, the covariance matrix C,, (t) may be expressed as
co (o) (o)(o)] c~(t) = 3.
(12)
i=4,7
(o) c~ (o)(o) ](o) (o) c~ (o)] [(o) (o)(o) c~]
where C o , C~,, C~ ,and C~ are the 3x3 covariance
Finding a reasonable estimate of the covariance matrix Cm(t) is rather difficult because the
matrices of the vectors
correlations in space and time between the attitude angles, yaw ~,, pitch 0, roll ~, their first and
and [B1,B2, B 3
second derivatives, ~,, ~),~, ~ , 0, ~, and the lever arm components, BI, B2, B3, have to be taken into account. Our aim is now to formulate a coherent rationale for the covariance matrix used, by means of qualitative considerations. First of all, the lever arm components are constant, and result from terrestrial precise measurements carried out once. They are therefore not correlated with attitude angles at all. The attitude angles at a given time, for their part, are determined by measuring the coordinates of baselines connecting the 4 antennas of the positioning system. Because the GPS antenna are close to each other, they receive highly correlated GPS signals, experiencing the same ionospheric and tropospheric perturbations, and, to a certain extent,
(13)
[ 0 , 0 , 0 ] r,
[~,~,~)]r,
respectively.
Covariance matrix CB is simply equal to:
CB - o - ~2i 3 , where
o-2
components matrix. Such
is
the
common
(14)
variance
of the
B1, Be, B3, and I3 is the 3x3 identity
covariance
matrices
as
Q,a=O,p',()
contain the variances of quantities
a, h,//
(the
diagonal terms), and also the covariances between each other (non-diagonal terms). The numerical computation of h at time t, can be performed, with sufficient precision for our application, using ~t - a(t + cYt) - a(t - cYt) , 26t
(15)
813
814
B. de Saint-Jean. J. Verdun. H. Duquenne • J. P. Barriot • S. Melachroinos • J. Cali
where 6t corresponds to a small, constant time increment• According to our previous assumptions, the quantities a at time t - fit and t + c~t, are postulated to be uncorrelated. In that case, the variance o-a2 of c~ may be expressed as a function of the variance O- a2 of a as" 2
2
o-a .
2 O"a
.
2
[21
can be
2
oa 4tit 4"
(17)
The covariance matrices Ca, a - 0,~,,~b may be
O"B2
2 o-~2 + 5.0x 10-9 o-,.
(22) (23)
In order to achieve 1 mOal precision for the determination of terms 4 and 7, o-r4,,, and crrT,,p plane, equations (22) and (23) represent two ellipses of semi-major axis and semi-minor axis for equation (22)" d.r4=
0.3x10 -5
B x/2.2 x 10 -6 = 2[ram] d.~ = 0.3 x 10 -5
finally expressed as" 1
c~=~
2"80-2 + 2.2 x 10 -6
have to be smaller than 0.3 mOal. In the (o-B,o-~)
1 oa 2dr 2 2b't 2
2b't 2
O"742,up --
(16)
deduced from o-2 and o-2 by: O"a
(21)
[m]
the standard deviations of the lever arm along the axis of NF (Navigation Frame), we obtain, for the variance of the vertical components (in the Navigation Frame) of terms 4 and 7:
0..2j~,,,p __ 8 . 9 X 1 0 - 5
2 O"a
. . . 46t 2 26t 2
As a consequence, the variance o-2 o f / /
2
O"B -- O-B1 -- O"B2 -- O"B3
o
0
0
~
o
2~t 2
0
.
(18)
(24)
and for equation (23):
1
0
= O. 18 x 10 -5 [rad] ~ 0.4"
&Br7 _ 0.3 x 10 -5 4.2
Numerical
The
Application
covariance
matrices
and
Cr~
Cr~
were
calculated using Maxima again. A numerical computation can then be performed using the attitude parameters values given in equation (9). Although these values apply only for airborne gravimetry surveys, they can be used to assess the influence of attitude variations compared with lever arm effects on term 4 end 7 uncertainties. Given the following values:
x / 5 x l 0 -9 = 42 [ram] &~i = 0.3x10 -5 x/8.5x10 -5 = 3.2 x 10 -4 [rad] ~ 66"
t ) - 0.005 0-0.0012
[iv///]
the respective conditions" o-1~,,,, < 0.3 reGal cry,,,p < 0.3 m G a l
[rad / s] [ r a d / s 2]
- g) - 0.050
[rad]
- ~' - 0.0070
[rad/s]
- ~ - 0.0070
[rad / S 2 ]
(19)
and denoting o-~ - o-o - o-O - o-~
[rad]
(26)
The results suggest that in the case of a lever arm exactly known (orB=0), the precision required for attitude angle determination to fulfil 1 reGal precision is 0.4" for term 4 and 66" for term 7.
[rad]
0 - 0.00087
(25)
These two ellipses define two regions in the (o- e, cra ) plane in which the couples (o-,, o-~ ) fulfil
• B 1 - B 2 - B 3 - 1/%/-3
•
(20)
the standard deviations of the attitude angles, and
Obviously, such precisions on angle determination are out of reach of any current navigation systems; for instance, our GPS system can measure angles at 0.2 ° precision. This suggests that 1 mGal precision on term 4 and term 7 cannot be achieved using only isolated, pointwise correction values. In contrast, filters are commonly used to eliminate zero-mean disturbing accelerations, as is
Chapter 115 • Fine Analysisof Lever Arm Effects in Moving Gravimetry
the Term 4 (but not the Term 7). This involves groups of successive samples, and not isolated, pointwise samples. In fact, filters are crucial for reducing the variance of noise, but at the same time, they unavoidably attenuate the target gravitational signal. Thus, a compromise has to be found between precision and resolution. 5
Conclusions
)
d 2 A ]t4 dt 2
d 2 A ]t4 d o) ,vF/ ,: = ~ + ~ ~IF d t 2 / xt: dt
2o)~w/w A
~
A (O)I\T/IF A A~dl)
+O)NF/IF
dt
A A2t4 + / M,
(A6)
ix,:
The first term of the right-hand side of (A6) is computed using (A2) and the Body Frame B F , in which M is hold fixed: )
The method presented defines crucial values for the determination of the attitude of the gravity measuring system. Among the two terms studied, term 4 sets the precision needed for the platform's attitude. A reduction in the length of the lever arm will allow a larger tolerance. Thus, a lever arm of 20 cm would increase the value of d-24 by a factor of three. This tolerance should be significantly increased by means of suitable filtering methods. More work has to be done, since this study is based on navigation models and it has to be tested on field values.
d2 A M dt 2
We give the derivation of equation (3), which expresses the acceleration of a proof mass M with respect to an inertial frame. The first and second derivatives of any vector a with respect to a frame F1 are computed from its derivatives with respect to a frame F2 by the basic formulas: dg
dg
,
= --
d t /FI
d2~
d2~ = - -
d t 2 /FI
d t 2 /F2 ,
+2o)~:2//: ~ A
(A1)
) d o ) F2/F 1 + - -
dl
d~
AO
/p2 ,(
+ O) I/2/FI A
d t /F2
)
)
)
) (AS)
We derive (A8) with respect to the navigation frame" )
)
d o ) NF/W
)
do) eHw /N~
dt
d o ) :vF/e~ /x~
dt
/N~
We transform the first term of the right-hand side by (A1), considering the earth rotation a constant, and we multiply by Aa/l. We obtain the second term of the right-hand side of (A6):
.)
(
do)NF / I/ dt
/ NF
do)NF / E~ ~
A
(A9)
AM
/ NF
To compute the third term of the right-hand side of (A6), we split up the rotation vector by (A8) and we transform the derivative using (A1), regarding
(A2) #Aag
dt
.
.
/ NF
+ 2o)Ne/2 A (O)ee/Ne AAM)
)
(al0)
(A3)
and the formulas of the double cross product:
(a
)
CO NF/IF -- o ) EF/IF + o ) NF/EF
We will also make use of the addition formula for instantaneous rotation vectors: O) R3/RI -- (O R3/R2 -lt- (-01¢2/1¢1
O) BF I NF A
AM as a constant vector with respect to the body frame"
, ) (O I:2/FI A CI
A AM / ut,
Applying (A3) to the instantaneous rotation vector of the Navigation Frame N F with respect to the Inertial Frame I F , we get:
dt
d t /F2
dt
(a7)
+
+ CO F2/F1 A 0
/ xt,
+ O) BF I NF A
dt
Appendix
- do)BF/"rF
(A4)
The last term of the right-hand side of (A6) is developed using (A3) and the distributivity property: O)NF //F A
(
(
+(OEF/I F A
From (A2), the second derivative of the lever arm vector AM with respect to the inertial frame I F is"
v) '
(_ONF//F A A
ONF/E F A A]~J
- O)EF //F A
)
(
+ONF/E F A
O)EF //F A A
(
(_OEF/1F A
M) A
M)
815
816
B. de Saint-Jean. J. Verdun. H. Duquenne. J. P. Barriot • S. Melachroinos. J. Cali
(A7), (A9),
Putting leads to:
a ~A M df 2
=
aco~,i( ~
~IF
/ BF
O)BF/NF
A
Combining this equation with the well known equation of the vector moving gravimetry"
AAM
dt
(
+O)BF/N F
(A6)
(A10) and ( A l l ) into
m
A
act2 / ,F
M)(
-Jr- (-f)EF/NF
)
COEF/IF
A
dt
dt
/ EF / x~:
AM
A
+ (2COEF/,F + CO,;~/EF) A dO/i dt
-~ dco'vF/EF A AI]/[ + 2COEF/,F A(COBF/~,:F A A M dI / NF
(
+2CO,VF/EF
A
O)BF/NF
( (M)
--]-O)EF/IF
ONF/E F
A
+ co,,ve/ eF A
A
A
A
A M) M)
+ OEF/I F
+ O)NF/EF
A
(M) (M) OEF/I F
A
O)EF/]F
)
(
+ (_.OEF/IF A A
A
References
co,,ve/ EF A A
(A12) Applying (A4) to the 3 ~d term of the right-hand side term of (A12) and (A5) to the 9 th term gives:
( (I) EF/ NF' A
COEF / IF A
--
(I) NF/ EF
(I)
EF/ IF
--
A
,)
A
(# A
')
,
• (0
El
" (I)
NF/ EF
(I) EF/ IF
N o w we add these equations together. Two terms )
)
cancel each other because of COEH,VF----COXHEF • Then we use (A5) "backwards" and we obtain" (__OEF/NF
(
(-'OEF I IF A
A
O)EF/IF
A
A J ~ 2r- O.yF/EF
A
(.OEF/IF
A
A
--
(.ONF i EF A
That is nothing but the into"
8 th
term of (A12) which turns
d 2AM _- dcoBF/NI: A AM dt 2 / ~ dt /~
+ - df
(
+2col,vf,/E~ A C%,~/d,,~ A A M
)
Boedecker, G., Leismtiller, F., Spohnholtz, T., Cuno, J., and Neumayer, K. H. (1994). Accelerometer / gps integration for strapdown airborne gravimetry: First test results. In Sunkel, H. and Marson, I., editors, Gravity and G e o i d l A G Symposium, volume 113, pages 177-186. Springer. Bruton, M.-A. (2000). Improving the Accuracy and Resolution o f SINS/DGPS Airborne Gravimetry. PhD thesis, University of Calgary. Duquenne, H., Olesen, A.-V., Forsberg, R., and Gidskehaug, A. (2003). Improvement of the gravity field and geoid around Corsica by aerial gravimetry. In. 'Gravity and Geoid 2002', I.N. Tziavos (ed.), Ziti, Thessaloniki, Greece, pp. 167-172. Harlan, R. (1968). E6tv6s correction for airborne gravimetry. Journal o f Geophysical Research, 73:4675-4679. Jekeli, C. (2001). Inertial navigation systems with geodetic Applications. de Gruyter. Klingel6, E., Cocard, M., and Kahle, H.-G. (1997). Kinematic GPS as a source for airborne gravimetry reduction in the airborne gravity survey of Switzerland. Journal o f Geophysical Research, 102(B4):7705-7715. LaCoste, L., Ford, J., Bowles, R., and Archer, K. (1982). Gravity measurements in an airplane using state-of-the-art navigation and altimetry. Geophysics, 47:832-838. Melachroinos, S. (2004). Le d6veloppement d'un syst6me de gravim6trie mobile. M. sc. thesis, Paris Observatory. Rand, R. H. (2005). Introduction to Maxima. Available from:
http://maxima, sourceforge, net/docs/ int romax/intromax, html.
a A M + 2COEF/z~ a c%~/N~ a A M / NF
(.OEF/IF A
completes the derivation of (3).
A
A
/ EF
(
+ (oE,~/~,~ A COE,~/~,~A A
")
Valliant, V. (1991). The LaCoste and Romberg air/sea gravity meter: an overview. In Handbook o f Geophysical Exploration at Sea, volume 1, pages 141-176. CRC Press. Verdun, J., Klingel6, E., Bayer, R., Cocard, M., Geiger, A., and Kahle, H.-G. (2003). The Alpine Swiss-French airborne gravity survey. Geophysical Journal International, 152:8-19.
Chapter 116
Improving LiDAR-based Surface Reconstruction Using Ground Control Charles K. Toth Center for Mapping, The Ohio State University, 1216 Kinnear Road, Columbus, Ohio 43212, E-mail: [email protected] Nora Csanyi, Dorota A. Grejner-Brzezinska Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State University, 2070 Nell Avenue, Columbus, Ohio 43210 Abstract. Modern LiDAR systems offer ranging
1 Introduction
accuracy at 2-3 cm level at scanning rates of up to 100 kHz, assuring high point density that can support demanding mapping applications. However, the high complexity of the system may result in error sources that can degrade the accuracy of the points on the ground; thus, careful calibration and rigid connection among the system's components is crucial. Obviously, GPS/INS accuracy is required at the highest level to guarantee that the high ranging accuracy translates to a comparable accuracy of the collected ground points.
LiDAR systems are complex multi-sensor systems that incorporate three main sensors: the GPS and INS navigation sensors, and the laser-scanning device. Due to the complexity of the system, a number of potential error sources can degrade the accuracy of the acquired data. Baltsavias (1999) provides an overview of the possible error sources and error formulas. The errors in the coordinates of measured LiDAR points can result from sensor platform position and attitude errors, individual sensor calibration or measurement errors, and intersensor calibration errors or a misalignment between the different sensors. The sensor platform position and attitude errors are GPS/IMU related errors. Positioning errors are directly transferred to the coordinates of the measured ground points, while the effect of attitude errors on the calculated ground coordinates is strongly dependent on the flying height. Sensor calibration errors include scan angle error, range measurement error and other errors related to the movement of the rotating mirror. Inter-sensor calibration errors are errors in the measured lever arms between the three sensors, and any angular misalignment between the IMU and the laser sensor, the so-called boresight misalignment. The angular misalignments are more critical since any angular inaccuracy, unlike linear offsets, is amplified by the flying height, and therefore a small angular error can have significant effect on the LiDAR point accuracy. Any of the above error sources translates to an error in the calculated ground point coordinates as shown in Figure 1. LiDAR systems have significantly improved in recent years. In particular, the pulse rate frequency has increased drastically, the ranging accuracy has improved to the 2-3 cm level, and the availability of intensity signal has become common (Toth, 2004). Furthermore, the calibration methods have improved and several strip adjustment methods have been developed (Burman, 2002; Filin, 2003; Toth et al., 2002). These developments have resulted in
One way to achieve the high accuracy required for engineering scale mapping is to use absolute control information in the data. The usual method is to use fiat areas with known elevations or measure control profiles. This method, however, can refine only the vertical error, while the planimetric errors are usually more significant. A method proposed here rectifies both planimetric and vertical errors in LiDAR points by using well-identifiable LiDARspecific ground control targets. This paper discusses the design of the optimal LiDAR targets, including the shape, size, and signal response, as well as the actual performance based on two data sets collected in Ohio, over local highways, using the Optech ALTM 30/70 LiDAR system operated by the Ohio Department of Transportation. A detailed performance analysis investigating the achievable LiDAR data accuracy improvement using LiDAR-specific ground control targets is presented. The point accuracies before and after the LiDAR strip transformation based on the target data points are compared. The preliminary results indicate that LiDAR-specific targets can improve the typical 20 cm or larger planimetric accuracy of the surface points to below the 10 cm level and the vertical accuracy can approach the 2-3 cm level (basically, the laser ranging error).
818
C.K. Toth. N. Csanyi • D. A. Grejner-Brzezinska
improved data quality in terms of higher point density and better accuracy, which in turn have opened new application areas for LiDAR, and for the first time it has theoretically become possible to approach cm-level, geodetic-grade accuracy using LiDAR under well controlled circumstances. However, even after careful calibration of the system, errors can be detected in the data and typically navigation errors dominate, therefore the achievable accuracy depends considerably on the accuracy of the navigation solution. One solution to attain the desired level of accuracy required for engineering scale mapping is to use absolute control information in the data.
2 LiDAR Target Design, Positioning Accuracy and Processing Concept
rM,INS// YM
/."
I j,"
/'/
accuracy, the control targets have to be designed in a way that is optimal for LiDAR data and appropriate algorithms have to be developed for target determination. Therefore simulations were carried out for the optimal targets, and algorithms were developed to accurately determine their position in the LiDAR data. To investigate the optimal target density and distribution, two test flights were carried out with different target densities and distributions, and also with different LiDAR settings. The details of the optimal target design and achievable positioning accuracy are discussed in the first section, while the second section introduces the algorithms developed for target position determination as part of the whole processing workflow, and finally the last section provides a detailed performance analysis based on the two test flights.
rL k AZ
XM
Figure 1. LiDAR system components
The usual control information used in LiDAR data processing is horizontal planes with known elevations, or measuring control profiles across the strips. The problem with these types of control information is that they only provide vertical control. However, horizontal errors in LiDAR data are usually more significant (Vosselman and Maas, 2001) than vertical errors. Hence, to support applications that require cm-level accuracy of the data, 3-dimensional ground control information in the form of control targets is necessary. These ground control targets can then be used in the strip adjustment, or after the strip adjustment to correct any remaining absolute errors in the data. The achievable accuracy improvement using the 3D ground control targets depends on the following factors: (1) accuracy of GPS-determined ground control target coordinates (2) accuracy of determined target positions in the LiDAR data (3) ground control target density and distribution The high accuracy of the GPS-determined control coordinates can be provided by careful planning of the target survey. To facilitate the determination of the target positions in the LiDAR data with high
In designing an optimal LiDAR target the objective is to find a design that facilitates easy identification of the target in LiDAR data and provides highly accurate positioning accuracy in both horizontal and vertical directions. The target positioning accuracy is crucial since it determines the lower boundary for errors in the data that can be detected and corrected based on the LiDAR targets. After analyzing the characteristics of LiDAR data, it was found that the optimal LiDAR target is rotation invariant, circleshaped, and in order to reliably identify targets in elevation data, the target should be elevated from the ground. Furthermore, since newer LiDAR systems are capable of measuring intensity data, automatic target identification can further be facilitated if targets have a coating that provides substantially different reflectance than their surroundings. To determine the optimal target circle size and coating pattern that results in the best possible target positioning accuracy, extensive simulations were carried out. Since the designed targets are mobile targets, placed and surveyed before the LiDAR survey, economical aspects were also kept in mind. LiDAR points on the target circle were simulated in the case of different assumed circle radii and different coating patterns, such as one or twoconcentric-circle designs with different signal response coatings. LiDAR points were simulated with three different point densities, 0.25*0.25, 0.50*0.50, and 0.75*0.75 m. For the simulations, 10 cm (1 sigma) vertical accuracy and 25 cm footprint size of the LiDAR points were assumed. The major findings can be summarized as: (1) as expected, the
Chapter 1 1 6
larger the size the better the positioning accuracy, however, the results have shown that from about 5 pts/m 2 point density, a l m circle radius can already provide sufficient accuracy and further increasing the target size will not lead to significant improvements, (2) the two-concentric-circle design (the inner circle has half the radius of the outer circle) with different coatings results in significant accuracy improvements in the determined horizontal position since it provides an additional geometric constraint in contrast to the one-circle design, and (3) the optimal coating pattern is a special white coating for the inner circle and black for the outer ring. Based on the simulation results, targets were fabricated by the Ohio Department of Transportation (ODOT) to support performance validation experiments; a target is shown in Figure 2. The GPS antenna in the photo is not part of the LiDAR t a r g e t it is used to determine the ground coordinates of the target.
•
Improving LiDAR-BasedSurface Reconstruction Using Ground Control
followed by the extraction of the LiDAR points falling on the targets using elevation and intensity data. Filter target areas based on known target location and maximum expected error in LiDAR data
Extract target points based on elevation and intensity data
Horizontal and vertical positioning of targets in LiDAR data (Hough transform, mean elevation) Error computation and analysis at target locations (user interaction if needed) Applying correction to LiDAR strip by accepted transformation
Figure 3. Processing flow of target-based LiDAR strip correction After the LiDAR points on the target circle are identified, the horizontal and vertical target positions are found by separate algorithms. Since the targets are leveled, the vertical position is determined simply by averaging the elevations of the LiDAR points that fall on the target. The accuracy of the computed target height can be determined by error propagation:
O'vertical_pos
=0" Z /%/-H, where n is the number of
points on the target, and cr z is the vertical coordinate
Figure 2. LiDAR target fabricated by the Ohio Department of Transportation Based on the simulation results, the estimated positioning accuracies for the optimal target design at different LiDAR point densities, as obtained using the algorithms introduced in the next section, are shown in Table 1, where a 10 cm vertical LiDAR point accuracy and 25 cm footprint size has been assumed. Table 1. Estimate of positioning accuracies based on simulation results. LiDAR point density [m2]
LiDAR point spacing [m]
16 4 1.78
0.25*0.25 0.50*0.50 0.75*0.75
Accuracy of horizontal position of target circle [cml 2-3 5-10 10-15
Accuracy of vertical position of target circle [cm] 1.3 2.5 4.0
accuracy of the LiDAR points. The horizontal target position is found by an algorithm similar to the generalized version of the Hough-transform (Hough, 1959; Duda and Hart, 1972). The search is based on the known radius of the target circle, and the process finds all the possible locations of the target circle center as the intersection region of circles with the target radius around the LiDAR points. The implementation of the algorithm uses an accumulator array to find the intersection region. More details can be found in (Csanyi and Toth, 2004). In the case of one-circle design, Figure 4 illustrates the accumulator array (a) and the fitted circle (b) with center location area on a real example. The light grey patch shows all the possible locations of the circle center, the black circles are the corresponding circle positions and the light grey circle is the final accepted target circle position. 250 r. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
200 150
To automate the target-based LiDAR data quality check and correction, a software toolbox was developed. The main steps of the processing flow are shown in Figure 3. The processing starts with filtering the target areas from the LiDAR strips based on the known, GPS-measured target coordinates, and the maximum expected error in the LiDAR data. This is
100
0
50
1 O0
150
200
250
~~ ~
~
c-
a) b) Figure 4. Accumulator array (a) and fitted circle (b)
819
820
C.K. Toth. N. Csanyi • D. A. Grejner-Brzezinska 2
1.5 ,~11
1
li~.,ii +ii)
0.5
•320
0
-0.5
-1 -1
@I
+'r41q
#.,I
~iL-I~:,
~¢3
_.1~(.
!l
fil
L'411 ~,40
<3 if:+
•
-~8 "05 I
-0.5
0
0.5
1.5
One-circle design
"11
2
LiDAR points with intensity value
U'b.
U
U
1
1 '~
"
Two-concentric-circle design
Figure 5. Advantage of the two-concentric-circle-design
Obviously, in the case of the two-concentriccircle design the algorithm is more complex but the basic principle is the same. Figure 5 illustrates the advantage of the two-concentric-circle-design compared to the one-circle-design for the same example. The numbers next to the LiDAR points represent the LiDAR intensity values. The twoconcentric-circle design can clearly provide the horizontal center position with greater accuracy. The standard deviations of the determined horizontal target center coordinates are noticeably smaller for the two-concentric-circle design; the actual numbers are 10 cm vs. 5 cm. The target identification is followed by an interactive analysis of the LiDAR data accuracy. If errors are detected in the data at the target locations, the user can interactively select the optimal transformation for the LiDAR strip and decide whether to include all targets in the adjustment or exclude some of them. Depending on the complexity of the detected errors and the number of targets in the data, three different transformations can be applied to correct the data. The simplest transformation is a simple vertical shift, requiring at least one target. If errors are more complex than just a vertical shift, a 3-dimensional similarity transformation (requiring minimum 3 targets) or a 3-dimensional affine transformation (requiring minimum 4 targets) can be applied to the LiDAR strip based on the known and measured target positions. The corrections can be applied to each strip separately or, in more complex cases, strips can be subdivided to smaller segments and each segment can have different corrections.
3 Test Results with Varying Target Densities and Distribution To investigate the optimal target density and distribution, and to analyze the achievable accuracy improvement of LiDAR data using the designed targets, two test flights were carried out over local
highways in Ohio with an Optech ALTM 30/70 LiDAR system operated by the Ohio Department of Transportation.
3.1 Ashtabula Test Flight The first flight carried out in Ashtabula, Ohio, was aimed at transportation corridor mapping, and several strips were flown in opposite directions over a 23 km section of 1-90. The flight parameters are shown in Table 2. To support our investigations, 15 pairs of targets were placed at approximately 1600m intervals along the road, as indicated in Figure 6 and both elevation and intensity data were collected. To provide control coordinates for the targets, the target circle centers were GPS surveyed with a horizontal accuracy of about 1-2 cm, and a vertical accuracy of 2-3 cm. Table 2. Ashtabula test flight parameters Altitude (AGL) Scan angle Pulse rate
Scan frequency Point density
-620 m 14 ° 70 kHz 70 Hz 5 pts/m 2
As a quality check of the data, the targets were identified in all strips, and errors found at the target locations were analyzed as discussed in the previous section. For this paper, two opposite, about 8 km long, strips were chosen for detailed discussion; they are shown in Figure 6 together with the target locations. The strips will be denoted as strip #1 (light grey) and strip #2 (grey). As an example, Figure 7 illustrates a 3 m by 3 m area around target # 108 in strip # 1 in elevation data (a), intensity data (c), and the determined target circle position (b) as a result of the above described Hough-transform-based algorithm. This example is a good illustration of how well LiDAR points can be filtered from the LiDAR data using elevation and intensity information. It should be noted that in the
Chapter 1 1 6
figure, only for visualization purposes, the points are interpolated and shown in grey scale by elevation or intensity values, respectively.
110 309 109 308 108 307 107
Figure 6. Discussed LiDAR strips with target locations.
" m~o;;: 1 / / [~ [~ [~ [~ [~
1
Improving LiDAR-BasedSurface Reconstruction Using Ground Control
The first part of Table 3 (a) and (b) summarizes the detected errors at the target locations in the three coordinate directions, together with their standard deviations in strip # 1 and strip #2, respectively. The errors are the differences between the computed target coordinates from the LiDAR strip and the GPS-measured coordinates; the standard deviation values are provided by the target identification algorithm. The detected differences clearly show a vertical error in both strips, all the targets fall within their GPS-measured coordinates; the vertical error in strip #2 is even more significant, about 15 cm. The determined horizontal errors, however, are not significant for most of the targets; they fall below their determined standard deviation values, except for one or two targets.
310
7
•
"
~o4o~; ....
204.041 204.082 204.123 204.164 204.205 204.247 204.288 204.329 No D
204.082 204.123 204.164 204.205 204.247 204.288 204.329 204.37
.,,- .... ,
.¢. +
,. -
%,
:,.
,,
." • , . . . . . . ,'
,
ata
(a) elevation
(b) circle position
142.588-137.799 1 1
137.799- 233.01 233.01 - 328.22
1 [~ [~
328.22 - 423.431 423.431 - 518.642 518.642- 613.853
[~ [~ [~
613.853- 709.064 709.064- 804.275 804.275- 899.485
1
No D
"
ata
(c) intensity
Figure 7. Target in elevation (a) and intensity data (c) and identified target circle (b) Table 3. Errors at target locations in strip # 1 (a) and strip #2 (b) Target ID Easting 307 -0.01 108 0.05 308 0.05 109 0.13 309 -0.02 110 0.00 310 0.01 Mean 0.03 Std 0.05
Error [m| Northing Elevation
Target ID Easting 107 0.04 108 0.01 109 -0.05 110 0.10 310 -0.02 Mean -0.02 Std 0.06
Error [m] Northing Elevation
0.03 -0.06 -0.03 0.00 0.00 0.07 -0.13
-0.18 -0.06 -0.07 -0.05 -0.08 -0.05 -0.06
-0.02 0.06
-0.08 0.05
Standard Deviation [m] Easting Northing Elevation 0.07 0.06 0.08 0.06 0.08 0.04 0.04
0.07 0.08 0.05 0.08 0.07 0.05 0.05
0.02 0.02 0.02 0.02 0.02 0.02 0.02
Easting
Residual [m] Northing Elevation
-0.05 0.02 0.02 0.10 -0.04 -0.02 -0.01
0.03 -0.04 -0.01 0.03 0.03 0.11 -0.08
-0.03 0.04 0.01 0.02 -0.02 0.00 -0.02
0.00 0.05
0.01 0.06
0.00 0.03
(a)
0.10 -0.03 0.07 -0.02 0.18
-0.11 -0.13 -0.16 -0.12 -0.17
0.06 0.09
-0.14 0.03
Standard Deviation [m] Easting Northing Elevation 0.03 0.04 0.02 0.03 0.06
0.05 0.05 0.01 0.04 0.05
0.02 0.02 0.03 0.02 0.02
Easting
Residual [m] Northing Elevation
0.04 0.01 -0.05 0.10 -0.01
0.03 -0.10 0.00 -0.09 0.11
-0.01 0.02 -0.02 0.01 -0.01
0.02 0.06
-0.01 0.09
0.00 0.02
(b)
After analyzing the data, a 3-dimensional similarity transformation applied separately for both strips including all the targets was found to be adequate for the correction of the LiDAR data (more details can be found in Csanyi et al., 2005).
The last three columns of Table 3 (a) and (b) show the residuals at the target locations after the transformation. As expected, the horizontal coordinates did not change much at the targets where the detected differences were originally in the
821
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C.K. Toth. N. Csanyi • D. A. Grejner-Brzezinska
range of the horizontal coordinate determination accuracy. However, at the few targets where the determined errors were significant, an improvement was experienced. The transformation significantly decreased the vertical errors, and they are approximately in the range of the vertical accuracy of the determined target coordinates. As the flight was aimed at transportation corridor mapping, to analyze the impact of the target-based correction on the road surface extraction accuracy, two 5 m by 5m road surface areas were selected from the overlapping areas of the two strips, and their discrepancies were checked before and after the target-based correction of the strips. One area was selected in the vicinity of target # 109 (denoted as area#l), and the other one half way between target #110 and # 310 (area #2). The road areas were modeled by surface fitting, and Table 4 shows the discrepancies between the two strips at the two road areas before and after the target-based correction. Table 4. Elevation differences between strip# 1 and strip#2 before and after transformation
Table 5. Madison test flight parameters Altitude (AGL) Scan angle Pulse rate Scan frequency Point density
-700 m 10 °, 20 ° 33, 50,70 kHz 36-70 Hz Varying depending on settings
Again 15 sets of targets were placed symmetrically along the road, however, as shown in Figure 8, the targets were placed closer to each other than during the first test flight, and with varying density. In the middle of the strip the average distance was about 130 m between targets, and towards the ends, 450 m and 950 m, as seen in the figure. The targets were GPS surveyed with a horizontal accuracy of 1-2 cm, and a vertical accuracy of about 2-3 cm.
'Jilt
212
213
.~1
112
11:.t
I
1FI1
Road area
#1 #2
Elevation difference [m] Before After
-0.13 -0.14
-0.04 -0.05
For both road surface areas significant accuracy improvement was found after the target-based 3dimensional similarity transformation of the strips, the original 13-14 cm elevation difference decreased to 4-5 cm. The results of the first test showed that the specifically designed targets can indeed improve the surface extraction accuracy from LiDAR data. For this test the targets were placed with a relatively low density, and therefore to see how consistent the errors are in between the targets, another test flight was carried out with a high target density and varying distribution. 3.2
Madison
Test
Flight
The second test flight was dedicated to testing the consistency of the errors in the LiDAR strips, as well as the performance of the targets at different LiDAR point densities, different pulse rate frequencies, scan frequencies and scan angle settings. The test area was located in Madison County, Ohio, and a 7 km section of U.S. Route 40 was flown in both directions with a couple of cross strips. Both elevation and intensity data were collected. Table 5 shows the flight parameters.
Figure 8. LiDAR strips with targets
All strips were processed automatically as described above, and the errors at the target locations and their distribution were analyzed. As an example, Table 6 shows the errors - the differences between LiDAR determined coordinates and GPS surveyed coordinates, together with their standard deviations in the three coordinate directions for the 28 successfully identified targets in a selected strip. The settings for the strip were 50 kHz pulse rate, 63 Hz scan frequency, and 10 degrees scan angle. The horizontal errors cannot be considered significant as they are within their standard deviation values, except for only a few targets. However, there is a significant vertical error of about 10 cm that is consistent within the whole strip for all targets. It needs to be emphasized that there could be an error of less than 10 cm horizontally, but due to the accuracy of the horizontal positioning, it cannot be detected, while any vertical error larger than 2-3 cm can be detected using the targets. Three-dimensional-similarity transformation was again found to be adequate for the correction of the strip. The last three columns of Table 6 show the residual errors after the transformation. The horizontal errors did not change much since they
Chapter 116 • Improving LiDAR-Based Surface Reconstruction Using Ground Control
were
the
o f t h e d e t e c t e d e r r o r s i n d i c a t e s t h a t t h e e r r o r s in t h e
vertical errors significantly decreased, their m e a n v a l u e d e c r e a s e d f r o m - 1 0 c m to z e r o a n d t h e i r 2 c m
not
significant
originally,
however,
L i D A R strip w e r e v e r y stable, a n d a less d e n s e t a r g e t d i s t r i b u t i o n w o u l d h a v e b e e n e n o u g h to
s t a n d a r d d e v i a t i o n c o r r e s p o n d s to t h e v e r t i c a l t a r g e t position determination accuracy. The consistency
achieve comparable accuracy of the corrected data than using such a dense target distribution.
Table 6. Errors at target locations with their standard deviations and residuals after 3D similarity transformation Target ID 100 200 101 201 202 103 203 104 204 105 205 106 206 107 207 108 208 109 209 110 210 111 211 112 212 113 114 214 Mean Std
Easting -0.01 -0.12 0.04 -0.06 -0.05 -0.03 0.08 -0.03 -0.03 0.08 -0.06 -0.04 -0.03 0.00 -0.07 0.01 -0.01 0.07 0.03 0.14 0.01 0.07 0.02 -0.03 -0.02 -0.02 0.07 0.01 0.00 0.06
E r r o r [m] Northing Elevation -0.12 -0.13 0.05 -0.09 -0.16 -0.10 0.04 -0.08 0.02 -0.10 -0.18 -0.10 -0.06 -0.09 -0.09 -0.11 0.05 -0.09 -0.15 -0.08 0.01 -0.09 -0.02 -0.08 0.06 -0.10 -0.09 -0.13 0.07 -0.10 -0.04 -0.12 0.04 -0.11 0.05 -0.12 0.02 -0.11 -0.07 -0.10 0.03 -0.12 -0.02 -0.11 0.12 -0.10 -0.04 -0.12 -0.02 -0.11 -0.08 -0.08 -0.09 -0.07 -0.01 -0.13 -0.02 -0.10 0.08 0.02
S t a n d a r d Deviation [m] Easting Northing Elevation 0.08 0.06 0.03 0.10 0.09 0.03 0.09 0.08 0.03 0.04 0.06 0.03 0.09 0.08 0.03 0.05 0.08 0.03 0.09 0.07 0.03 0.08 0.05 0.03 0.05 0.08 0.03 0.10 0.08 0.03 0.05 0.07 0.03 0.09 0.07 0.03 0.07 0.05 0.03 0.09 0.06 0.03 0.07 0.09 0.03 0.05 0.04 0.03 0.04 0.09 0.03 0.09 0.09 0.03 0.10 0.07 0.03 0.13 0.08 0.03 0.07 0.08 0.03 0.06 0.08 0.03 0.05 0.07 0.03 0.08 0.06 0.03 0.10 0.07 0.03 0.08 0.09 0.03 0.08 0.08 0.03 0.10 0.06 0.03
Easting 0.06 -0.05 0.08 -0.02 -0.02 -0.01 0.10 -0.01 -0.01 0.10 -0.05 -0.03 -0.02 0.00 -0.06 0.02 0.00 0.07 0.03 0.13 0.01 0.06 0.01 -0.05 -0.04 -0.05 0.02 -0.04 0.01 0.05
Residual [m] Northing Elevation -0.08 -0.03 0.09 0.01 -0.13 0.00 0.07 0.02 0.05 0.00 -0.15 0.01 -0.04 0.01 -0.07 -0.01 0.08 0.01 -0.13 0.03 0.04 0.01 0.01 0.02 0.09 0.00 -0.06 -0.02 0.10 0.00 -0.02 -0.01 0.06 -0.01 0.07 -0.01 0.04 -0.01 -0.04 0.01 0.05 -0.02 0.00 0.00 0.14 0.00 -0.02 -0.02 -0.01 0.00 -0.07 0.02 -0.08 0.03 0.00 -0.03 0.00 0.00 0.08 0.02
Table 7. Mean vertical target elevation errors in the different Madison strips
Strip ID
P RF lkHz]
4 2b 4b 5b 8b 7 8 15 19 11 18 l0 13 14 17 2 5 12 9
70 70 70 70 70 70 70 70 70 50 50 50 50 50 50 33 33 33 33
Scan F r e q IHz] 70 70 70 70 70 50 50 50 50 63 63 44 44 44 44 51 51 51 36
Scan Angle ldeg] 10 10 10 10 10 20 20 20 20 10 10 20 20 20 20 10 10 10 20
M e a n Target Elevation Difference lm] -0.20 -0.20 -0.11 -0.13 -0.15 -0.12 -0.12 -0.11 -0.13 -0.10 -0.10 -0.12 -0.10 -0.07 -0.08 -0.05 -0.03 0.00 -0.06
Std Elevation Difference lm] 0.017 0.018 0.014 N/A N/A 0.020 0.014 0.032 0.022 0.017 N/A 0.018 0.016 0.017 N/A 0.018 0.017 0.015 0.015
N u m b e r of Targets in Strip 30 30 29 2 2 29 25 3 4 28 2 19 22 20 2 23 19 27 11
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824
C.K. Toth. N. Csanyi• D. A. Grejner-Brzezinska
All strips were similarly processed, and Table 7 summarizes the average vertical errors together with their standard deviation for each strip calculated from the vertical errors found at all identified targets in each strip. The horizontal errors were found to be within their standard deviation, and therefore are not listed here. The table also shows the number of targets identified in the strips, the pulse rate, scan frequency and the scan angle settings. It should be noted that those strips where only two-four targets were identified are cross strips, as shown in Figure 8. The strips are grouped by pulse rate frequency since a noticeable relationship was found between the pulse rate frequency setting for the strips and the mean elevation error found at the targets. This issue requires further investigation.
4
Conclusions
This paper has discussed the optimal ground control target design to facilitate surface reconstruction to cm-level accuracy from LiDAR data. Based on extensive simulations, the optimal LiDAR target was found to be circular-shaped having 1 m radius and elevated from the ground. Furthermore, for improved horizontal positioning accuracy, the target has a two-concentric-circle design (with 0.5 m inner circle radius) with different coatings. For accurate target position determination in LiDAR data using elevation and intensity data, and for the correction of LiDAR data based on the identified targets, efficient algorithms were developed. The test results have shown that the specifically designed LiDAR targets can indeed validate or improve cm-level LiDAR data accuracy and thereby improve surface reconstruction from LiDAR data. Using the designed targets 10 cm horizontal accuracy and 2-3 cm vertical accuracy of the extracted surface can be achieved at a LiDAR point density of 5 points per m 2. To provide this high level of accuracy, a dense and well-distributed network of targets is needed. The target processing is automated, and the developed software toolbox
provides robust processing and is ready for normal mapping operations. Acknowledgements
The authors would like to thank the Ohio Department of Transportation for manufacturing the LiDAR targets and providing essential data for this research. References
Baltsavias, E.P. (1999). Airborne laser scanning: basic relations and formulas. ISPRS Journal of Photogrammetry & Remote Sensing Vol. 54, pp. 199-214. Csanyi N, C. Toth and D. Grejner-Brzezinska, 2005. Using LiDAR-specific ground targets: a performance analysis, ISPRS WGI/2 Workshop on 3D Mapping from InSAR and LiDAR, Banff, Alberta, Canada, 7-10 June, CD-ROM. Csanyi N, C. Toth, D. Grejner-Brzezinska and J. Ray, 2005. Improving LiDAR data accuracy using LiDAR-specific ground targets, ASPRS Annual Conference, Baltimore, MD, March 7-11, CD-ROM. Csanyi N., and C. Toth., 2004. On using LiDAR-specific ground targets, ASPRS Annual Conference, Denver, CO, May 23-28, CD-ROM. Duda R. O. and P.E. Hart, 1972. Use of the Hough Transformation to detect lines and curves in pictures, Graphics and lmage processing, 15:11-15.
Filin, S., 2003. Analysis and implementation of a laser strip adjustment model. International Archives of Photogrammetry and Remote Sensing, 34 (Part 3/W13): 65-70. Maas, H.-G., 2001. On the use of pulse reflectance data for laserscanner strip adjustment. International Archives of Photogrammetry, Remote Sensing and Spatial Information Sciences, 33 (Pal~ 3/W4): 53-56.
Hough P.V.C., 1959. Machine analysis of bubble chamber pictures, international Conference on High Energy Accelerators and Instrumentation, CERN. Toth C., N. Csanyi and D. Grejner-Brzezinska. 2002. Automating the calibration of airborne multisensor imaging systems, Proc. A CSM-ASPRS Annual Conference, Washington, DC, April 19-26, CD ROM.
Toth, C. 2004: Future Trends in LiDAR, Proc. ASPRS 2004 Annual Conference, Denver, CO, May 23-28, CD-ROM. Vosselman, G. and H.-G. Maas, 2001. Adjustment and filtering of raw laser altimetry data. Proceedings OEEPE Workshop on Airborne Laserscanning and Interferometric SAR for Detailed Elevation Models. OEEPE Publications
no. 40: 62-72.
Chapter 117
The Use of G PS for Disaster Suspension Bridges
Monitoring
of
Dr Gethin Wyn Roberts, Dr Xiaolin Meng Institute of Engineering Surveying and Space Geodesy, The University of Nottingham, University Park, Nottingham, NG7 2RD, UK Chris Brown School of Engineering and Design, Brunel University West London, Uxbridge, Middlesex, UB8 3PH, UK
The use of kinematic GPS for deformation monitoring and now deflection monitoring of structures is an ongoing topic of research. It is now possible to measure the 3D deflections of discrete points upon a structure at a rate of up to 100Hz with GPS alone. The precisions of these results are typically sub centimetre, and real time is possible. However, with the integration of GPS with other sensors such as accelerometers the data acquisition rate can be increased up to 2,000Hz or so. The GPS based deflection measurements have the distinct advantage over other sensors in that GPS doesn't drift, uses precise time and can be used in all weather. The following paper discusses some of the background to this work as well as showing three case studies whereby GPS has been used on bridges to measure the deflection and natural frequencies of the bridges. The bridges investigated are the Nottingham Wilford Suspension Bridge, The London Millennium Bridge and the Humber Bridge. The paper also uses this and previous knowledge to discuss the possible use of such a GNSS based deflection monitoring strategy and combining this with Finite Element Modelling of such a structure to allow a disaster monitoring tool to be developed. Abstract.
Keywords. GPS, deflection monitoring.
1 Introduction The following paper details the ongoing research in deflection monitoring and deformation monitoring of structures, notably bridges. The use of kinematic GPS is being used for this and the work has been ongoing for about a decade at the University of Nottingham in collaboration with Brunel University (Ashkenazi et al, 1996, 1997,
Brown et al, 1999). It is possible to measure 3-D deformations of discrete points upon the bridge at rates of up to 100Hz using GPS alone. It is also possible to measure sub-centimetre precisions over the short baselines used for this work. Typically the baseline lengths from the reference to rover receivers are approximately lkm or so. Research into the use of sub centimetre kinematic GPS for bridge deflection monitoring has been ongoing at the University of Nottingham for over a decade. The initial trials were carried out with Brunel University upon the Humber Bridge in the North East of England. The partnership between the University of Nottingham and Brunel University came about through The Humber Bridge Board. Brunel University were given a research grant by the Humber Bridge Board to create a Finite Element Model (FEM) of the Bridge. In order to validate the model, the Humber Bridge Board required a method for measuring true movements to enable the model to be compared to. At the same time, The University of Nottingham had an Engineering and Physical Sciences (EPSRC) research grant investigating the development of kinematic GPS and looking at engineering applications. One such application was identified as being the deflection monitoring of large suspension bridges. The University of Nottingham and Brunel University were brought together and carried out a number of trials upon the Humber Bridge investigating the use of GPS to measure the deflections and hence establish the frequency of the movements using kinematic GPS. Initial trials were carried out on the Humber Bridge and since then a number of trials have been carried out on other bridges including the Wilford Suspension Bridge in Nottingham, the Millennium Bridge in London and more recently the Forth Road Bridge in Scotland. During the trials, survey grade dual frequency GPS receivers are used, however, more recently research has also
826
G.W.Roberts.X.Meng.C.Brown been carried out investigating the use of single frequency survey grade receivers (code and carrier) (Roberts, et al, 2004, Cosser et al, 2003). Further to this, trials have been carried out investigating the use of cheap hand held GPS receivers as it is now possible to output the carrier phase from these receivers (Cosser, et al, 2004a). The work itself investigates the use of kinematic GPS as well as comparing this to the modelling of such structures. The overall aim is to be able to use a fixed number of GPS receivers located upon the bridge at pre-determined discrete locations and comparing the movements at these points with the FEM. Once agreement between the real data and the FEM established, then it is then possible to use the FEM to model how the remainder of the bridge moves based upon this real data.
2 GPS Integrated With Accelerometers When this project initially started approximately 10 years ago, the GPS receivers available at the time were capable of gathering raw data at a rate of 5 or maybe 10 Hz (Roberts et al, 2004). This was adequate for large bridges that have a low frequency, however, for bridges with higher frequencies this was not sufficient. The Nyquist theorem states that the data-gathering rate has to be at least twice as quick as the frequencies detected. Therefore, this means that gathering data at 10Hz only allows movements with frequencies of up to 5Hz to be detected. Due to this, the p r o j e c t initially i n t e g r a t e d accelerometers with GPS. GPS has the benefit in that the coordinates do not drift over time and absolute 3-D coordinates are achieved, however, it did have a relatively low data rate of up to 10Hz. Today, the Javad JNS100 will gather data at up to 100Hz. Accelerometers can gather data at up to 1,000s of Hertz, however, they do drift over a short period of time. By integrating GPS and accelerometers by using Adaptive Filtering (AF) (Roberts, et al, 2002), it was shown that the benefits of the two different sensors could be used for deflection monitoring. AF has been extensively used throughout this work for integrating GPS with other sensors, as well as being used for multipath tropospheric error mitigation. Multipath occurs when the GPS signal reaches the GPS receiver through more than one path. This occurs when the signal is reflected off of adjacent objects and causes noise in the results.
Multipath is one of the limiting factors when trying to achieve sub-centimetre precision when using GPS. Multipath for a static point repeats itself on a daily basis, as the satellite constellation repeats itself at approximately 1 lhrs and 58mins, therefore at approximately 23hrs and 56mins the constellation seen will repeat itself. As long as the reflecting surfaces don't move and as long as the GPS receiver is in the same position then the multipath characteristics on a daily basis will repeat. Considering a roving GPS receiver upon a bridge, as long as the bridge doesn't move too much then the multipath characteristics will repeat on a daily basis. The overlying infrastructure of the bridge can introduce a lot of multipath and is not an ideal situation to be gathering GPS data underneath. Therefore, a lot of research has gone into this and is still required. In the same way, if the reference receiver and rover receivers are located at different altitudes, the troposphere introduces an error. Again, AF techniques have been developed to effectively mitigate these problems.
3 Internet RTK GPS Novel use of the Internet to transfer reference receiver to rover receiver data has been developed at Nottingham. In addition to this, the use of the Internet to allow the user to view the results in real time has also been developed. This means that the bridge engineer does not have to be located at the structure, and can be anywhere in the world viewing real time information about the deflection of the bridge.
4 Pseudolites Pseudolites are pseudo-satellites, which are ground based transmitters of GPS signals. They can help to overcome the problems with outages with the current GPS constellation. Pseudolites are capable of improving the availability, reliability and accuracy of the GPS solution, especially in the vertical component. Successful trials on bridges both in the UK and Australia through collaboration with the University of New South Wales have been carried out (Cosser et al, 2004b, Barnes et al, 2002, Barnes et al, 2003). Figure 1 illustrates a sky plot of the GPS constellation in London on the 24 November 2000 over a 24 hour period.
Chapter 117 • The Use of GPS for Disaster Monitoring of Suspension Bridges
Print...
ObsDucti . . . . . .
Help
_d ._LzJ
Fig. 1 A 24 hour GPS Constellation sky plot Over London on the 24 November 2000.
It can be seen from this that there is a big hole in the constellation where there will never be a satellite. This introduces a bad DOP value in this direction. Therefore in the UK the East-West component is the most precise followed by NorthSouth followed by height. The use of pseudolites has been investigated to improve this phenomena in the UK as well as to improve the height component and increase the availability of satellites. Figure 2 illustrates a pseudolite trials at Nottingham and at Parsley Bay in Sydney, Australia. The work in Nottingham concentrated on obtaining an indication on the precision obtainable from pseudolites whereas the Parsley Bay trials concentrated on monitoring the deflection of a small suspension bridge using pseudolites and GPS.
.
.
.
.
-
........
..
<
...-
.
..
,
.
.
.
.
Fig. 2 Pseudolite trials at Nottingham and Sydney.
5 Adaptive Filtering Results Figure 3 shows the results from using adaptive filtering on two consecutive days at the same time. It can be seen in the figure that the two top graphs show the height movement from the Wilford Bridge in Nottingham. It can be seen that there are common signature to the graphs, which could be mistaken for deflections to the untrained eye.
However, these apparent movements are due to multipath error, and hence repeat every day. The signal out in Figure 3 illustrates the uncommon element of the two signals, and the common part illustrates the common element of the two signals. The common part in these two sets is due to the multipath, and the resulting signal out is multipath free.
827
828
G.W. Roberts. X. Meng. C. Brown Ad~ptive Filtering (AF) from Two Days Reference Stations (Data:D=blb2-20, X=bl b2-21, H in BCS, 10Hz)
-0.02 F ' ~ - ~ - T ' ~ ~ ' T " 0 1000
. . . . . . . . . . . . i ...................................] ............................................. [ ......................................... I 2000 3000 4000 5000 6000
.....................
n,' 0 0 ~.02
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[_____~._~~=.L_
2000 ....
3000 I
4000 1
.............J 5000 !
with each other in various set-ups where there is a common element and uncommon element. The common element could be successive days' multipath or movement of two receivers located close to each other on a structure.
6000 1
~0 L ~ = ~ i ~ = = . ~ l , ~ = a ~ ~ = , ~ _ ~ , ~ , ~ , a ~ ~ . ~ . J f.02
0
1000
2000
3000
4000
5000
::::::::::::::::::::::::::::::::::::::::::::::: 0
1000
2000
3000
4000
6000
:,:;1 5000
6000
NO of Sample Points
Fig. 3 Multipath mitigattion through the use of AF to compare consecutive days' GPS data.
Figure 4 shows the real time troposphere delay estimation using a series of GPS receivers. Again the results are analysed using the adaptive filtering. This time the height results from one GPS receiver processed relative to another at two different heights, approximately 60m different is compared to a second GPS receiver processed relative to a second reference station. The two reference GPS receivers were placed a few metres apart, as were the two static rover receivers. Multipath error is already accounted for through AF.
Fig. 4 The use of AF for Troposphere Error Mitigation in GPS data.
Figure 5 illustrates the two roving stations. Adaptive Filtering is therefore used for all sorts of data comparison, either integrating the GPS and accelerometer data, or comparing GPS results
Fig. 5 The Wilford Suspension Bridge with two reference GPS receivers in the foreground.
Figure 6 illustrates the spectral analysis of some raw height results for the Wilford Suspension Bridge. It can be seen here that there are no obvious fundamental frequencies. Figure 7 illustrates the GPS results having had the AF used to mitigate the multipath. The bottom figure illustrates the frequency from using the accelerometer alone. It can be seen that the two techniques pick up the same natural frequency, illustrating that the AF has indeed mitigated multipath noise and left behind a realistic signal. This however, is only useful for structures that do not move too much. Large suspension bridges can not use this technique as the bridges can move a metre or more due to wind and traffic loading, meaning that the multipath characteristics would differ on a daily basis. However, the multipath and movement could be separated by comparing two closely located GPS receivers upon a bridge. These should be close enough as to experience the same bridge deflections, but far enough apart as to experience different multipath. An AF could then be used to separate the two signals.
Chapter 117 • The Use of GPS for Disaster Monitoring of Suspension Bridges x 10 4
5
Signal Spectrum Before AF Applied
bridges in close proximity to Nottingham, therefore this bridge is ideal for carrying out preliminary trials before trying the techniques out on longer bridges. Various trials have been carried out on the bridge. Details about these trial and the results can be found in (Cosser et al, 2003).
.~3 eot m
1
41;:. ' ' ',i1~ ' i=] :,
I
1
2
0
i
3
4 Frequency (Hz~
Fig. 6 Spectral Analysis of the raw data
Figure 8 illustrates the deflection on two consecutive days and illustrates the multipath characteristics analysed through using the AF. Under close inspection Figure 9 illustrates that the fundamental frequency of the deflection is 2.117 Hz. This compares very favourably with the accelerometer results of 2.116Hz.
I
Signal Spectrum Comparison Between Recursive AF Output with Acceleration (height, BCS)
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Fig. 7 Spectral Analysis of the AF cleaned GPS data (top) and accelerometer data (bottom)
o00':0r,................. :i:
No of Sample Points
x 104
Fig. 8 Multipath Mitigation using AF on the Wilford Bridge. Two successive days' of GPS data compared.
6 Case Studies 6.1 Case Study 1; The Wilford S u s p e n s i o n Bridge
°°~/
"
0.0~
The Wilford Suspension Bridge is located approximately 4km from the University of Nottingham's Campus and is held by two sets of suspension cables restrained by two massive masonry anchorages. The Bridge has a span of 68m in length and 3.65m in width. It consists of a steel deck covered by a floor of wooden slats. Underneath the deck there are three gas and one water pipe laid underneath the deck transferring the utilities across the River Trent. It is possible to obtain several centimetres of movement under normal loading and this has been used by the IESSG since 2000. There are no long suspension
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Fig. 9 Fundamental Frequency extracted from the GPS time series for the Wilford Bridge.
829
830
G.W. Roberts.X. Meng.C.Brown
approximately £18m and the modification costs were approximately £5m. During its closure and before the refurbishment, the University of Nottingham and Brunel University were allowed to place GPS receivers upon the bridge to gather data to analyse the movement characteristics. Figure 10 illustrates the location of the antennas as well as the time of the antennas being placed at the survey points on the bridge. Due to the fact that the University of Nottingham only had four Leica SR530 dual frequency GPS receivers at the time, the reference station and the GPS receiver at point B were constantly in place. The other two GPS receivers were shared between points A, C, D and E as illustrated in Figure 10. These trials were carried out towards the end of November 2000.
6.2 Case Study 2; The Millennium Bridge.
The Millennium Bridge was designed by architect Sir Norman Foster, Sculptor Anthony Caro and Engineers Arup. The bridge was opened on the 10 June 2000, and closed on the 12 June 2000 following violent and un-predicted movements of the structure during a sponsored walk being carried out over it. The movement was rectified by placing dampers underneath the bridge. The bridge was closed for refurbishment and reopened on the 27 February 2002. The bridge has an overall length of 330m, width of 4m and lies at a height of 10.8 m above the River Thames at high tide. The bridge's piers are made of concrete and steel and the cables are 120mm lock coiled, with an aluminium decking. The construction cost for the bridge was
II
Date
Session No
Start Time
End Time
Antenna Location
22/11/00
1 1 1 1
14.35.42 14.36.03 14.35.52 15.08.12
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Having processed the data AF was used to mitigate the multipath, the results in Figure 11 were found showing the lateral dynamics at the midspan. The prediction for the natural frequencies at this point were found to be 0.5 and 0.95 Hz and the actual frequencies obtained by the GPS were 0.55Hz and 0.95Hz. Figure 12 illustrates the lateral dynamics at the South Span. Again the prediction of the fundamental natural frequency is 0.77 Hz and the actual from GPS is 0.75Hz
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Chapter 117 • The Use of GPS for Disaster Monitoring of Suspension Bridges
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Fig. 12 Lateral Dynamics at South Span of the Millennium Bridge.
6.3 C a s e S t u d y 3; The Humber Bridge A whole series of trials have been carried out on the Humber Bridge. Figure 13 illustrates the Humber Bridge with a midspan length of 1.4km,
Fig. 14 A GPS receiver and weather station located on the Humber Bridge.
and an overall length of 2.22 km. Figure 14 illustrates a GPS receiver and a weather station located on the Bridge's handrail. Extensive trials were carried out in March 2004, whereby 13 GPS receivers were used. Two receivers were located upon the Humber Bridge Control building, which is approximately at the same altitude as the bridge deck, one GPS receiver was placed at the estuary which is lower than the bridge and one receiver was placed at the top of the northern towers, which is approximately 155.5 m higher than the estuary, Figure 15.
Fig. 13 The Humber Bridge. 1 singlefrequency
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Layout of the receivers for the Humber Bridge Trial, March 2004 Tower
2 referencelocations on top of the HumberBridge Board building. 1 dual frequency and
1 dual frequency A receiver
I
1 Estuary
Fig. 15 GPS Layout on the humber Bridge, March 2004.
1 singlefrequency
831
832
G.W. Roberts. X. Meng. C. Brown
Figure 16 illustrates the height deflections of points 1 and 7. It is clear from here that the two receivers located at opposite sides of the same part of the bridge experience similar movements. However, under closer inspection it can be seen that the difference in height between these two points does vary, illustrating that there is a torsional movement.
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Figure 17 illustrates the results from carrying out a spectral analysis on the GPS results. It can be seen that the results of this are 0.117 Hz. The first vertical vibration frequency predicted by an FEM created by Brunel University is 0.116Hz. Vibration Frequency Detected by GPS r
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This paper has illustrated the work carried out over the years in investigating the use of GPS for deflection monitoring of structures. Three case studies have been shown here, illustrating that GPS can successfully be used to measure the deflections and frequencies of the bridges' movements. Research is underway making the system into real- time. Near real-time integration with FEM and other models is also underway, resulting in an integrated system comprising of the prediction and actual data in harmony with each other. One of the areas of research required is to assess how accurate the models and GPS data needs to be in order to detect any significant changed or damage in the structure. This could well improve the safety and increase the life of the structure.
References
i l!
300
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44.1
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.......
Fig. 18 Height deflections for the humber Bridge.
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Figure 18 illustrates the height of a point on the bridge on 3 consecutive days. On the third day the temperature was warmer, and it can be seen that the overall height of the bridge deck is lower. On all days the height gradually drops over a period of hours. This is due to the heating effect causing the steel cables to expand.
Ashkenazi, V., Dodson, A. H., Moore, T., and G. W. Roberts (1996) Real Time OTF GPS Monitoring of the Humber Bridge, Surveying World, May/June 1996, Vol. 4, Issue 4, ISSN 0927-7900, pp 26-28. Ashkenazi V. and G. W. Roberts (1997) Experimental Monitoring the Humber Bridge with GPS. In: Proc. Institution of Civil Engineers; Civil Engineering, Nov 1997, vol 120, Issue 4., pp. 177-182. ISSN 0965 089 X. Barnes, J., Wang, J., Rizos, C., and T. Tsujii (2002). The performance of a pseudolite-based positioning system for deformation monitoring. In: Proc 2nd Syrup. on Geodesy for Geotechnical & Structural Applications, 21-24 May 2002, Berlin, Germany.
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Chapter 117 • The Use of GPS for Disaster Monitoring of Suspension Bridges
Barnes, J., Wang, J., Rizos, C., Small, D., Voight, G., and N. Gambale (2003). Guidelites: Intelligent synchronised pseudolites for cmlevel stand-alone positioning. 1 l th Int. Assoc. of Institutes of Navigation (1AIN) World Congress, 21-24 October 2003, Berlin, Germany. Brown, C. J., Karuna, R., Ashkenazi, V., Roberts, G. W. and R. Evans (1999) Monitoring of Structures using GPS, In: Proc Institution of Civil Engineers', Structures, February 1999, pp 97 - 105, ISSN 0965 092X. Cosser, E., Roberts, G. W., Meng, X., and A. H. Dodson (2003) The Comparison of Single Frequency and Dual Frequency GPS for Bridge Deflection and Vibration Monitoring. In: Proc of the Deformation Measurements' and Analysis', 11 th International Symposium on Deformation Measurements, International Federation of Surveyors (FIG), May 2003, Commission 6 - Engineering Surveys, Working Group 6.1, Santorini, Greece. Cosser, E., Hill, C.J., Roberts, G. W., Meng, X., Moore, T. and A. H. Dodson (2004a) Bridge Monitoring with Garmin Handheld Receivers. In: Proc of the 1+'tFIG International Symposium on Engineering Surveys for Construction Works and Structural Engineering, June 2004, Nottingham, UK.
Cosset, E., Meng, X., Roberts, G. W., Dodson, A. H., Barnes, J. and C. Rizos (2004b) Precise Engineering Applications of Pseudolites Augmented GNSS. in: Proc of the 1`~tFIG International Symposium on Engineering Surveys for Construction Works and Structural Engineering, June 2004, Nottingham, UK. Roberts, G. W., Meng, X., and A. H. Dodson (2002) Using Adaptive Filtering to Detect Multipath and Cycle Slips in GPS/Accelerometer Bridge Deflection Monitoring Data, in: Proc XXII international Congress of the FIG, TS6.2 Engineering Surveys for Construction Works and Structural Engineering II, April 1 9 - 26, 2002, Washington DC, USA. Roberts, G. W., Meng, X., Cosser, E. and A. H. Dodson (2004) The Use of Single Frequency GPS to Measure the Deformations and Deflections of Structures. In: Proc of the FIG Working Week, May 2004, Athens. Roberts, G. W., Meng, X. and A. H. Dodson (2004) integrating a Global Positioning System and Accelerometers to Monitor the Deflection of Bridges. Journal of Surveying Engineering, American Society of Civil Engineers, May 2004,pp 65 - 72, Vol 130, No 2, ISSN 07339453.
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Part VIII Atmospheric Studies Using Space GeodeticTechniques Chapter 118
The Impact of Mapping Functions for the Neutral Atmosphere Based on Numerical Weather Models in GPS Data Analysis
Chapter 119
Validation of Improved Atmospheric Refraction Models for Satellite Laser Ranging (SLR)
Chapter 120
Correlation Analyses of Horizontal Gradients of Atmospheric Wet Delay Versus Wind Direction and Velocity
Chapter 121
Ionospheric Scintillation Effects on GPS Carrier Phase Positioning Accuracy at Auroral and Sub-Auroral Latitudes
Chapter 122
The Impact of Severe Ionospheric Conditions on the GPS Hardware in the Southern Polar Region
Chapter 123
Preliminary Research on Imaging the Ionosphere Using CIT and China Permanent GPS Tracking Station Data
Chapter 118
The impact of mapping functions for the neutral atmosphere based on numerical weather models in GPS data analysis J. Boehm, P.J. Mendes Cerveira, H. Schuh Institute of Geodesy and Geophysics, Vienna University of Technology, Gusshausstrasse 27-29, 1040 Vienna, Austria P. Tregoning Research School of Earth Sciences, The Australian National University, Canberra, ACT, Australia
Abstract. Since troposphere modelling is one of the major error sources in the geodetic applications of Global Positioning System (GPS) and Very Long Baseline Interferometry observations, mapping functions have been developed in the last years which are based on data from numerical weather models. This paper presents the first results with the Vienna Mapping Functions 1 (VMF1) implemented in a GPS software package (GAMIT/GLOBK). The analysis of a global GPS network from July 2004 until July 2005 with VMF1 and the Niell Mapping Functions (NMF) shows that station heights can change by more than 10 mm, in particular from December to January in the Antarctic, Japan, the northern part of Europe and the western part of Canada, and Alaska. The application of the VMF1 (instead of NMF) also improves the repeatability of the geodetic results and reduces seasonal signals in the station height time series.
Keywords: Mapping function, GPS, numerical weather model, VMF 1
1 Introduction One of the major error sources in the analyses of Global Positioning System (GPS) and Very Long Baseline Interferometry (VLBI) observations is modelling path delays in the neutral atmosphere of microwave signals transmitted by satellites or emitted from astronomical radio sources. The common concept of troposphere modelling is based on the separation of the path delay, AL, into a hydrostatic and a wet part (Davis et al. 1985).
AL(e): AL~ "mh (e)+ AL~w.row(e)
(1)
In Equation 1, the total delays AL(e) at an elevation angle e are made up of a hydrostatic (index h) and a wet (index w) part, and each ot these terms is the product of the zenith delay (ALhz or ALl) and the corresponding mapping function mh or row. These mapping functions, which are independent of the azimuth of the observation, have been determined for the hydrostatic and the wet part separately by fitting the coefficients a, b, and c of a continued fraction form (Marini 1972) to standard atmospheres (e.g. Chao 1974), to radiosonde data (Niell 1996), or recently to numerical weather models (NWMs) (Niell 2001, Boehm and Schuh 2004). The hydrostatic zenith delays, ALhz, can be determined from the total pressure p and the station coordinates at a site (Saastamoinen 1973), and the hydrostatic and wet mapping functions are assumed to be known. The wet zenith delays ALw~ are estimated within the least-squares adjustment of the GPS or VLBI observations. The Vienna Mapping Functions 1 (VMF1), as introduced by Boehm and Schuh (2004) and updated by Boehm et al. (2006), are based on raytracing through the NWMs at an initial elevation angle of 3.3 °. It has been shown for VLBI analyses (Boehm et al. 2006) that VMF1 yields significantly better results in terms of baseline length repeatabilities than the Niell Mapping Functions (NMF) (Niell 1996) (which uses an empirical function dependent on only the day of year and station latitude and height) and that its application will influence the terrestrial reference frame (TRF). The investigations presented here show the first GPS results with the VMF1 implemented in a GPS software package (GAMIT/GLOBK) (King and Bock, 2005; Herring, 2005). We use solutions computed with the NMF as a reference because it is most often used in GPS and VLBI analysis, and thus it is the
838
J. Boehm • P. J. Mendes Cerveira • H. Schuh • P.Tregoning
basis for the present realization of the terrestrial reference frame. Several investigations (e.g., Boehm et al. 2006) have shown that there is no significant station height change resulting from differences between the VMF1 and NMF wet mapping functions. In contrast, there are significant differences between the hydrostatic mapping functions at low elevations which cause apparent station height changes. There exists a "rule of thumb" to estimate the approximate height change from a difference in the hydrostatic mapping function (Niell et al. 2001): "The change of the station height is
difference at the lowest elevation included in the analysis. (Station heights increase with increasing mapping functions.)" This rule of thumb shall be illustrated by one example: Figures 1 and 2 show the hydrostatic delays at 7 ° elevation calculated using the N M F and VMF1 for station OHI2 (O'Higgins, Antarctica), and the corresponding station heights obtained from GPS analysis, respectively. In January, when the difference between VMF1 and N M F is at a maximum (30 mm), differences of the station heights are also a maximum (10 mm).
approximately one third of the tropospheric delay 16.92
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Fig. 1. Hydrostatic delays at 7° elevation determined with NMF and VMF1 at station OHI2 (O'Higgins), Antarctica. There is good agreement between the mapping functions in July and August, but the disagreement reaches ~ 30 mm at 7 ° elevation in the Antarctic summer (from December through to February).
Fig. 2. Station heights of OHI2 determined from GPS with
2 Simulation
hydrostatic delay differences at 7 ° can be applied to assess the apparent station height changes when using VMF1 instead of NMF with the rule of thumb mentioned above. Figure 3a shows these simulated station height changes (VMF1 minus NMF) for January 2001 with large positive values for stations south o f - 4 5 ° latitude and also for those in Japan and north-eastern China. On the other hand, the changes are negative over the northern part of Europe, the western part of Canada, and Alaska. In contrast, there are hardly
studies
Based on monthly mean values from 40 years reanalysis data (ERA40) provided by the European Centre for Medium-Range Weather Forecasts (ECMWF) in 2001, differences between the two hydrostatic mapping functions, N M F and VMF1, have been determined on a global grid (30 ° in longitude by 15 ° in latitude) for an elevation angle of 7 ° . Multiplied by the hydrostatic zenith delay which is taken from the ERA40 data, the
NMF or VMF1 with an elevation cutoff angle of 7 °. The annual variation is plotted for both time series. The differences of the hydrostatic delays (Figure 1) are mirrored in the station height differences. The station height difference in January 2005 is about 10 mm, which is approximately one third of the hydrostatic delay difference at 7° elevation. The amplitude of the annual variation becomes significantly smaller when using VMF 1 instead of NMF.
Chapter
118 • The Impact of Mapping Functions for the Neutral Atmosphere Basedon Numerical Weather Models in GPS Data Analysis
any differences in June through August (Figure 3b), indicating that there is a much better agreement between NMF and VMF1 at that time (except for Antarctica). Since there is a clear annual signal in the differences of the mapping functions, it can be expected that the apparent station height changes for 2001 would be very
a
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-8
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similar in other years. In other words, Figures 3a and 3b show the errors that have been introduced into the estimates of GPS or VLBI station heights that have been obtained previously when using the NMF based on a common assumption for the global weather.
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Fig. 3a. GPS station height changes (in mm) simulated from ERA40 data for January 2001 when using VMF1 instead of NMF. From these simulations, large positive station height changes (>10 mm) can be expected for Antarctica and around Japan and negative height changes can be expected for the northern part of Europe. Fig. 3b. GPS station height changes (in mm) simulated from ERA40 data for July 2001 when using VMF1 instead of NMF. In July 2001, there are only relatively small height changes which can be expected in other years, too. Fig. 3e. Station height differences from GPS analysis using either NMF or VMF1 for January 2005. Black bars indicate positive height changes, grey bars negative height changes. It is evident that these analysis data confirm the simulations from ERA40 data for January 2001 (Figure 3a), i.e. positive station height changes can be found in the southern hemisphere and around Japan, and negative station height changes occur at stations in northern Europe, the western part of Canada, and Alaska. Fig. 3d. Station height differences from GPS analysis using either NMF or VMF1 for July 2005. Clearly, these analyses confirm the simulations from ERA40 data for July 2001 (Figure 3b), i.e. the estimated station height changes are moderate compared to January (see Figure 3c).
3 Vienna Mapping Functions in the GAMIT software
A global network of more than 100 GPS stations was analysed with the software package GAMIT software Version 10.21 (King and Bock, 2005) applying first the NMF and then the VMF1 mapping functions. We analysed a full year of global observations from July 2004 until July
2005, producing a fiducial-free global network for each day analysed. The elevation cutoff angle was set to 7 ° and no downweighting of low observations was applied. Atmospheric pressure loading (tidal and non-tidal) (Tregoning and van Dam, 2005) was applied along with ocean tide loading and the IERS2003 solid Earth tide model (IERS Conventions 2003). We estimated satellite orbital parameters, station coordinates, zenith tropospheric delay parameters every 2 hours, and
839
840
J. B o e h m • P. J. M e n d e s Cerveira • H. Schuh • P. T r e g o n i n g
and Figures 3b/3d). This confirms that the NMF has temporal deficiencies, with a maximum around January, especially at high southern latitudes, for Japan, the northern part of Europe, the western part of Canada, and Alaska.
resolved ambiguities where possible. We used-60 sites to transform the fiducial-free networks into the ITRF2000 by estimating 6-parameter transformations (3 rotations, 3 translations) (Herring 2005). For the investigations described below the time series of estimated station heights at 133 sites were used. Each of these sites has more than 300 daily height estimates. The site distribution is shown in Figure 3c. Amplitudes A and phases doyo (in terms of day of year) of annual periodic signals were estimated by the method of least-squares for all station heights from the NMF and VMF1 time series (see Figure 2). Then these sinusoidal functions (Equation 2) were used to calculate the station height differences on 1 January 2005 and 1 July 2005, respectively (Figures 3c and 3d). h - A . sin l, doy - doy o . 2 rc)
Figure 4 shows the amplitudes and phases for all 133 GPS stations. (Figure 5 shows amplitudes and phases in Europe for clarity.) Generally, the VMF 1 reduced the amplitudes of annual variations on around 50% of sites. However, there is a systematic improvement (reduction of amplitude larger than 5 mm, see Figure 2) at sites situated below 45 ° S, indicating deficiencies of the NMF at higher southern latitudes. The agreement between the amplitudes and phases when changing from NMF to VMF1 is rather good; however, at some stations, especially in the southern hemisphere and in northern Europe, large discrepancies occur. This may be due to the short time series that has been used in this analysis (only one year). If significant, these changes in amplitudes of annual signals might influence the determination of normal modes of the Earth according to Blewitt (2003).
(2)
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A comparison of the estimated height differences from GPS with those predicted from the NWMs shows a very high correlation (cf. Figures 3a/3c
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Chapter 1 1 8
The Impact of Mapping Functions for t h e
•
Neutral
Based on Numerical Weather Models in GPS Data Analysis
Atmosphere
deviation of the daily station heights with regard to the annual signal is smaller for 117 of the 133 stations, and the average relative improvement is about 6%. Thus, using the VMF1 not only changes the terrestrial reference frame but it also improves considerably the precision of the GPS analysis. The progression from the old N M F to the new mapping functions based on N W M s influences the terrestrial reference frame by changing the heights of some stations - in particular, in Japan and in some regions of the northern hemisphere (Figure 7). Thus, there will be a distortion of the whole frame and rather likely a general shift along the zaxis. As radio-wave techniques play an important role in the realization of the International Terrestrial Reference Frame (ITRF), a significant influence on the next ITRF can be expected if weather-based mapping functions are used in the analysis of the GPS and VLBI observations.
Fig. 5. Amplitudes A and phases doyo (see Equation 2) of annual variation in the station height time series determined from GPS analyses using NMF or VMF1 in Europe. The grey bars correspond to NMF, the black bars to VMF1. The phase angles doyo are counted from north (January) to east (April). The standard respect to decreases for compared to
The results presented here are derived from analyses where no elevation-dependent weighting of the observations has been performed. Very similar results are obtained when such weighting is used, although the influence of the more accurate tropospheric mapping functions is reduced.
deviation of the station heights with the sinusoidal functions clearly almost all stations using the VMF1 N M F (Figure 6). The standard
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841
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J. Boehm • P. J. Mendes Cerveira • H. Schuh • P. Tregoning
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4 Conclusions For the first time, the Vienna Mapping Functions 1 (VMF1) based on data from a numerical weather model have been applied in global GPS analysis. Significant improvements in the precision of geodetic results are found compared to using the Niell Mapping Function (NMF) based on a very general assumption about the global weather. After removing an annual signal, the standard deviation of the residual station heights decreases for more than 87% of the stations, and the average relative improvement is about 6% compared to NMF, with values as high as 30% at some stations. Furthermore, the application of the Vienna Mapping Functions 1 will change the terrestrial reference frame by changing station heights. The m a x i m u m station height differences (up to 20 mm) when changing from the N M F to VMF1 occur in January, especially in Antarctica, Japan, the northern part of Europe, the western part of Canada, and Alaska.
Acknowledgements We would like to thank ZAMG in Austria for providing us access to the ECMWF data and the Austrian Science Fund (FWF) (project P 16992-N 10) for supporting this work. We are also grateful to the IGS for providing the global
geodetic data. The inclusion of the VMF1 into the GAMIT software was funded in part by the Fonds National de la Recherche du Luxembourg grant FNR/03/MA06/06. The GPS analyses were computed on the Terrawulf linux cluster belonging to the Centre for Advanced Data Inference at the Research School of Earth Sciences, The Austrian National University.
References Blewitt G (2003). Self-consistency in reference frames, geocenter definition, and surface loading of the solid Earth, J o u r n a l o f G e o p h y s i c a l R e s e a r c h , Vol. 108, NO. B2, 2103, doi: 10.129/2002JB002082 Boehm J, Schuh H (2004). Vienna Mapping Functions in VLBI analyses, G e o p h y s . R e s . Lett. 31(1):L01603, DOI: 10.1029/2003GL018984 Boehm J, Werl B, Schuh H (2006). Troposphere mapping functions for GPS and very long baseline interferometry from European Centre for Medium-Range Weather Forecasts operational analysis data, J Geophys Res, Vol. 111, B02406, DOI: 10.1029/2005JB003629 Chao CC (1974) The troposphere calibration model for Mariner Mars 1971, J P L T e c h n i c a l R e p o r t 32-1587, NASA JPL, Pasadena CA Davis JL, Herring TA, Shapiro II, Rogers AEE, Elgered G (1985). Geodesy by Radio Interferometry: Effects of Atmospheric Modeling Errors on Estimates of Baseline Length, R a d i o S c i e n c e 20(6): 1593-1607
Chapter 118 • The Impact of Mapping Functions for the Neutral Atmosphere Based on Numerical Weather Models in GPS Data Analysis
Herring TA (2005)GLOBK Global Kalman Filter VLBI and GPS analysis program, version 10.1, Mass. Inst. of Technol., Cambridge, MA King RW, Bock Y (2005). Documentation for the GAMIT GPS processing software Release 10.2, Mass. Inst. of Technol., Cambridge, MA McCarthy, DD, Petit G (2004). IERS Conventions (2003), IERS Technical Note, No. 32, Verlag des Bundesamtes ffir Kartographie und Geodfisie Marini JW (1972). Correction of satellite tracking data for an arbitrary tropospheric profile, Radio Science, Vol. 7, No. 2, pp. 223-231 Niell AE (1996). Global mapping functions for the atmosphere delay at radio wavelengths, J. Geophys. Res. 101 (B2):3227-3246
Niell A.E. (2001). Preliminary evaluation of atmospheric mapping functions based on numerical weather models, Phys. Chem. Earth, 26, 475-480 Niell AE, Coster AJ, Solheim FS, Mendes VB, Toot PC, Langley RB, Upham CA (2001). Comparison of Measurements of Atmospheric Wet Delay by Radiosonde, Water Vapor Radiometer, GPS, and VLBI, Journal of Atmospheric and Oceanic Technology 18:830850 Saastamoinen J (1973). Contributions to the Theory of Atmospheric Refraction, Part II, Bulletin Geodesique, Vol. 107, pp. 13-34 Tregoning P, van Dam T (2005). Atmospheric pressure loading corrections applied to GPS data at the observation level, Geophys. Res. Lett., 32, L22310, doi: 10.1029/2005GL024104
843
Chapter 119
Validation of improved atmospheric refraction models for Satellite Laser Ranging (SLR) G. Hulley, E.C. Pavlis Joint Center for Earth Systems Technology (JCET), University of Maryland Baltimore County, Baltimore, MD, USA V.B. Mendes Laboratorio de Tectonofisica e Tectonica Experimental and Departamento de Matematica, Faculdade de Ciencias da Universidade de Lisboa, Lisbon, Portugal Abstract. Atmospheric refraction is an important accuracy-limiting factor in the use of satellite laser ranging (SLR) for high-accuracy science applications. In most of these applications, and particularly for the establishment and monitoring of the Terrestrial Reference Frame (TRF), of great interest is the stability of its scale and its implied height system. A new zenith delay (ZD) model developed by Mendes and Pavlis, [2004] called FCULzd, along with new mapping functions [Mendes et al., 2002] are now the adopted standard for refraction modeling. The model is valid for a wide spectrum of wavelengths and has showed submillimeter accuracies from 0.355 ~tm to 1.064 ~tm. Using 3-d ray tracing and globally distributed satellite data from the Atmospheric Infrared Sounder (AIRS) instrument on NASA's AQUA platform, as well as 3-d analysis fields from the European Center for Medium Weather Forecasting (ECMWF) and the National Center for Environmental Prediction (NCEP), we assess the new zenith delay model and mapping function both spatially and temporally. We also investigate the effects of horizontal refractivity gradients on the atmospheric delay, specifically with respect to seasonal changes and station location.
component of the model), and a - 1 mm bias in the computation of the zenith delay. The mapping functions developed by Mendes et al. [2002] represent a significant improvement over the builtin mapping function of the Marini-Murray model and other known mapping functions. The new mapping functions can be used in combination with any zenith delay model. The assumption made by all current atmospheric delay models of a spherically symmetric atmosphere with uniform refractive index shells is inherently flawed. Spherical shell models do not take into account the presence of horizontal gradients, and this is most likely the largest error source in current refraction modeling techniques.
2 AIRS data
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The modeling of atmospheric refraction in the analysis of SLR data comprises the determination of the delay in the zenith direction and subsequent projection to a given elevation angle, using a mapping function (MF). In the past the most common approach to modeling the effects of atmospheric refraction was to use the MariniMurray model [Marini and Murray, 1973 ]. However Mendes et al., [2002] pointed out some limitations in the Marini-Murray model used in SLR since its introduction in 1973, namely, the modeling of the elevation dependence (the mapping function
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We will be using meteorological data sets from NASA's Atmospheric Infrared Sounder (AIRS see mission concept above) in order to validate the new ZD and MF models, and to develop new models that include variations in horizontal refractive indices. The AIRS Level-2 support product gives profiles of temperature, pressure and
Chapter 119
• Validation of
water vapor from the surface to the top of the atmosphere (TOA) in 100 standard pressure levels. The pressure levels extend from 1100 mb up to 0.1 mb. AIRS obtains temperature profiles to an accuracy of 1o K for every 1o km in the troposphere and 1o K for every 4 km in the stratosphere up till 40 km. The temperature profile accuracy in the troposphere matches that achieved by radiosonde launches from ground stations. The advantage of AIRS is that it provides rapid and global coverage of the earth. AIRS also obtains humidity profiles to an accuracy of 10% in 2 km layers in the lower troposphere and -50% in the upper stratosphere. The AIRS data is retrieved in the form of "granules". One granule contains 6 minutes of data and is approximately 1600 (E-W) x 2300 (N-S) km in spatial extent with a 50 km resolution within the granule. One day of data yields 240 granules.
3 AIRS, NCEP and ECMWF Ray Tracing (ART, NRT and ERT) We have developed a ray tracing algorithm specifically tailored for AIRS, ECMWF and NCEP data that will calculate the atmospheric delay directly by integrating all the values through which the ray traverses, independent of any mapping function. In order to perform the ray tracing, the data is first processed and grouped into 10°xl0 ° latitude/longitude grids up to 0.1 mb in order to build 3-d atmospheric profiles around each operational ILRS-SLR tracking station. ART, ERT and NRT are used to calculate both the total atmospheric delay as well as the delay due to horizontal gradients in refractivity. We should also point out the most significant errors in current ray tracing techniques: accuracy of the data itself, effects of hydrometeors such as rain, snow, ice, and horizontal gradients which will be discussed in more detail later. The first two errors however can be very difficult to quantify exactly, as they are highly variable and depend on many factors.
Improved Atmospheric Refraction Models for Satellite Laser Ranging (SLR)
zenith hydrostatic delay, in meter units is as follows"
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3 Model Development 3.1 Zenith Delay Model The new zenith delay model, FCULzd derived in detail by Mendes and Pavlis [2004] uses new formulations for the group refractivity of visible and near-infrared waves as described by Ciddor [1996] and Ciddor and Hill [1999]. The final form of the
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845
846
G. Hulley. E. C. Pavlis. V. B. Mendes
4 Horizontal Refractivity Gradients Including the atmospheric delay due to horizontal refractivity gradients in the total atmospheric delay is essential in minimizing errors in SLR measurements, especially at low elevations. Past studies of atmospheric gradients by MacMillan [1995], Chen and Herring [1997], and MacMillan and Ma [1997] were all based on their impact on the analysis of VLBI and GPS geodetic data in microwave frequency bands. Chen and Herring [ 1997] found north-south gradients with average values of up to 20 mm of delay when averaged over a month, and for an elevation angle of 10° . The gradients were calculated from 3-d weather analysis fields from NCEP. Chen and Herring [1997] have also developed a parametric form for the gradient delay along with a mapping function (MTT) that can be adapted to analyze various space geodetic data. The delay due to horizontal refractivity gradients can be written:
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5 Analysis and Results 5.1 Model Validation We applied our ray-tracing technique to calculate the total atmospheric delay at 12 SLR stations: Herstmonceux, Monument Peak, Hartebeesthoek, Yarragadee, Mt Stromlo, Haleakala, Arequipa, McDonald, Graz, Matera, Zimmerwald and Greenbelt during February and August 2004. We do not include the effects of horizontal gradients here since the new model (FCULzd) does not account for
them. Figure 2 shows box-and-whisker plots for Herstmonceux and Greenbelt. The box has lines at the lower quartile, median and upper quartile values. Whiskers are lines extending from each end of the box to show the extent of the rest of the data. Outliers are values beyond the whiskers and are represented by crosses. ART has a mean bias of 0.3 mm and an RMS of 7.4 mm at 10 ° elevation at Herstmonceux during Aug. 2004 (Figure 2). ERT and NRT have larger means of 8.3 mm and 12.8 mm but have a smaller scatter about the mean. ART has a much larger scatter about the mean mainly due to errors in surface measurements. For instance, approximately only 67% of AIRS profiles below 3 km are valid over land. At Greenbelt, ART has a larger mean bias of 4 mm with NRT having a smaller bias (7.4 mm) than ERT (10 mm). This makes sense since we should expect the NCEP data to be more accurate over North American sites and ECMWF data to be more accurate for the European sites. Above 20 ° elevation we see biases on the order of a few millimeters, and sub-millimeter differences in the zenith direction at all stations.
5.2 Gradient Delay Results We have found that the largest variations in the gradient delay results arise due to seasonal changes from summer to winter. The mean NS and EW gradients showed no discernable trends for the global set of stations, but vary primarily from factors such as station location and surface topography. The fluctuation between summer and winter gradients is more noticeable when stations are situated near the ocean, resulting in large temperature gradients or near varying topographical features that results in larger pressure gradients. During the summer we also have larger diurnal fluctuations and the temperatures in the boundary layer are more unstable due to convection, resulting in the gradients being more variable. The time periods chosen fall in the middle of the winter and summer seasons for the northern and southern hemisphere sites respectively. The following indepth discussion of the results from selected stations is summarized in Tables 1-5. Figure 3 shows comparisons between ART, ERT and NRT gradient delay results at Greenbelt and Herstmonceux.
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G. Hulley. E. C. Pavlis. V. B. Mendes 5.2.1
Herstmonceux
Herstmonceux in southern England, has a climate that is influenced mainly by its close proximity to water. As a result the weather is normally damp and subject to frequent changes. The mean NS gradient delay is -3.7 mm during the winter and increases to -5.2 mm during the summer month (see Table 5). The RMS (note: R M S ~ = M e a n ~ + S t d ~ ) remained constant at 6 mm for both seasons, implying that the scatter about the mean was largest during the winter month. The EW gradient delay increased from 0.2 mm to -1.6 over the same time period with the RMS remaining constant at 2.2 ram. In Figure 3 we see similar trends between AIRS, ECMWF and NCEP results for February 2004.
5.2.2
Monument Peak
Monument Peak in Mount Laguna, CA has larger EW gradients than NS gradients during summer and winter with a mean of 5.0 mm (RMS 5.2 ram) during the summer and 4.0 mm (RMS 4.5 mm) during the winter. The NS gradients stayed relatively constant from summer to winter with means of only -2.5 mm. The primary factor contributing to the larger EW gradients are temperature fluctuations from land to the cooler Pacific Ocean directly west of the station. Monument Peak is also located in a mountainous region at 1800 m above sea-level, resulting in temperature inversions during the summer and larger than expected pressure gradients. It is also interesting to note that the gradients are relatively constant with RMS values of only a few tenths of a millimeter larger than the means. 5.2.3
Yarragadee
Yarragadee, situated in southwestern Australia is also an interesting case in that during the winter, the NS gradients (3.3 mm) are larger than the EW (1.4 mm), while during the summer the EW gradients (3.3 mm) are larger than the NS (-0.5 mm). As with Monument Peak, the Indian Ocean to the west of the station has a large effect on the EW gradients which double from winter to summer. Average temperatures in this region range from 28 ° to 32 ° with fluctuations from 20 ° to 38 ° during the summer resulting in large horizontal temperature gradients from land to the cooler ocean during this season.
5.2.4
Hartebeesthoek
Hartebeesthoek is located 50 km west of Johannesburg, South Africa and is good example of what type of gradients to expect when the station is surrounded by land. This region has a very mild climate with dry, cool winters and warm, wet summers. Seasonal changes are flexible, one sometimes running into the next resulting in no distinguishable gradient trends from summer to winter. In fact, the NS gradient delays remain constant from summer to winter with an RMS of 2.2 mm. The EW gradients also stay relatively constant with an RMS of 1.9 mm during the summer and 2.1 mm during the winter.
5.2.5
Arequipa
Situated in southern Peru with the Pacific Ocean to the west and the Andes mountain range to the east, Arequipa is yet another good example of a station with large gradients due to ocean and topographical effects. In fact Arequipa has the largest NS and EW gradients of all the stations being investigated in our analyses. The station is situated to the west of the Andes mountain range that breaks the country into two parts: the desert coast and Amazon jungle. The winds from the jungle therefore are unable to reach the coast, resulting in the coast being temperate and dry. These climate extremes result in large gradients in both directions, but primarily in the EW direction. It is interesting to note that the EW gradients are in fact identical when averaged out over November 2003 and May 2003, with a mean of 6.0 mm and RMS of 6.1 mm. The NS gradients are largest in the winter month with an RMS of 6.2 mm and this can be explained by the icy Andes mountain range to the northeast during the winter. Other interesting cases include Zimmerwald, which has large EW gradient delays exceeding 4 mm during the summer and winter, and larger N S and EW delays during the winter month. This can be attributed to Zimmerwald being situated in a region with varying topography and the Swiss Alps to the south also having an effect on the gradients. Matera, in southern Italy and surrounded by the Mediterranean has large NS gradient delays during the summer, while during the winter the NS and EW mean delays are relatively small with much larger RMS values indicating larger variations during the winter.
Chapter 119 • Validation of Improved Atmospheric Refraction Models for Satellite Laser Ranging (SLR)
Table 1. Statistics showing gradients expressed as the delay at 10 ° elevation for Arequipa during May and November 2003.
Table 2. Statistics for Monument Peak during February and August of 2004.
Station
Date
Method
NS Gradient Mean RMS mm mm
Arequipa
May
ART
-5.9
6.2
6.0
6.1
ERT
-5.8
5.8
3.4
NRT
-3.9
4.0
ART
-4.8
ERT NRT
Nov
EW Gradient Mean RMS mm mm
Station
Date
Method
NS Gradient Mean RMS mm mm
Monument Pk
Aug
ART
-2.8
3.7
5.0
5.2
3.4
ERT
-0.5
0.7
1.9
1.9
4.2
4.8
NRT
0.7
1.4
2.3
2.9
4.9
6.0
6.1
ART
-2.5
3.2
4.0
4.5
-6.0
6.0
3.2
3.2
ERT
-1.5
1.8
2.6
2.9
-2.6
2.7
2.7
2.9
NRT
-1.3
2.1
3.3
3.1
Table 4. Statistics Yarragadee during February and August of 2004.
Table 3. Statistics for Hartebeesthoek during February and August of 2004.
Station
Date
Method
NS Gradient Mean RMS mm mm
Hartebeesthoek
Aug
ART
1.3
2.2
-0.9
2.1
ERT
1.1
1.2
-0.7
NRT
2.5
2.7
ART
1.1
ERT NRT
Feb
EW Gradient Mean RMS mm mm
Station
Date
Method
NS Gradient Mean RMS mm mm
Yarragadee
Aug
ART
-0.5
6.0
-1.6
2.2
0.7
ERT
-3.6
4.0
-0.8
1.1
0.4
0.7
NRT
-2.0
2.3
0.1
0.9
2.2
-1.4
1.9
ART
-3.7
5.7
0.2
2.1
0.4
0.4
-0.4
0.5
ERT
-3.3
4.7
0.1
1.2
0.3
0.9
0.1
0.3
NRT
-2.6
2.9
0.6
1.1
Table 5. Statistics for Herstmonceux during February and August of 2004.
Station
Date
Method
NS Gradient Mean RMS mm mm
Herstmonnceux
Aug
ART
-5.2
6.0
-1.6
2.2
ERT
-3.6
4.0
-0.8
1.1
NRT
-2.0
2.3
0.1
0.9
ART
-3.7
5.7
0.2
2.1
ERT
-3.3
4.7
0.1
1.2
NRT
-2.6
2.9
0.6
1.1
Feb
Feb
EW Gradient Mean RMS mm mm
EW Gradient Mean RMS mm mm
Feb
EW Gradient Mean RMS mm mm
849
850
G. Hulley. E. C. Pavlis. V. B. Mendes
I
6 -
I
Greenbelt
-e'4 - (a) North-South E2
AIRS ECMWF
.... x .... NCEP
_
E
>,0
, , .IN:. . . .
"
-
"
.
N.,
,,
,'
~'
',,
......
%
~-2 m -4
0
I
I
I
I
I
I
5
10
15
20
25
30
I
I
i
35
Aug 2004
I
6
Greenbelt
4
(b) East-West
I
I
I
AIRS
_
-~-ECMWF . . . . . . . . NCFP
x,,
-4 -6 t 0
-
I
I
I
I
I
I
5
10
15
20
25
30
35
Aug 2004
I
I
I
I
I
_ Herstmonceau×
I
AIRS
- e - - ECMWF
(al North-South _
,,,,x .... NCEP
~o
..,~,,,~., ,,
.
.."
'
..... ~ .
e-
I"
"x ....
,
,,~ . . . . . . \
X
.........
~ /,
.
~e
.,',~.~
-]
...~1.
J -10
_
I
I
5
I0
I
I
I
15 Feb 2004
I
I
I
20
25
I
I
Herstmonceau×
:30
I
I
(b) East-West
AIRS
-e-ECMWF .... x .... NCEP
_
E
5
J
~, 0
-10 I
I
I
I
I
5
10
15
20
25
Feb 2004
Fig. 3. AIRS, ECMWF and NCEP horizontal gradient delay comparisons showing (a) North-South and (b) East-West gradients at 10 degrees elevation for Greenbelt (top) and Herstmonceux (bottom) during August and February 2004.
:30
Chapter 119 • Validation of Improved Atmospheric Refraction Models for Satellite Laser Ranging (SLR)
5 Conclusions The current accuracy of the new zenith delay model, FCULzd has sub-millimeter differences when compared with ray-tracing of radiosonde data at 180 stations for a full year [Mendes and Pavlis, 2004]. In our results (Figure 2) we see that all three methods, ART, ERT and NRT have sub-millimeter mean biases in the zenith with RMS values not exceeding 1 mm. We have found similar values at other stations and during different times of the year, using a wide range of wavelengths from 0.355 1.064 /~m. At lower elevations, the largest biases occur due to errors in the mapping function, FCULa which has a two-year average RMS of 7 mm (model minus ray-tracing through radiosonde data) at 10°elevation [Mendes et al., 2002]. Our ART results show sub-centimeter biases with the model at 10 ° elevation at all stations, but with a large scatter about the mean. ERT results also show subcentimeter biases, but only at European stations such as Herstmonceux, Graz and Zimmerwald. We see differences above the centimeter level at other stations in Australia and North America. NRT on the other hand has sub-centimeter biases at all North American sites, but has biases at the 12 mm level at all other stations.
from l0 ° to 20 ° elevation, and are at the submillimeter level in the zenith direction. By raytracing at 12 different azimuth angles around stations with varying climates and in mountainous regions, we found noticeable gradient delay differences in each azimuth direction. This was expected since horizontal gradients are largely inhomogeneous and the delay should be dependent on azimuth angle as a result. In conclusion, our study addresses the validation of the latest atmospheric delay model as well as the contribution of horizontal refractivity gradients to the computation of the total atmospheric delay. In order to accomplish this we have developed a robust 3-d ray-tracing program with input atmospheric fields from AIRS, ECMWF and NCEP that will calculate the total atmospheric delay including the contribution from horizontal gradients at selected azimuths, elevation angles and at any given location on the globe.
Acknowledgments.
We
gratefully
acknowledge the support from NASA's Cooperative Agreement with JCET NCC5-339 and NGA's Grant NURI NMA201-01-BAA-2002. ECMWF
for
providing
us
We also thank
with
data
during
February and August 2004 in order to make
Although we only have made comparisons at 12 globally distributed stations and for two months during 2004, we have found that FCULzd does perform well when compared with our ray-tracing methods and for a wide range of wavelengths. However, the use of a mapping function at low elevations still produces errors at sub-centimeter levels.
References
We have found that the NS and EW refractivity gradients are affected significantly by changes in climate from summer to winter seasons at the majority of SLR stations. Sites situated in mountainous regions had larger horizontal pressure gradients while a station's proximity to a large body of water resulted in larger horizontal temperature gradients. No significant non-hydrostatic gradients were found, with maximum wet delays only reaching a few tenths of a millimeter at stations such as Greenbelt. We have found that NS horizontal gradient delays can have month-long average values of up to 6 mm at 10 ° elevation. Gradient delays decrease by approximately half
Ciddor, P.E., (1996), Refractive index of air: New equations for the visible and near infrared, Applied @tics, 35, No. 9, pp. 1566-1573. Ciddor, P.E. and R. J. Hill, (1999), Refractive index of air. 2. Group index, Applied @tics, 38, pp. 1663-1667. Chen, G.E. and T.A. Herring, (1997), Effects of atmospheric azimuthal asymmetry on the analysis of space geodetic data, J. Geophys. Res., 102, No. B9, pp. 20,489-20,502. MacMillan, D.S. and C. Ma, (1997), Atmospheric gradients and the VLBI terrestrial and celestial reference frames, J. Geophys. Res.,24, No.4, pp. 453-456.
comparisons with AIRS results, and also Scott Hannon
from
UMBC
for
processing
and
distributing the data to us.
851
852
G. Hulley. E. C. Pavlis. V. B. Mendes
MacMillan, D.S., (1995), Atmospheric gradients from very long baseline interferometry observations, Geophys. Res. Lett., 22, 1041-1044. Marini, J.W., and C.W. Murray, (1973), Correction of laser range tracking data for atmospheric refraction at elevations above 10 degrees, NASA Rep. X-591-73-351, Goddard Space Flight Cert., Greenbelt, MD.
Mendes, V. B., G. Prates, E. C. Pavlis, D. E. Pavlis, and R. B. Langley, (2002), Improved Mapping Functions for Atmospheric Refraction Correction in SLR, Geophys. Res. Lett., 29(10), 1414, doi: 10.1029/2001GL014394. Mendes, V.B., and E. C. Pavlis, (2004), HighAccuracy Zenith Delay Prediction at Optical Wavelengths, Geophys. Res. Lett., 31, L14602, doi: 10.1029/2004GL020308.
Chapter 120
Correlation Analyses of Horizontal Gradients of Atmospheric Wet Delay versus Wind Direction and Velocity Torao TANAKA Department of Environmental Science and Technology Meijo University, 1-501 Shiogamaguchi, Tenpaku-ku, Nagoya 468-8502, Japan
Abstract.
Total water vapor contents were
phenomenon found by Mousa was real and not
monitored along lines of sight at such low
due to instrumental origin, setting of instruments
elevation angles as 10 and 15 degrees with two
nor
water vapor radiometers, WVR1100 TM, WVR05
coefficients used for data transformation from
and 06. They were installed in two directions of
observed brightness temperatures with Water
improper
determination
of
retrieval
N-S and E-W in Uji city, southwest Japan in the
Vapor Radiometers, WVR1100 TM WVR05 and 06
period from 1997 to 99. Results show that
produced by Radiometrics Corp., to the excess
differences of wet delays between N and S
path delay of micro waves, namely wet delay
directions, which correspond to the gradient of
(hereafter WD) (Tanaka, 2003).
wet delays of microwaves, sometimes reach to
Since the amount of water vapor is the largest
3cm or more and continue to exist stably for a
in the lowest part of the troposphere
few days or longer.
It is ascertained that the
decreases exponentially with height, WD gives
and
horizontal gradient of water vapor distribution in
larger directional differences at lower elevation
the N-S direction is caused by atmospheric
angles of GPS satellites. The most probable
conditions, especially by wind direction and
sources of such spatial inhomogeneous WD
velocity, and also probably by sunlight. Similar
distribution, namely the horizontal gradient, are
correlations are apparent between E-W gradients
the topography around an observation site and the
of wet delay and wind velocity.
However, the
meteorological condition such as air temperature,
data is not enough to draw definite conclusions on
wind direction and velocity, air pressure, humidity.
the E-W component.
and sunlight. Aonashi et al. (2004) estimated the horizontal
Keywords.
wet
delay,
wind
velocity,
positioning, water vapor radiometer
scales of the gradient of precipitable water vapor content using three water vapor radiometers. They concluded that the horizontal scale was less than
1 Introduction
10km. Although their aim was to have statistical
Inhomogeneous distribution of water vapor in
information on temporal variations of the gradient.
the atmosphere causes errors of the precise
the temporal variations themselves are very
positioning
important
and
surveying
techniques such as GPS.
with
the
space
information
to
correct positioning
Mousa(1997) found a
errors in observations of stepwise or transient
stable inhomogeneous water vapor distribution
ground movements with GPS due to spatially
continuing for one day or longer at Shionomisaki
inhomogeneous
Promontory in southwest Japan. Tanaka et al.
GPS-derived water vapor and horizontal wind
(2004) carried out similar observations in Uji city
was obtained from Doppler radar-derived radial
located
such
wind data above Kanto Plain in Japan by Seko et
another
al. (2004). However, the spatial scale is too large
at
inhomogeneous
inland
to
investigate
distributions
geographical setting.
at
They ascertained that the
WD
distributions.
Mesoscale
to apply their results to correct positioning errors
854
T.
Tanaka
Pl~isht
(rn}
~.~:;:'.:....~ ~." ..z-- ..'..~.,"~.-,
:: ,L; .~.I;'.,,,'::?,:. •
~_ ~i...
[
~:.,"~:~' ~i~.
.:.. ..
,,,.~
!~:..,~.i~'-./Z-'~.-I
Z
. ,,,,,
".,'.:'S,"i-'~ ,~ --...",~.
o~~.~
• ~: .":i.. J..'
~,:, ~
I
--',
,'~,~ " " L.
"
~_ __ # t ,
~
,
~-~--.
.....
,.
'.~.,~."-...-"-,, ~
I
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::':;.:J0. . ~,~- .
-
i~.
..~,
.. . . . .. . . . .
,o , 4 ~;:!.!:'-.:::;IS',.~',
,"
.~; ":.-~:~::i ...."4'~
".
r:.. -t, r :-,;,,,,~'~4" ~.~ .......:..-,,_~ .: . -. ..... .
!, '
:
I;~'
..,~:}.. ri.,-;....;':~
i -,:,..,:~i-'-... ' ;'~
I
.
.-
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•
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Ill :: .,,," •
• "'~:ii,, ~'i':';:" ~" :~~',~i '.:i:- ::,': "'~'~!:;~:
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8
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.~..: ....,
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:.,,,.. -:.:::.~,-~
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' ,;~..,',"-!".:.'.~',,,i~"~-;:"7'.,~;;.:. ':..i " *~'"-:-'""' .~'~:'!i":'~h,:!: ,;.~:.:~.?::";'-:,.-:.!.> ,,-...- "-~;!..,..::~._:~ .... . ,,,:.........~"~' I
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--'
I I
, I
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I
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v, :; . ~, ,' ..'it',
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;:.'.,'.~.: '
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:'.,.~;.?:~ ' '.~_ ..'~.. ;~
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.: - .
'...
:. -
,.",-,...-....i .....
..-. 480
i
100
a:
o 0
W
O~nc~
(m}
E
Fig. 1. The geographical setting around UJI. The solid circle indicates the observation site. Topographic profiles of N-S and E-W directions through the site are also shown.
in GPS measurements.
E and W.
If we could find any clear causality and process between
the
horizontal
gradient
and
meteorological and/or topographical conditions,
Although the sampling interval of
WVRs from a northward observation to the next southward
observation
for
example,
is
approximately 1-2 minutes, we neglected the
we will be able to reduce the positioning errors
interval and calculated the differences assuming
except random noise from observational data, and
that the two
observations were
carried out
raise the accuracy to extract more reliable results
concurrently, since there were no large changes of
such as transient changes of target positions.
the WD differences in a few minutes except rough weather condition.
2 Observations
and
Data
The observation site at UJI was at the roof of opposite
the building of Research Center for Earthquake
directions, N and S as well as E and W at several
Prediction, Disaster Prevention Research Institute
elevation angles such as 10, 15, 30, 50, 70 and
of
location
and
90(zenith) degrees cyclically by changing the
topographic profiles are indicated in Fig.1.
The
angle of reflectors of the two WVRs. Since the
latitude and longitude of UJI are 34.9N and
We
observed
WDs
toward
two
Kyoto
University.
The
propagation path of microwaves in the boundary
135.8E, respectively. Because UJI is located at the
layer
angle
eastern edge of a basin, the site is surrounded by
observations, larger horizontal gradients of WD
low hills on the north and south sides. On the
are observed at lower elevation angles.
Thus we
other hand low hills are near to the east as shown
calculated the WD differences of 10 degrees
in the E-W profile. Although the basin opens
between two opposite directions of N and S, and
toward the west, several trees occur to the west at
is
longer
at
lower
elevation
Chapter120 • CorrelationAnalysesof HorizontalGradientsof AtmosphericWet DelayVersusWind Directionand Velocity
a few hundred meters from the site. Thus the hills to the east and trees to the west might give
Firstly we have calculated the correlation coefficients between the hourly temporal changes
some multipath influence due to reflection by the
of wind velocity at Kyoto projected to the six
hills and trees (Radiometrics Corporation, 1990).
directions to N, NEN, ENE, E, ESE, SE and the
The Uji river flows from SE to NW and the
hourly S-N and E-W components observed with
distance from the site to the river is approximately 500m to SW (Fig.l). Although this river will
WVR05
and
record,
under
influence the observed WD gradient at UJI, we
meteorological
cannot
through the whole period.
distinguish
this
effect
from
other
topographical factors at present. by
Kyoto
the
the whole period of each assumption
condition
did
that not
the
change
The maximum
correlation coefficients and the wind direction for
We have used meteorological data observed at Kyoto
06 for
Local
Meteorological
the maximum value are given on the second and third columns of Table 1 and 2.
Unfortunately
Observatory of Japan Meteorological Agency
we could not use proper covers to prevent the
(hereafter Kyoto) through its Website. Hourly
WVR redomes from getting wet with rain, and the
values of pressure, temperature, relative humidity,
data
wind direction and velocity, precipitation, sunlight
calculating correlation coefficients due to the
and snow depth are obtained on the site.
large data fluctuations.
Kyoto
during
rainfalls
cannot
be
used
for
Therefore next we
is situated approximately 12km north of UJI and
excluded the period of rainfalls from the records
the Observatory is located at the northern part of a
based on the precipitation data at Kyoto. By
basin. Accordingly its meteorological condition is
considering rainfall duration at Kyoto including a
expected to be different from that at UJI. A
few hours before and after the rainfall and the
temporal lag or advance may occur between them.
distance between UJI and Kyoto, we divided
Therefore we should compare the meteorological
records into two or three parts, and calculated
data at Kyoto with the horizontal gradient of WD
correlation coefficients for each period. The
observed
the
results thus obtained are also given on the fourth
differences in precipitations and wind conditions
to sixth columns in Table 1 and 2 for the S-N and
between Kyoto and UJI are essential. Thus the
E-W directions, respectively.
at UJI
carefully.
Especially
The records are
differences should be considered. We carried out a
arranged in the monthly order, not in the
preliminary analysis and obtained the following
temporally serial order, for checking any seasonal
result. WD gradient and wind velocity sometimes
tendency of the correlations on the column
give higher correlation by assuming one hour
"Correlation 2".
delay or advance between them. We should take
seasonal changes of the correlations.
the
differences
into
the
analysis
of
We cannot
find any clear
the
Naturally, the shorter is the length of divided
observational data. However, we neglect the
record, the higher correlation coefficients than
temporal difference here as a first step.
those for the whole period are expected. This will
Since we had carried out several campaigns and
mean that the wind direction is comparatively
special experiments with WVRs and had some
constant in the case of analyzing short data.
instrumental troubles in the period from 1997 to
Accordingly
99, the numbers of available records in the present
between the wind velocity and the horizontal
analysis are 14 and 3 in the case of WVR05 and
gradient of WD might be obtained for the analysis
06, respectively as summarized in Tables 1 and 2
of shorter data. At present we cannot identify the
for S-N and E-W directions, respectively.
actual
3
Correlation
Horizontal Wind
Coefficients
Gradients
between
of Wet Delay and
cause
higher
of
the
correlation
change
of
coefficient
correlation
coefficients. In other words, we cannot determine whether it is a real change caused by the wind, or an apparent change due to wrong comparison
855
856
T. Tanaka
between an irrelevant pair of data, or the two
southwest Japan.
We concluded that the gradient
results generated from an unknown origin.
was
by
Here we
show three
cases
of correlation
generated
analysis from Table land one from Table 2 in the from Jan.15 to 18 in 1999. The WD differences,
2
the wind velocity, precipitation, and sunlight in
1
this period are given in the figure. The correlation
..A..JiL
"
1
25
the small change of WD difference. Another
BSunlight (hr)
N
49
WD diff. (cm)
o
Wind vel. (m/s)
73
Hour
46 to 52 hours due to rainfall as mentioned above,
small coefficient of 0.10 is probably results from
local
4
'1
IrlF,[,.Y -
-2-11~I~'~ IL ,..t /
E3
excluded the period from 1 to 26 hours and from
direction E (Table 1 and Fig.3). The reason of the
the
[]
coefficient is 0.099 for the whole period. Next we
for the wind direction N and 0.35 for the wind
under
Wet delay d i f f e r e n c e ( S - N ) , Wind v e l o c i t y ( N W ) , Preci. & Sunlight; C O R = - 0 . 0 9 9 , d a n . 1 5 - 1 8 , 1999 3 2 _- _- . Prec. (mm)
following. Fig.2 is the record of WVR05 observed
and calculated the correlation coefficients as 0.10
wind
topographic condition. The sunlight under
Fig.2. WD differences between S and N at UJ1, SE wind velocity, precipitation, and sunlight at Kyoto. The correlation coefficient in the whole period between S-N gradient of WD and the component of wind velocity is its maximum 0.099, at the wind direction of SE.
correlation coefficient 0.35 in the later period in Wet delay d i f f e r e n c e ( S - N ) , Wind velocity(N & E), Preoi. & Sunlight; COR=0.10 & 0.35, Jan.15-18, 1999
Fig.3 might be generated by the larger change of WD difference as seen in Fig.2, and is as large as
2
:--2::;Prec.
other divided records in Table 1. Fig.4
gives
the
relationship
(om)
i
between
the
gradients of WD and wind direction for the whole
~E ~_'~,
~
BSunlight (hr)
0
-~-o - o e-
o
-1
Wind vel. (m/s)
period of April 10-21, 1997. This period was no -2
precipitations condition.
and
under
the
stable
weather
It is apparent that the wind from
north gives the high correlation coefficient with the S-N horizontal gradient. In addition, Fig.4 indicates that the gradual increase of the S-N gradient of WD is caused by the calm condition of
Wd diff. (crn)
1
25
49
73
Hour
Fig.3. Comparison of the WD gradient in the S-N direction with the N and E wind velocities for the early and later periods, respectively. The two horizontal bars show the periods for which the correlation coefficients have been calculated.
the wind during the fine days, in other words, under the long sunlight.
Wet delay difference(S-N), Wind velocity(N), Preci. & Sunlight; COR=0.63; Apr.10-16, 1997
Fig.5 is an example of the WD difference in the E-W direction and the wind velocity. The E-W
.
10
8
~Sunlight (hr)
gradients show large correlation coefficients as those of the S-N gradients shown in Table 1.
~-~
However, we cannot draw definite conclusions
"~ ~ - 5
from the present data due to the small number of the records.
4 Summary This paper has calculated the correlation coefficients of the horizontal gradients of WD observed with water vapor radiometers to the wind velocity at the site in an inland basin in the
0
4 2
-10
0 1
25
49
73 97 Hour
[]
WD diff. (cm) Wind vel.
(m/s)
121 145
Fig.4. WD differences between S and N at UJI, N wind velocity, precipitation, and sunlight at Kyoto. The correlation coefficient in the whole period between S-N gradient of WD and the component of wind velocity is its maximum 0.63, at the wind direction of N. There were no rains in this period.
Chapter 120 • Correlation Analyses of Horizontal Gradients of Atmospheric Wet Delay Versus Wind Direction and Velocity Table 1.
Maximum correlations of the WD differences between the south and north directions observed with WVR05 to the wind
velocity. Period 1 and 2 are the whole length and divided lengths of each record, Correlation 1and 2 are the maximum correlation to the wind velocity to the direction given in the third and sixth column, respectively. Wind scale is as follows; la: light air (0.3-1.5 m/s), lb: light breeze (1.6-3.3), gb: gentle breeze (3.4-5.6)
Period 1
Wind
Period 2
Correlation 1
direction 1
(hour)
Correlation 2
direction 2
Wind scale
Jan. 15- 18, 1999 (66)
0.1
SE
27-45
0.1
S
1b
Mar. 14- 19, 1997 (132)
0.18
WNW
2 0 - 25, 1997 (142)
0.04
E
(hour)
2 6 - 31, 1997 (142)
0.1
SES
Wind
53-81
0.35
E
1b
l - 19
0.32
SE
1b
70-145
0.32
WNW
1b
1-34
0.29
E
1b
54-143
0.31
NW
1b
1- 16
0.65
NE
1b
34-74
0.25
ESE
gb
101-143
0.32
NWN
gb
17-36
0.23
SE
1b
Mar. 25 - 28, 1999 (67)
0.05
WSW
63-84
0.16
SWS
1b
Apr.
1-
6, 1997 (131)
0.13
NE
1-37
0.28
NW
1b
Apr.
2-
3, 1999 (30)
0.13
EWE
23-44
0.42
SES
1b
7-
Jul.
9, 1997 (38)
0.69
E
no rain
1a
10- 16, 1997 (149)
0.63
N
no rain
gb
1 7 - 2 5 , 1997 (199)
0.27
W
4-
1-110
0.36
WSW
1b
150-200
0.26
NE
gb
SW
18-81
0.29
SWS
gb
8,1997(95)
0.18
Nov. 10- 14, 1998 (97)
0.32
W
no rain
1b
18- 19, 1998 (27)
0.25
WSW
no rain
1b
0.5
ESE
Dec. 22 - 26, 1998 (83)
Table 2.
15-45
0.18
W
1a
57-98
0.48
ESE
1b
Maximum correlations of the WD differences between the east and west directions observed with WVR06 to the wind
velocity. The legends are same as in Table 1.
Period 1
Wind
Period 2 (hour)
(hour)
Correlation 1
direction 1
Apr. 16 - 21, 1998 (119)
0.5
NWN
Nov. 11 - 14, 1997 (70)
Dec. 22 - 24, 1998 (40)
0.17
0.05
SES
WSW
Wind Correlation 2
direction 2
Wind scale
17-47
0.55
SES
1b
77-120
0.59
SWS
1b
17-45
0.41
N
1a
56-76
0.21
E
1b
15-46
0.27
ESE
gb
857
858
T. Tanaka
calm wind condition also seems to increase
Wet delay difference(E-W), Wind velocity(SES), Preci. & Sunlight; COR=-0.50, Apr.16-21, 1998
horizontal WD gradients. We have to investigate not
only
the
correlation
between
the
WD
12
8
10
gradients and wind velocity but the effect of the
8
topography around the site in more detail.
Acknowledgements I wish to thank to Dr. H.Aoki, Director and the staffs of Tono Research Institute of Earthquake
- ' " " : Prec. (mm)
~s~Nr~0r)
6
~ ~o
4 2 -4
0 25
Science, Association for the Development of Earthquake
49
73
.
W D diff
(cm) (ms)
Wind veL
97
Hour
Prediction for their valuable suggestions. I also would like to thank Dr. Kazuaki Hori at Meijo University for his valuable advices and proofreading the manuscript.
Fig. 5. Comparison of the E-W gradient of WD and the NWN wind velocity. We excluded the period of two
References Aonashi, K., T. Iwabuchi, Y. Shoji, R. Ohtani, R.
precipitations observed between 48 and 76 hr and
Ichikawa (2004): Statistical study on precipitable
from 17 to 47 and 77 to 120, obtaining 0.55 and 0.59 for
water
SES and SWS winds, respectively.
content
variations
observed
with
calculated the correlation coefficients for the two periods
ground-based microwave radiometers, Journal of the Meteorological Society of Japan, Vol.82, No. 1B, 269-275. Mousa, A. E. (1997): Characteristics of wet tropospheric delay deduced from water vapor radiometer data and their implications for GPS Thesis,
Tanaka, T. (2003): On the retrieval coefficients
Graduate School of Science, Kyoto University,
used in observations with water vapor radiometers
122p.
(Second report: An evaluation of errors due to
Radiometrics Corporation (1990): Water Vapor
retrieval coefficients), Report of Tono Research
Radiometer, March 1990, 34p.
Institute of Earthquake Science, Seq. No.12,
Seko, H, T. Kawabata, T. Tsuyuki, H. Nakamura,
November 30, 2003, 109-113 (in Japanese).
K. Koizumi, T. Iwabuchi (2004): Impacts of
Tanaka, T., Y. Hoso, M. Harada, T. Hayashi, A. E.
GPS-derived
Mousa, K. Hirahara (2004): Gradients of water
baseline
solution
measured
by
prediction
of
water
accuracy.
vapor
Doppler
Doctoral
and
Radar
radial on
wind
numerical
vapor distribution in the troposphere observed
the
with water vapor radiometers in Uji, southwest
Meteorological Society of Japan, Vol.82, No.IB,
Japan, Journal of the Geodetic Society of Japan,
473-489.
50, 2, 67-79
precipitation,
Journal
of
Chapter 121
Ionospheric Scintillation Effects on GPS Carrier Phase Positioning Accuracy at Auroral and Subauroral Latitudes M. Aquino, A. Dodson, J. Souter and T. Moore Institute of Engineering Surveying and Space Geodesy- IESSG The University of Nottingham, University Park, Nottingham, UK, NG7 2RD
Abstract. Ionospheric scintillation may present significant effects on GPS, mainly in equatorial and auroral regions, and during times of high solar flux. In the auroral regions scintillation occurrence mostly relates to geomagnetic activity and can affect GNSS users even at sub-auroral (and potentially mid-latitude) regions, with impact ranging from degradation of accuracy to loss of signal tracking. Recent work at Nottingham investigated the impact of ionospheric scintillation and Total Electron Content (TEC) gradients on GNSS users, through a network of four GPS Ionospheric Scintillation Monitors set up in the UK and Norway. Statistical analyses of the scintillation and TEC data, aiming to characterise ionospheric scintillation over Northern Europe, were also carried out. Critically to GNSS users these studies covered, in particular, aspects of availability and integrity, through the assessment of occurrence of loss of lock on GPS satellites due to high scintillation levels. However, accuracy aspects have also been investigated, through the analysis of standalone GPS, DGPS, EGNOS aided DGPS and carrier phase errors, which have been correlated with observed scintillation levels and geomagnetic indices. Horizontal errors in GPS C/A code pointpositioning were seen to correlate to enhancement in the background TEC observed during times of occurrence of high scintillation. DGPS positioning accuracy was seen to be affected by TEC gradients occurring at auroral and sub-auroral latitudes, especially under enhanced geomagnetic activity. Missing corrections in the EGNOS ionospheric grid during periods of occurrence of high phase scintillation suggested an inability of the EGNOS reference stations to track one or both of the GPS signals of some satellites. In this paper the main focus is on carrier phase positioning experiments, which revealed an increase in the measurement noise and positioning accuracy degradation
significantly correlated with the occurrence of high phase scintillation. Keywords. Global Positioning System (GPS), Global Navigation Satellite System (GNSS), accuracy, ionospheric scintillation
1 Background and Introduction The ionisation in the Earth's upper atmosphere is highly variable, and during high sunspot activity can become disturbed and turbulent. This turbulence creates irregularities in the ionosphere which can interfere with the wave front of transmitted satellite signals, causing fluctuations in both the intensity (amplitude) and phase of the received signal. These are referred to here as scintillations and are mainly observed in three regions. In the first, the lowlatitude region, strongest levels of scintillation have been historically recorded, which occur in the postsunset period, with peaks observed around-15 ° and +15 ° magnetic latitude. The second region covers the night-side auroral oval and dayside cusp (the auroral region), whereas the third corresponds to the polar cap region. Generally, more moderate levels of scintillation are observed in the nightside auroral oval, as compared to the polar cap (Rodrigues et al, 2004). It is also interesting to note that in a recent study Mitchell et al (2005) discussed the association of scintillations with high TEC values observed in the Northern European Arctic (Ny Alesund, -79°N), concluding that these high TEC values originated from convection of plasma from the North American sector through the polar cap. As part of a study aiming to investigate the effects of ionospheric scintillation on GPS measurements over Northern Europe, a network of specialised GPS scintillation monitor receivers was deployed in the UK and Norway. It was anticipated that scintillations would be observed, given that this network operated during the peak of the solar cycle,
860
M. Aquino • A. Dodson. J. Souter • T. Moore
in particular in conjunction with periods of high geomagnetic activity. Expecting that high levels of scintillation would adversely affect GPS/GNSS users, the main aim of the study was to assess how significant the effects would be and how warning/mitigation mechanisms could be devised. For that purpose, in parallel with analyses of positioning and measurement errors, statistical analyses were undertaken (Rodrigues et al, 2004) with the objective of characterising scintillation occurrence from a climatological point of view. No attempt was made to describe the statistical results from a geophysical perspective, but rather to convert the historical data into probabilities that could be of direct use to assess potential effects on users. For instance it was confirmed, based on two years of records of the widely used amplitude ($4) and phase (the 60 seconds c~ or Phi60) scintillation indices, that levels of phase scintillation are significantly higher than amplitude scintillation in Northern Europe. Moderate to strong amplitude scintillation ($4>0.6) barely occurred at our monitor stations, whereas moderate to strong phase scintillation occurrence (%~>0.5) was significantly higher, with values reaching that level for 5% of the time on many days at our northern-most stations, especially during the night time sectors.
Also, long term comparisons between the Kp geomagnetic index (from the USA NOAA SEC) and scintillation activity were carried out, statistically corroborating the existence of significant correlation between the two. GPS phase scintillation at high latitudes has a daily pattern mainly controlled by the receiver location moving into the aurora oval (Aarons, 1997). During magnetic quiet periods the aurora oval has the largest latitudinal extension at magnetic midnight, as a consequence of its natural expansion in the anti-sunward direction, and phase scintillation activity is greatest around that time. Occurrence of phase scintillation during daytime is mostly due to the oval's expansion during geomagnetic storms. Establishing the correlation between scintillation occurrence and geomagnetic events is of particular relevance to the development of scintillation prediction and warning mechanisms. We analysed the correlation between the Kp and scintillation activity recorded at our monitoring stations, in particular Bronnoysund (-65°N) and Hammerfest (-71°N), these being the most likely stations to be affected by the oval expansion/contraction. For
instance, at Bronnoysund we observed strong correlation of the maximum daily occurrence of ~,p>0.5 for 3-hour time bins with the maximum daily value of Kp (which is computed for each 3-hours) for every day of 2002 and 2003. Correlation was particularly significant during 2003, when a greater number of Kp values over 6 occurred (this value of Kp corresponds to a major geomagnetic storm according to NOAA's classification). It was also seen, through similar analysis, that geomagnetic, more than solar activity, was predominant in controlling scintillation occurrence at Hammerfest in 2003 (the reader is referred to Aquino et al, 2005a for more details of these analyses).
However, to assess how detrimental the effects of scintillation may be to a GPS user, it is necessary to investigate how many satellites are likely to be simultaneously affected by strong scintillation, in particular phase scintillation. Whether a certain level of scintillation (defined as moderate or strong) will have a direct influence on the quality of the measurement made at the receiver, or lead to loss of lock on the satellite, an estimate of how many satellites may be affected at the same time is of relevance, as this will impact on the user positioning accuracy or the user ability to calculate position at all. For that analysis, the probability of a number of GPS satellites simultaneously observing different levels of phase scintillation was calculated. It was shown that at our station in Hammerfest there was a probability of about 0.1% of 2 satellites being simultaneously affected by c~>0.5 in 2002. This probability increased to 0.25% in 2003, amounting to about 22 hours within that one year. This may pose a concern if lock on satellite is lost due to this level of scintillation, especially for applications requiring high availability, such as in civil aviation. A similar analysis for station Bronnoysund indicated probabilities of 0.06% and 0.1%, respectively for the years of 2002 and 2003. More importantly, when analysing the severe geomagnetic storm that occurred during October 29th and 30th, 2003, probabilities of about 3% and 2.5% of 2 satellites observing ~ > 0 . 5 simultaneously, respectively at Hammerfest and Bronnoysund were found. It was also seen that under such severe storm conditions 3 satellites observed c~>0.5 simultaneously with probabilities of 1% and 0.75%, respectively at Hammerfest and Bronnoysund.
Loss of lock on satellites possibly correlated with high levels of scintillation is thus a primary concern
Chapter 121 • Ionospheric Scintillation Effects on GPS Carrier Phase Positioning Accuracy at Auroral and Sub-Auroral Latitudes
for GNSS users in Northern Europe, in particular due to the greater susceptibility of the L2 to loss of lock in the presence of high phase scintillation. A codeless receiver exhibited up to 50% of L2 data loss during a severe geomagnetic storm and L1 or L2 data from up to 7 satellites was not present in the corresponding RINEX file on some occasions. On several occasions loss of lock was observed only on the L2 signal, with a direct implication for SBAS reference stations, as they must track both GPS signals in order to compute and disseminate ionospheric delay corrections. Further analyses on loss of lock involved studying its dependence on elevation angle, and loss was found to be more evident for elevations lower than 20 °. Losses of satellite lock correlated with phase/amplitude scintillation even occurred with our GPS scintillation monitors, highlighting the potential impact of ionospheric scintillation in the tracking performance of conventional GPS receivers. In this paper, however, we mainly concentrate on experiments aimed to investigating possible accuracy degradation on carrier phase positioning related to scintillation effects. In the next section we review some of our overall positioning accuracy results when analysing GPS positioning errors concurrent with scintillation occurrence. The following sections describe and discuss results of experiments performed in particular with carrier phase positioning. In section 3 we deal with static carrier phase positioning, whereas in section 4 we discuss results of using a pseudo-kinematic carrier phase technique. Section 5 contains our conclusions.
series of normalised horizontal positioning errors (the dark triangles), with the observed values of the phase scintillation index for all satellites shown in the background. We use the normalised horizontal errors, which are given by the horizontal errors divided by the HDOP (Horizontal Dilution of Precision) of each epoch, in order to enable a meaningful comparison between epochs as the effect of the geometry is thereby neutralised. Although the errors are greater during the period of high scintillation levels, an epoch by epoch comparison does not show a direct correlation between the two. The bottom plot shows that in fact the errors correlate better with the background TEC, which is shown in the plot by the (non-calibrated) slant TEC values for all satellites being observed.
B r o n n o y s u n d , 30 O c t o b e r 2 0 0 3
•
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.
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2 Summary of Previous Work 2 , 0
Our investigations of GPS C/A code point positioning errors have shown there to be no significant accuracy degradation which can be strongly correlated with increased levels of scintillation, but rather any degradation in standalone code positioning occurred as a result of increased levels in the background TEC. A representative example is given in figure 1, which refers to C/A code stand-alone positioning errors computed for every 1 minute at our station in Bronnoysund (--65°N) during a period of increasing scintillation levels (given by the 60 seconds cy~ index), between 14:00UT and 24:00UT on 30 October 2003. In the top plot we show the time
16
18
20
22
0
Universaltime
Fig.
1 Scintillation
Effects
on
C/A
code
stand-alone
positioning
Due to the nature of the algorithms used to compute pseudorange corrections in a conventional DGPS, where clock, orbit and atmospheric errors are combined together, it is expected that varying ionospheric conditions will lead to spatial decorrelation between reference station and user.
861
862
M. Aquino • A. Dodson. J. Souter • T. Moore
Varying ionospheric conditions between reference and user may take place due to north-south TEC gradients that occur at sub-auroral and auroral latitudes during the solar high, in particular under active geomagnetic conditions. This was also investigated and when comparing the 2drms errors from a (approximately) north-south baseline with those from an (approximately) east-west oriented baseline in the UK, during a severely disturbed period (5 th to 8th November 2001), a degradation of about 30% in the horizontal positioning accuracy was observed, confirming the potential influence of TEC gradients on DGPS positioning (Moore et al, 2002). A way of improving stand-alone or DGPS user accuracy is to access and apply location specific ionospheric corrections disseminated by Augmentation Systems such as the European Geostationary Navigation Overlay Service (EGNOS). Such corrections would in principle resolve the problem of spatial decorrelation due to strong TEC gradients. However, the computation of the ionospheric correction by these systems relies on dual frequency GPS data from their reference stations, that may also be affected by scintillation during periods of unfavourable geomagnetic conditions. In particular the acquisition of the less robust L2 observations may be compromised. We have observed missing corrections in the EGNOS ionospheric grid during periods of occurrence of high values of phase scintillation which are likely to relate to the inability of the EGNOS reference stations to track one or both of the GPS signals of some satellites. Users at mid-latitudes opting to avoid satellites not monitored by the EGNOS ionospheric grid were shown to, thereby, indirectly suffer the effects of scintillation due to ionospheric irregularities occurring further northwards. On occasion such users were left with only 4 satellites to compute position, which has been shown to cause problems in the least squares solution due to lack of redundant measurements (Aquino et al, 2005b). This could be an issue if users do not have a choice to include (non-monitored) satellites in their solution.
Experiments were carried out to analyse possible position and measurement accuracy degradation when using carrier phase observations during periods of observed high levels of scintillation. These are described in the next two sections.
3 Static Carrier Phase Experiments Initial experiments with static carrier phase positioning have been reported in Aquino et al (2005b). Carrier phase data from the 30 th of October 2003 was analysed. On that day significant enhancement in phase scintillation (a~) values was observed even at our mid-latitude station in Nottingham (~53°N). The increased levels began quite markedly during the night time hours, with values changing from low during the day to moderate and tending to high (figure 2 top plot). For our analyses, the 24 hours RINEX file of our scintillation monitor was split in 2 hours sessions for processing.
o
Nottingham,
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1
30 October 2003
o--
o.~--
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,~-----~l 08:00
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18:00
20:00
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Fig. 2 Phase scintillation values for all satellites recorded at Nottingham, 30 October 2003
Each session was processed using IGS (International GPS Service) precise orbits and three IGS permanent stations with data available in the region, to form a network and solve for the coordinates of our scintillation monitor. The observable used was the ionosphere corrected L1 double difference carrier phase. Estimated coordinates were compared with accurate ground truth coordinates, with results for each 2 hours session shown in figure 2 (bottom plot). Temporal correlation of the RMS of the recorded phase scintillation values with, respectively, the 3D errors
Chapter 121 • IonosphericScintillationEffectson GPSCarrier Phase Positioning Accuracyat Auroral and Sub-Auroral Latitudes
and the RMS of the measurement residuals was then analysed for every 2 hours session. Results are shown in the first row of table 1. NSF06 indicates our scintillation monitor located in Nottingham. An entirely similar analysis was conducted involving data from a permanent Ashtech ZXII semicodeless GPS receiver (station IESG), whose antenna is located just a few meters from our scintillation monitor's, with accurately known coordinates. Resulting correlation coefficients are seen in the second row of table 1. A comparison in the performance of the two receivers, not shown on the table, revealed that the scintillation monitor overall provided better positioning accuracy than the Ashtech semicodeless receiver, especially when scintillation levels were low. However when scintillation was at its peak on that day, the latter seemed to have been less affected. Influence of scintillation on the RMS residuals did not show any appreciable difference between the two receivers (notice correlation coefficients of 0.96 and 0.98 respectively), however both receivers indicate a strong correlation between phase scintillation and the quality of their measurements. These results clearly deserve further investigation, which is however outside the scope of this paper. We also analysed the performance of our monitor in Bronnoysund using a similar approach, during the same day. Resulting correlation coefficients are shown in the third row of table 1, where NSF03 indicates our monitor located in Bronnoysund.
shortening the time span of the processing session to 15 minutes. Also, a baseline was compared with a network approach in order to assess whether the latter could improve the solution in the presence of strong scintillation. The baseline was formed by our Bronnoysund receiver (NSF03) and EUREF station VILO (Vilhelmina), approximately 240km to the east, which was held fixed while solutions were obtained for NSF03. The network solutions included another 2 permanent IGS stations with data available in the region. Figure 3 shows the results. The top plot represents the baseline solutions, with the RMS of the 3D errors of 0.90m, mainly due to the poor performance (spikes in the plot) during the sessions where high scintillation occurred. Average values of the ~ index for all satellites are shown in the bottom plot. The middle plot shows the network solutions, with the RMS of the 3D errors of 0.43m. In both cases the correlation between the errors and the phase scintillation values is clearly visible, and the results also indicate that the network solution leads to a significant improvement in accuracy, as expected due to the greater redundancy and geometrical strength given by the network solution. It is noted however that the poorer solutions during the periods of stronger scintillation remain.
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It must be mentioned that the levels of the RMS o~ are markedly higher at that station than at Nottingham and significantly more degradation in positioning accuracy and higher residual errors are observed (Aquino et al, 2005b). Our latest experiments involved expanding the analyses during the October 2003 geomagnetic storm, by processing static carrier phase solutions for a longer period of time (29 th to 31 st October) and by
Fig. 3 Comparison of baseline and network static carrier phase solutions and correlation with phase scintillation values
The correlation of errors with the o~ index suggests the possibility of de-weighting carrier phase measurements with the use of a scintillation related parameter. These findings require further investigations, but are nevertheless encouraging and give scope for the potential development of
863
864
M. Aquino • A. Dodson. J. Souter • T. Moore
warning/mitigation mechanisms that could be based on phase scintillation indices.
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The correlation between phase scintillation levels and position accuracy degradation observed in section 3 prompted more detailed investigations. Although correlation was seen when 2 hour and 15 minute sessions were used, ideally a temporal resolution compatible with the time span which the scintillation indices referred to (1 minute) should be used for comparisons. We used the Bernese software, of the Astronomical Institute of the University of Berne, to shorten the processing interval to 1 minute, through their 'pseudokinematic' solution. This is a kinematic solution whereby epoch-wise corrections to the a priori coordinates of the rover receiver are computed, under the assumption that its displacement is small, so that it 'remains within the linear regime of the partial derivatives' (Hugentobler et al, 2001). In our case this solution was indeed suitable, as in all of our experiments the 'rover' station was actually static. That therefore allowed us to assess more directly the relationship between positioning accuracy and its degradation under the influence of increased levels of phase scintillation, in our case estimated by the 60 seconds cy~ index. The result may be assessed by examining the plots of figure 4. We processed the same baseline as above (i.e. between our monitor in Bronnoysund and EUREF station VILO) using Bernese's epochwise (1 minute) pseudo-kinematic solution. Again station VILO was held fixed and solutions were obtained for NSF03. The time span was again 29 th to 31 st October 2003. Besides the ionosphere corrected L1 double difference carrier phase solution (bottom plot) another two different observables were used in an attempt to assess and compare the effects respectively on the L1 C/A code (middle plot) and on the phase smoothed L1 C/A code (top plot) solutions. For each plot the dark line represents the epochwise 3D positioning errors, sorted in ascending order. The small dots are the corresponding average values of the 60 seconds phase scintillation index for all satellites tracked within the corresponding epochs, as measured at Bronnoysund. It can be seen that correlation is visible on the top and bottom plots, where scintillation values begin to increase as the errors are greater, towards the right of the plots.
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s o l u t i o n for b a s e l i n e V I L O - N S F 0 3
(see text for details)
A less clear correlation is seen on the middle plot, confirming that the C/A code solution does not seem strongly affected by phase scintillation (as noted for instance by Conker et al, 2003, that 'phase scintillations have little effect on code tracking errors'). However, simply introducing the carrier phase data to smooth the code measurements causes the positioning accuracy to degrade, as seen in the top plot. Also, in 9.6% of the epochs an ionospheric free solution was not possible, due to missing L2 data (extreme right of the bottom plot). Analyses at the lower latitude of Nottingham (-53°N) were also undertaken, using the same pseudo-kinematic approach, in order to further the investigations of section 3. The same time span (29 th to 31 st October 2003) was analysed with two different baselines. The first baseline was between our scintillation monitor (NSF06) and station IESG of section 3, whose antenna is separated by about 3m from our scintillation monitor's antenna. The second baseline was between NSF06 and the permanent Ordnance Survey of Great Britain (OSGB) station NOTT, located about 2km to the north, where an Ashtech ZXII receiver is also
Chapter 121 • Ionospheric Scintillation Effects on GPS Carrier Phase Positioning Accuracy at Auroral and Sub-Auroral Latitudes
installed. In both cases the baselines were processed to obtain a solution for NSF06 coordinates, with the opposite end of the baseline fixed. The aim was to carry out the analysis when all stations were, in all likelihood, subjected to similar, if not identical, levels of scintillation. In the previous analyses involving our Nottingham monitor (section 3) a regional network had been formed, with other stations probably not sensing the same scintillation effects. Results are seen in figures 5 and 6, where the dark dots show the deviation of the 3D positioning errors from the mean error and the light crosses are the average phase scintillation values for all satellites within each 1 minute epoch. Firstly, we focus on figure 5. Despite the slight departure from the mean near epoch 2800 (late night on the 3 0 th October and early morning of 31 st October), in correspondence with the peak in scintillation levels, no significant increase in the errors is observed that could be clearly correlated with the higher scintillation values. Analysing the pseudo-kinematic solution file it was possible to verify that, although over the 3 days there were 1.8% of epochs when no solution was possible, these epochs did not concentrate particularly around the period of high scintillation.
separated by only about 3m, implying that even in the presence of small scale irregularities (tens of meters), it is much likely that the link from each satellite to the receivers traverse the same electron density, causing the effect to cancel out. When analysing figure 6, it is possible to see that for the 2km baseline a significant effect takes place in line with the higher scintillation levels. A visible degradation in accuracy correlates with the peak in the 60 seconds c~ values. In contrast with the shorter baseline now the percentage of epochs without solution nearly doubles, to 3.2%. The pseudo-kinematic solution, clearly, was computed in post-processing. When no solution is possible Bernese interpolates the values of the coordinates using the surrounding valid solutions. It can be seen that next to epoch 2881 (beginning of 31 st October) the errors increase quite remarkably in line with the increase in the phase scintillation levels. During that period of time many of the solutions seen in the plot are the result of interpolations (sparse dark dots), however these (intermediate) results can only be obtained because for some of the epochs a solution was actually computed.
-2km ~3m b a s e l i n e
NSF06-1ESG
3.2%
1.8% of epochs no solution was possible
~n > co
0.42
baseline
NSF06-NOTT
of e p o c h s no s o l u t i o n was possible
I
:.
0.42
6
.°
5 ,.¢o
_
0.28
0.28
,._
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og
oo
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~
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.
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~
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= = 0.14
e E
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0.28
961
1441
1921
2401
2881
-
3361
0
--
3841
-o
1 minute epochs past midnight 481
961
1441
1921
2401
2881
3361
3841
1 minute epochs past midnight
Fig. 6 Pseudo-kinematic solution for baseline NOTT-NSF06 Fig. 5 Pseudo-kinematic solution for baseline IESG-NSF06
and correlation of errors with phase scintillation values
and correlation of errors with phase scintillation values
If the quality of the carrier phase measurements was somehow degraded during that period, both receivers were affected to a similar degree, as only very marginal positioning accuracy degradation could be seen during that period. Given the short separation of about 3m between the two antennas, the corresponding ionospheric pierce points are also
The conclusion is that for the epochs where the solution was possible (e.g. the top-most dark dot) the phase measurements used in the computation seem to have been corrupted to the point of notably degrading the positioning accuracy. In this particular case the maximum error observed in the chosen period was over 7m. Clearly the two receivers involved are not the same and differences in the robustness to scintillation by individual receivers is
865
866
M. Aquino • A. Dodson. J. Souter • T. Moore
well documented (e.g. Skone et al, 2001), so this may be partly responsible for the effect not cancelling out. For this baseline length, the separation of the pierce points for the two receivers is just under 2km, implying that in the presence of small scale irregularities, which could be smaller than 2km, the effect may not be expected to cancel out. This may have implications for applications that rely on the spatial correlation of errors over short baselines, such as RTK GPS, with potential adverse effects on carrier phase ambiguity resolution. 5
Conclusions
Studies carried out based on long term ionospheric scintillation data recorded in Northern Europe revealed that the worst case scenario for a GPS/GNSS user is the loss of lock on satellites due to strong scintillation. Statistical analyses on the data showed that during periods of severe geomagnetic activity the probability of 2 satellites being simultaneously affected by strong scintillation may be as much as 3% at auroral latitudes, with implications in particular for safety critical applications such as civil aviation. In this paper we concentrated on effects that occur when degradation of measurement quality and positioning accuracy take place as a result of scintillation occurrence, rather than on the implications of loss of satellite lock. Clearly these effects will be the more harmful the greater the number of satellites that are subjected to strong scintillation. Effects on C/A code standalone positioning were seen not to show a direct correlation with scintillation occurrence and conventional DGPS positioning was seen to be affected by TEC gradients that develop under adverse geomagnetic conditions. It was also shown that EGNOS ionospheric corrections may falter when they are most needed, potentially due to scintillation occurrence affecting its reference stations. We analysed in particular carrier phase data and resulting positioning performance during periods of severe geomagnetic activity, with which strong scintillation is normally associated at sub-auroral and auroral regions. Correlation was seen between phase scintillation and static and pseudo-kinematic carrier phase data processing results, both in terms of 3D errors and RMS of residuals. A baseline of approximately 240km was analysed using 15 minute sessions on a carrier phase static solution, where errors were seen to correlate with
scintillation occurrence. Also in this case a network approach was shown to improve positioning accuracy. The adverse influence of phase scintillation was also seen when this baseline was processed using a (1 minute) epochwise pseudokinematic solution. More importantly it was verified that effects may be significant at the sub-auroral latitude of Nottingham (-53°N). Analyses carried out on a 2km baseline in that area showed the occurrence of errors not observed on a significantly shorter baseline, of approximately 3m in length. In the 2km baseline, errors reached values over 7 meters on a (1 minute) epochwise pseudo-kinematic solution, suggesting that scintillation may cause problems for RTK GPS, with possible adverse effects on carrier phase ambiguity resolution.
References
Aarons, J. (1997). Global Positioning System Phase Fluctuations at Auroral Latitudes. Journal of Geophysical Research, 102, A8, 17219-17231. Aquino, M., F. S. Rodrigues, J. Souter, T. Moore, A. Dodson and S. Waugh (2005a). Ionospheric Scintillation and Impact on GNSS Users in Northern Europe: Results of a 3 Years Study. Accepted .for publication on the Space Communications Journal, 10S Press. Aquino, M, T. Moore, A. Dodson, S. Waugh, J Souter and F. S. Rodrigues (2005b). Implications of Ionospheric Scintillation for GNSS Users in Northern Europe (2005b). The Journal of Navigation, 58, pp 241-256. Conker, R. S., M. B. E1-Arini, C. Hegarty and T. Hsiao (2003). Modelling the Effects of Ionospheric Scintillation on GPS/SBAS Availability, Radio Science, 38, 1. Hugentobler, U., S. Schaer and P. Fridez (2001). Bernese GPS Software Version 4.2. Astronomical Institute, University of Berne. Mitchell, C., L. Alfonsi, G. De Franceschi, M. Lester, V. Romano and A. Wernik (2005). GPS TEC and Scintillation Measurements from the Polar Ionosphere during the October 2003 Storm. Geophysical Research Letters, 32, L 12S03, doi."10.. 1029/2004GL021644 Moore, T., M. Aquino, A. Dodson, S. Waugh and C. Hill (2002). Evaluation of the EGNOS Ionospheric Correction Model under Scintillation in Northern Europe, In: Proc. of ION GPS 2002 Conference, Copenhagen, Denmark, pp1297-1306. Rodrigues, F. S., M. Aquino, A. Dodson, T. Moore and S. Waugh (2004). Navigation, Journal of the Institute of Navigation, 51, 1, pp 59-76. Skone, S., K. Nudsen and M. de Jong (2001). Limitations in GPS Receiver Tracking Performance under Ionospheric Scintillation Conditions, Physics and Chemistry of the Earth (A), 26, 6-8, pp 613-621.
Chapter 122
The impact of severe ionospheric conditions on the GPS hardware in the Southern Polar Region D. A. Grejner-Brzezinska, C-K. Hong, P. Wielgosz
Satellite Positioning and Inertial Navigation (SPIN) Lab Department of Civil and Environmental Engineering and Geodetic Science The Ohio State University, 470 Hitchcock Hall, Columbus, OH 43210-1275, USA L. Hothem United States Geological Survey 521 National Center, 12201 Sunrise Valley Dr., Reston, VA 20192 USA
Abstract. The primary objective of this paper is to present results of an experiment to determine the effects of moderate and severe ionospheric conditions on the GPS signal tracking by different L1/L2 receivers operating in the Southern Polar region. In this study, data collected by the Ohio State University (OSU) and the U.S. Geological Survey (USGS) joint team within the TAMDEF (Transantarctic Mountains Deformation) network were used together with the IGS and UNAVCO Antarctic stations. Seventeen Antarctic stations equipped with different dual-frequency GPS hardware were selected, and data were evaluated for two 24-hour periods of severe ionospheric storm (October 29 th, 2003) and active ionospheric conditions (moderate storm of November 11th, 2003). The UNAVCO QC software was used to carry out the analyses. Depending on the data sampling rate and the elevation mask angle, the expected number of observations per receiver/satellite was compared to the actual number of measurements collected during the ionospheric storms, with a special emphasis on L2 data. Depending on the receiver model, the number of lost measurements during the severe ionospheric conditions ranged from 0.5% to 30.0%. In addition, the number of cycle slips (CS) per number of observations as a function of receiver model was computed; it shows great variation for different hardware. The possible variability of the ionospheric conditions at some of these sites (due to their separation) is considered in the conclusions. The results indicate that depending on the severity of ionospheric conditions, there is a significant difference in the impact on the operations of different hardware models. Thus, careful hardware selection is needed to assure data quality/continuity when observations may be affected by severe ionospheric disturbances.
Keywords. Ionospheric effects on GPS, GPS hardware, GPS signal tracking conditions
1 Introduction The ionosphere is one of the primary error sources that affect the GPS observables. The carrier phase is subject to phase advance while the pseudorange is subject to a code delay. The group delay/phase advance effect is due to the propagation through an ionized medium where the refractive index changes throughout the medium, while the increased signal dynamics accounts for total electron content (TEC) fluctuations. Under disturbed geomagnetic conditions, the error due to ionosphere may reach tens or even hundreds of meters, and still under relatively benign conditions this effect amounts to meters or tens of meters, depending on the geographic location of the GPS station, the time of day (solar angle) and the 11year Sun-spot cycle. Another ionospheric effect is the signal loss of lock, due to radio wave scintillations that are amplitude and phase fluctuations originated by small-scale electron density irregularities. The regions where ionospheric scintillations are more present are low, auroral and polar latitudes in a geomagnetic sense. Rapid ionospheric disturbances may cause cycle slips (CS) and losses of signal lock, since a Doppler shift of the signal caused by ionospheric disturbances may exceed the bandwidth of the phase tracking loops (note that tracking loops are tunable to account for Doppler shift within certain limits). A loss of code lock can occur as a result of an increase in fading amplitude and a drop in the signal-to-noise (SNR) ratio. Since the ionosphere is a dispersive medium, the L2 signal with a wavelength of 24.45 cm is subject to larger ionospheric code delays than the L1 signal with a wavelength of 19.04 cm. By design, the L2 signal
868
D.A. Grejner-Brzezinska • C-K. Hong. P.Wielgosz • L. Hothem
is weaker compared to L1; the transmitted power levels are 23.8 dBW and 19.7 dBW for the encrypted P-code signals on L1 and L2, respectively, and 28.8 dBW for the L1 C/A-code signal (Langley, 1998). Moreover, due to L2 signal encryption, it is tracked using codeless and semicodeless techniques, which are more likely to give rise to loss of lock due to the lack of despreading gain. Most of the receivers provide a hybrid tracking technique, where the L1 carrier phase is reconstructed by code-correlation using the C/A code, while a codeless or semi-codeless technique is used to acquire the L2 carrier phase. The codeless technique assumes no knowledge of the Y-code, while the semi-codeless method relies on the knowledge of the known P-code, and the fact that the Y-code is a modulo-2 sum of the P-code and the W-code, making it possible to remove the P-code component of the modulation using a locally generated replica of the P-code (see, Woo, 1999 for more details on semi-codeless and codeless tracking). The codeless techniques, such as signal squaring, and cross correlation, result in SNR changes by -30 dB and-27 dB, respectively, while the corresponding numbers for the semicodeless techniques (code correlation plus squaring, and Z-tracking) are -17 dB and-14 dB, respectively (Ashjaee and Lorenz, 1992; Woo, 1999; Hofman-Wellenhof et al., 2001). In addition, the L2 phase tracking loops apply a narrower bandwidth, compared to L1. As a result, L2 is often lost, while the L 1 signal may still be tracked. Thus, the tracking performance of a GPS receiver depends not only on the state of the ionosphere (and thus, the amount of the ionospheric disturbances), but it is also a function of the receiver tracking capabilities. Therefore, the purpose of the study presented here is to determine the level of differences in the receiver tracking performance under severe ionospheric conditions for different types of geodetic-grade receivers. The motivation behind this study is to determine the most suitable GPS hardware for operational deployments in Antarctica, where ionospheric conditions are much more severe, as compared to mid-latitudes, and thus, the hardware tracking continuity and signal quality are of major concern. It should be noted that the comparison was performed for receivers significantly spread in latitude and longitude, under the assumption that the ionospheric conditions were roughly the same on average for the analyzed periods. Due to general ionospheric structure dynamics, it is a
reasonable assumption; however, more appropriate would be to test the receivers connected to the same antenna, to assure no time and space dependency as a function of the storm evolution.
2
Experimental
data
and
test
conditions
Seventeen GPS stations in Antarctica (Figure 1), where nine different types of geodetic-grade receivers are deployed, were analyzed. Several stations of the TAMDEF (Transantarctic Mountains Deformation) network were used together with the IGS and UNAVCO Antarctic stations; TAMDEF is a joint project between The Ohio State University and the US Geological Survey, to measure crustal motion in the Transantarctic Mountains of the Southern Victoria Land (http ://www.geology.ohiostate.edu/TAMDEF/). Figure 2 illustrates examples of the antenna location at some of the analyzed sites. The test datasets include the ionospheric superstorm of October 29 th, 2003 and a moderate ionospheric storm of November 11 th, 2003", the Kp index for both days is plotted in Figure 3. The UNAVCO QC (hereinafter QC) version 3.0 software was used for the analyses; the ionospheric effects based on L2 data are presented, with a special emphasis on the number of cycle slips and amount of data loss under increased ionospheric activity; an elevation cut-off of 10 ° was used. The series of geomagnetic and radiation storms, including the superstorm of October 29 th, 2003 represents a very interesting phenomenon. Solar Cycle 23 began its 11-year cycle in May 1996, and reached its peak in April 2000. While the level of ionospheric activity was low at the beginning of October 2003, by the end of the month, massive sunspot groups developed on the solar surface, leading to the most severe solar activity observed in recent years. For example, 17 major solar flares occurred between October 19 th and November 5 th, 2003; geomagnetic storms observed at that time reached extreme (G5) levels, and solar radiation storms reached severe ($4) levels, according to NOAA's Space Weather Scales (http ://www. sec.noaa.gov/NOAAscales/), and the 6th most intense geomagnetic storm since 1932, and the 4th most intense radiation storm since 1976 occurred on October 29 th, 2003. The largest solar flare recorded occurred on November 4 th, 2003 (U.S. Department of Commerce, 2004).
Chapter 122 • The Impact of Severe Ionospheric Conditions on the GPS Hardware in the Southern Polar Region
/
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Fig. 1 Test site locations.
0
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18
21
24
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S u m Kp = 40-
8
6
o o
3
6
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12 15 UT hour
Fig. 3 Kp index, October 29 th, 2003 (top) and November th 11 , 2003 (bottom).
Fig. 2 Examples of antenna locations/mounts for selected sites (from the top" CRAR, PALM, A M U N and MCM4).
869
870
D.A. Grejner-Brzezinska • C-K. Hong. P. Wielgosz • L. Hothem
Table 1. Hardware characteristics and grouping into common categories; L2 tracking techniques: C = codeless, SC = semicodeless. Receiver Characteristics Station Index
.-i =
Station Name
Antenna
REC #
Type
Version
CR1
MT312404634
TPS LEGACY
2.3
2
CRAR
MT310943221
TPS LEGACY
3
CAS 1
C 122U
DAV1
1
ANT #
Type
C
-Unknown-
Trimble Choke ring
C 1, L1, L2, P 1, P2, D 1, D2
2.3
C
RRA078
JPSREGANT_ DD_E
C1, L1, L2, P1, P2, D1, D2
AOA ICS-4000Z ACT
00.02.05 / 3.3.32.3
C
259
AOAD/M_T AUST
L 1,L2,P 1, P2,S 1,$2
C119U
AOA ICS-4000Z ACT
00.17.12 / 3.3.32.4
C
193
AOAD/M_T AUST
L 1, L2, P 1, P2, S1, $2
MAW 1
C 119U
AOA ICS-4000Z ACT
00.12.06 / 3.3.32.4
C
275
AOAD/M_T AUST
L 1,L2,P 1, P2,S 1,$2
A
4
B
5
Observation Types
6
C
MCM4
R1111
AOA SNR-12 ACT
3.3.32.2
C
363
AOAD/M_T JPLA
L 1, L2, P 1, P2, C1, S 1, $2
7
D
OHI2
T-394U
AOA SNR-8000 ACT
00.01.31 / 3.3.32.3
C
414U
AOAD/M_T DOME
C 1,L 1,L2, P1,P2
AMUN
LP01937
ASHTECH Z-XII3
1L00 1D04
SC
CR1369
ASH700936D _M SCIS
C1, L1, L2, P1, P2, D1, D2
9
PALM
RS00178
ASHTECH Z-XII3
CD00 1D02
SC
CR14107
ASH700936D _M SCIS
C1, L1, L2, P1, P2, D1, D2
10
DAVR
ZX00106
ASHTECH Z 18
0064
SC
193
AOAD/M_T AUST
C1, L1, L2, P1, P2, D1, D2
11
OHI3
ZX199909119
ASHTECH Z18
0064
SC
CRL 2199911002
ASH701941.B SNOW
C1, P1, P2, L1, L2, D1, D2
12
CONZ
3751A21424
TRIMBLE 4000SSI
7.19B
C
0220194270
TRM33429.20 +GP
C1,L1,L2,P1,P 2,S1,$2
13
ROY0
3753A21565
TRIMBLE 4000SSI
NP 7.19 / SP 3.04
C
0220067539
TRM29659.00
C1,L1,L2,P1,P 2
14
SYOG
13343
TRIMBLE 4000SSI
Nav.7.19/SIG 3.04
C
476
AOAD/M_T DOME
15
VESL
3733A20280
TRIMBLE 4000SSI
Nav 7.29 SIG 3.07
C
022008
TRM29659.00 TCWD
C1,L1,L2, P2
8 E
F
C1,L1,L2,
G
P1,P2,D1, S1,$2
16
H
HOOZ
0220219345
TRIMBLE 4700
1.37
C
0220195127
TRM33429.20 +GP
C1,L1,L2,P1,P 2
17
I
CR2
nsj 02210016
DL4-Novatel
2.110
SC
-Unknown-
Trimble Choke ring
C 1, L 1, L2, P2, D1, D2
As a result, TV and radio satellite services experienced serious problems maintaining routine operations. The loss of the Japanese ADEOS-2 spacecraft is attributed to these storms, while Mars Explorer Rover entered "sun idle" mode due to excessive star tracker events and the l~iars th Odyssey had a memory error on October 29 ; the MARIE instrument on Mars Odyssey suffered from a temperature red alarm and had to be powered off on October 28 th. The instrument is not ext~ected to recover (U.S. Department of
Commerce, 2004). Numerous Earth orbiters experienced computer problems, communication disruptions and data loss during downloads; WAAS services were declared unsuitable for commercial aircraft precision approach for 15 hours on October 29 th, 2003 and 11 hours on October BO th . Antarctic stations suffered due to no HF radio communication between McMurdo and Amundsen-Scott South Pole Station for almost a week; as a result numerous local and to/from Antarctica flights were cancelled (Kuenning,2003)
Chapter 122 • The Impact of Severe Ionospheric Conditions on the GPSHardware in the Southern Polar Region
3 Results and analyses Table 1 lists the characteristics of the hardware tested; stations are grouped into nine categories (A,B,C,D,E,F,G,H,I) according to the type of receiver deployed. It should be noted that identical receivers listed in the table may use different antennas, but they are still considered within the same category. Table 2 provides additional receiver characteristics; notice differences in the number of tracking channels and observation types. However, a relative measure of tracking capability is used here, namely, an expected number of L2 measurements is computed and compared with the actual L2 GPS tracking, as shown in Figures 4a and b, which illustrate the data collection rates for all the receivers and the prescribed categories. Note that a 12-channel tracking capability was assumed in QC for all receivers tested. Receivers 1, 13, 16 and 17 did not have data collected on October 29, 2003. Another receiver tracking performance indicator is the number of cycle slips in the carrier phase data. Two cases were analyzed separately:
Table
2. Hardware tracking characteristics.
Station Index 1
Station Name
Group Index
CR1
Data interva 1 [sec] 15
A 2
CRAR
15
3
CAS1
30 30 30
4
DAV1
5
MAW1
B
6
MCM4
C
30
7 8 9
OHI2 AMUN PALM
D
30 30 30
10
DAVR
11
OHI3
30
12 13 14 15 16 17
CONZ ROY0
30 30
E
30 F
SYOG
VESL HOOZ CR2
G H I
30
30 15 15
Total # of tracking channels
36 (GPS + GLONASS) 36 (GPS + GLONASS) 28 (GPS only) 28 (GPS only) 28 (GPS only) 28 (GPS only) 28 (GPS only) 28 (GPS only) 28 (GPS only) 33 (GPS + GLONASS) 36 (GPS + GLONAS) 28 (GPS only) 28 (GPS only) 28 (GPS only) 28 (GPS only) 28 (GPS only) 28 (GPS only)
(1) cycle slips due to ionosphere and (2) cycle slips due to multipath. The ionospheric slip occurs when the rate of ionospheric delay variations is too high for the receiver tracking loop bandwidth, as already explained. In the analyses presented here it was assumed that if the rate of change of the ionospheric delay on L 1 (referred to as IOD in QC) exceeds 400 cm/min, a cycle slip due to the ionosphere is considered. Figures 5 and 6 illustrate the results of the ionospheric cycle slip analysis. QC also keeps track of the multipath slips. The multipath running average, using a 50-data point window, and the corresponding RMS for each satellite, are computed. The expected mean RMS (per station) for the P-code multipath is 50 cm (for P l multipath, denoted as MP1) and 65 cm (for P2 multipath, denoted as MP2). It is used in conjunction with the multipath slip sigma threshold of 4 to determine multipath slips. If the multipath RMS value is less than the expected one, this value is multiplied by the sigma to obtain the amount by which the multipath must change to be considered a slip. If the current running RMS is greater, then it will be used instead of this value. Jumps in multipath large enough to be considered as slips occur when both the L1 and L2 phase values get reset to zero after a loss of lock. Since both L 1 and L2 change, a slip does not show up in the ionospheric value (UNAVCO, 1994). Figures 7 and 8 provide the summary statistics for CSs due to multipath and ionosphere for all receivers (Figure 7) and receiver categories (Figure 8).
Data collection percentage 105
[~1
100
Severe Moderate
_ 95-90-85. 8O 75 70 65 60
0
2
4
6
8
10
12
14
16
18
Station Index
Fig. 4a Data collection rates per individual receiver (black bars refer to the October 29th, 2003 superstorm; white bars represent a moderate storm of November 11 th, 2003).
871
872
D.A. Grejner-Brzezinska • C-K. Hong. P. Wielgosz • L. Hothem Data collection percentage 105 m I
100
Severe I Moderate m
95 90
,....,
85 80
Due to the space limitations, the complete statistics of MP1 and MP2 R M S are not shown here, However, it should be ment i oned that receivers 3 and 4 (two out of three in category B) display the smallest MP1 and MP2 R M S values, while receivers 6 and 15 show the largest. While the majority of the receivers show a m e a n MP1 R M S b e l o w the 50 cm threshold, its value for
receivers 1, 2, 8 (only for the superstorm), 13 and 17 exceeds the threshold value. The MP2 RMS threshold of 65 cm is exceeded by receivers 1, 2,
75 70 65
6, 8 (only for the superstorm), 13, 14, 15, and 17.
6O
A
B
C
D
E F Group Index
G
H
I
Fig. 4b Data collection rates per receiver category (black bars refer to the October 29 th, 2003 superstorm; white bars represent a moderate storm of November 11 th, 2003).
Slips per observation (IOD or MP)
Slips per obser~tion (IOD)
- - -
[
~ I
[
~ I
Severe I Moderate
Severe I Moderate
~4 4-
3
3
0 0
ii L 2
I1_ m_ m_ • 4 6
i i:Iti--i8 10 Station Index
12
14
16
0 0
18
Fig. 5 Cycle slip statistics due to ionosphere per receiver type (black bars refer to the October 29th, 2003 superstorm; white bars represent a moderate storm of November 11 th, 2003).
,, L 2
,,~ .,.. |~ I,4 6
8 10 Station Index
I 12
14
16
18
Fig. 7 Cycle slip statistics due to ionosphere and multipath per receiver type (black bars refer to the October 29th, 2003 superstorm; white bars represent a moderate storm of November 11 th, 2003).
Slips per observation (IOD or MP)
Slips per observation (IOD) Severe ]Moderate
I
8
~ [
==
Severe I Moderate
5
~4 4. 3
0
•
, A
m,_ B
m, C
----ii----i
D
E F Group Index
G
H
]--I I
Fig. 6 Cycle slip statistics due to ionosphere per receiver category (black bars refer to the October 29th, 2003 superstorm; white bars represent a moderate storm of November 11th, 2003).
A
B
C
D
E F Group Index
G
H
I
Fig. 8 Cycle slip statistics due to ionosphere and multipath per receiver category (black bars refer to the October 29th, 2003 superstorm; white bars represent a moderate storm of November 1l t h 2003).
Chapter 122 • The Impact of Severe Ionospheric Conditions on the GPS Hardware in the Southern Polar Region
Table 3. MP slip statistics (number of CS) as a function of satellite elevation angle.
Station Index
3 4 5 6 7
10 11 12 13 14 15 16 17
Station Name CR1 CRAR CAS1 DAV1 MAW1 MCM4 OHI2 AMUN PALM DAVR OHI3 CONZ
ROY0 SYOG VESL HOOZ CR2
Group Index A
C D
G
H
MP1 slips
MP2 slips
10/29
11/11
10/29
11/11
<25 ° 1>25 o
<25 ° 1>25 o
<25 ° 1>25 o
<25 ° 1>25 o
N,'A 18 10 9 123 71 109 220 37 338 123 46 N/A 136 247 N/A N/A
2~)6 19 10 10 128 73 101 46 4 467 100 77 162 97 117 6 270
N/A 17 10 8 210 453 85 119 25 366 137 50 N/A 156 283 N/A N/A
74 14 6 10 215 303 80 23 3 472 106 89 194 133 164 4 276
N/A 28 1 3 2 4 0 479 5 552 1311 164 N/A 407 269 N/A N/A
14 2 0 2 0 3 0 0 0 806 1040 107 111 185 73 13 13
N/A 34 1 3 2 18 0 288 6 563 1316 167 N/A 410 286 N/A N/A
10 2 0 2 0 8 0 3 0 812 1037 112 114 190 74 13 12
Table 4. Ionosphere-related slip statistics (number of CS) as a function of satellite elevation angle.
L1 slips Station Index
4 6 7
10 11 12 13 14 15 16 17
Station Name CR1 CRAR CAS1 DAV1 MAW1 MCM4 OHI2 AMUN PALM DAVR OHI3 CONZ
ROY0 SYOG VESL HOOZ CR2
Group Index A
B C D
G
H I
10/29 <25 °
1>25o
N/A 0 9 8 92 61 1381 511 7 0 133 38 N/A 49 0 N/A N/A
N/A 0 0 0 0 4 888 1337 0 0 1288 1 N/A 5 0 N/A N/A
L2 slips 11/11
10/29
11/11
<25 ° I >25
<25 ° I >25
<25 ° I >25
0 0 6 11 94 50 147 4 7 0 110 44 52 53 0 41 331
0 0 0 0 0 4 2 0 0 0 1034 2 0
1 0 6 86
N/A 0 9 8 92 60 1381 785 75 0 133 97 N/A 187 14 N/A N/A
N/A 0 0 0 0 4 888 1474 0 0 1288 167 N/A 407 21 N/A N/A
0 0 6 10 94 50 147 181 14 0 110 128 153 149
1 32 366
0 0 0 0 0 4 2 9 0 0 1034 109 111 185 4 7 33
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D.A. Grejner-Brzezinska • C-K. Hong. P. Wielgosz • L. Hothem
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TEC over Antarct~a 302/2003 00:00
TEC over Antarctincoa 302/2003 22:00
1 80°W
1 80°W
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4
8
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2J8
3f2
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1
44
Fig. 9 Examples of Total electron content (TEC) over Antarctica, October 29th, 2003 (DOY 302).
To complete the CS statistics, Tables 3 and 4 compare the numbers of CSs as a function of the satellite elevation angle. In general, more slips happen at lower elevation (below 25°); however, there are receivers where significantly larger numbers of CSs occur for the signals observed above 25°; e.g., receiver 8, 11 (only on October 29 th) and 14 (for MP2 and L2 slips). Based on the test results presented here, it is clear that significant differences can be found in the level of impact of the ionospheric conditions on different geodetic-grade GPS hardware. The levels of data loss and cycle slips vary among the receiver categories. For example, 5-8 percent of CSs and 15-30 percent of data loss were observed for categories D and F. However, receivers in group F did not behave similarly- receiver 11 is significantly worse than receiver 10 (this may be contributed to different antenna type). Groups A and B have the smallest amount of data loss of 13 percent and the CS rate is less than 1 percent. No significant difference between the effects of the superstorm and moderate storm was found for groups A, B, C and F, while groups D, E and G display the largest differences between the two events (0.3 and-~8 percent of CS, respectively, and 12 and 29 percent of data loss, respectively, for group D; 0.1 percent and 1.8 percent of CS, respectively, and 3 and 11 percent of data loss, respectively, for group E; 1.0 percent and 2.2 percent of CS, respectively, and 8 and 14 percent of data loss, respectively, for group G. Group !
performed well under moderate conditions (with 3.0 of percent data loss); no comparison was done to the superstorm conditions, as no data were collected by this station on October 29 th. In general, more CSs were caused by the rapidly varying ionosphere, as compared to multipath slips. Since the receiver performance strongly depends on the L2 tracking technology, with codeless receivers generally expected to perform worse than the semi-codeless ones, due to the more significant loss of the signal strength, as compared to the semi-codeless approach, as already explained (Woo, 1999). Based on the results presented here, a correlation of the overall receiver performance with the tracking technique is evident; however, location-specific factors also play a significant role. For example, receivers from categories A and B (both codeless) showed similarly very good tracking performance during both storms, while categories F (semi-codeless) and G (codeless) showed similarly weak tracking capabilities. Receiver 7 (codeless) displayed the largest difference in signal tracking for both events, as already mentioned, and a semi-codeless receiver 11 was subject to largest losses of signal in both cases.
4 Summary and conclusions The tests presented here quantify the differences in the tracking abilities among a number of
Chapter 122 • The Impact of Severe Ionospheric Conditions on the GPS Hardware in the Southern Polar Region
geodetic-grade GPS receivers deployed in Antarctica, under disturbed ionospheric conditions. Two ionospheric storms, with significantly different magnitudes, were tested. Substantial differences were found among the hardware types, indicating the importance of a careful selection of GPS receivers deployed in Antarctica, where the ionosphere is more active, as compared to mid-latitudes. Differences in receiver performance are a result of several factors. Apart from the L2 tracking technique, tracking loop bandwidth, firmware version, the in-receiver processing algorithms, and the antenna gain pattern make a significant difference in the overall performance of a given receiver. The antenna environment is a contributing factor, too. For example, different antenna settings and types might be a reason for a considerably different performance of the same receiver type (categories F (semi-codeless) and G (codeless)). A similar study, performed by Skone and Knudsen (2000), reports better performance of the semi-codeless receivers. In this study, semicodeless receivers 9, 10 and 17 showed very good performance, even under very severe ionospheric disturbances, while semi-codeless receiver 11 displayed the worse performance among the receivers tested. Codeless receivers 1, 2, 3, 4 and 5 demonstrated a very good performance during both storms. On average, however, the performance of the codeless receivers was worse, as compared to the semi-codeless ones. It should be mentioned, however, that the comparison was performed for receivers significantly spread in latitude and longitude, and thus, the effects of the ionospheric storms analyzed might not have been identical in space and time for all receivers tested (see Figure 9). A better approach would be to test the receivers connected to the same antenna, or at least, antennas located in a close vicinity to each other, to assure no time and space dependency as a
function of the storm evolution. In such a case, the data loss and reacquisition times could be compared. However, this type of data is not available to the authors at this time. In summary, it should be pointed out, however, that some receivers tested here were in fact located in close proximity to each other, and thus, were subject to virtually identical (or very similar) ionospheric conditions. For example, receivers 1 and 2 (both TPS Legacy with an identical version of the firmware) were separated by about only 17 m, and still show different levels of data loss (by ~3percent), most probably due to the difference in the antennas used. Receiver 6 (AOA SNR-12), located within ~1000 m from receivers 1 and 2, displays a significantly lower data collection percentage, compared to receivers 1 and2. References
Ashjaee J. and Lorenz R. (1992). Precision GPS Surveying after Y-code. Proc. ION GPS-92, pp. 657-659. Hofmann-Wellenhof, B., Lichtenegger, H., and Collins, J., (2001). Global Positioning System. Theory and Practice. 5 th edition, Springer-Verlag,Wien, 382 pages. Kuenning, K. (2003). Great balls of fire, in Antarctic Sun, published at McMurdo Station, Antarctica, for the United States Antarctic Program; N o v e m b e r 9th, http ://www.polar.org/antsun/oldissues20032004/Sunl 10903/index.htm. Langley R. B. (1998). Propagation of the GPS Signals and GPS Receivers and the Observables, in GPSfor Geodesy by Teunissen P. J. and Kleusberg A. (editors), Springer, 2 nd edition, pp.112-185. Skone, S. and Knudsen K. (2000). Impact of ionospheric scintillations on SBAS performance, Proceedings of the ION GPS 2000, Salt Lake City, Utah, September 19-22, pp. 284-293. UNAVCO (1994). QC v3 Users Guide, March 1994. U.S. Department of Commerce (2004). Service Assessment on Intense Space Weather Storms, October 19-November 07, 2003. Woo, K.T (1999). Optimum Semi-Codeless Carrier Phase Tracking of L2. Proceedings of the ION GPS 1999, Nashville, TN, September 14-17, CD ROM, pp. 289-305.
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Chapter 123
Preliminary research on imaging the ionosphere using CIT and China permanent GPS tracking station data Y.B. Yuan 1, D.B. Wen 1'2, J.K. Ou ~' 2, X.L. Huo ~, R.G. Yang 1'2 1 Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan 430077, China; 2 Graduate School of Chinese Academy of Sciences, Beijing, 100039, China K.F. Zhang 3, R. Grenfel 3 3 School of Mathematical and Geospatial Sciences, RMIT University, Australia
A GNSS-based platform for imaging ionosphere and atmosphere (GPFIIA) is being developed at the institute of geodesy and geophysics, Chinese Academy of Sciences, and School of Mathematical and Geospatial Sciences, RMIT University. The GPFIIA will be an important part of a scientific software platform for processing and analyzing a large amount of different types of data associated with ever increasing number of GNSS and low-orbit earth observation satellites. Currently, GPFIIA can combine a voxel-based ionosphere tomographic technique with ground-based and space-based GPS data to image the ionospheric electron density (IED) distribution over the area of coverage, to study the vertical structures and related properties of the ionosphere, as well as improving the accuracy of ionospheric delay correction. Dual frequency GPS data selected from the Crustal Movement Observation Network of China and some simulated GPS/MET data on 1 March 2003 are used to invert a time series of the IED profiles over China using tomographic algorithms. The reconstructed IED profiles were partially validated with those obtained using the IRI2001 model. The inverted results illustrate some active characteristics of the lED. The difficulties yet to be solved for using GPS data to invert the lED profiles in China are discussed and some suggestions are given for further investigation. Abstract.
Keywords. Global Positioning System computerized ionospheric tomography ionospheric electron density (lED).
1
(GPS), (CIT),
Introduction
The ionosphere is one of the main error sources in GPS positioning and navigation. On the other hand, GPS is an important tool to monitor and investigate
the properties and activities of the ionosphere by extracting and analyzing high accuracy total electron content (TEC) information from precise dual frequency data. Determining high precision ionospheric TEC is beneficial to improving GPS positioning and navigation performance for single frequency users as well as investigating the earth's ionosphere especially for the ionosphere electron density (IED). Better knowledge of the temporal and spatial variations of the lED is helpful not only for studying the fine structure of the ionosphere but also for determining high precision ionospheric TEC to serve for scientific and application fields such as GPS/GNSS geodesy and surveying, satellite communication and altimetry. Therefore, investigation of the fine structures and other properties of the ionosphere, and precise and reliable determination of the ionospheric TEC and its effects on radio signals using high accuracy dual frequency GPS data have attracted much attention in both GPS and ionosphere research fields (eg. Klobuchar, 1987; EL-Arini et al., 1994; Skone, 1998; Skone & Knudsen, 2000;Yuan and Ou, 2002; etc). One of the key approaches applied in the above research is the so-called computerized tomography. Inverting IED has been implemented using the computerized ionosphere tomography (CIT) technique and the observations obtained from some low-orbit navigation satellite systems over the past few decades (Austen et al., 1988; Raymund et al., 1990,1995; Na and Lee, 1991; Fremouw and James, 1992). The advent of GPS has opened a new avenue for monitoring and investigating the ionospheric activities due to its many advantages such as high precision, real time and high resolution. Global and regional ground-based GPS networks (e.g. IGS) and space-based GPS (e.g. GPS/MET and CHAMP etc) observations provide unprecedented opportunity for
Chapter123 • PreliminaryResearchon Imagingthe IonosphereUsingCITand ChinaPermanentGPSTrackingStationData
further improving the CIT technique (Raymund et al., 1990; Hajj et al., 1994; Raymund, 1995; Rius et al., 1997; Komjathy, 1997; Hajj and Romans, 1998; Howe and Runciman, 1998; Ruffini et al., 1998; Gao and Liu, 2002; Hernandez-pajares et al., 1998, 2000; Michell and Spencer, 2003, Bust et al., 2004; Liu, 2004; etc). Research indicates that it is feasible to reconstruct global, regional or local IED profiles by combining GPS data with the CIT technique. It is well known that the variations of the IED over China are very complicated since China spans a large range in both longitude and latitude directions (Yuan and OU, 2001, 2002; Huo et al., 2005). The GPS Crust Movement Observation Network of China (CMONOC) has been established and currently 25-30 stations (to be increased to 250 in the second stage) are operational (Yuan and Ou, 2002). The CMONOC also provides high precision dual-frequency GPS data for studying the activities of the ionosphere over China. In view of this, to apply more and more abundant ground (and space) based GNSS data to efficiently explore the ionosphere and atmosphere and solve their effects on the radio signals that pass through them, a GNSS-based platform for imaging the ionosphere and atmosphere (GPFIIA) is being developed by the GPS group at the Institute of Geodesy and Geophysics (IGG), Chinese Academy of Sciences (CAS), and School of Mathematical and Geospatial Sciences, RMIT University. The GPFIIA will be an important part of a scientific software platform for processing and analyzing a large amount of different types of GPS/GNSS data. Since the GPFIIA is under development, some important algorithms investigated by our group and other researchers for imaging ionosphere and atmosphere need yet to be incorporated in the software in the future. In this paper, scientific backgrounds and application goals, operational requirements and development progress of the GPFIIA are briefly introduced in sections 2 and 3. In section 4, the tomographic algorithm applied in the current GPFIIA and dual frequency GPS data selected from the CMONOC network and some simulated space-based GPS data similar to GPS/MET data are employed to invert a time series of the lED profiles over China for the date selected. The inversed results are also preliminarily validated with the IRI2001 model. Since the research of imaging ionosphere using the GPS data and the CIT technique is in its early stage in China, the precision and reliability of the inverted IED profiles may not
be satisfactory, and need to be further improved. For this reason, difficulties and problems in this research are presented for future research and suggestions are also given to push the research forward in depth in sections 4 and 5, respectively.
2 Imaging the ionosphere atmosphere using GPFIIA
and
The GPFIIA together with other parts of the overall platform are expected to provide various valuable products for related areas of Earth science. This includes: (1) Ionosphere high resolution 3D ionospheric imaging, ionospheric structure and dynamics, ionosphere-thermosphere-atmospheric interactions, onset-evolution-prediction of space storms, travelling ionospheric disturbances and global energy transport; (2) Atmosphere--climatic change and weather modeling, global profiles of atmospheric density, pressure, temperature and geopotential height, atmospheric winds, tropospheric water vapor distribution, structure and evolution of surface/atmosphere boundary layer, etc; (3) Geodesy--precise orbit determination, static gravity model, temporal gravity model, etc; (4) Oceans--significant wave height, ocean geoids and global circulation, surface winds and sea state, etc; (5) Solid e a r t h - - e a r t h rotation, polar motion, vertical motion of crust and lithosphere, location and motion of the geocenter, gross mass distribution, structure and evolution of the deep interior, etc. The GPFIIA focuses on the first two parts, and is required to be able to simultaneously compute a large amount of ground-based GPS data over different scales of coverage such as local, regional and global (e.g. the IGS and the CMONOC) together with space-based GPS data obtained in low-orbit earth's satellites (e.g. GPS/MET and CHAMP ) at different altitudes over the earth's surface. It will provide various functions for modeling ionosphere and atmosphere in different modes like (horizontal or vertical) 2D, 3D and 4D by jointly or individually employing different means such as single (or multiple) layer modeling, occultation inversion, tomography imaging and so on. The above differential technical procedures can be implemented to provide efficient services for high accuracy operational systems used in geodesy (e.g. global or regional crust monitoring networks), surveying (e.g. Network RTK for different regional or local areas) and navigation (e.g. WAAS, WADGPS, etc), as well as available products for the scientific research mentioned previously, in
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Y.B. Yuan. D. B. Wen. J. K. Ou. X. L. Huo. R. G. Yang. K. E Zhang • R. Grenfel
real-time or post-processing scenario. The GPFIIA must have an ability to simultaneously perform efficient data-preprocessing for a large amount of GPS data including test and repair of cycle slips for phase data and check and correction of outliers for both phase and code data through combing many effective approaches. Also, the GPFIIA will provide some reliable and feasible test methods (e.g. the IRI2001 model, incoherent scatters data and ionosonde measurements, absolute (or relative) positioning technique with precise known coordinates (or baselines) for validating the reconstruction results. In addition, the GPFIIA needs to have a strong computational efficiency for meeting the different application requirements by applying sound mathematical methods.
Hence, the GPFIIA makes, in an order, differences between the phase measurements to every two epochs spanning a time interval within a continuous observation arc of the given pair of transmitter and receiver, in order to eliminate the above constant terms. The current GPFIIA employs the Quasi-Accurate Detection of Gross Errors method (Ou, 1999) to check the quality of GPS phase data L4, during which the key issue is to test, clean and repair possible outliers and cycle slips in the GPS phase data, before inverting lED profiles.
3.3 lED discretization and its Voxel-based inversion equation
3 CIT algorithms used in the GPFIIA 3.1 Outline of the CIT algorithms Since the GPFIIA is under development, this paper will only provide some preliminary ionosphere inversion results obtained using the CIT algorithms incorporated in the present GPFIIA software package. Currently, the GPFIIA can implement the following lED inversion algorithms and related complementary methods: (1) a radio occultationbased reconstruction method with Abel transform based on a spherical asymmetric assumption; (2) a voxel-based tomography model with a refined algebraic reconstruction algorithm (ARA) for improving the computational efficiency and overcoming the related underdetermined problem, (3) an improved empirical ionosphere model for providing better constraints for more efficiently conducting the reconstruction using the CIT, (4) efficient GPS data quality test for effectively performing the data-processing, (5) a joint inversion method for imaging lED using ground-based and space-based GPS data, and the above models and algorithms.
In essence, the so-called GPS CIT technique is an inversion procedure whereby an unknown electron density distribution is reconstructed from a set of known ray paths and some TEC data derived from GPS observations. Since the relative position of any point with respect to the sun does not vary with time in the sun-fixed reference frame, the variation of solar dark radiation is small. So, the ionosphere over China is discretized into a set of cells in the sun-fixed reference frame. Assuming that the ionospheric electron density is constant in a cell. Since the integer ambiguities term in L4 is a constant within each arc of continuous carrier phases the difference of the L4 observations between the time t iand t k is made for a satellite and receiver pair to cancel this term. Thus, the following equation (Hernandez-pajares et al., 1998) can be obtained -
•
(ASz,m," -- ASZ,m,,) l
tn
n
(1)
[,m,n
where AL - L4(t i ) - L4(t k ); o~= 1.05x 10-17mS/el;
l, m, n are the indices for each cell along longitude, 3.2 Basic observation and data-processing method for conducting the CIT Zero-differenced geometry-free dual frequency GPS phase linear combination observation, usually referred to the L4 combination by forming the difference between L1 and L2 (L1-L2), which eliminates the effects of geometry range, clock bias and tropospheric delay, is used to conduct the CIT technique. It is well known that the instrumental bias can be considered as a constant over one day and the unknown ambiguity is also a constant within an arc of continuous phase data in the observation L4.
latitude and height directions. Ne is the electron density in a cell. Astl'm'n is the length of the ray path in the cell (1, m, n) at time t. Without loss of generality, equation (1) can be further expressed in a matrix notation as follows: YMxl = a " A M × N X N x l
(2)
where Y is a column vector of M measurements; X is a column vector of N unknown parameters; A is a M x N matrix and its elements Aij are the length of the ith ray in the jth pixel.
Chapter 123 • Preliminary Research on Imaging the Ionosphere Using CIT and China Permanent GPS Tracking Station Data
3.4 A n improvement algorithm
of
the
ARA
Since only part of the GPS data is available due to its geometry limitation, Eq. (2) is underdetermined. GPS/MET occultation data are therefore simulated to make up the gaps of GPS rays for the related voxels. However, even after adding the occultation data, some of the voxels still do not have observation ray through them. So a GPS-based improved IRI2001 model is used to further provide more simulated GPS rays for completely "filling" the data gaps. In the mean time, it is difficult for normal PC computers to efficiently implement the computation task since the Matrix A in Eq (2) is tremendous and sparse. Taking computational efficiency into account, the ARA algorithm (Austen et al., 1988) is used to solve equation (2) for improving reconstruction. The basic idea of the inverse algorithm based on the ARA method is: (1) to give initial values for X ; (2) to improve X step by step by performing an iterative process during which X is corrected according to the differences between the newest two adjacent projections; (3) to determine the final solution of the X when it converges as much as possible to the true value of the equation (2). Since the implementation of the ARA needs to select a set of initial values for the model parameter X, the GPS-based improved IRI2001 model is also used to give a set of relatively precise initial values for better implementing the ARA algorithm.
4.2 Outline of experiment Inverting the ionospheric electron density is performed using the L4 (L1-L2) phase observations and the method mentioned above. A one-hour observation session is adopted: an inverted result is individually calculated using each hour of GPS data for the day. Since the selected GPS data set was collected in March 2003 (during high solar activity) and China lies in mid-latitude and low-latitude areas, the size of the voxels used here must be small enough to ensure that the actual ionospheric electron density in a cell changes as little as possible around the constant value assumed in the CIT technique. The inverted space ranges from 70 ° to 140 ° in longitude, from 10 ° to 55 ° in latitude, and from 100 km to 1000 km in height. The sizes of the cells used are 5 ° (longitude), 2.5 ° (latitude) and 20km (height), respectively. The inverted values are restricted to be non-negative when they are less than zero. i
i
i
i
i
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90
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110
120
130
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Fig. 1 The Sketch map of CMONOC contribution
4 P r e l i m i n a r y results and a n a l y s e s 4.1 GPS data As an example, dual frequency GPS carrier phase data of 23 reference stations of the CMONOC network on 1 March 2003 is used. The time series of the ionospheric electron densities is obtained over China using the improved CIT method mentioned above. These 23 CMONOC stations include: BJFS, HUN, DLHA, DXIN, GUAN, HLAR, JIXN, KMIN, LHAS, LUZH, QION, SUIY, TAIN, TASH, URUM, WUHN, WUSH, XIAA, XIAG, XIAM, XNIN, YANC, YONG. The sample interval is 30 seconds. The elevation angle of 10 degrees is adopted. When performing the CIT process using GPS data, the vertical resolution of the inverted IED profiles is strongly limited if only ground-based GPS data is used. Some simulated spaced-based GPS data similar to the GPS/MET data is used to improve the vertical resolution of the inverted results together with the ground-based GPS data.
4.3 Result, validation and analysis 4.3.1 Reconstructed lED Images A time series of the inverted ionospheric electron densities over China for the day is obtained. This includes a series of 24 hourly electron density snapshots. Due to the limitation of length for this paper, only 12 images are given here, which are taken at the following reference epochs: 1:00, 3:00, •"-, 21:00, and 23:00UT, respectively. The electron density is expressed in units of 1011 el/m 3. Note that the above inverted results were shown in the earth-fixed reference frame while they were calculated in the sun-fixed reference frame. Fig.2 shows the 12 snapshots of the inverted ionospheric electron densities corresponding to the above reference epochs. Each snapshot is plotted with respect to latitude and height in the meridian plane
879
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Y.B. Yuan. D. B. Wen. J. K. Ou. X. L. Huo. R. G. Yang. K. F. Zhang • R. Grenfel
(D ~0 ~D 10
15
20
25
30
35
10
15
20
25
30
35
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10
15
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25
30
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35
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.,..~ (D
zz
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Latitude (degree) Fig. 2 A time series of the ionospheric electron density (IED) snapshots over China for March 1, 2003 The 12 sub-figures from left to right corresponding to 1:00, 3:00, "", 23:00UT for the day, respectively (The unit of the color bar is l0 ll el/m 3)
130
14070
80
90
100
110
120
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140
Longitude (degree) Fig. 3 A time series of the inverted average ionospheric electron density over China in the sub-layer from 340 km to 360 km above the earth's surface for 1 March 2003. The 12 sub-figures from left to right corresponding to 1:00, 3:00, "--, 23:00UT for the day, respectively (The unit of the color bar is 1011el/m3)
4.3.2 Properties
of 110°E. Fig. 3 shows that the inverted snapshots in the sub-layer from 340 km to 360 km over China. Fig. 4 compares the vertical ionospheric electron density profiles inverted using the GPS data with those provided by the IRI 2001 model.
120
of the inverted
lED profiles
Comparing all sub-figures in Fig 2, it can be seen that the peak height of the ionospheric electron density gradually grows during the time period from 1 to 7 UT and then begins to fall during the nexttime period, and the peak height falls from 350km to
Chapter 1 2 3
•
Preliminary Research on Imaging the Ionosphere Using CIT and China Permanent GPS Tracking Station Data
300km at about 15UT. Meanwhile, the peak lED starts to move southward. At about 21 UT (ie 5 BT(Beijing time)), the minimum IED appears over China. This roughly reflects typical characteristics of the vertical variations of the IED for the day over China. Fig. 3 illustrates the average variations of the IED over China in the sub-layer from 340 to 360 km above the earth's surface for the day. From Fig. 3, it can be found that the inverted average IED values grow from small to large first, and then fall from large to small, which is consistent with the earth's rotation from west to east. This is also consistent with the normal change laws in daytime and nighttime over China as well as the fact that the
--
-
--
-
ground-based GPS reconstruction space-based and ground-based GPS reconstruction I R 1 - 2 0 0 1 model
characteristics of IED depends mainly on the dark solar activity. Figures 2 and 3 also show that there are larger differences between the characteristics of the IEDs in mid-latitude and low-latitude areas, and the values of IED over the north of China are smaller than those over the south of China as a whole. This indicates a strong correlation of the variation of IED with latitude. Figures 4a and 4b show the above three types of vertical IED profiles at two locations (110°E, 45°N) and (110°E, 30°N) respectively at 1 UT on the day. Fig. 4a suggests that the vertical IED profiles inverted using the CIT technique agree well with those calculated by the IRI-2001 model except around the peak density, where the reconstruction technique yields slightly lower density estimates than the IRI-2001 model, while the heights of the peak densities of the three profiles are almost the same. From Fig. 4b, it can be seen that although the peak height of the reconstructed profile is higher than that of the model profile, the two kinds of IED profiles agree well each other. Fig. 4a and 4b also show that the characteristics and values of top side and bottom side densities of all profiles also agree well. This indicates that the preliminarily inverted results are, to some extent, validated and verified using the IRI-2001 model.
200 m
4.3.3 Limitation factors 0
'
I
0
'
I
2
'
I
4
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'
I
6
profiles
8
at (110°E,
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I
'
10
density
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I 15
at
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(110°E, 30°N)
Fig. 4 Various vertical electron density profiles over two different locations
The main factors limiting the reconstruction accuracy and resolution using the present GPFIIA include: (1) Limitations of observation geometry: The available CMONOC GPS data is usually insufficient for tomography determination due to the limitation of the spatial distribution of the GPS signal propagation paths; (2) Limitations of a priori information: although some space-based data is processed together with the ground-based GPS data, the data used is still insufficient when inversing the ionospheric electron density using the CIT technique. Some other types of TEC data such as the ionosonde TEC data may be incorporated as much as possible to improve the vertical resolution of the imaged electron densities; (3) Limitations of inversion method and CIT model: Although the improved ARA process preliminarily resolves the difficulties caused due to the underdetermined matrix A, better methods for more efficiently handling the matters due to the matrix A need to be investigated. In addition, some more reasonable CIT models also need to be investigated to make more GPS data available for the reconstruction.
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Handling well the above problems will be advantageous to further improving the CIT-based IED inversion efficiency using GPS/GNSS data.
5 Summary and discussion The above results suggest that the CIT method in the GPFIIA and the CMONOC GPS data can be employed to monitor and investigate variations of the ionosphere electron density with respect to longitude, latitude, height, and time over China. Further research for improving the inversion accuracy and resolution should be done from the following aspects: (1) Incorporate other types of ionospheric measurements such as ionosonde data for the inversion; (2) Augment the CMONOC GPS network in terms of density and coverage (3) Establish a more efficient CIT model and refine its inversion method. It can be seen that the number of CMONOC GPS observation stations is increasing and becomes more evenly distributed. In addition, space-based GPS data is also becoming more and more abundant, with GALILEO and other global navigation satellite system to be implemented. All of these will open a better opportunity to further investigating the properties of the ionosphere. So, it is intended that the upcoming complete GPFIIA software will play a significant role in many application fields that need high accuracy ionosphere correction or other related services, such as geodesy, GPS/GNSS high precision positioning and surveying, satellite altimetry, communication and navigation.
Acknowledgement: Support from the National Nature Science Foundation of China (Grant Nos. 40204001 and 40474009) and Australia Research Council (LP0455170) are gratefully acknowledged.
References Austen, J. R., S. J. Franke and C. H. Liu (1988). Ionospheric imaging using computerized tomography. Radio Science, 23(3), pp. 299-307. Bust, G. S., T. W. Garner and T. L. Gaussiran (2004). Ionospheric data assimiliation 3 dimensional (IDA3D): a global, multisensor, electron density specification algorithm, Journal of Geophysical Research-Space Physics, 109 (A1 1), doi: 10.1029/2003JA010234 E1-Arini, M. Bakry. and J. Klobuchar (1994). Comparison of Real-Time Ionospheric Algorithms for a GPS Wide-Area Augmentation System (WAAS), Navigation, 41(4), 1994-1995, pp. 393-413. Fremouw, E. J., J. A. Secan and B. M. Howe (1992). Application of stochastic inverse theory to ionospheric
tomography. Radio Science, 27(5), pp. 721-732. Gao Y. and Z.Z. Liu (2002). Precise Ionosphere Modeling Using Regional GPS Network Data. Journal of Global Positioning Systems, vol. 1, No. 1, pp. 18-24 Hajj, G. A., R. Ibanez-Meier and E. R Kursinski and L. J. Romans (1994). Imaging the ionosphere with the global positioning system. International Journal of imaging systems and Technology, 5, pp. 174-184. Hajj, G. A. and Romans, L. J (1998). Ionospheric electron density profiles obtained with the global positioning system: results from the GPS/MET experiment. Radio Science, 33(1), pp. 175-190. Hernandez-pajares, M., J. M. Juan and J. Sanz (1998).Global observation of the ionospheric electronic response to solar events using ground and LEO GPS data, Journal of Geophysical Research-Space Physic), 103 (A9), pp. 207 89-20796 Hernandez-pajares, M., J. M. Juan, J. Sanz and O. L. Colombo (2000). Application of Ionospheric Tomography to Real-Time GPS carrier Phase Ambiguities Resolution, at Scales of 400-1000 km and with High Geomagnetic Activity, Geophysical Research Letters, 27(13), pp. 2009-2012. Howe, B. M. and K. Runciman and J. A. Secan (1998). Tomography of the ionosphere: Four-dimensional simulations. Radio Sciences, 33(1), pp. 109-128. Huo, X. L., Y. B. Yuan, J. K. Ou, D.B. Wen and X. W. Luo (2005). The diurnal variations, seminal annual and winter anomalies of the ionospheric TEC based on GPS data in China, Progress in Natural Science, 15 (1), pp.56-60 Klobuchar J.A. (1987). Ionospheric Time Delay Algorithm for Single Frequency GPS Users. 1EEE transactions on erospace and electronic systems, 23(3), pp. 325-331 Komjathy, A (1997). Global Ionospheric Total Electron Content Mapping Using the Global Positioning System, Ph.D thesis, Technical Report No.188, Department of Geodesy and Geomatics Engineering, the University of New Brunswick, New Brunswick Liu Z.Z (2004). Ionosphere Tomography Modeling and Applications Using Global Positioning System (GPS) Measurements. Ph.D thesis, UCGE Reports Numbers 20198, Department of Geomatics engineering, the University of Calgary, Calgary. Mitchell, C. N. and P. S. Spencer (2003). A threedimensional time-dependent algorithm for ionospheric imaging using GPS, Annals" of Geophysics, 46(4), pp.687 -696 Na, H. and Lee, H (1991). Orthogonal decomposition technique for ionospheric tomography. International Journal of Imaging System and Technology, 3, pp. 354-365. Ou, J .K (1999). Quasi-accurate Detection of Gross Errors (QUAD), ACTA GEODETICA et, CARTOGRAOHICA SINICA, 28(1),pp. 12-20 Raymund, T.D (1995). Comparisons of several Ionospheric Tomography Algorithms. Annales Geophysicae, 13(2), pp.1254- 1262. Raymund, T. D., J.R. Austen and S.J. Franke, C. H. Liu, J. A. Klobuchar and J. Stalker (1990). Application of computerized tomography to the investigation of ionospheric structure. Radio Science, 25, pp.771-789. Rius, A., G. Ruffini and L. Cucurull (1997). Improving the vertical resolution of ionospheric tomography with GPS occultation. Geophysical research letter, 24(18), pp. 2291-2294.
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Chapter 123 • Preliminary Research on Imaging the Ionosphere Using CIT and China Permanent GPS Tracking Station Data
Ruffini, G., A. Flores and A. Rius (1998). GPS tomography of the ionospheric electron content with a correlation function. IEEE Transactions on Geosciences and remote sensing, 36(1), pp.143-153. Skone, S (1998). Wide Area Ionosphere Grid Modeling in Auroral Region. Ph.D thesis, UCGE Reports Numbers 20123, Department of Geomatics engineering, the University of Calgary, Calgary. Skone S. and K. Knudsen (2000). Impact of ionospheric Scintillations on SBAS Performance, In: Proceedings of
Institute of Navigation, ION GPS-2000, Salt Lake City, September, pp. 284-293. Yuan, Y. B. and J. K Ou (2001). The First Study of Establishing China Grid Ionospheric Model, In: Proceedings of Institute of Navigation, ION GPS-2001, Salt Lake City, September, pp. 2516-2524 Yuan, Y. B. and J. K Ou (2002). Differential Areas for Differential Stations (DADS): A New Method of Establishing Grid Ionospheric Model. Chinese Science Bulletin, 47(12), pp.1033-1036
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Part IX Geodesy of the Planets Chapter 124
Gravity, Rotation, and Interior of the Terrestrial Planets from Planetary Geodesy: Example of Mars
Chapter 125
Mars Long Wavelength Gravity Field Time Variations: a New Solution from MGS Tracking Data
Chapter 126
Potential Capabilities of Lunar Laser Ranging for Geodesy and Relativity
Chapter 124
Gravity, rotation, and interior of the terrestrial
planets from planetary geodesy: example of Mars V. Dehant, Royal Observatory of Belgium, Av. Circulaire 3, B-1180, Brussels, Belgium; email: [email protected], T. Van Hoolst, Royal Observatory of Belgium, Av. Circulaire 3, B-1180, Brussels, Belgium; email: [email protected].
Abstract. Information on planetary interiors can be obtained from planetary geodesy measurements of global parameters such as the gravity field and the rotation variations (including precession). These properties can be derived from radio science experiments in which Doppler shifts of radio signals between the Earth, planetary landers, and orbiters are measured. A radio link between the Earth and one lander on Mars has been successfully used to constrain the rotation and the precession, and consequently to obtain the moment of inertia of the planet, the global mass repartition, and its seasonal variations. A radio link between the Earth and one orbiter has also been used to determine the gravity field and its time variations. In particular, the tidal Love number k: has been determined, from which the Martian core has been shown to be at least partially liquid. New missions that involve a space geodesy experiment addressing these topics are presented. A network of landers could probe the interior of the planet through seismic monitoring, magnetic sounding, and measurements of its rotational dynamics by radio science. This would allow the determination of the overall interior structure, including crust, mantle, and core divisions, and the state of the core (liquid/solid, density). Also composition, mineralogy, density, and temperature profiles could be inferred. Information on the interior of terrestrial planets (and large moons) other than Mars has also been determined by using radio tracking of an orbiter. For example, the tidal Love number k: of Venus was deduced from Magellan's radio science experiment. Future missions such as Messenger and BepiColombo have dedicated onboard planetary geodesy instruments and will provide improved information on the interior of Mercury from Doppler measurements.
Keywords. Mars tides, Mars rotation, Mars gravity field, planetary interior, radio science, planetary geodesy
1 Introduction This paper concerns the application of radio science to planetary physics. It is an overview of the present advances in planetary geodesy and it addresses the Earth geodesy community to show that the application of geodesy to other terrestrial planets is an exciting and promising field. We have chosen to concentrate mostly on the planet Mars because it is the planet for which we have the most recent data and because similar approaches for studying rotation and gravity can be used as for the Earth. Space radio science uses radio links between a spacecraft and the Earth, a spacecraft and a lander, and/or a lander and the Earth. From the observed Doppler effects on the signal the relative motions between the two objects can be determined, the trajectory or the orbit of the spacecraft can be reconstructed, and the velocity, the acceleration, and/or the position of the transmitter with respect to the antenna observing the spacecraft or the lander can be derived. It will be shown that such space geodesy experiments allow obtaining geophysical information related to the static gravity field, the time variable gravity field, and the rotation.
2 Static gravity field observation and interpretation The static gravity field can be deduced from its effects on a spacecraft orbit. From the accumulation of measurements such as those from the US Mars Global Surveyor (MGS) spacecraft performed over a long period of time and covering the whole planet, the gravity field developed in spherical harmonics can be obtained (see Lemoine et al., 2001, Yuan et al., 2001). These data can be compared with the topography as obtained from an altimeter such as MOLA on MGS (Smith et al., 2003). Regions with high correlation between topography and the free-air gravity anomaly as well as areas with very low correlation occur on
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V.Dehant.T.VanHoolst Mars (Zuber et al., 2000). A high correlation such as in the Tharsis region means that the topographic load is not fully isostatically compensated. Flexure models can then be used to determine properties of the lithosphere and the crust, such as the crustal thickness, the lithospheric rigidity, and the density of the crust. High gravity anomalies that are not explained completely by the topography are expected to be partially due to internal loading. Generally speaking, the gravity field follows Kaula's law: the degree variances of the harmonic development of the field scale as c/l 2, where 1 is the degree of the gravity coefficients and c is a scaling that depends on the planet. Kaula's law is verified for the Earth, Mars and Venus. The scaling coefficient from one planet to the other is based on the hypothesis that the terrestrial planets support the same stresses. The density anomalies are then scaled by 1/g, and as the gravity coefficients are normalized by GM/R, one has a general scaling of 1/g/(GM/R) (where g is the gravity at the surface, R is the radius, and M the mass of the planet). In the literature, one finds scalings of 1/g 2 (Kaula, 1993, Vincent and Bender, 1990, Wu et al., 1995, 1997) or 1/g (Milani et al., 2001). Line-of-sight accelerations of the spacecraft can also be deduced from the radio science Doppler m e a s u r e m e n t s . E x a m p l e s of l i n e - o f - s i g h t acceleration over the Tharsis region on Mars from the ESA Mars Express spacecraft can be found in Beuthe et al. (2005). In the case of Tharsis, the gravity anomalies are very high and can be explained with a rather high density for the crust and/or a rather high rigidity for the lithosphere and/or with an additional internal loading in phase with the topography effect (Belleguic et al., 2005). This is not in contradiction with the recent discovery of the geologically recent volcano activity on Mars as discovered by the camera team of Mars Express (Neukum et al., 2004, 2005).
3 Time variable gravity field v a t i o n a n d its i n t e r p r e t a t i o n
obser-
Changes in the gravity can be induced by changes in the mass anomalies. Especially fluid motion can lead to mass redistribution on time scales that are within reach for space missions. For Mars for instance, the seasonal changes in the icecaps change the gravity field. The sublimation and condensation of the polar caps can involve up to about one third of the total mass of the atmosphere and induces relative gravity changes on the order of 10 -9- which can be compared with the second-order gravity field coefficient J2 = 0.00195545 (Lemoine et al., 2001) - with largest
changes for the l o w - d e g r e e zonal gravity coefficients. This signature can be detected in the orbit parameters, and the variations in the lowestdegree zonal gravity coefficients have been determined (Smith et al., 2001, Yoder et al., 2003, Karatekin et al., 2005, see also Balmino et al., present issue). The induced changes in the orbit of a spacecraft are due to a linear combination of the odd coefficients and a linear combination of the even coefficients, called lumped coefficients. With a single orbiter, these lumped coefficients can be derived from radio science experiments. Therefore, the combination of observations from different orbiters with different orbit characteristics is very promising (see Karatekin et al., 2005). Another source of time-variable gravity is the tides. The solid planet deforms due to the gravitational attraction of the Sun and possibly the other planets and moon(s), and there is an associated mass redistribution inside the planet. This induces in turn a change in the gravitational field exerted on the spacecraft and therefore, changes in the orbital parameters of the spacecraft. Yoder et al. (2003) have used these observed changes to deduce the planet response to the tides, represented by the k: Love number. They obtained a value of 0.153 +/- 0.017 for k2. This number depends on the global structure of the planet and in particular, when the core of a planet is liquid, the Love number is higher than when the core is solid. The value also depends on the dimension of the core. For the Earth, the Love number k2 is about 0.30, for Mars about 0.1. The presence of a liquid core is visible on these values at more than a few percents. For Mars, one finds in the literature values around 0.07 for a solid core and values ranging between [0.1, 0.2] for a liquid core depending on the dimension of the core; the larger the core, the larger the Love number (see Van Hoolst et al., 2003). The allowed range of models for Mars interior is limited by the total moment of inertia, which is also deduced from observation. It has been determined from an analysis of Mars Global Surveyor tracking and Doppler and range measurements to the Mars Pathfinder and Viking Landers to be 0.3650 +/- 0.0012 (Yoder et al., 2003). From their observed values of k2, Yoder et al. (2003) inferred a core radius between 1520 and 1840 kilometers. Balmino et al. (2005, this issue) have also determined the value of the Love number from the same geodetic observations, indicating a liquid core. On the other hand, research and development on modeling the interior of Mars have been performed and more realistic models of the Martian mantle m i n e r a l o g y have been
Chapter 124 • Gravity, Rotation, and Interior of the Terrestrial Planets from Planetary Geodesy: Example of Mars
constructed (Verhoeven et al., 2005). Coupled with an iron core model (containing 14% of the light element sulfur), these models allow computing theoretical Love numbers. For a liquid core and by constraining with the observed moment of inertia, the values are ranging from [0.1, 0.16] for core radius between [1500km, 1850km]. The dimension and state of the core has important implications on the evolution and present constitution and state of Mars (Breuer et al., 1997, Spohn et al., 2001). The evolution of a planet and the possibility to generate a magnetic field are highly dependent on its ability to develop convection in the core and in the mantle. In particular, a core hydrodynamo is related to the presence or absence of a solid inner core, which is in turn related to the percentage of light elements in the iron alloy of the core. Interior structure largely sets the stage for how the mantle may flow and transfer mass and heat. Mantle d y n a m i c s is also essential in forming the g e o l o g i c a l e l e m e n t s of the surface and in sustaining plate tectonics (Spohn et al., 1998). The dimension of the core has also in addition an implication on the possible mantle convection scenarios and in particular on the presence of an olivine phase transition at the bottom of the mantle.
4 Rotation and its interpretation Periodic rotation variations of planets can most easily be determined from a direct radio link between the Earth and landers on the surface of the planet (see Figure 1). Nevertheless, rotation variations with surface displacements on the order of a few centimeter can also be observed from a combination of a radio link between the Earth and an orbiter and a radio link between the orbiter and landers. Alternatively, an Earth-orbiter radio link can be used together with camera or altimeter observations from the same orbiter of spots on the planet' s surface. The idea is that, given the very accurate knowledge of the orientation of the Earth in space (at the milliarcsecond level, i.e. 3cm at the surface of the Earth), one computes the orbit from the Earth-orbiter radio link, and Mars' orientation and rotation in space from the additional radio link between the lander and the orbiter. Simulations for Mars (Yseboodt et al., 2003) have shown that we may expect to get the meter precision on one measurement and even better (a few centimeters) on periodic components (Yseboodt et al., 2003).
Fig. 1. Representation of the radio links between the Earth and the orbiter, the orbiter and landers on the surface of Mars. The Earth's orientation in space is considered to be almost perfectly known with r e s p e c t to an inertial frame (at the milliarcsecond level). Radiolinks between the Earth and the orbiter and between the orbiter and the landers provide relative positions of these objects. From these space geodesy measurements, Mars orientation can be deduced with respect to an inertial space. Rotation variations are represented by several parameters: precession and nutation, libration, polar motion, and l e n g t h - o f - d a y variations. Together, these motions constitute the orientation of the surface of the planet in space and the rotation speed of the planet. Precession and nutation are induced by the tidal gravitational torque on an oblate planet; this is the case for Mars and for the Earth for instance. The long-term component., precession, has a period of about 91000 Martian years or 170000 Earth years. Both motions are very interesting for studying the deep interior of Mars, precession because it constraints the moment of inertia of the planet, nutations mainly because they are different for a planet with a liquid core than for a planet with a solid core. Similarly, raw (liquid) and cooked (solid) eggs rotate differently. F r o m the observation of nutation, it can be determined whether Mars has a liquid core or a solid core (see Figure 2). As for the Earth, the position of Mars' rotation axis varies with time due to the gravitational attraction exerted by the Sun and the Mars' natural satellites Phobos and Deimos. Because of the existence of an equatorial bulge (like the Earth, Mars is flattened at the poles), the Sun's attraction
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continuously tends to tilt Mars' equatorial plane towards the orbital plane. The rotating Mars reacts to this force as a gyroscope, and Mars' rotation axis d e s c r i b e s a b r o a d cone a r o u n d the perpendicular to the orbital plane. This forced motion is called precession (see Figure 3).
very small enhancement (which is the case at that frequency) provides a large effect in terms of the amplitude contribution (see Figure 4).
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Fig. 5. Nutation amplification for three interior structure models of Mars with different core dimensions. The differences reflect the shift in the FCN frequency.. The vertical lines indicate the Martian nutation frequencies.
Because the relative positions of the Sun and Mars periodically change with time, precession also shows periodic variations called nutations (see Figure 3) (also Phobos, and Deimos, the two little moons of Mars can be involved). The internal structure of Mars influences the nutations; in particular a resonance effect in the nutations could be seen if Mars has a liquid core. The resonant enhancement of nutation when the core is liquid is due to a normal mode called the free core nutation (Dehant et al., 2000b, Roosbeek, 2000). This mode is related to the existence of a flattened fluid core inside a deformable mantle. It exists only when the core is liquid and when the core is flattened. For Mars it has a period of around 250 days in space, which is very close to the ter-annual nutation (see Dehant et al., 2000a, 2000b, Van Hoolst et al., 2000a, 2000b) (see Figure 4). The prograde semi-annual nutation is not close to the FCN frequency but has a very high amplitude. A
The precision of the observation of the orientation in space must reach the 5cm level as this is the level of the core state contribution. Observing the nutations at a better level provides the answer to the question: has Mars a solid or a liquid core. Additionally, the dimension of the core has an influence on the resonance frequency (see Figure 5) and one will be able to constrain the dimension of the core. Polar motion is the motion of the rotation axis in a frame tied to the planet. Mars polar motion (see Figure 6) contains the seasonal effects of the atmosphere as well as a resonance to a normal mode of the planet, the Chandler Wobble (CW), related to the fact that an oblate planet that does not rotate around its principle moment of inertia undergoes a wobbling. The period and damping of this mode are very interesting as they are related to the interior structure of the planet. The CW period depends mainly on the dynamical flattening of the
Chapter 124 • Gravity, Rotation, and Interior of the Terrestrial Planets from Planetary Geodesy: Example of Mars
planet and provides information on the planet's elasticity (at the level of 11 days), anelastic behaviour (effect of up to 7 days), and the existence of a fluid core (at the level of 1.5 days).. Polar Motion CIII 40
Fig. 8. Sublimation/condensation process related to length of day variation.
5 Present and future missions
Fig. 6. Computed polar motion of Mars. The large CW component is computed from a random atmospheric excitation and the seasonal components from the periodic changes of mass in the atmosphere. Length-of-day variations, presented in Figure 7, are deviations from the uniform rotation speed of the planet. They are mostly related to the geophysical fluids entering to the system (core, ocean, atmosphere, hydrosphere.., if they exist). For Mars for i n s t a n c e , the seasonal condensation/sublimation of the icecaps induce a large change in the length-of-day at the seasonal periods (Cazenave and Balmino, 1981, Chao and Rubincam, 1990, Defraigne et al., 2000, Van den Acker et al., 2002).
Length-Of-Day (LOD) Variations
Fig. 7. Length-of-day variations computed from the annual and semi-annual excitation of the atmosphere. The changes with respect to the uniform rotation induce position changes at the level of 10 meters on the equator. The main part of the signal is due to the moment of inertia changes induced by the mass repartition (see Figures 7 and 8). Length-ofday variations can be computed from general circulation models (GCM) (see e.g. Defraigne et al., 2000, Van den Acker et al., 2002, using the GCM from Forget et al., 1995, 1998) and have been determined from Viking lander data (Yoder and Standish, 1997, Folkner et al., 1997).
The interior of the terrestrial planets is presently addressed by several spacecrafts, and a lot of new science will also be done in the future in the frame of space geodesy. The NASA satellites MGS and Mars Odyssey, and the ESA satellite Mars Express (MEX) are continuously tracked from Earth, which has provided unprecedented accuracy on the Martian gravity field. MGS was mostly used for determining the spherical harmonics coefficients of the gravity field development (see Lemoine et al., 2001, Yuan et al., 2001). MEX has recently been used for further improvement. The altitude of Mars Express at pericenter, between 265 and 330 km, is significantly lower than the minimum altitude (370 km) of Mars Global Surveyor (MGS) during its Gravity Calibration Orbit (GCO) and Mapping phases. Short wavelength orbital perturbations due to gravity are thus significantly larger on MEX than on MGS, so that MEX gravity data are useful to improve our knowledge of short wavelength gravity anomalies on the observed targets. The gravity studies performed with the radio science experiment MaRS of MEX use Doppler data along the line-of-sight near pericenter above selected target areas of geophysical interest (P~itzold et al., 2004). The average coherence MEX/topography is high down to a wavelength corresponding to the altitude of the spacecraft at pericenter. This analysis of the coherence shows that MEX can be used as an independent check of the quality of the existing gravity solutions in the target regions analyzed (Beuthe et al., 2005). The Mars Reconnaissance Orbiter (MRO), which is on its way to Mars, could complementary be used. Recent results on the time variable gravity field and on the k2 Love number are presented and discussed in the present issue (see Balmino et al.). They follow a study published by Yoder et al. (2003), in which it was shown that the k2 Love number from MGS corresponds to a large liquid core. In situ geophysical investigations on Mars with landers would provide a large step forward.
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GEP (Geophysical and Environment Package, a Mars long-lived surface package) is a long lived geoscience observatory on Mars, which will consist of a permanent network of fixed stations on the planet, operating for a decade. It will be based on a piggyback approach and will use future Mars missions for its deployment. Space geodesy by means of radio science is foreseen on these stations. The first opportunity for this launch could be in the ExoMars mission within the ESA AURORA program. Concerning the other planets, the past Magellan mission to Venus has provided us with unprecedented results about Venus' gravity field (Barriot et al., 1998, Konopliv et al., 1996, 1999). Magellan has answered many questions about Venus geophysics and geological history but it has also raised new questions. A new opportunity to study the gravity field of Venus will be provided by Venus Express (VEX), and in particular the radio science experiment VeRa (Venus Express Radioscience experiment) (Hfiusler et al., 2005). Given that the VEX pericenter altitude is about 250 km, one can expect an improvement of this wavelength resolution up to about 250 kin, similarly as for Mars Express (see above and Beuthe et al., 2005). In turn, the understanding of the lithospheric structure under targeted areas might be improved on the basis of the analysis of the relationship between the topography and the gravity with respect to previous studies as the one of Leftwich et al. (1999). Rotation variations for a planet with ellipsoidal equatorial shape such as for Mercury, the Galilean satellites, and the Moon, are usually called librations. As an example, Mercury is in a 3:2 spin-orbit resonance and has an oblate equator. Consequently, the Sun exerts a torque on the planet and tends to bring the bulge of the equator into its direction, producing either a slow-down or an acceleration of the rotation. Superposed upon its uniform rotation, the planet is then twisting periodically around its rotation axis. The main forced libration of the planet Mercury has a period of about 88 days and its amplitude reaches a few hundred meters at the equator. A solid core would participate in the libration, but when the core is liquid and the core-mantle boundary is spherical, the core does not participate in the rotation, and the libration amplitude is about twice the libration amplitude for a rigid core (see Peale, 1976, Peale et al., 2002, Wu et al., 1995, 1997, Rambaux and Bois, 2004). The amplitude depends mainly on the moment of inertia of the mantle and on the equatorial flattening. The influence of core-mantle
coupling is very small (Peale et al. 2002, Rambaux et al., 2005). By observing repeatedly with a camera spots at the surface of the planet, by tracking the orientation of the camera with respect to the stars, and by determining the spacecraft's position by radio-tracking, it is possible to obtain the libration of Mercury (Milani et al., 2001). Alternatively, the libration can be determined from the planetary topographic and gravitational shape (Solomon et al, 2001). The future NASA mission MESSENGER is on its way to Mercury and will reach Mercury in 2009. The ESA cornerstone mission BepiColombo will be launched in 2013. These missions will determine a highly accurate gravity field and a precise characterization of the libration of Mercury. Acknowledgements. This work was financially supported by the Belgian PRODEX program and benefited from the support of the European C o m m u n i t y ' s Improving Human Potential Programme under contract RTN2-2001-00414, MAGE. We are thankful to Bill Folkner for fruitful discussions and reading the manuscript. References
Balmino G., Duron J., Marty J.C., and Karatekin O., 2005, Mars long wavelength gravity field time variations. A new solution from MGS tracking data, present issue. Barriot J.-P., Val6s N., Balmino G., and Rosenblatt P., 1998, A 180th degree and order model of the Venus gravity field from Magellan line of sight residual Doppler data, Geophys. Res. Letters, 25(19), 3743-3746. Belleguic V., P. Lognonn6, and M. Wieczorek, 2005, Constraints on the Martian lithosphere from gravity and topography data, J. Geophys. Res., ll0(Ell), Ell005, DOI: 10.1029/2005JE002437. Beuthe, M., Rosenblatt P., Karatekin O., Dehant V., P~itzold M., Haeusler B., Le Maistre S., Van Hoolst T., and Barriot J.-P. , 2005, Assessment of the Martian Global Gravity Field at Short Wavelength with Mars Express, Geophys. Res. Letters, submitted for publication. Breuer, D., Yuen D.A., and Spohn T., 1997, Phase transitions in the Martian mantle: Implications for partially layered convection, Earth Planet. Sci. Letters, 148(3-4), 457-469, DOI: 10.1016/S0012-821 X(97)00049-6. C a z e n a v e A. and B a l m i n o G., 1981, Meteorological effects on the seasonal variations of the rotation of Mars, Geophys. Res. Letters, 8,245-248.
Chapter 124 • Gravity, Rotation, and Interior of the Terrestrial Planets from Planetary Geodesy: Example of Mars
Chao B.F., and Rubincam D.P., 1990, Variations of Mars gravitational field and rotation due to seasonal CO2 exchange, J. Geophys. Res., 95, 14755-14760. Defraigne P., de Viron O., Dehant V., Van Hoolst T., and Hourdin F., 2000, Mars rotation variations induced by atmospheric CO2 and winds, J. Geophys. Res., 105(El0), 2456324570. Dehant V., Defraigne P., and Van Hoolst T., 2000a, Computation of Mars' transfer function for nutation tides and surface loading, Phys. Earth planet. Inter., 117,385-395. Dehant V., Van Hoolst T., and Defraigne P., 2000b, Comparison between the nutations of the planet Mars and the nutations of the Earth, Survey Geophys., 21, 1, 89-110. Folkner W.M., Yoder C.F., Yuan D.N., Standish E.M., and Preston R.A., 1997, Interior Structure and Seasonal Mass Redistribution of Mars from Radio Tracking of Mars Pathfinder, Science, 278(5344), 1749. Forget F., Hourdin F., and Talagrand O., 1995, The sensitivity of the Martian surface pressure and atmospheric mass budget to various parameters: A comparison between numerical simulations and Viking observations, J. Geophys. Res., 100(E3), 5501-5523. Forget F., Hourdin F., and Talagrand O., 1998, CO2 Snowfall on Mars: Simulation with a General Circulation Model, Icarus, 131 (2), 302316. Harri A.M., Marsal O., Leppelmeier G.W., Lognonn6 P., Glassmeier K.-H., Angrilli F., Banerdt W.B., Barriot J.P., Bertaux J.L., Berthelier J.J, Calcutt S., Cerisier J.C., Crisp D., Dehant V., Di Pippo S., Giardini D., Guerrier D., Jaumann R., Kumpulainen K., Langevin Y., Larsen S., Menvielle M., Musmann G., Polkko J., Pommereau J.P., Runavot J., Schumacher W., Siili T., Simola S., and Tillman J.E., 1999, Network Science Landers for Mars, Advances in Space Research, 23(11), 1915-1924. H~iusler, B., P~itzold M., Tyler G.L., Simpson R.A., Hinson D., Bird M.K., Treumann R.A., Dehant V., Eidel W., Remus S., and Selle J., 2005, Atmospheric, Ionospheric, Surface, and Radio Wave Propagation Studies with the Venus Express Radio Science Experiment VeRa, ESA Scientific Publication, ESA-SP, in preparation. Karatekin O., Duron J., Rosenblatt P., Dehant V., Van Hoolst T., Barriot J.P., 2005, Martian Time-Variable Gravity and its Determination; Simulated Geodesy Experiments, J. Geophys. R e s., 1 10 ( E 6 ) , E 0 6 0 0 1, DOI: 10.1029/2004JE002378.
Kaula W.M., 1993, Higher order statistics of planetary gravities and topographies, Geophys. Res. Letters, 20(23), 2583-2586. Konopliv, A.S., and Yoder C.F., 1996, Venusian k2 tidal Love number from Magellan and PVO tracking data, Geophys. Res. Letters, 23(14), 1857-1860, DOI: 10.1029/96GL01589. Konopliv, A.S., Banerdt W.B., and Sjogren W.L., 1999, Venus gravity: 180 th degree and order, Icarus, 139, 1, 3-18. Leftwich, I.E., von Frese R.R.B., Kim Hyung R., Noltimier H.C., Potts L.V., Roman D.R., and Tan Li, 1999, Crustal analysis of Venus from Magellan satellite observations at Atalanta Planitia, Beta Regio, and Thetis Regio, J. Geophys. Res., 104(E4), 8441-8462, DOI: 10.1029/1999JE900007. Lemoine, F.G., Smith D.E., Rowlands D.D., Zuber M.T., Neumann G.A., Chinn D.S., and Pavlis D.E., 2001, An improved solution of the gravity field of Mars (GMM-2B) from Mars Global Surveyor, J. Geophys. Res., 106 (El0), 2335923376. Milani A., Rossi A., Vokrouhlicky D., Villani D., and Bonnano C., 2001, Gravity field and rotation state of Mercury from the BepiColombo Radio Science Experiments, Planet. Space Sci., 49, 1579-1596. Neukum, G., Jaumann R., Hoffmann H., Hauber E., Head J.W.III, Basilevsky A.T., Ivanov B.A., Werner S.C., van Gasselt S., Murray J.B., McCord T., HRSC Co-Investigator team, 1994, Mars as seen by the HRSC experiment: Recent and episodic volcanic, hydrothermal, and glacial activity, American Astronomical Society, DPS meeting #36, #31.01. Neukum, G., Jaumann R., Hoffmann H., Hauber E., Head J.W.III, Basilevsky A.T., Ivanov B.A., Werner S.C., van Gasselt S., Murray J.B., McCord T., HRSC Co-Investigator team, 2005, Mars: Recent and Episodic Volcanic, Hydrothermal, and Glacial Activity Revealed by the Mars Express High Resolution Stereo Camera (HRSC), 36th Annual Lunar and Planetary Science Conference, March 14-18, 2005, in League City, Texas, abstract no.2144. Peale S. J., 1976, Does Mercury have a molten core, Nature, 262, 765-766. Peale S.J., Phillips R.J., Solomon S.C., Smith D.E., and Zuber M.T., 2002, A procedure for determining the nature of Mercury's core, Meteoritics Planet. Sci., 37(9), 1269-1283. P~itzold, M., Neubauer F.M., Carone L., Hagermann A., Stanzel C., H~iusler B., Remus S., Selle J., Hagl D., Hinson D.P., Simpson R.A., Tyler G.L., Asmar S.W., Axford W.I.,
893
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Hagfors T., Barriot J.-P., Cerisier J.-C., Imamura T., Oyama K.-I., Janle P., Kirchengast G., and Dehant V., 2004, MARS: Mars Express Orbiter Radio Science, in: Mars Express- the scientific payload, ESA Special Publication SP1240, 141-163. Rambaux, N., and Bois E., 2004, Theory of Mercury's spin-orbit motion and analysis of its main librations, Astron. Astrophys., 413, 381393, DOI: 10.1051/0004-6361:20031446. Rambaux, N., Van Hoolst T., Dehant V., and Bois E., 2005, Inertial core-mantle coupling and libration of Mercury, Astron. Astrophys., submitted. Roosbeek F., 2000, Analytical developments of rigid Mars nutation and tide generating potential series, Celest. Mech. Dynam. Astron., 75, 287300. Smith D.E., Zuber M.T., and Neumann G.A., 2001, Seasonal Variations of Snow Depth on Mars, Science, 294(5549), 2141-2146, DOI: 10.1126/science. 1066556. Smith, D.E., G. Neumann, R. E. Arvidson, E. A. Guinness, and S. Slavney, 2003, Mars Global Surveyor Laser Altimeter Mission Experiment Gridded Data Record, MGS-M-MOLA-5MEGDR-L3-V1.0, NASA Planet. Data System, Washington, D. C. Solomon, S.C, and 20 co-authors, 2001, The MESSENGER mission to Mercury: scientific objectives and implementation, Planet. Sp. Sci. 49, 1445-1265. Spohn T., Sohl F., and Breuer D., 1998, Mars, Astron. Astrophys. Rev., 8(3), 181-235. Spohn T., Acufia M.H., Breuer D., Golombek M., Greeley R., Halliday A., Hauber E., Jaumann R., and Sohl F., 2001, Geophysical Constraints on the Evolution of Mars, Space Sci. Rev., 96(1/4), 231-262. Van den Acker E., Van Hoolst T., de Viron O., Defraigne P., Dehant V., Forget F., and Hourdin F., 2002, Influence of the winds and of the CO2 mass exchange between the atmosphere and the polar ice caps on Mars' orientation parameters, J. Geophys. Res., 10.1029/2000JE001539. Van Hoolst, T., Dehant V., Roosbeek F., and Lognonn6 P., 2003, Tidally induced surface displacements, external potential variations, and gravity variations on Mars., Icarus, 161, 281296, DOI: 10.1016/S0019-1035(2)00045-2. Van Hoolst T., Dehant V., and Defraigne P., 2000a, Sensitivity of the Free Core Nutation and the Chandler Wobble to changes in the interior
structure of Mars, Phys. Earth planet. Inter., 117,397-405. Van Hoolst T., Dehant V., and Defraigne P., 2000b, Chandler Wobble and Free Core Nutation for Mars, Planet. Space Sci., 48, 1214, 1145-1151. Van Hoolst, T., and Jacobs C., 2003, Mercury's tides and interior structure, J. Geophys. Res., 108(Ell), 7-11, CiteID 5121, DOI 10.1029/2003JE002126. Vincent M.A. and Bender P.L., 1990, Orbit determination and gravitational field accuracy for a Mercury transponder satellite, J. Geophys. Res., 95, 21357-21361. Wu X., Bender P.L., and Rosborough G.W., 1995, Probing the interior structure of Mercury from an orbiter plus single lander, J. Geophys. Res., 100, 1515-1525. Wu X., Bender P.L., Peale S. J., Rosborough G.W., and Vincent M.A., 1997, Determination of Mercury's 88 day libration and fluid core size from orbit, Planet. Space Sci., 45, 15-19. Verhoeven O., Rivoldini A., Vacher P., Mocquet A., Choblet G., Menvielle M., Dehant V., Van Hoolst T., Sleewaegen J., Barriot J.-P., and Lognonn6 P., 2005, Interior structure of terrestrial planets. I. Modeling Mars' mantle and its electromagnetic, geodetic and seismic properties, J. Geophys. Res., 110(E4), E04009, DOI: 10.1029/2004JE002271. Yoder C.F. and E.M. Standish, 1997, Martian moment of inertia from Viking lander range data, J. Geophys. Res., 102 (E2), 4065-4080. Yoder C. F., Konopliv A.S., Yuan D.N., Standish E.M., and Folkner W.M., 2003, Fluid Core Size of Mars from Detection of the Solar Tide, Science, 300(5617), 299-303. Yseboodt M., Barriot J.P., and Dehant V., 2003, Analytical modeling of the Doppler tracking between a lander and a Mars orbiter in term of rotational dynamics, J. Geophys. Res., 108(E7), 5076, DOI: 2003JE002045. Yuan, D.-N., Sjogren W.L., Konopliv A.S., and Kucinskas A.B., 2001, Gravity field of Mars: A 75th Degree and Order Model, J. Geophys. Res., 106 (El0), 23377-23401. Zuber, M.T., Solomon S.C., Phillips R.J., Smith D.E., Tyler G.L., Aharonson O., Balmino G., BanerdtW.B., HeadJ.W., JohnsonC.L., Lemoine F.G., McGovern P.J., Neumann G.A., Rowlands D.D., and Zhong S., 2000, Internal Structure and Early Thermal Evolution of Mars from Mars Global Surveyor Topography and Gravity, Science, 287(5459), 1788-1793.
Chapter 125
Mars Long Wavelength Gravity Field Time Variations: A New Solution from MGS Tracking Data G. Balmino, J.C. Marty Centre National d'Etudes Spatiales, 18, Avenue Edouard Belin, 31401 Toulouse Cedex 9, France J. Duron (1~, O. Karatekin Observatoire Royal de Belgique, 3, Avenue Circulaire, 1180 Brussels, Belgium (~ Presently at CNES, Toulouse
A b s t r a c t Several modern solutions of the Mars gravity field have already been obtained from the Mars Gobal Surveyor (MGS) mission, by different NASA teams working at GSFC and at JPL, which also have shown that degree two and three lumped zonal coefficients exhibit time variations related to the seasonal cycle of carbon dioxide exchange between the planet surface and its atmosphere. A new solution of these time variations has been obtained by a third team working in Europe with a totally independent software. Five years of one and two way Doppler, and range tracking data collected by the Deep Space Network have been processed in three day arcs, taking into account all disturbing forces of gravitational and non gravitational origins; for each arc the state vector, drag and solar pressure model multiplying factors, and angular momentum dump parameters are adjusted. The zonal harmonics up to degree five and the k2 Love number are solved for. The zonals are estimated every ten days, or every thirty days in some variants, with a priori uncertainties either on their values or on their changes. The lumped C20 and C30 coefficients show similar patterns as in the anterior US solutions, also in accordance with the variations estimated from the output of a Global Circulation Model and from the HEND instrument on board Mars Odyssey. Annual and semiannual terms have been derived for comparisons and future evaluation in terms of global constraints put on such planetary mass transfer. Finally the values found for k2 (in the range 0.10 - 0.15 depending on the solution strategy) are discussed.
Keywords. Mars - Global gravity field modeling Time variations of gravity
1 Introduction The Mars Global Surveyor (MGS) mission has for the first time enabled the direct determination of the predicted time variations of the long wavelength gravity field of Mars (Smith et al. (1999)). These variations are due to mass transfers between the atmosphere and the polar caps at seasonal time scales, which result from the condensation, sublimation and precipitation of mostly carbon dioxide. Those phenomena also induce variations of the polar cap size and thickness which have been detected by the Mars Orbiter Laser Altimeter on MGS (Smith et al. (2001)), as well as length of day perturbations - already seen long ago in the Viking atmospheric pressure measurements (Cazenave et al. (1981)). Several solutions of these time variations, limited to those of the second and third degree and zero order coefficients of the gravity field spherical harmonic expansion, have been produced by the American teams at the Goddard Space Flight Center (Smith et al., (2001); Smith et al. (2003)) and at the Jet Propulsion Laboratory (Yoder et al. (2003)), by different methods and software. The GSFC solution was completely numerical and used MGS tracking data from February 1999 to May 2001, whereas the JPL solution was based on an analytical formulation of the perturbations and the re-analysis of the MGS orbital elements history between February 1999 and April 2002. Both solutions also estimated the second degree Love number, but the published values at the time of the above referenced articles were quite different. We presently have at our disposal a much larger data set, covering about five terrestrial years, or three martian years, a good reason to re-do this type of analysis, moreover with an independent software. Our approach and the used data set will be briefly
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described, several sets of results corresponding to various strategies and assumptions will be presented and discussed, and finally we will show comparisons with gravity variations derived from a general circulation model (GCM) and from the High Energy Neutron Detector (HEND) instrument on-board Mars Odyssey.
2 A complex problem The retrieval of the gravity coefficient time variations from the MGS orbital perturbations is a difficult process. First of all, the determination of gravity harmonics (Kzm in complex notation) from observations of a single spacecraft is hampered by the high correlations between coefficients of the same order m and different degrees l, which appear in the solution of the inverse problem; this is rooted in the disturbing function ~ itself which is expanded as:
GM~~a
L
L
+oc
~
m=o j = - L q=-oc
L
~
1
l=m l-k:even
K l m f lm,(l_k)/2 ( I ) a l , ( l _ k ) / 2 , q (e)
where IJiIkmq =
(R)
t exp[i W~mq]
k(co+M) + m(f2 - O) + qM
a, e, I, f2, co, M are the usual orbital elements, R is the planet equatorial radius (3397 km), GM is the product of the gravitational constant by its mass, the F's and G's are the Kaula's functions of the inclination and eccentricity respectively, see Kaula (1966), L is the truncation degree of the expansion. The perturbation analysis being viewed as a kind of Fourier analysis (with arguments qJ) it is clear that only the linear combinations in brackets {...} can be reliably determined. Then in the case of the zonal harmonics Cio (Cto = Kto ), using the Lagrange planetary equations, it is easy to show that the major perturbations affect the longitude of node f2 on one hand (unbounded secular effect due to the even degree zonals), and the eccentricity e and argument of periapsis co on the other hand (long period effects due to the odd degree harmonics); from here the time variations of the so-called lumped coefficients 6Cev~n and 6Coddcan be derived (Karatekin et al. (2005)), with:
6ce~,~ = f: 6C2o + f~ 6C~o + . . .
6co~ =// f~ ( f~ 6C~o + f , 6c, o + ...) For the mapping phase of MGS we have a = 3796 km, e = 0, I = 92.8 ° , and therefore: f2 = 1, f4 = 1.4932, f: = - 0 . 2 4 7 0 , f5 =-0.2419 The magnitude of the contribution of the higher degree zonals compared to 6C2o and 6CLio is rather large and consequently one should not try to retrieve them separately from MGS alone. In what follows and for sake of convenience we will use the notations 6C20 and 6C30 in place of 6Ceven and 6Co~ respectively. Another aspect of the inverse problem is the sensitivity of the observables to the perturbations on f2, e and co. As noted by several authors (for instance (Yoder et al. (2003)), it is geometrically obvious that the plane perturbations are much better seen (in the Doppler and range measurements) when the orbit is seen face-on, favoring the determination of 6C20; however it comes at the cost of a reduced ability to directly sense the in track non-conservative forces as well as a weakening of the period estimation, both spilling over into the nodal changes at some level. On the contrary, perturbations on (e, co) are better mapped onto the observables when the orbit is seen edge-on, thus improving the determination of 6C30. In a pure numerical approach, as used in the present work, all variations in the orbital elements are analyzed, however it is implicitly those on f2, e, co which bear more signal. Finally another factor may weaken the retrieval of 6C20: the second degree Love number, k2, produces annual and semi-annual perturbations (mostly on the orbital plane) which may be in phase with those originating from 6C20; it is therefore necessary to estimate k2 simultaneously, keeping in mind that correlation will again decrease the accuracy with which both parameters are obtained.
3 Data used, processing characteristics and parameterization We have analysed all tracking data collected by the Deep Space Network (DSN) since the beginning of the orbital mission until March 2005. Of these, only the GCO plus mapping phase parts (since Feb. 20, 1999) were retained for the present work. The data consist in one and two-way ramped Doppler and also range measurements, in X band, acquired by the Deep Space Network stations at the three sites of Goldstone, Madrid and Canberra. The averaging time for the Doppler data was 10 seconds
Chapter 125 • Mars Long Wavelength Gravity Field Time Variations: a New Solution from MGS Tracking Data
over the selected period. In total we kept 285353 one-way and 5 5 6 5 7 4 two-way Doppler measurements, plus 5528 range observations, processed in arcs of approximately three days duration each. The precise orbit determination (POD) and the retrieval of the physical parameters are based on a full dynamical approach using the numerical integration of the equations of motion (and associated variational equations) combined with a least squares adjustment of the unknowns which enter in the linearized observation equations. The processing of the observations and the orbit dynamic modeling are done according to the following: - DE403 planetary ephemerides (Standish et al. (1995)), IAU 1991 planetary constants (Davies et al. (1992)) and martian nutations (Reasenberg et al. (1979)), Earth kinematics, polar motion, solid and ocean tide loading effects according to the IERS standards, see Mc-Carthy (1996); Earth atmosphere loading effect at tracking stations uses the Gegout (1996) formulation with the 6-hour pressure grids from the ECMWF; gravitational forces include the GMM-2B Mars gravity field model complete to degree and order 80 (Lemoine et al. (2001)), third body attraction due to the Sun, Moon and planets (using DE403 and its constants) and also due to the natural martian satellites Phobos and Deimos (using the Bureau des Longitudes ephemeris; Chapront-Touz6 (1990), and solid tide effect due to k2 (with initial value of 0.1); - surface forces include the atmospheric drag evaluated with a semi-empirical Drag Temperature Model (DTM; Bruinsma et al. (2002)), direct solar pressure, indirect reflected solar radiation from Mars and radiation pressure from the Mars thermal emission according to Lemoine (1992); empirical accelerations estimated over the duration of each angular momentum desaturation (AMD) event, according to information provided by the project: the spacecraft fires from time to time pairs of thrusters in order to desaturate its momentum wheels; due to non perfect balance between these thrusters, the spacecraft center of mass is subject to an unknown acceleration which magnitude must be adjusted; the effect is spread over the closest integer number of stepsizes (each of 20 sec duration in our numerical integration) covering the AMD event and without integrator restarting at the thrust boundaries; relativistic effects on the measurements (i.e. delays on ranges and differenced-range Doppler, and frequency modification) and on the spacecraft -
-
-
dynamics (e.g. the Schwarzschild effect) are based on the parameterized post-Newtonian (p.p.n.) formulation; time t r a n s f o r m a t i o n s , b e t w e e n coordinate and atomic time scales, are done numerically as in Moyer (2000) with the utmost precision; - surface forces modeling uses a macro-model of the MGS spacecraft decomposed into a bus (parallelepiped with six plates) and two solar panels (four plates for the front and back sides), with coefficients for the specular and diffuse properties of each surface elements; the attitude of the spacecraft and of its articulated panels in inertial space is known from sets of quaternions provided by the project- or sometimes modeled analytically when the information is missing (Lemoine, personal communication); other corrections to the measurements include: the tropospheric delay which makes use of the meteorological data collected every half-hour at the DSN sites, antenna offsets which are different for each of the four low-gain antennas and for the highgain antenna mounted on the spacecraft- for the latter quaternions are used for modeling its position (after April 1, 1999 when it was deployed); these corrections include center of mass offsets provided with respect to the spacecraft body-fixed coordinate system. -
Finally the parameters which may be solved for are: - the state vector components at epoch for each arc; - e m p i r i c a l multiplying factors, FD, for the drag acceleration (one per revolution in this work); - e m p i r i c a l multiplying factors, Fs, for the solar radiation pressure (here one per arc only); - empirical accelerations at the AMD epochs; - one bias per arc for o n e - w a y D o p p l e r measurements; - one bias per DSN station and per arc for the other measurements; - gravity harmonics up to degree and order 5, at 10 day intervals (with the possibility to solve for a subset every 10 or 30 days, depending on the strategy). - Love number k2. Correlations exist between many of these parameters (Pi) as well as lack of observability in some instances; this is specially true for the FD'S per revolution. Therefore a priori variances oi are input (as additional equations in the least squares procedure) to stabilize the inversion at the POD stage and at the gravity parameters retrieval stage as well.
897
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G. Balmino.J. C.Marty.J. Duron• O. Karatekin 4 How to comparisons findings
evaluate the with a GCM
solutions" and HEND
The mass estimates of C02 exchanged by the atmosphere and the surface can be used to compute the variations of the coefficients 8C20 and 8C30 (Karatetin et. al. (2005)). Such estimates currently come from GCMs' outputs. A GCM is a powerful tool for the simulation of the global scale atmospheric dynamics of a planet. The fluid dynamics equations are used with various physical parameterizations such as, in the case of Mars, the albedo and emissivity of polar regions. Much tuning is then performed to reproduce the seasonal surface pressure variations measured by landers. The GCM solution of NASA-AMES (Haberle et al. (1999)), which has a surface resolution of 7.5 ° in latitude and 9 ° in longitude, has been used in this study. Another estimate of seasonal mass variations comes from the HEND instrument on-board Mars Odyssey which measures the neutron flux from the martian surface (Litvak et al. (2004)). However HEND solutions are G C M - d e p e n d e n t since models of subsurface layers and atmospheric CO2 variations are needed in order to retrieve the time variable CO2 snow depth from the data. The seasonal variations of atmospheric CO2 contents given by the NASA-AMES GCM were used in the HEND data analysis, which could explain the remarkable agreement between the HEND and GCM solutions, as the reader will see it in figure 3 below.
more realistic than using the absolute value criterion. The solution covering the longest time period used arcs from Feb. 4, 1999 to March 2, 2005; however the behaviour of the drag coefficients for the first months looked so weird that we decided to solve for arcs starting only in mid-May 1999. Even more, from several of the above criteria, this "longest" solution, which will be called full period solution, was further restricted to Aug. 7, 1999 to Feb. 24, 2005. Two classes of sub-solutions were then derived, each class corresponding to a subset of the arcs used in the full period solution and comprising different solutions obtained with different constraints or a priori's. The first class (which will be called Period 1 solutions) uses arcs from Aug. 7, 1999 to July 16, 2002; the second class (Period 2 solutions) extends from Nov. 1, 2002 to Feb. 24, 2005 - in between MGS was in solar conjunction. As seen on figure 1, the geometry of the Period 1 solutions comprises two face-on ([3 = 10 deg.) and one edge-on ([3 = 90 deg.) sub-periods, while the Period 2 solutions only have one edge-on ([3 = 85 deg.) and one face-on ([3 = 30 deg.) not so favorable geometries. Figure 2 shows the FD and FS coefficients, and the rms of the two-way and oneway Doppler residuals over the full time span (range data have been used too, but are much more sparse). Apart from the first year, the one-way Doppler data clearly show much higher residuals than the two-way's, which has not been explained.
5 Results A large number of solutions have been computed using different types of constraints on the estimated gravity coefficients, and with the following criteria for judging their quality: (i) rms values of the m e a s u r e m e n t residuals, and overlapping orbit differences at times, (ii) values of the FD coefficients (the Fs showed a good stability, in the range 0.5 to 1.5 approximately), (iii) realistic values for the Czo (1 = 2, 3, 4, 5) or the 6C20 and 6 C 3 0 coefficients, (iv) the geometry of the orbit with respect to Earth (i.e. the so-called out-of-plane angle, [3, which is the angle between the Earthspacecraft vector and the orbit normal). Condition (ii) was applied with severe thresholds and resulted in a drastic reduction of the number of retained arcs (175 out of a total of 700). This will likely need additional investigation in future work; for instance editing based on relative changes of FD could be
o
jaw
sc, ~Period~
~[[ Period2"-
(13
4O
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90 d e ~ m e a n s t h e orbit is seen edge o n f r o m the E a r t h c'l r~99
2000
ZIX~L
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20(14
2035
2006
Fig. 1 Out of plane angle ([3) variation over the period of analysis. The orbit is seen face-on when [3 is small and edge-on when [3 is close to 90 degrees. The upper x axis indicates when Martian Ls is zero related to civil year.
Chapter 125 • Mars Long Wavelength Gravity Field Time Variations: a New Solution from MGS Tracking Data Period 1
Period
2
(al)
:.. __i___:i_::_:_:=_;_:_.~,.~ I~o ,,, * " ' ~ ',,~: .. ~ !~ ~,V.-:~,...~ . ~ -...... ~-~ ~. "
I
:1 -
.
~, x
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2000
200 [
2002
2003
2.004
2005
2006 ,. ,:; . . . .
~
~ .~
.~. ;:
I :
'
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....
,; . . . . .
...,,
.
.,..
.,,.
-
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' ~............
.: ,,. ,. . . . .
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'" -t- -2", ........ ~ '~,.
i! ~-:.::..,:' '~-
. . . . . . . . . . . . .
.
.,..
1o5
1 0.5 1999
,% -.:.:
-~1
-.
2.000
• • ° "T 2001
%
I
I
I
"I
2002
2.003
2004
2005
{bt~
~.b2}
2.006
:1
8C30
_
. o
i 2wa, resi ua's I
..
o
-~)999
2.000
"
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2002
,¢, ~t" 2003
" ,,.,#.-~'~,~
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:::i.....
.~:..~"~""~" .>-
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;4
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2006 ,.~...
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', . . . .
.., ..
. ;..
.',,.
.,..
:• .
,,..
I "way residuals 1
0.15 I O.l
,..,
0.05 0 1999
2000
200 t
2002
2003
2.004
2005
'
•
-.,:,',:,.z
-.,:,~:,.:,
l
........ ........
2006
Fig. 2 History of the POD quality fits, estimated via the variations of the empirical drag (FD) and solar pressure (Fs) coefficients, and r.m.s, of the Doppler measurement residuals here given in hertz (multiply by 35 and 17.5 for one and two-way Doppler respectively to obtain residuals in mm/sec). The mean and r.m.s, values of F D are respectively 0.10, 0.30 for period 1 and 0.46, 0.79 for period 2; Fs has more stable mean and r.m.s, values of 0.99, 0.15 for period 1 and 0.99, 0.11 for period 2.
Three main types of solutions, derived at ten day intervals (coverage permitting) for 6C20 and 6C30, have been obtained: the first type uses continuity constraints in the form of a priori o's on the difference between two consecutive values of these parameters; handling of large gaps between some arcs is delicate and these solutions are not presently satisfactory; - in the second type a priori o's on the parameter values themselves are introduced; after several attempts we took a unique value of 3 . 1 0 -9 which corresponds to the magnitude of the variations predicted by G C M simulations (Smith et al. (1999)); these solutions will be refered to as simply
regularized; - the last type introduces an a priori annual (period of 687 terrestrial days) and semi-annual behaviour of the gravity coefficients with given o's of 3.10 -9 o n the sine and cosine factors of the periodic terms; these solutions will be refered to as forced periodic. In all solutions the Love number k 2 w a s solved for, without any constraint. Figure 3 shows the time variations of 6C20 and 6C30 for different periods and different types, with the fitted annual and semiannual terms (adjusted a posteriori). Also shown are the variations predicted from the NASA-AMES GCM and those derived from the HEND instrument on Mars Odyssey (see section 4).
,-:-.:,.:.
....
2,:,,:,,:,
=.,:,t:,,
2,:,':,--,
...,..,_,~.. - ' - -'
_:,.. . . . . ......
. . . .
Fig. 3 Time variations of 6C20 and 6C30 for different solution classes and types. (al) , (bl): forced periodic, period 1;(a2) , (b2): forced periodic, period 2; (c) and (d): forced periodic, full period. The error bars correspond to formal errors (non calibrated). Also shown are results from GCM and HEND (periodized). Table 1 lists the amplitudes or. and phases c9. of the periodic terms and the k 2 value in each case. For sake of comparison with Yoder et al. (ibid.) the periodic terms are here written as: 0% sin(n MMar~ + Cgn) with MMars being the mean anomaly of Mars and with n=l and 2 for the annual and semi-annual variations respectively. We did not solve for terannual or higher order terms for we do not believe (from the graphs of the 6C20 and 6C30 variations and from the magnitude of the formal errors) that they could be reliably estimated. We do not show the simply regularized solutions for periods 1 and 2 separately for they look very close to the corresponding parts in the full period case. For all solutions the phase values come with large formal errors, which precludes any physical interpretation. We believe that the scattering of the Table 1 amplitude values is in line with the true errors (which we cannot estimate and which are probably much larger than the formal ones) and is indeed
899
900
G. Balmino. J. C. Marty. J. Duron • O. Karatekin
related to the geometry and to the quality of the POD which result in more or less observability of some of these p a r a m e t e r s and/or significant correlations. For instance the clear increase of the formal error bars on 6C20 between mid-2000 and the first quarter of 2001 (see fig. 3) corresponds to an edge-on geometry. In an attempt at mitigating some of these effects we derived monthly solutions under similar conditions, but these do not show significant differences with the ten day solutions apart from some (expected) reduction of the amplitudes of some periodic terms. We found that the simply regularized solutions for ~C20 and ~C30 have smaller annual amplitudes and that ~C20 has a weaker semi-annual signal compared to the HEND and the GCM estimates; h o w e v e r the phases of 6C30 are in fairly good agreement despite the large formal errors as pointed out above. The forced periodic solution over period 1 (Fig. 3- a l and b l) looks comparable to previous results such as those of Yoder et al. (ibid.) which is no surprise since these authors solved explicitly for periodic terms; the fact that an apparently fiat signal for 6C20 in the simply regularized solutions t r a n s f o r m s into an o r g a n i s e d signal can be explained by the strong decorrelation brought by the forcing equation. This is also clear from the solution over the full period (Fig. 3-c). It leads us to favor the forced periodic solutions. Our amplitudes of the periodic terms fitted to ~C30 are significantly smaller than those found by Yoder et al. even in the forced periodic solution over period 1 (close to the period analyzed by these
authors); the 6C20 variations are closer, and the semi-annual term is also in reasonable agreement with the value given by Smith et al. (2001). But our strategy is different: Yoder evaluates the 6 C 2 0 , 6C30 and k 2 coefficients separatly whereas we estimate all parameters simultaneously. The forced periodic solutions for 6C30 (Fig 3- b2 and d) show similar phases and smaller amplitudes than those of HEND and GCM, as in the case of the simply regularized solution, but for period 1 (Fig.3b 1) the annual amplitude is in good agreement with the HEND and GCM d a t a - but not the phase. The agreement between forced 6C20 variations is also best over this period (Fig. 3-a 1). The 6C20 forced periodic signal obtained from period 2 (Fig 3-a2) has larger annual and semi-annual components than those of the signal from the full period (Fig. 3-c) and there is a large phase difference compared to the period 1 results, as well as to the HEND and GCM solutions (which look almost anti-correlated). Values of k2 are also scattered. The smallest one (0.099) is obtained with the period 1 forced periodic solution (with a better 6C20 match up), where Yoder et al. (ibid.) find a high value of 0.153 (_+ 0.017). Our largest value (0.154), close to the value found by these authors, comes with the period 2 solution. The value of 0.130 found in between with a small formal error of 0.006 uses the full period data set, comes with forced periodic ~C20, 6 C 3 0 and should be the most reliable from the dynamical viewpoint; it is compatible with the hypothesis of a fluid Martian core (Yoder et al. (ibid.).
Table 1. Annual (A) and semi-annual (SA) amplitudes (in units of 10 -9) and phases (in degrees) of ~C20 and 6C30, and values of the k 2 Love number for the most significant solutions, and comparisons with Smith et al. (2001) and Yoder et al. (2003) solutions, s.regul, are simply regularized solutions, forced per. are forced periodic solutions as explained in the text.
Solution Full period s. reguL Full period forcedper. Period 1 forcedper. Period 2 forcedper. Smith et aL,
2001, forced per. Yoder et aL, 2003, forced per.
eoeff. 6C20 ~C30 ~C20 6C30 ~C20 ~C30 6C20 6C30
A
A
SA
SA
amplitude
phase
amplitude
phase
0.3 + 0.3 0.4 +0.3 1.8+0.4 0.6+0.5 1.9 + 1.4 1.3+ 1.1 2.9 + 1.0 0.4 + 0.3
13+61 43 ±40 4+63 40+ 42 09 + 28 65 + 28 7 + 80 13 + 68
0.1 +0.2 0.05 ± 0.3 0.7 +0.3 0.1 +0.3 1.3+ 1.6 0.4 + 0.8 1.4 + 1.0 0.07 + 0.1
67+34 -11 +45 63+33 -9 + 48 -87 + 25 -94 + 28 21 + 55 10 + 67
6C20 ~C20 ~C30
1.96 + 0.69 1.81 + 1.02 6.59 + 0.28
12 -7
2.32 + 0.94 1.34 + 0.26
Love number kz 0.113 + 0.004 0.130 + 0.006 0.099 + 0.007 0.154 + 0.021 0.055 + 0.008
-3 -15
0.153 + 0.017
Chapter 125 • Mars Long Wavelength Gravity Field Time Variations:a New Solution from MGS Tracking Data 6 Conclusions
References
Seasonal variations of the Martian gravity field at very large scale have been confirmed from the analysis of five years of MGS tracking data. The behaviour of the lumped zonal harmonics of degrees two and three shows annual and semiannual patterns, which are more or less clear depending on the analyzed period and orbital geometry. The amplitudes of these terms, considering their formal errors, are compatible with those found earlier by other authors. The solutions depend on the period of the data set, which was processed in two sub-sets, then in one set. Regularized and forced periodic solutions give similar results for 6C30 but not for dC20. Comparisons with external data (GCM and HEND) are best generally for the forced periodic solutions and especially over the first sub-set period, and the match up is satisfactory for 6C30. The degree two Love number value is in the range 0.10-0.15 depending on the solutions. The intermediate value of 0.130 _+ 0.006, obtained with the full data set and a forced periodic solution for the zonal terms, may be considered the most probable; it supports the hypothesis of a fluid core of Mars. However the forced period fits for 6C20 show some dependence o n k 2 and disagree significantly with the models, casting some question on the k2 recovery.
Bruinsma, S., and F. G. Lemoine (2002), A preliminary semiempirical thermosphere model of Mars: DTM-Mars, J. Geophys. Res., 107(E 10), doi: 10.1029/2001JE001508. Cazenave, A., and G. Balmino (1981), Meteorological effects on the seasonal variations of the rotation of Mars, Geophys. Res. Lett., 8, 245-248. Chapront-Touz6, M. (1990), Orbits of the martian satellites from ESAPHO and ESADE theories, Astro. Astrophys., 240, 159-172. Davies, M. E., et al. (1992), Report of the I A U / I A G / C O S P A R working group on cartographic coordinates and rotational elements of the planets and satellites, Celestial Mech. Dyn. Astron., 53, 377-397. Haberle, R. M., et al. (1999), General circulation model simulations of the Mars Pathfinder atmospheric structure investigation/meteorology data, J. Geophys. Res., 104(E4), 8957-8974. Karatekin, (), J. Duron, P. Rosenblatt, T. Van Hoolst, V. Dehant, and J. P. Barriot (2005), Mars'time-variable gravity and its determination: Simulated geodesy experiments, J. Geophys. Res., l l O(E06001), doi: 10.1029/2004JE002378. Kaula, W. M. (1966), Theory of Satellite Geodesy, Blaisdell, Waltham, Mass. Lemoine, F. G., D. E. Smith, D. D. Rowlands, M. T. Zuber, G. A. Neumann, D. S. Chinn, and D. E. Pavlis (2001), An improved solution of the gravity field of Mars (GMM-2B) from Mars Global Surveyor, J. Geophys. Res., 106(ElO), 23,359-23,376. Litvak, M. L., et al. (2004), Seasonal carbon dioxide depositions on the Martian surface as revealed from neutron measurements by the HEND instrument onboard the 2001 Mars Odyssey spacecraft, Sol. Syst. Res., 38, 167-177, doi: 10.1023/B: SOLS.0000030856.83622.17. McCarthy, D. D. (Ed.), (1996), IERS Technical Note 21, U.S. Naval Obs., Washington D. C. Moyer, T. D. (2000), Formulation for observed and computed values of Deep Space Network data types for navigation, Monograph 2, Deep Space Communications and Navigation series. Reasenberg, R. D. , and R. W. King (1979), The Rotation of Mars, J. Geophys. Res., 84(B11), 6231-6240. Smith, D. E., M. T. Zuber, R. M. Haberle, D. D. Rowlands, and J. R. Murphy (1999), The Mars seasonal CO2 cycle and the time variation of the gravity field: A general circulation model
Acknowledgments
This study used the GINS software and DYNAMO libraries developped by the Satellite Geodesy team at CNES. We acknowledge F.G. Lemoine and D.D. Rowlands (GSFC) for their help in making comparisons between their own software (GEODYN) and GINS, for providing assistance and software for handling some delicate aspects of the spacecraft attitude. MGS Radio Science team members and especially Dick Simpson (Stanford University) are thanked for their help in providing data, advises and encouragements. M. Litvak from SRI and R. Haberle from NASA Ames are acknowledged for providing the HEND and GCM data. This research was supported by the CNES Programme Directorate and the Toulouse Space Center on one hand, and by the Observatoire Royal de Belgique on the other hand. It benefited from PRODEX and Action-2 grants of the Belgian Science Federal Policy and from the support of the E u r o p e a n ' s Community Improving Human Potential Programme, under contract RTN-200100414, MAGE.
901
902
G. Balmino. J. C. Marty. J. Duron • O. Karatekin
simulation, J. Geophys. Res., 1 04(El), doi: 10.1029/1998JE900024. Smith, D. E., M. T. Zuber, and G. A. Neumann (2001), Seasonal variations of snow depth on Mars, S c i e n c e , 294, 2141-2146, doi:10.1126/science. 1066556. Smith, D. E., and M. T. Zuber (2003), Seasonal changes in the masses of the polar ice caps of Mars as derived from Mars Global Surveyor gravity, Mars Atmosphere Modelling and Observations Workshop, Cent. Natl. d'Etudes
Spatiales, Granada, Spain, 13-15 Jan. Standish, M. E., X X Newhall, J. G. Williams, and W. M. Folkner (1995), JPL planetary and lunar ephemerides DE403/LE403, JPL Interoff. Memo. 314.10-127, Jet Propul. Lab., Pasadena, Calif., May 22. Yoder, C. F., A. S. Konopliv, D. N. Yuan, E. M. Standish, and W. M. Folkner (2003), Fluid core size of Mars from detection of the solar tide, Science, 300, 299-303, doi: 10.1126/science. 1079645.
Chapter 126
Potential Capabilities of Lunar Laser Ranging for
Geodesy and Relativity Jfirgen Mtiller Institut ffir Erdmessung, University of Hannover, Schneiderberg 50, 30167 Hannover, Germany [email protected] James G. Williams, Slava G. Turyshev Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109, USA Peter J. Shelus University of Texas at Austin, Center for Space Research, 3925 W. Braker Lane, Austin, TX 78759, USA
Abstract. Lunar Laser Ranging (LLR), which has been carried out for more than 35 years, is used to determine many parameters within the Earth-Moon system. This includes coordinates of terrestrial ranging stations and that of lunar retro-reflectors, as well as lunar orbit, gravity field, and its tidal acceleration. LLR data analysis also performs a number of gravitational physics experiments such as test of the equivalence principle, search for time variation of the gravitational constant, and determines value of several metric gravity parameters. These gravitational physics parameters cause both secular and periodic effects on the lunar orbit that are detectable with LLR. Furthermore, LLR contributes to the determination of Earth orientation parameters (EOP) such as nutation, precession (including relativistic precession), polar motion, and UT1. The corresponding LLR EOP series is three decades long. LLR can be used for the realization of both the terrestrial and selenocentric reference frames. The realization of a dynamically defined inertial reference frame, in contrast to the kinematically realized frame of VLBI, offers new possibilities for mutual cross-checking and confirmation. Finally, LLR also investigates the processes related to the Moon's interior dynamics. Here, we review the LLR technique focusing on its impact on Geodesy and Relativity. We discuss the modern observational accuracy and the level of existing LLR modeling. We present the near-term objectives and emphasize improvements needed to fully utilize the scientific potential of LLR. Keywords. Lunar Laser Ranging, Relativity, EarthMoon dynamics
1
Motivation
Being one of the first space geodetic techniques, lunar laser ranging (LLR) has routinely provided ob-
servations for more than 35 years. The LLR data are collected as normal points, i.e. the combination of round trip light times of lunar returns obtained over a short time span of 10 to 20 minutes. Out of ~ 1019 photons sent per pulse by the transmitter, less than 1 is statistically detected at the receiver Williams et al. (1996); this is because of the combination of several factors, namely energy loss (i.e. 1 / R 4 law), atmospherical extinction and geometric reasons (rather small telescope apertures and reflector areas). Moreover, the detection of real lunar returns is rather difficult as dedicated data filtering (spatially, temporally and spectrally) is required. These conditions are the main reason, why only a few observatories worldwide are capable of laser ranging to the Moon. Observations began shortly after the first Apollo 11 manned mission to the Moon in 1969 which deployed a passive retro-reflector on its surface. Two American and two French-built reflector arrays (transported by Soviet spacecraft) followed until 1973. Since then over 16,000 LLR measurements have by now been made of the distance between Earth observatories and lunar reflectors. Most LLR data have been collected by a site operated by the Observatoire de la Cgte dAzur (OCA), France. The transmitter/receiver used by OCA is a 1.5m altaz Ritchey-Chr&ien reflecting telescope. The mount and control electronics insure blind tracking on a lunar feature at the 1 arcsec level for 10 minutes. The OCA station uses a neodymium-YAG laser, emitting a train of pulses, each with a width of several tens of picoseconds). LLR station at the McDonald Observatory in Texas, USA is another major provider of the LLR data. The McDonald Laser Ranging Station (MLRS) is built around a computer-controlled 0.76m x-y mounted Cassegrain/Coudd reflecting telescope and a short pulse, frequency doubled, 532nm, neodymium-YAG laser with appropriate computer, electronic, meteorological, and timing interfaces. Until 1990, Haleakala laser ranging station
904
J. MLiller. J. G. Williams. S. G. Turyshev • P. J. Shelus
on the island of Maui (Hawaii, USA) contributed to LLR activities with its 0.40 m telescope. Single lunar returns are available from Orroral laser ranging station in Australia (closed 1 November 1998) and the Wettzell Laser Ranging System (0.75 m) in Germany. Other modem stations have demonstrated lunar capability, e.g. the Matera Laser Ranging Station (0.50 m) in Italy and Hartebeesthook Observatory (0.762 m) in South Africa. A new site with lunar capability is currently being built at the Apache Point Observatory (New Mexico, USA) around a 3.5 m telescope. This station, called APOLLO, is designed for mm accuracy ranging (Williams et al. 2004b). Today MLRS and OCA are the only currently operational LLR sites achieving a typical range precision of 18-25 mm. Fig. 1 shows the number of LLR normal points per year since 1970. As shown in Fig. 2, the range data have not been accumulated uniformly; substantial variations in data density exist as a function of synodic angle D, these phase angles are represented by 36 bins of 10 degree width. In Fig. 2, data gaps are seen near new Moon (0 and 360 degrees) and full Moon (180 degrees) phases. The properties of this data distribution are a consequence of operational restrictions, such as difficulties to operate near the bright sun in daylight (i.e. new Moon) or of high background solar illumination noise (i.e. full Moon).
600
._.q500 rn L Q Q.
t 400 O
300 e~
o ? 200 e~
E z 100
30
60
90
120 150 180 210 240 270 300 330 360 Synodic Angle D [degrees]
Fig. 2. Data distribution as a function of the synodic angle D.
range measurements. 25
.........................................................................................................................................................................................
20
..i . . . . . . . . . . . . . . . . . . . . . . . .
~. . . . . . . . . . . . . . . . . . . . . . . . .
~. . . . . . . . . . . . . . . . . . . . . . . .
: ........................
~. . . . . . . . . . . . . . . . . . . . . . . . .
~. . . . . . . . . . . . . . . . . . . . . . . . .
- ........................
, ....
.....
15 1 ....................... ~......................... i................................................. i................................................ i........................ i
lo
i ........................ i .............
5[',
........................
........... ~........................
; ........................ ',........................ i ........................ i .....
i ........................
i ........................
i
i ........................
~...................
i ........................
.....
i ........................ ~
1200
0
1000
I
1970
1975
1980
1985
1990
1995
2000
2005
years 800
Fig. 3. Weighted residuals (observed-computed Earth-Moon distance) annually averaged.
6 0 0
400
200
OLm.m 1970
1975
1980
1985
years
1 ggo
1995
2000
2005
Fig. 1. Lunar observations per year, 1970 - 2005.
While measurement precision for all model parameters benefit from the ever-increasing improvement in precision of individual range measurements (which now is at the few cm level, see also Fig. 3), some parameters of scientific interest, such as time variation of Newton's coupling parameter ( ~ / G or precession rate of lunar perigee, particularly benefit from the long time period (35 years and growing) of
In the 1970s LLR was an early space technique for determining Earth orientation parameters (EOP). EOP data from LLR are computed for those nights where sufficient data were available (approximately 2000 nights over 35 years). Although, the other space geodetic techniques (i.e., SLR, VLBI) dominate since the 80s, today LLR still delivers very competitive results, and because of large improvements in ranging precision (30 cm in 1969 to 2 cm today), it now serves as one of the strongest tools in the solar system for testing general relativity. Moreover, parameters such as the station coordinates and velocities contributed to the International Terrestrial Reference Frame ITRF2000, EOP quantities were used in combined solutions of the International Earth Rotation and Reference Systems Service IERS (~ = 0.5 mas).
Chapter 126 • Potential Capabilities of Lunar Laser Ranging for Geodesy and Relativity
2
LLR Model
The existing LLR model has been developed to compute the LLR observables the round trip travel times of laser pulses between stations on the Earth and passive reflectors on the Moon (see e.g. Mfiller et al. 1996, Mfiller and Nordtvedt 1998 or MLiller 2000, 2001, Mfiller and Tesmer 2002, Williams et al. 2005b and the references therein). The model is fully relativistic and is complete up to first post-Newtonian (1/c 2) level; it uses the Einstein's general theory of relativity- the standard theory of gravity. The modeling of the relativistic parts is much more challenging than, e.g., in SLR, because the relativistic corrections increase the farther the distance becomes. The modeling of the 'classical' parts has been set up according to IERS Conventions (IERS 2003), but it is restricted to the 1 cm level. Based upon this model, two groups of parameters (170 in total) are determined by a weighted least-squares fit of the observations. The first group comprised from the so-called 1Newtonian' parameters such as - g e o c e n t r i c coordinates of three Earth-based LLR stations and their velocities; a set of EOPs (luni-solar precession constant, nutation coefficients of the 18.6 years period, Earth's rotation UT0 and variation of latitude by polar motion);
-
- selenocentric reflectors;
coordinates
of
four
retro-
- rotation of the Moon at one initial epoch (physical librations); orbit (position and velocity) of the Moon at this epoch;
-
The second group of parameters used to perform LLR tests of plausible modifications of general theory of relativity (these parameter values for general relativity are given in parentheses): - geodetic de Sitter precession ~dS of the lunar orbit (_~ 1.92"/cy); space-curvature parameter 7 (= 1) and nonlinearity parameter ~ (= 1); time variation of the gravitational coupling parameter GIG (= 0 yr -1) which is important for the unification of the fundamental interactions; -
s
t
equivalence principle (EP) parameter, which for metric theories is r/ - 4~ - 3 - 7 r
o
n
g
(=0); -
-
EP-violating coupling of normal matter to 'dark matter' at the galactic center; coupling constant c~ (= 0) of Yukawa potential for the Earth-Moon distance which corresponds to a test of Newton's inverse square law;
- combination of parameters ~1 - ~0 - 1 (= 0) derived in the Mansouri and Sexl (1977) formalism indicating a violation of special relativity (there: Lorentz contraction parameter ~1 = 1/2, time dilation parameter ~0 = - 1 / 2 ) ; -- OL1 (= 0) and c~2 (= 0) which parameterize 'pre-
ferred frame' effects in metric gravity. Most relativistic effects produce periodic perturbations of the Earth-Moon range n
ArEM -- E Ai cos(w/At + ~i).
(1)
i=1
orbit of the Earth-Moon system about the Sun at one epoch;
-
- mass of the Earth-Moon system times the gravitational constant; -
t
h
lowest mass multipole moments of the Moon; e
- lunar Love number and a rotational energy dissipation parameter; -
lag angle indicating the lunar tidal acceleration responsible for the increase of the Earth-Moon distance (about 3.8 cm/yr), the increase in the lunar orbit period and the slowdown of Earth's angular velocity.
Ai, czi, and ¢i are the amplitudes, frequencies, and phases, respectively, of the various perturbations. Some example periods of perturbations important for the measurement of various parameters are given in Table 1.Note: the designations should not be used as formulae for the computation of the corresponding periods, e.g. the period 'sidereal-2-annual' has to be calculated as 1/(1/27.32 d - 2/365.25 d) ~ 32.13 d. 'secular + emerging periodic' means the changing orbital frequencies induced by GIG are starting to become better signals than the secular rate of change of the Earth-Moon range in LLR. Fig. 4 represents the sensitivity of the Earth-Moon distance with respect to a possible temporal variation of the gravitational constant in the order of 8 . 1 0 -13,
905
906
J. Mailer. J. G. Williams. S. G. Turyshev. P. J. Shelus
10 e
,
,
,
10 r 10 4
o c
, E -~
+ ~, ., C~l
em
II
¢N
in
~
in lii
m
10 2
~
•"o
*'-
~
E
u~
"o
© ~
,~
o
, E E,-®
u~ ~ ,
= ILl ~.lll
,,
Q~ ~IN10 °
10 .2 n
lO 4
10 "s
0
2000
4000 6000 8000 10000 Days since December 2 7 , 1 9 6 9
12000
10-el 10 -4
Fig. 4. Sensitivity of LLR with respect to
AG/G -
14000
........
,
. . . . . . . .
10 .3
i
. . . . . . . .
10 .2 F r e q u e n c y [1 / d]
i
10 "1
. . . . . . . .
10 °
GIG assuming
8.10 -13 yr -1.
Fig. 6. Power spectrum of a possible equivalence principle violation assuming A(raa/m±) ~ 10-13
10 s
with different amplitudes, so that they can be distinguished and separated from the effects investigated•
,~,E } 10 4
U)
~
o~ "=
E o
= ~
"
10 2
~
II)
==
"10
= I/ j[
O4 o
~o
Table 1. Typical periods of some relativistic quantities, taken
from Mfiller et al. (1999). 10 °
?
Parameter
~'10-2 n
~h-~:o-1
10 -4
6ggalactic
al
10 "s
10 .8 10 .4
........
i 10 -3
........
i 10 .2
Frequency
........
i 10 -1
a2 O/G
........ 100
Typical Periods synodic (29 d 12h44m2.9s) annual (365.25d) sidereal (27d7h43m11.5s) sidereal, annual, sidereal-2.annual, anomal. (27d 13h 18m 33.2s)+annual, synodic 2.sidereal,2-sidereal-anomal., nodal (6798a) secular + emerging periodic
[1 / d]
Fig. 5. Power spectrum of the effect of GIG in the EarthMoon distance assuming AG/G - 8.10 -13 yr -1.
3
the present accuracy of that parameter. It seems as if perturbations of up to 9 meters are still caused, but this range (compared to the ranging accuracy at the cm level) can not fully be exploited, because the lunar tidal acceleration perturbation is similar. The largest periods for dl/G are shown in Fig. 5 and for the EP-parameter in Fig. 6. Obviously many periods are affected simultaneously, because the perturbations, even if caused by a single beat period only (e.g. the synodic month for ~7), change the whole lunar orbit (and rotation) and therefore excite further frequencies. Nevertheless these properties can be used to identify and separate the different effects and to determine corresponding parameters (note that relativistic phenomena show up with typical periods). Other effects, like the asteroids cause similar orbit perturbations (i.e. with the same frequencies) but
The global adjustment of the model by least-squaresfit procedures gives improved values for the estimated parameters and their formal standard errors, while consideration of parameter correlations obtained from the covariance analysis and of model limitations lead to more 'realistic' errors. Incompletely modeled solid Earth tides, ocean loading or geocenter motion, and uncertainties in values of fixed model parameters have to be considered in those estimations. For the temporal variation of the gravitational constant, GIG (6 + 8 ) - 1 0 -13 has been obtained, where the formal standard deviation has been scaled by a factor 3 to yield the given value. This parameter benefits most from the long time span of L L R data and has experienced the biggest improvement over the past years (cf. Mfiller et al. 1999). In contrast, the EP-parameter r/ (-- (6 + 7). 10 -4) benefits most from highest accuracy over a sufficient
Results
-
Chapter 126
long time span (e.g. one year) and a good data coverage over the synodic month, as far as possible. Its improvement was not so big, as the LLR RMS residuals increased a little bit in the past years, compare Fig. 3. The reason for that increase is not completely understood and has to be investigated further. In combination with the recent value of the space-curvature parameter ")/Cassini (")/ -- 1 - (2.1 + 2.3). 10 -5) derived from Doppler measurements to the Cassini spacecraft (Bertotti et al. 2003), the non-linearity parameter ~ can be determined by applying the relationship r/ = 4 / 3 - 3 - ")/Cassini- One obtains / 3 - 1 - (1.5 4-1.8)- 10 -4 (note that using the EP test to determine parameters r/and/3 assumes that there is no composition-induced EP violation). Final results for all relativistic parameters obtained from the IfE (Institute ftir Erdmessung) analysis are shown in Table 2. The realistic errors are comparable with those obtained in other recent investigations, e.g. at JPL (see Williams et al. 1996, 2004a, 2004b, 2005b). Table 2. Determined values for the relativistic quantities and their realistic errors.
Parameter
Results
diff. geod. prec. f~GP - f~deSit ["/cy]
Yuk. coupl, const. O~A=4.105 km spec. relativi. ¢1 ~0 1 intl. of dark matter ~9galactic [cm/s 2 ] 'preferred frame' effect Ctl
(6±io).io -3 (4 4- 5). 10 - 3 (--2 ± 4). 10 -3 (1.5 ± 1.8). 10 -4 (6-q- 7)" 10 -4 (6 4- 8). 10 -13 (3 4 - 2 ) . 10 -11 ( - 5 ± 12). 10 - 5 (4 ± 4) • 10 -14 ( 7+9).I0 -5
'preferred frame' effect c~2
(1.8 4- 2.5) • 10-5
metric par. 7 1 metric par. ~ 1 (direct) and from r/= 4/~ 3 ")'Cassini equiv, principle par. r/
time var. grav. const. G/G [yr-1]
4
•
Potential Capabilities of Lunar Laser Ranging for
Geodesy
and Relativity
timated simultaneously in the standard solution, contribute to the realization of the international terrestrial reference frame, e.g. to the last one, the ITRF2000. 3. Earth rotation: LLR contributes, among others, to the determination of long-term nutation parameters, where again the stable, highly accurate orbit and the lack of non-conservative forces from atmosphere (which affect satellite orbits substantially) is very convenient. Additionally UT0 and VOL (variation of latitude) values are computed (e.g. Dickey et al. 1985), which stabilize the combined EOP series, especially in the 1970s when no good data from other space geodetic techniques were available. The precession rate is another example in this respect. The present accuracy of the longterm nutation coefficients and precession rate fits well with the VLBI solutions (within the present error bars), see (e.g. Williams et al. 2005a,c) 4. Relativity: In addition to the use of LLR in the more 'classical' geodetic areas, the dedicated investigation of Einstein's theory of relativity is of special interest. With an improved accuracy the investigation of further effects (e.g. the Lense-Thirring precession) or those of alternative theories become possible. 5. Lunar physics: By the determination of the libration angles of the Moon, LLR gives access to underlying processes affecting lunar rotation (e.g. Moon's core, dissipation), cf. Williams et al. (2005a). A better distribution of the retroreflectors on the Moon (see Fig. 7) would be very helpful.
Further Applications
In addition to the relativistic phenomena mentioned above, more effects related to lunar physics, geosciences, and geodesy can be investigated. The following items are of special interest: 1. Celestial reference frame: A dynamical realization of the International Celestial Reference System (ICRS) by the lunar orbit is obtained (~ = 0.001") from LLR data. This can be compared and analyzed with respect to the kinematical ICRS from VLBI. Here, the very good longterm stability of the orbit is of great advantage. 2. Terrestrial reference frame: The results for the station coordinates and velocities, which are es-
6. Selenocentric reference frame: The determination of a selenocentric reference frame, the combination with high-resolution images and the establishment of a better geodetic network on the Moon is a further big item, which then allows accurate lunar mapping. The LLR frame is used as reference in many application, e.g. to derive gravity models of the Moon (e.g. Konopliv et al. 2001). 7. Earth-Moon dynamics: The mass of the EarthMoon system, the lunar tidal acceleration, possible geocenter variations and related processes as well as further effects can be investigated in detail.
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8. Time scales: The lunar orbit can also be considered as a long-term stable clock so that LLR can be used for the independent realization of time scales. Those features shall be addressed in the future.
Fig. 7. Distribution of retro-reflectors on the Moon surface. To use the full potential of Lunar Laser Ranging, the theoretical models as well as the measurements require optimization. Using the 3.5 m telescope at the new Apollo site in New Mexico, USA, millimeter ranging becomes possible. To allow the determination of the various quantities of the LLR solution with a total gain of resolution of one order of magnitude, the models have to be up-dated according to the IERS conventions 2003, and made compatible with the IAU 2000 resolutions. This requires, e.g., to better model • higher degrees of the gravity fields of Earth and Moon and their couplings; • the effect of the asteroids (up to 1000); • relativistically consistent torques in the rotational equations of the Moon; • relativistic spin-orbit couplings; • torques caused by other planets like Jupiter; • the lunar tidal acceleration with more periods (diurnal and semi-diurnal); • ocean and atmospheric loading by updating the corresponding subroutines; • nutation using the recommended IAU model;
• the tidal deformation of Earth and Moon; • Moon's interior (e.g. solid inner core) and its coupling to the Earth-Moon dynamics. Besides modeling, the overall LLR processing shall be optimized. The best strategy for the data fitting procedure needs to be explored for (highly) correlated parameters. Finally LLR should be prepared for a renaissance of lunar missions where transponders (e.g. Degnan 2002) or new retro-reflectors may be deployed on the surface of the Moon which would enable many pure SLR stations to observe the Moon. NASA is planning to return to the Moon by 2008 with Lunar Reconnaissance Orbiter (LRO, 2005), and later with robotic landers, and then with astronauts in the middle of the next decade. The primary focus of these planned missions will be lunar exploration and preparation for trips to Mars, but they will also provide opportunities for science, particularly if new reflectors are placed at more widely separated locations than the present configuration (see Fig. 7). New installations on the Moon would give stronger determinations of lunar rotation and tides. New reflectors on the Moon would provide additional accurate surface positions for cartographic control (Williams et al. 2005b), would also aid navigation of surface vehicles or spacecraft at the Moon, and they also would contribute significantly to research in fundamental and gravitational physics, LLR-derived ephemeris and lunar rotation. Moreover in the case of co-location of microwave transponders, the connection to the VLBI system may become possible which will open a wide range of further activities such as frame ties. 5
Conclusions
For the IERS, LLR has contributed to the realization of the International Terrestrial Reference Frame ITRF2000 and to combined solutions of Earth Orientation Parameters. Additionally, LLR has become a technique for measuring a variety of relativistic gravity parameters with unsurpassed precision. No definitive violation of the predictions from general relativity are found. Both the weak and strong forms of the EP are verified, while strong empirical limitations on any inverse square law violation, time variation of G, and preferred frame effects are also obtained. LLR continues as an active program, and it can remain as one of the most important tools for testing Einstein's general relativity theory of gravitation if appropriate observations strategies are adopted and
Chapter 126 • Potential Capabilities of Lunar Laser Ranging for Geodesy and Relativity
if the basic L L R m o d e l is further e x t e n d e d and imp r o v e d d o w n to the m i l l i m e t e r level o f accuracy. Additional r a n g i n g devices on the M o o n w o u l d have benefits for lunar science, f u n d a m e n t a l physics, control n e t w o r k s for surface m a p p i n g , and navigation. D e m o n s t r a t i o n o f active devices w o u l d prepare the w a y for very accurate r a n g i n g to Mars and other solar s y s t e m bodies. Acknowledgments. Current LLR data is collected, archived and distributed under the auspices of the international Laser Ranging Service (ILRS). All former and current LLR data is electronically accessible through the European Data Center (EDC) in Munich, Germany and the Crustal Dynamics Data Information Service (CDDIS) in Greenbelt, Maryland. The following web-site can be queried for further information: http://ilrs.gsfc.nasa.gov. We also acknowledge with thanks, that the more than 35 years of LLR data, used in these analyses, have been obtained under the efforts of personnel at the Observatoire de la Cote dAzur, in France, the LURE Observatory in Maui, Hawaii, and the McDonald Observatory in Texas. A portion of the research described in this paper was carried out at the Jet Propulsion Laboratory of the California Institute of Technology, under a contract with the National Aeronautics and Space Administration. References
Bertotti, B., less, L., and Tortora, R: A test of general relativity using radio links with the Cassini spacecraft. Nature 425, 374-376, 2003. Dickey, J., Newhall, X.X., and Williams, J.G.: Earth Orientation from Lunar Laser Ranging and an Error Analysis of Polar Motion Services, JGR, Vol. 90, No. B 1 l, R 93539362, 1985. Degnan, J.J.: Asynchronous Laser Transponders for Precise Interplanetary Ranging and Time Transfer. Journal of Geodynamics (Special Issue on Laser Altimetry), 551-594, 2002. IAU Resolutions 2000: http://danof.obspm.fr/ IAU_resolutions/Resol-UAI. htm IERS Conventions 2003. IERS Technical Note No. 32, D.D. McCarthy and G. Petit (eds.), Frankfurt, BKG, 2004. Please find electronic version at http ://www. iers. org/iers/products/conv/ Konopliv, A.S., Asmar, S.W., Carranza, E., Yuan, D.N., and Sjogren, W.L.: Recent gravity models as a result of the lunar prospector mission, Icarus, 150, 1-18,2001. Website of the Lunar Reconnasissance Orbiter (LRO) 2005: http://lunar, gs fc. nasa. gov/missions/. Mansouri, R.M., and Sexl, R.U.: A test theory of Special Relativity. General Relativity and Gravitation, 8, No. 7,497-513
(Part i); No. 7, 515-524 (Part ii); No. 10, 809-814 (Part iii), 1977. Mtiller, J.: FESG/TUM, Report about the LLR Activities. ILRS Annual Report 1999, M. Pearlman, L. Taggart (eds.), R 204-208, 2000. Mfiller, J.: FESG/TUM, Report about the LLR Activities. ILRS Annual Report 2000, M. Pearlman, M. Torrence, L. Taggart (eds.), R 7-35 - 7-36, 2001.
Mtiller, J., and Nordtvedt, K.: Lunar laser ranging and the equivalence principle signal. Physical Review D, 58, 062001, 1998. Mfiller, J., Nordtvedt, K., Schneider, M., and Vokrouhlickg, D.: improved Determination of Relativistic Quantities from LLR. In: Proceedings of the l lth International Workshop on Laser Ranging Instrumentation, held in Deggendorf, Germany, Sept. 21-25, 1998, BKG Vol 10, R 216-222, 1999.
Mfiller, J., Nordtvedt, K., and Vokrouhlick), D.: Improved constraint on the c~1 PPN parameter from lunar motion. Physical Review D, 54, R5927-R5930, 1996.
Mfiller, J., and Tesmer, V.: Investigation of Tidal Effects in Lunar Laser Ranging. Journal of Geodesy, Vol. 76, R 232237, 2002. Williams, J.G., Newhall, X.X. and Dickey, J.O.: Relativity parameters determined from lunar laser ranging. Physical Review D, 53, 6730, 1996. Williams, J.G., Turyshev, S.G., and Boggs, D.H.: Progress in lunar laser ranging tests of relativistic gravity. Phys. Rev. Lett., 93, 261101, 2004a, gr-qc/0411113. Williams, J.G., Turyshev, S.G., and Murphy, T.W., Jr.: Improving LLR Tests of Gravitational Theory. (Fundamental Physics meeting, Oxnard, CA, April 2003), International Journal of Modem Physics D, V13 (No. 3), 567-582, 2004b, gr-qc/0311021. Williams, J.G., Boggs, D.H., and Ratcliff, J.T.: Lunar Fluid Core and Solid-Body Tides. Abstract No. 1503 of the Lunar and Planetary Science Conference XXXVI, March 14-18, 2005a. Williams, J.G., Turyshev, S.G., and Boggs, D.H.: Lunar Laser Ranging Tests of the Equivalence Principle with the Earth and Moon. In proceedings of 'Testing the Equivalence Principle on Ground and in Space', Pescara, Italy, September 20-23, 2004, C. Laemmerzahl, C.W.F. Everitt and R. Ruffini (eds.), to be published by Springer Verlag, Lect. Notes Phys., 2005b, gr-qc/0507083. Williams, J.G., Turyshev, S.G., Boggs, D.H., and Ratcliff, J.T.: Lunar Laser Ranging Science: Gravitational Physics and Lunar Interior and Geodesy, In proceedings of "35th COSPAR Scientific Assembly," July 18-24, 2004, Paris, France. Accepted, to be published in Advances of Space Research, 2005c, gr-qc/0412049.
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