International Association of Geodesy Symposia Michael G . Sideris, Series Editor
International Association of Geodesy Symposia Michael G . Sideris, 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 110: 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 Its 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 … Symposium 131: Geodetic Deformation Monitoring: From Geophysical to Engineering Roles Symposium 132: VI Hotine-Marussi Symposium on Theoretical and Computational Geodesy Symposium 133: Observing our Changing Earth Symposium 134: Geodetic Reference Frames Symposium 135: Gravity, Geoid and Earth Observation
Geodesy for Planet Earth Proceedings of the 2009 IAG Symposium, Buenos Aires, Argentina, 31 August - 4 September 2009
Edited by Steve C. Kenyon Maria C. Pacino Urs J. Marti
Editors Steve Kenyon National Geospatial-Intelligence Agency SN L-41 Vogel Rd. 3838 63010-6238 Arnold Montana USA Urs Marti Federal Office of Topography swisstopo Geodesy Department Seftigenstrasse 264 3084 Wabern Switzerland
Maria Cristina Pacino Universidad Nacional de Rosario (UNR) Facultad de Ciencia Exactas Ingenierı´a y Agrimensura (FCEIA) Av. Pellegrini 250 2000 Rosario Argentina
ISBN 978-3-642-20337-4 e-ISBN 978-3-642-20338-1 DOI 10.1007/978-3-642-20338-1 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011941192 # Springer-Verlag Berlin Heidelberg 2012 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
The International Association of Geodesy IAG2009 “Geodesy for Planet Earth” Scientific Assembly was held 31 August to 4 September 2009 in Buenos Aires, Argentina. The theme “Geodesy for Planet Earth” was selected to follow the International Year of Planet Earth 2007–2009 goals of utilizing the knowledge of the world’s geoscientists to improve society for current and future generations. The International Year started in January 2007 and ran thru 2009 which coincided with the IAG2009 Scientific Assembly, one of the largest and most significant meetings of the Geodesy community held every 4 years. The IAG2009 Scientific Assembly was organized into eight Sessions with SubSessions in five of them. Four of the Sessions of IAG2009 were based on the IAG Structure (i.e. one per Commission) and covered Reference Frames, Gravity Field, Earth Rotation and Geodynamics, and Positioning and Applications. Since IAG2009 was taking place in the great Argentine city of Buenos Aires, a Session was devoted to the Geodesy of Latin America. A Session dedicated to the IAG’s Global Geodetic Observing System (GGOS), the primary observing system focused on the multidisciplinary research being done in Geodesy that contributes to important societal issues such as monitoring global climate change and the environment. A Session on the IAG Services was also part of the Assembly detailing the important role they play in providing geodetic data, products, and analysis to the scientific community. A final Session devoted to the organizations ION, FIG, and ISPRS and their significant work in navigation and earth observation that complements the IAG. This volume contains the proceedings of the eight Sessions which are listed below: Session 1: Reference frames implementation for geosciences’ applications: From local to global scales Convenors: Zuheir Altamimi, Claudio Brunini Session 2: Gravity of the Planet Earth Convenors: Yoichi Fukuda, Pieter Visser Session – 2.1: Physics and Geometry of Earth: Focus on satellite altimetry and InSAR Convenors: Cheinway Hwang, Jose´ Luis Vacaflor Session – 2.2: Gravity – An Earth Probing Tool: Focus on CHAMP/GRACE/ GOCE missions, relative/absolute/superconducting gravimetry, and their applications Convenors: Roland Pail, Leonid F. Vitushkin
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Session – 2.3: Modern Height Datum: Focus on definition and realization of GPS-levelling and gravity-base height datum Convenors: Sı´lvio R.C. de Freitas, Dru A. Smith Session – 2.4: Gravity and Geoid Modelling: Focus on global and regional gravity and geoid modelling Convenors: Urs Marti, Yan Ming Wang Session 3: Geodesy and Geodynamics: Global and Regional Scales Convenors: Mike Bevis, Sylvain Bonvalot Session – 3.1: Rotation of the Planet Earth Convenors: Richard Gross, Rodrigo, Abarca del Rio Session – 3.2: Sea level changes and post-glacial rebound Convenors: Juan Fierro, Michael Bevis Session – 3.3: Ocean loading and global water distribution / geophysical fluids Convenors: Tonie van Dam, Richard Gross Session – 3.4: Geodesy, crustal motions and geodynamic processes Convenors: Juan Carlos Baez, Sylvain Bonvalot, Arturo Echalar Session – 3.5: Geodesy and the near-field solid earth response to cryospheric mass changes Convenors: Jim Davis, Gino Casassa Session 4: Positioning and remote sensing of land, ocean and atmosphere Convenors: Sandra Verhagen, Pawel Wielgosz Session – 4.1: Technology and land applications Convenors: Dorota Grejner-Brzezinska, Xiaoli Ding Session – 4.2: Modelling and remote sensing of the atmosphere Convenors: Marcelo Santos, Cathryn Mitchell, Jens Wickert Session – 4.3: Multi-satellite ocean remote sensing Convenors: Shuanggen Jin, Ole Baltazar Andersen Session 5: Geodesy in Latin America Convenors: Denizar Blitzkow, Claudia Tocho Session 6: JOINT ION/FIG/ISPRS session on Navigation and Earth Observation Convenors: Dorota Grejner-Brzezinska, Charles Toth Session – 6.1: Navigation (FIG, ION) Convener: Dorota Grejner-Brzezinska Session – 6.2: Earth Observation (ISPRS)C onvener: Charles Toth Session 7: The Global Geodetic Observing System: Science and Applications Convenors: Richard Gross, Hans-Peter Plag, Luiz Paulo Fortes Session – 7.1: Past Progress and Future Plans Convenors: Hans-Peter Plag, Richard Gross, Luiz Paulo Fortes Session – 7.2: Science and Applications Convenors: Richard Gross, Hans-Peter Plag, Luiz Paulo Fortes Session 8: The IAG International Services and their role for Earth observation Convenors: Ruth Neilan, Rene Forsberg The number and quality of contributions for the eight Sessions clearly demonstrated the important and vital role that Geodesy plays in understanding the earth and its dynamical processes. Satellite, airborne, and terrestrial systems and networks are
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continually measuring and analyzing the earth for global change. For this reason the name of these proceedings is “Geodesy for Planet Earth” which reflects the role fourdimensional geodesy plays in understanding the changes to Planet Earth. The 2009 Assembly attracted nearly 500 oral and poster presentations from 370 geodesists from 45 countries and clearly shows the interest and importance of geodesy globally. The approximately 130 papers that are included in these proceedings (about 25% of the total) are intended to cover much of the latest research and projects on-going in the field. These proceedings would not be possible without the tremendous work by the Convenors of each of the Sessions and Sub-Sessions. They devoted a enormous amount of time and energy in organizing the reviews and final acceptance of the papers for their Sessions. We are very grateful to IAG Secretary General Hermann Drewes and IAG President Michael Sideris for all their guidance and help with these proceedings. The Local Organizing Committee in Buenos Aires was invaluable in helping arrange a very memorable Assembly and provided essential support in the development of these proceedings. And lastly, sincere thanks go out to all the participating scientists and graduate students who made the IAG 2009 “Geodesy for Planet Earth” Scientific Assembly and these proceedings a tremendous success. Steve Kenyon Maria Cristina Pacino Urs Marti
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Contents
Session 1
Reference Frames Implementation for Geoscience’s Applications: From Local to Global Scales Convenors: Z. Altamimi, C. Brunini 1
Improved Analysis Strategy and Accessibility of the SIRGAS Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 C. Brunini, L. Sanchez, H. Drewes, S. Costa, V. Mackern, W. Martı´nez, W. Seemuller, and A. da Silva
2
Improved GPS Data Analysis Strategy for Tide Gauge Benchmark Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alvaro Santamarı´a-Go´mez, Marie-Noe¨lle Bouin, and Guy Wo¨ppelmann
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A Dense Global Velocity Field Based on GNSS Observations: Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Bruyninx, Z. Altamimi, M. Becker, M. Craymer, L. Combrinck, A. Combrink, J. Dawson, R. Dietrich, R. Fernandes, R. Govind, T. Herring, A. Kenyeres, R. King, C. Kreemer, D. Lavalle´e, J. Legrand, L. Sa´nchez, G. Sella, Z. Shen, A. Santamarı´a-Go´mez, and G. Wo¨ppelmann Enhancement of the EUREF Permanent Network Services and Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Bruyninx, H. Habrich, W. So¨hne, A. Kenyeres, G. Stangl, and C. Vo¨lksen
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Can We Really Promise a mm-Accuracy for the Local Ties on a Geo-VLBI Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ulla Kallio and Markku Poutanen
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Recent Improvements in DORIS Data Processing at IGN in View of ITRF2008, the ignwd08 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Willis, M.L. Gobinddass, B. Garayt, and H. Fagard
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Towards a Combination of Space-Geodetic Measurements . . . . . . . . . . . A. Pollet, D. Coulot, and N. Capitaine
8
Improving Length and Scale Traceability in Local Geodynamical Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Jokela, P. H€akli, M. Poutanen, U. Kallio, and J. Ahola
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How to Fix the Geodetic Datum for Reference Frames in Geosciences Applications? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Drewes Transforming ITRF Coordinates to National ETRS89 Realization in the Presence of Postglacial Rebound: An Evaluation of the Nordic Geodynamical Model in Finland . . . . . . . P. H€akli and H. Koivula Global Terrestrial Reference Frame Realization Within the GGOS-D Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Angermann, H. Drewes, and M. Seitz Comparison of Regional and Global GNSS Positions, Velocities and Residual Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Legrand, N. Bergeot, C. Bruyninx, G. Wo¨ppelmann, A. Santamarı´a-Go´mez, M.-N. Bouin, and Z. Altamimi
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GPS Metrology: Bringing Traceable Scale to a Local Crustal Deformation GPS Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 H. Koivula, P. H€akli, J. Jokela, A. Buga, and R. Putrimas
14
Impact of Albedo Radiation on GPS Satellites . . . . . . . . . . . . . . . . . . . . . . . . 113 C.J. Rodriguez-Solano, U. Hugentobler, and P. Steigenberger
Session 2 Gravity of the Planet Earth Convenors: Y. Fukuda, P. Visser 15
On the Determination of Sea Level Changes by Combining Altimetric, Tide Gauge, Satellite Gravity and Atmospheric Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 G.S. Vergos, I.N. Tziavos, and M.G. Sideris
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Arctic Sea Ice Thickness in the Winters of 2004 and 2007 from Coincident Satellite and Submarine Measurements . . . . . . . . . . . . 131 J. Calvao, J. Rodrigues, and P. Wadhams
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The Impact of Attitude Control on GRACE Accelerometry and Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 U. Meyer, A. J€aggi, and G. Beutler
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Using Atmospheric Uncertainties for GRACE De-aliasing: First Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 L. Zenner, T. Gruber, G. Beutler, A. J€aggi, F. Flechtner, T. Schmidt, J. Wickert, E. Fagiolini, G. Schwarz, and T. Trautmann
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Challenges in Deriving Trends from GRACE . . . . . . . . . . . . . . . . . . . . . . . . . 153 A. Eicker, T. Mayer-Guerr, and E. Kurtenbach
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AIUB-GRACE02S: Status of GRACE Gravity Field Recovery Using the Celestial Mechanics Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 A. J€aggi, G. Beutler, U. Meyer, L. Prange, R. Dach, and L. Mervart
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Comparison of Regional and Global GRACE Gravity Field Models at High Latitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 B.C. Gunter, T. Wittwer, W. Stolk, R. Klees, and P. Ditmar
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A New Approach for Pure Kinematical and Reduced-Kinematical Determination of LEO Orbit Based on GNSS Observations . . . . . . . . . 179 A. Shabanloui and K.H. Ilk
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Pure Geometrical Precise Orbit Determination of a LEO Based on GNSS Carrier Phase Observations . . . . . . . . . . . . . . . . . . . . . . . . . . 187 A. Shabanloui and K.H. Ilk
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On a Combined Use of Satellite and Terrestrial Data in Refined Studies on Earth Gravity Field: Boundary Problems and a Target Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 P. Holota and O. Nesvadba
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Moho Estimation Using GOCE Data: A Numerical Simulation . . . . . 205 M. Reguzzoni and D. Sampietro
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CHAMP, GRACE, GOCE Instruments and Beyond . . . . . . . . . . . . . . . . . 215 P. Touboul, B. Foulon, B. Christophe, and J.P. Marque
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The Future of the Satellite Gravimetry After the GOCE Mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 P. Silvestrin, M. Aguirre, L. Massotti, B. Leone, S. Cesare, M. Kern, and R. Haagmans
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Future Satellite Gravity Field Missions: Feasibility Study of Post-Newtonian Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 R. Mayrhofer and R. Pail
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Local and Regional Comparisons of Gravity and Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 C. Jekeli, O. Huang, and T.L. Abt
30
Combination of Local Gravimetry and Magnetic Data to Locate Subsurface Anomalies Using a Matched Filter . . . . . . . . . . . . 247 T. Abt, O. Huang, and C. Jekeli
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On the Use of UAVs for Strapdown Airborne Gravimetry . . . . . . . . . . 255 Richard Deurloo, Luisa Bastos, and Machiel Bos
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Updating the Precise Gravity Network at the BIPM . . . . . . . . . . . . . . . . . 263 Z. Jiang, E.F. Arias, L. Tisserand, K.U. Kessler-Schulz, H.R. Schulz, V. Palinkas, C. Rothleitner, O. Francis, and M. Becker
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Precise Gravimetric Surveys with the Field Absolute Gravimeter A-10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 R. Falk, Ja. M€ uller, N. Lux, H. Wilmes, and H. Wziontek
34
Reconstruction of a Torsion Balance and the Results of the Test Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 L. Vo¨lgyesi and Z. Ultmann
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The Superconducting Gravimeter as a Field Instrument Applied to Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 C.R. Wilson, H. Wu, L. Longuevergne, B. Scanlon, and J. Sharp
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Local Hydrological Information in Gravity Time Series: Application and Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 M. Naujoks, S. Eisner, C. Kroner, A. Weise, P. Krause, and T. Jahr
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Signals of Mass Redistribution at the South African Gravimeter Site SAGOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 C. Kroner, S. Werth, H. Pflug, A. G€ untner, B. Creutzfeldt, M. Thomas, H. Dobslaw, P. Fourie, and P.H. Charles
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Gravity System and Network in Estonia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 To˜nis Oja
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Evaluation of EGM2008 Within Geopotential Space from GPS, Tide Gauges and Altimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 N. Dayoub, P. Moore, N.T. Penna, and S.J. Edwards
40
Fixed Gravimetric BVP for the Vertical Datum Problem . . . . . . . . . . . . 333 ˇ underlı´k, Z. Fasˇkova´, and K. Mikula R. C
41
Realization of the World Height System in New Zealand: Preliminary Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 R. Tenzer, V. Vatrt, and M. Amos
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Comparisons of Global Geopotential Models with Terrestrial Gravity Field Data Over Santiago del Estero Region, NW: Argentine . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 L. Galva´n, C. Infante, E. Laurı´a, and R. Ramos
43
Intermap’s Airborne Inertial Gravimetry System . . . . . . . . . . . . . . . . . . . . 357 Ming Wei
44
Galathea-3: A Global Marine Gravity Profile . . . . . . . . . . . . . . . . . . . . . . . . . 365 G. Strykowski, K.S. Cordua, R. Forsberg, A.V. Olesen, and O.B. Andersen
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Dependency of Resolvable Gravitational Spatial Resolution on Space-Borne Observation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 P.N.A.M. Visser, E.J.O. Schrama, N. Sneeuw, and M. Weigelt
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A Comparison of Different Integral-Equation-Based Approaches for Local Gravity Field Modelling: Case Study for the Canadian Rocky Mountains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 R. Tenzer, I. Prutkin, and R. Klees
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Global Topographically Corrected and Topo-Density Contrast Stripped Gravity Field from EGM08 and CRUST 2.0 . . . . . . . . . . . . . . . 389 R. Tenzer, Hamayun, and P. Vajda
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Local Gravity Field Modelling in Rugged Terrain Using Spherical Radial Basis Functions: Case Study for the Canadian Rocky Mountains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 R. Tenzer, R. Klees, and T. Wittwer
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A Sensitivity Analysis in Spectral Gravity Field Modeling Using Systems Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 Vassilios D. Andritsanos and Ilias N. Tziavos
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Investigation of Topographic Reductions for Marine Geoid Determination in the Presence of an Ultra-High Resolution Reference Geopotential Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 C. Tocho, G.S. Vergos, and M.G. Sideris
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Effects of Hypothetical Complex Mass-Density Distributions on Geoidal Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 Robert Kingdon, Petr Vanı´cˇek, and Marcelo Santos
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Evaluation of Gravity and Altimetry Data in Australian Coastal Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 S.J. Claessens
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Development and User Testing of a Python Interface to the GRAVSOFT Gravity Field Programs . . . . . . . . . . . . . . . . . . . . . . . . . . 443 J. Nielsen, C.C. Tscherning, T.R.N. Jansson, and R. Forsberg
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Progress and Prospects of the Antarctic Geoid Project (Commission Project 2.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 Mirko Scheinert
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Regional Geoid Improvement over the Antarctic Peninsula Utilizing Airborne Gravity Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 J. Schwabe, M. Scheinert, R. Dietrich, F. Ferraccioli, and T. Jordan
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Auvergne Dataset: Testing Several Geoid Computation Methods . . . 465 P. Valty, H. Duquenne, and I. Panet
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In Pursuit of a cm-Accurate Local Geoid Model for Ohio . . . . . . . . . . . 473 K.R. Edwards, Dorota Grejner-Brzezinska, and Dru Smith
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Adjustment of Collocated GPS, Geoid and Orthometric Height Observations in Greece. Geoid or Orthometric Height Improvement? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 I.N. Tziavos, G.S. Vergos, V.N. Grigoriadis, and V.D. Andritsanos
Session 3 Geodesy and Geodynamics: Global and Regional Scales Convenors: M. Bevis, S. Bonvalot 59
Regional Geophysical Excitation Functions of Polar Motion over Land Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 J. Nastula and D.A. Salstein
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Geophysical Excitation of the Chandler Wobble Revisited . . . . . . . . . . 499 Aleksander Brzezin´ski, Henryk Dobslaw, Robert Dill, and Maik Thomas
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On the Origin of the Bi-Decadal and the Semi-Secular Oscillations in the Length of the Day . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 S. Duhau and C. de Jager
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Future Improvements in EOP Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 W. Kosek
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Determination of Nutation Coefficients from Lunar Laser Ranging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 L. Biskupek, J. M€ uller, and F. Hofmann
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A Set of Analytical Formulae to Model Deglaciation-Induced Polar Wander . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 W. Keller, M. Kuhn, and W.E. Featherstone
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Stabilization of Satellite Derived Gravity Field Coefficients by Earth Orientation Parameters and Excitation Functions . . . . . . . . . 537 Andrea Heiker, Hansjo¨rg Kutterer, and J€ urgen M€uller
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The Statistical Characteristics of Altimetric Sea Level Anomaly Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 T. Niedzielski and W. Kosek
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Testing Past Sea Level Reconstruction Methodology (1958–2006) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 J. Viarre and R. Abarca-del-Rı´o
68
Precise Determination of Relative Mean Sea Level Trends at Tide Gauges in the Adriatic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 M. Repanic´ and T. Basˇic´
69
Quantile Analysis of Relative Sea-Level at the Hornbæk and Gedser Tide Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 S.M. Barbosa and K.S. Madsen
70
Assessment of the FES2004 Derived OTL Model in the West of France and Preliminary Results About Impacts of Tropospheric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 F. Fund, L. Morel, and A. Mocquet
71
Gravimetric Time Series Recording at the Argentine Antarctic Stations Belgrano II and San Martı´n for the Improvement of Ocean Tide Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 Mirko Scheinert, Andre´s F. Zakrajsek, Lutz Eberlein, Reinhard Dietrich, Sergio A. Marenssi, and Marta E. Ghidella
72
Mass-Change Acceleration in Antarctica from GRACE Monthly Gravity Field Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 Lo´ra´nt Fo¨ldva´ry
73
Mass Variations in the Siberian Permafrost Region from GRACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 Holger Steffen, J€ urgen M€ uller, and Nadja Peterseim
Contents
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74
Seasonal Variability of Land Water Storage in South America Using GRACE Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 Claudia Tocho, Luis Guarracino, Leonardo Monachesi, Andre´s Cesanelli, and Pablo Antico
75
Water Storage Changes from GRACE Data in the La Plata Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613 A. Pereira, S. Miranda, M.C. Pacino, and R. Forsberg
76
Second and Third Order Ionospheric Effects on GNSS Positioning: A Case Study in Brazil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 H.A. Marques, J.F.G. Monico, G.P.S. Rosa, M.L. Chuerubim, and Ma´rcio Aquino
77
Advanced Techniques for Discontinuity Detection in GNSS Coordinate Time-Series. An Italian Case Study . . . . . . . . . . . . 627 A. Borghi, L. Cannizzaro, and A. Vitti
78
Traditional and Alternative Network Adjustment Approach for the TAMDEF GPS in Antarctica . . . . . . . . . . . . . . . . . . . . . . . 635 G. Esteban Va´zquez, Dorota A. Grejner-Brzezinska, and Burkhard Schaffrin
79
Impact of Loading Phenomena on Velocity Field Computation from GPS Campaigns: Application to ResPyr GPS Campaign in the Pyrenees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 J. Nicolas, F. Perosanz, A. Rigo, G. Bliguet, L. Morel, and F. Fund
80
Comparison of the Coordinates Solutions Between the Absolute and the Relative Phase Center Variation Models in the Dense Regional GPS Network in Japan . . . . . . . . . . . . . . . 651 S. Shimada
81
The 2009 Horizontal Velocity Field for South America and the Caribbean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 H. Drewes and O. Heidbach
82
New Estimates of Present-Day Crustal/Land Motions in the British Isles Based on the BIGF Network . . . . . . . . . . . . . . . . . . . . . . 665 D.N. Hansen, F.N. Teferle, R.M. Bingley, and S.D.P. Williams
83
GURN (GNSS Upper Rhine Graben Network): Research Goals and First Results of a Transnational Geo-scientific Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673 M. Mayer, A. Kno¨pfler, B. Heck, F. Masson, P. Ulrich, and G. Ferhat
84
Determination of Horizontal and Vertical Movements of the Adriatic Microplate on the Basis of GPS Measurements . . . . . . 683 M. Marjanovic´, Zˇ. Bacˇic´, and T. Basˇic´
85
Determination of Tectonic Movements in the Swiss Alps Using GNSS and Levelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689 E. Brockmann, D. Ineichen, U. Marti, S. Schaer, A. Schlatter, and A. Villiger
xvi
Contents
86
A Compilation of a Preliminary Map of Vertical Deformations in New Zealand from Continuous GPS Data . . . . . . . . . . . . . . . . . . . . . . . . . . 697 R. Tenzer, M. Stevenson, and P. Denys
87
Detection of Vertical Temporal Behaviour of IGS Stations in Canada Using Least Squares Spectral Analysis . . . . . . . . . . . . . . . . . . . . 705 James Mtamakaya, Marcelo C. Santos, and Michael Craymer
Session 4 Positioning and Remote Sensing of Land, Ocean and Atmosphere Convenors: S. Verhagen, P. Wielgosz 88
Positioning and Applications for Planet Earth . . . . . . . . . . . . . . . . . . . . . . . . 713 S. Verhagen, G. Retscher, M.C. Santos, X.L. Ding, Y. Gao, and S.G. Jin
89
Report of Sub-commission 4.2 “Applications of Geodesy in Engineering” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719 G. Retscher, A. Reiterer, and G. Mentes
90
A Fixed-s Digital Representation of a Random Scalar Field . . . . . . . . 725 K. Becek
91
The Impact of Adding SBAS Data on GPS Data Processing in Southeast of Brazil: Preliminary Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 W.C. Machado, F. Albarici, E.S. Fonseca Junior, J.F.G. Monico, and W.G.C. Polezel
92
First Results of Relative Field Calibration of a GPS Antenna at BCAL/UFPR (Baseline Calibration Station for GNSS Antennas at UFPR/Brazil) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739 S.C.M. Huinca, C.P. Krueger, M. Mayer, A. Kno¨pfler, and B. Heck
93
Medium-Distance GPS Ambiguity Resolution with Controlled Failure Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 D. Odijk, S. Verhagen, and P.J.G. Teunissen
94
Toward a SIRGAS Service for Mapping the Ionosphere’s Electron Density Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753 ´ ngel, C. Brunini, F. Azpilicueta, M. Gende, A. Arago´n-A M. Herna´ndez-Pajares, J.M. Juan, and J. Sanz
95
Assisted Code Point Positioning at Sub-meter Accuracy Level with Ionospheric Corrections Estimated in a Local GNSS Permanent Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761 M. Crespi, A. Mazzoni, and C. Brunini
96
Semi-annual Anomaly and Annual Asymmetry on TOPEX TEC During a Full Solar Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769 F. Azpilicueta, C. Brunini, and S.M. Radicella
97
Numerical Simulation and Prediction of Atmospheric Aerosol Extinction Using Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . 775 J. Shin, S. Lim, C. Rizos, and K. Zhang
Contents
xvii
98
Impact of Atmospheric Delay Reduction Using KARAT on GPS/PPP Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781 Ryuichi Ichikawa, Thomas Hobiger, Yasuhiro Koyama, and Tetsuro Kondo
99
Modelling Tropospheric Zenith Delays Using Regression Models Based on Surface Meteorology Data . . . . . . . . . . . . . . . . . . . . . . . . . . 789 Tama´s Tuchband and Szabolcs Ro´zsa
100
Calibration of Wet Tropospheric Delays in GPS Observation Using Raman Lidar Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795 P. Bosser, C. Thom, O. Bock, J. Pelon, and P. Willis
101
Generation of Slant Tropospheric Delay Time Series Based on Turbulence Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801 M. Vennebusch and S. Scho¨n
102
Fitting of NWM Ray-Traced Slant Factors to Closed-Form Tropospheric Mapping Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809 Landon Urquhart, Marcelo Santos, and Felipe Nievinski
103
Estimation of Integrated Water Vapour from GPS Observations Using Local Models in Hungary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817 Sz. Ro´zsa
104
GNSS Remote Sensing in the Atmosphere, Oceans, Land and Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825 Shuanggen Jin
105
Mean Sea Surface Model of the Caspian Sea Based on TOPEX/Poseidon and Jason-1 Satellite Altimetry Data . . . . . . . . . 833 S.A. Lebedev
Session 5 Geodesy in Latin America Convenors: D. Blitzkow, C. Tocho 106
Combination of the Weekly Solutions Delivered by the SIRGAS Processing Centres for the SIRGAS-CON Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845 L. Sa´nchez, W. Seem€uller, and M. Seitz
107
Report on the SIRGAS-CON Combined Solution, by IBGE Analysis Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853 S.M.A. Costa, A.L. Silva, and J.A. Vaz
108
Processing Evaluation of SIRGAS-CON Network by IBGE Analysis Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859 S.M.A. Costa, A.L. Silva, and J.A. Vaz
109
ProGriD: The Transformation Package for the Adoption of SIRGAS2000 in Brazil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 869 Marcos F. Santos, Marcelo C. Santos, Leonardo C. Oliveira, Sonia A. Costa, Joa˜o B. Azevedo, and Maurı´cio Galo
xviii
Contents
110
The New Multi-year Position and Velocity Solution SIR09P01 of the IGS Regional Network Associate Analysis Centre (IGS RNAAC SIR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877 W. Seem€ uller, M. Seitz, L. Sa´nchez, and H. Drewes
111
Analysis of the Crust Displacement in Amazon Basin . . . . . . . . . . . . . . . 885 G.N. Guimara˜es, D. Blitzkow, A.C.O.C. de Matos, F.G.V. Almeida, and A.C.B. Barbosa
112
The Progress of the Geoid Model for South America Under GRACE and EGM2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893 D. Blitzkow, A.C.O.C. de Matos, J.D. Fairhead, M.C. Pacino, M.C.B. Lobianco, and I.O. Campos
113
Combining High Resolution Global Geopotential and Terrain Models to Increase National and Regional Geoid Determinations, Maracaibo Lake and Venezuelan Andes Case Study . . . . . . . . . . . . . . . . 901 E. Wildermann, G. Royero, L. Bacaicoa, V. Cioce, G. Acun˜a, H. Codallo, J. Leo´n, M. Barrios, and M. Hoyer
114
Evaluation of a Few Interpolation Techniques of Gravity Values in the Border Region of Brazil and Argentina . . . . . . . . . . . . . . . 909 R.A.D. Pereira, S.R.C. De Freitas, V.G. Ferreira, P.L. Faggion, D.P. dos Santos, R.T. Luz, A.R. Tierra Criollo, and D. Del Cogliano
115
RBMC in Real Time via NTRIP and Its Benefits in RTK and DGPS Surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917 S.M.A. Costa, M.A. de Almeida Lima, N.J. de Moura Jr, M.A. Abreu, A.L. de Silva, L.P. Souto Fortes, and A.M. Ramos
Session 6 Joint ION/FIG/ISPRS Session on Navigation and Earth Observation Convenors: D.A. Grejner-Brzezinska, C.K. Toth 116
Bootstrapping with Multi-frequency Mixed Code Carrier Linear Combinations and Partial Integer Decorrelation in the Presence of Biases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925 P. Henkel
117
Real Time Satellite Clocks in Precise Point Positioning . . . . . . . . . . . . . 935 R.J.P. van Bree, S. Verhagen, and A. Hauschild
118
Improving the GNSS Attitude Ambiguity Success Rate with the Multivariate Constrained LAMBDA Method . . . . . . . . . . . . . . 941 G. Giorgi, P.J.G. Teunissen, S. Verhagen, and P.J. Buist
119
An Intelligent Personal Navigator Integrating GNSS, RFID and INS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949 G. Retscher
120
Integration of Image-Based and Artificial Intelligence Algorithms: A Novel Approach to Personal Navigation . . . . . . . . . . . . . 957 Dorota A. Grejner-Brzezinska, Charles K. Toth, J. Nikki Markiel, Shahram Moafipoor, and Krystyna Czarnecka
Contents
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121
Modernization and New Services of the Brazilian Active Control Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967 L.P.S. Fortes, S.M.A. Costa, M.A. Abreu, A.L. Silva, N.J.M Ju´nior, K. Barbosa, E. Gomes, J.G. Monico, M.C. Santos, and P. Te´treault
122
magicSBAS: A South-American SBAS Experiment with NTRIP Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973 I. Alcantarilla, J. Caro, A. Cezo´n, J. Ostolaza, and F. Azpilicueta
Session 7 The Global Geodetic Observing System: Science and Applications Convenors: R. Gross, H.-P. Plas, L.P. Forles 123
Scientific Rationale and Development of the Global Geodetic Observing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987 G. Beutler and R. Rummel
124
GGOS Bureau for Standards and Conventions: Integrated Standards and Conventions for Geodesy . . . . . . . . . . . . . . . . 995 U. Hugentobler, T. Gruber, P. Steigenberger, D. Angermann, J. Bouman, M. Gerstl, and B. Richter
125
VLBI2010: Next Generation VLBI System for Geodesy and Astrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999 W.T. Petrachenko, A.E. Niell, B.E. Corey, D. Behrend, H. Schuh, and J. Wresnik
126
The New Vienna VLBI Software VieVS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007 J. Bo¨hm, S. Bo¨hm, T. Nilsson, A. Pany, L. Plank, H. Spicakova, K. Teke, and H. Schuh
127
Estimating Horizontal Tropospheric Gradients in DORIS Data Processing: Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013 P. Willis, Y.E. Bar-Sever, and O. Bock
Session 8
The IAG International Services and their Role for Earth Observation Convenors: R. Neilan, R. Forsberg 128
The BIPM: International References for Earth Sciences . . . . . . . . . . 1023 E.F. Arias
129
Development of the GLONASS Ultra-Rapid Orbit Determination at Geodetic Observatory Pecny´ . . . . . . . . . . . . . . . . . . . . . 1029 J. Dousa
130
AGrav: An International Database for Absolute Gravity Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037 H. Wziontek, H. Wilmes, and S. Bonvalot
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043
.
Session 1 Reference Frames Implementation for Geoscience’s Applications: From Local to Global Scales Convenors: Z. Altamimi, C. Brunini
.
1
Improved Analysis Strategy and Accessibility of the SIRGAS Reference Frame C. Brunini, L. Sanchez, H. Drewes, S. Costa, V. Mackern, W. Martı´nez, W. Seemuller, and A. da Silva
Abstract
The SIRGAS reference system is at present realized by the SIRGAS Continuously Operating Network (SIRGAS-CON) composed by about 200 stations distributed over Latin America and the Caribbean. SIRGAS member countries are improving their national reference frames by installing continuously operating GPS stations, which have to be consistently integrated into the continental network. As the number of these stations is rapidly increasing, the analysis strategy of the SIRGAS-CON network is based on two hierarchy levels: a) A core network with homogeneous continental coverage and stable site locations ensures the long-term stability of the reference frame. This network is processed by DGFI (Germany) as the IGS RNAAC SIR. b) Several densification sub-networks (corresponding to the national reference networks) improve the accessibility to the reference frame in the individual countries. Currently, the SIRGAS-CON stations are classified in three densification sub-networks (a southern, a middle, and a northern one), which are processed by the SIRGAS Local Processing Centres CIMA (Argentina), IBGE (Brazil), and IGAC (Colombia). These four Processing Centres deliver loosely constrained weekly solutions for the assigned sub-networks, which are integrated in a unified solution by the SIRGAS Combination Centres (DGFI and IBGE). The main SIRGAS products are: loosely constrained weekly solutions in SINEX format for further combinations of the
C. Brunini (*) Universidad Nacional de La Plata (UNLP), La Plata, Argentina e-mail:
[email protected] L. Sanchez H. Drewes W. Seemuller Deutsches Geod€atisches Forschungsinstitut (DGFI), Munich, Germany S. Costa A. da Silva Instituto Brasileiro de Geografia e Esta´tistica (IBGE), Rio de Janeiro, Brazil V. Mackern Universidad Nacional de Cuyo, Mendoza (CIMA), Argentina W. Martı´nez Instituto Geogra´fico Agustı´n Codazzi (IGAC), Bogota, Colombia S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_1, # Springer-Verlag Berlin Heidelberg 2012
3
4
C. Brunini et al.
network, weekly positions aligned to the ITRF as reference for GPS positioning in Latin America; and multi-year solutions (positions + velocities) for practical and scientific applications requiring time-dependent coordinates. This paper describes the analysis of the SIRGAS-CON network as the current realization of the SIRGAS reference system, its quality and consistency, as well as the planned activities to continue improving this reference frame.
1.1
Introduction
SIRGAS (Sistema de Referencia Geoce´ntrico para las Ame´ricas) as a reference system is defined identical with the ITRS (International Terrestrial Reference System). It is realized by means of a regional densification of the global ITRF (International Terrestrial Reference Frame) in Latin America and the Caribbean. The SIRGAS reference frame is in the same way extended to the countries through national densifications, which provide accessibility to the reference frame at national and local levels (Sanchez and Brunini 2009). SIRGAS has three realizations: two by means of episodic GPS campaigns and one by means of a network of continuously operating GPS stations. The first realization of SIRGAS (SIRGAS95) refers to the ITRF94, epoch 1995.4. It is given by a high-precision GPS network of 58 points distributed over South America (SIRGAS, 1997). In 2000, this network was re-measured and extended to the Caribbean, Central and North American countries. This second realization (SIRGAS2000) includes 184 GPS stations and refers to the ITRF2000, epoch 2000.4 (Drewes et al. 2005). The third realization of SIRGAS is the SIRGAS Continuously Operating Network (SIRGAS-CON), which is at present composed by more than 200 permanently operating GPS sites. The SIRGAS-CON network is weekly computed by the SIRGAS Analysis Centres; main products of this computation are: loosely constrained weekly solutions for station positions to be included in the IGS (International GNSS Service) global polyhedron and in multi-year solutions of the network; weekly station positions aligned to the ITRF for further applications in Latin America; and multi-year solutions providing station positions and velocities for high-precise practical and scientific applications. The SIRGASCON weekly positions refer to the observation epoch and to the current frame in which the GPS satellite orbits (i.e. IGS final orbits, Dow et al. 2009) are
given, at present the IGS05, the IGS realization of the ITRF2005 (see IGSMAIL 5447, http://igscb.jpl.nasa. gov/). The coordinates of the multi-year solutions refer to the latest available ITRF and to a specified epoch, e.g. the most recent SIRGAS-CON multi-year solution SIR09P01 refers to IGS05, epoch 2005.0 (Seemuller et al. 2009). The relationship between the different SIRGAS realizations is given by the transformation parameters between the corresponding ITRF solutions they refer and by taking into account the station position variations with time through a velocity (deformation) model (Drewes and Heidbach 2005). In this way, realizations or densifications of SIRGAS associated to different ITRFs and reference epochs materialize the same reference system and, after reducing them to the same frame and epoch, their positions are compatible at the mm-level. The present paper summarizes the improvement of the SIRGAS realization by means of the SIRGASCON network, the applied procedures for generating weekly solutions of this reference frame, quality and consistency of the obtained coordinates, as well as ongoing activities to avoid some disadvantages of the actual analysis strategy. The multi-year solutions for the SIRGAS-CON reference frame are presented by e.g. Seemuller et al. 2008, 2009.
1.2
The SIRGAS-CON Reference Frame
The initial realizations of SIRGAS based on pillars have been replaced by an increasing number of continuously operating GPS stations (Fig. 1.1), which all together constitute the SIRGAS-CON network (Fig. 1.2). 48 of these stations belong to the IGS global network, while the others (about 160) correspond to the national reference frames.
1
Improved Analysis Strategy and Accessibility of the SIRGAS Reference Frame
Fig. 1.1 SIRGAS continuously operating stations since 2000
5 IGS
No. of stations
Regional
Total
200 150 100 50 0 2000
2001
2002
2003
2004
2005
2006
2007
2008 06/2009
Year
To guarantee the consistency of the national reference frames with the global reference frame in which the GPS orbits are computed, the national reference stations are integrated into the SIRGAS-CON network and all together are processed in a common way. This provides homogeneous consistency and accuracy of their coordinates on a continental level. Until GPS week 1495 (August 2008), the Deutsches Geod€atisches Forschungsinstitut (DGFI, Germany), as the IGS RNAAC SIR (IGS Regional Network Associate Analysis Centre for SIRGAS), processed the entire SIRGAS-CON network in one block only (Seemuller and Drewes 2008). However, given the large number of SIRGAS-CON stations, this usual one-block processing became unfeasible and it was necessary to redefine the analysis strategy of the network. The new analysis strategy is based on (1) defining a core continental network (SIRGAS-CON-C) as the primary densification of the ITRF in Latin America, and (2) improving the geographical density of this core network by means of densification sub-networks (SIRGASCON-D). The core network ensures the long-term stability of the continental reference frame, while the densification sub-networks make it available at national and local levels. Although, they appear as two different categories, core and densification stations match requirements, characteristics, performance, and quality of the ITRF stations. The SIRGAS-CON-D sub-networks shall correspond to the national reference frames, i.e., as an optimum there shall be as many sub-networks as countries in the region. Since at present not all of the countries are operating a Processing Centre, the existing stations are classified in three densification sub-networks (Fig. 1.2): a northern one covering Mexico, Central America, the Caribbean, Colombia, and Venezuela; a
middle one comprising stations installed on Brazil, Ecuador, Bolivia, Suriname, French Guyana, Guyana, Peru, and Bolivia; and a southern one including the stations located in Uruguay, Paraguay, Argentina, Chile, and Antarctica. Each densification sub-network includes a minimum number of IGS and SIRGASCON core stations as overlapping points for the combination.
1.3
Analysis of the SIRGAS-CON Network
The SIRGAS-CON-C network is computed by DGFI. The densification sub-networks are processed by the active SIRGAS Local Processing Centres until new ones become operational. At present, they are: Centro de Procesamiento Ingenierı´a-Mendoza-Argentina at the Universidad Nacional del Cuyo (CIMA, Argentina), Instituto Brasileiro de Geografia e Estatistica (IBGE, Brazil), and Instituto Geogra´fico Agustı´n Codazzi (IGAC, Colombia). These four Processing Centres apply a common procedure established by SIRGAS (in agreement with the standards of the IGS and the IERS – International Earth Rotation and Reference Systems Service) to generate loosely constrained weekly solutions for station positions (see e.g. Natali et al. 2009; Seemuller and Sanchez 2009). In these solutions satellite orbits, satellite clock offsets, and Earth orientation parameters are fixed to the final weekly IGS solutions (Dow et al. 2009) and all station positions are constrained to 1 m. The individual contributions are integrated in a unified solution by the SIRGAS Combination Centres DGFI and IBGE (Fig. 1.3). The DGFI combinations are provided to the users as the SIRGAS official
6
C. Brunini et al.
Fig. 1.2 SIRGAS-CON network (status August 2009)
products (Sanchez et al. 2011), while the IBGE combinations assure redundancy and control for those products (Costa et al. 2009). At present, all SIRGAS Analysis Centres use the Bernese GPS Software (Dach et al. 2007) for processing the individual sub-networks and for their combination. Before combining the individual solutions, the constraints included in the delivered normal equations
are removed and the solutions are separately aligned to the IGS05 reference frame. The obtained standard deviations are analysed to establish the quality of the individual solutions and to determine variance factors, when it is necessary to compensate differences in the stochastic models of the Processing Centres. The station positions computed from each solution are compared by means of a similarity transformation to the
1
Improved Analysis Strategy and Accessibility of the SIRGAS Reference Frame
7
Fig. 1.3 Data flow in the weekly processing of the SIRGASCON reference frame
IGS weekly values and to each other to identify possible outliers. Once inconsistencies and outliers are reduced from the individual free normal equations, a combination for a loosely constrained weekly solution for station positions (all of them constrained to 1 m) is computed. This solution is submitted in SINEX format to IGS for the global polyhedron and it is stored to be included in the next multi-year solution of the SIRGAS-CON network. A solution aligned to the IGS05 reference frame is also computed to provide weekly positions of all SIRGAS-CON stations for further applications (Fig. 1.4). Different criteria are applied to establish the quality of the contributing solutions delivered by the SIRGAS Processing Centres. The first one relates to the determination of mean standard deviation of station positions by solving the individual normal equations with respect to the IGS05 frame. These standard deviations represent the formal errors of the individual solutions. Secondly, the analysis of station position time series allows ascertaining the consistency of the individual contributions from week to week (repeatability). Then, the comparison by means of a similarity transformation of the individual solutions referring to the IGS05 with the IGS weekly positions provides information about their compatibility with the IGS global network. Figure 1.5 presents the mean values
Fig. 1.4 Combination procedure applied to generate the weekly solution of the SIRGAS-CON reference frame
for each criterion and for each sub-network for the period between the GPS weeks 1495 and 1538. These results indicate that the individual solutions are at the same level of precision: the formal error of the station positions is about 1.6 mm and the repeatability of the weekly coordinates is estimated to be ~ 2.0 mm for the horizontal component and 4.0 mm in the height.
1.4
Weekly Processing of the SIRGAS-CON Reference Frame
Regional and national reference frames supporting GNSS positioning must be consistent with the reference frame in which the GPS orbits are determined. For that reason, the IGS RNAAC SIR yearly generates a new multi-year solution referred to the current ITRF realization and including the SIRGAS-CON stations operating more than 2 years (e.g. Seemuller et al 2008, 2009). The latest solution SIR09P01 contains 128 stations with positions and velocities referring to IGS05, epoch 2005.0 (Seemuller et al. 2009). However, as mentioned before, the SIRGAS-CON network
8
Fig. 1.5 Evaluation of the solutions computed for the SIRGASCON individual sub-networks (mean values for the period GPS weeks 1495–1538)
is composed by more than 200 stations and those stations (about 80) that are not included in SIR09P01 can be used as reference points only, if their weekly positions linked to the ITRF (i.e. IGS05) are available. In this way, weekly solutions of the SIRGAS-CON network aligned to the IGS05 frame are necessary. Usually, epoch solutions (daily, weekly, multi-year) of regional reference networks are aligned to the ITRF using a set of fiducial stations with known positions and constant velocities; i.e. they consider linear coordinate changes only. However, GPS stations show significant seasonal position variations (mainly in the up component) resulting from a combination of geophysical loading and systematic errors. Ignoring these seasonal variations at reference stations can introduce systematic errors in the datum realization and the reference networks can be significantly deformed. These effects are larger in regional networks than in global ones, especially in zones with strong seasonal variations as the SIRGAS region. In this way, with the objective of minimizing the influence of seasonal
C. Brunini et al.
variations in the weekly realization of the SIRGASCON frame, the SIRGAS Working Group I (Reference System) analyzed different strategies for the datum definition taken into account the minimal network deformation, the weekly repeatability of station positions, and the consistency with the IGS weekly solutions for the global network. This analysis basically consisted of solving the same free normal equations applying two different sets of reference coordinates for the datum definition: the first one corresponds to the IGS05 positions at epoch 2000.0 extrapolated to the observation epoch using the ITRF2005 constant velocities. The second set corresponds to the weekly positions determined for the IGS05 reference stations within the IGS weekly combination (igsyyPwwww. snx). After comparing the loosely constrained solutions (in which the network is not deformed) with the constrained ones, the main conclusion shows that applying constant velocities to the reference coordinates introduces the largest distortions (more than 5 mm) into the station positions, mainly at the fiducial points (Fig. 1.6). This is a consequence of constraining a seasonal signal to be a linear trend. In this way, the SIRGAS-CON weekly solutions are aligned to the IGS05 by constraining the positions of the reference IGS05 stations to the values resulting of the IGS weekly combinations (Sanchez et al. 2011). The quality control of the SIRGAS-CON weekly solutions takes into account (1) the mean standard deviation for station positions to estimate the formal error of the final values; (2) the coordinate repeatability after combining the individual solutions to evaluate the internal consistency of the combined network; (3) station position time series analysis to determine the consistency of the combined solutions from week to week; and the (4) comparison with the IGS weekly coordinates and the IBGE weekly combinations to ascertain the reliability of the weekly solutions as well as to guarantee the required redundancy for the generation of the final SIRGAS-CON weekly positions. Table 1.1 summarizes the mean values resulting from the evaluation criteria for the period covering the GPS weeks 1495–1538. The mean standard deviation of the combined solutions is very similar to those obtained for the individual contributions (Fig. 1.5), i.e. their quality is maintained and their combination does not generate distortions in the SIRGAS-CON weekly realization. The weekly repeatability of the resulting
1
Improved Analysis Strategy and Accessibility of the SIRGAS Reference Frame
Fig. 1.6 3D residuals derived after comparing the primary loosely constrained solutions for the SIRGAS-CON network with the same solutions aligned to the IGS05 frame applying two different sets of reference coordinates for the datum definition. In (a) the IGS05 station positions at 2000.0 are
9
extrapolated to the observation epoch by means of the ITRF2005 constant velocities. In (b) the station positions computed within the IGS weekly combinations for the IGS05 stations are directly introduced as reference coordinates. Values presented on the maps are mean values for 117 weeks
Table 1.1 Evaluation of the SIRGAS-CON weekly realizations positions provides an estimate of the internal consis(mean values for the period between GPS weeks 1495–1538) tency of approximately 0.8 mm in the horizontal Criteria Component Value in components and 2.5 mm in the vertical one. The [mm] RMS values derived from the time series for station Mean standard deviation 1.64 coordinates and with respect to the IGS weekly 0.61 Mean RMS of residuals for coordinate N positions indicate that the external accuracy of the repeatability in the weekly combination E 0.87 network is about 1.5 mm in the horizontal position Up 2.51 and 3.8 mm in the height. Total 2.73 Mean RMS of residuals derived from N 1.50 time series of station positions E 1.36 Up 3.80 1.5 Closing Remarks and Outlook Total 4.33 RMS of residuals wrt IGS weekly N 1.39 The processing strategy described in this paper for the solutions E 1.75 SIRGAS-CON network is applied since GPS week Up 3.69 1495. As already mentioned, before (since June 1996 Total 4.35 to August 2008), the entire SIRGAS-CON network RMS of station coordinate differences N 1.10 was computed by DGFI in one adjustment. In order between DGFI and IBGE combinations E 1.10 to establish the consistency of the current combined Up 1.40 solutions with the previous computations, residual Total 2.20
position time series were generated from the weekly
10
solutions available between January 2000 and January 2009. Discontinuities or jumps at the epoch in which the analysis strategy was changed (last week of August 2008) are not identifiable. Results show that the current weekly combined solutions are at the same accuracy level and totally consistent with the previous computations (when the network was calculated in one block). Nevertheless, the present sub-network distribution has two main disadvantages: (1) Not all SIRGASCON stations are included in the same number of individual solutions, i.e., they are unequally weighted in the weekly combinations, and (2) since there are not enough Local Processing Centres, the required redundancy (each station processed by at least three processing centres) is not fulfilled. Therefore, SIRGAS promotes the installation of more Local Processing Centres hosted by Latin American countries. In this frame, institutions interested to install a SIRGAS Processing Centre shall pass a test period of one year. In this period, they have to align their processing strategies with the SIRGAS guidelines and meet the delivering deadlines. At present, there are five Experimental Processing Centres: Instituto Geogra´fico Militar of Ecuador (IGM, Ecuador), Laboratorio de Geodesia Fı´sica y Satelital at the Universidad del Zulia (LGFS-LUZ, Venezuela), Servicio Geogra´fico Militar of Uruguay (SGM, Uruguay), Instituto Nacional de Estadı´stica y Geografı´a (INEGI, Mexico), and Instituto Geogra´fico Nacional de Argentina (IGN, Argentina). They will become Official Processing Centres in the near future and a redistribution of the SIRGAS-CON stations between the operative SIRGAS Analysis Centres will allow including each regional station in the same number of individual solutions. This will significantly improve the reliability and quality control of the weekly solutions for the SIRGAS-CON reference frame.
References Costa SMA, da Silva AL, Vaz JA (2009) Report of IBGE Combination Centre. Period of SIRGAS-CON solutions:
C. Brunini et al. from week 1495 to 1531. Presented at the SIRGAS 2009 General Meeting. Buenos Aires, Argentina. September. Available at www.sirgas.org. Dach R, Hugentobler U, Fridez P, Meindl M (eds) (2007) Bernese GPS Software Version 5.0 – Documentation. Astronomical Institute, University of Berne, p 640 Dow JM, Neilan RE, Rizos C (2009) The International GNSS Service in a hanging landscape of Global Navigation Satellite Systems. J Geodesy 83:191–198. doi:10.1007/s00190-0080300-3 Drewes H, Kaniuth K, Voelksen C, Alves Costa SM, Souto Fortes LP (2005) Results of the SIRGAS campaign 2000 and coordinates variations with respect to the 1995 South American geocentric reference frame. In: Sanso F (ed) A window on the future of geodesy, vol 128, IAG symposia. Springer, Heidelberg, pp 32–37 Drewes H, Heidbach O (2005) Deformation of the South American crust estimated from finite element and collocation methods. In: Sanso F (ed) A window on the future of geodesy, vol 128, IAG Symposia. Springer, Heidelberg, pp 544–549 Natali MP, M€uller M, Ferna´ndez L, Brunini C (2009) CPLat: the pilot processing center for SIRGAS in Argentina. In: Drewes H (ed) Geodetic reference frames, vol 134, IAG Symposia. Springer, Heidelberg, pp 179–184 Sanchez L, Brunini C (2009) Achievements and challenges of SIRGAS. In: Drewes H (ed) Geodetic reference frames, vol 134, IAG symposia. Springer, Heidelberg, pp 161–166 Sanchez LW, Seem€uller MS, Seitz M (2011) Combination of the weekly solutions delivered by the SIRGAS Processing Centres for the SIRGAS-CON reference frame. In: Kenyon S et al (eds) Geodesy for planet Earth, Buenos Aires Argentina, vol 136, IAG symposia. Springer, Heidelberg Seemuller W, Drewes H (2008) Annual Report 2003–2004 of IGS RNAAC SIR. In: IGS 2001–02 Technical Reports, IGS Central Bureau, (eds) Jet Propulsion Laboratory, Pasadena, CA. Available at http://igscb.jpl.nasa.gov/igscb/resource/ pubs/2003-2004_IGS_Annual_Report.pdf Seemuller W, Krugel M, Sanchez L, Drewes H (2008) The position and velocity solution DGF08P01 of the IGS Regional Network Associate Analysis Centre for SIRGAS (IGS RNAAC SIR). DGFI Report No. 79. DGFI, Munich. Available at www.sirgas.org Seemuller W, Seitz M, Sanchez L, Drewes H (2009) The position and velocity solution SIR09P01 of the IGS Regional Network Associate Analysis Centre for SIRGAS (IGS RNAAC SIR). DGFI Report No. 85. DGFI, Munich. Available at www.sirgas.org Seemuller W, Sanchez L (2009) SIRGAS Processing Centre at DGFI: report for the SIRGAS 2009 General Meeting. Presented at the SIRGAS 2009 General Meeting. Buenos Aires, Argentina, September. Available at www.sirgas.org SIRGAS (1997). SIRGAS Final Report; Working Groups I and II IBGE, Rio de Janeiro, p 96. Available at www.sirgas.org
2
Improved GPS Data Analysis Strategy for Tide Gauge Benchmark Monitoring Alvaro Santamarı´a-Go´mez, Marie-Noe¨lle Bouin, €ppelmann and Guy Wo
Abstract
The University of La Rochelle (ULR) TIGA Analysis Center (TAC) completed a new global reprocessed solution spanning 13 years with more than 300 GPS permanent stations, 216 of them being co-located with tide gauges. A state-of-theart GPS processing strategy was applied, in particular, the station sub-networks used in the daily processing were optimally built. Station vertical velocities were estimated in the ITRF2005 reference frame by stacking the weekly position estimates. Outliers, offsets and discontinuities in time series were carefully examined. Vertical velocities uncertainties were assessed in a realistic way by analysing the type and amplitude of the noise content in the residual position time series. The comparison shows that the velocity uncertainties have been reduced by a factor of 2 with respect to previous ULR solutions. The analysis of this solution and its by-products shows the high geodetic quality achieved in terms of homogeneity, precision and consistency with respect to other top-level geodetic solutions.
2.1
A. Santamarı´a-Go´mez (*) Instituto Geogra´fico Nacional, c/ General Iban˜ez Ibero 3, 28071, Madrid, Spain Institut Ge´ographique National, LAREG/GRGS, 6–8 av. Blaise Pascal, 77455, Champs-sur-Marne, France e-mail:
[email protected] M.-N. Bouin Institut Ge´ographique National, LAREG/GRGS, 6–8 av. Blaise Pascal, 77455, Champs-sur-Marne, France CNRM/CMM, Me´te´o France, 13 rue du Chatellier, CS 12804, 29228, Brest, France G. W€oppelmann Universite´ de La Rochelle-CNRS, UMR 6250 LIENSS, 2 rue Olympe de Gouges, 17000, La Rochelle, France
Introduction
In order to estimate long-term geocentric sea level rise, tide gauges trends must be corrected for the long-term vertical displacements of the land upon which they are settled. In addition, for proper satellite altimeter calibration purposes, tide gauges trends must be referred to a common, global and stable reference frame, such as the latest realization of the International Terrestrial Reference Frame (ITRF) (Altamimi et al. 2007). These long-term vertical displacements can be corrected by modelling geological processes as the Global Isostatic Adjustment (GIA) (e.g. Douglas 2001) or directly from continuous geodetic observations at or near tide gauges. This second method should be preferred as it takes into account local displacements (geological, anthropogenic or whatever), not accounted
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_2, # Springer-Verlag Berlin Heidelberg 2012
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12
Fig. 2.1 Number of daily available stations (grey) and percentage of daily resolved ambiguities (black)
for in the GIA models. Within the different geodetic techniques used for this purpose (GPS, DORIS and absolute gravity), GPS is the most widespread. Recent studies (W€ oppelmann et al. 2009, Bouin and W€oppelmann 2010) have shown that correcting the tide gauge trends using continuous GPS stations (cGPS@TG) improves the consistency of the sea level rates. To this aim, the International GNSS Service (IGS) Tide Gauge Benchmark Monitoring Pilot Project (TIGA) was established in 2001 (Sch€ one et al. 2009). Since 2002, the ULR consortium contributes to the TIGA project as an Analysis and Data Center (W€oppelmann et al. 2004). Several global vertical velocity field solutions (ULR solutions hereafter) were released with different station networks, time spans and processing strategies (W€oppelmann et al. 2007, 2009). In this paper, we present the fourth ULR solution based on an homogeneous reprocessing of a larger global network of 316 stations, spanning an increased period of 13 years (January 1996 to December 2008). This solution comes out with a new data analysis strategy, including a new sub-network design and combination. The troposphere and ocean tide modelisation were also improved. Both GPS processing and vertical velocity estimation strategies are described; realistic uncertainties are estimated by analysing the noise content of time series. Finally, the quality of the solution is assessed and discussed.
2.2
Data Analysis Strategy
2.2.1
Data
The global tracking network consists of 316 GPS stations. 216 of them are cGPS@TG, including 81 stations committed to TIGA. Also 124 of them are
IGS reference frame (RF) stations used for realizing the reference frame (Kouba et al. 1998) and for improving the network geometry. This network was processed over the period 1st January 1994 to 31st December 2008. Small RINEX files (less than 5 h of observation) were rejected. This quality check procedure yielded a number of daily available stations between a minimum of 25 in 1994 (53 in 1996) and a maximum of 239 in 2006 (grey line in Fig. 2.1). 1994 and 1995 were finally not retained in the solution due to a lack of fixed ambiguities and therefore quality (black line in Fig. 2.1) and they will not be further considered.
2.2.2
Improved Network Geometry
GPS processing time increases exponentially with the number of stations. To overcome this limitation, it is usual to split the whole network in several subnetworks, to process each sub-network independently and then to combine the sub-network solutions into a unique daily solution. Historic ULR solutions (ULR1 to ULR3 solutions) used five global, manually-selected, permanent subnetworks over the entire data span (“static subnetworks” hereafter). Using this approach, the a priori stations included in each sub-network were always the same, whether or not their data were available for a specific day, making the geometry worse when their data were missing, and therefore, possibly yielding an unnecessary large number of sub-networks in the processing (always five). This static configuration was changed in the ULR4 solution into a new station distribution approach resulting in global, automatic, daily-variable sub-networks (“dynamic sub-networks” hereafter), with up to 50 stations per sub-network.
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Improved GPS Data Analysis Strategy for Tide Gauge Benchmark Monitoring
Shorter baselines improves ambiguity resolution (Steigenberger et al. 2006). With the dynamic approach, all daily available stations were distributed into the strictly necessary number of sub-networks, ensuring optimal dense sub-networks. Thus, the number of dynamic sub-networks used grows from 1 in 1996 to 6 in 2003. Moreover, to obtain global geometrically well-distributed sub-networks for optimal orbit estimation, each station is assigned to the sub-network where it is more isolated, i.e. reducing the baselines. In this way, “deserted” areas of each sub-network are iteratively being “populated”. In addition, six daily-variable common IGS RF stations, with more than 12 h of observation, are included in each dynamic sub-network to combine the solutions. Northernmost and southernmost stations are always selected and then four other globally welldistributed stations are added. Static versus dynamic approaches were compared by processing two solutions using the same stations and processing strategy except for the stations distribution. Figure 2.2 shows that using dynamic sub-networks clearly increases the percentage of resolved ambiguities as the number of available stations decreases, up to 20% in 1997 (Fig. 2.2). The 10% offset in the percentage of resolved ambiguities observed at the end of 1999 for both approaches is related to the use of code bias corrections (see Sect. 2.2.3), only available for post2000 year period when the test was performed.
2.2.3
Models and Parameterization
Double-differenced ionosphere-free carrier phase data is analysed using GAMIT software version
Fig. 2.2 Resolved ambiguities for static (grey), dynamic subnetworks (top black) and the difference (bottom black)
13
10.34 (Herring et al. 2006a). The elevation cut-off angle is set to 10 , avoiding mismodelling of lowelevation troposphere and phase center variations (PCV) of relative-to-absolute antenna calibration. Sampling rate is set to 3 minutes. Carrier phase observations are weighted in two iterations: by elevation angle first and then by elevation angle and by station, accounting for the station phase residuals from the first iteration. Code bias corrections are applied for the whole period using monthly tables from the Astronomical Institute of the University of Bern (AIUB) [IGSMAIL-2827 (2000) at http:// igscb.jpl.nasa.gov/mail/). Real-valued double differenced phase cycle ambiguities are adjusted except when they can be resolved confidently. In this case, they are fixed using the MelbourneW€ubbena wide-lane to resolve L1–L2 cycles and then estimation to resolve L1 and L2 cycles. For satellite antennas, satellite-specific z-offsets (Ge et al. 2005) and block-specific nadir angledependent absolute PCV (Schmid et al. 2007) are applied. For receiver antennas, L1/L2 offsets and azimuth-dependent, when available, and elevationdependent absolute PCV are applied. A priori zenith hydrostatic (dry) delay values are extracted by station from the ECMWF meteorological model through the VMF1 grids (Boehm et al. 2006). Residual delays are adjusted for each station assuming mostly dominated by the wet component and parameterized by a piecewise linear, continuous model with 2 h intervals. Both dry and wet VMF1 mapping functions are used. One gradient is estimated for each day and each station. Solid Earth tides are corrected following IERS Conventions (2003) (McCarthy and Petit, 2004). Ocean tide loading is corrected using FES2004 model (Lyard et al. 2006). No atmospheric tide nor non-tidal corrections were applied. Earth orientation parameters (EOP) are daily estimated as a piecewise, linear model with a priori values from IERS Bulletin B. UT1–UTC offsets are highly constrained to their a priori values. Satellite positions and velocities are adjusted in 24 h arcs taking IGS final orbits (Dow et al. 2005) as a priori. Solar radiation pressure parameters are estimated using the Berne model (Beutler et al. 1994).
A. Santamarı´a-Go´mez et al.
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2.2.4
Data Processing Scheme and Reference Frame
Each dynamic sub-network is processed independently using GAMIT software. The daily sub-network solutions are combined into a daily solution (by estimating only translations and rotations) using GLOBK (Herring et al. 2006b) by means of the estimated orbital parameters, the estimated positions of the six common stations and their estimated zenith tropospheric path delays. Daily loose solutions are constrained by no-net-rotation (NNR) constraints with respect to ITRF2005 and combined into a weekly solution using CATREF software (Altamimi et al. 2007). These weekly solutions are aligned to ITRF2005 using NNR constraints with all IGS RF stations available, whereas inner constraints (Altamimi et al. 2007) are used for scale and translation, in order to preserve the weekly apparent geocenter motion information. All the weekly solutions for the whole period (GPS weeks 0834–1512), are then combined into a longterm solution using CATREF. This long-term solution (ULR4) is aligned to ITRF2005 using minimal constraints over all the transformation parameters with a selected set of IGS RF stations called datum. The 68 stations retained in the datum were selected based on their data availability (at least present in 80% of the whole processed period) and their quality as follows. Firstly, stations with known or suspected velocity discontinuities were rejected, and secondly, in an iterated process, stations showing large position and velocity residuals with respect to ITRF2005 values were also rejected. Thresholds for positions were set to 0.5 cm in horizontal and 1.5 cm in vertical. The larger value in the vertical component is due to the fact that ITRF2005 GPS coordinates were estimated with a relative PCV model. Station differences using the absolute PCV model are estimated to be within this range. Thresholds for velocity residuals were set to 1.5 mm/year and 2 mm/year respectively. The residual position time series of each station were visually examined. To avoid biased velocities, all discontinuities (significant offsets and velocity changes) were detected, identified if possible, and removed using ITRF2005 discontinuities as a priori. Then, all outliers were removed in an iterative process, from bigger to smaller magnitude (depending on the time series noise), down to a minimum of 2 cm for residuals and 4 for normalized residuals.
2.3
Results
2.3.1
Vertical Rates
The vertical velocity fields of ULR4 and ULR3 (W€oppelmann et al. 2009)) solutions were compared using a common set of 170 stations with more than 4.5 years of data. Figure 2.3 shows that most of the velocity differences are below 1 mm/year (RMS of 0.8 mm/year), except some stations for which larger differences are due to different discontinuities on their time series. The mean difference between both velocity fields is 0.16 0.06 mm/year which is related to the different datum used to aling the solutions. This misalignment is under the internal precision of the ITRF2005. From the complete ULR4 solution, 224 stations with more than 4.5 years of data were retained. For these stations, their estimated velocities are confidently not influenced by seasonal signals (Blewitt and Lavalle´e 2002). Nevertheless, the rate uncertainties estimated with a standard least squares algorithm (based on a Gaussian white noise process) are clearly optimistic by a factor of 3–11 (Zhang et al. 1997; Mao et al. 1999). More realistic uncertainties of the estimated velocities must account for correlated noise present in the time series. A noise analysis was performed using the Maximum Likelihood Estimation (MLE) technique (CATS software, (Williams, 2008)). Vertical velocity uncertainties were estimated using a white noise plus power law noise model. To avoid biased adjustments, time series were previously examined for periodic signals. Besides the annual and semi-annual terms, we also found and removed up to six harmonics of
Fig. 2.3 Vertical velocity difference between ULR3 and ULR4. Dashed lines represent 1 mm/year
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Improved GPS Data Analysis Strategy for Tide Gauge Benchmark Monitoring
15
Fig. 2.4 Histogram of estimated uncertainties for ULR4 (grey) and ULR3 (black) solutions and their median values
the GPS “draconitic” period described by Ray et al. (2007). Figure 2.4 shows the histogram of the realistic vertical velocity uncertainties of the ULR4 solution with respect to the realistic uncertainties estimated for the ULR3 solution also using CATS. The improvement is close to a factor of 2. Also the factor of optimism of the formal uncertainties with respect to the realistic ones is 2–3, quite smaller than the abovementioned values. This is due to the improvement and consistency of the processing strategy presented here, which results in a noticeable reduction of the correlated noise content for the ULR4 solution compared to previous solutions.
2.3.2
Weekly Repeatability
The internal quality of the ULR4 solution was assessed by analysing the repeatability of the weekly position solutions. Figure 2.5 shows the repeatability of the time series (mean values of the weighted RMS of the weekly positions with respect to the long-term combined positions) for ULR4 and ULR3 solutions. Horizontal and vertical repeatabilities are improved in the ULR4 solution. Moreover, for the whole reprocessed period vertical repeatabilities are more stable, showing the improved ULR4 time consistency. ULR4 repeatability values are between 1 and 3 mm for the horizontal and between 4 and 6 mm for the vertical component (3D weighted RMS between 2 and 4 mm). These values are fully consistent with those of the IGS
Fig. 2.5 Horizontal (bottom) and vertical (top) weighted RMS of the weekly solutions with respect to the long-term solution for both ULR4 (black) and ULR3 (grey) solutions
combined solution (Altamimi and Collilieux, 2008), showing that ULR4 solution is comparable in quality with the ITRF2005.
2.3.3
Origin and Scale
As a satellite technique, GPS estimated origin should be coincident with the Earth’s center of mass. However this affirmation is not completely fulfilled due to remaining GPS-specific systematic errors, as the modelling of the solar radiation pressure coefficients or the unaccounted effect of higher ionospheric orders (Herna´ndez-Pajares et al. 2007). We have estimated here apparent geocenter motion using the network shift or geometric approach
A. Santamarı´a-Go´mez et al.
16
(Lavalle´e et al. 2006). Figure 2.6 shows the translation and scale parameters of the weekly solutions with respect to the long-term combined solution aligned to the ITRF2005. Translation trends are not significant, showing the consistency of the secular origin definition with respect to the ITRF2005. The scale shows no trend either, as this parameter is completely dependent on the ITRF2005 scale definition through the satellites antenna z-offset corrections. For intercomparison purposes, an annual signal was estimated for each transformation parameter (Table 2.1).
Compared to SLR results (Collilieux et al. 2009), the annual amplitudes of the equatorial components (X and Y) and the scale are fully consistent. However, the amplitude of the Z component is twice larger. Regarding the annual phase, the scale parameter is fully consistent, but all translational parameters show a shift of about 137º (4.5 months). Compared to other GPS results (Lavalle´e et al. 2006), the amplitude of the Z component and both equatorial phases are consistent. The phase of Z component exhibits larger solution-dependent variations. Both issues point probably at the above-mentioned GPS systematic errors and also at the poor performance of the network shift method used with a not-well distributed global network (Lavalle´e et al. 2006).
2.3.4
Orbits
The estimated ULR4 orbits were compared with the current official non-reprocessed IGS final orbits (Dow et al. 2005). A classic 7-parameter Helmert transformation was applied between both 24 h-arc sets. 1D RMS differences (the average of the three RMS components) were estimated for each common observed satellite and then the median daily RMS value was extracted and traced (black line, Fig. 2.7). We show that ULR and IGS orbits are in good agreement with each other, from 8.5 cm in 1996 to 1.5 cm in 2009. The same range of differences was obtained between IGS orbits and reprocessed orbits from SIO/SOPAC IGS Analysis Center (light grey line). Some smaller differences were obtained with
Fig. 2.6 Weekly translation and scale parameters with respect to the ITRF2005. Also their trends and annual signal are traced
Table 2.1 Annual signal of apparent geocenter and scale TX TY TZ Scale
Amplitude (mm) 2.3 0.2 4.2 0.3 9.9 0.8 1.8 0.1
Phase (deg) 164.6 5.4 122.2 3.5 171.3 3.5 243.2 1.6
Fig. 2.7 7-day smoothed daily RMS between final IGS orbits and ULR (black), SIO/SOPAC (light grey) and CODE/AIUB (dark grey) reprocessed orbits
2
Improved GPS Data Analysis Strategy for Tide Gauge Benchmark Monitoring
reprocessed CODE/AIUB IGS Analysis Center (dark grey line) for the post-2000 period. This demonstrates that the ULR4 orbits are of the same quality as the reprocessed orbits of some of the IGS Analysis Centers.
2.4
Concluding Remarks
The new ULR4 solution is based on an homogeneous reprocessing of a global GPS network of 316 stations spanning up to 13 years of data. The processing strategy was improved with respect to past ULR solutions. Special attention was paid to the sub-network geometry distribution, which clearly improves the quality of the reprocessing by increasing the number of resolved ambiguities. The analysis of the results and by-products of this solution (vertical velocities, repeatability, transformation parameters and orbits) shows the high geodetic quality achieved. The stateof-the-art GPS processing strategy implemented fulfils the IGS requirements and recommendations. Thereby, in addition to the IGS TIGA project, the ULR consortium is participating with its latest solution to the first IGS reanalysis campaign, enabling an invaluable extension of IGS and ITRF reference frames towards tide gauges. Also, the ULR consortium is contributing to the Working Group on Regional Dense Velocity Fields of the International Association of Geodesy Subcommision 1.3. (Bruyninx 2011). Further studies will be carried out in order to assess the geophysical usefulness of this solution. For example, this global and accurate vertical velocity field may be used to separate vertical land motion trends from relative sea level trends as recorded by tide gauges. Acknowledgements The authors acknowledge two unknown reviewers who contributed to an improved paper. We also thank the invaluable technical support given by Mikael Guichard, Marc-Henri Boisis-Delavaud and Frederic Bret from the IT centre of the University of La Rochelle (ULR). The ULR computing infrastructure used for the reprocessing of the GPS data was partly funded by the European Union (Contract 31031-2008, European regional development fund). This work was also feasible thanks to all institutions and individuals worldwide that contribute to make GPS data and products freely available.
17
References Altamimi Z, Collilieux X (2008) IGS contribution to ITRF. J Geod 83(3–4):375–383 Altamimi Z, Collilieux X, Legrand J, Garayt B, Boucher C (2007) ITRF2005: a new release of the International Terrestrial Reference Frame based on time series of station positions and Earth Orientation Parameters. J Geophys Res 112:B09401 Beutler G, Brockmann E, Gurtner W, Hugentobler U, Mervart L, Rothacher M (1994) Extended orbit modeling techniques at the CODE Processing Center of the International GPS Service for Geodynamics (IGS): theory and initial results. Manuscripta Geodaetica 19:367–386 Blewitt G, Lavalle´e D (2002) Effect of annual signals on geodetic velocity. J Geophys Res 107:B02145 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 111:B02406 Bouin M-N, W€oppelmann G (2010) Land motion estimates from GPS at tide gauges: a geophysical evaluation. Geophys J Int 180:193–209 Bruyninx C (2011) A dense global velocity field based on GNSS observations: preliminary results. In: Kenyon S et al (eds) Geodesy for planet Earth. Springer, Heidelberg Collilieux X, Altamimi Z, Ray J, van Dam T, Wu X (2009) Effect of the satellite laser ranging network distribution on geocenter motion estimation. J Geophys Res 114:B02145 Douglas B (2001) Sea level change in the era of the recording tide gauge, vol 75, International geophysics series. Academic, San Diego, CA Dow JM, Neilan RE, Gendt G (2005) The International GPS Service: celebrating the 10th anniversary and looking to the next decade. Adv Space Res 36:320–326 Ge M, Gendt G, Dick G, Zhang F, Reigber C (2005) Impact of GPS satellite antenna offsets on scale changes in global network solutions. Geophys Res Lett 32:L06310 Herna´ndez-Pajares M, Juan JM, Sanz J, Oru´s R (2007) Secondorder ionospheric term in GPS: Implementation and impact on geodetic estimates. J Geophys Res 112:B08417 Herring TA, King RW, McClusky SC (2006a) GAMIT: Reference Manual Version 10.34. Internal Memorandum, Massachusetts Institute of Technology, Cambridge Herring TA, King RW, McClusky SC (2006b) GLOBK: Global Kalman filter VLBI and GPS analysis program Version 10.3. Internal Memorandum, Massachusetts Institute of Technology, Cambridge Kouba J, Ray J, Watkins M (1998). IGS Reference Frame realization. IGS 1998 Analysis Center Workshop – Proceedings Darmstadt. p 139 Lavalle´e D, van Dam T, Blewitt G, Clarke P (2006) Geocenter motions from GPS: A unified observation model. J Geophys Res 111:B05405 Lyard F, Lefevre F, Letellier T, Francis O (2006) Modelling the global ocean tides: modern insights from FES2004. Ocean Dyn 56:394–415 Mao A, Harrison CGA, Dixon TH (1999) Noise in GPS coordinate time series. J Geophys Res 104:2797–2816
18 McCarthy D, Petit G (2004) IERS Technical Note 32 – IERS Conventions (2003) Technical report, Verlag des Bundesamts fur Kartographie und Geodasie, Frankfurt am Main, Germany Ray R, Altamimi Z, Collilieux X, Van Dam T (2007) Anomalous harmonics in the spectra of GPS position estimates. GPS Solut 12(1):55–64 Schmid R, Steigenberger P, Gendt G, Ge M, Rothacher M (2007) Generation of a consistent absolute phase-center correction model for GPS receiver and satellite antennas. J Geod 81:781–798 Sch€one T, Sch€on N, Thaller D (2009) IGS Tide Gauge Benchmark Monitoring Pilot Project (TIGA): scientific benefits. J Geod 83:249–261 Steigenberger P, Rothacher M, Dietrich R, Fritsche M, R€ulke A, Vey S (2006) Reprocessing of a global GPS network. J Geophys Res 111:B05402 Williams SDP (2008) CATS: GPS coordinate time series analysis software. GPS Solut 12(2):147–153
A. Santamarı´a-Go´mez et al. W€oppelmann G, McLellan S, Bouin M-N, Altamimi Z, Daniel L (2004) Current GPS data analysis at CLDG for the IGS TIGA Pilot Project. Cahiers du Centre Europe´en Ge´odynamique & de Sismologie 23:149–154 W€oppelmann G, Martı´n Mı´guez B, Bouin M-N, Altamimi Z (2007) Geocentric sea-level trend estimates from GPS analyses at relevant tide gauges world-wide. Glob Planet Change 57(3–4):396–406 W€oppelmann G, Letretel C, Santamarı´a A, Bouin M-N, Collilieux X, Altamimi Z, Williams S, Martı´n Mı´guez B (2009) Rates of sea-level change over the past century in a geocentric reference frame. Geophys Res Lett 36:L12607 Zhang J, Bock Y, Johnson H, Fang P, Williams S, Genrich J, Wdowinski S, Behr J (1997) Southern California Permanent GPS Geodetic Array: error analysis of daily position estimates and site velocities. J Geophys Res 102:18035–18056
3
A Dense Global Velocity Field Based on GNSS Observations: Preliminary Results C. Bruyninx, Z. Altamimi, M. Becker, M. Craymer, L. Combrinck, A. Combrink, J. Dawson, R. Dietrich, R. Fernandes, R. Govind, T. Herring, A. Kenyeres, R. King, C. Kreemer, D. Lavalle´e, J. Legrand, L. Sa´nchez, G. Sella, Z. Shen, A. Santamarı´a-Go´mez, and €ppelmann G. Wo
Abstract
In a collaborative effort with the regional sub-commissions within IAG subcommission 1.3 “Regional Reference Frames”, the IAG Working Group (WG) on “Regional Dense Velocity Fields” (see http://epncb.oma.be/IAG) has made a first attempt to create a dense global velocity field. GNSS-based velocity solutions
C. Bruyninx (*) Royal Observatory of Belgium, Av. Circulaire 3, 1180 Brussels, Belgium e-mail:
[email protected] Z. Altamimi Institut Ge´ographique National, Service de la recherche/ LAREG, c/o ENSG, 6 - 8 Ave. Blaise Pascal, 77455 Champssur-Marne, France M. Becker Institute of Physical Geodesy, Technische Universit€at Darmstadt, Petersenstr, 13, 64287 Darmstadt, Germany M. Craymer Geodetic Survey Division, Natural Resources Canada, 615 Booth Str., Ottawa Ontario K1A 0E9, Canada L. Combrinck HartRAO, PO Box 443, 1740 Krugersdorp South Africa, Canada A. Combrink HartRAO, PO Box 443, 1740 Krugersdorp South Africa, Canada J. Dawson Geoscience Australia, Cnr Jerrabomberra Ave. and Hindmarsh Drive, Symonston, ACT 2609, Australia R. Dietrich TU Dresden, Institut f€ ur Planetare Geod€asie, Mommsenstr, 13, 01062 Dresden, Germany R. Fernandes University of Beira Interior, IDL, R. Marqueˆs d’Aacute;vila e Bolama, 6201-001 Covilha˜, Portugal and Delft University of Technology, DEOS – PSG, PO Box 5058, 2600 GB Delft, The Netherlands
R. Govind Geoscience Australia, Cnr Jerrabomberra Ave. and Hindmarsh Drive, Symonston, ACT 2609, Australia T. Herring Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA A. Kenyeres FOMI Satellite Geodetic Observatory, POBox 585 1592 Budapest, Hungary R. King Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA C. Kreemer Nevada Bureau of Mines and Geology, and Seismological Laboratory, University of Nevada, 1664 N. Virginia Str., MS178, Reno, NV 89557, USA D. Lavalle´e Nevada Bureau of Mines and Geology, and Seismological Laboratory, University of Nevada, 1664 N. Virginia Str., MS178, Reno, NV 89557, USA J. Legrand Delft University of Technology, Faculty of Aerospace Engineering, DEOS – PSG, PO Box 5058 2600 GB Delft, Netherlands L. Sa´nchez Royal Observatory of Belgium, Ave. Circulaire 3, 1180, Brussels, Belgium G. Sella Deutsches Geod€atisches Forschungsinstitut, Alfons-Goppel-Str. 11, 80539 M€unchen, Germany
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_3, # Springer-Verlag Berlin Heidelberg 2012
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for more than 6,000 continuous and episodic GNSS tracking stations, were proposed to the WG in reply to the first call for participation issued in November 2008. The combination of a part of these solutions was done in a two-step approach: first at the regional level, and secondly at the global level. Comparisons between different velocity solutions show an RMS agreement between 0.3 and 0.5 mm/year resp. for the horizontal and vertical velocities. In some cases, significant disagreements between the velocities of some of the networks are seen, but these are primarily caused by the inconsistent handling of discontinuity epochs and solution numbers. In the future, the WG will re-visit the procedures in order to develop a combination process that is efficient, automated, transparent, and not more complex than it needs to be.
3.1
Introduction
The Working Group “Regional Dense Velocity Fields” has been created in 2007 at the IUGG (International Union of Geodesy and Geophysics) General Assembly in Perugia, Italy. It is embedded within IAG (International Association of Geodesy) sub-commission 1.3 on “Regional Reference Frames” where it co-exists with the regional sub-commissions for Europe, South and Central America, North America, Africa, South-East Asia and Pacific, and Antarctica (Drewes et al. 2008). The long-term goal of the Working Group is to provide a globally referenced dense velocity field based on GNSS observations and linked to the multitechnique global conventional reference frame, the ITRF (International Terrestrial Reference Frame, Altamimi et al. 2007a).
Z. Shen NOAA-National Geodetic Survey (NGS), 1315 East-West Hwy, Silver Spring, MD 20910, USA A. Santamarı´a-Go´mez University of California, Department of Earth and Space Sciences, 595 Charles E Young Drive, Los Angeles, CA 900951567, USA G. W€oppelmann Instituto Geogra´fico Nacional, c/ General Iban˜ez Ibero 3, 28071 Madrid, Spain and Institut Ge´ographique National, Service de la recherche/LAREG, c/o ENSG, 6–8 Ave. Blaise Pascal, 77455 Champs-sur-Marne, France
3.2
Working Group Objectives and Work Plan
3.2.1
Objectives
The Working Group on “Regional Dense Velocity Fields” joins the efforts of groups processing local/ regional/global CORS or repeated GNSS campaigns and set up the following action items: – Define specifications and quality standards for the regional SINEX solutions and relevant meta-data (e.g. description of GNSS equipment and position/ velocity discontinuities) – Collect SINEX solutions and their meta-data – Study in-depth the individual strengths and shortcomings of local/regional and continuous/ epoch GNSS solutions to determine site velocities – Define optimal strategies for the combination of regional and global SINEX solutions – Provide dense regional velocity fields – Provide the densification of the ITRF2005 (or its successors) – Encourage participation in related symposia – Implement a web site in order to provide information on the activities and access to the products of the WG – And prepare recommendations and a comprehensive final report on the WG activities at the next IUGG General Assembly in 2011
3.2.2
Work Plan
The work plan of the WG has been divided into two major parts. During the first part, covering 2007–2009, the WG set up the initial strategy and submission
3
A Dense Global Velocity Field Based on GNSS Observations: Preliminary Results
guidelines, collected a first set of test solutions, and performed a first preliminary velocity combination. The working group closely links its activities with the regional sub-commissions within IAG sub-commission 1.3. Their expertise, coordination role for their region, and their capability to generate a unique cumulative solution for their region including velocity solutions from third parties (even campaigns) is essential for the WG. The initial WG strategy consisted therefore in a two-step approach. First, region coordinators (one for each region corresponding to the regions of the different regional sub-commissions) gathered sub-regional velocity solutions for their region (in accordance with the WG requirements) and combined these with, where available, the velocity solution from the regional subcommissions (e.g. EUREF, SIRGAS. . .) in order to produce one regional combined velocity solution in the SINEX format. Secondly, two combination coordinators -T. Herring (MIT, US) and D. Lavalle´e (TU Delft, Netherlands)- combined these regional SINEX solutions with the long-term solutions from global networks to generate a preliminary velocity solution tied to the ITRS. The main goal of this preliminary solution was to identify the problems that would arise and help to set strategic choices and guidelines for the future. These guidelines will be used to issue a new solution in the second part of the WG term, 2010–2011. The results of the 2007–2009 period will be presented in this paper. As mentioned in the introduction, the WG accepts velocity solutions based on CORS and repeated GNSS campaigns (under specific conditions, see Sect. 3.1). One of the strategic choices the WG group had to make from the start was to decide whether to (1) stack weekly combined regional and global (position) SINEX solutions to compute the velocities or to (2) combine cumulative regional and global (position + velocity) SINEX solutions. Considering that the WG does not have access to the weekly SINEX of many cumulative velocity solutions, it was decided to go for approach (2). This will allow us combining, if necessary, only velocities (without the positions). In addition, it will allow us a stepwise combination of regional and global solutions; it will also facilitate meta-data management and outlier detection, as these will be done at regional level. And finally, it perfectly fits in the initial frame (using region and combination coordinators) that was set up. The disadvantages of combining cumulative (position+) velocity solutions are
21
however that no coordinate time series will be available to the WG and that it will be necessary to consistently handle discontinuities, especially on frame-attachment sites.
3.3
Call for Participation
3.3.1
Initial Submission Guidelines
In order to allow inclusion of a maximum number of velocity solutions, the WG set up the following guidelines for the contributing solutions: – Minimum 2 years of continuous data or 2 campaign epochs over a 4 year period – Minimum 2.5 years of continuous data if significant seasonal signals are present – Significant number of “frame-attachment” sites, preferably observed over a period exceeding 5 years – Position/velocity discontinuities should be identical to the ones used by the (IGS) International GNSS Service (Dow et al. 2009) – Velocity constraints should be minimal or removable – SINEX format should contain full covariance information (an exception is allowed for PPP solutions only providing correlations between individual station coordinates) The detailed submission guidelines are available from the Working Group web site: http://www. epncb.oma.be/IAG/.
3.3.2
Call for Participation
A first Call for Participation (CfP) was issued at the end of 2008. Analysts, producing regional and global velocity solutions, were invited to submit their SINEX files to the Working Group. Figure 3.1 shows the map with the sites for which solutions that have been proposed following this CfP (black dots); in total more than 6,000 sites were proposed.
3.4
Input for Preliminary Combination
A preliminary velocity combination has been computed in the summer of 2009. This solution contained contributions from the region coordinators as well as one global solution.
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Fig. 3.1 Map with the sites for which velocity solutions have been proposed to the Working Group up to July 2009 (black dots). In total more than 6,000 sites have been proposed. In red: sites used in the first preliminary combination (see Sect. 3.4)
3.4.1
Regional Contributions
Several of the region coordinators prepared a velocity solution for their region to be included in the preliminary combination. The African and South & Central American contributions are based on the contribution of a single analysis center, while the solutions from Europe and South-East Asia & Pacific are combined solutions based on input from several analysis groups. More details are given below: – Africa (see Fig. 3.2): The solution includes 93 CORS and covers the period from Jan. 1996 till June 2009 (Fernandes et al. 2007). The GNSS data analysis, has been done using GIPSY/OASIS II (Zumberge et al. 1997) by applying the PPP strategy with ambiguity resolution (Blewitt 2008). GIPSY tools were also used to derive the velocity solutions. – Europe (see Fig. 3.3): The solution includes the velocities estimated from a reprocessing of the EUREF Permanent Network (EPN), maintained by the regional sub-commission for Europe (Bruyninx et al. 2009), complemented with several sub-regional velocity solutions. In total, velocity estimates for 525 sites were obtained, which is more than twice the number of the sites presently included in the EPN. All of the contributing sub-regional solutions were available in the SINEX format and the combination was done with the CATREF software (Altamimi et al. 2007b). The main problem encountered during the combination was the fact that some of the submitted sub-regional solutions did not use any discontinuities at all (more about this in Sect. 3.6).
Fig. 3.2 Sites contributing to the African velocity solution
Fig. 3.3 Sites and solutions included in the European solution: reprocessed EPN (‘96-’09) in black, AGNES (‘98-09’) in pink, AMON (‘01-’09) in green, ASI (’97-’09) in red, IGN (‘98-’09) in brown, and CEGRN (‘94-’07) blue triangles
– South and Central America (see Fig. 3.4): The solution includes about 128 CORS from the
3
A Dense Global Velocity Field Based on GNSS Observations: Preliminary Results
23
In addition, as indicated in Fig. 3.5, not all solutions were available in full SINEX format.
3.4.2
Fig. 3.4 Sites contributing to the South and Central America velocity solution
One global solution was included in this first test combination. This solution, from the ULR consortium (Universite´ de La Rochelle and IGN/LAREG) is based on a reprocessing of 299 CORS from Jan. 1996 till Jan. 2009 using the GAMIT software (Herring et al. 2007), see Fig. 3.6. Its main objective is the correction of vertical land movements that affect the tide gauge records (W€oppelmann et al. 2009). A key issue to achieve the accuracy requirement of the sea-level application (submm/year) is the realization of a stable and accurate reference frame. The ULR solution therefore includes a global set of reliable reference frame stations from the IGS. It includes three additional years of data and an improved data analysis strategy with respect to the previous solution (W€oppelmann et al. 2009). See details in Santamaria-Gomez et al. (this issue). The stacking of the solutions was done with CATREF. The ULR network has several sites common to the regional solutions.
3.5 Fig. 3.5 Sites and solutions contributing to the East-Asia and Pacific velocity solution. Solutions with full SINEX information are indicated with circles: PCGIAP (‘97-’06) in dark blue, SW Austr. seism. zone (‘02-’06) in light blue, and GeoScience Australia TIGA (‘97-’09) in black. The triangles indicate the stations belonging to networks providing velocity-only solutions: Tibet (‘98-’04) in green, Asia (‘94-04) in pink, Global (‘95-’07) in orange, and Indonesia (‘91-’01) in red
SIRGAS network (Seem€ uller et al. 2009) and covers the period from 2000 till 2009. The GNSS data processing, as well as the cumulative SINEX solution, have been computed using the Bernese V5.0 software (Beutler et al. 2007). – South-East Asia and Pacific (see Fig. 3.5): The solution comprises 1,156 sites resulting of a combination of several sub-regional networks. The combination was done using the CATREF software. In this solution, ensuring the consistent use of station names, particularly four-character identifiers, was a major struggle for many stations.
Global Contribution
Test Combination
The submitted regional networks and the global network were combined using two different approaches. First approach was a step-wise one: first a combination of sites that are present in at least three solutions was done. Based on the common sites from this combination, re-weighting factors for each SINEX file were estimated. The solution is then iterated with the final weights coming from the w2 of the individual solution velocity estimates with respect to the combination, using sites common to at least two solutions. The variance weights vary from about 200 to 1.6. The differences are most probably caused by the usage of different software packages and will be investigated in more detail in the future. The second approach consisted in attaching each regional network to the global network (ULR) using frame-attachment sites. This means that the global sites are not changed by the attachment of the regional sites but the regional networks are adjusted. More details on this approach are given in Davis and Blewitt (2000).
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Fig. 3.6 Sites contributing to the global ULR velocity solution
Figure 3.7 and Table 3.1 present the first comparisons between each of the regional velocity solutions and the global ULR solution. The comparison was done after performing a Helmert transformation on both positions and velocities (estimating translations and scale, together with their rate). The results clearly show that the European and South-Asia & Pacific solutions agree better with the ULR solution than the SIRGAS and African solutions. This does not necessarily mean that the quality of the latter solutions is worse than the first ones, but reflects more the fact that inconsistent discontinuity epochs and solution numbers are used between the last two solutions and the ULR. This is confirmed by the fact that the SIRGAS and African solutions also have a significantly larger position RMS w.r.t. to the ULR solution compared to the European and South-Asia & Pacific solution. Detailed maps of the comparison between the different solutions are available from the WG web site at http://epncb.oma.be/IAG/
3.6
Difficulties
Not all sites included in the contributing solutions have official DOMES numbers assigned by the IERS (International Earth Rotation and Reference Systems Service) and this can make SINEX combination software fail. As a large number of these sites are third party sites without detailed monumentation information, it is impossible to request official IERS DOMES numbers for them. Therefore, the WG implemented a coordinated approach for attributing virtual DOMES
Fig. 3.7 Differences between global ULR velocity solution and each of the regional velocity solutions after a Helmert transformation. In purple: vertical velocity differences and in blue: horizontal velocity differences. Top: Europe, middle-left: Africa, middleright: South and Central America, bottom: East-Asia and Pacific
numbers. Moreover, in the case of duplicate station names, a new station identification and virtual DOMES number was assigned in a coordinated way,
3
A Dense Global Velocity Field Based on GNSS Observations: Preliminary Results
avoiding overlaps and inconsistencies between the different regions. Typically, when a position change occurs at a CORS, two different positions are estimated: one before the discontinuity epoch and one after it. Independently of the fact that the position change is associated with a velocity change or not, most stacking is done in a way to estimate, in addition to the two position solutions, also two velocity solutions. Separate velocities may also be estimated but can be linked through constraints if they are statistically compatible, as usually the case for instrument-related discontinuities. This principle is illustrated in Fig. 3.8. When discontinuities occur at reference frame sites or at sites common to different solutions, it is imperative that the same discontinuity epochs and solution numbers are applied during the analysis Table 3.1 Agreement between global ULR velocity solution and each of the regional velocity solutions after a Helmert transformation Solution
RMS Pos. (mm) Hor. Up. Europe 1.68 2.58 Africa 4.54 4.14 South & Central America 3.85 4.41 South-Asia & Pacific 2.12 3.83 1997
1998
1999
2000
North-component
15 7.5 0 -7.5 -15
# Common (excluded sites) 43(10) 12(2) 25(3) 26(13)
2001
2002
before combining the solutions. As in this first test, cumulative position + velocity solutions have been combined, the WG asked the analysts producing these solutions to apply the station discontinuities identified by the IGS/ITRF (ftp://macs.geod.nrcan.gc.ca/ pub/requests/sinex/discontinuities/ALL.SNX). However, the treatment of discontinuities and velocity changes are subject to interpretation and analysis groups may not necessarily agree on discontinuity epochs. In practice, the different contributors did not strictly follow the IGS discontinuities, obviously influencing the estimated positions as well as velocities and most probably causing some of the outliers seen in Fig. 3.8. This exercise demonstrated the need to come up with a consensus on the discontinuities. To test a collaborative approach, a first attempt was made to merge the discontinuities reported by the different groups in one file which can be edited and maintained by the different groups. In addition, to the “bookkeeping” problems described above, some sub-regional solutions consisted of precise velocity estimates with only approximate coordinates. The implication is that inter-site correlations (not always negligible) are neglected which caused failure of some combination software. Other numerical instabilities were seen due to the equating (or heavily constraining) of velocities before and after a position jump. 2003
2004
2005
2006
2007
2008
2009
15 7.5 0 -7.5 -15
Discontinuity Epoch
[mm]
[mm]
1996
15 7.5 0 -7.5 -15
RMS Vel. (mm/year) Hor. Up 0.28 0.44 0.92 1.24 0.74 1.26 0.22 0.47
25
East-component
15 7.5 -7.5 -15
25
25
0
0
[mm]
-25
-25
-50
Position Solution 1
-50
Position Solution 2
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-75
-100
Up-component
-125 850
900
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Heavily constrained velocities
-125 1050
1100
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GPS WEEK
Fig. 3.8 Principle of the introduction of discontinuity epochs and solution numbers
1350
1400
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1500
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3.7
C. Bruyninx et al.
Summary and Outlook
The IAG Working Group on “Regional Dense Velocity Fields” preformed a first test combination of a set of cumulative velocity solutions from regional and global networks in order to identify the main problems when producing a dense velocity field based on multiple cumulative position and velocity solutions. The test identified the urgent need for a consensus on the attribution of discontinuity epochs for stations common to several solutions. Due to the use of different analysis strategies and software packages by the individual contributors, finding such a consensus is a challenge as most probably not the same discontinuities are seen by different people. In addition, the treatment of the post-seismic signals is also subject to interpretation. A possible way to go ahead for the Working Group could be to combine solutions at the weekly level and only deal with the attribution of the discontinuity epochs at the combination level. This would mean that in a first step the weekly global solutions would be combined with to generate a global core network with reliable velocity solutions. In a second step, weekly combined regional solutions (including the sub-regional solutions providing weekly contributions) could be added to this global core network on a weekly basis resulting in a densified core network which could then be used for velocity estimation using one single set of agreed-upon discontinuities. Finally, the remaining cumulative velocity solutions (for which the WG does not have access to weekly position solutions) could be attached to the cumulative densified core network. This approach is one of the alternative procedures which are presently under discussion within the WG. Acknowledgements The authors of this paper would like to thank the groups who submitted velocity solutions. The full list of contributors is available from http://epncb.oma.be/IAG/
References Altamimi Z, Collilieux X, Legrand J, Garayt B, Boucher C (2007a) ITRF2005: a new release of the international terrestrial reference frame based on time series of station positions and earth orientation parameters. J Geophys Res 112: B09401. doi:10.1029/2007JB004949
Altamimi Z, Sillard P, and Boucher C (2007b). CATREF software: Combination and analysis of terrestrial reference frames. LAREG, Technical, Institut Ge´ographique National, Paris, France Beutler G, Bock H, Brockmann E, Dach R, Fridez P, Gurtner W, Habrich U, Hugentobler D, Ineichen A, Jaeggi M, Meindl L, Mervart M, Rothacher S, Schaer R, Schmid T, Springer P, Steigenberger D, Svehla D, Thaller C, Urschl R, Weber (2007) Bernese GPS software version 5.0. ed. Urs Hugentobler, R. Dach, P. Fridez, M. Meindl, Univ. Bern Blewitt G (2008) Fixed point theorems of GPS carrier phase ambiguity resolution and their application to massive network processing: Ambizap. J Geophys Res 113:B12410. doi:10.1029/2008JB005736 Bruyninx C, Altamimi Z, Boucher C, Brockmann E, Caporali A, Gurtner W, Habrich H, Hornik H, Ihde J, Kenyeres A, M€akinen J, Stangl G, van der Marel H, Simek J, S€ohne W, Torres JA, Weber G (2009). The European Reference Frame: Maintenance and Products, IAG Symposia Series, “Geodetic Reference Frames”, Springer, vol 134, pp 131–136, DOI: 10.1007/978-3-642-00860-3_20 Davis Ph, Blewitt G (2000) Methodology for global geodetic time series estimation: A new tool for geodynamics. J Geophys Red 105(B5):11083–11100 Dow JM, Neilan RE, Rizos C (2009) The International GNSS Service in a changing landscape of Global Navigation Satellite Systems. J Geod 83:191–198. doi:10.1007/s00190008-0300-3 ´ da´m J, Ro´zsa S (eds) (2008). The Drewes H, Hornik H, A geodesist handbook 2008. Journal of Geodesy, Springer, 82 (11):661–846 Fernandes RMS, Miranda JM, Meijninger BML, Bos MS, Noomen R, Bastos L, Ambrosius BAC, Riva REM (2007) Surface velocity field of the Ibero-Maghrebian Segment of the Eurasia-Nubia plate boundary. Geophys J Int 169:315–324. doi:10.1111/j.1365-246X.2006.03252.x Herring TA, King RW, McClusky SC (2007) Introduction to GAMIT/GLOBK, Release 10.3, Mass. Instit. of Tech., Cambridge Santamaria-Gomez A, Bouin M-N, and W€oppelmann G. An improved GPS data analysis strategy for tide gauge benchmark monitoring. (Submitted) Seem€uller W, Seitz M, Sa´nchez L, Drewes H (2009) The position and velocity solution SIR09P01 of the IGS Regional Network Associate Analysis Centre for SIRGAS (IGS RNAAC SIR). DGFI Report No. 85. DGFI, Munich. Available at http://www.sirgas.org/index.php?id¼97 W€oppelmann G, Letetrel C, Santamaria A, Bouin M-N, Collilieux X, Altamimi Z, Williams SDP, Martin Miguez B (2009) Rates of sea-level change over the past century in a geocentric reference frame. Geophys Res Lett 36:L12607. doi:10.1029/2009GL038720 Zumberge J, Heflin M, Jefferson D, Watkins M, Webb F (1997) Precise point positioning for the efficient and robust analysis of GPS data from large networks. J Geophys Res 102:5005–501
4
Enhancement of the EUREF Permanent Network Services and Products €hne, A. Kenyeres, G. Stangl, C. Bruyninx, H. Habrich, W. So €lksen and C. Vo
Abstract
This paper describes the EUREF Permanent Network (EPN) and the efforts made to monitor and improve the quality of the EPN products and services. It is shown that the EPN is becoming a multi-GNSS tracking network and that the EPN Central Bureau and the Analysis Centers are preparating to include the new satellite signals in their routine operations. Thanks to the EPN Special Project on “Reprocessing”, set up early 2009, EPN products with much better quality and homogeneity will be generated. The Special Project on “Real-time analysis” will improve the reliability of the EPN real-time data streams and develop new EPN real-time products.
4.1
C. Bruyninx (*) Royal Observatory of Belgium, Av. Circulaire 3, 1180 Brussels, Belgium e-mail:
[email protected] H. Habrich W. S€ ohne Bundesamt f€ur Kartographie und Geod€asie, Richard-StraussAllee 11, 60598 Frankfurt am Main, Germany A. Kenyeres FOMI Satellite Geodetic Observatory, P.O. Box 585, 1592 Budapest, Hungary G. Stangl Department of Satellite Geodesy, Institute of Space Research, Schmiedlstrasse 6, 8042 Graz, Austria C. V€olksen Bayerische Kommission f€ ur die Internationale Erdmessung, Bayerische Akademie der Wissenschaften, Alfons-Goppel-Str. 11, 80539 M€unchen, Germany
Introduction
The IAG (International Association of Geodesy) Regional Reference Frame sub-commission for Europe, EUREF, is responsible for defining, providing access and maintaining the European Terrestrial Reference System (ETRS89), which is recommended by the European Commission for use in all EU member states (Bruyninx et al. 2009a). In 1996, EUREF created the EUREF Permanent Network (EPN) based on a partnership with site operators of permanent GNSS sites who are willing to share their data with the public. The EPN cooperates closely with the International GNSS Service (IGS, Dow et al. 2005); EUREF members are participating to the IGS Real-Time Pilot Project, the IGS GNSS Working Group, the IGS Antenna Calibration Working Group and the IGS Infrastructure Committee. The EPN network contains more than 220 stations (as of October 2009) and its GNSS observations are used extensively by the public, national mapping agencies, and researchers (Bruyninx 2004). EPN
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_4, # Springer-Verlag Berlin Heidelberg 2012
27
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C. Bruyninx et al.
stations provide daily (mandatory) and hourly (93% of the stations) data in the RINEX format as well realtime data (48% of the stations) in RTCM, RTIGS or raw formats where 15-min high-rate RINEX files are derived from. Two Regional Data Centres (at BKG and OLG) provide access to all hourly and daily data, one data centre (at EPN Central Bureau) to provides access to historical daily EPN observation data, and one regional broadcaster (www.euref-ip.net at BKG) broadcasts all real-time EPN data streams. 16 Local Analysis Centers (LAC) deliver weekly (some also daily and hourly) site position estimates in SINEX format and hourly zenith path delays. These results are the basis to generate the following EPN products: – Weekly combined site positions (in SINEX format) – Daily combined site zenith path delays – Cumulative site positions and velocities, in both the ITRS (International Terrestrial Reference System) and ETRS89, and a discontinuity table – Fully analysed residual position time series The day-to-day management of EPN is performed by the EPN Central Bureau (http://epncb.oma.be). In this paper we describe the efforts recently undertaken within the EPN to keep the pace with new user demands and GNSS modernization, and to continuously improve the EPN products and services.
4.2
Modernization of the EPN
4.2.1
Tracking Network
Since 1996, the EPN has grown from a network of continuously observing dual frequency GPS stations to a network where 49% of the stations (Oct. 2009) are observing GPS as well as GLONASS satellites. In addition, 48 EPN stations have the GPS L2C tracking capability and 31 are equipped with an L5-capable receiver. The new observation types imply switching to RINEX 3 (e.g. ftp://ftp.unibe.ch/aiub/rinex/rinex300. pdf) and an update of the quality checking routines. Since Dec. 2006, each antenna/radome introduced in the EPN (new stations or replacements at existing stations) must have true absolute calibrations. Thanks to this requirement, the number of antenna/radome combinations without true absolute calibrations decreased from 31 to 18%. In addition, more than 90% of the new antennas installed within the EPN are multi-GNSS antenna. From these, 75% are designed for observing GPS and GLONASS signals
and 25% are in addition Galileo-ready. Taking into account that due to local site conditions and near-field multipath antenna/radome replacements continue to introduce jumps in the station positions (even with absolutely calibrated antenna/radome combinations) and that the large majority of new EPN antennae are multi-GNSS capable, the EUREF Technical Working Group (TWG) -which is the EUREF Steering Committee- strongly recommends all station managers, in the case an antenna has to be replaced, to replace it directly with a multi-GNSS antenna.
4.2.2
Data Analysis
The large majority of the EPN Analysis Centers uses data analysis software from third parties and therefore depends on the multi-GNSS policy of these software providers. In practice, 15 of the 16 LAC use GNSSanalysis software that is GLONASS-ready. However, even while EPN stations have been observing GLONASS signals since several years, only 4 of these 15 EPN Analysis Centers presently include GLONASS observations in their data analysis. But, the majority of the LACs plan to start with GPS +GLONASS data analysis within the next year. To guarantee consistency with the IGS, EUREF asks the EPN LACs to use IGS products (e.g., satellite orbits & clocks, and antenna calibrations) within their data analysis. However, to accommodate the missing combined GPS/ GLONASS IGS orbits, most of the four LACs processing GLONASS use the fully consistent combined GPS/GLONASS orbit files from individual IGS ACs, such as CODE (Dach et al. 2009). In addition, contrary to the IGS, the EPN has allowed the introduction of individual receiver antenna calibrations. They allow to introduce new antenna-specific calibrations while, to maintain the consistency of the reference frame, the type calibrations cannot always be updated.
4.3
Quality of the EPN Products
4.3.1
IGS Global Combination of Regional Networks
The Massachusetts Institute of Technology (MIT), Cambridge, USA is an IGS Analysis and Associate Analysis Centre. It generates the global MIT T2 solution as a combination of eight IGS ACs solutions
4
Enhancement of the EUREF Permanent Network Services and Products
(Herring 2008). In addition, MIT also generates the MIT T2 RNAAC solution by adding three regional solutions (EPN, SIRGAS -Sistema de Referencia Geoce´ntrico para Las Ame´ricas- and GSI –Geographical Survey Institute-) and one global solution (CNES Centre National d’Etudes Spatiales- Toulouse) to the MIT T2 solution. The delay of the publication of the mentioned two products is 8 weeks after the end of observations and it requires publishing the weekly EPN combination well in time. The global solution from CNES largely overlaps with the MIT T2 solution. The EPN, SIRGAS and GSI solutions densify the IGS network in Europe, South America and Japan. The coordinate comparison between the EPN and the MIT T2 solutions may be considered as a kind of accuracy estimate and interpreted as a measure of the global alignment of the EPN network. The computation of such r.m.s. numbers for 10 weeks (week 1,511–1,520) confirm that the global alignment is in the range from 2 to 4 mm.
4.3.2
Residual Position Time Series
The main target of the EPN coordinate time series analysis and monitoring is to strengthen the EPN as a geodetic reference network and to offer various products for geodesists and geophysicists. Using the CATREF software (Altamimi et al. 2007b), each 15 weeks, updated EPN cumulative solutions are released (while each 5 weeks an internal solution is created) based on the weekly combined EPN SINEX solutions. During the time series analysis all station specific events (coordinate outliers and discontinuities), confirmed by the log files are identified and considered. A cumulative solution is associated with the following products: – The EPN cumulative solution in SINEX format tied to the actual ITRS reference frame realization (currently ITRF2005) using minimum constraints – EPN station positions and velocities, as the most accurate and up-to-date solutions for the EPN sites. They are used for the maintenance of the regional densification of the ITRFyy between two releases and also for the maintenance of the ETRS89 (see Sect. 4.2) – An up-to-date list of station discontinuities fully harmonized with the IGS/ITRF discontinuity table – Residual coordinate time series as the Helmert difference between the positions in the cumulative
29
solution and the ones in the weekly input SINEX solutions – Harmonic analysis of the time series to detect seasonal coordinate variations – Sophisticated statistical analysis of the position residuals using the CATS software (Williams 2008) to estimate reliable velocity uncertainties and station-specific noise characteristics More information on these products is available from the EPN CB web page at http://www.epncb.oma.be/ _dataproducts/products/timeseriesanalysis/ (Fig. 4.1).
4.3.3
Tropospheric Zenith Path Delays
Since June 2001 (GPS week 1108) the EPN LACs are delivering the estimated Zenith Path Delay (ZPD) parameters to the BKG Data Centre. Daily files (in the so-called SINEX TRO format) are produced on a weekly basis, together with the weekly coordinate solution. These ZPD-estimates were originally combined using a procedure based on the strategy developed for the IGS (Gendt 1997), but since then several changes have been introduced regarding, e.g., reference frame realization, software versions, processing options, absolute antenna calibration, etc. The results have been improved step by step, leading to an internal precision of 2–3 mm ZPD (Fig. 4.2). The most significant progress can be seen from November 2006 on (GPS week 1400) when the absolute antenna models were introduced, especially for stations in the south of Europe. Thanks to the Memorandum of Understanding between EUREF and EUMETNET (Pottiaux et al. 2009) EUREF has now access to radiosonde observations and synoptic data. Applying these data, GNSS processing and analysis may be improved in the future. Moreover, a validation of the GNSS-derived ZPD parameters by those derived from radiosonde observations and vice versa can be performed. Figure 4.3 shows the time series of the differences between ZPD parameters derived from radiosonde data and the EPN combined solution for the station YEBE. One can see that the jump in November 2006 reduced the bias between both observables and the scattering increased during summer time. In 2008 the EPN Special Project “Troposphere Parameter Estimation” was closed and the ZTD processing moved towards routine operation.
30
C. Bruyninx et al.
Fig. 4.1 Residual position time series of the EPN station CHIZ. Top: raw time series; bottom: cleaned time series after removal of seasonal signals and introduction of multiple position and velocity estimates
4.4
Maintenance of and Access to ETRS89
4.4.1
Historical EUREF Campaigns
Immediately after defining the ETRS89 and realizing the ETRF89 (with ETRF, the EUREF Terrestrial Reference Frame), European countries started the
densification of the ETRF using GPS campaigns. These campaigns had typically a minimal duration of three full days so that the comparison of the daily position estimates allowed to get an impression of the precision of the positions. Based on this, EUREF campaigns are divided in three classes: Class A is (only for permanent stations) requiring a 1 cm r.m.s. for the position over a time period exceeding the
4
Enhancement of the EUREF Permanent Network Services and Products
Fig. 4.2 Daily ZPD bias time series from EPN combination for YEBE
Fig. 4.3 Time series of differences between radiosonde and GNSS ZPD parameters for YEBE
campaign period. A class B campaign requires an r.m.s. of 1 cm for the estimated positions at the epoch of observations, whereas Class C means the precision is worse than 1 cm. Not all of these campaigns have been accepted by EUREF, but most of them have been adopted by the National Mapping Agencies of the different countries (Fig. 4.4). Some countries have in the mean time replaced the old campaigns of the 1990s by newer ones, using a set of permanent stations. These pseudo-campaigns are usually based on 1 week of permanent observations. Campaign coordinates are valid for the mean epoch of the measurements within the then used reference frame, e.g., ETRF2000. The advantage of campaigns is that a number of national markers get coordinates validated by an international organization.
4.4.2
Densification of the ETRF
Only a selected number of EPN sites (mostly the ones belonging to the IGS) have coordinates included in recent ITRF realizations released by the IERS
31
(International Earth Rotation and Reference Systems Service). In addition, the latest realization of the ITRS, ITRF2005 (Altamimi et al. 2007a) is based on observations from space geodetic techniques (GNSS, DORIS, VLBI, SLR) up to December 2005 and does not take into account any of the IGS/EPN data gathered after Jan 1st, 2006. The problem of relative antenna models, the limited number of stations, and the lack of frequent updates consequently restricts the use of the ITRF for EUREF densifications The antenna modeling problem will be resolved with the release of the ITRF2008 (expected for 2010) which will be compatible with absolute GNSS antenna models. To take full advantage of the EPN and its most recent GNSS observation data, the EUREF TWG decided, to release regular official updates of the ITRS/ETRS89 positions/velocities of the EPN stations. A first step in this process consisted in a densification of the ITRF2005 using all EPN data up to Dec. 2005 (the same observation period as covered by the ITRF2005). Since early 2009, this realization is updated each 15 weeks. The advantage of the regular update is that the most recent EPN results are taken as much as possible into account. In order to provide the most reliable products, the EPN stations are categorized taking the station quality and the length of available observation span into account (Kenyeres 2009) (Fig. 4.5): – Class A stations with positions at the 1 cm precision and velocities at the 1 mm/yr precision at all epochs. – Class B stations with positions at the 1 cm precision at the epoch of minimal position variance of each station; where no velocities are provided. Exclusively the Class A stations can be used as reference stations for the computation of EUREF campaigns and densification of the ETRF (see Bruyninx et al. 2009b).
4.5
Recent Initiatives
4.5.1
Real-Time Analysis Project
At the end of 2007, the EUREF-IP pilot project for real-time data streaming was successfully transferred to routine operation. Within this pilot project the EPN
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C. Bruyninx et al.
Fig. 4.4 Geographical distribution of accepted EUREF campaigns Fig. 4.5 In green: EPN class A stations; in red: EPN class B stations (status Sept. 2009)
guidelines for real-time data have been developed (Dettmering et al. 2006) and disseminating streams via NTRIP (Weber et al. 2005) became a widely used and implemented feature. From Jan. 2006 to
May 2009 the number of EPN real-time stations grew from 16 to 103 and the number of registered users at EUREF-IP broadcaster at BKG grew from 500 to 1,250 . To pursue this success, the EUREF
4
Enhancement of the EUREF Permanent Network Services and Products
TWG launched a new Special Project “EPN RealTime Analysis” with the primary goals to develop and set-up a re-dissemination of GNSS real-time data/products in Europe via NTRIP broadcasters, validate satellite orbit and clock correctors to broadcast ephemeris and establish backups for all critical realtime service components. In that frame, a new realtime dissemination concept was developed (S€ohne et al. 2009). While today EPN stations stream realtime data to EUREF-IP broadcaster maintained by BKG, the goal is to maintain a series of top-level casters distributing the real-time data and sharing workload. This activity will be supported by a number of relay casters distributing real-time data to, e.g., special communities. In the final stage it is envisaged that almost every reference station is streaming realtime data to at least two different broadcasters to overcome the “single point of failure” issue.
4.5.2
Reprocessing Special Project
Reprocessing of regional networks (V€ olksen 2008) or the complete EPN became an important issue during recent years due to the availability of reprocessed orbit and clock products (Steigenberger et al. 2006). A reprocessing of the complete EPN has been done in 2008 by the MUT (Warsaw) and ROB (Brussels) LAC, clearly demonstrating an improvement of the EPN time series which were previously inconsistent due to analysis and modeling changes (Kenyeres et al. 2009). In order to coordinate the reprocessing of the full EPN between all EPN LAC a new EPN Special Project was created at the EUREF TWG meeting held in Budapest on Feb. 26–27, 2009 (V€ olksen 2009). During its pilot phase, from 2009 till 2010, a test reanalysis of the year 2006 will be performed in order to test and develop new strategies and standards for the data analysis. It is expected that each LAC setup the facilities for the reprocessing of the 2006 data before the end of 2009. At the end of the pilot phase, it is expected to have identified suitable sets of reprocessed products, which will be generated during re-analysis as well as the day-to-day analysis. The next step will then consist of a re-analysis of the complete EPN data set (1996-200x) applying the new strategies and standards. The combination of the different solutions by the analysis coordinator will provide
33
reports and feedback to the working group and the LACs.
4.6
Summary
In response to evolving user needs and new satellite signals, the EPN is continuously improving its tracking network, products and services. Examples are the new EPN real-time analysis and reprocessing special projects, and the fact that new cumulative EPN coordinates are now updated each 15 weeks. While the EPN tracking network is preparing to track new and modernized GNSS signals, monitoring and processing the new and modernized GNSS signals imposes the usage of updated exchange formats, antenna calibrations, consistent multi-GNSS satellite orbits and clocks, and requires additional software developments. For several reasons, the EPN Central Bureau (resp. Analysis Centers) are not yet ready for a real multi-GNSS data monitoring resp. analysis, but this situation is expected to improve dramatically in the next year.
References Altamimi Z, Collilieux X, Legrand J, Garayt B, Boucher C (2007a) ITRF2005: a new release of the International Terrestrial Reference Frame based on time series of station positions and Earth Orientation Parameters. J Geophys Res 112:B09401. doi:10.1029/2007JB004949 Altamimi Z, Sillard P, Boucher C (2007b) CATREF software: combination and analysis of terrestrial reference frames. LAREG, Technical, Institut Ge´ographique National, Paris, France Bruyninx C (2004) The EUREF Permanent Network: a multidisciplinary network serving surveyors as well as scientists. GeoInformatics 7:32–35 Bruyninx C, Altamimi Z, Boucher C, Brockmann E, Caporali A, Gurtner W, Habrich H, Hornik H, Ihde J, Kenyeres A, M€akinen J, Stangl G, van der Marel H, Simek J, S€ohne W, Torres JA, Weber G (2009a) The European reference frame: maintenance and products, IAG Symposia Series, “Geodetic Reference Frames”, vol 134. Springer, Heidelberg, pp 131–136. doi: 10.1007/978-3-642-00860-3_20 Bruyninx C, Altamimi Z, Caporali A, Kenyeres A, Lidberg M, Stangl G, Torres JA (2009b) Guidelines for EUREF Densifications, ftp://epncb.oma.be/pub/general/Guidelines_ for_EUREF_Densifications.pdf Dach R, Brockmann E, Schaer S, Beutler G, Meindl M, Prange L, Bock H, J€aggi A, Ostini L (2009) GNSS processing at CODE: status report. J Geodesy 83(3–4):353–366. doi:10.1007/s00190-008-0281-2
34 Dettmering D, Weber G, Bruyninx C, v.d.Marel H, Gurtner W, Torres J, Caporali A (2006) Real-time GNSS in routine EPN operations: concept. http://epncb.oma.be/_organisation/ guidelines/EPNRT_WhitePaper.pdf Dow JM, Neilan RE, Gendt G (2005) The International GPS Service (IGS) celebrating the 10th anniversary and looking to the next decade. Adv Space Res 36(3):320–326. doi:10.1016/j.asr.2005.05.125 Gendt G (1997) IGS Combination of Tropospheric Estimates (Status Report). In: Proceedings of the IGS Analysis Centre Workshop, Pasadena, CA Herring T (2008) RNAAC combinations in the IGS, presented at IGS Analysis Center Workshop 2008, Miami Beach, FL, 2–6 June 2008 Kenyeres A (2009) Maintenance of the EPN ETRS89 coordinates. In: Minutes of EUREF TWG meeting, Budapest, Hungary, 26–27 Feb 2009. http://www.euref.eu/ Kenyeres A., Figurski M, Legrand J, Kaminski P, Habrich H (2009) Homogeneous reprocessing of the EPN: first experiences and comparisons. Bull Geod Geomatics 3:207–218 Pottiaux E, Brockmann E, .S€ ohne W, Bruyninx C (2009) The EUREF – EUMETNET collaboration: first experience and potential benefits. Bull Geod Geomatics 3:269–288
C. Bruyninx et al. S€ohne W, St€urze A, Weber G (2009) Increasing the GNSS stream dissemination capacity for IGS and EUREF. http://epncb.oma. be/_dataproducts/data_access/real_time/BroadcastConcept.pdf Steigenberger P, Rothacher M, Dietrich R, Fritsche M, R€ulke A, Vey S (2006) Reprocessing of a global GPS network. J Geophys Res 111(B5):B05402. doi:10.1029/ 2005JB003747 V€olksen C (2008) Reprocessing of a regional GPS network in Europe. In: Sideris MG (eds) Observing our changing Earth: proceedings of the 2007 IAG General Assembly, Berlin, pp 57–64 V€olksen C (2009) Draft Charter for the EUREF Working Group on reprocessing of the EPN. http://epn-repro.bek.badw.de/ Documents/charter_repro.pdf Weber G, Dettmering D, Gebhard H (2005) Networked Transport of RTCM via Internet Protocol (NTRIP). In: Sanso F (ed) Proceedings of the IAG symposia ‘A window on the Future of Geodesy’, vol 128. Springer, Heidelberg, pp 60–64 Williams SDP (2008) CATS: GPS coordinate time series analysis software. GPS Solut 12(2):147–153. doi:10.1007/s1029-0070086-4
5
Can We Really Promise a mm-Accuracy for the Local Ties on a Geo-VLBI Antenna Ulla Kallio and Markku Poutanen
Abstract
For the next-generation geodetic VLBI network a 1 mm positioning accuracy is anticipated. The accuracy should be site-independent consistent, reliably controlled, and traceable over long time periods. There are a number of remaining limitations. These include random and systematic components of the delay observable itself, various antenna-related errors, and especially a proper handling of local ties at multi-technique sites. At the Mets€ahovi Fundamental Station operated by the Finnish Geodetic Institute there are a CGPS and Glonass receivers, both in IGS network, a SLR (currently under renovation), a DORIS beacon, a superconducting gravimeter, and a 14.5 m radio telescope owned by Mets€ahovi Radio Observatory of the Helsinki University of Technology. Between five and eight geodetic VLBI sessions are conducted annually. We tested a method to simultaneously measure the tie of the VLBI antenna to the GPS network by tracking the movement of two GPS antennas attached to the radio telescope during geodetic VLBI sessions. We used kinematic trajectory solutions of the two GPS antennas to calculate the orientation and the reference point of the VLBI antenna. In this paper we describe the data acquisition, calculation model, some error sources and test results of four campaigns. The position of the reference point is time, temperature, antenna elevation and azimuth dependent. We propose that in the future, the position should be tracked permanently during geo-VLBI campaigns with attached GPS antennas.
5.1
U. Kallio M. Poutanen (*) Department of Geodesy and Geodynamics, Finnish Geodetic Institute, Geodeetinrinne 2, 02431 Masala, Finland e-mail:
[email protected]
Introduction
The IERS Working Group on co-location and local ties gives recommendations and guidelines for tie vector measurements, computation and transformation for multi-technique sites in the global reference frame (IERS 2005, 2009). The idea of local tie measurements is to solve for the reference points and orientations of the all instruments in a fundamental station with
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_5, # Springer-Verlag Berlin Heidelberg 2012
35
36
respect to one common marked point or solve for the reference point and orientation of one instrument relative to another instrument in a local reference frame. The tie vector must then be transformed to the global reference frame. Local ties should be determined at regular intervals. There are different practices and recommendations on the repeat rate, ranging from twice per year up to several years. How often ties can be completed in practice depends on the workload and difficulties of the measurement. It is a totally different question, how often such ties should be made to reliably detect possible movement or deformation of an instrument. The measurement methods in different ties depends on the case and the instruments used in tracking have varied from tachymeters and levelling instruments or laser trackers to GPS (Dawson et al. 2007; IERS 2005; ITRF 2009; Sarti et al. 2004; L€ osler and Haas 2009). The methods have been space intersection with two or more theodolites or kinematic or static GPS measurements, and the use of a laser tracker. As pointed out e.g., in (Sarti et al. 2008), there are basically three surveying approaches: a direct method, a hybrid method and an indirect method. In the direct method the reference point of an instrument is an actual geodetic marker which can be surveyed directly. In the hybrid method (mainly applicable to some VLBI and SLR telescopes) the reference point itself is not identified by a geodetic marker but defined by the axes of rotation of the telescope. Some surveying targets are fixed on the telescope to make the axis visible for pointing of the surveying instrument. The indirect method is entirely based on observed positions of targets fixed on solid structure of the telescope. The positions of the targets are measured with surveying techniques when the telescope is turned in different positions. In our approach, the indirect method was applied. Actually there is the same local tie problem with reference points and orientations of the measuring instruments as there is with the VLBI antenna itself. One needs a local reference network around the fundamental station. Another question is how to connect the local network to the global reference frame reliably with controlled error estimates, and how to combine different measurements together. Additionally, a local geoid model is needed to tie the local measurements in a global frame.
U. Kallio and M. Poutanen
In our measurements the main goal was to determine the vector between the VLBI antenna reference point and the IGS GPS point. Beside that we tied these two points to the benchmarks of the local network. We also determined the orientation and the offset of axes of the radio telescope.
5.2
Modelling the Antenna Rotation
The IVP (Invariant Point) of a VLBI antenna is defined as a fixed point (in a reference frame) to which VLBI observations are referred. The IVP is the intersection of the primary axis with the shortest vector between the primary and secondary axis (Dawson et al. 2007). The radio telescope is an instrument with two rotation axes, and like the other instruments with axes (e.g., tachymeters, laser trackers, cameras, SLR) has errors such as eccentricity or offset between axes and axes misalignment. In general the directions of axes are not perpendicular. The secondary axis (elevation) rotates about the primary axis (azimuth) and they don’t intersect. In this case, the reference point is the projection of the secondary axis on the primary axis. IVP is not a marked point and no line of sight from outside the telescope to the reference point is possible. The determination of the IVP coordinates must be done indirectly by tracking the positions of some points on the antenna structure in different antenna positions. Let’s assume that the antenna is rotated around its axes. If the azimuth is kept fixed then a point in the antenna structure rotates about the elevation axis and the track forms a circle in space. Similarly, if the elevation is locked then the point rotates about the azimuth axis. In an ideal case, all the tracks together form a sphere. In recent determinations of the IVP, the model of three dimensional circles has been the most popular. (Dawson et al. 2007; ITRF 2009) The order of observations has been planned so that the circle fitting is possible. Observing all circles which are needed for reliable fitting means a large number of VLBI antenna positions. The whole process is typically very time consuming. A different approach for calculating the reference point was presented by L€osler (L€osler and Hennes 2008; L€osler and Haas 2009). In the L€osler’s model
5
Can We Really Promise a mm-Accuracy for the Local Ties on a Geo-VLBI Antenna
37
one computes the transformation of the tracked coordinates from the instrument coordinate system to the VLBI-antenna coordinate system. It uses the telescope angle readings as observations but the order of observations is arbitary. Therefore the model can be applied also to the sparse and scattered data and it includes antenna axes errors.
5.3
€hovi Model The Metsa
We developed a model to be used in connection of kinematic GPS measurements which were made during the normal use of the VLBI telescope. GPS data were collected during an ordinary 24-h geodetic VLBI session. Therefore, paths of the two GPS-antennas attached on the radio telescope dish do not form such kind of circles that could be used in three dimensional circle fitting model to solve for the azimuth and the elevation axes and the IVP coordinates. The idea of using telescope angles as observations is the same as in L€ osler’s model, but we reformulated the movement of the antenna points and the rotation matrices. Our model is suitable also for different types of telescope mounting other than the elevationazimuth mount described here. The basic assumptions of our model are that points in antenna structure rotate about the elevation axis and the elevation axis rotates about the azimuth axis. The axes need not intersect or be perpendicular. The position vector of the GPS-antenna X (in an arbitrary reference frame) is the sum of three vectors: the position vector of the reference point X0, the axis offset vector (E-X0) rotated by angle a about the azimuth axis a and a vector from the eccentric point E to the antenna point p rotated about the elevation axis e by angle b and about the azimuth axis by angle a (Fig. 5.1). Unknown parameters are X0, E, a, e, and p. Observations are coordinates X for each antenna point and epoch, and VLBI antenna angle readings a and b for every epoch. The estimated values of E, e and p are those of an antenna initial position which may be zero for both angles. The basic equation of our model is X0 þ Ra;a ðE X0 Þ þ Ra;a Rb;e p X ¼ 0
(5.1)
Fig. 5.1 The model parameters in a local reference frame. IVP denotes the Invariant Point, to which VLBI observations are referred. Vectors X, p1 and p2 denote the position vectors of the two GPS antennas attached on side of the VLBI antenna
Rotation matrices Ri,j are formed by applying the Rodrigues’ rotation formula which uses rotation axis and angle. The rotated X is Xr ¼ cosðaÞX þ ð1 cosðaÞÞaaT X þ sinðaÞa X (5.2) The rotation matrix is then 0
1 Rða; aÞ ¼ cosðaÞ@ 0 0 0 xx þ ð1 cosðaÞÞ@ xy xz 0 0 z þ sinðaÞ@ z 0 y x
1 0 0 1 0A 0 1 1 xy xz yy yz A yz zz 1 y x A 0
(5.3)
where a is the rotation angle, and the components of the unit vector of a rotation axis a are x, y and z. In our model the angles in the rotation matrices are the VLBI antenna azimuth and elevation angle readings. Azimuth and elevation axes are described with unknown unit direction vectors. There are four parameters in rotation matrix: three for axis and one
38
U. Kallio and M. Poutanen
for the angle, although only three are independent parameters because the axis is a unit vector. Therefore, we need two condition equations between parameters in our adjustment, one for each axis. aT a 1 ¼ 0
(5.4)
e e1¼0
(5.5)
T
The other two conditions between parameters are needed for finding the shortest vector between the azimuth and elevation axes. The offset vector between the axes is perpendicular to both axes. ðE X0 ÞT a ¼ 0
(5.6)
ðE X0 ÞT e ¼ 0
(5.7)
We do not need any condition equation between the GPS-antenna points: the model itself includes the condition of the constant distance between the points. The number of unknown parameters in the model is u ¼ 12 þ m 3
(5.8)
and the number of observations n¼tm3þt2
1 0 0 0 0 0 FðX0 ; E ; a ; e ; p ; ai ; bi ; X1 Þ C B .. yi ¼ @ A . 0
0
(5.10)
The solution of unknown parameters is reached by iteration in a linearized least squares mixed model with conditions between the parameters: 0 t h 11 i P T 1 T 1 T x D A ðB P B Þ A i i i i i A ¼@ i k h D 0 0 t h i1 P T 1 T Ai ðBi P1 i Bi Þ yi A @ i GðX0 0 ; E0 ; a0 ; e0 ; p0 Þ (5.11)
0
0
0
(5.12)
0
FðX0 ; E ; a ; e ; p ; ai ; bi ; Xm Þ
The matrix G includes condition equations with approximate values of parameters. 0
1 g1 ðX0 0 ; E0 ; a0 ; e0 ; p0 Þ B g2 ðX0 0 ; E0 ; a0 ; e0 ; p0 Þ C C G¼B @ g3 ðX0 0 ; E0 ; a0 ; e0 ; p0 Þ A g4 ðX0 0 ; E0 ; a0 ; e0 ; p0 Þ
(5.13)
Jacobian matrices A and B for epoch i are 0 B Ai ¼ B @
(5.9)
where t is the number of epochs and m is the number of points to be tracked. The degree of freedom in our mixed model least squares adjustment with four conditions between parameters is r ¼ 3m t þ 4 u
where xh is the correction to the approximate values of parameters after h iteration and yi includes the basic equation for all points in epoch i with approximate values of parameters. The apostrophes here denote the approximate value.
0 B Bi ¼ B @
@F @X0
@F @X0
@F @azi
@F @azi
@F @E
@F @a
.. .
@F @e
@F @p1
@F @E
@F @a
@F @e
@F @eli
@F @X1
.. .
.. .
@F @eli
0
.. .
0
1 0 .. C . C A (5.14) @F @pm
1 0 .. C . C A
(5.15)
@F @Xm
and D¼
@G @G @G @G @G @G ... @X0 @E @a @e @p1 @pm
(5.16)
The weight matrix of the epoch i is 0
s2a @ Pi ¼ 0 0
0 s2b 0
11 0 0A Ci
(5.17)
s2a and s2b are variances of azimuth and elevation observations, respectively, and Ci is the covariance matrix of coordinate observations in epoch i.
5
Can We Really Promise a mm-Accuracy for the Local Ties on a Geo-VLBI Antenna
5.4
Tie Measurements Using Kinematic GPS
5.4.1
Installation of the GPS Antennas
Due to restrictions on telescope time and the anticipated time-consuming procedure with traditional surveying methods or static GPS, we ended up testing a kinematic approach. We had two Ashtech Dorne Margolin-type GPS antennas attached to the edge of theVLBI antenna and we gathered data with two Ashtech Z-12 receivers during the following geo-VLBI campaigns: EUROPE-96 (27–28. 11.2008), IVS-T2059 (16–17.12.2008), EUROPE-97 (19–20.1.2009), EUROPE-98 (25–26.3.2009), and IVST2061 (7–8.4.2009). The GPS-antennas were attached to rotating holders on both sides of the radio telescope dish, and they had the counterweights that forced them to point to the zenith regardless of the position of the radio telescope. The GPS antenna reference point (ARP) was in the rotation axis of the antenna holder (Fig. 5.2). The antennas were assembled so that the rotation axis of the holder of a GPS-antenna pointed to the second antenna and this direction was approximately the same as the elevation axis. We measured the angle between the antenna north arrow and the direction of the elevation axis. It is possible to calculate the GPS antenna orientation and phase centre corrections for every epoch. For this we can use the results of an individual antenna calibration.
5.4.2
39
same time we gathered GPS data on six pillar points to form a local network. In this paper we concentrate on the tie to the IGS point. Kinematic co-ordinate solutions were computed with the Trimble TTC software on the fly strategy. We accepted only the fixed solutions with the pdop values less than 8 and the standard deviations of coordinates less than 0.1 m. If the left hand side GPS antenna was rejected we also rejected the right hand side of the same epoch for symmetry. Because the mutual distance of the antennas did not change, we rejected both observations where the distance between the antennas differed more than 0.05 m from the median value. The height difference between the antennas was allowed to differ from the median not more than 0.1 m. Azimuth and elevation angles of the VLBI antenna were dumped into binary disk files every 15 s and v tagged with UTC time, one file for each epoch. Subsequently, those files were afterwards combined into one ASCII file. Due to a system error during the first campaign, the actual VLBI antenna information was missing. The telescope angle readings are used as observations. The trajectory co-ordinate data of the GPSantennas and the VLBI-antenna angle readings must be combined. The synchronization error of one second in time may cause up to 0.01 (depending on the angular velocity) orientation error to the estimated elevation axis. We obtained good data only when the VLBI-antenna was tracking a radio source. When it changed the source the movement was too quick for fixed solutions.
Preparation of the Kinematic Data
The GPS observation interval was 30 s. This is also the interval for the Mets€ahovi IGS GPS receiver which was our main target for the tie measurement. At the
Fig. 5.2 GPS antenna holder on the VLBI antenna
5.4.3
GPS Antenna Phase Center Variation
The orientation of the GPS antennas on the VLBI antenna was changed when the VLBI antenna rotated. We used individually calibrated antennas to eliminate the effect of antenna phase center variation in the computation. Only in one azimuth position the orientation of the GPS antennas was the same as used in the calibration. If we were using the existing antenna calibration tables in the kinematic solution in the ordinary way it would cause systematic errors in the coordinates of GPS antennas. This would propagate
40
U. Kallio and M. Poutanen
to the solution of the reference point coordinates, and especially to the axis offset. One possible solution for the problem is to use only elevation dependent tables for phase variations and let the north and east components of the offset vectors be zeros. If the distribution of VLBI antenna positions over azimuth angles is homogeneous and number and directions of satellites is equal to every direction, then the systematic error will be canceled in the calculation of the reference point. However, it affects still in the solution of axis offset. Unfortunately the distribution of VLBI antenna positions was not equal because of the distribution of the tracked radio sources during the geodetic VLBI session. The VLBI antenna dish also blocked observations so that it was not possible to get fixed solution for trajectory points in every azimuth and elevation positions. We tested the possibility to apply antenna correction tables from the absolute calibration directly to the RINEX data and then use the antenna calibration tables with zero offsets and zero phase variations during the GPS data processing. The GPS antenna orientation was calculated for every epoch from known angle between elevation axis and the north arrow of the GPS antenna and the azimuth angle reading of the telescope. The antenna offset in the direction of a satellite was then subtracted in the RINEX code and phase data.
about 2/3, but we had still data enough for the final adjustment (Table 5.1).
5.4.4
Table 5.2 The local tie vectors between METS GPS and VLBI reference point and the axis offsets in the four campaigns in meters
Computing and Outlier Detection
We programmed the linearized least squares mixed model with conditions between parameters using Octave language. Over two iterations the residuals were checked and outliers removed. Because there still existed some bad data after the preprocessing and filtering we rejected those observations which had associated residuals larger than 1 m. Also angle readings with residuals larger than 0.1 were rejected. After two more iterations we repeated the rejection procedure with limit values 0.2 m and 0.001 . After this we repeated again the adjustment and rejected those observations which had a standardized residual larger than 3. For symmetry we rejected observations from both sides of the VLBI antenna. The loss of the data during the process was
5.5
Results
The maximum difference between the solutions of the reference point in the four campaigns was 2 mm in each vector component. The dependence of the axis offset values on the GPS antenna orientation changes is clear. The VLBI axis offset vector E-X0 rotates about the azimuth axis at the same time and the same amount as the GPS antenna phase center offset vector. In our basic equation it is impossible to distinguish these offsets from each other. In the Trimble TTC processing the control point was METS with coordinates in ITRF 2005 at epoch 1.1.2009. In the results presented in the Table 5.2. the Table 5.1 The loss of the data during the processing Campaign IVS-T2059 EUROPE-97 EUROPE-98 IVS-T2061
GPS antenna Left Right Left Right Left Right Left Right
IVS- T2059
Trajectory points 2,495 2,674 2,642 2,810 2,676 2,823 2,807 2,787
Points in the final adjustment 809 809 642 642 698 698 499 499
EUROPE-97 EUROPE-98 IVS- T2061
N 37.6086 2 37.6080 3 37.6071 3 37.6094 3 E 122.4009 2 122.4006 2 122.4019 2 122.3995 3 U 14.6781 4 14.6780 5 14.6776 6 14.6786 8 Axis 0.0045 6 0.0050 7 0.0052 9 0.0020 11 offset Differences to ITRF2005 (2009.0) N 0.0021 0.0027 0.0035 0.0012 E 0.0005 0.0008 0.0005 0.0019 U 0.0239 0.0239 0.0234 0.0245 Differences to the determination by Jokela N 0.0061 0.0067 0.0076 0.0053 E 0.0032 0.0028 0.0041 0.0018 U 0.7262 0.7262 0.7267 0.7256
Single numbers denote the standard error, in 0.0001 m. We also show a comparison to the vectors from ITRF2005 (2009.0), and to the earlier determination by Jokela
5
Can We Really Promise a mm-Accuracy for the Local Ties on a Geo-VLBI Antenna
GPS antenna phase center variations have been corrected in the RINEX data epoch by epoch before the kinematic trajectory computing. For comparison, the vector from the METS GPS ARP to the Mets€ahovi VLBI reference point in ITRF 2005 coordinates at epoch 1.1.2009 was calculated from the ITRF data set. Differences to our results are presented in Table 5.2. The agreement in horizontal components is good but the vertical component differs more than 2 cm. The comparison to the earlier direct determination by Jorma Jokela in 2004 (Jokela et al. 2009) is shown in Table 5.2. The vertical component differs more than 70 cm because the directly achievable point in Jokela’s determination was not the intersection point of the axes but the pinhole on the floor of the upper platform of the VLBI telescope. Beside the local tie vector we computed the antenna axis offset and orientation. The orientation was modeled as unit direction vectors of the axes. The orientation angles of the telescope, non-perpendicularity of axes and the tilt of the azimuth axis was calculated. Because our model doesn’t take into account epoch-wise variation of the azimuth axis tilt, the direction vector of azimuth axis varies between campaigns. The antenna axes offset differs from the offset determined by Leonid Petrov in 2007: 0.0051 0.0037 m (Petrov 2007). The absolute value from the first three campaigns is about the same, but the sign is opposite to that of Petrov.
41
expect more reliable results for the axes orientation and alignment. We have also used traditional terrestrial surveying methods combined with space intersection technique by measuring reflecting targets on antenna construction. This method was very time consuming and we had telescope for that purpose only during bad weather conditions (e.g., heavy rain, snowing) when it was not in normal use. The results will be published in a future paper. The kinematic method has proved to be suitable for monitoring the IVP of a radio telescope simultaneously during a VLBI session. Especially for sites without a radome, there will be less noisy data. Based on our experience, we may propose that GPS tracking should be done permanently during VLBI sessions. This will give an instantaneous tie to the co-located CGPS antenna. This technique should be considered especially for the VLBI2010 plan. Acknowledgements We would like to thank T. Lindfors, A. Mujunen, M. Tornikoski, J. Kallunki, E. Oinaskallio and H. R€onnberg of the Mets€ahovi Radio Observatory of the Aalto University for their invaluable help. Elevations and azimuths of GPS satellites were computed using “wheresat” of GPStk. (http://www.gpstk.org/bin/view/ Documentation/WebHome). The Mets€ahovi model was programmed with Octave language (http://www.octave.org). This work was partly supported by the Academy of Finland project 134952.
References 5.6
Conclusions and Future Actions
The test shows that a millimeter accuracy is possible to achieve in local tie vector determination with kinematic GPS. Our calculation model is suitable for sparsely scattered data as well as for the data with preplanned circles. The angle readings of the telescope must be synchronized carefully with the GPS observations. Systematic errors like GPS antenna phase center variations must be taken into account although they are smaller than the accuracy of the trajectory point in kinematic solution. Future studies will continue with the analysis of the static GPS observation with two hour sessions of each 90 pre-planned VLBI antenna position. Observations that have been made in March and April 2009. We
Dawson J, Sarti P, Johnston GM, Vittuari L (2007) Indirect approach to invariant point determination for SLR and VLBI systems: an assessment. J Geodesy 81(6–8):433–441 IERS (2005) In: Richter B, Schwegmann W, Dick WR (eds) Proceedings of the IERS workshop on site co-location, Matera, Italy, 23–24 Oct 2003. IERS technical note No. 33. Verlag des Bundesamts f€ur Kartographie und Geod€asie, Frankfurt am Main, p 148 IERS (2009) http://www.iers.org/MainDisp.csl?pid¼68-38. (8.10.2009) ITRF (2009) Co-location survey: online reports, in ITRF web site: http://itrf.ensg.ign.fr/local_surveys.php. (8.10.2009) Jokela J, H€akli P, Uusitalo J, Piironen J, Poutanen M (2009) Control measurements between the Geodetic Observation Sites at Mets€ahovi. In: Drewes H (ed) Geodetic reference frames. IAG Symposium Munich, Germany. Springer, Heidelberg, pp 101–106, 9–14 Oct 2006 L€osler M, Haas R (2009) The 2008 local-tie determination at the Onsala Space Observatory. In: Proceedings of the EVGA, 2009, Bordeaux, France, 24–25 Mar 2009
42 L€osler M, Hennes M (2008) An innovative mathematical solution for a time-efficient ivs reference point determination. In: Measuring the changes, 2008. 4th IAG Symposium on Geodesy for Geotechnical and Structural Engineering, Lisbon, Portugal, 12–15 May 2008 Petrov L (2007) VLBI antenna axis offsets. http://geminig. sfc. nasa.gov/500/oper/solve_save_files/2007c.axo (27.10.2009)
U. Kallio and M. Poutanen Sarti P, Sillard P, Vittuari L (2004) Surveying co-located space geodetic instruments for ITRF computation. J Geod 78(3):210–222 Sarti P, Abbondanza C, Vittuari L (2008) Terrestrial Surveying Applied to Large VLBI Telescopes and Eccentricity Vectors Monitoring. 13th FIG symposium on deformation measurement and analysis. LNEC, Lisbon
6
Recent Improvements in DORIS Data Processing at IGN in View of ITRF2008, the ignwd08 Solution P. Willis, M.L. Gobinddass, B. Garayt, and H. Fagard
Abstract
In preparation for the computation of ITRF2008, the DORIS IGN analysis center has undertaken the task of a complete reprocessing of all DORIS data from 1993.0 to 2009.0, using all available DORIS data as well as the most recent models and estimation strategies. We provide here a detailed description of the major improvements recently made in the DORIS data processing, mainly in terms of solar radiation pressure, atmospheric drag, gravity field, and tropospheric correction. We address here the impact of the new IGN time series (ignwd08) on geodetic products using comparison to the previous IGN solutions (ignwd04). In particular, previous artifacts, such as 118-day or 1-year periodic errors in the TZ-geocenter solution or in the vertical component of high latitude DORIS tracking stations, have now disappeared, leading to more precise and reliable time series of DORIS station coordinates. Finally, possible future improvements are discussed proposing new investigations for the future.
6.1
P. Willis (*) Institut Ge´ographique National, Direction Technique, 2 avenue Pasteur, 94165 Saint-Mande´, France Institut de Physique du Globe de Paris, PRES Sorbonne Paris Cite´, UFR STEP, 35 rue He´le`ne Brion, 75013 Paris, France e-mail:
[email protected] M.L. Gobinddass Institut Ge´ographique National, LAREG, 6-8 avenue Blaise Pascal, 77455 Marne-la-Valle´e, France Institut de Physique du Globe de Paris, PRES Sorbonne Paris Cite´, UFR STEP, 35 rue He´le`ne Brion, 75013 Paris, France B. Garayt H. Fagard Institut Ge´ographique National, Service de Ge´ode´sie et de Nivellement, 2 avenue Pasteur, 94165 Saint-Mande´, France
Introduction
DORIS (Doppler Orbitography and Radiopositioning Integrated on Satellite) is one of the four geodetic techniques participating in the realization of the International Terrestrial Reference System (ITRS) (Willis et al. 2006). Figure 6.1 presents the current DORIS permanent network, demonstrating a dense and homogenous geographical distribution of 57 tracking stations (Fagard 2006). Since 2003, an International DORIS Service (IDS) was created in order to foster international cooperation (Tavernier et al. 2002; Willis et al. 2010a). The Institut Ge´ographique National (IGN) is one of the seven IDS Analysis Service, providing products on a weekly basis (Willis et al. 2010b). In preparation of ITRF2008 (Altamimi and Collilieux 2010), a new DORIS time
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_6, # Springer-Verlag Berlin Heidelberg 2012
43
44
P. Willis et al. Table 6.1 Main differences in models and analysis strategies used for the two latest DORIS/IGN weekly solutions Model/strategy Gravity field Tropospheric mapping function Elevation cut-off Solar radiation pressure coefficient Atmospheric drag parameter
ignwd04 GGM01C Lanyi 15 estimated Reset/6h
ignwd08 GGM03S GMF 10 fixed Reset/1 h
Fig. 6.1 DORIS permanent tracking network (as of September 2009)
series (ignwd08) was reprocessed using data from 1993.0 to 2009.0. Since then, new IGN weekly solutions using the same processing strategy are regularly delivered at the IDS data center, on average once a week. The goal of this article is to present the major differences in terms of data analysis between this new reprocessed solution (ignwd08) and the previous DORIS solution (ignwd04), and to provide an overview of current available geodetic products from the IGN Analysis Center (AC): weekly time series of station coordinates, velocity field, terrestrial reference frame and polar motion.
6.2
DORIS Data Analysis
Table 6.1 summarizes the main differences between the ignwd04 and the ignwd08 analysis strategies. A more recent GGM gravity was used (Tapley et al. 2005). C21 and S21 rates were corrected to follow the IERS 2003 conventions (McCarthy and Petit 2004). No annual correction was taken into account in the gravity field coefficients. Using the more recent GMF mapping function (Boehm et al. 2006) allowed us to use DORIS data at a lower elevation without any drawback. Following recent investigations (Gobinddass et al. 2009a, 2009b), solar radiation pressure models were rescaled using a constant empirical parameter per satellite. Atmospheric drag parameters were reset every 1-hour for the SPOT and Envisat satellites (Gobinddass et al. 2010). This new strategy can be used for all days, even during high geomagnetic activity (Willis et al. 2005a). For the previous ignwd04 solution, a specific strategy was earlier required
(resetting the drag parameter for estimation every minute instead of every 6 h). This involved some non-automated processing for these few days, while the new procedure is fully automated.
6.3
Time Series of ignwd08 Station Coordinates
Unlike other IDS Analysis Centers, DORIS data are processed automatically as soon as they appear at the IDS data centers. Weekly station coordinates in SINEX format are available at these data centers within a few hours, and are provided in free-network (loosely constrained) for the IDS combination but also after projection and transformation in the latest ITRF solution (currently ITRF2005 but soon in ITRF2008 as this transformation is straightforward and does not require any DORIS data processing). These results are freely available to the scientific community at the following URL address for further geophysical investigations: http://ids.cls.fr/html/doris/ids-stationseries.php3. This technique provides SINEX results for geodesists (including full covariance information in a loosely constrained terrestrial reference frame). It also provides tabulated results in STCD format (Noll and Soudarin 2006) for geophysicists directly expressed in ITRF2005 (Altamimi et al. 2007). Figure 6.2 provides an example of such results for the Rio Grande station in Argentina. Different colors indicate the different occupations of this station, as related to equipment upgrades. The positions of successive occupation at the same site were tied together using geodetic local tie information. Smaller scatter after 2002.4 indicates an improvement in repeatability when 4 or 5 DORIS satellite are available. No large discontinuity can be noticed, showing a good
6
Recent Improvements in DORIS Data Processing at IGN in View of ITRF2008
45
Fig. 6.2 Weekly time series of ignwd08 Rio Grande station coordinates expressed in ITRF2005 (from STCD files). In North (a) and Up (b) (in mm)
agreement between the DORIS results and the geodetic local tie vectors. However a closer inspection may indicate possible problem after first occupation or after data gap (vertical). The East component (not displayed here) is noisier due to the North-South tracks of the sun-synchronous satellites (SPOTs and Envisat), especially for mid-latitude stations. Other authors already showed that previous artifacts in the DORIS time series at 118 days (TOPEX draconitic period) are no longer visible in the vertical component for the high-latitude station when solar radiation pressure models are empirically rescaled (Amalvict et al. 2009; Kierulf et al. 2009). The previous periodic effects visible in the ignwd04 time series were due to an improper handling of the solar radiation pressure (Gobinddass et al. 2009a, 2009b) and were observed in previous time series (le Bail 2006; Williams and Willis 2006) but were never fully appreciated. These problems were rather serious for altimetry as they affected mostly the Zcomponent of the stations, which is the major factor for mean sea level determination (Morel and Willis 2002, 2005; Beckley et al. 2007). Table 6.2 provides some information on annual signals present in DORIS/ IGN time series in the polar region. Table 6.2 demonstrates that the previous annual signals around 10 mm in the ignwd04 time series of vertical coordinates of high-latitude station are not present anymore in the new ignwd08 time series. Currently observed annual signals around 3 mm could easily be explained by real geophysical reasons, as geocenter motion is estimated to be at this level.
Table 6.2 Amplitude of annual signals in DORIS vertical time series (in mm) Station SPJB THUB ADEA ROTA SYOB BEMB
Lat (deg) 78 550 76 320 66 400 67 340 69 000 77 520
ignwd04 13.7 11.6 3.4 6.6 6.5 11.2
ignwd08 2.7 2.7 1.9 3.0 3.9 5.0
A similar problem was also previously detected at 1 year (draconitic periods of SPOT and Envisat, being sun-synchronous satellites) and also disappeared in this new ignwd08 solution (Willis et al. 2010b).
6.4
ign09d02 Derived Velocity Field
At regular intervals (every 6 months to 1 year), we also stack all the available DORIS weekly solutions and provide a cumulative position and velocity solution, making full use of geodetic local ties between successive DORIS occupations, using proper a priori variance, as provided by IGN. The latest IGN velocity field (ign09d02) available at both IDS data centers (CDDIS in USA and IGN in France). In most cases, these results can be explained by plate tectonics (Soudarin and Cre´taux 2006; Argus et al. 2010). However, in other cases, such as the Socorro Island (Mexico), the station displacement is not linear and can be attributed to local volcanic deformations (Briole et al. 2009). While GPS is the
46
P. Willis et al.
key player for geodynamics due to its easy densification, DORIS may still have a role to play in a few cases such as Africa where the geodetic infrastructure is still sparse (Nocquet et al. 2006; Argus et al. 2010). We do not estimate the DORIS-derived velocity more often than every 6 months, as a large number of DORIS observations is already available (16 years) For most stations, formal errors of 0.15–0.30 mm/yr are typical.
6.5
Terrestrial Reference Frame
As our DORIS weekly solutions are provided in free-network form, we can compute for each week the 7-parameters for the TRF (origin, orientation and scale), looking at the best transformation fit into the ign09d02 position/velocity solution already aligned on ITRF2005, but including all DORIS stations. Figure 6.3 displays results for the ignwd08 weekly scales with respect to ITRF2005. No antenna map correction was used for DORIS (Willis et al. 2005b) to map these results toward any ITRF. While the weekly scatter is small, a significant drift can be seen toward ITRF2005 realization. This is currently visible in all results from all DORIS Analysis Centers, sometimes with lower values (Valette et al. 2010) and no convincing explanation has yet been proposed.
Fig. 6.3 Weekly determination of the TRF scale between the ignwd08 solution and ITRF2005 (through ign09d02)
No significant discontinuity can be seen in these results. In our opinion, the discontinuities detected at the end of 2004 by Altamimi and Collilieux (2010) and Valette et al. (2010) may be related to results from other DORIS ACs that could map into the IDS-3 combination, which is the DORIS combination submitted for ITRF2008. Such problems could be related to the end of the TOPEX/DORIS data or to a software modification in the Envisat satellite (Willis et al. 2005b, 2007). The IGN weekly solutions also contain more DORIS stations than the IDS-3 solution, as a preselection was done for IDS-3, based on recommendation from DPOD2005 (Willis et al. 2009). Finally, when transforming into ITRF2005, we do not use the original ITRF2005 coordinates and velocities but our internal ign09d02 long-term solution. We also disregard some DORIS stations (six in total) on a week-byweek basis, taking into account possible temporary problems with these data, as done recently in a more sophisticated way using genetically modified networks (Coulot et al. 2009). It is certainly too early to have a definite conclusion on this difficult problem and more tests are required to understand the exact nature of such possible discontinuities. In Fig. 6.3, a different behavior may be observed for the very early data. This is still under investigation but it could be linked to data availability (only two DORIS satellites : TOPEX/Poseidon and SPOT-2) or
.
6
Recent Improvements in DORIS Data Processing at IGN in View of ITRF2008
to a preprocessing problem as detected earlier in the case of SPOT4 for most of the 1998 data and some early 1999 data (Willis et al. 2006).
6.6
Polar Motion
We also update every week polar motion estimation derived using the DORIS data. Tables 6.3 and 6.4 display direct comparisons of these daily DORIS results with the JPL/GPS time series. When considering the full data set (from 1993.0) in Table 6.3, a clear improvement can be seen for the most recent ignwd08 solution. When considering only the best DORIS results, when four or five satellites are available after 2002.4, the improvement is even more pronounced and RMS of 0.5 mas are now achievable (Table 6.4), even if a small bias of 0.3 mas in X still needs to be explained. This is a significant improvement when compared to earlier determination (Gambis 2006): RMS of 1.74 mas for XPole and 0.99 mas for YPole, with a 0.24 offset for recent data (2000.0–2004.0). Part of these improvements is due to the fact that no daily polar rates are estimated in the recent ignwd08 solution. Another improvement is related to systematic errors at the 5.2 day period (SPOT sub-cycle), linked again to the solar radiation pressure estimation as demonstrated in Willis et al. (2010).
Table 6.3 DORIS polar motion external precision compared to IGN/JPL time series (mean removed) XPole RMS (mas) ignwd04 1.864 ignwd08 1.228
YPole RMS (mas) 1.440 1.290
XPole mean (mas) 0.287 0.126
YPole mean (mas) 0.164 0.373
6.7
47
Discussion on Future Improvements
While the ignwd08 solution is still very new and regularly updated, future possible improvements are already considered: • Early analysis of Jason-2 data showed that it could improve the realization of the terrestrial reference frame (Zelensky et al. 2010) and that no effect related to the South Atlantic Anomaly (Willis et al. 2004) is observed in these data. • New DORIS satellites will be launched soon (Cryosat-2 from ESA and Altika from India and France). Direct use of the new DORIS phase and pseudorange (Mercier et al. 2010) should be investigated. • While only minor improvement is expected at the altitude of the DORIS satellite from a GOCEderived gravity field (Visser et al. 2009), timevarying effects , new tide models and AOD (Atmospheric and Ocean De-aliasing) corrections could improve the DORIS orbits. • For the tropospheric correction, VMF (Boehm 2004) could be used instead of GMF. Early tests showed that horizontal tropospheric gradients could be considered for DORIS data processing as well (Flouzat et al. 2009). • Previous studies using Laser data also demonstrated possible time tagging problems in the DORIS data files (Zelensky et al. 2010). • Finally, the recent inter-comparisons between the 7 IDS Analysis Centers (Valette et al. 2010) should certainly lead to new investigations, for example for problems related to the South Anomaly (Bock et al. 2010, Stepanek et al. 2010).
6.8
Conclusions
Period 1993.0–2002.4
Table 6.4 DORIS polar motion external precision compared to IGN/JPL time series (mean removed) XPole RMS (mas) ignwd04 1.387 ignwd08 0.584 2002.4–2008.7
YPole RMS (mas) 0.740 0.525
XPole mean (mas) 0.003 0.289
YPole mean (mas) 0.287 0.012
In conclusion, the new ignwd08 solution is a clear improvement over the previous ignwd04 solution. In particular, due to a better analysis strategy concerning the solar radiation pressure, previous artifacts at 118 days and 1 year have now disappeared in the Zgeocenter as well in the vertical component of highlatitude station time series. Significant improvements were also obtained for polar motion for which 0.5 mas
48
comparison with GPS results can be observed, when 4 or more DORIS satellites are available. Finally, with the use of more recent satellites (Jason-2, Cryosat-2, and Altika) equipped with new digital DGXX equipments, more improvements are already foreseen and currently under investigation. Acknowledgement This work was supported by the Centre National d’Etudes Spatiales (CNES). It is based on observations with DORIS embarked on SPOTs, TOPEX/Poseidon, ENVISAT and Jason satellites. This paper is IPGP contribution number 2593.
References Altamimi Z, Collilieux X (2010) DORIS contribution to ITRF2008. Adv Space Res 45(12):1500–1509 Altamimi Z, Collilieux X, Legrand J, Garayt B, Boucher C (2007) ITRF2005, a new release of the International Terrestrial Reference Frame based on time series of station positions and Earth orientation parameters. J Geophys Res 112(B9), art. B09401 Amalvict M, Willis P, W€ oppelmann G, Ivins ER, Bouin MN, Testut L (2009) Isostatic stability of the East Antarctica station Dumont d’Urville from long-term geodetic observations and geophysical models. Polar Res 28 (2):193–202 Argus DF, Gordon R, Heflin M, Ma C, Eanes R, Willis P, Peltier WR, Owen S (2010) The angular velocities of the plates and the velocity of the Earth’s center from space geodesy. Geophys J Int 180(3):916–960. doi:10.1111/j.1365246X.2009.04463 Beckley BD, Lemoine FG, Luthcke SB, Ray RD, Zelensly NP (2007) A reassessment of global and regional mean sea level rends from TOPEX and Jason-1 altimetry ased on revised reference frame and orbits. Geophys Res Lett 34(14):L14608 Bock O, Willis P, Lacarra M, Bosser P (2010) An intercomparison of tropospheric delays estimated from DORIS and GPS data, Adv Space Res 46(12):1648–1660 Boehm J (2004) Vienna mapping functions in VLBI analysis. Geophys Res Lett 31:L01603 Boehm J, Niell A, Tregoning P, Schuh H (2006) Global Mapping Function (GMF), a new empirica mapping function based on numerical weather model data. Geophys Res Lett 33(7), art. L07304 Briole P, Willis P, Dubois J, Charade O (2009) Potential applications of the DORIS system. A geodetic study of the Socorro Island (Mexico) coordinate time series. Geophys J Int 178(1):581–590 Coulot D, Collilieux X, Pollet A, Berio P, Gobinddass ML, Soudarin L, Willis P (2009) Genetically modified networks. A genetic algorithm contribution to space geodesy, application to the transformation of SLR and DORIS EOP time series. In: European Geocience Union meeting, Vienna, Austria, EGU2009-7988 Fagard H (2006) Twenty years of evolution for the DORIS permanent network, from its initial deployment to its renovation. J Geod 80(8–11):429–456
P. Willis et al. Flouzat M, Bettinelli P, Willis P, Avouac JP, Heriter T, Gautam U (2009) Investigating tropospheric effects and seasonal position variations in GPS and DORIS time series from the Nepal Himalaya. Geophys J Int 178(3):1246–1259 Gambis D (2006) DORIS and the determination of the Earth’s polar motion. J Geod 80(8–11):649–656 Gobinddass ML, Willis P, de Viron O, Sibthorpe AJ, Zelensky NP, Ries JC, Ferland R, Bar-Sever YE, Diament M (2009a) Systematic biases in DORIS-derived geocenter time series related to solar radiation pressure mis-modelling. J Geod 83 (9):849–858 Gobinddass ML, Willis P, de Viron O, Sibthorpe A, Zelensky NP, Ries JC, Ferland R, Bar-sever YE, Diament M, Lemoine FG (2009b) Improving DORIS geocenter time series using an empirical rescaling of solar radiation pressure. Adv Space Res 44(11):1279–1287 Gobinddass ML, Willis P, Diament M, Menvielle M (2010). Refining DORIS atmospheric drag estimation in preparation of ITRF2008. Adv Space Res 46(12):1566–1577 Kierulf HP, Pettersen BR, MacMillan DS, Willis P (2009) The kinematics of Ny-Alesund from space geodetic data. J Geodyn 48(1):37–46 le Bail K (2006) Estimating the noise in space-geodetic positioning, the case of DORIS. J Geod 80(8–11):541–565 McCarthy D, Petit G (eds) (2004) IERS 2003 Conventions. In: IERS Techn Note 32, Frankfurt-am-Main, Germany Mercier F, Cerri L, Berthias JP (2010) Jason-2 DORIS phase measurement processing. Adv Space Res 45 (12):1441–1454. doi:10.1016/j.asr.2009.12.002 Morel L, Willis P (2002) Parameter sensitivity of TOPEX orbit and derived mean sea level to DORIS stations coordinates. Adv Space Res 30(2):255–263 Morel L, Willis P (2005) Terrestrial reference frame effects on global sea level rise determination from TOPEX/Poseidon altimetric data. Adv Space Res 36(3):358–368 Nocquet JM, Willis P, Garcia S (2006) Plate kinematics of Nubia-Somalia using a combined DORIS and GPS solution. J Geod 80(8–11):591–607 Noll C, Soudarin L (2006) On-line resources supporting the data, products, and information infrastructure for the International DORIS Service. J Geod 80(8–11):419–427 Soudarin L, Cre´taux JF (2006) A model of present-day tectonic plate motions from 12 years of DORIS measurements. J Geod 80(8–11):609–624 Stepanek P, Dousa J, Filler V (2010) DORIS data analysis at Geodetic Observatory Pecny using single-satellites and multisatellites geodetic solutions. Adv Space Res 46 (12):1578–1592 Tapley B, Ries J, Bettadpur S, Chambers D, Cheng M, Condi F, Gunter B, Kang Z, Nagel P, Pastor R, Pekker T, Poole S, Wang F (2005) GGM02, an improved Earth gravity field model from GRACE. J Geod 79(8):467–478 Tavernier G, Soudarin L, Larson K, Noll C, Ries J, Willis P (2002) Current status of the DORIS Pilot experiment. Adv Space Res 30(2):151–156 Valette JJ, Lemoine FG, Ferrage P, Yaya P, Altamimi Z, Willis P, Soudarin L, (2010) IDS contribution to ITRF2008 Adv Space Res 46(12):1614–1632 Visser PNAM, van den IJssel J, van Helleputte T et al (2009) Orit determination for the GOCE satellite. Adv Space Res 43 (5):760–768
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Williams SDP, Willis P (2006) Error analysis of weekly station coordinates in the DORIS network. J Geod 80 (8–11):429–456 Willis P, Haines B, Berthias JP, Sengenes P, Le Mouel JL (2004) Behavior of the DORIS/Jason oscillator over the South Atlantic Anomaly. CR Geosci 336(9):839–846 Willis P, Deleflie F, Barlier F, Bar-Sever YE, Romans L (2005a) Effects of thermosphere total density perturbations on LEO orbits during severe geomagnetic conditions (Oct – Nov 2003). Adv Space Res 36(3):522–533 Willis P, Desai SD, Bertiger WI, Haines BJ, Auriol A (2005b) DORIS satellite antenna maps derived from long-term residuals time series. Adv Space Res 36(3):486–497 Willis P, Jayles C, Bar-Sever YE (2006) DORIS, from altimeric missions orbit determination to geodesy. CR Geosci 338 (14–15):968–979 Willis P, Haines BJ, Kuang D (2007) DORIS satellite phase center determination and consequences on the derived scale of the Terrestrial Reference Frame. Adv Space Res 39 (10):1589–1596 Willis P, Ries JC, Zelensky NP, Soudarin L, Fagard H, Pavlis EC, Lemoine FG (2009) DPOD2005: realization of a DORIS
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terrestrial reference frame for precise orbit determination. Adv Space Res 44(5):535–544 Willis P, Fagard H, Ferrage P, Lemoine FG, Noll CE, Noomen R, Otten M, Ries JC, Soudarin L, Tavernier G, Valette JJ (2010a) The International DORIS Service, toward maturity. Adv Space Res 45(12):1408–1420. doi:10.1016/j.asr.2009. 11.018 Willis P, Boucher C, Fagard H, Garayt B, Gobinddass ML (2010b) Contributions of the French Institut Ge´ographique National (IGN) to the International DORIS Service. Adv Space Res 45(12):1470–1480. doi:10.1016/j.asr.2009.09.019 Zelensky NP, Berthias JP, Lemoine FG (2006) DORIS time bias estimated using Jason-1, TOPEX/Poseidon and Envisat orbits. J Geod 83(9):497–506 Zelensky NP, Berthias JP, Lemoine FG (2006) DORIS time bias estimated using Jason-1, TOPEX/Poseidon and ENVISAT orbits. J Geod 80(8–11):497–506 Zelensky NP, Lemoine FG, Chinn DS, Rowlands DD, Luthcke SB, Beckley D, Pavlis D, Ziebart A, Sibthorpe A, Willis P, Luceri V (2010) DORIS/SLR POD modeling improvements for Jason-1 and Jason-2. Adv Space Res 46(12): 1541–1558
7
Towards a Combination of Space-Geodetic Measurements A. Pollet, D. Coulot, and N. Capitaine
Abstract
The International Terrestrial Reference Frame (ITRF), the Earth Orientation Parameter (EOP) time series, and the International Celestial Reference Frame are obtained separately and may thus present inconsistencies. To solve this problem, a first step has been made with the latest ITRF realization (ITRF2005), which has been computed, for the first time, with consistent EOP time series. Another approach to better understand this issue is to directly estimate, in the same process, station positions and EOP time series, from all the space-geodetic measurements. In the framework of the French Groupe de Recherche de Ge´ode´sie Spatiale (GRGS) activities, this latter approach has been studied for several years. For this purpose, the observations of VLBI, SLR, GPS, and DORIS techniques are combined using the same models and software for all the individual data processing. In this paper, we study methodological issues regarding the definition and the consistency of the weekly combined terrestrial frames.
7.1
Introduction
The International Earth rotation and Reference systems Service (IERS) provides different geodetic products as the International Terrestrial Reference Frame (ITRF), the Earth Orientation Parameters (EOP), and the International Celestial Reference Frame (ICRF), the EOP providing the link between
A. Pollet (*) D. Coulot Institut Ge´ographique National, LAREG & GRGS, 6-8 Avenue Blaise Pascal, 77455 Champs-sur-Marne, Marne-la-Valle´e, France e-mail:
[email protected] N. Capitaine SYRTE, Observatoire de Paris, CNRS, UPMC & GRGS, 61 Avenue de l’Observatoire, 75014 Paris, France
ITRF and ICRF. These products are computed separately; this may introduce inconsistencies. The latest ITRF realization, ITRF2005 (Altamimi et al. 2007), has been a major step toward the consistency between IERS products. Indeed, for the first time, the ITRS Product Center (PC) has provided the ITRF2005 together with consistent EOP time series. Another approach is under investigation, namely the combination of space-geodetic techniques (DORIS/GPS/SLR/VLBI) at the measurement level. This method permits us to use the same models and software for all techniques in order to obtain consistent results. Furthermore, such a combination enables the introduction of new common parameters and technique links, in addition to the local ties. It should thus allow us to evidence the possible systematic errors of each technique in order to understand and
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_7, # Springer-Verlag Berlin Heidelberg 2012
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A. Pollet et al.
Table 7.1 Configurations of GPS, VLBI, DORIS, and SLR processing Data
GPS Double differences
Satellites
GPS constellation
Sampling rate Elevation cutoff angle Network
10 min 10 10 150 stations (IGS core sites + colocated stations) Absolute ITRF2005 IERS 05 C04 (Bizouard and Gambis 2009) IERS conventions (McCarthy and Petit 2004) FES2004 (Lyard et al. 2006) FES2004 (Lyard et al. 2006) Not modeled ECMWFa (Guo and Langley 2004) 48 h with a 24-h overlapping
Antenna model Stations EOP Solid Earth tides Ocean tides Ocean loading Atm. loading Tropospheric model Mapping function Orbital arc length a
VLBI IVS-R1, IVS-R4, intensive sessions
DORIS
SLR
SPOT 2,4,5 ENVISAT Jason 1
LAGEOS 1 LAGEOS 2
12 8 All available stations
ITRF2005 rescaled
30 h with a 6-h overlapping
(Mendes and Pavlis 2004) (Mendes et al. 2002) 9 days with a 2-day overlapping
http://www.ecmwf.int
to reduce them. Several studies have been carried out on this subject. Andersen (2000) has applied this kind of combination with a stochastic approach and a square root formation filtering and smoothing to VLBI sessions. Thaller et al. (2007) have combined GPS and VLBI data and got promising results regarding EOP and Zenithal Tropospheric Delays (ZTD). A proof of the great interest aroused by such combination is the creation of a new IERS working group (COL, for Combination at the Observation Level) in 2009. The present study is the continuation of Coulot et al. (2007), who have combined DORIS, GPS, SLR, and VLBI normal systems to evidence the interest of this approach for EOP (and, in particular, Universal Time – UT). We focus here on the definition and the homogeneity of the combined terrestrial frame. In the first section, we present the data used and the different parameters estimated during the combination. Then, we test different possible approaches to combine the observation systems with a particular emphasis on the consistency of the combined frame. Finally, we underline the major role of the local ties in the computation and we provide some conclusions and prospects.
7.2
Data, Software, and Parameters
We carry out tests over nearly 3 months of data obtained by the four space-geodetic techniques, available at the IVS, ILRS, IGS, and IDS data centers (between January 9 and March 19, 2005). The CNES1/GRGS software GINS provides the observation systems for each technique. Indeed, this software, which is also designed for gravity field determination (Bruinsma et al. 2009), has the capability to process DORIS, GPS, SLR, and VLBI data (cf. Bourda et al. 2007). It is used by the IDS LCA-CNES/CLS, the IGS CNES/CLS, and the ILRS GRGS Analysis centers (AC), and by the Bordeaux Observatory for VLBI data processing. Table 7.1 details the processing configurations for each technique. These configurations are close to those applied by the AC for the satellite techniques and by the Bordeaux Observatory team for VLBI. The LAREG/GRGS LOCOMOTIV software combines weekly individual observation systems, uses the degree of freedom method (Sillard 1999) in
1
Centre National d’Etudes Spatiales, French institute.
7
Towards a Combination of Space-Geodetic Measurements
order to provide optimal weights for each satellite technique (one weight per satellite observation set) and each VLBI session observation set in the combination, and estimates the following parameters: – Weekly station positions, at the middle of the GPS week. – Daily EOP (polar motion and UT), at noon. – Orbital parameters in agreement with the three GRGS AC configurations. – ZTD, every 2 h, except for SLR. – Technique specific biases in agreement with the three GRGS AC configurations and with the Bordeaux Observatory VLBI configuration. In addition, depending on the model applied for the combination, we estimate or not estimate Helmert parameters (translations and scale factor for each satellite technique and only scale factor for VLBI, cf. Sect. 7.2).
7.3
Combination, Referencing, and Consistency
Coulot et al. (2007) directly combined the normal equation systems derived from the technique observations. In this work, only the EOP were used as common parameters. The minimum constraints were applied over four sub-networks (one per technique). These minimum constraints were related to the parameters evidenced, per technique, in loose constrained combined solutions, by the reference system effect criterion (Sillard and Boucher 2001). This combination provided heterogeneous combined frames. Indeed, each technique realized its own reference frame. This may be caused by the lack of links between the technique networks; common EOP indeed only link techniques regarding orientation. In the next sections, we thus introduce the local ties and we carry out different combinations (from C1 to C4) in order to obtain a solution for which the systematic errors of the techniques are well handled.
7.3.1
Model for an Ideal Case
We introduce local ties in the combination through observation equations:
53
X01 þ dXc1 ðX02 þ dXc2 Þ ð12Þ
¼ LocTieX
ðSLocTie12 Þ ;
(7.1)
with X01 (resp. X02 ) being the station 1 (resp. 2) a priori positions, dXc1 (resp. dXc2 ) the estimated position ð12Þ the local tie offsets of station 1 (resp. 2), LocTieX vector between the stations 1 and 2 and SLocTie1–2 its variance-covariance matrix. Theoretically speaking, each technique is sensitive to the scale of the terrestrial frame. Each satellite technique is sensitive to the terrestrial frame origin (the geocenter), via the dynamical orbits of its dedicated satellites. Through local ties, we can transmit this definition of the origin to the VLBI station network. Finally, we must conventionally define the orientation of the frame; no technique is sensitive to this orientation (EOP are estimated). We should thus theoretically obtain a homogeneous combined solution by gathering the normal equation systems per technique, with local ties and three minimum constraints (one per rotation) applied over a GPS sub-network to define the orientation of the combined frame w.r.t. ITRF2005. However, as shown in Table 7.2 (C1 test), we still have inconsistent results. Indeed, we estimate the transformation parameters between different networks in the Fc combined frames (all the stations and DORIS, GPS, SLR, and VLBI sub-networks) and ITRF2005 in order to evaluate the homogeneity of the combined solutions. Each technique network realizes its own reference frame. For example, we notice a 3D discrepancy of about 9.6 mm between the DORIS and GPS translations and a scale difference of about 0.7 ppb (4 mm) between SLR and VLBI. Two reasons may explain these heterogeneities. Systematic errors exist between the techniques, due to problems in the models used, in particular the models specific to a particular technique [antenna model for GPS (Ge et al. 2005), solar radiation pressure for DORIS (Gobinddass et al. 2009), etc.] and/or introducing local ties on a weekly basis may be problematic. Indeed, on a weekly basis, we can get poor network distributions, especially regarding the co-located station networks for VLBI or SLR. To investigate these inconsistencies, we directly introduce Helmert parameters to take into account possible mismodellings.
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A. Pollet et al.
Table 7.2 Transformation parameters – mean value (formal error of the mean value) standard deviation value in mm for translations and scale factor Tx, Ty, Tz, and D and m as for rotations Rx, Ry, and Rz – estimated between different networks in the Fc combined frames (all the stations and DORIS, GPS, SLR, and VLBI sub-networks) and ITRF2005, for C1, C2, C3, and C4 combinations over 3 months of data (see text) Combination C1 C2 Helmert parameters No Yes Local tie handling Equation (7.1) Equation (7.1) Local tie weights Provided by ITRF PC Provided by ITRF PC Transformation parameters between all stations and ITRF2005 Tx 5.2 (0.7) 3.5 0.1 (0.2) 0.1 Ty 1.5 (0.5) 1.6 0.0 (0.2) 0.1 Tz 6.9 (0.9) 5.3 0.2 (0.3) 0.3 D 2.1 (0.4) 1.1 0.0 (0.3) 0.3 Rx 40 (5) 28 2 (3) 2 Ry 36 (17) 34 2 (5) 2 Rz 32 (15) 26 2 (6) 3 Transformation parameters between DORIS network and ITRF2005 Tx 2.6 (0.7) 3.5 5.9 (0.6) 2.1 Ty 0.1 (0.7) 2.5 1.7 (0.6) 1.0 Tz 0.8 (0.8) 3.7 2.9 (0.7) 3.0 D 5.9 (0.5) 3.7 0.5 (0.7) 3.3 Rx 63 (22) 60 29 (27) 73 Ry 105 (25) 72 298 (37) 139 Rz 175 (28) 93 146 (24) 58 Transformation parameters between GPS network and ITRF2005 Tx 5.1 (0.7) 3.2 0.1 (0.3) 0.1 Ty 2.3 (0.7) 2.2 0.0 (0.2) 0.0 Tz 9.8 (1.0) 5.3 0.1 (0.3) 0.1 D 1.3 (0.4) 0.9 0.0 (0.2) 0.1 Rx 20 (16) 31 2 (2) 0 Ry 29 (16) 30 1 (2) 0 Rz 25 (14) 24 3 (3) 0 Transformation parameters between SLR network and ITRF2005 Tx 3.9 (0.7) 3.9 0.5 (0.7) 1.6 Ty 2.4 (0.5) 1.9 3.4 (0.6) 1.6 Tz 0.8 (0.9) 4.1 3.9 (0.7) 0.6 D 16.2 (0.6) 2.4 2.4 (0.7) 1.2 Rx 34 (23) 85 36 (24) 86 Ry 16 (26) 92 29 (27) 91 Rz 47 (24) 66 51 (19) 57 Transformation parameters between VLBI network and ITRF2005 Tx 3.8 (0.9) 3.5 1.5 (0.9) 3.2 Ty 5.5 (0.9) 4.8 6.9 (0.9) 4.6 Tz 7.2 (0.9) 3.5 10.8 (1.0) 4.1 D 12.1 (0.7) 3.8 6.2 (0.9) 4.4 Rx 126 (25) 75 126 ( 25) 62 Ry 76 (28) 94 36 (31) 95 Rz 87 (38) 172 64 (43) 181
C3 No Equation (7.7) Equation (7.8)
C4 Yes Equation (7.7) Equation (7.8)
5.3 (0.7) 3.6 1.9 (0.5) 1.5 5.9 (0.8) 5.3 3.3 (0.4) 1.5 36 (5) 27 41 (18) 39 36 (16) 30
0.1 (0.2) 0.2 0.0 (0.2) 0.2 0.4 (0.3) 0.4 0.1 (0.2) 0.3 5 (3) 6 2 (5) 3 1 (7) 5
4.9 (0.6) 3.0 1.8 (0.6) 2.0 5.3 (0.8) 4.0 3.1 (0.4) 1.3 26 (19) 42 42 (17) 36 31 (16) 30
0.3 (0.4) 0.6 0.7 (0.4) 0.5 0.3 (0.4) 0.7 0.5 (0.4) 1.5 4 (13) 22 8 (8) 8 1 (12) 16
5.1 (0.7) 3.3 0.1 (0.6) 1.4 7.3 (0.9) 5.0 1.8 (0.4) 1.0 30 (15) 28 33 (17) 34 29 (15) 26
0.0 (0.2) 0.1 0.0 (0.2) 0.0 0.0 (0.2) 0.0 0.0 (0.2) 0.1 0 (2) 0 0 (2) 0 0 (2) 0
5.2 (0.6) 3.2 1.2 (0.5) 1.7 0.3 (0.8) 3.6 12.0 (0.4) 2.4 13 (17) 40 36 (17) 39 40 (16) 39
1.4 (0.4) 0.3 0.2 (0.4) 1.2 0.4 (0.4) 1.0 0.6 (0.5) 1.2 9 (14) 33 13 (14) 29 10 (11) 25
5.0 (0.7) 2.8 1.5 (0.7) 2.5 2.0 (0.8) 3.3 6.3 (0.5) 1.8 6 (22) 58 64 (19) 44 51 (19) 45
0.9 (0.5) 1.3 1.5 (0.5) 1.5 1.5 (0.5) 1.2 2.2 (0.5) 1.8 8 (19) 43 32 (19) 45 24 (18) 39
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Towards a Combination of Space-Geodetic Measurements
7.3.2
55
3 dXc 6 dEOPc 7 7 6 6 a 7 7 6 7 Y ¼ B6 6 yDORIS 7; 6 yGPS 7 7 6 4 ySLR 5 yVLBI 2
Improved Model
A possible way to ensure the homogeneity of the combined solution is to introduce the Helmert (transformation) parameters in the observation systems of each technique. We consider the following observation system: 2
3 dX Y ¼ A4 dEOP 5; a
(7.2)
with Y being the pseudo-observations, A the design matrix, dX the station position offsets w.r.t. a frame Fk, dEOP the EOP offsets consistent with the orientation of Fk and a the other parameter offsets such as orbital parameters, ZTD, etc. We then introduce the transformation parameters, using the following equations: dX ¼ dXc þ T þ DX0 þ RX0 ; dEOP ¼ dEOPc þ R0 ;
(7.3)
with dXc being the station position offsets w.r.t. Fc, the combined frame, T, D, R, and R’ the scalars, vectors, and matrices related to the transformation parameters between the frames Fc and Fk, X0 the a priori station positions, and dEOPc the EOP offsets consistent with the orientation of Fc. In practice, no technique is sensitive to the orientation of Fc (we estimate EOP). The estimated EOP offsets thus align w.r.t. the orientation we define (ibid regarding the frame origin for the VLBI technique). We thus estimate translations and scale factor for satellite techniques: dX ¼ dXc þ T þ DX0 ; dEOP ¼ dEOPc ;
where ytech (tech corresponding respectively to DORIS, GPS, SLR or VLBI) is the vector of the transformation parameters per technique (a scale factor for each technique and three translation parameters for each satellite technique) and B is the new design matrix, deduced from the matrix A. These transformation parameters are estimated in addition to all the other parameters. Due to the lack of information regarding the Fc combined frame definition, the normal system deduced from this observation system presents rank deficiency. As local ties link the technique networks, this rank deficiency correspond to the seven parameters needed to define the combined frame. This definition is obtained by using constraints: either constraints on the estimated transformation parameters and/or minimum constraints. Indeed, we can apply seven minimum constraints w.r.t. ITRF2005 to define Fc. We can also take advantage of the Helmert parameters. For example, if we want to define the origin of Fc as the SLR origin, we do not estimate the SLR translations. In this case, we define the origin of the combined frame independently of ITRF.
7.3.3
Numerical Tests
(7.4)
and only scale factor for VLBI technique: dX ¼ dXc þ DX0 ; dEOP ¼ dEOPc :
(7.6)
(7.5)
We thus obtain the following global observation system:
We compare here two combinations: the “ideal case” combination, called C1 (cf. Sect. 7.3.1), and a similar one with Helmert parameters, called C2 (cf. Sect. 7.3.2). For local ties, we use the values and the associated standard deviations provided by the ITRF PC.2 More than 50 local ties are used per week. For the
2
See tab.
http://itrf.ensg.ign.fr/ties/ITRF2005/ITRF2005-localties.
56
A. Pollet et al.
combination C1, we apply three minimum constraints w.r.t. ITRF2005 over a GPS sub-network to define the orientation of Fc. Consequently, Fc has the orientation of ITRF2005 but not necessarily the same origin and scale. For the combination C2, we use seven minimum constraints w.r.t. ITRF2005 over the same GPS subnetwork to define Fc, which is thus theoretically fully expressed in ITRF2005. To check the homogeneity of the combined solutions, we compute the transformation parameters between different networks in the Fc combined frame (all the stations, and DORIS, GPS, SLR, and VLBI sub-networks) and ITRF2005. Table 7.2 shows that the estimation of the Helmert parameters in the combination increases the homogeneity of the solution. For example, the 3D discrepancy between the DORIS and GPS translations decreases from 9.6 to 6.7 mm. However, there are still some heterogeneities. The VLBI and SLR networks present the largest ones, probably due to poor co-located networks with a weekly sampling.
7.4
A Possible Approach of Referencing
In order to obtain an homogeneous solution, we present another way of introducing local ties in the combination by the use of equality constraints between estimated co-located station position offsets: dXc1 dXc2 ¼ 0 ðSTie12 Þ :
(7.7)
with dXc1 (resp. dXc2 ) the estimated position offsets of station 1 (resp. 2). Doing so, we consider that the value of the local tie is the difference between the ITRF2005 a priori station ð12Þ positions [X01 X02 ¼ LocTieX – cf. (7.1)]. We thus switch from rescaled ITRF2005 to ITRF2005 for SLR a priori station positions, in order to get consistent values. The standard deviations sTie12 (on which STie12 is based) are computed from the variancecovariance matrix of the ITRF2005 a priori station positions X01 andX02 , at the considered epoch t:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Var X01 ðtÞ X02 ðtÞ ¼ Var X01 ðtÞÞ þ VarðX02 ðtÞ 2 cov X01 ðtÞ; X02 ðtÞ
sTie12 ¼
i X0i ðtÞ ¼ X0i ðt0 Þ þ dt X_ 0 ðt0 Þ with dt ¼ t t0 hence : Var X0i ðtÞ ¼ Var X0i ðt0 Þ þ dt2 Var X0i ðt0 Þ i þ 2 dt cov X0i ðt0 Þ; X_ 0 ðt0 Þ and : cov X01 ðtÞ; X02 ðtÞ ¼ cov X01 ðt0 Þ; X02 ðt0 Þ 1 2 2 þ dt2 cov X_ 0 ðt0 Þ; X_ 0 ðt0 Þ þ dt cov X01 ðt0 Þ; X_ 0 ðt0 Þ 1 þ dt cov X_ 0 ðt0 Þ; X02 ðt0 Þ
(7.8) In practice, sTie12 sLocTie12 . Furthermore, this approach is ITRF-dependent: if an event has changed a co-located station position after 2005, this kind of local ties cannot be used for the considered co-location site. A secular solution, estimated from our data and used as the a priori solution, could avoid this dependence. Due to our short data span, here we have used the ITRF2005 to evaluate these constraints. We compute two combinations based on this approach, C3 and C4. Regarding the estimated parameters and the minimum constraints used, C3 (resp. C4) is similar to C1 (resp. C2). In addition to C1 and C2, Table 7.2 also provides the statistics for the transformation parameters for the C3 and C4 combinations. The homogeneity of the solutions is improved for C3 and C4 but only the C4 combination gives a real consistent combined solution. Indeed, the heterogeneities for the scale and Tz still exist in the C3 combination. Regarding C4, these heterogeneities are embedded in the estimated Helmert parameters. For example, the difference between the VLBI and SLR estimated scale factors is about 0.7 ppb over the 3 months of studied data (Mean of estimated VLBI/SLR scale factor: 2.1/2.8 ppb). In the same way, we find a 3D discrepancy of about 10.2 mm between the DORIS and GPS translations in the estimated Helmert parameters. They consequently do not disturb the referencing of the solution anymore. On the one hand, the local ties are essential to link the technique networks in the combination but, on the another hand, they must be used carefully as they have a great impact.
7
Towards a Combination of Space-Geodetic Measurements
7.5
Conclusions and Prospects
Through the direct estimation of transformation parameters in the combination process, we get an homogeneous combined solution. With our new approach, the problem of datum inconsistency evidenced by (Coulot et al. 2007) is solved. But the introduction of the colocation information on a weekly basis appears to be problematic and it can twist the combined frame, due to poor weekly co-located networks. From the tests carried out in the present study, we recommend to use equality constraints between estimated co-located station position offsets together with a rigorous weighting [cf. (7.7) and (7.8)] to obtain an homogeneous result. Even with this method, the definition of the combined frame will still inevitably depend on this link between the technique networks. Consequently, the introduction of supplementary common parameters and links could be helpful to decrease this dependence. At a terrestrial level, we could use common geodynamic signals (such as loading effects, for instance), ZTD, etc., and, at a space level, the use of multi-technique satellites should be of great interest. With our combination model, the inconsistencies between the techniques are embedded in the estimated Helmert parameters. The understanding of these inconsistencies should help to improve some models. Gradually, such improvements should lead to more consistency and, consequently, to definitions of combined frame more independent of any external terrestrial reference frame. However, as long as technique discrepencies exist, the estimation of the Helmert parameters is essential to insure the homogeneity of the combined solutions.
References Altamimi Z, Collilieux X, Legrand J, Boucher C (2007) ITRF2005: a new release of the international terrestrial reference frame based on time series of station positions and earth orientation parameters. J Geophys Res 112:B09401. doi:10.1029/2007JB004949 Andersen PH (2000) Multi-level arc combination with stochastic parameters. J Geod 74:531–551. doi:10.1007/s001900000115
57 Bizouard C, Gambis D (2009) The combined solution C04 for Earth orientation parameters consistent with International Terrestrial Reference Frame 2005. In: Proceedings of IAG Symposia, 134, pp 265–270. doi: 10.1007/978-3-64200860-3 Bourda G, Charlot P, Biancale R (2007) GINS: a new tool for VLBI geodesy and astrometry. In: Proceedings of the 18th European VLBI for Geodesy and Astrometry (EVGA) Working Meeting, 79, Vienna, pp 59–63, ISSN 1811–8380 Bruinsma S, Lemoine J-M, Biancale R, Vale`s N (2009) CNES/GRGS 10-day gravity field models (release 2) and their evaluation. Adv Space Res 45(2010):587–601. doi:10.1016/j.asr.2009.10.012 Coulot D, Berio P, Biancale R, Loyer S, Soudarin L, Gontier A-M (2007) Towards a direct combination of space-geodetic techniques at the measurement level: methodology and main issues. J Geophys Res 112:B05410. doi:10.1029/ 2006JB004336 Ge M, Gendt G, Dick G, Zhang FP, Reigber C (2005) Impact of GPS satellite antenna offsets on scale changes in global network solutions. Geophys Res Lett 32:L06310. doi:10.1029/2004GL0222241414 Gobinddass ML, Willis P, De Viron O, Sibthorpe A, Zelensky NP, Ries JC, Ferland R, Bar-Sever Y, Diament M, Lemoine FG (2009) Improving DORIS geocenter time series using an empirical rescaling of solar radiation pressure models. Adv Space Res 44(11):1279–1287. doi:10.1016/j. asr.2009.08.004 Guo J, Langley RB (2004) A new tropospheric propagation delay mapping function for elevation angles down to 2 degrees. In: Proceedings of ION GPS 2003, Portland, OR Lyard F, Lefevre F, Letellier T, Francis O (2006) Modelling the global ocean tides: modern insights from FES2004. Ocean Dyn 56:394–415. doi:10.1007/s10236-006-0086-x McCarthy DD, Petit G (2004) IERS conventions (2003). IERS technical note 32, Verlag des Bundesamts f€ur Kartographie und Geod€asie, Frankfurt, ISBN 3-89888-884-3 Mendes VB, Pavlis EC (2004) High-accuracy zenith delay prediction at optical wavelengths. Geophys Res Lett 31: L14602. doi:10.1029/2004GL020308 Mendes VB, Prates G, Pavlis EC, Pavlis DE, Langley RB (2002) Improved mapping functions for atmospheric refraction correction in SLR. Geophys Res Lett 29(10):1414. doi:10.1029/ 2001GL014394 Sillard P (1999) Mode´lisation des syste`mes de re´fe´rence terrestres. Contribution the´orique et me´thodologique. PhD thesis, Observatoire de Paris, Paris Sillard P, Boucher C (2001) A review of algebraic constraints in terrestrial reference frame datum definition. J Geod 75:63–73 Thaller D, Kr€ugel M, Rothacher M, Tesmer V, Schmid R, Angermann D (2007) Combined Earth orientation parameters based on homogeneous and continuous VLBI and GPS data. J Geod 81:529–541
.
8
Improving Length and Scale Traceability in Local Geodynamical Measurements J. Jokela, P. H€akli, M. Poutanen, U. Kallio, and J. Ahola
Abstract
Traceability is a feature that is required more frequently in local geodetic highprecision measurements. This basic term of metrology, a measurement science, describes the property of a measurement result whereby the result can be related to a reference through a documented unbroken chain of calibrations, each contributing to the measurement uncertainty (BIPM International vocabulary of metrology – basic and general concepts and associated terms (VIM). JCGM 200:2008. Joint Committee for Guides in Metrology, 2008b). GPS measurements are widely used in local geodynamical research. From the viewpoint of metrology, their traceability is uncontrollable because the scale cannot be unambiguously conducted based on the definition of the metre. In particular, atmospheric effects on a GPS signal cannot be modelled or calibrated along the path of the signal. We are testing a method to bring the traceable scale to small GPS networks using high-precision electronic distance measurement (EDM) instruments, the scales of which have been corrected and validated in calibrations at the Nummela Standard Baseline. The traceable scale of EDM is expected to explain the annual scale variations that have been found in GPS time series and to improve results of episodic GPS campaigns. The scale of a standard baseline is validated and maintained through regular interference measurements with the V€ais€al€a interference comparator, in which a quartz gauge conveys the traceable scale. The results from the interference measurements in 2005 and 2007 in Nummela are presented here together with a brief description of the present state of the renowned measurement standard. A standard uncertainty of 0.08 ppm was obtained again for the baseline length of 864 m, and the results confirm the good long-term stability of the baseline. The scale is transferred further to geodetic and geophysical applications by using calibrated high-precision EDM instruments as transfer standards.
J. Jokela (*) P. H€akli M. Poutanen U. Kallio J. Ahola Department of Geodesy and Geodynamics, Finnish Geodetic Institute, Geodeetinrinne 2, 02430 Masala, Finland e-mail:
[email protected] S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_8, # Springer-Verlag Berlin Heidelberg 2012
59
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Local geodynamical measurements will profit from the reduced and accurately estimated uncertainty of the measurement, and therefore we seek further innovations to improve their traceability. We present here a topical example of calibrations and scale transfer for a baseline and monitoring network around a nuclear power plant. We compare simultaneously measured EDM and GPS results and show a scale bias of approximately 1 ppm between them. By using a traceable length in the network, the bias could be reduced, e.g. by improving the processing strategy of GPS observations. This paper focuses on the metrological part of EDM. Some related results and analysis of GPS measurements are discussed in another paper in this volume (Koivula et al. GPS metrology – bringing traceable scale to local crustal deformation GPS network. IAG Scientific Assembly “Geodesy for Planet Earth”, Buenos Aires, Argentina, 2010).
8.1
The Nummela Standard Baseline
The length and scale traceability is based on the Nummela Standard Baseline of the Finnish Geodetic Institute (FGI). The baseline is widely known as the longest, most accurate and stable measurement standard in the world for traceable geodetic length measurements. It serves in national and international scale transfer measurements for other national or accredited calibration services for EDM instruments, as well as for scientific purposes. The accuracy of the baseline, in terms of standard uncertainty, is better than 0.1 mm/km according to the interference measurements with the V€ais€al€a comparator. Since 1997 the traceable scale has been transferred from Nummela to about 20 baselines or test fields in more than ten countries. The FGI is one of the National Standards Laboratories in Finland.
8.1.1
The 80-Years History
The 864-m geodetic baseline in Nummela, northwest of Helsinki, Finland, was established in 1933, replacing an older measurement standard in Santahamina, southeast of Helsinki. Since 1947, when the first successful interference measurement with the wellknown V€ais€al€a (white light) interference method was performed, it has been called the Nummela Standard Baseline. Originally it was used for the calibration of invar wires for triangulation, and later on for EDM instruments. The scale of the Finnish first-order triangulation network was conducted from Nummela.
The importance of the V€ais€al€a interference method was already recognized in an IAG motion in 1951 and in an IUGG resolution of 1954, which recommended the use of similar methods for assuring a uniform scale in geodetic networks. Since then, the FGI has measured 12 similar baselines on almost all continents. Interference measurements of a standard baseline are still performed using the classical V€ais€al€a comparator. Even today, the method is superior to modern techniques in terms of accuracy for outdoor baselines up to a few kilometres. The scale is traceable to the definition of the metre through a quartz gauge system. The lengths of one-metre-long quartz gauges are known from comparisons and absolute calibrations with better than 40 nm standard uncertainty and multiplied with the V€ais€al€a comparator. Also, the history of the quartz metres and their comparisons and calibrations dates back more than 80 years.
8.1.2
The Present State
Between 1947 and 2007 the Nummela Standard Baseline has been measured 15 times using the V€ais€al€a method, complemented by regular comparisons and absolute calibrations of quartz gauges. The latest absolute calibrations have been performed by the PTB, Braunschweig, in Germany, and MIKES, Helsinki, in Finland (Lassila et al. 2003), showing equal results with smaller than 40 nm standard uncertainties for a set of quartz gauges. Standard uncertainties of comparisons, performed at the Tuorla Observatory of the University of Turku and, since 2005, by the FGI, are a few nm (Fig. 8.1).
8
Improving Length and Scale Traceability in Local Geodynamical Measurements
61
151,5 mm + 151,0 1m 150,5 1950
1960
1970
1980
1990
2000
2010
Year
Fig. 8.1 Length of quartz metre no. VIII, which conducts the scale in interference measurements at the Nummela Standard Baseline, from comparisons at Tuorla. The black spot at the year 2000 is the latest absolute calibration of this particular gauge Table 8.1 Baseline lengths at the Nummela Standard Baseline from the 15 interference measurements from 1947 to 2007 Epoch 1947.7 1952.8 1955.4 1958.8 1961.8 1966.8 1968.8 1975.9 1977.8 1983.8 1984.8 1991.8 1996.9 2005.8 2007.8
024 mm + 24 m – – – – – – – – 33.28 0.02 33.50 0.02 33.29 0.03 33.36 0.04 33.41 0.03 33.23 0.04 33.22 0.03
072 mm + 72 m – – – – – – – – 15.78 0.02 15.16 0.02 15.01 0.03 14.88 0.04 14.87 0.04 14.98 0.04 14.95 0.02
0216 mm + 216 m – – – – – – – – 54.31 0.02 53.66 0.04 53.58 0.05 53.24 0.06 53.21 0.04 53.20 0.04 53.13 0.03
0432 mm + 432 m 95.46 0.04 95.39 0.05 95.31 0.05 95.19 0.04 95.21 0.04 95.16 0.04 95.18 0.04 94.94 0.04 95.10 0.05 95.03 0.06 94.93 0.06 95.02 0.05 95.23 0.04 95.36 0.05 95.28 0.04
0864 mm + 864 m 122.78 0.07 122.47 0.08 122.41 0.09 122.25 0.08 122.33 0.08 122.31 0.06 122.37 0.07 122.33 0.07 122.70 0.08 – 122.40 0.09 122.32 0.08 122.75 0.07 – 122.86 0.07
The number following the symbol is the numerical value (in mm) of the combined standard uncertainty
The results of our latest interference measurements in 2005 and 2007 are presented in Table 8.1, together with the previous results. In calibrations they are used as true values with a known uncertainty in the traceability chain. They are lengths between underground benchmarks, to which (and from which, for calibrations) the lengths between reference points on less permanent observation pillars are projected using precision tacheometry and mechanical plumbing and probing. To maintain sub-mm uncertainties, optimal measurement geometry is essential in these studies. The standard (1-s) uncertainties of the lengths from 24 to 864 m ranged from 0.02 to 0.07 mm in the 2007 measurements, which means they were about the same as previous measurements. The difference between the first interference measurement in 1947 and the latest one in 2007 is 0.08 mm for the full length of 864 m, and the variation during the
60-year time span is 0.6 mm. This proves in excellent fashion the stable location of the baseline on a forested non-frozen sandy ridge. The small, but sometimes significant variations in the results are impossible to detect with any other metrological or geodetic method in field conditions, and they are small enough not to disturb calibrations of the most precise EDM instruments. The variations are caused by the settling of the markers after they have been cast in the ground, by later construction projects and other disturbing activities in the neighbourhood, and by slightly improved methods in processing the measurements. Results for the distances up to 216 m before 1977 have not been published. In 1983 and 2005 measurements for the length 864 m were not possible because of unfavourable weather conditions. The latest publications giving more detailed information on the baseline are by Jokela and Poutanen (1998), Jokela et al. (2009) and Jokela and H€akli (2010).
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Working conditions and security at the unique measurement standard were greatly improved by construction projects in 2004. The old office and store buildings were replaced with a new house. A part of the 0.12 km2 baseline property was fenced, including all observation pillars and underground benchmarks. Roofs were built over all observation pillars. Most of the observation pillars were reconditioned before the interference measurements in 2007, and a new subsurface drainage system keeps the soil dry around the comparator house, preventing possible “floods” caused by melting snow. The old baseline and the interference method were acknowledged again in 2008 when the FGI, together with eight other European institutes, entered a joint research project called “Absolute long distance measurement in air” (Wallerand et al. 2008). This project, which is a part of the European Metrology Research Programme (EMRP), is financially supported by the European Union. The baseline will be utilized in testing and validating new absolute distance measurement (ADM) techniques and instruments, including improved determination of refraction. New instruments are expected to improve distance measurement traceability up to several kilometres, which will allow for more reliable measurements in local deformation networks or other geodetic applications requiring a traceable scale.
measured - true, mm
measured - true, mm
0.0
–0.5
216
432
648
2008-10-31... 2008-11-06
1.0
0.5
0
Calibration of Transfer Standards
High-precision EDM instruments (typically a Kern Mekometer ME5000, since other suitable instruments are even fewer) are used as transfer standards from the Nummela Standard Baseline to other applications. The principles for how to perform EDM observations and make the necessary reductions and corrections to them are well known, and only a few details noteworthy in our scale transfer measurements are discussed here. All measurements are performed in field conditions. The calibration of EDM equipment for a scale transfer takes a few days, including several measurements of all available baseline distances in both directions. This allows for possible daily changes in the equipment and makes the estimation of uncertainty more reliable. Projection measurements at the Nummela Standard Baseline are performed before and after every important calibration, usually a few times a year. More frequent projections are needed during interference measurements, which typically last about three months. Calibrations are performed immediately before and after the transfer, which reveals possible changes in the equipment during the procedure (Fig. 8.2). In addition to the calibration measurements, similar possible sources of uncertainty have to be taken into account at the transfer site. One of them is the centring
2008-08-28... 2008-09-03
1.0
–1.0
8.2
864
m
Fig. 8.2 Example of calibration of transfer standard (Kern ME5000) before and after a scale transfer. Scale correction (+0.151 mm/km 0.049 mm/km, 1-s) is determined from the
0.5
0.0
–0.5
–1.0
0
216
432
648
864
m
differences between measured and true distances. The additive constant (+0.079 mm 0.014 mm, 1-s) has been corrected. Variation and corrections for this equipment are especially small
8
Improving Length and Scale Traceability in Local Geodynamical Measurements
of instruments. Common commercial fixing methods are widely used with many kinds of surveying instruments and also at geodetic baselines, but in integrated geodynamical measurements self-made solutions are often needed. The GPS antenna is often centred and adjusted with a standard forced-centring plate or directly attached to a pillar. But does the uncertainty really remain smaller than 1 mm? Local monitoring networks and tie measurements that include reference points of, for example, SLR or VLBI are even more challenging. Weather correction is another major source of uncertainty. At local networks the distances to be measured may be short, but at least ambient temperature, air pressure and relative humidity must be observed with sufficient accuracy and observed distances corrected with proper formulas, e.g. as recommended by the IAG (1999). The influences of all sources of uncertainty in the traceability chain must be carefully estimated and summarized in the estimate of combined uncertainty as described in metrological regulations (BIPM 2008a). This also includes of course uncertainties in the quartz gauge system and in interference measurements. In favourable conditions 0.5 to 1.0 mm/km values of expanded (k ¼ 2) uncertainty can be expected at the transfer site, depending on the application. By using traceable baseline lengths to adjust the geodetic networks, we can extend these values to a traceable scale of the network, as was done in triangulation. The method as such is probably not reasonable for GPS networks, though it is an interesting addition for estimating the uncertainty of measurements.
8.3
Traceable Scale in Local Geodynamical Measurements
GPS is an excellent tool for crustal deformation research. There is no need for inter-station visibility, distances are not limited to local measurements and accuracies are generally superior to the traditional methods, especially over longer distances. Metrological traceability is a property of a measurement result whereby the result can be related to a reference through a documented unbroken chain of calibrations, each contributing to the measurement uncertainty. The basic requirement of calibrations
63
under specified conditions (BIPM 2008b) makes the traceability and scale of a GPS measurement uncontrollable from the viewpoint of metrology. Heights and height changes are more difficult to monitor than horizontal motions. This is mainly due to the influence of the atmosphere on the GPS signal and the effect of the observing geometry. Some sources of uncertainty are random variables, and can be estimated more reliably by increasing the number of observations or the length of time series. Others are systematic effects, which cannot be quantified, thus causing bias. Although the scale is based on time and the frequency carried by the GPS signal, one is unable to unambiguously conduct the measured distances from the definition of the metre. This is mainly due to the effects of the atmosphere and local conditions, such as multipath. There is no unique standardized set of parameters to process GPS observations (and to obtain the specified conditions), which leads to slightly different results. A more severe limitation is that there is no uniquely traceable documented chain from the computed distances to the definition of the metre. In geodynamical research, one is in many cases mainly interested in changes. In such cases the absolute scale is not critical, but it is sufficient to assume that the scale remains constant between measurements. However, this cannot be guaranteed in GPS observations, and under severe ionosphere conditions scale variations up to 1 ppm have been observed in the Olkiluoto network (Ollikainen et al. 2004). Such results, especially in episodic campaigns, beg for a proper scale in GPS measurements in order to obtain as reliable results as possible. Epoch to epoch variation in scale may lead to misinterpretation in deformation research and one cannot reliably determine the size of this deformation. One solution is to bring the scale into the network using high-precision EDM instruments. Local ties at multi-technique space geodetic sites are another interesting application of dimensional metrology. Quite often these ties are (at least partly) done using GPS, and current demands for sub-mm accuracies in the ties require calibrated EDM baselines for proper scale. Adjacent research projects on GPS/ EDM metrology in the FGI (local ties and another baseline) are reported in other papers in this volume (Kallio and Poutanen 2011; Koivula et al. 2011).
64
Olkiluoto Deformation Network
The FGI and Posiva, an expert organisation responsible for the final disposal of spent nuclear fuel, started GPS measurements in the 10-station control network at the Olkiluoto nuclear power plants in 1995 (Fig. 8.3). The purpose is to monitor possible local crustal deformations with regularly repeated GPS observations. The original network of 10 observation pillars has been slightly changed and expanded through adding new pillars, due to construction projects in the area (Chen and Kakkuri 1995, 1996; Ollikainen et al. 2004; Kallio et al. 2009). The extent of the network, 2 km 4 km, is quite optimal for combining EDM and GPS measurements. The interpretation of results from repeated GPS measurements has been complicated by obvious variations in scale, which are often explained by deficiencies in ionospheric modelling. Since autumn 2002 most GPS campaigns give 1 ppm longer distances than the spring campaign of the same year (Fig. 8.4). Annual variation is also visible in the 196 km long vector between the permanent GPS stations at Olkiluoto and Mets€ahovi (Fig. 8.5). To better monitor the scale and variations to it, the 511-m line between the observation pillars GPS7 and GPS8 was cleared and equipped with a geodetic baseline for high-precision EDM. The visibility for terrestrial observations is not obtainable between the other pillars. Between 1995 and 2009 the network has been measured with episodic GPS campaigns 28 times. Dual-frequency Ashtech Z-12 and Ashtech mZ receivers and Dorne Margolin choke ring antennas have been used to collect data and, to reduce uncertainties caused by antennas, the same antennas have been used at the same stations in each campaign. All observation campaigns have been processed using Bernese software (the latest with version 5.0) and with equal processing principles, e.g. by using network solution, independent L1 and L2 observables to obtain lower measurement noise, and a local ionosphere model computed from the observations (e.g. Kallio et al. 2009). Observation sessions have taken a minimum of 24 h, with an observing interval of 30 s. Since 2002, GPS and EDM observations have been simultaneously conducted twice a year, when possible, for about one week in April and October. The expected movements are extremely small. The largest detected, but
Fig. 8.3 The network for deformation research around the nuclear power plants at Olkiluoto
511258,0 Olkiluoto baseline
GPS
511257,5
EDM
511257,0
Length [mm]
8.4
J. Jokela et al.
511256,5 511256,0 511255,5 511255,0 2002
2003
2004
2005 2006 Year
2007
2008
2009
Fig. 8.4 The GPS and EDM results for the baseline between pillars GPS7 and GPS8 at Olkiluoto
statistically significant, movements are 0.18 mm/a, with a standard deviation of 0.06 mm/a (Kallio et al. 2009). The scale for EDM in the 15 measurements between 2002 and 2009 has been determined in 13 calibrations at the Nummela Standard Baseline. In the 13 measurements between 2002 and 2008 the result for EDM measurements has 12 times been shorter than the result for GPS measurements (Fig. 8.4), with the maximum differences being about 1 mm and the average about 1 ppm. The GPS result is longer in October than in April in six cases out of seven. The scale difference between the GPS and high-precision EDM is obvious. The probable reasons causing such behaviour in the GPS results are uncertainties in GPS antenna calibration values, atmospheric modelling and site-specific effects like multipath conditions. Thus far, the EDM scale has been used for comparison only. By applying the EDM scale directly to the
8
Improving Length and Scale Traceability in Local Geodynamical Measurements
Fig. 8.5 A typical GPS time series between two permanent GPS stations (Mets€ahovi and Olkiluoto). From top to bottom: change in height, East and North component. One can see the annual variation in the vector components. Vertical scale is in millimetres (Kallio et al. 2009)
GPS analysis, the traceable scale can be brought into the GPS network too. Some further research is currently being conducted to better understand the reason for the GPS/EDM difference, and to eliminate any site or antenna dependency (Koivula et al. 2011).
8.5
Conclusions and Further Activities
Scale transfer from a standard baseline (a national geodetic measurement standard) is a method for improving length and scale traceability, which can also be used for local geodynamical measurements. The traceability chain includes the maintenance of the quartz metre system, baseline measurements with the V€ais€al€a interference comparator, baseline maintenance with projection measurements, the calibration of transfer standards (high-precision EDM) and
65
measurements at the transfer site. Most of these stages are included in the customary maintenance of the measurement standard and calibration service, and the effort needed for applications is simpler. The components of measurement uncertainty at every stage are estimated and, by combining them, an important part of a final measurement result, the uncertainty of measurement in the traceability chain, relative to the definition of the metre, is obtained. Up to 0.5 mm/km expanded uncertainty is obtained for distances of about 1 km. New results for the world-class measurement standard, the Nummela Standard Baseline of the FGI, are now available. They are frequently needed for scientific research and for scale transfers to other geodetic baselines and test fields and also for local geodynamical monitoring networks. The GPS/EDM network at the Olkiluoto nuclear power plants is presented as an example. Monitoring one GPS baseline with traceable EDM measurements cannot explain the scale variation in GPS results but can probably improve the processing strategy of episodic GPS campaigns in Olkiluoto. The scale difference between GPS and EDM has also provided motivation for further research. Measurements at Olkiluoto will be continued and there is a plan to expand the network. The amount of continuous GPS measurements may be increased; at the moment there is only one permanent station. A newly planned EastWest EDM/GPS baseline, in addition to the current North-South baseline, could be used to obtain better control over possible azimuth dependency. Acknowledgement For the scale transfer, we have used the Kern Mekometer ME5000 of Helsinki University of Technology’s (TKK) Laboratory of Geoinformation and Positioning Technology as a transfer standard. We thank Professor Martin Vermeer and the rest of the staff there for their willing cooperation.
References BIPM (2008a) Evaluation of measurement data – guide to the expression of uncertainty in measurement (GUM). JCGM 100:2008. Joint Committee for Guides in Metrology. http:// www.bipm.org/ BIPM (2008b) International vocabulary of metrology – basic and general concepts and associated terms (VIM). JCGM 200:2008. Joint Committee for Guides in Metrology. http:// www.bipm.org/
66 Chen R, Kakkuri J (1995) GPS Work at Olkiluoto for the Year of 1994. Work Report PATU-95-30e, Teollisuuden Voima Oy. Helsinki, p 11 Chen R, Kakkuri J (1996) GPS Operations at Olkiluoto, Kivetty, and Romuvaara in 1995. Work report PATU-96-07e, Posiva Oy. Helsinki, p 68 IAG (1999) IAG Resolutions adopted at the XXIIth General Assembly in Birmingham, 1999. http://www.gfy.ku.dk/~ iag/HB2000/part2/iag_res.htm Jokela J, H€akli P (2010) Interference measurements of the Nummela Standard Baseline in 2005 and 2007. Publications of the Finnish Geodetic Institute, no 144, p 85 Jokela J, Poutanen M (1998). V€ais€al€a baselines in Finland. Publications of the Finnish Geodetic Institute, no 127, p 61 Jokela J, H€akli P, Ahola J, Bu¯ga A, Putrimas R (2009) On traceability of long distances. XIX IMEKO World Congress Fundamental and Applied Metrology, Lisbon, Portugal, pp 1882–1887, 6–11 Sept 2009. http://www.imeko2009.it.pt/ Papers/FP_100.pdf Kallio U, Poutanen M (2011) Can we really promise a mmaccuracy for the local ties on a geo-VLBI antenna? (Scientific Assembly “Geodesy for Planet Earth”, Buenos Aires, Argentina, Aug 31–Sept 4) Kallio U, Ahola J, Koivula H, Jokela J., Poutanen M (2009) GPS operations at Olkiluoto, Kivetty and Romuvaara in 2008.
J. Jokela et al. Working Report 2009–75. Posiva, Olkiluoto, p 216. http:// www.posiva.fi/en/databank/working_reports Koivula H, H€akli P, Jokela J, Buga A., Putrimas R (2011) GPS metrology – bringing traceable scale to local crustal deformation GPS network. (IAG Scientific Assembly “Geodesy for Planet Earth”, Buenos Aires, Argentina, Aug 31–Sept 4) Lassila A,. Jokela J, Poutanen M, Xu J (2003) Absolute calibration of quartz bars of V€ais€al€a interferometer by white light gauge block interferometer. XVII IMEKO World Congress, Dubrovnik, Croatia, pp 1886–1889, 22–27 June 2003. http://www.imeko.org/publications/wc-2003/PWC-2003TC14-026.pdf Ollikainen M, Ahola J, Koivula H (2004). GPS operations at Olkiluoto, Kivetty and Romuvaara in 2002–2003. Working Report 2004–12. Posiva, Olkiluoto, p 268. http://www. posiva.fi/en/databank/working_reports Wallerand J-P, Abou-Zeid A, Badr T, Balling P, Jokela J, Kugler R, Matus M, Merimaa M, Poutanen M, Prieto E, van den Berg S, Zucco M (2008) Towards new absolute long distance measurement systems in air. 2008 NCSL International Workshop and Symposium, Orlando (USA), Aug 2008. http://www.longdistance project.eu/files/towards_ new_absolute.pdf
9
How to Fix the Geodetic Datum for Reference Frames in Geosciences Applications? H. Drewes
Abstract
Geodetic parameters always correspond to a reference system defined by conventions and realized by a reference frame through materialized points with given coordinates. For the coordinate estimation one has to fix the geodetic datum, i.e. the origin and directions of the coordinate axes, and the scale unit. In geosciences applications, e.g. for geodynamics and global change research, the datum has to be fixed over a very long time period in order to refer time-dependent parameters to one and the same reference frame. The paper focuses on the methodology how to fix the datum by parameters independent of the measurements and deformations of the reference frame, and to hold it over a long time span. It is shown that transformations between reference frames at different epochs are not suited to realize the datum parameters because systematic network deformations may affect it. Independent parameters are in particular the first degree and order harmonic coefficients of the gravity field for fixing the origin, and external calibration for fixing the scale. The long-term stability is achieved by the permanent fixing of the datum parameters. Regional reference frames must refer to the global datum by using epoch station coordinates as fiducial values.
9.1
Introduction
The geodetic datum provides the origin, orientation and scale unit of a coordinate system with respect to the body of the Earth. All geodetic parameters refer to a given datum. Coordinates cannot be estimated from geodetic measurements without fixing the datum; there is a rank defect in the observation equation
H. Drewes (*) Deutsches Geod€atisches Forschungsinstitut, Alfons-Goppel-Str. 11, 80539 M€unchen, Germany e-mail:
[email protected]
systems equal to the number of necessary datum parameters (datum defect). In a three-dimensional Cartesian coordinate system, the defect is seven: three to fix the position of the origin, three to fix the orientation of the coordinate axes, and one to fix the scale unit. The datum parameters cannot be measured; they must be given with the definition of the reference system (Latin “datum” ¼ given). If the geodetic datum is not unique or it changes in parameter estimations, the results cannot be compared among each other; e.g. the position coordinates referring to the origin in one location cannot be used together with coordinates referring to an origin in another location.
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_9, # Springer-Verlag Berlin Heidelberg 2012
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H. Drewes
The most challenging task of modern geodesy is the accurate measurement and the reliable representation of parameters of global phenomena and processes within the Earth System for applications and interpretations in geosciences, e.g. geodynamics and global change research. As these processes are very slow and long-lasting, the parameters must refer to one and the same datum over very long time intervals for the unique and unequivocal representation (stability over decades).
9.2
Terminology
In order to avoid misunderstandings we shall use the following nomenclature in this paper: – Datum parameters fix the origin, orientation and the scale unit of a coordinate system. – Network (or reference frame) motions are named shifts, rotations and dilatation given with respect to one and the same geodetic datum. – Network (similarity) transformation parameters provide the translations, orientation changes and scale factor of one network with respect to another one with a different datum. We further shall consider the following facts: – Transformation of one network to another changes the geodetic datum of the transformed network with respect to the original one. – Transformation between two different epochs of a geodetic network with identical datum provides the average motion of the whole network (shifts, rotations and dilatation) and not a datum change. – A deformed network is in general mathematically not similar to the original network, i.e. the deformed network cannot generally be expressed as a similarity transformation of the original one. An example is the change of the flattening of the Earth which requires affine transformations.
9.3
Current Status of Datum Realization
The International Terrestrial Reference System (ITRS) is defined as follows (McCarthy and Petit 2004):
– The origin of the coordinate system is in the Earth’s centre of mass (geocentre). – The orientation of coordinate axes follows the Earth Orientation Parameters of the BIH 1984.0. – The scale unit is the metre consistent with TCG. The International Reference Frame (ITRF) is a realization of the ITRS. In global and regional analyses, solutions are often aligned to the ITRF by transformations of observed station networks to a preexisting reference frame or to a previously observed network, e.g. by the conditions of “no net translation”, “no net rotation”, and/or “no net scale” (NNT, NNR, NNS) (e.g. Thaller 2008; Sa´nchez et al. 2011). Doing so between different epochs of an identical network, the transformation parameters provide the average motion of the network and not different realizations of the datum. The transformation parameters may not be taken as a datum correction. Coordinates transformed in this way cannot be used for applications in geosciences because common motions have gone to the transformation parameters and can no longer be identified in the coordinate differences. Let us demonstrate this effect by two examples: The Earth’s surface undergoes long-term global deformations. The largest vertical deformations are caused by post-glacial rebound (glacial isostatic adjustment), and horizontal ones caused by tectonic processes (plate tectonics, intra- and inter-plate deformations). The station network for observing these deformations is very inhomogeneous (e.g. Collilieux et al. 2009); we have an agglomeration of stations in North America and Europe, and sparse station distribution in other regions such as Africa and Asia (Fig. 9.1). To detect the deformations, we need long-lasting measurements and a reference frame with a stable geodetic datum over decades. Under these prerequisites we discuss the effect of a similarity transformation between the global station network observed at time t and the same network at time t + Dt, and its relevance for geosciences applications.
9.3.1
Vertical Deformation
We start with the example of an uplift caused by postglacial rebound, which produces changes in vertical station coordinates (heights h). If the vertical motions
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How to Fix the Geodetic Datum for Reference Frames in Geosciences Applications?
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Fig. 9.1 ITRF2005 station distribution (http://itrf.ensg.ign.fr)
Fig. 9.2 Vertical motions appear as transformation parameters
are the same in all observation points, they appear as a scale factor (a) in transformation parameters, not as displacements of individual stations (Fig. 9.2, above). If they are different in the northern and southern
hemispheres, they appear as a translation and a scale factor. Due to the inhomogeneous station distribution there may also appear changes in orientation (Fig. 9.2, below), although no network rotation occurred, because the transformation parameters become correlated if the origins of the coordinate systems to be transformed (geocentre) do not coincide with the centres of the networks. What is said here for displacements holds also for velocities in case of their transformation. The described effect can be seen in the results of ITRF2005 computations by different approaches. The computation at Institut Ge´ographique National (IGN), Paris (Altamimi et al. 2007) is based on similarity transformations of the epoch coordinate solutions (weekly from satellite techniques or daily from VLBI) to the combined solution, where the difference in epoch is taken into account by the estimated velocities. The Deutsches Geod€atisches Forschungsinstitut (DGFI), M€unchen, (Angermann et al. 2007, 2009) accumulates datum-free normal equations of the measurements. The comparison of both solutions shows systematic differences. These are predominantly negative in the northern and positive in the southern hemisphere (Fig. 9.3). The differences can be approximated by a sine function of latitude ’ (Fig. 9.4), which is equivalent to the Z-component (DZ ¼ Dh · sin ’). The associated differences in the Z-component are shown in Fig. 9.5. They are – as expected – predominately
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Fig. 9.3 Differences of vertical velocities between ITRF2005 computations at DGFI and IGN
Fig. 9.4 Latitude dependence of vertical velocity differences and fit by a sine function, which indicates a translation of the network in Z-direction
negative with an average of 1.4 mm/a, and show a longitude dependence. There are, however, accumulated effects affecting the Z-component, which will be discussed below. Figure 9.6 shows the longitude dependence of the Z-component of velocity differences. We again see a sine function with even larger amplitude. The interpretation is more difficult, because there are additional effects from horizontal deformations.
9.3.2
Horizontal Deformation
Horizontal deformations in a global scope are mainly caused by plate tectonics. The kinematic datum of the ITRF is defined by the condition of no net rotation with regard to horizontal tectonic motions over the whole
Earth (McCarthy and Petit 2004). It is realized by transforming the geodetic velocities to those derived from plate tectonic models. The ITRF2005 computation at IGN uses the geophysical model NNR NUVEL-1A (DeMets et al. 1990, 1994). This model does not fulfil the NNR condition at present, because it is based on observations over geological time scales, includes only 16 rigid plates, and does not consider the extended deformation zones along the plate boundaries. Therefore, DGFI uses the Actual Plate Kinematic and Crustal Deformation Model (APKIM2005) based on the structure of the geophysical model PB2002 (Bird 2003) with 52 (micro-) plates, the ten major ones being identical with NUVEL-1A, and 13 deformation zones. The plate velocities are computed from present-day geodetic observations (Drewes 2009b) and show significant differences with respect to the geophysical model
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How to Fix the Geodetic Datum for Reference Frames in Geosciences Applications?
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Fig. 9.5 Differences of the Z-component of station velocities from ITRF2005 computations at DGFI and IGN
Fig. 9.6 Longitude dependence of the Z-component of station velocity differences, indicating a shift (1.4 mm/a) and a rotation of the network around an axis in the equatorial plane
(Fig. 9.7). The differences of the horizontal velocities from both ITRF computations based on these models are shown in Fig. 9.8 (mind the different velocity scales of Fig. 9.7 and Fig. 9.8). We clearly see the rotation around an axis from the Northern Atlantic to the Southern Indian Ocean which is nearly identical with the differences between NNR NUVEL-1A and APKIM2005. The global motions of all tectonic plates, which are modelled as rotations on a sphere, have a dominant trend towards northeast (Fig. 9.7). This causes a shift of the whole network with an average dX/dt ¼ 10 mm/a, dY/dt ¼ +13 mm/a, dZ/dt ¼ +15 mm/a (Drewes 2009b). This shift enters completely into the station velocities and does not affect the datum, if the origin of the reference frame is fixed to the geocentre and the NNR condition of the plate model is fulfilled. If we perform a transformation of the discrete geodetic
network with inhomogeneous point distribution to a slightly rotating geophysical plate model (like NNR NUVEL-1A with oX ¼ 0.04 mas/a, oY ¼ +0.03 mas/a, oZ ¼ 0.02 mas/a.), then the residual rotations appear also in the translation transformation parameters. The effect can be seen in the systematic change of all the ITRF translation parameters and its rates (Altamimi et al. 2002, 2007), in particular in the Y and Z components.
9.3.3
Regional Deformation
The examples presented so far refer to effects of global deformations on the global datum realization. Regional reference frames are normally defined as densifications of the ITRF, i.e. they are subject to the
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H. Drewes
Fig. 9.7 Comparison of the geophysical plate kinematic model PB2002 and the geodetic model APKIM2005
Fig. 9.8 Differences of horizontal velocities between ITRF2005 computations at DGFI and IGN
same effects. In addition, there are regional deformations due to tectonic, isostatic, sedimentary, atmospheric, hydrospheric, and other processes. Large effects are caused by seasonal loading. Figure 9.9 shows height variations of stations distributed over all the South American continent (from ’ ¼ 5 to ’ ¼ 35 ) computed from observations in the South
American reference frame (SIRGAS, Sa´nchez et al. 2011; Seem€uller et al. 2009). They clearly demonstrate a systematic periodic (seasonal) variation up to 2 cm. The datum of a regional reference frame is often realized by NNR, NNT, and NNS (i.e. similarity) transformations of the regional network to ITRF
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How to Fix the Geodetic Datum for Reference Frames in Geosciences Applications?
73
KOUR ( =5°, =-53°)
IMPZ ( =-5°, = - 47°)
AREQ ( =-16°, = - 71°)
CIUB ( =-16°, = -56°)
BRAZ ( =-16°, = - 48°)
LPGS ( =-35°, = -58°)
Fig. 9.9 Station height variations in South America [cm]
(or IGS) coordinates extrapolated with linear velocities from the ITRF (or IGS) reference epoch to the regional reference epoch. If the reference stations do not move linearly (Fig. 9.9), the extrapolated coordinates do not refer to actual geocentric station positions at every time. We get then the same type of effects as described in chapter 3, and we cannot use coordinate variations
estimated in this way for geosciences applications. To avoid this effect, the reference frames in some regional projects are realized by transforming the epoch (weekly) coordinates to the weekly IGS solutions (Craymer et al. 2007; Sa´nchez et al. 2011). This procedure accounts for the regional seasonal effects, but still includes the global motions of the network (secularly
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due to post glacial rebound, seasonally due to different atmosphere and hydrosphere loads in the northern and southern hemispheres).
9.4
How to Fix the Geodetic Datum?
9.4.1
Fixing the Origin
The realization of the geodetic datum by similarity transformation to other networks or reference frames is not a suitable approach, because network deformations in the reference stations selected for the transformation change the datum definition. Fixing the origin to the geocentre must be done by a gravimetric approach as a network-independent method (Drewes 2009a). The Earth’s centre of mass is defined by (M ¼ total mass of the Earth) X0 X dm=M Y0 Y dm=M Z0 Z dm=M:
9.4.2 (9.1)
The first degree and order spherical harmonic coefficients of the Earth’s gravity field express the position of the geocentre with the semi-major axis a as a scaling factor: C11 X dm=a M S11 Y dm=a M C10 Z dm=a M:
do not change the geodetic datum. They must be interpreted as orbit errors and be reduced by improved orbit modelling. The geocentric coordinates of the ground stations are derived from the geocentric coordinates of the satellites using, e.g. SLR distance measurements. In differential approaches, like double differencing GNSS or Doppler methods, most of the relation to the geocentre is lost in the differencing. It can therefore not strictly be realized. As a consequence, SLR measurements should be included for any datum realization, also in regional reference frames. The fixing of the geocentre through the gravity field holds for any time. Thus it is clear that there is no time evolution of the origin. It remains always in the geocentre. If it is realized in this way, it guarantees the long-term stability of the reference frame for geosciences applications.
(9.2)
Using a gravity field model with C11¼S11¼C10¼0 in the coordinate estimation with dynamic satellite methods, as customary in satellite geodesy, fixes the origin of the coordinate system automatically to the geocentre. Satellite orbits are always geocentric unless additional constraints are introduced. Such constraints may enter by fixing the coordinates of tracking stations or an average of them (e.g. NNT constraints). Therefore, global orbit computations for reference frame determinations must not fix any terrestrial coordinates; the necessary minimum datum constraint is given by the gravity field. Wrong modelling of other gravitational or non-gravitational forces (e.g. solar radiation pressure) introduces additive constraints, too. They cause systematic errors in the orbits, but
Fixing the Orientation
The orientation of the coordinate system could also be realized by gravimetric methods through the principal axes of inertia, which are expressed by the second degree spherical harmonics of the gravity field (C21, S21, S22). These coefficients, however, can at present not be determined with the required accuracy. This may change by including data from future gravity field and global navigation satellite missions. Until these are available, the orientation must further be fixed conventionally (e.g. BIH84). The time evolution of the orientation of the coordinate system is characterized by the high correlation between station velocities and Earth Orientation Parameters (EOP). Systematic velocity changes may interchange with EOP. It is therefore indispensable to estimate velocities and EOP simultaneously in a global adjustment, and to use a present-day no-netrotation constraint by models derived from geodetic measurements instead of geologic-geophysical models. The orientation resulting from network transformations refers then to the same coordinate axes in velocities and EOP.
9.4.3
Fixing the Scale
The scale unit has to be fixed to the definition of the unit of length (metre) by calibrating the measuring
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How to Fix the Geodetic Datum for Reference Frames in Geosciences Applications?
instruments and reduction of atmospheric effects. There is no time evolution of the scale; it is always the same unit of length. We must never perform a similarity transformation between networks at different epochs with a scale factor as a parameter. This would only absorb the network dilatation and exclude geosciences interpretations (e.g. to prove or disprove the expansion theory of the Earth, or to detect global postglacial rebound). Conclusions
If we realize the geodetic datum by network transformations, we change the datum between the original and the transformed reference frame. A geocentric frame is then loosing the geocentricity. This behaviour is often discussed as geocentre motion. Doing so, one has to keep in mind that it is a motion w. r. t. the always deforming crust-based reference frame. There is no stable reference. A consequence is that the non-geocentric reference frame is no longer consistent with the satellite orbit computations, which normally use a geocentric gravity field (C11 ¼ S11 ¼ C10 ¼ 0). One would then have to re-compute all SLR, GPS and DORIS orbits with the changed lower spherical harmonic coefficients. For users of such a moving reference frame, one would have to provide the corresponding time-dependent gravity fields together with the station coordinates. It is by far more practicable to keep the origin fixed in the geocentre and to include all the global motions in the station velocities. Experiences with inconsistent global reference frames were reported, e.g. from satellite altimetry studies on global sea level changes (e.g. Beckley et al. 2007; Morel and Willis 2005). There are considerable differences in the interpretation of results when using different ITRF realizations as a reference frame. The datum of regional reference frames has strictly to be realized as a densification of the global reference frame in order to be consistent with the satellite orbits. True coordinates (ITRF or IGS) have to be used as fiducial values, and not the coordinates extrapolated from a reference epoch with linear velocities. In epoch solutions (e.g. weekly), the coordinates of the same epoch have to be taken from the superior frame. As the global
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network may undergo a common motion w. r. t. the geocentre, one has to fix the geocentric datum by external gravimetric parameters (C11 ¼ S11 ¼ C10 ¼ 0). This should be done by including always station coordinates derived from SLR in a combined network adjustment.
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(B10) 2214:19. doi:10.1029/2001JB000561 Altamimi Z, Collilieux X, Legrand J, Garayt B, Boucher C (2007) ITRF2005: a new release of the International Terrestrial reference frame based on time series of station positions and Earth Orientation Parameters. J Geophys Res 112:19. doi:10.1029/2007JB004949, B09401 Angermann D, Drewes H, Kr€ugel M, Meisel B (2007) Advances in terrestrial reference frame computations, vol 130, IAG Symposia. Springer, Heidelberg, pp 595–602 Angermann D, Drewes H, Gerstl M, Kr€ugel M, Meisel B (2009) DGFI combination methodology for ITRF2005, vol 134, IAG Symposia. Springer, Heidelberg, pp 11–16 Beckley BD, Lemoine FG, Luthcke SB, Ray RD, Zelensky NP (2007) A reassessment of global and regional mean sea level trends from TOPEX and Jason-1 altimetry based on revised reference frame and orbits. Geophys Res Lett 34:14608. doi:10.1029/2007GL030002, 5pp Bird P (2003) An updated digital model for plate boundaries. G3 – Geochem Geophys Geosyst 4(3):52. doi:1010.1029/ 2001GC000252 Collilieux X, Altamimi Z, Ray J, Van Dam T, Wu X (2009) Effect of the satellite laser ranging network distribution on geocenter motion estimation. J Geophys Res 114:B04402. doi:10.1029/2008JB005727, 17pp Craymer MR, Piraszewski M, Henton JA (2007) The North American Reference Frame (NAREF) project to densify the ITRF in North America. Proceedings ION GNSS 20th International Technical Meeting of the Satellite Division, pp 2145–2154 DeMets C, Gordon RG, Argus DF, Stein S (1990) Current plate motions. Geophys J Int 101:425–478 DeMets C, Gordon R, Argus DF, Stein S (1994) Effect of recent revisions to the geomagnetic reversal time scale on estimates of current plate motions. Geophys Res Lett 21:2191–2194 Drewes H (2009a) Reference systems, reference frames, and the geodetic datum – basic considerations, vol 133, IAG Symposia. Springer, Heidelberg, pp 3–9 Drewes H (2009b) The Actual Plate Kinematic and crustal deformation Model (APKIM2005) as basis for a non-rotating ITRF, vol 134, IAG Symposia. Springer, Heidelberg, pp 95–99 McCarthy DD, Petit G (2004) IERS Conventions 2003. IERS Technical Note No. 32 Morel L, Willis P (2005) Terrestrial reference frame effects on sea level rise determined by TOPEX/Poseidon. Adv Space Res 36:358–368. doi:10.1016/j.asr.2005.05.113
76 Sa´nchez L, Seem€uller W, Seitz M (2011) Combination of the weekly solutions delivered by the SIRGAS Processing Centres for the SIRGAS-CON Reference Frame. In: Kenyon S et al (eds) Geodesy for Planet Earth, IAG Symposia. Springer, Heidelberg Seem€uller W, Seitz M, Sa´nchez L, Drewes H (2009) The position and velocity solution SIR09P01 of the IGS Regional
H. Drewes Network Associate Analysis Centre for SIRGAS (IGS RNAAC SIR). DGFI Report No. 85 Thaller D (2008) Inter-technique combination based on homogeneous normal equation systems including station coordinates, Earth orientation and troposphere parameters. GFZ Potsdam Sci Rep STR08/15
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Transforming ITRF Coordinates to National ETRS89 Realization in the Presence of Postglacial Rebound: An Evaluation of the Nordic Geodynamical Model in Finland P. H€akli and H. Koivula
Abstract
The IAG Reference Frame Sub-Commission for Europe (EUREF) created the European Terrestrial Reference System 89 (ETRS89) and fixed it to the Eurasian plate in order to avoid time evolution of the coordinates due to plate motions. However, the Fennoscandian area in Northern Europe is affected by postglacial rebound (PGR), causing intraplate deformations with respect to the stable part of the Eurasian tectonic plate. The Nordic countries created their national ETRS89 realizations in the 1990s and have adopted them as the basis for geospatial data. As the most accurate GNSS processing is done in ITRS realizations, an accurate connection to national ETRS89 realizations is required. If the official EUREF transformation is used, residuals are up to 10 cm in the Nordic countries. Therefore, the Nordic Geodetic Commission (NKG) has created a 3-D intraplate velocity model NKG_RF03vel over Fennoscandia and a new transformation procedure to correct for the deformations caused by PGR. This paper evaluates the NKG approach and compares it to the current recommendation given by EUREF with a 100-point ETRS89 realization in Finland. The results show that, by using a high-quality intraplate velocity model, the transformation residuals are reduced to the cm-level.
10.1
Introduction
National reference frames are usually fixed to a global or regional terrestrial reference frame (TRF) that in most cases is one of the ITRS realizations, ITRFyy. ITRF solutions are the most accurate global realizations, but the coordinates are also time-
P. H€akli (*) H. Koivula Department of Geodesy and Geodynamics, Finnish Geodetic Institute, Geodeetinrinne 2, 02430 Masala, Finland e-mail:
[email protected]
dependent due to site velocities. Site velocities are needed if the Earth’s dynamics (e.g., plate tectonic motion) are to be taken into account at the cm-level. Time variable coordinates, however, are not useful for geodetic practice. To avoid time evolution of the coordinates, the IAG Reference Frame Sub-Commission for Europe, EUREF, has created the European Terrestrial Reference System 89 (ETRS89), which is fixed to the stable part of the Eurasian tectonic plate and coincides with the ITRS at the epoch 1989.0 (Boucher and Altamimi 1992). However, the Fennoscandian area in Northern Europe is affected by postglacial rebound (PGR), causing intraplate
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78
deformations. The uplift part of the PGR in Fennoscandia has been observed for centuries, first with tide gauges, later with repeated levellings and gravity observations and, recently, with permanent GPS stations as well. Intraplate deformations in Europe and Fennoscandia have been studied, for example, by Nocquet et al. (2001, 2005), Johansson et al. (2002) and Lidberg et al. (2007). The land uplift has a maximum of approx. 10 mm/year (from the reference ellipsoid). The horizontal part of the PGR, from the time series of permanent GPS stations, is up to 2.5 mm/year. These deformations can be seen, for example, in EPN (EUREF Permanent Network) velocities expressed in ETRS89, see Figs. 10.1 and 10.2. The figures were plotted from the EPN cumulative solution up to GPS week 1,540 of class A stations (EPN 2010). Class A stations are categorized as highquality fiducial stations meaning that the velocity estimates are known better than 0.5 mm/yr (Kenyeres 2010). The Nordic countries created their national ETRS89 realizations in the 1990s using the ITRF of that time (Denmark ITRF92, Norway ITRF93, Finland ITRF96 and Sweden ITRF97). The ETRS89 realizations have been adopted as the basis for geospatial data. The Finnish ETRS89 realization, called EUREF-FIN, was measured in 1996–1997 and realized through ITRF96(1997.0) with official
formulas provided by EUREF (Boucher and Altamimi 2008), yielding ETRF96 coordinates in 1997.0. The formulas correct the rigid plate motion to epoch 1989.0, but for intraplate deformations the epoch remains in 1997.0, which can be considered the reference epoch for EUREF-FIN. Since the most accurate GNSS processing has to be done in adequate ITRFyy, accurate transformations between ITRF and national ETRS89 realizations are needed. The connection between different threedimensional reference frames is generally presented with a linear 7-parameter similarity transformation. EUREF provides a 14-parameter similarity transformation approach (including time evolution of the parameters) from ITRS to ETRS89 that takes into account the rigid plate motion of the Eurasian tectonic plate (Boucher and Altamimi 2008). In the context of a GNSS campaign, EUREF does not recommend using velocity models. However, any similarity transformation alone cannot depict the influence of PGR since it deforms the crust unequally. Therefore, the EUREF transformation does not describe discontinuities and intraplate deformations; they remain as transformation residuals. In the Fennoscandian area, the EUREF approach is therefore not cm-level accurate when transforming from present ITRF to national ETRS89 realizations. The residuals are already up to 10 cm within a 10-year time span.
340
˚ 35 40˚ 0˚ 0˚ 10˚ 20˚ 30˚
70 ˚
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10mm/yr
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mm/yr 10
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0 2
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˚ 350˚
4
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Fig. 10.1 Horizontal velocity field in ETRF2000 (EPN 2010)
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© FGI/PH 2010
10
Transforming ITRF Coordinates to National ETRS89 Realization in the Presence
ITRF2000 to ETRF96 is done in two steps. The first step transforms ITRF2000(tc) coordinates to ITRF96(tc), keeping the epoch of coordinates untouched. The transformation parameters are given at reference epoch t0 (for ITRF2000 t0 ¼ 1997.0) and the parameters have to be converted to the epoch of observations (tc) with the change rates of the parameters. In the second step, ITRF96(tc) coordinates are transformed to ETRF96(tc). Since no intraplate or epoch correction is recommended, the epoch of coordinates remains at tc.
To overcome this problem, the Nordic Geodetic Commission (NKG) has created a 3-D velocity model, NKG_RF03vel, and a procedure that also takes into account intraplate deformations. This study evaluates the NKG approach and compares it to the transformation recommended by EUREF. Transformations were evaluated using a 100-point network that is defining the Finnish ETRS89 realization, EUREF-FIN. EUREF-FIN was originally measured in 1996–1997 and the same network was re-measured in 2006. The re-measurement campaign was processed with Bernese 5.0 and the coordinates for fiducial stations at the central epoch of observations, tc ¼ 2006.5, were obtained from the official IERS ITRF2000 solution (ITRF 2010a). The resulting campaign coordinates were transformed using EUREF and NKG approaches and compared to the official EUREF-FIN coordinates.
10.2
10.2.2 Official NKG Transformation The NKG transformation was done according to the recommendations given by the NKG working group for positioning and reference frames (NKG WG). The NKG transformation takes intraplate deformations into account using velocity model NKG_RF03vel. The model corrects both horizontal and vertical intraplate deformations. The horizontal part originates from the GIA model by Milne et al. (2001) that was rotated to the GPS-derived velocities by Lidberg et al. (2007) (Fig. 10.3). The vertical part, NKG2005LU (ABS) model by NKG working group for height determination, is constructed from tide gauge, levelling and
Transformation Approaches
10.2.1 Official EUREF Transformation The EUREF transformation was made according to the recommendation and parameters given by Boucher and Altamimi (2008). The transformation from 340
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8
5 4 2
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2
50 ˚
40 ˚
12
4
55 ˚
˚
35 −2
0 −2
−2
˚ 350˚
0˚
10˚
Fig. 10.2 Vertical velocity field in ETRF2000 (EPN 2010)
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340
14
6
6
12
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65
6
8
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© FGI/PH 2010
P. H€akli and H. Koivula
80
72˚
0˚ 4˚ 8 36˚ 4 ˚ 12˚ 16 ˚ 20˚ 24˚ 28˚ 32˚
70˚
72˚ mm/yr 2.5
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58˚ 56˚
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1mm
4˚
0.5
58˚
8˚
12˚
16˚
20˚
24˚
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36˚
0.0
54˚ 40˚
Fig. 10.3 NKG_RF03vel model for horizontal intraplate deformations in Fennoscandia (Nørbech et al. 2006)
permanent GPS data (Fig. 10.4) (Nørbech et al. 2006; Lidberg 2008). The Nordic ETRS89 realizations are based on different ITRF solutions and reference epochs leading up to discrepancies of a few cm. In order to take into account these discrepancies, a common reference frame was needed. The common Nordic frame, NKG_RF03, was realized with a NKG2003 GPS campaign and computed in ITRF2000 at epoch 2003.75. The complete description of NKG_RF03 is given by Jivall et al. (2005). The transformation is determined through the NKG_RF03 frame in order to determine the transformation for each Nordic country. For Finland, the transformation consists of three steps: 1. ITRF2000(tc) ! ITRF2000(2003.75) (NKG_ RF03) 2. ITRF2000(2003.75) ! ITRF2000(2003.75)1997.0 (¼intraplate corrected to 1997.0) 3. ITRF2000(2003.75)1997.0 ! EUREF-FIN The first step is to transform ITRF2000 coordinates at the epoch of observations to ITRF2000 at epoch 2003.75 in which the NKG_RF03 is expressed. The
rigid plate motion during the period is reduced with the ITRF2000 rotation pole for the Eurasian plate (given by Boucher and Altamimi 2008). In addition to the rigid plate motion, also intraplate deformations during the period have to be taken into account in order to simulate the ITRF2000 coordinates at epoch 2003.75 accurately. For this purpose the NKG_RF03vel intraplate velocity model is used. In the second step, the resulting coordinates in ITRF2000(2003.75) are further corrected for intraplate deformations to the reference epoch (tr) of the national ETRS89 realization, in Finland tr ¼ 1997.0. This step corrects the internal geometry of the network at the epoch 2003.75 to the one at the reference epoch of EUREF-FIN. In the third step these intraplate corrected ITRF2000(2003.75) coordinates (we are using notation ITRF2000(2003.75)1997.0) are transformed to EUREF-FIN with a 7-parameter similarity transformation. The parameters were determined between the intraplate corrected NKG_RF03(tr) coordinates and the Nordic ETRS89 realizations by the NKG WG.
10
Transforming ITRF Coordinates to National ETRS89 Realization in the Presence 0˚ 8˚ 12 6˚ 4 ˚ 16˚ 20˚ 24˚ 28˚ 32˚ 3
4˚
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81
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0
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1
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9
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mm/yr 10
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0
4˚
8˚
12˚
16˚
−1
20˚
24˚
28˚
32˚
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40˚
Fig. 10.4 NKG_RF03vel model for vertical intraplate deformations in Fennoscandia (Nørbech et al. 2006)
The standard error of unit weight for the parameter estimation with 12 fitting points in Finland was 1.6, 1.5 and 3.2 mm for North, East and the up component, respectively. The formulas, parameters and more detailed explanation are given by Nørbech et al. (2006).
10.3
Evaluation of the Transformations
10.3.1 EUREF Transformation Residuals of the official EUREF transformation for the vertical coordinates are up to 7 cm (Fig. 10.6) and show the neglected effect of the PGR between 1997.0 and 2006.5 (compare to the contours of the land uplift model NKG_RF03vel, Fig. 10.4). The horizontal residuals of the transformation (Fig. 10.5) have approximately the same direction, but there is a large difference in magnitude between Southern and
Northern Finland. The residuals vary from a couple of mms to 3 cm (see also Table 10.1). In order to understand this feature, and despite the fact that the EUREF does not recommend using intraplate velocities in the context of a GPS campaign, we subtracted the effect of PGR between 1997.0 and 2006.5 from the transformed coordinates. After removing the effect of PGR, horizontal residuals are more equal in size and direction throughout Finland. All residuals are approximately 15–20 mm and the direction is N-NW (and slightly rotating from the North to the West when moving from Southern to Northern Finland). This bias likely originates from different plate models used in ITRF2000 and ITRF96. Altamimi et al. (2002) report a significant disagreement between ITRF2000 velocities and those predicted by NNR-NUVEL-1A that were used in ITRF96. The difference between ITRF2000PMM (plate motion model) and NNR-NUVEL-1A, illustrated by Altamimi and Boucher (2002) and
P. H€akli and H. Koivula
82 Fig. 10.5 Horizontal residuals of official EUREF transformation
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30˚
28˚
70˚
JOLL
NUHP
70˚
TSUA
ERTE KEVO
TSHU
KARR
20
OTSA JAKA
JERS
NUNA
20 25
68˚
TSAR KAUN
KORS
68˚
KORV KANG
mm
KOIT
25
KUMP
ROUT
AALI KIRI
50 45
20
25
SODA ORAT
TAKK
ISOP
HYYP
40
HOPI
66˚
VAMM
KAAK
KARI
20
HERV KEMI
10mm
PYHI
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KUUS IIVA
35
LATV PALJ
30
10
PETA
OULU VIIN PITK
PESI
25
15
RAAH
15
20
KEST
64˚
ROMU
NISU KALL
64˚
PIHJ
KATI JAUH
10
TAIN
10
KOLI
HONK PETT
KIVE
5
10
PASS
ISOL
KASK
TAHK
10
LEHT VAAS
15
5
PARN
KOTA
NUMM
JOEN
5
62˚ SAMM
PUOL
SIIK
5
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5
KAIP KUVA
0
SAUK
ISOV POHJ
LAAJ
OLKI
MUST
HEVO
GETA
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5
VAAT
KARH
PORL
PERN
DEGR KOKA
DRAG
YLIJ KYMI VIRO
ROKO
TUOR
METS HELS BRED
HARK
60˚
HANK
© FGI/PH 2009
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Table 10.1 Statistics of transformation residuals from ITRF2000 to EUREF-FIN with different approaches (mm) n ¼ 95 Mean Std Rms Min Max
EUREF N 10.8 8.0 13.4 3.3 26.0
E 1.6 5.5 5.7 13.9 8.6
U 37.7 18.8 42.1 1.1 75.4
NKG-1 N 3.7 4.1 5.5 5.5 14.6
Altamimi et al. (2003), shows a strong agreement with our horizontal residuals after intraplate correction for the Fennoscandian area. Vertical residuals are reduced to a couple of cm when intraplate deformations were removed.
E 3.0 3.5 4.6 4.9 10.8
U 16.9 9.5 19.4 39.8 0.0
NKG-2 N 2.2 4.0 4.5 8.3 12.1
E 0.7 3.3 3.3 7.0 9.0
U 3.5 8.1 8.8 22.2 15.0
10.3.2 NKG Transformation The ITRF2000(2006.5) coordinates were transformed according to the official NKG transformation (solution labelled NKG-1). The solution includes intraplate
10
Transforming ITRF Coordinates to National ETRS89 Realization in the Presence
Fig. 10.6 Vertical residuals of official EUREF transformation
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83
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ERTE KEVO JOLL
NUHP
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TSHU
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KARR
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JAKA
JERS
TSAR KAUN
5KORS 0
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KORV KANG
mm
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KUMP
40 SODA ORAT
TAKK
ROUT
AALI KIRI
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ISOP
HYYP
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HOPI
VAMM
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KAAK
KARI
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HERV
KEMI
PYHI
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LATV PETA
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50
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PALJ
OULU VIIN 60PITK
PESI
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RAAH KEST
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NISU KALL
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KATI
JAUH
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20TAIN
LEHT VAAS
KOLI
HONK
PETT
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KIVE ISOL
NUMM
30 SAMM
PUOL
SIIK
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KOTA
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TAHK
KAIP KUVA
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100 90 80 70 60 50 40 30 20 10 0 −10 −20 −30 −40 −50 −60 −70 −80 −90 −100
SAUK
ISOV
10LAAJ
POHJ OLKI
MUST
VAAT
GETA
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HEVO
DEGR KOKA
PORL
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METS HELS BRED
YLIJ KYMI VIRO
ROKO
TUOR
HARK
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HANK
© FGI/PH 2009
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corrections of approx. 3–9 mm/year vertical and up to 2 mm/year horizontal velocities in Finland. The residuals that are at the level of a few cm are clearly smaller than those of the EUREF transformation without intraplate corrections. The horizontal residuals have a random nature, but the vertical residuals are systematically biased by an average of 16.9 mm (see Table 10.1). This is caused mainly by biased NKG_RF03 coordinates. The NKG2003 campaign was computed in ITRF2000(2003.75), but in Finland the vertical coordinates of NKG_RF03 differ approx. 13 cm from the ITRF2000 coordinates computed with positions and velocities published by the IERS (these were used in evaluation). The difference is of
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the same magnitude as the residuals (however, comparison was possible only at 4 Finnish EPN stations: METS, JOEN, VAAS and SODA). The NKG2003 campaign and resulting NKG_RF03 frame was computed using IGS00 cumulative solution. The NKG_RF03 was processed using GIPSY-OASISII, GAMIT/GLOBK and Bernese software and the final solution is a combination of these solutions. Constraining to ITRF2000 was done with GIPSY and GAMIT/GLOBK solutions, meaning that the NKG2003 campaign has a global alignment, through transformation parameters, to ITRF2000 (Jivall et al. 2005). The bias is partly caused by the cumulative IGS00 coordinates, but in Finland these are equal within a
P. H€akli and H. Koivula
84 Fig. 10.7 Horizontal residuals of NKG2 transformation
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70˚ ERTE KEVO JOLL
NUHP
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TSHU
5
KARR
OTSA NUNA
JAKA
JERS
TSAR KAUN
KORS
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KORV
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KUMP
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TAKK
ROUT
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AALI KIRI
ISOP
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HYYP
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HOPI
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KARI
KEMI
HERV
PYHI
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LATV
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10mm
10
10
KAAK
PALJ
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PETA
OULU VIIN PITK
PESI
25
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RAAH
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KEST
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ROMU
NISU KALL
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PIHJ
KATI
5 LEHT
VAAS
TAHK
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KOLI
KIVE ISOL
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PARN
KOTA
NUMM
JOEN
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HONK
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PETT
PASS KASK
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JAUH
SAMM
PUOL
SIIK
KAIP 10KUVA
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ISOV
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SAUK
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POHJ
LAAJ
OLKI
MUST
HEVO VAAT GETA
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PORL
PERN
DEGR KOKA
DRAG
YLIJ KYMI VIRO
ROKO
TUOR
METS HELS BRED
HARK
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© FGI/PH 2009
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few mm to the IERS-published ITRF2000 coordinates at epoch 2003.75. Instead, aligning the regional solution with transformation parameters to global ITRF2000 may lead to a biased solution since this method is sensitive to station selection (network effect), see e.g., Altamimi (2003) and Legrand and Bruyninx (2008). The ITRF positions and velocities published by the IERS are widely used, even if the positions usually need to be extrapolated outside of the temporal extent of the data used in the ITRF solution. Therefore, misaligned NKG_RF03 coordinates cannot be used for transformation, even if the solution itself is accurate. As a consequence, either new parameters from
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the IERS-published coordinates to national realization or an additional step from NKG_RF03 to ITRF2000 (2003.75) is needed. We solved for new national transformation parameters using the IERS-published positions and velocities. Figures 10.7 and 10.8 show the residuals for this solution (labelled NKG-2). This corrects most of the vertical bias in the NKG-1 solution. The accuracy (rms) of this transformation is better than 1 cm. The residuals of different transformations are summarized in Table 10.1. The NKG-2 transformation was also evaluated with ITRF2005, using only permanent FinnRef stations. The transformation has an additional step from ITRF2005(tc) to ITRF2000(tc) that was made
10
Transforming ITRF Coordinates to National ETRS89 Realization in the Presence
Fig. 10.8 Vertical residuals of NKG-2 transformation
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85
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NUHP
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TSUA
−10
ERTE KEVO JOLL
TSHU
KARR OTSA
0
NUNA
JAKA
JERS
−1
0
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TSAR KAUN
KORS
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KORV
KANG
SODAORAT
mm
0
KOIT
KUMP
TAKK
ROUT
AALI KIRI
ISOP
HYYP HOPI
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0
VAMM
−10KARI
−10
KAAK
PYHI
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KUUS IIVA
−10
0
KEMI
HERV
LATV PALJ
PETA
OULU VIIN PITK 1
0
PESI
0
RAAH
NISU
−1
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−10
KEST
0
KALL
ROMU
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PIHJ
KATI
−1 0
−10
JAUH
0
−1
PASS
KASK
PETT
KIVE
−2
0
PARN
KOTA
ISOL
JOEN
SAUK
0
ISOV POHJ
−10
LAAJ HEVO
VAAT
60˚
KARH 0
PERN DRAG
MUST
0 YLIJ
KYMI VIRO
ROKO
TUOR
DEGR KOKA
PORL
10
GETA
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0
0
PUOL
SIIK
KAIP KUVA
−1
−2
SAMM
TAIN
KOLI
HONK
NUMM
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OLKI
TAHK
0
0
LEHT
VAAS
100 90 80 70 60 50 40 30 20 10 0 −10 −20 −30 −40 −50 −60 −70 −80 −90 −100
METS HELS BRED
HARK
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HANK
© FGI/PH 2009
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Table 10.2 Comparison of ITRF2005/ITRF2000 transformations to EUREF-FIN (mm) n ¼ 12 Mean Stdev Rms Min Max
ITRF2000(2006.50) N E U 1.1 1.9 1.8 2.7 2.3 5.2 2.8 2.9 5.3 3.0 2.0 12.0 4.7 5.2 3.8
ITRF2005(2008.56) N E U 1.3 4.3 3.1 4.2 1.3 7.6 4.2 4.4 7.9 11.7 7.1 18.6 3.1 2.4 7.1
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plate motion with the ITRF2000 rotation pole and intraplate deformations with the NKG_RF03vel model. Table 10.2 summarizes the residuals for 12 permanent FinnRef stations. The accuracy of the ITRF2005 transformation was slightly worse, as expected. However, an accuracy of 1 cm (rms) can also be achieved from ITRF2005 to EUREF-FIN.
10.4 according to EUREF’s MEMO. The IERS-published ITRF2005 coordinates were used for the fiducial station METS (ITRF 2010b). The epoch of ITRF2005 coordinates was 2008.56, so the transformation includes an additional 2.06 year correction for rigid
24˚
Conclusions and Discussion
We have studied several ways of transforming ITRF coordinates back to national ETRS89 realization in Finland, where the reference frame is under the influence of postglacial rebound (PGR). The official
86
EUREF transformation led to transformation residuals up to 7 cm in vertical and up to 3 cm in horizontal components. The pattern of vertical residuals is consistent with the effect of PGR and the magnitude is too large for accurate geodetic purposes. The NKG has created a common reference frame NKG_RF03, expressed in ITRF2000 at epoch 2003.75, and a transformation through it back to the Nordic ETRS89 realizations. The official NKG transformation includes a high-quality intraplate velocity model for Fennoscandia and corrects most of the PGR effect. However, a bias of 16.9 mm was found in vertical residuals and was identified as being the consequence of a network effect. NKG_RF03 was created using minimum constraints approach that is known to be sensitive for site selection. By using IERS-published official ITRF2000 coordinates instead of NKG_RF03, this bias was greatly reduced. This NKG-2 solution gives transformation residuals below 5 mm for horizontal and 8.8 mm (rms) for vertical coordinates, which is acceptable for most geodetic purposes. The main outcome of the study is that there are several influencing factors (e.g., plate rigid rotation, intraplate deformations and constraining approach to the ITRF) that need to be taken into account appropriately in order to get cm-level transformation accuracies back to national ETRS89 realizations. Our results verify that we need a velocity model to map coordinates with cm-level accuracy from the current ITRF to the Finnish realization of ETRS89. However, guidelines for using such models are missing or their use is not recommended. There are several different ways to implement such models. Therefore, guidelines for incorporating deformation models in a unified and standardized way or even a pan-European velocity model would be desirable.
References Altamimi Z (2003) Discussion on how to express a regional GPS solution in the ITRF, EUREF Publication No. 12. Verlag des Bundesamtes f€ ur Kartographie und Geod€asie, Frankfurt am Main, pp 162–167 Altamimi Z, Boucher C (2002) The ITRS and ETRS89 Relationship: New Results from ITRF2000. EUREF Publication No. 10. Verlag des Bundesamtes f€ ur Kartographie und Geod€asie, Frankfurt am Main, pp 49–52 Altamimi Z, Sillard P, Boucher C (2002) ITRF2000: a new release of the International Terrestrial Reference Frame for
P. H€akli and H. Koivula earth science applications. J Geophys Res 107(B10):2214. doi:10.1029/2001JB000561 Altamimi Z, Sillard P, Boucher C (2003) The impact of a nonet-rotation condition on ITRF2000. Geophys Res Lett 30 (2):1064. doi:10.1029/2002GL016279, 2003 Boucher C, Altamimi Z (1992) The EUREF Terrestrial Reference System and its First Realization. Report on the Symposium of the IAG Subcommission for the European Reference Frame (EUREF) held in Florence 28–31 May 1990. Ver€offentlichungen der Bayerischen Kommission Heft 52, M€unchen Boucher C, Altamimi Z (2008) Memo: specifications for reference frame fixing in the analysis of a EUREF GPS campaign. Version 7: 24-10-2008 EPN (2010) EPN cumulative solution GPS weeks 860–1540. ftp://epncb.oma.be/pub/station/coord/EPN/ EPN_A_ETRF2000_C1540.SSC. Accessed 16 Mar 2010 ITRF (2010a) Primary ITRF2000 solution. http://itrf.ensg. ign. fr/ITRF_solutions/2000/sol.php. Accessed 16 Mar 2010 ITRF (2010b) ITRF2005. http://itrf.ensg.ign.fr/ITRF_solutions/ 2005/ITRF2005.php. Accessed 16 Mar 2010 Jivall L, Lidberg M, Nørbech T, Weber M (2005) Processing of the NKG 2003 GPS Campaign. LMV-rapport 2005:7, Reports in Geodesy and Geographical Information Systems, G€avle 2005. Available at http://www.lantmateriet.se/templates/ LMV_Page.aspx?id¼2688. Accessed 16 Mar 2010 Johansson JM, Davis JL, Scherneck H-G, Milne GA, Vermeer M, Mitrovica JX, Bennett RA, Jonsson B, Elgered G, Elo´segui P, Koivula H, Poutanen M, R€onn€ang BO, Shapiro II (2002) Continuous GPS measurements of postglacial adjustment in Fennoscandia 1. Geodetic results. J Geophys Res 107(B8):2157. doi:10.1029/2001JB000400 Kenyeres A (2010) Categorization of permanent GNSS reference stations. Bolletino di Geodesia e Scienze Affini Legrand J, Bruyninx C (2008) EPN Reference Frame Alignment: Consistency of the Station Positions. EUREF 2008 Symposium, Brussels, Belgium, 18–21 June 2008 Lidberg M (2008) Geodetic Reference Frames in Presence of Crustal Deformations. Integrating Generations, FIG Working Week 2008, Stockholm, Sweden, 14–19 June 2008 Lidberg M, Johansson JM, Scherneck H-G, Davis JL (2007) An improved and extended GPS-derived 3D velocity field of the glacial isostatic adjustment (GIA) in Fennoscandia. J Geodesy 2007(81):213–230. doi:10.1007/s00190-006-0102-4 Milne GA, Davis JL, Mitrovica JX, Scherneck H-G, Johansson JM, Vermeer M, Koivula H (2001) Space-geodetic constraints on glacial isostatic adjustments in Fennoscandia. Science 291:2381–2385 Nocquet J-M, Calais E, Altamimi Z, Sillard P, Boucher C (2001) Intraplate deformation in western Europe deduced from an analysis of the ITRF97 velocity field. J Geophys Res 106 (B6):11239 Nocquet J-M, Calais E, Parsons B (2005) Geodetic constraints on glacial isostatic adjustment in Europe. Geophys Res Lett 32:L06308. doi:10.1029/2004GL022174, 2005 Nørbech T, Engsager K, Jivall L, Knudsen P, Koivula H, Lidberg M, Madsen B, Ollikainen M, Weber M (2006) Transformation from a Common Nordic Reference Frame to ETRS89 in Denmark, Finland, Norway, and Sweden – status report. Also Paper C in Lidberg M (2007) Geodetic Reference Frames in Presence of Crustal Deformations. Ph.D. Thesis, Chalmers University of Technology, G€oteborg, Sweden
Global Terrestrial Reference Frame Realization Within the GGOS-D Project
11
D. Angermann, H. Drewes, and M. Seitz
Abstract
The GGOS-D terrestrial reference frame has been computed in a common adjustment of station positions and velocities together with the Earth orientation parameters and the quasar coordinates (celestial reference frame). The data were processed as datum-free normal equations from homogeneously generated VLBI, SLR and GPS observation time series using identical standards for the modelling and parameterization. A major focus was on the analysis of the station position time series, investigations regarding seasonal variations in station motions and on the combination methodology for the terrestrial reference frame computation.
11.1
Introduction
The project GGOS-D was funded by the German Federal Ministry of Education and Research (BMBF) in the GEOTECHNOLIEN-Programme under the topic “Observation of System Earth from Space”. GGOS-D involved four German institutions: Deutsches GeoForschungsZentrum Potsdam (GFZ), Bundesamt f€ur Kartographie und Geod€asie (BKG), Institut f€ur Geod€asie und Geoinformation, Universit€at Bonn (IGG-B) and DGFI. An overview of the project is given by Rothacher et al. (2010). The major goals and challenges of this project were (1) the definition and implementation of common GGOS-D standards and a unique modelling and parameterization in the different software packages,
D. Angermann (*) H. Drewes M. Seitz Deutsches Geod€atisches Forschungsinstitut (DGFI), AlfonsGoppel-Strasse 11, 80539 M€ unchen, Germany e-mail:
[email protected]
(2) the generation of homogeneously re-processed observation time series from the different space geodetic observation techniques, (3) the computation of the GGOS-D terrestrial and celestial reference frames, and (4) the generation of consistent, high-quality time series of geodetic-geophysical parameters describing the Earth System. In this paper, we focus on the computation of the terrestrial reference frame (TRF). The input data comprise observation time series from the space techniques Very Long Baseline Interferometry (VLBI), Satellite Laser Ranging (SLR) and the Global Positioning System (GPS) based on identical standards for the modelling and parameterization. The data were processed as datum-free normal equations in two major steps (1) Analysis and accumulation of time series normal equations per technique, (2) Intertechnique combination and computation of the final TRF solution. A major focus was thereby on the investigation of seasonal variations in station positions and on the development of advanced combination methods.
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_11, # Springer-Verlag Berlin Heidelberg 2012
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D. Angermann et al.
Data Used for TRF Computation
The computation of the GGOS-D terrestrial reference frame is based on homogeneously processed VLBI, SLR and GPS observation time series.Table 11.1 gives an overview of the input data. The time series of the different techniques were provided as unconstrained datum-free normal equations. The GGOS–D input data also comprise the local tie information, which is available at the ITRS Centre at http://itrf.ensg.ign.fr/ local_survey.php. The common models used in the various software packages were not only implemented but also validated in detail by intensive comparisons. Based on the experiences gathered a refined set of standards was agreed upon the participating institutions and was subsequently implemented for the second project phase (Rothacher et al. 2007; Steigenberger et al. 2010). Most recent state-of-the models were taken into account, e.g., the use of 6-hourly ECMWF grids for computing the hydrostatic delay and the use of the hydrostatic mapping function VMF1 (B€ ohm et al. 2006) for VLBI and GPS, the correction for higherorder ionospheric terms for GPS, and the modelling of thermal deformation for the VLBI telescopes (Nothnagel 2008). In order to allow for a comparison and validation of the results, for each of the techniques two solutions were computed with different software packages. Based on the individual observation time series, intratechnique combinations were performed for VLBI and SLR. For GPS such a combination was not performed since the EPOS solution was not fully consistent
Table 11.1 GGOS-D data used for the global TRF computation. For each technique two series were computed using two different software packages. In case of GPS the second series computed with EPOS was used for comparisons only, since the time resolution of this solution is 1 week and there are also small differences w.r.t. the GGOS-D standards Institution Software Institution Software Time interval resolution No. of stations
GPS GFZ Bernese GFZ EPOS 1994–2007 1/7 days 240
SLR DGFI DOGS GFZ EPOS 1993–2007 7 days 71
VLBI DGFI OCCAM IGGB Calc/Solve 1984–2007 1 h sessions 49
with the Bernese solution. For the same reason, DORIS (Doppler Orbitography and Radiopositioning Integrated by Satellite) data were not included in the TRF computation, since the standards and models applied for the DORIS processing were not fully consistent with the GGOS-D standards.
11.3
Combination Methodology for the TRF Computation
The methodology applied in the GGOS-D project was based on combining datum-free normal equations of the VLBI, SLR and GPS observation time series. The tropospheric parameters and low-degree spherical harmonic coefficients, which were also included in the time series normal equations, were reduced and presently not considered for the TRF computations. The station positions and velocities of the observing stations (TRF) were combined in a common adjustment with the quaser coordinates (CRF), and the Earth Orientation Parameters (EOP). The TRF computation has been performed with the DGFI Software package DOGS. The general procedure of the combination methodology is given in Fig. 11.1.
Input: Datum-free normal equations (NEQ) Epoch 1
VLBI NEQ
SLR NEQ
GPS NEQ
Epoch 2
VLBI NEQ
SLR NEQ
GPS NEQ
Epoch n
VLBI NEQ
SLR NEQ
GPS NEQ
Accumulation of time series Multi-year NEQ‘s
VLBI NEQ
SLR NEQ
GPS NEQ
Inter-technique combination TRF (station positions and velocities), EOP, and quasar coordinates)
Fig. 11.1 Combination methodology for the terrestrial reference frame computation
11
Global Terrestrial Reference Frame Realization Within the GGOS-D Project
11.3.1 Accumulation of Time Series per Technique In the first part of the terrestrial reference frame computation, the time series of the daily/weekly normal equations were accumulated to a multi-year normal equation for each technique. The procedure comprises two major steps (1) The generation and analysis of station position time series to detect non-linear behaviours, and (2) the computation of the cumulative multi-year normal equations per technique. The time series analysis and accumulation was performed in an iterative procedure. At first, constant velocity parameters were set up in the epoch normal equations to represent linear station motions and the epoch station positions were transformed to positions at the reference epoch 2000.0. From the analysis of position time series discontinuities were identified for many stations, which are often caused by equipment changes or geophysical events (e.g., earthquakes). The discontinuities were parameterized by setting up new position and velocity parameters after the jump. The total number of discontinuities could significantly be reduced compared to ITRF2005, in particular for GPS stations. The ITRF2005 solution computed at DGFI (ITRF2005-D) comprises 221 discontinuities for 332 stations (67%), whereas 124 discontinuities for 240 stations (52%) were identified within GGOS-D. This improvement was mainly achieved by the homogeneuosly processed GGOS-D data sets and, in the case of GPS, by the implementation of absolute antenna phase center corrections. Figure 11.2 shows the position time series for the GPS station Yuzhno-Sakhalin (YSSK), Russia, located in a geodynamically very active region, the Sakhalin seismic belt. Two large earthquakes both with a magnitude of 8.3, at Hokkaido (25 September 2003) and at Kuril island (15 November 2006) caused discontinuities of about 1–2 cm in the position time series, in particular in the north and east component. The station velocity in the north component was changed by about 5 mm/year after the first earthquake, changing again to its nominal value after about 2 years. The precision of repeated station positions obtained from the accumulation of the time series solutions for each space technique was used to estimate scaling factors for the technique-specific normal equations. For stations with discontinuities separate positions
89
were estimated for each segment. Statistical tests were applied to decide whether the estimated velocities can be equated or not.
11.3.2 Computation of the TRF Solution Input for the combination of different techniques were the accumulated intra-technique normal equations for GPS, SLR and VLBI. The parameters include station positions, velocities, daily EOPs, and (for VLBI) also quasar coordinates (right ascension and declination). The connection of the different techniques’ observations is given by geodetic local tie measurements between the instruments’ reference points at colocation sites. The selection of suitable local ties is a critical issue because the number and the spatial distribution of “high quality” co-location sites is not optimal. Furthermore, there are discrepancies between the difference vectors derived from the space geodetic techniques and the local ties, as shown for example in Fig. 11.3 for VLBI and GPS co-locations. The results are given for the GGOS-D TRF computation in comparison with ITRF2005-D (Angermann et al. 2007, 2009), which is in excellent agreement with ITRF2005 (Altamimi et al. 2007). The agreement of the space geodetic solutions with the local ties is better for most stations of the GGOS-D computation, in particular for those located on the southern hemisphere. In the ideal case (without systematic errors) the EOP estimates must be (statistically) identical for all space techniques. Thus, their estimates are used as a criterion to validate the selected local ties and to stabilize the inter-technique combination as additional “global ties”. The selection and implementation of local ties for the inter-technique combination as well as the equating of station velocities at co-location sites was done in an iterative procedure (see Kr€ugel and Angermann 2007). The selection of local ties is performed on the basis of two criteria (1) the pole coordinates as common parameters provide an ideal basis to measure the consistency: A first combination of station coordinates is computed using a set of local ties, while the EOP are not combined. The mean offsets between the estimated coordinates of the pole are then used to quantify the consistency of the TRF. (2) The r.m.s. values of the 7-parameter similarity transformations between the combined TRF solution and the single-technique
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2001
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0 –5 –10 –15 –20 25
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20 15 10 5 0
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JD2000 [Tage] Fig. 11.2 Position time series for the GPS station Yuzhno-Sakhalin (YSSK), Russia 25 ITRF 2005 GGOS D
20 15 10
HOB2
TIDB
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Fig. 11.3 Comparison of the GGOS-D results with ITRF2005D. The 3D difference vectors (mm) between the VLBI and GPS solutions and the terrestrial difference vectors are given for 21
co-location sites. The stations located in the southern hemisphere are highlighted by yellow (grey) color
solutions are a measure of the deformation of the station networks (caused by the local tie implementation). The two above mentioned criteria were applied in various test computations by satisfying also that an optimal number of co-location sites with a good spatial distribution is selected.
To identify the best (most suitable) set of co-location sites, different solutions were computed, varying the co-locations and the assumed accuracy of the introduced local ties. As an example, the results for VLBI and SLR co-locations are given in Table 11.2. Shown are the mean pole offsets between the VLBI
11
Global Terrestrial Reference Frame Realization Within the GGOS-D Project
Table 11.2 Mean pole offsets and network deformation for VLBI and GPS co-locations obtained from the GGOS-D intertechnique combination in comparison with the DGFI solution for ITRF2005 (ITRF2005-D) Selected VLBI-GPS co-locations Mean pole difference (mas) Network deformation Epoch 2000.0 (mm)
GGOS-D 19
ITRF2005-D 13
0.035 0.3
0.041 1.0
and GPS estimates as a measure for the consistency. Furthermore, the r.m.s. position differences of a 7-parameter similarity transformation between the VLBI-only and the combined TRF solution (expressed at the reference epoch 2000.0) as a measure for the network deformation are shown. The GGOS-D results are compared with those obtained from ITRF2005-D. It shall be emphasized that the number of co-locations significantly increased for the GGOS-D computations, accompanied with a better agreement of the VLBI and GPS pole coordinates and a smaller network deformation. Major differences of the GGOS-D solution w.r.t. ITRF2005 are (1) an improved modelling of the individual space techniques, (2) a homogeneous re-processing by applying unified standards and models for all techniques, and (3) the inclusion of two more years of recent data until the end of 2007. Other tasks of the inter-technique combination include the weighting of the different techniques and the equating of station velocities of co-located instruments. The weighting was done by estimating variance factors for the normal equations based on the precision of repeated (daily/weekly) station positions. The station velocities of co-located instruments were estimated as separate parameters. The velocities were equated, if the differences are statistically not significant. To generate the TRF solution, minimum datum conditions were added to the combined normal equations, and the complete normal equation system was inverted. The origin was realized by SLR observations. As shown by R€ ulke et al. (2008), the network translation parameters derived from GPS are affected by orbital perturbances and cannot be used for the datum definition. The scale of the TRF solution was defined by SLR, VLBI and GPS observations. This was justified, since a comparison of the techniquespecific station networks w.r.t. the scale showed no significant differences. The orientation of the TRF was
91
defined by a No-Net-Rotation (NNR) condition w.r.t. ITRF2005. The kinematic datum of the TRF solution was given by an actual plate kinematic and crustal deformation model (APKIM) derived from observed station velocities (Drewes 2009). The GGOS-D combination results comprise the terrestrial reference frame (station positions and velocities), daily EOP and quasar coordinates, which were estimated in a common adjustment. The terrestrial reference frame results provide the basis for the generation of consistent, high-quality time series of geodetic-geophysical parameters describing the Earth System (Nothnagel et al. 2010).
11.4
Analysis of Seasonal Station Motions
In specific GGOS-D studies, the time series of co-located VLBI and GPS stations were analysed and compared (Tesmer et al. 2009). As an example, Fig. 11.4 shows the VLBI and GPS height time series for two co-location sites: Wettzell (Germany) and Ny-Alesund (Spitsbergen, Norway). The VLBI antenna of Wettzell has the most dense height series (two observations per week), which agrees rather well with GPS. In case of Ny-Alesund there are larger discrepancies between the VLBI and GPS height time series, which may be caused by solutionand/or technique-specific effects. The results given by Tesmer et al. (2009) show a rather good agreement of the height time series for most of the VLBI and GPS colocation sites. The strongest signals in the time series have mostly annually repeating patterns. The station height time series were compared with geophysical model results (Seitz and Kr€ugel 2009). It was found, that a large part of the observed annual signals can be explained by loading, which was computed from atmospheric, hydrospheric and non-tidal oceanic loading variations. Unfortunately, the geophysical models are not as accurate as necessary to reduce loading from the variations of the station positions. Thus, the reduction of loading effects from the original observations is problematic. However, these seasonal station motions will affect the terrestrial reference frame computations, if the temporal variations of station positions are described only by constant velocities, as it is done currently. The consequences are (1) Deviations of the station motions from a linear model (e.g., seasonal variations) will
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D. Angermann et al. WTZR vs. WETTZELL: wmeans (each 7 days for −+35 days)
dR [cm]
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time [year] NYAL vs. NYALES20: wmeans (each 7 days for −+35 days)
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Fig. 11.4 Homogeneously re-processed VLBI (stars) and GPS (circles) height time series at two co-located sites: Wettzell (Germany) and Ny-Alesund (Spitsbergen, Norway). The figures illustrate 90-days moving weighted means computed each 7 days
produce errors in the combination results; (2) seasonal variations will affect the velocity estimations, in particular for stations with relatively short observation time spans (i.e., <2 years); and (3) the alignment of epoch solutions to a reference frame with positions and constant velocities is affected by non-linear station motions. To have a closer look to the shape of the seasonal variations we computed a mean averaged signal from the original height time series for each station. This procedure also makes sense from a geophysical point of view, as many of the observed signals are driven by annually repeating processes. Figure 11.5 shows two examples for the mean averaged shape of such annual variations. The annual signal can explain most of the variability of the Brasilia station, while this is not the case for the Ankara station, for which at least a semi-annual signal must also be removed. Thus, the averaged annual motions of both stations can mathematically rather well be represented by annual and semi-annual sine functions. However, the computation of a mean (averaged) annual motion is problematic, in particular if the seasonal variations are different over the observation time span. Furthermore, the additional parameters will affect the stability of the solutions, which is in particular a problem for stations with short observation time spans. Thus, the handling of
seasonal variations in station positions is a challenge for future ITRF computations.
11.5
Conclusions and Outlook
The GGOS-D global terrestrial reference frame (TRF) was computed in a common adjustment with the EOP and quasar coordinates (CRF). The results were obtained from homogeneously re-processed VLBI, SLR and GPS time series based on unified standards for the modelling and parameterization. Most recent, state-of-the-art models were implemented in the different software packages. It was shown for VLBI and SLR co-locations, that the agreement of the space geodetic solutions with the local ties is better for most stations of the GGOS-D computations compared to ITRF2005-D. Another focus was on the analysis of seasonal station motions and their impact on the terrestrial reference frame results. The comparisons of the height time series of co-located VLBI and GPS stations showed mostly a rather good agreement between both techniques, in particular for the annually repeated patterns. A large part of the seasonal height variations can be explained by loading effects (i.e., atmospheric and hydrospheric loading). The consideration of seasonal signals in station positions is a challenge and it is
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Global Terrestrial Reference Frame Realization Within the GGOS-D Project 15
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Brasilia
Ankara 10 height component [mm]
height component [mm]
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Fig. 11.5 Shape of the “averaged” annual height signal for two GPS stations. The fitted curves represent the mathematical approximation by annual and semi-annual sine/cosine functions
one of the most important tasks for future TRF realizations. It is recommended to use the geophysical models, but to estimate the actual parameters from the geodetic observations. Acknowledgements The work within the GGOS-D project has been funded by the German Ministry of Education and Research (BMBF) in the programme GEOTECHNOLOGIEN under the topic “Observation of System Earth from Space”, grant 03F0425.
References Altamimi Z, Collilieux X, Legrand J, Garayt B, Boucher C (2007) ITRF2005: a new release of the International terrestrial reference frame based on time series of station positions and earth orientation parameters. J Geophys Res 112: B09401. doi:10.1029/2007JB004949 Angermann D, Drewes H, Kr€ ugel M, Meisel B (2007) Advances in terrestrial reference frame computations. IAG symposia, vol. 130. Springer, Berlin, pp 595–602 Angermann D, Drewes H, Gerstl M, Kr€ ugel M, Meisel B (2009) DGFI combination methodology for ITRF2005 computation. In: Drewes H (ed) Geodetic reference frames. IAG symposia, vol. 134. Springer, Berlin, pp 11–16. B€ohm 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 111: B02406. doi:10.129/2005JB003629 Drewes H (2009) The APKIM2005 as basis for a non-rotating ITRF. In: Drewes H (ed) Geodetic reference frames. IAG symposia, vol. 134. Springer, Berlin, pp 95–99 Kr€ugel M, Angermann D (2007) Frontiers in the combination of space geodetic techniques. IAG symposia, vol. 130. Springer, Berlin, pp 158–165
Nothnagel A (2008) Conventions on thermal expansion modelling of radio telescopes for geodetic and astrometric VLBI. J Geod. doi:10.1007/s00190-008-284-z Nothnagel A, Artz T, B€ockmann S, Panafidina N, Rothacher M, Seitz M, Steigenberger P, Thaller D (2010) GGOS-D consistent and combined time series of geodetic/geophysical parameters. In: Flechtner F, Gruber T, G€untner A, Mandea A, Rothacher M, Sch€one T, Wickert J (Eds) Observation of Earth System from Space, Springer Rothacher M, Drewes H, Nothnagel A, Richter B (2007) Integration of space geodetic techniques as the basis for a Global Geodetic-geophysical Observing System (GGOS-D): an overview. GEOTECHNOLOIEN science report, no. 11, ISSN 1619–7399 Rothacher M, Drewes H, Nothnagel A, Richter B (2010) Integration of space geodetic techniques as the basis for a Global Geodetic-geophysical Observing System (GGOS-D). In: Flechtner F, Gruber T, G€untner A, Mandea A, Rothacher M, Sch€one T, Wickert J (Eds) Observation of Earth System from Space, Springer R€ulke A, Dietrich R, Fritsche M, Rothacher M, Steigenberger P (2008) Realization of the terrestrial reference system by a reprocessed global GPS network. J Geophys Res. doi:10.1029/2007JB005231 Seitz F, Kr€ugel M (2009) Modeling vertical site displacements due to surface loads in consideration of crustal inhomogenities. In: Drewes H (ed) Geodetic reference frames. IAG symposia, vol. 134. Springer, Berlin, pp 23–29 Steigenberger P, Artz T, B€ockmann S, Kelm R, K€onig R, Meisel B, M€uller H, Nothnagel A, Rudenko S, Tesmer V, Thaller D (2010) GGOS-D consistent, high-accuracy techniquespecific solutions. In: Flechtner F, Gruber T, G€untner A, Mandea A, Rothacher M, Sch€one T, Wickert J (Eds) Observation of Earth System from Space, Springer Tesmer V, Steigenberger P, Rothacher M, B€ohm J, Meise B (2009) Annual deformation signals from homogeneously reprocessed GPS and VLBI height time series. J Geod. doi:10.1007/s00190-009-0316-3
.
Comparison of Regional and Global GNSS Positions, Velocities and Residual Time Series
12
€ppelmann, J. Legrand, N. Bergeot, C. Bruyninx, G. Wo A. Santamarı´a-Go´mez, M. -N. Bouin, and Z. Altamimi
Abstract
More than 10 years (1996–2008) of weekly GPS solutions of 299 globally distributed stations have been used to quantify the impact of the reference frame definition and especially the size of the network on the estimated station positions, velocities, and residual position time series. For that purpose, weekly regional solutions (covering the European region) and global solutions have been respectively stacked to obtain regional and global station positions, velocities, and residual position time series. In both cases, the estimated long-term solutions have been tied to the ITRF2005 under minimal constraints using a selected set of reference stations. This study shows that: (1) regional position and velocity solutions can present biases with respect to each other and to global solutions, while in comparison, global solutions are much more stable; (2) the obtained residual position time series are affected by the size of the network with significantly reduced periodic signals in the regional networks, e.g. a 27% reduction of the annual signals in the height component.
J. Legrand (*) N. Bergeot C. Bruyninx Royal Observatory of Belgium, Avenue Circulaire 3, 1180 Brussels, Belgium e-mail:
[email protected] G. W€oppelmann UMR LIENSS, Universite´ de La Rochelle-CNRS, 2 rue Olympe de Gouges, 17000 La Rochelle, France A. Santamarı´a-Go´mez LAREG/IGN, 6-8 Avenue Blaise Pascal, 77455 Marnela-Valle´e, France Instituto Geogra´fico Nacional, c/General Iban˜ez Ibero 3, 28071 Madrid, Spain M.-N. Bouin CNRM/Centre de Me´te´o Marine, 13 rue du Chatellier, 29604 Brest, France Z. Altamimi LAREG/IGN, 6-8 Avenue Blaise Pascal, 77455 Marnela-Valle´e, France
12.1
Introduction
Different GNSS reprocessing performed during the last years (Steigenberger et al. 2006; W€oppelmann et al. 2007, 2009; Kenyeres 2009) have shown that significant improvements in the quality and homogeneity of the estimated parameters can be obtained. Within the EUREF Permanent Network (EPN; Bruyninx 2004), recently, a new Special Project (SP) dedicated to the coordination of the EPN re-processing has been set up (V€olksen 2009). This SP will use the consistent high quality GNSS products (e.g. orbits, clocks and Earth rotation parameters) issued from the International GNSS Service (IGS; Dow et al. 2009) re-processing campaign, to re-process the EPN data. During the project Pilot Phase, optimal
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processing strategies are investigated such as the need to add (or not) global IGS stations to the EPN reprocessing. Indeed, with the improving computing facilities and GNSS data analysis, it has become less demanding to perform a global analysis and regional networks may consider this approach. In Legrand and Bruyninx (2009), global and regional station positions solutions were compared and it was demonstrated that positions obtained from global network solutions are less sensitive to the reference frame definition compared to regional solutions. In Legrand et al. (2009), it was shown that when expressing a GNSS solution in the ITRF2005 (Altamimi et al. 2007a) using minimal constraints, the network effect (due to the size of the GNSS network and the choice of the reference stations) significantly influences the estimated velocity field and consequently might cause incorrect geodynamical interpretations. Consequently, when sub-mm/year accuracy is required, e.g. for a proper interpretation of intraplate deformations or vertical velocities, a global approach should be considered. In this paper, these studies have been repeated using an enhanced GPS re-processed solution with an enlarged network and, in addition, they were extended to also investigate the impact of the size and the geometry of the network on the station residual position time series.
12.2
Input Data
More than 10 years (1996–2008) of weekly GPS solutions produced by the ULR consortium (Universite´ de la Rochelle and IGN/LAREG) as its contribution to the Tide Gauge Benchmark Monitoring project of the IGS (TIGA) have been used throughout this paper. The ULR weekly solutions provide station coordinates together with their covariance information for 299 globally distributed continuously observing GNSS stations (Fig. 12.1) from which 265 stations have more than 3.5 years of data. The same parameterization and observation modeling were used over the whole 13-year period, estimating station coordinates, satellite orbits, Earth orientation parameters, and zenith tropospheric delay parameters every 2 h. IGS absolute phase centre corrections for both the tracking and transmitting
Fig. 12.1 Global (black triangles) and regional (white triangles) networks used in this study. The dashed area corresponds to Fig. 12.2
antennas were applied (see Santamaria et al. in this issue, for further details on the GNSS reprocessing strategy). In order to investigate the impact of using regional network instead of a global network, we elaborated several long-term solutions by varying the geographical extension of the network and the reference stations used in the alignment to the ITRF2005. First, regional weekly solutions have been created from the ULR global weekly solutions by extracting the 74 GNSS stations located in Europe, all of them are included in the EUREF Permanent Network (EPN). Then, global and regional cumulative solutions (positions, velocities, and residual time series) were computed by stacking both sets of weekly (regional and global) solutions. The stacking was performed with CATREF (Altamimi et al. 2007b) and the position and velocity combined solutions (regional and global) were tied to the ITRF2005 under minimal constraints using 14 transformation parameters (translations, rotations, scale and their rates) using a selection of ITRF2005 reference stations. The minimal constraints approach has the advantage of preserving the intrinsic characteristics of the stacked solution (Altamimi 2003) while avoiding any internal distortion of the original network geometry. The selected reference stations were chosen in such a way that they have a station observation history of at least 3 years in the ITRF, as well as in the ULR time series, show a good agreement with the ITRF2005 solution, and are optimally distributed over the network. To evaluate the impact of the reference stations on the global and regional velocity fields, several sets of
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Comparison of Regional and Global GNSS Positions, Velocities and Residual Time Series
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Fig. 12.2 Stations used for the reference frame alignment of regional solutions. Selection A: stations are indicated with black and white triangles; selection B: stations are indicated with white triangles
regional and global reference stations were tested. In the global case, we selected 100 reference stations geographically well-distributed over the globe. While in the regional case, the set of reference stations covered only the European region. In order to highlight the instability observed in a regional case, based on the same set of 74 regional stations, two different regional solutions were computed using two sets of reference stations (Fig. 12.2) having a large probability of being used in Europe: • Selection A: 30 reference stations • Selection B: 19 stations, subset of selection A with stations located only on the European continent
Table 12.1 Statistics on position and velocity differences between Regional A (resp. Regional B) and global solution for the 74 common stations Position differences (mm) Horizontal Regional A – Global Regional B – Global Vertical Regional A – Global Regional B – Global Velocity differences (mm/year) Horizontal Regional A – Global Regional B – Global Vertical Regional A – Global Regional B – Global
RMS 0.9 1 1.7 2.1 RMS 0.3 0.5 0.6 0.6
Max. 2.5 3.2 7.8 6.8 Max. 0.6 1.1 1.4 1.7
12.3.1 Positions and Velocities
12.3
Results
In a first step, the cumulative positions and the velocities of the global solution have been compared to the cumulative positions and the velocities of the two regional solutions (at epoch 2003.0). In a second step, a similar comparison was done for the residual position time series. For the common stations, the different cumulative solutions are based on identical weekly positions.
The differences between the positions obtained using the different networks (see Table 12.1) can reach 3 mm in the horizontal component and 8 mm in the vertical component. They are due to the network effect and entail a systematic effect on both the horizontal and vertical positions. Nevertheless, these differences are much smaller than in Legrand and Bruyninx (2009). Indeed, in the previous paper, the authors used a 1-year cumulative solution and found differences reaching 8 mm in the horizontal
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and 2 cm in the vertical. In this paper, the use of solutions based on a longer time span, 13 years instead of 1 year, allows to stabilize the reference frame definition and entails a better (more stable and reliable) alignment of the solution to the ITRF2005. Similar systematic effects also affect the horizontal and the vertical velocities (Table 12.1). They are highlighted in Fig. 12.3, which shows the horizontal velocity differences between the global solution and the two regional solutions (top: regional A, bottom: regional B). These differences (as well as the ones 290°
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(2009), where the differences between the regional velocity fields and the global velocity field could reach up to 1.3 mm/year in the horizontal and 2.9 mm/year in the vertical. The smaller differences obtained in this paper are most probably due to the availability of a larger number of reference stations in good agreement with the ITRF2005. W€ oppelmann et al. (2008) investigated the influence of using different sets of reference stations to express a global solution in a given frame and concluded that the best
results were obtained using a large global distribution of reference stations mitigating the individual problems at each of the reference stations. Similarly, in a regional network, more reliable velocities are obtained using a larger number of reference stations. The comparison between regional A and B solutions shows that the disagreement between the global and regional solutions (both positions and velocities) is amplified when the reference stations cover a smaller geographical area.
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12.3.2 Time Series In addition to the observed position and velocity differences, the size of the network also affects the residuals position time series. These residual position time series are obtained when removing the estimated site velocity from the weekly positions and provide information on the non-linear site (e.g. seasonal) motions. The residual time series from regional A and from regional B are identical. Indeed, they depend on the size of the network and stations in the solution, and not on the selected reference stations. Consequently, in the following, only the residual time series from the regional A solution are compared with the global residual time series. During the stacking, discontinuities have been introduced to account for jumps in the timeseries. A new station position is estimated after each discontinuity and the velocities are usually constrained to be equal before and after a discontinuity. As, only a linear motion was assumed, the RMS of a residual time series reflects the noise, but also the seasonal signals which affect the GNSS stations. In average, the RMS of the regional residual time series is reduced by about 20% compared to the global residual time series. Figure 12.5 shows, for each European station, the difference between the RMS of the global residual time series and the regional residual time series. The mean RMS reduction is 0.75 mm on the vertical component and is maximal in the North-East of Europe.
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On the horizontal component, the RMS reduction is about 0.25 mm. Figure 12.6 illustrates the RMS reduction for the station GLSV (Kiev, Ukraine). The RMS of the height time series obtained with the global network is 5.8 mm compared to 4.3 mm for the regional one; this means that the RMS was reduced by 1.5 mm. The amplitude of the annual (resp. semi-annual) signal is 5.2 mm (resp. 0.9 mm) for the global and 3.8 mm (resp. 0.8 mm) for the regional. Table 12.2 gives the mean amplitudes of the annual and semi-annual terms simultaneously fitted on the residual time series; it evidences an amplitude reduction for all the components when a regional network is considered instead of a global one. For the horizontal components, the annual amplitude decreases by 8% for the east component and 15% for the north component. The semi-annual amplitude is decreased by 9% for both the east and the north components. The up component is the most affected by this reduction: the annual amplitude is decreased by 27% (see Figs. 12.7 and 12.8 for details) and the semi-annual amplitude is reduced by 15%. Figure 12.9 (resp. Fig. 12.10) shows the histograms of the phases of the annual signal in the up component for the global network (resp. for the regional network). In the global network, the predominant phase is around 180 , while within the regional network the phases are more randomly distributed. This study shows that our regional network is not able to reliably reconstruct the spatially correlated
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Fig. 12.6 Residual time series of GLSV: Up component. Top: global, bottom: regional A
Table 12.2 Mean amplitudes and standard deviations of the annual and semi-annual terms estimated from the residual time series Mean amplitude (mm)
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Annual 0.66 0.46 0.74 0.68 2.37 1.17
annual and semi-annual signals. Indeed, during the stacking, these common signals are absorbed by the transformation parameters estimated to align each individual solution to the final combined solution. Conclusion
We investigated the influence of the reference frame definition in terms of reference station selection and network extension on the cumulative positions, velocities, and residual position time series obtained from a GNSS network which was tied to the ITRF2005 using minimal constraints. It was shown that, based on identical sets of weekly positions, the estimated long-term positions and velocities can differ (up to 2 mm in the horizontal and 8 mm in the vertical for the positions and
Regional Semi-annual 0.23 0.12 0.29 0.16 0.93 0.58
Annual 0.61 0.44 0.63 0.62 1.73 1.04
Semi-annual 0.21 0.11 0.22 0.14 0.79 0.47
up to 0.5 mm/year in the horizontal and 2 mm/year in the vertical for the velocities) due to a network effect which depends on the selection of the reference stations. The disagreement between the global and regional solutions (both positions and velocities) is amplified when the regional reference stations cover a smaller geographical area. In a regional network, the absorption of the common mode signals induces several effects on the residual position time series. The mean RMS of the regional residual time series shows a reduction of about 20% compared to the global residual time series. This RMS reflects not only the noise but also the seasonal signals in the time series. We demonstrated that the regional network underestimates the amplitude of the annual (27% reduction of the
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Fig. 12.7 Histogram of the annual amplitudes observed in the up component time series in Europe with our global network
Fig. 12.8 Histogram of the annual amplitudes observed in the up component time series in Europe with our regional network
annual signal in the height component) and semiannual signals in all components. In addition, the phase of the annual and semi-annual signals is altered: while in a global network the predominant
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Fig. 12.9 Histogram of the annual phases observed in the up component time series in Europe with our global network
Fig. 12.10 Histogram of the annual phases observed in the up component time series in Europe with our regional network
phase is 180 , this is not the case anymore in a regional network. Consequently, a geophysical interpretation of the seasonal signals observed using a regional network can be more challenging.
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References Altamimi Z (2003) Discussion on How to Express a Regional GPS Solution in the ITRF. EUREF Publication No. 12, Verlag des Bundesamtes f€ ur Kartographie und Geod€asie, Frankfurt am Main, pp 162–167 Altamimi Z, Collilieux X, Legrand J, Garayt B, Boucher C (2007a). ITRF2005: A new Release of the International Terrestrial Reference Frame based on Time Series of Station Positions and Earth Orientation Parameters. J Geophys Res 112: B09401, doi:10.1029/2007JB004949 Altamimi Z, Sillard P, Boucher C (2007b) CATREF software: combination and analysis of terrestrial reference frames. LAREG technical. Institut Ge´ographique National, Paris, France Bruyninx C (2004) The EUREF Permanent Network; a multidisciplinary network serving surveyors as well as scientists. GeoInformatics 7:32–35 Dow JM, Neilan RE, Rizos C (2009) The International GNSS Service in a changing landscape of Global Navigation Satellite Systems. J Geod 83:191–198. doi:10.1007/s00190-0080300-3 Kenyeres A, Legrand J, Figurski M, Bruyninx C, Kaminski P, Habrich H (2009) Homogenous Reprocessing of the EPN: First Experiences and Comparisons. Bulletin of Geodesy and Geomatics 3:207–218 Legrand J, Bruyninx C (2009) EPN reference frame alignment: consistency of the station positions. Bull Geod Geomat LXVIII(1):20–34
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Legrand J, Bergeot N, Bruyninx C, W€oppelmann G, Bouin M-N, Altamimi Z (2010) Impact of regional reference frame definition on geodynamic interpretations, Journal of Geodynamics 49(3–4):116–122 Santamarı´a-Go´mez, A, Bouin M-N, W€oppelmann G (2011) Improved GPS data analysis strategy for tide gauge benchmark monitoring, pp 11–18 Steigenberger P, Rothacher M, Dietrich R, Fritsche M, R€ulke A, Vey S (2006) Reprocessing of a global GPS network. J Geophys Res 111:B05402. doi:10.1029/2005JB003747 V€olksen C (2009) Draft Charter for the EUREF Working Group on Reprocessing of the EPN, http://epn-repro.bek.badw.de/ Documents/charter_repro.pdf W€oppelmann G, Martin Miguez B, Bouin M-N, Altamimi Z (2007) Geocentric sea-level trend estimates from GPS analyses at relevant tide gauges world-wide. Global Planet Change 57:396–406 W€oppelmann G, Bouin M-N, Altamimi Z (2008) Terrestrial reference frame implementation in global GPS analysis at TIGA ULR consortium. Physics and Chemistry of the Earth, Parts A/B/C Volume 33, Issues 3–4, Observing and understanding sea level variations, pp 217–224 W€oppelmann G, Letetrel C, Santamaria A, Bouin M-N, Collilieux X, Altamimi Z, Williams SDP, Martin Miguez B (2009) Rates of sea-level change over the past century in a geocentric reference frame. Geophys Res Lett 36:L12607. doi:10.1029/2009GL038720
.
GPS Metrology: Bringing Traceable Scale to a Local Crustal Deformation GPS Network
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H. Koivula, P. H€akli, J. Jokela, A. Buga, and R. Putrimas
Abstract
A constant scale difference between GPS solutions and traceable electronic distance measurement (EDM) results was found during semi-annually repeated campaigns performed in Olkiluoto, Finland. Since EDM results are very accurate and uncertainties are well-defined, this leads to an assumption that the GPS solution is biased. At the Kyvisˇke˙s test field in Lithuania, the true lengths with traceable uncertainties between observation pillars were measured using a Kern ME5000 Mekometer as a scale transfer standard. GPS observations were processed using individual and type calibrated antenna tables, a local and global ionosphere model, and three different cut-off elevation angles, and several linear combinations and were then compared with the EDM results. The results show that the ambiguity resolution strategy and antenna calibration model play a significant role compared to the cut-off elevation angle and ionosphere model. Individual antenna calibration is required for the best metrological accuracy by means of the best agreement with traceable EDM results. The best metrological agreement was obtained with an L1 solution and individually calibrated antennas. The rms and maximum difference to the true (EDM) values were 0.3 and 0.7 mm, respectively. However, a clear distance dependency of 0.5 ppm was also evident. In particular, linear combinations with type calibrated tables caused variations up to 4 mm from the true value, even when high quality choke ring antennas were used. With individually calibrated antennas, all solutions were within 1 mm of the true value.
13.1 H. Koivula (*) P. H€akli J. Jokela Department of Geodesy and Geodynamics, Finnish Geodetic Institute, P.O. Box 15, 02431 Masala, Finland e-mail:
[email protected] A. Buga R. Putrimas Vilnius Gediminas Technical University, Institute of Geodesy, Sauletekio al. 11, 10223 Vilnius, Lithuania
Introduction
The Finnish Geodetic Institute (FGI) investigates crustal deformations in Olkiluoto, at a proposed disposal site for nuclear waste in Finland. The size of the research area is approximately 2 4 km2 and consists of ten concrete pillars attached to the bedrock. Since 1995, the network has been measured semi-annually with GPS campaigns. The observation sessions, lasting
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at least 24 h, have been measured using dual frequency receivers and choke ring antennas. Data is processed with Bernese software using a local ionosphere model, solving for L1 and L2 ambiguities with a 15 cut-off angle and relative antenna calibration tables. Lower cut-off angles are not possible to use since some of the sites are located within a nature preserve where cutting down the trees is prohibited. The maximum change rate of a GPS vector in the network has been (0.20 0.03) mm/year, based on 24 campaigns over 13 years (Ahola et al. 2008). The scale of the GPS network has varied from one campaign to another by up to 0.8 ppm. In order to control the scale of the network, since 2002 a 511-m GPS baseline has been simultaneously measured with an EDM using a Kern ME5000 Mekometer. The Mekometer has been regularly calibrated at the Nummela Standard Baseline, which is regularly measured using the V€ais€al€a white light interference comparator with a total standard uncertainty of 0.09 mm/km in the traceability chain to the definition of the meter (Jokela et al. 2009). This brings a well-defined metrological traceable scale based on measurement standards to the GPS network. The standard uncertainty of the scale transfer to the Olkiluoto baseline is approximately 0.3 mm. The estimation of uncertainty in length measurements is computed according to the GUM (BIPM 2008). The length of the 511-m Olkiluoto baseline with GPS differed from that of EDM by an average of 0.64 mm (more than 1 ppm) between the years 2002 and 2007 and the difference is systematic in nature; GPS gives longer distances. The same baseline has been measured with GPS 24 times since 1995 and the rms of the baseline length is 0.5 mm, indicating relatively stable GPS results (Ahola et al. 2008). However, the EDM results are even more accurate, uncertainties well-defined and the length traceable to the definition of the meter. Thus, it is assumed that the GPS solution is biased. Systematic GPS errors may be of site-specific origin or related to biases, e.g., in antenna calibration values, site-specific multipath conditions and modeling of the atmosphere. The error budget of GPS is generally well known. However, when the highest accuracy is needed, the site and antenna related biases may play a dominant role. Long observations at permanent GPS stations show that antenna changes often introduce offsets to time series. This may happen even if antennas of the
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same type are used. Wanninger (2009) states that shifts are mainly caused by changing carrier-phase multipath effects and errors in antenna calibration values. Individual antenna calibration is a way to improve the situation, but even then the site-specific effects remain. GPS antennas may be calibrated relatively with respect to a reference antenna (Mader 1999) or absolutely in an anechoic chamber or by using a calibration robot (W€ubbena et al. 2000). G€orres et al. (2006) showed that calibration using a robot and in an anechoic chamber agree at a 1-mm level. Relative calibration cannot provide such a homogenous distribution of observations with regard to the antenna hemisphere as the absolute methods since it is based on the true static GPS observations. The same is true for calibration values at low elevation angles (G€orres et al. 2006). In Olkiluoto, relative type calibration tables have been used under the assumption that there should be no practical difference between relative or absolute antenna calibrations when baseline lengths are short. In Olkiluoto, a cut-off elevation angle of 15 is mandatory since cutting down trees is prohibited. In this study, lower cut-off elevation angles are also used to verify whether or not the constant offset is related to the elevation angle. An unmodelled ionosphere introduces a scale bias to the network, especially when L1 or L1 and L2 observations are used. The scale bias is proportional to the total electron content at the time of the observations. The ionosphere may be modeled locally using dual-frequency GPS data or globally using a GIM (Global Ionosphere Map) provided by numerous processing centres. To study the problem more thoroughly, all of the GPS antennas were sent to Geo++ for an individual absolute antenna calibration and additional EDM and GPS measurements were carried out at another length standard: Kyvisˇke˙s calibration baseline and test field in Lithuania in 2008. The results of this experiment are described in this paper. In Kyvisˇke˙s, we are able to do the comparison for baseline lengths between 20 and 1,320 m in ideal conditions (open sky, stable monumentation, etc.). The baseline was proven to be very stable through repeated calibrations (Buga et al. 2008). Different lengths allow us to verify whether the systematic difference between GPS and EDM seen at Olkiluoto is distance-dependent or constant.
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GPS Metrology: Bringing Traceable Scale to a Local Crustal Deformation GPS Network
The EDM results will be considered as true values since they are traceable to the definition of the meter and more accurate than GPS results. The GPS results were processed with Bernese 5.0 software using several different processing strategies, including different ambiguity resolution strategies and ionosphere models. Also, the influence of individual antenna calibration was examined. The GPS and EDM results were compared to find the optimum GPS processing strategy in the metrological sense.
13.2
EDM Measurements
13.2.1 Kyvisˇke˙s Calibration Test Field The Kyvisˇke˙s calibration baseline was established by the Institute of Geodesy of Vilnius Gediminas Technical University (VGTU) in 1996. It originally consisted of six observation pillars in line for surveying instruments at 0, 100, 360, 1,120, 1,300 and 1,320 m. A seventh pillar, constructed in the year 2000, extends the baseline to a triangle-shaped test field (Fig. 13.1). The diameter of the concrete pillars, which extend to a depth of 3 m in the mostly sandy ground, is 450 mm. The pillars have a metal forced-centring plate on the top, about 1 m above ground level, and they are insulated from the supporting structures (Jokela et al. 1999). The environment is treeless grassland an airfield (Fig. 13.2). Height differences are less than 6 m.
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13.2.2 Calibration of the Kyvisˇke˙s Test Field The Kyvisˇke˙s baseline has been calibrated three times in a co-operative effort between the VGTU and the FGI: in June 1997, October 2001, and August 2007. Congruent results from the four calibrations show that the baseline is stable, and that the repeatability of scale transfers is good (Buga et al. 2008). All calibrations have been performed using a scale transfer with high precision EDM instruments from the Nummela Standard Baseline of the FGI. The primary transfer standard has been a Kern ME5000 Mekometer EDM instrument with a prism reflector. Accurate results and traceability to the definition of the meter with well-defined uncertainty can be achieved by using V€ais€al€a interference measurements (including quartz gauge system) and calibration of the transfer standard at Nummela Standard Baseline (Jokela et al. 2009).
Fig. 13.1 Kyvisˇke˙s calibration baseline (thick line) and test field which also includes pillar 7
Fig. 13.2 Pillars 5 and 6 at Kyvisˇke˙s test field. The environment is excellent for satellite positioning, offering unobstructed visibility at most pillars
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Table 13.1 The components of the combined standard uncertainty at Kyvisˇke˙s calibration test field Component Nummela standard baseline Interference measurements of Nummela Calibration of the transfer standard Projection measurements at Nummela Determination of scale correction Additive constant At Kyvisˇke˙s test field Temperature observations Air pressure observations Determination of relative humidity Random errors from adjustment
Uncertainty 0.09 mm/km 0.07 mm 0.05 mm/km 0.02 mm 0.30 mm/km 0.07 mm/km 0.02 mm/km 0.03–0.22 mm
In order to study the scale of GPS baselines, a new calibration of the Kyvisˇke˙s calibration baseline was performed in 2008. Each of the 21 distances between the seven baseline pillars were measured from both ends (double-in-all-combinations) before and after the GPS observation session. Weather data was observed using Assmann psychrometers (for dry and wet temperatures) at both ends of each interval and two Thommen aneroid barometers (for air pressure) simultaneously with the EDM measurements. Only calibrated weather instruments were used. The combined standard uncertainty of the scale transfer with EDM from the Nummela Standard Baseline to the Kyvisˇke˙s calibration test field is summarized in Table 13.1. The results of the interference measurements of Nummela with 0.09 mm/km standard uncertainty have been used as true values when the scale transfer standard (EDM equipment) was calibrated. The sources of uncertainty in the calibration of the transfer standard are projection measurements at Nummela 0.07 mm, determination of scale correction 0.05 mm/km and additive constant 0.02 mm. At the Kyvisˇke˙s test field, as is typical with EDM, changing atmospheric conditions caused the largest uncertainty component in the estimation of total uncertainty. The standard uncertainty due to temperature observations was estimated to be 0.30 mm/km, due to air pressure observation 0.07 mm/km and due to determination of relative humidity 0.02 mm/km. These values are based on analyses of weather data at several baseline measurements, which the FGI performed in autumn 2008. Random errors, from least-squares adjustments of test field, ranged from 0.03 to 0.22 mm. The expanded uncertainties (2 s) of scale transfer at
Table 13.2 Results of the calibrations of the Kyvisˇke˙s baseline, slope distances (mm) between the top surfaces of observation pillars with expanded uncertainties (2 s) Pillars 1–2 1–3 1–4 1–5 1–6 1–7 2–3 2–4 2–5 2–6 2–7
Slope distance 100163.4 0.2 360177.1 0.3 1120386.7 0.8 1300483.7 0.9 1320495.1 0.9 841814.4 0.8 260013.8 0.3 1020223.3 0.7 1200320.4 0.8 1220331.8 0.8 775244.5 0.8
Pillars 3–4 3–5 3–6 3–7 4–5 4–6 4–7 5–6 5–7 6–7
Slope distance 760209.6 0.5 940306.7 0.7 960318.1 0.7 644380.7 0.7 180098.0 0.5 200110.0 0.5 804747.3 0.8 20012.6 0.2 933821.8 0.9 949189.6 0.9
Kyvisˇke˙s ranged from 0.2 to 0.9 mm. The results are summarized in Table 13.2 and used as true values in comparison with the GPS results.
13.3
GPS Observations
GPS observations with a 30 s observing interval were collected using Ashtech Z-XII3 GPS receivers and ASH7000936C_M choke ring antennas. Two 24-h sessions of GPS data were processed with Bernese 5.0 software (Dach et al. 2007) using different processing methods. All solutions were processed using the IGS precise orbits. A priori coordinates for the pillars were estimated using data from Vilnius (VLNS) IGS station, located within 15 km of the test field. Every baseline was processed independently without network adjustment, keeping one end fixed. The troposphere was handled using Dry Niell as the a priori model for the dry component. Since the troposphere parameters are highly correlated over short baselines, only relative troposphere parameters (the wet component) were estimated. A Wet Niell mapping function was used for estimating the parameters every 2 h with respect to the a priori model parameters of the fixed end of the baseline. Four different ambiguity resolution strategies were tested: an L1 and L1&L2 ambiguity resolution using a sigma-dependent ambiguity resolution as well as QIF (Quasi IonosphereFree) and narrow-lane strategies. With Bernese software, the narrow-lane method is performed in several steps. First, wide-lane ambiguities are resolved with the sigma-dependent method keeping the coordinates
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GPS Metrology: Bringing Traceable Scale to a Local Crustal Deformation GPS Network
fixed to the accurate a priori coordinates. In a second run, the ionosphere free L3 linear combination is processed; wide-lane ambiguities are introduced as known and narrow-lane ambiguities are resolved. All cases were processed using three different cutoff elevation angles (3, 10 and 20 ) and two different ionosphere models. The first ionosphere model is a local one (“loc” in figures and tables) which was generated using geometry free linear combination of undifferenced dual-frequency data from GPS observations at pillar 4. It is a single layer model in which all free electrons are assumed to be in a layer of infinitesimal thickness. The global ionosphere model (“glo” in figures and tables) offered by CODE (Centre for Orbit Determination in Europe) was used. Relative and absolute antenna calibration tables of the IGS and individual calibration tables from Geo++ were used during processing. Relative and absolute calibration tables by IGS give values for type calibrated GPS antennas; the ASH700936C_M was used in this study. An individual calibration gives independent absolute values for each antenna. When comparing the phase center offset, it is noticeable that individual calibrations of our seven antennas differed by up to 1.5 mm in a NE direction and by 2.5 mm in an up component from the absolute type calibration. When Schmitz et al. (2002) report that the standard deviation for robot calibration is in the order of 0.2–0.3 mm, leading to a position uncertainty of 1 mm, it is evident that there are significant differences in antennas of the same type.
13.4
Metrological Accuracy of GPS
Here the term metrological accuracy refers to the agreement of the GPS results with the traceable scale transfer results at Kyvisˇke˙s test field (Table 13.2). Results with different cut-off elevation angles and local and global ionosphere models are almost identical. Lowering the cut-off elevation angle from 20 to 3 decreased the rms of the results less than 0.1 mm (Table 13.3). However, in case of relative calibration tables, the use of a low cut-off elevation angle is debatable since the calibration values are not given for lower than 10 . The choice of the ionosphere model had even less of an influence on the results (Table 13.4). However,
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Table 13.3 Rms values in mm of the whole network when the Mekometer results from Table 13.2 are used as true values Cut-off Absolute Relative Individual
L1 3 0.7 0.7 0.3
10 0.7 0.7 0.3
20 0.7 0.7 0.3
QIF 3 1.9 1.9 0.5
10 2.0 2.0 0.6
20 2.0 2.0 0.6
Results are shown with different cut-off angles using a local ionosphere model Table 13.4 Rms values in mm of the whole network when Mekometer results from Table 13.2 are used as true values
Absolute Relative Individual
L1 loc 0.7 0.7 0.3
glo 0.7 0.7 0.3
L1&L2 loc glo 0.4 0.4 0.4 0.4 0.3 0.3
QIF loc 1.9 1.9 0.5
glo 1.9 1.9 0.5
Narrow loc glo 1.9 1.9 1.9 1.9 0.5 0.5
Only solutions with a 3 cut-off angle are shown. Results with local (loc) and global (glo) ionosphere models give identical results
a local ionosphere model may be useful during high solar activity. Tables 13.3 and 13.4 show that the major improvement was obtained by using individually calibrated antennas and that the L1 and L1&L2 solutions are better than QIF and narrow-lane solutions. Figure 13.3a–b show the results for absolute type and absolute individual antenna calibration solutions with a global ionosphere model and a 3 cut-off elevation angle. The results show the differences between the mean of two 24-h GPS sessions and the Mekometer true values (Table 13.2). The uncertainty bars represent standard uncertainties of the Mekometer scale transfer. Absolute type calibrations (Fig. 13.3a) give similar results as relative type calibration, which is not shown here. Figure 13.3b shows results using individual calibration. One antenna caused significant discrepancies (with the individual calibration table only) and is therefore left out of all the results. Results of this antenna deviated up to 4.3 mm from the true EDM values. This and some additional tests suggest biases in the calibration values for this antenna. Further studies for this antenna will be performed. Notice that Figs. 13.3a and b have different scales, since the deviation of results decreases significantly. Table 13.5 shows that the maximum deviations from the true values were up to 3.9 mm when type calibration tables were used.
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Fig. 13.3 The deviation of GPS results from the Mekometer results with (a) absolute type, and (b) individual antenna calibrations. One antenna was rejected (see text) from both series. Note that the scale of a and b is different. The results were obtained using a global ionosphere model and a 3 cut-off angle. The uncertainty bars are standard uncertainties of Mekometer solutions
Table 13.5 Minimum, maximum and mean in mm when GPS is compared to true values from Table 13.2 Relative
Absolute
Individual
Mean Min Max Mean Min Max Mean Min Max
L1 0.1 1.3 1.0 0.1 1.3 1.0 0.1 0.6 0.4
L1&L2 0.1 0.7 0.7 0.1 0.7 0.7 0.1 0.5 0.3
QIF 0.0 3.9 2.8 0.0 3.9 2.8 0.1 0.9 0.8
Narrow 0.0 3.9 2.8 0.0 3.9 2.8 0.1 0.9 0.8
Figure 13.4a–b show the L1 and QIF solutions of Fig. 13.3b, where individual antenna calibrations were used. The uncertainties represent the repeatability of the two 24-h GPS solutions. L1&L2 and narrowlane solutions are not shown because the results are similar to L1 and QIF results, respectively. The uncertainties of the L1 solutions are clearly distance dependent, although the ionosphere model was used. The uncertainties of QIF are smaller and not distancedependent. However, the QIF results deviate more from the true values than the L1 solutions. This may indicate small uncertainties in antenna calibration or site-specific multipath, etc., effects when the linear combinations of L1 and L2 are created. The difference between daily GPS results shows that L1 and L1&L2 solutions have a distance dependency of 0.4–0.6 ppm (Fig. 13.5) with both a global
and a local ionosphere model when all the baselines are considered. Elevation cut-off angle did not have a significant effect on the distant-dependent scaling error. QIF and narrow-lane techniques do not have a statistically significant distance dependency. The distance dependency might be partly caused by possible rapid fluctuations in tropospheric delays and by ionospheric effects that were not successfully estimated. In Fig. 13.6 the metrological accuracy of the GPS solutions is shown. When type calibrated antennas were used, L1&L2 gives the best results with rms of 0.4–0.5 mm. The rms of L1 is 0.7 mm and of QIF and narrow-lane between 1.9 and 2.0 mm. When individually calibrated antennas were used, the rms values decreased to 0.3 mm for L1 and L1&L2 solutions and to 0.5–0.6 mm for QIF and Narrow-lane solutions. The ionosphere model or cut-off elevation angle did not have a significant influence on the results. Figure 13.7 shows the precision of GPS solutions by means of the repeatability of the subsequent processing days. QIF and narrow-lane solutions have the best repeatability with rms of 0.2 mm. The rms of L1 solutions is 0.4–0.5 mm and of L1&L2 0.5–0.6 mm. By using the global ionosphere model, the rms of L1 and L1&L2 solutions is 0.1 mm smaller. Conclusions
When the best possible metrological accuracy is required, the use of individually calibrated GPS antennas is strongly recommended. In particular,
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GPS Metrology: Bringing Traceable Scale to a Local Crustal Deformation GPS Network
Fig. 13.4 L1 and QIF solutions from Fig. 13.3b, with the uncertainties derived from the repeatability of two GPS solutions. The figures clearly show a distance dependent Fig. 13.5 The distance dependency (in ppm) of the repeated GPS solutions for different processing strategies using a global ionosphere model
Fig. 13.6 Accuracy of GPS. The rms values of the different processing strategies with a global ionosphere model. The Mekometer results (Table 13.2) have been used as true values
Fig. 13.7 Repeatability of GPS with a global ionosphere model. The rms values of differences between subsequent GPS solutions indicating the precision of GPS solutions
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increase of uncertainty in the L1 solution. The uncertainty of the QIF solutions is not distance dependent, but the results deviate more from Mekometer results than the L1 solutions
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if the linear combinations of L1 and L2 are used, the daily GPS results with type calibrated (high quality choke ring) antennas may differ nearly 4 mm from the true values. With individually calibrated antennas, all daily solutions were within 1 mm of the true values. At the Kyvisˇke˙s test field, with baseline lengths between 20 and 1,320 m, metrologically the best GPS accuracy was obtained with the L1 solution and individually calibrated GPS antennas. The rms and maximum difference with respect to the true values were 0.3 mm and 0.7 mm, respectively. If only type calibrations are available, the best accuracy is obtained with an L1&L2 solution (rms was 0.4 mm and max. difference 0.7 mm). The cut-off angle did not have a significant effect on the results and there were no difference if global or local ionosphere model was used. The QIF and Narrow-lane results deviate more (rms 1.9 and max difference 3.9 mm) from the true values than the L1 or L1&L2 solutions. Small uncertainties in antenna calibration or site-specific multipath affect the phase centre of a linear combination more and the problem is clearly visible. However, the precision of QIF and narrow-lane is very high and it seems that already on distances of 1 km they fall within the expanded uncertainties of the Mekometer results, when at the same time the uncertainties of L1 and L2 become larger due to the ppm-effect. None of the linear combinations reach the accuracy of the Mekometer on baselines shorter than 500 m. At the Olkiluoto deformation network in Finland, a relative antenna calibration model and an L1&L2 solution with a local ionosphere model were used. According to this study, the method gives the smallest rms if an individual calibration table is not available. Part of the offset of 0.64 mm may be caused by the 0.5 ppm distance dependent effect shown in this study. The other part may be related to antenna calibration, but this cannot be verified yet since the rejected antenna (because of large
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discrepancies when the individual calibration table was used) is the one used at the Olkiluoto baseline. Acknowledgements This project was partly funded by the Academy of Finland; Decision number 122822. Prof. Martin Vermeer and his group from the Helsinki University of Technology (Laboratory of Geoinformation and Positioning) are acknowledged for letting us to borrow their Mekometer Kern ME5000.
References Ahola J, Koivula H, Poutanen M and Jokela J (2008) GPS Operations at Olkiluoto, Kivetty and Romuvaara in 2007. Working Report 2008–35. Posiva Oy. p 189 BIPM (2008) Evaluation of measurement data – guide to the expression of uncertainty in measurement (GUM). JCGM 100:2008. Joint Committee for Guides in Metrology. p 120 Buga A, Jokela J and Putrimas R (2008) Traceability, stability and use of the Kyvisˇke˙s calibration baseline – the first 10 years. In Cygas, D. and K.D. Froehner (eds.): The 7th International Conference Environmental Engineering, Selected Papers, vol 3, p 1274–1280. Vilnius, Lithuania, May 22–23, 2008 Dach R, Hugentobler U, Fridez P, Meindl M (eds) (2007) Bernese GPS Software Version 5.0. Astronomical Institute, University of Bern, Switzerland G€orres B, Campbell J, Becker M, Siemes M (2006) Absolute calibration of GPS antennas: laboratory results and comparison with field and robot techniques. GPS Solutions 10:136–145 Jokela J, Petrosˇkevicˇius P, Tulevicˇius V (1999) Kyvisˇke˙s Calibration Baseline. Reports of the FGI, 99:3, p 15 Jokela J, H€akli P, Ahola J, Buga A, Putrimas R (2009) On traceability of long distances. In: Proceedings of XIX IMEKO World Congress, Fundamental and Applied Metrology, September 6–11, 2009, Lisbon, Portugal, pp 1882–1887, IMEKO, ISBN 978-963-88410-0-1 Mader GL (1999) GPS antenna calibration at the National Geodetic Survey. GPS Solutions 3:50–58 Schmitz M, W€ubbena G, Boettcher G (2002) Tests of phase center variations of various GPS antennas, and some results. GPS Solutions 6:18–27 Wanninger L (2009) Correction of apparent position shifts caused by GNSS antenna changes. GPS Solutions 13:133–139 W€ubbena G, Schmitz M, Menge F, B€oder V, Seeber G (2000) Automated absolute field calibration of GPS antennas in real time. Proceedings of ION GPS 2000, 19–22 September, Salt Lake City, Utah, USA
Impact of Albedo Radiation on GPS Satellites
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C.J. Rodriguez-Solano, U. Hugentobler, and P. Steigenberger
Abstract
GPS satellite orbits available from the International GNSS Service (IGS) show a peculiar pattern in the SLR residuals at the few centimeter level that is related to radiation pressure mismodeling. Part of the mismodeling may be attributed to neglecting the solar radiation reflected and reemitted from the Earth, the albedo radiation, as most IGS analysis centers do not yet take into account this radiation pressure component. In this study the relative importance of different albedo model constituents is analyzed. The impact of nine albedo models with increasing complexity is investigated using 1 year of global GPS data from the IGS tracking network. The most important model components are the solar panels of the satellites while different Earth radiation models have a minor impact on orbits at GPS altitudes. Albedo radiation has the potential to remove part of the anomalous SLR residual pattern observed by Urschl et al. (Calibrating GNSS orbits with SLR tracking data. Proceedings of the 15th International Workshop on Laser Ranging, 2008) in a Sun-fixed reference frame.
14.1
Introduction
Satellite Laser Ranging (SLR) to the two GPS satellites SVN35 and SVN36 that are equipped with laser retro reflectors arrays (LRA) shows a consistent bias relative to the IGS final orbits of 4–5 cm which has been called the GPS-SLR orbit anomaly. Urschl et al. (2005, 2008) and Ziebart et al. (2007) found indications that this bias could be due to the Earth radiation impacting the satellites, an effect that is
C.J. Rodriguez-Solano U. Hugentobler (*) P. Steigenberger Institute for Astronomical and Physical Geodesy, Technische Universit€at M€unchen, Arcisstraße 21, 80333 Munich, Germany e-mail:
[email protected]
currently not yet included in the computation of most analysis centers contributing to the IGS final orbits. Consequently there is an increasing interest in the scientific community that uses GPS for very precise applications to understand the impact that the Earth radiation has on GPS satellites orbits. There is an interest in developing models that are of sufficient fidelity to represent the characteristics of the Earth albedo effect on GPS orbits but simple enough for easy implementation and handling. In the literature different ways to construct Earth radiation and satellite models are given. Analytical descriptions of the Earth radiation acting on artificial satellites have been developed, see Borderies and Longaretti (1990). The mathematical model used in our study was developed by Knocke et al. (1988). This model assumes that the Earth reflects and emits
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_14, # Springer-Verlag Berlin Heidelberg 2012
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radiation in a purely diffuse way like a Lambertian sphere, and any specular reflection is assumed to have a small contribution to the irradiance received by a satellite. The model was constructed for latitudedependent terrestrial surface reflectivity and emissivity but can easily be adapted to constant surface properties or to properties derived from satellite measurements. Using such a model and Earth reflectivity and emissivity satellite data from the CERES [Clouds and the Earth’s Radiant Energy System, (Wielicki et al. 1996)] and ERBE [(Earth Radiation Budget Experiment, (Barkstrom 1984)] missions, Ziebart et al. (2004) computed the irradiance of the Earth that reaches the GPS satellites. More sophisticated models, like the one of Martin and Rubincam (1996) used information from the ERBE mission to consider a more accurate phase function for radiation reflected on the Earth’s surface than the Lambertian scattering law. Regarding the satellite models, one can find in the literature many works dealing with the interaction of the satellites with solar radiation pressure, see, e.g., Sibthorpe (2006). These existing models can be adapted for the case of Earth radiation interacting with GPS satellites. For example Fliegel et al. (1992) and Fliegel and Gallini (1996) have made public the optical properties and dimensions of the different GPS satellite blocks, together with the physical description of the effect of radiation on the surface of the satellites. Ziebart et al. (2005) have developed sophisticated satellite models for GPS satellites. In a study based on several days of global GPS tracking data Ziebart et al. (2007) demonstrated the impact of albedo radiation, including microwave antenna power thrust, on GPS satellite orbits and showed a reduction of the GPS-SLR anomaly – the anomalous SLR range bias to IGS orbits – by 2 cm. In a different approach based on the adjustment of GPS measurements for constructing radiation pressure models, we find the empirical models developed by Springer et al. (1999) and Bar-Sever and Kuang (2004). This study aims not at the development of the most sophisticated albedo models but at the identification of the relative importance of different albedo model constituents. The impact of nine albedo models with increasing complexity was investigated using 1 year of global GPS data (2007) from the IGS tracking network. These models include different Earth radiation models starting from a simple analytical representation of the Earth’s visible and infrared radiation to a
C.J. Rodriguez-Solano et al.
radiation model derived from satellite measurements, and different satellite models from simple cannon-ball to box-wing models with different surface properties. Details on the models may be found in RodriguezSolano (2009).
14.2
Models
14.2.1 Earth Radiation Models The irradiance received by an artificial satellite, due to the Earth’s reflected (visible) and emitted (infrared) radiation, is calculated by introducing three main assumptions: the Earth behaves like a Lambertian sphere, the radiation is reflected or emitted at its surface and there is a global conservation of energy, i.e., all the energy received by the Earth from the Sun must also leave it (as reflected or emitted radiation). To compute the irradiance received by the satellite, first the solar irradiance received by each surface element of the Earth visible by the satellite is determined. Then the irradiance received by the satellite based on that element’s reflectivity and emissivity is computed. Finally, integration over all surface elements provides the irradiance at the location of the satellite. Reflectivity and emissivity coefficients of surface elements of the Earth were obtained from monthly satellite data from NASA’s CERES project (Kusterer 2009) to construct our most sophisticated model of the Earth radiation. A simplified version of the model considers only a latitude-dependency of the coefficients obtained from CERES data. Finally, the most simple model considers a globally constant albedo of a ¼ 0.3 and an emissivity of e ¼ 1 – a for the entire Earth. Assuming that the Earth’s irradiance reaches the satellite just in radial direction (a reasonable approximation if the satellite’s distance is much larger than the Earth’s radius) allows us to perform the integration over the visible illuminated surface of the Earth analytically. This most simple albedo model is very easy to implement by one line of source code. In total four Earth radiation models with increasing complexity were tested. E1: a model based on analytical integration over the Earth’s surface assuming a constant albedo; E2: a model based on numerical integration over the Earth’s surface considering constant albedo; E3: a numerical model with latitudedependent reflectivity and emissivity of the Earth’s
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surface elements; E4: a numerical model adopting time-dependent reflectivity and emissivity from CERES satellite data.
Ziebart et al. (2004), which gives an extra acceleration in the radial direction, comparable in magnitude to the effect of Earth radiation.
14.2.2 GPS Satellite Models
14.3
Different satellite models were constructed to describe the interaction with the radiation coming from the Earth. These models are mainly based on the work of Fliegel et al. (1992) and Fliegel and Gallini (1996) that provide dimensions and reflectivity n and specularity m for the surface elements of GPS satellites. Since the irradiance consists of a visible and an infrared component, the optical properties of the surfaces must be known also for the infrared. For this study the infrared properties were assumed to be n ¼ 0.2 and m ¼ 0.5 for all types of satellites and surfaces. This choice is justified by the necessity of the satellites to dissipate heat into space, the surfaces thus having a high emissivity for thermal radiation. The infrared specularity was adopted based on the assumption that the surfaces reflect equally in a diffuse and in a specular way. These assumptions were necessary since the properties reported by Fliegel et al. (1992) and Fliegel and Gallini (1996) are just for radiation in the visible part of the spectrum. The simplest adopted model was a cannon-ball type satellite model assuming constant cross-section and average optical properties. A simple analytical box-wing model was constructed adopting nominal attitude with the satellite’s solar panels oriented perpendicular to the Sun and the navigation antenna pointing to the center of the Earth. The acceleration for this analytical box-wing model assumes that irradiance from the Earth has only a radial component. Finally, a box-wing model was constructed where the irradiance was not only considered to be radial but coming from the full disc of the Earth seen by the satellite. This was called the numerical box-wing model, since no analytical expressions can be obtained. To summarize, three satellite models were tested: S1: the cannon-ball model; S2: the analytical boxwing model considering only radial albedo irradiance; S3: the numerical box-wing model. We also distinguish between satellite models with average optical properties and properties equal to the published blockspecific values (B). Finally we took into account the thrust of the navigation antenna (A) as reported by
A number of models were available for tests. The developed Earth radiation and satellite models were combined to obtain a sequence of models with increasing complexity, see Table 14.1. No a priori direct solar radiation pressure model was used for most experiments but five solar radiation pressure parameters were estimated for each orbit determination step. Nevertheless, tests including the ROCK models (see Fliegel et al. 1992; Fliegel and Gallini 1996) were done with the most sophisticated albedo model (ALB-9) as well as – for comparison – without any albedo model (ALB-R). Finally, tests including antenna thrust were executed (ALB-8 and ALB-9).
Results
14.3.1 Acceleration on the Satellites Figure 14.1 shows the radial, along track and cross track accelerations for the models ALB-1 to ALB-8 as a function of the longitude along the orbit, measured from the sub-solar point. The angle Du ¼ 0 thus corresponds to the point along the orbit that is closest to the direction to the Sun. The figure covers one revolution of SVN36 for an elevation of the Sun above the orbital plane of b ¼ 20.2 . For the cannon-ball models ALB-1 and ALB-2 the radial acceleration is simply proportional to the albedo irradiance. The maximum is reached with the satellite Table 14.1 Selection of earth radiation and satellite models Test # ALB-R ALB-0 ALB-1 ALB-2 ALB-3 ALB-4 ALB-5 ALB-6 ALB-7 ALB-8 ALB-9
Abbreviation E0-S0-R E0-S0 E1-S1 E2-S1 E2-S2 E3-S2 E4-S2 E4-S2-B E4-S3-B E4-S3-BA E4-S3-BA-R
Complexity change No albedo, ROCK a priori No albedo, no ROCK a priori Simplest albedo models Num. (const. albedo) model Box-wing analytical model Latitude-dependent albedo Incorporating CERES data Block-specific properties Box-wing numerical model Including antenna thrust With a priori ROCK model
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Fig. 14.1 Acceleration for selected models (b ¼ 20.2 ), SVN36
above the side of the Earth that is illuminated by the Sun while the minimum is found above the night side of the Earth. Adding the solar panels to the satellite model (ALB-3) changes the picture drastically as the satellite’s cross section as seen from the Earth varies much during one satellite revolution. The minima are reached when the geocentric directions to the satellite and to the Sun are at a right angle, where exposure of the solar panels to Earth radiation is minimal. A secondary maximum is found above the night side of the Earth where the solar panels are maximally exposed to the Earth’s infrared radiation.
Further improvement of the Earth radiation models including latitude-dependent albedo (ALB-4) or timeand position-dependent reflectivity and emissivity from CERES data (ALB-5) has no significant impact on the acceleration. The use of block-specific optical properties (ALB-6) has a noticeable impact while the use of the numerical box-wing model (ALB-7) changes the curves just around Du ¼ 90 and Du ¼ 270 , eliminating the sharp cusps of the analytical box-wing model. Finally, as expected, the antenna thrust (ALB-8) causes a radial offset of the acceleration.
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Impact of Albedo Radiation on GPS Satellites
For the along track and cross track components of the acceleration we can observe differences between models, too. The albedo acceleration in these two components is, however, an order of magnitude smaller than in the radial acceleration. Finally, note that the magnitude of the total acceleration is around 1 109 ms2 and thus of the same order as the Y-bias effect, see, e.g., Springer et al. (1999).
14.3.2 Impact on the Orbits After evaluating the acceleration induced by albedo radiation for a single orbital revolution of GPS satellites, the impact on orbits estimated using global GPS tracking data was studied. The processing scheme of the CODE (Center for Orbit Determination in Europe) was employed with the Bernese GPS Software (Dach et al. 2007) to analyze 1 year (Jan. to Dez. 2007) of GPS tracking data of about 190 globally distributed IGS stations (Dow et al. 2009). In fact, cleaned single difference files with fixed ambiguities from the CODE contribution to the IGS reprocessing (Steigenberger et al. 2011) were used to determine daily GPS orbits Table 14.2 Mean and standard deviation of orbit differences for different albedo models over 1 year for SVN36 in mm Difference ALB-1 – ALB-0 ALB-2 – ALB-1 ALB-3 – ALB-2 ALB-4 – ALB-3 ALB-5 – ALB-4 ALB-6 – ALB-5 ALB-7 – ALB-6 ALB-8 – ALB-7
Radial 16.4 0.4 1.5 0.1 0.9 2.0 0.3 4.7
Fig. 14.2 Orbit differences for models ALB-8 – ALB0 for satellite SVN36 for the year 2007
1.6 1.0 3.6 0.8 1.2 1.1 0.8 0.7
Along track 0.6 2.3 0.5 2.1 3.8 5.5 0.4 2.2 0.0 2.5 1.0 2.4 0.1 2.0 0.1 2.0
Cross track 0.2 0.9 0.0 0.6 0.0 5.2 0.0 0.6 0.0 1.2 0.0 1.1 0.0 0.3 0.0 0.1
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for each of the described albedo models estimating six orbital parameters and five empirical solar radiation pressure parameters as well as one stochastic pulse in the middle of the arc. For each run one of the albedo models listed in Table 14.1 was implemented. The orbit differences between successive solutions are appropriate to identify the factors that are essential for a proper modeling of the albedo effect for GPS satellites. Table 14.2 lists differences between orbits computed with different albedo models in radial, along track, and cross track directions in millimeters for SVN36. Mean and standard deviation of the differences were computed for 1 year. Figure 14.2 shows the differences between the orbits based on the box-wing model with CERES Earth radiation model (ALB-8) and orbits determined with no albedo model (ALB-0) for SVN36 for the entire year 2007. As one prominent feature we observe the radial offset (ALB-1 – ALB-0) that is common to all albedo models with respect to orbits computed without albedo model. As already noted by Ziebart et al. (2007) this effect reduces the aberrant SLR-GPS anomaly by 1–2 cm. The reason is that GPS measurements, being essentially angular measurements due to required clock synchronization, mainly determine the mean motion of the satellite. As a matter of fact, a constant positive radial acceleration (equivalent to a reduction of GM) decreases the orbital radius according to Kepler’s third law. The second important feature are the increased differences per revolution in all three components between orbits determined with a cannon-ball (ALB-2) and with a box-wing model (ALB-3) exhibiting themselves by a significantly increased standard deviation in Table 14.2 and a temporally correlated pattern displayed in Fig. 14.2 that is mainly due to the boxwing model. As already observed for the accelerations,
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Fig. 14.3 Radial residuals (in cm) between the models ALB-8 and ALB-0 for SVN36 in a Sun-fixed reference frame
the box-wing satellite model represents an essential key element of any albedo model. Refinement of the Earth radiation model from constant albedo (ALB-3) to latitude-dependent albedo (ALB-4) or to CERES derived surface properties (ALB-5), on the other hand, does not have an important impact on the orbits. Obviously GPS satellites are high enough such that varying surface albedo is averaged out over the visible illuminated surface area of the Earth. We may, finally, represent the radial orbit differences in a Sun-fixed reference frame as a function of longitude Du along the orbit, measured from the sub-solar point and the elevation b of the Sun above the orbital plane, see Fig. 14.3 for the same example represented in Fig. 14.2. We observe a significant radial deformation of the orbits as a function of satellite position with respect to the position of the Sun that resembles the pattern observed by Urschl et al. (2008) for SLR residuals of GPS satellites on the night-time side of the Earth. The amplitude of the effect is, however, only about 2 mm compared to the 5 cm effect found by Urschl et al. (2008). Note, however, that such a pattern is not present for a cannon-ball satellite model. Finally, an external validation of the different orbit solutions can be accomplished by performing an SLR validation for the satellites SVN35 and SVN36 that are equipped with LRAs. Results in form of mean, root mean square (RMS) and standard deviation of SLR residuals over the entire year 2007 are presented in Table 14.3. The results confirm what was already found from Table 14.2. The solutions with albedo (ALB-8 resp. ALB-9) show a radial bias with respect to solutions without albedo (ALB-0 resp. ALB-R) that is about 16 mm reduced. About 5 mm thereof are due to
Test # ALB-R ALB-0 ALB-1 ALB-2 ALB-3 ALB-4 ALB-5 ALB-6 ALB-7 ALB-8 ALB-9
Mean [cm] 3.24/3.45 2.36/2.61 0.80/1.07 0.92/1.13 0.99/1.38 0.98/1.35 0.92/1.29 1.12/1.46 1.15/1.48 0.68/1.01 1.51/1.84
RMS (cm) 4.83/5.02 3.62/4.16 2.86/3.42 2.95/3.48 2.91/3.49 2.90/3.49 2.89/3.48 2.96/3.55 2.97/3.55 2.82/3.38 3.37/4.02
Sigma (cm) 3.58/3.64 2.75/3.24 2.75/3.25 2.81/3.29 2.73/3.21 2.73/3.22 2.74/3.23 2.74/3.23 2.74/3.23 2.74/3.23 3.02/3.57
the antenna thrust. It is interesting to note that the solutions involving the ROCK a priori solar radiation model (ALB-R and ALB-9) show a larger bias as well as a larger standard deviation for both satellites. This is an indication that there is a problem with the ROCK model for Block II/IIA satellites, a conclusion that requires further investigation. A small change in bias is observed when implementing block-specific instead of typical optical properties. As expected Table 14.3 shows that the introduction of a box-wing model slightly reduces the standard deviation of the residuals while no significant change is observed when changing the Earth radiation model. Even the simplest analytical model (ALB-1) behaves astonishingly well in terms of standard deviation of SLR residuals. This is probably an indication that a significant fraction of albedo (apart from a bias) can be absorbed by the estimated empirical and stochastic parameters. Conclusions
The acceleration caused by Earth radiation pressure has a non-negligible effect on the orbits of GPS satellites. The acceleration has a similar order of magnitude as the so-called Y-bias. The effect of this acceleration on the GPS orbits is mainly a mean reduction of the orbit radius by about 1 cm. The radial orbit differences obtained by considering an albedo model based on a box-wing satellite model show a prominent dependency of the satellite’s position with respect to the direction of the Sun. The corresponding pattern has similarities to the patterns found by Urschl et al. (2008) in SLR residuals for SVN35 and SVN36. The size of the effect is, however more than a magnitude smaller.
14
Impact of Albedo Radiation on GPS Satellites
Nevertheless, albedo may have the potential to explain part of this behavior. The results of the study, based on 1 year of GPS tracking data, clearly indicate – consistently with the findings of Ziebart et al. (2007) – that albedo radiation as well as antenna thrust should be considered for high precision GPS orbit determination. Mandatory is the inclusion of the solar panels into the satellite model. The details of the Earth radiation seem, however, to have a minor effect at GPS satellite altitude.
References Barkstrom BR (1984) The earth radiation budget experiment (ERBE). Bull Am Meteorol Soc 65(11):1170–1185 Bar-Sever Y, Kuang D (2004) Improved solar-radiation pressure models for GPS satellites. NASA Tech Briefs (NPO-41395) Borderies N, Longaretti PY (1990) A new treatment of the albedo radiation pressure in the case of a uniform albedo and of a spherical satellite. Celestial Mech Dyn Astron 49:69–98 Dach R, Hugentobler U, Fridez P, Meindl M (2007) Bernese GPS Software, Version 5.0. Astronomical Institute, University of Bern Dow JM, Neilan RE, Rizos C (2009) The International GNSS Service (IGS) in a Changing Landscape of Global Navigation Satellite Systems. J Geodes 83(3–4):191–198 Fliegel H, Gallini T (1996) Solar force modelling of block IIR global positioning system satellites. J Spacecraft Rockets 33 (6):863–866 Fliegel H, Gallini T, Swift E (1992) Global positioning system radiation force model for geodetic applications. J Geophys Res 97(B1):559–568 Knocke PC, Ries JC and Tapley BD (1988) Earth radiation pressure effects on satellites. Proceedings of AIAA/AAS Astrodynamics Conference, pp 577–587
119 Kusterer J (2009) CERES Data and Information. http://eosweb. larc.nasa.gov/PRODOCS/ceres/table_ceres.html. Accessed 30 Sep 2009 Martin C, Rubincam D (1996) Effects of earth albedo on the LAGEOS I satellite. J Geophys Res 101(B2):3215–3226 Rodriguez-Solano CJ (2009) Impact of albedo modelling on GPS orbits. Master Thesis, Technische Universit€at M€unchen Sibthorpe AJ (2006) Precision non-conservative force modelling for low earth orbiting spacecraft. PhD Thesis, University College London Springer T, Beutler G, Rothacher M (1999) A new solar radiation pressure model for GPS satellites. Adv Space Res 23(4):673–676 Steigenberger P, Hugentobler U, Lutz S, Dach R (2011) CODE contribution to the IGS reprocessing. Tech. Rep. 1/2011. URL http://www.iapg.bv.tum.de/mediadb/1352924/ 1352925/CODE_Repro1.pdf 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 Urschl C, Beutler G, Gurtner W, Hugentobler U and Schaer S (2008) Calibrating GNSS orbits with SLR tracking data. Proceedings of the 15th International Workshop on Laser Ranging, pp 23–26 Wielicki BA, Barkstrom BR, Harrison EF, Lee RB III, Smith GL, Cooper JE (1996) Clouds and the earth’s radiant energy system (CERES): an earth observing system experiment. Bull Am Meteorol Soc 77(5):853–868 Ziebart M, Edwards S, Adhya S, Sibthorpe A, Arrowsmith P, Cross P (2004) High precision GPS IIR orbit prediction using analytical non-conservative force models. Proc ION GNSS 2004:1764–1770 Ziebart M, Adhya S, Sibthorpe A, Edwards S, Cross P (2005) Combined radiation pressure and thermal modeling of complex satellites: Algorithms and on-orbit tests. Adv Space Res 36(3):424–430 Ziebart M, Sibthorpe A, Cross P (2007) Cracking the GPS – SLR Orbit Anomaly. Proc. of ION GNSS 2007, pp 2033–2038
.
Session 2 Gravity of the Planet Earth Convenors: Y. Fukuda, P. Visser
.
On the Determination of Sea Level Changes by Combining Altimetric, Tide Gauge, Satellite Gravity and Atmospheric Observations
15
G.S. Vergos, I.N. Tziavos, and M.G. Sideris
Abstract
The determination, monitoring and understanding of sea level change at various spatial and temporal scales has been the focus of many studies during the past decades. The advent of satellite altimetry and the multitude of unprecedented in accuracy and resolution observations that it offers allowed, in combination with tide gauge data, precise determinations of sea level variations. The realization of the GRACE mission and the forthcoming GOCE data offer new opportunities for the estimation of sea level trends at regional and global scales and the identification of seasonal signals. In such studies, even though the data combination and processing strategies have been carried out carefully with proper control, a point that has been given little attention is error propagation and variance component estimation of the data variance-covariance matrices. The latter two are of significant importance in heterogeneous data combination studies, since on one hand error propagation can provide reliable estimates of the output signal error while variance component estimation allows for a rigorous control of the data covariance matrices and subsequent sound decisions on statistical testing of hypotheses involving least-squares residuals and the estimated deterministic parameters. This work presents some new ideas towards the estimation of sea level change through the combination of altimetric, tide gauge, atmospheric, and GRACE- and GOCEtype observables. The combination scheme is based on a hybrid deterministic and stochastic treatment of the data errors and an estimation of sea level changes through least-squares collocation with considerations for glacial isostatic adjustment and continental water outpour effects. The deterministic model parameters treat datum and geophysical correction model inconsistencies in the data used, while the stochastic part allows for a simultaneous determination of stochastic parameters included in the data in terms of residual signals. Within this mixed adjustment scheme with stochastic parameters, variance component estimation is
G.S. Vergos (*) I.N. Tziavos Department of Geodesy and Surveying, Aristotle University of Thessaloniki, University Box 440, 541 24 Thessaloniki, Greece e-mail:
[email protected] M.G. Sideris Department of Geomatics Engineering, University of Calgary, Calgary, AB, Canada S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_15, # Springer-Verlag Berlin Heidelberg 2012
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carried out using the iterative almost unbiased estimator method. The analytical equations for the prediction of the adjusted input and output signals are presented along with possible modifications of the observation equation for the determination of solely steric and atmospheric-driven sea level changes.
15.1
Introduction
Sea level changes at global and regional scales are triggered by a number of factors that take place within system Earth. These natural processes originate from variations in the physical properties of the ocean water and from water mass transport between the Earth’s oceans, continents and the atmosphere. The former, refers to variations in the seawater density triggered by salinity and temperature changes and results in so-called steric sea level changes. On the other hand, water mass transport due to changes in the continental reservoirs (river run-off), glacial and ice caps mass variations (melting) and atmospheric water vapor changes (precipitation and evaporation), trigger the socalled non-steric (eustatic) sea level changes (Chen et al. 2005; Chambers et al. 2004; Garcia et al. 2007). A traditional tool to monitor sea level changes at local, national and regional scales has been the deployment of tide-gauge (TG) stations along the coastline and their unification in a common vertical datum in order to refer to the same zero plane. The main disadvantages of such tide-gauge networks were and still are the inability to provide global estimates for the mean sea level change, the operational costs, the cumbersome nature of the measurements and the non-unified accuracy of the final sea level estimates. On the other hand, since tide-gauge stations exist for long periods of time, their record of sea level variations extends to many decades. The advent of satellite altimetry in the early 1980s resulted in the availability of sea surface height measurements with global coverage, homogeneous accuracy and resolution. These observations form an unprecedented database to monitor variations of the sea level at global scales, without the limitations of ground-based observations. Moreover, the original exact repeat missions (ERM) of ERS1 and TOPEX/ Poseidon were followed by ERS2/ENVISAT and JASON1/JASON2 respectively, which are set at the exact same orbit as their predecessors (Chelton et al. 2001). In that way, a 20-year record of observations
for the sea level is available from altimeters on-board satellites. The only disadvantages in the altimetric measurements for sea level monitoring are that a) their record is not as long as that of the tide-gauges, so that no definite conclusions can be drawn w.r.t. long-term predictions and trends for the sea level variations and b) due to the scattering of the radar pulse close to the coastline and in shallow-water regions their correlation with TG measurements not trivial. It should be pointed out that the TG and altimetric observations record both components, steric and non-steric, of the sea level change. Therefore if one would like to determine only the steric component then oceanographic observations about the salinity and temperature need to be employed. On the other hand, eustatic sea level changes imply the use of some models for water mass changes within the ocean, continents and atmosphere system. This situation arrived at a landmark in March 2002, when the Gravity Recovery and Climate Experiment (GRACE) mission was launched. The twin GRACE satellites operating in SST-ll (low-low satellite to satellite tracking) mode, managed to provide invaluable observations of the spatio-temporal variations of the Earth’s gravity field. These are utilized through the 10-day and monthly gravity field models generated from GRACE data, so that their differences with a mean (static) gravity field yield gravity variations. The variations are expressed as lateral mass changes under the assumption that they are caused by redistributions of the continental, oceanic and atmospheric water mass (Garcia et al. 2006). Based on its instrumentation, orbit and measurement precision, GRACE quantifies vertical integrated water mass changes with an accuracy of a few mm for spatial scales of ~400 km. Water mass changes are expressed as geoid height anomalies with respect to a static field, by evaluating time-dependent (dCnm ðtÞ and dSnm ðtÞ) spherical harmonic coefficients (Lombard et al. 2007; Swenson and Wahr 2002). The latter, result as differences between the temporal and mean
15
On the Determination of Sea Level Changes by Combining Altimetric
expansions of the Earth’s gravity field in spherical harmonics coefficients. From the previous analysis it becomes evident that new prospects, as far as sea level change monitoring is concerned, are offered. A combination scheme that utilizes satellite altimetry, GRACE and tide gauge data would in fact manage to lead to the optimal estimation of both steric and non-steric sea level anomalies (SLA) at variable spatial and time scales. If these observations are coupled with the forthcoming GOCE measurements of the second order derivatives of the Earth’s potential, which will result to more accurate estimates of the static geoid, then a heterogeneous data combination methodology can be outlined in order to estimate both sea level changes as well as the time-varying and quasi-stationary sea surface topography. In modern day geodetic research one of the most versatile, among others (Sanso` and Sideris 1997; Tziavos et al. 1998), and rigorous combination methods is that based on least-squares collocation (Moritz 1980). This study focuses on the presentation of a combination scheme based on a hybrid deterministic and stochastic treatment of the data errors and an estimation of sea level changes through leastsquares collocation (LSC). The deterministic model parameters treat datum and geophysical correction model inconsistencies in the data used (satellite altimetry and GRACE observations), while the stochastic part allows for a simultaneous determination of stochastic parameters included in the data in terms of residual signals. Analytical equations for the prediction of the adjusted input and output signals are presented along with reliable estimates of the output signal error covariance matrices. Finally, within this mixed adjustment scheme with stochastic parameters, a note on variance component estimation is carried out either using minimum norm quadric unbiased estimation or iterative almost unbiased estimator methods.
15.2
Data and Observation Equations
The analysis presented herein is based on the assumption that satellite altimetry, GRACE, GOCE and tidegauge observations are available. Furthermore, we assume that all time-varying observations refer to the same epoch, thus we neglect the variable of time t in the observation equations. For multi-epoch data,
125
e.g.,ti fi ¼ 1; 2; :::; ng; n-times more observations equations would be available.
15.2.1 Satellite Altimetry Observations Altimeters on-board satellites practically measure the instantaneous height of the sea surface from a reference ellipsoid. The observation equation for the altimetric measurements can be written as: MSS þ valt ; hSSH ¼ hSLA alt þ h
(15.1)
where hSSH are the altimetric sea surface heights (SSHs), hSLA alt denotes the sea level anomaly (SLA) as observed by the altimeter, hMSS denotes the mean sea surface (MSS) and valt are the errors of the observations, which are estimated to be ~4 mm (Chelton et al. 2001). Note that hSLA alt , contrary to GRACE measurements, contains both the steric an the non-steric components of the sea level variations. Some considerations for the altimetric measurements are a) all geophysical and instrumental corrections need to be applied, b) they also need to be corrected for the inverse barometric (IB) effect, so that the total ocean mass signal will be accounted for (Garcia et al. 2007; Vergos et al. 2005), and c) the contribution of the glacial isostatic adjustment (GIA), which reduces the global mean sea level by ~0.3 mm/y needs to be accounted for in order to explain the altimeter sea level rise due to climate factors (Lombard et al. 2007). Note that the IB correction should be exercised with care, since GRACE data observe the real water mass signal so in order to be consistent with that altimetric observations should not be IB-corrected in that case. On the other hand if we are interested on the total ocean mass signal then the IB correction should be applied to altimetry data and restored to GRACE observations. In (15.1) hSLA alt contains both the steric and non-steric SLA, so it can be written as altSLA altSLA hSLA alt ¼ hsteric þ hnonsteric where the steric effect equals to that of the time-varying sea surface topography (SST) dz. Knowing that the MSS height can be expressed in terms of the geoid height N and the quasistationary SST zc we can re-write (15.1) as: c alt hSSH ¼ hSLA alt þ N þ z þ v :
(15.2)
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If multi-mission satellite altimetry data are to be used, bias aalt and tilt balt parameters can be introduced in (15.2) to account for orbital errors, multi-mission MSL deviations, remaining tidal-effects, datum inconsistencies, etc. Therefore the final observation equation for altimetric data can be formed c alt alt dt þ valt : (15.3) hSSH ¼ hSLA alt þ N þ z þ a þ b
15.2.2 Tide-Gauge and GPS Observations For the incorporation of tide-gauge observations we assume that a local TG is available so that a locally MSS determined MSS HTG from its series of measurements is available. Moreover, it is equipped with a GPS receiver so that its ellipsoidal height hTG can be determined. Then we can write the initial observation equation for the TG measurements as: c TG hTG ¼ HMSS þ vTG ; TG þ N þ z þ a
(15.4)
where vTG are the errors of the observations and aTG is a bias parameter to account for the deviation of the local MSL, determined by the TG station, compared to the global one determined by altimetry. Note that this bias parameter can be also regarded as the deviation between a so-called local geoid (local equipotential surface Wo determined by the TG measurements) and a global geoid. It is acknowledged that the incorporation of GPS observations to TG data limits their record, but it is required in order to relate them MSS to geoid heights. The term HTG can be further decomposed in order to include the instantaneous ISL and the SLA observations of the TG station HTG MSS ISL hSLA , since HTG ¼ HTG þ hSLA . Note that hSLA is no more the local SLA observed by the TG, but the so-called global one since the bias term aTG has been included in the observation equation. If we further decompose the errors of the observations in those due to the GPS measurements and the ISL and SLA measurements of the TG we arrive at the final observation equation for the TG records:
15.2.3 GOCE Observations GOCE, was launched in March 17, 2009 and through its satellite gradiometry measurement principle will provide observations of the disturbing potential and its second order derivatives. It is foreseen that the geopotential models that will result from GOCE, in the form of spherical harmonic expansions of the Earth’s potential, will have a cumulative geoid accuracy of 1 cm to degree and order 200. For GOCE observations to be included in the combination scheme, we assume that (a) data for the disturbing potential T and its second order derivatives Trr are available and (b) these observations are processed ones, i.e., they represent the static gravity field as sampled by GOCE. The observation equations can then be written as: T ¼ TGOCE þ vT
h
=
HISL TG
þh
SLA
;
(15.6)
GOCE
:
(15.7)
þ vTrr Trr ¼ TGOCE rr
In (15.6) and (15.7) vT and vTrr are prediction errors of T and Trr with given covariance and cross-covariance structure.
15.2.4 GRACE Observations As it has already been mentioned, GRACE data are available in terms of spherical harmonics expansion of the geoid as static 10-day and monthly fields. Since we are interested in mass variations we will utilize the differences between a variable and the static model in order to have available at hand geoid height (actually potential) changes. These can be translated into water thickness equivalent, thus allowing the determination of sea level changes. It should be once again stressed that GRACE data over the oceanic domain observe only the variations of the sea level due to ocean mass change and not the steric part of it. Lets assume that we take the difference between a GRACE monthly solution w.r.t. to a mean field then the geoid height variation N(t) can be expressed as dN(t) ¼ R
TG
GOCE
1 X n X
fPnm ðcos yÞ
n¼1 m¼0
dCnm ðtÞ cos m l þ dSnm ðtÞ sin m l ;
þN HISL TG
þ zc þ aTG þ vhTG þ v
þ vh
SLA
:
(15.5)
(15.8)
15
On the Determination of Sea Level Changes by Combining Altimetric
where d{.} denotes difference and all other terms in (15.8) are known and need not be defined (Heiskanen and Moritz 1967). Then surface mass variations can be determined (Garcia et al. 2006) Ds ¼
1 X n RrE X 2n þ 1 Pnm ðcos yÞ; 3 n¼1 m¼0 1 þ kn (15.9)
½dCnm cos ml þ dSnm sin mlg where rE is a mean density of solid Earth and kn denotes load love numbers. Finally, utilizing the mass variations Ds GRACE inferred SLA can be determined: hSLA GRACE ¼
Ds ½mm; 1:029
(15.10)
If we combine the GRACE SLA observations with the corresponding ones from altimetry, we can write the final observation equation for GRACE data SLA
SLA SLA hGRACE hSLA ; GRACE ¼ halt hsteric þ v
(15.11)
Some notes for the GRACE data and resulting SLA should be pointed out. GRACE spherical harmonic expansions do not include degree1 harmonics, therefore in order to be consistent with the altimetric data (T/P, JASON1/2, etc.) the ones used for the latter should be employed. Furthermore, during GRACE data processing, atmospheric and barotropic models are used to eliminate atmospheric and ocean loading effects. But, since we are interested in the full ocean signal (see also the discussion in Sect. 15.2.1 for the altimetry data), these need to be restored in consistency with the ones applied for the altimetric data processing (Garcia et al. 2007). Finally, GRACE data need to be corrected for the GIA where a linear term of ~1.7 mm/y is sufficient (Lombard et al. 2007).
15.3
Data Combination
Having outlined the observation equations for the input data to be used, (15.3), (15.5), (15.6), (15.7) and (15.11) we can compact them in the following form based on LSC with parameters
127
y ¼ Ax þ s þ v;
(15.12)
where the bold faced letters denote vectors, y is the vector of observations, x is the vector of parameters, A is the design matrix, s is the vector of signals and v is the vector errors. Based on the analysis of the observations to be used we can write the full observation vector SSHT y ¼ hm1
T
hTTG
hSLA GRACE
q1
k1
TT
p1
TTrr
T ; (15.13)
p1
where we assume to have available m altimetric, q TG, k GRACE and p GOCE observations and the vector of deterministic parameters as 22
3 ::: 6 6 aalt 7 6 7 x¼6 4 4 balt 5 ::: m1
3T aTG q1
7 0ðkþ2pÞ1 7 5
(15.14)
The signals must all be consistently related through one and the same covariance function, so we can write the matrix of signals (Barzaghi et al. 2009) 3 c SLA g T þ z þ halt 7 61 6 g T þ zc þ HISL TG 7 7 61 s ¼ 6 dT dz 7: 7 6 g 5 4T Trr 2
1
(15.15)
Note that in (15.15) instead of hSLA steric we have used the signal of the time-varying SST dz which is equivSLA alent. The same holds for halt steric as it is already mentioned in Sect. 15.2.1. If we assume that: (a) the noises of the input data are independent from one another and from T, (b) dz has some isotropic covariance (CV) function over the area under study, (c) dz is independent of T and from all other noises and errors, (d) the errors v are noises with either given variances or prediction errors with known CV structure, (e) T has some isotropic covariance function, (f) cn(T) are the degree variances of the geopotential model up to its maximum degree nmax and we further denote sn(T) the degree variances from nmax to 1, then we can write the covariance function of T as
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CTT ðP; QÞ ¼
!nþ1 nmax
GM 2 X R2 cn ðTÞ Pn cos cPQ rQ rP R n¼0
GM 2 þ R
1 X n¼nmax þ1
s n ð TÞ
R2B rQ rP
!nþ1
sn ð dTÞ ¼ R2
n¼1 m¼0
Pn cos cPQ ;
(15.16)
where RB denotes depth to Bjerhammar sphere and the degree variances sn(T) can be determined by the Tscherning and Rapp (1974) model. Having our fundamental CV function we can determine all auto- and cross-covariance function of all other functionals through covariance propagation (see Barzaghi et al. 2009 for analytical expressions). In complete analogy we can define the covariance function of the quasi-stationary SST as C z c zc ¼
1 X n¼0
2ðnþ1Þ RB c sn ð z Þ Pn ðcos cÞ; (15.17) R
where sn ðzc Þare the degree variances of zc. Note that in all cases the analytical covariance function models should agree to the empirical values available for the area under study in order to represent the local statistical characteristics of the signal under consideration, i.e., the quasi-stationary SST in this case (Knudsen 1991). For the description of the behaviour of the degree variances given in (15.16), Knudsen (1987, 1992, 1993) and Knudsen and Tscherning (2006) use a 3rd degree Butterworth filter so that the degree variances of the MDT are given as:
k23 k13 : ðsn ð z Þ Þ ¼ b 3 k2 þ n3 k13 þ n3 c
2
nmax X n n o X ðdCnm Þ2 þ ðdSnm Þ2 ;
(15.18)
The factors b, k1, k2 and RB are determined so that the analytic model fits the empirical values describing the statistical characteristics of the quasi-stationary SST in the area under study and more precisely the variance and the correlation length. Various such models of the quasi-stationary SST are available and can be employed as for instance the one by Rio (Rio and Hernandez 2004). For the GRACE data covariance function, (15.16) is to be used with the only difference that the signal degree variances are now expressed as
(15.19) The final covariance function that we need to determine in order to have the complete problem set-up and proceed to the estimation of the signals is that of the time-varying SST. In Knudsen (1991) a model of the signal degree variances of the dz has been introduced, so that since dz varies with time, time-dependency should enter in the computation of Cdzdz in a way that temporal correlations are given in the same way as spatial ones. Therefore, for some time-separated points Dt ¼ jt t0 j the covariance function can be expressed as: Cdzdz ðc; DtÞ ¼ 8 9 1
> < P ðsn ð d zÞÞ2 Pn cos c þ kdz Dt for c þ kdz Dt p > = ; n¼0
> > : for c þ kdz Dt > p ; 0
(15.20)
where kdz is a conversion factor representing in the case of the time-varying SST the correlation time of the signal. This should be studied and determined in each region under study, since the characteristics of dz vary significantly for each area and in open or closed sea regions (Knudsen and Tscherning 2006). The degree variances sn ðd zÞ are determined as in (15.17) in order to fit the local characteristics of the time-varying SST. Since the time-varying SST is triggered by salinity and temperature variations, in-situ oceanographic data and climatology models can be used for the fit of the analytical model to empirical values. Given the analytical expressions for the covariance functions of all observations, it is possible to proceed to the simultaneous estimation of the deterministic parameters and of the signals z c, dz, T, hSLA nonsteric , etc., along with their prediction errors as:
1 ^x ¼ AT C1 A AT C1 yy yy y;
(15.21)
^zc ðPÞ ¼ Czs ðP; ÞC1 y Ax ^ ; yy
(15.22)
n o s^2c ðPÞ ¼ Czc zc ðP; PÞ Czc s C1 C1 AN1 AT C1 Cszc : yy yy yy z
(15.23)
15
On the Determination of Sea Level Changes by Combining Altimetric
^hSLA ðPÞ ¼ C SLA ðP; ÞC1 ðy A^ xÞ; hSTERIC s STERIC yy
(15.24)
SLA ðP; PÞ ChSLA s2^hSLA ðPÞ ¼ ChSLA STERIC hSTERIC STERIC s STERIC n o 1 1 T 1 C1 : yy Cyy AN A Cyy CshSLA STERIC
(15.25) The estimation of all other signals and prediction errors can be done according to (15.22)–(15.25). A short note on the estimation of variance components within this optimal combination scheme will be given. Given the general observation equation for LSC estimation with parameters and knowing that v ~ (0, Vy ¼ Cvv) we can write Cvv in a form as to depend on a unknown set ofPso-called variance components yi. Therefore, Vy ¼ yi Vi and the matrices Vi come from the original datai error CV matrices. Within the present combination scheme Vy would take the form " Vy ¼ yhalt Qhalt þ yhTG 1m mm
2
6 4
# yHISL TG q3
QhTG
0 QHISL TG
0
yhSLA 3 7 5
QhSLA
3q3q
þ yhSLA QhSLA þ yT QT þ yTrr QTrr GRACE GRACE 1k
kk
1p pp
1p pp
(15.26) The variance components can then be estimated by an iterative procedure such as the iterative almost unbiased estimation-IAUE (Rao and Kleffe 1988). According to IAUE, we first have to compute matrix W as T 1 1 T 1 1 W ¼ C1 A Cvv ; vv Cvv A A Cvv A
(15.27)
and then estimate the variance components ^y y ¼ Jþ k depending on according to ^y ¼ J1 k or ^ whether matrix J can be inverted. Matrices J and k can be analytical determined from the data error CV matrices Jij ¼ tr WCvi vi WCvj vj ;
(15.28)
1 ki ¼ ^ vT C1 v: vv Cvi vi Cvv ^
(15.29)
129
After the first variance components have been estimated, the iterations can be carried out according to ^y
a i
¼
^ya1 ^vT C1 Cv v C1 ^v i i i vv vv : trfWCvi vi g
(15.30)
where ^y a1 is the previous estimate and y^ ai is the i new one. When their difference is smaller than a a1 ^ ^ a certain threshold e, so that y y < e, then convergence is achieved, the variance components are estimated and the iterations can stop. Conclusions
A detailed combination scheme of satellite altimetry, tide-gauge, GRACE and GOCE data for the determination of various signals both static and variable has been presented. The latter, can be the disturbing potential, its second order derivatives, geoid heights, steric and non-steric sea level variations and the stationary and time-variable sea surface topography. It is clear that the versatility of LSC with parameters allows including other observations too, like gravity anomalies, salinity and temperature from oceanographic measurements, hydrological models, etc. Therefore, the proposed combination scheme can be extended easily as new data sources become available, so that better estimates will be derived. Another advantage of the combination strategy outlined is that rigorous signal estimation errors can be derived along with the possibility of variance component estimation so that the covariance matrices and errors models can be calibrated.
References Barzaghi R, Tselfes N, Tziavos IN, Vergos GS (2009) Geoid and high resolution sea surface topography modelling in the mediterranean from gravimetry, altimetry and GOCE data: evaluation by simulation. J Geod 83(8):751–772 Chambers DP, Wahr J, Nerem RS (2004) Preliminary observations of global ocean mass variations from GRACE. Geophys Res Lett 31:L13310. doi:10.1029/ 2004GL0204 61 Chelton DB, Ries JC, Haines BJ, Fu LL, Callahan PS (2001) Satellite altimetry. In: Fu LL, Cazenave A (eds) Satellite altimetry and earth sciences: a handbook of techniques and applications, vol 69, International Geophysics Series. Academic Press, San Diego, CA, pp 4–131 Chen JL, Wilson CR, Tapley BD, Famiglietti JS, Rodell M (2005) Seasonal global mean sea level change from satellite
130 altimeter, GRACE and geophysical models. J Geod 79:532–539 Garcia D, Chao BF, Rio JD, Vigo I (2006) On the steric and mass-induced contributions to the annual sea level variations in the Mediterranean Sea. J Geophys Res 111:C09030. doi:10.1029/2005JC002956 Garcia D, Ramillien G, Lombard A, Cazenave A (2007) Steric sea-level variations inferred from combined Topex/Poseidon altimetry and GRACE gradiometry. Pure Appl Geophys 164:721–731 Heiskanen WA, Moritz H (1967) Physical geodesy. W.H. Freeman, San Fransisco, CA Knudsen P (1987) Estimation and modelling of the local empirical covariance function using gravity and satellite altimeter data. Bull Geod 61:45–160 Knudsen P (1991) Simultaneous estimation of the gravity field and sea surface topography from satellite altimeter data by least-squares collocation. Geophys J Inter 104(2):307–317 Knudsen P (1992) Estimation of sea surface topography in the Norwegian sea using gravimetry and Geosat altimetry. Bull Ge´od 66:27–40 Knudsen P (1993) Integration of gravity and altimeter data by optimal estimation techniques. In: Rummel R, Sanso` F (eds) Satellite altimetry for geodesy and oceanography, vol 50, Lecture Notes in Earth Sciences. Springer, Berlin, pp 453–466 Knudsen P, Tscherning CC (2006) Error Characteristics of dynamic topography models derived from altimetry and GOCE Gravimetry. In: Tregoning P, Rizos C (eds) Dynamic Planet 2005 – Monitoring and Understanding a Dynamic Planet with Geodetic and Oceanographic Tools, IAG Symposia, vol 130. Springer, Berlin, pp 11–16
G.S. Vergos et al. Lombard A, Garcia D, Ramillien G, Cazenave A, Biancale R, Lemoine JM, Fletcher F, Schmidt R, Ishii M (2007) Estimation of steric sea level variations from combined GRACE and Jason-1 data. Earth Planet Sci Lett 254:194–202 Moritz H (1980) Advanced physical geodesy. Wichmann, Karlsruhe Rao CR, Kleffe J (1988) Estimation of variance components and applications. North-Holland Series in Statistics and Probability, vol 3 Rio MH, Hernandez F (2004) A mean dynamic topography computed over the world ocean from altimetry, in-situ measurements and a geoid model. J Geoph Res 109(12): C12032 Sanso` F, Sideris MG (1997) On the similarities and differences between systems theory and least-squares collocation in physical geodesy. Boll di Geod e Scie Aff 2:174–206 Swenson S, Wahr J (2002) Methods for inferring regional surface-mass anomalies from GRACE measurements of time-variable gravity. J Geophys Res 107(B9):2193 Tscherning CC and Rapp RH (1974) Closed covariance expressions for gravity anomalies, geoid undulations, and deflections of the vertical implied by anomaly degreevariance models. Reports of the Department of Geodetic Science, 208, The Ohio State University, Columbus, OH Tziavos IN, Sideris MG, Forsberg R (1998) Combined satellite altimetry and shipborne gravimetry data processing. Mar Geod 21:299–317 Vergos GS, Tziavos IN and Andritsanos VD (2005) On the determination of marine geoid models by least-squares collocation and spectral methods using heterogeneous data. In: Sanso´ F (ed) IAG Symposia, A Window on the Future of Geodesy, vol 128. Springer, Berlin, pp 332–337
Arctic Sea Ice Thickness in the Winters of 2004 and 2007 from Coincident Satellite and Submarine Measurements
16
J. Calvao, J. Rodrigues, and P. Wadhams
Abstract
The L3H phase of operation of ICESat’s laser in the winter of 2007 coincided for about two weeks with the cruise of the British submarine “Tireless” which was equipped with upward-looking sonars. This provided a rare opportunity for a simultaneous determination of the sea ice freeboard and draft in the Arctic Ocean.
16.1
Introduction
The determination of the thickness of the sea ice layer in the Arctic Ocean and surrounding seas is an essential component of the study of the climate of the Arctic. On the one side, the Arctic environment as a whole and, in particular, its sea ice cover, respond quickly to global climatic changes due to some amplification mechanisms that tend to be present in the polar regions, like the well-known albedo effect. As such, a decline in the volume of sea ice can be considered as one of the best indicators of the warming of our planet. On the other hand, because the sea ice acts as a regulator of heat and moisture transfer between the ocean and the atmosphere, its thickness is likely to affect significantly the climate of the Arctic and nearby regions. Basin-wide measurements of the thickness of the sea ice in the Arctic Ocean began in 1958 when the
J. Calvao (*) Lattex, IDL, Faculdade de Cieˆncias da Universidade de Lisboa, Lisboa, Portugal e-mail:
[email protected] J. Rodrigues P. Wadhams Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK
U.S. submarine “Nautilus” reached the North Pole for the first time. Since then, regular Arctic cruises by U.S. submarines have collected a vast amount of sea ice draft data, mostly in the so-called SCICEX box, which roughly coincides with the portion of the Arctic Ocean outside international waters (but including the region north of Alaska). Most of these data sets are available through the National Snow and Ice Data Center (Boulder, CO) archive and have been extensively analysed by Rothrock et al. (1999, 2008). In the early 1970s British submarines started cruising in the Arctic Ocean. For more than three decades they have been taking ice thickness data in regions rarely visited by U.S. boats, such as Fram Strait and the waters north of Greenland. The latter are of special importance in the understanding of the large scale sea ice dynamics of the Arctic Ocean. Through Fram Strait passes most of the ice, fresh water and heat exchanged between the Arctic Ocean and the rest of the world oceans. The complex and variable sea ice conditions in Fram Strait partly reflect the diverse origins of that ice; some of it was transported all the way from the coasts of East Siberia by the Transpolar Drift, other parts may have been advected from the west or the east, and some is formed locally. On the other hand, the region north of Greenland and Ellesmere Island is known to have the thickest ice in
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the Arctic. The last data collected by British submarines suggest that, unlike the rest of the Arctic, there is no decline in ice thickness here in recent years. Data from earlier cruises have been processed at the University of Cambridge and the results published in several papers. Wadhams (1990) provides the first direct evidence of the thinning of the sea ice north of Greenland. Later, he and, independently, Rothrock, observed a significant overall thinning of the Arctic sea ice by comparing results from cruises in the mid 1970 s and in the mid 1990 s (Rothrock et al. 1999, Wadhams and Davis 2000). The Polar Oceans Physics group of the University of Cambridge is currently in the last stages of the statistical analysis of the data gathered during the last two Royal Navy submarine cruises, in the winters of 2004 and 2007. Data from the March 2007 cruise will be particularly relevant because they were taken in several regions of the Arctic with very different ice regimes, some of which would later become ice-free during the exceptional summer of 2007. In this paper we show some results from these two cruises. The quality and scientific importance of the submarine data are unquestionable. However, the use of different sonar equipment in different cruises, the frequent impossibility of an independent evaluation of the accuracy of the measurements, the scarsity of the voyages, the non-coincidence of the tracks, the varied times of the year of the cruises, and the difficulty in merging British and U.S. data in a single global sonic data bank on Arctic sea ice thickness, suggest that some caution is needed when they are used to derive long-term trends. The first use of satellite altimetry to retrieve sea ice freeboard (elevation of the ice floes above the sea level) and thickness is due to Laxon et al. (2003). They used data from the European Space Agency (ESA) satellites ERS-1 and ERS-2 in order to explore the correlation between ice thickness and the length of the ice season. More recently, the same group analysed other radar altimetry data, this time from ESA’s Envisat, obtained between the winters of 2002/2003 and 2007/2008 (Giles et al. 2008). They found a significant reduction in sea ice thickness in the region south of 81.5 N after the record minimum ice extent of September 2007 but no particular trend (in the winter season) between 2003 and 2007. The launch of NASA’s ICESat in January 2003, equipped with its high accuracy Geoscience Laser
J. Calvao et al.
Altimeter System (GLAS), allowed the first determinations of the central Arctic sea ice thickness from freeboard retrievals. Methods to apply laser altimetry to measurements of ice freeboard have been developed by Forsberg and Skourup (2005), Kwok et al. (2007) and Zwally et al. (2008), among others. In a thorough analysis of ten ICESat campaigns between 2003 and 2008, Kwok et al. (2009) derived the evolution of the Arctic sea ice thickness during this five year period, and were able to separate the contribution of first and multi-year ice. They concluded that the average winter ice thickness in the Arctic decreased from 3.3 m in February-March 2005 to less than 2.5 m in February-March 2008 and showed that this decline is essentially due to the disappearance of multi-year ice which, in the winter of 2008, was responsible for only one third of the total volume of ice in the Arctic Ocean. During a period of two weeks in March 2007 the L3H phase of operation of ICESat’s laser coincided with the cruise of the British submarine HMS “Tireless”. Large scale simultaneous observations of the Arctic Ocean’s sea ice thickness are very rare events. To our knowledge, this is only the second occurrence of this type. The first one was reported by Kwok et al. (2009), who used U.S. submarine data collected during a cruise in October/November 2005. The simultaneous measurement of the sea ice freeboard and draft offers the possibility of a cross-check (hopefully crossvalidation) between the results for Arctic sea ice thickness obtained independently by the complex satellite and submarine measurements. In this paper we present results for the sea ice thickness in the Arctic Ocean in the winter of 2007 (which preceded the extraordinary minimum ice extent of September 2007) and in the winter of 2004 obtained from ICESat and submarine data. In Sect. 16.2 we briefly describe the methods used in the retrieval of ice freeboard from ICESat and present overall results for the ice freeboard during the L2B and L3H phases. In Sect. 16.3 we give an overview of the techniques used to calculate ice draft from submarine observations and give average draft values for each section of the two cruises. In Sect. 16.4 we compare the ice thickness obtained by the two platforms and identify the main changes between the ice distributions in 2004 and 2007. We end up with a discussion on the possible causes of the observed discrepancies between ICESat and submarine results.
16
Arctic Sea Ice Thickness in the Winters of 2004 and 2007 from Coincident Satellite
16.2
ICES at Measurements
The laser altimeter system GLAS carried on board the ICESat satellite, especially dedicated to the observation of Polar Regions, generates profiles of the surface topography along the tracks with an accuracy of 15 cm. The beam width is 110 mrad and the pulse rate emission is 40 Hz, sampling the Earth surface with a footprint of 70 m diameter, spaced at 170 m intervals. Small-scale features inside the footprint modulate the amplitude and character of the waveforms, which can be used to determine sea-ice freeboard heights using a “lowest level” filtering scheme: specular laser returns from open water in leads between ice floes are used to define an ocean reference surface. The procedure applied to obtain the ice freeboard F ¼ h-N-MDT (where h is ICESat’s ellipsoidal height estimate of the surface, N is the geoid undulation and MDT is the ocean mean dynamic topography) for the whole Arctic basin (with the exception of points beyond 86 N) consisted of a high-pass filtering of the satellite data to remove low frequency effects due to the geoid and ocean dynamics (the geoid model ArcGP with sufficient accuracy to allow the computation of the freeboard was very recently made available (Forsberg et al. 2007). The original tide model was replaced by the tide model AOTIM5 and the tide loading model TPXO6.2. The inverse barometer correction was applied. As there are no MDT models with enough accuracy, the definition of the ocean surface was done through the identification of areas of open water (or thin ice, typically less than 30 cm thick) in each 20 km segment of the tracks to allow the interpolation of the ocean surface. Several solutions were tested to define the ocean surface and the computed freeboard values were interpolated on a 5 5 minute grid, where the submarine track was interpolated. Some waveform derived parameters in the ICESat data products were used to filter unreliable elevation estimates: surface reflectivity (i_reflctUncorr), detector gain (i_gain_rcv) and surface roughness (i_SeaIceVar). Figure 16.1a, b show the spatial distribution of the sea ice freeboard during the L2B (2004) and L3H (2007) phases of ICESat. Some comparisons between the ice freeboards in the two campaigns are now in order. The mean ice
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freeboard decreased between 2004 and 2007 in some regions of the central Arctic Ocean such as north of the Barents and Kara Seas in the Russian Sector, and between 120 W and 180 in the Canadian/Pacific sectors. Assuming that the depth of the snow on sea ice is similar in the two periods, this indicates a decrease in the sea ice thickness. The volume of ice in Fram Strait is much larger in 2007 than in 2004. There appears to be more thick ice in the vicinity of the north coasts of Greenland and Ellesmere Island in 2007 than in 2004. These observations are consistent with the strengthening of the Transpolar Drift in recent years reported by several authors, with large quantities of ice being transported across the central Arctic Ocean from the Chukchi and Beaufort Seas. Part of this ice is advected to the Atlantic through Fram Strait while the remaining piles up along the north coasts of Greenland and Ellesmere Island.
16.3
Submarine Measurements
In March 2007 the British submarine HMS “Tireless” carried out the route shown in Fig. 16.1b. After entering the Arctic Ocean through Fram Strait, it rounded northeast Greenland, thence cruised north of Greenland and Ellesmere Island roughly along the 85 N parallel until approximately 92 W, when it started heading SW towards the S Beaufort Sea. In the vicinity of (85 200 N, 64 080 W) the boat ran a set of parallel lines at low speed and depth with total length of 200 km under an area that was simultaneously used as a base for experiments of the EU DAMOCLES project. The outgoing part of the cruise ended at the location of the SEDNA ice camp (73 070 N, 143 440 W), where several other ice thickness measurements were performed (Hutchings et al. 2008). The submarine returned to the UK following almost exactly the same route as in the outbound journey and also collected draft data. However, because it was travelling at higher speed and depth, the quality of the data was somewhat inferior and will not be considered here. The April 2004 track, also shown in Fig. 16.1a, included a transect to and from the Pole and a diversion to (85 N, 62 W) in order to survey under a region which a month later was used for an ice camp experiment as part of the EU GreenICE project. Draft data were collected between the marginal ice zone in Fram Strait and the westernmost point of the
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Fig. 16.1 (a) L2B freeboard grid (m): max ¼ 0.97, mean ¼ 0.13, std ¼ 0.13. The track of the “Tireless” is shown in red. (b) L3H freeboard grid (m): max ¼ 0.84, mean ¼ 0.14, std ¼ 0.13. The track of the “Tireless” is shown in red
transect, including about 250 km under the GreenICE camp. Shortly after this survey the sonar stopped recording, except for a short period in the vicinity of the North Pole. In both cruises sea ice draft was measured with two upward-looking single-beam echosounders: the analogue system known as Admiralty Type 780 and the digital system known as Admiralty Type 2077. Because the latter malfunctioned during most of the 2007 voyage, in this paper we limit ourselves to the data obtained with AT780. Though the entire return from each ping is recorded in a paper chart, with the darkness of the trace a function of the intensity of the echo, we shall only take into account the first return from the insonified area (which is typically a circle with a diameter of a few metres for the 3 nominal beam width of the AT780 and ordinary cruise depths). This is a standard procedure in the analysis of submarine data: it is more reliable than to average over the entire pulse and it ensures compatibility with digital systems (where only the first return is recorded). Details of the techniques used to process this type of data can be found in Wadhams (1981).
Once the navigation data are included, the final product is the ice draft as a function of the position of the submarine or the along-track distance for the entire cruise. It is traditional to divide the full track into 50 km sections, as it has been shown that with such a length there is reliable statistics and a low probability of finding different ice regimes within one single section. Figure 16.2a, b show the mean ice draft obtained from the AT780 mounted in the “Tireless” for each of the sections of the 2004 and 2007 cruises. Notice that the 2004 cruise (which took place in April) did not coincide in time with ICESat’s winter campaign. However, because both measurements were made at the end of the winter, when the ice thickness is at or near its annual maximum, a comparison between them is, in our view, legitimate. This is also the reason why we used the freeboard values obtained throughout the whole ICESat winter campaign in 2007. A different approach, which the authors may follow in future work, would be to consider only the portion of the ICESat tracks that coincide in time and position with the track of the submarine.
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Fig. 16.2 (a) Mean ice drafts in the 2004 submarine cruise. (b) Mean ice drafts in the 2007 submarine cruise
16.4
Results
The sea ice thickness hi can be calculated from either the freeboard f (which includes the snow layer of depth hs because the laser pulse is reflected at the snow-air
interface if there is snow on top of the sea ice) or from the ice draft d through the relations hi ¼
rw r rs f w hs rw ri rw ri
(16.1)
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hi ¼
rw r d s hs ri ri
which follow from the assumption of isostatic equilibrium. Here rw ¼ 1024 kg=m3 , ri ¼ 920 kg=m3 and rs ¼ 360 kg=m3 are our assumed densities of the sea water, sea ice and snow, respectively. Values for the density of sea ice ranging between 850 and 925 kg/m3 have been reported (the later is used by Kwok et al. (2009)) while the density of the snow at the end of the winter may range from 300 to 360 kg/m3. Our choices have been widely used in the literature. While the snow term in the second equation can be safely neglected, this is not the case in the retrieval of the ice thickness from the ice (plus snow) freeboard measured with altimetry techniques, where the snow depth plays a crucial role. Unfortunately, the algorithms that attempt to extract snow depth from passive microwave satellite imagery (such as the NASA Team algorithm used to process raw AMSRE data) suffer from severe limitations. As such, in this paper we choose to work with values observed during recent winter field campaigns in the Beaufort and Lincoln Seas, which, typically, vary from zero to 30 cm. Here we use the upper value of 30 cm. In this case, it should be understood that the minima of the freeboard used to define a reference level as explained in Sect. 16.2 correspond not to zero elevation but to 30 cm elevation. Consequently, the value of the freeboard used to retrieve the sea ice thickness from (16.1) is the value that appears in the maps shown in Fig. 16.1a, b with 30 cm added. In fact, it is reasonable to assume that at the end of the winter the minima do not coincide with open water but with refrozen leads, which would have some snow on top. Tables 16.1a and 16.1b show the mean ice thicknesses obtained from satellite and submarine measurements for different regions of the Arctic visited by the submarine. Central (North) Fram Strait is the portion of the strait between 80 N and 82 N (82 N and 84 N); North Greenland 1 is the path of the submarine in 2004 along the 85 N parallel roughly between 20 W and 70 W, which coincides with the path of 2007; North Greenland 2 is the portion of the trajectory of the boat between 85 N and 86 N when it was heading northeastwards. Note that within each section, or region, the amount of invalid data differs for the satellite and the
Table 16.1a Mean sea ice thickness in the winter of 2004 Region
Sections
Central Fram Strait W Central Fram Strait E North Fram Strait Northeast Greenland North Greenland 1 North Greenland 2
01–18 30–33 34–37 38–43 44–57 59–62
Thickness (ICESat) 2.28 3.73 3.67 4.03 4.41 4.14
Thickness (Sub) 3.37 2.13 2.95 3.70 5.90 5.42
Table 16.1b Mean sea ice thickness in the winter of 2007 Region
Sections
Central Fram Strait North Fram Strait Northeast Greenland North Greenland N Ellesmere Island Canadian Basin Beaufort Sea
06–10 11–14 15–18 19–32 35–36 38–52 53–64
Thickness (ICESat) 2.85 2.88 2.78 4.65 4.65 2.90 2.27
Thickness (Sub) 4.43 3.93 3.77 5.95 7.00 5.42 3.73
submarine, which may explain some of the differences in the retrieved ice thickness. Figure 16.3a, b show the mean ice thickness for each of the 50nmjh sections into which the transects of the submarine were divided. ICESat values for each section are taken from a “square” box whose boundaries are the extreme values of the latitude and longitude of the submarine within that section.
16.5
Discussion
As it can be seen from Fig. 16.3a, b, although the results obtained from satellite and submarine data processing have good correlation concerning the shape of the curves representing sea ice thickness, there are visible discrepancies. The mean ice thickness computed from submarine data is consistently higher, with differences that can reach 1.5 m. Such differences occur mostly in the heavily ridged zone north of Greenland and Ellesmere Island (sections 44–57 in 2004 and 19–36 in 2007). There is reasonably good agreement in the Southern Beaufort Sea (last sections in 2007), where the submarine measured the thinnest ice of the whole transect.
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Fig. 16.3 (a) Mean ice thickness in the winter of 2004 for each section of the submarine cruise. (b) Mean ice thickness in the winter of 2007 for each section of the submarine cruise
Several reasons contribute to these results: (a) The comparison relates instantaneous data obtained from submarine draft data with gridded freeboard data computed from almost one month of satellite data, both converted into thickness, which means that this latter data set is a mean value for that period (moreover, the processing scheme produces some kind of attenuation of the freeboard grid). (b) The size of the footprints of the sonar and laser are quite different, which means that the integrated value and the resolution of the draft and freeboard are correspondingly different. (c) Beamwidth sonar correction were not performed, so the draft tends to be overestimated. Because of the comparatively narrow beam of the AT780 echosounder, such corrections have been considered
negligible in previous publications. However, a recent study by one of the authors suggests that this may not always be the case, namely in regions with a high density of pressure ridges. This might explain why the largest differences were found north of Greenland and Ellesmere Island. (d) A more sophisticated treatment of the snow may be required. We look forward to the release of version V12 of the NASA Team algorithm where, hopefully, some of the problems in the retrieval of the snow depth have been sorted. These should be considered as preliminary results. Clearly, more research is needed to fully understand the causes of the differences in the values of the sea ice thickness extracted from submarine sonars and satellite altimetry.
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References Forsberg R, Skourup H (2005) Arctic Ocean gravity, geoid and sea ice freeboard heights from ICESat and Grace. Geophys Res Lett 32:L215502. doi:10.1029/2005GL023711 Forsberg R, Skourup H, Andersen OB, Knudsen P, Laxon SW, Ridout A, Johannsen J, Siegismund F, Drange H, Tscherning CC, Arabelos D, Braun A, Renganathan V (2007) Combination of spaceborne, airborne and in-situ gravity measurements in support of Arctic Sea ice thickness mapping, Danish National Space Center, Technical report Nº 7, 2007 Giles KA, Laxon SW, Ridout AL (2008) Circumpolar thinning of Arctic sea ice following the 2007 record ice extent minimum. Geophys Res Lett 35:L22502. doi:10.1029/ 2008GL035710 Hutchings J et al (2008) Role of ice dynamics in the sea ice mass balance. Eos Trans AGU. 89(50). doi:10.1029/ 2008EO500003 Kwok R, Cunningham GF, Zwally HJ, Yi D (2007) Ice, Cloud, and land Elevation Satellite (ICESat) over Arctic sea ice: retrieval of freeboard. J Geophys Res 112:C12013. doi:10.1029/2006JC003978 Kwok R, Cunningham GF, Wensnahan M, Rigor I, Zwally HJ, Yi D (2009) Thinning and volume loss of the Arctic Ocean
J. Calvao et al. sea ice cover: 2003–2008. J Geophys Res 114:C07005. doi:10.1029/2009JC005312 Laxon S et al (2003) High interannual variability of sea ice thickness in the Arctic region. Nature 425:947–950 Rothrock DA, Yu Y, Maykut GA (1999) Thinning of the Arctic sea-ice cover. Geophys Res Lett 26(23):3469–3472. doi:10.1029/1999GL010863 Rothrock DA, Percival DB, Wensnahan M (2008) The decline in Arctic sea-ice thickness: Separating the spatial, annual, and interannual variability in a quarter century of submarine data. J Geophys Res 113:C05003. doi:10.1029/ 2007JC004252 Wadhams P (1981) Sea ice topography of the Arctic Ocean in the region 70 W to 25 E. Philos Trans Roy Soc Lond A302: 45–85 Wadhams P (1990) Evidence for thinning of the Arctic ice cover north of Greenland. Nature 345:795–797 Wadhams P, Davis N (2000) Further evidence of ice thinning in the Arctic Ocean. Geophys Res Lett 27:3973–3976 Zwally HJ, Yi D, Kwok R, Zhao Y (2008) ICESat measurements of sea ice freeboard and estimates of sea ice thickness in the Weddell Sea. J Geophys Res 113, C020S15. doi:10.1029/2007JC004284.
The Impact of Attitude Control on GRACE Accelerometry and Orbits
17
U. Meyer, A. J€aggi, and G. Beutler
Abstract
Since March 2002 the two GRACE satellites orbit the Earth at relatively low altitude (500 km in 2002, still close to 460 km mid of 2009). GPS-receivers for orbit determination, star cameras and thrusters for attitude control, accelerometers to observe the surface forces, and a very precise microwave link (K-band) to measure the inter-satellite distance with micrometer accuracy are the principal instruments onboard the satellites. Determination of the gravity field of the Earth including its temporal variations from the satellites’ orbits and the inter-satellite measurements is the main goal of the mission. The accelerometers are needed to separate the gravitational acceleration from the surface forces acting on the satellites. They collect a wealth of information about the atmospheric density at satellite height as a by-product. These accelerations have not yet been analyzed thoroughly, because their interpretation is complicated due to numerous thruster spikes. We outline a method to model the thruster spikes and to clean the time series of the accelerations. The isolated effect of the modeled thruster pulses on the satellite orbits is studied and a first interpretation of the cleaned accelerations is given. A correlation between K-band residuals and regions of high atmospheric fluctuations was not observed, which is probably due to time variable signals of hydrological origin that dominate the residuals.
17.1
Introduction
The Gravity Recovery And Climate Experiment (GRACE, Tapley et al. 2004) satellites orbit the Earth at a height, where the atmospheric drag, in addition to solar radiation pressure and albedo effects,
U. Meyer (*) A. J€aggi G. Beutler Astronomical Institute, University of Bern, Sidlerstrasse 5, 3012, Bern, Switzerland e-mail:
[email protected]
still plays a major role in orbit dynamics. When estimating the Earth’s gravity field, these surface forces are considered as disturbing forces that either have to be modeled or preferably directly observed with an accelerometer onboard the satellite. In order to separate the surface forces from gravity, the accelerometer has to be located at the center of mass of the satellite. The position of the accelerometer relative to the center of mass is checked regularily by dedicated shaking maneuvers. If necessary, the center of mass is adjusted with movable masses inside the satellite.
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Direct solar pressure mainly depends on the satellite’s cross-section exposed to the sun and therefore varies twice per revolution. The influence of albedo is quite small in comparison to the other forces. As both satellites are identical and follow each other at a distance of 220 km or a timelag of 28 s on average in the same orbit, the influence of solar radiation pressure on the satellites will be greatly reduced in the difference of the accelerations between the two satellites. The highly variable part of atmospheric drag is the main signal remaining in the acceleration differences. Simultaneous acceleration differences are a measure for the spatial variability of the atmosphere’s density, at a distance of 220 km. If the timelag of the trailing satellite is taken into account when forming the difference, one obtains a measure for the temporal variability of the atmosphere’s density at one and the same location within 28 s. Both kinds of acceleration differences are studied subsequently. The accelerations originally measured in the satellite-fixed frames have to be transformed into a common reference frame before forming the differences. The transformation into a co-rotating frame with its axes pointing radially outward (R), normally to the orbital plane of the satellites (W), and orthogonally to these axes completing a right-handed frame (S, approximately in the direction of flight of the satellites) is performed using the attitude observations of the star cameras and the reduced dynamic satellite
orbits (J€aggi 2007). The interpretation of the resulting acceleration differences is complicated by numerous spikes (see Fig. 17.1), which usually may be attributed to one of the two satellites and which are highly correlated with thruster firings for attitude control, as recorded in the THR1B-files (Case et al. 2002). The thruster fire should, in principle, not be recorded by the accelerometers at all, because for each event two thrusters fire simultaneously and the corresponding thrusters are positioned in such a way, that their combined thrusts should only cause angular accelerations on the satellite. The observed linear accelerations may have two explanations: – The thrusters of a pair do not fire perfectly symmetrically and cause real linear accelerations. – The accelerometer is not positioned exactly in the center of mass of the satellite and therefore measures some artefacts due to thruster firings. As the thrusters fire frequently (500–800 times per day), it is necessary to develop a model for the thruster pulses to clean the observed accelerations.
17.2
Modeling Thruster Pulses
Six thruster pairs for attitude control and one thruster pair for orbit maneuvers are mounted on each satellite (the latter is not studied here). Time and duration of the thruster pulses are recorded in the THR1B-files.
R [μm/s2]
0.05 0 −0.05 240
260
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detail in Wu et al. (2006). Flury et al. (2008) discuss artefacts in the ACC1A data and their removal by the low-pass filter. The accelerometers do not sense the entire pulse. Therefore, we estimate scaling factors between the simulated and the actually measured accelerations for each thruster pair and each axis (R, S and W) of the reference frame. From the huge number of daily pulses we select only those which do not heavily overlap (which are separated by half the fit interval length of the low pass filter, i.e., by 70 s) and which have a certain minimal thrust duration (here 50 ms). The simulated accelerations by each of these pulses are fitted to the acceleration differences between the two satellites (the leading satellite shifted by 28 s) by estimating a bias, drift, and scale factor. The estimation of bias and drift parameters is necessary, because the signal of the thruster fire interferes with accelerations caused by atmospheric density variations. Because some of the pulses cannot be fitted very well, a screening step is needed. For all pulses of one and the same thruster pair a mean fit (RMS) is computed and badly fitted pulses are removed by a one sigma criterion. Mean scale factors are determined from the remaining pulses. Table 17.1 shows the results using data of the year 2003 and Table 17.2 for data of the year 2007, respectively. The thruster pairs 2 and 4 fire less frequently than the other thrusters and the estimation of scaling factors therefore is difficult (not shown). For all other thruster
The nominal thrust is 10 mN and the duration per thrust varies between 20 and several hundred milliseconds. To model that part of the thruster pulses, that is mapped to the measured accelerations, it is assumed that a constant scaling factor exists for each thruster pair and each axis of the reference frame. The processing procedure from the raw accelerations, measured onboard the satellites, to the ACC1B accelerations has to be understood first. An analog Butterworth filter of order 3 with a cut-off frequency of 3 Hz is applied onboard the satellites to the raw data to prevent aliasing. A shift by 0.14 s, called Butterworth-delay, is an unwanted artefact of this filter. Afterwards the data is sampled at a rate of 10 Hz, resulting in ACC1A-data. In the next steps the time scale is converted from On Board Data Handling (OBDH) to GPS time, the Butterworth-delay is removed and the data resampled at 1 Hz using a Charge Routing Network (CRN) class digital low-pass filter with a fit interval of 140.7 s and a target low-pass bandwidth of 0.035 Hz (Thomas 1999). This filter dampens the sharp spikes due to thruster firings and massively widens their shape. The effect of this transformation on orbit integration is not important, because the energy of the pulses is preserved. Together with the masses of the satellites, available in the MAS1B-files, we are now able to model the acceleration caused by a sharp 10 mN pulse on the satellites as in the ACC1A- or ACC1B-data (see Fig. 17.2). The filtering procedure is described in
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Table 17.1 Scaling factors of four thruster pairs, their error and the number of pulses considered, estimated from ACC1Baccelerations of 2003 2003 R Thruster 1 Thruster 3 Thruster 5 Thruster 6 S Thruster 1 Thruster 3 Thruster 5 Thruster 6
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Table 17.2 Scaling factors of four thruster pairs, their error and the number of pulses considered, estimated from ACC1Baccelerations of 2007 2007 R Thruster 1 Thruster 3 Thruster 5 Thruster 6 S Thruster 1 Thruster 3 Thruster 5 Thruster 6
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pairs results of similar quality are obtained for 2003 and 2007. The scaling factors therefore can indeed be considered as constant over time (the change in sign in the S-direction from 2003 to 2007 is caused by the satellites’ switch of position in December 2005). All the pulses from the THR1B-files can now be modeled using the estimated scaling factors in the R, S, W-tripod. The results are promising (see Fig. 17.3), except for the W component. This poor quality in W is probably explained by the reduced sensitivity of the accelerometers in cross-track direction (Frommknecht et al. 2003). The accelerations in W are not further considered here, because they have the least influence on the inter-satellite distance and therefore on the gravity field estimation (J€aggi et al. 2011).
17.3
Effect on Satellite Orbits
We are now in a position to study the isolated impact of the thrusters on the satellite orbit and to compare it with the effect of the cleaned surface forces and with the combined effect of the original ACC1Baccelerations. Therefore, we determine orbits in four different ways: (a) Without ACC1B-DATA (b) With the complete, original ACC1B-data (c) Only with the modeled pulses of the thruster firings (d) With the accelerations reduced by the thrusterinduced effects The orbits were determined using the Celestial Mechanics Approach (CMA, Beutler 2005). Piecewise constant accelerations, set up for every 15 min interval, were constrained to zero with a sigma of 3.109 m/s2 and subsequently estimated together with all other parameters. This approach is very flexible, the pseudo-stochastic accelerations are able to absorb the missing non-gravitational accelerations to a large extent. To limit the model error by the static gravity field, AIUB-GRACE02S (J€aggi et al. 2011) was used as background model. AIUB-GRACE02S has been estimated from GRACE GPS and K-Band range-rate data of the years 2006 and 2007. Figure 17.4 shows the resulting K-band range-rate residuals. To separate the effects of interest from errors in the background models, the differences of the residuals from experiments (b) to (d) with respect to those of experiment (a) are shown (the model errors cancel out). The orbit perturbations caused by thruster firings and those caused by the cleaned surface forces are of comparable size. The simulated effect of the thruster fire on the orbit, obtained by direct numerical integration of the modeled accelerations and subsequent high-pass filtering (by subtraction of a moving mean over 3 min) is included in Fig. 17.4. The correlation with the K-band residuals is clearly visible. The differences are explained by the pseudostochastic accelerations. Therefore, we conclude that the thrusters significantly perturb the satellite orbits. Figure 17.5 shows the original range-rate residuals of experiments (b) and (d) and their difference due to the missing thruster accelerations in (d). The systematic effects are at least by a factor of three larger than the thrusters-induced effects. The thrusters (or the thruster
The Impact of Attitude Control on GRACE Accelerometry and Orbits
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Fig. 17.3 Original (black) and cleaned (gray) acceleration differences between GRACE B and A, GRACE B shifted by 28 s
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mismodeling) are thus not responsible for these systematic effects. In a last orbit experiment we study the interaction between the linear thruster accelerations and the pseudo-stochastic pulses of the CMA using a Keplerian orbit. The Kepler ellipse, disturbed by the modeled thruster accelerations, is then introduced as “observed” orbit (3D-positions at 30 s intervals). These positions are used in an orbit determination, where pseudo-stochastic pulses are set up at 15-min
intervals. The resulting velocity residuals are compared with the range-rate residuals of the real orbit determination process in Fig. 17.6. The simulated and the real residuals are of the same order of magnitude. The jumps in the residuals every 15 min are due to the stochastic pulses. The Kepler orbit is a useful tool to assess the thruster-induced orbit perturbations in a qualitative way. We have not yet answered the question, whether the observed linear accelerations due to the thruster
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actuations are artefacts or real. GRACE orbits are determined according to the experiments (a) to (d) for 60 days of September and October 2007 to answer this question. Figure 17.7 shows daily RMSvalues of the range-rate residuals for all four cases. The results from Fig. 17.4 are confirmed, i.e., the thrusters do significantly perturb the orbits. It becomes clear by now, that their inclusion in the accelerometer files leads to a considerable improvement of the orbit fit. The linear thruster accelerations are thus mainly caused by non-symmetric firings of the thrusters.
17.4
Interpretation of Accelerations and Correlation with K-Band Residuals
Let us now use the cleaned accelerations to study the atmosphere. We limit our analysis to the acceleration differences in the direction of flight S of the satellites, which characterize the spatial or temporal variability of the atmosphere’s density. The state of the atmosphere mainly depends on the insolation, which is why sun-related coordinates are selected to represent the accelerations. The first coordinate is b, the angle
The Impact of Attitude Control on GRACE Accelerometry and Orbits
Fig. 17.7 Daily RMS of the range-rate residuals without using nongravitational accelerations (black), using the complete ACC1B-data (gray), considering only the modeled effect of the thrusters (black, dashed) or the cleaned effect of the surface forces (gray, dashed)
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Conclusions
We developed a method to model the linear accelerations on the satellites caused by the frequent thruster pulses. The impact of these pulses on the satellite orbits was studied and it was shown that the observed spikes represent real accelerations, which have to be taken into account for orbit determination. The method can be used to separate the surface forces from the thruster-induced accelerations for the radial and along-track direction. The analysis and interpretation of the cleaned accelerations, which mainly represent the atmospheric conditions at satellite height, is still at an early stage. To study the impact of atmospheric density fluctuations on orbit determination and on subsequent gravity field estimation, we plan to estimate and remove the strong seasonal variations of hydrological origin in the near future to be able to look for correlations between surface forces and K-band residuals. Acknowledgements The authors gratefully acknowledge the generous financial support provided by the Institute for Advanced Study (IAS) of the Technische Universit€at M€unchen.
β Fig. 17.9 Acceleration differences between the two GRACE satellites at the same time (separated 220 km in space), thruster fire removed
and exit are visible extremely well. Inclined, linear structures in the lower part of the figure are not yet explained. Strong atmospherical density variations can be observed close to the Earth’s north pole (around u ¼ +90 ), caused by interactions of the atmosphere with the magnetosphere. The corresponding variations close to the south pole (around u ¼ 90 ) are much less pronounced, because the southern atmosphere gets less insolation during the time period shown. The smoothness of the acceleration differences in Fig. 17.8 indicates that the signals visible in Fig. 17.9 really are atmospheric density fluctuations and not just noise. Currently, we cannot answer the question whether the atmosphere density variations shown in Fig. 17.9 have an impact on the orbit quality. The K-band rangerate residuals show a strong signal related to hydrology, which has to be removed before addressing the issue of the density variations.
References Beutler G (2005) Methods of celestial mechanics. Springer, Berlin. doi:10.1007/b138225 Case K, Kruizinga G, Wu S (2002) GRACE level 1B data product user handbook. D-22027. JPL, Pasadena, CA Flury J, Bettadpur S, Tapley B (2008) Precise accelerometry onboard the GRACE gravity field satellite mission. Adv Space Res 42:1414–1423. doi:10.1016/j.asr.2008.05.004 Frommknecht B, Oberndorfer H, Flechtner F, Schmidt R (2003) Integrated sensor analysis for GRACE – development and validation. Adv Geosci 2003(1):57–63 J€aggi A (2007) Pseudo-stochastic orbit modeling of low earth satellites using the global positioning system. Geod€atischgeophysikalische Arbeiten in der Schweiz, vol 73. doi:9783-908440-17-8 J€aggi A, Beutler G, Meyer U, Prange L, Dach R, Mervart L (2011) AIUB-GRACE02S – status of GRACE gravity field recovery with the celestial mechanics approach. In: Kenyon S et al (eds) Geodesy for planet earth. Springer, Heidelberg Tapley BD, Bettadpur S, Ries JC, Thompson PF, Watkins M (2004) GRACE measurements of mass variability in the Earth system. Science 30:5683. doi:10.1126/science.1099192 Thomas JB (1999) An analysis of gravity-field estimation based on Intersatellite Dual-1-way biased ranging. JPL, Pasadena, CA, pp 98–115 Wu SC, Kruizinga G, Bertiger W (2006) Algorithm theoretical basis document for GRACE level-1B data processing V1.2. D-27672. JPL, Pasadena, CA
Using Atmospheric Uncertainties for GRACE De-aliasing: First Results
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L. Zenner, T. Gruber, G. Beutler, A. J€aggi, F. Flechtner, T. Schmidt, J. Wickert, E. Fagiolini, G. Schwarz, and T. Trautmann
Abstract
In standard gravity field processing, short-term mass variations in the atmosphere and the ocean are eliminated in the so-called de-aliasing step. Up to now the background models used for de-aliasing have been assumed to be error-free. As the accuracy assessed prior to launch could not yet be achieved in the analysis of real GRACE data, the de-aliasing process and related geophysical model uncertainties have to be considered as potential error sources in GRACE gravity field determination. The goal of this study is to identify the impact of atmospheric uncertainties on the de-aliasing products and on the resulting GRACE gravity field models. The paper summarizes the standard GRACE de-aliasing process and studies the effect of uncertainties in the atmospheric (temperature, surface pressure, specific humidity, geopotential) input parameters on the gravity field potential coefficients. Finally, the impact of alternative de-aliasing products (with and without atmospheric model errors) on a GRACE gravity field solution is investigated on the level of K-band range-rate residuals. The results indicate that atmospheric model uncertainties are small in terms of the associated spherical harmonic coefficients. The effect in terms of K-band observation residuals is negligible compared to other modeling errors.
18.1 L. Zenner (*) T. Gruber Institut f€ur Astronomische und Physikalische Geod€asie, Technische Universit€at M€ unchen, 80290, M€ unchen, Germany e-mail:
[email protected] G. Beutler A. J€aggi Astronomisches Institut, Universit€at Bern, 3012, Bern, Switzerland F. Flechtner T. Schmidt J. Wickert GeoForschungsZentrum, 14473, Potsdam, Germany E. Fagiolini G. Schwarz T. Trautmann Deutsches Zentrum f€ ur Luft- und Raumfahrt, 82234, Weßling, Germany
Introduction
Mass redistributions inside, on, and above the Earth’s surface are responsible for time variable gravity field forces, which directly act on a satellite orbit. If these variations cannot be directly measured by repeated observations within short periods, they have to be removed during the gravity field determination in order to avoid aliasing due to undersampling. Shortterm mass variations in the atmosphere and in the oceans are part of the variations which cannot be measured with GRACE due to the inadequate spacetime sampling of the GRACE mission. To deal with this problem, the high frequency atmospheric and
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oceanic signals are modeled and “removed” prior to or during the GRACE data processing. The “removal” of these high-frequency mass variations is called dealiasing (Flechtner 2007). As the GRACE results are still not meeting the expected pre-launch baseline accuracy, the de-aliasing process and related geophysical model uncertainties are considered here as a possible limiting factor for the GRACE gravity field determination. Therefore we explore ways to improve the de-aliasing process. In a first step, the de-aliasing process and its fundamental formulas, with the main focus on the processing sequence of the atmosphere, is reviewed (Sect. 18.2). As the standard processing scheme assumes error-free atmospheric and oceanic parameters and as it is well known that in areas with sparse observations the atmospheric models are degraded in quality (Salstein et al. 2008), there is reason to assume that one could improve the de-aliasing product and consequently the gravity field solution by taking into account uncertainties of the atmospheric parameters. This is why a mathematical model of error propagation of atmospheric model uncertainties into the gravityfield de-aliasing coefficients was developed. In order to study the impact of atmospheric model uncertainties on intermediate and final gravity field results, a test environment based on a real GRACE data set, data from the European Center for Medium Range Weather Forecast (ECMWF) atmospheric analysis (ECMWF 2009), and from the Ocean Model for Circulation and Tides (OMCT) (Dobslaw and Thomas 2007) was set up. The error scenarios for the atmosphere are defined in Sect. 18.2.2. The ocean is assumed as error-free in this study. For the time being, only atmospheric uncertainties provided by ECMWF were used to get a first insight into the effect of model errors on the de-aliasing coefficients (AOD) and GRACE. In Sect. 18.3.1 the impact of the error assumptions on the estimated de-aliasing coefficients and the geoid is studied. Differences between the error-scenarios are computed and compared to the GRACE error predictions. Finally, the newly computed de-aliasing coefficients are used for a GRACE gravity field determination. In order to identify their impact on the resulting gravity field, observation residuals are investigated in detail in Sect. 18.3.2. Sect. 18.4 summarizes the results and draws conclusions.
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18.2
Theory: The De-aliasing Process
18.2.1 Fundamental Formulas A short overview of the fundamental formulas of the current atmospheric and oceanic de-aliasing processing sequence is provided below. The following formulas outline the computation of the harmonic coefficients Cnm , Snm due to atmospheric and oceanic variations. Pkþ1=2 ¼ akþ1=2 þ bkþ1=2 Ps
(18.1)
Tv ¼ ð1 þ 0:608SÞT
(18.2)
Hkþ1=2 ¼ Hs þ
kmax X Pjþ1=2 RTv ln g Pj1=2 j¼kþ1
Z Z Cnm a2 ð1 þ kn Þ ¼ ½In þ PO ð2n þ 1ÞMg Snm Pnm ðcos yÞ Z0 In ¼ Ps
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Pk+1/2 ¼ pressure at half-levels ak+1/2, bk+1/2 ¼ model dependent coefficients Tv ¼ virtual temperature R ¼ gas constant for dry air Hk+1/2 ¼ geopotential height at half-levels g ¼ mean gravity acceleration a ¼ semi major axis of reference ellipsoid M ¼ earth mass kn ¼ loading love numbers Po ¼ ocean bottom pressure In ¼ vertically integrated atm. pressure Pnm ¼ assoc. norm. legendre polynomials N’ ¼ mean geoid height above sphere r ¼ a Four input parameters are needed for the determination of the atmospheric potential coefficients: Surface pressure Ps, surface geopotential height Hs, temperature T and specific humidity S at the 91 atmospheric model levels (full-level). After a few steps, the vertical integration of the atmospheric pressure In is performed in (18.5). The atmospheric pressure In is
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Using Atmospheric Uncertainties for GRACE De-aliasing: First Results
then combined with the ocean bottom pressure Po in order to obtain the combined atmospheric and oceanic potential. The combined atmospheric and oceanic pressure is subsequently called PAo (18.6). In a final step (18.4), the numerical integration is performed and the atmosphere and ocean de-aliasing product (AOD) is obtained. A few comments should be made concerning the outlined procedure: Usually, a mean field is subtracted before-hand from the vertically integrated atmospheric pressure In and from the ocean bottom pressure Po, i.e., PAo contains the residual atmospheric and oceanic pressure. The mean fields are subtracted in order to analyze gravitational variations. Since the mean mass distribution of the atmosphere and ocean by definition refers to the static part of the gravity field, only the deviations from the mean value have to be taken into account. In the currently realized GRACE gravity field processing as well as in our investigations, mean fields obtained from the years 2001/2002 are used. For details we refer to Flechtner (2007).
18.2.2 Changing the Current De-aliasing Process: Two Error Scenarios Temperature, specific humidity, surface pressure, and surface geopotential height have so far been assumed to be error-free, although it is known that there are large uncertainties in the atmospheric input parameters, particularly in the surface pressure (Ponte and Dorandeau 2003). The goal of this study was to find out to what extent the atmospheric model uncertainties affect AOD and GRACE results. To answer this question, the atmospheric field errors are propagated into the vertically integrated atmospheric pressure In in a first step. As mentioned previously, for the time-being, the ocean is regarded as error-free. In a second step, in order to take the error of In into account and to further propagate it into the AOD coefficients Cnm , Snm , the present approach of numerical integration (18.4) was replaced by a least-squares adjustment (Gauß-Markov, model). Equation (18.4) was used as an observation equation: Mg X 2n þ 1 X Pnm ðcos yÞ a2 n 1 þ k n m ½Cnm cos ml þ Snm sin ml;
PAO ¼
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where the combined residual atmospheric and oceanic pressure fields PAO are used as observations and the error of PAO as individual (inverse) weights. The error of PAO equates to the error of as the ocean In is assumed as error-free. In this way we obtain the AOD coefficients as a function of the errors of the input parameters, i.e., as a function of the weighting. In order to see whether the atmospheric model uncertainties significantly affect AOD and GRACE results, two so-called error scenarios were studied: “error-free AOD”: The uncertainties of the four atmospheric parameters are not taken into account. “full-error AOD”: The uncertainties of the four atmospheric parameters are taken into account. The “error-free” experiment thus assumes that all observations PAO have the same weight. The “fullerror” scenario assumes that the observations PAO are weighted individually using the inverse error of PAO on the diagonal of the weighting matrix. After performing the least-squares adjustment for these two scenarios, one obtains two sets of AOD products. By comparing them (Sect. 18.3), one gets insight into the effect of atmospheric uncertainties on the atmospheric and oceanic potential coefficients Cnm , Snm . The following results were achieved using data provided by the ECMWF (Operational archive, Atmospheric model, Analysis and Errors in analysis; ECMWF 2009).
18.3
Results: Impact of Atmospheric Uncertainties on AOD and GRACE
18.3.1 Effect of Model Uncertainties on AOD Figure 18.1 shows the difference between the AOD product for the “full-error” and the “error-free” experiment in terms of geoid heights. The atmospheric model uncertainties affect the geoid in the range of 0.8 to 0.3 mm. The largest differences are visible in regions where the weights for the observations are extreme (high or low). This result seems to be clear: The deviation between the two error-scenarios is only caused by the different weighting of the observations PAO. In the “error-free” case the combined atmospheric and oceanic pressure has the same weights,
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Fig. 18.1 Geoid difference between the “error-free” and “fullerror” scenario. 2007-08-01 00h. Unit (mm)
whereas in the “full-error” scenario the observations are weighted using the inverse error of PAO (which is equal to the vertically integrated atmosphere error since the ocean is assumed as error-free). Therefore, the error of the atmospheric pressure is reflected in the difference between the two error-scenarios. The effect of atmospheric uncertainties on AOD is in the sub-mm domain, which seems to be small. Figure 18.2 shows the difference between the two error-scenarios in terms of spherical harmonic degree variances. The difference between the two error-scenarios is presented in the dotted line marked with circles. Compared to the GRACE baseline (Kim 2000; lowest line in Fig. 18.2) the effect of atmospheric model uncertainties clearly is in the sensitivity range of GRACE. Therefore we conclude that an effect should be visible in the gravity field solution as well. This aspect will be studied in Sect. 18.3.2.
18.3.2 Effect of Model Uncertainties on K-Band-Residuals The K-band range-rate (KBRR) residuals are obtained from an extended orbit-determination problem, where one tries to minimize the differences between the computed K-band range-rate observations based on a priori models and the real observations made by GRACE by only adjusting orbit parameters. In theory, the resulting KBRRresiduals would become zero, if all force models (tides, static gravity field, AOD-model,. . .) as well
as the real observations were error-free. As it is known, this is not the case. Thus differences between the modeled and the real world occur in the KBRRresiduals. The residuals contain all model errors, measurement errors, and unmodeled forces. By applying the new de-aliasing coefficients, which take the atmospheric model uncertainties into account, we hope to reduce these residuals. This would imply that the new (“full-error”) AOD product is more realistic than the standard or “error-free” AOD. For this purpose the Bernese GPS Software (Dach et al. 2007) and the Celestial Mechanics Approach (J€aggi et al. 2011) is used to perform an orbit determination based on kinematic positions and K-band range-rate observations by using the gravity field model AIUB-GRACE02S (J€aggi et al. 2011) as known and using the different sets of AOD coefficients. By leaving all other settings unchanged, one gains insight into the effect of atmospheric model errors. For detailed parametrization information, we refer to J€aggi et al. (2011). In addition to the two error-scenarios described in Sect. 18.2.2, the overall effect of AOD by not applying AOD products for orbit determination was included as well. The impact of the three experiments on the KBRR residuals is shown in Fig. 18.3. Applying (dark grey “area” in Fig. 18.3) or not applying AOD (light grey “area”) clearly affects the KBRR-residuals. Removing the high frequent atmospheric and oceanic mass variations significantly reduces the RMS over 1 year of KBRR-residuals by about 10%. This implies that the inclusion of the short-term atmospheric and oceanic variations for gravity field solutions leads to more realistic results. Figure 18.3 shows that the amplitudes of the prominent peaks are reduced, but not removed, when applying AOD. One reason for that might be due to the fact that AIUB-GRACE02S is a static field where the time-variable gravity field signals are not yet modeled. Thus, the KBRRresiduals contain apart from other model errors mainly the unmodeled time-variable gravity forces (in particular hydrology). However, in general the signal in the residual series is reduced. Is the newly computed full-error AOD product able to further improve de-aliasing, i.e., to further reduce KBRR-residuals? Figure 18.3 (bold black line) shows that this is not the case. This is contrary to our
18
Using Atmospheric Uncertainties for GRACE De-aliasing: First Results
Fig. 18.2 Spherical harmonic degree variance differences between the “error-free” and “full-error” scenario in terms of geoid heights. 2007-08-01 00h. Unit (m)
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18.4
Conclusion and Discussion
Figure 18.4 is identical to Fig. 18.2, except that it also contains the achieved (as opposed to the expected prelaunch) accuracy of gravity field determination with GRACE (curve with asterisk). As the differences
“error-free” minus “full-error” are clearly below the curve “achieved GRACE baseline”, it is clear that the impact of the refined AOD model studied here is not visible in gravity field determination. It must be pointed out, however, that our results are based on atmospheric error fields taken from the operational analysis of the ECMWF. The results shown are therefore only true for the ECMWF error-fields. The effect of alternative uncertainty estimates from other sources [e.g., NCEP (National Centers for Environmental Prediction)] has not been investigated yet.
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Fig. 18.4 Degree variance differences of error-free AOD and full-error AOD in terms of geoid heights (m) for 2007-08-01 00h, and the predicted and current GRACE error estimates
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Also, one should take into account that the used errorfields might be too optimistic as they result from the assimilation model (ECMWF 2009). More work has to be carried out concerning the determination of representative error parameters (e.g., Schmidt et al. 2008). Previous investigations have shown that the surface pressure error has the most significant effect on the vertically integrated atmospheric pressure. It is therefore essential to determine reliable and reasonable surface pressure values and use them as input for the upcoming investigations. Furthermore, it must be emphasized that so far the ocean has been assumed to be error-free. To determine representative error-fields of ocean bottom pressure and to take them into account during AOD processing will be one of the next steps.The goal of this study was to improve the de-aliasing process by taking atmospheric model uncertainties into account. It was shown that atmospheric model uncertainties are not able to improve the de-aliasing process or GRACE orbit determination, because the actual error of GRACE is above the predicted baseline accuracy. It should be kept in mind, however, that model uncertainties will become more important if the achieved accuracy of GRACE can be further improved towards the assumed baseline. Acknowledgements This study was conducted as part of the IDEAL-GRACE project with the support of the German Research Foundation (Deutsche Forschungsgemeinschaft) and within the SPP1257 priority program “Mass transport and Mass Distribution in the System Earth”. The International Graduate School for Science and Engineering of the Technische Universit€at M€unchen also supported this work.
References Dach R, Hugentobler U, Fridez P, Meindl M (eds) (2007) Bernese GPS software version 5.0. Astronomical Institute, University of Bern, Bern Dobslaw H, Thomas M (2007) Simulation and observation of global ocean mass anomalies. J Geophys Res 112:C05040. doi:10.1029/2006JC004035 ECMWF (2009) MARS user guide. Technical Notes, p 5. http:// www.ecmwf.int/publications/manuals/mars/guide/MarsUser Guide.pdf. Accessed date 17 August 2011 Flechtner F (2007) AOD1B product description document. GRACE project documentation, JPL 327–750, Rev. 1.0. JPL, Pasadena, CA. 17 August 2011 J€aggi A, Beutler G, Meyer U, Prange L, Dach R, Mervart L (2011) AIUB-GRACE02S: status of GRACE gravity field recovery using the celestial mechanics approach. In: Kenyon S et al (eds) Geodesy for planet earth. Springer, Heidelberg Kim J (2000) Simulation study of a low-low satellite-to-satellite tracking mission. Technical report, University of Texas at Austin, Austin, TX Ponte RM, Dorandeau J (2003) Uncertainties in ECMWF surface pressure fields over the ocean in relation to sea level analysis and modeling. J Atm Ocean Technol 20(2):301. doi:10.1175/1520-0426(2003)020 Salstein DA, Ponte RM, Cady-Pereira K (2008) Uncertainties in atmospheric surface pressure fields from global analyses. J Geophys Res 113:D14107. doi:10.1029/ 2007JD009531 Schmidt T, Wickert J, Heise J, Flechtner F, Fagiolini E, Schwarz G, Zenner L, Gruber T (2008) Comparison of ECMWF analyses with GPS radio occultations from CHAMP. Ann Geophys 26:3225–3234. http://www.ann-geophys.net/26/ 3225/2008/. Accessed date 17 August 2011
Challenges in Deriving Trends from GRACE
19
A. Eicker, T. Mayer-Guerr, and E. Kurtenbach
Abstract
The following contribution addresses some of the problems involved with the determination of long-term gravity field variations from GRACE satellite observations. First of all the choice of the time span plays a very important role, especially since it generally is a hard task to derive secular trends from only a few years of satellite data. Another issue, when one is interested in a single trend phenomenon, is the reduction of all other geophysical effects causing long-term gravity field variations. This paper uses the example of trends in continental hydrological water masses for the case of the High Plains aquifer to demonstrate some of the challenges implicated by trend analysis from GRACE.
19.1
Introduction
During the last years, the satellite mission GRACE (Tapley et al. (2004)) has provided significant improvement in the knowledge of the Earth’s gravity field, in the static as well as in the time variable component. The time variabilities derived from GRACE deliver information about short periodic variations such as the seasonal hydrological cycle (e.g. Schmidt et al. 2008), occasional incidents such as large earthquakes (Han et al. 2006), and also long-term trends. The investigations to be presented in this contribution are dedicated to the analysis of these long-term processes from GRACE observations. Examples are the melting of the ice sheets and glaciers in Greenland and Antarctica (Horwath and Dietrich 2009), rebound
A. Eicker (*) T. Mayer-Guerr E. Kurtenbach Institute of Geodesy and Geoinformation, University of Bonn, Nussallee 17, 53115 Bonn, Germany e-mail:
[email protected]
effects due to glacial isostatic adjustment (GIA), see e.g. Steffen et al. (2008), or trends in the continental hydrological water masses due to water withdrawal or climatic changes, as outlined in Rodell et al. (2009). When trying to derive long-term trends from GRACE gravity field observations, one is faced with a number of challenges. First of all, the time span used in the analysis is very important because it is very difficult to derive long-term gravity field variations from only 7 years of GRACE data or even less. The other important aspect that has to be taken into account is that when the focus is on one single phenomenon, all other effects causing long-term gravity field variations have to be removed. This is generally performed by the application of geophysical models, with the drawback that inaccuracies or uncertainties of these models are introduced into the trend analysis. Some confusion might be caused by the fact that these models are not only used for signal separation but also, with a different intention, in the GRACE data analysis (linearization, de-aliasing). The latter point is discussed in
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_19, # Springer-Verlag Berlin Heidelberg 2012
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Sect. 19.2 while Sect. 19.4 is dedicated to the problem of signal separation. The High Plains aquifer (marked in white in Fig. 19.2), located in the central United States with an area of 450,000 km2, serves as an example in this article to demonstrate some of the issues that have to be addressed when deriving long-term gravity field variations from GRACE. This investigation area was chosen because of an assumed mass loss trend due to ground water depletion caused by large-scale irrigation, see Strassberg et al. (2009). It is subject to discussion, whether GRACE is able to detect such human ground water withdrawals, see Rodell (2002) and Strassberg et al. (2009). However, this paper does not intend to give quantifications of potential mass loss or to draw qualitative conclusions whether such investigations are indeed possible with GRACE. The main intention is to point out some of the issues that have be considered when such investigations shall be performed. They are not specific for the example area but are valid for every investigation dealing with the interpretation of GRACE data.
19.2
Gravity Field Processing
In order to interpret the GRACE solutions and to derive, for example, trend information out of the GRACE products it is necessary to understand some details about the processing procedures. During the processing of the original GRACE data (level 1B), the observations y are expressed as non-linear functionals y ¼ f (x) of the unknown gravity field parameters x. In the examples described below, these parameters are given as a set of spherical harmonics coefficients for each monthly solution. After a linearization process the coefficients can be estimated by means of a least-squares adjustment. The linearization is given by: y ¼ f ðx0 Þ þ
@f ðx x0 Þ: @x x0
(19.1)
Here x0 describes the approximate solution, functionals of which are reduced from the original observations. This means that the least squares adjustment delivers a solution for Dx ¼ x x0. In order to keep the linearization error small, the approximate
solution should be as close as possible to the real gravity field signal in the corresponding month as possible. Therefore, the following models are taken into account by the processing centers: a static gravity field solution, a trend for the lower spherical harmonics, tides (direct tides, Earth and ocean tides), periodic changes of the centrifugal potential due to polar motion (pole tide and ocean pole tide), and mass variations of the atmosphere and the ocean. Apart from the linearization, these models serve also for the reduction of short-term mass variations that cannot be modelled by monthly representations (dealiasing). To obtain the complete gravity field signal for a monthly solution, the components removed by these models should be restored according to: x ¼ x0 þ Dx:
(19.2)
But in case of the official GRACE products, this is not carried out in a consistent manner. Only the static solution is re-added, but the trend and all other models are not present in the solutions. The atmosphere and ocean model, however, is delivered as an additional product in terms of monthly mean values. The consideration of these background models differs slightly from one analysis center to the next, for example by a different treatment of the permanent tides and different trend models for the lower harmonic coefficients. Details can be found in Flechtner (2007), Bettadpur (2007), and Watkins and Yuan (2007). Another aspect that has to be taken into account when dealing with GRACE solutions is the fact that the coefficients of degree n ¼ 1 cannot be determined from GRACE observations, and are therefore set to zero. This corresponds to the realization of a coordinate system in the center of mass (CM) of the complete Earth system (solid Earth including ocean and atmosphere and all other subsystems). The CM varies in comparison to the center of the solid Earth (CE) or the center of figure (CF). A good discussion of this topic can be found in Blewitt (2003). While there exist models to account for the seasonal variations of these changes, the longterm variations are still under investigation (Klemann and Martinec 2009). It is commonly known that the accuracy of the GRACE monthly solutions decreases with increasing spherical harmonic degree, i.e. with increasing resolution. Therefore, the solutions have to be filtered in order to allow a meaningful interpretation of the
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Challenges in Deriving Trends from GRACE
results. The choice of the filtering technique has a significant influence of the solution. As there have been a number of investigations on this topic in the past, see for example Steffen et al. (2009) and Werth et al. (2009), we will not go into further detail at this point.
19.3
Time Span
Some of the geophysical phenomena causing longterm gravity field variations have been going on for time spans of decades to millenia and even longer. In contrast to this, the measurement period of GRACE with only a few years of observations is very short. Taking into account the strong inter-annual gravity field variations, it is very difficult to derive representative trends from such a short time series. It has been shown, that especially when only parts of the GRACE period are evaluated, the estimated trend changes significantly, an example given by Horwath and Dietrich (2009) for the case of ice mass loss in Western Antarctica. In the study at hand, we perform a similar analysis for the temporal evolution in the High Plains aquifer, using the GFZ-RL04 monthly gravity field series (Flechtner et al. 2009) for the time span January 2004 to December 2008. The monthly gravity field solutions provided by the CSR (Bettadpur 2007), the JPL (Watkins and Yuan 2007) and the ITG-Grace03s (Mayer-G€urr et al. 2007) time series were evaluated as well, all providing similar results to those presented below. But for reasons of clarity and because the ITGGrace solutions are so far only available until April 2007, only the results for the GFZ solutions will be displayed. The monthly gravity field models are smoothed by a 500 km Gaussian filter and integrated over the area of the High Plains aquifer. Figure 19.1 displays the mass variations of the GFZ-RL04 monthly gravity field solutions in terms of geoid heights. If only the time span until the end of December 2006 is considered, the linear trend (black line on the left) shows a strongly negative behavior suggesting an interpretation as mass loss in the area. This is the time span under investigation in Strassberg et al. (2009), but the examination of the complete time series up to today shows that statements about such trends are critical from only a few years. When introducing the next 2 years of GRACE data,
155
the trend for 2007–2008 (black line on the right) shows a different behavior by being slightly positive. The overall linear trend for the whole time span (indicated by the gray line) is therefore slightly positive as well. Of course the results raise the question, whether a linear function is the correct representation of the long-term variability in this specific area. Alternatively, one could think about the use of basis functions changing slowly in time or applying low pass filters to remove short-term variations.
19.4
Separation of Trend Signals
GRACE can only observe the integrated mass signal without being able to distinguish between different mass sources. In order to separate different mass signals external information is required, which is commonly obtained from geophysical models. More precisely this implies that if one is interested in only one specific geophysical phenomenon (e.g. continental water variations), the GRACE solutions must be reduced by models of all other mass variations. At this point it has to be emphasized that this task of signal separation is in principle independent of the linearization and dealiasing process described above. This is especially important when considering the different error sources. During the linearization process, the introduction of approximate values does not lead to any additional model errors, apart from linearization errors which are negligible in case of GRACE. Therefore, after readding all models that are used in the processing, the result is the complete mass signal which is only influenced by the observation errors of GRACE. This solution should be independent of the inaccuracies in the models as they are only used as approximation values. In contrast to this, the model errors play a very important role for the task of signal separation, as the quality of the separation can only be as good as the accuracy of the models. In case of GRACE it is difficult to distinguish between signal separation and linearization as the restore step is not carried out in a consistent manner, as described in Sect. 19.2. In the context of our example of mass changes in the High Plains aquifer, the main focus is on the determination of changes in the continental water budget. Therefore, all other effects have to be removed from the GRACE signal. There is a wide variety of different mass variations that have to be considered.
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Fig. 19.1 Monthly solutions as provided by the GFZ-RL04 time series averaged over the High Plains aquifer region. One overall trend has been calculated for the entire time span (gray
line) and individual trends for the years 2004–2006 and 2007–2008 (black lines)
The first ones are the variabilities of the atmospheric masses and the effects on the ocean caused by these variabilities. These short- and long-periodic changes are included in the so-called atmosphere and ocean dealiasing product (AOD1B; see Flechtner 2005) which is reduced during the analysis process especially to prevent the solutions from aliasing errors caused by short periodic signals and is not restored afterwards. This means that the separation of the atmosphere masses has already been performed during the data processing step. Nevertheless, if a more suitable atmospheric model (e.g. a local model) is available, it might be a good choice to re-add the global model and remove a more accurate local model which is better tailored to the specific area. Another source for long-term variations might be the long-periodic part of the tidal forces (e.g. the 18.6year cycle), of the Earth tides and the ocean tides which appear as trends when only a few years of data are analyzed. However, those tidal effects are also taken into account by models during the GRACE gravity field analysis process from all of the different analysis centers and can therefore be regarded as nonexistent in the GRACE monthly solutions. These assumptions are only valid, of course, to the extent
of the accuracy of the given models, as errors in the modeling of these long-term variations might be wrongly interpreted as originating from a different source. It has to be emphasized that the AOD1B and tide models are used twice, each time with a different objective: for linearization but also for signal separation. A different issue is the handling of the trend which is removed during the GRACE processing. Most analysis centers use rates given for the lower spherical harmonic coefficients (dot coefficients) as proposed by the IERS conventions (McCarthy and Petit 2004). While these rates might be sufficient as approximation values, they are not suitable for signal separation. They were derived during the development of the EGM96 gravity field model by a combination of models and observations. The intention of these rates was to account for the integrated mass trend. It is not possible to distinguish which subsystems contribute to the dot coefficients. There is certainly some part of the glacial isostatic adjustment effect present, but a complete GIA model cannot be described by only a few coefficients. Furthermore, we do not know for certain whether there are also some hydrological effects comprised in those coefficients. Therefore, we
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Challenges in Deriving Trends from GRACE
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Fig. 19.2 Spatial distribution of different phenomena that cause long-term trends in GRACE data, displayed in terms of geoid heights per year in mm. Top: dot coefficients, middle: glacial isostatic adjustment, bottom: secular variations in pole tide model. The region of the High Plains aquifer in the Central United States is illustrated in white
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believe that it is preferable to restore these rates and to subtract the different models according to present knowledge afterwards. In order to give an impression of the order of magnitude and the spatial pattern,
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After re-adding the mass trend proposed by the IERS dot coefficients, the next step is to remove the known long-term mass change phenomena. As a first aspect in this context the glacial isostatic adjustment has to be mentioned, causing a mass redistribution due to the uplift of the land mass as a reaction to the deglaciation after the last ice age. This phenomenon can be observed especially in the Scandinavian and Northern Canadian area, but the effects can be witnessed further south as well. The spatial distribution of the GIA is displayed in the middle part of Fig. 19.2 as a rate in terms of millimeter of geoid height per year, the corresponding model was provided by Klemann and Martinec (2009). In the area of the High Plains aquifer positive as well as negative rates can be found in the GIA model, with the positive trend prevailing. The second aspect that has to be considered is the effect of the change in the centrifugal potential caused by polar motion (pole tide). While all of the analysis centers reduce a pole tide model during their gravity field estimation process, this model only contains the short-term variations of the pole tides, but not the secular mass changes. The spatial distribution of these secular variations, calculated according to the IERS conventions, can be found in the bottom part of Fig. 19.2. This effect is rather small globally, but in the investigation area in the central United States a comparably large positive trend can be observed nevertheless. The influence that the effects described above have on the estimation of the trend in the investigation area can be observed in Fig. 19.3. The original GFZ-RL04 time series is plotted in gray, where only the effects given by the de-aliasing product and the different tidal models have been removed. The remaining effects have been integrated over the High Plains region and the corresponding rates are shown as well. The effect caused by GIA is indicated by the orange line, here the positive trend can be noticed. The pole tide trend is given in green, even though this effect is quite small globally, in the region at interest a significant positive trend is present. The IERS rates are plotted in the black line and as already concluded from the spatial distribution, a large negative trend is visible when integrating over the High Plains region. Re-adding these rates and subtracting the GIA and pole tide model results in the reduced time series illustrated in red with the corresponding linear trend. In contrast to the original time series, the overall trend
A. Eicker et al.
is now negative. After having taken into account the variations of atmosphere and ocean, of tides and solid Earth mass transports, the remaining variations can be interpreted as changes in the hydrological budget. Here it has to be kept in mind that the changes in the gravitational potential (expressed in terms of geoid heights) does not only contain the changes in the water masses but also the reaction of the Earth’s crust due to loading. These effects can be separated using a loading theory. Specifically this implies the multiplication of the spherical harmonic coefficients with the load love numbers, see Wahr et al. (1998). Within this step it is reasonable to convert the resulting trend signal from potential to the generating water masses in terms of equivalent water heights. This conversion would have been mathematically possible at an earlier stage as well, but it is difficult to interpret effects such as GIA in terms of water heights. The resulting trend in terms of equivalent water heights of the original and the reduced time series is shown in Fig. 19.4. In contrast to the analysis in terms of geoid height, the trend is slightly positive. It is doubtful whether this trend can be regarded as significant. It is extremely difficult to make a reliable statement in this matter, as not only the currently achievable accuracy of GRACE has to be taken into account, but also errors in the background models used for signal separation. Especially for the geophysical models plausible error estimations are hardly available.
19.5
Conclusions
This article shows some of the pitfalls occurring when analyzing and interpreting long-term trends derived from GRACE data. The first important conclusion is the need of long time series to derive trends especially in areas with strong inter-annual variability. In order to give reliable statements concerning secular mass changes longer time series are inevitable, which emphasizes the importance of a GRACE follow-on mission without any gap. The second issue deals with the separation of signals originating from different geophysical processes. For example, if one is interested in changes in the hydrological water masses, the contributions of atmosphere, oceans, tides, and solid Earth have to be taken into account. These effects are usually removed by models. It has to be
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Challenges in Deriving Trends from GRACE
Fig. 19.3 Influence of the different background models on the trend in the High Plains aquifer: original GFZ-RL04 time series (gray), GIA model (orange), pole tide trend (green), rates
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proposed by the IERS (black). Reduced time series with corresponding linear trend: red
Fig. 19.4 The original (gray) and the reduced (red) time series expressed in terms of water heights in mm
kept in mind that in case of GRACE some part of the separation step has already taken place during the data analysis, while further models have to be reduced by the user. Other models are only used for linearization or de-aliasing purposes and are not suitable for signal
separation. This article summarizes the correct use of the different models and demonstrates their importance for the interpretation of trend signals from GRACE data using the example of the High Plains aquifer region.
160 Acknowledgments The authors would like to thank Volker Klemann from the GFZ for providing the GIA model. The support by the DFG (Deutsche Forschungsgemeinschaft) within the frame of the special priority program SPP1257 “Mass transport and mass distribution in the Earth system” is gratefully acknowledged.
References Bettadpur S (2007) UTCSR level-2 processing standards document for level-2 product release 0004. GR-03-03. CSR, Austin, TX Blewitt G (2003) Self-consistency in reference frames, geocenter definition, and surface loading of the solid Earth. J Geophys Res 108(B2):2103. doi:10.1029/2002JB002082 Flechtner F (2005) GRACE AOD1B product description document (Rev. 2.1) Flechtner F (2007) GFZ level-2 processing standards document for level-2 product release 0004. GR-GFZ-STD-001. GFZ, Potsdam Flechtner F, Dahle Ch, Neumayer KH, Koenig R, Foerste Ch (2009) The release 04 CHAMP and GRACE EIGEN gravity field models. In: Flechtner F, Gruber T, Guentner A, Mandea M, Rothacher M, Wickert J (eds) Satellite geodesy and earth system science – observation of the earth from space. Springer, Berlin (in preparation) Han S-C, Shum CK, Bevis M, Ji C, Kuo C-Y (2006) Crustal dilatation observed by GRACE After the 2004 SumatraAndaman Earthquake. Science 313:658–662. doi:10.1126/ science.1128661 Horwath M, Dietrich R (2009) Signal and error in mass change inferences from GRACE: the case of Antarctica. Geophys J Int 177(3):849–864. doi:10.1111/j.1365-246X.2009.04139.x Klemann V, Martinec Z (2009) Contribution of glacial-isostatic adjustment to the geocenter motion. Tectonophysics. doi:10.1016/j.tecto.2009.08.031 Mayer-G€urr T, Eicker A, Ilk KH (2007) ITG-Grace03 gravity field model. http://www.geod.uni-bonn.de/itg-grace03.html
A. Eicker et al. McCarthy DD, Petit G (2004) IERS conventions 2003. IERS technical notes, 32 Verlag des Bundesamts fuer Kartographie und Geod€asie, Frankfurt am Main Rodell M (2002) The potential for satellite-based monitoring of groundwater storage changes using GRACE: the High Plains aquifer, Central US. J Hydrol 63:245–256. doi:10.1016/ S0022-1694(02)00060-4 Rodell M, Velicogna I, Famiglietti JS (2009) Satellite-based estimates of groundwater depletion in India. Nature 460: 999–1002. doi:10.1038/nature08238 Schmidt R, Petrovic S, G€untner A, Barthelmes F, W€unsch J, Kusche J (2008) Periodic components of water storage changes from GRACE and global hydrology models. J Geophys Res 113:B08419. doi:10.1029/2007JB005363 Steffen H, Denker H, M€uller J (2008) Glacial isostatic adjustment in Fennoscandia from GRACE data and comparison with geodynamic models. J Geodyn 46(3–5):155–164. doi:10.1016/j.jog.2008.03.002 Steffen H, Petrovic S, M€uller J, Schmidt R, W€unsch J, Barthelmes F, Kusche J (2009) Significance of secular trends of mass variations determined from GRACE solution. J Geodyn 48(3–5):157–165. doi:10.1016/j.jog.2009.09.029 Strassberg G, Scanlon BR, Chambers D (2009) Evaluation of groundwater storage monitoring with the GRACE satellite: case study of the High Plains aquifer, central United States. Water Res Int 45:W05410. doi:10.1029/2008WR006892 Tapley BD, Bettadpur S, Watkins M, Reigber C (2004) The gravity recovery and climate experiment: mission overview and early results. Geophys Res Lett 31:L09607 Wahr J, Molenaar M, Bryan F (1998) Time variability of the Earth’s gravity field: hydrological and oceanic effects and their possible detection using GRACE. J Geophys Res 103(B12):30, 20530, 229 Watkins M, Yuan D-N (2007) JPL level-2 processing standards document for level-2 product release 04. ftp://podaac.jpl. nasa.gov/pub/grace/doc/ Werth S, G€untner A, Schmidt R, Kusche J (2009) Evaluation of GRACE filter tools from a hydrological perspective. Geophys J Int 179(3):14991515
AIUB-GRACE02S: Status of GRACE Gravity Field Recovery Using the Celestial Mechanics Approach
20
A. J€aggi, G. Beutler, U. Meyer, L. Prange, R. Dach, and L. Mervart
Abstract
The gravity field model AIUB-GRACE02S is the second release of a model generated with the Celestial Mechanics Approach using GRACE data. Intersatellite K-band range-rate measurements and GPS-derived kinematic positions serve as observations to solve for the Earth’s static gravity field in a generalized orbit determination problem. Apart from the normalized spherical harmonic coefficients up to degree 150, arc-specific parameters like initial conditions and pseudo-stochastic parameters are solved for in a rigorous least-squares adjustment based on both types of observations. The quality of AIUB-GRACE02S has significantly improved with respect to the earlier release 01 due to a refined orbit parametrization and the implementation of all relevant background models. AIUB-GRACE02S is based on 2 years of data and was derived in one iteration step from EGM96, which served as a priori gravity field model. Comparisons with levelling data and models from other groups are used to assess the suitability of the Celestial Mechanics Approach for GRACE gravity field determination.
20.1
Introduction
The Gravity Recovery And Climate Experiment (GRACE) mission, launched on March 17, 2002, has significantly improved our knowledge of the Earth’s gravity field in terms of accuracy, spatial and temporal resolution. The time-varying part of the Earth’s gravity field has been inferred with unprecedented accuracy from space by analyzing GPS, accelerometer, and
A. J€aggi (*) G. Beutler U. Meyer L. Prange R. Dach Astronomical Institute, University of Bern, Sidlerstrasse 5, 3012, Bern, Switzerland e-mail:
[email protected] L. Mervart Institute of Advanced Geodesy, Czech Technical University, Thakurova 7, 16629, Prague, Czech Republic
inter-satellite K-band observations (Tapley et al. 2004). For the static part GRACE models report an accuracy of about 3 cm in terms of geoid heights at a spatial resolution of 200 km (F€orste et al. 2008a), which is an improvement by about one order of magnitude compared to the high-resolution model EGM96 (Lemoine et al. 1997). Apart from the official GRACE models derived by the Center for Space Research (CSR) of the University of Texas at Austin, the Jet Propulsion Laboratories (JPL), and the GeoForschungsZentrum Potsdam/Groupe de Recherche de Ge´ode´sie Spatiale (GFZ /GRGS), alternative state-of-the-art GRACE gravity field models have been computed, e.g., at the Institut f€ur Geod€asie und Geoinformation of the University of Bonn (Mayer-G€urr 2008).
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_20, # Springer-Verlag Berlin Heidelberg 2012
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Gravity field recovery from satellite data has also been initiated at the Astronomical Institute of the University of Bern (AIUB). The so-called Celestial Mechanics Approach (CMA) has been applied to both the high-low satellite-to-satellite tracking (hlSST) data of CHAMP and the low-low (ll) SST data of GRACE. Consolidated results for CHAMP and first results for GRACE may be found in Prange et al. (2009) and J€aggi et al. (2010), respectively. This article focuses on updates of the CMA, which were implemented to improve the combined processing of GRACE hl-SST and ll-SST data of the K-band ranging system (Sect. 20.2). Various gravity field recovery experiments illustrate the sensitivity of the CMA on, e.g., the orbit parametrization or the background modeling (Sect. 20.3). A significantly improved gravity field model AIUB-GRACE02S has been derived from GRACE data covering the years 2006 and 2007 (Sect. 20.4). The quality of this 2-year solution is assessed and perspectives for future developments are presented.
20.2
Celestial Mechanics Approach
GRACE gravity field determination using the CMA is based on the analysis of Level 1B inter-satellite Kband measurements (Case et al. 2002) and GPSderived kinematic positions (J€aggi et al. 2009). The static part of the gravity field is modeled with a series of normalized spherical harmonic (SH) coefficients (Heiskanen and Moritz 1967) from degree 2 up to a maximum degree of 150. The time variable part is currently not yet modeled. Based on a priori orbits derived from the kinematic positions of both GRACE satellites and the inter-satellite measurements (Sect. 20.1) normal equations for both types of observations are set up on a daily basis for the unknown gravity field coefficients and for additional arc-specific parameters, i.e., two normal equation systems based on the kinematic positions of GRACE A and B (Sect. 20.2), and one normal equation system based on the ll-SST data of the K-band ranging system (Sect. 20.3). The resulting daily normal equations are then combined into one system for each daily arc. Finally, arc-specific parameters are pre-eliminated and the combined daily normal equations are accumulated into monthly, annual, and multi-annual systems, which are eventually inverted to solve for the SH coefficients
without applying any regularization. Details about the general procedure may be found in (J€aggi et al. 2010).
20.2.1 A Priori Orbit Generation As opposed to the processing scheme described by J€aggi et al. (2010), improved a priori orbits are used here for gravity field recovery: Kinematic positions and K-band measurements are already taken into account for the a priori orbit generation. Using a specific force model (a priori gravity field model, accelerometer data, ocean tide model,. . .), only the weighted kinematic positions of both GRACE satellites are fitted in a first step by numerically integrating the corresponding equations of motion (Beutler 2005) and by adjusting the arc-specific orbit parameters. Efficient numerical integration techniques are applied to solve the so-called variational equations (Beutler 2005) in order to obtain the needed partial derivatives with respect to the orbit parameters. The normal equations of this first step are stored as they are subsequently needed for the generation of the “final” a priori orbits. Apart from the initial osculating elements constant and once-per-revolution empirical accelerations are set up, which act over the entire arc in the radial, along-track, and cross-track directions, and constrained piecewise constant accelerations acting over 15 min intervals in the same three directions. Additional polynomial coefficients up to degree 3 are set up per arc in the along-track direction in order to compensate for instrument drifts when using accelerometer data. For numerical reasons the arc-specific parameters are not set up separately for the two GRACE satellites. Transformed parameters representing the mean values of the corresponding parameters of both GRACE satellites, which are mainly determined by the kinematic positions, and half of the differences, which are mainly determined by the K-band observations are used. Based on the orbits of GRACE A and B from the first step, K-band measurements are used to set up the K-band normal equations for the same arc-specific parameters. The three normal equation contributions are then combined into one system for each daily arc, which is eventually inverted to solve for the arcspecific orbit parameters. The a priori orbits used for gravity field recovery are finally obtained by propagating the improved state vectors by numerical integration. As these orbits represent the K-band
20
AIUB-GRACE02S: Status of GRACE Gravity Field Recovery
observations already at a rather comfortable level even when using an a priori gravity field model from the pre-GRACE era, the gravity field solution should not be harmed by linearization effects (see Sect. 20.3).
20.2.2 Daily Normal Equations from Positions Based on the a priori orbits of GRACE A and B gravity field recovery from orbit positions is set up as a generalized orbit improvement process for each GRACE satellite as described by J€aggi et al. (2010). The actual orbit of one satellite is expressed as a truncated Taylor series with respect to the unknown arc-specific orbit parameters and the unknown SH coefficients about the a priori orbit. In analogy to Sect. 20.2.1 the partial derivatives of the a priori orbit are computed with respect to all parameters. These partials eventually allow it to set up the daily normal equations based on kinematic positions for all parameters according to a standard least-squares adjustment. In analogy to Sect. 20.2.1 the arc-specific parameters are set up as the sum and the difference of the original parameters in order to avoid numerical problems.
20.2.3 Daily Normal Equations from II-SST Based on the a priori orbits of GRACE A and B gravity field recovery from K-band measurements is set up as a differential orbit improvement process as described by J€aggi et al. (2010). The actual orbit difference is expressed as a truncated Taylor series with respect to the unknown parameters about the a priori orbit difference. The same techniques as in Sect. 20.2.2 are applied to solve the variational equations separately for both GRACE satellites. As K-band observations only contain information about the line-of-sight orbit difference, a projection of the partial derivatives on the line-of-sight direction is required. These projected partials eventually allow it to set up the daily normal equations based on K-band measurements for all parameters. The transformed arc-specific parameters are set up for both GRACE satellites, which implies that the daily normal equation matrices would be singular if only K-band data were used.
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20.3
Gravity Field Recovery Studies
Gravity field models from GRACE kinematic positions and K-band range-rate data of the year 2007 were derived using the methodology outlined in Sect. 20.2. Due to the computational burden of gravity field recovery, only static fields up to degree 120 were estimated. EGM96 served as a priori model up to the same degree. A spacing of 15 min for the piecewise constant accelerations turned out to be sufficient to account for various model shortcomings. In order to illustrate the sensitivity of the CMA on orbit parametrization (Sect. 20.3.1), data weighting (Sect. 20.3.2), linearization effects (Sect. 20.3.3), and background models (Sect. 20.3.4), different solutions were made, which are subsequently compared with the gravity field model ITG-GRACE03S (Mayer-G€urr 2008) in terms of the square-roots of the degree difference variances up to degree 120.
20.3.1 Orbit Parametrization As the two GRACE satellites are only separated by about 30 s on the same orbital trajectory, they experience almost the same perturbations. An orbit and gravity field determination tailored to the GRACE configuration may therefore use this information and introduce additional, relative constraints between the estimated piecewise constant accelerations of both GRACE satellites. A series of gravity field models was thus generated to study the impact of relative constraints, which have been chosen to be 100 times tighter than the absolute constraints imposed on the piecewise constant accelerations. Figure 20.1 shows the impact of such constraints. The largest part of the improvement w.r.t. ITG-GRACE03S is already obtained by applying relative constraints only for the along-track direction. Additional constraints for the radial direction have a small positive impact, as well, whereas relative constraints for the cross-track direction do not significantly change the solution. The agreement (consistency) with the reference model for degree 2 is, however, degraded when relative constraints are applied for the radial and the along-track directions. Comparisons with EIGEN-GL04C (F€orste et al. 2008a) even revealed a degradation when relative constraints are only applied for the along-track
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164 Fig. 20.1 Square-roots of degree difference variances of annual recoveries when using different constraining options
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direction (not shown). Further investigations are necessary to better understand this mechanism.
20.3.2 Data Weighting According to J€aggi et al. (2010) a nominal weighting ratio of 1:108 between the GPS (L1) carrier phase
observations (error propagation taken into account by epoch-wise covariance information) and the K-band range-rate observations has been used for the determination of the gravity field solutions shown in Fig. 20.1. It turned out, however, that the agreement with ITG-GRACE03S may be slightly improved if stronger ratios are used, e.g., 1:109 or even 1:1010 (not shown). Despite the fact that a weighting ratio of 1:1010 is
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AIUB-GRACE02S: Status of GRACE Gravity Field Recovery
165
unrealistic in view of the a posteriori RMS errors, which are about 1 mm for the (L1) GPS carrier phase observations, and about 0.23 mm/s for the K-band range-rate observations (see Fig. 20.4), the latter was chosen for all subsequent solutions.
piecewise constant accelerations are able to compensate for the unmodeled non-gravitational accelerations to a great extent. Note that the cross-track component of the accelerometer data has not been taken into account for the solutions presented in this article, as almost no impact could be detected. The differences between the nominal and the experimental solutions are visible only up to about degree 90 because of an inadequate implementation of the ocean tide model used in the CMA at that time (see Sect. 20.3.4.3), and because of the omission errors due to the limited maximum degree of 120.
20.3.3 Linearization Gravity field recovery may depend on the a priori gravity field model used to set up the (linearized) observation equations. In order to rule out such dependencies for the CMA, EGM96 served as a priori gravity field model for all solutions presented in this article. Figure 20.2 shows the agreement of this baseline solution (iteration 1) with ITG-GRACE03S. A full iteration cycle was performed in order to assess the magnitude of linearization effects. For that purpose the previously obtained gravity field model (iteration 1) was used as a priori model for an additional iteration step (iteration 2). Figure 20.2 confirms that the agreement with ITG-GRACE03S is almost identical for both iteration steps. For most degrees the differences between the two iterations are about one order of magnitude smaller than the differences w.r.t. ITGGRACE03S. Gravity field recovery with the CMA is thus considered as highly independent of the a priori gravity field.
20.3.4 Background Modeling Since the results presented by J€aggi et al. (2010) all relevant background models are now implemented in the CMA. Their impact on static GRACE gravity field solutions is studied in the following subsections.
20.3.4.1 Non-gravitational Accelerations Accelerometer data are taken into account for GRACE gravity field recovery to separate the non-gravitational accelerations from the gravitational signal. Figure 20.3 compares the agreement of this nominal solution and of an experimental solution obtained without accelerometer data with ITG-GRACE03S. As expected, GRACE gravity field recovery clearly benefits from using accelerometer data. It is, however, remarkable that solutions of quite good quality may be obtained with the CMA even if accelerometer data are left out from the processing. Obviously, the estimated
20.3.4.2 Atmospheric and Oceanic Dealiasing Figure 20.3 also shows the impact of the non-tidal atmosphere and ocean short-term mass variations (Flechtner et al. 2006) on a static gravity field solution. Although the atmospheric and oceanic dealiasing (AOD) products are important for the generation of GRACE monthly solutions and clearly improve the K-band range-rate residuals (see Fig. 20.4 or Zenner et al. (2011) for more details), only a relatively small effect is visible in a static 1-year solution. 20.3.4.3 Ocean Tides The implementation of the ocean tide model in the CMA has been revised. Figure 20.3 shows that the improved implementation mainly had a positive effect on the high degrees above 90. The analysis of ocean tides also included the implementation of the models FES2004 (Lyard et al. 2006) and EOT08a (Savcenko and Bosch 2008) in the CMA. As no large differences on a static one-year solution have been found when using FES2004 (not shown) or EOT08A, the more recent EOT08a ocean tide model has been selected for the generation of the AIUB-GRACE02S model.
20.4
AIUB-GRACE02S
The AIUB-GRACE02S model was derived from kinematic positions and K-band range-rate data covering the years 2006 and 2007 using the methods from Sects. 20.2 to 20.3. As opposed to Sect. 20.3, the maximum degree was increased from 120 to 150. EGM96 thus also served as a priori model up to degree 150.
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166 Fig. 20.3 Square-roots of degree difference variances of annual recoveries when using different processing options
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20.4.1 Comparison with Other Models Figure 20.5 shows the agreement of the two solutions obtained from the 2006 and 2007 data sets and of the combined 2-year solution AIUB-GRACE02S w.r.t. ITG-GRACE03S. Compared to the figures from the previous section, clear improvement results for the higher degrees, but also the medium to high degrees
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show a better agreement with ITG-GRACE03 due to the increased maximum degree. Figure 20.5 even suggests that a maximum degree larger than 150 might have been chosen for the 2-year solution. The quality of the lower degrees improved slightly as well. Figure 20.6 shows the agreement of the three satellite-only models AIUB-GRACE02S, EIGEN-5S (F€orste et al. 2008b), and GGM03S (Tapley et al.
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AIUB-GRACE02S: Status of GRACE Gravity Field Recovery
Fig. 20.5 Square-roots of degree difference variances of annual recoveries and the combined 2-year solution
167
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2007) with ITG-GRACE03S. Although AIUBGRACE02S is based on only about half of the data volume of the other models, the agreement with ITGGRACE03S is comparable to that of EIGEN-5S and GGM03S for most degrees. Deficiencies in the AIUB-GRACE02S solution are only visible for degrees below 20. Longer data spans and, in particular, the estimation of time variable signals will further improve the low degree SH coefficients. Figure 20.4
ITG−GRACE03S GGM03S EIGEN−5S AIUB−GRACE02S
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150
indicates that the time variability of the Earth’s gravity field has a clear impact on the residuals of gravity field recovery.
20.4.2 Validation with External Data T. Gruber from the Institut f€ur Astronomische und Physikalische Geod€asie of the Technische Universit€at
A. J€aggi et al.
168 Table 20.1 RMS of differences (cm) between levelling and model geoid heights up to a maximum degree of 120 Levelling data EUREF GPS BRD EUVN BRD GPS Canada GPS 1998 Canada GPS 2007 Australia GPS Japan GPS USA GPS
2007 Solution 22.2 04.4 05.0 20.9 15.3 25.0 11.2 34.2
AIUB-GRACE02S 22.2 05.3 05.7 20.2 14.5 24.6 10.8 33.6
M€unchen compared the geoid heights derived from AIUB-GRACE02S, ITG-GRACE03S, EIGEN-5S, and GGM03S with geoid heights derived from different sets of recent levelling data using the method described in (Gruber 2004). Table 20.1 shows the RMS values of the differences around the mean values between the filtered levelling geoid heights and geoid heights from the different models up to degree 120. The solution AIUB-GRACE02S is of the same quality as the other satellite-only models. More pronounced differences between the models are only visible for some of the high-quality data sets, e.g., the BRD EUVN or BRD GPS data sets. Compared to the solution obtained from the GRACE data of the year 2007, AIUB-GRACE02S has improved for all data sets except for BRD EUVN and BRD GPS. The reason is yet unclear.
20.5
Conclusions
We used the CMA for gravity field determination with GPS-derived kinematic GRACE positions and Level 1B K-band range-rate data to generate the static gravity field model AIUB-GRACE02S with data covering the years 2006 and 2007. We showed that it is possible to start the recovery with EGM96 as a priori model and that the solution may be obtained in one iteration step. Accelerometer data are important for the quality of the resulting gravity field model, but models of remarkable quality may also be obtained with the CMA without accelerometer data. The overall quality of AIUB-GRACE02S was found to be comparable to that of other well-known satellite-only gravity fields, which is also confirmed by the validation with terrestrial measurements.
ITG-GRACE03S 21.9 03.7 04.4 19.6 15.0 24.3 10.3 33.4
EIGEN-5S 22.6 05.9 06.3 19.9 15.5 24.5 11.8 33.3
GGM03S 22.5 05.4 06.2 19.7 14.8 24.2 10.6 33.3
The quality of the very low degrees of AIUBGRACE02S is not yet optimal and should be improved in a future release. Part of the degradation might be due to too strong relative constraints applied to the piecewise constant accelerations. This aspect has to be further studied. The unmodeled time variability of the Earth’s gravity field, which clearly shows up in the a posteriori K-band range-rate residuals, could also contribute to the not yet optimally estimated low degree SH coefficients. Further refinements of the CMA will include the estimation of time variable signals for the low degree SH coefficients. This and the use of a longer data span will further improve GRACE gravity field recovery with the CMA. Acknowledgements The authors gratefully acknowledge the generous financial support provided by the Swiss National Science Foundation and the Institute for Advanced Study (IAS) of the Technische Universit€at M€unchen.
References Beutler G (2005) Methods of celestial mechanics. Springer, Berlin Case K, Kruizinga G, Wu S (2002) GRACE level 1B data product user handbook. D-22027. JPL, Pasadena, CA Flechtner F, Schmidt R, Meyer U (2006) De-aliasing of shortterm atmospheric and oceanic mass variations for GRACE. In: Flury J, Rummel R, Reigber C, Rothacher M, Boedecker G, Schreiber U (eds) Observation of the earth system from space. Springer, Heidelberg, pp 83–97 F€orste C, Schmidt R, Stubenvoll R, Flechtner F, Meyer U, K€onig R, Neumayer H, Biancale R, Lemoine JM, Bruinsma S, Loyer S, Barthelmes F, Esselborn S (2008a) The GeoForschungsZentrum Potsdam/Groupe de Recherche de Ge´ode´sie Spatiale satellite-only and combined gravity field models: EIGEN-GL04S1 and EIGEN-GL04C. J Geod 82:331–346 F€orste C, Flechtner F, Schmidt R, Stubenvoll R, Rothacher M, Kusche J, Neumayer KH, Biancale R, Lemoine JM,
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Barthelmes F, Bruinsma S, K€ onig R, Meyer U (2008b) EIGEN-GL05C – a new global combined high-resolution GRACE-based gravity field model of the GFZ-GRGS cooperation. Geophysical Research Abstracts, vol. 10. EGU, Vienna Gruber T (2004) Validation concepts for gravity field models from satellite missions. In: Proceedings of second international GOCE user workshop “GOCE, The Geoid and Oceanography”, ESA-ESRIN, Frascati Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman, San Francisco, CA J€aggi A, Dach R, Montenbruck O, Hugentobler U, Bock H, Beutler G (2009) Phase center modeling for LEO GPS receiver antennas and its impact on precise orbit determination. J Geod 83:1145–1162 J€aggi A, Beutler G, Mervart L (2010) GRACE gravity field determination using the celestial mechanics approach – first results. In: Mertikas S (ed) Gravity, geoid and earth observation. Springer, Berlin, pp 177–184 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) IAG symposia: gravity, geoid and marine geodesy. Springer, Berlin, pp 461–469
Lyard F, Lefevre F, Letellier T, Francis O (2006) Modelling the global ocean tides: insight from FES2004. Ocean Dyn 56:394–415 Mayer-G€urr T (2008) Gravitationsfeldbestimmung aus der Analyse kurzer Bahnb€ogen am Beispiel der Satellitenmissionen CHAMP und GRACE. Schriftenreihe 9, Institut f€ur Geod€asie und Geoinformation, University of Bonn, Bonn Prange L, J€aggi A, Dach R, Bock H, Beutler G, Mervart L (2009) AIUB-CHAMP02S: the influence of GNSS model changes on gravity field recovery using spaceborne GPS. Adv Space Res 45:215–224 Savcenko R, Bosch W (2008) EOT08a – empirical ocean tide model from multi-mission satellite altimetry. DGFI report 81, Deutsches Geod€atisches Forschungsinstitut, Munich Tapley BD, Bettadpur S, Ries JC, Thompson PF, Watkins M (2004) GRACE measurements of mass variability in the Earth system. Science 305(5683):503–505 Tapley B, Ries J, Bettadpur S, Chambers D, Cheng M, Condi F, Poole S (2007) The GGM03 mean earth gravity model from GRACE. Eos Trans AGU 88(52) Zenner L, Gruber T, Beutler G, J€aggi A, Flechtner F, Schmidt T, Wickert J, Fagiolini E, Schwarz G, Trautmann T (2011) Using atmospheric uncertainties for GRACE de-aliasing – first results. In: Kenyon S et al (eds) Geodesy for planet earth. Springer, Heidelberg
.
Comparison of Regional and Global GRACE Gravity Field Models at High Latitudes
21
B.C. Gunter, T. Wittwer, W. Stolk, R. Klees, and P. Ditmar
Abstract
In this study we address the question of whether regional gravity field modeling techniques of GRACE data can offer improved resolution over traditional global spherical harmonic solutions. Earlier studies into large, equatorial river basins such as the Amazon, Zambezi and others showed no obvious distinction between regional and global techniques, but this may have been limited by the fact that these equatorial regions are at the latitudes where GRACE errors are known to be largest (due to the sparse groundtrack coverage). This study will focus on regions of higher latitude, specifically Greenland and Antarctica, where the density of GRACE measurements is much higher. The regional modeling technique employed made use of spherical radial basis functions (SRBF), complete with an optimal filtering algorithm. Comparisons of these regional solutions were made to a range of other publicly available global spherical harmonic solutions, and validated using ICESat laser altimetry. The timeframe considered was a 3 year period spanning from October 2003 to September 2006.
21.1
Introduction
The launch of the Gravity Recovery and Climate Experiment (GRACE) in 2002 started a new wave of research into the Earth’s mass transport processes. The measurements from the mission’s twin satellites have enabled the multi-year tracking of many large scale processes, such as continental hydrology and ice mass changes in the cryosphere. While these first studies have produced some truly excellent results, there is always the desire to push the boundaries of what
B.C. Gunter (*) T. Wittwer W. Stolk R. Klees P. Ditmar Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, 2629HS Delft, The Netherlands e-mail:
[email protected]
GRACE can observe, in terms of spatial and temporal resolution. Previous studies have demonstrated that the current processing standards of GRACE data provide mass change accuracies on the order of 2 cm of equivalent water height (EWH) over spatial scales of 400 km and time intervals of 1 month (Klees et al. 2008a). This analysis was done by comparing the performance of a range of different GRACE processing strategies, including both regional and global methods, over selected river basins and other “dry” regions where little to no hydrological signal is expected. The global methods tested primarily involved traditional spherical harmonic solutions from various processing centers (CSR, GFZ, JPL, CNES, DEOS), but with various spatial filters applied. The regional solutions examined included the “mascon” approach (Luthcke et al. 2006) as well as
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solutions computed using spherical radial basis functions. In short, the overall conclusion of this earlier study was that there was no clear advantage to using regional techniques over global methods for the river basins studied. In fact, it turned out that the choice of spatial filter was the most important aspect in the comparisons; however, one of the limitations of this particular study was that most of the regions studied were at relatively low latitudes, where the density of GRACE measurements is the lowest. For higher latitude regions, it is possible that the increased data density might offer a higher signal-to-noise ratio that regional techniques might be able to better exploit. As a result, a follow-on study was conducted (Stolk 2009) to perform a similar analysis over regions such as Greenland and Antarctica, to see if the conclusions would be the same. This paper will provide an overview of the methods and conclusions of this follow-on study.
21.2
Spherical Radial Basis Functions
The focus of the regional techniques for the highlatitude regions involved the application of spherical radial basis functions (SRBF). The general concept behind this approach is to use a distribution of spacelocalizing functions to represent any complex spherical shape, such as the Earth’s gravity field. The functions can be constructed using a number of different methods, although the kernel adopted for the current study makes use of Poisson wavelets of order three (Holschneider et al. 2003; Wittwer 2009). The shape and spatial distribution of the SRBFs are determined by the depth (i.e., bandwidth) and the level (i.e., spacing on a Reuter grid) assigned to each SRBF, as illustrated in Fig. 21.1. As with spherical harmonic solutions, SRBF solutions suffer from north-south error patterns (i.e., “stripes”), which require the application of a suitable filter. The anisotropic, non-symmetric (ANS) filter developed by Klees et al. (2008b) offers a number of benefits over other traditional filtering techniques, such as destriping or Gaussian smoothing, primarily because use is made of the full statistical information of the solution (i.e., signal and noise variance-covariance matrices are used). For example, if spherical harmonics are used to parameterize the time-variable gravity field, and we let N^ x ¼ b represent the normal
equations for a monthly GRACE solution, the ANS filter W can be applied as follows: ^xw ¼ W^x ¼ ðN þ D1 Þ1 b
(21.1)
where N is the normal matrix, D the signal covariance matrix (i.e., the auto-covariance matrix of the vector ^x), ^x the estimated parameter vector, and b the rightside vector. The matrix N is determined from the partial derivatives of the system dynamics; however, the auto-covariance matrix, D, must be determined empirically. This is done through an iterative process whereby the (time-independent) variances of the signal from the actual time series of monthly solutions (e.g., 36 months for this study) are computed at the nodes of an equal-angular grid and then transformed back to the spherical harmonic domain to form D. This signal covariance information has the effect of suppressing spurious noise in regions that typically do not have much mass variations (e.g., oceans and deserts), while also allowing the solution to adjust more freely in areas where the mass change signal has larger variations (e.g., river basins). Since this signal covariance matrix is computed from the time series of GRACE solutions, it is particular to the solution technique. A straightforward generalization of this concept to a SRBF parameterization is obtained when the relationship between spherical harmonic coefficients, x, and SRBF coefficients, a, is exploited. This relationship can be written as x ¼ Qa
(21.2)
Hence, given the auto-covariance matrix in the spherical harmonic domain, D, we can obtain the corresponding auto-covariance matrix in the SRBF domain according to a ¼ Qþ x ) Da ¼ Qþ DðQþ ÞT
(21.3)
where Q+ ¼ (QTQ)1QT is the pseudo-inverse of the Q matrix. This approach, however, fails because the spectrum of a given SRBF parameterization comprises spherical harmonic degrees, which may exceed the maximum degree of a given GRACE monthly solution (the number of harmonic coefficients in x is often much larger than the number of SRBF coefficients in a).
21
Comparison of Regional and Global GRACE Gravity Field Models at High Latitudes
173
Fig. 21.1 Example spherical radial basis functions
Therefore, the optimal filter needs to be designed directly in the SRBF domain. If f is the time-variable gravity signal in terms of equivalent water heights, and a comprises the SRBF coefficients, we write the SRBF synthesis as f ¼ Ba
(21.4)
Using the pseudo-inverse of B, B+ ¼ (BTB)1BT, we write a ¼ Bþ f
(21.5)
and obtain the auto-covariance matrix in the SRBF domain, Da, as Da ¼ Bþ D ðBþ ÞT
(21.6)
Hence, if Na a ¼ ba is the system of normal equations in terms of SRBFs, the equivalent expression of (21.1) is aW ¼ Wa a ¼ ðNa þ Da Þ1 ba
(21.7)
With these relationships, the signal covariance matrix now can be computed, and the ANS filter applied to the SRBF coefficients. Note that since the computation of the signal covariance matrix is done iteratively, an initial set of values must first be chosen. The standard deviations chosen for this initial signal variance covariance matrix are essentially arbitrary, although proper choices might reduce the number of iterations needed. For the current study, the initial standard deviations were set to 50 mm globally. This initial standard deviation is propagated from the spatial domain to the frequency domain using (21.6), then a new signal variance matrix is created from the
filtered solution. Iteration is halted when the difference in equivalent water height between two consecutive iterations for each grid point is less than 35 mm (chosen experimentally to balance convergence speed and the determination of accurate signal variability). The determination of the optimal values for the level and depth of the SRBF solutions depends on the spatial variations, and noise content, of the data involved. Placing a dense grid of functions at a relatively shallow depth (i.e., small bandwidth) may result in noisy solutions, especially for GRACE data. The general approach used here was to employ a level high enough to represent what was believed to be the signal content in the data, and to place these functions as deep as possible in an attempt to smooth out the noise in the data. Many combinations of level and depth were evaluated, with the determination that a level 90 (i.e., ~220 km Reuter grid spacing), depth 900 km parameterization offers the highest quality solutions for Greenland and Antarctica.
21.3
Comparisons
Having finalized the optimal parameterization and filtering of the SRBF solutions, the next step was to compare the results of the mass change estimates derived from these solutions to those derived from other techniques, over Greenland and Antarctica. Since the goal of the study was simply to compare global versus regional techniques, only a limited set of spherical harmonic solutions were involved, and included those from the Center for Space Research (CSR) and the Delft Institute of Earth Observation and Space Systems (DEOS), who now produce a set of publically available monthly gravity solutions called the DEOS Mass Transport (DMT-1) models
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Table 21.1 Descriptions of the various global and regional solutions used for comparison Model name CSR DS400
Solution type Global
CSR DS0 DMT-1
Global Global
SRBF global
Regional
SRBF regional
Regional
Description Spherical harmonic solution to 60x60 derived from CSR RL04 data (see http://podaac.jpl.nasa.gov/ grace) ; destriped; 400km Gaussian smoothing applied; SLR C20 values; degree 1 coefficients taken from Swenson et al. (2008) Similar to above, except without Gaussian smoothing applied DEOS Mass Transport models (see http://www.lr.tudelft.nl/psg/grace); spherical harmonic solutions to 120x120 generated from KBR L1B data using the range-combination approach (Liu et al. 2009); anisotropic non-symmetric (ANS) filter applied (Klees et al. 2008b) Spherical radial basis function approach using Poisson wavelets, in which a global distribution of nodes with a Reuter grid spacing of level 90 and depth of 900km is used. Low level data derived from the same KBR L1B data as the DMT-1 solution, with a similar anisotropic non-symmetric filter applied (adapted for use with SRBF’s) Similar to the SRBF global approach, but using only regional data (i.e., within a 30 extended boundary from the target region)
(Liu et al. 2009). A summary of the solutions used in the comparisons is provided in Table 21.1. In short, the CSR solutions used both un-filtered and Gaussian filtered solutions, with an additional destriping filter applied similar to that of Swenson et al. (2008) (see Gunter et al. (2009) for further details of the CSR solution processing). The DMT-1 solutions are global spherical harmonic solutions computed using the acceleration approach (Ditmar and van Eck van der Sluijs 2004; Liu 2008), and with the ANS filter applied. Two types of SRBF solutions were tested, one using a global distribution of functions (SRBF global), and one using a more regional distribution (SRBF regional) in which a latitudinal buffer of 30 was used to reduce edge effects. For each solution, both the long-term linear trends (with bias and annual/ semi-annual terms included) and monthly variations in the signals were examined. The timeframe considered was a 3 year period spanning from October 2003 to September 2006. Some selected results from the comparisons are shown in Figs. 21.2 and 21.3. Figure 21.2 shows the geographical plot of the linear trends for the CSR400, DMT-1 and SRBF regional solutions over Greenland and Antarctica. Figure 21.3 is a plot of the maximum amplitude of the annual signal variation which is coestimated along with the linear trend parameter. This is useful to visualize where the largest fluctuations in mass change exist. The first observation that can be made from looking at these two figures is that the resolution for the ANS filtered solutions is much higher than those of the CSR DS400 solution, particularly for Greenland. The
DMT-1 and SRBF solutions are quite similar, but differences do exist. It is also interesting to note that the amplitude plots for the DMT-1 and SRBF regional solutions show subtle difference as well. For example, the SRBF solution shows a noticeable variation at the tip of the Antarctic Peninsula, where the DMT-1 solution does not. Similarly, in the Amundsen Sea sector (SW Antarctica), the SRBF solutions show two distinctive peaks, where the DMT-1 solutions show only one.
21.4
Validation
The determination of whether the differences seen in Figs. 21.2 and 21.3 represent genuine improvements in the signal recovered by the SRBF solutions is a difficult question to answer, and is a topic of current and future research efforts. One attempt made in this study to do this utilized surface elevation change data from the Ice, Cloud and Land Elevation Satellite (ICESat), a laser altimetry mission launched in 2003. ICESat observes the volume changes due to ice mass changes, which are naturally correlated to the mass changes observed by GRACE. The spatial resolution of ICESat is also much higher than that of GRACE, so a test was developed whereby the ICESat data was smoothed using a full-width Gaussian filter [as opposed to the traditional half-width filter normally used in geodesy, e.g., Jekeli (1981)] at intervals ranging from 0 to 2,500 km. Trend maps over the 3-year time period for both GRACE and ICESat were computed and each map was individually normalized. The normalization was needed because the ICESat
Comparison of Regional and Global GRACE Gravity Field Models at High Latitudes DMT–1
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Fig. 21.2 Geographical plot of the 3-year trend computed from selected global and regional solutions, in units of equivalent water height CSR DS400
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Fig. 21.3 Geographical plot of the estimated annual amplitude variations computed from selected global and regional solutions, in units of equivalent water height
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map represents physical height changes (dh/dt, in cm/ yr), whereas GRACE maps represent annual changes in EWH (also in cm/yr). As these are not the same quantities, the normalization allows a more direct comparison of the two data types under the assumption that a strong change in volume directly corresponds to a strong change in mass (and vice-versa). For each smoothing increment, correlations were computed between the smoothed ICESat map and the corresponding GRACE map. A peak in the resulting correlation curve would give an indication of the spatial resolution of the GRACE solution tested. The results of this test for all of the GRACE solutions mentioned in Table 21.1 are provided in Fig. 21.4. For Greenland, the correlations with ICESat for the ANS filtered solutions (DMT-1 and the SRBF solutions) peak at around 1,300 km (full-width) 1
Correlation
0.8
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0.8 0.6 CSR DS400 CSR DS0 DMT–1 SRBF global SRBF regional
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ICESat Gaussian smoothing full width [km] Fig. 21.4 Results of the spatial correlation test to ICESat data for Greenland (top) and Antarctica (bottom), for the various GRACE solutions
Gaussian smoothing, where the CSR solutions peak in the 2,200–2,500 km range. This implies that the ANS filter is the driving force for the accuracy levels in Greenland, and not necessarily the solution technique. For Antarctica, the situation is slightly different. Here, the correlation peak of the SRBF regional solution is approximately 5–10% higher than the SRBF global solution and the unfiltered CSR solution. This would suggest that the SRBF regional approach is achieving slightly better spatial resolution than the other global approaches. Conclusions
The results of the analysis for this study supports the earlier conclusions by Klees et al. (2008a) that the choice of the spatial filter used in the GRACE processing has the largest impact on the comparisons. When compared to the standard destriping and Gaussian filter approach (i.e., DS400), the anisotropic, non-symmetric (ANS) filter offers many benefits in terms of improved spatial resolution. That said, there were other indications that the choice of solution method may also offer some improvements, although to a much smaller degree. For Antarctica, the SRBF regional solution had the best spatial correlation when compared to the corresponding height change data from ICESat (Fig. 21.4), and was the only solution to observe annual variations in the Antarctic Peninsula (Fig. 21.3). For Greenland, all ANS filtered solutions (global and regional) performed essentially the same, with all of them offering substantial improvements over the corresponding CSR fields (DS400 and DS0). This is primarily due to the fact that the CSR fields have inherently lower resolution (with maximum degree and order 60), and because a Gaussian filter was applied (equivalent ANS filtered CSR solutions were not possible since the monthly noise covariance matrices are not publicly available). Regardless, the results suggest that, at a minimum, the regional SRBF techniques are equivalent to other global spherical harmonic solutions (i. e., DMT-1), but that there is also the possibility that a 5–10% improvement might be gained, depending on the region. Future studies will attempt to verify these results with more extensive comparisons with independent data sets, such as in-situ glaciological measurements or other satellite measurements.
21
Comparison of Regional and Global GRACE Gravity Field Models at High Latitudes
Acknowledgements The authors would like to thank Tim Urban at the UT-Austin Center for Space Research for providing the ICESat crossover height change data.
References Ditmar P, van Eck van der Sluijs AA (2004) A technique for modeling the Earth’s gravity field on the basis of satellite accelerations. J Geodesy 78:12–33 Gunter BC, Urban T, Riva REM, Helsen M, Harpold R, Poole S, Nagel P, Schutz B, Tapley B (2009) A comparison of coincident GRACE and ICESat data over Antarctica. J Geodesy 83(11):1051–1060. doi:10.1007/s00190-009-0323-4 Holschneider M, Chambodut A, Mandea M (2003) From global to regional analysis of the magnetic field on the sphere using wavelet frames. Phys Earth Planet Inter 135:107–124 Jekeli C (1981) Alternative methods to smooth the Earth’s gravity field. Technical Report 327, Ohio State University, Department of Geodetic Science and Surveying, December 1981 Klees R, Liu X, Wittwer T, Gunter BC, Revtova EA, Tenzer R, Ditmar P, Winsemius HC, Savenije HHG (2008a) A comparison of global and regional GRACE models for land hydrology. Surv Geophys 29(4–5):335–359
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Klees R, Revtova EA, Gunter BC, Ditmar P, Oudman E, Winsemius HC, Savenije HHG (2008b) The design of an optimal filter for monthly GRACE gravity models. Geophys J Int 175:417–432 Liu X (2008) Global gravity field recovery from satelliteto-satellite tracking data with the acceleration approach. Ph.D. Thesis, Netherlands Geodetic Commision, Publication on Geodesy 68, Delft, The Netherlands Liu X, Ditmar P, Siemes C, Slobbe DC, Revtova EA, Klees R, Riva R, Zhao Q (2009) DEOS Mass Transport model (DMT1) based on GRACE satellite data: methodology and validation. Geophys J Int 181(2):769–788 Luthcke S, Rowlands D, Lemoine F, Klosko S, Chinn D, McCarthy J (2006) Monthly spherical harmonic gravity field solutions determined from GRACE inter-satellite range-rate data alone. Geophys Res Lett 33:L02402 Stolk W (2009) An evaluation of the use of radial basis functions for mass change estimates at high latitudes. Msc Thesis, Delft University of Technology, The Netherlands Swenson S, Chambers D, Wahr J (2008) Estimating geocenter variations from a combination of GRACE and ocean model output. J Geophys Res 113, B08410, doi:10.1029/ 2007JB005338 Wittwer T (2009) Regional gravity field modeling with radial basis functions. Ph.D. Thesis, Netherlands Geodetic Commision, Publication on Geodesy 72, Delft, The Netherlands
.
A New Approach for Pure Kinematical and Reduced-Kinematical Determination of LEO Orbit Based on GNSS Observations
22
A. Shabanloui and K.H. Ilk
Abstract
The geometrical point-wise satellite positions of a Low Earth Orbiter (LEO) equipped with a Global Navigation Satellite System (GNSS) receiver can be derived by GNSS analysis techniques based on hl-SST (high-low Satellite to Satellite Tracking) observations. In the geometrically determined LEO orbit, there is no connection between subsequent positions, and consequently, no information about the velocity and the acceleration or in general kinematical information of the satellite is available. If the kinematical parameters which consistently connects positions, velocities and accelerations are determined by a best fitting process based on the observations, we perform a pure Kinematical Precise Orbit Determination (KPOD). In addition, the proposed approach has a capability to use certain dynamical constraints based on the dynamical force function model. In this case, we introduce a Reduced-Kinematical Precise Orbit Determination (RKPOD) of a specific level depending on the strength of the “dynamical constraints”. The various possibilities and the corresponding results of CHAMP orbits based on GNSS observations are presented.
22.1
Introduction
Estimated positions based on GNSS techniques are purely geometric and there is no connection between subsequent positions, consequently no information about the velocity and the acceleration of the LEO is available. Therefore, to describe the time dependency of the satellite’s motion, it is necessary to provide a properly constructed function which consistently connects positions, velocities and accelerations. Such
A. Shabanloui (*) K.H. Ilk Institute of Geodesy and Geoinformation, University of Bonn, Nussallee 17, 53115, Bonn, Germany e-mail:
[email protected]
a continuous function can be provided by a least squares adjustment process based on a table of geometrical positions. It should be pointed out that the orbit determination in this investigation is restricted to short arcs. In order to keep the accumulated effects of the disturbing forces on LEO as small as possible, satellite arcs can be divided into short arcs. Especially, in case of the measurement of LEO’s surface forces (e.g. CHAMP or twin-GRACE) or its compensation (e.g. GOCE), it is preferable to reduce the residual effects of an incomplete compensation by selecting short arcs. In this paper, the kinematical orbit is represented by a sufficient number of approximated parameters, including the boundary values of the short arc. These parameters are determined in such a way that
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the hl-SST observations (or geometrical positions as pseudo-observations) are approximated in the best possible way w.r.t. a properly selected norm. The approach is based on approximated parameters, which have also a clearly defined relation to the dynamical model of the satellite’s motion. If the kinematical representation parameters are determined by a best fitting process based on the hl-SST observations or geometrical positions, then a pure kinematical POD is realized. If instead all orbit representation parameters are determined by a model of the forces acting on the LEOs, then a Dynamical Precise Orbit Determination (DPOD) is introduced. In addition, there is also a possibility to use certain dynamical constraints based on the dynamical force function models. Therefore, a reduced-kinematical precise orbit determination of a specific level depending on the strength and contribution of the “dynamical restrictions” can be realized. It should be pointed out that this formulation of a POD problem allows a smooth transition from pure KPOD to RKPOD and finally fully DPOD of a LEO (Shabanloui 2008). In addition to the kinematic and the dynamic orbit determination technique, J€aggi (2007) also introduced a further type of methods called reduced dynamic methods which is located somewhere in between the kinematic and purely dynamic methods. This group of methods is characterized by the fact that additionally to the parameters of the dynamic methods, stochastic parameters of various types are introduced at intervals of a few minutes along the orbit. In this way the total orbit is divided into short arcs with different continuity properties at the arc boundaries depending on the type of the stochastic parameters. There are various articles which demonstrate the advantages of this method (e.g. J€aggi 2007). On the other hand, Sˇvehla and Rothacher (2003) proposed a reduced-kinematic POD which is here defined as a GPOD with a-priori low accuracy dynamical information between successive epochs in radial, cross and along-track directions. Their proposed reduced-kinematic method is only used to improve the characteristics of the purely GPOD by a considerable reduction of spikes and jumps. It should be mentioned that the proposed reduced-kinematical (geometrical) POD by Sˇvehla and Rothacher (2003) reduces geometrical information, while our new proposed reduced-kinematical POD in this investigation
A. Shabanloui and K.H. Ilk
reduces kinematical information in the LEO orbit (Shabanloui 2008).
22.2
Methodology
The precise orbit determination procedure is formulated as a Boundary Value Problem (BVP) of Newton-Euler’s equation of motion in the form of an integral equation of Fredholm type (Sect. 22.2.1). The solution of a Fredholm integral can be formulated in a semi-analytical way, either as a series of Fourier coefficients (Sect. 22.2.2), or as a series of Euler and Bernoulli polynomials (Sect. 22.2.3). The kinematical LEO orbit can be represented based on the combination of Euler–Bernoulli polynomials and Fourier series (Sect. 22.2.5). With introducing “dynamical restrictions” to the Fourier coefficients, the reduced kinematical POD is realized (Sect. 22.2.6).
22.2.1 Equation of Motion Based on the Linear Extended Newton Operator The basic idea was proposed as a general method for artificial satellite orbit determination by Schneider in 1968 (Schneider 1968). Before applying this method in the orbit determination, it was modified for the Earth gravity field determination by Reigber (1969). This orbit determination technique has been further developed and modified to various applications in satellite geodesy, especially to recover the Earth’s gravity field based on POD methods (Ilk 1977). A technique for the numerical solution of two BVP of Newton–Euler’s equation of motion as well as of Lagrange’s equation of motion based on Schneider’s proposed method was developed in the same publication. Later, this technique has been successfully applied to recover the Earth gravity field based on high–low SST and low–low SST observations (Mayer-G€urr 2006), but was not applied to produce the kinematical precise orbit of a LEO. At that time, orbit observations were sparse and not suited for precise kinematical orbit computation. The availability of GNSS changed the situation dramatically. The mathematical–physical model for densely tracked LEO orbits is based on the formulation of the equation of motion as (Ilk 1977)
22
A New Approach for Pure Kinematical and Reduced-Kinematical Determination
€rðtÞ ¼ aðt; r; r_ Þ
(22.1)
where the vectors r, r_ and €r denote the position, velocity and acceleration of the LEO, respectively. The function a indicates the specific (mass-related) force function acting on the LEO. The solution of Newtons equation of motion formulated as a BVP with the boundary values rA :¼ rðtA Þ;
rB :¼ rðtB Þ
ð1 rðtÞ ¼ rðtÞ T
K II ðt; t0 ÞaII ðt0 ; r; r_ Þdt0 (22.2)
t0 ¼0
with the starting time tA, the end time tB, the normalized time variable t and the arc length T t tA t :¼ ; T
with t 2 ½tA ; tB ;
T :¼ tB tA
K II ðt; t0 Þ ¼
0 1 sinmð1tÞsinmt ; sinm m 0 1 sinmð1t Þsinmt ; sinm m
sinðupt0 ÞaII ðt0 ; r; r_ Þdt0 :
t0 ¼0
(22.6)
It can be shown (Ilk (1977)), that the solution series (22.5) contains a generalized Fourier series of the difference function dðtÞ : ¼ rðtÞ rðtÞ 1 X ¼ du sinðuptÞ :¼ d1 F ð tÞ
(22.7)
u¼1
du ¼ 2
0
tt:
sinmð1 tÞ sinmt (22.3) rA þ rB sinm sinm qffiffiffiffiffiffi with the mean anamoly m ¼ GM a3 T, where a is the semi-major axis of the LEO orbit and GM denotes the standard Earth’s gravitational constant. The reduced force function acting on the satellite reads m2 rðtÞ: T2
(22.4)
The solution of the BVP (Fredholm’s integral equation in (22.2)) reads rðtÞ ¼ rðtÞ þ dðtÞ 1 X du sinðuptÞ ¼ r ð tÞ þ
dðt0 Þsinðupt0 Þdt0 :
(22.8)
t0 ¼0
r ð tÞ ¼
u¼1
ð1
ð1
t0 t
The Keplerian orbit as the reference motion reads
aII ðt; r; r_ Þ ¼ aðt; r; r_ Þ
2T 2 du ¼ 2 2 u p m2
with the Fourier coefficient
as well as the the integral kernel (
with the sine coefficient
22.2.2 Interpretation of the Solution of Fredholm’s Integral Equation as Fourier Series
reads according to Schneider (1968)
2
181
(22.5)
If the difference function d (t) is continued to an odd periodic function with the period 2T or to the normalized interval [–1, 1], then we get because of property of sine function dðtÞ ¼ dðtÞ
(22.9)
and all cosine terms vanish (i.e. special case of Fourier series). Therefore, all properties of the Fourier series hold for the solution of the differential function. If we restrict the upper summation indices of the Fourier series to finite number n then the Fourier series in (22.7) taking the Fourier remainder function (RF (t)) into account reads n d1 F ðtÞ ¼ dF ðtÞ þ RF ðtÞ n 1 X X ¼ du sinðuptÞ þ du sinðuptÞ: u¼1
u¼nþ1
(22.10)
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When neglecting the Fourier remaining term RF (t) in (22.10), then Fourier coefficients from discrete not necessary equidistant positions can be estimated by a least square adjustment (Shabanloui 2008).
J X 2 ð1Þjþ1
du ¼
2jþ1
ðupÞ
j¼1
ð1Þu d½2j ð1Þ d½2j ð0Þ
þ zðb; J; uÞ: (22.13)
22.2.3 From Fourier Series to Series of Euler–Bernoulli Polynomials
Introducing (22.13) into (22.7) results in
If (22.8) is partially integrated, then the first integration by parts reads
d(tÞ ¼
ð1
2jþ1
ð1Þu dj2jj ð1Þ
ð u pÞ j2jj d ð0Þ þBðb; J; uÞ sin ðuptÞ: u¼1
j¼1
(22.14)
dðt0 Þsinðupt0 Þdt0
du ¼ 2 t0 ¼0
¼ 2dðt0 Þ þ2
1 J X X 2ð1Þjþ1
1 up
1 1 cosðupt0 Þj0 up ð1 d½1 ðt0 Þcosðupt0 Þdt0 :
(22.11)
dðtÞ ¼
t0 ¼0
1 X
du sinðuptÞ
u¼1
By inserting the limits and considering that the difference function at the boundaries is zero, then we get 1 du ¼ 2 up
If we separate the inner sum of (22.14) in terms of even and odd indices u we get,
¼
½1
0
0
2jþ1
j¼1
þ
ð1
J X 2 ð1Þjþ1
J X 2 ð1Þj j¼1
0
d ðt Þcosðupt Þdt : þ
t0 ¼0
ð2pÞ
1 X
ðpÞ2jþ1
d½2j ð1Þ d½2j ð0Þ
d½2j ð1Þ þ d½2j ð0Þ
1 X sinð2uptÞ u2jþ1 u¼1
1 X sinð2u 1Þpt u¼1
ð2u 1Þ2jþ1
zðb; J; uÞsinðuptÞ:
u¼1
(22.15)
After 2J + 2 integrations by parts (Klose 1985), the general expression reads du ¼
J X
d½2j ðt0 Þ
j¼1
þb
The terms in (22.15) can be replaced by the absolutely and uniformly continuous series expansions of the Euler polynomials (Abramowitz and Stegun 1972)
2 ð1Þjþ1
1 cosðupt0 Þj0 2jþ1
ðupÞ ð1
2 ðupÞ2Jþ2
d½2Jþ2 ðt0 Þsinðupt0 Þdt0
t0 ¼0
E2j ðtÞ ¼
1 4 ð1Þj ð2jÞ! X sinð2u 1Þpt 2jþ1 2jþ1 p u¼1 ð2u 1Þ
(22.16)
(22.12) and the Bernoulli polynomials
with b¼
þ1 1
J ¼ 1; 4; 5; 8; 9; 12; . . . ; J ¼ 2; 3; 6; 7; 10; 11; . . . :
Evaluating the integration limits in the first term of (22.12) and denoting the remaining terms as x (b, J, u) we get
B2jþ1 ðtÞ ¼
1 2 ð1Þjþ1 ð2j þ 1Þ! X sinð2uptÞ : (22.17) u2jþ1 ð2pÞ2jþ1 u¼1
If (22.16) and (22.17) are inserted into (22.15), then we get:
22
A New Approach for Pure Kinematical and Reduced-Kinematical Determination
d ð tÞ ¼
1 X u¼1
¼
J X j¼1
þ
22.2.5 Kinematical Precise Orbit Determination
du sinðuptÞ ¼ d1 F ð tÞ e2j E2j ðtÞ þ
J X
b2jþ1 B2jþ1 ðtÞ
j¼1
1 X u¼1
zðb; J; uÞsinðuptÞ ¼ dJP ðtÞ þ RP ðtÞ (22.18)
with the term RP (t) as the Euler–Bernoulli truncation error term from degree J + 1 to infinity and the coefficients of the Euler polynomials (Shabanloui 2008) 1 ½2j d ð1Þ þ d½2j ð0Þ 2ð2jÞ!
e2j ¼
(22.19)
and the coefficients of the Bernoulli polynomials b2jþ1 ¼
1 d½2j ð1Þ d½2j ð0Þ : ð2j þ 1Þ!
183
(22.20)
In Shabanloui (2008), it was demonstrated that the upper degree J of the series in terms of Euler– Bernoulli polynomials should be very high to achieve an approximation accuracy at the sub-millimeter level. Also, it was shown that a fit of a series in terms of Euler–Bernoulli polynomials to the geometrically determined LEO’s short arc at the reasonable upper degree Jmax ¼ 4 should guarantee a sufficient determination of the Euler–Bernoulli coefficients corresponding to sufficiently precise arc derivatives at the boundaries (refer to (22.19) and (22.20)). Therefore, the resulting residual sine series after removing the Euler–Bernoulli terms should show a fast convergence and small residuals when compared to the true ephemerides (Shabanloui 2008). The satellite arc is presented kinematically by the selected reference motion, the Euler–Bernoulli polynomial up to degree Jmax and the residual Fourier series up to index n as rðtÞ ¼ rðtÞ þ
Jmax X
e2j E2j ðtÞ
j¼1
22.2.4 Determination of the Euler–Bernoulli Polynomials Coefficients If we restrict the upper summation indices of the Euler–Bernoulli polynomials to finite number J, then the LEO short arc can be represented based on GNSS observations according to (22.18) by omitting RP (t) as dðtÞ ’
J X j¼1
e2j E2j ðtÞ þ
J X
b2jþ1 B2jþ1 ðtÞ: (22.21)
j¼1
It should be mentioned that an Euler polynomial of degree 2j and a Bernoulli polynomial of degree 2j + 1 belong together as a pair. Based on the selected reference motion (refer to (22.3)), the LEO’s short arc can be approximated with the Euler–Bernoulli polynomial of degree Jmax in the space or spectral domain. We may note that the approximation quality increases with increasing upper degree J and the convergence of the Euler–Bernoulli polynomials requires a large upper index J (Shabanloui 2008).
þ
Jmax X j¼1
b2jþ1 B2jþ1 ðtÞ þ
n X
u sinðuptÞ: d
u¼1
(22.22) u denotes the residual Fourier coefwhere the vector d ficient. Based on the (22.22), in the first step, Euler–Bernoulli coefficients up to degree Jmax ¼ 4 are estimated from discrete GNSS observations. In the second step, the short arc boundary values as well as the residual Fourier coefficients are determined based on GNSS observations which are reduced w.r.t. the Euler–Bernoulli term up to degree Jmax ¼ 4.
22.2.6 Reduced-Kinematical Precise Orbit Determination The kinematical orbit parameters contain no dynamical information of the force function model; they are only based on observations taken at discrete epochs of the satellite motion along the orbit. In the following, we will extend the hybrid case of the kinematical orbit determination procedures as treated in Sect. 22.2.5 in
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such a way that dynamical restrictions can be introduced in the orbit determination procedure. This dynamical ~u information is contained in the orbit coefficients d which are related to the force function according to ~j (i and j as ~ i to d (22.6). If the dynamical quantities d start and end indices) from (22.6) are considered as a-priori dynamical information with the corresponding ~i Þ to Cðd ~j Þ, then the observavariance-covariance Cðd tion equation taking dynamical restrictions into account reads as ¼
A1 0
A2 I
x1 x2
(22.23)
with (pseudo) observations,
Diff. (m)
l1 ¼ ð rðt1 Þ . . . rðtK Þ ÞT ~ T ~ l2 ¼ d . . . dj i where l1 and l2 denote geometrically determined LEO ~i to d ~j which positions and dynamical restrictions d are derived from (22.6), respectively. The terms A1 and A2 are design matrices w.r.t. the boundary values and the Fourier series coefficients which are computed based on (22.3) and (22.5), and x1 and x2 are the unknown boundary values and the Fourier series coefficients, respectively. The a-priori variancecovariance information reads C¼
C1
0
d
x
dy
10
dz
20
30
20
30
20
30
Minute 0.0004 0 -0.0004
0
10 Minute
0 C2 C1 ¼ diagðCðt1 Þ CðtK ÞÞ ~ ~ C2 ¼ diag Cðd i Þ Cðdj Þ
0.02 0.01 0 -0.01 -0.02 0
Diff. (m/s)
Numerical Tests
The orbit determination approach is tested based on simulated GNSS observations for a 30 min arc of a simulated CHAMP orbit with a sampling rate of 30 s. The geometrical positions of CHAMP are estimated based on the simulated hl-SST carrier phase observations which are contaminated with a white noise of 2 cm.
(22.24)
where C1 and C2 denote LEO position variancecovariance matrix derived from GPOD procedure and variance-covariance of dynamical restriction, respectively. Based on the estimated boundary values and the Fourier coefficients from (22.23), the Euler–Bernoulli polynomials up to degree Jmax ¼ 4 by a least squares adjustment are estimated (for more details refer to Shabanloui 2008). The coefficients of the residual Fourier series up to degree n from (22.22) are determined based on the estimated boundary
Diff. (m/s2)
l1 l2
22.3
2E-005 0 -2E-005
0
10 Minute
Coef. (m)
values, the Euler–Bernoulli coefficients and the Fourier coefficients. Therefore, the orbit parameters are determined in such a way that the kinematical parameters are constrained according to the specified level of “dynamical restrictions”.
0.08 0.04 0 -0.04 -0.08 -0.12
dv,x
dv,y
20
dv,z
40
60
Index
Fig. 22.1 Position differences, velocity differences, acceleration differences and the coefficients of the residual Fourier series for the kinematical POD, respectively
Diff. (m)
22
A New Approach for Pure Kinematical and Reduced-Kinematical Determination 0.02 0.01 0 -0.01 -0.02
d
0
x
dy
10
dz
20
30
20
30
20
30
Diff. (m/s)
Minute 0.0008 0.0004 0 -0.0004 -0.0008
0
10
Diff. (m/s2)
Minute 4E-005 2E-005 0 -2E-005 -4E-005
0
10
Coef. (m)
Minute dv,x
0.08
dv,y
dv,z
185
To verify results, kinematical ephemerides are determined based on estimated orbit representation parameters at the interval of 10 s and compared with the true ephemerides. The kinematical position, velocity and acceleration differences as well as the Fourier coefficients are shown in Fig. 22.1. It should be pointed out that the kinematical position differences are in range of the given white noise (2 cm). To test the proposed RKPOD procedure, the same short arc of CHAMP as used in the kinematical case, ~ 1 to d ~5 has been selected. The Fourier coefficients d are determined based on EGM96 and have been introduced to the observation equations as dynamical restrictions. The reduced-kinematical position, velocity and acceleration differences at the interval 10 s as well as the residual Fourier coefficients for the reduced-kinematical case are shown in Fig. 22.2. The comparison of the RKPOD with the KPOD shows some improvements in the results, because of introducing dynamical restrictions to estimation procedure (refer to Table 22.1).
0 -0.08
Conclusions
Fig. 22.2 Position differences, velocity differences, acceleration differences and the coefficients of the residual Fourier series for the reduced-kinematical POD case, respectively
The new proposed kinematical and reducedkinematical POD procedures open a wide window to represent low-flying orbits. The method is very flexible with the possibility of smooth transition from pure kinematical to fully dynamical POD.
Table 22.1 3D RMS of positions, velocities and accelerations for kinematical and reduced-kinematical POD numerical tests
Acknowledgment We gratefully acknowledge the financial support of the BMBF under the project ”REAL-GOCE”.
20
40
60
Index
Method KPOD RKPOD
Pos.(m) 0.011339 0.008873
Vel.(m/s) 0.000351 0.000337
Acc.(m/s2) 0.000015 0.000016
References The geometrically estimated positions are used as pseudo-observations in case of the kinematical and reduced kinematical orbit determination procedures. To demonstrate the kinematical POD procedure, the Keplerian orbit is used as reference motion. A series of Euler–Bernoulli polynomials up to degree Jmax ¼ 4 has been fitted to the difference function d(t). Based on geometrical determined LEO positions, the Keplerian orbit and the estimated Euler–Bernoulli polynomials coefficients, the residual Fourier coefficients up to the index n ¼ 30 are estimated.
Abramowitz M, Stegun IA (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover, New York Klose U (1985). Beitr€age zur L€osung einer Integralgleichung vom HAMMERSTEINschen Typ. Diploma Thesis, TUM, Germany Ilk KH (1977) Berechnung von Referenzbahnen durch L€osung selbstadjungierter Randwertaufgaben, DGK, Reihe C, Heft 228, Munich, Germany J€aggi A (2007) Pseudo-stochastic orbit modeling of low earth satellites using the GPS. Ph.D. Thesis, University of Bern, Switzerland Mayer-G€urr T (2006) Gravitationsfeldbestimmung aus der Analyse kurzer Bahnb€ogen am Beispiel der Satellitenmission
186 CHAMP und GRACE. Ph.D. Thesis, IGG, University of Bonn, Germany Reigber C (1969) Zur Bestimmung des Gravitationsfeldes der Erde aus Satellitenbeobachtungen, DGK, Reihe C, Heft Nr. 137, Munich, Germany Schneider M (1968) A general method of orbit determination. Ph.D. Thesis, Ministry of Technology, Farnborough, England
A. Shabanloui and K.H. Ilk Shabanloui A (2008) A new approach for a kinematic-dynamic determination of low satellite orbits based on GNSS observations. Ph.D. Thesis, IGG, University of Bonn, Germany Sˇvehla D, Rothacher M (2003) Kinematic, reduced-kinematic, dynamic and reduced-dynamic precise orbit determination in the LEO orbit, 2nd CHAMP Science Meeting, Potsdam, Germany
Pure Geometrical Precise Orbit Determination of a LEO Based on GNSS Carrier Phase Observations
23
A. Shabanloui and K.H. Ilk
Abstract
The interest in a precise orbit determination of Low Earth Orbiters (LEOs) especially in pure geometrical mode using Global Navigation Satellite System (GNSS) observations has been rapidly grown. Conventional GNSS-based strategies rely on the GNSS observations from a terrestrial network of ground receivers (IGS network) as well as the GNSS receiver on-board LEO in double difference (DD) or in triple difference (TD) data processing modes. With the advent of precise orbit and clock products at centimeter level accuracy provided by the IGS centers, the two errors associated with broadcast orbits and clocks can be significantly reduced. Therefore, higher positioning accuracy can be expected even when only a single GNSS receiver is used in a zero difference (ZD) procedure. Along with improvements in the International GNSS Services (IGS) products in terms of Global Position System (GPS) satellite orbits and clock offsets, the Precise Point Positioning (PPP) technique based on zero (un-) differenced carrier phase observations has been developed in recent years. In this paper, the zero difference procedure has been applied to the CHAllenging Minisatellite Payload (CHAMP) high–low GPS Satellite to Satellite Tracking (hl-SST) observations, then the solution has been denoted as Geometrical Precise Orbit Determination (GPOD). The estimated GPOD CHAMP results are comparable with results of other groups e.g. Sˇvehla at TUM (Sˇvehla D, Rothacher M (Sˇvehla and Rothacher 2002) and Bock at Bern (Bock 2003) but because of different outliers detection and data processing strategies, the GPOD results presented here are more or less different than the other groups’ results. The estimated geometrical orbit of CHAMP is point-wise and its accuracy relies on the geometrical status of the GNSS satellites and on the number of the tracked GNSS satellites as well as on the GNSS measurement accuracy in the data
A. Shabanloui (*) K.H. Ilk Institute of Geodesy and Geoinformation, University of Bonn, Nussallee 17, 53115, Bonn, Germany e-mail:
[email protected] S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_23, # Springer-Verlag Berlin Heidelberg 2012
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A. Shabanloui and K.H. Ilk
processing. The position accuracy of 2–5 cm of CHAMP based on high–low GPS carrier phase observations with zero difference procedure has been achieved. These point-wise absolute positions can be used to estimate kinematical orbit of the LEOs.
23.1
Introduction
Among many possibilities to determine precise orbit of LEOs based on GNSS observations, zero differenced of high–low GPS-SST observations makes possible to be independent of the ground GNSS stations. In other words, geometrical LEO orbit can be estimated point-wise at the tracked epochs with only high–low observations which connect the GPS satellites to LEO with high precision. With the proposed method, only table of absolute positions at the desired epochs are estimated and subsequently no velocities and no accelerations and other kinematical parameters can be directly estimated in this procedure. In order to estimate kinematical parameters (e.g. velocities) specified functions or polynomials have to be fitted to the determined geometrical absolute positions of a LEO based on high–low GPS-SST observations. In this paper, methodology, numerical tests for the CHAMP satellite and some conclusions for the proposed geometrical precise orbit determination are presented.
the outliers in the high–low GPS-SST observations have to be flagged in the pre-processing process and excluded from the main GPS data processing. The detection methods are described in Sect. 23.2.2 in more detail.
23.2.1 Zero Differenced High–Low GPS-SST Observations In the zero (un-)differenced observations (refer to Fig. 23.1), always only the geometrical connection between GPS satellites (as sender) and GPS receiver on-board LEO (as receiver) is used to estimate the LEO geometrical orbit (Shabanloui 2008). The zero differenced carrier phase GPS-SST observations between the GPS satellite s and the LEO satellite r at frequency i with respect to the ambiguity parameter and all the error terms can be written as (Shabanloui 2008) si
23.2
E e3
Methodology
Zero differenced procedure of GPS-SST observations is used to estimate the geometrical absolute positions and the clock offset of a LEO. The approximated LEO absolute positions and the clock offsets can be determined epoch-wise based on code pseudo-range GPS-SST observations. The geometrical LEO absolute positions are improved with the accurate screened carrier phase observations. It has to be mentioned that the estimator, based on the least squares assuming standard normal distribution is not robust. Therefore, any outliers in the high–low GPS-SST observations, prior variances or other kinds of random parameters could dramatically affect the estimates of the unknown parameters and the variance components. Therefore,
Geocenter
rsi(t – –r tsi)
r
r(t)
E e2 E e1
Fig. 23.1 Zero differenced geometrical orbit determination
23
Pure Geometrical Precise Orbit Determination of a LEO s Fsr;i ðtÞ ¼ li fsr;i ðtÞ ¼ rsr ðtÞ cdtr ðtÞ þ li Nr;i
þ xsr;Fi ðtÞ þ eFi ðtÞ
(23.1)
s s where at epoch t, Fr,j and fr,j are the observed carrier phase between the GPS satellite s and the LEO GPS receiver r at frequency of i in unit of length and cycle respectively, rr,s is the true geometrical distance between the corresponding GPS and LEO, cdtr is the s LEO GPS receiver clock offset, Nr,j is the ambiguity parameter, li is the wavelength of the given GPS signal at frequency of i. The terms xsr,Фi and eФi represent the summation of all error effects on the LEO and the GPS satellites and the remaining error that cannot be modeled in the carrier phase observations, respectively (Shabanloui 2008). In the zero differenced principal, the error terms either have to be modeled with accurate specified models or have to be eliminated in the data processing procedure. For example, to eliminate the largest part of ionospheric effect on the carrier phase observations, the linear combination (L3) of the carrier phase observation at two frequencies (L1) and (L2) has been used. Consequently, the ionosphere free carrier phase observations at the epochs t can be written as (Shabanloui 2008):
s þ xsr;F3 ðtÞ þ eF3 ðtÞ Fsr;3 ðtÞ ¼ rsr ðtÞ cdtr ðtÞ þ l3 Nr;3
(23.2) with considering the GPS signal sending time and the Sagnac effect on the carrier phase observation, the observation equation can be rewritten as (Shabanloui 2008), Fsr;3 ðtÞ ¼ RZ ðoe tsr Þrs ðt tsr Þ rr ðtÞ s þ l3 Nr;3 cdtr ðtÞ þ xsr;F3 ðtÞ þ eF3 ðtÞ
(23.3) or the observation equation reads as (Leick 1995), s Fsr;3 ðtÞ ¼ rsm ðt tsr Þ rr ðtÞ þ l3 Nr;3 cdtr ðtÞ þ xsr;F3 ðtÞ þ eF3 ðtÞ
(23.4)
where oe, tsr and Rz(oetsr) represent rotation rate of the Earth, GPS signal travel time between sender and receiver and rotation matrix of the ITRF around the Z axis by the angle oetsr, respectively. The term rs(ttsr) represents the absolute position of the GPS
189 s satellite s at the sending time, rm (ttrs) is the GPS satellite position after applying the GPS signal travel time and the Sagnac effect and finally rr(t) is LEO receiver r at the receiving time, respectively.
23.2.2 Data Pre-processing For all applications of GPS, an efficient pre-processing and data screening of GNSS observations are necessary. It is particularly an important issue for the processing of the high–low GPS-SST observations of space-borne GNSS receivers on-board LEOs to determine precise orbit of the LEOs. In other words, any orbit determination procedure based on high–low GPS-SST observations depends crucially on the ability to remove invalid or degraded observations from the estimation procedures. It is clear that code pseudo-range GPS-SST observations are only used to determine initial geometrical absolute positions of a LEO. The procedure applied to detect outliers in the code observations is based on the “Majority voting”, which used the estimated LEO clock offset at every epoch (Bock 2003). In other words, the Majority voting is based on an epoch-wise processing of the GPS-SST observations. Therefore, at the first step, the receiver clock is synchronized with the GPS time, then based on the Majority voting algorithm, the bad observations are flagged to be excluded from GPS data processing. Based on the iterative least squares data processing, these flagged observations may be detected and subsequently be excluded from the main GPS data processing. The same Majority algorithm can be applied to the subsequent time differenced carrier phase observations between two epochs to detect outliers in the high–low carrier phase observations (Bock 2003). But the key factor limiting the performance of the pre-screening procedure of the carrier phase observations is the quality of the a-priori LEO absolute positions. Therefore, the rejection threshold should be set not too small. Consequently, a second screening procedure has to be applied to the carrier phase observations to find all bad observations which deteriorate the solution. The second issue, in case of carrier phase observation, is the cycle slips in the high–low GPS-SST observations. In order to obtain high precision orbit based on the carrier phase observations, cycle slips in the carrier phase observations have to be detected,
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identified and repaired at the pre-processing stage. A slip of only few cycles can influence the measurements such that a geometrical precise orbit cannot be achieved. Therefore, the detection of the cycle slips in the carrier observations is the major issue in the geometrical precise orbit determination procedure. Repairing process in this case needs to determine the size of cycle slips at the every carrier phase frequency, but sometimes it is very difficult to identify the size of an every frequency part. Therefore, after identification of the cycle slips, to avoid the repairing of the carrier phase, as an alternative, the new ambiguity parameters can be introduced to detected carrier phases in the GPS main processing. In order to detect the cycle slips in carrier phase observation, the observations or combination of observations have to be used sensitively to the cycle slips. Therefore, the detection of cycle slips in this case can be performed based on the quantities which based on the geometrical free combination of carrier phase observation at Li frequencies (i.e. fi, i ¼ 1,2) and code pseudo-range observations. The geometrical free combination (frs, GF(t) of carrier phase observation can be formulated as (Hofmann-Wellenhof et al. 2001):
GPS ambiguity terms, but is non-linear with respect to LEO absolute position. Therefore, to estimate geometrical absolute position, the observation equation has to be linearized. The linearized ionosphere-free GPSSST carrier phase observation reads (Xu 2007),
f1 s f ðtÞ f2 r;2 f1 s bion ðtÞ f2 s ¼ Nr;1 Nr;2 ð1 12 Þ þ ef ðtÞ f2 f1 f2 (23.5)
which xr(t), yr(t) and zr(t) represents the coordinates of the LEO absolute position at time t. The linearized observation can be rewritten as (Shabanloui 2008),
fsr;GF ðtÞ ¼ fsr;1 ðtÞ
the left side of the geometrical free carrier phase observations at two frequencies shows that only time-varying quantity is the ionosphere term bion. This equation shows that the ionosphere effect has been reduced with respect to the original carrier phase observations (Shabanloui 2008). Now, if there are no cycle slips, the temporal variations of the geometrical free of carrier phase observations would be small for normal ionosphere conditions. The geometrical free observations have disadvantages to determine the cycle slips on L1 and L2 or both, but with introducing new ambiguities, this problem can be solved.
23.2.3 Solution Obviously, the carrier phase GPS-SST observation is linear with respect to the LEO clock offset and the
Fsr;3 ðtÞ ¼ Fsr;3;0 ðtÞ þ
@Fsr;3 ðtÞ ðx x0 Þ @x
(23.6)
where x is a vector of the unknown parameters which contains the LEO absolute position, the LEO clock offset and GPS ambiguity terms at the time t respectively; and x0 is a vector of the initial values of the unknown parameters as follows, 0
1 xr ðtÞ B yr ðtÞ C B C B zr ðtÞ C B C B C x :¼ B cdtr ðtÞ s1 C; B l3 Nr;3 C B C B .. C @ . A sn l3 Nr;3
x0 :¼ xj0
(23.7)
DFsr;3 ðtÞ ¼ asr ðtÞDxðtÞ; wsr ðtÞ ¼
s20 cosðzsr ðtÞÞ: s2Fs
(23.8)
r;3
The terms wrs(t), s2Fs and zrs(t) represent the obserr;3 vation weight, carrier phase observation variance and zenith distance of the GPS satellite s from the GPS receiver r, respectively. If we assume that, a number n of GPS satellites si, i¼1, . . ., n are available at the time t. The Gauss-Markov model for all observed carrier phase observations corresponding to weight matrix with considering the unit vector between the GPS satellite s and LEO on-board receiver r reads (Shabanloui 2008), rsm ðt tsr Þ rr ðtÞ esri ðtÞ ¼ rs ðt ts Þ rr ðtÞ m r i i esr;y esr;zi ; ¼ esr;x
(23.9)
23
Pure Geometrical Precise Orbit Determination of a LEO
1 1 DFsr;3 ðtÞ C B .. A @ . n ðtÞ DFsr;3 1 esr;y ðtÞ esr;z1 ðtÞ 1 .. .. .. . . . n esr;y ðtÞ esr;zn ðtÞ 1
23.3
0
0
1 esr;x ðtÞ B .. ¼@ . n ðtÞ esr;x
191
1 .. . 0
1 0 0 .. .. C: . .A 0 1
0
1 Dxr ðtÞ B Dyr ðtÞ C B C B Dzr ðtÞ C B C B C :B DcdtrsðtÞ C þ «ðtÞ; 1 B l3 Nr;3 C B C B C .. @ A . sn l3 Nr;3 Wl ðtÞ ¼ diagðwsr1 ðtÞ wsrn ðtÞÞ:
(23.10)
Obviously, at the first epoch, the number of observations are smaller than the number of the unknowns (under determined problem). Therefore, the carrier phase observations of the other epochs have to be summed up and the observation equation has to be solved in a batch processing for all desired epochs. The observation equations for all observed epochs can be written as, « þ Dl ¼ ADx; Wl
(23.11)
by a least square adjustment, the unknowns read 1 T D^x ¼ AT Wl A A Wl Dl:
Numerical Tests
To verify the proposed geometrical orbit determination procedure, four 30 min arcs (short arcs) of CHAMP have been selected. As a pre-processing step, the outliers and cycle slips in carrier phase observations are removed with the “Majority voting” technique and different combination form of the observations. Finally, to minimize or mitigate the effect of multi-path effect nearby weighting the carrier phase observations with respect to the GPS zenith distance, a 15 cut-off angle has been applied to the observations. The ground tracks of four CHAMP short arcs are shown in Fig. 23.2. Because of the fact that the accuracy of GPS-SST carrier phase observations are in the range of some millimeters, one can expect that the millimeter level of accuracy for the GPS receiver is achievable. In case of GPS receivers at the ground stations, the problematic modeling of the atmosphere parameters can reduce the accuracy of GPS estimated positions. The situation is less critical for GPS receiver on-board LEO. There is no tropospheric effect in the altitude of LEOs (e.g. 430 km); but at this altitude, the ionosphere can dramatically affect the GPS-SST observations. The multipath effect is another major problem in the data processing of LEOs. To validate the estimated geometrical CHAMP orbit externally, the CHAMP PSO dynamical orbits provided by GFZ-Potsdam have been used. In Fig. 23.3, the differences between estimated CHAMP absolute positions and PSO dynamical orbit for the four short arcs (cases a–d) are shown. Figure 23.4 represents the carrier phase GPS-SST observations for four short arcs. It should be pointed
(23.12)
The estimated unknowns are the corrections to the LEO absolute positions, the LEO clock offsets and the GPS ambiguity terms. Because of the non-linearity of the observations, the convergence is achieved after few iterations e.g. after i iterations; subsequently the unknowns vector and variance-covariance read,
90°N
d
60°N 30°N
c
0° 30°S
b
a
60°S
^xi ¼ ^xi1 þD ^ xi ;
1 C^xi ¼ AT Wl A
(23.13)
the estimated unknowns are the LEO absolute positions, the LEO clock offsets and the float GPS ambiguity terms at all desired epochs.
90°S 180°W
120°W
60°W
0°
60°E
120°E
180°
Fig. 23.2 The ground tracks of four 30 min short arcs for the time: (a) 2002 03 21 13h 30m 0.0s – 14h 00m 0.0s, (b) 2002 07 20 12h 48m 0.0s – 13h 18m 0.0s, (c) 2003 03 21 17h 20m 0.0s – 17h 50m 0.0s, (d) 2003 03 31 17h 00m 0.0s – 17h 30m 0.0s
a
A. Shabanloui and K.H. Ilk 0.12
dx
dy
dz
Diff. (m)
0.08 0.04 0 -0.04
a
0.02
Residuals (m)
192
0.01 0 -0.01 -0.02
-0.08
52354.565
52354.565 52354.57 52354.575 52354.58 MJD (days) 0.08
dx
dy
dz
Diff. (m)
0.04 0
52354.575
52354.58
MJD (days)
b
0.02
Residuals (m)
b
52354.57
0.01 0 -0.01
-0.04 -0.02
-0.08
52475.535
MJD (days)
c
0.2
dx
dy
dz
Diff. (m)
0.1 0
c
0.012
Residuals (m)
52475.535 52475.54 52475.545 52475.55 52475.555
0.008
-0.1
0 -0.004 -0.008 52719.725 52719.73 52719.735 52719.74 MJD (days)
MJD (days) dx
dy
dz
Diff. (m)
0.1 0
d
0.012
Residuals (m)
52719.725 52719.73 52719.735 52719.74
d
52475.55
0.004
-0.2
0.2
52475.54 52475.545 MJD (days)
0.008 0.004 0 -0.004 -0.008 -0.012
-0.1
52729.71 52729.715 52729.72 52729.725 52729.73 MJD (days)
-0.2 52729.71 52729.715 52729.72 52729.725 52729.73 MJD (days)
Fig. 23.3 Absolute position differences between estimated geometrically orbit and PSO dynamical CHAMP orbit for the case (a), case (b), case (c) and case (d)
out that the residuals for all four short arcs are in the range of 2 cm. The geometrical absolute positions of CHAMP can be estimated with an accuracy of 2–5 cm. In the cases (c) and (d), the residuals in some epochs show zero values which means availability of minimum four GPS satellites at these epochs. To show real precision of estimated CHAMP orbits, polynomial of degree five seems to be a proper degree to fit to the differenced ephemeris. Therefore, a polynomial of degree five is fitted to the differences between the
Fig. 23.4 Carrier phase GPS-SST observation residuals for case (a), case (b), case (c) and case (d)
estimated CHAMP orbit and the PSO dynamical orbit. After removing trend with the polynomial of degree five, RMS values of orbits is shown in Table 23.1. Table 23.1 RMS values of the estimated CHAMP orbits w.r.t CHAMP PSO dynamical orbit after removing trend with polynomial degree five Case a b c d
X(m) 0.0047 0.0024 0.0140 0.0087
Y(m) 0.0096 0.0062 0.0171 0.0073
Z(m) 0.0122 0.0075 0.0085 0.0105
23
Pure Geometrical Precise Orbit Determination of a LEO
23.4
Conclusions
Conventional GNSS-based POD strategies rely on the GNSS observations from a terrestrial network of ground receivers (IGS network) as well as the GNSS receiver on-board LEO in different difference data processing modes. Advent of GNSS orbits and clock offset, the geometrical precise orbit of LEO has been realized based on high–low GPS-SST observations. Zero differenced procedure provides an efficient possibility to estimate geometrical orbit of a LEO. The proposed geometrical LEO POD strategy could be characterized as pure geometrical type, point-wise and three-dimensional. In other words, only geometrical parameters (e.g. absolute positions) can be estimated based on the geometrical observations. An efficient data pre-processing and data screening procedure are very important to achieve precise geometrical LEO orbits based on GPS-SST observations. Specially, the detection, identification and removing (repairing) procedure are the most important issues in the main data processing. In this paper, real case results demonstrate that an accuracy of centimeter has been achieved for the geometrical orbit of CHAMP. It has to be mentioned that the estimated geometric precise LEO orbits can be used for further investigations e.g. determination of kinematical (continuous) LEO orbit for space borne recovery of the Earth gravity field based on GPS-SST observations. It is clear that,
193
geometrical or kinematical POD of a LEO have not been affected by the dynamical information from the Earth gravity field. Acknowledgment We gratefully acknowledge the financial support of the “German Federal Ministry for Education and Research” (BMBF) under the project “LOTSE-CHAMP/ GRACE”.
References Bock H (2003) Efficient methods for determining precise orbits for low earth orbiters using the global positioning system (GPS). PhD thesis, Astronomical Institute, University of Bern, Switzerland Hofmann-Wellenhof B, Lichtenegger H, Collins J (2001) GPS, theory and practice. Springer, New York Leick A (1995) GPS satellite surveying, 3rd edn. Wiley, New York Shabanloui A (2008) A new approach for a kinematic-dynamic determination of low satellite orbits based on GNSS observations. PhD thesis, Department of Astronomical, Physical and Mathematical Geodesy, Institute of Geodesy and Geo-information, University of Bonn, Germany Sˇvehla D, Rothacher M (2002) Kinematic orbit determination of LEOs based on zero or double difference algorithms using simulated and real SST data. In: Adam J, Schwarz K-P (eds) Vistas for geodesy in the new millennium, vol 125. Springer, Berlin, pp 322–328 Xu G (2007) GPS, theory, algorithms and applications, 2nd edn. Springer, Berlin
.
On a Combined Use of Satellite and Terrestrial Data in Refined Studies on Earth Gravity Field: Boundary Problems and a Target Function
24
P. Holota and O. Nesvadba
Abstract
The purpose of the presentation is to discuss the combination of terrestrial and satellite gravity field data and to show its spectral and space domain interpretation. Potential theory is of key importance in this field. However, the problems discussed are overdetermined by nature. Therefore, methods typical for the solution of boundary-value problems are used together with an optimization concept, target functions and regularization techniques. Two cases are treated. They are motivated by the use of a satellite-only model of the gravity field of the Earth or by data coming from satellite missions (especially GOCE) in common with terrestrial gravity measurements. For the results reached in the spectral domain summation techniques are applied in order to find the interpretation of the results in terms of kernel (Green’s) functions related to the particular combination scheme. This enables to show the tie between the global and the local modelling of the gravity field. In order to demonstrate the efficiency of the procedure results of numerical simulations are added. Differences of the closed loop simulation are very promising.
24.1
Introduction
In this paper a concept is discussed enabling combination of terrestrial gravity measurements with a satellite-only model of the Earth’s gravity field or with data coming from satellite missions, in particular GOCE [treated in terms of the space-wise approach, see e.g.
P. Holota (*) Research Institute of Geodesy, Topography and Cartography, 250 66 Zdiby 98, Praha-vy´chod, Zdiby, Czech Republic e-mail:
[email protected] O. Nesvadba Land Survey Office, Pod Sı´dlisˇteˇm 9, 182 11, Praha 8, Czech Republic e-mail:
[email protected]
Migliaccio et al. (2004)]. The paper ties to Holota (2007) and Holota and Nesvadba (2006, 2007, 2009). In the sequel OTOPO means a domain bounded by two surfaces, from above by a geocentric sphere of radius Re and from below by a surface GTOPO , which is star-shaped at the origin and about as irregular as the surface of the Earth. Moreover, let h be the height of the boundary point x 2 GTOPO above a geocentric sphere of radius Ri . We will consider the following two problems D T ¼ 0 in jxj
@T þ 2T ¼ jxjDg @jxj
O TOPO for
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_24, # Springer-Verlag Berlin Heidelberg 2012
x 2 GTOPO
(24.1) (24.2)
195
196
P. Holota and O. Nesvadba
T ¼ t for
jxj ¼ Re
(24.3)
where t means the input from an available satelliteonly model of the Earth’s gravity field and D T ¼ 0 in jxj
@T þ 2T ¼ jxjDg @jxj @2 T @ j x j2
¼G
O TOPO for
for
x 2 GTOPO jxj ¼ Re
(24.4) (24.5)
(24.6)
where G represents the input from satellite gradiometry. Note also that DT means Laplace’s operator applied on T, Dg is the gravity anomaly, x ðx1 ; x2 ; x3 Þ, jxj ¼ ðS3i¼1 x2i Þ1=2 and x1 ; x2 ; x3 are the usual rectangular Cartesian co-ordinates in threedimensional Euclidean space R3 . Our starting point will be the transform of the domain O TOPO given by x1 ¼ ½ r þ oðrÞ hð’; lÞ cos ’ cos l
(24.7)
x2 ¼ ½ r þ oðrÞ hð’; lÞ cos ’ sin l
(24.8)
x3 ¼ ½ r þ oðrÞ hð’; lÞ sin ’
(24.9)
where ’ and l are the spherical coordinates of the point x 2 O TOPO (geocentric latitude and longitude, respectively), r is defined by jxj ¼ r þ oðrÞ hð’; lÞ and oðrÞ is a suitably chosen and twice differentiable auxiliary function of r 2 ½Ri ; Re . The transformation, as defined in Holota and Nesvadba (2009), is a oneto-one mapping between O TOPO and a domain O, O f x; x 2 R3 ; Ri < jxj
h f ¼ 1þ Dg; Ri
(24.10)
valid for r ¼ Ri . A somewhat more complicated problem is to express Laplace’s operator of T in terms of the
coordinates r; ’; l. Nevertheless, using tensor calculus, we can follow Sokolnikoff (1971), so that after some algebra DT ¼
h 1þo r
2
½DS T dðT; h; oÞ
(24.11)
where D S T has the well-known structure of Laplace’s operator (applied on T) expressed in spherical coordinates, i.e. r; ’; l are formally interpreted as spherical coordinates, @T @ 2T þ A2 2 þ @r @r (24.12) 1 @ 2T 1 @2T þ A3 þ A4 r @r@’ r cos ’ @r@l
dðT; h; oÞ ¼ A1
and Ai are topography-dependent coefficients, see Holota and Nesvadba (2009). The solution T has to be a harmonic function in the original solution domain O TOPO , i.e. D T ¼ 0 in O TOPO . Hence D S T ¼ dðT; h; oÞ
in
O
(24.13)
in view of (24.11) and our two combination problems attain the following form. In the first case D S T ¼ g in
O
@T 2T þ ¼ f ; r ¼ Ri and T ¼ t; r ¼ Re @r Ri
(24.14)
(24.15)
In the second case D S T ¼ g in
O
(24.16)
@T 2T @ 2T þ ¼ f ; r ¼ Ri and ¼ G; r ¼ Re (24.17) @r Ri @r 2 Recall that g ¼ dðT; h; oÞ and f ¼ ð1þ h=Ri Þ Dg. As is well-known, for the problems above, we can split the solution into two parts, i.e., T ¼ T þ T , where T is a harmonic function and T is a particular solution of Poisson’s equation DS T ¼ g in O. In reality, however, g depends on T and T is a value of an operator K applied on T, i.e., T ¼ KT.
24
On a Combined Use of Satellite and Terrestrial Data in Refined Studies on Earth Gravity Field
Therefore, T ¼ T þ KT, which formally has a structure of Fredholm’s equation of the second kind. It may be solved by means of the method of successive approximations
T ¼ lim Tk ;
Tk ¼ T þ KTk1
k
f and tn are the respective where q ¼ Ri =Re , while Dg n f harmonic components of Dg and t, i.e. Dgð’; lÞ ¼ P1 P1 f Dg ð’; lÞ and t ð’; lÞ ¼ tn ð’; lÞ. Hence n¼0
where T0 is a starting approximation, e.g. T0 ¼ T , see Holota and Nesvadba (2009).
n
n¼0
h i f þ ðn þ 2Þ qn tn =Dn T~nðiÞ ¼ Ri Dg n
(24.25)
h i f ðn 1Þ tn =Dn T~nðeÞ ¼ Ri qnþ1 Dg n
(24.26)
(24.18)
197
and Dn ¼ ðn þ 2Þð 1 þ q2nþ1 Þ 3 is the determinant of the system.
24.2
Gravimetry and a Satellite-Only Model
In this case the k - th iteration step can be given the following structure Tk ¼ T~ þ T~ , where ð
1 T~ ¼ 4p
t r¼Re
@G 1 dS þ @r 4p
ð
f G dS (24.19) Dg
f ¼ Dg @ðDgÞk1 h (24.20) Dg @r
@Tk1 2 Tk1 @r Ri
Here also Tk ¼ T~ þ T~ with T~ generally given by (24.22) and T~ represented by the first of (24.20), but ðiÞ ðeÞ now T~n and T~n result from
where Gn are the respective spherical harmonics in P Gð’; lÞ ¼ 1 n¼0 Gn ð’; lÞ. Thus h i f þ R2 ðn þ 2Þqn Gn Dg T~nðiÞ ¼ Ri nðn 1ÞDg n e n (24.29)
(24.21) h f T~nðeÞ ¼ Ri ðn þ 1Þðn þ 2Þqnþ1 Dg n R2e ðn 1ÞGn Dgn
see (Holota and Nesvadba 2009). Alternatively, for the harmonic part T~ we can also write T~ ¼
ðiÞ
1 nþ1 X Ri T~nðiÞ ð’; lÞ r n¼0 1 n X r T~nðeÞ ð’; lÞ þ R e n¼0
(24.22)
24.4
ðeÞ
qnþ1 T~nðiÞ þ T~nðeÞ ¼ tn
(24.30)
and Dgn ¼ nðn 1Þ2 þ ðn þ 1Þðn þ 2Þ2 q2nþ1 is the determinant of the system.
where T~n and T~n are the respective surface spherical harmonics. For any individual n [in view of (24.15)] they can be obtained from f ðn 1ÞT~nðiÞ ðn þ 2Þqn T~nðeÞ ¼ Ri Dg n
(24.27)
ðn þ 1Þðn þ 2Þqnþ1 T~nðiÞ þ nðn 1ÞT~nðeÞ ¼ R2e Gn (24.28)
and ðDgÞk1 ¼
Gravimetry and Gradiometry
f ðn 1ÞT~nðiÞ ðn þ 2Þqn T~nðeÞ ¼ Ri Dg n
r¼Ri
G is the respective Green’s function, @Tk1 oh; T~ ¼ @r
24.3
(24.23) (24.24)
Optimization in H2
In both the cases DS Tk ¼ 0 for r 2 ½Re e; Re in view of oðrÞ ¼ 0 for r 2 ½Re e; Re and the smoothness of oðrÞ, see Holota and Nesvadba (2009). In consequence Tk and T~ represent a part of the same branch of a harmonic function for r 2 ½Re e; Re . The problem, however, is that in general the continuation of T~ for r>Re is not regular at infinity, for r ! 1 it does not decrease as c=r (c is a constant) or faster. This is
198
P. Holota and O. Nesvadba
caused by measurement errors. The data given for r ¼ Ri are sufficient to determine a harmonic function in Oext fx 2 R3 ; r>Ri g and thus in O Oext . The data for r ¼ Re are excess data and give rise to “interðeÞ nal” terms ðr=Re Þn T~n that are not regular at infinity. In solving this overdetermined problem, we will look for a harmonic function f , which is regular at infinity and minimizes the functional ð Fðf Þ ¼
ð ðf ; gÞ1 ¼
hgrad f ; grad gi dx
(24.36)
Oext ð1Þ
We look for a function f 2 H2 ðOext Þ that minimizes the functional ð Cðf Þ ¼ jgradðf T~ Þj2 dx
(24.37)
O 2
ðf T~ Þ dx
(24.31)
O
In particular we suppose that f 2 H2 ðOext Þ, where H2 ðOext Þ is a space of harmonic functions with inner product ð ðf ; gÞ
In this case the function f is defined by the following integral identity ð
ð
hgrad f ; gradvidx ¼ hgrad T~ ; gradvidx (24.38)
O
O ð1Þ
r 2 fg dx
(24.32)
valid for all v 2 H2 ðOext Þ. In consequence f is given by (24.34), but now with an ¼ 0 for all n.
Oext
Hence, assuming F has its minimum at a point f 2 H2 ðOext Þ, we know that Gaˆteaux’ differentials of F equals zero at the point f . This yields ð
ð fv dx ¼ T~ v dx for all v 2 H2 ðOext Þ
O
(24.33)
O
24.6
ð1Þ
Optimization in H2 -Traces ð1Þ
Considering the fact that functions from H2 ðOext Þ have precisely defined traces on the boundary @O of ð1Þ
O, we can also look for f 2 H2 ðOext Þ that minimizes the functional ð
and we easily obtain that
Yðf Þ ¼
1 nþ1 h i X Ri T~nðiÞ þ an T~nðeÞ f ¼ r n¼0
2 ð f T~ Þ dS
(24.34)
Again, Y attains its minimum and ð
with
ð fv dS ¼
an ¼
ð2n 1Þð 1 q2 Þ n2 q 2ð 1 q2n1 Þ
@O
(24.35)
see Fig. 24.1. In particular a0 ¼ ð1 þ qÞ=2q and lim an ¼ 0 as n ! 1, see Holota and Nesvadba (2006, 2007).
Optimization in ð1Þ
ð1Þ H2
Let H2 ðOext Þ be the space of harmonic functions on Oext equipped with inner product
T~ v dS
(24.40)
@O
ð1Þ
valid for all v 2 H2 ðOext Þ is the respective condition for Y to have a minimum at the point f . Hence, the function f can be represented by (24.34) too, but with an ¼ bn , n ¼ 1; 2; . . . 1 and bn ¼
24.5
(24.39)
@O
1 þ q n1 q 1 þ qn1
(24.41)
see Fig. 24.1. One can show that b0 ¼ 1 and that lim bn ¼ 0 as n ! 1.
24
On a Combined Use of Satellite and Terrestrial Data in Refined Studies on Earth Gravity Field
Fig. 24.1 Values of an and bn for Ri ¼ 6; 378 km and two cases of Re : Re ¼ Ri þ 250 km and Re ¼ Ri þ 400 km, i.e., for q ¼ 0:96228 and q ¼ 0:94099, respectively
199
1,2
· a n for q = 0, 96228
1,0
a n for q = 0, 94099
0,8
+ b n for q = 0, 96228
0,6
× b n for q = 0, 94099
0,4 0,2 degree n 0,0 0
24.7
AðiÞ n ¼
Optimized Solution: Target Function Fðf Þ
In order to show more clearly how the optimization works we have to return to the structure of the ðiÞ ðeÞ harmonics T~n and T~n . For the combination of gravimetry and a satellite-only model they are given by (24.25) and (24.26). Thus the optimization leads to f ¼
1 nþ1 X Ri n¼0
r
AðiÞ n
Ri f Dg þ AðeÞ n tn n1 n
25
(24.42)
with nþ1 Þ=Dn AðiÞ n ¼ ðn 1Þð 1 an q
(24.43)
n AðeÞ n ¼ ½ðn þ 2Þq þ an ðn 1Þ =Dn
(24.44)
f¼
100
125
AðiÞ n
r
Ri f R2e Dgn þ þAðeÞ Gn n n1 ðn þ 1Þðn þ 2Þ
AðeÞ n ¼
ðn þ 1Þðn þ 2Þ ½ðn þ 2Þqn þ an ðn 1Þ Dgn (24.47)
illustrated in (comparative) Fig. 24.3. For a greater range of degree n the plots shown in Figs. 24.2 and 24.3 can be seen in Holota and Nesvadba (2006, 2007).
24.8
(24.45)
150
n1 nðn 1Þ an ðn þ 1Þðn þ 2Þqnþ1 g Dn (24.46)
Optimization – Comparizon: Target Functions Fðf Þ, Cðf Þ and Yðf Þ
Similarly as for the optimization based on the target function Fðf Þ we can proceed in case of target functions Cðf Þ and Yðf Þ. The optimized solution is again given by (24.42) for the first problem and ðiÞ ðeÞ (24.45) for the second problem, but with An and An illustrated in Figs. 24.2 and 24.3.
1 nþ1 X Ri n¼0
with
75
and
and
illustrated in (comparative) Fig. 24.2. For the combination of gravimetry and satellite ðiÞ ðeÞ gradiometry T~n and T~n are given by (24.29) and (24.30). Thus the optimization leads to
50
24.9
Space Domain Interpretation: Terrestrial Term
Our aim is now to sum the series
200
P. Holota and O. Nesvadba
Φ(f)
Ψ (f)
Θ (f )
2,0
2,0
2,0
• q = 0, 96228
q = 0, 94099 1,5
1,5
1,5 (i)
(i)
1,0
(i)
An
An
0,5
An
1,0
1,0
0,5
0,5
(e)
An
0,0
0,0
0,0 (e)
(e)
An
An
degree n -0,5
-0,5 0
2
-0,5 0
50
25
50
0
25
50
ðeÞ Fig. 24.2 Combination of gravimetry and a satellite-only model: Coefficients AðiÞ n and An for target functions Fðf Þ, Cðf Þ, Yðf Þ and for q ¼ 0:96228 and q ¼ 0:94099, i.e., for Re ¼ Ri þ 250 km and Re ¼ Ri þ 400 km
Φ( f )
Ψ (f)
2,0
Θ (f)
2,0
2,0
1,5
1,5
• q = 0, 96228
q = 0, 94099 1,5 (i)
1,0
0,5
(i)
(i)
An
An
An 1,0
1,0
0,5
0,5
(e)
An
0,0
0,0
0,0 (e)
-0,5
-0,5 0
25
(e)
An
degree n 50
An -0,5
0
25
50
0
25
50
ðeÞ Fig. 24.3 Combination of gravimetry and satellite gradiometry: Coefficients AðiÞ n and An for target functions Fðf Þ, Cðf Þ, Yðf Þ and for q ¼ 0:96228 and q ¼ 0:94099, i.e., for Re ¼ Ri þ 250 km and Re ¼ Ri þ 400 km
fterr ¼
1 nþ1 X Ri n¼0
r
AðiÞ n
Ri f Dg n1 n
(24.48)
representing the terrestrial term in (24.42) or (24.45) for the target function Fðf Þ. To our opinion the function combines the effect of boundary data and the smoothness in constructing the optimized solution optimally. For the first problem we obtain
R2i 1q f R3 2 ð1 þ qÞq f Dg1 Dg0 þ 2i r 2ð2q 1Þ r 6q2 ð Ri f ds þ S ðr; cÞ Dg 4p s (24.49) 1 X 2n þ 1 Ri nþ1 AðiÞ Pn ðcos cÞ S ðr; cÞ ¼ n r n1 n¼2
fterr ¼
(24.50)
24
On a Combined Use of Satellite and Terrestrial Data in Refined Studies on Earth Gravity Field
201
15 Classical Stokes' function S(psi) = S(1,psi) 12 S* Kernel, terrestrial data and a satellite-only model, q = 0.94099 9
S* Kernel, terrestrial data and a satellite-only model, q = 0.96228 S* Kernel, gravimetry and satellite gradiometry, q = 0,94099
6 S* Kernel, gravimetry and satellite gradiometry, q = 0.96228 3 0 -3 -6 psi ® -9 0°
30°
60°
90°
120°
150°
180°
Fig. 24.4 Kernel functions in the integral representation of the terrestrial term (Stokes’ function drawn for comparison) ðiÞ
with An given by (24.43). Note that c is the angle between the computation and the running point of the integration. Similarly, for the second problem we have fterr ¼
R2i 1 þ q f R3 1 þ q f Dg0 2i Dg1 r 4q r 6q2 ð Ri f ds þ S ðr; cÞ Dg 4p s
ð1Þ
fsat ¼
1 nþ1 X Ri n¼0
r
AðeÞ n tn
(24.52)
and (24.51)
where the kernel S ðr; cÞ appears again, but with ðiÞ coefficients An given by (24.46). It is clear that for the space domain interpretation of the terrestrial term the dependence of S ðr; cÞ on the angle c is of considerable instructive value. For r ¼ Ri the diagrams were derived (indirectly) from summation of about 300–450 first terms of the difference between the series expression of Stokes’ function and the series in (24.50), see Fig. 24.4. The figure shows that in fterr the influence of distant zones is considerably suppressed.
24.10 Space Domain Interpretation: Satellite Term Of course, our aim also is to sum the satellite term
ð2Þ fsat
¼
1 nþ1 X Ri n¼0
r
AðeÞ n
R2e Gn ðn þ 1Þðn þ 2Þ
(24.53)
in (24.42) and (24.45), respectively. After some algebra we have ( ) ð ð1Þ ð1Þ S ðr; cÞ 0 t 1 fsat ¼ ds ð2Þ ð2Þ S ðr; cÞ G 4p s 0 fsat (24.54) 1 X Ri nþ1 AðeÞ Pn ðcos cÞ Sð1Þ ðr; cÞ ¼ n ð2n þ 1Þ r n¼0 (24.55) ðeÞ
with An given by (24.44) and Sð2Þ ðr; cÞ ¼
1 X n¼0
ðeÞ
AðeÞ n
nþ1 2n þ 1 Ri Pn ðcos cÞ ðn þ 1Þðn þ 2Þ r
with An given by (24.47). The kernels Sð1Þ and Sð2Þ are illustrated in the following figure (Fig. 24.5).
202 Fig. 24.5 Kernel functions in the integral representation of the satellite terms
P. Holota and O. Nesvadba 8000
10
7000
9
S^(1) Kernel: gravimetry and a satellite only model, Re = Ri + 400km
6000 5000
8 7
S^(1) Kernel: gravimetry and a satellite only model, Re = Ri + 250km
4000
S^(2) Kernel: gravimetry and satellite gradiometry, Re = Ri + 400km S^(2) Kernel: gravimetry and satellite gradiometry, Re = Ri + 250km
6 5
3000 4 2000
3
1000
2
0
1
psi →
-1000
psi →
0 0°
30°
60°
90°
0°
30°
60°
90°
Fig. 24.6 (Above) fterr and (below) fsat for r ¼ Ri in case of the combination of gravimetry and a satellite-only model (r ¼ 6; 378 km and Re ¼ Ri þ 250 km, i.e. q ¼ 0:96228)
24
On a Combined Use of Satellite and Terrestrial Data in Refined Studies on Earth Gravity Field
203
Fig. 24.7 (Above) fterr and (below) fsat for r ¼ Ri in case of the combination of gravimetry and a satellite gradiometry (r ¼ 6; 378 km and Re ¼ Ri þ 250 km, i.e. q ¼ 0:96228)
24.11 Numerical Simulations and Final Comments The procedure discussed here was tested by means of simulated data. For W we took the EGM08 potential from Pavlis et al. (2008) and U was the potential of the Somigliana-Pizzetti normal gravity field with parameters of the GRS1980 system. f was simulated at more than The anomaly Dg 10,000,000 points of a “regular” grid on the sphere of radius Ri ¼ 6; 378 km given by the tenth level of the icosahedron refinement. The hierarchically created grids were also exploited for Romberg’s integration method in calculating the integrals in (24.49), (24.51) and (24.54), see Nesvadba et al. (2007).
Similarly, also the data t and G were simulated at more than 10,000,000 points of a “regular” grid, but on the sphere of radius Re ¼ Ri þ 250 km. The restrictions of T ¼ W U and @ 2 T=@r 2 were used for this purpose. The case of gravimetry and a satellite-only model was treated first. The terrestrial term represented by (24.49) and the satellite term given by the first of (24.54) were computed for r ¼ Ri and Re ¼ Ri þ 250 km. They are shown in Fig. 24.6. Figure 24.7 concerns the combination of gravimetry and satellite gradiometry. The fterr term is given by (24.51) and fsat by the second of (24.54). The computation was done for r ¼ Ri and Re as above. In both the cases the composition (optimized solution) f ¼ fterr þ fsat was compared with T as obtained
204
directly from W and U, and restricted to r ¼ Ri . The results of the comparison are nearly the same in both the cases considered. Globally the RMS of the differences expressed in GeoPotential Units (1 GPU 1 m2 s2 ) does not exceed 0.06 GPU (max. difference smaller than 0.9 GPU) and excellently conforms to an apriori estimate of the error obtained from Romberg’s integration method. The results well confirm our earlier conclusions derived from numerical simulations based on the gravity field model EGM96, see Holota and Nesvadba (2006, 2007). In addition the dependence of the kernels Sð1Þ and Sð2Þ on the angle c, shown in Fig. 24.5, offers a suitable springboard for investigations on polar cap problem in using satellite data. Acknowledgements The work on this paper was done within ESA GOCE Project ID: 4311. The presentation of the paper at the IAG 2009 Scientific Assembly “Geodesy for Planet Earth”, Buenos Aires, Argentina, August 31 to September 4, 2009, was sponsored by the Ministry of Education, Youth and Sports of the Czech Republic through Projects No. LC506. All this support is gratefully acknowledged.
References Holota P (2007) On the combination of terrestrial gravity data with satellite gradiometry and airborne gravimetry treated in terms of boundary-value problems. In: Tregoning P and
P. Holota and O. Nesvadba Rizos C (eds) Dynamic Planet. IAG Symp., Cairns, Australia, 22–26 Aug 2005. IAG Symposia, vol 130, Springer, Berlin etc., Chap 53, pp 362–369 Holota P and Nesvadba O (2006) Optimized solution and a numerical treatment of two-boundary problems in combining terrestrial and satellite data. Proc. 1st Intl. Symp. of the IGFS, 28 Aug-1 Sept, 2006, Istanbul, Turkey. Spec. Issue: 18, Genl. Command of Mapping, Ankara, 2007, pp 25–30 Holota P and Nesvadba O (2007) A regularized solution of boundary problems in combining terrestrial and satellite gravity field data [CD-ROM]. Proc. ‘The 3rd International GOCE User Workshop’, ESA-ESRIN Frascati, Italy, 6–8 Nov 2006 (ESA SP-627, Jan 2007), pp 121–126 Holota P and Nesvadba O (2009) Domain transformation, boundary problems and optimization concepts in the combination of terrestrial and satellite gravity field data. In: Sideris M (ed) Observing our changing earth, Proc. 2007 IAG Gen. Assembly, Perugia, Italy, July 2–13, 2007. IAG Symposia, vol 133, Springer, Berlin etc. pp 219–228 Migliaccio F, Raguzzoni M, Sanso` F (2004) Space-wise approach to satellite gravity field determination in the presence of coloured noise. J Geod 78:304–313 Nesvadba O, Holota P and Klees R (2007) A direct method and its numerical interpretation in the determination of the Earth’s gravity field from terrestrial data. In: Tregoning P and Rizos C (eds) Dynamic Planet. IAG Symp., Cairns, Australia, 22–26 Aug 2005. IAG Symposia, vol 130, Springer, Berlin etc., Chap. 54, pp 370–376 Pavlis NK, Holmes SA, Kenyon SC and Factor JK (2008) An Earth Gravitational Model to Degree 2160: EGM2008. Presented at the 2008 General Assembly of the European Geosciences Union, Vienna, Austria, April 13–18, 2008 Sokolnikoff IS (1971) Tensor analysis, theory and applications to geometry and mechanics of continua. Nauka Publishers, Moscow (in Russian)
Moho Estimation Using GOCE Data: A Numerical Simulation
25
Mirko Reguzzoni and Daniele Sampietro
Abstract
The GOCE mission, exploiting for the first time the concept of satellite gradiometry, promises to estimate the Earth’s gravitational field from space with unprecedented accuracy and spatial resolution. Also inverse gravimetric problems can get benefit from GOCE observations. In this work the general problem of estimating the discontinuity surface between two layers of different density is investigated. A possible solution based on a local Fourier analysis and Wiener deconvolution of satellite data (such as gravitational potential and its second radial derivative) is proposed. Moreover a numerical method to combine in an efficient way gridded satellite data with sparse ground data, like gravity anomalies, has been implemented. Numerical simulations on different synthetic Moho profiles have been carried out. Finally a two-dimensional simulation on realistic data over the Alps has been set up. The results confirm that GOCE data can significantly contribute to the detection of geophysical structures, leading to a much better determination of the signal long wavelengths (up to about 200 km). The use of local ground data improves the satellite-only estimate, making possible the recovery of higher resolution details.
25.1
Introduction
The crust-mantle boundary, the so called Moho discontinuity, has been studied with profitable results by means of seismic profiles and ground gravity anomalies
M. Reguzzoni Department of Geophysics of the Lithosphere, OGS, c/o Politecnico di Milano, Polo Regionale di Como, Via Valleggio 11, 22100 Como, Italy e-mail:
[email protected] D. Sampietro (*) DIIAR, Politecnico di Milano, Polo Regionale di Como, Via Valleggio 11, 22100 Como, Italy e-mail:
[email protected]
(e.g. Grad and Tiira 2009). However there are areas where, due to the lack of observations, the crustal structure has not been yet determined by geophysical sounding. In such areas the Moho surface can be estimated using isostatic theories, but especially young orogenic regions are not necessarily in isostatic equilibrium. Since satellite data are collected worldwide, they can be profitably used to overstep these problems and correct isostatic models even where ground observations are not available. Studies on this subject have been done using GRACE data (see e.g. Shin et al. 2007) or simulated GOCE gradiometric data (see e.g. Benedek and Papp 2009). In our solution, the gravitational potential and its second radial derivative at
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_25, # Springer-Verlag Berlin Heidelberg 2012
205
206
M. Reguzzoni and D. Sampietro
satellite altitude, as observed from a GOCE like mission, are combined with point-wise observations at ground level (e.g. gravity anomalies) to model the Moho surface. This inverse gravimetric problem is in general ill-posed: in fact the solution is not unique (i.e. the space of all solutions corresponding to a fixed potential is infinite-dimensional) and not stable because the Newton operator is a smoother and its inverse is unbounded (Sanso´ 1980). In this work the non-uniqueness has been treated by taking peculiar hypotheses on the shape of the density discontinuity. In fact it has been proved that to guarantee the uniqueness of the solution it is possible to consider a two layer density discontinuity, in planar or spherical approximation, where the two different densities are known (see e.g. Sampietro and Sanso´ 2009 and the references therein). Note that for a lot of geophysical problems, included the one considered in this work, these kinds of approximations are commonly adopted and in general well satisfied (see e.g. Gangui 1998). In the first part of the paper the proposed algorithm is theoretically illustrated (Sect. 25.2) and numerically assessed (Sect. 25.3) on the basis of one-dimensional Moho profiles in planar approximation. Then the algorithm is generalized to the two-dimensional case and tested on a realistic simulation in the Alpine area (Sect. 25.4). The results obtained in both cases are presented in the paper, drawing conclusions and discussing future perspectives.
ð ð ð z0 þDðxÞ
TðxÞ ¼
Tz ðxÞ ¼
z0 hðxÞ
@ @z
ð ð ð z0 þDðxÞ z0 hðxÞ
Methodology
The analysis is developed accordingly to the inversion algorithm described in Reguzzoni and Sampietro 2008 and to the theory of partitioned (or stepwise) collocation solution (see Tscherning 1974). Let us recall here the main concepts. We consider a Cartesian coordinate system, where the (x, y) plane is at satellite level, x is oriented along track and z is pointing downward. The topography profile h(x) and the Moho profile D(x) are both defined with respect to the ground level z ¼ z0 ¼ 250 km. In order to solve the gravimetric inverse problem the following deterministic models have to be inverted (Heiskanen and Moritz 1967):
(25.1)
G r dxd dz qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ðx xÞ þ þ ðz zÞ
z¼z0
(25.2)
Tzz ðxÞ ¼
@ @z2 2
ð ð ð z0 þDðxÞ z0 hðxÞ
G r dx d dz qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx xÞ2 þ 2 þ ðz zÞ2
z¼0
(25.3) where T is the anomalous potential, G is the gravitational constant, r the density contrast and x, , z are integration variables in the x, y, z directions respectively (planar approximation). Modelling the Moho as DðxÞ ¼ D þ dDðxÞ, linearizing (25.1)–(25.3) around D in the z direction and finally integrating with respect to over a bounded interval, we get:
dTðxÞ ¼
dTz ðxÞ ¼
25.2
G r dx d dz qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx xÞ2 þ 2 þ z2
dTzz ðxÞ ¼
ð þ1 1
ð þ1 1
ð þ1 1
kT ðx xÞ dDðxÞ dx
(25.4)
kTz ðx xÞ dDðxÞ dx
(25.5)
kTzz ðx xÞdDðxÞ dx
(25.6)
where dT(x), dTz(x) and dTzz(x) are the residual gravitational effects, i.e. the actual effect minus the effects due to the topography and to the mean Moho D, while kT, kTz and kTzz are convolution kernels (for details on the form of these kernels we refer again to Reguzzoni and Sampietro 2008). To estimate the Moho we have to solve the system obtained by inverting (25.4)–(25.6) degraded with the corresponding observation noise. As for the noise in (25.4) and (25.6), none of the two quantities is a direct observation of the GOCE
25
Moho Estimation Using GOCE Data: A Numerical Simulation
mission: in fact the potential T is derived from GPS tracking data, for example by applying the so called energy integral approach (Jekeli 1999; Visser et al. 2003), while the second radial derivatives Tzz are obtained by preprocessing the gradiometer observations taken in the instrumental reference frame (Cesare 2002; Pail 2005). However the resulting potential is known to have an almost white error, while the second radial derivatives have a time-correlated error with spectral characteristics almost identical to the original observations (Migliaccio et al. 2004a). As for the ground gravity anomalies Tz, a hypothesis of white noise is generally adopted. At this point a remark is due: the three different observations have complementary spectral characteristics and spatial distribution. In particular T and Tzz contain the low and medium frequencies of the signal because they are observed at satellite altitude; they are regularly sampled along the orbit and therefore are given on a regular grid when inverting a Moho profile; note that, in the two-dimensional case, T and Tzz can be anyway predicted on a grid (Migliaccio et al. 2007) thanks to the good spatial coverage of the GOCE data. On the other hand, Tz contains the highest frequencies of the gravimetric signal because it is observed at ground level; Tz observations are generally available at sparse points and cannot be reasonably gridded due to their typical dishomogeneous spatial distribution, e.g. in mountain areas. For these reasons it is important to combine, in an optimal way, the point-wise gravity data with the gridded satellite data. This can be achieved by means of collocation, treating dD as a random signal: dD^ ¼ CTy;dD C1 y;y y
(25.7)
207
the covariance matrix between a and b plus the error covariance matrix Cna ;nb . Since we consider the noise of the three observations uncorrelated, the error covariances are present only on the diagonal blocks of the matrix Cy,y. It can be noticed that all the needed covariance matrices, with the exception of the error covariance matrices, can be computed by propagating CdD,dD through the corresponding convolution kernels. In this way the resulting covariance matrices describe the different spectral contents of the observables. The solution of (25.7) can be obtained by using a partitioned method separating the gridded observations from the sparse ones:
dD^ ¼ mT
dTzz þ lT dTz dT
(25.8)
where: 2 6 m¼4
CTzz ;Tzz
CTzz ;T
CT;Tzz
CT;T
31 02 B6 @4
7 5
3 CTzz ;dD
7 6 54
CT;dD
2 3T 2 C C 6 Tzz ;Tz 7 6 Tzz ;Tzz B 5 4 l ¼ @CTz ;Tz 4 CT;Tz CT;Tzz 2
6 B @CTz ;dD 4
CTzz ;Tz
3T 2 7 6 5 4
CT;Tz
CTzz ;Tz
7 C 5lA;
CT;Tz
0
0
3 1
2
CTzz ;T
31 2
311 C 6 Tzz ;Tz 7C 4 5A CT;Tz
7 5
CT;T
CTzz ;Tzz
CTzz ;T
CT;Tzz
CT;T
31 2 6 4
7 5
31 CTzz ;dD
7C 5A:
CT;dD
The solution can be rearranged in the following way: T T dTzz T dTzz ^ þ l dTz m2 dD ¼ m 1 dT dT
(25.9)
with: where: 2
3
2
3
CTzz ;dD CTzz ;Tzz CTzz ;T CTzz ;Tz 7 7 6 6 Cy;dD ¼ 4 CT;dD 5; Cy;y ¼ 4 CT;Tzz CT;T CT;Tz 5; CTz ;dD CTz ;Tzz CTz ;T CTz ;Tz 3 2 dTzz 7 6 y ¼ 4 dT 5; dTz dD is the unknown Moho depth, y are the observations, Ca,b is the covariance matrix between a and b and Ca;b
2 m1 ¼ 4 2 m2 ¼ 4 0
CTzz ;Tzz
CTzz ;T
CT;Tzz
CT;T
CTzz ;Tzz
CTzz ;T
CT;Tzz
CT;T 2
B l ¼ @CTz ;Tz 4
31 2 5
31 2 5
CTzz ;Tz CT;Tz
4
3T
4
CTzz ;dD CT;dD CTzz ;Tz
3 5;
3
5; CT;Tz 11 0
5 m C 2A
2
B @CTz ;dD 4
CTzz ;Tz CT;Tz
3T
1
5 m C 1 A:
The matrices m1 and m2 are the two heaviest terms to be computed, because there are in general much
208
more satellite data than ground observation points and, therefore, the covariance matrix of the satellite data is much larger than the one of the ground data. In other words, after computing m1 and m2, the determination of l is easy from the numerical point of view. Note that the first term of the sum in (25.9) is exactly the collocation estimate of the Moho when considering as observations T and Tzz only. Since both these observations are regularly sampled, and the Moho is estimated on the same gridded points, the matrix m1 can be efficiently computed in terms of Multiple Input Single Output (MISO) Wiener filter in the frequency domain (Papoulis 1977; Sideris 1996). The other onerous term, namely m2, can be seen as the collocation transfer operator from satellite to ground data. Naturally, if the ground data were regularly sampled too, this term could be also efficiently computed in the frequency domain. In reality, as we explained before, ground data are generally sparse and cannot be conveniently gridded. Our idea is first to compute the term m2 as if the ground data were available on a regular grid, thus exploiting the properties of the Fourier transform; this leads to a discrete convolution kernel, which is interpolated by splines to derive a continuous model. By applying a principle of stationarity (invariance by translations), this continuous kernel is shifted to each ground observation point and then resampled over the satellite data grid to get the corresponding row of the matrix m2. Of course this
M. Reguzzoni and D. Sampietro
matrix is not Toeplitz even if it is computed according to the stationarity principle, because ground data are not regularly distributed. Note that the harmonicity of Tz predicted by m2 is in general not guaranteed when splines are used to model the continuous kernel; however this is not critical here, because all covariances and cross-covariances are anyway consistent with one another through the Newton integral and the use of a unique covariance of dD.
25.3
Tests on Synthetic Models
The proposed methodology has been assessed on different synthetic models. The aim of the first model is to illustrate the complementary behaviour of the three kinds of observations used. The reference Moho is taken as a simple profile with non-zero mean, a linear trend and a spike modelled by a Gaussian function, see Fig. 25.1. The Moho covariance function and the corresponding spectrum, which are computed disregarding the trend in the data, have a Gaussian shape as shown in Fig. 25.2. Even if it is quite simplistic, this model can be thought as the Moho counterpart of a mountain loading the crust. The depth of the root extends from 32 to 38 km (we fixed D ¼ 30 km) and the density contrast between the root and the underlying mantle is 0.6 g/cm3. The simulation is performed on nearly 150
Fig. 25.1 Estimated Moho using only T, only Tzz or combining T and Tzz. Reference Moho in black solid line
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Moho Estimation Using GOCE Data: A Numerical Simulation
209
Fig. 25.2 Error spectra using only T, only Tzz or combining T and Tzz. Moho signal spectrum in black solid line. To convert timefrequency into space-frequency divide by the mean satellite velocity of about 7 km/s
Fig. 25.3 Error spectra comparing a single-track solution versus a ten-track solution. Error spectrum using only gridded ground Tz. Moho signal spectrum in black solid line. To convert time-frequency into space-frequency divide by the mean satellite velocity of about 7 km/s
points, which are regularly spaced at a distance of 7 km, consistently with the sampling rate of the GOCE observations. Concerning the data noise along the satellite orbit, the potential is degraded by a white noise with a standard deviation of 0.3 m2/s2, while the second radial derivatives have a coloured noise with a spectrum
derived from a realistic simulation of the GOCE mission (Catastini et al. 2007). The result of the inversion is shown in Figs. 25.1 and 25.2. It can be seen that low frequencies (such as biases and trends) are completely lost when using Tzz observations, but are recovered from T observations. On the other hand, the high resolution details
210
M. Reguzzoni and D. Sampietro
Fig. 25.4 Estimated Moho using satellite T and Tzz or combining satellite data with sparse ground Tz observations. Reference Moho in black solid line
(the spike in our simulation) are caught by Tzz observations. Obviously, combining the two solutions it is possible to estimate both the low and the high frequencies of the signal. This analysis is confirmed by the comparison between the signal and error spectra shown in Fig. 25.2. Here we consider as “well reconstructed” those frequencies for which the error spectrum is lower than 20% of the signal spectrum. According to this threshold, it can be stated that the maximum resolvable resolution with GOCE data only is about 250 km. This result can be improved by considering more than one satellite orbit flying over the Moho profile. Assuming independent observations for each orbit and considering for example ten orbits (corresponding to a reasonable repeat period of two months for the GOCE mission), the maximum resolution becomes 200 km, see Fig. 25.3. Ground gravity data add information at higher frequencies: a numerical simulation shows that if also ground gravity anomalies were regularly sampled as the GOCE observations, then the corresponding error spectrum could be computed (see Fig. 25.3), making possible to reconstruct Moho details up to about 70 km. A second simulation scenario is set up in order to investigate the improvement in the result due to the combination of gridded satellite data and sparse ground gravity data. The algorithm described in Sect. 25.2 is used. A high resolution detail, i.e. a
smaller Gaussian spike, is added to the previously simulated Moho model and we assume to have few (40) gravity anomaly observations at random points at ground level concentrated over this new feature (see Fig. 25.4). A white noise with a standard deviation of 1 mGal is used for these observations. As one can see, these additional data locally improve the Moho estimation coming from satellite-only data; in particular the estimation error decreases from 0.36 to 0.16 km. On the other hand it has been tested that, if the ground data are far from the higher resolution detail, they have no impact on the final solution, thus showing the numerical stability of the method. Finally a test on a realistic Moho profile over the Alpine region has been studied. In this case the reference Moho has been computed starting from the actual topography and applying the Airy isostasy theory (see Watts 2001). The result ranges from 30 to 46 km and clearly shows the presence of the root due to the Alps (see Fig. 25.5). Since gravity anomaly observations are usually placed in flat areas, we simulate a set of 40 ground observations in the Po valley. Again we can notice that local ground data contribute to improve high frequencies: in fact where the point-wise ground observations are present the error r.m.s. significantly decreases (from 0.72 to 0.36 km), while where they are not available the satellite-only solution and the combined solution have practically the same smooth behaviour.
25
Moho Estimation Using GOCE Data: A Numerical Simulation
211
Fig. 25.5 Estimated Moho profile over the Alps using satellite T and Tzz or combining satellite data with sparse ground Tz observations. Reference Moho in black solid line
25.4
A Two-Dimensional Numerical Example in the Alpine Area
In the previous section, some synthetic Moho profiles have been used to test the performances of the proposed inversion algorithm. However the final goal is to estimate the Moho discontinuity surface and not only a profile along the satellite orbit. For this reason the method has been generalized to the two-dimensional case (Sampietro 2009) and has been tested on a realistic numerical example in a region of 10 7 in the centre of Europe (latitude between 44 and 51 North and longitude between 5 and 15 East). This area presents a complex topography characterized by the presence of the Alps. Again, the Airy isostasy theory is used to simulate a reference Moho over the region of interest. Starting from this surface, the gravitational potential and its second radial derivatives have been numerically computed at satellite altitude on a grid of 0.2 0.2 by using (25.1)–(25.3), see Table 25.1. Note that these simulated observables are realistic as for the Moho contribution, but do not represent the full gravity signal. In this experiment, the noise is derived by applying the so called space-wise approach (Migliaccio et al. 2004b; Reguzzoni and Tselfes 2009) to realistic alongtrack simulated GOCE observations (Catastini et al.
Table 25.1 Statistics on simulated signal and corresponding errors on a grid at satellite altitude T signal [m2/s2] T error [m2/s2] Tzz signal [mE] Tzz error [mE]
Min 0.57 0.19 12.62 1.70
Max 1.24 0.29 17.68 1.86
Mean 0.17 0.02 0.14 0.10
Std 0.43 0.10 7.57 0.70
2007), obtaining grids of potential and second radial derivatives as intermediate results. Some statistics of the error of these grids are reported in Table 25.1. Note that the GOCE signal generated by the simulated Moho over the selected Alpine area is about one order of magnitude larger than the grid error, especially in the case of the second radial derivatives. In order to apply a Fourier analysis, it is required that the error covariance matrices of the gridded data have a Toeplitz–Toeplitz structure (Grenander and Szeg€o 1958). This is not the case because the gridding is performed by least squares collocation in spherical approximation (Tscherning 2004). For this reason the error covariance matrix has been approximated by averaging covariances along diagonals for each block of the Toeplitz–Toeplitz structure, namely: Cout i;iþk ¼
N k 1 X Cin N k j¼1 j;jþk
i ¼ 1; 2; :::; N k k ¼ 0; 1; :::; N 1 (25.10)
212
where Cin is the original error covariance matrix, Cout is the approximated Toeplitz–Toeplitz error covariance matrix and N is the block dimension depending on the grid definition. The differences between the original and the approximated matrix are about one order of magnitude smaller than the original covariances, numerically justifying this approximation.
M. Reguzzoni and D. Sampietro
The estimated Moho with and without the aid of additional ground gravity anomalies is shown in Fig. 25.6. These ground observations are located in the centre of the study area, i.e. close to the Alps and along the valleys in order to make the simulation more realistic. Note that the region covered by satellite data is generally larger due to the smoothing effect of
Fig. 25.6 Estimated Moho surface over the Alps using satellite data only (top) or combining satellite data and ground gravity data (bottom)
25
Moho Estimation Using GOCE Data: A Numerical Simulation
the gravity field with increasing altitude. As expected, the satellite data fix the long wavelengths of the Moho, while the ground data improve the details, In particular, the error r.m.s. is reduced from 0.95 to 0.86 km in the area where ground data are available.
25.5
Conclusions and Perspectives
The inverse gravimetric problem of reconstructing the shape of the Moho using GOCE satellite observations has been studied, as well as the integration of additional ground data in a numerically efficient way. In this work we assume a simple two layer model (to guarantee the uniqueness of the solution) with constant densities of mantle and crust in planar approximation; under these hypotheses, positive results have been obtained. In particular the long wavelengths are well recovered thanks to the use of the gravitational potential, while higher resolution details come from gradiometric observations. Sparse gravity anomaly observations at ground level further improve the resolution of the final model. All in all, the estimated Moho over the Alpine area presents errors of less than 1 km for a grid of 0.2 0.2 resolution. However, it has to be stressed that the GOCE-only solution, without integrating any ground observations, is just 10% worse than the combined solution. This shows that reasonable Moho models can be computed at lower resolution also in areas where ground observations are not available, just using GOCE data. Beyond the positive results there are still open issues that need to be solved. First of all, the hypothesis of planar approximation has to be replaced by the spherical one in order to apply the inversion algorithm over larger areas. This implies that the whole theory, including filter design, has to be revisited. Furthermore, it seems interesting to proceed with the research by including additional geophysical information, such as geological models and bounds. This should contribute to get a more reliable solution from the physical point of view. Last but not least, the application of the method to real GOCE data, being the mission in its operational phase. However this requires to face the main problem of every inversion algorithm, that is how to disentangle the different gravimetric signals mixed up into the observed data. The results presented in this paper
213
are based on the assumption that the only source of the gravimetric signal was the Moho discontinuity. As a matter of fact modelling crustal dishomogeneities, as well as unwrapping the contributions of large deep features from those closer to the surface, is a problem that can be solved only with additional geological information. Acknowledgements The present research has been partially funded by the Italian Space Agency (ASI) through the GOCEITALY project.
References Benedek J, Papp G (2009) Geophysical inversion of on board satellite gradiometer data – a feasibility study in the ALPACA region, Central Europe. Acta Geodaetica et Geophysica Hungarica 44(2):179–190 Catastini G, Cesare S, De Sanctis S, Dumontel M, Parisch M, Sechi G (2007) Predictions of the GOCE in-flight performances with the End-to-End System Simulator. In: Proc. of the 3rd International GOCE User Workshop, 6–8 November 2006, Frascati, Italy, pp 9–16 Cesare S (2002) Performance requirements and budgets for the gradiometric mission, Technical Note, GOC-TN-AI-0027, Alenia Spazio, Turin, Italy Gangui AH (1998) A combined structural interpretation based on seismic data and 3-D gravity modeling in the Northern Puna, Eastern Cordillera, Argentina. Dissertation FU Berlin, Berliner Geowissenschaftliche Abhandlungen, Reihe B, Band 27, Berlin, Germany Grad M, Tiira T (2009) The Moho depth map of the European Plate. Geophys J Int 176(1):279–292 Grenander U, Szeg€o G (1958) Toeplitz forms and their applications. University of California Press, Berkeley, Los Angeles Heiskanen WA, Moritz H (1967) Physical geodesy. Springer, Wien, Austria Jekeli C (1999) The determination of gravitational potential differences from satellite to satellite tracking. Celestial Mech Dyn Astron 75:85–101 Migliaccio F, Reguzzoni M, Sanso´ F, Zatelli P (2004a) GOCE: dealing with large attitude variations in the conceptual structure of the space-wise approach. In: Proc. of the 2nd International GOCE User Workshop 8–10 March 2004, Frascati, Italy Migliaccio F, Reguzzoni M, Sanso´ F (2004b) Space-wise approach to satellite gravity field determination in the presence of coloured noise. J Geod 78(4–5):304–313 Migliaccio F, Reguzzoni M, Sanso´ F, Tselfes N (2007). On the use of gridded data to estimate potential coefficients. Proc. of the 3rd International GOCE User Workshop, 6–8 November 2006, Frascati, Rome, Italy, pp 311–318 Pail R (2005) A parametric study on the impact of satellite attitude errors on GOCE gravity field recovery. J Geod 79 (4–5):231–241 Papoulis A (1977) Signal analysis. McGraw-Hill, New York
214 Reguzzoni M, Sampietro D (2008) An inverse gravimetric problem with GOCE data. In: IAG Symposia “Gravity, Geoid and Earth Observation”, Mertikas SP (ed), vol 135, Springer, Berlin, pp 451–456 Reguzzoni M, Tselfes N (2009) Optimal multi-step collocation: application to the space-wise approach for GOCE data analysis. J Geod 83(1):13–29 Sampietro D (2009) An inverse gravimetric problem with GOCE data. PhD Thesis, Doctorate in Geodesy and Geomatics, Politecnico di Milano, Italy Sampietro D, Sanso´ F (2009) Uniqueness theorems for inverse gravimetric problems. Proc. of the VII Hotine-Marussi Symposium, 6–10 June 2009, Rome, Italy (in press) Sanso´ F (1980) Internal collocation. Memorie dell’Accademia dei Lincei. vol XVI, N. 1 Shin YH, Xu H, Braitenberg C, Fang J, Wang Y (2007) Moho undulations beneath Tibet from GRACE-integrated gravity data. Geophys J Int 170(3):971–985
M. Reguzzoni and D. Sampietro Sideris MG (1996) On the use of heterogeneous noisy data in spectral gravity field modeling methods. J Geod 70 (8):470–479 Tscherning CC (1974) A FORTRAN IV Program for the Determination of the Anomalous Potential Using Stepwise Least Squares Collocation. Reports of the Department of Geodetic Science, No. 212, The Ohio State University, Columbus, Ohio Tscherning CC (2004) Testing frame transformation, gridding and filtering of GOCE gradiometer data by Least-Squares Collocation using simulated data. In: IAG Symposia “A Window on the Future of Geodesy”, Sanso´ F (ed), vol 128, Springer, Berlin, pp 277–282 Visser PNAM, Sneeuw N, Gerlach C (2003) Energy integral method for gravity field determination from satellite orbit coordinates. J Geod 77(3–4):207–216 Watts AB (2001) Isostasy and flexure of the lithosphere. Cambridge University Press, Cambridge
CHAMP, GRACE, GOCE Instruments and Beyond
26
P. Touboul, B. Foulon, B. Christophe, and J.P. Marque
Abstract
The electrostatic accelerometers of the CHAMP satellite as well as of the GRACE two ones have provided the necessary information to distinguish the satellite actual trajectories from the pure gravitational orbits. By providing the measurements of the satellite non-gravitational forces, one can distinguish the position or velocity fluctuations of the satellite due to the Earth gravity anomalies from those due to the drag fluctuations. In-orbit calibration and validation of onboard instruments, bandwidth, bias stability and resolution proof the success of the mission scientific geodesic return. The basic principle of these sensors stays on the servo-control of one solid mass, maintained motionless from the instrument highly stable structure. Care is paid for the mass motion detection, down to tenth of Angstrom, and to the fine measurement of the servo-controlled forces applied on the mass through electrostatic pressures. With the same concept and technologies, the GOCE inertial sensors have been designed, produced and tested to reach even better performances in order to deal with the milli-E€otv€os gradiometer objectives. The performance of the instrument and the interest of the obtained measurements do not only depend on the sensor accuracy itself but also on the onboard environment (magnetic, thermal, vibrational. . .), on the satellite attitude motions and on the in-orbit configuration and aliasing aspects. Future missions will have also to consider these aspects, especially when envisaging cryogenic electrostatic sensors which can exhibit better self accuracy or when considering satellite to satellite laser tracking.
26.1
Introduction
Space gradiometry missions have been already envisaged in the 1980s with different types of instruments demonstrating the interest of low altitude
P. Touboul (*) B. Foulon B. Christophe J.P. Marque ONERA - The French Aerospace Lab, F-91761 Palaiseau, France e-mail:
[email protected]
satellites with drag compensation systems to eliminate the effects of the non-gravitational forces on the satellite orbital motion [1, 2]. To that aim the concept and the technology of a three axis linear electrostatic accelerometer have been developed in view of providing both the measurement of the satellite drag and the Earth gravity gradient effect [3]. From the experience acquired with the three flights of the ASTRE instrument, developed for the on-board shuttle microgravity environment survey [4], the
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STAR ultra-sensitive space accelerometer has been designed, based on the accurate electrostatic levitation of a parallelepiped proof-mass [5]. This accelerometer has been integrated at the center of mass of the CHAMP satellite, first geodesic mission of the 2000 decade. For now quite 10 years, STAR performs the measurement of the non-gravitational accelerations of the satellite which perturb its low altitude trajectory at less than 500 km. The satellite orbit fluctuations are finely determined taking advantage of the on-board GPS receiver, aiming at the accurate recovery of the low spherical harmonics of the Earth’s gravity field. The concept of operation, the design and the technology of the accelerometer have been especially selected for space applications with the potential to optimize the definition for different mission performance and on-board accommodation requirements. So, the Super-STAR instrument has been designed in view of the two GRACE satellites with a reduced full scale range and a resolution ten times better leading to 1010 ms2 over 1 Hz bandwidth. The launch of the GRACE satellite in 2002 has allowed for the first time to compare the in orbit outputs of two accelerometers quite in same operational conditions: both spacecrafts are identical, flying on the same orbit with a 200 km distance from each other, corresponding to less than half a minute, the radiation pressure and the atmospheric drag being thus similar. The difference of velocities between the two satellites is provided with an accuracy of about 1 mm/s by a microwave Doppler device which links the two satellites and is finely analyzed in conjunction with the accelerometer outputs. This leads to the determination of the Earth’s gravity potential, every month and even more, allowing us to evaluate the secular and seasonal fluctuations of large mass distribution like the Antarctic and Greenland ices or the hydrologic basins of large rivers like Mississippi or Amazon [6]. Another way of improving the accuracy of the static Earth field with a much better geographical resolution is the accommodation of a space gradiometer on-board a drag-free satellite at an altitude lower than 300 km, like in the recently launched GOCE mission [7]. The gravity gradiometer is composed of several identical sensors mounted on a rigid and stable structure. In addition to the linear and angular acceleration of the satellite, it provides the fine measurements of the three diagonal components of the Earth’s gravity gradient tensor for the determination of the higher
P. Touboul et al.
harmonics of the potential. Taking advantage of the fine active thermal control of the instrument case and of the drag-free compensation system of the satellite, the full range of the sensors is limited to a few 106 ms2 to the benefit of the resolution of better than 2 1012 ms2Hz1/2 in the frequency measurement bandwidth, from 5 103 to 0.1 Hz. A resolution of a few milli-E€otv€os is thus expected at an altitude as low as 260 km. After the in-orbit switch on of the six accelerometers in April 2009, first gradiometric measurements have been recently obtained showing new space signatures of the Earth gravity field that will provide absolute reference for ocean circulation studies, geophysics, water or atmospheric changes [8]. With nine electrostatic space accelerometers, now in orbit on board four satellites at altitudes between 450 and 260 km, the performance of the concept, its robustness and flexibility to different functioning environment is clearly demonstrated to the benefit of the observation of the Earth gravity field, which is not only the static reference for many other disciplines but also the variation of mass distribution and transportation. Furthermore, improvements of the instrument can be expected as well as improvements of its accommodation on board future satellites and of involved space measurement techniques. So, beyond these three missions, other solid Earth missions are envisaged but also applications in fundamental physics or planetology.
26.2
The STAR and Super STAR Accelerometer for the CHAMP and GRACE Mission
The German CHAMP mission (CHAllenging Microsatellite Payload for geophysical research and application) was dedicated to Earth’s observation: global magnetic and gravity fields mapping. The satellite has been launched on July 15th, 2000 from the cosmodrome Plesetsk by a Russian COSMOS rocket at an altitude of 454 km in a circular orbit with a 87.3 inclination. CHAMP performed for the first time the combination of uninterrupted three dimensional high low tracking of its low orbit perturbations by the satellites of the GPS constellation and a high-precision three-axes measurement of the satellite surface forces: residual drag, solar and Earth radiation pressures and attitude manoeuvre thrusts are measured by the STAR
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CHAMP, GRACE, GOCE Instruments and Beyond
217
(Space Three-axis Accelerometer for Research) accelerometer integrated at the centre of mass of the satellite. A by-product of the accelerometer measurements is the determination of the atmospheric density variations during the decade of the mission. STAR is a six-axis accelerometer providing the three linear accelerations along the instrument sensitive axes and the three angular accelerations about these axes (see Fig. 26.1). STAR presents a measurement range of 104 ms2 and exhibits a resolution of better than 3 109 ms2 for the y and z axes and 3 108 ms2 for the x axis within the measurement bandwidth from 104 to 101 Hz. The measurements are integrated over 1 s before delivery to the satellite data bus. The configuration of the instrument is compatible with ground tests which demand specific characteristics of the less accurate x-axis for the operation under 1 g gravity field. For example, measurements of residual low frequency vibrations on a specific testing pendulum platform have been performed along the y and z axes of the accelerometer controlled horizontal with a resolution of 108 ms2 Hz1/2 at frequencies lower than 0.1 Hz corresponding to the requirement of 3 109 ms2 rms in the measurement bandwidth. The accelerometer low level of bias (less than 105 ms2) of the same y and z sensitive axes has also been verified in free fall in the Bremen drop tower [9]. The following GRACE mission put in evidence the temporal variability of the Earth’s gravity field. In
addition to the High-Low GPS satellite tracking and to the accelerometer surface force measurement, the GRACE mission is based on low–low K-band tracking between two identical satellites separated by about 220 km on the same quasi circular orbit. The twin GRACE satellites were launched on March 17th 2002 from Plesetsk by Rockot launch vehicle at a initial altitude of 500 km on a quite polar orbit (89.0 inclination). The configuration of the two SuperSTAR accelerometers is quasi identical to STAR and takes advantage of the CHAMP mission experience. The concept of electrostatic servo-controlled accelerometer is well suited for space applications: the electrostatic forces give the possibility to generate very weak but accurate accelerations while the capacitive sensing offers a high position resolution with negligible backaction. The accelerometer proof-mass is fully suspended with six servo-control loops acting along its six degrees of freedom, suppressing any mechanical contact to the benefit of the resolution and yielding to a six-axis accelerometer. The internal cage of the accelerometer, constituted by three electrode-plates made of silica glass, surrounds the solid parallelepiped proof-mass, of 4 4 1 cm sides and 72 g weight, made in Chromium coated Titanium alloy (see Fig. 26.2). The accelerometer performance depends mainly on the geometrical accuracy and the stability of the core mechanical assembly. In operation, the mass is maintained motionless at the centre of the cage with
Fig. 26.1 STAR sensor unit before integration in the CHAMP satellite
Fig. 26.2 GRACE accelerometer mechanical core
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stability up to a few 1012 m/Hz1/2. The fine measurements of the applied voltages on the electrodes provide the accelerometer outputs. Independent analyses performed on the in orbit data confirm that the instrument specifications are achieved for both GRACE accelerometers in particular, along the sensitive axes Y and Z, the bias being less than 2 106 ms2 [10] and the noise PSD: S(f) ¼ 1020 (1 + 5 103/f) m2s4/Hz [11]. In fact, the instrument is so sensitive that the on board environment has to be precisely decorrelated from the self behavior of the instrument to exploit the sensor limit of performance.
26.3
From GRACE Accelerometer to GOCE Gradiometer
The aim of ESA’s Gravity Field and Steady-State Ocean Circulation Explorer mission is also to generate accurate global mapping of the Earth’s gravity field for oceanographic, geophysical, hydrological or climatologic applications. Compare with previously launched missions, GOCE seeks particularly to better understand the highest spherical harmonics of the gravity field model. Launched on March 17th 2009 from Plesetsk, the GOCE mission exploits for the first time a three-axis gradiometer consisting of six electrostatic accelerometers offering an outstanding resolution of 2 1012 ms2Hz1/2. The accelerometers were successfully activated on April 6th 2009. The satellite is on a sun-synchronous, quasi circular and quasi polar (96.5 ) orbit at an altitude near 260 km. Weighting 1 ton with a length of 5 m, the spacecraft has a very rigid structure without moving parts and exhibiting a limited cross section (1 m diameter) in order to reduce its drag. Its drag compensation system counteracts all non- gravitational forces acting on the spacecraft along its velocity vector through two ion thrusters. The electrostatic gravity gradiometer is composed of three pairs of accelerometer sensor head including the Platinum test masses, on one Carbon–Carbon ultra stable structure, three front end electronics units for the masses sensing and actuations, one gradiometer interface unit for the experiment control and the mass control laws, contained in three accurately thermal controlled stages. The outputs delivered by each pair of accelerometers are combined to provide the gravity
gradient along the gradiometer axis: this is the main science data, to be separated from residual angular accelerations of the satellite. This angular acceleration about the three axes, and by integration the angular rate, can be deduced from the transverse axes of the accelerometers, symmetrically distributed around the gradiometer centre which is also the satellite centre of inertia. In addition the common measured acceleration provides the external forces applied on the satellite to be cancelled along the velocity vector by the drag compensation system. The GOCE sensors are similar to the CHAMP and GRACE ones except the very dense proof-mass made in PtRh10 alloy, the eight pairs of electrodes instead of six for redundancy, plus the digital control loops instead of the previous analogue circuits. The main characteristics of the three accelerometers are compared in Table 26.1. On April 6th 2009, the six accelerometers were switched on and the proof-masses were immediately all levitated at the centre of their cage. The next day, the six accelerometers passed in Science Mode with a more limited full range and better resolution. On May 29th 2009, the GOCE satellite was successfully commanded in Drag-Free mode demonstrating the performing association of the accelerometer measurements and the electric ion propulsion system [12]. The residual along-track external acceleration (red spectrum on Fig. 26.3) is around 109 ms2Hz1/2 in the measurement bandwidth, more than one order of magnitude less than the specified value. Estimation of the accelerometer noise can be deduced from redundant sensor measurement (blue spectrum of Fig. 26.3) and is perfectly in line with the predicted noise computed from on ground tests and evaluations (black curve): the coherence of the two curves between 1 and 5 Hz, where the position sensing source is the main contributor confirms the right operation of the capacitive sensors; similarly, the coherence of the curves between 10 and 100 mHz where the electrostatic actuation is the main contributor, confirms the right operation of the electrode voltage driving amplifiers. This result, associated to the coherence of the data provided by the 36 channels of the six accelerometers is a very good clue for the success of the future data processing of the mission. The very low satellite altitude, facilitated by the Sun’s exceptionally weak activity and associated in consequence to a longer
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CHAMP, GRACE, GOCE Instruments and Beyond
219
Table 26.1 Comparison between the mains characteristics of the CHAMP, GRACE and GOCE accelerometers Mission PM material PM mass Gap X Gap Y,Z Vd Vp Ge Acceleration sensitivity Y,Z Position sensitivity Y,Z Measurement range Y,Z Specified resolution Y,Z Specified freq. bandwidth
CHAMP TA6V 0.072 60 75 5 20 1.8 104 0.4 104 <108 104 $ 0.1
Fig. 26.3 GOCE drag-free system performance and accelerometer noises estimation (# ESA/GOCE core team)
duration of the mission should lead to even better gravity field resolution.
26.4
Future Missions and Instrument Improvements
These recent satellite gravity missions have run successfully. The measured Earth’s gravity field provides the unique integral measurements of the mass distribution and redistribution in the complete Earth system that are indicators of phenomena such as ice sheet mass balances and isostatic adjustments, terrestrial water storage changes, sea level rises and ocean circulations, tectonic and seismologic activities. . . Today,
GRACE TA6V 0.072 60 175 5 10 1.7 105 0.33 5 105 1010 104 $ 0.1
GOCE PtRh10 0.320 32 299 7.6 7.5 9.8 107 1.7 6.5 106 2 1012 5 103 $ 0.1
Unit kg mm mm Vrms V ms2/V mV/nm ms2 ms2/Hz1/2 Hz
the continuity of these measurements is of peculiar importance in many scientific and societal fields and the operation of a constellation of several satellites flying in formation like GRACE is much preferred to solve the observed limitations induced by signal aliasing considering the satellite actual track associated with the Earth’s rotation and the space–time mass fluctuations. The better separation of the sources of mass redistribution is indeed of major importance as well as the integration of the space gravity observations with those of surface displacements in conjunction with a priori models. Thus, future missions are already considered as well as the improvement of the on-board inertial sensors in term of performance, easiness of its accommodation and data exploitation. Many avenues can be envisaged in different domains. The instrument exhibits a frequency domain depending of the proof mass electrostatic servo-control bandwidth. Provided data at 10 Hz frequency and even larger can be envisaged corresponding to less than 1 km satellite displacement along its orbit. Analogue electronics and in loop digital sampling and control laws can be managed in consequence. This will have nevertheless to be considered in regard to the satellite natural structural modes that may be excited by thermal effects, in particular in presence of eclipse, or by the drag. A better a priori calibration of the instrument is interesting for efficient comparison with other data and models: estimates of a priori sensitivity and bias can be envisaged. The selected anisotropic configuration with a parallelepiped mass allows to operate in laboratory the two ultra-sensitive axes in their flight conditions and to project on them the normal gravity with a calibrated inclination. The bias evaluation can
220
be done in free fall tests and the recent implementation of a catapult in the ZARM (Centre of Applied Space technology and Microgravity, at Bremen University) drop tower has been already exploited to observe space accelerometer outputs at the summit of the capsule parabolic motion. The thermal sensitivity of the instrument can also be reduced by selecting the test mass material with quite null coefficient of thermal expansion like gold coated silica or ULE (Ultra Low Expansion glass from Corning). The geometry of the electrostatic configuration is then very steady with fixed gaps between the mass and the electrodes; remains only the electronics sensitivity that can be reduced with selected components and controlled unit temperature. GRACE type instruments could be considered after demonstration of the resistance to launch vibrations. Much improved resolution can also be expected with low temperature instrument. Sensors operating at cryogenic temperature have been developed based on the superconducting magnetic suspension of the mass and its displacement detection by SQUID. The in-orbit operation is much more complex, it requires in addition to the magnetic system an electrostatic device to centre the mass and to damp its motion [13, 14]. Such concept does not benefit of the servo-control of the mass position and attitude. This leads to changes in the mass-instrument cage geometry susceptible of mass motion electrostatic disturbances. In opposite, the thermal limit of the resolution of the space electrostatic accelerometers can be overcome by simply operating the levitation in a cryogenic Dewar without changing the room temperature electronics and the sensor basic configuration. In liquid Helium temperature, radiometric and radiation pressure effects on the proof-mass are much smaller as well as Nyquist noise induced by the mass motion damping. By reducing the diameter of the gold wire used for the mass charge control, thermal noise can be limited to a few 1015 ms2Hz1/2 [15]. Photoelectric techniques with UV sources can also be used instead of gold wire. Then, the instrument limitation should come from the measurement pickup electronics driven by the ratio between the instrument full range and the expected resolution. So, electrostatic space gradiometer is a fine compromise in particular because the mission limitation should certainly be the residues in the measured gravity gradients of the effects of the satellite attitude and orbital motion. Furthermore, the mission duration
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is limited by the Helium consumption and so the Dewar size. As shown with the CHAMP, GRACE and GOCE instrument configuration, room temperature accelerometer can be optimised considering the mass motion disturbances, the capacitive position sensing resolution and back action, the performance of the pickup measurement electronics, the electrostatic actuator noise being rejected by the loop gains as in GOCE electronics configuration. Increasing the mass, the gaps and the accuracy and stability of the core geometry, suppressing the gold wire, reducing the full-range are possible improvements of sensor resolution. This is in particular possible when the satellite tracking is performed in drag-free mode: gravitational sensors are sufficient for the drag-free and pointing system. No acceleration measurements are needed relaxing some configuration constraints. Laser interferometric sensors can also be now envisaged for the mass positioning of electrostatic accelerometer limiting the sensing signal back action on the mass motion.
26.5
Conclusion and Perspective
The sensor core configuration of the electrostatic accelerometers of the CHAMP, GRACE and GOCE missions have been especially designed for space applications and so optimized in regard to the weak level of acceleration to sustain and measure on-board the satellites depending on the orbit altitude and the presence of drag compensation system or not. This configuration is rather robust, simple to operate inorbit and nevertheless compatible with specific ground tests. The first in-flight results obtained with the GOCE instrument and the long series of data from the CHAMP and GRACE mission have definitively demonstrated the capability of such an instrument for gravity missions which should now operate continuously to the benefit of the Earth mass transportation survey interesting many disciplines like solid Earth physics, hydrology, oceanography, sea level and geodesy, climate change. . . By exhibiting resolution lower than the pico-g, these results will help to consider even better instruments to be proposed for future Earth gravity missions but also planet exploration or space fundamental physics experiments. Moon or Mars gradiometry missions can be considered with smaller and lighter configuration of the sensor to be
26
CHAMP, GRACE, GOCE Instruments and Beyond
compatible with interplanetary spacecraft [16]. Aeronomy missions are also envisaged for Mars [17] and other planets like Jupiter and its moons. Fundamental physics experiments aiming at the test of the gravity law, the research of new interactions or at the direct observation of gravity waves, demand measurements of very weak accelerations that can only be performed in a very quiet and steady environment that could be provided by dedicated satellites. This is the case of the LISA mission which envisages the triangle formation of three drag-free satellites in heliocentric orbit [18]. The test of the equivalence principle with an accuracy of 1015 is also under development in the frame of the MICROSCOPE mission [19]. The satellite payload is composed of two similar differential accelerometers including each two test masses: the concept and the technology are very similar to the GOCE accelerometer in spite of the cylindrical configuration, well suited for implementation of concentric sensors [20]. This mission will also exploit a drag-free satellite to be launched in 2014 and demonstrate the possibility of femto-g acceleration measurements. Long term stability and corrected bias are the major requirements of the accelerometer for the ODYSSEY mission, recently proposed for solar system navigation [21]. This demonstrates that beside the present developments of cold atom interferometers, also proposed for these future applications [22], mature technology instrument can be sufficient as a first step in the field.
References Rummel R, Colombo OL (1985) Gravity field determination from satellite gradiometry. Bull Geod 59:233–246 Touboul P, Bernard A, Barlier F, Berger C (1991) Air drag effect on gradiometer measurements. Manuscripta Geodaetica 16 (2):73–91 Silvestrin P, Bernard A, Touboul P, Foulon B, Gay M, Le Clerc GM (1994) Development of ultra-sensitive spaceborne accelerometers, Preparing for the future, ESA, Vol. 4, N 2, McPherson KM, Nati M, Touboul P, Sch€ utte A, Sablon G (1999) A summary of the quasi-steady acceleration environment on-board STS-94 (MSL-1). AIAA, Reno
221 Reigber C (1995) CHAMP a challenging micro-satellite payload for geophysical research and application, GFZ Final Report, Postdam Germany Tapley B, Bettadpur S, Ries J, Thompson M, Watkins M (2004) GRACE measurements of mass variability in the earth system. Science 305(5683):503–505 Gravity Field and Steady-State Ocean Circulation Mission, ESA SP-1233(1), ESA 1999 http://news.bbc.co.uk/go/pr/fr/-/2/hi/science/nature/8408957.stm, 10:06 GMT, 24 December 2009 Touboul P, Foulon B (1998) Space accelerometer development and drop tower experiments. Space Forum 4: 145–165 Van Helleputte T, Doornbos E and Visser P, (2009) CHAMP and GRACE accelerometer calibration by GPS-based orbit determination. Adv Space Res 43 Flury J, Bettadpur S and Tapley BD (2008) Precise accelerometry on board the GRACE gravity field satellite mission. Adv Space Res 42 Allasio A, Muzi D, Vinai B, Cesare S, Catastini G, Bard M and Marque JP (2009) GOCE: space technology for the reference Earth gravity field determination, EUCASS 2009, Versailles Moody MV, Paik HJ, Canavan ER (2002) Three-axis superconducting gravity gradiometer for sensitive gravity experiments. Rev Sci Instrum 73:3957 Sumner TJ et al (2007) STEP (satellite test of the equivalence principle). Adv Space Res 39:254–258 Lafargue L, Rodrigues M, Touboul P (2002) Towards low temperature electrostatic accelerometry. Review of Scientific Instrument 73(1):196–202 Foulon B, Christophe B and Marque JP (2009) GREMLUN: a miniaturized gravity gradiometer for planetary and small bodies exploration, IAC-08-A.3.5.9, Glasgow Chassefie`re E (2004) Dynamo: a Mars upper atmosphere package for investigating solar wind interaction and escape processes and mapping martian fields. Adv Space Res 33:2228–2235 Merkowitz S et al. (2009) ESA technology status summary LISA-MSE-RP-0001, issue1.0 Touboul P (2009) The MICROSCOPE mission and its uncertainty analysis. Space Science Reviews, Vol. 148, Issues 1–4 Chhun R, Hudson D, Flinoise P, Rodrigues M, Touboul P, Foulon B (2007) Equivalence principle test with MICROSCOPE: laboratory and engineering models preliminary results for evaluation of performance. Acta Astronautica 60 (10–11):873–879 Christophe B et al (2009) Odyssey: a solar system mission. Exp Astron 23:529–547 Wolf P et al (2009) Quantum physics exploring gravity in the outer system: SAGAS. Exp Astron 23:651–687
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The Future of the Satellite Gravimetry After the GOCE Mission
27
P. Silvestrin, M. Aguirre, L. Massotti, B. Leone, S. Cesare, M. Kern, and R. Haagmans
Abstract
Launched on March 17th 2009 from the Plesetsk Cosmodrome (Northern Russia), GOCE maps the Earth’s gravity field with unprecedented accuracy and resolution and will be of benefit for many branches of Earth science. This paper gives an overview of the European Space Agency’s (ESA) recent technical developments and activities going beyond the GOCE mission and its technology. It describes the outcome of the recent Laser SST concept studies, the Laser metrology concept and the objectives of the ongoing parallel Next Generation Gravimetry Mission studies, together with an overview on the latest technology development studies on atomic clocks and atom interferometry for possible future gravity sensing.
27.1
Introduction: ESA Studies for Gravity Sensing
ESA has a long tradition in supporting activities in relation to gravity field research and technical development. In recent years a number of studies have been initiated to establish scientific mission requirements (Enabling Observation Techniques 2004), to select and develop appropriate measurement techniques and metrology concepts (Laser Doppler 2005; Laser Interferometry 2008; System Support 2008), to advance in modelling and monitoring individual sources of mass distribution (Monitoring and Modelling 2008) and,
P. Silvestrin M. Aguirre L. Massotti B. Leone M. Kern R. Haagmans (*) ESA-ESTEC, Keplerlaan 1, 2201, AZ Noordwijk ZH, The Netherlands e-mail:
[email protected] S. Cesare Thales Alenia Space Italia, Strada Antica di Collegno 253, 10146, Turin, Italy
most recently, to define the system scenarios for a “Next-Generation Gravimetry Mission” (NGGM) (Assessment of a Next Generation Gravity Mission to Monitor the Variations of the Earth’s gravity field: ongoing activity). Based on the scientific requirements established in these study reports and workshop reports (see Koop and Rummel 2007), a NGGM needs to satisfy the requirements of geophysical applications (solid Earth science, glaciology, hydrology, oceanography, atmosphere circulation, etc.). It shall provide temporal variations of the Earth’s gravity field over a long time span (>6 years) with high spatial resolution (comparable to that provided by GOCE) and high temporal resolution (weekly or better, so to reduce the level of aliasing of the high frequency phenomena found in the time series of the Earth’s gravity field variation provided by GRACE), and it must improve the separability of the observed geophysical signals (Koop and Rummel 2007). In one of the preparatory studies carried out for ESA by Thales Alenia Space I (TAS-I) (Laser Doppler 2005), a resolution of
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0.1 mm/year of the geoid height variation rate at spherical harmonic degree ¼ 200 (or ~100 km spatial resolution) was preliminarily identified satisfying the most stringent geophysical performance requirement.
27.2
FD2
D2 g2
Satellite 1
FD1
D1
Δd = ΔdG + ΔdD
g1
Earth
Measurement Techniques and Mission Scenario
Fig. 27.1 Principle of the LL-SST technique
The monitoring of the temporal variations of the Earth’s gravity field over a long period of time requires flying at an altitude higher than that of GOCE, in order to compensate the drag forces with an affordable amount of propellant. Since the gravitational potential U of degree rapidly decreases with the orbit radius r, i.e. U‘ / r ð‘þ1Þ ;
d
ZJ2000 S2
ZO 1
XO 1 YO 1
S1
(27.1)
it would be necessary to increase proportionally the gradiometer baseline (and/or the sensitivity of the accelerometers) in order to maintain the same signalto-noise ratio (see Cesare et al. 2009). Considering also that the signal of the temporal variations of the gravity field is at least one order of magnitude weaker than that of the static geopotential, for altitudes above ~300 km at the moment the most consolidated technique to measure the Earth’s gravity field is the “LowLow Satellite-to-Satellite Tracking” (LL-SST), which exploits the satellites themselves as the “proof masses” immersed in the Earth’s gravity field, as in the GRACE mission. Assuming a loose formation of two satellites, the distance variation between their centres of mass (Dd, produced by both gravitational and non-gravitational forces together) is gauged via length metrology. The drag accelerations (D1, D2) experienced by the two satellites are separately measured by means of accelerometers. From of Dd€D ¼ D1 D2 , the non-gravitational component RR the distance variation is obtained ðDdD ¼ Dd€D Þ and subtracted from Dd to isolate the component (DdG) produced by Earth’s gravity field only (see Fig. 27.1). Thanks to the separation between the satellites, this “measurement instrument”, which can be regarded as a kind of one-dimensional gradiometer with a very long baseline, has a higher sensitivity for the phenomena being targeted than a gradiometer embarked on a single satellite.
sun
i
γ = 90°
XJ2000
line of nodes
YJ2000
orbit
Fig. 27.2 Basic mission scenario
The simplest scenario implementing LL-SST consists of a loose formation of two co-orbiting satellites. In particular, the main parameters of the mission scenario defined in Laser Doppler study (2005) are (see Fig. 27.2): • Circular orbit with mean altitude of 325 km • Orbit inclination is 96.78 (sun-synchronous) • Local time of the ascending node: 6 am (dawn-dusk orbit) or 6 pm (dusk-dawn orbit) • Inter-satellite distance d ¼ 10 km These parameters and the requirements on the fundamental observables utilized for the determination of the Earth’s gravity field (Dd and Dd€D , as shown in Fig. 27.3), have been so chosen to fulfill the requirements on the geoid variation rate (0.1 mm/ year) at 100 km spatial resolution. They have been derived using analytical models and numerical simulations of the gravity field determination through the LL-SST technique, with the support of geodesy institutes (Laser Doppler 2005; Cesare et al. 2006).
27
The Future of the Satellite Gravimetry After the GOCE Mission
Fig. 27.3 Requirements on Dd€D (above) and inter-satellite (or “relative”) Dd (below) measurement error spectral density. These requirements are strictly applicable to the frequency
Although this basic mission scenario still does not meet all the objectives of a NGGM, in particular with respect to the improvement of the temporal resolution and the separability of the geophysical signals, it can provide the “building block” for the realization of more complex scenarios that can potentially fulfill these objectives. For instance, a dense and uniform coverage of the Earth surface that allows generating gravity field solutions in less than 1 week can be obtained by flying two pairs of co-orbiting satellites
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range [1100 mHz], the identified measurement bandwidth (MBW) of the NGGM
in two circular orbits with altitude at 312 km and with 90 and 62.7 inclination respectively, as in Bender et al. (2008).
27.2.1 Laser Metrology System The metrology system designed for the NGGM (Laser Doppler 2005; Laser Interferometry 2008) is based on a Michelson-type heterodyne laser interferometer (see
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Fig. 27.4 Functional scheme of the laser metrology
Fig. 27.4). A measurement laser beam produced by a frequency-stabilized Nd:YAG source (wavelength ¼ 1,064 nm, output optical power ¼ 750 mW) is emitted by satellite 1 (follower) towards satellite 2 (leader). Here it is retro-reflected by a corner-cube optical system towards satellite 1, where it is superimposed to a
reference laser beam generated by the same source but frequency shifted relatively to the measurement beam. The measurement beam is amplitude modulated in on-off mode with a period corresponding to the roundtrip time to avoid the occurrence of spurious signals and non-linearity caused by the unbalance between the
27
The Future of the Satellite Gravimetry After the GOCE Mission
optical power of the strong reference beam and of the weak returning measurement beam. The distance variation is obtained from the phase variation of the interference signal formed by the measurement and the reference laser beams. A breadboard of such an interferometer has been realized (as in Fig. 27.5) and subject to functional and performance tests up to ~90 m distance. The test results shown in Fig. 27.6 prove that the requirement (as depicted in Fig. 27.3) can be met in the MBW. The laser metrology system is completed by: • An Angle Metrology for measuring the angles of the two satellites relative to the laser beam
Fig. 27.5 Breadboard of the laser interferometer Fig. 27.6 Results of the performance test (distance measurement error spectral density)
227
• A Beam Steering Mechanism (BSM), i.e. a device in charge of the acquisition and maintenance of the optical link between the satellites and of the laser beam fine pointing during the measurement phase • A Lateral Displacement Metrology for measuring the lateral displacement of the satellite 2 relative to the beam sent by the satellite 1 and driving in closed loop the BSM The Angle Metrology is merged with the Lateral Displacement Metrology and is realized by three small telescopes focusing the collected light on three Position Sensing Detectors (PSDs). Each PSD measures the position and the energy of the laser beam spot focused by the optics on the detector plane. The orientation and the lateral shift of satellite 2 relative to the laser beam are derived from the spot position and from the optical power measured by the three PSDs respectively. Breadboards of the Angle and Lateral Displacement Metrology and of the BSM have been realized (Fig. 27.7) and tested by TAS-I (Laser Doppler 2005; System Support 2008), including a test of the whole laser beam pointing system (BSM driven in closedloop by the Lateral Metrology). The elements of the laser metrology system described above are arranged on an optical bench, shared with the accelerometers utilized for the drag accelerations measurement (as shown in Fig. 27.8).
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satellite 2 can take its role, after the two satellites exchange positions.
27.2.2 Control Requirements
Fig. 27.7 Breadboard of the BSM
Angle Metrology
Beam Steering Mechanism Retro Reflectors
Interferometer core Accelerometers Angle/Lateral Metrology
Fig. 27.8 Configuration of the optical bench installed on each satellite
Two accelerometers, which can be similar to those of GOCE, are used on each satellite, arranged symmetrically about the centre of mass (COM). So doing, it is possible to measure the drag acceleration of the satellite COM, leaving the space in proximity of the COM free for the accommodation of the laser retro-reflector. The same optical bench is replicated on both satellites even if the laser is emitted only by satellite 1. Then in case of failure of the interferometer on satellite 1,
A complete assessment of the control requirements was done in Laser Doppler (2005), and revised and updated along the following studies: the control boundaries were conceived for the baseline configuration of two satellites chasing each other in a drag-free environment and flying in loose formation flying. The modeled metrology is the consolidated LL-SST technique, enhanced by the BSM. More in detail, in order to operate the accelerometers at the performance level required for the NGGM, each satellite must be endowed with a drag-free control system to reduce the level of the non-gravitational 2 accelerations below 106m/s and below a spectral 2 pffiffiffiffiffiffi 8 density level of 10 m s Hz between 1 and 10 mHz, along each axis. The drag compensation is realized using ion thrusters in the same way as in GOCE. The formation control is in charge of keeping satellite 2 in the “control box” established by the working range of the optical metrology system: DdX ¼ 500 m (with respect to the nominal distance, d ¼ 10 km); DdY, DdZ ¼ 50 m along the Y, Z axes of the satellite 1 Local Orbital Reference Frame. The challenge of the formation control design for this mission consists in keeping the relative motion within these boundaries without interfering with the scientific measurements, which require that the satellites must be “free” to move under the effect of the gravity field over time scales of 1,000 s, and also without spoiling the drag-free environment (formation control accelerations must fulfil the drag-compensation requirements too), while minimizing thrusters use in terms of dynamic range, propellant consumption. Control architecture and recommended actuator technology is elaborated in Cesare et al. (2009).
27.3
Parallel Studies
27.3.1 NGGM All the illustrated studies have constituted a solid background for the preparation of a new pre-Phase A study, so called “Assessment of a Next Generation
27
The Future of the Satellite Gravimetry After the GOCE Mission
Gravity Mission to Monitor the Variations of the Earth’s Gravity Field” (ESA ITT 1-5914/09/NL/CT). Thus, the ESA Future Missions Division has started two parallel studies led by two competitive consortia (Astrium D and ThalesAlenia Space I) in the second quarter of 2009, with the main objective of establishing mission architecture aimed at the optimal recovery of the Earth variable gravity field, through satellite-to-satellite tracking observation techniques.
27.3.2 Mass Transport Study The “Mass transport study” (Monitoring and Modelling Individual 2008) aimed at monitoring and modeling individual sources of mass distribution and transport in the Earth mass model (11 years, 0.5 0.5) was compiled including mass fields stemming from atmosphere, oceans, continental water, ice (ice sheets and glaciers) and the solid Earth. This model was used to optimize the ability to observe changes in mass over the continents and oceans. Detailed and realistic close-loop simulations were performed to study separability and aliasing characteristics inherent in different satellite mission concepts. One of the conclusions was that temporal aliasing is intrinsic to observing gravity field changes by satellites, but e.g. leads to relatively smaller distortions for hydrology than for oceanography. Furthermore, the combination of LL-SST and orbit observables does not allow the precise determination of the spherical harmonic degree 1 terms or geo-centre variations. Finally, flying more than one pair of LLSST satellites can significantly reduce the gravity field retrieval errors.
27.4
Parallel Technology Developments
27.4.1 Atomic Clocks (ESA studies) The European Space Agency is currently carrying on two parallel activities on atomic clocks. The ACES (Atomic Clock Ensemble in Space) mission on the International Space Station fosters research on coldatom sensor technology for space, in view of applications in different domains, e.g. fundamental physics tests (violation of special relativity, search for drifts of fundamental constants) and relativistic
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geodesy (till 10 cm accuracy) with a time and frequency metrology at 1016 stability and accuracy level. ACES will be ready for launch in 2013. Secondarily, a Pre-phase A study of an atomic clock ensemble in space based on the optical transitions of strontium and ytterbium atoms has started in 2006. The so-called SOC (Space Optical Clocks, ESA-AO-2004-100) will take advantage of the ACES heritage and will push stability and accuracy of atomic frequency standards down to the 1018 regime.
27.4.2 Atom Interferometers The European Space Agency has been funding a number of technology development activities and studies in the area of atom interferometry in view of the everincreasing requirements for future inertial satellite platforms. Recently, a study has been initiated to explore the “Applications and Implementations of Atom-Based Inertial Quantum Sensors” in future space missions, in the fields of Space Science, Earth Observation, Fundamental Physics, Microgravity, and Navigation. As part of this ongoing study, the fundamental limits of atom interferometry were translated into the expected performance of a realistic gravity gradiometer payload for a future gravity mission. Gravity gradients can be measured by comparing the time evolution of two clouds of atoms (106 to 108 or more atoms), suitably cooled and prepared by means of lasers as well as other (time varying) electromagnetic fields, and separated by a given baseline. The time evolution of each cloud is measured by sequentially inducing suitable atomic transitions by means of lasers. This constitutes the atom interferometric measurement (see Peters et al. 1999). Comparison of the two atomic clouds is also achieved optically by ensuring the optical phase locking of the interferometric measurements effected on the two clouds. A major advantage compared to a macroscopic gravity gradiometer, such as the one onboard GOCE, is that the absolute local value of gravity is measured for each atomic cloud. Therefore, the atom interferometry gravity gradiometer does not suffer from drift but exhibits constant performance across the measurement frequency band. The acceleration sensitivity of an atom interferometer increases, amongst others, with the square of the interrogation time. This time is effectively limited on
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the ground to the achievable free fall time of an atomic cloud, which amounts to a few 100 ms in a typical vacuum chamber. By contrast, space operation, characterized by the intrinsic microgravity environment, would enable interrogation times of several tens of seconds leading to a sensitivity improvement of four orders of magnitude. Assuming a baseline of 1 m, this would amount to sensitivities of the order of 1 mE/Hz in space. This is one order of magnitude better than the GOCE specifications for measurements at frequencies above ~1 mHz, and, thanks to the absence of drift, well below, by several orders of magnitude, for frequencies below 1 mHz. The above estimate, however, does not yet represent a fundamental limit. Performance also increases with the magnitude of the probing laser wave vector and the signal-tonoise ratio, which is a function of the number of atoms in the cloud. Both these quantities can be effectively increased using quantum entanglement techniques. Although these still need to be demonstrated, the potential sensitivity for the same 1 m baseline atom interferometer gravity gradiometer would then be in the order of 1mE/Hz. In order to become suitable for future space mission, atom interferometry technology requires further development in the area of miniaturization and integration of both optical and laser components, as well as vacuum chambers. Furthermore, space worthiness also needs testing. These developments and investigations are currently underway in a number of European institutes (see Schmidt et al. 2009; Stern et al. 2009).
27.5
Conclusions
Since 2003 a number of studies have been initiated by ESA focusing on various technological, scientific and mission aspects of a potential future gravity field mission. It is important that the technology, not only related to the payload but also to satellite and/or constellation control, reaches the right readiness level. In addition, tools need to be developed to allow full mission performance analysis. Besides LL-SST other interesting technologies are under development but at different levels of maturity. The road towards availability of these techniques for future missions is still long and full of challenges.
Acknowledgements The authors gratefully acknowledge Prof. Achim Peters, Head, and Malte Schmidt, of the Humboldt University of Berlin, for their expert advice on atom interferometry.
References Bender PL, Wiese DN, and Nerem RS (2008) “A Possible DualGRACE Mission with 90 Degree and 63 Degree Inclination Orbits”, In: Proceedings of the 3 rd International Symposium on Formation Flying, Missions and Technologies, Noordwijk (NL) Cesare S, Sechi G, Bonino L, Sabadini R, Marotta AM, Migliaccio F, Reguzzoni M, Sanso` F, Milani A, Pisani M, Leone B, Silvestrin P (2006) “Satellite-to-satellite laser tracking mission for gravity field measurement”. In: Proceedings of the 1st International Symposium of IGFS, 28 Aug–1 Sept. 2006, Istanbul, Turkey. Harita Dergisi, Special Issue 18, pp 205–210 Cesare S, Allasio A, Aguirre M, Leone B, Massotti L, Muzi D, and Silvestrin P (2009) The Measurement of Earth’s Gravity Field after the GOCE Mission. In: Proceedings of 60th International Astronautical Congress, Daejeon, Korea, 2009, IAC-09.B1.2.7 Enabling Observation Techniques for Future Solid Earth Missions (2004) ESA Contract No: 16668/02/NL/MM, Final Report, Issue 2, 15 July 2004 Koop R, Rummel R (2007) The Future of Satellite Gravimetry, Final Report of the Future Gravity Mission Workshop, 12–13 April 2007 ESA/ESTEC, Noordwiik, Netherlands Laser Doppler Interferometry Mission for determination of the Earth’s Gravity Field (2005), ESA Contract 18456/04/NL/ CP, Final Report, Issue 1, 19 December 2005 Laser Interferometry High Precision Tracking for LEO (2008), ESA Contract No. 0512/06/NL/IA, Final Report, July 2008 Monitoring and Modelling Individual Sources of Mass Distribution and Transport in the Earth System by Means of Satellites (2008), ESA Contract No. 20406/06/NL/HE, Final report, November 2008 Peters A, Chung KY, Chu S (1999) Measurement of gravitational acceleration by dropping atoms. Nature 400:849–852 Schmidt M, Senger A, Gorkhover T, Grede S, Kovalchuk E, and Peters A (2009) “A Mobile Atom Interferometer for High Precision Measurements of Local Gravity”, In: Frequency Standards and Metrology – Proceedings of the 7th Symposium pp 511–516 Stern G, Battelier B, Geiger R, Varoquaux G, Villing A, Moron F, Carraz O, Zahzam N, Bidel Y, Chaibi W, Pereira Dos Santos F, Bresson A, Landragin A, Bouyer P (2009) Lightpulse atom interferometry in microgravity. Eur Phys J D 53:353–357 System Support to Laser Interferometry Tracking Technology Development for Gravity Field Monitoring (2008), ESA Contract No. 20846/07/NL/FF, Final report, September 2008
Future Satellite Gravity Field Missions: Feasibility Study of Post-Newtonian Method
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Abstract
Modern satellite gravity field recovery missions use accelerometric, intersatellite tracking or gradiometric observables for deducing gravity field related data. In this study an alternative observable type for gravity field recovery, the relativistic frequency shift, is investigated. As Einstein stated in his general theory of relativity, gravity can be considered as attribute of space-time. In this view mass alters the geometric shape of the metric tensor. Moreover mass, respectively gravity, has effects on electromagnetic wave propagation [Einstein (Annalen der Physik 35:898–908 1911)]. Although these relativistic effects are quite small and difficult to measure, with upcoming atomic clocks which have sufficient accuracy and short-term stability it will be possible to derive meaningful gravity related information. Since relativistic effects are used this method is called Post-Newtonian method. The main target of this paper is to demonstrate the validity of the derived relativistic equations. The scientific quality of the relativistic frequency shift observed by means of highly accurate atomic clocks is investigated. In our basic scenario a low earth orbit (LEO) sends an electromagnetic wave to a receiver. The reference station determines the frequency shift of the signal, which is connected to the time dilatation between the atomic clock of the satellite and an identical atomic clock nearby the receiver. A simplified, mathematical model for numerical simulations of this configuration is presented. The effect of different error sources are investigated by numerical closed-loop simulations. Thus, the performance requirements of atomic clocks, position and velocity determination and limiting factors for deducing earth’s gravity field can be derived.
28.1
R. Mayrhofer (*) R. Pail Institute of Navigation and Satellite Geodesy, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria e-mail:
[email protected]
Introduction
In this study the principle of the Post-Newtonian method for earth gravity field determination is presented. The observable for gravity field reconstruction is the frequency shift of an electromagnetic signal transmitted from a satellite to a receiver station. This frequency shift is caused by relativistic time dilatation
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_28, # Springer-Verlag Berlin Heidelberg 2012
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which is related to the gravity potential. By numerical simulations the spatial and spectral performance of a LEO satellite mission equipped with atomic clocks, which will be available within the next tow decades, is investigated. There have been done some studies which are related to future satellite gravity field missions and general relativistic effects. There is for example the Einstein Gravity Explorer mission proposal (Schiller et al. 2009) in which testing methods of relativistic effects and physical constants based on atomic clock measurements are investigated. M€ uller et al. (2007) explored the impact of relativity on various geodetic topics like the geoid, reference systems, geodynamics, Global Positioning System (GPS), Satellite Laser Ranging (SLR), and Very Long Base Interferometry (VLBI). Gulkett (2003) investigated relativistic effects on GPS and LEO and showed how to implement them correctly within a relativistic framework. The IAU already included relativistic effects for frame transformations (Soffel et al. 2003). For being able to realise a satellite mission as proposed in this paper, atomic clocks with sufficient quality will be needed. Actual atomic clocks achieve a short term stability of 1016 s (between two measurement epochs) on earth (Schiller 2007) and are expected to achieve 1018 s within the next 15 years. According to Cacciapuoti (2006), actual space-borne atomic clocks achieve an accuracy of 1015 s. Thus, a satellite mission with an atomic clock with 1018 s stability should be possible within the next 30 years. In this study, the equations for the Post-Newtonian method are derived in Chaps. 28.2 and 28.3. In Chap. 28.4 the simulation setup is described in more detail. Chapter 28.5 shows the simulation results, a performance analysis, and the analysis of the spectral and spatial error behaviour. Finally a conclusion and outlook is given in Chaps. 28.6 and 28.7.
28.2
Relativistic Time Dilatation
The metric tensor gmn describes the curvature of spacetime. Thus, it can be used for deducing a description of the relativistic frequency shift. The line element ds can be described by ds2 ¼ gmn ðxÞdxm dxn
(28.1)
Here Einstein’s tensor convention has been applied (Einstein 1916). Double upper and lower indices describe a summation. The gradient dxk ¼ ½c dt dx1 dx2 dx3 T of the scalar vector field x ¼ ðxk Þ contains position and time information. c is the speed of light in vacuum, dt the time element of a chosen time system and dxi describe the three dimensional coordinate elements. The metric tensor gmn ðxÞ is a function of xk ðm; n; k ¼ ½0; 1; 2; 3Þ, which means that the curvature of space-time depends on time and position. A solution of Einstein’s field equations delivers the elements of the metric tensor. A series expansion representation of the tensor elements is (Soffel et al. 2000) 2 FðxÞ 2 F2 ðxÞ þ Oðc5 Þ c2 c4 4 Fi ðxÞ g0i ¼ þ Oðc5 Þ c3 2 FðxÞ 2 F2 ðxÞ þ Oðc6 Þ gij ¼ dij 1 þ c2 c4 (28.2)
g00 ¼ 1 þ
where FðxÞ is the gravity potential and i; j ¼ ½1; 2; 3. The earth’s static gravity field potential is usually expressed by a spherical harmonic series expansion (Heiskanen and Moritz 1967): FE ðr; y; lÞ ¼
1 lþ1 GM X R R l¼0 r
l X
ðClm cos ml þ Slm sin mlÞ Plm ðcos yÞ
m¼0
(28.3) Here r, y and l are spherical coordinates, R is the earth’s reference radius, GM the gravitational constant times mass of the earth, l and m are the degree and order of the fully normalized spherical coefficients Clm ; Slm , and Plm ðcos yÞ represents the fully normalized Legendre function. Equation (28.3) describes the functional model for setting up the design matrix for least squares adjustment in our simulation environment. The spherical coefficients Clm ; Slm are the system parameters, while the gravity potential F, which is derived from relativistic frequency shifts is the observable of our system.
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Future Satellite Gravity Field Missions: Feasibility Study of Post-Newtonian Method
The local line element dsclock of a clock moving within a gravity field affects the displayed time dtclock and frequency uclock of the clock. dtclock ¼
1 uclock
¼
dsclock 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ gmn ðxÞdxm dxn (28.4) c c
x ¼ ðxk Þ and dxk are related to the coordinates of the clock. The coordinate elements of a moving clock can be described by dxk ¼ ½ c dt
v1 dt
v2 dt
v3 dt
(28.5)
Here, vi ¼ vi ðxÞ denotes the velocity of the clock and is related to its coordinates, too. Merging (28.2), (28.4) and (28.5) and omitting elements smaller then c3 leads to a description of the inherent time of a moving body: 2 FðxÞ v2 ðxÞ dt2 dt2 ¼ 1 þ þ c2 c2
(28.6)
v ðxÞ is the local scalar velocity of the clock. The asterisk ‘*’ is used for underlining that v is scalar and preventing to mix it up with the velocities from (28.5). dt represents a virtual time of an non-moving, gravityfree (inertial) body located at infinite distance.
28.3
Functional Model
A LEO satellite transmits an electromagnetic signal via microwave link to a receiver station. This receiver station could be a geostationary satellite or a reference station located on earth’s surface. By comparing the local frequency of the transmitted signal with the local frequency of the received signal the time dilation between receiver and transmitter is defined. By using (28.6) and again omitting elements smaller then c3 the ratio of receiver and transmitter frequency is obtained by uR dtT ¼ uT dtR vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u1 þ 2F2 T þ vT2 c c t ¼ þ Oðc4 Þ v2R 2FR 1 þ c2 þ c2
DuRT ¼
(28.7)
The lower indices R and T describe a receiver, respectively a transmitter related variable. The gravity
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potentials are related to the receiver or transmitter position FR ¼ FðxkR Þ; FT ¼ FðxkT Þ. This equation is used for synthesizing relativistic frequency shifts in our simulation environment. Moreover it is the fundamental equation for the Post-Newtonian approach. As the relativistic time dilation, which is not modeled by the Newtonian framework, is taken into account, the nomenclature ‘Post-Newtonian’ has bee chosen to describe this method. The gravity potential FT deduced from the frequency shift DuRT is the prime observable for the Post-Newtonian method. A general description of (28.7) is DuRT
uR dtT ¼ ¼ ¼ uT dtR
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gmn ðxT ÞdxmT dxnT gmn ðxR ÞdxmR dxnR
(28.8)
By combining (28.8) with the description of the metric tensor elements in (28.2), (28.7) can be achieved. Based on this function the gravity potential FT at the satellite position can be calculated from a measured frequency shift DuRT . After some reformulations a function, which is used in our simulation environment as functional model for deducing the gravity potential at transmitter position from the frequency shift DuRT , is achieved: 2 v2T c 2 FT ¼ DuRT OR 2 þ 1 c 2
(28.9)
Here a support variable OR has been introduced. It contains all position, velocity and gravity potential information of the receiver station: OR ¼ 1 þ
2 FR v2R þ 2 : c2 c
Equations (28.9) and (28.10) describe two relativistic effects. The first one is time dilatation caused by relative movement of receiver and transmitter, the second one time dilatation caused by the gravity potential difference between transmitter and receiver location. The gravity potential FðxÞ is composed of all occurring gravity potentials. As a first approximation in our simulations, all non-earth gravity potentials and tide signals have been neglected. Beside the special relativistic and general relativistic effects the Doppler shift is the third large effect which influences the
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frequency observations of the receiver station. As a satellite in a LEO achieves large velocities, the radial velocity between receiver and transmitter cause a Doppler frequency shift which has to be modeled. The Doppler shift (Doppler 1842) is defined by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c vTR uR ¼ uT c þ vTR
(28.10)
where vTR is the scalar radial velocity between transmitter and receiver. The Doppler shift is applied to the measured frequency at the receiver station. It has to be mentioned that as we are working in a relativistic framework, the radial velocity has to be derived in a coordinate system located at the center of the receiver station.
28.4
Simulation Setup
Figure 28.1 shows the schematic set-up of our simulation software. In a configuration file the orbit properties, computation switches and observation noise types are defined. The software computes based on this configuration, the orbit positions and the frequency related effects by using (28.7) and (28.10). In our simulation environment (28.9) has been used to calculate the gravity potential at the satellite position from the synthesized frequency shifts. The simulation environment has been designed to determine the influence of data noise on the reproduced gravity field model. Therefore it is possible to manually add realistic noise on frequency and velocity measurements. Following effects on the frequency shift observable have been simulated:
• Special relativistic frequency shift caused by relative velocity of receiver and transmitter clock (28.7). • General relativistic frequency shift caused by relative potential difference at receiver and transmitter clock positions (28.7). • Doppler Effect caused by relative radial velocity of receiver and transmitter clock (28.10). For every effect a realistic stochastic noise signal was added on noise-free frequency and velocity measurements. Shin et al. (2008) suppose a coloured noise for H-maser clocks with increasing amplitudes at low and high frequencies and linear behaviour in-between. Figure 28.2 shows the power spectrum density function of the atomic clock noise with amplitude 1017 and 1018 s generated for our simulations based on this information. For the velocity error, white noise has been assumed. In the frame of gravity field adjustment, the stochastic models, which define the metric of the normal equation system, have been consistently incorporated by correspondingly designed digital filters applied to both, the observation time series and the columns of the design matrix (Schuh, 2001). A nearly polar (89.5 inclination), circular repeat orbit with 25 days and 403 cycles at 300 km mean height with 10 s sampling interval has been chosen for all simulations.
28.5
28.5.1 Performance Analysis The main observable of the Post-Newtonian method is the frequency shift. Equation (28.7) shows that beside the atomic clock noise the velocity determination noise
Orbit Definition
Simulation Definition
Fig. 28.1 Schematic presentation of the simulation environment used in this study
Simulation Results
Orbit Synthesis
Signal Synthesis
Noise Definition
Gravity Field Coefficients equation (28.3) least squares adjustment
Gravity Field Analysis equation (28.9)
equation (28.7), (28.10)
Potential Reconstruction
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Future Satellite Gravity Field Missions: Feasibility Study of Post-Newtonian Method
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Fig. 28.2 Smoothed noise amplitude spectrum of coloured clock noise with amplitude 1017 s and 1018 s
Fig. 28.3 Degree error median plot of simulations with frequency noise with 1017 s and 1018 s amplitude and velocity noise with 105 m/s and 106 m/s white noise
of the transmitter is a stochastic variable, too and is the second limiting factor for the Post-Newtonian method. Based on the frequency shift the transmitter potential can be determined by using (28.10). Thus, first a noise free data set has been defined, and the calculation method has been verified by using closed-loop computations. Next, realistic coloured clock noise (Fig. 28.2) with amplitudes of 1016 up to 1018 s has been applied to the synthetic observations. Finally, signals including white velocity noise at amplitudes of 104 to 106 m/s have been used instead. Figure 28.3 shows the degree error median of the by least squares adjustment reproduced gravity field
coefficients up to d/o (degree and order) 150. It can be seen that the velocity noise of 105 m/s has an effect on the gravity field reconstruction error which is comparable to 1017 s clock noise, while 106 m/s velocity noise has a similar influence as 1018 s clock noise. Moreover it can be seen that future atomic clocks (Cacciapuoti 2006; Schiller 2007) with an expected short term stability of 1018 s clock noise, and positioning precision of 106 m/s velocity noise it would be possible to deduce earth’s gravity field up do d/o 120. It has to be mentioned that following effects, which will in practice have additional contributions on the total error budget, have been neglected in our simulations:
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• • • • • •
Tidal and non-tidal temporal variable effects. Non-conservative potentials Relativistic effects on frame transformations. Non-uniformly rotating earth. Non-inertial potentials (satellite rotation). Receiver position related errors (position, velocity and the gravity potential at receiver position are assumed to be error-free). • Dispersive atmosphere related effects. However, it can be expected that the error terms included in this study are the dominant ones.
Fig. 28.4 Geoid height error of simulation with 1018 s clock noise applied on frequency observations. A homogeneous and isotropic error structure is provided
Fig. 28.5 Degree error median plot of Doppler velocity error with 104 m/s noise simulation. Compared to the simulations shown in Fig. 28.3 the influence of Doppler noise is negligibly small
R. Mayrhofer and R. Pail
28.5.2 Spatial and Spectral Error Structure As the gravity potential at the transmitter position, which is deduced from frequency measurements, is the observable for the least squares adjustment, the error structure of the recovered earth gravity field corresponds to the error structure of direct potential observations. Thus, in the case of white noise, the error amplitude increases with higher degree and order of deduced spherical coefficients and the slope of the degree error median is related to the chosen orbit height. Figure 28.4
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Future Satellite Gravity Field Missions: Feasibility Study of Post-Newtonian Method
shows the homogeneous and isotropic spatial error distribution of a simulation configuration with 1018 s clock noise. Simulations with white velocity noise superposed show similar spatial error structures.
28.5.3 Doppler Shift A simulation setup has been defined, where the Doppler shift has been applied on synthetic frequency observations based on noise-free data with which relativistic influences were modeled before. White noise with amplitude of 104 m/s has been superposed on the noise-free velocities (28.10). Figure 28.5 shows that the Doppler shift can be modeled very well even at high velocity noise amplitudes. Moreover, the influence on the recovered gravity field is negligibly small up to d/o 250. So the Doppler shift is no limiting factor. The reason for this is because compared to (28.9), the radial velocity in (28.10) is not scaled by c2 .
28.6
Conclusions
It has been shown that the Post-Newtonian method is a feasible method for reproducing earth’s gravity field for lower and medium frequencies, provided that the technological development of space-borne atomic clocks proceeds in the future. Additionally it has been shown that the derived equations are valid within the defined mission scenario. The two dominant error contributions of this method are the atomic clock noise and the satellite velocity error of the precise orbit determination. The Doppler-effect, which also influences the frequency measurements, can be modeled with sufficient precision, so its error does not leak into the recovered gravity coefficients. It has to be mentioned that the simulations done here should be seen as a concept study and some more realistic simulations will be done to provide more information about the behaviour of the method. A velocity determination precision up to 106 m/s and an atomic clock short term stability of 1018 s is required to resolve the gravity field up to d/o 120. The main advantage of this method is its homogeneous and isotropic spatial error structure.
28.7
237
Outlook
With upcoming atomic clocks below 1016 s short term stability and improving positioning and velocity determination methods, the presented method can be an additional piece for a global earth gravity field monitoring framework in a not too far future. Beside the single-satellite mission presented in this study, various satellite constellations and formations can be designed. In what extent satellite formations like Pendulum, Cartwheel or LISA-like lead to improved precision still has to be investigated by numerical simulations. There is no doubt that multi-satellite missions would underline the possible power of the Post-Newtonian method. One, two or three geostationary satellites could be used as reference stations. Additional rover satellites could be placed in different orbit types. A dense network of satellites could improve the time resolution, so the time variable gravity field could be optimally mapped. These satellites could be equipped with two or more RF-antennas, so one satellite could establish a connection to two or more other satellites, which would further increase the measurement density of the network. As the equations used in this study are strongly simplified, the influences of other effects have to be further investigated. First real orbits and non-conservative forces have to be modeled. Next, the influence of satellite rotation has to be investigated. Additional attitude information from star-tracker measurements and its noise behaviour have to be simulated. All observations and calculations have to be done in a relativistic framework. So the influences of frame transformations in this relativistic framework have to be investigated. All other effects listed in Chap. 28.5.1 will be further investigated in upcoming simulations. This will lead to a much more complex mathematical description which will be harder to linearize, but will also be closer to reality. The main target of upcoming simulations will be to set up a more realistic environment and to design multi-satellite missions that support the advantages of the Post-Newtonian method. Moreover there will be done simulations concerning the time variable gravity field. It will be investigated how different mission design affects the quality of the static and time variable gravity field and if there is
238
a possibility to deduce models with higher spatial and temporal resolution. Acknowledgements We would like to thank L. Vitushkin and an anonymous reviewer for their valuable comments which helped to improve the manuscript.
References Cacciapuoti L (2006) Atomic Clocks in Space. ESA-ESTEC (SCI-SP), Frascati Doppler C (1842) Ueber das farbige Licht der Doppelsterne und einiger anderer Gestirnen des Himmels. Abhandlungen der k. b€ ohm. Gesellschaft der Wissenshaften Folge V Band 2, In Commision bei Borrosch & Andre´, Prag € Einstein A (1911) Uber den Einfluß der Schwerkraft auf die Ausbreitung des Lichtes. Annalen der Physik 35, Verlag von Johann Ambrosius Barth, Leipzig pp 898–908 Einstein A (1916) Die Grundlage der allgemeinen Relativit€atstheorie. Annalen der Physik 49, Verlag von Johann Ambrosius Barth, Leipzig pp 769–821 Gulkett M (2003) Relativistic effects in GPS and LEO. Rapport for the Cand. Scient. degree, University of Copenhagen, Department of Geophysics, The Niels Bohr Institute for Physics, Astronomy and Geophysics, Denmark Heiskanen WA, and Moritz H (1967). Physical Geodesy. W. H. Freeman & Co Ltd, San Francisco, ISBN-13 978–0716702337
R. Mayrhofer and R. Pail M€uller J, Soffel M and Klioner SA (2007) Geodesy and relativity. Journal of Geodesy 82 Number 3, Springer, Berlin, ISSN 0949–7714, pp 133–145 Schiller S (2007) Gravimetry with optical clocks. Workshop on The Future of Satellite Gravimetry 12–13 April 2007, ESTEC Schiller S, Tino GM, Gill P, Salomon C, Sterr U, Peik E, Nevsky A, G€orlitz A, Svehla D, Ferrari G, Poli N, Lusanna L, Klein H, Margolis H, Lemonde P, Laurent P, Santarelli G, Clairon A, Ertmer W, Rasel E, M€uller J, Iorio L, L€ammerzahl C, Dittus H, Gill E, Rothacher M, Flechner F, Schreiber U, Flambaum V, Ni, Wei-Tou, Liu, Liang, Chen, Xuzong, Chen, Jingbiao, Gao, Kelin, Cacciapuoti L, Holzwarth R, Heß MP, Sch€afer W (2009) Einstein gravity explorer – a medium class fundamental physics mission. Exp Astron 23 (2):573–610 Shin MY, Park C and Lee SJ (2008) Atomic Clock Error Modeling for GNSS Software Platform. Position, Location and Navigation Symposium, IEEE/ION Plans 2008, pp 71–76, 1-4244-1537-03/08 Schuh W-D (2001) Improved modeling of SGG-data sets by advanced filter strategies. ESA-Project (Hg.): From E€otv€os to mGal+, WP2, Midterm-Report, pp 113–181 Soffel M, Klioner A, Petit G, Wolf P, Kopeikin SM, Bretagnon P, Brumberg VA, Capitaine N, Damour T, Fukushima T, Guinot B, Huang T-Y, Lindegren L, Ma C, Nordtvedt K, Ries JC, Seidelmann PK, Vokrouhlicky D, Will CM, Xu C (2003) The IAU 2000 resolution for astrometry, celestial mechanics, and metrology in the framework: explanatory supplement. Astron J 126:2687–2706, The American Astronomical Society. USA
Local and Regional Comparisons of Gravity and Magnetic Fields
29
C. Jekeli, O. Huang, and T.L. Abt
Abstract
A long recognized connection between the gravitational gradients of the Earth’s crust and its magnetic anomalies, known as Poisson’s relationship, is the object of investigation in this paper. We develop the mathematical and theoretical basis of this relationship in both the space and frequency domains. Anomalies of the magnetic field thus implied by the gravitational gradients (or other derivatives of the gravitational potential) are called pseudo-magnetic anomalies; and, they assume a linear relationship between the mass density of the source material and its magnetization induced by the Earth’s main magnetic field. Tests in several regions of the U.S. that compare gravitational gradients derived from the highresolution model, EGM08, and a continental magnetic anomaly data base reveal that the correlation implied by Poisson’s relationship is not consistent. Some areas exhibit high positive correlation at various frequencies, while others have even strong negative correlation. Therefore, useful applications of Poisson’s relationship depend on the validity of the underlying assumptions that, conversely, may also be investigated and studied using a combination of gradiometric and magnetic data.
29.1
Introduction
The gravitational and magnetic fields of the Earth are connected by the fact that both are generated in part by the same source material, gravitationally because this material has mass density, and magnetically because it is magnetized by the Earth’s main magnetic field due to its liquid outer core. At the microscopic level, the atomic structure of the material has both mass and
C. Jekeli (*) O. Huang T.L. Abt Division of Geodetic Science, School of Earth Sciences, The Ohio State University, 125 South Oval Mall, Columbus, OH 43210, USA e-mail:
[email protected]
electric dipoles that respond to an applied magnetic field and, in turn, generate an induced field. Mathematically, both fields satisfy Laplace’s differential equation and the solutions are similar, one involving a Green’s function for a distribution of monopoles, the other a Green’s function for a distribution of dipoles. The putative relationship between gravity and magnetic fields was developed already by Poisson (1826), was popularized by Baranov (1957), and essentially posits that magnetic anomalies are gravitational gradients, that is, higher-frequency signals. Mathematical elaborations were carried out by Gunn (1975) and Klingele et al. (1991), and others. The idea that gravitational gradients are intimately connected to the magnetization of the Earth’s crust has motivated
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_29, # Springer-Verlag Berlin Heidelberg 2012
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C. Jekeli et al.
geophysicists to use more easily obtained magnetic data to make gravimetric-type interpretations (Fedi 1989), and to combine (e.g., Dindi and Swain 1988; Ates and Keary 1995) or contrast (Briden et al. 1982) magnetic and gravimetric data for an improved characterization of subsurface geologic structures. While most previous investigations used gravimetric data to model the gravitational gradients, the correlation between gravitation and magnetism can now also take advantage of various gradiometric data sets (e.g., Jekeli et al. 2008). The purpose of this paper, however, is to investigate this correlation on the basis of the new very-high-degree gravitational model, EGM08 (Pavlis et al. 2008), from which gravitational gradients are readily computed.
29.2
Basic Theory
The gravitational potential, V, at some point, x, due to a mass point (monopole) with mass, m, located at x0 , is well known and given by V ðxÞ ¼ G
m ; jx x0 j
(29.1)
where G ¼ 6:674 1011 m3 /(kg s2 Þ is Newton’s gravitational constant. For the (anomalous) magnetic field, the most elemental source is a dipole, modeled as arising from a microscopic loop current. It can be shown (Telford et al. 1990) that the corresponding magnetic potential, A, is AðxÞ ¼
m0 p e ; 4p jx x0 j2
(29.2)
where m0 ¼ 4p 107 kg m/ ðamp sÞ2 is the magnetic permeability of free space, p is the magnetic dipole moment (vector), and e is the unit vector from x0 to x. For a continuous volume distribution of mass monopoles, we define the mass density as rðx0 Þ ¼ dm=dv; and, analogously, for a continuous distribution of magnetic dipole moments, we define the magnetic dipole moment density as z ðx0 Þ ¼ dp=dv. This density is also called the magnetic polarization or the magnetization intensity. The field is given in each case above, according to the law of superposition, by
ððð V ð xÞ ¼ G v
m A ð xÞ ¼ 0 4p
ððð v
rðx0 Þ dv; jx x0 j z ðx 0 Þ e jx x0 j 2
dv;
(29.3)
(29.4)
where dv ¼ dx1 0 dx2 0 dx3 0 . It is easily verified that the scalar magnetic potential is also expressed as A ð xÞ ¼
m0 4p
ðð ð
z ðx0 Þ:rx
v
1 dv jx x0 j
(29.5)
The magnetization, z, of the crust material, in part, is induced by the main field flux density, H0 , that is generated by the Earth’s outer liquid core, and includes a remnant (remanent) magnetization left from the time of rock formation: z ðx0 Þ ¼ wðx0 ÞH0 ðx0 Þ þ z R ðx0 Þ;
(29.6)
where w is the (presumably scalar) magnetic susceptibility. For present purposes, we assume that the Koenigsberger ratio, Q ¼ z R =wH 0 , is sufficiently small so as to make z R negligible. Furthermore, locally, H0 is almost constant in magnitude and direction. For a given volume of material (x0 2 v), we assume that the mass density and the magnetic susceptibility are linearly related: wðx0 Þ ¼
w0 rðx0 Þ; r0
(29.7)
where r0 and w0 are constants; then z ðx0 Þ ¼ k0
z0 rðx0 Þ; r0
(29.8)
where z 0 ¼ w0 H0 , H0 ¼ H0 k0 , and k0 ¼ ð cos cos x sin cos x
sin x ÞT ;
(29.9)
and where x is the dip angle (inclination) with respect to the local horizon, and is the strike angle (azimuth, or declination) with respect to local geodetic north. With (29.8), the magnetic potential becomes the pseudo-magnetic potential:
29
Local and Regional Comparisons of Gravity and Magnetic Fields
m z AðxÞ ¼ 0 0 k0 rx 4p r0
ððð v
rðx0 Þ dv; jx x0 j
(29.10)
where k0 rx is the directional derivative in the (constant) direction of the magnetization of the material. Combining (29.3) and (29.10), we thus have Poisson’s relation: A ð xÞ ¼
m0 z0 k0 rx V ðxÞ; 4pGr0
(29.11)
or, in terms of the force fields, g ¼ rV, F ¼ rA (the signs are conventional, and the subscript on the gradient operator is now redundant): F ð xÞ ¼
m0 z0 rgT ðxÞk0 : 4pGr0
(29.12)
241
Therefore, from (29.13), we have DB ¼
m0 z 0 T k GðxÞk0 ; 4pGr0 0
which directly relates magnetometry and gravity gradiometry. Technically, the right side of (29.16) is called the pseudo-magnetic anomaly. Under the assumption of a flat Earth, it is possible to relate the gravitational and magnetic fields locally also in the spatial frequency domain. Consider a potential that satisfies Laplace’s equation in the exterior space, z0 <0 (z0 is positive downwards): 1 ð
1 ð 1 ð
U ð xÞ ¼ 1 1 0
Defining the gravitational gradient tensor as G ¼ rgT , we finally have FðxÞ ¼
m0 z0 GðxÞk0 ; 4pGr0
(29.13)
showing that under certain conditions (density and magnetization of the source material are linearly related) the magnetic field is the gravitational gradient. The magnetic anomaly is defined as the difference in magnitudes of the total field, B ¼ B0 þ F, and that of the main field generated by the inner core, B0 : DB ¼ jBj jB0 j:
(29.14)
From Fig. 29.1, it is readily seen that the magnetic anomaly at x is the field due to the magnetization field, F, projected in the direction, k0 , of the main field, provided jFj jB0 j; that is, DBðxÞ ¼ k0 FðxÞ:
B
(29.16)
sðx0 Þ 0 0 0 dz dx dy ; ‘
(29.17)
for some arbitrary source function, sðx0 Þ, where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ‘ ¼ ðx x0 Þ2 þ ðy y0 Þ2 þ ðz z0 Þ2 . With respect to the x,y variables, U is a convolution of s and the reciprocal distance: 1 ð
U ð xÞ ¼
ðs LÞðx; y; z; z0 Þdz0 ;
(29.18)
0
where L is the reciprocal distance function: 2 1=2 : L ¼ x2 þ y2 þ ðz z0 Þ
(29.19)
Let U ðu; v; zÞ be the 2-D Fourier transform of U with respect to x and y, at level z: U ðu; v; zÞ
$
U ðx; y; zÞ;
(29.20)
(29.15) where the planar (radian) frequencies are u and v. It is well known that for harmonic functions, F
U ðu; v; zÞ ¼ U ðu; v; 0Þewz ;
z < 0;
(29.21)
|B| − |B0| B0 κ0
Fig. 29.1 The geometry of the magnetic field vectors
and, the derivative operators have the following Fourier transforms: @ =@x $ iu ;
@ =@y $ iv;
@ =@z $ w;
(29.22)
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C. Jekeli et al.
where w2 ¼ u2 þ v2 . The Fourier transform of the reciprocal distance function, L, is given by Lðu; v; z z0 Þ ¼
2p ðzz0 Þw e ; w
w 6¼ 0:
(29.23)
Hence, by the convolution theorem applied to (29.18),
U ðu; v; zÞ ¼
1 ð
2p w
0
S ðu; v; z0 Þeðzz Þw dz0 ;
(29.24)
0
where S ðu; v; z0 Þ is the 2-D Fourier transform of the source function, s. Applying this result to the gravitational potential, (29.3), we have 2pG V ðu; v; zÞ ¼ w
1 ð
0
Rðu; v; z0 Þeðzz Þw dz0 ;
(29.25)
With the transforms (29.22) and (29.29) offers a pseudo-magnetic anomaly for each of the derivatives of the gravitational potential.
29.3
Data Comparisons
Poisson’s relationship, either (29.16) that requires the full gravitational gradient tensor or the alternatives in the frequency domain generated by (29.29), can be tested easily in areas containing both types of data (e.g., Jekeli et al. 2008). Here we investigate a broader spectrum using the model, EGM08, to compute the gradient tensor and a national data base of magnetic anomalies. The gravitational model is given by a spherical harmonic expansion of the potential in terms of spherical polar coordinates, ðr; y; lÞ: V ðr; y; lÞ ¼
0
(29.30)
0
0
where Rðu; v; z Þ is the 2-D Fourier transform of rðx Þ at level, z0 . Assuming (29.8) and the local constancy of k0 , the 2-D Fourier transform of the magnetization, zðx0 Þ, at level, z0 , is Z ðu; v; z0 Þ ¼
nmax X n nþ1 GM X a Cnm Ynm ðy; lÞ; a n¼0 m¼n r
w0 H 0 Rðu; v; z0 Þ: r0
(29.26)
Defining the directional derivative, k0 rx ¼ @ =@k0 : @ @ @ @ ¼a þb þc ; @k0 @x @y @z
(29.27)
with corresponding directional cosines, a, b, c, given by (29.9), the 2-D Fourier transform of the magnetic potential, (29.5), is Aðu; v; zÞ ¼ ðiðau þ bvÞ þ cwÞ 1 ð m0 0 Z ðu; v; z0 Þeðzz Þw dz0 2w
(29.28)
where nmax ¼ 2160 (for EGM08), a is the radius of the bounding (Brillouin) sphere (often approximated by the semi-major axis of the mean-Earth normal ellipsoid), the Cnm are the (unit-less) Stokes coefficients, and the Ynm ðy; lÞ are surface spherical harmonics defined by Ynm ðy; lÞ ¼ Pn;jmj ðcos yÞ
1 4p
ðð
m0 (29.31) m<0
ðYnm ðy; lÞÞ ds ¼ 1 for all n; m; 2
(29.32)
s
where s represents the unit sphere. The local gravitational gradients in a Cartesian, south-east-up coordinate system are given by Gxx ¼
Combining (29.15) and (29.25)–(29.28) yields m0 w 0 H 0 V ðu; v; zÞ 4p Gr0 (29.29)
cos ml; sinjmjl;
The functions, Pn;jmj , are associated Legendre functions, fully normalized so that
0
DBðu; v; zÞ ¼ ðiðau þ bvÞ þ cwÞ2
Gxy ¼ Gyx ¼
@2T 2
ðr@yÞ
þ
1 @T ; r @r
(29.33)
@2T cot y @T ; ðr@yÞðr sin y@lÞ r r sin y@l (29.34)
29
Local and Regional Comparisons of Gravity and Magnetic Fields
Gxz ¼ Gzx ¼
Gyy ¼
@2T 1 @T ; ðr@yÞð@r Þ r r@y
@ 2T ðr sin y@lÞ
Gyz ¼ Gzy ¼
2
þ
cot y @T 1 @T þ ; r r@y r @r
(29.35) Area 1
(29.36)
Area 3
Area 2
@2T 1 @T ; (29.37) ðr sin y@lÞð@r Þ r r sin y@l Gzz ¼
@2T ð@r Þ2
:
(29.38)
The magnetic anomalies for North America have been compiled from airborne surveys over the last several decades by the North American Magnetic Anomaly Group (NAMAG 2002) and are available from the Regional Geospatial Service Center of the University of Texas at El Paso (website: http://gis. utep.edu/). These data were combined by NAMAG with respect to a common reference and subjected to a 500 km high-pass filter to remove regional biases. The magnetic anomalies have a resolution of about 1 km, but a simple interpolation onto a 5 arcmin grid yields adequate consistency in resolution with the EGM08 model. Analogously, removing harmonic degrees n ¼ 80 and lower effectively filters the wavelengths greater than 500 km. Three areas in the U.S. were considered as shown in Fig. 29.2. Each area has a particularly characteristic feature of the gravity field. Area 1 with relatively flat terrain nevertheless features a strong anomaly due to the mid-continent rift. Area 2 exhibits a typically moderate field, as opposed to Area 3 that straddles the transition from the Great Plains to the Rocky Mountains with corresponding changes in the gravity field. Using the National Geospatial-Intelligence Agency (NGA) software program, GEOMAG.FOR,1 the values of the main field, B0 , and its inclination, x, and declination, , according to the World Magnetic Model 2005 (WMM2005) were obtained for each area in Fig. 29.1. The values ðB0 ; x; Þ on a 50 50 grid were used for the pseudo-magnetic anomaly, (29.16), but were averaged and assumed constant over each area for the pseudomagnetic anomaly derived by Fourier transforms (29.29). Although the NAMAG data generally refer to
1
243
http://www.ngdc.noaa.gov/geomag/WMM/DoDWMM.shtml.
Fig. 29.2 Areas where magnetic anomalies are compared to pseudo-magnetic anomalies
epochs in the latter half of the twentieth century, it was assumed that the changes in the main field since then do not significantly affect the present analyses. The scale of the pseudo-magnetic anomaly depends on the values of the magnetic susceptibility and mass density; we assumed constants, w0 ¼ 0:05 (S.I:Þ and r0 ¼ 2:67 g/cm3 . Figure 29.3 shows the magnetic anomaly and three versions of the pseudo-magnetic anomaly computed from EGM08 (degrees 81–2,160) in Area 1. The pseudo-magnetic anomalies were re-scaled to be consistent with the magnetic anomalies. The prominent feature of the pseudo-magnetic anomalies in the eastern half of the area reflects the gravity anomaly associated with the mid-continent rift. It is also seen in the magnetic anomaly. With lower derivatives of the gravitational potential, the pseudo-magnetic anomaly appears smoother, despite the application of the derivative transforms (29.22). The numerical derivatives of the EGM08 values on a 5 arcmin grid are not able to recover the same detail as the analytic derivatives of this model. Therefore, we may conclude that measured gravitational gradients should be transformed to pseudo-magnetic anomalies, since they reside in the same part of the spectral domain as the magnetic anomaly [see (29.16)]. Similar comparisons between the magnetic anomaly and the pseudo-magnetic anomaly from (only) the full gravitational gradient tensor (29.16) are shown in Figs. 29.4 and 29.5 for Areas 2 and 3. The correlation between the anomalies is not strong. In particular, the distinct transition from the Great Plains to the Rocky mountains (Area 3) reflected in the gravitational gradients is almost absent in the magnetic anomaly. Similarly, the elongated magnetic feature on the
244
Fig. 29.3 Magnetic anomaly (top left) in Area 1, and pseudomagnetic anomalies using the full gravity gradient tensor (top right), the gravity anomaly (bottom right), and the gravitational potential (bottom left). All pseudo-magnetic anomalies
C. Jekeli et al.
are based on EGM08 from degrees 41 to 2,160. The pseudomagnetic anomalies were scaled to the same range as the magnetic anomalies, between 2,000 and 3,000 nT
Fig. 29.4 Magnetic anomaly (left) in Area 2, and pseudo-magnetic anomaly using the full gravity gradient tensor from EGM08 (right). The scale is the same as in Fig. 29.3
29
Local and Regional Comparisons of Gravity and Magnetic Fields
245
Fig. 29.5 Magnetic anomaly (left) in Area 3, and pseudo-magnetic anomaly using the full gravity gradient tensor from EGM08 (right). The scale is the same as in Fig. 29.3
correlation at medium and short wavelengths, in others the correlation is hardly evident or even strongly negative. Similar results (not shown) are found in the other areas. Conclusion
Fig. 29.6 Profiles of the magnetic anomaly and pseudo-magnetic anomaly from the EGM08 gradient tensor in Area 1 (45 latitude)
eastern side of Area 2 is not repeated in the pseudomagnetic anomaly. Figure 29.6 shows profiles of the magnetic anomaly and the pseudo-magnetic anomaly from the EGM08 gradient tensor across the middle of Area 1. They are seen to correlate variously in both space and frequency domains. While in some parts there is high positive
This paper develops the mathematical and theoretical connection between gravitational gradients and the magnetic anomalies due to induced magnetization of the Earth’s upper crust material, leading ultimately to Poisson’s relationship for the pseudomagnetic anomaly, formulated in both the space and frequency domains. This connection is founded on the assumption of a linear relationship between mass density and magnetization (with constant direction) of the source body. Preliminary tests of this relationship on a regional basis using the high-resolution gravitational field model, EGM08, and a high-resolution magnetic anomaly data base yield mixed results. The correlation between gravitational gradients and magnetic anomalies is evident at short to medium scales, but not consistently. In areas where the correlation is low or even highly negative, the assumption of a linear relationship between mass density and magnetization apparently does not hold. Such areas would be of particular interest for further geophysical study.
246
References Ates A, Keary P (1995) A new method for determining magnetization direction from gravity and magnetic anomalies: application to the deep structure of the Worcester Graben. J Geol Soc 152:561–566 Baranov V (1957) A new method for interpretation of aeromagnetic maps: Pseudo-gravimetric anomalies. Geophysics 22:359–383 Briden JC, Clark RA, Fairhead JD (1982) Gravity and magnetic studies in the Channel Islands. J Geol Soc 139:35–48 Dindi EW, Swain CJ (1988) Joint three-dimensional inversion of gravity and magnetic data from Jombo Hill alkaline complex, Kenya. J Geol Soc 145:493–504 Fedi M (1989) On the quantitative interpretation of magnetic anomalies by pseudo-gravimetric integration. Terra Nova 1: 564–572 Gunn PJ (1975) Linear transformations of gravity and magnetic fields. Geophys Prospect 23:300–312
C. Jekeli et al. Jekeli C, Erkan K, Huang O (2008) Gravity vs pseudo-Gravity: a comparison based on magnetic and gravity gradient measurements. In: Proceedings of the International Symposium on Gravity Geoid and Earth Observations, GGEO2008, 23–27 July 2008, Chania, Greece, Springer Klingele EE, Marson I, Kahle H-G (1991) Automatic interpretation of gravity gradiometric data in two dimensions: vertical gradient. Geophys Prospect 39:407–434 North American Magnetic Anomaly Group (NAMAG) (2002): Magnetic anomaly map of North America. U.S. Department of the Interior, Washington, DC Pavlis NK, Holmes SA, Kenyon SC, and Factor JK (2008) An earth gravitational model to Degree 2160: EGM2008. Presented at the General Assembly of the European Geosciences Union, Vienna, Austria, April 13–18, 2008 Poisson SD (1826) Me´moire sur la the´orie du magne´tisme. Me´moires de l’Acade´mie Royale des Sciences de l’Institut de, France, pp 247–348 Telford WM, Geldart LP, Sheriff RE (1990) Applied geophysics, 2nd edn. Cambridge University Press, Cambridge
Combination of Local Gravimetry and Magnetic Data to Locate Subsurface Anomalies Using a Matched Filter
30
T. Abt, O. Huang, and C. Jekeli
Abstract
The detection of mass anomalies in the near subsurface is an important problem in many areas of interest, such as archeology, construction, and hazard analysis. Assuming that the approximate geometry of an anomaly is known, its possible location can be determined by applying a matched filter to observations of gravity, gravity gradient, and magnetic anomalies, as well as to electro-magnetic data. We analyze the specific combination of gravity, gravity gradient, and magnetic data in order to determine their relative strengths and weaknesses in the detection problem. Poisson’s Relation is used to model the magnetic signals generated by the source to be detected, and the mutual covariances of the background geologic noise that may contaminate the observations. Simulations show that the magnetic data can improve the detection using the matched filter, especially with limited gravity gradients from a typical ground gravity gradiometer. Further analyses using actual data over a known local anomaly illustrate the enhancements as well as limitations of the gravimetry, gradiometry, and magnetic data combinations.
30.1
Introduction
The main idea of this project is to detect local mass anomalies in the shallow subsurface, such as a cavity filled with air or water. This general task becomes useful in a wide range of different applications, for example, in archeology, mining, hazard analysis, and road constructions (FHWA 2005). Gravity and magnetics are fundamental properties that can be observed in geophysical research. In contrast to, for example, seismology or ground-penetrating
T. Abt (*) O. Huang C. Jekeli Division of Geodetic Science, School of Earth Sciences, The Ohio State University, 125 South Oval Mall, Columbus, OH 43210, USA e-mail:
[email protected]
radar, gravimetry and magnetometry do not require an active sensor and are, therefore, simple and cost efficient techniques. In the following, it is assumed that a profile of gravity, gravity gradient and magnetic field data is provided. This setup can in future be extended from a single profile to a full area.
30.2
Theoretical Background
The Matched Filter (MF) locates a given signal within a noisy data set. This requires that the sought signal s, which is generated by the local mass anomaly, is known. The Matched Filter is the convolution of its filter function h and the observation vector z (30.1) (Dumrongchai 2007).
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_30, # Springer-Verlag Berlin Heidelberg 2012
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yr ¼
X
hT xr xj zj
(30.1)
j
The filter function h (30.2) is designed to maximize the filter output y at the location of the sought signal. h is scaled by l, where l2 represents the maximum signal-to-noise ratio (30.3). 1 X 1 f ðj; kÞsðxk Þ h xj ¼ l k l2 ¼
XX j
sT xj f1 ðj; kÞsðxk Þ
(30.2)
Poisson’s Relation will here only be considered for the anomalous magnetic and gravitational field where the main reference field of the Earth has been subtracted. The first derivative of Poisson’s Relation (30.5) leads to the relation between the magnetic field B and the gravitational gradient tensor G (30.6). rVmag ¼
m0 M d rVgrav 4p Gr dk0
(30.5)
m0 M d g 4p Gr dk0
(30.6)
B¼
(30.3)
k
The matrix F consists of the covariances between profile points and describes the noise of the background field, which interferes with the sought signal. The indices j,k, and r account for the particular points of the profile with coordinate x. The above equations also hold in the two-dimensional case (multiple profiles). The mix of data (gravity, gravity gradients, magnetic field) can easily be included in the observation vector but requires some modeling of the covariance matrix in the filter function. Poisson’s Relation (30.4) takes advantage of the similar structure of the magnetic potential Vmag and the gravitational potential Vgrav (Blakely 1996). It allows for modeling the required covariance functions in the Matched Filter. Vmag ¼
m0 M d Vgrav 4p Gr dk0
(30.4)
with k0 ¼ ½ cos D cos I sin D sin I sin D T The directional derivative of the gravity vector g accounts for the deviation of the magnetic field direction from the geodetic coordinate system, i.e., the inclination I with respect to local horizon and declination D with respect to local geodetic North (Fig. 30.1). The magnitude of the magnetic field B in direction of magnetization is finally given by (30.7). B¼
m0 M T k Gk0 4p Gr 0
(30.7)
with dkd 0 ¼ k0 r The direction of the magnetization is approximated by the direction of the Earth’s main magnetic field. This is a legitimate approximation if no remanent magnetization occurs since the main magnetic field is much larger than the anomalous field. Further details are provided in (Jekeli et al. 2009).
m0 ¼ magnetic permeability of free space M ¼ magnetization G ¼ Newton’s gravitational constant
North
r ¼ mass density k0 ¼ unit vector in direction of magnetization Poisson’s Relation holds under the two conditions that first, the source that generates the magnetic field is identical to the source that generates the gravitational field and second, that the magnetization to density ratio of the source is constant under constant direction of magnetization. If the first condition is violated it is practical to subdivide the source into smaller parts that each fulfill the condition. Due to those restrictions
α
D
I
β
γ
Down
Fig. 30.1 Direction of magnetization
East B |B|
Combination of Local Gravimetry and Magnetic Data to Locate Subsurface Anomalies
30.3
Matched Filter Simulations
Γ33 [E] 100
200
90
150
80
100
70 50
60 0
50 40
–50
30
–100
20
–150
10 –200
20
40
60
80
100
x [m]
Fig. 30.3 Gravity gradient background
Bt [nT] 100
100
90 80
50
70 60 y [m]
These simulations validate the advantages and disadvantages of the Matched Filter approach as well as further analyze the results of the various sensor combinations. The sought mass anomaly is represented by a threedimensional prism (Fig. 30.2) of a constant mass density contrast Dr and a constant magnetization contrast DM to the background field. The three-dimensional prism is considered to be a good first approximation for many real features and it has the advantage that closed analytic expressions for its generated gravity, gravity gradients, and magnetic field exist. These define the sought signal s. In this setup, a 100 m profile with a point interval of 1 m crosses the mass anomaly perpendicularly after 30 m. The dimensions of the prism are 1 m in width, 100 m in length, and 2 m in height. The length is set arbitrarily long in order to reduce the setup to a onedimensional problem. The top of the prism is at a depth of 2 m. The covariance matrix F of the observation points is derived from a local analytical model for gravity disturbances based on reciprocal distances (Jekeli 2003). The derivatives of this model provide the covariance functions for gravity and gravity gradients. The corresponding covariance functions of the magnetic field are computed by applying Poisson’s Relation. The observation vector z consists of a superposition of three signals: (1) The sought signal generated by the anomaly at 30 m of the profile, (2) the background field in which the sought signal is buried, and (3) random instrument noise. The gravity and gravity gradient background fields are derived from power spectral densities that correspond to the same analytic covariance model as introduced above. A set of normally distributed random numbers synthesizes the
249
spectrum and generates a random background field on a 100 m 100 m grid with a point interval of 1 m. Two examples are given in Fig. 30.3 for the gravity gradient G33 and in Fig. 30.4 for the magnetic field B. The standard deviation for the first is sG33 70E and for the latter sB 40nT. While the gravity and gravity gradient fields are directly based on the covariance model, the magnetic field is again derived by utilizing Poisson’s Relation. Out of each random background field a 100 m profile is chosen (horizontal track in Fig. 30.3) and added to the observations. The simulated instrument noise is white noise with the respective standard deviations: sg ¼ 0:01mgal, sG ¼ 3E, and sB ¼ 1nT. Figure 30.5 summarizes the
y [m]
30
0
50 40
–50
30 20 –100 10 20
40
60 x [m]
Fig. 30.2 Profile above anomaly
Fig. 30.4 Magnetic field background
80
100
250 Fig. 30.5 MF observation
T. Abt et al. Matched Filter Simulation for G33
200 150 100 50 0 –50
Sought Signal Background Field
–100
Instrument Noise
–150
0
10
20
Fig. 30.6 MF output
30
40
50 [m]
60
70
80
90
100
80
90
100
Matched Filter Simulation for G33 200 150 100 50 0 –50 Observation MF Output*50
–100 –150
0
superposition of the three signals that enter the observation vector in case of, for example, the gravity gradient G33 . The resulting background field in Fig. 30.6 is so strong that it is impossible to locate the sought signal by visual inspection. However, the matched filter output clearly shows a maximum at 30 m and has, therefore, correctly detected the prism. In this way the matched filter has been run 1,000 times; each time containing a different random background profile. Those runs are used to analyze the different combinations of data. The results are presented in Table 30.1.
10
20
30
40
50 [m]
60
70
Table 30.1 Simulation results G33,G13 G33,B G33,G13, B 24.4% 84.0% 95.5% 95.2% 98.0% 98.6% 1.57 13.16 19.48 18.74 22.90 25.12 2.44 3.71 4.40 4.36 4.79 5.02 g3
Detection SNR Mean ymax
G33
B
Detection is given as the percentage of how many times the prism was located correctly. In a direct comparison between the gravity component g3 , the gradient component G33 , and the magnetic field B, it becomes clear that gravity is the weakest data while magnetics
30
Combination of Local Gravimetry and Magnetic Data to Locate Subsurface Anomalies
simulations except that the standard deviations of the instrument noise are altered. It is striking that the gravity gradient G33 and the magnetic field B show a similar satisfying performance while the gravity component g3 , on the other hand, requires an unrealistic signal-to-instrument-noise ratio of 200 before it reaches the 90% detection mark.
performs better than gravity gradients. In general, the more data that are combined, the better the detection. This encourages the combination of gradiometry and magnetometry. The combination of more data components has also the effect that the maximum signal-to-noise ratio (SNR) as well as the maximum filter output increase. This is an important fact for a statistical interpretation of the matched filter output. The major reason why the gravity-only simulations have such a low success rate in detection is due to the relatively high instrument noise of the gravimeter. This is further analyzed by computing the signal-toinstrument-noise ratio (30.8), which is defined as the maximum (or minimum) sought signal feature divided by the standard deviation of the respective instrument noise. SNInstrument R ¼
maxksðxÞk sInstrument
251
30.4
Field Tests
The first actual data acquisition for this project has been carried out on the oval of the Ohio State University campus. Figure 30.8 is a map of the oval showing an underground utility conduit (horizontal arrow), which is perpendicularly crossed by a brick walk (vertical arrow). Below are pictures (Fig. 30.9) of the brick walk as well as the inside of the utility conduit, respectively. Each brick has a width of approximately 0.1 m and is used to define the observation point interval Dx. The entire profile measures 20 m and is centered over the utility conduit. The dimensions of the conduit are 1.68 m in width, a few hundred meters in length, and 1.64 m in height. The depth of the top of the conduit is approximately 0.8 m. The profile is measured three times with a CG-5 Autograv relative gravimeter. The gravity observations are carried out one time in full resolution, i.e., 0.1 m point intervals, and the other two times with a 0.2 m and a 0.4 m point interval. The gravimeter
(30.8)
The extrema generated by the prism are in the following three cases: maxkg3 ðxÞk ¼ 0.024mgal maxkG33 ðxÞk ¼ 86.107E maxkBðxÞk ¼ 45:309nT Figure 30.7 plots the detection success rate with respect to the signal-to-instrument-noise ratio. All input parameters remain the same as in the previous
MF Performance based on Data Type 100 90
Detection [%]
80 70 data: g3
60
data: B
50
data: Γ33
40 30
Fig. 30.7 Comparison of sensors
20
0
50
100 150 200 Signal-to-Instrument Noise Ratio
250
252
T. Abt et al. g3 Profile on Oval (measured) 0.01
[mgal]
0 –0.01 –0.02
Δx = 0.4m Δx = 0.2m Δx = 0.1m Simulation
–0.03 –0.04 0
5
10 Distance [m]
15
20
Fig. 30.10 Gravity on oval
Fig. 30.8 Utility conduits under oval
station is measured after every 20 observation points. However, since the magnetic measurements are fast, occurring drifts are negligible. The observation point interval is 0.2 m. Figure 30.10, illustrates the results of the gravity measurements on the oval. Each profile of measured gravity is reduced by its mean value. The solid line is the signal generated by a three-dimensional prism that approximates the utility conduit and assumes a density contrast of air within soil (Dr ¼ 1900kg m3 ). Since no gravity gradiometer was available the gravity gradient component G31 is derived from the gravity measurements in the following way (30.9). The profile is almost level so that the height is considered to be constant throughout the profile. G31
Fig. 30.9 Brick walk and conduit
applies tidal corrections based on the input date and coordinates. Furthermore, an instrument drift calibration has been carried out prior to the measurements. In order to reduce residual tidal and drift errors the first observation point of the profile is defined as base station and is remeasured every half an hour. The magnetic field is measured with a portable proton precession magnetometer. As the instrument does not take into account any corrections, a base
g3 ðx2 Þ g3 ðx1 Þ Dx
(30.9)
The results of the three computed gravity gradients profiles are presented in Fig. 30.11. The solid line is again the simulated signal. The g3 as well as the G31 plots show that the profile of the 0.1 m interval matches the extrema of the simulated signal the closest but high oscillations occur on the sides. The observations with a 0.2 m and 0.4 m interval appear to be similar. This raises the question for an optimal point interval. The magnetic field is simulated by applying Poisson’s Relation to the gravity gradients of the prism under the assumption of the following
30
Combination of Local Gravimetry and Magnetic Data to Locate Subsurface Anomalies Γ31 Profile on Oval (derived)
200
253 B Profile on Oval
2000
Δx = 0.4m Δx = 0.2m Δx = 0.1m Simulation
1000 0 [nT]
[E]
100
0
–1000 Observation Simulation*30
–2000
–100
–3000
5
10 Distance [m]
15
20
–4000
0
5
10 Distance [m]
15
20
Fig. 30.11 Gravity gradient on oval
Fig. 30.12 Magnetic field on oval
parameters for magnetic susceptibility contrast, magnetic field intensity, inclination, and declination:
brings the shape of the simulated signal much closer to the observed magnetic field.
Dk ¼ 0:001 Hmag ¼ 42:57A=m
)
Conclusions
M ¼ DkHmag
I ¼ 68:093 D ¼ 6:699 The magnetometer measures the magnitude of the total magnetic field. Since only the anomalous magnetic field is here of interest the mean has been removed from the observations. According to Fig. 30.12, the measured magnetic anomaly is unexpectedly strong. In order to bring the measured magnetic field and the simulated field (derived with Poisson’s Relation) into the same scale, the simulated signal needs to be multiplied by a factor of 30. Possible reasons why the utility conduit generates such a high magnetic field are the metal pipes and electrical wires inside the conduit that are difficult to model. The simulations, on the other hand, are computed under the assumption that the conduit is filled with air only. A further aspect, which has been neglected so far, is that other local anomalies might influence the observations. Recent studies (I. Prutkin personal communication and report in progress, 2009) suggest that the magnetic observations in this case are better interpreted by three anomalies associated with the structure of the conduit instead of just one. This
The simulations show that the Matched Filter is a good technique to locate a mass anomaly even if its signal is weaker than the signals of the surrounding environment. The main advantage of the Matched Filter in comparison to other filters is thereby that it maximizes the signal-to-noise ratio. A major drawback of the Matched Filter is the fact that the sought signal must be known. However, in many applications of interest a three-dimensional prism provides a good approximation. Analyzing the performance of the three different sensors (gravity, gravity gradients, and magnetic field), the gravity observations have the least contribution. If the sought signal is as dominant as in the field tests on the oval, the gravimeter is a helpful technique to map the mass anomaly. However, if the sought signal is difficult to detect as in the simulated background fields the gravity gradient data perform much better. The magnetic field data present a slightly better performance than the gravity gradients but in general, the magnetic field is more difficult to interpret. Other sources easily interfere with the sought signal of the main source. Since the magnetometer is a simple and low-cost device, it is still always advisable to include it in a measurement survey.
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In conclusion, gravity and magnetic data based on Poisson’s Relation are a reasonable and beneficial combination, as demonstrated by the simulations, for local applications due to the potentially high theoretical correlation. The results of the local field tests seem to support this correlation, although the magnetic data cannot distinguish between the cavity and the metal within the cavity. Future field tests are in preparation above a completely airfilled conduit where the matched filter would be exercised to detect less strong gravitational and magnetic signals.
T. Abt et al.
References Blakely RJ (1996) Potential theory in gravity & magnetic applications. Cambridge University Press, New York, NY Dumrongchai P (2007) Small anomalous mass detection from airborne gradiometry. The Ohio State University, Report 482 FHWA – Federal Highway Administration – (2005) Subsurface Imaging of Lava Tubes – Roadway Applications. Publication No. FHWA-CFL/TD-05-005 Jekeli C (2003) Statistical analysis of moving base gravimetry and gravity gradiometry. The Ohio State University, Report 446 Jekeli C, Huang O, Abt T (2009) Local and regional comparisons of gravity and magnetic fields. In: Proceedings for the IAG 2009 Scientific Assembly “Geodesy for Planet Earth”. Springer, Heidelberg
On the Use of UAVs for Strapdown Airborne Gravimetry
31
Richard Deurloo, Luisa Bastos, and Machiel Bos
Abstract
Airborne gravimetry is a cost-effective technique to complement the gravity field information from satellite missions such as GRACE and, in the near future, GOCE. Measurements can be collected over regional areas in a relatively short time. Moreover, it is an especially useful method in remote areas and coastal water zones where terrestrial and ship-borne gravity measurements are difficult. One drawback of airborne gravimetry is aircraft availability and the associated cost of flight time. If these problems can be reduced or minimised, airborne gravimetry can be made more accessible and cost-effective. A possible option discussed here is based on the use of Unmanned Autonomous Vehicles (UAVs). The use of UAVs has increased in a large number of fields and is proving to be a viable and cost-effective option. However, the performance of UAVs is considerably different from regular fixed-wing aircraft and will have a significant impact on the performance of a gravimetry system. Also, due to the dimensions and operational requirements of spring gravimeter systems, these systems cannot be used inside UAVs. This paper discusses how parameters, such as flight speed, endurance, and flight dynamics, can affect the determination of gravity anomalies and definition of the geoid using strapdown gravimetry systems on board of UAVs. The study is limited to the use of the so-called Light UAVs which have a Maximum Take-Off Mass (MTOM) of less than 150 kg, since those can be relatively inexpensive. Some typical parameters used here for this range of UAVs are a cruise speed of 30 m/s, an endurance of several hours and a payload mass of several tens of kilograms.
R. Deurloo L. Bastos (*) Observato´rio Astrono´mico, Faculdade de Cieˆncias da Universidade do Porto, Alameda do Monte da Virgem, 4430–146 V.N. Gaia, Portugal Centro Interdisciplinar de Investigac¸a˜o Marinha e Ambiental, Rua dos Bragas, 289, 4050–123 Porto, Portugal e-mail:
[email protected] M. Bos Centro Interdisciplinar de Investigac¸a˜o Marinha e Ambiental, Rua dos Bragas, 289, 4050–123 Porto, Portugal S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_31, # Springer-Verlag Berlin Heidelberg 2012
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First results, based on PSD (Power Spectral Density) analysis, indicate that, for a navigation grade inertial system, the use of UAVs can be an advantage in recovering the short wavelength (<5 km) information of the gravity field.
31.1
Introduction
Satellite-based sensors can be a useful tool to determine the global gravity field. However, since high spatial resolution information about the gravity field is attenuated more at high altitudes than low resolutions, satellite observations alone cannot provide a full and accurate map of the Earth’s gravity field in all wavelengths. For example, the much anticipated GOCE mission (recently launched) is expected to provide an accuracy of 1 mGal (105 m/ s2) down to a spatial resolution (half-wavelength) of 100 km (Johannessen and Aguirre–Martinez 1999). For areas such as resource exploration and mineral prospecting 1 mGal accuracy for spatial resolutions smaller than 10 km is required and for cm-level geoid determination spatial resolutions down to about 5 km are required (Bruton 2000). There are various possibilities to locally augment the low resolution information (>150 km) from satellite missions with medium (5–150 km) to high (<5 km) resolution observations. But while methods, such as terrestrial or ship-borne gravimetry, are timeconsuming and limited to specific areas, airborne gravimetry is an efficient method to survey regional areas in a relatively short time. Moreover, airborne gravimetry is especially useful in remote areas, where terrestrial gravity measurements are difficult, and in transition regions, such as coastal water zones, where other methodologies may not be practical. To reduce the cost of airborne mapping, such as airborne gravimetry, the Astronomical Observatory of the Faculty of Science of the University of Porto has developed a low-cost strapdown GPS/INS system based on an INS (Inertial Measurement System) and one or more GPS (Global Positioning System) receivers. Specifically a Litton LN-200 strapdown IMU (Inertial Measurement Unit) was used. The system has been shown to provide 5–10 mGal accuracy for the 10–100 km resolution bandwidth (Tome´ 2002; Bastos et al. 2000). But although the GPS/INS concept can significantly reduce the cost of a survey, problems of aircraft
availability and cost of flight time remain. If these can be reduced, airborne gravimetry can be made more cost-effective. One of the possible options for doing this may be to use UAVs (Unmanned Autonomous Vehicles). The performance of UAVs is however considerably different from regular fixed-wing aircraft. Here we show that the reduced flight speed of UAVs has a significant impact on the spectral window and signalto-noise ratio of the airborne gravimetry system. For GPS/INS systems based on a navigation grade inertial system the reduced speed is an advantage; for a tactical grade inertial system it is a disadvantage. The reduced flight speed also affects the flight dynamics and reduces the phugoid period, allowing the recovery of spatial resolutions down to less than 1 km. We also show that to augment the above mentioned GOCE mission with spatial resolutions up to 100 km, the endurance of a UAV should be 3.7 h or higher.
31.2
Unmanned Autonomous Vehicles
The use of UAVs has increased in a large number of fields and is proving to be a viable and cost-effective option. As a result, a wide variety of UAVs is currently available (DoD 2002). For the purpose of this discussion we limit ourselves to so-called Light UAVs. This particular type of UAVs has a Maximum Take-Off Mass (MTOM) of less than 150 kg. This is a typical limit in current civil aviation regulations (e.g. European Civil Aviation Regulation 216/2008). Other parameters used here for this range of UAVs: a cruise speed of 30 m/s, an endurance of several hours and a payload mass of several tens of kilograms. Due to the dimensions and operational requirements of spring gravimeter systems, such as the LaCoste & Romberg Air-Sea gravimeters, these systems cannot be used inside Light UAVs. This discussion is therefore limited to strapdown GPS/INS systems.
31
On the Use of UAVs for Strapdown Airborne Gravimetry
31.3
Strapdown Airborne Gravimetry
31.3.1 Gravity Disturbance The well-known model for the gravity disturbance can be derived from Newton’s second law of motion and Newton’s law of gravitation. It can be shown that combining both laws and converting to a local-level reference frame the gravity disturbance vector dgl can be obtained using the equation (e.g. Jekeli 2001): dgl ¼ v_l C1i ai þ ð2O1ie þ O1ei Þ vi gi
(31.1)
where the disturbance vector was defined as: dgi ¼ gl gi
(31.2)
In the above gl is the gravity vector and g‘ is the normal gravity vector. The vector al is the specific force acting on the vehicle, v_l is the vehicle’s (kinematic) acceleration, and vl is the vehicle’s velocity. The rotation matrix Cli indicates the attitude of the local-level frame with respect to the inertial frame. The matrices Olis and Olsl are the skew-symmetric matrix representations of, respectively, the Earth-rate rotation vector and the rotation vector as a result of the vehicle’s motion along the Earth’s surface. Both rotation vectors are a function of position rl and velocity vl . Using the radii of curvature in meridian and prime vertical, RM and RN , the Earth’s angular speed, oE , and the latitude ’, (31.1) can be written in its North-, East- and Up-components:
257
The two horizontal terms of this equation are small compared to the vertical term. Determining all three components is the goal of vector gravimetry. In scalar gravimetry only the vertical term is considered, which is generally assumed to be equal to the magnitude of the gravity disturbance vector.
31.3.2 GPS/INS Error Spectral Density In a typical strapdown GPS/INS system the position, velocity and kinematic acceleration are determined from GPS measurements, while the specific force vector and attitude are determined by the INS. By combining the measurements from both instruments, the gravity disturbance is determined according to (31.1). As a result the accuracy of the estimated gravity disturbance (vector) will be dependent on the errors of the GPS and INS. To show how the errors of these two instruments affect the estimation of the gravity spectrum, Schwarz et al. (1994) and Schwarz and Li (1996) defined models for the Power Spectral Densities (PSDs) of the instrument errors. It is stated therein that for scalar gravimetry the estimation error is mainly due to the errors in the GPS-derived accelerations v_ l and the INSsensed specific force al . For vector gravimetry the errors in the INS attitude Cli also play an important role as small attitude errors can cause large errors in the horizontal gravity components. The spectral model from Schwarz and Li (1996) for the inertial system is repeated here for later use in Sect. 31.4.1. The model is defined as:
ve ve vn dgn ¼ v_n an þ 2oE cos ’ þ gn tan ’ þ R RM N ve ðvu vn tan ’Þ gg dgg ¼ v_ g ag þ 2oE cos ’ þ RN ve vn 2 ve þ dgu ¼ v_u au þ 2oE cos ’ þ gu RN RM
SSNSg ðf Þ ¼
g2
g2 2ba 2 b þ d 2þ sa;c 2 þ Qa;w a 2 2 g 2 2 2 2 2 2 2 2 2 2 4p f 2 ba 2 R 4p f ð4p f os Þ ð4p f os Þ SSNS;S ðf Þ ¼
2ba sa;c 2 þ Qa;w þ ba 2
4p2 f 2
(31.3)
(31.4a)
(31.4b)
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Navigationa Gyroscopes: dg
p=h ffiffiffiffiffiffi Hz
Accelerometers: ba mGal p ffiffiffiffiffiffi Hz Qa;w mGal2 Hz sa;c ðmGalÞ 1 ðsÞ b
0.001
Tacticalb 3.0
10.0
200.0
1.0
1225.0
10.0 7200.0
20.0 175.0
a
a
Schwarz et al. (1994) b Empirically derived for a Litton LN-200
where g is the mean gravity acceleration, R is the mean Earth radius, and oS is the Schuler frequency. The model in (31.4a) is the PSD for the INS noise affecting the horizontal term in vector gravimetry. It is composed of second-order Gauss-Markov models for gyro drift dg and accelerometer bias ba , a first-order Gauss-Markov model for accelerometer coloured noise sa;c (with correlation time 1=ba ), and accelerometer white noise Qa;w . Equation (31.4b) is the PSD model of the INS noise affecting scalar gravimetry and the vertical term of vector gravimetry. It is only composed of the coloured and white noise. The values for model parameters are shown in Table 31.1. The table shows two types of inertial systems: a navigation grade system as in Schwarz et al. (1994), and a less accurate tactical grade system such as the Litton LN-200. The PSD for the GPS derived accelerations that is used in Sect. 31.4.1 is an empirical model based on dualfrequency double-differenced phase measurements. For more details about this model we refer to Schwarz and Li (1996).
31.4
Impact of Using UAVs
The effects of flight altitude will not be discussed here, since the attenuation of the gravity signal has limited impact up to an altitude of 2.5 km (Schwarz et al. 1994; Schwarz and Li 1996), which is well above the flight altitude of the type of UAVs discussed here.
31.4.1 Cruise Speed The cruise speed is one of the main factors that will affect the performance of the GPS/INS gravimetry system. Since the gravity measurements are performed on a moving vehicle, the temporal frequency spectrum of the gravity signal will depend on the vehicle’s speed. As a result, the cruise speed affects the signalto-noise ratio (SNR) of the gravity disturbance signal and the GPS/INS system noise. Figures 31.1 and 31.2 show the PSDs for the GPS/ INS system noise affecting the vertical component (scalar gravimetry component) and the horizontal components (vector gravimetry components) of the gravity disturbance. For the combined GPS/INS spectra the following relation was used: SINS ðf Þ ¼ SINS ðf Þ þ SGPS ðf Þ
Figure 31.1 shows the PSDs for the navigation grade inertial system and Fig. 31.2 shows the PSDs λ(km)
10 10 10 10 10 10 10 10
Light UAVs have a much lower cruise speed than regular fixed-wing aircraft. The effect of this is discussed below using the PSD models defined above. In addition, the effect of a UAV’s flight dynamics and the impact of a UAV’s endurance will be briefly discussed.
(31.5)
CPS
PSD (mGal2/Hz)
Table 31.1 Model parameters for the power spectral densities of a navigation grade and tactical grade inertial system
9
500
200 100
50
20
10
5
2
INS/GPS (scalar) INS/GPS (vector) Gravity (flat) Gravity (mountain)
8 7 6 5 4 3 2 1
10 -4 10
10
-3
10
-2
f (Hz)
Fig. 31.1 INS/GPS system noise PSDs and gravity disturbance signal PSDs for a navigation grade inertial system at a cruise speed of 70 m/s (252 km/h)
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On the Use of UAVs for Strapdown Airborne Gravimetry
259
λ(km)
PSD (mGal2/Hz)
10 10 10 10 10 10 10
500
200 100
50
20
λ(km)
10
5
2
INS/GPS (scalar) INS/GPS (vector) Gravity (flat) Gravity (mountain)
8 7 6 5 4
10 10
PSD (mGal2/Hz)
10
9
3
10 10 10 10
2
10
1
10 -4 10
10
9
200 100
50
20
10
5
2
1
INS/GPS (scalar) INS/GPS (vector) Gravity (flat) Gravity (mountain)
8 7 6 5 4 3 2 1
10
-3
10
10 -4 10
-2
10
-3
f (Hz)
for the tactical grade inertial system (see Table 31.1). Both figures also show two empirical PSDs for the gravity disturbance signal: one for flat areas and one for mountainous areas. Details of these models can also be found in Schwarz and Li (1996). For the conversion of the gravity signals a typical cruise speed for a gravity survey with a regular fixed-wing aircraft was used, i.e. 70 m/s. As expected, the figures show that the tactical grade system has much higher noise and that the weaker gravity signal in flat areas drives the system’s accuracy. Figure 31.2 shows that for this particular IMU for vector gravimetry the signal power of the gravity disturbance is much lower than the system noise, making recovery of the gravity signal practically impossible. On the other hand, for scalar gravimetry the gravity disturbance signal power is slightly higher than the system noise power and recovery of the gravity signal is still possible. Frequencies higher than 102 Hz are problematic for both systems due to high frequency system noise. This is mainly due to the fact that GPS receiver noise is amplified when determining the kinematic accelerations (Schwarz and Li 1996). In Figs. 31.3 and 31.4 the same PSDs are shown, but now the typical cruise speed of the Light UAV (30 m/s) is used to convert the gravity signals. The reduced cruise speed shifts the gravity curves towards the lower frequencies. For the navigation grade system this seems to be an advantage. The short wavelengths
-2
f (Hz)
Fig. 31.3 INS/GPS system noise PSDs and gravity disturbance signal PSDs for a navigation grade inertial system at a cruise speed of 30 m/s (108 km/h)
10 10
PSD (mGal 2/Hz)
Fig. 31.2 INS/GPS system noise PSDs and gravity disturbance signal PSDs for a tactical grade inertial system at a cruise speed of 70 m/s (252 km/h)
10
10 10 10 10 10 10
9
200 100
λ (km) 20 10
50
5
2
1
INS/GPS (scalar) INS/GPS (vector) Gravity (flat) Gravity (mountain)
8
7 6
5
4 3
2
1
10 -4 10
10
-3
10
-2
f (Hz)
Fig. 31.4 INS/GPS system noise PSDs and gravity disturbance signal PSDs for a tactical grade inertial system at a cruise speed of 30 m/s (108 km/h)
of the gravity disturbance signal (<5 km) now become better observable. This is important for areas such as resource exploration. On the other hand, the lower frequencies (<6 104 Hz) or longer wavelengths (>50 km) move out of the observable range, but only for vector gravimetry. This is mainly due to low frequency INS attitude noise. The shift of the gravity curves also means that for lower speeds high frequency GPS errors (>102 Hz) are no longer a problem for the estimation of the gravity signal.
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For the tactical grade system it can be seen that for the lower UAV speed a large portion of the gravity disturbance signal falls below the system noise level, indicating that the recovery of the gravity becomes more difficult for the reduced speed and the system’s performance degrades. In particularly, the short to medium wavelengths (<30 km) are affected. This means that Light UAVs with a tactical grade system are less suitable to use for cm-level geoid determination or resource exploration.
31.4.2 Flight Dynamics Due to their different size, shape and mass UAVs respond differently to external forces than regular aircraft. While it is difficult to model what external forces may affect a particular UAV, these force will excite the natural modes of a UAV. For airborne gravimetry the phugoid (pitch mode) is the most important natural mode, as it is known to interfere with the measurements (Bruton 2000; Tome´ 2002). Table 31.2 shows, for different cruise speeds, the approximate phugoid periods, obtained with equation: pffiffiffi v Tph ¼ p 2 g
(31.6)
where Tph is the phugoid period, v is the cruise speed, and g is the gravity acceleration. It can be seen that with lower speed, the wavelength corresponding to the phugoid period becomes shorter. For UAV speed (30 m/s) the corresponding phugoid wavelength falls below 1 km spatial wavelength. It should be noted that the above equation is less accurate for smaller aircraft. Nonetheless the values in Table 31.2 provide a good indication of the impact of reducing the cruise speed. It suggests that for the reduced cruise speed of a UAV filtering out the
Table 31.2 Phugoid period and corresponding frequency and spatial wavelength for different vehicle speeds v ðm=sÞ 30.0 70.0
v ðkm=hÞ 108.0 252.0
Tph ðsÞ 13.6 31.7
fph ðHzÞ 0.0736 0.0315
lph ðkmÞ 0.4076 2.2192
phugoid interference still allows the recovery of the gravity signal to less than 1 km resolution.
31.4.3 Endurance From the PSD analysis of Sect. 31.4.1 it can be seen that the recovery of long wavelength (low frequency) information of the gravity signal is limited by the performance of the INS. But the INS is not the only limiting factor. The maximum wavelength that can be recovered is also related to the size of the surveyed area (Rayleigh criterion). For UAVs this is defined by the UAV’s flight range. The range R follows from: R ¼ vE
(31.7)
where E is the endurance and v is the cruise speed. It is assumed that for logistical reasons the UAV would need to return to its original starting location. The maximum flight line length is therefore half of the UAV’s range. The flight line length sets the size of the surveyed area. Table 31.3 shows the approximate maximum recoverable wavelength for different endurances and a cruise speed of 30 m/s. Note that for the calculation other factors such as the time to fly to the survey area and the time to climb to surveying altitude were not taken into account. For example, to augment the GOCE mission mentioned in the introduction with spatial resolutions up to 100 km and to cover a survey area of 200 km by 200 km, a UAV with an endurance higher than 3.7 h could be used; although the vehicle would need to return for refuelling after each flight profile. If the flight lines are separated by 10 km (Schwarz et al. 1994), ten or more flights would be required to cover such a survey area. If a UAV’s endurance is a multiple of 3.7 h it will be able to perform multiple
Table 31.3 Maximum recoverable gravity wavelength for different endurances (v ¼ 30 m/s) Endurance ðhÞ 1 3 5 7
Range ðkmÞ 108 324 540 756
Flight line ðkmÞ 54 162 270 378
lmax ðkmÞ 54 162 270 378
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flight profiles, making it more practical for actual use in a survey.
can induce non-linearity effects into the INS error. This subject remains to be investigated. Although the endurance of a UAV sets limits to the range and therefore limits the maximum recoverable wavelength of the gravity signal, it does not pose any problems. UAVs with endurance larger than 3.7 h can already be used for local geoid augmentation up to the 200 km wavelength.
Conclusions
The effects of using strapdown gravimetry systems on board of Light UAVs is discussed by analysing the impact of the Light UAV’s flight speed (30 m/s), endurance (several hours), and flight dynamics. A PSD analysis of the GPS/INS system errors was performed to show effect of a Light UAV’s low cruise speed. Two types of systems are considered: one based on a navigation grade inertial system, and one based a tactical grade system. Both systems are assumed to use dual-frequency doubledifferenced GPS phase measurements. The analysis suggests that for the navigation grade inertial system a Light UAV’s low cruise speed can be an advantage in recovering the short wavelengths (<5 km) of the gravity field. This is especially true for the case of scalar gravimetry, which is less limited by the low frequency INS attitude errors. The PSDs for the tactical grade system (such as a Litton LN-200) show that the low speed is a disadvantage. It limits the recovery of the short to medium wavelengths (<30 km) of the gravity field. A tactical grade system is therefore less suitable for use with Light UAVs in cmlevel geoid determination or resource exploration. Analysis of the flight dynamics suggests that the use of Light UAVs in airborne gravimetry is advantageous. The low speed results in a short phugoid period (approximately 13.6 s). Even after filtering out the phugoid interference it is still possible to recover the gravity signal to less than 1 km resolution. Possible effects of higher dynamics due to, for example, turbulence were not considered. These effects may be important since higher dynamics
Acknowledgement Richard Deurloo is supported by a Ph.D. grant (SFRH/BD/25618/2005) from the Fundac¸a˜o para a Cieˆncia e Tecnologia (FCT). This funding is greatly acknowledged.
References Bastos L, Tome P, Cunha T, Cunha S (2000) Gravity anomalies from airborne measurements – experiments using a low cost IMU. In: Proceedings of Gravity, Geoid and Geodynamics, IAG Symposium 123, Springer, New York, pp 253–259 Bruton, A.M. (2000). Improving the accuracy and resolution of SINS/DPGS airborne gravimetry. Ph.D. Thesis, Department of Geomatics Engineering, The University of Calgary, December 2000 DoD (2002). Unmanned Aerial Vehicle Roadmap 2002–2027. Office of the Secretary of Defense, Department of Defense, United States of America, December 2002 Jekeli C (2001) Inertial navigation systems with geodetic applications. Walter de Gruyter, Berlin Johannessen, J.A., and M. Aguirre–Martinez (1999). The Four Candidate Earth Explorer Core Missions – Gravity Field and Steady-State Ocean Circulation. Reports for mission selection 1, European Space Agency, July 1999, ESA SP–1233 Schwarz KP, Li YC (1996) What can airborne gravimetry contribute to geoid determination? J Geophys Res 101 (B8):17873–17881 Schwarz KP, Li YC, Wei M (1994) The spectral window for airborne gravimetry and geoid determination. In: Proceedings of the international symposium on kinematic systems in geodesy, geomatics, and navigation, Banff, Canada, pp 445–456, August 30 – September 2 Tome´ P (2002) Integration of inertial and satellite navigation systems for aircraft attitude determination. Ph.D. Thesis, Faculty of Sciences, University of Porto, Porto, Portugal, January 2002
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Z. Jiang, E.F. Arias, L. Tisserand, K.U. Kessler-Schulz, H.R. Schulz, V. Palinkas, C. Rothleitner, O. Francis, and M. Becker
Abstract
The goal of maintaining an accurate gravity network at the BIPM headquarters is twofold: firstly to support the International Comparison of Absolute Gravimeters (ICAG), and secondly to support the BIPM watt balance (WB) project, which aims at determining the Planck constant h or realizing a future new definition of the kilogram based on a fixed value of h. In addition the absolute gravity measurements, Relative Gravity Campaign (RGC) is organized as part of each ICAG. The BIPM gravity network is characterized by its small size, indoor laboratory conditions, three-dimensional structure, and large number of parallel absolute gravity determinations. Over the last 3 decades, repeated precise horizontal and vertical ties have been measured using relative and absolute gravimeters, and precision leveling has been undertaken regularly to monitor the deformation of the terrain. The ICAG is held every 4 years; the 8th ICAG took place in mid-2009 and for the first time, it was organized as a metrological key comparison as defined by the CIPM MRA. Its results will such constitute a precise and consistent gravity
Z. Jiang (*) E.F. Arias L. Tisserand International Bureau of Weights and Measures (BIPM), Pavillon de Breteuil 92312 SEVRES CEDEX, France e-mail:
[email protected] K.U. Kessler-Schulz H.R. Schulz Angewandte Gravimetrie (AG), Rosengarten, Germany V. Palinkas Geodetic Observatory Pecny (GOP), Research Institute of Geodesy, Topography and Cartography, Ondrejov, Czech Republic C. Rothleitner O. Francis University of Luxembourg (UL), Luxembourg, Luxembourg M. Becker (*) Institute of Physical Geodesy, Darmstadt University of Technology (IPGD), Darmstadt, Germany e-mail:
[email protected] S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_32, # Springer-Verlag Berlin Heidelberg 2012
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reference system in SI units which can be used as the global basis for geodetic and geophysical observations. Additionally, to support the WB project, the network was extended to the BIPM WB laboratory. In this paper, we briefly recall the background of the ICAG/RGC2009 and outline the new characteristics of the updated network, the organization and the performance of the measurements. Finally we present preliminary results from the RGC2009.
32.1
Introduction
Notation: Gal ¼ 1 cm s2 g: Absolute gravity acceleration value in mGal (minus a constant value of 980 900 000 mGal); RG/AG: Relative/Absolute gravimeter; dg: Difference of g measured by RG; dg/dH: Vertical gravity gradient; KC/PS: Key Comparison/Pilot Study [1, 2]; Site, station and point: A site is comprised of one or several stations located in an isolated indoor laboratory. The three sites at the BIPM headquarters used in ICAG2009 are: A, B and WB (Fig. 32.4); a station is comprised of 3–5 points vertically aligned and is marked by a benchmark (Figs. 32.2, 32.3 and 32.4); a point is the location at 30 cm, 90 cm, 130 cm, 155 cm or 170 cm vertically above the benchmark of a station (Fig. 32.1a). Simple schedule: the basic RG measurement set designed for the KC; Full schedule: a strengthened RG measurement schedule designed for the PS with more redundant measurements and closure constraints; WB schedule: additional RG measurements at the watt balance site.
32.1.1 Background An International Comparison of Absolute Gravimeters (ICAG) takes place every 4 years at the BIPM headquarters in Se`vres, France. The 8th ICAG (2009) was organized in accordance with a proposal made at the 3 rd Joint Meeting of the Working Group on Gravimetry of the Consultative Committee for Mass (CCM WGG) and the SGCAG 2.1.1 of the International Association of Geodesy (IAG), held on 24 August 2007 [1, 3]; it was agreed that the main part of
ICAG2009 should be considered a key comparison (KC, CCM.G-K1 [2]) under the terms of the Mutual Recognition Arrangement of the International Committee for Weights and Measures (CIPM MRA, www.bipm.org/en/convention/mra). This decision constitutes an important step in accurate gravimetry applications as it means that the resulting ICAG2009, traceable to the International System of Units (SI), can be used as the global basis for geodesic and geophysical observations. Measurements not forming part of the KC itself were undertaken in the framework of a pilot study (PS). The associated RGC2009 was therefore adapted to better support the new functions of the ICAG [3] and the WB project. The result is an extended, updated and SI-traceable gravity network at the BIPM. The technical specifications of the RGC2009 were discussed during two meetings of the steering committee held at the BIPM headquarters in Se`vres on 21 Nov. 2008 and in Prague on 11–12 May 2009. A number of essential points were agreed during these two meetings: 1. The main role of RGC2009 is to support the KC and PS of the ICAG2009; the results are the set of gravity distributions above the gravity stations (with their uncertainties) which allow the individual absolute determinations to be reduced to the same reference to permit their comparison. 2. The result of the RGC2009 should be independent of the absolute result of ICAG2009, i.e., the RGConly computation does not require calibration from the ICAG. 3. The data recording and submission should be fully digital. 4. The measurements should be made using at least 5 gravimeters following carefully designed schedules. 5. The measurements should be closely synchronized with the ICAG2009.
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a 11 13
170cm
b
Simple schedule : 1-7 Full schedule: 1-10 WB schedule : 1-14
130cm
90cm
30cm
3
6
9 12
24
1 5 7
8
10
14
d
Ground
c 170cm
Sensor 30cm
15cm 25cm
40cm
60cm
Fig. 32.1 (a) Vertical dg measurements at a station following simple, full and WB schedules for a Scintrex CG gravimeter. (b) The BIPM tripods strengthened with removable legs for the measurements performed at heights of 155 cm and 170 cm. (c) Set-up of the BIPM fixed-level tripod measurements using a Scintrex CG5 at a height of 170 cm at stations W1 and W2 (watt balance laboratory). (d) Set-up for measurements with a ZLS Burris gravimeter at a height of 155 cm at stations W1 and W2
6. Additional measurements should be organized in the framework of the BIPM watt balance (WB) project. 7. Measurements should be made using strengthened tripods to increase the stability (Fig. 32.1b) and to enable measurements at heights of 155 cm and 170 cm as required by the WB project.
Based on the above considerations and the experience gained in the previous RGCs [4, 5], it was decided that the best performing relative gravimeters should be used, for their advantages in calibration and digital data management. Nine selected Scintrex CG and ZLS BURRIS gravimeters were invited to participate.
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Table 32.1 List of the participants in the RGC2009 No. 1
Institute/abbreviation BIPM
Gravimeter Scintrex CG5
No.# S348
3/4
University of Luxembourg (UL)
Scintrex CG5
5 6/7
LNE-SYRTE (LS) Bureau de Recherches Ge´ologiques et Minie`res (BRGM)
Scintrex CG5 Scintrex CG5
S008 S010 S105 S028 S539
8 9
Finnish Geodetic Institute (FGI) Angewandte Gravimetrie (AG) RIGTC, Geodetic Observatory Pecny (GOP)
Scintrex CG5 ZLS Burris ZLS Burris
10
32.1.2 Organization of RGC2009 Unlike the previous ICAGs, the ICAG2009 consisted of two parts: a key comparison (KC) and a pilot study (PS) as described in detail in [1–3]. The measurement schedules and the data-processing strategy of the related “relative” campaign RGC2009 are described below. Matthias Becker (IPG DTU) and Zhiheng Jiang (BIPM) were charged with organizing the measurements and data processing.
32.1.3 Participants of RGC2009 Seven organizations with 9 gravimeters took part in RGC2009 (Table 32.1). The measurements were made during two periods: in August 2009 with 5 gravimeters (from BIPM, UL, AG and GOP) and in October 2009 with the other 4 gravimeters (from LS, BRGM and FGI). The fourth column in Table 32.1 (No.#) shows the last three digits of the serial number of each gravimeter. Note that gravimeter No. 2 is not listed, as finally it was unable to participate.
32.2
Design of the Vertical and Horizontal dg Measurement Schedules
The measurement schedule was designed to yield the lowest possible uncertainty in dg under the BIPM laboratory conditions, given an achievable amount of work. The measurement scheme had a closure-based sequence with short and symmetrical time-distance
S052 B025
Operator L. Tisserand Z. JIANG O. Francis C. Rothleitner S. Merlet P. Jousset D. Lequin F. Dupont J. M€akinen K.U. Schulz, H.R.Schulz
B020
V. Palinkas
intervals so as to minimize the influence of the uncertainties due to gravimeter zero-drift (reference point displacement with time), set-up of the gravimeters, and displacement and environmental influences. Level fixed tripods were used for the vertical dg measurements to avoid errors in the height measurements. The main point of each station is defined at 90 cm vertically above the ground surface marker to reduce the near ground non-linear variation in g. The ZLS RGs are always set up to be oriented to the north. The RG sensor is vertically above the benchmark and close to the height of the defined point to minimize the eccentricity. A typical occupation takes about 5 min, including setting up the gravimeter, allowing 30 s for stabilization, then making two readings averaged out of two samples of 100 s recordings separated by a 10 s pause. As described above, there were three categories of schedule; the simple schedule designed for the KC, the full schedule for the PS and for further scientific studies where more redundant measurements and closure constraints are needed, and an additional WB schedule to investigate the local field in order to determine the g-value at the center of test mass of the WB. Raw digital recordings were reported to the BIPM, without any corrections such as for the Earth tide or zero-drift corrections. In comparison with the previous RGCs, hold in 2001 [4] and 2005 [5], the main novel features of the RGC2009 are that: 1. Nine of the best performing gravimeters in Europe were invited to participate, including 7 Scintrex CG and 2 ZLS Burris. All have digital recording facility and owner calibration. The out-door sites C1 and C2 [5] were used to verify the owner calibrations.
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2. The number of the total occupations of a meter for RGC2005 was 157 which took 2–3 days. In 2009 there were 96 occupations in the simple schedule and 163 in the full schedule. 3. Instead of simply increasing the repeated measurements over the same ties, more geometry or closing conditions are designed in the full schedule, such as additional horizontal ties at heights of 30 cm and 130 cm, producing a 3D grid instead of a simple horizontal net at 90 cm as before. This redundancy allows the constraints in the adjustment to be strengthened and gives more conditions to estimate the uncertainty and to improve the results of the ICAG, e.g., to directly compare the FG5 at 130 cm height. 4. A dedicated WB schedule was organized on the WB site with 101 additional occupations. The purpose of these measurements was to map the local g distribution at the site of the WB before and after setting up the watt balance, using the so-called remove-restore technique. Repeated precision leveling measurements have also been carried out between the benchmarks of all the stations. The leveling network is linked to the French national height reference (IGN69).
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for the full schedule. The measurements followed two schemes as illustrated in Fig. 32.2: the “odd” scheme for the odd-numbered gravimeters (1, 3, 5, 7 and 9 in Table 32.1) and the “even” scheme for the even-numbered gravimeters (4, 6, 8 and 10 in Table 32.1). There were 10 occupations at a height of 90 cm in the simple schedule and 10 occupations at each of 30 cm, 90 cm and 130 cm in the full schedule. WB site: Horizontal dg ties at the WB site were measured between W1 and W2 as illustrated in Fig. 32.3. W1 is the gravity station and W2 is the location of the BIPM watt balance for which the reference height will be about 1.5 m (not confirmed as yet). The gravity pillar W1 is founded 1.5 m deep in the B6
B2
B3
B
B1
B4
B5
Odd schema: B-B1-B5-B-B2-B6-B-B3-B4-B for BIPMS348, UL-S008, LS-S105, BRGM-S539 and AG-B25 B2
B6
B3
32.2.1 Vertical dg Ties Figure 32.1a illustrates the schedule of the vertical dg measurements for the gradient determination at each station. These measurements were realized with the help of a set of fixed-level tripods. The simple-schedule included 7 occupations at each station, and the full schedule included 10 occupations. The measurements were made at the 8 stations A, B, B1, B2, B3, B4, B5 and B6, giving a total of 56 occupations in the simple schedule and 80 in the full one. In the WB schedule, additional points at 155 cm height for the ZLS meters and 170 cm height for the Scintrex meters were measured at the stations W1 and W2 so as to coincide with the reference height of the BIPM watt balance. Figures 32.1c, d show the set-up used for the vertical dg measurements.
32.2.2 Horizontal dg Ties B site: Horizontal dg 3D-grid measurements at site B were performed at a height of 90 cm for the simple schedule and at heights of 30 cm, 90 cm and 130 cm
B
B1
B4
B5
Even schema: B-B1-B2-B-B3-B6-B-B4-B5-B for UL-S010, BRGM-S028, FGI-S052 and GOP-B020
Fig. 32.2 Horizontal dg grid measurement schedules at site B performed separately for heights of 30 cm, 90 cm and 130 cm
W2
W1
Fig. 32.3 Horizontal dg tie measurements scheduled at site WB for heights of 30 cm, 90 cm, 130 cm, 155 cm and 170 cm. W1 is the gravity station and W2 is the BIPM watt balance location
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Site A
W3
W10
Site WB
100cm
70cm
W4
W8
Site B
W2
Fig. 32.4 Outdoor horizontal dg measurements scheduled between the sites A, B and WB (simple schedule at a height of 90 cm; full schedule at heights of 30 cm, 90 cm and 130 cm
ground and has a top surface of 1.5 m 1.5 m which is 6 cm lower than the surrounding ground. The WB pillar also has a foundation of 1.5 m in the ground and a top surface of 4 m 2.5 m which is at the same level as the surrounding ground. There were 5 occupations at each station for each height of 30 cm, 90 cm, 130 cm, 155 cm and 170 cm. Outdoor ties between the sites A, B and WB: Outdoor horizontal dg measurements were scheduled between the sites A, B and WB as shown in Fig. 32.4 at a height of 90 cm for the simple schedule and at heights of 30 cm, 90 cm and 130 cm for the full schedule. There were 10 occupations for the simple schedule and 21 occupations for the full schedule with 7 occupations at each height. W2 station: In order to set up a precise model of the local gravity field around the BIPM watt balance, a 3D dg grid was constructed based on measurements made at five horizontal levels at heights of 30 cm, 90 cm, 130 cm, 155 cm and 170 cm above the station W2 (Fig. 32.5). The scheme consisted of four separated triangles, each triangle being closed at the point W2. To reduce the total number of measurements required, other ties, e.g., between W3, W5, W7 and W9, were not measured. Similarly, the heights of 155 cm and 170 cm were not measured by both of the models (Scintrex and ZLS). In total there were thus 52 occupations at heights of 30 cm, 90 cm, 130 cm, 155 cm and 170 cm in the WB schedule. WB site: To investigate the gravity variation in the WB laboratory, a grid consisting of the dg profiles was measured. The horizontal dg was measured at the
W9
W6 W5
W7
Fig. 32.5 Scheme of the horizontal 3D dg grid measurements scheduled at station W2 at heights of 30 cm, 90 cm, 130 cm, 155 cm (for ZLS Burris) and 170 cm (for Scintrex CG5)
nodes of the grid with a cell – size of 2 m 2 m, 130 cm above the ground. The grid consists of four independent closures as illustrated in Fig. 32.6. In total 42 occupations were organized for the WB schedule. Table 32.2 displays the total number of occupations included in the different schedules. An adapted schedule was printed and distributed to the operators of the gravimeters. It was important that the schedules be tightly respected so that the designed uncertainty would be attained and all the results would be identifiable from the raw recordings. The latter is particularly important when identifying the causes of an outlying data point, which might arise due to measurement errors or simply typing errors in a point name. In total there were 96 and 163 occupations in the simple and full schedules, and 101 additional occupations in the WB schedule.
32.3
Uncertainty Evaluation for a dg
The measurement uncertainty was estimated assuming ideal indoor air-conditioned laboratory conditions, characterized by small values of dg, small and symmetric time intervals and short distances between the points, use of fixed-height tripods and respect of the carefully designed closure-based measurement schedules. Table 32.3 lists the standard uncertainty budget of a dg obtained during one measurement by a single RG [1, 5]. Here “Eccentricity of gravimeter sensor” refers
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Fig. 32.6 The horizontal dg grid measured in the WB schedule at the nodes of the grid on the WB site of which the cell surface is 2 m 2 m and 130 cm above the ground in the BIPM watt balance laboratory
6m W2 2.5m
4m 9.3m 1.5m W1
Table 32.2 Number of total occupations per gravimeter involved in each schedule dg Vertical dg Horizontal dg at B Horizontal dg between W1 and W2 Horizontal dg at W2 pillar Horizontal dg grid at the WB room Outdoor dg between A, B and W1 Total
Simple schedule 70 9 7
Full schedule 100 27 15
WB schedule – – 7
– –
– –
52 42
10
21
–
96
163
101
Table 32.3 Estimated standard uncertainty of a dg (gravimeter-measurement) No.
Source of uncertainty
1 2 3 4 5 6 7 8 9 10 11
Resolution of gravimeter readout Scale factor Feedback and non-linearity Un-leveling effect Environmental effects (e.g., Temperature) Transport/displacement Atmosphere pressure correction Eccentricity of gravimeter sensor Tidal corrections Zero-drift correction Others TOTAL
Std. Uncertainty /mGal 1.0 0.5 0.5 1.0 1.5 1.0 0.1 1.5 0.5 1.5 2.0 3.8
to the uncertainty due to the gravimeter sensor not being located exactly on the required point. The total combined standard uncertainty is about 4 mGal. Assuming there are M RGs and each has N measurements, the uncertainty of the mean value of M N measurements is 4.0/(M N), which is 2 mGal for M N 4. In the simple schedule for the vertical gravity gradient measurements, M N 16, i.e., the uncertainty of the gradient correction 4.0/(M N)/0.4 m 2.5 mGal/m.
32.4
Preliminary Result
Figure 32.7 shows histograms of the residuals of an adjustment for five of the gravimeters participating in the measurement campaign. The standard uncertainties of the ZLS Burris B020 and Scintrex CG5 S348 gravimeters are 1.3 mGal and 1.6 mGal respectively. Tables 32.4 and 32.5 give preliminary results. Table 32.4 presents the polynomial coefficients of the gradients and Table 32.5 the gravity differences at a height of 90 cm between the BIPM network stations. Conclusion
ICAG2009 was the 8th in the series of International Comparisons of Absolute Gravimeters, but the first metrological key comparison, the results of which will be used for the BIPM watt balance project. The accompanying campaign of relative gravity
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Fig. 32.7 Histograms of the adjusted residuals for the gravimeters B020 and S348. The vertical axis in each case shows the residuals in mGal and the horizontal axis is the frequency of a residual falling into the related interval
0
10
20
30
40
–7 –6
2
–5
0
–4
0
–3 μgal
ZLS Burris B020 σ = 1.3 μGal
1
–2
6 28
–1
35
0 18
1 2
7
3
0
4 0
10
20
10
0
20
30 30
40 40
50
–6
–4
Scintrex CG5 S348 σ = 1.6 μGal
4 3
–2
21 32
μgal
41
0 32 17
2 5 3
4
6 0
10
20
30
40
50
Table 32.4 Polynomial fitting of the gravity gradients, g(H) ¼ a+bH + cH, with H in m, a in mGal, b in mGal/m and c in mGal/m2
Table 32.5 Adjusted gravity differences in mGal between the ICAG points at a height of 90 cm
Station A B B1 B2 B3 B4 B5 B6 W1 W2
St. A B B1 B2 B3 B4 B5 B6 W1 A 0.0 B 2316 0.0 B1 2310 -6.1 0.0 B2 2295 -21.1 -15.0 0.0 B3 2300 -16.2 -10.1 4.9 0.0 B4 2312 -3.4 2.7 17.7 12.8 0.0 B5 2317 1.3 7.4 22.4 17.5 4.7 0.0 B6 2296 -19.8 -13.7 1.3 -3.6 -16.4 -21.1 0.0 W1 769 -1546 -1540 -1525 -1530 -1543 -1548 -1527 0.0 W2 693 -1622 -1616 -1601 -1606 -1619 -1624 -1603 -75.8
a 25982.11 28288.56 28275.37 28256.38 28273.67 28291.90 28289.87 28264.56 26714.51 26635.89
b -313.77 -301.73 -296.70 -291.97 -304.53 -312.67 -302.27 -299.53 -269.17 -271.13
c 4.667 2.833 6.000 5.667 4.333 6.667 3.417 5.583 0.417 6.083
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measurements (RGC2009) was adapted correspondingly and the BIPM network was updated to follow the new developments. New features were that only selected relative gravimeters were used, the network was extended to the watt balance laboratory, new points were measured at heights of 150 cm and 170 cm and strengthened tripods were employed to improve the stability of the measurement set-up. The resulting network is a 3D-grid. In addition, new precision leveling was performed at all the gravity stations. Measurement schedules employing two types of relative gravimeters were carefully designed to increase redundancy and constraints in order to reduce the uncertainty of the dg and enable to study the performance of individual gravimeters. The performance of ZLS and Scintrex gravimeters was compared and our preliminary analysis indicates that they agree very well within the estimated uncertainty. Detailed discussions will be given in a separate paper including the final results, the uncertainty analysis, comparisons with the results of the earlier ICAGs and that of the absolute determinations as well as the gravimeter behaviors etc. Accurate knowledge of the BIPM gravity network provides a basis for monitoring the stability of the local gravity field at the BIPM.
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References 1. BIPM, Technical Protocol of the 8th International Comparison of Absolute Gravimeters ICAG-2009 2. CIPM MRA Key Comparison CCM.G-K1. http://kcdb.bipm. org/appendixB/KCDB_ApB_result.asp?cmp_idy¼935& cmp_ cod¼&search¼&cmp_cod_search¼&page¼&met_ idy¼ &bra_idy¼&epo_idy¼&cmt_idy¼&ett_idy_org¼ &cou_ cod¼ 3. BIPM, 1st Circular letter about Relative Gravity Campaign 2009. ftp://tai.bipm.org/ICAG/2009/RGC/FirstLetter/ 4. Vitushkin L, Becker M, Jiang Z, Francis O, van Dam TM, Faller J, Chartier J-M, Amalvict M, Bonvalot S, Debeglia N, Desogus S, Diament M, Dupont F, Falk R, Gabalda G, Gagnon CGL, Gattacceca T, Germak A, Hinderer J, Jamet O, Jeffries G, K€aker R, Kopaev A, Liard J, Lindau A, Longuevergne L, Luck B, Maderasl EN, M€akinen J, Meurers B, Mizushima S, Mrlina J, Newell D, Origlia C, Pujol ER, Reinhold A, Richard Ph, Robinson IA, Ruess D, Thies S, van Camp M, van Ruymbeke M, de Villalta Compagni MF, Williams S (2002) Results of the Sixth International Comparison of Absolute Gravimeters ICAG-2001. Metrologia 39:407–24 5. Jiang Z, Becker M, Francis O, Germak A, Palinkas V, Jousset P, Kostelecky J, Dupont F, Lee CW, Tsai CL, Falk R, Wilmes H, Kopaev A, Ruess D, Ullrich MC, Meurers B, Mrlina J, Deroussi S, Me´tivier L, Pajot G, Pereira Dos Santos F, van Ruymbeke M, Naslin S, Ferry M (2009) Relative Gravity Measurement Campaign during the 7th International Comparison of Absolute Gravimeters (2005). Metrologia 46:214–226
.
Precise Gravimetric Surveys with the Field Absolute Gravimeter A-10
33
€ller, N. Lux, H. Wilmes, and H. Wziontek R. Falk, Ja. Mu
Abstract
The A-10 field absolute gravimeter produced by the company Micro-g LaCoste Inc. was found to comply well with the producer specification of an uncertainty of 100 nm s2 for the determination of gravity acceleration. Repeated observational checks at a known reference station and careful calibration of instrumental standards demonstrated that the gravity measurements quality could further be enhanced. This opens new applications for precise gravimetry like the establishment of reference networks for monitoring global change processes where uncertainties of a few 10 nm s2 are of high value. The results and experiences from two extensive field campaigns using an A-10 gravimeter are presented.
33.1
Introduction
The Federal Agency for Cartography and Geodesy (BKG) has been performing basic measurements for the gravity reference system in Germany using the FG5-101 absolute gravimeter since 1993. This absolute gravimeter (AG) is regularly checked through the participation in international comparisons at the Bureau International des Poids et Mesures (BIPM), thereby representing a component of the international gravity reference system. Periodical parallel measurements of the FG5-101 in Bad Homburg with a continuously registering superconducting gravity meter (SG) and comparative measurements with other AGs form the
Note: The mention of any commercial products and company names is for information only, and does not constitute an endorsement by BKG. FG5 and A-10 are trademarks of Micro-g LaCoste Inc., CG5 is a trademark of SCINTREX Ltd. and WEO100 is a trademark of Winters Electrooptics Inc. R. Falk Ja. M€uller N. Lux H. Wilmes (*) H. Wziontek Federal Agency for Cartography and Geodesy, Richard-StraussAllee 11, 60598 Frankfurt/M, Germany e-mail:
[email protected]
basis of the BKG gravimetric reference station (Wilmes et al. 2003). FG5 measurements require stable environmental conditions and are preferably performed at inside locations with observation piers. A use of the instrument in the open field would only be possible with a great deal of logistical support (e.g. temporary measurement container or tent, preparation of a measurement pier). For this reason, BKG supported the development of a field absolute gravity meter at Micro-g LaCoste Inc. (MGL), purchased prototype A-10#002 in 1999 and contributed to the instrument development with a study of practical use and areas of application (Holweg 2001). After essential modifications by MGL the instrument was again at the disposal of BKG in 2004 and in the same year BKG purchased a second instrument, A-10#012 from the newly launched series production. Both gravimeters have been used in projects since then, vide e.g. (Flury et al. 2007). Schmerge and Francis (2006) performed measurements with another A10 at the Underground
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_33, # Springer-Verlag Berlin Heidelberg 2012
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Laboratory for Geodynamics in Walferdange (Luxembourg) over a period of about 1 month to investigate the instrument accuracy. These laboratory results were clearly better than the specifications of the producers. Earlier, promising investigations by Liard and Gagnon (2002) with an A-10 prototype, but using an external iodine stabilized laser, also evoked optimism. However, information on the long term stability of the A-10 in the field is not yet published and is becoming available now, through two nation-wide projects performed by BKG.
33.2
Theory of Operation
The A-10 absolute gravimeter (Fig. 33.1) is a portable, high precision, free fall instrument for measuring vertical acceleration of gravity (g) in the field. A test mass is dropped by a mechanical device in a vacuum chamber over an average distance of 7 cm. The A-10 uses a laser, an interferometer, a long period inertial isolation device (super spring) and an atomic clock to accurately determine the position of the free-falling test mass as it accelerates due to gravity. The acceleration of the test mass is calculated directly from the measured trajectory (Micro-g LaCoste 2008). The producer specification for the determination of gravity is 100 nm s2. The A-10 is capable of measuring
Fig. 33.1 A-10 Schematic (courtesy of Micro-g LaCoste Inc.)
gravity at a rate of one drop per second, so that the statistically required data quantity for a station is available within less than 1 h. The gravimeter consists of two units which are placed at the site on top of each other and are connected to a controller unit and laptop PC. For field campaigns, the gravimeter is transported in a measuring vehicle and only the two measurement units need to be placed to the site during observations. For continuous power supply and temperature control all components remain connected during transports.
33.3
Calibration of Instrument Standards and Instrument Accuracy
The functional status of the A-10 gravimeter needs to be monitored on a continuous basis. This can be done through repeated calibration of the two physical standards, i.e. the stabilised laser and the rubidium atomic clock in the instrument. However, only comparison measurements with a more accurate instrument (e.g. FG5) or observations at a site with well investigated gravity, a gravimetric reference station, can monitor the correct function and long-term stability of the A-10 or reveal error sources. The rubidium normal in the A-10 as a time scale showed aging of about 1 1010 per year which, if neglected, leads to apparent gravity change of 2 nm s2 per year. The A-10 employs a HeNe polarization-stabilized laser (ML-1) as length standard with a stabilisation principle which results from two laser modes with orthogonal linear polarization. The two laser modes (“red-shifted” and “blue-shifted”) are alternately used during the gravity measurement with the A-10 gravimeter and therefore both wavelengths need to be calibrated carefully. The stabilisation principle of the laser assumes that the frequency shifts of the two laser lines, induced by temperature changes and aging, balance to a major part (Micro-g LaCoste 2008). The ML-1 laser is rugged and powerful and well-adapted to the A10, but can cause significant offsets in the measured gravity if not monitored repeatedly. Studies performed by Niebauer et al. (1988) as well as by M€akinen and Stahlberg (1998) report long term drifts of more than 1 MHz per year corresponding to apparent gravity changes of more than 20 nm s2 per year if neglected.
33
Precise Gravimetric Surveys with the Field Absolute Gravimeter A-10
In the first years of application at BKG, calibration of the A-10 laser was only performed during maintenance checks at the instrument manufacturer or during a comparison campaign at BIPM in 2005. In cooperation with the Technical University Darmstadt in 2007 a comparison unit was set up to check the instrumental standards used for absolute gravity measurements. The device is based upon a commercially available laser heterodyne system (Winters 2004) and an iodine-stabilized laser WEO100 (both produced by Winters Electro-Optics Inc.). The calibration unit uses an optical fibre adapter which can be placed upon the cover of the lower A10 unit to couple in the laser beam of the gravimeter (Fig. 33.2). Together with the use of a fibre combiner this simplified the adjustment effort and calibration work significantly. (Hahn 2007). Prior to and following each campaign, an A-10 measurement was carried out at the BKG gravimetric reference station in Bad Homburg (Wilmes and Falk 2006). This site is situated in the cellar of the castle of Bad Homburg and is equipped with four stable pillars of concrete. Figure 33.3 shows the combined residual gravity time series from continuous superconducting gravimeter (SG) recordings (continuous line) and monthly FG5 absolute gravimeter observations (blank dots) since 2006 together with the results of the A-10 reference measurements at Bad Homburg. Hereby the FG5 measurements define the absolute reference for the time series of the superconducting (relative) gravimeters. A comparison of the combined FG5 and SG time series with the repeated A-10#12 observations demonstrated that in 2008 it was not sufficient to use
Fig. 33.3 Observations of A10#012 (filled dots) at BKG gravimetric reference station Bad Homburg versus combined residual time series of SG (continuous line) and FG5-AG (blank dots) measurements between 2006 and 2009. All AG measurements are referred to site AA (@ 1.25 m) and are reduced by 9,810,550,450 nm s2
275
the calibration parameter provided after the maintenance in 2006. A yearly drift of 44 nm s2 deduced from this comparison agrees quite well with the results derived with the above mentioned laser calibration
Fig. 33.2 Comparison unit for physical standards during calibration of A-10 laser at BKG with (1) Caesium frequency standard , (2) Control electronics of Iodine-stabilised laser, (3) Spectrum analyzer and frequency counter, (4) Laser beam adapter (5) Fibre optics, fibre combiner and photodiode, (6) Iodine-stabilised laser head
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unit. The results from these frequency determinations are shown in Fig. 33.5. It also is clearly visible that each laser shows individual characteristics. The filled circles in Fig. 33.3 mark the A-10#012 reference measurements, reprocessed with the appropriate laser frequencies valid for the specific Fig. 33.4 Histogram of differences between A-10 observations and combined residual time series of SG at reference station Bad Homburg (cf. Fig. 33.3)
measurement epoch. More than 95% of all measurements agree within the instrument accuracy with the gravity reference, leading to an rms of 60 nm s2 (cf. Fig. 33.4). Through the unique possibilities at the gravity reference station Bad Homburg – continuously recording Deviation of A-10 measurements versus SG residual curve 30
Count
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drift of center frequency : –13 MHz year–1
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340 2010 390
A-10#012
–380
380
–390
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–400
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–410 –420 2004
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drift of center frequency : –22 MHz year–1
2005
2006
Fig. 33.5 Laser calibrations from determinations between 2004 and 2009 at different institutions: BIPM (2005), MGL (2004, 2006 and 1/2007) and since 4/2007 TUD and BKG. Red shifted laser mode frequencies are plotted with triangles (left-hand-axes) and blue shifted laser mode frequencies are plotted with squares (right-hand-axes). Drift of center frequency (mean of red shifted and blue shifted frequencies), plotted as
2007
2008
2009
340 2010
Blue shifted mode [MHz]
–370
Blue shifted mode [MHz]
Red shifted mode [MHz]
Red shifted mode [MHz]
Deviation [nm s–2]
circles, has the same scale, but arbitrary vertical position. (Red and blue shifted frequencies are given relative to Frequency component i of transition R (127) 11–5 in Iodine 127 (Mise en pratique of the definition of the metre, Recommendation 1 (CI1997) of the CIPM, 86st meeting, 1997): fi ¼ 473,612,214,705 MHz)
33
Precise Gravimetric Surveys with the Field Absolute Gravimeter A-10
SG and monthly repeated FG5 observations providing a combined residual gravity time series – the accuracy and long time stability of the A-10 gravimeters can be evaluated. We proved that the instrument A-10#012 can provide an accuracy of about 60 nm s2 at a laboratory site. As indicated from the results presented in Chap. 33.5, A-10#012 can reach an accuracy of better than 100 nm s2 (manufacturers specification) at field sites, assuming that instrument standards are checked on a regular basis (at least every 6 months).
33.4
Observation Procedure at Field Points
Typical field campaigns at BKG involve observations of about 10–20 points within 2 or 3 weeks. At each field-site (cf. Fig. 33.6) the absolute gravity value is observed using two independent instrument setups with opposite instrument orientation (“north” and “south”). Thus, setup errors and other bad influences (e.g. strong magnetic fields) can be identified early. Each setup is followed by at least six sets with 175 single drops each (between successive sets the laser mode lock was switched, cf. Sect. 33.3).
Fig. 33.6 A-10#012 at a field site of project “DHHN”
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During the measurements, the operator performs a rough quality analysis for the following parameters: 1. Standard deviations for single drops should not exceed 1 mm s2 2. rms from a least squares fit of the drop parabola should not exceed 1 nm 3. Standard deviation for single sets should not exceed 60 nm s2 4. Differences between the two independent setups should not exceed 80 nm s2 After the A-10 observation the local vertical gravity gradient is measured using a Scintrex CG5 relative gravimeter. The local vertical gravity gradient is essential for a benchmark tie because the gravity value measured by the A-10 absolute gravimeter is related to a height of about 70–80 cm above setup surface, depending on the use of tripods. A special gradient tripod, assembled at BKG, can be placed at the same positions as the A-10 gravimeter and has proved very helpful (cf. Fig. 33.7). For adverse installations, the tripod tubes can also be used as extension legs for the A-10.
33.5
Field Campaigns and Results
Within the two projects – “GOCE GRAND II” and “Renewal of the German first order height network (DHHN)” – 180 field points were observed with A-10#012 between 2006 and 2009. In order to check and improve the quality of the terrestrial gravity data base, within the project “GOCE-GRAND II” spotchecks at 94 existing stations evenly distributed over Germany were performed. The measurements had the purpose of detecting systematic errors and level differences within the existing terrestrial gravity data (mainly within the first order gravity network), because all further measurements are dependent on this hierarchy (Ihde et al. 2010). The measurements in the GOCE GRAND II project were made on very diverse types of platforms, on the street or in the open field. For the renewal of the German first order height network 100 new pillars were built and are presently being observed with A-10 until the end of 2010; half of these measurements have been completed in 2009. The main conclusions drawn from the experience of these campaigns in regard to accuracy requirements are:
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Fig. 33.7 Determination of vertical gravity gradient over height differences of about 50 cm with SCINTREX CG5 relative gravity meter using a special tripod with a footprint spacing of A-10 dimensions and extension legs of A-10
R. Falk et al.
Fig. 33.8 Spatial distribution of differences between A-10#012 observations and the official gravity values of 94 stations of DHSN96 first order German gravity network, values related to the top of the benchmarks 40 30 25 20 15 10 5 0 650
250
150
50
–50
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1. At each field-site, two independent instrument setups positioned in opposite directions (north and south) are used to increase the reliability and detect gross errors. The differences between both orientations will not exceed 80 nm s2 if the instrument setups have been carried out carefully. 2. In order to transfer a measured absolute gravity value to the benchmark of a certain field point it is necessary to determine the local vertical gravity gradient. Using the free-air gradient (3.086 mm s2 m1) instead of the local value would lead to systematic errors of up to 500 nm s2. The local vertical gravity gradient can be severely disturbed by the topographically induced mass distribution, e.g. at hilltops or in valleys. Figures 33.8 and 33.9 depict the differences between A-10 measurements taken within project “GOCE-GRAND II” and the gravity values of the
occurence in %
35
nm s–2 Fig. 33.9 Histogram of differences between A-10 observations and DHSN96 first order network gravity values
first order gravity network DHSN96 (German Main Gravity Network 1996). This was established as a relative gravity network and later transformed to the level of absolute gravity determination using identical absolute gravity sites. The deviations underscore the quality of the A-10#012 and confirm the first order
33
Precise Gravimetric Surveys with the Field Absolute Gravimeter A-10
gravity network. The displayed differences do not only mirror the measurement error of the A-10 but also contain measurement errors and adjustment effects within the first order gravity network, reduction errors in the relative measurements as well as changes in gravity over time. Systematic offsets were not detected.
33.6
Conclusions
In two field projects the A-10#012 gravimeter has proved of value under indoor and field conditions. The instrument gives results significantly better than the producer specification of 100 nm s2 and the repeatability within hours and weeks is mostly better than 60 nm s2. Precise gravity observations comprising two independent A-10 setups (to avoid and check for gross errors) and determination of the local vertical gravity gradient (for precise benchmark ties) take about 4 h in the field. Within the framework of the “GOCE-GRAND II” project the gravity network values were confirmed within the accuracy of the instrument and the network quality. Repeated observations at a gravity reference station are highly recommended in order to monitor instrument behaviour over a longer time span. Regular checks of the laser frequencies are necessary and enable significant enhancement of accuracy. For the A-10 gravimeters operated at BKG a measured gravity value at one and the same site would decrease by 26 nm s2 year1 for A-10#002 and 44 nm s2 year1 for A-10#012 respectively, if laser frequencies of red shifted and blue shifted mode lock remain unchecked. A monitoring unit for physical standards was established at BKG in 2007 and has proved to work well since then. For this reason the expectations for the accuracy of results of the still ongoing project “DHHN” are in the range of 60–80 nm s2 (absolute gravity value at 75 cm height above benchmark) and 100–120 nm s2 (absolute gravity value at the benchmark, errors from absolute and relative measurement) respectively. Applications of precise gravimetric surveys with the field absolute gravimeter A-10 are seen in installation and monitoring of gravimetric reference networks, investigations of gravity changes due to
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height and mass variations for studying global change processes on local and regional scales as well as geoid model determination and improvement. Finally, the A-10 opens new, not only economical, possibilities for the community applying gravity meters. Acknowledgments The project “GOCE-GRAND II” was supported within the framework of the GEOTECHNOLOGIEN program of the German Bundesministerium f€ur Bildung und Forschung (BMBF), Grant: 03F0422A. The helpful comments of an anonymous reviewer are gratefully acknowledged.
References Flury J, Peters T, Schmeer M, Timmen L, Wilmes H, Falk R (2007) Precision gravimetry in the new Zugspitze gravity meter calibration system. In: Proceedings of IAG Gravity Field Service Symposium. Istanbul, pp 401–406 Hahn R (2007) Frequenzbestimmung und Stabilit€atsuntersuchung eines HeNe Lasers, Bachelor Thesis, Technical University Darmstadt Holweg D (2001) Systemanalyse des Feld-Absolutgravimeters A-10#b002, Diploma Thesis, Technical University Darmstadt Ihde J, Wilmes H, M€uller Ja, Denker H, Voigt C, Hosse M (2010) Validation of satellite gravity field models by regional terrestrial data sets. In: System Earth via Geodetic-Geophysical Space Techniques, Springer, pp 277–296 Liard J, Gagnon C (2002) The new A-10 absolute gravimeter at the 2001 International Comparison of Absolute Gravimeters. Metrologia 39:477–484 M€akinen J, Stahlberg B (1998) Long-term frequency stability and temperature response of a polarization-stabilized He-Ne laser. Measurement 24:179–185 Micro-g LaCoste Inc (2008) A-10 portable gravimeter user’s manual. http://www.microglacoste.com, accessed on Nov. 10, 2009 Niebauer TM, Faller JE, Godwin HM, Hall JL, Barger RL (1988) Frequency stability measurements on polarizationstabilized He-Ne lasers. Appl Optic 27(7):1285–1289 Schmerge D, Francis O (2006) Set standard deviation, repeatability and offset of absolute gravimeter A10-008. Metrologia 43:414–418 Wilmes H and Falk R (2006) Bad Homburg – a regional comparison site for absolute gravity meters. In: international comparison of absolute gravimeters in Walferdange (Luxembourg) of November 2003. Francis O and van Dam T (eds), Cahiers du Centre Europen de Geodynamique et de Seismologie (EGCS), Luxembourg, vol 26, pp 29–30 Wilmes H, Richter B, Falk R (2003) Absolute gravity measurements: a system by itself. In: Gravity and Geoid 2002 – Proc. of 3rd Meeting of the International Gravity and Geoid Commission, Tziavos IN (ed), Editions Ziti, pp 19–25 Winters M (2004) Laser Heterodyne System, http://www. winterseo.com/hetero.html, Accessed on Nov. 10, 2009
.
Reconstruction of a Torsion Balance and the Results of the Test Measurements
34
€lgyesi and Z. Ultmann L. Vo
Abstract
During recent investigations concerning geodetic applications of the torsion balance measurements several problems arose, which required performing new torsion balance measurements. For that reason an E€otv€os-Ryba´r (Auterbal) torsion balance, which has been out of operation for many decades, was reconstructed and modernized. The scale reading has been automated and its accuracy has been improved by using CCD sensors. Calibration and processing of field measurements were computerized to meet today’s requirements. Test measurements have shown that this instrument was able to work according to the expectations of our age.
34.1
Introduction
The first torsion balance field measurements were made between 1901 and 1903 by Lorand E€ otv€ os and his colleagues. Afterwards, until 1960s about 60,000 points were measured on the flat and semi hilly areas for geophysical prospecting by the Hungarian-American Oil Company (MAORT), Lorand E€ otv€ os Geophysical Institute of Hungary (ELGI) and the National Oil and Gas Co. Ltd. (OKGT) (Polcz 2003). Because of the
L. V€olgyesi (*) Department of Geodesy and Surveying, Faculty of Civil Engineering, Budapest University of Technology and Economics, PO Box 91, 1521 Budapest, Hungary Research Group of Physical Geodesy and Geodynamics of the Hungarian Academy of Sciences, PO Box 91, 1521 Budapest, Hungary e-mail:
[email protected] Z. Ultmann Department of Geodesy and Surveying, Faculty of Civil Engineering, Budapest University of Technology and Economics, PO Box 91, 1521 Budapest, Hungary
former measurements were made for geophysical purposes, so only the horizontal gradients Wzx, Wzy were used, and the curvature gradients Wyx and WD which are very important for geodesy left unprocessed. Unfortunately some data of the former measurements are lost, but many of them are available as on original field books or maps. From the year 1995 the experts of the Lorand E€otv€os Geophysical Institute make lots of efforts to save those data to computer database by the financial support of the Department of Geodesy and Surveying of the Budapest University of Technology and Economics. At present more, than 36,500 valuable gradient data are waiting in this database for the further processing mainly for geodesy.
34.2
Geodetic Applications of the Torsion Balance Measurements
Lorand E€otv€os made simultaneously with his first measurement a special computation method to determine the change of deflection of the vertical between two points used the curvature gradients WD and Wxy
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(E€otv€os 1906). If we know the observed values of deflection of the vertical in some points of an area by astrogeodetic method, than values of the deflection of the vertical can be interpolated on every torsion balance points. At the same time using the interpolated values of the deflection of the vertical, applying the method of astronomic leveling, it is possible to determine the geoid heights (V€ olgyesi 2001a), so using torsion balance measurements we are able to determine the fine structure of the geoid. Nowadays the computation method of E€ otv€ os was successfully improved with the aid of the modern mathematical and computational methods (V€ olgyesi 1993, 1995, 2005; V€olgyesi et al. 2005) and we are able to determine the deflection of the vertical and the local geoid shape even more accuracy. Improvements of the new computational methods give new possibilities for geodetic application of all elements of the E€ otv€ os tensor. Besides the geodetic application of the curvature data the horizontal gradients of gravity measured by torsion balance can be used for geodetic purposes too. Because the knowledge of the real gravity field of the Earth has a great importance in physical geodesy, the possibility and the need for the usage of these horizontal gradients are important. Using these gradients combined with gravity or gravity anomalies the components of the local gravity field especially the low-degree components can be reproduced (V€ olgyesi et al. 2004). Knowledge of the vertical gradients is very important for different kind of purposes in geodesy too, but according to our researches, the real value of this vertical gradients significantly differ from the normal one (Csapo´ and V€ olgyesi 2004). Moreover this is the only component of the E€ otv€ os-tensor which is not observable by torsion balance. Because the classical determination of the vertical gradients directly by
Fig. 34.1 Geodetic application of gravity gradients
€lgyesi and Z. Ultmann L. Vo
gravimeters is rather time consuming and expensive process so another more simply and not so expensive method is necessary. Torsion balance measurements give new possibility to determine vertical gradients by an interpolation method. Starting from curvature and horizontal gradients of gravity measured by torsion balance, the Tzx, Tzy horizontal gradients and the TD ¼ Tyy-Txx, 2Txy curvature data of the disturbing potential T ¼ W – U can be formed, and according to Haalck method (1950) the vertical gradient Tzz can be determined from these value (V€olgyesi et al. 2004). This method, similarly to the astronomical leveling, generates differences of vertical gradients at least between three points measured by torsion balance. For this interpolation it is necessary to know the real (observed) value of vertical gradients in some points of the area. Another new important geodetic application of torsion balance measurements is the 3D inversion reconstruction of gravity potential based on gravity gradients. This new inversion method gives opportunity to determine the function of gravity potential and their all first and second derivates (the components of gravity vector and the elements of the full E€otv€os tensor – including the vertical gradient) (Dobro´ka and V€olgyesi 2008). Comparing the elements of the computed E€otv€os tensor to the gravity gradients measured by torsion balance gives a good opportunity to control the inversion. Hereby an opportunity presents itself for the analytical determination of the potential surfaces. All the geodetic applications of torsion balance measurements are summarized on Fig. 34.1. On the left-hand side of the figure the elements of E€otv€ostensor are arranged to three groups: curvature data are indicated with light-gray shading, horizontal gradients
34
Reconstruction of a Torsion Balance and the Results of the Test Measurements
of gravity are marked by dark-grey shading (these can be measured directly by torsion balance) and the crossed element (the vertical gradient) on the right hand side of the E€ otv€ os tensor, is not measurable directly by torsion balance. On the right-hand side of Fig. 34.1 the types of the possible geodetic applications are shortly summarized.
34.3
Necessity of Further Measurements
In the latest time some new special problems arose in our researches about the geodetic application of gravity gradients, which necessitates new torsion balance measurements. The most important reason is coming from the determination of vertical gradient. As we mentioned a new interpolation method is developed based on the Haalck’s (1950) idea to determine Wzz using torsion balance measurements (To´th et al. 2005; To´th 2007). We have controlled our method by synthetic data, but the full control with real data is still missing. For this purpose such points are needed, where both torsion balance measurements and vertical gradient observations are available. Unfortunately vertical gradients were not measured on those points where torsion balance was used; on the other hand coordinates of earlier torsion balance points are not known the required accuracy. A good opportunity presents itself in the Ma´tya´s cave in Budapest, where a special microbase network of the Lorand E€otv€os Geophysical Institute of Hungary can be found. This network contains 14 points where former vertical gradient measurements are available, and the area is extremely suitable for torsion balance measurements (Csapo´ 1991).
34.4
Preparation of the Instruments
Recently there are two types of former manufactured torsion balances which are still capable for field measurements. One of them is the E€ otv€ os-Ryba´r (Auterbal) balance, which was developed to the end of 1920s; the other is the improved E54 type manufactured in 1954. A short basics about torsion balance can be found in (V€ olgyesi 2001b). The E54 torsion balance of the Lorand E€otv€os Geophysical Institute has been renewed by the specialists of the institute. Unfortunately it turned out
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that the narrow measuring range of E54 instrument was not suitable for the measurements in the Ma´tya´s cave because of the huge gravity gradient values. So only the Auterbal torsion balance with wider measuring range has remained the applicable instrument for the observation in the Ma´tya´s cave. An Auterbal torsion balance could be found in the museum of the Department of Geodesy and Surveying of the Budapest University of Technology and Economics. After a long field work this instrument was donated to the department by the Lorand E€otv€os Geophysical Institute in 1964 for educational purposes. Later this torsion balance went wrong and it has been located in the museum of the department. Many years after, in 2008 we examined the condition of the instrument, and we have found the nearly 80 years old Auterbal torsion balance to suffer several problems, but the torsion wires were not broken. We did not know anything about the main spring, which has been spanned over 40 years. Repairing seemed to be a serious problem because there was not any description about this instrument. The first challenge was to guess the steps of dismantling and assembling, and to find out the functions of the many screws fixing elements and parts. We should have to be very careful at the first steps of dismantling because of the spanned main spring (which helps the turning of the torsion balance to the different azimuts) might caused serious damages. For our fortune after separating the balance box and the middle part of the instrument (see Fig. 34.2) at the beginning of the dismounting the release of the spanned main spring was succeeded and considering carefully all the other steps of dismantling at last we were able to remove the damaged clockwork. The prepared clockwork can be seen on Fig. 34.3. After changing damaged parts and repairing and refashioning of the clockwork further several improvements were carried out too. CCD sensors were supplied on both reading microscopes for automatic readings and special strong LED light was installed for lighting the scale (see Fig. 34.4). Controlling and processing the CCD readings special computer software was developed under Linux operation system. Applying the new reading system new possibilities was born for very fast readings which were impossible before, e.g. detailed investigation of attenuation of the oscillation and studying of the long scale drift became possible.
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Fig. 34.4 CCD camera for scale reading
Fig. 34.2 The Auterbal torsion balance
Fig. 34.5 Inner construction of the Auterbal balance
Fig. 34.3 The clockwork for turning the torsion balance
34.5
Fine Tuning of the Instruments and the Test Measurements
If we want to understand the essence of the fine tuning, the inner construction of the torsion balances has to be known. Both Auterbal and E54 instruments contain two independent antiparallel torsion balances inside the balance box (see Fig. 34.5). The two antiparallel balances are being enclosed in a double walled
heat-insulating box. The aluminum balance bars are connected to the torsion wires through reading mirrors. Diameter of torsion wires are 0.017 mm (diameter of a most thin hair is about 0.02 mm). In Fig. 34.5 the brick-shape upper mass can be seen on the left side balance bar while on the other right side parallel balance bar the fixing place of the lower suspended mass can be seen. For the purpose of the exact identification one of the two antiparallel torsion balances are marked by circle (“O”) and the other by square (“[]”). The oscillation range of balances can be regulated by the limit plates, as it can be seen on the Fig. 34.5, the angle of the oscillation amplitude is smaller than 2 . Before the first field observations some special test measurements was required to control the functioning and behavior of the repaired and modernized torsion balances. The most important question was the drift of the torsion wires. (Drift is the small continuous
34
Reconstruction of a Torsion Balance and the Results of the Test Measurements
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Fig. 34.6 Shape of the drift curve after the renovation of the Auterbal torsion balance
changing of scale reading because of the small oneway twisting of a torsion wire.) Drift can be explained based on the knowledge of the solid state physics. If a torsion wire is in excellent condition, the balance bar of the torsion balance remains at the same position for a long time, in other cases the balance bar is turning a little bit in time, so scale reading is changing permanently. Reduction of the drift can be achieved by the heat treatment and by spanning (hanging) the torsion wires for a long time. After the renovation of the torsion balance, the drift was controlled over 30 days by the readings ten times a day in the same azimuth. Results on the first day were alarming: as the [] torsion wire seemed to be in excellent condition, than the drift of the O wire was enormous: readings were fallen more than 20 scale unit in the first 24 h. After a few days the situation become not so bad, the incline of the drift curve started to be smaller and smaller, which is shown on the Fig. 34.6. Two weeks later the reading was changed just 1–2 scale units a day, but a new adjustment of the torsion wire was necessary because the zero position of the balance bar turned extremely towards to the lower limit. (Adjustment of a torsion balance means a very small turning or moving up and down of the balance bar to reach the middle/ zero position.) At this time the O balance bar have been a little beat overturned, because further permanent decreasing drift was expected (see Fig. 34.6).
Because the permanent drift really stayed further on, so the torsion wire was taken out from the instrument for a purpose of a detailed investigation. We have found a problem about the fixing of the torsion wire so we resoldered it to the right position. The permanent drift was stopped by a certain extent opposite turning with a tiny loading and than by a long time suspension. The treatment was successful the smooth drift curves can be seen on the Fig. 34.7 after the treatment of the O torsion wire. Important experiences came from the investigations of the relation between the temperature changing and the reading data. The main task was to determine the necessary waiting time for the first reading after the release of the balances in case of taking out the instrument from the storage and setup on the field with different temperature. Result of this investigation can be seen on Fig. 34.8. Based on the figure it can be stated, that the applied 40 min waiting time for the first reading after releasing is not enough because the thermal equilibrium, when the drift become already linear ensures during 90 min. Further investigations were executed referring to the correction of the readings depending on the variation of the temperature. These investigations have not been finished yet further examinations are necessary. Torsion balance reacts very sensible to several tenth degree temperature variations. According to our experiments while the slow
€lgyesi and Z. Ultmann L. Vo
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Fig. 34.7 The drift curve after the treatment of the torsion wire
Fig. 34.8 Relation between the temperature changing and the scale readings
variations are caused by the temperature sensitivity of torsion wires than the quick variations came mainly from the thermal expansion of reading arms
34.6
Measurements in the Ma´tya´s Cave
As we mentioned in the Part 3 the torsion balance measurements were made on the gravity microbase points of the Geodynamical Laboratory of Lorand E€otv€os Geophysical institute in the Ma´tya´s cave. The map of the Geodynamical Laboratory in the cave can
Fig. 34.9 The torsion balance microbase points
34
Reconstruction of a Torsion Balance and the Results of the Test Measurements
Fig. 34.10 Results of the torsion balance measurements
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€lgyesi and Z. Ultmann L. Vo
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be seen in Fig. 34.9, where the places of the 14 microbase points and the gravity basepoint of Hungary are shown. First we tried to use the E54 torsion balance, but after some attempt it turned out that this instrument was not suitable for us, because its measuring range (O: 0–170; []: 200–370 scale unit) not enough wide to measure the very large variation of gravity gradients in the Ma´tya´s cave. Because the measuring range of the Auterbal torsion balance is much wider (O: 0–280; []: 280–560) so this instrument was applied. During our measurements many different problems occurred. The permanently extant 11 C temperature was advantageous for the drift but that humid and could air was very adverse for the turning mechanism and the clockwork of the torsion balance. The O and [] torsion balances should be adjusted asymmetrical several times because of the extremely high values of gravity gradients. On three points near to the entrance of the cave the wider measuring range of the Auterbal torsion balance was not even enough to measure the very large gravity gradients. Results of our measurements started from the gravity base point 82 to the endpoint of the gravity microbase 82/11 are summarized in Fig. 34.10. On the upper part of the figure the horizontal gradients of gravity Wxz and Wyz, on the lower part the curvature data WD and Wxy can be seen. As it was mentioned above there are two independent antiparallel torsion balances inside the balance box 7 cm distance from each other, so we get two independent observations for each gravity gradients and curvature data at each torsion balance station. Supposing errorless measurements the two independent observations should be the same, and the occasional differences give information about the reliability of the measurements. Both the measured data of the O and the [] balance and their mean values marked by dashed line can be well distinguished on Fig. 34.10. Analyzing the differences of the O and the [] independent balances it can be find that the repeated observations on the gravity base point 82 the same gradient values was observed by the O and the [] balances, but going towards to the entrance of the cave bigger and bigger differences can be measured by the two balances. The gravity base point can be found in the middle of a big hall relatively far from masses, but going towards to the entrance the microbase points are closer and closer to the disturbing masses, and the gradients are so large here, which
gives measurable differences of gravity gradients by the O and the [] balances (Ultmann 2009).
34.7
Summary
Before the end of the 1960s approximately 60,000 torsion balance measurements were made in Hungary. Recent research on the field of their geodetic applications showed the need for new observations. After an interruption of half a century, torsion balances are operated again in Hungary. Following the renovation and modernization of an Auterbal and an E54 balance, measurements are taking place in the Ma´tya´s cave. Gravity gradiometry was introduced by Lorand E€otv€os by creating his famous instrument in the beginning of the 1900s, but nowadays it is playing important role in geodesy again. Torsion balance measurements will be important and indispensable data source for the determination of small wavelength gravity field and geoid features in Hungary. A group of professionals has been formed again who have the intent to continue the scientific efforts of Lorand E€otv€os, are able to renovate and modernize obsolete instruments, and have the field experience to continue torsion balance measurements. Acknowledgements Our investigations are supported by the National Scientific Research Fund (OTKA K-76231).
References Csapo´ G (1991) Dg and vertical gradient measurements by LCR gravimeres and observations by E54 torsion balance on the microbase of ELGI in the Ma´tya´s cave. (ELGI datastore) (in Hungarian) Csapo´ G, V€olgyesi L (2004) New measurements for the determination of the local vertical gradients. Magyar Geofizika 45, 2:64–69 (in Hungarian) Dobro´ka M, V€olgyesi L (2008) Inversion reconstruction of gravity potential based on gravity gradients. Math Geosci 40(3):299–311 E€otv€os R (1906) Bestimmung der Gradienten der Schwerkraft und ihrer Niveaufl€achen mit Hilfe der Drehwaage. Verhandl. d. XV. allg. Konferenz der Internat. Erdmessung in Budapest Haalck H (1950) Die vollst€andige Berechnung €ortlicher gravimetrisher St€orfelder aus Drehwaagemessungen. Ver€offentlichungen des Geod€atischen Institutes Potsdam, Nr. 4, Potsdam Polcz I (2003) The history of the Lorand E€otv€os Geophysical Institute I. ELGI (k€ul€on kiadva´ny) (in Hungarian)
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Reconstruction of a Torsion Balance and the Results of the Test Measurements
To´th Gy (2007): Vertical gravity gradient interpolation in a grid of E€otv€os torsion balance measurements. Geomatikai K€ozleme´nyek X 29–36 (in Hungarian) To´th Gy, V€olgyesi L, Csapo´ G (2005) Determination of vertical gradients from torsion balance measurements. IAG Symposia Vol 129, Gravity, Geoid and Space Missions C, Jekeli L, Bastos J, Fernandes (eds.), Springer, pp 292–297 Ultmann Z (2009) Investigation of the gradients of gravity in the Ma´tya´s cave. Diplomawork, BME E´pı´to˝me´rn€ oki Kar (in Hungarian) V€olgyesi L (1993) Interpolation of deflection of the vertical based on gravity gradients. Periodica Polytechnica Civ Eng 37(2):137–166 V€olgyesi L (1995) Test Interpolation of deflection of the vertical in hungary based on gravity gradients. Periodica Polytechnica Civ Eng 39(1):37–75
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V€olgyesi L (2001a) Local geoid determinations based on gravity gradients. Acta Geodaetica et Geophysica Hung 36 (2):153–162 V€olgyesi L (2001b) Geodetic applications of torsion balance measurements in Hungary. Reports on Geodesy, Warsaw University of Technology, 57(2), pp 203–212 V€olgyesi L (2005) Deflections of the vertical and geoid heights from gravity gradients. Acta Geodaetica et Geophysica Hungarica 40(2):147–159 V€olgyesi L, To´th Gy, Csapo´ G (2004) Determination of gravity anomalies from torsion balance measurements. Gravity, Geoid and Space Missions GGSM 2004. Springer, Heidelberg, 129:292–297 V€olgyesi L, To´th Gy, Csapo´ G, Szabo´ Z (2005) The present state of geodetic applications of Torsion balance measurements in Hungary. Geode´zia e´s Kartogra´fia, 57(5), 3–12 (in Hungarian)
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The Superconducting Gravimeter as a Field Instrument Applied to Hydrology
35
C.R. Wilson, H. Wu, L. Longuevergne, B. Scanlon, and J. Sharp
Abstract
We describe development of a transportable version of the Superconducting Gravimeter (SG) and its test in a field experiment to monitor storage in a karst (limestone) aquifer in central Texas. The SG is contained within two aluminum enclosures, one holding the SG in its 35 l helium dewar, plus electronics; and the second for refrigerator and power supply. In the field test, the SG was supported on threaded steel rods cemented into limestone, and surrounded by weatherprotecting sheds. The steel rod design was not completely satisfactory, and in most field settings a concrete floor will probably be required. Field operation requires wired electric power, but is managed remotely using wireless internet. The experiment south of Austin Texas was designed to monitor ground water level, precipitation, and other variables, and observe mass variations associated with storage changes in the aquifer. Drought conditions prevailed, limiting conclusions about the aquifer, but the experiment demonstrated the feasibility of remote unattended operation for periods of many months.
35.1
C.R. Wilson (*) L. Longuevergne Department of Geological Sciences, Jackson School of Geosciences, University of Texas, Austin, TX, USA Bureau of Economic Geology, Jackson School of Geosciences, University of Texas, Austin, TX, USA e-mail:
[email protected] H. Wu J. Sharp Department of Geological Sciences, Jackson School of Geosciences, University of Texas, Austin, TX, USA B. Scanlon Bureau of Economic Geology, Jackson School of Geosciences, University of Texas, Austin, TX, USA
Introduction
The Superconducting Gravimeter (SG) consists of a niobium 2.54 cm spherical proof mass, levitated in magnetic fields, and contained within a liquid helium bath in a dewar (Prothero and Goodkind 1968). Early papers (Prothero and Goodkind 1972) drew interest within the geophysics community, and led to establishment of a company (GWR) to undertake commercial production. Goodkind (1999) reviews SG principles, and a summary of history, design, performance, data analysis, and applications is given by Hinderrer et al. (2007). GWR gives precision for current SG’s as 0.01 mGals (0.1 nm s 2) or better, and drift below 1 mGal (10 8 m s 2) per month. Several dozen SG’s in service (Crossley et al. 1999) have confirmed that local groundwater variations are
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Fig. 35.1 Interior of enclosure E1, containing UPS and refrigerator. Lightning surge protector and thermostatic control for the top-mounted cooling fan are located at mid-level. Refrigeration hoses (15 m length) and cold head drive cable extend to the cold head mounted on the dewar in E2
readily detectable, though this source of gravity change is usually considered a nuisance. The motivation for developing a transportable SG has been to allow installations at locations where groundwater or other subsurface fluid changes are the signal of interest.
35.2
Configuration for Transport and Field Use
Development began with a standard single sphere SG (serial number 047) delivered to the University of Texas in 2007. SG047 components were packaged within Enclosures 1 and 2 (E1 and E2), shown in Figs. 35.1 and 35.2. Each is constructed of aluminum, with dimensions ~1.5 0.8 1 m, and mass ~250 kg including SG equipment. The two enclosures provide physical protection, facilitate control of temperature
Fig. 35.2 Interior of enclosure E2 showing helium dewar, cold head and cold head frame with rack mounting of electronics at right. Electronics and computer are those of a standard single sphere observatory-class SG as supplied by GWR in 2006. UPS electric power is supplied via a 10 m cable from E1
and humidity, and simplify handling during transport. Figure 35.1 shows E1 with refrigerator compressor, power supply (UPS), and lightning protector. Figure 35.2 shows E2 with dewar, refrigerator cold head frame, and electronics (data logger, levitation and sensor controls, barometer, GPS receiver, power supply, and computer). E1 and E2 are connected by electric power lines, data cables, and helium refrigeration hoses, and are separable by up to 10 m. In a field setting, a weather station connects to E2 and provides a mast for the system timing GPS antenna. E1 is cooled via a top-mounted exhaust fan and requires ambient temperatures below 35 C following the manufacturer’s (Sumitomo Heavy Industries) specifications for refrigerator compressor operations. Temperature control requirements for E2 are similar, but heat production is much lower. During laboratory trials, we tested a 0.4 kW cooling unit mounted on the interior of E2 as a possible way to regulate
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The Superconducting Gravimeter as a Field Instrument Applied to Hydrology
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(Fig. 35.3) bear the weight. After securing the dewar, E2 is raised above the aluminum pillars with hand lifts and the pillars are removed. Then E2 is lowered and secured to a wheeled dolly, and moved to the new location. After relocation the proof mass is re-centered to adjust for gravity differences at the two sites, and any helium lost in transit (while the refrigerator is off) is replenished by liquefying gas from a cylinder.
35.3 Fig. 35.3 Securing the base of the dewar to the cold head frame with brackets allows E2 to be transported without disassembly
temperature. However, the cooling unit fans cycled on and off every few minutes while keeping temperature in regulation. This time scale of temperature cycling caused oscillations in the level control system (which uses a thermal mechanism). Therefore, E2 was cooled using circulated air during the field trial. The cold head frame is secured to the base of E2 with vibration isolation mounts, and the dewar containing the SG sensor is supported on three aluminum pillars. When installing on a concrete floor, the pillars are bolted to an aluminum plate. Installation involves raising E2, sliding the pillar-plate assembly beneath it, and lowering E2 as the three pillars pass through clearance holes to support the dewar. During the field trial, the pillar plate assembly was replaced by braced threaded rods cemented into outcropping limestone (Fig. 35.5) with the three aluminum pillars threaded onto the rods. A portable air conditioning unit supplied cool air to the base of E1, while hot air was exhausted from the top of E1 to the exterior of the shed. A fan and duct connecting the two instrument sheds circulated cool air to E2. These systems worked well, although additional reflecting insulation on the exterior of the sheds was required to reduce heat load from the sun. Relocation is accomplished with the proof mass levitated. Power cables and refrigeration lines are first disconnected. Then the dewar is attached to the base of the cold head frame using brackets shown in Fig. 35.3. The cold head lifting arm can also be secured to the dewar neck using four machine screws, giving resistance to lateral motion at the top of the dewar during transport, while the lower brackets
Laboratory and Field Trials
Figure 35.4 shows four time series of residual gravity variations over the period May 2007 to June 2009, taken during three laboratory and one field trial. Tidal variations, pole tide, and atmospheric mass effects were removed from all series. The first period (May–July 2007) (University of Texas Geology Building), shows a very high drift rate, and SG047 was returned to GWR for repairs after this. The second period (October 2007–March 2008) (University of Texas Geology Building) shows that after repairs, drift rate was reduced to ~10 mGals per month, still a high rate. The third period (March–October 2008 at the University of Texas Research Campus) followed a test of relocation procedures, and included integration of the weather station. The fourth period (November 2008–June 2009) shows data from the field site south of Austin, Texas (Figs. 35.5 and 35.6). This site is in the recharge zone of the Edwards Aquifer, adjacent to a well with a level recorder. Wireless internet allowed remote monitoring and data transfer. No permanent structure was permitted at the site, requiring the monument design shown in Fig. 35.5. The Edwards Aquifer (EA) is a major water resource both for the City of San Antonio, and regions to the north and west. As a karst (limestone) aquifer, groundwater is stored in small pores, fractures, and dissolution features (voids, tunnels, caves) that are spatially heterogeneous and not accessible to observation. There is no direct way to assess the volume of water stored in this aquifer. However, when both a well level record and precise observations of gravity at the surface are available, a reliable estimate of effective porosity (specific yield), should be possible. Observations of the EA in most locations show that periods of heavy rainfall cause water levels in wells to increase rapidly, by 10 m or more over periods of few days to a few weeks. An event of this magnitude might
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Gravity variations –2 [nm.s ]
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5000 4000 3000 2000 1000 0 –1000 –2000
Geology building Research campus Edwards aquifer
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Fig. 35.4 Four time series of SG047 residual gravity for three laboratory and one field trial from May, 2007 to June 2009. Voltage was converted to gravity using a calibration factor estimated from earth tides and tidal loading calculated from tide model FES2004. For hydrologic applications of the SG, a more precise calibration method (Van Camp et al. 2000) is not
Fig. 35.5 Edwards Aquifer field site monument installation. Plates and long rods were removed after the epoxy cement had cured. Shorter rods were installed and the plates were replaced to form bracing for the vertical rods. Finally, aluminum pillars were threaded to the top of the steel rods, and E2 lowered onto them. The last step involved erecting the instrument shed around E2
increase gravity by ~20–40 mgals, and be detectable as a transient event, even with a large SG drift rate. It was hoped that at least one major recharge event would occur during the field trial, but the entire period was one of exceptional drought, with no significant water level changes. Although little was learned about EA hydrologic properties the field trial provided a useful test of operations. Constraints on monument design were
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essential, as it typically differs from a tidal calibration by less than 1%. Predicted tides, tidal loading, and local barometric pressure effects (empirically determined admittance of 3.3 nm s 2 mbar 1) were removed. The text describes the four separate time series. The last time series is from the Edwards Aquifer field trial
Fig. 35.6 Instrument shed containing E2, during installation November, 2008. A similar shed behind this contains E1. A portable air conditioner in the E1 shed provided cool air for the refrigerator compressor, and was ducted to the E2 shed to cool electronics and the computer. Sheds were covered in reflective insulation to reduce heat load from the sun
revealed by two major anomalies in the time series (Fig. 35.4). The first, appearing soon after installation in November, 2008, was caused by a sagging wooden floor in the instrument shed (Fig. 35.6), allowing the cold head frame to touch the dewar neck. Manual adjustment of the cold head frame solved this problem. (The problem also arose, to a lesser degree in May 2009). The second anomaly, in March 2009, followed a small rainfall event, insufficient to recharge the aquifer, but sufficient to wet clay-filled joints in the limestone. The joints expanded causing the monument (Fig. 35.5) to tilt beyond the range of automatic compensation. A site visit was required to re-level. Both
35
The Superconducting Gravimeter as a Field Instrument Applied to Hydrology
problems would be resolved if a poured concrete floor had been in place, and this appears to be a requirement for most field installations. Side-by-side operation with a spring-type survey gravimeter during the field trial demonstrated the high stability of the SG in the presence of temperature variations in the shed, which were often in the range of 15 C during the day. Conclusions
It is feasible to use the SG as a transportable field instrument with remote unattended operation and wireless internet data transfer and system monitoring. At most sites, a concrete foundation will be required. Even under apparently ideal conditions (outcropping massive limestone) a cemented braced steel rod monument was not sufficiently stable. Climate control is a challenge in a field setting, but can be dealt with using portable cooling equipment. The relatively high drift rate during laboratory and field trials was unrelated to the transportable configuration, and the problem is to be corrected at the GWR factory prior to a forthcoming field experiment in the Arizona desert. A main motivation for developing a transportable SG is to monitor subsurface fluids. In this role, the SG can serve alone, or support observations with portable relative or absolute gravimeters (Pool and Eychaner 1995; Naujoks et al. 2007). For this purpose, the SG has better precision than the best portable instruments (Kroner and Jahr 2006), but a multiplicity of local and distant gravity sources make 1 mGal precision a practical goal. In addition to sensing local fluid storage changes, the SG may be able to identify gravity ‘noise’, not due to local subsurface fluids. Noise at the level of several to tens of mGals occurs over a range of time scales and a variety of sources, including atmospheric attraction not predictable from local barometric pressure, regional and distant water loads, and non-tidal ocean mass redistribution. (Boy and Hinderer 2006). In support of
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portable gravimeters, the SG might provide a local ‘Gravity Noise Correction’ time series similar to an Earth Tide Correction. This could improve precision of surveys with portable absolute and relative meters. Practically, this would require an estimate of SG drift (via absolute gravimeter co-location) and careful monitoring of offsets (tares). Acknowledgments This work was supported by National Science Foundation Instrumentation and Facilities grant EAR03-45864. Additional support provided by the Geology Foundation of the University of Texas at Austin.
References Boy J-P, Hinderer J (2006) Study of the seasonal gravity signal in superconducting gravimeter data. J Geodyn 41:227–233 Crossley DJ, Hinderer G, Casula O, Francis H-T, Hsu Y, Imanishi B, Meurers J, Neumeyer B, Richter K, Shibuya TS, Van Dam T (1999) Network of superconducting gravimeters benefits a number of disciplines: EOS, Transactions. Am Geophys Union 80(121):125–126 Goodkind JM (1999) The superconducting gravimeter. Rev Sci Instrum 70(11):4131–4152 Hinderrer J, Crossley D, Warburton R (2007) Superconducting gravimeter, book chapter 9. In: Herring T, Schubert G (eds) Treatise on geophysics, vol 3. Elsevier, Amsterdam Kroner C, Jahr T (2006) Hydrological experiments around the superconducting gravimeter at Moxa Observatory. J Geodyn 41:268–275 Naujoks MA, Weise C, Kroner, and Jahr T (2007). Detection of small hydrologic variations in gravity by repeated observations with relative gravimeters. J Geodesy, December 2007, doi:10.1007/s00190-007-0202-9 Pool D, Eychaner J (1995) Measurements of aquifer storage change and specific yield using gravity surveys. Ground Water 33(3):425–432 Prothero W, Goodkind JM (1968) A superconducting gravimeter. Rev Sci Instrum 39:1257–1262 Prothero W, Goodkind JM (1972) Earth-tide measurements with a superconducting gravimeter. J Geophys Res 77:926–937 Van Camp M, Wenzel H, Schott P, Vauterin P, Francis O (2000) Accurate transfer function determination for superconducting gravimeters. Geophys Res Lett 27:37–40
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Local Hydrological Information in Gravity Time Series: Application and Reduction
36
M. Naujoks, S. Eisner, C. Kroner, A. Weise, P. Krause, and T. Jahr
Abstract
Hydrological variations of up to some 10 nm/s2 are significant and broadband signals in temporal gravity observations. On the one hand they need to be eliminated from the data as they interfere with geodynamic signals. On the other hand they can be used to improve the understanding of hydrological process dynamics and to evaluate distributed hydrological models. Compared to satellite observations which are affected by global and regional hydrological variations continuous recordings from superconducting gravimeters (SGs) additionally may contain extractable information on local changes. To compare terrestrial data to satellite observations and to regional/global hydrological models, a local hydrological impact on the observations must be quantified and appropriately reduced first. To investigate the local hydrological impact on gravity of the hilly and geologically heterogeneous surroundings of the SG at the Geodynamic Observatory Moxa, Germany, interdisciplinary research has been carried out. For an area of about 1.5 1.5 km2 a hydrological catchment model was combined with a gravimetric 3D model, including heterogeneities of the subsoil and topography in detail. A reduction of the local hydrological signal in the SG recordings was developed. About 30% of the local hydrological effect in the SG data originate from an area within a radius of 90 m around the observatory. The contribution of areas above the SG level is about 85% of the total local effect. After the local hydrological signal is separated, the SG data become suitable to be interpreted with regard to changes in continental water storage as found in GRACE satellite observations and in global hydrological models. The evaluation of the local
M. Naujoks (*) A. Weise P. Krause T. Jahr Institute of Geosciences, Friedrich Schiller University, Jena, Germany e-mail:
[email protected] S. Eisner Center for Environmental Systems Research, University of Kassel, Kassel, Germany C. Kroner Physikalisch-Technische Bundesanstalt Braunschweig, Potsdam, Germany S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_36, # Springer-Verlag Berlin Heidelberg 2012
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hydrological model basing on the gravimetric modelling and the SG data highlights approaches for further enhancement of the internal hydrological process representations.
36.1
Introduction
Temporal gravity data are affected by every mass change taking place in the earth system. The integral impact of broadband hydrology-related mass changes up to 100 nm/s2 is known for many years (e.g. M€akinen and Tattari 1988; Kroner 2001; Abe et al. 2006; Neumeyer et al. 2006; Sato et al. 2006; Kroner et al. 2007; Naujoks et al. 2008). Both, local and largescale hydrological variations impact the gravity field. From the hydrological point of view, this impact can be considered as a valuable data source as it augments traditional meteorological and hydrological point measurements. It can be used for hydrological process studies and to calibrate and validate hydrological models. From the geophysical view point, hydrological changes represent a disturbing influence in high-resolution gravity observations because they superimpose geodynamic signals. Thus, the gravity data require an a priori elimination of this impact. In particular, local hydrological fluctuations have a crucial effect on gravity (e.g. Llubes et al. 2004; Boy and Hinderer 2006; Harnisch and Harnisch 2006; Imanishi et al. 2006; Kroner and Jahr 2006; Van Camp et al. 2006; Virtanen et al. 2006; Hokkanen et al. 2007; Meurers et al. 2007; Creutzfeldt et al. 2008). Most of these investigations regarding local hydrological effects have been based on simplified models of both geological context and water flux description. In this study a combined hydrogravimetric analysis is used based on interdisciplinary research. The local topographic, geological, and hydrological situation in the surroundings of the SG at the Geodynamic Observatory Moxa, Germany is considered in detail. At Moxa, gravity field variations are monitored continuously with the stationarily operating SG GWR CD034 for more than 10 years. They show a significant hydrological impact in an order of magnitude of some 10 nm/s2 (Kroner 2001; Kroner et al. 2004, 2007). As the observatory is built close to a steep slope, most of the local hydrological changes occur above gravimeter level. Thus, an anti-correlation between gravity and the
hydrological situation exists. At rain events most of the water mass is first stored above the gravimeter level, leading to a fast gravity decrease. While the water is moving downwards below gravimeter level, gravity successively increases. In order to compare terrestrial gravity data to GRACE satellite observations and gravity variations derived from regional or global hydrological models, local hydrological effects need to be separated from large-scale changes. To identify, quantify, and eliminate the local effects in the SG data at Moxa, very precise repeated measurements of gravity differences on a local network were carried out (Naujoks et al. 2008) and a combined hydrological and gravimetrical model was developed (Naujoks et al. 2010). This offers the possibility to constrain ambiguities and to localise the different hydrological compartments affecting gravity in detail.
36.2
Combined Hydrogravimetric Modelling
For a local hydrological modelling of the Silberleite catchment the fully distributed process-oriented hydrological model J2000 was applied (Krause et al. 2009). It implements the processes evapotranspiration, snow accumulation and melt, interception, infiltration, soil water movement, and groundwater recharge as conceptual approaches. The Silberleite catchment was partitioned into 337 hydrological response units (HRUs) which were delineated according to the relevant physio-geographical input information (topography, landuse, soil types, geology; Fig. 36.1). At the observatory measured meteorological data precipitation, temperature, solar radiation, relative humidity, and wind speed were used as driving forces. Each HRU is represented by different water storages (surface depression storage, snow storage, interception storage, medium and large pore soil storage, and groundwater storages) which interact with each other. Lateral water movement is simulated by topological connections
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Fig. 36.1 Combined hydrogravimetric modelling area covering 4 km2 and HRUs in the close vicinity of the observatory. The HRUs 191, 193, 209, and 210 (light blue) comprise an area around the observatory with a radius of about 90 m; the other HRUs (dark blue) are within a distance of 250 m around the observatory
between the HRUs. The model was applied in hourly time steps for the period from 03/10/23 to 07/05/27. It was calibrated with streamflow records and validated against in-situ soil moisture measurements (Krause et al. 2009). The comparison of the simulated to the observed streamflow at the observatory yields a NashSutcliffe efficiency value of 0.8 (Nash and Sutcliffe 1970) showing that the hydrological dynamics of the catchment is reproduced fairly well. The actual contents of all storages of each HRU of the hydrological model – mass variations in hourly time steps – were converted to density changes of the subsoil bodies of a three-dimensional (3D) gravimetric model. This model was set up for the observatory surroundings utilising the software package IGMAS (Interactive Gravity and Magnetic Application System; G€otze and Lahmeyer 1988).
Extensive geological, tectonical, hydrological and geophysical constraints ensured a realistic modelling and limited ambiguities. The topography as well as local geological and hydrological structures were implemented in detail to correctly represent mass changes due to the hydrological situation. Besides the bedrock a soil layer and a layer of disaggregated bedrock was implemented for each geological unit of the model. The model consists of 28,729 triangles resulting in a resolution of up to 5 m in the direct vicinity of the observatory. A detailed model description is given in Naujoks (2008) and Naujoks et al. (2008). Up to a distance of 250 m from the observatory the soil layer and the disaggregated bedrock were divided into the respective HRUs of the local hydrological model. For each HRU, the modelled water mass
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storage variation was applied as density change of the respective body of the gravimetric model. For the soil bodies the sum of the variations in the medium and large pore soil storage, interception storage, snow storage, and depression storage was applied. For the bodies of disaggregated bedrock the variations of the groundwater storage were used. In distances between 250 m and 1 km from the observatory the hydrological dynamics have less impact on gravity and can be assumed to be largely homogeneous regarding hydrological properties as derived from measurements. Hence, spatially averaged water storage variations of all HRUs of the hydrological model were applied to the soil layer and disaggregated bedrock.
36.3
Results
The output of the combined modelling is compared to the observed gravity changes. Hereby, the temporal and spatial variability of local hydrological variations in the gravity data is analysed. The different hydrological compartments in the vicinity of the SG are studied for both, their time-variable water content and their impact on gravity. The gravity data are reduced for all known influences besides hydrology (earth and ocean tides, 3D air mass variations, polar motion, and instrumental drift).
36.3.1 Influence of the Slope and the Valley According to Newton’s law hydrological variations in the immediate vicinity of the SG have the biggest impact on gravity. Thus, the hydrological compartments in the observatory valley and at the steep slope directly to the east of the instrument are studied in detail. The HRUs 191 and 209 are located in the observatory valley around the instrument, the HRUs 193 and 210 at the steep slope, likewise close to the SG. They comprise an inner area with a radius of about 90 m around the observatory (cf. Fig. 36.1). In Fig. 36.2a the storage contents of the HRU 191 in the valley are given. The variations of the HRU 209 are very similar. The variations in the storage contents illustrate the high temporal dynamics of the hydrological state variables in the model. The medium pore storage which represents the available water capacity
of the soil is the most important one in terms of water storage. It is saturated for some periods in particular in winter. The large pore soil storage and the quick groundwater storage show an immediate reaction on rain events followed by a quick decay. The variations of the medium pore storage are more attenuated and show a clear seasonality as of course the snow storage does as well. This seasonality is even more pronounced in the curve of the total sum of all storages. The storages in the HRUs at the steep slope to the east of the gravimeter (e.g. HRU 193, Fig. 36.2b) show in general similar characteristics, but with an amplitude of about 30 l/m2 less. While in the HRUs in the valley a summed water content between 30 and 120 l/m2 is obtained, the water content in the HRUs at the steep slope is in the range between 15 and 80 l/m2. Compared to the valley floor, soils at the steep slope are shallow and, thus, the water storage capacity of the soil column is smaller. Additionally, water is transported speedily to the valley due to the strong topographic gradient (cf. Kroner and Jahr 2006). The time-varying hydrological effect on gravity of the four HRUs closest to the SG is given in Fig. 36.3. The sum amounts to 16 nm/s2 in winter 2004. The variations show a maximum effect of 6 nm/s2 from the HRU 193 which covers a large area of about 3,200 m2 and is situated completely above the level of the SG. Because HRU 210 covers an area of only 2,300 m2 and is in some distance to the gravimeter (cf. Fig. 36.1) a smaller effect of maximum 3 nm/s2 emerges. The HRUs 191 (area: 2,800 m2) and 209 (area: 3,700 m2) are located closest to the gravimeter. Because their area covers regions below and above the level of the SG, the hydrological effects partially compensate causing an overall negative effect.
36.3.2 Contribution from Areas Around the SG The contributions from different zones around the SG to the local hydrological effect in gravity are given in Fig. 36.4 for the period from December 2003 to May 2007. A seasonal signal can be identified in the modelled hydrological effect. The maximum amplitudes in winter are mainly caused by high soil water content and snow cover, they amount to up to 54 nm/s2. The effect has a negative sign because most of the surroundings of the observatory are above the gravimeter level.
Local Hydrological Information in Gravity Time Series: Application and Reduction
Fig. 36.2 Water content of the hydrological mass storages (a) in the HRU 191 in the observatory valley, and (b) in the HRU 193 at the steep slope east of the observatory
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–18 Dec-03
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Fig. 36.3 Contributions to the local hydrological effect in gravity from the HRUs 191, 193, 209, and 210 as well as the sum of these HRUs (cf. Fig. 36.1), which build up a zone of approximately 90 m around the SG at Moxa
The effect of the HRUs within a radius of 90 m around the gravimeter is approximately 30% of the total local hydrological effect considered in this study. The area
of the HRUs in a zone between 90 and 250 m around the observatory takes a proportion of 37% from the total effect and shows the greatest impact with
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Fig. 36.4 Contributions to the local hydrological effect in gravity from different zones around the superconducting gravimeter at Moxa observatory
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Fig. 36.5 (a) Local hydrological effect derived from the combined hydrogravimetric modelling for the SG at Moxa; (b) gravity residuals of the SG without and with local hydrological reduction comprising the areas above and below the gravimeter level and the complete area
maximum 20 nm/s2. The area in the zone between 250 and 1,000 m contributes a maximum effect of 18 nm/ s2 to the total local effect, which corresponds to 33%.
36.3.3 Influence of Areas Above/Below the SG The total hydrological effect on gravity in the local catchment (up to a distance of 1,000 m from the gravimeter) for the period from May 2004 to December 2006 is given in Fig. 36.5a for the areas above the SG, below
the SG and for the total area. The maximum amplitude of the total local hydrological effect for the complete model amounts to 54 nm/s2 in winter compared to a dry situation during summer. In the local effect seasonal variations can be identified with maximum amplitudes in winter, which are mainly caused by high soil water content and snow. The maximum amplitude of the hydrological effect from the area below the SG level is 11 nm/s2, corresponding to about 15% of the total local effect. The contributions of the areas above the instrument, which amount to maximum 65 nm/s2, correspond to about 85% of the entire local effect.
36
Local Hydrological Information in Gravity Time Series: Application and Reduction
The reduction of the total local effect in the SG data (Fig. 36.5b) reveals a seasonal signal in the gravity data which was masked by local hydrology. This seasonal signal with maximum amplitude in winter and minimum in summer is in general correspondence to GRACE satellite observations and to global hydrological models (Weise et al. 2009; Naujoks et al. 2010). If only the part of the hydrological signal originating from below or above the SG level were reduced, there still remains a local hydrological signal in the SG data which is not the case for satellite data or the gravity variations derived from global hydrological models. Thus, a complete reduction of local hydrological effects seems to be necessary here.
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terrestrial gravity records to satellite data and to gravity effects derived from global hydrological models reasonable. In future a discussion is required about a consistent reduction of local hydrology from highprecision superconducting and absolute gravity data to make them more available for comprehensive largescale hydrological studies. Acknowledgements The authors are indebted to Manfred Fink, Norbert Kasch, Wernfrid K€uhnel, Matthias Meininger, Martin Rasmussen and Stefanie Zeumann from FriedrichSchiller-University-Jena for their help in the extensive field work. We gratefully acknowledge Hans-J€urgen G€otze and Sabine Schmidt from Chrisitan-Albrechts-University Kiel for fruitful discussions and for providing the software IGMAS. The authors thank the German Research Foundation (DFG) for their funding of this research.
Conclusions
The local hydrological impact on gravity of the hilly and geologically heterogeneous surroundings of the superconducting gravimeter at the Geodynamic Observatory Moxa is studied by combined hydrogravimetric modelling. The local topography, geology, and hydrological processes are considered in detail. It is shown that the changes in the hydrological state variables are very heterogeneous in the different HRUs. Thus, a simplified description of the local hydrological processes (e.g. homogeneous layers and subsoil) would not be successful for the surroundings of the SG at Moxa. In the observatory valley soil profiles are deeper and, thus, the soil water content is generally by about 30% higher than at the steep slope. Most of the hydrological contributions are due to changes in the soil water content of the upper soil layer and the disaggregated heterogeneous bedrock, only a minor part due to ground water level changes. About 30% of the local hydrological effect originate from an area within a radius of 90 m around the observatory, and 37% from the zone between 90 and 250 m around the instrument. The contribution of areas above the SG level at Moxa is about 85% of the total local effect leading to a reverse correlation between gravity and local hydrological changes. After the local hydrological signal is removed, the SG data become suitable to be interpreted with regard to changes in continental water storage as found in GRACE satellite observations and in global hydrological models. Only a complete reduction of local hydrological signals seems to make a comparison of
References Abe M, Takemoto S, Fukuda Y, Higashi T, Imanishi Y, Iwano S, Ogasawara S, Kobayashi Y, Takiguchi H, Dwipa S, Kusuma DS (2006) Hydrological effects on the superconducting gravimeter observation in Bandung. J Geodyn 41(1–3):288–295. doi:10.1016/j.jog.2005.08.030 Boy J-P, Hinderer J (2006) Study of the seasonal gravity signal in superconducting gravimeter data. J Geodyn 41 (1–3):227–233. doi:10.1016/j.jog.2005.08.035 Creutzfeldt B, G€untner A, Kl€ugel T, Wziontek H (2008) Simulating the influence of water storage changes on the superconducting gravimeter of the Geodetic Observatory Wettzell, Germany. Geophysics 73(6):WA95–WA104. doi: 10.1190/1.2992508 G€otze H-J, Lahmeyer B (1988) Application of three-dimensional interactive modelling in gravity and magnetics. Geophysics 53(8):1096–1108. doi:10.1190/1.1442546 Harnisch G, Harnisch M (2006) Hydrological influences in long gravimetric data series. J Geodyn 41(1–3):276–287. doi:10.1016/j.jog.2005.08.018 Hokkanen T, Korhonen K, Virtanen H, Laine EL (2007) Effects of the fracture water of bedrock on superconducting gravimeter data. Near Surf Geophys 5(2):133–140 Imanishi Y, Kokubo K, Tatehata H (2006) Effect of underground water on gravity observation at Matsushiro, Japan. J Geodyn 41:221–226. doi:10.1016/j.jog.2005.08.031 Krause P, Naujoks M, Fink M, Kroner C (2009) The impact of soil moisture changes on gravity residuals obtained with a superconducting gravimeter. J Hydrol 373(1–2):151–163. doi:10.1016/j.jhydrol.2009.04.019 Kroner C (2001) Hydrological effects on gravity data of the Geodynamic Observatory Moxa. J Geod Soc Jpn 47(1): 353–358 Kroner C, Jahr T (2006) Hydrological experiments around the superconducting gravimeter at Moxa Observatory. J Geodyn 41(1–3):268–275. doi:10.1016/j.jog.2005.08.012 Kroner C, Jahr T, Jentzsch G (2004) Results of 44 months of observations with a superconducting gravimeter at
304 Moxa/Germany. J Geodyn 38(3–5):263–280. doi:10.1016/j. jog.2004.07.012 Kroner C, Jahr T, Naujoks M, Weise A (2007) Hydrological signals in gravity – foe or friend? vol 130, IAG symposia series. Springer, Berlin, pp 504–510 Llubes M, Florsch N, Hinderer J, Longuevergne L, Amalvict M (2004) Local hydrology, the Global Geodynamics Project and CHAMP/GRACE perspective: some case studies. J Geodyn 38(3–5):355–374. doi:10.1016/j.jog.2004.07.015 M€akinen J, Tattari S (1988) Soil moisture and groundwater: two sources of gravity variations. Bull Inf Mare´es Terr 63:103–110 Meurers B, Van Camp M, Petermans T (2007) Correcting superconducting gravity time-series using rainfall modelling at the Vienna and Membach station and application to Earth tide analysis. J Geod 81(11):703–712. doi:10.1007/ s00190-007-0137-1 Nash JE, Sutcliffe JV (1970) River flow forecasting through conceptual models part I – a discussion of principles, J Hydrology 10(3):282–290. doi:10.1016/0022-1694(70)90255-6 Naujoks M (2008) Hydrological information in gravity: observation and modelling. PhD thesis, Institute of Geosciences, Friedrich-Schiller-University Jena. http://www.db-thueringen. de/servlets/DerivateServlet/Derivate-16661/Naujoks/Dissertation.pdf, Access date 25 August 2011 Naujoks M, Weise A, Kroner C, Jahr T (2008) Detection of small hydrological variations in gravity by repeated observations with relative gravimeters. J Geod 82:543–553. doi:10.1007/s00190-007- 0202-9
M. Naujoks et al. Naujoks M, Kroner C, Weise A, Jahr T, Krause P, Eisner S (2010) Evaluating local hydrological modelling by temporal gravity observations and a gravimetric three-dimensional model. Geophys J Int. 182:233–249 Neumeyer J, Barthelmes F, Dierks O, Flechtner F, Harnisch M, Harnisch G, Hinderer J, Imanishi Y, Kroner C, Meurers B, Petrovic S, Reigber C, Schmidt R, Schwintzer P, Sun H-P, Virtanen H (2006) Combination of temporal gravity variations resulting from superconducting gravimeter (SG) recordings, GRACE satellite observations and global hydrology models. J Geod 79(10–11):573–585. doi:10.1007/s00190-005-0014-8 Sato T, Boy J-P, Tamura Y, Matsumoto K, Asari K, Plag H-P, Francis O (2006) Gravity tide and seasonal gravity variation ˚ lesund, Svalbard in Arctic. J Geodyn 41(1–3): at Ny-A 234–241. doi:10.1016/j.jog.2005.08.016 Van Camp M, Vanclooster M, Crommen O, Petermans T, Verbeeck K, Meurers B, van Dam T, Dassargues A (2006) Hydrogeological investigations at the Membach station, Belgium, and application to correct long periodic gravity variations. J Geophys Res 111:B10403. doi:10.1029/ 2006JB004405 Virtanen H, Tervo M, Bilker-Koivula M (2006) Comparison of superconducting gravimeter observations with hydrological models of various spatial extents. Bull Inf Mare´es Terr 142:11361–11368 Weise A, Kroner C, Abe M, Ihde J, Jentzsch G, Naujoks M, Wilmes H, Wziontek H (2009) Gravity field variations from superconducting gravimeters for GRACE validation. J Geodyn 48(3–5):325–330. doi:10.1016/j.jog.2009.09.034
Signals of Mass Redistribution at the South African Gravimeter Site SAGOS
37
€ntner, B. Creutzfeldt, M. Thomas, C. Kroner, S. Werth, H. Pflug, A. Gu H. Dobslaw, P. Fourie, and P.H. Charles
Abstract
The superconducting gravimeter (SG) operating at the South African Geodynamic Observatory Sutherland (SAGOS) is one of the few instruments installed in the southern hemisphere and presently still the only one of its kind on the African continent. SAGOS is located in the Karoo, a semi-arid area with an average annual precipitation of 200–400 mm. The distance to the ocean is approx. 220 km. A local hydrology-related seasonal effect on gravity is clearly seen in the SG record. Its general order of magnitude is estimated to be about 4–10 nm/s2. A large-scale hydrological influence in a similar order of magnitude or even larger (up to 60 nm/s2 peak-to-peak) is inferred from global hydrological models for the years 2003–2007. Significant contributions are found for the southern coast, the central Cape region, and the basin of the Orange river. Contributing basins with larger distance comprise the areas of Okavango/Sambesi, Congo, and eastern Africa. Between SG data, temporal GRACE gravity field solutions, and the gravity effect derived from global hydrological models clear differences exist. Among others, the deviations between the hydrological models can be traced to deviations in the gravity effect originating from the Okavango basin and the central Cape region.
C. Kroner (*) Physikalisch-Technische Bundesanstalt Braunschweig, Bundesallee 100, 38116 Braunschweig, Germany e-mail:
[email protected] S. Werth Institute of Earth and Environmental Science, University of Potsdam, Karl-Liebknecht-Str. 24-25, 14476 Potsdam-Golm, Germany H. Pflug A. G€untner B. Creutzfeldt M. Thomas H. Dobslaw Helmholtz Centre Potsdam GFZ German Research Centre for Geosciences, Telegrafenberg, 14473 Potsdam, Germany P. Fourie P.H. Charles South African Astronomical Observatory, Observatory Road, Observatory, 7925, South Africa S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_37, # Springer-Verlag Berlin Heidelberg 2012
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Gravity residuals reduced for changes in continental water storage are compared to the gravity effect caused by non-tidal oceanic mass changes. A rudimentary correlation between observed variations and modeled effect is found. The peak-to-peak amplitude of the modeled effects amounts to 15 nm/s2 for the years 2001–2008. After reducing the SG data for this oceanic effect the variation of the residuals decreases by 9%. The present findings indicate the suitability of the SG observations at Sutherland for studies on mass transport phenomena in the South African region.
37.1
Introduction
Terrestrial gravity observations are a versatile tool to study processes of mass redistribution on various spatial and temporal scales as they fully contain the changes in all compartments in contrast to hydrological point observations. Processes of interest relate to variations in continental water storage as well as tidal and non-tidal mass transport in the oceans. Recently, efforts have been made to combine superconducting gravimeter (SG) observations for Europe (Crossley et al. 2005; Hinderer et al. 2006; Neumeyer et al. 2008; Weise et al. 2009). A main issue regarding the terrestrial data is the separation of local (distances a few km from the instrument) and regional hydrological impacts. This can be achieved by appropriate local hydrological measurements and local to regional models. On the other hand the dependency on local phenomena can be deployed for the study of smallscale hydrological processes as associated with rain events and the subsequent percolation. Furthermore main impact areas of non-tidal ocean loading effects can be identified. Based on recent experiences the application of terrestrial gravity observations to studies of mass transport phenomena will be extended to the region of southern Africa. At the South African Geodynamic Observatory Sutherland (SAGOS; 32.38 S, 20.81 E, 1,759 m above msl) the superconducting gravimeter CD037 was operating between spring 2000 and summer 2008. In August 2008 the gravimeter was replaced by the OSG-052. The shortest distance of SAGOS to the coast is about 220 km. The dominating country rock is dolerite which is covered by a weathering layer of several 10 cm thickness. The annual precipitation is in the range of 200–400 mm. The vegetation cover (low shrubs and grasses) is expected to be in the range of 15–20% (Essler et al. 2006).
37.2
Variations in Continental Water Storage
37.2.1 Local Hydrological Effect The impact of changes in local hydrology (typically zone < 1 km from the instrument) on gravity depends on the time-scale and local topography. The observatory is located on a mountain plateau which rises approximately 150–200 m above the surrounding area. For temperature screening the observatory is covered by a soil-gravel layer. On a time-scale of up to a couple of days soil moisture variations related to rain events are predominantly visible in the gravity record (Fig. 37.1). Due to the coverage and the lowered gravimeter pier most of the topography in the immediate vicinity of the instrument is above sensor level which leads to the observed anti-correlation between the gravity residuals and hydrological variations. In the gravity record rain events are visible as distinct peaks. 1 mm rain roughly corresponds to 0.5 nm/s2 gravity decrease. Variations of the groundwater table (Fig. 37.2) have a smaller gravity effect because of the depth of the water table (about 60 m below the gravimeter) and the limited spatial correlation length of water level variations at these short time scales. At longer time scales the area increases in which coherent changes in water storage occur, which thus leads to a significant impact on gravity. This becomes visible when comparing gravity residuals with water table variations at a site in the valley 2–3 km away from the gravimeter and with filtered air temperature data as an indicator for seasonal soil moisture variations (Fig. 37.2). The water table variations are characterized by a seasonal variation. Around February/ March 2005 there was a dry spell and substantially
Signals of Mass Redistribution at the South African Gravimeter Site SAGOS
Fig. 37.1 Gravity residuals, precipitation, soil moisture (depth 0–0.4 m) and relative water level changes (distance ~ 50 m) in the SG vicinity during a period of strong rain events. The grey line is for better comparison
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increased water pumping which led to a decrease of the water level by about 2 m. These groundwater dynamics may be a good indicator of soil moisture and groundwater depletion in a larger area around the gravimeter and, thus, a highly correlated gravity decrease is seen for this period. An empirical reduction of the water table effect with a regression coefficient leads to gravity residuals which still exhibit, among other inter-annual and seasonal signals, such as a pronounced about annual variation of 10–15 nm/s2 peak-to-peak-amplitude. Variations in the gravity residuals reduced by the empirical method cannot be explained by changes in regional continental water storage, because its seasonal contribution is opposite to the observed residuals. This indicates that gravity signals originating from the immediate gravimeter vicinity still exist which are not captured, most likely due to soil moisture variations.
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In Fig. 37.3 the impact of large-scale changes in continental water storage at SAGOS is shown exemplarily estimated from two global hydrological models, the WaterGAP Global Hydrological Model oll et al. 2003; (WGHM, spatial resolution 0.5 ; D€ G€untner et al. 2007) and the Global Land Data Assimilation System (GLDAS, spatial resolution 1 ; Rodell et al. 2004), both with daily resolution. A main difference between the two hydrological models exists in the hydrological compartments considered. In contrast to GLDAS the WGHM includes the components surface and groundwater table storage. The loading effect is computed by use of Green’s
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Fig. 37.2 Gravity residuals, relative water level changes observed at one site in the valley filtered air temperature (25 h) and precipitation between Jan. 2001 and Dec. 2006
function (Farrell 1972; Francis and Dehant 1987) assuming an earth structure according to PREM (Dziewonski and Anderson 1981).
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The gravity residuals given in Fig. 37.3 are reduced for local hydrology. The local hydrological reduction was realized based on the water level observations given in Fig. 37.3 which reflect soil moisture variations, the dominant source of local hydrological effects. The SG data are detided, detrended and reduced for atmospheric mass changes using 3d meteorological data, for polar motion (solid earth + ocean) and LOD-effect. Details are found in Kroner et al. (2009) and Chen et al. (2009). Substantial discrepancies exist between the time series with regard to amplitude and features. There is approximately a factor of 5 between the peak-to-peakamplitudes of the variations based on the hydrological models. The modeled gravity changes are in both cases in an order of magnitude that they should be detectable in the observed gravity variations without any difficulty. The GRACE-derived gravity changes are again larger than the SG variations and the modeled hydrological effects. Presently no obvious agreement between the data sets is evident. The existing discrepancies emphasize the necessity for
research with respect to hydrological processes in the southern African region. A separation of the hydrological impacts at Sutherland according to major river and drainage basins yields main contributions from the regions of Okavango/Sambesi (4), Congo (6), and eastern Africa (3) (Fig. 37.4) in addition to the area (5) in which SAGOS is located. Considering the area of southern Africa (Fig. 37.5), substantial effects emerge from the southern coast (55), Namibia/Bots-wana (57) and the area of the Orange river (56). The order of magnitude of the peak-to-peak amplitude depends on the hydrological model and is in the range of 2–4 nm/s2 for the remote river and drainage basins considering the period from January 2003 to June 2006. From the areas closer to the SG site effects in the order of 3–7 nm/s2 emerge. Figure 37.6 illustrates the differences in the impact the individual cells have at SAGOS depending on the hydrological model. Larger deviations in the rmsvalues of about 6 10 3 and 8 10 3 nm/s2 occur in the western and central Cape region and the Okavango basin. The deviations in the contributions of the various basins to gravity changes at SAGOS will be used to work towards an improvement of the hydrological models in this region.
37.3
Non-Tidal Ocean Loading
In addition to changes in continental water storage gravity variations are induced by tidal and non-tidal mass redistribution in the oceans. At SAGOS gravity changes in the range of 15 nm/s2 peak-to-peak occur related to non-tidal mass transport (Kroner et al. 2009) which are in a similar order of magnitude as loading effects due to oceanic tides. The estimates are based on the ‘Ocean Model for Circulation and Tides’ (OMCT, Dobslaw and Thomas 2007). Both, shortterm and seasonal contributions exist. The seasonal contribution amounts to 8 nm/s2 peak-to-peak. The effects are sufficiently large to be detectable in the SG observations. Due to its order of magnitude this influence should be eliminated in the gravity data for hydrology-related studies. A rudimentary correlation exists between the SG residuals reduced for hydrological effects and the
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Signals of Mass Redistribution at the South African Gravimeter Site SAGOS
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Fig. 37.4 Gravity variations at SAGOS derived from monthly values of global hydrological models for various main catchment areas and regions, Jan. 2003–Jun. 2007 (different scaling). A local zone of 1 km (upper right graph) around the gravimeter is omitted in the computations of the effect from the cell in which SAGOS is located assuming that in this area local effects dominate
OMCT-derived gravity variations (Fig. 37.7). From the point of view of the gravity observations the effect of non-tidal ocean loading appears to be even underestimated regardless which global hydrological
model is applied for the reduction of large-scale changes in continental water storage. The reduction of the gravity observations for non-tidal ocean loading leads to a decrease in the rms-value by 9%.
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Fig. 37.5 Gravity variations at SAGOS derived from monthly values of global hydro-logical models for sub-basins of basin 5 which covers southern Africa, Jan. 2003–Jun. 2007 (different scaling)
Two regions can be identified (Fig. 37.7) from which the non-tidal loading contributions at SAGOS mainly originate: the immediate coastal area west of SAGOS with a peak-to-peak amplitude of 4 nm/s2 and a substantially larger region east of South Africa with a peak-to-peak amplitude of 9 nm/s2. The zones of similar oceanic mass variations are derived from fh-contour lines which are computed from the latitudedependent Coriolis parameter f and the ocean depth h. Due to the sensitivity of gravity regarding mass changes in specific oceanic areas the SG observations at Sutherland can be deployed as constraint in modeling of non-tidal the in these areas provide the basis towards an assessment and thus improvement of these models for the southern African region. Further work needs to be done in order to obtain a better agreement between the terrestrial observations and GRACE gravity field solutions. Due to the small order of magnitude of mass redistribution signals
compared e.g. to regions in North or South America or Asia ambiguities in modeling and data processing become more apparent in the data sets covering the southern African region than in the previously mentioned ones. In addition to an impact from large-scale hydrological mass changes a non-negligible effect induced by non-tidal oceanic loading on gravity is found at SAGOS on both, short-term and seasonal scale. The peak-to-peak amplitude of the effect is in the range of 15 nm/s2 based on the years 2001–2008. From a comparison with the SG data it even appears that the effect is underestimated. The immediate coastal area west of SAGOS and a larger region east of South Africa are identified as the zones from which together more than 85% of the effect originate. The sensitivity of the gravimeter at SAGOS to mass variations in these regions can be used as approach for an improvement of the model on non-tidal ocean loading.
37
Signals of Mass Redistribution at the South African Gravimeter Site SAGOS
311
Fig. 37.6 RMS-values of gravity effect at SAGOS originating from various cells of the global hydrological models WGHM, LaD, GLDAS-Noah, and average over the three models (monthly values), Jan. 2003–Jun. 2007
ConclusionsFrom estimates based on global hydro-
logical models a gravity effect of regional changes in continental water storage emerges in an order of magnitude that should be detectable in the record of the superconducting gravimeter at SAGOS. With 10–60 nm/s2 depending on model and time resolution it is in a similar order of magnitude or even
significantly larger than the impact of local hydrological mass changes. Presently, no appropriate agreement between the estimates on the regional hydrological impact, temporal GRACE-derived gravity field variations, and SG data is found. Specific areas from which main contributions arise can be identified. The amount of the contributions from
312
C. Kroner et al. 30
total OMCT-dffect
nm/s2
10
–10
–30
6
gravity residuals
Jan-01 Jan-02 Jan-03 Jan-04 Jan-05 Jan-06 Jan-07
nm/s2
2 50° –2 0° –6 Jan-01
Jan-03
Jan-05
Jan-07
6
Jan-09 –50°
–150°
–100°
–50°
0°
50°
100°
150° 6
nm/s2
2
–2 nm/s2
2
–6
–2 Jan-01
Jan-03
Jan-05
Jan-07 6
Jan-09 6
–6 Jan-01
Jan-03
Jan-05
Jan-07
Jan-09
nm/s2
2
nm/s2
2
–2
–2
–6
–6 Jan-01
Jan-03
Jan-05
Jan-07
Jan-09
Jan-01
Jan-03
Jan-05
Jan-07
Jan-09
Fig. 37.7 Total effect of non-tidal oceanic mass redistribution and contributions from different zones to gravity variations at SAGOS. The computations are based on the OMCT (Dobslaw and Thomas 2007). For comparison the gravity residuals at SAGOS are given. In the SG data shown local and large-scale/based on GLDAS) hydrological effects are eliminated
these basins differs between the hydrological models considered. The differences in these areas provide the basis towards an assessment and thus improvement of these models for the southern African region. Further work needs to be done in order to obtain a better agreement between the terrestrial observations and GRACE gravity field solutions. Due to the small order of magnitude of mass redistribution signals compared e.g. to regions in North or South America or Asia ambiguities in modeling and data processing become more apparent in the data sets covering the southern African region than in the previously mentioned ones. In addition to an impact from large-scale hydrological mass changes a non-negligible effect induced by non-tidal oceanic loading on gravity is
found at SAGOS on both, short-term and seasonal scale. The peak-to-peak amplitude of the effect is in the range of 15 nm/s2 based on the years 2001 to 2008. From a comparison with the SG data it even appears that the effect is underestimated. The immediate coastal area west of SAGOS and a larger region east of South Africa are identified as the zones from which together more than 85% of the effect originate. The sensitivity of the gravimeter at SAGOS to mass variations in these regions can be used as approach for an improvement of the model on non-tidal ocean loading. In studies of either of the two phenomena, the impact of the reduction of the second effect on the results should be considered. From present findings it appears the influence of the uncertainties in the models is small. In geodynamic studies both
37
Signals of Mass Redistribution at the South African Gravimeter Site SAGOS
effects ought to be eliminated in the gravity data in order to avoid contamination of the results. Acknowledgements We thank P. D€ oll, the GGFC Special Bureau for Hydrology, and NOAA for providing data of the global hydrological models used in this study. The provision of the weekly GRACE gravity field and MASCON solutions by GFZ-ISDC and NASA is gratefully acknowledged. Our thanks also go to two anonymous reviewers for their support.
References Chen X-D, Kroner C, Sun H-P, Abe M, Zhou J, Yan H, Wziontek H (2009) Determination of gravimetric parameters of the gravity pole tide using observations recorded with superconducting gravimeters. J Geodyn 48(3–5):348–353 Crossley D, Hinderer J, Boy J-P (2005) Time variation of the European gravity field from superconducting gravimeters. Geophys J Int 161(2):257–264 Dobslaw H, Thomas M (2007) Simulation and observation of global ocean mass anomalies. J Geophys Res 112:C05040 D€oll P, Kaspar F, Lehner B (2003) A global hydrological model for deriving water availability indicators: model tuning and validation. J Hydrol 270:105–134 Dziewonski AM, Anderson DL (1981) Preliminary reference earth model (PREM). Phys Earth Planet Int 25(4):297–367 Essler KJ, Milton SJ, Dean WRJ (2006) Karoo Veld – ecology and management. BRIZA Publications, Pretoria, 214
313
Farrell WE (1972) Deformation of the earth by surface loads. Rev Geophys 10:761–797 Francis O, Dehant V (1987) Recomputation of the Green’s functions for tidal loading estimations. Bulletin d’Information des Mare´es Terrestres 100:6962–6986 G€untner A, Stuck J, Werth S, D€oll P, Verzano K, Merz B (2007) A global analysis of temporal and spatial variations in continental water storage. Water Resour Res 43(5):W05416 Hinderer J, Andersen O, Lemoine F, Crossley D, Boy JP (2006) Seasonal changes in the European gravity field from GRACE: A comparison with superconducting gravimeters and hydrology model predictions. J Geodyn 41(1–3):59–68 Kroner C, Thomas M, Dobslaw H, Abe M, Weise A (2009) Seasonal effects of non-tidal oceanic mass shifts in observations with superconducting gravimeters. J Geodyn 48(3–5):354–359 Neumeyer J, Barthelmes F, Kroner C, Petrovic S, Schmidt R, Virtanen H, Wilmes H (2008) Analysis of gravity field variations derived from superconducting gravimeter recordings, GRACE satellite and hydrological models at selected European sites. Earth, Planets Space 60:1–14 Rodell M, Houser PR, Jambor U, Gottschalck J, Mitchell K, Meng C-M, Arsenault K, Cosgrove B, Radakovich J, Bosilovich M, Entin JK, Walker JP, Lohmann P, Toll D (2004) The global land data assimilation system. Bull Am Meteorol Soc 85(3):381–394 Weise A, Kroner C, Abe M, Ihde J, Jentzsch G, Naujoks M, Wilmes H, Wziontek H (2009) Terrestrial gravity observations with superconducting gravimeters for validation of satellite-derived (GRACE) gravity variations. J Geodyn 48(3–5):325–330
.
Gravity System and Network in Estonia
38
To˜nis Oja
Abstract
Preparations to establish a new accurate gravity network in Estonia were initiated in 2001. Since then several LCR (LaCoste&Romberg) G-type and Scintrex CG-5 relative gravimeters have been used to determine gravity differences precisely. The calibration functions of those relative instruments have been repeatedly checked at the calibration lines in Estonia and in Finland. Since the beginning of the 1990s absolute gravity values have been determined three times in Estonia: in 1995 at three stations with JILAg-5 by the Finnish Geodetic Institute (FGI), in 2007 at two stations with FG5-220 by the Institut f€ur Erdmessung (IfE), University of Hannover, and a year later at seven stations with FG5-221 again by FGI. On the ground of collected absolute and relative gravity data, a new realization (network GV-EST) of the Estonian gravity system (EGS) is currently being established. However, before the completion of the network, several issues should be solved, including the calibration of relative gravimeters, the corrections of readings and setup of the functional model, the weighting of observation data and selection of statistical tests, the short and long term changes of the gravity field, the choice of the epoch. In the current paper I introduced the concept of EGS as well as the methodology to solve the afore-mentioned issues. Since the estimated uncertainties of gravity values from network adjustment stayed below 10 mGal (1 mGal ¼ 108 m/s2) it was concluded that the selected methods had been efficient.
38.1
T. Oja (*) Department of Geodesy, Estonian Land Board, Mustam€ae tee 51, Tallinn 10621, Estonia e-mail:
[email protected]
Introduction
In the modern world, the gravity system and network are the essential part of the geodetic reference system and frame of every country. The geodetic reference frame which realises the geodetic system consists of geodetic, levelling and gravity networks. In Estonia (see Fig. 38.1), these networks are developed and maintained by the Estonian Land Board (ELB). The establishment of nationwide gravity network is an
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_38, # Springer-Verlag Berlin Heidelberg 2012
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Fig. 38.1 The location of Estonia in Northern Europe
expensive, time-consuming, and technically complicated work. However, modern gravity network with accurately determined gravity accelerations serves as a basis for many practical applications, including geoid computation, the realization of the height system, geological mapping, the improvements of metrological standards. The output of these applications can be influenced by random and especially systematic errors in the gravity values of the network. Thus, when establishing the reference gravity network, a great deal of attention should be paid to such issues like gravimeter’s calibration, the reductions of raw data, the adjustment of the network, temporal gravity changes and so forth. The latest official solution of gravity network in Estonia was presented in 1999. That solution was based on the gravity data measured mainly with GAG-2 relative gravimeters during the years 1970–1990 (Sildvee 1998). Occasional observations were taken later with some LCR G-type instruments as well. The network itself was connected, although quite poorly, to three absolute points where JILAg-5 gravimeter was used in 1995 to determine absolute gravity accelerations by the team of the Finnish Geodetic Institute (FGI). On the basis of these gravity campaigns only arithmetic means of gravity values were estimated for the points in the network solution of 1999 (Oja and
Sildvee 2004). No rigid adjustment of the network with proper weighting, statistical tests etc. were done due to the lack of knowledge and software at that time. Besides, the available, but outdated instrumentation (e.g. GAG-2 meters) did not allow to perform accurate gravity measurements to connect the network reliably to the absolute points.
38.2
New Gravity System and Network
Preparations for establishing a modern and accurate gravity network were initiated in 2001 since NGA (National Geospatial-Intelligence Agency, at that time NIMA) from US loaned three LCR G-type gravimeters to Estonia for the period of 2001–2004. The participation in the BGI/ICET 2002 Summer School and in the gravity campaign along the Fennoscandian Land Uplift Gravity Line 63 N organised by FGI in 2002 gave indispensable knowledge and practical skills necessary for performing high precision relative gravimetry in Estonia. At the same time software for the processing of gravity data and for an adjustment computation of the network were acquired from KMS (National Survey and Cadastre, Denmark) and FGI. By the end of 2003 the whole Estonian gravity network of 2nd order had been measured with LCR
38
Gravity System and Network in Estonia
317
Fig. 38.2 First, second and third order points of the Estonian gravity network. Three absolute (first order) points have been observed twice (KURE) and three times (SUUR, TORA), in 1995 by FGI, in 2007 by IfE and in 2008 by FGI again. Other first order points have been measured once in 2008. Ties between second order points (in 19 closed loops) have been
observed with three LCR G-type relative gravimeters in 2003. Additional measurements were carried out with LCR G and with Scintrex CG-5 gravimeters in 2001–2002 and 2004–2006, respectively. Dashed lines indicate relative gravity ties between the Latvian and Estonian gravity networks observed in 2006. Gray lines show gravity calibration lines in Estonia
G-type gravimeters (Fig. 38.2). In 2004 LCR meters were returned to NGA and since then two modern Scintrex CG-5 gravimeters have been successfully used for supplementary measurements. All these gravimeters have been repeatedly tested along the calibration lines in Estonia and on the Masala-Vihti line of FGI in Finland. In 2004 the Estonian gravity system (EGS) was defined and enforced as a part of the Estonian geodetic system by the regulation of the Government (see Ministry of Environment 2008). According to the regulation the EGS is realised through the gravity network (named GV-EST), which is divided into first, second and third order networks. The scale and level of the network have been secured through absolute gravity determinations, performed with JILAg, FG-5 gravimeters according to IAGBN (International Absolute Gravity Base Station Network) standards. In 2007–2008 four new first order points were additionally built and today the network consists of seven points suitable for high precision absolute
gravimetry. Five of them are spatially integrated with continuously operating GNSS reference stations (within 1 km radius). At present three absolute gravity campaigns have been carried out at the points of first order network. As mentioned before, the first one was carried out in 1995 by FGI. In 2007 two stations were re-occupied by the team of IfE (Institut f€ur Erdmessung, Leibniz Universit€at Hannover, Germany) who applied modern FG5-220 gravimeter for accurate gravity determinations (Oja et al. 2009). In 2008 FG5-221 instrument was used to observe absolute gravity at all seven stations by FGI team. Thus, reliable and accurate absolute gravity datum has nowadays been realized in Estonia through these determinations. Those repeated absolute gravity campaigns help also to constrain the rates of gravity change due to postglacial rebound (PGR) in Northern Europe, the phenomenon which is studied in cooperation with the Nordic Geodetic Commission (NKG). Few Estonian absolute gravity stations (SUUR, TORA, KURE, see Fig. 38.2) belong to the Nordic
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Geodetic Observing System (NGOS) (Poutanen et al. 2006). In 2006 with the help of the Latvian Geospatial Information Agency the precise relative gravity measurements were carried out with Scintrex CG-5 gravimeters of ELB between the points of the Latvian and Estonian national gravity networks. As a result of the campaign, the gravity scale and level of both neighboring countries were checked and unified. On the ground of collected absolute and relative gravity data, a new realisation of the EGS is currently being established. The data processing and the adjustment of the network are currently being completed (see hereinafter).
38.3
Network Densification
Concurrently with the development of higher order networks also third order gravity network was established with relative gravity measurements on the points of the national geodetic network. To date, more than 250 gravity points of third order have been observed with LCR and Scintrex gravimeters (Ellmann et al. 2009), see also Fig. 38.2. Relative gravity measurements for such densification purposes will be continued on the points of the geodetic network and along the lines of high-precision levelling network.
38.4
Data Processing and Adjustment of the Network
The first step in data processing was a conversion of raw gravity data into the units of CGS (centimetregram-second system) according to the calibration function of the gravimeter’s manufacturer. In order to meet high accuracy requirements, the additional calibration correction function was applied. The parameters of that function were exclusively estimated from the measurements of calibration lines. A more detailed description of how to set up the function and estimate the parameters can be found later. It is critical to remove any known disturbing environmental effects from the observation data before the calibration and network adjustment. Today there are many state-of-the-art models available to compute correction values for the readings of relative
gravimeters. It should be pointed out that the IAGBN standards for data processing in absolute gravimetry (see Boedecker 1988 for details) are a useful base also for the processing of relative gravity observations. For the accurate tidal corrections the ETGTAB software (Wenzel 1994) was modified and integrated into our software using Tamura’s tidal potential development with 1,200 waves (Tamura 1987). Local amplitude factors d and phase lags DF for the wave groups were interpolated from the global grid (Timmen and Wenzel 1994). The permanent tide was treated according to the zero system concept, i.e. for M0S0 d ¼ 1.0 (see IAG Resolution 1984). For the atmospheric correction local air pressure, normal pressure from DIN 5450 formula, and the coefficient 0.3 mGal/hPa (1 mGal ¼ 10 nm/s2) were used. To reduce the gravity value from the observation elevation to the top of the benchmark, equal to hred, (see Fig. 38.3), free air correction with vertical gravity gradient (0.31 mGal/mm or local observed value) was applied. For such correction one should know the approximate position of the gravimeter’s sensor (hsys) from the top plate of the meter. For LCR G-type gravimeter it is about 0.16 m (M€akinen et al. 1986) and for Scintrex CG-5 the same value is close to 0.21 m (CG-5 manual). Heights hinst and hbase were observed during the field campaigns. The gravity acceleration of the Estonian gravity network points are also affected by glacial isostatic adjustment (GIA) and postglacial rebound (PGR). Although the resulting gravity change (g˙ ¼ dg/dt) is estimated to be less than 1 mGal/year in Estonia (see Fig. 38.4), it is biasing systematically the gravity values of the network. To avoid such interference, ˚ gren and the vertical uplift model NKG2005LU (A
Fig. 38.3 Position of the gravimeter’s sensor relative to the benchmark
38
Gravity System and Network in Estonia
319
Fig. 38.4 Contours represent g˙ isolines (mGal/year), derived ˚ gren and Svensson 2007). from the NKG2005LU uplift map (A The eustatic rise of mean sea level (+1.32 mm/year), the uplift
of the geoid (about 6%) were added and then scaled by value 0.2 mGal/mm (see M€akinen et al. 1986)
Svensson 2007) was used to derive the time rates of gravity change for every point in the network. These rates were then applied to reduce the observation data to a common epoch T0 (currently T0 ¼ 1995.8). The corrected readings of the gravimeter were introduced into functional model as independent readings
are computed for LCR G-type as well as for Scintrex CG-5 gravimeters. FPer (function of periodic errors) is necessary only for LCR instruments (CG-5 has no periodic errors)
yðt; T0 Þ ¼ gðT0 Þ þ N0 þ DðtÞ þ FðzÞ;
(38.1)
where y(t, T0) is the corrected reading of the gravimeter at the observation time t (reduced to T0), g(T0) is the gravity value (at the epoch T0), D(t) is the polynomial drift function, N0 is the relative instrument level and F (z) the calibration correction function of a counter reading z (in counter units, C.U. or in mGal). The calibration correction function F(z) can be divided into two: FðzÞ ¼ FPol ðzÞ þ FPer ðzÞ; where FPol (low degree polynomial) FPol ðzÞ ¼
Xm k¼1
Yk z k :
(38.2)
FPer ðzÞ ¼
Xn l¼1
El sinð2p z=Pl þ ’l Þ;
where Y, E, ’ are unknown and estimable parameters of the calibration correction function, P is the known periodic term in counter units (Torge 1989). After the setup of the model, all observations were collected into the system of linear equations which was then solved by well-known least squares (LS) method to get optimal unbiased estimation of the vector of unknown parameters. For the linearization of function FPer the addition theorem of the sine function and the conversion from polar (E, ’) to Cartesian coordinates (x, y) were used. The correction of readings, the model setup and an adjustment of observations were efficiently done with the programs GRREDU2 and GRADJ2. However, decisions on when and how (e.g. what degree) to use drift, calibration correction and time variation functions were made by an experienced user after
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several computation test runs. More details on the gravity data processing and the adjustment of the network can be found in Oja (2008).
38.5
Preliminary Results
After the reduction of relative gravity data with the models described before, the calibration function of every relative gravimeter was verified. It became obvious from the results of calibration lines that the calibration correction function (38.2) needs to be determined for every LCR G gravimeter via LS adjustment (see results in Table 38.1 and Fig. 38.5). For example, amplitude for the wave with period about 71 C.U. was estimated to be about 53 mGal which is remarkably high value. No significant calibration errors for Scintrex CG-5 gravimeters were found, which proves good manufacturer calibration and reliability of that kind of instrument.
After the correction of readings according to the parameters of calibration function in Table 38.1, LS adjustment of the second order network with the model (38.1) was performed. The computation was done step by step: first drift functions and jumps (tares) were described and outliers were removed (after careful testing), then the weight value for every gravimeter was deduced from the adjustment of instrument’s dataset with unit weights. The standard deviation of single reading (as a unit weight) was used for that purposes. The common adjustment of the cleaned and weighted gravity data of all gravimeters resulted in the standard deviations of gravity values (of the second order points) as well as the RMS of the residuals of all readings below 10 mGal. The points of first order network were fixed in adjustment but only the absolute gravity determinations from 1995 were used. Due to the incomplete data processing, the results of more recent absolute gravity campaigns (in 2007–2008) were not included yet into
Table 38.1 Estimated parameters of calibration correction function (see (38.2)) for LCR gravimeters. Only statistically significant estimates (that passed t-test) are shown Gravimeter
G-4
G-113
G-115
FPol, Y1 (105)
377.2 4.9
16.4 2.6
3.8 2.3
FPer P (C.U.) 1.0000 3.9412 7.8824 35.4706 70.9412
E (mGal) 4.4 1.1 – – 4.5 1.0 6.0 1.2 4.3 1.7
Fig. 38.5 Modeled periodical errors of LCR G-113. For the wave with period about 71 C.U. an amplitude is estimated to be roughly 53 mGal (see also Table 38.1)
’ ( ) 322 – 166 79 57
14 – 16 14 23
E (mGal) 2.6 2.3 6.3 5.8 52.5
0.6 0.6 0.6 0.7 0.8
’ ( ) 235 196 268 7 326
14 13 5 6 1
E (mGal) 1.7 – 3.7 11.1 8.7
0.5 – 0.6 0.6 0.7
’ ( ) 69 – 53 212 12
17 – 9 3 6
38
Gravity System and Network in Estonia
the current adjustment of the network. Thus the adjustment presented here should still be regarded as a preliminary solution and the final results with full statistics were not showed here. Nevertheless, the estimated accuracy (below 10 mGal) proves to a certain extent that the above-described methods and models worked well in our case. The observation data of third order network have currently been processed and adjusted as described beforehand. Although, the second order points (with lower weights than the first order points) were kept fixed too. The standard deviations of adjusted gravity values stayed within the limits of 50 mGal.
38.6
Applications of the Gravity Network
Based on the new realisation of national gravity system, the scale and level of historical networks have been re-evaluated. For that purpose relative gravity ties were measured between the points of new and old networks if no common point was available. It became evident that several realisations of Potsdam gravity system established from 1930 to 1960-ies differ +12.9. . .+15.4 mGal from the modern gravity datum in Estonia. In case of IGSN71 realisations, offsets about +0.07. . .+0.16 mGal could be traced (Oja 2007). Lately the gravity values of the network were used as control data to test the quality of the datasets of historic gravity surveys which have been extensively used in geoid computations (see J€ urgenson 2003; Ellmann 2004) as well as for geological mapping purposes (All et al. 2004). The comparison based on the spatial interpolation revealed clear systematic discrepancies in several places in Estonia, with an average bias up to 3–4 mGal (Ellmann et al. 2009). The conlcusion of that study was that erroneous historic data should be replaced with new survey data as soon as possible. For that purpose a cooperation project between ELB, TUT (Tallinn University of Technology), EULS (Estonian University of Life Sciences) and GSE (Geological Survey of Estonia) was started and for the moment several 100 new survey points have been observed along the roads and the high-precision levelling lines as well as on large icecovered lakes. The gravity values of contemporary
321
second and third order networks were serving as a reliable base for collecting such data. A couple of years ago, the geologists from GSE converted their bulky gravity dataset (about 1.3 105 points) from the IGSN71 level to the current GV-EST datum. In 2007 the national office of metrology, AS Metrosert requested accurate gravity determinations in their laboratories in Tallinn and Tartu. The sites were directly connected to the first order points of gravity network. Conclusion
The setup of a modern gravity system with the establishment of accurate gravity network includes many issues which need to be considered and solved. In this paper, the lately defined national gravity system, the EGS, and its newest realisation in Estonia, network GV-EST was introduced. The methods for the calibration of the gravimeters, the corrections of raw data, the functional model setup, etc. were introduced. The presented results show an reliable procedure to achieve modern and accurate national gravity network. New accurate gravity accelerations at the points of the Estonian gravity network are available for geodesists, geologists, metrologists and for other national and international users. Acknowledgements Part of this work has been supported by the Estonian Science Foundation grant ETF 7356. Maps were created with GMT – The Generic Mapping Tools (http://gmt. soest.hawaii.edu/). Programs GRREDU2, GRADJ2 have been originally developed by R. Forsberg. The author thanks two anonymous reviewers whose comments helped to improve the paper.
References All T, Puura V, Vaher R (2004) Orogenic structures of the Precambrian basement of Estonia as revealed from the integrated modelling of the crust. Proc Estonian Acad Sci Geol 53(3):165–189 ˚ gren J, Svensson R (2007) Postglacial land uplift model and A system definition for the new swedish height system rh 2000. LMV-Rapport 2007:4 Boedecker G (1988) International Absolute Gravity Basestation Network (IAGBN). Absolute gavity observations data processing standards and station documentation. BGI Bull Inf 63:51–57 CG-5 Gravity Meter Operation Manual, part # 867700 Rev. 4, http://www.scintrexltd.com (2008-10-01)
322 Ellmann A (2004) The geoid for the Baltic countries determined by the least squares modification of Stokes’ formula. Doctoral Disseration, Royal Institute of Technology, Stockholm, 80 Ellmann A, All T, Oja T (2009) Toward unification of terrestial gravity data sets in Estonia. Estonian J Earth Sci 58:229–245 IAG Resolution (1984) IAG Resolution No.16. Hamburg 1983, Bulletin Ge´ode´sique, 58, p 321 J€urgenson H (2003) Eesti t€appisgeodi arvutus. (Determination of Estonian precise geoid). PhD Thesis, Estonian Agricultural University, Tartu (in Estonian) Ministry of Environment (2008) Geodeetiline s€ usteem. Keskkonnaministri m€a€arus no 26, 30 juuni 2008.a. (Geodetic system. Regulation No 26 of the Minister of Environment, June 30, 2008), Tallinn, (in Estonian) ˚ , Remmer O (1986) The M€akinen J, Ekman M, Midtsundstad A Fennoskandian Land Uplift Gravity Lines 1966–1984. Reports of Finnish Geodetic Institute, 85:4, Helsinki Poutanen M, Knudsen P, Lilje M, Nørbech T, Scherneck H-G (2006) NGOS, the Nordic Geodetic Observing System. Proceedings of the 15th NKG General Assembly, Copenhagen, Denmark Oja T, Timmen L, Gitlein O (2009) Determination of the gravity acceleration at the Estonian stations Suurupi and To˜ravere
T. Oja with the absolute gravimeter FG5-220 in 2007. J Geod 39:16–27 (in Estonian, with abstract in English) Oja T (2008) New solution for the Estonian gravity network GVEST95. The 7th International Conference, Environmental Engineering Selected Papers, vol III, pp 1409–1414 Oja T (2007) Gravimeetrilised s€usteemid ja vo˜rgud Eestis: IGNS71 ja EGS ajavahemikus 1975–2007 (Gravity systems and networks in Estonia: IGSN71 and EGS from 1975 to 2007). Geodeet 35:11–20 (in Estonian) Oja T, Sildvee H (2004) Gravity Base Network of Estonia. In: Proceedings of the Joint Baltic and Nordic Geoid Meeting in Tallinn in 2000, Estonian Agricultural University, pp 27–31 Sildvee H (1998) Gravity measurements of Estonia. Report of the Finnish Geodetic Institute, 98:3, Masala Tamura Y (1987) A harmonic development of the tide generating potential – Bulletin d’Informations Mare´es Terrestres, 99, 6813–6855 Timmen L, Wenzel HG (1994) Worldwide synthetic gravity tide parameters available on internet. BGI Bull Inf 75:32–40 Torge W (1989) Gravimetry. Walter de Gryter, p 465 Wenzel HG (1994) Earth tide analysis package ETERNA 3.0. Bulletin d’Informations des Mare´es Terrestres, No 118, pp 8719–8721
Evaluation of EGM2008 Within Geopotential Space from GPS, Tide Gauges and Altimetry
39
N. Dayoub, P. Moore, N.T. Penna, and S.J. Edwards
Abstract
The new global Earth gravitational model EGM2008 has been evaluated within geopotential space by comparison with its predecessor EGM96 and the GRACE combination model EIGEN-GL04C. The methodology comprises establishing geodetic coordinates of mean sea level (MSL) from GPS observations, tide gauge (TG) time series and levelling. The gravity potential at MSL was estimated at each TG location by utilising the ellipsoidal harmonic coefficients of the adopted gravity field models to their maximum degree and order. This study uses data from 23 TGs around the Baltic Sea, nine in the UK and one in France. Comparison involves testing the agreement between geopotential values for each country as gravity potentials at MSL are supposed to be consistent for regions where mean dynamic topography (MDT) does not differ significantly. Results show significant improvement with the EGM2008 model compared against its counterparts. The study shows the effect of omission errors on the solution by limiting the EGM2008 model to maximum degree and order 360 in the regional study. In addition to the regional study, EGM2008 was also evaluated globally using MSL derived from altimetric data. The global study shows that W0, the potential value on the geoid, is not affected by high degree terms of the EGM2008.
39.1
Introduction
A new ultra high degree combination Earth gravitational model EGM2008 (Pavlis et al. 2008) has been released by the EGM development team of the National Geospatial-Intelligence Agency (NGA). This model consists of a complete set of spherical
N. Dayoub (*) P. Moore N.T. Penna S.J. Edwards School of Civil Engineering and Geosciences, Newcastle University, Newcastle NE1 7RU, UK e-mail:
[email protected]
harmonics up to degree/order 2159/2159 (4670000 coefficients) and is further extended by additional harmonics to degree 2190 and order 2159. The model yields gravity effects of features larger than 9 km. EGM2008 is based on gravitational information from GRACE and incorporates 50 50 global gravity anomalies. Coefficients of this model are freely available (http://earth-info.nga.mil/GandG/). In this study, EGM2008 has been tested within geopotential space against the European Improved Gravity model of the Earth by New techniques (EIGEN-GL04C) (F€orste et al. 2006) and the EGM96 model (Lemoine et al. 1998) which consist
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of coefficients only up to degree and order 360. Evaluation involves establishing the geodetic coordinates of MSL at TGs and estimating the gravity potential at these points. The gravity potentials at MSL should be consistent in countries or regions where the effects of MDT are small. This is particularly applicable when oceanographic and meteorological effects such as ocean currents, atmospheric pressure and winds are similar such as in a semi-enclosed sea. As proof that MDT is small over the study region, 120 monthly means of the Simple Ocean Data Assimilation Analysis (SODA) MDT model (Carton et al. 2000) were analysed. This showed that MDT does not vary by more than 10 cm within any of the individual countries involved in this study. Thus, the improvement of the gravity model can be detected from the increased agreement between the results for each country. It is noted that a global gravity model (GGM) to degree and order 360 would only cover gravity features larger than 55 km. This may not provide a sufficiently accurate solution for local or regional work as a result of the omission of coefficients of degree greater than 360. In addition to the regional approach, EGM2008 has also been evaluated globally using MSL derived from satellite altimetry. In particular, the effect of omission errors and the use of a global MDT model are investigated.
39.2
Data: Regional Analysis
The TGs used for the regional analysis are shown in Fig. 39.1. 23 of the TGs are situated around the Baltic Sea, namely four in Germany (GER), eight in Finland (FIN), two in Lithuania (LIT), two in Poland (POL) and seven in Sweden (SWE). These TGs are connected to the geocentre by means of episodic GPS observations which were collected in 1993.4 and 1997.4 as a part of the Baltic sea level project (Poutanen and Kakkuri 1999), with geocentric coordinates given in Ardalan et al. (2002). All GPS heights were used from the later campaign except for Swinoujscie in Poland the 1993.4 value was used as the site was not occupied in 1997.4. Unlike the other Baltic Sea TGs, three of the German TGs are located on the North Sea with the other in the Danish channel. This study also uses data from nine UK TGs and one in France (Brest), all of which are co-located with continuously operating GPS (CGPS) receivers. Precise levelling was used to connect the GPS and TG datums.
To calculate the geodetic coordinates of the UK and Brest CGPS stations, 1 year of GPS data from 2006 were analysed using GIPSY-OASIS 5.0 software in precise point positioning mode (Zumberge et al. 1997). The data were processed in 24 h sessions: JPL reprocessed non-fiducial orbits, satellite clocks and Earth rotation parameters were held fixed. Models were applied for absolute transmitter and receiver antenna phase centre variations (Schmid et al. 2007); Earth body tides according to the IERS 2003 conventions (McCarthy and Petit 2004); and ocean tide loading using the FES2004 model (Lyard et al. 2006), computed via www.oso.chalmers.se/~loading. Wet tropospheric zenith delays and north-south and east-west horizontal gradients were estimated every 5 min while the VMFI mapping function was used (Boehm et al. 2006) together with an elevation angle cut-off 10 degrees. Ambiguities were fixed via ambizap (Blewitt 2008). The non fiducial daily coordinates were transformed to ITRF2005 with the final height estimates taken as the mean values for the whole of 2006. Mean monthly data files for the UK/France TGs were selected in Revised Local Reference (RLR) format from the Permanent Service for Mean Sea Level (PSMSL). The chosen stations have a minimum of 30 full years of sea level data, which is needed to determine secular MSL changes precise to 0.5 mm/year (Woodworth et al. 1999), while 50 years of data increase the precision to 0.3 mm/year (Douglas 1991). For each TG time series the mean and trend were estimated along with annual, semiannual and 18.6 year tidal signals. The mean was moved to year 2006.5 to correspond to the GPS epoch using (39.1) TGRt ¼ TGR0 þ ðt t0 Þ tnd
(39.1)
where t corresponds to 2006.5, TGRt is the value of the TG record at the time t, t0 refers to the reference time for TGR0 , and tnd the annual MSL trend at the TG. It is assumed here that the principal change in the TG time series is caused by secular changes in the sea level and land movement due to global isostatic adjustment (GIA). Both can be considered secular on a short time scale as in this study although one can not exclude the possibility of non-linear changes due localised movement and an acceleration in sea level change.
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Fig. 39.1 TG locations used in the regional analysis. UK/ France: Lerwick (LWTG), Aberdeen (ABER), North Shields (NSTG), Lowestoft (LOWE), Sheerness (SHEE), Portsmouth (PMTG), Newlyn (NEWL), Liverpool (LIVE), Brest (BRST). For Baltic sites see Poutanen and Kakkuri (1999)
39.3
Methodology: Regional Analysis
The methodology involves constructing the geodetic coordinates of MSL at the TG. The geodetic latitude and longitude were obtained directly from the GPS analysis, while the geoid heights were obtained from (39.2) which is illustrated in Fig. 39.2 N ¼ h þ TGR ðDH þ HTG Þ:
(39.2)
In (39.2) MDT has been neglected; thus MSL approximate the geoid. The potential value on the geoid is given by W0 ð’; l; NÞ ¼ Vð’; l; NÞ þ F ð’; NÞ
(39.3)
where V is the gravitational potential, F the centrifugal potential and j, l and N the geodetic latitude, longitude and height of the point corresponding to MSL. The World Geodetic Datum 2000 (WGD2000) was used as the reference ellipsoid with a ¼ 6378136.701 m and b ¼ 6356751.661 m in the mean tide system (Ardalan and Grafarend 2000). The GGMs were also transformed into the mean tide system as far
as the permanent tide was concerned to maintain consistency with the reference ellipsoid. As this work is part of a more extensive study using the normal gravity field we made use of the representation of the Earth’s gravity field in terms of ellipsoidal coefficients (Ardalan et al. 2002). Although the study could have use the standard spherical harmonics we note here that ellipsoidal harmonics do not suffer from the ultra high degree problem (Jekeli et al. 2007). Accordingly, the ellipsoidal harmonics used were derived from the GGM spherical harmonics and referenced to the WGD2000 datum. In practice, the choice of datum is arbitrary as long as the TG ellipsoidal height and ellipsoidal harmonics of the GGM are treated in a consistent manner. The coefficient rates provided with each GGM were used to move the model to the year of interest.
39.4
Results: Regional Analysis
With reference to the Baltic data, the results presented in Figs. 39.3, 39.4 and 39.5 show significant improvement on using the EGM2008 model. The scatter of the geopotential values is relatively large with EGM96
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Fig. 39.2 Geoid height from GPS and TG
GPS
TG
h: Ellipsoidal height; N: Geoid height; H: Height difference between GPS and TG; HTG: Height of TG above the TG zero level; TGR: Tide gauge reading.
H
MSL
h
HTG N
TGR
TG Zero
Sea floor
Ellipsoid
Fig. 39.3 W0: Baltic TGs from EGM96: Ustka (Poland) circled
Fig. 39.4 W0: Baltic TGs from EIGEN-GL04C
and EIGEN-GL04C, while the results become more consistent with EGM2008. In particular, there is enhanced agreement when comparing values for each country separately. The results are particularly improved at Ustka in Poland which is highlighted with a circle in Fig. 39.3. This was considered as an outlier with the EGM96 model and with a clear offset
of 4 m2s2 from the other Polish station on using the EIGEN-GL04C model. In Fig. 39.5, the German stations possess higher geopotential values than the other Baltic countries. This is due to the German stations being exposed to North Sea MDT that is distinct in its nature from the sites in the enclosed Baltic Sea.
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Fig. 39.5 W0: Baltic TGs from EGM2008 to maximum degree 2160
Fig. 39.6 W0: UK and France TGs from EGM96
Fig. 39.7 W0: UK and France TGs from EGENGL04C
A similar improvement accrued when processing the UK and Brest (BRST) data which were referenced to year 2006.5. Here, the UK is divided into three regions according to the underlying height datum, UK mainland (England, Scotland and Wales), Lerwick (LWTG) and Stornoway (SWTG). As shown in
Figs. 39.6, 39.7 and 39.8 the EGM2008 model has given consistency to all geopotential values over the entire area especially for the Stornoway value which was considered as an outlier with the other gravity models. Furthermore, results from EGM2008 over the UK and Baltic areas seem to depart from the mean by
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Fig. 39.8 W0: UK and France TGs from EGM2008 to maximum degree 2160
Table 39.1 Mean gravity potential (with 95% confidence level error estimate) for Finland, Sweden, UK/France
Country UK/France Finland Sweden
Gravity Potential – 62636850 m2s2 EGM96 EIGEN-GL04C EGM2008 7.12 0.99 7.52 1.00 7.84 0.37 4.78 1.06 4.70 0.86 4.35 0.32 5.25 0.96 5.01 0.67 4.79 0.32
less than 1 m2s2 (10 cm in a metric sense). This is understandable in terms of the MDT effects in a relatively small region. Table 39.1 summarises the mean gravity potential results for Finland, Sweden and UK/ France which possess the largest number of stations. In this table, the standard errors were reduced by a factor of two or more when EGM2008 was used illustrating the improvement of EGM2008 against its counterparts.
39.5
Omission Errors: Regional Scale
To investigate the role of omission errors on the improvement of the results, the geopotential values were re-computed but now limiting EGM2008 to degree/order 360/360. Results, presented in Figs. 39.9 and 39.10, show that EGM2008 to degree and order 360/360 does not perform substantially better than EGM96 or EIGEN-GL04C. Although some improvement can still be seen, especially for the German and some UK stations, the offsets from the mean are large for the other stations. This confirms that the higher frequency part of the EGM2008 model is responsible for most of the improvement of the results, which also shows the significance of the omission errors on the regional scale solution.
39.6
EGM2008 and Omission Errors: Global Scale
To investigate the significance of omission errors and to evaluate EGM2008 globally, the geopotential value was computed from a global dataset using the aforementioned GGMs. The MSSCLS01 (Hernandez and Schaeffer 2001) (for brevity CLS01) was used as the global MSL surface. This model supplies MSL covering the latitude domain 82 /80 N/S. The CLS01 model was established from 7 years of TOPEX/ poseidon data (1993–1999), 5 years of ERS-1/2 altimetry between 1993 and 1999, GEOSAT 1987–1988 altimetry and altimetry from the geodetic phase of ERS-1 between 1994 and 1995. CLS01 is supplied as a continuous surface with the EGM96 geoid used to complete the model over land and a cosine tapering performed to smooth the connection between land and sea values. For this work, data over land and from the interpolation zone was excluded. CLS01 yields coordinates of MSL which is different from the geoid by MDT, as shown in Fig. 39.11. l; hÞ on MSL with Thus, to compute W0, a point Pð’; an ellipsoidal height (h) has to be moved to the corresponding point P0 ð’; l; NÞ on the geoid via the MDT value. MDT was obtained from the ECCO2 (Estimating the Circulation and Climate of the Ocean) oceanographic model which has a near global latitude domain 78 /78 N/S (Roemmich et al. 2004). CLS01 and ECCO-2 together provide geodetic coordinates of points on the geoid surface. It is noted here that ECCO-2 is reference frame neutral with MDT ¼ 0 equivalent to an equipotential surface of the Earth’s gravity field. Further details of
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Fig. 39.9 W0: Baltic TGs from EGM2008 to maximum degree 360
Fig. 39.10 W0: UK and France TGs from EGM2008 to maximum degree 360
Table 39.2 The effect on W0 of using different GGMs and maximum degree (n) (with 95% confidence level error estimate) based on CLS01 (70 /70 N/S) with/without correction for MDT from ECCO-2
Fig. 39.11 Geoid height (N) from ellipsoidal height (h) and MDT
oceanographic model reference frames are given in (Hughes and Bingham 2008). Data between 70 /70 N/S were employed for this study. We computed geopotential values on a 1 1 latitude/longitude grid by expanding EGM96, EIGEN-GL04C and EGM2008 to degree/order 360/360 and EGM2008 to
GGM
n
EGM96 EIGEN-GL04C EGM2008 EGM2008
360 360 360 2,160
W0 – 62636850 m2s2 CLS01 CLS01 and ECCO-2 4.30 0.07 4.34 0.03 4.27 0.07 4.30 0.03 4.25 0.06 4.29 0.02 4.25 0.06 4.29 0.02
degree/order 2160/2160. As before the GGMs were transformed into the mean tide system. The gravity potential was determined at each grid point of the CLS01 model, with the equi-area weighted average used to estimate W0. Table 39.2 shows W0 values with 95% confidence level error estimation before and after accounting for MDT. The results, summarised in Table 39.2, show that the global value of W0 is essentially invariant with
330 9 W0-62636850 (m2/sec2)
Fig. 39.12 Dependence of W0 on maximum degree n of the GGM (GGM: EGM2008, MSS: CLS01)
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8 7 6 5 4
the GGM. Furthermore, EGM2008 to degree/order 2160/2160 gave exactly the same value W0 as that from the field truncated at degree and order 360. Although removing MDT has only a small effect on W0, consideration of MDT has halved the standard errors for EGM96 and EIGEN-GL04C, and reduced the standard error by two thirds for EGM2008. Furthermore, EGM2008 has the lowest standard error which reflects an improvement in this model. The consistency between the GGMs and the agreement between the full and truncated EGM2008 fields shows that omission errors after a certain degree/ order do not influence W0 globally. Accordingly, a high resolution GGM is not necessary for estimating W0. To find the minimum degree required of the GGM after which the omission errors are not significant, W0 was estimated with EGM2008 with truncation at various degrees (n). Figure 39.12 shows that the geopotential values converge approximately at degree 80-100, while after n ¼ 120, there is practically no difference in W0. Thus, a GGM to degree 120 is sufficient to estimate W0 at the global scale. This enables the possibility to determine W0 from a satellite-only Earth gravity field model such as EIGENGL04S1 (F€orste et al. 2006). It is noted, however, that Sanchez (2008) showed that W0 is dependent on the latitude band over which W0 was estimated. Conclusions
The performance of the EGM2008 gravity field model was evaluated within geopotential space over five Baltic countries, the UK and France. It appears that, the use of EGM2008 has significantly increased the consistency of the gravity potentials
0
100
200 Degree (n)
300
400
at MSL for the countries involved (see Table 39.1). It was seen that omission errors are the main reason for the large offsets between the geopotential values at local and regional scales. Additional studies extending the regional network used here is necessary to provide further validation of EGM2008. Globally, all GGMs give essentially the same results within the standard errors (see Table 39.2). However, of more significance is the use of a MDT model. The results show that, at the global scale, the high frequency part of EGM2008 has a negligible effect on W0. Acknowledgements The authors would like to thank the following institutions for supplying data for this study: NASA JPL for GIPSY software and the provision of orbital products, NERC BIGF for GPS data at UK tide gauge sites and EUREF/IGS for Brest GPS data.
References Ardalan A, Grafarend E, Kakkuri J (2002) National height datum, the Gauss-Listing geoid level value W0 and its time variation W0 (Baltic Sea Level project: epochs 1990.8, 1993.8, 1997.4). J Geodesy 76(1):1–28 Ardalan AA, Grafarend EW (2000) Reference ellipsoidal gravity potential field and gravity intensity field of degree/order 360/360 (manual of using ellipsoidal harmonic coefficients ellipfree.dat and ellipmean.dat). http://www.uni-stuttgart.de/ gi/research/index.html#projects Blewitt G (2008) Fixed point theorems of GPS carrier phase ambiguity resolution and their application to massive network processing: Ambizap. J Geophys Res B Solid Earth 113(12)
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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 B Solid Earth 111(2) Carton JA, Chepurin G, Cao X, Giese B (2000) A simple ocean data assimilation analysis of the global upper ocean 1950-95. Part I: methodology. J Phys Oceanogr 30(2):294–309 Douglas BC (1991) Global sea level rise. J Geophys Res 96 (C4):6981–6992 F€orste C, Flechtner F, Schmidt R, K€ onig R, Meyer U, Stubenvoll R, Rothacher M, Barthelmes F, Neumayer H, Biancale R (2006) A mean global gravity field model from the combination of satellite mission and altimetry/gravimetry surface data: EIGEN-GL04C. Geophys Res Abstr 8:03462 Hernandez F, Schaeffer P (2001) The CLS01 mean sea surface: a validation with the GSFC00 surface. press, CLS Ramonville StAgne, France Hughes CW, Bingham RJ (2008) An oceanographer’s guide to GOCE and the geoid. Ocean Sci 4(1):15–29 Jekeli C, Lee JK, Kwon JH (2007) On the computation and approximation of ultra-high-degree spherical harmonic series. J Geodesy 81(9):603–615 Lemoine FG, Kenyon SC, Factor JK, Trimmer RG, Pavlis NK, Chinn DS, Cox CM, Klosko SM, Luthcke SB, Torrence MH (1998) The Development of the Joint NASA GSFC and the National Imagery and Mapping Agency(NIMA) Geopotential Model EGM 96. NASA, (19980218814)
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Lyard F, Lefevre F, Letellier T, Francis O (2006) Modelling the global ocean tides: modern insights from FES2004. Ocean Dyn 56(5–6):394–415 McCarthy DD, Petit G (2004) IERS Conventions 2003. IERS Technical Note, 32, p 127 Pavlis N, Kenyon S, Factor J, Holmes S (2008) Earth gravitational model 2008. In SEG Technical Program Expanded Abstracts, vol 27:761–763 Poutanen M, Kakkuri J (1999) Final results of the Baltic Sea level 1997 GPS Campaign. Rep Finnish Geodetic Institute. 99(2) Roemmich D, Riser S, Davis R, Desaubies Y (2004) Autonomous profiling floats: workhorse for broadscale ocean observations. J Mar Technol Soc 38:31–39 Sanchez L (2008) Approach for the establishment of a global vertical reference level. In: Proceedings of the VI HotineMarussi Symposium, Springer, May 2008 Schmid R, Steigenberger P, Gendt G, Ge M, Rothacher M (2007) Generation of a consistent absolute phase-center correction model for GPS receiver and satellite antennas. J Geodesy 81(12):781–798 Woodworth PL, Tsimplis MN, Flather RA, Shennan I (1999) A review of the trends observed in British Isles mean sea level data measured by tide gauges. Geophys J Int 136(3):651–670 Zumberge JF, Heflin MB, Jefferson DC, Watkins MM, Webb FH (1997) Precise point positioning for the efficient and robust analysis of GPS data from large networks. J Geophys Res 102:5005–5017
.
Fixed Gravimetric BVP for the Vertical Datum Problem
40
R. Cˇunderlı´k, Z. Fasˇkova´, and K. Mikula
Abstract
This paper discusses advantages of the fixed gravimetric boundary value problem (FGBVP) for precise gravity field modeling that is necessary for a unification of local vertical datums (LVDs). Our objective is to show how inconsistencies of input gravity data due to shifts and tilts of LVDs can influence precise solutions. Such systematic errors can backward affect estimations of the shifts and tilts of LVDs. This drawback completely vanishes in case of FGBVP. Terrestrial gravimetric measurements accompanied by the precise 3D positioning by GNSS yield globally consistent surface gravity disturbances that are fully independent from any LVD (assuming the same gravity datums). Since terrestrial gravity data from the past are related to LVDs, we try to reconstruct their ellipsoidal heights using available geoid/quasi-geoid models as well as shifts and tilts of LVDs modeled from GPS/Leveling data. In this way we simulate consistent surface gravity disturbances that represent oblique derivative boundary condition for FGBVP. In the numerical experiments we deal with (i) the global gravity field modeling using the boundary element method (BEM), and with (ii) the continental modeling using the finite volume method (FVM). In both cases we compare the numerical solutions obtained with and without taking into account corrections from the shifts and tilts of LVDs in the input data. It shows how an absolute precision of vertical positions of terrestrial gravity data influence precise numerical solutions.
40.1
Introduction
A realization of the global vertical reference system and unification of LVDs is performed on the basis of the global geopotential models, cf. e.g., Bursˇa et al. R. Cˇunderlı´k (*) Z. Fasˇkova´ K. Mikula Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, Radlinske´ho 11, 813 68, Bratislava, Slovakia e-mail:
[email protected]
(2004, 2007). While their low frequency part is very precisely determined from satellite missions, modeling of the high frequency part requires reliable terrestrial and marine/altimetry-derived gravity data. In case of the terrestrial gravimetric measurements, a precision of their positions, especially their vertical components, is of the same importance as the precision of gravity itself. Therefore inconsistencies due to shifts and tilts of LVDs can mislead precise solutions. In contrary, the precise 3D positioning by GNSS has brought a striking advantage that all terrestrial
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gravity data can have consistent vertical information. Such benefit is promising for precise gravity field modeling in future, especially for purposes of the vertical datum problem, and motivates to solve the fixed gravimetric boundary value problem. At present, modern global and local geoid/quasigeoid models are usually fixed to the low frequency part obtained from the satellite only geopotential models that are independent from LVDs. However, one has to take care that inconsistencies in vertical information of the terrestrial gravity data fully constrain high frequencies or residual parts of the input boundary condition (BC). This can lead to systematic tendencies in the precise gravity field models that are later used for estimating shifts and tilts of LVDs. This drawback completely vanishes in case of FGBVP. Surface gravity disturbances based on the precise 3D positioning by GNSS provide consistent BC that are fully independent from leveling, i.e., from any LVD. Obviously, we realize that coverage of such data on lands is, so far, not realistic, since a majority of terrestrial gravimetric measurements collected for decades have been accompanied by leveling, or have had their positions estimated from topographic maps. Therefore we try to reconstruct ellipsoidal heights of available discrete terrestrial gravimetric measurements (mainly in North America and Australia) using the national geoid/quasigeoid models, the EGM2008 global geopotential model (Pavlis et al. 2008) as well as corrections from the shifts and tilts of LVDs estimated from GPS/Leveling data. Remark: in case of historic gravity data, errors of horizontal positions need to be considered, that are usually of two sorts: (1) a gravity point with leveling but poorly determined horizontal position, (2) a gravity point without leveling, whose height is scaled off of a contour map. While the first case has almost negligible effect on BVP, the second source can result in height errors that are considerably larger than error from LVD. Therefore all available gravimetric measurements require verification before using as input data. Our objective in this paper is to demonstrate how inconsistencies of vertical positions of input gravity data due to the shifts and tilts of LVD can influence precise numerical solutions that are based on integrating over the whole Earth’s surface. We compare the numerical solutions with and without considering the corrections from the shifts and tilts of LVDs. We perform two types of experiments (1) a global
numerical solution to FGBVP using the boundary element method, and (2) a continental modeling in Australia using the finite volume method, where the solution is fixed to the satellite only geopotential model on artificial upper and side boundaries.
40.2
Global Gravity Field Modeling by BEM
The linearized FGBVP represents an exterior oblique derivative problem for the Laplace equation, cf. Koch and Pope (1972), Bjernhammer and Svensson (1983), Heck (1989) or Grafarend (1989). In order to solve such a problem we use the direct BEM formulation and collocation with linear basis functions, for more ˇ underlı´k et al. (2008); Cˇunderlı´k and details see C Mikula (2010). It allows us to get a global numerical solution with a resolution up to 0.1 deg that is based on the refined integration over the whole Earth’s surface. At first we approximate the Earth’s surface by a triangulated surface. Vertices of this triangulation represent the collocation points. Their horizontal positions are generated by the algorithm developed in Cˇunderlı´k et al. (2002). The mesh size of the triangular elements is 0.1 deg in latitude, i.e., 4,860,002 collocation points regularly distributed over the Earth’s surface. Vertical positions of the collocation points are interpolated from the DNSC08 mean sea surface (Andersen et al. 2008) at oceans, and SRTM30PLUS-V5.0 global topography model (Becker et al. 2009) added to EGM-96 (Lemoine et al. 1998) on lands. The input surface gravity disturbances as the oblique derivative BC are generated from the DNSC08 gravity field model (Andersen et al. 2008). In order to reduce large memory requirements we eliminate far zones interactions using the ITGGRACE03S satellite only geopotential model (Mayer-G€urr 2007) up to degree 180 and the iterative procedure described in Cˇunderlı´k and Mikula (2010). In the case, we have original terrestrial gravity data at disposal, we replace positions and the surface gravity disturbances of the corresponding collocation points by original values (Fig. 40.1). Table 40.1 shows the type and source of available gravity data. It is distinguished between two types of gravity data: 1. Discrete gravimetric measurements (North America, South Africa, Spain, central Europe and central part
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Fig. 40.1 Collocation points with input data from discrete terrestrial gravimetric measurements and grided gravity anomalies Table 40.1 Available datasets of discrete terrestrial gravimetric measurements and grided gravity anomalies Area USA Canada Globe Australia ArticGP
Source NGS NRC BGI AUSLIG ArticGP
Data Observed gravity Observed gravity Observed gravity Simple Bouguer Anomaly (grid) Free-air Gravity Anomaly (grid)
of Andes), for which we reconstruct the ellipsoidal heights by adding orthometric/normal heights related to LVDs to the local geoid/quasigeoid models. Then we get the surface gravity disturbances dg from the relation dg ¼ gOBS gWGS84 ðN LocalGeoid þ H Orth: Þ;
(40.1)
where gOBS is the observed gravity, gWGS84 is the normal gravity value related to the WGS84 reference ellipsoid (NIMA 2001), N is the geoidal height and H is the orthometric height. In case of the normal Molodensky heights, the quasigeoidal heights are considered. 2. Free-air gravity anomalies Dg (Australia, Arctic area), for which an approximate transformation is applied dg ¼ Dg þ 0:30855 NLocalGeoid ;
½mGal
Due to an insufficient distribution of the original gravity data in a majority of lands, we focus on areas of North America and Australia. In North America we use discrete data (Table 40.1) and the national geoid models, namely, GEOID03 in USA (Roman et al. 2004) and CGG2000 in Canada (Ve´ronneau 2001). In Australia we generate the input data from the simple Bouguer gravity anomalies (Table 40.1). First they are transformed into the free-air gravity anomalies using SRTM30PLUS-V5.0 and then into surface gravity disturbances applying (40.2) and the AUSGeoid98 national geoid model (Featherstone et al. 2001). The surface gravity disturbances computed by (40.1) and (40.2) still include systematic errors due to the shifts and tilts of LVDs. Hence, the numerical solution obtained by BEM is affected by these errors. Comparing the BEM numerical solution with EGM2008 up to degree 2,160 one can see an evident depression more than 1 m in Noth America and the slight uplift in Australia (Fig. 40.4a). Therefore we correct the ellipsoidal heights in order to simulate 3D positions that we would get using the GNSS positioning. We evaluate corrections at GPS/ Leveling points using EGM2008 and the value W0 ¼ 63,626,856.0 m2.s2 adopted by the Bursˇa et al. (2004).
(40.2) Corr ¼ HEGM H Leveling ;
using the local geoid/quasigeoid models and assuming the normal gravity gradient of 0.30855 mGal/m. H EGM ¼ ðW0 U GPS T EGM2008 Þ=g þ ðz NÞ;
(40.3) (40.4)
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Fig. 40.2 (a) Shift and tilt of NAVD88 evaluated at GPS/ Leveling points using EGM2008 for W0 ¼ 63626856.0 m2.s2, (b) geopotential on the DNSC08 mean sea surface around North America evaluated from EGM2008 (the constant 63626800.0 m2.s2 is removed)
where HLeveling is the orthometric/normal height at the GPS/Leveling point transformed from the mean-tide into the tide-free system, UGPS is the normal potential at this point, TEGM2008 is the disturbing potential evaluated from EGM2008 and g is the mean normal gravity along the normal plumb line. The term (z -N) represents a difference between geoid and quasigeoid. This term is not considered in case that HLeveling is the normal Molodensky height. The corrections evaluated at GPS/Leveling points in North America represent the shift and tilt of the North American Vertical Datum 1988 (NAVD88) (Fig. 40.2a). For a comparison, Fig. 40.2b shows the geopotential on the DNSC08 mean sea surface around North America evaluated from EGM2008. Since we
do not have at disposal the GPS/Leveling data in Australia, we simulate the corrections (Fig. 40.3a) according to the geopotential on the DNSC08 mean sea surface around Australia evaluated from EGM2008 (Fig. 40.3b) and considering the tilt and shift of the Australian Vertical Datum (AHD) as a result of the GPS/Leveling test (Fig. 40.3c) discussed in Claessens et al. (2009). The obtained corrections allow us to make the orthometric/normal heights related to the same equipotential surface W0. In order to get consistent ellipsoidal heights, they have to be added to geoidal/ quasigeoidal heights of a globally consistent gravity field model like EGM2008. Then we are able to simulate the consistent surface gravity disturbances from
40
Fixed Gravimetric BVP for the Vertical Datum Problem
Fig. 40.3 (a) Simulated corrections of AHD with respect to EGM2008 and W0 ¼ 63626856.0 m2.s2, (b) geopotential on the DNSC08 mean sea surface around Australia evaluated from EGM2008 (the constant 63626800.0 m2.s2 is removed), (c) GPS/Leveling test in Australia at 254 GPS–AHD points using EGM2008 (Claessens et al. 2009)
337
338 Fig. 40.4 Comparison between the numerical solution by BEM and EGM2008 (a) with vertical positions related to LVDs, (b) with 3D positions corrected from shifts and tilts of LVDs, (c) comparison between both solutions
R. Cˇunderlı´k et al.
40
Fixed Gravimetric BVP for the Vertical Datum Problem
the discrete gravimetric measurements accompanied by leveling. The global numerical solution recomputed by BEM with corrected input BC in North America and Australia is again compared with EGM2008 (Fig. 40.4b). It is evident that the depression in North America (Fig. 40.4a) significantly decreases, while the uplift in Australia slightly increases. It is probably due to the fact that our simulated corrections in Australia should be applied to the SRTM-30PLUS-V5.0 topography model in order to reconstruct original heights used for developing datasets of the simple Bouguer anomalies. Figure 40.4c depicts a comparison between both numerical solutions showing how the corrections of ellipsoidal heights contribute to a change of the global solution. It also indicates how accuracy of the 3D positions of input gravity data is important for precise numerical solutions.
40.3
Local Gravity Field Modeling by FVM
In this numerical experiment we present how inconsistencies of input gravity data due to the shifts and tilts of LVDs can influence local quasigeoid modeling that is fixed to the satellite only geopotential model. Here we use FVM to solve the geodetic BVP with mixed BC, i.e., with the oblique derivative BC (surface gravity disturbances) on the Earth’s surface (bottom boundary) and with the Dirichlet BC on artificial upper and side boundaries (Fig. 40.5), for more details see Fasˇkova´ et al. (2007) and Fasˇkova´ (2008).
Fig. 40.5 An artificial domain of the geodetic BVP with mixed BC used for local quasigeoid modeling by FVM
339
The disturbing potential as the Dirichlet BC on the artificial boundaries is evaluated from ITGGRACE03S satellite only geopotential model up to degree 180. The upper boundary is located at altitude of 500 km above the Earth’s surface. The surface gravity disturbances in our chosen area, in Australia, are generated in the same way as described in the previous chapter. Again, we consider two cases, with and without considering simulated corrections from the shift and tilt of AHD (Fig. 40.3a). The comparison between both numerical solutions (Fig. 40.6c) shows an influence of the simulated corrections to local solutions. In addition, both of them are compared with EGM2008 (Fig. 40.6a, b) to illustrate our results. Figure 40.6c shows that corrections of ellipsoidal heights from 0.1 m up to 0.8 m (Fig. 40.3a) result in an increase of the quasigeoid undulations up to 8 cm. Such change, significant for precise geoid/quasi-geoid modeling, is partly reduced by (1) the fixing of the numerical solutions to the satellite only geopotential model on the upper and side artificial boundaries, and due to (2) a size of the land area, since the altimetryderived data at oceans are the same in both cases.
40.4
Discussion and Conclusions
The presented numerical experiments demonstrate the need of consistent terrestrial gravity data for precise gravity field modeling. The precise 3D positioning by GNSS has brought an opportunity to make new gravimetric data independent from local LVDs. In our opinion, precise gravity field models based on such data can estimate the shifts and tilts of LVDs more precisely detecting also latent errors that GPS/ Leveling tests (using the same leveling as for input data) can omit. This can be challenging for a realization of the global vertical reference system. The global numerical solution obtained by BEM shows that the systematic shift and tilt of NAVD88 up to 2.5 m leads to a depression of more than 1 m (Fig. 40.4a) in North America. Although the modern geoid/quasigeoid models are fixed to satellite only models from CHAMP and GRACE (GOCE in future), input data without corrected vertical positions can misled “cm-accurate” geoid/quasigeoid modeling. The local numerical solution obtained by FVM in Australia indicates that 0.8 m error in ellipsoidal heights leads to discrepancies up to 8 cm, although
340 Fig. 40.6 Comparison between the local numerical solution by FVM in Australia and EGM2008 (a) with vertical positions related to LVDs, (b) with 3D positions corrected from shifts and tilts of AHD, (c) comparison between both solutions
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40
Fixed Gravimetric BVP for the Vertical Datum Problem
the solution is fixed to ITG-GRACE03 satellite only model. All the presented results represent numerical solutions to FGBVP, where input data are more or less simulated. It is due to the fact that, so far, we do not have at disposal any dataset based only on terrestrial gravimetry with 3D positioning by GNSS. Anyhow, we hope that near future will overcome this drawback and sophisticated solutions of FGBVP will be able to contribute to a precise realization of the world height system. Acknowledgements Authors gratefully thank to the financial support given by grants: VEGA 1/0269/09, APVV-LPP-0216-06 and APVV-0351-07.
References Andersen OB, Knudsen P, Berry P, Kenyon S (2008) The DNSC08 ocean wide altimetry derived gravity anomaly field. Presented at the 2008 General Assembly of EGU, Vienna, Austria, 2008 Becker JJ, Sandwell DT, Smith WHF, Braud J, Binder B, Depner J, Fabre D, Factor J, Ingalls S, Kim S-H, Ladner R, Marks K, Nelson S, Pharaoh A, Sharman G, Trimmer R, vonRosenburg J, Wallace G, Weatherall P (2009) Global bathymetry and elevation data at 30 arc seconds resolution: SRTM30_PLUS. Mar Geodesy 32(4):355–371 Bjerhammer A, Svensson L (1983) On the geodetic boundaryvalue problem for a fixed boundary surface – satellite approach. Bull Ge´od 57:382–393 Bursˇa M, Kenyon S, Kouba J, Sˇ´ıma Z, Vatrt V, Vojtı´sˇkova´ M (2004) A global vertical reference frame based on four regional vertical datums. Stud Geophys Geod 48(3):493–502 Bursˇa M, Kenyon S, Kouba J, Sˇ´ıma Z, Vatrt V, Vı´tek V, Vojtı´sˇkova´ M (2007) The geopotential value W0 for specifying the relativistic atomic time scale and a global vertical reference system. J Geod 81(2):103–110 Claessens SJ, Featherstone WE, Anjasmara IM (2009) Is Australian data really validating EGM2008, or is EGM-2008 just in/validating Australian data? In: Proceedings of GGEO2008, Chania, Greece Cˇunderlı´k R, Mikula K (2010) Direct BEM for high-resolution gravity field modelling. Stud Geophys Geod 54(2):219–238
341 Cˇunderlı´k R, Mikula K, Mojzesˇ M (2002) 3D BEM application to Neumann geodetic BVP using the collocation with linear basis functions. In: Proceedings of ALGORITMY 2002, Conference on Scientific Computing, Podbanske´, pp 268–275 Cˇunderlı´k R, Mikula K, Mojzesˇ M (2008) Numerical solution of the linearized fixed gravimetric boundary-value problem. J Geod 82:15–29 Fasˇkova´ Z (2008) Numerical methods for solving geodetic boundary value problems. Ph.D. Thesis, Svf STU Bratislava, Slovakia Fasˇkova´ Z, Cˇunderlı´k R, Jana´k J, Mikula K, Sˇprla´k M (2007) Gravimetric quasigeoid in Slovakia by the finite element method. Kybernetika 43(6):789–796 Featherstone WE, Kirby JF, Kearsley AHW, Gilliland JR, Johnston GM, Steed J, Forsberg R, Sideris MG (2001) The AUSGeoid98 geoid model of Australia: data treatment, computations and comparisons with GPS-leveling data. J Geod 75(5–6):313–330 Grafarend EW (1989) The geoid and the gravimetric boundaryvalue problem. Rep 18 Dept Geod. The Royal Institute of Technology, Stockholm Heck B (1989) On the non-linear geodetic boundary value problem for a fixed boundary surface. Bull Ge´od 63:57–67 Koch KR, Pope AJ (1972) Uniqueness and existence for the geodetic boundary value problem using the known surface of the earth. Bull Ge´od 46:467–476 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) EGM96 – the development of the NASA GSFC and NIMA Joint Geopotential Model. NASA Technical Report TP-1998-206861 Mayer-G€urr T (2007) ITG-Grace03s: The latest GRACE gravity field solution computed in Bonn. Presentation at GSTM + SPP, 15–17 Oct 2007, Potsdam NIMA (2001) Department of Defense World Geodetic System 1984, its definition and relationships with local geodetic systems. Third Edition, National Geospatial-Intelligence Agency. Technical Report TR8350.2 Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2008) An earth gravitational model to degree 2160: EGM2008. Presented at the 2008 General Assembly of EGU, Vienna, Austria, 2008 Roman DR, Wang YM, Henning W, Hamilton J (2004) Assessment of the New National Geoid Height Model, GEOID03. In: Proceedings of the American Congress on Surveying and Mapping 2004 meeting Ve´ronneau M (2001) The Canadian gravimetric geoid model of 2000 (CGG2000). Geodetic Survey Division, Natural Resources Canada, Ottawa, Canada
.
Realization of the World Height System in New Zealand: Preliminary Study
41
R. Tenzer, V. Vatrt, and M. Amos
Abstract
We utilize the geopotential value approach to determine the average offsets of 12 major local vertical datums (LVDs) in New Zealand (NZ) relative to the world height system (WHS). The LVD offsets are estimated using the EGM2008 global geopotential model coefficients complete to degree 2159 of spherical harmonics and the GPS-levelling data. WHS is defined by the adopted geoidal geopotential value W0 ¼ 62636856 m2s2. Our test results reveal that the average offsets of 12 major LVDs situated at the South and North Islands of NZ range from 0.01 m (Wellington 1953 LVD) to 0.37 m (One Tree Point 1964 LVD). The geopotential value of the tide-gauge station used as the origin for the LVD Wellington 1953 is thus almost the same as the geoidal geopotential value W0. EGM2008 and GPSlevelling data are further used to compute the differences between the NZGeoid05 regional quasigeoid model and the EGM2008 global quasigeoid model. The same analysis is done for NZGeoid2009 which is the official national quasigeoid model for NZ. The systematic bias of about 0.56 m is found between NZGeoid05 and EGM2008. A similar systematic bias of about 0.51 m is confirmed between NZGeoid2009 and EGM2008.
41.1
R. Tenzer (*) School of Surveying, Faculty of Sciences, University of Otago, 310 Castle Street, Dunedin, New Zealand e-mail:
[email protected] V. Vatrt Military Geographic and Hydrometeorogic Office, Cs. odboje 676, 518 11 Dobrusˇka, Czech Republic M. Amos Land Information New Zealand, PO Box 5501, Wellington 6145, New Zealand
Introduction
The vertical reference system in the North and South Islands of New Zealand (NZ) was realized by 12 major local vertical datums (LVDs). Since gravity observations were not made along the precise levelling lines, these LVDs were defined in the approximate normal-orthometric height system using precise levelling from 11 different tide-gauge stations. The Dunedin-Bluff 1960 LVD was defined by fixing the heights of two benchmarks with heights in terms of the Dunedin 1958 and Bluff 1955 LVDs instead of using the tide-gauge station as the origin. The normalorthometric height is defined based on the normalgeopotential number, which is the result of the levelled
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_41, # Springer-Verlag Berlin Heidelberg 2012
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height differences and the computed normal gravity values along the levelling lines. Moreover, the normalorthometric heights in NZ are only approximate due to the fact that the cumulative normal-orthometric corrections to the levelled height differences were computed approximately using a truncated form of the GRS67 normal-orthometric correction formula (cf. Rapp 1961). Amos and Featherstone (2009) applied the iterative gravimetric approach to unify the NZ LVDs using a regional gravimetric quasigeoid model and GPSlevelling data on each LVD. The principle of this method is based on an iterative quasigeoid model where the LVD offsets computed from an earlier model are used to apply additional reductions to the gravity observations within each LVD. They showed that the solution converges after only three iterations. The estimated offsets of 13 LVDs at the North, South and Stewart Islands range between 24 and 58 cm with the estimated standard deviation of about 8 cm. The New Zealand Quasigeoid 2005 (NZGeoid05) was the first official national quasigeoid model in NZ. NZGeoid05 was computed jointly by the Land Information of New Zealand (LINZ) and the Western Australian Centre for Geodesy – Curtin University of Technology (see Amos 2007). NZGeoid05 was determined using a combination of different heterogeneous data sets: the ground, seaborne and satellite-based gravity data and digital terrain models. The New Zealand Quasigeoid 2009 (NZGeoid2009) is the current official national quasigeoid model for NZ (Claessens et al. 2009). NZGeoid2009 is provided on a 1 1 arc-min geographical grid over NZ and its continental shelf. The main difference between NZGeoid05 and NZGeoid2009 is the different global geopotential models (GGMs); NZGeoid05 was compiled using EGM96, while EGM2008 was used in the compilation of NZGeoid2009. The NZGeoid2009 is shown in Fig. 41.1. The offsets of 12 major LVDs situated at the North and South Islands of NZ from the NZGeoid05 and NZGeoid2009 quasigeoid models are summarized in Table 41.1. The geopotential value approach (Bursˇa et al. 1999) is utilized in this study to compute the average LVD offsets relative to a world height system (WHS). The principle of the geopotential value approach is provided in Sect. 41.2. The LVD offsets relative to WHS and the vertical displacements of the
R. Tenzer et al. 165°
170°
175°
180°
–35°
–35°
–40°
–40°
–45°
–45°
165° –10
170° –5
0
5
10
175° 15
20
25
180° 30
35
40
Fig. 41.1 The NZGeoid2009 quasigeoid model (Claessens et al. 2011). The units are in meters
Table 41.1 The LVD offsets from the NZGeoid05 and NZGeoid2009 quasigeoid models LVD One Tree Point 1964 Auckland 1946 Moturiki 1953 Gisborne 1926 Napier 1962 Taranaki 1970 Wellington 1953 Nelson 1955 Lyttelton 1937 Dunedin 1958 Dunedin-Bluff 1960 Bluff 1955
Offset to NZGeoid05 0.26 0.50 0.32 0.59 0.32 0.45 0.51 0.26 0.35 0.48 0.26 0.37
Offset to NZGeoid2009 0.06 0.34 0.24 0.34 0.20 0.32 0.44 0.29 0.47 0.49 0.38 0.36
The units are in meters
NZGeoid05 and NZGeoid2009 regional quasigeoid models relative to the EGM2008 global quasigeoid model are computed and results are presented in Sect. 41.3. The summary and conclusions are given in Sect. 41.4.
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Realization of the World Height System in New Zealand: Preliminary Study
41.2
Methodology
The principle of the geopotential value approach utilizes Molodensky’s concept for the definition of the normal heights. According to this concept, the normal gravity potential UðH N ; OÞ evaluated in the point at the telluroid equals the actual gravity potential Wðh; OÞ evaluated in the point at the Earth’s surface (cf. Molodensky et al. 1960) UðH N ; OÞ ¼ Wðh; OÞ;
(41.1)
where h is the geodetic height above the level ellipsoid, H N the normal height, and O ¼ ðf; lÞ the geocentric direction with the geocentric spherical latitude f and longitude l. In general, however, the normal heights of LVD are not realized with respect to the geoidal geopotential value W0 but they are referred to the mean sea level (MSL) at the tide-gauge station which is used as the LVD origin. The normal heights of LVD are thus refereed to the particular geopotential value W0;LVD (i.e., W0;LVD 6¼ W0 ). The difference of the normal gravity potential evaluated in the point at the telluroid and the actual gravity potential evaluated in the point at the Earth’s surface gives the geopotential difference dW0;LVD at the GPS-levelling point (cf. Bursˇa et al. 1999). Hence dW0;LVD ¼ W0 W0;LVD ¼ UðHN ; OÞ Wðh; OÞ:
(41.2)
From (41.2), the LVD offset evaluated at the GPSlevelling point is defined as (ibid.) dH0;LVD ¼ ¼
dW0;LVD g UðH N ; OÞ Wðh; OÞ ; g
(41.3)
where g is the integral mean of the normal gravity along the normal plumbline between the level ellipsoid and the telluroid. The gravity potential W in (41.3) is computed in the point at the Earth’s surface from the GGM coefficients using the well-known expressions for the spherical harmonic analysis of the gravity field (see e.g., Heiskanen and Moritz 1967, Chaps. 2–5). The normal gravity U in the point
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at the telluroid can be computed using the Somigliana’s formula (see e.g., Heiskanen and Moritz 1967, eq. 2-62). The mean value of the normal gravity g in (41.3) can be computed using the expression given in Heiskanen and Moritz (1967, eq. 4-42).
41.3
Numerical Realization
The testing network consists of 2,241 GPS-levelling points situated at the South and North Islands of NZ. The number of GPS-levelling testing sites for each of 12 LVDs varies from 62 to 590 (see Table 41.2). The geographical configuration of the GPS-levelling testing network is shown in Fig. 41.2. The configuration of the GPS-levelling points is very irregular and does not cover all parts of NZ. The geodetic heights above the GRS80 reference ellipsoid are referred to the New Zealand Geodetic Datum 2000 (NZGD2000). The NZGD2000 is aligned to the International Terrestrial Reference Frame 1996 (ITRF1996) at the reference epoch of January 1st 2000 (LINZ 2007). Depending on whether the direct, direct and indirect or no tidal corrections are applied to the observed values, the harmonic gravity field analysis is realized in the zero-tide, tide-free or mean-tide system (IAG SC3 Report 1995; see also Vatrt 1999). Since the tidal corrections were not applied to the levelling observations in NZ, the normal-orthometric heights are assumed to be in the mean-tide system. The geodetic heights above the GRS80 reference ellipsoid realized in the NZGD2000 geodetic datum are defined in the tide-free system. All computations in this study Table 41.2 The number of the GPS-levelling points for each of 12 LVDs in NZ LVD One Tree Point 1964 Auckland 1946 Moturiki 1953 Gisborne 1926 Napier 1962 Taranaki 1970 Wellington 1953 Nelson 1955 Lyttelton 1937 Dunedin 1958 Dunedin-Bluff 1960 Bluff 1955
Number of testing sites 108 590 333 101 62 77 80 176 340 94 188 92
.
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Fig. 41.2 The GPS-levelling testing network in NZ consisting of 12 LVDs
are realized in the tide-free system. The differences due to using two different permanent tide systems for a realization of the normal-orthometric and geodetic heights are not taken into consideration. We note here that the differences between the heights defined in the mean-tide and tide-free system computed at the GPS-levelling testing sites in New Zealand vary between 0.0 cm (f ¼ 35:1243 deg ) and 4.1 cm (f ¼ 46:7950 deg ). As follows from the principle of the geopotential value approach, the normal heights are used to compute the normal gravity in the point at the telluroid. To compute rigorously the geopotential differences dW0;LVD at the GPS-levelling points according to (41.2), the approximate normal-orthometric heights should be first converted into the normal heights. In this study, the differences between these two types of heights are, however, neglected. The values of the gravity potential W in the points at the Earth’s surface are computed using the tide-free EGM2008 spherical harmonic coefficients (Pavlis et al. 2008) complete to degree and order 2159. The permanent tide systems are defined by the four fundamental parameters of the Earth: the geocentric gravitational constant GM, the mean angular velocity of the Earth’s rotation o, the geoidal geopotential value W0, and the second zonal parameter c2,0. Whereas the parameters GM, o, and W0 are not dependent on the
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tide, the second zonal parameter c2,0 is tide dependent. Alternatively, the permanent tide systems can be defined by the set of the following four parameters: GM, o, a, and f; where a and f is the semi-major axis and linear flattening of the level ellipsoid, respectively. In our computations we adopted the following set of constants: GM ¼ 398600441.8 106 0.8 106 m3s2 (Ries et al. 1992); o ¼ 7292115 1011 rad/s (IAG SC3 Report 1995); c2,0 ¼ 1082626.7 109 0.1 109 (IAG SC3 Report 1995); W0 ¼ 62636856 0.5 m2s2 (Bursˇa et al. 1997). The values of the normal gravity potential U in the points at the telluroid are computed only approximately using the available normal-orthometric heights HNO instead of using rigorously the normal heights H N . The parameters of the level ellipsoid in the tidefree system are defined by the following values: a ¼ 6378136.52 m, and 1/f ¼ 298.260310 (cf. Bursˇa et al. 1999). The geopotential differences dW0;LVD at the GPS-levelling points are computed from the values of the EGM2008 gravity potential at the Earth’s surface and the normal gravity potential at the telluroid using (41.2). The corresponding offsets dH0;LVD at the GPS-levelling points are computed using (41.3). The geopotential differences and offsets are then averaged for each LVD. The estimated average geopotential differences dW0;LVD for each of 12 major LVDs in NZ and the corresponding estimated average LVD geopotential values W0;LVD are summarized in Table 41.3. The estimated average LVD offsets dH0;LVD relative to WHS and the standard deviations sdH of the LVD offsets are summarized in Table 41.4. The average offsets of 12 major LVDs at the South and North Islands of NZ vary from 0.01 m (Wellington 1953) to 0.37 m (One Tree Point 1964). The estimated standard deviations of the average LVD offsets vary from 4 cm (Wellington 1953) up to 19 cm (Dunedin 1958). The inaccuracy of the estimated average LVD offsets is mainly due to the EGM2008 errors. The errors up to several centimetres are expected due to inaccuracies within the GPS and levelling measurements. The EGM2008 omission errors, the inaccuracies due to the tectonic and other vertical deformations of the levelling networks, and the errors due to disregarding the differences between the normal and normal-orthometric heights are unknown.
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Realization of the World Height System in New Zealand: Preliminary Study
347
Table 41.3 The estimated average geopotential values W0;LVD and the geopotential differences dW0;LVD of 12 major LVDs in NZ LVD
W0;LVD
One Tree Point 1964 Auckland 1946 Moturiki 1953 Gisborne 1926 Napier 1962 Taranaki 1970 Wellington 1953 Nelson 1955 Lyttelton 1937 Dunedin 1958 Dunedin-Bluff 1960 Bluff 1955
62636852.4 62636854.8 62636854.1 62636855.0 62636853.7 62636854.9 62636855.9 62636854.1 62636854.7 62636855.3 62636853.7 62636854.3
The geopotential differences are taken W0 ¼ 62636856 m2s2. The units are in m2s2
dW0;LVD 3.6 1.2 1.9 1.0 2.3 1.1 0.1 1.9 1.3 0.7 2.3 1.7 relative
to
Table 41.4 The estimated average offsets dH0;LVD and their standard deviations sdH of 12 major LVDs in NZ. The LVD offsets are taken relative to WHS (defined based on W0 ¼ 62636856 m2s2) LVD One Tree Point 1964 Auckland 1946 Moturiki 1953 Gisborne 1926 Napier 1962 Taranaki 1970 Wellington 1953 Nelson 1955 Lyttelton 1937 Dunedin 1958 Dunedin-Bluff 1960 Bluff 1955
dH0;LVD 0.37 0.12 0.21 0.10 0.23 0.11 0.01 0.19 0.13 0.07 0.23 0.18
Fig. 41.3 The vertical displacements of the NZGeoid05 regional quasigeoid model taken relative to the EGM2008 global quasigeoid model computed at the GPS-levelling testing network in NZ. The units are in meters
sdH 0.06 0.04 0.10 0.04 0.05 0.07 0.04 0.09 0.11 0.19 0.07 0.05
The units are in meters
The vertical displacements of the NZGeoid05 and NZGeoid2009 regional quasigeoid models taken relative to the EGM2008 global quasigeoid model are computed at the points of the GPS-levelling testing network. The results are shown in Figs. 41.3 and 41.4. The small discrepancies due to using two different tide systems for computing the EGM2008 global quasigeoid model (defined in the tide-free system) and the NZGeoid05 and NZGeoid09 regional
Fig. 41.4 The vertical displacements of the NZGeoid2009 regional quasigeoid model taken relative to the EGM2008 global quasigeoid model computed at the GPS-levelling testing network in NZ. The units are in meters
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quasigeoid models (both defined in the zero-tide system) are neglected. We note here that the differences between the quasigeoid undulations defined in the zero-tide and tide-free system computed at the GPS-levelling testing sites in NZ vary from 0.0 cm (f ¼ 35:1243 deg ) to 1.8 cm (f ¼ 46:7950 deg ). The differences between NZGeoid05 and EGM2008 are positive and vary from 0.24 to 0.81 m with the mean of 0.56 m. The differences between NZGeoid09 and EGM2008 are again positive and vary from 0.27 to 1.14 m with the mean of 0.51 m. The range of differences between NZGeoid2009 and EGM2008 at the GPS-levelling testing network is 0.87 m. The corresponding range of differences between NZGeoid05 and EGM2008 is 0.57 m. The range of differences between NZGeoid2009 and EGM2008 is thus about 66% larger than the range of differences between NZGeoid05 and EGM2008.
geopotential numbers which require gravity measurements along the levelling lines. We also note that the realization of a WHS in NZ requires a proper treatment of the tide systems, because of existing inconsistencies in applied tidal corrections to levelling, GPS and gravity data. The EGM2008 coefficients and the GPS-levelling data have been used to compute the vertical displacements of NZGeoid05 and NZGeoid2009 relative to EGM2008. The results at the testing network of GPS-levelling points revealed the presence of the systematic bias of about 0.56 m between NZGeoid05 and EGM2008. A similar systematic bias of about 0.51 m was confirmed between NZGeoid2009 and EGM2008. The range of differences between NZGeoid05 and EGM2008 (0.87 m) is about 66% larger than the range of differences between NZGeoid05 and EGM2008 (0.57 m).
References 41.4
Summary and Conclusions
We have applied the geopotential value approach to estimate the average LVD offsets in NZ using the EGM2008 coefficients and the GPS-levelling data. The LVD offsets are taken relative to a WHS. The WHS is defined by the adopted geoidal geopotential value W0 ¼ 62636856 m2s2. The estimated average offsets of 12 major LVDs situated at the North and South Islands of NZ range from 0.01 m (Wellington 1953) to 0.37 m (One Tree Point 1964). The Wellington 1953 coincides best with the WHS. The accuracy of the estimated average LVD offsets varies significantly. The error analysis revealed that the accuracy of the estimated LVD offset of Wellington 1953 is about 4 cm in terms of the standard deviation. The highest uncertainty of about 19 cm was found in the estimated LVD offset of Dunedin 1958. The errors of the estimated LVD offsets are largely due to inaccuracies of the EGM2008 coefficients. Large errors are also expected due to existing systematic distortions of the LVDs, and tectonic and other vertical movements. Since the LVDs in NZ are defined in the normal-orthometric height system, additional systematic errors are expected due to using the normal-geopotential numbers instead of the actual
Amos MJ (2007) Quasigeoid modelling in New Zealand to unify multiple local vertical datums, Ph.D. Thesis, Curtin University of Technology, Perth, Australia Amos MJ, Featherstone WE (2009) Unification of New Zealand’s local vertical datums: iterative gravimetric quasigeoid computations. J Geod 83:57–68 Bursˇa M, Radej K, Sˇ´ıma Z, True SA, Vatrt V (1997) Determination of the geopotential scale factor from TOPEX/POSEIDON satellite altimetry. Studia Geoph Geod 41:203–216 Bursˇa M, Kouba J, Kumar M, M€uller A, Radej K, True SA, Vatrt V, Vojtı´sˇkova´ M (1999) Geoidal geopotential and world height system. Studia Geoph Geod 43:327–337 Claessens S, Hirt C, Featherstone W, Kirby J (2009) Computation of a new gravimetric quasigeoid model for New Zealand. Technical report prepared for Land Information New Zealand by Western Australia Centre for Geodesy, Curtin University of Technology, Perth, p 39 Claessens S, Hirt C, Amos MJ, Featherstone WE, Kirby JF (2011) NZGeoid09 quasigeoid model of New Zealand. Surv Rev 43(319):2–15 Heiskanen WH, Moritz H (1967) Physical geodesy. W.H. Freeman and Co., San Francisco, CA IAG SC3 Report (1995) Travaux de L’Association Internationale de Ge´ode´sie, Tome 30, Paris LINZ (2007) Standard for New Zealand Geodetic Datum 2000, LINZS25000, Land Information New Zealand, Wellington. Available at: www.linz.govt.nz Molodensky MS, Yeremeev VF, Yurkina MI (1960) Methods for study of the external gravitational field and figure of the Earth. TRUDY Ts NIIGAiK, 131, Geodezizdat, Moscow. English translatation: Israel Program for Scientific Translation, Jerusalem 1962, p 248
41
Realization of the World Height System in New Zealand: Preliminary Study
Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2008) An Earth Gravitational Model to Degree 2160: EGM2008, presented at the 2008 General Assembly of the European Geosciences Union, Vienna, Austria, 13–18 April 2008 Rapp RH (1961) The orthometric height. MS Thesis. Department of Geodetic Science and Surveying, Ohio State University, Columbus, OH
349
Ries JC, Eanes RJ, Shum CK, Watkins MM (1992) Progress in the determination of the gravitational coefficient of the Earth. Geophys Res Lett 19(6):529–531 Vatrt V (1999) Methodology of testing geopotential models specified in different tide systems. Studia Geoph Geod 43:73–77
Comparisons of Global Geopotential Models with Terrestrial Gravity Field Data Over Santiago del Estero Region, NW: Argentine
42
L. Galva´n, C. Infante, E. Laurı´a, and R. Ramos
Abstract
The recent improvements in satellite tracking data processing, the availability of new surface gravity data sets, and the availability of a new mean sea surface height model from altimetry processing gave rise to the generation of several new global gravity field models. However, to know their potentiality for using in practical situations, we understood that it was necessary their applications in a limited regions. This paper we compare recent geopotential models with gravimetric data over leveling points of Argentinean National Geografhical Institute (ANGI) vertical network in Santiago del Estero region, northwestern Argentine. We have highlighted the most important information, we have established the future expectations to continue with such applications. Results of comparisons are presented.
42.1
Introduction
The development diverse Global Geopotential Models (GGM) occurred in the last decades have shown successive increase in the spatial resolution and accuracy. These improvements have been due essentially to the incorporation of better quality data coming from diverse sources for all the Earth. In order to evaluate what of these released models is the well adapted one for the Santiago del Estero region, a set of recent GGM have been compared. In this first stage of
L. Galva´n (*) C. Infante Faculty of Exact Sciences and Technology. National University of Santiago del Estero, Av. Belgrano (S) 1912. (4200), Santiago del Estero, Argentine e-mail:
[email protected] E. Laurı´a R. Ramos National Geographical Institute, Av. Cabildo 381 (1426), Buenos Aires, Argentine
work, the Free-Air Gravity Anomalies (FAGA) extracted of models EIGEN-05C, GGM03C and EGM2008, everyone developed up to degree and order 360, and EGM2008 complete development up to degree 2190 and order 2159. These GGM were compared with the calculated FAGA from measures of gravity carried out by ANGI. (Amos and Featherstone 2005, 2006). This comparison has been realized on points ANGI. The GRS used is WGS 84. The tide_system used is free_tide. In order to determine the model that better adjusts to the region we used a series of statistical on the resulting residuals, such as: minimums (MIN), maximums (MAX), average (AVER) and root mean square (RMS). The RMS is adopted to evaluate the GGM that better adjustment to the terrestrial data. Consequently the GGM that presents the smaller RMS as a result of the contrast with the terrestrial data is the one that better adjusts to the region.
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_42, # Springer-Verlag Berlin Heidelberg 2012
351
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352 Table 42.1 Global geopotential models Model GGM03C EIGEN-5C EGM2008 EGM2008
42.2
Year 2009 2008 2008 2008
Degree 360 360 2190 360
Data S (Grace), G, A S (Grace, Lageos), G, A S (Grace), G, A S (Grace), G, A
Reference Tapley et al. (2007) F€orste et al. (2008) Pavlis et al. (2008) Pavlis et al. (2008)
P
Description and Results
ζ
The zone of study has been defined between 25 to 30 latitude and 60.5 to 65.5 longitude. It corresponds approximately with the territory of the province of Santiago del Estero, Northwest of Argentine. The adopted Global Geopotential Models for the present study appear in Table 42.1. These Models are derived from the combination of satellite data (S), observations of terrestrial gravity (G) and satellite altimetry data (A). In Table 42.1 they specify the year of liberation, the Degree of harmonic coefficients, the origin of the data, and the reference. GGM03C Model released in 2009 is sustained by the previous GGM02 model that is based on the analysis of data coming from 363 observation days of the mission GRACE, from April 4th, 2002 until December 31th, 2003. It is developed up to degree and order 360 and incorporates gravity and altimetry data. The model EIGENGL-05C was released in September 29th, 2008. It is developed up to degree and order 360. It is base in a combination of data from the mission GRACE and LAGEOS (January 2002–December 2006), gravimetric data 0.5 0.5 degrees and altimetry data. Its, space resolution is about 55 km on the terrestrial surface. EGM2008 Model has been released by the National Geospatial-Intelligence Agency (NGA). This gravitational model is complete for degree and order 2159 and contains additional coefficients up to the degree 2190 and order 2159. 750 land gravity data performed by ANGI over 19 leveling lines have been used for the present work over Santiago del Estero region. The gravity measures were obtained on 741 normal benchmark (BMK) and 9 nodal ones of the High Precision Leveling Network for the province. The BMK have geocentric latitude and longitude in WGS84 system with variable accuracy. The values of gravity are referred IGSN71 system. (ANGI 1979, 1983).
earth’s surface S
Q
H*
h
Q0
telluroid ∑
ellipsoid E
Fig. 42.1 Telluroid, ellipsoid and earth surface. (HofmannWellenhof and Moris 2006)
There is no adequate information about ellipsoidal height for the BMK for that reason is realized the comparison on gravimetric anomalies (Del Cogliano 2006). Free air gravity anomalies have calculated for terrestrial gravity data using the equation in the context of Molodensky’s theory (Hofmann-Wellenhof and Moris 2006; Heiskanen and Moritz 1967; Zakatov 1997): Dg ¼ gP gQ
(42.1)
Where Dg is the free – air gravity anomalies, gP is the gravity observed in the terrestrial surface and gQ is the normal gravity in the telluroid. See Fig. 42.1. The equations (42.2), (42.3), (42.4) and (42.5) and parameters (Table 42.2) used for the calculation of gQ are (Hofmann-Wellenhof & Moris 2006; Heiskanen and Moritz 1967; Zakatov 1997): gQ ¼ g 2ga =a½1 þ f þ m þ ð3f þ 5=2mÞsin2 ’h þ 3ga =a2 h2 (42.2) g ¼ ga ð1 þ f sin2 ’ 1=4f 4 sin2 2’Þ
(42.3)
42
Comparisons of Global Geopotential Models
353
Table 42.2 Parameters and derived constants of the WGS84 (Hofmann-Wellenhof and Moris 2006) WGS 84 Parameter a b f GM ga gb M m
Description Semimajor axis of the ellipsoid Semiminor axis of the ellipsoid Flattening of the ellipsoid Geocentric gravitational constant of the earth (including the atmosphere) Normal gravity at the ecuator Normal gravity at the pole Mass of the earth (includes atmosphere) m ¼ o2a2b/(GM)
Value 6378137 m 6356752.3142 m 1/298.25722356 3986004.418 108 m3seg2 9.780325336 m2seg2 9.832184938 m2seg2 5.9733328,1024kg 0.003449787
FAGA
FAGA GGM 2008 up 2159 [mGal]
50 40 30 20 10 0 -30
-20
-10
0
10
20
30
40
50
60
70
-10 -20 FAGA ANGI [mGal]
FAGA
Fig. 42.2 Correlation of the free-air anomalies with height for Santiago del Estero region
f ¼ ðgb ga Þ=ga
(42.4)
f 4 ¼ 1=2f 2 þ 5=2fm
(42.5)
On the other hand free – air gravity anomalies of each one of the GGM specified more above (Table 42.1) for the zone of study have been determined. From the comparison of the values of FAGA between the GGM and Argentinean National Geographical Institute points, the residuals for the points of leveling of the ANGI have been obtained (42.6). DgRES ¼ DgANGI DgGGM
(42.6)
Where DgRES is the free – air gravity anomalies residual, DgANGI is ANGI points free – air gravity
anomalies and DgGGM is GGM free – air gravity anomalies. The statistics are in Table 42.3. It is possible to be observed that EGM2008 model presents the smaller RMS of 8.996 mGal reason why is inferred that this model presents the best adjustment for the zone of study. A good general correlation in both magnitudes can be observed, with greater emphasis in the positive anomalies (Fig. 42.2). The dispersion of the magnitudes is greater in correspondence with the major heights of the region. Two leveling lines are emphasized in the analysis because their different behavior: the line 313 is comprised between the points number 720 and 760 and the line 180 between the points 1 and 57. Their graphic representation related to heights ANGI, the
L. Galva´n et al.
354
gravimetric anomalies and the resulting residuals are in the Fig. 42.3 (Introcaso, 1997, 2006). The line 313 is characterized to present mean sea level heights between 150 and 220 m (Fig. 42.3a). The variation of the gravimetric anomaly of the terrestrial data is in agreement to this variation the anomaly of the model (Fig. 42.3b). The FAGA residuals present low values pointing out a good adjustment between ANGI data and EGM2008 (Fig. 42.3c). The magnitude of the residuals are in the interval 5 mGal. This line has a main directorate from the north-west to south-east in the North zone of the province. It is characterized for being a plain with slopes in the same direction. On the other hand the line 180 is characterized by a strong variation in height, with values between 110 and 630 m (Fig. 42.3d). It is
b
Gravity Anomalies. Line 313. FAGA-ANGI FAGA-GGM
34
Gravity Anomalies Residuals. [mGal]
Gravity Anomalies [mGal]
a
29 24 19 14 720
730
740
750
possible to emphasize that the variation of the gravimetric anomaly of the terrestrial data is in agreement to this variation whereas the anomaly of the model presents a smoothed curve (Fig. 42.3e). The residuals take values from 40 to 10 mGal, where the bigger values are in correspondence with the bigger heights (Fig. 42.3f). This line is developed between the mountain ranges of Sumampa and Ambargasta to the south of the province with a North-South direction. FAGA and residuals are shown in Fig. 42.4a, b (Corchete & Pacino 2007). Figure 42.4a shows free air gravimetric isoanomalies curves of EGM2008 up 2190 model over an image SRTM 90 m of the territory of the province (Galva´n et al. 2009). It could be observed that in the south of the province the anomalies grow in
760
Bench Marks
c
400 300 200 100 10
20
30
40
50
60
Gravity Anomalies [mGal]
IGM Heights [m]
500
Bench Marks
IGM Heights [m]
730
735
740
745
750
Height Line 313
200
180
160
730
740 Bench Marks
750
760
755
760
-3 Residuals-Line 313
-5 Bench Marks
Gravity Anomalies. Line 180. FAGA-ANGI FAGA-GGM
75 65 55 45 35 25 15 5 -5 0 -15
10
20
30
40
50
60
Bench Marks
f
Height Line 313
Gravity Anomalies Residuals [mGal].
e
140 720
725
85
600
220
-1720
d
Height
0
1
-7
Height Line 180 700
Residuals - Line 313 3
Residuals- Line 180 Residuals-Line180
35 25 15 5 -5 0
10
20
30
40
50
-15 Bench Marks
Fig. 42.3 Heights (a, d), AAL (b, e) and Residuals (c, f) of Leveling Lines 313 and 180 for Santiago del Estero region
60
42
Comparisons of Global Geopotential Models
355 Table 42.3 Statistical comparisons between FAGA residuals of GGM and terrestrial data MODEL
MAX (mGal) GGM03C 43,747 EIGEN-GL05C 44,783 EGM2008 38,443 EGM 2008 45,202 up to 360
MIN MEAN (mGal) (mGal) 24,856 3,024 24,511 2,396 25,379 2,322 24,649 2,647
RMS (mGal) 9,884 9,403 8,996 9,303
agreement with the Mountain ranges of Ambargasta and Sumampa and in the west agrees with the Mountain ranges of Guasaya´n. The rest of the territory characterizes by a surface with general slope in sense the north-west to south-east. In this sector the curves of isoanomalies take positive and negative values. Also the lines of leveling of the ANGI are represented by sequence of points. Figure 42.4b shows space distribution of the differences between the FAGA of the EGM2008up2190 model with the terrestrial data. It is possible to be observed that in ample zones of the territory small residues of the order of 5 mGal appear. The highest residues are positioned in the zones of mountainous areas already-mentioned and they coincidence with the majors values of anomalies. Conclusions
Fig. 42.4 Map of Gravity Anomalies (FAGA) (a) and Map of Gravity Anomalies Residuals of EGM2008 up 2190 Model (b) for Santiago del Estero region
The results of comparisons of recent selected GGM in the present work (EIGEN-GL05C, GGM03C and EGM2008 upto 360 and EGM2008 complete) with terrestrial gravity data of ANGI show the EGM2008 upto 2190 model presents the better fits to the investigated region. For the comparison free air gravity anomalies calculated in 750 ANGI points and data of free air gravity anomaly of each one of the models have been used. They were calculated the respective residuals for each GGM. The small mean residuals (2.322 mGal) and RMS were found for the EGM2008 model. Last model is the one that has been used for the space analysis. Two zones with high residual values of FAGA have been detected that correspond with the mountain ranges of Sumampa and Ambargasta to the south and the mountain range of Guasaya´n to the west of the province. The rest of the territory presents FAGA residuals in the range of about 5 [mGal].
356 Acknowledgements The present work has been developed within the framework of the Investigation Project “Evaluation of the elevation global and geopotentials models for the province of Santiago del Estero”, financed by the Council of Investigation of Science and Technical (CICYT) of the National University of Santiago del Estero (UNSE). To Cristina Pacino by the original idea.
References Amos MJ, Featherstone WE (2005) Comparisons of recent global geopotential models with terrestrial gravity field data over New Zealand and Australia. Geomatics Research Australasia Amos MJ, Featherstone WEA (2006) Comparison of gridding Techniques for Terrestrial Gravity Observations in New Zealand. Geomatics Research Australasia Corchete V, Pacino MC (2007) The first high–resolution gravimetric geoid for Argentina: GAR. Physics of the earth and planetary interiors Del Cogliano D (2006) Modelado del Geoide con GPS y Gravimetrı´a. Caracterizacio´n de la Estructura Geolo´gica de Tandil. Universidad Nacional de Rosario F€orste C, Flechtner F, Stubenvoll R, Rothacher M, Kusche J, Neumayer K-H, K€ onig R, Barthelmes F, Raimondo JC, Biancale R, Bruinsma S, Lemoine JM, Dahle C (2008) EIGEN-5C – the new GeoForschungsZentrum Potsdam/ Group de Recherche de Geodesie Spatiale combined gravity field model, AGU 2008 Fall Meeting (San Francisco 2008)
L. Galva´n et al. Galva´n L, Infante C, Goitea I, Laita´n H, Duro J, Pirola M, Luna JP, Laurı´a E, Ramos R (2009) Evaluacio´n del modelo SRTM 90m en alturas IGM para la provincia de Santiago del Estero. V Jornadas de Ciencia y Tecnologı´a de Facultades de Ingenierı´a del NOA. Facultad de Ingenierı´a. Universidad Nacional de Salta Heiskanen WA, Moritz H (1967) Physical geodesy. W.H.Freeman, San Francisco, CA Hofmann-Wellenhof B, Moris H (2006) Physical geodesy, 2nd edn. Springer Wien, New York, NY Instituto Geogra´fico Militar (ANGI) (1979) 100 An˜os en el Quehacer Cartogra´fico del Paı´s. p 112 Instituto Geogra´fico Militar (ANGI) (1983) Guı´a de la Repu´blica Argentina para Investigaciones Geogra´ficas. pp 119–120 Introcaso A (1997) Gravimetrı´a. UNR Editora. Editora de la Universidad Nacional de Rosario Introcaso A (2006) Geodesia Fı´sica. Boletı´n del Instituto de Fisiografı´a y Geologı´a, Rosario Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2008) An Earth Gravitational Model to Degree 2160: EGM2008, Presented at the 2008 General Assembly of the European Geoscience Union, Vienna, Austria, April 13–18, 2008. Available at: http://earthinfo.nga.mil/GandG/wgs84/gravitymod/egm2008/ NPavlis&al_EGU2008.ppt Tapley B, Ries J, Bettadpur S, Chambers D, Cheng M, Condi F, Poole S (2007) The GGM03 Mean Earth Gravity Model from GRACE, Eos Trans. AGU, 88(52), Fall Meet. Suppl., Abstract G42A-03 Zakatov PS (1997) Curso de Geodesia Superior. Editorial Rubin˜os
Intermap’s Airborne Inertial Gravimetry System
43
Ming Wei
Abstract
The airborne gravimetry system provides an efficient tool to collect the homogeneous airborne gravity data over large areas. For this purpose Intermap has developed a new Airborne Inertial Gravimetry System (AIGS), based on the GPS/INS components of Intermap’s Interferometric SAR (IFSAR) system and the airborne gravity process software, called StarGrav. The state-of-the art in the acquisition of airborne gravity data at Intermap will be discussed and the process in gravity determination will be described. The paper presents recent airborne gravity results for different topography and scenarios. The airborne gravity measurements by Intermap’s StarGrav system are compared to the upward continued ground gravity data and to the independent airborne gravity results provided by NGS. The results demonstrate that the accuracy of 2–3 mGal (1s) for the airborne gravity measurements by Intermap airborne gravity mapping system can be achieved. The geoid determined using the airborne gravity data could have the relative accuracy of 5 cm (1s) when compared with an independently determined geoid reference.
43.1
Introduction
Space-borne gravity missions are in the process of measuring the low and medium resolution features of the gravity field of the Earth at an unprecedented rate and accuracy for most of our planet. However, in order to meet many of the requirements of determination of precise regional geoid and other geodetic applications, there remains a significant challenge in accurate recovery the high-resolution features of the gravity field
M. Wei (*) Intermap Technologies Corp., #1200, 555 – 4th Avenue, SW, Calgary, Alberta, Canada, T2P 3E7 e-mail:
[email protected]
(i.e. those with wavelengths shorter than 100–200 km) over large areas which could be effectively and homogenously measured by the airborne gravimetry. The objective of this paper is to present the ability of Intermap Technologies Corporation (“Intermap”) to provide high-resolution airborne gravity data at a high accuracy for very large areas of the Earth. Currently two different technologies are available for the large scale airborne gravimetry: The airborne gravimeter based on the gimbal platform and the airborne inertial gravimetry system (AIGS) based on the precise inertial system, particularly the strapdown inertial system. The typical airborne gravimeter of the first technology is the spring-based relative gravimeter, e.g. the LaCoste and Romberg (L&R) air-sea gravimeter (currently the Micro-g LaCoste gravity
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_43, # Springer-Verlag Berlin Heidelberg 2012
357
358
system), see [1–5]. The AIGS normally uses the precise strapdown inertial system as gravity sensors. Using their leading edge IFSAR mapping technologies, Intermap is in the process of digitally remapping entire countries and building unprecedented national databases of highly accurate digital topographic maps in a series of programs called NEXTMap, i.e. NEXTMap USA for entire contiguous USA and NEXTMap Europe for west Europe. By making use of the IMU and GPS data collected as part of the NEXTMap programs, and by using the airborne gravity determination technology, Intermap could provide an accurate representation of the high-resolution features of the gravity field for up to nation-wide scales. This innovative solution to the airborne gravity measurement is particularly attractive because of the homogeneity of the gravity measurements collected by the NEXTMap data acquisition program (the radar acquisition over the entire continuous USA), and especially because of the cost-effectiveness implied by the data having already been acquired as part of the NEXTMap program. The Intermap’s NEXTMap programs including NEXTMap USA and NEXTMap Europe is described in Sect. 43.2. This includes the coverage of the airborne IFSAR mapping area and the characteristics of airborne data acquisition. Using this information we can see the potential contribution of the airborne gravity measurements from the NEXTMap program to the national wide airborne gravity program. The state of the art in the acquisition of airborne gravity data will be discussed in Sect. 43.3, and the performance of the AIGS is demonstrated by sharing the airborne gravity results in different topography and scenarios in Sect. 43.4. The airborne gravity measurements from the NEXTMap USA program are compared to the upward continued ground gravity data and to the independent airborne gravity results provided by NGS and other agencies. The results demonstrate that the accuracy of 1–3 mGal (1s) for the airborne gravity measurements from the AIGS can be achieved. The drawback of the airborne inertial gravimetry system for the application of the national wide airborne gravity measurements is also discussed in the paper. Based on the analysis of the airborne gravity measurement characteristics of two different airborne gravimetric technologies (The airborne relative gravimeter and the airborne inertial gravimetry system) an approach to combine the airborne gravity
M. Wei
measurements from the two airborne gravimetric technologies is proposed to timing- and costeffectively collect the airborne gravity measurements.
43.2
Intermap’s NEXTMap Program
In 2002/2003 Intermap has launched NEXTMap mapping program for the countrywide 3D maps including digital elevation data and the ortho-rectified radar image. Currently the main focus of the NEXTMap program is the NEXTMap USA and NEXTMap Europe 3D mapping programs to remap the continental USA and the countries mainly in west Europe using Intermap IFSAR technology. The NEXTMap products include the high resolution radar orthoimage and digital elevation data (DEM). The data acquisition of the NEXTMap USA and NEXTMap Europe programs has been completed in 2009 as announced by Intermap. Figure 43.1 shows the acquisition areas and blocks of the NEXTMap USA program. As shown in Fig. 43.1 the imagery data collected for NEXTMap USA covers the 48 continuous states (more than 8 million km2). The DGPS and INS (high performance Honeywell H-770 IMU) data collected by the airborne interferometric SAR system for the NEXTMap program can be used to generate the airborne gravity measurements. As shown in Sect. 43.4, the accuracy of the airborne gravity measurements calculated based on the DGPS/INS data of the airborne IFSAR system is at the similar level of what from the traditional airborne gravimeters. This could open an opportunity using the airborne gravity measurements from the existing DGPS/INS data as a supplement to the national wide airborne gravity data. The flight altitude of the airborne data acquisition for the NEXTMap USA program is at 8–10 km. The airborne GPS/INS data are mainly collected along the parallel primary lines, and also at many tie lines perpendicular to the primary line. The space between the primary lines is about 8–10 km in the flat areas and 5–7 km in the mountain areas. Because of the airborne data acquisition restriction the airborne GPS/INS data are mainly collected at the flight altitude. Thus the airborne gravity results are normally not directly tied to the ground gravity reference. The airborne GPS/INS data were collected from one block to other block. The airborne data of one
43
Intermap’s Airborne Inertial Gravimetry System
359
Fig. 43.1 Area coverage and acquisition blocks of NEXTMap USA
block consist of regular primary lines and many tie lines. They can be used to create the crossover points (grid points) for the crossover adjustment of the airborne gravity measurements. As shown in Fig. 43.1, the size of many blocks in the north part of USA is significant large which gives the advantage of the airborne gravity measurements for the least squares adjustment of the airborne gravity data. Combined with the airborne gravity data of the few control lines additionally collected by using the L&R gravimeter, the airborne gravity measurements calculated from the airborne GPS/INS data of the NEXTMap USA program can be re-adjusted and tied to the ground reference as discussed in the conclusions.
for the airborne gravimetry are its size, operational efficiency and stable performance over the large area. The airborne inertial gravimetry system consists of an aircraft mounted inertial navigation system (INS), a GPS receiver and a data collection system on the aircraft. A second GPS receiver at a ground control point is also required for the differential GPS process. Using an inertial navigation system and DGPS the gravity disturbance dg can be determined based on Newton’s equation of motion in the gravitational field of the Earth. The principle of airborne scalar gravimetry is described by the following equation dg ¼ fu v_ u þ
43.3
Airborne Inertial Gravimetry System
The concept of using strapdown inertial systems as an airborne gravity system has been discussed in [5–8]. The major advantages of using strapdown technology
v2e v2n þ Rn þ h Rm þ h
þ 2ve oe cos ’ g
(43.1)
where fu is the upward component of the specific force f l the local-level frame, ve , vn , vu are the east, north and up components of vehicle velocity computed from GPS, ’ and hare the geodetic latitude and height,
360
M. Wei
Rm and Rn are the meridian and prime vertical radii of curvature, oe is the earth rotation rate, g is the normal gravity. The sum of the third to fifth terms at the right side is often called E€ otv€ os correction. Equation (43.1) gives the basic principle of the airborne inertial gravimetry. When using an integrated DGPS/INS system as an airborne gravity system, the specific force in the local-level frame can be provided by the inertial system (INS). The vehicle kinematics ’; h and ve , vn , vu , and the vehicle acceleration v_u are obtained by DGPS. The E€ otv€ os correction and normal gravity are normally computed from DGPS position and velocity results. In many case the gravity anomaly Dg is required. The gravity disturbance dg can be converted to the gravity anomaly using the following equation Dg ¼ dg þ
@g N @h
(43.2)
where N is the geoid undulation. Figure 43.2 shows the block diagram of the AIGS, where f b and f l are specific force vector in the body frame and the local-level frame, and Rlb is the transformation matrix from the b-frame to the l-frame. Determining the gravity disturbance at the flying altitude consists of three steps. In the first step, the attitude, position and velocity of the airborne gravity system are computed using the GPS/INS navigation software. In this step the gyro drifts and accelerometer biases are estimated by the Kalman filtering of the DGPS/INS integration process. Using INS attitude solution, the specific force of inertial sensors (the acceleration observed by IMU) is transformed from the IMU body frame to the local-level frame. In the second step, the vehicle accelerations are derived and the E€ otv€ os correction is calculated from GPS navigation solution. The airborne gravity disturbance is calculated by the difference between the specific force measured by inertial sensors, the vehicle accelerations and the E€ otv€ os correction. In order to reduce the effect of the measurement noise on the airborne gravity data in the high frequency bandwidth, a low-pass filtering is applied to DGPS and INS results. In the third step a crossover adjustment technique for the airborne gravity measurements at crossover points is applied to estimate the long term error of the airborne gravity data due to the inertial measurement bias and drift.
Attitude
fb
Rbl
fl
+ g +
IMU – vu
DGPS
Eötvös Correction
+ + –
γ
Fig. 43.2 Airborne inertial gravimetry system
43.4
Airborne Gravity Test Results
In the past 15 years a series of successful airborne gravity flights with the airborne inertial gravimetry systems have been accomplished using different strapdown inertial navigation systems (INS) and the differential GPS (DGPS). The airborne gravity results from the AIGS of high performance strapdown inertial systems have shown the similar accuracy of the gravity measurements from the L&R airborne gravimeters. The average accuracy of airborne gravity measurements from the high performance strapdown inertial gravity system is about 1–3 mGal (1s) when using a low-pass filter of 1/120 s.
43.4.1 Airborne Gravity Tests by the University of Calgary In 1995 and 1996 the University of Calgary conducted two flight tests to investigate the possibility of using strapdown INS/DGPS for airborne gravity determination. The integrated system consists of a strapdown inertial system and two GPS receivers for the differential GPS process. The strapdown inertial system used for the airborne gravity test is the Honeywell LASEREF II system, a high performance strapdown INS with GG1342 RLG and QA2000 accelerometers.
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Intermap’s Airborne Inertial Gravimetry System
The flight on June 1995 was an airborne test with repeat lines crossing the Rocky Mountains in east-west direction. The second airborne test was carried out in Sept. 1996 over the Rocky Mountains. It consists of flight lines in perpendicular directions, one is in eastwest directions and other is in north-south direction, to generate crossover points. The flight lines are above the Rocky Mountain with high variability of the gravity disturbance, see [9–11]. In 1998 the same strapdown INS/DGPS system has been used for an airborne gravity test in the Disko Bay area off the west coast of Greenland by the University of Calgary, in cooperation with the Danish National survey and Cadastre (KMS). A major purpose of this flight test was to compare different airborne gravity systems side by side, e.g. the LaCoste and Romberg (L&R) airborne gravimeter and the airborne inertial gravimetry system, for more details see [12]. In the spring of 2000, an airborne gravity campaign was carried out with the goal of comparing the airborne gravity systems based on the three available scalar airborne gravimetry concepts. The data was collected as part of the Airborne Gravity for Exploration and Mapping (AGEM) project of the GEOIDE, a national wide Network Center of Excellence program in Canada. The airborne data was collected at the lower altitude of 575 m for significant gravity signals, see [13]. The airborne gravity measurements are compared to the upwards continued ground gravity data or shipborne gravity data. The accuracy of these four airborne tests is listed in Table 43.1, for detailed analysis see [9–13].
43.4.2 Airborne Gravity Results from the NEXTMap USA Program Due to the success of the airborne gravity tests conducted by the University of Calgary, Intermap has developed an AIGS based on H-770 strapdown IMU and DGPS components of the Internap’s airborne Interferometric SAR system (the START system) and the airborne gravity process software, StarGrav under the support of the University of Calgary. The Intermap’s STAR AIGS consists of the Honeywell H-770 strapdown system with the performance of 0.8 nm/h and the GPS receivers. The most advantage of H-770 is that the output of IMU box is three raw acceleration measurements with the output
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rate at 1,200 Hz in the body frame. This provides an advantage to develop an effective anti-dither filtering algorithm for high rate inertial measurements. The airborne GPS receiver is an Ashtech Z-12 receiver. The STAR acquisition system synchronizes INS recording times with GPS times and records both the H-770 IMU raw measurements and airborne GPS data. In order to assess the performance of the Intermap’s AIGS and test the process software StarGrav, Intermap has selected two test areas from the NEXTMap data: Sacramento valley in California and Baltimore/Washington DC area, see [14–16]. Both areas are well controlled by the ground gravity anomaly data which could be used as the gravity reference. The airborne gravity data of Baltimore/Washington DC area have been delivered to NGA as a research project and to OSU for the independent comparison and evaluation. In Oct 2006 Intermap has been awarded a contract with NGS to calculate airborne gravity measurements based on the NEXTMap DGPS/INS data. Two areas of airborne gravity results are selected by NGS: one is in California mixed with mountains and valley and other is in Mexico Gulf. The first area has significant variability of gravity signal for the evaluation while the second area has independent airborne gravity results from NRL L&R airborne gravimeter. The ground gravity data in both areas are also available for the independent comparison. The airborne gravity results are delivered to NGS for an assessment, see [17, 18]. Table 43.1 summarizes the airborne gravity results of the total eight flight tests when compared to the upward continued ground or shipborne gravity data. As shown in Table 43.1, the accuracy of airborne gravity measurements from the DGPS/INS data of the Intermap’s AIGS is about 1–3 mGal (from the last four tests ) when using a low-pass filter of 1/120 s. On other hand the statistics of airborne gravity difference at the crossover points could be used for the accuracy assessment. As shown in [15–18], the RMS of the gravity difference at the crossover points is about 1–3 mGal for last four airborne tests in Table 43.1. Figure 43.3 shows a typical airborne gravity anomaly difference between the airborne gravity data from the Intermap’s AIGS and the upward continued ground data in Mexico Gulf. The randomly distributed errors in Figure 43.3 indicate no long term or medium term variation in the airborne gravity data
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Table 43.1 Airborne gravity test results Airborne tests Test in June 1995 by the University of Calgary (RMS) Test in Sept. 1996 by the University of Calgary (RMS) Test in Greenland, in 1998 by U of C (RMS, filter of 1/200 s) Airborne test in Canada, in 2000 by U of C (1 s) Airborne gravity in Sacramento, USA, by Intermap (1 s) Airborne gravity in Baltimore/Washington, USA, by Intermap (1 s) Airborne gravity in California, USA, by Intermap (1 s) Airborne gravity in Mexico Gulf, USA, by Intermap (1 s)
Low-pass filter of 1/120 s 3.0–3.5 2.0–4.0 1.5–3.0 1.5–2.4 1.0–3.0 1.0–2.0 2.2 1.5
Low-pass filter of 1/90 s 3.5–4.5 3–5 / 2.2–3.7 1.2–3.7 / / /
Fig. 43.3 Airborne gravity anomaly difference
and the homogenous accuracy of the gravity measurements from the Intermap’s NEXTMap USA DGPS/INS data. Conclusions
The airborne gravity results of the airborne tests by the University of Calgary and that from four different areas of Intermaps’s NEXTMap USA DGPS/ INS data show: • The airborne gravity measurements from a high performance strapdown inertial system with raw
data output could provide the similar performance as the L&R gravimeter (currently Micro-G Air-Sea Gravity system), i.e. 1–3 mGal accuracy at flight altitude. • The airborne inertial gravimetry system (AIGS) based on H-770 inertial system gives slightly better performance than that from LASEREF III system due to the higher output rate of the raw data of H-770 strapdown inertial system. • One advantage of the airborne inertial gravimetry system is that AIGS could provide very stable and
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reliable airborne gravity measurements due to the robustness of the high performance strapdown inertial system, such as Honeywell H-770 inertial system. • Because of the data acquisition procedure, the airborne gravity measurements from the NEXTMap USA program are not tied to the ground gravity network. In order to mitigate the above limitation of the airborne gravity measurements from the NEXTMap USA program a solution combing the available airborne gravity measurements of the NEXTMap USA program and the airborne gravity data from the airborne relative gravimeter, such as L&R gravimeter could be considered • In order to control the bias between different blocks the additional control lines are introduced. They should be principally perpendicular to the primary lines of the existing gravity data from the NEXTMap program and could be flexible in terms of the line configuration, length and spacing. The airborne gravity data along the control lines are collected using the airborne gravimeter, e.g. L&R gravimeter, which is normally tied to the ground gravity network. Applying the least squares adjustment for the crossover points between the primary lines of the existing gravity data and the control lines of the gravity data from the L&R gravimeter, the gravity bias of the existing gravity data could be estimated and corrected.
Reference 1. Schwarz KPO, Colombo G. Hein and Knickmeyer ET (1991): Requirements for airborne vector gravimetry, Proc. of IAG Symposia 110 From mars to Greenland: Charting Gravity with Space and Airborne Instruments, Vienna 1991, Springer 2. Brozena JM (1991) The Greenland aerogeophysics project: Airborne gravity, topographic and magnetic mapping of an entire continent, International Association of Geodesy Symposium No. 110, Springer 1992 3. Forsberg R (1993) Impact of Airborne gravimetry on geoid determination – the Greenland example. Bull Int Geoid Ser 2:32–43
363 4. Brozena JM and Peters MF (1994) State-of-art airborne gravimetry, International Association of Geodesy Symposia No. 113, Springer pp 187–197 5. Gleason DM (1992) Extracting gravity vectors from the integration of Global Positioning System and Inertial navigation System data. J Geophys Res 97(B6):8853–8864 6. Schwarz KP and We M (1994): Some unsolved problems in airborne gravimetry, IAG Symposium “Gravity and Geoid”, Graz, Austria, Sept. 11–17, 1994, pp.131–150 7. Jekeli C (1995) Airborne vector gravimetry using precise, position-aided inertial measurement units. Bulletin Ge´ode´sique 69(1):1–11 8. Wei M and Schwarz KP (1996) Comparison of different approaches to airborne gravimetry by strapdown INS/GPS, Proc. of International Symposium on Gravity, Geoid and Marine geodesy, Tokyo, Sep. 30–Oct. 5, 1996 9. Wei M, Schwarz KP (1998) Flight test results from a strapdown airborne gravity system. J Geod 72(6):323–332 10. Glennie CL and Schwarz KP (1997) Airborne gravity by strapdown INS/DGPS in a 100 km by 100 km area of the Rocky Mountains, Proc. of International Syposium on Kinematic Systems in Geodesy, Geometics and Navigation (KIS97), Banff, Alberta, Canada, June, 1997, pp 619–624 11. Glennie CL, Schwarz KP (1999) A comparison and analysis of airborne gravimetry results from two strapdown inertial/ DGPS systems. J Geod 73(6):311 12. Glennie CL, Schwarz KP, Bruton AM, Forsberg R, Olesen AV, Keller K (2000) A comparison of stable platform and strapdown airborne gravity. J Geod 74(5):383 13. Bruton AM, Schwarz KP, Ferguson S, Hammada Y, Halpenny J, Wei M (2000): A comparison of inertial platform, damped 2-axis platform and strapdown airborne gravimetry. In: Proceedings of International Symposium on Kinematics Systems in Geodesy, Geomatics and Navigation (KIS2001), ,June 5–8, 2001, Banff, Canada 14. Tennant JK, Wei M, Schwarz KP and Glennie C (1998) STAR-3i gravity mapping – California test results, Proc. 20th Remote Sensing Symposium, Calgary, Canada, May 10–13, 1998, pp 23–28 15. Wei M and Tennant JK (1999) Star-3i airborne geoid mapping system, Proc. of IAG Symposium, IAG General Assembly, Birmingham, UK, July 18–30, 1999 16. Wei M and Tennant JK (2000) Star-3i airborne gravity and geoid mapping system, International Symposium on Gravity, Geoid and Geodynamics 2000, Banff, Canada, July 31August 4, 2000 17. Airborne Gravity Data Production Report & Process Report: m3317 – California area, Intermap internal report (Airborne Gravity Data Production Report delivered to NGS), Oct, 2006. 18. Airborne Gravity Data Production Report & Process Report: m3419 – Mexico Gulf area, Intermap internal report (Airborne Gravity Data Production Report delivered to NGS), Nov., 2006
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Galathea-3: A Global Marine Gravity Profile
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G. Strykowski, K.S. Cordua, R. Forsberg, A.V. Olesen, and O.B. Andersen
Abstract
Between Aug 14, 2006 and Apr 24, 2007, and enjoying a considerable interest from the Danish authorities, the Danish public and the Danish media, the scientific expedition Galathea-3 circumnavigated the globe. Its domestic purpose was to attract the Danish youth to science. DTU Space, Technical University of Denmark, participated in the expedition with two experiments. From Perth, Western Australia to Copenhagen, Denmark the exact position and movements of the ship were monitored using a combination of GPS, INS and laser measurements. The purpose was to measure the instantaneous sea surface topography. This paper reports on the second experiment in which a continuous marine gravity profile along the ship’s route was measured. The focus of the paper is on the practical aspects of such large scale world wide operation and on the challenges of the data processing. Furthermore, the processed free-air gravity values are compared to three global models: EGM96, EGM08 and DNSC08. Even though the along-track resolution of marine data is higher than the resolution in any global gravity model (which influences the direct comparison of the collected marine data to the model) the statistics for the residual free-air gravity anomalies show, that EGM08 and DNSC08 are better models than EGM96 for all Galathea-3 legs. Some areas along the ships route are quite challenging for modellers.
44.1
Introduction
Between Aug 14, 2006 and Apr 24, 2007, the Danish Galathea-3 expedition circumnavigated the globe by sailing more than 60,000 km, see Fig. 44.1. This was
G. Strykowski (*) K.S. Cordua R. Forsberg A.V. Olesen O.B. Andersen DTU Space, National Space Institute, The Technical University of Denmark, Juliane Maries Vej 30, 2100 Copenhagen, Denmark e-mail:
[email protected]
the largest Danish scientific expedition in more than 50 years following two great expeditions of the past; Galathea-1 (1845–1847) and Galathea-2 (1950–1952). The overall political aim was to attract the Danish youth to the natural sciences. The scientific part of Galathea-3 consisted in total of 71 projects on board the navy surveillance vessel Vædderen (‘The Ram’) which was modified for the expedition. There were some 35 scientists on-board, a dozen of journalists, photographers and TV crew members, two high-school students and
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_44, # Springer-Verlag Berlin Heidelberg 2012
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Fig. 44.1 The Galathea-3 route from Copenhagen and around the world
a teacher, and a crew of 50 from The Royal Danish Navy.1 DTU Space participated in the Galathea-3 expedition from Perth, Western Australia to Copenhagen, Denmark measuring gravity and sea level heights along the ship’s route. The results of sea level height measurements using GPS, INS and laser were reported by Andersen et al. (2010). The gravity anomaly variations along the route mainly reflect the changes of the depth to the sea bottom. These depths are not always well known in parts of the world’s oceans; e.g. in the southern Pacific Ocean. In an independent Galathea-3 project the Danish Hydrographic Office charted the depths to the sea bottom and made the raw depth data available to the gravity project The timing of the Galathea-3 expedition was just prior to the release of the new global Earth gravity model EGM08 (Pavlis et al. 2008) and the associated background products. One such product is DNSC08 (Andersen and Knudsen 2009); a global gravity model from satellite altimetry for the marine areas which also includes a high resolution global bathymetry/topography model. Figure 44.1 shows that the ship’s route, e.g. in South America, is in near coastal areas; i.e. an area which is particularly challenging in constructing the
1
Web 1: http://www.galathea3.dk/uk
global gravity models. Consequently, such direct and independent measurements of gravity can serve as a valuable validation tool for the global models. Another possible use of such long marine gravity profiles is the possibility to validate the old marine gravity survey data. Usually, the data from each marine survey are processed independently and linked to some known reference gravity value on land through a harbor tie. If this reference gravity value is wrong or uncertain, or if the data processing is wrong, the whole gravity survey is biased or tilted. By crossing the old marine surveys with a consistent and modern marine survey, the relative biases of the old marine surveys with respect to each other can be detected. Section 44.2 contains a short description of the setup of the gravity system on-board Galathea-3. The semi-automatic procedure for the data collection and the handling of the data gaps in both gravity and GPS data are briefly described in Sect. 44.3. In Sect. 44.4 we shortly report on the harbor ties. In particular, we mention the practical aspects of collecting such measurements in remote parts of the world. Section 44.5 discusses the importance of the ship’s navigation, which in case of Galathea-3 expedition was not optimal. In Sect. 44.6 we briefly describe the changes in the marine gravity software that were implemented. In Sect. 44.7 we compare the results of the gravity data processing to three global gravity models: EGM96, EGM08 and DNSC08.
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Fig. 44.3 Two Javad GPS antennas (indicated by arrows) were installed on the roof of the bridge Fig. 44.2 The LC&R S-38 marine gravity meter in “the hangar”. The LC&R land gravity meter G-867, which was used for the harbour ties, is stored in an aluminium box and strapped to the lower shelf next to the marine meter
44.2
Gravity Measurements: The Setup
The Technical University of Denmark, DTU Space, The National Space Institute, participated in the Galathea-3 expedition with a gravity project. From Perth, Western Australia to Copenhagen, Denmark a ZLS Ultrasys LaCoste & Romberg (LC&R) SeaAir gravimeter, S-38 (a long-term loan from NGA, USA) was operated. It resulted in an almost continuous (see Sect. 44.3) marine gravity transect from the southern to the northern hemisphere. The meter occupied a small area, see Fig. 44.2, in the corner of “the hangar” on the main deck, i.e. a place where the helicopter could be stored when the vessel was operating as a naval surveillance ship. During the Galathea-3 expedition, the vessel was modified to accommodate the research facilities and “the hangar” was used as a meeting and lecture area. Horizontally, the gravimeter was placed as close as possible to the location of the centre of mass (CM) of the ship. Vertically, the gravimeter was on the main deck, i.e. above the CM of the vessel. Other locations were considered, but the placing of the meter closer to the CM in the vertical was not practical. Even though the navigation of the ship was constantly monitored by the central GPS antenna a set of two independent Javad antennas were placed on the roof of the bridge (see Fig. 44.3). The signal from the GPS antennas was collected on the bridge and merged
with the bathymetric signal provided by The Danish Hydrographic Office, see Sect. 44.1.
44.3
The Data Collection and the Data Gaps
The logistics of the expedition and various scientific projects on-board were organized in legs, i.e. in parts of the route between two harbour stays. A project span could be one or more of such legs. It is during the short and busy harbour stays that the crew could be replaced, the new project scientists could enter the vessel and the old projects could disembark. Any delay could affect other projects. It is under these conditions that the hardware for the two DTU Space projects was mounted in Fremantle, Western Australia. It took some time to make the system work and to do the harbour tie. Although prior to the Galathea-3 expedition there was a trial cruise in Denmark where the equipment could be tested, not everything could be planned ahead. Under such strict time constraints it required skill and a lot of experience to install the hardware and to solve the data collection problems. Concerning the gravity project, the gravimeter data and the navigation/bathymetry data were collected separately. The time stamp for the gravimeter data was the UTC time while for the navigation and bathymetry data it was the GPS time. There was at the time of expedition a time difference of 14 s between the two clocks. Furthermore, there is a time
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delay between the timing of the individual gravity measurements and the timing of their output as an averaged gravity value, see Sect. 44.6. The requirement was to take all the above into consideration when e.g. deciding on the sampling rate for the data collection. Ideally, for both DTU Space Galathea-3 projects, the data collection was semi-automatic, i.e. the data could be collected without the presence of DTU Space personnel. During the short harbour stay the system could be switched on and off, a harbour gravity tie could be made (see Sect. 44.4) and the data from the finished leg could be downloaded and stored. Thus, unlike for most other Galathea-3 projects, the main working load for the DTU Space projects was in the harbours. In practice, the DTU space personnel boarded the vessel roughly every second leg. For the “unmanned” survey legs the colleagues from The Danish Hydrographic Office were trained to download the data and to manage the switching off and restarting the system. In case of problems they could communicate by email with the trained personnel from DTU Space. Despite all these efforts to ensure the continuity in the data collection in a semi-automatic mode, there were problems in running the system with untrained surveillance which, unfortunately, led to some undesirable data gaps, see below. In the survey leg from Fremantle, Australia to Hobart, Australia, there is a data gap in the first part of the leg caused by the limited installation time in Fremantle. On leaving the harbour, the ad-hoc navigation collection system was not yet fully operational. Fortunately, the marine gravimeter was running all the time. For the “unmanned” leg between Sydney, Australia and Gizo, The Solomon Islands, the gravimeter was not collecting data at all. The software stopped collecting the data shortly after leaving the Sydney harbour without anybody noticing it. For the leg along Antarctica in the southern Pacific Ocean, and without the presence of DTU Space personnel, the gravimeter was clamped in the Argentinean territorial waters, because the expedition had no formal permission to collect data there. Crossing the Gulf Stream on a leg from St. Croix, The West Indies to Boston, USA, and without any DTU Space personnel on-board, the data were not collected. Crossing the Atlantic, from Boston, USA to Copenhagen, Denmark the gravimeter was switched off by accident.
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Furthermore, there were frequently drop outs in the collection of GPS data and in merging the depth data. In fact, the DTU Space staff when present frequently checked whether the GPS collection system and the merging with the depth data were running as expected. Consequently, a successful semi-automatic data collection certainly requires a more stable data collection system and, perhaps, more time for the installation. Also, the presence of trained personnel is desirable. The standard sampling rate for the gravimeter was 10 s, except in the West Indies; between the islands of St. Croix and St. Thomas where it was 1 s. The sampling rate in the West Indies was denser because the gravity data could support a high-resolution seismic project there.
44.4
The Harbour Ties
One important aspect of a marine gravity survey are the harbour ties, i.e. a calibration measurement which relates the marine gravimeter reading (in counter units, CU) to the absolute gravity value transferred from a known gravity station on shore. For a gravity survey circumnavigating the globe, such measurements are particularly challenging. They are difficult (if not impossible) without the help from the local authorities and local colleagues. To ensure the harbour ties along Galathea-3 route DTU Space has contacted a number of national and international institutions (Geoscience Australia (formerly AGSO); LINZ, New Zealand; Instituto Geogra´fico Militar, Chile) and few individuals who coordinate large scale gravity data collection (Steve Kenyon, NGA, USA; Denizar Blitzkow, Univ. Sa˜o Paulo, Brazil). With their help, the harbour ties could be measured in such remote places as the Solomon Islands and the Galapagos Islands. The harbour ties were constrained in time to the short harbour stay, see Sect. 44.2. A DTU Space LC&R land gravity meter G-867, see Fig. 44.2, was used. The meter was permanently stationed on the vessel for the duration of the Galathea-3 expedition. In Chile the measuring of the harbour ties was quite easy, but the high quality gravity stations in Valparaiso and Antofagasta were inside the military areas. Thus, in order to measure there a special permission was needed, which was facilitated by the Instituto Geogra´fico Militar. Other places it was
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Galathea-3: A Global Marine Gravity Profile
sometime unclear whether permission was needed and how to obtain it. In the Galapagos Islands a colleague from DTU Space almost got arrested for making a harbour tie without a formal permission. In Gizo, The Solomon Islands, the situation was also difficult, but mostly from the technical point of view. Firstly, the vessel was not anchored at the harbour but in the middle of a bay. Secondly, the reference land gravity station for a harbour tie was on a neighbouring island. It took 2 days to transfer the gravity value to both sides of the bay. A simple interpolation technique based on distance weighting to the two stations was used to make a harbour tie.
44.5
The Navigation
The modern marine gravity surveys are often conducted in connection with seismic surveys. The advantage is, that the marine seismic survey navigates in “straight lines” (i.e. in sections with a constant azimuth) and at a constant and moderate speed. This type of navigation is optimal for the gravity surveys because the E€otvos correction (see e.g. Torge 1989, Eq. 7–21a) is constant. The navigation of Galathea-3 expedition was mainly governed by the needs of other projects; i.e. holding still for taking samples of the sea bottom or sea water, or for fishing. At other times the vessel speeded up to catch up with the time delay. Another characteristic of Galathea-3 navigation was that there were no crossings points, i.e. that the vessel return to the same location at some later time. In marine surveys where gravity and seismic data are collected there is usually a substantial number of crossing line segments. The advantage of such configuration is that it provides a more robust estimation of the linear drift model for the marine gravimeter. If no such crossing points are available, the estimation of the drift model is based solely on the harbor ties corrected for the tidal signal. As stated in Sect. 44.2, in the Galathea-3 experiment two GPS antennas were mounted on the roof of the bridge of the vessel to yield an independent system. The antennas were mounted to the port side and the star side of the ship and placed such that no objects on the ship could shade for the antennas, see Fig. 44.3. This antenna configuration was more relevant for the sea surface topography project than for the gravity
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project. The idea was to monitor not only the movement of the centre of mass of the vessel, but also the tilt, roll and yaw of the ship and to correct for the deviation of the laser beam from the verticality (Andersen et al. 2010). In practice, for the Galathea-3 gravity project, and after some experiments in harbours, we found it redundant to use the star side antenna as it doubled the amount of input data without improving the gravity and sea surface topography information.
44.6
The Data Processing
The challenge of the Galathea-3 expedition forced us to revise the software previously used e.g. in processing survey data from Greenland and around the Faroe Islands. The main difference is that data previously had to be organized in line segments associated with “the seismic lines”. Each segment was individually inspected, corrected and went into the adjustment. The processing was quite laborious. Such a procedure was not applicable in the present case. The whole survey is just one long line, often without any crossing points, and stretching in time over many days. In the software bookkeeping we use Julian days, JD, for the time sequence, and, thus, precisely like in the standard Ultrasys LC&R data format for the raw gravimeter readings. The new software uses as input all the marine gravity data in one file ordered by the (UTC) time and the navigation/ depth data in another file ordered by the (GPS) time. Also, for each survey leg, the harbour ties are input into the software in a separate file. The software output is a file with all the processed marine gravity data for each survey leg. The new software can also process marine data from the old type of survey based on seismic lines (see above). The time difference between the two clocks (GPS and UTC, see Sect. 44.3) is handled by a time shift and the large data gaps in both the gravity and the GPS navigation data are identified. The gap in the GPS navigation data, while the gravimeter is still running, is not that serious. One can still assume the same instrumental drift model for the marine gravimeter across the gap. Due to the lack of the navigation data (i.e. the positions for the gravity stations) there will be no output data for this part of the survey, but the continuity of the gravity data across the gap is valid.
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One additional, problem is that the GPS navigation data are also used for modelling the E€ otvos correction (see below) filtered consistently with the gravity and bathymetry data. It takes few initial healthy GPS data before the filtered E€ otvos corrections can be used. More severe problems with data gaps are the gravimeter malfunction at sea, see Sect. 44.3. This violates the continuity in the gravity data collection and breaks the implicit assumption of a constant linear gravimeter drift between the harbours. The problem can to some degree be overcome (bias and tilt correction) using a comparison with the existing knowledge about other gravity in the area. The problem is similar to the problem of ensuring the results of a marine survey even when one or two harbour ties are missing. The drawback is that the survey results and the other existing gravity data from the area are no longer independent. After the data gaps were identified, the remaining healthy navigation and depth data were interpolated to the (UTC) timing of the marine data leading to a consistent data set: time, location, depth (if available), gravity reading. Subsequently, the navigation data and the time were used to model the fictitious accelerations affecting the gravimeter reading (the E€ otvos correction, see Torge 1989). One difficulty is that the standard Ultrasys output gravity data are averages in time so that the timing of the gravity output does not reflect the gravity reading for this navigation position but is an averaged value for prior positions. Thus, the output gravity data must be shifted backward in time to the correct location and the E€otvos correction must be filtered consistently. The filtering is done by a sequence of digital filters. The depth data, if known, must be filtered accordingly to yield the Bouguer anomaly. If the depth data are available the output is the free-air anomalies and Bouguer anomalies, if not, only free-air gravity anomalies are output. The Galathea-3 navigation data included raw depth records provided by The Danish Hydrological Office. The data transfer was sometimes unstable and the data themselves were often very noisy. A preparation of such data for the Bouguer gravity processing requires a cumbersome data cleaning the removal of the spikes, the smoothing, and often the interpolation. At the present, it was only done to the data from the leg
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between St Thomas and St Croix in the West Indies. We found it useful to use the global bathymetry models, e.g. DNSC08 (Andersen et al. 2008), as a reference. Especially when there was a doubt of which of the different (and shifted in the vertical) pieces of the raw depth information was correct. The Danish Hydrological Office in their project will provide a clean set of depth data. In practice, in the new software, there are some limitations to the size of the input data files. The processed Galathea-3 gravity data were processed leg by leg, and for some legs even split into smaller units (related to the data gaps). The data sampling was the standard 10 s, except for the St Thomas -St Croix leg when it was 1 s.
44.7
Comparison with the Global Models
The Galathea-3 free-air gravity anomalies were compared to three global gravity models (GM): EGM96 (Lemoine et al. 1998), EGM08 (Pavlis et al. 2008) and DNSC08 (Andersen et al. 2008). For practical reasons, i.e. because the number of data is quite large (1,261,833), we’ve computed GM gravity values using a dense grid (spacing NS WE: 0.025 0.025 ) for each survey leg. For each GM we used the maximal truncation degree Nmax (EGM96: Nmax ¼ 360; EGM08: Nmax ¼ 2,160). The model values DgGM were then interpolated from grids to the location of the gravity data Dg and subtracted. Table 44.1 shows the residual statistics for Dgres , where Dgres Dg DgGM . The marine data have a higher along-line spatial resolution than any GM. If the gravity field in some area varies a lot large values of min and max for the residuals can be expected. The GMs are by default too coarse to model such details. However, the GMs themselves, e.g. EGM96 vs. EGM08 or DNSC08, can also differ. This is clearly seen in e.g. the St Thomas– St Croix leg, a relatively small area where the EGM96 residual statistics are extremely bad, while the EGM08 and DNSC08 statistics look fine and consistent with Galathea-3 data. Thus, for this area EGM08 and DNSC08 improve the gravity information compared to EGM96 and Galathea-3 marine data confirm this independently.
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Table 44.1 Statistics of the residual free-air gravity anomalies (in mgals) for each Galathea-3 survey leg DNSC08 Mean Std. Perth-Hobart DgGM 10.8 26.9 Dgres 1.1 7.7 Hobart-Sydney DgGM 16.0 36.9 Dgres 0.9 6.5 Sydney-Gizo No data 11.1 37.2 Gizo-Christchurch DgGM Dgres 1.3 10.4 ChristchurchDgGM 7.7 31.5 Valparaiso Dgres 10.4 20.9 ValparaisoDgGM 59.5 75.0 Antofagasta Dgres 5.6 7.9 AntofagastaDgGM 45.3 54.8 Galapagos Dgres 0.2 8.3 Galapagos-St. Thomas DgGM 9.0 41.9 Dgres 1.7 8.4 St Thomas-St Croix DgGM 30.7 80.2 Dgres 0.2 12.6 St Croix – Boston DgGM 25.9 20.0 0.1 7.2 Dgres Boston-Copenhagen DgGM 13.9 24.7 Dgres 0.3 5.1
EGM08 Min Max Mean 100.5 62.1 10.7 51.7 66.3 1.0 92.9 52.5 16.2 31.1 37.6 1.1 No data 104.4 157.1 11.4 60.7 190.6 1.1 100.9 116.0 8.7 96.0 89.0 11.4 178.0 75.8 59.3 49.7 53.8 5.7 248.8 86.2 45.3 60.9 70.1 0.1 155.7 130.8 9.6 133.3 129.9 1.1 171.9 111.0 31.0 50.3 55.7 0.0 199.0 57.2 25.7 67.2 128.7 0.2 66.1 134.6 13.9 35.2 48.7 0.3
In general, the residual statistics are better for survey legs when DTU Space personnel were on-board. This could be improved in the future by providing a more robust data collection system. The Galathea-3 marine data are now ready for a detailed inspection, and will in the near future be enhanced with the depth information provided by the Danish Hydrographic Office so that the Bouguer gravity anomalies can also be computed.
Conclusions
In this paper we report on the practical aspects of collecting a very long marine gravity transect on a global scale, and the subsequent data processing. The navigation of the Galathea-3 expedition was not optimal for the gravity data collection. Nevertheless, useful gravity information was collected and processed. The measured data cross many interesting and challenging areas for the global gravity models and the processed data can be used as a valuable verification tool.
Std. 26.8 7.8 36.7 6.5 37.1 10.1 32.2 22.3 75.0 7.8 54.9 8.3 42.6 8.6 80.2 13.5 20.1 7.2 24.5 5.1
EGM96 Min Max Mean 99.6 63.6 8.3 53.2 66.2 1.4 91.7 52.7 13.2 30.5 35.3 2.0 No data 85.2 154.9 9.6 61.9 187.9 2.8 101.2 117.8 7.6 99.6 88.3 10.3 177.5 75.8 61.4 50.4 43.9 3.7 247.9 84.8 48.4 60.0 69.7 3.3 156.0 129.8 10.9 133.1 129.8 0.2 174.1 109.4 20.6 52.8 58.2 51.6 199.6 59.0 25.3 65.6 127.9 0.6 64.2 131.5 14.0 36.0 50.4 0.2
Std. 20.2 18.0 37.8 14.5 24.9 25.5 26.4 24.8 78.9 23.2 49.6 21.7 38.4 24.0 20.8 83.8 19.3 12.6 21.8 13.1
Min Max 96.2 42.6 56.7 81.2 76.3 48.2 46.0 31.4 51.2 70.8 88.8 120.2 172.5 101.3 211.1 103.6 98.1 181.8 26.0 184.5 201.0 86.3 63.2 49.2
95.4 151.7 76.3 103.4 93.3 53.5 94.2 90.0 112.2 124.3 69.2 143.9 22.6 123.1 76.4 110.7
Acknowledgements Dansk Expeditions fond for including DTU Space’s “Tyngdeprojektet” as a part of Galathea-3 expedition. This is an official Galathea-3 publication P51.
References Andersen OB, Knudsen P (2008) The DNSC08MSS global Mean Sea surface and mean dynamic topography from multi-mission radar altimetry, EGU2008-A-08267, Vienna, Austria Andersen OB, Knudsen P (2009) The DNSC08 mean sea surface and mean dynamic topography. J Geophys Res 114, C11, doi:10.1029/2008JC005179, 2009 Andersen OB, Olesen AV, Forsberg R, Strykowski G, Cordua K, Zhang X (2010) Ocean Dynamic Topography from GPS – Galathea-3 First results. Proceedings IAG International Gravity Symposium, Chania, Greece, 23–27 June 2008, IAG Symposia, Vol. 135, S.P. Mertikas (ed) Springer Verlag, pp 239–246 Lemoine et al (1998) The development of the Joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) Geopotential Model EGM96, NASA/TP-1998-206861 Pavlis NK, Holmes S, Kenyon SC and Factor JK (2008) An earth gravitational model to degree and order 2160: EGM2008, presented EGU, Vienna, April Torge W (1989) Gravimetry, de Gruyter
Dependency of Resolvable Gravitational Spatial Resolution on Space-Borne Observation Techniques
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P.N.A.M. Visser, E.J.O. Schrama, N. Sneeuw, and M. Weigelt
Abstract
The so-called Colombo-Nyquist (Colombo, The global mapping of gravity with two satellites, 1984) rule in satellite geodesy has been revisited. This rule predicts that for a gravimetric satellite flying in a (near-)polar circular repeat orbit, the maximum resolvable geopotential spherical harmonic degree (lmax) is equal to half the number of orbital revolutions (nr) the satellite completes in one repeat period. This rule has been tested for different observation types, including geoid values at sea level along the satellite ground track, orbit perturbations (radial, along-track, cross-track), low-low satellite-to-satellite tracking, and satellite gravity gradiometry observations (all three diagonal components). Results show that the Colombo–Nyquist must be reformulated. Simulations indicate that the maximum resolvable degree is in fact equal to knr + 1, where k can be equal to 1, 2, or even 3 depending on the combination of observation types. However, the original rule is correct to some extent, considering that the quality of recovered gravity field models is homogeneous as a function of geographical longitude as long as l max < nr/2.
45.1
Introduction
Colombo (1984) has indicated that for exact satellite circular repeat orbits and for continuous space-borne gravimetric observations, the normal matrix of gravity field spherical harmonic (SH) coefficients becomes block-diagonal when organized per SH order. The
P.N.A.M. Visser (*) E.J.O. Schrama Faculty of Aerospace Engineering, Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, 2629 HS, Delft, The Netherlands e-mail:
[email protected] N. Sneeuw M. Weigelt Institute of Geodesy, University of Stuttgart, GeschwisterScholl-Str. 24D, 70174, Stuttgart, Germany
correlation between different orders is zero as long as one can avoid overlapping frequencies, which is generally guaranteed if the maximum resolvable SH degree (lmax) is less than half the number of orbital revolutions nr which the satellite completes in a repeat period of nd nodal days, or lmax < nr/2 (Schrama 1990). Although Sneeuw (2000) has pointed out that avoiding overlapping frequencies is fundamentally a restriction on the maximum SH order. Nevertheless this has led to the rule-of-thumb that the maximum resolvable degree is equal to nr/2, referred to as the Colombo–Nyquist rule. This rule has major implications for the design of future gravity field missions, where several trade-offs have to be made, such as temporal and spatial resolution, the observation/decoupling of different sources of gravity field
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changes, etc. (Bender et al. 2008; Reubelt et al. 2010; Visser and Schrama 2005). Also, this rule has implications for designing efficient gravity field estimation schemes taking advantage of the structure of normal matrices (Schrama 1991). It has to be noted that the maximum resolvable degree is defined as the maximum SH degree for which also all coefficients with SH orders complete to this maximum degree can be resolved. It is thus not precluded that certain individual coefficients with a higher SH degree can be resolved, however with a SH order that is not higher. The Colombo–Nyquist rule-of-thumb has been tested for a number of mission scenarios, i.e. different repeat orbits and combinations of observables. It is shown that this rule needs to be reformulated. The selected mission scenarios are outlined in Sect. 45.2. The method used for establishing the maximum resolvable degree for these mission scenarios is briefly described in Sect. 45.3. Results are presented in Sect. 45.4 and summarized in Sect. 45.5.
45.2
Mission Scenarios
The selected repeat orbits and observable types are listed in Table 45.1. The orbits are polar to ensure global coverage. A repeat orbit is specified by the number of revolutions nr that is completed in nd nodal days, where nr and nd do not have common prime factors (except 1). Short repeat periods ranging from 1 to 3 days have been selected to limit the computational burden. These short repeat periods are however sufficient to test the validity of the Colombo–Nyquist rule. Different parities for nr and nd were selected to assess the possible impact on the maximum resolvable degree of the number of distinct equator crossings. For nr nd even the number of
Table 45.1 Selected polar repeat orbits and observation techniques. The time interval between observations is always taken equal to 1 s Repeat period nd (days) Number of revolutions nr Height (km) 1 15 554.25 2 31 404.35 3 46 453.41 3 47 356.16 Observation technique Precision level Geoid 1 cm Orbit 1 cm ll-SST 1 mm SGG 0.01 E
equator crossings is equal to nr, whereas this is 2nr for nr nd odd (Fig. 45.1). The observable types include geoid values at sea level along the satellite ground path (closely related to altimeter observations), orbit perturbations in the radial, along-track and cross-track direction, low-low satellite-to-satellite tracking (ll-SST) range observations, and satellite gravity gradient (SGG) observations (the diagonal components, where the gradiometer instrument is aligned with the radial, alongtrack, and cross-track direction). The observations are assumed to be provided continuously with a constant time step of 1 s. The relation between SH gravity field coefficients and observations is given by wellestablished and tested transfer functions (e.g. Schrama 1991; Sneeuw 2000; Visser 1992, 2005; Visser et al. 1994, 2001, 2003). These transfer functions are used to set up the observation equations, which are to be solved by the weighted least-squares method (Sect. 45.3). The observations are assigned weights in accordance with the precision levels listed in Table 45.1.
Fig. 45.1 Ground track pattern for polar repeat orbits, where nr/nd is equal to respectively 15/1 (left) and 31/2 (right)
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375
Fig. 45.2 Structure of normal matrix for gravity field coefficients complete to degree and order 40 for a nr/nd ¼ 15/1 polar repeat orbit based on ll-SST observations (“kite matrix”). Zero values are indicated by white and non-zero by gray color
Fig. 45.3 Condition number of the normal equations (left) and global RMS formal geoid error as a function of the maximum retrieved spherical harmonic degree. Use is made of geoid observations at sea level
45.3
Estimating the Maximum Resolvable Spherical Harmonic Degree
For a repeat orbit, Colombo (1984) indicated that when a least-squares estimation method is used and if a continuous time series of observations is obtained with constant time interval, the normal matrix for the SH coefficients will become block-diagonal when
organized per order, and correlations between different orders will be equal to zero as long as the maximum resolvable degree is below nr/2. For higher degrees, different orders get correlated and the normal matrix adopts a Kite-like structure (e.g. Fig. 45.2). The question is addressed if still a stable gravity field solution can be obtained in the presence of these correlations, thereby assuming that no use is made of prior knowledge and/or regularization. This is tested by computing the condition number of this matrix
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lmax = 7
lmax = 8
Fig. 45.4 Formal geoid error as a function of the geographical location for geoid observations along a nr/nd ¼ 15/1 repeat orbit Table 45.2 Formal global geoid error (mm) (RMS, minimum and maximum) and the ratio of maximum and minimum geoid error at the equator (req) for nr/nd ¼ 15/1 and 31/2 repeat orbits. For lmax < nr/2 the error is always constant as a function of longitude RMS Obs. lmax nr/nd ¼ 15/1-repeat Geoid 7 0.2632 Geoid 8 0.3976 Geoid 15 1.7342 ll-SST 7 0.0023 ll-SST 8 0.0027 ll-SST 15 0.0076 nr/nd ¼ 31/2-repeat Geoid 15 0.3810 Geoid 16 0.4092 Geoid 32 1.4984 ll-SST 15 0.0033 ll-SST 16 0.0036 ll-SST 31 0.0108
req
Minimum
Maximum
1.00 1.08 12.54 1.00 1.00 1.37
0.1333 0.1419 0.1924 0.0013 0.0014 0.0059
0.3200 0.9105 6.4213 0.0028 0.0034 0.0109
1.00 1.00 1.00 1.00 1.00 1.00
0.1350 0.1392 0.1922 0.0013 0.0014 0.0041
0.4724 0.5314 4.9795 0.0038 0.0042 0.0139
(ratio of maximum and minimum eigenvalue) and by computing the Root-Mean-Square (RMS) of the cumulative global formal geoid commission error for the estimated SH coefficients. The formal geoid errors were taken from the inverse (if the normal matrix is invertible) of the weighted normal matrix. In all cases, normal equations were set up for all SH coefficients from degree 2 to a certain maximum degree lmax. Thus the impact of omission and/or aliasing of unmodeled gravity field sources are not taken into account. The exercises described in this paper only address the issue of observability of a static gravity field complete to the maximum SH degree solved for.
45.4
Gravity Field Observability
As a first test case, the condition numbers of the normal matrix and associated geoid error were computed for nr/nd ¼ 46/3 and nr/nd ¼ 47/3 repeat orbits using geoid observations along the ground track. The condition numbers display a large jump at lmax ¼ nr (Fig. 45.3, left) and in fact the normal matrix could not be inverted for higher degrees (no formal geoid errors could be estimated, Fig. 45.3, right). For lmax ¼ nr/2, a small jump in the condition number occurs due to the additional correlations between different SH orders, but this does not lead to an unstable normal matrix. Also, the slope of the geoid error increases for lmax > nr/2. Based on these results, it can already be concluded that the maximum resolvable degree can be as big as nr and does not depend on the parity of nr and nd. It is interesting to note that as long as lmax < nr/2, the geoid error is only latitude dependent and does not change with longitude, whereas for lmax > nr/2 the correlations between different orders cause the geoid error to change as a function of longitude as well (Fig. 45.4). The variation of the geoid error as a function of latitude and longitude depends on the observable. For a nr/nd ¼ 15/1 repeat orbit and geoid observations, the minimum and maximum formal geoid error is equal to 0.19 and 6.42 mm for a gravity field recovery complete to degree and order 15, i.e. a ratio of 34, compared 0.0059 and 0.0109 or a ratio of 1.8 for ll-SST observations (Table 45.2). Figures 45.5 and 45.6 display the condition numbers of the normal matrix and formal geoid error
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377
nr /nd =15/1 polar repeat orbit
nr /nd =31/2 polar repeat orbit
Fig. 45.5 Condition number of the normal equations as a function of the maximum retrieved spherical harmonic degree and the observation technique (the minimum degree is equal to 2)
estimates for gravity field recoveries up to lmax ¼ 50 for the nr/nd ¼ 15/1 repeat orbit, i.e. lmax > 3nr + 1, and up to lmax ¼ 65 for the nr/nd ¼ 31/2 repeat period, i.e. lmax > 2nr +1. The observable types include (1) geoid values, (2) radial orbit perturbations, (3) alongtrack orbit perturbations, (4) orbit perturbations in all directions (3D), (5) along-track diagonal gravity gradient component (Гxx), (6) cross-track diagonal gravity gradient component (Гyy), (7) radial diagonal gravity gradient component (Гxx), and (8) all three diagonal gravity gradient components (Гxx+yy+zz). It can be observed that for one-directional observables, such as geoid values, radial orbit perturbations, along-track perturbations, and one diagonal of the gravity gradients, the condition numbers display in general small jumps at nr/2 and large jumps
at nr + 1. The same can be observed for the associated geoid error estimate (provided the normal matrix was invertible). In other words, for such one-directional observables it seems like the maximum resolvable SH degree is equal to the number of revolutions nr + 1 in a repeat period. When using ll-SST observations, combinations of orbit perturbations (3D) or combinations of SGG diagonal components, the normal matrix is stable up to at least lmax ¼ 2nr + 1. For the 3D combination of orbit perturbations, the condition number and associated formal geoid error estimate stays stable for lmax + 1 up to 3nr + 1, whereas for the combination of all three diagonal SGG components, this is still 2nr + 1. Two questions that might now immediately be raised is why this is not 3nr + 1 for the combination
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nr /nd =15/1 polar repeat orbit
nr /nd =31/2 polar repeat orbit
Fig. 45.6 Global RMS formal geoid error from the inverse of the normal equations as a function of the maximum retrieved spherical harmonic degree and the observation technique (the minimum degree is equal to 2)
of three SGG components as well and why it is 2nr + 1 for ll-SST observations, which is a one-directional observation type, namely along the line-of-sight between two trailing satellites. Concerning the SGG observations, it can be argued that the three diagonal components are not independent because the gravitational potential satisfies the Laplace equation, or Gxx + Gyy + Gzz ¼ 0. Thus one diagonal SGG components can always be written as a linear combination of the other two. Thus, in fact only two independent components remain. Concerning the ll-SST observations, it can be argued that these observations are a modulated combination of along-track and radial orbit perturbations (Visser 2005), assuming the two associated satellites fly in the same orbital plane.
Conclusions
Computations have shown that the Colombo–Nyquist rule in satellite geodesy, which predicts that the maximum resolvable degree is equal to half the number of orbital revolutions nr in a repeat period of nd nodal days, requires revision. Colombo’s rule is correct in the sense that block-diagonal matrices are formed when lmax < nr/2 and when organized per SH order, with no correlations between the orders. Colombo’s rule is in general too pessimistic to infer statistical significance of SH coefficients in a gravity field model, i.e. solutions are possible where lmax nr/2 as is discussed in this paper. If the maximum degree of
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Dependency of Resolvable Gravitational Spatial Resolution
estimated SH coefficients is larger than nr/2, the gravity field solution will however no longer be homogeneous in the longitude direction for even parities of nr and nd. However, the Colombo–Nyqyist rule can be considered to be correct to some extent. Namely, as stated in the previous paragraph, the quality of recovered gravity field models is always homogeneous as a function of geographical longitude as long as lmax < nr/2. It was also found that the maximum resolvable degree does not depend on the parity of the number of revolutions and nodal days in a repeat orbit, but that the recovery error as a function of longitude does vary due to the increasing ground track density when traveling away from the equator. Finally, the maximum resolvable degree depends on the (combination of) observable type(s). In case of combinations of independent observables, this maximum degree can be one, two or three times the number of orbital revolutions in a repeat period (plus 1 if the minimum SH degree is taken equal to 2). Fortunately, in general gravity satellites carry a complement of observing instruments, including always GPS receivers in addition to for example ll-SST instruments or a gradiometer.
References Bender PL, Wiese DN, Nerem RS (2008) A possible dualGRACE mission with 90 degree and 63 degree inclination orbits. In: ESA (ed) 3rd International symposium on formation flying, missions and technologies, 23–25 April 2008, ESA/ESTEC, Noordwijk, pp 1–6
379 Colombo OL (1984) The global mapping of gravity with two satellites, vol 7, no 3, Publications on geodesy, New Series. Netherlands Geodetic Commission, Delft Reubelt T, Sneeuw N, Sharifi MA (2010) Future Mission Design Options for Spatio-Temporal Geopotential Recovery. International Association of Geodesy Symposia. Vol. 135, Springer, pp 163–170 Schrama EJO (1990) Gravity field error analysis: applications of GPS receivers and gradiometers on low orbiting platforms, NASA technical memorandum 100679. GSFC, Greenbelt, MD Schrama EJO (1991) Gravity field error analysis: applications of GPS receivers and gradiometers on low orbiting platforms. J Geophys Res 96(B12):20041–20051 Sneeuw N (2000) A semi-analytical approach to gravity field analysis from satellite observations. PhD dissertation, Deutsche Geod€atische Kommission, Reihe C, Nr. 527 Visser PNAM (1992) The use of satellites in gravity field determination and model adjustment. PhD dissertation, Delft University of Technology, Delft Visser PNAM (2005) Low-low satellite-to-satellite tracking: applicability of analytical linear orbit perturbation theory. J Geod 79(1–3):160–166 Visser PNAM, Schrama EJO (2005) Space-borne gravimetry: how to decouple the different gravity field constituents? In: Jekeli C et al (eds) Gravity, geoid and space missions, vol 129, International association of geodesy symposia. Springer, Berlin, pp 6–11 Visser PNAM, Wakker KF, Ambrosius BAC (1994) Global gravity field recovery from the ARISTOTELES satellite mission. J Geophys Res 99(B2):2841–2851 Visser PNAM, van den IJssel J, Koop R, Klees R (2001) Exploring gravity field determination from orbit perturbations of the European Gravity Mission GOCE. J Geod 75(2/3):89–98 Visser PNAM, Sneeuw N, Gerlach C (2003) Energy integral method for gravity field determination from satellite orbit coordinates. J Geod 77(3/4):207–216
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A Comparison of Different IntegralEquation-Based Approaches for Local Gravity Field Modelling: Case Study for the Canadian Rocky Mountains
46
R. Tenzer, I. Prutkin, and R. Klees
Abstract
We compare the accuracy of local gravity field modelling in rugged mountains using three different discretised integral equations; namely (1) the single layer approach, (2) Poisson’s integral approach, and (3) Green’s integral approach. The study area comprises a rough part of the Canadian Rocky Mountains with adjacent plains. The numerical experiment is conducted for gravity disturbances and for topographically corrected gravity disturbances. The external gravity field is parameterized by gravity disturbances (Poisson’s integral approach) and disturbing potential values (Green’s integral approach), both discretised below the data points at the same depth beneath the Bjerhammar sphere. The point masses in the single layer approach are discretised below the data points on a parallel surface located at the same depth beneath the Earth’s surface. The accuracy of the gravity field modelling is assessed in terms of the STD of the differences between predicted and observed gravity data. For the three chosen discretisation schemes, the most accurate gravity field approximation is attained using Green’s integral approach. However, the solution contains a systematic bias in mountainous regions. This systematic bias is larger if topographically corrected gravity disturbances are used as input data.
46.1
Introduction
For the gravimetric geoid/quasigeoid modelling from regional gravity data, the two-step approach is often used in practice (e.g., Bjerhammar 1962, 1987;
R. Tenzer (*) School of Surveying, Faculty of Sciences, University of Otago, 310 Castle Street, Dunedin, New Zealand e-mail:
[email protected] I. Prutkin R. Klees Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, 2629 Delft, The Netherlands
Martinec 1996). It involves the inverse of the Poisson integral equation and consequently the Stokes/Hotine integration. An alternative method of computing the disturbing potential from regional gravity data was formulated in Nova´k (2003). This method combines the solution of the first and second/third boundaryvalue problems and directly relates observed gravity with the disturbing potential values by means of Green’s integrals. We call this method Green’s integral approach. Nova´k et al. (2003) applied this method to geoid modelling from airborne gravity data. Alberts and Klees (2004) compared various integral-equationbased approaches with least-squares collocation in the context of quasigeoid modelling from airborne gravity
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data. The objective of this study is to investigate the performance of the single layer, Poisson’s integral and Green’s integral approaches in rugged mountain areas. In particular, we want to answer the question whether topographically corrected gravity data provide better solutions than uncorrected gravity data. Moreover, edge effects are investigated. The study area and data sets are specified in Sect. 46.2. The discretised integral equations are defined in Sect. 46.3. The results of the numerical analysis re presented and discussed in Sect. 46.4. The conclusions are given in Sect. 46.5.
46.2
Study Area and Data Sets
The target area is situated in the Canadian Rocky Mountains with adjacent plains, bounded by the parallels of 50 and 60 arc-deg Northern spherical latitude and the meridians of 240 and 250 arc-deg Eastern spherical longitude. The 5 5 arc-min mean orthometric heights are computed by spatial averaging from the 3 3 arc-sec detailed digital terrain model (provided by of the Geodetic Survey Division of Natural Resources of Canada). The heights vary from 174 to 2,898 m (see Fig. 46.1). The three different data areas are chosen to investigate the edge effects extending the target area in all directions by 1, 2 and 3 arc-deg. The data sets consist of mean gravity disturbances (Fig. 46.2) and mean topographically corrected gravity
Fig. 46.1 Topography of the target area
disturbances (Fig. 46.3), both provided on a 5 5 arcmin grid located at the Earth’s surface. The 5 5 arcmin mean gravity disturbances are compiled from the gravity data of the Geodetic Survey Division of Natural Resources of Canada. The topographically corrected gravity disturbances are obtained from the gravity disturbances after subtracting the total contribution of topography assuming homogeneous mass density of 2670 kg/m3. The procedure of computing the total contribution of topography on gravity is described in Vajda et al. (2008).
46.3
Discretised Integral Equations
After linearization and spherical approximation, the discretised integral equations are formulated for the gravity disturbance dg at the data point r in the following form dgðrÞ ¼
I X
bðr0 i Þ Cðr; r0 i Þ;
(46.1)
i¼1
where fCðr; r0 i Þ : i ¼ 1; 2; :::; Ig are the discretised integral equation functionals, and bðr0 i Þ are the coefficients which parameterize the gravity field at the positions r0 i . In the single layer approach, Cðr; r0 i Þ is a function of the radial derivative of Newton’s kernel:
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A Comparison of Different Integral-Equation-Based Approaches
383
Fig. 46.2 The mean gravity disturbances at the Earth’s surface (statistics: min ¼ 126.8 mGal, max ¼ 120.3 mGal, mean ¼ 10.2 mGal, STD ¼ 24.2 mGal)
Fig. 46.3 The mean topographically corrected gravity disturbances at the Earth’s surface (statistics: min ¼ 339.6 mGal, max 159.3 mGal, mean ¼ 219.2 mGal, STD ¼ 36.7 mGal)
0
Cðr; r i Þ ¼ G
T r0 jrj jr0 i j ^ r ^ jr r0 i j3
jr0 i j2 DOðr0 i Þ;
(46.2)
where G denotes Newton’s gravitational constant; jr r0 i j is the Euclidean spatial distance; and 0 ^ r ¼ r=jrj and ^ r ¼ r0 =jr0 j are unit vectors in the direction of r and r’, respectively; DOðr0 i Þ is the surface
element of the unit sphere, DO ¼ cos ’ Dl D’; where D’ and Dl are discretisation steps in latitude and longitude, respectively. The parameterization of the gravity field is given by the point masses discretised at the positions r0 i located at a constant depth of 10 km below the data points. In Poisson’s integral approach, Cðr; r0 i Þ is a function of the Poisson kernel (e.g. Kellogg 1929):
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Cðr; r0 i Þ ¼
1 jrj2 jr0 i j2 0 2 jr i j DOðr0 i Þ: 4p jrj jr r0 i j3
(46.3)
The parameterization is given by the gravity disturbances discretised at the positions r0 i below the data points at a constant depth of 10 km beneath the Bjerhammar sphere. The radius of the Bjerhammar sphere is set equal to 6,371 km. The depth 10 km approximately equals the mean distance between the data points. The choice of the depth is usually done empirically as a trade-off between fit to the data and smoothness of the solution. As we demonstrated in Tenzer and Klees (2008), the optimal depth should be found based on the analysis of accuracy of the gravity field approximation. In Green’s integral approach, Cðr; r0 i Þ is a function of the negative radial derivative of the Poisson kernel (Nova´k 2003, see also Starostenko 1978):
accuracy is very similar for all three data area extensions. The STD fit of gravity disturbances is 2.3 mGal, and the STD fit of topographically corrected gravity disturbances is 1.4 mGal. The application of the topographical correction to the gravity data and consequently the smoothing of the high-frequency part of gravity signal thus improved the STD fit by 39%. The geographic plot and histogram of the residuals of uncorrected gravity disturbances at the data points are shown in Fig. 46.4. We observe a strong correlation of residual amplitudes with the topography. The largest residuals are located in mountainous regions and attain absolute values up to 23 mGal. The accuracy over the flat terrain, where gravity is smooth, is significantly better; the residuals are below 5 mGal.
" T 1 jrj2 jr0 i j2 3jr0 i j r0 Cðr; r i Þ ¼ jrj jr0 i j ^ r ^ 5 0 4p jr r i j 0
2jrjjr0 i j jr r0 i j3
# DOðr0 i Þ:
(46.4)
The parameterization is given by the disturbing potential values discretised at the positions r0 i below the data points at a constant depth of 10 km beneath the Bjerhammar sphere. The number of unknown parameters is identical to the number of input gravity data. To reduce the size of the design matrix, the system of discretised integral equations is formed only for the near zone, while the far-zone contribution is disregarded. The systems of discretised integral equations for the near zone are solved using the Jacobi iteration scheme (e.g., Young 1971). The optimal regularization parameter is estimated using the minimization of the RMS differences between predicted and observed values, and the regularization matrix is the identity matrix.
46.4
Numerical Experiment
The accuracy of gravity field approximation is assessed at the data points within the target area. The results of the single layer approach show that the
Fig. 46.4 Geographic map and histogram of the gravity disturbance residuals at the data points for the 3 arc-deg data area extension and the single layer approach (statistics: min 22.7 mGal, max ¼ 22.0 mGal, mean ¼ 0.0 mGal, STD ¼ 2.3 mGal)
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Fig. 46.5 Geographic map and histogram of the residuals of topographically corrected gravity disturbances at the data points for the 3 arc-deg data area extension and the single layer approach (statistics: min ¼ 11.0 mGal, max ¼ 10.7 mGal, mean ¼ 0.0 mGal, STD ¼ 1.4 mGal)
The STD fit of topographically corrected gravity disturbances at the data points is shown in Fig. 46.5. The largest residuals (up to 11 mGal) are attained in mountainous regions. Over flat terrain, the residuals do not exceed 2.5 mGal. The results of Poisson’s integral approach show that the accuracy of the gravity field approximation at the data points is very similar for all three data area extensions. The STD fit of gravity disturbances is 3.6 mGal, and the STD fit of topographically corrected gravity disturbances is 2.0 mGal. The application of the topographical correction to the observed gravity data improves the STD fit by 44%. This is a slightly better improvement of the STD fit than for the single layer approach (39%). For the approximation of
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gravity disturbances at the data points, the STD fit of the single layer approach is 36% better than the STD fit of Poisson’s integral approach. Similarly, a 30% improvement of the STD fit of topographically corrected gravity disturbances at the data points is attained when using the single layer approach. This significant improvement of the accuracy is due to the choice of discretisation at a constant depth of 10 km below the data points. The diagonal components of the design matrix are equal and consequently all unknown parameters (point masses) are estimated with a similar accuracy. When the same discretisation is used as for Poisson’s integral approach (i.e., below the data points at a constant of 10 km beneath the Bjerhammar sphere), the accuracy of the gravity field approximation for both types of the integral equation approaches is very similar. We explain this by the fact that the behaviour of the Poisson kernel and the radiallydifferentiated Newton’s kernel is almost identical. The STD fit of gravity disturbances at the data points is shown in Fig. 46.6. The largest residuals are located in mountainous regions where they attain absolute values up to 42 mGal. The accuracy over the flat terrain is considerably better; the residuals are mostly below 10 mGal. The STD fit of topographically corrected gravity disturbances at the data points is shown in Fig. 46.7. The largest residuals are again located in mountainous regions where they reach up to 20 mGal and decrease substantially to less than 4 mGal over the flat terrain. The accuracy of Green’s integral approach for three different data area extensions is summarized in Tables 46.1 and 46.2. Green’s integral approach better approximates uncorrected than topographically corrected gravity disturbances. For the 3 arc-deg data area extension for instance, the application of topographical correction to the gravity data worsens the STD fit by 50%. This is in contrary to what we found when using the single layer and Poisson’s integral approaches where the application of the topographical correction to the gravity data smoothed the high-frequency part of gravity signal and consequently improved the STD fit. There is also evidence for a systematic bias in the approximation of uncorrected gravity disturbances. This systematic bias is due to the non-uniqueness of the solution of Green’s integral equation, caused by transforming gravity to potential and consequently introducing a low-frequency component of which the radial
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Fig. 46.6 Geographic map and histogram of the gravity disturbance residuals at the data points for the 3 arc-deg data area extension and Poisson’s integral approach (statistics: min ¼ 34.6 mGal, max ¼ 41.3 mGal, mean ¼ 0.0 mGal, STD ¼ 3.6 mGal)
Fig. 46.7 Geographic map and histogram of the residuals of topographically corrected gravity disturbances at the data points for the 3 arc-deg data area extension and Poisson’s integral approach (statistics: min ¼ 19.5 mGal, max ¼ 15.2 mGal, mean ¼ 0.0 mGal, STD ¼ 2.0 mGal)
Table 46.1 Statistics of the residuals of gravity disturbances for three data area extensions; Green’s integral approach
Table 46.2 Statistics of the residuals of topographically corrected gravity disturbances for three data area extensions; Green’s integral approach
Data area extension 1 arc-deg 2 arc-deg 3 arc-deg
Min (mGal) 1.4 2.5 3.9
Max (mGal) 1.2 2.3 3.7
Mean (mGal) 0.1 0.1 0.2
STD (mGal) 0.2 0.2 0.3
derivative equals zero. The systematic bias further magnifies after applying the topographical correction to the gravity data due to introducing a large longwavelength signal; the magnitude of the mean value increases from 10.2 mGal (uncorrected gravity disturbances) to 219.2 mGal (topographically corrected gravity disturbances). For both types of gravity disturbances, the systematic bias increases
Data area extension 1 arc-deg 2 arc-deg 3 arc-deg
Min (mGal) 2.8 4.3 5.8
Max (mGal) 0.9 0.1 0.7
Mean (mGal) 0.4 1.1 1.9
STD (mGal) 0.3 0.5 0.6
with increasing data area. The approximation of gravity disturbances at the data points using Green’s integral approach for the 3 arc-deg data area extension is shown in Fig. 46.8. The largest residuals are located in mountainous regions where they have some systematic pattern but reach not more than 4 mGal. The accuracy over the flat terrain is better
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A Comparison of Different Integral-Equation-Based Approaches
Fig. 46.8 Geographic map and histogram of the gravity disturbance residuals at the data points for the 3 arc-deg data area extension and Green’s integral approach (statistics see Table 46.1)
than 1 mGal. The approximation of topographically corrected gravity disturbances at the data points for the 3 arc-deg data area extension is shown in Fig. 46.9. All residuals in mountainous regions are below 6 mGal in magnitude and have a systematic negative trend which decreases over the flat terrain.
46.5
Summary and Conclusions
We have compared the accuracy of gravity field approximation using three integral-equation-based approaches. Poisson’s and Green integral equations were discretised at the same depth of 10 km beneath the Bjerhammar sphere. The point masses (single layer approach) were discretised at a constant depth of 10 km beneath the Earth’s surface. This type of
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Fig. 46.9 Geographic map and histogram of the residuals of topographically corrected gravity disturbances at the data points for the 3 arc-deg data area extension and Green’s integral approach (statistics see Table 46.2)
discretisation improved the STD fit of the single layer approach comparing to Poisson’s integral approach particularly in mountainous regions. It is superior to the traditional approach of using a spherical surface as support of the single layer density. The most accurate approximation of the gravity field was however attained when using Green’s integral approach. Tenzer and Klees (2008) demonstrated that almost the same accuracy of local gravity field modelling can be achieved for different types of the integral kernels if the depth of the parameterization surface is chosen optimally. The selection of the optimal depth always needs to be done with care for every integral kernel facilitating the analysis of accuracy of the gravity field approximation at a set of control or data points. As shown in Klees et al. (2007) and
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Tenzer and Klees (2008), it can be done by either Generalized Cross Validation or RMS minimization techniques. The application of topographical correction to the gravity data significantly improved the STD fit of Poisson’s and single layer approaches. In contrary, Green’s integral approach better approximates the uncorrected gravity disturbances than the topographically corrected gravity disturbances. This is due to the presence of a systematic bias particularly in mountainous regions which further magnifies after applying the topographical correction to gravity data. This systematic bias is due to the non-uniqueness of the solution of Green’s integral equation which transforms gravity into potential values.
References Alberts B, Klees R (2004) A comparison of methods for the inversion of airborne gravity data. J Geod 78:55–65 Bjerhammar A (1962) Gravity reductions to a spherical surface. Royal Institute of Technology, Division of Geodesy, Stockholm
R. Tenzer et al. Bjerhammar A (1987) Discrete Physical Geodesy. Report 380, Dept. of Geodetic Science and Surveying, The Ohio State University, Columbus Kellogg OD (1929) Foundations of potential theory. Springer, Berlin Klees R, Tenzer R, Prutkin I, Wittwer T (2007) A data-driven approach to local gravity field modelling using spherical radial basis functions. J Geod 82:457–471 Martinec Z (1996) Stability investigations of a discrete downward continuation problem for geoid determination in the Canadian Rocky Mountains. J Geod 70:805–828 Nova´k P (2003) Geoid determination using one-step integration. J Geod 77:193–206 Nova´k P, Kern M, Schwarz KP, Sideris MG, Heck B, Ferguson S, Hammada Y, Wei M (2003) On geoid determination from airborne gravity. J Geod 76:510–522 Starostenko VI (1978) Stable computational approaches in gravimetry problems. Kiev, Naukova dumka Tenzer R, Klees R (2008) The choice of the spherical radial basis functions in local gravity field modelling. Stud Geoph Geod 52:287–304 Vajda P, Ellmann A, Meurers B, Vanı´cˇek P, Nova´k P, Tenzer R (2008) Global ellipsoid-referenced topographic, bathymetric and stripping corrections to gravity disturbance. Stud Geoph Geod 52(1):19–34 Young D (1971) Iterative solutions of large linear systems. Academic, New York
Global Topographically Corrected and Topo-Density Contrast Stripped Gravity Field from EGM08 and CRUST 2.0
47
R. Tenzer, Hamayun, and P. Vajda
Abstract
We compute globally the topographically corrected and topo-density contrast stripped gravity disturbances and gravity anomalies taking into account the major known density variations within the topography. The topographical and topo-density contrast stripping corrections are applied to the EGM08 gravity field quantities in two successive steps. First, the gravitational contribution of the topography of constant average density 2,670 kg/m3 is subtracted. Then the ice, sediment, and upper crust topo-density contrast stripping corrections are applied to the topographically corrected gravity field quantities in order to model the gravitational contribution due to anomalous density variations within the topography. The coefficients of the global geopotential model EGM08 complete to degree 180 of spherical harmonics are used to compute the gravity disturbances and gravity anomalies. The 5 5 arc-min global elevation data from the ETOPO5 are used to generate the global elevation coefficients. These coefficients are utilized to compute the topographical correction with a spectral resolution complete to degree and order 180. The 2 2 arc-deg global data of the ice, sediment, and upper crust from the CRUST 2.0 global crustal model are used to compute the ice, sediment, and upper crust topo-density contrast stripping corrections with a 2 2 arc-deg spatial resolution. All data are evaluated globally on a 1 1 arc-deg grid at the Earth’s surface.
47.1 R. Tenzer (*) School of Surveying, Faculty of Sciences, University of Otago, 310 Castle Street, Dunedin, New Zealand e-mail:
[email protected] Hamayun Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, 2629, HS Delft, The Netherlands P. Vajda Geophysical Institute, Slovak Academy of Sciences, Du´bravska´ cesta 9, 845 28, Bratislava, Slovak Republic
Introduction
The constant average topographical density is commonly assumed when modelling either the longwavelength topographical correction or the far-zone contribution of the topography-generated gravitational field (e.g., Nova´k et al. 2001; Tenzer et al. 2003; Nova´k and Grafarend 2005). Recently, lateral topographical density distribution models have been used more often in detailed modelling of the near-zone contribution and the constant average topographical density has been adopted for the far-zone contribution
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(e.g., Martinec et al. 1995; K€ uhtreiber 1998; Huang et al. 2001; Hunegnaw 2001; Sj€ oberg 2004). In regions with variable geological structures, large sedimentary basins and continental ice sheet, however, the effect of the anomalous density variations on the gravitational field from the far-zone topography can still be significant. Therefore, we expect that the currently available global crustal model CRUST 2.0 (Bassin et al. 2000) allows computing the far-zone contributions more realistically taking into the account the ice, sediment and upper crust density contrasts within the topography. Nevertheless, large errors in modelling the global no-topography gravity field are still expected mainly due to inaccuracies of the CRUST 2.0 model. The global topographical and topo-density contrast stripping corrections are computed in Sect. 47.2. The global topographically corrected and topo-density contrast stripped gravity disturbances and gravity anomalies are presented in Sect. 47.3. The influence of the anomalous topo-density distribution on the global geoid is not investigated in this study. The summary and conclusions are given in Sect. 47.4. The application of isostatic compensation or topographic condensation to the gravity field quantities for reducing a large primary indirect topographical effect on the geoid is out of the scope of this study. For more details we refer the readers to Heck (2003) and Martinec (1998).
47.2
Global Topographical and Topo-Density Contrast Stripping Corrections
The 5 5 arc-min global elevation data from the ETOPO5 (provided by the NOAA’s National Geophysical Data Centre) are used to generate the Global Elevation Model (GEM) coefficients. These coefficients are utilized to compute globally the topographical corrections to the gravity field quantities with a spectral resolution complete to degree and order 180. The average topographical density 2,670 kg/m3 is adopted (cf. Hinze 2003). The gravitational potential V t generated by the topography of constant average density ro is computed using the following expression
V t ðr; OÞ ¼ 2p G ro n h X
N X
1 2n þ1 n¼0
i 2R Hn;m þ ð1 nÞ H2n;m Yn;m ðOÞ;
m¼n
þ 2p G ro
N H ðO Þ X n R n¼1 2n þ 1
n h i X 2R Hn;m þ ð1 nÞ H2n;m Yn;m ðOÞ
m¼n
þ p G ro
n h X
N H ðOÞ2 X n ð n 1Þ 2 R 2n þ 1 n¼2
i 2R Hn;m þ ð1 nÞ H2n;m Yn;m ðOÞ;
m¼n
(47.1) where G is Newton’s gravitational constant, HðOÞ is the height of the computation point, R is the mean earth’s radius, Hn;m and H2n;m are the GEM coefficients, Yn;m ðOÞ are the surface spherical functions (e.g., Heiskanen and Moritz 1967), and N is the upper summation index for computing the topography-generated gravitational field (i.e., N ¼ 180). The 3-D position is defined by the geocentric spherical coordinates r, ’, l; r is the geocentric radius, ’ and l are the spherical latitude and longitude, O ¼ ð’; lÞ. Similarly, the topography-generated gravitational attraction gt reads gt ðr; OÞ ¼ 2p G ro
N 1 X n R n¼1 2n þ 1
n h i X 2R Hn;m þ ð1 nÞ H2n;m Yn;m ðOÞ m¼n
N H ðOÞ X n ðn 1Þ 2 R n¼2 n h i X 2R Hn;m þ ð1 nÞH2n;m Yn;m ðOÞ 2n þ 1
þ 2p G ro
m¼n N H ðOÞ2 X n ðn 1Þðn 2Þ 3 2n þ 1 R n¼3 n h i X 2RHn;m þð1nÞH2n;m Yn;m ðOÞ:
þ p G ro
(47.2)
m¼n
Alternative expressions for modelling the topography-generated gravitational field from the spectral
47
Global Topographically Corrected and Topo-Density Contrast Stripped
coefficients can be found for instance in Vanı´cˇek et al. (1995), Nova´k and Grafarend (2006), and Nova´k (2010). The topographical correction to gravity disturbances is defined as the negative direct topographical effect. It is computed on a 1 1 arc-deg geographical grid at the Earth’s surface and the result is shown in Fig. 47.1. The complete topographical correction to gravity anomalies comprises not only the direct topographical effect but also the secondary indirect topographical effect. The secondary indirect topographical effect is defined as the topographygenerated gravitational potential V t multiplied by the term 2r1 , where r is the geocentric radius of the computation point. The complete topographical correction to gravity anomalies computed at the Earth’s surface is shown in Fig. 47.2. The discrete data of the ice thickness and elevations with a 2 2 arc-deg geographical resolution from the CRUST 2.0 are used to generate the Global Ice-Thickness Model (GITM) coefficients and the Global Lower-Bound Ice Model (GLBIM) coefficients. These coefficients are utilized to compute globally the ice topo-density contrast stripping corrections with a spectral resolution complete to degree and
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order 90. The mean value of the ice density contrast 1,757 kg/m3 is adopted. It is defined as the difference between the mean ice density 913 kg/m3 and the average topographical density 2,670 kg/m3. The gravitational potential and attraction generated by the ice density contrast are computed according to (47.1) and (47.2) modified for the GITM and GLBIM coefficients. The ice topo-density contrast stripping corrections to gravity disturbances and gravity anomalies are shown in Figs. 47.3 and 47.4. The 2 2 arc-deg discrete data of the sediment thickness and density from the CRUST 2.0 are used to compute globally in a spatial representation the sediment topo-density contrast stripping corrections. The sediment density contrast is defined relative to the average topographical density of 2,670 kg/m3. The sediment topo-density contrast stripping corrections to gravity disturbances and gravity anomalies are shown in Figs. 47.5 and 47.6. The 2 2 arc-deg discrete data of the density and thickness of the upper crust from the CRUST 2.0 are used to compute globally in a spatial representation the upper crust topo-density contrast stripping corrections. The upper crust density contrast is defined
Fig. 47.1 The global topographical correction to gravity disturbances computed with a spectral resolution complete to degree and order 180
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Fig. 47.2 The global topographical correction to gravity anomalies computed with a spectral resolution complete to degree and order 180
180°
240°
300°
0°
60°
120°
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60°
60°
0°
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-60°
-60°
180°
240°
300°
0°
60°
120°
180°
250
300
mGal 0
50
100
150
200
Fig. 47.3 The global ice topo-density contrast stripping correction to gravity disturbances computed with a spectral resolution complete to degree and order 90
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Global Topographically Corrected and Topo-Density Contrast Stripped 180°
240°
300°
0°
393 120°
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-60°
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240°
300°
0°
120°
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180° mGal
-50
50
0
100
150
200
Fig. 47.4 The global ice topo-density contrast stripping correction to gravity anomalies computed with a spectral resolution complete to degree and order 90
180°
240°
300°
0°
60°
120°
180°
60°
60°
0°
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-60°
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240°
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0°
10
20
60°
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180° mGal
-20
-10
0
30
40
50
60
Fig. 47.5 The global sediment topo-density contrast stripping correction to gravity disturbances computed with a 2 2 arc-deg spatial resolution
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0°
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180°
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-50
-40
240°
-30
300°
-20
-10
0°
0
120°
60°
10
20
30
180°
40
mGal 50
Fig. 47.6 The global sediment topo-density contrast stripping correction to gravity anomalies computed with a 2 2 arc-deg spatial resolution
relative to the average topographical density of 2,670 kg/m3. The upper crust topo-density contrast stripping corrections to gravity disturbances and gravity anomalies are shown in Figs. 47.7 and 47.8. Statistics of the global topographical and topodensity contrast stripping corrections to gravity disturbances and gravity anomalies are summarized in Tables 47.1 and 47.2. Statistics of the global secondary indirect topographical and topo-density contrast stripping effects are given in Table. 47.3.
47.3
Global Topographically Corrected and Topo-Density Contrast Stripped Gravity Field
The global geopotential coefficients taken from the EGM08 complete to degree 180 of spherical harmonics are used to compute the gravity disturbances dg and the gravity anomalies Dg. The computation is realized globally on a 1 1 arc-deg geographical grid at the Earth’s surface. The expressions for
computing the gravity field in terms of spherical harmonics can be found for instance in Heiskanen and Moritz (1967) (Chaps. 2–17). The statistics of the EGM08 gravity field quantities are given in Tables 47.4 and 47.5. The topographically corrected gravity disturbances dgT and the topographically corrected gravity anomalies DgT are obtained from the EGM08 gravity field after applying the corrections due to the topography of average topographical density 2,670 kg/m3. The results are shown in Figs. 47.9 and 47.10. The topographically corrected and topo-density contrast stripped gravity disturbances dgNT and the topographically corrected and topo-density contrast stripped gravity disturbances DgNT are obtained from the corresponding topographically corrected gravity field quantities after applying the ice, sediment and upper crust topo-density contrast stripping corrections. The results are shown in Figs. 47.11 and 47.12. Statistics of the step-wise topographically corrected and topo-density contrast stripped gravity disturbances and gravity anomalies computed at the earth’s surface are summarized in Tables 47.4 and 47.5.
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Global Topographically Corrected and Topo-Density Contrast Stripped 180°
300°
240°
0°
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395 120°
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-60°
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-3
240°
0
300°
0°
60°
3
6
9
120°
180° mGal 15
12
Fig. 47.7 The global upper crust topo-density contrast stripping correction to gravity disturbances computed with a 2 2 arc-deg spatial resolution 180°
240°
300°
0°
120°
60°
180°
60°
60°
0°
0°
-60°
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0
2
240°
4
6
300°
0°
8
10
60°
12
120°
14
16
180°
18
mGal 20
Fig. 47.8 The global upper crust topo-density contrast stripping correction to gravity anomalies computed with a 2 2 arc-deg spatial resolution
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Table 47.1 Statistics of the topographical and topo-density contrast stripping corrections to gravity disturbances computed at the Earth’s surface Correction Topographical Ice Sediment Upper crust
Min (mGal) 619.1 2.5 24.9 3.1
Max (mGal) 71.4 300.3 59.1 14.4
Mean (mGal) 15.7 21.9 3.1 0.7
STD (mGal) 91.5 56.2 4.4 1.3
Table 47.2 Statistics of the complete topographical and topodensity contrast stripping corrections to gravity anomalies computed at the eEarth’s surface Correction Topographical Ice Sediment Upper crust
Min (mGal) 373.9 53.6 39.6 0.6
Max (mGal) 261.8 199.2 47.0 18.4
Mean (mGal) 104.1 1.4 4.4 2.1
STD (mGal) 65.3 36.7 3.6 1.6
Table 47.3 Statistics of the secondary indirect topographical and topo-density contrast stripping effects computed at the Earth’s surface Secondary indirect effect Topographical Ice Sediment Upper crust
Min (mGal) 74.2 109.8 16.1 0.9
Max (mGal) 278.4 10.0 4.8 4.0
Mean (mGal) 119.8 23.3 7.5 1.4
STD (mGal) 36.7 23.0 2.3 0.4
47.4
Summary and Conclusions
We have computed the global gravity field quantities, the global topographical corrections for the average topographical density, and the global topo-density contrasts stripping corrections due to the major known density contrasts within the topography. The CRUST 2.0 global crustal model and the global
Table 47.4 Statistics of the step-wise topographically corrected and topo-density contrast stripped gravity disturbances computed at the Earth’s surface Gravity disturbances Min (mGal) dg 303.2 548.7 dgT dgNT 530.5
Max (mGal) 292.6 302.9 307.3
Mean (mGal) 0.7 16.4 9.3
STD (mGal) 29.4 94.2 56.4
Table 47.5 Statistics of the step-wise topographically corrected and topo-density contrast stripped gravity anomalies computed at the Earth’s surface Gravity anomalies Dg DgT DgNT
Min (mGal) 282.3 275.8 277.5
Max (mGal) 286.2 375.4 364.4
Fig. 47.9 The global topographically corrected gravity disturbances computed at the Earth’s surface
Mean (mGal) 0.5 103.6 99.9
STD (mGal) 23.7 64.5 41.9
47
Global Topographically Corrected and Topo-Density Contrast Stripped
397
Fig. 47.10 The global topographically corrected gravity anomalies computed at the Earth’s surface
Fig. 47.11 The global topographically corrected and topo-density contrast stripped gravity disturbances computed at the Earth’s surface
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Fig. 47.12 The global topographically corrected and topo-density contrast stripped gravity anomalies computed at the Earth’s surface
geopotential model EGM08 complete to spherical harmonic degree 180 were used as the input data. The topo-density contrasts of the ice, sediment and upper crust are taken relative to adopted value of the average topographical density 2,670 kg/m3. The topo-density contrast stripping corrections of these CRUST 2.0 model components have been applied to the topographically corrected gravity field in order to obtain a more realistic representation of the global no-topography gravity field. As seen from results in Figs. 47.9–47.12, the application of the topo-density contrast stripping corrections considerably changed the topographically corrected gravity disturbances and gravity anomalies. The largest changes up to several hundreds of mGals at the regions with the polar ice sheet of Greenland and Antarctica are due to applying the ice topo-density contrast stripping correction. The smaller changes up to several dozens of mGals at the regions with large continental sediment basins and variable geological structure are due to applying the sediment and upper crust topo-density contrast stripping corrections.
References Bassin C, Laske G, Masters G (2000) The current limits of resolution for surface wave tomography in North America. EOS. Trans AGU 81:F897 Heck B (2003) On Helmert’s methods of condensation. J Geod 77(3–4):155–170 Heiskanen WH, Moritz H (1967) Physical geodesy. W.H., Freeman and Co, San Francisco Hinze WJ (2003) Bouguer reduction density, why 2.67? Geophysics 68(5):1559–1560 Huang J, Vanı´cˇek P, Pagiatakis SD, Brink W (2001) Effect of topographical density on the geoid in the Canadian Rocky Mountains. J Geod 74:805–815 Hunegnaw A (2001) The effect of lateral density variation on local geoid determination. Bollettino di geodesia e scienze affini 60(2):125–144 K€uhtreiber N (1998) Precise geoid determination using a density variation model. Phys Chem Earth 23(1):59–63 Martinec Z (1998) Boundary-value problems for gravimetric determination of a precise geoid. Springer, Berlin Martinec Z, Vanı´cˇek P, Mainville A, Veronneau M (1995) The effect of lake water on geoidal height. Manusc Geod 20:193–203 Nova´k P (2010) High resolution constituents of the Earth gravitational field. Surv Geoph, 31(1), pp. 1–21, doi:10.1007/ s10712-009-9077-z
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Nova´k P, Grafarend EW (2005) The ellipsoidal representation of the topographical potential and its vertical gradient. J Geod 78(11–12):691–706 Nova´k P, Grafarend EW (2006) The effect of topographical and atmospheric masses on spaceborne gravimetric and gradiometric data. Stud Geoph Geod 50(4):549–582 Nova´k P, Vanı´cˇek P, Martinec Z, Veronneau M (2001) Effects of the spherical terrain on gravity and the geoid. J Geod 75 (9–10):491–504
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Sj€oberg LE (2004) The effect on the geoid of lateral topographic density variations. J Geod 78(1–2):34–39 Tenzer R, Vanı´cˇek P, Nova´k P (2003) Far-zone contributions to topographical effects in the Stokes-Helmert method of the geoid determination. Stud Geoph Geod 47(3):467–480 Vanı´cˇek P, Najafi M, Martinec Z, Harrie L, Sj€oberg LE (1995) Higher-degree reference field in the generalised Stokes-Helmert scheme for geoid computation. J Geod 70:176–18
.
Local Gravity Field Modelling in Rugged Terrain Using Spherical Radial Basis Functions: Case Study for the Canadian Rocky Mountains
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R. Tenzer, R. Klees, and T. Wittwer
Abstract
We analyze the performance of a least-squares approximation of the gravity field using Spherical Radial Basis Functions (SRBFs) in rugged mountains. The numerical experiment is conducted for the gravity disturbances and for the topographically corrected gravity disturbances, both provided on a 5 5 arc-min grid located at the Earth’s surface. The target area is a rough part of the Canadian Rocky Mountains with adjacent planes. The data area and the parameterization area extend the target area in all directions by 3 arc-deg. The accuracy of the gravity field approximation is investigated using a SRBF parameterization (Poisson wavelet of order 3) on different spherical equal-angular grids with step sizes varying between 6 and 12 arc-min. For every choice of grid, the optimal depth of SRBFs bellow the Bjerhammar sphere is found using the minimization of the RMS differences between predicted and observed values. The results of the numerical experiment reveal that the application of the topographical correction to the observed gravity data reduces the number of SRBFs by more than 56%, and improves the fit to the data by 12%. Unfortunately, it also introduces a systematic bias in the adjusted gravity disturbances.
48.1
Objective
Various types of spherical radial basis functions (SRBFs) were adopted for a parameterization of the earth’s gravity field such as the point mass kernel (Weightman 1965), the radial multipoles of different
R. Tenzer (*) School of Surveying, Faculty of Sciences, University of Otago, 310 Castle Street, Dunedin, New Zealand e-mail:
[email protected] R. Klees T. Wittwer Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, 2629 Delft, The Netherlands
orders (Marchenko 1998), Poisson wavelets of different orders (Holschneider et al. 2003), and the Poisson kernel. Tenzer and Klees (2008) demonstrated that almost the same accuracy of gravity field modelling can be achieved for different types of SRBFs if the bandwidth of the SRBFs is chosen optimally. Klees et al. (2007) developed a novel least-squares approach for local gravity field modelling using SRBFs. This approach utilizes a data-driven strategy to select automatically the horizontal positions and depths of the SRBFs using Generalized Cross Validation (GCV) (Wahba 1977). Variance Component Estimation (VCE) techniques are utilized for observation group weighting and selection of the optimal regularization parameter (Kusche 2003). The application of this approach to real data in
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Germany and the Netherlands showed a very good performance of SRBFs in flat terrain with a smooth gravity signal (cf. Klees et al. 2007; Tenzer et al. 2009). In this study we investigate the performance of this approach in rugged mountains with large gravity signal variations. In particular, we want to investigate the effect of applying a topographic correction on the number of SRBFs and the quality of the SRBF representation of the data. The study area and input data sets are specified in Sect. 48.2. The results of the numerical experiment are presented and discussed in Sect. 48.3, and the conclusions are given in Sect. 48.4.
48.2
Data Sets
The target area is located in the Canadian Rocky Mountains with adjacent planes to the East, bounded by the parallels of 50 and 60 arc-deg Northern spherical latitude and the meridians of 240 and 250 arc-deg Eastern spherical longitude. The 5 5 arc-min mean orthometric heights are computed by spatial averaging from the 3 3 arc-sec detailed digital terrain model (provided by of the Geodetic Survey Division of Natural Resources of Canada). The heights vary from 174 to 2,898 m (see Fig. 48.1). To reduce edge effects, the data area and the parameterization area extend the target area in all directions by 3 arc-deg. The data sets consist of mean gravity
Fig. 48.1 Topography of the target area
disturbances and mean topographically corrected gravity disturbances, both provided on a 5 5 arc-min grid located at the earth’s surface. The 5 5 arc-min mean gravity disturbances are compiled from the gravity data of the Geodetic Survey Division of Natural Resources Canada. The topographically corrected gravity disturbances are obtained from the gravity disturbances after subtracting the total contribution of topography of average topographical density 2,670 kg/m3 (cf. Hinze 2003). The computation of the topographical correction on gravity is described in Vajda et al. (2008). Each data set comprises 36,864 mean gravity values. Within the target area, the mean gravity disturbances vary from 126.8 to 120.3 mGal; the mean is 10.2 mGal and the standard deviation is 24.2 mGal (Fig. 48.2). The topographically corrected gravity disturbances vary from 339.6 to 159.3 mGal; the mean is 219.2 mGal and the standard deviation is 36.7 mGal (Fig. 48.3). The correlation of gravity disturbances and topographically corrected gravity disturbances with the topography is shown in Figs. 48.4 and 48.5, respectively. Pearson’s (1896) correlation coefficient is used to compute the correlation between gravity data and topography within the target area. The correlation of gravity disturbances with the topography is 0.62; the correlation of topographically corrected gravity disturbances with the topography is 0.88. This high negative correlation is due to the isostatic
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Fig. 48.2 The mean gravity disturbances at the earth’s surface
Fig. 48.3 The mean topographically corrected gravity disturbances at the earth’s surface
compensation in the Canadian Rocky Mountains. To quantify the smoothing effect of the topographical correction on the high-frequency part of gravity signal, we compare horizontal gravity gradients on a 5 5 arc-min grid. The mean horizontal gradient of gravity disturbances is 3.3 mGal/km, while only 1.6 mGal/km for the topographically corrected gravity disturbances. Hence, the topographically corrected gravity disturbances have a higher signal standard deviation
than the uncorrected ones, but are significantly smoother.
48.3
Numerical Experiment
The accuracy of the SRBF least-squares approximation of the gravity field is investigated using a SRBF parameterization on different spherical equal-angular
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Fig. 48.4 The correlation between the gravity disturbances at the earth’s surface and the topography
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grids with step sizes varying between 6 and 12 arcmin. We use the Poisson wavelets of order 3 (Holschneider et al. 2003). The number of SRBFs for different grid sizes is summarized in Table 48.1. The number of SRBFs represents between 18% (12 arc-min grid) and 70% (6 arc-min grid) of the number of data points. For each grid size, the optimal depth of the SRBFs bellow the Bjerhammar sphere is determined using the minimization of the RMS differences between predicted and observed gravity values. The radius of
the Bjerhammar sphere is set equal to 6,371 km. No regularization is applied. The optimal depths for each SRBF parameterization are shown in Fig. 48.6. There is an almost linear relationship between SRBF grid size and optimal depth down to a grid spacing of 6 arc-min. As the grid gets finer, the SRBFs are located at a shallower depth. Further reducing the grid size may lead to two problems: (1) an ill-conditioned normal equation matrix, because neighboured SRBFs overlap too much or (2) an overoptimistic fit of the model to the data and
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consequently a poor solution between the data points. Figure 48.6 also shows that the optimal depth is almost the same for topographically corrected and uncorrected gravity disturbances, though the latter contain more energy at the higher frequencies than the former. We explain this by the fact that these frequencies are not completely captured by the SRBF grid, but would need additional SRBFs to be located in areas with significant residual signal (cf. Klees et al. 2007). The numerical experiment also reveals that the RMS fit of the model to the data is very sensitive with respect to the choice of the depth. Therefore, looking for the optimal depth requires a dense sampling of the search space of candidate depths. This is different from what Table 48.1 Distance between SRBFs (grid spacing) and number of SRBFs. The parameterization area extends the target area in all directions by 3 arc-deg Distance between SRBFs (deg) 0.20 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.10
Fig. 48.6 The optimal depths of SRBFs below the Bjerhammar sphere as a function of the distance between the SRBFs for gravity disturbances dg and topographically corrected gravity disturbances dgT
Number of SRBFs 6,480 7,225 7,921 9,025 10,100 11,449 13,225 15,252 17,956 21,316 25,760
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has been reported by Tenzer and Klees (2008) for experiments with gravity data in flat to hilly terrain, where the fit of the model to the data was found to be relatively robust with respect to the choice of the depth; fixing the depth within an interval of a few kilometres does not significantly change the RMS of the least-squares residuals. Figure 48.7 shows the RMS of least-squares residuals as a function of the SRBF grid size. The best solution in terms of the RMS of least-squares residuals of gravity disturbances is attained for the 6 arc-min grid size. Decreasing the grid size below 6 arcmin does not further improve the gravity field solution. The corresponding RMS of the least-squares residuals is 4 mGal. A grid of 6 arc-min contains 25,760 SRBFs (cf. Table 48.1), which is about 70% of the number of data points. This is an unfavourable ratio of number of SRBFs to the number of data points. It can be explained by the roughness of gravity disturbances; more SRBFs are needed to improve the fit of the model to the data over the areas with large high-frequency gravity signal variations. For gravity data in flat to hilly regions, Tenzer and Klees (2008) found typical ratios of 25–30%. In this study, the optimal depth of the SRBFs is 40.5 km below the Bjerhammar sphere; the corresponding bandwidth of the Poisson wavelets of order 3 at this depth is about 12 km. When using topographically corrected gravity disturbances as input data, the optimal grid size is 9 arc-min. Hence, the number of SRBFs (11,449) is
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Fig. 48.7 RMS of leastsquares residuals of gravity disturbances dg and topographically corrected gravity disturbances dgT at the data points as a function of the distance between the SRBFs
only 31% of the data points, more favourable than without topographic correction. Moreover, Fig. 48.7 also reveals that the model fit to the data is more robust against the choice of the grid size than when using uncorrected gravity disturbances. The RMS of leastsquares residuals for the 9 arc-min grid is 3.5 mGal; this is slightly smaller than for uncorrected gravity disturbances (4.0 mGal). The depth of the SRBFs is 69.4 km, which corresponds to a bandwidth of about 21 km for the Poisson wavelet of order 3. The increased bandwidth of the SRBFs for topographically corrected gravity disturbances is in agreement with what can be expected for a gravity field which is smoother than a gravity field without topographic correction. As the gravity field gets smoother, the optimal bandwidth of the SRBFs will get broader and vice versa. The comparison of the least-squares approximation of uncorrected and topographically corrected gravity disturbances shows that the application of the topographical correction to gravity data significantly reduces the number of SRBFs, by about 56% and improves the fit of the model to the data by about 12%. We also note that the same accuracy of about 4.6 mGal and the same optimal depth of SRBFs of about 44 km are attained for both types of gravity disturbances when the parameterization is realized by the 7 arc-min SRBF grid (cf. Fig. 48.7). A geographic plot of the least-squares residuals for uncorrected gravity disturbances and the 6 arc-min
grid is shown in Fig. 48.8. We observe a strong correlation of residual amplitudes and topography. As the terrain gets rougher, the residuals are larger. Note that the number of SRBFs is already 70% of the number of data points, which limits the space for significant improvements by locating additional SRBFs in these areas. Test computations confirm this: adding more SRBFs has only little effect on the model fit to the data. For example, when we further increased the number of SRBFs up to 80% of the number of data points, the RMS of least-squares residuals improved less than 0.1 mGal. This is an argument in favour of the application of the topographic correction when SRBFs are used in gravity field modelling. Over the flat areas, the least-squares residuals are mostly bellow 5 mGal. Figure 48.9 shows a geographic plot of the leastsquares residuals of topographically corrected gravity disturbances for the 9 arc-min solution. The largest residuals are again located in mountainous regions where they reach up to a few dozen mGals. Despite the fact that the model fit to the data over the flat terrain is significantly better compared with uncorrected gravity disturbances, systematic bands in the residuals are evident over the whole target area, which are not visible in the solution based on uncorrected gravity disturbances. This is because the topographically corrected gravity disturbances have more power at long wavelengths. In particular, the application of the topographic correction increased the magnitude
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Fig. 48.8 Geographic map and histogram of the leastsquares residuals for uncorrected gravity disturbances and a 6 arc-min grid of SRBFs (statistics: min ¼ 35.5 mGal, max ¼ 39.8 mGal, mean ¼ 0.1 mGal, RMS ¼ 4.0 mGal)
of the mean value from 10.2 mGal (uncorrected gravity disturbances) to 219.2 mGal (topographically corrected gravity disturbances)! This large long-wavelength signal introduced when applying the topographical correction can be avoided if isostatic compensation is taken into account or if another topographic correction such as the residual terrain model (RTM) method (Forsberg and Tscherning 1981) is used. Another feature is a negative systematic bias in the residuals for topographically corrected gravity disturbances of about 2.6 mGal. We believe that this bias can also
be attributed to the topographical correction. The topographical bias in gravity data can be largely reduced by subtracting a higher-degree reference gravity field from the gravity data.
48.4
Summary and Conclusions
We have investigated the performance of a SRBF parameterization for local gravity field modelling in rugged terrain using gravity disturbances with and
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Fig. 48.9 Geographic map and histogram of the leastsquares residuals for topographically corrected gravity disturbances and a 9 arc-min grid of SRBFs (statistics: min ¼ 32.1 mGal, max ¼ 27.8 mGal, mean ¼ 2.6 mGal, RMS ¼ 3.5 mGal)
without topographic correction. The main result of the study is that the topographic correction reduces the number of SRBFs by about 56%, which is a consequence of the smoothing effect of the topographic correction on the high-frequency part of the gravity signal. At the same time, the fit of the model to the data is improved by about 12% in terms of RMS. Moreover, we found that in rugged terrain, the model fit to the data is very sensitive with respect to the choice of the depth down to the level of several 100 m. Therefore, when choosing the optimal depth of the SRBFs, one has to search the space of candidate depths sufficiently dense
to find the optimal solution. We also found that the topographic correction has to be handled with care: depending on the level of isostatic compensation, long-wavelength signals in the gravity disturbances are introduced when applying topographic correction, which makes it more difficult to find a suitable parameterization using SRBFs. A possible solution to this problem is the use of a high-degree global reference gravity model in the remove-restore procedure in combination with the RTM method. This would also reduce the bias we found in the adjusted topographically corrected gravity disturbances.
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References Forsberg R, Tscherning CC (1981) The use of height data in gravity field approximation by collocation. J Geoph Res 86:7843–7854 Hinze WJ (2003) Bouguer reduction density, why 2.67? Geophysics 68(5):1559–1560 Holschneider M, Chambodut A, Mandea M (2003) From global to regional analysis of the magnetic field on the sphere using wavelet frames. Phys Earth Planet Int 135:107–124 Klees R, Tenzer R, Prutkin I, Wittwer T (2007) A data-driven approach to local gravity field modelling using spherical radial basis functions. J Geod 82:457–471 Kusche J (2003) A Monte-Carlo technique for weight estimation in satellite geodesy. J Geod 76:641–652 Marchenko AN (1998) Parameterization of the Earth’s gravity field: point and line singularities. Lviv Astronomical and Geodetical Society, Lviv Pearson K (1896) Mathematical contributions to the theory of evolution. III – Regression, heredity and panmixia. Philos Trans R Soc Lond Ser A 187:253–318
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Tenzer R, Klees R (2008) The choice of the spherical radial basis functions in local gravity field modelling. Stud Geophys Geod 52:287–304 Tenzer RR, Klees Prutkin I, Wittwer T, Alberts B, Schirmer U, Ihde J, Liebsch G, and Sch€afer U (2009) Comparison of techniques for the computation of a height reference surface from gravity and GPS-levelling data. In: Sideris M (ed), Observing our Changing Earth, International Association of Geodesy Symposia, vol 133, Springer, Berlin, pp 263–274 Vajda P, Ellmann A, Meurers B, Vanı´cˇek P, Nova´k P, Tenzer R (2008) Global ellipsoid-referenced topographic, bathymetric and stripping corrections to gravity disturbance. Stud Geoph Geod 52(1):19–34 Wahba G (1977) A survey of some smoothing problems and the method of generalized cross-validation for solving them. In: Krishnaiah PR (ed) Applications of statistics, Amsterdam, The Netherlands: North-Holland, pp 507–523 Weightman JA (1965) Gravity, geodesy and artificial satellites. A unified approach. Proc. Symp. on the use of Artificial Satellites for Geodesy, Athens, Greece
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A Sensitivity Analysis in Spectral Gravity Field Modeling Using Systems Theory
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Vassilios D. Andritsanos and Ilias N. Tziavos
Abstract
The Input–Output System Theory (IOST) method is primarily based on the spectral combination of heterogeneous data taking into account their statistical properties and approximating the Power Spectral Density (PSD) functions of the signals and their errors. In this study a Multiple Input–Multiple Output System (MIMOS) is proposed, where the input measurements as well as the input and output signals are different gravity field observables. The work presents a sensitivity analysis of the IOST method towards the input data noise and the effect of integral kernel modifications to the error prediction estimates. Simulated input noise fields along with the contribution of the input data resolution to the output prediction errors are investigated towards the analysis of the 2D error covariance functions. Special attention is paid to the contribution of different kernels to the error prediction estimates and 2D planar and spherical discrete kernels are tested considering the overall prediction improvement of the spectral procedure. The MIMOS system is finally assessed by a number of numerical tests and conclusions are drawn in terms of the optimal modelling of the input data noise and the significance of the spherical kernels in the improvement of the prediction results.
49.1
Introduction
The spectral techniques have been widely used in gravity field modeling applications during recent decades and more specifically for the efficient evaluation of
V.D. Andritsanos Geodesy Laboratory, Technological and Educational Institute of Athens, Athens, Greece e-mail:
[email protected] I.N. Tziavos (*) Department of Geodesy and Surveying, Aristotle University of Thessaloniki, University Box 440, 541 24 Thessaloniki, Greece e-mail:
[email protected]
convolution integrals. The main advantage of the analysis in the spectral domain is the possibility of handling large amounts of data simultaneously with a considerable saving in computer time and memory. The efficiency of spectral methods led to the development of appropriate algorithms for terrain correction, geoid and deflections of the vertical computations, as well as the evaluation of Molodensky’s series (see, e.g., Sideris 1984, 1987; Forsberg and Sideris 1993; Haagmans et al. 1993; Tziavos 1995; Liu et al. 1997). In order to overcome the limitations of the typical FFT (Fast Fourier Transform)-based procedures related to the utilization of homogeneous and noise-free gridded data, the Input–Output System Theory (IOST)
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_49, # Springer-Verlag Berlin Heidelberg 2012
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methods were properly transformed and extensively used in physical geodesy applications over the past 2 decades. These methods can handle heterogeneous data given on the same grid and propagate data errors into the results (see, Bendat and Piersol 1986; Sideris 1996). The systems theory solution is based on the assumption that both the input signals and their errors are stochastic variables. Additionally, Power Spectral Density (PSD) functions of both signals and their errors are known and the solution depends on the ratio of the error PSD and the signal PSD. In the case that the inputs are correlated, the systems theory solution is formally equivalent to the classical Least Squares Collocation (LSC) approach or to its stepwise alternative (Sideris 1996; Sanso` and Sideris 1997). The generalization of the IOST to the Multiple Input–Multiple Output System Theory (MIMOST) for geodetically oriented applications was proposed by Andritsanos and Tziavos (2000). Extensive derivations of the output signals and their accuracy estimations both with numerical tests on the determination of gravity field observables in land and sea areas are given by Andritsanos et al. (2000, 2001). The discussion in this paper is directed towards the efficient use of the IOST based methods to several fundamental problems of gravity field modeling through numerical tests and examples. First, the propagation of the input data errors to the output data sets is investigated. In a second test the effect of different kernels of gravity field observables to the results obtained by spectral methods is studied. Finally, the advantages of the IOST methods in combination gravity field prediction schemes are analyzed both in terms of input data resolution and different kernel representations.
Fig. 49.1 The representation of a Multiple Input–Multiple Output System (MIMOS) with noise
V.D. Andritsanos and I.N. Tziavos
49.2
Theoretical Background
49.2.1 Multiple Input–Multiple Output System with Noise A single input–single output system is a spectral method where both signal and noise PSDs are required for obtaining an optimal solution. The inherent transfer functions in such a system are modified by factors dependent on the noise-to-signal ratio. Consequently, the noise of the input data is filtered out and an error estimation can be derived for the output (predicted) signals (see Sideris 1996). The extension of the afore mentioned single input–single output system to a MIMOS one is depicted in Fig. 49.1, where yi0 are the input observations, yi are the pure input signals, mi are the input noises, xj are the unknown output signals and ej are the output noises. The total number of input data (i) may be equal or different to the number of output signals (j). The system of Fig. 49.1 can be described using matrix notation (see, e.g., Andritsanos et al. 2000; Andritsanos and Tziavos 2000). According to Andritsanos et al. (2000) the input and output vectors can be written as follows: 2 2 3 3 Y1 þ M1 X1 E 1 6 Y2 þ M 2 7 6 X2 E2 7 6 6 7 7 Y0 ¼ 6 7 X0 ¼ 6 7; .. .. 4 4 5 5 . . Yq þ Mq
Xw E w
(49.1) where capital letters stand for the spectra of the respective quantities. The transfer function matrix is of the form 2 3 Hx1o y1o Hx1o y2o Hx1o yqo 6 Hx2o y1o Hx2o y2o Hx2o yqo 7 6 7 Hxo yo ¼ 6 . .. 7 (49.2) .. .. 4 .. . 5 . . Hxwo y1o Hxwo y2o Hxwo yqo
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that clearly shows the dependence of each output on all inputs. Andritsanos et al. (2000) showed that the input–output (I/O) error PSD matrix can be computed only if the input noise PSD matrix is known: Pxy ¼ Hxy Pxy ¼ Hxy Pyo yo Pmm :
(49.3)
In the above equation Hxy stands for the matrix containing the theoretical operators that connect every input with the output signals. In the case that the input signal is gravity anomaly and the output signals are geoid heights and deflections of the vertical, then the above mentioned matrix elements are the Stokes and Vening-Meinesz operators, respectively. The final solution and the estimation of the output error PSD in matrix notations are as follows (Andritsanos et al. 2000): ^ x y Yo ¼ Hxy Py y Pmm P1 Yo ; (49.4) ^o ¼ H X o o yo yo o o ^ x y Py y P^e^e ¼ Hxy Pyo yo Pmm H o o o o T ^ ^ T HT xy Hxy þ Hxo yo Pmm Hxy
(49.5)
More information about the explicit derivation of the above equations can be found in Sideris (1996) and Andritsanos et al. (2000).
49.2.2 Input Noise Considerations In geodetic applications of MIMOST special care should be paid towards the efficient evaluation of PSD functions. Following Marple (1987), PSD estimations can be derived from classical nonparametric and modern parametric methods. The classical methods are divided into the socalled periodogram and correlogram approaches. Both techniques are based on the application of FFT properties. In the periodogram approach the PSD is estimated directly from the available data. On the other hand, in the correlogram approach PSD is approximated by the direct transform of the autocorrelation function. For more details on the classical PSD evaluation methods the interested reader should consult (Marple 1987; Li 1996; Sideris 1996; Andritsanos and Tziavos 2000).
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The modern parametric methods are based on the parametric model description of the data used in the PSD evaluation. The parameters of the model are estimated in a least-squares sense. In addition, the degree of expansion of each model depends on the data properties and can be computed using certain criteria, mainly oriented to the minimization of the input data errors (see Marple 1987; Kay 1987). As it is commented in Sect. 49.2.1 the I/O signal PSD can be evaluated only if the input error PSD matrix Pmm is known. In geodetic applications each measurement is usually followed by a variance representing the accuracy of some repeated procedure. If the errors themshelves are known, then the input error PSD can be estimated using, e.g., the periodogram approach. The weakness of the direct input noise PSD estimation is the major difficulty for IOST application in gravity field modeling. Nevertheless, some techniques are used in order to efficiently overcome this drawback: 1. Noise simulation: In simulation studies noise fields can be used and introduced into MIMOS. These fields can be randomly generated following specific distribution (normal, uniform, etc.). The variance of the field can be selected from some a-priori information related to the observation accuracy. The input error PSD is then computed using the periodogram approach by FFT. 2. PSD models: Instead of the assumption made for the errors themselves, one can consider some models for the input noise PSD directly. White noise PSD models (constant value noise PSD) as well as colored noise models (non-constant value noise PSD) can be used. 3. Repeated data information: This is the case for data from multi-mission satellite altimetry as well as those from the recent gravity field dedicated satellite missions. These data sets provide the opportunity to estimate input signal and noise PSDs from repeated satellite track information. The original idea is due to Sailor (1994), while analytical derivations of the input PSD estimations are given in Andritsanos et al. (2001).
49.2.3 Kernel Representations The theoretical operators that connect the input and output signals are in convolution form and can be
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properly evaluated by FFT techniques. This is the basic advantage of the IOST, i.e., heterogeneous input signals can be combined in order to approximate multiple outputs and their respective estimation errors. Each input is connected with each output through the theoretical operator mentioned above. This operator can be analyzed using analytically defined or discrete kernels as described in Sideris (1987) and Tziavos (1995). In the specific case that the input signal is gravity anomaly and output signal geoid height the specta of the input and output signals are connected by the Stokes operator in the frequency domain either in its planar (49.6) or spherical form (49.7–49.9): 1 1 FfDggFf‘g; ‘ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F Npl ¼ 2 2pg x þ y2
(49.6)
R FfDg cos ’m gFðSÞ F Nsph ¼ 4p g
(49.7)
1 S ¼ 4 6s þ 10s2 3 6s2 ln s þ s2 s
(49.8)
c s ¼ sin : 2
(49.9)
In the above equations x and y are the horizontal coordinates, ‘ and c are the planar and spherical distances between the computational and the running point, respectively, S is the Stokes function and F denotes the direct Fourier transform.
49.3
Numerical Investigation
A number of numerical examples are performed in order to investigate the flexibility of MIMOST with respect to the magnitude of input error, the effect of planar and spherical kernels, the data spatial resolution and the impact of global geopotential models in such combination schemes. Furthermore, applications with simulation and real data are carried out in order to investigate the sensitivity of the systems designed for the specific tests.
49.3.1 Sensitivity Studies The area bounded by the limits 45 < j < 55 and 55 < l < 45 is first selected in order to investigate the effect of the input errors to the output estimates. A two-input-two-output system is designed using as input signals gravity anomalies and geoid heights with a 5 arcmin resolution and reduced to EGM96 geopotential model. The input signal fields are depicted in Fig. 49.2. Simulated input noise fields following either the normal or uniform distribution and having different standard deviation (sd) values are produced by random data generators and finally added to the original signals. The sd of the input noise fields are chosen at the level of 3, 5 and 10 mGal and 3, 5, 10 cm for the input gravity and geoid signals, respectively. The noise fields are added to the input signals in the frequency domain and the synthetic observations are used towards the optimal approximation of the frequency response function as described in Sect. 49.2. The input signal and error PSDs are computed using the periodogram approach. The scope of this numerical test is twofold. First, an attempt is made to reconstruct each signal in the frame of the system adopted and then to check the degree of filtering of the input noise and its final effect to the output signals. In Table 49.1 the statistics of differences between the input and output signals are tabulated in terms of minimum, maximum and sd values, along with all other numerical information concerning the input signals and their noises. The efficient filtering of the input noise can be observed in the results of Table 49.1. Considering all input noise combinations for gravity anomalies and geoid heights, it is obvious that the input noise is considerably filtered out at a level of approximately 78% for the geoid field and 67% for the gravity field. Obviously, these mean values reflect also the degree of reconstruction of the two signals in terms of the output estimates. Furthermore, the analysis of the results of this test shows that the most sufficient scheme for the input errors is 10 cm for the geoid heights and 3 mGal for gravity anomalies in terms of sd. In this case the degree of filtering and reconstruction as well reaches the level of 88% and 86% for the geoid and gravity anomaly field, respectively. Based on the results of this simulation test one can realize the sensitivity of MIMOST to the input noise.
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Fig. 49.2 Gravity anomaly and geoid height input signal fields
The fact that most of the input noise is filtered and only a small portion is propagated into the results shows the advantages of systems theory to a wide number of geodetic applications. As far as the effect of the input noise distribution is concerned, it seems that it is not significant, since the results obtained are almost identical either using normal or uniform distribution.
49.3.2 Sensitivity of Error Covariance Approximation The gravity anomaly observation field of the previous test is introduced into a single-input–single-output system, where geoid heights are the output estimates. Different sd values of the input noise are used in order to investigate their effect to the estimation of the geoid height error covariance function. The 2-D error covariance can be derived by the so-called inverse correlogram approach, where the inverse FFT is applied to the error PSD. The estimated geoid error covariance function is depicted in Fig. 49.3
considering an input error for gravity anomalies equal to 5 mGal. The representation of this covariance function is close to an exponential model and its correlation length is quite small (5 arcmin). These characteristics of the estimated covariance function are expected due to the Dirac form of the simulated input error field. The error variances for the input and output signals of this test are presented in Table 49.2. It is concluded that when the variance of the noise of the input gravity anomaly field increases the variance of the output geoid field increases respectively. In satellite altimetry applications the shape of the output error covariance function would be different taking into account the repeated track information used for the estimation of the input PSD. For the PSD estimation using altimetry data the interested reader should consider Andritsanos et al. (2001). It is also noticed that the geoid error covariance function cannot be determined when the variance of the input noise is not available. In this case the optimal frequency response function coincides with the theoretical Stokes operator.
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Table 49.1 Differences between input signals and output estimates using different input errors and distributions
49.3.3 The Effect of Kernel Representation
SD of input errors Max Min Normal distribution – Output geoid heights (cm) 3 cm/3 mGal 2.8 5.2 3 cm/5 mGal 3.4 6.3 3 cm/10 mGal 4.3 7.1 10 cm/3 mGal 4.7 5.9 10 cm/5 mGal 6.4 7.9 10 cm/10 mGal 7.9 9.4 Uniform distribution – Output geoid heights (cm) 3 cm/3 mGal 3.2 5.2 3 cm/5 mGal 3.3 6.1 3 cm/10 mGal 4.7 6.9 10 cm/3 mGal 4.6 7.2 10 cm/5 mGal 5.8 8.7 10 cm/10 mGal 7.4 12.0 Normal distribution – Output gravity anomalies (mGal) 3 cm/3 mGal 6.5 7.1 3 cm/5 mGal 8.2 8.9 3 cm/10 mGal 10.6 11.8 10 cm/3 mGal 8.5 8.7 10 cm/5 mGal 10.2 11.5 10 cm/10 mGal 15.0 16.3 Uniform distribution – Output gravity anonalies (mGal) 3 cm/3 mGal 7.4 6.9 3 cm/5 mGal 10.3 9.2 3 cm/10 mGal 11.3 11.5 10 cm/3 mGal 7.8 9.0 10 cm/5 mGal 13.0 12.3 10 cm/10 mGal 17.2 18.3
In order to investigate the effect of planar and spherical Stokes kernels (49.6–49.9) towards the error covariance estimation in the frame of a singleinput–single-output system, gravity anomalies on a 5 arcmin grid are used in an area bounded by the limits 34 < j < 40 and 18 < l < 26 . Simulated noise fields (normal distribution) of 1, 3, 5, 10 and 20 mGal in terms of sd are introduced into the system and their effect to the prediction of the output geoid error is investigated. Both planar and spherical kernels are used in order to evaluate the I/O signal PSD. As it is observed from the statistics of Table 49.3, where the sd of the output geoid error is tabulated along with the sd of the input gravity noise, the results obtained when using spherical kernel are slightly better to those derived by the use of planar kernel. This improvement is directly related to the output error estimation, which is a criterion of the internal accuracy of the method used and reflects the sensitivity of the I/O system in connection with the kernel used. The sd of the geoid
Fig. 49.3 Estimated geoid 2-D error covariance function
SD 0.7 0.9 1.0 1.2 1.5 1.9 0.7 0.9 1.0 1.2 1.5 1.9 1.2 1.4 1.7 1.4 1.8 2.3 1.2 1.4 1.7 1.4 1.8 2.4
Table 49.2 Error variances of the (a) input gravity anomaly noise and (b) the output geoid height field Gravity anomaly (mGal2) 9 25 100
Geoid height (cm2) 0.6 2.0 11.8
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A Sensitivity Analysis in Spectral Gravity Field Modeling Using Systems Theory
Table 49.3 Geoid prediction error with respect to the gravity anomaly input noise for (a) planar and (b) spherical kernel SD of input noise (mGal) 1 3 5 10 20
Planar (cm) 0.4 0.9 1.5 3.2 6.9
Spherical (cm) 0.1 0.5 0.9 2.4 5.6
Table 49.4 The effect of the input field resolution Resolution (arcmin) 5 10 25
Output error variances Planar kernel Spherical kernel 2.5 2.3 9.2 8.3 145.4 144.8
Unit: [cm2]
error from 0.4 cm drops to 0.1 in the case of 1 mGal input noise and from 6.9 to 5.6 cm in the case of 20 mGal input noise. The more the input noise increases the spherical kernel leads to a better internal accuracy comparing to the results achieved by the planar kernel.
49.3.4 The Effect of Data Resolution The impact of the resolution of the input gravity field to the prediction accuracy is studied in the same test area as before. For this reason the original gravity anomaly signal of 5 arcmin is averaged in order to produce two coarser gravity grids with resolutions equal to 10 and 25 arcmin, respectively. The input noise is kept stable at the level of 5 mGal sd (normal distribution) in all cases. The error variances of the predicted geoid are shown in Table 49.4. The results show that the high resolution of the input data results in better accuracy estimates in a MIMOS procedure. Additionally, higher prediction accuracy is observed using the spherical instead of the planar kernel for the I/O PSD estimation.
49.3.5 Global Geopotential Models and IOST In order to investigate the effect of different kernels in conjunction with the advent of the newly available
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Table 49.5 Statistics of differences between geoid heights derived by a single input–output system and corresponding heights from GPS/leveling at 37 benchmarks Model/kernel form EGM96/planar EGM2008/planar EGM2008/spherical
SD before tilt 0.43 0.34 0.32
SD after tilt 0.14 0.10 0.09
Unit: [m]
global geopotential model EGM2008 (Pavlis et al. 2008), a test area with available 5 arcmin gravity anomalies is selected in northern Greece (40 .25 < j < 41 and 22 .50 < l < 24 ). The observed gravity anomalies are reduced to EGM96 (Lemoine et al. 1998) and EGM2008 global geopotential models and the effect of the topography is also taken into account through the RTM reduction. The reduced gravity field is introduced into a single I/O system and a noise field of 3 mGal in terms of sd is assumed. The final geoid heights are estimated by restoring the effects of the geopotential model and the terrain. The assessment of the accuracy of the predicted geoid model is realized by comparing the computed geoid heights with corresponding heights from GPS/leveling at 37 benchmarks. The statistics of the results of this comparison are tabulated in Table 49.5 before and after the application of a bias and tilt model to the computed differences. It is obvious that EGM2008 improves significantly the geoid estimates comparing to EGM96 model. A slight improvement is also observed with the spherical kernel utilization to the I/O PSD estimation, but this is due to the limited extent of the area. Conclusions
The application of the MIMOST to geoid modeling is presented through a number of numerical tests in different areas using real or simulation data. Various distribution noise fields of different sd are introduced in properly designed systems and the level of the prediction accuracy is discussed. It should be mainly noticed the sensitivity detected in the I/O procedures towards the filtering of the input noise. The main part of this noise is filtered out and only a small portion of it is propagated into the estimation results. Consequently, the signal reconstruction is achieved at a level of more than
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85%, when proper input noises for geoid and gravity signals are considered (10 cm and 3 mGal, respectively). Special attention is necessary to be paid in the preprocessing of the original data through the reference to global geopotential models and the subtraction of the effect of the topography. The numerical tests with observed gravity data show a significant improvement towards the external accuracy of the systems methodology, when the newly developed EGM2008 geopotential model is used as the reference field in these combination schemes. The effect of the input data resolution to the final prediction accuracies is also investigated in conjunction with the adequate kernel form for the I/O PSD evaluation. The use of different kernels indicates the outperformance of the spherical over the planar kernel approximation in the output estimates either considering single or multiple systems.
References Andritsanos VD, Tziavos IN (2000) Estimation of gravity field parameters by a multiple input/output system. Phys Chem Earth 25(A):39–46 Andritsanos VD, Sideris MG, Tziavos IN (2000) A survey of gravity filed modeling applications of the Input-Output system Theory (IOST). Int Geoid Serv Bull 10:1–17 Andritsanos VD, Sideris MG, Tziavos IN (2001) Quasistationary sea surface topography estimation by the multiple input-output method. J Geod 75:216–226 Bendat JS, Piersol AG (1986) Random data – analysis and measurements procedures, 2nd edn. Wiley, New York, NY
V.D. Andritsanos and I.N. Tziavos Forsberg R, Sideris MG (1993) Geoid computations by the multi-band spherical FFT approach. Manuscripta Geodaetica 18:82–90 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 Kay SM (1987) Modern spectral estimation. Prentice Hall, Englewood Cliffs, NJ Lemoine FG et al. (1998) The development of the join NASA GSFC and NIMA geopotential model EGM96, NASA Technical Paper, 1998 – 206861 Li J (1996) Detailed marine gravity field determination by combination of heterogeneous data. UCSE Rep. no 20102, University of Calgary, Canada Liu QW, Li YC, Sideris MG (1997) Evaluation of deflections of the vertical on the sphere and the plane: a comparion of FFT techniques. J Geod 71:461–468 Marple SL (1987) Digital spectral analysis with applications. Prentice Hall, Englewood Cliffs, NJ Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2008) An earth gravitational model to degree 2160: EGM2008, presented at the 2008 General Assembly of the European Geosciences Union, Vienna, Austria, 13–18 April 2008 Sailor RV (1994) Signal processing techniques. In: Vanicek P, Christou NT (eds) Geoid and its geophysical interpretations. CRC, Ann Arbor, MI, pp 147–185 Sanso` F, Sideris MG (1997) On the similarities and differences betweeen systems theory and least-squares collocation in physical geodesy. Bollettino di Geodesia e Scienze Affini 2:174–206 Sideris MG (1984) Computations of gravimetric terrain corrections using fast Fourier transform techniques. USCE Rep. no 20007, University of Calgary, Canada Sideris MG (1987) Spectral methods for the numerical solution of Molodensky’s problem. USCE Rep. no 20024, University of Calgary, Canada Sideris MG (1996) On the use of heterogeneous noisy data in spectral gravity field modelling methods. J Geodesy 70 (8):470–479 Tziavos IN (1995) Comparisons of spectral techniques for geoid computations over large regions. J Geodesy 70:357–373
Investigation of Topographic Reductions for Marine Geoid Determination in the Presence of an Ultra-High Resolution Reference Geopotential Model
50
C. Tocho, G.S. Vergos, and M.G. Sideris
Abstract
During the last decade, the realization of the satellite gravity missions of CHAMP and GRACE, the acquisition of new gravity data and the development of novel processing methodologies has led to the determination of more accurate and higher in resolution global geopotential models. The spatial scale of ~110 km that EGM96 could represent has improved today with EGM2008 to the level of ~16 km (full wavelength). This advance in the representation of higher frequencies by the geopotential models may signal the need to reassess the methodologies and techniques traditionally used for local and regional geoid determination. The traditional procedure followed is that of the remove-computerestore method. The input functionals related to the Earth’s gravity field are first reduced to a reference geopotential model, then the topographic effects are taken into account through one of the available reduction methods, computations follow using the reduced observations, and finally the contribution of the global geopotential model and the topographic indirect effects are added back to the computed reduced geoid values. One crucial point to this operation is that the attraction of the masses considered with a topographic reduction scheme is supposed to represent the medium and high frequencies in the gravity field, which still remain in the data, in principle even after they have been reduced to a geopotential model. Given that the best available digital depth models have a resolution of 30 arcsec, which translates to roughly 1 km spatial wavelength, it becomes apparent that the contribution of such a model to the reduction of gravity and geoid data, when a high resolution geopotential model is used as reference, is questionable or should be at least investigated. This final point is the main goal of
C. Tocho (*) Facultad de Ciencias Astrono´micas y Geofı´sicas, La Plata, Argentina e-mail:
[email protected] G.S. Vergos Department of Geodesy and Surveying, Aristotle University ofThessaloniki, Thessaloniki 54124, Greece M.G. Sideris Department of Geomatics Engineering, University of Calgary, 2500 University Drive N.W, Calgary, AB, Canada T2N 1N4 S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_50, # Springer-Verlag Berlin Heidelberg 2012
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this paper, i.e., to investigate the contribution of the available digital depth models to the reduction of gravity anomalies and geoid heights when a geopotential model with the resolution of EGM2008 is used. To this extent, marine gravity anomalies and satellite altimetry sea surface heights are used off-shore the Atlantic coast of Argentina. EGM2008 is used as a reference surface to reduce the available gravimetric and altimetric observations, and the latest bathymetry model from the Scripps Institute of Oceanography group (SIOv11.1) is employed in order to compute topographic reductions based on the Residual Terrain Model (RTM) scheme. The results acquired are validated in terms of the reduction they provide to the available input data, both the mean and the standard deviation of the residuals, as well as in terms of the spectral content of the residual signal spectrum. Conclusions and recommendations on the use of topographic reductions and the treatment of topographic effects for geoid modelling in the presence of a high-resolution geopotential model are also drawn so as to ensure the consistency between data used and results acquired.
50.1
Introduction
The most popular scheme used during the last years for geoid modeling is based on the well-known remove-compute-restore (RCR) method (Forsberg 1993, Sideris 1994). This method is based on removing the long wavelengths by a Global Gravity Model (GGM) while the short-wavelengths are supposed to be modeled by available Digital Topography and Bathymetry Models (DTMs and DBMs respectively). To that extent, the available DTMs and DBMs should contain enough high-resolution information and be accurate enough in order to represent frequencies shorter than those of the GGM, for a rigorous use of the RCR method. With the advent of the CHAMP, GRACE and GOCE missions and the realization of the EGM2008 GGM (Pavlis et al. 2008), the available GGM have much more power up to very-high degrees and increasing accuracy. EGM2008 has been recently released to public by the U.S. Geospatial-Intelligence Agency (NGA) EGM Development Team. It presents a spherical harmonics expansion of the geopotential to degree and order 2159, while additional spherical harmonics coefficients to degree 2190 and order 2159 are also available. The full degree and order of EGM2008 (2159) translates to a spatial resolution of ~5 arcmin, but in the present study it has been used only up to degree 1834 since above that the signalto-noise ratio is smaller than 1 (see Fig. 50.1).
Contrary to the best available DTMs today (SRTM-class), which have a spatial resolution of 3 arcsec, the corresponding DBMs have a spatial resolution of 30 arcsec (best case scenario). This arises some questions as to whether their spatial resolution is enough in order to contemplate that of the EGM2008 model. Another possible limiting factor in the use of the currently available DBMs, for marine gravity and geoid modeling, is their accuracy. Errors in the DBMs will introduce errors in the estimated terrain effects thus deteriorating the quality of computed terrain reductions. These considerations where the source that set the main goal of the present study that is to evaluate the performance of the currently best available DBM towards marine geoid modeling employing the RCR method in the presence of an ultra-high degree GGM.
50.2
Computation Strategy and Results
In order to evaluate the performance of the currently best available DBM towards marine geoid modeling employing the RCR method, ERS1GM Sea Surface Heights (SSHs) and satellite altimetry derived marine free-air gravity anomalies from the Danish National Space Agency DNSC08 (Andersen and Knudsen 2008) high resolution (1 arcmin) model are used as input data. The satellite altimetry data were 70510 Corrected Sea Surface Height (CORSSHs)
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Investigation of Topographic Reductions for Marine Geoid Determination
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Fig. 50.1 By degree EGM2008 signal and error power
measurements from the Geodetic Mission (GM) of the European Remote-sensing Satellite 1 (ERS1), which are generated by the CLS Space Oceanography Division and provided by Archiving, Validation and Interpretation of Satellite Oceanographic data (AVISO 1998). The study is carried out off-shore the Atlantic coast of Argentina, limited by 34 S to 55 S in latitude and 70 W (290 E) to 56 W (304 E) in longitude. Within the RCR frame, EGM2008 complete to degree and order 1834 is used as a reference geopotential model and the effect of bathymetry is taken into account through a Residual Terrain Model (RTM) reduction using the Scripps Institute of Oceanography SIOv11.1 (Smith and Sandwell 1997) bathymetry model (see Fig. 50.2). The RIO Dynamic Ocean Topography (DOT) model (Rio and Hernandez 2004) is used to reduce the altimetric SSHs to the geoid. For the RTM reduction, a reference elevation model is constructed from the fine one with 6 arcmin resolution (corresponding to degree 1834) by taking simple moving averages. In all computations the detailed DBM has been used, both for the near-zone and far-zone effects, since with the compute power available today there is little need to use coarser resolution terrain grids for the distant effects. The resulting residual geoid heights and gravity anomalies are evaluated both in terms of their statistics, compared to the EGM2008 reduced fields, as well as in terms of their spectra. Table 50.1 presents the statistics of the available ERS1 SSHs, the DOT as computed on the ERS1
sub-satellite points, the EGM2008 contribution on the same points and reduced field Nred, which is the difference between the ERS1 SSHs minus the DOT, minus EGM2008. Note that in Table 50.1 Nalt denotes the DOT corrected ERS1 SSHs. The statistics of the DNSC08 and EGM2008 gravity anomalies as well as their differences can be seen in Table 50.2. Following Forsberg (1984) the RTM reduction for gravity anomalies is computed as: dgRTM ¼ 2 p D rðh href Þ
(50.1)
where H is the bathymetric depth given by a global bathymetry model, href is the depth of a smooth mean reference surface (the 6 arcmin model in this case as previously described) and Dr is the density contrast between Earth’s crust and seawater. For all terrain effects computations the GRAVSOFT (Tscherning et al. 1992) suite has been used to create the reference bathymetric grid and estimate the RTM reduction on geoid heights and gravity anomalies. The primary use of the RTM reduction is to obtain residual SSHs, so that after that step prediction and interpolation can be performed with a smoother field. The statistics of the RTM effects computed using different density contrasts are presented in Table 50.3 together with the residual Sea Surface Heights that represent the medium wavelengths of the geoid heights and can be considered as residual geoid heights (Nres). Comparing the results acquired from the RTM reduction of the ERS1 SSHs (see Tables 50.1 and
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Fig. 50.2 The area under study and its bathymetry according to the SIOv11.1 model
Table 50.1 Statistics of the available ERS1 SSHs, DOT, EGM2008 contribution and reduced fields ERS1SSHs DOT NEGM2008 Nalt Nred
Max 19.010 1.807 18.995 17.732 0.912
Min 0.589 0.955 1.014 0.52 3.21
Mean 11.259 1.482 11.071 9.777 1.294
RMS 11.673 1.494 11.442 10.208 1.316
STD 3.079 0.192 2.890 2.933 0.239
Unit: [m]
50.3), it becomes evident that, whatever the density contrast used, there is no gain both in terms of the mean and the std of the field. In most cases, apart from the one that a density contrast of 1.3 gr/cm3 has been
Table 50.2 Statistics of the available DNSC08 gravity anomalies, EGM2008 contribution and reduced fields DgDNSC08 DgEGM2008 Dgred
Max 135.07 135.96 21.36
Min 134.66 137.63 29.81
Mean 3.57 3.57 0.00
RMS 25.82 25.97 3.67
STD 25.57 25.72 3.67
Unit: [mGal]
used, the residual Nres field has larger mean and std. These signal that the available bathymetry model has insufficient resolution to depict more detailed bathymetric features than those included in EGM2008. Moreover, given that the SIOv11.1 DBM has been estimated by inverting satellite altimetry data, it can
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Investigation of Topographic Reductions for Marine Geoid Determination
Table 50.3 Statistics of the RTM-effects and residual ERS1 geoid heights for various density contrasts Dr ¼ NRTM Nres Dr ¼ NRTM Nres Dr ¼ NRTM Nres Dr ¼ NRTM Nres Dr ¼ NRTM Nres Dr ¼ NRTM Nres Dr ¼ NRTM Nres Dr ¼ NRTM Nres
Max 2.67 g/cm3 1.233 0.955 2.47 g/cm3 1.083 0.948 2.3 g/cm3 0.955 0.943 2.2 g/cm3 0.880 0.939 2.1 g/cm3 0.805 0.936 1.6 g/cm3 0.429 0.920 1.4 g/cm3 0.278 0.913 1.3 g/cm3 0.265 0.908
Min
Mean
RMS
STD
1.490 3.164
0.080 1.374
0.191 1.409
0.174 0.309
1.308 3.172
0.071 1.365
0.168 1.396
0.152 0.296
1.153 3.179
0.062 1.357
0.148 1.386
0.134 0.286
1.063 3.183
0.058 1.352
0.136 1.380
0.124 0.280
0.972 3.187
0.053 1.347
0.125 1.375
0.113 0.275
0.518 3.206
0.029 1.323
0.067 1.347
0.060 0.252
0.336 3.214
0.019 1.313
0.044 1.336
0.039 0.246
0.324 3.218
0.015 1.302
0.033 1.324
0.029 0.242
Unit: [m]
be concluded that such DBMs need to be augmented by soundings in order to determine collocated solutions with higher resolution and accuracy (Smith and Sandwell 1997). Of course, echo soundings are scarce, especially over such large regions, therefore the improvement in the DBMs by combined solutions would probably be only local. The behaviour of EGM2008 is quite peculiar (see Table 50.1), since the reduced filed has a very large mean value (1.3 m). This, is not reduced by the RTM reduction, which indicates that some signal(s) remain in the field which should be modelled. Similar mean value results have been achieved for EGM96 (Tocho et al. 2005a, b) and may be due to the truncation of EGM2008 to degree 1834. On the other hand this behaviour might be due to the incorporation of the DOT model for the reduction of the ERS1 SSHs to the geoid. Notice that the DOT has a mean value of 1.5 m which is almost equal (with opposite sign) to that of the Nres field. In fact if the ERS1 SSHs are not reduced for the DOT, the EGM2008 reduced SSHs
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Table 50.4 Statistics of the RTM-effects and residual gravity anomalies for various density contrasts Max Dr ¼ 1.6 g/cm3 DgRTM 16.62 Dgres 50.43 Dr ¼ 1.5 g/cm3 DgRTM 13.77 Dgres 46.34 Dr ¼ 1.4 g/cm3 DgRTM 10.84 Dgres 42.26 Dr ¼ 1.3 g/cm3 DgRTM 7.91 Dgres 39.22
Min
Mean
RMS
STD
53.07 21.15
0.13 0.12
1.81 3.92
1.80 3.91
48.97 18.97
0.10 0.09
1.53 3.81
1.53 3.81
44.86 16.65
0.07 0.06
1.27 3.72
1.27 3.72
40.76 14.97
0.05 0.04
1.01 3.65
1.01 3.65
Unit: [mGal]
have a mean value of 0.19 m only, though the std increases to 0.33 m. This may signal that part of the DOT of the area is included in EGM2008 spectrum, maybe due to the altimetry data used in its development. It should be noted though that when altimetric data are used for marine geoid modelling, their reduction for the DOT is mandatory whether else the surface determined is not the geoid but the mean sea surface. In similar studies performed in other areas of the world like the Mediterranean and off-shore Newfoundland such behaviour has not been observed either for EGM96 and EGM2008 (Vergos et al. 2005a, b, 2007). The same holds for this particular area under study when other GGMs have been used (Tocho et al. 2007). This behaviour of EGM2008 remains to be investigated in future work. Table 50.4 shows the statistics of both the RTM effect on gravity computed with (50.1), using various density contrasts, and the residual gravity anomalies computed using (50.2): Dgres ¼ DgFA 2pGrðh href Þ DgGM
(50.2)
where DgFA are the free-air satellite gravity anomalies from DNSC08 model reduced by the residual terrain model reduction and the geopotential model. It should be noted that all density contrasts as in the case of ERS1 geoid heights (Table 50.3) have been tested as well, but only the ones with the best statistics after the reduction are reported. For the reduction of the DNSC Dg similar results are obtained. Only with a density contrast of 1.3 gr/cm3
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Fig. 50.3 Signal PSDs of the original gravity data (top left), EGM2008 (nmax ¼ 1,834) contribution (top right), reduced gravity (bottom left) and residual field after the RTM reduction (bottom right)
a reduction in the standard deviation (std) was achieved by 0.02 mGal only (last line in Tables 50.2 and 50.4), which is clearly insignificant. EGM2008 performs very well in the contribution to Dg, since the reduced field has a zero mean and a std at the 3.7 mGal level. The RTM reduction did not manage to provide significant improvement, which signals that EGM2008 contains all the power that the bathymetry has to offer. Therefore, higher-resolution DBMs should be employed, whether else the reduction of marine data for the bathymetry, within gravity field modeling and gravimetric geoid studies, may not have any meaning. From the signal PSDs of the gravity data depicted in Fig. 50.3, it is clear that EGM2008 has almost the same power as the original data. But, the two sidelobes in the EGM2008 PSD (circles) indicate that the geopotential model has some of its power in higher degrees and larger correlation length than the original Dg. These side-lobes are at wavelengths of harmonic degrees ~85-90 (~244 km), so they may indicate the influence of GRACE data in EGM2008. In any case
they should be further investigated, since in similar tests in other areas (Mediterranean Sea) such effects are absent (Tziavos et al. 2010). From the signal PSDs of the geoid heights shown in Fig. 50.4, it is clear that EGM2008 has almost the same power as the original data. In the geoid height contribution of EGM2008 no side-lobes are observed, which is probably due to the fact that less power, compared to Dg, of the geoid height spectrum is contained in higher degrees. After the reduction to EGM2008, the remaining field does not present clear random characteristics (noise) since the mean value remains quite large. This is evident by the fact that the signal gain is reduced but not significantly, even after the RTM reduction. One factor that can explain that is the presence of part of the DOT signal in it, since the spectrum of the latter has power up to ~150 km in the area under study so its contribution will have impact on the residual field (bottom right in Fig. 50.4). Finally, it is worth mentioning that in related studies over continental regions where the DTM resolution is higher compared to that of the DBMs
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Investigation of Topographic Reductions for Marine Geoid Determination
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Fig. 50.4 Signal PSDs of the original ERS1 SSHs (top left), EGM2008 (nmax ¼ 1,834) contribution (top right), reduced geoid heights (bottom left) and residual field after the RTM reduction (bottom right)
(3 vs. 30 arcsec), even when EGM2008 is used as a reference, a significant reduction of the mean and std of the residual field can be achieved (Tziavos et al. 2010). Conclusions
From the results acquired, it can be concluded that the spatial resolution of the currently available DBMs is not enough in order to provide higher frequency content information compared to EGM2008 for marine gravity field and geoid modeling. Therefore, the methodology followed during the RCR scheme should be revised, at least when EGM2008, or other ultra-high degree geopotential models, are used as reference. Such high-resolution geopotential models representing the geoid can be used from now on for DOT and time-varying DOT modeling in combination with altimetry, GOCE- and GRACE-type of data. Unless the available global DBMs, which in most cases come from the inversion of altimetric observations, do not increase their spatial resolution, then they
should be used in marine gravity field and geoid modeling with caution. If topographic reductions at sea are to be used for the latter, then higher-resolution DBMs should be developed from combination techniques (altimetry & soundings). Moreover, a new geodetic mission from altimetry, which will improve the across-track spacing of the currently available multi-mission altimetric record, may improve the currently available DBM resolution. Acknowledgement The terrain reductions presented in the paper have been computed with the GRAVSOFT package (Tscherning et al. 1992). We extensively used the Generic Mapping Tools (Wessel and Smith 1998) in displaying our results.
References Andersen O, Knudsen P (2008) The DNSC08 ocean wide altimetry derived gravity field. Presented EGU-2008, Vienna, Austria, April 2008 AVISO User Handbook (1998) Corrected Sea Surface Heights (CORSSHs) AVI-NT-011-311-CN Edition 31
426 Forsberg R (1984) A study of terrain corrections, density anomalies and geophysical inversion methods in gravity field modelling. Rep. No. 355, Department of Geodetic Science and Surveying, The Ohio State University, Columbus, OH Forsberg R (1993) Modelling the fine-structure of the geoid: methods, data requirements and some results. Surv Geophys 14(4–5):403–418 Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2008) An Earth Gravitational Model to Degree 2160: EGM2008, presented at the 2008 General Assembly of the European Geosciences Union, Vienna, Austria, 13–18 April 2008 Sideris MG (1994) Regional geoid determination. In: Vanicek P, Christou NT (eds) Geoid and its geophysical interpretations. CRC, Boca Raton, FL, pp 77–94 Smith WHF, Sandwell DT (1997) Global sea floor topography from satellite altimetry and ship depth soundings. Sci Mag 277(5334) Rio MH, Hernandez F (2004) A mean dynamic topography computed over the world ocean from altimetry, in-situ measurements and a geoid model. J Geoph Res 109(12): C12032 Tocho C, Vergos GS, Sideris MG (2005a) Optimal marine geoid determination in the Atlantic coastal region of Argentina. In: Sanso F (ed) A window on the future of geodesy, vol 128, International Association of Geodesy Symposia. Springer, Berlin, pp 380–385 Tocho C, Vergos GS, Sideris MG (2005b) A new marine geoid model for Argentina combining altimetry, shipborne gravity data and CHAMP/GRACE-type EGMs. Geod Cartogr 54(4):177–189 Tocho C, Vergos GS, Sideris MG (2007) Estimation of a new high-accuracy marine geoid model offshore Argentina using
C. Tocho et al. CHAMP- and GRACE-derived geopotential models. Presented at the XXIV General Assembly of the IUGG (IUGG2007), Perugia, Italy, 2–13 July Tscherning CC, Forsberg R, Knudsen P (1992) The GRAVSOFT package for geoid determination. In: Holota P, Vermeer M (eds) 1st Continental Workshop on the Geoid in Europe, pp 327–334 Tziavos IN, Vergos GS, Grigoriadis VN (2010) Investigation of topographic reductions and aliasing effects to gravity and the geoid over Greece based on various digital terrain models. Surv Geophys 31(3):23–67. doi:10.1007/s10712-009-9085-z Vergos GS, Grigoriadis V, Tziavos IN, Sideris MG (2007) Combination of multi-satellite altimetry data with CHAMP and GRACE EGMs for geoid and sea surface topography determination. In: Tregoning P, Rizos C (eds) Dynamic planet 2005 – monitoring and understanding a dynamic planet with geodetic and oceanographic tools, vol 130, International Association of Geodesy Symposia. Springer, Berlin, pp 244–250 Vergos GS, Tziavos IN, Andritsanos VD (2005a) On the determination of marine geoid models by least-squares collocation and spectral methods using heterogeneous data. In: Sanso´ F (ed) A window on the future of geodesy, vol 128, International Association of Geodesy Symposia. Springer, Berlin, pp 332–337 Vergos GS, Tziavos IN, Andritsanos VD (2005b) Gravity data base generation and geoid model estimation using heterogeneous data. In: Jekeli C, Bastos L, Fernandes J (eds) Gravity geoid and space missions 2004, vol 129, International Association of Geodesy Symposia. Springer, Berlin, pp 155–160 Wessel P, Smith WHF (1998) New improved version of generic mapping tools released. EOS Trans Am Geophys Union 79(47):579
Effects of Hypothetical Complex Mass-Density Distributions on Geoidal Height
51
Robert Kingdon, Petr Vanı´cˇek, and Marcelo Santos
Abstract
Geoid computation according to the Stokes-Helmert scheme requires accurate modelling of the variations of mass-density within topography. Current topographical models used in this scheme consider only horizontal variations, although in reality density varies three-dimensionally. Insufficient knowledge of regional three-dimensional density distributions prevents evaluation from real data. In light of this deficiency, we attempt to estimate the order of magnitude of the error in geoidal heights caused by neglecting the depth variations by calculating, for artificial but realistic mass-density distributions, the difference between results from 2D and 3D models. Our previous work has shown that for simulations involving simple massdensity distributions in the form of planes, discs or wedges, the effect of massdensity variation unaccounted for in 2D models can reach centimeter-level magnitude in areas of high elevation, or where large mass-density contrasts exist. However, real mass-density distributions are more complicated than those we have modeled so far, and involve multiple structures whose effects might mitigate each other. We form a more complex structure by creating an array of discs that individually we expect to have a very significant effect, and show that while the contribution of such an array to the direct topographical effect on geoidal height is sub centimeter (0.85 cm for our simulation), the resulting primary indirect topographical effect may reach several centimeters or more (5 cm for our simulation).
51.1
R. Kingdon (*) P. Vanı´cˇek M. Santos Department of Geodesy and Geomatics Engineering, University of New Brunswick, P.O. Box 4400, Fredericton, NB, Canada E3B 1L6 e-mail:
[email protected]
Introduction
Forward modeling of gravitational effects of threedimensionally varying density distributions has a long history. Evaluation of the Newton kernel over three-dimensionally varying mass distributions is an important task in geophysics, and later geodesy. Early attempts decompose crustal masses into prisms, over which the Newton integral is evaluated analytically, to determine either the effect of the masses on gravity
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(Mollweide 1813) or gravity potential (Bessel 1813). More complex representations of crustal density bodies model crustal masses as polyhedrons (Paul 1974) or tesseroids (Seitz and Heck 2001), and may allow vertical density variation within the bodies (e.g. Poha´nka 1998). In terms of geodesy, recent efforts have relied heavily on forward modeling of crustal mass effects, most prominently in creating synthetic gravity models (e.g. Baran et al. 2006) useful for testing geoid computation techniques. However, three dimensional crustal density effects have not yet been incorporated in Stokes-Helmert geoid modeling. The Stokes-Helmert method of geoid modeling requires determination of effects of all topographical masses, i.e. crustal masses above the geoid. These calculations have traditionally used a constant value of topographical density (e.g. Vanı´cˇek and Kleusberg 1987), but numerous investigations have shown that to obtain a precise geoid the effects of density variations within topography must also be calculated (e.g. Martinec 1993; Pagiatakis et al. 1999; Huang et al. 2001). These efforts have almost exclusively focused on horizontal density variations. Since the actual topographical density varies with depth, two dimensional topographical density models (2DTDMs) cannot exactly model the real density distribution. Martinec (1993) suggests a method of dealing with threedimensionally varying density by averaging density in each topographic column to derive a laterallyvarying density distribution, and this approach is applied by Martinec et al. (1995) to find effects of lake waters on the geoid. However, in most situations the information required to construct a 2DTDM using averaged data along columns of topography is not available, and so some other method is used, such as assigning surface density values to a whole column of topographical density (Huang et al. 2001), or applying Monte Carlo methods (Tzaivos and Featherstone 2001). Investigation into effects of three-dimensionally varying density has been limited, because a three dimensional topographical density model (3DTDM) has not yet been developed with a high enough resolution and over a large enough area to be suitable for geoid modeling (Kuhn 2003). This is because the 3D density structure of the topography is known to a high resolution only over small areas (for local geophysical
R. Kingdon et al.
studies or prospecting), or to very coarse resolutions over large areas (e.g. the CRUST 2.0 model developed by Bassin et al. 2000). A more complete discussion on the difficulties of creating a 3D density model for geodetic purposes is given in Kingdon et al. (2009). Despite the lack of suitable 3DTDMs for geoid modeling, we can still guess some things about the shortcomings of 2DTDMs. Kingdon et al. (2009) recently showed that in the presence of a single body of topographical density not accounted for in the 2DTDM, using only a 2DTDM might introduce errors of up to several centimeters in areas of high topography. In reality, topography does not contain only a single body of anomalous density, but is a complex arrangement of bodies of varying densities. Thus, effects of a single body might be mitigated by the effects of the bodies around it. In this effort, we try to discover whether in extreme cases adjacent density bodies mitigate each other’s effects on the geoid. If effects of adjacent masses cancel each other even in a hypothetical situation created so that they are unlikely to do so, it is unlikely that the less extreme situations existing in reality will be of any concern. However, if the effects of the adjacent masses remain significant then more work is necessary to define situations where 3DTDMs are needed. Once we know what constitutes a distribution where 3DTDMs are needed, will we be ready to choose some real data sets where these situations exist, for further testing. Our investigation is done within the framework of the Stokes-Helmert scheme of geoid modeling, following the methodology discussed in Sect. 51.2. Section 51.3 will show and discuss our results using this methodology, and finally we will present the conclusions derived from our results and make recommendations for future work in Sect. 51.4.
51.2
Methodology
51.2.1 3D Density Modeling in the Stokes-Helmert Context Stokes-Helmert geoid computation requires a model of topographical density both for calculating the transformation of gravity anomalies to the Helmert
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Effects of Hypothetical Complex Mass-Density Distributions on Geoidal Height
space (called the Direct Topographical Effect or DTE), and for calculating the transformation of the Helmert co-geoid back to the real space (the Primary Indirect Topographical Effect or PITE) after the Stokes integration (Martinec and Vanı´cˇek 1994a, b). Existing models normally consider topography of constant density, r0 (usually 2,670 kg m3), and may additionally include laterally variations of density, dr with respect to r0. For our modeling, we will consider the variation of topographical density from the laterally varying values, in a three dimensional sense. We label this anomalous topographical density dr. It can be considered as a residual density term, such that: rðr; OÞ ¼ r0 þ drðOÞ þ drðr; OÞ;
(51.1)
where r is the geocentric radius of a point where density is being represented, and O is a geocentric direction, representing the point’s geocentric latitude and longitude Each of the transformations required in the StokesHelmert method comprise an evaluation of the difference between the effect of real and of condensed anomalous topographical density at the location of each gravity anomaly. Here, we follow the approach outlined in Kingdon et al. (2009), which is a generalization of the approach given by Martinec (1998), and uses Helmert’s second condensation method (Martinec and Vanı´cˇek 1994a, b). The DTE on gravity is calculated by the integral formula: edr DTE ðr; OÞ ðð ¼ 0
O 2O0
2
@ 6 4 @r
0 rt ðO ðÞ
0
0
0
0
02
0
drðr ; O ÞKðr; O; r ; O Þr dr
429
The PITE on gravitational potential, edr PITE ðrg ðOÞ; OÞ, is calculated by the formula: edr PITE ðrg ðOÞ; OÞ 2 0 rt ðO ðð ðÞ @ 6 2 ¼ drðr 0 ; O0 ÞKðr; O; r0 ; O0 Þr 0 dr0 4 @r O0 2O0
r 0 ¼rg ðO0 Þ
#
0
0
0
dsðO ÞKðr; O; rg ðO Þ; O Þrg ðO Þ dO0 : 2
0
(51.3) Notice that the DTE for a particular computation point is evaluated at the topographical surface, since it is applied to gravity anomalies at the topographical surface. The PITE is evaluated for a point on the geoid surface, which we approximate for the evaluation of the Newton kernels as a sphere of radius R ¼ 6371008.144 m, the mean radius of the Earth. The condensation density referred to in these formulas is calculated for a 3DTDM according to:
1 dsðOÞ ¼ 2 R
rt ðOÞ ð
r 0 drðr0 ; OÞdr 0 : 2
(51.4)
r 0 ¼r g ðOÞ
For our investigation, we convert the effects in (51.2) and (51.3) into effects on geoidal heights. In the case of the DTE, the effect can be computed by applying Stokes integration to the DTE on gravity: : dr dNDTE ðOÞB¼
1 4pgðOÞ
ðð
0 0 Sðc½O; O0 Þedr DTE ðO ÞdO ;
O0 2O0
0
r0 ¼rg ðO Þ
(51.5)
dsðO0 ÞKðr; O; rg ðO0 Þ; O0 Þrg ðO0 ÞdO0 (51.2) where edr DTE is the DTE on gravity at a point, and dr is the anomalous density given by a 3DTDM for the integration point at coordinates r0 , O0 . rt(O0 ) and rg(O0 ) are the surface of the topography and the geoid, respectively. The function K(r,O;r0 ,O0 ) is the Newton kernel, equal to the inverse distance between the computation and integration points.
dr is the DTE on geoidal height, g (O) is the where dNDTE normal gravity on the surface of the reference ellipsoid, and S(c[O,O0 ]) is the Stokes kernel. In the case of the PITE, the effect on geoidal height is found by applying Bruns’s formula: dr dNPITE ðOÞ ¼
edr PITE ðOÞ : gðOÞ
(51.6)
These effects on geoidal height allow us to compare the effects of masses to some meaningful tolerance
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to determine whether they are significant. In this case, where we are looking for a 1 cm geoid, we will consider any effect over 0.5 cm significant.
51.2.2 Numerical Considerations The Newton kernel and its integrals and derivatives in (51.2) and (51.3) can be computed numerically in various ways. For our computation, we use the prismoidal method as summarized by Nagy et al. (2000, 2002) for integration near to the computation point, and 2D numerical integration (Martinec 1998) farther from the computation point. Applying the prismoidal method to a 3DTDM, the anomalous topographical masses are divided into blocks, and the integral of the Newton kernel in planar coordinates is evaluated over each block analytically. The same procedure was applied in a slightly different context by Kuhn (2003). This formulation captures very well the behavior of the Newton kernel near to the computation point, and in that region is superior to 2D numerical integration (Heck and Seitz 2007), even though the 2D integration normally uses the more accurate spherical expression of the Newton kernel. The accuracy improvement from evaluating the kernel analytically near to the computation points outweighs the accuracy benefit of using a spherical formulation (Nagy et al. 2000). The prismoidal formula is inherently faster than other analytical methods such as the polyhedral method, and is also faster than the tesseroidal method near to the computation point. The tesseroidal method is very fast when only the zero-order and second-order terms of its Taylor series representation are used, but these do not provide sufficient accuracy near to the computation point (Heck and Seitz 2007). Comparison of the planar Newton kernels used in the prismoidal approach and the spherical Newton kernels shows that the kernels used to evaluate the DTE are more than 1% different beyond 5 arc-minutes from the computation point, and that those used to evaluate the PITE are more that 1% different beyond 15 from the computation point. Fortunately, even for computation points within these ranges of the computation point, the 2D numerical integration provides identical results to the slower prismoidal approach, and so these differences are moot.
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The 2D integration employs the radial integrals of the Newton kernel developed by Martinec (1998) to perform the radial integration of the Newton kernel over the vertical anomalous density variations in a given topographical column, discretized as segments of the column, thus evaluating the radial component of the 3D integral analytically. The horizontal integration is performed by summing the radial integral over each particular column, and then summing the products of the values of the integrands of (51.2) and (51.3) at the cell centers with the cell areas. The 2D horizontal integration is suitable beyond about 5 arc-minutes of the computation point for the DTE, and beyond 1 of the computation point for the PITE. For our evaluations of the PITE, we use the prismoidal formula within 5 of the computation point, to take greater advantage of its superior accuracy near the computation point. For the DTE, we use the prismoidal formula only within 5 arc-minutes of the computation point. Beyond these limits, 2D integration is used. With both of the methods we have chosen above, a discretization error is present since the actual mass distribution of the topography is represented as a series of rectangular prisms of varying height. Such discretization errors will be present unless the topographical density model exactly reflects the topographical density distribution, and is difficult to quantify since its behavior changes for different mass distributions. For example, discs of different size will have different discretization errors. We can decrease the error by using a smaller cell or prism size in our integration procedures. By testing we have found that for the discs used here a resolution of 1 1 arc-second is sufficient. Using a higher resolution than 1 1 arc-second did not significantly change the results (<0.01 mGal), while going from 1 1 arc-second to 3 3 arc-second step affected the results by up to 0.21 mGal. To validate our computational procedures in this investigation, we have tested our numerical integration for the case of a single disc of anomalous density, against results from analytical formulas for the DTE on gravity and PITE on gravity potential at the center of the disc, similar to the approach of Heck and Seitz (2007). This is the point where the PITE caused by the disc is greatest, and a point where the DTE is very large. In the test, we find errors of up to 18% in the numerical integration for the DTE and 5% for the PITE in extreme cases, but normally less than 5% for
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Effects of Hypothetical Complex Mass-Density Distributions on Geoidal Height
the DTE and 1% for the PITE. The larger errors occur when the disc is very small, and consequently do not indicate a large magnitude of overall error. In terms of magnitude, the largest errors in the DTE are only 0.4 mGal, and in PITE on geoidal height only 0.5 mm. Errors in DTE and PITE were usually less than 0.15 mGal and 0.3 mm in magnitude respectively. We consider such errors admissible for determining the order of magnitude of differences between results from 3DTDMs and 2DTDMs.
51.2.3 Proposed Tests Our question is: can the effect on geoidal height of an anomalous density body, unaccounted for by a 2DTDM, be mitigated by the presence of adjacent bodies of different anomalous density? To investigate this, we take two extreme cases, each involving an array of anomalous masses. In case A, we choose masses that individually are known to have a particularly large DTE, and investigate the effect of the conglomerate of these masses on geoidal height. In case B, we choose masses known to have a large PITE. In each case, we integrate three dimensionally over the masses considered unaccounted for in a 2DTDM, so that the results of the integration will give us the deficiency of the 2DTDM. If the effects mitigate each other significantly, it is an indicator that we can expect the same in any less extreme case. In both cases A and B, we use an array of vertical cylinders as our density model. The upper part of the cylinders are assigned alternating density contrasts of positive or negative 600 kg m3, considered anomalous relative to a laterally-varying density model. The anomalous density outside the cylinders is zero; i.e., the 2DTDM is considered accurate outside of the cylinders. Our past work on individual mass bodies has shown that the DTE and PITE are greatest when: 1. The topography involved is thick 2. Anomalous density is distributed away from the geoid, and 3. There is a large density contrast Regarding the horizontal size of the bodies, for the PITE the larger the body the greater its effect will be, although the rate at which the effect increases becomes very low for bodies beyond about 110,000 m wide. Therefore, we use a width of 110,000 m for the
431
discs in case B. Of course, the largest PITE would be for a spherical shell, but such a model would not allow us to test whether adjacent masses mitigate each other, and so we have instead used a disc that induces much of the effect that a spherical shell of the same thickness would, but whose effect still might be mitigated by adjacent discs of opposite anomalous density. For the DTE, by contrast, there is a range of disc widths of about 3,300 m where the effect is greatest. This is because the DTE is the difference between the effect of a real anomalous mass on gravity at a computation point, and the effect of the anomalous mass condensed onto the geoid. At about 3,300 m from the computation point, for the case of a disc as tested here, these effects become similar, and thereafter the condensed density of a given mass has a greater effect on gravity at the computation point than its real density, so that the overall DTE becomes smaller. Thus, we use 3,300 m as the disc diameter for case A. In order to accommodate items 1 and 2 in the list above, we choose flat topography 2,000 m thick, and use discs extending from the surface of the topography to 500 m depth. These can be thought of as anomalous masses relative to a 2DTDM that accurately portrays the density below the masses. An illustration of our model, including possible cylinder densities that might result in the +/600 kg m3 anomalous densities used in our experiment, is given in Fig. 51.1. We calculate results over a 1 by 1 area for our case A simulation, and a 2 by 2 area for our case B
500 m
2200 kg m–3 (–600 kg m–3) 3300 kg m–3 (+600 kg m–3) 2800 kg
1500 m
m–3
2700 kg m–3 w
Fig. 51.1 Topographical density distribution used for testing. w ¼ 3,300 m for DTE tests, and w ¼ 110,000 m for PITE tests
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simulation, both described in Sect. 51.3 below. We use a radius of 2 for Stokes integration, and so our array of cylinders in case A extended over a 5 by 5 area to capture most of their effect.
It thus remains to find the maximum effect that masses can have in a simulation like our own.
51.3.2 Case B
51.3
Results
51.3.1 Case A The DTE on gravity for case A, as described in Sect. 51.2.3, is given in Fig. 51.2. The adjacent density anomalies do not significantly mitigate each other’s effects on the DTE, which reaches +/ 16 mGal. This is not surprising, since the derivative of the Newton kernel, used to calculate these effects, decreases very rapidly with distance from the source masses. However, we are ultimately interested in the DTE on geoidal height, resulting from the Stokes integration (given by (51.5)) over the DTEs on gravity, and shown in Fig. 51.3. Under the smoothing influence of the Stokes kernel, the adjacent masses attenuate each other’s contributions to the DTE on the geoidal height, which reaches about +/ 0.85 cm. This may not be the case for density anomalies with greater horizontal extent, since in such cases the Stokes integral would do less to mitigate the effects of the anomaly on gravity at its center, where the effect is largest.
Fig. 51.2 DTE on gravity for case A
The PITE on gravity for case B, as described in Sect. 51.2.3, is given in Fig. 51.4. We see that for such large cylinders, the effect of adjacent cylinders of opposite anomalous density is minimal. Here the effects reach +/ 5 cm, but for larger discs the magnitude would be somewhat greater and the
Fig. 51.3 DTE on geoidal height for case A
Fig. 51.4 PITE on geoidal height for case B
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Effects of Hypothetical Complex Mass-Density Distributions on Geoidal Height
attenuation even less significant, since it is hardly significant even for the discs tested here. It is thus likely that the PITE will only increase for wider cylinders. Conclusions
The DTE on gravity and the PITE on geoidal height for the anomalous masses (not modeled in 2DTDM) in our simulations are not significantly diminished by the presence of adjacent anomalous masses, even when there is an extreme density contrast. The PITE still reaches about 5 cm, and the DTE reaches about 16 mGal. This demonstrates that the error in the PITE resulting from only using a 2DTDM may be large even in the presence of adjacent mass anomalies. The DTE on geoidal height resulting from using a 2DTDM is significantly diminished by the presence of adjacent masses, though it still approaches 1 cm level. However, this is entirely a result of Stokes integration, which was not considered in the development of our extreme case scenarios. With this in mind, larger bodies of anomalous density that still have a significant impact on the DTE on gravity may produce significantly larger effects on geoidal heights.
References Baran I, Kuhn M, Claessens SJ, Featherstone WE, Holmes SA, Vanı´cˇek P (2006) A synthetic Earth gravity model designed specifically for testing regional gravimetric geoid determination algorithms. J Geod 80:1–16 Bassin C, Laske G, Masters G (2000) The current limits of resolution for surface wave tomography in North America. EOS Trans AGU 81:F897 Bessel FW (1813) Auszug aus einem Schreiben des Herrn Prof. Bessel. Zach’s Monatliche Correspondenz zur Bef€ orderung der Erd- und Himmelskunde, XXVII, pp 80–85 Heck B, Seitz K (2007) A comparison of the tesseroid, prism and point-mass approaches for mass reductions in gravity field modelling. J Geod 81:121–136 Huang J, Vanı´cˇek P, Pagiatakis S, Brink W (2001) Effect of topographical density variation on geoid in the Canadian Rocky Mountains. J Geod 74:805–815
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Kingdon R, Vanı´cˇek P, Santos M (2009) First results and testing of a forward modelling approach for estimation of 3D density effects on geoidal heights. Canad J Earth Sci 46 (8):571–585. doi:10.1139/E09-018 Kuhn M (2003) Geoid determination with density hypotheses from isostatic models and geological information. J Geod 77:50–65 Martinec Z (1993) Effect of lateral density variations of topographical masses in improving geoid model accuracy over Canada. Contract Report for Geodetic Survey of Canada, Ottawa Martinec Z (1998) Boundary value problems for gravimetric determination of a precise geoid (Lecture Notes in Earth Sciences). Springer, New York Martinec Z, Vanı´cˇek P (1994a) Direct topographical effect of Helmert’s condensation for a spherical approximation of the geoid. Manuscripta Geodaetica 19:257–268 Martinec Z, Vanı´cˇek P (1994b) The indirect effect of topography in the Stokes-Helmert technique for a spherical approximation of the geoid. Manuscripta Geodaetica 19:213–219 Martinec Z, Vanı´cˇek P, Mainville A, Ve´ronneau M (1995) The effect of lake water on geoidal height. Manuscripta Geodaetica 20:199–203 Mollweide KB (1813) Aufl€osung einiger die Anziehing von Linien Fl€achen und K€opern betreffenden Aufgaben unter denen auch die in der Monatl Corresp Bd XXIV. S, 522. vorgelegte sich findet. Zach’s Monatliche Correspondenz zur Bef€orderung der Erd- und Himmelskunde, XXVII, pp 26–38 Nagy D, Papp G, Benedek J (2000) The gravitational potential and its derivatives for the prism. J Geod 74:552–560. doi:10.1007/s001900000116 Nagy D, Papp G, Benedek J (2002) Corrections to “The gravitational potential and its derivatives for the prism”. J Geod 76:475. doi:10.1007/s00190-002-0264-7 Pagiatakis, S., D. Fraser, K. McEwen, A. Goodacre and M. Ve´ronneau (1999). Topographic mass density and gravimetric geoid modelling. Bollettino di Geofisica Paul MK (1974) The gravity effect of a homogeneous polyhedron for three dimensional interpretation. Pure Appl Geophys 112:553–561 Poha´nka V (1998) Optimum expression for computation of the gravity field of a polyhedral body with linearly increasing density. Geophys Prospect 46:391–404 Seitz K, Heck B (2001) Tesseroids for the calculation of topographic reductions. IAG 2001 Scientific Assembly, Budapest, Hungary, September 2–7 Tzaivos IN, Featherstone WE (2001) First results of using digital density data in gravimetric geoid computation in Australia. In: Sideris MG (ed) Gravity, geoid and geodynamics. Springer, Berlin, pp 335–340 Vanı´cˇek P, Kleusberg A (1987) The Canadian geoid – stokesian approach. Manuscripta Geodaetica 12:86–98
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Evaluation of Gravity and Altimetry Data in Australian Coastal Regions
52
S.J. Claessens
Abstract
Satellite altimetry in near-coastal marine areas is notoriously problematic, and gravity anomalies derived from various satellite-altimetry-derived gravity models differ significantly near the coast. In this paper, gravity anomalies from the DNSC08 and Sandwell & Smith v18.1 (SS18) models are compared and ‘validated’ against shipborne and airborne gravity anomalies around the Australian coast. Due to the scarcity of high-quality gravity observations just off the coast, a true validation of the models cannot be achieved. However, DNSC08 conforms slightly better to both shipborne and airborne gravity observations closest to the coast in selected test areas, although the standard deviation of differences between the models and the test data barely exceeds the estimated test data accuracy.
52.1
Introduction
Over the past two decades, satellite radar altimetry has proven a valuable tool for the computation of marine gravity anomalies. Several marine gravity models have been computed from satellite altimetry data. Two of the most recent models are the DNSC08 model (Andersen et al. 2008) and the Sandwell & Smith v18.1 model (SS18; Sandwell and Smith 2009). It is well-known that the accuracy of satellitederived gravity data degrades close to the coast (e.g., Andersen and Knudsen 2000; Deng et al. 2002). The main reasons for this are the poorer accuracy of applied corrections, most notably tide
S.J. Claessens (*) Western Australian Centre for Geodesy, The Institute for Geoscience Research, Curtin University of Technology, GPO Box U1987, Perth, WA 6845, Australia e-mail:
[email protected]
corrections, near the coast, and the fact that the altimeter loses track of the sea surface close to the coast. The poorer accuracy of altimeter-derived gravity data near the coast provides a challenge for geoid modelling in coastal areas, because it leaves a gap between high-quality land gravity observations (where available) and high-quality marine gravity data from satellite altimetry further away from the coast. High-quality ship-borne gravity data is also scarce close to the coast in many areas. In some regions, the gap between altimetry derived gravity and land gravity also displays a significant ‘jump’ of several tens of milliGals (e.g. in the Perth region of Western Australia, Claessens et al. 2001). Airborne gravity data can fill this gap, because it provides a seamless coverage over land and sea (e.g. Olesen et al. 2002; Hwang et al. 2006), but it is not available everywhere. In Australia, very little airborne data is available, and geoid modelling relies almost entirely on satellite
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S.J. Claessens
altimetry data in marine coastal areas. Comparisons between the AUSGeoid98 geoid model over Australia (Featherstone et al. 2001) and the EGM2008 global gravity model (Pavlis et al. 2008) have shown that the largest differences between these two models in terms of (quasi)geoid height (up to ~1 m) exist in several marine coastal regions (Claessens et al. 2009). The majority of these differences can be attributed to differences between marine gravity models used in the computation of AUSGeoid98 and EGM2008. AUSGeoid98 is based on the Sandwell & Smith v7.2 model (Sandwell and Smith 1997) and EGM2008 is based on the DNSC07 model (Pavlis et al. 2008). In view of continuous efforts to improve the Australian geoid model (e.g. Featherstone et al. 2007), the DNSC08 and SS18 marine gravity models are compared against each other, as well as against shipborne and airborne gravity data, where available, around the Australian coast. While both models are largely based on the same altimeter data (ERS-1 and Geosat/GM), different computational techniques and gridding algorithms are used in their computation, giving rise to significant differences near the coast.
52.2
Description of Data
52.2.1 Satellite Altimetry Models Both the DNSC08 and the SS18 model have a resolution of 10 (~1–2 km). DNSC08 uses marine geoid heights (corrected for mean dynamic topography) to compute gravity anomalies. The reference global gravity model used is PGM2007B, a test version of EGM2008. Interpolation is achieved through leastsquares collocation (Andersen et al. 2008). SS18 uses a conversion of vertical deflections via integration of Laplace’s equation to compute gravity anomalies. The reference global gravity model used in the remove-restore-compute process is EGM2008. Biharmonic spline interpolation is used to compute the residual vertical deflection grid. Furthermore, in the computation of SS18 the applied tide model is lowpass filtered with a cut that begins at 26.8 km wavelength, has 0.5 gain at 14.6 km, and zeros at 10 km (Sandwell and Smith 2009). For further information about both models, refer to the respective references provided.
52.2.2 Shipborne Gravity Data Ship-track gravity data around Australia (Symonds and Willcox 1976; Mather et al. 1976) was extracted from the July 2007 version of the Australian national gravity database (Murray 1997). The marine gravity observations were removed from subsequent versions of the database by Geoscience Australia based on words of caution in Featherstone (2009), who showed the existence of many gross errors (up to 900 mGal). These ship-track gravity data had not been crossover adjusted. Petkovic et al. (2001) readjusted them, but the observations were constrained to Sandwell and Smith’s altimeter-derived gravity anomalies (version unknown), which renders the re-adjusted ship-track data unfit for the purpose of this study. The shiptrack data from the July 2007 version of the Australian national gravity database used in this study are the non-crossover-adjusted ones, which are thus independent from any satellite altimetry model. Featherstone (2003) shows that the quality of the ship-track gravity data around Australia overall is not of sufficient quality to be used in validation of satellite altimetry-derived gravity models. However, in some regions around the Australian coast, such as on the north-west shelf off Western Australia’s Pilbara coast, the ship-track data is of better-thanaverage quality (cf. Claessens et al. 2009), and validation may be more successful in small nearcoastal regions.
52.2.3 Airborne Gravity Data The airborne gravimetry dataset from the Barrier Reef Airborne Gravity Survey (BRAGS’99) (Sproule et al. 2001), carried out in 1999, contains the only airborne gravity data available over Australian coastal waters. This survey covers an area over the shallow waters of the Great Barrier Reef to the north-east of Australia. The flight altitude was ~500 m and low-pass filtering was applied with filter parameters set such that the survey has a spatial resolution of 8 km. Sproule et al. (2001) performed a crossover analysis and found a standard deviation of 2.8 mGal, which is used as an estimate of the noise level of the data.
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Evaluation of Gravity and Altimetry Data in Australian Coastal Regions
52.3
Results
SS18 is provided on a regular Mercator projection grid and DNSC08 is provided on a regular latitude/ longitude grid, the grid nodes of the two models were first made to coincide. This was achieved through a conversion of Mercator grid coordinates to latitude and longitude using the Generic Mapping Tool (GMT) (Wessel and Smith 1998), followed by bicubic interpolation of SS18 gravity anomalies to a regular latitude/longitude grid. The differences between DNSC08 and SS18 are significantly smaller than the differences between the KMS01 model (an earlier version of DNSC08) and the Sandwell & Smith v9.2 model found by Featherstone (2003), which is a testament to the improvements of the newer models compared to their predecessors. However, significant differences can still be seen
Firstly, the DNSC08 and SS18 models are compared to one another to evaluate the differences between the two models. Then a validation is attempted by use of shipborne and airborne gravity data, in areas where high-quality data is available and where large differences occur between DNSC08 and SS18.
52.3.1 Comparison of Satellite Altimetry Models Differences between altimetry-derived gravity anomalies around the Australian coast from the DNSC08 and SS18 models are shown in Fig. 52.1. Because
110° –10°
120°
437
130°
140°
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Fig. 52.1 Differences between gravity anomalies computed from DNSC08 and SS18 (Mercator projection; units in mGal)
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(Fig. 52.1). They are largest near the coast and over the Great Barrier Reef. In many near-coastal regions, ‘stripes’ in north-south direction can be seen with a wavelength of ~20–30 arcmin (~40–60 km) and amplitude up to ~15 mGal. These ‘stripes’ are not very regular due to additional distortions, but they are visible nonetheless. They are most profound off the northern coasts of Western Australia (see insets), but can also be seen in the Gulf of Carpentaria (~140 E, 17 S) and over the Great Barrier Reef (~148 E, 20 S). The reason for the appearance of these ‘stripes’ is not obvious, but is possibly an artefact of the processing strategy applied in one of the models. The differences between DNSC08 and SS18 cannot be explained by the fact that both use a different reference global gravity model (PGM2007B and EGM2008, respectively). Differences between these two reference models are smaller in magnitude and do not show the striping effect. Large differences between DNSC08 and SS18 also occur off the central Queensland coast (~150 E, 22 S). This is a challenging region for satellite altimetry due to its shallow sea and abundance of small islands. It is also a region where Sandwell and Smith (2009) found a large RMS deviation of along-track slopes from retracked Geosat GM data with respect to slopes derived from gravity field estimation. Unfortunately, in this region no independent data in the form of shipborne or airborne gravity data is available.
52.3.2 Comparisons with Shipborne Gravity Data One of the few areas where ship-track data of reasonable quality and coverage is available is the north-west shelf off the Pilbara coast of Western Australia. This is also the region where the ‘stripes’ in the differences between DNSC08 and SS18 gravity anomalies are the clearest. For the comparison of DNSC08 and SS18 to the shipborne gravity anomalies, the original gridded anomalies from the models were bicubically interpolated to the positions of the shipborne data. Thus, SS18 gravity anomalies were interpolated straight from the Mercator grid to shipborne data positions, not to a regular latitude/longitude grid first. This ensures a fair comparison of both models.
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Unfortunately, the ship-tracks are not close enough to the shoreline to determine the cause of the ‘stripes’ with any certainty. However, from visual inspection, DNSC08 gives a slightly better agreement with the ship-track data than SS18 (Fig. 52.2). For example, at (~120 E, 19.7 S), the ship-track data agrees better with DNSC08 than with SS18. However, due to the limited accuracy of the shiptrack data this observation must be treated with care. Figure 52.2 more clearly and not surprisingly shows that the quality of the shipborne data, even in this area, is of insufficient quality to properly validate either model.
52.3.3 Comparisons with Airborne Gravity Data Molodensky-type free-air gravity anomalies were computed from the raw gravity observations taken at flight altitude during the BRAGS’99 survey. These gravity anomalies were ‘downward-continued’ to mean sea level by adding the difference between EGM2008 ellipsoidal (linearly approximated) gravity anomalies at sea level (approximated by the quasigeoid computed from EGM2008) and at flight altitude. This gave a closer agreement between the airborne data and all tested models than free-air downward continuation. Figure 52.3 shows the differences between the downward continued airborne gravity data and EGM2008, DNSC08 and SS18. For this purpose, EGM2008 gravity anomalies were computed through synthesis of the spherical harmonic coefficients in ellipsoidal approximation. DNSC08 and SS18 gravity anomalies were bicubically interpolated from the original model coordinates to the airborne gravity observation coordinates. A similar pattern can be seen in all three plots of Fig. 52.3, especially further away from the coast, indicating that the differences here are primarily due to errors in the airborne data. The magnitudes of these differences are as expected. Closer to the coast, however, differences can be seen. For example in Princess Charlotte Bay (~144 E, 14 S), EGM2008 and DNSC08 agree significantly better with the airborne data than SS18, and a similar pattern can be seen in some other areas. For example, a large difference with SS18 is seen at ~145 E, 14.5 S, while the
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Fig. 52.2 Differences between gravity anomalies from DNSC08 and SS18 (top), ship-track observations and DNSC08 (centre), and ship-track observations and SS18 (bottom) (Mercator projections; units in mGal)
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Fig. 52.3 Differences between gravity anomalies from the BRAGS’99 airborne gravity survey and EGM2008 (left), DNSC08 (centre) and SS18 (right) (Mercator projections; units in mGal)
comparison with DNSC08 does not show a large difference there. A numerical analysis on the differences was performed by dividing the airborne gravity data in three regions based on their distance from the coast: less than 20 km from the coast, 20–50 km from the coast and more than 50 km from the coast. These values were selected based on the finding by Deng et al. (2002) that altimeter data around Australia is poorest within 8–22 km from the coast. The accuracy of the airborne gravity data is assumed to be largely independent of distance from the coast. GPS positioning of the aircraft may be less accurate further from the coast due to the larger distance from base stations, but the effect on gravity anomalies is relatively small and opposite to the expected trend of poorer accuracy near the coast displayed by the altimetry models. Statistics of differences between airborne data and the altimetry models and EGM2008 in the three regions are shown in Table 52.1. It can be seen that the differences between the airborne data and DNSC08 show little dependence on proximity to the coast, since the standard deviations of the differences are similar in each region. SS18 has a very similar standard deviation >50 km from the coast, where the differences are
Table 52.1 Statistics of differences with airborne gravity at various distances from the coast after 3s outlier removal (from top to bottom: standard deviation, mean, absolute maximum, number of outliers) Model All 0–20 km # Points 6,726 1,626 Standard deviation (mGal) EGM2008 3.751 2.816 DNSC08 3.754 3.609 SS18 4.125 4.447 Mean (mGal) EGM2008 2.267 1.871 DNSC08 2.401 1.857 SS18 2.606 2.244 Absolute maximum (mGal) EGM2008 22.458 12.202 DNSC08 18.731 16.318 SS18 26.182 26.182 Number of outliers EGM2008 62 17 DNSC08 85 32 SS18 92 48
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likely to be dominated by errors in the airborne data, but closer to the coast the standard deviations are larger, indicating a degraded accuracy close to the
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coast. Interestingly, EGM2008 shows the smallest standard deviation closest to the coast, performing better than the altimetry models despite its lower spatial resolution. Conclusions
Comparison between the DNSC08 (Andersen et al. 2008) and Sandwell & Smith v18.1 (SS18) (Sandwell and Smith 2009) satellite-altimetryderived gravity anomaly models shows differences around the Australian coast up to and greater than 20 mGal. The differences are mainly of shortwavelength nature and show ‘stripes’ in northsouth direction in several areas near the coast. The cause of the differences between DNSC08 and SS18 cannot be determined with certainty from a validation with Australian data. Shipborne gravity data around Australia has insufficient coverage and quality <20 km from the shoreline to validate satellite altimetry models. The best dataset in Australia for validation of satellite altimetry models is the BRAGS’99 airborne gravity survey over a small area off northern Queensland. DNSC08 agrees better with airborne gravity data than SS18 <20 km from the coast (standard deviation of 3.6 mGal vs. 4.4 mGal respectively), but not as well as EGM2008 (2.8 mGal). At >50 km from the coast, DNSC08 and SS18 perform equally well (3.7 mGal), and better than EGM2008 (4.2 mGal). However, it should be noted that these standard deviations do not exceed the estimated accuracy of the airborne data (2.8 mGal standard deviation) by much.
Acknowledgements All suppliers of data for this study are gratefully acknowledged.
References Andersen OB, Knudsen P (2000) The role of satellite altimetry in gravity field modelling in coastal areas. Phys Chem Earth 25(1):17–24 Andersen OB, Knudsen P, Berry P, Kenyon S (2008) The DNSC08 ocean wide altimetry derived gravity field. Presented to EGU-2008, Vienna, Austria, April, 2008
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Claessens SJ, Featherstone WE, Barthelmes F (2001) Experiences with point-mass modelling in the Perth region, Western Australia. Geomatics Research Australasia 75:53–86 Claessens SJ, Featherstone WE, Anjasmara IM, Filmer MS (2009) Is Australian data really validating EGM2008, or is EGM2008just in/validating Australian data? Newton’s Bull 4:207–251 Deng XL, Featherstone WE, Hwang C, Berry PAM (2002) Estimation of contamination of ERS-2 and POSEIDON satellite radar altimetry close to the coasts of Australia. Mar Geod 25(4):249–271. doi:10.1080/01490410290051572 Featherstone WE (2003) Comparison of different satellite altimeter-derived gravity anomaly grids with ship-borne gravity data around Australia. In: Tziavos IN (ed) Gravity and Geoid 2002, Department of Surveying and Geodesy, Aristotle University of Thessaloniki, 326–331 Featherstone WE (2009) Only use ship-track gravity data with caution: a case-study around Australia. Austr J Earth Sci 56 (2):191–195. doi:10.1080/08120090802547025 Featherstone WE, Kirby JF, Kearsley AHW, Gilliland JR, Johnston GM, Steed J, Forsberg R, Sideris MG (2001) The AUSGeoid98 geoid model of Australia: data treatment, computations and comparisons with GPS-levelling data. J Geod 75(5–6):313–330. doi:10.1007/s001900100177 Featherstone WE, Claessens SJ, Kuhn M, Kirby JF, Sproule DM, Darbeheshti N, Awange JL (2007) Progress towards the new Australian geoid-type model as a replacement for AUSGeoid98.In: Proceedings of SSC 2007, Hobart, May 2007 Hwang C, Guo J, Deng X, Hsu H-Y, Liu Y (2006) Coastal gravity anomalies from retracked Geosat/GM altimetry: improvement, limitation and the role of airborne gravity data. J Geod 80:204–216. doi:10.1007/s00190-0060052-x Mather RS, Rizos C, Hirsch B, Barlow BC (1976) An Australian gravity data bank for sea surface topography determinations (AUSGAD76), Unisurv G25. School of Surveying, University of New South Wales, Sydney, pp 54–84 Murray AS (1997) The Australian national gravity database. AGSO Journal of Australian Geology & Geophysics, 17: 145-155. Also see http://www.ga.gov.au/minerals/research/ methodology/geophysics/ngdpage.jsp Olesen AV, Andersen OB, Tscherning CC (2002) Merging of airborne gravity and gravity derived from satellite altimetry: test cases along the coast of Greenland. Studia Geophysica et Geodaetica 46(3):387–394. doi:10.1023/A:1019577232253 Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2008) An Earth Gravitational Model to Degree 2160:EGM2008, Presented to EGU-2008, Vienna, Austria, April, 2008. Also see http:// earth-info.nga.mil/GandG/wgs84/grav-itymod/egm2008/ index.html Petkovic P, Fitzgerald D, Brett J, Morse M, Buchanan C (2001) Potential field and bathymetry grids of Australia’s margins, Proc ASEG 15th Geophysical Conference and Exhibition, Brisbane, August [CD-ROM] Sandwell DT, Smith WHF (1997) Marine gravity anomaly from Geosat and ERS-1 altimetry. J Geophys Res 102 (B5):10039–10054
442 Sandwell DT, Smith WHF (2009) Global marine gravity from retracked Geosat and ERS-1 altimetry: Ridge segmentation versus spreading rate. J Geophys Res 114:B01411. doi:10.1029/2008JB006008 Sproule D, Kearsley AHW, Olesen A, Forsberg R (2001) Barrier Reef Airborne Gravity Survey (BRAGS’99), in:
S.J. Claessens Geoscience and Remote Sensing Symposium, 2001. IGARSS’01, IEEE2001 International, 7: 3166-3168 Symonds PA, Willcox JB (1976) The gravity field offshore Australia. BMR J Aust Geol Geophys 1:303–314 Wessel P, Smith WHF (1998) New, improved version of generic mapping tools released. EOS Trans Am Geophys Union 79(47):579
Development and User Testing of a Python Interface to the GRAVSOFT Gravity Field Programs
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J. Nielsen, C.C. Tscherning, T.R.N. Jansson, and R. Forsberg
Abstract
The GRAVSOFT suite of Fortran programs enables gravity field modeling using 3D or 2D Least-Squares Collocation and Fourier techniques, the computation of topographic effects, the evaluation of high-degree spherical harmonic series and several other functions. It has been developed since the early 1970s with a lineoriented DOS-interface. Sponsored by the Geodetic Survey of Malaysia a modern graphical interface has been designed using Python (www.python.org) and the widget toolkit Tk, following the Apple Design Guidelines. A prototype was designed and tested at a geoid workshop in Malaysia. An iteration of this was then tested at the International Geoid School, Como and a third iteration during a graduate course at the University of Copenhagen. The result is two main redesigns; the GRAVSOFT Launcher Interface and the browsing Interface. User evaluation showed high satisfaction with the Interface, but identified the error/help support as dissatisfying. However 1 in 4 found it difficult to learn to use the programs. Difficulties in learning is correlated with participants educational level, showing that when applications – which have been used in research – target other user groups, redesign and user testing is required.
53.1 J. Nielsen Copenhagen Business School, Center for Applied ICT, Howitzvej 60, Frederiksberg, Denmark C.C. Tscherning (*) University of Copenhagen, Juliane Maries Vej 30, Copenhagen Oe, Denmark e-mail:
[email protected] T.R.N. Jansson University of Copenhagen, Juliane Maries Vej 30, Copenhagen Oe, Denmark Schlumberger, Titangade 15, Copenhagen N, Denmark R. Forsberg DTU-Space, Juliane Maries Vej 30, Copenhagen Oe, Denmark
Introduction
As a support for a Height Modernization Project carried out by the Geodetic Survey of Malaysia, a height reference surface (geoid) was computed in the summer of 2008. This was done using the GRAVSOFT suite of FORTRAN programs. These programs have been developed continuously since 1970 by the staff of the National Space Center, Danish Technical University (DTU-Space) [earlier a part of the Danish National Survey and Cadastre (KMS)] in cooperation with the Niels Bohr Institute, University of Copenhagen (UCPH).
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_54, # Springer-Verlag Berlin Heidelberg 2012
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The program software was constructed with a DOS interface (see e.g. Tscherning et al. 1992). This is difficult to use for users who are accustomed to graphical interfaces. On request by the Geodetic Survey of Malaysia a modern user-friendly system was developed (Sect. 53.3). UCPH had earlier cooperated with the research group on HumanComputer Interaction (HCI) at Copenhagen Business School (CBS) on an EU project on a distributed visualization tool for virtual collaboration (Nielsen et al. 2003). The task of developing GRAVSOFT with a graphical interface became an interdisciplinary undertaking in collaboration with CBS and two design principles were specified; Use and User – Minimize cognitive load on users; Interface – Logically related application programs and functions in program modules must be placed together. As tools to create the new interface the programming language Python and the widget toolkit Tk were selected (Sect. 53.4). These languages permit the creation of cross-platform interfaces (Windows, Linux, Mac) and are open-source products. http:// www.python.org/ http://en.wikipedia.org/wiki/Tk_%28framework%29 In the following we introduce the background of GRAVSOFT, and describe the design process, specifying the four phases in the iterative process. We then turn to the resulting design, the PYTHON/ Tk application, and give guidelines on installation and launch. We show and describe the Launcher interface and an example of the computation of geoid heights in a grid over Denmark based on the EGM96 Earth Gravity Model. In a final paragraph we focus on the empirical data from user testing and evaluation which is presented and discussed.
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GRAVSOFT
The GRAVSOFT programs have been used in the daily work of KMS, DTU-Space and UCPH both for gravity field modeling and teaching. The current suite of programs also contain some useful service programs for example for map-transformations and conversion of GRAVSOFT grid files to files acceptable for mapping software like Surfer (Golden Software 2002) or GMT (http://gmt.soest.hawaii.edu/ ), (see Fig. 53.2).
The programs are written and maintained in FORTRAN77, with one exception which is a FORTRAN 95 version of the program GEOCOL, which use multiprocessing when solving systems of equations or when evaluating a series of spherical harmonics. This was excluded in the redesign because many scientific communities do not have access to Fortran95 and multiCPU computers. Researcher who have access to a FORTRAN compiler, may change the source-codes if errors are found, and re-compile the programs. The programs are copyright by the authors, but distributed freely to non-commercial users, see (Forsberg and Tscherning 2008).
53.3
Interface Design: An Iterative Process
The interface design has been developed following Apple Design Guidelines (1987) and has evolved during an iterative lifecycle process of pre-analysis, conceptual design, product design and test (Preece et al. 2004). The Life Cycle is a dynamic model that combines the progression of an interaction design process over time with the returning activities of each iteration during the process from the initial project start, to the final product deliverance (cf. Fig. 53.1 below). In the Pre-analysis phase focus is on the users. Who are the users, what is their educational background, are they computer experts or novices, are there culture specific aspects to be addressed etc? What exactly does the client mean when requesting a user friendly interface? What kind of computers and software does the client use? Once this data is analyzed and put into a coherent picture the Idea Generation and Conceptual Design work is undertaken. In this phase it is important not to work under restrictions but let ideas flourish. Some ideas are more viable than others and will be conceptualized in designs which may be sketches, a list of needed functionalities, initial interface design (which may be tested with users or experts) etc In the Product Design phase, the actual development begins. This is where the interaction is designed, where a first running version of the program and the graphical interface is constructed, tested and redesigned. Test and evaluation is carried out both on conceptual design, e.g. sketches or interface drafts, and on
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Fig. 53.1 The LifeCycle model of the Interface design process
Fig. 53.2 GRAVSOFT Launcher
the product design and may be both expert tests and user tests. The conceptual design resulted in several interface sketches which were tested by experts. This led to redesign of the GRAVSOFT Launcher Interface (see Fig. 53.2) dividing it into three groups; 3d Applications; 2D Applications and Service Programs with sky blue background color to distinguish the group headings from the specification of the programs which are on grey background. After two iterations a high fidelity prototype of GRAVSOFT with new graphical interface was developed and initially tested by users who participated in the geoid workshop in Malaysia. This led to an iteration and the second version of GRAVSOFT was tested by participants in the
International Geoid School in Como. Finally a third iteration was tested during a graduate course on gravity field modeling at the University of Copenhagen. In the following we describe the resulting PYTHON/Tk interface design. We then focus on the Test and Evaluation which have been part of the launching of the program where data was collected during the geoid workshops.
53.4
Python/Tk Application
The interface design was implemented using Python and Tk. A primary reason is that the two software are under open source license and free to use, and that they
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can run on multiple platforms. This gives a few restrictions, but many other valuable features are implemented, e.g. Python itself finds the relevant Linux or Windows executable. One restriction is that each interface for each module must fit on one screen, no scrolling is possible. Also it was decided to only use GRAVSOFT data formats, i.e. data records on the form id#, latitude, longitude (both in decimal degrees), altitude (m) and a number of data columns and grid data also only in the GRAVSOFT format. Modules are available in the Service Programs section to convert to other grid-formats.
53.4.1 Installation and Launch Windows users must download and install Python from http://www.python.org. The python programs, the FORTRAN executables and the test-data must be extracted from a ZIP-file (provided on the UCPH ftpserver) using Winzip to a directory pyGravsoft which contains source code, binaries and test data. Detailed instructions are found in (Nielsen et al. 2008). The interface can be started up either by clicking on the program launcher.py or by writing launcher.py in a window shell. This will create the initial screen which also gives an overview over the programs (Fig. 53.2). The Launcher window shows the Interface layout and the division of the programs into 3 groups: 3D applications, 2D applications and Service programs. The last group of programs is primarily used for data-manipulation or coordinate transformation as explained above. Nearly all the 2D programs use a planar approximation to the surface of the Earth and makes naturally necessary corrections related to this into account. The 3D programs operate either without any approximations (global calculations) or uses spherical approximations for local or regional calculations.
53.4.2 Use The program may be run by clicking on the “Run program” button. This will create an ASCII input file named < program name>.inp which is used by the executable. The result of the run will be seen on the screen, and it is also stored automatically in a file < program name>.log. If only the input file is
Fig. 53.3 Python interface to GEOEGM
needed, clicking on the “Write setting” button is sufficient. The files are over-written if the program is run a second time. The file must be renamed if the user wants to save it. In order to use a program, the user must click on the appropriate button with the name of the program, e.g. 3D Application, GEOEGM. This will launch a window with slots where the user must key information, e.g. the values of parameters. File selection can be done graphically. Some default values are provided in order to aid the user in understanding what is asked for, but in most cases clicking on a helpbutton will give detailed information (Fig. 53.3). A link to a general Help (documents on the internet or in the sub-directory doc) is also provided. The example shows the computation of height anomalies in a grid over Denmark based on the EGM96 Earth Gravity Model. The run as shown requires that the user has created a sub-directory called “result”, where the grid file is stored. Note in Fig. 53.3 that default values for non-grid application are dimmed.
53.5
User Testing and Evaluation
The questionnaire (see Fig. 53.4) is part of the test and evaluation circle of the Gravsoft software interface. It is designed as an electronic user feed back survey and
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the survey is anonymous. It is accessible from http:// www.cctscherning.dk/survey . There are 25 questions divided into the following four themes • General aspects of usability satisfaction • Interface design, subdivided into: screen, terminology and system information, system capabilities and learning (to use) • Usefulness and ease of use • Background information The questionnaire is designed with a feedback in percentage so the users know how far they are in the survey. The themes are constructed using the Likert scale (Likert 1932) and most themes need to be answered before the user can move on to the next theme. Prior to the test all participants took part in a workshop/course in geoid determination and the data was gathered during three workshops taking place in Malaysia, in Como and in Copenhagen with participants coming from Malaysia, Europe mainly and Denmark.
Fig. 53.4 Example page from survey
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53.5.1 Age and Education of Users Approximately 28% are between 40–50 years old, 32% are between 30–40 years and a little more than 39% are between 20–30 years old. Around 80% are males and 20% are females. Most are university graduates, typically in engineering, surveying, geodesy, mathematics and geophysics, a few have a bachelor, some have other education.
53.5.2 Knowledge and Competence As part of background information we asked about participants’ knowledge of English, understanding of mathematics, work experience and knowledge of physical geodesy. The picture is very homogenous, participants primarily rate themselves as proficient or competent (score 4 and 5) in English, in knowledge of computers, and in mathematics. However, their work experience with GPS positioning, remote sensing, digital mapping and precise levelling showed a
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more diverse pattern as did the pattern concerning their background knowledge and familiarity with physical geodesy (gravity field, – potential, – anomaly etc). Knowledge level of physical geodesy shows that 75% are proficient or competent whereas only 25% see themselves as advanced beginners.
the needed file. However a new user test showed that this did not fully solve the problem, as it created another problem. Sometimes the file names would have too many letters for the browser to function optimal. In the third iteration of the design we rewrote the programs to permit file names up to 128 characters and the problem was solved.
53.5.3 Error Message and Browsing Functionality
53.5.4 Usability Satisfaction
The user evaluation and testing showed a general satisfaction with the software, with two important exception, see Fig. 53.5a–f. The error-messages were often considered confusing and not informative. The main reason for this is related to the user writing erroneous file-names or using wrong input values. Also the browsing functionality created problems. The user was required to know and remember the file names. These names were long and often a confusing mix of letters, numbers and signs and users asked for a more user friendly solution. A solution was to change the design allowing the user to browse in order to find
The evaluation showed that users found the interface easy and quick to use, screen layout, characters and colors easy to understand, interaction flexible and easy, but error/ help messages were found dissatisfying. However 1 in 4 found it difficult to learn to use the programs These difficulties in learning showed a correlation with participants’ educational level, pointing to the need for addressing this in the learning design of the schools or workshops. There is a limit to how far design can take us, new users also have to engage in a learning process.
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Concluding Remarks
We ascribe the high score on user satisfaction with the design to the iterative user centered design process applied. However, the problem which remains is that 1 in 4 found it difficult to learn to use the GRAVSOFT programs despite the redesign. This shows that when research software is being made available to users outside the research community then usability, redesign and user testing have to be addressed. This is a challenge to the geodetic community.
References Apple Human Interface Guidelines (1987) The Apple Desktop Interface by Inc. Apple Computer Forsberg R, Tscherning CC (2008) An overview manual for the GRAVSOFT Geodetic Gravity Field Modelling Programs. 2 edn. Contract report for JUPEM
449 Golden Software: Surfer User’s Guide (2002) http://www. goldensoftware.com Likert R (1932) A technique for the measurement of attitudes. Archives of Psychology 140: 1–55 Nielsen Janni, Lone Dirckinck-Holmfeld, Oluf Danielsen (2003) Dialogue Design-with mutual learning as guiding principle. International Journal of Human-Computer Interaction Nielsen J, Jansson TRN, Tscherning CC (2008) Creating a user interface to GRAVSOFT. Report prepared for Geodesy Section Dep. of Surveying and Mapping Malaysia, 2008. Available cct.gfy.ku.dk/publ_cct/cct1932.pdf Preece J, Rogers Y, Sharp H (2004) Interaction design, Apogeo Editore Tscherning CC, Forsberg R, Knudsen P (1992) The GRAVSOFT package for geoid determination. Proc. 1. Continental Workshop on the Geoid in Europe, Prague, May 1992, pp 327–334, Research Institute of Geodesy, Topography and Cartography, Prague
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Progress and Prospects of the Antarctic Geoid Project (Commission Project 2.4)
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Mirko Scheinert
Abstract
The Antarctic Geoid Project (AntGP) aims at the improvement of terrestrial observations of the gravity field in Antarctica and, eventually, at the improvement of the Antarctic geoid. Until present, vast areas of Antarctica are still unexplored with regard to gravity measurements. The polar data gap due to the deflection from a polar inclination of the respective satellite and the limitation to a certain harmonic degree of resolution prevent a complete, high-resolution data coverage to be obtained from the dedicated gravity satellite missions only. In this context, the International Polar Year (March 2007 to February 2009) provided a framework for broad international and interdisciplinary collaboration, which opened also an excellent opportunity for the realization of new gravity surveys. Especially, there was a focus on airborne gravimetry which provides the most powerful technique to carry out observations in vast and remote areas. The paper will review the present situation and will give an outlook to further activities. The feasibility of the regional geoid improvement in Antarctica will be discussed, utilizing the heterogeneous gravity data available from different surveys and techniques.
54.1
AntGP as the IAG Commission Project 2.4
Adopted at the IUGG General Assembly in Sapporo, 2003, it is the first time that within IAG a special group is dedicated to the determination of the gravity field in Antarctica. This entity is named IAG Commission Project 2.4 “Antarctic Geoid” (short: AntGP) and takes its place within Commission 2 (Gravity Field).
M. Scheinert (*) TU Dresden, Institut f€ ur Planetare Geod€asie, 01062 Dresden, Germany e-mail:
[email protected]
The main goal of AntGP is to compile already existing gravity data and to support new gravity surveys in Antarctica in order to complement and to densify the data from the dedicated gravity satellite missions. Since terrestrial measurements are limited in extension and resolution due to the complicated logistics and the hostile environmental conditions in Antarctica, airborne gravimetry provides the only powerful method to survey extended areas. In this respect, collaboration exceeds the field of geodesy: An interdisciplinary cooperation has been established, which is also reflected in the membership of AntGP. Currently, AntGP has 22 full and corresponding members from all over the world. During the first 4-year period (2003–2007) of AntGP a great step forward has been
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made concerning the establishment of cooperation and of close linkages between the different scientific disciplines working in Antarctica (Scheinert 2005, 2007). At the IUGG General Assembly in Perugia, 2007, it was decided to continue AntGP. Since then, the gravity data coverage in Antarctica has been continuously improved realizing new surveys, and a number of further collaborative activities have taken place. An important linkage is being maintained to the Scientific Committee on Antarctic Research (SCAR) and the respective group on Geodetic Infrastructure in Antarctica (GIANT), where the author chairs the project “Physical Geodesy”. This linkage provides a marvellous opportunity to improve the cooperation within Antarctic earth sciences.
54.2
Current Status of the Gravity Field in Antarctica
The state-of-the-art high-resolution model of the Earth’s gravity field in a global scale is given by the model EGM2008 (Pavlis et al. 2008). It provides the gravity field information in form of spherical harmonic coefficients complete up to degree and order 2160. This model’s view of the gravity field in Antarctica in terms of gravity disturbances is shown in Fig. 54.1a. Nevertheless, the information content for Antarctica is limited. This is especially true for EGM2008 over the Antarctic continent, where the field shows only a long-wavelength behavior in contrast to the much finer resolution visible outside Antarctica (Fig. 54.1a). Either terrestrial gravity data could not be included for the Antarctic continent at all – and then the satellite (GRACE) provides the source of information – or there is only very sparse data which entered the computation of a combined solution. In order to reach the grid density for gravity anomalies necessary to compute the combined solution at the anticipated resolution, at those regions (like Antarctica) data were “filled in” (hence computed by geophysical prediction or densification of satellite data). The present knowledge of the Antarctic gravity field can be best illustrated when the difference between EGM2008 and another recent combined gravity field model like EIGEN-5C (F€ orste et al. 2008) is taken. Fig. 54.1c shows this difference in terms of gravity disturbances, and Fig. 54.1d in terms
of the quasigeoid. Over the Antarctic continent, these differences are at the level of 50 mGal for gravity disturbances and of 1 m for the quasigeoid (maximum values are ca. 100 mGal and 3 m, respectively). The picture is much better over the Antarctic ocean, since there information from satellite altimetry could be incorporated into the global models (see e.g. Andersen and Knudsen 2009). Another problem arises from the fact that the inclination of the gravity satellite missions is not exactly 90 (CHAMP: 87.3 , GRACE: 89.5 , GOCE: 96.5 ). The resulting data gaps are shown in Fig. 54.1b. Whereas for GRACE the data gap is relatively small – but the resolution (half wavelength) of the static gravity field yielded by GRACE is limited to about 200 km (spherical harmonic degree 100) – the data gap for GOCE, which shall deliver a much higher resolution of the static gravity field, is comparably large: The anticipated resolution is about 100 km (half wavelength, spherical harmonic degree 200), whereas the polar gap not covered by GOCE data has a radius of about 700 km corresponding to a spherical harmonic degree of about 30 (in terms of half-wavelength resolution). Certainly, the data gap is not exactly a spherical cap with 700 km radius, since the satellite senses the integrated effect of the mass distribution in the region it is currently passing by. Nevertheless, the mission altitude is about 250 km only, so that the principal size of the data gap remains. From these arguments it becomes obvious that there is a need for “real” gravity data in Antarctica. Of course, a number of gravity surveys have been already realized, either conducting terrestrial measurements (using a spring-type gravity meter along profiles or at grid points covering a limited area, see e.g. Fig. 54.2) or airborne measurements (utilizing an adapted gravity meter like LaCoste & Romberg type S together with an ensemble of aerogeophysical instrumentation on board an aircraft). Complementing the continental gravity surveys a number of shipborne gravity measurements in the Antarctic ocean can be found. The Antarctic gravity data so far available in the database at the author’s institution are plotted in Fig. 54.3. It has to be stated that these data, originating from different sources, exhibit a large heterogeneity. Reasons for this can be found in a different gravimetric datum, in different reductions and/or corrections applied to the data. The accuracy of the data is
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Progress and Prospects of the Antarctic Geoid Project (Commission Project 2.4)
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Fig. 54.1 Gobal gravity field models in Antarctica (for references see text)
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Fig. 54.4 Selected projects of the International Polar Year comprising gravity observations in Antarctica (IPY chart from www.ipy.org)
Fig. 54.2 Current status of the gravity anomaly data holdings for Antarctica at TU Dresden. The inset circles denote the borders of the polar data gap of the satellite missions (CH: CHAMP, G: GRACE, GO: GOCE), cf. Fig. 54.1b. For the ICECAP project in East Antarctica only profiles are shown (#University of Texas, Institute of Geophysics, Polar Studies Group, http://www.ig.utexas.edu/research/projects/icecap)
Fig. 54.3 Terrestrial gravity survey in Antarctica (photography: M. Scheinert)
different due to the respective technique, hence there is also a different level of redundancy. It is anticipated to mitigate these problems and to homogenize the data in order to finally produce a gridded gravity data compilation for Antarctica. In the recent years a number of new surveys have been conducted. Cooperations were managed in the
framework of SCAR and especially during the International Polar Year (IPY, March 2007 to February 2009). There was a number of approved IPY projects which planned to incorporate a noteworthy activity on gravity observations, either to be realized by terrestrial measurements (surveys along profiles and absolute observations at stations) or by airborne gravimetry. The following IPY projects should be mentioned: Project 67 “Origin, evolution and setting of the Gamburtsev subglacial highlands (AGAP)”, project 97 “Investigating the Cryospheric Evolution of the Central Antarctic Plate (ICECAP)”, project 42 “Subglacial Antarctic Lake Environments (SALEUNITED)”, project 152 “Trans-Antarctic Scientific Traverses Expeditions (TASTE-IDEA)”, project 185 “Polar Earth Observing Network (POLENET)”, see Fig. 54.4 displaying the chart of IPY projects overlaid by the names of the above mentioned projects. First results and a description of the projects can be found on the internet e.g. for AGAP1 and for ICECAP,2 for the coverage of the latter also see Fig. 54.3. In West Antarctica, the progress made by conducting aerogeophysical surveys can be kept track on e.g. in the papers of Ferraccioli et al. (2007), Diehl et al. (2007, 2008), Jordan et al. (2007, 2009).
1 2
www.ldeo.columbia.edu/mstuding/AGAP/. www.ig.utexas.edu/research/projects/icecap/.
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Progress and Prospects of the Antarctic Geoid Project (Commission Project 2.4)
54.3
Regional Geoid Improvement
The terrestrial gravity data form the major data source to perform a regional geoid improvement. Nevertheless, as already mentioned above, first one has to clarify which reductions and corrections were applied. Moreover, the results of airborne gravimetry were often referred to as gravity anomalies, whereas in the geodetic understanding these data are gravity disturbances (Hackney and Featherstone 2003). The geodetic analysis is in general done according to Molodensky’s theory. Following the Remove-ComputeRestore (RCR) technique one normally reduces the long-wavelength signal by applying a global gravity field model (like EGM2008 or EIGEN-5C) and the short-wavelength part by applying a certain kind of topographic reduction. For the latter, an appropriate topography model is needed. Either it could be provided utilizing data acquired during the same aerogeophysical survey where also the gravimetry data were recorded or one has to use a continental model. In the first case, the radio echo sounding (RES) method is used, which allows to compute the ice-surface topography as well as bedrock (subglacial) topography by analyzing the travel time of the electromagnetic waves reflected by the different density boundaries. In the latter case, one can make use of the BEDMAP topographic model (Lythe et al. 2001), which, however, features larger inaccuracies and deficiencies (see e.g. Schwabe et al. 2011). The group at the author’s institution performed several case studies for a regional geoid improvement in Antarctica. In the region of the Prince Charles Mountains we could make use of homogeneously sampled data both for gravity and for topography (Damaske and McLean 2005; Damm 2007). The processing strategy and the results were described in detail by Scheinert et al. (2008a), where the RemoveCompute-Restore techniques together with leastsquares collocation were applied. In the final regional geoid solution the dominant features of the bedrock topography were clearly visible, especially the graben structure of the Lambert glacier system. Based on the collocation which was applied to infer the regional geoid the accuracy of the solution was estimated to be at the level of 15 cm. This reflected the potential level of accuracy to be reached in Antarctica. In the region of Dronning Maud Land (DML, Atlantic sector of East Antarctica) the author’s
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institution is maintaining a long-term cooperation with the Alfred Wegener Institute for Polar- and Marine Research (AWI), Bremerhaven, Germany (Scheinert et al. 2008b). In the framework of a joint research project AWI conducted a number of aerogeophysical surveys in DML (Riedel and Jokat 2007). The project work on the geoid improvement in this region is still in progess, first results were published by M€uller et al. (2007). In the region of the Antarctic Peninsula gravity data acquired by the British Antarctic Survey (BAS) could be utilized (Ferraccioli et al. 2007). First results on the regional geoid improvement in that region were published by Scheinert et al. (2007), and discussed in detail in this volume by Schwabe et al. (2011). It is needless to emphasize that this work shall be continued and extended with the final goal of an improved geoid solution for entire Antarctica.
54.4
Summary
AntGP forms the IAG Commission Project 2.4 “Antarctic Geoid”. It could be shown that in the recent time substantial progress has already been made in order to improve the coverage of terrestrial gravity data in Antarctica. Nevertheless, still remarkable data gaps need to be filled in. An improved data coverage – and thus also closing the polar data gap in gravity resulting from the satellite missions – will contribute substantially to the improvement both of global gravity field models as well as of regional geoid solutions in Antarctica. Acknowledgements The members of AntGP and all colleagues we are cooperating with in Antarctic research are gratefully acknowledged for their support. Especially I like to thank Detlef Damaske, Fausto Ferraccioli, Rene´ Forsberg, Wilfried Jokat, Steve Kenyon and German Leitchenkov for their active commitment.
References Andersen OB, Knudsen P (2009) DNSC08 mean sea surface and mean dynamic topography models. J Geophys Res 114 (C11):c11001. doi:10.1029/2008JC005179 Damaske D, McLean M (2005) An aerogeophysical survey south of the Prince Charles Mountains, East Antarctica. Terra Antarct 12(2):87–98 Damm V (2007) A subglacial topographic model of the southern drainage area of the Lambert Glacier/Amery Ice Shelf
456 System – results of an airborne ice thickness survey south of the Prince Charles Mountains. Terra Antarct 14(1):85–94 Diehl TM, Blankenship DD, Holt JW, Young DA, Jordan TA, Ferraccioli F (2007) Locating subglacial sediments across West Antarctica with isostatic gravity anomalies. In: Cooper A, Raymond C (eds) Antarctica: a keystone in a changing world, Online Proceedings for the 10th ISAES, USGS OpenFile Report 2007-1047. doi:10.3133/of2007-1047.ea107 Diehl TM, Holt JW, Blankenship DD, Young DA, Jordan T, Ferraccioli F (2008) First airborne gravity results over the Thwaites Glacier catchment, West Antarctica. Geochem Geophys Geosystems 9(4). doi:10.1029/2007GC001878 Ferraccioli F, Jones PC, Leat P, Jordan TA (2007) Airborne geophysics as a tool for geoscientific research in Antarctica: some recent examples. In: Cooper A, Raymond C (eds) Antarctica: a keystone in a changing world, Online Proceedings for the 10th ISAES, USGS Open-File Report 2007-1047. doi:10.3133/of2007-1047.srp056 F€orste C, Flechtner F, Stubenvoll R, Rothacher M, Kusche J, Neumayer H, Biancale R, Lemoine J, Barthelmes F, Bruinsma S, K€ onig R, Dahle C (2008) EIGEN-5C: the new GeoForschungsZentrum Potsdam/Groupe de Recherche de Geodesie Spatiale combined gravity field model. Presentation/Abstract 32, Ocean Surface Topography Science Team Meeting, Nice, Nov 10–12, 2008 Hackney RI, Featherstone WE (2003) Geodetic versus geophysical perspectives of the ‘gravity anomaly’. Geophys J Int 154:35–43 Jordan TA, Ferraccioli F, Jones PC, Smellie JL, Ghidella M, Corr H, Zakrajsek AF (2007) High-resolution airborne gravity imaging over James Ross Island (West Antarctica). In: Cooper A, Raymond C (eds) Antarctica: a keystone in a changing world, Online Proceedings for the 10th ISAES, USGS open-file report 2007-1047. doi:10.3133/of2007047.srp060 Jordan TA, Ferraccioli F, Vaughan DG, Holt JW, Corr HF, Blankenship DD, Diehl TM (2009) Aerogravity evidence for major crustal thinning under the Pine Island Glacier region (West Antarctica). GSA Bullet 122(5–6):714–726. doi:10.1130/B26417.1 Lythe MB, Vaughan DG, The BEDMAP Consortium (2001) BEDMAP: a new ice thickness and subglacial topographic model of Antarctica. J Geophys Res 106(B6):11,335–11,351 M€uller J, Riedel S, Scheinert M, Horwath M, Dietrich R, Steinhage D, Ansch€ utz H, Jokat W (2007) Regional geoid
M. Scheinert and gravity field from a combination of airborne and satellite data in Dronning Maud Land, East Antarctica. In: Cooper A, Raymond C (eds) Antarctica: a keystone in a changing world – Online Proceedings for the 10th ISAES, USGS open-file report 2007-1047. doi:10.3133/of2007-1047.ea022 Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2008) An earth gravitational model to degree 2160: EGM2008. Presentation at EGU general assembly, Vienna, April 13–18, 2008 Riedel S, Jokat W (2007) A compilation of new airborne magnetic and gravity data across Dronning Maud Land, Antarctica. In: Cooper A, Raymond C (eds) Antarctica: a keystone in a changing world – Online Proceedings for the 10th ISAES, USGS open-file report. doi:10.3133/of2007047.ea149 Scheinert M (2005) The Antarctic geoid project: status report and next activities. In: Jekeli C, Bastos L, Fernandes J (eds) IAG symposia, vol 129, Gravity, geoid and space missions. Springer, Berlin, pp 137–142 Scheinert M (2007) The Antarctic geoid project (AntGP). Poster presentation, XXIV IUGG general assembly, Perugia, July 2–13, 2007 Scheinert M, Ferraccioli F, M€uller J, Jordan T, Dietrich R (2007) Modelling recent airborne gravity data over the Antarctic Peninsula for regional geoid improvement. In: Cooper A, Raymond C (eds) Antarctica: a keystone in a changing world – Online Proceedings for the 10th ISAES, USGS open-file report 2007-1047. doi:10.3133/of20071047.ea014 Scheinert M, M€uller J, Dietrich R, Damaske D, Damm V (2008a) Regional geoid determination in Antarctica utilizing airborne gravity and topography data. J Geod 82(7): 403–414. doi:10.1007/s00190-007-0189-2 Scheinert M, M€uller J, Riedel S, Horwath M, Ansch€utz H, Bayer B, Eberlein L, Groh A, Steinhage D, Oerter H, Dietrich R, Jokat W, Miller H (2008b) Die Kombination von bodengebundenen, flugzeuggest€utzten und Satellitendaten zur Bestimmung von Schwerefeld, Magnetfeld, Eismassenhaushalt und Krustenstruktur in DronningMaud-Land, Antarktis: Ergebnisse des Forschungsprojektes VISA. Poster presentation at 23rd International Polar Meeting, M€unster, March 10–14, 2008 Schwabe J, Scheinert M, Dietrich R, Ferraccioli F, Jordan T (2011) Regional geoid improvement over the Antarctic Peninsula utilizing airborne gravity data. In: Kenyon S et al (eds) Geodesy for Planet Earth. Springer, Heidelberg
Regional Geoid Improvement over the Antarctic Peninsula Utilizing Airborne Gravity Data
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J. Schwabe, M. Scheinert, R. Dietrich, F. Ferraccioli, and T. Jordan
Abstract
We present an improved quasigeoid of the Palmer Land Region, Antarctic Peninsula, derived from recent aerogravimetry profiles provided by the British Antarctic Survey (BAS). Special focus is given to the treatment of the ice layer covering the bedrock topography, the latter one being regarded as the boundary surface. The remove-compute-restore technique (RCR) with least-squares collocation (LSC) and a point mass modeling, respectively, are applied and compared. In addition to previous studies, an alternative strategy regarding downward continuation has been introduced. Furthermore, the Residual Terrain Model (RTM) has been enhanced to incorporate the individual densities of water, ice and bedrock.
55.1
Introduction
Over Antarctica, global potential models (GPM) still do not depict the mid and short wavelengths of the gravity field accurately due to the polar data gap of the satellite missions and the lack of terrestrial gravity data. The differences between the models can exceed 50 mGal or 1 m (Scheinert 2011). As can be seen from Table 55.1, even the state-of-the-art model EGM2008 does not offer any improvement as no surface observations of the continent have been included at all (Pavlis et al. 2008).
J. Schwabe (*) M. Scheinert R. Dietrich TU Dresden, Institut f€ ur Planetare Geod€asie, 01062 Dresden, Germany e-mail:
[email protected] F. Ferraccioli T. Jordan British Antarctic Survey, High Cross, Madingley Road, Cambridge CB3 0ET, UK
During the recent years more and more aerogravimetric surveys have been conducted in Antarctica. The technique uniquely matches the harsh polar conditions, as stated by many authors (e.g. Forsberg et al. 2000; Scheinert et al. 2008). Furthermore, it provides a homogeneous data coverage well suited for geoid prediction. However, extended Antarctic regions remain to be filled in with gravimetric data. The Antarctic Geoid Project (AntGP) aims to close these gaps and to provide a homogeneous compilation of gravity and a high-resolution geoid model (Scheinert 2005, 2011). In 2002/2003, BAS accomplished another airborne campaign in the region of the Antarctic Peninsula (Ferraccioli et al. 2006) and contributed collected airborne and prior surface gravity data to the AntGP database hosted at TU Dresden. In the following sections the properties and the processing of the data will be discussed.
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Table 55.1 Complete airborne dataset reduced for selected global models up to degree l GPM EGM96 EIGEN-GL04C EIGEN-5C EGM2008 EIGEN-GL04S EIGEN-5S EGM96 EIGEN-GL04C EIGEN-5C EGM2008 EGM2008
lGPM 150
360
2190
Mean 0.8 12.9 10.5 10.2 11.0 9.2 1.8 13.3 10.3 9.7 10.3
STD 57.9 43.9 42.7 42.1 44.1 42.1 57.2 42.9 41.6 40.6 42.1
Min 153.9 129.1 130.6 134.8 135.3 133.6 148.2 121.7 122.4 124.8 134.9
Max 179.5 164.8 169.5 156.0 155.3 150.7 182.7 170.5 169.0 147.1 155.0
Gravity disturbances at flight altitude (mGal)
55.2
Data Analysis
55.2.1 Airborne Gravity Data The airborne survey was conducted at 2,800 m nominal altitude employing a modified LaCoste & Romberg S-83 sea gravity meter aboard a Twin Otter aircraft. Flight lines (Fig. 55.1) follow the main shape of the peninsula in a narrow strip north to south (1996 mission data), with additional east west profiles having a slightly denser spacing of 5–10 km (2002/2003 mission data). This rectangular patch was chosen for setting up the processing scheme, hence forming the subset treated in the subsequent quasigeoid computations. The dataset contains preprocessed and leveled freeair anomalies at median flight altitude of 2,777.3 m (unpublished meta information), corrected for kinematic accelerations and providing a mean accuracy of 3 mGal. Taking GPS navigation into account, we can actually regard the anomalies as gravity disturbances in terms of geodesy (Hackney and Featherstone 2003).
55.2.2 Cross-over and Spectral Analysis At this stage, the following investigation of the data quality could only be done with strongly edited profiles, therefore the nominal mean accuracy of 3 mGal was kept during the latter processing. Though the survey comprised both N–S and E–W profiles, main parts of the N–S profiles have been deleted in the area of investigation, leaving some short tracks connecting more distant E–W profiles. Hence,
cross-overs could only be estimated roughly at some mostly T-shaped intersections. Although the lines obviously have been smoothed and adjusted, as at first glance no signal variation is noticeable in subsequent points having a mean spacing of ca. 100 m (Fig. 55.2), the analysis surprisingly revealed larger cross-over differences and even some gross errors (Table 55.2). However, the alignment of the profiles delivered did not allow for further editing except deleting apparently erroneous segments. In the future, a reprocessing of the raw data might be necessary. Additionally, along-track empirical autocovariance functions and spectra were calculated. According to the stride of the Palmer Land mountains, the covariance half-length mainly ranges from 8 to 15 km while amplitudes imply a filter cut-off at ca. 10 km wavelength (Figs. 55.3 and 55.4). Thus, in spite of covering a much denser along-track spacing, the dataset can be regarded isotropically sampled in the spectral domain. The airborne data were checked against surface gravity for consistency. The obtained residuals from 3D LSC (Fig. 55.5) indicate that the surface data are highly heterogeneous. Naturally, land observations differ from airborne measurements in respect of bandwidth and sampling. However, biases, arising from datum realization, the use of different instruments and the application of non-uniform reductions, are also likely to be inherent in terrestrial gravity data of Antarctica, especially if they were collected over decades like in this case.
55.3
Quasigeoid Improvement Using LSC
55.3.1 Processing Scheme In large part, the computation strategy follows previous works at TU Dresden (M€uller et al. 2007; Scheinert et al. 2008). Employing the GRAVSOFT software (Forsberg and Tscherning 2008), the RCR was applied following Molodenski’s theory in order to 1. Remove the known contributions of a global potential model (GPM, long-wavelength part) and the residual terrain (RTM, short-wavelength part). 2. Use LSC to predict height anomalies from the band-limited residual field. 3. Restore the GPM and RTM contributions in terms of height anomalies.
55
Regional Geoid Improvement over the Antarctic Peninsula Utilizing Airborne Gravity Data
Fig. 55.1 BAS gravity data of the Antarctic Peninsula. Lines: airborne profiles (light: season 1996, dark: season 2002/2003); Triangles: surface gravity locations. Dashed lines indicate main faults of the George VI Sound graben (after Edwards 1979)
–64˚
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Fig. 55.2 W–E profile at approx. 71 southern latitude
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The suitable boundary surface to account for is the bedrock topography,1 so that the gravitational effect of
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Or, in case of negative heights, the ellipsoid.
the covering ice layer was reduced in advance by means of a complete Bouguer reduction. Indirect effects both on gravity and quasigeoid were neglected. Hence, strictly speaking, the presented solution resembles some kind of “co-quasigeoid”. In this
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Table55.2 Cross-over analysis of the airborne gravity data (mGal) n
1996 Alone 2002/2003 Alone 1996 vs. 2002/2003
10 49 34
Mean modulus 11.0 6.8 6.2
r.m.s.
Max. modulus 30.0 31.0 29.9
14.1 9.3 9.6
Suspected gross errors of 60.2 and 133.9 mGal were excluded from the statistics of the items “2002/2003 alone” and “1996 vs. 2002/2003”, respectively
40
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mountaineous region the removed ice layer is thin and dominated by small scale variations, therefore no long-wavelength contribution was reduced in the GPM. Some major modifications were made in this study: • The ice below zero level remaining after the Bouguer reduction is causing an additional density contrast in the RTM reduction scheme similar to the “marine convention” implemented in the TC program. So far, this matter could be handled simply by modifying the respective density value. Since the Antarctic Peninsula is a coastal zone, the program was modified to take an additional density grid into account. • In their study, Scheinert et al. (2008) gridded the airborne gravity, computed vertical gradients and used these for downward continuation (by FFT) prior to the actual remove step and the solution of the boundary value problem. To obtain more realistic prediction errors, an alternative strategy was tested where the residual height anomalies at
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Fig. 55.5 Surface gravity minus prediction from airborne profiles. For the comparison, points sufficiently close to the profiles were chosen by means of the LSC prediction error. A point was rejected if the ratio of a posteriori and a priori standard deviation was larger than one
the boundary surface are predicted directly from the residual observations at flight altitude, making use of 3D collocation. As the latter procedure simultaneously performs analytic continuation and conversion of gravity to quasigeoid, we subsequently refer to these different methods as the two-step and one-step approach, respectively. • A variety of solutions based on selected cut-off wavelengths of GPM and RTM, respectively, were produced to investigate the impact of the topography information on the solution and the consistency of the modeled signal contents (Sect. 55.3.2).
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Table 55.3 Standard deviations (mGal) of selected residual fields (observed minus ice layer minus listed contributions) Altitude (approach)
lGPM
lRTM (km)
– GPM
Flight Altitude (One-step) Boundary Surface (Two-step)
120 180 360 120 180 360
300 200 100 300 200 100
36.1 32.7 30.8 39.0 37.9 37.1
Table 55.4 Mean values of one-step computations showing the datum shift of the restored quasigeoid caused by RTM lGPM 120 180 360 120 180 360 120 180 360
lRTM (km) 300 200 100 100
–
dgres (mGal) 30.7 22.7 13.6 16.4 16.0 13.6 13.5 13.1 10.8
zres (m) 0.81 0.71 0.42 0.43 0.33 0.42 0.31 0.18 0.28
zres+RTM (m) 5.20 3.92 2.00 2.01 1.91 2.00
zrestored (m) 2.74 1.48 +0.39 +0.46 +0.53 +0.39 +2.15 +2.26 +2.11
55.3.2 Impact of the Topography Model Unlike the case in other airborne surveys, no accurate high-resolution subglacial bedrock topography model obtained from Radio Echo Sounding was available for this campaign. For Antarctica, until now there exists only one continental-wide model which comprises both ice and bedrock surfaces. However, this BEDMAP database (Lythe et al. 2001) suffers from a rather crude resolution2 of 5 km and an empirical uncertainty of about 200 m. At least for the fully reduced part of the ice layer, one has to live with the fact that errors in the BEDMAP model will completely map into the obtained solution. The rigorous evaluation of these uncertainties will be subject to further investigations. First, the utilization of BEDMAP did not yield a significant smoothing of the residual field (Table 55.3). Furthermore, different cut-off wavelengths, both equal for GPM and RTM respectively, resulted in a systematic shift of the final quasigeoid, irrespective of the applied approach (Sect. 55.3.1).
2
And even sparser in terms of initial data coverage.
GPM RTM 30.4 32.4 25.9 50.0 50.3 40.2
Ratio 16% 1% 16% +28% +33% +8%
Table 55.5 Differences one-step minus two-step approach lGPM 120 180 360
Mean 0.10 0.24 0.10
STD 0.11 0.12 0.09
r.m.s. 0.15 0.27 0.14
Min. 0.32 0.10 0.20
Max. 0.37 0.59 0.38
RTM was not applied. Restored height anomalies (m)
However, this effect did not show up when fixing the RTM wavelength and only varying the wavelength of the GPM (Table 55.4). This suggests that the contributions from GPM and RTM in fact do not correspond. Whereas the LSC deals on the residual signal parts adequately, the RTM effects seem suspicious.
55.3.3 Comparison of the Two Approaches In the last section it was found that the choice of neither one- or two-step approach does change the behaviour of the RTM. Likewise, it could be observed that the RTM does not significantly affect the consistency of the two approaches. For one certain set of RTM and GPM cut-off wavelengths, both strategies yielded similar results at the decimeter level (Table 55.5).
55.3.4 Final Solution In the following, the results of the main processing steps according to the one-step approach are outlined. Prior to the actual remove step, the gravitational effect at flight altitude of the ice layer above the chosen boundary surface was reduced by taking the difference of virtual ice topographies built up by the ice and bedrock surface, respectively. This slightly roughened the subsequent residual field. Then, the contributions of the then up-to-date EIGEN-GL04C model (F€orste et al. 2006), complete
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Table 55.6 Impact of the ice layer effect on residual fields Max. 159.6 55.8 84.4 82.8 154.4 160.7 94.1
800
Covariance (mGal²)
Field/contribution Mean STD Min. Observed 33.5 40.8 72.6 Contributions Ice layer effect 14.2 11.1 0.6 EIGEN-GL04C 30.0 32.4 30.5 RTM (100 km) 2.3 17.5 34.8 Residual fields, case 1: ice layer removed Ice layer 18.8 39.3 79.1 EIGEN-GL04C 10.8 30.8 143.2 RTM 13.6 25.9 131.5 Residual fields, case 2: ice layer not removed EIGEN-GL04C 3.5 27.0 127.8 RTM 1.1 23.6 116.1
1000
600 400 200 0
166.1 98.3
Gravity disturbancies at flight altitude (mGal). Subsequently removed contributions are indicated by “–”. The bold row denotes the residual field used in the final solution
to degree and order 360, and the RTM, with corresponding wavelength of 100 km, were subtracted. The statistics of all reduced contributions are shown in Table 55.6. In view of the effects induced by the RTM (Sect. 55.3.2) and the fact that there is no way to calibrate the gravimetric quasigeoid using GPS leveling due to the lack of data, we decided to skip the RTM in the final solution. Thus, the prediction was performed using the dataset reduced for ice and GPM, except that only for the covariance estimation a bilinear trend was subtracted in order to obtain stationarity. The parameters of the commonly used covariance model as proposed by Tscherning (2008) were computed to RB–RE ¼ 2.60 km (depth of Bjerhammar sphere), C0 ¼ 914.91 mGal2 (data variance at zero altitude) and a ¼ 6.21 (scale factor of the GPM error-degree variances) employing the GRAVSOFT programs EMPCOV and COVFIT. The suitability of the chosen model for such noisy data as in the present case was not explicitely inspected. However, the estimated covariance function shows a considerably good fit to the empirical one (Fig. 55.6). The height anomalies on the chosen boundary surface were predicted in a 0.1 (in latitude) by 0.3 (in longitude) grid using GEOCOL16. The observations were assigned the referenced standard deviation of 3 mGal. Under this assumption, formal prediction errors amount to about 5 cm in the densely covered center and 10–15 cm near the edges.
0.0
0.1
0.2
0.3
0.4
0.5
Spherical distance (deg) Fig. 55.6 Empirical (dots) and tabulated model covariances (line) of the detrended residual field, reduced for ice and GPM only
The restored quasigeoid using EIGEN-GL04C as background model is shown in Fig. 55.7. The improved model contains more details and depicts the graben structure of George VI Sound which recent GPM were not able to resolve properly.
55.4
Point Mass Modeling
55.4.1 Framework A convenient mathematical description of the potential field can alternatively be obtained by means of source representations. Being multipoles of zeroth order, point masses can be used in a sequential iterative adjustment optimizing horizontal position and depth as demonstrated by Claessens et al. (2001). In the following, we will only outline the main conclusions of the algorithm applied to airborne data so far. For details, the reader may refer to the above cited publication. It shall be shown that the point mass method provides a feasible strategy for regional geoid modeling in Antarctica. Nevertheless, further investigations are necessary.
55.4.2 Results From Newton’s law of attraction it follows that the depth of a point mass governs the shape of the signal at the surface, or, in other words, the spectrum, while the
55
Regional Geoid Improvement over the Antarctic Peninsula Utilizing Airborne Gravity Data –66˚
–68˚
–70˚
–64˚
–62˚
–60˚ –70˚
4
Fig. 55.7 Final improved quasigeoid from airborne profiles. Height anomalies (m). Background: EIGENGL04C
463
8
4
2
0
6
4
0
–71˚
4
0
–2
0
–72˚
2
–73˚
–6
–2
0
2
6
4
–66˚
–68˚
10
12
–64˚
0.8 0.8
0.4
0.8
4
–71˚
−0.4
0.
–62˚
0
0
−0.4
−0.4 0
8
0.4
Fig. 55.8 Height anomaly differences (m) between point mass modeling and the selected final solution from 3D LSC
–4
–72˚
−0.8
modulus scales the magnitude of this pattern. Thus, too shallow masses may strongly influence prediction points that reside very close. To ensure that the model can be used for predicting surface quantities from airborne observations at the desired approximation accuracy, minimum and maximum depths should be chosen that are in keeping with the bandwidth of the data. In the simulations, a stable solution could be obtained allowing point masses to be positioned in a depth range of 30–200 km. Also, different scenarios for the horizontal allocation of both airborne data and point masses were investigated with regard to height anomalies predicted from the obtained representation. First, some sort of constraint is necessary to assure a proper mean level. Second, data from outside the area have to be included to prevent edge effects. It was found that zero-padding of a GPM reduced field provides a simple and feasible way to meet both aspects. The transition at the edges is
−0.4
0.0
0.4
0.8
smooth with only minor distortions. Moreover, this strategy is convenient when working with residual data, as in that case the padded areas inherently represent the global model. The extent of the zero-padding was set to about 100 km, for now incorporating only observations of an area of about 100 by 200 km due to computation speed. To ensure the stability of the inversion, a nearly equidistant subset of the observations was created by simply thinning the airborne profiles in order to have a similar sampling along and across the flight lines. The iteration finished with 256 point masses when the r.m.s. of the well distributed residuals reached 5 mGal. The obtained point mass model adjusted to the residual field was used to predict height anomalies, which could then be compared with the LSC solution. The main topographic features are visible and the mean level is reasonable, however, some longwavelength deviations up to 1 m occur (Fig. 55.8).
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As point masses generate full spectra, these deviations are assumed to be caused by the limited area adopted so far or by the chosen maximum depth. With the full dataset included and fine-tuned parameters, the longer waves should be correctly fixed. Until now, only gravity observations are implemented. If the adjustment is extended to combine different types of observations, the zero-padding will potentially be more effective introducing height anomalies around the airborne gravity profiles.
55.5
Conclusions and Outlook
Least squares collocation as well as point mass adjustment were applied in a remove-restore method to derive an improved quasigeoid of Palmer Land region, Antarctic Peninsula, from recent airborne gravity provided by BAS. Prediction errors from LSC indicate an accuracy of the estimated residual height anomalies of about 10–15 cm. However, due to the noisy airborne data and the imperfect topography model available, we assume a value of up to 50 cm to be a conservative estimate for the overall uncertainty of the final quasigeoid. The feasibility of the applied methods could be demonstrated. The analysis will be extended to comprise all available data and to elaborate the optimal data combination strategy for the geoid improvement in Antarctica. The fundamental questions are related to the proper conceptual treatment of boundary surface, gravimetric datum and topography in view of the ice coverage. Acknowledgement We would like to emphasize our gratefulness the authors of GRAVSOFT for contributing and maintaining the software.
References Claessens SJ, Featherstone WE, Barthelmes F (2001) Experiences with point-mass gravity field modelling in the Perth Region, Western Australia. Geom Res Aust 75:53–86
Edwards CW (1979) New evidence of major faulting on Alexander Island. Br Antarct Surv Bull 49:15–20 Ferraccioli F, Jones PC, Vaughan APM, Leat PT (2006) New aerogeophysical view of the Antarctic Peninsula: more pieces, less puzzle. Geophys Res Lett 33:L05,310. doi:10.1029/2005GL024,636 Forsberg R, Tscherning C (2008) An overview manual for the GRAVSOFT geodetic gravity field modelling programs, 2nd edn. Contract report for JUPEM. http://www.gfy.ku.dk/cct/ cct1936.pdf Forsberg R, Olesen A, Bastos L, Gidskehaug A, Meyer U, Timmen L (2000) Airborne geoid determination. Earth Planets Space 52(10):863–866 F€orste C, Flechtner F, Schmidt R, K€onig R, Meyer U, Stubenvoll R, Rothacher M, Barthelmes F, Neumayer H, Biancale R, Bruinsma S, Lemoine JM, Loyer S (2006) A mean global gravity field model from the combination of satellite mission and altimetry/gravimetry surface data – EIGEN-GL04C. Presentation at EGU Gen. Assembly, Vienna, 3–7 April 2006 (Abstr. 03462) Hackney RI, Featherstone WE (2003) Geodetic versus geophysical perspectives of the ‘gravity anomaly’. Geophys J Int 154:35–43 Lythe MB, Vaughan DG, The BEDMAP Consortium (2001) BEDMAP: a new ice thickness and subglacial topographic model of Antarctica. J Geophys Res 106(B6):11,335–11,351 M€uller J, Riedel S, Scheinert M, Horwath M, Dietrich R, Steinhage D, Ansch€utz H, Jokat W (2007) Regional geoid and gravity field from a combination of airborne and satellite data in Dronning Maud Land, East Antarctica. In: Cooper A, Raymond C (eds) Antarctica: a keystone in a changing world – online proceedings for the 10th ISAES, USGS open-file report 2007-1047, doi: 10.3133/of2007-1047.ea022 Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2008) An earth gravitational model to degree 2160: EGM2008. Presentation at EGU general assembly, Vienna, April 13–18, 2008 Scheinert M (2005) The Antarctic Geoid Project: status report and next activities. In: Jekeli C, Bastos L, Fernandes J (eds) IAG symposia, vol 129, Gravity, geoid and space missions. Springer, Berlin, pp 137–142 Scheinert M, M€uller J, Dietrich R, Damaske D, Damm V (2008) Regional geoid determination in Antarctica utilizing airborne gravity and topography data. J Geod 82(7):403–414. doi:10.1007/s00190-007-0189-2 Scheinert M (2011) Progress and prospect of the Antarctic Geoid Project (Commission Project 2.4). In: Kenyon S et al. (eds), Geodesy of Planet Earth. Springer, Heidelberg Tscherning C (2008) Geoid determination by 3D least-squares collocation. http://www.gfy.ku.dk/cct/cct1935.pdf
Auvergne Dataset: Testing Several Geoid Computation Methods
56
P. Valty, H. Duquenne, and I. Panet
Abstract
In 2004, the French Institut Ge´ographique National (IGN), upon the request of the steering committee of the European Gravity and Geoid Project, prepared a dataset to test geoid computation methods. It consists of a set of about 240,000 gravity points, three digital terrestrial models (an accurate one, a low-resolution one and a filtered one) and 75 GPS/levelling points to evaluate the quality of the computed geoid models (Duquenne, A data set to test geoid computation methods. In: Dergisi H (eds.), Proceedings of the 1st international symposium of the international gravity field service “gravity field of the earth”, pp 61–65, 2006). In this paper, we compared the following geoid computation methods using the Auvergne dataset: the remove-compute-restore method using the unmodified Stokes’ kernel, the deterministic (Wong and Gore) and stochastic (KTH) modifications of Stokes’ kernel. For each method, we tested different choices of the parameters (radius of integration of Stokes’ anomalies, degree of modification of Stokes’ kernel, radius of integration of terrain effect, etc.). We analysed the results in order to find out which method performs the best and how the geoid modelling method impacts the results, considering the presence of errors in the dataset. The question that this work intends to answer is whether we should put our efforts rather on the theoretical investigations of geoid modelling methodologies, or on the acquisition of gravity measurements.
56.1
Introduction
A dataset for the French area of Auvergne has been created by Duquenne (2006). The aim was to have a stable set of gravity data which could be used as a basis for testing geoid computation methods. Although
P. Valty (*) H. Duquenne I. Panet IGN/LAREG, 6-8 Av Blaise Pascal, Champs sur Marne 77420, France e-mail:
[email protected]
it is centred on the semi-mountainous area of Auvergne (where the final geoid is computed), in particular it is to be composed of gravity measurements coming from very different geographic areas, the high mountains of Alps, the semi-mountainous areas of Massif Central or the plains of Aquitaine and Bassin Parisien (Fig. 56.1). These roughly 244,000 gravity points, shown in Fig. 56.2, have been obtained from a validation and correction of the French gravimetric network. Three DTM (refined, filtered and coarse) are also provided in order to have the same reference for computing terrain effects. Their spatial coverage is shown
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_56, # Springer-Verlag Berlin Heidelberg 2012
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Fig. 56.1 Distribution of the data used
in Fig. 56.1. The most accurate one has been obtained from the IGN database and is coming from topographical maps or aerial surveys. Its resolution is of 4.500 in. latitude and 600 in. longitude. Its vertical uncertainty is estimated to 5 m in flat terrain and up to 15 m in rough topography areas. The two other DTMs have been computed from this accurate one. The coarse one was obtained by under sampling and the filtered one was obtained by filtering out the lower wavelengths of the topography (degrees higher than the GGM maximal degree). A set of 75 GPS/levelling points is also added so as to test the quality of the computed geoid models. Their accuracy is estimated between 2 and 3 cm.
The original gravity data have been directly extracted from the database of the Bureau Gravime´trique International. Their geographic distribution is quite homogeneous, except in the most mountainous areas and in a few very isolated regions (south of Auvergne). The geographic coverage is 43 to 47 N/1 W to 7 E, with a mean density of 0.59 point/km. These data were converted into the IGSN71 system. Although they have already been checked by the BGI, some points are still obviously wrong. Some points exhibit heights inconsistent with the DTM. Moreover, the spatial positioning of many points seems inaccurate. Indeed, most of them do not match the road and railway network, where they are in fact
56
Auvergne Dataset: Testing Several Geoid Computation Methods
467
Fig. 56.2 Map of gravity points distribution
located. The accuracy of the gravity values was evaluated to be 0.25–0.75 mGal, but in reality the consistency between two gravity campaigns can be worse than 2 mGal. As gravity campaigns were realized mostly in the 1970s, the accuracy of the East and North coordinates of these points can be worse than 300 m (mean of 50–100 m). The altimetry positioning is good, since most of these points were measured during levelling campaigns. The reader is referred to Duquenne (2006) for a more detailed description of the Auvergne dataset. The performance of various geoid modelling methods has been tested for the Auvergne area. A reference geoid was computed using the Gravsoft package (Tscherning et al. 1992) and the unmodified Stokes’ function. Modifications of Stokes’ kernel using deterministic or stochastic methods have been tested. More precisely, the Wong and Gore and the KTH method (Sj€oberg 2003) have been used. For each geoid modelling method, we tested several input parameters, such as the global gravity models (EGM08 and GL05C) or the size of the kernels for the Stokes’ integration (up to 3 ). The quality of the computed
geoid models should be a good indicator of how mathematical modifications can be a solution to the problems of geoid computations.
56.2
Computation Using the Unmodified RemoveCompute-Restore Method
We first computed geoid models using the removecompute-restore method and the Gravsoft package (Tscherning et al. 1992) using the residual terrain anomalies. This method has been applied on the same dataset by Duquenne (2006). The computed geoid height (before restoring the contribution of the terrain) can be written as (Vincent and Marsh 1974): R N¼ 4pg þ
ðð s0
SL ðcÞðDg DgGGM Þds
Dmax R X 2 Dgn 2g n¼2 n 1
(56.1)
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P. Valty et al.
SL ðcÞ ¼ SðcÞ ¼
1 X 2n þ 1 n¼2
n1
Pn ðcos ’Þ
(56.2)
or: SL ðcÞ ¼ SðcÞ ¼
1 6 sinðc=2Þ þ 1 5 cosðcÞ sinðc=2Þ 3 cosðcÞ In (sin(c=2ÞÞ þ sin2 ðc=2Þ (56.3)
47 53 52.5 52
48
47.5
46.5
51.5 51
48.5 48
49
.5
49
49
48
46
50
45.5
.5
50 . 51 5
.5
50
50.5 50
50
49.5 49 48.5
51
.5
50
Various values for the integration radius of the terrain corrections and the Stokes’ kernel have been tested in order to find the most appropriate ones. Furthermore, various global geopotential models have been tested to evaluate their performance in the long to medium wavelengths of the gravity field spectrum EGM2008 (Pavlis et al. 2008) up to 360 and GL05C (F€orste et al. 2008). As we need a grid of residual gravity anomalies to compute a numerical Stokes’ integration, we applied the kriging interpolation method for the gridding as it is known to give good results. The estimated precision of the interpolation is 2.2 mGal. We also used for the interpolation a low noise variance of 0.2 mGal, close to the a-priori data precision (standard deviation of around 0.45 mGal, corresponding to a variance of 0.2 mGal). This value leads to a better agreement between the grid and the values of residual point
51
n¼2
the global geopotential model and Dg the free-air gravity anomaly corrected for the terrain effects. Dmax is the maximum degree of expansion of the global gravity field model used. Here SL(c) is the unmodified Stokes’ function, and can be written as:
gravity anomalies. The main changes are summarized in Table 56.1. The best results were obtained using the following parameters: • The long wavelengths of the geoid were computed using the GL05C global field model (F€orste et al. 2008) up to 360 . • The terrain corrections are integrated up to a radius of 75 km for the low-resolution DTM and 20 km for the accurate DTM, in order to take into account all gravity points. • The gravity anomalies are integrated to a radius of 2.5 while the 1D-FFT spherical Stokes’ convolution has been used for the computation of the residual geoid heights. To check the quality of the computed models, we compared them with the geoid heights at the GPS/levelling points. We estimated and then removed a linear trend which is a function of the latitude and the longitude (5 parameters), and the obtained residuals are shown in Fig. 56.4. The mean value and the
.5
Dgn is the gravity anomaly from
49
DP max
49
where DgGGM ¼
52
45 1
1.5
2
2.5
3
3.5
4
4.5
5
48 47.5 47 46.5
Fig. 56.3 Geoid model obtained using GL05C global field and a 2.5 Stokes’ integration (m)
Table 56.1 Statistics (bias and standard deviation) obtained on geoid models computed with the remove-compute-restore method and unmodified Stokes’ kernel Global field Radius of integration of refined terrain anomalies Radius of integration of Stokes’ anomalies Noise variance (for interpolation of residual gravity anomalies) (mGal) Bias (cm) Standard deviation (cm)
EGM08 (360 ) 20 km 2 2.3
EGM08 (360 ) 20 km 2.5 2.3
EGM08 (360 ) 20 km 2.5 0.2
EGM08 (360 ) 20 km 3 2.3
20 km 2 2.3
20 km 2.5 2.3
21 4.9
24.5 4.8
20.7 5.9
29.4 4.7
18.1 4.4
19.0 4.3
GL05C GL05C
Noise variance is computed as the difference between the detrended data variance and the least-squares estimated covariance at the origin
56
Auvergne Dataset: Testing Several Geoid Computation Methods
standard deviation of these residuals are given in Tables 56.1–56.3.
56.3
Test of Stokes’ Modified Kernel Methods
Equation (56.1), in Sect. 56.2, is the generalization of Stokes’ scheme by Vanicek and Sj€ oberg (1991). In the classic remove-compute-restore method, the Stokes’
46.5
46
469
function is unmodified, but in the modified approach, SL can be expressed as: SL ðcÞ ¼ SðcÞ
L X 2k þ 1 k¼2
2
sk Pk ðcosðcÞÞ
(56.4)
where the sk coefficients are the modification parameters of the Stokes’ function. We compared two kinds of modifications of the Stokes’ formula: deterministic methods and stochastic methods (Ellmann 2004). The deterministic modification reduces the effect of the high-frequency contribution of the neglected integration area (when c > c0), whereas the stochastic ones also take into account the errors of the global field model and of the gravity data, and proceed towards minimization of the influence of these errors (see Sect. 56.3.1).
56.3.1 Deterministic Kernel Modifications
45.5
2
2.5
3
3.5
4 – 0.0781 to –0.05 – 0.05 to –0.02 – 0.02 to 0.02 0.02 to 0.05
0.05 to 0.1 0.1 to 0.105
Fig. 56.4 Residuals (in m) obtained on the GPS/levelling points with the geoid computed with the GL05C model, the Stokes’ radius set to 2.5 and the refined terrain integration radius set to 20 km
We tested the Wong and Gore modification of the Stokes’ kernel (Wong and Gore 1969) on the Auvergne dataset. In (56.4), the parameters sk can thus be expressed as: sk ¼
2 kþ1
(56.5)
An important parameter to set for the methods employing modification of Stokes’ kernel is the higher degree of modification L. A good choice is to set L such as SL(c) ¼ 0 for c ¼ c0, where c
Table 56.2 Statistics (bias and standard deviation) obtained on geoid models computed with the Wong and Gore modified Stokes’ kernel Modification used : Wong Gore L Radius of integration of Stokes’ anomalies Bias (cm) Standard deviation (cm) Modification used : Wong Gore L Radius of integration of Stokes’ anomalies Bias (cm) Standard deviation (cm)
EGM08 (360 ) EGM08 (360 ) EGM08 (360 ) 115 92 67 2 2.5 2 11.9 0.1 11.8 4.5 4.3 5.0 EGM08 (360 ) EGM08 (360 ) 20 100 2 2 16.6 11.4 5.1 4.7
EGM08 (360 ) 67 2.5 10.6 4.7 EGM08 (360 ) 360 2 13.4 7.1
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€berg Table 56.3 Statistics (bias and standard deviation) obtained on geoid models computed with the KTH method and the Sjo biased modification, for several parameters and global geopotential models Modification used : Sj€ oberg biased L Dmax Radius of integration for Stokes’ anomalies Variance of gravity data (mGal) Atmospheric and ellipsoidal corrections Bias (cm) Standard deviation (cm) Standard deviation for Wong–Gore modification (with the same parameters) Standard deviation for classic remove-compute-restore method (with the same parameters)
EGM08 (360 ) 90 90 2 4 No 14.4 4.4 4.5
EGM08 (360 ) 72 72 2.5 4 No 19.7 4.2 4.3
4.9
4.8
EGM08 (360 ) 72 72 2.5 1 No 19.4 4.2
EGM08 (360 ) 60 60 3 4 No 33.6 3.8 4.0
GL05C GL05C GL05C 90 90 2 4 No 13.3 4.2 4.4
72 72 2.5 4 No 14.1 4.0 4.1
4.7
4.4
4.3
72 72 2.5 4 Yes 13.2 4.0
To allow comparisons, results obtained with Wong and Gore and unmodified method are recalled in the two last rows
is the Stokes’ integration radius (Ellmann 2004). The index Dmax is independent of L. Better results were obtained using Dmax ¼ 360, but other values (especially Dmax ¼ L) have been tested. The Stokes’ integration is carried out on the residual gravity anomalies. The bias of the computed geoid is smaller than that of the unmodified Stokes’ kernel, and the precision is also slightly better, especially when L is chosen as explained above. To illustrate the effects of the different parameters, we present in Table 56.2 the results obtained with EGM08. Other results obtained with GL05C can be found in Table 56.3.
56.3.2 Stochastic Modifications of Stokes’ Kernel Following the studies of Ellmann (2004), we tested the KTH (Stockholm Royal Institute of Technology) method with Sj€oberg stochastic modifications of the Stokes’ formula. The two main characteristics of this approach are: • The Stokes’ integral is computed with gravity anomalies without removing the global field • The coefficients sk are computed by least-squares, so as to minimise the mean square error of the difference between the estimator of (56.6) and the “true” geoid height expressed by (56.7) (see below). The modelled geoid height (Sj€ oberg 2003) and the “true” geoid height (Heiskanen and Moritz 1967) can be written as:
Nmodelled
R ¼ 4pg
ðð s0
SL ðcÞDgds þ
Dmax R X bn Dgn 2g n¼2
(56.6) Ntrue ¼
1 R X 2 Dgn 2g n¼2 n 1
(56.7)
The expected mean square error m of the difference between (56.6) and (56.7) is given by (56.8): ðð 1 m2 ¼ E ðNmodelled Ntrue Þ2 ds 4p s0 2 X 2 R Dmax 2 R ¼ bn dcn þ 2g n¼2 2g " 2 # 1 X 2 L 2 L ðbn Qn sn Þ cn þ Qn sn s2n n 1 n¼2 (56.8) The coefficients sk are computed by solving the equation: @m2 ¼0 @sk
(56.9)
The coefficients sk can be evaluated as functions of the degree variances cn and the error degree variances dcn of the global gravity field model on the one hand, and of the error degree variance of gravity data sn on the other hand. sn is the function of the a priori variance of gravity data C(0). Note that the coefficients of (56.8) also contain parameters which depend on the
Auvergne Dataset: Testing Several Geoid Computation Methods 47
0.19
0.07
0.05
0.17 0.15
11 0.
0.0
0.13
9
0.11 0.09
0. 07
46.5
0.09 0.07
1
1 0.
0.09
Fig. 56.5 Discrepancies between geoid models computed with Sj€oberg biased modification and unmodified Stokes’ kernel (Sect. 56.2), with GL05C global field and a 2 radius of Stokes’ integration
471
0.0 9
56
46
0.07 0.05
03
0.
0.03
– 0.01
0.05
45.5
.01
–0
0.0
1
0.0
0.01 – 0.01
0.11 .09 7 0 0 .0 0 .0 5 3 0.0
0.01
– 0.03 – 0.05 – 0.07
1
– 0.09 – 0.11
45 1
1.5
radius of Stokes’ integration. For more details see Ellmann (2004) and Sj€ oberg (2003). We compared the biased and the unbiased Sj€oberg modification (Sj€ oberg 1984). In the latter method, the least-squares estimated parameters sk are equal to the bk in (56.6). But we found difficulties to invert the normal matrix for the computation of the coefficients sk with the unbiased method. Indeed, the matrix was strongly ill-conditioned and, for example, had only 5 or 6 acceptable singular values out of 67. Consequently this was not enough to be inverted without losing the meaning of the method. Although the unbiased method is considered the best (Ellmann 2004), we finally used the biased one. Several values for the variance of gravity data and for the degree of modification of the Stokes’ formula were tested. As for the other methods, Dmax is not necessarily equal to L, but, here, we must have Dmax L, because of the presence of bk in the second term of (56.6). Table 56.3 summarizes the main results. In all computation cases, modifying the Stokes’ kernel by the Sj€oberg method seems to improve a little the quality of the geoid model developed: the standard deviation of the GPS/levelling points is reduced by one millimeter at least, as compared to the other methods. The bias is also reduced as compared to the unmodified remove-compute-restore method. The precision of the best geoid models is better than 4 cm. Atmospheric (Sj€ oberg 1998) and ellipsoidal corrections (Sj€oberg 2002) do not clearly improve the precision of the models, but they allow a reduction in the bias. Figure 56.5 shows the discrepancies between the models computed with the biased Sj€ oberg method and
2.5
2
3
3.5
4
4.5
5
46.5
46
45.5
2
2.5
3
3.5
4 – 0.0785 to –0.05 – 0.05 to –0.02 – 0.02 to 0.02 0.02 to 0.05 0.05 to 0.1 0.1 to 0.1065
Fig. 56.6 Residuals in meters on GPS/levelling points obtained with Sj€oberg biased modification (with GL05C, Stokes’ integration at 2 )
with the unmodified remove-compute-restore method (with the same parameters). Discrepancies are quite small, but they are higher on the eastern and southwestern boundaries. However, the differences of standard deviation between the computation methods are however not large enough to have a clear conclusion about the influence of the Sj€oberg modification.
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be improved. We will also work on the dataset in order to improve the accuracy of the spatial (2D) positioning of the gravity points, using the database of roads and railways and DTMs.
46.5
References
46
45.5
2
2.5
3
3.5
4 –0.0845 to – 0.05 –0.05 to – 0.02 –0.02 to 0.02 0.02 to 0.05 0.05 to 0.09131
Fig. 56.7 Residuals in meters on GPS/levelling points obtained with unmodified Stokes’ kernel (with GL05C, Stokes’ integration at 2 )
Finally, we further compared the residuals on the GPS/levelling points. The analysis of Figs. 56.6 and 56.7 shows that the residuals at GPS/levelling points are reduced by the Sj€ oberg modification on a large western side, even if we note some higher residuals in the North-Eastern corner. These residuals are computed after removing bias and trend. Before removing the bias, residuals obtained with the Sj€oberg modification are much smaller. Conclusions
This study contributes to our efforts for improving geoid modelling in France. The best results were obtained with the Sj€ oberg biased modification of Stokes’ kernel, with an average precision of 4.11 cm and a best fit obtained at 3.8 mm. Nevertheless, the Wong and Gore approach performs almost as well, with an average precision of 4.26 cm. The classical method is a bit worse (4.62 cm), but still provides acceptable results. In the future, we will continue this work by trying to better understand why the condition numbers of the unbiased Sj€oberg method are so low and if they can
Duquenne H (2006) A data set to test geoid computation methods. In: Dergisi H (eds.), Proceedings of the 1st international symposium of the international gravity field service “gravity field of the earth”, International gravity field service meeting, Istanbul, Turkey, pp 61–65 Ellmann A (2004) The geoid for the Baltic countries determined by least-squares modification of Stokes’ formula. Doctoral Dissertation in Geodesy, Royal Institute of Technology, Stockholm, Sweden F€orste C, Flechtner F, Schmidt R, Stubenvoll R, Rothacher M, Kusche J, Neumayer KH, Biancale R, Lemoine J-M, Barthelmes F, Bruinsma S, K€onig R, Meyer U (2008) EIGEN-GL05C – a new global combined high-resolution GRACE-based gravity field model of the GFZ-GRGS cooperation, Geophysical Research Abstracts, vol 10, EGU2008A-03426 Heiskanen WA, Moritz H (1967) Physical geodesy. WH Freeman, San Francisco, CA Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2008) An earth gravitational model to degree 2160: EGM2008, Session G3: “GRACE Science Applications”, EGU Vienna Sj€oberg LE (1984) Least-squares modification of Stokes and Vening-Meinesz formulas by accounting for errors of truncation and potential coefficients errors. Manuscripta Geodetica 9:209–229 Sj€oberg LE (1998) The atmospheric geoid and gravity corrections. In: Proceedings of the 2nd continental workshop on the geoid in Europe, Report of the Finnish Geodetic Institute 98:4 Sj€oberg LE (2002) The ellipsoidal correction to Stokes’ formula. In: “Gravity and Geoid 2002”, proceedings of the 3rd meeting of the ICGC, Ziti editions Sj€oberg LE (2003) A general model of modifying Stokes’ formula and its least-squares solution. J Geodesy 77:459–464 Tscherning CC, Forsberg R, Knudsen P (1992) The GRAVSOFT package for geoid determination. In: Proceedings of 1st continental workshop on the geoid in Europe, Prague Vanicek P, Sj€oberg LE (1991) Reformulation of Stokes’ theory for higher than second-degree reference field and modification of integration kernels. J Geophys Res 96 (B4):6529–6339 Vincent S, Marsh J (1974) Gravimetric global geoid. In: Proceedings of the international symposium on the use of artificial satellites for geodesy and geodynamics, National Institute of Technology, Athens, Greece Wong L, Gore R (1969) Accuracy of geoid heights from the modified Stokes’ kernels. Geophys J Roy Astron Soc 18:81–91
In Pursuit of a cm-Accurate Local Geoid Model for Ohio
57
K.R. Edwards, Dorota Grejner-Brzezinska, and Dru Smith
Abstract
As part of its strategic plan for 2008–2018 [The NGS Ten Year-Plan, Mission, vision and strategy, 2008–2018], the National Geodetic Survey has resolved to engage in activities which would allow for the development of a 1-cm accurate national gravimetric geoid for the conterminous US. In this regard, the Ohio Department of Transportation has been collaborating with the OSU SPIN Laboratory in height modernization activities for the state of Ohio. Presented in this paper are the results of an investigation used to evaluate the quality of gravity and height data needed to produce a cm-accurate geoid in Ohio. In this study a local geoid model over Ohio was computed in a remove-restore geoid determination procedure using EGM2008 [Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2008) An earth gravitational model to degree 2160: EGM2008, presented at the 2008 General Assembly of the European Geosciences Union, Vienna, Austria, April 13–18], publicly-available surface gravity data from the PACES website and the GTOPO30 DEM. Terrain corrections (subject to a planar approximation) were evaluated using a 2D FFT algorithm [Forsberg R (1985) Gravity field terrain effect computations by FFT. Bull Ge´ode´sique 59:342–3601; Forsberg R (1997) Terrain effects in geoid computations. Lecture notes. International School for the Determination and Use of the Geoid, Rio de Janeiro, Sept 1997]. Ohioan terrain being substantially flat (on average about 330 160 m AMSL) produced terrain corrections which were, for the most part, at the sub-mGal level. However, these translated into a geoid contribution of about 0.039 0.038 m in the local model. A 1D FFT technique [Haagmans R, de Min E, von 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] was used to evaluate the Stokes’
K.R. Edwards (*) D. Grejner-Brzezinska Satellite Positioning and Inertial Navigation (SPIN) Laboratory, The Ohio State University, Columbus, OH 43210, USA e-mail:
[email protected] D. Smith National Oceanic and Atmospheric Administration/National Geodetic Survey, 1315 E-W Highway, Silver Spring, MD 20910, USA S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_57, # Springer-Verlag Berlin Heidelberg 2012
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integral [Heiskanen WA, Moritz H (1967) Physical geodesy, W.H. Freeman, San Francisco, CA] over a 5 5 region on a 50 50 grid encompassing the state of Ohio and its environs. This local geoid model was used as a reference solution for statistical comparisons made to subsequently computed geoid determinations (over the same region) in which the latter had been evaluated using surface gravity and height data sets subjected to simulated zero-mean Gaussian-distributed random errors of homogeneous spatial distribution. While EGM2008 was assumed to be perfect, the standard deviation of the errors applied to the surface gravity ranged from 0.1 to 5 mGal while those associated with the height data (of both the gravity and the DEM) ranged from 0.5 to 20 m. For each Gaussian dispersion utilized, 100 simulated error-prone data sets were selected and their associated geoid solutions determined using the same Stokes/FFT algorithms which were used to evaluate the aforementioned reference geoid model. Summary statistics were evaluated for each set of the 100 “randomized” geoid models relative to the reference solution, allowing for an evaluation of the potential impact of random errors present in the input height and gravity data on the local geoid solution in Ohio. It was found that simulated height errors which were 10 m or less produced a 1 cm (1s) accurate local geoid while those in excess did not. RMS differences of 1–1.6 cm occurred on application of gravity errors of 3 to 5 mGal, prohibiting the possibility of achieving a cm-accurate geoid. Based on the results of this study, it was concluded that minimum requirements for a cm accurate geoid determination in Ohio would be a combination of gravity and height data accurate up to about the 3 mGal and 10 m levels, respectively. Future studies will be conducted using updated gravity and height data sets. In addition the geoid height error analysis would account for (1) the implementation of a spatially-heterogeneous error modeling scheme based on surface gravity data density and (2) simulated random errors in the global geopotential model used in concert with the surface gravity and height data sets.
57.1
Data
57.1.1 PACES Gravity Data Approximately 43,100 spatially-inhomogeneous spot gravity data throughout Ohio and its environs (cf. Fig. 57.1) were downloaded from the Pan American Center for Earth and Environmental Studies (PACES) GeoNet gravity database. As can be seen from Fig. 57.1 there is no gravity data over the Great Lakes region. Also noteworthy is the fact that some areas evidence gravity data density which is as low as 1 point per 4 km2 to as many as 10 points per km2 (such as in the northwest and south-west corners of the state of Ohio). Given that the local gravimetric geoid was computed using a
50 50 spacing, there are a few grid cells in the nonlake region that are data-bankrupt.
57.1.2 GTOPO30 A local digital elevation model (cf. Fig. 57.2) extracted from the 3000 3000 GTOPO30 model was used to compute the terrain corrections. DEM heights in the region vary from about 0 to 2,000 m with an average elevation of 294.109 186.894 m. GTOPO30’s heights were compared to the orthometric heights associated with the PACES gravity data in order to validate GTOPO30’s nominal RMSE of 18 m. The average difference between them was
57
In Pursuit of a cm-Accurate Local Geoid Model for Ohio
475
–31.5
42
–32.5
–33.5
40
–34.5
–35.5
38
–36.5
-85
-83
-81
Fig. 57.3 EGM2008 over Ohio and its environs [m]
(cf. Fig. 57.3). Gravity in the 50 50 grid cells which are gravity data-less will automatically be implied by the EGM2008-derived gravity anomalies.
Fig. 57.1 PACES gravity data in the region of interest
1900 44
1700 1500 42
57.2
Local Geoid and Reference Model Formulation
1300 1100
57.2.1 Terrain Correction Computations
40
900 700
38
In this study, terrain reductions were performed using the classical terrain correction, cP, given by:
500 300
36
1 ð
100 -88
-86
-84
-82
-80
1
-78
Fig. 57.2 GTOPO30 over Ohio and its environs [m]
found to be 2.973 19.940 m (1s), essentially validating the 18 m error estimate. Notably, no bathymetry in the Great Lakes region was used.
57.1.3 EGM2008 The WGS84-referenced geoid undulations and gravity anomalies of EGM2008 (the global geopotential model complete to spherical harmonic degree and order 2159) [5] contributed medium to long wavelength geoid features to the local geoid determination
ð ðh
cP ¼ G r
z hP dxdydz r3
(57.1)
hP
where G is the gravitational constant, r is the mass density, (xP, yP, hP) refers to position of the computation point, (x, y, z) refers to the integration point and r2 ¼ ðxP xÞ2 þ ðyP yÞ2 þ ðhP zÞ2 . Given that Ohioan topography is predominantly flat the approximation: r2 ðxP xÞ2 þ ðyP yÞ2 ¼ r20
(57.2)
can be used, giving rise to the planar approximation cP: cP
ymax xmax X Gr dx dy X ðh hP Þ2 : 2 r30 x¼xmin y¼y min
(57.3)
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Therefore, in keeping with common practice, the local terrain corrections (shown in Fig. 57.4) were computed using a 2D FFT algorithm [1,2]. The difference between the terrain-corrected and non-terrain corrected local geoid solution is shown in Fig. 57.5. Note that in spite of Ohio’s flatness, the neighboring Appalachian mountain range contributes a terrain effect which, in Ohio, ranges from as little as 1 mm to about 7 cm (which is not negligible). The difference is understandably most significant in the SW corner of Fig. 57.5 due to the West Virginian mountain chain where the terrain correction is as large as 30 mGal. 45
5
44
4.5
43
4
42
3.5
41
3
40
2.5
39
2
38
1.5
37
1
36
0.5
35 -88
-86
-84
-82
-80
-78
0
Fig. 57.4 Terrain corrections using GTOPO30 [mGal]
57.2.2 Local/Reference Geoid Determination The local geoid model used as the reference solution (MCG0 in Fig. 57.6) for the Monte Carlo analysis (which is described in Sect. 57.3) was computed using the well-known remove-restore technique. There are three main contributors to the local geoid height, N: N ¼ NEGM þ NStokes þ NIndirect Effect :
– NEGM is the height anomaly contribution of EGM2008 from degree and order 0 to 2159. – NStokes is the result of the Stokes’ integral [4] defined by: Stokes
N
R ¼ 4pg
ðð DgFaye res Sð cÞd s
–0.03
(57.5)
s
where R is the earth’s radius, g is the normal gravity of the GRS80 ellipsoid, S(c) is the Stokes function (a gravity-weighting function which is inversely proportional to the spherical distance c between the computation and integration points) and DgFaye is res essentially the gravity anomaly from which the EGM2008 effect has been removed. In this study, the Stokes’ integral was implemented using the 1D FFT [3].
–0.01 42
(57.4)
-30.5 42
-31.5
–0.05
-32.5
–0.07
40
40 –0.09
-33.5
–0.11 –0.13
38
-34.5 38
-35.5
–0.15
-85
-83
-81
Fig. 57.5 Impact of terrain corrections on the local geoid evaluation [m]
-85
-83
-81
Fig. 57.6 Reference Geoid model (MCG0) over region of interest [m]
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In Pursuit of a cm-Accurate Local Geoid Model for Ohio
477
– NIndirect Effect compensates for mass re-distribution techniques used and terrain correction assumptions made and was computed using:
Monte Carlo geoid grids referred to as MCGk (where k ¼ 1, 2, . . ., n ¼ 100). 3. Finally the statistical indicators (defined below as DMCG and rmseMCG) used to compare the Monte Carlo geoid grids (MCGk) to the reference solution (referred to as MCG0 in Fig. 57.6) were evaluated for each grid cell (i, j) as follows: (a) DMCGði; jÞ ¼ meanMCGði; jÞ MCG0 ði; jÞ
NIndirect Effect ¼
pGrH2 : g
(57.6)
Given the gravity data distribution (shown in Fig. 57.1), the local gravimetric geoid was computed using a 50 50 grid spacing.
57.3
100 P MCGk ði; jÞ where: meanMCGði; jÞ ¼ n1 k¼1 (b) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½MCGk ði; jÞ MCG0 ði; jÞ2 rmseMCGði; jÞ ¼ n
Monte Carlo Study
The main objective of this study was to evaluate the extent to which errors in the input height and observed gravity data translate into discrepancies in the local geoid model. To this end, a Monte-Carlo analysis of the geoid computation process was performed as follows: 1. Zero-mean uncorrelated Gaussian errors of selected standard deviations (expressed by sHgrid, sHspot and sg) were applied to GTOPO30 heights, spot orthometric heights and the gravity datasets, respectively. 2. Using the normally-distributed error-prone datasets, multiple (n ¼ 100) processing runs of the geoid process were executed to create the
57.4
Results
As can be seen from Table 57.1 and Figs. 57.7–57.10, it was found that cm-accurate results occur under the following conditions: – If only DEM errors are applied (cf. Fig. 57.7a, b), the DEM should be accurate up to 10 m. Note that the GTOPO30 extract used in this study demonstrated accuracy of about 19 m in the region of interest. – If only spot height errors are applied (cf. Fig. 57.8a, b), the height observation technique should be better than 15 m. However, given the ubiquitous use
Table 57.1 Comparison of Monte Carlo Geoids to Reference Geoid for Multiple Error Combinations (Results of highlighted rows are also demonstrated in Figs. 57.7–57.10) Standard deviation of applied errors sHgrid [m] sHspot [m] sg [mGal] 5 10 15 5 10 15 0.5 1 3 10 3 2 10 3 1 10 2 1
Summary statistics for DMCG grid Mean [m] STD [m] Min [m] Max [m] 0.001 0.001 0.001 0.003 0.004 0.002 0.001 0.008 0.009 0.002 0.001 0.015 0.000 0.000 0.001 0.001 0.000 0.001 0.003 0.002 0.000 0.001 0.004 0.005 0.000 0.000 0.001 0.001 0.000 0.000 0.001 0.001 0.000 0.001 0.005 0.004 0.004 0.002 0.001 0.009 0.004 0.001 0.000 0.008 0.004 0.002 0.002 0.008
Summary statistics for rmseMCG grid Mean [m] STD [m] Min [m] Max [m] 0.004 0.000 0.003 0.006 0.010 0.001 0.006 0.013 0.016 0.002 0.009 0.021 0.003 0.001 0.001 0.007 0.007 0.002 0.002 0.013 0.010 0.003 0.002 0.020 0.002 0.001 0.000 0.003 0.003 0.001 0.001 0.007 0.010 0.003 0.002 0.021 0.012 0.002 0.006 0.019 0.010 0.001 0.006 0.014 0.010 0.001 0.006 0.014
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Fig. 57.7 (a) DMCG grid [in meters] (b) rmseMCG grid [in meters] where sHgrid ¼ 10 m, sHspot ¼ 0 m, sg ¼ 0 mGal
Fig. 57.8 (a) DMCG grid [in meters] (b) rmseMCG grid [in meters] where sHgrid ¼ 0 m, sHspot ¼ 15 m, sg ¼ 0 mGal
of GPS for real-time positioning, future gravity surveys should be able to achieve heighting accuracy at the dm (or better) level. – If only gravity errors are applied (cf. Fig. 57.9a, b) then the gravity data accuracy should be better than 3 mGal. In addition: – Figures 57.8b and 57.9b demonstrate that the areas evidencing the lowest rmseMCG values
(cf. Fig. 57.8b) coincide with those of a highest gravity data density, as well as, the data-deficient Great Lakes region. Conversely, geoid height accuracy degrades (as evidenced by the higher rmseMCG pixel values) in areas of sparse gravity data distribution (Great Lake region exclusive). – Other combinations of errors which produced cm-accurate results include those indicated by the last three rows of Table 57.1 and Fig. 57.10.
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In Pursuit of a cm-Accurate Local Geoid Model for Ohio
479
Fig. 57.9 (a) DMCG grid [in meters] (b) rmseMCG grid [in meters] where sHgrid ¼ 0 m, sHspot ¼ 0 m, sg ¼ 3 mGal
Fig. 57.10 (a) DMCG grid [in meters] (b) rmseMCG grid [in meters] where sHgrid ¼ 10 m, sHspot ¼ 3 m, sg ¼ 2 mGal
57.5
Conclusions and Future Work
This Monte Carlo study was an attempt to intuitively evaluate the quality of height and gravity data needed to produce a cm-accurate gravimetric geoid – a goal expressed in NGS Ten-Year Plan for 2008–2018 [6]. It was found that gravity data accuracy should be better than 3 mGal, while the accuracy of the DEM and spot height data should be better than 10 and
15 m respectively. This implies that an alternative, higher accuracy (and likely higher resolution) DEM and more accurate spot gravity heighting techniques (e.g. employing real-time GPS observations) should be used for the terrain reductions. In addition, further consideration should be given to the impact of gravity data density on the solution as areas of higher density gravity data demonstrated higher accuracy geoid results.
480
Without a doubt, inclusion of gravity data over the Great Lakes region (which is consistent with the onshore data) will impact positively on the solution. It should be noted, though, that results in this study seem to suggest that EGM2008 has provided and can provide a reasonable substitute until such time as this Great Lakes data deficiency can be met. In addition, future work should evaluate the extent to which errors in EGM2008 and issues like topographic and density variability of terrain affect one’s ability to achieve cm-accuracy in the geoid. Acknowledgements This study is one aspect of Height Modernization investigations being conducted for the state of Ohio and being supported by the Ohio Department of Transportation (ODOT). We are grateful to ODOT’s Mr. John Ray for his support of this project.
K.R. Edwards et al.
References 1. Forsberg R (1985) Gravity field terrain effect computations by FFT. Bull Ge´ode´sique 59:342–360 2. Forsberg R (1997) Terrain effects in geoid computations. Lecture notes. International School for the Determination and Use of the Geoid, Rio de Janeiro, Sept 1997 3. Haagmans R, de Min E, von 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 4. Heiskanen WA, Moritz H (1967) Physical geodesy. W.H. Freeman, San Francisco, CA 5. Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2008) An earth gravitational model to degree 2160: EGM2008, presented at the 2008 General Assembly of the European Geosciences Union, Vienna, Austria, April 13–18 6. The NGS Ten Year-Plan, Mission, vision and strategy, 2008– 2018. http://www.ngs.noaa.gov/INFO/NGS10yearplan.pdf
Adjustment of Collocated GPS, Geoid and Orthometric Height Observations in Greece. Geoid or Orthometric Height Improvement?
58
I.N. Tziavos, G.S. Vergos, V.N. Grigoriadis, and V.D. Andritsanos
Abstract
The combined adjustment of GPS/Levelling observations on benchmarks with gravimetric geoid heights has been the focus of extensive research both from the theoretical and practical point of view. Up until today, with few exceptions, the main blame for the inconsistencies/disagreement between these three types of heights has been put to the geoid heights due mainly to their poorer accuracy. With the advent of the new CHAMP- and GRACE-based global geopotential models and the realization of EGM2008 the achievable cumulative geoid accuracy has improved significantly so that its differences to GPS/Levelling heights reach the few cm level. In Greece, GPS observations on BMs are very scarce and cover only small parts, in terms of spatial scale, of the country. Recently, an effort has been carried out to perform new GPS measurements on levelling BMs, so that reliable GPS/Levelling and gravimetric geoid height adjustment studies can be carried out. This resulted in part of North-Western Greece to be covered with reliable observations within an area extending 3 in longitude and 1 in latitude. Therefore, some new potential for the common adjustment of the available geometric, orthometric and geoid heights, using various parametric surfaces to model and interpret their differences, are offered. These are used to come to some conclusions on the accuracy of the various geoid models used (both global geopotential and local gravimetric models), while an extensive outlook is paid to the questionable behaviour of the orthometric heights. The latter is especially important for the Greek territory since the available benchmarks are delaminated in so-called “map-leaflets” and a common adjustment of the entire vertical network has not been carried out so far. It is concluded that even between neighbouring “map-leaflets” large biases in the adjusted GPS/Levelling and gravimetric geoid heights exist, which indicates distortions in the Greek vertical datum as this is realized by the levelling benchmarks. Given that the latter are commonly used for everyday surveying purposes, conclusions and proposals on the determination of adjusted orthometric heights are finally drawn.
I.N. Tziavos G.S. Vergos (*) V.N. Grigoriadis V.D. Andritsanos Department of Geodesy and Surveying, Aristotle University of Thessaloniki, University Box 440, 54124, Thessaloniki, Greece e-mail:
[email protected] S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_58, # Springer-Verlag Berlin Heidelberg 2012
481
482
58.1
I.N. Tziavos et al.
Introduction
During the last two decades and since the advent of GNSS positioning, the combined adjustment of GPS ellipsoidal heights (h) with orthometric heights (H) from conventional levelling and gravimetric geoid heights (N) has gained increasing importance (Featherstone 1998). This refers both to the scientific treatment of the combination problem as well as to every day surveying campaigns. The purely scientific treat of the combination of these three height types dealt mainly with efforts to model and interpret the height residuals at stations where collocated GPS/Levelling and geoid observations were available. The differences between them were, and still are, explained as datum biases, long-wavelength geoid errors and random errors remaining to all height types. In most cases, the blame for the large discrepancies was put to gravimetric geoid heights due to the inadequacy, in both resolution and accuracy, of the historical gravimetric databases and the unavailability of satellite observations to boost the accuracy of global geopotential models (GGMs) to higher degrees of expansion. On the other hand, GPS and levelling observations were considered to contribute little to the total error budget due to the accuracy of the former in differential static measurements at levelling benchmarks (BMs) and the unanimous knowledge that spirit levelling is indeed the most accurate means for orthometric height determination. Within this frame, collocated observations of h, H and N are used to: (a) assess the external accuracy of gravimetric geoid models (Featherstone et al. 2001), (b) construct socalled corrector surfaces in an area of study, so that the transformation between either of the three can be made (Sideris et al. 1992), and (c) substitute conventional spirit levelling by GPS/Levelling during which there is no need to measure orthometric heights since they are determined by GPS measurements and gravimetric geoid heights (Fotopoulos et al. 2001; Vergos and Sideris 2002). A distinction has to be made at this point concerning the terms scientific and everyday surveying purposes mentioned previously. As far as the former is concerned, we are mostly interested in the absolute differences between h, H and N using statistical measures as the range, mean and standard deviation (std) to assess the performance of (mainly) the available gravimetric geoid model and/or GGM. Relative differences are important as well, but as an
additional measure of the achievable accuracy. Due to the need for high-accuracy in an absolute sense, almost all available GGMs and gravimetric geoid models, until recently, did not manage to provide rigorous results for point (c) above. On the other hand, for everyday surveying purposes, where a pair of GPS receivers is used and the base is set at a reference benchmark, the need for high absolute accuracy is not mandatory. This is so because even with an EGM96-class of GGM, the long-wavelength and other errors in the geoid heights are removed by computing essentially relative height differences between the measuring point that the rover and the benchmark of the base is set to. With the recent gravity-field dedicated missions of CHAMP, GRACE and GOCE and the realization of EGM2008 (Pavlis et al. 2008), the available GGMs have much more power up to very-high degrees and increasing accuracy. EGM2008 was released to public by the U.S. Geospatial-Intelligence Agency (NGA) EGM Development Team and presents a spherical harmonics expansion of the geopotential to degree and order 2,159. The availability of such GGMs poses new potentials in order to validate available orthometric heights and subsequently correct blunders in the levelling databases. This is of special importance in countries like Greece where: (a) the vertical reference network, realized through the network of levelling BMs, has not been commonly adjusted in a unified frame, (b) in various parts of the country the zero-point w.r.t. which the heights of the BMs have been determined, varies and is set to coincide with a local tide-gauge station, (c) the levelling BMs are delaminated in so-called “mapleaflets” which often have horizontal and vertical distortions. The latter creates significant problems to everyday GPS surveying applications when levelling BMs from neighbouring “map-leaflets” are used in a single traverse. The main goal of the present study stems from the aforementioned problems for the Greek territory and has two main goals. The first one is to investigate whether blunders in the orthometric heights can be identified and corrected when collocated GPS and geoid observations are available. The second one is to evaluate the performance of GGMs and regional gravimetric geoid models in terms of the differences between h, H and N during their combined adjustment. For that purpose recent observations collected over Northern
58
Adjustment of Collocated GPS, Geoid and Orthometric Height Observations in Greece
21˚00' 41˚00'
21˚30'
22˚00'
22˚30'
23˚00'
23˚30' 41˚00'
483
aTi x ¼ x0 þ x1 cos ’i cos li þ x2 cos ’i sin li þ x3 sin ’i ; (58.3) aTi x ¼ x0 þ x1 cos ’i cos li þ x2 cos ’i sin li þ x3 sin ’i þ x4 sin2 ’i ;
40˚30'
40˚30'
aTi x ¼ 40˚00'
40˚00'
M X N X
(58.4)
xq ð’i ’0 Þn ðli l0 Þm cosm ’i :
m¼0 n¼0
(58.5)
39˚30' 21˚00'
21˚30'
22˚00'
-1600 -1200 -800 -400
0
39˚30' 23˚30' m 800 1200 1600 2000 2400 2800 22˚30'
400
23˚00'
Fig. 58.1 The distribution of the available GPS/Levelling BMs in Northern Greece (triangles)
Greece in a network of 43 benchmarks (see Fig. 58.1) are used.
58.2
Data and Observation Equations
Given the availability of collocated GPS, levelling and gravimetric geoid heights one can write the vector of observations ‘i and the observation equations for their combined adjustment as: GPS=Lev
‘i ¼ hi Hi Ngr i ¼ Ni
Ngr i
In matrix notation the system of observation equations and the solution are written b ¼ Ax þ v
(58.6)
1 ^x ¼ AT PA AT Pb:
(58.7)
and
In (58.7), matrix P is the weight matrix, i.e., the inverse of the variance-covariance matrix C of the observations. Throughout this study we have assumed that (a) the observations and the errors are uncorrelated for all height types and (b) no correlation exists for the same height type among different observation stations i. Therefore the minimization principle and the corresponding weight matrix take the form (Kotsakis and Sideris 1999): P ¼ ðChGPS þ CHLEV þ CNgrav Þ1
(58.1)
(58.8)
and
and ‘i ¼ aTi xi þ vi :
(58.2)
where the elements aTi of the design matrix A and the unknowns xi depend on the parametric model chosen to describe the differences between the triplet of heights. In (58.1) and (58.2), hi, Hi and Nigr denote the available GPS, levelling and gravimetric geoid GPS=Lev heights at station i, and Ni ¼ hi Hi are the so-called GPS/Levelling geoid heights. For the parametric model to be used, various choices have been tested, namely the well-known four- and fiveparameter similarity transformation models and first, second and third order polynomial ones, as presented in (58.3)–(58.5) respectively (Fotopoulos 2003)
1 T 1 T vThGPS C1 hGPS vhGPS þ vH CH vH þ vN grav CN grav vN grav ¼ min
(58.9) where v and C denote residuals and variance-covariance matrices of the GPS, levelling and gravimetric geoid height observations. Based on the parameter estimation ^ and N ^ grav can be ^ H; in (58.7), adjusted observations h; estimated as well along with adjusted residuals ^v and ^ h; C ^ H ; and C ^N adjusted variance-covariance matrices C (see Fotopoulos 2003). Within the frame of the objectives set, first an evaluation of the available parametric models is performed using EGM2008 geoid heights in order to determine the one that provides the best fit. The
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one selected, is then employed to detect blunders in the orthometric heights and estimate new corrected values. A new adjustment using these corrected orthometric heights is performed in order to assess the improvement achieved. Then, an investigation of the influence of the observation input errors on the results of the adjustment is carried-out. Therefore, the fit achieved, when using the local gravimetric geoid model and the other GGMs, is compared to the results provided by EGM2008. The GGMs employed in this study in order to investigate their fit to the GPS/ Levelling geoid heights are EGM2008 (Pavlis et al. 2008), EGM96 (Lemoine et al. 1998), GGM03c, GGM03s (Tapley et al. 2007) and EIGEN5c (Reigber et al. 2005) representing the latest satellite-only and combined models. The final part is devoted to some examples of the biases that exist between neighbouring “map-leaflets” in the adjusted GPS/Levelling and gravimetric geoid heights.
58.3
Combined Adjustment Results
The first set of tests deals with the improvement that each parametric model offers in the adjusted height residuals. All five models have been tested employing the 43 GPS/Levelling observations, geoid heights from EGM2008 and a uniform accuracy of 1 cm for all height types. It is worth mentioning that higher-order polynomial models have also been tested but their parameters have been proven insignificant. From Table 58.1, where the results are summarized, it becomes evident that the best fit is achieved when the third order polynomial model is employed to model the residuals. After the fit, a reduction by 8 cm of the std is achieved while the range reduces also by ~66 cm. The performance of the third order
Table 58.1 Statistics of the differences NGPS/Lev-NEGM08 before and after the fit. Unit: (m) Before 4-param 5-param Firts pol. Second pol. Third pol.
Max 1.314 0.456 0.360 0.400 0.340 0.320
Min 0.268 0.649 0.634 0.813 0.619 0.598
Mean 0.750 0.000 0.000 0.000 0.000 0.000
rms 0.786 0.178 0.168 0.200 0.163 0.156
Std 0.234 0.178 0.168 0.200 0.163 0.156
polynomial model is 1–5 cm better (1s) than the others, which gives good evidence that it is the one to be used for all subsequent fit investigations. Examining the residuals before the fit, the large mean and std of the height differences is noticing. Even though the mean can be attributed to some datum bias, which is treated by the parametric model, the std of the differences is outside the range of the performance of EGM2008, at least for European areas. The latter is expected to reach ~16–17 cm according to the EGM2008 validation performed during its development (Pavlis et al. 2008). Plotting the height differences for all stations (see Fig. 58.2), the results achieved for two of these (pointed with a circle in Fig. 58.2) indicate that they probably contain blunders. This was concluded based on a 2 rms criterion applied to the residuals before the fit (see first line in Table 58.1). Given that the EGM2008 accuracy can be regarded uniform for small areas like the one under investigation and that no blunders are included in the GPS geometric heights, the blame can be put to the orthometric heights for the benchmarks under question. In order to computed adjusted orthometric heights for the two BMs, a new fit was carried out, using the remaining 41 stations and a third order polynomial as a parametric model. Then, employing the estimated parameters from (58.7), adjusted orthometric heights have been determined by applying corrections of 0.506 and 0.115 m. Following the determination of the adjusted orthometric heights a new common adjustment of all 43 stations, similar to the previous one, has been carried out with the results reported in Table 58.2. Comparing the residuals before the fit (first line in Tables 58.1 and 58.2), when the new adjusted orthometric heights are employed, an improvement in the std by ~6 cm is achieved. This signals that the estimated adjusted orthometric heights for the two BMs successfully manage to provide smaller residuals. Moreover, the initial “formal” values for the heights of the levelling benchmarks clearly contain errors which would be propagated to any surveying observations if used. This is important too when a validation of a gravimetric geoid model is performed with such faulty orthometric height observations, since the conclusions drawn would be misleading. In any case, from the results presented in Table 58.2, the superior performance of the third order polynomial model is once again evident, since the std drops by ~5 cm compared
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Fig. 58.2 Differences between GPS/Levelling and EGM2008 geoid heights at available BMs Table 58.2 Statistics of the differences NGPS/Lev-NEGM08 before and after the fit using the adjusted orthometric heights for the two BMs. Unit: (m) Before 4-param 5-param First pol. Second pol. Third pol.
Max 1.118 0.337 0.319 0.323 0.291 0.244
Min 0.238 0.338 0.372 0.389 0.387 0.324
Mean 0.741 0.000 0.000 0.000 0.000 0.000
rms 0.786 0.137 0.135 0.160 0.135 0.123
std 0.176 0.137 0.132 0.160 0.135 0.123
to the differences before the fit and the range by ~79 cm. Notice that the incorporation of the adjusted orthometric heights for the two BMs improves the fit as well, since the std and the range after the fit with the third order polynomial model improve by 3.3 and 35 cm respectively (last row in Tables 58.1 and 58.2). The next set of tests performed refers to the investigation of the influence that the data input error would have on the adjusted residuals. To this extent three separate cases have been identified: (a) The first one assumes that all height types have a uniform accuracy of 1 cm, so that the covariance matrices are all equal to the identity matrix I, (b) A-priori standard
deviations (sh ¼ 2 cm, sH ¼ 3 cm and sN ¼ 4 cm) are assigned to the observations assuming that the accuracy of the geometric heights is the highest, with the orthometric and geoid heights following, and (c) The input error for the geometric heights was that from the GPS data processing, the error of the orthometric heights was the formal one provided by the Hellenic Military Geographic Service and the geoid height error was set again to a standard deviation sN ¼ 4 cm. All these cases will be identified herein as caseA, caseB and caseC respectively. It should be noted that the mean error for caseC was 0.3 and 0.5 cm for the ellipsoidal and orthometric heights, respectively. In all cases the adjustment took place by employing the third order polynomial model, which provided the best results in the previous test, and geoid heights from EGM2008 to represent the gravimetric geoid model. Table 58.3 presents the results achieved after the fit for the three scenarios examined. It is clear that no improvement is achieved when employing the most rigorous caseC for the data covariance matrices, even compared to caseA where the input errors are set equal to 1 cm for all height types. The reduction of the std of the
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Table 58.3 Statistics of the differences NGPS/Lev-NEGM08 before and after the fit using different input error models. Unit: (m) Before caseA caseB caseC
Max 1.118 0.244 0.244 0.243
Min 0.238 0.324 0.324 0.324
Mean 0.741 0.000 0.000 0.000
rms 0.786 0.123 0.123 0.122
std 0.176 0.123 0.123 0.122
differences by 1 mm for caseC is clearly insignificant and signals that, as far as the fit is concerned, the input errors for the observations seem to play little role. This is of course not the case when testing and scaling the supplied covariance matrices, calibrating geoid error models and assessing/evaluating the accuracy of the orthometric heights. In such cases the input errors and variance component estimation can prove a useful and significant tool (Fotopoulos 2003). The final set of tests performed, incorporated the other available GGMs as well as a local gravimetric geoid model developed for the Greek territory. The objectives were twofold. First to investigate and assess the improvement that EGM2008 brings compared to older GGMs and secondly to determine its performance w.r.t. a local geoid model. A brief overview of the latter, with emphasis on the treatment of the topographic effects is given in Tziavos et al. (2010). Once again a third order polynomial model has been employing to describe the differences between ellipsoidal, orthometric and geoid heights while caseC, the most rigorous of the three, has been used to describe their errors. Table 58.4 presents the results acquired for all geoid models, both before and after the fit, with the ones for EGM2008 reported in Tables 58.2 and 58.3. From the results presented in Table 58.4 it is clear that EGM2008 outperforms all other GGMs, since the std of 12.2 cm it provides after the fit is ~13 cm better than that of the others. Of course, this is expected since GGM03s is a satellite only model, while the others are complete to degree and order 360, rather than 1,834 where EGM2008 was truncated. This is a clear indication of the significant improvement that this recently released GGM brings to all geosciences and especially geodetic and oceanographic research. One further note for the superior performance of EGM2008 is the std of the differences before the fit (17.6 cm) which is better that the std of the fitted residuals for the other models. Comparing the performance of the local gravimetric geoid model,
Table 58.4 Statistics of the differences between GPS/levelling and geoid heights from the local model and the GGMs before and after the fit. Unit: (m) Max Min grav local Differences with N Before 0.220 0.714 After 0.198 0.237 Differences with NGGM03c Before 1.256 0.423 After 0.772 0.336 Differences with NEIGEN5c Before 1.209 0.603 After 0.771 0.317 Differences with NGGM03s Before 2.413 1.953 After 0.830 0.433 Differences with NEGM96 Before 0.860 0.784 After 0.758 0.293
Mean
rms
std
0.452 0.000
0.471 0.104
0.133 0.104
0.159 0.000
0.408 0.255
0.376 0.255
0.040 0.000
0.378 0.252
0.376 0.252
0.212 0.000
1.709 0.268
1.438 0.268
0.124 0.000
0.383 0.250
0.362 0.250
Table 58.5 Statistics of the differences between GPS/levelling and geoid heights from the local model and EGM2008 for neighbouring map-leaflets. Unit: (m) Map id NEGM08 Ngrav local Map id NEGM08 Ngrav local Map id NEGM08 Ngrav local
Mean 132 0.748 0.247 85 0.848 0.478 26 0.540 0.092
std 0.118 0.109 0.060 0.050 0.141 0.112
Mean 12 0.816 0.445 166 0.604 0.318 303 0.298 0.103
std 0.084 0.058 0.084 0.042 0.166 0.129
it can be concluded that it gives better results than EGM2008 by 4 and 2 cm (1s) before and after the fit, respectively. Moreover, the range of the differences for the local gravimetric geoid model is smaller by ~42 and ~12 cm before and after the fit. This is a good indication that even in the presence of high-resolution and high-accuracy GGMs, like EGM2008, local and regional gravimetric geoid models have still to offer and need not to be abandoned. A final note refers to some examples of the biases that exist between neighbouring “map-leaflets” in the adjusted GPS/Levelling and gravimetric geoid heights for the Greek levelling network. Table 58.5 presents the mean and std of the differences between GPS/Levelling and NEGM08/Ngravlocal geoid heights
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for neighbouring “map-leaflets”. Note that in principle trigonometric BMs between neighbouring “map-leaflets” can be used in everyday surveying applications as known stations for traverses, so that any datum shifts between them will introduce unrealistic miss-closure errors. From Table 58.5, where the different “map-leaflets” are distinguished by their id, it can be concluded that significant biases ranging from 5 to 25 cm exist between levelling BMs residing in neighbouring “map-leaflets”, which is a clear indication that, un-modelled, datum shifts exist in the Greek datum. The differences in the std range between 0.8 and 3 cm which can be regarded as normal as far as random errors in the vertical datum are concerned, especially for long-levelling traverses (the shortest distance between the levelling BMs for neighbouring “map-leaflets” is ~30 km in the present study). In any case, a safe conclusion can be drawn at this point, i.e., that since a common adjustment of the entire Greek vertical network has not been carried out, traverses employing BMs from more than one “map-leaflet” should be dealt with care.
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matrices is needed. EGM2008 provided the best fit when compared to the other recent GGMs signalling the significant improvement that this model brought to modern-day geodetic research. It is worth mentioning that even the std of the differences, before the fit, that EGM2008 provided was smaller than that of the other GGMs after the fit of the parametric model. Regional and local gravimetric geoid model development has still to offer, since it provided better results by ~3 cm (1s) compared to EGM2008, which provides evidence that even ultra-high degree GGMs, at least until today, cannot depict the local peculiarities of the Earth’s gravity field. Finally, some problems arising from the fact that the Greek vertical network has not been commonly adjusted for the entire country have been demonstrated. This lack of a common adjustment introduces significant biases in the orthometric heights of the order of 5–25 cm when levelling BMs from neighbouring “map-leaflets” are used in a traverse. Therefore, such operations should be exercised with caution and control.
Conclusions
A detailed scheme for the combined adjustment of ellipsoidal, orthometric and geoid heights over a network of 43 benchmarks in Greece has been presented. Various parametric models were tested, in order to model the residual differences, along with different choices for the data input errors. From the results acquired, it was concluded that orthometric height validation and blunder detection is feasible when high-accuracy GGMs and local geoid models are available. When blunders are detected and adjusted orthometric heights are determined then improved residuals by ~6 cm are achieved. These can then be used to improve local gravimetric geoid fit to GPS/Levelling heights. In all cases the selection of a third polynomial as a parametric model provided the best results for the fitted residuals, since it reduced the std and the range, compared to the other models, by ~5 and ~79 cm respectively. From the analysis of the influence of the errors of the observations, it was concluded that practically no improvement in the fitted residuals was achieved when either the identity or proper covariance matrices were employed. This conclusion holds for the specific set of tests and not when, e.g., the calibration of the covariance
References Featherstone W (1998) Do we need a gravimetric geoid or a model of the Australian height datum to transform GPS Heights in Australia? Austr Survey 43(4):273–280 Featherstone WE, Kirby JF, Kearsley AHW, Gilliand JR, Johnston J, Steed R, Forsberg R, Sideris MG (2001) The AUSGeoid98 geoid model of Australia: data treatment, computations and comparisons with GPS/levelling data. J Geod 75(5–6):313–330 Fotopoulos G (2003) An analysis on the optimal combination of geoid, orthometric and ellipsoidal height data. UCGE Rep Nr 20185, Calgary AB, Canada Fotopoulos G, Kotsakis C, Sideris MG (2001) How accurately can we determine orthometric height differences from GPS and geoid data? J Surv Eng 129(1):1–10 Kotsakis C, Sideris MG (1999) On the adjustment of combined GPS/levelling/geoid networks. J Geod 73(8):412–421 Lemoine GG, KenyonSX, Factim JK, Trimmer RG, Pavlis NK, Chinn DS, Cox CM, Klosko SM, Luthcke SB, Torrence MH, Wang YM, Williamson RG, Pavlis EC, Rapp H, Olson TR (1998) The development of the joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) Geopotential Model EGM 96. Pub. Goddard Space Flight Center Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2008) An Earth Gravitational Model to Degree 2160: EGM2008, presented at the 2008 General Assembly of the European Geosciences Union, Vienna, Austria, April 13–18, 2008
488 Reigber Ch, Jochmann H, W€ unsch J, Petrovic S, Schwintzer P, Barthelmes F, Neumayer KH, K€ onig R, F€ orste Ch, Balmino G, Biancale R, Lemoine JM, Loyer S, Perosanz F (2005) Earth gravity field and seasonal variability from CHAMP. In: Reigber Ch, Luhr H, Schwintzer P, Wickert J (eds) Earth observation with CHAMP: results from three years in orbit. Springer, Berlin, pp 25–30 Sideris MG, Mainville A, Forsberg R (1992) Geoid testing using GPS and levelling (or GPS testing using levelling and the geoid?). Aust J Geod, Photogramm Surv 57:62–77 Tapley B, Ries J, Bettadpur S, Chambers D, Cheng M, Condi F, Poole S (2007) The GGM03 mean earth gravity model from
I.N. Tziavos et al. GRACE. Eos Trans. AGU 88(52), Fall Meet Suppl, Abstract G42A-03 Tziavos IN, Vergos GS, Grigoriadis VN (2010) Investigation of topographic reductions and aliasing effects to gravity and the geoid over Greece based on various digital terrain models. Surv Geophs 31(1):23–67. doi:10.1007/s10712009-9085-z Vergos GS, Sideris MG (2002) Evaluation of geoid models and validation of geoid and GPS/levelling undulations in Canada. IGeS Bull 12:3–17
Session 3 Geodesy and Geodynamics: Global and Regional Scales Convenors: M. Bevis, S. Bonvalot
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Regional Geophysical Excitation Functions of Polar Motion over Land Areas
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Abstract
Here we estimate hydrological polar motion excitation functions over land areas regionally from hydrological models and from Gravity and Climate Recovery Experiment (GRACE) gravity fields. The models include equivalent water heights fields determined from groundwater, soil moisture and snow estimates on continents. In this regard, we consider land data from the Climate Prediction Center (CPC) hydrological model and from the surface modelling system Global Land Data Assimilation System (GLDAS), both of which produce monthly estimates. Also we used satellite -gravimetry data, in the form of the GRACE RL04 equivalent water heights from Center for Space Research. The mass effects of the ocean and atmosphere and postglacial rebound are being removed, so in this way hydrological excitation of polar motion can be estimated from the gravimetric data. The monthly step of the data restricts our analysis to seasonal signals only. Large hydrological variability in equivalent water thickness occurs in the lower latitude Southeast Asia, South Asia, and the South American Amazon regions, and remain important in polar excitation even after multiplication by polar motion transfer functions, with the exception of the band very close to the equator. Differences among models and GRACE related values are still considerable, and need to be reconciled to form the best estimates of hydrological variability. Additionally, variations from the atmosphere are determined over land areas from NCEP/NCAR reanalyses; they are noted to be strongly dependent on variability over the high topography regions of Eurasia and North America.
59.1
J. Nastula (*) Space Research Centre of the PAS, Bartycka 18a, Warsaw, Poland e-mail:
[email protected] D.A. Salstein Atmospheric and Environmental Research Inc, Lexington, MA, USA
Introduction
Motion of the Earth’s pole may be excited by variability in the mass distribution in the geophysical fluids. The importance of atmospheric and oceanic angular momentum signals for polar motion excitation at monthly and longer periods is well established (Brzezin´ski et al. 2005; Gross et al. 2003; Gross 2007). Similarly, the variability that originates as a result of hydrological mass variations over lands may be
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_59, # Springer-Verlag Berlin Heidelberg 2012
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important as well as polar motion excitations. The role of the continental hydrologic signals, from land water, snow, and ice, on polar motion excitation (hydrological angular momentum-HAM), however, is less well known, especially at periods shorter than annual. Estimates of HAM have been made from several models of global hydrology based upon the observed distribution of surface water, snow, and soil moisture (Chen and Wilson 2005, 2008). Recently the GRACE mission monitoring Earth’s time variable gravity field has allowed us to determine polar motion excitation mass functions and to compare them with mass terms of the geodetic excitation functions (Chen and Wilson 2005, 2008; Nastula et al. 2007; Seoane et al. 2009). The gravimetric observations are a new approach to evaluate the effect of land hydrology on polar motion. The polar motion excitation is the measure of the total sum of contributions from mass variations over the globe. But who-even with a global quantity already on hand-would not desire to learn the processes that are at the roots of it? Determinations of which regions in the geophysical fluids in which time, in which spectral band, are the main sources of polar motion excitation due to mass-related geophysical fluids will lead to elucidating underlying mechanism. Here, we investigate regional excitation signals from variations in the mass of the two geophysical fluids; atmosphere and land hydrology, according to both region and spectral band. Variations of polar motion excitation functions from the atmosphere have been determined on a number of time scales, from subseasonal to interannual, and their regional origins have been noted to be strongly dependent on variability over the high topography areas of Eurasia and North America (Nastula and Salstein 1999; Nastula and Kolaczek 2002; Nastula et al. 2009). Our main objective is to compare patterns of polar motion excitation functions from land area computed from different sets of observations: atmospheric angular momentum, GRACE gravity field solutions, hydrological models
59.2
Data
The equatorial components of polar motion excitation functions available for transfer of the Geophysical Fluids Angular Momentum of the solid Earth have
been formalized as w1 and w2, components towards longitudes 0 and 90 E, respectively, of the excitation functions of polar motion. In this study we compared regional contributions to polar motion excitations determined separately from each of three kinds of geophysical data: atmospheric pressure, equivalent water heights (EWH) estimated from hydrological models and equivalent water thickness estimated from the Gravity Recovery and Climate satellite experiment. Hydrological polar motion excitation functions of polar motion are estimated here using the formula given by Chen and Wilson (2005) from the two latitude–longitude grids of water storage: both available from the IERS Special Bureau for Hydrology: ftp://ftp.csr.utexas.edu/pub/ggfc/water/: • Land Data Assimilation System (LDAS) monthly water storage determined on 1 1 latitude– longitude grid with monthly frequency in the period Jan 1948–Dec 2007. The LDAS is one of the land surface models developed at NOAA Climate Prediction Center (CPC) and is forced by observed precipitation, derived from CPC daily and hourly precipitation analyses, downward solar and longwave radiation, surface pressure, humidity, 2-m temperature and horizontal wind speed from NCEP reanalysis (Fan et al. 2003). • GLDAS: NASA Global Land Data Assimilation System determined on a 1 1 latitude–longitude grid with monthly frequency in the period Jan 1979–Jul 2009. The GLDAS generates optimal fields of land surface states and fluxes by integrating satellite- and ground-based observational data products, using advanced land surface modelling and data assimilation techniques (Rodell et al. 2004). The water storage is the sum of soil moisture, snow water equivalent and canopy surface water not counting changes in groundwater below the depth defined by the model. Usually, the polar motion excitation functions based on gravimetric observations is derived from the DC21 and DS21 coefficients of the gravity field solution (see, for example, Chen et al. 2004; Nastula et al. 2007). In this paper, we estimate GRACE-based excitation in the same way like the hydrological excitation namely from the formula given by Chen and Wilson (2005) and using equivalent water height maps processed by D.P. Chambers (2006) and available at (http://grace.jpl.nasa.gov).
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The 1 1 grids implemented with the destriping technique, calibrate results against sea surface height corrected for climatological steric effects. The mass of the atmosphere is removed during processing using ECMWF fields, so these grids do not reflect atmospheric variability over land, except for errors in ECMWF. Additionally the EWH data are corrected with the Post-Glacial Rebound (ICE-5 G deglaciation model). In this study we use the EWH grids determined from Center for Space Research (CSR) RL04, monthly solution in the period Oct 2002–Jul 2009. Regional values of atmospheric excitation functions of polar motion were computed on the basis of the NCEP–NCAR reanalysis (Salstein et al. 1993; Kalnay et al. 1996). The basic data include 6-hourly surface pressure data pressure data given in 2.5 2.5 latitude–longitude grid, provided by the NOAA Operational Model Archive Distribution System (http:// nomad3.ncep.noaa.gov). The pressure data were averaged to a 30-day solution for consistency with monthly
hydrological and gravimetric data. Next regional atmospheric excitation functions were computed from the surface pressure at grid point from the classical formula by Barnes et al. (1983). Our comparisons are restricted to the common period Oct 2002–Dec 2008.
Fig. 59.1 Maps of amplitudes of the annual oscillation of complex-valued components of polar motion excitation functions; (a) atmospheric pressure polar motion excitation function, in 2.5 2.5 grids; (b) gravimetric polar motion
excitation function, from GRACE CSR RL04 solution in 1 1 grids; from hydrological polar motion excitation function in 1 1 grids from two models (c) CPC, (d) GLDAS (units – mas per grid)
59.3
Analyses and Results
In the paper seasonal oscillations are estimated by least square fitting the model, comprised of the 1st order polynomial and a sum of sinusoids with periods 1, 1/2, 1/3 years to the w1 and w2 components at every grid point. Figure 59.1 compares the variability of annual oscillations in terms of maps of square root of sum of squares of amplitudes computed for w1 and w2 separately. It is necessary to notice that ocean areas are masked in the case of atmospheric data and GRACE
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gravity fields-based maps. The structure of this annual variability of AAM relates largely to the high altitude topography of Asia and North America. The Himalayan Mountains, for example, form a small minimum surrounded by larger values of covariance in a ringlike formation. The structure of this variability, here seen in finer spatial resolution, is reminiscent of the first mode of the EOF analysis shown in our earlier study with atmospheric excitation of polar motion in coarser sectors (Nastula et al. 2003, 2009). Atmospheric variations of polar motion excitation are generally an order of magnitude larger than hydrological variations. In the gravimetric maps and hydrological maps we note that the prominent annual signals are due to the monsoonal climates (Fan and van den Dool 2004) situated at latitudes lower than 30 . The regions of maxima are located in the Amazon, Central Africa, South Africa, North Australia, and India. We observe that the CPC model provides stronger amplitudes
predictions than GLDAS. Additionally in the patterns computed from the CPC models some signals over western Eurasia and northwest North America are also seen. Figure 59.2 shows patterns of semi-annual oscillations of polar motion excitation in terms of maps of square root of sum of squares of amplitudes computed for w1 and w2. from atmospheric pressure, GRACEbased gravity fields, and the hydrological models. The semi-annual oscillation of atmospheric excitation has an order lower amplitude than annual though uncertainties are relatively large (Fig. 59.2). These patterns of AAM are dominated by the formation reaching from Eastern Europe to the high topography region of central Eurasia. Some maxima are also situated in Spain, North Africa, and North America. Similar structures of this semi-annual variability were shown in our earlier study of polar motion based on coarser sectors (Nastula et al. 2003, 2009). Patterns of semi-annual oscillation computed from the GRACE data and from hydrological models are
Fig. 59.2 Maps of amplitudes of semiannual oscillation of complex-valued components of polar motion excitation functions; (a) atmospheric pressure polar motion excitation function, in 2.5 2.5 grids; (b) gravimetric polar motion
excitation function, from GRACE CSR RL04 solution in 1 1 grids; hydrological polar motion excitation function in 1 1 grids from two models (c) CPC, (d) GLDAS (units – mas per grid)
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markedly different. In the hydrological maps based on GLDAS-data, regions of maxima are situated in Northern America and Eurasia while these regions seems not to contribute to the variability in CPC hydrological maps. The distribution of semi-annual oscillation signals based on the GRACE data is comparable to both the CPC-based and GLDASbased patterns. We observe common maxima located in Amazon, Central Africa, South Africa, North Australia, and India. Generally semi-annual oscillations of gravimetric and hydrological polar motion excitation functions have amplitudes smaller than those of atmospheric excitation function but on the same order. Figure 59.3 shows patterns of residuals remaining after removing from the polar motion excitation functions the models’ seasonal signals, comprised of the 1st order polynomial and a sum of sinusoids with periods 1, 1/2, 1/3 years from every gridpoint.
Residuals of atmospheric excitation have values larger than values of semi-annual oscillations and they have a longitudinal formation mostly across the Eurasian continent. Residuals remaining after removing seasonal signals from gravimetric and hydrological excitation functions generally repeat patterns in the seasonal variations but have small amplitudes. Finally, to determine a quantitative estimation of interactions between regional and global signals in the excitation function, correlation coefficients between them were computed (Fig. 59.4). Note that, for 50 independent points, values of the correlation coefficient equal to 0.268 are statistically significant at 99% confidence. If we assume that due to removing an seasonal model from the data only 25 points are independent the critical values of the correlation coefficient is equal to 0.278 at 99% confidence level. It is seen in Fig. 59.4 that correlation coefficients between regional and global atmospheric excitation functions respectively reach significant high values of about 0.7,
Fig. 59.3 Maps of amplitudes of standard deviations of complex-valued components of polar motion excitation functions; after removing seasonal model (a) atmospheric pressure polar motion excitation function, in 2.5 2.5 grids; gravimetric
polar motion excitation function, from GRACE CSR RL04 solution in 1 1 grids; hydrological polar motion excitation function in 1 1 grids from two models (c) CPC, (d) GLDAS (units – mas per grid)
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Fig. 59.4 Maps of complex correlation coefficient magnitude of between complex-valued regional and global components of polar motion excitation functions; after removing seasonal model (a) atmospheric pressure polar motion excitation
0.8 across the longitudinal formation across the Eurasian continent. In fact it is not surprising and confirms known fact of existence of subseasonal oscillation giving an impact to the global variation in the area (Nastula et al. 2009). In the case of gravimetric and hydrological data maxima of correlation coefficient are scattered over continents. Only the map of correlation between hydrological GLDAS data reveals formation of high correlation areas in Eurasia. Conclusions
Excitations functions of polar motion from GRACE gravity fields and from the hydrological models are dominated by seasonal, mostly annual, variability. Prominent maxima are situated over the equatorial monsoonal regions of the Amazon, India, central and southern Africa, and northern Australia. In the patterns computed from the CPC model some signals over western Eurasia are also seen.
J. Nastula and D.A. Salstein
function, in 2.5 2.5 grids; (b) gravimetric polar motion excitation function, from GRACE CSR RL04 solution in 1 1 grids; hydrological polar motion excitation function in 1 1 grids from two models (c) CPC, (d) GLDAS (per grid)
Semi-annual oscillations of polar motion from GRACE gravity fields and the hydrological models have lower amplitudes than annual, though uncertainties are relatively large. Residuals remaining after removing seasonal signals from gravimetric and hydrological excitation functions generally repeat patterns in the seasonal variations but have small amplitudes. Amplitudes of seasonal oscillations and standard deviations of seasonal excitation computed from GRACE gravity fields are of the order of those computed from the CPC and GLDAS. Regarding variations of atmospheric excitations of polar motion, they are typically an order of magnitude higher than hydrospheric. The annual scale regional patterns are dominated by maxima in central Asia, with the Himalayan Mountains forming a relative minimum surrounded by larger values in a ring-like formation.
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Regional Geophysical Excitation Functions of Polar Motion over Land Areas
Semi-annual oscillation of polar motion estimates from the AAM are dominated by high topography region of Eastern Eurasia. Residuals remaining after removing seasonal signals are dominated by variations over Western Eurasia. The results revealed that atmospheric excitation over land regions in the northern middle latitudes, especially over Asia, is exceptionally strong, a point generally noted in previous studies, though without identifying detailed features. Acknowledgements The research reported here was supported by the Polish Ministry of Scientific Research and Information Technology, through project N526140735, and by U.S. National Science Foundation project ATM-0913780. We are grateful to Barbara Kolaczek for her suggestions.
References Barnes RTH, Hide R, White AA, Wilson CA (1983) Atmospheric angular momentum fluctuations, length-of-day changes and polar motion. Proc R Soc Lond A387:31–73 Brzezinski A, Nastula J, Kołaczek B, Ponte RM (2005) Oceanic excitation of polar motion from intraseasonal to decadal periods. In: Sanso` F (ed) Proceedings a window on the future of geodesy. IAG Symposia, vol 128. Springer, Berlin, pp 591–596 Chambers DP (2006) Evaluation of new GRACE time-variable gravity data over the ocean. Geophys Res Lett 33(17):LI7603 Chen JL, Wilson CR, Tapley BD, Ries JC (2004) Low degree gravitational changes from GRACE: Validation and interpretation. Geophys Res Lett 31:L22607, doi:10.1029/ 2004GL021670 Chen JL, Wilson CR (2005) Hydrological excitation of polar motion, 1993–2002. Geophys J Int 160:833–839. doi:10.1111/j.1365-246X.2005.02522 Chen JL, Wilson CR (2008) Low degree gravity changes from GRACE, Earth rotation, geophysical models, and satellite laser ranging. J Geophys Res 113:B06402. doi:1029/ 2007JB005397 Fan Y, van den Dool H, Mitchell K, Lohmann D (2003) NWSCPC’s monitoring and prediction of US soil and moisture and associated land surface variables: land data reanalysis. In: Proceedings of the Climate Diagnostics Workshop,
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National Centers for Climate Prediction, Camp Springs, MD, 21–25 Oct 2002 (CD-ROM) Fan Y, van den Dool H (2004) Climate Prediction Center global monthly soil moisture data set at 0.5 resolution for 1948 to present. J Geophys Res 109:D10102, doi:10.1029/ 2003JD004345 Gross RS (2007) Earth rotation: long-period variations. In: Herring GTA (ed) Treatise of geophysics, vol 3. Elsevier, Oxford Gross RS, Fukumori I, Menemenlis D (2003) Atmospheric and oceanic excitation of the Earth’s wobble during 1980–2000. J Geophys Res 108:B82370. doi:10.1029/ 2002JB002143 Kalnay E, Kanamitsu M, Kistler R, Collins W, Deaven D, Gandin L, Iredell M, Saha S, White G, Woollen J, Zhu Y, Chelliah M, Ebisuzaki W, Higgins W, Janowiak J, Mo KC, Ropelewski C, Wang J, Leetmaa A, Reynolds R, Jenne R, Joseph D (1996) The NCEP/NCAR 40-year reanalysis project. Bull Am Meteorol Soc 77:437–471 Nastula J, Kolaczek B (2002) Seasonal oscillations in regional and global atmospheric excitation of polar motion. Adv Space Res 30(2):381–386, 2002 COSPAR. Published by Elsevier Science, Ltd Nastula J, Salstein DA (1999) Regional atmospheric momentum contributions to polar motion. J Geophys Res 104:7347–7358 Nastula J, Salstein DA, Ponte RM (2003) Empirical patterns of variability in atmospheric and oceanic excitation of polar motion. J Geod 36:383–396, doi:10.1016/S0264-3707(03) 00057-7 Nastula J, Ponte RM, Salstein DA (2007) Comparison of polar motion excitation series derived from GRACE and from analyses of geophysical fluids. Geophys Res Lett 134:L11306. doi:10.1029/2006GL028983. 375 Nastula J, Salstein DA, Kolaczek B (2009) Patterns of atmospheric excitation functions of polar motion from high-resolution regional sectors. J Geophys Res 114:B04407. doi:10.1029/2008JB005605 Rodell M, Houser PR, Jambor U, Gottschalck J, Mitchell K, Meng C-J, Arsenault K, Cosgrove B, Radakovich J, Bosilovich M, Entin JK, Walker JP, Lohmann D, Toll D (2004) The global land data assimilation system. Bull Am Meteorol Soc 85:381–394 Salstein DA, Kann DM, Miller AJ, Rosen RD (1993) The subbureau for atmospheric angular momentum of the International Earth Rotation Service: a meteorological data center with geodetic applications. Bull Am Meteorol Soc 74:67–80 Seoane L, Nastula J, Bizouard C, Gambis D (2009) The use of gravimetric data from GRACE mission in the understanding of polar motion variations. Geophys J Int: L11306. doi:10.1111/j.1365-246X.2009.04181
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Geophysical Excitation of the Chandler Wobble Revisited
60
Aleksander Brzezin´ski, Henryk Dobslaw, Robert Dill, and Maik Thomas
Abstract
The 14-month Chandler wobble is a free motion of the pole excited by geophysical processes. Several recent studies demonstrated that the combination of atmospheric and oceanic excitations contains enough power at the Chandler frequency and is significantly coherent with the observed free wobble. This paper is an extension of earlier studies by Brzezin´ski and Nastula (Adv Space Res 30:195–200, 2002), Brzezin´ski et al. (Oceanic excitation of the Chandler ´ da´m J, wobble using a 50-year time series of ocean angular momentum. In: A Schwarz K-P (eds) Vistas for geodesy in the new millennium. IAG Symposia, vol 125. Springer, Berlin, pp 434–439, 2002) using the same method of analysis but other available estimates of atmospheric and oceanic excitation of polar motion. We also try to assess the role of land hydrology in the excitation balance by taking into account the hydrological angular momentum estimates. Our results generally confirm earlier conclusions concerning the atmospheric and oceanic excitation. Adding the hydrological excitation is found to increase slightly the Chandler wobble excitation power, while the improvement of coherence depends on the geophysical models under consideration.
60.1
Introduction
The Chandler wobble, which is the largest component of polar motion, is a free motion of the pole. It has been observed as a quasi-circular motion of the pole in
A. Brzezin´ski (*) Faculty of Geodesy and Cartography, Warsaw University of Technology, Warsaw, Poland Space Research Centre, Polish Academy of Sciences, Warsaw, Poland e-mail:
[email protected] H. Dobslaw R. Dill M. Thomas Section 1.5: Earth System Modelling, Deutsches GeoForschungsZentrum GFZ, Potsdam, Germany
prograde (counterclockwise) direction. During 110 years of observation, the mean period of the wobble was 433 days and the mean amplitude about 170 milliarcseconds (mas). The amplitude is variable but it does not exhibit any permanent decaying trend. This proves that there must exist a process, or a combination of processes exciting this free wobble and maintaining it against energy dissipation. There were many attempts in the past to explain the excitation mechanism of the Chandler wobble; see Brzezin´ski (2005) and the references therein for review. Certainly, the best observed mechanism was the redistribution of air masses and changes in wind patterns causing variations of atmospheric angular momentum (AAM). However, most of the excitation
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_60, # Springer-Verlag Berlin Heidelberg 2012
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studies using the available meteorological observations and models indicated that AAM variations provide less than half of the power needed to explain the observed free wobble. One possible candidate for explaining the remaining large gap in the Chandler wobble excitation balance was the non-tidal variation of the ocean angular momentum (OAM). Its significant part expressing the ocean response to the overlying air pressure variations has long being modeled by adding the so-called inverted barometer (IB) correction to the pressure term of AAM. Estimation of the remaining dynamic component of OAM was much more difficult because it required three-dimensional modeling of the global ocean dynamics. Only recently there have been successful attempts to develop the high-quality ocean general circulation models and estimate the corresponding OAM data. The excitation studies, e.g., by Gross (2000), Brzezin´ski and Nastula (2002), using the OAM and AAM data from the improved global models could demonstrate that during 1985–1996 the Chandler wobble was mostly driven by the irregular angular momentum transfer from the coupled atmosphereoceans system to the solid Earth. The most important excitation mechanisms were the ocean-bottom pressure and atmospheric pressure fluctuations. The next attempt of Brzezin´ski et al. (2002) using a 50-year time series of OAM yielded significantly worse results than those of Brzezin´ski and Nastula (2002). The purpose of this research is to repeat the estimation by the method developed by Brzezin´ski and Nastula (2002) using other available time series of the geophysical excitation of polar motion. An important question which we want to address here concerns the role of land hydrology in the Chandler wobble excitation balance. Therefore, we include in the analysis estimates of the hydrological angular momentum (HAM). Of particular interest is a new consistent set of 20-year excitation series AAM, OAM and HAM based on the ERA Interim reanalysis fields and corresponding simulations from the hydrological model LSDM and the ocean model OMCT.
60.2
The Model of Geophysical Excitation
Geophysical excitation of the Chandler wobble is governed by the following first-order differential equation (e.g., Brzezin´ski 1992)
p_ i sc p ¼ i sc w;
(60.1)
pffiffiffiffiffiffiffi in which i ¼ 1 denotes the imaginary unit, p ¼ xp iyp describes the change of the terrestrial direction of the Celestial Intermediate Pole (CIP), that is polar motion; and w ¼ w1 þ iw2 is the forcing (excitation) function of the geophysical fluid (in the present case w equals AAM, OAM or HAM). The complex angular frequency of the Chandler resonance is sc ¼ 2pFc ð1 þ i=2Qc Þ. We adopt here the resonant frequency Fc ¼ 0.843 cycles per year (cpy) corresponding to the period Tc ¼ 433 days and the quality factor Qc ¼ 179, following Wilson and Vicente (1990). The underlying terrestrial reference system is geocentric with its z-axis pointing towards the North pole, the x-axis towards the Greenwich meridian and the y-axis towards 90 East longitude. We assume that the excitation function w can be adequately modeled as a complex-valued stochastic process with a power spectral density (PSD) function Sw ðf Þ changing smoothly in the vicinity of the resonant frequency f ¼ Fc . For our empirical data w we remove first all known components which do not contribute to the Chandler wobble (seasonal sinusoids and polynomial trend) and then express the residual series as a realization of an autoregressive (AR) process. Parameters of the AR model are estimated by the maximum entropy method (MEM) algorithm developed by Brzezin´ski (1995). The optimum AR order, based on the final prediction error criterion, was found relatively low, between 8 and 15. As we will see in the plots, the corresponding PSD is smooth indeed in the vicinity of Fc. The MEM algorithm used here yields an estimate S^w ðf Þ of the excitation power spectrum which is realistic in a sense that its integral over the entire Nyquist frequency interval 1=2Dt
60
Geophysical Excitation of the Chandler Wobble Revisited
Given the time series of polar motion observations p(t), we compute the corresponding “geodetic” excitation based on (60.1). Next, we perform a consistent initial reduction of both the geodetic and geophysical excitations. By an unweighed least squares fit, we estimate parameters of the model comprising the sum of complex sinusoids with periods +1, 1/2, 1/3 years, where the sign +/ indicates the prograde/retrograde motion, and the 4-th order polynomial accounting for low-frequency variation. This polynomial-harmonic model is then removed and the residual excitation series are simultaneously smoothed and interpolated at uniform 10-days intervals by the Gaussian low-pass filter with full width at a half of maximum equal to 20 days. A first step of comparison is in the time domain, by visual inspection of the plots and standard overall variance-correlation analysis. Then comparison is done in the frequency domain focusing attention on the resonant frequency f ¼ Fc . We follow the scheme described by Brzezin´ski (1995). We estimate the MEM power spectra S^wo ðf Þ; S^wg ðf Þ and crosspower spectrum S^wo wg ðf Þ of the observed (geodetic) and geophysical excitations wo, wg. These spectral estimates, which are analytical functions of frequency f, are then integrated numerically over the frequency interval of length Df centered at Fc giving I^wo ðDf Þ; I^wg ðDf Þ and I^wo wg ðDf Þ, respectively. The last quantities can be interpreted as the variances and covariance of the input excitation series after the narrow band-pass filtering over ðFc Df =2; Fc þ Df =2Þ. The corresponding correlation coefficient I^wo wg ðDf Þ ^ Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi CðDf I^wo ðDf Þ I^wg ðDf Þ
(60.2)
will be called the coherence. The geophysical excitation function wg can be considered responsible for the observed Chandler wobble if the excitation powers are equal at the resonant frequency, S^wo ðFc Þ ¼ S^wg ðFc Þ, ^ Þ has the magnitude equal to 1 and the coherence CðDf and the argument equal to 0 for small Df.
60.3
Data Analysis
The geodetic excitation of polar motion has been computed by the time domain deconvolution procedure applied to the observed Earth Orientation Parameters
501
(EOP). We used the EOP series IERS C04 over 1962.0–2009.6 (Bizouard and Gambis 2009) based on a combination of different types of measurements. The series is available from the IERS Earth Orientation Center, http://hpiers.obspm.fr/iers/eop/. The atmospheric excitation of polar motion is expressed by a time series of AAM over 1948.0– 2009.5 estimated by the procedure developed by Salstein and Rosen (1997) on the basis of the output fields of the U.S. NCEP-NCAR reanalysis project (Kalnay et al. 1996). The series is available from the IERS Special Bureau for the Atmosphere, ftp://ftp.aer. com/pub/anon_collaborations/sba/. The nontidal oceanic excitation of polar motion is expressed by two OAM series estimated from the ECCO ocean model forced by atmospheric surface wind stress, heat and freshwater fluxes taken from the output fields of the NCEP-NCAR reanalysis. The series, which are available from the IERS Special Bureau for the Ocean, ftp://euler.jpl.nasa. gov/sbo/, will be denoted here as ECCO1 – model without data assimilation, c20010701 run over 1980.0– 2002.2 (Gross et al. 2003), and ECCO2 – dataassimilating model, kf066b run over 1993.0–2009.0 (Gross 2008). The hydrological excitation of polar motion is expressed by the HAM series NCEP-water over 1948.0–2009.0, computed from NCEP-NCAR reanalysis soil moisture and snow accumulation data, excluding contributions from Antarctica and Greenland (Chen and Wilson 2003). The series is available from the IERS Special Bureau for Hydrology, ftp://ftp.csr. utexas.edu/pub/ggfc/ham/. In addition, we used in the analysis a new set of geophysical excitation series AAM, OAM and HAM. The AAM series is based on the ERA Interim reanalysis from ECMWF (Uppala et al. 2008) with 1 1 regular grids and 37 vertical pressure levels. The HAM estimate is computed from output of the hydrological model LSDM (Dill 2008) with 0.5 0.5 spatial resolution. The model is forced by precipitation, evaporation and 2m-temperatures. The OAM series is computed from the Ocean Model for Circulation and Tides (OMCT; Dobslaw and Thomas 2007). Discretized on a regular 1.875 1.875 grid with 13 vertical layers, the model is forced by wind stress, atmospheric pressure, 2m-temperatures, and freshwater fluxes from both atmosphere and continental hydrosphere.
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60.4
Results and Conclusions
2
PSD in decibels (mas /cpy)
χ1+50
χ2−50 1997
1998
1999 Time (years)
2000
0
12 10
−90 8 6 −1.5
−1
−0.5
0
0.5
Frequency (cpy)
1
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2
CPSD magnit. in db (mas /cpy)
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CW freq.
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16 14
CPSD: C04 vs. AAM+OAM+HAM 180 90 0
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CPSD phase (degrees)
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2002
CPSD: C04 vs. AAM+OAM 180 CPSD phase (degrees)
CPSD magnit. in db (mas /cpy)
CPSD: C04 vs. AAM 18
2001
2
1996
CPSD magnit. in db (mas /cpy)
1995
18 17 16 15 14 13 12 11 10 9
CW freq.
80 60 40 20 0 −20 −40 −60 −80
CW freq.
Observed and modeled excitations of polar motion are compared in Figs. 60.1–60.3 and in Tables 60.1–60.3. Note that the tabulated estimates of power, correlation and coherence are different not only due to the difference in underlying geophysical models but also depend on the time span of analysis. This comparison confirms several conclusions from the earlier excitation studies. First, the atmosphere alone contains only less than a half of the power needed to explain the observed Chandler wobble. In our estimation, atmospheric contribution was found at the level of 25– 40% of the observed excitation power. Second, adding OAM to the AAM brings the modeled excitation close to the observed one and increases significantly the coherence at the resonant frequency. The only significant deficit of the excitation power is found in case of the ECCO1 combination, at the level of 25%. The highest coherence at the Chandler frequency is obtained for the ECCO2 combination AAM+OAM, with its magnitude equal to 0.805 and phase difference of 2 . The most efficient excitations mechanisms are atmospheric pressure and ocean bottom-pressure variations, each with power of up to about 12 mas2/cpy. The contributions from wind and ocean current terms, as well as from the land hydrology are all considerably smaller, at the level of 3–5 mas2/cpy.
CW freq.
Excitation function (mas)
After initial reduction of all the excitation series, as described in Sect. 60.2, we compared the observed excitation C04 to the following combinations of geophysical excitations AAM, AAM+OAM, and AAM +OAM+HAM: AAM – NCEP-NCAR rean., OAM – ECCO1, HAM – NCEP-water, time span: 1980.0–2002.2; AAM – NCEP-NCAR rean., OAM – ECCO2, HAM – NCEP-water, time span: 1993.0–2009.0; AAM – ERA-interim rean., OAM – OMCT, HAM – LSDM, time span: 1989.0–2009.0. Each of these combinations will be identified in the rest of paper by the name of the contributing OAM model. Let us make the following remarks concerning the geophysical excitations: (1) Among the three combinations listed above only the last one is consistent in a sense of mass conservation in the corresponding models. As the excitation of the Chandler wobble is shown to be dominated by processes associated with a mass redistribution, this problem can be very important when considering the free wobble excitation balance. (2) The NCEPwater series consists only of the mass term. The LSDM data contains both the mass and the motion terms, however the last one is extremely small and completely negligible in the context of the present analysis.
−180 1.5
Frequency (cpy)
Fig. 60.1 Comparison of the observed (C04) and modeled (combination ECCO1) excitations of polar motion. Period of analysis: 1980.0–2002.2
Geophysical Excitation of the Chandler Wobble Revisited
503 PSD in decibels (mas /cpy)
2
χ2−50 1997
1998
1999 Time (years)
2000
10 −90 8 6 −1.5
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16 90 14 0
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CPSD magnit. in db (mas /cpy)
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CPSD: C04 vs. AAM+OAM+HAM 180
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CPSD: C04 vs. AAM+OAM 180 CPSD phase (degrees)
CPSD magnit. in db (mas /cpy)
CPSD: C04 vs. AAM 18
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A A+O A+O+H C04
18 17 16 15 14 13 12 11 10 9
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CPSD magnit. in db (mas /cpy)
Excitation function (mas)
60
−180 1.5
χ2−50 1998 1999 Time (years)
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CW freq. −1
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CPSD: C04 vs. AAM+OAM 180
18
CPSD phase (degrees)
CPSD magnit. in db (mas /cpy)
CPSD: C04 vs. AAM
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CW freq.
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CPSD magnit. in db (mas /cpy)
1995
A A+O A+O+H C04
18 17 16 15 14 13 12 11 10 9
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PSD in decibels (mas /cpy)
χ1+50
80 60 40 20 0 −20 −40 −60 −80
CW freq.
Excitation function (mas)
Fig. 60.2 Same as in Fig. 60.1 but for the combination ECCO2. Period of analysis: 1993.0–2009.0
−180 1.5
Fig. 60.3 Same as in Fig. 60.1 but for the combination OMCT. Period of analysis: 1989.0–2009.0
First results concerning the role of the land hydrology in the excitation of the Chandler wobble are not conclusive. In case of the OMCT combination adding the HAM-LSDM series to the atmospheric-oceanic excitation significantly improves the coherence in both the magnitude and the argument, while increasing only a little the total excitation power. When adding the HAM series from the NCEP-water model to the ECCO combinations AAM+OAM there is also a small increase of the excitation power while the coherence becomes worse (ECCO2) or a little bit better in magnitude but worse in argument (ECCO1).
The atmospheric and oceanic excitations from the combination ECCO1, shown in Table 60.1, correspond directly to the results reported in Sect. 5 of (Gross et al. 2003). Gross et al. used the same AAM and OAM data over slightly shorter time interval 1980.0–2001.0. They derived the observed excitation from the polar motion series COMB2000, and applied spectral and cross-spectral analysis based on the classical Fourier approach. Two basic differences between our results and those of Gross et al. are the following: (1) They found the contribution from the OAM mass term to be clearly dominant, by 50% larger than the
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Table 60.1 Comparison of the observed (C04) and modeled (combination ECCO1) excitations of polar motion Geophys. excitation vs. C04 A-pressIB A-wind A O-mass O-motion A+O H A+O+H
Correlation overall
Coherence at Chandler frequency
0.561
1
0.498
–24
0.757
–3
0.573
–4
0.771
–4
0.642
–13
PSD at Chandl. fr. mod./obs. 6.7/31.8 4.9/31.8 7.8/31.8 6.1/31.8 2.5/31.8 21.6/31.8 2.7/31.8 24.9/31.8
Complex correlation and coherence coefficients are shown as magnitude and argument, PSD unit is mas2/cpy. Period of analysis: 1980.0–2002.2 Table 60.2 Same as in Table 60.1 but for the combination ECCO2 Geophys. excitation vs. C04 A-pressIB A-wind A O-mass O-motion A+O H A+O+H
Correlation overall
Coherence at Chandler frequency
0.574–3
–3
0.663
12
0.847
–4
0.805
–2
0.842
–5
0.800
–9
PSD at Chandl. fr. mod./obs. 6.9/42.6 3.1/42.6 11.9/42.6 10.4/42.6 3.6/42.6 44.4/42.6 3.0/42.6 49.1/42.6
Period of analysis: 1993.0–2009.0 Table 60.3 Same as in Table 60.1 but for the combination OMCT Geophys. excitation vs. C04 A-pressIB A-wind A O-mass O-motion A+O H A+O+H
Correlation overall
Coherence at Chandler frequency
0.585
0
0.388
3
0.782
0
0.590
–9
0.787
2
0.784
5
PSD at Chandl. fr. mod./obs. 11.6/51.5 4.8/51.5 20.0/51.5 10.6/51.5 3.4/51.5 48.7/51.5 4.0/51.5 54.9/51.5
Period of analysis: 1989.0–2009.0
contribution from the AAM pressure (IB) term. In our estimation the second contribution is larger by about 10%. (2) Gross et al. estimated the total atmospheric plus oceanic excitation to be larger than the observed one, while we detected a 25% deficit of power. These discrepancies are presumably caused by differences in algorithms used for the spectral estimation. The results derived from the OMCT combination (Table 60.3) should be compared to those from the ECCO2 (Table 60.2) due to the better time coincidence than with ECCO1. There is a good agreement of the excitation power contributed by the ocean and
land hydrology. The AAM components of the OMCT combination have larger power but there is also a similar increase of the observed power coming from the period 1989.0–1993.0. The coherence of AAM and AAM+OAM with the observed excitation is definitely worse in case of the OMCT combination. However, adding the contribution of HAM from the LSDM model to the atmospheric and oceanic excitations increases the coherence to almost the same level as in the ECCO2 combination. As a general conclusion we should say that results of this study support a thesis that the angular momentum
60
Geophysical Excitation of the Chandler Wobble Revisited
exchange between the solid Earth and external fluid layers, the atmosphere, the oceans and the land hydrology, is capable to explain the whole observed Chandler wobble. However, as there are still differences between the results based on different excitation series, further improvements of geophysical models are necessary before considering the excitation balance to be closed. Acknowledgments This research has been supported by the Polish national science foundation under grant No. N526 037 32/3972 as well as by Deutsche Forschungsgemeinschaft within the research unit “Earth rotation and dynamic processes” under grant TH864/7-1.
References Bizouard C, Gambis D (2009) The combined solution C04 for Earth orientation parameters consistent with International Terrestrial Reference Frame 2005. In: Drewes H (ed) Geodetic reference frames, vol 134, IAG Symposia. Springer, Berlin, pp 265–270 Brzezin´ski A (1992) Polar motion excitation by variations of the effective angular momentum function: considerations concerning deconvolution problem. Manuscripta Geodaetica 17:3–20 Brzezin´ski A (1995) On the interpretation of maximum entropy power spectrum and cross-power spectrum in earth rotation investigations. Manuscripta Geodaetica 20:248–264 Brzezin´ski A (2005) Review of the Chandler Wobble and its excitation. In: Plag H-P, Chao B, Grossand R, van Dam T (eds) Proceedings of the workshop “Forcing of polar motion in the Chandler frequency band: a contribution to understanding interannual climate variations”. Cahiers du Centre Europe´en de Ge´odynamique et de Se´ismologie, vol 24. Luxembourg, pp 109–120 Brzezin´ski A, Nastula J (2002) Oceanic excitation of the Chandler wobble. Adv Space Res 30:195–200
505 Brzezin´ski A, Nastula J, Ponte RM (2002) Oceanic excitation of the Chandler wobble using a 50-year time series of ocean ´ da´m J, Schwarz K-P (eds) Vistas angular momentum. In: A for geodesy in the new millennium, vol 125, IAG Symposia. Springer, Berlin, pp 434–439 Chen JL, Wilson CR (2003) Low degree gravitational changes from earth rotation and geophysical models. Geophys Res Lett 30(24):2257. doi:10.1029/2003GL018688 Dill R (2008) Hydrological model LSDM for operational earth rotation and gravity field variations. Scientific Technical Report 08/09, Helmholtz Centre Potsdam, Deutsches GeoForschungsZentrum GFZ, Potsdam, Germany, p 37 Dobslaw H, Thomas M (2007) Simulation and observation of global ocean mass anomaly. J Geophys Res 112:C05040. doi:10.1029/2006JC004035 Gross RS (2000) The excitation of the Chandler wobble. Geophys Res Lett 27:2329–2332 Gross RS (2008) An improved empirical model for the effect of long-period ocean tides on polar motion. J Geodesy. doi:10.1007/s00190-008-0277-y Gross RS, Fukumori I, Menemenlis D (2003) Atmospheric and oceanic excitation of the Earth’s wobble during 1980–2000. J Geophys Res 108(B8):2370. doi:10.1029/2002JB002143 Kalnay E, Kanamitsu M, Kistler R, Collins W, Deaven D, Gandin L, Iredell M, Saha S, White G, Woollen J, Zhu Y, Chelliah M, Ebisuzaki W, Higgins W, Janowiak J, Mo KC, Ropelewski C, Wang J, Leetmaa A, Reynolds R, Roy J, Dennis J (1996) The NMC/NCAR 40-year reanalysis project. Bull Am Meteorol Soc 77:437–471 Salstein D. A. and R. D. Rosen (1997). Global momentum and energy signals from reanalysis systems. In: Proceedings of the 7th conference on climate variations, American Meteorological Society, Boston, MA, pp 344–348 Uppala S, Dee D, Kobayashi S, Berrisford P, Simmons A (2008) Towards a climate data assimilation system: status update of ERA Interim. ECMWF Newsl 115:12–18 Wilson CR, Vicente RO (1990) Maximum likelihood estimate of polar motion parameters. In: McCarthy DD, Carter WE (eds) Variations in Earth rotation, vol 59, Geophysical Monograph Series. AGU, Washington, DC, pp 151–155
.
On the Origin of the Bi-Decadal and the Semi-Secular Oscillations in the Length of the Day
61
S. Duhau and C. de Jager
Abstract
It is presently believed that the liquid core motions that lead to variations in the geomagnetic field and in the Earth’s rotation rate in scales at and above the bi-decadal originate internally. Although, the length of the day (LOD) variations bears some relationship with solar activity and, therefore, either solar activity is exciting some modes of oscillations in the Earth, or these modes have the same external origin in the two bodies. We have introduced a suitable wavelet base function that allows for splitting the modulation in the Hale solar cycle in the Gleissberg cycle and two quasiharmonic oscillations. They have periodicities in the lower Gleissberg band, 40–60 years and in the Hale, 20–30 years, bands and are baptized semi-secular (SS), and bi-decadal (BD) oscillations, respectively. Here we find that these two modes of oscillations are also visible in LOD variations. But while the bi-decadal modes in sunspot number (R) and in LOD are synchronic and follow each other linearly, the SS modes are non-linearly related, with a time lag in LOD, of 94 years. We compare the SS mode of oscillations in LOD with oscillations in the angular velocity of precession of the solar orbit in the inertial reference frame (Daxym). The lengths of Daxym and LOD SS oscillations vary slightly with their amplitudes, but the relationship between them is always exactly 3/4. The 178.7 year cycle in the modulations of Daxym oscillations is also followed by the SS oscillation in LOD. The wavelet components that correspond to the semi-secular oscillation are in the upper band of the modes of torsional oscillations in the liquid core as inferred from geomagnetic variations. The large time lag of LOD semi-secular oscillation with respect to the exciting force indicates that the torsional
S. Duhau (*) Laboratorio de Meca´nica Computacional, Departamento de Fı´sica, Facultad de Ingenierı´a, Av. Paseo Colo´n 850, 2do Piso, 1063 Buenos Aires, Argentina e-mail:
[email protected] C. de Jager Royal Netherlands Institute for Sea Research, P.O. Box 59, 1790 AB Texel, The Netherlands S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_61, # Springer-Verlag Berlin Heidelberg 2012
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oscillations originate deep inside the core. This is consistent with the fact that torsional oscillations have been found to originate near the surface of a cylinder that is tangent to the equator of the inner core. In view to the above findings we discuss a mechanism by which torsional oscillations in the liquid core might be externally excited but internally powered.
61.1
Introduction
It is well established that the origin of the shortperiodic variations in the Earth’s rotation rate is the exchange of angular momentum between the atmosphere and the solid Earth’s core and evidences have been found that some of them may be excited by solar activity (Gribbin and Plagemann 1972; Abarca del Rio et al. 2003 and references therein). Instead, in the longer time scale, it is believed that they originate in torsional oscillations in the Earth’s liquid core that in turn are related to the geomagnetic field variations (Courtillot and Le Moue¨l 1984, and references therein; Zatman and Bloxham 1997; Dickey and Viron 2009; Buffett et al. 2009). An internal origin is assumed for LOD variations for periods above the bi-decadal periods. Namely: liquid core torsional oscillations are coupled, either electromagnetic (Bullard et al. 1950), or topographically (Hide 1969, 2005) to the mantle so leading to variations in the rotation rate of the Earth, as is indicated by the fact that LOD is well correlated with the variations in the east–west derivative of the geomagnetic field (Vestine 1953; Le Mo€ uel et al. 1981; Duhau and Martı´nez 1997; Buffett et al. 2009). A close relationship between LOD and the 11 years running mean of the sunspot number, R11, does exist with LOD lagging behind R11 by 94 year (Duhau and Martı´nez 1995). This large time lag of the LOD signals with respect to the same signal in the sunspot number indicates that the source of LOD variations is seated very deep in the Earth core. So it is consistent with the finding of Zatman and Bloxham (1997) and Buffett et al. (2009) that much of the excitation that causes LOD variations appears to originate near the surface of a cylinder that is tangent to the equator of the inner core. A possible mechanism (Duhau and Martı´nez 1995) by which solar activity might drive torsional oscillations in the liquid core are the Lorenz forces
that would produce the current induced by diffusion inside the liquid core by impulsive changes in geomagnetic storm time variations. However, model computations (Arreghini and Duhau 1997) indicate that these geomagnetic variations cannot penetrate further than a thin layer into the liquid core at the mantle core boundary, which is inconsistent with a source located near the inner core boundary. Therefore, it is unlikely that solar activity could be the cause of the excitation of deep liquid core motions. As an alternative explanation we search here for evidences of a common source for geodynamo and solar dynamo motions excitations in the solar system. Beginning with the investigations of Wolf and Carrington that searched for a relationship between solar orbital parameters and the sunspot number time series, a growing body of evidence indicates that a relationship does exist between the modulation of sunspot cycle maxima and solar orbital motion parameters (Fairbridge and Shirley 1987, and references therein; Landscheidt 1999; Palus et al. 2007; for a review see Mackey 2007). If the geodynamo and solar dynamo motions were driven both by planetary orbital motions the respective dynamo fields might be sharing some modes of oscillations and also with orbital motions. In the present paper we focus in the semi secular oscillations and its first quasi-harmonic as found by De Jager and Duhau (2010).
61.2
The Non Linear Relationship Between LOD and R
To take into account tidal friction, postglacial rebound and atmospheric tides, a linear rate of change of 2.11 10 2 (Merriam 1988) has been subtracted from LOD in Fig. 61.1. The strong semi-secular oscillation in R11 around 1810 that led to the Dalton
61
On the Origin of the Bi-Decadal and the Semi-Secular Oscillations
Minimum is seen in LOD 94 years later (see Fig. 61.1). While the two signals follow each other closely in the interval 1712–1878, they do not before and after. Figure 61.2 compares the wavelet scalogram of LOD and R. It may be seen that a given mode of 0
–6
R11
LOD shifted 94 yr backward
60
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LOD (ms)
180 –3
1878 1800 Time (yr.)
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oscillation in LOD follows a similar oscillation in R but with a different time delay, amplitude and periodicity. This circumstance leads to the apparent intermittent lack of coherence between the two signals, which is seen in Fig. 61.1. Each mode may be built up by superposing all the nearby wavelet components that contains the time variable periodicities of the given mode (De Jager and Duhau 2010). In the following we will analyze the semi-secular oscillation and its first quasi harmonic, that are reconstructed by adding all wavelet component with Fourier periods in the bands 17–34 and 35–70 year, respectively.
2000
Fig. 61.1 LOD yearly values shifted 94 year backward (thick line) and sunspot number 11 year running mean (thin line) for the intervals 1650–1997 and 1550–2000, respectively. LOD data in the intervals 1650–1831, 1832–1961 and 1962–1997 are from Mc Carthy and Babcock (1986), Gross (2001) and Rathcliff and Gross (2009), respectively. Sunspot number prior and after 1705 is from Hoyt and Schatten (1998) and ftp://ngdc. noaa.gov/STP/solar_data, respectively
61.2.1 The Bi-Secular and the Semi-Secular Oscillations In Fig. 61.3 we plot the semi secular and the bidecadal oscillations as split from the sunspot number, R, (thin line) and LOD (thick line), respectively. The
Fig. 61.2 Normalized Morlet Wavelet scalogram of Rz (upper panel) and LOD (lower panel) along years 1650–1943 and 1743–1997, respectively
510
S. Duhau and C. de Jager
methodology applied to perform this splitting is summarized in De Jager and Duhau (2010). A good fit between the two bi-secular oscillations is found all along the observed interval with LOD being synchronic and in an opposite phase to R. (Fig. 61.3 top). The two signals have the same periodicity. Each bursting in LOD is synchronic with the same bursting in R but substantially shorter in LOD, that have larger quiet intervals between bursting. On the other hand (Fig. 61.3 bottom) it appears that while the length of the semi-secular cycle in LOD is quite regular, it changes suddenly in R in such a way that the signals lost coherence prior to 1712 and after 1858. The intermittent loss of coherence between the two signals is due to the phase transitions (Duhau and de Jager 2008) by which the sun is reacting to the common forcing agent. The bi-decadal oscillations are synchronic, but the semi-secular oscillations are not, being LOD oscillations lagging R oscillations by 94 year. This difference indicates that the mechanisms that is relating the two modes of oscillations in the two time series are different. It is worth emphasizing that in spite of this quite notorious difference in the relative phase of the two modes, the right and left scales in the top diagram are the same as in the bottom one. This means that the coefficient that relates the amplitude of LOD with R oscillations is the same for the two modes of oscillations.
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The Semi-Secular Oscillation in LOD and the 179 Year Precessional Cycle in the Solar Orbit
To quantify the precessional motion of the solar orbit in the inertial frame, Fairbridge and Sanders (1987) defined the Axym angle as the angle between the tangents to the two successive loops of the solar orbit that intercept near the barycentre (see Fig. 1 in Fairbridge and Shirley 1987). Axym is linked with the Saturn–Jupiter lap cycle in such a way that each successive conjunction of these two bodies occurs each 19.96 year and 117.4 away from the previous one, returning to approximately the same part of the range in three lap cycles, i.e. each 59.9 year. In Fig. 61.4 we have plotted Daxym, defined as the difference between the Axym angle of two successive orbits. The time interval plotted includes three full 178.7 year cycles in the precessional orientation parameter (Jose 1965; Fairbridge and Shirley 1987). The 178.7 year cycle is seen in the modulation of Daxym amplitude (see upper panel in Fig. 61.4). There are four Daxym and three LOD oscillations in each 179.7 year cycle in Daxym amplitude modulation (see bottom panel in Fig. 61.4). Therefore Daxym and LOD periodicities are 44.9 years and 59.9 year, respectively. The last is equal to the Jupiter Saturn lap cycle length. Also, the amplitude of the oscillations in LOD follows the 178.7 year cycle in Daxym modulation. We conclude that LOD follows linearly solar system motions.
LOD (ms)
LOD (ms)
2
61.3
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1858
–2 –40
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1760 Year
1840
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Fig. 61.3 The bi-decadal (top) and the semi-secular (bottom) quasi-harmonic oscillations in LOD (thick) and R (thin line). In the bottom panel LOD has been shifted backwards by 94 year
1600
1700
1800 Year
1900
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Fig. 61.4 The difference between the Axym angles of two successive astral orbits (Daxym) (Fairbridge and Shirley 1987) and the semi-secular cycle in LOD shifted backward by 94 year (thick line in bottom panel)
61
On the Origin of the Bi-Decadal and the Semi-Secular Oscillations
61.4
Summary and Discussion
Variations in LOD and the amplitude modulation of the 11-year sunspot cycle are closely related. The modulation of the 11-year sunspot cycle undergoes sudden changes of phase and length (Duhau and de Jager 2008) in time scales at and above the semisecular which lead to an intermittent loss of coherence with respect to Daxym. Instead in historical times LOD follows linearly Daxym. As a consequence an intermittent loss of coherence between the semisecular oscillation in R and LOD occurs. The bi-decadal oscillations in LOD and R are synchronic, in opposite phase and have the same periodicity. With a 94 year time delay the semi secular oscillations in the LOD follow linearly the Jupiter–Saturn lap cycle and the 178.7 year cycle in its amplitude modulation, as seen in Daxym oscillation being this last the difference between two successive Axym angles as defined by Fairbridge and Shirley (1987). As the bi-decadal oscillations in LOD and R are synchronic, but the semi-secular cycle in LOD. is delayed by nearly a century with respect to that oscillation in R, they cannot originate in the same layers of the geodynamo. In fact evidences have been forwarded (Gribbin and Plagemann 1972; Abarca del Rio et al. 2003 and references therein) that signals in the decadal band and below are excited in the mantle by atmospheric motions that, in turn, appear to be driven by solar activity. Instead, torsional oscillations are found to have their sources at the outer–inner core boundary (Zatman and Bloxham 1997; Buffett et al. 2009). We remark that the fact that the semi-secular oscillation in LOD is linearly related to cycles in solar orbital parameters does not mean that there is a causal relationship of LOD variations with respect to solar orbital motion. This is so because the solar orbital size is many orders of magnitude smaller than the Sun–Earth distance and so the perturbations of the Earth’s motion by the solar orbital motion are negligible. The semi-secular oscillation in LOD has a periodicity equal to the Jupiter Saturn Lap cycle, 59.9 year, and its amplitude follows the 178.7 year cycle in the modulation of the lap cycle amplitude as seem in the precessional motion of the solar orbit. These relationships strongly support the suggestion that the semi secular oscillation in LOD is excited by planetary orbital motions.
511
From geomagnetic variation four modes of outer core torsional oscillations have been found (Dickey and Viron 2009 and reference therein) being the periods of modes 2 and 3 inside the band (34–72 year) of the Fourier periods of the wavelet components that conform the semi-secular cycle. The century time lag of the semi-secular cycle in LOD with respect to Daxym indicates that torsional oscillations are first excited very deep. This is consistent with the fact (Zatman and Bloxham 1997; Buffett et al. 2009) that a large part of the excitation of torsional oscillations appears to originate near the surface of a cylinder that is tangent to the equator of the inner core, being the restoring force provided by the magnetic field. We conclude that there are robust evidences that the semi-secular oscillations in LOD are the result of torsional oscillations in the liquid core that are excited 94 year before at the bottom of this layer by planetary motions. To fulfill the free fall motion principle (Shirley 2006) the total angular momentum of the Earth must be preserved. Therefore as the motions in the liquid core are first excited at its boundary with the solid core, this last must undergoes a change of impulse of the opposite sign. In fact which is likely to be first excited is inner core motion, since the differential action of the planetary system on the Earth’s layers cause the periodic observed relative displacements and the relative turns between the inner core and the mantle (Bakin and Vilke 2004 and references therein).
References Abarca del Rio RD, Gambis PN, Dai A (2003) Earth rotation and solar variability. J Geodyn 36:443–443 Arreghini M, Duhau S (1997) The induction produced by a non periodic field in a spherical three layered earth’s model, II: application to a geomagnetic storm. Actas 19ª Reunion Cientifica de la Asoc. Arg Geofis, Geodes, pp 134–138 Bakin YUV, Vilke VG (2004) Celestial mechanics of planets shells. Astr Astrophys Trans 23:533 Buffett BA, Mound J, Jackson A (2009) Inversion of torsional oscillations for the structure and dynamics of the earth’s core. Geophys J Int 177:878–890 Bullard EC, Freedmann C, Gellman M, Nixon J (1950) The westward dift of the earth’s magnetic field. Phys Trans Roy Astr Soc 243:67–74 Courtillot V, Le Moue¨l JL (1984) Geomagnetic secular variation impulses. Nat Rev Paper 311:709–716
512 De Jager C, Duhau S (2010) The variable solar dynamo and the forecast of solar activity; influence on terrestrial surface temperatures; in ‘Global Warming in the 21th Century’. Nova, Hauppauge, NY Dickey J and Viron O (2009) Leading modes of torsional oscillations within the earth’s core. Geophys Res Lett L15302, doi:10.1029/2009GL0 386 Duhau S, de Jager C (2008) The solar dynamo and its phase transitions during the last millennium. Solar Phys 250:1–15 Duhau S, Martı´nez EA (1995) On the origin of the length of day and the geomagnetic field on the decadal time scale. Geophys Res Lett 22:3283–3288 Duhau S and Martı´nez EA (1997) The secular variation of the geomagnetic westward drift in the last 1000 years. Actas 19ª Reunion Cientifica de la Asoc. Arg. Geofis, Geodes. 91–95 Fairbridge R, Sanders JE (1987) The sun’s orbit A.D. 750–2050: basis for new perspectives on planetary dynamics and earthmoon linkage. In: Rampino MR, Sanders JE, Newman WS, Konigsson LK (eds) Climate: history, periodicity, and predictability. Van Nostrand Reinhold, New York, pp 446––471 Fairbridge RW, Shirley JH (1987) Prolonged minima and the 179 yr cycle of the solar inertial. Motion Solar Phys 110:91–220 Gribbin J, Plagemann S (1972) Discontinuous change in earth’s spin rate following great solar storm of august 1972. Nature 244:416–217 Gross RS (2001) A combined length-of-day series spanning 1832–1987: Lunar97. Phys Earth Planet Int 123:65–76 Hide R (1969) Interactions between the Earth’s liquid core and solid mantle. Nature 222:1055–1056 Hide R (2005) The topographic torque on a bounded surface of rotating gravitating fluid and the excitation by core motions
S. Duhau and C. de Jager of decadal fluctuations in the earth’s rotation. Geophys Res Lett 22:961–964 Hoyt DV, Schatteen KH (1998) Group sunspot numbers: a new solar activity reconstruction. Solar Phys 181:491–512 Jose PD (1965) Sun’s motion and sunspots. Astronom J 70:193 Mc Carthy D, Babcock AK (1986) The length of day since 1659. Phys Earth Planet Int 44:292–235 Mackey R (2007) Rhodes fairbridge and the idea that the solar system regulates the earth’s climate. J Coast Res ICS2007 Proceedings Australia SI, 50:955–968 Merriam JB (1988) Planetary scale flow in the earth and geodetic observations. In: structure and dynamics of the earth’s deep interior. Geophys Monogr 46, IUGG, 73–77 Landscheidt T (1999) Extrema in sunspot cycle linked to sun motions. Solar Phys 189:413–424 Le Mo€uel JL, Madden T, Ducruix J, Courtillot V (1981) Decade fluctuations on geomagnetic westward drift and earth’s rotation rate. Nature 290:763–76 Palus M, Kurths J, Schwarz U, Seehafer N, Novotna D, Cha´rva´tova I (2007) The solar activity cycle is weakly synchronized with the solar inertial motion. Phys Lett A 365:421–428 Rathcliff JT and Gross RS (2009) Combinations of Earth Orientation Measurements: SPACE2007, COMB2007,and POLE2007. JPL Publication, pp 9–18 Shirley J (2006) Axial rotation, orbital revolution and solar spinorbit coupling. Monthly Not. of the Royal Astron. Soc, vol 368, pp 280–282 Vestine EH (1953) Variations of the geomagnetic field, fluid motions and the rate of earth rotation. J Geophys Res 58:127–145 Zatman T, Bloxham J (1997) Torsional oscillations and the magnetic field within the earth’s core. Nature 388:76–7
Future Improvements in EOP Prediction
62
W. Kosek
Abstract
The Earth orientation parameters (EOP) are determined by space geodetic techniques with very high accuracy corresponding to a few millimeters on the Earth’s surface. However, the accuracy of their prediction, even for a few days in the future, is several times lower and still unsatisfactory for certain users. Wavelet decomposition of the EOP data and prediction of their different frequency components reveals that the increase of x, y pole coordinate and UT1-UTC data prediction errors up to about 100 days in the future are mostly caused by irregular short period oscillations with periods less than half a year. These irregular short period variations in x, y pole coordinates data are mostly excited by the equatorial components of atmospheric and ocean excitation functions while in UT1-UTC data are excited mostly by axial component of atmospheric excitation function. The main problem of each prediction technique is to predict simultaneously long and short period oscillations of the EOP data. The nature of short period oscillations in EOP data is mostly stochastic and longer period seasonal oscillations can be modeled using deterministic method. It has been shown that the combination of the prediction methods which are different for deterministic and stochastic part of the EOP can provide the best accuracy of prediction. Several prediction techniques, involving the least-squares extrapolation for prediction of the deterministic part and autoregressive method to predict short period stochastic part are good candidates for the prediction algorithms of the EOP data. The main problem of each prediction technique is to predict simultaneously long and short period oscillations of the EOP data and this problem can be solved by the combination of wavelet transform decomposition with the autocovariance prediction method.
62.1
W. Kosek (*) Space Research Centre, Polish Academy of Sciences, Bartycka 18A, 00-716 Warsaw, Poland e-mail:
[email protected]
Introduction
The increase of accuracy of space geodetic techniques during the last decades caused increase of determination accuracy of Earth orientation parameters which are x, y pole coordinates, universal time UT1-UTC and celestial pole offsets. However, the prediction errors
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_62, # Springer-Verlag Berlin Heidelberg 2012
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of x, y and UT1-UTC data are much greater than their determination errors. Now the determination errors of the EOP data correspond to about 3–4 mm on the earth’s surface. The ratio of the EOP prediction errors to their determination errors increases with the prediction length and even for a few days in the future the EOP prediction errors are hundred times greater than their determination errors. The future EOP data are needed to compute real-time transformation between the celestial and terrestrial reference frames. This transformation is for example important for the NASA Deep Space Network, which is an international network of antennas that supports: interplanetary spacecraft missions, radio and radar astronomy observations and selected Earth-orbiting missions. EOP predictions are published by the International Earth Rotation and Reference Systems Service (IERS) Rapid Service/Prediction Centre (RS/PC). Pole coordinates data in the IERS RS/PC are predicted by the combination of the least-squares (LS) and autoregressive (AR) method (e.g., Kosek et al. 2004) and UT1-UTC data are predicted using the forecast of axial component of atmospheric angular momentum AAM and AR process (e.g., Johnson et al. 2005). The importance of the EOP prediction to organize the Earth Orientation Parameters Prediction Comparison Campaign (EOP PCC) which started on October 2005 and ended in February 2008. The aim of this campaign was to join scientists who work on the EOP predictions and then compare results of different prediction techniques and algorithms provided by them during this campaign using equal and well-defined rules (Kalarus et al. 2010). Some of the prediction algorithms including Kalman filter (KF) (Gross et al. 1998), fuzzy interface system (Akyilmaz and Kutterer 2004), autocovariance prediction (Kosek 2002), combination of the LS extrapolation and AR prediction (Kosek et al. 2004, 2008) as well as combination of the discrete wavelet transform with autocovariance (DWT+AC) prediction (Kosek and Popin´ski 2006) were involved in the EOP PCC. Recently, Niedzielski and Kosek (2008) applied a multivariate AR model comprising length of day and axial component of atmospheric angular momentum time series and gained the improvement of UT1-UTC predictions especially during the ENSO (El Nino Southern Oscillation) events.
W. Kosek
62.2
Data
The following data sets were used in the analysis: 1. x, y pole coordinates data, universal time UT1-UTC and length of day D data from the IERS: EOPC04_IAU2000.62-now from 1962.0 to 2009.6 with the sampling interval Dt ¼ 1 day, http://hpiers. obspm.fr/iers/eop/eopc04_05/, 2. Equatorial and axial components of atmospheric angular momentum (AAM) from NCEP/NCAR, aam.ncep.reanalysis.* from 1948.0 to 2009.3 with the sampling interval Dt ¼ 0.25 day, ftp://ftp.aer. com/pub/anon_collaborations/sba/. These data were interpolated with 1 day sampling interval using the boxcar window with the length of one day. 3. Equatorial components of ocean angular momentum: (OAM): c20010701.oam in Jan. 1980 – Mar. 2002 with the sampling interval Dt ¼ 1 day and ECCO_kf066b.oam from Jan. 1993 to Dec. 2008 with the sampling interval Dt ¼ 1 day, http://euler. jpl.nasa.gov/sbo/sbo_data.html,
62.3
Prediction Algorithms
To compute the prediction of pole coordinates data the combination of LS extrapolation and AR prediction was applied, denoted as LS+AR combination in this paper. In this LS+AR prediction algorithm first the LS model which consists of the Chandler circle, annual and semi-annual ellipses and linear trend is fit to the complex-valued pole coordinates data. The difference between pole coordinates data and its LS model is equal to the LS residuals. Prediction of pole coordinates data is the sum of the LS extrapolation model and the AR prediction of the LS residuals. Figure 62.1 shows the mean prediction errors of x, y pole coordinates data computed by the LS method and the LS+AR combination. In the LS method different lengths of pole coordinates data were used to fit the LS model. The length of the AR model to fit the LS residuals was equal to 850 days. The smallest mean prediction errors of the LS+AR combination are obtained when the length of pole coordinates data to fit the LS model is equal to 10 years. To compute the prediction of UT1-UTC data the same LS+AR combination was applied to the length of
62
Future Improvements in EOP Prediction
arcsec 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 0 arcsec 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 0
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Fig. 62.1 Mean prediction errors of pole coordinates data computed by the LS (dots or circles) and LS+AR (thin or dashed line) methods in 1984–2009. To fit the LS model 10 years (circles or thin line), 6 years (dots) and 4 years (dashed line) of pole coordinates data were used ms 120 100 80 60 40 20 0
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350
Fig. 62.2 Mean prediction errors of UT1-UTC data computed by the LS+AR method in 1984–2009. To fit the LS model 5 years (dots), 10 years (thin line) or 15 years (triangles) of DdD data were used
day D data from which the model of tidal oscillations dD was removed (McCarthy and Petit 2004). To obtain such non-tidal DdD data leap seconds were removed from UT1-UTC data to get UT1-TAI data. First difference of UT1-TAI time series is equal to length of day data D. In this prediction algorithm the LS model consists of annual, semi-annual 18.6, 9.3 year oscillations and linear trend. The prediction of UT1-UTC data is computed by integrating the prediction of DdD data and adding tide model and leap seconds. Figure 62.2 shows the mean prediction errors of UT1-UTC data computed by the LS+AR combination. In the LS method different lengths of DdD data were used to fit the LS model. The length of the AR model to fit the LS residuals was equal to 1.5 years.
The smallest mean prediction errors of the LS+AR combination are obtained when the length of day DdD data to fit the LS model is equal to 10 years. To compute the prediction of UT1-UTC data by the DWT+AC combination (Kosek and Popin´ski 2006) first we need to compute as in the previous method DdD data. In this prediction method the DdD data are decomposed into frequency components using the discrete wavelet transform band pass filter (DWT BPF) (see Fig. 62.8). This filter enables decomposition of the signal into frequency components in such a way that their sum is exactly equal to the input time series. In the DWT+AC prediction method each frequency component is predicted separately by the autocovariace (AC) prediction (Kosek 2002) and the prediction of DdD is the sum of predictions of all the frequency components. Prediction of UT1-UTC data were computed from prediction of DdD data. This DWT+AC method was used to compute weekly UT1-UTC prediction during the EOP PCC. Figure 62.3 shows comparison of three mean absolute errors of UT1-UTC data for 30 day predictions computed from DWT+AC, AR predictions provided by L. Zotov and Kalman filter predictions provided by R. Gross in the EOP PCC (Kalarus et al. 2010). The best prediction results of UT1-UTC data were obtained when the Kalman filter was applied (Gross et al. 1998), however other two methods give comparable results, especially for short term predictions.
62.4
Geophysical Causes of the EOP Prediction Errors
To examine the influence of fluid excitation functions on prediction errors of x, y pole coordinates data first the pole coordinates model data were computed from the equatorial components of these excitation
516
W. Kosek arcsec 0.1 0.08 0.06 0.04 0.02 0
days in the future
x (IERS) 200 0 200 0
x (AAM)
x (OAM)
200 0 x (AAM+OAM)
days in the future
200 0 200 0
x
0.03 0.02 0.01 0.00
200 0 1980 1984 1988 1992 1996 2000 2004 2008 y (IERS) 200 0
arcsec
0
50
100
150
200
250
300
350
50
100
150
200
250
300
350
y 0.03 0.02
y (AAM)
0.01
y (OAM)
0.00 y (AAM+OAM)
Fig. 62.4 The LS+AR prediction errors of IERS pole coordinates data and of pole coordinates model data computed from AAM, OAM and AAM+OAM excitation functions
functions. The differential equation of polar motion is given by the following formula (Brzezin´ski 1992): _ i mðtÞ=s ch þ mðtÞ ¼ wðtÞ
0
days in the future
200 0 1980 1984 1988 1992 1996 2000 2004 2008 YEARS
(62.1)
in which m(t) ¼ x(t) i y(t) are the pole coordinates data to be computed, w(t) ¼ w1(t) + iw2(t) are equatorial components corresponding to AAM and OAM excitation functions, sch ¼ [1 + i / (2Q)]2p/Tch is the complex-valued Chandler frequency, Tch ¼ 433 days and Q ¼170 is the quality factor. Solution of this equation in discrete time moments can be obtained using the trapezoidal rule of numerical integration: mðtþDtÞ¼mðtÞexpðisch DtÞ isch Dt½wðtþDtÞþwðtÞexpðisch DtÞ=2; (62.2) where Dt is the sampling interval of data. The maps in Fig. 62.4 shows the time varying LS+ AR prediction errors up to one year in the future of pole coordinates data or the pole coordinates model data computed from atmospheric, ocean and join atmospheric-ocean excitation function. Note that the large prediction errors in 1980–1982 can be explained
Fig. 62.5 The mean LS+AR prediction errors of IERS pole coordinates data (thin line) and of pole coordinates model data computed from AAM (circles), OAM (triangles) and AAM +OAM (dots) excitation functions
by ocean excitation function and the large prediction errors in 2006–2007 can be explained by joint atmospheric-ocean excitation. Figure 62.5 shows the corresponding mean prediction errors of pole coordinates data and of the pole coordinates model data computed from atmospheric, ocean and joint atmospheric-ocean excitation functions. The contributions of atmospheric or ocean excitation to the mean prediction errors of pole coordinates data from 1 to about 50–100 days in the future is of the same order and explain about 60% of the total prediction error. When the prediction length increases then the mean prediction errors caused by ocean excitation become greater than these caused by atmospheric excitation. The mean prediction errors of the model data computed from joint atmospheric-ocean excitation function explain about 80–90% of the total prediction errors of the IERS pole coordinates data.
62.5
Contribution of Different Frequency Bands to the EOP Prediction Errors
In order to find a contribution of different frequency bands of the EOP data on their prediction errors the DWT BPF was applied. Figure 62.6 shows an example
62
Future Improvements in EOP Prediction
Fig. 62.6 Frequency components of the IERS x pole coordinate data computed by the DWT BPF with the Shannon wavelet function
arcsec 0.050 0.000 -0.050
0
0.040 0.000 -0.040
1
0.040 0.000 -0.040
2
0.040 0.000 -0.040
3
0.200 0.000 -0.200
517
4
0.020 0.000 -0.020
5
0.010 0.000 -0.010
6
0.007 0.000 -0.007
7
0.003 0.000 -0.003
8
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9
0.001 0.000 -0.001
10
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11 1986
1989
1992
1995
1998
2001
2004
2007
YEARS
of the reconstruction of x pole coordinate data into frequency components using the DWT BPF with the Shannon wavelet function (Benedetto and Frazier 1994; Kosek et al. 2009). The sum of these frequency components is exactly equal to the x pole coordinate data. The frequency component with index k ¼ 4 correspond to the sum of the Chandler and annual oscillations and the frequency component with index k ¼ 5 corresponds to the semiannual oscillation. Components with frequency indices k < 4 and k > 5 correspond to longer and shorter period oscillations, respectively. The pole coordinates model data were computed by summing the chosen frequency components but always including the component k ¼ 4 corresponding to the sum of the Chandler and annual oscillations. Figure 62.7 shows the mean prediction errors of the IERS x, y pole coordinates data up to 30 days in the future, and of the pole coordinates model data computed by summing the chosen DWT BPF frequency components. If pole coordinates data model
are composed of Chandler, annual and shorter period variations then the mean prediction errors are almost equal to the mean prediction error of the IERS pole coordinates data. If pole coordinates model data are composed of the Chandler, annual and longer period variations then the mean prediction error are very small. It means that the Chandler and annual oscillations with variable amplitudes and phases have meaningless influence on the short term prediction errors of pole coordinates data. The short term mean prediction errors for a few days in the future of the pole coordinates model data obtained after removal two frequency components k ¼ 10 and k ¼ 11 with the highest frequencies are several times smaller than the mean prediction errors of the IERS pole coordinates data. In order to find a contribution of different frequency bands of UT1-UTC data on their prediction errors the DWT BPF using the Meyer wavelet function (Popin´ski and Kosek 1995) was applied. Figure 62.8 shows an example of the reconstruction of D dD
518
W. Kosek arcsec
0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0.000
x
0
5
10
5
10
15
20
25
30
15 20 days in the future
25
30
y
0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0.000 0
Fig. 62.7 The mean LS+AR prediction errors of the IERS pole coordinates data (thin line) and the pole coordinates model data computed by summing the frequency components corresponding to the Chandler, annual and shorter period variations (triangles), frequency components from 0 to 9 (dashed line) and the Chandler, annual and longer period variations (circles)
s -0.00040 -0.00080 0.00120 0.00000 -0.00120 0.00020 0.00000 -0.00020
13
12 11
0.00020 0.00000 -0.00020
10
0.00030 0.00000 -0.00030
9
0.00040 0.00000 -0.00040
8
0.00040 0.00000 -0.00040
7
0.00030 0.00000 -0.00030
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5
0.00020 0.00000 -0.00020
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3
0.00012 0.00000 -0.00012
2
0.00006 0.00000 -0.00006
1 1986
1989
1992
1995
1998
2001
2004
2007
YEARS
Fig. 62.8 Frequency components of the IERS D dD data computed by the DWT BPF with the Meyer wavelet function
62
Future Improvements in EOP Prediction
519
ms 5 4 3 2 1 0
0
5
10
15 20 days in the future
25
30
Fig. 62.9 The mean LS+AR prediction errors of the IERS UT1-UTC data (thin line) and the model UT1-UTC data computed by summing the frequency components corresponding to the sum of annual, semi-annual and shorter period variations (triangles), frequency components from 3 to 13 (dashed line) and the annual, semi-annual and longer period variations (circles)
data into the frequency components using the DWT BPF with the Meyer wavelet function. The frequency component with the frequency index j ¼ 8 corresponds to the annual oscillation and with the frequency index j ¼ 7 corresponds to the semi-annual oscillation. Frequency indices j > 8 and j < 7 correspond to shorter and longer period variations, respectively. The sum of these frequency components is exactly equal to the D dD data. Figure 62.9 shows the mean prediction errors of UT1-UTC data and UT1-UTC model data up to 30 days in the future. If UT1-UTC data are composed of the annual, semiannual and shorter period oscillations then the mean prediction errors are almost equal to the mean prediction errors of the IERS UT1UTC data. If the model UT1-UTC data are composed of the annual, semi-annual and longer period variations then the mean prediction errors are very small. It means that the annual and semi-annual oscillations with variable amplitudes and phases have meaningless influence on the prediction errors of UT1-UTC data. The mean prediction errors for a few days in the future of the model UT1-UTC data obtained after removal two highest frequency components with indices j ¼ 1 and j ¼ 2 are several times smaller than the mean prediction errors of the IERS UT1-UTC data. Conclusions
Short term prediction errors of pole coordinates data are caused by wideband short period oscillations in joint atmospheric-ocean excitation functions. Some big prediction errors of pole coordinates data in 1981–1982 are caused by
wideband oscillations in OAM excitation functions and in 2006–2007 are caused by wideband oscillations in joint atmospheric-ocean excitation functions. The recommended prediction method for pole coordinates data is the LS+AR combination and Kalman filter is recommended for prediction of UT1-UTC data. Short term prediction errors of the EOP data are not caused by variable amplitudes and phases of the most energetic oscillations in these data. Short term EOP prediction errors are much smaller when these data are smoothed by removing the frequency components computed by the DWT BPF. Since short term EOP prediction errors are mostly caused by atmospheric and ocean excitation functions these functions together with their predictions should be involved in the future EOP prediction algorithms using Kalman filter which takes advantage of the AAM forecast or multivariate AR techniques which uses AAM as the input data Acknowledgements The research was supported by Polish Ministry of Science and Higher Education through the grant no. N N526 160136 under leadership of Dr Tomasz Niedzielski.
References Akyilmaz O, Kutterer H (2004) Prediction of Earth rotation parameters by fuzzy inference systems. J Geod 78:82–93 Benedetto JJ, Frazier MW (1994) Mathematics and applications. LRC Press, Boca Raton, FL, pp 221–245 Brzezin´ski A (1992) Polar motion excitation by variations of the effective angular momentum function: considerations concerning deconvolution problem. Manuscripta Geodaetica 17:3–20 Gross RS, Eubanks TM, Steppe JA, Freedman AP, Dickey JO, Runge TF (1998) A Kalman filter-based approach to combining independent Earth-orientation series. J Geod 72:215–235 Johnson T, Luzum BJ, Ray JR (2005) Improved near-term Earth rotation predictions using atmospheric angular momentum analysis and forecasts. J Geodyn 39:209–221 Kalarus M et al (2010) Achievements of the earth orientation parameters prediction comparison campaign. J Geod 84 (10):587–596 Kosek W (2002) Autocovariance prediction of complex-valued polar motion time series. Adv Space Res 30(2):375–380 Kosek W, McCarthy DD, Johnson TJ, Kalarus M (2004) Comparison of polar motion prediction results supplied by the IERS Sub-bureau for Rapid Service and Predictions and results of other prediction methods. Proceedings of the Journees 2003, pp 164–169
520 Kosek W, Popin´ski W (2006) Forecasting of pole coordinates data by combination of the wavelet decomposition and autocovariance prediction. Proceedings of the Journees 2005, pp 139–140 Kosek W, Kalarus M, Niedzielski T (2008) Forecasting of the Earth orientation parameters – comparison of different algorithms. Proceedings of the Journees 2007, pp 155–158 Kosek W, Rzeszo´tko A, Popin´ski W (2009) Contribution of wide-band oscillations excited by the fluid excitation
W. Kosek functions to the prediction errors of the pole coordinates data. Proceedings of the Journees 2008, pp 168–171 McCarthy DD, Petit G (eds) (2004) IERS Conventions 2003 IERS Technical Note No. 32 Niedzielski T, Kosek W (2008) Prediction of UT1-UTC, LOD and AAM w3 by combination of least-squares and multivariate stochastic methods. J Geod 82:83–92 Popin´ski W, Kosek W (1995) Discrete Fourier and wavelet transforms in analysis of Earth rotation parameters. Proceedings of the Journees 1995, Warsaw, pp 121–124
Determination of Nutation Coefficients from Lunar Laser Ranging
63
€ller, and F. Hofmann L. Biskupek, J. Mu
Abstract
It was just July 20, 1969 when the first retro-reflector for Lunar Laser Ranging (LLR) was deployed on the Moon by the Apollo 11 crew. From this day on, LLR is carried out to measure the distance between Earth and Moon. The complete set of observations is analysed and various parameters of the Earth–Moon system are determined by least-squares adjustment. Because of the long time span of data, long-term lunisolar nutation coefficients of the 18.6-year period (and the precession rate) can be determined well. But also other periods (182.62-day, 9.3-year, 365.26-day) can be fitted. The nutation coefficients were determined from LLR based on the models for precession and nutation according to the IAU Resolution 2006 and compared to the MHB2000 model of Mathews et al. (2002). In this paper, the corresponding preliminary results are discussed.
63.1
Introduction
For almost 40 years, measurements of the round-trip travel times of laser pulses between stations on the Earth and retro-reflectors on the Moon have been performed. The signal-to-noise ratio is rather weak, because of, e.g. energy loss, atmospheric extinction and geometric reasons. This makes LLR tracking challenging and only a few observatories worldwide are capable to perform these measurements. The lunar returns obtained over a short period of 15–20 min (only a few minutes for the new APOLLO site) are combined to so-called Normal Points (NP). About 17,000 NP until 2009 serve as observations for the
L. Biskupek (*) J. M€ uller F. Hofmann Institut f€ur Erdmessung, Leibniz Universit€at Hannover, Schneiderberg 50, 30167 Hannover, Germany e-mail:
[email protected]
data analysis. Based on these data, various investigations of the Earth–Moon system can be carried out, e.g. dedicated to Earth orientation parameters (EOP) like pole coordinates, variation of latitude (VOL) (Biskupek and M€uller 2009), UT0-UTC, precession and nutation. Also studies about the interior of the Moon (Williams et al. 2001) or tests of general relativity can be made, for further details see Soffel et al. (2008), and M€uller and Biskupek (2007).
63.2
Analysis Model
The existing model to analyse NP from LLR at the Institut f€ur Erdmessung (IfE) is based on Einstein’s theory of gravity. It is fully relativistic and complete up to the first post-Newtonian (1/c2) level, e.g. M€uller et al. (2008). The simplified observation equation for the station-reflector distance d is
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_63, # Springer-Verlag Berlin Heidelberg 2012
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t d ¼ c ¼ rEM rstation þ rreflector þ cDt; 2
(63.1)
where c is the speed of light and t the pulse travel time. The vector rEM connects the geocenter and selenocenter, rstation represents the geocentric coordinates of the observatory and rreflector the selenocentric coordinates of the retro-reflector, see Fig. 63.1. Dt describes corrections of the pulse travel time, such as atmospheric delays or the gravitational time delay. In order to apply (63.1), all vectors have to be transformed from their respective reference frames (the terrestrial (TRF) or selenocentric (SRF) reference frame) into the inertial frame, the Barycentric Celestial Reference Frame (BCRF), by rstation ¼ Rearth ðtÞ rTRF station rreflector ¼ Rmoon ðtÞ rSRF reflector:
(63.2)
Here, the EOP are used in matrix Rearth(t) for the Earth and the libration angles, computed by numerical integration, in matrix Rmoon(t) for the Moon. The Earth–Moon vector rEM in (63.1) is obtained by numerical integration of the corresponding equation of motion (here again in simplified version) :: ¼ GMEþM r þ b rEM EM Newtonian þ bRelativity ; 3 rEM (63.3) :: where rEM is the relative acceleration vector between Earth and Moon, GME+M is the gravitational constant
times the Earth–Moon mass, rEM is the Earth–Moon distance, bNewtonian comprises all further Newtonian terms like the gravitation of the solar system bodies, bRelativity includes all relativistic terms, i.e. those of the Einstein–Infeld–Hoffmann (EIH) equations. Based on the LLR model, two groups of parameters for the Earth–Moon system (ca. 180 in total) are determined by a weighted least-squares adjustment of the observations. The first group consists of the so-called Newtonian parameters, e.g. initial position, velocity and physical librations of the Moon, coordinates of LLR observatories and retro-reflectors, mass of Earth–Moon system, precession rate, long-periodic nutation coefficients and the lag angle, indicating the lunar tidal acceleration. The second group of parameters is used to perform LLR tests of the general theory of relativity, like temporal variation of the gravitational constant (M€uller and Biskupek 2007) and metric parameters. It is also possible to address the strong equivalence principle and preferred-frame effects (Soffel et al. 2008). Figure 63.2 shows the annually averaged weighted post-fit residuals of the LLR analysis with data from Jan. 1970 to Dec. 2007 (16,230 normal points). It reflects the precision of the LLR measurement and analysis model, about 20–30 cm up to the mid 1980s. From 1985 on, more stations started to observe the Moon and the residuals decreased. In the last years, only two observatories, one with reduced performance, tracked the Moon, so that the residuals increased again. For more details about the LLR network, see M€uller and Shelus (2007).
50 45
WRMS [cm]
40 35 30 25 20 15 10 5 0 1970
Fig. 63.1 Measurement setup
1975
1980
1985
1990 year
1995
2000
2005
Fig. 63.2 Weighted post-fit residuals (observed minus computed Earth–Moon distance) annually averaged
63
Determination of Nutation Coefficients from Lunar Laser Ranging
The post-fit residuals can be further investigated for Earth rotation parameters like VOL (Biskupek and M€uller 2009) and UT0-UTC using the daily decomposition method (Dickey et al. 1985).
63.3
Nutation
63.3.1 Nutation Model As mentioned in Sect. 63.2, all EOP are needed for the transformation of station coordinates from TRF into the inertial frame. In the following, we only discuss nutation. The IAU 2000 nutation model is described in the IERS Conventions 2003 (McCarthy and Petit 2004) as a series for nutation in longitude Dc and obliquity De, referred to the mean ecliptic of date: with Dc ¼
n X
ðAi þ Ai 0 tÞ sinðARGi Þ
i¼1 00
(63.4)
000
þðAi þ Ai tÞ cosðARGi Þ De ¼
n X
ðBi þ Bi 0 tÞ cosðARGi Þ
i¼1 00
t is measured in Julian centuries from epoch J2000 and n defines the number of terms the model is composed of, 678 lunisolar and 687 planetary terms with in-phase (first part of the sum in (63.4) and (63.5)) and out-of-phase (second part of the sum) coefficients. Nij denote multipliers for the respective Delaunay variables Fj. This series is based on the REN2000 nutation solution (Souchay et al. 1999) for the rigid Earth, which is convolved to the nutation model MHB2000 for the non-rigid Earth by the transfer function from Mathews et al. (2002). The MHB2000 model is constructed on the solution of linearised dynamical equations for each forcing frequency, adding contributions from non-linear terms and other effects not included in the linearised equations. The theoretical formulation improved the IAU 1980 nutation theory (Wahr 1981; Seidelmann 1982) by the incorporation of mantle anelasticity, ocean tide effects and electromagnetic couplings of the mantle and the solid inner core to the fluid outer core. Table 63.1 gives the values for the non-time-dependent coefficients in the MHB2000 model for the periods with the largest contribution to the nutation angles. These periods are: 18.6-year, 182.62-day, 13.66-day, 9.3-year and 365.26day, sorted in order of their largest contribution.
(63.5)
000
þðBi þ Bi tÞ sinðARGi Þ
63.3.2 Nutation Coefficients from LLR Data
With ARGi ¼
5 X
Nij Fj :
(63.6)
j¼1
Table 63.1 Values for nutation coefficients of different period in the MHB2000 model 18.6-year 182.62-day 13.66-day 9.3-year 365.26-day
523
Ai (mas) 17,206.42 1,317.09 227.64 207.46 147.59
Bi (mas) 9,205.23 573.03 97.85 89.75 7.39
A00i (mas) 3.34 1.37 0.28 0.07 1.18
B00i (mas) 1.54 0.46 0.14 0.03 0.19
From the analysis of LLR observations as part of the global adjustment, amplitudes of the long-term nutation coefficients are determined. Particularly, the non-time-dependent coefficients Ai,A00i , Bi and B00i of the periods with the largest contribution to the nutation angles (Sect. 63.3.1) were fitted. The 13.66-day period is difficult to determine from LLR data, because of the inhomogeneous distribution of the NP. The coefficients for the other four periods were fitted from LLR data in a least-squares adjustment for the time span from 1970 to 2007 and compared with the values of the MHB2000 model. Table 63.2 gives the preliminary results. Precession rate was fitted
Table 63.2 Values and their standard deviation for nutation coefficients of different periods from LLR data 18.6-year 182.62-day 9.3-year 365.26-day
Ai (mas) 17,201.75 1,316.87 207.08 146.70
0.42 0.15 0.32 0.22
Bi (mas) 9,203.59 572.91 90.75 7.82
0.20 0.06 0.15 0.09
A00i (mas) 3.83 3.13 1.38 0.21
0.31 0.14 0.32 0.16
B00i (mas) 3.92 1.02 0.19 0.57
0.18 0.06 0.16 0.07
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together with the nutation coefficients. The result is 1.96 0.02 mas/y. The correlation to some of the nutation coefficients is up to 30%.
63.4
Discussion
For the comparison of nutation coefficients from LLR data to the MHB2000 model, the differences between these two solutions are given in Table 63.3. The largest differences are in the coefficients of the 18.6-year period: i.e. the in-phase coefficient for Dc and the Table 63.3 Differences for nutation coefficients: MHB2000 minus LLR results 18.6-year 182.62-day 9.3-year 365.26-day
Ai (mas) 4.67 0.22 0.38 0.88
Bi (mas) 1.65 0.13 1.00 0.44
A00i (mas) 0.49 1.76 1.45 0.97
B00i (mas) 2.38 0.56 0.16 0.38
Amplitude [mas]
a 2
0
−2 1970
Amplitude [mas]
b
1980
1990 years Δy sin(ε0)
2000
in-phase as well as the out-of-phase coefficients for De. One reason could be the way of determining nutation coefficients in the MHB2000 model, where some of the parameters of the theory are obtained from an adjustment to VLBI data. In contrast to VLBI, LLR is very sensitive to the lunar orbit, where the 18.6-year period is linked to the ascending node of the Moon. Another reason could be insufficient modelling of the lunar orbit and rotation or inconsistencies between the coordinate frames of the MHB2000 model and LLR analysis. Differences of more than one mas are also present in the Dc out-of-phase coefficients for 182.62-day and 9.3-year. Here, the reason is still not understood. Similar differences in the nutation coefficients are also seen in the analysis of other groups (Williams 2008 private communication). Also the resulting nutation angles Dc sin(e0) and De were calculated once from the MHB2000 model and once from the LLR solution to illustrate the full difference, shown in Fig. 63.3. They are between 2.5 and 3.7 mas for Dc sin(e0) (see Fig. 63.3a) and between 4.5 and 4.1 mas for De (see Fig. 63.3b). In both figures, the 18.6-year period is dominant, as expected from the numbers in Table 63.3. But also differences in the other frequencies are obvious. It is also clear, that the differences in the in-phase and outof-phase components do not compensate each other. Zerhouni and Capitaine (2009) determined celestial pole offsets DX and DY from LLR data. Then they analysed these offsets to derive nutation coefficients and compared their values to corresponding results from VLBI data. They obtained a better agreement between the LLR and VLBI results than our analysis. But the two LLR solutions are not directly comparable, because of different processing approaches.
4
Conclusions
2
0
−2
−4 1970
1980
1990 years Δε
2000
Fig. 63.3 Differences in nutation angles: MHB2000 minus LLR results
A 38-years LLR data set has been analysed to determine long-periodic nutation coefficients from LLR analysis. The results show differences from the MHB2000 model for some of the coefficients. The differences cannot be completely explained yet and must be further investigated. In a next step, the results from LLR shall be improved by better controlling the consistency between the different reference systems of the analysis, refining the modelling of the lunar interior and the asteroids (up to now only a few major asteroids are
63
Determination of Nutation Coefficients from Lunar Laser Ranging
included). Also a direct determination of the nutation coefficients in a combined LLR + VLBI solution is planned.
Acknowledgments Current LLR data are collected, archived and distributed under the auspices of the International Laser Ranging Service (ILRS) (Pearlman et al. 2002). We acknowledge with thanks, that the more than 38 years of LLR data, used in these analyses, have been obtained under the efforts of personnel at the Observatoire de la Cote d’Azur in France, the LURE Observatory in Maui, Hawaii, the McDonald Observatory in Texas as well as the Apache Point Observatory (APOLLO) in New Mexico. We would also like to thank the DFG, the German Research Foundation, which funded this study within the research unit FOR584 “Earth rotation and global dynamic processes”.
References Biskupek L, M€uller J (2009) Lunar laser ranging and Earth orientation. In: Soffel M, Capitaine N (eds) Proceedings of the Journe´es 2008 “Syste`mes de re´fe´rence spatiotemporels”, pp 182–185 Dickey JO, Newhall XX, Williams JG (1985) Earth orientation from lunar laser ranging and an error analysis of polar motion services. J Geophys Res 90:9353–9362 Mathews, P.M., Herring, T.A. and Buffett, B.A. (2002). Modeling of nutation and precession: New nutation series for nonrigid Earth and insights into the Earth’s interior. J Geophys Res 107(B4):ETG 3-1. 10.1029/2001JB000390.
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McCarthy DD, Petit G (eds) (2004) IERS Conventions (2003), IERS Technical Note No. 32. Verlag des Bundesamts f€ur Kartographie und Geod€asie, Frankfurt am Main M€uller J, Biskupek L (2007) Variations of the gravitational constant from lunar laser ranging data. Class Quantum Grav 24:4533–4538. doi:10.1088/0264-9381/24/17/017 M€uller J, Shelus PJ (2007) Lunar Laser Ranging (LLR) Network. ILRS Report 2005/2006 In: Noll C, Pearlman M (eds) NASA pp 2-2–2-3. M€uller J, Williams JG, Turyshev SG (2008) Lunar laser ranging contributions to relativity and geodesy. In: Dittus H, L€ammerzahl C, Turyshev SG (eds) Lasers, clocks and drag–free control: exploration of relativistic gravity in space, vol 349. Springer, Berlin, pp 457–472 Pearlman MR, Degnan JJ, Bosworth JM (2002) The International Laser Ranging Service. Adv Space Res 30:135–143. doi:10.1016/S0273-1177(02)00277-6 Seidelmann PK (1982) 1980 IAU theory of nutation – the final report of the IAU Working Group on nutation. Celest Mech Dyn Astr 27:79–106 Soffel M, Klioner S, M€uller J, Biskupek L (2008) Gravitomagnetism and lunar laser ranging. Phys Rev D 78:024033 Souchay J, Loysel B, Kinoshita H, Folgueira M (1999) Corrections and new developments in rigid earth nutation theory – III Final tables “REN-2000” including crossednutation and spin-orbit coupling effects. Astronomy and Astrophysics Supplement Series, vol 135. pp 111–131 Wahr JM (1981) The forced nutations of an elliptical, rotating, elastic and oceanless Earth. Geophys J R Astron Soc 64:705–727 Williams JG, Boggs DH, Yoder CF, Ratcliff TJ, Dickey JO (2001) Lunar rotational dissipation in solid body and molten core. J Geophys Res 106(E11):27933–27968 Zerhouni W, Capitaine N (2009) Celestial pole offsets from lunar laser ranging and comparison with VLBI. Astron Astrophys 507:1687–1695
.
A Set of Analytical Formulae to Model Deglaciation-Induced Polar Wander
64
W. Keller, M. Kuhn, and W.E. Featherstone
Abstract
Traditionally, deglaciation-induced polar wander changes are modelled using a saw-tooth-shaped function for the time-history of ice sheets and spherical caps to express their spatial extent. In this contribution we present a set of analytical formulae that allow for a more realistic temporal evolution as well as spatial distribution of current ice masses and the corresponding sea level change when partly or completely melted. Starting with the linearized Liouville equations we develop closed-form time-domain solutions via the Laplace-domain, which are based on the assumption of a piecewise linear time-history of the perturbation of the inertia, which do not require the solution of convolution integrals. As being a central aspect of polar wander modelling we also revisit perturbation of the moment of inertia changes due to arbitrary surface loading due to changes in ice and ocean water masses and compare them with the result of the more simplistic models of spherical ice caps and a uniform sea level change. Finally, the correctness of the developed formulae is checked by various numerical checks based on more simplistic models and numerical integration techniques.
64.1
Introduction
It is well known that past, current and future changes of the cryosphere leave their fingerprint in several geophysical parameters such as global sea level (e.g. Farrell and Clark 1976; Conrad and Hager 1997; Tamisiea et al. 2001; Mitrovica and Milne 2003;
W. Keller Institute of Geodesy, Universit€at Stuttgart, Geschwister-SchollStr. 24D, 70174 Stuttgart, Germany M. Kuhn (*) W.E. Featherstone Western Australian Center for Geodesy and The Institute for Geoscience Research, Curtin University of Technology, Perth, Australia e-mail:
[email protected]
Cazenave and Nerem 2004; Kuhn et al. 2010), the Earth’s gravity field including the center of mass (e.g. Chao and O’Connor 1987; Mitrovica and Peltier 1989, 1993; Tamisiea et al. 2001) and rotation vector (e.g. Nakiboglu and Lambeck 1980; Chao and O’Connor 1987; Mitrovica and Milne 1998; Gross 2007). The latter is the result of a change in the Earth’s moments of inertia (e.g. Munk and MacDonald 1960; Lambeck 1980) caused by the corresponding surface mass load changes resulting in a change of the rotational speed (e.g. expressed in length of day, LOD, changes) and the movement of the rotational pole relative to the Earth’s surface (polar wander). Up until now the majority of studies on long-term Earth’s rotation changes look back in time (e.g. Pleistocene deglaciation) with the aim to infer information on the
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_64, # Springer-Verlag Berlin Heidelberg 2012
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viscosity distribution within the Earth’s mantle (e.g. see the list of references in Gross 2007). Based on some early studies on GIA-induced long-term Earth’s rotation changes (e.g. Wu and Peltier 1984) past icemass distributions are frequently modelled by spherical caps of constant thickness of which only the total mass but not the geometry changes over time. This shortcoming has been addressed by some authors ((e.g. James and Ivins 1997; Mitrovica et al. 2001)) and long-term polar wandering modelling has been done using the gravitationally self-consistent sealevel equation. Of course for a more realistic scenario no closed formulae for the perturbations of inertia are available. Therefore we develop and follow a numerical integration approach to model polar wander due to current and future melting of currently ice covered areas. Furthermore, we present the Laplace transform solution of various ice history functions suitable for polar-wander modelling due to future ice melt. As in the simple spherical case the visco-elastic response of the Earth’s rotation to any mass re-distribution is given by the solution of the Liouville equations (e.g. Munk and MacDonald 1960; Lambeck 1980). Looking over geological time-spans the Liouville equations are frequently linearized and solved via the Laplace transform domain, thus the solution in the time-domain is given as a convolution. While our approach is still based on the linearized Liouville equations we follow two avenues for their solutions: (1) the closed solution of the simplified Liouville equations via the Laplace transform of various piecewise linear ice histories and (2) the solution via a numerical integration approach allowing for sophisticated mass distributions and ice histories. With the so developed tools we investigate a number of scenarios in order to demonstrate the sensitivity of polar wander on the chosen mass distributions. This is done by comparing the analytical solutions based on the Laplace Transformation of the linearized Liouville equations with the more rigorous numerical solution.
{ _ þm¼c m sr
with the complex variable m ¼ m1 þ lm2 combining the two polar wander components m1 and m2. Hereby, the real-valued component m1 points along the Greenwich Meridian and the imaginary-valued component m2 towards the 90 E meridian. The frequency sr ¼ O(CA)/A is the well-known Chandler frequency of a rigid Earth with A and C denoting the equatorial and axial principal moments of inertia, respectively and O is the mean value of the Earth’s angular velocity. The r.h.s. of (64.1) is the so-called excitation function c, which can be expressed by c¼
The Linearized Liouville Equations
We start here with the ‘polar wander part’ of the linearized Liouville equations for a spherical, stratified viscoelastic Earth, given in complex notation by (Wu and Peltier 1984)
1 CA
I13 þ
I_23 O
I_13 þ { I23 O
(64.2)
with Iij being the perturbations of inertia. The perturbations of inertia are decomposed in a loading and rotational term through Iij ¼ IijL þ IijR ¼ 1 þ k2L IijR þ IijRot
(64.3)
where * denotes a convolution in time and k2L is the surface load Love-number of degree 2. Here, the pertubation of the moments of inertia due to a surface load s(#, l) are denoted by IijR . The first term ð1 þ k2L Þ IijR indicates the direct and indirect effect of the surface load onto Iij and the term IijRot is related to the change of moments of inertia due to moving rotation axis. For an analytical solution of (64.1) it is of benefit to decompose the excitation function through c ¼ cLcRot in a loading part cL and rotational part cRot. Following the same decomposition as in (64.3) the loading part can be expressed as c ¼ "
cL1
L
64.2
(64.1)
þ
{ cL2
1 þ k2L ¼ ! CA
R I_ R I13 þ 23 O
R I_ R 13 þ { I23 O
!#
(64.4)
and the rotational part is coupled to the polar wander components by (e.g. Wu and Peltier 1984) cRot
k2T m; kf
(64.5)
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A Set of Analytical Formulae to Model Deglaciation-Induced Polar Wander
with k2T the tidal Love number and kf being the socalled fluid Love number both of degree 2, which is given by kf ¼
3G ðC AÞ; a5 O2
(64.6)
where G is the gravitational constant and a the Earth’s mean radius. For the definition and practical determination of the Love numbers k2L , k2T and kf for a stratified visco-elastic Earth model the interested reader is referred to e.g. Wu and Peltier (1984). Finally, introducing (64.4) and (64.5) into (64.1) yields an alternative expression of the linearized Liouville equations by (cf. Sabadini and Vermeersen 2002, eq. 45) { k2T _ þ 1 (64.7) m ¼ cL : m kf sr In the remainder of this paper we will construct a closed solution of (64.7) via the Laplace-domain.
64.3
Solutions in the Laplace Domain
The transformation of (64.7) into the Laplace domain yields " !# ~T { ~ k 2 ~ sþ 1 (64.8) m ¼ cL ; kf sr where () denotes the Lapalce transform of the corresponding time-domain function. An equivalent expression to (64.8) is obtained through the introduction of the Laplace transform of the tidal Love number T k~ (cf. Wu and Peltier 1984, eq. 64)
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assumed that the time derivatives of the perturbations of inertia can be neglected, implicitly assuming that only the magnitude but not the geometry of the considered masses (e.g. ice and ocean) does change over time. With the Laplace transform f~ðsÞ of the timehistory function f(i) the above assumptions lead to a simplified form of (64.9) given by ! 8 R 0 B0 X a O j R ~ ¼ f~ðsÞ; I þ {I23 A þ þ m s n þs As0 13 j¼1 j (64.11) where s0 denotes the Chandler wobble of a homogeneous deformable Earth. Here the numerical values nj and aj, A0 , B0 have been obtained through the numerical values for the parameters rj, sj, tj taken from Wu and Peltier (1984). For our numerical studies we will investigate four combinations in terms of a fixed or variable geometry of the ice masses and in terms of the time-history function used, listed below with increasing levels of complexity: Time-history 1: fixed geometry and one phase complete deglaciation Time-history 2: fixed geometry and one phase incomplete deglaciation Time-history 3: fixed geometry and two phase incomplete deglaciation All four cases use a linear glaciation phase. The first two cases only serve as pre-steps to solve the more interesting last two case and will be used for a sensitivity analysis in order to examine the dependency of the time-domain solution on the choice of geometry and time-history function.
2
~L {sr c ~ ¼ h i: m P tj s 1 {skfr N j¼1 sj ðsþsj Þ
(64.9)
The Laplace transformation of k2T assumes a stratified viscoelastic Earth with N normal modes. A time-domain solution of (64.9) is frequently obtained by the introduction of the assumption (e.g. Wu and Peltier (1984))
64.3.1 Fixed Geometry and One Phase Complete Deglaciation
(64.10)
Here we assume the classical saw-tooth shaped timehistory of glaciation and deglaciation ( t 0 t
that the perturbations of inertia change over time in the same way as the assumed time-history function f(t) of the glaciation deglaciation process. Furthermore, it is
with a glaciation period of G and a subsequent complete deglaciation within the period of D as illustrated in Fig. 64.1.
IijR ðtÞ ¼ IijR f ðtÞ
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Fig. 64.1 Time-history for a complete deglaciation in one phase (top) and or an incomplete deglaciation in one phase (bottom)
esG esG D sD esD 1 e þ 2 þ 2 s D s s s
The Laplace transform of fc (t, G, D) is obtained through: f~c ðsÞ : ¼
ð1 0
f ðtÞest dt
ð GþD t G st 1 e dt D 0 G ð ðD 1 G st t st e dt ¼ te dt þ eGs 1 G 0 D 0 ð 1 t 1 st G e dt est þ ¼ G s s 0 ð D ðD 1 Gs st st þe e dt te dt D 0 0 1 G 1 G esG 2 ½est 0 ¼ G s Gs 1 Gs þe est jD 0 s D # ð 1 t st 1 st e dt e þ D s s 0 1 ¼ G
ðG
¼
test dt þ
1 1 ¼ eGs 2 esG 1 s Gs
1 1 þ esG eDs þ s s ) 1 t st 1 st D e 2e D s s 0 1 1 esG eðGþDÞs ¼ eGs þ 2 2 s Gs Gs s
1 esG eðGþDÞs eðGþDÞs þ 2 2 Gs s s Gs þ
esðGþDÞ esG 2 Ds2 Ds
1 esG esG esðGþDÞ 2 þ Ds Ds2 Gs2 Gs2 sG sG 1 1 e e esðGþDÞ þ ¼ 2 G D D s G ¼
(64.13)
Inserting the Laplace transformation of the timehistory into (64.11) and performing the back-transfor2 mation using the Laplace transform relations s13 $ t2 1 and s2 ðsþbÞ $ b12 ðebt þ bt 1Þ yields the time-domain solution to O R B0 2 R t mc ðt; G; DÞ ¼ ðI13 þ {I23 Þ A0 f ðtÞ þ 2G As0 B0 ðG þ DÞ B0 ðt ðG þ DÞÞ2 ðt GÞ2 þ 2D 2GD 8 X aj nj t þ ðe þ nj t 1Þ Gn2j j¼1
aj ðG þ DÞ nj ðtGÞ ðe þ nj ðt GÞ 1Þ GDn2j
aj nj ðtGDÞ ðe þ nj ðt G DÞ 1Þ þ Dn2j
!# :
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A Set of Analytical Formulae to Model Deglaciation-Induced Polar Wander
531
64.3.2 Fixed Geometry and One Phase Incomplete Deglaciation In analogy to the previous case also for the case of a non-complete deglaciation but still using a fixed geometry the time-domain solution of (64.11) can be derived. Hereby the time-history function is given by 8 t 0 t G < G; tG fin1 ðt; G; D; pÞ ¼ 1 ð1 pÞ D ; G t G þ D : p; GþD t (64.14) which is illustrated in Fig. 64.1. The underlying assumption of such time-history function is that current deglaciation processes might be stopped as the climate stabilizes after the time-period D, so that a certain percentage 0 p 1 of the maximum ice masses remain still frozen. Following the same derivation principle then in the previous section the timedomain solution for the polar wander components is given by
O R B0 R ðI13 þ {I23 Þ A0 f ðtÞ þ t2 min1 ðt;G;D;pÞ ¼ 2G As0 0 B ðð1 pÞG þ DÞ ðt GÞ2 2GD B0 þ ð1 pÞ ðt ðG þ DÞÞ2 2D 8 X aj nj t þ ðe þ nj t 1Þ 2 Gn j j¼1
aj ðð1 pÞG þ DÞ nj ðtGÞ ðe þ nj ðt GÞ 1Þ GDn2j
aj þ 2 ð1 pÞðenj ðtGDÞ þ nj ðt G DÞ 1Þ Dnj
Fig. 64.2 Time-history for an incomplete deglaciation in two phases
masses of the pleistocene (past) and the recent ice shields, respectively. Phase 2: An acceleration of the deglaciation during the next time-period D2 from present onwards melting (1 p) percent of the recent ice masses. Phase 3: A consolidation phase after phase 2, where the melting stops. Obviously, the two-phase deglaciation time-history fin2 can be represented as the superposition of two onephase time histories. Using the relation ð1 pÞ ¼
D 2 Mp 1 D 1 Mr
(64.15)
we obtain the representation of the new time-history to !) :
64.3.3 Fixed Geometry and Two Phase Incomplete Deglaciation In this case we now drop the assumption that the deglaciation is going to happen with a uniform speed since the last glaciation maximum, but assume three phases (cf. Fig. 64.2). Phase 1: Deglaciation of Mr/Mp percent of the ice massed during the time-period D1 since the last glaciation maximum. Here Mp and Mr are the
fin2 ðt; G; D1 ; D2 ; pÞ ¼ fc ðt; G; D1 Þ Mr fin1 ðt G; D1 ; D2 ; pÞ: þ Mp (64.16) Correspondingly, the time-domain solution in the two-phase case is the superposition of the timedomain solutions of the two one-phase solutions mc and min1 such as min2 ðt; G; D1 ; D2 ; pÞ ¼ mc ðt; G; D1 Þ Mr þ min1 ðt G; D1 ; D2 ; pÞ: Mp (64.17)
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W. Keller et al.
64.3.4 Varying Geometry and Two Phase Incomplete Deglaciation In the previous three cases we have implicitly assumed that during glaciation and deglaciation the ice covered area remained the same and that only the amount of ice-mass changed. This is of course far from reality as in the past not only the amount of the ice masses but also their geographic distribution has changed (e.g. advance and retreat of ice shields). Therefore, we introduce an improved model assuming a linear change both in geometry and in mass density. In this case the perturbations of inertia do not follow any more the simple relation given in (64.10) but become a more general function with time. Like for the timehistory function fin2 (t, G, D1, D2, p) in the previous case the perturbations of inertia have now four phases: Phase 1: During the glaciation period G the ice shields build up and the glaciation occurs at all places that are ice covered at the glaciation maximum. So the geometry does not change in this phase only the mass changes. Phase 2: During the first phase D1 which extends here from the glaciation maximum to the time being, the inertia changes linearly from their pleistocene to their present values. In this phase both geometry and mass are changing from the pleistocene configuration to the recent configuration. Phase 3: The third phase is an accelerated melting phase due do global warming which starts at the time being and lasts over the time-period D2. Here again the geometry is assumed to remain unchanged and only the ice mass changes. This means the current ice distribution remains the same but the ice mass density reduces linearly. Phase 4: The last phase is an assumed climate consolidation, where p percent of the recent ice masses remain still frozen. In this more general case the linearized Liouville equations (cf. 64.11) have to be solved for the temporal variation of the disturbances of inertia given in (64.18), Ii3 ðt;G;D1 ;D2 ;pÞ ¼ 8 p tIi3 > > G ; > > < tGD1 p tG r D1 Ii3 þ D1 Ii3 ; > r > Ii3 ð1 ð1 pÞ tGD1 Þ; > D2 > : r pIi3 ;
0 t
p r Here Ii3 and Ii3 are the perturbations of inertia at the time of maximal glaciation and at recent time, respectively. The former can be computed from ice records the latter from information on the current ice cover such as provided by the JGP95E model (Lemoine et al. 1998). This equation has to be transformed into the Laplace domain. The Laplace transform of (64.18) is obtained through ð1 ~ Ii3 ðtÞest dt I i3 ðsÞ ¼ 0
Ip ¼ i3 G
ðG
tedt dt
0 p ð GþD1 Ii3
D1
ðt G D1 Þest dt
G
ð Ir GþD1 þ i3 ðt GÞest dt D1 G ð GþD1 þD2 t G D1 r 1 ð1 pÞ þ Ii3 D2 GþD1 est dt ð1 r þ pIi3 ¼
p Ii3 G
ðG 0
p Ii3 D1
GþD1 þD2
est dt
tedt dt
ð D1
ðt D1 Þest dteGs
0
ð Ir D1 st Gs þ i3 te dte D1 0 ð D2 t r est dt þ Ii3 1 ð1 pÞ D 2 0 eðGþD1 Þ2 ð1 r þ pIi3 est dteðGþD1 þD2 Þs ¼
0 p Ii3 Gs Gs e e Gs2
Gs 1
p Ii3 eGs eD1 s þ D1 s 1 2 D1 s Ir þ i3 2 eðGþD1 Þs eD1 s D1 s 1 D1 s Ir þ i3 2 eðGþD1 þD2 Þs D2 s Ds e 2 ðD2 s ð1 pÞÞ D2 ps þ ð1 pÞ
þ
þ
r Ii3 p ðGþD1 þD2 Þs e s s2
64
A Set of Analytical Formulae to Model Deglaciation-Induced Polar Wander
p r p p 1 Ii3 Ii3 Ii3 Ii3 Gs ¼ 2 þe D1 G D1 s G p I Ir Ir þ eðGþD1 Þs i3 i3 i3 ð1 pÞ D1 D1 D2 Ir þeðGþD1 þD2 Þs i3 ð1 pÞ D2
the back-transformation into the time-domain can be given as mi ðtÞ ¼ (64.19)
Ir þ i3 ð1 pÞ’ðt G D1 D2 Þ D2
(64.24)
The special choice of the following values p ¼ 0;
r Ii3
¼
p Ii3
1
D1 D1 þ D 2
(64.25)
(64.20)
Inserting (64.19) into (64.20) and introducing the abbreviations r Ii3 Ip Ip i3 i3 ; D1 G D1 Ip Ir Ir Yi ¼ i3 i3 i3 ð1 pÞ D1 D1 D2
Xi ¼
assume, that the ratio of the perturbations of inertia do not change during the whole glaciation-deglaciation cycle. This brings us back to the special case in Sect. 64.3.3. And indeed for this choice we obtain
(64.21) p Xi ¼ Ii3
the linearized Liouville equations in the Laplacedomain can be written component wise as ~ i ðsÞ ¼ m
OA0 Ii3 ðtÞ As0
p O Ii3 ’ðtÞ þ Xi ’ðt GÞ þ As0 G þ Yi ’ðt g D1 Þ
In order to drive a time-domain solution we first reformulate the linearized Liouville equations in the Laplace-domain (cf. 64.11), which results in ! 8 0 X a O B j ~ mðsÞ ¼ ðI~13 ðsÞ A0 þ þ s n þs As0 j¼1 j þ { I~23 ðsÞÞ
533
O 0~ A Ii3 ðsÞ As0 ( ! p 8 O Ii3 B0 X aj þ þ As0 G s3 j¼1 s2 ðnj þ sÞ ! 8 aj B0 X þ Xi 3 þ eGs 2 ðn þ sÞ s s j j¼1 ! 8 0 aj B X eðGþD1 Þs þ Yi 3 þ 2 ðn þ sÞ s s j j¼1 ! 8 r aj Ii3 B0 X þ ð1 pÞ 3 þ D2 s s2 ðnj þ sÞ j¼1 o eðGþD1 þD2 Þs
G þ D1 þ D2 ; GðD1 þ D2 Þ
Yi ¼ 0
(64.26)
which when introduced into (64.24) yields a solution, which in its structure is identical to the unchanging geometry solution with two deglaciation phases (cf. 64.15). The auxiliary function ’ is defined by ’ðtÞ :¼ B0
8 aj nj t t2 X ðe þ nj t 1Þ: þ 2 j¼1 n2j
(64.27)
p r and Ii3 are the perturbations of inertia at the Here Ii3 time of maximal glaciation and at recent time, respectively. The former can be computed from ice records the latter from information on the current ice cover such as provided by the JGP95E model (Lemoine et al. 1998) and the quantities Xi, Yi are given by
(64.22) Using the Laplace transform relation 8 8 2 X B0 X aj aj 0t $ B þ þ 3 2 s s ðnj þ sÞ 2 j¼1 n2j j¼1
ðenj t þ nj t 1Þ (64.23)
r Ii3 Ip Ip i3 i3 ; D1 G D1 Ip Ir Ir Yi ¼ i3 i3 i3 ð1 pÞ: D1 D1 D2
Xi ¼
(64.28)
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W. Keller et al.
64.4
Table 64.1 Description of the past Fennoscandian, Laurentide and Antarctic ice shields by the parameters of spherical ice caps
Perturbation of Inertia Due to Surface Loading
The perturbations of inertia due to an extra surface mass density s (#, l) is given by ð ¼
IijR
sð#; lÞðR2 dij xi xj Þds
(64.29)
S
where the integration is performed over the complete surface of the Earth S, R is the mean Earth radius, xi, i ¼ 1,2,3 are three-dimensional Cartesian coordinates and # and l indicate the spherical co-latitude and longitude, respectively. Introduction of the spherical harmonic expansion
Fennoscan. 0.56 1019 9.5 25.5 25
Antarctic 0.7 1019 20 180 n/a
Table 64.2 Relative differences between the numerical and the analytical results. n/a means the value is numerically zero s211/s212 I13/I23
Laurent. n/a/0.1% n/a/0.03%
Fennoscan. 0.12%/0.13% 0.13%/0.14%
Antarctic n/a/n/a 0%/0%
hicerice and soc ¼ rwathoc, where rice and rwat are the mean densities of ice and fresh water, respectively.
64.4.1 Validation Tests (64.30)
of the surface mass density in (64.29) yields the following relation between the perturbations of inertia and low-degree spherical harmonic coefficients of s(#, l) (e.g. Wu and Peltier 1984). 4 R ¼ pR2 s211 ; I13 5 R ¼ I33
4 R I23 ¼ pR2 s212 5
8 s201 : s000 5 3
(64.31)
(64.32)
For a given surface density s its spherical harmonichs coefficients can be obtained by standard numerical quadratur formulas on the sphere. Here we only look at changes of the cryosphere sice and the oceans soc. In order to conserve masses the integral over all surface mass densities ð
ð s¼
S
Laurent. 2 1019 15 30 270
1 X n X
ðsnm1 cosml þ snm2 sinlÞ n¼0 m¼0 Pm n ðcos#Þ
sð#; lÞ ¼
Mass M (kg) Radius a ( ) co-latit. # ( ) longit. l ( )
sice soc ds ¼ 0;
(64.33)
S
must hold Ð or equivalently the total ice mass change equal the total ocean water mass Mice ¼ Ssice must Ð change Moc ¼ S soc. If hice (#,l) and hoc (#,l) are the elevation functions of the ice masses and the global sea level change, respectively the corresponding surface mass densities can be expressed by sice ¼
In order to validate the effectiveness of the numerical procedure described in the previous section we performed a numerical test for the spherical cap model, which can be also expressed by a closed formula. Furthermore, we test the accuracy of the ocean function coefficients by the derivation of the same coefficients based on a more recent and more detailed data set of the world’s land-ocean distribution. We derive the spherical harmonic coefficients sice nm1 and sice nm2 and the disturbances of inertia (64.31 and 64.32) for the past Fennoscandian, Laurentide and Antarctic ice shields. While the characteristic parameters are provided in Table 64.1, the numerical results for the spherical harmonic coefficients as well as the disturbances of inertia based on the numerical technique and the closed formulae are listed in Table 64.2. This validation test indicates that all differences are well below 1%, thus confirming the correctness of the numerical results. Conclusion
We developed closed-form time-domain solutions via the Lapalce-domain for four piecewise linear time-history functions. The time-domain solutions are closed-form solutions in that they do not contain convolution integrals. We have confirmed the correctness and successful implementation of our
64
A Set of Analytical Formulae to Model Deglaciation-Induced Polar Wander
derivations through a comparison of our results with results from closed-form solutions. The validation showed that our developments are well suited for polar-wander modelling using more realistic iceocean geometries and time-history functions.
References Cazenave A, Nerem RS (2004) Present-day sea level change: observations and causes. Rev Geophys 42(RG3001):1–20 Chao BF, O’Connor WP (1987) Effect of a uniform sea-level change on the Earth’s rotation and gravitational filed. Geophys J Int 93:191–193 Conrad CP, Hager BH (1997) Spatial variations in the rate of sea level rise caused by the present-day melting of glaciers and ice sheets. Geophys Res Lett 24:1503–1506 Farrell WE, Clark JA (1976) On postglacial sea level. Geophys J R Astr Soc 46:647–667 Gross RS (2007) Earth rotation variations – long period. In: Herring TA (ed) Treatise on geophysics, vol 3. Elsevier, Oxford, pp 239–294 James TS, Ivins ER (1997) Global geodetic signature of the Antarctic ice sheet. J Geophys Res 102(B1):605–633 Kuhn M, Featherstone WE, Makarynskyy O, Keller W (2010) Deglaciation-Induced Spatially Variable Sea Level Change: A Simple-Model Case Study for the Greenland and Antarctic Ice Sheets. International Journal of Ocean and Climate Systems 1(2):67–83 Lambeck K (1980) The earth’s variable rotation: geophysical causes and consequences. Cambridge University Press, London
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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 NASA GSFC and National Imaginary and Mapping Agency (NIMA) Geopotential Model EGM96, Rep. NASA/TP-1998-206861. NASA, Maryland Mitrovica JX, Peltier WP (1989) Pleistocene deglaciation and the global gravity filed. J Geophys Res 94 (B10):13,651–13,672 Mitrovica JX, Peltier WP (1993) Present-day secular variations in the zonal harmonics of the Earth’s geopotential. J Geophys Res 98(B4):4,509–4,526 Mitrovica JX, Milne GA (1998) Glaciation-induced pertubations in the Earth’s rotation: A new appraisal. J Geophys Res 103:985–1005 Mitrovica JX, Milne GA (2003) On post-glacial sea level: I. General Theor Geophys J Int 154:253–267 Mitrovica JX, Milne GA, Davis JL (2001) Glacial isostatic adjustment on a rotating Earth. Geophys J Int 147:562–578 Munk WH, MacDonald GJF (1960) The rotation of the earth. Cambridge University Press, London Nakiboglu SM, Lambeck K (1980) Deglaciation effects on the rotation of the Earth. Geophys J Royal astro Soc 62:49–58 Tamisiea ME, Mitrovica JX, Milne GA, Davis JL (2001) Global geoid and sea level changes due to present-day ice mass fluctuations. J Geophys Res 106:30,849–30,863 Sabadini R, Vermeersen BLA (2002) Long-term rotation instabilities of the earth: a reanalysis, ice sheets, sea level and the dynamic earth. Geodyn Ser 29:51–67 Wu P, Peltier WR (1984) Pleistocene deglaciation and the Earth’s rotation: a new analysis. Geophys J R Astr Soc 76:753–791
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Stabilization of Satellite Derived Gravity Field Coefficients by Earth Orientation Parameters and Excitation Functions
65
€rg Kutterer, and Ju €rgen Mu €ller Andrea Heiker, Hansjo
Abstract
The time variable gravity field of the Earth is determined by GRACE and SLR. Different gravity field solutions reveal some discrepancies in the low degree coefficients, especially C20. The second degree gravity field coefficients are directly related to the Earth’s unknown tensor of inertia as well as the mass terms of the excitation functions, which describe the effects of atmosphere and ocean on Earth rotation. A further relationship exists between the Earth orientation parameters (polar motion and length of day), the motion terms of the excitation functions and the tensor of inertia. Up to now these interdependencies are not used for the calculation of the gravity field coefficients. They can therefore be used to validate the various parameter groups mutually. More reliable second degree gravity field coefficients can possibly be obtained if the Earth orientation parameters and the excitation functions are taken into account. This paper presents a novel method to integrate Earth orientation parameters, excitation functions and gravity field coefficients in a least-squares adjustment model with additional condition equations. This leads to consistent time series.
65.1
Introduction
The second degree gravity field coefficients (GFC) are functionally related to the tensor of inertia, which also connects the Earth orientation parameters (EOP) and the excitation functions (see Sect. 65.2). These relations are not taken into account during the calculation of the gravity field coefficients. So we use them
A. Heiker (*) H. Kutterer Geod€atisches Institut, Leibniz Universit€at Hannover, Nienburger Straße 1, 30167 Hannover, Germany e-mail:
[email protected] J. M€uller Institut f€ur Erdmessung, Leibniz Universit€at Hannover, Schneiderberg 50, 30167 Hannover, Germany
to integrate the three different parameter groups into a least-squares adjustment model described in Sect. 65.3. The expected results (Sect. 65.4) are consistent, validated time series for all parameters. The variations of the gravity field, Earth rotation and the excitation functions are caused by various geophysical processes. The displacement of masses affects the gravity field. The motion of masses inside and outside of the Earth affects the Earth rotation in two ways. First, the displacement of masses changes the tensor of inertia. Second, the motion induces changes in the relative angular momentum (Moritz and Mueller 1987). These two effects are described by the mass and motion terms of the excitation functions, which are linearly dependent on variations of the tensor of inertia and the relative angular momentum, respectively.
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Following aspects complicate the integration of the data in the adjustment model: • The distribution and motion of masses are not easily accessible. There exist a few models which calculate variations of the tensor of inertia and the relative angular momentum caused by atmosphere, ocean and hydrology. Some of the models do not provide absolute references but relative values. Hence unknown biases exist which have to be estimated within the least-squares adjustment. • The available time series for the EOP, the GFC and excitation functions differ in temporal resolution. If values with different temporal resolution shall be compared with each other, a common temporal representation is required for each epoch. This is achieved by implementing a filter. • The stochastic model is incomplete or unknown. Hence, we have to find a workaround with empirical variance covariance matrices (VCM) and autoand cross correlation coefficients. Various authors have compared Earth orientation parameters, second degree gravity field coefficients and excitation functions (e.g. Gross 2007; Nastula et al. 2007; Chen and Wilson 2008; Dobslaw and Thomas 2007). To our knowledge consistent time series have not been derived from these comparisons. An earlier publication (Heiker et al. 2008) describes the method of mutual validation of EOP and gravity field coefficients. Meanwhile the functional and stochastic models described in Heiker et al. (2008) were improved. The adjustment model was revised, it includes now a variance component estimation. The exemplary numerical results are obtained by introducing the following data in the adjustment model: • Daily Earth orientation parameters: EOP 05 C04, (Bizouard and Gambis 2007). • Daily excitations functions based on the operational ECMWF fields (atmosphere) and OMCT (ocean model driven by the ECMWF fields); see Thomas (2002), Dobslaw and Thomas (2007) Excitation functions for hydrology are not taken into account here. The neglected hydrology is discussed in Sect. 65.4. • Time variable ITG-Grace03 GSM+GAC gravity field model given by quadratic splines with a nodal point distance of half a month (Mayer-G€urr 2006). The GRACE C20 coefficients are not substituted by corresponding SLR solutions to keep consistency. Note, our goal is the stabilization
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of (somewhat insufficiently determined) gravity field coefficients.
65.2
Geophysical Model
The Earth can be considered as a gyro. The relation between the Earth’s tensor of inertia, the relative angular momentum caused by motions and the Earth’s rotation vector is described by the nonlinear Euler-Liouville equations (Gross 2007; Moritz and Mueller 1987). In the following it is assumed, that the deviations of the tensor of inertia from the principal tensor of inertia (Dij,k for j,k ¼ 1,2,3) and the total relative angular momentum h1, h2 and h3 are solely caused by atmosphere and ocean. Considering the linear approximation of the Euler-Liouville equations (Gross 2007), excitation functions w1, w2 and w3 are defined by i h þ O½1 þ ðk0 2 þD k0 an ÞDi p_ ¼ ; s0 ðC A0 þ A0 m þec Ac Þs0 DL h3 þ O½1 þ a3 ðk0 2 þD k0 an ÞDi33 w3 ¼ ¼ kr : OCm L0 (65.1)
w1 þ iw2 ¼ p þ
The equatorial equations of the Euler-Liouville equations are combined in a complex notation. Table 65.1 denotes the parameters in (65.1). Note that the equatorial equations require the time derivatives of polar motion. The excitation functions can be split in two parts: The mass terms of the excitation functions contain the terms with the deviations of the tensor of inertia and the motion terms contain the relative angular momentum. Gross (2007) lists the numerical values of the geophysical constants in (65.1) describing the Earth’s model. If the origin of the Earth’s reference frame (here ITRF 2005) is identical with the Earth’s center of mass, the second degree gravity field coefficients are connected to the tensor of inertia by (HofmannWellenhof and Moritz 2006) A þ B 2C Di11 þ Di22 2Di33 ¼ 2Ma2 2Ma2 Di13 Di23 C21 ¼ S21 ¼ Ma2 Ma2 B A Di22 Di11 Di12 C22 ¼ S22 ¼ : 4Ma2 4Ma2 2Ma2 (65.2)
C20
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Stabilization of Satellite Derived Gravity Field Coefficients by Earth Orientation
539
Table 65.1 Geophysical parameters decribing the Earth M a A, B, C A0 ¼ AþB 2
Mass of Earth Semi-major axis of Earth Principal moments of inertia Average principal moment of inertia (if a rotation symmetric Earth is assumed) Principal moments of inertia of the Earth’s mantle and core Complex polar motion Changes of length of day Nominal length of day Complex Chandler frequency Mean rotation velocity Complex relative angular momentum Complex deviations of the tensor of inertia Modified degree-2 load Love number due to anelastic mantle Factor modifying degree-2 load Love number due to core decoupling Factor due to the ellipticity of the core surface Factor due to the rotational deformation on lod
A0m , AC, Cm p ¼ xiy DL L0 s0 O h ¼ h1 + ih2 Di ¼ Di13 þ iDi23 0 k20 þ Dkan a3 ec kr
Neither (65.1) nor (65.2) allow the determination of the trace of the tensor of inertia. Thus, the trace is a degree of freedom. Therefore an additional condition equation is required. According to Rochester and Smylie (1974) the trace of the tensor of inertia is invariant for deformations as long as mass conservation is considered. The GRACE GSM+GAC product includes all effects caused by atmospheric and oceanic masses. Hence the assumption of a constant trace seems reasonable. The deviations of the tensor of inertia caused by ocean and atmosphere are small compared to the principal moment of inertia. Therefore one can assume, that the trace is the constant sum of the principal moments of inertia c ¼ A + B + C. This leads to one condition equation for each epoch Di11 þ Di22 þ Di33 ¼ 0:
65.3
(65.3)
Gauss–Markov Model
In order to benefit from the redundant determination of the Earth’s tensor of inertia, a least-squares adjustment is performed based on a Gauss–Markov model (GMM). The GMM consists of the functional and stochastic models described below. The difficulties mentioned in Sect. 65.1 are considered as follows. The functional model is given by the equations in Sect. 65.2. The different temporal resolution of the
data and the condition (65.3) cause a separation of the functional model into an observation part and a restriction part. If a time series with a high temporal resolution is compared with a time series with a low resolution, one has to find comparable representative values. This can be achieved by filtering the time series with the higher temporal resolution or by interpolating the time series with the lower resolution. We choose a cubic splines filter because cubic splines allow the analytic determination of the time derivatives of polar motion. The observed EOP and excitation functions are linearly dependent on the unknown spline coefficients. The observed GFC depend on the tensor of inertia (65.2). Further unknown parameters are the six biases of the excitation functions mentioned in Sect. 65.1. Denoting the observation with 1, the residuals with v and the design matrix with A, the linear observation part of the functional model is described by 2
3 2 3 2 32 3 0 xSplines vEOP AEOP 0 lEOP 4 lw 5 þ 4 vw 5 ¼ 4 Aw 0 0 5 4 xBias 5 : 0 0 A lGFC vGFC xInertia GFC |fflfflfflffl{zfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} ¼:l
¼:v
¼:A
¼:x
(65.4) The restriction part of the functional model is described by (65.1), by the dependency of the mass terms on the tensor of inertia and by the condition
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(65.3). Design matrices of the condition are denoted as B, this leads to 2
B1S 4 B2S 0
B1B B2B 0
32
3
B1I xSplines B2I 54 xBias 5 ¼ Bx ¼ 0: B3I xInertia
AT S1 A B
BT 0
T 1 x A S l : ¼ l 0
0.4 0.2
(65.6)
l denotes the Lagrange correlates introduced through the condition equations.
65.4
0.8 0.6
(65.5)
The results of an adjustment model do not only depend on the functional model but also on the stochastic model S. As the C04 time series provides only the variances of the EOP, a diagonal VCM is chosen for the EOP. The modeled excitation functions are taken as pseudo observations with an empirically derived VCM. For the GFC, a fully occupied VCM is available epoch by epoch. Temporal correlations are not taken into account in any parameter group. Since the three parameter groups originate from different sources they are not correlated with each other. A best invariant quadratic estimation (BIQUE) of the variance components is included in order to determine the weights between the EOP, excitation functions and the GFC (Koch 1999). The variance components are obtained by an iterative algorithm. A paper about the variance component estimation is in preparation. The least squares minimization vTS1v !min considering the restriction Bx ¼ 0 leads to the normal equation system
1.0
Results and Discussion
The partial redundancies shown in Fig. 65.1 measure the reliability of the adjusted observations. Observations with a partial redundancy of ri ¼ 1 are fully redundant. They do not support the parameter estimation. A partial redundancy of ri ¼ 0 implies, that this particular observation is not checked by other observations. The parameter estimation depends completely on this observation. The averaged partial redundancies of the gravity field coefficients C22 and S22 are approximately zero. This can be explained as follows. If a constant trace is
0.0
ERP x
y lod
mass motion χ1 χ2 χ3 χ1 χ2 χ3
GFC C20 C21 S21 C22 S22
Fig. 65.1 Averaged partial redundancies per spline (EOP and excitation function) or per observation (GFC)
assumed, the element Di33 of the tensor of inertia is only dependent on C22 but not on C22. Hence, C22 as well as S22 are not redundantly determined since the tensor elements Di11, Di22 and Di12 are not contained in the linear approximation of the Euler-Liouville equations according to (65.1) or in the excitation functions. The residuals for C22 and S22 are therefore not caused by any redundancy but by the given covariances between C22 and S22 on the one hand and gravity field coefficients on the other hand. In order to obtain consistent gravity field coefficients, the residuals of C22 and S22 have to be considered, but their values are not reliable due to the lack of redundancy. Figure 65.2 shows the amplitude spectra of the residuals of polar motion and lod. The cubic spline filter is a low pass filter. Hence, it is expected that the residuals of the EOP contain monthly and shorter periods. Frequencies with a period lower than a month indicate inconsistencies between the EOP, the excitation functions and the second degree gravity field coefficients. The polar residuals contain various prograde and retrograde frequencies with periods between 13.7 and 34.3 days. Lower frequencies are barely present. The amplitude spectrum for length of day (lod) contains as well long periods (longer than 1 month) as short periods (between 11 and 30 days). An exact description of annual and longer periods is not possible due to the shortness of the time series. The long periods in the lod residuals can be explained as follows: Decadal periods in lod are caused by coremantle coupling (Gross et al. 2005). As long as they are not modeled by excitation functions they remain in the residuals of lod. Annual and semiannual periods in the lod residuals indicate unmodeled components in the excitation functions (Chen 2005). In fact the results shown here suffer from inconsistencies due to
Stabilization of Satellite Derived Gravity Field Coefficients by Earth Orientation
541
Motion Term
0.6
Mass Term
3
prograde
χ1 [10−7]
retrograde 0.4
0 −3 −6
0.2
−0.2
−0.1 0.0 0.1 Frequency [1/day]
0.2
0.3
χ2 [10−7]
3
0.0 −0.3
0 −3 −6
0.04 χ3 [10−9]
Amplitude lod [ms/day]
Amplitude pole [mas]
65
0.03 0.02 0.01 0.00
0.0
0.1
0.2
0.3
10 5 0 −5 −10 −15
observed adjusted 2003
2004
2005
2003
2004
2005
Fig. 65.3 Observed and adjusted excitation functions
Frequency [1/day]
Fig. 65.2 Amplitude spectra of the residuals of polar motion (top) and length of day (bottom)
neglected hydrology. GRACE gravity field coefficients contain the continental water masses, which are not reflected by dedicated hydrological excitation functions here. Hydrological excitation functions will be considered in future work. The observed and adjusted excitation functions are displayed in Fig. 65.3. High frequencies of the observations are averaged out by the adjusted splines. The estimated bias parameters differ significantly from zero. The existence of equatorial bias parameters can be explained by different rotation axes: polar motion, the gravity field coefficients C21 and S21 and the tensor elements Di13 and Di23 correspond to each other. The instantaneous pole given by the IERS time series rotates about a secular pole (McCarthy and Petit 2003). C21and S21 are referenced to the secular pole itself (Mayer-G€urr 2006) The excitation functions are calculated on a sphere with the rotation axis identically to the z-axis (Maik Thomas 2009, personal communication). The bias parameters of the w1 and w2 mass terms result from the difference between the secular pole given by C21/S21 and the z-axis. The obtained numerical values for w1/w2 mass biases are about 85% and 105% of the secular pole values published in McCarthy and Petit (2003. Biases in the w1 and w2 motion terms occur, if the secular pole, about which the instantaneous pole rotates, and the secular pole realized by C21/S21 differ. One reason
for a bias parameter in w3 mass and motions terms might be the violation of the assumption due to the trace of the tensor of inertia. If the invariant trace corresponds to a different constant c 6¼ A + B + C, as assumed in (65.3), bias parameters in w3 mass and motion terms occur. So our approach can only stabilize the variations of the gravity field coefficients but not their absolute values. This results from the fact that the bias parameters contain all constant offsets between the parameter groups. The observed and adjusted gravity field coefficients C20, C21 and S21 are shown in Fig. 65.4. The adjusted C20 have an averaged standard value of 4.0 1011; the averaged standard value of C21 and S21 is 1.6 1011. The values differ slightly. In C20 the adjusted values seem to be smoother than the observed values. If the GMM shall be used to obtain validated gravity field coefficients for further applications, it has to be extended to gravity field coefficients of higher degrees due to existing covariances. The covariances between the second degree gravity field coefficients and those of higher degrees induce changes of the higher degree coefficients.
65.5
Summary and Outlook
In this paper, a method for the mutual validation of the Earth orientation parameters, atmospheric and oceanic excitation functions and second degree gravity field coefficients has been presented. Data with different
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understanding of the quality of the different gravity field solutions and the models for the excitation functions.
C20 [10−4]
−4.841688 −4.841690 −4.841692
Acknowledgement The results presented have been derived within the work on the project “Mutual validation of EOP and gravity field coefficients” within the research unit Earth Rotation and global geodynamic processes funded by the German Research Foundation (DFG FOR584: http://www.erdrotation.de). This is gratefully acknowledged.
−4.841694 −4.841696
C21 [10−10]
1 0
observed adjusted
−1 −2 −3
References
−4
S21 [10−10]
17 16 15 14 13 12
Jan 03
Jan 04
Jan 05
Fig. 65.4 Observed and adjusted gravity field coefficients C20, C21 and S21. The averaged standard deviations of the adjusted GFC are 4.0 1011 for C20 and 1.6 1011 for C21 and S21
temporal resolution, quality and origin have been integrated by means of a least-squares adjustment. Inconsistencies between the different data were modeled by bias parameters. The results of the mutual validation are consistent time series for the Earth orientation parameters, excitation functions, gravity field coefficients and the Earth’s tensor of inertia. These time series contain low frequencies only. The numerical results shown here agree well. Stabilized second degree gravity field coefficients are obtained by taking the EOP and excitation functions into account. Further improvements are planned. The covariances in the time domain not yet considered shall be approximated by synthetic variance-covariance matrices. A better knowledge of the model behind the excitation functions may improve the still incomplete understanding of the bias parameters. In near future, different time series (different GRACE solutions, SLR, other excitation functions) will be tested in different combinations. Hydrological excitation functions will be considered to achieve mass conservation. This work will then lead to a better
Bizouard C, Gambis D (2007) The combined solution C04 for Earth orientation parameters consistent with International Terrestrial Reference Frame 2005. Online February 2010. http://hpiers.obspm.fr/eoppc/eop/eopc04_05/C04_05.guide. pdf Chen J (2005) Global mass balance and the length-of-day variation. J Geophys Res 110: B08404. http: //dx.doi.org/10.1029/ 2004JB003474 Chen JL, Wilson CR (2008) Low degree gravity changes from GRACE, Earth rotation, geophysical models, and satellite laser ranging. J Geophys Res 113:B06402. http://dx.doi.org/ 10.1029/2007JB005397 Dobslaw H, Thomas M (2007) Simulation and observation of global ocean mass anomalies. J Geophys Res 112:C05040. http://dx.doi.org/10.1029/2006JC004035 Gross R (2007) Earth rotation variations – long period. In: Herring T (ed) Treatise on geophysics, vol 3. Elsevier, Amsterdam Gross R, Fukumori I, Menemenlis D (2005) Atmospheric and oceanic excitation of decadal-scale Earth orientation variations. J Geophys Res 110:B09405. http://dx.doi.org/ 10.1029/2004JB003565 Heiker A, Kutterer H, M€uller J (2008) Combined analysis of Earth orientation parameters and gravity field coefficients for mutual validation. In: Observing our changing Earth, vol 133. Springer, Berlin, pp 853–859. http://dx.doi.org/ 10.1007/978-3-540-85426-5_98 Hofmann-Wellenhof B, Moritz H (2006) Physical geodesy, 2nd edn. Springer, Wien Koch KR (1999) Parameter estimation and hypothesis testing in linear models, 2nd edn. Springer, Berlin Mayer-G€urr T (2006) Gravitationsfeldbestimmung aus der Analyse kurzer Bahnb€ogen am Beispiel der Satellitenmissionen CHAMP und GRACE. Ph.D. thesis, Universit€at Bonn. http:// hss.ulb.uni-bonn.de/diss_online/landw_fak/2006/mayerguerr_torsten/0904.pdf McCarthy DD, Petit G (eds) (2003) IERS Conventions. IERS technical notes No. 32. International Earth Rotation and Reference Systems Service (IERS), Verlag BKG, Frankfurt (Main) (2004). http://www.iers.org/MainDisp.csl?pid¼ 46-25776
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Stabilization of Satellite Derived Gravity Field Coefficients by Earth Orientation
Moritz H, Mueller II (1987) Earth rotation. The Ungar Publishing Company, New York, NY Nastula J, Ponte RM, Salstein DA (2007) Comparison of polar motion excitation series derived from GRACE and from analyses of geophysical fluids. Geophys Res Lett, 34, L11306, doi:10.1029/2006GL028983
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Rochester MG, Smylie DE (1974) On changes in the trace of the earth’s inertia tensor. J Geophys Res 79:4948–4951 Thomas M (2002) Ocean induced variations of Earth’s rotation – results from a simultaneous model of global circulation and tides. Ph.D. thesis, University of Hamburg
.
The Statistical Characteristics of Altimetric Sea Level Anomaly Time Series
66
T. Niedzielski and W. Kosek
Abstract
This paper provides a review on statistical properties of sea level fluctuations, both in global and regional scales. Accurate information on up-to-date dynamics of sea level variability can be obtained from satellite altimetry, in particular from TOPEX/Poseidon and Jason-1 missions. A global-scale analysis is based on a single time series, however a regional-scale investigation employs multiple data sets corresponding to dissimilar geographic locations. The statistical characteristics of long-, medium-, and short-term components of sea level fluctuations computed from TOPEX/Poseidon and Jason-1 altimetric measurements are discussed. The new finding of this paper is that a few statistical measures of sea level variability in central equatorial Pacific reveal similar spatial patterns.
66.1
Introduction
Recent environmental hazards include the sea level rise problem which is linked to climate change and global warming. There is a great debate as to the actual role of human impact on the increase in sea level. In order to properly understand sea level change and address the issue of reasons behind its recent rise one needs to handle long time series records. As sea level
T. Niedzielski Space Research Centre, Polish Academy of Sciences, Bartycka 18A, 00-716 Warsaw, Poland Oceanlab, University of Aberdeen, Main Street, Newburgh, Aberdeenshire, AB41 6AA, UK W. Kosek (*) Space Research Centre, Polish Academy of Sciences, Bartycka 18A, 00-716 Warsaw, Poland Environmental Engineering and Land Surveying, University of Agriculture in Krako´w, Balicka 253a, 30-198 Krako´w, Poland e-mail:
[email protected]
measurement history is very short and dates back to 1880s, it is often argued that it is impossible to unequivocally differentiate between a current rate of sea level change and a rate typical for the past. However, even short time series of modern and accurate sea level records produced by tide-gauges and satellite altimetry, if analysed statistically, can partially serve the purpose of prediction and building scenarios for the future. Sea level variability reveals different properties depending on temporal and spatial scales. Temporal variability ranges from short- through medium- to long-term changes. Spatial patterns of sea level change vary from local, through regional to global coverage. Long-term sea level changes are usually expressed by linear trends which probably can be perceived as fragments of very long-period harmonic oscillations. Investigations based on tide-gauge and altimetric measurements provide different estimates of sea level change rates. For instance, Gornitz and Lebedeff (1987) found that global mean sea level rate was
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1.2 0.3 mm/year, Douglas (1991) got 1.8 0.1 mm/ year, Leuliette et al. (2004) obtained 2.8 0.4 mm/ year, and Ablain et al. (2009) got 3.1 0.6 mm/year. In contrast, rates of sea level change at different geographic locations may vary considerably (from less than 20 to more than 20 mm/year) as shown by many researchers (e.g. Kosek 2001; Cazenave et al. 2003; Niedzielski and Kosek 2010). Long-term sea level changes may also comprise an irregular El Nin˜o/Southern Oscillation (ENSO) signal revealing periods from 2 to 7 years. This inter-annual signal is rather weakly represented in global sea level change but can be extremely powerful in certain regions of the Pacific or the Indian Ocean. This may lead to a significant rise in sea level and hence can impact the corresponding trends. It is well known that sea level change in the tropical Pacific during ENSO is driven by the dynamics of the equatorial Kelvin and Rossby waves (Cane 1984). In particular, the thermosteric sea level change recorded in the equatorial Pacific reveals ENSO-driven properties (Cabanes et al. 2001; Cazenave et al. 2003; Antonov et al. 2005; Lombard et al. 2005). Medium-term sea level variability is associated with regular harmonic oscillations exhibiting intraannual periods. Typical oscillations range from 31 to 365 days but the most powerful ones are annual and semiannual terms (Kosek 2001). The magnitudes of such oscillations depend on geographic locations. Thus, they are different for global ocean and for specific ocean areas (Niedzielski and Kosek 2009). Short-term sea level fluctuations are often referred to as residuals or stochastic components. They are usually very irregular as indicated by many authors (e.g. Niedzielski and Kosek 2005; Iz 2006; Niedzielski and Kosek 2009). It is difficult to unequivocally address the issue of stochastic models controlling short-term sea level variations as they differ for various spatial and temporal scales applied. The aim of this paper is to focus on sea level observations obtained by satellite altimetry TOPEX/ Poseidon (Fu et al. 1994) and Jason-1 (Lafon 2005). The time series gained during these projects serve well the purpose of studies aiming at large-scale ocean dynamics investigations (Nerem et al. 1999). To capture dynamic sea level fluctuations, sea level anomaly (SLA) time series are considered. The review on statistical characteristics of sea level change obtained from TOPEX/Poseidon and Jason-1 missions is provided as well as a few new results in this field for
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central equatorial Pacific are incorporated. The new results are obtained by processing the gridded (1 1 ) TOPEX/Poseidon data limited to the rectan gle (70 –180 W 10 S–10 N) spanning the time interval from 10.01.1993 (cycle no. 12) to 01.08.2002 (cycle no. 364). The data are obtained from the Center for Space Research, University of Texas at Austin, USA.
66.2
Trends from Satellite Altimetry
Trends in sea level variations are usually equated with rates of sea level change. The latter are calculated by retrieving slope parameters from linear trend equations. Such formula are determined by fitting straight lines to the time series. The most common fitting techniques are based upon least-squares and robust approaches. A final estimate of a rate of sea level change is controlled by properties of a method and data time spans. Let us recall that the commonly cited estimates based on TOPEX/Poseidon and Jason-1 altimetry are: 2.8 0.4 mm/year (Leuliette et al. 2004), 3.4 0.4 mm/year (Beckley et al. 2007), and 3.1 0.6 mm/year (Ablain et al. 2009). A different rate, approximately 1.5 mm/year, is calculated by Kosek (2001), who uses a robust technique to fit a trend to TOPEX/Poseidon global SLA time series. Recent studies indicate that the global mean sea level rate is reduced now, and the lower rates have been reported from early 2000s (Cazenave et al. 2008; Ablain et al. 2009). The statistically significant trend in sea level rise is noticeable after 4.3 years of TOPEX/Poseidon data collection (Niedzielski and Kosek 2007). Rates of sea level change are different for dissimilar geographic locations. Analyses of TOPEX/Poseidon data suggest that the highest values, about 20 mm/ year, are obtained for instance in western Pacific region, whereas the lowest rates, around -20 mm/ year, are typical for selected mid-oceanic regions, e.g. mid-Indian Ocean and mid-Pacific Ocean (Kosek 2001; Cazenave et al. 2003). Rates of sea level change in ENSO-vulnerable east equatorial Pacific region are quite low or even declining (Niedzielski 2010). Figure 66.1a shows rates of sea level change for central and east equatorial Pacific. The mean trend estimates for 10 S–10 N and 70 W–180 W are juxtaposed below in Table 66.1, whereas for 10 S–10 N and 70 W–110 W are shown in Table 66.2. The local spatial minimum rate of sea level change is less than 6 mm/year and is
1
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Fig. 66.1 Maps showing sea level change statistics: rate in cm per year (a), standard deviation in cm (b), skewness (c), and kurtosis (d)
located slightly south of the Equator between east and central Pacific. In the study area, rates become positive far west, namely at 160 –180 W (Fig. 66.1a). According to Table 66.1, mean rates of sea level change are minimum at locations very near to the Equator (5 S–5 N) and tend to be lower in the South ern Hemisphere (5 S–0 ) than in the Northern Hemi sphere (0 –5 N).
66.3
547
For site-specific approaches, the power of the above-mentioned oscillations is different depending on location. Mean amplitudes of 183-, 120-, 62-, 37-, and 30-day components of SLA time series depend on geographic locations. For example, enormous magnitudes are typical for annual and semiannual terms during El Nin˜o events in the east equatorial Pacific. In what follows, the study by Niedzielski and Kosek (2009) shows that both components are responsible for up to 34 cm contribution to the sea level rise during El Nin˜o 1997/1998 event. In addition, Barbosa et al. (2006) investigate seasonal modes of Atlantic Ocean sea level variability and link the results for tropical zone to the trade winds regime.
1.5 200
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The Statistical Characteristics of Altimetric Sea Level Anomaly Time Series 10
66
Oscillations from Satellite Altimetry
The time series obtained from TOPEX/Poseidon and Jason-1 altimetry can be applied to detect harmonic oscillations in sea level variations. Indeed, there is a wide range of periodic signals present within these data. For global SLA time series, the most powerful oscillations are annual and semiannual as shown by Niedzielski and Kosek (2005).
66.4
Stochastic Fluctuations from Satellite Altimetry
In order to consider irregular sea level variability, one needs to subtract trends and seasonal oscillations from original SLA data and determine residuals. In the equatorial Pacific, they comprise many information on fluctuations driven by ENSO. Irregular stochastic variations in global SLA residuals are investigated by Niedzielski and Kosek (2005) and are shown to follow autoregressive models. Such models, however, do not necessarily guarantee an improvement of predictions of global sea level residuals (Niedzielski and Kosek 2009). Irregular variability of SLA residuals depends on geographic location. Indeed, statistical characteristics of underlying probability distributions vary between regions. The analysis for central and east equatorial Pacific (70 –180 W) is presented below and is latter compared with the results for east equatorial Pacific (70 –110 W). In order to evaluate statistical properties of SLA residuals around a deterministic signal (gained by removing linear trends as well as harmonic terms, here annual and semiannual oscillations), a few momentbased measures are employed, i.e. standard deviation, skewness and kurtosis. Similar approach, based on slightly different SLA residuals, was used by Niedzielski (2010) and Niedzielski and Kosek (2010). However, unlike the aforementioned papers, the single-satellite TOPEX/Poseidon data are processed in this paper. Standard deviations around a deterministic signal of sea level change in equatorial Pacific increase westward from South America towards mid-ocean regions
548
(Fig. 66.1b). Skewness calculated for residuals around a deterministic signal decreases westward towards central Pacific and attains zero for longitudes between 140 and 180 W (Fig. 66.1c). The pattern of zeroskewness contour in Fig. 66.1c spatially coincides with the area where standard deviation gradient plummets. This is also spatially correlated with areas revealing no trend in sea level change (Fig. 66.1a). In addition, kurtosis also declines from South America towards west and reaches 3 roughly at the same Pacific areas where skewness yields zero (Fig. 66.1d). It is difficult to draw a parallel between the abovementioned spatial agreement and ENSO spatial extent in the Pacific. However, a few statistical measures clearly show that in central equatorial Pacific: (1) the huge SLA variability plummets (standard deviation analysis), (2) the probability distribution changes from non-Gaussian to normal probability law (skewness and kurtosis analysis). The similar results can be found for the combined TOPEX/Poseidon and Jason-1 residuals determined by removing not only annual and semiannual terms but also the alias-type 62-days oscillation (Niedzielski 2010; Niedzielski and Kosek 2010). In order to give a comprehensive insight into the spatial variability of statistical parameters of sea level change in the study area, mean values of various statistics are computed for specific sections of the Pacific Ocean. The following near-equatorial zones are selected: 5 S–5 N, 10 S–10 N, and 10 S–6 S combined with 6 N–10 N. These zones are subdivided into south and north sections in respect to the Equator. Mean standard deviations of sea level change residuals are the highest near the Equator (5 S–5 N) (Table 66.1). They significantly decline towards south and north of the Equator, however it is tough to find a difference in rates between those sections. The similar implication holds for mean skewness which is the highest (exceeds 1) in the vicinity of the Equator and decreases towards 10 S and 10 N parallels (Table 66.1). For mean kurtosis, however, no meaningful decline towards those parallels can be noticed. Indeed, the greatest values are again placed in the narrow section between 5 S and 5 N parallels but the decline of mean kurtosis with latitude can only be noticed for the Southern Hemisphere. In contrast, in the Northern Hemisphere mean kurtosis increases slightly towards 10 N parallel. The similar exercise is performed for east equatorial Pacific, hence for the rectangle 10 S–10 N
T. Niedzielski and W. Kosek
and 70 W–110 W. The corresponding results are shown in Table 66.2. It is apparent from this table that SLA data in the vicinity of the Equator are highly non-Gaussian. Both skewness and kurtosis indicate departures from the normal distribution for two zones under study, i.e. (1) 5 S–5 N and (2) 10 S–6 S combined with 6 N–10 N. However, the closer distance to the Equator, the higher non-Gaussian behaviour of SLA time series exists. It is also shown that the departure from the normal distribution for east equatorial Pacific is greater in the south of the Equator than in the north of the Equator (Table 66.2). The comparison of the results for 70 –180 W and 70 –110 W shows that mean standard deviations of SLA residuals are only slightly smaller for the first Table 66.1 Mean statistics (R-rate; SD-standard deviation; S-skewness; K-kurtosis) of sea level variation for equatorial Pacific (70 –180 W) Region Central 5 S–5 N 10 S–10 N 10 –6 (S+N) South 5 S–0 10 S–0 10 S–6 S North 0 –5 N 0 –10 N 6N –10 N
R cm/year
SD cm
S
K
0.40 0.32 0.23
7.06 6.04 4.93
1.02 0.81 0.58
5.42 5.31 5.19
0.42 0.33 0.23
7.20 6.08 4.77
1.05 0.81 0.53
5.55 5.25 4.90
0.38 0.31 0.22
7.13 6.20 5.06
1.00 0.83 0.63
5.32 5.39 5.49
Table 66.2 Mean statistics (R-rate; SD-standard deviation; Sskewness; K-kurtosis) of sea level variation for equatorial east Pacific (70 –110 W) Region Central 5 S–5 N 10 S–10 N 10 –6 (S+N) South 5 S–0 10 S–0 10 S–6 S North 0 –5 N 0 –10 N 6N –10 N
R cm/year
SD cm
S
K
0.29 0.26 0.22
7.42 6.56 5.63
2.10 1.95 1.79
7.90 7.30 6.65
0.28 0.23 0.17
7.51 6.39 5.12
2.18 2.01 1.83
8.26 7.36 6.36
0.30 0.29 0.28
7.46 6.86 6.14
2.01 1.89 1.74
7.56 7.28 6.94
66
The Statistical Characteristics of Altimetric Sea Level Anomaly Time Series
study area than for the second region. More significant differences are found for mean skewness and mean kurtosis. Indeed, mean skewness SLA residuals at east and central equatorial Pacific is lower by at least 1 than the corresponding statistics for east Pacific. In addition, mean kurtosis is also lower for 70 –180 W than for 70 –110 W. Geophysical interpretations of departures from the normal distribution of SLA residuals in the tropical Pacific were given by Niedzielski and Kosek (2010) and Niedzielski (2010). Conclusions
The paper gathers the selected results aiming at a recognition of statistical properties of sea level anomaly data obtained by TOPEX/Poseidon and Jason-1 satellite altimetry. A few published results focusing on both global and site-specific spatial scales are discussed. Different temporal domains are in the scope of this paper, including long-, medium, and short-term components of sea level variability. The particular emphasis is put on central and east equato rial Pacific between 10 S and 10 N. It is found that ENSO signal ceases in a specific zone of central Pacific determined by decreasing standard deviation gradient as well as switching to the normal distribution for sea level variation residuals. Acknowledgements The research was financed by Polish Ministry of Science and Higher Education through the grant no. N N526 160136 under leadership of Dr Tomasz Niedzielski at the Space Research Centre of Polish Academy of Sciences. The first author is also supported by EU EuroSITES project. The authors thank the Center for Space Research, University of Texas at Austin, USA for TOPEX/Poseidon and Jason-1 time series. The authors of R 2.9.0 - A Language and Environment and additional packages are acknowledged.
References Ablain M, Cazenave A, Valladeau G, Guinehut S (2009) A new assessment of the error budget of global mean sea level rate estimated by satellite altimetry over 1993–2008. Ocean Sci 5:193–201 Antonov JI, Levitus S, Boyer TP (2005) Thermosteric sea level rise, 1955–2003. Geophys Res Lett 32:L12602. doi:10.1029/ 2005GL023112 Barbosa SM, Silva ME, Fernandes MJ (2006) Multivariate autoregressive modelling of sea level time series from TOPEX/Poseidon satellite altimetry. Nonlin Process Geophys 13:177–184
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Beckley BD, Lemoine FG, Luthcke SB, Ray RD, Zelensky NP (2007) A reassessment of global rise and regional mean sea level trends from TOPEX/Poseidon and Jason-1 altimetry based on revised reference frame orbits. Geophys Res Lett 34:L14608. doi:10.1029/2007GL030002 Cabanes C, Cazenave A, Le Provost C (2001) Sea level rise during past 40 years determined from satellite and in situ observations. Science 294:840–842 Cane MA (1984) Modeling sea level during El Nin˜o. J Phys Oceanogr 14:1864–1874 Cazenave A, Cabanes C, Dominh K, Gennero MC, Le Provost C (2003) Present day sea level change: observations and causes. Space Sci Rev 108:131–144 Cazenave A, Lombard A, Llovel W (2008) Present-day sea level rise: a synthesis. C R Geoscience 340:761–770 Douglas BC (1991) Global sea level rise. J Geophys Res 96:6981–6992 Fu L-L, Christensen EJ, Yamarone CA Jr, Lefebvre M, Me´nard Y, Dorrer M, Escudier P (1994) TOPEX/POSEIDON mission overview. J Geophys Res 99(C12):24369–24381 Gornitz V, Lebedeff S (1987) Global sea-level changes during the past century. In: Nummedal et al. (ed) Sea-level fluctuation and coastal evolution. Soc Econ Paleontol Miner Spec Publ 41:3–16 Iz HB (2006) How do unmodeled systematic mean sea level variations affect long-term sea level trend estimates from tide gauge data? J Geod 80:40–46 Kosek W (2001) Long-term and short period global sea level changes from TOPEX/Poseidon altimetry. Artif Satel 36:71–84 Lafon T (2005) JASON 1: lessons learned from the development and 1 year in orbit. Acta Astron 56:45–49 Leuliette EW, Nerem RS, Mitchum GT (2004) Calibration of TOPEX/Poseidon and Jason Altimeter data to construct a continuous record of mean sea level change. Mar Geod 27:79–94 Lombard A, Cazenave A, Le Traon P-Y, Ishii M (2005) Contribution of thermal expansion to present-day sea-level change revisited. Global Planet Change 47:1–16 Nerem RS, Chambers DP, Leuliette EW, Mitchum GT, Giese BS (1999) Variations in global mean sea level associated with the 1997–1998 ENSO event: implications for measuring long term sea level change. Geophys Res Lett 26:3005–3008 Niedzielski T, Kosek W (2005) Multivariate stochastic prediction of the global mean sea level anomalies based on TOPEX/Poseidon satellite altimetry. Artif Satel 40:185–198 Niedzielski T, Kosek W (2007) A required data span to detect sea level rise. In: Weintrit A (ed) Advances in marine navigation and safety of sea transportation. Gdynia Maritime University, Poland, pp 367–371 Niedzielski T, Kosek W (2009) Forecasting sea level anomalies from TOPEX/Poseidon and Jason-1 satellite altimetry. J Geod 83:469–476 Niedzielski T (2010) Non-linear sea level variations in the eastern tropical Pacific. Artif Satel 45:1–10 Niedzielski T, Kosek W (2010) El Nino’s impact on the probability distribution of sea level anomaly fields. Pol J Environ Stud 19:611–620
.
Testing Past Sea Level Reconstruction Methodology (1958–2006)
67
J. Viarre and R. Abarca-del-Rı´o
Abstract
Understanding present-day global sea level rise requires a correct evaluation of past sea level field variability. We use sea level height fields obtained by satellite altimeters between 1992 and 2006, sea level height fields from recent reanalyses of oceanic circulation (SODA) and worldwide tide gauges series for the time interval 1958–2006, to investigate the limitations inherent in reconstructing the past ~50 years of sea level variation using empirical orthogonal function (EOF) decomposition. To understand some of the weaknesses we found, we tested the influence of the spatial distribution of tide gauges as well as the ability to properly reconstruct sea level in certain frequencies bands. The presence of the particularly strong 1997–1998 El Nin˜o event, and the short time span of the base period (1992–2006), limits the determination of other spatial teleconnections and then the reconstruction over preceding epochs. More particularly, during the presatellite era and outside the tropics, the non-stationary characteristics of heat transport at interannual time scales and other low frequencies oscillations associated with sea level height fields undermine the methodology.
67.1
Introduction
Numerous motivations exist to determine precisely the rate of sea level rise. This includes assessing its impact on coastal vulnerability, and investigating its connection to global warming. Ocean covers about 2/3 of the earth surface, and thus participates actively in climate variability. The prominent role of the ocean in the global exchange of energy (Trenberth and Fasullo 2010), along with availability
J. Viarre R. Abarca-del-Rı´o (*) Departamento de Geofı´sica (DGEO), Universidad de Concepcio´n, Concepcio´n, Chile e-mail:
[email protected]
of different and complementary modern observation network and satellite mission data (see Milne et al. 2009 or Cazenave and Llovel 2010 for recent reviews on the subject), led to an increase of interest on sea level variability as exhibited by the number of recent studies (Church et al. 2004, 2006; Jevrejeva et al. 2006; Woodworth et al. 2004; Holgate 2007; Beckley et al. 2007; Wunsch et al. 2007; Berge-Nguyen et al. 2008; Domingues et al. 2008; Woodworth et al. 2009; Leuliette and Miller 2009; Milne et al. 2009; W€oppelmann et al. 2009; Cazenave and Llovel 2010). Resolving the various contributions to sea level rise is a complex problem. However, it is well documented that thermal expansion of the ocean and freshwater inflow from continents and glaciers explains the largest part of sea level variability at
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_67, # Springer-Verlag Berlin Heidelberg 2012
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J. Viarre and R. Abarca-del-Rı´o
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secular time scales (see Cazenave and Llovel 2010). Altimeter satellite mission fields provide evidence of sea level variation, well defined spatially but available only since 1992. Nevertheless, to quantify precisely the effects of each phenomenon at decadal to secular time scales, longer time spans of sea level observations are necessary. These are available lightly since the last quarter of nineteenth century (Woodworth and Player 2003) using a sparse network of tide gauge records. However, they are affected by land movement (GIA), which requires a rigorous correction (Peltier 2001). To overcome the problem of the heterogeneous distribution of in situ records, global reconstruction’s based in the temporal and spatial auto covariance properties of the sea level fields has been implemented (Church et al. 2004 (CW2004 in the following), Berge-Nguyen et al. 2008 [BN2008 in the following]). This consists roughly in projecting tide gauge data into spatial teleconnections maps obtained from satellite altimeter fields of sea level heights. The analysis of globally averaged tide gauges measurements results in a global average trend over the recent century of 1.8 0.4 mm/year from 1870 to 2004 (Church et al. 2004). It is almost similar to the one estimated by Holgate (2007) over 1904–2003 (1.74 0.16 mm/year) or by W€ oppelmann et al. (2009) (1.61 0.19 mm/year). The latter is weaker than the one obtained directly from the satellite altimeter fields over 1993–2007 (3.1 0.4 mm/year, Beckley et al. 2007). In the presence of significant low frequency variability, trend estimates tend to oscillate (Jevrejeva et al. 2006 [J2006 in the following]), so long-term rate determination remains a challenge. Certainly, given some of the differences found (Caccamise et al. 2005) construction of a well-defined and correct global reference frame (Bevis et al. 2002) for determination of vertical station velocities and its correction is crucial (W€ oppelmann et al. 2009). To quantify and hopefully reduce the uncertainties about accelerations in sea level rise, it is necessary to understand the dynamics of the processes controlling sea level fluctuations and the limitations inherent in our approach to data analysis. Our primary focus here is to test our methodology. However, although the problems linked to the tide gauges are well documented (Woodworth and Player 2003), the problems related to statistical methods based on the singular value decomposition are not yet well
established. For example, we try to reconstruct past variability based in the spatial teleconnections obtained over the recent decade or so (14 years), which are then projected onto the tide gauge information available over longer intervals of time. How much of the past (temporal and spatial) variability can we really reconstruct? In other words, what are the limits of the reconstruction methods we are employing? Is it possible to derive others, or perfect those already in use? As a result, we will test the main methodology used for reconstructing the global mean sea level variability. First, test the spatial efficiency with various tide gauges repartitions. Second, the temporal structure, by reconstructing separately different frequency bands of the spatial sea level variations.
67.2
Processing Data
Sea surface height can mainly be extracted from three distinct sources, which all present some limitations for the interpretation of long term sea level variability. Satellite altimeter data set presents an accurate spatial definition but covers only the period 1992 to present. Tide gauge measurements provide a better temporal coverage (almost since 1870) but are sparsely distributed in time and space and require major corrections (adjustment due to the post glacial rebound and atmospheric pressure forcing). Finally, ocean circulation model runs supply for a correct spatial coverage and temporal definition. However, as ocean circulation interacts with varied climate forcing, and is subject to diverse sources of water inflow (ice caps, ice sheets, continental storage), the knowledge of its varying 3-D temporal structure is still complex. Recent reconstruction of sea surface height mainly employed satellite altimeter and/or tide gauge data set (CW2004, J2006, BN2008) but also happens to take in account the spatial structure of long term ocean model runs (LLovel et al. 2009). When reconstruction through singular value decomposition is used, the spatial relations of sea level variation (the Empirical Orthogonal Teleconnections [EOT]) are computed from satellite altimeter fields with statistical modes. The temporal variability is obtained through the respective principal components (PC’s). Assuming spatial stationary for reconstructing past variability, the EOT are projected on the tide gauges series for obtaining the respective PC’s, which are then used for
67
Testing Past Sea Level Reconstruction Methodology (1958–2006)
reconstructing the time evolution of the whole global sea level. In our case we will use three complementary data sets: (a) sea level fields as computed by the Simple Ocean Data Assimilation (SODA) reanalysis model (1958–2006) (Carton and Giese 2008) to test the different reconstructions on longer time scales; (b) Sea level anomalies computed by the Topex/Poseidon and Jason-1 satellite altimeter (1993–2006); (c) a subset of the tide gauge network (1900–2006). These are described more precisely in the following section.
67.3
Data
67.3.1 Simple Ocean Data Assimilation For our simulations, we used the 49-year-long time series (1958–2006) of monthly Sea Surface Height (SSH) extracted from a 0.5 0.5 grid (75.250 S to 89.250 N, 0.250 to 0.250 W) of the Simple Ocean Data Assimilation (Carton and Giese 2008). We employed only SSH data in the latitude band 70 S–70 N, avoiding overestimation in the heat transport intensity of the Antarctic Circumpolar Current at Drake Passage (Carton and Giese 2008). A climatology was computed over 1992–2006 to allow the use of Sea Level Anomalies (SLA). Because our interest is only located in large scale sea level variations, these data were adjusted in a 1 1 grid and annually averaged. As explained previously, the SODA reanalysis data are used to simulate the sea level information available from T/P and J-1 altimeter and tide gauge records. Two sets of data are thus extracted: – A global set over the period 1992–2006 traducing satellite altimeter information. – A sparse network of coastal time series over 1958–2006 traducing tide gauge records. Various configurations are implemented to test the influence of tide gauge distribution in the method efficiency and to evaluate the quality of the reconstruction if records all over the coast and islands could be known: (C1). Coastal time series regularly distributed covering the entire Southern Hemispheric coast (1 1 grid) (C2). Coastal time series regularly distributed covering the entire Northern Hemispheric coast (1 1 grid)
553
Fig. 67.1 Map of the spatial distribution of tide gauges (256 stations) and their linear trends (red circles for positive significant trends, red solid circles for positive non-significant trends, blue circles for negative significant trends and blue solid circles for negative non-significant trends)
(C3). Coastal time series regularly distributed covering the entire coasts (1 1 grid) (C4). 256 coastal time series irregularly distributed but traducing the configuration of real tide gauge network used (Fig. 67.1).
67.3.2 Satellite Altimeter Dataset As a reference to the sea level variability, we used the corrected Topex/Poseidon and Jason-1 satellite altimeter data set available from 1993 to 2008. Merged Sea Surface Anomalies (MSSA), initially extracted from a 0.5 0.5 grid (0–360 E, 70 S to +70 N) were adjusted on a 1 1 grid and annually averaged. The altimeter products were produced by Ssalto/Duacs and distributed by Aviso (personal communication).
67.3.3 In Situ Tide Gauges Data We employed the 1,169 Revised Local Reference (RLR) time series of annual average (over 1900–2006) from the Permanent Service for Mean Sea Level (PSMSL, Woodworth and Player 2003). Following recent studies (CW2004, J2006, BN2008), various corrections were carried out, reducing the number of tide gauges records used to 256. We first suppressed the records which don’t present 10 years of data (continuous or not, 444 time series), while the rest, when there were gaps spanning less than 2 years, were interpolated (spline). These records were then spatially adjusted into the same 1 1 grid of TP and J-1 altimeter data set. If various records were available
J. Viarre and R. Abarca-del-Rı´o
554
for the same location, the corresponding time series were completed, after verifying their correlation. One station was suppressed since its latitude was not included in the spatial domain of the altimeter measurements (70 S to +70 N). The final statistical corrections consisted of suppressing records were 10% of gauged amplitudes exceeded the “mean 2 standard deviation” value of each time series, and records that exhibited an excessive level of variance. Additionally, we corrected the resulted estimations from air pressure forcing and land motion. Effects of the Post Glacial rebound were corrected using the predictions suggested by Peltier (2001). Concerning the air pressure forcing, we applied an inverted barometer correction using the Met Office Hadley Centre’s mean sea level pressure (MSLP) data set (HadSLP2), available on a 5 latitude–longitude grid from 1850 to 2004 (Allan and Ansell 2006). To complete this data set to cover the period 1850–2006, we used the monthly SLP produced by the NCEP/NCAR reanalysis project (1948–present), interpolated over the same grid. The differences between the two data sets over high latitudes of the Southern Hemisphere for the period 1950–1970 (Ponte 2006) has been avoided here since we consider only the NCEP/NCAR data from 2004 to 2006. Nevertheless, before computing the inverted barometer adjustment, the variance of the NCEP/NCAR data was adjusted at each location to respect the one calculated from the HadSLP2 data set. Finally, we subtracted for each time series, the respective annual mean computed with all estimations. Figure 67.1 exhibits linear trends from each instrumental record. The non-parametric Mann-Kendall statistic test ensures the significance of the computed trends.
67.4
Methodology
A global reconstruction using a Simplified Reduced Space Optimal Interpolation (SRSOI) is applied in this research, as explained in the following.
67.4.1 Reconstruction Approach The reconstruction based on the Reduced Space Optimal Interpolation (RSOI) aim to extract, from the modern global data, the spatial teleconnections of
sea level variability expressed in Empirical Orthogonal Teleconnections (EOT). These spatial interactions are expressed with statistical modes, where each mode explains a part of sea level variability. Considering this spatial structure stationary, we find, for each mode and each time step, the amplitude of these EOT’s, using a sparse network of tide gauge data. The reconstructed field is obtained by summing all these modes. We briefly detail here the reconstruction method applied (for more details, see the reconstruction of sea level pressure carried out by Kaplan et al. 2000).
67.4.2 Initial Process The variations of mean sea level at secular time scales determine the largest signal variance and prevent an accurate statistical determination and understanding of annual to multi-decadal fluctuations. To avoid this predominance, we removed the global sea level trends (estimation by least square in each point grid) computed from 1993 to 2006 before computing the EOT. Its determination is described in Sect. 67.4.5.
67.4.3 Determination of Spatial Interactions Using Singular Value Decomposition Let us consider the modern global data and the sparse network of in situ data expressed by the data matrix RAW and M respectively. As explained above, spatial connections are extracted from modern global data using a Singular Value Decomposition, which determines singular values and singular vectors of the matrix RAW: RAW ¼ USV t
(67.1)
where each column of the orthonormal vector U expresses a mode of spatial variation, while the corresponding row of the orthonormal vector V expresses the time series of the amplitude of the mode. S is a diagonal matrix containing the singular values of the data matrix D. A major concern of this method is to determine the number of modes necessary for the reconstruction, while avoiding the use of higher-order modes explaining lower part of variance dominated by
67
Testing Past Sea Level Reconstruction Methodology (1958–2006)
noise. In our research, we will arbitrarily determine the number of modes computed by selecting the percentage of variance expressed. We called Empirical Orthogonal Teleconnections (EOT), the rescaled vector UKSK, containing the K first modes which explain the variance elected.
67.4.4 Principal Components To complete the reconstruction, we need to identify for each time step the principal component alpha of the modes (called the reduced space optimal interpolation solution), and compute the reconstructed field as the projection of these PCs on the K-dimensional space of leading EOT’s: RECðx; y; tÞ ¼ EOTðx; yÞ:aðtÞ
(67.2)
Following the method introduced by Kaplan et al. (2000), this is possible by minimizing, for each mode and each year, the cost function: SðaÞ¼ðH:EOT:aMÞt :R1 :ðH:EOT:aMÞþat :L1 :a (67.3) where H is a sub matrix of the identity matrix, which terms equal to 1 at the location and the time step where in situ data is available; M is the matrix of in situ data; lambda is the diagonal matrix containing the singular values of the covariance matrix computed from the modern global data RAW; R is the data error covariance matrix given by the sum of the instrumental error and the error enclosed in the exclusion of higher-order modes in the reconstruction: R ¼ Rins þ K:EOT 0 :L0 :EOT 0 :K t t
(67.4)
where EOT’s are the Empirical Orthogonal Teleconnections omitted during the reconstruction. One simplification induced by Smith and Reynolds (2004) for reconstructing sea surface temperature, consists in excluding the second term in (67.3) (simplification done in our study) and omitting instrumental as methodological errors considering the data error covariance matrix R as the identity matrix. In our case, we took in account also these error estimations, by evaluating the instrumental error of tide gauges records as described in Sect. 67.4.6. The final cost function becomes:
555
SðaÞ ¼ ðH:EOT:a MÞt :R1 :ðH:EOT:a MÞ (67.5)
67.4.5 Processing Global Sea Level Trend Church et al. (2006) evaluated the non uniform increase in the mean sea level (initially removed from the altimeter field) by adding a spatial field constant in height called EOF0 and determining its amplitude resolving the cost function. This technique (applied in our study) is similar to regress by least squares the temporal evolution of global sea level expressed in the PC0 directly to the tide gauge records.
67.4.6 Estimation of Instrumental Error Rins The accuracy of satellite altimeters observations allows using it as the reference of the global sea level field. A way to estimate the error enclosed in the in situ estimations is to compare the variance of tide gauges time series spanning 1993–2006 with those computed with the closest grid point of the altimeter field. This comparison leads to the determination of an average ratio of 1.021 between the altimeter and in situ variance. Therefore, we define the data matrix Rins as the diagonal matrix containing a constant covariance error r ¼ 1.021 mm.
67.5
Simulation
In this section, after justifying the use of SODA data set to simulate reconstruction’s possibilities, we develop briefly the various methods employed.
67.5.1 Justifying the Use of SODA Dataset to Simulate Method’s Possibilities SODA 1.4.2 reanalysis data provide a simulation of spatial and temporal variations of the steric and mass component of the sea level rise over 1958–2006 (Carton and Giese 2008). The similar properties (patterns and amplitudes of sea level variance) observed in the last improved version of the SODA
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reanalysis project and the global satellite altimeter observations ensure the reliability of our approach (Carton and Giese 2008). Moreover, improvements in the reanalysis efficiency lead to an average correlation of 0.7 between SODA coastal time series and a set of 20 annual sea level tide gauges (lightly weaker to the one calculated with satellite altimeter time series’ records). These characteristics pointed out, it appears well-founded to take advantage of a global field over a larger period. With this in mind, both the spatial teleconnections (over 1993–2006) and principal components (over 1958–2006) will be computed using a global field and a sparse network of coastal records extracted from the last version of SODA reanalysis. Results will be interpreted in terms of method’s efficiency.
67.5.2 SIM1. Testing Influence of an Hemispheric Repartition of Tide Gauge Records Tide gauge repartition clearly connotes a hemispheric dependence, leading to a weak spatial coverage of South Hemisphere amplified by a diminution of variance of southern records (Fig. 67.1). We test here the effect of this non-homogeneity, in terms of reconstructed variability. Four reconstructions are thus completed, respectively using the four virtual tide gauges configurations presented in section Data – (a). Reconstructions using the configurations C1, C2, C3 and C4 are then compared by plotting the spatial correlation maps between the original and reconstructed global field over 1958–1992 and 1993–2006 respectively. We choose here to show 1958–1992 rather than the whole period (1958–2006) as it is done usually. Using the same data set to simulate in situ records and altimeter field strongly overestimates the method’s efficiency over 1993–2006 and therefore will mask whatever the drawbacks of the methodology over precedent epochs. Our aim here is precisely to investigate these.
67.5.3 SIM2. Filtering Data Using a Discrete Wavelet Transform The phenomena responsible for sea level variability occur at different spatial and temporal scales, which
complicate seriously their separation in distinct statistical modes. Dommenget and Latif (2002), suggested a careful interpretation of these modes of variability, which could correspond to a superposition of physical phenomenon or create virtual patterns of sea level variability. Here, we used a Discrete Wavelet Transform (DW) (Craigmile and Percival 2002) to filter the global field (1993–2006) as well as coastal time series (1958–2006) in frequency bands and thus consider only phenomenon developing over the same frequency bands. The initial global field and coastal time series are lengthened to avoid the creation of virtual signal at both sides of time series. We employed the classic discrete Morlet’s mother wavelet considering that the use of an advanced wavelet leads to the extraction of similar patterns (Kumar and Foufoula-Georgiou 1997). First details D1, D2, reveal the high frequency of sea level variation (~2–8-year periods), D3 supplies decadal times (~8–13-year periods) and A3 provides the other low frequencies (interdecadal oscillation) as well as the secular component (trends). After filtering, global and coastal time series are reconstructed over 1958–2006 and 1992–2006 respectively, and each detail component (D1, D2, D3 and A3) represents a global but frequency delimited sea level field variation. Final reconstruction is obtained by summing all the details’ reconstructed fields. For similar reasons as presented in the previous section, we concentrate our interpretation into the 1958–1992 time-interval.
67.6
Results
67.6.1 SIM1 For each configuration (C1, C2, C3, C4) of virtual in situ records, we show in Fig. 67.2 the resulting spatial correlation averaged over latitude bands (1958–1992 Fig. 67.2 left, and 1993–2006 Fig. 67.2 right) respectively. The mean correlation is later represented over 5 latitude bands to allow a better visualization. Figure 67.2 clearly indicates a strong spatial dependence in term of percentage of reconstructed variability over 1958–1992, pointing out an efficient reconstruction only in the Equatorial zone (10 S–10 N, maximum correlation of 0.658 for the latitude 2 S) and revealing the inability to reproduce the initial strong variance situated in the convergence zones of oceanic streams (40 S–50 S, 40 N–50 N). According to the spatial
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Testing Past Sea Level Reconstruction Methodology (1958–2006)
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60 40 20 0 –20 –40 –60 0
0.1
0.2
0.3
0.4
0.5
0.6 0.7
0.8
0.9
1
Fig. 67.2 Averaged spatial correlation in latitude bands over 1993–2006 (to the right, cyan) and 1958–1992 (to the left) using tide gauges configurations C1 (blue), C2 (green), C3 (red), C4 (grey)
coverage of the reconstructed field (70 S–70 N) and the period of the EOT computation (1993–2006), clearly the large scale variability of the 1997/98 El Nin˜o event severely influenced the determination of spatial interactions (Mode 1–4, not shown, 43.2% of variance). This explains the excellent reconstruction of the Equatorial zone in comparison to higher latitude bands.
67.6.2 SIM2 Large scale sea level variability in the major basin’s results from the modulation of the principal climatic oscillations occurring at inter-annual to decadal time scales. For example, Chambers et al. (1999) drew attention to the fact that the largest part of the Indian Ocean variability at inter-annual to decadal time scales was determined by the El Nin˜o Southern Oscillation. While Marcos and Tsimplis (2007), revisiting the different climatic influences in the European mean sea level, pointed out the largest role of the North Atlantic Oscillation and pressure effects in the determination of the North Atlantic steric component as well as water mass variations. In fact different time scales dominates sea level variability (Milne et al. 2009; Cazenave and Llovel 2010), whatever its climatic source (Woodworth et al. 2009) owing the necessity to separate them well, to investigate their influence on the reconstruction method. To separate each of these climatic influences
Fig. 67.3 Correlation maps 1993–2006 (up) and 1958–1992 (bottom) for (a) high frequency (mode D1, 2–3 years periods), (b) inter-annual (mode D2, 3–8 years periods), (c) decadal (mode D3, 8–13 years periods) and secular variability (mode A3)
on a global scale, or the different frequencies participating in, we filtered (see Sect. 67.5.3) all the time series and sea level heights fields to supply a frequency delimited reconstruction. For each one of the frequencies delimited voices, the correlation between the filtered reconstructed and initial field (over 1958–1992 and 1993–2006 respectively) are shown in Fig. 67.3. These estimations of spatial correlations are then averaged over 50 latitude bands (Fig. 67.4). The main result of this analysis can be resumed as follows. 1. Reconstruction of the high frequency of the interannual variability (voice D1, 2–3 years periods) appears to be clearly inefficient over 1958–1992, excepting the Equatorial Pacific basin 2. Reconstruction of lower frequency of inter-annual variability (voice D2, 3–8 years periods) presents higher spatial correlations for the Indian basin and Northern latitude (North Atlantic over 500 N) and a correlation mean of 0.1 elsewhere (although accurate reconstruction of the Pacific Equatorial basin)
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558 80
D1 A3 D3 D2
D3 D2 A3 D1
60 40
20 0 –20 –40 –60 –80 –0.1
1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Fig. 67.4 Averaged zonal correlation in latitude bands over 1993–2006 (right) and 1958–1992 (left) for the different modes of the discrete wavelet filtering
3. Reconstruction of decadal variability (voice D3, 8–13 years periods) connotes deficiencies over the Eastern Equatorial Pacific basin over 1993–2006 4. Reconstruction of secular variability is better pronounced, leading to an efficient reconstruction of Northern–Eastern Atlantic (over 400 N) and Southern variability (correlation of 0.4 over 600 S) but exhibits heavy loss of variability over the Indian and North Pacific oceanic basins
67.7
Summary and Conclusions
Recent reconstruction of sea surface height employing satellite altimeters and/or tide gauge data (CW2004, J2006, BN2008) as well as ocean model runs (LLovel et al. 2009) allowed for interpreting the increase of sea level height over recent decades and the past century. We investigate the limitation for the principal component method when reconstructing past sea level fields assuming a short base period. For example the use the altimetry sea level fields over the recent decade (1993–2006), as a base field, with the help over longer time scales of the distribution of tide gauges over lands, to reconstruct sea level fields over that past five decades or more. This was tested by using the sea level height over 1958–2006 from the oceanic reanalyzes of SODA (Carton and Giese 2008) that we corroborate over the overlapping times with the altimetric
fields (1993–2006). The analysis showed that the reconstruction over 1958–1992 (the one that does not overlap with the altimetry time span) is not spatially correct, reproducing essentially the variability over the equatorial band. To understand the origin of this disparity, we first band pass filtered the tide gauges, sea level height fields from altimetry and oceanic reanalyzes over the same frequency bands (interannual, decadal and other low frequencies), before computing the reconstruction. It’s not surprising given the length (14 years) of the “base” 1993–2006 field that secular (and other low frequencies in) and decadal components explains part of the disparity. Interestingly the interannual time scales presents also lacks of reconstruction. It’s the case particularly for the high frequency part (periods 2–3 years), excepting within the equatorial band. Instead, the lower part (periods 3–8 years) although not being particularly more efficient over extra tropical bands present better compatibility’s over some regions within the northern hemisphere. As interannual time’s scales teleconnections are in principle resolved by the time span of the base period. The reason for this disparity can be partly associated with mass and heat transport from the equator to extra tropics, and vice versa, that are not being taken into account efficiently by the methodology. This is most probably associated with non stationary characteristics of this transport (Boucharel et al. 2009). We also tested the influence of the spatial distribution of tide gauges (rather than the number) on the quality of the spatial reconstruction. Only the equatorial variability (off the coasts), whatever the distribution performed here, is correctly reproduced. Results do not appear to be sensitive to the distribution of tide gauge sites, outside those over the equator (20 S–20 N). Another final justification for this assumption; if we compare two reconstruction’s computed over 1993–2006, one with only coastal records located in the equatorial band (20 S–20 N) with another complete (70 S–70 N, not shown), similar correlations are found for all zonal bands. Definitely, in this case, supplying more geographical information than the equatorial band appears useless. As a conclusion, apart from the determination of the instrumental error (which leads to minor uncertainties), a deficiency in the reconstruction method could mainly be due to the lack (distribution) of information, or due to the stationary character
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Testing Past Sea Level Reconstruction Methodology (1958–2006)
of the empirical Orthogonal functions method. For this case, we should also add first the time span (roughly 14 years) of the base fields from which the teleconnections are extracted that limits the determination of teleconnections with longer time scales. Secondly the presence of a particularly strong climate signal clearly limits the method capabilities; the very strong 1997–1998 El Nin˜o event, which propagates first along the equatorial and tropical pacific controls the large scale variability of world oceans and world’s interannual climate, associated to its non stationary characteristics, severely influences the determination of spatial interactions by the PCs and then the past “global” sea level field reconstruction. Acknowledgements We thank B. Lagos from the Statistics Department of the University of Concepcio´n, for his very useful comments. Helpful suggestions by M. Bevis improved the manuscript. The merged altimeter sea level fields were obtained from AVISO (CLS-CNES, Toulouse, France). Tide gauge data came from the PSMSL (NOC-NERC, Liverpool, UK). Research presented here was part of the third year engineer’s internship (ENSEEITH, Toulouse-France) of Mr. J. Viarre within our department.
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Precise Determination of Relative Mean Sea Level Trends at Tide Gauges in the Adriatic
68
M. Repanic´ and T. Basˇic´
Abstract
Relative mean sea level (MSL) trends for nine tide gauges in the Adriatic have been determined with high precision using common inter-annual sea level variability. Annual MSL values are free from seasonal changes and short-periodic tidal effects. Moreover, one can assume that inter-annual sea level changes, driven by climate variations and long-periodic tidal effects, are very similar for close sites, especially in enclosed seas such as the Adriatic. Accordingly, common interannual variation of sea level can be determined for the tide gauges in question from annual means, under assumption that residual variations at each site are close to random. Through common adjustment of nine tide gauge time series of annual means from PSMSL common inter-annual variability as well as relative MSL trends and MSL for each tide gauge have been determined. For 50-year period relative trends have been determined with standard deviations from 0.1 to 0.3 mm/year (later refers to tide gauges with shorter records). Values of common inter-annual variability are determined with standard deviations of less than 4 mm.
68.1
Introduction
Today, a rise of mean sea level is frequently in focus of public attention. Unlike satellite altimetry that provides absolute sea level variations (measured relative to a reference ellipsoid), tide gauges data provide sea level variations relative to a specific point on the Earth’s crust and consequently include the effects of vertical land movements. In spite of absolute
M. Repanic´ (*) T. Basˇic´ Croatian Geodetic Institute, Savska c. 41/XVI, p.p.19, 10144 Zagreb, Croatia e-mail:
[email protected]
character, global coverage and high precision of altimetry data, tide gauges are still crucial in measuring sea level and determining its variability. While satellite altimetry data is available for nearly two decades, some tide gauges records cover the time span of more than a century. Moreover, a network of high quality tide gauges is used to remove satellite bias and drift from altimetry data (Pugh 2004). If collocated with continuous GPS stations, vertical land movements can be removed from tide gauge data and absolute sea level trends can be determined. However, reliable measurements of trends from GPS may take several decades (Pugh 2004). This paper deals with precise determination of relative mean sea level (MSL) trends in enclosed sea
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_68, # Springer-Verlag Berlin Heidelberg 2012
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from tide gauge records. Annual and inter-annual sea level changes, which are driven by climate variations, make it difficult to calculate precisely the sea level trends. Averaging over a year removes seasonal changes. However, systematic effects from changes of meteorological parameters (e.g. steric effect, air pressure effects) and associated ocean circulation that last for longer than a year are maintained, as well as long-periodic tidal effects (e.g. nodal tide with 18.6 year period, Chandler wobble). Thus, in order to precisely determine a sea level trend, it is necessary to remove periodic and irregular variations from sea level records. One can assume that such inter-annual changes are very similar for sites in small enclosed seas and residual variations for each site are close to random. Accordingly, a mathematical model that distinguishes between inter-annual variations of sea level common for all sites in some small enclosed sea and linear trend effects specific for each site can be defined. Sea level trends determined in such a way are precise, but relative. Beside absolute rise of sea level, such sea level trends include effects of regional and local vertical land movements.
68.2
Data
For the purpose of this study annual MSL values from Permanent Service for Mean Sea Level (PSMSL) have been used. Nine tide gauges in Adriatic with significantly long records have been selected. These are Venezia – Punta Della Salute and Trieste in Italy; Koper in Slovenia; Rovinj, Bakar, Split – Rt Marjana, Split – Harbour and Dubrovnik in Croatia and Bar in Montenegro (Fig. 68.1). Respective annual MSL values are plotted in Fig. 68.2, while information on records is summarised in Table 68.1. Since the distance between the two farthest tide gauges in question is about 600 km and all sites are in the small enclosed sea, one can assume that inter-annual sea level changes, driven by climate variations and longperiodic tidal effects, are highly coherent. High correlation between annual MSL values is apparent from Fig. 68.2. If calculated directly from each MSL record independently according to the least squares (LSQ) principle, relative MSL trends can be obtained with standard deviations from 0.3 up to 0.6 mm/year (Table 68.2).
Fig. 68.1 Tide gauges involved in the study
68.3
Mathematical Model
In order to determine values of common inter-annual variability for each year and relative MSL trends for each tide gauge, all the data have been adjusted simultaneously according to well known Gauss–Markov model and LSQ principle. Trend line at each tide gauge has been modelled by two unknown parameters: TLtTG ¼ aTG þ bTG ðt t0 Þ;
(68.1)
where TLtTG is a trend influence at the tide gauge TG at the time t (in years), t0 reference time (set to middle year 1981), and aTG and bTG are unknown trend line parameters. Accordingly, measurement equitation for annual MSL at the tide gauge TG for the year t is: MSLtTG þ vtTG
¼ TLtTG þ V t ¼ aTG þ bTG ðt t0 Þ þ V t
(68.2)
where vtTG is respective measurement residual and V t inter-annual variability for the year t common to all tide gauges. Thus, for nine tide gauges and time interval from the year 1956 to 2006 there are 69 unknown parameters (2 trend line parameters for each tide
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Precise Determination of Relative Mean Sea Level Trends at Tide Gauges in the Adriatic
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Table 68.1 Tide gauges data overview Tide gauge
PSMSL code Venezia – Punta Della 270/054 Salute Trieste 270/061 Koper 279/002 Rovinj 280/006 Bakar 280/011 Split – Rt Marjana 280/021 Split – Harbour 280/031 Dubrovnik 280/081 Bar 281/011
Time span No. of annual MSLs 1956–2000 43 1956–2006 1962–1991 1956–2006 1956–2006 1956–2006 1956–2006 1956–2006 1965–1990
51 26 50 51 49 51 49 26
Table 68.2 Relative MSL trends from each record Tide gauge Venezia – Punta Della Salute Trieste Koper Rovinj Bakar Split – Rt Marjana Split – Harbour Dubrovnik Bar
Fig. 68.2 Annual MSL at tide gauges
gauge and common inter-annual variability value for 51 years). The model (68.2) has also been used by Buble et al. (2010). The parameter aTG corresponds to an offset of each tide gauge’s RLR datum, as defined by PSMSL (URL1), from common inter-annual variability, while bTG can be seen as a scale factor. Since there is no reference offset and scale defined, there is a datum defect in the model, that, if compared with free network adjustment (Feil 1990), accounts for translation and scale freedom of movement. Consequently, in
MSL trend (mm/year) 0.93 0.87 0.22 0.49 0.81 0.63 0.44 0.96 1.26
St. dev. (mm/year) 0.37 0.27 0.58 0.27 0.32 0.30 0.29 0.28 0.63
the course of adjustment pseudo inversion has been applied. In the absence of the data on the basis of which the quality of the records and annual MSL values in particular could be quantified, all the annual MSL values are presumed to be of equal accuracy. Described model is valid under assumption that residual interannual variations at each site are close to random.
68.4
Results
The applied mathematical model resulted in significant decrease in standard deviations of estimated relative MSL trend values (Table 68.3) as compared to independently estimated trends (Table 68.2). Specifically, standard deviations of estimated trends at all tide gauges amount 0.1 mm/year, except for Koper and Bar, which have only 26 annual MSLs available (Table 68.1). Moreover, relative trend values estimated in this way correspond to the same time
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564 Table 68.3 Relative MSL trends from common adjustment Tide gauge Venezia – Punta Della Salute Trieste Koper Rovinj Bakar Split – Rt Marjana Split – Harbour Dubrovnik Bar
MSL trend (mm/year) 1.29 0.87 1.04 0.52 0.81 0.53 0.44 0.85 2.67
St. dev. (mm/year) 0.12 0.09 0.23 0.09 0.09 0.10 0.09 0.10 0.27
Fig. 68.3 Common inter-annual variability 3 standard deviations
interval, although not all the tide gauge records cover the entire time span. The values of common inter-annual variability are determined with standard deviations of less than 4 mm (Fig. 68.3). In order to validate the assumption on randomness of residual MSL variations, Lilliefors normality test (Lilliefors 1967) at the 5% significance level has been applied on the residuals as a whole and the residuals from each tide gauge in particular. Accordingly, the residuals from Split – Rt Marjana and Split – Harbour tide gauges, as well as residuals as a whole are not normally distributed. However, unlike two other cases, the residuals from Split – Rt Marjana tide gauge pass the test if slightly lover significance level is applied (4.8%). Moreover, non-normal distribution of the residuals from Split – Harbour tide gauge probably causes non-normality of residuals as a whole. Consequently, one can presume that residual inter-annual variations at each site are close to random, with the
exception of Split – Harbour tide gauge. In addition, the t-test at the 5% significance level has been applied, in order to determine whether the groups of the residuals have a mean of zero. Accordingly, the residuals as a whole and the residuals from each tide gauge in particular have a zero mean. Histograms of the residuals for respective tide gauge are plotted in Fig. 68.4.
68.5
Discussion
The applied model presumes vertical land movements linear in time and residual inter-annual variations close to random at each site. Non-normality of the residuals from Split – Harbour tide gauge can be caused by non-compliance with either of two postulations. In addition, non-normality can be caused by tide gauge’s instrumental errors (e.g. datum shifts). The fact that Split – Harbour tide gauge is only 4 km apart from tide gauge Split – Rt Marjana, whose residuals are close to random can be significant. Further study is necessary in order to ascertain the cause of non-normality. However, it probably reflects specificities of the site rather than inadequacy of the method. Although the applied mathematical model resulted in relative MSL trends of high precision, one have to consider that MSL values for consecutive years are physically correlated. Consequently, because such correlation is not comprised by the stochastic model, derived standard deviations of relative trends and common inter-annual variability are probably too favourable. Though, the decrease in standard deviations of relative MSL trend values estimated in common adjustment (Table 68.3) as compared to independently estimated trends (Table 68.2, probably also too optimistic for the same reason) emphasizes the significance of the applied model. The applied mathematical model yields relative trend values for the period over which the common inter-annual variability has been determined (i.e. 1956–2006), even though not all the tide gauge records cover the entire time span. One has to keep in mind that relative trends are not necessary constant in time. Consequently, relative (and absolute) MSL trends depend on time interval over which they are determined.
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Precise Determination of Relative Mean Sea Level Trends at Tide Gauges in the Adriatic
Fig. 68.4 Histograms of residuals for respective tide gauges
A number of authors have determined relative MSL trends and related vertical land movements in the Adriatic. Orlic´ and Pasaric´ (2000) have analysed
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relative trends at four Croatian tide gauges over a 30year sliding window after correcting the MSL values for air-pressure anomalies. They found that the middle and south Adriatic coast rises relatively to the north Adriatic cost at a 1 mm/year speed, while Bakar area is characterized by anomalous crustal movement. Woodworth (2003) applied “buddy checking” on 11 MSL time series in the northern Mediterranean, including nine tide gauges involved in this study. He determined MSL trends relative to Trieste tide gauge. Though, the trends determined by “buddy checking” refer to different time intervals. Buble et al. (2010) used observations from tide gauges and collocated continuous GPS stations to investigate crustal deformation and sea level changes at six form nine tide gauges involved in this study. The results of this study (Table 68.3) do not confirm pattern of vertical land movement found by Orlic´ and Pasaric´ (2000). Though, if the trend value at Trieste tide gauge is subtracted, the trends are in a considerable agreement with Woodworth’s (2003) relative trends for records with matching time intervals. In addition, relative relation among the trends determined in this study is in the agreement with one determined by Buble et al. (2010). Namely, relative MSL trends determined in this study indicate that there is no significant vertical land movement of Split area, as compared to Rovinj area. Split and Rovinj are probably rising as compared to other tide gauge sites. Moreover, there is no significant difference in relative MSL trends at Trieste, Koper, Bakar and Dubrovnik tide gauges, which are about 0.4 mm/year higher than at Rovinj and Split tide gauges. Appreciably higher relative MSL trend is determined at Bar tide gauge, which is in the area of high seismic activity. In addition, relative MSL trend at Venice tide gauge is higher to some extent, which is in agreement with previously reported subsidence due to abstraction of groundwater until 1970s (Woodworth 2003; Pugh 2004). The change in vertical land motion about the year 1970 at Venice tide gauge (Woodworth 2003; Pugh 2004) probably resulted in fairly higher value of standard deviation of estimated trend (Table 68.3), as compared to other tide gauges with about 50 annual means used (Table 68.1). Estimated common inter annual variability (Fig. 68.3) should account for different systematic effects common to all sites such as effects from changes of meteorological parameters, associated ocean circulation, long-periodic tidal effects as well as certain anthropogenic influences. Such influences in
M. Repanic´ and T. Basˇic´
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the Adriatic have been described by a number of authors (e.g. Orlic´ and Pasaric´ 2000; Vilibic´ et al. 2005). However, the interpretation of the derived common inter-annual variability requires further study and is beyond the scope of this paper. Conclusion
The adjustment model enables the estimation of relative MSL trends with up to three times smaller standard deviations as compared to independently estimated trends, without the aid of models for sea level variability, such as ocean circulation models, steric effect models or models for tidal effects. Moreover, all estimated trends are related to a common time interval, even though not all the tide gauge records cover the same time span. Full significance of the model can be reached when estimated trends shall be associated with measurements of vertical land movements from collocated continuous GPS stations. From such measurements at one or more tide gauges, the scale factor for adjustment model (discussed in Chap. 3) can be defined. Consequently, absolute MSL trends for all tide gauges can be estimated. However, in order to reliably determine vertical
land movement and the scale factor, sufficient time interval of continuous GPS measurements is needed.
References Buble G, Bennett RA, Hreinsdo´ttir S (2010) Tide gauge and GPS measurements of crustal motion and sea level rise along the eastern margin of Adria. J Geophys Res 115: B02404 Feil L (1990) Theory of errors and least squares adjustment – part two (in Croatian). Faculty of Geodesy, University of Zagreb, Zagreb Lilliefors HW (1967) On the Kolmogorov-Smirnov test for normality with mean and variance unknown. J Am Stat Assoc 62:399–402 Orlic´ M, Pasaric´ M (2000) Sea level changes and crustal movements recorded along the east Adriatic coast. Nuovo cimento della societa` italiana di fisica 23(4):341–364 Pugh D (2004) Changing sea levels: effects of tides, weather and climate. Cambridge University Press, Cambridge URL1: http://www.pol.ac.uk/psmsl/datainfo/psmsl.hel Vilibic´ I, Orlic´ M, Cˇupic´ S, Domijan N, Leder N, Mihanovic´ H, Pasaric´ M, Pasaric´ Z, Srdelic´ M, Strinic´ G (2005) A new approach to sea level observations in Croatia. Geofizika 22 (1):21–57 Woodworth PL (2003) Some comments on the long sea level records from the northern Mediterranean. J Coast Res 19 (1):212–217
Quantile Analysis of Relative Sea-Level at the Hornbæk and Gedser Tide Gauges
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S.M. Barbosa and K.S. Madsen
Abstract
The quantification of the long-term variability of relative sea-level is a fundamental problem in geodesy. In the present study, quantile regression is applied for characterising the long-term variability in relative sea-level at the Gedser and Hornbæk tide gauges, in the North Sea–Baltic Sea transition zone. Quantile regression allows to quantify not only the rate of change in mean sea-level but also in extreme heights, providing a more complete description of long-term variability. At Gedser the lowest relative heights are increasing at a rate approximately 40% higher than the mean rate, while at Hornbæk the relative sea-level slopes are stable across most of the quantiles. A 30-year running window analysis shows that the linear trends display considerable decadal variability over the twentieth century for both stations.
69.1
Introduction
The variability of mean sea-level is a fundamental problem in geodesy and also a key issue in global and regional climate change. Of particular interest is the quantification of low-frequency variability from records of sea-level observations, but the task is far from trivial (e.g. Caccamise et al. 2005; Barbosa et al. 2008).
S.M. Barbosa (*) University of Lisbon, IDL, Campo Grande, Edifı´cio C8, 1749-016 Lisboa, Portugal e-mail:
[email protected] K.S. Madsen Danish Meteorological Institute, COI, Lyngbyvej 100, 2100 Copenhagen OE, Denmark National Environmental Research Institute, Aarhus University, Aarhus, Denmark
Long-term variability of mean sea-level is usually described by linear slopes, derived by the fitting of a linear model to tide gauge records of sea-level heights. Such linear trends derived by ordinary least squares reflect the temporal evolution of the mean. However, of equal or even greater interest is the temporal evolution of features of the data distribution other than the mean, such as the minima and maxima. These can be particularly important for navigation and protection of coastal areas. While the classical linear regression approach is only able to provide information on the rate of change of the mean of the data distribution, quantile regression (Koenker and Basset 1978) allows to determine the rate of change at a given quantile of the data distribution, thus providing a more complete and robust description of long-term variability. In this work, quantile regression is applied for characterising the long-term variability in relative sea-level at the Gedser and Hornbæk tide gauges, in
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the North Sea–Baltic Sea transition zone. Barbosa (2008) applied quantile regression for examining the long-term variability of relative sea-level in the Baltic area and concluded that the rate of change of the upper quantiles is significantly higher than the rate of change of the mean sea-level. However, these results were based on monthly tide gauge data, while atmospheric events and strong storms anticipated to affect extreme sea-levels occur on sub-monthly scales, typically of a few days (Stigebrandt 1984). Furthermore, the quantile trends derived for most of the stations in the Baltic area reflected both sea-level change and land uplift due to glacial isostatic adjustment. Thus in the present study the objective is to perform quantile analysis on hourly (rather than monthly) tide gauge records and to focus on the North Sea–Baltic Sea transition area, farther from the uplift center of Fennoscandia. A further objective of this work, taking advantage of the length of the considered tide gauge records, is to assess the stability in time of the derived trends. The tide gauge data and the quantile regression approach are briefly described in Sect. 69.2. The quantile analysis and stability assessment results are presented in Sect. 69.3, with concluding remarks given in Sect. 69.4.
69.2
Data and Methods
processing of the data has been performed, i.e. tides and the seasonal cycle have not been removed. The rationale is that the tides in the area are very small (e.g. Feistel et al. 2008) and the interest is on trends in relative sea-level rather than in relative sea-level anomalies, since it is the long-term variability in relative sea-level, including seasonality, that is of practical interest, for example for coastal populations.
69.2.2 Quantile Regression Quantile regression was introduced in the seminal work of Koenker and Basset (1978) and has been extensively described, mainly in econometric contexts (Koenker and d’Orey 1994; Koenker and Hallock 2001; Koenker 2005) and a few ecological applications (e.g. Cade and Noon 2003). A brief overview of the main ideas of the method is given here. The reader is referred for further details to the original references, and in particular the overview in the monograph of Koenker (2005). Quantile regression is a unified statistical methodology for estimating models of conditional quantile functions. Given a random variable Y of measurements for some population, the quantile is defined to be the value QY(t) satisfying PfY QY ðtÞg ¼ t
69.2.1 Tide Gauge Data
0t1
(69.1)
The analysed tide gauge records are the hourly values of sea-level observations from two Danish stations in the North Sea–Baltic Sea transition area, Gedser and Hornbæk (Fig. 69.1), for the period from January 1891 to December 2005 (Hansen 2007). The corresponding time series plots are shown in Fig. 69.2. No further 57° North Sea Hornbæk
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54° 10°
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Fig. 69.1 Study area and tide gauge locations
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Fig. 69.2 Tide gauge records of hourly relative sea-level heights for Gedser (top) and Hornbæk (bottom) and sea-level trends at quantiles 0.05 (dotted line), 0.5 (solid line) and 0.95 (dashed line)
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Quantile Analysis of Relative Sea-Level at the Hornbæk and Gedser Tide Gauges
whereas the quantile function is defined as QY ðtÞ ¼ F1 Y ð tÞ
(69.2)
with FY(y) being the cumulative distribution function of the population. Considering the conditional distribution of Y given X ¼ x, the conditional quantile function QY|X(t;x) satisfies P Y QYjX ðt; xÞjX ¼ x ¼ t
(69.3)
While classical ordinary least squares regression is based on the conditional mean function, the mean of the response variable Y conditional on x, denoted by E [Y|X ¼ x], quantile regression is based on the conditional quantile function (69.3). Thus, whereas classical regression is based on the minimisation of the residuals X
ðyi E½YjX ¼ xi Þ2
(69.4)
Quantile regression is based on the minimisation of the sum of asymmetrically weighted absolute residuals X
rðtÞ yi QYjX ðt; x ¼ xi Þ
(69.5)
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(median) and 95% of the sorted sea-level observations are represented in Fig. 69.2 and displayed in Table 69.1. For comparison, the slope derived using the usual ordinary least squares (OLS) approach is also included in Table 69.1. Both classical and quantile regression give different relative sea-level rates for the two tide gauge stations, around 1 mm/year for Gedser and 0.25 mm/ year for Hornbæk. The median slope (quantile 0.5) is similar to the mean-based (OLS) slope, indicating that the distribution is close to symmetric and that the ordinary least squares slope is in this case a robust measure of the rate of change of the centre of the data distribution. The lower quantile slope for Gedser is considerably higher than the median and mean slopes, while for Hornbæk there is consistency between the slopes derived for the different quantiles. For a more complete description of long-term variability in relative sea-level heights at Gedser and
Table 69.1 Quantile slopes and corresponding standard errors t ¼ 0.05 t ¼ 0.5 t ¼ 0.95 OLS
Gedser 1.37 (0.020) 1.00 (0.0074) 1.13 (0.018) 1.09 (0.0072)
Hornbæk 0.25 (0.016) 0.25 (0.0072) 0.29 (0.023) 0.27 (0.0071)
where t is the tilted absolute value function (Koenker and Hallock 2001). In this work, quantile regression is performed using the R-language implementation (package quantreg).
69.3
Results
69.3.1 Quantile Trends Quantile slopes are computed from each tide gauge record using a modified version of the Barrodale and Roberts algorithm (Koenker and d’Orey 1994). Standard errors for the quantile slopes are obtained by computing a Huber sandwich estimate (Huber 1967) using a local estimate of the sparsity (Koenker and Machado 1999). Note that since sample variation increases away from the centre of the distribution, the standard errors are larger for the more extreme quantiles. For illustration, the slopes at quantiles 0.05, 0.5 and 0.95, corresponding respectively to the lowest 5, 50
Fig. 69.3 Quantile slopes (black points) and corresponding standard errors (vertical error bars) for Gedser (top) and Hornbæk (bottom). The horizontal solid line indicates the ordinary least squares slope (same for all quantiles)
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Hornbæk from quantile regression, the quantile slopes are computed for quantiles 0.02–0.96 (in steps of 0.02) and represented along with corresponding standard errors in Fig. 69.3. The diagrams show a marked distinction between the two locations: while for Gedser the lower quantiles display a slope considerably higher than for the middle of the data distribution, at Hornbæk the quantile slopes are much closer to the usual ordinary least squares slope and more similar across the data distribution.
S.M. Barbosa and K.S. Madsen
69.4
Concluding Remarks
In order to assess the temporal stability of the derived relative sea-level slopes, the analysis is repeated in 30-year running windows (with a 10-year overlap). The temporal evolution of the slopes at quantiles 0.05, 0.5 and 0.95 as well as of the OLS slopes is shown in Figs. 69.4 and 69.5 for Gedser and Hornbæk, respectively. For both stations there is a very good agreement between the OLS and median slopes over the whole period. Furthermore, the median slope displays considerable decadal variability at both stations, with lower values occurring in the 1930s and 1960s. The lower and upper quantile slopes display even stronger decadal variability, particularly at Hornbæk (Fig. 69.5).
In this work quantile regression is applied for quantifying the long-term variability of relative sealevel. The computation of quantile trends allows to extract more information from the tide gauge records and to describe long-term variability not only in terms of the rate of change of the mean but also in terms of the rate of change of other features of the data distribution, including extreme heights. The results show a very good agreement between the usual ordinary least squares slope and the slope for quantile 0.5 (median), indicating that the data distribution is symmetric and the estimates are robust. Since these OLS and median slopes have been derived independently, the similarity on the limiting case of Gaussian symmetric data gives confidence on the good performance of the quantile regression implementation. The quantile regression results show that at Gedser the lowest relative heights are increasing at a rate approximately 40% higher than the mean rate, while a similar behaviour is not observed for the highest relative levels. At Hornbæk the relative sea-level slopes are stable across most of the quantiles and closer to the ordinary least squares slope. At both stations the linear trends for the different quantiles show considerable decadal variability over the 20th century, with lower rates of change of relative sea-level heights in the 1930s and 1960s. These
Fig. 69.4 Relative sea-level slopes computed over 30-year running windows for the Gedser record. Upper plot: OLS slope (black) and median slope (grey). Middle plot: quantile t ¼ 0.05. Lower plot: quantile t ¼ 0.95
Fig. 69.5 Relative sea-level slopes computed over 30-year running windows for the Hornbæk record. Upper plot: OLS slope (black) and median slope (grey). Middle plot: quantile t ¼ 0.05. Lower plot: quantile t ¼ 0.95.
69.3.2 Temporal Stability
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features are likely related to variability in North Atlantic atmospheric circulation and regional wind patterns, and deserve further investigation. An important caveat concerns the estimation of realistic error bars taking into account the autocorrelation in the data (e.g. Lee and Lund 2004). In this work the standard errors were estimated using the standard approach for independent data and therefore are very likely to be optimistic. Despite this caveat, this work shows the advantages of applying quantile regression for the quantification of long-term variability in relative sea-level, and the approach is expected to be very useful for the analysis of other geodetic time series such as GPS records. Acknowledgements Thanks are due to R. Koenker for providing the R-package quantreg, to the R development core team for the R software, and to P. Wessel and W.H.F. Smith for the GMT software. This work is supported by FCT (contract under Programme Ciencia 2008).
References Barbosa SM (2008) Quantile trends in Baltic sea-level. Geophys Res Lett 35:L22704 Barbosa SM, Silva ME, Fernandes MJ (2008) Time series analysis of sea-level records: characterising long-term variability. Lect Notes Earth Sci 112:157–173
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Caccamise DJ II, Merrifield MA, Bevis M, Foster J, Firing YL, Schenewerk MS, Taylor FW, Thomas DA (2005) Geophys Res Lett 32:L03607 Cade B, Noon B (2003) Gentle introduction to quantile regression for ecologists. Front Ecol Environ 1:412–420 Feistel R, Nausch G, Wasmund N (2008) State and evolution of the Baltic Sea, 1952–2005: a detailed 50-year survey of meteorology and climate, physics, chemistry, biology, and marine environment. Wiley, Hoboken, NJ Hansen L (2007) Hourly values of sea level observations from two stations in Denmark. Hornbæk 1890–2005 and Gedser 1891-2005. DMI Technical Report, No07-09 Huber PJ (1967) The behavior of maximum likelihood estimates under nonstandard conditions, in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, pp 221–223 Koenker R (2005) Quantile regression. Cambridge University Press, New York Koenker R, Basset G (1978) Regression quantiles. Econometrica 46:33–50 Koenker RW, d’Orey V (1994) Computing regression quantiles. Appl Stat 43:410–414 Koenker R, Hallock K (2001) Quantile regression. J Econ Perspect 15:143–156 Koenker R, Machado AF (1999) Goodness of fit and related inference processes for quantile regression. J Am Stat Assoc 94:1296–1310 Lee J, Lund R (2004) Revisiting simple linear regression with autocorrelated errors. Biometrika 91:240–45 Stigebrandt A (1984) Analysis of an 89-year-long sea-level record from the Kattegat with special reference to the barotropically driven water exchange between the Baltic and the sea. Tellus 36A:401–408
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Assessment of the FES2004 Derived OTL Model in the West of France and Preliminary Results About Impacts of Tropospheric Models
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F. Fund, L. Morel, and A. Mocquet
Abstract
Ocean Tide Loading displacements in the West of France are among the largest in the world. In order to contribute to the improvement of tidal models for GPS processing, we estimate ocean tides in this area with static GPS. Observations recorded over at least 1.5 years are used from ~60 stations with typical separation distances of 50–80 km. Previous studies showed the consistency of the FES2004 model of OTL displacements with GPS estimates of the main semi-diurnal tides in the North of Brittany. We extend this study over a larger spatial scale and investigate the ten main tides. The largest differences between GPS estimates and modeled values are obtained for the M2, K1, and K2 tides. We attribute the discrepancies of K1 and K2 tides to multipath effects. Preliminary results about the impact of using current annual empirical tropospheric models instead of 6 h-period data derived from ECMWF data and provided by TU Vienna are presented.
70.1
Overview
The West of France is a region where ocean tides are of the largest in the world. The tidal range due to the M2 wave can reach 4 m and the sum of all the mean waves leads to tidal ranges in excess of 11 m.
F. Fund (*) L. Morel Laboratoire de Ge´omatique et de Ge´ode´sie (L2G), ESGT/ CNAM, 1 Boulevard Pythagore, 72000, Le Mans, France e-mail:
[email protected] A. Mocquet Laboratoire de Plane´tologie et Ge´odynamique, Universite´ de Nantes, Nantes Atlantique Universite´, CNRS, UMR 6112, UFR des Sciences et des Techniques, 2 rue de la Houssinie`re, BP 92208, Nantes Cedex 3 44322, France
The impact of such tides on ground displacements amounts to several centimeters in the height component (Fig. 70.1). Melachroinos et al. (2007) and Vergnolle et al. (2008) presented the results of a 12 GPS station campaign conducted in 2004 with a 100-day dataset. They concluded that (1) static GPS measurements are able to detect OTL displacements as well as shallow-water tides caused by the complexity of the French coast along the English Channel, (2) the quality of the FES2004 model (Lyard et al. 2006) leads to only 1–3 mm biases with GPS results on the height component, (3) a 100-day long dataset is not sufficient to discriminate between all tidal periods, and (4) the observations of partial tides like K1 and K2, whose periods are very closed to integer multiples of the GPS orbital period, are corrupted by daily multipath effects.
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Fig. 70.1 M2 wave displacements according to the FES2004 model in mm
Here, we present (1) preliminary results of a new local assessment of the FES2004 model over a larger area and with longer times series of at least 1.5 year duration and (2) the detection of high frequency signals that had not been taken into account in previous studies when estimating OTL displacements using GPS. In this regard, we present preliminary results about the impact of using annual empirical tropospheric models (GPT, Boehm et al. 2007 and GMF, Boehm et al. 2006a) instead of 6-h period data derived from ECMWF (Simmons and Gibson 2000) data provided by TU Vienna (ZHD and VMF1, Boehm et al. 2006b). A way to detect multipath effects is suggested in the conclusion.
70.2
GPS Processing
Two kind of networks are available (bottom of Fig. 70.2) (1) a network where PCV are corrected with absolute values derived from relative ones and limited to a 10 cutoff angle (black triangles), and (2) a network where absolute corrections are directly available and observations are limited down to a 5 cutoff angle (white diamonds). They are respectively called ELEV and AZEL networks. In both cases, GPS observations are processed by overlapping 2 h sessions every hour and the networks are stabilized in the ITRF2005 with common regional IGS-core stations (top of Fig. 70.2; black squares). FES2004 derived OTL displacements are applied to IGS coordinates to reduce the impact of large scale OTL displacements when computing translation
Fig. 70.2 The GPS networks. Up: the IGS network. Down: the ELEV network with absolute PCV corrections derived from relative ones and limited to a 10 cutoff angle (black triangles) and the AZEL network with absolute PCV corrections (white diamonds)
Table 70.1 Parameterizations used to compute the AZEL and ELEV networks Parameter Software Sessions Orbits/EOPs Stabilization Tide loading Troposphere PCV
Cutoff Nb. Stations Data set
Network AZEL Network ELEV GAMIT-GlobK Overlapping 2 h sessions every hour Fixed (final data) ITRF2005 corrected for OTL All European IGS-core stations Atmospheric: Yes Oceanic: No One by session No gradient Absolute derived Real absolute from relative ones (azimuth and (only elevation) elevation) 10 5 ~10 ~50 2003–2006 2007–2008
parameters during the stabilization procedure (Vergnolle et al. 2008). Briefly, GPS observations are processed in a classical way (Table 70.1) and processing between both networks differ by the PCV models and consequently by the cutoff angle. The data duration is ~4 years for the AZEL network and ~1.5 year for the ELEV one (installed by the end of 2007). ELEV is used to assess the FES2004 model and AZEL to test the impact of troposphere models when using a low cutoff angle. Both networks are only corrected at the observation level from the non-tidal atmospheric loading presented in Tregoning and Van Dame (2005). Due to the presence of remaining partial tides at S1 and S2 frequencies (Tregoning and Watson 2009), we do not correct for tidal atmospheric loading. Displacements caused by the ten OTL main tides (M2, S2, N2, K2, K1, O1, P1, Q1, Mf, and Mm) are estimated, in terms of amplitudes and Greenwich phase lags, from coordinate time series using least squares spectral analysis developed in the t_tide software (Pawlowicz et al. 2002).
70.3
Assessment of the FES2004 Model
Observations of the ELEV network are computed with 6-h period tropospheric data (ZHD and VMF1) in order to achieve the highest precision within the GPS time series (Boehm et al. 2006b).
575 RMS East (mm) RMS North (mm)
Assessment of the FES2004 Derived OTL Model in the West
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Fig. 70.3 RMS misfits of the difference OTL(ZHD + VMF1) – FES2004 in mm (see detail of the computation in Melachroinos et al. 2007)
With more than 1 year of data and according to the Rayleigh criteria (Simon 2007) we expect to separate ten main tides of the FES2004 model. Figure 70.3 presents the RMS misfit over the whole ELEV network between the estimated components (amplitudes and Greenwich phase lags) of a specific tide and its components according to the FES2004 model (see details of the computation in Melachroinos et al. 2007). The largest RMS value is obtained for the M2 tide when looking at North, East, and Up components together. It is ~2 mm on the Up component and ~0.5 and ~1 mm in the North and East directions, respectively. Other semi-diurnal tides like S2 and N2 reach ~1 and 0.5 mm in the Up component. These values are in equal proportion of the amplitude of the tides in the region. RMS of diurnal and long tides are generally negligible, but K1 and K2 tides present large misfits, by 1–2 mm in the Up component, which is ~30% of K1 displacements in the West of France. These RMS values are consistent with the results of Melachroinos et al. (2007) and can be caused by multipath effects and/or errors in the GPS orbits. Phasor plots of M2 and K1 tides on the Up component are presented in Fig. 70.4. Results for M2 illustrate typical local amplitude differences (with respect to FES2004) for semi-diurnal tides. Largest results are reached in the Western Brittany, more than 3 mm, where OTL displacements are maximum in the West of France (see Fig. 70.1) while inland sites present amplitude differences of ~1 mm. According to our results, FES2004 underestimates OTL displacements in the Western Brittany. On the other hand, estimated Greenwich phase lags are globally consistent (differences are less than a few degrees)
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with those predicted by the FES2004 model. Our amplitude and Greenwich phase lag estimates need to be validated in a classical GPS processing (24 h batch sessions) in order to quantify any reduction of spurious signals in position time series (Penna et al. 2007). K1 phasor plot illustrate typical “random” amplitude and Greenwich phase lag estimates for K1 and K2 tides. No coherence, over the whole ELEV network, with the FES2004 model is found, showing further investigations about multipath effects must be conducted to improve K1 and K2 estimates. As observations have been corrected from non-tidal atmospheric loading with data presented in Tregoning
Fig. 70.4 Phasor plots of M2 (left) and K1 (right) on the Up component. Greenwich phase lags are unit vectors. White and grey arrows represent computed and predicted (FES2004) phase lags, respectively. Amplitude differences (in mm) are presented with grey-shaded triangles
F. Fund et al.
and Van Dame (2005), S2 estimates are not totally corrected from atmospheric loading because of remaining partial tides, which could not be corrected when using tidal plus “partial” non-tidal atmospheric tides (Tregoning and Watson 2009). The results obtained at S1 and S2 frequencies might be aliased by such effects. The impact of tidal plus non-tidal atmospheric loading corrections (Tregoning and Watson 2009) on OTL estimates needs further investigations.
70.4
Impacts of Tropospheric Models
In this section, we study the impact of changing a priori ZHDs and mapping functions from 6-h period data to annual empirical models. We compute (1) the differences between ZHD values derived from GPT (ZHD_GPT) and the gridded ZHDs derived from ECMWF data (called Vienna ZHDs; VZHDs), and (2) the differences between GMF and the gridded VMF1 (both hydrostatic and wet parts). The results obtained for 14 years of VZHD and VMF1 data from 1994 to 2008 are presented in Fig. 70.5 and expressed in terms of equivalent height errors with the approximation that the height error is ~1/5 of the path delay at 5 elevation (Boehm et al. 2006b). Even if mean differences (not shown here) in our area are closed to zero, except for the hydrostatic mapping functions (~0.5 mm), standard deviations of the differences are in the interval 1–4 mm. This is
Fig. 70.5 Standard deviations of the differences between ZHD_GPT and VZHD data and between GMF and VMF1 data respectively, in mm. Results are expressed in terms of equivalent height error (see Boehm et al. 2006b)
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Fig. 70.6 A typical periodogram of VZHD and VMF1 time series in the West of France in mm. Results are expressed in terms of equivalent height error (see Boehm et al. 2006b)
a consequence of the natural variations of the 6h period gridded data set, which depends on meteorological observations (ECMWF data). Any misfit of the annual GPT and GMF terms can also explain these standard deviations. The contribution of high-frequency variations to these standard deviations is evaluated through a spectral analysis of VZHD and VMF1 data (calculated with a Fast Fourier Transform) over the 14 years period (Fig. 70.6). Though the amplitudes at S1 and S2 periods are significantly over the noise level for VZHD and VMF1H time series, they reach very small values. Consequently, they should not influence OTL estimates by GPS in the West of France. VMF1W time series are the noisiest with still significant peaks at S1 and S2 periods. Their spatial variations are displayed in Fig. 70.7. Mean amplitudes are 0.5 and 0.2 mm for S1 and S2, respectively. This might induce spatial variations of ZTD estimates and station heights due to the correlation between both ZTD and height parameters (Niell 2000). The differences between tropospheric models have significant impacts on positioning for elevation angles below 10 elevation (above, mapping functions differences are not significant; Niell 1996; Fund et al. 2010). A priori tropospheric corrections within the AZEL network are either VZHD + VMF1 or ZHD_ GPT + GMF. The resulting RMS differences between both computations are plotted in Fig. 70.8. The RMS misfit for the S2 tide (Up component) is consistent with the results presented in Fig. 70.6. Diurnal and long tides present maximum differences of ~0.3 mm on the Up component (especially Mm and Mf tides). This result shows that meteorological
Fig. 70.7 S1 and S2 amplitudes in the VMF1W time series in France. Results are expressed in terms of equivalent height error (see Boehm et al. 2006b)
RMS Up (mm) RMS East (mm) RMS North (mm)
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Fig. 70.8 RMS misfits of the difference OTL(ZHD_GPT/ VMF1) – OTL(VZHD/VMF1) in mm
variations at synoptic scales described by VZHD and VMF1 and smoothed by the GPT and GMF models (only annual variations are parameterized) have significant impacts. Differences on tides closed to S1 and S2 have been also identified. The main difference on the three components (North, East, and Up) is on the M2 tide, ~0.5 mm on the Up component. We attribute partially this difference to the 6 hourly sampling of VZHD and VMF1 which does not well describe diurnal meteorological variations (Ponte and Ray 2002). Tregoning and Watson (2009) showed that significantly different results may be obtained depending on the chosen mapping function. Further studies are required to test this interpretation. The RMS differences that we identified on the ten main tides are significant. Despite of the very good agreement between ZHD_GPT and VZHD, and between VMF1 and GMF at the annual frequency, using empirical models can change GPS assessment of tide loading (atmospheric, oceanic, hydrologic. . .) at the millimeter
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level (i.e. the sum of the ten tide differences). Also, any mismodelling of ocean tide could lead to spurious signals when processing 24 h batch solutions (Penna et al. 2007). These results cannot explain the RMS values between the ELEV network and the FES2004 model (Fig. 70.3) because it is composed of six times more stations than the AZEL network, and uses a high (10 ) cutoff elevation angle. However, it shows that using different tropospheric models can lead to significantly different OTL estimates when using low cutoff angles. We consider that this result is important for further OTL studies using low cutoff angles thanks to the availability of actual absolute phase center corrections (this was not possible for the ELEV network due to antenna PCV corrections). Tests on the impact of cutoff angles will become possible when actual absolute phase center corrections will also be available for the stations of the ELEV network.
70.5
Conclusion and Future Prospects
GPS derived estimates of OTL displacements in the West of France present significant differences with respect to the FES2004 model. Largest differences are obtained for the M2 and K1/K2 tides. We found that the FES2004 model underestimates amplitudes of the M2 tides on the western part of Brittany, where displacements are maximum. Amplitudes differences are about 3 mm, which is approximately 10% of the mean M2 OTL amplitude in the region. No coherence between K2 and K1 estimates has been found. We attribute these results to multipath effects because K2 and K1 periods are integer multiples of the GPS satellite orbital period. So, improving the detection of multipath effects is needed. We suggest stacking slant path residuals (Shoji et al. 2004) to detect any systematic residuals and problems in the field of the antenna. A typical result of stacking is plotted in Fig. 70.9. It reveals constant mean values (~20 mm) at low elevations, which might be the result a combination of PCV and multipath problems. When using GPS data to study OTL displacements, the impact of using empirical tropospheric annual model instead of 6-h period gridded data may
Fig. 70.9 A typical example of stacking residuals in the West of France (LPPZ station at the extreme West point of the ELEV network near GUIP station; see Fig. 70.2) in mm
introduce changes in the estimation of tide components at the millimeter level. This is partially due to presence of significant amplitudes at S1 and S2 frequencies in tropospheric series, especially in the wet part of VMF1. We find that RMS for the M2 tide can change by about 0.5 mm on the Up component, but it might be corrupted by partial tides remaining in the non-tidal atmospheric loading model that we use. Even if using 6-h period gridded tropospheric data misfit diurnal and semi-diurnal meteorological variations (Ponte and Ray 2002), they include variations at the synoptic scales. Neglecting them when using empirical tropospheric models like GPT and GMF might introduce errors in Mf and Mm estimates, by 0.3 mm on the Up component. The stability of the preliminary results that were presented here may be tested following two tracks of further investigation (1) the impact of using a global GPS data set instead of a regional one during the stabilization procedure, and (2) the implementation of non-tidal atmospheric loading corrections (Tregoning and Watson 2009) in order to better separate the respective contributions of atmospheric tide effects and tropospheric models (Steigenberger et al. 2009). Acknowledgements We thank the editor RS Gross and two anonymous reviewers for their constructive comments on previous versions of the manuscript. FF thanks the “Ordre des Ge´ome`tres Expert” and the “Re´gion des Pays de la Loire” for their financial support.
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Assessment of the FES2004 Derived OTL Model in the West
References Boehm J, Niell A, Tregoning P, Schuh H (2006a) Global mapping function (GMF): a new empirical mapping function based on numerical weather model data. Geophys Res Lett 33:L07304. doi:10.1029/2005GL025546 Boehm J, Werl B, Schuh H (2006b) Troposphere mapping functions for GPS and very long baseline interferometry from European Center for Medium-Range Weather Forecasts operational analysis data. J Geophys Res 111: B02406. doi:10.1029/2005JB003629 Boehm J, Heinkelmann R, Schuh H (2007) Short note: a global model of pressure and temperature for geodetic applications. J Geod. doi:10.1007/s00190-007-0135-3 Fund F, Morel L, Mocquet A, Boehm J (2010) Assessment of ECMWF derived tropospheric delay models for Europe with GPS data. GPS Solut. doi:10.1007/s10291-010-0166-8 Lyard F, Lefevre F, Letellier T, Francis O (2006) Modeling the global ocean tides: insights from FES2004. Ocean Dyn. doi:10.1007/s10236-006-0086-x Melachroinos SA, Biancale R, Llubes M, Perosanz F, Lyard F, Vergnolle M, Bouin M-N, Masson F, Nicolas J, Morel L, Durand S (2007) Ocean tide loading (OTL) displacements from global and local grids: comparisons to GPS estimates over the shelf of Brittany, France. J Geod. doi:10.1007/ s00190-007-0185-6 Niell AE (1996) Global mapping functions for the atmospheric delay at radio wavelengths. J Geophys Res 101:3227–3246 Niell AE (2000) Improved atmospheric mapping functions for VLBI and GPS. Earth Planets Space 52:699–702 Pawlowicz R, Beardsley B, Lentz S (2002) Classical tidal harmonic analysis including error estimates in MATLAB using T_TIDE. Comput Geosci 28:929–937
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Penna NT, King MA, Stewart MP (2007) GPS height time series: Short-period origins of spurious long-period signals. J Geophys Res 112:B02402. doi:10.1029/2005JB004047 Ponte RM, Ray RD (2002) Atmospheric pressure corrections in geodesy and oceanography: a strategy for handling air tides. Geophys Res Lett 29(24):2153. doi:10.1029/2002GL016340 Shoji Y, Nakamura H, Iwabuchi T, Aonashi K, Seko H, Mishima K, Itagaki A, Ichikawa R, Ohtani R (2004) Tsukuba GPS dense net campaign observation: improvement in GPS analysis of slant path delay by stacking one-way postfit phase residuals. J Meteorol Soc Jpn 82(1B):301–314 Simmons AJ, Gibson JK (2000) The ERA-40 project plan. ERA-40 project report series no. 1, European Center for Medium-Range Weather Forecast, Reading Simon B (2007) La mare´e: la mare´e oce´anique coˆtie`re, Ed. Institut oce´anographique, Fondation Albert 1er, prince de Monaco, 433 pp Steigenberger P, Boehm J, Tesmer V (2009) Comparison of GMF/GPT with VMF1/ECMWF and implications for atmospheric loading. J Geod. doi:10.1007/s00190-009-0311-8 Tregoning P, Van Dame T (2005) Effects of atmospheric pressure loading and seven-parameter transformations on estimates of geocenter motion and station heights from space geodetic observations. J Geophys Res 110:B03408. doi:10.1029/2005JB003334 Tregoning P, Watson C (2009) Atmospheric effects and spurious signals in GPS analysis. J Geophys Res 114:B09403. doi:10.1029/2009JB006344 Vergnolle M, Bouin M-N, Morel L, Masson F, Durand S, Nicolas J, Melachroinos SA (2008) GPS estimates of ocean tide loading in NW-France: determination of ocean tide loading constituents and comparison with a recent ocean tide model. Geophys J Int. doi:10.1111/j.1365-246X.2008.03734.x
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Gravimetric Time Series Recording at the Argentine Antarctic Stations Belgrano II and San Martı´n for the Improvement of Ocean Tide Models
71
Mirko Scheinert, Andre´s F. Zakrajsek, Lutz Eberlein, Reinhard Dietrich, Sergio A. Marenssi, and Marta E. Ghidella
Abstract
As a joint project, the Instituto Anta´rtico Argentino (IAA) and TU Dresden (TUD) carried out gravimetric time series recordings at the Argentine Antarctic stations Belgrano II and San Martı´n. At both stations gravimetric data were recorded for about 1 year, using LaCoste&Romberg gravity meters. At Belgrano II the observations were carried out from February to November 2007, and at San Martı´n from February 2008 to January 2009. The set-up of the gravimetric stations and the instrumentation utilized are shown. We discuss and compare the data gained at the two stations as well as the analyses in order to solve for the tidal constituents. Generally, tidal gravimetry has the potential to provide independent data to validate and to improve ocean tide models at the Antarctic seas, especially since in-situ data to be used for the establishment of the models are sparse and satellite altimetry has limitations in high latitudes and over ice-covered regions. We discuss the feasibility of the obtained data to pursue this goal of an improvement of ocean tide models. Finally, an outlook for further investigations is given.
71.1
Motivation
Ocean tides are one of the major geodynamic phenomena. They have to be taken into consideration when analyzing many different observations referring to changes in the system Earth. For instance, information on ocean tides have to be utilized when computing de-aliasing products for the analysis of satellite
M. Scheinert (*) L. Eberlein R. Dietrich TU Dresden, Institut f€ ur Planetare Geod€asie, 01062 Dresden, Germany e-mail:
[email protected] A.F. Zakrajsek S.A. Marenssi M.E. Ghidella Instituto Anta´rtico Argentino, Direccio´n Nacional del Anta´rtico, Cerrito 1248, C1010AAZ Buenos Aires, Argentina
gravimetry data, especially from GRACE. Since the Antarctic continent is entirely surrounded by the ocean, the ocean signal at all (not only the tides) is otherwise a major source for aliasing the estimates of ice mass change (Horwath and Dietrich 2009). The effect of ocean tidal loading has to be reduced in precise GPS analyses for the study of the reference system realization and of crustal motions (R€ulke et al. 2008; Penna et al. 2007). Eventually, ocean tides directly affects observation techniques based on satellite height measurements at the ocean – hence also at the ice-covered Antarctic ocean – as obtained e.g. by ICESat (Padman and Fricker 2005) or CRYOSAT-2.1
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http://www.esa.int/esaLP/LPcryosat.html
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_71, # Springer-Verlag Berlin Heidelberg 2012
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To count for such reductions, ocean tide models are used. Nevertheless, current ocean tide models lack accuracy and reliability in Antarctic seas. This is partly due to the fact that the major data source for setting up ocean tide models is provided by satellite altimetry, which due to limited latitude range and/or due to limitations in the analysis of the reflected signal over sea-ice and ice shelves can only insufficiently contribute to an improvement of ocean tide models in Antarctica. This has been discussed extensively by King and Padman (2005). It was shown that the largest discrepancies between different ocean tide models can be found for the semidiurnal constituents M2 and S2 at the Filchner-Ronne ice shelf, and for the diurnal constituents O1 and K1 at the Ross ice shelf. In-situ data can help to improve this situation by a direct validation of ocean tide models and, furtheron, by assimilating the data into the modeling. On the one hand, GPS offshore measurements provide direct observations of ocean tides (King and Aoki 2003; King et al. 2005), but the K1 and K2 constituents are subject to systematic biases in GPS coordinate time series. On the other hand, onshore tidal gravimetry can provide independent observations and is thus complementary to GPS. The principle of tidal gravimetry is – of course – not new; it has been successfully applied world-wide (Melchior and Francis 1996). Tidal gravimetry may not be regarded a direct observation of ocean tides, since it measures mainly the tides of the solid earth and comprises the effect of ocean tides through the attraction of the tidal waters (direct effect) and their exerted load (indirect effect). The principles of the elastic load theory have been treated in detail in literature based on the work by Farrell (1972) and shall not be repeated here. Several authors offer offline or online software to compute ocean tidal loading (Agnew 1996; Bos and Scherneck 2007), and the computed effects can be utilized in further analyses. Against this background, Instituto Anta´rtico Argentino (IAA) and TU Dresden (TUD) started a cooperation to set up gravimetry observatories at the Argentine Antarctic stations Belgrano II and San Martı´n. This cooperation was a contribution to the project 185 “Polar Earth Observing Network” (POLENET) of the International Polar Year (IPY, March 2007–February 2009). The goal of POLENET was to deploy autonomous observatories at remote polar sites including GPS, seismology, gravimetry and tide gauges. Closing observational gaps in polar
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regions POLENET contributed substantially to investigate polar geodynamics and to gain deeper insight into the interactions between cryosphere, solid earth, oceans and atmosphere.2 The TUD group gained a lot of experience in carrying out geodetic surveys in polar regions and also in realizing tidal gravimetry under polar conditions (Dietrich et al. 1998; Scheinert et al. 1998, 2008d). Joining this experience with the expertise of the IAA group we successfully realized the gravimetric time series observations at Belgrano II and at San Martı´n. The progress of this joint project was already presented in several papers or presentations (Scheinert et al. 2007, 2008a, b, c). In the following, we will describe the set-up and realization of the gravimetric time series observations as well as the main results.
71.2
Realization of the Gravimetric Time Series Observations
The location of the two Argentine stations in Antarctica is shown in Fig. 71.1. The station Belgrano II is situated at 77 520 3000 S and 34 370 1800 W at the south-eastern
Fig. 71.1 Map of Antarctica with the location of the Argentine Antarctic stations Belgrano II and San Martı´n (map drawn using GMT 4.2 (Wessel and Smith 1998) and ADD 4.0 (ADD Consortium 2000))
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Belgrano II: Small hut in the foreground
San Martín: Usina Emergencía (foreground)
Fig. 71.2 The gravimetry observatories at Belgrano II and San Martı´n
edge of the Weddell Sea, East Antarctica. At the southern bank of the Vahsel Bay the station sits on a small bedrock outcrop, the Bertrab Nunatak (Kleinschmidt and Boger 2008). For the gravimetry observatory, a small, especially adapted hut could be used. Separately from the foundation of the hut a concrete pillar was erected and grounded more than 1 m into the permafrost soil in order to reach a maximum level of mechanical uncoupling. The situation at Belgrano II is shown in Fig. 71.2a. The height of the gravimetry observatory above the ocean is about 250 m, its shortest distance to the coast of the Vahsel Bay amounts to only about 1 km where the ocean tide signal is subject to a certain damping due to the glacier covering the Vahsel Bay, which is freely floating only to a certain extent (Scheinert et al. 2007), and the distance to the coast of the Weddell Sea is about 25 km. In contrast, the station San Martı´n, at 68 070 4900 S and 67 060 2000 W, is situated at Barry Island, Marguerite Bay, west side of the Antarctic Peninsula, and thus directly at the coast. The Usina Emergencı´a (emergency power station) served as the gravimetry observatory, see Fig. 71.2b. Its height above the ocean is only about 6 m, and its distance to the coast 20 m. For the gravimeter installation also a concrete pillar was constructed, which was, however, directly connected to the foundation of the building. At both observatories, for the installation of the gravity meters the aforementioned pillar could be used. We installed the TUD gravity meter L&R D193, which runs in connection with the L&R digital feedback DFB-144. Utilizing the digital feedback for
the measurement, the gravity meter itself is set to a fixed position of its internal gear, i.e. the coarse dial has to be fixed only once. The fine screw stays fixed as long as the measurements are within the range of the feedback. It needs a re-adjustment due to the gravimeter’s drift only after several weeks. The feedback itself was calibrated at a vertical calibration line in Hannover, Germany. From this calibration the scale factor 0.450 for the conversion of feedback units to gravity units was inferred. The IAA gravity meter L&R G-748 was additionally installed as a second instrument, utilizing the CPI output to record the data. This instrument served as a backup for the case, that the L&R D-193 failed to operate. Since the latter was not the case, in the following we discuss only the recordings of the gravity meter L&R D-193. Furthermore, a digital barometric pressure gauge SETRA 370 served to register air pressure variations at both observatories. A small GPS receiver (“GPS Mouse”) was installed on the roof of the respective gravimetry hut in order to provide the system with UTC time signals. Thus, a reliable time synchronization could be reached. The power supply was realized for all instruments by a 12-voltage-system, utilizing a 100 A h sealed battery in conjunction with a 220V/ 12V-7A charging device. A schematic diagram of the instruments and connections as set up in San Martı´n is given in Fig. 71.3a, and a principal photography in Fig. 71.3b. In Belgrano II the set up of the instrumentation was done in a similar way. To improve the thermal insulation a box made of polyurethane was installed housing the entire equipment. However, due
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Fig. 71.3 Installation of the gravity meters and auxiliary instrumentation at San Martı´n
to the Antarctic conditions (low temperatures and high winds especially in the winter, high temperature differences between day and night time) and due to the fact that the huts used for the gravimetry observatories were not insulated very well, it could not be circumvented that the temperature inside the thermal insulation box varied by several degrees. The data were registered with the help of a data logging program running under the Linux operating system on an ordinary laptop. At Belgrano II data were recorded during 2007 (February to December), and at San Martı´n during 2008 (February 2008 to January 2009).
71.3
Tidal Adjustment
In the following, we present the processing based on the gravity data recorded by the gravity meter L&R D193 together with the L&R digital feedback DFB-144. The shortest possible sampling rate enabled by the DFB-144 is 3 min. We only used the unfiltered output, thus the data were given at a 3 min sampling interval free of any time lag. Also filtered data were recorded (with a filter length of 2 min), however, these were not utilized. The data which were originally output in terms of feedback units were transformed to gravity units (nm/s2) using a calibration factor of –0.450 for the unfiltered feedback channel. These data were then
analyzed with the help of the software package ETERNA 3.4 (Wenzel 1997), applying a removerestore type processing scheme. First, a known a priori earth tide model and a model air pressure effect (using the air pressure data together with a linear regression coefficient of –3.0 nm/s2/hPa) were removed from the data, leaving as a result the unresolved earth tide signal together with the residual air pressure effect, the ocean tide loading effect, instrumental drift, data gaps, outliers and spikes, and noise. Then, short data gaps were closed, and outliers and spikes were removed. The model signals (earth tides, air pressure) were added back. The resulting gravity time series cleaned in this way were then filtered and resampled to hourly data. For this, a new finite impulse response (FIR) filter was designed to perform only one single filter step (from 3-min data to 1-h data), since an appropriate filter was not provided with ETERNA 3.4. These cleaned, hourly gravity time series are shown in Fig. 71.4 for Belgrano II and in Fig. 71.5 for San Martı´n. First of all, we notice a relatively strong instrumental drift, which occasionally can turn from positive to negative for a certain time, and several data gaps. These features were likely to be caused mainly by the temperature variations and by strong winds or even storms, latter occurring especially during the Antarctic winter. If much better conditions for
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Fig. 71.4 Time series of the gravity data at Belgrano II (hourly data, spikes removed)
Fig. 71.5 Time series of the gravity data at San Martı´n (hourly data, spikes removed)
the gravity recording can be maintained it could be shown that the drift is much smaller and of much less variability (Scheinert et al. 1998). Given the available facilities and logistics in Antarctica, one has to live with these hostile environmental conditions, nevertheless, it has been proven that reliable data could be recorded. Finally, the hourly data comprises 258 days in the year 2007 for Belgrano II, and 188 days in the year 2008 for San Martı´n. The longest data block amounts to 84 days for Belgrano II and to 63 days for San Martı´n. To count for the instrumental drift, ETERNA 3.4 offers two approaches: first, an approximation
technique utilizing Tschebyscheff polynomials, and second a high-pass filtering. Due to the large and highly varying drift we decided to high-pass filter the data (ETERNA filter N60M60M2 with 167 coefficients, with Hanning window applied). In the final step of ETERNA 3.4 the parameters (gravimeter factor and phases) for the tidal constituents as well as a linear air pressure regression coefficient were computed by least-squares adjustment. Especially the gravimeter factor is adjusted by a comparison of the observed to the theoretical amplitude. (The amplitude of the corresponding vertical deformation caused by the tide generating potential can roughly be estimated
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by multiplying the theoretical (gravity) amplitude with the factor of 0.19 mm/nm/s2. Thus, a theoretical gravity amplitude of 100 nm/s2 corresponds to 19 mm amplitude of the vertical deformation as to be measured by e.g. GPS.) For eight main constituents the results of the least-squares adjustment are shown in Tables 1 and 2. The mean standard deviation of the adjustment is 36.4 nm/s2 for Belgrano II and 17.8 nm/ s2 for San Martı´n, the largest residual is 246 nm/s2 for Belgrano II and 91 nm/s2 for San Martı´n. These results demonstrate again the larger noise due to the measurement conditions which were far from being optimal. For the air pressure regression we obtained values of 11.1 nm/s2/hPa for Belgrano II and of 7.1 nm/s2/hPa for San Martı´n. These results are in contradiction to the value which normally would be expected to be in the level of 3 . . .5 nm/s2/hPa. For instance, for the Antarctic coastal station Forster/Novolazarevskaya (central Dronning Maud Land, East Antarctica) we got a regression coefficient of –3.7 nm/s2/hPa (Dietrich et al. 1998). The presently obtained values cannot be interpreted in terms of geodynamics or geophysics. Therefore, after the instrument was transferred back home again, it was checked for an insufficient instrumental pressure insulation utilizing a pressure chamber. Also, a calibration of the meter against controlled air pressure values in the range of 875 to 1,025 hPa was carried out. The regression yielded a value of 7.5 nm/s2/hPa. If this regression coefficient is used to correct for the air pressure variation this effect could be almost entirely removed without changing the adjusted tidal parameters. Thus, it turned out that this instrumental insufficiency was the reason for the unexpected results of the air pressure regression adjustment, so that the obtained regression coefficients cannot be used for further interpretation.
Meanwhile, the instrument underwent a major check, and the instrumental pressure insulation was repaired.
71.4
Comparison with Ocean Load Tides
As discussed in Sect. 71.1, global ocean tide models lack information for Antarctic seas. Nevertheless, it was shown that for the circum-Antarctic seas the model TPXO6.2 (Egbert and Erofeeva 2002) performs best of the models tested (King et al. 2005; King and Padman 2005). These statements were principally confirmed by Thomas et al. (2008). It was also stressed that for West Antarctica (and hence the Antarctic Peninsula) larger differences between the predictions of the different models were observed, due to the complex coastline – and one has to add, also due to the complex and not well known bathymetry. For the discussion of the ocean load tide effect we restricted ourselves to the model TPXO6.2. The resulting statements can be regarded as generally valid in terms of a qualitative discussion. Nevertheless, further investigations are needed taking the various recent ocean tidel models into account, especially when a model improvement is anticipated. For the computation of the ocean load tide effects from the model TPXO6.2 the SPOTL software was used (Agnew 1996). The predicted ocean load tide vectors for the eight constituents were compared with the residuals computed as the vector difference between the adjusted earth tide vector and the a priori earth tide vector. The results are given in Tables 71.3 and 71.4 for Belgrano II and San Martı´n, resp., as well as in Figs. 71.6a, b and 71.7a, b for Belgrano II and San Martı´n, resp.
Table 71.1 Results of the tidal adjustment for Belgrano II
Table 71.2 Results of the tidal adjustment for San Martı´n
Tide
Tide Theoretical amplitude (nm/s2)
Delta factor Adjusted Error
Q1 O1 P1 K1 N2 M2 S2 K2
1.35 1.30 1.23 1.27 0.96 1.37 1.22 1.31
Q1 O1 P1 K1 N2 M2 S2 K2
Theoretical Delta factor amplitude Adjusted Error (nm/s2) 24.5 127.8 59.5 179.7 6.4 33.2 15.4 4.2
1.43 1.33 1.28 1.31 2.15 2.25 2.62 2.74
0.11 0.03 0.07 0.02 0.22 0.04 0.08 0.23
Local phase Adjusted ( ) 17.9 2.0 7.4 1.1 1.2 7.9 19.7 17.3
Error ( ) 4.5 1.1 3.1 0.9 5.9 1.1 1.8 4.8
41.2 215.1 100.1 302.3 20.0 104.4 48.6 13.2
0.03 0.01 0.02 0.01 0.04 0.01 0.01 0.04
Local phase Adjusted Error ( ) ( ) 1.2 1.3 0.4 0.3 1.0 0.7 1.9 0.2 21.0 2.4 15.1 0.3 6.8 0.7 0.1 1.9
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Table 71.3 Comparison of residual vector with ocean load tide vector from TPXO6.2 for Belgrano II
Table 71.4 Comparison of residual vector with ocean load tide vector from TPXO6.2 for San Martı´n
Tide Residual vector Local Amplitude phase ( ) (nm/s2) Q1 14.0 50.6 O1 43.1 8.1 P1 18.8 31.2 K1 56.3 4.5 N2 7.3 2.3 M2 42.0 14.1 S2 26.4 31.1 K2 7.6 26.7
Tide Residual vector Local Amplitude phase ( ) (nm/s2) Q1 14.5 4.2 O1 63.8 1.8 P1 23.4 5.2 K1 82.5 9.0 N2 7.2 106.2 M2 50.6 47.7 S2 12.2 34.7 K2 4.1 0.3
Ocean load tide vector Amplitude Local (nm/s2) phase ( ) 7.4 23.0 35.6 8.1 10.1 1.9 33.5 2.0 6.3 8.5 41.8 10.8 25.5 28.1 7.2 31.6
a
Ocean load tide vector Amplitude Local (nm/s2) phase ( ) 8.1 27.4 40.1 14.4 12.2 4.7 38.8 4.0 8.3 111.3 44.5 63.6 10.1 23.7 2.9 28.9
b
Diurnal constituents
Semidiurnal constituents
Fig. 71.6 Residual gravity and ocean load tide vectors at Belgrano II (gravity unit: nm/s2, 3s error ellipse at same scale)
For Belgrano II we got an astonishingly good agreement, especially for the semidiurnal constituents. Except for K1 the individual ocean load vectors are within the 3s error ellipse of the adjusted tide parameters. For San Martı´n, the comparison did not yield such a good agreement. For the diurnal constituents, the residual amplitudes are explained by the modeled ocean load vector only by about 50–60%, and there are larger differences in the phases for nearly all constituents. With a latitude of about 68 S and 78 S of San Martı´n and Belgrano II, resp., the observatories are situated relatively close to the south
pole. Thus, especially the semi-diurnal earth tides are already relatively small (amplitudes reach a magnitude of only 12% of its theoretical maximum for San Martı´n, and of 6% for Belgrano II) and the predicted ocean load tide amplitudes are in the same order of magnitude. The large discrepancy between the residual and ocean load tide vectors for San Martı´n have to be explained to a large extent by the insufficient capture of the ocean tides by the model. Especially for this coastal station the bathymetry, coastline and the spatial gridding of the water columns within the ocean
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a
b
Diurnal constituents
Semidiurnal constituents
Fig. 71.7 Residual gravity and ocean load tide vectors at San Martı´n (gravity unit: nm/s2, 3s error ellipse at same scale)
tide model – both on local and regional scale – play a crucial role for the performance of the respective model. These quantities need to be provided in a finer resolution, which could not be accomplished by a global ocean tide model. Hence, this is one motivation for further research on a model improvement at a regional scale. For Belgrano II the insufficient capturing of the regional features of the ocean tides turned out to be not that severe, since in contrast to San Martı´n the station is situated in a larger distance to the coast, and at least the coastline does not exhibit such high variability.
71.5
Summary
To realize gravimetric tide observations a joint Argentine-German project was initiated, which was a contribution to the project POLENET of the International Polar Year 2007/2008. For about 1 year each, gravimetric time series recordings were carried out at the Argentine Antarctic stations Belgrano II (2007) and San Martı´n (2008) utilizing L&R gravity meters. Affected by the harsh environmental polar conditions the data showed a larger noise and a noticeable large
instrumental drift in contrast to data taken under laboratory conditions. Nevertheless, the data analyses yielded feasible results. The mean standard deviations of the tidal adjustment were 36.4 nm/s2 for Belgrano II and 17.8 nm/s2 for San Martı´n. For eight tidal constituents the resulting residual vectors were compared to the respective ocean load tide vectors predicted on basis of the ocean tide model TPXO6.2. Besides investigations by other authors showing that all ocean tide models lack information content and thus reliability for the circum-Antarctic seas, we got a very good agreement for Belgrano II (especially for the four semi-diurnal constituents). For San Martı´n the comparison yielded major discrepancies. These can be explained to a large extent by the insufficient capture of the ocean tides by the model. Information on bathymetry and coastline as well as the spatial gridding in the ocean tide modelling need to be improved. Tidal gravimetry provides an independent source of in-situ data. Hence, it is planned to utilize the recorded gravimetric time series data in order to carry out a regional improvement of the ocean tide model, especially in the region of the station San Martı´n, where also further data sources could be opened up.
71
Gravimetric Time Series Recording at the Argentine Antarctic Stations
Acknowledgement We like to thank all institutions and people in Argentina who supported the project especially with regard to logistics and to the set-up and maintenance of the gravimetry observatories: Comando Conjunto Anta´rtico, Comando Anta´rtico de Eje´rcito, Direccio´n Nacional del Anta´rtico and the overwintering teams at Belgrano II and San Martı´n, especially at LaBel (Laboratorio Belgrano II) and LaSan (Laboratorio San Martı´n). The project was supported by the bilateral scientific-technological cooperation program of SECyT (grant AL/ PA/05-AVII/009), Argentina, and BMBF (grant ARG-05/Z04), Germany.
References Agnew DC (1996) SPOTL: Some Programs for Ocean-Tide Loading. SIO Ref. Ser. 96-8, p 35, Scripps Institution of Oceanography, La Jolla, CA (version 3.2.1 of 26 July 2005) Bos MS, Scherneck HG (2007) Free ocean tide loading provider. http://www.oso.chalmers.se/ loading ADD Consortium (2000) Antarctic Digital Database, Version 3.0. Database, manual and bibliography. Scientific Committee on Antarctic Research, Cambridge, manual and bibliography, 93 pp Dietrich R, Dach R, Korth W, Polzin J, Scheinert M (1998) Gravimetric Earth Tide Observations in Dronning Maud Land/Antarctica to Verify Ocean Tidal Loading. In: Ducarme B, Paquet P (eds) Proc. 13th Intl. Symp. on Earth Tides, Brussels, July 22–25, 1997, Obs. Royal de Belgique, Brussels, pp 529–536 Egbert GD, Erofeeva L (2002) Efficient inverse modeling of barotropic ocean tides. J Atm Ocean Techn 19 Farrell WE (1972) Deformation of the earth by surface loads. Rev Geophys Space Phys 10(3):761–797 Horwath M, Dietrich R (2009) Signal and error in mass change inferences from GRACE: the case of Antarctica. Geophys J Int 177(3):849–864. doi:10.1111/j.1365-246X.2009.04,139.x King M, Aoki S (2003) Tidal observations on floating ice using a single GPS receiver. Geophys Res Lett 30(3):1138. doi:10.1029/2002GL016,182 King MA, Padman L (2005) Accuracy assessment of ocean tide models around Antarctica. Geophys Res Lett 32:L23,608, doi:10.1029/2005GL023,901 King M, Penna NT, Clarke PJ (2005) Validation of ocean tide models around Antarctica using onshore GPS and gravity data. J Geophys Res 110:B08,401, doi:10.1029/2004JB003,390 Kleinschmidt G, Boger SD (2008) The Bertrab, Littlewood andMoltke Nunataks of Prinz-Regent-Luitpold-Land (Coats Land) – Enigma of East Antarctic Geology. Polarforschung 78(3):95–104, (published 2009) Melchior P, Francis O (1996) Comparison of recent ocean tide models using ground-based tidal gravity measurements. Mar Geod 19:291–330
589
Padman L, Fricker HA (2005) Tides on the Ross Ice Shelf observed with ICESat. Geophys Res Lett 32:L14,503, doi:10.1029/2005GL023,214 Penna NT, King MA, Stewart MP (2007) GPS height time series: short-period origins of spurious long-period signals. J Geophys Res 112:B02402. doi:10.1029/2005JB004047 R€ulke A, Dietrich R, Fritsche M, Rothacher M, Steigenberger P (2008) Realization of the terrestrial reference system by a reprocessed global GPS network. J Geophys Res B08403. doi:10.1029/2007JB005231 Scheinert M, Dietrich R, Schneider W (1998) One Year of Gravimetric Earth Tide Observations in Kangerlussuaq / West Greenland. In: Ducarme B, Paquet P (eds) Proc. 13th Intl. Symp. on Earth Tides, Brussels, July 22–25, 1997, Obs. Royal de Belgique, Brussels, pp 201–208 Scheinert M, Zakrajsek AF, Eberlein L, Marenssi SA, Ghidella M, Dietrich R, R€ulke A (2007) Gravimetry and GPS Observations at Belgrano II Station to Test Ocean Tidal Models. Rev Asoc Geol´ogica Argentina 62(4):646–651 Scheinert M, Zakrajsek AF, Dietrich R (2008a) Gravimetric tide observations and ocean tides in Antarctica: From historic measurements to the IPY project POLENET. Presentation at Earth Tide Symposium 2008 “New Challenges in Earth’s Dynamics”, Jena, September 1–5, 2008 Scheinert M, Zakrajsek AF, Dietrich R, Marenssi SA, Eberlein L (2008b) Long-time gravimetric recordings at the Argentine Antarctic stations Belgrano II and San Mart´n: A contribution to the IPY project POLENET. Presentation at 23rd Intl. Polar Meeting, M¨unster, March 10–14, 2008 Scheinert M, Zakrajsek AF, Eberlein L, Dietrich R, Marenssi SA, Ghidella ME (2008c) Gravimetric time series recording at the Argentine Antarctic stations Belgrano II and San Mart´n: A contribution to the IPY project POLENET. Presentation at Earth Tide Symposium 2008 “New Challenges in Earth’s Dynamics”, Jena, September 1–5, 2008 Scheinert M, Zakrajsek AF, Marenssi SA, Dietrich R, Eberlein L (2008d) Tidal gravimetry in polar regions: an observation tool complementary to continuous GPS for the validation of ocean tide models. In: Capra A, Dietrich R (eds) Geodetic and geophysical observations in Antarctica – an overview in the IPY perspective. Springer, Berlin, pp 267–280 Thomas ID, King MA, Clarke PJ (2008) A validation of ocean tide models around Antarctica using GPS measurements. In: Capra A, Dietrich R (eds) Geodetic and geophysical observations in Antarctica – an overview in the IPY perspective. Springer, Berlin, pp 211–236 Wenzel HG (1997) Earth Tide Data Processing Package ETERNA 3.4. (Program Manual) Wessel P, Smith WHF (1998) New, improved version of the generic mapping tools released. EOS Trans AGU 79:579
.
Mass-Change Acceleration in Antarctica from GRACE Monthly Gravity Field Solutions
72
€ldva´ry Lo´ra´nt Fo
Abstract
In a warming climate, it is critical to accurately estimate ice-sheet mass balance to quantify its contribution to present-day sea-level rise. In this study temporal mass variations in Antarctica are investigated based on monthly GRACE gravity solutions. In order to diminish the effect of large uncertainties in glacial isostatic adjustment (GIA) models, an approach is developed to estimate the acceleration of the ice-sheet mass, assuming the presence of accelerated melt signal in the GRACE data. Though the estimate of accelerated melt does not provide an absolute value for the volume of the melting ice, it is a viable tool for characterizing the present-day ice-sheet mass balance.
72.1
Introduction
Antarctica is one of the most challenging areas of ice mass balance studies due to the many unknown processes resulting in an unambiguous interpretability of the detected mass and/or surface variations. Measurements of the radar or laser altimetry generally lacks on the information of the density variations of snow and ice content, while gravity measurements are highly infected by the GIA processes (Wahr et al. 2000; Shum et al. 2008). By any means, misinterpretation of the physical content of the observed signal is strongly affecting the estimation of the present-day ice mass loss presumably occurring due to the recently observed climate variations (or global warming). This huge amount of ice (consisting
L. F€oldva´ry (*) HAS-BUTE Research Group for Physical Geodesy and Geodynamics, Muegyetem rkp 3, Budapest 1111, Hungary e-mail:
[email protected]
about 90% of the world’s ice content) manifests such a large signal coupled with lack of information on the precipitation, water/snow/ice displacement and density variation (ice melting, snow collapse into ice, etc.) resulted in a large uncertainty in the knowledge of physical processes in Antarctica. Recently several methods have been developed by different groups to estimate ice-sheet mass balance. Certain studies estimate the mass variation in a direct modelling method (Chen et al. 2006) in most cases however the classical inverse approach is used (Swenson and Wahr 2002). Regardless of the applied technique, numerical results differ notably on the choice of the GIA model, on the spatial smoothing and on the choice of gravity field solutions by various data centres. In this study at first a modelling of the ice-sheet mass balance is performed using a simple method with the most commonly used parameters by other studies. This is done in order to have a solution to provide us a base, which is not expected to serve as the best possible solution, instead as a conceptually acceptable solution. Subsequently, an alternate method is developed
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_72, # Springer-Verlag Berlin Heidelberg 2012
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€ldva´ry L. Fo
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which is expected to provide a new insight on the present state of the Antarctic ice-sheet mass variation. The proposed method provides an estimate on the tendency of the mass change rather than estimating the mass loss properly. The usefulness of the acceleration of the mass has already been recognized and applied in some way in other studies, however these studies either use the temporal variation of the trend as a subsidiary tool for the trend estimation (e.g. M€uller et al. 2009), or applying a fundamentally different method (e.g. Heki and Matsuo 2009).
72.2
Estimation of Surface Mass Variations
f ðtÞ ¼ A1 sinðo1 t þ ’1 Þ (72.1)
but most recently other frequencies are also included, such as 161 day (Peltier 2009; M€ uller et al. 2009) and 2.5 year variation (M€ uller et al. 2009). GRACE provides monthly averaged data, which means that extreme values are smoothed in the average. Using (72.1) yields an unavoidable underestimation of Trend [kg/m2/yr] 0
0
0
0
40
30
0 –1 –20
50
0
10
Fig. 72.1 Linear trend of the Antarctic surface mass
A ¼ AN
1 ; sincð1=NÞ
(72.2)
where AN is the estimated amplitude using the averaged data, and A is the “real” amplitude. For large number of N the difference is negligible, however for the semiannual period it already means a 5% underestimation. Hence we suggest the use of the combination of (72.1) and (72.2) for determining mass variations, so
Usually the linear trend of the surface mass variation is determined simultaneously with the periodic variations (e.g. Chen et al. 2006). Typically two frequencies, the annual and the semi-annual are considered, as follows:
þ A2 sinðo2 t þ ’2 Þ þ A3 t;
the amplitude of the periodical variations. The amount of the underestimation for a time series averaged into N samples is (Bath 1974):
1 sinðo1 t þ ’1 Þ f ðtÞ ¼ A1 sinc 12 1 sinðo2 t þ ’2 Þ þ A3 t: þA2 sinc 6
(72.3)
Surface mass variations were determined following the scheme of Swenson and Wahr (2002). Gravity field coefficients of CSR RL (Release) 04 were used up to the maximum degree and order (60) for the period of day 112, 2002 through day 139, 2009 (Bettadpur 2003). The obtained mass variations were smoothed by a Gaussian filter with a radius of 500 km (Jekeli 1981; Swenson and Wahr 2002). The ICE5G GIA model (Peltier 2004) was used to correct for the GIA effect in the GRACE data. In order to provide an overview of the typical magnitude of the detected variations, area-weighted mean of them has been determined. The average amplitude of the annual variation is found to be 11.19 kg/m2 (or 11.19 mm in equivalent water thickness), that of the semi-annual variation is 5.06 kg/m2, while the trend is 6.31 kg/m2/ year. Figure 72.1 shows the estimated linear trend. The estimated surface mass variation trend consists of several geophysical causes and modelling errors. The mass sources are provided by temporal variations of hydrological processes (snow accumulation, mass redistributions by ice flows, ice melting, etc.), redistributions of the atmospheric mass, and that of the solid Earth (elastic loading due to present-day mass variations and visco-elastic GIA due to the extreme mass loss over the continent since the Last Ice Age). Modelling errors are generated on both the geophysical signal side (leaking of the atmospheric, hydrologic and oceanic mass variations from outside
Mass-Change Acceleration in Antarctica from GRACE Monthly Gravity Field Solutions
of the area of interest) and on the observation/ processing side (GRACE measurement errors, gravity model errors). For ice mass balance investigations the effect of the melting and snow accumulation is the point of interest. GRACE solutions have been corrected for the atmosphere loading effect. Recent investigations could not detect significant error for this correction (Gruber et al. 2009). The elastic loading is taken into account by the standard manner using elastic Love numbers (c.f. Swenson and Wahr 2002). The mass effect of the surrounding area is found to be generally negligible (Wahr et al. 2000). Therefore surface mass variations contain contributions mainly from the ice melting, snow accumulation, horizontal ice flows, gravity field model errors and GIA modelling errors. In this study we propose the use of a differential method in order to eliminate constant contributions of the mass trend. Horizontal ice flows in Antarctica at most places are known to be steady over long time scales (Wahr et al. 2000, according to Oerlemans 1981), so most of its effect is anticipated not to significantly influence the difference of two subsequent epochs. Partially snow accumulation is also eliminated, since it has a strong century-scale rate, about 18 cm/year (Wahr et al. 2000), though there are interannual and interdecadal variations of accumulation as well. These are unlikely remaining in a differential solution. Generally visco-elastic GIA can be assumed to be constant within a decade, or so. Therefore temporal variation of the trend consists relevant contribution from the ice melting and the snow accumulations, furthermore measurement errors, gravity modelling errors, modelling errors of elastic GIA. We assume that the systematic part of the later error effects is eliminated by a differentiation, thus generally their contribution is reduced. Inter-annual and inter-decadal variations are still in the recovered surface mass variation signal, and cannot be separated adequately from the trend using a time series of 7 years or so.
72.3
Estimation of the Trend by Moving Windows
Temporal variations of the trend were determined by estimating the trend for shorter periods, and moving the estimation period window epoch by epoch. Each
593
30 20 10 trend [kg/m2/yr]
72
0 –10 –20 –30
2 yr 3 yr 4 yr 5 yr
–40 –50 2003
2004
2005
2006
2007
2008
2009
time [yr]
Fig. 72.2 Time series of the trend estimated by moving a temporal window along the mass variations Table 72.1 Statistics of the trend estimation Window size (year) 2 3 4 5
Trend and RMS 6.48 21.34 kg/m2/year 8.09 13.00 kg/m2/year 8.56 6.93 kg/m2/year 8.32 2.77 kg/m2/year
trends then referred to the middle of the time interval, resulting in a time series of trends with a length of number of epochs minus length of the window. For the size of the moving window 2, 3, 4 and 5 years were used. The time series for a randomly chosen point are displayed in Fig. 72.2. The consistency of the trend estimation is determined by simple statistics (Table 72.1). For each window size average by area and time of the trends was derived for determining a typical trend for the continent, and the standard deviation of the trends in a similar manner to determine the consistency of the trend estimation. According to the table, with the exception of the 2 year windowing, the typical trends are close to each other. If we have a look on the temporally averaged trends (not shown), the pattern is also very similar for each window sizes (and also similar to Fig. 72.1). The consistency, however, varies a lot. The large inconsistency of the trend estimations suggests a strong dependence on the timing of the data, which has been used for trend estimation. The standard deviation decreases by increasing window size, suggesting the obvious fact that longer data series provide more reliable trend estimation.
€ldva´ry L. Fo
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Interestingly, the standard deviation of the 5 year windowed trend estimation (2.77 kg/m2/year) is in fairly good agreement with precipitation model based estimate by Wahr et al. (2000), which was found to be 3 kg/m2/year. In that pre-GRACE study the dependence of the hydrology signal (considered to be the observable by GRACE) on the timing of the sampling of a 5-year period (i.e. the mission period) was estimated in a similar moving window manner, as is done in this study. The similarity of the estimated and the observed consistency can be a pure coincidence, but also can be a validation of either the method used in this study, or the hydrology model used by Wahr et al. (2000).
a
72.4
Time series of the trend in each point manifest a periodic and a linear characteristic (c.f. Fig. 72.2 in a randomly chosen point). The linear temporal variation of the trend is named in this study as trend rate. This is obtained by a simple linear trend fit to each time series in each point. The resulted in an independent trend rate estimates in each point for each window sizes are shown in Fig. 72.3. In Fig. 72.3 all trend rate estimates have similar pattern, though the exact values of the trend rates differ. A weighted average of the trend rate estimates has been determined using the variances of the trends
b
Trend rate [kg/m2/yr2], window size: 2 yr
Estimation of the Trend Rate
Trend rate [kg/m2/yr2], window size: 3 yr
–5
0
5
0 5
5
0 –5
5
–1 5
–5
0
0 –1
–5
–5
0
–10
d
Trend rate [kg/m2/yr2], window size: 4 yr
Trend rate [kg/m2/yr2], window size: 5 yr 5
5
5
5
–10
5
0
c
5
0
0
–5
–10
0
–15
0
–5
–5
–5
0
–15
0
–5
0
–1
–5
Fig. 72.3 Trend rate estimates for (a) 2 year, (b) 3 year, (c) 4 year, and (d) 5 year moving windowed trend time series
72
Mass-Change Acceleration in Antarctica from GRACE Monthly Gravity Field Solutions
595
30
Average trend rate [kg/m2/yr2]
5
–5
20
–10
trend [kg/m2/yr]
0
10
–5
–5
–10
0
0 –10 –20 2 yr 3 yr 4 yr 5 yr
–30
0
–10
–5
–5
Fig. 72.4 Trend rates over Antarctica in the period of 2002.112–2009.139 derived as in the text
(cf. Table 72.1) for weights. The weighted average trend rate is shown in Fig. 72.4. The accuracy of the trend rate estimates have been determined by the confidence intervals of the trend rate estimates at 95% confidence level. Student t-distribution statistics of the regression coefficients (i.e. the trend rate), rc were determined, and the confidence interval has been defined as: rc 2 ½rcestimated s tN2 ; rcestimated þ s tN2 , where tN-2 is the t-distribution coefficient at a number of epochs, N, reduced by 2, at 95% confidence level, and s is the standard deviation of the regression coefficient, i.e. vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 PN 2 u i¼1 ðyi yÞ : s ¼ tN2 PN 2 i¼1 ðxi xÞ It is turned out that the trend rate field in Fig. 72.4 lies within the 95% confidence intervals in each point.
72.5
–40 2003
2004
2005
2006 time [yr]
2007
2008
2009
Fig. 72.5 Area-weighted average series of the trend estimations as on Fig. 72.2
(c.f. Fig. 72.2) was determined, and the obtained average series (c.f. Fig. 72.5) was further investigated. A periodical time series of mass variation was sought for, which provides similar pattern to Fig. 72.5 after the windowed trend estimation is applied in exactly the same manner as for the GRACE-derived surface mass variations. The best fit at about 3.8 year with 40 kg/m2 amplitude has been found. This period provides a nice interpretation of the periodic variations of the 2- and the 3-year windowed average series. However 4- and 5-year averages differ, suggesting variations at longer periods, which cannot be detected due to the shortness of the time span. This is an unlikely feature, which probably has nothing to do with real mass variations. The existence of this periodical variation is probably due to the areaweighted averaging – this periodic variation cannot be found in each point, it is only visible on the areaweighted average, therefore this period should be an integrated effect of various variations.
72.6
Discussion and Conclusions
The average of the mass variation trend (cf. Fig. 72.1)
Estimation of the Periodic Variation is 6.31 kg/m2/year. This is relatively “small” comof the Trend pared to the huge amount of the ice in Antarctica, so
Since periodic variations of the trend time series are of less interest for us, the periodic variations were only approximately investigated. Instead of doing it pointwisely, area-weighted average of trend time series
no convincing answer on the effect of the global warming can be provided. In contrary, the mass loss of Greenland, using exactly the same method and parameterization as in this study, was found to be 64.30 kg/m2/year (Nemeth 2009), which is more
€ldva´ry L. Fo
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than 10 times larger impact on the more than 10 times smaller ice content, and rather can be assumed to be an irreversible tendency. Moreover, the estimated Antarctic trend is notably influenced by the choice of the GIA model. The differences of the ICE5G (Peltier 2004) and IJ05 (Ivins and James 2005) models were determined for the area. The area-weighted average of the differences of these models was found to be 2.39 kg/m2/year, which is in the same order of the mass variation estimate. Thus the determination of the mass variation of the Antarctic ice is not convincing on continental scale. This led us to determine the acceleration of the melting. This way the constant trend is not investigated, therefore several temporally constant component of the trend is not concerned (including GIA). Technically for trend estimation purposely none of the most elaborated methods was used (e.g. Guo et al. 2010; Chen et al. 2006), but a simple one, in order to avoid numerical manoeuvres involving elusive, hardly definable error propagation. It also means that the result (Fig. 72.4) is probably not the most optimal one, instead along with the corresponding accuracy tests it is a validation of the methodology. Still, a comparison of Fig. 72.4 with Fig. 72.1 suggests that at the largest area of Antarctica, where both figures have a negative sign, the melting is accelerated. At places where the trend is positive but the acceleration is negative, such as at Enderby Land, it means that the mass accumulation is slowed down. There are two locations where the acceleration has a positive sign: these are Queen Maud Land and the Ross Ice Shelf. At Ross Ice Shelf the mass is generally decreasing, so here a slowing of the melting could be detected. In case of the Queen Maud Land making estimations with different parameters for smoothing radius and using different GIA models, the results became very divers (not shown). So no emphatic mass change could be defined in this area, still a mass accumulation is expected on longer time spans. The area-weighed averaged ice mass accelerated melt estimate is 3.16 5.15 kg/m2/year2. It is suggesting a melting tendency in the future. However, the standard deviation shows that this is still an initial estimate, which can be a good basis for future refinement of this technique.
Acknowledgement This research has been supported by the Bolyai-Kelly scholarship. The contributions of CK Shum, Junyi Guo, Hyongki Lee and Zhenwei Huang are gratefully acknowledged.
References Bath M (1974) Spectral analysis in geophysics. Developments in solid Earth geophysics. Elsevier, Amsterdam Bettadpur S (2003) Level-2 gravity field product user handbook. GRACE project material Chen JL, Wilson CR, Blankenship DD, Tapley BD (2006) Antarctic mass change rates from GRACE. Geophys Res Lett 33:L11502. doi:10.1029/2006GL026369 Gruber T, Zenner L, J€aggi A (2009) Impact of atmospheric uncertainties on GRACE de-aliasing and gravity field models. Presented paper at ‘Geodesy for Planet Earth’, IAG 2009 Buenos Aires Guo JY, Duan XJ, Shum CK (2010) Non-isotropic Gaussian smoothing and leakage reduction for determining mass changes over land and ocean using GRACE data. Geophys J Int 181:290–302 Heki K, Matsuo K (2009) Ice loss versus uplift: current mass balance in Asian high mountains from satellite gravimetry. Presented paper at ‘Geodesy for Planet Earth’, IAG 2009, Buenos Aires Ivins ER, James TS (2005) Antarctic glacial isostatic adjustment: a new assessment. Antarct Sci 17(4):541–553 Jekeli C (1981) Alternative methods to smooth the Earth’s gravity field. OSU Report Series, vol. 327 M€uller J, Peterseim N, Steffen H (2009) Mass variations in the Siberian permafrost regions from GRACE. Presented paper at ‘Geodesy for Planet Earth’, IAG 2009, Buenos Aires Nemeth D (2009) Analysis of ice mass redistribution in Greenland based on GRACE gravity models. Proceedings of the Scientific Students Conference, Budapest University of Technology and Economics, Budapest (accepted) Oerlemans J (1981) Effect of irregular fluctuations in Antarctic precipitation on global sea level. Nature 290:770–772 Peltier WR (2004) Global glacial isostasy and the surface of the ice-age Earth: the ICE-5G (VM2) model and GRACE. Annu Rev Earth Planet Sci 32:111–149 Peltier WR (2009) Closure of the budget of global sea level rise over the GRACE era: the importance and magnitudes of the required corrections for global glacial isostatic adjustment. Quatern Sci Rev. doi:10.1016j.quascirev.2009.04.004 Shum CK, Kuo C, Guo J (2008) Role of Antarctic ice mass balances in present-day sea level change. Polar Sci. doi:10.1016/j.polar.2008.05.004 Swenson S, Wahr J (2002) Methods for inferring regional surface-mass anomalies from GRACE measurements of timevariable gravity. J Geophys Res 107(B9):2193 Wahr J, Duncan W, Bentley C (2000) A method of combining ICESat and GRACE satellite data to constrain Antarctic mass balance. J Geophys Res 105(B7):16,279–16,294
Mass Variations in the Siberian Permafrost Region from GRACE
73
€rgen Mu €ller, and Nadja Peterseim Holger Steffen, Ju
Abstract
After 7 years in orbit, the GRACE satellite mission now facilitates the detection of smaller secular trends of mass variations as well as long-periodic signals. In this study, we focus on changes of the permafrost regime in Siberia, Russia, using GRACE monthly solutions from the three main analysis centres GFZ, CSR and JPL. The results show that observed positive trends of mass changes are related to large Siberian rivers such as Ob, Lena and Yenisei. Two major trends of about 0.7 mGal/a can be clearly identified. The first concerns the upper Ob River. It includes, depending on the specific GRACE solution centre, the Angara River drainage basin, which is part of the Yenisei River system. The second trend is centred on the upper Lena River north-east of Lake Baikal and is also clearly determined, but with different solution-dependent values. All these significant trends seem to be caused by long-term hydrological changes, especially since no other reasonable geophysical explanation is found yet. Similar features can be found in the trend of the GLDAS hydrology model. Removing the hydrological contribution positive mass changes of about 0.8 mGal/a appear in the Central Siberian Plateau and the Kolyma River drainage basin, which may be related to changes in permafrost. However, further investigations are needed to really understand such mass changes and attribute them to the related physical processes.
73.1
H. Steffen Department of Geoscience, University of Calgary, 2500 University Drive NW, Calgary, AB, Canada T2N1N4, e-mail:
[email protected] J. M€uller (*) N. Peterseim Institut f€ur Erdmessung, Leibniz Universit€at Hannover, Schneiderberg 50, 30167 Hannover, Germany e-mail:
[email protected]
Introduction
The Gravity Recovery and Climate Experiment (GRACE) satellite mission is now in orbit for more than 7 years. A number of studies regarding periodic signals as well as long-term trends have been carried out so far. For the trends, research is/was focused on regions with large signals such as Greenland and Antarctica for ice melt (see Horwath and Dietrich 2009, for a summary) and North America and Fennoscandia for glacial rebound-induced mass variations (see, e.g., Ivins and Wolf 2008). The long
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_73, # Springer-Verlag Berlin Heidelberg 2012
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data timespan now also allows for the detection of smaller secular trends of mass variations as well as long-periodic signals in several regions that are still insufficiently investigated. In our study, we have selected the permafrost regime in Siberia, Russia, for further investigation. Climate over Siberia has experienced significant changes during the past few decades (Yang et al. 2002), with considerable effects such as permafrost thawing (Pavlov 1994) due to global warming, which is also expected to be particularly significant in Arctic regions (Papa et al. 2008). As an example, Yang et al. (2002) found that the average annual discharge of fresh water from the six largest Eurasian rivers to the Arctic Ocean has increased by 7% over the last century. The permafrost envelopes an area in north-eastern Siberia ranging from the lower Yenisei River in the west to Kamchatka in the east (Fig. 73.1a), whereas permafrost south of 60 N is mainly related to high mountain areas. The three largest rivers, the Ob, Yenisei and Lena rivers, flow into the Arctic Ocean, having a mean discharge of 395, 610 and 532 km3/year, respectively, which in sum results in approximately 46% and thus almost a half of total river inflow into the Arctic Ocean (Berezovskaya et al. 2004). Their river basins are located in different parts of the permafrost area (Fig. 73.1b). While the Ob River basin is mainly located west of the permafrost, the Lena River basin completely covers permafrost areas. In contrast, only eastern tributaries of the Yenisei River flow through permafrost areas. Several studies have investigated hydrological changes in, e.g., river runoff, precipitation and snow
H. Steffen et al.
cover in Siberia (e.g., Berezovskaya et al. 2004, 2005; Yang et al. 2002, 2003, 2004a, b). Interestingly, Berezovskaya et al. (2004) already noted an increase in river runoff which is in contrast to an observed precipitation decrease. As permafrost occupies about 80% of Siberia, reaching the greatest thickness of over 1 km in Yakutia, central Siberia (Duchkov 2006), it is expected that permafrost thawing will have an impact on the water balance especially in the large rivers. It will thus lead to mass changes that can be detected by GRACE. Hence, the aim of this paper is to analyse the long-term trend component determined from GRACE monthly solutions on signals induced by hydrological changes in Siberia that can be related to permafrost thawing. In the next section, we briefly describe our analysis approach followed by a description and discussion of the results. Finally, we summarise our main findings.
73.2
GRACE Processing
GRACE monthly solutions from the three main analysis centres Helmholtz-Zentrum Potsdam, Deutsches GeoForschungsZentrum (GFZ), University of Texas at Austin, Center for Space Research (CSR), and the Jet Propulsion Laboratory (JPL), Pasadena, are used to determine gravity changes in Siberia. As gravity field variations of the Earth result from the integral effect of oceanic, atmospheric and hydrological mass movements and those caused by dynamics in the Earth’s interior, adequate pre-processing and appropriate filter techniques have to be applied to extract the hydrological contribution. Fortunately, it
Fig. 73.1 (a) Modified figure of Yang et al. (2002) showing the permafrost distribution in Siberia. (b) figure of Berezovskaya et al. (2004) showing the Ob, Yenisei and Lena river basins in Siberia
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Mass Variations in the Siberian Permafrost Region from GRACE
can be assumed that in Siberia the long-term signal is solely generated by hydrological changes, which includes runoff, snow cover and permafrost changes. In addition to these variations, the monthly solutions may also contain residual signals from insufficient pre-processing. We use 69 monthly gravity field solutions from CSR and JPL and 68 from GFZ for the time window from July 2003 to March 2009. January 2004 is missing in the GFZ dataset. All analysis centres have reduced the atmospheric and oceanic contributions as well as the tidal effects during the standard GRACE processing by applying corresponding global models. Each monthly solution consists of a set of Stokes coefficients Clm and Slm up to degree and order 120 (GFZ, JPL) or 60 (CSR). Calibrated errors are provided by CSR and GFZ only. The gravity field solutions require smoothing to reduce the effects of errors present in shortwavelength components. If the smoothing radius is chosen too small, these errors manifest themselves in maps of surface mass variability as long, linear features generally oriented north to south (so-called stripes, see Swenson and Wahr 2006, for more information). Due to the mentioned errors, the spherical harmonic coefficients are only considered up to degree and order 50 (which in principle corresponds itself to a rectangular box filtering), followed by different filter techniques, which will be discussed below. The gravity values dg (’, l, t) are computed on a 1 1 grid for each monthly solution. We simultaneously fit a constant, a linear trend, an annually and 2.5-yearly varying term. The latter is an average period obtained by a basin-related and filter-dependent frequency analysis according to Schmidt et al. (2008). In addition, a 161 day period is included to reduce effects that may result from an insufficient ocean tide correction (aliasing), particularly in high latitude areas (Ray et al. 2003). The accuracy of the determined parameters is about 0.1 mGal/a for the secular trend, when applying a Gaussian filter with 400 km radius. From the several filter techniques published in the last years (see Kusche 2007, for a summary), we have depicted the commonly used Gaussian filter (Wahr et al. 1998) and additionally tested the so-called destriping filter (Swenson and Wahr 2006). While the Gaussian filter depends on the spherical harmonic degree l only, the destriping filter is a non-isotropic
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filter for decorrelation of GRACE coefficients. The spectral signature of the correlated errors is examined and removed using polynomials, which clearly reduces the presence of stripes. Meanwhile a 400 km filter radius is mainly applied (Steffen et al. 2009). Nevertheless, we tested several filter radii. Figures 73.2a–e show the trend estimates from GFZ monthly solutions using five different filter radii between 350 and 800 km. More detailed features are visible when using a smaller filter radius such as 350 or 400 km. There are prominent minima between 40 and 50 N as well as two main regions of positive mass change around 60 N. In turn, the stripe feature shows up with smaller radii, probably resulting in several non-realistic minima south of 50 N. This feature vanishes with larger radii of at least 450 km. Using the destriping filter (Fig. 73.2f), NS-stripes signatures are almost completely removed. Unfortunately, real signals may removed with this filter as well (Swenson and Wahr 2006). As the large rivers in Siberia flow from south to north, possible hydrological changes are strongly affected by this filter technique and likewise are removed. Thus, for further investigation, we select the Gaussian filter with 400 km filter radius. Applying it, the maximum areas become isolated and the signature of the stripes has either a minor contribution or is actually a real mass change.
73.3
Long-Term Gravity Changes in Siberia
Figure 73.3c shows the long-term gravity change in Siberia as determined with a Gaussian filter of 400 km from the GFZ solutions. A clear signal of about 0.7 mGal/a is visible in an area between the upper Ob-Irtysh drainage system and the Angara River, a tributary to the Yenisei River. A second maximum with a value of less than 0.6 mGal/a appears more to the east in the upper Lena River. In the same range, another maximum is found in the north-east close to the Indigirka and Kolyma rivers. Minima occur around the Caspian Sea, east of the Aral Sea, in Mongolia and north-eastern China. They range from 1 to 0.6 mGal/a. The CSR solution (Fig. 73.3a) also shows clear signals, but larger for the maxima and smaller for the minima. Furthermore, the doublepeak feature for the Ob-Yenisei area as seen by GFZ is coalesced to an elliptic shape. The maximum in the
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(c) 450 km, (d) 600 km, (e) 800 km, (f) destriping filter from Swenson and Wahr (2006) with additional 400 km Gaussian filter. Units in mGal/a
upper Lena river is more prominent, reaching now more than 0.6 mGal/a. Compared to the maxima found with these two solutions, slightly smaller signals in extension and magnitude are obtained for the JPL solution (Fig. 73.3b). More negative trends are seen compared to CSR. The differences between the three GRACE centres are caused by different processing (e.g., use of different reduction models), see Steffen et al. (2009).
Comparing the GRACE results to the distribution of permafrost in Siberia (Fig. 73.1a), the maximum is found south-west of the permafrost area. The Ob River is not affected by changes in permafrost as there permafrost is only found close to the Obmouth. The Angara River is also located far from permafrost. Thus, this maximum cannot be related to permafrost thawing in Siberia and has to have its origin in other hydrological changes (or residual atmospheric
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Fig. 73.3 Secular variation after Gaussian filtering with 400 km radius in Siberia determined from GRACE monthly solutions as provided by CSR (a), JPL (b), and GFZ (c), as
well as from the GLDAS hydrology model (Rodell et al. 2004) (d); (e) Difference of (c) and (d). Units in mGal/a.
contribution). Changes in the geosphere such as uplift or mass movements in the Earth’s mantle (Steffen et al. 2008) are unlikely. The second maximum (in the east) is located in the permafrost area between the Vilyuy River, a tributary to the Lena River, and the Lena River itself. An effect from permafrost changes is thus more likely. However, before discussing both maxima in more detail, a comparison to known hydrological
phenomena is indispensable. We compare the GRACE-derived trends to the Global Land Data Assimilation System (GLDAS, Rodell et al. 2004), an almost global hydrology model (spatial extent is all land north of 60 S) provided in monthly fields with 1 1 resolution. It is generated by optimal fields of land surface data such as soil, vegetation and elevation, and forced by multiple datasets derived from satellite measurements and atmospheric analyses.
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We use monthly fields from July 2003 to March 2009 to be consistent with GRACE. The comparison (Fig. 73.3a–d) shows several similarities especially for the two maximum regions and for some of the minima south of 50 N. Generally, the only maxima found in the GLDAS trend are at the two locations also found with GRACE, though smaller in the west and slightly larger in the east. In Figure 73.3e we subtract the hydrology trend from the GFZ GRACE result, which should actually result in quite small values in the range of the errors (0.1 mGal/a) when assuming that (a) GRACE only reflects hydrological changes in Siberia and (b) GLDAS is correct. In contrast, the GRACE-derived maxima and almost all minima (except the one in the Caspian Sea) vanish and new maxima appear. The most prominent one is found in the Central Siberian Plateau north of Lake Baikal reaching a bit more than 0.8 mGal/a. Other maxima of about the same value show up around the Kolyma River, in Kamchatka Peninsula, and around the upper Amur rivers. All of these maxima are already indicated in GLDAS as slight or quite large minima. In combination with GRACE they change sign to prominent maxima. In view of changes in the permafrost regime two maxima are interesting, the one in the Central Siberian Plateau and the one in the Kolyma River drainage basin. Several tributaries of the Yenisei and Lena rivers rise in the Central Siberian Plateau. However, both river basins do not show an increase and remain more or less constant in their mass balance. There are only slight variations within the errors and in total no change is determined. But, the Kolyma River experiences a strong mass increase which in turn is in contrast to hydrological studies that observed a long-term decrease in discharge of up to 11% (Mahji and Yang 2008). GLDAS also indicates a small negative trend which is in agreement with this result. As GLDAS does not include a permafrost contribution (Rodell et al. 2004), our result may point to permafrost changes in the Kolyma River drainage basin. The maximum in the Central Siberian Plateau can be induced by changes in the water balance, which includes snow cover, discharge and groundwater, but also sub-permafrost and supra-permafrost groundwater. Besides hydrological effects, the maximum could principally also be related to the growth of vegetation, mining (although this is not expected to cause such large anomalies) or other sources. However, up to now it is not possible to distinguish these effects from
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GRACE data alone. Furthermore, shortcomings of GLDAS modelling in Siberia might be responsible for in-correct trend signals and some misinterpretation. These discrepancies are subject to further investigation.
73.4
Summary and Outlook
In this study, we investigated temporal gravity variations in the Siberian permafrost regime based upon GRACE monthly gravity field solutions. The GRACE data, after testing different filter approaches, clearly show temporal gravity variations in Siberia with two maxima of about 0.7 mGal/a in the upper Ob and Lena rivers as well as in the Angara River drainage basin. To further reduce “known” signals, we subtracted the long-term trend calculated with the GLDAS hydrology model from the GRACE result. New maxima of about 0.8 mGal/a in the Central Siberian Plateau and the Kolyma River drainage basin arose that might partly be related to permafrost changes such as thawing, or other several reasonable sources. In the future, a comparison to water balance estimates and independent hydrological data in Siberia may help to improve the determination of permafrost thawing signals from GRACE data and to identify possible reasons for mass changes in Siberia. Acknowledgments We would like to thank the GRACE science team for overall support, GFZ, CSR, and JPL for providing the GRACE monthly solutions. We would also like to thank Julia Boike (AWI Potsdam), Sean Swenson (University of Colorado) and Wouter van der Wal (TU Delft) for helpful discussions. This research was partly funded by the DFG (research grant MU1141/8-1).
References Berezovskaya S, Yang D, Kane DL (2004) Compatibility analysis of precipitation and runoff trends over the large Siberian watersheds. Geophys Res Lett 31:L21502. doi:10.1029/ 2004GL021277 Berezovskaya S, Yang D, Hinzman L (2005) Long-term annual water balance analysis of the Lena River. Glob Planet Change 48:84–95. doi:10.1016/j.gloplacha.2004.12.006 Duchkov AD (2006) Characteristics of permafrost in Siberia. In: Advances in the geological storage of carbon dioxide. Lombardi S et al. (eds). NATO Sci. Series IV: Earth and Environmental Sciences, 8191, Springer, Berlin
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Horwath M, Dietrich R (2009) Signal and error in mass change inferences from GRACE: The case of Antarctica. Geophys J Int 177(3):849–864. doi:10.1111/j.1365-246X.2009.04139.x Ivins ER, Wolf D (2008) Glacial isostatic adjustment: New developments from advanced observing systems and modeling. J Geodyn 46:69–77. doi:10.1016/j. jog.2008.06.002 Kusche J (2007) Approximate decorrelation and non-isotropic smoothing of time-variable GRACE-type gravity field models. J Geod 81(11):733–749, doi:10.1007/s00190-0070143-3 Mahji I, Yang D (2008) Streamflow characteristics and changes in Kolyma Basin in Siberia. J Hydrometeorol 9:267–279. doi:10.1175/2007JHM845.1 Papa F, Prigent C, Rossow WB (2008) Monitoring flood and discharge variations in the large Siberian rivers from a multisatellite technique. Surv Geophys 29:297–317. doi:10.1007/ s10712-008-9036-0 Pavlov AV (1994) Current change of climate and permafrost in the Arctic and subarctic of Russia. Permafrost Periglacial Processes 5:101–110 Ray RD, Rowlands DD, Egbert GD (2003) Tidal models in a new era of satellite gravimetry. Space Sci Rev 108 (1-2):271–282, doi:10.1023/A:1026223308107 Rodell M, Houser PR, Jambor U, Gottschalck J, Mitchell K, Meng C-J, Arsenault K, Cosgrove B, Radakovich J, Bosilovich M, Entin JK, Walker JP, Lohmann D, Toll D (2004) The global land data assimilation system. Bull Am Meteor Soc 85(3):381–394. doi:10.1175/BAMS-85-3-381 Schmidt R, Petrovic S, G€ untner A, Barthelmes F, W€ unsch J, Kusche J (2008) Periodic components of water storage
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changes from GRACE and global hydrology models. J Geophys Res 113:B08419. doi:10.1029/2007JB005363 Steffen H, Denker H, M€uller J (2008) Glacial isostatic adjustment in Fennoscandia from GRACE data and comparison with geodynamic models. J Geodyn 46:155–164. doi:10.1016/j.jog.2008.03.002 Steffen H, M€uller J, Denker H (2009) Analysis of mass variations in northern glacial rebound areas from GRACE data. In: Sideris M (ed) Observing Our Changing Earth. Springer, IAG Symp. Ser. vol. 133, 501–510, doi:10.1007/ 978-3-540-85426-5_60 Swenson S, Wahr J (2006) Post-processing removal of correlated errors in GRACE data. Geophys Res Lett 33: L08402. doi:10.1029/2005GL025285 Wahr J, Molenaar M, Bryan F (1998) Time variability of Earth’s gravity field: Hydrological and oceanic effects and their possible detection using GRACE. J Geophys Res B12:30.205–30.230 Yang D, Kane DL, Hinzman L, Zhang X, Zhang T, Ye H (2002) Siberian Lena River hydrologic regime and recent change. J Geophys Res 107(D23):4694, doi:10.1029/2002JD002542 Yang D, Robinson D, Zhao Y, Estilow T, Ye B (2003) Streamflow response to seasonal snowcover extent changes in large Siberian watersheds. J Geophys Res 108 (D18):4578 Yang D, Ye H, Kane DL (2004a) Streamflow changes over Siberian Yenisei River Basin. J Hydrol 296:59–80. doi:10.1016/j.jhydrol.2004.03.017 Yang D, Ye B, Shiklomanov A (2004b) Discharge characteristics and changes over the Ob River Watershed in Siberia. J Hydrometeorol 5(4):69–84
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Seasonal Variability of Land Water Storage in South America Using GRACE Data
74
Claudia Tocho, Luis Guarracino, Leonardo Monachesi, Andre´s Cesanelli, and Pablo Antico
Abstract
The objective of the present study is to analyze seasonal variability of land water storage in South America from GRACE data. High precision estimate of temporal variations in the Earth’s gravity are obtained using monthly Release-04 (RL04) gravity field coefficients provided by the Center for Space Research (CSR) of the University of Texas at Austin. Water mass anomalies, as equivalent height of water, are calculated based on the direct relationship between gravity and mass. To remove the effects of the noise observed in the equivalent-water thickness solutions at high harmonic degrees, an optimized smoothing technique is applied. Finally, temporal distributions of land water storage are compared to monthly mean precipitation data extracted from in-situ rain gauge records in order to identify, correlate and understand patterns of water movement at continental scale in South America.
74.1
Introduction
Variations of water storage at regional and continental scales are not known with sufficient accuracy due to the lack of appropriate direct observations and limitations in global hydrological models. An alternative method to quantify variations of water storage is based on satellite observations of the Earth’s time variable gravity field from the Gravity Recovery and
C. Tocho (*) Facultad de Ciencias Astrono´micas y Geofı´sicas, La Plata, Argentina e-mail:
[email protected] L. Guarracino L. Monachesi A. Cesanelli P. Antico Facultad de Ciencias Astrono´micas y Geofı´sicas, La Plata, Argentina Consejo Nacional de Investigaciones Cientı´ficas y Te´cnicas, Buenos Aires, Argentina
Climate Experiment (GRACE). GRACE is the first remote-sensing satellite mission which is directly applicable to the assessment of land water storage under all types of terrestrial conditions, and represents a promising contribution to global hydrology. The gravity field varies with time due to mass redistribution on both solid Earth and fluid envelopes (atmosphere, oceans, ice caps and water reservoirs). Satellite measurements of the time-variable gravity can be used to study a wide variety of geophysical problems that involve mass redistribution, like, e.g., continental water storage (Schmidt et al. 2006), global sea level (Chambers et al. 2004), polar ice sheet mass balance (Chen et al. 2008), crust deformations produced by earthquakes (Chen et al. 2007), etc. The GRACE satellite mission is a joint project between the National Aeronautics and Space Administration (NASA) and the Deutsches Zentrum f€ur Luft und Raumfahrt (DLR). It was launched on March 17,
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_74, # Springer-Verlag Berlin Heidelberg 2012
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2002, and since that time the GRACE Science Data System (SDS) has produced and distributed estimates of the Earth’s gravity field. Each gravity field solution is derived in terms of Stokes coefficients of a spherical harmonic expansion. GRACE can observe changes in the Earth’s gravity field that are caused by changes in mass distribution. These changes are due to variations in the distribution of mass in the atmosphere, ocean, and water storage in hydrologic reservoirs. The main objective of this study is to investigate how the differences of GRACE observations of two different time periods are caused by water storage change. The mass of the atmosphere and ocean are removed during the processing of GRACE data using models, and if these models were perfect, the data left would reflect the land hydrology water storage as the main contributor of the temporal gravity variations. In this paper we describe the post-processing procedures that are used to convert GRACE data into maps of equivalent water thickness for South America. The results obtained were correlated with precipitation data. This external type of data is based on rain gauge observations.
74.2
Data Used
74.2.1 Grace Data The exterior potential of the Earth System, or geopotential, includes its entire solid and fluid (oceans and atmosphere) components. The gravitational potential at a point P exterior to the Earth system may be represented using an infinite spherical harmonic series:
Legendre polynomials of degree n and order m, and Cnm ; Snm are the fully-normalized spherical harmonic coefficients of the geopotential. The spherical harmonic expansion of the geopotential requires an infinite series of harmonics, but in practice the summation is limited to a maximum degree Nmax. The independent time variable (t) is introduced in (74.1), through Cnm ðtÞ; Snm ðtÞ . This means that the geopotential at a fixed location is variable in time due to mass movement and exchange between the Earth system components. Therefore we have to treat the spherical harmonic coefficients of the geopotential as a function of time. GRACE data consists of a set of spherical harmonic coefficients, called Stokes coefficients, that describe the Earth’s gravity field and they are averaged over a period of approximately 1 month. We have used the CSR’s RL04 Level-2 GSM files computed by the Center for Space Research of The University of Texas at Austin (Bettadpur 2007), which is one of the three centers of SDS. The Level-2 GSM files contain monthly estimates of spherical harmonic coefficients for the Earth’s gravity field up to degree and order Nmax ¼ 60. The data used in this study consist of 75 files of monthly spherical harmonic coefficients from April, 2002 to September, 2008 (June and July, 2002 and June, 2003 are missing). The data has been corrected by using appropriate atmospheric and ocean models during processing. For further details of the CRS data, see Bettadpur (2007). The GRACE C20 coefficient was used, and the degree 1 coefficients were set to 0 since the origin of the reference frame coincides with the center of mass of the entire Earth system, including its solid component and fluid envelopes.
GM Vðr;y;l;tÞ ¼ 74.2.2 Precipitation Data r " # Nmax X n X 1þ Pnm ðcosyÞðCnm ðtÞcosmlþSnm ðtÞsin mlÞ The monthly precipitation data set provided by the Global Precipitation Climatology Centre (GPCC) n¼0 m¼0 (74.1) was used in this research. These data consist of monthly precipitation grids with a spatial resolution where r, y, l represent the geocentric radius, colatitude of 1 1 , which are based on rain gauge and longitude of point P, respectively; GM is the observations. We used full data monthly precipitation product of the gravitational constant and mass of the (mm/month) grids from January 2002 to December Earth; a is the mean equatorial radius (or a scale 2007 (Schneider et al. 2008; Rudolf and Schneider distance); Pnm are the fully-normalized associated 2005).
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74.3
Computation Strategy
The computational methodology to obtain mass distribution from the time-variable harmonic gravity solutions was described in detailed in Wahr et al. (1998), and it is the one we follow in this paper. The temporal variations of mass distribution will produce variations in the geoid that can be expressed as: DNðy; l; tÞ ¼ Nðy; l; tÞ N0 ðy; lÞ
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74.4 Pnm ðcos yÞðDC~nm ðtÞ cos ml
where re is the average density of the Earth, rw is the density of the water and kn is the Love number for degree n. The Love numbers take into account the Earth’s elastic compensation of superficial load due to mass redistribution. We used the kn values proposed by Wahr and Molenaar (1998) using structural parameters from the Preliminary Reference Earth Model (PREM) of Dziewonski and Anderson (1981). Unfortunately, the maps of equivalent water thickness obtained using the technique described above suffer from errors for high degrees and orders. These systematic errors present a different tendency between odd and even degree coefficients for the same order. They propagate into north–south bands called “stripes” when the maps are computed. The simple truncation of the spherical harmonic series at low degrees causes the loss of an unacceptably large portion of the signal. Therefore, before converting the GRACE gravity coefficients into maps of equivalent water thickness the data was filtered using the procedure described in Swenson and Wahr (2006). The idea of this filter is to leave unchanged the lower 8 8 coefficients and fit a second order polynomial to the remaining coefficients, different for odd and even degrees. This fit can be considered as an estimation of the systematic errors present in the GRACE data. The filtered coefficients will be the original coefficients minus the fit. After the de-stripping filtering was applied, an additional Gaussian filter was used to smooth the Stokes coefficients (Chambers 2006):
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where (Cnm ðtÞ,Snm ðtÞ) are the Stokes coefficients for a 0 0 specific month, and ðCnm ; Snm Þ, are the spherical harmonic coefficients for the static geoid. We can compute water storage anomalies h(t) over the land as:
DC~nm ðtÞ DS~nm ðtÞ
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Results
Before converting the GRACE gravity coefficients into maps of equivalent water thickness, the data was filtered using the procedure described in Sect. 74.3. In order to enhance the seasonal pattern of the hydrological signal, the GRACE Stokes coefficients,
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e.g., for all the January months (2003–2008) were averaged. The same computation was done for the remaining 11 months. In this way, we obtained 12 monthly solution grids of land water in South America, corresponding to each of the months of the year. These grids represent the water storage variations where the seasonal behavior of the data was summed in phase while all other information in the data with a period different from the annual was attenuated. They consist of 0.5 by 0.5 grids ranging from latitudes 15 N to 60 S and longitudes 30 W to 85 W. The units of the “equivalent water thickness” grids are shown as equivalent height of water in millimetres. The water storage anomalies maps for the 12 months are depicted in Fig. 74.1. We can observe from these maps than the amplitudes vary from 350 to 350 mm. Figure 74.1 shows a clear seasonal pattern of the water mass variations in South America, which are confined mostly to tropical latitudes. We identify two regions within tropical South America: Amazonia, which extends from the equator to 20 S, and Orinoco, which extends north of the equator. The water storage reaches its maximum in Amazonia during March–June, and its minimum during September–December. Conversely, the water storage has its maximum in Orinoco during July–October and its minimum during January–April. Seasonal variations in water storage are lesser than those of Amazonia. This occurs because the available surface for water storage in Orinoco is smaller than in Amazonia. Results from GRACE were compared with monthly precipitation data from GPCC at 1 1 resolution. We computed the total average of the dataset for the period 2002–2007 and then it was subtracted from each one of the 12 monthly means. Thus, we obtained 12 panels that show the seasonality of precipitation in South America as they are shown in Fig. 74.2. In both figures, units are mm of equivalent water height. The highest variation in monthly precipitation is found within the Tropics north of 20 S, with values ranging on average from 350 to +350 mm. They describe a pattern with opposite rainy seasons in each hemisphere. Thus, Amazonia has most of its annual precipitation during January–March, when the dry season occurs in the Orinoco. The dry season in Amazonia extends during June–August, while the rainy season in Orinoco occurs during May–August.
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Other regions are found in South America with less variation in precipitation amounts. One is in the subtropics (20–35 S) east of the Andes and another one along the Andes between 35 and 45 S.
74.5
Conclusions and Future Work
The methodology presented seems adequate to analyze seasonal water storage variations based on GRACE measurements in continental scales such as South America. Mass variations produce changes in gravity that were converted into an equivalent change in water storage. However the technique can not distinguish between water masses stored above or below Earth’s surface. Variations of water storage at regional and continental scales are not known with sufficient accuracy due to the lack of appropriate direct observations and limitations in global hydrological models. GRACE data can be used to observe and monitor water storage changes in large basins with difficult access. The observed GRACE anomalies were shown as equivalent height of water in millimetres and were compared with precipitation data. We identify similar patterns for both the precipitation and the water storage variation. The main observed anomalies coincide with two large basins: Amazon and Orinoco. The analysis was based in the definition of four-month periods with maximum and minimum values both for water storage and precipitation. It was found that the water storage maximum occurs three months later than the previous rainy season, both in Amazonia as in Orinoco. A similar behaviour was observed for the minimum water storage in Amazonia and the previous dry season. In contrast, the minimum water storage in the Orinoco occurs just one month after the previous dry season. The patterns between water storage and precipitation are similar. But they are not in phase (2–3 months). These can be due to the velocity of the flux of groundwater is very slow in the continent. We expect to improve the data resolution, eliminating data noise still present using different types of filters like a non-isotropic smoothing decorrelation filter (Kusche et al. 2008), and we will also investigate how the 10-day temporal resolution of the GRGS (Bruinsma et al. 2010) impact on the computation of the water storage variations.
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Seasonal Variability of Land Water Storage in South America Using GRACE Data
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March
April
May
June
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Fig. 74.1 GRACE water storage variations maps. Units: mm
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February
March
April
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May
June
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Fig. 74.2 GPCC precipitation data. Units: mm
December
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Seasonal Variability of Land Water Storage in South America Using GRACE Data
References Bettadpur S (2007) CSR level-2 processing standards document for product release 04, GRACE 327–742. The GRACE project, Center for Space Research, University of Texas, Austin Bruinsma S, Lemoine JM, Biancale R, Vale`s N (2010) CNES/ GRGS 10-day gravity field models (release 2) and their evaluation, FEB 15 2010, Advances in Space Research. 45 (4):587–601 Chambers DC (2006) Converting release-04 gravity coefficients into maps of equivalent water thickness. http://grace.jpl. nasa.gov/files/GRACE-dpc200711_RL04.pdf. Accessed date 17/08/2011 Chambers DP, Wahr J, Nerem RS (2004) Preliminary observations of global ocean mass variations with GRACE. Geophys Res Lett 31:L13310. doi:10.1029/ 2004GL020461 Chen JL, Wilson CR, Tapley BD, Grand S (2007) GRACE detects coseismic and postseismic deformation from the Sumatra-Andaman earthquake. Geophys Res Lett 34(13): L13302. doi:10.1029/2007GL030356 Chen JL, Wilson CR, Blankenship BD, Young D (2008) Antarctic regional ice loss rates from GRACE. Earth Planet Sci Lett 266:140–148 Dziewonski A, Anderson DL (1981) Preliminary reference Earth model. Phys Earth Planet Int 25:297–356 Kusche J, Schmidt R, Petrovic R, Rietbroek R (2008) Decorrelated GRACE time-variable gravity solutions by
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GFZ, and their validation using a hydrological model. J Geod. doi:10.1007/s00190-009-0308-3 Ramillien G, Cazenave A, Reigber CH, Schmidt R, Schwintzer P (2005) Recovery of global time-variations of surface water mass by GRACE geoid inversion. International Association of Geodesy Symposia, vol. 12. In: Jekeli C, Bastos L, Fernandes J (eds) Gravity geoid and space missions 2004. Springer, Berlin, pp 310–315 Rudolf B, Schneider U (2005) Calculation of gridded precipitation data for the global land-surface using in-situ gauge observations. Proceedings of the 2nd workshop of the international precipitation working Group IPWG, Monterey October 2004, EUMETSAT, ISBN 92-9110-070-6, ISSN 1727-432X, 231–247 Schmidt R, Schwintzer P, Flechtner F, Reigber CH, G€unter A, D€oll P, Ramillien G, Cazenave A, Petrovic S, Jochmann H, W€unsch J (2006) GRACE observations of changes in continental water storage. Glob Planet Change 50:112–126 Schneider U, Fuchs T, Christoffer A, Rudolf B (2008) Global precipitation analysis products of the GPCC. Global Precipitation Climatology Centre (GPCC) Internet Publikation, pp 1–12. http://gpcc.dwd.de Accessed July 2009 Swenson S, Wahr J (2006) Post-processing removal of correlated errors in GRACE data. Geophys Res Lett 33: L08402. doi:10.1029/2005GL025285 Wahr, J, Molenaar M, Bryan F (1998) Time variability of the Earth’s gravity field: Hydrological and oceanic effects and their possible detection using GRACE. J Geophys Res 103:30205–30229
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Water Storage Changes from GRACE Data in the La Plata Basin
75
A. Pereira, S. Miranda, M.C. Pacino, and R. Forsberg
Abstract
Aquatic environments perform important functions in nature such as the control of climate, floods and nutrients, and they provide goods and services for humanity. To monitor these environments at large spatial scales, the satellite gravity mission GRACE provides time-variable gravity field models that reflect the Earth’s gravity field variations due to mass transport processes like continental water storage variations. The La Plata Basin is the second largest one in South America and it is a sample of the abundance, variety and quality of natural resources and the possibilities offered in connection with the production of goods and services. In this work the GRACE capability to monitor the water storage over La Plata Basin will be analyzed, using the solutions provided by the four different GRACE processing centers: CSR, GFZ, JPL and BGI. Afterward the calculated hydrological signal will be used to estimate four mass change models over this hydrographic system’s area using a generalized inversion method on the gravity trends. Also, preliminary results from ENVISAT altimetry data are presented and compared with GRACE solutions. All the solutions detected the significant mass changes of the area, thought there are some discrepancies between the four GRACE processing centers.
75.1 A. Pereira (*) M.C. Pacino Facultad de Ciencias Exactas, Ingenierı´a y Agrimensura de la Universidad Nacional de Rosario, CONICET, Rosario, Argentina e-mail:
[email protected] S. Miranda Facultad de Ciencias Exactas, Fı´sicas y Naturales, Universidad Nacional de San Juan, Rivadavia, Argentina R. Forsberg National Space Institute, Technical University of Denmark, Copenhagen, Denmark
Introduction
Recent gravity satellite campaigns represent a great improvement to several applications related to the gravity field modeling; moreover, they provide valuable information about the geodynamic behavior of our planet since they offer the temporal variations of the gravity field. GRACE (Gravity Recovery And Climate Experiment) mission can map the mass distribution by measuring the changes in the Earth’s gravity field. Most of the monthly gravity changes detected by this campaign
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can be associated with the variations in terrestrial water storage, which includes hydrological reservoirs changes, groundwater, soil moisture, lakes, streams, snow, ice and glaciers (Ramillien et al. 2005; Rodell et al. 2007; Velicogna and Wahr 2005, 2006; Wahr et al. 2004). A few years ago, these measurements were only achieved with ground-based instrumentation, obtaining results of low precision and for small areas. The La Plata Basin is one of the largest in the world with an area of about 3,100,000 km2, which is equivalent to the 17% of South America’s area. Also, in the La Plata basin lies a big part of the second largest fresh groundwater reservoir of the world (the Guaranı´ Water-bearing System), thus knowing its behavior in space and time is of extreme importance for being a great environmental, economic and strategic resource. The satellite gravity mission GRACE offers one such possibility. Temporal and spatial variations of water storage in the area covered by the basin are difficult to measure using ground data because of the hydrographic system’s size; but with the present advance of technologies represented by GRACE data it is possible to detect the monthly spatial changes in the distribution of water masses in these regions. Furthermore, GRACE can map water storage changes to a height of about 1 cm for areas ranging in size on the order of 400 km. In this paper the water storage variability in the La Plata Basin is estimated from GRACE solutions provided by different processing centers using the inversion method proposed by Sandberg Sørensen and Forsberg (2010). In addition, ENVISAT altimetry data distributed in the basin’s area were used to compare with GRACE results between January 2003 and December 2007.
75.2
Data and Methodology
75.2.1 Study Area The La Plata basin covers five countries: Brazil (1,415,000 km2), Argentina (920,000 km2), Paraguay (410,000 km2), Bolivia (205,000 km2), and Uruguay (150,000 km2). Considering that the basin occupies a very large area, a great variety of characteristics and physical and environmental conditions can be found.
Fig. 75.1 La Plata Basin and its main rivers
This basin is located in subtropical and midlatitude areas of South America and in between The Andes Mountain Chain in the West and the Brazilian Plateau and “Sierra del Mar” in the Northeast and East. In general, there is a clear predominance of plains in the basin’s area. The main rivers that are part of this hydrographic system are the Parana´, Uruguay, Paraguay, and La Plata (Fig. 75.1); and they are also the main subbasins of the La Plata basin. The Parana´ basin is the largest one with an area of 1,510,000 km2 and a longitude of 4,000 km. It is usually divided into three sub-basins: the Upper, Middle and Lower Parana´ (Berbery and Barros 2002). The climate characteristics of the basin depend on both latitude and relief. In general, the main precipitations coincide with the Parana´, Uruguay and Paraguay rivers headwaters; and the mean annual rainfall records decreases from North to South and from East to West. Berbery and Barros (2002) have extensively analyzed the hydrological cycle of the La Plata Basin. They concluded that the amplitude of the mean annual cycle is small due to different precipitation regimes that contribute with both spatial and temporal differences.
75.2.2 GRACE Data The GRACE data used for this analysis are Level-2 (Bettadpur 2007) data spanning the period April 2002 to August 2008. They are provided by a number of
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Water Storage Changes from GRACE Data in the La Plata Basin
Table 75.1 Description of the data sets used in this study (n, m)max is maximum degree and order of the spherical harmonic expansions Release Epochs Start End (n, m)max
CSR RL04 74 4-2002 7-2008 60
GFZ RL04 68 8-2002 8-2008 120
JPL RL04.1 70 4-2002 4-2008 120
BGI RL01 202 8-2002 5-2008 50
processing centers around the world. Particularly, we have used data computed at CSR (Center for Space Research, University of Texas), JPL (Joint Propulsion Laboratories, California), GFZ (geoForschungsZentrum, Potsdam) and BGI-CNES (Bureau Gravime´trique International, Centre National d’E´tudes Spatiales, Tolouse)1. The data sets are shortly described in Table 75.1. They consist of spherical harmonic expansions (or Stokes coefficients) of the Earth’s gravity potential. The time-variability of each set is represented by a sequence of 30 day approximately, except for the BGI solutions, where the data is given every 10 days and it is based on the running average of three 10-day periods (Lemoine et al. 2007, 2008). The analysis centers follow different data preprocessing, processing and post-processing strategies, which cause differences in the sets of spherical harmonic coefficients (Klees et al. 2008). Therefore, one of the goals of this study is to quantify the differences between the different solutions of spherical harmonic models of terrestrial water mass change in the La Plata region.
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dg ðtÞ ¼ a þ bt þ c cosðtÞ þ d sinðtÞ
Figure 75.2 displays the calculated gravity trends at satellite altitude (500 km) based on CSR, JPL, GFZ and BGI epochs. To allow a fair comparison and limit the errors introduced to the high order coefficients, in all cases we have filtered the data by truncating the spherical harmonic expansions. Following Sandberg Sørensen and Forsberg (2010), the truncation was performed at degree and order 30. The plots indicate some differences among the GRACE solutions, but they do agree on a positive trend over north La Plata and a negative trend over the south, indicating a mass gain and a mass loss, respectively.
75.3
Mass Change Results
We derive four mass change models of the La Plata Basin using a generalized inversion method on the gravity trends that can be stated as a linear problem according to (75.2) (Sandberg Sørensen and Forsberg 2010): y ¼ A : x
The continental water storage changes of the La Plata basin region will be estimated from the gravity trend in the area. First we determine the change in time of the gravity fields, dg, by means of a 4-parameter trend analysis of the gravity disturbances. In order to do that a bias, trend and two yearly seasonal terms are estimated in each grid point (75.1):
1
These solutions are actually computed by the GRGS- CNES (Groupe de Recherche de Ge´ode´sie Spatiale, Centre National d’E´tudes Spatiales, Tolouse).
(75.2)
The observation vector y (75.3) consists of the gravity trend at satellite altitude determined over a grid that covers the area under study (Fig. 75.2): y ¼
75.2.3 Gravity Disturbance Trend
(75.1)
dgi ; i ¼ 1; . . . ; n dt
(75.3)
A is the response matrix, and x is the model parameter vector (75.4) which contains the point masses m. The solution domain is a grid defined by the area of the La Plata Basin. x ¼ fmj g; j ¼ 1; . . . ; m
(75.4)
For each point mass element the gravity trend can be related by the form:
dgi ¼
X
(
)
Gmj ðR2 r R3 cos cij Þ 3
j
ðr2 þ R2 2Rr cos cij Þ2
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Fig. 75.2 Gravity trends at satellite altitude (500 km) given in mgal per year for the period 2002 to 2008, based on BGI-RL01, JPLRL04.1, GFZ-RL04 and CSR-RL04 epochs
where R is the Earth radius, r ¼ R + 500 km the GRACE orbit altitude, and c the spherical distance. The problem is solved by using Tychonov generalized inversion with regularization with: x ¼ ½AT A þ lI1 AT y Here I is the unit matrix and l is a regularization factor, needed to obtain a regularized inversion
problem. The l-factor determines the necessary tradeoff between model smoothness and residuals; the total mass change of the water will be affected only to a small degree by the choice of l; the areal shape of the modelled mass change is, however, strongly affected by this choice. The total mass change is estimated by the summation of all the point masses m. This inversion method was successfully applied to estimate the total mass loss in the Greenland ice sheet
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Fig. 75.3 Estimated water storage variations expressed in km3 water equivalent from GRACE, and altimetry in meters from ENVISAT (preliminary results)
by Sandberg Sørensen and Forsberg (2010) and Forsberg and Reeh (2006). Figure 75.3 presents the monthly water storage changes computed from each monthly gravity field as predicted by BGI, GFZ, CSR and JPL epochs given in km3 water equivalent. The estimated monthly mean water mass changes derived from BGI, CSR and GFZ seem to fit very well. There are practically no amplitude and phase differences in the annual and semi-annual cycle (around of 0.9 and 3.4 months, respectively). The phases of the JPL model differ from the phases of the other models, especially in the annual cycle (2.39 months). Furthermore, the JPL estimated amplitudes are always lower than GFZ, BGI and CSR amplitudes. In order to compare the obtained results from GRACE with independent data, ENVISAT altimetry information was included in Fig. 75.3. The time series of water levels of the main rivers in the La Plata Basin were obtained from the LEGOS center (Laboratoire d’Etudes en Ge´ophysique et Oce´anographie Spatiales, France). That satellite data were only used where complete time series of river levels can be derived from satellite altimetry. Crude ENVISAT signals (not corrections applied) in Fig. 75.3 show variations in both amplitude and phase of the seasonal signal that can be seen from year to year. A comparison of the annual signal from different GRACE products and ENVISAT data was obtained by computing temporal correlations for the period 2003–2007. The temporal correlation between ENVISAT and BGI, CSR and GFZ products is about 0.7, and close to 0.6 for JPL mass estimates.
75.4
Discussion and Conclusions
It can be concluded from the analysis of the figures that GRACE detected the significant mass changes in the area corresponding with La Plata Basin. From the trend figures, it could be seen that there are some discrepancies between the four GRACE processing Centers, and the JPL is the one showing the highest differences. There is a strong negative gravity trend in the southern area of the Basin, which can be related with the Parana´ River behavior and its seasonal variations. The time series figure shows that the four different GRACE product Centers (BGI, CRS, GFZ, JPL) detected that water storage in La Plata Basin has a dominant seasonal signal peaking in late April; and also that the lowest water storage level is in late October. GRACE products differences are currently being studied in terms of resolution and pre-processing. In the comparison between GRACE mass estimates and ENVISAT altimeter data, it must be taken into account that GRACE satellites observe the integrated regional mass change; and ENVISAT measures only the variation in surface water storage at a particular location. Nonetheless, the correlation between these data is very high for the years 2003, 2006 and 2007, varying from 0.75 (JPL) to 0.9 (BGI, CSR, GFZ). Future work will include all necessary corrections to ENVISAT data. In a recent study, Klees et al. (2008) have evaluated global and regional GRACE models in some basins around the world. For the La Plata Basin, they found
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large differences among GRACE models for the period February 2003 to February 2006. These differences were explained in connection with complex hydrological cycle. We have not found such large differences among global GRACE solutions maybe due to the simple filtering method that we used (e.g., truncating of harmonic series). Therefore, this subject will be carefully explored in order to find the optimum filtering method for the La Plata Basin. This will have to be verified by independent sources such as river gauges observations, rainfall data, hydrological models and satellite altimetry. Acknowledgements This paper was partially supported by Agencia Nacional de Promocio´n Cientı´fica y Tecnolo´gica (ANPCyT) – PICT 2006 – 01590.
References Berbery EH, Barros VR (2002) The hydrologic cycle of the La Plata Basin in South America. J Hydrometeorol 3:630–645 Bettadpur S (2007) Level-2 gravity field product user handbook (Rev. 2.3, February 20, 2007). Center for Space Research, University of Texas at Austin Forsberg R, Reeh N (2006) Mass change of the Greenland ice sheet from GRACE. Proceedings of the first meeting of the
A. Pereira et al. International Gravity Field Service: Gravity Field of the Earth 2006, Springer Klees R, Liu X, Wittwer T, Gunter BC, Revtova EA, Tenzer R, Ditmar P, Winsemius HC, Savenije HHG (2008) A comparison of global and regional GRACE models for land hydrology. Surv Geophys 29:335–359. doi:10.1007/s10712-008-9049-8 Lemoine JM, Bruinsma S, Biancale R (2008) 10-Day gravity field solutions inferred from GRACE data. Proceedings of the IAG International Symposium on Gravity, Geoid and Earth Observation 2008 Lemoine J-M, Bruinsma S, Loyer S, Biancale R, Marty J-C, Perosanz F, Balmino G (2007) Temporal gravity field models inferred from GRACE data. Adv Space Res 39:1620–1629. doi:10.1016/j.asr.2007.03.062 Ramillien G, Frappart F, Cazenave A, G€untner A (2005) Time variations of land water storage from an inversion of 2 years of GRACE geoids. Earth Planet Sci Lett 235(2005):283–301 Rodell M, Chen J, Kato H, Famiglietti JS, Nigro J, Wilson CR (2007) Estimating groundwater storage changes in the Mississippi River basin (USA) using GRACE. Hydrogeol J 15:159–166 Sandberg Sørensen L, Forsberg R (2010) Greenland Ice Sheet Mass Loss from GRACE Monthly Models. Proc. IAG Symposium on Gravity, Geoid and Earth Observation, (135), Part 7, 527–532, doi: 10.1007/978-3-642-10634-7_70, Springer Velicogna I, Wahr J (2005) Greenland mass balance from GRACE. Geophys Res Lett. doi:10.1029/2005GL023955 Velicogna I, Wahr J (2006) Acceleration of Greenland ice mass loss in spring 2004. Nature. doi:10.1038/nature05168 Wahr J, Swenson S, Zlotnicki V, Velicogna I (2004) Timevariable gravity from GRACE: first results. Geophys Res Lett 31:L11501. doi:10.1029/2004GL019779
Second and Third Order Ionospheric Effects on GNSS Positioning: A Case Study in Brazil
76
H.A. Marques, J.F.G. Monico, G.P.S. Rosa, M.L. Chuerubim, and Ma´rcio Aquino
Abstract
The Global Positioning System (GPS) transmits signals in two frequencies. It allows the correction of the first order ionospheric effect by using the ionosphere free combination. However, the second and third order ionospheric effects, which combined may cause errors of the order of centimeters in the GPS measurements, still remain. In this paper the second and third order ionospheric effects, which were taken into account in the GPS data processing in the Brazilian region, were investigated. The corrected and not corrected GPS data from these effects were processed in the relative and precise point positioning (PPP) approaches, respectively, using Bernese V5.0 software and the PPP software (GPSPPP) from NRCAN (Natural Resources Canada). The second and third order corrections were applied in the GPS data using an in-house software that is capable of reading a RINEX file and applying the corrections to the GPS observables, creating a corrected RINEX file. For the relative processing case, a Brazilian network with long baselines was processed in a daily solution considering a period of approximately one year. For the PPP case, the processing was accomplished using data collected by the IGS FORT station considering the period from 2001 to 2006 and a seasonal analysis was carried out, showing a semi-annual and an annual variation in the vertical component. In addition, a geographical variation analysis in the PPP for the Brazilian region has confirmed that the equatorial regions are more affected by the second and third order ionospheric effects than other regions.
76.1
H.A. Marques (*) J.F.G. Monico G.P.S. Rosa M.L. Chuerubim Departamento de Cartografia, Universidade Estadual Paulista, Roberto Simonsen 305, Presidente Prudente 19060-080, Sa˜o Paulo, Brazil e-mail:
[email protected] M. Aquino University of Nottingham, IESSG, University Park, Nottingham, UK
Introduction
Global Navigation Satellite Systems (GNSS), especially the GPS, represent one of the most advanced technologies which are currently available for geodetic positioning. GPS has been widely used both for geodetic positioning and scientific research work that are demanding more and more precise coordinate determination. After the deactivation of SA (Selective Availability), the ionospheric effect is one of the main factors that cause limitations to the accuracy of positioning with single
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frequency receivers, also causing difficulties to resolve the carrier phase integer ambiguities. When GPS receivers capable of collecting double frequency data are available, it is possible to obtain the ionosphere free linear combination from L1 and L2 signals in order to correct for first order ionospheric effects. However, the second (2nd) and third (3rd) order ionospheric effects (also called higher order ionospheric effects), which may cause errors of the order of centimeters in the GPS measurements, still remain. In recent years the international scientific community has driven more attention to this kind of effects and some works have shown that for high accuracy GNSS positioning these effects have to be taken into consideration (Herna´ndez-Pajares et al. 2007). In this paper the 2nd and 3rd order ionospheric effects, which were taken into account in the GPS data processing in the Brazilian region, were investigated. The GPS observables were corrected from these effects using an in-house software called “RINEX_HO” (see Sect. 76.3). The mathematical models used to compute these effects will be presented, as well as the transformations involving the Earth magnetic field and the use of TEC from Global Ionospheric Maps or calculated from GPS pseudorange measurements. The corrected and not corrected GPS data were processed in the relative and PPP approaches, respectively, using Bernese V5.0 software and PPP software (GPSPPP) from NRCAN. The main goal of the exercise was to analyze the impact of accounting for higher order effects in the modeling of the ionosphere when processing GNSS data in the Brazilian region. For the relative positioning case, a Brazilian network with long baselines was processed in a daily solution considering a period involving the year 2006. For the PPP case, the analysis was accomplished using data collected in the IGS FORT station, considering the period from 2001 to 2006. A seasonal analysis was also carried out. In addition, a geographical variation analysis using PPP results for the Brazilian region was undertaken.
(2007), among other authors. The phase ( fLi ) and pseudorange (PRLi ) observations equations in the band Li (i ¼ 1, 2) can be written as: 1 ð2Þ 1 ð3Þ fLi ¼ r0 Ið1Þ g Ig Ig þ NLi þ ufLi 2 3 ð2Þ ð3Þ PRLi ¼ r0 þ Ið1Þ þ I þ I g g g þ uPDLi ; (76.1) where, r0 is the geometric distance between the satellite and receiver added by non-dispersive effects such ð2Þ as troposphere and clocks. The components Ið1Þ g , Ig ð3Þ and Ig denote, respectively, the first, second and third order ionospheric effects of the group. The phase ambiguity is represented by NLi and the components ufLi and uPLi are, respectively, unmodeled phase and pseudorange effects. From (76.1) it is noticeable that the ionospheric effects are similar for phase and group, differing only by the sign and the factors 2 and 3, respectively, for the 2nd and 3rd order ionospheric effects. The 2nd and 3rd order ionospheric effects for the group in the frequency fLi (i ¼ 1, 2) can be computed by (Bassiri and Hajj 1993; Odijk 2002): Ið2Þ g ¼
Second and Third Order Ionospheric Effects
The development of the equations to compute the 2nd and 3rd ionospheric effects can be found in Bassiri and Hajj (1993), Odijk (2002), Herna´ndez-Pajares et al.
(76.2)
where, A ffi 80:6 m3 =s2 ; e ¼ 1.60218.1019 C for the electron charge; me ¼ 9.10939.1031 kg for the electron mass; kBk denotes the magnitude of the geomagnetic induction vector B and y the angle between the wave propagation direction and the geomagnetic field vector. The product kBkjcos yj has to be evaluated for the computation of the 2nd order effect what can be
S
||B||.|sin θ| B
76.2
eA kBk jcos yj TEC: f 3Li 2 pme
r
θ ||B||.|cos θ|
Single layer
φ
Fig. 76.1 Geomagnetic induction vector B. Adapted from Odijk (2002)
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Second and Third Order Ionospheric Effects on GNSS Positioning
done based on the inner product of the vector B (geomagnetic induction) and the unit vector J (propagation direction of the signal) at the height of the single ionospheric layer (Fig. 76.1): kBk jcos yj ¼ kBk kJk jcos yj ¼ Bt J
(76.3)
The inner product Bt J is more easily obtained in a geomagnetic reference system, where it is possible to use a dipolar geomagnetic system or a more realistic geomagnetic field, such as that of the International Geomagnetic Reference Model (IGRM). In general, the approximation by dipolar model has an accuracy of approximately 75% (Bassiri and Hajj 1993). Then, the geodetic receiver coordinates must be transformed to geomagnetic coordinates and, after that, transformed to the geomagnetic local system (Em, Nm, Um), as exemplified in Fig. 76.2. Then, the inner product Bt J can be computed as a function of the receiver and satellite positions (Odijk 2002; Kedar et al. 2003): 0 0 Bt J ¼ cos fm sin zm cos am 2sin fm cos zm 3 (76.4) Re Beq Re þ hion where, am and zm denote, respectively, the geomagnetic azimuth and zenithal angle of the satellite in the
621 0
local geomagnetic system (Fig. 76.2), fm is the geomagnetic latitude of the pierce point, Re is Earth equatorial radius, hion is the height of the ionospheric layer, and Beq is the amplitude of the equatorial magnetic field at the Earth’s surface (~3.12 105 T). The 3rd order ionospheric effect can be computed by (Odijk 2002): Ið3Þ g ¼
3 A2 Ne;max TEC 8 f 4Li
(76.5)
The 3rd order ionospheric effect is very similar to the 2nd order effect once it is given as a function of the TEC. However, it is a function of the maximum electron density Ne;max and the factor Z, which can be approximated by a constant value of 0.66 (Odijk 2002). An expression to compute Ne;max as a function of the TEC is given based on studies made by Brunner and Gu (1991): ð20:0 6:0Þ:1012 :TEC: Ne;max m3 ¼ ð4:55 1:38Þ:1018
(76.6)
76.2.1 TEC from Pseudorange The Total Electron Content in the satellite-receiver path can be computed using pseudorange (PRLi) measurements:
S
TEC ¼
J
f 2L1 f 2L2 40:3 f 2L2 f 2L1 ½PRL1 PRL2 c(DCBr þ DCBs Þ þ eL1L2 (76.7)
Em em am Nm r
Um
Fig. 76.2 Geomagnetic local reference system. Adapted from Odijk (2002)
where, fLi (i ¼ 1, 2) is the GPS frequency, DCBr and DCBs (in units of seconds) are the Differential Code Bias, hardware delays between the two frequencies, respectively, in the receiver and satellite. The speed of light in vacuum is represented by c and eL1L2 represents all remaining unmodeled effects. Considering that the standard deviations of the pseudoranges (sPRL1 and sPrL2 , respectively, for PRL1 and PRL2) and of the DCBs (sDCBr and sDCBs , respectively for receiver and satellite) are known (in units of meters), it is possible to estimate the TEC variance, by the covariance propagation law, from (76.7):
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s2TEC ¼
f 21 f 22 40:3ðf 21 f 22 Þ
2 s2PDL1 þs2PDL2 þs2DCBr þs2DCBs 76.3 (76.8)
The TEC can also be computed based on the pseudorange smoothed by the phase or from Global Ionosphere Maps (GIMs). The GIMs are available at CODE (Center for Orbit Determination for Europe)’s home page, along with complementary information such as the daily and monthly DCBs.
76.2.2 The Earth Magnetic Field The internal magnetic field can be approximated by an earth centered dipole in ionosphere heights. The dipole axis crosses the Earth’s surface in two points (north and south poles) which change with time because of the secular variations of the Earth magnetic field. The geomagnetic south pole is located approximately at the geographic latitude 79 S and longitude 110 E, while the geomagnetic north pole at approximately 79 N and 70 W (Davies 1990). The relation between the dipolar coordinates and the corresponding geographic coordinates can be found in Davies (1990) and a way to update the pole coordinates in function of MJD (Modified Julian Date) can be found in Hapgood (1992). The Earth’s magnetic field can be represented more accurately when the Earth’s scalar potential is expanded in spherical harmonics, which involves an adjustment of the coefficients in certain intervals of time because of the intrinsic changes of the magnetic field. The responsibility of this task is of International Geomagnetic Reference Field (IGRF). The IGRF model consists of a set of global spherical harmonics coefficients that are available for users on the internet in addition to the corresponding subroutines to perform the transformations between geographic and geomagnetic systems, as for example in the package GEOPACK (Tsyganenko 2005). In this paper the Corrected Geomagnetic Model (CGM) was used, whose Fortran subroutines are available at the PIM (Parameterized Ionospheric Model) home page (PIM 2001). The CGM presents a grid with corrected geomagnetic latitudes and longitudes that were created based on a more realistic magnetic field, i.e. the DGRF and IGRF (Gustafsson et al. 1992).
Results
The GPS data processing was accomplished based, in turn, on data that was corrected and not corrected from 2nd and 3rd order ionospheric effects. Results were then compared, in order to analyze the impact of accounting or not for the higher order ionospheric effects. The corrections were applied based on the in-house C++ software called RINEX_HO. This software reads a RINEX file and applies the corrections to the GPS observations, creating a corrected RINEX file. The TEC can be interpolated from GIM (Global Ionospheric Maps) or computed from pseudorange measurements. The transformation involving terrestrial and geomagnetic systems is performed with the subroutines that apply the Corrected Geomagnetic Model (PIM 2001).
76.3.1 TEC Analysis Using the DCBs data available at CODE, their standard deviation (SD) was computed considering the data of the year 2002. The IGS FORT station was chosen and the computed precision for DCBr (P1-P2) was 2.07 TECU. Then choosing a typical satellite (PRN 31) and applying the same procedure the computed precision for DCBs (P1-P2) was 0.4 TECU. These computed precisions from CODE data were inserted in (76.8) to compute the precision of our calculated TEC. In this case the values adopted for the precisions of P1 and P2 observables were, respectively, 0.60 and 0.80 m. The result from the covariance propagation with (76.8) gives the value of 10.05 TECU. The computed precision of the TEC was used for the covariance propagation into the second order ionospheric effect. The covariance propagation equation was developed from (76.2), where the projection of the geomagnetic induction vector onto the propagation path was considered as constant. Figure 76.3 shows the computed precision of the 2nd order ionospheric effects (L1 and L2) for PRN 31 as a function of the computed TEC precision (10.05 TECU). From Fig. 76.3, it is possible to see that for low satellite elevation angles the precision reaches values near 1 mm for L1, showing that the TEC with an uncertainty of the order of 10 TECU can be used to
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Fig. 76.4 Geomagnetic latitude differences (degrees) Fig. 76.3 Precision of the 2nd order effect (SD of TEC 10.05 TECU)
calculate the 2nd order ionospheric effects. These results show that it is possible to use the TEC either from GIM or from pseudorange, once the precision of the TEC in both cases is near 10 TECU (Ciraolo et al. (2007)).
76.3.2 Dipolar Versus CGM The geomagnetic latitudes were calculated using the dipolar and the Corrected Geomagnetic Model. The differences can be seen in Fig. 76.4. The difference in geomagnetic latitude between dipolar and CGM reached up to 18 for Equatorial regions. The second order ionospheric effects (L1) for some IGS stations on the day 48 of 2007 were computed using the CGM and dipolar models, respectively. The differences for each IGS stations (Fig. 76.5) were then calculated using the max values (in mm) during the day. The differences are from 4 up to 4 mm, which corresponds to about 50% of the total effect considering the CGM.
Fig. 76.5 Second order ionospheric differences (dipolar versus CGM)
NAUS IMPZ
BOMJ CUIB
BRAZ
VICO PPTE
0 km
500 km 1000 km 1500 km 2000 km 2500 km
76.3.3 Relative Network Processing
Fig. 76.6 Brazilian station used in the relative processing
The relative GPS data processing was accomplished using data from a Brazilian network (Fig. 76.6) with the Bernese software in a free adjustment and a posteriori normal equations solution. The results were generated with data corrected and not corrected in a daily solution for the year 2006.
The daily vertical discrepancies (DU) due to accounting and not accounting for the higher order ionopsheric effects, in a local geodetic system for the BRAZ station, are shown in the Fig. 76.7. Figure 76.7 shows that the vertical discrepancies can reach the order of up to 4 mm. However, there are
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DU (mm)
4 2 0 2 –4 2006 2006.1 2006.2 2006.3 2006.4 2006.5 2006.6 2006.7 2006.8 2006.9 2007 Years
Fig. 76.7 Vertical daily differences (station BRAZ) 2
DU (mm)
1 0 BOMJ
BRAZ
CUIB
MPZ
NAUS
PPTE
VICO
–1 –2 –3 Stations
Fig. 76.8 Mean difference in the vertical component (year 2006) for all station (relative processing)
Fig. 76.9 Second- and third-order effect in PPP (geographical variation – mm)
some discontinuities in the time series which must be analyzed for future works. The mean discrepancies in the vertical component for all stations considering the entire period (year 2006) can be seen in Fig. 76.8. The largest difference was for the CUIB station as can be seen in the Fig. 76.7, reaching the order of 3 mm. For the remaining stations the mean discrepancies were of the order of 1–2 mm.
76.3.4 Precise Point Positioning The PPP was accomplished using the software GPSPPP from NRCAN. Dual frequency GPS data were used to form the ionosphere free combination and the following corrections were applied: Zenital Tropospheric Delay estimated as a random walk process; Earth body tide; Ocean Tide Loading, Absolute Phase Center Variations, Phase Windup, among others (NRCAN 2004). The coordinates are presented in terms of daily solutions. It is important to mention that the data was processed using precise ephemeris and satellite clock corrections from IGS which are not corrected from higher order ionospheric effects. The influence of the higher order ionospheric effects on PPP was analyzed geographically (Fig. 76.9) considering the day 80 of 2003, when kp (the planetary geomagnetic index) reached a maximum of 5, indicating a moderate to active ionosphere (Davies 1990).
Fig. 76.10 Time series of PPP errors caused by higher order ionospheric effects
The discrepancies due to accounting for 2nd and 3rd order ionospheric effects in the PPP reach up to 8 mm near the Equatorial region, as can be seen in Fig. 76.9. A time series of these discrepancies, is shown in Fig. 76.10. The values are in millimeters. In this case GPS data from station FORT (latitude: 3.52 ; longitude: 38.25 ) was used, considering the years 2001–2006. A seasonal variation of the vertical component (DU) is shown in Fig. 76.11, which is based on a decomposition time series using the statistical software Minitab (Minitab 2009). From the detrended data (Fig. 76.11) an annual and semi-annual variation caused by the 2nd and 3rd order ionospheric effects on PPP can be observed. Considering all years involved in the processing the vertical discrepancies (Fig. 76.10) reached up an average of approximately 4 mm with standard deviation of 2 mm. Concerning to the years 2001 and 2002 (maximum of
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shown annual and semi-annual variations. The relative network processing, using or not using higher order ionospheric corrections, has presented variations of the order of 4 up to 4 mm in the stations coordinates. This confirms that for high precision GNSS positioning, either relative or absolute, the 2nd and 3rd order ionospheric effects must be taken into account.
0 –2 –4
DU (mm)
625
–6 –8 – 10
0
Years
Fig. 76.11 Time series decomposition for DU
the solar cycle) the vertical discrepancies reached up the order of 12 mm. These results indicate that for high accuracy GNSS PPP these effects must be taken into consideration.
76.4
References
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Conclusions
In this paper the second and third order ionospheric effects, which were taken into account in the GPS data processing in the Brazilian region, were investigated. The GPS observables were corrected from these effects using an in-house software called “RINEX_HO”. The corrected and not corrected data was processed in the relative and PPP approaches. The main goal was to analyze the impact of accounting for higher order ionospheric effects in GNSS data processing for the Brazilian region. The discrepancies in the PPP due to the consideration of the 2nd and 3rd order ionospheric effects in the data processing reached up to the order of centimeters during an active ionosphere period (maximum of the solar cycle) and the PPP time series has
Bassiri S, Hajj GA (1993) Higher-order ionospheric effects on the global positioning systems observables and means of modeling them. Manuscr Geod 18:280–289 Brunner F, Gu M (1991) Manuscr Geod 16:205–214 Ciraolo L, Azpilicueta F, Brunini C, Meza A, Radicella SM (2007) J Geod 81(2):111–120 Davies K (1990) Ionospheric radio. Peter Peregrinus Ltd., London, 580 pp Gustafsson G, Papitashvili NE, Papitashvili VO (1992) A revised corrected geomagnetic system for epochs 1985 and 1990. J Atm Terr Phy 54(11/12):1609–1631 Hapgood MA (1992) Space physics coordinate transformations: a user guide. Planet Space Sci 40(5):711 Herna´ndez-Pajares M, Juan JM, Sanz J (2007) Second-order ionospheric term in GPS: implementation and impact on geodetic estimates. J Geophys Res 112:B08417 Kedar S, Hajj A, Wilson BD, Heflin MB (2003) Geophys Res Lett 30(16):1829 Minitab. Minitab Quality Companion (2009) MINITAB – statistical software. http://www.minitab.com. Acessed in March of 2009 NRCAN (2004) On-line precise point positioning: ‘how to use’ document, 2004. http://www.geod.nrcan.gc.ca/userguide/ index_e.php. Acessed in March of 2009. Odijk D (2002) Fast precise GPS positioning in the presence of ionospheric delays, 242 pp. PhD dissertation, Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft PIM (2001) Parametrized ionospheric model: user guide, 2001. http://www.cpi.com. Acessed in March of 2009. Tsyganenko N (2005) Geopack: a set of fortran subroutines for computations of the geomagnetic field in the Earth’s magnetosphere. Universities Space Research Association, Washington, DC
.
Advanced Techniques for Discontinuity Detection in GNSS Coordinate Time-Series. An Italian Case Study
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A. Borghi, L. Cannizzaro, and A. Vitti
Abstract
Since the past decade, geodetic techniques are widely used to gain important information for the monitoring and modeling of the deformation of the Earth at different length and time scales. Although the GNSS derived estimates of the Earth crust velocity are becoming more and more reliable, advanced data analysis techniques are needed to recognize geophysical features in the GNSS time-series, e.g., non linear behaviors, discontinuities in the signal and in its derivative, i.e., in the velocity. Unfortunately these phenomena are often hidden in the time-series noise and external information, as seismic events, are not always known. The main focus of this work is the detection of signal discontinuities in GNSS timeseries through the use of advanced analysis techniques: the wavelets, the Bayesian and the variational methods. The Mumford and Shah (Commun Pure Appl Math 42:577–685, 1989) and the Blake and Zisserman (Visual reconstruction, 1987) variational models for signal segmentation can detect signal discontinuities in an explicitly way. The Blake and Zisserman (Visual reconstruction, 1987) model can also detect discontinuities of the signal first derivative, i.e., velocity abrupt changes can be detected. At first, to prove and assess the capability to detect discontinuities correctly, the methods have been applied to some Cascadia (North America) time-series, characterized by well known aseismic deformations. A second test area has been taken into account: the Calabrian Arc subduction zone, in southern Italy. The analyzed Italian GNSS time-series are characterized by very weak and noisy signals and the geodynamic of the area is mostly unknown. When present, discontinuities are expected to be very small and
A. Borghi OGS – Italian Institute of Oceanography and Applied Geophysics, c/o Politecnico di Milano, P.zza Leonardo da Vinci 32, 20133 Milan, Italy L. Cannizzaro DIIAR – Politecnico di Milano, P.zza Leonardo da Vinci 32, 20133 Milan, Italy A. Vitti (*) DICA – Universita` di Trento, via Mesiano, 77, 38100 Trento, Italy e-mail:
[email protected] S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_77, # Springer-Verlag Berlin Heidelberg 2012
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compatible with the signal noise. This motivates the use of advanced data analysis techniques to investigate the presence of discontinuities. At the moment, the analysis of the Italian time-series has revealed several discontinuities which nature cannot be labeled easily as geophysical or geodetic.
77.1
Introduction
Continuous geodetic measurements are being widely used in geophysical applications for the monitoring of subduction zones, revealing transient deformations consistent with creep along the deeper plate interface. In these zones, coordinate time-series of continuous GPS stations (CGPS) are used to study the occurrence of slow slip events. Good and well know examples of this kind of applications are the surveys and studies performed in Japan (Miyazaki et al. 2003) and on the Cascadia subduction zone (Dragert et al. 2001) as well. Szeliga et al. (2008), by using the Wavelet transform, identified precisely the creep epochs in GPS coordinate time-series where the transient deformations appeared as signal discontinuities. In general, aseismic slips have a strong influence on the moment budget of faults which needs to be quantified for accurate seismic hazard estimates, because aseismic slip also redistributes stress in the crust affecting the locations of future earthquakes. In Italy, the Calabria region is located in the AfricaEurasia plate boundary and presents one of the higher seismic risk of the Country. This region is characterized by the subduction zone of the Calabrian Arc and, in the northern part, by the lasting quiescent Pollino-Castrovillari normal fault (Cinti et al. 2002), where GPS and DInSAR velocity inversions suggest fast aseismic creeping along all depths of the fault plane (Sabadini et al. 2009). To better understand the earthquake cycle and to evaluate the seismic hazard of this active fault during the inter-seismic phase, the Calabrian Arc subduction zone is currently under study and the occurrence of deformations consistent with creep along the subduction zone is under investigation. The Calabrian Arc GPS time-series do not present clear discontinuities: the slip events, if present, are hidden by the noise, being both of the same magnitude. This represents an extreme study case with respect to the Cascadia. Given the nature of such signals, advanced
methods for the detection of signal discontinuities have been applied to obtain results as much meaningful as possible. The focus of this work is mainly on the methodology aspects of the discontinuity research methods rather than on the geophysical implication of the results, even because of the very short available time-series. The methods taken into account in this work are the Wavelet transform, by analogy with the work of Szeliga et al. (2008) in the Cascadia subduction zone, the Bayesian and the variational methods. Each method has been tested in the well studied Cascadia time-series, to evaluate their potentiality. The paper presents, with different details, some introductory elements of the three analysis techniques. A short description of the Cascadia subduction zone follows. Results of the application of the variational method to some Cascadia time-series are presented in order to show the features of the variational solutions. The results of the analysis of the Calabrian Arc GPS time-series are then discussed. Eventually, some remarks and conclusions are given.
77.2
Advanced Methods for the Detection of Signal Discontinuities
In this section the three analysis techniques used to detect discontinuities in GPS time-series are introduced. This part is quite schematic and it is intended to describe the most relevant features of the methods rather than to give a theoretical treatment of their mathematical properties. To get a deeper knowledge please see the references.
77.2.1 The Bayesian Method We refer hereafter to the work of de Lacy et al. (2008) where the Bayesian method has been presented to face the problem of detecting and correcting cycle-slips in GPS phase data. The method is quite general (Koch
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1990) and can be applied to a generic time-series under the assumption that the data can be modeled locally by a smooth regression. Discontinuities are modeled as finite jumps in the polynomial regression. In this case, the observation equation can be written in a vectorial form as: y0 ¼ Ax þ k ht þn;
(77.1)
where the elements of the design matrix A are the base functions of the regression; x is the vector of the unknown regression parameters; t is the discontinuity epoch; k is the discontinuity amplitude; n is the observation noise and ht is the Heaviside function. The role of these terms is sketched in Fig. 77.1. The observables, i.e., the GPS residuals time-series after detrending for linear regression and periodic signals, and also the unknown parameters are all modeled as random variables. By introducing a probability distribution (the prior) for each parameter, the Bayesian theorem leads the determination of the conditional posterior distributions of the parameters given the observations. In this approach, the observation noise is considered as a Gaussian white noise with zero mean and unknown variance, all the parameters are considered a-priori stochastically independent, and non-informative priors are considered (Box and Tiao 1992). The discontinuity epoch is detected by computing the marginal posterior distribution of the parameter t and then by selecting theepoch with the highest a posteriori probability. The discontinuity amplitude is then estimated from its conditional posterior, given the data and the estimate of the discontinuity epoch, by
Fig. 77.1 Example of one discontinuity in a polynomial regression
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exploiting again the maximum a-posteriori (MAP) principle.
77.2.2 The Variational Method The variational approach is based on the minimization of a functional, proposed by Mumford and Shah (1989), where three terms interact to produce a so called segmentation of the data. Here, segmentation can be considered as the partitioning of the data into disjoint and homogeneous regions by smoothing the data while detecting the region boundaries and simultaneously preserving them from the smoothing. The Mumford and Shah functional involves a term that penalizes the distance between the approximating solution and the data; a term that penalizes the wiggle of the approximating solution; a term that penalizes the measure of the set of the region boundaries. The unknowns are the smooth approximation u of the data g, and the set K of the discontinuity points of g, i.e., the boundaries of the homogeneous regions. In one dimension, the Mumford and Shah functional is: ð MSðu; KÞ ¼
h OnK
i 2 ðu gÞ2 þ lðu0 Þ dx þ a#ðKÞ; (77.2)
where O is the domain of the data g, l and a are positive coefficients and #(K) is the counting measure of the discontinuity set K, i.e., the number of the discontinuity points. The problem is to find, over the admissible class of pairs (u, K), the minimum of MS (u, K). From the mathematical viewpoint, this problem presents many difficulties which are briefly described hereafter. The first term in (77.2) is a line integral whereas the second term is the counting measure of a set, i.e., a term of dimension lower of one order of magnitude. The second term depends on the discontinuity set K, which is not fixed a-priori and it is one of the main unknown of the problem. The domain of the line integral depends on the unknown set K too. This particular class of minimum problems are known as Free Discontinuity Problems (FDP) after the initial studies of De Giorgi et al. (1989). In general, minimum problems are also known as variational problems because the existence of the solution, i.e., of the minimum, is typically proved in the mathematical framework of the Calculus of Variations. The space
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of Special functions of Bounded Variations (SBV) plays a very essential role when the solution of a FDP is searched (see Ambrosio et al. 2000). The last issue is related again to the term #(K), being the numerical treatment of this term very difficult. Suitable methods for the numerical minimization of the functional exist and are based on the use of the theory of the -Convergence to define a variational approximation of (77.2) that can be implemented numerically (see Braides 1998, 2002). Ambrosio and Tortorelli (1992) proposed the first and most used approximation. The key idea underling this approximation is the use of an auxiliary discontinuity function that mimics the characteristic function of the solution u, i.e., that “sees” the discontinuities of u. An important extension of the Mumford and Shah (1989) functional has been proposed by Blake and Zisserman (1987) who introduced a second order term to control the smoothness of the solution u and a specific term to handle the discontinuity points of the first derivative of u. The Blake and Zisserman functional is: ð BZðu; K; DÞ ¼
OnðKþDÞ
h i 2 ðu gÞ2 þ gðu00 Þ dxþ þa#ðKÞ þ b#ðDÞ; (77.3)
77.3
Analysis of Some Cascadia Time-Series
The Cascadia subduction zone stretches from northern California to southwestern British Columbia. A dense permanent GPS network is operating over the entire area and GPS data are used for both monitoring and research purposes. Before applying the three analysis techniques introduced in the previous section to the Calabrian Arc GPS time-series, a subset of the Cascadia GPS time-series has been analyzed to test the behavior and the performance of the methods on well known signals. GPS time-series have been downloaded from the SOPAC archive (http://garner. ucsd.edu/pub/timeseries) for the ALBH, NEAH, SC02, SC03, SC04, and SEDR sites. Each of the three methods has detected correctly the most part of the known discontinuities. As an example, in Fig. 77.2 the results obtained for the East residual component time-series of the ALBH station are showed. Figure 77.3 shows the locations in time of the discontinuities commonly detected by all of the three methods on the East residual components timeseries of the Cascadia stations considered in this work. The known high temporal correlation existing between the discontinuity events (Szeliga et al. 2008) is clearly observable.
where K is again the set of discontinuity points of u and D is the set of the discontinuity points of u0 . A variational approximation, which involves also a first derivative discontinuity function that “sees” the discontinuities of u0 , is need to implement numerically the functional (77.3), (see Vitti 2008).
77.2.3 Wavelet Transform Wavelet transform is widely used in signal analysis to identify jumps in the signal. In bi-dimensional applications, jumps are often called “edges” (Addison 2002). In this work, the Wavelet transform has been used to reproduce the results presented by Szeliga et al. (2008) on the Cascadia time-series. The standard MATLAB implementation of the Wavelet transform has been used, in particular the time-series residuals have been transformed by Daubechies wavelet transformation (Daubechies 1992) of order 1 (db1).
Fig. 77.2 East residual component (mm) time-series of the ALBH station and the discontinuities detected by the Bayesian method (top), by the Wavelet method (middle) and by the Blake and Zisserman method (BZ): the auxiliary function that “sees” the discontinuities is plotted (bottom)
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The East residual component time-series of the SC02 station and its variational segmentation obtained with the Mumford and Shah model is plotted in Fig. 77.4; the Blake and Zisserman segmentation is given in Fig. 77.5, the auxiliary function that “sees” the discontinuities of the first derivative of the solution u, i.e., changes in the velocity values, is also plotted. The Mumford and Shah segmentation presents more discontinuities and it is more “flat” than the Blake and Zisserman segmentation. These facts are consistent with the different theoretical properties of the two models (e.g., see Vitti 2008).
77.4
Fig. 77.3 East residual components (mm) of some Cascadia stations and the discontinuities commonly detected by the Bayesian, the Wavelet and the variational methods
Analysis of the Calabrian ARC Time-Series
The permanent GPS network taken into account consists of nine stations set up by UNAVCO and operating in the Calabrian Arc from November 2006 only. Although GPS data are available for a very short
Fig. 77.4 Top – Mumford and Shah segmentation (black) of the East components (mm) (gray) of the SC02 station. Bottom – The discontinuity function
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Fig. 77.5 Top – Blake and Zisserman segmentation (black) of the East components (mm) (gray) of the SC02 station. Bottom – the discontinuity function (black) and the first derivative discontinuity function (gray)
time span, we investigate probable transient that could beattributed to a geophysical process. The GPS data collected by the permanent GPS network have been processed with the Bernese Software v.5.0 (Dach et al. 2007). The time-series have been analyzed according to Cannizzaro (2008) to estimate velocities by least square method; time correlation between daily coordinates has been taken into account. Clear discontinuities have not been detected in the residuals processed in this work. Reasonably, this result depends on the noise level of the timeseries. Slip events, if present, may be hidden by the noise. The analysis of the Calabrian Arc GPS timeseries did not highlight any significant time correlation between the detected discontinuity events. Figure 77.6 shows the positions in time of the discontinuities commonly detected by all of the three methods on the East residual component time-series of some UNAVCO stations operating on the Calabrian Arc. Analogous results have been obtained for the North and Up residuals components.
Conclusions
The Bayesian, the variational, and the Wavelet methods have been successfully applied to detect discontinuities in coordinate time-series coming from permanent GPS stations operating along the Calabrian Arc in Italy and in the well known Cascadia subduction zone. These methods can reveal the presence of slow events occurred in subduction zones. The Cascadia and Calabrian Arc areas are characterized by different signatures. The first one presents discontinuities clearly detectable. The Calabrian Arc area represents an extreme case where discontinuities are not visible to nakedeye and where information about their presence is not available. The application of all the techniques produced very consistent results. The Cascadia known transient deformations have been detected by all the methods and the results are in good agreement with the literature. The analysis of the Calabrian Arc has
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Fig. 77.6 East residual components (mm) of some UNAVCO stations operating on the Calabrian Arc and the discontinuities commonly detected by the Bayesian, the Wavelet and the variational methods
been performed on very short time-series as a first investigation of creep phenomena undergoing on the zone. Despite the noise of the Calabrian Arc GPS time-series, discontinuities have been detected by the three analysis techniques. Nevertheless, it has not been possible to observe any significant time correlation between the detected events. The nature of these events cannot be identified clearly as geodetic or geophysical. At the moment, many of the discontinuities might be due to the presence of undetected outliers. The Bayesian and the variational methods are both general and powerful tools and they could be useful in a wide range of geodetic and geophysical applications. In the Bayesian method significant constraints can be enforced on the parameter priors to make the detection more effective (de Lacy et al. 2008). The Bayesian method estimates the epochs, the a-posterior probability and the amplitudes of the discontinuities; a graphical interpretation of the results is not strictly needed. The Blake and Zisserman model appears very promising due to its capability to explicitly handle velocity discontinuities (see Vitti 2008).
Acknowledgment A. Borghi thanks support from the program: S1-RU5.01 INGV-DPC 2007–2008. This material is based on data, equipment and engineering services provided by the UNAVCO Facility with support from the National Science Foundation (NSF) and National Aeronautics and Space Administration (NASA) under NSF Cooperative Agreement No. EAR-0735156.
References Addison PS (2002) The illustrated wavelet transform handbook. Introductory theory and applications in science, engineering, medicine and finance. IoP Publishing, London Ambrosio L, Tortorelli VM (1992) On the approximation of free discontinuity problems. Bollettino dell’Unione Matatematica Italiana 6(B):105–123 Ambrosio L, Fusco N, Pallara D (2000) Functions of bounded variations and free discontinuity problems. Oxford University Press, Oxford Blake A, Zisserman A (1987) Visual reconstruction. The MIT Press, Cambridge Box GEP, Tiao GC (1992) Bayesian inference in statistical analysis. Wiley, New York Braides A (1998) Approximation of free-disconti-nuity problems. Lecture notes in mathematics 1694. Springer, Berlin Braides A (2002) Gamma-convergence for beginners. Oxford lecture series in mathematics and its applications, vol 22. Oxford University Press, London
634 Cannizzaro L (2008) Analysis of temporal correlations in GPS time series: comparison between different methods at different space scale. PhD Thesis in Geodesy and Geomatics, Politecnico di Milan, Italy Cinti FR, Moro M, Pantosti D, Cucci L, D’Addezio G (2002) New constraints on the seismic history of the Castrovillari fault in the Pollino gap (Calabria, southern Italy). J Seismol 6:199–217 Dach R, Hugentobler U, Fridez P, Meindl A (2007) BERNESE GPS Software Version 5.0. Astronomical Institute, University of Bern Daubechies I (1992) Ten Lectures on Wavelets. CBMS-NSF Lecture Notes 61 De Giorgi E, Carriero M, Leaci A (1989) Existence theorem for a minimum problem with free discontinuity set. Arch Rational Mech Anal 108:195–218 de Lacy MC, Reguzzoni M, Sanso` F, Venuti G (2008) The Bayesian detection of discontinuities in a polynomial regression and its application to the cycle-slip problem. J Geod 82 (9):527–542
A. Borghi et al. Dragert H, Wang K, James TS (2001) A silent slip event on the deeper Cascadia subduction interface. Science 292:1525–1528 Koch KR (1990) Bayesian Inference with geodetic applications. Lecture Notes in Earth Sciences, vol 31, Springer, Berlin Miyazaki S, McGuire JJ, Segall P (2003) A transient subduction zone slip episode in southwest Japan observed by the nationwide GPS array. J Geophys Res 108:B2 Mumford D, Shah J (1989) Optimal approximation by piecewise smooth functions and associated variational problems. Commun Pure Appl Math 42:577–685 Sabadini R, Aoudia A, Barzaghi R, Crippa B, Marotta AM, Borghi A, Cannizzaro L, Calcagni L, Dalla Via G, Rossi G, Splendore R, Crosetto M (2009) First evidences of fast creeping on a long lasting quiescent earthquake fault in Italy. Geophys J Int. doi:10.1111/j.1365-46X.2009.04312 Szeliga W, Melbourne T, Santillan M, Miller M (2008) GPS constraints on 34 slow slip events within the Cascadia subduction zone, 1997–2005. J Geophys Res 113:B04404 Vitti A (2008) Free discontinuity problems in image and signal segmentation. PhD thesis in Environmental Engineering, Universita` di Trento, Italy
Traditional and Alternative Network Adjustment Approach for the TAMDEF GPS in Antarctica
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G. Esteban Va´zquez, Dorota A. Grejner-Brzezinska, and Burkhard Schaffrin
Abstract
A network adjustment analysis was derived for a GPS network called TAMDEF (Trans Antarctic Mountains Deformation), located in Victoria Land, Antarctica. The network adjustment strategy involved the careful selection and application of the appropriate approach to process the TAMDEF network. Therefore, for the first part of the presented study, four cases denoted as Cases I–IV were investigated for the TAMDEF network processing with respect to the IGS (International GNSS Service) sites inside and outside the Antarctic continent. Here, the GPS data processing relied on the PAGES (Program for Adjustment of GPS Ephemerides) software, which was set up to run using the Least-Squares adjustment with Stochastic Constraints (SCLESS). The second part of the study focus in considering alternative network adjustment approaches: the Minimum-Norm LESS adjustment (MINOLESS); the Partial Minimum-Norm LESS (Partial-MINOLESS) and the Best LInear Minimum Partial-Bias Estimation (BLIMPBE) to validate results from the SCLESS approach (Case I) for IGS sites inside the Antarctica. Based on the applied network adjustment approaches within the Antarctic tectonic plate, it can be demonstrated that the GPS data used is clean of bias after properly taken care of ionosphere, troposphere and some other sources that affect GPS positioning.
78.1
G.E. Va´zquez (*) Earth Science School, The Autonomous University of Sinaloa, Culiacan, Sinaloa, Me´xico e-mail:
[email protected] D.A. Grejner-Brzezinska Satellite Positioning and Inertial Navigation (SPIN) Laboratory, The Ohio State University, Columbus, OH 43210, USA B. Schaffrin Division of Geodesy and Geospatial Science, The Ohio State University, Columbus, OH 43210, USA
Introduction
The TAMDEF (Trans Antarctic Mountains Deformation) network (http://www.geology.ohio-state. edu/TAMDEF/) is a GPS array deployed on bedrock, consisting of 25-campaign sites, 6-quasicontinuous sites and 2-continuous sites located in the Trans Antarctic Mountains of the southern Victoria Land and on the islands in the adjacent Ross Sea (Fig. 78.1). TAMDEF is the Ohio State University (OSU) and the United States Geological Survey (USGS) joint project sponsored by the National Science Foundation (NSF), initiated in 1996 with
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636 Fig. 78.1 TAMDEF Network, Victoria Land Antarctica
the primary objective to measure vertical and horizontal crustal deformation. This paper focuses on the selection and application of the appropriate network adjustment approach to process the TAMDEF network with respect to other IGS sites inside and outside Antarctica. Therefore, the McMurdo (MCM4) IGS site was adopted as part of the TAMDEF network, since it allows the link to the ITRF for this region. Thus, in order to verify and quantify how TAMDEF behaves to the Antarctic tectonic plate and other tectonic plates, it was processed with respect to all possible Antarctic IGS sites, and with at least four IGS stations outside Antarctica, located on the
South American, African and Australian continents. Next, to validate results from the Antarctic tectonic plate, alternative network adjustment strategy were took into account.
78.2
Data Processing and Network Adjustment Strategy
Dual-frequency geodetic-grade GPS receivers were used for the GPS data collection in static mode at the TAMDEF sites. The National Geodetic Survey (NGS) PAGES software (Eckl et al. 2001; Mader et al. 1995;
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Traditional and Alternative Network Adjustment Approach for the TAMDEF GPS in Antarctica
Schenewerk et al. 2001, http://www.ngs.noaa.gov/ GRD/GPS/DOC/pages/pages.html) was used to process the data by means of the ionosphere-free, double-difference carrier phase observation. In terms of tropo estimates, the Marini mapping function was used to compute wet zenith delay in a 3-h mode. Other specifications for the processing were: elevation mask of 15 , data sampling interval of 30 s, and use of precise ephemeris from IGS, http://igscb.jpl.nasa.gov/. Also, the carrier phase ambiguity parameters were fixed at the rate of 97–100% in all solutions. The network adjustment strategy followed in this research involved the careful selection and application of the appropriate approach to process the TAMDEF network, based on the data from the campaign, quasi-continuous and continuous trackers. In order to accomplish this, the PAGES software was set up to automatically design the connections or baselines, which form a minimal spanning tree (i.e,. there is only one path between any two sites). Furthermore, the software allows for changes in the network from session to session. For the first part of the study, variants of frame realization were testing inside and outside the Antarctic tectonic plate, and they are as follows. Case I: The GPS data from the TAMDEF stations, six IGS stations: Casey, CAS1; Davis, DAV1; Kerguelen Islands, KERG; Mawson, MAW1; Syowa, SYOG; Veleskarvet, VESL, and one non-IGS station (Palmer, PALM) in Antarctica, were processed as GPS network I. Case II: The GPS data from the TAMDEF stations, five IGS stations: Antuco, ANTC; Copiapo´, COPO; Iquique, IQQE; Punta Arenas, PARC; Santiago, SANT, and one non-IGS station (Puerto Williams, PWMS) in South America were processed as GPS network II. Case III: The GPS data from the TAMDEF stations and four IGS stations: Gough Island, GOUG; Hartebeesthoek, HARB; Hartebeesthoek Rao, HRAO; and Sutherland, SUTH in Africa were processed as GPS Network III. Case IV: The GPS data from the TAMDEF stations and four IGS stations: Alice Spring Avoir, ALIC; Ceduna AU019, CEDU; Karratha AU013, KARR; and Perth, PERT in Australia were processed as GPS Network IV. The decision to select the stations (IGS and non-IGS) for the four cases above was made based on GPS data availability and consistency; the data with the
637
extended periods of simultaneous GPS logging times with respect to the TAMDEF data were selected. The coordinates and velocities, with their corresponding variances, for all the involved IGS sites (ITRF00 at epoch 1997.0) were obtained from http://itrf.ensg.ign. fr/ITRF_solutions/2000/results/ITRF2000.php. Thus, these coordinates were fixed in the processing for the four cases described above. The results generated from this processing are expected to provide verification and quantification of the TAMDEF network’s behavior with respect to these four tectonic plates. For the second part of the study, to validate results from the SCLESS approach (Case I only) for IGS sites within the Antarctic tectonic plate, alternative network adjustment approaches were proposed and they are described in Sect. 78.4.
78.3
Traditional Least-Squares Adjustment
78.3.1 Least-Squares Adjustment in the Gauss–Markov Model with Stochastic Constraints (SCLESS) The SCLESS approach used in this research involves the use of prior information for the parameters, coordinates and velocities, with their corresponding variances for the IGS sites. Here, the positive definite weight matrix P0 is formed by the a priori variances obtained from a previously performed least-squares adjustment. For a detailed derivation of the formulas, see Schaffrin and Snow (2007). The Gauss–Markov with stochastic constraints in its linear form is given by: yn1 ¼ Anm xm1 þ en1 z0l1 ¼ Klm xm1 þ e0l1
(78.1)
where y is the (nx1) observation vector, A is the (nxm) design matrix, x is the (mx1) unknown parameter vector, e is the (nx1) random error vector and z0 is the vector with the stochastic constraint. With rankðAÞ ¼ q fm; ng; rankðKÞ ¼ l m q; rank ð½ AT K T Þ ¼ m, and with the stochastic model for e and e0 described by:
e e0
1 0 P ; s20 0 0
0 P1 0
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78.4
Alternative Network Adjustment Approaches
To strengthen the network adjustment component of this research, alternative network adjustment approaches (e.g., MINOLESS, Partial-MINOLESS and BLIMPBE) ought to be considered and compared with the SCLESS, and used in order to test algorithms and software developed in this area (by Snow and Schaffrin 2004). In general, in the network adjustment scenario (e.g., the TAMDEF GPS network, which is derived exclusively from the observed GPS baseline vectors) the estimation of the coordinates from a (weighted) LESS will not be unique, even though the adjusted baseline vectors are unique (Snow and Schaffrin 2004). In such a case, if one attempts to achieve uniqueness without affecting the adjustment, two alternatives are recommended (Snow and Schaffrin 2004): (1) introduce a minimum set of constraints for the position coordinates “datum”, or (2) apply a specific objective function on the set of LESSs that fulfills the “normal equations”. In both alternatives, bias control and minimization for some (or all) coordinates were taken into account, and investigated for the TAMDEF network.
78.4.1 Singular Least-Squares Solutions (SLESS) in a Rank-Deficient Gauss–Markov Model For those geodetic networks that are derived from the observed GPS baseline vectors (such as TAMDEF), the rank-deficient (singular) least-squares adjustment (SLESS) was employed because of the presence of an inherent datum deficiency of three, due to the unknown translation/shift parameters. This type of adjustment will lead to a 3-D hyperspace of the LESS for the traditional normal equations (Schaffrin and Iz 2002; Kuang 1996). The rank-deficient Gauss–Markov linear model for the analysis of GPS networks (with datum deficiency) is given by: y ¼ A n1
x þ e
nm m1
n1
(78.2)
where y, A, x and e were described before, with the stochastic model for e described by: e ð0; s20 P1 Þ
In order to overcome the rank-deficiency problem and to affect minimization of the bias for certain coordinates of the TAMDEF network a rigorous examination and investigation of further extensions of the minimum-norm solution in the least-squares solution space were performed.
78.4.2 Minimum-Norm Least-Squares Solution (MINOLESS) This approach was used to perform a free network
2
adjustment. The target function given by ^x ¼ ^xT ^x ¼ min fN ^x ¼ cg will guarantee that the vector ^ x
containing the coordinate changes possesses the minimum norm. Furthermore, Snow and Schaffrin (2004) proved that the MINOLESS adjustment will generate a minimum Mean Square Error (MSE) risk on average. Another reason for using MINOLESS as a network adjustment alternative for TAMDEF is that this method belongs to the larger class of LESSs. Thus, the adjusted observations will be an unbiased estimate of the “true” observables. According to the rankdeficiency Gauss–Markov linear model given by (78.2), the estimated parameters, based on:
2
^
x ¼ ^xT ^x ¼ min fN ^x ¼ cg : ^ x
(78.3)
^xMINOLESS ¼ NðNNÞ c ¼ N þ c;
but NðNNÞ 6¼ N þ , and N þ denotes the MOORE PENROSE or pseudo-inverse, N þ 2 Nrs , where Nrs is a reflexive symmetric g-inverse.
78.4.3 Partial Minimum–Norm Least-Squares Solution (Partial-MINOLESS) This approach is referred to as the S-weighted MINOLESS, and it was also used to adjust the TAMDEF network. The selection matrix (S) was strategically constructed to allow for the selection of the primary points; if the TAMDEF sites FTP1, MCM4 and ROB1 are selected, then therankðSÞ ¼ 9.
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Traditional and Alternative Network Adjustment Approach for the TAMDEF GPS in Antarctica
Snow and Schaffrin (2004) proved that the S-weighted MINOLESS is uniquely defined whenever the matrix (S þ N) is invertible. An additional motivation for using this approach is that “the Partial-MINOLESS, where S :¼ Diagð1; . . . ; 1; 0; . . . ; 0Þ provides linear minimum bias estimates collectively for all those coordinates that do not participate in the “partial minimum norm” process” (Corollary 7 of Snow and Schaffrin 2004). Among all the LESSs, it also minimizes the partial trace of the dispersion matrix associated with the selected parameters (Koch 1999). However, this approach will generally not turn out to be “best” (in this class) because of the overall Mean Squared Error risk. Also, according to (78.2), the estimated parameters, based on
2
^ ^T ^ x ¼ cg :
x ¼ x Sx ¼ min fN ^ S
^x
(78.4)
1 ^x T PMINOLESS ¼ ðN þ SE ESÞ c
where ðN þ SET ESÞ1 is a symmetric g-inverse of N, with maximum rank, and thus not reflexive.
78.4.4 Best Linear Minimum Partial–Bias Estimation (BLIMPBE) Schaffrin and Iz (2002) developed the BLIMPBE estimator which, generally, cannot be considered as a LESS. It is a more robust estimator and, as the “partial” term implies, it is characterized by securing a “minimum bias” for a given subset of estimated parameters (e.g., for a certain group of point coordinates). Furthermore, BLIMPBE relies on the appropriate choice of the selection matrix (S), which ought to be constructed so that all the secondary points are selected, hence minimizing their bias. As can be seen, this was already done in the previous approach (Partial-MINOLESS); however, the new solution generated by the BLIMPBE will be the “best” among other minimum partial-bias solutions in terms of its MSE-risk. In other words, if we compare the trace of the dispersion matrix generated with the BLIMPBE, it will be smaller than the one generated with the Partial-MINOLESS. The adjusted observations of the BLIMPBE will also differ, in general, from those of the SLESS solutions, such as the MINOLESS or
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Partial-MINOLESS. BLIMPBE will simply reproduce coordinates of those points that were not selected, returning a zero variance for these points. This is what is called the reproducing or zero variance property of the BLIMPBE. Based on the rank-deficient Gauss–Markov linear model given by (78.2), the estimated parameters (BLIMPBE of x): ^^ SNÞ N Sc xBLIMPBE ¼ ½SNðN SN
(78.5)
with S as suitable “selection matrix”.
78.5
Test Results and Analysis
The results for the first part of the study based on the traditional network adjustment approach (SCLESS) were generated in terms of the time-series for the adjusted north (dn), east (de) and up (du) coordinate components for the three tested TAMDEF sites: FTP1, MCM4 and ROB1 (year 2000–2005). The time-series for the up coordinate component for MCM4 site is illustrated in Fig. 78.2 as an example. The statistics of the time-series for dn, de and du coordinate components (Case I–IV) are also illustrated in Table 78.1 for the sites tested. Even though the maximum and minimum values are not shown in Table 78.1, the scatter is smaller for the dn, de and du for the FTP1 site for Case I (ranges within ~50 to ~70 mm), and bigger scatter was obtained in Case III (ranges within ~100 to ~200 mm). A positive ~ +60 mm bias, a negative ~40 mm bias and a positive ~+20 mm could be inferred in dn, de and du, respectively, in Case I. Case II reported mean values close to zero for the dn and de in comparison with the other three cases. The MCM4 site seems more stable with respect to Case I (ranges within ~60 to ~100 mm). Similarly to the FTP1 site, the less homogeneous solution for the MCM4 site occurs when processed with respect to Case III (ranges within ~200 to ~200 mm) in all three coordinate components. However, although the du with respect to Case III looks much dispersed, its mean value was the smallest (2.8 mm), as compared to the other three cases. Finally, the solutions of the dn, de and du for the ROB1 site which, in comparison to the previous two sites (FTP1 and MCM4), looks more stable with respect to Case I (ranges within ~80 to ~60 mm).
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640 Fig. 78.2 Time-series for the up coordinate component at the MCM4 site (Cases I–IV)
DUp_MCM4
100
du [mm]
50
0
–50
–100 2000
2000.5
2001
2001.5
2002
IGS_Antarctica IGS_South America IGS_Africa IGS_Australia
2002.5
2003 year
2003.5
2004
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Table 78.1 Statistics for the time-series (coordinate components) when processing TAMDEF network with respect to the four cases for the tested sites Site
Coord. Comp. dn FTP1 de du dn MCM4 de du dn ROB1 de du
Case I Mean (mm) 58.9 37.1 20.0 2.6 8.8 4.6 41.9 35.8 1.2
Case II St. Dev. (mm) Mean (mm) 3.3 21.7 6.2 54.3 16.5 42.8 6.1 38.7 10.9 64.6 24.9 54.8 7.7 15.7 12.6 43.5 15.8 44.8
St. Dev. (mm) 15.6 11.7 27.9 14.6 18.0 50.5 21.1 16.5 32.9
This ROB1 site is also the least stable when processed with respect to Case III (ranges within ~100 to ~120 mm). A positive ~+35 mm bias could be inferred in both dn and de coordinate components, respectively, in Case I. Similarly to the other two sites, the best values in terms of the standard deviation for the three coordinate components were also found for the most consistent solution (Case I). Overall, it can be concluded from the results from the traditional network adjustment approach that the MCM4 is the site that behaves more stable in the three coordinate components with respect to Antarctic tectonic plate (Case I). On the other hand, the biggest scatter of the time-series occurs when processing TAMDEF network with respect to IGS Africa (Case
Case III Mean (mm) 14.8 27.1 53.2 47.9 23.0 2.8 3.7 13.9 9.0
St. Dev. (mm) 26.9 25.8 62.0 51.3 45.2 98.4 31.5 32.1 70.5
Case IV Mean (mm) 93.6 74.8 25.7 34.1 38.6 7.5 72.4 68.8 11.3
St. Dev. (mm) 12.4 12.6 23.9 14.7 16.8 30.3 14.4 12.8 24.9
III). This was expected, as in this case, longest baselines were processed. Still, the analyses presented here are useful, considering that variants of frame realization were testing inside and outside the Antarctic tectonic plate allowing ITRF connection of the TAMDEF network. Furthermore, in order to account for the quality of the time-series obtained from the traditional approach, PAGES allows computing the overall root mean squared (overall RMS deviations) for the processed GPS data. According to Schenewerk et al. (2001) a good rule-of-thumb is that the overall RMS deviations should be less than or equal to 0.015 meters for long baselines. In Fig. 78.3 the mean values for the overall RMS deviations (Cases I–IV) are shown for the option
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Traditional and Alternative Network Adjustment Approach for the TAMDEF GPS in Antarctica
Fig. 78.3 Overall RMS for deviations in the TAMDEF GPS data
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OVERALL RMS
20
ORMS [mm]
15
10
5
0 2000
2000.5 IGS IGS IGS IGS
2001
2001.5
Table 78.2 Statistics for overall RMS deviations in the TAMDEF GPS data IGS sites Antarctica South America Africa Australia
2002
Antarctica South America Africa Australia
Mean (mm) 8.7 7.7
Min (mm) 0.4 1.7
Max (mm) 23.3 16.7
St. Dev.
(mm) 2.4 2.7
10.0 11.4
0.4 5.2
23.0 24.1
4.8 2.2
with fixed ambiguity, where the mean RMS ranges from 7.7 to 11.4 mm, a convincing result in view of the (mostly) very long baselines. In addition, the statistics of the computed overall RMS deviations are shown in Table 78.2, where the mean and the standard deviation values confirm that the results for Case I are more stable, in comparison to Cases II–IV. The results in terms of coordinates for the second part of the study based on the alternative network adjustment approaches (MINOLESS, Partial-MINOLESS and BLIMPBE) were compared against those obtained from the traditional approach (SCLESS), specifically and only for Case I. To be consistent with the proposed scenario for Case I, a GPS network with 6 IGS stations and one non-IGS was constructed and analyzed. The a priori coordinates for the GPS stations involved in the adjustment came from a previously performed adjustment using the PAGES software; here it was assumed that the GPS baseline components should only vary at the random noise level (Table 78.3).
2002.5 2003 2003.5 Epoch (year)
2004
2004.5
2005
2005.5
2006
Table 78.3 Coordinate comparison between the traditional (SCLESS) vs. alternative (MINOLESS, Partial-MINOLESS and BLIMPBE) network adjustment approaches Dif. in X Dif. in Y Coord. (mm) Coord. (mm) SCLESS vs. MINOLESS FTP1 3.0 +3.0 MCM4 +3.0 3.0 ROB1 3.0 1.0 SCLESS vs. PARTIAL-MINOLESS FTP1 4.0 +1.0 MCM4 +5.0 5.0 ROB1 5.0 +3.0 SCLESS vs. BLIMPBE FTP1 2.0 +2.0 MCM4 +4.0 4.0 ROB1 4.0 +2.0 Site
Dif. in Z Coord. (mm) 3.0 3.0 +2.0 2.0 1.0 +4.0 2.0 2.0 +1.0
The results from the traditional network adjustment approach (SCLESS), are very comparable (at the mm level) with respect to those obtained from the alternative network adjustment approaches (MINOLESS, Partial-MINOLESS and BLIMPBE). It can be pointed out that the results from MINOLESS should guarantee that the vector of coordinate changes would be the smallest with respect to the other two solutions; however, results from both partial-MINOLESS and BLIMPBE are very similar to those from MINOLESS (showing 1–2 mm differences), and it seems that there is no reason to suspect any bias among the resulting coordinates. Thus, GPS data used for the TAMDEF
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network are consistent to the point that various network adjustment methods gave equivalent results. Conclusions
From the network adjustment process, when analyzing the TAMDEF network connection to ITRF (Case I) and outside (Cases II–IV) the Antarctica, it can be concluded that the time-series results look more stable (horizontal and vertical) when processing TAMDEF network with respect to the Antarctic tectonic plate. Hence, it can be demonstrated that the GPS data used is clean of bias after properly taken care of ionosphere, troposphere and some other sources that affect GPS positioning, for the Antarctic tectonic plate results. Additionally, the results from the traditional network adjustment approach (SCLESS) compare at the mm level to those obtained from the alternative network adjustment approaches (MINOLESS, partial-MINOLESS and BLIMPBE). In other words, the comparison SCLESS vs. MINOLESS, partial-MINOLESS and BLIMPBE are within 1–2 mm differences. In addition, the alternative methodology and experimental algorithms in this research regarding the TAMDEF network can also be considered as a good choice when performing a least-squares adjustment for other GPS networks.
References Eckl MC, Snay RA, Soler TA, Cline MW, Mader GL (2001) Accuracy of GPS-derived relative positions as a function of interstation distance and observation-session duration. Journal of Geodesy 75:633–640 http://igscb.jpl.nasa.gov/ http://itrf.ensg.ign.fr/ITRF_solutions/2000/results/ITRF2000.php http://www.geology.ohio-state.edu/TAMDEF/ http://www.ngs.noaa.gov/GRD/GPS/DOC/pages/pages.html Koch KR (1999) Parameter estimation and hypothesis testing in linear models, 2nd edn. Springer, Berlin Kuang S (1996) Geodetic network analysis and optimal design. Ann Arbor Press, Chelsea, MI Mader GL, Schenewerk MS, Ray JR, Kass WG, Spofford PR, Dulaney RL, Pursell DG (1995) GPS orbit and earth orientation parameter production at NOAA for the International GPS Service for Geodynamics for 1994. In: Zumberge JF et al (eds) International GPS Service for Geodynamics, 1994 Annual Report, 197–212, Jet Propulsion Lab., California Institute of Technology, Pasadena, CA Schaffrin B, Snow K (2007) Adjustment computations, Part II. Lecture Notes, School of Earth Science, The Ohio State University, Columbus, OH Schaffrin B, Iz HB (2002) BLIMPBE and its geodetic ´ da´m J, Schwartz KP (eds) Vistas for applications. In: A geodesy in the new millennium, 125th edn, IAG Symposium Series. Springer, Berlin, pp 377–381 Schenewerk MS, Marshall J, Dillinger W (2001) Vertical ocean loading deformations derived from a global GPS network. Journal of the Geodetic Society of Japan 47(1):237–242 Snow K, Schaffrin B (2004) GPS-network analysis with BLIMPBE: a less biased alternative to least-squares adjustment. In: Miller M, Arnold J (eds) Proceedings of IONMeeting, Dayton, OH, pp 614–625
Impact of Loading Phenomena on Velocity Field Computation from GPS Campaigns: Application to ResPyr GPS Campaign in the Pyrenees
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J. Nicolas, F. Perosanz, A. Rigo, G. Le Bliguet, L. Morel, and F. Fund
Abstract
To quantify the present-day tectonic deformation in the Pyrenees mountain range area where the seismic activity is continuous and moderate, different GPS ResPyr campaigns were performed between 1995 and 2008. Considering the expected rate of deformation of about 1 mm/yr, we can wonder what would be the impact of the different loading effects on velocity field computed from GPS campaigns and therefore on the characterization of the deformation. The data are processed using the GINS software package and we estimated the effect of the loading phenomena on the velocity field resulting from the different campaigns. The final solution uncertainties are of the same order of the expected displacements and we demonstrated the non negligible impact of the various loading phenomena on the velocity field. Indeed, the accumulated loading effect (except tidal ocean load) can reach 0.8 mm/yr on horizontal velocity estimates, which is at the level of the searched signal.
79.1 J. Nicolas (*) L. Morel F. Fund L2G/ESGT/CNAM, 1 Boulevard Pythagore, 72000 Le Mans, France e-mail:
[email protected] F. Perosanz CNES/GRGS, 18 Avenue Edouard Belin, 31401 Toulouse Cedex 9, France A. Rigo DTP UMR5562, Universite´ de Toulouse – CNRS – Observatoire Midi – Pyre´ne´es, 14 Avenue Edouard Belin, 31400 Toulouse, France G. Le Bliguet L2G/ESGT/CNAM, 1 Boulevard Pythagore, 72000 Le Mans, France DTP UMR5562, Universite´ de Toulouse – CNRS – Observatoire Midi – Pyre´ne´es, 14 Avenue Edouard Belin, 31400 Toulouse, France
Introduction
The seismic activity throughout the Pyrenees mountain range (boundary between France and Spain), which corresponds to the convergent suture between Eurasian and Iberian plates, is continuous and moderate (Mw 5) (Souriau and Pauchet 1998). This area is the most seismically active region in France. The surface deformation is expected to be at a maximum of ~1 mm/yr (Nocquet and Calais 2003). This very small expected rate of deformation requires periodic observations over a long period of time. In order to quantify the present-day tectonic deformation in this region, different GPS ResPyr campaigns were performed in 1995, 1997, and 2008 over the entire belt. For the 2008 campaign, about 40 of 85 ResPyr sites were observed based on 48-h session length, on both sides of the border. Figure 79.1 displayed the
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_79, # Springer-Verlag Berlin Heidelberg 2012
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Fig. 79.1 European permanent stations used in this study. The square indicates the Pyrenees area
43°N 30 43°N 42°N 30 0003
42°N 2°W
1°W30 1°W 0°W30 W0°E 0°E30
1°E 1°E30
2°E
2°E30
It is expected that loading effects may impact the estimated velocities. Indeed, Blewitt and Lavalle´e (2002) demonstrated that annual signals, mainly due to hydrological and atmospheric loading, can significantly bias site velocity estimates. They recommend a minimum data span of 2.5 years to derive reliable and unbiased velocity solutions for tectonic interpretation. In another hand, Collilieux et al. (2009) analyzed a posteriori loading corrections on positioning time series of permanent sites and demonstrated significant velocity differences for stations without enough observations. In our case, we consider sparse campaigns and the studied area with high relief and located between the Atlantic Ocean and the Mediterranean Sea may be particularly sensitive to such loading effects. This estimation will ensure better accuracy on the velocity field derived from different GPS campaigns which are very used for geodynamical studies. Thus, we estimated the effect of the various loading phenomena on the velocity field resulting from the different campaigns and therefore on the characterization of the deformation of the Pyrenees. This document shortly presents the ResPyr campaign data processing and the resulting positioning quality. Afterwards, the different loading effects are considered and we give an estimate of their impact on the velocity computation. Finally, these results are compared and discussed.
3°E
Fig. 79.2 2008 ResPyr campaign sites
European permanent IGS (Dow et al. 2009) GPS stations used in this study as well as the Pyrenees area location. Figure 79.2 indicates the different sites monitored during the 2008 ResPyr campaign. Considering the expected rate of deformation in this area (maximum of ~1.3 cm over the 13 years between the first and the last campaigns), we can wonder what would be the impact of the different processing parameters on velocity field computed from sparse GPS campaigns. In particular, in this study we focus on the impact of the different loading effects. Indeed, atmospheric, oceanic, and hydrological loading phenomena due to mass redistributions can induce crustal deformations up to several centimeters of amplitude with different time periods (annual, semiannual, diurnal, sub-diurnal. . .).
79.2
Processing Data and Positioning Quality
The data are processed using the GINS software package developed by CNES/GRGS. Today, GINS is exploited by several scientific groups for a wide variety of geoscience’s applications and is routinely operated in the framework of the CNES-CLS IGS Analysis Center activities (www.igsac-cnes.cls.fr). In this study we determine accurate coordinates at each site over a time window of more than 10 years. It was therefore demonstrated, for the first time, the capability to treat such measurements of time spaced campaigns with this software package. Both Precise Point Positioning (PPP) and double differencing (DD) strategies were used in a two step approach. In a first step, the PPP mode was used for data editing and a priori coordinates computation for the
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Impact of Loading Phenomena on Velocity Field Computation from GPS Campaigns
campaign stations with a site by site and day by day approach. The standard deviation of this solution is of about 1.4 cm, 0.7 cm, and 1.2 cm for X, Y, and Z components, respectively. Then, the DD mode was used with a 12 h-slidewindow overlapped every 6 h for each ResPyr station computed independently. The solution validation was performed by analyzing the coordinates obtained for the 2008 campaign with different processing strategy parameters: network configuration, baseline construction, session duration, parameter constraints. . . Baseline construction method can induce differences up to 1 cm. The number of reference stations used can have an effect of 2-8 mm. The final reference frame used (see Fig. 79.1) is constituted by 18 IGS core stations available for all the different ResPyr campaigns to ensure the same reference frame for all the data processing. The standard deviation is really improved up to 48 h session duration but not for longer sessions. This indicates that the observation strategy for the last ResPyr campaign
79.3
Loading Effects
Spatial and temporal variability of loads may change the velocity field determination and so its geodynamical interpretation. To evaluate the impact Table 79.1 Mean amplitude of the various loading effects for the different ResPyr sites over the different campaigns Loading effect Tidal ocean Non tidal ocean + atmosphere Hydrology
5
5
0
1995.55
1995.56
1995.57
–5 1997.54 1997.55
1997.56
1997.57
Date 10
5
5
5 N (mm)
10
N (mm)
10
– 10 1995.54
0
–5
1995.55
1995.56
Date
1995.57
Up (mm) 20.5 0.5 1.8
–5 2008.48 2008.49 2008.5 2008.51 2008.52
Date
–5
N (mm) 4.1 <0.1 0.9
0
Date
0
E (mm) 2.4 0.2 0.7
E (mm)
5 E (mm)
10
E (mm)
10
0
N (mm)
was correct. The final coordinate standard deviations are 1–4 mm in horizontal and 3-16 mm in vertical. Processing GPS geodetic data anterior to 2000 can be tricky. The first results of the 1997 campaign have standard deviations of about 1-2 cm. The treatment of 1995 data still needs some software development.
10
–5 1995.54
645
– 10 1997.54
0
–5
1997.55 Date
1997.56
1997.57
– 10 2008.48 2008.49 2008.5 2008.51 2008.52 Date
Fig. 79.3 Ocean tide loading deformation for the horizontal components (in mm) of the 0003 station during the different ResPyr campaigns
646
of the different loading effects (atmosphere, ocean, and hydrology) on the velocity determination, loading corrections were applied a priori at the observation level or a posteriori using external models. In this case, we computed loading time series from models and evaluated the impact on the campaign station coordinates at each ResPyr campaign epoch and thus on the velocity estimates. The mean amplitude of each loading effect for ResPyr sites over the different campaigns are displayed in Table 79.1. First of all, we considered the Ocean Tide Loading (OTL) from FES2204 model (Lyard et al. 2006) with 1 h sample. This effect can affect vertical and horizontal coordinates at the centimeter level even at 400 km from the coast. As an example, Fig. 79.3 shows the time variability of the signal for station 0003 located in the eastern part of the network (Fig. 79.1) during each campaign. For ResPyr sites, the mean OTL displacements amount about 2, 4, and 20 mm for East, North, and Up components, respectively. This effect is well known and implemented in most geodetic software packages, which is not the case of the other loading phenomena considered in this study. The atmospheric loading displacements are derived each 6 h on a 2.5 2.5 deg grid from NCEP (National Centers for Environmental Prediction) surface pressure reanalysis dataset using a modified inverted barometer hypothesis over the oceans (vanDam and Wahr 1987; van Dam et al. 1994). This effect induces horizontal deformation with a relative effect of 2–4 mm for different stations from a regional network and 1–2 mm for a given station at different epochs (Fig. 79.4). In addition, the non-tidal ocean loading (NTOL) (van Dam et al. 1997) displacement time series were computed each 12 h on a 1 1 deg grid from ECCO (Estimating the Circulation and Climate of the Ocean) model (JPL 2008). Considering the ResPyr campaigns, the mean accumulated effect of atmospheric loading and NTOL is lower than 1 mm for each component. Monthly GLDAS (Global Land Data Assimilation Systems) dataset was interpolated to weekly to compute hydrological loading displacements on a 1 1 deg grid (van Dam et al. 2001). NTOL and hydrological loadings can reach the millimeter level effect on the North component as shown in Figs. 79.5 and 79.6. From these modeled amplitude of the various loading effects for the different ResPyr station
J. Nicolas et al.
Fig. 79.4 Horizontal deformation (in cm) induced by atmospheric loading at 4 IGS stations at the same epoch (on the top) and at Toulouse at different periods (on the bottom)
coordinates, we computed the loading impact on velocities. The results are displayed in Table 79.2. Even if the impact is essentially on the vertical component, it is not negligible on the horizontal component. NTOL, hydrology, and atmosphere accumulated loading effects on the horizontal velocities are at the level of 0.8 mm/year which is the level of the expected tectonic signal.
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Impact of Loading Phenomena on Velocity Field Computation from GPS Campaigns
647
10 0003
N (mm)
5
0
–5
– 10 1995
2000
2005
2010
Date 10
5
5
5
N (mm)
N (mm)
10
N (mm)
10
0
0
0
–5
–5
–5
1995.541 1995.551 1995.561 1995.57
1997.541 1997.551 1997.561 1997.57
Date
Date
2008.48
2008.5
2008.52
Date
Fig. 79.5 Cumulative effect of non tidal ocean loading and atmospheric loading deformation for the north component (in mm) of the 0003 station
79.4
Conclusion and Prospects
This study is the first evaluation of the capability of GINS to treat such sparse campaign measurements. However, the final solution uncertainties are of the same order of the expected displacements. This study equally shows that it is essential to keep up the observation campaigns to have a better understanding of the Pyrenees range dynamics. For 2008 campaign, we obtained positioning precision of ~5 mm in horizontal and ~1 cm in vertical. However, some improvements can be achieved to increase the result quality. For instance we will treat all the campaign stations together and not each one separately as done in this study. 1995 and 1997 campaign data will also be processed more precisely.
Moreover, we demonstrated that the various loading phenomena may have a non negligible impact on the velocity field resulting from the different campaigns and therefore on the characterization of the Pyrenees deformation. Indeed, the NTOL, hydrology, and atmosphere accumulated loading effect on horizontal velocity estimates can reach 0.8 mm/yr, which is at the level of the searched signal of 1 mm/yr. Furthermore, these loading effects are not yet taken into account in routine GPS data processing. So, we recommend performing a priori loading effect correction when it is possible and at least applying a posteriori corrections on the coordinate time series. More refined loading models would also be welcome to go further in this kind of study. It also shows that it is essential, but not sufficient, to perform the surveys at the same period
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Fig. 79.6 Hydrological loading deformation for the north component (in mm) of the 0003 station
Table 79.2 Mean impact of the various loading effects over the different ResPyr campaigns on velocity Loading effect Tidal ocean Non tidal ocean + atmosphere Hydrology
2D (mm/yr) 3.7 0.7 0.1
Up (mm/yr) 16.5 1.3 0.2
3D (mm/yr) 16.9 1.5 0.2
Acknowledgements We gratefully acknowledge Tonie van Dam from University of Luxembourg for providing series of loading deformation models. Thanks to all the people who contributed to the ResPyr campaigns used in this study. The ResPyr campaign and this study were supported by the 3F program of the French CNRS/INSU.
References (summer in this case). Then, it may be necessary to estimate the impact of the loading effects on the velocity field obtained from GPS time series and particularly for sparse campaigns. This estimation will ensure better accuracy on the velocity field derived from different GPS campaigns which are usually used for geodynamical studies.
Blewitt G, Lavalle´e D (2002) Effect of annual signals on geodetic velocity. J Geophys Res 107(B7). doi: 10.1029/ 2001JB000570 Collilieux X, Altamimi Z, Coulot D, van Dam T, Ray J (2009) Impact of loading effects on determination of the International Terrestrial Reference Frame. Adv Space Res. doi:10.1016/j.asr.2009.08.024 Dow JM, Neilan RE, Rizos C (2009) The International GNSS Service in a changing landscape of Global Navigation
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Impact of Loading Phenomena on Velocity Field Computation from GPS Campaigns
Satellite Systems. J Geodesy 83:191–198. doi:10.1007/ s00190-008-0300-3 JPL (2008) Ecco ocean data assimilation. http://ecco.jpl.nasa. gov/ Lyard F, Lefe`vre F, Letellier T, Francis O (2006) Modelling the global ocean tides: a modern insight from FES2004. Ocean Dyn 56:394–415 Nocquet JM, Calais E (2003) Crustal velocity field of western Europe from permanent GPS array solutions, 1996–2001. Geophys J Int 154:72–88 Souriau A, Pauchet H (1998) A new synthesis of Pyrenean seismicity and its tectonic implications. Tectonophysics 290:221–244
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van Dam TM, Wahr J (1987) Displacements of the earth’s surface due to atmospheric loading: effects on gravity and baseline measurements. J Geophys Res 92:1281–1286 van Dam TM, Blewitt G, Heflin M (1994) Detection of atmospheric pressure loading using the Global Positioning System. J Geophys Res 99:23939–23950 van Dam TM, Wahr J, Chao Y, Leuliette E (1997) Predictions of crustal deformation and geoid and sea level variability caused by oceanic and atmospheric loading. Geophys J Int 99:507–517 van Dam T, Wahr J, Milly PCD, Shmakin AB, Blewitt G, Lavallee D, Larson K (2001) Crustal displacements due to continental water loading. Geophys Res Lett 28:651–654
.
Comparison of the Coordinates Solutions Between the Absolute and the Relative Phase Center Variation Models in the Dense Regional GPS Network in Japan
80
S. Shimada
Abstract
Coordinates solutions are compared between the absolute and the relative phase center variation (PCV) models using the dense regional GPS network data in Japan before 1400 GPS week. In the result for the regional network sites and the western Pacific IGS fiducial network sites during 2005 and 2006 (GPS weeks 1303 through 1399) the coordinates repeatabilities applying the absolute PCV models are better than those adopting the relative PCVs, although the advantages are not significant for the period between 1996 and 1999 (GPS weeks 869 through 1042). The former result indicates the solutions applying the absolute PCV models are more precise than those adopting the relative PCVs even for the weeks when IGS precise orbits are calculated using the relative PCV models. The latter result may be caused by the sparse distribution of the IGS fiducial sites in the region, and the impreciseness of IGS precise orbits caused by the immature regional reference frame.
80.1
Introduction
GPS precise orbits by the International GNSS Service (IGS) were calculated applying the relative antenna phase center variation (PCV) models compared with the Dorn Margolin choke ring antenna before GPS week 1400 (Nov 4, 2006), and changes to the absolute PCV models after 1,400 week. It is well known that there occurre the scale gaps in the time series of the GPS network site coordinates between the solutions applying the relative and the absolute PCV models [Herring (2007) personal communication]. To prevent
S. Shimada (*) Earthquake and volcano research unit, National Research Institute for Earth Science and Disaster Prevention (NIED), 3-1 Tennodai, Tsukuba-Shi, Ibaraki-Ken 305-0006, Japan e-mail:
[email protected]
the gap occurrence, we analyze the GPS network observations using the absolute PCV models even for the data before 1,400 week for the Japanese regional dense network data with the IGS fiducial network data in and around the western Pacific region. We examine to compare the repeatabilities of the site coordinates solutions with those calculated applying the relative PCV models.
80.2
Analysis and Crustal Events
For the Japanese regional network sites, we adopt the dense permanent network sites in the Tokai area, central Japan (Fig. 80.1). The Tokai dense GPS network consists about 95 GEONET sites and five NIED sites. GEONET is the Japanese nation-wide permanent dense GPS network established and operational by
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_80, # Springer-Verlag Berlin Heidelberg 2012
651
652
S. Shimada
Fig. 80.1 Permanent GPS network in the Tokai area with the location of the major crustal events
the Geographical Survey Institute of Japan (GSI) (Miyazaki et al. 1998). GEONET became operational since September 1996. Thus we analyze the data between September 1996 and November 2006 (until GPS week 1399). For the period there occurred two major crustal events in the area. The first event is a seismo-volcanic event (SVE) occurred from July to September 2000 in the northern Izu Island (Nishimura et al. 2001), accompanied with large crustal deformation in the eastern part of the network. The second is a slow earthquake event (SEE) beginning in mid-2000 around or after the SVE event and ceased in mid-2005 (Ozawa et al. 2002; Miyazaki et al. 2006). The SEE event is the thrust motion in the plate boundary zone of the subducted slab under the Tokai area, and associated with the crustal movements in the western part of the network. The epicentral regions of those events are also shown in Fig. 80.1. In the analysis, we use IGS sites in and around the western Pacific region. Figure 80.2 shows the distribution of the IGS sites. The site codes of the IGS sites operational and available since 1996 are enclosed by square. The total number of the fiducial sites is 26, but the number of the sites available from the beginning of the analysis is only 13.
We use GAMIT/GLOBK 10.34 (Herring et al. 2008) for the analyzing software. We adopt ITRF 2005 reference frame, and the coordinates and the velocities of the fiducial sites are adopted according to Altamimi et al. (2007). We use the IGS final orbit as precise ephemeris. For the PCV models for IGS and NIED sites, we adopt the IGS relative and absolute PCV antenna models (IGS05). Because the GEONET sites adopt unique radomes which affect PCV considerably, we adopt the relative and the absolute PCV models determined by GSI (Hatanaka et al. 2001a, b). For all of the GEONET sites antennas were exchanged from micro-strip antenna to the Dorn Margolin choke ring antenna in 2003 (date of antenna change differs site by site during GPS weeks 1207 and 1224). We calculate the weekly network solutions from the daily solutions and we use the weekly solution in this study.
80.3
Result
Because the motions of the Tokai network sites during 2000 and 2005 are complex and the velocities are not constant caused by the two crustal events mentioned above, we examine the coordinates repeatabilities of
80
Comparison of the Coordinates Solutions Between the Absolute
653
Fig. 80.2 IGS fiducial sites adopted in the analysis. The sites with squared site code are operational and available since 1996, the beginning of the analyzing period
Number of Stations
N-S component
E-W component Relative PCV
U-D component
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
0
0 0 1 2 3 4 5 6 7 8
Number of Stations
the analyzing solutions for the following two periods to prevent the true crustal motions contaminating in the repeatability examination: The first is the period during September 1996 and the end of 1999 before SVE event (GPS weeks 869 through 1042). The second is the period from the beginning of 2005 when the SEE event are almost ceased to November 2006 just before the IGS analysis condition has changed (GPS weeks 1303 through 1399). Figure 80.3 shows the histograms of the repeatabilities for sites in the Tokai area during September 1996 and the end of 1999 applying the relative and the absolute PCV models. We fit linear trend to calculate the repeatabilities. For each component the repeatabilities for both models are not seen the significant difference. Table 80.1 summarizes the average and the standard deviation of the repeatabilities for each component for the same period. Figure 80.4 shows the histograms of the repeatabilities for IGS fiducial sites during September 1996 and the end of 1999 applying the relative and the absolute PCV models. We fit a linear trend before calculating the repeatabilities. For each component
0 0 1 2 3 4 5 6 7 8 Absolute PCV
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
0
0 0 1 2 3 4 5 6 7 8 mm
0
4
8
12 16
0
4
8 mm
12 16
0 0 1 2 3 4 5 6 7 8 mm
Fig. 80.3 The histograms of the repeatabilities for the Tokai sites for the period of September 1996 and the end of 1999 applying the relative and the absolute PCV models
the repeatabilities for both models are also not seen the significant difference. Table 80.2 summarizes the average and the standard deviation of the
654
S. Shimada
E-W component 2.4 0.5 mm
5.5 0.8 mm
2.7 0.5 mm
6.0 1.2 mm
E-W component
U-D component
40
40
30
30
30
20
20
20
10
10
10
0
0 0 1 2 3 4 5 6 7 8
0 0 1 2 3 4 5 6 7 8
40
40
40
30
30
30
20
20
20
10
10
10
0
0
3
3
3
0
0 0 1 2 3 4 5 6 7 8
0 1 2 3 4 5 6 7 8 mm
0 0 1 2 3 4 5 6 7 8
0
4
0
4
8
12
16
0
4
8
12
16
0
4
8 12 mm
16
8 12 mm
16
Absolute PCV 6
6
6
3
3
3
0
0
0 0 1 2 3 4 5 6 7 8 mm
Fig. 80.4 The histograms of the repeatabilities for the IGS fiducial sites for the period of September 1996 and the end of 1999 applying the relative and the absolute PCV models
Table 80.2 The average and the standard deviation of the repeatabilities for the IGS fiducial sites for the period of September 1996 and the end of 1999 applying the relative and the absolute PCV models E-W component
U-D component
2.8 1.6 mm
7.1 2.9 mm
3.6 2.1 mm
7.9 3.4 mm
repeatabilities for each component for the same period. Figure 80.5 shows the histograms of the repeatabilities for sites in the Tokai area from the beginning of 2005 to November 2006 applying the relative and the absolute PCV models. We also fit linear trend to calculate the repeatabilities. In this case for every component the repeatabilities applying the absolute PCV models are better than those adopting the relative PCVs. Table 80.3 summarizes the average and the standard deviation of the repeatabilities for each component for the same period. Figure 80.6 shows the histograms of the repeatabilities for IGS fiducial sites from the
0 0 1 2 3 4 5 6 7 8 mm
Fig. 80.5 The histograms of the repeatabilities for the Tokai sites for the period of the beginning of 2005 and November 2006 applying the relative and the absolute PCV models Table 80.3 The average and the standard deviation of the repeatabilities for the Tokai sites for the period of the beginning of 2005 and November 2006 applying the relative and the absolute PCV models N-S component Relative PCV 2.7 0.6 mm Absolute PCV 1.6 0.7 mm
Number of Stations Number of Stations
6
6
N-S component Relative PCV 2.4 1.0 mm Absolute PCV 2.6 1.4 mm
U-D component
40
Relative PCV 6
0 1 2 3 4 5 6 7 8 mm
E-W component Relative PCV
Absolute PCV
N-S component Number of Stations Number of Stations
U-D component
Number of Stations
N-S component Relative PCV 2.0 0.6 mm Absolute PCV 2.1 0.6 mm
N-S component Number of Stations
Table 80.1 The average and the standard deviation of the repeatabilities for the Tokai sites for the period of September 1996 and the end of 1999 applying the relative and the absolute PCV models
E-W component
U-D component
2.6 0.4 mm
7.2 0.8 mm
1.9 0.5 mm
4.8 1.2 mm
N-S component 6
6
3
3
0
0 0 1 2 3 4 5 6 7 8
E-W component Relative PCV
U-D component 6 3 0
0 1 2 3 4 5 6 7 8
0
4
8
12
16
0
4
8 mm
12
16
Absolute PCV 6
6
6
3
3
3
0
0 0 1 2 3 4 5 6 7 8 mm
0 0 1 2 3 4 5 6 7 8 mm
Fig. 80.6 The histograms of the repeatabilities for the IGS fiducial sites for the period of the beginning of 2005 and November 2006 applying the relative and the absolute PCV models
beginning of 2005 to November 2006 applying the relative and the absolute PCV models. We also fit linear trend to calculate the repeatabilities. In this case also the repeatabilities applying the absolute PCV models are better than those adopting the relative PCVs. Table 80.4 summarizes the average and the
80
Comparison of the Coordinates Solutions Between the Absolute
Table 80.4 The average and the standard deviation of the repeatabilities for the IGS fiducial sites for the period of the beginning of 2005 and November 2006 applying the relative and the absolute PCV models N-S component Relative PCV 3.6 1.4 mm Absolute PCV 2.4 1.1 mm
E-W component
U-D component
4.5 2.5 mm
10.5 3.8 mm
2.9 1.6 mm
6.7 2.4 mm
standard deviation of the repeatabilities for each component for the same period.
80.4
Discussion
For the period of the beginning of 2005 and November 2007, the site coordinates solutions adopting the absolute PCV models are generally more precise than those applying the relative PCV models, judging the histograms of the repeatabilities for both models shown in Fig. 80.5 for the Tokai sites and in Fig. 80.6 for the IGS fiducial sites, although the advantage in the average values of the repeatabilities applying the absolute PCVs are not significant compared with those adopting the relative PCVs for the 95% confidence level according to the standard deviations. All Tokai sites and most of the IGS sites adopt the Dorn Margolin choke ring antennas, and the absolute PCV values are measured accurately for choke ring antenna in the IGS05 model, and it is thought to bring the improvements for the solutions applying the absolute PCV models. On the other hand for the period between September 1996 and the end of 1999, the repeatabilities of the site coordinates solutions adopting the absolute PCV models are almost same level as those applying the relative PCV models, judging the histograms of the repeatabilities for both models shown in Fig. 80.3 for the Tokai sites and in Fig. 80.4 for the IGS fiducial sites for the period. For the Tokai sites, the GEONET antennas in this period are micro-strip antennas, thus the result of the little advantage may be caused by the less accurate absolute PCV models of the micro-strip antennas compared with the choke ring antennas in the IGS05 models and/or the GSI models. But the result is also same for the IGS sites which antennas are the Dorn Margolin choke ring antennas in all sites. Thus it seems that other factors hide the advantage of the
655
absolute PCV antennas. We think that this is caused by the less accurate precise orbit over the area and the less dense IGS fiducial site distribution in and around the western Pacific region, only half number of sites compared with the number of the sites in the 2005–2006 period shown in the Chap. 2. Those two factors are coupled that the sparse distribution of the IGS sites causes less accurate regional reference frame and reduces the accuracy of the precise orbit above the region. It is reasonable that the recent improvement of the accuracy of the IGS precise orbit is caused by the establishment of the precise global reference frame by the denser IGS network in various regions in the world, compared with the situation in 1990s when the regional reference frames are accurate enough only in western European and North American regions. Conclusion
The repeatabilities of the site coordinates solutions are examined for both the relative and the absolute PCV models applied for the GEONET and NIED dense permanent network data in the Tokai area, central Japan, with the IGS fiducial sites in and around the western Pacific region for the period before GPS week 1400 when IGS has changed the analyzing method adopting the absolute PCV models behalf of the relative PCVs. To prevent the disturbance by the complex motions and that the velocity of the Tokai sites are not constant caused by two major crustal events in the area, we examine two periods: during September 1996 and the end of 1999 (GPS weeks 869 through 1042), and the period from the beginning of 2005 and November 2006 (GPS weeks 1303 through 1399). The repeatabilities of the site coordinates adopting the absolute PCV models are better than those applying the relative PCVs for the period of the beginning of 2005 and November 2006 for both the Tokai sites and the IGS fiducial sites dataset. On the other hand the repeatabilities adopting the absolute PCV models are almost same level as those applying the relative PCVs for the period of September 1996 and the end of 1999 also for both dataset. This is thought to be caused that the following two factors cover the advantage of the absolute PCVs; the less accurate precise orbit over the area, and the less dense IGS sites which bring less accurate regional reference frame and reduce the accuracy of the IGS precise orbit over the area.
656
In conclusion, even before GPS week 1400 when IGS calculated the precise orbit applying the relative PCV models, the GPS data analysis should adopt the absolute PCV models which measures more accurately in the IGS05 models. Acknowledgements Kurt Feigl reviews the paper and gives some kind comments. The GEONET RINEX files and the GEONET PCV models are provided by the Geographical Survey Institute of Japan. We use GMT program by Wessel and Smith (1998) to draw figures.
References Altamimi Z, Collilieux X, Legrand J, Garayt B, Boucher C (2007) ITRF2005: a new release of the International Terrestrial Reference Frame based on time series of station positions and Earth Orientation Parameters. J Geophys Res 112:B09401. doi:10.1029/2007JB004949 Hatanaka Y, Sawada M, Horita A, Kusaka M (2001a) Calibration of antenna-radome and monument-multipath effect of GEONET – part 1: measurement of phase characteristics. Earth Planets Space 53:13–21
S. Shimada Hatanaka Y, Sawada M, Horita A, Kusaka M, Johnson JM, Rocken C (2001b) Calibration of antenna-radome and monument-multipath effect of GEONET – part 2: evaluation of the phase map by GEONET data. Earth Planets Space 53:23–30 Herring TA, King RW, McClusky SC (2008) Documentation for the GAMIT/GLOBK GPS analysis software. Department of Earth, Atmospheric, and Planetary Science, Massachusetts Institute of Technology Miyazaki S, Hatanaka Y, Sagiya T, Tada T (1998) The nationwide GPS array as an Earth observation system. Bull Geogr Surv Inst 44:11–22 Miyazaki S, Segall P, McGuire JJ, Kato T, Hatanaka Y (2006) Spatial and temporal evolution of stress and slip rate during the 2000 Tokai slow earthquake. J Geophys Res 111: B03409. doi:10.1029/2004JB003426 Nishimura T, Ozawa S, Murakami M, Sagiya T, Tada T, Kaidzu M, Ukawa M (2001) Crustal deformation caused by magma migration in the northern Izu Islands, Japan. Geophys Res Lett 28:3745–3748 Ozawa S, Murakami M, Kaidzu M, Tada T, Sagiya T, Hatanaka Y, Yarai H, Nishimura T (2002) Detection and monitoring of ongoing aseismic slip in the Tokai Region, Central Japan. Science 298:1009–1012 Wessel P, Smith WHF (1998) New, improved version of the generic mapping tools released. EOS Trans AGU 79:579
The 2009 Horizontal Velocity Field for South America and the Caribbean
81
H. Drewes and O. Heidbach
Abstract
Station velocities derived from space geodetic measurements in Central and South America were processed by the finite element method using a geophysical model and by a least squares collocation approach with empirical correlation functions for computing a continuous velocity field of the South American and the Caribbean crust. Velocities of the reference frame for the Americas (SIRGAS), and of various geodynamic networks (CASA, SNAPP, CAP, SAGA, and seismic gap projects) are used as input data. In general, the results present good agreement with previous models. Moreover, there are significant improvements, particularly in areas with new data (northern and central Andes, southern Tierra del Fuego).
81.1
Introduction
A main objective of geodesy is the determination and representation of the geometry of the Earth surface and its variation with time, e.g. for the realization of reference frames, and the study of geodynamics and global change effects. The geophysical background model widely used for this purpose is the plate kinematic model NUVEL-1A (DeMets et al. 1990, 1994). However, this model represents only rigid plates and does not include deformation zones (such as the Andes). The geologic-geophysical model PB2002 (Bird 2003)
H. Drewes (*) Deutsches Geod€atisches Forschungsinstitut, Alfons-Goppel-Str. 11, 80539 M€unchen, Germany e-mail:
[email protected] O. Heidbach GFZ German Research Centre for Geosciences, Telegrafenberg, 14473 Potsdam, Germany
includes deformation zones and a large number of (micro-) plates, but not all of these are confirmed by present-day geodetic measurements. In order to model the motions of the Earth crust for geodetic purposes we therefore need a realistic present-day crust deformation model. The deformation of the South American crust is mainly due to the subduction of the Nazca Plate under the South America Plate (e.g. Espurt et al. 2008). This convergence of plates develops a broad deformation belt, as expressed by the growth of the Andes in the past 10 Ma (Heidbach et al. 2008). For geodetic purposes, e.g. for the transformation of station coordinates of observation sites without known velocities from one epoch to another, a continuous contemporary velocity field is essential. Such a field was first computed for the South American crust in 2003 (Drewes and Heidbach 2005). Since then, additional data sets from various geodetic and geodynamics projects have become available. They are used in the new computation presented here.
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_81, # Springer-Verlag Berlin Heidelberg 2012
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81.2
H. Drewes and O. Heidbach
Input Data
The input data for the present computation of the velocity field are taken from the following projects: • One twenty eight station velocities (Fig. 81.1) of the multi-year GPS solution SIR09P01 of the International GNSS Service’s (IGS) Regional Network Associate Analysis Center (RNAAC) for the Geocentric Reference System for the Americas (SIRGAS) at DGFI (Seem€ uller et al. 2009) form the reference frame of the complete model. In addition, selected data of the following geodetic and geodynamics networks are used: • Fifty three coordinate differences of the SIRGAS GPS campaigns 1995 and 2000, covering the entire South American continent (Drewes et al. 2005) • Thirty one velocities of the CASA geodynamics project, Venezuela, 1988. . .2002 (Kaniuth et al. 2002a)
Fig. 81.1 Reference model SIR09P01 (Seem€ uller et al. 2009)
• Forty four velocities of the CASA geodynamics project, Costa Rica, Panama, Colombia, and Ecuador, 1991. . .1998 (Trenkamp et al. 2002) • Twenty nine velocities of a local network within CASA around Cali, Colombia, 1996. . .2003 (Trenkamp et al. 2004) • Sixty nine velocities of the integrated CAP-SNAPP project, Peru and Bolivia, 1993. . .2001 (Kendrick et al. 2001, 2003) • Sixty eight velocities of the CAP geodynamics project, central Andes, 1993. . .2001 (Brooks et al. 2003) • Thirty eight velocities of the SAGA geodynamics project, northern Chile, 1996. . .1997 (Khazaradze and Klotz 2003) • Seventy nine velocities of the SAGA geodynamics project, central Chile, 1994. . .1996 (Klotz et al. 2001) • Ten velocities of a geodetic project, central Chile, 2006. . .2008 (Baez et al. 2007)
81
The 2009 Horizontal Velocity Field for South America and the Caribbean
• Forty four velocities of a seismic gap network, southern Chile, 1996. . .2002 (Ruegg et al. 2009) • Fifty four velocities of a deformation network, central Chile, 2004. . .2006 (Vigny et al. 2009) • Twenty velocities of a Scotia-South America plate project, 1998. . .2001 (Smalley et al. 2003).
81.3
Data Processing
The velocities of all the regional data sets refer to different kinematic datums, i.e. they used different ITRF realizations or individual station velocities as the reference. Therefore, in the first step of data processing, velocities of all data sets were transformed to the continental solution SIR09P01 in the ITRF2005 datum by estimated spherical rotation vectors using identical points (most projects include IGS stations). They were then reduced to the South American plate by its plate rotation parameters in the ITRF2005 (Drewes 2009). If no identical points with SIR09P01
Table 81.1 Velocities used for the modelling Project SIR09P01 as the reference frame SIRGAS 1995. . .2000 differences CASA (Venezuela) CASA (Costa Rica . . . Ecuador) CASA (Cali, Colombia) CAP-SNAPP (Peru, Bolivia) CAP (Central Andes) SAGA 1996. . .1997 (Northern Chile) SAGA 1994. . .1996 (Central Chile) Scotia-South American plate boundary Seismic gap (Southern Chile) Chile (others) Total
No. velocities 95 28 21 31 17 54 60 32 68 19 65 6 496
were available, stations overlapping with other projects were used. Identical stations in different projects were analysed w.r.t. reliability (number and length of observation periods, total time interval covered), and only one velocity per site was accepted. Doubtful velocities 290˚
280˚
270˚ 20˚
659
300˚
310˚
320˚ 20˚
2 cm/a
10˚
10˚
0˚
0˚
−10˚
−10˚
−20˚
−20˚
−30˚
−30˚
−40˚
−40˚
−50˚
Fig. 81.2 Velocity field from finite element model
−50˚
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H. Drewes and O. Heidbach
found in the comparisons were eliminated. In areas where earthquakes with considerable co-seismic station displacements occurred during the observation period, e.g. at Cariaco, Venezuela (Kaniuth et al. 2002a) and Arequipa, Peru (Kaniuth et al. 2002c), the seismic deformations were excluded from the velocity computations by using only pre-seismic observations. The complete set of final input data is listed in Table 81.1.
1,0
North, deformation zones
0,8 0,6
b = –0.01
0,4 0,2 0,0
0
1,0
100 200 300 400 500 600 700 800 900 [km] East, deformation zones
0,8
b = –0.008
0,6 0,4 0,2 0,0
0
1,0
100 200 300 400 500 600 700 800 900 [km] North, stable zones
0,8
b = –0.007
0,6 0,4 0,2 0,0
0
1,0
100 200 300 400 500 600 700 800 900 [km]
81.4
One possibility to estimate the continuous velocity field is to set up a geo-mechanical model with the observed velocities as boundary conditions. As the distribution of observation sites is very irregular, it is reasonable to use the finite element method for solving the numerical problem of partial differential equations of the equilibrium of forces. The two-dimensional model is approximated on a sphere with a radius of 6,371 km. The only further boundary condition is that the four corner sides are fixed in order to avoid rotation of the whole model. In contrast to the previous approach for that area presented in Drewes and Heidbach (2005), the implementation of the rightlateral Bocono´ – El Pilar fault system of Venezuela and Colombia is neglected because the velocities observed are purely from the inter-seismic phase, i.e. the fault is locked in this period. Thus, the implementation of a sliding fault does not meet the used data set. However, the impact of these local effects is rather small and does not change the results significantly. The rheology of this geo-mechanical model is a homogeneous, isotropic, elastic (Hooke) material with Young’s modulus E ¼ 70 GPa and Poisson’s ratio u ¼ 0.25. The discretization of the model area consists of 500,000 plane strain linear elements, over which the deformation e is computed in northern (N) and eastern (E) direction from the stress s derived from the geodetic observations. The basic equations read
East, stable zones
0,8 0,6
b = –0.005
0,4 0,2 0,0
0
1,0
100 200 300 400 500 600 700 800 900 [km]
Finite Element Model
eN ¼ 1=EðsN usE Þ
(81.1a)
eE ¼ 1=EðsE usN Þ:
(81.1b)
To solve the numerical problem, the commercial finite element software package ABAQUS, version
North-East deformation
0,8 0,6
b = –0.05
0,4
Table 81.2 Typical parameters of empirical covariance functions for North- and East velocity components and NorthEast cross-correlation
0,2 0,0
0
100 200 300 400 500 600 700 800 900 [km]
Fig. 81.3 Typical correlation functions of velocity North and East components and North-East cross correlations in deformation and stable zones
North East N–E
Deformation zones b a (cm/a)2 0.10 0.010 0.60 0.008 0.11 0.053
Stable zones a (cm/a)2 0.01 0.03 0.01
b 0.007 0.005 0.007
81
The 2009 Horizontal Velocity Field for South America and the Caribbean
6.9, was used. The result of the velocity field is displayed on a regular 1 grid and shown in Fig. 81.2.
81.5
Least Squares Collocation Approach
The least squares collocation approach was applied as a vector prediction using empirical covariance functions. The predicted continuous velocities are a function of the observed station velocities and the correlations between them: vpred ¼ cTin Cij 1 vobs ;
(81.2)
where vobs is the observed velocity vector, vpred the predicted vector, Cij the auto-covariance matrix between the observed and cin the covariance matrix between observed and predicted velocity vectors. The elements of covariance matrices are taken from empirical isotropic covariance functions: one for each of the north and east velocity components and one for the
cross correlation between the north and east components. Empirical correlations are computed in distance (d) classes between the points and approximated by simple exponential functions a exp (b d). The distances d are computed on a sphere with a radius of 6,371 km. Typical examples of covariance functions are shown in Fig. 81.3 and summarized in Table 81.2. The parameters a and b vary for individual regions up to about 50% around the given mean value. The figures clearly demonstrate that a uniform covariance function for all components cannot be applied. The correlation length in stable zones is significantly longer than in deformation zones, and the East component has a longer correlation than the North component. As expected, the cross-correlations between North and East components in deformation zones are very small. The use of these functions guarantees that the covariance matrices are always positive definite. To get a sufficiently dense input velocity field, a wide-spaced grid is first interpolated in areas with sparse observations; in particular in the central part of the stable South American plate (Brazil). Only station 290˚
280˚
270˚ 20˚
661
300˚
310˚
320˚ 20˚
2 cm/a
10˚
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−10˚
−10˚
−20˚
−20˚
−30˚
−30˚
−40˚
−40˚
−50˚
−50˚
Fig. 81.4 Velocity field from least squares collocation
270˚
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H. Drewes and O. Heidbach
velocities of this region are used for this interpolation. These interpolated values are then included in the final prediction procedure. A 1 1 grid covering the entire South American and Caribbean area is predicted by using estimated individual covariance functions from the surrounding observed velocities up to a distance of 2,000 km. The result is shown in Fig. 81.4. The formal prediction error of velocities varies, dependent on the region, from less than 1 mm/a up to 9 mm/a in the areas with sparse observation coverage.
81.6
Comparison of Finite Element and Least Squares Collocation Results
where the velocity variation is greater, we have deviations between 6 and +6.3 mm/a and an r.m.s. deviation of 1.4 mm/a. A graphical impression of the differences is given in Fig. 81.5. The discrepancies are largest in zones with poor observation data, i.e., in the Caribbean Sea, where only a few islands have been observed, and the jungle areas in the eastern Andes of Colombia, Peru and Bolivia, as well as in Patagonia. The different interpolation methods of the finite element and least squares collocation methods become effective here. A decision regarding the superior reliability of the two methods cannot be made.
81.7
The comparison of results of the two approaches shows an agreement better than the formal precision of the individual methods. The deviations vary in the North component from 5 to +3.5 mm/a with an r.m.s. deviation of 0.8 mm/a. In the East component,
The main differences of the velocity field here presented and the formerly computed velocity field for South America (Drewes and Heidbach 2005) are the increased number of observations (496 instead of 290˚
280˚
270˚ 20˚
Comparison with the Previous Model
300˚
310˚
320˚ 20˚
2 cm/a
10˚
10˚
0˚
0˚
−10˚
−10˚
−20˚
−20˚
−30˚
−30˚
−40˚
−40˚
Fig. 81.5 Differences between finite elements and least squares collocation models
−50˚
270˚
−50˚ 280˚
290˚
300˚
310˚
320˚
81
The 2009 Horizontal Velocity Field for South America and the Caribbean
Fig. 81.6 Differences between 2009 and 2003 velocity fields
290˚
280˚
270˚ 20˚
663 300˚
310˚
320˚ 20˚
2 cm/a
10˚
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0˚
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329), the better quality of measurements due to an increasing number of continuously observing GPS stations (included in SIR09P01), and the extension to the Caribbean and Tierra del Fuego (southern Argentina and Chile). The differences of the two models shown in Fig. 81.6 vary between 8 and +3 mm/a in the North component and 9 to +18 mm/a in the East component. The r.m.s. deviations are 1.3 mm/a in the North and 3.2 mm/a in the East components. The largest discrepancies are found in the central Andes and in Colombia, where a large number of continuously observing GPS stations has recently come online. Conclusions
The horizontal velocity model computed by finite element and least squares collocation approaches provides a continuous deformation model over the South American continent and the Caribbean. It can be used for interpolating point velocities arising
280˚
290˚
300˚
310˚
320˚
from plate tectonic motions and Earth crust deformations, e.g. for transforming coordinates of newly installed geodetic stations from the observation epoch to the reference epoch of a given reference frame. The velocities shown here refer to the South America plate. For transformation to the ITRF2005 reference frame one has to add its global plate rotation (Drewes 2009). The corresponding plot and data file is available for practical use of interpolation at http://www.sirgas.org. The velocity model presented here is not appropriate for any sophisticated analysis of geodynamic features and processes because no detailed crust and mantle models were set up. Furthermore, all the geodetic velocities included in the computations are inter-seismic data and, thus, do not necessarily represent the long-term average velocities needed for interpreting the tectonic evolution in the area. The vertical velocity component was not included in the model, because its modelling cannot
664
be done in a continental scale without very detailed regional and local geophysical models. There are significant differences in vertical velocities over short distances, e.g. caused by fluid withdrawal (e.g. Drewes 1980) and/or subsidence of sediment basins (e.g. Kaniuth et al. 2002b; Kaniuth and Stuber 2005). Thus, the correlation length in least squares collocation is extremely short for vertical velocities and the finite element model requires a very dense network of discretization, variable Hooke parameters and boundary conditions. Such models on a local or regional scale are still to be developed and established.
References Baez JC, de Freitas SRC, Drewes H, Dalazoana R, Luz RT (2007) Deformations control for the Chilean part of the SIRGAS 2000 frame. IAG Symposia, Springer, Berlin, vol 130, 660–664 Bird P (2003) An updated digital model for plate boundaries. Geochem Geophys Geosyst 4 No. 3, p 52, doi:1010.1029/ 2001GC000252 Brooks BA, Bevis M, Smalley R, Kendrick E, Manceda R, Lauria E, Maturana R, Araujo M (2003) Crustal motion in the southern Andes (26 –36 S): Do the Andes behave like a microplate? Geochem Geophys Geosyst GC000505 DeMets C, Gordon RG, Argus DF, Stein S (1990) Current plate motions. Geophys J Int 101:425–478 DeMets C, Gordon R, Argus DF, Stein S (1994) Effect of recent revisions to the geomagnetic reversal time scale on estimates of current plate motions. Geophys Res Lett 21:2191–2194 Drewes H (1980) Precise Gravimetric Networks and Recent Gravity Changes in Western Venezuela. Dt. Geod. Komm., Reihe B, Nr. 251, M€ unchen Drewes H (2009) The Actual Plate Kinematic and crustal deformation Model (APKIM2005) as basis for a nonrotating ITRF. Springer, IAG Symposia, vol 134, 95–99 doi:10.1007/978-3-642-00860-3_15 Drewes H, Kaniuth K, Voelksen C, Alves Costa SM, Fortes LPS (2005) Results of the SIRGAS campaign 2000 and coordinates variations with respect to the 1995 South American geocentric reference frame. IAG Symposia, Springer, Berlin, vol 128, 32–37 Drewes H, Heidbach O (2005) Deformation of the South American crust estimated from finite element and collocation methods. IAG Symposia, Springer, Berlin, vol 128, 544–549 Espurt N, Funiciello F, Martinod J, Guillaume B, Regard V, Faccenna C, Brusset S (2008) Flat subduction dynamics and deformation of the South American plate: Insights from analog modeling. Tectonics (27) TC3011, doi:10.1029/ 2007TC002175 Heidbach O, Iaffaldano G, Bunge H-P (2008) Topography growth drives stress rotations in the central Andes:
H. Drewes and O. Heidbach Observations and models. Geophys Res Lett (35) L08301, 6pp, doi:10.1029/2007GL032782 Kaniuth K, Drewes H, Tremel H, Stuber K, Kahle H-G, Geiger A, Hernandez JN, Hoyer MJ, Wildermann E (2002a). Interseismic, co-seismic and post-seismic deformations along the South American-Caribbean plate boundary from repeated GPS observations in the CASA project. Proceedings of the IAG Symposium “Recent crustal deformations in South America and surrounding area”, Santiago, Chile Kaniuth K, Haefele P, Sanchez L (2002b) Subsidence of the permanent GPS station Bogota. IAG Symposia, Springer, Berlin, vol 124, 56–59 Kaniuth K, Mueller H, Seemueller W (2002c) Displacement of space geodetic observatory Arequipa due to recent earthquakes. Zeitschr f€ur Verm 127:238–243 Kaniuth K, Stuber K (2005) Apparent and real local movements of two co-located permanent GPS stations at Bogota, Colombia. Zeitschr f€ur Verm 130:41–46 Kendrick E, Bevis M, Smalley R, Brooks B (2001) An integrated crustal velocity field for the central Andes. Geochem Geophys Geosyst 2:2001GC000191 Kendrick EC, Bevis M, Smalley R (2003) The Nazca – South America Euler vector and its rate of change. J South Am Earth Sci 16:125–131 Khazaradze G, Klotz J (2003) Short- and long-term effects of GPS measured crustal deformation rates along the south central Andes. J Geophys Res 108(B6):5, 15 Klotz J, Khazaradze G, Angermann D, Reigber C, Perdomo R, Cifuentes O (2001) Earthquake cycle dominates contemporary crustal deformation in Central and Southern Andes. Earth Planet Sci Lett 193:437–446 Ruegg JC, Rudloff A, Vigny C, Madariaga R, de Chabalier JB, Campos J, Kausel E, Barrientos S, Dimitrov D (2009) Interseismic strain accumulation measured by GPS in the seismic gap between Constitucio´n and Concepcio´n in Chile. Phys Earth Planet Int 175:78–85 Seem€uller W, Seitz M, Sa´nchez L, Drewes H (2009) The position and velocity solution SIR09P01 of the IGS Regional Network Associate Analysis Centre for SIRGAS (IGS RNAAC SIR). DGFI Report No. 85, available at http:// www.sirgas.org/ and this volume Smalley R, Kendrick E, Bevis MG, Dalziel IWD, Taylor F, Lauria E, Barriga R, Casassa G, Olivero E, Piana E (2003) Geodetic determination of relative plate motion and crustal deformation across the Scotia-South America plate boundary in eastern Tierra del Fuego. Geochem Geophys Geosyst (4) No. 9, doi:10.1029/2002GC000446 Trenkamp R, Kellogg JN, Freymueller JT, Mora HP (2002) Wide margin deformation, southern Central America and northwestern South America, CASA GPS observations. J South Am Earth Sci 15:157–171 Trenkamp R, Mora H, Salcedo E, Kellogg JN (2004) Possible rapid strain accumulation rates near Cali, Colombia, determined from GPS measurements (1996–2003). Earth Sci Res J 8:25–33 Vigny C, Rudloff A, Ruegg J-C, Madariaga R, Campos J, Alvarez M (2009) Upper plate deformation measured by GPS in the Coquimbo Gap, Chile. Phys Earth Planet Int 175:86–95
New Estimates of Present-Day Crustal/Land Motions in the British Isles Based on the BIGF Network
82
D.N. Hansen, F.N. Teferle, R.M. Bingley, and S.D.P. Williams
Abstract
In this study we present results from a recent re-processing effort that included data from more than 120 continuous Global Positioning System (CGPS) stations in the British Isles for the period from 1997 to 2008. Not only was the CGPS network dramatically densified from previous investigations by the authors, it now also includes, for the first time, stations in Northern Ireland, providing new constraints on glacio-isostatic processes active in the region. In our processing strategy we apply a combination of re-analysed satellite orbit and Earth rotation products together with updated models for absolute satellite and receiver antenna phase centers, and for the computation of atmospheric delays. Our reference frame implementation uses a semi-global network of 37 stations, to align our daily position estimates, using a minimal constraints approach, to ITRF2005. This network uses a combination of current IGS reference frame stations plus additional IGS stations in order to provide similar network geometries throughout the complete time span. The derived horizontal and vertical station velocities are used to investigate present-day crustal/land motions in the British Isles. This first solution provides the basis for our contribution to the Working Group on Regional Dense Velocity Fields, 2007–2011 of the International Association of Geodesy Subcommission 1.3 on Regional Reference Frames.
82.1
Introduction
There are two large-scale geophysical processes known to cause crustal motions in the British Isles. The first, the motion of the Eurasian plate due to plate
D.N. Hansen (*) F.N. Teferle R.M. Bingley Institute of Engineering Surveying and Space Geodesy, University of Nottingham, Nottingham NG7 2TU, UK e-mail:
[email protected] S.D.P. Williams Proudman Oceanographic Laboratory, Joseph Proudman Building, 6 Brownlow Street, Liverpool L3 5DA, UK
tectonics, predominantly acts on the horizontal coordinate components with negligible effect on the vertical. The second, known as glacio-isostatic adjustment (GIA), is the on-going viscous response of the solid Earth to past changes in ice sheets and sea level. This process contributes a signal in both the vertical and horizontal components (e.g. Milne et al. 2006; Bradley et al. 2009). The plate tectonics signal in the British Isles, which is considered part of the rigid interior of the Eurasian plate, is seen as motion in a northeasterly direction of approximately 23 mm/yr. The horizontal component of the GIA signal is primarily associated with the
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_82, # Springer-Verlag Berlin Heidelberg 2012
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D.N. Hansen et al.
Laurentide ice sheet and is predicted to be in a northwesterly direction, with a magnitude that varies across the region and is thought to be of the order of 1–2 mm/yr. For the vertical component, the dominant GIA signal is associated with the British-Irish ice sheet and the adjacent, significantly larger Fennoscandian ice sheet, resulting in subsidence on Shetland, uplift in most areas of Scotland and Northern Ireland, and subsidence in large areas of England, Wales, and Ireland on the order of 1–2 mm/yr.
82.2
The BIGF Network
The British Isles continuous GNSS Facility (BIGF) is funded by the UK’s Natural Environment Research Council (NERC) and is operated at the University of Nottingham by the Institute of Engineering Surveying and Space Geodesy (IESSG). The continuous GNSS data are supplied to BIGF by a number of collaborators (see Acknowledgments). Although, the current BIGF network holds data for over 150 CGPS stations, with some dating back to 1996, only a few of these stations have been established with precise geodetic or geophysical applications in mind. Furthermore, most of these stations have only been installed since about 2004 and others have been frequently moved or shutdown. In view of this, only about 30 stations were considered suitable for geophysical applications in recent publications which considered data up to 31 December 2005 (Bradley et al. 2009; Teferle et al. 2009; Woodworth et al. 2009). Furthermore, these 30 stations provided a fairly inhomogeneous station distribution, with a lack of stations in the geophysically interesting areas of Scotland and Northern Ireland. During the recent re-processing effort the BIGF network, seen in Fig. 82.1, was used. All stations with more than 2.5 years of data by 31 December 2008 were processed and this large data set was then reduced to a subset of stations which were thought to represent crustal/land motion. The selection procedure for these stations was based on a database of surface and bedrock geological data, obtained from the EDINA national academic data centre (see Acknowledgments), site photographs and monumentation data in the station log files. The selected stations were first chosen based on their time series length, greater than 6 years of continuous data or combined data for stations which have undergone
small location changes. The selection was then restricted to stations connected directly to bedrock, or stations on buildings or structures which are connected to bedrock. Unfortunately, the majority of stations in Scotland have fewer than 6 years of data, and since this region is one of the most geophysically interesting areas of the British Isles, a dual-CGPS station analysis (Teferle et al. 2002) was carried out in order to include some of these stations. A full description of the station classification scheme can be found in (Hansen et al. 2009). The result of the classification scheme was that 46 stations were concidered suitable for geophysical applications.
82.3
Re-processing of the BIGF Network
Daily double differenced solutions were computed from 1 January 1997 to 31 December 2008 using an in house modified Bernese GPS Software Version 5.0 (Dach et al. 2007). These solutions were obtained using the re-processed products (including satellite orbits and Earth orientation parameters) produced by (Steigenberger et al. 2006), as these were the only complete set of re-processed products available at the time of processing. Additionally, the processing strategy included the modeling of first order ionospheric effects, incorporated absolute satellite and receiver antenna phase centre models (Schmid et al. 2007; Cardellach et al. 2007), and employed an a-priori tropospheric model based on standard pressure and the Global Mapping Function (B€ohm et al. 2006). Furthermore, corrections for solid Earth tides and ocean tide loading were made based on the IERS 2000 standards and coefficients from the FES2004 model respectively. Finally, we used 37 IGS stations, as shown in Fig. 82.2, with well determined coordinates to align our daily coordinate results to ITRF2005 using a no-net translation minimal constraints approach (Altamimi et al. 2007). The daily coordinate estimates (normal-equation files) were then combined to create a set of weekly SINEX solutions whose coordinates are computed for the mid-epoch of the week’s data at each station. The resulting weekly coordinate time series were then analyzed using CATS (Williams 2008) and Maximum Likelihood Estimation (MLE). This provides realistic uncertainties for all estimated parameters, e.g. linear
82
New Estimates of Present-ay Crustal/Land Motions in the British Isles
60°
667
yr 0 2 4 6 8 10 12
LWTG LERW SUM1
THUR BUT1 SWTG
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THUS
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MORP KLRE NEWC PORK NSTG CARL OMGH KIRK WEAR BEES LOFT AMBL RICH BELF IOMN SCAR FLA1 IOMS GIGG YEAR SWAN BLAK LEED EASI LYN1 LIVE MANC HOOB DARE HOLY LINC SKEN LEEK ASAP NOTT WEYB IESG SHRE ABEA HEMS KING LICH PETE GORE MACY CLAW ABYW NEWR LOWE DROT NORT SHOB WAT T ABEP ALDB NEOT CARM BREC PERS CARD COLH STEV OXFR STRO CARF AMER ANGX BARKSHOE NEWB NAS1 SUNB HORT MARG WARI NPLDSHEE NFO1 FARN APPL DVTG WEIR TAUT OSHQ POOL HARD DUNG DUNK HERS PADS HURN PMTG EASTHERT PMTH EXMO SAND PORT CAMB
54°
52°
NEWL
50°
352°
PRAW
LIZA
356°
0°
4°
Fig. 82.1 BIGF network stations included in this re-processing effort. The colour-coded circles indicate the length of the coordinate time series up to 31 December 2008
rate, by accounting for both random and timecorrelated noise in the time series. Example results for IGS station HERS can be seen in Fig. 82.3.
It should be noted that great care was taken in order to account for all official coordinate offsets as reported in the IGS discontinuities file and for any extra
D.N. Hansen et al.
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THU1 THU2 KELY THU3
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668
POL2 KIT3
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Fig. 82.2 The 37 IGS stations used for the reference frame implementation in this re-processing effort. The colour-coded circles indicate the length of the coordinate time series up to 31 December 2008, for comparison with Fig. 82.1
80 40 0 -40 -80
Rate = 16.5 ± 0.1 mm/yr
80 40 0 -40 -80
Rate = 16.4 ± 0.1 mm/yr
40
Rate= 0.7 ± 0.2 mm/yr
Up [mm]
North [mm]
bl00rw01
East [mm]
Station: hers
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WRMS= 1.5 mm
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2000
2002 2004 Epoch [yr]
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Fig. 82.3 Example of weekly coordinate time series for IGS station HERS. The dashed vertical lines indicate epochs at which an offset was modelled
equipment changes from IGS station information logs that produced a visible discontinuity in the coordinate time series.
82.4
Crustal/Land Motion Estimates
Crustal motion estimates can be obtained from the estimates of station velocity under the assumption that the CGPS stations only experience displacements due to the motions of the Earth’s crust. If a station is affected by local and/or regional processes, e.g. the deformation of a man-made structure onto which the GPS antenna is mounted and/or natural compaction, then the estimates of station velocity can only provide land motion estimates, unless the effects of all active processes can be quantified. Therefore, vertical land motion can be considered being a combination of GIA and on-going natural compaction. These were assured through the station classification scheme outlined in Sect. 82.2. Furthermore, for the Nottingham region, the ground deformation pattern from a Persistent Scatterer Interferometry (PSI) analysis was also utilized to identify one station with an anomalous vertical station velocity (Leighton et al. 2011) and a few other stations with visibly anomalous vertical station velocities were also removed. This lead to a final set of 46 CGPS stations that were identified as experiencing crustal/land motions. A list of these
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New Estimates of Present-ay Crustal/Land Motions in the British Isles
a
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b GPS ITRF2005
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Fig. 82.4 Absolute (a) and residual (b) horizontal station velocities for the 46 selected CGPS stations. The residual station velocities were obtained by subtracting the ITRF2005 plate motion model velocities from the absolute horizontal station velocities
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Fig. 82.5 Maps of vertical crustal motions based on the 46 selected CGPS stations for (a) CGPS in ITRF2005 and (b) CGPS in ITRF2005 then aligned to AG
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stations together with their coordinates and velocity estimates can be obtained from the authors and is not given here for briefness.
82.4.1 Horizontal Motions As can be seen from Fig. 82.4, overall, the apparent horizontal motions follow the predicted motions of the ITRF2005 plate motion model, with residual velocities being generally smaller than 1 mm/yr with a RMS of 0.6 mm/yr. The residual velocities are apparently random with no discernible internal deformation and, as their magnitude is similar to the predicted horizontal motions associated with the GIA process, it is at this stage not possible to draw any conclusions without further investigations.
82.4.2 Vertical Motions As can be seen from Fig. 82.5, in general, the apparent vertical crustal motions confirm the expected pattern of subsidence on Shetland, uplift in most areas of Scotland and Northern Ireland, and subsidence in large areas of England and Wales. This suggests that the pattern of present-day vertical crustal/land motions based on geodetic data is largely consistent with the pattern of vertical crustal/land motions based on Holocene sea-level data. Furthermore, the inclusion of the CGPS stationsin Northern Ireland, not available in Bradley et al. (2009) or Teferle et al. (2009), defines the westerly boundary of the GIA-induced uplift over Scotland and Northern Ireland, which improves the geodetic constraints for future GIA models for the British Isles. Figure 82.5 shows maps of vertical crustal motions derived from the vertical station velocities as obtained in ITRF2005 and from those in ITRF2005 then aligned to absolute gravity (AG) following Teferle et al. (2009). In particular, this alignment is achieved by computing the weighted mean difference between the vertical station velocity estimates from CGPS and AG which is then subtracted from the CGPS estimates to form AG-aligned CGPS estimates of vertical crustal motions. These are considered as giving a more realistic picture of the uplift/subsidence pattern due to the uncertainties associated with the reference frame implementation of this study and the systematic
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effects in the ITRF2005 vertical velocity estimates (Altamimi et al. 2007). Conclusions
We have re-processed the CGPS data from over 120 stations in the BIGF network for the period from 1997 to 2008 and produced associated weekly SINEX solutions. The analysis of the derived coordinate time series showed that currently only 46 CGPS stations provide station velocity estimates that can be interpreted as present-day crustal/land motions. This represents an increase in the number of CGPS stations in the British Isles considered useful for geophysical applications and an improvement in the homogeneity of the network in Scotland and Northern Ireland compared to previous studies. Further work is required to analyze the horizontal velocity field and to move from a semi-global to a global reference frame implementation. For now, however, the cumulative SINEX solution from this re-processing effort will form BIGF’s first contribution to the Working Group on Regional Dense Velocity Fields, 2007–2011 of the International Association of Geodesy Subcommission 1.3 on Regional Reference Frames. Acknowledgements The work presented was partly funded through the UK Department for Environment, Food and Rural Affairs (Defra), the Environment Agency and the Natural Environment Research Council Strategic Ocean Funding Initiative grant NE/F012179/1. BIGF (http: //www.bigf.ac.uk) acknowledges its data providers: Defra, Environment Agency, IESSG, Land and Property Services Northern Ireland, Leica Geosystems Ltd., Met Office, National Physical Laboratory, NERC Proudman Oceanographic Laboratory, NERC Space Geodesy Facility, Newcastle University, and Ordnance Survey of Great Britain. The authors are also thankful to the IGS community for provision of all other CGPS data and related GPS products (Beutler et al. 1999), and EDINA JISC national academic data centre based at the University of Edinburgh, and the British Geological Survey.
References Altamimi Z, Collilieux X, Legrand J, Garayt B, Boucher C (2007) ITRF2005: a new release of the International Terrestrial Reference Frame based on time series of station positions and Earth orientation parameters. J Geophys Res 112:B09401. doi:09410.01029/02007JB004949 Beutler G, Rothacher M, Schaer S, Springer TA, Kouba J, Neilan RE (1999) The International GPS Service (IGS): an
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interdisciplinary service in support of earth sciences. Adv Space Res 23(4):631–653 B€ohm J, Niell A, Tregoning P, Schuh H (2006) Global Mapping Function (GMF): a new emperical mapping function based on numerical weather model data. Geophys Res Lett 33(7): L07304 Bradley SL, Milne GA, Teferle FN, Bingley RM, Orliac EJ (2009) Glacial isostatic adjustment of the British Isles: new constraints from GPS measurments of crustal motion. Geophys J Int 178(1):14–22 Cardellach E, Elosegui P, Davis JL (2007) Global distortion of GPS networks associated with satellite antenna model errors. J Geophys Res 112:B07405. doi:07410.01029/ 02006JB004675 Dach R, Hugentobler U, Fridez P, Meindl M (eds) (2007) Bernese GPS Software Version 5.0. Astronomical Institute, University of Bern, Bern, pp 612 Leighton J, Sowter A, Tragheim D, Bingley RM, Teferle FN (2011) Land Motion in the Urban Area of Nottingham Observed by ENVISAT-1. Int J Remote Sens, accepted Milne GA, Shennan I, Youngs BAR, Waugh AI, Teferle FN, Bingley RM, Bassett SE, Cuthbert-Brown C, Bradley SL (2006) Modelling the glacial isostatic adjustment of the
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UK region. Phil Trans R Soc 364:931–948. doi:10.1098/ rsta.2006.1747 Schmid R, Steigenberger P, Gendt G, Ge M, Rothacher M (2007) Generation of a consistent absolute phase-center correction model for GPS receiver and satellite antennas. J Geod 81(12):781–798 Steigenberger P, Rothacher M, Dietrich R, Fritsche M, R€ulke A, Vey S (2006) Reprocessing of a global GPS network. J Geophys Res 111:B05402. doi:10.1029/2005JB003747 Teferle FN, Bingley RM, Dodson AH, Baker TF (2002) Application of the dual-CGPS concept to monitoring vertical land movements at tide gauges. Phys Chem Earth 27 (32–34):1401–1406 Teferle FN, Bingley RM, Orliac EJ, Williams SDP, Woodworth PL, McLaughlin D, Baker TF, Shennan I, Milne GA, Hansen DN (2009) Crustal motions in Great Britain: evidence from continuous GPS, absolute gravity and Holocene sea level. Geophys J Int 178(1):23–46 Williams SDP (2008) CATS: GPS coordinate time series analysis software. GPS Solut 12:147–153, 110.1007/s1029110007-10086-10294 Woodworth PL, Teferle FN, Bingley RM, Shennan I, Williams SDP (2009) Trends in UK mean sea level revisited. Geophys J Int 176(1):19–30
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GURN (GNSS Upper Rhine Graben Network): Research Goals and First Results of a Transnational Geo-scientific Network
83
€pfler, B. Heck, F. Masson, P. Ulrich, and G. Ferhat M. Mayer, A. Kno
Abstract
The Upper Rhine Graben (URG) is a north-northeast trending rift system belonging to the European Cenozoic Rift System. Today, the southern part of the URG is seismically still active. Earthquakes of magnitude five have a recurrence time of approximately a few decades. In order to monitor and to determine recent crustal displacements in the URG area, the transnational cooperation GURN (GNSS Upper Rhine Graben Network) was established in September 2008. Within GURN geo-scientific research is carried out. The focus is on processing and analysing of observation data of continuously operating GNSS (Global Navigation Satellite Systems, e.g. GPS) sites.
83.1
Introduction
The Rhine Graben is the central, most prominent segment of the European Cenozoic rift system which extends from the North Sea through Germany and France to the Mediterranean Sea over a distance of some 1,000 km (Ziegler 1992; Bourgeois et al. 2007). GURN (GNSS Upper Rhine Graben Network) will focus geographically on the Upper Rhine Graben (URG). The URG is a 300 km long and 40 km wide SSW-NNE trending rift, extending from Basel (Switzerland) to Frankfurt (Germany). It is bounded to the west by the Vosges mountains and to the east
M. Mayer A. Kn€ opfler B. Heck (*) Geodetic Institute, Karlsruhe Institute of Technology (KIT), 76128 Karlsruhe, Germany e-mail:
[email protected] F. Masson P. Ulrich G. Ferhat Institut de Physique du Globe de Strasbourg, CNRS Strasbourg University, 5 rue Rene´ Descartes, 67084 Strasbourg Cedex, France
by the Black Forest. The northern limit is the uplifted area of the Rhenish Massif. To the south, the Leymen, Ferrette and Vendlincourt folds represent the northernmost structural front of the Jura fold and thrust belt. This thin-skinned compressive deformation front propagates 30 km further to the north up to Mulhouse (France). Preceded by late Cretaceous volcanism, the rifting was initiated during late Eocene to early Miocene (42–31 Ma) starting with broadly east-west or ENE-WSW extension and lasted until Aquitanian time (20 Ma). Today, the southern end of the Rhine Graben is characterized by small uplift and subsidence rates and by a quasi-compressive, left-lateral strike-slip tectonic regime with a maximum stress-axis oriented NW-SE. The URG is considered to be the most seismically active region of northwest Europe with significant probability for the occurrence of large earthquakes (Meghraoui et al. 2001; see Fig. 83.1). For a better understanding of the processes that lead to seismic activity, e.g. in the URG, it is necessary to study not only the location of the faults but also their kinematics. Seismic hazard assessment in
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_83, # Springer-Verlag Berlin Heidelberg 2012
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Fig. 83.1 Seismotectonic framework of the Lower and Upper Rhine Graben. Squares depict the instrumental seismicity from 1910 to 1990 (1 < M < 5.5). Circles correspond to the historical seismicity (Meghraoui et al. 2001)
this region is hindered by a lack of information on the time-dependent behaviour of active structures. GNSSbased deformation analyses in the URG region (e.g. Tesauro et al. 2005) are suffering from the little number of sites. GURN will contribute to remedy this regional deficiency by a first comprehensive and consistent analysis of data of permanently operating GNSS sites. In Sect. 83.2 the recent status of the GNSS Upper Rhine Graben Network will be described. Sect. 83.3 gives an overview on the first steps within GURN. In Sect. 83.4 some first GURN results are presented. The paper is closing with an outlook (Sect. 83.5).
83.2
GNSS Upper Rhine Graben Network
In September 2008, the E´cole et Observatoire des Sciences de la Terre (EOST, CNRS and Strasbourg University, France) and the Geodetic Institute of Karlsruhe University (Germany, GIK) established a
transnational cooperation called GURN. Within the GURN initiative these institutions are cooperating in order to carry out geo-scientific research in the framework of the transnational project TOPO-WECEP (Western and Central European Platform; link: http:// www.topo-wecep.eu/; Cloetingh et al. 2007), which succeeds the former project URGENT (Upper Rhine Graben Evolution and NeoTectonics; link: http:// comp1.geol.unibas.ch/; Behrmann et al. 2005) of the EUCOR universities (European Confederation of Upper Rhine Universities). The research is actually based on GNSS (Global Navigation Satellite Systems) in order to establish a highly precise and highly sensitive terrestrial network for the detection of recent crustal movements in the URG region. In the future additional sensors (e.g. InSAR) will be included. Geodetic measurements using satellite techniques have a long tradition in the URG, e.g. within the project EUCOR/URGENT GPS campaigns have been carried out in 1999, 2000, and 2003. These campaigns were suffering from the small number of occupied sites (approximately 30) as well as from poor and inhomogeneous spatial resolution and from a poor amount of GPS data (2 24 h), especially. But the results (e.g. movement rates, precision) and experiences gained were promising (Rozsa et al. 2005a, b). Therefore, in order to continue and intensify the work of the project EUCOR/URGENT, GURN was established.
83.3
First Steps of GURN
At the beginning of this initiative, GURN included German, French and Swiss continuously operating GNSS sites. Most data of the German sites were provided from SAPOS®-Baden-W€urttemberg. ® SAPOS is a service hosted by the German state surveys, see Wegener and Stronk (2005). Most sites were recently equipped to track GLONASS data. All SAPOS® antennas are calibrated on absolute and individual level. The data of the French sites have several origins: RENAG (network hosted by universities and research institutes, including EOST; http://webrenag. unice.fr/), RGP (network of Institut Ge´ographique National), Teria (private company, see Gaudet and Landry (2005)), and Orpheon (private company). The antennas of the French sites are not calibrated individually. Additionally, other continuously operating sites were included, e.g. IGS (Dow et al. 2005) resp. EPN
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GURN (GNSS Upper Rhine Graben Network): Research Goals
(Bruyninx 2004) sites HUEGelheim and ZIMMerwald as well as the GNSS site BFO1 of the Black Forest Observatory (Luo and Mayer 2008). In June 2009, GURN was extended to the north and to the south when SAPOS®-Rheinland-Pfalz (http:// www.lvermgeo.rlp.de/indexsapos.html) and the Federal Office of Topography swisstopo (http://www. swisstopo.ch) joined GURN. The resulting network covers the whole URG region homogeneously. The mean distance between the network sites is 40–60 km. GURN actually consists of approximately 75 permanently operating reference sites, see Fig. 83.2, which are delivering data in near real-time to the GURN server. SAPOS®Baden-W€urttemberg archives GNSS data since 2002, while SAPOS®-Rheinland-Pfalz collects data since 2004. Table 83.1 gives an overview of the data history of GURN. Fig. 83.2 Sites of GURN, status: July 2009
83.4
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First Results of GURN
The sites of GURN were established for scientific as well as for business purposes. According to e.g. Tesauro et al. (2005), Nocquet and Calais (2003), Ziegler (1992), and Zippelt and M€alzer (1981), maximum expected horizontal (approximately 0–1 mm/a) resp. vertical (approximately 1–2 mm/a) movement rates are very small. Thus, within a first step the quality of the data had to be verified with respect to parameters like monumentation (e.g. pillars, roof tops), multipath effects, and amount of data loss. This had to be done in order to derive later on meaningful statements of recent crustal motions, due to the fact that e.g. in the 2009 realisation of GURN only 10% of the sites taken into account were established on pillars anchored in the bedrock.
676 Table 83.1 History of GURN observation data Year 2002 2003 2004 2005 2006 2007 2008 Number of 13 17 42 44 46 48 65 GURN sites
Preliminary checks of the quality of the GNSS data were carried out using TEQC software (UNAVCO; Estey and Meertens 1999) based on code-related multipath effects as well as daily percentages of missing data of each site. This enables a preliminary quality classification of the GURN sites. In addition, the time series of L3 phase observation residuals were analysed using WaSoft/Multipath (Wanninger and Wildt 1997) to detect phase-related multipath effects on observation sites in order to improve the code-based quality classification of the GURN sites. Based on this analysis it was found that approximately 65%/20%/15% of the GURN sites show good/medium/bad multipath related signal quality (Kn€opfler et al. 2010). The GNSS data are routinely checked using the PPP (Precise Point Positioning) module of the Bernese GPS Software (Dach et al. 2007). This enables the generation of preliminary time series of all GURN sites independently from data of neighbouring sites. This leads to an improvement of the quality classification, e.g. by means of analysing the scattering of PPP time series. Based on this analysis related to daily solutions approximately 40%/50%/10% of the GURN sites show low/ medium/high coordinate scattering. In order to obtain precise coordinate estimates as well as velocities, the GURN data were analysed by the EOST working group using the GAMIT/GLOBK software Version 10.34 (Herring et al. 2006) in a three step approach. During the first step loose a priori constraints were applied to all parameters and double differenced GPS phase observations were used to estimate daily site coordinates, tropospheric zenith delay at each site for intervals of 2 h as well as orbital and Earth orientation parameters (EOP). Within this analysis step the observations of ~19 IGS-EUREF sites were included in order to link the regional GURN to global GNSS networks (see Fig. 83.3). In the second step, a consistent set of coordinates and velocities is estimated using the daily loosely constrained estimates of sites coordinates, orbits and EOP and their precision information (covariance matrix) as quasi-observations in a Kalman filter. In a third step, generalized
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constraints (Dong et al. 1998) were applied while estimating a six-parameter transformation (rate of change of translation and rotation). To derive velocities relative to stable Eurasia, the trend of the Eurasian plate defined by Altamimi et al. (2002) is removed from the resulting velocity field in the ITRF2000 reference frame. For all sites, two different preliminary time series (daily, 10-day-mean) are generated and analysed. Some sites (STUT, IFFE, FREI, GEIS, TAUB, BIBE, KARL, BSCN) are characterized by large semi-annual signals caused especially by the type of monumentation, see Fig. 83.4 (left). Some sites (e.g. STUT, HEID, SCHW, TAUB, OFFE) are characterized by discontinuities in the time series, which are mostly related to instrumentation or site modifications and have to be eliminated. Most of the French sites of GURN installed in 2007–2008 show short but good quality time series indicating that these sites (mostly established on pillars) are well connected to the bedrock, see the time series of the site WELSchbruch (Fig. 83.4, right) for example. Due to the fact that GURN is a project aiming for the long-time monitoring of slow motions (sub-mm per year; Illies 1977, 1979) in the URG region, a special focus has to be set on sites showing significant horizontal or vertical displacements. The behaviour of these sites has to be checked in detail in order to distinguish between local and regional displacement rates. In contrast to the EOST data processing strategy, the GIK uses the Bernese Software V5.0 (Dach et al. 2007) and applies the processing characteristics shown in Table 83.2. Within this analysis the observations of ~9 IGS-EUREF sites were included in order to link the regional GURN to the ITRF2005 (see Fig. 83.5). In order to derive coordinate time series which are easily interpretable, the Eurasian trend is eliminated and the absolute coordinates are transformed into local topocentric coordinate increments. Figure 83.6 is showing the detrended (Eurasian trend free) preliminary time series (northing, easting, up) of site TAUB, calculated by the GIK working group as an example for discontinuities of coordinate time series due to instrumentation replacement. A step within the three coordinate time series at the beginning of the year 2004 is clearly visible. Such steps have to be removed in order to obtain a continuous data base. Therefore, the detailed analysis of the history of the GURN sites is necessary.
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GURN (GNSS Upper Rhine Graben Network): Research Goals
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Fig. 83.3 Map of the sites used in the EOST processing of the GURN data
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Fig. 83.4 Eurasian trend free preliminary time series (10-day mean; northing, easting, up) of the sites KARL (left; significant annual signal) and WELS (right; low noise level)
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Table 83.2 Characteristics of the GIK processing strategy Parameter Sampling rate Observation weighting Elevation cut-off angle Satellite orbits and Earth rotation parameters Neutrospheric prediction Mapping function Neutrospheric parameters Ambiguity resolution Antenna calibration Baseline selection
Characteristics 30 s sin2e 10 Precise final IGS products, reprocessed (Steigenberger et al. 2006) Model Niell (dry) Niellwet Time span: 2 h QIF (Individual) absolute calibration Automatic, OBSMAX
Fig. 83.5 Frame realization of the GIK working group. Site selection based on geometrical distribution, data history, and data quality
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Fig. 83.6 Eurasian trend free preliminary time series (GIK, daily solution; northing, easting, up) of the site TAUB
The primary goal of the GURN project as a long term project is to obtain precise and reliable estimates of horizontal and vertical point motions based on permanently operated GNSS sites in the area under research. The GNSS-related short-term research aims within GURN are (1) Generation of highly precise daily and reliable GURN solutions based on inter-workinggroup comparisons, and (2) Automated web-based presentation of results in near real-time. Long-term goals of GURN are (1) Automated generation of high resolution regional water vapour fields, and (2) Revised geodynamic model of the URG area. Within the GURN project the two analysis centres (EOST, GIK) are using different GNSS software. This enables the GURN initiative to monitor its own and other networks (e.g. state survey SAPOS® networks) precisely and reliably. At the GIK a special GURN focus will be on the inter-technique validation and integration of different geodetic sensors (levelling, InSAR, GNSS). This will enable the GURN initiative e.g. to find geodynamically representative GNSS sites, see Heck et al. (2010) for further details. Within the GURN area various research projects are aiming on precise and reliable determination of displacement rates, e.g.
Sigmaringen Freiburg
Memmingen
Waldshut Frauenfeld
Sonthofen
Basel Zürich
Luzern
Fig. 83.7 Vertical displacement rates after Zippelt and Dierks (2006)
0.8 - 0.6 0.6 - 0.4 0.4 - 0.2 0.2 - 0.0 0.0 - – 0.2
– 0.2 - – 0.4 – 0.4 - – 0.6 – 0.6 - – 0.8 – 0.8 - – 1.0 – 1.0 - – 3.0
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Zippelt and Dierks (2006) analysed repeated precise levelling campaign data and proved that these linerelated data are suitable to determine vertical recent crustal movements, even in times of point-wise static high-precision GNSS measurements. Zippelt and Dierks (2006) demonstrated subsidence of the southern Rhine Graben and the Dinkelberg area with respect to the crystalline Black Forest, whereas the area of Hegau in the north of Lake Constance presents variable and mostly non-significant movement rates (Fig. 83.7). Additionally, it was shown that the Swiss molasses basin is a stable area with very small and alternating rates of subsidence and uplift. The analysis of highly precise levelling data will be extended to the northern URG starting in 2010. EOST will focus on the tectonic interpretation of the determined velocity field, including comparison with the local seismicity recorded by the ReNaSS seismic network (see http://renass.u-strasbg.fr/), and medium-term deformations estimated from geomorphological and paleoseismological studies. Acknowledgements We thank all of our data providers for supplying data of their permanently operating GNSS networks: RENAG (France), RGP (France), Teria (France), Orpheon (France), SAPOS®-Baden-W€ urttemberg (Germany), SAPOS®Rheinland-Pfalz (Germany), swisstopo (Switzerland), European Permanent Network, and IGS.
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(B10):2214. doi:10.1029/2001JB000561 Behrmann J, Ziegler P, Schmid S, Heck B, Granet M (2005) The EUCOR-URGENT project. Int J Earth Sci (Geol Rundsch) 94(4):505–506 Bourgeois O, Ford M, Diraison M, Le Carlier de Veslud C, Gerbault M, Pik R, Ruby N, Bonnet S (2007) Separation of rifting and lithospheric folding signatures in the NW-Alpine foreland. Int J Earth Sci (Geol Rundsch) 96:1003–1031. doi:10.1007/s00531-007-0202-2 Bruyninx C (2004) The EUREF permanent network: a multidisciplinary network serving surveyors as well as scientists. GeoInformatics 7:32–35 Cloetingh SAPL, Ziegler PA, Bogaard PJF, Andriessen PAM, Artemieva IM, Bada G, van Balen RT, Beekman F, BenAvraham Z, Brun J-P, Bunge HP, Burov EB, Carbonell R, Facenna C, Friedrich A, Gallart J, Green AG, Heidbach O, Jones AG, Matenco L, Mosar J, Oncken O, Pascal C, Peters G, Sliaupa S, Soesoo A, Spakman W, Stephenson RA, Thybo H, Torsvik T, de Vicente G, Wenzel F, Wortel
M. Mayer et al. MJR, TOPO-Europe Working Group (2007) TOPOEUROPE: the geoscience of coupled deep Earth-surface processes. Glob Planet Change 58(1–4):1–118. doi:10.1016/j.gloplacha.2007.02.008 Dach R, Hugentobler U, Fridez P, Meindl M (2007) Bernese GPS Software Version 5.0. User manual of the Bernese GPS Software Version 5.0 Dong D, Herring TA, King RW (1998) Estimating regional deformation from a combination of space and terrestrial geodetic data. J Geod 72:200–214 Dow JM, Neilan RE, Gendt G (2005) The International GPS Service (IGS): celebrating the 10th anniversary and looking to the next decade. Adv Space Res 36:320–326. doi:10.1016/ j.asr.2005.05.125 Estey LH, Meertens CM (1999) TEQC: the multi-purpose toolkit for GPS/GLONASS data. GPS Solut 3(1):42–49 Gaudet A, Landry JC (2005) TERIA: the GNSS network for France. Proceedings of the Pharaohs to geoinformatics, FIG Working Week 2005 and GSDI-8, Cairo, 16–21 April 2005 Heck B, Mayer M, Westerhaus M, Zippelt K (2010) Karlsruhe integrated displacement analysis approach: towards a rigorous combination of different geodetic methods. Proceedings of the FIG Congress 2010, Facing the Challenges – Building the Capacity, Sydney, 11–16 April 2010 Herring T, King B, McClusky S (2006) Introduction to GAMIT/ GLOBK. Reference manual. Global Kalman filter VLBI and GPS analysis program. Release 10.3. EAPS, MIT, Cambridge Illies JH (1977) Ancient and recent rifting in the Rhine graben. Geologie en Mijbouw 56:320–350 Illies JH (1979) Rhinegraben: shear controlled vertical motions of the graben floor. Allgemeine Vermessungs-Nachrichten 86:364–367 Kn€opfler A, Masson F, Mayer M, Ulrich P, Heck B (2010) GURN (GNSS Upper Rhine Graben Network) – status and first results. Proceedings of the 95th Journe´es Luxembourgeoises de Ge´odynamique, Echternach, Grand Duchy of Luxembourg, 9–11 November 2009 Luo X, Mayer M (2008) Automatisiertes GNSS-basiertes Bewegungsmonitoring am Black Forest Observatory (BFO) in Nahezu-Echtzeit. Zeitschrift f€ur Geod€asie, Geoinformatik und Land management (ZfV) 133(6):283–294 Meghraoui M, Delouis B, Ferry M, Giardini D, Huggenberger P, Spotke I, Granet M (2001) Active normal faulting in the Upper Rine Graben and paleoseismic identification of the 1356 Basel earthquake. Science 293:2070–2073 Nocquet JM, Calais E (2003) Crustal velocity field of Western Europe from permanent GPS array solutions, 1996–2001. J Geophys Res 154:72–88 Rozsa S, Mayer M, Westerhaus M, Seitz K, Heck B (2005a) Towards the determination of displacements in the Upper Rhine Graben area using GPS measurements and precise antenna modelling. Quatern Sci Rev 24:425–438 Rozsa S, Heck B, Mayer M, Seitz K, Westerhaus M, Zippelt K (2005b) Determination of displacements in the upper Rhine Graben area from GPS and leveling data. Int J Earth Sci 94:538–549 Steigenberger P, Rothacher M, Dietrich R, Fritsche M, R€ulke A, Vey S (2006) Reprocessing of a global GPS network. J Geophys Res 111:B05402. doi:10.1029/2005JB003747
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Tesauro M, Hollenstein C, Egli R, Geiger A, Kahle H-G (2005) Continuous GPS and broad-scale deformation across the Rhine Graben and the Alps. Int J Earth Sci 94(4):525–537 Wanninger L, Wildt S (1997) Identifikation von Mehrwegee infl€ussen in GPS-Referenzstationsbeobachtungen. Allgemeine VermessungsNachrichten 104:12–17 Wegener V, Stronk M (2005) SAPOS® – a satellite positioning service of the German state survey. GeoInformatics 8:40–43
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Ziegler PA (1992) European cenozoic rift system. Tectonophysics 208:91–111 Zippelt K, Dierks O (2006) Auswertung von wiederholten Pr€azisionsnivellements im s€udlichen Schwarzwald, Bodenseeraum sowie in angrenzenden schweizerischen Landesteilen. NAGRA, NAB 07–27 Zippelt K, M€alzer H (1981) Recent height changes in the central segment of the Rhinegraben and its adjacent shoulders. Tectonophysics 73:119–123
.
Determination of Horizontal and Vertical Movements of the Adriatic Microplate on the Basis of GPS Measurements
84
M. Marjanovic´, Zˇ. Bacˇic´, and T. Basˇic´
Abstract
The paper describes the determination of horizontal and vertical movements of the Adriatic microplate on the basis of GPS measurements carried out in the period between 1994 and 2005 within the frame of the 21 measuring campaigns organized at the research territory. The role of geodetic measurement methods particularly GPS method is essential in applications that requires high accuracy and precision as in the velocity field estimation of tectonic plates. The processing of GPS data as well as computation of the coordinates of points and their velocities was performed by Bernese GPS Software Ver. 5.0 based on 140 daily solutions. The mean standard deviations of estimated coordinates from repeatability of daily solutions and combined solution are s’ ¼ 2.0 mm, sl ¼ 2.2 mm and sh ¼ 5.6 mm. In purpose to determine and present the trend of height component the comparison of the data obtained by monitoring the change of sea level and the results of GPS measurement data was investigated. On the basis of the computed relative velocities of points as related to the Euro-Asian plate, the parameters of Euler rotation vector and Euler pole of the Adriatic microplate have been calculated and compared with other solutions. Also, the kinematic research area model has been determined on the basis of the combined solution results and compared with global kinematic models NNR-NUVEL-1A and APKIM2000.
84.1
Introduction
In recent years GPS become one of the most used methods in applications that requires high positioning precision such as in velocity field estimation M. Marjanovic´ (*) Zˇ. Bacˇic´ State Geodetic Administration, Grusˇka 20, 10000 Zagreb, Republic of Croatia e-mail:
[email protected] T. Basˇic´ Croatian Geodetic Institute, Savska cesta 41/XVI, 10000 Zagreb, Republic of Croatia
and computation of plate tectonic models. The coordinates of geodetic points on Earth’s surface change with time due to plate tectonics and therefore they are dependent on epoch of their determination. If we have measurements at least in two epochs it is possible to compute the change of point coordinates as a function of time. The Adriatic microplate is a plate or lithosphere block which includes the area of Adriatic Sea, eastern part of Italy, river Po valley and the area of western Dinarides. The microplate is situated on the border between two major tectonic plates, EuroAsian and African plate. The main cause for the
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deformation processes in whole area is the moving of African plate in north direction (Krijgsman 2002). The most part of Adriatic microplate is covered by the Adriatic sea and it is not possible to perform field measurements directly on the microplate, because of that the investigation of microplate movements have to be carried out based on the measurements made in surroundings area. At the beginning of the 1990s, Croatia starts to participate in international geodynamic projects and GPS campaigns, but the measurements were carried out just on one or two points. In year 1994 the project CRODYN started in cooperation of Faculty of Geodesy in Zagreb, State Geodetic Administration of Croatia and Institute of Applied Geodesy from Frankfurt (Germany) for the geodynamic research of Adriatic microplate based on GPS method (Cˇolic´ et al. 1996). The main objective of paper is the computation of coordinates and velocities of points and computation of Euler rotation vector and Euler pole of the Adriatic microplate based on relative velocities of points to the Euro-Asian plate. The processing of GPS data as well as computation of the coordinates of points and their velocities was performed by Bernese GPS Software Ver. 5.0.
84.2
Data Set
The computation of coordinates and velocities of points were performed based on 21 GPS campaigns observed in the period between 1994 and 2005 (Table 84.1). In the processing of the data 140 sessions (24 h) of 81 points were included. The GPS campaigns were carried out within international and national geodynamic projects, EUREF projects and national projects for establishment reference GPS networks. Also, 15 IGS points were included for datum definition and control of data processing.
84.3
Data Processing
The software used for the processing of GPS data was Bernese GPS Software Ver. 5.0 (Dach et al. 2007). The strategy of data processing followed: • Recommendations of EUREF Technical working group (Boucher and Altamimi 2001)
Table 84.1 GPS campaigns No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Campaign CEGRN94 CEGRN95 CEGRN96 CEGRN97 CEGRN99 CEGRN01 CEGRN03 CEGRN05 CRODYN94 CRODYN96 CRODYN98 CROSLO94 CROREF95 CROREF96 CROREF05 EUVN97 SLOVEN02 SLOVEN03 RTREAT03 RTREAT04 RTREAT05
Sessions 5 5 6 5 6 6 6 6 3 3 3 4 7 6 2 7 4 3 22 25 7
Year 1994 1995 1996 1997 1999 2001 2003 2005 1994 1996 1998 1994 1995 1996 2005 1997 2002 2003 2003 2004 2005
Points 8 11 9 14 13 16 13 15 21 38 34 20 28 44 50 17 12 14 24 24 16
Table 84.2 Computation of combined solution Number of input NEQ files Number of points Number of vectors Number of observations Number of parameters s0(mm)
140 81 2,148 40,236,729 81,592 1.4
• Specifications for the computation of EUREF/EPN network [URL 1) • Guidelines for using of IGS products (Kouba 2003) The computation of combined solution based on all 140 session solutions was done in ITRF2000, at the middle epoch of all measurements 2000.04 (Table 84.2). The IGS point Graz was used for datum definition for coordinates as well as for velocities (Fig. 84.1). The mean standard deviation of 21 GPS campaigns derived from the comparison of daily solutions of each GPS campaign shows very good quality and accuracy of computed coordinates what is very important for the computation of velocities (Fig. 84.2).
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Fig. 84.1 Computed velocity vectors
84.4
Results
84.4.1 Euler Rotation Vector and Euler Pole of the Adriatic Microplate
Fig. 84.2 Repeatabilites of daily solutions
The mean standard deviations of estimated coordinates from repeatability of 140 daily solutions and combined solution are s’ ¼ 2.0 mm, sl ¼ 2.2 mm and sh ¼ 5.6 mm.
The relative movement or rotation of one plate respect to the another plate is described by relative kinematic plate model or with rotation for angle O around the point on Earth’s surface or Euler pole. Each tectonic plate included in some kinematic plate model (absolute or relative) has altogether six parameters: Euler rotation vector (OX, OY, OZ) and Euler pole (’, l, O). Using the estimated relative velocities and coordinates of GPS points it is possible to estimate relative plate kinematic model of Adriatic microplate respect to Euro-Asian plate in least square adjustment process (Perez et al. 2003). The Euler theorem gives the linearized observation equation for one station in which we have relation between estimated (VX, VY, VZ, X, Y, Z) and unknown values (OX, OY, OZ)
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2
3 2 VX 0 4 VY 5 ¼ 4 Z Y VZ
Z 0 X
3 32 Y OX X 54 OY 5 0 OZ
(84.1)
if nf ¼ n u > 0
(84.2)
where n is number of measurements and u is number of unknown parameters. The computed parameters of Euler rotation vector are OX ½rad=milyears ¼ 0:0023 OY ½rad=milyears ¼ 0:0003
(84.3)
OZ ½rad=milyears ¼ 0:0023 and then the parameters of Euler pole are 0
1
OZ C B ’ ¼ tan 1 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA ¼ 45:4 2 2 OX þ OY OY ¼ 8:1 l ¼ tan 1 OX qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi O ¼ O2X þ O2Y þ O2Z ¼ 0:2 =milyears
84.4.2 Comparison of Computed and Global Plate Kinematic Models The kinematic plate model of research area computed based on the results and processing of GPS measurements was compared with two global plate kinematic models NNR-NUVEL-1A and APKIM2000 (DeMets et al. 1994; Drewes and Angermann 2001) of the EuroAsian tectonic plate parameters (Table 84.4). Using the parameter values from Table 84.4 and (84.1) the velocities of points can be computed for all three kinematic plate models (e.g. Brusnik, Table 84.5). The comparsion of three models (Tables 84.4 and 84.5) shows good agreement if we know that global models are based on geological and geophysical data (NNR-NUVEL-1A) and various satellite geodesy methods (APKIM2000), while this regional model was determined on GPS measurements carried out in the period between 1994 and 2005.
84.4.3 GPS Results and Sea Level Changes Data
s’ ¼ 0:1 ; sl ¼ 0:1 ; sO ¼ 0:0003 =milyears: (84.4) The estimated Euler pole of Adriactic microplate was compared with the results of investigations carried out between 1987 and 2005 (Table 84.3). The values in Table 84.3 shows good agreement, although different type of measurements and computation methods were used.
The monitoring of sea level changes and data analysis helps in determination of vertical movements of the Earth’s surface (Lambeck et al. 2004). The change of sea level is caused by eustatic, glacial and tectonic effects. The most interesting from geodynamic point of view are tectonic effects which have regional character caused by tectonic processes. During the planning of CRODYN project five GPS points (GPS) were established on or near tide gauges (TG) which are also included in GPS campaigns and data processing (Tables 84.6 and 84.7). Table 84.4 Kinematic plate model parameters OXYZ ( /mil years)
Table 84.3 Comparison of Euler pole parameters Solution 1 2 3 4 5 6
’ ( ) 45.8 44.5 46.8 45.3 46.7 45.4
l ( ) 10.2 9.5 6.3 9.1 9.7 8.1
O ( /mil years) – 0.3 0.3 0.5 0.4 0.2
1 Anderson and Jackson (1987); 2 Westaway (1990); 3 Ward (1994); 4 Calais et al. (2002); 5 Weber et al. (2005a, b); 6 This study
Model 1 2 3
OX 0.0010 0.0003 0.0003
OY 0.0024 0.0024 0.0020
OZ 0.0032 0.0038 0.0046
Table 84.5 Velocities of point Brusnik VNEU Model 1 2 3
VN (mm) 13.0 14.1 13.0
VE (mm) 21.3 21.3 21.7
1 NNR-NUVEL-1A; 2 APKIM2000; 3 This study
VU (mm) 0.0 0.0 0.0
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Determination of Horizontal and Vertical Movements of the Adriatic Microplate
Table 84.6 Tide gauges and GPS points
7160
TG/GPS BAKAR DUBROVNIK ROVINJ SPLIT ZADAR
7140
GPS (campaigns/sessions) 5/17 5/17 6/19 5/17 4/11
7120 7100 7080 mm
TG (years) 62 47 48 50 10
687
7060 7040 7020 7000
Table 84.7 Tide gauge and GPS point velocities TG/GPS BAKAR DUBROVNIK ROVINJ SPLIT ZADAR
VU (mm/year) 1.2 1.2 0.4 2.1 0.9
6980
sVU (mm/year) 0.8 0.9 0.4 0.7 0.9
6960 1930 1940 1950 1960 1970 1980 1990 2000 2010 year
Fig. 84.3 MSL of Bakar Tide Gauge 182.405 182.4
P1 r ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi P2 P3
(84.5)
where
182.395 182.39 m
For the determination of vertical movements of every tide gauge and GPS points the data of sea level changes and the values of ellipsoidal heights were analysed (e.g. Bakar, Figs. 84.3 and 84.4). In purpose to investigate the relationship between two series of data the correlation coefficients were computed based on both data series using
182.385 182.38 182.375 182.37 1994
1996
1998
2000 year
2002
2004
2006
Fig. 84.4 Ellipsoidal height of Bakar GPS Table 84.8 Correlation of TG and GPS data
P1 ¼
Tide gauge/GPS point BAKAR DUBROVNIK ROVINJ SPLIT ZADAR
m X n X Amn A Bmn B i¼1 i¼1
m X n X 2 Amn A P2 ¼ i¼1 i¼1
m X n X 2 P3 ¼ Bmn B
r 0.99 0.99 0.99 0.99 0.99
(84.6)
i¼1 i¼1
m and n are number of each data series with condition m ¼ n, Amm and Bmm are matrices of data and A and B matrices of mean values (Table 84.8). The trends of sea level changes and computed ellipsoidal heights on all tide gauges and GPS points as well as sign of determined VU velocities shows their relative movement in vertical sense, the rise of sea level or descending of coast. Also, the values of correlation coefficients in Table 84.8 shows perfect correlation between two series of data in time (r 1).
Conclusion
The determined parameters of Euler pole and plate kinematic model of research area computed on the basis of GPS measurements have good agreement with the results of previous investigations and global plate kinematic models NNR-NUVEL-1A and APKIM2000. The analysis of mean sea level data of five tide gauges and computed ellipsoidal heights of corresponding GPS points made possible to determine the direction of relative movement in
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vertical component by two independent methods. At the beginning of December 2008 the CROPOS (CROatian POsitioning System) was launched. The system has 30 reference GNSS stations and apart from applying CROPOS system for the state survey and cadastre, the data also will be used for further geodynamic research.
References Anderson H, Jackson J (1987) Active tectonics of the Adriatic Region. Geophys J 1987:937–983 Boucher C, Altamimi Z (2001) Specifications for reference frame fixing in the analysis of a EUREF GPS campaigns, IERS Memo Version 5, Observatoire de Paris, 2001 Calais E, Nocquet J-M, Jouanne F, Tardy M (2002) Current strain regime in the Western Alps from Global Positioning System measurements, 1996–2001. Geology 2002:651–654 Cˇolic´ K, Basˇic´ T, Seeger H, Gojcˇeta B, Altiner Y, Rasˇic´ Lj, Medic´ Z, Pribicˇevic´ B, Medak D, Marjanovic´ M, Prelogoviæ E (1996) Croatia in EUREF 94 and CRODYN project, Geod J 4, Zagreb, 331–351 Dach R, Hugentobler U, Fridez P (2007) Bernese GPS Software Version 5.0 Tutorial. Astronomical Institute University of Bern, Bern DeMets C, Gordon R, Argus D, Stein S (1994) Effects of recent revisions to the geomagnetic reversal time scale on estimates of current plate motions. Geophys Res Lett 21:2191–2194
M. Marjanovic´ et al. Drewes H, Angermann D (2001) The actual Plate Kinematic and Crustal Deformation Model (APKIM2000) as a Geodetic Reference System, IAG 2001 Scientific Assembly September 2–8, 2001, Budapest Kouba J (2003) A guide to using International GPS Service (IGS) Products, Geodetic Survey Division, Natural Resources Canada Krijgsman W (2002) The Mediterranean: Mare Nostrum of Earth Sciences. Earth Planet Sci Lett 205:1–12 Lambeck K, Antonioli F, Purcel A, Silenzi S (2004) Sea-level change along the Italian coast for the past 10,000 yr. Quat Sci Rev 23:1567–1598 Perez JAS, Monico JFG, Chaves JC (2003) Velocity field estimation using GPS precise point positioning: the South America plate case. J Glob Position Syst 2:90–99 URL 1: http://www.epncb.oma.be, EUREF/EPN Ward SN (1994) Constraints in the seismotectonics of the central mediterranean from very long baseline interferometry. Geophys J Int 117:441–452 Weber J, Vrabec M, Stopar B, Pavlovcˇicˇ-Presˇeren P, Dixon T (2005a) Active Tectonics at the NE Corner of the Adria-Europe Collision Zone (Slovenia and Northern Croatia): GPS Constraints on the Adria Motion and Deformation at the Alps-Dinarides-Panonian Basin Junction, 7th Alpine Workshop, Opatija, 2005 Weber J, Vrabec M, Stopar B, Pavlovcˇicˇ-Presˇeren P, Dixon T (2005b) New GPS constraints on Adria microplate kinematics, dynamics, and rigidity from the Istria peninsula (Slovenia and Croatia). Geophys Res Abstr 7:2005 Westaway R (1990) Present-day kinematics of the plate boundary zone between Africa and Europe, from Azores to the Aegean. Earth Planet Sci Lett 96(1990):393–406
Determination of Tectonic Movements in the Swiss Alps Using GNSS and Levelling
85
E. Brockmann, D. Ineichen, U. Marti, S. Schaer, A. Schlatter, and A. Villiger
Abstract
The Federal Office of Topography swisstopo is responsible for the maintenance of the coordinate reference frames in Switzerland. Beside the static reference frames, used for national surveying, the development of a kinematic model, mainly used for scientific investigations, is under development since many years. For the determination of vertical movements the analysis of more than 100 years of levelling observations showed a significant alpine uplift of maximally 1.5 mm/ year relative to an arbitrarily chosen bench mark in Aarburg (at the south of the Jura mountains). For the determination of horizontal movements the various GPScampaigns, measured since 1988, are the basis. With the Automated GNSS Network of Switzerland (AGNES), which operates permanently since 1998, valuable information can be extracted for 30 sites. The paper shows that the time series of AGNES nowadays allow statements concerning possible vertical movements. The potential of additional information sources, such as local tie surveys, are discussed. Comparisons of the results of the two independent measuring techniques GNSS and levelling are the main topic of the paper.
85.1
Permanent GNSS Network AGNES
85.1.1 Network and Infrastructure The Automated GNSS Network of Switzerland AGNES was created in 1997. After a 2-year pilot phase with only 11 stations, the network was completed
E. Brockmann (*) D. Ineichen U. Marti S. Schaer A. Schlatter Geodesy Division, Swiss Federal Office of Topography swisstopo, Seftigenstrasse 264, 3084 Wabern, Switzerland e-mail:
[email protected] A. Villiger ETH Zurich, Institute for Geodesy and Photogrammetry, Schafmattstr. 34, 8093 Z€ urich, Switzerland
between 2000 and 2002 to a nationwide coverage by adding 20 stations. This configuration of a total of 31 stations has been operating permanently since January 2003. AGNES serves various purposes such as realtime positioning (swipos®), national surveying, and scientific applications like monitoring of tectonic movements and GNSS-meteorology. The stations were selected fulfilling different criteria. Three different classes of stations were built in view of the stability of a station. Only ten sites fulfill class A standard with concrete monumentation on bed rock. Class C stations, the majority of the AGNES stations, reflect stations mainly designated for positioning applications. They are often installed on buildings using an aluminium tube. Other stations were selected with a connection to sites of MeteoSwiss
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(collocation with meteorological measurements) or to sites of the Swiss Seismological Service (collocation with seismic measurements). Since 2006, all major manufacturers of geodetic GNSS receivers have been designing combined receivers for both GPS and GLONASS. In order to keep step with this development, swisstopo adapted its network AGNES to the new technical demands in 2007. Since AGNES is a multifunctional reference network, swisstopo had to find a compromise between the continuity of the observations and the rapid alignment to the demands and developments of the surveying market. To assure continuity, already existing GPS receivers operate simultaneously with new GPS/ GLONASS receivers on ten AGNES stations (Brockmann et al. 2007; Ineichen et al. 2007). The rest of the 21 AGNES stations were converted to GNSS operating during summer 2007. To guarantee highest precision, all of the new GNSS antennas were calibrated by the specialized firm Geo++ in Germany (W€ ubbena et al. 2006). Absolute azimuth- and elevation-depending antenna phase center variations were derived for GPS (individual corrections for each antenna according) and for GLONASS (one group correction). Previously, the used GPS antennas were individually calibrated relative to a reference antenna using a swisstopo-owned test field consisting of a dozen markers, evenly separated by a distance of 10 m.
85.1.2 Operational Analysis of AGNES Data Since the beginning of AGNES, the RINEX data were routinely analyzed on a daily basis together with neighbouring IGS and EUREF stations using the Bernese GNSS software (Dach et al. 2007). These processing procedures are almost identical to the weekly SINEX file contribution of swisstopo as one of the first analysis centers within the European Permanent Network of EUREF (EPN). IGS final precise orbits and, since November 2006, the combined GPSGLONASS orbits produced by the Center for Orbit Determination CODE were used. In the history of the data processing different versions of the Bernese software were applied. Most of the software changes, as well as changes of the computer hardware or the operating system, did not have any impact on the resulting coordinates. More
E. Brockmann et al.
critical are changes in the used models (e.g. mapping functions, ocean loading model, and elevation cutoff). A prominent date for model changes is GPS week 1400 (November 5, 2006) when, beside the introduction of a new reference frame ITRF2005, an absolute antenna phase center variation (PCV) model was used in the global and European analyses of permanent networks with influences on the vertical coordinates of up to several centimeters. swisstopo followed the recommended processing options but also continued with daily and weekly solutions based on the previously used models in order to guarantee consistency of the long-term time series. These additional solution types will be continued as long as reprocessed solutions, using homogeneous processing options, are unavailable. Solutions based on the analyses of hourly data are part of the operational analyses of the AGNES data since 2000. The detection of possible station instabilities is the main focus. A web-based monitoring system displays the actual status of the network in colors of a traffic light in addition to SMS messages sent to the operator in case of anomalies (Brockmann et al. 2006). Furthermore, zenith total delay estimates (ZTD) are derived from the hourly processing which are suitable to support numerical weather prediction models in Switzerland and Europe. For the determination of velocities these hourly monitoring solutions are presently not yet used. Nevertheless, they are added to the long-term monitoring plots published on the web in order to bridge the time between the last weekly results and the actual time.
85.1.3 Multi-annual Analysis of AGNES Data Presently, more than 10 years of weekly normal equations are in our data archives. Every week, an updated multi-annual combination including velocity estimation is computed from all available AGNES normal equations starting with GPS week 973 (August 1998). For consistency reasons, the input normal equations are still based on relative antenna phase center variation (PCV) models. A multi-annual combination is generated based on roughly 700,000,000 GPS observations (sampled at 180 s) collected from about 220 station setups. For stations with observations longer than 1 year velocities are solved for.
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Determination of Tectonic Movements in the Swiss Alps Using GNSS and Levelling
60
40
50 30 40 30
20
20 10 10 0
0-2
2 - 4 4 - 6 6 - 8 8 - 10 > 10 interval length [years]
repeatability improvement [%]
number of stations [–]
Different station setups due to equipment changes are handled in a way that for each period coordinates and velocities are solved for with the restriction that the velocities are forced to be identical (relative constraints of 0.001 mm/year). These heavy constraints are necessary because of the long time series and the (formally) very high accuracies of the estimates. The determination of heights with GPS is due to the geometry and due to the high correlation with ZTD parameters and PCV antenna models much less precise than the determination of plane coordinates. The same holds true also for the first derivative of the coordinates – the velocities of a station. A detailed analysis with respect to the possible vertical movements in the Alps was performed using the AGNES sites and the alpine sites in Austria and Germany close to the Swiss border – a total of 57 sites. Jumps needed to be setup in several cases due to equipment changes, so that in total 110 station setups are available to determine vertical velocities. A histogram showing the length of the timespan of each of the station setup is given in Fig. 85.1. In average we have two station setups per station, which means one equipment change per station. The enhancement of the AGNES network with combined GPS-GLONASS antennas mid-2007 is a typical case for such an additional station setup, which explains also the high percentage of station setup intervals smaller than 2 years. Introduced jumps weaken the velocity estimates even when forcing identical velocity estimates for consecutive intervals. Velocities derived from uninterrupted and continuous time series pffiffiffiffi gain from longer time periods by a factor of t3 (Brockmann 1997), contradictionary to coordinates,
0
Fig. 85.1 Number of stations (gray) and improved repeatability (black) as a function of the interval length of a station setup
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which gain from longer time intervals only with the square root law. Time is “working very much” for a reliable velocity estimation if no antenna equipment is changed. Therefore, it is planned to operate and analyse the GPS-only equipped part of the double stations till the end of lifetime. Most of the selected double stations have uninterrupted time series longer than 8 years. To assess the significance of the vertical velocity estimates, the repeatabilities of station coordinates were compared between cumulative solutions with and without vertical velocity estimation. Whereas the repeatability of the horizontal coordinates does not change due to the additional vertical velocity estimation, repeatabilities with a reduced standard deviation for the vertical component are achieved. The improvement is depending on the length of the time interval as shown in Fig. 85.1. Obviously, there is a strong signal of vertical velocities in the time series of stations with a long time interval, which may improve the repeatabilities of up to 30%. Repeatabilities were not analysed for the 54 site setups with intervals smaller than 2 years (velocities are estimated for sites with intervals of larger than 1 year). The datum of the velocity estimates was defined by constraining the ITRF2005 velocities of site ZIMM, whereas the datum of the coordinates was realized by adding minimum constraints with respect to several ITRF2005 stations. In the ITRF2005 reference frame ZIMM rises with 2.3 mm/year. The results for the vertical velocities, generated with the combination of normal equations till GPS week 1541 (July 25, 2009) are shown in Fig. 85.2. Using this datum definition, the rising of the Alpine sites is clearly visible, whereas sites located on the stable part of Europe show only very small vertical movements, and sites in the Italian region of the river Po even seem to show subsidence. The vertical velocity difference of the two stations WTZR in Germany (+1.5 mm/year) and ZIMM (+2.3 mm/year) is in the ITRF2005 frame 0.8 mm/ year. The presented solution shows for the difference of these two stations a considerably bigger difference of +2.5 mm/year, which is, however, much closer to the recent published EPN densification solution of the ITRF in Europe, which states a vertical velocity difference of +1.5 mm/year for these two stations.
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Fig. 85.2 Vertical velocities in the ITRF2005 frame derived from the permanent GNSS Network AGNES. White arrows indicate uplift, black arrows indicate subsidence
85.2
National Height Network 95 (LHN95) and the Kinematic Model RCM04
Precise levelling measurements are carried out in Switzerland since more than 100 years. They follow major traffic arteries, pass through important tunnels and cross over main alpine passes. This network include approximately 8,000 benchmarks whose heights serve as the vertical reference for most observations in Switzerland. The measurements were first made between 1903 and 1945 and have since been repeated at least once. On average, the measuring of the levelling lines is repeated every 50–60 years. In March 2005, the final adjustment of the vertical reference frame (LHN95) was carried out and published based on a kinematic adjustment of the measurements between 1,600 selected stations along approximately
10,000 km of repeatedly observed precision levelling lines (Schlatter 2007). At the same time the geoid model CHGeoid2004 was released, which is based, amongst other input, on GPS-levelling results (Marti and Schlatter 2005). The reference point for the investigation of vertical movements was arbitrarily chosen in Aarburg (Canton of Aargau) at the south foot of the Jura Mountains, 50 km north-east of Zimmerwald. All vertical changes, estimated from levelling data, are therefore relative uplifts or subsidences with respect to this point. The formal precision of the vertical movements from the adjustment is of the order of 0.1–0.5 mm/year (95% significance level) depending on the distance to the reference point. A subset of 1,100 stations was used to interpolate the regular grid model RCM04, describing the recent crustal movements on the territory of Switzerland. The grid with a resolution of 1 km was generated
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Determination of Tectonic Movements in the Swiss Alps Using GNSS and Levelling
using the kriging method implemented in the software SURFER. The RCM04 model, given in Fig. 85.3, shows uplift rates of the central part of the Alps of roughly 1.5 mm/ year. Observations in the Jura Mountains are more likely to show subsidences. The Central Plateau seems to stay more or less stationary with the tendency of a tilt: subsidence in the west, uplift in the east. Even though the amounts seem small, the Alps are growing up to 15 cm in 100 years.
85.3
Comparison of Vertical Rates
Figure 85.3 compares the vertical rates derived from GNSS and levelling at the AGNES stations. Due to the fact that only a subset of AGNES sites is linked to levelling, the RCM04 model is used, even if the distance to the next levelling line might be large. Furthermore, the GNSS-derived vertical movements are shifted by 1.2 mm/year to adjust the different used datum definitions. This shift is determined from
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the average vertical motion of all sites located in the central Plateau and the pre-Alpine region (see also Table 85.1). The vertical rates derived from GNSS and from levelling agree quite well. The rise of the Alpine arc with respect to the Central Plateau is clearly visible also in case of GNSS. The monumentation of many AGNES sites might not be optimally suited to derive vertical velocities. In order to quantify the level of agreement, the AGNES stations were grouped in 4 different regions, shown in Fig. 85.4. The sites in Austria and Germany close to the Swiss border were grouped accordingly to this scheme. The results of two sites (high alpine sites JUJO and ZERM) were regarded as outliers. As shown in Table 85.1, mean regional vertical rates derived from GNSS can be determined from several sites with a standard deviation of roughly 0.4 mm/year. The interpolated RCM04 model, stemming from an interpolation of individual levelling results, allows the determination of mean regional vertical rates with a standard deviation of 0.2–0.3 mm/year.
mm/yr
Aarburg
ZIMM
Fig. 85.3 Comparison of vertical velocities between GNSS and the levelling-derived model RCM04 expressed in the Swiss system (Aarburg zero vertical velocity). White arrows indicate GNSS rates, black arrows indicate RCM04 model values at the AGNES sites. The background contour surface shows the RCM04 model
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Table 85.1 Mean regional vertical movements in the Swiss system Region North West Centera Alps a
No. of sites 7 5 15 17
Mean vertical movement and std (mm/year) GNSS RCM04 1.1 0.1 0.1 0.1 0.5 0.3 0.1 0.2 0.0 0.4 0.2 0.1 0.8 0.4 0.9 0.3
Diff. (mm/year) 0.9 0.4 0.2 0.1
And pre-Alps
5.775
North
5.774
Center + Pre-Alps Alps
height [m]
West
2008
5.773 5.772
2003
logfile
5.771
1998
5.770
Center + Pre-Alps
Fig. 85.4 Regions showing similar vertical velocities
Good agreement is achieved for the totally 32 sites of the Central part and the Alps. Systematic biases between GNSS and levelling seem to exist relative to the western region (0.4 mm/year) and the northern region (0.9 mm/year). Looking at the mean vertical movements between the northern and Alpine region, we see a differential vertical velocity of 1.9 mm/year derived from GNSS compared to 1.0 mm/year from levelling. This over-estimation is quite significant.
85.4
Local Ties
For the tectonically well monumented AGNES stations (mostly double stations) local tie information from a small local network consisting of several well monumented markers on the ground is collected. In case of possible movements, repeated local tie observations might answer the question, whether the GNSS antenna or the complete local network is moving. According to a 5-year cycle the local tie in Zimmerwald was re-observed in July 2008 (Ineichen and Brockmann 2008).
5.769 5.768 1994
1995.2 2004
1999
2009
time
Fig. 85.5 Local tie results between the SLR reference point (physical reference point) and the GPS antenna reference point (ZIMM 14001M004) for the vertical component. The error bars represent the 95% confidence interval. For comparison purposes the value of the ZIMM logfile is displayed
Compared to the previous determination of the local ties a more robust network geometry and improvements concerning the physical realization of the GNSS antenna reference points (ZIMM, ZIM2) without removing the antenna were introduced. The results of the vertical local tie between SLR and GPS as a function of time are shown in Fig. 85.5. A difference of roughly 5 mm is visible since 1995. The stability of the SLR point and also of several surface points over time could be proved using the local tie network. Therefore, the detected movement is most probably due to a small instability of the ZIMM reference point on top of a 9-m mast. Part of this difference might be temperature effects on the 9-m mast. 5 mm within 13 years is equivalent to a mean annual vertical movement of roughly 0.4 mm/ year, a value which is, compared to the relatively small vertical movements in Switzerland, not negligible.
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Determination of Tectonic Movements in the Swiss Alps Using GNSS and Levelling
Conclusions
The analysis of 10 years of permanent GNSS data impressively shows an Alpine uplift which is close to the uplift derived from 100 years of repeated levelling (three epochs). The general behavior of the vertical movements of different regions is well approved. Nevertheless, the GNSS-derived vertical rates seem to be significantly higher. This holds true especially in the differential velocity between the northern and Alpine region. Due to the fact that the analysis of the Swiss permanent network is embedded in the European EPN/IGS network, the movements with respect to the stable part of the Eurasian plate can be determined. The results have shown that Switzerland is rising in average by 1.2 mm/year. Not all AGNES sites are anchored directly in rock or on buildings, which are known to be stable and well-founded. Therefore, also local tie information, as shown for the reference station Zimmerwald, may help to properly determine the movements of the stable ground by removing local effects of the GNSS antenna. Local tie information will be collected on “class A”/double stations. The determination of tectonic motions in Switzerland will be a key issue also in the year 2010, where it is planned to re-observe roughly 200 GPS points for the fourth time since the first GPS campaigns took place in 1988.
695
References Brockmann E (1997) Combination of solutions for geodetic and geodynamic applications of the global positioning system (GPS), vol 55. Swiss Geodetic Commission, Zurich Brockmann E, Gr€unig S, Ineichen D, Schaer S (2006) Monitoring the automated GPS network of Switzerland AGNES. In: Torres JA, Hornik H (eds) Subcommission for the European Reference Frame (EUREF). EUREF, Riga Brockmann E, Kistler M, Marti U, Schlatter A, Vogel B, Wiget A, Wild U (2007) National report of Switzerland: new developments in Swiss national geodetic surveying. In: Torres JA, Hornik H (eds) Subcommision for the European Reference Frame (EUREF). EUREF, London Dach R, Hugentobler U, Meindl M, Fridez P (eds) (2007) The Bernese GPS Software Version 5.0. Astronomical Institute, University of Bern, Bern Ineichen D, Brockmann E (2008) Fundamentalstation Zimmerwald: Lokale Einmessung 2008. swisstopo-report 08–23. Internal swisstopo, Wabern Ineichen D, Brockmann E, Schaer S (2007) Enhancing the Swiss permanent GPS network (AGNES) for GLONASS. In: Torres JA, Hornik H (eds) Subcommision for the European Reference Frame (EUREF). EUREF, London Marti U, Schlatter A (2005) Festlegung des H€ohenbezugsrahmens LHN95 und Berechnung des Geoidmodells CHGeo2004. Geomatik Schweiz, 08/05 August 2005 Schlatter A (2007) Das H€ohensystem der Schweiz LHN95. Geod€atisch-geophysikalische Arbeiten in der Schweiz, Band 72. Schweizerische Geod€atische Kommission, Institut f€ur Geod€asie und Photogrammetrie, Eidg. Technische Hochschule Z€urich, Z€urich W€ubbena G, Schmitz M, Boettcher G and Schumann C (2006) Absolute GNSS Antenna Calibration with a Robot: Repeatability of Phase Variations, Calibration of GLONASS and Determination of Carrier-to-Noise Pattern. Proceedings of the IGS Workshop 2006 Perspectives and Visions for 2010 and beyond, May 8–12, ESOC, Darmstadt, Germany
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A Compilation of a Preliminary Map of Vertical Deformations in New Zealand from Continuous GPS Data
86
R. Tenzer, M. Stevenson, and P. Denys
Abstract
The NZGD2000 is the official 3-D semi-dynamic geodetic datum for New Zealand that incorporates a deformation model to correct the horizontal coordinates and survey observations for the effect of regional-scale tectonic movements of the Earth’s crust. These horizontal tectonic deformations are up to a few centimetres per year. Except for the Southern Alps (central South Island) and the Taupo volcanic zone (North Island), the currently available information indicates that the vertical tectonic deformations are an order of magnitude smaller than the horizontal components. In this study we compile a preliminary map of vertical deformations in New Zealand using the GPS time series. The leastsquares linear regression analysis is used to estimate the vertical velocities at GPS sites. After applying outlier detection, the vertical deformations are investigated with respect to the tectonic setup of New Zealand. The results reveal that the uplift of the Southern Alps at the currently established GPS sites reaches 6 mm/year. The largest regional-scale tectonic subsidence, at approximately 9 mm/year, is detected in the lower and central part of the North Island. The estimated vertical tectonic deformations are compared with evidence from geochronological data and results of previous studies.
86.1
Introduction
New Zealand is situated directly over the boundary between the Pacific and Australian lithospheric plates. This location results in a unstable tectonic environment with varying magnitudes of both horizontal and vertical tectonic deformations present throughout the country. To the north of New Zealand and beneath the
R. Tenzer (*) M. Stevenson P. Denys Faculty of Sciences, School of Surveying, University of Otago, 310 Castle Street, Dunedin, New Zealand e-mail:
[email protected]
eastern North Island, the (thin and dense) Pacific plate moves under the (thicker and lighter) Australian plate in a process known as subduction (e.g., Burchfiel 1980). Beneath the south-western South Island, the Australian plate is forced below the Pacific plate, while in the central South Island the plates are colliding as oblique strike slip. In this region the tectonic plate margin is marked by the Alpine Fault. The horizontal deformations in New Zealand have been studied extensively in Bourne et al. (1998), Beavan et al. (1999, 2002, 2007), Beavan and Blick (2005). Beavan and Haines (2001) produced presentday horizontal velocity and strain rate fields of the Pacific-Australian plate boundary zone throughout
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_86, # Springer-Verlag Berlin Heidelberg 2012
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New Zealand. These data were used to compile the horizontal deformation model utilized in a definition of the NZGD2000 semi-dynamic datum (Blick 2003; Blick et al. 2003). The vertical motion is generally described as a motion relative to a chosen reference surface. There are several models used to define the vertical motion; the physical model, the geometric model and the hybrid model. In this study, we use a hybrid model with the GPS vertical motions aligned to the ITRF2000. Blewitt (2004) identified several issues when using hybrid models, which include whether the geocenter motion has been modelled and if the reference stations are the same as those used in studies for which comparisons are made. The Southern Alps of New Zealand have been subject to a significant number of surface deformation and geophysical studies due to their close proximity to the Pacific and Australian plate boundary (see e.g., Wellman 1979; Bull and Cooper 1986). More recently, Beavan et al. (2004) investigated the rate of relative vertical movement across the central Southern Alps. They estimated that the highest vertical rates are about 6–7 mm/year relative to stations located on the Pacific tectonic plate. These results were obtained with average uncertainties better than 0.5 and 1 mm/year at continuous and semi-continious GPS stations, respectively. The aim of this study is to estimate the vertical deformations in New Zealand using a regionallyfiltered GPS time series. The input data sets are specified in Sect. 86.2. The methodology of modelling the vertical deformations using GPS data is defined in Sect. 86.3. The results are presented and discussed in Sect. 86.4. The summary and conclusions are given in Sect. 86.5.
86.2
Data Sets
To date, the network of permanent GPS stations consists of 129 sites (Fig. 86.1), established by various geodetic projects over the past decade. The majority of continuous GPS sites were set up as part of the GeoNet project since 2004. The North Island network consists of 99 continuous GPS sites, while only 30 GPS sites are located in the South Island. Certain areas of geophysical significance have led to dense concentrations of GPS stations, such as a small profile across the Southern Alps and throughout the fault littered regions
Fig. 86.1 The network of permanent GPS stations in New Zealand (year 2009)
of the central and lower North Island. Whereas the spatial coverage in the North Island varies considerably, the spatial coverage within the South Island is reasonably constant with average distance between GPS sites of about 110 km. The average distance between GPS sites in the lower and central North Island regions is roughly 30 km, while only about 85 km in the upper North Island. The occupation of GPS sites is either continuous or semi-continuous and the time periods during which data have been collected vary significantly. The longest occupation time is 3,360 days, the shortest is 83 days, and the average is 1,436 days. The processing strategy of GPS data is described in Beavan (2009). It involves the daily processing of the GPS station RINEX files using the Bernese v5.0 software. The product of this initial processing is GeoNet’s raw GPS time series. The raw GPS data are contaminated by noise due to a number of conditions such as offsets caused by equipment changes or earthquakes and seasonal signal variations. In addition, a generally constant spatially correlated noise might be present due to large-scale sources such as broad-scale weather patterns, atmospheric loading and errors in satellite positions (cf. Beavan 2005). The spatially correlated noise signal can be considered relatively constant over large regions and can therefore be removed using a
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A Compilation of a Preliminary Map of Vertical Deformations in New Zealand
regional filtering procedure. The filtering involves outlier detection and modelling of the spatially correlated noise signal. The estimated noise signal is then subtracted from the raw GPS time series. The resulting data sets represent the regionally-filtered GPS time series used in this study to estimate the vertical velocity rates at permanent GPS stations. The outlier detection and regional filtering procedure are described in Beavan (2005, 2009). He utilized the methodology of Williams et al. (2004) to model the spatially correlated noise signal in GPS data.
699
The normal matrix N reads: N ¼ AT S1 d A;
(86.5)
where Sd is the data noise variance-covariance matrix. The variance-covariance matrices of estimated ^ ¼ A ^x read: parameters ^x and adjusted observations d 1 S^x ¼ s2o AT S1 ; d A
(86.6)
1 T Sd^ ¼ s2o A AT S1 A ¼ A S^x AT ; d A
86.3
Methodology
A linear regression function: hi;j ¼ aj þ bj ðti;j t1;j Þ
(86.1)
is used to approximate the vertical component of GPS time series fhi;j : i ¼ 1; 2; ::: ; Ij ; j ¼ 1; 2; ::: ; Jg at time epochs ti;j , where J is the total number of GPS sites and Ij is the total number of data collected at the j-th GPS station. The parameters aj and bj in (86.1) define the offset and linear trend of the j-th GPS time series. The system of observation equations in (86.1) is written in the following vector-matrix form: d ¼ A x v;
(86.2)
where d is the vector of observations hi;j , Ais the design matrix, x is the vector of unknown parameters aj and bj , and v is the vector of residuals. Adopting the Gauss–Markov model (i.e., E f d g ¼ A x and D f d g ¼ Sd ; where E and D denote expectation and dispersion operators, respectively) and applying the linear estimation with respect to the data noise variance-covariance matrix norm:
v =r is the variance factor; r is where s2o ¼ ^vT S1 d ^ the redundancy number, and the vector of leastsquares residuals reads ^v ¼ A ^x d. The redundancy number is the difference between the number of observed values and the number of unknown parameters. When the noise model of the regionally-filtered GPS time series is unknown, the data noise variancecovariance matrix becomes the identity matrix (i.e., Sd ¼ I). The least-squares analysis is then simplified and the linear regression parameters aj and bj of the j-th GPS time series are estimated from T T ^xj ¼ N1 j Aj dj ; Nj ¼ Aj Aj ðj ¼ 1; 2; ::: ; J Þ (86.8)
where 0 0 1 1 1 0 h1;j B 1 t2;j t1;j C B h2;j C a^j B B C C ^xj ¼ ^ ; Aj ¼ B . C; dj ¼ B .. C (86.9) .. bj @ .. @ . A A . 1 tIj t1;j hIj The inverse normal matrix N1 j reads 0
^ v
T
S1 d
^ v ! min;
(86.3) N1 j
the estimation of unknown parameters in the parameter vector x is given by solving the system of normal equations: 1
^ x¼N
A
T
S1 d
d:
(86.4)
(86.7)
1 B B ¼ B det Nj @
Ij P
2
ðti;j t1;j Þ
i¼1 Ij P
Ij P i¼1
ðti;j t1;j Þ
1 ðti;j t1;j Þ C C C; A Ij
i¼1
det Nj ¼Ij
Ij X i¼1
2
ðti;j t1;j Þ
Ij X i¼1
!2 ðti;j t1;j Þ
: (86.10)
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The product of ATj dj equals: 0 B B ATj dj ¼ B Ij @P
Ij P
1 hi;j
i¼1
hi;j ðti;j t1;j Þ
C C C: A
(86.11)
i¼1
The precision of estimated parameters a^j and b^j is computed using the following expression: S^xj ¼
¼
^ vTj ^ vj T 1 Aj A j Ij 2
^vTj ^vj ðIj 2Þ det Nj 0 Ij P ðti;j t1;j Þ2 B B i¼1 B Ij @ P ðti;j t0;j Þ
Ij P i¼1
1 ðti;j t1;j Þ C C C: A Ij
i¼1
(86.12)
86.4
Results
The least-squares linear regression analysis (86.8) is applied to estimate the vertical velocity rates using the regionally-filtered vertical component of time series at 129 permanent GPS stations. The results reveal large uncertainties in estimated values of vertical velocity at GPS sites with a short occupation time. The correlation between the standard deviations of the vertical velocities and the length of the site occupation is shown in Fig. 86.2. The GPS sites with short occupation times are clearly shown, since these sites have (significantly) large uncertainties. The precision improves exponentially with increasing occupation time. While almost all GPS sites with occupation times shorter than 1 year have vertical velocity standard deviations greater than 1 mm/year, the standard deviations at GPS sites with 2 years and longer occupation time are better than 0.5 mm/year. Since the noise model of the regionally-filtered GPS time series is not currently available (Beavan 2009), the precision of vertical velocities is estimated only from the leastsquares residuals of linear regression analysis using
Fig. 86.2 The correlation between the standard deviation of vertical velocity and the duration of time occupation. The average uncertainty of estimated vertical velocities at the initial network of 129 GPS sites is 0.6 mm/year, and the maximum standard deviation is 10.2 mm/year
(86.12). The actual errors are thus more likely to be larger than the estimated values here. To remove GPS sites with a short occupation time, we apply outlier detection based on an analysis of the estimated standard deviations of vertical velocity. The results of the outlier detection for three different selection criteria are summarized in Table 86.1. The chosen selection criteria are based on removing the GPS sites with the uncertainties one, two and three times larger than the mean standard deviation (i.e., 1-s, 2-s and 3-s selection criteria). The mean standard deviation is computed by averaging the estimated standard deviations of vertical velocity at GPS sites. After applying the outlier detection based on 2-s selection criterion, a total of 27 GPS sites with an occupation of less than 591 days are removed from the initial network. In addition, the GPS site HOLD (730 days occupation time) with a standard deviation 0.9 mm/ year, was removed. The average accuracy of the vertical velocities at the remaining 101 GPS sites is 0.2 mm/year and the maximum standard deviation is better than 0.4 mm/year. The vertical velocities of the network consisting of 101 GPS sites (28 GPS sites in the South Island and 73 GPS sites in the North Island) after applying the outlier detection based on 2-s selection criterion are shown in Fig. 86.3. The corresponding standard deviations of the vertical velocity at these GPS sites are shown in Fig. 86.4. The statistics of estimated
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A Compilation of a Preliminary Map of Vertical Deformations in New Zealand
701
Table 86.1 The mean standard deviation of vertical velocity, the minimum occupation time and the total number of rejected GPS sites after applying the outlier detection based on 1-s, 2-s and 3-s selection criteria Selection criteria 1-s 2-s 3-s
Mean STD (mm/year) 0.1 0.2 0.3
Minimum occupation time (days) 1025.6 591.5 350.5
15 10 5 0 [mm/year]
Fig. 86.3 The estimated vertical velocities at the network consisting of 101 GPS sites in New Zealand (statistics see Table 86.2)
vertical velocities at 101 GPS sites in the South and North Islands are summarized in Table 86.2. The map of vertical deformations shows that the northern part of the North Island is without the presence of significant tectonic deformations; the maximum velocity rates are below 2 mm/year. In contrast, the subduction of the Australian tectonic plate beneath the Pacific plate results in large negative systematic vertical deformations at the central volcanic zone and southeast part of the North Island. The largest magnitude of the vertical velocities detected at GPS sites in the North Island is 16 mm/year (GPS site RGMK, data collected over 1,600 days). At this moment, the reason for the extreme rate of movement occurring at this GPS site is unknown, but it is most likely related to some sort of very local subsidence (J. Beavan, personal communication). Despite mostly negative
Number of rejected GPS sites 62 28 17
0.4 0.2 0.0 [mm/year]
Fig. 86.4 The estimated standard deviations of vertical velocities at the network consisting of 101 GPS sites in New Zealand
vertical deformations along the central volcanic zone of the North Island, two GPS sites TAUP and TGOH have large positive velocity rates of 4 and 5 mm/year, respectively. We expect that this large uplift is related to the regionally active subsurface volcanic process. Along the south-east coast of the North Island the negative velocity rates indicate that the subduction is not more than 9 mm/year. The vertical velocity rates of GPS sites at the northern part of the South Island have a small negative systematic trend between 3 and 1 mm/year. The positive velocity rates at GPS sites of the profile that crosses the Southern Alps (SAGENZ) indicates uplift up to 6 mm/year. The vertical velocity rates at GPS sites in the western part of the North Island do not indicate any significant vertical tectonic deformations; the estimated vertical velocity rates are within 1 and 2 mm/year. Since the spatial coverage within the South Island is relatively low (cf. Sect. 86.2), systematic vertical deformations
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Table 86.2 Statistics of the vertical velocities at 101 GPS sites after applying the outlier detection based on 2-sigma selection criterion Area North Island South Island New Zealand
Minimum (mm/year) 16 2 16
Maximum (mm/year) 5 6 6
Mean (mm/year) 1.9 0.3 1.3
Table 86.3 Statistics of the vertical velocities at 101 GPS sites for the adopted single-boundary configuration of the Pacific and Australian tectonic plates in New Zealand Tectonic plate Pacific Australian
Minimum (mm/year) 1 16
especially along the Southern Alps and the Alpine Fault could still remain unrevealed. The plate boundary between the Pacific and Australian tectonic plates in New Zealand does not have a single location. Instead, it is a broad zone of deformation covering much of the country as well as the offshore area of the Tasman Sea along the west coast of the South Island and the offshore area of the Pacific Ocean along the east coast of the North Island. Despite this broad band of deformation, the Alpine Fault in the South Island can be considered as the boundary between these two tectonic plates, while the position of the whole North Island can be attributed to the Australian tectonic plate (cf. Wallace et al. 2004, 2007). For this particular single-boundary tectonic setup of New Zealand, the network consisting in total of 101 permanent GPS stations is divided into 16 and 85 GPS sites located on the Pacific and Australian tectonic plates, respectively. Statistics of the vertical velocities at GPS sites on the Pacific and Australian tectonic plates are summarized in Table 86.3. The average vertical velocity at GPS sites on the Pacific tectonic plate is 0.5 mm/year. The average vertical velocity at GPS sites on the Australian tectonic plate is 1.7 mm/year. As shown in Fig. 86.2, the relative accuracy of vertical velocities at GPS sites depends on the duration of site occupation. Most of the permanent GPS stations in the South Island and the northern part of the North Island have long occupation times and consequently the estimated vertical rates have higher accuracy. On the other hand, some GPS sites in the central and eastern part of the North Island have lower accuracy due to the fact that these sites were established more recently under various projects dedicated to specific geophysical studies.
Maximum (mm/year) 6 5
86.5
Mean (mm/year) 0.5 1.7
Summary and Conclusions
We have investigated the vertical tectonic deformations in New Zealand using the regionallyfiltered vertical component of GPS time series. The results reveal that the northern regions of the North Island is without the presence of significant tectonic deformations. The systematic tectonic subsidence of about 4–9 mm/year is detected at GPS sites in the central and southern regions of the North Island regions with local uplift of about 4–5 mm/year at two GPS sites in the central volcanic zone of the North Island. In agreement with the results of Beavan et al. (2004), the positive velocity rates at GPS sites along the profile across the Southern Alps (western part of the South Island) indicates the uplift to about 6 mm/year. Wellman (1979) used geochronological data to compile a map of vertical tectonic deformations for the whole of the South Island. From his results, the maximum uplift along the Southern Alps reaches 10 mm/year with the velocity rates between 1 and 7 mm/year along the Alpine Fault further decrease to less than 1 mm/year in the south-east and north-west parts of the South Island. The uplift rates estimated by Wellman (1979) throughout the Southern Alps agree with more recent estimates from the geochronological data by Tippett and Kamp (1993), but are larger than our results from GPS time series. Elsewhere in the South Island the vertical velocity rates at GPS sites are similar to the results from the geochronological data (Wellman 1979; Tippett and Kamp 1993). A more detailed comparison is still restricted by the currently low spatial coverage of GPS permanent stations within the South Island.
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Acknowledgement We are thankful to Dr. John Beavan from GNS Science for a valuable discussion. The data used in this study are provided by the New Zealand GeoNet project and its sponsors EQC, GNS Science, and LINZ.
References 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 110:1–13 Beavan J (2009) Notes about GPS processing and creation of GPS time series. GNS Science, Wellington Beavan J, Blick G (2005) Limitations in the NZGD2000 deformation model. In: Tregoning P, Rizos C (eds) Dynamic planet. Springer, Berlin, pp 624–630 Beavan J, Haines J (2001) Contemporary horizontal velocity and strain rate fields of the Pacific-Australian plate boundary zone through New Zealand. J Geophys Res 106:741–770 Beavan J, Moore M, Pearson C, Henderson M, Parsons B, Bourne S, England P, Walcott D, Blick G, Darby D, Hodgkinson K (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. J Geophys Res 104 (B11):25233–25255 Beavan J, Tregoning P, Bevis M, Kato T, Meertens C (2002) Motion and rigidity of the Pacific Plate and implications for plate boundary deformation. J Geophys Res 107(B10):ETG 19/1-15 Beavan J, Matheson D, Denys P, Denham M, Herring T, Hager B, Molnar P (2004) A vertical deformation profile across the Southern Alps, New Zealand, from 3.5 years of continuous GPS data. Cahiers de Centre Europeen de Geodynamique et Seismologie, Proceedings of the workshop: the state of GPS vertical positioning: separation of earth processes by space geodesy, vol 24, pp 111–123
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Beavan J, Ellis S, Wallace L, Denys P (2007) Kinematic constraints from GPS on oblique convergence of the Pacific and Australian Plates, Central South Island, New Zealand. Geophys Monogr 175:75–94 Blewitt G (2004) Fundamental ambiguity in the definition of vertical motion. Cahier du Centre Europe´en de Ge´odynamique et de Se´ismologie 23:1–4 Blick G (2003) Implementation and development of NZGD2000. N Z Surveyor 293:15–19 Blick G, Crook C, Grant D, Beavan J (2003) Implementation of a semi-dynamic datum for New Zealand. In: Sanso` F (ed) A window on the future of geodesy. Springer, Berlin, pp 38–43 Bourne S, Arnadottir T, Beavan J, Darby D, England P, Parsons B, Walcott R, Wood P (1998) Crustal deformation of the Marlborough fault zone in the South Island of New Zealand: geodetic constraints over the interval 1982–1994. J Geophys Res 103:147–165 Bull W, Cooper A (1986) Uplifted marine terraces along the Alpine fault, New Zealand. Science 234:1225–1228 Burchfiel BC (1980) Plate tectonics and the continents: a review. Studies in geophysics: continental tectonics. National Academy of Sciences, Washington, DC Tippett JM, Kamp PJJ (1993) Fission track analysis of the late Cenozoic vertical kinematics of continental Pacific Crust, South Island, New Zealand. J Geophys Res 136 (B9):16119–16148 Wallace LM, Beavan J, McCaffrey R, Darby D (2004) Subduction zone coupling and tectonic rotations in the North Island, New Zealand. J Geophys Res 109:B12406 Wallace LM, Beavan J, McCaffrey R, Berryman K, Denys P (2007) Balancing the plate motion budget in the South Island, New Zealand, using GPS, geological and seismological data. Geophys J Int 168(1):332–352 Wellman H (1979) An uplift map for the South Island of New Zealand, and a model for uplift of the Southern Alps. Bull R Soc NZ 18:13–20 Williams SD, Bock Y, Fang P, Jamason P, Nikolaidis R, Prawirodirdjo LL, Miller M, Johnson DJ (2004) Error analysis of continuous GPS position time series. J Geophys Res 109:B03412
.
Detection of Vertical Temporal Behaviour of IGS Stations in Canada Using Least Squares Spectral Analysis
87
James Mtamakaya, Marcelo C. Santos, and Michael Craymer
Abstract
Unambiguous, consistent and homogeneous GPS station coordinates are the fundamental requirement in the appropriate determination of geodetic velocities that are often used for the derivation of geodetic and geophysical models for a variety of applications [Segall and Davis, Ann Rev Earth Planet Sci 23:201–336, 1997]. Because of this, there have been significant efforts to improve the modeling and parameterization of global GPS solutions in order to get stable and homogeneous positions and velocities. This paper presents a study aiming at detecting least-squares spectral peaks present at the best available (at the time) IGS weekly vertical component time series of five permanent stations in Canada. These peaks are the result of short and long term effects of mismodelled and unmodelled geophysical phenomena on the height. The LSSA approach is used. Results show strong periodic constituents in the LSSA spectrum below or at the 1 year window but most notably constituents with periods longer than a year.
87.1
Introduction and Motivation
Since early 1990s, geodetic coordinate time series have been generated from continuous observing GPS stations and used for many geodetic and geophysical applications that include the derivation of input velocities to geophysical models. Examples of them include, among others, modeling of plate boundary dynamics, postglacial rebound, surface mass loading
J. Mtamakaya (*) M.C. Santos Department of Geodesy and Geomatics Engineering, University of New Brunswick, P.O. Box 4400, Fredericton, NB, Canada, E3B 5A3 e-mail:
[email protected] M. Craymer Geodetic Survey Division, Natural Resources Canada, 615 Booth Street, Ottawa, ON, Canada, K1A 0E9
and other deformations of the solid Earth, Earth rotation, variations in the hydrosphere as well as satellite orbit determination and time and frequency transfer (Segall and Davis 1997). However, operational GPS time series are known to be inconsistent and inhomogeneous for a number of reasons such as changes in reference frame, atmosphere biases, biases due to Earth Rotational Parameters (ERP), and phase center variations. Significant efforts have been underway in the last decade to improve the modeling and parameterization of global GPS solutions in order to get stable and homogeneous positions and velocities. One of the latest improvements is the availability of new absolute phase center variations models that have been adopted by the International GNSS Service (IGS) in all their products since November 5, 2006 (GPS Week 1400) (Schmid et al. 2007). This adoption has caused changes in the IGS solution processing strategy and
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87.2
YELL (Yellowknife), DRAO (Penticton) and STJO (Saint John’s). Our analysis is based on 10 years (1999–2009) of unequally spaced weighted height time series, all of them shown in Figs. 87.2–87.4. Least Squares Spectral Analysis (LSSA) is based on the developments by Vanı´cˇek (1969, 1971) and HEIGHT TIME SERIES FOR ALGONQUIN 2001-2009 Height (m)
1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010
Height (m)
-19.25
HEIGHT TIME SERIES FOR CHURCHILL 2001-2009
-19.30 -19.35 -19.40 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010
Day of Year(DOY)
Fig. 87.2 Height time series for the IGS stations Algonquin (top) and Churchill (bottom)
Height (m)
541.90
HEIGHT TIME SERIES FOR DRAO 2001-2009
541.88 541.86 541.84 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 152.86
HEIGHT TIME SERIES FOR STJO 2001-2009
152.85 152.84 152.83 152.82 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010
Data and Computational Tool
The Least Squares Spectral Analysis (LSSA) technique has been applied to vertical time series of five IGS stations in Canada, shown in Fig. 87.1. The data sets are from the IGS weekly coordinate time series for stations ALGO (Algonquin), CHUR (Churchill),
200.94 200.92 200.90 200.88 200.86
Height (m)
necessitated the reprocessing of all of the historical GPS data. This ongoing effort by the IGS will generate the so-called REPRO1 solution, not available by the time of this work. The primary objective of the research presented in this paper is to investigate the height component of the GPS solutions of a few IGS stations in Canada. It looks into short and long term periodic effects within and beyond 1 cycle per year, caused by either mismodelled or unmodeled phenomena, in the official (and best available by the time of this study) IGS weekly coordinate time series from IGS public archives. It should be noted that the data used in this analysis is not the REPRO1 solution since it had not been made available yet. It is our intention to perform a similar and more comprehensive study using the REPRO1 data set, including a thorough analysis. Therefore, the work presented in this paper can be considered as an initial step towards a more comprehensive study. Past and recent spectral studies of GPS position time series have shown the existence of significant variation in the respective spectrum. They include studies by Blewitt and Lavellee (2002), Penna and Stewart (2003), Agnew and Larson (2007), Collilieux et al. (2007), Ray et al. (2007) and Fritsche et al. (2009). Most of them were based on solutions not yet using absolute phase center variation models and were limited to annual variations (except the last study). These studies attributed the variations to both known and unknown errors arising from different mathematical models and parameters.
Day of Year(DOY)
Fig. 87.3 Height time series for the IGS stations DRAO (top) and STJO (bottom)
HEIGHT TIME SERIES FOR YELLOWKNIFE 2001-2009
Height (m)
180.87
180.86
180.85
Fig. 87.1 The five (5) Canadian IGS stations used in this study
2000
2002
2004 2006 Day of Year(DOY)
2008
2010
Fig. 87.4 Height time series for the IGS station Yellowknife for the period of 1999–2009
Detection of Vertical Temporal Behaviour of IGS Stations in Canada Using Least Squares
Processing, Results and Assessment
LSSA is applied in a stepwise mode, in which consecutive runs can be performed. In each run, the most prominent (and statistically significant) spectral peak(s) estimated in the previous run can be enforced (removed) in the following one, allowing other peak(s) to be detected. The least squares spectra of the GPS time series for each station were detected in this way, first determined without enforcing any periodic constituents. Significant peaks from these spectra were then identified and enforced in subsequent spectra determinations until all statistically significant peaks were identified at the 99% confidence level. The procedure is repeated in successive analysis stages in a manner which would be meaningful through the observation of the significant reduction of the ChiSquared (w2) test on the variance and the quadratic norm of the residuals as well as the Chi-Squared goodness-of-fit test of the histogram of the residual. The results of the LSSA for all five stations are shown from Figs. 87.5–87.14. The letters “EF” followed by a number represent the number of enforced periods in that particular LSSA run. The vertical axis of the figures is in units of power spectral density (PSD) and the horizontal axis is frequency in units of cycles per year (CPY). The results show there exist at least six different groups of strong periodic constituents in the LSSA spectrum window of 0.05–2.5 cycles per year, but only a few of them can be explained (Dong et al. 2002). The group ranges are subject to minor shifts yet to be verified, that could be caused by regional and site dependent effects.
PSD (m2/cpy)
4000
← 1904.5d
3000 2000 1000 0
0
0.5
1.0
1.5
2.0
2.5
Spectral power of height T/S-ALGO FOR 2001-2009 + EF1 PSD (m2/cpy)
87.3
707
Spectral power of height T/S-ALGO FOR 2001-2009 + EF0
100
← 875.7d
← 373.6d
50
0
0
0.5
1.0 1.5 Frequency (cpy)
2.0
2.5
Fig. 87.5 Power spectral density for station ALGO. Top plot is with no periods enforced. The bottom plot is after enforcement of period 1,904.5 day
Spectral power of height T/S-ALGO FOR 2001-2009 + EF2
40
PSD (m2/cpy)
later improvements and implementation by Wells et al. (1985) and Pagiatakis (1998). LSSA has been adopted as the main tool of analysis because of the software capabilities that allow analysis of data time series with known and unknown a-priori variance factor. The analyzed data may also be correlated or uncorrelated. LSSA can also handle unequally spaced time series without a pre-processing requirement, rigorous analysis of systematic noise without shifts in the spectral peaks and the ability to test the statistical significance of the spectral peaks in the spectrum (Pagiatakis 1998).
30
667.0d →
20
← 471.7d
10 0 -10
0
0.5
1.0
1.5
2.0
2.5
Spectral power of height T/S-ALGO FOR 2001-2009 + EF3
40
PSD (m2/cpy)
87
30
← 1249.5d
20
← 230.7d
10 0 -10
0
0.5
1.0
1.5
2.0
2.5
Frequency (cpy)
Fig. 87.6 Power spectral density for station ALGO (Algonquin). Top plot is after enforcement of periods 1,904.5, 875.7 and 373.6 days. Bottom plot is after enforcement of periods 1,904.5, 875.7, 373.6, 667.0 and 471.8 days
The first group corresponds to a known constituent with periodicities between 177 and 200 days. Based on Melchior (1983), this group of periodic signals could be the impact of semi-annual solar waves.
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J. Mtamakaya et al. Spectral power of height T/S-CHUR FOR 2001-2009 + EF0
2000 1500
← 1760.5d
1000 500
10000
0.5
1.5
2
0
2.5
PSD (m2/cpy)
200 100
← 374.0d
475.2d → 0.5
1 1.5 Frequency (cpy)
2
500
PSD (m2/cpy) 2
30
339.9 d →
10
0
0.5
← 293.3 d
1 1.5 Frequency (cpy)
2
1 1.5 Frequency (cpy)
2
2.5
100
966.7 d ↓
50
582.5 d ↓
0
0.5
1
1.5
2
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Spectral power of height T/S-DRAO FOR 2001-2009 + EF3
0 -10
0.5
0
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1249.5 d ↓
20
← 361.7 d
150
Spectral power of height T/S-CHUR FOR 2001-2009 + EF3
40
PSD (m2/cpy)
1.5
PSD (m2/cpy)
PSD (m2/cpy)
← 582.5 d 1
2.5
Spectral power of height T/S-DRAO FOR 2001-2009 + EF2
← 966.7d
0.5
2
Fig. 87.9 Power spectral density for station DRAO. Top plot is with no periods enforced. Bottom plot is after enforcement of period 2,573.7 day
150
0
← 1388.9 d
0
Spectral power of height T/S-CHUR FOR 2001-2009 + EF2
0
1.5
0
2.5
Fig. 87.7 Power spectral density for station CHUR. Top plot is with no periods enforced. Bottom plot is after enforcement of period 1,760.5 day
50
1
1000
0 0
0.5
Spectral power of height T/S- DRAO FOR 2001-2009 + EF1
300
100
0
Spectral power of height T/S-CHUR FOR 2001-2009 + EF1
400
PSD (m2/cpy)
1
← 2573.7 d
5000
0 0
Spectral power of height T/S- DRAO FOR 2001-2009 + EF0
15000 PSD (m2/cpy)
PSD (m2/cpy)
2500
2.5
Fig. 87.8 Power spectral density for station CHUR (Churchill). Top plot is after enforcement of periods 1,760.5, 475.2 and 374.0 days. Bottom plot is after enforcement of periods 1,760.5, 475.2, 374.0, 996.7 and 582.5 days
The second group corresponds to a known constituent with periodicities between 200 and 400 days with most of them closer to the sidereal year (365.25 days) and 351.2 days, which is the time taken by the GPS constellation to repeat its inertial orientation with
100 50
419.0 d ↓
← 333.9 d
0 0
0.5
1 1.5 Frequency (cpy)
179.8 d ↓
2
2.5
Fig. 87.10 Power spectral density for station DRAO (Penticton). Top plot is after enforcement of periods 1,388.9 and 361.7 days. Bottom plot is after enforcement of periods 966.7 and 582.5 days
respect to the sun also known as the GPS year (Agnew and Larson 2007). Subject to proper quantification in the ongoing research, possible reasons for them could be the effects of annual solar (elliptical) waves and the systematic errors related to satellite
Detection of Vertical Temporal Behaviour of IGS Stations in Canada Using Least Squares
PSD (m2/cpy)
400
← 2091.7d
300 200 100
← 1815.8 d
1500 1000 500 0
0 0
0.5
1
1.5
2
0
2.5
0.5
200
← 1127.8d
100 50
1.5
2
2.5
588.8 d →
← 349.8 d
150 100 50 0
0 0
0.5
1 1.5 Frequency (cpy)
2
0
2.5
Fig. 87.11 Power spectral density for station STJO. Top plot is with no periods enforced. Bottom plot is after enforcement of period 2,091.7 days
600.8d → 20
483.5d ↓
← 375.8d
177.7d ↓
0 0.5
1 1.5 Frequency (CPY)
2
2
2.5
200 852.2 d ↓
100 0
2.5
0
182.0 d ↓
← 442.4 d
0.5
1
1.5
2
2.5
Spectral power of height T/S-YELL FOR 2001-2009 + EF3 PSD (m2/cpy)
Fig. 87.12 Power spectral density for station STJO after enforcement of periods 2,091.7 and 1,127.8 days
orbits such as orbit mismodeling and varying satellite geometry and the local multipath effects. Other possible reasons are the long periodic effect due to unmodeled tidal effects in diurnal and semi diurnal waves as well as the impact of hydrological and atmospheric loading (Van Dam et al. 2001). The third to sixth groups correspond to periodic constituents with long frequencies of, respectively, 400–600, 600–1,000, 1,000–2,000 and over 2,000 days. The physical causes of the periodic constituents for these groups have not been established nor has the extent of their bias in the present solutions.
1 1.5 Frequency (cpy)
Spectral power of height T/S-YELL FOR 2001-2009 + EF2
300 PSD (m2/cpy)
40
0.5
Fig. 87.13 Power spectral density for station YELL. Top plot is with no periods enforced. Bottom plot is after enforcement of period 1,815.8 day
Spectral power of height T/S- STJO FOR 2001-2009 + EF2 PSD(m2/cpy)
1
Spectral power of height T/S- YELL FOR 2001-2009 + EF1 PSD (m2/cpy)
PSD (m2/cpy)
Spectral power of height T/S- STJO FOR 2001-2009 + EF1
0
709
Spectral power of height T/S- YELL FOR 2001-2009 + EF0
Spectral power of height T/S- STJO FOR 2001-2009 + EF0
500
PSD (m2/cpy)
87
100 1186.4d ↓
50
465.0 d ↓
0 0
0.5
1 1.5 Frequency (cpy)
2
2.5
Fig. 87.14 Power spectral density versus frequency for station YELL. Top plot is after enforcement of periods 1,815.9, 588.8 and 349.8 days. Bottom plot is after enforcement of periods 1,815.85, 588.79, 349.831, 852.2, 442.4 and 182.0 days
In other words, even though interesting periods longer than one year have been detected, we do not have a convincing explanation for them yet.
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Conclusions
The primary objective of the work presented in this paper is to detect periodic components in the vertical component of GPS solutions for five IGS stations in Canada using the Least Square Spectral Analysis. It uses the IGS weekly coordinate time series (1999–2009) of stations ALGO, CHUR, DRAO, STJO and YELL. The LSSA results indicate the existence of significant periodic frequencies between 0.05 and 2.5 cycles per year. They have different spectral power with periodicities ranging from about 160 to over 2,000 days that are statistically significant at the 99% confidence level. They may be the effects of the temporal behavior of unmodeled geophysical phenomena and the accumulated impact of earth tides that are not properly modeled. Besides verifying the results of similar past and recent spectral studies, our results have also indicated the existence of a number of significant long periodic signatures in the LSSA spectra for all stations under investigation. The probable causes of the long periodic signatures (longer than 1 year) have not been discussed and are still under investigation. As part of future work, there is the intention to replicate this study using the final reprocessed IGS weekly coordinate solutions that include the new absolute antenna phase centers (REPRO1) once they become available. Similar effort will be made using the residuals provided in REPRO1. In this case, spectral corresponding to position and residual domain will be generated and will go through a full investigation and analysis of their causes, i.e., identifying any eventual unmodelled or mismodelled effect still present. Those results will also be compared to results for the previous IGS weekly solutions based on relative antenna phase centers in order to quantify the impact of the phase center models. Acknowledgements Special thanks go to the Government of Tanzania for sponsoring James Mtamakaya’s PhD studies at
J. Mtamakaya et al. UNB. We also thank Remi Ferland, the IGS Reference Frame Coordinator, for providing the data used in this analysis.
References Agnew D, Larson K (2007) Finding the repeat times of the GPS constellation. GPS Solut 11:71–76 Blewitt G, Lavellee D (2002) Effect of annual signals on geodetic velocity. J Geophys Res 107(B7):2145 Collilieux X, Altamimi Z, Coulot D, Ray J, Sillard P (2007) Comparison of very long baseline interferometry, GPS, and satellite laser ranging height residuals from ITRF2005 using spectral and correlation methods. J Geophys Res 112: B12403 Dong D, Fang P, Bock Y, Cheng M, Miyazaki S (2002) Anatomy of apparent seasonal variations from GPS-derived site position time series. J Geophys Res 107(B4):2075 Fritsche M, Dietrich R, R€ulke A, Rothacher M, Steigenberger P (2009) Low-degree earth deformation from reprocessed GPS observations. GPS Solut. doi:10.1007/s10291-009-0130-7 Melchior P (1983) The tides of the planet Earth. Pergamon, Oxford Penna NT, Stewart MP (2003) Aliased tidal signatures in continuous GPS height time series. Geophys Res Lett 30 (23):2184. doi:10.1029/2003GL018828 Pagiatakis S (1998) Stochastic significance of peaks in the leastsquares spectrum. J Geodesy 73(2):67–78 Ray JR, Altamimi Z, Collilieux X, van Dam T (2007) Anomalous harmonics in the spectra of GPS position estimates. GPS Solut. doi:10.1007/s10291-007-0067-7 Segall P, Davis JL (1997) GPS applications for geodynamics and earthquake studies. Ann Rev Earth Planet Sci 23:201–336, 1997 Schmid R, Steigenberger P, Gendt G, Ge M, Rothacher M (2007) Generation of a consistent absolute phase center correction model for GPS receiver and satellite antennas. J Geodesy 81(12):781–798 Van Dam T, Wahr J, Milly P, Shmakin A, Blewitt G, Lavallee D, Larson K (2001) Crustal displacements due to continental water loading. Geophys Res Lett 28:651–654 Vanı´cˇek P (1969) Approximate spectral analysis by leastsquares fit. Astrphys Space Sci 4:387–391 Vanı´cˇek P (1971) Further development and properties of the spectral analysis by least squares. Astrophys Space Sci 12:10–33 Wells D, Vanı´cˇek P, Pagiatakis S (1985) Least squares spectral analysis revisited. Department of Geodesy and Geomatics Engineering Technical Report No. 84, University of New Brunswick, Fredericton, NB, Canada
Session 4 Positioning and Remote Sensing of Land, Ocean and Atmosphere Convenors: S. Verhagen, P. Wielgosz
.
Positioning and Applications for Planet Earth
88
S. Verhagen, G. Retscher, M.C. Santos, X. L. Ding, Y. Gao, and S.G. Jin
Abstract
GNSS, InSAR and LIDAR are identified as important techniques when it comes to monitoring and remote sensing of our planet Earth and its atmosphere. In fact, these techniques can be considered as key elements of the Global Geodetic Observing System. Examples of applications are: environmental monitoring; volcano monitoring, land slides, tectonic motion, deforming structures, atmosphere modeling, and ocean remote sensing. Hence, it concerns applications at local and regional scales, as well as at global scales. The main issues can be summarized as: need for a better understanding of processes, leading to better models; need for observational material; and adequate modeling techniques.
88.1
Introduction
Recognising the central role that Global Navigation Satellite Systems (GNSS) play in many applications like engineering, mapping and remote sensing, the work of Commission 4 of the International
S. Verhagen (*) DEOS, Delft University of Technology, Delft, The Netherlands e-mail:
[email protected] G. Retscher Vienna University of Technology, Vienna, Austria M.C. Santos University of New Brunswick, Fredericton, NB, Canada X.L. Ding Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Y. Gao University of Calgary, Calgary, AB, Canada S.G. Jin Center for Space Research, University of Texas, Austin, TX, USA
Association of Geodesy (IAG) focuses on several GNSS-based techniques, taking into account GPS, Glonass, Galileo and Beidoe. These techniques include precise positioning, but extending beyond the applications of reference frame densification and geodynamics, to address the demands of precise, real-time positioning of moving platforms. Precise kinematic GNSS positioning technology itself (alone or in combination with other positioning sensors) is a key topic as well as its applications in surveying and engineering. Recognising the role of continuously operating GPS reference station network, the non-positioning applications of such geodetic infrastructure is considered as well, such as atmospheric sounding. The commission also deals with geodetic remote sensing, using (differential) InSAR, and GNSS as a remote sensor with land, ocean and atmosphere applications. This contribution aims to summarize the most important applications of GNSS, InSAR and LIDAR in the field of monitoring and remote sensing of our
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_88, # Springer-Verlag Berlin Heidelberg 2012
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planet Earth and its atmosphere, either at local or regional scales or even at a global scale. The techniques can be considered as key elements of the Global Geodetic Observing System (GGOS), which is the flagship of IAG (Drewes 2007).
88.2
Geotechnical and Structural Engineering
88.2.1 Monitoring and Alert Systems for Local Geodynamic Processes Nowadays extended multi-sensor deformation measurement systems consisting of terrestrial geodetic and geotechnical measurement as well as hydrological and meteorological instrumentation completed by the InSAR technique are mainly employed for multi-scale monitoring of landslide prone areas. Thereby InSAR is used for large-scale detection of landslide prone areas as well as for deformation measurements of the investigated landslide area. Such a complete measurement system is very suitable for the investigation of the kinematic behaviour of landslides and together with other (e.g. hydrological, meteorological, etc.) parameters for the study of the dynamics of landslides (Mentes 2008a). The observation data is usually collected in GIS (see e.g. Lakakis et al. 2009b; Mentes 2008a, b) and used to develop Spatial Decision Support Systems (SDSS) (e.g. Lakakis et al. 2009a) and Early warning systems. The Dunaf€oldva´r test site in Hungary is monitored by terrestrial and InSAR measurement techniques. Figure 88.1 show the test site before and after the
Fig. 88.1 Beginning of landslide in 2007 (left), and landslide after February 12, 2008 (right)
landslide on February 12, 2008. The high bank on this area was sliding slowly with increasing velocity since September of 2007 till 12 of February 2008. On this day there was an abrupt sliding. About 500,000 m3 loess was sliding toward to the river Danube. The whole sliding process was monitored. The study of the movement is a good possibility to understand the kinematics and dynamics of the slope. The monitoring will be continued in the future to study the after´ jva´ri et al. 2008, 2009). sliding processes (U
88.2.2 Application of Artificial Intelligence in Engineering Geodesy In the last years, Artificial Intelligence (AI) has become an essential technique for solving complex problems in Engineering Geodesy. AI is an extremely broad field – the topics range from the understanding of the nature of intelligence to the understanding of knowledge representation and deduction processes, eventually resulting in the construction of computer programs which act intelligently. Especially the latter topic plays a central role in applications (Reiterer and Egly 2008). Current applications using AI methodologies in engineering geodesy are: • Geodetic data analysis • Deformation analysis • Navigation • Deformation network adjustment • Optimization of complex measurement procedures An example highlighted in the following is a new deformation analysis system based on AI techniques. Here AI shall be used for the task of deformation assessment and deformation interpretation. The main task of the AI component is to transfer the information of a deformation analysis in a useable form for an automatic deformation interpretation. Therefore different approaches from the AI field are used and joined to a case-based reasoning system. The simplified measurement and analysis procedure, including the deformation assessment component is shown in Fig. 88.2. Although this deformation assessment is mainly developed for a terrestrial laser scanner and imagebased measurement system, an extension to other data acquisition systems is possible. The deformation assessment developed at the Vienna University of Technology is based on the idea of supporting the expert who performs the interpretation by former
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Positioning and Applications for Planet Earth
Fig. 88.2 Simplified system architecture of the deformation assessment component (after Lehmann et al. 2007)
deformation cases. The case base includes the reason for the former deformations (Lehmann and Reiterer 2008).
88.3
Modelling and Remote Sensing of the Atmosphere
88.3.1 Ionosphere The past years have seen an increasing effort in the collection of experimental data for monitoring of TEC and ionospheric scintillation studies. This effort has resulted in the deployment of dedicated networks of ground GNSS and scintillation receivers, at high and mid latitudes. One of such networks is the Canadian High Arctic Ionospheric Network (CHAIN) composed of GPS receivers collocated with ionosondes. This configuration of instruments (collocated ionosondes and GPS receivers) will have an added advantage in the tomographic imaging of the electron density structures in the polar cap and calibration of the GPS data (Jayachandran et al. 2009). Similar effort is taking place in the Antarctica (Alfonsi et al. 2008). Other networks exist. There is also effort by means of satellite missions. For example, in situ measurements from GRACE K-Band ranging and CHAMP planar Langmuir probe (PLP) have been used for the validation of the International Reference Ionosphere (IRI); and FORMOSAT-3/COSMIC occultation data used in combination with GNSS and satellite altimetry aiming at a combined global VTEC model (Jakowski et al. 2007; Alizadeh et al. 2008; Todorova et al. 2008; Mayer and Jakowski 2009). There has been effort put on enhancements in the spatial and temporal representation of TEC and or VTEC, globally, regionally or locally. Algorithms for spatial representation are a function of the area size
715
and can be spherical harmonics, trigonometric B-splines, endpoint-interpolating B-splines, Chapman functions. For the temporal representation, there is a choice among empirical orthogonal functions, B-splines and Fourier series representation. An a-priori regularization procedure is usually needed to handle existing data gaps, even though multivariate adaptive regression splines have been shown to be capable of dealing with rare observations without regularization (Schmidt et al. 2007, 2008). Near- and real-time applications require the dissemination of predicted values of TEC. This brings to mind the SBAS (WAAS, MSAS and EGNOS, the later has just made available its “Open Service”), based on continental networks but regional or local systems may also support these applications. Investigation into multi-GNSS constellation and higher order (e.g., 3rd) determination TEC seem to be gaining momentum. Higher order ionospheric delay terms, which have been mostly disregarded in the dual-frequency world, can be taken into account in a multi-frequency reality. The cm-level contribution of the ionospheric 2nd and 3rd order terms (in the cubic and quadratic inverse of the frequencies) can be more easily modelled in a triple- (or multi-) frequency system (Hoque and Jakowski 2008). The inclusion of higher order ionospheric terms in GNSS processing can potentially lead to an increase in accuracy at a global level by a few millimetres (Herna´ndezPajares et al. 2007).
88.3.2 Troposphere The increasing use of Numerical Weather Models (NWM) has helped enhancing the prediction of neutral atmospheric models (Boehm et al. 2006). It has also become a source of neutral atmospheric delay that can be directly applied in GNSS processing, including PPP (Hobiger et al. 2008a). If from one side NWMs contain a more realistic temporal representation of the delay than prediction models, from the other side the extraction of this information requires ray-tracing through the neutral atmosphere, a time consuming task if done properly (Nievinski 2009). Fast and accurate algorithms are of fundamental necessity (Hobiger et al. 2008b). How can NWM be used in practice
716
vis-a`-vis computational cost “vs.” accuracy? Is it necessary to extract all information contained in a NWM to obtain a more accurate delay than that provided by prediction models, especially at low elevation angles? These still seem to be open questions. There has been an increasing emphasis of neutralatmosphere delay monitoring by ground GNSS and satellite missions, with radio occultation consolidating itself as a solid technique (Wickert et al. 2009). There is a continuing effort towards enhancements in the spatial and temporal representation of the neutral-atmosphere including its azimuthal asymmetry. Several models incorporating gradients, spherical harmonics, tomography, have been further tested including information from NWMs (Urquhart et al. 2011; Ghoddousi-Fard et al. 2009; Rohm and Bosy 2009). The theory of turbulence seems to have gained more attention recently in the modeling of GNSS observations either at functional or stochastic levels (Sch€on and Brunner 2008; Nilsson et al. 2009).
S. Verhagen et al.
88.4
Applications of Satellite and Airborne Imaging Systems
Synthetic Aperture Radar (SAR) and Light Detection And Ranging (LiDAR) systems are very useful for geodetic applications, such as monitoring local area ground surface deformations due to volcanic and seismic activities, and ground subsidence associated with city development, mining activities (e.g. Ge et al. 2007; Ng et al. 2008), ground liquid withdrawal, and land reclamation (e.g. Ding et al. 2004). InSAR is a very active field of research in the geodetic research communities. The current research issues include the development of more effective methods/ algorithms for InSAR solutions, the quality control and assurance of InSAR measurements, the study and mitigation of biases in InSAR measurements such as the atmospheric effects (e.g. Li et al. 2006), integration of InSAR and other geodetic technologies such as GPS, and new and innovative applications of the technology in geodetic studies (Fig. 88.3).
Fig. 88.3 Deformation observed with InSAR after earthquake in L’Aquila, Italy on 6 April 2009 with magnitude of 6.3
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Positioning and Applications for Planet Earth
88.5
GNSS Reflectometry
The signals of Global Navigation Satellite Systems (GNSS), including US Global Positioning System (GPS), Russia GLONASS, Europe Galileo and China Compass, propagate the atmosphere with the delay and reach users with the multipath error. For a long time, such indirect GNSS signals are considered as a nuisance, i.e. error sources, and now can be transferred into useful products, e.g. the water vapor, temperature, and pressure, electron density of the atmosphere and surface characteristics of land and oceans (Jin and Komjathy 2010). Surface multipath delay from the GNSS signal reflecting from the sea and land surface, could be used as a new tool in ocean, coastal, wetlands, Crater Lake, landslide, soil moisture, snow and ice remote sensing (e.g. Komjathy et al. 2004). Together with information on the receiving antenna position and the medium, associating with the surface properties of the reflecting surface, the delay measurement can be used to determine such factors as wave height, wind speed, wind direction, and even sea ice conditions. MartinNeira (1993) first proposed and described a bistatic ocean altimetry system utilizing the signal of GPS. Recently, a number of applications have been implemented using GPS signals reflected from the ocean surface, such as determining wave height, wind speed and wind direction of ocean surface, ocean eddy, and Sea surface conditions (Rius et al. 2002; Komjathy et al. 2004; Gleason et al. 2005). The estimated wind speed using surface-reflected GPS data is consistent with independent wind speed measurements derived from the TOPEX/Poseidon altimetry satellite and Balloon at the level of 2 m/s, and the estimated wind direction using surface-reflected GPS data is almost consistent with results obtained from a buoy at the level of 10 (Garrison et al. 2002; Komjathy et al. 2004). Acknowledgements This contribution is the result of a collaborative effort of IAG Commission 4. See: http://enterprise.lr.tudelft.nl/iag/iag\_comm4.htm. The research of Sandra Verhagen is supported by the Dutch Technology Foundation STW, applied science division of NWO and the Technology Program of the Ministry of Economic Affairs.
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References Alfonsi L, Kavanagh E, Amata P, Cilliers E, Correia E, Freeman M, Kauristie K, Liu R, Luntama J-P, Mitchell CN, Zherebtsov GE (2008) Probing the high latitude ionosphere from ground-based observations: the state of current knowledge and capabilities during IPY (2007–2009). J Atmos Solar Terr Phys 70(18):2293–2308 Alizadeh MM, Schuh H, Todorova S, Schmidt M (2008) Towards 4D ionosphere model combining GNSS and satellite altimetry with Formosat-3/COSMIC. Geodetic Week, Bremen Boehm J, Niell A, Tregoning P, Schuh H (2006) Global mapping function (GMF): a new empirical mapping function based on numerical weather model data. Geophys Res Lett 33:L07304. doi:10.1029/2005GL025546 Ding XL, Liu GX, Li ZW, Li ZL, Chen YQ (2004) Ground settlement monitoring in Hong Kong with satellite SAR interferometry. Photogramm Eng Remote Sens 70(10):1151–1156 Drewes H (2007) Science Rationale of the Global Geodetic Observing System (GGOS). International Association of Geodesy Symposia, 2007, Volume 130, Part VI, 703–710, doi: 10.1007/978-3-540-49350-1_101 Garrison JL, Komjathy A, Zavorotny VU, Katzberg SJ (2002) Wind speed measurement from forward scattered GPS signals. IEEE Trans Geosci Remote Sens 40(1):50–65 Ge L, Chang H-C, Rizos C (2007) Mine subsidence monitoring using multi-source satellite SAR images. Photogramm Eng Remote Sens 73(3):259–266 Ghoddousi-Fard R, Dare P, Langley RB (2009) Tropospheric delay gradients from numerical weather prediction models: effects on GPS estimated parameters. GPS Sol 13:281–291 Gleason S, Hodgart S, Sun Y, Gommenginger C, Mackin S, Adjra M, Unwin M (2005) Detection and processing of bistatically reflected GPS signals from low earth orbit for the purpose of ocean remote sensing. IEEE Trans Geosci Remote Sens 43(6):1229–1241 Herna´ndez-Pajares M, Juan JM, Sanz J, Oru´s R (2007) Secondorder term in GPS: implementation and impact on geodetic estimates. J Geophys Res 112:B08417 (a correction appeared in Vol. 113, B06407) Hobiger T, Ichikawa R, Takasu T, Koyama Y, Kondo T (2008a) Ray-traced troposphere slant delays for precise point positioning. Earth Planets Space 60(5):e1–e4 Hobiger T, Ichikawa R, Kondo T, Koyama Y (2008b) Fast and accurate ray-tracing algorithms for real-time space geodetic applications using numerical weather models. J Geophys Res 113(D203027):1–14 Hoque MM, Jakowski N (2008) Estimate of higher order ionospheric errors in GNSS positioning. Radio Sci 43:RS5008. doi:10.1029/2007RS003817 Jakowski N, Wilken V, Mayer C (2007) Space weather monitoring by GPS measurements on board CHAMP. Space Weather 5:S08006. doi:10.1029/2006SW 000271 Jayachandran PT, Langley RB, MacDougall JW, Mushini SC, Pokhotelov D, Hamza AM, Mann IR, Milling DK, Kale ZC,
718 Chadwick R, Kelly T, Danskin DW, Carrano CS (2009) The Canadian high arctic ionospheric network (CHAIN). Radio Sci 44:RS0A03. doi:10.1029/2008 RS004046 Jin SG, Komjathy A (2010) GNSS reflectometry and remote sensing: new roles and progresses. Adv Space Res 46(2):111–117 Komjathy A, Armatys M, Master D, Axelrad P, Zavorotn V, Katzberg S (2004) Retrieval of ocean surface wind speed and wind direction using reflected GPS signals. J Atmos Oceanic Technol 21(3):515–526 Lakakis K, Charalampakis M, Savaidis P (2009a) A spacial decision support system for highway infrastructure. Fifth international conference on construction in the 21st century (CITC-V), Collaboration and Integration in Engineering, Management and Technology, 20–22 May, Istanbul, pp 1–8 Lakakis K, Charalampakis M, Savaidis P (2009b) A landslide definition by an integrated monitoring system. Fifth international conference on construction in the 21st century (CITCV), Collaboration and Integration in Engineering, Management and Technology, 20–22 May, Istanbul, pp 1–8 Lehmann M, Reiterer A (2008) Case-based deformation assessment – a concept. In: Reiterer A, Egly U (eds) Application of artificial intelligence in engineering geodesy. In: Proceedings of the first workshop on AIEG 2008, pp 91–98. http://info. tuwien.ac.at/ingeo/Downloads/AIEG2008_Proceedings.pdf Lehmann M, Reiterer A, Kahmen H (2007) Deformation classification in high density point clouds. Optical 3-D measurement techniques VIII, vol. I, Zurich Li ZW, Ding XL, Huang C, Wadge G, Zheng DW (2006) Modeling of atmospheric effects on InSAR measurements by incorporating terrain elevation information. J Atmos Solar Terr Phys 68:1189–1194 Martin-Neira M (1993) A passive reflectometry and interferometry system (PARIS): application to ocean altimetry. ESA J 17(4):331–355 Mayer C, Jakowski N (2009) Enhanced E-layer ionization in the auroral zones observed by radio occultation measurements onboard CHAMP and Formosat-3/COSMIC. Ann Geophys 27:1207–1212 Mentes G (2008a) Investigation of different possible agencies causing landslides on the High Loess Bank of the River Danube at Dunaf€ oldva´r, Hungary. In: Proceedings of the measuring the changes, 13th FIG international symposium on deformation measurements and analysis, 4th IAG symposium on geodesy for geotechnical and structural engineering, LNEC, Lisbon, CD, 12–15 May, pp 1–10 Mentes G (2008b) Investigation of micro-movements by borehole tiltmeters on the High Loess Bank of the River Danube at Dunaf€oldva´r in Hungary. In: Proceedings of the INGEO 2008 – 4th International conference on engineering surveying, Slovak University of Technology, Bratislava, ISBN 978-80-227-2971-0, p 11 Ng AH, Chang H, Ge L, Rizos C, Omura M (2008) Radar interferometry for ground subsidence monitoring using ALOS PALSAR data. In: Proceedings of the XXI congress,
S. Verhagen et al. The International Society for Photogrammetry and Remote Sensing, Beijing, 3–11 July 2008 Nievinski FG (2009) Ray-tracing options to mitigate the neutral atmosphere delay in GPS. Technical Report No. 262, Department of Geodesy and Geomatics Engineering, Fredericton, NB Nilsson T, Davis JL, Hill EM (2009) Using ground-based GPS to characterize atmospheric turbulence. Geophys Res Lett 36:L16807. doi:10.1029/2009 GL040090 Reiterer A, Egly U (eds) (2008) Application of artificial intelligence in engineering geodesy. In: Proceedings of the first workshop on AIEG 2008, 116 pp. http://info.tuwien.ac.at/ ingeo/Downloads/AIEG2008_Proceedings.pdf Rius A, Aparicio JM, Cardellach E, Martin-Neira M, Chapron B (2002) Sea surface state measured using GPS reflected signals. Geophys Res Lett 29(23). doi:10.1029/ 2002GL015524 Rohm W, Bosy J (2009) Local tomography troposphere model over mountains areas. Atmos Res 93:777–785 Schmidt M, Bilitza D, Shum CK, Zeilhofer C (2007) Regional 4-D modeling of the ionospheric electron content. Adv Space Res. doi:10.1016/j.asr.2007.02.050 Schmidt M, Karslioglu MO, Zeilhofer C (2008) Regional multidimensional modeling of the ionosphere from satellite data. In: Proceedings of the TUJK annual scientific meeting, 14–16 Nov 2008, Ankara Sch€on S, Brunner FK (2008) Atmospheric turbulence theory applied to GPS carrier-phase data. J Geod 82:47–57 Todorova S, Hobiger T, Schuh H (2008) Using the global navigation satellite system and satellite altimetry for combined global ionosphere maps. Adv Space Res 42:727–736. doi:10.1016/j.asr.2007.08.024 ´ jva´ri G, Mentes G, Theilen-Willige B (2008) Detection of U landslide prone areas on the basis of geological, geomorphological investigations – a case study. In: Proceedings of the measuring the changes, 13th FIG International symposium on deformation measurements and analysis, 4th IAG symposium on geodesy for geotechnical and structural engineering, LNEC, Lisbon, CD, 12–15 May 2008, pp 1–9 ´ jva´ri G, Mentes G, Ba´nyai L, Kraft J, Gyimo´thy A, Kova´cs J U (2009) Evolution of a bank failure along the River Danube at Dunaszekcso˝, Hungary. Geomorphology 109:197–209. doi:10.1016/j.geomorph.2009.03.002 Urquhart L, Santos MC, Nievinsk FG (2011) Fitting of NWM Ray-traced slant factors to closed-form tropospheric mapping functions. In: Kenyon S et al (eds) Geodesy for planet earth. Springer, Heidelberg Wickert J, Schmidt T, Michalak G, Heise S, Arras C, Beyerle G, Falck C, K€onig R, Pingel D, Rothacher M (2009) GPS radio occultation with CHAMP, GRACE-A, SAC-C, TerraSARX, and FORMOSAT-3/COSMIC: brief review of results from GFZ. In: Steiner AK, Pirscher B, Foelsche U, Kirchengast G (eds) New horizons in occultation research. Springer, Berlin, pp 3–15. doi:10.1007/978-3-642-00321-9
Report of Sub-commission 4.2 “Applications of Geodesy in Engineering”
89
G. Retscher, A. Reiterer, and G. Mentes
Abstract
Rapid developments in engineering, microelectronics and the computer sciences have greatly changed both instrumentation and methodology in engineering geodesy. To build higher and longer, on the other hand, have been key challenges for engineers and scientists since ancient times. Now, and for the foreseeable future, engineers confront the limits of size, not merely to set records, but to meet the real needs of society minimizing negative environmental impact. Highly developed engineering geodesy techniques are needed to meet these challenges. The SC will therefore endevour to coordinate research and other activities that address the broad areas of the theory and applications of engineering geodesy tools. The tools range from conventional terrestrial measurement and alignment technology (optical, RF, etc.), Global Navigation Satellite Systems (GNSS), geotechnical instrumentation, to software systems such as GIS, decision support systems, etc. The applications range from construction engineering and structural monitoring, to natural phenomena such as landslides and ground subsidence that have a local effect on structures and community infrastructure. The SC will carry out its work in close cooperation with other IAG Entities, as well as via linkages with relevant scientific and professional organizations such as ISPRS, FIG, IEEE, ION. The major objectives of the SC are: – To monitor research and development into new – To study advances in monitoring and alert systems technologies that are applicable to the general for local geodynamic processes, such as landslides, ground subsidence, etc. field of “engineering geodesy”, including hardware, software and analysis techniques. – To study advances in geodetic methods used on large construction sites. – To study advances in dynamic monitoring and data evaluation systems for buildings and other – To study advances in the application of artificial intelligence techniques in engineering geodesy. manmade structures. – To document the body of knowledge in this field, and to present this knowledge in a consistent frame G. Retscher (*) A. Reiterer work at symposia and workshops. Institute of Geodesy and Geophysics, Vienna University of – To promote research into several new technology Technology, Gusshausstrasse 27-29, 1040 Vienna, Austria areas or applications through the SC 4.2 Working e-mail:
[email protected] Groups. G. Mentes The achievements of the work of the SC in the past Geodetic and Geophysical Research Institute, Hungarian 2 years will be presented and discussed in this paper. Academy of Science, Csatkai u. 6-8, 9400 Sopron, Hungary S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_89, # Springer-Verlag Berlin Heidelberg 2012
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Introduction
The main tasks of the SC 4.2 are to study and enhance technologies for applications of engineering geodesy in order to address the objectives set in the Terms of References. To start with, a website has been set up to provide information on SC 4.2 activities, a professional meeting calendar, contact information of the WG members, etc. The address of the website is http://info.tuwien.ac.at/ingeo/sc4/ sc42.html. Initially it was planned to establish the following four Working Groups: – WG 4.2.1: Measurement Systems for the Navigation of Construction Processes Chair: Wolfgang Niemeier (Technical University Braunschweig, Germany) Co-Chair: TBD – WG 4.2.2: Dynamic Monitoring of Buildings Chair: Gethin Roberts (IESSG, Nottingham University, UK) Co-Chair: TBD – WG 4.2.3: Application of Artificial Intelligence in Engineering Geodesy Chair: Alexander Reiterer (Vienna University of Technology, Austria) Co-Chair: Uwe Egly (Vienna University of Technology, Austria) – WG 4.2.4: Monitoring of Landslides and System Analysis Chair: Gyula Mentes (Geodetic and Geophysical Research Institute of HAS, Hungary) Co-Chair: Paraskevas Savvaidis (University of Thessaloniki, Greece) The reports of the activities of WG 4.2.3 and WG 4.2.4 can be found below. This two WGs are very active. WG 4.2.3 has currently 8 members and WG 4.2.4 23 members. WG 4.2.4 has changed its title recently to “Investigation of Kinematic and Dynamic Behaviour of Landslides and System Analysis”. WG 4.2.1 and WG 4.2.2, however, are still in the process of establishment. We have received a proposal of Prof. Dr. Jose Bittencourt for the establishment of a new WG on Pavement Mapping. This proposed WG will be merged with the WG 4.2.1 under a new chair. Jose Bittencourt has started recently to invite people to join the new working group. He has arranged a first meeting of WG members at the 6th International Symposium on Mobile Mapping Technology (MMT’09)
which took place in Presidente Prudente, Sa˜o Paulo, Brazil, from 21 to 24 July 2009 (see http://www4.fct. unesp.br/simposios/mmt09/ingles/). Dr. Gethin Roberts is very active in FIG and will take over as a chair of FIG Commission 6 in 2010. He found it to be difficult to work actively in IAG as well. Therefore he has asked us to be replaced by someone else in his role of Vice-Chair of SC4.2 and Chair of WG 4.2.2. The secretary of SC 4.2 Dr. Michaela HaberlerWeber has left Vienna University of Technology at the end of last year and does not want to continue her work in our commission. Hopefully a new Secretary can overtake her role soon. In the last 2 years SC 4.2 was involved in the organization of the following conferences: 1. 8th Conference on Optical 3-D Measurement Techniques July 9–12, 2007 in Zurich, Switzerland http://www.photogrammetry.ethz.ch/optical3d/ 2. 4th IAG Symposium on Geodesy for Geotechnical and Structural Engineering and 13th FIG Deformation Measurement Conference May 12–15, 2008 in Lisbon, Portugal http://measuringchanges.lnec.pt/ 3. 9th Conference on Optical 3-D Measurement Techniques July 1–3, 2009 in Vienna, Austria http://info.tuwien.ac.at/ingeo/optical3d/o3d.htm 4. 6th International Symposium on Mobile Mapping Technology (MMT’09) July 21–24, 2009 in Presidente Prudente, Sa˜o Paulo, Brazil http://www4.fct.unesp.br/simposios/mmt09/ingles/ The established WGs have supported these four conferences and were represented by WG members and/or chairs. The sub-commission is also involved in the organization of the following upcoming meetings: 1. 5th IAG Symposium on Geodesy for Geotechnical and Structural Engineering and 14th FIG Deformation Measurement Conference November 2–4, 2011 in Hong Kong, P.R. China http://dma.lsgi.polyu.edu.hk/ 2. 10th Conference on Optical 3-D Measurement Techniques 2011 in Zurich, Switzerland The sub-commission will continue its work in the next 2 years and will encourage the active members to
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participate in the upcoming meetings to present and discuss their research. In the following sections the reports of the activities of WG 4.2.3 and WG 4.2.4 are presented.
89.2
Activity Report of IAG WG 4.2.3 “Application of Artificial Intelligence in Engineering Geodesy”
In the last years, Artificial Intelligence (AI) has become an essential technique for solving complex problems in Engineering Geodesy. AI is an extremely broad field – the topics range from the understanding of the nature of intelligence to the understanding of knowledge representation and deduction processes, eventually resulting in the construction of computer programs which act intelligently. Especially the latter topic plays a central role in applications. In 2008, the Working Group 4.2.3 was reorganized and extended from “Application of Knowledge-Based Systems” to “Application of Artificial Intelligence”. The reason behind this restructuring was to open the working group to researchers working on all sorts of problems concerning AI-techniques and engineering geodesy. Current applications using AI methodologies in engineering geodesy are: – Geodetic data analysis – Deformation analysis – Navigation – Deformation network adjustment – Optimization of complex measurement procedures The work of the WG 4.2.3 in 2008 can be summarized as follows: – Networking and knowledge exchange between members of the WG. – Organisation of a first meeting (in form of an international workshop): 1st Workshop on Application of Artificial Intelligence in Engineering Geodesy (AIEG 2008), December 1, 2008, Vienna, Austria Proceedings of the workshop may be downloaded at: http://info.tuwien.ac.at/ingeo/Downloads/ AIEG2008_Proceedings.pdf – Public relation in form of an website: http://info.tuwien.ac.at/ingeo/sc4/wg423/wg_423. html
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In 2009 the WG has organized a special session at the 9th Conference on “Optical 3-D Measurement Techniques” in Vienna. The session “Applications of Artificial Intelligence in Optical 3D-Systems” has presented research work of different origin and content, e.g. basic research, application oriented research, etc. Furthermore, the WG has begun to plan the “2nd International Workshop on Application of Artificial Intelligence in Engineering Geodesy – AIEG” which will be organized at the end of 2009/beginning of 2010. For an easy communication within the WG a central data exchange unit (ftp-server) and a mailing list have been installed. The existing website will be extended to a WIKI. Figure 89.1 shows a simplified system architecture of the developed deformation assessment component and Fig. 89.2 an example for the visualization of a deformation pattern.
Fig. 89.1 Simplified system architecture of the deformation assessment component (after Lehmann et al. 2007)
Fig. 89.2 Visualization of deformation pattern
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Activity Report of IAG WG 4.2.4 “Investigation of Kinematic and Dynamic Behaviour of Landslides and System Analysis”
In the frame of the working group the participants laid a great stress on multi-scale monitoring landslide prone areas (Kahmen et al. 2007). For the investigations several test site were used in China (Baota test site), in Hungary (High Danube banks at Dunaf€oldva´r and Dunaszekcso˝), in Greek (Touzla overpass, Kristallopigi landslide, Basilikos landslide, Gkrika Cuts, Prinotopa site, Anthohori entrance, the Big Cut), in Italy (Corvara test site). The different types of landslide areas make it possible to investigate the influence of geological, geomorphological, hydrological, meteorological, etc. factors and their role in triggering landslides. All the participants collect their data in GIS (see e.g. Lakakis et al. 2009b; Mentes 2008a, b) and use these data to develop Spatial Decision Support Systems (SDSS) (e.g. Lakakis et al. 2009a) and Early warning systems. Such systems consist of the following main modules: – Extended multi-sensor deformation measurement system based on geodetic, geophysical, geotechnical, hydrological, meteorological instrumentation. – A knowledge-based system that analysis all data and makes a rough risk assessment, triggering the alarm for possible immediate failures. – An alarm system to ensure the instant/direct authority action. – The overall assessment of the results and the final decision level due to geo-informatics solutions. The extended multi-sensor deformation measurement system consists of terrestrial geodetic and geotechnical measurements completed by InSAR technique. This latter is used for large-scale detection of landslide prone areas as well as for deformation measurements of the investigated landslide area. This complete measurement system is very suitable for the investigation of the kinematic behaviour of landslides and together with other (e.g. hydrological, meteorological, etc.) parameters for study the dynamics of landslides (Mentes 2008a). In the frame of a close cooperation between the Wuhan University of Technology (China) and the University of Braunschweig (Germany) the Baota test site (Riedel and Heinert 2008; Riedel and Walther 2008) and in a co-operation between University of Braunschweig and the Geodetic
and Geophysical Research Institute of the Hungarian Academy of Sciences in Hungary the Dunaf€oldva´r test site is monitored by terrestrial and InSAR measurement techniques. Berlin University of Technology, Institute of Applied Geosciences, Department of Hydrogeology and Bureau of Applied Geoscientific Remote Sensing takes also intensively part in the investigation of the Dunaf€oldva´r test site using remote sensing data. In the frame of the latter cooperation the role of tectonic movements in triggering of slope failures was also revealed (Mentes et al. 2008). In this period of the activity of the working group the most characteristic test site was the high loess bank of the Danube at Dunaszekcso˝ in Hungary. The high bank on this area was sliding slowly with increasing velocity since September of 2007 till 12 of February 2008. On this day there was an abrupt sliding. About 500,000 m3 loess was sliding toward to the river Danube (Figs. 89.3–89.6). The whole sliding process was monitored. The study of the movement is a good possibility to understand the kinematics and dynamics of the slope. The monitoring will be continued in the ´ jva´ri et al. future to study the after-sliding processes (U 2008, 2009). The University of Tessaloniki developed very intensively Spatial Decision Support Systems and applied it on several test sites (Lakakis et al. 2009a, b). The Institute of Geodesy and Geophysics of the Vienna University of Technology works on development of multi-sensor measurement systems (Kahmen et al. 2007) and in co-operation with the Geodetic and Geophysical Research Institute of the Hungarian Academy of Sciences develops measurement methods
Fig. 89.3 Landslide at High Danube banks at Dunaf€oldva´r and Dunaszekcso˝
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Fig. 89.6 New landslide prone area in Dunaszekcso˝
Fig. 89.4 Beginning of landslide in 2007
and their mathematical background for detecting very small displacements (Mentes 2008b, c). The working group participants are in connection with each other via internet and if it is necessary a workshop will be organised in spring of 2010 to discuss the problems and results.
89.4
Fig. 89.5 Landslide after February 12, 2008
Concluding Remarks and Outlook
This paper presented the activity report of the IAG Sub-Commission 4.2. in the period of 2007–2009. Two Working Groups, i.e., WG 4.2.3 on “Application of Artificial Intelligence in Engineering Geodesy” and WG 4.2.4 on “Investigation of Kinematic and Dynamic Behaviour of Landslides and System Analysis”, have worked successfully in the reported period. Their detailed report is described in the paper. The activities of these two groups shall be continued in the upcoming years. Two other planned working groups, i.e., WG 4.2.1 on “Measurement Systems for the Navigation of Construction Processes” and WG 4.2.2 on “Dynamic Monitoring of Buildings”, could not be successfully launched in the reported period. A decision has to be made if other working groups should be established instead of the two proposed ones. One proposal on a new working group on “Pavement Mapping” has been received. Now it has to be decided if such a group can be established and start their work successfully. Finally it can be said that the SC 4.2 want to continue its work in the upcoming years.
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References Kahmen H, Eichhorn A, Haberler-Weber M (2007) A multiscale monitoring concept for landslide disaster mitigation. In: Tregoning P, Rizos C (eds) Dynamic planet monitoring and understanding a dynamic planet with geodetic and oceanographic tools. IAG symposium, Cairns, 22–26 Aug 2005 Series, International Association of Geodesy Symposia, vol 130, pp 769–775 Lakakis K, Charalampakis M, Savaidis P (2009a) A spacial decision support system for highway infrastructure. Fifth international conference on construction in the 21st century (CITC-V), Collaboration and Integration in Engineering, Management and Technology, Istanbul, 20–22 May 2009, pp 1–8 Lakakis K, Charalampakis M, Savaidis P (2009b) A landslide definition by an integrated monitoring system. Fifth international conference on construction in the 21st century (CITC-V), Collaboration and Integration in Engineering, Management and Technology, Istanbul, 20–22 May 2009, pp 1–8 Lehmann M, Reiterer A, Kahmen H (2007) Deformation classification in high density point clouds. Optical 3-D measurement techniques VIII, vol I, Zurich Mentes G (2008a) Investigation of different possible agencies causing landslides on the High Loess Bank of the River Danube at Dunaf€ oldva´r, Hungary. Proceedings of the measuring the changes, 13th FIG international symposium on deformation measurements and analysis, 4th IAG symposium on geodesy for geotechnical and structural engineering, LNEC, Lisbon, CD, 12–15 May 2008, pp 1–10 Mentes G (2008b) Investigation of micro-movements by borehole tiltmeters on the High Loess Bank of the River Danube at Dunaf€oldva´r in Hungary. Proceedings of the INGEO 2008 – 4th International conference on engineering surveying,
G. Retscher et al. Slovak University of Technology, Bratislava, 2008, ISBN 978-80-227-2971-0, p 11 Mentes G (2008c) A new method for dynamic testing of accelerometers. INGEO 2008 – 4th International conference on engineering surveying, Slovak University of Technology, Bratislava, 2008, ISBN 978-80-227-2971-0, p 10 ´ jva´ri G Mentes G, Theilen-Willige B, Papp G, Sı´khegyi F, U (2008) Investigation of the relationship between subsurface structures and mass movements of the High Loess Bank along the River Danube in Hungary. J Geodyn. doi:10.1016/j.jog.2008.07.0005 Reiterer A, Egly U (eds) (2008) Application of artificial intelligence in engineering geodesy. Proceedings of the first workshop on AIEG 2008, 116 pp. http://info.tuwien.ac.at/ingeo/ Downloads/AIEG2008_Proceedings.pdf Riedel B, Heinert M (2008) An adapted support vector machine for velocity field interpolation at the Baota Landslide. In: Reiterer A, Egly U (eds) Application of artificial intelligence in engineering geodesy. Proceedings of the first workshop on AIEG 2008, pp. 101–115. http://info.tuwien.ac.at/ ingeo/Downloads/AIEG2008_Proceedings.pdf Riedel B, Walther A (2008) InSAR processing for the recognition of landslides. Adv Geosci 14:189–194 ´ jva´ri G, Mentes G, Theilen-Willige B (2008) Detection of U landslide prone areas on the basis of geological, geomorphological investigations – a case study. Proceedings of the measuring the changes, 13th FIG international symposium on deformation measurements and analysis, 4th IAG symposium on geodesy for geotechnical and structural engineering, LNEC, Lisbon, CD, 12–15 May 2008, pp 1–9 ´ jva´ri G, Mentes G, Ba´nyai L, Kraft J, Gyimo´thy A, Kova´cs J U (2009) Evolution of a bank failure along the River Danube at Dunaszekcso˝, Hungary. Geomorphology 109:197–209. doi:10.1016/j.geomorph. 2009.03.002
A Fixed-s Digital Representation of a Random Scalar Field
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K. Becek
Abstract
In this paper, “fixed-s” refers to a stochastic process that has a constant standard deviation. “Digital representation” refers to a discrete representation of a stochastic process. A good example of a digital representation of a stochastic process field is the earth’s surface, with elevation as the random scalar, commonly referred to as a digital terrain model (DTM). It is widely know that the roughness of terrain is a factor that inhibits DTM accuracy. This implies that accuracy of DTM varies from pixel to pixel; it is not fixed across the landscape because it varies according to the terrain’s vertical variations. Such a nonfixed-s DTM is not a proper source of topographic data. However, its regular character (pixels are of equal size) is required as a prerequisite for procedures involved in the DTM data acquisition, processing and dissemination. The data acquisition methods may include InSAR and LiDAR technology, and raster techniques for producing a picture on a screen. In this paper, we discuss the possibility of modelling any type of terrain using a DTM which would be characterised by a limiting or fixed standard deviation. A starting point for these considerations is a recently published result of investigations of estimates of the target-induced error of the Shuttle Radar Topography Mission (SRTM) dataset. This target induced-error model connects the standard deviation of the disparities (DTM versus reference data), pixel size and slope of the terrain in an original and straightforward framework. An obvious consequence of such a fixed-s DTM is the variable pixel size. This paper formulates a number of questions regarding the feasibility for such a DTM and potential advantages or disadvantages. For example, it appears that such a pixelvariable but fixed-s arrangement of a DTM would better serve many of its purposes and provide opportunities to increase the efficiency of digital image storing and processing. The discussion includes a potential and required change in the data acquisition strategy and feasibility from systematic sampling to an adaptive sampling of the earth’s surface. A new set of algorithms for such a DTM is required to calculate DTM-derived parameters, including aspect and
K. Becek (*) Environmental Studies Program, Universiti Brunei Darussalam, Jalan Tungku Link, Gadong BE1410, Brunei Darussalam e-mail:
[email protected] S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_90, # Springer-Verlag Berlin Heidelberg 2012
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slope. The proposed fixed-s digital representation of a random scalar field is not restricted in any way to only DTMs. It can be used in many other fields that use digital imagery.
90.1
Introduction
Modelling of a continuous phenomenon using a discrete set of points is routinely performed in almost all branches of sciences and their applications (Ayeni 1982). This operation is necessary for allowing computational studies and particular numerical solutions of real-life problems. This act of modelling is referred to as digitisation or sampling. One of the most important aspects of digitisation is selection of the degree of simplification of the modelled phenomena. This simplification is often referred to as resolution. As a selection criterion for the selection of the degree of simplification, the desired accuracy should be used. However, in many cases of upsampling (decreasing resolution) and downsampling (increased resolution) of a continuous data process, this criterion is ignored, mostly because of a lack of a clear quantitative rule for selection of resolution or sampling frequency or a pixel size. Rather, a rule of thumb is used instead – “higher resolution, more details, better accuracy”. Also some marketing cliche´s are at work mainly exposing the resolution of a digital data product over other parameters of digital imagery. For example, “. . .the pixel size of our imagery is just 61 cm!” In this paper the sampling frequency of a random continuous scalar field (F) versus an accuracy measure (a) of its discrete representation is closely examined. It is well understood that the above dependency, i.e., a1RðFÞ
(90.1)
where R is a certain digital representation of a continuous random field F, a is an accuracy measure, is controlled by the properties of F, and the way F is represented by R. In particular, these properties include the amplitude and frequency of changes occurring in F as well as the pixel size. In other words, the accuracy of a digital representation of a random field F depends on the “roughness” of the field on one hand and on the size of the pixel chosen to represent the field on the other. Naturally, the properties of the field
are beyond any control. Therefore, a key question one may formulate is how to find the pixel size to represent a continuous random field in such a way that the pixels may be used to represent the continuous field with an accuracy that is above an assumed threshold. There are numerous reports of studies developed in various branches of sciences on selecting a suitable pixel size for a given random field. Only few notable examples are referred to here. The issues of digitization of a random field were comprehensively investigated in the last century within the information, telecommunication and signal-processing theories. A solution was found in a form known as the sampling theorem or the Whittaker–Kotel’nikov–Shannon (WKS) sampling theorem or simply sampling theorem. The theorem simply identifies a sampling frequency based on the highest frequency found within the random process being digitised. It states also that the samples may be used for errorless reconstruction of the original process. The theorem applies to not only a one-dimensional case as it was initially proved but can be easily extended for more than one dimension. The theorem works also for non-equidistance samples (Abdul 1977). An example of the application of the sampling theorem in studies of a digital elevation model was provided by Smith and Sandwell (2003). A difficulty in applying the sampling theorem in studies on topography where spatial frequency is an uncommon parameter used lies in the fact that no maximum frequency in a terrain cross-section is known until a DEM is produced, i.e., the frequency may be calculated a posteriori. This is not the case, for example, in a telecommunication channel, i.e., the maximum signal frequency is known a priori. Some progress in overcoming the above difficulties in estimating spatial frequency was made by Borkowski (2002) who showed that the sampling rate for topography may be derived from the slope of terrain. It is worthwhile noting that the sampling theorem-based approach to digitisation of the continuous random field inherently assumes an errorless interpolation of the continuous field, hence no accuracy
90
A Fixed-s Digital Representation of a Random Scalar Field
assessment of the final result was offered. Thus, the fact of only an approximate character of the estimate of maximum spatial frequency and its influence on the accuracy of the digital model of a continuous field was ignored. An underlying idea for this study is a statistical method for calculating the accuracy of a digital representation of a continuous random field proposed in Becek (2008). The method allows for estimating a local accuracy measure based on the variations of the values of the field. In other words, the method may be used to calculate the local accuracy measure of the digitised field using values of the field within a preselected neighbourhood of a point and the size of the neighbourhood (pixel). It may be noted that one of the conclusions from the above is that the uniformly spaced digital representation of a random field leads to inconsistent results in terms of the accuracy of the digital model. This is because the accuracy varies from point to point. Naturally, this is not a welcomed result. In this report we demonstrate that it is possible to construct a digital representation of a continuous random field with an accuracy parameter at each point not larger than a preset limit. A need for this kind of digital representation is one of the fundamental requirements of a digital model. Yet, this requirement has been seamlessly overlooked, for instance, in all reports dealing with the accuracy assessment of DEMs. Many more examples might be given it this respect. The proposed approach to the digitisation of a continuous field may be used in many situations, including image compression techniques and others. This approach may be termed digitisation with a variable sampling step or pixel size, which is controlled by the set accuracy requirement of the model and the local characteristics of the field.
90.2
An Error of Discretisation of a Random Field
727
collection of all possible realisations ZS that might occur over the surface S. To each point p(x,y) on the surface S, a random number taken from the realisation Z is assigned. The collection of points on the surface S and the mapping operation of random numbers to the points are known as a continuous random scalar field. In this report only real values or scalars will be taken as the random numbers. This model may be used to represent any surface, including the earth’s surface, termed here as a continuous elevation model (CEM). A discrete representation of a continuous random field is attained by selecting a finite number of points from surface S. The discretisation may be performed in an orderly fashion leading to a uniformly distributed collection of points. Other possibilities do exist, including a network constructed of triangles, which is known as Delaney triangulation. Any other shape of the smallest element of the discrete model of a random field is acceptable as long as a collection of such elements can be used to cover the entire surface without gaps. Such an element is referred to as pixel. Let, S0 (d) denote a discrete representation of the continuous random field, where d is a parameter related to the area of a pixel. The discretisation requires determining values of pixels. This can be done by averaging the values of the field within each pixel or by selecting any value from within a pixel. The latter method is known as decimation and is seldom utilised in geosciences because it provides a biased estimator of the value for pixels (Becek 2007). Yet, it was used to upsample the SRTM data from a 1-arc-second to 3-arc-second pixel (Becek 2008). It should be noted that the determination of values of pixels is in fact a rounding-off operation in which a range of values found within the pixel is replaced by a single new value. The variation or the seconddegree moment of the round-off operation may be calculated from (90.2) (Gray and Davisson 2003; Smith 2007). Dh=2 Z
s ¼ 2
For the purpose of this study, a random field is defined as a one-dimensional stochastic process that is mapped on a set of points in two-dimensional (2D) space. This can also be expressed in the following manner. Let, pðx; yÞ 2 S R Rbe a point on a surface S representing a fragment of the 2D space. Let, z 2 ZS R be a realisation of a random process, taken from a
h2 pðxÞdx
(90.2)
Dh=2
where Dh is the value range of the field within a pixel, p(x) is the probability density function (pdf) of a random variable h, and h is the value of the field. Assuming a uniform distribution of the field values,
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K. Becek
h within each pixel (90.2) yields an expression for variance of the round-off error as: s2 ¼
Dh2 12
Dh d
(90.4)
Equation (90.4) is a familiar formula representing the slope a of the field at a given point of the field. Substituting dh in (90.3) by (90.4), a final expression for the variance of the error caused by the discretisation can be written as (Becek 2008): s2 ¼
d 2 tan2 ðaÞ 12
d2
(90.3)
The variations of the field values within a pixel can be related to the size of the pixel. Let d be the length of side of a square pixel. Hence, tanðaÞ ¼
error, the maximum pixel size may be calculated as follows:
(90.5)
The above considerations assume that the field is differentiable at every point. Therefore, the slope a is simply a gradient of the field at a given point. In addition, it should be noted that the gradients exist for every point of the field S. The collection of the gradients can be considered as a vector field S0 : rS ¼ ðdz=dx; dz=dyÞover the scalar field S. It is worthwhile noting that (90.5) does not represent the total variance of pixel’s value because the variance of the measurement of the field values should also be considered. For the purpose of this study, however, the variance of the measurements may be assumed as unchanged.
12s2 tan2 ðaÞ
where a is the slope, and s is the assumed maximum variance of the pixel’s value. Equation (90.6) exists for 0 < a < p/2 only. It expresses a commonly accepted fact that a small pixel size is needed to capture highfrequency variations of the field. This rule is followed by photogrammetrists, for instance, who increase the rate of spot elevation data acquisition in rough terrain and decrease the rate for flat topography. The resulting digital representation of a random field possesses an interesting property of adjusted resolution according to the property of the underlying field. Because of this fact, it may be considered as a “better” model of reality in comparison to a representation using pixels of a uniform and arbitrary size. Figure 90.1 shows the relationship between the pixel size and the slope for various levels of the root mean squared error (RMSE) of the discrete field values. The plot allows us to conclude that, in the case of a DTM with the RMSE of elevation at s ¼ 5 m and pixel size of d ¼ 100 m, the topography with slopes less than 10 is over-represented. The opposite is true for slopes larger than 10 in which the topography is under-represented. In both cases the digital representation of the field offers an inconsistently distorted model of the field.
103 ± 0.1m ± 0.5m ±1m ±3m ±5m
A Variance-Constant Representation of a Field
Pixel size (m)
102
90.3
(90.6)
101
100
Equation (90.5) relates the variance of the error of discretisation (s), a property of the field in a form of slope (a), and chosen pixel size (d). Therefore, (90.5) may be used to construct a digital representation of the field S with a constant variance at any given point. As expected, due to variations in the field values, a tradeoff of this constraint would be a variable pixel size. For an assumed maximum variance of the pixel’s
10-1 0 10
101 Slope (degrees)
102
Fig. 90.1 Log–log plot of the pixel size versus slope for various levels of the root mean squared error (RMSE) of the discrete field values. Pointers for reference slope of 10 and s ¼ 5 m are shown
A Fixed-s Digital Representation of a Random Scalar Field
90.4
729
Example
In order to demonstrate the above proposal at work, a DTM covering an area of 1,536 by 1,536 m of land located on the Gold Coast, Australia (27 570 1000 S, 153 180 0800 E), was chosen. The DTM pixel size was 3 by 3 m. Hence, the dimensions of the DTM were 512 by 512 pixels, and the total number of pixels used was 218. The test area is covered by a dry tropical rainforest of various densities. The elevation ranged from 46 to 204 m. The DTM was developed from the 1-m contours, which were interpolated from spot elevations, and the spot elevations. The spot elevations were acquired using the standard photogrammetric method. Due to the vegetation cover the spot elevations were taken as permitted by the visibility of the ground. The standard deviation of the spot elevations was around 0.3 m. However, spot elevations taken from the contour lines may have a larger standard deviation due to interpolation errors. Hence, it is assumed that the standard deviation of the elevation of points taken for the development of the DTM is around 0.4 m. In total, some 72,000 elevation points fairly evenly distributed were used, which translates to about 3 points per 100 m2. Figure 90.2 shows a sun-shadow representation of the DTM and a histogram of the slopes calculated from the DTM. The histogram of slopes allows concluding that the topography of the test site can be classified as rough. A quadtree decomposition (De Berg et al. 2000) of the DTM was performed using a procedure provided by the MathLab program. The procedure at each stage divides the DTM into four 2N 2N (N 0) pixel blocks. Homogeneity of each block is assessed in a reference to a threshold value, which is calculated for the allowed variance of the elevation error from (90.6). Therefore, the maximum allowable elevation difference Dh for a given variance s2 can be estimated from (90.7): pffiffiffiffiffi Dh s 12
(90.7)
Figure 90.3 shows the results of the quadtree decomposition of the DTM for various assumed values (0.5 to 5 m) of the maximum allowable RMSE of the DTM. The density of the grid provides a qualitative assessment of the roughness of the terrain, e.g., denser grid indicates larger slopes (rough terrain) and vice versa.
0.05 Mean slope = 17.3° Standard Deviation = ± 6.6°
0.04
Frequency
90
0.03
0.02
0.01
0
0
10
20 30 40 Slope (Degree)
50
60
Fig. 90.2 A sun-shadow visualisation of DTM used in the experiment (top panel). The pixel size is 3 m by 3 m, and the DTM is 512 by 512 pixels. The elevation range is 46–204 m. Histogram of slopes, which were calculated from the DTM (bottom panel). The mean slope is 17.3 and its standard deviation is 6.6
Beside an accuracy-consistent DTM, this quadtree representation of terrain dramatically reduces the number of pixels in the model. Figure 90.4 shows an empirical relationship between an assumed RMSE of the tested DTM and percentage of pixels needed in the quadtree DTM in relation to the number of pixels in the reference DTM. The relationship depends on the roughness of the terrain. For more moderate topography and for a given accuracy of the model, the percentage would be even smaller.
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Fig. 90.3 A fixed-s quadtree of the test area: (Top-left to bottom-right) s ¼ 0.5 m, s ¼ 1.0 m, s ¼ 3.0 m, and s ¼ 5.0 m
Conclusion 100 Relative Count of Pixels (%)
90 80 70 60 50 40 30 20 10 0 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
RMSE (m)
Fig. 90.4 Percentage of pixels in the quadtree decomposed DTM in relation to pixels in a reference DTM versus the assumed RMSE of the quadtree decomposed DTM. This graph was prepared for the test DTM
In this report, a new technique for a discrete representation of a scalar random field was introduced. The main feature of this discretisation technique is an accuracy-consistent digital representation of the field. This technique allows eliminating an inherent drawback of the equally spaced digital models of random fields, which is the dependency of the accuracy the model on the parameters of the field. A natural way to achieve such an accuracy-consistent digital model is to use the quadtree decomposition of the field. An additional and important ‘byproduct’ of this technique is the dramatic reduction in the number of pixels of various sizes needed to store the digital model of a random field. More studies are needed to develop procedures for deriving geomorphic characteristics from such a quadtree decomposed DTM (Ahmadzadeh and Petrou 2001).
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References Abdul JJ (1977) The Shannon sampling theorem – its various extensions and applications: a tutorial review. Proc IEEE 65:1565–1595 Ahmadzadeh MR, Petrou M (2001) Error statistics for slope and aspect when derived from interpolated data. IEEE Trans Geosci Remote Sens 39(9):1823–1833 Ayeni OO (1982) Optimum sampling for digital terrain models: a trend towards automation. Photogr Eng Remote Sens 48:1687–1694 Becek K (2007) Comparison of decimation and averaging methods of DEM’s resampling. In: Proceedings of the MapAsia 2007 conference, Kuala Lumpur. http://www. gisdevelopment.net/technology/ip/ma07267.htm. Accessed Dec 2008
731 Becek K (2008) Investigating error structure of shuttle radar topography mission elevation data product. Geophys Res Lett 35:L15403. doi:10.1029/2008GL034592 Borkowski A (2002) On the optimal grid cell size for digital terrain models interpolated from contour lines maps, Scientarum Polonorum. Geodesia et descriptio Terrarum 1(1–2):15–22 De Berg M, van Kreveld M, Overmars M, Schwarzkopf O (2000) Computational geometry, 2nd edn. Springer, Heidelberg. ISBN 3-540-65620-0 Gray RM, Davisson LD (2003) An introduction to statistical signal processing. Cambridge University Press, Cambridge Smith JO (2007) Fourier theorems for the DFT. In: Mathematics of the discrete fourier transform (dft) with audio applications [electronic], 2nd edn. W3K, Menlo Park, CA. http://ccrma. stanford.edu/_jos/mdft/Fourier_Theorems_DFT.html Smith B, Sandwell D (2003) Accuracy and resolution of shuttle radar topography mission data. Geophys Res Lett 30(9)
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The Impact of Adding SBAS Data on GPS Data Processing in Southeast of Brazil: Preliminary Result
91
W.C. Machado, F. Albarici, E.S. Fonseca Junior, J.F.G. Monico, and W.G.C. Polezel
Abstract
Nowadays, L1 SBAS signals can be used in a combined GPS+SBAS data processing. However, such situation restricts the studies over short baselines. Besides of increasing the satellite availability, SBAS satellites orbit configuration is different from that of GPS. In order to analyze how these characteristics can impact GPS positioning in the southeast area of Brazil, experiments involving GPSonly and combined GPS+SBAS data were performed. Solutions using single point and relative positioning were computed to show the impact over satellite geometry, positioning accuracy and short baseline ambiguity resolution. Results showed that the inclusion of SBAS satellites can improve the accuracy of positioning. Nevertheless, the bad quality of the data broadcasted by these satellites limits their usage.
91.1
Introduction
In order to improve the reliance in GPS navigation over a wide area, geostacionaries satellites are used to transmit corrections and information about GPS
W.C. Machado (*) Polytechnic School, University of Sao Paulo, Sao Paulo, Brazil Brazilian Institute of Geography and Statistics (IBGE), UESC, Floriano´polis, Brazil e-mail:
[email protected] F. Albarici E.S. Fonseca Junior Polytechnic School, University of Sao Paulo, Sao Paulo, Brazil J.F.G. Monico Department of Cartography, Sao Paulo State University – UNESP, Campus of Presidente Prudente, Presidente Prudente, SP, Brazil W.G.C. Polezel Graduate Program on Cartographic Science, Sao Paulo State University – UNESP, Campus of Presidente Prudente, Presidente Prudente, SP, Brazil
integrity to the users, forming the so-called SBAS (Satellite Based Augmentation System). The existent SBAS are the American WAAS (Wide Area Augmentation System), the European EGNOS (European Geostationary Navigation Overlay Service), the Indian GAGAN and the Japanese MSAS (Multi-functional Satellite Augmentation System). SBAS data are transmitted to the users through the same L1 frequency used by GPS satellites (1,575.42 MHz), in which, information about its position and clock bias are broadcasted. Such information allows including SBAS data into a combined processing with GPS data (Wanninger 2007). Although all these SBAS are available, signals from only one WAAS and two EGNOS satellites can be tracked in the southeast region of Brazil. Table 91.1 summarizes some information about these satellites. The aim of this research is to investigate the impact of adding SBAS data on GPS data processing in the southeast area of Brazil. In order to perform such analysis, GPSeq software (Monico et al. 2006) was
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Table 91.1 Information about SBAS tracked Name AOR-E ARTEMIS ANIK
PRN code 120 124 138
Longitude 15.5 W 21.5 E 107.3 W
Table 91.2 Baselines information SBAS usage EGNOS EGNOS WAAS
Baseline K J I H G F E D C B A
City Marı´lia Tupa˜ O. Cruz Assis Marı´lia Sa˜o Paulo N. Vene´cia N. Vene´cia N. Vene´cia N. Vene´cia Assis
Length (km) 12 9.5 2.8 2.7 1.8 0.45 0.04 0.04 0.04 0.04 0.005
Number of GPS satellites 6 7 9 10 8 7 8 7 8 8 10
SBAS WAAS WAAS WAAS WAAS WAAS WAAS EGNOS EGNOS EGNOS EGNOS WAAS
Table 91.3 SBAS azimuth and elevation angle City
Fig. 91.1 Location of tracked baselines
adapted to permit such a combined processing. Comparisons between GPS+SBAS and GPS-only single point and relative solutions were carried out. Preliminary results showed that the satellites geometry is improved, but the bad quality of the data broadcasted by SBAS satellites limits their usage.
91.2
Data Set
Using Promark500™ and Promark3™ receivers, 11 baselines were tracked between April 4 and June 28 of 2009 in six different cities and states of southeast region of Brazil: Nova Vene´cia (Espı´rito Santo State) and Sa˜o Paulo, Marı´lia, Tupa˜, Osvaldo Cruz and Assis (Sa˜o Paulo State). Figure 91.1 shows the localization of these cities. It can be observed that the data set comprises near the extreme east, north and south boundaries of the Brazilian southeast region. Table 91.2 shows the location, the baseline length and the number of simultaneous GPS and SBAS satellites tracked in each baseline. Six baselines present their length shorter than 1 km and five with length between 1 and 12 km. Baselines
N. Vene´cia Sa˜o Paulo Marı´lia Assis Tupa˜ Osvaldo Cruz
PRN 120 Azimuth 55 57 62 62 62 62
Elevation 54 46 43 42 43 42
PRN 138 Azimuth 278 283 284 284 284 284
Elevation 13 19 21 22 22 23
with different lengths were surveyed to make possible an assessment on distance-dependent errors behavior. Even thought it can be possible to track PRN 120 and PRN 138 simultaneously in the southeast region of Brazil, there was no data from these two satellites recorded by the two receivers in all tracked baselines. Table 91.3 presents the mean azimuth and elevation of SBAS 120 and 138 in the region where the data were recorded.
91.3
SBAS Broadcast Ephemeris Evaluation
In order to evaluate the behavior of the broadcast ephemeris elements from PRN 120 and PRN 138, geocentric distances and satellite clock bias were analyzed from the data tracked during the period between June 22 and June 26, as shown in Figs. 91.2 and 91.3, respectively. The behavior of PRN 120 geocentric distances is quite different from that of PRN 138, suggesting that
91
The Impact of Adding SBAS Data on GPS Data Processing in Southeast of Brazil Interpolated
138
Geocentric distance (km)
120
Geocentric distance (km)
42900
42700
42500
42300
22
23
24 June day
25
120
138
60 0 –60 Dts (m)
42176 42174 42172 42170 42168 418350
419350
420350 ToW (s)
421350
26
Fig. 91.2 PRN 120 and PRN 138 geocentric distance
–120 –180 –240 22
23
24 June day
25
26
Fig. 91.3 PRN 120 and PRN 138 clock bias
there are discontinuities between days in the predicted orbit of the first satellite. Figure 91.3 shows that the clock bias broadcasted by PRN 120 had a constant value of about 286 m during all the period of this analysis, while at PRN 138 it varied from –10 to +22 m.
91.4
Broadcasted
42178
Fig. 91.4 Interpolated broadcasted orbits (PRN 120)
42100
–300
735
GPS+SBAS Data Processing
GPSeq software was adapted to process SBAS data together with GPS data. GPSeq is an academic GNSS software for short baseline data processing under development at Universidade Estadual Paulista
(Unesp) (Monico et al. 2006). LAMBDA (Least squares Ambiguity Decorrelation Approach) method is used to solve the ambiguities to their integer number (Teunissen 1993; De Jonge and Tiberius 1996). SBAS data are transmitted in the same L1 frequency as GPS, hence no adaptation was needed neither in the functional mathematical model, which employ carrier-phase and pseudorange double differences as observables, nor in stochastic mathematical model. Instead of transmitting keplerian elements for satellite coordinates computation, SBAS satellites broadcast, in a predefined time interval, their Cartesian coordinates, velocities and accelerations. In such situation, as well as for GLONASS, a fourth-order Runge–Kutta numerical integration can be applied. Nevertheless, in this research, a five order polynomial interpolation model was employed. This model is represented by Horemuz and Anderson (2006): pðtÞ ¼ a1 tn þ a2 tn1 þ ::: þ an t þ an1
(91.1)
in which: p is the coordinate to be interpolated, t is the time, ai (i ¼ 1, n + 1) are the coefficients of the model and n is the polynomial order. The quality of the interpolation method was measured through the comparison between interpolated and broadcasted satellite coordinates. Figures 91.4 and 91.5 show the temporal behavior of geocentric distance computed by both interpolated and broadcasted coordinates from PRN 120 and 138, respectively. It can be seen that the interpolation method used fits quite well the orbit of SBAS satellites, with maximum difference in geocentric distance of the order of
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Geocentric distance (km)
Interpolated
Broadcasted
Table 91.4 PDOP improvement
42159,80
Improvement (%) 42159,60
Max Min Average
I 19 12 16
II 14 8 12
III 8 4 6
I ¼ GPS + 120 + 138, II ¼ GPS + 138 and III ¼ GPS + 120 42159,40
42159,20 418350
419350
420350 Tow (s)
GPS+138
421350
GPS
40
2.5 cm. Hence, applying this model of interpolation does not deteriorate the data processing quality.
91.5
Data Processing and Results
GPS-only and combined GPS+SBAS solutions were compared by processing the data in a single point positioning mode using C/A code pseudoranges as well as in a relative positioning mode with L1 code and carrier-phase. The aim of the first type of processing is to analyze the impact over satellite geometry and accuracy of the positioning, while the second had the purpose of assessing the quality of the baseline ambiguity resolution.
91.5.1 Impact over PDOP The results presented in this section refer to a station in which both PRN 120 and PRN 138 satellites were tracked simultaneously, allowing an investigation about the improvements in PDOP caused by different combination of satellites. Assuming the GPS-only solution as reference, Table 91.4 shows the improvement produced by the possible combinations between the GPS and SBAS satellites. It can be noted that the inclusion of SBAS satellites improves the geometry of satellites, as expected. In theory, it means that more accurated positioning could be achieved. In addition, the combination GPS + 120 + 138 gives the best improvement while the combination GPS + 120, the worst.
3-D Accuracy (m)
Fig. 91.5 Interpolated broadcasted orbits (PRN 138) 30
20
10
0 418350
419350
420350 Tow (s)
421350
Fig. 91.6 Three-dimensional accuracy of single point positioning (GPS + 138 and GPS-only)
91.5.2 Impact over Absolute Positioning Cartesian coordinates computed by a home made single point positioning software were compared with the results of the on-line Precise Point Positioning (PPP) application provided by IBGE (Brazilian Institute of Geography and Stathistics) (IBGE 2009). Three-dimensional accuracy of station coordinates obtained by processing GPS + 138 data (black line) and GPS-only data (gray line) is shown in Fig. 91.6. It can be observed that the three-dimensional accuracy produced by GPS-only solution is approximately 16 m better than GPS + 138 over 75% of time. However, it is important to report that it was the worst case among all tests carried out. Most of results presented differences of up to 2 m. Furthermore, the improvement on GPS-only solution, since epoch 975, occurred when a low elevation GPS satellite stopped being tracked. Figure 91.7 shows the three-dimensional accuracy for the same station, but in this case with
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The Impact of Adding SBAS Data on GPS Data Processing in Southeast of Brazil GPS+120+138
737
GPS
GPS+120
GPS+SBAS 18%
0%
70000
Success 3-D Accuracy (m)
Fail
60000 82%
100
Fig. 91.8 Ambiguity resolution success
50000
40000
F
K
650 550
420350 ToW (s)
421350
Fig. 91.7 Three-dimensional accuracy of single point positioning (GPS + 138 + 120 and GPS + 120)
GPS + 120 + 138 (black line) and GPS + 120 (gray line) data processing. Comparing the accuracy shown in both Figs. 91.6 and 91.7, it can be noted that the results became quite worse when PRN 120 was included, deteriorating the accuracy by more than 66 km. At this region, the PRN 138 satellite elevation varies from 13 to 23 . Thus, its observables are strongly affected by atmospheric and multipath effects, and their inclusion on data processing turns the quadratic form of the residuals 14 times worse. When data from PRN 120 are inserted such value goes to 106, showing that the mathematical model is corrupted by bundle errors. In fact, as shown in Fig. 91.3, the broadcasted positions may not fit the real position of this satellite.
Improvement %
419350
450 350 250 150 50 –50 –150
Ambiguity resolution is the key for precise GNSS positioning. However, solving them to the wrong integer number will produce coordinates with good precision but with lack of accuracy. Figure 91.8 shows a comparison between GPS and GPS+SBAS ambiguity resolution success. The inclusion of SBAS data decreased the success of ambiguity resolution by 18%. The ambiguity resolution has failed for baselines C and D, in which PRN 120 was the SBAS satellite used.
240
480 720 Seconds
960
1200
50 30 10 –10 –30 –50
91.5.3 Impact over Relative Positioning
0
Fig. 91.9 Best and worst ratio improvement
Mean ratio difference
30000 418350
A
B
E
F
G H Baseline
I
J
K
Fig. 91.10 Difference between ratio mean value of GPS +SBAS and GPS-only solutions
During the baseline estimation process the integer values of the ambiguities were analyzed epoch by epoch. In such case, 67% of the baselines processed had their ambiguities solved correctly in the first epoch for both GPS-only and GPS+SBAS data processing. However, the baseline K GPS-only solution had their ambiguities solved 2.5 min before than GPS+SBAS,
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showing that the combined GPS+SBAS can slow down the ambiguity resolution process. Due to the fact that in most of the cases the ambiguity resolution was achieved near the same epoch by both GPS-only and GPS+SBAS solutions, an analysis on ratio test (Teunissen 1998) was carried out. Figure 91.9 presents the behavior of the improvement with the addition of SBAS satellites over the ratio for the best (baseline F) and worst (baseline K) cases. Figure 91.9 shows that a short baseline solution can have its ratio improved by more than 600% if SBAS satellites data are processed together with GPS, as for baseline F. On the other hand it can also turn the ratio worse, as it was the case of baseline K. Note that the worst result occurs on the longest baseline and with less GPS satellites availability. The last case may be the main reason. In order to present a general result about the impact of adding SBAS satellite over the ratio, difference between the ratio mean value of GPS+SBAS and GPS-only solution of each baseline that had their ambiguities correctly solved was computed and analysed. When SBAS satellite was included, baselines B and F had their ratio mean value improved by approximately 200 and 500%, respectively. Although seven baselines presented worse results, only baselines A showed significant difference, resulting in a decreasing of about 72% in the ratio mean value. It is important to stand out that similar results over ambiguity resolution were obtained when the SBAS satellites were introduced as double difference base satellite.
91.6
Comments and Conclusions
Experiments on combined processing between SBAS and GPS data, either on absolute and relative positioning, in a southeast area of Brazil were carried out. Results showed that the inclusion of SBAS data improves the geometry of satellites, as expected. However, either on three-dimensional accuracy of absolute positioning or on baseline ambiguity resolution, the inclusion of SBAS data did not offer significant improvements.
It was shown that PRN 120 (EGNOS) ephemeris information presented a constant clock bias during the period of the data tracking, as well as a variation on orbital movement much larger than PRN 138 (WAAS), indicating discontinuities between days. When data from PRN 120 were used, the accuracy of absolute positioning became worse and some baselines had their ambiguities resolution failed, probably due to the ephemeris quality. It can be concluded that the inclusion of SBAS data can produce better solutions, however, due to bad quality of data broadcasted by these satellites the positioning using these data were degradeted. It is important to stand out that this paper presents the first experience employing GPS+SBAS in this area. Thus, deeper research must be conducted to produce more solid conclusions. Further studies can be conducted using another source of SBAS orbit. Acknowledgements IBGE for supporting his PhD Program at EPUSP – PTR EPUSP – PTR for supporting a part of the surveys FCT – UNESP for allowing using GPSeq software Alezi Teodoline for the receivers used in Sa˜o Paulo State Deborah Valandro de Souza for data tracking at Nova Vene´cia
References De Jonge P, Tiberius CCJM (1996) The LAMBDA method for integer ambiguity estimation: implementation aspects, T.U. Delft – internal report, Delft Horemuz M, Anderson JV (2006) Anderson polynomial interpolation of GPS satellite coordinates. GPS Solut 10:67–72 IBGE (2009) Posicionamento por Ponto Preciso (PPP) – Geodesy coordination. Available at http://www.ibge.gov.br/ home/geociencias/geodesia/ppp/default.shtm. Accessed 25 Aug 2009 Monico JFG, Souza EM, Polezel WGC, Machado WC (2006) GPSeq manual. Available at http://gege.prudente.unesp.br. Accessed 16 Apr 2007 Teunissen PJG (1993) Least-squares estimation of the integer GPS ambiguities. Invited lecture, section IV theory and methodology. In: IAG General Meeting, Beijing, China Teunissen PJG (1998) GPS carrier phase ambiguity fixing concepts. In: Teunissen PJG, Kleusberg A (eds) GPS for geodesy, 2nd edn. Springer, Berlin, pp 319–388 Wanninger L (2007) Combined processing of GPS, GLONASS, and SBAS code phase and carrier phase measurements. In: ION GNSS 2007, Proceedings of ION GNSS 2007. Fort Worth, USA, pp 866–875
First Results of Relative Field Calibration of a GPS Antenna at BCAL/UFPR (Baseline Calibration Station for GNSS Antennas at UFPR/Brazil)
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€pfler, and B. Heck S.C.M. Huinca, C.P. Krueger, M. Mayer, A. Kno
Abstract
The precise and accurate determination as well as the usage of individual calibration values of GNSS (Global Navigation Satellite Systems) antennas are of fundamental importance for state-of-the-art GNSS positioning at millimeter accuracy level, especially concerning the precise determination of the height component. Calibration values can be determined in various ways (relative vs. absolute, chamber vs. field). Within the transnational research project “PROBRAL: Precise positioning and height determination by means of GPS – Modeling of errors and transformation into physical heights”, the first relative antenna calibration field of Latin America was established. One main goal of this cooperation was to establish a relative receiving antenna calibration field for GNSS instrumentations on the roof top of the “Astronomical Laboratory Camil Gemael” called BCAL/UFPR (http://www. lage.ufpr.br/), situated on the Polytechnic Campus of the Federal University of Parana´ in Curitiba (Brazil). The status of the calibration field as well as recent results are going to be presented within this paper, e.g. a TRM22020.00+GP antenna was calibrated in three sessions of 24 h, with a data sampling rate of 15 s. The software WaSoft/ Kalib was used to process the GPS phase observations. Carrier phase center offset values and carrier phase center variation values were determined relatively with respect to a Leica Choke Ring antenna.
S.C.M. Huinca (*) C.P. Krueger Department of Geomatics, Federal University of Parana´, Centro Polite´cnico, CP 19.001, CEP 81531-990 Curitiba, Parana´, Brazil e-mail:
[email protected];
[email protected] M. Mayer A. Kn€ opfler B. Heck Geodetic Institute, Karlsruhe Institute of Technology (KIT), Englerstraße 7, 76131 Karlsruhe, Germany e-mail:
[email protected];
[email protected];
[email protected]
92.1
Introduction
The NAVSTAR GPS (Navigation System with Timing and Ranging – Global Positioning System), popularly known as “GPS”, is able to contribute to various applications. One of them is to determine point coordinates with mm-precision. In order to fulfill this
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goal, systematic errors have to be taken into account. Within the evaluation of GPS measurements, the receiving antenna is a possible source of systematic errors. Within the functional model of GPS data estimation, the location of the frequency-dependant phase center for GPS antennas is divided in two parts: PCO (phase center offset) and PCV (phase center variations), details see Sect. 92.2. The subject “GNSS antenna calibration” has been investigated and analyzed by different international working groups for nearly two decades. Various studies, e.g. on receiver and satellite antenna phase center variations (e.g. Schmid et al. 2007) and on the influence of the dominant reflectors in the vicinity of the receiving antenna (e.g. Dilßner et al. 2008), have been carried out by the GNSS community recently. Due to the continuous improvement of the functional GNSS model (e.g. orbit determination, atmospheric modeling), site-specific effects are playing nowadays a more important role in GNSS positioning. The intention for establishing a GNSS antenna calibration basis in Brazil emerged because there was no antenna calibration service in Latin America existing. The establishment of this calibration basis in Brazil (BCAL/UFPR: Baseline Calibration Station for GNSS Antennas at UFPR) is therefore the first one and was made possible especially due to technology transfer from Germany within the international cooperation PROBRAL: Precise positioning and height determination by means of GPS – Modeling of errors and transformation into physical heights, which was funded by CAPES (Brazil) resp. DAAD (Germany). PROBRAL is a joint project between the Department of Geomatics (DGEOM), Federal University of Parana´ (UFPR), Curitiba (Brazil) and the Geodetic Institute (GIK), Karlsruhe University (TH) resp. Karlsruhe Institute of Technology (KIT), Karlsruhe (Germany). The aim of the research project is to validate and improve the quality of GNSS-based positioning, especially of the height component. The joint project, which started in 2006, is founded by the Brazilian academic exchange service CAPES and by the German academic exchange service DAAD. BCAL/UFPR was established at the Centro Polite´cnico on the campus of Parana´ Federal University in the city of Curitiba, on the roof top of the “Camil Gemael” astronomical observatory (Fig. 92.1), at the
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Fig. 92.1 BCAL/UFPR; antenna calibration field
annex auditory alongside of LAGE (Spatial Geodesy Laboratory) with respect to logistic aspects (e.g. power supply, security) and signal quality at the site (e.g. multipath effects). Due to the fact that, multipath effects can cause significant errors when determining GPS antenna calibration values, these effects have to be taken into account, when choosing an appropriate calibration site location.
92.2
Receiving Antenna Calibration Values
The location of satellite signal reception related to a GNSS antenna is called phase center. Dealing with phase centers, one has to distinguish between the two terms: component-related mechanical phase center and electrical phase center, whereas the electrical phase center is identical to the directionrelated signal reception point. This phase center is in most cases neither located at the antenna’s rotation axis nor necessarily coincident with the geometric center of the antenna. The PCO is a 3d displacement vector between the average frequency-dependant phase center and the antenna reference point (ARP), see P of Fig. 92.2. The PCV are additional systematic displacements of the phase center of the antenna. PCV are elevation-dependent, see Fig. 92.2. The phase center will vary as function of the satellite signal direction change. Ignoring the variations of the phase center
First Results of Relative Field Calibration of a GPS Antenna
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P5
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Astronomical Laboratory Camil Gemael
Auditorium
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Fig. 92.2 Modeling schema, phase center
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P2
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P3
Fig. 92.4 Location of the three pillars of the BCAL/UFPR and the six leveling control points (P1 . . . P6) Reference Antenna
Antenna to be calibrated
92.4
L
Fig. 92.3 Relative field calibration
can lead to significant vertical errors, which can even reach up to 10 cm (e.g. Menge et al. 1998).
92.3
Relative Field Calibration of GNSS Antennas
A short baseline, usually a few meters long, is chosen for the relative field calibration approach for GNSS receiving antennas in order to minimize orbital errors and atmospheric influences (Freiberger 2007). The baseline has to be well defined and determined with high precision. Both sites of the baseline should be affected with minimum multipath effects. Within the relative calibration approach two antennas are used (see Fig. 92.3): The reference antenna is usually of Dorne/Margolin Choke Ring antenna type. The calibration values (PCV, PCO) of the second antenna are determined with respect to the reference antenna relatively. This antenna type is used by the NGS (National Geographic Service, Silver Spring, MD, USA) for relative field calibration (e.g. Mader 1999).
Baseline Calibration Station for GNSS Antennas at UFPR
The research described here, was carried out at Centro Polite´cnico of Federal University of Parana´ in the city of Curitiba (Brazil) as part of a PROBRAL project, where transfer of technology is one main goal. The BCAL/UFPR (Baseline Calibration Station for GNSS Antennas at UFPR) is actually equipped with three pillars (1000, 2000, and 3000) on the roof top of the “Astronomical Laboratory Camil Gemael”. The monumentation is based on a core of steel and concrete with a height of approx. 1.3 m, measured from the base of the block. Based on experiences concerning the establishment and monumentation of geodetic network points gained at the Geodetic Institute of the Karlsruhe University (TH) three stable pillars could be constructed consisting of material with a long expected lifetime. In order to monitor the behavior of the building, a leveling line (control points P1. . .P6, see Fig. 92.4) was established and is observed at least two times per year. Until now, the differences between the leveling runs were considerably small.
92.5
Multipath Analysis of BCAL/UFPR
In order to analyze the multipath impact on potential calibration sites, measurements were carried out using different dual-frequency GPS receivers in 2008. The
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data were collected in several sessions of 24 h with a data rate of 15 s. The software WaSoft/Multipath 3.2 was used to detect and localize carrier phase multipath (Wanninger 2009a). Therefore, various GPS networks consisting of a maximum of five reference stations were analyzed. According to Wanninger and May (2000) and Wanninger (2005) the multipath detection algorithm is based on the following characteristics and assumptions: • Dominant multipath periods range from 10 to 45 min and depend mainly on the distance between reflector and receiver antenna. • Ionospheric refraction is eliminated using the ionosphere-free linear combination.
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• The software WaSoft/Multipath 3.2 was developed to check multipath effects of GNSS reference stations. Due to the location of the receiver antennas (e.g. top of roofs), it could be assumed that all reflectors are situated at low elevation angles. Thus, it is appropriate to expect that satellite signals with low elevation angles (e.g. below 50 ) are affected and signals received from higher elevations (e.g. above 50 ) show little or no multipath effects. • Due to the fact that multipath effects are sitespecific, multipath effects of different network stations are uncorrelated. • In detail, the software is based on undifferenced residuals of the ionosphere-free linear combination
Fig. 92.5 Representative results of multipath case study of four sites (UFPR, pillars 1000, 2000 and 3000)
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First Results of Relative Field Calibration of a GPS Antenna
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L3, calculated for 20 min blocks. Using undifferenced residuals time series, double-differences of simulta- 92.6 Results of the First Relative Field Calibration Experiments neous observations of the highest and a low satellite of one site are calculated and compared based on correlation coefficients to the corresponding First calibration experiments were carried out in order observations collected at other sites of networks to gain calibration values for GPS TRM22020.00+GP consisting of at least three sites. By means of averag- antennas. Based on the above described multipath case ing techniques carrier phase ambiguities, remaining study, pillar 1000 (pillar 2000) was chosen as location tropospheric effects, and the influence of orbit errors for the reference antenna (antenna to be calibrated). The antenna to be calibrated was rotated with the soare reduced. The program WaSoft/Multipath 3.2 needs 24 h called DRB device. DRB was developed by the Technidata, with an interval not bigger than 60 s. In this cal University of Dresden (Germany) to meet the needs case study, RINEX data sets collected with different of scientific experiments, especially in GNSS antenna dual-frequency GPS instrumentations were used. calibration (Frevert et al. 2003). This automatic device Broadcast ephemerides were introduced, too. The net- rotates the antenna to be calibrated scheduled. Within work limits are smaller than 100 km. The coordinates this case study the antennas were rotated within 60 s of the stations are known in the WGS84 reference from north direction (measurement: T ¼ 0 s) to south frame with a precision in the range of a few direction (measurement: T ¼ 15 s). Fifteen seconds later the antenna pointes to west direction (measurecentimeters (Wanninger 2005). A subset of the analyzed network consisting of ment: T ¼ 30 s) and finally at T ¼ 45 s the antenna four stations (UFPR, pillars 1000, 2000, 3000) has points to east direction and is than rotated back to the been selected and WaSoft/Multipath was used to ana- north direction. This cycle is repeated each minute. The collected data was post-processed using the lyze the data in terms of multipath conditions (see software Wasoft/Kalib (Wanninger 2009b), which Fig. 92.5). The used symbols of Fig. 92.5 are relating to multi- was developed to determine GNSS antenna parameters for relative calibration consisting of PCO and PCV. path impact A two-frequency geodetic equipment (Leica • Blank: no observation data. • Dots: no multipath effects detected (standard devi- GPS1200 receiver, Leica Choke Ring antenna (AT504), see Fig. 92.6, left) was used as reference ation based on L3 < 5 mm). • Small squares: minor multipath effects (standard antenna. The antenna to be calibrated (Trimble TRM22020.00+GP) is shown in Fig. 92.6 (right) as deviation based on L3: 5–15 mm). • Completely black: major multipath effects (stan- well as the DRB device. The measurements were dard deviation based on L3 > 15 mm). Figure 92.5 shows highest carrier phase multipath effects resulting from signals with a low angle of incidence from east directions and from an elevation of 0 to 50 . All other signals show minor or no multipath errors. Pillar 3000 seems to be more affected by multipath effects than the pillars 1000 and 2000. Nowadays, all pillars are less affected by multipath due to the fact that obstacles – detected within the above described case study – were eliminated. With this analysis it could be ensured that the BCAL/UFPR was built at an appropriate location for relative calibration of Fig. 92.6 Relative field calibration at the BCAL/UFPR, left: GNSS antennas. reference antenna, right: DRB device and calibration antenna
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Table 92.1 PCO values (north, east and up components) and corresponding standard deviations PCO (mm) Antenna: TRM22020.00+GP (SN 12347) L1 Standard L2 Standard deviation deviation 1st experiment – GPS Day 245-2008 North component 0.1 0.0 0.3 0.1 East component 2.1 0.0 2.7 0.1 Up component 53.0 0.1 60.0 0.2 2nd experiment – GPS Day 247-2008 North component 0.2 0.0 0.5 0.1 East component 2.0 0.0 2.5 0.1 Up component 53.1 0.0 59.9 0.2 3rd experiment – GPS Day 248-2008 North component 0.3 0.0 0.6 0.1 East component 2.0 0.0 2.6 0.1 Up component 52.8 0.0 59.5 0.2
carried out in three sessions of 24 h, with a data recording interval of 15 s. The resulting PCO values (north component, east component, up component) and the standard deviations for L1 and L2 of the antenna TRM22020.00+GP (SN 12347) can be found in Table 92.1. The same antenna shows slightly different PCO behavior for both L1 and L2. The largest variations are on L2 carrier, due to lower transmitted energy (Freiberger 2007). For L1, the largest variation found is 0.2 mm for the experiments 1–3 in the north component. For L2, the largest variation found is 0.3 mm for the experiments 1–3 in the north component. In the up component for L2, the largest variation found is 0.5 mm. Comparing the north, east and up components, for both carriers (L1 and L2), the largest variations show up for the L2 carrier. All results show good repeatability.
92.7
Conclusions and Future Work
First results of relative field calibrations of GPS antennas at BCAL/UFPR (Baseline Calibration Station for GNSS Antennas at UFPR) were obtained. Such values can now be provided for Brazil and all countries of Latin America for the first time. Analyzing the experiment’s results, small variations were found, which proof that the calibration method was applied correctly.
It is intended to do further calibration experiments in different seasons of the year in order to verify the behavior of north, east and up PCO components. In addition, the PCV are going to be analyzed. Experiments using absorber material in order to reduce pillar-reflected multipath are planned. Investigations on the influence of meteorology on phase multipath effects will be carried out, too. Acknowledgements PROBRAL – the recent cooperation between the Geodetic Institute, University Karlsruhe (TH) resp. Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany and the Department of Geomatics, Federal University of Parana´, Curitiba, Brazil – is funded by the German academic exchange service DAAD and the Brazilian academic exchange service CAPES. We thank Prof. L. Wanninger (TU Dresden, Germany) for providing licenses for the WaSoft programs Kalib and Multipath.
References Dilßner F, Seeber G, W€ubbena G, Schmitz M (2008) Impact of near-field effects on the GNSS position solution. International technical meeting, ION GNSS-08, Savannah, Georgia, USA Freiberger J Jr (2007) Investigac¸o˜es para a calibrac¸a˜o relativa de antenas de receptores GNSS. Thesis (Doutorado em Cieˆncias Geode´sicas), Departamento de Geoma´tica, Universidade Federal do Parana´, Curitiba, Parana´, Brazil Frevert V, Nuckelt A, St€ocker D (2003) Beschleunigte Feldkalibrierung von GPS-Antennen. DGON-symposium POSNAV 2003, Dresden, 18./19.3.2003. Schriftenreihe des Geod€atischen Instituts, TU Dresden (Germany). Heft 3:353–359 Mader G (1999) GPS antenna calibration at the national geodetic survey. GPS Solut 3(1):50–58 Menge F, Seeber G, V€olksen C, W€ubbena G, Schmitz M (1998) Results of absolute field calibration of GPS antenna PCV. In: Proceedings of the 11th international technical meeting of the satellite division of the institute of navigation ION GPS98, September 15–18, Nashville, Tennessee Schmid R, Steigenberger P, Gendt G, Ge M, Rothacher M (2007) Generation of a consistent absolute phase center correction model for GPS receiver and satellite antennas. J Geodesy 81(12):781–798 Wanninger L, May M (2000) Carrier phase multipath calibration of GPS reference stations. In: Proceedings of ION GPS2000, Salte Lake City, UT, 132–144 Wanninger L (2005) Carrier phase multipath detection and localization in GPS reference station networks. http://www. wasoft.de/e/mltp/index.html Wanninger L (2009a) Anleitung Wasoft/Multipath. Ingenierb€uro Wanninger. http://www.wasoft.de Wanninger L (2009b) Anleitung Wa1/Kalib. Ingenierb€uro Wanninger. http://www.wasoft.de
Medium-Distance GPS Ambiguity Resolution with Controlled Failure Rate
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D. Odijk, S. Verhagen, and P.J.G. Teunissen
Abstract
The goal of Network RTK is to provide users with precise ionospheric corrections in order to conduct fast GPS ambiguity resolution and to get cm-level positioning results over medium-distance baselines. In this paper it is shown that a Network RTK user should apply the ratio test with fixed failure rate, having a threshold value that depends on the model at hand, as to test whether the estimated integer solution can be accepted with sufficient more likelihood than the second-best integer solution. Application of the traditional ratio test (with a fixed threshold value) may namely result in too many wrong fixes and consequently severe positioning errors. However, in the paper it is also demonstrated that the ratio test with fixed failure rate should be applied with care, since its correct performance depends on the correctness of the underlying model.
93.1
Introduction
Carrier phase ambiguity resolution is definitely the key to fast and rapid GNSS positioning. It is well known that the distance between rover and reference receiver is a limiting factor for these RTK applications, since for medium baselines of tens of
D. Odijk Department of Spatial Sciences, Curtin University of Technology, Perth, Australia S. Verhagen (*) Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Delft, The Netherlands e-mail:
[email protected] P.J.G. Teunissen Department of Spatial Sciences, Curtin University of Technology, Perth, Australia Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Delft, The Netherlands
kilometers (and longer) differential ionospheric and tropospheric errors may seriously hamper successful ambiguity resolution. To solve for the ionosphere, in many parts of the world permanent GNSS networks have been set up: using the data of these reference stations and based on the smooth spatial behavior of the ionosphere the relative ionospheric delays can be predicted (interpolated) and disseminated to users in the form of ionospheric corrections. The tropospheric errors are usually dealt with by the user by estimating a zenith tropospheric delay parameter. The sketched technique is known as Network RTK, see e.g. (Vollath et al. 2000), or Wide Area RTK (Herna´ndez-Pajares et al. 2004). The success of this technique largely depends on the quality of the ionospheric corrections: when they would perfectly match the true (unknown) ionospheric delays, the probability of successful ambiguity resolution will be close to 1, however if residual ionospheric delays remain present, a lower success rate may be expected. As part of the ambiguity
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resolution procedure in GPS RTK practice usually a ratio test is carried out, in order to test whether the integer solution may be accepted or not. In this paper the performance of the ratio test with fixed failure rate (the so-called FFRatio test, see Teunissen and Verhagen (2009)) is investigated. In contrast to traditional ratio tests, the FFRatio test has the advantage that the user has a priori control of the probability of wrong ambiguity fixing. The FFRatio test will be applied in combination with the LAMBDA method for baselines up to 40 km, for which ionospheric corrections are available from a network of permanent stations. The paper is set up as follows. Section 93.2 reviews the ionosphere-weighted model, which forms the starting point of medium-distance GPS ambiguity resolution. In Sect. 93.3 the applied ambiguity resolution procedure is briefly described, while test results are presented in Sect. 93.4. Finally, in Sect. 93.5 the conclusions of the paper are presented.
93.2
The Ionosphere-Weighted GPS Model
For medium-distance GPS baselines the differential ionospheric delays cannot be neglected, but need to be incorporated into the observation equations. As a more general formulation, we consider the ionospheric delays as stochastic variables, which can be constrained or weighted in the processing. The ionosphere-weighted model of GPS observation equations then reads (Odijk 1999): 2 3 g w w G C M 6 7 Qw 0 E ¼ ¼ 4 i 5; D i i 0 I 0 0 Qi a (93.1) with E(.) the expectation and D(.) the dispersion operator. Vector w contains the observed-minus-computed double difference (DD) code and carrier phase observations, while vector i represents the DD ionospheric delays. At the parameter side, vector g contains the unknown baseline components and -if necessary- tropospheric (zenith) delays, while the unknown integer ambiguities are captured by vector a. Matrices G, C, M and I (with I the identity matrix) capture the functional relations between the
observations and parameters. The precision of the DD code and phase observations is described by the variance matrix Qw, while Qi denotes the variance matrix of the ionospheric observations or constraints. The inverse of this latter matrix can be considered as ionospheric weight matrix, hence the model is referred to as the ionosphere-weighted model. As extremes of the ionosphere-weighted model, if we set Qi ¼ 0, we have the ionosphere-fixed model (the traditional shortbaseline model), in which the ionospheric variables are not unknown but deterministic, whereas if Qi ¼ 1, the ionosphere-float model is obtained, in which the ionospheric variables are considered as completely unknown parameters (the long-baseline model).
93.3
Multi-Carrier Ambiguity Resolution
The integer ambiguities are estimated using the integer least-squares algorithm implemented in the LAMBDA method (Teunissen 1995). Mathematically, the integer estimation can be described using the following projection S : Rn 7!Zn : ǎ ¼ Sð^ aÞ
(93.2)
with aˆ the float ambiguity vector and aˇ the integer vector. Input for the LAMBDA method are the float ambiguity vector plus its variance matrix, denoted as Qaˆ. To accept the integer least-squares ambiguity solution as obtained by the LAMBDA method, the Fixed Failure-rate (FF) Ratio test (Teunissen and Verhagen 2009) is executed, in order to test whether the integer solution is sufficiently more likely than the secondbest solution. The integer least-squares solution aˇ is only accepted if and only if: ka^ ǎk2Qa^ ka^ ǎ2 k2Qa^
bm
(93.3)
with the squared norm defined as kk2Q ¼ ðÞT Q1 ðÞ and where aˇ2 is the second-best integer ambiguity solution and m the critical value of the ratio test. If the FFRatio test is not passed, the integer solution should not be used, and one has to be satisfied with the float solution.
Medium-Distance GPS Ambiguity Resolution with Controlled Failure Rate
Test Results
As source of test data we have used GPS data of the Southern California Integrated GPS Network (SCIGN). This dense permanent GPS network consists of high-end dual-frequency GPS receivers, of which the data are freely available through the Internet (SCIGN 2008). From this network we selected four GPS receivers (i.e. NOPK-VNPS-CLAR-OAT2), acting as permanent network from which ionospheric corrections are generated, see Fig. 93.1. The master reference station of this network is station NOPK. As can be seen from the figure, the maximum distance between these reference stations is approximately 70 km. and OAT2 at 1,113 m. The four GPS receivers are all Ashtech Z-XII3 geodetic receivers. The two SCIGN stations FXHS and WLSN (also equipped with Ashtech Z-XII3 receivers) have been selected as two rovers stations as to test the performance of network RTK. The baseline NOPK-FXHS is 11 km, so we refer to this as a short baseline, while the
Rover station VNPS
62 k m
OAT2
WLSN
km
CLAR
38
km
93.4
Reference station
45
The FFRatio test differs from the more traditional ratio test, see e.g. (Leick 2003), by the choice of the threshold value m. In traditional ratio tests, this value is set to a fixed value (e.g. 1/2 or 1/3), irrespective of the model at hand. Using the FFRatio test, the threshold value depends on the GNSS model at hand (0< m 1). As shown by Teunissen and Verhagen (2009), using the FFRatio test it is guaranteed that the probability of wrong integer estimation (i.e. the failure rate) is lower than a fixed user defined threshold (e.g. 0.001), provided that the underlying models are correct. A user however does not have control of the failure rate when using the ratio test with fixed threshold value. The model-driven threshold value, as we use for this paper, depends on the actual variance matrix of the float ambiguities Qaˆ and a practical approach to determine it is by using look-up tables, generated by simulations, giving the threshold value m as function of a certain integer least-squares failure rate for varying n (number of ambiguities), see Teunissen and Verhagen (2009). The integer least-squares failure rate is here defined as 1 minus the integer least-squares success rate, which can be approximated by computing the success rate of integer bootstrapping based on the conditional standard deviations of the LAMBDAdecorrelated ambiguities (Teunissen 2005).
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FXHS
11 km
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61 km
NOPK (master)
Fig. 93.1 Simulated permanent network and rover stations
38-km baseline NOPK-WLSN is referred to as a medium baseline. GPS data of 1 January 2000 were selected as test data set. This date falls in a period of reasonable high solar activity, where the planetary Kp index, a measure for geomagnetic activity affecting the ionospheric activity, has increased levels (Kp > 4) for some part of the day. The maximum value the Kp can take on, is 9.
93.4.1 Permanent Network Processing The GPS data of the four permanent stations have been processed using in-house software based on a Kalman filter implementation. Because of its recursive character, this Kalman filter is suitable for real-time processing. In the time update of the filter it is assumed that the DD ambiguities are constant from one epoch to the next (provided that no cycle slips occur). The GPS data of the four receivers are processed using a true network solution (taking the mathematical correlations into account) of the ionosphere-weighted model, using the following settings: • Dual-frequency phase and code data (L1, L2, C1, P2) • Data sampling: 30 s; thus 2,880 epochs • Cut-off elevation: 10 ; there are 5–10 satellites in view during the day
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93.4.2 Rover’s Baseline Processing
0.5
0
–0.5
500 1000 1500 2000 2500
epoch
DD ionospheric delay [m]
DD ionospheric delay [m]
Ionospheric corrections have been generated based on the approximate locations of the two rover stations, FXHS (at 11 km from NOPK) and WLSN (at 38 km
from NOPK). The GPS data of these two rover stations are processed as independent baselines both having NOPK as reference. The ionosphere-weighted model serves as basis of the processing of the rover data, having the network ionospheric corrections as ionospheric observations (see (1)). The choice of the ionospheric standard deviation (modeled in Qi) is crucial here. It should match the quality of the ionospheric corrections: a low standard deviation if the corrections closely resemble the true ionospheric delays, and a higher standard deviation if discrepancies can be expected between interpolated and true delays. The discrepancies between true and interpolated delays are basically a function of the state of the ionosphere, the distances between the network stations and the length of the rover baseline. The baseline length dependence is confirmed for the two rover baselines, see Fig. 93.3, which shows the DD ionospheric delays estimated from the GPS data themselves and the residuals after applying the network ionospheric corrections. It can be seen that after applying the corrections for the 11-km baseline the (absolute) residuals are all below 5 cm, but for the 38-km baseline these can be up to 12 cm. It is emphasized that the ionospheric residuals could be computed here since we know the true ionospheric delays (from postprocessing). For truly real-time applications an approach to assess the quality of the ionospheric corrections is to predict the ionosphere not only for the rover but also for a reference station that is within the coverage of the network but not included in the correction generation. For such a station the differences between predicted and true ionospheric delays can be continuously monitored and information on the quality of the ionospheric corrections can be determined and disseminated to rovers. For this paper the ionospheric
0.5
0
–0.5
500 1000 1500 2000 2500
epoch
DD ionospheric delay [m]
• Standard deviation phase: 2 mm, code: 20 cm (undifferenced; in local zenith) • Ionospheric observations with zero values and standard deviations of 10 cm (undifferenced) • Observations are elevation-dependent weighted using an exponential function • No parameterization of receiver positions • Parameterization of one zenith tropospheric delay per network station • Satellite positions based on precise predicted IGS orbits • Integer ambiguities are estimated after 30 epochs of initializing the filter and also after 30 epochs for the integer ambiguities of a newly risen satellite Figure 93.2 shows the DD ionospheric delays, based on the fixed integer ambiguities, for the three ‘baselines’ in the network (all relative to NOPK). From the figures the daily cycle of the ionosphere is clearly visible: during local night time the DD ionospheric delays are small (few cm), while during day time these may rise up to 8–10 mm per km baseline length. Another feature of the ionosphere that can be seen in the figures is the strong spatial correlation between the DD ionospheric delays of the three baselines. The presence of this spatial correlation is crucial for the performance of network RTK since the ionospheric corrections for the rover receivers are interpolated (based on Kriging) from these spatially correlated network DD ionospheric delays.
0.5
0
–0.5
500 1000 1500 2000 2500
epoch
Fig. 93.2 Ambiguity-fixed DD ionospheric delays for all satellites in the permanent network data: (left) 61-km baseline NOPKCLAR, (middle) 62-km baseline NOPK-VNPS, (right) 45-km baseline NOPK-OAT2
0.2 0.1 0 –0.1 –0.2 1000
1500 epoch
2000
0.2 0.1 0 –0.1 –0.2 1000
standard deviations empirically derived from the residuals are 5 mm for the short baseline and 1 cm for the medium baseline (both values are undifferenced and apply to zenith). For both baselines the rover processing will be carried out on an epoch-by-epoch basis, as to investigate the performance of the fastest method of ambiguity resolution. Thus the ambiguities are resolved instantaneously (using LAMBDA and the FFRatio test), based on only the data of the current epoch, without taking any information of previous epochs into account. In all computations the failure rate of the FFRatio test has been set to 0.001 (so for 0.1% of the epochs a wrong integer solution is expected).
93.4.3 Short-Baseline Performance As the traditional GPS processing model for short baselines is based on completely neglecting the differential ionospheric delays, in addition to the ionosphere-weighted processing, the 11-km baseline data are also processed using the ionosphere-fixed model, in absence of using any ionospheric corrections. The same settings are applied to the ionosphere-fixed processing, i.e. instantaneous ambiguity resolution based on LAMBDA and FFRatio test using a fixed failure rate of 0.001. Figure 93.4 (top)depicts the outcomes of the FFRatio test for the ionosphere-fixed epoch-by-epoch processing. For 4.5% of the 2,880
1500 2000 epoch
0.1 0 –0.1 –0.2 500
1000
500
1000
2000
2500
1500 2000 epoch
2500
0.1 0 –0.1 –0.2
2500
1
1
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0
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ratio
500
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diff. DD ion est. & corr. [m]
DD ionospheric delay [m]
500
ratio
Fig. 93.3 Ambiguity-fixed DD ionospheric delays in short (11-km; top) and medium (38-km; bottom) baselines: without any correction (left) and after subtracting the network corrections (right)
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diff. DD ion est. & corr. [m]
Medium-Distance GPS Ambiguity Resolution with Controlled Failure Rate
DD ionospheric delay [m]
93
500 1000 1500 2000 2500
epoch
0.5
0
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epoch
Fig. 93.4 Ratio vs. critical value for the ionosphere-fixed (left) and ionosphere-weighted (right) epoch-by-epoch processing of the 11-km baseline data. If the ratio exceeds the critical value (blue), it is marked red, otherwise green
epochs the FFRatio tests exceeds its critical value, and these rejections are mainly occurring from epoch 2,000 towards the end of the day. For these epochs in Fig. 93.3 (top left) it can be seen that there are large DD ionospheric delays, exceeding 10 cm in absolute value. Further inspection of the integer solutions revealed that for 0.8% of the epochs the FFRatio tests is rejected unnecessarily; the integer solutions for these epochs actually correspond to the correct ones. Even more severe, for 7.0% of the epochs the FFRatio test turned out to accept the wrong integer solution! And this is in complete disagreement with the fixed failure rate set to execute the FFRatio tests of 0.1%. In this context it is emphasized that the performance of FFRatio test depends on the correctness of the underlying models. Hence, these results demonstrate that the traditional ionosphere-fixed model is not suitable for the current baseline as due to the
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93.4.4 Medium-Baseline Performance For the processing of the 38-km baseline the ionosphere-weighted model is applied using the ionospheric corrections from the network assuming an undifferenced standard deviation of 1 cm. Figure 93.5 shows for the 2,880 epochs the ratios vs. their critical values applying the FFRatio test. The FFRatio tests fails for 3.8% of the epochs, which are almost all false alarms, since for 3.5% of the day the FFRatio test is rejected unnecessarily. For 4 epochs during the day the wrong integer ambiguities are estimated, which corresponds to a failure rate of 0.1%. This percentage is in good agreement with the a priori set fixed failure rate in order to execute the FFRatio tests, as it should be when the underlying model is appropriate. For this processing the correct integer solutions were accepted for 96.1% of the epochs. Table 93.1 summarizes the empirical failure rates (wrongly accepted epochs), success rates (correctly
1
ratio
significant differential ionospheric delays for a considerable part of the day. In a next step, the 11-km baseline data have been reprocessed but including the ionospheric corrections using the ionosphere-weighted model and an undifferenced ionospheric standard deviation of 5 mm. Figure 93.4 (right) shows the results of the FFRatio tests for this processing. Comparison with Fig. 93.4 (left) not only demonstrates that the ratios are generally lower, but also the critical values differ between both processing strategies, as they are depending on the choice of model. Now the FFRatio test is rejected for 1.3% of the epochs. Like in the ionosphere-fixed processing, in 0.8% of the cases the FFRatio test is rejected unnecessarily. In contrast to the ionosphere-fixed processing, for this ionosphereweighted processing there are no longer epochs with wrongly accepted integer solutions, and this is in agreement with the fixed failure rate, which accepts at most 0.1% wrong solutions. This result demonstrates that the FFRatio test performs adequately when the processing model underlying is correct. With respect to the success rate (epochs with correctly accepted integer solutions), this is 88.5% for the ionosphere-fixed processing, but increases to 98.7% in case of an ionosphere-weighted processing.
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0.5
0
500
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1500
2000
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Fig. 93.5 Ratio vs. critical value for the ionosphere-weighted epoch-by-epoch processing of the 38-km baseline data. If the ratio exceeds the critical value (blue), it is marked red, otherwise green Table 93.1 Empirical instantaneous ambiguity resolution probabilities (using a fixed failure rate of 0.1%)
Failure Success False alarm
11-km ionospherefixed (%) 7.0 88.5 0.8
11-km ionosphereweighted (%) 0 98.7 0.8
38-km ionosphereweighted (%) 0.1 96.1 3.5
Table 93.2 Empirical instantaneous 95% horizontal and vertical position errors
95% HPE 95% VPE
11-km ionospherefixed 60.6 cm 74.8 cm
11-km ionosphereweighted 1.7 cm 3.9 cm
38-km ionosphereweighted 2.7 cm 6.2 cm
accepted epochs) and false alarm rates (wrongly rejected epochs) of the processing strategies applied to the 11-km and 38-km baselines. It is noted that the ‘correct rejection rate’ then follows from subtracting the first three rates from 100%. Table 93.2 summarizes the empirical 95% Horizontal and Vertical Position Errors (HPE and VPE) corresponding to the processing strategies applied to the 11-km and 38-km baselines. These position errors are computed using the precisely known coordinates of stations FXHS and WLSN. The large position errors for the ionosphere-fixed processing are due to the
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Medium-Distance GPS Ambiguity Resolution with Controlled Failure Rate
many wrong ambiguity fixes. The position errors for both ionosphere-weighted strategies are at sub-dm level. Conclusions
Multi-Carrier Ambiguity Resolution (MCAR) for GNSS-RTK applications should be a combination of LAMBDA and the Fixed Failure-rate (FF) Ratio test. While the LAMBDA method optimizes the success of correct integer estimation, the FFRatio test enables the user to have control of the rate of wrong fixes, this in contrast to the traditional ratio tests with fixed threshold values. However, it was shown that successful performance of this FFRatio Test for short to medium distance GPS applications depends on correctness of underlying model. In case the ionospheric delays cannot be neglected, even for short (~10 km) baselines, accurate network ionospheric corrections should be applied, while their uncertainty should be modeled appropriately through the ionosphere-weighted model. Acknowledgements Professor Peter J.G. Teunissen is the recipient of an Australian Research Council Federation Fellowship (project number FF0883188). This support is greatly acknowledged. The research of Sandra Verhagen is supported
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by the Dutch Technology Foundation STW, applied science division of NWO and the Technology Program of the Dutch Ministry of Economic Affairs.
References Herna´ndez-Pajares M, Juan JM, Sanz J, Oru´s R, Garcı´aRodrı´guez A and Colombo O (2004) Wide Area Real-time Kinematics with Galileo and GPS signals, Proc. ION-GNSS 2004, Long Beach, CA, 21–24 Sep. 2004, 2541–2554 Leick A (2003) GPS satellite surveying, 3rd edn. Wiley, New York Odijk D (1999) Stochastic modelling of the ionosphere for fast GPS ambiguity resolution. Geodesy beyond 2000. The challenges of the first decade, IAG General Assembly, Vol. 121, Birmingham, July 19–30 (1999), pp. 387–392 SCIGN (2008): http://www.scign.org Teunissen PJG (1995) The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation. J Geod 70:65–82 Teunissen PJG (2005) Integer aperture bootstrapping: a new GNSS ambiguity estimator with controllable fail-rate. J Geod 79:389–397 Teunissen PJG, Verhagen S (2009) The GNSS ambiguity ratiotest revisited: a better way of using it. Surv Rev 41 (312):138–151 Vollath U, Deking A, Landau H, Pagels C, and Wagner B (2000) Multi-base RTK positioning using Virtual Reference Stations. Proc. ION GPS-2000, Salt Lake City, UT, 19–22 September 2000
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Toward a SIRGAS Service for Mapping the Ionosphere’s Electron Density Distribution
94
C. Brunini, F. Azpilicueta, M. Gende, A. Arago´n-A´ngel, M. Herna´ndez-Pajares, J.M. Juan, and J. Sanz
Abstract
SIRGAS is responsible of the terrestrial reference frame of Latin America and the Caribbean. To fulfil this commitment it manages a continuously operational GNSS network with more than 200 receivers. Although that network was not planed for ionospheric studies, SIRGAS attempted to exploit it by establishing, in early 2008, a regular service for computing regional maps of the vertical Total Electron Content. This paper describes an effort for developing a new SIRGAS product, concretely, a 4-dimensional (space and time) representation of the free electron distribution in the ionosphere. The working methodology is based on the ingestion of dual-frequency GNSS observations into a global electron density model in order to update its parameters. Preliminary results are presented and their quality is assessed by comparing the electron density computed with the methodology here described and the one estimated from totally independent observations. A preliminary analysis reveals that the performance of the electron density model improves by a factor greater than 2 after data ingestion.
94.1
C. Brunini (*) F. Azpilicueta M. Gende Facultad de Ciencias Astrono´micas y Geofı´sicas de la Universidad Nacional de La Plata, Paseo del Bosque s/n, La Plata 1900, Argentina Consejo Nacional de Investigaciones Cientı´ficas y Te´cnicas, Buenos Aires, Argentina e-mail:
[email protected] ´ ngel M. Herna´ndez-Pajares J.M. Juan J. Sanz A. Arago´n-A Research Group of Astronomy and Geomatics, Department of Applied Mathematics IV, Universitat Polite`cnica de Catalunya, Barcelona, Spain
Introduction
Dual-frequency GNSS observations have become a valuable source of information for a variety of studies related to upper atmosphere, space weather, satellite navigation, telecommunications, etc. Several services for providing GNSS-based ionospheric information have been established by different organizations around the world. Outstanding examples are the Global Ionospheric Maps (GIMs) of the International GNSS Service (IGS) (Dow et al. 2005), which provide a global view of the vertical Total Electron Content (vTEC) distribution (i.e. the electron density integrated along the vertical) with a time resolution of 2-h, based on the GNSS observations from the IGS network (Herna´ndez-Pajares et al. 2009).
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_94, # Springer-Verlag Berlin Heidelberg 2012
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As the International Association of Geodesy (IAG) Sub-Commission 1.3b, SIRGAS (Sistema de Referencia Geoce´ntrico para las Ame´ricas; Sa´nchez and Brunini 2009) is responsible of the terrestrial reference frame of Latin America and the Caribbean according to the International Earth Rotation and Reference System Service (IERS) standards. In order to fulfil its commitment, SIRGAS manages a continuously operational network (SIRGAS-CON) including more than 200 dual-frequency GNSS receivers, which provides the best data coverage currently available in the region. In 2005 SIRGAS initiated a pilot project aimed to exploit the SIRGAS-CON infrastructure for atmospheric studies. Soon after, the Facultad de Ciencias Astrono´micas y Geofı´sicas of the Universidad Nacional de La Plata (Argentina) established an experimental service for computing hourly maps of vTEC for SIRGAS (Brunini et al. 2008). After a validation period of 2 years this experimental service was declared operational and started the distribution, through http://www.sirgas.org, of regional maps of vTEC for SIRGAS. Among other applications, these maps have been used for assessment studies performed in the framework of an International Civil Aviation Organization (ICAO) project for establishing a GNSS augmentation system in the region (http://www. rlasaccsa.com); and for evaluation of the International Reference Ionosphere (IRI) model (http://modelweb. gsfc.nasa.gov/ionos/iri.html). This paper describes an effort for upgrading the ionospheric maps elaborated by SIRGAS, concretely: the development of a methodology for imaging the 4dimensional (space and time) distribution of free electrons in the ionosphere. That methodology relies upon the use of a global electron density model whose parameters are updated using dual-frequency GNSS observations. The second section of this paper describes the models used in this work for the electron density and for the GNSS observations; the third section describes the procedure for ingesting the GNSS observations into the electron density model and updating its parameters; the fourth section presents some preliminary results and assesses its quality by comparing the electron density computed with the methodology here presented and the one estimated from totally independent observations; and the last section provides the main conclusion of the work.
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94.2
Models Used in this Work
94.2.1 Electron Density Model The NeQuick model is used as the basis for describing the global electron density distribution (Radicella and Leitinger 2001). Among other applications, NeQuick is being implemented by the Galileo GNSS for correcting the ionospheric range delay error for single frequency operation. NeQuick describes the vertical profile of electron density by means of the superposition of six semiEpstein functions that represent the lower and upper parts of the E and F1 layers, the lower part of the F2 layer, and the topside. Each function is tied to an anchor point defined by the electron density, Nm , and the height, hm , of the corresponding layer peak, which can be either measured (with ionospheric sounders) or modelled. Besides, the thickness of each layer is modelled with different functions for its lower and upper part and the topside thicknesses function, in particular, depends on the altitude. Despite the existence of three anchor points in the NeQuick formulation, the shape of the profile is dominated by the F2 parameters, Nm F2 and hm F2. In the absence of measurements NeQuick proceeds in the same way as the IRI model and computes the values of these parameters based on the International Telecommunication Union-Radiocommunication Sector (ITU-R) database, also known as Comite´ Consultatif des Radiocommunications (CCIR) database, ITU-R (1997). This database allows computing monthly mean values of the critical frequency, f0 F2, and the transfer parameter, M3000 F2, and from them NeQuick computes Nm F2 using the simple relation Nm F2½electron=m2 ¼ f0 F22 ½MHz=80:6, and hm F2 using the Dudeney formula in connection with M3000 F2 and the f0 F2=f0 E ratio. f0 F2 and M3000 F2 are computed form the ITU-R database following a numerical procedure developed by Jones and Gallet (1965): Oð’; l; tÞ ¼ a0 ð’; lÞ þ
" # J X aj ð’; lÞ cos jtþ j¼1
bj ð’; lÞ sin jt
(94.1)
where O is the parameter to be computed, ’ and l are the geographic latitude and longitude, t is the UT; J is
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Toward a SIRGAS Service for Mapping the Ionosphere’s Electron Density Distribution
the maximum number of harmonics for mapping the diurnal variation (J ¼ 6 for Of f0 F2 and J ¼ 4 for OM M3000 F2); and the geographically dependent Fourier coefficients are: XK aj ð’; lÞ ¼ U G ð’; lÞ; j 0 k¼0 2j;k k XK U G ð’; lÞ; j 1 bj ð’; lÞ ¼ k¼0 2j1;k k
(94.2)
where Uf ;i , i ¼ 0, . . ., 988, for Of , and UM;i , i ¼ 0, . . . , 450, for OM , are numerical coefficients whose values are computed from the ITU-R database. The explicit form of the Gk functions depends on the k index (K ¼ 75 for Of and K ¼ 49 for OM ). The computation of the Gk functions requires the use of the so-called modified dip (modip), m, defined by: tan m ¼ I
pffiffiffiffiffiffiffiffiffiffiffi cos ’
(94.3)
where I is the magnetic dip at 350 km above the Earth’s surface (computed with the spherical harmonics expansion of the geomagnetic field). The ITU-R database was established using observations collected from 1954 to 1958 by a network of around 150 ionospheric sounders with uneven global coverage. It contains monthly mean values of the Uf ;i and UM;i coefficients for low (R12 ¼ 0) and high (R12 ¼ 100) solar activity, R12 being the 12-month running mean value of the monthly mean sunspot number. For intermediate solar activity a linear interpolation is recommended. In summary, NeQuick is a rather complex analytical function for computing the electron density, NNQ , at any given location ’ and l, altitude, h, and time, t, that depends, through (94.1) and (94.2), on T Uf ¼ Uf ;1 Uf ;2 Uf ;988 and UM ¼ UM;1 UM;2 UM;450 ÞT , which are computed from the ITU-R database. Symbolically: NNQ ¼ fNQ ’; l; h; t; Of Uf ; OM ðUM Þ
(94.4)
Consequently, the slant Total Electron Content (sTEC), sNQ , along any given line-of-sight (LOS), Gð’; l; hÞ, can be computed as: ð sNQ ¼
NNQ dg Gð’;l;hÞ
(94.5)
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94.2.2 GNSS Model GNSS observations are processed with the La Plata Ionospheric Model (LPIM) (Azpilicueta et al. 2005). It relies upon the ionospheric observable, LI, which is directly related to the sTEC, sLP, along the LOS from the satellite, S, to the receiver, R: LI ¼ L1 L2 ¼ sLP þ bR þ bS þ u
(94.6)
where L1 and L2 are the carrier-phase observations in two different frequencies depurated from cycle slips and ambiguities; bR and bS are the receiver and satellite inter-frequency biases (IFB); and u accounts for the observational noise effects. LPIM approximates the whole ionosphere with a spherical shell of infinitesimal thickness located at 350 km of altitude and uses the following relation: sLP ¼ secðz0 Þ vLP
(94.7)
where z0 is the zenith angle of the LOS at the point where it crosses the spherical shell (the piercing point), and vLP is the vTEC at the piercing point. The spatial and temporal variability of the vTEC is represented with a spherical harmonics expansion with time dependent coefficients:
vLP ðm; t0 ; tÞ ¼
" # 15 X 15 X Alm ðtÞ cos mt0 þ l¼0 m¼0
Blm ðtÞ sin mt0
Plm ðsin mÞ (94.8)
where m is the modip (see (94.3)), t0 is the LT of the piercing point, t is the UT of the observation, and Plm ðsin mÞ are the Legendre’s associated functions. The time-dependent coefficients are assumed to be constant for every 2h UT interval: Alm ðti Þ ¼ Almi ; 2ði 1Þ
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s^LP sNQ ¼ e
solved with the Least Square method and the estimated ^ , are replaced in (94.6): ^ and b IFB, b R S ^ b ^ : s^LP ¼ LI b R S
(94.10)
94.2.3 Improved Abel Inversion Model The retrieval of electron density profiles from radio occultation measurements is a powerful tool to study vertical ionospheric 4-D structures at global scale, providing reliable density profiles up to the height of the Low Earth Orbiter (LEO) satellite that is in occulting geometry with respect a GPS one. It is possible to derive such electron densities from the change in the delay and bending of the signal path between the GPS and the LEO satellites induced by the inter-medium by means of the classical Abel inversion. Nevertheless, the spherical symmetry assumption lying within the hypothesis of use of the classical Abel inversion is one of the main miss modelings of this technique, since it implies that no horizontal gradients of electron density are considered, i.e. the only electron densities dependence is on height. In Herna´ndezPajares et al. (2000), the separability concept was introduced: the electron density was expressed by considering externally provided vTEC information at a given geographical location, ’ and l, and time, t, an unknown function, Fðh; tÞ, which is called shape function and assimilated the height dependence: NIA ¼ vTECð’; l; tÞ Fðh; tÞ
(94.11)
For the current study, the inversion procedure to retrieve electron densities from radio occultation scenarios is the Abel inversion implementing separa´ ngel bility to L1 phase excess as described in Arago´n-A et al. (2009). The required vTEC information has been computed from ground receiver data, being spatially and temporally extrapolated at each required geographical position and time from final IGS GIM.
94.3
Data Ingestion Procedure
If Gð’; l; hÞ in (94.5) is identical to the LOS from the satellite S to the receiver R in (94.6), the following condition applies:
(94.12)
where the term e arises from errors in the GNSS observations, in the LPIM formulation, in the NeQuick formulation and in the F2-peak parameters computed from the ITU-R database. Several problems may contribute to the last type of errors: the outdating of the ITU-R database, the fact that it provides monthly mean averages that do not account for the day-by-day variation, the linear interpolation used to account for the solar activity, and the intrinsic errors of the database. This work attempts to attenuate the effects of these kinds of errors by computing corrections DUf for the Uf coefficients so that an improved set of coefficients Uþ f ¼ Uf þ DUf can be used in connection with (94.1) and (94.2) for comþ puting an improved value of Oþ f ¼ Of ðUf Þ. Inserting this value in (94.4): þ ¼ fNQ ’; l; h; t; Of Uþ NNQ f ; OM ðUM Þ ffi NNQ þ
@fNQ @Of DUf @Of @Uf
(94.13)
And further inserting (94.13) in (94.5): ð
sþ NQ ¼ sNQ þ
Gð’;l;hÞ
@fNQ @Of DUf dg @Of @Uf
(94.14)
Finally (94.12) can be re-written:
ð s^LP sNQ Gð’;l;hÞ
s^LP sþ NQ ¼ @fNQ @Of DUf dg ¼ eþ @Of @Uf
(94.15)
Equation (94.14) constitutes the equation of condition for ingesting the LPIM sTEC into the NeQuick model. It allows computing, via the Least Squares method, the corrections DUf that minimize (in the Least Squares sense) the differences between LPIM and NeQuick sTEC. Using (94.1) and (94.2), (94.14) can be written:
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Toward a SIRGAS Service for Mapping the Ionosphere’s Electron Density Distribution
sLP sNQ ¼
K X
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H0;k DUf ;0;k
k¼0
þ
J X
cos jt
j¼1
þ
J X
K X
H2j;k DUf ;2j;k
(94.16)
k¼0
sin jt
j¼1
K X
H2j1;k DUf ;2j1;k
k¼0
where the Hi , i ¼ 0, . . . , 988, numerical constant coefficients can be computed from the Gk ð’; lÞ functions and the derivative and integral of the NeQuick function: ð H0;k ¼ Gk ð’; lÞ ð H2j;k ¼ Gk ð’; lÞ Gð’;l;hÞ
ð
H2j1;k ¼ Gk ð’; lÞ Gð’;l;hÞ
94.4
Gð’;l;hÞ
@fNQ dg @Of
@fNQ dg; j 0 @Of
(94.17)
@fNQ dg; j 1 @Of
Preliminary Results
The preliminary results that will be shown in this section are based on 24 h of observations for January 6, 2007, from a global network of 311 GNSS receivers whose distribution is shown with white circles in (94.1). That figure illustrates the behaviour of the correction to the critical frequency of the F2-peak at 11.5h UT of that particular day. More specifically, it shows the relative value of the correction in percentage of the value computed from the ITU-R database, O i.e. 100 Oþ O . Dashed white lines repref f f sent the modip parallels of 60º, 30º and 0º that roughly delimit the high, mid and low latitude ionosphere. The solar terminator is also depicted with a continuous white line. This paper does not intend to discus the physical significance of the correction computed for a particular day. Nevertheless, some general comments regarding the outstanding behaviours noticed in a set of figures (similar to Fig. 94.1) describing the variation of that correction at different UT, may be interesting at this point:
–50
–40
–30
–20
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0
10
20
30
40
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Fig. 94.1 Relative value of the correction to the critical frequency of the F2-peak at 11.5h UT for January 6, 2007, in percentage of the ITU-R value
• The correction shows a South-to-North general trend from ~+12 to 63%, with a global average of 18% • The outstanding feature in the daytime sector is a longitudinal variation – more pronounced in the region from modip ~60 to +30 – with largest values close to the local noon and lower values close to the solar terminators • The outstanding feature in the night-time sector is a longitudinal variation in the region from modip ~0º to +60º with largest values at ~2h after mid-night and lower values shorter after sunrise and before sunset. It is worth to underlay that what Fig. 94.1 shows is not the critical frequency of the F2 peak, but the correction to that parameter computed for a particular day and time from the ITU-R database. With the aim of validate the results obtained with the previously described methodology, the electron density computed with NeQuick before and after LPIM sTEC ingestion are compared to the corresponding electron density computed by using Improved Abel Inversion applied to GPS observations collected from the FORMOSAT-3/COSMIC constellation satellites. Figure 94.2 shows the geographical distribution of the ~2.5 105 data samples collected from 507 occultations happened on January 6, 2007. Figure 94.3 shows a few examples extracted from the 507 occultations analysed in this work. Each panel of this figure shows the electron density values (in the x-axis, in logarithmic scale and in electron/m2) as a function of the altitude (y-axis, in kilometres) for a particular satellite occultation. The left-handside panels show night-time occultations while the
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right-hand-side one show daytime occultations. From top to bottom the panels correspond to high, mid and low modip ocultations. The points represent the electron density from Improved Abel Inversion, and the
dotted and solid lines represent the electron density from NeQuick before and after the LPIM sTEC ingestion. As Fig. 94.3 shows the ingestion of the LPIM sTEC significantly reduces the discrepancies between the NeQuick and the Improved Abel Inversion electron density. In order to quantify the level of discrepancy the so-called “index of discrepancy” is introduced:
8 > < > :
PKi k¼1 NNQ;i;k NIA;i;k Dhi;k Di ¼ 100
PK i k¼1 NIA;i;k Dhi;k Dhi;k ¼
Fig. 94.2 Geographical distribution of the FORMOSAT-3/ COSMIC GPS occultations Fig. 94.3 Electron density values; the x-axis is in logarithmic scale and corresponds to electron density in electron/m2; the y-axis corresponds to altitude in kilometres; points are for Improved Abel Inversion; dashed and solid lines are for NeQuick before and after the LPIM sTEC ingestion
Dhi;1 ¼ hi;2 hi;1 hi;kþ1 hi;k1 ;k ¼ 2; . . . ; Ki 1 Dhi;Ki ¼ hi;Ki hi;Ki1 (94.18)
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night time - high latitude
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94
Toward a SIRGAS Service for Mapping the Ionosphere’s Electron Density Distribution
where i ¼ 1, . . . , 507 identifies the satellite occultation, k ¼ 1, . . . , Ki identifies the data sample for the occultation i; NIA;i;k is the electron density computed by Improved Abel Inversion at the point ’i;k , li;k , hi;k þ and time ti;k ; and NNQ;i;k (and hence Di ) or NNQ;i;k (and þ hence Di ) are the corresponding electron densities computed with NeQuick before or after the LPIM sTEC ingestion. It is worth to remark the presence of the absolute value in the numerator of (94.18), which avoids compensation between positive and negative differences. Table 94.1 shows the indexes of discrepancy for the occultations showed in Fig. 94.3. According to it, the LPIM sTEC ingestion is very effective from the point of view of reducing the discrepancies between the NeQuick and the Improved Abel Inversion electron density. For night-time, and also for daytime at high and mid modip, the index of discrepancy is reduced by a factor close and even greater than 2. For daytime at low modip the data ingestion is not so effective.
Table 94.1 Approximated modip, m, LT, and indexes of discrepancy, D and Dþ , before and after LPIM sTEC ingestion Pass Night-high Night-mid Night-low Day-high Day-mid Day-low
m 65.6º 37.9º 6.7º 60.7º 40.6º 27.6º
D (%) 360 201 238 158 87 47
LT (h) 2.2 0.2 21.8 10.2 10.5 9.0
Dþ 55 84 83 56 54 44
759
The histogram in Fig. 94.4 comprises the 507 occultations analysed in this work and confirms the affectivity of the data ingestion procedure presented in this paper. It shows the percentage of occultations (x-axis) that falls below a given index of discrepancy (y-axis). The dashed and solid lines correspond to NeQuick before and after the LPIM sTEC ingestion. It can be appreciated a general reduction of the index of discrepancy by a factor greater than 2 (e.g. the 95 percentile – depicted in the figure – reduces from 350 to 120%). Conclusions
This paper presented a methodology for ingesting dual-frequency GNSS observations into a global electron density model. The data ingestion allows computing a geographically and time depending correction to the critical frequency of the F2-Peak that improves the modelled electron density. The quality of the results was assessed by comparing the model results before and after data ingestion to values computed by Improved Abel Inversion on FORMOSAT-3/COSMIC GPS occultations. A preliminary analysis based on one day of data showed a general improvement of the model by a factor greater than 2. Further investigations are being carried out for a more exhaustive validation of the data ingestion methodology and for its implementation in the context of SIRGAS. Acknowledgments The authors wish to thanks the IGS for making readily available high quality GNSS observations.
600 500
References
D D+
D (%)
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0
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passess (%)
Fig. 94.4 Histogram of the index of discrepancy for the 507 occultations passes analysed in this work
´ ngel A, Herna´ndez-Pajares M, Juan JM, Sanz J (2009) Arago´n-A Improving the Abel transform inversion using bending angles from FORMOSAT-3/COSMIC. GPS Solut. doi:10.1007/s10291-009-0147-y Azpilicueta F, Brunini C, Radicella SM (2005) Global ionospheric maps from GPS observations using modip latitude. Adv Space Res 36:552–561 Brunini C, Meza A, Gende M, Azpilicueta F (2008) South American regional maps of vertical TEC computed by GESA: a service for the ionospheric community. Adv Space Res 42:737–744. doi:10.1016/j.asr.2007.08.041 Dow JM, Neilan RE, Gendt G (2005) The International GPS Service (IGS): celebrating the 10th anniversary and looking to the next decade. Adv Space Res 36(3):320–326. doi:10.1016/j.asr.2005.05.125
760 Herna´ndez-Pajares M, Juan JM, Sanz J (2000) Improving the Abel inversio´n by adding ground data LEO radio occultations in the ionospheric sounding. Geophys Res Lett 27(16):2743–2746 Herna´ndez-Pajares M, Juan JM, Sanz J, Orus R, Garcia-Rigo A, Feltens J, Komjathy A, Schaer SC, Krankowski A (2009) The IGS VTEC maps: a reliable source of ionospheric information since 1998. J Geodes 83(3–4):263–275 ITU-R (1997) Recommendation ITU-R P.1239, ITU-R reference ionospheric characteristics. International Telecommunications Union, Radio-Communication Sector, Geneva
C. Brunini et al. Jones WB, Gallet RM (1965) The representation of diurnal and geographical of ionospheric delay by numerical methods. Telecomm J 32:18 Radicella S, Leitinger R (2001) The evolution of the DGR approach to model electron density profiles. Adv Space Res 27(1):35–40 Sa´nchez L, Brunini C (2009) Achievements and challenges of SIRGAS. IAG Symposia 134:161–166. doi:10.1007/978-3642-00860-3
Assisted Code Point Positioning at Sub-meter Accuracy Level with Ionospheric Corrections Estimated in a Local GNSS Permanent Network
95
M. Crespi, A. Mazzoni, and C. Brunini
Abstract
It is well know that GNSS permanent networks for real-time positioning with stations spaced at few tens of kilometers in the average were mainly designed to generate and transmit products for RTK (or Network-RTK) positioning. In this context, RTK products are restricted to users equipped with geodetic-class receivers which are continuously linked to the network processing center through Internet plus mobile phone. This work is a first step toward using a local network of permanent GNSS stations to generate and make available products devoted to ionospheric delay correction that could remarkably improve positioning accuracy for C/A receiver users, without forcing them to keep a continuous link with the network. A simple experiment was carried out based on data from the RESNAPGPS network (w3.uniroma1.it/resnap-gps), located in the Lazio Region (Central Italy) and managed by DITS-Area di Geodesia e Geomatica, University of Rome “Sapienza”. C/A raw observations were processed with Bernese 5.0 CODSPP module (single point positioning based on code measurements) using IGS precise ephemeris and clocks. Further, the RINEX files were corrected for the Differential Code Biases (DCBs) according to IGS recommendations. One position per epoch (every 30 s) was estimated from C/A code; the vertical coordinate errors showed a typical signature due to the ionospheric activity: higher errors for day-time (up to 5 m) and smaller ones for night-time (around 1.5 m). In order to improve the accuracy of the solution, ionospheric corrections were estimated using the La Plata Ionospheric Model, based on the dual-frequency observations from the RESNAP-GPS network. This procedure allowed to reduce horizontal and vertical errors within 0.5 m (CE95) and 1 m (LE95) respectively. Finally, the possibility to predict the ionospheric model for few hours was preliminary checked. Our approach shows the possibility of a novel use of the measurements collected by
M. Crespi A. Mazzoni (*) DITS-Area di Geodesia e Geomatica, Facolta` di Ingegneria, Universita` di Roma, La Sapienza, Rome, Italy e-mail:
[email protected] C. Brunini Facultad de Ciencias Astrono´micas y Geofı´sicas, Universidad Nacional de La Plata and CONICET, La Plata, Argentina S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_95, # Springer-Verlag Berlin Heidelberg 2012
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GNSS permanent networks designed for real-time positioning services, which can assist and remarkably improve the C/A code real-time positioning supplying offline predicted ionospheric corrections, acting as a local Ground Based Augmentation System.
95.1
Introduction
In the past years many efforts were dedicated to implement Augmentation Systems, mainly devoted to support real time point positioning with low-cost receivers for the extremely wide market connected to the applications requiring real-time positioning at meter and sub-meter accuracy level. In this respect, it is well known that the ionosphere modeling is one of the crucial items, in order to reduce the positioning error remarkably below the level achievable in stand-alone point positioning. Nevertheless, if we consider the present situation about augmentation systems such as the EGNOS in Europe and the WAAS in North America (Seynat et al. 2009; Eldredge 2009), we may realize that the positioning error is not better than few meters (horizontal error < 3 m and vertical error < 4 m, for at least 99% of the time according to Seynat et al. 2009; 3D error within 1–2 m at 95% level according to Flament and Seynat 2008), even when the augmentation correction data are available in real time, what may be difficult in many situations due to the elevation of the transmitting geostationary satellites (2 GEOs, Seynat et al. 2009), as it happens, for example, in urban canyons or in a hilly/mountainous context. On the other hand, all over Europe, there is a wide availability of national and regional GNSS permanent networks, with permanent stations spaced at few tens of kilometers in the average, which might give a significant contribution to improve the ionosphere model at a local level. In this respects, it has to be also considered that, where these kind of networks are devoted to realize positioning services, the ionospheric model is routinely computed (together with other correction models) and delivered to the users connected by mobile phone/Internet in realtime through NTRIP protocol in RTCM 3.x format (SAPOS network in Germany, Ordnance Survey Net in UK, GPS LOMBARDIA in Italy, and many others). Nevertheless, the continuous connection with the processing center of the GNSS permanent network is
required as well, whilst no assistance to non connected user is presently supplied by these well distributed GNSS permanent networks. Overall, the goal of the paper is to demonstrate that these networks can give a significant support also to the users which cannot be continuously linked to them for several reasons (logistic, budget, . . .), by estimating, possibly predicting over few hours and then making available in advance (for example through a new dedicated web service) a local refined ionospheric model. In such a way, these permanent networks can act as local Ground Based Augmentation Systems. In particular, the main issues related to the model estimations, the achievable point positioning accuracy level and the accuracy of the predicted ionospheric model are addressed. In this respect, a preliminary experiment was carried out in order to establish an off-line procedure which could be used in real time. Data from four permanent stations included into the RESNAP-GPS network (Central Italy) were used: AQUN (L’Aquila, Abruzzo), FOLI (Foligno, Umbria), M0SE (Roma, Lazio) and RIET (Rieti, Lazio) (Biagi and Sanso` 2007). For each station, three consecutive days of data at 30 s sampling rate were used in the test (year 2008, doys 167, 168 and 169) together with IGS precise orbits and clocks and CODE Earth Rotation Parameters. All stations were equipped with Leica GX1230 receivers, supplying C1 and P2 codes. All the daily data-files, for every step of the following investigated strategy, were processed with Bernese 5.0 GPS software (Dach et al. 2007) CODSPP module (single point positioning based on code measurements) in order to obtain the epoch-by-epoch position.
95.2
Standard CODSPP Processing
At first, a standard processing was carried out with the main CODSPP parameters over three Permanent Stations (AQUN, FOLI, RIET); throughout the paper
Assisted Code Point Positioning at Sub-meter Accuracy Level with Ionospheric Corrections
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RMSE2N þ RMSE2E
LE95 ¼ 1:96RMSEU
4 2 0 –2 L1 - 167 L1 - 168 L1 - 169
–4 –6
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8
10 12 14 16 18 20 22 24 Hours
REIT - Vertical Errors 8
(95.1)
6 4
(95.2)
where RMSEE,N,U are computed over the mentioned errors. In Fig. 95.1 vertical errors vs. time are shown for each station; three daily solutions are stacked in the same graph. LE95 and CE95 respectively are at the level of 7 m and of 4 m (Table 95.1). Very strong correlations both in time and space are also clearly displayed; each station has very similar signature for different days and these signatures are very similar between different stations. Figure 95.2 displays the errors in terms of North and East components; for each station; three daily solutions are stacked in the same graph. Also in this case remarkable space–time correlations are highlighted. The strong space–time correlations between the errors (Fig. 95.1, Table 95.2) were hypothesized to be mainly due to satellite clocks Differential Code Biases (DCBs).
95.3
6
L1 Differential Code Biases CODSPP Processing
Meters
2:4477 CE95 ¼ pffiffiffi 2
763
AQUN - Vertical Errors
8
Meters
we just use the acronyms L1 to point out this processing. All others parameters, for these processing and also for the following ones, were set to default values; in particular, Saastamoinen tropospheric model was used, with cut-off angle set at 10 . For each processing, doy and station, kinematic coordinate errors with respect to the known station IGS05 positions (used as reference) were computed. The standard following statistical indexes were used in order to represent the horizontal and vertical accuracy respectively:
2 0 –2 L1 - 167 L1 - 168 L1 - 169
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95
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Fig. 95.1 Standard CODSPP processing – vertical errors
Table 95.1 Standard CODSPP processing Linear error LE95 (meters) Circular error CE95 (meters)
Matlab# routines were implemented in order to modify the native RINEX files with the CODE DCB P1C1 and P1P2 corrections (P1C10806 and P1P20806 DCB), following the recommendations reported in Schaer (2008).
Doy 167 168 169
AQUN 6.7 7.0 6.7
FOLI 6.7 7.0 6.7
RIET 6.6 7.0 6.6
AQUN 3.8 3.7 3.8
FOLI 3.7 3.7 3.6
RIET 3.6 3.6 3.6
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L1 - 167 L1 - 168 L1 - 169
4 North (Meters)
Table 95.3 DCBs CODSPP processing
AQUN Horizontal Errors
6
2 0 –2 –4 –6 –6
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2 0 East (meters)
4
6
AQUN Doy B R Linear error LE95 (meters) 167 6.0 6.0 168 6.2 6.2 169 5.8 5.8 Circular error CE95 (meters) 167 1.7 1.7 168 1.7 1.7 169 1.5 1.5
FOLI B
R
RIET B
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Fig. 95.2 Standard CODSPP – horizontal errors
In order to test the correct DCBs implementation in the RINEX files, two processing were carried out (Table 95.3, Fig. 95.3 and 95.4) with: • L1B – modified P1C1 RINEX (frequency L1) plus P1P20806.DCB file as input in CODSPP • L1R – modified P1C1 and P1P2 RINEX (frequency L1) as input in CODSPP Considering that CODSPP accepts only one DCB file a time. The first evidence is that the two processing led to equal results (Table 95.3), what confirms that DCBs corrections were correctly implemented in the native RINEX files by the Matlab# routines. In Fig. 95.3, the vertical component errors show, for every station and every doy, the typical signature due to the ionospheric activity: higher errors for daytime (up to 5 m) and smaller for night-time (around 1.5 m). In the same time, these signatures also indicates that the DCBs effects seem to be correctly removed from the observations; the same holds for the horizontal components (Fig. 95.4) (not so influenced by ionospheric activity effects), which are now less disperse w.r.t. the NO-DCB standard L1 processing Overall, the simple use of DCBs corrections remarkably increases the accuracies.
Table 95.2 Vertical errors space–time correlations Space correlations AQUN-FOLI AQUN-RIET RIET-FOLI Time correlations 167–168 167–169 168–169
167 0.92 0.93 0.97 AQUN 0.84 0.74 0.86
168 0.92 0.90 0.96 FOLI 0.87 0.74 0.85
169 0.92 0.92 0.97 RIET 0.86 0.74 0.87
95.4
Ionospheric Corrections
In order to further improve L1 processing accuracy, ionospheric corrections for the considered stations were estimated using Ionos New Generation Software (Implementation of La Plata Ionospheric Model LPIM) (Azpilicueta et al. 2005) on the basis of the GNSS permanent network stations observations.
95
Assisted Code Point Positioning at Sub-meter Accuracy Level with Ionospheric Corrections AQUN - Vertical Errors
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RIET - Horizontal Errors
RIET - Vertical Errors
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Fig. 95.4 DCBs CODSPP processing – horizontal errors
Fig. 95.3 DCBs CODSPP processing – vertical errors
LPIM is based on the ionospheric observable: LI;arc ¼ sTEC þ BR þ BS þ Carc þ eL ;
1
(95.3)
where LI,arc is the carrier-phase ionospheric observable – the sub-indices arc refers to every continuous arc of carrier-phase observations, which is defined as a group of consecutive observations along which carrier-phase ambiguities do not change; BR and Bs are
the so-called satellite and receiver inter-frequency biases (IFB) for carrier-phase observations; Carc is the bias produced by carrier-phase ambiguities in the ionospheric observable; and eL is the effect of noise and multi-path. All terms of (95.3) are expressed in Total Electron Content units (TECu), being 1 TECu equivalent to 1016 electrons per square meters. LPIM relays on the thin-shell and the mapping function approximations. Based on those approximations, (95.3) is written in the following way:
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LI;arc ¼ secðz0 Þ vTEC þ garc þ eL ;
(95.4)
where sec(z0 ) is the mapping function, z0 being the zenith distance of the satellite as seen from the point where the signal crosses the thin-shell (the so-called piercing point) that, in the case of LPIM, is located 450 km above the Earth’s surface; vTEC cos(z’) · sTEC, is the equivalent vTEC at the piercing point; and garc BR þ Bs þ Carc , is a calibration constant that encompasses, all together, receiver and satellite IFBs and the ambiguity term. The calibration, i.e. the estimation of garc for every continuous arc, is performed independently for every observing receiver in the network. To accomplish this task, the equivalent vTEC is approximated by a bilinear expansion dependent on the piercing point coordinates: vTEC aðtÞ þ bðtÞ ðlI l0 Þ cosðmI Þ þ cðtÞ ðmI m0 Þ;
(95.5)
where t is the Universal Time of the observation, lI and the geographic longitude and the modip latitude of the piercing point and l0 and m0 the geographic longitude and the modip latitude of the observing receiver. The dependence on time of the expansion coefficients is approximated with ladder functions: aðtÞ ¼ ai ; for ti t < ti þ dt; i ¼ 1;2;:::;
(95.6)
where a represents anyone of the coefficient a, or c; and dt is the interval of validity of every planar approximation (5m in the case of LPIM). Merging all together the observations gathered by a receiver in a given interval Dt (Dt >> dt), and arranging appropriately (95.4)–(95.6), LPIM forms an overdetermined linear system of equation of observations that contains, as unknowns, the calibration constants for all observed continuous arcs, garc (arc ¼ 1,2,. . .,narc), and the constant coefficients of the planar fits, a1,. . .,am, b1,. . .,bm, c1,. . .,cm (m ¼ Dt/dt). Since we are not interest on the coefficients, they are reduced from the system by means of a Gaussian elimination process and then, the system is solved by the Least Square methods, hence estimating the narc calibration constants. Finally, the observations are calibrated and the equivalent vTEC are estimated from (95.7):
Table 95.4 DCBs and ionospheric corrections CODSPP processing Linear error LE95 (meters) Circular error CE95 (meters) Doy 167 168 169
AQUN 0.9 0.7 0.9
FOLI 1.0 1.0 1.2
^ vTEC þ vT EC
95.5
RIET 0.8 0.8 1.1
AQUN 0.6 0.5 0.6
FOLI 0.5 0.5 0.6
eL LI;arc ^garc ¼ : secðz0 Þ secðz0 Þ
RIET 0.4 0.4 0.4
(95.7)
Final CODSPP Processing
For each station, daily ionospheric corrections values (epoch-by-epoch and satellite-by-satellite) for both frequencies L1 and L2 were stacked in a correction file. Again, Matlab# routines were implemented in order to modify RINEX files adding the ionospheric correction values to the observation epoch-by-epoch. The final CODSPP processing (Table 95.4, Figs. 95.5 and 95.6) was carried out with L1I – modified P1C1 and P1P2 RINEX and ionosperic corrections as input in CODSPP. Results clearly evidenced the success of the procedure: the combined use of DCBs and ionospheric corrections led to horizontal and vertical accuracies at 0.5 m (CE95) and 1 m (LE95), respectively. The space correlations between errors of different stations remarkably decrease (Table 95.5); their remaining parts might be caused by possible DCBs errors and should be investigated in the future.
95.6
Spatial Interpolation Test
In order to investigate the possibility to estimate ionospheric corrections within a GNSS Network, a simple interpolation test was carried out for doy 167 the basis of three stations (M0SE, FOLI and AQUN) L1 estimated corrections, values for RIET station (approximately located in the centre of the triangle AQUN, FOLI, M0SE) were spatially linearly interpolated, neglecting the ionospheric travelling disturbances which were also detected with similar signatures but in slightly different intervals for the
95
Assisted Code Point Positioning at Sub-meter Accuracy Level with Ionospheric Corrections AQUN - Vertical Errors
AQUN - Horizontal Errors 1.5
1
1 North (meters)
1.5
Meters
0.5 0 –0.5 L1I - 167 L1I - 168 L1I - 169
–1 –1.5
0
2
4
6
0.5 0 –0.5 –1
8
–1.5 –1.5
10 12 14 16 18 20 22 24 O Hours
L1I - 167 L1I - 168 L1I - 169
–1
FOLI - Vertical Errors 1
1 North (meters)
1.5
Meters
0.5 0 –0.5
–1.5
L1I - 167 L1I - 168 L1I - 169
0
2
4
6
–1.5 –1.5
10 12 14 16 18 20 22 24 n Hours
1 North (meters)
1 Meters
0.5 0 –0.5 L1I - 167 L1I - 168 L1I - 169
2
4
6
–1
10 12 14 16 18 20 22 24 Hours
Fig. 95.5 DCBs and ionospheric corrections CODSPP processing – vertical errors
three stations. The interpolation vas carried out w.r.t. longitude and latitude, with the standard model: aðtÞ CðtÞ ¼ bðtÞ cðtÞ lM0SE ¼ lAQUN lFOLI
fM0SE fAQUN fFOLI
–0.5 0 0.5 East (meters)
1
1.5
0.5 0 –0.5 –1
8
1.5
L1I - 167 L1I - 168 L1I - 169
REIT - Horizontal Errors 1.5
0
1
–0.5
RIET - Vertical Errors
–1.5
1.5
0
1.5
–1
1
0.5
–1 8
–0.5 0 0.5 East (meters) FOLI - Horizontal Errors
1.5
–1
767
1 1 IðtÞM0SE 1 IðtÞAQUN 1 IðtÞ FOLI
(95.8)
–1.5 –1.5
L1I - 167 L1I - 168 L1I - 169
–1
–0.5 0 0.5 East (meters)
Fig. 95.6 DCBs and ionospheric corrections CODSPP processing – horizontal errors Table 95.5 DCBs and ionosperic corrections CODSPP processing – vertical errors space–time correlations Space correlations AQUN-FOLI AQUN-RIET RIET-FOLI Time correlations 167–168 167–169 168–169
167 0.20 0.45 0.39 AQUN 0.34 0.28 0.25
168 0.14 0.37 0.34 FOLI 0.68 0.55 0.63
169 0.25 0.52 0.46 RIET 0.27 0.27 0.28
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IðtÞRIET ¼ aðtÞlRIET þ bðtÞ’RIET þ cðtÞ
(95.9)
where a(t), b(t), c(t) are the time dependent interpolation coefficients to be computed on the basis of the modeled ionospheric delay at permanent stations AQUN, FOLI, M0SE (95.8) and I(t)RIET is the interpolated ionospheric delay at RIET position. Interpolated values were compared to L1 ionospheric corrections estimated on the basis of the RIET station observations. The RMSEs of the differences were computed satellite-by-satellite and their values show that spatial interpolation accuracy is generally within 15 cm.
implementation of the ionospheric correction and its quality assessment; in this respect a simple low degree polynomial model for satellite arc could be effective for few hours prediction at 10–20 cm level. Finally, since at present only measurements acquired by geodetic class receivers working at permanent stations were considered, the proposed procedure should be tested over low-cost C/A receivers, in order to evaluate the overall achievable accuracy accounting also the higher noise level of such equipments.
References 95.7
Final Remarks
This work shows the possibility of a novel use of the measurements collected by GNSS permanent networks designed for real-time positioning services, which can assist and remarkably improve the C/A code real-time positioning supplying off-line predicted ionospheric corrections, acting as a local Ground Based Augmentation System. Horizontal and vertical accuracies at 0.5 m (CE95) and 1 m (LE95) are respectively achievable, provided DCBs and ionospheric corrections are duly considered. The work is still preliminary in the sense that some questions arose and have to be addressed in the next future: • The investigation of the possible causes of the still present space–time correlations in errors time series • The refinement of the ionospheric corrections interpolation with a space–time model, in order to account for the ionospheric travelling disturbances propagation The replacement of the interpolation with a prediction space–time model, in order to enable the real-time
Azpilicueta F, Brunini C, Radicella SM (2005) Global ionospheric maps from GPS observations using modip latitude. Advances in Space Research, JASR 7882, 8p Biagi L, Sanso` F (eds) (2007) Un libro bianco su “I servizi di posizionamento satellitare per l’e-government”, Geomatics workbooks, vol 7 Dach R, Hugentobler U, Fridez P, Meindl M (eds) (2007) Bernese GPS Software Version 5.0 – documentation. Astronomical Institute, University of Berne, Switzerland, p 640 Eldredge L (2009) GPS augmentation systems status. In: Proceedings of the 22nd international technical meeting of the Satellite Division of the Institute of Navigation (ION GNSS 2009), Savannah, GA, September 2009, pp 3437–3456 Flament D, Seynat C (2008) EGNOS status update. In: Proceedings of the 21st international technical meeting of the Satellite Division of the Institute of Navigation (ION GNSS 2008), Savannah, GA, September 2008, pp 1060–1088 http://www.gpslombardia.it/ http://www.ordnancesurvey.co.uk/oswebsite/gps/ http://www.sapos.de Schaer S (2008) Differential code biases (DCB) in GNSS analysis. IGS workshop 2008 Seynat C, Flament D, Brocard D (2009) EGNOS Status Update. In: Proceedings of the 22nd international technical meeting of the Satellite Division of the Institute of Navigation (ION GNSS 2009), Savannah, GA, September 2009, pp 3457–3483
Semi-annual Anomaly and Annual Asymmetry on TOPEX TEC During a Full Solar Cycle
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F. Azpilicueta, C. Brunini, and S.M. Radicella
Abstract
During the first decades of ionospheric research, the physical description of the ionospheric free electron vertical density was mainly given by the Chapman theory in which the main driving parameters were the solar irradiance level and the solar zenith distance from the observation point. Any new observed phenomenon that could not be explained by the Chapman theory was considered an ‘anomaly’. After more than 50 years of continuous aeronomic research, many of these phenomena then called ‘anomalies’ were physically explained but some of them are still open to discussions, like the so-called Semi-annual Anomaly that produces global mean TEC values larger for equinoxes than for solstices; and the Annual Asymmetry that causes larger mean global TEC during the December than the June solstice (far larger than the 7% that would be expected from the change on the Sun–Earth relative distance). Using the high-precision TEC 13-year data series provided by the TOPEX/Poseidon mission, the main finding of this work is the characterization of the annual variation of the ionospheric daily mean TEC that reflects the combined effects of the both mentioned anomalies. The analysis of this annual pattern allows a precise quantification of the level of the effects of both anomalies, and suggests that the semi-annual anomaly does not have a half-year period, instead might be considered as another annual anomaly with two maxima separated by 220 days.
96.1 F. Azpilicueta (*) C. Brunini Facultad de Ciencias Astrono´micas y Geofı´sicas de la Universidad Nacional de La Plata, Paseo del Bosque s/n, La Plata 1900, Argentina Consejo Nacional de Investigaciones Cientı´ficas y Te´cnicas, Buenos Aires, Argentina e-mail:
[email protected] S.M. Radicella Aeronomy and Radio-propagation Laboratory, Abdus Salam International Centre for Theoretical Physics, Trieste, Italy
Introduction
After the 1930s and for more than 4 decades the principal instrument for studying and modelling the ionosphere was the vertical ionosonde (Rishbeth and Setty 1961; King and Smith 1968; Torr and Torr 1973); but since the availability of TEC observations primarily coming from the Global Positioning System (GPS) permanent stations and secondly from scientific satellite missions, a great effort has been done by the aeronomy community in using these measurements as the baseline for new scientific findings.
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In recent papers, several authors have developed new ideas to define an innovative set of indices based on TEC measurements in order to characterize the ionospheric behaviour over global, hemispherical, or even smaller geographical regions, and to build a new framework for correlation studies between the TEC and other geophysical parameters. Just to mention the most recent publications, we can cite (Mendillo et al. 2005; Liu et al. 2007; Zhao et al. 2007). In the majority of these contributions, the main TEC parameters studied are derived from GPS measurements, particularly from the Global Ionospheric Maps (GIMs) that, for example, the Jet Propulsion Laboratory (JPL) computes on a daily-basis since 1998. It is worth to mention that the work that the JPL has been doing for producing the GIMs since the end of 1998, has made the JPL-GIMs a standard within the ionospheric scientific community. For details about the JPL-GIMs the interest reader is referred to (Mannucci et al. 1998). As it is well presented in Mendillo et al. (2005), during the first decades of ionospheric research in the twentieth century, the description of the ionospheric free electron vertical density (and its associated parameters like the TEC, i.e. the lineal integral of the electron density along the vertical) was mainly dominated by Chapman’s theory of atmospheric ionozation (Chapman 1931) in which the main driving parameters were the solar irradiance level and the solar zenith distance from the observation point. Chapman‘s theory was successful to explain several characteristics of the electron density vertical profiles measured by the ionosondes. Nevertheless, whatever new observed behaviour that could not be explained by Chapman’s theory was considered as an ‘anomaly’. From this historical reason, nowadays many effects that many researchers have explained and modelled with great effort are still called ‘anomalies’. The clearest example of this might be the so-called Equatorial Anomaly also known as Appleton Anomaly (Appleton 1946). However, there are some phenomena that were called anomalies at the beginning and remain as such until the present. For example the semi-annual anomaly, that produces global mean TEC values (or equivalently global mean F2 peak) larger for equinoxes than for solstices. In the same category is the annual anomaly (also called annual asymmetry to distinguish it from the seasonal anomaly) that is described by a mean global TEC (equivalently global mean F2 peak) larger during the December than June solstice, far larger than that 7% that would be
F. Azpilicueta et al.
expected considering the change in the Sun–Earth distance. Several recent articles study and analyze the behaviour of these two phenomena, (Fuller-Rowell 1998; Mendillo et al. 2005; Rishbeth and MullerWodarg 2006; Liu et al. 2007, Zhao et al. 2007). From these papers appears evident that their characterization and explanation are still open to discussion. Precisely these two anomalies are the subject of this work extending the studies reported in the articles mentioned before. For this we have analyzed the behaviour of a daily mean global TEC using the outstanding hi-precision TOPEX TEC data series provided by the TOPEX/Poseidon (T/P) (Fu et al. 1994). The data series correspond to the whole mission and spans from mid 1992 to mid 2005, i.e. for more than a full 11-year solar cycle. As it will appear in the following paragraphs, one important result is that the combined effect of both anomalies is modulated by the 11-year solar cycle. After the modulation is modelled and removed from the daily mean global TEC, it is possible to define clearly the mean annual pattern, which includes the semi-annual anomaly and the annual asymmetry. This pattern gives a better definition of both anomalies that could reach values as high as 20% between solstices (December higher than June) and 40% between the equinoxes and the June solstice. Another important issue coming from the annual pattern is the precise moment (more precisely the day of year) that the maxima and minima occur, indicating that they cannot be directly associated to the occurrence of the equinoxes and solstices.
96.2
Data Processing Strategy
TOPEX/Poseidon was a joint effort of the National Space and Aeronautic Agency (NASA) of the United Stated of America and the Centre National d’E´tudes Spatiales (CNES) of France. It was an outstanding mission that lasted for almost 13 years (1992–2005), whose main objective was to measure the sea topography for ocean circulation studies. At the moment of its launch it represented the state-of-the-art after more than 30 years of satellite oceanography. The importance of this unique mission from the aeronomic point of view is that it was the first satellite to fly a dualfrequency altimeter and nowadays, thanks to this, the aeronomic community counts with a continuous
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Semi-annual Anomaly and Annual Asymmetry on TOPEX TEC During a Full Solar Cycle
13-year data series of high-precision TEC. The satellite orbital characteristics were: 1,336 km of altitude, 66º orbital plane inclination, 112 min of orbital period, and 1 sec sampling rate. Panel (a) of Fig. 96.1 shows the ground track (equivalent to the sampling pattern) of T/P during one day; while the panel (b) shows an
Fig. 96.1 (a) T/P ground track after one day and (b) TOPEX TEC samples when the satellite crossed the Equatorial Anomaly
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example of TOPEX TEC sample when the satellite crosses the Equatorial Anomaly. In this work we have used the complete 13-year TOPEX/Poseidon data series, in such a way that the 1-sec TOPEX TEC values were processed so to compute a mean global TEC, mTEC, for every day. Due to several design considerations, the TOPEX instrument only provided TEC measurements over the sea surface. Thus, with the purpose of balancing the asymmetric land and sea distribution between the Northern and Southern hemisphere (that produces an asymmetry in the number of samples within each hemisphere), a geographic filter was applied for the computation of the daily mean TEC, using only the measurements within the sectors defined by (150 , 240 ) and (300 , 360 ) in longitude; and (60 , 60 ) in latitude. The dots in Fig. 96.2 show the obtained mTECdata series against the Modified Julian Date, mjd, and as a reference the two vertical dashed lines indicate the time period that corresponds to the year 2001. Figure 96.2 is quite illustrative to appreciate the sensitivity of the daily TEC parameter to the Sun’s radiation activity. Although it is not presented in this paper, the mTECvalues shown in Fig. 96.2 correlates perfectly well with the solar Extreme Ultra Violet (EUV) irradiance, reflecting variations from the 11-year solar cycle variation up to the 27-day solar rotation period. Since we are interested in the annual and semiannual signals, a function of the form given by (96.1) was adjusted, with period P0 ¼ 11:2x365:25 days.
Fig. 96.2 The dots represent the daily mean global TEC depicted against time for the 13-year data series. The black curve represents the 11.2-year periodic signal adjusted to the data series
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2p TEC0 ðmjdÞ ¼ A0 cos mjd þ ’0 P0
þ b0
(96.1)
After the least squares (fitting the following estimates for the parameters were obtained: A0 ¼ 14:4 TEC units; ’0 ¼ 1:9 rad and b0 ¼ 25:2 TEC units. The resulting function is represented by the dashed curve in Fig. 96.2.
96.3
Results
For this study, the most important information comes from the residuals that remain after the least squares adjustment, i.e.: dTECðmjdÞ ¼ mTECðmjdÞ TEC0 ðmjdÞ
(96.2)
Figure 96.3, panel (a) shows dTEC data series against mjd. A close look at the behaviour of these residuals reveals a semi-annual and annual patterns, closely associated to the anomalies that are the subject of this study; but another clear issue coming from the figure is a modulation of the amplitude of this signal by an almost 11-year periodic function that perfectly correlates with the daily mean TEC shown in Fig. 96.2. In order to reduce the effects of the amplitude modulation in the signal, the so-called relative residuals are computed by: rTECðmjdÞ ¼ dTECðmjdÞ=TEC0 ðmjdÞ
(96.3)
These relative residuals are shown in Fig. 96.3, panel (b), against mjd.
Fig. 96.3 (a) Residuals of the daily mean global TEC after removing the 11.2-year trend, represented against time and (b) relative residuals represented time. A 20-day moving average filter was applied so to reduce the dispersion
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Semi-annual Anomaly and Annual Asymmetry on TOPEX TEC During a Full Solar Cycle
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Fig. 96.4 Relative residuals represented against the day of year (every year has a different intensity of gray)
Semi-annual anomaly effect
Annual asymmetry effect
Fig. 96.5 Annual pattern, the dots represent the mean relative residuals depicted against the day of year; the black line represents what would be expected according to the relative change of the Sun–Earth distance
From the comparison of both panels of Fig. 96.3, it can be seen that the effect of the modulation is significantly reduced, allowing to quantify the relative variations that the anomalies under study produce. Then, under the hypothesis that the anomalies are recursive, in the sense that they synchronously repeat every year, it is useful to produce a composite graphic and represent the relative residuals against the day of year (doy). Figure 96.4 shows such a graphic, where for every doy there are 13 dots, each one with a different intensity of gray corresponding to a different year. The main feature that comes out from this figure is an annual structure with two maximums and a clear minimum.
Figure 96.5 shows the behaviour of the mean relative residuals along one nominal year (i.e. representative for every year). The annual pattern that is seen in the figure represents the combined effects of both the semi-annual anomaly and the annual asymmetry that cause the “anomalous” behaviour of the mean global TEC. The dotted curve in Fig. 96.5 presents two maximums, at doy 80 and at doy 305, a local minimum at doy 5 and absolute minimum close to doy 200. The two maximums can be associated to the semi-annual anomaly and the relative increment on the order of 20% with respect to the mean yearly value is in complete agreement with other author’s findings (e.g.
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Fuller-Rowell 1998). The two minimums can be associated to the so-called annual asymmetry and the relative difference of approximate 20% in favour of the December solstice is in perfect agreement with what is reported in recent articles, for example (Mendillo et al. 2005). For the purpose of comparison, the light gray line depicted in Fig. 96.5 indicates the expected behavior of the relative difference along the year, if only the yearly change of the Sun–Earth distance is considered. As can be inferred from this comparison, the annual asymmetry cannot be explained by just the annual change of the Sun–Earth distance. Figure 96.5 shows, for the first time in the specialized literature, the temporal evolution along the year of the combined effects of both anomalies, obtained with a data series that spans for more than a solar cycle. The annual pattern could be indicating that the direct association of the semi-annual anomaly with the equinoxes is only an approximation, because although the first maximum corresponds to doy 80, very close to doy 81 (doy of March equinox), the second maximum occurs at doy 305, 39 days later than doy 266 (doy of September equinox). This fact may indicate that the physical phenomenon behind this effect does not have a periodicity of half a year. Conclusions
Using the outstanding TOPEX TEC 13-year data series provided by the TOPEX/Poseidon mission, we presented a technique to compute the relative variation pattern along a standard year, called the annual pattern of the daily mean global TEC. The annual pattern was obtained using direct TEC measurements and with the any mathematical assumption of a 11.2 years periodic function for the TEC component. This can be physically justified by considering the variation of the solar irradiance along a solar cycle (particularly with the solar EUV irradiance). The annual pattern shown in Fig. 96.5 indicates the way in which the combined effect of the semiannual anomaly and the annual asymmetry evolves
F. Azpilicueta et al.
with time, along the year, to produce the relative variation on the ionospheric mean global TEC. A first analysis of the pattern suggests that the phenomenon known as semi-annual anomaly, does not occur with a period of half a year, but shows two maxima at doy 81 and 305. Thus, the effect should be considered as having an annual periodicity with two maxima at the doys mentioned. Acknowledgments The TOPEX TEC data series were obtained from the Physical Oceanography Distributed Active Archive Center (PODAAC) at the NASA Jet Propulsion Laboratory, Pasadena, CA. http://podaac.jpl.nasa.gov.
References Appleton EV (1946) Two anomalies in the ionosphere. Nature 157:691 Chapman S (1931) The absorption and dissociative or ionizing effect of monochromatic radiation in an atmosphere on a rotating Earth. Proc Phys Soc 43:26–45 Fu L-L, Christensen E, Yamarone C, Lefebvre M, Menard Y, Dorrer M, Escudier P (1994) TOPEX/POSEIDON mission overview. J Geophys Res 99(C12):24369–24381 Fuller-Rowell TJ (1998) The “thermospheric spoon”: a mechanism for the semi-annual density variation. J Geophys Res 103(A3):3951–3956 King JW, Smith PA (1968) The seasonal anomaly in the behaviour of the F2-layer critical frequency. J Atmos Terr Phys 30(9):1707–1713. doi:10.1016/0021-9169(68)90019-6 Liu H, L€uhr H, Watanabe S (2007) Climatology of the equatorial thermospheric mass density anomaly. J Geophys Res 112:A05305. doi:10.1029/2006JA012199 Mannucci A, Wilson B, Yuan D, Ho C, Lindqwister U, Runge T (1998) A global mapping technique for GPS-derived ionospheric total electron content measurements. Radio Sci 33(3):565–582. doi:10.1029/97RS02707 Mendillo M, Huang C, Pi X, Rishbeth H, Meier R (2005) The global ionospheric asymmetry in total electron content. J Atmos Terr Phys 67:1377–1387 Rishbeth H, Muller-Wodarg I (2006) Why is there more ionosphere in January than in July? The annual asymmetry in the F2-layer. Ann Geophys 24:3293–3331 Rishbeth H, Setty CS (1961) The F-layer at sunrise. J Atmos Terr Phys 21:263–276 Torr MR, Torr RL (1973) The seasonal behavior of the F2-layer of the ionosphere. J Atmos Terr Phys 35:2237–2252 Zhao B, Wan W, Liu L, Mao T, Ren Z, Wang M, Christensen AB (2007) Features of annual and semiannual variations derived from the global ionospheric maps of the electron content. Ann Geophys 25:2513–2527
Numerical Simulation and Prediction of Atmospheric Aerosol Extinction Using Singular Value Decomposition
97
J. Shin, S. Lim, C. Rizos, and K. Zhang
Abstract
The remote sensing problem of poly-dispersed aerosols in the single scattering approximation is a classical example of the first kind Fredholm integral equation. Assuming that the prediction errors due to arbitrarily small perturbations in the complex aerosol refractive index or the upper radius bounds are negligible, one can form the signal-to-noise ratio (SNR) of the kernel matrix in terms of the singular value of the kernel matrix and the number of measurement wavelengths. The smoothness of the kernel matrix and the information potentialities vary, depending on the choice of a combination of sounding channels. The optimal choice is the one that provides the largest SNR. A numerical simulation with 11 samples of possible combinations is conducted in order to demonstrate that the prediction of aerosol extinction measurements using singular value decomposition is comparable with reference values. If two similar prediction results (e.g. one with SNR 2.019 and the other 2.132) are obtained, the higher value is apparently better, however, a drawback in this case is that the prediction errors increase with the increasing number of sounding channels used. In conclusion, it is noted that the information on the smoothness and potentialities of the kernel matrix has to be factorized in order to increase the success rate of the prediction. Fortran source code is available by its authors upon request.
97.1
J. Shin School of Physics, The University of New South Wales, Sydney, NSW 2052, Australia S. Lim (*) C. Rizos School of Surveying and Spatial Information Systems, The University of New South Wales, Sydney, NSW 2052, Australia e-mail:
[email protected] K. Zhang School of Mathematical and GeoSpatial Sciences, RMIT, Melbourne, VIC 3001, Australia
Introduction
Light scattering by atmospheric aerosols, based on the Mie theory, provides a physical framework for remote sensing of the aerosol micro-structures from optical data measured at multiple frequencies. The problem of remote sensing of poly-dispersed aerosols in the single scattering approximation can be formulated in terms of a Fredholm integral equation of the first kind that is expressed in the following form: bext ðlÞ ¼
ð1 0
pr 2
Qext ðr; m; l; sÞ dRn ðrÞ dr gn ðrÞ dr
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_97, # Springer-Verlag Berlin Heidelberg 2012
(97.1)
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J. Shin et al.
where bext ðlÞ is the extinction measurement at a wavelengthl; Qext is the extinction efficiency factor which is dependent on the aerosol radius r, the complex aerosol refractive index m, and the shape of the aerosol s; dRn ðrÞ=dr is the aerosol size distribution; and gn ðrÞ depends on the type of the aerosol size distribution (Dubovik and Kin 2000). Equation (97.1) is a classical example of the first kind Fredholm integral with a square integrable kernel K ðr; lÞ which can be rewritten: bext ðlÞ ¼
ð rmax rmin
K ðr; lÞ
dRn ðrÞ dr; lmin l lmax dr (97.2)
where the discretized version of the kernelKd : L2 ½rmin ;rmax ! L2 ½lmin ;lmax is a compact operator from the Hilbert space L2 ½rmin ;rmax onto the Hilbert space L2 ½lmin ;lmax . Equation (97.2) can be reformulated in a matrix-vector form: Kd x ¼ y
(97.3)
where x 2 L ½rmin ;rmax and y 2 L ½lmin ;lmax . The inverse of the compact operator Kd is always unbounded and ill-posed, and the unknown input parameter x in the inverse problem is a discontinuous function of optical measurements (Bockmann 2001; Bockmann and Sarkozi 1999; Hansen 1998). Let the kernels Kl1 ; Kl2 ; . . . ; Klm be a basis K for the vector space Rm over a field f , then theoretically any vector Kx in Rm is a linear combination of vectors in Rm in a unique way: 2
ax Kx þ
m X
2
ali Kli ¼ 0; ali 2 f ; Kli 2 Rm
(97.4)
i¼1;li 6¼x
However, the presumed linear independence in the equation above is not valid in practice, largely owing to partial overlapping of the kernels. In this regard, (97.4) does not hold because of the fact that the discrete set of the kernels that corresponds to the limited set of optical data available is hardly a m generating set of the vector space m R . This raises the P ali Kli can be made, question of how close to zero l i in which case, a representation of Kx in Rm in the basis Kx ¼ fKl1 ; Kl2 ; . . . ; Klm gli 6¼x can be uniquely determined by the prediction kernel Kx and the basis Kx (Twomey 1996).
97.2
Selection of Coordinates of Prediction Kernel Based on Eigen-Decomposition
The prediction of extinction measurements can be made so that its prediction error sp becomes accurate within a priori upper bound. This a priori upper bound is confined in part by the machine-dependent precision of the singular value decomposition used and in part by the number of sounding channels that makes up an optimal combination with the largest signal-to-noise ratio (SNR). The first step towards making the prediction of measurements with a reasonable degree of accuracy is to find the coordinates ali of the prediction kernel, say Kx , that meet some constraints. It is often the case that the covariance matrix of the kernels (Yin et al. 1996) defined by (97.5) is real symmetric positive definite: max ð
Kðr; lÞK T ðr; lÞdr
C¼
(97.5)
min
where finding the coordinates ali of the prediction kernel Kx amounts to solving the eigen-system Ca ¼ La subject to the constraint aT a ¼ I. The eigenvectors a of the kernel covariance matrix are orthonormal and the corresponding eigenvalues are on the main diagonal elements of L (Twomey 1996; Yin et al. 1996). One of the necessary and sufficient conditions for the kernel covariance matrix C to be positive definite is aT Ca>0 for all eigenvectors a. An orthodox approach to the problem of finding the coordinates ali of the prediction kernel Kx is to minimize a quadratic form qðaÞ ¼ aT Ca ¼ ha; Cai subject to the constraint aT a ¼ I. In this regard, solving the eigenT system Ca ¼ La is equivalent to minimizing aaTCa a by choosing the first normalized eigenvector associated with the smallest eigenvalues (Strang 1988; Twomey 1996). However, there exists a situation where the abovementioned quadratic form becomes positive or even negative semi-definite, depending on the problem formulation in question. In addition, the numerical rank deficiency of the kernel covariance matrix and ill-posedness of the overall linear system have to be factored in, when dealing with not only the inverse problem, but also the direct problem.
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Numerical Simulation and Prediction of Atmospheric Aerosol Extinction
97.3
Selection of Coordinates of Prediction Kernel Based on Singular Value Decomposition
Any m-by-n kernel matrix K, either full-rank or rankdeficient, can be decomposed into: K ¼ ULV T
(97.6)
where the columns of m-by-m orthogonal matrix U are the eigenvectors fui g of ðKK T Þ, the columns of n-by-n orthogonal matrix V are the eigenvectors fvi g of ðKK T Þ; and the non-negative singular values fmi g on the main diagonal of m-by-n matrix L are the square roots of the eigenvalues of both ðKKT Þ and ðK T KÞ (Bockmann 2001). The orthonormal columns of U and V span the four fundamental spaces. The first re columns of U span the column space RðKK T Þ of K and the next m re columns of U form a basis for the leftnull space RðKÞ? of K where e represents the random relative measurement error. The first re columns of V span the row space RðK T KÞ of K and the next n re columns of V form a basis for the null space NðKÞ of K (Strang 1988). The orthonormal eigenvectors fui g; i ¼ 1; 2; . . . ; re of ðKK T Þ form a basis for the column space RðKK T Þ of the kernel matrix K and span the space of the data of true optical measurements, bext (Harsdorf and Reuter 2000): bext ¼
re X
hbext ; ui iui ; bext 2 RðKK T Þ
(97.7)
i¼1
In addition, the right singular vector fvi g of ðK T KÞ is mapped onto the left singular vector fui g of ðKK T Þ via the corresponding non-negative singular values fmi g on the main diagonal of m-by-n matrix L: K T Kui ¼ mi Kvi ¼ m2i ui ¼ li ui
(97.8)
where K T K ¼ UL2 U T is a compact self-adjoint linear operator mapping from y 2 L2 ½lmin ;lmax onto itself using the same notation as in (97.3). And (97.8), either in the form of K T Kui ¼ li ui or K T K ¼ UL2 U T , provides the basis for the selection of the coordinates ali of the prediction kernel Kx . In this regard, the fact that the numerical rank deficiency of the kernel covariance matrix and the smoothness thereof seem plausible
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deciding factors in making the optimal selection of the coordinates of the prediction kernel necessitates the following section. The numerical e rankðre Þ of the kernel covariance matrix is the number of its singular values greater than a predefined tolerance level, meaning that the same number of rows or columns of the kernel covariance matrix are guaranteed to be linearly independent within a degree of accuracy equivalent to the preset tolerance level (Hansen 1998). In this context, the question of how small is small is not just a side issue. Accordingly, the selection of the coordinates m P a li K li ali of the prediction kernel Kx that makes li
as close to zero as reasonable should be made based on or at least verified by more rigorous reasoning.
97.4
Analysis of Singular Value Spectrum in Relation to Selection of Coordinates of Prediction Kernel
The smoothness of the kernel covariance matrix defined by (97.5) is higher but the information potentialities thereof are lower as compared to that of the kernel matrix (Zuev and Naats 1982). A loss of information occurs when the non-smooth and high frequency components of the unknown input parameter x in (97.3) in the context of the inverse problem are smoothed out by the discrete kernel matrix Kd , amounting to which an arbitrarily small perturbation in the optical measurements is amplified and results in the highly unstable solution. The smoothness of the kernel covariance matrix determines the rate at which the singular value spectrum thereof decays and in turn provides one with a plausible standard by which the credibility of the selection of the coordinates of the prediction kernel is judged. The most widely accepted measure of the smoothness of the kernel covariance matrix is the reciprocal condition number (Sorensen et al. 1995). One might look at the angular difference between the space spanned by a collection of computed left singular vectors and the corresponding true space. The choice of the singular values and the corresponding coordinates of the prediction kernel could act as a regularization parameter in the possibly highly unstable problem of making the prediction of measurements.
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Incorporating A Priori Knowledge of Prediction Errors
It can be deduced that the solution to (97.3) comprises three parts: firstly, the undisturbed low frequency components spanning the true solution space: x¼
re X
hxi ; vi ivi ; x 2RðK T KÞ
(97.9)
i¼1
Secondly, the high frequency components of the solution which constitutes the noise; and lastly, the blind components perpendicular to the kernel function space which are filtered out by the measurement kernel (Yin et al. 1996). Prediction errors due to errors in the blind components of the solution cannot be reflected upon in the prediction of optical measurements. Assuming that the prediction errors due to an arbitrarily small perturbation in either the complex aerosol refractive index (m) or the upper radius bound rmax is negligible, the prediction errors due to those errors in the visible components of the solution needs to be filtered out by appropriate means. In this context, we can deduce that the choice of the singular values and the corresponding coordinates of the prediction kernel act as a regularization parameter in the form of the cutoff point between the high frequency noise and undisturbed low frequency parts of the prediction of optical measurements. The Generalized Cross Validation method provides a way of choosing the singular values and the corresponding coordinates of the prediction kernel in such a way that those coordinates of the prediction kernel that are likely to cause high error amplification are being discarded (Muller et al. 1999).
97.6
Prediction of Extinction Measurements
97.6.1 Channel Selection Based on Signal-to-Noise Ratio The smoothness of the kernel matrix and the information potentialities thereof vary, depending on the choice of a combination made from a group of measurement-wavelengths available. This is partly because, firstly, the measurement error of one channel in a multi-frequency setting can be transferred to
its adjacent channels largely owing to the partial overlapping of the kernels; and, secondly, the loss of necessary information may results as the number of available measurement-wavelengths is limited to a finite number. Listed are the measurement-wavelengths in mm used in this study: [0.414, 0.500, 0.615, 0.670, 0.862]. The prediction of extinction measurements is made at the measurement-wavelength of 0.615 mm. Assuming that the minimum number of sounding channels allowed in a combination is three and each combination should include the prediction wavelength of 0.615 mm, there are a total of 11 possible combinations to choose from. Assuming that the prediction errors due to an arbitrarily small perturbation in the complex aerosol refractive index (m) is negligible; and the prediction errors due to errors in the blind components of the solution is also negligible, then the signal-to-noise ratio (SNR) of the kernel matrix may take the following form (Yin et al. 1996): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !1 n n n X 1 X 1 X 2 e m2 n3 i¼1 i i¼1 mi 2 i¼1 i
(97.10)
where fmi g is the singular value of the kernel matrix and n is the number of measurement-wavelengths in the combination, and ei is the random relative measurement error of channel i. The optimal choice of combination of sounding channels is one that gives the largest signal-to-noise ratio (SNR).
97.6.2 Prediction of Measurements and Introduction of Scaling Factors Presented in this section is a kek2 -based method for making the prediction of extinction measurements which amounts to choosing the coordinates ali of the prediction kernel Kx and thescaling factors a* and b* P ali ax Kli such that the residual norm a Kx b l6¼x 2
is equal to a priori upper bound de for the error norm kek2 , where kek2 de ¼ dtrue Ddp , dp is the prediction error; and the scaling factors are defined by dbea a ¼ ebdc b2 ac ; b ¼ b2 ac where variables {a, b, c, d, e} are computed in such a way that the residual difference between de and di at each iteration is minimized in a root-mean-square sense (Heintzenberg et al. 1981).
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Numerical Simulation and Prediction of Atmospheric Aerosol Extinction
A priori upper bound for the error norm kek2 is confined in part by the machine-dependent precision (e) of the singular value decomposition and in part by the number of sounding channels that makes up an optimal combination with the largest signal-to-noise ratio and can be written as de ¼
97.7
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eðno.of wavelengthsÞðlargestSingularValueÞ
Numerical Simulation
The possible sources of errors in the problem of making the prediction of extinction measurements stem from the fact that an arbitrarily small perturbation either in the complex aerosol refractive index (m) or in the upper radius bound rmax is likely to be amplified primarily owing to the oscillatory nature of the optical efficiency factor in addition to the higher degree of smoothness of its corresponding kernel matrix Kd defined by (97.3). In this study, either dust-like or maritime aerosols are assumed. As suggested in the paper by Logan et al. (2002), the aerosol radius of such particles is assumed to fall 0.04 and 1.40 mm. Likewise, the real and imaginary parts of refractive index of such aerosols relative to the surrounding medium are kept constant at 1.53 and 0.0078 respectively. The degree of ill-posedness of the problem defined by (97.1) increases with the increasing absorption degree in which the larger absorption degree is accompanied by the larger value of the imaginary part of the complex aerosol refractive index (m). The prediction errors due to an arbitrarily small perturbation in the complex aerosol refractive index (m) are assumed to be negligible. It is often the case that the uncertainty in the choice of the upper radius bound rmax is a limiting factor in the correct computation of light scattering by aerosols defined by (97.1). One way to minimize the above-mentioned uncertainty is to divide the kernels by rk , where k is the kernel order and r is the aerosol radius in mm. The purpose of having the order of the kernel is to decrease the contribution of the big particle radii (Veselovskii et al. 2004). The kernel order can change according to the type of aerosols in question. The kernel order is varied between 0 for slower convergence and 2 for faster convergence. The usage of the excessive number of grid points in the interval ½rmin ; rmax does not
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necessarily give the better prediction results. Instead, it often leads to the rank-deficiency and singularity. The smoothness of the kernel covariance matrix defined by (97.5) is higher but its information potentialities are lower as compared to that of the kernel matrix partly because the matrix multiplication of two highly ill-conditioned matrices may lead to unnecessary inaccuracies. The number of grid points used to form the matrices ðKK T Þ and ðK T KÞ is preferably chosen in a way that neither of the kernel matrices K or K T is rank deficient. The problem of making the prediction of extinction measurements falls into the category of direct problem wherein the prediction of measurements is made from the first kind Fredholm integral system defined by (97.2), provided that the input aerosol size distribution is assumed to be already known within the reasonable degree of accuracy. In this study, the log normal distribution is assumed, where the volume geometric mean radius is 0.1588, the geometric standard deviation is 1.6113, and the volume constant per unit area is set to 0.0353, assuming that aerosols in question are either dust-like or maritime. Out of a group of four different types of aerosol size distributions, namely, volume; surface-area; radius; and number size distributions, the choice of the volume size distribution as the input gives better prediction results in general. In this study, a set of extinction measurement values produced by computer simulation is used to test the validity of this prediction method. The smoothness of the kernel matrix and the information potentialities thereof vary according to the choice of a combination of sounding channels made. The optimal choice of combination is supposedly the one that gives the largest signal-to-noise ratio (SNR) defined by (97.10). In Table 97.1, the optimal combination is the one with the largest SNR of 2.807. The prediction of extinction measurement (dimensionless) of near-zero made at the prediction wavelength of 615 nm is better than the prediction made via linear interpolation (0.3882). If one has to choose between the two equally reasonable prediction results, say SNR of 2.019 and SNR of 2.132, those with the larger number of elements would be a better choice, simply because there is more information available. However, a drawback of this choice is that the prediction errors increases with the increasing number of sounding channels used as is the case in Table 97.1.
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Table 97.1 Sample prediction result Ch1 (nm) 414 414 414 414 0 0 414 0 414 0 414
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Ch 2 (nm) 500 500 500 0 500 0 0 500 0 500 500
Ch 3 (nm) 615 615 615 615 615 615 615 615 615 615 615
Ch 4 (nm) 670 0 670 670 670 670 670 670 0 0 0
Concluding Remarks
Presented in this study is a kek2 -based method for making the prediction of measurements which amounts to choosing the coordinates of the prediction kernel and the scaling factors such that the residual norm is equal to a priori upper bound for the error norm. The prediction error is accurate to within a priori upper error bound and is confined in part by the machine-dependent precision of the singular value decomposition used and in part by the number of sounding channels used. The prediction of extinction measurement made at the prediction wavelength of 615 nm is better than the prediction made via linear interpolation. The possibility of extension of this study to other measurement-wavelengths hinges on the smoothness of the kernel matrix in question and the information potentialities thereof. The partial overlapping of the kernels would be one limiting factor. It is up to the individual to decide on the feasibility of the extension of this study to any other measurement-wavelengths of his choice. However, the information on the smoothness and information potentialities of the kernel matrix has to be factored into the problem formulation if one is to expect a successful outcome.
References Bockmann C (2001) Hybrid regularization method for the illposed inversion of multi wavelength lidar data in the retrieval of aerosol size distributions. Appl Opt 40:1329–1342
Ch 5 (nm) 862 862 0 862 862 862 0 0 862 862 0
SNR 1.939 2.293 1.805 1.905 2.019 2.132 1.821 1.881 2.807 2.613 2.304
Prediction at 615 nm 0.7808 0.5162 0.4976 0.5470 0.4399 0.2793 0.3180 0.2569 0.3238 0.2384 0.4693
Bockmann C, Sarkozi J (1999) The ill-posed inversion of multi wavelength lidar data by a hybrid method of variable projection. SPIE 3816:282–293 Dubovik O, Kin MD (2000) A flexible inversion algorithm for retrieval of aerosol optical properties from Sun and sky radiance measurements. J Geophys Res 105(16):673–696 Hansen PC (1998) Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion. SIAM 6 (19):46–49 Harsdorf S, Reuter R (2000) Stable deconvolution of noisy lidar signals. EARSeL eProceedings 1:88–95 Heintzenberg J, Muller H, Quenzel H, Thomalla E (1981) Information content of the optical data with respect to aerosol properties: numerical studies with a randomized minimization-search-technique inversion algorithm. Appl Opt 20:1308–1315 Logan A, Higurashi A, Nakajima T (2002) Tropospheric aerosol optical thickness from the GOCART model and comparisons with satellite and sun photometer measurements. J Atmos Sci 59:461–483 Muller D, Wandinger U, Ansmann A (1999) Microphysical particle parameters from extinction and backscatter lidar data by inversion with regularization: theory. Appl Opt 38:2346–2357 Sorensen D et al (1995) LAPACK users’ guide. Society for Industrial and Applied Mathematics, Philadelphia, PA Strang G (1988) Linear algebra and its applications. Harcourt Brace Jovanovich, San Diego, CA Twomey S (1996) Introduction to the mathematics of inversion in remote sensing and indirect measurements. Dover, Mineola, NY Veselovskii L, Kolgotin A, Griaznov V, Muller D, Franke K, Whiteman DN (2004) Inversion of multi wavelength Raman lidar data for retrieval of bimodal aerosol size distributions. Appl Opt 43:1180–1195 Yin Q, Zhang Z, Kuang D (1996) Channel selection of atmospheric remote sensing. Appl Opt 35:7136–7143 Zuev VE, Naats IE (1982) Inverse problems of lidar sensing of the atmosphere. Springer, Berlin, p 67
Impact of Atmospheric Delay Reduction Using KARAT on GPS/PPP Analysis
98
Ryuichi Ichikawa, Thomas Hobiger, Yasuhiro Koyama, and Tetsuro Kondo
Abstract
We have been developing a state-of-art tool to estimate the atmospheric path delays by ray-tracing through meso-scale analysis (MANAL data) data, which is operationally used for numerical weather prediction by Japan Meteorological Agency (JMA). The tools, which we have named “KAshima RAytracing Tools (KARAT)”, are capable of calculating total slant delays and ray-bending angles considering real atmospheric phenomena. The KARAT can estimate atmospheric slant delays by an analytical 2-D ray-propagation model by Thayer and a 3-D Eikonal solver. The biases of slant delay estimates between the Thayer model and the modern mapping functions, which are ranging from 18 to 90 mm, are considered to be a deficiency of the mapping functions. We compared PPP solutions using KARAT with that using the Global Mapping Function (GMF) and Vienna Mapping Function 1 (VMF1) for GPS sites of the GEONET (GPS Earth Observation Network System) operated by Geographical Survey Institute (GSI). In our comparison 57 stations of GEONET during the year of 2008 were processed. The KARAT solutions are slightly better than the solutions using VMF1 and GMF with linear gradient model for horizontal and height positions. Our results imply that KARAT is a useful tool for an efficient reduction of atmospheric path delays in radio based space geodetic techniques such as GNSS and VLBI.
R. Ichikawa Space-Time Standards Group, Kashima Space Research Center, National Institute of Information and Communications Technology (NICT), 893-1 Hirai, Kashima, Ibaraki, Japan e-mail:
[email protected] T. Hobiger Y. Koyama National Institute of Information and Communications Technology (NICT), 4-2-1 Nukui-Kitamachi, Koganei, Tokyo, Japan T. Kondo National Institute of Information and Communications Technology (NICT), 4-2-1 Nukui-Kitamachi, Koganei, Tokyo, Japan Department of Space Survey and Information Technology, Ajou University, Suwon, Republic of Korea
98.1
Introduction
Radio signal delays associated with the neutral atmosphere are one of the major error sources of space geodesy such as GPS, GLONASS, GALILEO, VLBI, In-SAR measurements. The recent geodetic analyses are carried out by applying the modern mapping functions based on the numerical weather analysis fields with horizontal gradient model for the purpose of a better modeling of these propagation delays, thereby improving the repeatability of site coordinates. The Global Mapping Function (GMF) (Boehm et al. 2006a), and Vienna Mapping Function 1 (VMF1)
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_98, # Springer-Verlag Berlin Heidelberg 2012
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(Boehm and Schuh 2004; Boehm et al. 2006b) have been successfully applied to model the zenith hydrostatic delay in the recent years. In addition, the lateral spatial variation of wet delay is reduced by linear gradient estimation (MacMillan 1995; Chen and Herring 1997). The anisotropic mapping function is also a powerful tool for removing or calibrating the effects of horizontal variability of atmosphere within GNSS and VLBI analyses. Atmospheric gradients are assumed to have a simple linear form which can be modeled by the anisotropic mapping function. However, it has been suggested that this assumption is not always appropriate in the context of intense mesoscale phenomena such as the passage of a cold front, heavy rainfall, or severe storms. Based on prior work by Ichikawa et al. (1995), we have developed a state-of-art tool to obtain atmospheric slant path delays by ray-tracing through the meso-scale analysis data from numerical weather prediction with 10 km horizontal resolution provided by the Japan Meteorological Agency (JMA) (Hobiger et al. 2008a, b). The tool, which we have named “KAshima RAytracing Tools (KARAT)”, is capable of calculating total slant delays and ray-bending angles considering real atmospheric phenomena. Hobiger et al. (2008a) preliminarily compared precise point positioning (PPP) estimates using KARAT with that using the GMF based on GPS data of GEONET (GPS Earth Observation Network System) operated by Geographical Survey Institute (GSI). Under the various atmospheric conditions the results imply that the performance of KARAT is almost equal to the solution which is obtained by applying the GMF with gradients. In our study, we have compared PPP processed position solutions using KARAT with those using state-of-the-art mapping functions in order to evaluate the present KARAT potential for longer time periods. In our comparison 57 stations of GEONET data during the year 2008 were processed.
98.2
KARAT and Modern Mapping Functions
The KARAT have been developed at the National Institute of Information and Communications Technology (NICT), Japan and are capable of calculating total slant delays and ray-bending angles.
The JMA meso-scale analysis data (which will be called “JMA MANAL data” hereafter) which we used in our study provides temperature, humidity, and pressure values at the surface and at 21 height levels (which vary between several tens of meters and about 31 km), for each node in a 10 km by 10 km grid that covers Japan islands, the surrounding ocean and East Asia (Saito et al. 2006). The 3-hourly operational products are available from JMA since March, 2006. A linear time interpolation is implemented in KARAT to obtain results at arbitrary epochs which allows also to evaluate temporal changes of estimates. Further details of KARAT are described in Hobiger et al. (2008a, b). KARAT can estimate atmospheric slant delays by two different calculation schemes as shown in Fig. 98.1. These are (1) a piece-wise linear propagation, (2) an analytical 2-D ray-propagation model by Thayer (1967), and (3) a 3-D Eikonal solver (Hobiger et al. 2008b). Though the third scheme can include small scale variability of atmosphere in the horizontal component, it has a significant disadvantage due to the massive computational load. In this paper we discuss estimations using the second and the third schemes since we would like to focus on both sophisticated methods. On the other hand, modern mapping functions such as the Niell Mapping Function (NMF) (Niell 1996), the Isobaric Mapping Function (IMF) (Niell 2001), GMF, and VMF1 aid in the estimation of zenith delays in the GNSS and VLBI processing in the past several years. The lateral spatial variation of atmospheric delay is usually reduced by linear gradient estimation. These mapping functions are considered to be globally available for reduction of atmospheric path delays within GNSS and VLBI processing. In order to assess the accuracy of these mapping functions in the East Asia region, we compared KARAT-based slant delays for the whole azimuth range from 0 to 360 in step of 1 at an elevation of 5 with each mapping function values at 16 IGS stations. Here, the Thayer model is applied for KARAT calculation. We examined both the mean and standard deviation of slant delay residuals by the equation DSL ¼ SLKARAT mf ðyÞ ZTDKARAT
(98.1)
where, DSL is a slant delay residual, SLKARAT is a KARAT-based slant delay, and mf ðyÞ is the mapping function. ZTDKARAT is a KARAT-based zenith total delay which means a direct numerical integration of
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Table 98.1 Mean bias and standard deviations (in millimeter) of slant delay residuals at elevation of 5 for 16 IGS stations in the East Asia region IGS YSSK MIZU TSKB KSMV KGNI MTKA USUD BJFS DAEJ SUWN AIRA GMSD CCJM TWTF TCMS TNML
VMF1-KARAT –8.3 18.7 16.1 43.6 12.3 48.8 8.9 45.0 20.7 44.9 17.9 44.1 56.9 32.1 13.6 37.7 16.4 64.6 –0.6 54.0 –78.5 65.4 6.5 50.2 15.8 65.0 21.5 79.7 63.5 93.3 63.5 93.3
GMF-KARAT –10.8 30.6 21.9 50.1 22.6 52.7 19.0 49.6 27.3 49.7 25.4 48.4 1.6 35.1 14.6 42.4 18.9 68.7 4.0 59.4 –93.9 63.1 0.5 53.4 2.5 66.0 5.3 80.0 53.8 92.9 53.9 92.9
NMF-KARAT –14.1 32.3 31.3 51.0 38.4 53.4 34.7 50.2 43.7 50.1 41.9 48.8 12.1 35.5 24.8 42.9 36.5 69.7 20.0 59.3 –82.1 63.0 25.2 54.9 28.4 66.7 42.5 81.4 92.9 94.1 93.0 94.1
Fig. 98.2 Histogram of residuals in slant delays at elevation of 5 from the end of June to September, 2007. Values between VMF1 and KARAT (upper), those between GMF and KARAT (middle), and those between NMF and KARAT (lower) are shown, respectively. The KARAT estimates are obtained using the Thayer model
Fig. 98.1 Three schemes of KARAT calculation [(1) piecewise linear, (2) Thayer model, (3) Eikonal solution strategy] (see Hobiger et al. (2008b) in detail)
the atmospheric refractivity along a zenith path. The mean value of DSL represents a bias associated with each mapping function, whereas the standard deviation value of DSL represents the scatter due to horizontal variability of atmosphere (mainly caused by water vapor variation). The results are summarized in Table 98.1 and one example histogram of the residuals at Kashima is shown in Fig. 98.2. Comparisons between KARAT-based slant delay and typical mapping
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functions indicate large biases ranging from 18 to 90 mm. If we assume that MANAL delays can represent the real atmosphere, these biases suggest systematic errors of hydrostatic delay estimates by using the modern mapping functions. On the other hand, large values of standard deviation, which are considered as effects of water vapor variability, are also shown in Table 98.1. Since the scatterings exceed the biases, it is difficult to discuss the significance of the biases.
98.3
Precise Point Positioning Results for GEONET Stations
In order to compare KARAT processing and modern mapping functions we analyzed data sets of GEONET, which is a nationwide GPS network operated by GSI. In our comparison 57 stations from GEONET of the year 2008 were considered for processing. We selected the stations which were not affected by crustal deformations caused by seismic activities. Figure 98.3 shows the locations of the selected stations in our study. Since these stations are distributed over the whole Japan islands evenly, we can investigate effects of various weather conditions on the processing. In addition, we can avoid uncertainties due to the individual difference of equipments in term of the same type of antenna-receiver set in GEONET. At first, precise point positioning (PPP) estimates covering the whole period shown above were obtained
Fig. 98.3 The GEONET stations processed in this study. The boundary of JMA meso-scale analysis (MANAL) data is also shown (blue line). The two triangles denote the location of Tsukuba and Koganei GEONET stations, respectively (see Figs. 98.4 and 98.5)
for all sites using GPSTOOLS (Takasu and Kasai 2005). The troposphere delays have been modeled by dry (using the Saastamoinen (1972) model) and wet constituents. The latter one was estimated as unknown parameters using the GMF and VMF1 together with linear gradients (Chen and Herring 1997). Process noise values of zenith delays and linear gradients were set to 0.1 mm and 0.01 mm, respectively. The elevation cutoff angle was set to 10 and downweighting at lower elevation angles was applied. The ocean loading correction based on NAO.99b model was applied (Matsumoto et al. 2000) and no atmospheric loading was applied. The a-priori hydrostatic zenith delays were computed from the Saastamoinen (1972) model based on standard atmosphere values with station height correction. The Kalman-filter estimation interval was set to 300 s, without overlapping data from consecutive days. The daily position estimates from these solutions act a reference to which the ray-traced solutions can be compared. In our comparison, PPP estimations using the GMF and VMF1 without linear gradient were also performed. We have not yet applied the Eikonal equation method for reducing the atmospheric delays from GPS data sets due to heavy computational and time-consuming load previously (see Sect. 98.2). According to our preliminary computations, slant delay differences between the Eikonal calculation and the Thayer model reach values of up to 5 mm at an elevation of 5 . In addition, the Eikonal calculation can predict smaller scale perturbation which is not retrieved using both Thayer and linear models. These results suggest that such higher order (mainly horizontal) variations of slant delays can be reduced from the GPS data using JMA MANAL data. Thus we performed KARAT calculation using both, the Thayer model and the Eikonal solver, in this study. Figure 98.4 represents two examples of station positions time series at Tsukuba and Koganei during the year 2008. In this figure two cases of solutions for each component are shown, i.e. KARAT solution using Eikonal solver with respect to the VMF1 solution and VMF1 with gradient solution with respect to the VMF1 solution. The station movements due to plate motion were already subtracted from the time series prior to the comparison. The large amplitudes due to high water vapor variability, which mean poorest repeatabilities, are presented during summer season (from June to August). The time series at both
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Fig. 98.4 Time series of station position differences at Tsukuba (upper) and Koganei (lower) during the year of 2008 (Station locations are indicated in Fig. 98.3). The position differences for solutions using Eikonal solver and VMF1 with gradient model with respect to the VMF1 solutions are represented, respectively
stations agree very well in both amplitude and phase through 1 year in spite of such high variability. The differences of both time series for each station are >0.2 mm in all coordinates. This means slant delay estimations using Eikonal solver and those using VMF1 with gradient are almost identical. In order to examine the position error magnitude the monthly averaged repetabilities for each coordinate component at both stations are displayed in Fig. 98.5. In this figure five cases of solutions (i.e. KARAT solution using Eikonal solver, KARAT solution using the Thayer model, VMF1 solution with gradient, VMF1 solution without gradient, GMF with gradient) are shown. The results of VMF1 without gradient reveal the largest repeatability value for all components at both stations during the summer season (July, August, and September), as one would expect. Tsukuba and Koganei have undergone severe heavy rainfall event during August 26–31, 2008. Especially, the total rainfall around Tsukuba was about 300 mm during these 6 days. The north–south position
Fig. 98.5 Monthly averaged repeatabilities of station positions at Tsukuba (upper) and Koganei (lower) during year of 2008
errors were caused by steep water vapor gradient associated with an EW rain band which lies around both stations. Such large position errors are partly reduced using the modern mapping functions with gradient model as shown in Fig. 98.5. On the other hand, the results of KARAT solutions (both the Eikonal solver and the Thayer model) are much better for the north–south component at the both station during the July and August. These suggest that the both KARAT solutions are quite competitive to the modern mapping functions with gradient model. In addition, Fig. 98.6, which shows the histogram of 57 stations repeatabilities for each station coordinate component (i.e. the north, east, and vertical errors) for five cases, also indicates that the north–south component of both KARAT solutions (Eikonal solver and Thayer model) are better than the modern mapping functions (GMF and VMF1) with gradient solutions.
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Fig. 98.7 Averaged repeatability of station position during year of 2008 for 57 GEONET stations shown in Fig. 98.3 Table 98.2 Summary of repeatabilities in millimeter for each coordinate component (north, east, and vertical) Eikonal solver (KARAT) Thayer model (KARAT) VMF1 with gradient VMF1 w/o gradient GMF with gradient GMF w/o gradient
EW 6.3 6.2 6.7 6.9 6.7 6.9
NS 4.6 4.6 4.8 5.3 4.8 5.3
UD 7.0 6.7 7.8 8.1 7.9 8.2
GMF without gradient) are represented. It indicates that both KARAT solutions are slightly better than the modern mapping functions with gradient solution. However, there are no significant differences between the Eikonal solver and the Thayer model. These characteristics are also summarized in Table 98.2. One has to consider that the time-resolution of the JMA 10 km MANAL data is 3 h, whereas the PPP processing including gradient estimation was performed for 300 s interval. Under the extreme atmospheric condition such as a severe rainfall event the 3 h time spacing and the 10 km horizontal resolution of the JMA MANAL data may not be always sufficiently accurate to reduce atmospheric path delay effects. Fig. 98.6 Distributions of repeatabilities for each station coordinate component (north, east, and vertical) for each solution
Finally, Fig. 98.7 shows the averaged repeatabilities for all 57 stations. In this figure the results for each coordinate component for all six solutions (i.e. Eikonal solver, Thayer model, VMF1 with gradient, VMF1 without gradient, GMF with gradient, and
98.4
Summary
We have assessed the performance of ray-traced atmospheric delay correction by comparison between the precise point positioning (PPP) solutions using the ray-tracing tool “KARAT” through the JMA MANAL data with those using the modern mapping
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Impact of Atmospheric Delay Reduction Using KARAT on GPS/PPP Analysis
functions based on numerical weather models. In our comparison 57 stations of GEONET during the year of 2008 were processed. The KARAT solutions are slightly better than the solutions using VMF1 and GMF with a linear gradient model for both horizontal and height positions. On the other hand, there were no significant differences between the two KARAT solutions, i.e. Eikonal solver and Thayer model. We need further investigations to evaluate the capability of KARAT to reduce atmospheric path delays under various topographic and meteorological regimes. One advantage of KARAT is that the reduction of atmospheric path delay will become more accurate each time the numerical weather model are improved (i.e. time and spatial resolution, including new observation data). In spite of the present model imperfectness and coarse time resolution, we think that KARAT will help to support station position determination by improving the numerical stability due to a reduction of unknown parameters. Acknowledgments We would like to thank the Geographical Survey Institute, Japan for providing GEONET data sets. We also thank the Japan Meteorological Agency for providing data and products. This study was supported by a Grant-in-Aid for Scientific Research A (No. 21241043) from the Japan Society for the Promotion of Science.
References Boehm J, Schuh H (2004) Vienna mapping functions in VLBI analyses. Geophys Res Lett 31:L01603. doi:10.1029/ 2003GL018984 Boehm J, Niell A, Tregoning P, Schuh H (2006a) Global mapping function (GMF): a new empirical mapping function
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based on numerical weather model data. Geophys Res Lett 33:L07304 Boehm J, Werl B, Schuh H (2006b) Troposphere mapping functions for GPS and very long baseline interferometry from European centre for medium-range weather forecasts operational analysis data. J Geophys Res 111:B02406 Chen G, Herring TA (1997) Effects of atmospheric azimuthal asymmetry on the analysis of space geodetic data. Geophys Res Lett 102:20489–20502 Hobiger T, Ichikawa R, Takasu T, Koyama Y, Kondo T (2008a) Ray-traced troposphere slant delays for precise point positioning. Earth Planets Space 60:e1–e4 Hobiger T, Ichikawa R, Koyama Y, Kondo T (2008b) Fast and accurate ray-tracing algorithms for real-time space geodetic applications using numerical weather models. J Geophys Res 113(D203027):1.14 Ichikawa R, Kasahara M, Mannoji N, Naito I (1995) Estimations of atmospheric excess path delay based on three-dimensional, numerical prediction model data. J Geod Soc Jpn 41:379–408 MacMillan DS (1995) Atmospheric gradients from very long baseline interferometry observations. Geophys Res Lett 22:1041–1044 Matsumoto K, Takanezawa T, Ooe M (2000) Ocean tide models developed by assimilating TOPEX/POSEIDON altimeter data into hydrodynamical model: a global model and a regional model around Japan. J Oceanograph 56:567–581 Niell AE (1996) Global mapping functions for the atmosphere delay at radio wavelengths. J Geophys Res 101(B2): 3227–3246 Niell AE (2001) Preliminary evaluation of atmospheric mapping functions based on numerical weather models. Phys Chem Earth 26:475–480 Saito K et al (2006) The operational JMA nonhydrostatic mesoscale model. Monthly Weather Rev 134:1266–1298 Saastamoinen J (1972) Contributions to the theory of atmospheric refraction, part 2. Bull G’eod’esique 107:13–34 Takasu T, Kasai S (2005) Evaluation of GPS precise point positioning (PPP) accuracy. IEICE Tech Rep 105 (208):40–45 Thayer GD (1967) A rapid and accurate ray tracing algorithm for a horizontally stratified atmosphere. Radio Sci 1 (2):249–252
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Modelling Tropospheric Zenith Delays Using Regression Models Based on Surface Meteorology Data
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Tama´s Tuchband and Szabolcs Ro´zsa
Abstract
The tropospheric zenith delay (ZTD) of GPS observations is closely related to the integrated water vapour (IWV) content of the atmosphere. The scale factor between the IWV and the ZWD is a function of the mean temperature of the water vapour, which can be computed by a linear regression equation based on the surface temperature. A similar linear regression can be used to compute the IWV from surface water vapour density. In this paper we show a formula derived from more than 10,000 radiosonde observations in Hungary. Using this relation, it is possible to estimate the IWV content of the atmosphere, which could be scaled down to tropospheric zenith wet delay. Two time intervals are used for the validation of the results. The first one was a stormy summer period, while the other was a dry winter period. The results show that this approach provides slightly better coordinate RMS than the Niell or the Hopfield model. Moreover the coordinate solutions are relatively stable during the summer period as well, when a heavy storm caused unstable weather conditions. Studying the performance of the local regression model, the Hopfield and the Niell (SaastamoinenþNiell mapping function) model it could be seen that the local regression model gave the best a priori tropospheric delays for the processing.
99.1
Introduction
Nowadays users demand better accuracy with the realtime precise point positioning (PPP) technique. There are different ways to improve the accuracy. We have to review all effects what disturb the signals. Two of these effects are caused by the atmosphere. The first
T. Tuchband (*) S. Ro´zsa Department of Geodesy and Surveying, Budapest University of Technology and Economics, Muegyetem rkp 3, Budapest 1111, Hungary e-mail:
[email protected]
effect is the influence of the ionosophere. This delay can be eliminated by combining observations collected at more than one frequency. The second effect is caused by the troposphere (Brunner and Welsch 1993). Tropospheric delay can be disassembled into two parts, the hydrostatic part which is about 90% and the wet part, which is about 10% of the total delay. The hydrostatic part is easily modelable. It depends on the thickness of the air column between the receiver and the upper border of the troposphere, and the air pressure. It is more difficult to model the wet part, since it depends on many factors (Beutler et al. 1989). There are several models that allow the computation of the tropospheric delay. There are global and local
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models. Global models can be used anywhere around the world. To use local models, additional parameters have to be measured. In this paper two widely used global models have been compared to a local regression model. The computed delays were used as a priori values for the processing.
99.2
The Applied Models
In the context of this paper, we refer to Niell model as the model formed by the Saastamoinen model and Niell mapping function. This global model is one of the most widely used models: 1255 þ 0:05 e ; ZTD ¼ 0:002277 p þ T (99.1) where ZTD is the Zenith Total Delay, p the air pressure in mbar, T the temperature of the water vapour in Kelvin, while e is the water vapour pressure (Seeber 1993). The zenith total delay in the Hopfield model is: ZTD ¼
106 5 3 2 p 77:64 ½40136 þ 148:72 ðT 273:16Þþ 7 6 T 7; 6 5 4 e e 5 12:96 þ 3:718 10 2 11000 T T (99.2)
where ZTD is the Zenith Total Delay, p the air pressure in mbar, T the temperature of the water vapour in Kelvin, while e is the water vapour pressure (Hofmann-Wellenhof et al. 1998). Moreover a local tropospheric model has also been investigated in this study. It consists of two parts, the hydrostatic part computed according to the Saastamoinen model and the wet part, which is computed using a local regression model (Ro´zsa et al. 2007): ZTD ¼ ZHD þ ZWD;
(99.3)
where ZTD is the Zenith Total Delay, ZHD the Zenith Hydrostatic Delay, while ZWD is the Zenith Wet Delay.
The hydrostatic part from the Saastamoinen model is: ZHD ¼ 0:002277 p;
(99.4)
where p is the air pressure. The wet part can be computed from: Rd k3 ZWD ¼ IWV Rv k1 þ k2 þ 106 ; Rv Tm (99.5) where IWV is the Integrated Water Vapour, Tm the average temperature of the water vapour, Rd, Rv are the specific gas constants, while k1, k2, k3 are empirical constants (Thayer 1974; Bevis et al. 1992). The integrated water vapour (IWV) can be estimated using surface meteorological data. In Budapest two radiosonde observations are carried out daily. The launch site is located in the center of the applied network of permanent GNSS stations. After processing more than 10,000 radiosonde observations a linear regression model was developed between the IWV and the observed surface water vapour density. Thus the IWV is computed by: IWV ¼ a rW þ b;
(99.6)
where, a and b are the parameters of the regression line found in Table 99.1, while rw is the surface water vapour density. Figure 99.1 shows the measured data and the annually computed regression line. The parameters of (99.6) have been computed both annually and monthly. In the latter case the observations were categorized according to the date of the observation, and the parameters were computed accordingly. Since the IWV has to be transformed to zenith wet delay, a scale coefficient (Q(Tm)) should be computed as a function of the mean temperature of the water vapour (Bevis et al. 1992): QðTm Þ ¼ Rv
Rd k3 k1 þ k2 þ 106 Rv Tm
(99.7)
The mean temperature of water vapour can be computed with a simple linear regression equation as a function of the observed surface temperature: Tm ¼ c Ts þ d;
(99.8)
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Modelling Tropospheric Zenith Delays Using Regression
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Table 99.1 Linear regression parameters and statistics between surface water vapour density (g/m3) and the integrated water vapour (mm)
Table 99.2 Linear regression parameters and statistics between surface temperature ( C) and the mean temperature of the water vapour (K)
Month 1 2 3 4 5 6 7 8 9 10 11 12 Year
a 2.50 2.43 2.45 2.28 2.18 2.13 2.24 2.21 2.43 2.50 2.47 2.50 2.39
R2 0.53 0.65 0.73 0.78 0.72 0.64 0.65 0.63 0.66 0.70 0.62 0.54 0.84
b –0.24 –0.78 –0.79 0.06 0.86 2.16 2.08 2.50 –0.72 –1.13 –0.38 –0.05 –0.14
s 2.90 2.37 2.34 2.26 2.93 3.75 3.79 3.86 3.63 3.83 3.32 3.07 3.27
N 849 787 862 837 847 828 864 857 831 865 829 859 10115
c 0.531 0.610 0.596 0.614 0.510 0.508 0.449 0.426 0.480 0.610 0.651 0.567 0.675
Month 1 2 3 4 5 6 7 8 9 10 11 12 Year
d 267.1 265.7 265.4 265.5 269.3 270.6 272.5 273.5 270.5 268.3 267.1 266.8 266.6
R2 0.37 0.49 0.50 0.58 0.48 0.45 0.41 0.41 0.41 0.54 0.55 0.40 0.80
s 3.16 3.39 3.03 2.88 2.86 2.99 2.75 2.75 2.84 3.07 2.92 3.09 3.21
N 849 787 862 837 847 828 864 857 831 865 829 859 10115
a and b are the regression parameters, R is the coefficient of determination, s is the standard deviation, while N is the number of samples
c and d are the regression parameters, R is the coefficient of determination, s is the standard deviation, while N is the number of samples
kg/m2
K 300 295 290 285 280 275 270 265 260 255 250
50 y = 2.3864 x - 0.1383 R2 = 0.8432
40 30 20 10 0 0
5
10
15
20 g/m3
Fig. 99.1 Integrated water vapour as a function of surface water vapour density (annual model)
where Ts is the observed surface temperature in C and c and d are the parameters of the regression line found in Table 99.2. Figure 99.2 shows the measured data and the annually computed regression line. To compute the Q(Tm) scale coefficient other equations can be used, too (Emardson and Derks 2000).
99.3
Data
In this study the GPS observations taken in the Hungarian active permanent GNSS network were used. In the summer period four stations have been used
y = 0,6753x + 266,64 R2 = 0,7952
-20
0
20
40
C Fig. 99.2 Average temperature of the water vapour as a function of surface temperature (annual model)
around Budapest, while in the winter period six stations were processed in the same area (Fig. 99.3). The surface meteorological observations such as the temperature, the relative humidity and the air pressure were provided by the Hungarian Meteorological Service. The observations were processed with the Bernese GPS Software V5.0 (Dach et al. 2007) using a precise point positioning scheme. To model the satellite orbit and clock errors the final products of the IGS (International GNSS Service) have been used. The effect of the ionosphere was eliminated by the iono-free linear combination (L3) of the L1 and L2 frequencies.
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Table 99.3 Coordinate differences [mm] between the Hopfield-and the Niell daily solutions, as well as the Regression and the Niell daily solutions in North, East and Up components 2006.08. JASZ MONO SZFV TATA
Hopfield N E 0.4 1.3 0.2 0.7 0.3 0.6 0.3 1.2
U 27.8 26.4 25.6 25.2
Regression model N E 0.5 1.4 0.3 0.6 0.5 0.5 0.1 1.4
U 8.2 7.3 5.1 6.0
Summer period
Fig. 99.3 The location of the permanent stations around Budapest (in 2006 the studied network consisted of: JASZ, MONO, SZFV, TATA)
Using this approach, daily coordinate solutions were computed for each station. Two seasonally different time intervals were used to evaluate the models: a wet summer period in August 2006 (including the evolution of a severe weather front) and a dry winter period in February 2007.
2007.02 JASZ KECS MONO PENC SZFV TATA
RM
Results
The coordinate solutions were computed using all the aforementioned troposphere models (Niell, Hopfield and Regression model) using surface meteorological data provided by the Hungarian Meteorological Service. In the case of the Niell model the Saastamoinen model was used as an a priori troposphere model including the hydrostatic and the wet part as well. During the processing the residual zenith wet delay estimation was done using the wet Niell mapping function. In the second case, the Hopfield model was used as the a priori model for the tropospheric delays. In this case the residual wet delay has been estimated using the Hopfield mapping function. In the third case the regression model was used for the computation of the a priori values of the tropospheric delays and the residual wet delay estimation was done using the wet Niell mapping function. The a priori values were computed using the monthly regression parameters. It must be noted that the residual wet delays were estimated for every 2 h for all the models. In order to evaluate the performance of the local regression model, the coordinate solutions have been
Hopfield N E 0.7 0.5 0.3 0.3 0.5 0.4 0.6 0.9 0.9 0.5 0.4 5.6
U 13.7 16.1 13.3 11.1 11.5 29.3
Regression model N E 0.2 0.2 0.1 0.1 0.2 0.1 0.1 0.2 0.1 0.3 0.1 0.1
U 3.3 3.5 2.9 0.8 3.2 2.7
Winter period
NIELL
HOPFIELD
2.5 2.3 RMS [mm]
99.4
Table 99.4 Coordinate differences [mm] between the Hopfield and the Niell daily solutions, as well as the Regression and the Niell daily solutions in North, East and Up components
2.1 1.9 1.7 1.5
JASZ
MONO
SZFV
TATA
Fig. 99.4 RMS of the up component using different troposphere models. Summer period
compared with the solutions obtained using the Niell model. The same comparison was made between the Hopfield and the Niell model, too. The results can be found in Table 99.3 for the summer period, while the results of the winter period can be seen in Table 99.4. The tables show that the regression model provides a better performance than the Hopfield model. It can be seen that the Hopfield model gives a significantly worse performance in the stormy summer period. To compare the performance of the three models, the RMS (Root Mean Square) of the coordinate solutions were also used. In Figs. 99.4 and 99.5 the summer and winter 3-day average RMS of the up coordinate
99
Modelling Tropospheric Zenith Delays Using Regression RM
NIELL
793
HOPFIELD
RM
2.5
Niell
Hopfield
Estimated
2.55
ZTD [m]
RMS [mm]
2.3 2.1 1.9
2.5
2.45
1.7 1.5
JASZ
KECS MONO PENC
SZFV
Fig. 99.5 RMS of the up component using different troposphere models. Winter period RM
NIELL
2.4
TATA
0
4
8 12 16 20 0
4
8 12 16 20 0
4
8 12 16 20
Fig. 99.7 A priori ZTD values and the estimated values from 19 to 21 August, 2006 (summer period) at station JASZ RM
HOPFIELD
Niell
Hopfield
Estimated
2.4
120.0 103.7 ZTD [m]
Corrections [mm]
100.0 80.0
2.35
60.0 36.4
40.0 20.0 0.0
20.2 22.0
Summer
2.3
19.0
0
4
8 12 16 20 0
4
8 12 16 20 0
4
8 12 16 20
14.0 Winter
Fig. 99.8 A priori ZTD values and the estimated values from 26 to 28 February, 2007 (winter period) at station JASZ
Fig. 99.6 Average residuals between the a priori and the estimated total zenith tropospheric delays
Conclusions
components are shown for each station using the different models. It can be seen that the local regression model provides coordinate solutions with the same, or in some cases with slightly better RMS values. Figure 99.6 shows the mean of the estimated residual zenith wet delays for all the models in the summer as well as in the winter period. It can be seen that these delays are slightly better for the regression model than the Niell model. The large average residual wet delay in case of the Hopfield model might be caused by the strong weather front in the summer period, since it cannot be seen in the significantly drier and calmer winter period. The computed total zenith delays, and the values provided by the various “a priori” models can be seen in Fig. 99.7 for the summer period and Fig. 99.8 for the winter period for the station JASZ. It can be seen that the regression model performs slightly better compared to the Niell model in summer as well as in the winter period.
The results show that the local regression model can be used to model the zenith wet delay. The regression model provides similar coordinates compared to the Niell model, while it does not show the relatively high coordinate differences of the Hopfield model during the summer period either. The model provides slightly better RMS values than the others in summer and it has better values in winter than the Niell model. It must be noted that surface meteorological data have been used for the computation of the a priori tropospheric delays in all cases. The coordinate differences between the regression model and the Niell model are smaller than the differences between the corresponding coordinates obtained by the Hopfield and Niell models. The reason for this could be that the same mapping function was used for the estimation of the residual ZWD in case of the Niell and the local regression model and only the a priori models were different. The corrections of the a priori ZWD are smaller using the regression model than for the others. This
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means that the a priori values are closer to the real ones, thus a better accuracy can be achieved when the local regression models are used in real-time PPP applications. Comparing the results in the summer and the winter period it can be clearly seen that the residual wet delays are smaller in the winter period. This can be explained by the fact that the troposphere contained a low amount of water vapour in the winter period and the weather conditions were more stable compared to the summer period. Acknowledgements This investigation is supported by the Hungarian Institute of Geodesy, Cartography and Remote Sensing, and the Hungarian Meteorological Service. I would like to thank the constructive comments of the two anonymous reviewers which helped to significantly improve this paper.
References Beutler G, Bauersima I, Gurtner W, Rothacher M, Schildknecht T, Geiger A (1989) Atmospheric refraction and other
T. Tuchband and S. Ro´zsa important biases in GPS carrier phase observations. In: Brunner Fk (ed) Monograph 12, atmospheric effects on geodetic space measurements, School of Geomatic Engineering, The University of New South Wales, 15–44 Bevis M, Businger S, Herring TA, Rocken C, Anthes A, Ware R (1992) GPS meteorology: remote sensing of atmospheric water vapor using the global positioning system. J Geophys Res 97:15787–15801 Brunner FK, Welsch WM (1993) Effect of the troposphere on GPS measurements. GPS World 4(1):42–51 Dach R, Hugentobler U, Fridez P, Meindl M (2007) Bernese GPS Software, Version 5.0, Astronomical Institute, University of Bern Emardson TR, Derks HJP (2000) On the relation between the wet delay and the integrated precipitable water vapour in the European atmosphere. Meteorol Appl 7:61–68 Hofmann-Wellenhof B, Lichtenegger H, Collins J (1998) GPS theory and practice, 4th edn. Springer, Vienna Ro´zsa Sz, Dombai F, Ne´meth P, Ablonczy D (2007) Estimation of integrated water vapour from GPS observations. Geomatikai K€ozleme´nyek 12(1):187–196 (in Hungarian) Seeber G (1993) Satellite geodesy: foundations, methods & applications. Walter de Gruyter, Berlin, 531p Thayer GD (1974) An improved equation for the radio refractive index of air. Radio Sci 9:803–807
Calibration of Wet Tropospheric Delays in GPS Observation Using Raman Lidar Measurements
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P. Bosser, C. Thom, O. Bock, J. Pelon, and P. Willis
Abstract
Water vapor measurements from a Raman lidar developed conjointly by the IGN and the LATMOS/CNRS are used for documenting the water vapor heterogeneities and correcting GPS signal propagation delays in clear sky conditions. We use data from four 6 h-observing sessions during the VAPIC experiment (15 May–15 June 2004). The retrieval of zenith wet delays (ZWDs) from our Raman lidar is shown to agree well with radiosonde (0.6 2.5 mm) and microwave radiometers (6.6 1.2 and 6.0 3.8 mm) retrievals. ZWDs estimated from GPS data present a good consistency too (2.0 2.7 mm) but they are still shown to not represent properly the fast evolutions with high frequency variations correlation about 0.12. Part of the errors is also due to multipath and antenna phase center variations. Within this framework, methodologies for integrating of zenith lidar observations into the GPS processing are described. They include also a correction for multipath and antenna phase center variation. The best results are obtained when the lidar ZWDs are used for a priori correcting the GPS phase observations: discrepancies between lidar and
P. Bosser (*) C. Thom Institut Ge´ographique National, LOEMI, 73 avenue de Paris, 94165 Saint-Mande´, France e-mail:
[email protected] O. Bock Institut Ge´ographique National, LAREG, 6-8 Avenue Blaise Pascal, 77455 Marne-la-Valle´e, France Institut Pierre Simon Laplace, LATMOS, 4 place Jussieu, 75005 Paris, France J. Pelon Institut Pierre Simon Laplace, LATMOS, 4 place Jussieu, 75005 Paris, France P. Willis Institut Ge´ographique National, Direction Technique, 2 avenue Pasteur, 94165 Saint-Mande´, France Institut de Physique du Globe de Paris, PRES Sorbonne Paris Cite´, UFR STEP, 35 rue He´le´ne Brion, 75013 Paris, France S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_100, # Springer-Verlag Berlin Heidelberg 2012
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GPS estimates are then reduced to 1.1 1.4 mm. It is shown also that mapping function derived from the lidar vertical profiles performs nearly as well as the VMF1 mapping function.
100.1 Introduction It is now well established that the troposphere has a significant impact on the accuracy of GPS height estimation, see for example Kleijer (2004). The troposphere impact is denoted as tropospheric delay and is currently modeled following this formula in GPS analysis software (Kleijer 2004): DLðt; e; aÞ ¼ DLzh ðtÞ mh ðt; eÞ þ DLzw ðtÞ mw ðt; eÞ þ Gðt; e; aÞ (100:1Þ where DL(t, e, a) represents the total tropospheric delay at epoch t for an elevation e and an azimuth a, dLzh ðtÞ and dLzw ðtÞ denote respectively the hydrostatic and wet zenith delays, and mh(t, e) and mw(t, e) are respectively the hydrostatic and wet mapping functions that describe the elevation dependency of the hydrostatic and wet delays. At last, (t, e, a) represents the horizontal gradients that describe the azimuthal asymmetry of the troposphere. Recent developments in troposphere modeling have been made using information from Numerical Weather Models (Boehm et al. 2006). These developments induce significant improvement on space geodetic results, especially on height accuracy (Boehm et al. 2006; Tesmer et al. 2007). Numerical weather models enable the consideration of synoptic scale disturbances that typically affect daily or weekly GPS solutions. However, it has been shown in previous study that small scale vapor heterogeneities could impact vertical determination at shorter time-scales and cannot be properly modeled (Bock et al. 2001a). In light of this, a first approach consists in the use of Water Vapor Radiometers (WVR) for water vapor correction. Despite of good result (Ware et al. 1993) calibration errors on this technique are still an important issue (Bock et al. 2001b; Haefele et al. 2004). An alternative approach consists in using the Raman lidar technique for the water vapor correction in GPS measurements (Bock et al. 2001b). This paper is organized as follows. First, we briefly present principles of water vapor measurements by
Raman lidar. Then, we focus in ZWD measurements during a field experiment that we organized. In the last part, we present results from GPS data analysis using the lidar data.
100.2 Lidar Measurements The Raman lidar technique is based on the detection of frequency shifted backscattered light from specific molecules. In our case, light is transmitted at 355 nm and collected at 387 nm (Nitrogen) and 408 nm (water vapor). In order to eliminate common and usually unknown parameters in the lidar signals, the water vapor mixing ratio (WVMR) is derived from the ratio between these two signals (Whiteman et al. 1992). Two important limitations of this technique have to be mentioned. First, a calibration is still required to relate the formed ratio of signals to physical parameters (water vapor and air densities). The lidar calibration constant is determined from collocated radiosonde WVMR observations in the 1–3 km atmospheric layer; the accuracy is expected to be around 5%. Moreover, since solar radiation may exceed Raman backscattering and clouds would attenuate the transmitted signal, night-time and clear sky measurements are also mandatory, which unfortunately drastically limits the observation availability of this technique. Lidar WVMR profiles are retrieved with a typical space-time resolution of 15 m 5 min in the lowest layers and up to 500 m 30 min in the highest layers (7–8 km above the ground). Pseudo-slant wet delays (PSWDs) and ZWD are derived from lidar zenith pointing measurements assuming a spherical symmetric atmospheric refractivity field and using a raytracing algorithm inspired from Rocken et al. (2001). The WVMR profiles from the lidar are completed with collocated radiosoundings above the lidar range, and pressure and temperature profiles from radiosoundings are used throughout the troposphere. Radiosoundings are therefore assumed to be vertical by neglecting the horizontal drift of the balloons. A lidar mappingfunction is also derived from the ratio of the PSWD to
Calibration of Wet Tropospheric Delays in GPS Observation Using Raman Lidar Measurements
ZWD estimates. This mapping-function takes into account the vertical structure of the troposphere and its temporal evolution at the time-resolution of the lidar WVMR retrievals (5 min).
100.3 Comparison of ZWD Measurements During the VAPIC Campaign Here we are interested in measurements from the VAPIC field experiment which took place in Palaiseau, France (near Paris) in May–June 2004. ZWD are retrieved by various instruments. Zenith lidar measurements are available over four 6 h session (from 18 to 25 May). Humidity profiles are calibrated using collocated radiosoundings and ZWDs are estimated using retrieved humidity profiles and collocated radiosounding for complement as previously explained. Two or three radiosoundings were also available during each session. Two WVRs (called Hatpro and Drakkar) were operated on site. They provide integrated water-vapor (IWV) estimates at high time-resolution (5–60 s) that are converted into ZWDs using the formula proposed by Bevis et al. (1992). Even if IWV retrievals from WVRs may exhibit biases due to particularities in the temperature and humidity profiles (Revercomb et al. 2003) they can usually properly capture the short-term fluctuations in IWV (5 min to 1 h). A GPS station was installed close to the Raman lidar system (called Rameau). A GPS solution was produced with the GIPSY/OASIS II v 5.0 software in Precise Point Positioning (PPP) mode (Zumberge et al. 1997). Relative antenna models were used for consistency with the JPL legacy orbits and satellite clocks (covering 30 h, centered around midday on the day of interest) for 2004. IERS2003 recommendations for solid Earth tides model (MacCarthy and Petit 2003) and FES2004 model for the ocean tide loading effect (Lyard et al. 2006) were also applied. We fixed the cut-off angle to 7 and the uncertainty in phase observations to 10 mm. The solution was produced with the VMF1 mapping functions (Boehm et al. 2006). A priori ZHDs were computed with the revised formulation of Saastamoinen formula (Saastamoinen 1972) proposed by Bosser et al. (2007), and hourly surface pressure measurements. The value for the a priori wet delays was fixed to 100 mm. The wet delay parameters and horizontal gradients were modeled
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with a time resolution walk propffiffiffiof 5 min as random pffiffiffi cesses (using 5 mm= h and 0:5 mm= h, respectively, as time constrains) and a constant offset in zenith wet delay (500 mm), as recommended by Kouba (2008); the ZWD parametrization is also shown to be consistent with observed variations from Raman lidar measurements. The estimated positions were transformed into the ITRF2005 (Altamimi et al. 2007) using a 7-parameter transformation. Figure 100.1 presents the evolution of ZWD from the different techniques during the 2004/05/19 session. This figure underlines the consistency between all the techniques even if a bias around 5 mm can be observed between lidar and Hatpro. Table 100.1 presents comparisons of ZWD from the different techniques over the four 6 h sessions. We also observe an excellent agreement between lidar, radisosoundings and GPS estimates both in terms of bias (1–2 mm) and standard deviation (about 2 mm). Radiometers present a significant bias with respect to lidar. This bias could be due to the calibration procedure of the radiometers and also to the conversion from IWV to ZWD using a standard formula. However, we observe that cross-correlation between lidar and radiometers is still good, especially for high frequency variations. We focus on ZWD high frequency variations presented in Table 100.1. We observe a pretty good consistency between lidar and radiometers variations. Comparisons are less conclusive between lidar and GPS high frequency variations. We also test relaxing the random-walk time constraint of ZWD evolution in the
20040519
90 ZWD [mm]
100
80
140.9
141.0 DOY
141.1
Fig. 100.1 Evolution of ZWD from the different techniques over one specific night (2004/05/19). The lidar data are plotted as circles, GPS as triangles, Drakkar data as squares and Hatpro data as inverted triangles and radiosoundings as diamonds
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140 145
Mean PFR correction – Applied Applied Applied
6
5.6
8 139
4 2
146 0 0.9
1.0 decimal day
1.1
Fig. 100.2 Evolution of GPS high frequency ZWD component (5 min) for the four nights. Black lines indicate a noticeable similarity with a time shift about 4 min per day between the four time series. An arbitrary offset of 4 mm is added to the time series
GPS analysis. Tests show that it does not improve GPS high frequency variations and that it degrades the solution. This highlights a limitation of GPS analysis procedure for retrieving short time-scale ZWD variations. Figure 100.2 present GPS ZWD high frequency variations over the four nights (18, 19, 24 and 25 of May). We observe that the high frequency variations present some similarities over all sessions with a time shift about 4 min/day as illustrated on this plot. The observed time shift is consistent with the shifting of the local time of passage of GPS satellites (4 min per day). We may suspect multipath or antenna phase center variations that seems to impact ZWD estimation.
100.4 GPS Analysis Using Zenith Lidar Measurements We still focus on GPS processing on the four 6 h nights. Changes in standard strategy are investigated. The first strategy (referred as #1) is the standard
#1
14 12 10 8 6 4 2 0 −2
Mapping function for ZWD estimation VMF1 VMF1 Lidar–MF Lidar–MF
See text for further details about the different strategies (ID). PFR denotes the post-fit residuals of the GPS analysis
RMS [mm]
δZWDHF [mm]
Bias (b, mm), standard deviation (s, mm), and frequency cross-correlations: overall (r), low (rLF, 60 min) and high (rHF, 5 min)
A priori ZWD 0.1 m 0.1 m 0.1 m Lidar
ID 1 2 3 4
#4
rHF NA 0.12 0.54 0.90
5.6
rLF NA 0.85 0.88 0.96
#3
0.6 2.0 6.0 6.6
r NA 0.77 0.88 0.96
5.6
Radiosounding GPS Drakkar Hatpro
s 2.5 2.8 1.2 3.8
#2
b
Table 100.2 Description of GPS processing strategy tests
6.8
Table 100.1 Comparison of ZWD from various techniques with lidar
Fig. 100.3 Average RMS post-fit residuals from the different processing strategies tested (see Table 100.2)
procedure we have studied previously. A second strategy (#2) aims at reducing the impact of multipath and antenna phase center variations on GPS data. We used therefore the procedure proposed by Shoji et al. (2004) by stacking line of sight post-fit phase residuals from the standard strategy over a 10 days period. The resulting mean residuals map issued for correcting a priori the GPS phase observations. In a third strategy (#3), we evaluate the impact of using the lidar derived mapping function. This mapping function is calculated using a ray-tracing algorithm from lidar zenith profile as it has been previously described (see Sect. 100.2). In a last strategy (#4), lidar PSWDs are used for a priori correcting the GPS phase observations; ZWD parameters are still estimated to adjust a possible residual bias in lidar retrievals. The different GPS processing strategies are summarized in Table 100.2. Output of the GPS analysis in terms of RMS post-fit residuals over the four 6 h-cases are presented on Fig. 100.3. When the mean residuals map is applied, we observe a decrease of RMS of phase post-fit residuals of about 1 mm. Lidar derived mapping function is shown to not induce any changes in RMS, which corroborates the accuracy of VMF1 on our
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Calibration of Wet Tropospheric Delays in GPS Observation Using Raman Lidar Measurements
with strategies #2 and #3. Strategy #4 considerably improves the consistency between lidar and GPS estimates fast variations. However discrepancies are still observed (for example around midnight). These discrepancies may be linked first to a residual multipath effect, that is not properly correct by the mean residuals maps. They could also be due to differences in the sensed volumes, since lidar observe along a lineof-sight and GPS analysis use an observation cone.
2 1 ΔZWD [mm]
799
0 –1 –2 –3 –4
#4
#3
#2
#1
–5
Conclusion
Fig. 100.4 Comparisons of ZWD estimates from GPS strategies 1, 2 and 3 with respect to lidar. Mean bias (bar) and temporal variability (error bar) of ZWD differences (GPS–lidar) over all four sessions from 18 pffiffiffito 25 May 2004. Standard deviation of bias estimates (s= n is about 0.1 mm for the four strategies.)
δZWDHF [mm]
20
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8
#2
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0
L
–4 140.9
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DOY
Fig. 100.5 Evolution of high frequency ZWD components computed from lidar estimates and the different GPS strategies for the sessions of 19 May 2004. The lidar estimates are plotted as solid lines with circles, GPS estimates from strategies #1 as triangles, #2 as inverted triangles, #3 as squares and #4 as diamond. An arbitrary offset of 4 mm is added to the time series
cases (strategy 2 and 3). Strategy 4 also induces no further changes in RMS post-fit residuals. Now we focus on ZWD estimates from strategy 1, 2, 3 and 4. Differences between GPS estimates and lidar observations are presented in Fig. 100.4. We can observe a reduction of the bias between lidar and GPS (about 1 mm) by introducing mean residuals maps. We can notice that the lidar derived mapping function induce no further changes on the estimates. Strategy #4 is shown to reduce standard deviation while bias keep unchanged comparing to strategies #2 and #3. Over one specific night (Fig. 100.5), discrepancies in high frequency ZWD variations are still observed
We have presented some evidence of improvement in GPS analysis using mean post-fit residuals maps and lidar measurements. Mean post-fit residuals map is shown to improve GPS ZWD estimates compared to lidar measurements by a decrease of the bias between the two techniques. The use of wet tropospheric delays retrieved from zenith lidar measurement during the GPS analysis further reduce discrepancies between GPS estimates and lidar retrieval, especially in terms of fast ZWD variations. We have also shown on our cases that current mapping functions provide similar results than lidar derived mapping functions. This result corroborates the accuracy of VMF1 on our cases. Impact of lidar measurements on positioning will be investigated in future study. Some limitations of our study have to be mentioned. First, it is difficult to generalize these results from such a limited case study (short series of four 6 h sessions). They have to be confirmed on a more larger and statistically significant data set. It has also to be mentioned that nighttime and clear sky observations are mandatory: lidar measurements may only be used as calibration points for one station or for punctual positioning. A spherical symmetric atmospheric refractivity field has been assumed to derive PSWDs. Water vapor heterogeneities in azimuthal directions are therefore neglected: in future work we plan to use slant lidar measurements for a four-dimension characterization of the water vapor field. However, from these results, it could be expected that Raman lidar would become a valuable technique either for long-term geodetic monitoring based on regular observations (permanent GPS network) or repeated short survey campaigns for measuring slow geodynamic phenomena.
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The VAPIC field experiment was conducted by M. Haeffelin and O. Bock with the support from the Programme National de Te´le´de´tection Spatiale (PNTS) of the Institut National des Sciences de l’Univers (INSU). All the plots have been made with the General Mapping Tool (GMT) software (Wessel and Smith 1991). This paper is IPGP contribution no. 2633.
References Altamimi Z, Collilieux X, Legrand J, Garayt B, Boucher C (2007) ITRF2005: a new release of the International Terrestrial Reference Frame based on time series of station positions and Earth Orientation Parameters. J Geophys Res 112:B09401 Bevis M, Bussinger S, Herring TA, Rocken C, Anthes RA, Ware RH (1992) GPS meteorology: remote sensing of atmospheric water vapor using the Global Positioning System. J Geophys Res 97:15787–15801 Bock O, Tarniewicz J, Thom C, Pelon J (2001a) Effect of smallscale atmospheric inhomogeneity on positioning accuracy with GPS. Geophys Res Lett 28:2289–2290 Bock O, Tarniewicz J, Thom C, Pelon J, Kasser M (2001b) Study of external path delay correction techniques for high accuracy height determination with GPS. Phys Chem Earth 26:165–171 Boehm J, Niell AE, Tregoning P, Schuh H (2006) The global mapping function (GMF): a new empirical mapping function based on numerical weather model data. Geophys Res Lett 33:L07304 Bosser P, Bock OJP, Thom C (2007) An improved mean gravity model for GPS hydrostatic delay calibration. Geosci Rem Sens Lett 4:3–7 Haefele P, Martin L, Becker M, Brockmann E, Morland J, Nyeki S, Malttzler C, Kirchner M (2004) Impact of radio water vapor measurements on troposphere and height estimates by GPS. In: Proceedings of the 17th international technical meeting of the satellite division of the Institute of Navigation Kleijer F (2004) Troposphere modeling and filtering for precise GPS leveling. PhD thesis, Delft University of Technology
P. Bosser et al. Kouba J (2008) Implementation and testing of the gridded Vienna Mapping Function 1 (VMF1). J Geodes 82:193–205 Lyard F, Lefevre F, Letellier T, Francis O (2006) Modelling the global ocean tides: insights from FES2004. Ocean Dynam 56:394–415 MacCarthy DD, Petit G (2003) IERS 2003 conventions. Technical report, IERS, Frankfurt-am-Main, Germany Revercomb HE, Turner DD, Tobin DC, Knuteson RO, Feltz WF, Barnard J, B€osenberg J, Clough S, Cook D, Ferrare R, Goldsmith J, Gutman S, Halthore R, Lesht B, Liljegren J, Linne´ H, Michalsky J, Morris V, Porch W, Richardson S, Schmid B, Splitt M, van Hove T, Westwater E, Whiteman D (2003) The ARM program’s water vapor intensive observation periods. Bull Am Meteorol Soc 84:217–236 Rocken C, Sokolovskiy S, Johnson JM, Hunt D (2001) Improved mapping of tropospheric delays. J Atmos Ocean Tech 18:1205–1213 Saastamoinen J (1972) Atmospheric correction for the troposphere and stratosphere in radio ranging of satellites, in the use of artificial satellites for geodesy. Geophys Monogr 15 (16):247–251 Shoji Y, Nakamura H, Iwabuchi T, Aonashi K, Seko H, Mishima K, Itagaki A, Ichikawa R, Ohtani Y (2004) Tsukaba GPS dense net campaign observation: improvement in GPS analysis of slant path delay by stacking one-way postfit phase residuals. J Meteorol Soc Jpn 82:301–314 Tesmer V, Boehm J, Heinkelmann R, Schuh H (2007) Effect of different tropospheric mapping functions on the TRF, CRF and position time-series estimated from VLBI. J Geodes 81:409–421 Ware R, Rocken C, Solheim F, Van Hove T, Alber C, Johnson J (1993) Pointed water vapor radio corrections for accurate global positioning system surveying. Geophys Res Lett 20:22635–22638 Wessel P, Smith WHF (1991) Free software helps map and display data. EOS Trans AGU 72:441 Whiteman DN, Melfi SH, Ferrare RA (1992) Raman Lidar system for the measurement of water vapor and aerosols in the Earth’s atmosphere. Appl Opt 31:3068–3082 Zumberge JF, Heflin MB, Jefferson DC, Watkins MM (1997) Precise point positioning for the efficient and robust analysis of GPS data from large networks. J Geophys Res 102: 5005–5017
Generation of Slant Tropospheric Delay Time Series Based on Turbulence Theory
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€n M. Vennebusch and S. Scho
Abstract
GNSS phase observations are well suited to study atmospheric refraction effects and to contribute to weather prediction. Current research activities focus on real time determination of slant tropospheric wet delays in order to determine the water vapor content. The long periodic variations of these delays are mainly caused by the steady state component of the refractive index field. In contrast, short periodic variations of the slant delays are induced by refractivity fluctuations along the signal’s path from the transmitter to the receiver. Focusing on higher temporal resolution of water vapor variations, this second component will be of special interest. Based on turbulence theory, Sch€on and Brunner (J Geodes 82(1): 47–57) developed a formulation of the variances and covariances of GNSS phase observations induced by refractivity fluctuations in the troposphere. In this paper, we will use this model to investigate the generation of slant delay time series based on a spectral decomposition of the simulated turbulence theory-based variance–covariance matrices. Using an exemplary GPS configuration, the impact of the model parameters (as e.g. the refractivity structure constant, the outer scale length, the effective tropospheric height, and the wind direction and magnitude) on the covariance matrix and the generated time series is analysed.
101.1 Motivation One of the main error effects on microwave signals on their way from the transmitter to the receiver is the delay caused by the so-called dry and wet parts of the troposphere. The long periodic behaviour of the “tropospheric delay” is mainly caused by the steady state
M. Vennebusch (*) S. Sch€ on Institut f€ur Erdmessung, Leibniz Universit€at Hannover, Schneiderberg 50, 30167 Hannover, Germany e-mail:
[email protected]
component of the refractive index field and has already been studied thoroughly for the different space geodetic techniques. The short periodic behaviour, caused by highfrequency fluctuations of the water vapor content and thus of the refractive index field within the atmospheric boundary layer, generate quasi-random phase fluctuations which set an irreducible accuracy threshold for microwave-based observation techniques. Turbulence theory describes adequately the impact of these refractivity fluctuations on electromagnetic waves (Wheelon 2001). This is motivated by the fact that, on the one hand, phase fluctuations disturb both
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_101, # Springer-Verlag Berlin Heidelberg 2012
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interferometer measurements and GNSS phase measurements and should thus be treated by appropriate means. On the other hand, phase fluctuations reveal information about the turbulent media through which the signal has travelled. Hence, turbulence theory not only describes the stochastics of microwave fluctuations but also of high-frequency variations of slant tropospheric delays. Sch€on and Brunner (2008) developed a formulation of the variances and covariances of GNSS phase observations caused by fluctuations of the refractive index field. Using this model, the main aim of this analysis is to describe and investigate the generation of slant delay time series based on the spectral decomposition of turbulence theory-based variance–covariance matrices of both real and artificial satellite-receiver geometries. The generated slant delay time series will be used to identify dominant model parameters and to determine their stochastic properties in terms of their average variations and the power law indices of their temporal structure functions. Hence, these analyses form a closed-loop test revealing dominant model parameters and forming a prerequisite for the analysis of real data.
101.2 Turbulence Theory-Based Stochastic Model of Slant Tropospheric Delays Atmospheric turbulence can be generated mechanically, thermally or inertially and is often described in terms of irregular swirls (or “eddies”) of different sizes and different motions (Stull 1988). Due to the complex, quasi-random motion of eddies a deterministic description of turbulence has not been found yet. The net effect of eddy motion, however, can be described statistically via the energy spectrum, which indicates the amount of total turbulence kinetic energy associated with each eddy scale. Among the different regimes of the turbulence spectrum the inertial subrange is of most relevance for the following investigations since in this part isotropic turbulence dominates and thus eddies behave randomly. The inertial part of the turbulence spectrum is usually expressed in terms of the Kolmogorov spectrum or the von Karman spectrum. The latter one reads (von Karman 1948):
11 0:033C2n / k 3 ; 0 < k < kS Fn ðkÞ ¼ 11 k2 þ k20 6
(101.1)
Here, C2n is the structure constant of refractivity and L0 and l0 are the outer and inner scale lengths, respec2p tively, with k0 ¼ 2p L0 and kS ¼ l0 expressing their corresponding wave numbers. In the optical range, C2n is related to temperature fluctuations, whereas in the microwave range C2n is dominated by water vapor variations. Typical values are in the range of 0.3 1014 5.76 1014 m2/3 (Wheelon 2001; Treuhaft and Lanyi 1987). Based on the von Karman spectrum and the assumptions of height-independent C2n , local isotropy, uniform wind speed and wind direction (Sch€on and Brunner 2008) developed a co-variance expression of GNSS phase observations. This expression quantifies the phase variations induced by refractivity fluctuations along the line-of-sight. Since only the variability of the refractivity in the troposphere is considered, this formulation can also be interpreted as co-variance model of two tropospheric slant delays TAi ðtA Þ and TBj ðtB Þ at two stations A and B, to two satellites i and j and at two epochs tA and tB. In its most general formulation the co-variance expression reads: D
E D E ’iA ðtA Þ; ’jB ðtB Þ ¼ TAi ðtA Þ; TBj ðtB Þ pffiffiffiffiffi 2 1 12 0:033 p3 k0 3 23 2 C ¼ 5 G 56 sin eiA sin ejB n ZH ZH 1 ðk0 dÞ3 K1 ðk0 dÞ dz1 dz2 ; 3
0
0
(101.2) with C2n being the refractivity structure constant, e the respective elevation angle, H the effective tropospheric height (integration height) and d ¼ |d| ¼ |r2 + r – vDt – r1| being the separation distance between the integration points along the signal propagation path. Here, r1 and r2 denote the respective line-ofsight vectors, r denotes the baseline vector between the observing stations, v indicates the 3D-wind vector and Dt is the temporal separation between the two measurements. Thus, this distance depends on the baseline length between the two stations involved, as
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Generation of Slant Tropospheric Delay Time Series Based on Turbulence Theory
well as on the prevailing wind speed v and the wind direction (azimuth) av. The Gamma function and the modified Bessel function of the second kind are indicated by Г and K, respectively. In addition, (101.2) represents a co-variance model for the two phase observations ’iA ðtA Þ and ’jB ðtB Þ. For further details the reader is referred to Sch€ on and Brunner (2008).
stationary even if the original time series is not stationary (Wheelon 2001). In (101.4), brackets h:i denote an ensemble average. Graphical visualisation of structure functions reveals information about the stochastic behaviour (including temporal or spatial correlation lengths) of the time series. For a time lag t, when DT reaches an asymptotic value the original time series is considered as being uncorrelated. Often, structure functions are expressed in terms of so-called power-law processes which are described by
101.3 Simulation of Slant Tropospheric Delay Variations Two common approaches for the simulation of correlated noise are based on the Cholesky decomposition or on the spectral decomposition of the predefined covariance matrix S. In the following, the spectral decomposition approach is used. Let S be the predefined positive-definite covariance matrix containing the stochastic properties of the correlated noise time series to be generated. The vector y of simulated correlated noise is given by Searle (1982): pffiffiffiffi y ¼ G Lx;
(101.3)
with the orthonormal matrix G containing the pffiffiffiffi eigenvectors of S, a diagonal matrix L containing the square roots of the eigenvalues of S on its main diagonal, and x being a vector of Gaussian random numbers with zero mean and unit variance.
101.4 Stochastic Behaviour of Simulated Slant Delay Time Series Many geophysical and atmospheric phenomena are described by non-stationary stochastic processes. Thus, (auto-)correlation functions and power spectral densities are not appropriate means to assess the temporal (or spatial) behaviour of these processes. A suitable measure is the temporal structure function DT(t) of a quantity T: D E DT ðtÞ ¼ ½Tðt þ tÞ TðtÞ2 :
(101.4)
The temporal differencing removes data trends and thus generates a difference process which is often
803
DT ðtÞ ¼ c ta ;
(101.5)
with a constant c and the power-law index a. Turbulence theory shows a smooth transition with a ¼ 2/3 for long baselines (or 2D turbulence) to a ¼ 5/3 for short baselines (or 3D turbulence) (Wheelon 2001). Similar expressions are used for spectral densities such as (101.1) which indicates a power law-process with a power law index of a ¼ 11/3 and which thus describes a non-stationary process.
101.5 Simulation Studies 101.5.1 Simulation Setup In order to simulate different satellite-receiver geometries together with different turbulent parameters sets, three geometrical configurations (scenarios 1–3, see Fig. 101.1) are used to generate variance– covariance matrices of slant tropospheric delays. For each scenario, various combinations of turbulence parameter values are chosen (see Table 101.1). Since
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Fig. 101.1 Scenarios used for simulation studies
€n M. Vennebusch and S. Scho
804 Table 101.1 Turbulence parameters used in the simulation studies (used in various combinations): C2n is the refractivity structure constant, L0 is the outer scale length, v is the wind speed, H is the height of the (wet) troposphere and av is the wind direction (azimuth) C2n (m2/3) 0.3 1014 5.76 1014 9.0 1014
L0 (m) 3,000 6,000
v (m/s) 8 15
H (m) 2,000 1,000
av ( ) 0 90 180 270
C2n only acts as a scaling factor [see (101.2)], in the following analyses an average refractivity structure constant of C2n ¼ 0:3 1014 m2=3 has been used. The parameter values in the first row of Table 101.1 represent average turbulence conditions (see e.g., Wheelon 2001, or Treuhaft and Lanyi 1987) and serve as reference parameters for the following comparisons. Exemplarily, Fig. 101.2 shows the obtained variance–covariance matrices (VCMs), correlation matrices, anti-diagonals of these matrices, simulated slant delay time series and their temporal structure functions for the three different scenarios and the reference turbulence parameter set, respectively.
101.5.2 Simulated Time Series For scenario 1 (fixed satellite and thus no geometry variations, 100 observations, 10 s sampling interval), (101.2) yields a variance–covariance matrix with a diagonal structure and with unique variances of 0.1 106 m2 for all slant delays (i.e. zenith delays, in this case). The corresponding correlation matrix shows identical correlation lengths of e.g. six epochs until a correlation of less than 0.2 is reached. Figure 101.2a also shows five realisations1 of simulated slant delays as well as their temporal structure functions. The amplitude of the simulated slant delays is in the range of 1 mm. The temporal structure functions show a power law behaviour with exponents which slowly decrease from 5/3 to 2/3 within the first approximately 150–180 s. For scenario 2 (low elevation scenario, 100 observations, 10 s sampling interval, see Fig. 101.2b) the variances range from 0.75 106 m2 for the
1
Each different set of five realisations has been generated using the same set of Gaussian vectors x [see (101.3)].
lowest elevations (at 10 elevation) to 6 2 0.48 10 m for the highest elevations (at 17 elevation). Thus, the variance–covariance matrix does not have a diagonal structure any longer. This is also reflected in the variability of the anti-diagonals. The correlations also show a slightly varying behaviour with correlations lengths (correlation below 0.2) of 13–18 epochs. The simulated slant delays show variations in the range of 2 mm. Here, a clear 5/3-power law behaviour for the first approximately 150–180 s of the structure function can be seen. Finally, for scenario 3 (rising satellite scenario, 1,000 observations, 10 s sampling interval, see Fig. 101.2c) the changing geometry leads to variance ranges of 0.75 106 m2 for the lowest elevations (at 10 elevation, as for scenario 2) and 0.1 106 m2 for zenith direction (as for scenario 1). Correlation lengths (correlation below 0.2) are between 5 and 16 epochs. The simulated slant delays show higher amplitudes (of approximately 2 mm) for low elevation directions and approximately 1 mm for observations near the zenith direction. Again, a slowly decreasing power law behaviour (with a long 5/3 behaviour section) can be seen for the first approximately 150–180 s of the structure function. Comparisons with time series of both estimated and measured (i.e., water vapor radiometer-derived) slant delays showed that the simulated variations best agree with estimated slant tropospheric delays (not explicitly shown here) (Bender et al. 2008).
101.5.3 Impact of Parameter Variations In order to identify dominant model parameters and to assess the impact of parameter variations on the simulated slant delay time series the turbulence parameter values have been varied within realistic ranges (see Table 101.1). In the following, only those scenarios with small geometry changes (i.e., scenarios 1 and 2) are considered so that the results are (almost) only affected by pure turbulence effects and are not superimposed by geometrical effects. First, the outer scale length L0 has been changed from L0 ¼ 3,000 m to L0 ¼ 6,000 m for both scenarios (see Fig. 101.3a). This increase results in both longer correlation lengths and higher variations in the simulated slant delay time series of both scenarios.
Generation of Slant Tropospheric Delay Time Series Based on Turbulence Theory
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Fig. 101.3 Impact of parameter variations on correlations lengths and average variations of simulated slant delay time series (a) variations of outer scale length L0, (b) variations of height of wet troposphere H, (c) variations of wind speed v, (d) variations of wind direction av
After decreasing the height of the effective wet troposphere H from H ¼ 2,000 m to H ¼ 1,000 m it can be observed (see Fig. 101.3b) that higher integration heights [see (101.2)] yield higher variations of simulated slant delays and only small increases of correlation lengths. Especially for the zenith scenario almost no impact on the correlation lengths can be recognised. The impact of increasing wind speed can be seen in Fig. 101.3c. For both scenarios a stronger wind acts decorrelating on slant delays but only for the low elevation scenario this also yields significantly higher variations of the simulated slant delays. Thus, the impact of wind speed depends on the current
satellite-receiver geometry. This can be explained by the wind dependency of the separation distance d ¼ |d| [cf. (101.2)]. For details, the reader is referred to Sch€on and Brunner (2007). Finally, changes in the wind direction (wind azimuth) av in steps of 90 from 0 to 270 clearly show the decorrelating effect of wind if it acts (nearly) orthogonal onto the lines-of-sight of a changing satellite-receiver geometry. For the low elevation scenario this also leads to different average variations of simulated slant delays: in those cases with nearly orthogonal wind directions (i.e., av ¼ 90 and av ¼ 270 ) the smaller correlation lengths also lead
Fig. 101.2 (continued) Example variance–covariance matrices, correlation matrices, simulated slant delay time series and their temporal structure functions for the three different scenarios (using the reference set of turbulence parameters) (a) fixed satellite in zenith direction, (b) low elevation satellite, (c) rising satellite
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Generation of Slant Tropospheric Delay Time Series Based on Turbulence Theory
to higher average variations of slant delays. For the zenith scenario, however, changes of wind directions neither affect correlation lengths nor the average variations of simulated slant delays. This can also be explained by analysing the separation distance d [see (101.2)]. Variations of C2n have been omitted since the refractivity constant only acts as a linear scaling factor [see (101.2)] and hence does not change any correlation coefficients. It does, however, change the (co-) variances (by the same amount) and thus affects the magnitudes of simulated slant delays.
101.6 Summary In conclusion, it was shown that • Using turbulence theory it is possible to generate variance–covariance matrices of slant tropospheric delays which take the physical properties of the atmosphere into account. • It is possible to simulate time series of slant tropospheric delays with variations in the range of 1–3 mm and with typical correlation lengths of approximately 180–200 s (valid for the particular set of turbulence parameters chosen here). • Over temporal ranges of several seconds to minutes, simulated slant tropospheric delays show higher variations (of approximately 2 mm) at low elevations and of approximately 1 mm at higher elevations. • Simulated slant tropospheric delay time series show a power law-behaviour of 5/3–2/3 (as expected). It was also observed that the superposition of geometric effects and atmospheric turbulence affects the
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interpretation of results and thus further investigations are needed. In the future, these results will be compared to the slant delays estimated from real GNSS phase measurements. Current activities concentrate on both the estimation of high-resolution (i.e., 1 Hz) ZTDs using Precise Point Positioning (PPP) approaches and on the analysis of double differenced phase observations measured in GNSS networks optimised for the investigation of atmospheric turbulence. Acknowledgements The authors thank the German Research Foundation (Deutsche Forschungsgemeinschaft) for its financial support (SCHO 1314/1-1).
References Bender M, Dick G, Wickert J, Schmidt T, Song S, Gendt G, Ge M, Rothacher M (2008) Validation of GPS slant delays using water vapour radiometers and weather models. Meteorol Zeitschrift 17(6):807–812 Sch€on S, Brunner FK (2007) Treatment of refractivity fluctuations by fully populated variance-covariance matrices. Proceedings of the 1st colloquium scientific and fundamental aspects of the Galileo Programme, Toulouse, Oct 2007 Sch€on S, Brunner FK (2008) Atmospheric turbulence theory applied to GPS carrier-phase data. J Geodes 82(1):47–57 Searle SR (1982) Matrix algebra useful for statistics. Wiley, New York, NY Stull RB (1988) An introduction to boundary layer meteorology. Springer, Berlin Treuhaft RN, Lanyi GE (1987) The effect of the dynamic wet troposphere on radio interferometric measurements. Radio Sci 22(2):251–265 von Karman Th (1948) Progress in the statistical theory of turbulence. Proc Natl Acad Sci U S A 34:530–539 Wheelon AD (2001) Electromagnetic scintillation-I. Geometrical optics. Cambridge University Press, Cambridge
.
Fitting of NWM Ray-Traced Slant Factors to Closed-Form Tropospheric Mapping Functions
102
Landon Urquhart, Marcelo Santos, and Felipe Nievinski
Abstract
In this paper we investigate the fitting of ray-tracing results to closed-form expressions. We focus on the variation of the delay with elevation angle and azimuth. For the elevation angle-dependence we compare the continued fraction form of Yan and Ping (Astron J 110(2):934–993, 1995) with that of Marini (Radio 11 Sci 7(2):223–231, 1972) (normalized to yield unity at zenith and found negligible differences between the two functional formulations for the hydrostatic case, while for the non-hydrostatic case, the Yan and Ping model performed marginally better. Since the ray-tracing results do not necessarily assume azimuthal symmetry, we have to account for the azimuth-dependence. For that we compare the linear gradient model of Davis et al. (Radio Sci 28(6):1003–1018, 1993) with the inclusion of second order terms (Seko et al., J Meteorol Soc Jpn 82 (1B):339–350, 2004) and arbitrary spherical harmonics. These functional forms performed very well for the hydrostatic case, although for the non-hydrostatic case there were some large biases, particularly in the spherical harmonics of order 1, degree 1 and the 2nd order polynomial case.
102.1 Introduction As electromagnetic signals propagate through the atmosphere they experience path delays due to the electrically neutral atmosphere. At the zenith, this delay roughly has a magnitude of 2.3 m (at sea level) and can grow to tens of meters near the horizon (Langley
L. Urquhart (*) M. Santos Department of Geodesy and Geomatics Engineering, University of New Brunswick, P.O. Box 4400, Fredericton, NB, Canada E3B 5A3 e-mail:
[email protected] F. Nievinski Department of Aerospace Engineering Sciences, University of Colorado, Boulder, CO 80309-0429, USA
1998). The ultimate goal in geodesy is to mitigate the tropospheric delay, in order to remove any bias from the resulting position estimates. To model the tropospheric delay, it is convenient to separate the delay into a hydrostatic and a non-hydrostatic component contributing to the delay experienced in the zenith direction. A mapping function then models the elevation dependence of the tropospheric delay: DLðeÞ ¼ kh ðeÞ DLzh þ knh ðeÞ DLznh
(102.1)
where DL is the total along path delay, kh and knh are the hydrostatic and non-hydrostatic mapping functions dependent only on elevation angle, DLzk and DLznk are the hydrostatic and non-hydrostatic zenith delay and e the elevation angle of the satellite.
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Various elevation angle-dependent mapping functions have been suggested in the past, many which were systematically tested by Mendes (1999), although under the assumption of a spherically symmetric atmosphere. With the improvement of space geodetic observations, the asymmetric nature of the atmosphere has been shown to have an impact on the resulting station position (MacMillan 1995). This has led to the development of mapping functions which attempt to model the variation of the delay, not only as a function of elevation angle, but also as a function of azimuth. The purpose of this contribution will be to compare the functional forms used in tropospheric delay modeling. Rather than focus on specific mapping functions, we work with the functional forms themselves in order to identify which ones provide the most realistic representation of the tropospheric delay. The end goal of this research is to identify the most realistic models of the tropospheric delay, expressed in a convenient closed-form manner, which can then be used to either predict or estimate the effect of tropospheric delay in space geodetic data analysis.
factor values with respect to the independent variables. This allows us to distinguish between the slant factor model or mapping function (denoted k) from a particular slant factor value (denoted k), resulting from the evaluation of the former at a specific epoch ðt ¼ t0 Þ, position ðf ¼ f0 ; l ¼ l0 ; h ¼ h0 Þ, and direction ðe ¼ e0 ; a ¼ a0 Þ: k ¼ kðt0 ; f0 ; l0 ; h0 ; e0 ; a0 Þ
It is helpful to further distinguish between the functional form (Boehm and van Dam 2009) and the realization of a mapping function. The functional form describes how the slant factor varies with respect to some parameter. The most common functional form in use today is Marini’s (1972) continued fraction 1 , normalized to yield unity at zenith, expansion of sinðeÞ as given by Herring (1992): 1þ
a 1
kðeÞ ¼
b 1
sinðeÞ þ sinðeÞ þ
102.2 Decomposing the Delay
(102.5)
c ... a
(102.6) b
sinðeÞ þ
c ...
Following Nievinski (2009), the tropospheric delay in its most general form, is a function of date and time (t), receiver location (latitude f, longitude l; and height h) and satellite location given in the topocentric frame (elevation angle e, and azimuth a):
The variation with respect to azimuth is, most of the time, neglected, and sometimes accounted for with a single main direction of asymmetry (Davis et al. 1993):
DL ¼ f ðt; f; l; h; e; aÞ:
dk ðe; aÞ ¼ k0 ðeÞ cotðeÞ ½GN cosðaÞ þ GE sinðaÞ
(102.2)
(102.7) The delay is most often decomposed as: DL ¼ DLz k
(102.3)
where DLz is the zenith delay, defined as: DLz ¼ f ðt; f; l; h; e ¼ 90 ; aÞ:
(102.4)
In this way, the variation of the slant delay to a DL given satellite is confined to the slant factor, k ¼ DL z , a unitless ratio. A mapping function can then be described as a model for the variation of the slant
where k0 represents the symmetric mapping function, GN and GE are the north and east coefficients describing the direction and magnitude of asymmetry exhibited by slant factors and a is the azimuth of the observation. The realization of the mapping functions described in (102.6) and (102.7) can take several different forms as shown in Niell (1996) and Boehm et al. (2006). The troposphere gradient terms, GN and GE can also be determined by using a profile method (Boehm and Schuh 2007) or three dimensional ray-tracing (Chen and Herring 1997) and it is also possible to have various gradient mapping functions k0 .
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Fitting of NWM Ray-Traced Slant Factors to Closed-Form Tropospheric Mapping Functions
So it can be seen that although the underlying functional forms for the mapping functions described above are the same, the realization of those models are quite different. The comparison can become even more complex if we consider mapping functions which require the use of external parameters directly in the mapping functions such as the generator function method described in Yan and Ping (1995). Due to these differences in the realization of the functional forms, the evaluation is typically of the mapping function itself, as is the case for Mendes (1999). In order to avoid this downfall, we use a homogenous approach to both derive the mapping functions based on the functional forms as well as in the evaluation.
102.3 Functional Formulations Various functional formulations have been developed over the years to describe both the elevation angle- and azimuth- dependence of the tropospheric delay. Table 102.1 shows the functional forms tested along with the number of coefficients to be estimated. For those functional forms which consider the atmosphere to by symmetrical, we have compared Marini’s (1972) continued fraction form, described in (102.6), for both the 3 coefficient expansion and the 4 coefficient expansion, Yan and Ping’s (1995) generator function method and the simplification of Marini’s (1972) expression using empirical values for the b and c coefficients as done in the Vienna Mapping Functions (VMF) described in Boehm et al. (2006). To ensure consistency with the truth values, the a coefficient in the VMF was calculated using the same ray-tracing scheme as the other models, although under the assumption of
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spherical osculating atmosphere, to be consistent with the approach used in Boehm et al. (2006). For the asymmetric delay, the standard, linear gradient formulation described in Davis et al. 1993 was compared to the 2nd order polynomial expansion developed by Seko et al. (2004) as well as the possibility of using spherical harmonics as described in Boehm and Schuh (2001).
102.4 Experiment Description The atmospheric parameters required for three dimensional ray-tracing were obtained through the Canadian Regional NWM, produced by the Canadian Meteorological Center (CMC). This NWM has a spatial resolution of 15 km horizontally, and contains 28 isobaric levels plus a surface level. Although at this resolution small scale perturbations which may be due to cloud like structures are not detectable, larger scale “gradient like” structures in the troposphere have been shown to be detectable (Davis et al. 1993). As these small scale perturbations fluctuate rapidly it would be more appropriate to treat them with a statistical model rather than attempt to model them in the functional form. That said, the use of a finer mesh NWM, may lead to further insights into the accuracy of the functional forms. A site-specific approach, similar to the approach used for the rigorous VMF1 site, was followed (Boehm et al. 2006). A total of 29 ray-traced observations at elevation angles of (3 , 4 , 6 , 8 , 14 , 30 , 70 and 90 ) and azimuths of (0 , 90 , 180 and 270 ) were used to estimate the coefficients through a least squares procedure. To evaluate the functional forms, truth values computed at a regularly spaced interval of 15 in
Table 102.1 Functional formulations and coefficients Formulation Marini 3 coefficient Marini 4 coefficient Yan and Ping Modified VMF Linear gradient model Spherical harmonics 2nd order polynomial
Number of unknowns 3 4 4 1 5 3 + harmonic coefficients 8
Coefficients a, b, c a, b, c, d d1, d2, d3, d4 a (UNB ray-tracing scheme to determine “a”) a, b, c, GN, GE a, b, c, a10, a11, b11, . . . , anm, bnm a, b, c, GN, GE, GNN, GEE, GNE
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102.5 Results and Discussion
a
2
Mean (mm)
The residuals of the least squares estimation for each formulation were sorted by elevation angle and the mean and standard deviation computed. For the modified VMF, there was no estimation involved so it is not included in the residual analysis. Figure 102.1 shows the mean of the residuals for the hydrostatic mapping functions. The largest bias of the symmetric mapping functions is present in the Marini four coefficient expression while the Yan and Ping expression exhibits a bias which is larger at higher elevation angles but seems to perform better than the Marini three coefficient model at the low elevation angles, although the differences are marginal. The standard deviations at each elevation angle, shown in Table 102.2, are very similar for all three symmetric cases which was expected as they are unable to model the azimuthal variation of the tropospheric delay. The asymmetric mapping functions exhibit biases of similar magnitude as the Marini 3 coefficient when binned by elevation. This is expected as the Marini 3 coefficient expression is part of the formulation used in the asymmetric mapping functions ðk0 Þ. The
1 0 -1 -2
3
4
6
8
14
30
70
Marini 4 Coeff.
Table 102.2 Standard deviations of the residuals for the fitting of the functional forms to the ray-traced hydrostatic delay (units ¼ mm) Formulation
Elevation angle 3 4 6 Marini 3 coefficient 57.2 41.8 24.0 Marini 4 coefficient 57.2 42.3 24.2 Yan and Ping 57.2 41.8 24.0 SH11 31.8 18.3 9.4 SH21 13.1 8.0 5.5 2nd order polynomial 5.1 4.0 6.0 Linear grad. 13.4 8.6 6.9
b
2 1
Yan and Ping
8 15.1 16.2 15.1 10.6 15.7 4.8 5.2
14 5.6 6.8 5.6 4.3 5.7 2.1 2.6
30 1.4 1.6 1.4 1.2 1.4 0.7 1.0
70 0.1 0.1 0.1 0.1 0.1 0.1 0.1
0 -1
3
-2
Elevation Angle (degrees) Marini 3 Coeff.
standard deviations shown in Table 102.2 shows that above 14 elevation angle there is really no benefit to using the asymmetric functional forms over the symmetric expressions. As we move to lower elevation angles, we start to see the improvement in the fit of the asymmetric expressions, especially in the standard deviation which is much smaller for the asymmetric cases. Next we consider the bias present between the truth observations and those computed using the derived mapping functions described above. Figure 102.2 shows that the Marini 3 coefficient model performs very well, exhibiting nearly a zero bias for the hydrostatic case. There appears to be no significant change to adding a fourth coefficient to the Marini expression. The mVMF, which uses empirical values for the 2nd and 3rd coefficient in the Marini expression, has a small bias but, even at 5 it is less than 1 cm, which again confirms the validity of using this approach for high accuracy applications. The biases are comparable for the asymmetric case. The real advantage of the asymmetric models can be seen in the standard deviations, shown in Table 102.3. For the spherical harmonics model, the standard deviation at the 5
Mean (mm)
azimuth and at elevation angles of 3 , 5 , 7 and 10 were computed. The truth values as well as the fitted values were determined for every 5th day of 2008 at epoch 00:00h UTC, for station CAGS located in Gatineau, Canada, totalling 73 days. Slant factors were computed, using (102.3) for the truth and fitted observations. The difference between the slant factors were then multiplied by a nominal 2.3 m for the hydrostatic and 0.22 m for the nonhydrostatic to obtain the error of the delay in units of meters.
4
6
8
14
30
70
Elevation Angle (degrees) SH11
SH21
Fig. 102.1 Hydrostatic Residuals binned by elevation angles: (a) symmetric; (b) asymmetric
2nd Order Poly.
Linear Grad.
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Fitting of NWM Ray-Traced Slant Factors to Closed-Form Tropospheric Mapping Functions
b Mean (mm)
Mean (mm)
a
813
15
-5
10
7 5 Elevation Angle (degrees)
Marini 3 Coeff.
Marini 4 Coeff.
-5
3
Yan and Ping
15
10
mVMF
SH11
7 5 Elevation Angle (degrees) SH21
2nd Order Poly
3
Linear Grad.
Fig. 102.2 Discrepancy between truth and mapped hydrostatic delay: (a) symmetric; (b) asymmetric Table 102.3 Standard deviation of the hydrostatic delay biases between the truth and mapped observations (units ¼ mm) Elevation angle 10 10.5 12.0 10.5 10.8 2.8 6.2 9.3 5.8
Marini 3 coefficient Marini 4 coefficient Yan and Ping mVMF SH11 SH21 2nd order polynomial Linear grad.
Mean (mm)
a
0.5 0 3
4
6
8
14
30
70
–0.5
7 19.4 20.0 19.3 19.8 7.9 4.9 19.0 12.2
5 32.2 32.3 32.2 32.8 17.1 6.9 37.3 26.2
b
100
Mean (mm)
Formulation
0 -100
3
4
6
3 58.8 58.9 58.8 59.5 36.7 17.6 104.1 82.2
8
14
30
70
-200 -300
–1
Elevation Angle (degrees)
Elevation Angle (degrees) Marini 3 Coeff.
Marini 4 Coeff.
Yan and Ping
SH11
sh21
2nd Order Poly
Linear Grad.
Fig. 102.3 Non-hydrostatic residuals binned by elevation angles: (a) symmetric; (b) asymmetric
elevation angle improves from 32.2 mm for the symmetric functional forms down to 6.9 mm for spherical harmonics of degree 2 and order 1 (SH21). Although the 2nd order polynomial performed well in the residual analysis, we see that it performed the worst out of all of the asymmetric formulations. This may be due to the number of azimuths used in the observation scheme. If the number of azimuths used to fit the models were increased from four to eight it is expected that we would see an improvement in the results. Figure 102.3 shows the non-hydrostatic results. For the symmetric mapping functions there is a bias of
less than 1 mm, which can be considered negligible. Although the Yan and Ping functional form was not originally intended for non-hydrostatic use it did perform well, having the smallest mean at most elevation angles. We did see a small improvement at low elevation angles as well as for the Marini 4 coefficient model, although the slight improvement does not justify the added complexity of a fourth coefficient in the estimation process. For the asymmetric case there was a large bias for spherical harmonics of, degree 1, order 1 (SH11) while the other mapping functions performed similar to the
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Table 102.4 Standard deviation of the residuals for fitting the functional forms to the non-hydrostatic delay (units ¼ mm) Formulation
Elevation angle 3 148.3 148.3 148.2 79.0 22.3 8.1 24.6
Marini 3 coefficient Marini 4 coefficient Yan and Ping SH11 SH21 2nd order polynomial Linear grad.
6 60.5 60.5 60.4 26.8 14.1 8.6 18.2
8 38.0 38.0 37.9 39.8 39.4 7.2 13.9
b
5
mean(mm)
mean (mm)
a
4 106.2 106.2 106.1 95.5 14.5 7.8 17.1
0 10
7
5
3
-5
14 13.9 13.9 13.8 13.3 14.3 7.0 6.0
30 3.1 3.2 3.1 2.8 3.2 3.4 1.6
70 0.3 0.6 0.3 0.1 0.1 0.6 0.2
600
0 10
7
5
3
-200
-10 Elevation Angle (degrees) Marini 3 Coeff.
Marini 4 Coeff.
Yan and Ping
Elevation Angle (degrees) mVMF
SH11
SH21
2nd Order Poly
Linear Grad.
Fig. 102.4 Discrepancy between truth and mapped non-hydrostatic delay: (a) symmetric; (b) asymmetric
symmetric mapping functions. The real improvement once again came when considering the standard deviations, shown in Table 102.4, which for the asymmetric mapping functions were much lower than the symmetric cases. The 2nd order model performed the best with an 8 mm standard deviation at 3 while the symmetric mapping functions exhibited a standard deviation of nearly 148 mm. Once again, for higher elevation angles above 15 there is no advantage to using the asymmetric models. For the non-hydrostatic case the bias for the three coefficient Marini expression based mapping functions are very similar to the mVMF (Fig. 102.4). The bias is typically around 7 mm. For the Yan and Ping formulation the bias is much better typically less than 1 mm. For the asymmetric expressions we see that now a bias appears in SH11 at low elevation angles. This is similar to the residual analysis which saw poor performance in the case of SH11. SH21 and the standard linear gradient formulation performed the best both having biases of about 1 cm at 3 elevation angle. The 2nd order polynomial, which performed very well in the residual analysis, was not able to adequately model the variation of the delay with respect to azimuth and large discrepancies were seen. Once again this may be a result of the low number of azimuths used in the estimation of the coefficients.
102.6 Conclusion and Future Work Closed-form mapping functions will continue to play an important role in the processing of space geodetic data for years to come. By using three dimensional ray-tracing through a NWP model it was possible to test some of the most common mathematical models in use today. In this contribution we choose to put the timevariation of the tropospheric delay out of the scope and focus on developing site and epoch specific mapping functions, similar to the manner used in the VMF1-site. Both symmetric and asymmetric formulations were tested. It was found that all of the symmetric mapping functions tested were able to adequately model the elevation angle dependence of the tropospheric delay above an elevation angle of 14 . Even using empirical values for several of the coefficients as done for the VMFs only introduced a slight bias and at elevation angles below 5 . However, since they do not consider the variation with respect to azimuth they possess large standard deviations. On the other hand, the use of spherical harmonics was shown to be an improvement over other models such as the standard linear gradient expression and the 2nd order polynomial model.
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Fitting of NWM Ray-Traced Slant Factors to Closed-Form Tropospheric Mapping Functions
For the non-hydrostatic delay, the Yan and Ping formulation performed very well exhibiting only a small bias in elevation angle where as the Marini expressions had a bias of nearly 8 mm. The asymmetrical models on the other hand experienced some difficulty and were not able to properly model the delay to the degree that was expected. Further research will need to be done to identify the problem, but by including more observations at different azimuths it is believed it would improve the fit of the models. Also experimenting with different gradient mapping functions, such as those suggested by Chen and Herring (1997) should be attempted. It is also necessary to expand the test locations to include more stations at various latitudes and elevations. Acknowledgements The authors would like to thank the Natural Sciences and Engineering Research Council of Canada for funding this research. The last author would also like to acknowledge funding provided by CAPES/Fulbright.
References Boehm J, Schuh H (2001) Spherical harmonics as a supplement to global tropospheric mapping functions and horizontal gradients. In: Behrend D, Rius A (eds) Proceedings of the 15th working meeting on European VLBI for geodesy and astrometry, Barcelona, 7–8 September, pp 143–148 Boehm J, Schuh H (2007) Troposphere gradients from the ECMWF in VLBI analysis. J Geod 81:403–408 Boehm J, Van Dam T (2009) Modelling deficiencies and modelling based on external data. Second GGOS unified analysis workshop, San Francisco, CA, 11–12 December Boehm J, Werl B, Schuh H (2006) Troposphere mapping functions for GPS and very long baseline interferometry
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from European Centre for Medium-Range Weather Forecasts operational analysis data. J Geophys Res 111:B02406 Chen G, Herring TA (1997) Effects of atmospheric azimuth asymmetry on the analysis of space geodetic data. J Geophys Res 102(B9):20489–20502 Davis JL, Elgered G, Niell AE, Kuehn CE (1993) Ground-based measurement of gradients in the “wet” radio refractivity of air. Radio Sci 28(6):1003–1018 Herring TA (1992) Modelling atmospheric delays in the analysis of space geodetic data. In: de Munck JC, Th. Spoelstra TA (eds) Proceedings of the symposium refraction of transatmospheric signals in geodesy, No. 36, Netherlands Geodetic Commission, The Hague, pp 157–164 Langley RB (1998) Propagation of the GPS signals. In: Teunissen PJG, Kleusberg A (eds) GPS for geodesy, 2nd edn. Springer, Berlin, pp 112–149 MacMillan DS (1995) Atmospheric gradients from very long baseline interferometry observations. Geophys Res Lett 95 (9):1041–1044 Marini JW (1972) Correction of satellite tracking data for an arbitrary tropospheric profile. Radio Sci 7(2):223–231 Mendes VB (1999) Modeling the neutral-atmosphere propagation delay in radiometric space techniques. PhD thesis, Department of Geodesy and Geomatics Engineering, Technical Report 199, University of New Brunswick, Fredericton, NB, April, 349 pp Niell AE (1996) Global mapping functions for the atmosphere delay at radio wavelengths. J Geophys Res 101(B2): 3227–3246 Nievinski, FG (2009) Ray-tracing options to mitigate the neutral atmosphere delay in GPS. MScE thesis, Technical Report No. 262, Department of Geodesy and Geomatics Engineering, University of New Brunswick, Fredericton, NB, 232 pp. hdl:1882/1050 Seko H, Nakamura H, Shimada S (2004) An evaluation of atmospheric models for GPS data retrieval by output from a numerical weather model. J Meteorol Soc Jpn 82 (1B):339–350 Yan H, Ping J (1995) The generator function method of the tropospheric refraction corrections. Astron J 110(2):934–993
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Estimation of Integrated Water Vapour from GPS Observations Using Local Models in Hungary
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Sz. Ro´zsa
Abstract
This paper studies the estimation of integrated water vapour (IWV) from the zenith tropospheric delay (ZTD). In order to evaluate the technique, six mathematical models are compared using a stormy summer period and a calm and dry winter period. The mathematical models include locally derived models using more than 10,000 radiosonde observations. The GPS derived IWV values are compared to radiosonde observations and a linear regression prediction of IWV using surface observations, too. Moreover the computations were carried out during a heavy storm in the summer period, when the estimated IWV distribution is compared to radar observations, too. The results show that the IWV values derived from GPS observations had an agreement with the radiosonde observations at the level of better than 2–3 mm in term of standard deviations. The results also show that GPS observations provide additional information to the estimation of IWV. Moreover the Hungarian Active GNSS Network was proved to be useful to monitor the water vapour content of the atmosphere during weather fronts as well.
103.1 Introduction With the application of the Global Navigational Satellite Systems (GNSS) it is possible to quantify the total signal delay caused by the troposphere. This delay can be separated into a dry (hydrostatic) and a wet part. The dry part can be modelled using local meteorological data, thus the wet tropospheric delay can be determined.
Sz. Ro´zsa (*) Department of Geodesy and Surveying, Budapest University of Technology and Economics, 1521 Budapest, P.O. Box 91 Hungary HAS-BME Research Group for Physical Geodesy and Geodynamics, 1521 Budapest, P.O. Box 91Hungary e-mail:
[email protected]
Using the wet delay the integrated water vapour in the troposphere can be estimated using various models. This integrated water vapour gives an upper bound of the precipitable water vapour. In Hungary some prior investigations have been made in this area (Borba´s 2000; Ba´nyai 2008), however the development of the Hungarian Active GNSS Network opened up new possibilities for the application of GNSS Meteorology. In this study various mathematical models are tested and evaluated for the feasibility of a continuous estimation of the water vapour from GNSS observations in Hungary. The estimations are validated using radiosonde observations provided by the Hungarian Meteorological Service.
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_103, # Springer-Verlag Berlin Heidelberg 2012
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103.2 The Estimation of IWV from the Tropospheric Delay
Applying (103.5) to the dry air and the water vapour and substituting the result into (103.3):
The tropospheric delay can be computed by:
ZTD ¼ ZHD þ ZWD,
Ds ¼ 106
ð
where: N ds;
(103.1) ZHD ¼ 106 k1 Rd
where N is the refractivity, which can be computed using the following formula (Haase et al. 2003): N ¼ k1
(103.6)
Pd e e þ k2 þ k2 2 ; TZd TZw T Zw
(103.2)
where Pd is the partial pressure of the dry air, e is the partial pressure of water vapour, T is the temperature, Zd and Zw are the compressibility coefficients for dry air and water vapour respectively and k1, k2 and k3 are experimental constants. For the values of these constants see (Thayer 1974 or Bevis et al. 1992). Substituting (103.2) into (103.1), the total tropospheric delay in the zenith direction can be computed using the following integral: ð zt ZTD ¼ 106 N dz ¼ 106 za ð zt Pd e e dz; k1 þ k2 þ k2 2 TZd TZw T Zw za (103.3)
ð zt
r dz;
(103.7)
za
and 6
Rd k2 k1 Rv
ZWD ¼ 10 Rv ð zt ð zt rv 6 rv dzþ 10 kg Rv dz za za T
(103.8)
The integrated water vapour is defined as: IWV ¼
ð zt
rv dz:
(103.9)
za
Introducing the mean temperature of the water vapour: R zt rv dz ; Tm ¼ R az zt rv az T dz
(103.10)
Equation (103.8) can be rewritten as: Rd ZWD ¼10 Rv k2 k1 IWV Rv IWV þ 106 k2 Rv : Tm 6
where za is the antenna height, while zt is the upmost level of the troposphere. The state equation of real gases can be written in a form of: m pV ¼ RZT; M
(103.4)
where p is the pressure, V is the volume, R is the universal gas constant, m is the total mass, and M is the molar mass of the gas. Express the pressure from (103.4): p¼
m R ZT ¼ ri Ri Zi T; V M
(103.5)
where ri is the density of the gas, Ri is the specific gas constant and Zi is the compressibility of the respective gas.
(103.11)
Thus the integrated water vapour can be estimated using the zenith wet delay by: IWV ¼
106 ZWD: Rv RRdv k1 þ k2 þ Tkm2
(103.12)
It can be seen that the zenith wet delay can be scaled to the IWV by applying a scaling factor (Q), which is a function of the mean temperature of water vapour: QðTm Þ ¼
106 : Rv RRdv k1 þ k2 þ Tkm2
(103.13)
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Estimation of Integrated Water Vapour from GPS Observations Using Local Models
The mean temperature of the water vapour can be estimated using a linear regression equation as a function of the surface temperature (Bevis et al. 1992): Tm ¼ 70:2 þ 0:72Ts :
(103.14)
Emardson and Derks (2000) derived a formula of Q as a function of the surface temperature using more than 120,000 observations of 38 European radiosonde stations: 1 þ a2 ðTf TÞ 2 ; (103.15) ¼ a0 þ a1 ðTf TÞ QðTf Þ where a0 ¼ 6.458 m3/kg, a1 ¼ 1.78102 m3/kg/K, a3 ¼ 2.2105 m3/kg/K e´s T ¼ 283:49 K:
103.3 The Studied Models 103.3.1 The Bevis Model This model is based on (103.12), and the mean temperature of the water vapour is estimated as a function of surface temperature using (103.14).
103.3.2 The Emardson–Derks Model This model is based on (103.12) as well, but Q is computed as a direct function of the surface temperature using (103.15).
103.3.3 Annual and Monthly Local Regression Model More than 10,000 local radiosonde observations were analyzed in Hungary, and the linear regression parameters have been computed between the mean temperature of water vapour and the surface temperature. The parameters have been computed on an annual basis, taking all the observations into account, as well as on a monthly basis, when the observations were separated into 12 groups according to the month of the observations. In this way the seasonal variation of the regression parameters could also be taken into account (Ro´zsa et al. 2009).
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Table 103.1 The monthly and annual regression parameters between the surface temperature ( C) and the mean temperature of the water vapour (K) Month 1 2 3 4 5 6 7 8 9 10 11 12 Annual
a 0.53 0.60 0.59 0.61 0.51 0.50 0.44 0.42 0.48 0.61 0.65 0.56 0.67
b 267.1 265.7 265.4 265.5 269.3 270.6 272.5 273.5 270.5 268.3 267.1 266.8 266.6
R2 0.37 0.49 0.50 0.58 0.48 0.45 0.41 0.41 0.41 0.54 0.55 0.40 0.80
s 3.2 3.4 3.0 2.9 2.9 3.0 2.8 2.8 2.8 3.1 2.9 3.1 3.2
N 849 787 862 837 847 828 864 857 831 865 829 859 10,115
Using all the observations, the following annual formula could be derived for the mean temperature of the water vapour: Tm ¼ 0:675Ts þ 82:3:
(103.16)
where TS and Tm are expressed in Kelvin. Altogether 12 monthly parameter sets have been also determined to take the seasonal variations into account. The parameters can be found in Table 103.1. In this case the regression equation can be written as: Tm ¼ aTs þ b:
(103.17)
where Ts is the surface temperature expressed in C. From Table 103.1 it can be seen that the monthly regression parameters have a smaller residual error than the annual models for the summer period only, when the troposphere contains more vapour than in the winter period. Assuming that the mean temperature of the water vapour can be determined with the accuracy of 3 K, using the law of error propagation the accuracy of the IWV would be 0.3 mm. Taking into account the accuracy of the ZWD estimation, the accuracy of the IWV is estimated to be 1.5–2.0 mm.
103.3.4 Regression Model Using Surface Water Vapour Density A similar set of linear regression parameters have been computed between the surface water vapour density
Sz. Ro´zsa
820 Table 103.2 The monthly, and annual regression parameters between the surface water vapour density and the integrated water vapour Month 1 2 3 4 5 6 7 8 9 10 11 12 Annual
a 2.50 2.43 2.45 2.28 2.18 2.13 2.24 2.21 2.43 2.50 2.47 2.50 2.39
b 0.24 0.78 0.79 0.06 0.86 2.16 2.08 2.50 0.72 1.13 0.38 0.05 0.14
R2 0.53 0.65 0.73 0.78 0.72 0.64 0.65 0.63 0.66 0.70 0.62 0.54 0.84
s 2.9 2.4 2.3 2.3 2.9 3.8 3.8 3.9 3.6 3.8 3.3 3.1 3.3
N 849 787 862 837 847 828 864 857 831 865 829 859 10,115
and the integrated water vapour using the aforementioned radiosonde observations: IWV ¼ arw þ b:
(103.17)
where rw is the surface water vapour density. The regression parameters can be found in Table 103.2. Studying the accuracy of the regression models it can be stated that this model has a lower accuracy than the previous models, which also utilize the ZWD stemming from the GNSS processing. On the other hand it should be also noted that although the accuracies are generally lower, the difference is much smaller in the winter period than in the summer period.
103.4 Data and Processing In order to evaluate the various models, two study periods have been chosen. One of them was a stormy summer period, when a heavy rainstorm hit the city of Budapest (August 19–21, 2006). The other period was chosen as a calm winter period February 26–28, 2007. In both cases the stations of the Hungarian Active GNSS Network have been used for the processing. In the first period altogether 17 stations have been chosen, while in the second period only 8 stations have been chosen for the processing. The estimated IWV values were compared to the radiosonde observations provided by the Hungarian Meteorological Service (HMS). Radiosondes are
launched in every 12 h in Budapest (0 and 12 h UTC). Since the closest GNSS station is approximately 10 km from the launchsite, an experimental GNSS station have been set up at the Observatory of the Hungarian Meteorological Service in the second period to study the effect of spatial decorrelation of the zenith wet delays. Surface meteorological observations have been used for computing the hydrostatic part of the tropospheric delay. Since only three of the Hungarian GNSS stations are equipped with meteorological sensors, the synoptic and climatic network of the HMS have been used to collect the meteorological data. The observations have been processed with the Bernese V5 processing software (Dach et al. 2007). Since in this case the mathematical models have been tested, a post-processing scheme was used, using IGS final orbits and IGS antenna phase center models. The effect of the ionosphere has been eliminated by processing the L3 ionosphere-free linear combination of the L1 and L2 frequencies. The coordinates of the stations have been computed by processing 7 days of observations of the prior GPS week. Afterwards these coordinates were fixed, and the tropospheric wet delay was estimated in every 20 min (2006) or 15 min (2007). The hydrostatic part of the troposphere was taken into account using surface meteorological data and the dry Niell model (the dry part of the Saastamoinen model using the Niell mapping function). Finally the IWV values have been estimated using the various methods mentioned before: • The Bevis model (BM) • Annual local regression model (ALRM) • Monthly local regression model (MLRM) • Emardson–Derks model (EDM) • Annual regression model using surface densities (ARMSD) • Monthly regression model using surface densities (MRMSD)
103.5 Results 103.5.1 Summer Period The observations of the 17 stations of the active GNSS network of Hungary had been processed and the IWV values were estimated at each station. Since the BUTE
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Estimation of Integrated Water Vapour from GPS Observations Using Local Models
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Table 103.3 The statistics of the comparison of measured IWV (radiosonde) and estimated IWV at the BUTE station in the summer period (values are in mm) IWV (radiosonde) BM ALRM MLRM EDM ARMSD MRMSD
Fig. 103.1 The estimated IWV values at the BUTE station (Emardson–Derks model) (dashed line) and the radiosonde observations (triangles)
station is the closest station to the radiosonde launch site, firstly the estimated IWV values stemming from this station were compared with the radiosonde
Minimum 27.1 4.2 3.9 4.0 3.9 2.6 3.1
Maximum 39.0 3.2 3.4 3.3 3.3 4.9 4.8
Mean 32.5 0.7 0.9 0.9 0.8 0.4 0.2
SD 5.19 2.87 2.84 2.89 2.83 3.15 3.13
observations. These observations can be seen on Fig. 103.1. This comparison has been made for all the six investigated models. The results of the comparison can be found in Table 103.3. It can be seen that the Emardson–Derks, and the local annual model had the smallest standard deviation. Moreover both of the models neglecting the information from GNSS (ARMSD and MRMSD) have significantly higher standard deviation.
Fig. 103.2 Radar images and IWV maps based on GPS observations on August 20, 2006. UTC 18.20 (left) and UTC 19.00 (right)
Sz. Ro´zsa
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Table 103.4 The statistics of the comparison of measured IWV (radiosonde) and estimated IWV at the OMSZ station in the winter period (values are in mm) IWV (radiosonde) BM ALRM MLRM EDM ARMSD MRMSD
Fig. 103.3 The estimated IWV values at the OMSZ station (Emardson–Derks model) (dashed line) and the radiosonde observations (triangles)
During this study period a heavy storm passed through the country. The storm formed a supercell in the vicinity of Budapest at 19.00 UTC on August 20, 2006 (this corresponds to 43 h from the beginning of the period). The effect of this storm and the heavy rainfall can be observed on Fig. 103.1, too. Since the estimation of the IWV was made on all the 17 stations of the active GNSS network, IWV maps could be created for the whole country. Finally these maps were also compared to the meteorological radar images created at the same time showing the intensity of the precipitation in the area. Figure 103.2 shows both the IWV map and the radar image for two epochs (18.20 UTC, and 19.00 UTC, when the storm hit the city of Budapest). On both of the figures a strong gradient can be observed on the IWV maps, exactly in the same location, where the highest rain rates were experienced. This example clearly shows the good time resolution of the IWV estimates using GNSS observations.
103.5.2 Winter Period In the winter period the observations of altogether 8 stations have been processed. Out of these stations the OMSZ station was a collocated station with the radiosonde launch site. After the estimation of IWV values, they were compared with the radiosonde observations at the OMSZ site. The results can be seen in Fig. 103.3 using the Emardson–Derks model,
Minimum 9.8 4.0 4.0 3.9 3.8 4.4 4.0
Maximum 15.0 0.0 0.0 0.0 0.1 4.5 5.0
Mean 12.6 2.5 2.4 2.4 2.3 1.2 0.9
SD 2.20 1.50 1.48 1.47 1.46 3.64 3.68
while the statistics of all the investigated models can be found in Table 103.4. From Table 103.4. it is visible that the accuracy of the estimations are better in the winter period compared to the summer period, and it reached the level of 1.5 mm for all of the models, which utilize the GNSS observations. It can be also observed that the models which neglect the GNSS observation perform worse in the comparison again. Conclusions
Our investigations proved again that the GNSS observations augmented by surface meteorological observations can be used for the estimation of integrated water vapour density. With this approach the frequency of IWV observations could be increased both in the space as well as in the time domain, since radiosonde launches are made in two places recently (Budapest and Szeged) in every 12 and 24 h respectively. The more detailed and more frequent IWV observations could enhance the forecasting of heavy storms. The studies in the summer period also showed that the GNSS based IWV estimations have a very short response time to real weather changes. On Fig. 103.2. it can clearly be seen that the estimations are very well in line with the evolution of the weather front in the area. Two regression models were also investigated, which neglect the additional information stemming from the GNSS observations. These models used surface water vapour density observations to predict the integrated water vapour in the troposphere. In both cases the performance of these models were poorer compared to the other models, which also proves the importance of GNSS observations in the estimation of the IWV in the troposphere.
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Estimation of Integrated Water Vapour from GPS Observations Using Local Models
However these latter models might be used for computing the effect of the troposphere using surface meteorological data, since they can be used to estimate the IWV, which might be scaled to the tropospheric zenith wet delay. This application is out of the scope of this paper, and will be investigated later. Acknowledgement The author acknowledges the help of the Satellite Geodetic Observatory of the Institute of Geodesy, Cartography and Remote Sensing for providing the GNSS observations, and the Hungarian Meteorological Service for providing the meteorological data for these studies.
References Ba´nyai L (2008) The Application of satellite positioning in Earth Sciences (in Hungarian). Geomatikai K€ ozleme´nyek, Sopron
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Bevis M, Businger S, Herring TA, Rocken C, Anthes A, Ware R (1992) GPS meteorology: remote sensing of atmospheric water vapor using the global positioning system. J Geophys Res 97:15787–15801 Borba´s E´ (2000) A new source of meteorological data: the Global Positioning System (in Hungarian). PhD thesis, Lora´nd E€otv€os University, Budapest Dach R, Hugentobler U, Fridez P, Meindl M (2007) Bernese GPS software, version 5.0. Astronomical Institute, University of Bern, Bern Emardson TR, Derks HJP (2000) On the relation between the wet delay and the integrated precipitable water vapour in the European atmosphere. Meteorol Appl 7:61–68 Haase J, Maorong G, Vedel H, Calais E (2003) Accuracy and variablity of GPS tropospheric delay measurements of water vapour in the Western Mediterranean. J Appl Meteorol 42:1547–1569 Ro´zsa Sz, Dombai F, Ne´meth P, Ablonczy D (2009) The estimation of integrated water vapour based on GNSS observations (in Hungarian). Geomatikai K€ozleme´nyek, XII, Sopron, pp 187–196 Thayer GD (1974) An improved equation for the radio refractive index of air. Radio Sci 9:803–807
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GNSS Remote Sensing in the Atmosphere, Oceans, Land and Hydrology
104
Shuanggen Jin
Abstract
The Global Navigation Satellite System (GNSS) provides a continuous, high precision, global, real-time, and all weather navigation and positioning, which powerfully contributes to all scientific questions related to precise positioning on Earth’s surface. Recently, the reflected and refracted signals from GNSS satellites resulted in many new applications in environmental remote sensing. The GNSS refracted signals from Radio Occultation satellites together with ground GNSS observations have produced wide applications in atmospheric remote sensing, including global monitoring of tropospheric water vapour, temperature and pressure, tropopause parameters and ionospheric parameters as well as ionospheric irregularities. The GNSS reflected signals from the ocean and land surface could determine the wave height, wind speed and wind direction of ocean surface and land surface conditions. In this paper, GNSS remote sensing applications in the atmosphere, oceans, land and hydrology are presented as well as recent results. With more and more GNSS satellite constellations in coming years, it is expected more and wider applications in various environment remote sensing.
104.1 Introduction Conventional radar techniques in the atmosphere, oceans and land remote sensing require the dedicated transmitters and receivers with large directional antennae top to achieve a high resolution. While the Global Navigation Satellite System (GNSS), including the Global Positioning System (GPS) in the United States, the Russian GLONASS, the European Galileo
S. Jin (*) Center for Space Research, University of Texas, Austin, TX, USA Shanghai Astronomical Observatory, CAS, Shanghai, China e-mail:
[email protected];
[email protected]
and Chinese Compass (or Beidou), is an interferometric and microwave technique with signals that are transmitted on microwave carriers (L-band) as well as continuous, global, all- weather observations (Jin and Komjathy 2010), it implies more and wider applications (Wagner and Kloko 2003). For a long time, the delay of GNSS measurements caused by the atmospheric refractivity was considered as an error source, and now has been widely used to determine the tropospheric water vapor, temperature, pressure and total electron content (TEC) and electron density profiles in the ionosphere (Jin et al. 2008). However, traditional observing instruments, e.g. water vapour radiometer (WVR), ionosonde, incoherent scatter radars (ISR), topside sounders onboard satellites, in situ rocket and satellite observations, are expensive and
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_104, # Springer-Verlag Berlin Heidelberg 2012
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also partly restricted to either the bottomside ionosphere or the lower part of the topside ionosphere (usually lower than 800 km), such as ground based radar ionospheric measurements. While GPS satellites in high altitude orbits (~20,200 km) are capable of providing details on the structure of the entire atmosphere, even the plasma-sphere. Moreover, GPS is a low-cost, all-weather, near real time, and hightemporal resolution (1–30 s) technique. Therefore, GPS has been widely used to investigate the ionospheric activities. In addition, surface multi-path is one of main error sources for GNSS navigation and positioning. It has recently been recognized, however, that the special kind of multi-path delay, that from the GNSS signal reflecting from the sea and land surface, could be used as a new tool in oceans, coastal, wetlands, landslide, Crater Lake, soil moisture, snow and ice remote sensing (Komjathy et al. 2000; Kostelecky´ et al. 2005). Furthermore, GNSS could compensate the defects of conventional radar remote sensing techniques. In this paper, some new applications and progresses of GNSS remote sensing in the atmosphere, oceans, land and hydrology are addressed as well as future opportunities.
104.2 GNSS Atmospheric Remote Sensing 104.2.1 Tropospheric Imaging
Lik;j ¼ lk fik;j i i ¼ r dion;k;j þ dtrop;j þ cðti tj Þ
(104.1)
104.2.2.1 2-D Ionospheric Imaging The ionospheric delay can be determined from the double frequency GPS phase and code (pseudorange) observations as
¼ 40:3ð
1 1 ÞFðzÞVTECðb; sÞ þ B4 (104.3) f12 f22
P4 ¼ Pi1j Pi2j 1 1 ¼ 40:3ð 2 2 ÞFðzÞVTECðb; sÞ þ b4 f1 f2
(104.4)
where FðzÞ is the mapping function, B4 is ðB4 ¼ l1 i i Þ þ l2 ðbi2j þ N2j ÞÞ, and b4 isðdq1j dq2j Þþ ðbi1j þ N1j i i ðdq1 dq2 Þ. The Differential Code Biases ðb4 Þ can be obtained through GPS carrier phase observations, and B4 can be obtained through the formula, N P ðp4 þ L4 b4 Þ=N, where N is the epoch of GPS i¼1
i i i þ dtrop;j þ cðti tj Þ þ dq;k Pik;j ¼ r þ dion;k;j
þ dq;k;j þ eij
104.2.2 Ionospheric Sounding
L4 ¼ fi1j fi2j
The GPS consists of a constellation of 24 operating satellites in six circular orbits 20,200 km above the Earth at an inclination angle of 55 with a 12-h period. The satellite transmits two frequencies of signals (f1 ¼ 1575.42 MHz and f2 ¼ 1227.60 MHz). The equations of carrier phase (L) and code observations (pseudorange P) of double frequency GPS can be expressed as follows:
i Þ lk ðbik;j þ Nk;j
where superscript i and subscript j represent the satellite and ground-based GPS receiver, respectively, r is the distance between satellite i and GPS receiver j, dion and dtrop are the ionospheric and tropospheric delays, respectively, c is the speed of light in vacuum space, t is the satellite or receiver clock offset, b is the phase delay of satellite and receiver instrument bias, dq is the code delay of satellite and receiver instrumental bias, l is the carrier wavelength, f is the total carrier phase between the satellite and receiver, N is the ambiguity of carrier phase, and e is the other residuals. With more than 4 satellites observations, the unknown parameters (e.g. GPS receiver coordinate and tropospheric delays) and their uncertainties can be obtained using the least square method (LS) with some algorithms. The tropopsheric delay has been widely used for meteorology and atmospheric research in the past two decades, e.g. numerical prediction model assimilation and water vapour and zenith tropospheric delay climatology (e.g. Jin et al. 2007).
(104.2)
observation (Jin et al. 2008). For the TEC representation, a single layer model (SLM) ionosphere approximation was used. SLM assumes that all the free
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GNSS Remote Sensing in the Atmosphere, Oceans, Land and Hydrology
electrons are contained in a shell of infinitesimal thickness at altitude H (generally 350 km above the Earth). A mapping function is used to convert the slant TEC into the vertical TEC (VTEC) as shown (Mannucci et al. 1998): FðzÞ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 R cosð90 zÞ=ðR þ HÞÞ
(104.5)
where R is Earth radius, H is SLM height, and z is satellite zenith angle. When using the above mapping function, F(z), one can obtain VTEC values at the ionosphere pierce points (IPPs). The GPS-derived TEC can correct ionospheric delay for microwave techniques and monitor space weather events. For example, Fig. 104.1 shows the
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VTEC time series (bottom panel) at the grid points of different latitudes, (15 S, 145 E), (30 S, 145 E) and (45 S, 145 E) during the geomagnetic storm from 19 to 21 Nov 2003 with the time series of the Dst index, and geomagnetic activity index Kp (upper panel). The Dst profile is the solid line and Kp indexes are the dash line. The Dst index is used to define the occurrence, duration and magnitude of a storm, whose unit is nT, nanotesla. And the Kp index is a disturbed level of geomagnetic field. The Kp indexes reach a value of 9 (close to its maximum, from 20:00 on 20 Nov and 4:00 UT on 21 Nov 2003, indicating a severely disturbed geomagnetic condition during these time. The Dst index also reaches the summit at 20:00 on 20 Nov, indicating a stronger geomagnetic activity. The TECs normally vary from day to night on 19 Nov 2003. But it suddenly reduces from 8:00 to 16:00 20 Nov 2003, and rarely varies from day to night on 21 Nov 2003. It is obvious to see that the TEC has a severe effect due to suffering a great geomagnetic storm.
104.2.2.2 3-D Ionospheric Tomography Reconstruction The continuous GPS observations provide a highresolution slant TEC of each ray-path, and the 3-D ionospheric electron density profiles are obtained through a tomography reconstruction algorithm (Jin et al. 2006). The tomography reconstruction algorithm can integrate the data from all available GPS receivers and all GPS satellites visible from each of these receivers. The unknown electron density profile is expressed in 4D (longitude–latitude-height and time) voxel basis functions over the following grid: longitude in m increments, latitude in n increments, altitude 100–1,000 km in 25 km increments and time: 0.5–1 h increments of linear change in the electron density per voxel. Each set of sTEC measurements along the ray paths from all observable satellites and from consecutive epochs are combined with the ray path geometry into a linear expression: Y ¼ Ax þ e
64
(104.6)
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Hour from 19 Nov 2003
Fig. 104.1 The time series of Dst index and geomagnetic activity Kp index on 19–21 Nov, 2003 (upper panel) and VTEC series at three grid points from 19 to 21 Nov 2003 (bottom panel)
where A is a matrix relating the ray paths to the voxels, Y is a column vector containing the observed sTEC values and x is the column vector of unknown coefficients of the basis functions. The inversion of the set of underdetermined linear equations is done
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using multiplicative algebraic reconstruction technique (MART) (Raymund et al. 1990). The inversion of this matrix gives the unknown coefficients of the electron density distribution from which the vertical electron density or vertical TEC at any grid can be inferred. The solution is constrained using a priori information from the IRI-2001 (Bilitza 2001) or ionosonde. For example, Fig. 104.2 shows comparison of the ionospheric electron density profiles derived from the ground-based GPS tomography reconstruction (solid line), ionosonde observation at Anyang stations (37.39N, 126.95E) (dashed line) and IRI-2001 estimation (dot line). It has shown that the GPS tomographically reconstructed density profile has a good agreement with ionosonde data from Anyang and IRI-2001 model, but is closer to the ionosonde data, especially at the electron density peak. In addition, a number of GPS Radio Occultation (RO) missions have been successfully applied in atmospheric and ionospheric detections and climate change related studies, e.g. the US/Argentina SAC-C, German CHAMP (CHAllenging Minisatellite Payload), Taiwan/ US FORMOSAT-3/COSMIC (FORMOsa SATellite mission - 3/Constellation Observing System for Meteorology, Ionosphere and Climate) satellites, and German TerraSAR-X satellites together with the European MetOp. These GPS RO satellites together have demonstrated wide applications in atmospheric remote
1000 GPS IRI-2001 Ionosonde
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sensing, including global monitoring of tropospheric water vapour, temperature and pressure, tropopause parameters and ionospheric parameters as well as its irregularities (e.g. Schmidt et al. 2008).
104.3 GNSS Ocean Remote Sensing The delay of the GPS reflected signal with respect to the rough reflected surface could provide information on the differential path between direct and reflected signals (see Fig. 104.3). Together with information on the receiving antenna position and the medium, the delay measurements associating with the surface properties of the reflecting ocean surface can be used to determine ocean surface wave height, wind speed, wind direction, and even sea ice conditions (Jin and Komjathy 2010). While conventional single radar remote sensing of the oceans requires dedicated transmitters and receivers with large directional antennae top to obtain a high resolution, GPS as a bistatic radar system requires only a receiver with two antennae, not a transmitter, and can receive freecost, global, real-time, continuous and all weather signals from GPS satellites. Therefore, potential applications of GPS reflected signals from the sea surface could compensate these defects of conventional radar techniques. Recently, a number of ocean remote sensing applications have been implemented using GPS signals reflected from the ocean surface (Martin-Neira et al. 2001; Rius et al. 2002; Germain et al. 2004; Komjathy et al. 2004; Thompson et al. 2005; Gleason et al. 2005). For example, Martin-Neira et al. (2001) showed the height of sea surface
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Fig. 104.2 Comparison of the electron density profiles derived from the ground-based GPS tomography reconstruction (solid line), ionosonde observation at Anyang stations (37.39 N, 126.95E) (dashed line) and IRI-2001 estimation (dot line)
Fig. 104.3 GPS reflection signals and geometry
GPS6
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GNSS Remote Sensing in the Atmosphere, Oceans, Land and Hydrology
determined from GPS reflected signals well coinciding with independent TOPEX/Poseidon data at 2 cm precision. Furthermore, GPS in a bistatic radar configuration can receive the power from a reflected signal for a variety of delays and Doppler values in a glistening zone surrounding a nominal specular reflection point (Garrison et al. 1998). The size and shape of the glistening zone are functions with the roughness of the ocean surface. Recently, using the Delay Mapping Receiver (DMR) to predict the interaction of the L1 GPS signal at 1575.42 MHz, it has been successful to estimate wind speeds and directions on the ocean surface with high accuracy. The estimated wind speed using surface-reflected GPS data was consistent with independent wind speed measurements derived from the TOPEX/Poseidon altimetry satellite and balloon measurements at the level of 2 m/s (e.g., Garrison et al. 2002). Moreover, the estimated wind direction using surface-reflected GPS data was also consistent with results obtained from a buoy at the level of 10 (Zavorotny and Voronovich 2000; Komjathy et al. 2004; Cardellach et al. 2003). In addition, due to complex and varying conditions of sea ice, e.g. an inaccessible environment and persistent cloud coverage, it is very difficult to monitor sea ice conditions using conventional instruments. Synthetic aperture radar (SAR) has sufficient spatial resolution to resolve detailed ice features, but repeat times of existing satellites are relatively long when compared to the change rate of open water fraction in the ice pack, although it may change with more satellites in the future. Moreover, SAR carries a substantial requirement in cost for acquisition and image processing. Space-borne passive microwave sensor provides more frequent coverage at several wavelengths, but has substantially lower spatial resolution. While optical and thermal sensors provide a middle ground in resolution and temporal sampling between SAR and passive microwave satellites, but are limited by cloud cover and visibility conditions. Nowadays, low-cost and all-weather GPS reflectometry technique can monitor the sea ice surface characteristics in any conditions. Komjathy et al. (2000) analyzed the experiment of GPS reflections from Arctic sea ice and over the ice pack near Barrow, Alaska, USA. Correlations from comparisons between RADARSAT backscatter and GPS forward scattered data indicate that the GPS reflected signals could estimate useful
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information about ice conditions, in addition to the basic ability to detect the presence of sea ice. For example, Melling (1998) determined the internal ice structure with the reflection coefficient of a frozen sea surface by the effective dielectric constant of ice depending on various factors, such as an ice composition, salinity, temperature, density, age, origin, morphology (Shohr 1998) and by the dielectric constant of the underlying water under some conditions. So the reflected GPS signals are particularly useful for investigations of the state of inaccessible and sea ice cover.
104.4 GNSS Land/Hydrology Remote Sensing The soil moisture content is an important parameter in hydrology, climatology, and agriculture (Jackson et al. 1996). In the past, the soil moisture was usually inferred using passive radiometers and active radar sensors. However, these two remote sensing techniques are more expensive and non-real-time. GPS measurements reflected from the land surface is similar with the multi-static radar system, with transmitters located at each GPS satellite and a separate receiver located above the surface of the earth. The GPS signal forward scattered from a land surface is analogous to scattering from an ocean surface, while the main differences are in the spatially and temporally varying dielectric constant, surface roughness, and possible vegetative cover. Masters et al. (2000, 2004) obtained soil moisture on USDA SCAN site, located on the Central Plains Experimental Range of Colombia, with peak power of the GPS reflections. Katzberg et al. (2006) has also successfully estimated soil reflectivity and dielectric constant with GPS reflected signals. The results indicate the potential for remotely sensing soil moisture content with bistatic GPS. Furthermore, the multi-path from ground GPS network has also retrieved fluctuations in near surface soil moisture from a 300 m2 area, closely matching soil moisture fluctuations in the top 5 cm of soil measured with conventional sensors (Larson et al. 2008). Therefore, the ground-based GPS multi-path signals indicate the potential for remotely sensing soil moisture content. In the future, the existing global GPS networks with more than thousands of GPS receivers operated around
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the world may be used to estimate soil moisture in near real-time for hydrology and climate studies. In addition, GPS reflected signals also could detect characteristics of land surface as a new remote sensing tool. Treuhaft et al. (2001) measured the Crater Lake surface height with 2-cm precision in 1 second with GPS signals reflected from the surface of Crater Lake. The strength of the reflected signal is also a discriminator between wet and dry ground areas and therefore could be applied to coastal and wetland mapping.
104.5 Concluding Remarks Traditional radar remote sensing techniques have some limitations, e.g. more-cost and low temporal resolution. Nowadays, GNSS resulted in many new applications and opportunities in environmental remote sensing as a low-cost, continuous, global, allweather, near-real-time and interferometric microwave (L-band) technique. The refracted signals from Radio Occultation satellites together with ground GNSS observations have produced high-resolution tropospheric and ionospheric parameters. The reflected GNSS signals useable on any type of aircraft, are able to determine the ocean surface wave height, wind speed and direction, salinity, sea ice cover conditions, soil moisture, wetland and Crater Lake conditions. It is expected to revolutionize various environmental remote sensing. Therefore, GNSS provides us with the unique opportunity as a new remote sensing tool at a global scale to infer various geophysical parameters as well as their applications in environment and climate. In the coming years, with more and more GNSS satellite constellations, the surface reflected and refracted GPS signals would soon become a more powerful tool for various environment remote sensing.
References Bilitza D (2001) International reference ionosphere 2000. Radio Sci 36:261–275 Cardellach E, Ruffini G, Pino D, Rius A, Komjathy A, Garrison JL (2003) Mediterranean Balloon Experiment: ocean wind speed sensing from the stratosphere, us GPS reflections. Rem Sens Environ 88:351–362
S. Jin Garrison JL, Katzberg SJ, Hill MI (1998) Effect of sea roughness on bistatically scattered range coded signals from the global positioning system. Geophys Res Lett 25(13): 2257–2260 Garrison JL, Komjathy A, Zavorotny VU, Katzberg SJ (2002) Wind speed measurement from forward scattered GPS signals. IEEE Trans Geosci Rem Sens 40(1):50–65 Germain O, Ruffini G, Soulat F, Caparrini M, Chapron B, Silversten P (2004) The Eddy Experiment: GNSS-R speculometry for directional sea-roughness retrieval from low altitude aircraft. Geophys Res Lett. 31, doi: 10.1029/ 2004GL020991 Gleason S, Hodgart S, Sun Y, Gommenginger C, Mackin S, Adjrad M, Unwin M (2005) Detection and processing of bistatically reflected GPS signals from low earth orbit for the purpose of ocean remote sensing. IEEE Trans Geosci Rem Sens 43(6):1229–1241 Jackson T, Schmugge J, Engman E (1996) Remote sensing applications to hydrology: soil moisture. Hydrol Sci 41(4): 517–530 Jin SG, Komjathy A (2010) GNSS reflectometry and remote sensing: new objectives and results, Adv Space Res 46(2): 111–117, doi: 10.1016/j.asr.2010.01.014. Jin SG, Park J, Wang J, Choi B, Park P (2006) Electron density profiles derived from ground-based GPS observations. J Navig 59(3):395–401 Jin SG, Park J, Cho J, Park P (2007) Seasonal variability of GPSderived Zenith Tropospheric Delay (1994–2006) and climate implications. Journal of Geophysical Research, 112:D09110, doi:10.1029/2006JD007772 Jin SG, Luo OF, Park P (2008) GPS observations of the ionospheric F2-layer behavior during the 20th November 2003 geomagnetic storm over South Korea. J Geodesy 82 (12):883–892. doi:10.1007/s00190-008-0217-x Katzberg S, Torres O, Grant MS, Masters D (2006) Utilizing calibrated GPS reflected signals to estimate soil reflectivity and dielectric constant: Results from SMEX02. Rem Sens Environ 100(1):17–28 Komjathy A, Zavorotny V, Axelrad P et al (2000) GPS signal scattering from sea surface: wind speed retrieval using experimental data and theoretical model. Rem Sens Environ 73:162–174 Komjathy A, Armatys M, Masters D, Axelrad P, Zavorotny V, Katzberg S (2004) Retrieval of ocean surface wind speed and wind direction using reflected GPS signals. J Atmos Ocean Technol 21(3):515–526 Kostelecky´ J, Klokocˇnı´k J, Wagner CA (2005) Geometry and accuracy of reflecting points in bistatic satellite altimetry. J Geod 79(8):421–430 Larson KM, Small EE, Gutmann E, Bilich A, Braun J, Zavorotny V (2008) Use of GPS receivers as a soil moisture network for water cycle studies. Geophys Res Lett 35: L24405 doi:10.1029/2008GL036013 Mannucci AJ, Wilson B, Yuan DN et al (1998) A global mapping technique for GPS-derived ionospheric total electron content measurements. Radio Sci 33(3):565–574 Martin-Neira M, Caparrini M, Font-Rosselo J et al (2001) The PARIS concept: an experimental demonstration of sea surface altimetry using GPS reflected signals. IEEE Trans Geosci Rem Sens 39:142–150
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Masters D, Zavorotny V, Katzberg S, Emery W (2000) GPS Signal Scattering from Land for Moisture Content Determination, Presented at IGARSS, July 24–28 Masters D, Axelrad A, Katzberg SJ (2004) Initial results of landreflected GPS bistatic radar measurements in SMEX02. Rem Sens Environ 92:507–520 Melling H (1998) Detection of features in first-year pack ice by synthetic aperture radar (SAR). Int J Rem Sens 19 (6):1223–1249 Raymund TD, Austen JR, Franke SJ (1990) Application of computerized tomography to the investigation of ionospheric structures. Radio Sci 25:771–789 Rius A, Aparicio JM, Cardellach E, Martin-Neira M, Chapron B (2002) Sea surface state measured using GPS reflected signals. Geophys Res Lett, 29(23), doi:10.1029/2002GL015524 Schmidt T, Wickert J, Beyerle G, Heise S (2008) Global tropopause height trends estimated from GPS radio occultation data. Geophys Res Lett 35:L11806
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Shohr ME (1998) Field observations and model calculations of dielectric properties of Arctic sea ice in the microwave C-band. IEEE Trans Geosci Rem Sens 36(2):463–478 Thompson DR, Elfouhaily TM, Garrison JL (2005) An improved geometrical optics model for bistatic GPS scattering from the ocean surface. IEEE Trans Geosci Rem Sens, 43(12):2810.2821. doi:10.1109/TGRS.2005.857895 Treuhaft RN, Lower ST, Zuffada C et al (2001) 2-cm GPS altimetry over Crater Lake. Geophys Res Lett 28(23):4343–4346 Wagner C, Kloko J (2003) The value of ocean reflections of GPS signals to enhance satellite altimetry: data distribution and error analysis. J Geod 74:128–138 Zavorotny AU, Voronovich AG (2000) Scattering of GPS Signals from the ocean with wind remote sensing application. IEEE Trans Geosci Rem Sens 38(2):951–964
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Mean Sea Surface Model of the Caspian Sea Based on TOPEX/Poseidon and Jason-1 Satellite Altimetry Data
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Abstract
Usually mean sea surface (MSS) is calculated by averaging altimetric measurements of sea surface height (SSH) over a given region and over a given time period. However, in the case of the Caspian Sea this represents a certain challenge and existing MSS models are unacceptable. One of the possible solutions is to analyse the Caspian Sea MSS as task of investigation of space-time variability of equipotential sea surface or SSH without seasonal and synoptic variability. Regional MSS model of the Caspian Sea (GCRAS08 MSS) was calculated according to the following scheme. For satellite altimetry data processing dry tropospheric corrections was calculated on atmospheric pressure from nearest weather stations located along the Caspian Sea costal line. From the TOPEX/Poseidon (T/P) and Jason-1 (J1) satellite altimetry data, the SSH synoptic and seasonal variations for all passes of each repeat cycle were eliminated. In last phase, the GCRAS08 MSS was constructed as a SSH function of latitude, longitude, and time with correction on climatic dynamic topography. For specified time interval SSH was interpolated on grid by radial basis function method. For the first time GCRAS08 MSS model allow to investigate space–time variability of the Caspian Sea level. According to the received results spatial variability of rate of the Caspian Sea level change well correlate with EGM96 gravity anomalies field and the greatest variability is observed in the zone of gravity anomalies gradient maximum.
105.1 Introduction
S.A. Lebedev (*) Department of Geoinformatic, Geophysical Center, Russian Academy of Sciences (GCRAS), 3 Molodezhnaya Str, 119296 Moscow, Russian Federation e-mail:
[email protected]
The Caspian Sea is the world’s largest isolated water reservoir, with only isolation being its significant dissimilarity from the open seas. Other features of the Caspian Sea including its size, depth, chemical properties, peculiarities of thermohaline structure, and water circulation allow it to be classified as a deep inland sea. At the present time its level is at 27 m (with respect to the Baltic altitude system) measured
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against the World Sea Level. The sea occupies an area of 392,600 km2, with mean and maximum depths being 208 and 1,025 m, respectively. The Caspian’s longitudinal extent is three times larger than its latitudinal one (1,000 vs. 200–400 km), resulting in great variability of climatic conditions over the sea. The isolation of the Caspian Sea from the ocean and its inland position dictate the significance of its outer thermohydrodynamic factors, specifically, heat and water fluxes through the sea surface, and river discharge over sea level variability, formation of its 3D thermohaline structure, and water circulation (Kosarev and Yablonskaya 1994). Over the past half-century, there was the Caspian Sea level regression until 1977 when the sea level lowered to 29 m (Fig. 105.1). It is obvious that man’s impact led to a more than 2.5 m regression of the Caspian Sea level caused by the creation of cascade reservoirs in the Volga and Kama Rivers. This drop is considered to be the deepest for the last 400–500 years (Kosarev 2005). In 1978 the water level started to rise rapidly, and now it has stabilized at the 27 m level. There has been increasing concern over the Caspian Sea level fluctuations. Significant of the Caspian Sea level fluctuations have serious consequences for the region, leading to displacement of thousands of shoreline inhabitants and damaging industrial constructions and infrastructure. Continuous weather-independent observations from satellite altimetry are a natural choice to complement the existing in situ sea level observations and to provide new information for open sea regions that has never been covered by direct sea level observations.
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Fig. 105.1 Interannual variations of the Caspian Sea level measured by sea level gauges (gray line) and satellite altimetry (black line) since 1837 till 2008
105.2 MSS Models of the Caspian Sea Usually MSS models are calculated by averaging altimetric measures over a given region and over a given time period. However, in the case of the Caspian Sea this represents a certain challenge. Firstly, it is storm surges, which depend on wind field and local physico-geographical conditions. The highest onsets are characteristic of the shallow-water North Caspian, where, in extreme cases, surges can reach heights of 3–4 m (Kosarev 2005; Kouraev et al. 2009). Secondly, there is the issue of sea ice. On the North Caspian ice formation begins in the middle of November and starts to decay in March in moderate winters. On average, the duration of the ice period is 120–140 days in the eastern part of the Northern Caspian and 80–90 days or less in the western part. On the eastern coast of the Middle Caspian, ice formation is possible in severe winters Thirdly, there is the issue of Caspian Sea water balance peculiarities. The most important of these are the Volga River discharge (more 80%), evaporation from the sea surface and the dynamics between the Caspian Sea and the Kara-Bogaz-Gol Bay. Thus for the period of time when satellite altimetry measurements were conducted from 1985 until 2008 water discharge in the Volgograd power station oscillated from 5609.25 m3/c (2006) to 1,136,983 m3/c (1996) (Kosarev 2005; Lebedev and Kostianoy 2005). In June 1992, the dam between the Caspian Sea and the Kara-Bogaz-Gol Bay was destroyed and the natural seawater runoff to the bay resumed. Up to the middle of 1996, the bay was rapidly filled with Caspian water (Kosarev and Kostianoy 2005; Lebedev and Kostianoy 2005). Fourthly, interannual changes of sea level (that are about 3 m for 1929–1977 and about 2.5 m for 1977–1995) here are sometimes much higher than seasonal ones (that are about 30 cm). Existing MSS models essentially differ according to the used information or temporal averaging interval (Fig. 105.2) (Lebedev and Kostianoy 2008). One of the possible solutions is to analyse the Caspian Sea MSS as task of investigation of spacetime variability of equipotential sea surface height or SSH without seasonal and synoptic variability.
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Fig. 105.2 Comparison of temporal intervals of average satellite altimetry data for principal MSS model and the Caspian Sea level variability from January 1985 to December 2001
105.3 Data and Methodology To analyze equipotential sea surface height variations and construction GCRAS08 MSS, measurements from both the T/P and J1 satellites were used for the following reasons. The position of T/P and J1 ground tracks (Kosarev and Kostianoy 2005) is optimal for analysis of sea level variations in the Caspian Sea. The precision of SSH measurements by T/P and J1 to the relative reference ellipsoid is 1.7 cm (Fu and Pihos 1994), which is higher than other altimetry missions (Chelton et al. 2001). At the same time, accuracy of sea level measurements is at ~4 cm (Birkett 1995; Nerem and Mitchum 2001) that allows adequate accuracy for studies to be conducted. The orbital repeat period (~10 days) enables analysis of interannual and seasonal variability of the sea level. The T/P data represent the longest time-series of satellite altimetry measurements (September 1992–August 2002 or 1–365 cycles) with the possibility of the data being extended by J1 data along the same tracks (August 2002–February 2009 or 1–259 cycles). We have analyzed T/P and J1 data since January 1993 until December 2008 satellite altimetry data from T/P merged geophysical data records B (Benada 1997) and J1 geophysical data record (Picot et al. 2006). Information and software of the Integrated Satellite Altimetry Data Base developed in the Geophysical Center of Russian Academy of Sciences
(Lebedev and Kostianoy 2005) have been used for data processing and analysis. SA data processing methods and analysis as well as obtained results on the SSH variations were described in detail in the book by Lebedev and Kostianoy (2005). All necessary corrections from satellite altimetry data-base: microwave radiometer wet tropospheric, smoothed dual-frequency ionosphere and sea state bias are used on data processing. Maximal tide height for the Caspian Sea is 2 cm in coastal zone so this correction isn’t used on data processing. In the framework of the NATO SfP Project “Multidisciplinary Analysis of the Caspian Sea Ecosystem” (MACE) three drifters have been deployed in the southern part of the Caspian Sea in October 2006. After 5 months two drifters were found on the periphery of the stationary anticyclonic eddy located in the Middle Caspian Sea. The trajectory of the last one revealed a complex vertical structure in the Southern Caspian Sea. Comparison dry tropospheric corrections calculated on atmospheric pressure model (ECMWF or NCEP) data and on atmospheric pressure data of MACE drifter experiment show the RMS differences between 2.25 and 3.86 cm (Fig. 105.3) (Table 105.1). Therefore for satellite altimetry data processing dry tropospheric corrections was calculated on atmospheric pressure from nearest weather stations located along the Caspian Sea costal line.
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data. (b) Difference (in cm) between dry tropospheric corrections on ECMWF (black line) and NCEP (gray line) model and 34,267 drifter data from 6 November 2006 to 17 February 2007 (c)
Table 105.1 Comparison dry tropospheric corrections calculated on atmospheric pressure drifter and model data
seasonal variations for all passes of each repeat cycle were eliminated. In last phase, the GCRAS08 MSS was constructed as a SSH function of latitude, longitude, and time with correction on climatic dynamic topography (Popov 2004) (Fig. 105.4). For specified time interval SSH was interpolated on grid by radial basis function method.
Difference between drifter and model, (cm) Drifter Model Min. Max. Mean RMS 34,265 ECMWF 5.0 5.0 0.10 3.86 NCEP 4.3 4.5 0.02 3.35 34,266 ECMWF 4.8 2.6 0.50 2.25 NCEP 4.8 2.6 0.40 2.29 34,267 ECMWF 6.4 4.9 0.40 2.94 NCEP 5.6 4.6 0.80 2.65
GCRAS08 MSS model of the Caspian Sea was calculated according to the following scheme. From the T/P and J1 SSH data correcting the synoptic and
105.4 Gravitational Field of the Caspian Sea Features of a gravitational field of the Caspian Sea can be estimated on EGM96 gravity model (Lemoine et al. 1998) with decomposition to 360 degrees.
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Fig. 105.4 The climatic or mean dynamic topography (cm) from the numerical hydrodynamic model (Popov 2004). Shades of gray correspond to negative values. Dashed lines show borders of the Northern-Middle and the Middle-Southern parts of the Caspian Sea
For the Northern Caspian geoid heights decrease from 4 to 16 m along an arch from the Kizlyar Bay to Urals river estuary (Fig. 105.5a). In northern part of the Middle Caspian geoid height decrease from 4 to 12 m in a direction from Makhachkala to the Fort of Shevchenko, and in southern part its decrease also from 10 to 18 m along an the sea axis in a direction to the Apsheron Swell and reach the minimum on border of the Middle-Southern parts of the Caspian Sea. The geoid height minimum about 21 m corresponds to depth drop near the Apsheron Peninsula. Then it’s raise in a direction to the western and southern coast and are repeating themselves of bottom relief change. East coast of the Southern Caspian in
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the Turkmen Bay has a second local geoid heights minimum (about 19 m). According to EGM96 model the minimum of the Caspian Sea gravity anomalies 148 mGal (Fig. 105.5b) located near Azerbaijan coast along the Apsheron Swell. Other local minimum gravity anomalies 105 mGal obtain in west part of the South Caspian depression. The Caspian Sea gravity anomalies have maximum 53 mGal near canal between the sea and the Kara-Bogaz-Gol Bay. In North Caspian gravity anomalies change weakly from 35 to 15 mGal. Minimum values is obtained in the Kizlyar Bay and Urals river estuary and maximum value located near the Volga river delta in a direction to Kulaly Island. In the Middle Caspian gravity anomalies have value from 80 to 60 mGal along west cost where it increase in a direction to the Apsheron Swell. Along southern-west cost of Mangyshlak Peninsula gravity anomaly increase in a direction from Fort Shevchenko to the Kara-Bogaz-Gol Bay from 20 to 0 mGal. Strong variability of gravity anomalies from 140 to 18 mGal obtain in the South Caspian Depression and it’s have good correlation with depth. Extreme of gravity anomalies gradient (more than 2 mGal/km) located near the Apsheron Swell along the sea axis and near southwestern coast of the South Caspian.
105.5 Results and Discussion We would like to illustrate capability of using GCRAS08 MSS Model for gravitational field investigation for the Caspian Sea area. This fact with the following data. We consider SSH variation along 092 tracks for the time interval 1993–2008 after filtration of the sea level synoptic and seasonal variability. It is apparent (Fig. 105.6), that a SSH maximum for the period from September 1992 to June 2004 at latitude 43.5 N corresponds to the Caspian Sea level maximum observable in the summer of 1995. Between 43 N and the boundary of the Northern and Middle Caspian Sea strong modification of SHH gradient along a 092 track is observed in spatial position isoline at 35.5 m (Fig. 105.6a). This SSH response is explicable due to this area depth changing
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Fig. 105.5 Map of (a) geoid heights (m) and gravity anomalies based (mGal) on EGM96 model (complete 360 ) for the Caspian Sea. Shades of gray correspond to negative values.
Dashed lines show borders of the Northern-Middle and the Middle-Southern parts of the Caspian Sea
from 10 to 50 m. Gravity anomaly increases from 11.4 to 22.6 mGal and also the gravity anomaly gradient along the track has maximum 0.27 mGal/km (Fig. 105.6b). At latitude 39.8 N SSH minimum vanish from 1994 to 1997, and then reappears. It is readily visible in the spatial position isoline at 46 m (Fig. 105.6a). The position of this minimum correlates well to about a gravity anomaly minimum of 98.8 mGal (Fig. 105.6c). The SSH gradient over time behaves differently. We would like to illustrate this with calculations of the annual SSH gradient over time or rate of the Caspian Sea level change along 092 tracks (Fig. 105.6b). Extreme of it are located in the Middle Caspian. Maximum values of rise change from 22 to 25 cm/year (1994–1995) and values of drop change from 21 to 24 cm/year (1996–1997) though gravity anomalies and it gradient have minimum value 60–0 mGal (Fig. 105.6c). Intense variability of level change rate are located to area of extreme value gravity anomaly gradient 0.34 and 0.25 mGal/km in the Middle
Caspian and 63 and 0.56 mGal/km in the Southern Caspian. According to temporal variation of the interannual Caspian Sea level change (Lebedev and Kostianoy 2008; Kouraev et al. 2009) we allocate five time period: strong rise (1993–1995), strong drop (1995–1997), slow rise (2002–2006), slow drop (1997–2002 and 2006–2008). For investigation space variability we construct annual MSS for each year from 1993 to 2008 with using GCRAS08 MSS Model data and interpolation on grid by radial basis function method. Map of rate of the interannual Caspian Sea level change calculated are calculated as difference from conformable annual MSS maps (Fig. 105.7). Results of calculated mean rate of the interannual Caspian Sea level change for each time period are represented in Table 105.2. For time period of strong sea level rise (1993–1995) mean value of level change rate is 9.2 cm/year. Maximum value (more than 12 cm/year) was located in the Middle and Southern Caspian and correlated with depths (Fig. 105.7a). Near the Kara-Bogaz-Gol Bay
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Fig. 105.6 Time variability: (a) of SSH without seasonal and synoptic variability or part of the GCRAS08 MSS Model (m), (b) annual GCRAS08 MSS gradient (cm/year) along the descending 092 track from September 1992 to December 2008 on the basis of satellite altimetry data of the T/P and J1,
(c) Variability of gravity anomaly (firm line) and gravity anomaly gradient (dashed line) calculated by EGM96 model, decomposing to 360 degrees. Shades of gray correspond to negative values. Dashed lines show borders of the NorthernMiddle and the Middle-Southern parts of the Caspian Sea
rate of sea level rise 8.1 cm/year was less then mean value because in this time the bay was destroyed and the natural seawater runoff to the bay resumed. In next time period (1995–1997) of strong sea level drop maximum value (more 15 cm/year) was obtained in the Northern Caspian and minimum (less 3 cm/year) are located near southwestern coast of the Southern Caspian (Fig. 105.7b). Similar regime of slow sea
level drop hold since 1997 till 2002 (Fig. 105.7c) only rate was less than preceding time period. Mean value of sea level change rate was 4.8 cm/year. In the neat time periods of slow drop (1997–2002) and slow rise (2002–2006) rate of sea level (Fig. 105.7d, e) change good correlate with gravity anomaly field (Fig. 105.5b), Extreme variation was obtained in the Middle and Southern Caspian near the Apsheron Swell.
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Fig. 105.7 Map of rate of the interannual Caspian Sea level change (cm/year) for five time period: (a) – strong rise (1993–1995), (b) – strong drop (1995–1997), (c) – slow drop (1997–2002), (d) – slow rise (2002–2006), (e) – slow drop
(2006–2008). Shades of gray correspond to negative values. Dashed lines show borders of the Northern-Middle and the Middle-Southern parts of the Caspian Sea
Table 105.2 The rate of the interannual Caspian Sea level change calculated on GCRAS08 MSS model
References
Time period (year) 1993–1995 1995–1997 1997–2002 2002–2006 2006–2008
Rate of change (cm/year) Average Minimum 9.2 2.4 0.4 12.4 3.3 0.9 4.8 1.4 1.8 5.2 0.7 3.9 6.8 0.8 4.9
Maximum 12.4 17.7 8.1 7.8 9.5
Benada RJ (1997) Merged GDR (TOPEX/POSEIDON). Generation B Users Handbook, Version 2.0, Physical Oceanography Distributed Active Archive Center (PODAAC), Jet Propulsion Laboratory, Pasadena, JPL D-11007, p 131 Birkett CM (1995) The contribution of TOPEX/ POSEIDON to the global monitoring of climatically sensitive lakes. J Geophys Res 100:25179–25204. doi:10.1029/95JC02125 Chelton DB, Ries JC, Haines BJ, Fu LL, Callahan PS (2001) Satellite altimetry. In: Fu LL, Cazanave A (eds) Satellite
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altimetry and earth sciences, a handbook of techniques and applications. Academic, San Diego, pp 1–131 Fu LL, Pihos G (1994) Determining the response of sea level to atmospheric pressure forcing using TOPEX/POSEIDON data. J Geophys Res 99:24633–24642. doi:10.1029/94JC01647 Kosarev AN (2005) Physico-geographical conditions of the Caspian Sea In: The Caspian Sea Environment, Kostianoy, A.G., and A.N. Kosarev (Eds.), Hdb Env Chem Vol. 5, Part P, Springer-Verlag, Berlin, Heidelberg, New York, 59–81, doi: 10.1007/698_5_002. Kosarev AN, and Kostianoy AG (2005) Kara-Bogaz-Gol Bay. In: The Caspian Sea environment. Kostianoy AG and Kosarev AN (eds), Hdb Env Chem vol 5, Part P, Springer, Berlin pp 211–221 doi:10.1007/698_5_011 Kosarev AN, Yablonskaya EA (1994) The Caspian Sea. SPB Academic Publishing, The Hague, p 259 Kouraev AV, Cre´taux J-F, Lebedev SA, Kostianoy AG, Ginzburg AI, Sheremet NA, Mamedov R, Zakharova EA, Roblou L, Lyard F, Calmant S and Berge-Nguyen M (2009) In: Vignudelli SAG, Kostianoy, Cipollini P and Benveniste J (eds), Coastal altimetry. Springer, Berlin (in press) Lebedev SA, and Kostianoy AG (2005) Satellite Altimetry of the Caspian Sea. Sea, Moscow, p 356 (in Russian)
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Lebedev SA, and Kostianoy AG (2008) Integrated using of satellite altimetry in investigation of meteorological, hydrological and hydrodynamic regime of the Caspian Sea. Terr Atmos Ocean Sci, 19, 1–2, 71–82 doi:0.3319/ TAO.2008.19.1-2.71(SA) 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, and Olson TR (1998) The development of the joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) Geopotential Model EGM96. NASA Technical Memorandum. NASA/TP-1998-206861 p 575 Nerem RS, Mitchum GT (2001) Sea level change. In: Fu LL, Cazenave A (eds) Satellite altimetry and earth sciences, a handbook of techniques and applications. Academic, San Diego, pp 329–350 Picot N, Case K, Desai S and Vincent P (2006) AVISO and PODAAC User Handbook. IGDR and GDR Jason Products, SMM-MU-M5-OP-13184-CN (AVISO), JPL D-21352 (PODAAC), 3rd edn, p 112 Popov SK (2004) Simulation of climatic thermohaline circulation of the Caspian Sea. Meteorol Hydrol 5:76–84 (in Russian)
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Session 5 Geodesy in Latin America Convenors: D. Blitzkow, C. Tocho
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Combination of the Weekly Solutions Delivered by the SIRGAS Processing Centres for the SIRGAS-CON Reference Frame
106
€ller, and M. Seitz L. Sa´nchez, W. Seemu
Abstract
The SIRGAS reference frame is given by more than 200 continuously operating stations (SIRGAS-CON network), which are classified in four sub-networks: a continental one (SIRGAS-CON-C) with about 100 stations homogeneously distributed over Latin America and the Caribbean, and three densification subnetworks (SIRGAS-CON-D) covering the northern part, the middle part, and the southern part of the SIRGAS region. Each sub-network is processed by one of the SIRGAS Processing Centres: DGFI (Germany) is responsible for the SIRGASCON-C network, IGAC (Colombia) for the northern densification sub-network, IBGE (Brazil) for the middle one, and CIMA (Argentina) for the southern one. These Processing Centres deliver loosely constrained weekly solutions, which are integrated in a unified solution by the SIRGAS Combination Centres operating at DGFI and IBGE. The DGFI (i.e. IGS RNAAC SIR) weekly combinations are delivered to the IGS Data Centres for the global polyhedron, and are made available for users as official SIRGAS weekly reference frame solution. The IBGE weekly combinations provide control and redundancy. This paper describes the combination strategy applied by DGFI, emphasizing the evaluation of the individual solutions and the quality control of the final weekly combinations. The reliability of the resulting coordinates is estimated by comparing them with those produced by IBGE and the weekly combinations of the IGS global network.
106.1 Introduction The densification of the ITRF in Latin America and the Caribbean is the SIRGAS Continuously Operating Network (SIRGAS-CON). This network comprises two hierarchy levels (Brunini and Sa´nchez 2008):
L. Sa´nchez (*) W. Seem€ uller M. Seitz Deutsches Geod€atisches Forschungsinstitut (DGFI), AlfonsGoppel-Str. 11, 80539 Munich, Germany e-mail:
[email protected]
core stations (SIRGAS-CON-C) providing the primary link to the global ITRF; and densification stations (SIRGAS-CON-D) containing all the fundamental GNSS sites of the national reference frames. The densification stations are further classified in three subnetworks covering the northern part, the middle part, and southern part of the SIRGAS region (Fig. 106.1). The core network ensures the long-term stability of the continental reference frame. The densification sub-networks improve the geographical density of the reference stations facilitating the accessibility to the reference frame in national and local levels.
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network. Afterwards, with the introduction of the core and densification sub-networks, as well as the installation of SIRGAS Processing Centres in Latin American institutions, DGFI is now responsible for (1) processing the SIRGAS-CON-C core network, (2) combining this core network with the densification sub-networks, and (3) making available the final SIRGAS products (i.e. loosely constrained weekly solutions, weekly station positions referred to the ITRF, and multi-annual solutions providing station positions at a reference epoch and the corresponding velocities). This paper summarizes the activities carried out by DGFI as SIRGAS Combination Centre.
106.2 DGFI Combination Strategy
Fig. 106.1 Core and densification sub-networks of the SIRGAS-CON reference frame
This operational infrastructure is possible thanks to the active participation of many Latin American and Caribbean institutions, who not only make available the measurements of their stations, but also are hosting SIRGAS Analysis Centres in charge of processing the observational data on a routine basis. As responsible for the IGS Regional Network Associate Analysis Centre for SIRGAS (IGS RNAAC SIR), DGFI delivers loosely constrained weekly solutions for the SIRGAS-CON network to the IGS. These solutions are combined together with those generated by the other IGS Analysis Centres to form the IGS polyhedron (Seem€ uller and Drewes 2008). The processing of the SIRGAS-CON network in the frame of the IGS RNAAC SIR also includes the computation of weekly position solutions aligned to the current ITRF and accumulative position and velocity solutions for estimating the kinematics of the network (e.g. Seem€uller 2009; Seem€ uller et al. 2009). Until GPS week 1,495, DGFI processed the entire SIRGAS-CON
The SIRGAS Processing Centres (Table 106.1) deliver loosely constrained weekly solutions for the assigned SIRGAS-CON sub-networks. In these solutions, satellite orbits, satellite clock offsets, and Earth orientation parameters are fixed to the final weekly IGS values, and positions for all sites are constrained to 1 m. These individual contributions are integrated in a unified solution by the SIRGAS Combination Centres: DGFI and IBGE. The combination strategy applied by DGFI is: 1. Individual solutions are corrected for possible format problems, station inconsistencies, utilization of erroneous equipment, etc. 2. Constraints included in the delivered normal equations are removed 3. Sub-networks are separately aligned to the IGS05 reference frame by applying the no net rotation (NNR) and no net translation (NNT) conditions. The reference values are the positions of the IGS05 stations included in the corresponding IGS weekly combination, i.e. files igsyyPwwww.snx (yy ¼ year, wwww ¼ GPS week) 4. Positions obtained in (3) for each individual solution are compared to the IGS weekly values and to each other to identify possible outliers 5. Stations with large residuals (more than 10 mm in the north or east components, and more than 20 mm in the height) are reduced from the corresponding normal equations. Steps (3), (4), and (5) are iterative 6. Variances obtained in the final computation of (3) are analyzed to estimate variance factors for
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Combination of the Weekly Solutions Delivered by the SIRGAS Processing Centres
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Table 106.1 SIRGAS Processing Centres and their solutions evaluated in this study Id
Processing centre
First week
SIRGAS official processing centres CIM Centro de Procesamiento Ingenierı´a-Mendoza-Argentina at the Universidad Nacional del Cuyo (CIMA, Argentina) DGF Deutsches Geod€atisches Forschungsinstitut (DGFI, Germany) IBG Instituto Brasileiro de Geografia e Esta´tistica (IBGE, Brazil) IGA Instituto Geogra´fico Augustı´n Codazzi (IGAC, Colombia) SIRGAS experimental processing centres ECU Instituto Geogra´fico Militar of Ecuador (IGM, Ecuador) LUZ Laboratorio de Geodesia Fı´sica y Satelital at the Universidad del Zulia (LGFS-LUZ, Venezuela) URY Servicio Geogra´fico Militar of Uruguay (SGM, Uruguay)
relative weighting of the individual solutions (see Sect. 106.4.1) 7. Once inconsistencies and outliers are reduced from the individual free normal equations, these equations are accumulated for a loosely constrained weekly combination, in which all station positions are constrained to 1 m. This combination is submitted to IGS for the global polyhedron and is stored to be included in the next multi-year solution of the network 8. Finally, a weekly solution aligned to the IGS05 frame is computed. The geodetic datum is defined by constraining the coordinates of the IGS05 reference stations (Fig. 106.1) to their positions computed within the IGS weekly combinations (igsyyPwwww. snx). To minimize network distortions, the reference coordinates are introduced with a weight inversely proportional to 1E-04m. This solution provides the final weekly positions for the SIRGAS-CON stations 9. The accumulation and solution of the normal equations are carried out with the Bernese GPS Software V.5.0 (Dach et al. 2007) Resulting products are (available at http://www.sirgas. org): SIRwwww7.SNX: SINEX file for the loosely constrained weekly combination. SIRwwww7.SUM: Report of weekly combination. siryyPwwww.snx: SINEX file for the constrained weekly combination. siryyPwwww.crd: Final SIRGAS-CON positions for week wwww.
Latest week
Sub-network
No. stations
1,495 1,538
Southern sub-network
50
1,495 1,538 1,495 1,538 1,495 1,538
Core network Middle sub-network Northern sub-network
107 94 94
1,513 1,538
Selected stations northern and middle sub-networks Northern sub-network
31
Selected stations southern and middle sub-networks
44
1,520 1,538 1,526 1,538
94
Before the weekly combinations computed by DGFI for the SIRGAS-CON network are published or made available for users, a quality control is carried out to guarantee consistency and reliability of the SIRGAS products. This quality control is described in Sect. 106.4 of this paper.
106.3 Evaluation of the SIRGAS Experimental Processing Centres SIRGAS promotes the installation of more Processing Centres hosted by Latin American institutions. Motivations for this are (Brunini et al. 2011): 1. SIRGAS member countries are qualifying their national reference frames by installing an increasing number of continuously operating GNSS stations and each country shall be able to process the data of its own stations 2. Since there are not enough Local Processing Centres, the required redundancy in the analysis of the SIRGAS-CON network is not fulfilled: not all SIRGAS-CON stations are included in the same number of individual solutions and they are unequally weighted in the weekly combinations. As an optimum, each SIRGAS-CON station shall be processed by the same number of Processing Centres (at least three) In this frame, institutions interested to install a SIRGAS Processing Centre shall pass a test period of one year. In this period, they have to align their processing strategies with the SIRGAS guidelines and
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meet the delivering deadlines. DGFI as a SIRGAS Combination Centre is also responsible for evaluating the weekly solutions generated by the so-called SIRGAS Experimental Processing Centres. This evaluation includes not only the analysis of accuracy and compatibility of the individual solutions with the official SIRGAS products, but also the revision of administrative issues such as meeting deadlines, observance of the SIRGAS guidelines, accordance with the log files information, etc. The evaluation of the solutions produced by the SIRGAS Experimental Processing Centres is carried out applying the same procedure used for the SIRGAS Official Processing Centres. Details of evaluation and results are presented in the following.
106.4 Quality Control of the SIRGAS-CON Network Weekly Combinations The generation of the weekly SIRGAS-CON products (i.e. loosely constrained combinations and station positions aligned to IGS05) includes a quality control at two levels: Firstly, the individual solutions delivered by the Processing Centres (official and experimental) are analysed to establish their quality and consistency. Once the individual solutions are reviewed and free of inconsistencies, their combination is carried out by applying the procedure summarized in Sect. 106.2. Then, the second quality control concentrates on the results of this combination. Here, the main objective is to ascertain the accuracy and reliability of the weekly solutions for the entire SIRGAS-CON network. It should be mentioned that the DGFI combinations made available for users include the solutions provided by the SIRGAS Official Processing Centres only. Combinations including solutions delivered by the Experimental Processing Centres are for internal control. In order to present the actual status of all Processing Centres, the procedures, analysis, and conclusions contained in this paper are based on all the weekly solutions summarized in Table 106.1.
106.4.1 Evaluation of Individual Solutions Identification of outliers: To avoid deformations in the combined network, those stations with very large outliers are reduced from the weekly normal equations.
L. Sa´nchez et al.
In this step, the individual loosely constrained weekly solutions are separately aligned to the IGS05 reference frame. Then, time series of weekly solutions are generated for each station included in each individual solution. In this way, if one station is processed by three Processing Centres, there will be three different time series available for the same station. By comparing the time series amongst one to another, it is easier to identify outliers and their possible causes: if outliers, jumps, or interruptions are identifiable in the different series, the problems may be specifically associated to the station (tracking deficiencies, equipment changes, failure of the data submission, earthquakes, etc.). If outliers, jumps, or interruptions are not present in all time series, the deficiencies may be associated to administrative issues of the Processing Centres (neglecting of stations, incomplete download of RINEX files, disagreement with the log files, etc.). After comparing the individual solutions, differences exceeding five times the mean RMS values derived from the position time series (i.e. N ¼ (5 2) mm, E ¼ (5 2) mm, H ¼ (5 4) mm) are assumed as outliers, and the corresponding station(s) is reduced from the respective individual solution(s) before combination. Quality control of the individual solutions: The relative weighting of individual solutions by means of variance factors is necessary to compensate possible differences in the stochastic models of the Processing Centres. To validate these models, we compare mean standard deviations of coordinates (obtained from solving the individual normal equations) with mean RMS values derived from the time series of station positions. The latter ones reflect the precision of the weekly position solutions. If the relation between the individual standard deviations is the same as the relation between the RMS values derived from the time series, the stochastic models are compatible and it is not necessary to apply relative weighting factors. To ensure that the RMS values are not dominated by individual stations that are not included in all solutions, they are computed in two different ways (Sa´nchez et al. 2008): (a) Evaluation of the time series of station positions per Processing Centre to ascertain the consistency of the individual solutions from week to week (i.e. weekly repeatability) (b) Comparison of the individual solutions with the weekly IGS positions (igsyyPwwww.snx) to validate their compatibility with the IGS global network
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Table 106.2 Variance factors (with respect to DGF values, i.e. SIRGAS-CON-C core network) for the individual normal equations generated by each SIRGAS Processing Centre Processing Approach centre/component (a) RMS residuals for weekly repeatability (mm) CIM N 1.9 E 1.6 Up 3.8 Total 4.7 DGF N 2.2 E 2.2 Up 5.0 Total 6.1 IBG N 2.6 E 2.6 Up 5.3 Total 6.8 IGA N 1.9 E 2.1 Up 4.2 Total 5.2 ECU N 1.5 E 1.4 Up 4.1 Total 4.7 LUZ N 1.4 E 1.5 Up 3.5 Total 4.6 URY N 1.6 E 1.2 Up 3.6 Total 4.5
(b) RMS residuals wrt IGS weekly solutions (mm) 2.1 2.4 4.6 5.7 1.9 2.2 4.2 5.2 2.0 2.1 4.7 5.6 1.6 1.9 4.5 5.2 2.0 2.0 4.8 5.7 1.6 2.0 4.7 5.4 1.4 1.3 2.9 5.1
Complementary, the mean standard deviations (item c in Table 106.2) are determined after solving the individual normal equation with respect to the IGS05 reference frame by means of minimum datum conditions (NNR and NNT). They represent the formal errors of the individual solutions. Table 106.2 summarizes the mean values for the described approaches over the total analysed period (Table 106.1 shows the number of GPS weeks included per Processing Centre). The variance factors are calculated with respect to the DGFI values, since they correspond to the major SIRGAS-CON-C core network. In general, the variance factors derived from the different RMS values (criteria a, b) are very similar and can be
(c) Mean standard deviation (mm)
Variance factors (a) (b) Mean of a, b
(c)
1.7
1.3
0.9
1.1
0.9
1.6
1.0
1.0
1.0
1.0
1.7
0.9
0.9
0.9
0.9
1.5
1.2
1.0
1.1
1.0
1.4
1.3
1.1
1.0
0.9
1.6
1.3
1.0
1.1
1.0
1.3
1.4
1.2
1.1
1.0
averaged. These mean values are then compared with the variance factors derived from the standard deviations (item c). They agree quite well. Consequently, and keeping in mind that all the Processing Centres are applying the same analysis strategy (double differences), the same software (Bernese), the same satellite orbits, satellite clock offsets, and Earth orientation parameters (final IGS products), as well as the same observations for the common stations (RINEX files with sampling rate of 30 s), we conclude that the parameters estimated by each of the contributing solutions are at the same precision level (i.e. there are no differences in the stochastic models) and a relative weighting of the Processing Centres is not necessary.
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106.4.2 Evaluation of Combined Solutions The weekly combined solutions for the SIRGAS-CON network are aligned to the IGS05 reference frame by constraining the positions of the IGS05 stations (Fig. 106.1) to the coordinates obtained within the IGS weekly combinations (see Sect. 106.2). The quality evaluation of these combined solutions is based on the following criteria: (a) Mean standard deviation for station positions after aligning the network to the IGS05 reference frame indicates the formal error of the final combination (b) The weekly repeatability of station positions after combining the individual solutions provides information about the internal consistency of the combined network (c) Time series analysis for station positions allows to determine the compatibility of the combined solutions from week to week (d) Comparison with the IGS weekly coordinates indicates the consistency with the IGS network (e) Comparison with the IBGE weekly combination (ibgyyPwwww.snx) fulfils the required redundancy to generate the final SIRGAS products. This comparison is carried out directly with the station positions (no 7-parameter similarity transformation is applied here) Figure 106.2 presents mean values of the different applied criteria for the period covering the GPS weeks 1,495–1,538. The mean standard deviation of the combined solutions agrees quite well with those computed for the individual contributions (Table 106.2), i.e. the quality of the individual solutions is maintained and their combination does not deform or damage the internal consistency of the entire SIRGAS-CON network. The position repeatability in the weekly
combinations indicates that the internal consistency of the SIRGAS-CON network is about 0.8 mm in the horizontal components and about 2.5 mm in the vertical one. The RMS values derived from the station position time series and with respect to the IGS weekly coordinates indicate that the accuracy of the weekly positions for the SIRGAS-CON stations is about 1.5 mm in the North and the East, and 3.8 mm in the height. The differences with respect to the IBGE combinations are about 1 mm for the three components (N, E, Up). Although, these differences are within the accuracy level of the weekly solutions, they are a bit larger, considering that DGFI and IBGE apply the same input normal equations for combination. This can be a consequence of the different combination strategies, especially the methodology used for the datum realization. A description about the IBGE combination procedure is given by Costa et al. (2009).
106.4.3 Correction of the Stochastic Model of the Combined Solutions The individual solutions contributing to the final combination of the SIRGAS-CON network include common stations and they are therefore highly correlated. In spite of this, they are initially treated as independent within the combination. The omission of that correlation conduces to an overestimation of the standard deviations by a factor of about the square root of the number of individual solutions including each station. To compensate this overestimation, the standard deviations have to be multiplied by this factor and the variance-covariance matrix by the square of the
Fig. 106.2 Evaluation of the weekly positions computed for the SIRGAS-CON stations
106
Combination of the Weekly Solutions Delivered by the SIRGAS Processing Centres
factor. If each station is included in exactly the same number of individual solutions, this procedure can easily be carried out. However, due to different causes, the station distribution between the SIRGAS Processing Centres is not homogeneous, i.e. not all stations are included in the same number of individual solutions. It implies that the stochastic model of the combined solution cannot be corrected by one (unique) factor. It is necessary to determine separately correction factors for the stations, depending on the number of individual solutions including them. At present, we are trying to implement a method to compute and apply these factors directly in the combination software. In the mean time, it is not possible and therefore, this study does not take into account corrections for the stochastic model of the combined solutions. A good alternative to avoid this procedure is to guarantee that each regional station is included in exactly the same number of individual solutions. For that, a redistribution of the stations between the different SIRGAS Processing Centres would be necessary. Conclusions
DGFI as a SIRGAS Combination Centre reviews, evaluates, and combines on a weekly basis the individual solutions delivered by the SIRGAS Analysis Centres: four Official Processing Centres (CIM, DGF, IBG, IGA) and three Experimental Processing Centres (ECU, LUZ, URY). The weekly solutions for the SIRGAS reference frame (i.e. loosely constrained weekly solutions and weekly station positions aligned to the IGS05) released by DGFI include the contributions of the Official Processing Centres only. Analyses containing solutions provided by the Experimental Processing Centres are for internal control. After analysing the individual solutions delivered for the period covered between the GPS weeks 1,495 and 1,538, the results permit to conclude that all SIRGAS Processing Centres (official and experimental) satisfy the administrative and quality requirements
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defined in the SIRGAS guidelines. Their weekly solutions are at the same precision level with respect to each other and with respect to the weekly combinations. In general, the precision (internal consistency) of results is about 0.8 mm for the horizontal position and 2.5 mm for the vertical one, while the realization accuracy with respect to the IGS05 frame (reliability) is about 1.5 mm for the horizontal components and 3.8 mm for the height.
References Brunini C, Sa´nchez L (eds) (2008) Reporte SIRGAS 2007–2008. Boletı´n Informativo No. 13. pp. 40 Available at http://www.sirgas.org Brunini C, Sa´nchez H, Drewes S, Costa V, Mackern W, Martı´nez W, Seem€uller A, da Silva (2011) Improved analysis strategy and accessibility of the SIRGAS Reference Frame. In: Geodesy for Planet Earth, Buenos Aires Argentina. August 31 to September 4, 2009. IAG Symposia, this volume Costa SMA, da Silva AL, Vaz JA (2009) Report of IBGE Combination Centre. Period of SIRGAS-CON solutions: from week 1495 to 1531. Presented at the SIRGAS 2009 General Meeting. Buenos Aires, Argentina. September. Available at http://www.sirgas.org Dach R, Hugentobler U, Fridez P, Meindl M, (eds) (2007). Bernese GPS Software Version 5.0 – Documentation. Astronomical Institute, University of Berne, January, 640 Sa´nchez L, Seem€uller W, Kr€ugel M (2008) Comparison and combination of the weekly solutions delivered by the SIRGAS Experimental Processing Centres. DGFI Report No. 80. DGFI, Munich. Available at http://www.sirgas.org Seem€uller W, Drewes H (2008) Annual Report 2003–2004 of IGS RNAAC SIR. In: IGS 2001–02 Technical Reports, IGS Central Bureau, (eds), Pasadena, CA: Jet Propulsion Laboratory. Available at http://igscb.jpl.nasa.gov/igscb/ resource/pubs/2003-2004_IGS_Annual_Report.pdf Seem€uller W (2009) The Position and Velocity Solution DGF06P01 for SIRGAS In: Drewes H (ed) Geodetic Reference Frames, IAG Symposia; Springer, Heidelberg vol 134:167–172 Seem€uller W, Seitz M, Sa´nchez L, Drewes H (2009) The position and velocity solution SIR09P01 of the IGS Regional Network Associate Analysis Centre for SIRGAS (IGS RNAAC SIR). DGFI Report No. 85. DGFI, Munich. Available at http://www.sirgas.org
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Report on the SIRGAS-CON Combined Solution, by IBGE Analysis Center
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S.M.A. Costa, A.L. Silva, and J.A. Vaz
Abstract
Since the last SIRGAS meeting in May, 2008, IBGE has been playing a role as Processing and Combination Center of SIRGAS-CON network, in order to provide support to the SIRGAS reference frame. The SIRGAS-CON densification network is composed now by 150 continuously operating GNSS permanent stations and this number is increasing continuously. In the same SIRGAS meeting some aspects related to the network coordination were discussed in order to correct several factors that were affecting the quality of solutions, e.g. equipment changes, which may cause a discontinuity in the coordinate time series. One of the tasks of the IBGE Analysis Center is the combination of weekly solutions computed by each SIRGAS-CON Processing Center and to generate a cumulative solution aligned to the IGS05 reference frame. In this paper four combination strategies were explored using the minimum constraints approach, preserving the original characteristics of the weekly solutions and providing the alignment to the IGS05 reference frame. The procedures adopted for the combination and statistical analysis of results are presented in this paper.
107.1 Introduction The SIRGAS Continuously Observing Network (SIRGAS-CON) contributes with weekly solutions of most Continuously Operating Reference Stations (CORS) in South and Central America, the Caribbean, as well as a few North American stations (SIRGAS 2009). One of the objectives of SIRGAS Working Group I is to produce coordinate solutions in IGS
S.M.A. Costa (*) A.L. Silva J.A. Vaz Brazilian Institute of Geography and Statistics, Av. Brasil 15671, Rio de Janeiro 21241-051, Brazil e-mail:
[email protected]
SINEX format, specifically, weekly combinations of sub-network solutions. In this paper are presented analyses of weekly solutions provided by three SIRGAS Local Processing Centers identified in this work as: CIM: Instituto de Geodesia y Geodina´mica de la Universidade Nacional de Cuyo IGG-CIMA, Argentina. This center is in charge of processing SIRGAS-CON stations from the southern SIRGAS-CON sub-network. IBG: Instituto Brasileiro de Geografia e Estatı´stica (IBGE), Rio de Janeiro, Brasil. This center is in charge of processing SIRGAS-CON stations from the central SIRGAS-CON sub-network. IGA: Instituto Geogra´fico Agustı´n Codazzi (IGAC), Bogota´, Colombia. This center is in charge of
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processing SIRGAS-CON stations from the northern SIRGAS-CON sub-network. Deutsches Geod€atisches Forschungsinstitut-DGFI, identified in this report as SIR, processes SIRGASCON data from a core sub-network which has stations in stable locations to ensure long-term stability of the reference frame. At the same time, IBGE started the combination task of the weekly solutions from the Local Processing Centers and DGFI. The combined solutions must be delivered 4 weeks after the date of observation. Results are available at IBGE FTP server in two types of weekly combined solutions: loosely constrained and constrained solutions (IBGE 2009). The combined solutions presented in this work the coordinates of 182 stations were estimated using as constraint the IGS05 Reference Frame coordinates at epoch 2009,0 (GPS week 1,513). The solutions provided by CIM, SIR, IBG and IGA span the period of week 1,495–1,531 (37 weeks, from October 2008 to May 2009), and are available as loosely constrained weekly solution in SINEX format. Four combination strategies were evaluated using the minimum constraints and constrained approach, preserving the original characteristics of the weekly solutions and providing the alignment to the IGS05 reference frame. To generate the final weekly solutions, fourteen IGS05 stations in the ITRF2005 were used for datum definition: BRAZ, CHPI, CONZ, CRO1, GOLD, ISPA, LPGS, MANA, MDO1, OHI2, PIE1, SANT, SCUB, UNSA, and VESL. The Bernese GPS Software v.5.0 (Dach et al. 2007) was used to combine SIRGASCON solutions. Figure 107.1 shows the Contributions to the SIRGAS-CON densification network for a period involving GPS weeks 1,495–1,531. Table 107.1 gives the number of common stations between all sub-networks from SIRGAS-CON combination. Table 107.2 summarizes the number of stations with redundant solutions. As can be seen in Table 107.2 the number of stations present in more than one solution is still very small. This redundancy is an important consideration in the combination for detection of outliers and to ensure reliable alignment. To improve the SIRGAS solutions in this aspect, new Local Processing Centers at American institutions were trained and will contribute to the SIRGAS realization as Official Local Centers after a period of evaluation.
S.M.A. Costa et al.
107.2 IBGE Combination Strategy As mentioned before, the present solutions from three sub-networks (north, south and middle) contribute to the SIRGAS-CON network, but only DGF processes the core stations that belong to the three sub-networks. The SIRGAS-CON network comprises South, Central, part of North America, and the Caribbean Region. The combination strategy of weekly solutions carried out at IBGE is: (1) Constraints are removed from the weekly solutions of each Processing Center, making the solution become like a free network solution; (2) The free network solution of each processing center is aligned to a set of stations that belong to IGS05 (2000.0) Reference network applying “no net rotation” and “no net translation” conditions. The IGS05 stations are: BRAZ, CHPI, CONZ, GOLD, ISPA, LPGS, FLOWS, MDO1, OHI2, PIE1, SANT, SCUB, UNSA and VESL. (3) The coordinates from step (2) of each processing center are compared with IGS05 coordinates propagated to week epoch and between themselves to identify possible high residuals. The stations with residuals exceeding 10 mm in horizontal components and 20 mm in the vertical component will be analyzed and possibly removed from the solution. In the case of station exclusion the steps and (2) will be repeated for the refinement of final solution and consequently the variance factor of the estimate. (4) The covariance matrix of each solution is scaled by the variance factor or scale factor. (5) The normal equations of each solution are combined to produce the loosely constrained weekly solution (IBGwwwwS.SNX) applying a weight of 1 m to all stations. (6) The normal equations of each solution are combined to produce the constrained solution (IBGyyPwwww.SNX) applying a weight of 1E-04 meters for IGS05 stations mentioned in step (2).
The flow diagram in Fig. 107.2 shows the sequence of combination strategy carried out by IBGE.
107.3 Evaluation of New Combination Strategies Four combination strategies were carried out for a period of 37 weeks (1,495–1,531 GPS week) in order to choose the best solution of SIRGAS-CON network. Table 107.3 presents the description of each strategy and datum definition.
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Fig. 107.1 Contributions to the SIRGAS-CON densification network for a period 1,495–1,531
Table 107.1 Number of stations processed by local processing centers Processing center CIMA DGFI IBGE IGAC
CIMA 52
DGFI 35 110
IBGE 44 61 115
IGAC 15 66 22 85
A set of selected IGS05 stations, called fiducial stations in all strategies are: BRAZ, CHPI, CONZ, GOLD, ISPA, LPGS, MANA, MDO1, OHI2, PIE1, SANT, SCUB, UNSA e VESL. The reference epoch is (the middle of the time interval):2009-01-07, 00:00:00 (2009.02), GPS week 1,513.
Table 107.2 Redundancy of solutions, from official processing centers
n of solutions n of stations 1 62 2 82 3 24 4 15 Total 182
107.4 Results and Comparison of Different Strategies Tables 107.4 and 107.5 present the transformation parameters between IGS05 and SIR solution, epoch 2009,02 (GPS week 1,513) and the four combination strategies, in order to check the external fit of each
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(4) Helmert TransfOuttler detection CIM×IGS CIM×DGF CIM×IBG CIM×IGA
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FINAL COMBINATION Daturn Definiton 14IGS05 stations NNR and NNT conditions
IGA×IGS IGA×CIM IGA×DGF IGA×IBG
Fig. 107.2 Diagram of combination strategy
Table 107.3 The proposed combination strategies Strategy Description (1) Minimum constraint conditions: the solution is aligned to a set of IGS stations, from IGS05 (IGS05_R.CRD) realization, applying the “no net rotation” and “no net translation” conditions. (2) Minimum constraint condition: Solution is aligned to a set of IGS stations, from IGS05 week (IGSyyPwwww.CRD) realization, applying the “no net rotation” and “no net translation” conditions. (3) Constraint solution: constrain coordinates of a selected set of IGS05 stations to their a priori coordinates for geodetic datum definition. The strength of the constraints is s ¼ 1E-06m in all components (4) Constraint solution: constrain coordinates of a selected set of IGS05 stations to their a priori coordinates for geodetic datum definition. The strength of the constraints is s ¼ 1E-06m in all components
Reference of coordinates IGS05_R.crd coordinates. propagated to week 1,513, using IGS05_R.vel. IGS weekly solution 1,513 (IGS09P1513.crd).
IGS05_R.crd coordinates. propagated to week 1,513, using IGS05_R.vel IGS week solution 1,513 (IGS09P1513.crd)
Table 107.4 Transformation parameters and rms of transformation between IGS05 weekly solution (week 1,513) and each combination strategy Strategy Tx/s (mm) Ty/s (mm) Tz/s (mm) Rot_X/s (00 ) Rot_Y/s (00 ) Rot_Z/s (00 ) Scl/s mm/km
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3.5/1.1 4.2/0.8 7.4/1.0 0.00005/ 0.00003 0.00005/ 0.00003 0.00016/ 0.00004 0.0002/ 0.0001
0.8/1.2 1.6/0.8 5.4/1.0 0.00003/ 0.00003 0.00005/ 0.00003 0.00013/ 0.00004 0.0002/ 0.0001
6.1/1.9 3.7/1.3 5.8/1.5 0.00021/ 0.00005 0.00000/ 0.00004 0.00023/ 0.00007 0.0002/ 0.0002
1.0/1.1 1.4/0.8 2.9/0.9 0.00009/ 0.00003 0.00002/ 0.00003 0.00005/ 0.00004 0.0000/ 0.0001
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Table 107.5 Transformation parameters and rms of transformation between between SIR weekly solution (week 1,513) and each combination strategy Strategy Tx/s (mm) Ty/s (mm) Tz/s (mm) Rot_X/s (00 ) Rot_Y/s (00 ) Rot_Z/s (00 ) Scl/s mm/km
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0.4/0.7 3.7/0.5 5.2/0.6 0.00000/0.00002 0.00003/0.00002 0.00007/0.00002 0.0003/0.0001
3.4/0.7 1.3/0.5 3.2/0.6 0.00002/0.00002 0.00004/0.00002 0.00005/0.00002 0.0003/0.0001
3.5/0.8 2.4/0.5 3.0/0.7 0.00013/0.00002 0.00006/0.00002 0.00015/0.00003 0.0001/0.0001
1.2/0.7 0.7/0.5 0.7/0.6 0.00004/0.00002 0.00006/0.00002 0.00002/0.00002 0.0003/0.0001
Table 107.6 RMS of coordinate residuals between each combination strategy and weekly IGS solution and SIR solution on week 1,513 Strategy (1) (2) (3) (4)
IGS N (mm) 1.4 1.4 3.1 1.1
E (mm) 2.0 2.0 2.2 1.5
U (mm) 3.6 3.6 6.2 3.9
solution to IGS05. As can be seen, rotation and scale are meaningless in the results: translations values are bigger in strategies (1) and (3). Table 107.6 shows that the four strategies proposed have a good consistency with IGS and SIR solution, but bigger root mean square (RMS) were found in strategy (3). Conclusions
It is still necessary to add more redundant solutions for as many stations as possible, especially those in the SIRGAS-CON network. Many SIRGAS-CON stations are still in only in one regional solution and therefore have no independent quality control check. It is important to mention that results are more consistent than the ones computed last year, this is mainly due to the new procedure established for updating the station information file, and the fact, that all centers are now using the correct information.
SIR N (mm) 1.4 1.4 1.9 1.5
E (mm) 1.6 1.6 1.6 1.5
U (mm) 4.0 4.0 4.6 3.9
There are several strategies to integrate a regional solution in the global ITRF frame, having different impacts on the results influenced by the weighting, network configuration and quality of observations. Analyzing the four adjustment strategies shown in the results section, the free network solution with minimum constraints approach, allows the integration of the SIRGAS2000 network in the IGS05, keeping its internal and original consistency.
References Dach R, Hugentobler U, Fridez P, Meindl M (eds) (2007) Bernese GPS Software Version 5.0 Documentation. Astronomical Institute, University of Berne, January, p 640 IBGE (2009). SIRGAS-CON Results, ftp://geoftp.ibge.gov.br/ SIRGAS SIRGAS (2009) Sistema de Referencia Geocentrico para las Americas, http://www.sirgas.org
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Processing Evaluation of SIRGAS-CON Network by IBGE Analysis Center
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S.M.A. Costa, A.L. Silva, and J.A. Vaz
Abstract
IBGE – The Brazilian Institute of Geography and Statistics became an Analysis Center for the SIRGAS-CON (Geocentric Reference System for the Americas), permanent GNSS network after getting experience as a Pilot Processing Center for 2 years. In the beginning, about 30 stations were processed. With the growing number of GNSS stations on the American continent, there are now about 100 stations being processed every week. Since week 1,495, IBGE officially shares the processing task of SIRGAS-CON network with three more Analysis Centers: Instituto Geogra´fico Agustı´n Codazzi – IGAC (Coloˆmbia), Instituto de Geode´sia y Geodinaˆmica de la Universidad Nacional del Cuyo, IGG-CIMA (Argentina) and Deutsches Geod€atisches Forschungsinstitut-DGFI (Germany). Each center is responsible for the processing of a group of stations. The purpose of this effort is to contribute to the IGS Regional Network Associate Analysis Center for SIRGAS (IGS RNAAC SIR) solution, with a densified network. We present the current status and efforts of IBGE as an official Processing Center for SIRGAS. The perspective is to increase the number of stations in 2009, with the inclusion of new stations from the Brazilian GNSS Permanent Network, RBMC – Rede Brasileira de Monitoramento Contı´nuo dos Sistemas GNSS. The processing strategy applied using the Bernese GNSS Software is presented, as well as relevant information for the development of activities. Results are evaluated and compared to the solutions provided by other institutions (DGFI, IGG-CIMA and IGS) and discrepancies are analyzed. Some important issues related to the maintenance of the national permanent GPS networks are shown in the coordinate time series.
108.1 Introduction
S.M.A. Costa (*) A.L. Silva J.A. Vaz Department of Geodesy, Brazilian Institute of Geography and Statistics – IBGE, Av. Brasil 15671, CEP, 21241-051 Rio de Janeiro, Brazil e-mail:
[email protected]
In 2005, the Geocentric Reference System for the Americas – SIRGAS in its realization 2000, became officially the new geodetic reference system for Brazil, as published in the IBGE Presidential Resolution, R.PR-1/2005 (IBGE 2005). Being a high precision reference system with Earth’s center of mass as origin,
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_108, # Springer-Verlag Berlin Heidelberg 2012
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the coordinate variation over time is an important issue for the maintenance of that system. Factors such as lithospheric plate movement and the subsidence of soil, influence the position of each of the stations that materialize this system and must be monitored. The permanent GNSS networks (Global Navigation Satellite System) are defined by a set of geodetic stations in areas of stable surface, materialized by a rigid structure, equipped with dual frequency GNSS receivers collecting data continuously. With the implementation of this new concept of geodetic stations, it becomes possible to systematically assess the changes in the realization of a geodetic reference system over time, thus enabling the establishment of new parameters for this system as well as improving the velocity models (DeMelts et al. 1994). Currently, the SIRGAS is materialized by a network called SIRGAS-CON, with about 200 GNSS stations in continuous operation, distributed in South America, Central America and the Caribbean, as shown in Fig. 108.1. More information can be found in SIRGAS 2009. The SIRGAS Processing Centers have been established with the purpose to systematically determine the coordinates of the SIRGAS-CON stations, as well as other information about the network, following preestablished criteria in order to support the maintenance of the system. This information is used for the evaluation and a future implementation of SIRGAS. Currently, there are three official SIRGAS processing centers: IBGE through the Coordination of Geodesy, Geographic Institute Agustı´n Codazzi – IGAC (Colombia), and the Institute of Geodesy and Geodynamics of the National University of Cuyo, IGG-CIMA (Argentina). Additionally, the Deutsches Geod€ atisches Forschungsinstitut – DGFI, responsible for the processing of the IGS regional network in South America, which is composed by the “core” stations of SIRGAS-CON network.
108.2 The Continuous GNSS Network SIRGAS-CON The number of continuous SIRGAS-CON stations has increased significantly each year. In July 2009 the number was already more than 200 stations. This increase meant that there was a need to divide the network into sub-networks, in a way that the processing centers would not be overburdened by the
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number of stations to be processed. Consequently, the SIRGAS-CON network was divided into four sub-networks: SIRGAS-CON C and SIRGAS-CON D north, central (Fig. 108.2) and south. With this new configuration of sub-networks, each processing center became responsible to process a given sub-network. The results of each sub-network are combined by the SIRGAS Combination Centers, a task currently carried out by IBGE and DGFI.
108.3 Processing Center SIRGAS: IBGE The SIRGAS processing center – IBGE, despite having officially started operations only in August 2008, has results of the processing of GNSS data collected since January 2003 (Costa et al. 2007). Responsible for the processing of sub-network SIRGAS-CON-D central (Fig. 108.2), it determines the coordinates of these stations, as well as information regarding the accuracy of processing. The results are sent to the Combination Centers, responsible for combining the results of all processing centers. Since the beginning of processing activities at IBGE, several stations of the SIRGAS-CON network have been disabled, modified or created. Natural phenomena such as earthquakes, lightning, as well as problems with equipment, contribute to the fact that stations do not remain in the processing over time. Station located in the Andes, like Bogota – BOGT, Arequipa – AREQ, are often affected by earthquakes, causing in many cases the deactivation of the station (Seem€uller et al. 2008). In the Amazon region, stations suffer frequent effects of lightning, damaging receiver and antenna. The GNSS processing software used by IBGE to carry out the activities of the SIRGAS Processing Center is the Bernese GPS Software, version 5.0, developed by the University of Berne in Switzerland (Hugentobler et al. 2006). Moreover, the BPE (Bernese Processing Engine) allows operation in automatic mode. The main characteristics of the processing performed by the IBGE using the Bernese software are presented in Table 108.1. The processing is performed 2 weeks after data collection, because the final IGS precise orbits used in the processing are available only 14 days after observation date. The results generated by each processing centers are available to the DGFI within an interval of 3 weeks after observations have been collected.
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Fig. 108.1 SIRGAS-CON network (SIRGAS 2009)
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Fig. 108.2 SIRGAS-CON D central (SIRGAS 2009)
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Table 108.1 Main processing characteristics
Method Observations Software Sampling interval Elevation cutoff angle Baseline strategy Orbit/EOP Troposphere model a priori Troposphere
Ambiguities Ocean loading model Phase center variations Coordinates and velocities Daily solutions Weekly solutions
The station information used in processing such as receiver and antenna types, domes number, antenna height, amongst others, are obtained from the station log files available on the DGFI and IGS servers: ftp://ftp.dgfi.badw-muenchen.de/pub/gps/DGF/ station/log/ ftp://igscb.jpl.nasa.gov/pub/station/log/ The results of the Processing Center SIRGAS – IBGE are generated daily, but the solution sent to the combination centers are weekly files obtained by combining the daily results in the SINEX format.
108.4 Results The systematic processing of the stations belonging to the SIRGAS-CON network generates time series for each station. These results allow to investigate the behavior of stations coordinates over time and detect possible problems that may have occurred and to evaluate seasonal effects, local geodynamic behavior and to determine the station velocities due to the movement of lithospheric plates. Because SIRGAS-CON stations are distributed in different parts of the continent, the temporal behavior can manifest itself in very distinct ways, as is shown in Figs. 108.3 and 108.4 for
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Strategy Double-difference Bernese 5.0 (BPE mode) 30 s 03 SHORTEST IGS final – IGS05 EOP week Niell dry component Zenith delay estimated each 2 h (12 daily corrections p/station) A priori sigmas applied with respect to Niell prediction model (wet component) First parameter +/ 5 m absolute and +/ 5 cm relative QIF strategy with GIM from CODE FES2004 Absolute (IGS_05) IGS05_R All stations constrained (s ¼ 1 m) OUTPUT FILES: SINEX Troposphere maps All stations constrained (s ¼ 1 m) OUTPUT FILES: SINEX
stations SANT (Santiago – Chile) and RIOD (Rio de Janeiro – Brazil), respectively. The East component of station SANT behaves quite differently from that presented by station RIOD. While the first moves East at a speed of about 2 cm/year, the second moves West by 0.3 cm/year in the west. This is due to the fact that the station SANT is located in the Andes, where the region is influenced by the Nazca plate. The temporal behavior of the stations allows the determination of velocity vectors, mainly due to the tectonics of lithospheric plates, as shown in Fig. 108.5. To calculate these vectors, we used time series with at least 1 year of GPS data. Other factors, such as subsidence and accommodation of the soil, as well as seasonal effects can be detected in the systematic processing of GNSS stations. An example is the altimetric behavior displayed by NAUS station located in Manaus city. The height component of this station displays seasonal variations of about 8 cm/year (Fig. 108.6) which are directly related to the water levels of the River Negro (Bevis et al. 2005). This is verified when comparing the results for station NAUS with the results from hydrometric stations along the banks of the River Negro (Fig. 108.7) (CPRM 2007).
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Fig. 108.3 Time series of Santiago station (SANT)
Fig. 108.4 Time series of Rio de Janeiro station (RIOD)
As the water level in the river increases, the station’s altitude determined by GPS decreases, reaching lowest altitude at times of peak water level. Inversely, at times
of lowest water levels in the river, the station reaches its highest altitude. The distance between the two stations, hydrometric and GPS is approximately 13 km.
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Fig. 108.5 Velocity vectors of the stations
Besides detecting effects related to displacements of a monument, the continuous processing of SIRGAS-CON stations, allow also to check for equipment problems, phase center, verticality of monument, among others. Figure 108.8 shows the temporal behavior of station POAL, where can be seen that after changing the antenna, a jump of 2 cm in horizontal components was detected. This occurred because the force centering
device at the top of monument was not perfectly placed on the vertical. In order to keep the station active, the old antenna was placed again on the monument. In order to evaluate the results computed by IBGE, the comparisons between the solutions with other three SIRGAS processing centers, as well as the IGS weekly solution are performed. Both comparisons show that IBGE results to be accurate, with RMS of about
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Fig. 108.6 Time series of Manaus station (NAUS)
Hydrometric station x GPS - (Normalized 1 and -1) 1.0
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Fig. 108.7 Hydrometric station x GPS NAUS
2 mm in horizontal component and 5 mm in height. Figure 108.9 shows the comparison of the weekly solutions between IBGE and IGS, respectively. Conclusions
The continuous processing of the active SIRGASCON stations is extremely important for the maintenance of SIRGAS. This processing ensures a quality control in each station that realizes this reference frame. Until August 2008, IBGE has worked as experimental basis in the processing of SIRGAS stations,
fulfilling all requirements concerning quality and commitment in the results presented. On occasion of the SIRGAS project meeting in May 2008 in Montevideo, it was decided that IBGE becomes an official SIRGAS processing center. Since then IBGE has been carrying out systematic processing of the SIRGAS-CON network, providing the results to the combination centers. The time series and velocities are still a first sample regarding the temporal behavior of the stations and therefore the South American plate. Several other studies will be conducted in order to monitor the permanent SIRGAS stations, ensuring the maintenance of that system. At this moment, we can say that the results obtained by the IBGE are accurate and consistent with the results obtained by other processing centers. Even though the velocity results obtained are in agreement with values determined by previous studies (Perez 2002; Costa 2001), but it is necessary to obtain longer periods of GPS observations, and hence, more reliable are the results. Some researchers consider a period of 5 years appropriate to initiate studies of Geodynamics (Costa 1999). With this purpose in mind, the intention is to continue the processing of the SIRGAS network and to evaluate the station’s velocities.
Fig. 108.9 Comparisons weekly solutions IBGE X IGS
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108 Processing Evaluation of SIRGAS-CON Network by IBGE Analysis Center 867
Fig. 108.8 Time series of Porto Alegre station (POAL)
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References Bevis M, Alsdorf D, Kendrick E, Fortes LP, Forsberg B, Smalley R, Becker J (2005) Seasonal fluctuations in the mass of the Amazon River system and Earth’s elastic response. Geophys Res Lett 32:L16308. doi:10.1029/ 2005GL023491 Costa SMA (1999) Integrac¸a˜o da Rede Geode´sica Brasileira aos Sistemas de Refereˆncia Terrestres, tese de doutorado, Universidade Federal do Parana´ – UFPR Costa SMA (2001) Estimativa do Campo de Velocidade a partir das Estac¸o˜es da RBMC. Congresso Brasileiro de Cartografia, Porto Alegre Costa SMA, Silva AL, Lago GN (2007) Primeiro Ano de Atividades do Centro de Processamento SIRGAS – IBGE. XXIII Congresso Brasileiro de Cartografia, Rio de Janeiro, Brasil CPRM (2007) Monitoramento Hidrolo´gico, Boletim nº 1, http:// www.cprm.gov.br/
S.M. Costa et al. DeMelts C et al (1994) Effect of recent revisions to the geomagmetic reversal time scale on estimates of current plate motions. Geophys Res Lett 21(20):2191–2194 Hugentobler U et al (2006) Bernese GPS Software Version 5.0. Astronomical Institute University of Berne, Berne IBGE, R.PR – 1/2005 (2005) Resoluc¸a˜o do Presidente do IBGE N 1/2005 – Altera a caracterizac¸a˜o do Sistema Geode´sico Brasileiro, http://www.ibge.gov.br Perez JAS (2002) Campo de Velocidade para as Estac¸o˜es da RBMC e do IGS Localizadas na Placa Sul-Americana: Estimativa A partir do Processamento de Dados GPS. Universidade Estadual Paulista – UNESP, Presidente Prudente, SP, Brasil Seem€uller W, et al. (2008) The position and velocity solution DGF08P01 of IGS regional network associate analysis centre SIRGAS (IGS RNAAC SIR), DGFI Report Nº 79, M€unchen, Germany SIRGAS (2009) Sistema de Referencia Geoceˆntrico para as Ame´ricas, http://www.sirgas.org
ProGriD: The Transformation Package for the Adoption of SIRGAS2000 in Brazil
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Marcos F. Santos, Marcelo C. Santos, Leonardo C. Oliveira, Sonia A. Costa, Joa˜o B. Azevedo, and Maurı´cio Galo
Abstract
Brazil adopted SIRGAS2000 in 2005. This adoption called for the provision of the relationships between SIRGAS2000 and the previous reference frames used for positioning, mapping and GIS, namely, the Co´rrego Alegre (CA) and the South American Datum of 1969 (SAD 69). Two programs were designed for this purpose. The first one, TCGeo, provided the relationships based on threetranslation Similarity Transformation parameters. TCGeo was replaced in December 2008, by ProGriD. ProGriD offers, besides the same similarity transformation as TCGeo, a set of transformations based on modelling the distortions of the networks used in the various realizations of CA and SAD 69. The distortion models are represented by a grid in which each node contains a transformation value in terms of difference in latitude and in longitude. The grid follows the same specifications of the NTv2 grid, which has been used in other countries, such as Canada, USA and Australia. This paper presents ProGriD and its main functionalities and capabilities.
109.1 Introduction M.F. Santos S.A. Costa J.B. Azevedo Coordenacao de Geode´sia, Instituto Brasileiro de Geografia e Estatistica, Av Brasil 15671, Parada de Lucas, Rio de Janeiro 21241-051, Brazil M.C. Santos (*) Department of Geodesy and Geomatics Engineering, University of New Brunswick, P.O. Box 4400, Fredericton, NB, Canada E3B 5A3 e-mail:
[email protected] L.C. Oliveira Sec¸a˜o de Ensino de Engenharia Cartogra´fica, Instituto Militar de Engenharia, Prac¸a General Tiburcio, 80-6 andar, Rio de Janeiro 22290-270, Brazil M. Galo Departamento de Cartografia, Universidade Estadual Paulista, Rua Roberto Simonsen, 305, Presidente Prudente, Brazil
Historically, two geodetic reference systems have been officially and widely used in Brazil in support of surveying and mapping. By ‘officially’ it is meant that they were regulated by specific legislation. The first one, the Co´rrego Alegre (CA), started to be developed in the 1950s and was used as the official system until 1983 when it was replaced by the South American Datum of 1969 (SAD 69). Different realizations of Co´rrego Alegre and SAD 69 exist. Both systems have co-existed for mapping applications. In 2005, Brazil adopted SIRGAS2000 as its official reference system (IBGE 2005). A period of 10 years, which started in 2005, was suggested. During this period all components of government and private sector should start using SIRGAS2000 in their activities and should start
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Fig. 109.2 Co´rrego Alegre 1970 + 1972
migrating into SIRGAS2000 all their databanks currently in either Co´rrego Alegre or SAD 69. Co´rrego Alegre is a classical datum, developed during the 1960s and the 1970s, which uses Hayford (Torge 2001) as its reference ellipsoid. There were three realizations of Co´rrego Alegre, in 1961, in 1970 and in 1972. Figures 109.1 and 109.2 show the coverage of Co´rrego Alegre 1961 and of Co´rrego Alegre 1970 and 1972 put together, respectively. It can be seen that Co´rrego Alegre 1961 was mostly present in the Southeast part of the country. Co´rrego Alegre 1970 and 1972 increased the southeastern coverage as well as grew towards the South and the Northeast. The North and the Center-West are big empties.
Fig. 109.4 SAD 69/96
SAD 69 is a classical datum, developed during the 1980s and the 1990s, which uses the GRS 67 as its reference ellipsoid (IAG 1971). There were two realizations of SAD 69, the original and another one released in 1996. Figure 109.3 and 109.4 show these two realizations. It can be noted the network is vastly enhanced, with several points in the Northern part of the country (the Amazon) most of them determined using NNSS receivers in absolute mode. The topocentric origin of both Co´rrego Alegre and SAD 69 are located in different geodetic markers, albeit very close to each other, in the state of Minas Gerais. The distance between the (non-geocentric)
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centers of each ellipsoid to the center-of-mass is around 150 m. There were instances of local datums being implemented in remote areas devoid of geodetic infrastructure in the past to serve as basis for hydrographic operations, and exploration of oil and other minerals. These numerous local datums are out of the scope of this paper, either for not being used any more, or for lack of documentation, or, as in the case of a couple of them, for requiring a specific treatment. The adoption of SIRGAS2000 by Brazil, to satisfy traditional activities related to surveying and mapping plus more recent ones such as GIS and Spatial Data Infrastructure, created the need for a consistent computational tool to help users in their transition efforts from the ‘old’ frames to SIRGAS2000. This need has been addressed initially with the release of a program named TCGeo, in 2005. TCGeo was capable of performing a three parameter similarity transformation between SAD 69 and SIRGAS2000, using the parameters published by IBGE (IBGE 2005). Finally, in 2008, a program named ProGriD was released, allowing for the modelling of the distortions remaining from the similarity transformation, resulting in a more accurate transformation. Besides SAD 69, ProGriD can also handle Co´rrego Alegre. For the sake of completeness, SIRGAS2000 network is shown in Fig. 109.5.
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109.2 ProGriD As said before, ProGriD is a computational tool that performs coordinate transformations among realizations of Co´rrego Alegre, SAD 69 and SIRGAS2000. ProGriD is based on grids that contain the shifts between pairs of realizations. These grids follow the National Transformation Version 2 (NTv2) format (Junkins 1998). This format was chosen because several other countries (e.g., Canada, the USA, and Australia) have done so before, and many GIS software packages are already capable of handling it. The shifts contained at the nodal points of the grids represent the value to be used in the transformation from Co´rrego Alegre and SAD 69 to and from SIRGAS2000. These transformation values are a result of a distortion model. Oliveira et al. (2008) described the effort of several research groups in developing different distortion models for the Brazilian situation. The outcome of this effort showed that the developed models yielded results similar to the NTv2 model (Junkins and Farley 1995; Nievinski 2006), in a certain way validating its choice. NTv2 relates two reference frames by applying a Helmert transformation and a “grid” calculation, the latter being a correction that models the distortions derived from the materializations of the reference systems. The distortions are computed according to an exponential weight function, which is a function of distance between point of interest and its neighbouring network points. This allows the creation of a grid of corrections (shifts) for latitude and longitude. A file containing these shifts in the form of a grid was created for each one of the 2D classical reference frames or interest, therefore, called as the shift grid files. To determine the shift values from the grid, a bi-linear interpolation is used. The realizations of Co´rrego Alegre and SAD 69 supported by ProGriD are: – Co´rrego Alegre 1961 (CA61). – Co´rrego Alegre 1970 and 1972 put together (CA70 + 72). – SAD 69 original including only the classical network (SAD 69). – 1996 realization of SAD 69 (SAD 69/96) including only the classical network. – SAD 69 points established by space geodetic techniques.
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Further explanation is required about the list presented above. During the data analysis it was realized that two of the realizations of the Co´rrego Alegre (1970 and 1972) were exactly the same, differing only in the coverage of the respective networks. Therefore, it was decided to merge them into a single one. During the data analysis it became clear that the classical points integrating the SAD 69 have very distinct distortion behaviour than those determined by geodetic space techniques (Doppler and GPS). Classical points are inherently two dimensional whereas space geodetic points are three dimensional quantities. If treated together it would make the distortion modelling very difficult to accomplish with the risk of resulting in an unrealistic model. For example, Fig. 109.6 shows the different level of distortions of triangulation and GPS points in the southern tip of Brazil in SAD 69/96 (distortion here defined as the difference between the SAD 69/96 coordinates transformed into SIRGAS2000 by a three-parameters transformation ‘minus’ the adjusted coordinates in SIRGAS2000). The long arrows show the distortion of the triangulation points. The dots show the distortion of the GPS points. In reality, the dots representing the distortion of the GPS points are arrows too, but their magnitudes are so much smaller than the distortion of the triangulation points that they can be barely identified as arrows. If looking carefully, it can also be seen that several “dots” fall in the middle of the “arrows.” Moreover, it should be considered that the users, in their surveying and mapping applications,
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would have used either classical (2D) points or space-determined (3D) points separately. These facts led to the conclusion that the 2D and the 3D points should be treated separately as two independent networks. The points determined via space geodetic techniques were then removed from the SAD 69, forming an independent group of points been referred to as SAD 69 Doppler or GPS Technique. In a nutshell, the networks of distinct nature (classical and space geodetic) are treated by ProGriD differently. The classical networks relates to SIRGAS2000 by grid shift files that model their distortions. The space network relates to SIRGAS2000 by three translation parameters.
109.3 Transformations Treated by ProGriD ProGriD was designed to handle a number of transformations between the geodetic frames presented before, allowing the use of different coordinate types. Figure 109.7 indicates the flow of transformations. One can relate any one of the reference frames through SIRGAS2000. For example, ProGriD permits coordinate transformation between Co´rrego Alegre 1961 (CA61) and SAD 69 via SIRGAS2000. It is important to say that the relation between CA61, CA70 + 72, SAD 69 and SAD 69/96 to SIRGAS2000 is 2D, being done via the NTv2 grid. The relation between the network of 3D points determined by space geodetic techniques (TE) and SIRGAS2000 is done using the official three translations as published by IBGE (IBGE 2005). Figure 109.8 illustrates the type of coordinates handled by two-dimensional transformations, i.e., transformations that involve any one of the classical
Fig. 109.7 Flow of transformations
Fig. 109.6 Different distortion levels of triangulation and GPS points
Fig. 109.8 Coordinates used in 2D transformations
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Fig. 109.9 Coordinates used in 3D transformations
reference frames. In this case, only geodetic latitude f and longitude l and UTM E and N coordinates can be used. Figure 109.9 portrays the type of coordinates handled by three-dimensional transformations, i.e., transformations that involve 3D points (included in SAD 69 Doppler or GPS Technique) and SIRGAS2000. In this case, not only geodetic latitude f and longitude l and UTM E and N coordinates can be used but also Cartesian coordinates and geodetic height h. In the case of mixed-type transformations (ones involving any one of the classical 2D frames and the 3D SAD 69 Doppler or GPS Technique) the geodetic height information will be either treated or ignored during the process. The user has the option to input the coordinates in any order he/she wants, in either decimal or in degrees, minutes and seconds, for the geodetic coordinates. As far as heights are concerned, ProGriD handles only geodetic heights. Orthometric height must be appropriately transformed before running ProGriD. Another program, MAPGEO2004, must be used to handle these height transformations. It is very important to realize that ProGriD does not identify a wrong input. If the wrong height type is used, ProGriD will treat it as geodetic height.
109.4 Uncertainties ProGriD handles the uncertainties associated with the transformations as follows. The standard deviation sG of the nodal points of the grid represents transformations involving only classical networks from/to SIRGAS2000. They come from a specific grid shift file. Each individual value sG is obtained as a weighted average among the standard deviation of neighbouring classical points around a particular nodal grid point. The weight is a function of distance between the classical point and the grid. A radius of 500 km was used. They reach maximum
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values of 0.7 m for Co´rrego Alegre 1961 and 1970 + 1072, and 0.5 m for most of the SAD 69, except for an area in the State of Amapa´, in the Amazon, where it reaches 6 m (an open electronic traverse). The standard deviation sT of the transformations involving SAD 69 Doppler or GPS Technique and SIRGAS2000 correspond to the estimated standard deviation of the three official translations. Its value is at cm-level for the three translations. If the transformation involves more than one network, it can be computed as: (a) If the transformation involves any one of the 2D classical networks and the 3D SAD 69 Doppler or GPS Technique network, it entails the uncertainty from a grid sG and the uncertainty from the transformation parameters sT. The uncertainty given by ProGriD is computed as: sPG ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2G þ s2T :
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(b) If the transformation involves any two of the 2D classical networks, it entails the uncertainty from a grid sG1 and a from a grid sG2. The uncertainty given by ProGriD is computed as: sPG ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2G1 þ s2G2 :
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ProGriD has no control over the uncertainty associated to the coordinates input by the users. In other words, the user cannot input the uncertainty associated with the coordinates desired to be transformed. If the final uncertainty of the transformation sF is desired, it must be computed by the user combining the uncertainty given by ProGriD sPG with the uncertainty of the user’s input coordinates sU, contained in the user’s data base, as: sF ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2PG þ s2U :
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109.5 Practical Considerations Users of ProGriD must realize that there are several decisions that depend on them. They include, for example, the proper choice of the network being
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dealt with, if either Co´rrego Alegre 1961 or Co´rrego Alegre 1970 + 1972, if either SAD 69 or SAD 69 realization 1996, and if either SAD 69 (2D) classical network or SAD 69 (3D) Doppler or GPS network. This is very important because the software cannot distinguish between networks. Erroneous solutions of up to a few metres will arise if users by mistake: – Treat Co´rrego Alegre 61 as 70 + 72 and vice-versa – Treat SAD 69 as SAD 69 realization 1996 and viceversa – Treat a “satellite” point as classical point and viceversa A common misconception that exists among users is that, somehow, the quality of their coordinates will improve when migrated from a classical network into SIRGAS2000. As a matter of fact, the transformation does not improve the quality of the original coordinates. Other examples of what ProGriD does not do are summarized in the paragraphs that follow. ProGriD does not distinguish vertical coordinate type. In other words, the user must make sure that geodetic heights are input instead of any other height type. ProGriD will always treat any input height as geodetic height. Until recently, the relationship between the classical frames used in Brazil and the early ones used in space geodesy (e.g., WGS72 and the earlier realization of WGS84) was treated by means of translation parameters. These parameters were official, i.e., they were published by IBGE. With the new paradigm created by the more rigorous distortion modelling handled by ProGriD there is no need to rely on those parameters any more. Therefore, ProGriD does not transform any of the classical networks into SIRGAS2000 using official translation parameters. ProGriD does not handle points outside the predefined limit areas. This limitation was implemented in order to avoid the use of the software for areas outside the continental Brazil. ProGriD does not handle non-official frames. As mentioned earlier in this paper, besides the two official classical 2D reference frames, Co´rrego Alegre and SAD 69, Brazil has had in its geodetic history many other ‘non-official’ frames were developed to satisfy different reasons. One of them is the frame Aratu. Aratu is a series of local frames joined together
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over the years, developed by the Brazilian Petroleum S.A. (Petrobras) in support of its exploration and production activities. Other local frame developed over the years is the one known as SICAD. This frame was developed and has been used in the mapping of Brazil’s Federal District, where the city of Brası´lia is located. Aratu, SICAD, or any other non-official frame must go through a particular treatment to establish the proper relation with SIRGAS2000.
109.6 Concluding Remarks This paper presents ProGriD, the transformation program package developed to handle coordinate transformation between realizations of Co´rrego Alegre and SAD 69 to SIRGAS2000 in Brazil. It overviews the major characteristics of ProGriD, including a brief discussion on the model used for the modelling of the distortions of the 2D frames, NTv2. It also describes that the 3D points were separated from their SAD 69 parent frame to form an independent network, which relates to SIRGAS2000 by means of three translation parameters. ProGriD provides error estimates of the transformations. It concludes with a few practical considerations dealing with situations where the misuse of ProGriD may cause unwanted errors. Currently, a desktop version of ProGriD can be downloaded from IBGE’s web site at ftp://geoftp.ibge. gov.br/programa/Transformacao_de_Coordenadas/. A web version is planned to be released early in 2010. The desktop version of ProGriD was released in December 2008. It has been widely used by the community at large, including the transformation of large data sets, such as the whole spatial database of the State of Espı´rito Santos (GEOBASES). Acknowledgments ProGriD was developed under the scope of the National Geospatial Framework Project (PIGN) http://www. pign.org funded by the Canadian International Development Agency – CIDA. This project is lead by the Brazilian Institute of Geography and Statistics (IBGE) and by the University of New Brunswick (UNB). The effort which led to the development of ProGriD had the involvement of other institutions, namely the Military Institute of Engineering (IME) and the State University of Sa˜o Paulo (UNESP). Thanks to Carlos Alexandre Garcia for the generation of the network maps.
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References IAG (1971) “Geodetic Reference System 1967.” IAG Special Publication No. 3, International Association of Geodesy, Paris IBGE (2005) Resoluc¸a˜o do Presidente 1/2005 – Altera a caracterizac¸a˜o do Sistema Geode´sico Brasileiro. Rio de Janeiro, Fundac¸a˜o Instituto Brasileiro de Geografia e Estatı´stica, 25/02/2005. Available on ftp://geoftp.ibge.gov. br/documentos/geodesia/pmrg/legislacao/RPR_01_25fev2005. pdf Junkins DR (1998) “NTv2 – Procedures for the Development of a Grid Shift File.” Geodetic Survey Division – Geomatics Canada Junkins DR and Farley SA (1995). “NTv2 – National Transformation Version 2 – User’s Guide.” Geodetic Survey
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Division – Geomatics Canada. Available on http://www. geod.nrcan.gc.ca/pdf/ntv2_guide_e.pdf Nievinski FG (2006) “NTv2 and the new Brazilian Frame: an alternative for distortion modeling.” Course Delivered in Presidente Prudente, SP, Brazil March 21–24, University of New Brunswick Oliveira, Leonardo C, Marcelo C. SANTOS, Felipe G. Nievinski, Rodrigo F. Leandro, Sonia M. A. Costa, Marcos F. Santos, Joa˜o Magna Jr., Mauricio Galo, Paulo O. Camargo, Joa˜o G. Monico, Carlos U. Silva, Tule B. Maia (2008) “Searching for the optimal relationships between SIRGAS2000, South American Datum of 1969 and Co´rrego Alegre in Brazil.” Observing our Changing Earth, In: Sideris M (ed) Proceedings of the 2007 IAG General Assembly, Perugia, Italy, 2 – 13 July, 2007, International Association of Geodesy Symposia, vol 133 Springer, Berlin, pp 71–79 Torge W (2001) Geodesy, 3rd edn, de Gruyter
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The New Multi-year Position and Velocity Solution SIR09P01 of the IGS Regional Network Associate Analysis Centre (IGS RNAAC SIR)
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€ller, M. Seitz, L. Sa´nchez, and H. Drewes W. Seemu
Abstract
The Deutsches Geod€atisches Forschungsinstitut (DGFI) acts as the IGS Regional Network Associate Analysis Centre for SIRGAS (IGS RNAAC SIR) since June 1996. Each week a loosely constrained position solution including all available observations of the SIRGAS Continuously Operating Network (SIRGAS-CON) is generated and delivered to the IGS Data Centres to be integrated into the IGS polyhedron. Based on these weekly solutions, DGFI also computes multi-annual solutions for station positions and velocities to estimate the kinematics of the SIRGAS reference frame. These multi-annual solutions are updated yearly and include those stations operating more than 2 years continuously. This paper describes the computation of the latest multi-annual solution of the SIRGASCON network. Identified as SIR09P01, it was released in June 2009 and contains all the weekly solutions provided by the SIRGAS Analysis Centres from January 2, 2000 (GPS week 1,043) to January 3, 2009 (GPS week 1,512). It refers to the IGS05 frame at the epoch 2005.0 and provides positions and velocities for 128 SIRGAS-CON stations. The accuracy of its positions at the reference epoch is estimated to be better than 0.5 mm in the horizontal component and 0.9 mm in the vertical one. The accuracy of the linear velocities is about 0.8 mm/a.
110.1 Introduction DGFI is in charge of the IGS Regional Network Associate Analysis Centre for SIRGAS (IGS RNAAC SIR, Seem€uller and Drewes 2008). Its main responsibility is to deliver loosely constrained weekly position solutions of the regional network SIRGAS-CON (SIRGAS Continuously Operating Network) in SINEX format to
W. Seem€uller M. Seitz L. Sa´nchez (*) H. Drewes Deutsches Geod€atisches Forschungsinstitut (DGFI), AlfonsGoppel-Str. 11, 80539 Munich, Germany e-mail:
[email protected]
the IGS. These solutions are combined with the results from the other IGS Analysis Centres to compute the IGS global polyhedron. The processing of the SIRGAS-CON network includes also the generation of weekly station coordinates aligned to the current ITRF realization for applications in Latin America, as well as accumulative (multi-annual) position and velocity solutions for estimating the kinematics of the network. These multi-annual solutions are updated yearly (e.g. Seem€uller et al. 2002, 2008; Seem€uller 2009). The input data for the SIRGAS-CON multi-year solutions are the same loosely constrained weekly solutions delivered to the IGS data centres. These weekly solutions were calculated by DGFI in only
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one adjustment for the entire network including all data up to 31 August 2008 (GPS week 1,495). After that, since in the last years the Latin American countries are qualifying their national reference networks by installing continuously operating GNSS stations, the number of SIRGAS-CON sites is rapidly increasing and the processing of the entire network in only one adjustment became impracticable. It was necessary to split the network into different clusters to be separately processed and then combined to a unified solution for the entire region (Brunini et al. 2011). In this way, the SIRGAS-CON network at present comprises one core network covering homogeneously Latin America and the Caribbean, and three densification sub-networks distributed on the northern, the middle, and the southern parts of the region
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(Fig. 110.1). These four sub-networks are individually calculated by four SIRGAS Processing Centres (see Sect. 110.2.1), which generate loosely constrained weekly solutions to be combined in a unified solution for the entire network (see Sect. 110.2.2). According to this, the weekly solutions after GPS week 1,495 included in the new multi-year solutions for the SIRGAS-CON network correspond to the combinations of the four mentioned sub-networks. This paper describes the input data, analysis strategy, and results of the first multi-annual solution for the SIRGAS-CON network computed after introducing the new analysis structure based on four sub-networks and six SIRGAS Analysis Centres (four Processing and two Combination Centres). Besides to obtain precise station positions and velocities, this study shall provide information about the consistency between the previous weekly solutions (when the network was not clustered) and the actual ones, guarantying a continuous reliability of the SIRGAS products.
110.2 New Structure of SIRGAS-CON
Fig. 110.1 Core and densification sub-networks within SIRGAS-CON
Since August 31, 2008, the SIRGAS-CON stations are classified in four sub-networks (Fig. 110.1): 1. One core network (SIRGAS-CON-C) with continental coverage and stabile site locations to ensure the long-term stability of the reference frame 2. Three densification sub-networks (SIRGAS-CON-D) covering the northern, the middle, and the southern part of the SIRGAS region to provide accessibility to the reference frame at national and local levels Although, they appear as two different levels, both kinds of stations (core and densification) match requirements, characteristics, performance and quality of the ITRF stations. The SIRGAS-CON-C network is processed by DGFI (Germany) as the IGS-RNAAC-SIR. The SIRGAS-CON-D sub-networks are computed by the SIRGAS Local Processing Centres according to the following distribution (Brunini et al. 2011): – Instituto Geogra´fico Agustı´n Codazzi, Colombia (IGAC) is in charge of the northern densification sub-network; it covers Mexico, Central America, the Caribbean, Colombia, and Venezuela. – Instituto Brasileiro de Geografia e Estatistica, Brazil (IBGE) processes the middle densification sub-network, which comprises stations operating in
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Brazil, Ecuador, Bolivia, Suriname, French Guyana, Guyana, Peru, and Bolivia. – Centro de Procesamiento Ingenierı´a-MendozaArgentina at the Universidad Nacional del Cuyo, Argentina (CIMA) computes the southern densification sub-network, which includes the stations located in Uruguay, Paraguay, Argentina, Chile, and Antarctica. These four processing centres deliver loosely constrained weekly solutions for the densification SIRGAS-CON-D sub-networks, which are combined with the continental SIRGAS-CON-C network to get homogeneous precision for station positions and velocities in a continental level.
110.2.1 Procedure to Generate Loosely Constrained Weekly Solutions for the SIRGAS-CON Sub-Networks The SIRGAS Processing Centres commonly apply basic solution standards established by SIRGAS (in accordance with the IGS and IERS standards) to generate the loosely constrained weekly solutions for the assigned sub-networks. The main characteristics of which are: 1. Elevation mask and data sampling rate are set to 3 and 30 s, respectively 2. IGS absolute calibration values for the antenna phase centre corrections are applied 3. Satellite orbits, satellite clock offsets, and Earth orientation parameters are fixed to the combined IGS weekly final solutions 4. Phase ambiguities for L1 and L2 are solved. At present, the four SIRGAS processing Centres apply the quasi ionosphere free (QIF) strategy of the Bernese software (Dach et al. 2007) 5. Periodic site movements due to ocean tide loading are modelled according to the FES2004 ocean tide model (Letellier 2004). The corresponding values are provided by M.S. Bos and H.-G. Scherneck at http://129.16.208.24/loading/ 6. At the moment, the four SIRGAS Processing Centres use the Niell (1996) dry mapping function to map the a priori zenith delay (~ dry part), which is modelled using the Saastamoinen model (1973). The wet part of the zenith delay is estimated at a 2 h interval within the network adjustment and it is mapped using the Niell wet mapping function
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7. Daily free normal equations are computed and combined to get a loosely constrained weekly solution for station positions (all of them are constrained to 1 m) 8. Stations with large residuals in the weekly combination (more than 20 mm in the N and E components, and more than 30 mm in the Up component) are reduced from the corresponding daily session(s) before the final normal equations are generated. Steps (7) and (8) are iterative 9. The four Processing Centres apply the Bernese GPS Software V.5.0 (Dach et al. 2007) The loosely constrained solutions in SINEX format are called CCCwwww7.SNX. CCC identifies the corresponding Processing Centre (i.e. CIM, DGF, IBG, IGA), wwww stands for the GPS week, and 7 for including the 7 days of the week. The individual solutions delivered by the SIRGAS Processing Centres are available at the SIRGAS FTP: ftp.sirgas.org.
110.2.2 Procedure to Combine the Loosely Constrained Weekly Solutions for the SIRGAS-CON Sub-Networks The individual contributions delivered by the Processing Centres are integrated in a unified solution by the SIRGAS Combination Centres: DGFI and IBGE. The DGFI combinations are provided to the users as the SIRGAS official products, while the IBGE combinations assure redundancy and control for those products. The main features of the combination strategy are (Sa´nchez et al. 2011): 1. Constraints included in the delivered normal equations are removed and the sub-networks are individually aligned to the IGS05 frame by applying the not net rotation (NNR) and no net translation (NNT) conditions 2. Station positions obtained in (1) for each Processing Centre are compared to the IGS weekly values and to each other to identify possible outliers 3. Stations with large residuals (more than 10 mm in the north or east components, and more than 20 mm in the height) are reduced from the corresponding individual normal equations 4. Variances obtained in the final computation of step (1) are analyzed to estimate scaling factors for relative weighting of the individual solutions
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5. Once inconsistencies and outliers are reduced from the individual free normal equations, a combination for a loosely constrained weekly solution for station positions (all of them constrained to 1 m) is computed. This solution is submitted to IGS for the global polyhedron and stored to be included in the next multi-year solution of the SIRGAS-CON network. The resulting SINEX file is called SIRwwww7.SNX. 6. Additionally, a weekly solution for station positions aligned to the IGS05 reference frame is computed based on the IGS weekly coordinates. Details are presented by Sa´nchez et al. (2011). This solution provides the final SIRGAS-CON weekly positions for further applications. It is identified with the name siryyPwwww.crd (yy: two last digits of the year). The loosely constrained combinations as well as the weekly SIRGAS-CON coordinates are available at the SIRGAS FTP site mentioned in Sect. 110.2.1.
110.3 The Multi-year Solution SIR09P01 110.3.1 Input Data The input data for the generation of the multi-year solution SIR09P01 are the loosely constrained weekly solutions of the SIRGAS-CON network between January 2, 2000 and January 3, 2009. As already mentioned, these weekly solutions were computed by DGFI in only one adjustment for the entire network containing data up to August 31, 2008 (GPS week 1,495). The loosely constrained weekly solutions for
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the later weeks correspond to the combination of the four SIRGAS-CON sub-networks (see Sect. 110.2). Weekly solutions from January 2000 (GPS week 1,043) to October 2006 (1,399) formerly computed with relative antenna phase centre corrections and referring to previous ITRF solutions have been reprocessed following the standards presented in Sect. 110.2.1, i.e. they include absolute phase centre corrections and the IGS05 as reference frame. This reprocessing provides homogeneously computed weekly solutions for the complete time span covered by the SIR09P01 solution and allows to improve reliability and accuracy of station positions and velocities. Figure 110.2 shows timetable and infrastructure used for processing and reprocessing of the loosely constrained weekly solutions included in SIR09P01. Reprocessed solutions are identified with the name SI1wwww.SNX to be distinguished from the old weekly solutions.
110.3.2 Processing Strategy The processing strategy for the SIRGAS reference frame SIR09P01 is given as a flow chart in Fig. 110.3. It is realized using the Bernese GPS Software V5.0 (Dach et al. 2007). The main parts of the analysis are: 1. Computation of time series and time series analysis to identify outliers and discontinuities (jumps, velocity changes) in station positions (see grey arrows in Fig. 110.3)
Fig. 110.2 Timetable and infrastructure used to generate the loosely constrained weekly solutions included in the multi-annual solution SIR09P01
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The New Multi-ear Position and Velocity Solution SIR09P01 of the IGS Regional Network
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the height (about the fourfold of the mean RMS). If outliers appear sporadically (without pattern), the station position is reduced from the normal equation for the corresponding week. If outliers correspond to a discontinuity, a new position is set up for the station. According to these criteria, discontinuities were identified for 11 stations and appropriate time spans were defined (see Seem€uller et al. 2009, page 6). The analysis of the residual position time series shows that the weekly combined solutions are at the same accuracy level and totally consistent with the previous computations (when the network was computed in one block); therefore, the series are appropriate to be accumulated into a multi-year solution.
Fig. 110.3 Processing strategy for the computation of the SIR09P01 multi-year solution
2. Combination of weekly normal equations (NEQ) to compute the SIRGAS reference frame (see black arrows in Fig. 110.3) Before starting with the computation preparative steps are necessary. Unconstrained (free) normal equations are reconstructed from the weekly solutions stored in SINEX format. Thereby, the station information, e.g. antenna and receiver types, is compared to the log files and corrected if necessary in order to guarantee consistency of the station information. So, the input data for the combination are unconstrained (condition free, non-deformed) normal equations and correct station information.
110.3.2.1 Computation of Time Series and Time Series Analysis The weekly normal equations are solved separately applying the NNR and NNT conditions with respect to IGS05. To generate residual position time series, the weekly solutions are transformed to an a priori SIRGAS reference frame (i.e. the actual SIRGAS reference frame SIR08P01, Seem€ uller et al. 2008) by a seven-parameter similarity transformation. The residual time series of station positions are analysed and the detected discontinuities and outliers are taken into account for the computation of the SIRGAS reference frame (see Sect. 110.3.2.2). The chosen thresholds for outliers are: 15 mm for north and east, and 30 mm for
110.3.2.2 Combination of Weekly Normal Equations According to Fig. 110.3, the weekly normal equations are combined to a multi-year solution setting up station velocities. Seasonal (e.g. loading) signals are not considered up to now. So, stations with observation time spans of less than 2 years of data are excluded (reduced) as the velocity estimation would be affected strongly by possible seasonal variations. The geodetic datum is realized by applying the NNR and NNT conditions with respect to IGS05 using a set of reliable and well distributed IGS stations in the SIRGAS region. The selected sites are: ASC1, BRAZ, CHPI, CRO1, GOLD, LPGS, MDO1, PIE1, SANT, SCUB, UNSA. The chosen reference epoch of the SIRGAS reference frame is 2005-01-01. After solving the first SIRGAS reference frame, step (1) and (2) are iterated: new station position residual time series are generated by transforming the weekly solutions to the computed SIRGAS reference frame. Discontinuity and outlier detection are repeated and the new information is introduced into the computation of a refined reference frame.
110.4 Results The final coordinates and velocities (Figs. 110.4 and 110.5) contained in the multi-year solution SIR09P01 refer to the IGS05, epoch 2005.0. It includes 128 stations. It is well known, that the formal errors (included in the SINEX file) estimated in the GPS observation analysis are too small because physical correlations between the GPS observations are not
882 Fig. 110.4 Horizontal velocities of the SIR09P01 solution
Fig. 110.5 Vertical velocities of the SIR09P01 solution
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The New Multi-ear Position and Velocity Solution SIR09P01 of the IGS Regional Network
accurately known and thus not fully considered. Therefore, standard deviations for station positions and velocities are derived from the residual position time series. According to this, the accuracy of coordinates at reference epoch is estimated to be 0.5 mm in the horizontal component and 0.9 mm in the height. The accuracy of velocities is about 0.8 mm/year. The SIR09P01 solution (coordinates, velocities, and SINEX file) is available at http://www.sirgas.org or at the SIRGAS FTP site (see Sect. 110.2.1). It should be noted that the reference epoch in the SINEX file is 2000-01-02 – the epoch of the first included GPS observation. As mentioned above, the standard deviations included in the SINEX file are not reliable. Realistic accuracy estimations for all the stations are given in Seem€uller et al. (2009), as well as on the SIRGAS web site. A loosely constrained version of this solution was delivered as the SIRGAS contribution to the IAG SC1.3 Working Group on Regional Dense Velocity Fields (see http://www.epncb.oma.be/IAG/index.php). Here the station positions for all sites were constrained to 10.0 m and velocities for all sites to 10.0 mm/year.
References Brunini C, Sa´nchez L, Drewes H, Costa S, Mackern V, Martı´nez W, Seem€uller W, da Silva A (2011) Improved analysis strategy and accessibility of the SIRGAS Reference Frame. In: Geodesy for Planet Earth, Buenos Aires Argentina. IAG Symposia, this volume
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Dach R, Hugentobler U, Fridez P, Meindl M (eds) (2007) Bernese GPS software version 5.0 – documentation. Astronomical Institute, University of Berne, Berne Letellier T (2004) Etude des ondes de mare´e sur les plateux continentaux. The`se doctorale, Universite´ de Toulouse III, Ecole Doctorale des Sciences de l’Univers, de l’Environnement et de l’Espace Niell AE (1996) Global mapping functions for the atmosphere delay at radio wavelength. J Geophys Res 101:3227–3246 Saastamoinen J (1973) Contribution to the theory of atmospheric refraction. Part II: Refraction corrections in satellite geodesy. Bull Ge´od 107:13–34 Sa´nchez LW, Seem€uller M. Seitz (2009). Combination of the weekly solutions delivered by the SIRGAS Processing Centres for the SIRGAS-CON reference frame. In: Geodesy for Planet Earth, Buenos Aires Argentina. IAG Symposia, this volume Seem€uller W, Drewes H (2008). Annual Report 2003–2004 of IGS RNAAC SIR. In: IGS 2001–02 Technical Reports, IGS Central Bureau, (eds), Pasadena, CA: Jet Propulsion Laboratory. Available at http://igscb.jpl.nasa.gov/igscb/ resource/pubs/2003-2004_IGS_Annual_Report.pdf Seem€uller W, Kaniuth K, Drewes H (2002) Velocity estimates of IGS RNAAC SIRGAS stations. In: Drewes H, Dodson A, Fortes LP, Sa´nchez L, Sandoval P (eds): Vertical reference systems, IAG Symposia, Springer, vol 124:7–10 Seem€uller W, Kr€ugel M, Sa´nchez L, Drewes H (2008) The position and velocity solution DGF08P01 of the IGS Regional Network Associate Analysis Centre for SIRGAS (IGS RNAAC SIR). DGFI Report No. 79. DGFI, Munich. Available at http://www.sirgas.org/ Seem€uller W (2009) The position and velocity solution DGF06P01 for SIRGAS. In: Drewes H (ed): Geodetic Reference Frames, IAG Symposia; Springer, vol 134: 167–172 Seem€uller W, Seitz M, Sa´nchez L, Drewes H (2009). The position and velocity solution SIR09P01 of the IGS Regional Network Associate Analysis Centre for SIRGAS (IGS RNAAC SIR). DGFI Report No. 85. DGFI, Munich. Available at http://www.sirgas.org/
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Analysis of the Crust Displacement in Amazon Basin
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G.N. Guimara˜es, D. Blitzkow, A.C.O.C. de Matos, F.G.V. Almeida, and A. C. B. Barbosa
Abstract
The analysis of the crust displacement in Amazon basin from the comparison between three data sources is the aim of this paper. The data involved are in-situ water level time series measured at ground-based hydrometric station of Ageˆncia Nacional de A´guas (ANA), vertically-integrated water height deduced from GRACE geoid (height anomaly) and a continuous monitoring GPS station of Instituto Brasileiro de Geografı´a e Estatı´stica (IBGE). Two analysis are carried out: the first comprehend the GPS vertical coordinate (UP) and in-situ daily data; the second is the 10-day interval of GRACE models, UP and in-situ data for a ~3-years period (January-2006 to December-2008). The GRACE models were computed by Groupe de Recherce de Ge´ode´sie Spatiale (CNES/GRGS) and the height anomaly was converted into Equivalent Water Height (EWH). The precise point positing (PPP) is the technique for absolute GPS processing. The Canadian Spatial Reference System (CSRS-PPP) service was used. The coordinate UP presents an annual cycle of vertical displacement with peak-to-peak amplitude of 80–100 mm. A correlation about 90% between in-situ and GRACE is detected. However, they have an inverse phase correlation with the vertical coordinate. This implies that the crust responds instantaneously to the hydrological loading cycle.
111.1 Introduction
G.N. Guimara˜es (*) D. Blitzkow A.C.O.C. de Matos F.G.V. Almeida Laboratory of Topography and Geodesy, Department of Transportation, University of Sa˜o Paulo, EPUSP-PTR, Postal Code 61548, CEP, 05424-970 Sa˜o Paulo, Brazil e-mail:
[email protected] A.C.B. Barbosa Astronomy, Geophysics and Atmospheric Science Institute of University, Sa˜o Paulo, Brazil
The Amazon region is known worldwide for the availability of water. It has the largest river system in the world, occupying a total of 6,110,000 km, (ANA 2006). Extends over seven countries where ~68% it belongs to Brazil and it represents a total of 55% of Brazilian territory. Solimo˜es and Amazon are the main rivers of the region. The global climate variations have affected the Amazon basin (Fig. 111.1). The study of the large floods that regularly occurs in the area is important to characterize the volume of water of ebb, the human occupation impact, the feasibility navigation and the
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_111, # Springer-Verlag Berlin Heidelberg 2012
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Fig. 111.1 Amazonas basin
displacement of the crust. The annual rainfall varies of 1,500–1,700 mm. At the mouth of the Amazon River it can reach 3,000 mm, and in Manaus city the average annual rainfall is 2,000 mm. The river level variation is a result of several important features, for example, the flow rate and extent of the riverbed, including wetlands. Tucci (1993) defines that the variation in the level or flow of a river depends basically on the climatic and physical characteristics of the concerned basin. The temporal and spatial distribution of rainfall is the main weather issues that change the level of a river. The crust displacement occurs when the surface of the earth oscillates in response to seasons fluctuations due to loadings imposed on the lithosphere by the atmosphere and, more importantly, by the hydrosphere. This vertical elastic response to environmental loading occurs at global (Blewitt et al. 2001), regional (Heki 2001) and local scales (Bevis et al. 2005). The aim of this paper is to analysis the crust displacement in Amazon basin from the comparison between three data sources.
111.2 Rainfall in the Amazon An important part of rain in the Amazon is supplied by evapotranspiration of the ecosystems, with an annual average contribution of 55–60% from the total precipitation. Nevertheless, it does not mean that all evapotranspiration generated in the Amazon basin is totally
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converted into precipitation over the region. During the rainy season, the rainfall rate is usually greater than twice the evapotranspiration rate, which means that most of the humidity necessary to generate the rain is brought from outside the region. The rain distribution from December through February shows a high precipitation region (exceeding 900 mm) located in the western and southern Amazon. The period from March through May is the rainy season in the central region near the equator. From June through August is the dry weather period for that region. In this way, the centre of maximum precipitation is shifted to the north, in Roraima and northern South America in the same epoch. During this period, the central part is predominantly influenced by Hadley cell (atmospheric circulation cell with upward winds in the North Tropical Atlantic and downward winds in the Amazon region); thus this dry period remains until approximately September/October in the southern Amazon and a month later in central part. Cutrim et al. (2000) showed that the north and east part of the Amazon basin is extremely affected by changes in the atmospheric circulation during El-Nin˜o and La-Nin˜a events. La-Nin˜a is an oceanicatmospheric phenomenon characterized by an anomalous cooling in the sea surface of Tropical Pacific Ocean. On the other hand, El-Nin˜o is the anomalous warming in the Tropical Pacific. El-Nin˜o episodes in the Equatorial Pacific Eastern seem to reduce the total rainfall in the Amazon basin, while La-Nin˜a events intensify the rain for the region. Many authors (Marengo 1992, 2004; Uvo et al. 1998; Ronchail et al. 2002) emphasized that some of the droughts periods at Amazon were due to: 1. The occurrence of intense El-Nin˜o events 2. The strong warming of sea surface of the tropical North Atlantic Ocean; or 3. Both of them (Marengo et al. 2008) Anomalies of Sea Surface Temperature (ASST) in the tropical Pacific are responsible for less than 40% of the rainfall variability in the Amazon basin (Marengo 1992; Uvo et al. 1998) and (Marengo et al. 2008). This fact suggests that other sources exist, such as the inter-hemispheric SST gradient pointing to the south of the Intertropical Atlantic Ocean, which affects mainly the north and central Amazon (Blitzkow et al. 2010).
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111.3 Data Set The data sets used consist of time series formed by: (1) Daily GPS file processed by the CSRS service online, available at http://www.geod.nrcan.gc.ca/online_data_e. php; (2) in situ water level and precipitation measurements provided by ANA; (3) 10-day GRACE data geoid variations computed by Biancale et al. (2006) and converted into EWH by Ramillien et al. (2005). The data sets are detailed in the following.
111.3.1 The Geodetic GPS Station The geodetic GPS station used in this study to check crust displacement is located on the northern of Manaus city (station 8 in Fig. 111.1), near the confluence of Rio Negro and Amazon rivers. It belongs to the RBMC (Brazilian Network for Continuous GPS Monitoring) and it is in operation since October 2005. Its coordinates are (SIRGAS2000): ’ ¼ 03 01’ 22.5108”, l ¼ 60 03’ 18.0599”, h ¼ 93.89 m. A daily file (24 h) is available at ftp://geoftp.ibge. gov.br/RBMC/dados/. These data (HATANAKA file) were submitted through the internet facility for processing at CSRS-PPP. The processing uses precise GPS orbit and clock information, referred to the International Terrestrial Reference Frame (ITRF). The processing is carried out in the absolute mode, and for this experiment L1 and L2 pseudo-range and carrier phase observations were used. According to the Geodetic Survey Division, available at http://198.103.48.76/ userguide/index_e.php, the quality of the results depends on the type of the equipment used, atmospheric dynamics and the observed session duration.
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reading the water level value on the limnimeter scale and the pluviometer. These data are collected by ANA and distributed through their website. In order to make the HS data sampling consistent with EWH series, running averages were performed over 10-day periods.
111.3.3 GRACE Data For each 10-day geoid, the Stokes coefficients were converted into EWH coefficients (Ramillien et al. 2004, 2005, 2006), and then 1 1 global grid of water thickness (in mm) were calculated. From the ground-base stations position, GRACE-based water heights were interpolated by applying a bi-linear algorithm to the gridded data. Thus, for the same position, satellite and ground-based data were compared. Each GRACE solution is shifted by 10 days from the previous one and computed as a weighted average of about 1 month of GRACE data using the 10-day factors 0.5/1/0.5. In this processing, the spectrum of the monthly solution is forced for harmonic degrees less than 30 to be less than a reference, which is empirically derived from the variance spectrum of the static gravity field (i.e. Kaula’s rule). The reference date of each grid is taken as the 15th day of the interval. A linear equation was assumed and fitted by leastsquare linear inversion (Almeida, 2009): YðtÞ ¼ aXðtÞ þ b
(111.1)
where X(t) is GRACE data, Y(t) is in situ data (ANA), a the transfer function slope coefficient and b is Y-axis intersection value.
111.4 Errors Analysis 111.3.2 In Situ Water Level and Precipitation Measurements from ANA Ground-Based Stations ANA has a few hundreds of limnimeter and pluviometer stations all over the country. The hydrological stations (HS) are coded according to their location. In this study, both stations are located at the Negro River. The methodology of acquisition is based on daily in situ water level and precipitation data, by visually
The estimated error in-situ on the limnimetric and pluviometric scale in the visual process of data collecting was up to 20 mm (ANA 2006). Three different sources of errors from GRACE data have been considered: (1) uncertainties on the Stokes coefficients; (2) spectrum truncation due to degree and order 50 imposed on the Stokes coefficients; (3) possible “leakage” of geophysical signals such as oceanic signals from surrounding regions. The average amplitudes of GRACE errors in the Stokes coefficients
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provided by GRGS are ~130 mm. This is consistent with results from previous studies (see for instance Wahr et al. 2006). Errors due to GRACE coefficients truncation at N ¼ 50 is 10–15 mm of equivalent-water thickness for a 450–500 km radius. To evaluate the leakage error, an inverse mask disk of 450 km radius created at the station location was used, with values of “0” inside the disk and “1” outside. The spherical harmonic A’nm and B’nm corresponding to this mask were computed up to degree 200. The averaged leakage effect D’v(t) was calculated using the formula (111.2): D0 vðtÞ ¼ 4pR2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XX ðD0 cnm ðtÞ A0 nm þ D0 snm ðtÞ B0 nm Þ n
m
(111.2)
where D’cnm(t) and D’snm(t) were the continental water storage WGHM (WaterGAP Global Hydrology Model) outputs (D€ oll et al. 2003) converted into spherical harmonics for the considered period t. In case of no leakage, D’v(t) should be close to zero. Attention was given to the leakage effects from continental water WGHM only, although other water mass reservoirs are known to create leakage errors, too (e.g., oceanic mass errors from the removal of oceanic and atmospheric models to geopotential fields). These errors can reach ~21 mm of equivalent-water height. These errors in the measurements degrade a and b coefficients precision, up to 1.5% for a (dimensionless) and 2 mm for b and can be found in Almeida (2009).
According to the GPS processing data, the average errors in the GPS coordinates are: ’ ~2 mm; l ~6 mm; h ~12 mm.
111.5 The Squared Pearson Coefficient (R) and the Slope Coefficient (a) Between GRACE and ANA Data Figure 111.2 shows ANA ground-based time-series database and GRACE-based equivalent water height, the scatter plot and time scale correlation. Manaus station shows a correlation equal of 90%. The squared Pearson coefficient (R) is related to the hydrogeology; high correlation is found in sedimentary basin and flooding areas. The Amazon basin is predominantly sedimentary with smooth relief (Almeida 2009). The coefficient a represents the relationship between the superficial water level and the water accumulated in the surface layer of the Earth. At Manaus station a is equal to 8.5. It shows that the accumulated water is strongly correlated with the dynamics of the superficial water.
111.6 Results and Discussion In order to detect and to verify the crust displacement two comparisons were made: GPS vs. water level and GPS vs. GRACE, Figs. 111.3 and 111.4, respectively. Moreover, it was possible to see the dry and the wet
Fig. 111.2 ANA ground-based time-series database and GRACE-based equivalent water height, the scatter plot and time scale correlation
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Analysis of the Crust Displacement in Amazon Basin
seasons in the analysis. From the wet to the dry season the Amazonas River can vary up to 10 m. The hydrological variation in Manaus is shown in both figures. Water level and GRACE present a significant correlation, as shown in Fig. 111.2 and Table 111.1. However, they have an inverse phase correlation with the vertical coordinate (Table 111.1, Figs. 111.3 and 111.4). The grey curve in Figs. 111.3 and 111.4 is the annual cycle of the elipsoidal height. The reason for this is that the ground-based hydrometric station detects water level and GRACE detects EWH. On the other hand, GPS data processing detects the variability of the vertical loading component applied on the crust. Table 111.1 Phase and amplitude of the annual cycle Phase ( ) Amplitude (mm) Water level (ANA) 139.08 5140.19 Equivalent water height (GRACE) 137.98 625.44 Elipsoidal heights (GPS PPP) 311.77 30.25
Fig. 111.3 Comparison between GPS height and water level
Fig. 111.4 Comparison between GPS height and GRACE
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GPS-PPP processing shows that the coordinate UP presents an annual cycle of vertical displacement with peak-to-peak amplitude of 80–100 mm. Negro River is the largest tributary of Amazonas River. The volume of the water of this river, added to the rainfall in Manaus region, results in an average change in water level of 10 m. The behavior of the GRACE data along the Solimo˜es and Amazonas Rivers was analyzed and it shows the wet and the dry season from 2005 to 2008 for May (wet) and October (dry) months (Fig. 111.6). The GPS stations in the Solimo˜es and Amazonas rivers are (Figs. 111.1 and 111.5): Tabatinga (1); Sa˜o Paulo de Olivenc¸a (2); Santo Antoˆnio de Ic¸a (3); Fonte Boa (4); C.Misso˜es (5); Itapeua´ (6); Manacapuru (7); P.Trapiche 15 ´ bidos (10); Santare´m (Manaus) (8); Parintins (9); O (11); Porto de Moz (12); Porto de Santana (13). The year 2005 was one of the driest in Amazon history and this was verified by GRACE data (solid line). Since then, rainfall is increasing in Manaus (Fig. 111.5). In
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890 Fig. 111.5 Comparison between precipitation and ANA
Fig. 111.6 GRACE behavior along Solimo˜es and Amazonas Rivers
2008, GRACE detected the maximum EHW along the rivers, especially in stations 5, 6, 7 and 8, the last one is located in Manaus (Fig. 111.6). This fact is due to the huge volume of the water concerning the Solimo˜es and Negro Rivers confluence. Conclusions
At Manaus station (number 1499000) a good correlation between GRACE-based and in-situ observations can be observed. On the other hand, the GPS height shows an inverse phase correlation between GRACE-based and in-situ observations. This implies that the crust responds instantaneously to the hydrological loading cycle. This can be observed in Figs. 111.3 and 111.4. PPP proved to be an efficient tool in the GPS processing. The GPS
height shows the vertical crust displacements due to the huge variation and amount of the water in Manaus. The Amazon basin is an ideal place to study the crust displacement associated with insitu water level, GPS and GRACE data, once has a high hydrological potential due to the variation in the volume of water in the dry and the wet season in Amazon basin and especially in Manaus due to the Negro and Solimo˜es River confluence. Further investigation, involving GPS stations, in-situ and GRACE data, should be carried out in order to verify the crust displacement throughout Amazon basin. Acknowledgements ANA agency for the hydrometric stations data, CNPq and Capes/Cofecub for supporting this research, Geopotencial Models from GRGS.
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References ANA – AGEˆNCIA NACIONAL DE AGUAS (2006) Bacı´a do Rio Amazonas: Informac¸o˜es sobre a bacia . Available:
. Accessed:22 Feb 2007 Almeida FGV (2009) Variac¸a˜o temporal do campo gravitacional detectada pelo sate´lite GRACE: Aplicac¸a˜o na bacia Amazoˆnica. PhD thesis – Escola Polite´cnica, Universidade de Sa˜o Paulo, Sa˜o Paulo, 146 p. Available online at: http://thesesups.ups-tlse. fr/583/1/Vaz-De-Almeida_Flavio-Guilherme.pdf Bevis M, Alsdorf D, Kendrick E, Fortes LP, Forsberg B, Smalley R, and Becker J (2005) Seasonal fluctuations in the mass of the Amazon River system and Earth’s elastic response; Geophys Res Lett, 32:L16308, doi:10.1029/ 2005GL023491 Biancale R, Lemoine J-M, Balmino G, Loyer S, Bruisma S, Pe´rosanz F, Marty J-C, and Ge´gout P (2006). 3 years of geoid variations from GRACE and LAGEOS data at 10-day intervals from July 2002 to March 2005, CNES/GRGS products Blewitt G, Lavalle´e D, Clark P, Nurutdinov K (2001) A new global mode of Earth deformation: Seasonal cycle detected. Science 294:2342–2345 Blitzkow D, Matos ACOC, Campos IO, Fonseca Jr, ES, Almeida FGV, Barbosa ACB (2010) Water Level Temporal Variation Analysis at Solimo˜es and Amazonas. In: Michael Sideres (ed) Gravity, Geoid and Earth Observation. Berlin: Springer Berlin Heidelberg 135:533–538 Cutrim EMC, Martin DW, Butzow DG, Silva IM, Yulaeva E (2000) Pilot analysis of hourly rainfall in Central and Eastern Amazonia. J Climate 13(7):1326–1334 D€oll P, Kaspar F, Lehner B (2003) A global hydrological model for deriving water availability indicators: model tuning and validation. J Hydrol 270:105–134
891 Heki K (2001) Seasonal modulation of interseismic strain build up in Northeastern Japan driven by snow loads. Science 293:89–92 Marengo JA (1992) Interannual variability of surface climate in the Amazon basin. Int J Climatol 12:853–863 Marengo JA (2004) Interdecadal variability and trends of rainfall across the Amazon basin. Theor Appl Climatol 78:79–96 Marengo JA, Nobre CA, Tomasella J, Oyama MD, Oliveira GS, Oliveira R, Camargo H, Alves LM, Brown IF (2008) The drought of Amazonia in 2005. J Climate 21:495–516 Ramillien G, Cazenave A, Brunau O (2004) Global timevariations of hydrological signals from GRACE satellite gravimetry. Geophys J Int 158:813–826 Ramillien G, Frappart F, Cazenave A, G€untner A (2005) Time variations of the land water storage from an inversion of 2 years of GRACE geoids. Earth Planet Sci Lett 235:283–301 Ramillien G, Lombard A, Cazenave A, Ivins ER, Llubes M, Remy F, Biancale R (2006) Interannual variations of the mass balance of the Antarctica and Greenland ice sheets from GRACE. Glob Planet Change 53:198–208 Ronchail J, Cochonneau G, Molinier M, Guyot J-L, Chaves AGM, Guimara˜es V, Oliveira E (2002) Interannual rainfall variability in the Amazon basin and sea-surface temperatures in the equatorial Pacific and the tropical Atlantic Oceans. Int J Climatol 22:1663–1686 Tucci CEM (1993) “Controle de enchentes”. In: Tucci CEM (ed) Hidrologia: cieˆncia e aplicac¸a˜o. Porto Alegre: UFRGS, Cap. 16, pp 621–658 Uvo CRB, Repelli CA, Zebiak SE, Kushinir Y (1998) The relationship between tropical Pacific and Atlantic SST and northeast Brazil monthly precipitation. J Climate 11:551–562 Wahr J, Swenson S, Velicogna I (2006) Accuracy of GRACE mass estimates. Geophys Res Lett 33:L06401. doi:10.1029/ 2005GL025305
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The Progress of the Geoid Model for South America Under GRACE and EGM2008
112
D. Blitzkow, A.C.O.C. de Matos, J.D. Fairhead, M.C. Pacino, M.C.B. Lobianco, and I.O. Campos
Abstract
The efforts to compute the geoid model for South America, limited by 15 N and 57 S in latitude and 30 W and 95 W in longitude, are presented. The terrestrial gravity data for the continent have been updated with the most recent measurements in Argentina, Brazil, Chile, Ecuador and Paraguay. An attention was also addressed to DTM, on the basis of SRTM (Shuttle Radar Topography Mission). The complete Bouguer gravity anomaly; the direct, secondary and primary indirect topographic effects; and direct and primary indirect atmospheric effects have been derived through the Canadian package SHGEO (Stokes-Helmert Geoid software). The short wavelength component was estimated via FFT with Featherstone, Evans and Olliver (1998) modified kernel. The geopotential model EGM2008 represents an important contribution to the long and medium wavelength component knowledge of the gravitational field and it has been used as a reference field restricted to degree and order 150. The model has been validated over 1,411 GPS observations on Bench Marks of the spirit leveling network, where the geoidal height was derived from the association of the geodetic and the orthometric heights.
D. Blitzkow (*) A.C.O.C. de Matos Laboratory of Topography and Geodesy, Department of Transportation, University of Sa˜o Paulo, EPUSP-PTR, Postal Code 61548, CEP, 05424-970 Sa˜o Paulo, Brazil e-mail: [email protected] J.D. Fairhead GETECH & University of Leeds, Kitson House, Elmete Hall, Elmete Lane LS8 2LJ, UK M.C. Pacino Facultad de Ciencias Exactas, Ingenieria y Agrimensura (UNR), Av. Pellegrini 250, (S2000BTP) Rosario – Rep, Argentina M.C.B. Lobianco Geodesy Coordination – Brazilian Institute of Geography and Statistics (IBGE), Av. Brasil, 15671, Parada de Lucas, CEP, 21241-051 Rio de Janeiro, Brazil I.O. Campos Faculty of Civil Engineering – Federal University of ´ vila, 2160 Campus Santa Uberlaˆndia, Av. Joa˜o Naves de A Moˆnica, Uberlaˆndia, Brazil S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_112, # Springer-Verlag Berlin Heidelberg 2012
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The height anomaly derived from EGM2008 (degree and order 2,159 with additional coefficients to degree 2,190 and order 2,159), EIGEN_05c (degree and order 360) and MAPGEO2004 (official geoid model in Brazil since 2004) have also been compared with the GPS points.
112.1 Introduction A new version of the geoid model for South America (Geoid09) was computed, limited by 15 N and 57 S in latitude and 30 W and 95 W in longitude. EGM2008 up to degree and order 150 as the reference field was used. The reduced Helmert mean gravity anomalies were estimated in blocks of 50 for continental area. Trident satellite altimetry model (Fairhead et al. 2009) was used for the Ocean. The digital terrain model SAM3s_v2 (Blitzkow et al. 2008), with a grid size of 300 300 (~90 90 m), was chosen for computing the related quantities. The processing of the modified Stokes integral was carried out using FFT, as mentioned. The new geoid is an advanced version with respect to MAPGEO2004 (IBGE 2004; Lobianco et al. 2005).
geoid model used more terrestrial gravity data than EGM08, principally areas in Argentina, Midwest and Southeast in Brazil, Southern Chile, central part of Ecuador, and Paraguay (Chaco).
112.2.2 Geopotential Models SHGEO solution of the Stokes boundary value problem employs a modified Stokes’s formula in conjunction with the low-degree contribution of a Global Geopotential Model (GGM). The new model EGM2008 was used. This GGM was released by Earth Gravitational Model (EGM) development team (Pavlis et al. 2008). This model is completed to degree and order 2,159 and contains additional spherical harmonic coefficients extending to degree 2,190 and order 2,159.
112.2 Data Set An important effort to improve the necessary data for this new version of the geoid was undertaken. These data set are described in the sequel.
112.2.1 Terrestrial Gravity Data Many different organizations, universities and research institutes in South America addressed their attention to the improvement on geoid data in the last few years. It is important to mention: 1. (Brazilian Institute of Geography and Statistics) 2. NGA (National Geospatial-Intelligence Agency) 3. GETECH Group plc 4. Civil and military institutions in several countries of South America Due to the big efforts undertaken there are available at the moment approximately 924,600 point gravity data in the continent, including Central America (Fig. 112.1). Due to the mentioned efforts the present
Fig. 112.1 Gravity data in South America
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The Progress of the Geoid Model for South America Under GRACE and EGM2008
EIGEN-05 C was also used for comparison and validation. It is an outcome of the joint data processing activities at GFZ Potsdam and GRGS Toulouse. It is a combination of GRACE (Gravity Recovery and Climate Experiment) and LAGEOS satellite missions plus 0.5o 0.5o grid of gravimetry and altimetry surface data and it is complete to degree and order 360 (F€orste et al. 2006).
112.2.3 Digital Terrain Model For the present study, a suitable gridded topography data with a grid size of 300 300 (approximately 90 90 m) from SAM3s_v2 was available (Blitzkow et al. 2008). This model consists of SRTM3 (Farr et al. 2007), but EGM96 (Lemoine et al. 1998a, b) height anomalies used in the SRTM3 was substituted by EIGEN-GL04C (F€ orste et al. 2006); in order to derive the orthometric height. The gaps were substituted by combination of digitising maps and DTM2002 (Saleh and Pavlis 2002; Blitzkow et al. 2007).
112.2.4 TRIDENT The three satellite gravity solutions: GETECH (Fairhead et al. 2004), Sandwell and Smith v16 (Sandwell and Smith 2009) and Danish National Space Centre (DNSC08, Andersen et al. 2008) have been shown by Fairhead et al. (2009) to be of comparable resolution. Since each solution has been generated from independent retracking (repicking the radar onset time), corrections (such as transmission, tidal and ocean current, etc.) and from the gravity field using ‘gradients to gravity’ and ‘geoid to gravity’ methods, then differences between the methods can be reasonably considered to result mainly from random rather than systematic error. Trident, the new satellite gravity model for the oceans, thus uses this random noise assumption and has stacked (or averaged) the grid. The resulting Trident grid, when compared with higher resolution marine gravity data (ground truth data test), shows a significant improvement. Individual solutions at 50% coherence give full wavelengths of 14–15 km whereas the Trident gives 13 km and the RMS for individual solutions is between 2.2 and 2.6 mGal whereas the Trident is 2.0 mGal. The amplitude is
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more difficult parameter to define due to the dependence of the wavelength (see Fairhead et al. 2009).
112.2.5 Control Points GPS observations carried out on benchmarks of the spirit levelling network (GPS/BM) for South America have been delivered under SIRGAS (Geocentric Reference System for Americas) project and contributed for testing the gravimetric determination of the geoid as well as the GGMs. A total of 1,411 GPS points were available.
112.3 Geoid Computations The SHGEO precise geoid determination software was employed to compute the geoid model. This package has been developed under the leadership of Prof. Petr Vanı´cˇek at the Department of Geodesy and Geomatics Engineering, University of New Brunswick (UNB), Canada. The software (Ellmann 2005a, b) uses StokesHelmert method. The gravity anomalies over the entire Earth are required for the geoid determination by the original Stokes formula. In practice, the area of integration is limited to some domain around the computation point, usually circular. The Stokes equation used to compute the geoidal heights (Ellmann and Vanı´cˇek 2007) is NðOÞ ¼
R 4pg0 ðfÞ
ðð Oc0
SM ðc0 ; cðO; O0 ÞÞDgðrt ; OÞdO0
þ
M R X 2 Dgh ðrg ; OÞ 2g0 ðfÞ n¼2 n 1 n
þ
dV t ðrg ; OÞ dV a ðrg ; OÞ þ g0 ðfÞ g0 ðfÞ (112.1)
The geocentric position (r, O) of any point can be represented by the geocentric radius r and a pair of geocentric coordinates O ¼ (f,l), where f and l are the geocentric spherical coordinates; R is the mean radius of the Earth. The modified Stokes kernel SM(c0,c(O,O0 )) was computed according to Featherstone and Sideris (1998), where c(O,O0 ) is the geocentric angle between the computation and the integration
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Mean 1.22 0.30 0.02 0.82 0.01 1.23
RMS 35.87 3.35 0.04 0.04 0.05 35.50
Max. 544.88 105.25 0.004 0.61 0.06 481.49
Min. 362.39 87.34 0.36 0.84 0.77 363.22
Table 112.2 Statistic analysis of the PIAE, PITE and new geoid’s geoidal heights Meter PIAE PITE Geoid09
Fig.112.2 Gravity anomaly referred to the geoid surface
points; dO0 is the area element of integration. That modification is Meissl (1971) applied to Vanı´cˇek and Kleusberg (1987) kernel. Meissl’s modification subtracts the value of the spherical Stokes kernel at the truncation radius c0, making the Fourier series of the truncation bias to converge to zero faster. The Vanı´cˇek and Kleusberg approach works to minimize the truncation bias, i.e., uses the low-frequency part of the geoid described by a global geopotential model and a spheroid of degree M as a new reference surface (Vanı´cˇek and Sj€ oberg 1991). The Featherstone and Sideris (1998) proposal is considered an advance upon the previous deterministic modification because it combines all the perceived advantages of each one (Lobianco et al. 2005). The upper limit (M) for the modified Stokes kernel used in this case was set up to 150. This option showed good agreement with GPS/levelling data. In the right-hand side of (112.1), the first term is the Helmert residual co-geoid. The long-wavelength contribution to the geoidal height (Heiskanen and Moritz 1967), i.e., the reference spheroid, must be added to the residual geoid (the second term). The sum of the first and second terms results in the Helmert co-geoid. The third term on the right-hand side of (112.1) is the primary indirect topographical effect (PITE; Martinec 1993) and the last term is the primary indirect atmospheric effect on the geoidal heights (PIAE; Nova´k 2000).
Mean 0.006 0.06 1.45
RMS 0.0001 0.18 16.99
Max. 0.005 0.01 48.22
Min. 0.007 1.96 69,11
The term Dg(rt,O), of (112.1) is the Helmert gravity anomaly referred to the Earth’s surface (HGES); it can be obtained by (Vanı´cˇek et al. 1999). Dgðrt ; OÞ ¼ Dgðrt ; OÞ þ dAt ðrt ; OÞ 2 dV t ðrt ; OÞ þ dAa ðrt ; OÞ þ rt ðOÞ (112.2) The first term on the right-hand side of (112.2) is free-air anomaly; the second and third terms are the direct topographic effect (DTE) and secondary indirect topographic effect (SITE). The last term is the direct atmospheric effect (DAE). The 50 50 grid of the mean free–air gravity anomalies was derived from point gravity data. Over the ocean, Trident with resolution 10 10 was used. The respective anomalies are represented in Fig. 112.2. The white areas are gaps with respect to gravity terrestrial information. Tables 112.1 and 112.2 show the results of the statistics analysis carried out with the new model and different informations. The SITE, DAE geoidquasigeoid and PIAE have small contribution. The computed geoid model is presented in Fig. 112.3.
112.4 Geoid Validation The GPS/BM contributed for testing the gravimetric determination of the geoid as well as Global Geopotential Models (GGMs). A total of 1,411 GPS/
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The Progress of the Geoid Model for South America Under GRACE and EGM2008
BM points were available in South America. These GPS/BM are in Argentina, Brazil, Chile, Equador,
Fig. 112.3 Model Geoid09
Fig. 112.4 Histograms of discrepancies in South America
Fig.112.5 Histograms of discrepancies in Brazil
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Uruguay and Venezuela. Most points are located in southeastern Brazil. Attention to the following GGMs has been addressed: EIGEN-GL05C (n ¼ m ¼ 360) and EGM2008 (n ¼ m ¼ 2,160). They are tide free model. The zero-degree term was considered null to Geoid09 and GGMs undulations. Figures 112.4 and 112.5 present the histograms of discrepancies among GPS/BM and Geoid09 geoidal heights, height anomalies of the GGMs in South America and in Brazil, respectively. The statistics analysis of South America is in Table 112.3 and the discrepancies between GPS/BM and Geoid09 are presented in Fig. 112.6. The RMS shows that EGM2008 is a slightly better than the Geoid09. Another comparison has been carried out for Brazil and the official geoid model for the country, MAPGEO2004, was used. The zero-degree term of 0.5 m was applied to this model. The related statistic analysis is shown at Table 112.4 and histograms of the discrepancies at Fig. 112.5. There are similar values for three of the models with worse value for MAPGEO2004.
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Table 112.3 Statistic analysis with GPS/BM in South America GPS/BM Geoid09 EIGEN_05C (360) EGM08 (2,160)
Mean 0.16 0.26 0.19
RMS 0.74 0.68 0.64
Max. 2.98 2.87 3.32
Min. 2.83 3.38 3.64
Acknowledgements The authors acknowledge Prof. Dr. Artur Ellmann (Tallinn University of Technology), Prof. Dr. Peter Vanı´cˇek and Prof Dr. Marcelo Carvalho dos Santos (University of New Brunswick) for SHGEO package; Foundation of the State of Sa˜o Paulo (FAPESP) for supporting the Thematic Project; NGA and the Civil and Military organizations in the South America (Argentina, Brazil, Chile, Colombia, Ecuador, Paraguay, Uruguay and Venezuela) for the efforts to provide data. The Activity has been partially undertaken with the financial support of Government of Canada provided through the Canadian International Development Agency (CIDA).
References
Fig. 112.6 Discrepancies between GPS/BM and Geoid09 Table 112.4 Statistic analysis with GPS/BM in Brazil GPS/BM Geoid09 EIGEN_05C (360) EGM08 (2,160) MAPGEO2004
Mean 0.35 0.37 0.27 0.51
RMS 0.62 0.69 0.60 0.84
Max. 2.98 2.87 3.11 3.97
Min. 2.83 3.38 3.64 4.13
Conclusion
The GGMs with maximum order and degree show small differences with respect to Geoid09. The highest differences are in points near or on high mountains. Geoid09 is better than MAPGEO2004 for Brazil and is similar to EGM08 in terms of RMS differences. The Geoid09 has approximately 12,000 more gravity terrestrial data information than EGM08. Despite of the efforts undertaken by different organizations, universities and research institutes in recent years to fill in the areas without terrestrial gravity data, there are still large gaps in South America. There are always inconsistencies in these areas.
Andersen OB, Knudsen P (1998) Global marine gravity field from the ERS-1 and GEOSAT geodetic mission altimetry. J Geophys Res 103(C4):8129–8137 Andersen et al. (2008) The DNSC08 ocean wide altimetry derived gravity field. Presented EGU-2008, Vienna, Austria, April Blitzkow D, Matos ACOC, Campos IO, Ellmann A, Vanı´cˇek P, Santos, MC (2007) An attempt for an Amazon geoid model using Helmert gravity anomaly. Earth: our changing planet,, IAG General Assembly at IUGG XXIV 2007, Perugia Italia, July 2–13 Blitzkow D, Pacino MC, Matos ACOC (2008) Activities in south America: gravity and geoid projects. Available online at: http://www.sirgas.org/fileadmin/docs/Boletines/Bol13/34_ Activities_in_South_America_Gravity_and_Geoid_Pacino_ Blitzkow.pdf. Model available online at: http://www.ptr. poli.usp.br/ltg/proj/proj26.htm Ellmann A (2005a) SHGEO software packages-an UNB application to Stokes-Helmert approach for precise geoid computation, reference manual I, p 36 Ellmann A (2005b) SHGEO software packages-an UNB application to Stokes-Helmert approach for precise geoid computation, reference manual II, p 43 Ellmann A, Vanı´cˇek P (2007) UNB application of StokesHelmert’s approach to geoid computation. J Geodyn 43:200–213 Fairhead JD, Green CM and Fletcher KMU, (2004) Hydrocarbon screening of the deep continentalmargins using nonseismic methods. First Break, v22, pp 59–63 Fairhead JD, Williams SE, Fletcher KMU, Green CM, Vincent K (2009) Trident – a new satellite gravity model for the oceans. Extended Abstract 6039, 71st EAGE Conference & Exhibition — Amsterdam, The Netherlands, 8–11 June 2009 Farr TG, Rosen PA, Caro E, Crippen R, Duren R, Hensley S, Kobrick M, Paller M, Rodriguez E, Roth L, Seal D, Shaffer S, Shimada J, Umland J, Werner M, Oskin M, Burbank D, Alsdorf D (2007) The shuttle radar topography mission. Rev Geophys, 45, RG2004, doi:10.1029/2005RG000183 Featherstone WE (2003) Software for computing five existing types of deterministically modified integration kernel for gravimetric geoid determination. Computer Geosci, 29:183–193. Software available at: http://www.iamg.org/CGEditor/index. html
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Featherstone WE, Sideris MG (1998) Modified kernels in spectral geoid determination: first results from Western Australia. In: Forsberg R, Feissl M and Dietrich R (eds). Geodesy on the move: gravity, geoids, geodynamics, and Antarctica, Springer, Berlin. International Association of Geodesy Symposia, vol 119,188–193 F€orste Ch, Flechtner F, Schmidt R, K€ onig R, Meyer U, Stubenvoll R, Rothacher M, Barthelmes F, Neumayer H, Biancale R, Bruinsma S, Lemoine JM, Loyer S (2006) A mean global gravity field model from the combination of satellite mission and altimetry/gravimetry surface data – EIGEN-GL04C. Geophys Res Abstr 8:03462 Heiskanen WA, Moritz H (1967) Physical geodesy. W.H. Freeman, San Francisco, p 364 Instituto Brasileiro de Geografia e Estatı´stica (2004) Modelo de ondulac¸a˜o geoidal – programa MAPGEO2004. IBGE, Available online at: http://www.ibge.gov.br Jana´k J, Vanı´cˇek P (2005) Mean free-air gravity anomalies in the mountains. Studia Geophysica et Geodaetica 49:31–42 Lemoine FG, Pavlis NK, Kenyon SC, Rapp RH, Pavlis EC, and Chao BF (1998a) New high-resolution model developed for Earth gravitational field, EOS, Transactions, AGU, 79, 9, March 3, No 113, 117–118 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 (1998b) The development of the joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) geopotential model EGM96, NASA/TP-1998-206861. National Aeronautics and Space Administration, Maryland, USA Lobianco MCB, Blitzkow ACOC, Matos (2005) O novo modelo geoidal para o Brasil, IV Colo´quio Brasileiro de Cieˆncias Geode´sicas,Curitiba, 16 a 20 de maio de 2005. Available online at: ftp://geoftp.ibge.gov.br/documentos/geodesia/ artigos/2005_O_novo_Modelo_Geoidal_para_o_Brasil.pdf Martinec Z (1993) Effect of lateral density variations of topographical masses in view of improving geoid model
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accuracy over Canada. Final Report of the contract DSS No. 23244-2-4356. Geodetic Survey of Canada, Ottawa Meissl P (1971) Preparations for the numerical evaluation of second-order Molodensky-type formulas, Report 163, Department of geodetic Science and Surveying, Ohio State university, Columbus, USA, p 72 Nova´k P (2000) Evaluation of gravity data for the Stokes–Helmert solution to the geodetic boundary-value problem. Technical Report No. 207, Department of Geodesy and Geomatics Engineering, University of New Brunswick, Fredericton Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2008) An Earth Gravitational Model to Degree 2160: EGM2008- Presentation given at the 2008 European Geosciences Union General Assembly held in Vienna, Austria, April13-18, 2008. Available in :http://earth-info.nga.mil/GandG/wgs84/gravitymod/ egm2008 Saleh J, and Pavlis NK (2002) The development and evaluation of the global digital terrain model DTM2002, 3 rd Meeting of the International Gravity and Geoid Commission, Thessaloniki, Greece Sandwell DT, Smith WHF (2009) Global marine gravity from retracked Geosat and ERS-1 altimetry: Ridge segmentation versus spreading rate. J Geophys Res 114:B01411. doi: 10.1029/2008JB006008 Vanı´cˇek P, Kleusberg A (1987) The Canadian geoid-Stokesian approach. Manuscripta Geodaetica 12(2):86–98 Vanı´cˇek P, Sj€oberg LE (1991) Reformulation of Stokes’s theory for higher than second-degree reference field and modification of integration kernels. J Geophys Res 96 (B4):6339–6529 Vanı´cˇek P, Huang J, Nova´k P, Pagiatakis SD, Ve´ronneau M, Martinec Z, Featherstone WE (1999) Determination of the boundary values for the Stokes–Helmert problem. J Geodesy 73:160–192
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Combining High Resolution Global Geopotential and Terrain Models to Increase National and Regional Geoid Determinations, Maracaibo Lake and Venezuelan Andes Case Study
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E. Wildermann, G. Royero, L. Bacaicoa, V. Cioce, G. Acun˜a, H. Codallo, J. Leo´n, M. Barrios, and M. Hoyer Abstract
The combination of global geopotential earth models with high resolution digital terrain models forms the main tool for geoid modelling and satisfies the requirements at a wide range of gravity field frequencies. The need for local and regional data can be reduced to minor areas around points of interest. The resolution gap between potential models and digital terrain models is narrowing – EGM08 and SRTM, for instance, going down to 50 - and 300 -spacing, respectively. The influence of regional gravity data or other field components is reduced to a limited area zone. This means a significant advantage particularly in the case of developing countries which normally demonstrate heterogeneous data quality, bad geographical distribution or complete lack of local data sources at all. Using as test area the Maracaibo Lake and the Venezuelan Andes region – an area with a 4,000 m height variation and some major height changes caused by natural or human based environmental hazards–, a recent GPS project dedicated to precise satellite observations over points of the first order leveling network in the area was performed. Applying most rigorous processing techniques, ellipsoidal height determinations at the 1–3 cm level were achieved at the 2008 campaign. Agreement between GPS/Leveling measurements and modeled geoid values improved. Comparing the new data with GPS observations dating from 1993 to 1998 – which formed a crucial part of the previous national geoid determination –, showed the limitations of the applied technique, partly explainable by height changes occurring in the area. An increasing epoch difference between new data sets and the original observations of the leveling network contributes in the differences, too.
113.1 Introduction
E. Wildermann (*) G. Royero L. Bacaicoa V. Cioce G. Acun˜a H. Codallo J. Leo´n M. Barrios M. Hoyer Laboratorio de Geodesia Fı´sica y Satelital, Universidad del Zulia, Maracaibo 4005, Venezuela e-mail: [email protected]
The decomposition of the Earth’s gravity field in the frequency domain has been extensively used for geoid determination purposes. One of the main applications for field description is its development by means of spherical harmonics, expressing the spectral long and medium band wavelength by Earth Gravity Models
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_113, # Springer-Verlag Berlin Heidelberg 2012
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(EGM), see Torge (2001). On the other end of the spectra, the high frequency part, caused mainly by nearby topography, can be evaluated using Newton’s Law of Attraction applying high (and very high) resolution Digital Elevation Models (DEM) to determine quite efficiently near point mass influences (Forsberg 1998). The low-frequency part of the gravity field spectrum has been improved significantly during the last decade. EGM96, introduced some 15 years ago, accurately resolved lower band frequencies for most regional purposes. New developments, mostly induced by the new satellite missions CHAMP and GRACE, have improved components up to harmonic degree l ¼ 150, and in combination with other data sources (mainly altimetry and terrestrial observations) various earth gravity models have been published (up to degree l ¼ 360) during the past years (F€ orste et al. 2008). A next step will be the incorporation of the GOCE observation data (GOCE-ESA 2008). Collecting new data at higher resolution with improvements at both observation and combination techniques led to the EGM08 development with harmonic series complete to degree and order of l ¼ 2,159 (Pavlis et al. 2008) resolving globally spectral components up to 50 (10 km) resolution. Continental DEM datasets increased rapidly in resolution from some 50 of ETOPO5, almost, a decade ago to 300 of SRTM (Rodriguez et al. 2005), and more recently, 100 of ASTER-GDEM (NASA/METI 2009), available globally for scientific purposes. At ocean areas the spatial resolution of available digital bathymetry models is somewhat lower, but satellite altimetry and combination processes have been providing new more detailed bathymetry, as can be seen in the latest developments like SRTM30+ (Becker and Sandwell 2007), DNSC08 (Andersen and Knudsen 2008) or GEBCO08 (GEBCO 2009), globally providing mean depth data with 3000 pixel length. All these advances, in combination with a nowadays available high efficiency of computer resources, allow strict computation of the near terrain effect at a very high resolution level – in spectral terms, these high frequencies can easily be considered as corresponding to degree values l 40,000. After some experimental procedures developed between 1985 and 1995, combined geoid determination efforts in Venezuela started with analyzing impact of available data sources. To homogenize and unify
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several data sources, nationwide GPS stations have been implemented at first order leveling points. An intense cooperation between governmental organizations and private institutes, under the leadership of Laboratorio de Geodesia Fı´sica y Satelital (LGFS), Zulia University, Maracaibo, occupied nearly 300 leveling points distributed nationwide covering nearly the entire existing leveling networks. Separation of points was at the order of 50 km. Processing of GPS data had been made by rigorous application of scientific software package Bernese to obtain high quality ellipsoidal height information on the occupied bench marks (Lopez 2005). Combinations with other data sources led to a couple of preliminary national models, such as VGM03 (Acun˜a and Bosch 2003) or MGV04 (Hoyer et al. 2004), which has been adopted in 2004 as the official geoid model for Venezuela by Instituto Geogra´fico Venezolano Simo´n Bolivar, Caracas. Meanwhile, incorporation of new DEM data has been performed leading to VENDEM05, a combined bathymetry/topography model based on altimetry and SRTM, respectively. In this analysis, the influences of new EGMs have been investigated, too (Leo´n and Codallo 2008). In the following, some detailed information about a new geoid08 GPS/Leveling campaign will be outlined. Application of EGM08 shows very clearly the globalregional behavior of the height anomaly field. Topographic reductions are obtained using new bathymetric and terrestrial combined DEM. Some problems of these combinations are presented. The resulting residual anomalies show a much smoother behavior than former solutions. Some problems of combining GPS and leveling measurements are addressed, too.
113.2 geoid08 GPS/Leveling Campaign and Results At September 2008, a GPS observation campaign was performed on 57 leveling benchmarks (BMs) distributed around Maracaibo Lake and the Venezuelan Andes, see Fig. 113.1. They belong to the national first order vertical network and, on the eastern coast of Maracaibo Lake, to a precise subsidence leveling network. Logistics and technical considerations were taken into account in a rigorous way to optimize available resources and to get highly satisfactory results
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Combining High Resolution Global Geopotential and Terrain Models to Increase National 11°N
P er ija M
oun
tain
s
Maracaibo
Maracaibo Lake
Andes Mountains
0mm
73°W
50mm 100mm
8°N 71.5°W
Fig. 113.1 Geographical location of geoid08 study area
especially at the vertical component, to conduct some geoidal studies in the area. Measurements were performed during days 239–244 of 2008, using double frequency geodetic GPS receivers and the permanent Maracaibo (MARA) station, which belongs to the SIRGAS permanent GPS network. According to project goals, the campaign observation parameters defined allowed acquisition of high precision GPS data. Most of the BMs had unfavorable GPS surveying conditions (vegetation coverage, proximity to frequent highway traffic, etc.), so auxiliary observation points had to be established for 47 stations within distances between 4 and 150 m, the mean distance to the center was 30 m. Relative leveling with an accuracy of 2 mm was performed to connect to the nearby BM. Destruction of various selected bench marks needed some in situ changes of the observation plans. Caused by its irregular geometry along main street lines and a quite difficult logistics, the GPS network was divided in six segments, measuring each one during one whole day working session. Interconnection between segments was obtained by using nodal points which were reoccupied during subsequent sessions. Short baselines up to 30 km were established alongside each segment assuming that some common error sources related to the observation technique could be treated easier. For processing the GPS observations the Bernese GPS Software v 5.0 (Dach et al. 2007) was employed, composing three principal steps within the calculations. The first one determined the actual coordinates of station MARA by directly linking it to the nearby IGS stations during the campaign duration. Estimated coordinates have an RMS of 1 mm for all three components. They were then used
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as reference for the following calculations. At the second step, high quality positions for nodal BMs, considered as segment base stations, were determined linking them to MARA, with RMS in the order of 2 mm. Finally, coordinates of the remaining BMs were calculated in radial mode from the internal base stations. The processing strategy allowed a quite relevant reduction of systematic effects in the measurements, although some residual errors are still present associated to high obstruction levels, oscillations due to traffic and huge topographic variations of the campaign segments observed simultaneously. This last problem affected mainly the troposphere correction models. A large number of tests and external control calculations (restringing satellite data, control stations, etc.) have been performed to establish a unified network accuracy at the 2 mm level.
113.3 Application of EGM08 Previous combined geoid determinations in Venezuela were based mainly on EGM96. Considering a submetric accuracy level, the main geoid structures were well described nationwide. At our study region, however, some discrepancies have been found principally at the flanks of the Andes Mountain chain and in the southern part of Maracaibo Lake, with large negative differences at observed GPS/Leveling stations. These results could be partially explained by poor GPS data quality – the observations were made in 1993 under a very difficult logistic and a not quite rigorous computation design had to be used. Physical reasons to explain these large discrepancies could not be found (Lopez 2005). Newer global gravity models showed some better agreement countrywide (Leo´n and Codallo 2008). To show EGM08 regional height anomaly patterns, a grid with 10 spacing has been estimated between longitudes 75 W to 69 W, and latitudes 8 N to 14 N, respectively. The resulting surface is showing height anomaly variations between 37 m, at the areas northeast limit in the Caribbean Sea, and peaks of +10 m at the Santa Marta Mountain and +15 m at the Andes chain near Pamplona, Colombia (Fig. 113.2). The Caribbean Sea part of the area is characterized by always large negative values. Level lines, at 2 m interval, are showing very clearly the Maracaibo Lake
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Fig. 113.2 EGM08 regional behavior (white lines indicating new Venezuelan border and coastline file) Fig. 113.3 Residual height anomalies at GPS/Leveling points after reduction of EGM08 and residual terrain model
5680m 5000m
13°N
2500m
0m
– 4667m
7°N
68°W
– 2500m 74°W
depression around 15 m anomaly level and ascending trends at the flanks on the surrounding mountain chains, Andes and Perija Mountains, raising up to 0 m height anomaly at the Venezuelan Andes in the southern part of the graph. An extreme anomaly gradient can be found at latitude 9 N and longitude 71 300 W, approximately, where a geoid change of nearly 12 m in less than 50 km of distance occurs. To the west of Lake Maracaibo depression a less step raising anomaly gradient can be seen in coincidence with the Perija/Santa Marta Mountains. At the problem zones mentioned previously EGM08 gives changes of height anomaly up to the 1 m level compared to older models used. The EGM08 agrees much better with observed geoid features of our studying zone. At all GPS/Leveling points within the study area, the EGM08 height anomalies have been calculated, too. On the western side of Maracaibo Lake, always negative differences between observed and EGM08 values up to 20 cm have been found (Fig. 113.3). The changes on the eastern coast of the lake are showing a not so clear trend, some values have negative signs up to 10 cm, but most stations give large positive differences. Considering the homogeneity of our GPS height results we suggest some problems with submitted leveling values. Influences by subsidence processes occurring at the area caused by the decade long oil exploration at Maracaibo Lake and shoreline could have produced same changes at the leveled heights (Paris 2008).
Fig. 113.4 Regional structure of digital elevation model GEBCO08-VENDEM05
The southern part of the GPS/Leveling network is clearly influenced by the Andes Mountain chain. EGM08 model “over”-reduces the observations by up to 60 cm at stations situated in the southern Maracaibo Lake basin lowlands. Within the Andes Mountain these tendencies mostly continue – with one exception occurring at our highest observation point (A281) located at nearly 4,100 m of altitude. Here a +10 cm difference has been found, quite astonishing as two other close stations (MUCU, A272) show clearly negative values, as shown in Fig. 113.3.
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Table 113.1 Comparison of heights from geoid08 stations and digital elevation models ASTER and GEBCO08-VENDEM05 DEM comparison HDEM_ASTER with HORTHOMETRIC (m) HDEM_GEBCO08-VENDEM05 with HORTHOMETRIC (m)
Mean 1.797 0.101
RMS 6.939 2.902
Min max 14.396 +12.907 8.272 +6.955
The trend with raising anomaly gradients at the Andes Table 113.2 Residual terrain effects of GEBCO08VENDEM05 on gravity field parameter using a 50 km influence flanks can be seen clearly. radius
113.4 DEM Development and Effects The gravitational behavior at a station is systematically influenced by nearby topographic variations, but this effect can be suitably reduced by using DEM information. DEM resolution in recent years has increased significantly, since SRTM was made available globally on continents with 300 spacing. Since the end of June ASTER-GDEM (NASA/METI 2009) is submitting heights at 100 pixel resolution. At the oceans, using principally altimetry data in combinations with ship soundings, bathymetry data resolution has increased, too, for instance, GEBCO08 gives now globally distributed bathymetry with a 3000 pixel resolution. At Venezuela, the last nationwide development for DEM was VENDEM05 combining principally SRTM with satellite bathymetry. Some filtering technique had been applied to the SRTM data mainly to eliminate cells with invalid data (rivers, lakes, etc.). Here, especially, the Maracaibo Lake had to be considered replacing unreal SRTM oscillations by ocean bathymetry. A new, more detailed coastline polygon has been developed, too, permitting a more precise separation of land and ocean areas. Significant displacements to globally available coastline data at the Venezuelan coast were found (Leo´n and Codallo 2008). The VENDEM05 model showed bathymetry values in the Maracaibo Lake up to 200 m – quite inconsistent with this really mostly shallow Lake Site. They were replaced by combing the original VENDEM05 dataset with GEBCO08. Near the coastline, some positive GEBCO08 values at the ocean and larger negative VENDEM05 values at land sites have been substituted by filtering techniques applied to the original data. Employing a grid filtering methodology implemented in the GMT suite (Wessel and Smith 2005), repeated Gaussian filtering with 1 km length has been applied to replace the data of affected cells and to combine the data sources. The resulting new model shows now a quite smooth transition from land to sea data
Difference H–HDEM (m) dg(magal) dx(00 ) dZ(00 ) dz(m)
Mean 9.00 13.22 0.19 + 0.33 0.03
RMS 5.00 26.37 1.49 1.44 0.05
Min max 17.00 + 8.00 117.27 +19.86 6.61 +4.60 1.43 +9.42 0.18 +0.21
(Fig. 113.4). The Maracaibo Lake results with depth up to 30 m. Applying first tests with ASTER-GDEM height data some surprising results were obtained using the available GPS/Leveling stations (Table 113.1). The mean discrepancy to orthometric heights is 1.797 m, data distribution here is quite smooth following a normal distribution. The standard deviation obtained is 6.9 m. Although spatial resolution of ASTER is 9 times higher than SRTM, the agreement between real height data and SRTM is quite better. The mean difference reduces to only 0.101 m remaining some larger discrepancies at the mountains. Thirtyseven stations show differences under the 2 m level. SRTM seems to represent the region heights much better than ASTER. Some further investigations are under way. Residual terrain model calculation (RTM) has been applied at all the observed GPS/Leveling stations using GRAVSOFT programs (Tscherning et al. 1994). As high resolution grid the described GEBCO08-VENDEM05 combination was used estimating a reference grid with 50 cells, which is in coincidence to EGM08 resolution. Terrain calculations have been extended to 50 km around station positions. Some statistical results are shown at Table 113.2. For height anomalies at the Andes Mountains, some extreme reduction values of up to +20 cm have been found; the most pronounced values occurring at the Andes flanks. At the lowland part south of Maracaibo Lake dz values in the order of 6–10 cm remained, influenced mainly by the mountain chain; to the north of our study zone effects of residual terrain model is decreasing rapidly to 0 m.
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Table 113.3 Residuals at GPS/leveling stations after reduction of EGM08 and RTM Difference NGPSniv – zEGM08 52 values NGPSniv – zEGM08+RTM 52 values
Mean 0.165 0.133
Table 113.4 Gravity anomaly behavior at geoid08 study area Data Mean Data range (mgal) DgOBSERVED ¼ 15427 (within area 08 N-11 N; 73 W-70 W) Dg10 10 26.439 156.579 +252.787 DgEGM08 25.857 156.027 +353.686 3.728 141.964 + 53.186 DgRTM;10 10 DgEGM08+DgRTM 29.585 158.984 +287.601 DgRESIDUAL +3.146 142.951 +128.196
Applying the combined EGM08- and RTMmodeling, differences to GPS/Leveling results decrease. At the Andes region differences at the 40 cm level remained, in the northern part some 10–20 cm discrepancies are found. The major part is showing negative sign. At the eastern zone of Maracaibo Lake the changing sign discrepancies between modeling and GPS/Leveling continued at the mentioned level. This had to be expected considering that these stations are situated at lowlands with only minor topographic variations in the area and a relative smooth EGM08 behavior (Fig. 113.3 and Table 113.3). Some statistical testing for gross-error, normal distribution, skewness and excess showed no significance. The residuals are showing a quite smooth uniform behavior over the whole data range
113.5 Some Final Discussions Until now, a collocation solution of residuals to fill-in the data gap between geopotential and terrain model has not been executed. Our main reason is the lack of data which could give us more information for frequencies between the global gravity model (50 or approximately 10 km) and the RTM (300 or 90 m). Only an approximate 12.5% of the area is filled by 10 mean gravity values. Looking for available data at our study area, some geophysical induced gravity field campaigns had been found, but only at areas of interest for petroleum exploration (near Maracaibo Lake). At the mountainous parts, gravity data distribution is quite sparse, only
RMS 0.201 0.174
Min max 0.597 +0.196 0.449 +0.189
very few stations situated at intervals of up to 30 km distance alongside mountain values; no data at all was found at extended areas in the high Andes and Perija mountains. The lowlands also show lack of uniform data distribution, or a completely absence in the case of the Venezuelan-Colombian border region. Using mean 10 gravity anomaly cells, most of region shows lack of available data, Table 113.4 resumes characteristics for this dataset. Most of the high differences after modeling occurring at former GPS/Leveling data in the southern part around Maracaibo Lake were reduced showing now a pronounced uniform regional trend. The GPS/Leveling points arose some doubts, too. A major problem seemed to be the large epoch difference between the recent satellite observations and the leveling of first order vertical network – sometimes dating from the 60ties and 70ties of last century. A lot of original marks are destroyed incrementing station separation to nearly 30 km and more. A significant number of points encountered is situated at road bridges and could be influenced by continuous traffic vibrations and lack of maintenance work; others are found near or beneath fast growing tropical forest canopy. Height changes caused by subsidence, too, could influence especially the stations situated at the eastern coast of Maracaibo Lake. Investigations of undergoing reasons are in progress. The geoid08 GPS/Leveling observation source will be incorporated into the new regional and national geoid determination by collocation and FFT techniques, in combination with the construction and application of a verified national gravity data base as a fundamental input-set to fill the gaps between the local and global spectra parts. Acknowledgments The authors would like to thank the LGFS students group for their remarkable efforts at data acquisition and processing phases. Staff and logistic help of Petroleos de Venezuela SA Geodetic Group; Puerto La Cruz, at geoid08 observation campaign is gratefully acknowledged. The helpful comments and suggestions of two anonymous reviewers are gratefully appreciated. Parts of the study were funded by CONDES, Universidad del Zulia, Maracaibo.
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References Acun˜a G, Bosch W (2003) Combining terrestrial marine and satellite gravity for geoid modeling in Venezuela. IUGG General Assembly 2003, Saporro, Japan Andersen OB, Knudsen P (2008) The DNSC08BAT Bathymetry developed from satellite altimetry. DTU-SPACE, ftp://ftp.spacecenter.dk/pub/BATHYMETRY Becker JJ, Sandwell DT (2007). SRTM30_PLUS: SRTM30 coastal & ridge multibeam estimated topography. http://topex. ucsd.edu/WWW_html/srtm30_plus.html Dach R, Hugentobler U, Fridez P, Meindl M (2007) Bernese GPS Software Version 5.0. Astronomical Institute, University of Bern, Bern Forsberg R (1998) Topographic corrections in geoid determinations. International Geoid School, Rio de Janeiro, Brasil F€orste C, Flechtner F, Schmid R, Stubenvoll R, Rothacher M, Kusche J, Neumayer H, Biancale R, Lemoine JM, Barthelmes F, Bruinsma S, K€ onig R, Meyer U (2008) A new global combined high-resolution GRACE-based gravity field model of the GFZ-GRGS cooperation. EGU General Assembly, 13–18 April 2008. Vienna, Austria GEBCO (2009) The general bathymetric chart of the oceans: The GEBCO_08 Grid, version 20081212. http://www.gebco.net GOCE-ESA (2008) Gravity field and steady-state Ocean Circulation Explorer (GOCE). ESA. Noordwijk, Netherlands. http://www.esa.int/esaLP/LPgoce.html Hoyer M, Wildermann E, Suarez H, Herna´ndez J (2004) Modelo Geoidal Combinado para Venezuela (MGCV04) Interciencia, vol 29,Caracas, pp 660–666
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Leo´n J, Codallo H (2008) Implementacio´n de un modelo geopotencial ma´s preciso y un modelo del terreno de alta resolucio´n en la determinacio´n del geoide Venezolano. Trabajo Especial de Grado, Facultad de Ingenierı´a, Universidad del Zulia. Maracaibo Lopez MG (2005) Procesamiento y ana´lisis de mediciones GPS para la determinacio´n del geoide en Venezuela. Trabajo Especial de Grado, Facultad de Ingenierı´a, Universidad del Zulia. Maracaibo NASA/METI (2009) ASTER Global Digital Elevation Model V001; ASTER GDEM is a product of METI and NASA Paris LC (2008) Aplicacio´n de la altimetrı´a satelital para el estudio de subsidencia en el Lago de Maracaibo. Trabajo Especial de Grado, Facultad de Ingenierı´a, Universidad del Zulia. Maracaibo Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2008) An earth gravitational model to degree 2160: EGM2008. General Assembly of the European Geosciences Union 2008, Vienna, Austria Rodriguez E, Morris CS, Belz JE, Chapin EC, Martin JM, Daffer W, Hensley S (2005) An assessment of the SRTM topographic products – Technical Report JPL D-31639. Jet Propulsion Laboratory. Pasadena, California Torge W (2001) Geodesy. New York, Berlin: de Gruyte Tscherning C, Knudsen P, Forsberg R (1994) Description of the GRAVSOFT package. Geophysical Institute Technical Report. 4th edn, University of Copenhagen Wessel P, Smith WHF (2005) The Generic Mapping Tools. Technical Reference & Cookbook. SOEST/NOAA
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Evaluation of a Few Interpolation Techniques of Gravity Values in the Border Region of Brazil and Argentina
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R.A.D. Pereira, S.R.C. De Freitas, V.G. Ferreira, P.L. Faggion, D.P. dos Santos, R.T. Luz, A.R. Tierra Criollo, and D. Del Cogliano
Abstract
Least Squares Collocation (LSC) and kriging are the most used techniques to predict gravity values as well as gravity anomalies. The limitations of LSC technique are mainly related in obtaining an adequate co-variance function. Moreover, LSC and kriging predictions depend strongly on known data distribution. Artificial Neural Network (ANN) is a promising tool to be applied in the interpolation problems. Even though, far from the deterministic ones, these techniques are presented as alternatives for interpolating due their good adaptation to several data distribution and easy implementation for fusion of different kinds of data basis. To test the performance of ANN in face of interpolation problems with respect to LSC and kriging, an experiment was developed in a region in the Brazil–Argentina border. Interpolated gravity values were obtained by LSC and kriging and compared with values obtained by ANN considering different data distributions and by using the same test points where gravity values are known. Considering the need of consistency of datum for predicting gravity related values, only a Brazilian data set was used in the present analysis. The smallest number of reference data for training and the low dispersion reveals the ANN as an alternative for LSC and kriging techniques for the usual poor gravity data distribution in South America.
114.1 Introduction
R.A.D. Pereira (*) S.R.C. De Freitas V.G. Ferreira P.L. Faggion D.P. dos Santos Centro Polite´cnico da Universidade Federal do Parana´, Jd. Das Ame´ricas – Curitiba, Brazil e-mail: [email protected] R.T. Luz Fundac¸a˜o Instituto Brasileiro de Geografia e Estatı´stica, Rio de Janeiro, Brazil A.R.T. Criollo Escuela Polite´cnica Del Eje´rcito de Equador, Quito, Ecuador D. Del Cogliano Universidad Nacional de La Plata, Paseo S/N Del Bosque, La Plata, Argentina
Due to the impossibility of observing gravity values all over the Earth’s surface, in some cases it is necessary to do a prediction of these values from two or more known data. Prediction of gravity values is very useful in Geodesy, when we use techniques such as those related to the solution of the Geodetic Boundary Value Problems (GBVP) where gridded data are needed; or for geopotential numbers computation along with leveling lines. This problem is present in the South American territory where most of vertical networks have only normal-orthometric corrections
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_114, # Springer-Verlag Berlin Heidelberg 2012
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and the vertical datums are only local ones because the poor distribution of gravity data necessary to an adequate GBVP solution. Interpolation of gravity values is affected by several effects. Most of them are very difficult to model because the need of considering different kinds of data and the fusion of local gravity data with different origin like Digital Topography Models and Global Geopotential Models. Interpolated values from a data set by deterministic or statistical methods in the space domain have some limitations in computations related with inadequate data distribution. The classic technique usually used when predicting the gravity values is the LSC which highly depends on the chosen covariance function and of data distribution. The problem is emphasized when we need to consider different kind of data source. However, with the development of the computing methods, other methodologies can be applied, in some cases, with advantages. We have the Artificial Neural Networks (ANNs) as an example of that. The used network for testing interpolation techniques has 34 benchmarks in the Brazilian side and other additional reference points. The PREDGRAV tool provided by SIRGAS/WG III (Drewes et al. 2002) based on LSC was used. The kriging technique was tested with basis in the SURFER™ package. ANNs were constructed with a radial basis function with distributed training points in the region. Several tests were realized. The best results with LSC points out a RMS of 1.57 mGal but most cases presented limitations regarding the data distribution. The ANN presented in the better case a RMS of 2.39 mGal in similar situation. But the ANNs have less limitation for data distribution, still working with poor data distribution. Also, the ANN allowed incorporating EGM2008 information which improved the prediction capability reducing the RMS to half. Kriging presented, in general, worse results even for the best data distribution.
114.1.1 Data Set There is a Bilateral Project (involving Brazilian and Argentine Institutes and Universities) for the connection between Brazilian and Argentine Fundamental Vertical Networks. This project has main purpose to build a vertical net based on geopotential numbers in
R.A.D. Pereira et al.
the Brazil–Argentina border. However, there are several problems, not discussed here, to put the Brazilian and Argentine data sets in a common basis. Aiming to generate a consistent data basis in the Brazilian–Argentine border region some bi-lateral campaigns were organized in the region of Corrientes and Rio Grande do Sul states in August and December, 2008. Each country has its data basis referred to different datums. In this sense we are considering only the Brazilian data set in this manuscript. The Brazilian gravity reference in the region is the Sa˜o Borja gravity station which is a point of the Brazilian Fundamental Gravity Network (ON 1986). Several gravity observations associated with GPS/RTK positioning were realized in the region, most of them over existing benchmarks. The gravity observations were performed with the LaCoste & Romberg G-372 gravimeter calibrated on the Brazilian Absolute Gravity Network (RENEGA) in 2007 and the SCINTREX CG3 with factory original calibration and which was submitted to static drift determination before each campaign. The GPS positioning was performed with a pair of Leica Geosystems 1,200 dual frequency GPS receivers equipped with RTK system. GPS/RTK positioning was used to improve the velocity of position determination for points until distances of 15 km of reference station. However most of points were processed with basis in static relative technique by using some local reference stations realized with reference in GNSS Continuous Monitoring Brazilian Network (RBMC) part of the SIRGAS network. The obtained mean precisions in position for static-relative positioning were: Horizontal: 3 mm; and Vertical 6 mm. In the RTK positioning the mean precisions were: Horizontal 20 mm; and Vertical 45 mm. In Fig. 114.1 is possible to see the range of heights and gravity anomalies variation at the studied region (Fig. 114.2). The apparent low correlation among free air gravity anomalies and heights points out that the relief information is not fundamental for gravity interpolation. It must be emphasized that the Bouguer anomalies seems to have a distribution of values as rough as the free air anomalies. In general, the gravity data on the whole South America is not adequate because the poor distribution of data. Therefore, the gravity prediction based on all possible related information is still a necessity, mainly because the lack of resources to cover the entire region with gravimetry in a terrestrial conventional form.
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–57
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Santo Tomé-São Borja
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Alvear-Itaqui
Fig. 114.3 Artificial neural network
Leveled Heights and Gravimetry - IGN
–29.5
Lev.Heights, Grav., GNSS- UFPR/IBGE
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GNSS and Gravimetry - UFPR/IBGE Brazilian Roads
Uruguaiana-Paso de Los Libres
BRAZIL SIDE –30
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–56.5
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–55.5
10
100 90 80 70 60 50 40 30 20 10 0 Heights (m)
5 0 –5 –10 –15
Anomalies (mGal)
Height (m)
Fig. 114.1 Gravity information after two observation campaigns in the studied area
–20 –25 Free-Air Anomalies (mGal)
Bouguer Anomalies (mGal)
In Fig. 114.3, W is an initial vector of weight which is adjusted through interactions with values obtained from the transference function f. b is a unit vector which is called bias and which increases the weight vector. The iteration process continues until the weights are adjusted so that we can get the desired values in a certain level of confidence. Several kinds of ANNs can be built by using different instruction routines.
114.2.2 Least Squares Collocation Least Squares Collocation (LSC) is a technique which serves both a prediction and a filtering of data. For example, a formula that uses the least squares collocation concept for prediction gravity anomalies and its details can be found in Tscherning (1974):
Fig. 114.2 Heights, free-air anomalies and Bouguer anomalies for the data set presented in Fig. 114.2
~ ¼ Dg P
n X
ai Dgi ;
(114.1)
i¼1
114.2 Interpolation Techniques 114.2.1 Artificial Neural Networks ANNs are computing instructions used mainly for the classification of groups. The basic idea came up from the development of the perceptron algorithms by McCulloch & Pits (Negnevitsky 2002). The perceptron allows, using only one function, to distinguish two groups which are linearly separable. However, when the groups are not linearly separable, different functions are used to allow such classification. The result of that classification, known as learning, has, as a final result, a matrix which can be used to make a prediction of different data which belong to the same initial group. An ANN (Fig. 114.3) can therefore, be used with latitude and longitude values to obtain a specific result, such as gravity value (Tierra Criollo and de Freitas 2005).
~ is where Dgi are gravity anomalies observed and Dg P the predicted gravity anomaly at point P. In HofmannWellenhof and Moritz (2005) the treatment of (1) can be found in a matrix form:
2
~ ¼ ½ CP1 Dg P
C11 6C 6 21 6 . 6 . 4 . Cn1
CPn 3 3 2 Dg1 C1n 1 6 Dg 7 C2n 7 7 6 27 7 7 6 .. 7 6 .. 7: 4 . 5 . 5 Cnn Dgn
CP2
C12 C22 .. .
.. .
Cn2
(114.2)
Hofmann-Wellenhof and Moritz (2005) consider that “for optimal prediction, we must know the statistical behavior of the gravity anomalies through the covariance function C”. The C functions are obtained, in general, from terms that depend of positions of points, from spherical distance between those points
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and other operators. For each different quantity, there is a different covariance function associated, which makes the LSC a not easy problem to solve. Because of many types of covariance functions developed for different situations, this paper used the PREDGRAV, a LSC tool provided by SIRGAS WGIII.
114.2.3 Kriging Kriging is a prediction process which uses the principle that closer points must presents more similar characteristics than farther away ones. There are many kinds of kriging. This work uses ordinary kriging. According to Trauth (2007), ordinary point kriging uses a weighted average of the neighboring points to estimate the value of an unobserved point: g~P ¼
n X
li gi ;
114.3 Prediction of Gravity Values For training the ANNs, it was used the data configuration shown in Fig. 114.4. These points were the same generated database for kriging and PREDGRAV prediction. Four kinds of ANNs were tested (Cases number 01, 02, 03 and 06). Several other tests can be found in Pereira (2009). In Case 01, latitude, longitude and height were used in the training to obtain gravity values. The architecture is presented in Fig. 114.5, with 3, 4, 3 and 1 neurons in a hyperbolic -tangent sigmoid, in a radial basis function, in a hyperbolic tangent sigmoid and in a linear transfer function, respectively. In Case 02 (Figs. 114.6 and 114.7), latitude and longitude were used. However, heights were not
(114.3)
i
where li are the weights which have to be estimated. The sum of the weights should be one to guarantee that the estimates are unbiased: n X
li ¼ 1:
(114.4)
i
The expected error of the estimation has to be zero, in this way we have: Eðg~P gP Þ ¼ 0;
(114.5)
where gP is the expected true, but unknown value. Fig. 114.5 Case 01 of ANN
Fig. 114.4 Database for the initial prediction 4 Layers
Input a
ϕ1 λ1 h1
+1 0 –1 a = tansig(n)
ϕn λn hn
Fig. 114.6 Case 02 of ANN
a 0.5 0.0
3
–0.833 +0.833 a = radbas(n)
n
+1
0
0
–1 a = tansig(n)
4
a
+1
n
a +1
a = purelin(n)
3
–1 a = tansig(n)
2
a
gn
1
0.5 0.0
–0.833 +0.833 a = radbas(n)
3
Output a
1.0 n
g1
–1
4 Layers
0
ϕ n λn –
a
1.0
Input ϕ 1 λ1
Output
a
+1
+1
0
0
–1 a = tansig(n)
2
g1 n
–1 a = purelin(n)
1
gn
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Evaluation of a Few Interpolation Techniques of Gravity Values in the Border Input
2 Layers a
ϕ1 λ1 h1
+1
+1
0
0
–1 a = tansig(n)
ϕn λn hn
2
Output
a
n
g1
–1 a = purelin(n)
3
gn
Fig. 114.7 Case 03 of ANN
used. The structure is similar to case 01. However, the numbers of neurons in each layer are not the same. For Case 04, it was used PREDGRAV, and in the case number 05, kriging.
Fig. 114.8 Grid from g observe.
114.4 Results As already mentioned, the RMS was calculated considering the local observations built in the campaigns as a reference. The next figures show the predicted grids from the techniques presented. In the all cases of ANN, the trainning’s goal was 0.02. The worst case of ANN reached the desired values after 504 epochs. In the following figures the isolines from gridding must be added by 979,000 mGal. Figure 114.8 shows the grid from local observations of g. The next ones (Figs. 114.9–114.13) shows the results reached. Another possibility offered by ANN is to use quantities derived from global geopotential models without modifications in the original routines. Case 06 related with this approach is showed in Fig. 114.14. Latitude, longitude and heights can be obtained from local frame and the learning can be improved with geoidal heights from EGM 2008 to compute gravity values. The worst Case (03) was used to test this hypothesis becaming Case 06. With the same points of initial database, the RMS computed for the Case 6 was 3.68 mGal, which confirmed the hipotesis. Table 114.1 summarizes the main obtained results in this work: Other interesting situation happen when the geometry and number of database points is changed. It must be emphasized that in PREDGRAV applied for the LSC, it is necessary at least 30 different points to generate the database. For the cases involving ANN it is shown that the number of training points could be a half part for applying the LSC technique. Another
Fig. 114.9 Isolines of predicted grid from ANN Case 01
Fig. 114.10 Isolines of predicted grid from ANN Case 02
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2 Layers a
ϕ1 λ1 h1 N1
5
Output
+1
+1
0
0
–1 a = tansig(n)
ϕn λn hn Nn
a
n
g1
–1 a = purelin(n)
1
gn
Fig. 114.14 Case 06 of ANN with learning based on geoidal heights Table 114.1 RMS of different predictions without restriction of data distribution Fig. 114.11 Isolines of predicted grid from ANN Case 03
Case ANN Case 01 ANN Case 02 ANN Case 03 PREDGRAV (Case 4) Kriging (Case 5) ANN Case 06
RMS (mGal) 4.08 2.39 6.64 1.57 3.10 3.68
Fig. 114.12 Isolines of predicted grid from PREDGRAV (Case 04)
Fig. 114.15 Database of ANN Case 03 where less of 30 points were used in the learning
Fig. 114.13 Isolines of predicted grid from kriging (Case 05)
aspect is that the ANN is less exigent about data distribution. Figure 114.15 shows a case where the database has less than 30 points and the computed RMS was 1.18 mGal with the ANN Case 03. Some these effects could be related to terrain corrections. However, if we calculate that quantity from Digital Elevation Model DTM2006 (ICGEM 2010) (Fig. 114.16), the magnitude of results shows that the differences in prediction data not comes from local effects once that the terrain corrections are, at least, two times smaller.
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In the Brazilian case, because of heterogeneities of height system, again the ANN can be used with advantages, once that gravity anomaly is strongly dependent of point’s height. In spite of positive results about of ANN, it must be considered that these results are referred to the studied region and therefore kriging and LSC concepts applied in other regions can furnish different results.
Fig. 114.16 Terrain Corrections for database used (mGal)
Acknowledgements The authors would like to thank Federal of Parana University and La Plata University, which lent the necessary equipment to project’s development and Brazilian (IBGE)/Argentina (IGM) institutes which allowed access to their database. The acquisition of database and the development of this work were sponsored by the following projects: CNPQCONICET 490245/2007-2, CNPQ 301797/2008-0, CNPQ Scholarship 560796/2008-0 and CNPQ Scholarship 143345/ 2009-5.
Conclusions
References
The ANNs are very easy to apply for gravity prediction, even considering the integration of different data basis. It must be emphasized that the worst case of interpolation with ANN (Case 03) could be improved, reducing the RMS to a half part by integrating EGM2008 geoid heights. This process is not trivial to implement in the LSC because the difficulties to establish the covariance function in this case. For the cases in that it is necessary to integrate many parameters in LSC; the central problem is to obtain the covariance functions. However, for builting an ANN, additional care must be taken into account. The number of layers, usually related to separable groups, is not observed in case of prediction of gravity values. There were examples in which the number of neurons is the same as the number of points used in the learning, but the RMS was one order higher than Case number 01 of ANN. It must be considered as a special case that the structure of Case 03 applied to few training points still works with good performance while LSC and kriging do not work.
Drewes H, Sa´nchez L, Blitzkow D, de Freitas SRC (2002) Scientific Foundations of the SIRGAS Vertical Reference System. In: Drewes H et al. (eds) Vertical reference systems (IAG Symposia, vol 124). Springer, Berlin Hofmann-Wellenhof B, Moritz H (2005) Physical geodesy. Springer, Bad V€oslau ICGEM (2010) International Centre for Global Earth Models. . Access in March-2010 Negnevitsky M (2002) Artificial intelligence - a guide to intelligent systems, 2nd edn. Addison-Wesley, Harlow ON 1986 Rede Gravime´trica Fundamental Brasileira, 1976–1986. Boletim. CNPq – Observato´rio Nacional, Departamento de Geofı´sica, Rio de Janeiro pp 16 Pereira RAD (2009) Conexa˜o das Redes Verticais Fundamentais do Brasil e da Argentina com base em nu´meros geopotenciais. Master Dissertation, Universidade Federal do Parana´, Curitiba Tierra Criollo A, de Freitas SRC (2005) Artificial neural network: a powerful tool for predicting gravity anomaly from sparse data. In: Jekely C, Bastos L, Fernandes J (eds) IAG series. Springer, Berlin, pp 208–213 Trauth MH (2007) MATLAB® recipes for earth sciences, 2nd edn. Springer, Berlin Tscherning CC (1974) A FORTRAN IV Program for the Determination of the Anomalous Potential Using Stepwise Least Squares Collocation. Reports of the Department of Geodetic Science No. 212, The Ohio State University, Columbus, Ohio
.
RBMC in Real Time via NTRIP and Its Benefits in RTK and DGPS Surveys
115
S.M.A. Costa, M.A. de Almeida Lima, N.J. de Moura Jr, M.A. Abreu, A.L. da Silva, L.P. Souto Fortes, and A.M. Ramos
Abstract
Currently, IBGE is working on providing new services together with the modernization of the RBMC, such as real-time services via Internet using NTRIP (Networked Transport of RTCM via Internet Protocol), called RBMC-IP and the computation of WADGPS (Wide Area Differential GPS) corrections. A NTRIP caster is in operation at IBGE and receives the streams of 26 stations established in the main cities of Brazil. It is expected to provide real-time data access to all users in the first half-year of 2009. In order to evaluate this new realtime service in terms of precision and accuracy, some tests were performed in Rio de Janeiro state using code and phase observables in static mode. Parameters like distance to the reference stations and the reliability of the connection in urban and rural areas were considered in this evaluation. Another test was performed by the Brazilian Navy during a bathymetric survey with the purpose to update the nautical cartography in an area south of the Brazilian coast, using RTK (RealTime Kinematic) corrections in RTCM3.01 (Real-Time GNSS data Transmission Standard). A comparison between NTRIP solution and standard RTK solution using radio link was performed. This paper presents the results of these two experiments and provides an analysis of the advantages, disadvantages and potentialities of this new solution for kinematic and static real-time surveys.
115.1 Introduction 1
RTCM means Radio Technical Commission for Maritime Services.
S.M.A. Costa (*) M.A. de Almeida Lima N.J. de Moura Jr M.A. Abreu A.L. da Silva L.P.S. Fortes Coordenac¸a˜o de Geode´sia, Instituto Brasileiro de Geografia e Estatı´stica - IBGE, Av. Brasil 15671, Rio de Janeiro 21241051, Brazil e-mail: [email protected] A.M. Ramos Setor de Geode´sia, Divisa˜o de Levantamentos, Centro de Hidrografia da Marinha-CHM, Nitero´i, Brazil
The Brazilian Institute of Geography and Statistics IBGE, has been working together with National Institute of Colonization and Land Reform–INCRA, on the expansion of the permanent GNSS networks RBMC (Brazilian Network for Continuous Monitoring of GNSS), managed by the IBGE) and RIBaC (Community Bases Network of INCRA, managed by the INCRA). These two networks will provide a larger national coverage with new operational characteristics. Part of this expansion was completed by the installation of 36 new stations and the equipment change in 13
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_115, # Springer-Verlag Berlin Heidelberg 2012
917
918
Fig. 115.1 Scheme of NTRIP components
RBMC stations. Another initiative was to provide satellite communication links to the most remote stations of the country, improving the availability of their data to the users. RBMC (Brazilian Network for Continuous Monitoring of GNSS) is today the main geodetic reference structure of the country, whose information is important for the scientific community and for practical purposes, providing to the users a direct link to the Brazilian Geodetic System (SGB) and the main link with international networks, like IGS and SIRGASCON (Fortes et al. 2007). At the moment, IBGE is working on the modernization of the RBMC, such as real-time services via Internet using NTRIP, called RBMC-IP. The BKG, in cooperation with the University of Dortmund, has developed a technique for streaming GNSS data over to internet. This technique establishes the open non-proprietary “Networked Transport of RTCM via Internet Protocol” (NTRIP) (RTCM 2004). The NTRIP is a HTTP protocol developed with the intention to have an additional possibility beyond the radio links by wireless Internet, for example using GPRS, GSM or 3G (Weber et al. 2005; Weber 2006). A NTRIP caster is in operation at IBGE receiving the streams of 26 stations established in the main cities of Brazil (Costa et al. 2008). RTK (Real Time Kinematics) or DGPS (Differential GPS) are positioning techniques based on the corrections of GNSS signals that are transmitted in real time from the reference station to a station whose coordinates need to be determined. Normally the corrections are transmitted to the rover receivers via UHF radio, which is installed together with the GNSS receiver in a station with known coordinates.
S.M. Costa et al.
NTRIP is implemented in three system software components (Fig. 115.1): NtripClients, NtripServers and NtripCasters. The NtripCaster is the actual HTTP server program whereas NtripClient and NtripServer are acting as HTTP clients (Dammalage et al. 2004). In order to evaluate this new real-time service in terms of precision and accuracy, some tests were carried out in Brazil using code and phase observables in static and kinematic mode. Parameters like distance to the reference stations and the reliability of connection in urban and rural areas were considered in this evaluation.
115.2 RBMC-IP Service It is a real-time positioning service from the RBMC, for users who make use of RTK or DGPS survey techniques. The transmission of the data is carried out the following way: a GNSS receiver continuously sends RTCM messages to the IBGE server where the NTRIP caster is installed. The user, with a client application, such as GNSS Internet Radio or BNC (BKG NTRIP Client) connects to the IBGE NTRIP caster by wireless Internet and chooses the station(s) of the RBMC-IP whose corrections he desires to receive. The corrections are received by users (to rover) either serial port or TCPIP port 2101 in such a way that allows the computation of the corrected position by the rover (Fig. 115.1). Currently, IBGE caster receives data from 26 stations located in the main capitals of the Brazilian states: Bele´m (BELE), Belo Horizonte (MGBH), Boa Vista (BOAV), Brası´lia (BRAZ), Campo Grande (MSCG), Cuiaba´ (CUIB), Curitiba (UFPR), Fortaleza (CEEU), Macapa´ (MAPA), Manaus (NAUS), Natal (RNNA), Recife (RECF), Rio Branco (RIOB), Porto Alegre (POAL), Porto Velho (POVE), Presidente Prudente (PPTE), Salvador (SAVO), Sa˜o Luı´s (SALU), Santa Maria (SMAR), Sa˜o Paulo (POLI), Rio de Janeiro (ONRJ and RIOD), Palmas (TOPL), Victoria (CEFE), Imbituba (IMBT) and Campos dos Goytacazes (RJCG) (IBGE 2009a). Six of the 26 stations contribute for the global real time network, RTIGS (Real-Time International GNSS Service) and nine also contribute for the global network IGS-IP. These nine stations are marked by a circle in Fig. 115.2.
115
RBMC in Real Time via NTRIP and Its Benefits in RTK and DGPS Surveys
919
Fig. 115.2 The configuration of the RBMC-IP network – August 2009
Access to the IBGE caster is free, however it is necessary that users fill a registration form in order to use the RBMC-IP service. Some restrictions of access are necessary in order to prevent congestion of traffic in the IBGE network, they are: • Users are only allowed to access three stations simultaneously. • The identification and password for access will be valid for a maximum period of 3 months. Some users are part of a special group, like Brazilian universities and public institutions. The UNESP (Universidade Estadual Paulista/Campus Presidente Prudente) and INPE (Instituto Nacional de Pesquisas Espaciais) are working together with RBMC data in order to generate numerical weather forecast models. At the present more than 500 users already registered for this service, and the most important ones are the representatives of receiver manufacturers.
115.3 Tests Performed The main purpose of the tests carried out was to show the confidence of NTRIP solutions in different survey modes, e.g., static and kinematic.
For static mode, two tests were realized, one using different reference stations in ten different periods each. The distances from the reference station vary from 12 to 669 km. The second test was realized in a local network in Rio the Janeiro state. The rover receiver occupied 11 geodetic stations with known high precision coordinates. In both cases the rover receiver was of double frequency type. The parameters analyzed in these two tests were: number of satellites observed, precision and accuracy. For the kinematic mode, a test was performed by the Brazilian Navy during a bathymetric survey with the purpose to update the nautical cartography in an area south of the Brazilian coast, using RTK corrections. A RTK survey was carried out by the same ship using two systems; NTRIP/RTK and the standard RTK solutions using radio link for the transmission of corrections. To validate the NTRIP solution a comparison between these two solutions was performed. A driving test was realized on the main road linking Rio de Janeiro to Sa˜o Paulo. The distance from the reference station reached 40 km. In this test three types of receivers were used: a navigator, single frequency and double frequency. The wireless communication
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S.M. Costa et al.
link used in all cases was wide band 3G. The driving speed reached up to 80 km per hour. The driving route passes through forests and mountains with many curves. The main objective of the driving test was to check the system performance under the following aspects: • Behavior of the wireless communication link while the rover receiver moves at a speed of up to 80 km per hour • System tolerance to an unstable wireless communication link • Compare real-time results with post-mission, like PPP
115.4 Static Results After ten sessions occupying each rover station, the final the solutions were achieved after 10–12 min. The horizontal components (latitude and longitude) have centimeter accuracy and precision for short distances (<80 km) from reference stations. For distances longer than 200 km from the reference stations, a submeter accuracy and precision was achieved. These results are presented in Table 115.1. In a Regional Network the results are similar to the first case, but tests were realized using shorter distances. The tests showed that ambiguities were solved for sessions which baselines are shorter than
80 km and the centimeter accuracy was obtained for horizontal and vertical components in most of cases. These results are presented in Table 115.2.
115.5 Kinematic Results The RTK/NTRIP system was used by the Navy for the first time during the bathymetric survey to update the nautical chart of Port of Laguna (SC), using RTK, RTCM v.3 corrections, generated from the reference station Imbituba (IMBT), belonging to the RBMC. For the validation of the NTRIP solution, as well as a standard RTK solution using radio link for the transmission of corrections, the horizontal precision according International Hydrographic Organization (IHO) was used. A comparison between these two solutions was performed. The rover receiver is a double frequency receiver and it was located 32 km from the NTRIP reference station and 4 km from the radio reference station. The positions provided by RTK NTRIP had been correlated with the positions of RTK UHF RADIO by means of the time registered in NMEA GGA messages. NMEA GGA messages of the RTK UHF RADIO had been recorded directly from the raw files generated by the program HYPACK, while NMEA files of RTK NTRIP system were stored in the memory card of the GPS equipment. Both messages had a
Table 115.1 The mean values after ten sessions of observations Dist. from rover (km) 12 235 325 669
s Horiz. (m) 0.01 0.21 0.22 0.41
s Vert. (m) 0.02 0.46 0.48 0.87
Type of sol. n of sat. used Fixed 10 Float 9 Float 9 Float 8
Horiz. Acc. Dif. (m) Vert. Acc. Dif. (m) 0.01 0.21 0.11 0.11 0.40 0.14 0.28 0.16
Table 115.2 Results of regional network Dist. from rover (km) 31 39 45 48 69 77 92 133 206
s Horiz. (m) 0.02 0.03 0.03 0.04 0.23 0.21 0.26 0.26 0.31
s Vert. (m) 0.03 0.03 0.05 0.03 0.32 0.38 0.29 0.33 0.61
Type of sol. Fixed Fixed Fixed Fixed Float Float Float Float Float
n of sat. used 9 9 13 10 8 9 11 9 7
Horiz. Acc. Dif. (m) 0.06 0.01 0.01 0.02 0.04 0.34 0.04 0.05 0.06
Vert. Acc. Dif. (m) 0.06 0.14 0.17 0.02 0.37 0.17 0.82 0.16 0.30
115
RBMC in Real Time via NTRIP and Its Benefits in RTK and DGPS Surveys
Table 115.3 Dates and periods where the analyses between UHF systems RTK NTRIP and RTK had been made
(m)
Date 17/11/2008 (JD 322) 17/11/2008 (JD 322) 19/11/2008 (JD 324) 20/11/2008 (JD 325)
0.4 0.3 0.2 0.1 0 –0.1 –0.2 –0.3 –0.4 16:30:00
Start time 11:22:15 16:42:43 10:39:22 15:58:55
End time 16:21:51 19:21:33 21:16:35 19:00:49
921
60%
44.4% 40% 35.8%
19.0%
20%
0.8%
0% 0.00 - 0.20 m 17:00:00
17:30:00 18:00:00 Day 20-11-2008
18:30:00
0.21 - 0.50 m
0.51 - 1.00 m
> 1.0 m
19:00:00
Fig. 115.3 Differences between a standard RTK survey (via Radio) and a RTK survey using NTRIP
sampling rate of 1 s. The periods considered in the tests are presented in Table 115.3. The horizontal differences between two GPS antennae, were measured with a centimetric tape and were corrected in the observations. The rover receiver presented position solutions with fixed ambiguities in both cases (via UHF radio and NTRIP). The differences between horizontal coordinates obtained from radio and NTRIP did not exceed 10 cm, during all period of the survey, as can be seen in Fig. 115.3. These results confirm the similar quality of RBMC-IP service by compared with radio solution. The results demonstrated the great potential of NTRIP in hydrographic surveys, resulting in cost reduction and simplified logistics, however, a wireless coverage (GSM, 3G, etc.) is necessary. The driving test was carried out on the main road that links Rio de Janeiro to Sa˜o Paulo and a double frequency receiver installed in the car. The distance to the reference station reached was approximately 50 km. The wireless communication link used was wide band 3G. The sampling rate used for this test was 1 s and the duration of the test was 30 min. The driving speed reached up to 80 km per hour. The main objective of the driving test was to check the system performance under the following aspects: • Behavior of the wireless communication link while the rover receiver moves at a speed of up to 80 km per hour
Fig. 115.4 Differences between RBMC-IP (NTRIP) and postprocessing (IBGE-PPP) results
• Compare real-time results with post-mission, provided by IBGE-PPP service (IBGE 2009b) Figure 115.4 shows the coordinates differences between RBMC-IP and PPP. The differences didn´t exceed 20 cm in 44% of the observations and only 0.8% exceed 1.0 m.
115.6 Advantages and Disadvantages of the Use of the NTRIP in RTK and DGPS Surveys The advantages and disadvantages of the use of the NTRIP in RTK and DGPS surveys are: • RTK receivers do not need special licenses to work with NTRIP, only a wireless Internet access is needed in the region where the survey is executed • GSM/GPRS or 3G communication is more economic than a UHF radio connection • It is not necessary to keep a receiver operator at the reference station • It is not necessary to install the reference station in a high place in order to increase the area in which corrections can be received • The reach of the Internet is greater than radio • Radio is hindered by obstructions. Internet is not hindered by obstructions between rover and the reference station The main disadvantages of the use of NTRIP lies in the fact that wireless Internet or services provided for
922
mobile communications only work in areas covered by CDMA, GPRS, GSM or 3G for Internet access.
115.7 Final Remarks In places where the reception of mobile communications is available NTRIP is a powerful tool for a variety of surveys, e.g., cadastre, mapping, GIS, etc. With the expansion of the RBMC and communication services of GSM, GPRS and 3G in the Brazilian cities, the NTRIP will be more present in the new future for real time. The NTRIP is based on the concept of open GNSS data from different sources through long distances, providing good data availability, accuracy and reliability compared to the radio connection. The tests carried out revealed that few centimeters of accuracy can be reached for distances up to 80 km from the reference station and a sub-meter accuracy is obtained for distances between 80 and 500 km from the reference station. Acknowledgements We would like to thank Georg Weber from BKG, which helped in the implementation of the NTRIP caster at IBGE.
S.M. Costa et al.
References Costa SMA, Lima MAA, Ju´nior NJM, Abreu MA, Silva AL, Fortes LPS (2008) RBMC em Tempo Real, Via NTRIP, e seus Benefı´cios nos Levantamentos RTK e DGPS. II Simpo´sio Brasileiro de Cieˆncias Geode´sicas e Tecnologias da Geoinformac¸a˜o, II SIMGEO, Recife, Brazil Dammalage TL, Srinuandee P, Samarakoon L, Susaki J, Srisahakit T (2004).Potential Accuracy and Practical Benefits of NTRIP Protocol Over Conventional RTK and DGPS Observation Method, http://www.gisdevelopment. net/technology/gps/ma06_102.htm Fortes LPS, Costa SMA, Abreu MA, Ju´nior NJM, Silva AL, Lima MAA, Moˆnico JFG, Santos MC (2007) Plano de Expansa˜o e Modernizac¸a˜o das Redes Ativas RBMC/RIBaC IBGE (2009a) RBMC-IP. http://www.ibge.gov.br/home/ geociencias/geodesia/rbmc/ntrip/ IBGE (2009b) Posicionamento Preciso por Ponto – PPP. http:// www.ibge.gov.br/home/geociencias/geodesia/ppp/default.shtm Radio Technical Commission for Maritime Services (2004) “RTCM Recommended Standards for Networked Transport of RTCM via Internet Protocol (Ntrip), Version 1.0”, RTCM Paper 200-2004/SC104-STD Weber G (2006) Streaming Real-Time IGS Data and Products Using NTRIP Proceedings Darmstadt IGS Workshop, ftp:// igscb.jpl.nasa.gov/pub/resource/pubs/06_darmstadt/IGS% 20WS%202006%20Papers%20PDF/6_Weber_IGS_Proceedings_Darmstadt_NTRIP.pdf Weber G, Dettmering D, Gebhard H, Kalafus R (2005) Networked transport of RTCM via Internet Protocol (NTRIP)-IP-Streaming for Real-Time GNSS Applications, ION GNSS 18th International Technical Meeting of the Satellite Division
Session 6 Joint ION/FIG/ISPRS Session on Navigation and Earth Observation Convenors: D.A. Grejner-Brzezinska, C.K. Toth
.
Bootstrapping with Multi-frequency Mixed Code Carrier Linear Combinations and Partial Integer Decorrelation in the Presence of Biases
116
P. Henkel
Abstract
Carrier phase measurements are extremely accurate but ambiguous. The reliability of integer ambiguity resolution is improving with Galileo which uses a Binary Offset Carrier (BOC) modulation, large signal bandwidths of up to 50 MHz and additional carrier frequencies. In this paper, a group of multi-frequency mixed code carrier linear combinations is derived which preserves geometry, eliminates the ionospheric delay and maximizes the ratio between wavelength and noise standard deviation of the combination. Moreover, a partial integer decorrelation is suggested to improve the robustness of ambiguity resolution over biases due to orbital errors, satellite clock offsets, and multipath. The proposed group of multi-frequency mixed code carrier linear combinations is characterized by a wavelength of more than 3 m, which makes this group of combinations an interesting candidate for both Wide Area Real Time Kinematics (RTK) and Precise Point Positioning.
116.1 Introduction Precise carrier phase positioning with Galileo and GPS requires the resolution of integer ambiguities. The reliability of ambiguity fixing can be improved by multi-frequency mixed code carrier linear combinations that eliminate the first order ionospheric delay, suppress the E1 code multipath by at least one order of magnitude, and maximize the ratio between wavelength and standard deviation of the combined noise.
P. Henkel (*) Institute of Communications and Navigation, Technische Universit€at M€unchen, Theresienstraße 90, 80333 M€ unchen, Germany e-mail: [email protected]
A new ionosphere-free linear combination of Galileo E1, E5, E5a, E5b and E6 code and carrier phase measurements is derived that is characterized by a wavelength of 3.939 m and a noise level of a few centimeters for a carrier to noise power ratio of 45 dB-Hz. This combination also suppresses the E1 code multipath by 18.0 dB. The integer ambiguities of the optimized linear combinations are fixed sequentially with bootstrapping (Blewitt 1989). An integer ambiguity transformation decorrelates the search space which improves the search efficiency dramatically but might result also in a substantial amplification of residual biases due to orbital errors, satellite clock offsets, and multipath. Therefore, a partial integer decorrelation is used for bootstrapping to achieve the optimum trade-off between variance reduction and bias amplification.
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_116, # Springer-Verlag Berlin Heidelberg 2012
925
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P. Henkel
It is shown that the probability of wrong fixing can be reduced by up to 10 orders of magnitude compared to a complete integer decorrelation for phase biases of only 0.05 cycles. The partial integer decorrelation also improves the poor performance of integer least squares estimation techniques in the presence of biases. Sequential ambiguity resolution based on multifrequency mixed code carrier widelane combinations and partial integer decorrelation are considered to be an interesting candidate for both Precise Point Positioning and Wide Area RTK. These combinations increase the wavelength to several meters while the noise level is still kept at a centimeter level by carrier smoothing (Hatch 1982). The smoothing period can also be included in the combination design.
lfku ¼
M X am lm fku;m þ bm rku;m m¼1
¼
M X
ðam þ bm Þ ruk þ druk þ Tuk
m¼1
þ
M X
ðam þ bm Þ c dtu dtk
m¼1
þ
M X
k am lm Nu;m
m¼1
M X
ðam bm Þ q21m Iuk
m¼1
M X þ am bkfu;m þ bm bkru;m m¼1
M X am ekfu;m þ bm ekru;m : þ
(116.3)
m¼1
116.2 Derivation of Multi-frequency Code Carrier Linear Combinations The carrier phase measurement on frequency m 2 f1; . . . ; Mg at user u ¼ {1, . . . , U} from satellite k ¼ {1, . . . , K} is modeled as lm fku;m
k q21m Iuk þ Tuk þ lm Nu;m
(116.1)
with the wavelength lm, the user-satellite range ruk , the projected orbital error druk , the user/satellite clock errors {cdtu, cdtk}, the ionospheric delay Iuk on L1, the ratio of frequencies qij ¼ fi/fj, the tropospheric delay k , the phase bias bkfu;m and Tuk , the integer ambiguity Nu;m ekfu;m
carrier phase noise including multipath. A similar model is used for the code measurements, i.e., rku;m ¼ ruk þ druk þ c dtu dtk þ Tuk þ q21m Iuk þ bkru;m þ ekru;m :
M X
ðam þ bm Þ ¼ 1;
(116.4)
m¼1
which preserves also the orbital error, the tropospheric delay and the clock offsets. A second constraint shall eliminate the ionospheric delay (IF constraint), i.e.,
¼ ruk þ druk þ c ðdtu dtk Þ þ bkfu;m þ ekfu;m ;
The linear combination preserves the range if
M X
ðam bm Þq21m ¼ 0:
(116.5)
m¼1
Moreover, the superposition of ambiguities should be an integer number of a common wavelength, i.e., M X
am lm Nm ¼ lN;
(116.6)
m¼1
which can be solved for N: (116.2)
A multi-frequency mixed code carrier linear combination weights the phase measurements by am and the code measurements by bm, i.e.,
N¼
M X ! am lm Nm 2 Z; l m¼1
(116.7)
where Z denotes the set of integer numbers. Equation (116.7) is fulfilled if
116
Bootstrapping with Multi-frequency Mixed Code Carrier Linear Combinations
jm ¼
am lm ! 2Z l
8m:
(116.8)
M X
am q21m
M X
m¼1
Solving for am yields am ¼
b1 ¼
927
¼
M X
bm q21m
m¼2
am q21m
1
m¼1
jm l : lm
(116.9)
M X
am b1
m¼1
M X
M X
! bm q212
m¼3
bm q21m :
ð116:15Þ
m¼3
The wavelength can be factorized into two components, i.e., l¼~ l0 wf ;
Replacing am by (116.9), using (116.10), and solving for b1 yields
(116.10) b1 ¼ s1 þ s2 wf þ
with
M X
sm bm ;
(116.16)
m¼3
~l0 ¼
1 M P m¼1
and wf ¼ 1 jm lm
M X
bm :
(116.11)
with
m¼1
s1 ¼
The M + 2 constraints of (116.4), (116.5) and (116.8) leave M – 2 degrees of freedom for the design of am and bm. Therefore, a fourth constraint has been introduced by Henkel et al. (2009) to maximize the ambiguity discrimination, i.e., lða1 ; : : : ; aM ; b1 ; : : : ; bM Þ ; D¼ max 2sn ða1 ; : : : ; aM ; b1 ; : : : ; bM Þ a1 ; : : : ; aM b1 ; : : : ; b M (116.12)
s2 ¼
sm ¼
q212 1 q212
(116.17)
M ~l X jm ðq212 þ q21m Þ 2 1 q12 m¼1 lm
(116.18)
q212 q21m 1 q212
(116.19)
8m 2 f3; . . . ; Mg:
Equation (116.16) is used to rewrite (116.14) as b2 ¼ 1 s1 ð1 þ s2 Þwf
M X
ð1 þ sm Þbm ;
m¼3
with
(116.20) vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u M M X uX a2m s2fm þ b2m s2rm : sn ¼ t m¼1
(116.13)
m¼1
The optimization of (116.12) consists of a numerical search over jm and an analytical computation of am and bm which is presented here. The code weight b2 is obtained from (116.4): b2 ¼ 1
M X m¼1
am b1
M X
bm ;
(116.14)
m¼3
and the code weight b1 is computed from the ionosphere-free constraint, i.e.,
which allows us to express D as a function of wf and bm, m 3: 0 ~l D ¼ wf :@~2 w2f þ 2
s1 þ s2 wf þ
þ
m¼3
!1=2 b2m s2rm
2 P with ~2 ¼ ~l M m¼1 nation is obtained by
!2 sm bm
m¼3
þ 1 s1 ð1 þ s2 Þwf M X
M X
M X
s2r1 !2
ð1 þ sm Þbm
s2r2
m¼3
ð116:21Þ j2m 2 s . l2m fm
The maximum discrimi-
928
P. Henkel
@D ¼ 0; @wf
s1 þ s2 wf tT A1 ðc þ bwf Þ s1 tT A1 ðc þ bwf Þ s2r1 þ 1 s1 ð1 þ s2 Þwf þ uT A1 ðc þ bwf Þ 1 s1 þ uT A1 ðc þ bwf Þ s2r2
(116.22)
and @D ¼0 @bm
8m 2 f3; . . . ; Mg:
(116.29)
Equation (116.23) is equivalent to sm s2r1
s1 þ s2 wf þ
M X
with t ¼ [s3, . . . , sM]T, u ¼ s + 1, and the diagonal matrix S that is given by
! sl bl ð1 þ sm Þ
2
l¼3
s2r2
þ ðc þ bwf ÞT AT SA1 ðc þ bwf Þ ¼ 0;
(116.23)
1 s1 ð1 þ s2 Þwf
M X
!
6 S¼6 4 0 0
ð1 þ sl Þbl
l¼3
þ bm s2rm
¼ 0;
A ½b3 ; . . . ; bM T þ b wf þ c ¼ 0;
..
0
0
.
3
7 7 0 5: s2rM
(116.30)
(116.29) can be simplified as the quadratic terms with w2f cancel, i.e., r1 þ r2 wf ¼ 0;
(116.25)
(116.31)
with
with
2 r1 ¼ s1 t T A1 c s2r1 2 þ 1 s1 þ uT A1 c s2r2
Am;l ¼ sl sm s2r1 þ ð1 þ sl Þð1 þ sm Þs2r2 þ s2rm dðm lÞ
þ cT AT SA1 c;
bm ¼ s2 sm s2r1 þ ð1 þ sm Þð1 þ s2 Þs2r2 cm ¼
s1 sm s2r1
ð1 þ sm Þð1
s1 Þs2r2 ;
(116.26)
and d (m – l) being 1 for m ¼ l and 0 otherwise. Solving (116.25) for bm yields ½b3 ; . . . ; bM T ¼ A1 ðc þ b wf Þ:
(116.27)
Constraint (116.22) is written in full terms as s1 þ s2 wf þ
M X
! sm bm
s1 þ
m¼3
1 s1 ð1 þ s2 Þwf
M X
! sm bm s2r1
m¼3 M X
!
ð1 þ sm Þbm
m¼3
0
(116.24)
which can be written also in matrix-vector notation as
þ
s2r3
1 s1
M X m¼3
!
ð1 þ sm Þbm s2r2 þ
M X m¼3
b2m s2rm ¼ 0: (116.28)
Replacing [b3, . . . , bM]T by (116.27) yields
(116.32)
and s1 tT A1 c t T A1 b þ s2 tT A1 b s1 tT A1 c s2r 1 þ 1 s1 þ uT A1 c uT A1 b 1 þ s2 uT A1 b 1 s1 þ uT A1 c s2r2 þ cT AT SA1 b þ bT AT SA1 c : (116.33)
r2 ¼
The optimum phase weighting wfopt ¼ r1 =r2 is used in (116.27), (116.20) and (116.16) to obtain the code weights. (116.10) provides the optimum wavelength for the computation of the phase weights with (116.9). The optimization of the multi-frequency linear combinations depends on the assumed phase and code noise variances. The latter ones are obtained from the Cramer Rao bound that is given by Kay (1993) as
116
Bootstrapping with Multi-frequency Mixed Code Carrier Linear Combinations
c Gm ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; R Es N0
2
(116.34)
2
ð2pf Þ jSm ðf Þj df
R
jSm ðf Þj2 df
with the speed of light c, the signal to noise power ratio NE0s , and the power spectral density Sm( f ) that has been derived by Betz (2001) for binary offset carrier (BOC) modulated signals. Table 116.1 shows the Cramer Rao lower bounds (CRB) of the Galileo signals of maximum bandwidth. Table 116.2 shows the weighting coefficients and properties of multi-frequency linear combinations of maximum ambiguity discrimination. The dual frequency E1-E5a combination is characterized by a noise level of 31.4 cm and a wavelength of 4.309 m which allows reliable ambiguity resolution within a few epochs. As only the E1 and E5a frequencies lie in aeronautical bands, this linear combination might be useful for aviation. Linear combinations that comprise the wideband E5 and E6 code measurements benefit Table 116.1 CRB for Es/N0 ¼ 45 dB–Hz E1 E5 E5a E5b E6
Signal MBOC AltBOC(15,10) BPSK(10) BPSK(10) BOC(10,5)
BW [MHz] 20 51 20 20 40
G [cm] 11.14 1.95 7.83 7.83 2.41
929
from a substantially lower noise level which turns into a larger ambiguity discrimination. It increases to 25.1 for the E1-E5 combination, to 39.2 for the E1-E5-E6 combination, and to 41.0 for the E1-E5a-E5b-E5-E6 combination. The large wavelength of these combinations makes them robust to the non-dispersive orbital errors and satellite clock offsets. The linear combination of measurements on 5 frequencies has the additional advantageous property of |bm| < 1 and |jm| 2 for all m.
116.3 Sequential Fixing with Partial Integer Decorrelation The reliability of ambiguity resolution is increased by using two linear combinations: a code carrier combination of maximum P discrimination and a code~ k only combination rku ¼ M m¼1 bm ru;m . The weighting coefficients of the latter one are chosen to preserve geometry, eliminate the ionospheric delay, and to minimize the noise amplification. Both linear combinations are written in matrix vector notation for U ¼ 1: T C ¼ lf11 ; . . . ; lfK1 ; r11 ; . . . ; rK1 ¼ Hj þ AN þ n þ b;
(116.35)
Table 116.2 Geometry-preserving, ionosphere-free, integer-preserving mixed code carrier widelane combinations of maximum discrimination for sf ¼ 1 mm and srm ¼ Gm jm am bm jm am bm jm am bm jm am bm jm am bm jm am bm
E1 1 17.2629 –0.0552 1 22.6467 –1.0227 1 18.5565 –0.2342 1 21.1223 –0.0200 1 23.4845 –0.0468 1 20.6978 –0.0159
E5a
–1 –16.9115 –3.7125 4 55.4284 –0.8502
E5b
E5 –1 –13.0593 –3.1484
–5 –71.0930 –0.8075 1 15.9789 –1.1422
1 17.5371 –0.1700 1 15.4562 –0.0578
E6
0 0.0000 –0.1615 0 0.0000 –0.0549
0 0.0000 –0.9084
–2 –34.2894 –0.6495 –2 –38.1242 –1.5191 –2 –33.6004 –0.5166
l
sn
D
3.285 m
6.5 cm
25.1
4.309 m
31.4 cm
6.9
3.531 m
133.3 cm
13.3
4.019 m
5.1 cm
39.2
4.469m
6.3cm
35.3
3.938m
4.8cm
41.0
930
P. Henkel
with the geometry matrix H, the unknown real-valued parameters j 2 R, the wavelength matrix A, the unknown integer ambiguities N 2 Z, additive white Gaussian noise n Nð0; SÞ and biases b. An integer least-squares estimator minimizes the weighted sum of squared errors, i.e., min k C Hj AN k2S1 x;N
s:t:
N 2 Z: (116.36)
The least-squares float solution of ambiguities is obtained by an orthogonal projection and by disregarding the integer nature of ambiguities, i.e., T 1 1 T 1 ^¼ A S A A S C; N
(116.37)
s2N^
kj1;...;k1
(116.38)
jj1; ... ;j1
j¼1
sN^kj1; ... ;k1 ;N^lj1; ... ;l1 ¼ 0
s2 N^
jj1; ... ;j1
; (116.42)
8k 6¼ l:
(116.43)
Consequently, the success rate of ambiguity resolution can be computed analytically Ps ¼
K Z Y k¼1
A ¼ P? H A;
s2N^k N^
which depends on the order of fixings. This sequential fixing minimizes the variances of the conditional ambiguity estimates. These N^kj1; ... ;k1 are uncorrelated, i.e.,
0
with
k1 X
¼ s2N^k
þ0:5
0:5
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ps2N^ kj1;...;k1
ðN^kj1; ... ;k1 bN^kj1; ... ;k1 Þ
2
exp@
2s2N^
1 Ad N^kj1; ... ;k1 ;
kj1;...;k1
(116.44)
and the orthogonal projection 1 T 1 T 1 P? H ¼ I PH ¼ I HðH S HÞ H S :
(116.39) The Least squares Ambiguity Decorrelation Adjustment (LAMBDA) method of Teunissen (1993) is in an integer least-squares estimator which uses an ambiguity transformation Z to decorrelate the search space space, and a search S for ambiguity fixing. The fixed ambiguity estimates are given by ^
N ¼ Z1 SðZN 0 Þ;
(116.40)
where Z and S are described in details in Teunissen (1995) and Jonge and Tiberius (1996). An alternative integer estimation method is bootstrapping (Blewitt 1989) which fixes the float ambiguity estimates sequentially, i.e., the k-th conditional estimate is given by N^kj1; ... ;k1 ¼ N^k
k1 X j¼1
sN^k N^jj1; ... ;j1 s2 N^
jj1; ... ;j1
ðN^jj1; ... ;j1 ½N^jj1; ... ;j1 Þ;
(116.41)
where [.] denotes the rounding operator and s2N^ is jj1;...;j1 the variance of the conditional ambiguity estimate. It is given by
where bN^kj1; ... ;k1 denote the biases in the conditional ambiguity estimates (Teunissen 2001). The success rate of bootstrapping can be substantially improved by an integer ambiguity transformation Z. This integer decorrelation transformation reduces the (co-) variances but also increases residual biases due to orbital errors, clock offsets and multipath. Therefore, a partial integer decorrelation with a reduced number L of decorrelation steps has been suggested by Henkel and G€unther (2009), i.e., Z ¼ Zð1Þ Zð2Þ . . . ZðLÞ ;
(116.45) ðnÞ
where each Z(n) includes an integer decorrelation Z1 ðnÞ and a permutation Z2 of two ambiguities, i.e., ðnÞ
ðnÞ
ZðnÞ ¼ Z1 Z2 :
(116.46)
116.4 Simulation Results This section shows the benefit of multi-frequency mixed code carrier widelane combinations for Galileo. Simulated measurements are generated for the Galileo Walker constellation with 27 satellites. The section is
116
Bootstrapping with Multi-frequency Mixed Code Carrier Linear Combinations
subdivided into two parts: First, double difference measurements are considered for Wide Area RTK (Hernandez-Pajares et al. 2000). The second subsection addresses Precise Point Positioning (Zumberge et al. 1997) with satellite–satellite single difference measurements.
116.4.1 Wide Area RTK Double difference measurements are used to estimate the baseline (once per epoch), the integer ambiguities (with bootstrapping and integer decorrelation), the tropospheric wet zenith delay (of first epoch), the temporal gradient of the tropospheric wet zenith delay, the ionospheric delays for all satellites (of first epoch) and the temporal gradient of ionospheric delays for all satellites. The latter two parameters are not estimated in the case of IF combinations. The measurement noise is assumed to a follow a Gaussian distribution with the variances given by the Cramer Rao bounds of Table 116.1. Figure 116.1 shows the benefit of the geometrypreserving, ionosphere-free E1-E5 code carrier combination of maximum ambiguity discrimination (Table 116.2) for ambiguity resolution. A geometrypreserving, ionosphere-free E1-E5 code-only combination is additionally used to exploit the redundancy of dual frequency measurements. The large wavelength of l ¼ 3.285 m reduces the probability of wrong fixing by more than 10 orders of magnitude compared
to uncombined measurements with wavelengths of l1 ¼ 19.0 cm and l2 ¼ 25.2 cm. Measurements from only three epochs (separated by 1 s) were considered. The temporal variations in the probability of wrong fixing are caused by a change in the satellite geometry over 1 day as seen from our institute at 48.1507 N, 11.5690 E. Figure 116.2 shows a comparison of different ambiguity resolution methods for both unbiased and biased measurements. The skyplot indicates the chosen satellite geometry with six visible satellites. An ionosphere-free smoothing with smoothing time constant t was applied to both the E1-E5 linear combination of maximum ambiguity discrimination and the E1-E5 code-only combination. The float ambiguity solution is then determined by least-squares estimation from the smoothed measurements of a single epoch. Obviously, a larger smoothing period results in a lower probability of wrong fixing. For unbiased measurements, the integer least-squares (ILS) estimation achieves the lowest error rate, followed by bootstrapping and rounding. The SD bias amplitudes are modeled by an elevation dependent, exponential profile (horizon: 10 cm for code, 0.1 cycles for phase measurements; zenith: ten times lower values than in horizon) and the bias signs are chosen to obtain a worst-case bias accumulation over satellites and frequencies in the conditional ambiguities (Henkel and G€unther 2009). Note that the bias amplitude is based on the satellite of lower elevation in the satellite–satellite SD pair, and that the
100
10–5
10
–9
10–10
10–15 IF code−carrier and IF code−only combinations no combinations over frequencies
Probability of wrong fixing
Probability of wrong fixing
100
10–2 1 0
10–4
−1 −1
0 Rounding, 1 unbiased Seq., cor., unbiased Seq., dec., unbiased ILS, unbiased Rounding, biased Seq., cor., biased Seq., dec., biased ILS, biased
10–6
–20
10
931
0
5
10 15 Time [h]
20
Fig. 116.1 Benefit of multi-frequency mixed code carrier linear combinations for ambiguity resolution: the increase in wavelength from 19.0 cm to 3.285 m enables a substantial reduction in the probability of wrong fixing
10–8
2
4 6 8 Smoothing constant τ [s]
10
12
Fig. 116.2 Comparison of integer estimation methods for unbiased and biased measurements
932 108 1 comb., no int. dec.
Number of halted searches
satellite of highest elevation is chosen as reference satellite. Additionally, the Cramer Rao bounds are scaled by 3 to include multipath. The optimality of the integer estimators is vice versa in the presence of worst-case biases, i.e., the integer ambiguity transformation results in an amplification of the biases which more than compensates for the gain obtained from the variance reduction. Note that the probabilities of wrong fixing for rounding and ILS were obtained from Monte Carlo simulations.
P. Henkel
1 comb., int. dec., 25 iter.
106
1 comb., int. dec., 50 iter. 1 comb., int dec., 210 iter. 2 comb., no int. dec.
104
102
100
1
2
3
116.4.2 Precise Point Positioning
9
10
Fig. 116.3 Reduction of halted searches by multi-frequency linear combinations
100
Probability of wrong fixing
The optimized linear combinations of Table 116.2 are also beneficial for Precise Point Positioning (Zumberge et al. 1997) with satellite–satellite single difference measurements. The previously used E1–E5 code carrier and code-only combinations are smoothed over 10 s, and then used to estimate the receiver position (once per epoch), ambiguities and tropospheric zenith delay (once per epoch). Figure 116.3 shows that an efficient integer search (LAMBDA) can be achieved either by the optimized code carrier combination with complete integer decorrelation, or by additionally using the code-only combination without integer decorrelation. Note that the first four ambiguities can always be fixed without frequent halted searches due to the four degrees of freedom for position and tropospheric wet zenith delay. However, the extremely low conditional variances of the 5th and further ambiguities result in frequent halted searches. The use of an integer decorrelation or of an additional linear combination flattens the ambiguity spectrum and, thereby, improves the search efficiency dramatically. Figure 116.4 shows that a partial integer decorrelation with a reduced number of L decorrelation steps enables a substantial improvement in ambiguity fixing for worst-case SD phase biases of only 0.05 cycles (on both E1 and E5) in the horizon. The following assumptions were made for the remaining parameters of the elevation dependent, exponential SD bias profiles: zenith: 0.01 cycles for phase and 0.01 m for code biases; horizon: 0.05 m for code biases. A 30 s ionosphere-free smoothing was used to reduce the noise level and, thereby increase the margin for biases.
4 5 6 7 8 Number of fixed ambiguities
10–5
10–10
L=1 L = 20 L = 50 L = 248
10–15
10–20
0
0.02
0.04
0.06
0.08
0.1
0.12
Maximum SD phase bias on E1 and E5 [cycles]
Fig. 116.4 Benefit of partial integer decorrelation for sequential ambiguity resolution
References Betz J (2001) Binary offset carrier modulations for radionavigation. Navigation 48(4):227–246 Blewitt G (1989) Carrier-phase ambiguity resolution for the Global Positioning System applied to geodetic baselines up to 2000 km. J Geophys Res 94:10187–10203 Hatch R (1982) The synergism of GPS code and carrier phase measurements. Proceedings of the 3rd international geodetic symposium on satellite doppler positioning, vol 2, New Mexico, pp 1213–1232 Henkel P, Gomez V, G€unther C (2009) Modified LAMBDA for absolute carrier phase positioning in the presence of biases. Proceedings of ION ITM, Anaheim, CA, pp 644–651
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Bootstrapping with Multi-frequency Mixed Code Carrier Linear Combinations
Henkel P, G€unther C (2009) Partial integer decorrelation: optimum trade-off between variance reduction and bias amplification. J Geod 84:51–63 Hernandez-Pajares K, Juan J, Sanz J, Colombo O (2000) Application of ionospheric tomography to real-time GPS carrier-phase ambiguities resolution, at scales of 400–1000 km, and with high geomagnetic activity. Geophys Res Lett 27:2009–2012 Jonge P, Tiberius C (1996) The LAMBDA method for integer ambiguity estimation: implementation aspects, LGR Ser (12):1–59. TU Delft. Kay S (1993) Fundamentals of statistical signal processing: estimation theory. Prentice Hall, Upper Saddle River, NJ, p 47
933
Teunissen P (1993) Least-squares estimation of the integer ambiguities. Invited lecture, Section IV, Theory and Methodology, IAG General Meeting, Beijing, China Teunissen P (1995) The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation. J Geod 70:65–82 Teunissen P (2001) Integer estimation in the presence of biases. J Geod 75:399–407 Zumberge J, Heflin M, Jefferson D, Watkins M, Webb F (1997) Precise point positioning for the efficient and robust analysis of GPS data from large networks. J Geophys Res 102: 5005–5017
.
Real Time Satellite Clocks in Precise Point Positioning
117
R.J.P. van Bree, S. Verhagen, and A. Hauschild
Abstract
Computing a position with Single Frequency Precise Point Positioning (SF-PPP) algorithms compels to the use of satellite clock corrections, and for use with realtime applications, only a limited set of sources for orbit and clock data is available, for example the predicted Ultra Rapid products of the International GNSS Service (IGS). Recently, real-time clock estimates have become available, for example the RETICLE clocks developed by GSOC/DLR. In this research, first a comparison is made in the satellite clock error domain as the real-time RETICLE and predicted Ultra Rapid corrections are compared to the Final IGS clock corrections. The empirical standard deviation of clock differences between Final and RETICLE clocks become less than 0.4 ns. Differences between Final and Ultra Rapid clocks lead to a standard deviation of around 2 ns. Secondly the single frequency precise point positioning position errors in the North, East and Up directions are investigated. With the use of the RETICLE clocks the empirical standard deviation of the position errors in the North and East directions are between 2 and 3 dm and in the Up direction around 5 dm. These results are comparable with the accuracies reached when using Final products.
117.1 Introduction This paper demonstrates the positioning accuracy which can be achieved with real-time clock estimates compared to predicted clocks for Single Frequency
R.J.P. van Bree S. Verhagen (*) Mathematical Geodesy and Positioning (MGP), Delft University of Technology, Kluyverweg 1, 2629 Delft, The Netherlands e-mail: [email protected] A. Hauschild Deutsches Zentrum f€ ur Luft-und Raumfart (DLR), German Space Operations Center (GSOC), Oberpfaffenhofen, 82234 Wessling, Germany
Precise Point Positioning (SF-PPP). The REal-TIme CLock Estimation (RETICLE) clock corrections, which have been developed at German Space Operations Center of the German Aerospace Center (GSOC/DLR), are the new satellite clock estimates under investigation in this research. In the first part of the paper the RETICLE clocks from DLR and the predicted Ultra Rapid clocks from IGS are directly compared to the IGS Final clocks, which serve as reference. The analysis of these data are done for several types of GPS satellites (block II, block IIR and block IIR-M) operating on Cesium (CS) and Rubidium (Rb) standards. It was found that the Ultra Rapid clock errors, as compared to the Final clock errors, increase linearly from the beginning of
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_117, # Springer-Verlag Berlin Heidelberg 2012
935
936
the predicted part of the Ultra Rapid data. When, for example, the predicted Ultra Rapid clocks from 6 h UTC become available at 9 h UTC, the “age” of the predicted clocks is already 3 h and will extend to 9 h until the new 12 h UTC file becomes available. It is found that between 3 and 9 h after the start of the predicted part, the clock errors can reach up to several nanoseconds. However for some older Cesium clocks even errors up to 20 ns (6.0 m) are detected. The new real-time RETICLE corrections, available as corrections on ephemerides data and as SP3 files, have greatly improved clock accuracies, up to subnanosecond level, which will lead to improvement in the position results. In the second part of the paper the effects of the RETICLE orbits and clocks on the accuracy of the SFPPP position solution will be evaluated. The position errors in the North, East and Up directions are investigated and compared to position results when using predicted Ultra Rapid and Final clock and orbit products.
117.2 Single Frequency Precise Point Positioning At present Single Frequency Precise Point Positioning (SF-PPP) is commonly known and used in the GNSS community. A detailed description of the Single Frequency Precise Point Positioning can be found in Heroux and Kouba (1995), Øvstedal (2002), Gao et al. (2006) and Le and Tiberius (2007). The best accuracy with SF-PPP is reached when precise GPS products are used, i.e. the final satellite orbits and clocks, final ionospheric delays and latest Differential Code Biases (DCB). For the mitigation of the tropospheric delay the Saastamoinen model is used. When a real-time position solution is requested predicted satellite orbits, clocks and ionospheric delays must be used, resulting in position solution with a much larger error. The newly developed Real Time SF-PPP software at Delft University is based on the in-house SF-PPP software, which uses un-differenced single frequency code and carrier phase observations and is calculated on an epoch by epoch base. The SF-PPP software uses several public GPS related products and models to account for the various errors sources. A quite extensive description of all these error sources is given by Kouba (2009).
R.J.P. van Bree et al.
117.2.1 Predicted Ultra Rapid IGS The Ultra Rapid clocks are one possible source of clock corrections for the SF-PPP algorithm. These clock files contain 48 h worth of orbits and clocks. The first 24 h contain orbits and clocks based on actual observations. The second 24 h contain predicted orbits and clocks. The error can get as large as 20 ns for the old Cesium clocks but even the newer clocks have errors as large as 5–6 ns. Of course in practice the Ultra Rapid clocks are updated every 6 h resulting in much lower clock errors. However, prediction errors in the order of several tens of nanoseconds still exist for the Cesium clocks. The real-time clock estimates from DLR, which have been used in our SF-PPP system as well, are described in the following section.
117.2.2 RETICLE DLR The REal-TIme CLock Estimation (RETICLE) system has been developed at GSOC/DLR. The system computes clock corrections for the entire GPS constellation in real-time, based on a world-wide network of reference stations. The reference stations transmit their measurements via the protocol NTRIP, which has been developed by BKG to enable real-time exchange of GNSS data (Weber et al. 2005). The dual-frequency measurements arrive with a typical latency of 2–3 s. The station network used for the clock estimation currently comprises 37 stations. The Kalman-Filter employed in the RETICLE system estimates the GPS satellites’ clock offsets- and drifts, the station clock offsets, the zenith-tropospheric delays and the carrier-phase ambiguities. The most recent available Ultra Rapid predicted orbits are automatically downloaded from the IGS server and held fixed during the clock estimation. The carrier-phase ambiguities are estimated as float values. Reference station positions are fixed and taken from recent IGS Sinex-files. Prior to processing, the observations are rigorously screened for outliers and corrupted measurements. During the filter measurement update, the ionosphere free linear combinations of pseudorange and carrier-phase observations are processed. The modeling of the observations comprises corrections for the tropospheric delay, antenna phase center offsetand variations, station motion due to earth-tides and
117
Real Time Satellite Clocks in Precise Point Positioning
ocean loading as well as carrier-phase wind-up. To make the clocks consistent with the P1/P2observations, corrections for the differential code biases between the C/A-code and the P1-code must be applied for certain receivers, which do not report the P1-code. A more detailed description of the filter setup and precise orbit determination results with RETICLE products are contained in Hauschild and Montenbruck (2008).
117.3 Test Description In this section the Real Time PPP Software will be described, together with the experiments that were setup to answer the research question. As mentioned in the previous section, the real-time SF-PPP software has been developed at Delft University from the SF-PPP software (Le 2004; Le and Tiberius 2007). For the static tests, the RINEX data is obtained from a JPS LEGACY receiver named dlft and located in Delft as a permanent reference station with its location known up to a few millimeters and therefore very suitable for the static tests (see http:// gnss1.lr.tudelft.nl/dpga/station/Delft.html). The used antenna is the type JPS REGANT DD E (dual-depth Choke Ring). The 1 day predicted Global Ionosphere Map (GIM) is used to correct for the ionospheric delays and is downloaded from the CODE Website (ftp://ftp.unibe.ch/aiub/CODE/) and updated every day. From this Website also the Differential Code Biases (DCBs) are downloaded. These DCBs must be applied because we use Single Frequency PPP (Kouba 2009). From the International Earth rotation and Reference systems Service (IERS) the earth rotation parameters are downloaded whenever new data are available. These parameters are used for the calculation of the solid earth tides. Finally for the satellite orbits and clocks there is the choice between the Ultra Rapid and the RETICLE. The first one is downloaded from the IGS Website (see ftp://cddis.gsfc.nasa.gov/ gps/products/). The latter is obtained by request from DLR. For the analysis of the real-time clocks two domains are studied, the satellite clock error domain and the position error domain. For the analysis in the clock domain only the Final clocks, the predicted Ultra Rapid clocks and the DLR RETICLE clock files are needed. To obtain sound statistics on the satellite clocks 163 days of Ultra
937
Rapid and Final data, and 68 days of RETICLE data are analysed (see Table 117.1). The test period of the RETICLE products is shorter, since the products have not been available for this analysis prior to day 135. In the first column of Table 117.1 the type of orbit/clock product is given. The second column states the days of the year in 2009 that were used and the number of days in the third column. In the fourth the spacing in time as used is given and in the last column the number of samples per satellite that are used in the calculations. Hence we can compute the clock difference between the Ultra Rapid and Final clocks using the Ultra Rapid from day of year 41–203 and the Final from the same period, both sampled at 15 min. The difference between RETICLE and Final clocks is calculated using the RETICLE SP3 files from day of year 135–202 and the Final from that period, both sampled at 30 s. In Table 117.2 the orbit/clock products that are used with the SF-PPP processing are given. The first column gives the clock/orbit product in the SF-PPP processing. The comment (rt) means that the position results are obtained using the Real Time PPP Software and results have a latency of a little bit over 1 min. If (pp) is stated, the results are obtained in postprocessing using the standard SF-PPP processing
Table 117.1 Overview of the clock products used Orbit/clock product Ultra Rapid IGS (sp3) RETICLE DLR (sp3) Final IGS (clk/clk_30s)
Day of year (2009) 41–203
No. of days 163
No. per Interval satellite 15 min 15,648
135–202
68
30 s
195,840
41–203 135–202
163 68
15 m 30 s
15,648 195,840
Table 117.2 Overview of the data used for position error analysis SF-PPP with Orbit/ clock products Ultra Rapid IGS (rt) Final IGS (pp) RETICLE DLR (pp) Final IGS (pp)
Day of year (2009) 40–113 40–113 135–181
No. of days 74 74 47
Interval clock 15 min 15 min 30 s
No. of epochs 5,048,930 5,947,269 3,389,443
135–181
47
30 s
3,813,765
rt real-time, pp post processing
938
R.J.P. van Bree et al.
software. The following columns are the same as described with Table 117.1. For ionosphere correction, Final GIM from IGS is used for processing with Final products, and the 1 day predicted Ultra Rapid GIM from CODE is used with the processing of the Ultra Rapid and RETICLE clocks. The DCB files and Earth Rotation Parameters of most recent date are used when processing was in real-time, and the final versions of these files are used when post-processing is performed.
117.4 Test Results This section discusses the results of the tests that are described in the previous section. The results of the static tests will be given, both the clock error domain and the position error domain results.
117.4.1 Clock Errors The Ultra Rapid and Final clocks have a different absolute reference time, and the RETICLE clocks are constrained to the Ultra Rapid clocks as discussed by Hauschild and Montenbruck (2008). Therefore, the effect of the different reference times, which manifests itself epoch wise in a mean clock offset of the entire constellation, must be removed in the direct comparison of the clock products. The results of the direct comparison is given in Table 117.3. In this table, four groups of satellites are distinguished, based on their on-board clock. For each group the mean, standard deviation (STD) and 95% value of the clock differences are given. The block IIA Cs clocks are Cesium based and are present on the oldest part of the GPS satellite gamma, resulting in large deviations from the Final clock products (see first row of
Table 117.3). However, a large improvement is clearly visible for the RETICLE product compared to the Ultra Rapid product. The 95% value of the clock error is reduced by a factor of 8. The block IIA Rubidium clocks are present on the latest versions of the block IIA satellite and operate on the at relative short time spans more accurate and stable Rubidium standard. Ninety-five percent values improve by a factor of 3 as compared to predicted Ultra Rapid products (see second row Table 117.3). The block IIR and IIR-M satellites, both with Rubidium clocks show results up to a factor of 4 of improvement in clock difference between Ultra Rapid and RETICLE clock products (see third and fourth row in Table 117.3). Most of the GPS satellites today are of IIR(-M) type, 18 out of 30, and as for the Rubidium clocks the score is even 23 out of 30. In the near future the Cs clock type satellites are phased out and replaced by Rubidium clocks, resulting in better clock performance. In the final row of Table 117.3 a weighted average of the clock differences is given. These values indicate that overall values improve by a factor of 5–6. IGS states that the standard deviation of their predicted Ultra Rapid clocks is around 1.5 ns, which is in good agreement with the value found here, however the reason why IGS temporarily removed the block IIA Cs clocks from their predicted Ultra Rapid products in July this year (IGS 2009) is understandable when looking at Table 117.3.
117.4.2 Position Errors The real improvement when using RETICLE clocks over predicted Ultra Rapid ones becomes visible with Precise Point Positioning. The more accurate clocks will produce more accurate position results, but not by
Table 117.3 Clock difference results between final and predicted ultra rapid clocks and between Final and RETICLE clocks Final–ultra rapid Satellite block type IIA Cs IIA Rb IIR Rb IIR-M Rb Weighted average
Final–RETICLE Mean (ns) 0.09 0.42 0.16 0.24 0.20
STDa (ns) 4.91 1.82 1.27 1.05 2.11
95% (ns) 9.74 3.48 2.32 2.10 4.09
No. of samples 99,424 75,653 187,114 93,742 455,933
Mean (ns) 0.21 0.21 0.04 0.32 0.16
STD (ns) 0.51 0.53 0.24 0.29 0.36
95% (ns) 1.20 1.14 0.48 0.80 0.82
No. of samples 1,132,972 774,641 1,993,492 997,722 4,898,827
Clocks are separated by block type and given are the mean, standard deviation and 95% value of the differences. Weighted averages weighted by the number of samples available per block type a Value of STD given by IGS is 1.5 ns
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Table 117.4 Mean, standard deviation and 95% position errors in the North, East and up direction from SF-PPP with four different sets of orbit and clock products North UR Final 40–113 40–113 Mean 0.03 0.06 (m) STD 0.63 0.25 (m) 95% 1.28 0.50 (m)
East
Up
RETICLE Final UR Final RETICLE Final UR Final RETICLE Final 135–181 135–181 40–113 40–113 135–181 135–181 40–113 40–113 135–181 135–181 0.01 0.05 0.02 0.03 0.01 0.04 0.00 0.04 0.15 0.13 0.25
0.25
0.45
0.22
0.21
0.20
1.18
0.43
0.42
0.46
0.49
0.50
0.90
0.43
0.41
0.41
2.33
0.84
0.86
0.92
Ultra Rapid over the days 40–113 (UR, 40–113) is compared with Final over the same period (Final, 40–113), and RETICLE (135–181) is compared with Final over the days 135–181
the factor as found in the previous section. When clock products are used, Ultra Rapid or RETICLE, always the associated orbits must be used. The position results which are presented here are compared to the very well known position of the receiver. The results of the SF-PPP processing are presented in Tables 117.2 and 117.4. During the period between day 40 (Feb 9) and 113 (April 23) the real-time software was running using Ultra Rapid clock products. To compare the results, the same days were post-processed using Final orbits, clocks and ionospheric maps. The only difference between real-time and post-processing lies in the actuality of the earth rotation parameters and DCBs. Secondly, the RETICLE results are compared with the results using Final products over the same period (day 135–181). This period is different from the period used for the evaluation of the Ultra Rapid product because the RETICLE clocks were only available from day 135 on. Evaluating the results in Table 117.4 shows that with the use of the RETICLE clocks the position results improve significantly over the results when using predicted Ultra Rapid products. Standard deviation and 95% errors improve by a factor of 2–3 in all directions, and certainly the improvement in the up direction is significant. When compared to the Final products the RETICLE products perform equally and sometimes even better in this research, which is a very good result. In the IGS Final products, PRN01 was included for several days of our test period between day 135 and 181. By intentionally excluding this new satellite (SVN49) in our processing, results with Final products seem to improve slightly (e.g. the up-component by 10%). So far, PRN01 has not been included in the RETICLE products.
It must be kept in mind that the evaluation site is located at 52 latitude where ionosphere activity is quite low. The period of observations lies between the months February and July, leading to different Final results between the first and second part of this period, days 40–113 versus days 135–181. When evaluating this in particular, the error in the up directions shows a larger bias during the second part of the period when the ionosphere tends to induce larger delays and to be more turbulent. The above result opens a lot of new possibilities for the use of real-time and single frequency PPP. The latter having the advantage of much cheaper receivers than the dual frequency opponent. Conclusions
A significant improvement of the position accuracy can be gained with the RETICLE orbit-/clock products over the predicted Ultra Rapid products in single frequency precise point positioning. Accuracies improve by a factor of 2–3, resulting in a standard deviation of ~30 cm for the horizontal coordinates and a standard deviation of ~40 cm for the vertical (95% values of ~50 and ~90 cm, respectively). These results are obtained evaluating 47 days of data, producing a sound statistical basis for these conclusions. The RETICLE clocks products are more accurate than the Ultra Rapid predicted clocks. The analysed predicted Ultra Rapid clocks products hold a standard deviation of around 2 ns, and the standard deviation of the differences of the RETICLE clocks with the IGS Final ones reach up to only 0.4 ns.
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The use of real-time satellite clocks opens up a lot of new possibilities for the usage of low cost real-time single frequency PPP in the market. Two major advantages of this method over the dual frequency PPP solutions are the cost of the receiver and the fast convergence time of the solution, especially when the order of accuracy that is required lies at the decimeter level. More research should be performed on the performance of the real-time SF-PPP with RETICLE clocks at other locations and especially at low latitudes. Also other sources of real-time clocks and improved models or maps for ionospheric delays form another topic of research within real-time SFPPP.
References Gao Y, Zhang Y, Chen K (2006) Development of a real-time single-frequency precise point positioning system and test results. Proceedings of ION GNSS 2006, Fort Worth, TX, 26–29 September 2006, pp 2297–2303
R.J.P. van Bree et al. Hauschild A, Montenbruck O (2008) Real-time clock estimation for precise orbit determination of LEO-satellites. In: Proceedings of the ION GNSS meeting 2008, Savannah, GA, 16–19 September 2008 Heroux P, Kouba J (1995) GPS precise point positioning with a difference. Geomatics’95, Ottawa, ON, 13–15 June 1995 IGS (2009) mail 5965 http://igscb.jpl.nasa.gov/mail/igsmail/ 2009/msg00090.html, and IGS mail 5969 http://igscb.jpl. nasa.gov/mail/igsmail/2009/msg00094.html Kouba J (2009) A guide to using international GPS service (IGS) products. http://igscb.jpl.nasa.gov/components/usage. html Le AQ (2004) Achieving decimeter accuracy with single frequency standalone GPS positioning. In: 17th international technical meeting of the Satellite Division of Navigation (ION GNSS), Long Beach, CA, 21–24 September 2004, pp 1881–1892 Le AQ, Tiberius CCJM (2007) Single-frequency precise point positioning with optimal filtering. GPS Solut 11(1):61–69 Øvstedal O (2002) Absolute positioning with single frequency GPS receivers. GPS Solut 5(4):33–44 Weber G, Dettmering D, Gebhard H (2005) Networked transport of RTCM via internet protocol (NTRIP). In: International Association of Geodesy symposia, vol 128, Springer, Berlin, pp 60–64
Improving the GNSS Attitude Ambiguity Success Rate with the Multivariate Constrained LAMBDA Method
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G. Giorgi, P.J.G. Teunissen, S. Verhagen, and P.J. Buist
Abstract
GNSS Attitude Determination is a valuable technique for the estimation of platform orientation. To achieve high accuracies on the angular estimations, the GNSS carrier phase data has to be used. These data are known to be affected by integer ambiguities, which must be correctly resolved in order to exploit the higher precision of the phase observables with respect to the GNSS code data. For a set of GNSS antennae rigidly mounted on a platform, a number of nonlinear geometrical constraints can be exploited for the purpose of strengthening the underlying observation model and subsequently improving the capacity of fixing the correct set of integer ambiguities. A multivariate constrained version of the LAMBDA method is presented and tested here.
118.1 Introduction Attitude determination is an important issue in remote sensing applications: the knowledge of the orientation of the platform which carries the sensors (radars, lasers, etc.) is required for the pointing procedures. Although the accuracy of a stand-alone GNSS attitude system might not be comparable with the one obtainable with other modern attitude sensors, a GNSS-
G. Giorgi S. Verhagen (*) P.J. Buist Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, PO Box 5058, 2600 Delft, The Netherlands e-mail: [email protected] P.J.G. Teunissen Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, PO Box 5058, 2600 Delft, The Netherlands Department of Spatial Sciences, Curtin University of Technology, GPO Box U1987, Perth, WA 6845, Australia
based system presents several advantages. The main assets are that it is driftless and it requires less maintenance. Many works investigated the feasibility and performance of GNSS Attitude Determination, see e.g. [1–6]. The key for a precise attitude estimation is the ambiguity resolution process, since only when the integers inherent to the GNSS carrier phase observations are correctly fixed one is able to exploit the carrier phase data, which are of two orders of magnitude more accurate than the GNSS code observations. In this contribution we focus on the problem of fixing the correct integer ambiguities for data collected on a frame of antennae firmly mounted on a rigid platform: the relative positions between the antennae are assumed to be known and constant. In such configurations, the baselines lengths and the angles between them are known, resulting in a set of nonlinear constraints posed on the baseline vectors which can be exploited to strengthen the underlying observation model. The set of GNSS phase and code observations is cast into a linearized system solvable
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_118, # Springer-Verlag Berlin Heidelberg 2012
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in a least-squares sense, where both the integerness of the ambiguities and the constraints on the baselines must be fulfilled. The method is based on an extension of the Integer Least Squares (ILS, [7]) principle, employed to solve in a least-squares sense a linear system of equations where some of the unknowns are integer-valued. A well-known mechanized implementation of the ILS principle is the Least squares AMBiguity Decorrelation Adjustment (LAMBDA) method [8], widely used for its high computational efficiency. This method has been recently extended to accommodate those baseline applications where the distance between the antennae is known and constant: the baseline Constrained LAMBDA method was introduced in [9, 10]. The inclusion of the baseline length constraint results in a large improvement in the success rate, as shown in [11–15]. We here present and test the performance of the multivariate generalization of the Constrained LAMBDA method [16], which solves the model where the full set of nonlinear geometrical constraints is taken into account, i.e. the different baseline lengths and their relative positions.
118.2 Modeling of the Multi-antennae GNSS Observations We consider a set of m þ 1 antennae (m independent baselines) simultaneously tracking the same n þ 1 GNSS satellites on a single frequency. The set of linearized Double Difference (DD) GNSS phase and code observations obtained on the m baselines can be cast into a Gauss–Markov model as follows: EðYÞ ¼ AZ þ GB Z 2 Znm ; B 2 R3m DðvecðYÞÞ ¼ QY
(118.1)
where E(·) is the expectation operator Y is the 2n by m matrix whose columns are the code and phase DD observations derived at each baseline: Y ¼ ½ y1
y2
...
ym
Z is the n by m matrix whose columns are the integervalued ambiguities for each baseline:
Z ¼ ½ z1
z2
. . . zm
B is the 3 by m matrix whose columns are the realvalued baseline coordinates: B ¼ ½ b1
b2
. . . bm
A is the 2n by n design matrix which contains the carrier wavelength G is the 2n by 3 design matrix of line-of-sight vectors D(·) is the dispersion operator QY is the 2nm by 2nm variance–covariance matrix of the vector of observations vec (Y) We make use of the vec operator, which stacks the columns of a matrix below each other, to define the variance–covariance matrix of the vector of observations vec (Y). We assume that the antennae are separated by short baselines, for which the atmospheric effects can be neglected, and the only real-valued unknowns to be estimated are the 3m coordinates of the baseline vectors. Also, the short baseline hypothesis allows us to make use of a unique matrix of line-of-sight vectors G for all the baselines. A Gaussian-distributed error is assumed on the observables Y. Aiming to estimate a platform’s orientation solely via GNSS measurements, two or more antennae are assumed to be firmly mounted on the platform, which is here considered as a rigid body. We introduce a system of body axes (local) u1u2u3 taken as to have the first axis u1 aligned with the first baseline, the second axis u2 perpendicular to u1 and lying in the plane formed by the first two baselines, and the third axis u3 oriented that u1u2u3 form an orthogonal frame (see Fig. 118.1, where fi is the ith baseline, and fij indicates the jth coordinate of the baseline i). The baseline coordinates expressed in the local frame are collected in matrix F, and the relationship between these and the coordinates B expressed in the global frame x1x2x3 is: B¼RF
(118.2)
where R is the orthogonal matrix which describes the relative orientation between the local and global frames, i.e. the attitude of the platform. For notational convenience, the rotation matrix and the local baseline coordinates F are defined as [16]
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subject to the integer constraint on Z. Via a modification of the LAMBDA method, the model (118.4) is solvable in a rigorous least-squares sense taking into account all the different constraints, which are the integer nature of the entries of Z and the orthogonality of the rotation matrix R. This is demonstrated in the following section.
118.3 Integer Least Squares
Fig. 118.1 The first baseline f1 (Main antenna – Aux1 antenna) defines the first body axis u1, while the second body axis u2, perpendicular to u1, lies in the plane formed by f1 and f2 (Main antenna – Aux2 antenna). u3 is taken as to form an orthogonal frame
(
2
m3 q¼3
( (
3 f11 f21 f31 fm1 7 6 : RF ¼ ½ r1 ; r2 ; r3 4 0 f22 f32 fm2 5 0 0 f33 fm3 f11 f21 : RF ¼ ½ r1 ; r2 0 f22
We aim to solve for the model (118.4) in a rigorous least-squares sense, minimizing the weighted squared norm of the residuals. The solution of the model (118.4) is derived with a three-steps procedure: first obtain a float solution, then search for the integer ambiguities and finally extract the orthogonal matrix R. In this section we describe each of these steps.
m¼2 q¼2 m¼1 q¼1
: RF ¼ ½ r1 ½ f11 (118.3)
where q is a parameter introduced to cope with the case of m < 3 baselines. The relation (118.2) is a linear transformation, which changes the unknowns of the problem: the estimation of the baseline coordinates B turns into the estimation of the components of the matrix R, of which only three are independent. Hence, in addition to the integer constraint on the matrix Z, also the orthogonality of the matrix R has to be considered. This allows to rewrite the set of baseline observations (118.1) as EðYÞ ¼ AZ þ GRF Z 2 Znm ; R 2 O3q DðvecðYÞÞ ¼ QY
(118.4)
This is the model that we aim to solve in a leastsquares sense. The standard LAMBDA method can be employed to solve the system when the constraint on the matrix R is disregarded, being the system solely
118.3.1 The Float Solution The float solution of (118.4) is the least-squares solution obtained disregarding both the integerness of the matrix Z and the orthogonality of R: ! ^ Im AT vecðZÞ Q1 ¼ N Y vecðYÞ T ^ F G vecðRÞ Im AT Q1 (118.5) N¼ Im A FT G Y T FG where is the Kronecker product and we made use of the property vecðX1 X2 X3 Þ ¼ X3T X1 vecðX2 Þ ^ and vecðRÞ ^ are the float estimators of Z and R; vecðZÞ their v–c matrices are obtained via the inversion of the normal matrix N: QZ^ QZ^R^ ¼ N 1 (118.6) QR^Z^ QR^ If one assumes that the matrix Z is known, the conditional solution of R (conditioned on the knowledge of the matrix Z) is obtained as ^ ^ ^ Q ^ ^Q1 vecðRðZÞÞ ¼ vecðRÞ RZ Z^ vecðZ ZÞ (118.7)
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^ The precision of the conditional solution RðZÞ is described by the v–c matrix ¼ QR^ QR^Z^Q1 QZ^R^ QRðZÞ ^ Z^
(118.8)
estimation of the matrix ZU involves a direct search inside a set of suitable integer candidates: o n ^ 2 w2 OU w2 ¼ Z 2 Znm j vecðZ ZÞ Q^ Z
(118.11)
118.3.2 The Search for the Integer Minimizer ^ the conditional solution Given the float estimator Z, ^ RðZÞ and their v–c matrices, we can write the sum-ofsquares decomposition of the weighted squared norm of the residuals of (118.4) as vecðYÞ ðIm AÞvecðZÞ ðFT GÞvecðRÞ 2 QY 2 2 ^ ^ ¼ vecðEÞ QY þ vec Z Z Q Z^ 2 ^ þ vec RðZÞ R Q ^ RðZÞ
(118.9) where E^ is the matrix of least-squares residuals. The decomposition shows that the last term can always be made zero for any value assumed by Z, by taking ^ R ¼ RðZÞ, if the orthogonality of R is disregarded. The minimization of the least-squares residuals then reduces to the well known case of finding the integer minimizer of the second term, and the standard LAMBDA method can be directly applied. When the orthogonality constraint on the matrix R is taken, the ^ last term generally differs from zero, since RðZÞ is usually non orthogonal: this leads to a modification of the search algorithm to be adopted, resulting in a multivariate constrained version of the LAMBDA method.
The set OU, which geometrically draws an hyper^ and size/shape driven by ellipsoid centered in vecðZÞ the entries of QZ^, is evaluated and the integer matrix Z that minimizes the squared norm (118.10) is extracted. The LAMBDA method is applied to perform the search in an efficient and fast way; it works by decorrelating the ambiguities performing an admissible (i.e. which preserves the integerness of the variables) transformation: the effect of the decorrelation is to have a reduced set of integer candidates, among which the matrix ZU is quickly extracted.
118.3.2.2 The Multivariate Constrained LAMBDA Method The full-constrained least-squares minimization of (118.9) is obtained by taking the minimization with respect both the matrix of ambiguities Z and the orthogonal matrix R: ZC ¼ arg min CðZÞ Z2Znm 2 ^ 2 þ vecðRðZÞ ^ CðZÞ ¼ vecðZ ZÞ RðZÞÞ Q^ Q^ Z
RðZÞ
(118.12) with vecðRðZÞ ^ RÞ 2 vecðRðZÞÞ ¼ arg min Q^ 3q R2O
RðZÞ
(118.13)
118.3.2.1 The LAMBDA Method Disregarding the orthogonality of R, the integervalued minimizer of (118.9) equals vecðZ ZÞ ^ 2 ZU ¼ arg min Q^ nm Z2Z
(118.10)
Z
The matrix ZU has the minimum distance from the float solution Z^ in the metric defined by QZ^: since no closed-form solution of (118.10) is known, the
where O3q is the class of 3 q orthogonal matrices, i.e. RTR ¼ Iq. The integer minimizer ZC weighs the sum of two terms: the first is the distance with respect to the float solution Z^ weighted by QZ^, and the second ^ is the distance between RðZÞ and the solution of the nonlinear constrained least-squares problem (118.13). The latter gives the orientation of the platform RðZÞ by minimizing in a least-squares sense the distance from ^ the matrix RðZÞ, subject to the orthogonal constraint. Note that for the single-baseline case (m ¼ 1) the
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Improving the GNSS Attitude Ambiguity Success Rate
problem reduces to the one addressed in [9, 10, 17]: the method discussed here is a multivariate generalization. The minimizer of (118.12) is searched in the set defined as
O w2 ¼ Z 2 Znm jCðZÞ w2
R: via a suitable parameterization of the rotation matrix, e.g. Euler angles or Quaternions [18], the orthogonality is implicitly fulfilled, and the leastsquares solution of (118.15) can be solved for example with the Newton method.
(118.14)
Minimizing the cost function (118.12) in the set O(w2) is a non-trivial task: the evaluation of C(Z) involves the computation of a nonlinear constrained least-squares problem, and if the set contains a large number of candidates the search is very timeconsuming. Hence, the choice for the scalar w is an important issue, since it strongly affects the time dedicated to the minimization process. To make the search more time-efficient, and to cope with both the problems of setting the value of w and computing (118.13) a large number of times, the two algorithms coined as the Expansion and the Search and Shrink approaches were developed [12–15]: by using two functions that provide a lower and an upper bound for the cost function C(Z) and that are easier to evaluate (i.e. the computation of (118.13) is avoided), the search for the integer minimizer ZC is computed efficiently and in a much faster way.
118.3.3 The Attitude Solution The two above mentioned search methods provide the ILS minimizer Z of the expression (118.9), respectively with (Constrained LAMBDA, Z ¼ ZC ) or without (LAMBDA, Z ¼ ZU ) considering the orthogonality of 6 ZC . Given the integer R. Note that in general ZU ¼ minimizer resolved, the conditional attitude solution ^ is obtained as in (118.7): the solution RðZÞ is characterized by a better accuracy (118.8), but it is in general non-orthogonal. In order to obtain the sought orthogonal attitude matrix, the following nonlinear constrained least-squares problem has to be solved: ZÞÞ ¼ arg min vecðRð ^ ZÞ RÞ 2 vecðRð Q^ 3q R2O
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RðZÞ
(118.15) where Z ¼ ZU for the LAMBDA method and Z ¼ ZC for the Constrained LAMBDA method. The nonlinearity of the problem comes from the orthogonality of
118.4 Testing the Method The method presented was tested with simulated data and with data collected during a kinematic experiment. All the data sets were processed with both the LAMBDA method and the Constrained LAMBDA method, in order to compare the different performance obtained in terms of single-epoch, single-frequency success rate.
118.4.1 Simulation Results Different sets of data were generated via a Monte Carlo simulation, reproducing the set of baseline observations according to the model (118.4). Table 118.1 summarizes the set-up of the simulations: from the actual GPS constellation on 22 January 2008 (as seen from Delft, The Netherlands), we selected five to eight satellites, with corresponding PDOP values between 4.2 and 1.8. Two baselines were simulated, of 1 and 2 m length, forming an angle of 100 . For each of the 24 scenarios, 105 samples were generated, aiming to extract an accurate estimation of the success rate, defined as the ratio of samples where the correct integer ambiguity matrix has been fixed and the total number of samples. The data sets were processed applying the LAMBDA and the Constrained LAMBDA methods as described in Sect. 118.3. Table 118.2 shows the single-frequency, single-epoch Table 118.1 Simulation set up Frequency Number of satellite (PRNs) 5/6/7/8 Undifferenced code noise sp (cm) Undifferenced phase noise sf (mm) Baselines fi (x1, x2, x3) Samples simulated
L1 Corresponding PDOP 4.19/2.14/ 1.92/1.81 30-15-5 3-1 ~ f 1 ¼ ½1; 0; 0m ~ f 2 ¼ ½0:35; 1:97; 0m 105
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Table 118.2 Simulation results: single-frequency, singleepoch success rates for the unconstrained and constrained LAMBDA methods. Success rates higher than 99.9% are stressed sf (mm) 3
1
sp (cm)
30
N
Two-baselines success rate, LAMBDA Two-baselines success rate, constrained LAMBDA 0.17 5.69 83.87 0.55 10.18 95.48 99.60 99.94 100 100 100 100 9.81 56.89 96.83 30.12 81.35 100 99.99 100 100 100 100 100 31.97 73.07 99.74 61.16 91.33 100 99.99 100 100 100 100 100 81.72 93.12 99.99 99.99 100 100 100 100 100 100 100 100
5 6 7 8
15
5
30
15
5
Table 118.3 Vessel experiment results: single-epoch, single frequency success rates for the LAMBDA and Constrained LAMBDA methods Baselines
Single-epoch, single frequency success rate (%)
LAMBDA
Constrained LAMBDA 99.96 8996 1–2 + 1–3 54.08 4867 9000 9000
success rates for the methods: the improvement is large especially for the weaker scenarios (lower number of satellite/higher levels of noise), where the difference between the methods is significant. For example, the weakest simulated data set, with five available satellites and high noise values, shows an improvement from a low 0.17–99.60%. The number of correctly fixed samples for the Constrained LAMBDA method is always higher than 99.6%: as expected, the strengthening of the underlying model due to the embedded geometrical constraints substantially affects the capacity of fixing the correct integer ambiguity vector. The two-baseline case shows success rates higher than 99% on all the data sets processed, obtaining a 100% success rate on 20 out of 24 data sets simulated.
(1) a choke-ring antenna connected to an Ashtech receiver; (2) an antenna connected to a Leica SR530 receiver; (3) an antenna connected to a Novatel OEM3 receiver. The baseline lengths between the antennae are 2 m (antennae 1–2) and 1.5 m (antennae 1–3), and the coordinates F in the local plane u1u2 are 1:5 1:89 f11 f21 ¼ ðmÞ F¼ 0 0:74 0 f22 The vessel sailed for about 2.5 h, collecting a total of 9,000 epochs of GPS observations. The number of tracked GPS satellite varied between seven and eight, except for the first thousands epoch, were only data from six GPS satellite were stored. The PDOP values were between 2.1 and 4. We processed both the baselines (1–2 þ 1–3) embedded in the model (118.4). The LAMBDA method gave a 54.08% single-epoch, single-frequency success rate, thus providing the correct attitude solution for about half the epochs processed. As expected, the number of correctly fixed epochs largely increased when the Constrained LAMBDA method was employed: only four epochs were incorrectly fixed, thus achieving 99.96% success rate. This is due to the stronger observation model obtained by including the known geometrical constraints on the different baselines. Each of the (either correctly or incorrectly) fixed ambiguity matrices were used to compute the full attitude of the vessel in the East-North-Up (ENU) frame according to (118.15). Figure 118.2 shows the Heading, Elevation and Bank angles computed epoch by epoch according to the output of the LAMBDA algorithm: the plot is rather scattered due to the low success rate (54.08%), which ZÞ. resulted in many wrong attitude solutions Rð The Constrained LAMBDA method, providing for 99.96% of the epochs the correct integer matrix, produces the plots of Fig. 118.3: as shown, the singleepoch full attitude solution was available for the entire (only four epoch missed) duration of the experiment. Conclusions
118.4.2 Experimental Results We tested the new method on a set of data collected onboard a vessel during a kinematic experiment held in Delft, The Netherlands. The vessel was equipped with three couples of antennae-receivers
In this contribution it is analyzed how to resolve the integer ambiguities in a rigorous ILS sense for GNSS Attitude Determination applications. It is firstly described how to model the set of GNSS phase and code observations collected on a frame of antennae mounted on the same platform. Then the
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a
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a
Heading angle
Heading angle
b
b
Elevation angle
Elevation angle
c
c
Bank angle
Bank angle
Fig. 118.2 Single-epoch/single-frequency full attitude solution, LAMBDA method
Fig. 118.3 Single-epoch/single-frequency full attitude solution, Multivariate Constrained LAMBDA method
Constrained LAMBDA method is introduced, which solves the problem of minimizing the norm of the least-squares residuals taking into account both the integerness of the ambiguities and the orthogonality of the rotation matrix. The latter constraint is derived assuming the relative positions of the antennae as known and constant. The strengthening of the observation model, due to the inclusion of the geometrical constraints, improves the capacity of fixing the correct set of integer ambiguities, as shown via simulations and as tested on data
collected during a kinematic experiment. The high fixing rates obtained from the tests suggest that for GNSS Attitude Determination applications the single-epoch, single-frequency ambiguity resolution is feasible when either the quality of the observation is high or the number of constrained baselines on the platform increases. During the kinematic test was proven that already with a three antennae/two baselines configuration the Constrained LAMBDA method is capable of providing the correct full attitude solution almost at every epoch.
948 Acknowledgements P.J.G. Teunissen is the recipient of an Australian Research Council Federation Fellowship (project number FF0883188): this support is gratefully acknowledged. The research of S. Verhagen is supported by the Dutch Technology Foundation STW, applied science division of NWO and the Technology Program of the Ministry of Economic Affairs.
References 1. Cohen CE (1992) Attitude determination using GPS. Ph.D. thesis 2. Crassidis JL, Markley FL, Lightsey EG (1997) A new algorithm for attitude determination using global positioning system signals. AIAA J Guid Contr Dynam 20(5):891–896 3. Dai L, Ling KV, Nagarajan N (2004) Real-time attitude determination for microsatellite by LAMBDA method combined with Kalman filtering. In: Proceedings of the 22nd AIAA international communications satellite systems conference and exhibit 2004 (ICSSC), Monterey, CA 4. Li Y, Zhang K, Roberts C, Murata M (2004) On-the-fly GPS-based attitude determination using single- and double-differenced carrier phase measurements. GPS Solut 8:93–102 5. Madsen J, Lightsey EG (2004) Robust spacecraft attitude determination using global positioning system receivers. J Spacecr Rocket 41(4):635–643 6. Psiaki ML (2006) Batch algorithm for global-positioningsystem attitude determination and integer ambiguity resolution. J Guid Control Dyn 29(5):1070–1079 7. Teunissen PJG (1994) Integer least-squares estimation of the GPS phase ambiguities. In: Proceedings of the international symposium on kinematic systems in geodesy, geomatics and navigation, pp 221–231
G. Giorgi et al. 8. Teunissen PJG (1993) Least squares estimation of the integer GPS ambiguities. In: Invited lecture, section IV theory and methodology, IAG general meeting, Beijing. LGR series No. 6, Delft Geodetic Computing Center, Delft University of Technology 9. Teunissen PJG (2006) The LAMBDA method for the GNSS compass. Artif Satell 41(3):89–103 10. Teunissen JG (2008) GNSS ambiguity resolution for attitude determination: theory and method. In: Proceedings of the international symposium on GPS/GNSS, Tokyo, Japan 11. Park C, Teunissen PJG (2003) A new carrier phase ambiguity estimation for GNSS attitude determination systems. In: Proceedings of international GPS/GNSS symposium, Tokyo 12. Buist PJ (2007) The baseline constrained LAMBDA method for single epoch, single frequency attitude determination applications. In: Proceedings of ION GPS 13. Park C, Teunissen PJG (2008) A baseline constrained LAMBDA method for integer ambiguity resolution of GNSS attitude determination systems. J Control Robotic Syst (Korean) 14(6):587–594 14. Giorgi G, Teunissen PJG, Buist PJ (2008) A search and shrink approach for the baseline constrained LAMBDA: experimental results. In: Proceedings of the international symposium on GPS/GNSS, Tokyo, Japan 15. Giorgi G, Buist PJ (2008) Single-epoch, single frequency, standalone full attitude determination: experimental results. 4th ESA workshop on Satellite Navigation User Equipment Technologies, NAVITEC 16. Teunissen PJG (2007) A general multivariate formulation of the multi-antenna GNSS attitude determination problem. Artif Satell 42(2):97–111 17. Teunissen PJG (2010) Integer least squares theory for the GNSS compass. J Geod 84:433–447 18. Battin RH (1987) An introduction to the mathematics and methods of astrodynamics. AIAA Education Series
An Intelligent Personal Navigator Integrating GNSS, RFID and INS
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G. Retscher
Abstract
Personal navigation services usually rely on GNSS positioning and therefore their use is limited to open areas where enough satellite signals can be received. If the user moves in obstructed urban environment or indoors, alternative location methods are required to be able to locate the user continuously. In our approach GNSS positioning is combined with a MEMS-based Inertial Measurement Unit for continuous position determination. In addition, Radio Frequency Identification (RFID) Location Methods are employed. In RFID positioning the location estimation can be based on signal strength measurements (i.e., received signal strength indication RSSI) which is a measurement of the power present in a received radio signal. Then the mobile receiver can compute its position using various methods based on RSSI. Three different methods have been developed and investigated, i.e., cell-based positioning, trilateration using ranges to the surrounding RFID transponders (so-called RFID tags) deduced from RSSI measurements, and RFID location fingerprinting. In most common RFID applications positioning is performed using cell-based positioning. In this case, RFID tags can be installed as active landmarks with known location. The user is carrying a RFID reader and is positioned using Cell of Origin (CoO). GNSS and RFID are then integrated with INS positioning for continuous position determination. INS measurements would be utilized to calculate the trajectory of the user based on the method of strap down mechanization. Since the INS components produce small measurement errors that accumulate over time and cause drift errors, the positions determined by RFID or GNSS are needed regularly to reduce the drift. All observations are integrated in a Kalman filter to estimate the user’s position and velocity. By integrating the above mentioned measurements into an intelligent software package the developed personal navigator will enable to determine the mobile user’s position continuously, automatically and ubiquitously.
G. Retscher (*) Institute of Geodesy and Geophysics, Vienna University of Technology, Gusshausstrasse 27–29, 1040 Vienna, Austria e-mail: [email protected] S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_119, # Springer-Verlag Berlin Heidelberg 2012
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119.1 Introduction In mobile positioning, alternative location methods for absolute positioning are needed in areas where no GNSS position determination is possible. In the presented approach the location of a pedestrian user in urban and indoor environments will be determined by an integration of radio-frequency identification (RFID) positioning and INS. Using RFID technology, data can be transmitted from RFID tags to a reader via radio waves even without line-of-sight contact. These data can include the position of the RFID tags in addition to their respective ID. RFID tags can be deployed as active landmarks or other points with known coordinates in the surrounding environment. The most common method for positioning using RFID is CoO. This algorithm determines the location of the user in a cell (signal coverage area) around the RFID tag with a size defined by the maximum read range of the RFID signals (see Fu and Retscher 2009c). The achievable positioning accuracy thereby depends on the size of the cell defined by the maximum read range of the signal. Using long range active RFID this read range can be up to 100 m. Higher positioning accuracies can be achieved using trilateration. For the signal strength to distance conversion several models have been developed and tested. It was found that a simple polynomial relationship between the signal strength and the range provides reasonable results. Then positioning accuracies on the few meter level can be achieved for a continuously moving user. In case of RFID location fingerprinting, RSSI is measured in a training phase at known locations and stored in a database. In the positioning phase, these measurements are used together with the current measurements to obtain the current location of the user. For the estimation of the current location different approaches have been employed and tested. Then similar positioning accuracies as in trilateration can be achieved (see Fu and Retscher 2009b). The disadvantage, however, of trilateration and fingerprinting is the required calibration in the off-line or training phase. Therefore a new method based on CoO was developed in the project. For continuous position determination of a mobile user we combine the new RFID CoO method with an Inertial Navigation System (INS) because RFID tags
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cannot necessarily be deployed in a sufficient density to cover all areas of interest, and the INS bridges the gaps between the individual RFID cells. The initial position can be determined using RFID CoO in the presence of at least one RFID tag or with GNSS in the presence of enough satellite signals for positioning. RFID CoO or GNSS absolute positioning are used for updating the relative positions derived by the INS and thus for mitigating INS-based drift errors. In this way, the trajectory of the user can be calculated continuously. For the trajectory determination using the INS we employ a strap down mechanization (see Titterton and Weston 2005). The estimated trajectory is corrected using the RFID position solution whenever the RFID reader detects the signal strength above a certain threshold from the nearest RFID tag in the surroundings. The location determined in this way ensures that the calculated position is closest to the true position of the RFID tag. We use a time-varying Kalman Filter (see Brown and Hwang 1997) to correct the position and velocity dynamics of the INS sensor. This filter is a generalization of the steady-state filter for time-varying systems with non-stationary noise covariance. The above concept has been implemented and tested in a real world environment. The test was conducted in an office building of the Vienna University of Technology and its surroundings. From the test results it can be concluded that an integrated RFID and INS positioning is suitable for continuous position determination of a mobile user in challenging indoor and urban environments.
119.2 Active RFID for Positioning of a Pedestrian User In RFID positioning of a pedestrian, the location estimation is based on RSSI which is a measurement of the power present in a received radio signal (Fu 2008). The receiver can compute its position using various methods based on RSSI. In total, three different methods have been employed, i.e., cell-based positioning, trilateration using ranges to the surrounding RFID tags deduced from received signal strength measurements and RFID location fingerprinting. These technologies can be employed depending on the density of the RFID transponders (so-called tags) in the surrounding
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environment. For positioning with RFID either readers or tags can be placed at known location in the surrounding environment. We have chosen a lowcost concept where the less expensive tags are deployed in the surrounding environment at active landmarks (i.e., known location) or at regular distances. The mobile user is carrying a reader in form of a PC-card, which can be plugged into the mobile device (e.g. a pocket PC or laptop). The most straightforward method is cell-based positioning. The maximum range of the RFID tag defines a cell of circular shape in which a data exchange between the tag and the reader is possible. Several tags located in the smart environment can overlap and define certain cells with a radius equal the read range. The accuracy of position determination is defined by the cell size. Using active RFID tags the positioning accuracy therefore ranges between a few meters up to tens of meters. However, the accuracy could be improved by using the so-called time-based CoO (see Fu and Retscher 2009c). In time-based CoO two improvements of the standard cell-based positioning have been made to get a higher positioning accuracy. First of all, a threshold value is set to reduce the size of the cell. Secondly, the mean value of the corresponding time is calculated for all signal strength measurements above the threshold. As a result, the positioning accuracy is improved. The RFID time for each detected ID is the mean value of the corresponding time. The location determined in this way ensures that the calculated position is closest to the true position of the RFID tag. The approach takes the fact into account that the received signal strength is highly variable in indoor environments with a large number of obstacles and moving objects which affect the propagation of the RFID signals. For verification of the RFID time-based CoO measurements a tool was developed under the environment of Microsoft Visual Studio 2008 (see Fu and Retscher 2009c). If the user passes by an RFID tag a marker can be set in the program by a simple mouse click capturing the system time. This is used as an indication for the user currently being nearest to the true location of the RFID tag. The verification tool is called “time data capture tool”. The known RFID tag coordinates are regarded as true positions at this point of time. The location determined by the integration of RFID and INS at the corresponding point of time is the estimated position. The differences between these two positions are the estimated errors.
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As an alternative to time-based CoO, trilateration and location fingerprinting have been investigated. Trilateration can be employed if the ranges to several tags in the surrounding environment can be determined. Then these ranges are used for intersection. The range from the antenna of the reader to the antenna of the tag is deduced from the conversion of signal strength into distances. Strategies for the conversion of the signal strength measurements into distances are distinguished between indoor and urban outdoor environment (see Fu and Retscher 2009b). The highest positioning accuracies can be obtained with location fingerprinting. Location fingerprinting, however, is more costly and complicated in comparison to cell-based positioning and trilateration. For this method different advanced approaches have been developed (see Fu 2008). For the creation of the database in RFID location fingerprinting interpolation methods can be used, in order to achieve a further improvement of the positioning accuracy. To test the different methods experiments have been conducted in a test bed near and in the university building of the Vienna University of Technology in downtown Vienna. The conducted experiments showed that these approaches are suitable to navigate the user with different positioning accuracies, i.e., lower positioning accuracies on the several meter level in outdoor environment using cell-based positioning and higher positioning accuracies on the one meter level in indoor environments with trilateration, fingerprinting and time-based CoO. The selection of the suitable location method depends on the current number of available RFID tags in the surrounding environment and is performed by a developed intelligent software package which makes use of a knowledge-based system (see Fu and Retscher 2008).
119.3 RFID and INS Integration If a continuous position determination of a pedestrian is required by the application, the RFID time-based CoO can be combined with the measurements of a low-cost MEMS-based INS (see Renaudin et al. 2007) in addition. Then the INS bridges the gaps between the individual RFID cells as it is not economical to deploy RFID tags in such a density that cells overlap. RFID positioning is used for absolute positioning and INS for relative positioning. With the INS
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the trajectory of the user can be calculated from a start position determined by RFID positioning using dead reckoning. For updating the INS drift errors at regular time intervals again RFID positioning is employed. A combined positioning solution is obtained using a time-varying Kalman filter. Figure 119.1 shows the calculation steps to obtain the trajectory of a pedestrian user from a combination of RFID and INS. The INS sensor delivers acceleration accx, accy, accz and orientation data (unit quaternions q0, q1, q2, q3 or Euler parameters f, y, c) in the INS body frame (see Shuster 1993). Firstly, the input data are used to calculate the free acceleration accx_b, accy_b, accz_b and the rotation matrix from the body to the navigation frame Rnb; whereby, in the free acceleration the gravity and centrifugal force are not included, and the rotation matrix is used for the transformation between the body frame b and the navigation frame n. In the second step, the position of the user px_n, py_n, pz_n in the navigation frame is computed by integrating acceleration and velocity over time. Then the relative positions from the INS and absolute positions from
Fig. 119.1 Process flow of the calculation of the trajectory of a pedestrian user
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RFID are introduced as observations in a time-varying Kalman filter. Using the filter an optimal estimate of current user’s position, velocity and orientation is obtained. Further details about the process flow and the mathematic derivations behind can be found in Retscher and Fu (2009).
119.4 Urban Outdoor Environment Test Results Our approach has been tested for the positioning of a mobile user in outdoor urban environment. Figure 119.2 shows the test bed in the surroundings of the Vienna University of Technology in the fourth district of the city of Vienna. The path is about 550 m long and leads from the subway stop ‘Karlsplatz’ to an university building. The first part of the path is in a park (i.e., the ‘Resselpark’) where GNSS is mostly available. Then the user is walking along a narrow street (i.e., ‘Karlsgasse’) with 5-storey buildings where GNSS is either not available or larger positioning errors occur. The end point of the trajectory is the
Fig. 119.2 Outdoor trajectory from subway stop ‘Karlsplatz’ to an office building of the Vienna University of Technology
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Fig. 119.3 INS trajectory along the path from subway stop ‘Karlsplatz’ to an office building of the Vienna University of Technology (the dots in the left figure represent the location of the RFID tags)
entrance to an office building at the Vienna University of Technology. Due to obstruction of the satellite signals caused by the buildings it is necessary to employ an alternative absolute positioning method in addition to GNSS positioning in this area to be able to update the continuous INS positioning. In the test bed a total of 15 RFID tags has been installed on active landmarks (e.g. on entrances to buildings and other points of interest such as monuments). Figure 119.3 shows the resulting INS trajectory of the pedestrian along the selected path. The RFID time-based CoO method has been employed to update the INS positioning and to reduce the sensor drifts. It can be seen that the pedestrian user can be positioned continuously with the required precision of a few meters. The user is positioned on the correct side of the road on the walkway along Karlsgasse. A further improvement of the positioning result may be achieved with smoothing of the resulting trajectory.
119.5 Indoor Environment Test Results The test was conducted on the third floor in an office building of the Vienna University of Technology. The coordinates of the tags in a local reference
system were known. The origin of the local reference system was located at the start point near the elevator. The x-axis ran parallel to the middle line of the corridor and the y-axis is orthogonal to the x-axis. The route was 39.9 m long in total and could be divided into three rectangular sections that are next to each other. The trajectory started in front of the elevator and continued towards a general teaching room along a corridor around the corner. The second part of the route ran along the middle line of the corridor with a length of 25.9 m. The last part is in the general teaching room. In total, nine RFID tags were mounted in the test bed (ID numbers 80–88; see Fig. 119.4). Tags were suspended from the ceiling at a height of 2.0 m above the ground. The first tag was located very close to the start point in front of the elevator, in order to be able to correct the trajectory determined by the INS shortly after the beginning. Two tags were located very close before and after the turning points between the orthogonal sections of the track to ensure that the determined position was correct when turning, since a sharp turn can lead to larger deviations in heading due to the high sensitivity of the INS gyros. Three further RFID tags were evenly spaced along the corridor, in order to update the INS measurement regularly.
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Fig. 119.4 Indoor trajectory on the third floor of an office building of the Vienna University of Technology (TP stands for turning point; the ID numbers 80–88 represent the RFID tags’ locations)
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Fig. 119.5 Signal reception in the test bed using RFID cellbased positioning (RSSI of ID numbers 80–86 are shown)
In the experiments conducted, cell-based positioning (i.e., RFID time-based CoO) was employed. Figure 119.5 shows the signal reception using RFID time-based CoO in the test bed. It includes the information of the RSSI, the time the tag was detected, and the ID of the tag. The user allows for rapid collection of the data while moving from one tag to the next. Each tag is identified by an ID presence or absence. As indicated by the circles, RSSI with a value higher than the threshold value of 46 dBm were selected. If the user passes by an RFID tag a marker was set by simple mouse click capturing the system time (socalled “time data capture tool”; see Fu and Retscher 2009c). This was done for verification of the RFID time-based CoO positioning method. Then the user has been nearest to the true location of the RFID tag. The known RFID tag coordinates are regarded as true positions at this point of time. The location determined
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by the integration of RFID and INS at the corresponding point of time is the estimated position. The differences between these two positions are the estimated errors. The maximal estimated error of the x-coordinate is 1.62 m, while the maximal estimated error of the y-coordinate is 0.69 m. In terms of the position accuracy, the best result was achieved when passing by tag 80; there the estimated error in the position was only 0.17 m. The largest error of 1.76 m was found in the measurements of the cell of tag 86. Generally, a mean accuracy of 0.80 m can be achieved for the position using RFID cell-based positioning (RFID time-based CoO). The position determined using RFID time-based CoO was utilized in order to update the trajectory calculated by using the measurements of the INS. Figure 119.6 shows the filtered INS trajectory. As can be seen in Fig. 119.6, the error in position increased over time with a maximum error in the y-direction of around 2.5 m and a maximum error in the x-direction of around 5.0 m. The error can be further reduced with forward and backward filtering of the resulting trajectory in a post processing step. Further testing has shown that then a positioning accuracy in the range of one to two meter can be achieved (see e.g. Fu and Retscher 2009c).
119.6 Concluding Remarks and Outlook This paper presented the integration of RFID and lowcost INS for continuous positioning in urban and indoor environments. In the project, a low-cost INS
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An Intelligent Personal Navigator Integrating GNSS, RFID and INS
of around one to two meter in the indoor environment and a few meters in outdoor urban areas. Further testing will be conducted in the near future on trajectories in urban outdoor and indoor environments as well as in the transition zone between indoor to outdoor environments and vice versa.
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Acknowledgments This research is supported by the research project P19210-N15 “Ubiquitous Carthograpy for Pedestrian Navigation (UCPNAVI)” founded by the Austrian Science Fund (Fonds zur F€orderung wissenschaftlicher Forschung FWF). The author would like to thank Mrs. Qing Fu who is a research assistant in the UCPNAVI project and also the students participating in the LBS course at the Vienna University of Technology in the summer term 2009 for the performance of the test measurements.
Fig. 119.6 Filtered INS trajectory along the indoor path in the office building of the Vienna University of Technology (TP stands for turning point; the ID numbers 80–88 represent the location of the RFID tags)
References
from Xsens, the MTi, was employed. For calculating the positions from the measured data of the sensor, the strap down mechanization was used. Furthermore, a time-varying Kalman filter was employed to correct the position and acceleration that resulted from the strap down mechanization. Additionally, cell-based positioning (i.e., RFID time-based CoO) was utilized to determine the current position of the user, when the RFID reader detected a signal from an RFID tag in the surrounding environment. This determined position was needed to update and correct the trajectory calculated by the INS, since the INS components produce small measurement errors that accumulate over time and cause drift errors. The algorithms have been integrated into an intelligent software package. Additionally it shall be investigated in the near future how a calibration of the DR observations from the INS (i.e., step length and step frequency from the MEMS-based IMU as well as the heading from the digital compass) under GPS signal blockage using the knowledge of the human locomotion model when GPS is available (see Grejner-Brzezinska et al. 2007) can lead to a further improvement of the navigation solution. Then the developed personal navigator will enable the mobile user to determine its position continuously, automatically and ubiquitously. The described experiments showed that the determination of the trajectory using an integration of RFID with INS achieved an accuracy
Brown RG, Hwang PYC (1997) Introduction to random signals and applied Kalman filtering, 3rd edn. Wiley, New York, 484 pp Fu Q (2008) Active RFID for positioning using trilateration and location fingerprinting based on RSSI. Papers presented at the ION GNSS conference, Savannah, GA, 16–19 September 2008. CD-Rom proceedings, 14 pp Fu Q, Retscher G (2008) Using RFID technology in pedestrian navigation for information transmission and data communication recording. Papers presented at the junior scientist conference, Vienna, 16–18 November 2008, 2 pp Fu Q, Retscher G (2009b) Active RFID trilateration and location fingerprinting based on RSSI for pedestrian navigation. J Navig 62(2):323–340 Fu Q, Retscher G (2009c) Integration of a RFID time-based CoO positioning with INS using a time data capture tool for verification. Papers presented at the ION GNSS conference, Savannah, GA, 22–25 September 2009. CD-Rom proceedings, 9 pp Grejner-Brzezinska D, Toth C, Moafipoor S (2007) Pedestrian tracking and navigation using an adaptive knowledge system based on neural networks. J Appl Geod 3(1):111–123 Renaudin V, Yalak O, Tome´ P (2007) Hybridization of MEMS and assisted GPS for pedestrian navigation. Inside GNSS January/February 2007, pp 34–42 Retscher G, Fu Q (2009) An intelligent personal navigator integrating GNSS, RFID and INS for continuous position determination. Papers presented at the 6th international symposium on mobile mapping technology, Presidente Prudente, Sao Paulo, 21–24 July 2009, 8 pp Shuster MD (1993) A survey of attitude representations. J Astronaut Sci 41(4):437–517 Titterton DH, Weston JL (2005). Strapdown inertial navigation technology, 2nd revised edition. Institution of Engineering and Technology, Stevenage
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Integration of Image-Based and Artificial Intelligence Algorithms: A Novel Approach to Personal Navigation
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Dorota A. Grejner-Brzezinska, Charles K. Toth, J. Nikki Markiel, Shahram Moafipoor, and Krystyna Czarnecka
Abstract
Navigation systems, such as the Global Positioning System (GPS) and inertial measurement units (IMUs) become miniaturized and cost effective, enabling their fusion in a portable, low-cost navigation device for individual users, supporting predominantly outdoor navigation. This paper presents an unconventional solution designed for indoor–outdoor navigation, based on integration of GPS, IMU, digital barometer, magnetometer compass, and human locomotion model handled by Artificial Intelligence (AI) techniques that form an adaptive knowledge-based system (KBS). KBS is trained during the GPS signal availability, and is used to support navigation under GPS-denied conditions. A complementary technique used in our solution, which supports indoor navigation, is the image-based technique that uses a Flash LADAR sensor. Navigation from 3D Flash LADAR scene reconstruction utilizes the range distance to static features common in images acquired from two separate locations, which allows for triangulating the user’s position. By combining Flash LADAR image with the IMU data, a linear feature-based algorithm that identifies common static features between two images, along with the error estimates, is facilitated. Since the algorithm is based on linear methodologies, it enables rapid processing while generating robust, accurate position and error estimation data. In this paper, system design, as well as a summary of the performance analysis in the mixed indoor–outdoor environments is presented.
120.1 Introduction
D.A. Grejner-Brzezinska (*) C.K. Toth J.N. Markiel S. Moafipoor Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State University, 2070 Neil Avenue, Columbus, OH 43210, USA e-mail: [email protected] K. Czarnecka Department of Geodesy and Cartography, Warsaw University of Technology, Warsaw, Poland
Personal navigation systems represent an active research area, experimenting with different sensors in a variety of potential applications. Many of them, aside from traditionally used Global Positioning System (GPS) and inertial measurement units (IMUs), include vision sensors. The vision modules are usually designed to provide navigation information by matching the captured images with the pre-registered images stored in a database (e.g., Fang et al. 2005;
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Kourogi et al. 2006). More practical solutions have been proposed by Veth and Raquet (2007) and Moafipoor (2006), based on the technique called feature tracking-based navigation. In this approach, when GPS/IMU data are available, the captured images are continuously georeferenced. Then, with the GPS signal blockage, the new images that overlap with the earlier georeferenced images are co-registered with the image sequence, providing the navigation solution. Another type of optical tracking systems, based on laser ranging, provides range measurements to active or passive targets (Toth et al. 2009). This method is well suited for measuring distances from several meters (with a centimeter level accuracy) to a few hundreds of meters (with a decimeter-level accuracy), and even considerably longer distances, and thus, it is suitable for both outdoor and indoor applications. Others proposed systems, designed for smart environments, use active tracking technology, such as RF signals or RFID tags (e.g., Cho et al. 2003; Brand and Phillips, 2003; Koide and Kato 2005; Kourogi and Kurata 2003; Kourogi et al. 2006). However, in many applications, such as military and emergency operations, no special infrastructure may exist. Similarly, ground-based RF systems, such as pseudolite systems (e.g., Barnes et al. 2003a, b), representing a promising technique for navigation (Kee et al. 2000), also operate in a pre-set environment. The GPS modernization program, including development of highsensitivity GPS receivers, or Assisted-GPS (A-GPS), enables operation with much weaker signals (even indoors) and has shown significant improvement in the past few years (Lachapelle et al. 2006). Yet, there are still situations where even A-GPS does not provide sufficiently accurate position fix within an acceptable time interval. Grejner-Brzezinska et al. (2008) summarizes the state-of-the-art in sensors and technologies used in personal navigation. Growing requirement for seamless transition between indoor and outdoor environments increasingly leads to multisensor solutions for pedestrian navigation (e.g., Retscher 2004a, b; Retscher and Thienelt 2004; Kourogi et al. 2006; Lachapelle et al. 2006) as well as military and emergency personnel (Grejner-Brzezinska et al. 2006a, 2007). Notice that personal navigation is understood in general as navigation of military and emergency personnel with a variety of dynamic activities, while pedestrian
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navigation refers to all other mobile user with regular dynamic, i.e., walking.
120.2 OSU Personal Navigation Prototype A prototype of a personal navigation system, referred to as OSU PN (Grejner-Brzezinska et al. 2007) is currently based on a sensor suite that includes a dual frequency Novatel OEM4 GPS receiver with a TRM22020.00+GP antenna, Honeywell tactical grade HG1700 IMU, micro-switches placed in shoe soles, used for timing the step events, PTB220A barometer, and a three-axis Honeywell HMR3000 magnetometer with an integrated pitch-roll sensor. The GPS carrier phase and pseudorange measurements in the double difference (DD) mode, barometric height, magnetometer heading, and the IMU-derived position and attitude information are integrated together in the tightly coupled Extended Kalman Filter (EKF) with 27 states (Grejner-Brzezinska et al. 2006b). The demonstration prototype is based on a backpack configuration, as shown in Fig. 120.1.
120.3 Artificial Intelligence-Aided Navigation A key constituent of the OSU PN Dead-Reckoning (DR) module is the use of human body as navigation sensor. Modeling the human dynamics is quite complex, as it depends on a number of individual parameters, such as height, weight, physical fitness, etc., as well as the surrounding environment. Due to the complexity of the human motion model, including high level of non-linearity and non-stationary stochastic properties, conventional analytical models are not effective and/or feasible, and other methods that are better suited should be applied. Recent developments in Artificial Intelligence (AI) techniques resulted in a rapid increase in engineering, social, biomedical, and a variety of other applications, primarily due to effectiveness of Machine Learning (ML) in forming highly complex and adaptive models nearly autonomously. In particular, AI techniques are suited for applications related to human motion
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120.3.1 ANN-Based Adaptive Knowledge Base System A single-layer artificial neural network with Radial Basis Function (RBF) is used in the ANN-based KBS. The number of RBF functions ranges between 30 and 40 in a single hidden layer, and one output parameter, step length (SL), is provided. More details on the design of ANN-based KBS are given in (e.g., Grejner-Brzezinska et al. 2008; Moafipoor et al. 2008).
120.3.2 FL-Based Adaptive Knowledge Base System
Fig. 120.1 Backpack prototype system of OSU PN
modeling, and are being increasingly used for this purpose, as they allow for better process control and more reliable prediction and modeling of the processes under consideration. Our implementation is based on Artificial Neural Networks (ANNs) and Fuzzy Logic (FL) methods that create a Knowledge Base System (KBS), which represents a simplified human dynamics model that consists of three basic parameters: step length (SL), step frequency (SF), and step direction (SD). Together, these parameters are used to facilitate DR navigation. The human dynamics model is calibrated when other sensors, primarily GPS, provide a continuous navigation solution, and the AI-based navigation is activated when GPS is significantly degraded or not available. The primary operational components, including the newly added image-based module, discussed in the sequel, are shown in Fig. 120.2.
Similarly to ANN, fuzzy systems are suitable for approximate reasoning, especially for systems with a mathematical model that is difficult to derive. Fuzzy sets and fuzzy logic are used as a means of representing, manipulating and utilizing uncertain information, and to provide a framework for handling uncertainties and imprecision in systems, such as SL modeling. Compared to ANN, the FL is easier to use and simpler to apply. Although FL allows estimating state parameters under incomplete or uncertain information, they cannot automatically acquire the rules which are being used to make those decisions. Therefore, fuzzy systems are restricted to applications where expert knowledge is available. The FL-based KBS supporting SL modeling combines various types of external information in the form of membership functions developed using the training datasets, acquired during the KBS design and training process (Grejner-Brzezinska et al. 2007, 2008; Moafipoor et al. 2008).
120.4 Flash LADAR-Based Navigation Flash LADAR camera is a device that measures the time of flight of a modulated laser signal, and recovers a distance measurement with respect to the speed of light (Lange 2000; Oggier and Lehmann 2003). The resulting outputs are the 3D coordinates for each pixel in the image together with their intensity information, which allows for direct reconstruction of the object
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Fig. 120.2 OSU PN prototype including the Flash LADAR module
space in 3D, which can be then used for navigation. For example, if the initial starting point of a mobile platform is known (e.g., from GPS/IMU), the position and orientation of features extracted from the 3D image can be established, since the range distances are known. When the platform moves to a new location, the same features can be used to determine the new position of the camera (platform), provided that these features can be identified in the subsequent image. In order to recover position coordinates, only static features can be used, as in case of moving features it would practically impossible to separate their motion from the camera motion. Consequently, separation of static and non-static features is a crucial task in image-based navigation. An example 3D image sequence is shown in Fig. 120.3. Our feature based navigation algorithm based on Flash LADAR consists of four primary modules (aside from image acquisition), as discussed in (Markiel et al. 2008); note that strong ambient radiation, such as sunshine or strong artificial light can easily overpower the sensor, so its use is currently limited to indoor applications. The basic assumptions of the algorithm are: (1) features exist in the acquired imagery, (2) sufficient correspondence exists for static features in two temporally spaced images, and (3) sufficient static features exist to enable the determination of position and orientation. The primary steps in the algorithm are (1) image acquisition, (2) image segmentation, (3) coordinate transformation, (4) image matching, and (5) determination of position and orientation that are fed to EKF as measurement updates. It is important to note that our objective is not to extract entities from the imagery as distinct physical objects; thus, the
algorithm does not require object recognition. It suffices to assure that features are static and matching between the consecutive images. A generic workflow for the method, including the four-step algorithm and navigation filter, is shown in Fig. 120.4. The first module presents an innovative method for image segmentation based upon the eigenvector signatures of linear features (Markiel 2007). Pixels with similar signatures are merged to create features; the edges of these features are converted to a binary image to enable rapid evaluation and processing of feature edges in later modules. A key innovation is the utilization of statistics drawn from the image data to drive thresholding heuristics. The two images are treated as samples drawn from a larger, unknown distribution of range values, and compared to verify the condition of homogeneity. After verification, the heuristics are dynamically adjusted based upon changes to the distribution of range values. Since the algorithm does not rely on a priori values to determine the segmentation characteristics, the program can operate on an automated basis. The second module converts the previous image to the same coordinate frame as the current image. This is accomplished by a two step process. First, information from the inertial system provides an initial estimate of the necessary adjustment. This initial transformation incorporates errors inherent to the inertial system and must be refined. Step two, an additional innovation is the implementation of a RANSAC (Fischler and Bolles 1981) style approach to finalize the transformation. The range of solution space is constrained by the error information from the initial inertial data, which enables the algorithm to determine
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INERTIAL SENSOR
ACQUISITION
SEGMENTATION
FINAL POSITION & ORIENTATION
EXTENDED KALMAN FILTER
COARSE ORIENTATION
FEATURE MATCHING
DETERMINE POSITION & ORIENTATION
Fig. 120.4 OSU Flash LADAR-based navigation system structure
Fig. 120.3 Flash LADAR indoor image sequence (subsampled)
the solution based exclusively upon the sensor data. After the quaternion based transformation of the previous image is complete, the image is motion compensated to keep only those pixels that cover the same object space. Matching features between images is essential to co-register imagery. The problem of locating n features from the initial image amongst m features in the current image is not trivial; in general, the problem is not well posed. The third module of the algorithm resolves the challenge of feature matching by comparing eigenvector signatures for feature edge pixels in each image. The algorithm again leverages statistics derived from the image to enable automated, heuristically based evaluation of matching features. The final module triangulates the position of the mobile unit based upon the known ranges to matched features. The information is then returned to the tightly-coupled EKF as the measurement update to assure continuous calibration of the inertial system errors. Thus, in the image-based navigation concept, the feature-based, triangulated position is utilized in place of GPS measurements, enabling the update of the covariance matrix for the corrected position at each iteration. This reflects the variable uncertainty due to differences in image matching results. After updating, the inertial unit returns coarse information for the relative change in pose during the acquisition
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Table 120.1 Image-to-image navigation results Y
Z
0.0040 0.0069
0.0175 0.0304
0.0026 0.0045
0.0017 0.0029
0.0014 0.0024
0.0012 0.0022
0.0062 0.0108
0.0014 0.0024
of the next image, and the algorithm initiates a fresh sequence of feature matching and position evaluation. The image-to-image performance results, indicating that drift errors remain below 1 cm for a sequence up to 30 images, are listed in Table 120.1
120.5 Combining AI-Aided and Range Image-Based Navigation Based on the encouraging performance results of Flash LADAR-based indoor navigation, the OSU PN prototype was extended to incorporate a Flash LADAR sensor to support DR navigation, as shown in Figs. 120.2 and 120.5. The adaptive EKF algorithm has been extended to handle image-based measurement update, as explained in the previous section.
120.6 Performance Assessments A number of multisensor (GPS/IMU/barometer/magnetometer/LADAR) data acquisitions were completed for both outdoor and indoor conditions, as described in Markiel et al. (2008). The outdoor tests included various terrain types and different operators with different walking patterns, where reference trajectory was established by GPS/IMU solution. To permit validation of the indoor navigation results it was necessary to establish a network of ground reference points (Fig. 120.6) with sub-cm-level accuracy using classical surveying techniques. Table 120.2 summarizes typical navigation performance results for outdoor
SR3000 Flash LADAR
Digital camera
Fig. 120.5 Flash LADAR extension of the OSU PN prototype Reference Trajectory DR with Fuzzy SL DR with ANN SL
Center for Mapping Height (m)
With motion compensation X Image 2–3 Position error 0.0066 Position s 0.0114 Image 3–4 Position error 0.0027 Position s 0.0047 Image 4–5 Position error 0.0045 Position s 0.0078 Image 31–32 Position error 0.0022 Position s 0.0039
211 210 209 10 15 20 25 30 –15
35 Easting (m)
–20 40
–25 Northing (m)
Fig. 120.6 Indoor reference system, ANN- and FL-based DR navigation
environments (no flash LADAR data used due to sensor limitations explained earlier). Table 120.3 lists example performance for indoor environment, with an emphasis placed on the comparison of navigation accuracy using different sources of heading and SL. Note that Circular Error Probable (CEP) 50 of 5 m is the target accuracy for indoor navigation with the OSU PN. An example of indoor navigation results of the extended OSU PN system (including Flash LADAR) for x coordinate, including trajectory and error estimates, are shown in Fig. 120.7. Similar results were obtained for y and z coordinates. Notice that inclusion of the imaging component significantly improves the indoor navigation accuracy.
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Table 120.2 Errors at target locations with their standard deviations and residuals after 3D similarity transformation Test data 1 2 3 4 5 6 7
Terrain type Flat Flat Flat Slope Slope Flat Flat
User S E A S S S E
SL modeling mean std (cm) 27 38 26 07 26 05 15
Step count 426 528 424 381 802 361 535
Total reference trajectory length (m) 277.7 341.0 355.4 343.4 679.7 287.1 378.2
Total fuzzy-based trajectory length (m) 260.7 361.4 349.0 336.8 692.0 287.4 391.8
Error % 4 3.8 1.5 1.7 1.5 0.1 2.5
Table 120.3 Errors statistical fit to reference indoor trajectory of 473 m (four indoor loops illustrated in Fig. 120.7) to DR trajectories generated using SL predicted with FL and ANN, with gyro and integrated gyro/compass heading SD modeling Gyro/magnetometer
SL modeling FL ANN FL ANN
Gyro
Mean (m) 5.12 5.23 5.32 5.38
Std (m) 4.23 4.15 4.34 4.41
Max (m) 4.97 5.13 4.87 4.94
CEP50 (m) 4.95 5.04 5.16 5.12
Position “X”
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End misclosure (m) 4.40 4.56 4.50 4.43
50 40 30 0
1
2
3
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5
6
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Uncertainity of Position Error “X”
0.035 Meters
0.03 0.025 0.02 0.015 0.01 0
1
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Fig. 120.7 X coordinate: indoor trajectory and error estimates based on Flash LADAR augmentation to OSU PN
Conclusions
This paper discussed a new system design and prototype implementation for indoor–outdoor personal navigation, based on the integration of GPS, IMU, digital barometer and magnetometer compass with human locomotion model handled by Artificial Intelligence techniques; the most recent augmentation that supports indoor navigation, based on the Flash LADAR image sequence matching, was also discussed.
The primary novelty of the multi-sensor approach presented here is the use of AI techniques and image-based navigation derived from the 3D Flash LADAR images to support dead reckoning navigation mode. The first component is based on supervised learning via the Knowledge Based System that models human locomotion, while the second one utilizes the range distance to static features common to images acquired from subsequent positions of a mobile user, for scene reconstruction,
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which allows for triangulation of the user’s position. By utilizing the Flash LADAR image supported by the IMU-based orientation, a linear feature-based algorithm is facilitated and a significant increase of indoor navigation accuracy is accomplished. The algorithm is particularly unique in that it does not require a priori information about the scene, nor does the algorithm require a significant a priori thresholding values to control the ongoing processing. While extensive opportunities exist to test various aspects of the algorithm, the current results indicate it to be a viable option to the problem of navigation in GPS challenged environments. More tests with the extended sensor configuration are necessary to fully assess the achievable indoor navigation accuracy. Acknowledgement This research is supported by the National Geospatial-Intelligence Agency 2004 NURI grant. The authors would like to acknowledge Dr. Jacob Campbell of the Air Force Research Laboratories in providing the Flash LADAR camera and supporting this research.
References Barnes J, Rizos C, Wang J, Small D, Voigt G, Gambale N (2003a). Locata: a new positioning technology for high precision indoor and outdoor positioning. In: Proceedings, ION GNSS, Portland, OR, September 9–12, CD ROM, pp 1119–1128 Barnes J, Rizos C, Wang J, Small D, Voigt G, Gambale N (2003b) High precision indoor and outdoor positioning using LocataNet. In: Proceedings of international symposium on GPS/GNSS, Tokyo, November 15–18, CD ROM, pp 9–18 Brand TJ, Phillips RE (2003) Foot-to-foot range measurements as an aid to personal navigation. In: Proceedings of the 59th annual meeting/CIGTF 22nd guidance, test symposium, ION, Albuquerque, NM, June 2003, pp 113–121 Cho SY, Lee KW, Park CG, Lee JG (2003) A personal navigation system using low-cost, MEMS/GPS/Fluxgate. In: Proceedings of the 59th annual meeting/CIGTF 22nd guidance, test symposium, ION, Albuquerque, NM, June 2003, pp 122–128 Fang L, Antsaklis JP, Montestruque LA, McMickell MB, Lemmon M, Sun Y, Fang H, Koutroulis I, Haenggi M, Xie M, Xie X (2005) Design of a wireless assisted pedestrian dead reckoning system-the NavMote experience. IEEE Trans Instrum Meas 54(6):2342–2358 Fischler M, Bolles R (1981) Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography. Comm ACM 24:381–395 Grejner-Brzezinska DA, Toth CK, Moafipoor S, Jwa Y (2006a) Multi-sensor personal navigator supported by human motion dynamics model. In: Proceedings of 3rd IAG symposium on geodesy for geotechnical and structural engineering/12th
D.A. Grejner-Brzezinska et al. FIG symposium on deformation measurements, Baden, Austria, May 2006, CD ROM Grejner-Brzezinska DA, Toth CK, Jwa Y, Moafipoor S (2006b) Seamless and reliable personal navigator. In: Proceedings ION technical meeting, Monterey, CA, 18–20 January 2006, pp 597–603 Grejner-Brzezinska DA, Toth CK, Moafipoor S (2007) Pedestrian tracking and navigation using adaptive knowledge system based on neural networks and fuzzy logic. J Appl Geod 1 (3):111–123 Grejner-Brzezinska DA, Toth CK, Moafipoor S (2008) Performance assessment of a multi-sensor personal navigator supported by an adaptive knowledge based system. In: ISPRS, XXXVII (B5), Beijing, pp 857–867 Kee C, Jun H, Yun D, Kim B, Kim Y, Parkinson B, Lenganstein T, Pullen S, Lee J (2000) Development of Indoor navigation system using asynchronous pseudolite. In: Proceedings of the 13th international technical meeting of the satellite division of the institute of navigation, ION GPS-2000, Salt Lake, UT, September, pp 1038–1045 Koide S, Kato M (2005) 3-D human navigation system considering various transition preferences. In: Systems, man and cybernetics, 2005 IEEE international conference, vol 1, October, 2005, pp 859–864 Kourogi M, Kurata T (2003) Personal positioning based on walking locomotion analysis with self-contained sensors and a wearable camera. In: Proceedings of the second IEEE and ACM international symposium on mixed and augmented reality, October 2003, pp 103–112 Kourogi M, Sakata N, Okuma T, Kurata T (2006) Indoor/outdoor pedestrian navigation with an embedded GPS/RFID/ self-contained sensor system. In: Proceedings of 16th international conference on artificial reality and telexistence (ICAT2006), Hangzhou, September 2006, pp 1310–1321 Lachapelle G, Godha S, Cannon ME (2006) Performance of integrated HSGPS-IMU technology for pedestrian navigation under signal masking. In: Proceedings of European navigation conference, Royal Institute of Navigation, ENCGNSS, Manchester, UK, May 2006, pp 1–24 Lange R (2000) 3D time-of-flight distance measurement with custom solid-state image sensors in CMOS/CCD-technology. PhD thesis, University of Siegen, Siegen Markiel JN (2007) Separation of static and non-static features from three dimensional datasets by implicit parametric equations. Master’s thesis, The Ohio State University, Columbus, OH Markiel JN, Grejner-Brzezinska D, Toth C (2008) An algorithm for the extraction of static features from 3D flash LADAR datasets: supporting navigation in GPS challenged environments. In: Proceedings, IEEE/ION PLANS meeting, Monterey, CA, 5–8 May 2008, CD ROM Moafipoor S (2006) Updating the navigation parameters by direct feedback from the image sensor in a multi-sensor system. In: Proceedings, ION GNSS meeting, Fort Worth, TX, September 26–29, pp 1085–1092 Moafipoor S, Grejner-Brzezinska DA, Toth CK (2008). Multisensor personal navigator supported by adaptive knowledge based system: performance assessment. In: IEEE/ION PLANS 2008 meeting, Monterey, CA, May 5–8, CD-ROM
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Oggier T, Lehmann M (2003) An all-solid-state optical range camera for 3D real-time imaging with sub-centimeter depth resolution (SwissRangerTM). Proc SPIE 5249:534–545 Retscher G (2004a) Multi-sensor systems for pedestrian navigation and guidance services. In: Proceedings, 4th symposium on mobile mapping technology, Kunming, March 29–31, CD ROM Retscher G (2004b) Multi-sensor systems for pedestrian navigation. In: Proceedings, ION GNSS 2004, Long Beach, CA, September 21–24, CD ROM, pp 1076–1088
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Retscher G, Thienelt M (2004) NAVIO – A navigation and guidance service for pedestrians. J Glob Position Syst 3 (1–2):208–217 Toth C, Grejner-Brzezinska DA, Wang X, Sun H (2009) Terrestrial laser scanning to support land navigation. In: ASPRS annual conference, Baltimore, MD, 9–13 March 2009, CD-ROM Veth M, Raquet J (2007) Fusing low-cost image and inertial sensors for passive navigation. J Inst Navig 54(1):11–20
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L.P.S. Fortes, S.M.A. Costa, M.A. Abreu, A.L. Silva, N.J.M Ju´nior, K. Barbosa, E. Gomes, J.G. Monico, M.C. Santos, and P. Te´treault
Abstract
The Brazilian Network for Continuous Monitoring of GNSS – RBMC is a national network of continuously operating reference GNSS stations. Since its establishment in December of 1996, it has been playing an essential role for the maintenance and user access of the fundamental geodetic frame in the country. In order to provide better services for RBMC, the Brazilian Institute of Geography and Statistics – IBGE and the National Institute of Colonization and Land Reform – INCRA are both partners involved in the National Geospatial Framework Project – PIGN. This paper provides an overview of the recent modernization phases the RBMC network has undergone highlighting its future steps. These steps involve the installation of new equipment, provide real time data from a group of “core” stations and compute real-time DGPS corrections, based on CDGPS (The real-time Canada-Wide DGPS Service) (The Real-Time CanadaWide DGPS Service. http://www.cdgps.com/ 2009a).
L.P.S. Fortes S.M.A. Costa (*) M.A. Abreu A.L. Silva N.J.M. Ju´nior Coordenacao de Geode´sia, Instituto Brasileiro de Geografia e Estatistica, Av Brasil 15671, Parada de Lucas, Rio de Janeiro 21241-051, Brazil e-mail: [email protected] K. Barbosa E. Gomes Coordenac¸a˜o de Cartografia, Instituto Nacional de Colonizac¸a˜o e Reforma Agra´ria, SBN Qd. 01 Bloco D - Edifı´cio Pala´cio do Desenvolvimento, Brası´lia, DF, Brazil J.G. Monico Departmento de Cartografia, Universidade Estadual Paulista, Rua Roberto Simonsen, 305, Presidente Prudente, Brazil M.C. Santos Department of Geodesy and Geomatics Engineering, University of New Brunswick, P.O. Box 4400, Fredericton, NB, Canada E3B 5A3 P. Te´treault Geodetic Survey Division, Natural Resources Canada, 615 Booth Street, Ottawa, ON, Canada K1A 0E9 S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_121, # Springer-Verlag Berlin Heidelberg 2012
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In addition to this, a post-mission Precise Point Positioning (PPP) service has been established based on the current Geodetic Survey Division of NRCan (CSRS-PPP) service. This service is operational since April 2009 and is in large use in the country. All activities mentioned before are 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. The Geodetic Survey Division of NRCan is also participating in this modernization effort under the same project. This infrastructure of 66 GNSS stations, the real time, post processing services and the potentiality of providing Wide Area DGPS corrections in the future show that the RBMC system is comparable to those available in USA and Europe.
121.1 Introduction In operation for more than 1 decade, the Brazilian Network for Continuous Monitoring of GNSS – RBMC (Fortes et al. 1998, 2006a), is an active geodetic network, constituting the main geodetic framework of the country. It provides Brazilian users with a precise link to the Brazilian Geodetic System – SGB, SIRGAS2000 (IBGE 2009b), fully compatible with GNSS technology. The purpose of RBMC is to provide data, without any cost, for several post-processing applications. In this respect, it is important to mention that RBMC represents a substantial economical, technological and scientific effort of several Brazilian federal, state and academic institutions (in close collaboration with some Canadian agencies) to build and maintain a modern geo-spatial infrastructure. A law of 2001 requiring all rural properties to be referred to the Brazilian geodetic system intensified the use of RBMC reference data. In order to provide better services for RBMC, the Brazilian Institute of Geography and Statistics – IBGE, and the National Institute of Colonization and Land Reform – INCRA, became partners involved in the National Geospatial Framework Project – PIGN (PIGN 2009). This paper provides an overview of the recent modernization phases of the RBMC network and the future steps. The first step was the acquisition of new equipment and network expansion providing a better national coverage and new characteristics of operation. The RBMC increased from 24 stations by the end of 2006 to 66 stations at the beginning of 2009. The modern GNSS receivers allow for remote control and real time transfer of observations over the Internet. Data is available on two websites, INCRA (2009)
providing hourly and IBGE (2009a) providing daily files with different sampling rates. In 2008, a Networked Transport of RTCM via Internet Protocol (NTRIP) caster was put in operation, providing real-time data of 26 stations. This new service called RBMC-IP is open for all users since May 2009, through a login and password. A challenge for the future is to compute Wide Area Differential GPS (WADGPS)-corrections to be transmitted, in real time, to users in Brazil and surrounding areas. For this purpose some receivers are working already with an external frequency standard. It is estimated that users will be able to achieve a horizontal accuracy between 0.5 and 1 m (1s) in static and kinematic positioning and better for dual frequency users. The availability of the WADGPS service will allow users to tie to the new SIRGAS2000 system for positioning and navigation applications in a more rapid and transparent way It should be emphasized that support to post-mission static positioning will continue to be provided to users interested in higher accuracy levels. In addition to the RBMC services, a post-mission Precise Point Positioning (PPP) (IBGE 2009c) is in operation since April 2009. The IBGE-PPP service is based on the Geodetic Survey Division of NRCan (CSRS-PPP) service (CSRS-PPP 2009b). Millimeter precision is achieved after 6 h of data for double frequency receivers, and sub-meter precision is achieved for single frequency receivers after 2 h of data.
121.2 Network Expansion After 3 years working on the expansion plan of RBMC \RIBac, the number of stations increased from 24 to 66 in 2009. The goal is provide a larger national coverage
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with additional capabilities, for example, real time data and an adequate infrastructure for collecting data from GPS and GLONASS, foreseeing the possibility of collecting GALILEO data in the future. Each RBMC station is equipped with a double frequency GPS receiver. At the end of each 24 h observing session, the collected data is automatically transferred to a server at the control center in Rio de Janeiro, through the Internet connection (cable or satellite). After transferred, data is checked and made freely available at the site ftp://geoftp.ibge.gov./ RBMC/ within 24 h after the observation date. After all these years of operation, the network has been largely used by the national and international community, as it is demonstrated by many projects
Fig. 121.1 RBMC status in August 2009
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and published papers, as in Fortes et al. (2006a, 2007). An important national project example is related to land formalization and regularization, under the responsibility of INCRA. The cooperation between IBGE and INCRA continues with the installation of more 30 stations that will be concentrated in the central and northern part of country. By the end of this year, 10 additional stations will be installed, extending the present network configuration from 66 to 76 stations. The data collected at new stations are released after a period of evaluation, when the official SIRGAS2000 coordinates are computed and data delivered is analysed. Figure 121.1 shows the distribution of the station’s network and Table 121.1 shows the current (August 2009) receivers being used.
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Table 121.1 Type and number of receivers in operation Receiver type NetRS NetR5 Trimble 4000SSi Leica GSX1200 Ashtech ZFX Ashtech UZ-12 Total no. of receivers
No. of receivers Mar/07 2 0 17 1 4 0 24
Aug/09 15 40 3 2 4 1 66
RBMC corresponds to the Brazilian contribution to many international initiatives, such as SIRGAS (2009), IGS (2009a), Real-Time IGS IGS (2009b), NTRIP and others related to world climate monitoring. The network maintenance and operation improved in the last 4 years with only 6% of outages per month and consequently the number of users increased more than 50% per month due to service credibility.
121.3 Real Time Differential GPS Corrections Service In May of 2009 the IBGE NTRIP caster was officially opened to Brazilian users, with 26 mount points (stations) installed in the main cities of the country where wireless internet is available. For a better control of the network traffic we are limiting the number of registered users to access stream data from three mount points in a maximum period of 3 months. For the Universities or Research Institutions all mount points are available without restrictions. This new service is called RBMC-IP and more information can be found at: http://www.ibge.gov.br/home/geociencias/ geodesia/rbmc/ntrip/. At present, Brazil contributed with five new IGS stations (POVE, SALU, SAVO, RECF and UFPR) and eight RTIGS stations (POVE, SALU, SAVO, RECF, UFPR, CEEU, BRAZ and ONRJ) with GNSS receivers belonging to a subset of RBMC-IP network. Other nine stations (POVE, SALU, SAVO, RECF, UFPR, ONRJ, NAUS, CEEU and BRAZ) contribute to the IGS-IP network providing data in real time as well. In a second step the real-time Wide Area Corrections service of the RBMC, corresponding to PIGN Demonstration Project #7, is under development together with the Geodetic Survey Division of NRCan
which is transferring its experience from the CDGPS – Canadian Differential GPS service. The availability of the WADGPS service will allow users to tie to the new SIRGAS2000 system in a more rapid and transparent way. Besides the group of real-time stations, the Master Active Control System (MACS), is the “heart” of the system. The hardware at the MACS are composed of: • Two HP-UX servers (running RTAP – Real Time Application) • Two Linux servers (storing and data management) • Time server with GPS receiver providing GPS time to the HP-UX servers The following parts of the system’s implementation are already concluded: 1. Connect a subset of receivers to the atomic clock. At least two stations can satisfy this requirement, CEEU (located in Fortaleza) is connected to a Hydrogen Maser and ONRJ (located in Rio de Janeiro) is connected to a Caesium clock that belongs to the Brazilian Time Service at the National Observatory 2. Operate a group of stations streaming data on real-time 3. Installation of Linux and HP-UX servers responsible for the data management and computation of corrections 4. Installation and configuration of software for data format conversion, scan and management, and the computation of corrections The real-time GPS data is transmitted using UDP/ IP over Internet to the Linux server, which then re-broadcasts this data using multicast UDP/IP over the internal IBGE network. The multicast data is received by the two HP-UX servers and processed to produce GPS corrections. The following steps will be completed until the end of this year: 1. The installation and configuration of a NTP time server, to slave the MACS server’s (CPU responsible for the DGPS corrections computation to the GPS time). 2. Near real time orbit computations. 3. Evaluation of ION grid corrections generated with Brazilian data. 4. The transmission of corrections to the users, through internet. The resulting configuration for this system is shown in Fig. 121.2.
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RBMC station1
RBMC station 2
station ONRJ
Atomic Clock
Internet
NTRIP caster
Server for data conversion
IBGE Network
Server for Corrections´ Computation
NTP Time
IBGE Time Server
Server for Corrections´ Computation
Real-Time GPS
Fig. 121.2 Configuration of real-time wide area corrections of the RBMC
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) (Rho et al. 2003).
121.4 IBGE-PPP Service The IBGE Precise Point Positioning (IBGE-PPP) is an on-line service for GPS data-processing. It allows the GPS users to get coordinates of good precision in the Geocentric Reference System for the Americas (SIRGAS2000) and the International Terrestrial Reference Frame (ITRF). In GPS positioning, the term Precise Point Positioning usually refers to the computation of a single station using carrier phase observation, obtained from dualfrequency receivers, together with IGS products. The IBGE-PPP service processes data collected in either static or kinematic modes, from single or dual frequency receivers. Only data collected after February 25, 2005, are accepted, this being the time of official adoption of the SIRGAS2000 frame in Brazil. The following information is required by the service:
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1. GPS data in RINEX or Hatanaka format, preferably compressed using Winzip, Gzip or Tar-Gzip. 2. Antenna type used to collect the data, following the IGS identification scheme, and the value of the antenna height in meters referred to the antenna reference plane (ARP). Besides the user supplied RINEX files, the IBGEPPP service uses other information for the processing, such as: orbit and clock (satellite) information from IGS (final and rapid), IGS antenna phase centre corrections for satellite and receiver antennas, transformation parameters between ITRF and SIRGAS2000 frames, FES04 ocean loading model parameters, velocity model VEMOS and the geoidal ondulation model – MAPGEO2004. The results are informed through a compressed archive containing five files, as following: readme, summary report, coordinate series for kinematic mode, Google Earth file (kml), and the complete report. This positioning service makes use of CSRSPPP developed by the Geodetic Survey Division of Natural Resources of Canada (NRCan) and can be accessed through the following webpage: http:// www.ibge.gov.br/home/geociencias/geodesia/ppp/.
121.5 Accuracy of IBGE-PPP After 2 h of data from double frequency receivers, the horizontal coordinates have an accuracy better than 4 cm and height better than 6 cm. After 6 h of data the accuracy stabilizes on 2 and 4 cm level, respectively. These results confirm the good quality of service. Figure 121.3 shows the positioning accuracy using a double frequency receiver in the static mode during 24 h of data. After 2 h of data from single frequency receivers, the horizontal coordinates have an accuracy better than 30 cm and height better than 70 cm. After 6 h of data the accuracy stabilizes on the 15 and 40 cm level, respectively. Figure 121.4 shows the positioning accuracy using a single frequency receiver in the static mode during 24 h of data.
121.6 Final Considerations The new structure, after full implementation, will have as main characteristics:
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• Provide 1-Hz real time data to IBGE NTRIP caster, located in Rio de Janeiro • Generate real-time WADGPS corrections (orbit, clocks and ionosphere) • Make corrections available to users in Brazil (and surrounding areas) through Internet • Offer a Precise Point Positioning (PPP) service (IBGE-PPP) to users • Collaborate with international GNSS networks such as IGS and RTIGS Acknowledgments The work described here has been carried out under the scope of the Projeto de Mudanc¸a de Referencial Geode´sico and the National Geospatial Framework Project http://www.pign.org, funded by the Canadian International Development Agency (CIDA).
CDGPS (2009a) The Real-Time Canada-Wide DGPS Service. http://www.cdgps.com/ CSRS-PPP (2009b) Precise Point Positioning Service. http://ess. nrcan.gc.ca/2002_2006/gnd/csrs_e.php Fortes LPS, Luz RT, Pereira KD, Costa SMA, Blitzkow D (1998) The Brazilian network for continuous monitoring of GPS (RBMC): operation and products. Advances in positioning and reference frames. In: Brunner FK (ed) International association of geodesy symposia, vol 118, pp 73–78 Fortes LPS, Costa SMA, Lima MAA, Fazan JA, Santos MC (2006a) Accessing the New SIRGAS2000 Reference Frame through a modernized Brazilian Active Control Network. In: Rizos C (ed) International association of geodesy symposia. IAG, IAPSO and IABO Joint Assembly “Dynamic Planet”, Cairns, Australia, Springer, 22–26 Aug 2005, pp 655–659 Fortes LPS, Costa SMA, Lima MAA, Fazan JA, Monico JFG, Santos MC and Te´treault P (2006b) Modernizing the Brazilian active control network. In: Proceedings of the 17th international technical meeting of the satellite division of the institute of navigation ION GPS/GNSS 2003. Fort Worth, Texas, 26–29 Sep, pp 2759–2768 Fortes LPS, Costa, SMA, Abreu MA, Junior NJM, Silva AL, Carvalho M, Monico JFG (2007) Plano de Expansa˜o e Modernizac¸a˜o das Redes Ativas RBMC/RIBAC. Rio de Janeiro. Technical Publication of XXIII Brazilian Carthography Meeting IBGE (2009a) Rede Brasileira de Monitoramento Contı´nuo. http://www.ibge.gov.br/home/geociencias/geodesia/rbmc/ rbmc.shtm. IBGE (2009b) Projeto Mudanc¸a do Referencial Geode´sico. http:// www.ibge.gov.br/home/geociencias/noticia_sirgas.shtm IBGE (2009c) Posicionamento Preciso por Ponto – PPP. http:// www.ibge.gov.br/home/geociencias/geodesia/ppp/default. shtm IGS (2009a) International GNSS Service Tracking Network. http://igscb.jpl.nasa.gov/network/netindex.html IGS (2009b) IGS Real Time Working Group. http://www. rtigs.net. INCRA (2009) Rede INCRA de Bases Comunita´rias do GPS. http://www.incra.gov.br/_htm/serveinf/_htm/_asp/ estacoes_dcn/default.asp PIGN (2009) National Geospatial Framework Project. http:// www.pign.org/ Rho H, Langley R, Kassam A (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 SIRGAS (2009) Sistema de Referencia Geoce´ntrico para las Ame´ricas. http://www.sirgas.org
magicSBAS: A South-American SBAS Experiment with NTRIP Data
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I. Alcantarilla, J. Caro, A. Cezo´n, J. Ostolaza, and F. Azpilicueta
Abstract (English)
The current map of satellite navigation systems includes global systems without integrity service (GPS and GLONASS), regional systems with integrity service (WAAS, EGNOS, MSAS), and future navigation systems providing integrity with global coverage (Galileo). The development and deployment of a complete satellite navigation system for a given region is a serious technological challenge that requires significant investment and a relatively long process, which may last many years. Systems like WAAS or EGNOS have required around a decade from their original concept design to the final achievement of the operational status. In this context, GMV (http://www.gmv.com), a reference Spanish company in the domain of GNSS ground segment computation facilities, has developed magicSBAS. magicSBAS takes advantage of the technology already developed both in the frame of satellite navigation and in the area of personal communications in order to propose alternatives for new regions to a full development of a new system. Moreover, it represents the first multi-constellation SBAS integrity provider by augmenting not just GPS, but GPS and GLONASS satellites. magicSBAS collects real-time pseudorange measurements and ephemeris in RTCM format from existing reference stations in the Internet via the NTRIP protocol (http://igs.bkg.bund.de/ntrip/ntriphomepage). Then magicSBAS computes corrections (SV orbits and clocks, ionosphere), integrity and all additional information required by a SBAS system in real-time (as specified in MOPS-C, Minimum Operational Performance Standards for Global Positioning System/ Wide Area Augmentation System Airborne Equipment. RTCA/DO-229C. 28 Nov 2001) and broadcasts this information to the final user using SISNET format (http:// www.egnos-pro.esa.int/sisnet/index.html). The SISNET broadcast information can be accessed via Internet or GPRS/3G technology by user receivers to navigate safely.
I. Alcantarilla J. Caro A. Cezo´n (*) J. Ostolaza GMV S.A., Madrid, Spain e-mail: [email protected] F. Azpilicueta Laboratorio GESA, Universidad de Universidad de La Plata, Argentina S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_122, # Springer-Verlag Berlin Heidelberg 2012
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magicSBAS has been adapted to the real-time processing of the NTRIP data in South America with excellent results. It will be shown how a SBAS service can be provided in South America in real time with the same performances as any other SBAS system. Accuracy, availability, continuity and integrity will be evaluated in South America with NTRIP data + magicSBAS. Moreover, the performances of the real-time SV orbits and clocks and ionosphere corrections will be shown. Abstract (Espan˜ol)
El mapa actual de sistemas de navegacio´n por sate´lite incluye sistemas globales sin provisio´n de integridad (GPS y GLONASS), sistemas regionales con provisio´n de integridad (WAAS, EGNOS, MSAS), y los futuros sistemas de navegacio´n con provisio´n de integridad y con cobertura mundial (Galileo). El desarrollo y despliegue de un sistema completo de navegacio´n por sate´lite para una determinada regio´n es un serio desafı´o tecnolo´gico que requiere una inversio´n importante y un proceso relativamente largo, que puede durar muchos an˜os. Sistemas como EGNOS o WAAS han requerido en torno a una de´cada desde su concepcio´n inicial hasta la consecucio´n del estatus operacional. En este contexto, GMV (http://www.gmv.com), una empresa espan˜ola de referencia en el a´mbito del segmento de tierra GNSS, ha desarrollado magicSBAS. magicSBAS aprovecha las ventajas de la tecnologı´a ya desarrollada, tanto en el marco de la navegacio´n por sate´lite y en el a´rea de las comunicaciones personales con el fin de proponer una alternativa para las regiones frente al desarrollo de un sistema completo. Adema´s, representa el primer SBAS multi-constelacio´n al aumentar no so´lo los sate´lites GPS sino los sate´lites GPS y GLONASS. magicSBAS recoge de Internet en tiempo real las medidas de pseudoco´digo y las efeme´rides en formato RTCM de las estaciones de referencia a trave´s del protocolo NTRIP (http://igs.bkg.bund.de/ntrip/ntriphomepage). Luego magicSBAS calcula correcciones SBAS (o´rbitas y relojes de sate´lites e ionosfera), junto con su integridad y toda la informacio´n adicional requerida por un sistema SBAS en tiempo real, especificado en MOPS-C (Minimum Operational Performance Standards for Global Positioning System/Wide Area Augmentation System Airborne Equipment. RTCA/DO-229C. 28 Nov 2001), y transmite toda esta informacio´n al usuario final utilizando el formato SISNET (http://www.egnos-pro.esa.int/sisnet/index.html). Esta informacio´n SISNET se puede acceder a trave´s de Internet o con la tecnologı´a GPRS/3G por receptores de usuario para poder navegar con total seguridad. magicSBAS se ha adaptado para procesar en tiempo real datos NTRIP en Sudame´rica con excelentes resultados. Se mostrara´ co´mo un servicio SBAS en tiempo real en Sudame´rica es posible con las mismas prestaciones que cualquier otro sistema SBAS. La precisio´n, disponibilidad, continuidad e integridad sera´n evaluadas para Sudame´rica.
122.1 Introduction A SBAS (Space Based Augmentation System) is in charge of augmenting the navigation information provided by different satellite constellations (such as
GPS or GLONASS and in the future Galileo) by providing ranging, integrity and correction information via geostationary satellites. Thus, the system is composed of:
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1. Ground infrastructure 2. SBAS geostationary satellites 3. SBAS receivers The ground infrastructure includes the monitoring and processing stations, which receive the data from the navigation satellites and compute integrity, corrections and ranging data which form SBAS signal-in-space (SIS). The SBAS satellites relay the SIS from the ground infrastructure to the SBAS user receivers, which determine position and time information from core satellite constellation(s) and SBAS satellites (via GEO). The SBAS receivers acquire the ranging and correction data and apply these data to determine the integrity and improve the accuracy of the derived position. The SBAS ground system measures the pseudorange between the satellites and a set of SBAS reference receivers at known location and provides separate corrections and levels of confidence for satellite position errors, satellite clock errors and ionospheric errors. The user will apply these corrections to improve its estimate of its position and its level of confidence.
122.2 What Is magicSBAS? The magicSBAS scheme is based on the collection of measurements and data from existing reference stations in the Internet in a protocol called NTRIP [2]. Then magicSBAS computes corrections, confidence levels and all additional information required by a SBAS system, using enhanced EGNOS [5, 6] algorithms, and broadcasts this information to the final user via Internet using the format SISNET [4]. Thus, the magicSBAS system is composed of: 1. NTRIP data + magicSBAS as ground infrastructure. 2. SISNET broadcasts over the Internet – which can be accessed via GPRS – replacing the SBAS geostationary satellites. 3. SBAS receiver processing SISNET format – such as GMV [1] I-10, Septentrio or standard nonSISNET receivers complemented with SW tools. In this way, magicSBAS does not require a dedicated space segment or deployed stations,¡ and the transmission can be achieved with full independence from other systems. This leads to a more efficient management and decision driving. The region keeps full control and sovereignty on the magicSBAS operations. Moreover, and since magicSBAS follows
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standard protocols, it can be easily complemented with dedicated receivers to enhance the performances in the target region. Those dedicated receivers, which should be capable of RTCM output, will be added as new NTRIP stations, to cover areas where the lack of stations makes magicSBAS offer worse performances. Figure 122.1 provides a graphical representation of magicSBAS elements. It can be seen that magicSBAS consists of just one PC with magicSBAS SW receiving data from stations in Internet through NTRIP casters. Then, it computes the corrections and integrity and provides the SBAS message to Internet for a later access through mobile technology. Dedicated receivers and magicSBAS monitoring are optional enhanced capabilities. The magicSBAS Monitor provides the necessary real-time and post-processing performance analyses tools (used in EGNOS validation): 1. teresa [7] is a software tool that fully implements User Receiver Algorithms providing real-time and post-processing GNSS performance (GPS, GLONASS and SBAS Systems) according to MOPS standards [3]. 2. eclayr [8] is a SBAS performance analyzer that provides very detailed post-processing performance analyses at range (SV and ionosphere) and user levels. eclayr is fully automatable. The real time NTRIP data available at present world-wide and shown in Fig. 122.2 are daily updated from (http://igs.bkg.bund.de/root_ftp/NTRIP/maps/ networks/All-World.png): The full list of the available NTRIP stations with their characteristics (receiver model, system, carrier, position, NTRIP broadcaster. . .) can also be seen at http://igs.bkg.bund.de/root_ftp/NTRIP/streams/ streamlist_world-wide.htm. Please note that the only restriction in the input data is that magicSBAS needs to be feed with 1 Hz real time continuous NTRIP Data and the type of the computed solution depends of the input data, i.e. you will need GPS+GLONASS data, therefore GPS+GLONASS receivers for a GPS+GLONASS magicSBAS solution. For a pre-evaluation of the number and geometry of the NTRIP stations, polaris software [9] is often used except when a very limited number of stations are available, that the maximum number of stations are to be used.
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Fig. 122.1 magicSBAS overview
Fig. 122.2 NTRIP stations world-wide
122.3 SBAS Performances Evaluation Augmentation system performances are defined with respect to the service level provided. Most of the analyses to characterize system performances are provided at user level, where the main concepts can be measured in a simplified way in the following terms:
1. Availability: Sufficient information is broadcast by the system to compute a valid navigation solution and the horizontal/vertical protection levels (HPL and VPL) do not exceed the alarm limits (HAL and VAL) for the corresponding service level. 2. Accuracy: Difference between estimated and real user position.
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3. Continuity: Service level declared available for the whole operation. 4. Integrity: Navigation error not exceeding the alarm limits. It is important to mention that the main performance indicator is the availability map (the larger the better) provided that the integrity of the system is maintained (accuracy is generally met and continuity generally not met).
Fig. 122.3 Eastern Europe magicSBAS availability
Fig. 122.4 New Zealand magicSBAS availability
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122.4 magicSBAS Performances Around the World Figures 122.3 and 122.4 provide the real-time availability maps for two cases. The first one corresponds to Europe including the ex-Eastern Bloc and some Islands where EGNOS (the European SBAS) does not provide service yet, while the second one is the New Zealand (NZ) case, where no SBAS is currently deployed.
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It is possible to observe from these figures that 99% APV-I availability is reached for the targeted zones which imply that magicSBAS performances are in line with those of the current SBAS in the world (e.g., WAAS, EGNOS).
122.5 magicSBAS Performances in South-America magicSBAS has been run with the available NTRIP data in South-America. The stations used were GPSonly, double frequency geodetic receivers and their location is shown in Fig. 122.5. The number of stations at the time of the demo was very limited in South America except for Brazil, and
Fig. 122.5 Ntrip-Stations used in South America
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the NTRIP data was frequently lost from some stations due to internet problems there, so the maximum number of stations available was configured for evaluation. Typical results with magicSBAS performances when most of the NTRIP stations were available are shown next. Similar results were obtained during other periods of the day, but due to the limited space to present results, they have been omitted. Should you require further analysis please send an e-mail to [email protected]. It is important to remark that the performances provided by ECLAYR are independent of the GPS measurements availability at receiver level and give an estimation of the levels of availability, continuity, accuracy and integrity that could be achieved by a fault-free receiver in the Service Area using only the
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GPS ephemera and the SBAS corrections broadcast by the corresponding GEO. Figure 122.6 represents for each user position, the percentage of time that the protection level is lower than the APV-I 99% alarm limit (HAL ¼ 40 m, VAL ¼ 50 m). This percentage has been computed respect to the monitored epochs.
Fig. 122.6 South-America magicSBAS availability
Fig. 122.7 South-America magicSBAS availability
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Performances figures highly depends on NTRIP station availability, as it can be seen in Fig. 122.7 where availability performances are analyzed in a different time period as presented for AAGG2009 congress. The following figures represent both the Horizontal and Vertical Accuracy measured at percentile 95%.
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Fig. 122.8 South-America magicSBAS horizontal accuracy
Fig. 122.9 South-America magicSBAS vertical accuracy
As it can be seen it is in the order of 1–2 m (Figs. 122.8 and 122.9). The Safety Index for each position represented in the figures below (Figs. 122.10 and 122.11) is defined as the division between the position error versus the
protection level for each epoch. As can be seen, integrity is preserved as the index is always below 1. Protection levels are about 5 times higher than user errors. Note that Safety Index also represents the margin between integrity and availability.
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Fig. 122.10 South-America magicSBAS horizontal integrity
Fig. 122.11 South-America magicSBAS vertical integrity
The following figure represents the Continuity Risk factor for each position. Continuity Risk is defined as the probability of having the service unavailable (PLs > ALs) during the aircraft landing operation provided the system was available at the beginning of the operation. Notice that continuity risks lower than 1e-5 are
better than the qualified EGNOS continuity risk and thus a major performance indicator of the SBAS service (Figs. 122.12 and 122.13). The figure below was obtained with TERESA in the Concepcio´n station (CONZ) located in Chile, inside the coverage area. After magicSBAS initialization
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Fig. 122.12 South-America magicSBAS continuity
which takes about 3,600 epochs, magicSBAS gets similar performances than a “GPS-only” system in terms of Navigation System Error (North, East and Vertical Error) – note that SBAS systems are not designed for the provision of accuracy performances but rather for the stringent integrity requirements, 1e-7-. TERESA also provides real-time Protection Levels information as can be seen in Fig. 122.14. Note that magicSBAS also provides other real-time products, such as ionosphere corrections and integrity. The next figure shows the differences in ionosphere corrections provided by magicSBAS when compared to the “igsg” IONEX [10] for the same day and locations. Ionoe Real Time Estimation error (RMS) is always below 0.75 m (4,5 TECUs) (Fig. 122.15). At magnetic equatorial latitudes (10 –15 from magnetic equator) the ionosphere activity can become a limitation on GNSS augmentation systems, and therefore special attention has to be paid to the estimation of ionospheric delays through SBAS-like algorithms and its performances. From the figures included in this paper, it can be observed that magicSBAS performances in the targeted zone are in line with any other operational SBAS (WAAS) or yet to be operational (EGNOS).
122.6 Summary It is shown in the paper that a SBAS service through magicSBAS product can be provided without the need for the deployment of a dedicated infrastructure, obviously as far as NTRIP data is available. The development and deployment of a complete satellite navigation system for a given region is an expensive technological challenge that requires a relatively long process. Having magicSBAS solution in the region opens the door to multiple developments of applications or scientific research in the region related to SBAS technology without the need of a great investment. A real-time SBAS is ALREADY running in SouthAmerica with about 25 Ntrip GPS-only Stations located in Brazil, 4 stations in Argentina, 1 in Chile and 1 in Venezuela. The APV-I 99% coverage area will be extended as soon as further NTRIP stations will be available (note that GNSS receivers all over South America are already available as can be seen at http:// www.sirgas.org/, but there is a need to upload the data to an NTRIP Caster to be able to be used by magicSBAS). magicSBAS performances are shown to be as good in terms of availability, accuracy, continuity
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Fig. 122.13 Navigation system error comparison
Fig. 122.14 Protection levels in CONZ
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Fig. 122.15 magicSBAS real-time ionosphere
and integrity as any other operational SBAS (WAAS, EGNOS) without the need to deploy a complete set of GNSS stations. Due to the characteristics of the region in terms of ionosphere perturbations, special attention has to be paid to this aspect. Therefore, for an operational SBAS, magicSBAS should be customized to the ionosphere of the region. For a real time monitoring of that SBAS service, please send an email to [email protected] for a username, password and instructions to access this real-time SBAS.1 Biography I. Alcantarilla, J. Caro, A. Cezo´n and J. Ostolaza are part of GMV GNSS team. GMV is responsible for the computing centers in EGNOS – CPFPS – and Galileo programmes – OSPF and IPF – in charge of the computation of the corrections for SV orbits and clocks and ionosphere and integrity. Acknowledgements The authors would like to thank F. Azpilicueta from La Plata University in Argentina for conducting the presentation and the valuable comments on the paper and presentation. http://www.esa.int/esaNA/egnos.html.
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Note that real-time SBAS demonstrator is continuously running from middle of July 2009.
References GMV: http://www.gmv.com Networked Transport of RTCM via Internet Protocol. http://igs. bkg.bund.de/ntrip/ntriphomepage Minimum Operational Performance Standards for Global Positioning System/Wide Area Augmentation System Airborne Equipment. RTCA/DO-229C. 28 Nov 2001 Signal-In-Space available over the Internet (SISNET). http:// www.egnos-pro.esa.int/sisnet/index.html European Geostationary Navigation Overlay Service (EGNOS) EGNOS publications. http://www.egnos-pro.esa.int/ Publications/fact.html TEsting Receiver for EGNOS using Software Algorithms (TERESA). http://magicgnss.gmv.com/ EGNOS Continuous Logging AnalYseR (ECLAYR): http:// www.eclayr.com Polaris. http://www.polarisgmv.com/ Ionospheric delays in IONEX format. ftp://cddisa.gsfc.nasa. gov/pub/gps/products/ionex/
Session 7 The Global Geodetic Observing System: Science and Applications Convenors: R. Gross, H.-P. Plag, L.P. Fortes
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Scientific Rationale and Development of the Global Geodetic Observing System
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G. Beutler and R. Rummel
Abstract
Before addressing GGOS issues we briefly introduce the modern understanding of geodesy and we review the development of geodesy as a science. This background is required to understand the motivation behind the development the International Association of Geodesy’s Global Geodetic Observing System (GGOS). The article then reviews the development of GGOS since the 1998 IAG Section II Symposium in Munich, which may be viewed as the GGOS date of birth. It introduces the GGOS mission and summarizes the milestones of the GGOS establishment. The current state of GGOS implementation is presented and the next steps of the GGOS deployment are discussed.
123.1 Introduction Modern geodesy is based on the three pillars (1) positioning on and near the Earth’s surface, (2) orbital and rotational motion of the Earth as a planet, (3) the Earth’s gravity field. The development of space geodesy with its elements Satellite Laser Ranging (SLR), Very Long Baseline Interferometry (VLBI), and Global Navigation Satellite Systems (GNSS) revolutionized positioning and the monitoring of Earth rotation in the second half of the twentieth century. In the 1990s it became clear that international collaboration in this
G. Beutler (*) Astronomical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland e-mail: [email protected] R. Rummel Institut f€ur Astronomische und Physikalische Geod€asie, Technische Universit€at M€ unchen, Arcisstrasse 21, 80333 M€unchen, Deutschland, Germany
field, realized by the IAG services and the IERS, reached a level of maturity allowing it to study all geodetic aspects related to the first two pillars in a system approach. It became clear, as well, that the planned missions CHAMP (CHAllenging Minisatellite Payload, Reigber et al. (2004)), GRACE (Gravity Recovery And Climate Experiment, Tapley et al. (2004)), and GOCE (Gravity field and steady-state Ocean Circulation Explorer Drinkwater et al. (2006)), would put gravity field research on a comparable level of accuracy, allowing it to study the system Earth on the part per billion level – including mass transport within and between land, oceans, atmosphere and ice fields. It was therefore logical to put international collaboration on a new basis by creating an overarching structure, which is called today the Global Geodetic Observing System (GGOS). The challenge of the coming decade, both in terms of theory and practice, is the integration and maintenance of the three pillars at the one part per billion (10 9) level. Only if this step is successful the metric basis for a prominent geodetic
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_123, # Springer-Verlag Berlin Heidelberg 2012
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component will exist in global change research with themes such as the melting of ice shields, trends of the global water cycle, sea level rise, mass and heat transport in the oceans, post-glacial isostatic adjustment.
123.2 Development of Geodesy as a Science GGOS views the Earth as a complex system. The system approach was already introduced into geodesy about 150 years ago. Since that time geodesy can be regarded as an independent discipline of science. General Baeyer’s memorandum about the size and figure € of the Earth “Uber die Gr€ osse und Figur der Erde”, (Baeyer 1861), may be seen as a starting point, even though important geodetic work had been done before by eminent scientists such as Newton, Laplace, Gauss, and Bessel. But their work was not referred to as geodesy and they did not think of themselves as geodesists. Baeyer’s initiative resulted in an extension and unification of existing triangulation and levelling networks covering Central Europe. This work was then expanded to the whole of Europe, prior to its transition to an international effort with the aim to determine the global figure of the Earth. Baeyer’s initiative eventually led to one of the first international projects in science and marks the root of what is today the International Association of Geodesy (IAG), cf. (Torge 2001). Modern space age began with the launch of Sputnik 1 on October 4, 1957 and shortly after of Sputnik 2. With Vanguard I the first geodetic satellite was launched on March 17, 1958. These satellites already had a fundamental impact on geodesy. Almost instantaneously a large part of 100 years of diligent geodetic work dedicated to the determination of the figure of the Earth became outdated. From the precession of satellites’ orbital planes the Earth’s flattening could be determined much more accurately than with classical astro-geodetic work, compare e.g. (King-Hele 1992). The measurement techniques used in the second half of the twentieth century differed substantially from those of the pre-space age. Astrometric observations were first complemented, then ruled out by the methods to measure distances (e.g., using SLR (Satellite Laser Ranging)) and distance differences (e.g., using VLBI (Very Long Baseline Interferometry)).
G. Beutler and R. Rummel
In the 1980s the partial deployment and open availability of the Global Positioning System (GPS) opened the door to ultra precise positioning and navigation with a modest terrestrial infrastructure. In the 1990s the institutions devoted to the scientific exploitation of the GPS joined forces in the International GPS Service (IGS), today called International GNSS Service (GPS), to deliver GPS ephemerides, GPS satellite clock predictions, Earth rotation parameters and a relatively dense network of site coordinates (and velocities) on an operational basis. For a brief IGS history we refer to Beutler et al. (2009). Positioning, gravity field determination, Earth rotation monitoring and geodetic remote sensing thus can be done much more accurately, completely and efficiently from space. Geodesy became truly global and three-dimensional. Oceans, a “terra incognita” in the classical times became an area of intense geodetic work in the satellite age. Classical geodetic techniques did not allow the accurate measurement of zenith angles, due to atmospheric refraction. From space the vertical dimension of the Earth’s surface can be determined almost as accurately as the horizontal components. Progress of space geodesy was rapid and had a great impact on geodesy and geodynamics. Hand in hand with the rapid development of the geodetic space techniques geosciences became more and more interested in and dependent on the geodetic work. Twenty years ago prominent geodesists, Earth scientists and physicists were invited to a workshop in Erice, Sicily. The workshop dealt with the interdisciplinary role of space geodesy and was organized by Ivan I Mueller and Susanna Zerbini (Mueller and Zerbini 1989). The introductory chapter of this report, due to WM Kaula, is particularly worthwhile to read. He mentions five fields of Earth science and the possible role of space-geodetic techniques for them. The five fields are (1) Earth rotation and core-mantle interaction, (2) mantle convection, (3) regional tectonics and earthquakes, (4) ocean dynamics and (5) Venus–Earth differences. Erice marks the beginning of the era of space geodesy as an autonomous discipline of Earth sciences. In recent years the general emphasis in Earth sciences and the public interest in our field has moved towards Climate Change and Earth System Science. Awareness grew that we need a much better understanding of the Earth as a system, of solar
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radiation as its driving force, of the thermal back radiation and how it is affected by even tiny changes in the chemical composition of the atmosphere, and last not least of the impact of man. One fundamental deficiency became particularly evident in the course of the preparation of the last report of the Intergovernmental Panel on Climate Change, see (Climate 2007), and has been addressed since in several articles in Science and Nature, see Hogan (2005) and the articles quoted there: There is a clear lack of observations. Space geodesy is able to provide important new and unique data to Global Change research by measuring mass and energy transport processes in the Earth system. Chao (2003) wrote: “After three decades and three orders of magnitude of advances, space geodesy is poised for prime time in observing the integrated mass transports that take place in the Earth system, from the high atmosphere to the deep interior of the core”. Geodesy is today in a position to provide “metric and weight” to Earth system research by merging geometry, Earth rotation, gravity and geoid.
123.3 GGOS Mission With this background the establishment of the GGOS was the right step at the right time for geodetic community. It is the GGOS objective to integrate the three pillars into one unified Earth-fixed reference system with a relative precision level of 10 9 and to keep this system stable over decades. Where does such a demanding requirement come from? In geosciences one usually deals with estimates accurate to only a few percent. Global change parameters are, however, small und their temporal changes are slow and even smaller. In general they cannot be observed directly, but have to be derived from a combination of several measurement systems and models. In order to be able to analyze them as a global process they have to be scaled relative to the dimension of the Earth. Sea level at an arbitrary tide gauge may, e.g., vary by a few meters, due to tides and storm surges. Measurement of sea level change with a precision of a few mm in decades requires therefore a relative precision of 10 3 at this particular station. Local sea level monitoring can be transformed into a global monitoring system by satellite systems such as altimetry and GPS. Only then a global process can be deduced
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from local tide gauge records. In order to achieve cm- or mm-precision with satellite systems globally, orbits and altimetric measurements have to be delivered with a relative precision of 1 ppb (part per billion). In order to meet the goals set for GGOS a series of fundamental geodetic problems have to be solved. The three pillars of geodesy, geometry, Earth rotation and gravity, have to be expressed in one and the same Earth-fixed reference system with mm precision and the stability (of the frame) has to be guaranteed over decades. This requires the space as well as the ground segments to function as one homogeneous entity, as if all observations were performed in one observatory encompassing the Earth. Each observation contains a superposition of a variety of effects, related to ionosphere, neutral atmosphere, oceans, ice shields and solid Earth. In order to use them for Earth system research, strategies have to be developed for their separation and quantification by analyzing their spatial, temporal and spectral characteristics. Satellite missions generate time series along their orbit. Via the Earth’s rotation and the choice of the satellites’ orbital elements these time series represent a special spatial and temporal sampling of the measured Earth’s properties. The reconstruction of the temporal and the spatial geophysical phenomena poses a complicated problem of aliasing and inversion. The current investigations of the global water cycle or of the ice mass balance in Greenland and Antarctica from GRACE gravimetry are problems of this type. The inclusion of terrestrial and airborne data, such as surface loading, ocean bottom pressure, tide gauges, gravimetry or altimetry may certainly help. However, this step is not a trivial one either, because terrestrial measurements are affected by local influences and exhibit a spectral sensitivity quite different from that of satellite observations. Probably the most effective support to de-aliasing and separation of geophysical phenomena is the inclusion of prior information, such as models of solid Earth and ocean tides, atmosphere, oceans, ice, hydrology or glacial isostatic adjustment, however, only if they are introduced consistently for all techniques of the observing system. Important work towards these goals is currently underway and we see geodetic techniques used much more widely in the various Earth disciplines than in previous centuries.
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123.4 GGOS History The GGOS is based to a large extent on the IAG services, in particular on the IERS, the IGS, the ILRS, the IVS, the IDS, and the IGFS. Table 123.1 contains the full names, the key objectives and a standard citation. Only the services which already contributed significantly to the GGOS are listed in Table 123.1. A complete list of the currently active services is contained in the IAG ByLaws (accessible, e.g., via the IAG homepage http://www.iag-aig.org/). The essential events related to the creation of GGOS are summarized in Table 123.2. The first event, which may be viewed as the birth date of GGOS, but also of the new IAG Structure, is the IAG Section II Symposium 1998 in Munich. It became clear at this symposium that IAG would not only need a new focus, but an entirely new structure, allowing Table 123.1 IAG Services active in GGOS Unit/start Name and description IERS/87 International Earth rotation and Reference systems Service. Provides data on Earth orientation, Celestial and Terrestrial reference systems/frames, and on geophysical fluids; maintains the IERS conventions (Dick and Richter 2005) IGS/94 International GNSS Service. Provides orbit and clock information for all GNSS satellites for which data are publicly available, derives dense global reference frame, EOPs, atmosphere information (Beutler et al. 2009) ILRS/98 International Laser Ranging Service. primary mission: to support, through satellite and lunar laser tracking data and related products, geodetic and geophysical research activities (Pearlman et al. 2002) IVS/99 International VLBI Service for geodesy and astrometry. Provides a service to support geodetic, geophysical and astrometric research and operational activities, interacts with users of VLBI products, promotes research and development in all aspects of the geodetic and astrometric VLBI technique. (Schl€uter and Behrend 2007) IDS/03 International DORIS Service. Provides a support, through DORIS data and products, to geodetic, geophysical, and other research and operational activities (Tavernier et al. 2006) IGFS/08 International Gravity Field Service. “Umbrella” service coordinating collection, validation, archiving and dissemination of gravity field related data, [. . .] unifying gravity products for the needs of the GGOS (Forsberg et al. 2005)
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the Association to focus on a central issue. This is why the GGOS project (called IGGOS at that time) and the creation of the new IAG structure cannot be separated. For more information concerning the early GGOS planning phase consult Rummel et al. (2000). At the IAG General Assembly in Birmingham in 1999, attached to the XXIInd IUGG General Assembly, the results of 4 years of analysis of the IAG were presented and discussed at the IAG symposium “structure to meet future challenges”. It was decided that an IAG Review Committee should be created and given the task to come up with a proposal for a new IAG structure at the IAG Scientific Meeting in 2001 in Budapest. The IAG Review Committee organized a retreat in February 2000, where experts from geodesy, Earth sciences, government organizations, etc., were invited to develop, together with the committee, a first draft for the new structure. The committee completed its work in a series of meetings, documented its findings in (Beutler et al. 2002), drafted new IAG statutes and ByLaws, and presented its work for approval to the IAG Executive Committee and the IAG Council in Table 123.2 GGOS chronicle of events Time Event 1998 IAG Section II Symposium in Munich: IAG needs a new focus, GGOS named as a candidate 1999 XXII IUGG General Assembly in Birmingham: IAG Review Committee set up 2001 IAG Scientific Assembly in Budapest, Hungary: Proposed changes to the IAG Statutes and ByLaws accepted by IAG Council, Planning group for a GGOS project set up 2003 XXIII IUGG General Assembly in Sapporo, Japan: New IAG Structure implemented, GGOS Planning Group set up (Chair Ch. Reigber, Secretary H. Drewes) 2005 IAG Scientific Assembly in Cairns, Australia: GGOS Planning Group presents GGOS Implementation Plan (Drewes et al. 2005), Creation of GGOS Steering Committee, Chair M. Rothacher, Vice-Chairs: H.-P. Plag, R. Neilan 2007 XXIV IUGG General Assembly in Perugia, Italy: GGOS accepted by IAG Council as Project of IAG 2008 Call for Participation for a GGOS Coordination Office, a Bureau for Network and Communication, a Bureau for Standards and Conventions, and a Bureau for Satellite Missions 2009 Completion of the document “Global Geodetic Observing System: Meeting the Requirements of a Global Society on a Changing Planet in 2020” (Plag and Pearlman 2009)
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Scientific Rationale and Development of the Global Geodetic Observing System
September 2001 in Budapest. The IAG Review Committee also proposed to create a planning group for the establishment of GGOS as IAG’s first project. This proposal was accepted at the IAG Scientific Assembly 2001 by the IAG Council and the IAG Executive Committee, together with the proposed new IAG Statutes and ByLaws (after minor modifications). In 2001 the IAG council approved, upon recommendation of the IAG Executive, a new structure with the GGOS project as the association’s flagship. Modern geodesy should be viewed from the global perspective and efforts of all branches of this science should be bundled to serve one and the same goal. The proposal put forward by the GGOS planning group to the IAG Executive Committee and the IAG Council at the IUGG General Assembly in Sapporo in 2003 contained definition, vision, and mission statements. Moreover, the objectives were specified, a science rationale provided, and the plan to implement GGOS was specified. The GGOS planning committee was created with Prof. Christian Reigber as Chair and Prof. Hermann Drewes as Secretary. In 2005 the GGOS planning committee was dissolved and the GGOS implementation committee created under the leadership of Prof. Markus Rothacher as Chair, with Prof. Hans-Peter Plag and Ms. Ruth Neilan as Vice-Chairs. For the GGOS development between 2000 and 2005 we refer to Beutler et al. (2004, 2005). At the XXIV IUGG General Assembly in Perugia, Italy, GGOS was accepted as a Project by the IAG Council and put on the same level as the Commissions and the Services in the IAG structure. The Chairs and Vice-Chairs of the now official project remained the same as those of the GGOS implementation phase. The highlights since 2007 are (a) the completion of the GGOS2020 reference document (Plag and Pearlman 2009) and (b) the Call for participation in planned new GGOS entities (see Sect. 123.5).
123.5 GGOS Implementation Several phases of the GGOS development were distinguished in Sect. 123.4. Currently, the implementation of GGOS is underway. Let us look more closely into the planned structure and at the deployment phase – using the material of Chap. 10 in (Plag and Pearlman
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2009). GGOS ideally should have the following highlevel components: – Terrestrial, technique-specific entities coordinating the worldwide collection and primary analysis of the observations and generating unique technique-specific products – Uninterrupted series of geodesy-related space missions to observe the time-varying gravity field, the time-varying sea-surface and ice-surface topography to maintain the geometric and gravimetric reference frames – Entities combining the technique-specific products and to come up with technique-independent, combined series of products – An entity proposing the geodetic (and geodesy related) space missions in collaboration with the major space agencies (including geodetic missions to the Moon and other planets) – An entity for Communications and Network coordination to design and continuously improve the GGOS network – A Bureau of Standards to deal with the conventions for the reference systems and frames and the geodetic standards – A Central Coordination Office to coordinate the activities of the above entities and to provide the interface to the GGOS user community and the political decision makers The technique-specific networks and the center for combination of the geometry-related products and Earth rotation, already exist. The GGOS elements in space related to the geometry pillar are in place, as well (GNSS and laser ranging satellites, including our Moon with its laser reflectors). There are, however, no consistent plans for monitoring the gravity field and the fluid components on a long term basis. GGOS must develop a master plan for missions monitoring the Earth’s gravity field, reference frames, and the geometry of oceans and ice sheets. This task has to be seen in analogy to monitoring the geometrical components of the geodetic reference system. This is why GGOS shall have an entity for proposing the geodetic space missions, closely cooperating with the space agencies. GGOS also needs a science panel composed of the leading experts in geodesy and (more generally) in Earth sciences to ensure the GGOS focus to remain on the relevant scientific and societal needs. The GGOS structure is visualized in Fig. 123.1 ((Plag and Pearlman 2009), Chap. 10).
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G. Beutler and R. Rummel
Observations
Steering
Mission teams
Coordiation
Combination IVS
Communication & Networks
Steering Committee
IERS Geometry-related Products
IGS
ILRS Executive Committee
Coordination Office
Geodetie Standards & Conventions
IDS
...
Science Panel
Satellite & Space Missions
IGFS Gravity & Surface Topography
GGP
PSMSL
GLOSS
Fig. 123.1 Proposed GGOS structure, (Plag and Pearlman 2009), Chap. 10
123.6 Embedding GGOS into GEO
123.7 Current Activities and Epilogue
The necessity to preserve the infrastructure for global Earth observation was recognized on the ministerial level. In 2003, the Group on Earth Observations (GEO, http://www.earthobservations. org/) was established as a result of a G8-meeting, and guided by a series of three ministerial-level Earth Observation Summits, GEO developed a plan for the implementation of the Global Earth Observation System of Systems (GEOSS). In 2005, GEO was established permanently, and currently, this group includes 65 member countries and 43 participating organizations. GEOSS will build on and add value to existing Earth-observation systems by coordinating their efforts, addressing critical gaps, supporting their inter-operability, sharing information, reaching a common understanding of user requirements, and improving delivery of information to users. The IAG, represented in GEO by GGOS Steering Committee members, is one of the very active participating organizations. GGOS clearly must become a crucial part of the GEOSS. GGOS may be viewed as the metrological basis of GEOSS.
An attempt was made to fill the gaps in the GGOS structure (see Fig. 123.1) by issuing a Call for Participation in July 2008 for – A Coordinating Office, – A Bureau for Network and Communication, – A Bureau for Standards and Conventions, – A Bureau for Satellite Missions. Three proposals were received, one for the Web Portal (part of the Coordinating Office), one for the Bureau for Networks and Communication, and one for the Bureau for Standards and Conventions. The GGOS Bureau on Networks and Communication (BNC) is hosted at NASA’s Goddard Space Flight Center. The Director of the BNC is Michael Pearlman, Harvard-Smithsonian Center for Astrophysics. The GGOS Bureau on Standards and Conventions (BSC) is jointly hosted by the Research Group on Satellite Geodesy in Germany. The Director of the BSC is Urs Hugentobler, Technical University of Munich. The GGOS Portal is hosted by the German Bundesamt f€ur Kartographie und Geod€asie (BKG) in Frankfurt. A prototype is under development. No
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proposals were received in 2008 for the Coordinating Office and the Bureau on Satellite Missions. This partial failure of the GGOS Call for Participation posed a serious problem. Therefore, the conference of November 1–2, 2009, in Frankfurt am Main (Germany) bringing together the principal agencies contributing to the IAG services and GGOS components, called by Prof. Dietmar Gr€ unreich, President of the German Federal Agency for Cartography and Geodesy (BKG), was timely and necessary. The outcome of the conference is remarkable. The key agencies – Signed up to the GGOS concept laid down in (Plag and Pearlman 2009), – Took note of the situation created by the partial failure of the GGOS Call for participation and started developing a strategy to fill the remaining gaps, – Set up a planning committee for a “GGOS Intergovernmental Committee (GIC)” to take over political responsibility for the GGOS long-term stability (sustainability). Let us conclude by stating that GGOS is based today on a convincingly simple concept uniting all relevant geodetic measurement and analysis techniques. GGOS approaches completion. More than 10 years of struggle to realize GGOS eventually converge to a sound solution enabling a permanent monitoring of the Earth as a complex geodetic system over decades. If the ongoing deployment is pursued with the same determination as in the past 10 year, GGOS will undoubtedly meet the requirements of a Global Society on a Changing Planet in 2020 – as advocated by Plag and Pearlman (2009).
References € Baeyer JJ (1861) Uber die Gr€ osse und Figur der Erde – Eine Denkschrift zur Begr€ undung einer mittel-europ€aischen Gradmessung. Georg Reiner Verlag, Berlin Beutler G, Brunner F, Dickey J, Feissel M, Forsberg R, Mueller II, Rummel R, Sanso F, Schwarz K-P (2002) The IAG review 2000–2001 – executive summary. International Association of Geodesy Symposia, vol 125:603–608 Beutler G, Drewes H, Verdun A (2004) The new structure of the International Association of Geodesy (IAG) viewed from the perspective of history. J Geod 77:566–575 Beutler G, Drewes H, Verdun A (2005) The Integrated Global Geodetic Observing System (IGGOS) viewed from the perspective of history. J Geod 40:414–431 Beutler G, Moore AW, Mueller II (2009) The International global navigation satellite systems service (IGS): development and achievements. J Geod 83:297–307
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Chao BF (2003) Geodesy is not just for static measurements anymore, EOS Transactions. Am Geophys Union 84 (16):145 Climate Change (2007) The physical science basis, contribution of Working Group 1 to the fourth assessment report of the IPCC, Cambridge University Press, Cambridge Dick WR, Richter B (eds) (2005) IERS Annual Report 2005. International Earth Rotation and Reference Systems Service, Central Bureau, Verlag des Bundesamts f€ur Kartographie und Geod€asie, 2007. p 175, ISBN 3-89888-838-X Drewes H, Reigber C, Plag H-P, Rothacher M, Rummel R, Beutler G (2005) IAG GGOS implementation plan, Technical report, GeoForschungsZentrumPotsdam, Germany Drinkwater, M, Haagmans R, Muzi D, Popescu A, Floberghagen R, Kern M, Fehringer M (2006) The GOCE gravity mission: ESA’s first core explorer. ESA SP-627, ESA Publication Division, pp 1–7 Forsberg R, Sideris M, Shum C-K (2005) The gravity field and GGOS. J Geodyn 40:387–393 Hogan J (2005) Warming debate highlights poor data. Nature 436:896 King-Hele D (1992) A tapestry of orbits. Cambridge University Press, Cambridge Mueller II, Zerbini S (eds) (1989) The interdisciplinary role of Space Geodesy, Lecture Notes in Earth Sciences, 22, Springer, Berlin Pearlman MR, Degnan JJ, Bosworth JM (2002) The international laser ranging service. Adv Space Res 30(2): 135–143 Plag H-P, Pearlman M (eds) (2009) Global geodetic observing system: meeting the requirements of a global society on a changing planet in 2020. Springer, Berlin Reigber C, Jochmann H, W€unsch J, Petrovic S, Schwintzer F, Barthelmes F, Neumayer KH, K€onig R, F€orste C, Balmino G, Biancale R, Lemoine JM, Loyer S, Pe´rosanz F (2004) Earth gravity field and seasonal variability from CHAMP. In: Reigber C, Schwintzer P, Wickert J (eds) Earth observation from CHAMP – results from three years in orbit. Springer, Berlin, pp 25–30 Rummel R, Drewes H, Bosch W, Hornik H (eds) (2000) Towards an Integrated Global Geodetic Observing System (IGGOS). International Association of Geodesy Symposia, vol 120:1–261 Rummel R, Drewes H, Beutler G (2002) Integrated global geodetic observing system (IGGOS): A candidate IAG project. Int Assoc Geod Symp 125:609–614 Schl€uter W, Behrend D (2007) The International VLBI Service for Geodesy and Astrometry (IVS): current capabilities and future prospects. J Geod 81(6–8):379–387 Tapley BD, Bettadpur S, Ries JC, Thompson PF, Watkins M (2004) GRACE measurements of mass variability in the Earth system. Science 305(5683) Tavernier G, Fagard H, Feissel-Vernier M, Le Bail K, Lemoine F, Noll C, Noomen R, Ries JC, Soudarin L, Valette JJ, Willis P (2006) The International DORIS Service: genesis and early achievements, in DORIS Special Issue, Willis P (ed), J Geod 80(8–11), pp 403–417 Torge W (2001) Geodesy, 3rd edition, de Gruyter, Berlin
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GGOS Bureau for Standards and Conventions: Integrated Standards and Conventions for Geodesy
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U. Hugentobler, T. Gruber, P. Steigenberger, D. Angermann, J. Bouman, M. Gerstl, and B. Richter
Abstract
The Global Geodetic Observing System (GGOS) is the contribution of the International Association of Geodesy (IAG) to the Global Earth Observing System of Systems (GEOSS). The implementation of common standards and conventions in all components of GGOS is of crucial importance to ensure highest accuracy of geodetic and geophysical products. Consistency with relevant external standards and conventions is mandatory to achieve interoperability with GEOSS. The Bureau for Standards and Conventions (BSC) complements the already existing GGOS structure. Its tasks include keeping track of the strict observance of adopted geodetic standards and conventions applied by the different GGOS components and assuring consistency of data sets released by the services. Complementary tasks include the interaction with international bodies engaged with standards and conventions and the promotion of geodetic standards in the broader scientific and user community and society in general.
124.1 Introduction IAG’s Global Geodetic Observing System (GGOS), geodesy’s contribution to the Global Earth Observing System of Systems (GEOSS, Battrick 2005) provides the metrological basis – in the form of well defined and accurate geodetic products – for measuring and interpreting global deformation and mass exchange processes in the Earth system. The observing system is built on the IAG Services and shall provide integrated
U. Hugentobler (*) T. Gruber P. Steigenberger Institute for Astronomical and Physical Geodesy, Technische Universit€at M€unchen, Arcisstraße 21, 80333 Munich, Germany e-mail: [email protected] D. Angermann J. Bouman M. Gerstl B. Richter Deutsches Geod€atisches Forschungsinstitut, Alfons-Goppel-Str. 11, 80539 Munich, Germany
and consistent products on an operational basis making use of the large variety of space-based and ground-based geodetic observation techniques (Plag and Pearlman 2009). Main initial products are a long-term stable, accurate and global terrestrial reference frame at the mm-level as well as the geopotential reference surface. This requires the integration of geometry, gravimetry, and Earth rotation into a coherent model of the Earth system. The implementation of common standards and conventions for the generation of geodetic and geophysical products is of crucial importance for GGOS. It is essential, that a combined and integrated analysis of the highly precise geodetic observation data is based on consistent and integrated standards and conventions to be applied by all GGOS components. GGOS standards and conventions have to be harmonized with those adopted by other components of
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GEOSS to achieve interoperability with similar observing systems like, e.g., the Global Ocean Observing System (GOOS, Johnson 2009), the Global Climate Observing System (GCOS, Spence and Townshend 1995), or the Global Terrestrial Observing System (GTOS, Kondratev 1995). Users of GGOS products have to be aware of the standards and conventions these products are based on, in order to fully exploit their accuracy and to allow for a coherent interpretation.
124.2 Standards and Conventions The work of the IAG Services is based on defined standards and conventions. The International Earth Rotation and Reference Systems Service (IERS), e.g., provides well developed conventions that are the basis for the work of the geometric Services (McCarthy and Petit 2004). A similar standards document is in preparation by the International Gravity Field Service (IGFS). Nevertheless there are inconsistencies and also gaps. Examples for necessary conventions include a global reference surface pressure (for modeling atmospheric pressure loading) and a global reference temperature (for modeling VLBI telescope deformation). Similar examples can also be found for gravimetric products that depend on numerous reductions of observed quantities which can be done under various assumptions. More complex examples include, e.g., a uniform handling of atmospheric forcing in ocean tide models. A potential source of confusion concerns the time and the tide systems as used by the different geodetic communities. International Astronomical Union (IAU) and International Union of Geodesy and Geophysics (IUGG) resolutions from 1991 require units to be consistent with the Geocentric Coordinate Time (TCG) scale. In practice, however, Terrestrial Time (TT) scale is commonly used since all geodetic measurements are time tagged with a time scale consistent with TT. The scale difference between the two systems amounts to 0.7 ppb (parts per billion) corresponding to 4.5 mm in height at the surface of the Earth. For analysis of geodetic data all constants and measurements have to refer to the same system. For orbit analysis in TT scale, e.g., the corresponding geocentric gravity parameter GM has to be adopted, see Table 124.1. Although the scale of the
U. Hugentobler et al. Table 124.1 Geocentric gravitational constant (GM) System TCG value TT value
Value 398.600 441 8 10 398.600 441 5 10
12
m3 s 12 m3 s
2 2
Comment IERS 2003 EGM96, . . .
International Terrestrial Reference Frame (ITRF) is TT-compatible for all realizations since ITRF2000, the GM value specified in the IERS Conventions 2003 (McCarthy and Petit 2004) corresponds to TCG scale. This may cause confusion. As an example we may note that the GM value specified in the Galileo Open Service Signal in Space Interface Control Document (Galileo OS SIS ICD 2006) corresponds to TCG scale while the broadcast information provided by the Galileo system is consistent with ITRF and thus corresponding to TT scale. Consequently, Galileo uses different scales for orbits and for the terrestrial network. The difference in orbit scale, amounting to 2 cm at satellite altitude, is insignificant in view of the broadcast accuracy. It is, however, a potential source of confusion. The Galileo Geodetic Service Provider (Gendt et al. 2009), on the other hand, uses a GM value that is consistent with TT. Orbits and the Galileo Terrestrial Reference Frame (GTRF) provided by this service for high precision users are thus consistent. Table 124.2 lists a selection of geodetic constants as defined in the Geodetic Reference System GRS80 (Moritz 2000) and IERS2003 (McCarthy and Petit 2004). The IERS Conventions specify uncertainties as given by Groten (2004). Further details on the interpretation of these uncertainties (formal errors? one-sigma values?) need to be specified. Finally, a source of confusion is the handling of the permanent tide that varies between the different geodetic communities. While the gravimetric Services Table 124.2 Selected geodetic constants with uncertainties in the last digits (in brackets) Quantity GRS80 (Moritz 2000) GM 398.600 5 10 12 J2 1082.63 10 6 ae 6378 137 1/f 298.257 22 o 7.292 115 10 W0 62 636 860.85
5
IERS2003 (McCarthy and Petit 2004) 398.600 441 8(8) 10 12 m3 s 2 1082.635 9(1) 10 6 m3 s 6378 136.6(1) m 298.256 42(1) m 7.292 115 10 5 rads 1 62 636 856.0(5) m2 s 2
2
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GGOS Bureau for Standards and Conventions
provide products in the zero-tide system, in agreement with IAG resolution 16 of the 18th General Assembly 1983, the geometric Services provide their products, e.g., the ITRF, in the tide-free system for historical reasons. Inconsistencies may arise if geometric and gravimetric products are combined without prior conversion into the same tide system. Eventually all GGOS components should strive for a consistent handling of permanent tides. As a shortterm goal it may, however, be difficult to reach this ideal state. It is more important to make users aware about the difference, e.g., by annotating products with the permanent tide system they are based on. A step may be, e.g. to define a corresponding mandatory keyword for the SINEX (Software Independent Exchange) format. Consistency of standards and conventions used within GGOS with similar standards and conventions employed by external organizations, in particular with the other observing systems contributing to GEOSS, is equally important as internal consistency. GEOSS strives for interoperability of the observing systems in order to provide ready and transparent access to products by users of GEOSS. Interoperability has two aspects, on the first hand data and metadata formats, interfaces and web portals have to be standardized. On the other hand, products exchanged within and provided by GEOSS have to base on consistent numerical constants, standards and conventions. It is this second task where the BSC shall engage to assure use of coherent geodetic standards and conventions throughout Earth observation disciplines..
124.3 Tasks of the GGOS Bureau for Standards and Conventions The Bureau for Standards and Conventions (BSC) complements the already existing GGOS structure. It is the successor of the GGOS Working Group on Conventions, Models, Analysis that was chaired by Hermann Drewes. The BSC is hosted and supported by the Institut f€ ur Astronomische und Physikalische Geod€asie (IAPG) of Technische Universit€at M€ unchen and the Deutsches Geod€atisches Forschungsinstitut (DGFI), both in Munich, Germany, under the umbrella of the Forschungsgruppe Satellitengeod€asie (FGS). This consortium, that includes in addition the Forschungseinrichtung Satellitengeod€asie (FESG) of
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Technische Universit€at M€unchen, the Bundesamt f€ur Geod€asie und Kartographie (BKG), Frankfurt, Germany, and the Institut f€ur Geod€asie und Geoinformation (IGG) of Universit€at Bonn, Germany, operates the geodetic observatory in Wettzell, Germany, and pursues research projects in space geodesy. According to the Terms of Reference the objectives of the BSC are: • To keep track of the strict observance of adopted geodetic standards, standardized units, fundamental physical constants, resolutions and conventions in the generation of the products issued by GGOS. • To review, examine and evaluate all standards, constants, resolutions and conventions adopted by IAG or its components and recommend their use or propose the necessary updates. • To identify gaps, inconsistencies, and deficiencies in standards and conventions and to initiate steps to remove them. • To propose the adoption of new standards and conventions where necessary. • To propagate standards and conventions to the wider scientific community and promote their use The work of the BSC is thus on one hand directed to the geodetic community to assure that a consistent set of standards and conventions is used and on the other hand to the broader scientific community and to society in general by promoting the use of such consistent standards. Tasks of the GGOS BSC include reviewing the relevant resolutions concerned with geodetic standards and conventions and drawing up an inventory of constants, standards, and conventions used across all IAG Services for the computation of GGOS products. They specifically also include assuring consistency with the relevant external standards and conventions employed by other Earth observation systems. The BSC shall provide the appropriate information to convert the products of these observation systems – e.g. atmospheric, hydrological and oceanographic fields – to those consistent with the GGOS standards and conventions. The BSC shall engage, e.g., with the Standards and Interoperability Forum (SIF) of GEOSS to support this aspect of interoperability. The BSC is already associated to the control body for the geodetic registry network of ISO/TC211 (International Organization for Standardization, standard technical committee Geographic information/ Geomatics). The Bureau shall facilitate the registration
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of geodetic standards and conventions through international standards organizations such as ISO. To evaluate the impact of inconsistent use of standards and conventions the Bureau has software packages available for analysis of data from the major space geodetic techniques, possesses GNSS reprocessing capability, and can draw experience from its deep involvement in global gravity field recovery missions like GRACE and GOCE. Furthermore the BSC shall stimulate and support the documentation of conventions similar to those of IERS for the gravity Services and their products (IGFS Standards and Conventions). To increase users’ appreciation of the standards and conventions that are the basis of the GGOS products provided by the IAG Services, the BSC advocates the development of corresponding product metadata. In fact, it is mandatory that each product is accompanied by a standards documentation sheet. Since the conversion from one tide system to the other is straightforward, validated and easy-to-use tools can be made available for the transformations. The BSC considers it as one of its tasks to develop such a toolbox and standardization of the transformations between the two systems for all affected quantities. Based on today’s best estimates the BSC should compile a new set of defining constants similar to GRS80 that may eventually result in a new Geodetic Reference System, e.g., GRS2011. To ensure global acceptance all relevant parties have to be engaged in such developments. To fulfill its objectives the BSC may set up working groups for specific tasks. Outreach activities will include the participation at conferences, submission of papers to journals in neighboring fields, organization of dedicated workshops, and the setup of a website at the GGOS portal. The BSC works closely together with experts in this field, inside and outside GGOS. The BSC maintains regular contact and establishes a strong interface with all the IAG Services and Commissions and international bodies involved in the adoption of standards, resolutions, and conventions, in particular with IERS, IAU, BIPM (Bureau International des Poids et Mesures), the GEOSS Standards and Interoperability Registry, CODATA (Committee on Data for Science and Technology), NIST (National Institute of Standards and Technology), and ISO/ TC211.
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124.4 Summary Consistent and integrated standards and conventions are of crucial importance for GGOS to make available the metrological basis for monitoring mass transport processes in the Earth system with long-term stability and high accuracy and thus to contribute to the understanding of Global Change in concert with GEOSS. The GGOS Bureau for Standards and Conventions was installed in spring 2009 with the objective to assure consistency of standards and conventions used by the different GGOS components, to guarantee interoperability with GEOSS, and to promote geodetic standards and conventions in the broader scientific community. Interfaces are being installed with national and international bodies engaged with standards and conventions. Everybody is invited to contribute to this effort. Acknowledgement We acknowledge the comments and contribution of additional aspects related to standards and conventions provided by the reviewers which was greatly appreciated.
References Battrick B (ed) (2005) Global Earth Observing System of Systems (GEOSS) – 10-year implementation plan reference document. GEO 1000R/ESA SP-1284. ESA Publications Division, Noordwijk Galileo OS SIS ICD (2006) Galileo open service signal in space interface control document, draft. European Space Agency/ Galileo Joint Undertaking Gendt G, Altamimi Z, Dach R, S€ohne W, Springer T (2009) GGSP: realisation and maintenance of the Galileo terrestrial reference frame. In: Proceedings of the 2nd international colloquium – scientific aspects of the Galileo Programme, Padua, 14–16 October 2009 Groten E (2004) Fundamental parameters and current (2004) best estimates of the parameters of common relevance to astronomy, geodesy, and geodynamics. J Geod 77(10–11): 724–731. doi:10.1007/s00190-003-0373-y Johnson M (2009) Implementing the global ocean observing system. WMO Bull 57(1):35–40 Kondratev K (1995) Global Terrestrial Observing System (GTOS) – detection and monitoring of continental systems. Earth Obs Remote Sens 12(3):468–473 McCarthy DD, Petit G (2004) IERS Conventions (2003). IERS Technical Note 32, BKG, Frankfurt Moritz H (2000) Geodetic reference system 1980. J Geod 74(1): 128–133. doi:10.1007/s001900050278 Plag HP, Pearlman M (2009) Global geodetic observing system – meeting the requirements of a global society on a changing planet in 2020. Springer, Berlin. ISBN 978-3-642-02686-7 Spence T, Townshend J (1995) The Global Climate Observing System (GCOS). Clim Change 31(2–4):131–134. doi:10.1007/BF01095141
VLBI2010: Next Generation VLBI System for Geodesy and Astrometry
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W.T. Petrachenko, A.E. Niell, B.E. Corey, D. Behrend, H. Schuh, and J. Wresnik
Abstract
The International VLBI Service for Geodesy and Astrometry (IVS) is well on the way to fully defining a next generation VLBI system, called VLBI2010. The goals of the new system are to achieve 1-mm position accuracy over a 24-h observing session and to carry out continuous observations, with initial results to be delivered within 24 h after taking the data. These goals require a completely new technical and conceptual design of VLBI measurements. Based on extensive simulation studies, strategies have been developed by the IVS to significantly improve its product accuracy through the use of a network of small (~12-m) fastslewing antennas, a new method for generating high precision delay measurements, and improved methods for handling biases related to system electronics, deformations of the antenna structures, and radio source structure. To test many of the proposed strategies, NASA is sponsoring a proof-of-concept development effort using IVS antennas near Washington, DC, and Boston, MA. Furthermore, as of Feb. 2009, the construction of ten new VLBI2010 sites has already been funded, which will improve the geographical distribution of geodetic VLBI sites and provide an important step towards a global VLBI2010 network.
125.1 Introduction
W.T. Petrachenko (*) Geodetic Survey Division, CCRS, Natural Resources Canada, 615 Booth St, Ottawa, ON K1A 0E9, Canada e-mail: [email protected] A.E. Niell B.E. Corey Haystack Observatory/Massachusetts Institute of Technology, Off Route 40, Westford, MA 01886-1299, USA D. Behrend NVI, Inc./Goddard Space Flight Center, Code 698, Greenbelt, MD 20771, USA H. Schuh J. Wresnik Institute of Geodesy and Geophysics, Vienna University of Technology, Gusshausstrasse 27-29, 1040 Vienna, Austria
Very Long Baseline Interferometry (VLBI) is an essential technique for the realization of reference frames, both terrestrial and celestial. It owes its significance primarily to the fact that it refers its observations to quasars, which are distant extragalactic radio sources that appear to be nearly fixed in angular position. VLBI determinations of the coordinates of a selected subset of these sources realize the International Celestial Reference Frame (ICRF), and VLBI determinations of the positions and velocities of about 30 globally distributed radio antennas contribute significantly to the realization of the International Terrestrial Reference Frame (ITRF), in particular to the
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definition of its scale. VLBI also measures the link between the ICRF and ITRF by determining the full set of Earth orientation parameters (EOP) including DUT1 and nutation which are provided uniquely. Because of its dual role with respect to the ICRF and ITRF, the International VLBI Service for Geodesy and Astrometry (IVS), which is the governing body for VLBI, is a joint service of the International Association of Geodesy (IAG) and the International Astronomical Union (IAU) (Schl€ uter and Behrend 2007). The current VLBI system was conceived and constructed mostly in the 1960s and 1970s. Aging antennas, increasing radio frequency interference (RFI) problems, obsolete electronics, and high operating costs make it increasingly difficult to sustain the current levels of accuracy, reliability, and timeliness. In September 2003 the IVS, recognizing the limitations of existing VLBI infrastructure and the increasingly demanding requirements of space geodesy, established Working Group 3 (WG3): VLBI2010 to investigate options for modernization (Behrend et al. 2009). Guided by emerging space geodesy science and operational needs (Schuh et al. 2002; Plag and Pearlman 2009), WG3 established challenging goals for the next generation VLBI system: • 1 mm position and 0.1 mm/year velocity accuracy on global scales • Continuous measurements for time series of station positions and EOP • Posting of initial geodetic results less than 24 h after observations are complete In its final report (Niell et al. 2006), WG3 proposed strategies to move toward the unprecedented 1 mm position accuracy target and broad recommendations for a next generation system based on the use of smaller (~12 m) fast-slewing automated antennas. To help make these recommendations more specific, the report additionally suggested a series of 13 studies and development projects. In order to encourage the realization of the recommendations of WG3, the IVS, in September 2005, established the VLBI2010 Committee (V2C) as a permanent body of the IVS. The V2C takes an integrated view of VLBI and evaluates the effectiveness of proposed system changes based on the degree to which they improve the final products. The V2C work goes hand-in-hand with the gradual establishment of the Global Geodetic Observing System
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(GGOS) (Plag et al. 2009) of the IAG. To realize the demanding goals for VLBI2010, the following strategies have been investigated by the V2C: • A reduction of the random component of the delay observable noise, e.g., the measurement error per observation, the stochastic properties of the clocks, and the unmodeled variations in the atmosphere • A reduction of the systematic errors, e.g., the thermal and gravitational deflection of the antenna, drifts of the electronics, and radio source structure • An increase of the number of antennas for geodetic VLBI and an improvement of their geographic distribution • An increase of the observation density, i.e. the number of observations per unit time • A reduction of susceptibility to external radio frequency interference This paper summarizes the work of the V2C to date, which has been described thoroughly in a recently published progress report (Petrachenko et al. 2009).
125.2 Monte Carlo Simulators Making rational design decisions for VLBI2010 requires a realistic understanding of the impacts of new operating modes on final products. These impacts are difficult to evaluate analytically due to complex interactions in the VLBI analysis process and are impractical to evaluate with real data due to the high cost of VLBI systems and operations. To fill this gap, Monte Carlo simulators based on the two VLBI analysis software packages Calc/Solve and OCCAM and a third simulator based on Precise Point Positioning (PPP) were developed. The concept of a Monte Carlo simulator involves the generation of several sets (typically 25 for the VLBI2010 studies) of input data, with each set driven by different random numbers. All data sets are then processed as if they were from real sessions, and the ensemble of output products is analyzed statistically to produce estimates of the bias and standard deviation of those products. The Monte Carlo simulators are only as realistic as the models used to generate the simulated input data. Efforts continue to improve those models (Nilsson and Haas 2008; MacMillan 2008; Wresnik et al. 2008). The simulators have been used to study the effects of the dominant
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median rms 3D position error [mm]
12 OCCAM PPP SOLVE
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Fig. 125.1 Median of the rms 3D position errors for uniform sky schedules with regular source-switching intervals ranging from 15 to 360 s. The delay measurement noise is 4 ps per baseline observation, the clock Allan standard deviation is 1 10 14 at 50 min, and the turbulence parameters are those driven by a turbulence model (Petrachenko et al. 2008)
VLBI random error processes (related to the atmosphere, the reference clocks, and the delay measurement noise) and the benefit of new approaches to reduce them, such as decreasing the sourceswitching interval and improving analysis and scheduling strategies. Of particular merit is the strategy to reduce the source-switching interval. Because of the directionality of the parabolic antennas used in VLBI, each antenna can observe only one source at a time. The antenna must then be mechanically redirected to observe the next source. Reducing the average time to complete the observe/redirect cycle was shown to result in a nearly proportionate improvement in station position accuracy (see Fig. 125.1). Regardless of the strategy employed, the simulators confirm that the dominant error source is the atmosphere. Research into better ways to handle the atmosphere will continue to be a priority for the IVS (Petrachenko et al. 2009).
125.3 System Considerations Based on the Monte Carlo studies, high priority is placed on finding strategies for reducing the sourceswitching interval. This entails decreasing both the on-source time needed to make a precise delay measurement and the time required to slew between
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sources. From these two somewhat competing goals, recommendations for the VLBI2010 antennas are emerging, e.g. use either a single 12-m diameter antenna with very high slew rates (e.g., 12 /s in azimuth and 4 /s in elevation) or a pair of 12-m diameter antennas, each with more moderate slew rates (e.g., 5 /s in azimuth and 1.5 /s in elevation) (Petrachenko et al. 2008). In order to shorten the on-source observing time, it is important to find a means for measuring the delay with the requisite precision even at a modest signal-tonoise ratio. To do this a new approach is being developed in which several widely spaced frequency bands are used to unambiguously resolve the interferometric phase. The new observable is being referred to as the broadband delay. To do this, a four-band system is recommended that uses a broadband feed to span the entire frequency range from 2 to 14 GHz. In order to be able to detect an adequate number of high-quality radio sources, a total instantaneous data rate as high as 32 Gbps and a sustained data storage or transmission rate of 8 Gbps are necessary. Since the broadband delay technique is new and untested, NASA is funding a proof-of-concept development effort to verify that it works and to gain experience with the new VLBI2010 systems. It is also recognized that reducing systematic errors plays a critical role in improving VLBI accuracy. For minimizing electronic biases, updated calibration systems and processes are being developed. For errors due to source structure, the application of corrections based on images derived directly from the VLBI2010 observations is under study. For antenna deformations, conventional surveying techniques continue to be refined, while the use of a small reference antenna for generating deformation models and establishing site ties is also under consideration (Petrachenko et al. 2009). The last concept is based on the idea that the reference antenna, being small, can be mechanically and thermally well understood. Connected element interferometry between the small antenna and the primary VLBI antenna can then be done to develop thermal and gravitational deformation models for the primary antenna. The small antenna being parabolic also has a well-defined phase center and experiences minimal multi-path. If it is sensitive to GNSS frequencies, joint observations can be carried out with a co-located GNSS antenna to develop in situ models for the GNSS
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antenna phase center deviations. In the process of carrying out these observations, the effective reference points of both the VLBI and GNSS antennas can be determined relative to the intersection of axes of the small reference antenna, the net result being a site tie not between nominal reference points for the antennas but between the effective reference points as they appear in the context of normal operations. This can be done with minimal complication or risk of misinterpretation compared to more classical approaches. The system development effort for VLBI2010 is significant. It involves nearly a complete reworking of the current S/X system. To give a feeling for the degree of change, a number of the new VLBI2010 characteristics are listed below: • The VLBI2010 feed is dual linearly polarized and spans the entire frequency range from 2 to 14 GHz. • The entire feed structure is cryogenically cooled. • The phase calibration unit is constructed using digital integrated circuits. • The amplitude calibration unit will operate in the 80-Hz synchronous mode. • The radio frequency (RF) signals are transported from vertex to control room via analog fiber optics. • The RF frequency of four dual-polarized bands can be independently selected using flexible up-down converters (UDC). • The outputs of the UDCs are high-frequency sampled with at least 8-bit resolution and all back end processing is performed digitally. • RFI mitigation strategies will be implemented in the digital back ends. • Data rates of up to 32 Gbps will be burst-mode acquired into RAM while the antenna is on source and will be recorded (or transmitted via eVLBI) at sustained rates of up to 8 Gbps as the antenna slews. • Correlation will be done in software.
through averaging, dense time series of station positions will perhaps more importantly prove valuable for revealing, understanding, and eventually reducing systematic errors. For Celestial Reference Frame (CRF), a larger, better-distributed global network improves u-v coverage, which is a prerequisite for generating higher quality images of radio sources, and also yields more uniform CRF quality between the northern and southern celestial hemispheres. In addition, a more uniform north-south distribution of stations leads to reduced coupling between global troposphere gradients and estimates of station latitude and source declinations. For EOP it is necessary that the VLBI2010 estimates be strongly coupled to the ITRF. Experience shows that VLBI EOP estimates include small network dependent biases and that those biases change somewhat with time. The systematic impact of any single station on VLBI EOP determinations can be expected to become smaller as the network size increases, making a larger network more robust against changes in network composition. It is recommended that a globally distributed network of at least 16 VLBI2010 antennas observe every day to determine EOP, and that other antennas be added as needed for the maintenance of the ICRF and ITRF. A subset of antennas with access to highspeed fiber networks is also required to enable daily delivery of initial IVS products in less than 24 h. For the observation of faint radio sources for densification of the ICRF at least four large radio telescopes per hemisphere are also recommended. High priority is placed on increasing the number of stations in the southern hemisphere.
125.4 Network Considerations
The key to developing a fully integrated observing strategy for CRF, Terrestrial Reference Frame (TRF), and EOP is to have a correlator capacity that can handle significantly more sites than is needed just for daily determination of EOP. Incorporating the added antennas into integrated observing schedules that overlap with the daily EOP schedules will enhance the connection to the stations that observe on a daily basis and hence have well established locations.
It is vital that the ITRF scale provided by VLBI2010 be accurate and be transferred as effectively as possible to the ITRF. A robust transfer requires a large total number of VLBI sites co-located with the other techniques, while a more stable scale estimate requires more frequent observations with a larger properly distributed network. Although more frequent observations can be expected to improve results
125.5 Operational Considerations
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Since IVS products must be delivered without interruption, a transition period to VLBI2010 operations is required in which there will be a mix of antennas with current and next-generation receiving systems. For this period a compatibility mode of operation has been identified and tested to a limited extent with the NASA proof-of-concept system. To preserve continuity in particular with respect to the strength of stable long-term time series of station positions, baseline lengths, and troposphere parameters, among other things, several existing radio telescopes are expected to continue VLBI observations for many years to come. In order to increase reliability and to reduce the cost of operations, enhanced automation will be introduced both at the stations and in the analysis process. Stations will be monitored centrally to ensure compatible operating modes, to update schedules as required, and to notify station staff when problems occur. Automation of the analysis process will benefit from the work of IVS Working Group 4, which is updating VLBI data structures and modernizing data delivery.
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125.6 Status at the End of 2009 VLBI2010 is now well on the way to the definition of requirements and recommendations for subsystem specifications. This is expected to be complete in 2010. At the same time, sponsored by NASA, full progenitor VLBI2010 signal paths have already been implemented on two antennas, the 18-m antenna at the Haystack Observatory in Westford, Massachusetts, and the 5-m MV-3 antenna at the NASA Goddard Space Flight Center (GSFC) in Maryland, a baseline of 597 km. The development effort is significant since the VLBI2010 system design involves nearly a complete reworking of the current S/X system. With the new systems, fringes are now routinely detected and the broadband delay technique is currently under test. A 12-m VLBI2010-style antenna has also been acquired for the GSFC site and will soon be installed. Worldwide, about ten VLBI2010 antennas have been funded and are currently in various stages of implementation. New antennas are coming up in Australia, Korea, New Zealand, Germany (Fig. 125.2), Spain/Portugal
Fig. 125.2 An artist’s conception of the VLBI2010 antenna being designed by Vertex Antennentechnik GmbH. A pair of these antennas will be installed at Wettzell Fundamental Station in Germany as part of the Twin Telescope Wettzell project. These antennas are fully VLBI2010 compliant
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Fig. 125.3 Proposed locations for antennas of the Spain/Portugal RAEGE Project for studying dynamics at the intersection of the North American, African and Eurasian Plates
Fig. 125.4 Prototype of a broadband 2–14 GHz “Eleven” feed for VLBI2010. The feed is being developed at Chalmers University in Sweden
(Fig. 125.3), and USA. The IVS has been approached by several other countries regarding VLBI2010, and a number of proposals for new sites are in various stages of preparation and approval. Beyond the NASA development effort, organizations in other countries are involved in system developments potentially applicable to VLBI2010. These include Australia, China, Finland, Germany, Italy, Japan, Norway, Sweden (Fig. 125.4), and possibly others. A small radio antenna that is potentially applicable as a reference antenna for VLBI antenna deformation modeling and site ties is under development in Japan (Ichikawa et al. 2008). The VLBI2010 concept also needs strategic and political support to be realized. In March 2009 a small VLBI2010 Project Executive Group (V2PEG) was established to move to the next level of defining
development and deployment schedules and soliciting contributions. There are a number of risks to the successful implementation of VLBI2010, the most significant of which follow, together with respective fallback options: • The higher slew rates and the smaller antennas for the VLBI2010 system will result in a significant increase in data volume and hence higher shipping and/or transmission costs are anticipated. It is expected that future technological advances will reduce these costs. In the interim less data-intensive operating modes may be employed. • Radio frequency interference (RFI) is an ever increasing problem in the VLBI2010 spectrum. The VLBI2010 system is being designed to be resilient against it. • The broadband delay technique has not been fully verified. Known risks come from RFI and source structure. The NASA proof-of-concept test is now in the process of testing the concept. In the event that problems are identified, less attractive but adequate fallback options, such as the use of multiple bands for group delay, have been defined.
125.7 Next Steps The development, realization, and implementation of the VLBI2010 system will continue in the next several months and years. Major items on the work schedule are: • Continue the NASA proof-of-concept effort. • Continue defining subsystem recommendations. • Improve algorithms for scheduling observations.
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• Develop VLBI2010 analysis strategies including automation. • Promote the expansion of the VLBI2010 network. • Develop VLBI2010 deployment schedules. Acknowledgments The work on the definition and design of VLBI2010 was supported by many IVS members and components. All of them are gratefully acknowledged.
References Behrend D, B€ohm J, Charlot P, Clark T, Corey B, Gipson J, Haas R, Koyama Y, MacMillan D, Malkin Z, Niell A, Nilsson T, Petrachenko B, Rogers A, Tuccari G (2009) Recent progress in the VLBI2010 development. In: Sideris M (ed) Observing our changing earth, vol 133. IAG Symposia, Perugia, pp 833–840 Ichikawa R, Ishii A, Takiguchi H, Kuboki H, Kimura M, Nakajima J, Koyama Y, Kondo T, Machida M, Kurihara S, Kokado K, Matsuzaka S (2008) Development of a compact VLBI system for providing over 10-km baseline calibration. In: Finkelstein A, Behrend D (eds) “Measuring the future”, proceeding of the fifth IVS general meeting. Nauka, St. Petersburg, ISBN:978-5-02-025332-2, pp 400–404 MacMillan D (2008) Simulation analysis of the geodetic performance of networks of VLBI stations. In: Finkelstein A, Behrend D (eds) “Measuring the future”, proceeding of the fifth IVS general meeting. Nauka, St. Petersburg, ISBN:9785-02-025332-2, pp 416–420 Niell A, Whitney A, Petrachenko B, Schl€ uter W, Vandenberg N, Hase H, Koyama Y, Ma C, Schuh H, Tuccari G (2006) VLBI2010: current and future requirements of geodetic VLBI systems. In: Behrend D, Baver K (eds) International VLBI Service for Geodesy and Astrometry 2005 annual report, NASA/TP-2006-214136, pp 13–40
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Nilsson T, Haas R (2008) Modelling tropospheric delays with atmospheric turbulence models. In: Finkelstein A, Behrend D (eds) “Measuring the future”, proceedings of the fifth IVS general meeting. Nauka, St. Petersburg, ISBN:978-5-02025332-2, pp 361–370 Petrachenko B, Boehm J, MacMillan D, Pany A, Searle A, Wresnik J (2008) VLBI2010 antenna slew rate study. In: Finkelstein A, Behrend D (eds) “Measuring the future”, proceedings of the fifth IVS general meeting. Nauka, St. Petersburg, ISBN:978-5-02-025332-2, pp 410–415 Petrachenko B, Niell A, Behrend D, Corey B, Boehm J, Charlot P, Collioud A, Gipson J, Haas R, Hobiger T, Koyama Y, MacMillan D, Malkin Z, Nilsson T, Pany A, Tuccari G, Whitney A, Wresnik J (2009) Design aspects of the VLBI2010 system – progress report of the IVS VLBI2010 Committee, NASA/TM-2009-214180, 58 pp Plag H-P, Pearlman M (eds) (2009) Global geodetic observing system meeting the requirements of a global society on a changing planet in 2020. Springer, Berlin. doi:10.1007/9783-642-02687-4. ISBN 978-3-642-02686-7 Plag H-P, Gross R, Rothacher M (2009) Global geodetic observing system for geohazards and global change. Geosci BRGM’s J Sustain Earth 9:96–103 Schl€uter W, Behrend D (2007) The International VLBI Service for Geodesy and Astrometry (IVS): current capabilities and future prospects. J Geod 81(608):379–387 Schuh H, Charlot P, Hase H, Himwich E, Kingham K, Klatt C, Ma C, Malkin Z, Niell A, Nothnagel A, Schl€uter W, Takashima K, Vandenberg N (2002) IVS Working Group 2 for product specification and observing programs. In: Vandenberg N, Baver K (eds) International VLBI Service for Geodesy and Astrometry 2001 annual report, NASA/TP2002-21000181, pp 13–45 Wresnik J, Boehm J, Pany A, Schuh H (2008) VLBI2010 simulations at IGG Vienna. In: Finkelstein A, Behrend D (eds) “Measuring the future”, proceedings of the fifth IVS general meeting. Nauka, St. Petersburg, ISBN 978-5-02025332-2, pp 421–426
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The New Vienna VLBI Software VieVS €hm, S. Bo €hm, T. Nilsson, A. Pany, L. Plank, J. Bo H. Spicakova, K. Teke, and H. Schuh
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Abstract
New VLBI (Very Long Baseline Interferometry) data analysis software (called Vienna VLBI Software, VieVS) is being developed at the Institute of Geodesy and Geophysics in Vienna taking into consideration all present and future VLBI2010 requirements. The programming language MATLAB is used, which considerably eases the programming efforts because of many built-in functions and tools. MATLAB is the high-end programming language of the students at the Vienna University of Technology and at many other institutes worldwide. VieVS is equipped with the most recent models recommended by the IERS Conventions. The parameterization with piece-wise linear offsets at integer hours in the leastsquares adjustment provides flexibility for the combination with other space geodetic techniques. First comparisons with other VLBI software packages show a very good agreement, and there are plans to add further features to VieVS, e.g. capabilities for Kalman filtering, phase delay solutions, and spacecraft tracking.
126.1 Introduction Until 2009, the Institute of Geodesy and Geophysics (IGG) of the Vienna University of Technology had been using the Occam software package (Titov et al. 2004) for the analysis of Very Long Baseline Interferometry (VLBI) observations. Occam (previously called Bonn VLBI Software System, BVSS) had been in use since the beginning of the 1980s, and many subroutines and source code lines have piled up which are obsolete and no longer necessary. Some
J. B€ohm (*) S. B€ ohm T. Nilsson A. Pany L. Plank H. Spicakova K. Teke H. Schuh Institute of Geodesy and Geophysics, Vienna University of Technology, Gusshausstrasse 27-29, 1040 Vienna, Austria e-mail: [email protected]
technical Fortran-related details date back to the 1980s, e.g. the use of COMMON blocks to exchange variables between the subroutines. Thus, the source code of Occam is rather difficult to read for persons who have not been involved in the development of the software. At Geoscience Australia (GA) and Deutsches Geod€atisches Forschungsinstitut (DGFI) efforts are being undertaken to modernize the existing Occam software package. As an Associated IVS Analysis Center, IGG Vienna mostly deals with research tasks and not with operational (routine) VLBI processing, e.g. by making modifications of the source code by introducing new models and algorithms. Most of the students at IGG are experts in MATLAB but are not experienced in writing Fortran source code, which makes the work very difficult and time consuming for them. This has
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been the main motivation for developing new VLBI software in MATLAB. Furthermore, MATLAB has many built-in tools and functions which make the writing of the code much faster, and the source code is significantly shorter and more concise. Examples for built-in tools are matrix operations, plotting and graphical interface tools, NetCDF readers and writers, or structure arrays which are very useful when storing scan-based information. Another reason for developing new VLBI software is that several updates of the VLBI software package Occam 6.1 were pending, and the accomplishment of these tasks was much easier in MATLAB than in Fortran. In particular, the software was made fully consistent with the most recent IERS Conventions (McCarthy and Petit 2004) (e.g., nonrotating origin and the corresponding partials for the nutation parameters), and for clarity all former models which were no longer necessary were removed (e.g. equinox-based transformation). Of course, there are also arguments against using MATLAB – the main of which is certainly that MATLAB is commercial software. However, many institutions worldwide have access to MATLAB and use it. Alternatively, we can provide executables of VieVS which run on any machine (without having MATLAB installed). Another possibility is the use of non-commercial counterparts of MATLAB like Octave which will be tested in the next months. If possible, we will modify VieVS in a way that it can be run with Octave or other counterparts. The other argument against MATLAB is that it is slower than Fortran or C/C++. This is certainly true, but tests showed that this is not critical in our case. Even if VieVS takes twice as long as e.g. the Fortran-based Occam, we think that this is fast enough for most of the research purposes, but also for the determination of global solutions. At the IGG Vienna, the classical Gauss–Markov least-squares adjustment of Occam 6.1 has been used which is based on piece-wise linear functions using rates for the representation of zenith wet delays, clocks, or Earth orientation parameters. These rates do not correspond to intervals between integer hours as their first epoch is arbitrary (epoch of first observation in the VLBI session). With the new software VieVS, we estimate the parameters as piece-wise linear offsets at integer hours (see below), which makes our results comparable with those from other space geodetic techniques like the Global Navigation Satellite Systems (GNSS) or Satellite Laser Ranging (SLR).
126.2 Concept We did not start from scratch when developing VieVS, but we relied heavily on the Occam 6.1 software package. At first, we made a ‘line by line’ translation from Fortran to MATLAB to guarantee getting identical results. In the second step we deleted all obsolete parts and comments, and we applied MATLAB tools to shorten and re-structure the source code considerably, with results still being identical to those of Occam. Finally, we implemented new models like the non-rotating origin (in parallel we removed the equinox-based transformation) and we made other changes to strictly follow the IERS Conventions (McCarthy and Petit 2004) or IVS Standards like the treatment of thermal deformation of VLBI radio telescopes (Nothnagel 2009). However, the disagreement of the reduced observations was still at the 1.5 mm level compared to Occam 6.1 (see Fig. 126.1). In order to clarify and remove those differences (not only between VieVS and Occam 6.1, but w.r.t. other VLBI software packages) a comparison campaign of computed delays has been initiated that is supported by the IVS Analysis Coordinator. One important change in the least-squares adjustment is that we are using piece-wise linear offsets for all parameters which can be estimated. These piecewise linear offsets are estimated at integer hours (e.g., at 18 UTC, 19 UTC, . . .), at integer fractions of integer hours (e.g., 18:20 UTC, 18:40 UTC, . . .) or at integer multiples of integer hours (e.g. 18:00 UTC, 0:00 UTC,
Fig. 126.1 Difference in mm of computed time delays from Occam 6.1 and VieVS plotted vs. days since 1 January 2000 for artificial observations every 30 min between WETTZELL and WESTFORD to the radio source 0642+449
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6:00 UTC, . . .). This representation is not only possible for troposphere zenith delays and gradients, station clocks, and Earth orientation parameters, but also for coordinates of selected stations and radio sources. Equation (126.1) shows the functional model of the troposphere delay L at one station represented by piece-wise linear offsets x1 and x2 of the zenith delays at the integer hours t1 and t2. m(t) denotes the mapping function at the epoch t of the observation which is in between the integer hours. The partial derivatives which have to be entered in the design matrix are shown in (126.2) and (126.3). This concept is similar for all parameters, and with this kind of parameterization future combinations (at the normal equation level) with other space geodetic techniques will be easier. LðtÞ ¼ mðtÞ x1 þ mðtÞ
t t1 ðx2 x1 Þ (126.1) t2 t1
dL t t1 ¼ mðtÞ mðtÞ t2 t1 dx1
(126.2)
dL t t1 ¼ mðtÞ dx2 t2 t1
(126.3)
Figure 126.2 shows a comparison of hourly UT1 estimates between VieVS and Calc/Solve (Artz et al. 2010) for the continuous VLBI campaign CONT08
which was observed in August 2008 (http://ivscc. gsfc.nasa.gov/program/cont08/). The standard deviation between the UT1 series is 5 ms and the maximum difference is 20 ms. Figure 126.3 shows a flowchart describing the present structure of the Vienna VLBI software VieVS. It is important to note that single sessions can be analyzed by setting the parameterization options with a graphical interface, but that global solutions can also be run in a batch mode without any user interactions. The Vienna VLBI Software (VieVS) comes at the right time, because it is strongly related to many current activities within the VLBI community and at IGG Vienna: (1) IVS Working Group 4 (http://ivscc.gsfc. nasa.gov/about/wg/wg4/) is developing new VLBI data structures. We will contribute to the definition of the format, and we plan to fully incorporate it in VieVS. With the new MATLAB software VieVS we are very flexible to consider any new development. E.g., as the new data structures will be based on NetCDF, built-in MATLAB tools will be available to read and write the data. (2) IGG Vienna has been contributing to the simulation studies for VLBI2010 (Petrachenko et al. 2009). We plan to continue these simulations with VieVS, and we will take advantage of the experience that we gained with our previous Occam simulations (Wresnik et al. 2009a, b). (3) We work on new scheduling software for geodetic VLBI
50 VieVS Calc/Solve
40 30
Δ DUT1 [μs]
20 10 0 -10 -20 -30 -40 -50 12
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18 20 22 Day of August 2008
24
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Fig. 126.2 Comparison of hourly UT1 estimates with Calc/Solve and VieVS during CONT08. For both time series the interpolated IERS 05 C04 values and the high frequency ocean tidal variations according to the IERS Conventions are removed. The standard deviation between both series is 5 ms, and the maximum difference is 20 ms
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Fig. 126.3 Flowchart of the Vienna VLBI Software VieVS as of November 2009
which will be closely related to VieVS. In particular, we plan to use the simulation capabilities of VieVS to determine and validate optimized schedules.
126.3 Outlook The Vienna VLBI Software (VieVS) will be further improved and extended in many aspects, but the main advantage is its simple applicability and that it can be easily modified or extended. Together with the National Institute of Information and Communications Technology (NICT, Japan), phase delay solutions will be implemented in VieVS and initial steps will be taken to equip the software with tools for spacecraft tracking and space VLBI. Furthermore, steps ‘backwards’ in the processing chain will be covered, such as group delay ambiguity resolution in order to close the complete software chain from the correlator output to ambiguity-free group delays in the Vienna VLBI software.
VieVS will also be equipped with the capability to set up global solutions for estimating not only terrestrial and celestial reference frames and Earth orientation parameters, but also geodynamical parameters like Love and Shida numbers amplitudes of tidally induced Earth rotation variations. In addition to the classical Gauss–Markov model, we will implement a Kalman filter solution which can be very beneficial for VLBI2010 when in future the observations will be available in near real-time. The Vienna VLBI Software (VieVS) will be made freely available to registered users to get feedback from as many groups as possible. Acknowledgements We would like to thank the IVS for providing the data (Schl€uter and Behrend 2007). Hana Spicakova is grateful to Mondi Austria Privatstiftung for financial support during her phd study at TU Vienna. Andrea Pany is recipient of a DOC-fForte fellowship of the Austrian Academy of Sciences at the Institute of Geodesy and Geophysics, Vienna University of Technology. Harald Schuh is grateful to the Austrian Science Fund (FWF) for supporting his work within
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project P21049-N10, and Tobias Nilsson to the Deutsche Forschungsgemeinschaft (DFG) (project SCHUH 1103/3-1).
References Artz T, B€ockmann S, Nothnagel A, Steigenberger P (2010) Sub-diurnal variations in the Earth’s rotation from continuous VLBI campaigns. J Geophys Res 115:B05404 McCarthy DD, Petit G (2004) IERS Conventions 2003. IERS Technical Note 32, Verlag des Bundesamtes f€ur Kartographie und Geod€asie Nothnagel A (2009) Conventions on thermal expansion modelling of radio telescopes for geodetic and astrometric VLBI. J Geod 83:787–792 Petrachenko B, Niell A, Behrend D, Corey B, Boehm J, Charlot P, Collioud A, Gipson J, Haas R, Hobiger T, Koyama Y, MacMillan D, Malkin Z, Nilsson T, Pany A, Tuccari G,
1011 Whitney A, Wresnik J (2009) Design aspects of the VLBI2010 system, progress report of the IVS VLBI2010 Committee, NASA/TM-2009-214180 Schl€uter W, Behrend D (2007) The International VLBI Service for Geodesy and Astrometry (IVS): current capabilities and future prospects. J Geod 81:379–387 Titov O, Tesmer V, Boehm J (2004) OCCAM v.6.0 software for VLBI data analysis. In: Vandenberg NR, Baver K (eds) International VLBI Service for Geodesy and Astrometry 2004 general meeting proceedings, NASA/CP-2004212255, pp 267–271 Wresnik J, Boehm J, Schuh H (2009a) VLBI2010 simulation studies. In: Drewes H (ed) Geodetic reference frames – IAG symposium, vol 134, Munich, Germany, 9–14 October 2006, pp 61–65 Wresnik J, Boehm J, Pany A, Schuh H (2009b) Towards a new VLBI system for geodesy and astrometry. Adv Geosci 13:167–180
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Estimating Horizontal Tropospheric Gradients in DORIS Data Processing: Preliminary Results
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P. Willis, Y.E. Bar-Sever, and O. Bock
Abstract
Estimating horizontal tropospheric gradients is a common practice in VLBI and GPS data analyses. We investigate here the possibility to do the same for DORIS. We reprocessed all 2007 DORIS data for all satellites, using exactly the same strategy as the latest ignwd08 solution (Willis et al., Adv Space Res 45(12): 1470–1480, 2010) but adding two new parameters per day to account for any asymmetry of the tropospheric delays. When averaged over the full year the DORIS north gradient estimates show a significant correlation with GPS estimates at 33 co-located sites. The east gradient is loosely determined with DORIS due to the north-south orientation of the satellites passes in 2007. Typical values are below 1 mm and North component shows a latitude dependency, negative values in the Northern hemisphere and positive values in the Southern hemisphere. The stacking of DORIS station weekly coordinates provides a more realistic value for a factor of unit weight when done using gradient estimation. Station coordinates also indicate a small improvement in internal consistency when compared to a 1-year position and velocity solution without estimating gradients. The DORIS-derived tropospheric gradients may still absorb other types of un-modeled errors, but estimation of such parameters should be investigated in more detail before reprocessing the entire DORIS data set in view of the next ITRF realization, following ITRF2008.
127.1 Introduction P. Willis (*) Institut Ge´ographique National, Direction Technique, 2 Avenue Pasteur, 94165 Saint-Mande´, France Institut de Physique du Globe de Paris, PRES Sorbonne Paris, Cite´, UFR STEP, 35 rue He´le`ne Brion, 75013 Paris, France e-mail: [email protected] Y.E. Bar-Sever Jet Propulsion Laboratory, California Institute of Technology, MS 238-600 Oak Grove Drive, Pasadena, CA 91109, USA O. Bock Institut Ge´ographique National, LAREG, 6-8 Avenue Blaise Pascal, 77455 Marne-la-Valle´e, France
DORIS (Doppler Orbitography and Radiopositioning Integrated by Satellite) is one of the four fundamental geodetic techniques participating in the realization of the International Terrestrial Reference System (ITRS) (Willis et al. 2006; Altamimi et al. 2007). Latest DORIS solution performed at IGN (ignwd08, Willis et al. 2010a) uses GMF (Boehm et al. 2006) as tropospheric mapping function. While some major improvements were recently made for this technique in data analysis, and while a complete data reprocessing was
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performed in preparation of ITRF2008 (Willis et al. 2010b; Valette et al. 2010), we investigate here potential improvements to the DORIS processing by testing the estimation of daily horizontal tropospheric gradients. While this technique has been used successfully for years in other techniques (MacMillan 1995 for VLBI, Hulley and Pavlis 2007) for Satellite Laser Ranging and Bar-Sever et al. 1998 for GPS), and recommended in the latest IERS conventions (McCarthy and Petit 2004), to our knowledge, it has not yet been fully investigated for the DORIS technique. We have recomputed a full year of DORIS data, using exactly the same strategy used for the latest ignwd08 solution (Willis et al. 2010a), but also estimating two daily horizontal tropospheric gradients (east and north) for each ground tracking station, to account for a possible asymmetry of the lower atmosphere above each station. First, we studied each time series of tropospheric gradients to detect significant deviations from a zero estimate. We then compared the DORIS estimates with GPS-based estimates at over 30 co-located sites. Finally, we compared time series of the DORIS-derived station coordinates to investigate if the estimation of such gradients improves the internal consistency of our results.
the station coordinates (Williams and Willis 2006). Figure 127.1 shows the DORIS tracking network available in 2007. This network possesses an excellent geographical distribution (Fagard 2006), making it a good candidate for participation in the Global Geodetic Observing System (GGOS) (Rummel et al. 2005). The DORIS data are freely available to all scientists through the data centers of the International DORIS Service (Noll 2010; Willis et al. 2010a). More than half of these 57 DORIS tracking beacons (33) are co-located with GPS permanent receivers.
127.2 Data Description
127.2.2 DORIS Data Processing
127.2.1 DORIS Data
In this study we used exactly the same strategy used in the latest IGN solution (ignwd08) using the GIPSY/OASIS software package (Zumberge et al. 1997). In particular, the major improvements were related to: (1) a better handling of solar radiation pressure (Gobinddass et al. 2009a, b), (2) a more frequent estimation of atmospheric drag coefficient (Gobinddass et al. 2010), allowing improvement during periods of high geomagnetic activity (Willis et al. 2005a), (3) a more recent GGM03S GRACE-derived gravity field (Tapley et al. 2005), (4) use of global mapping function (GMF, Boehm et al. 2006) with an elevation cutoff of 10 degrees. DORIS tropospheric corrections were estimated in a DORIS-multi-satellite solution, estimating parameter at start of new pass, and using time-constraints between successive parameters (Willis et al. 2010a). A priori values were computed only using altitude. Here, we now estimate two additional parameters (sine and cosine) per station and per day, using the
As it was not practical to reprocess the full DORIS data set since 1993.0, we chose to reprocess a full year of data. We selected 2007 as it is representative of the current satellite constellation: SPOT-2, SPOT-4, SPOT-5 and Envisat. We did not select 2008 to avoid dealing with the newest DORIS/Jason-2 data, as there is still some concern about its proper data processing, in particular concerning the adoption of the vector between the center of the phase of the on-board antenna and the center of mass of the satellite and its dependency with tropospheric mapping function (Zelensky et al. 2010). The solar radiation pressure model also needs to be properly tuned using more data (Gobinddass et al. 2009a, b as it is currently the largest source of discrepancy (Cerri et al. 2010). However, as all remaining DORIS satellites are sun-synchronous in 2007, with an almost polar orbit, degradation is expected in the east component of
Fig. 127.1 DORIS tracking network in 2007. Red symbols indicate an active co-location with a GPS receiver in 2007
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standard formulation of horizontal tropospheric gradients (East and North) (Bar-Sever et al. 1998). Direct comparisons between DORIS and GPS tropospheric zenith delays were recently performed, without gradient estimation, using the latest ignwd08 solution (Bock et al. 2010). Some improvement was observed relative to a previous analysis (using the older ignwd04 solution) during the CONT02 campaign (Snajdrova et al. 2006), mainly due to better DORIS data analysis, as discussed above.
127.2.3 GPS Data Processing For comparison purposes, we used GPS tropospheric gradients estimated at JPL for GPS precise orbit determination (POD), also using the same GIPSY/OASIS software package. Note that regular official products of the International GNSS Service (IGS) are different and derived using a Precise Point Positioning (PPP) approach (Byun and Bar-Sever 2009). For GPS, as continuous data are available from several satellites at the same time, the tropospheric gradients were estimated every 5 min using a random-walk approach and a 7.5 elevation cutoff.
127.3 Results and Discussion 127.3.1 DORIS-Derived Horizontal Tropospheric Gradients Figure 127.2 shows a typical example of DORIS daily estimation of tropospheric gradients with their daily formal errors. While the DORIS measurements are not sufficiently numerous to estimate 5-min results, Fig. 127.2 illustrates the internal precision of the daily values (with daily formal errors well below 1 mm). For Arequipa, the north gradient is significantly different from zero, and rather consistent on a dayby-day basis. For all stations, estimated values are usually small (less than 1 mm) and could indicate that no large asymmetry is detected at the DORIS stations on average. As the day-to-day repeatability of the gradient estimates seems reasonable, we then derive for each DORIS station a mean yearly value estimated over all
Fig. 127.2 Daily DORIS-derived horizontal North gradient at Arequipa station (ARFB), Peru
2007 results. Figure 127.3 shows histograms of these values for the North and East tropospheric gradients. It can be seen that for all DORIS stations, the estimated values are small and typically only a small fraction of a mm. However, for a few stations, larger values are observed. As most of these results are not significantly different from zero, we first focus here on the most extreme values (negative and positive).
127.3.2 Comparisons with GPS a Co-located Sites Tables 127.1 and 127.2 focus in the sites at the edges of these histograms, and compare them with GPS results to check if these largest values do really represent some physical phenomena. For some of the DORIS stations, no co-located GPS receivers are available (represented by a dash). However, we also provide these DORIS results as they might be of interest for future studies. In a few cases, the GPS instrument exists but the GPS data were not used at JPL in the POD estimation (noted as N/A ¼ not available). It can be seen that DORIS and GPS results are rather different for these extreme cases. For Everest, even though the DORIS gradient amplitude is much lower than GPS, the direction of the gradient is compatible between GPS (Flouzat et al. 2009). The GPS and DORIS stations are also rather far apart. This large East gradient could be linked to the fact that a high mountain range (Himalaya) stands North of both
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Fig. 127.3 Histogram over all sites of DORIS-derived mean tropospheric gradients (in mm). In North (left) and East (right) Table 127.1 Largest estimated values of DORIS and GPS horizontal tropospheric gradients. Mean values over 2007 of north component Station Everest Monument Peak Crozet Easter Island Arequipa Port Moresby
DORIS station EVEB MOOB
DORIS gradient 1.07 0.95
GPS station GUMBa MONP
GPS gradient 3 0.05
CRPB EASB
0.55 0.65
– ISPA
– 0.13
ARFB MORB
0.81 0.81
AREQ –
0.38 –
a
GPS results from GUMB are not part of this study but were derived from Flouzat et al. (2009)
Table 127.2 Largest estimated values of DORIS and GPS horizontal tropospheric gradients. Mean values over 2007 of east component Station Amsterdam Island Arequipa Amsterdam Island Futuna Papetee Marion Island Santiago Port Moresby Cachoeira
DORIS station AMTB
DORIS gradient 1.31
GPS station –
GPS gradient –
ARFB AMUB
1.00 0.84
AREQ –
0.39 –
FUTB PATB MATB SANB MORB CADB
0.73 0.68 0.65 0.53 0.59 0.73
FTNA THTI MARN SANT – CHPI
N/A 0.39 N/A 1.20 – 0.19
station and could impact the propagation of humidity in the lower atmosphere in these regions (Flouzat et al. 2009). It could also be related to multi-path effects. For Amsterdam Island (in Table 127.2), we note that the gradient estimation from the two successive DORIS beacons, corresponding to equipment upgrade and treated independently, are in rather close agreement showing one of the largest negative values. If we analyze the worst case of Arequipa (in Table 127.2), for which even the sign of the gradient is not the same between GPS and DORIS, we can see from Fig. 127.4 that the DORIS determination of the east component is noisier. In Fig. 127.4, the density of the GPS data (1 point every 5 min) shows a smooth signal, potentially linked with an expected annual signal. Here, the DORIS results are not compatible with GPS, even though the sites are only located 20 m apart (Fagard 2006). As expected, we notice that the agreement of the east gradient is the weaker one due to the specific northsouth orientation of the satellite passes (all sunsynchronous in 2007). Figure 127.5 shows a clear correlation in the north gradients between DORIS and GPS estimations. To estimate these yearly values, we averaged the best 70% of the daily data for each technique, looking at their relative formal errors. In Fig. 127.5, the estimated slope is 0.7 (black line) instead of 1.0, with a correlation of 63%. Figure 127.6 presents the same results plotted as a function of station latitude. A clear slope can be seen in results from both techniques. Positive values
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Fig. 127.4 Horizontal gradients as determined by GPS every 5 min (left) and by DORIS every day (right) for the Arequipa station. East component
are mostly found in the Southern hemisphere, while negative values are mostly found in the Northern hemisphere. Similar results were presented earlier in MacMillan (1995), showing a similar North-South pattern for VLBI with compatible magnitude. Physical explanation was proposed as follow: Warmer equator been observed always South from the Northern hemisphere and always North from the Southern hemisphere.
Fig. 127.5 Comparison of DORIS and GPS estimation of horizontal tropospheric gradients at 33 co-location sites (in mm)
Fig. 127.6 Yearly estimation of horizontal tropospheric gradients at 33 co-location sites (North component in mm). DORIS (red circles) and GPS (blue squares)
127.3.3 Impact on DORIS-Derived Station Coordinates In a second step, we looked at the internal consistency of the DORIS station coordinates. To provide better results, we first combined daily station positions into weekly station positions, using full covariance information. As these stations were estimated using a freenetwork approach (Willis et al. 2010a), results cannot be directly compared since the terrestrial reference frame is different from 1 week to the other. They first need to be expressed in the same terrestrial, estimating weekly 7-parameter Helmert-transform with regard to a reference position/velocity solution. As our latest ign09d02 position/velocity solution (Willis et al. 2010b) was based on the ignwd08 time series, it would not be wise to use this reference to test the consistency of the ignwd08 time series. We then compared our weekly solutions (ignwd08) and the new solution (using tropospheric gradients), using an internal position/velocity based only on their
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Table 127.3 Internal consistency of weekly station coordinates (from combined weekly SINEX solutions) with and without estimating horizontal tropospheric gradients Solution ignwd08 This study
Chi2/DOF 3.11 2.39
North (mm) 9.1 8.8
East (mm) 11.1 11.5
Up (mm) 9.2 9.0
respective 2007 data for each of them, and expressed in ITRF2005 using a standard 14-parameter transformation. Of course, the velocity itself is very weakly determined, but the position at the epoch of observation is correct, as no extrapolation is done. When performing a parallel stacking of the two 2007 weekly time series into a position and velocity file, we note from Table 127.3 that the Chi2/DOF significantly improves, getting closer to unity. This is an excellent indication that estimating tropospheric gradients provides more consistent positions from week-to-week. About 10 years ago, a typical value for this parameter was 6 or 7. Since then, each improvement in the physical models or in the DORIS analysis strategy led to a consistent decrease in this ratio. Table 127.3 also provides the mean values of all 2007 weekly weighted root mean squares (WRMS) of station coordinate components. Some small improvement can be seen in North and Vertical while the East repeatability shows some degradation. This is rather encouraging, because, as we estimate two additional parameters per day, the estimation system (least squares or filter in this case) becomes looser and one could expect a larger variability in the DORIS results. It demonstrates that these parameters were able to absorb some asymmetry in the lower atmosphere (as previously shown in the GPS comparisons). They may also have absorbed any other un-modeled geographically-correlated effect. It could be linked to multi-path, which is supposed to be small in the DORIS-case (Willis et al. 2005b) or at least signal degradation as detected recently for some stations (Yaya and Tourain 2010). It may also come from remaining (geographical gradient) errors in the gravity field. This last hypothesis will soon be tested using some newly available GOCE-derived static gravity field (Rummel et al. 2002). However, we do not propose at this point to change our DORIS analysis strategy. First because more work is needed. Reprocessing of older data when only two DORIS satellites were available could lead to a
different conclusion due to the limited number of available DORIS data by then. Furthermore, the new coordinates are more consistent on a week-by-week basis, but they are also different in a systematic way. We confirmed this point by testing our new weekly station coordinates with the ign09d02 based on the previous time series (ignwd08) and observed significant differences. Consequently, if only one IDS analysis center modifies its strategy alone, its results will be different from all the other ACs. Some technical discussion between all ACs must be undertaken, before such a decision, which impacts the station positions in a systematic way, can be taken. Conclusions
By reprocessing a complete year of DORIS data (2007), we were able to show that the horizontal tropospheric gradients (estimated once per day) have reasonable values (typically around 1 mm or below), even if the number of available DORIS measurements is still very limited compared to GPS. For a few DORIS stations, an average of these gradients over the full year shows a significant difference with zero-hypothesis. Comparisons of all available DORIS gradients with GPS at 33 co-located sites show a significant correlation for the North component, being the best determined component due to the DORIS satellite geometry. Finally, a closer look at the DORIS station coordinate time series shows that the internal agreement is better when estimating horizontal tropospheric gradients (once per day), as demonstrated in a significant improvement in the chi2/DOF factor as well as in slight improvement in station coordinates residuals (at least in North and Vertical). We suggest looking into more details by reprocessing older data (e.g. 1993–1994 data when much fewer DORIS data were available, as only TOPEX/Poseidon and SPOT-2 satellites were available by then). A discussion between all IDS Analysis Centers must also be conducted on this subject in order to avoid systematic errors between solutions if some groups are estimating such gradients and some do not. Acknowledgement Part of this work was supported by the Centre National d’Etudes Spatiales (CNES). Part of this work was carried out at the Jet Propulsion Laboratory, California
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Institute of Navigation, under a contract with the National Aeronautics and Space Administration. This paper is IPGP contribution number 2635.
References Altamimi Z, Collilieux X, Legrand J, Garayt B, Boucher C (2007) ITRF2005, A new release of the International Terrestrial Reference Frame based on time series of station positions and Earth orientation parameters. J Geophys Res 112(B9):B09401 Bar-Sever YE, Kroger PM, Borjesson JA (1998) Estimating horizontal gradients of tropospheric path delay with a single GPS receiver. J Geophys Res 103(B3):5019–5035 Bock O, Willis P, Lacarra M, Bosser P (2010) An inter-comparison of zenith tropospheric delays derived from DORIS and GPS data. Adv Space Res 46(12):1648–1660 Boehm J, Niell A, Tregoning P, Schuh H (2006) Global Mapping Function (GMF): a new empirica mapping function based on numerical weather model data. Geophys Res Lett 33(7):L07304 Byun SH, Bar-Sever YE (2009) A new type of troposphere zenith path delay product of the International GNSS Service. J Geod 83(3–4):367–373 Cerri L, Berthias JP, Bertiger WI, Haines BJ, Lemoine FG, Mercier F, Ries JC, Willis P, Zelensky NP, Ziebart M (2010) Precision orbit determination standards for the Jason series of altimeter missions. Mar Geod 33:379–418 Fagard H (2006) Twenty years of evolution for the DORIS permanent network: from its initial deployment to its renovation. J Geod 80(8–11):429–456 Flouzat M, Bettinelli P, Willis P, Avouac JP, Heriter T, Gautam U (2009) Investigating tropospheric effects and seasonal position variations in GPS and DORIS time series from the Nepal Himalaya. Geophys J Int 178(3):1246–1259 Gobinddass ML, Willis P, de Viron O, Sibthorpe AJ, Zelensky NP, Ries JC, Ferland R, Bar-Sever YE, Diament M (2009a) Systematic biases in DORIS-derived geocenter time series related to solar radiation pressure mis-modelling. J Geod 83 (9):849–858 Gobinddass ML, Willis P, de Viron O, Sibthorpe A, Zelensky NP, Ries JC, Ferland R, Bar-sever YE, Diament M, Lemoine FG (2009b) Improving DORIS geocenter time series using an empirical rescaling of solar radiation pressure. Adv Space Res 44(11):1279–1287 Gobinddass ML, Willis P, Diament M, Menvielle M (2010) Refining DORIS atmospheric drag estimation in preparation of ITRF2008. Adv Space Res 46(12):1566–1577. doi:10.1016/j.asr.2010.04.004 Hulley GC, Pavlis EC (2007) A ray-tracing technique for improving Satellite Laser Ranging atmospheric delay corrections, including the effects of horizontal gradients. J Geophys Res 112:B06417. doi:10.1029/2006JB004834 MacMillan DS (1995) Atmospheric gradients from Very Long Baseline interferometry observations. Geophys Res Lett 22 (9):1041–1044
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McCarthy D, Petit G (eds) (2004) IERS 2003 Conventions. IERS Technical Note 32, Frankfurt, Germany Noll CE (2010) The Crustal Dynamics Data Information System: a resource to support scientific analysis using space geodesy. Adv Space Res 45(12):1421–1440. doi:10.1016/j. asr.2010.01.018 Rummel R, Balmino G, Johannessen J, Visser P, Woodworth P (2002) Dedicated gravity field missions: principles and aims. J Geodyn 33(1–2):3–20 Rummel R, Rothacher M, Beutler G (2005) Integrated Global Geodetic Observing System (IGGOS): science rationale. J Geodyn 40(4–5):357–362 Snajdrova K, Boehm J, Willis P, Haas R, Schuh H (2006) Multitechnique comparisons of tropospheric delays derived during the CONT02 campaign. J Geod 79(10–11):613–623 Tapley B, Ries J, Bettadpur S, Chambers D, Cheng M, Condi F, Gunter B, Kang Z, Nagel P, Pastor R, Pekker T, Poole S, Wang F (2005) GGM02 – an improved Earth gravity field model from GRACE. J Geod 79(8):467–478 Valette JJ, Lemoine FG, Ferrage P, Yaya P, Altamimi Z, Willis P, Soudarin L (2010) IDS contribution to ITRF2008. Adv Space Res 46(12):1614–1632 Williams SDP, Willis P (2006) Error analysis of weekly station coordinates in the DORIS network. J Geod 80(8–11): 429–456 Willis P, Deleflie F, Barlier F, Bar-Sever YE, Romans L (2005a) Effects of thermosphere total density perturbations on LEO orbits during severe geomagnetic conditions (Oct-Nov 2003). Adv Space Res 36(3):522–533 Willis P, Desai SD, Bertiger WI, Haines BJ, Auriol A (2005b) DORIS satellite antenna maps derived from long-term residuals time series. Adv Space Res 36(3):486–497 Willis P, Jayles C, Bar-Sever YE (2006) DORIS, from altimeric missions orbit determination to geodesy. CR Geosci 338 (14–15):968–979 Willis P, Boucher C, Fagard H, Garayt B, Gobinddass ML (2010a) Contributions of the French Institut Ge´ographique National (IGN) to the International DORIS Service. Adv Space Res 45(12):1470–1480. doi:10.1016/j.asr.2009. 09.019 Willis P, Fagard H, Ferrage P, Lemoine FG, Noll CE, Noomen R, Otten M, Ries JC, Soudarin L, Tavernier G, Valette JJ (2010b) The International DORIS Service: toward maturity. Adv Space Res 45(12):1408–1420. doi:10.1016/j. asr.2009.11.018 Yaya P, Tourain C (2010) Impact of DORIS ground antennas environment on their radio signal quality. Adv Space Res 45 (12):1465–1469. doi:10.1016/j.asr.2010.01.031 Zelensky NP, Lemoine FG, Chinn DS, Rowlands DD, Luthcke SB, Beckley D, Pavlis D, Ziebart A, Sibthorpe A, Willis P, Luceri V (2010) DORIS/SLR POD modeling improvements for Jason-1 and Jason-2. Adv Space Res 46 (12):1541–1558 Zumberge JF, Heflin MB, Jefferson DC, Watkins MM, Webb FH (1997) Precise point positioning for the efficient and robust analysis of GPS data from large networks. J Geophys Res 102:5005–5017
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Session 8 The IAG International Services and their Role for Earth Observation Convenors: R. Neilan, R. Forsberg
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The BIPM: International References for Earth Sciences
128
E.F. Arias
Abstract
The Time, Frequency and Gravimetry Section of the International Bureau of Weights and Measures (BIPM) is charged with maintaining the conventional reference time scales on the basis of international coordination, and is responsible jointly with the U.S. Naval Observatory (USNO) for the Conventions Centre of the International Earth Rotation and Reference Systems Service (IERS). Another task of the Section relates to gravimetry; the gravity field is monitored both with gravimeters operated by the BIPM and with instruments from other institutes participating every 4 years in the International Comparisons of Absolute Gravimeters (ICAGs), organized jointly by the BIPM, the Consultative Committee for Mass and Related Quantities (CCM) and the International Association of Geodesy (IAG). The BIPM Time, Frequency and Gravimetry (TFG) Section is a service of the IAG, and works in coordination with other IAG services, including the IERS and the International GNSS Service (IGS).
128.1 Introduction The International Bureau of Weights and Measures (BIPM) is the intergovernmental organization charged with providing the basis for a single, coherent system of measurements throughout the world, traceable to the International System of Units (SI) (BIPM 2006). The BIPM was established by the Metre Convention, is financed by its Member States and operates under
E.F. Arias is an Astronomer associated to the Observatoire de Paris, France. E.F. Arias (*) International Bureau of Weights and Measures (BIPM), Pavillon de Breteuil, 92312 Sevres Cedex, France e-mail: [email protected]
the supervision of the International Committee for Weights and Measures (CIPM). A number of Consultative Committees have been established by the CIPM to provide guidance to the BIPM in the different fields of metrology. Four of the CIPM Consultative Committees are related to the work of the IAG: the Consultative Committee for Time and Frequency (CCTF), dealing with the reference time scales; the Consultative Committee for Mass and Related Quantities (CCM), with its Working Group on Gravimetry (CCM WGG); the Consultative Committee for Length (CCL), with competence in the definition and practical realization of the metre; and the Consultative Committee for Units (CCU), which makes recommendations concerning the definition and realization of the units.
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The scientific work at the BIPM is divided between five sections. Activities in the Time, Frequency and Gravimetry Section a have strong impact on the Earth sciences, and consequently on the activities of the IAG. The section’s activities include the maintenance of the international time scales, responsibility for the Conventions Centre of the International Earth Rotation and Reference Systems Service (IERS), shared with the U.S. Naval Observatory (USNO), the realization of the metre, measurement of the gravity field and comparisons of gravimeters, and justifies the designation of the BIPM TFG Section as a scientific service of the IAG. The BIPM has been mandated by its Member States to maintain and disseminate the international atomic time scales, International Atomic Time (TAI) and Coordinated Universal Time (UTC), which are computed and published monthly. TT(BIPM) is another realization of terrestrial time, which is more accurate and more stable than TAI in the long term; it is computed yearly. The BIPM maintains a network of sites for gravimetry measurements with the aim of characterizing the local gravity field for internal applications such as the watt balance, and at the same time contributing to the maintenance of the world’s gravity system. In support of the IAG, the BIPM has been responsible for the International Comparisons of Absolute Gravimeters (ICAGs) since 1981. The most recent comparison of absolute gravimeters, ICAG-2009, was designated as a key comparison within the structure of the CIPM Mutual Recognition Arrangement (CIPM MRA) (CIPM 1999). The BIPM also contributes to the Global Geodetic Observing System (GGOS), which relies on long-term maintenance of the Earth systems.
128.2 Time Scales International Atomic Time (TAI) is a realization of terrestrial time maintained since 1988 at the BIPM. TAI relies today on data from about 350 industrial atomic standards and a dozen primary frequency standards operated in almost 70 time laboratories distributed world-wide. It is calculated from 30-day batches of data and is published monthly with a latency of about 10 days following the last date of
E.F. Arias
data in the BIPM Circular T (BIPM monthly). The calculation algorithm has been designed to yield a time scale more stable and accurate than any of the individual participating standards. The frequency stability of TAI at 30 days is better than four parts in 1016, and its frequency accuracy is a few parts in 1016. In spite of its quality, long-term frequency instabilities of one to two parts in 1015 make TAI inadequate for some scientific applications requiring higher long-term stability. Another realization of terrestrial time, named TT(BIPM), is computed yearly based on all available measurements of primary frequency standards; the long-term stability of TT (BIPM) is better than that of TAI by a factor of between 2 and 3 (Petit 2003), and its accuracy is five parts in 1016. Coordinated Universal Time (UTC) is derived from TAI by the application of leap seconds. The dates of application of leap seconds are decided and announced by the IERS in the IERS Bulletin C (IERS bi-annual). About 70 laboratories contribute data for the calculation of TAI and UTC. These laboratories realize local approximations to UTC indicated by UTC(k). Each month in the BIPM Circular T the BIPM publishes the differences [UTC-UTC(k)] at 5-day intervals. Annual results are presented in the BIPM Annual Report on Time Activities (BIPM annual). The algorithm for the calculation of TAI is based on clock differences (Guinot and Thomas 1988; Audoin and Guinot 2001). Since the clocks are located in remote sites, we make use of techniques of time transfer with a statistical approach that minimizes the noise of the transfer allowing the determination of the clock differences. TAI relies at present on 68 participating time laboratories equipped with Global Navigation Satellite System (GNSS) receivers and/or operating two-way satellite time and frequency time transfer (TWSTFT) stations. The two techniques are independent and rather different. While GNSS time transfer relies today on a one-way time comparison between a clock in a laboratory and another on board a satellite of the GPS constellation, TWSTFT is a two-way comparison of clocks in two stations where the signals are conveyed by a telecommunications satellite. The GPS all-in-view method used today (Matsakis et al. 2006; Petit and Jiang 2008a, 2008b), has
The BIPM: International References for Earth Sciences
succeeded the classical common-views of GPS satellites (Allan and Weiss 1980) in the establishment of time links for the calculation of TAI at the BIPM. Its application has been possible by taking advantage of the increasing quality of the International GNSS Service (IGS) products (clocks and time scale of the IGS). Clock comparisons are possible with C/A code measurements from GPS single-frequency receivers; with dual-frequency, multi-channel GPS geodetictype receivers (P3); and two-way satellite time and frequency transfer through geostationary telecommunications satellites (TWSTFT). Following a successful pilot experiment started in April 2008 at the BIPM on the combined use of phase and code data from GPS signals, some links calculated with this method have been officially included in the monthly calculation of TAI since September 2009 (Petit and Jiang 2008a, 2008b). Time laboratories contribute GPS phase and code data and the BIPM uses the Precise Point Positioning technique (PPP) to generate monthly solutions. Time transfer constitutes the main component of the uncertainty of the local realizations of UTC indicated by UTC(k), the other component being the uncertainly of the participating clocks (Lewandowski et al. 2006, 2008). In metrology, we consider two components in any uncertainty budget: the statistical uncertainty referred to as the Type A uncertainty, and the systematic uncertainty referred to as Type B uncertainty (JCGM 2008). In the case of time transfer, the systematic uncertainty (Type B) is caused by the delay of the equipment, and cannot be reduced with statistical processes. The Type A uncertainty represents the noise of a time link, and can be minimized through appropriately adapted mathematical strategies. Figure 128.1 shows the Type A and B uncertainties (uA and uB respectively) for different techniques and methods of time transfer used in TAI. The Type A uncertainty is at the level of 1 ns or better for TWSTFT and for the links by dual-frequency receivers, free of the additional ionospheric delays (P3, PPP). Single-frequency observations, which are affected by the ionospheric delays, have Type A uncertainties of a few nanoseconds. The Type B uncertainty of TWSTFT is about 1 ns, while GPS equipment, still limited by some
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Fig. 128.1 Uncertainty of time transfer techniques and methods. SC (single-channel GPS receivers), MC (multichannel GPS receivers), CV (common-view), AV (all-inview), P3 (dual frequency GPS observations), PPP (GPS PPP), TW (TWSTFT)
hardware problems, limits the Type B uncertainty to 5 ns in the best cases.
128.3 The IERS Conventions Centre Since 2001 the IERS Conventions Centre is provided jointly by the BIPM and the USNO. The Conventions Centre provides models and procedures for astronomy and geodesy, with printed versions of the Conventions produced at intervals of about 5 years, when major changes are introduced. Updated versions are provided in electronic form, after approval of the IERS Directing Board. In the meantime, work on interim versions is also available through the internet at http://tai.bipm.org/iers. The IERS Conventions 2003 is the last printed version available (IERS 2004); a new one is expected to be published before the end of 2009.
128.4 Gravimetry at the BIPM The Working Group on the Intercomparison of Absolute Gravimeters of the International Gravity Commission (IGC) started in 1981 with the first International Comparison of Absolute Gravimeters (ICAG) at the headquarters of the BIPM in Se`vres. Since then,
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successive ICAGs have been organized at 4-year intervals by the BIPM, the CCM and the IAG Commission 2, “Gravity Field”. Modern gravimeters measure the acceleration of a free falling test mass using the SI unit of length (the metre), as realized by a wavelength of laser light, and SI the unit of time (the second) as defined and realized by an atomic transition. These gravimeters allow a very accurate determination (to a few mGal) of the absolute value of gravity, necessary for a number of applications related to the Earth modelling. This high accuracy is also required at the BIPM in the scope of the watt balance project, which will support the new definition of the kilogram in the next years. While a time series of data from a single instrument provides information on the stability of the instrument, simultaneous measurements at the same location by a number of different absolute gravimeters allow the characterization of the scatter in the value of gravity measured by the different instruments. The objective of the ICAGs is to provide estimates of the accuracy of absolute gravimeters. The BIPM provides the expertise in metrology which is necessary to accomplish this task. In addition, at each ICAG the lasers and frequency measurement of the rubidium clocks in the gravimeters can be verified in the laboratories of the TFG Section. In 1999 the CIPM MRA was created as a means of giving mutual recognition of national standards. The CIPM MRA is supported by key comparisons whose results are published in the BIPM key comparison data base (KCDB, http://kcdb.bipm.org). The ICAG-2009 was organized in accordance with the proposal made at the 3rd Joint Meeting of the CCM WGG and the SGCAG 2.1.1 of the IAG on 24 August 2007. ICAG2009 consists of a single comparison that includes a key comparison and a pilot study. The status of key comparison for the ICAG was approved by the CCM Working Group on Key Comparisons in November 2008. Only National Metrology Institutes that are signatories of the CIPM MRA and other officially designated institutes can participate in a key comparison; their measurements can contribute to the evaluation of the key comparison reference value (KCRV) and their degrees of equivalence can be published in the BIPM KCDB. Only the results of the absolute measurements will be used in the key comparison part of ICAG-2009 to evaluate the KCRV. This key
E.F. Arias
Site A
Site B
Site WB
Fig. 128.2 Location of the gravimetry sites A, B and WB at the BIPM, in Se`vres (France)
comparison has been designated CCM.G-K1 and the BIPM has been nominated the pilot laboratory. In parallel to ICAG-2009, a comparison of relative gravimeters has been coordinated by the BIPM with a small number of high quality instruments to support the absolute campaign (Jiang et al. 2011). Twenty-four absolute gravimeters participated in ICAG-2009, half of which fulfilled the conditions for participating in the key comparison, and the results of the other gravimeters contributing to the pilot study. The complete schedule of absolute measurements included traditional sites A and B at the BIPM, and for the first time, two sites in the room where the BIPM watt balance will be operated (site WB). The Fig. 128.2 shows the distribution of the gravimetry sites in ICAG-2009 at the BIPM headquarters. By November 2009, all measurements have been completed, and the BIPM team in coordination with the participants has started analysing the results in preparation for publication.
128.5 Summary The BIPM is the intergovernmental organization responsible for the world-wide coordination of physical and chemical measurements. Through its TFG Section, the BIPM is at the service of the IAG, providing the time references TAI, UTC and TT (BIPM) in coordination with other IAG services such as the IERS and the IGS. Since 2001 the BIPM and the USNO have jointly provided the IERS Convention Centre, the publication of the IERS Conventions and
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The BIPM: International References for Earth Sciences
regular updates of the Conventions website (http://tai. bipm.org/iers). The BIPM shares responsibility for the International Comparisons of Absolute Gravimeters (ICAGs) with the IAG Commission 2, “Gravity Field”. In November 2008 the CCM Working Group on Key Comparisons has approved ICAG-2009 as a key comparison in gravimetry. Absolute gravimeters participating in the key comparison and the pilot study making up ICAG-2009 realized measurements at the BIPM in mid 2009. A comparison of relative measurements which involved only ten selected relative gravimeters was run in parallel in support of the absolute campaign. The results will be available in the first semester of 2010.
References Allan DW, Weiss AM (1980) Proceedings of the 34th annual symposium on frequency control, pp 334–346 Audoin C, Guinot B (2001) The measurement of time. Cambridge University Press, Cambridge Bureau International des Poids et Mesures, BIPM circular T, monthly, 7p Bureau International des Poids et Mesures, BIPM annual report on time activities, Annual publication Bureau International des Poids et Mesures (2006) SI brochure, Le Syste`me international d’unite´s, 8th edn. Available at http://www.bipm.org/en/si/si_brochure Bureau International des Poids et Mesures (2009) Technical protocol of the 8th international comparison of absolute gravimeters ICAG-2009. Available at the BIPM TFG Section
1027 Comite´ International des Poids et Mesures (1999) Mutual recognition of national measurement standards and of calibration certificates issued by national metrology institutes, Bureau International des Poids et Mesures. Available at http://www.bipm.org/utils/en/pdf/mra_2003.pdf Guinot B, Thomas C (1988) Establishment of international atomic time, Annual report of the BIPM time section, 1, D1–D22 JCGM (2008) Evaluation of measurement data: Guide to the expression of uncertainty in measurement, JCGM 100:2008, available at http://www.bipm.org/utils/common/documents/ jcgm/JCGM_100_2008_E.pdf International Earth Rotation and Reference Systems Service. IERS Bulletin C, mailed every six months, available at http://www.iers.org/MainDisp.csl?pid¼44-14 International Earth Rotation and Reference Systems Service (2004) IERS conventions (2003), IERS technical note No. 32, McCarthy and Petit (eds), BKG Jiang Z, Arias EF, Tisserand L, Kessler-Schulz KU, Schulz HR, Palinkas W, Rothleitner C, Francis O, Becker M (2011) Updating the precise gravity network at the BIPM, this publication. pp 263–271 Lewandowski W, Matsakis D, Panfilo G, Tavella P (2006) The evaluation of uncertainties in [UTC–UTC(k)]. Metrologia 43 (3):278–286 Lewandowski W, Matsakis D, Panfilo G, Tavella P (2008) Analysis of correlations, and link and equipment noise in the uncertainties of [UTC–UTC(k)], UFCC, pp 750–760 Matsakis D, Arias EF, Bauch A, Davis J, Gotoh T, Hosokawa M, Piester D (2006) On optimizing the configuration of timetransfer links used to generate TAI, Proceedings of the 20th EFTF, pp 448–454 Petit G (2003) Proceedings of 35th PTTI, pp 307–316 Petit G, Jiang Z (2008a) GPS all in view time transfer for TAI computation. Metrologia 45(1):35–45 Petit G, Jiang Z (2008b) Precise point positioning for TAI computation, IJNO Article ID 562878
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Development of the GLONASS Ultra-Rapid Orbit Determination at Geodetic Observatory Pecny´
129
J. Dousa
Abstract
Observations from up to 51 GPS+GLONASS satellites are available as of September 2009. Mainly (near) real-time GNSS analyses, particularly navigation, warning systems or atmosphere monitoring, will benefit from the data from all these satellites. The International GNSS Service (IGS) has been providing precise GPS ultra-rapid orbits since 2000, but up to these days, due to a lack of contributing analysis centers, it does not provide GLONASS ultra-rapid orbit product. The Geodetic Observatory Pecny´ has been contributing to the IGS ultra-rapid orbits since 2004. In 2008/2009 an extension of the orbit determination procedure was prepared for the GLONASS system. Although the GLONASS global data coverage is far from optimal, we focused on a robust and satisfactory routine product already usable in (near) real-time GNSS analysis. We have tested the system for different schemes of processing – (1) common GNSS solution and (2) stand-alone GLONASS or GPS solutions. Resulting orbits and ERPs were evaluated with respect to the IGS final products. The use of the GLONASS ultrarapid orbits was demonstrated in near real-time water vapor monitoring using the European network of 38 GNSS stations.
129.1 Introduction The active constellation of Russian GLONASS (GLObalnaya NAvigatsionnaya Sputnikovaya Sistema) has reached 20 active satellites (Sept 2009). A fully operational system will consist of 30 satellites in 2011 as announced by A. Permikov, the head of the Russian Space Agency, see RIA Novosti (2008).
J. Dousa (*) Geodetic Observatory Pecny´, Research Institute of Geodesy, Topography and Cartography, CZ 251 65 Ondrˇejov, Czech Republic e-mail: [email protected]
Today, observations from up to 51 GNSS satellites are available for various applications – the main interest is in the field of real-time or near real-time analysis, which usually depend on predicted precise orbits. The benefit of GLONASS inclusion is particularly expected for navigation, warning systems or atmosphere monitoring. Since 2005, the International GNSS Service (IGS), see Dow et al. (2005), has been providing precise GLONASS final orbits, but nevertheless, due to a lack of contributing analysis centers, the IGS does not provide GLONASS ultra-rapid or even rapid orbit products so far. The Geodetic Observatory Pecny´ (GOP) has been contributing to the IGS ultra-rapid orbits as of 2004. Motivated by atmosphere monitoring, we initiated the
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extention of our orbit determination procedure towards GLONASS. Because GLONASS is still an incomplete system, our current effort has been to provide a satisfactory and robust result even though its global hourly data coverage is far from optimal. We briefly summarize the current orbit determination procedure at GOP in Chap. 2, we explain specific steps towards the successful extension to GLONASS in Chap. 3, we evaluate and discuss the results of orbits and the Earth rotation parameters (ERP) generated in different testing variants in Chap. 4, and finally, we demonstrate the use of the GLONASS orbits in near real-time GNSS atmosphere monitoring in Chap. 5.
129.2 Current GOP Near Real-Time GPS Orbit Determination Procedure During 2000, we developed a precise orbit determination procedure in near real-time mode at GOP, see Dousa and Mervart (2001). Later in 2002, we improved the procedure and started a routine processing of a global network providing precise ultra-rapid orbits (Dousa 2004). Since 2004, this product has been officially contributing to the International GNSS Service. Besides many scripts and programs of the system, the Bernese GPS Software V5.0 (Dach et al. 2007) is a core analysing tool applied for batch processing of double-difference observations. Due to limited resources, a highly efficient procedure, which uses only the last 6-h data batch for pre-processing, has been developed at GOP. The pre-processing is done in five individual clusters in parallel mode and from each of them the full information matrix (normal equations) is saved as the result. In the following step, these normal equations are combined in sequential least square adjustment to a unique global 6-h solution. At the end, this is followed by the combination of consecutive 6-h normal equations into a single 3-day global solution. The orbit parameters (as well as ERPs and coordinates) are stacked for the whole 3-day interval to provide the best 24 h orbit prediction. The procedure consists of two iterations of orbit improvements and it is initialized with merged predicted precise IGS orbits and broadcast orbits from global navigation files. Through analysing navigation messages, we are able to detect satellite manoeuvres and this information is used during the pre-processing as well as during the final orbit
J. Dousa
parameter stacking procedure. We do not need other a priori information for our system than the navigation messages and the ERPs from the IERS Rapid Service (IERS 2006). Another advanced feature of the procedure is that no satellite is excluded from the processing (it can be excluded during final product generation), but in specific cases the observations from problematic satellites are downweighted. Finally, special attention is given to the assessment of individual orbit quality by setting the appropriate accuracy code in output (SP3) file.
129.3 Extention Towards GNSS More variants were considered for developing the extension towards not yet complete GLONASS. The robustness was the priority under the current conditions. We intended to test whether we are able to routinely generate even a stand-alone GLONASS orbit product. We expected that with combined GNSS solution the better results for GLONASS orbits will be achieved, but, on the other hand, we needed to know if resulting GPS orbits benefit from GNSS data or if they degrade because of possible problems with data from the incomplete GLONASS system. In Sept 2009, the GLONASS constellation consists of 20 (+1) satellites (R09 is excluded due to incomplete data). Although the availability of GLONASS data (and it concerns especially hourly data files, which are used in our processing system) from the world-wide IGS stations has increased during last years, the global coverage is still not optimal. The situation is even worse because some receivers in the IGS global network do not record all available satellites. The use of additional global non-IGS stations providing real-time data streams was thus identified as necessary for the orbit determination as well as a careful selection of the stations for maximizing all GNSS observations. For this reason, we have implemented a support of real-time NTRIP GNSS data streams (Weber et al. 2006) of non-IGS stations in GOP data center. The number of observations is still significantly lower for GLONASS than GPS, especially when processing 6-h data batch only, see Fig. 129.1. Due to a different GLONASS orbit revolution time (11 h 15 min), the number of observations in batch fluctuates more
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Development of the GLONASS Ultra-Rapid Orbit Determination
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data from a single satellite, such observations are downweighted and they are not completely excluded from the processing. This is important also for further combination steps and orbit prediction generation. No satellite is thus missing in the core processing, but any satellite can be excluded (if requested) at the final step of official product generation. This isn’t usually done, but a high accuracy code is preferably set for satellites in a manoeuvre, for example.
129.4 GNSS Orbit and ERP Evaluation
Fig. 129.1 Number of observations for satellites GPS 22 (top) and GLONASS 22 (bottom) plotted for four 6 h batches during a day
frequently in GLONASS than in GPS (Fig. 129.1) and in some cases reaches even zero. We have prepared a robust procedure to concatenate global hourly GLONASS navigation files for initializing our orbit determination procedure. A checking procedure has been improved for GLONASS files to eliminate incorrectly collected messages. Individually prepared GPS and GLONASS orbit files are finally merged together. The original GOP processing scheme has been enhanced with a simple option for switching between GPS, GLONASS or GNSS analysis. Accordingly, all the individual processing steps were enhanced to use combined data as well as to apply specific options for GPS or GLONASS. For example, due to missing a general integer ambiguity resolution strategy for GLONASS in Bernese 5.0, the relevant steps are not performed for the GLONASS data. The global ionosphere model, estimated in near real-time for the ambiguity resolution is estimated from GPS code data only and it is not thus provided in a stand-alone GLONASS solution. Finally, the procedure of the long-arc orbit combination together with the assessment of accuracy codes for estimated orbits has been revised. Thanks to the capability of the system to identify problematic
The combined GNSS and stand-alone GPS, GLONASS orbit and ERP products were evaluated during two 60-day periods: Nov/Dec 2008 and May/ Jun 2009. When considering the GNSS solution as an official product, the most important result is that there is no negative impact of GLONASS data on GPS orbit determination, see Fig. 129.2. We can thus maintain a single rigorous GNSS solution and simply provide a fully consistent GNSS orbit and ERP product. Figure 129.3 demonstrates that the new solution resulted in slightly better statistics (especially due to the higher consistency in quality) compared to the operational GPS orbit product from GOP. This can be partly attributed to the simulation mode of the near real-time processing (particularly to more complete data) and to the specific development towards a robust stand-alone GLONASS solution. Figure 129.4 shows that the stand-alone GLONASS solution results in approx. 20% worse orbits compared to the GNSS solution. Obviously, it is due to the fact
Fig. 129.2 Position RMS for GPS satellites from stand-alone GPS and GNSS solution for 12 h-predicted and 24 h-fitted interval
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Fig. 129.3 Position RMS for GPS satellites from operational and new solution for 12 h-predicted and 24 h-fitted interval
Fig. 129.4 Position RMS for GLONASS satellites from stand-alone GLONASS and GNSS solutions – 12 h-predicted, 24 h-fitted
Fig. 129.6 Comparisons of the Earth pole motion, the rate and the length of day (LOD) for different solutions. Fitted and predicted values comparison is in the left and the right part of the figure, respectively
Fig. 129.5 Position RMS for GLONASS satellites from GNSS solutions during two periods: Nov–Dec 2008 and May–Jun 2009
that GLONASS solution still benefits from the common parameters estimated predominantly with GPS data (station coordinates, troposphere path delays, Earth rotation parameters). Figure 129.5 shows the comparison of two testing periods: Nov/Dec 2008 and May/Jun 2009. The latter period provides slightly better orbit results, where additional satellites and new available GNSS stations in global network were processed. As shown in Fig. 129.6, the GLONASS contribution is marginal especially in case of estimated Earth
rotation parameters (lower accuracy for the standalone GLONASS solution). Besides the data flow and coverage, our solution can be further improved especially by ambiguity fixing (needs to apply inter-frequency biases) and by improving the estimation of the solar radiation pressure parameters.
129.5 The Use of GLONASS Predicted Orbits in Near Real-Time Troposphere Monitoring In 1999, a near real-time GPS data processing (Dousa 2001) has been developed at GOP for the troposphere parameter estimation within the GPS-meteorology concept see Bevis et al. (1992). The precise IGS
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Fig. 129.7 Time-series for zenith total delay parameters estimated at stations ONSA (Onsala) using GPS, GLONASS or both data
orbit prediction was available only for 24–48 h, which was a motivation to develop a near real-time orbit determination procedure in 2000. Even today, the atmosphere monitoring applications play an important role in extending towards near real-time GLONASS application. One reason is the reduction of the high correlation between estimated position height and zenith troposphere parameters (ZTD), another reason is the fact that more information can be achieved from estimating the tropospheric gradients or slant tropospheric path delays. In 2009, we have extended our near real-time procedure of ZTD estimation routinely contributing to the COST-716 (Elegered 2001), TOUGH (Vedel et al. 2003) and to the E-GVAP (egvap.dmi.dk) projects towards the use of available GLONASS data. The stand-alone GPS or GLONASS solutions were enabled for testing purpose too. The GLONASS implementation itself consisted of some specific steps mainly for improving the robustness of the solution when an incomplete system is included. The testing was performed using a network of 38 European stations providing both, GPS and GLONASS, hourly data. Such network was processed during 30 days for GNSS as well as for the stand-alone GPS and GLONASS solutions. Common configuration and strategy were applied for all these solutions. To enable a simple ZTD comparison between solutions, the coordinates were kept fixed on values, which were estimated using GPS data only and using fixed integer ambiguities. Assuring the same strategy for GPS as for GLONASS, the float ambiguities were estimated together with ZTDs during the final adjustment. Dousa (2002) also found a small offset in ZTD (approx. 1 mm) between ambiguity-fixed and ambiguity-free solution even in the post-processing solution.
The ZTDs from each processing solution were compared to the post-processing results. A comparison of ZTDs proved the overall very good consistency of the GLONASS, GPS and GNSS near real-time solutions. The standard deviations of ZTDs of the last processing hour are in the range of 3–6 mm for all 38 stations of the processed network, see Figs. 129.7 and 129.8. We didn’t find a significant improvement in ZTD quality comparing GNSS to stand-alone GPS solution. On the other hand, surprisingly, the stand-alone GLONASS solution achieved already very good results, although the system is incomplete and the predicted orbits are of lower quality than GPS, see Fig. 129.7. Nevertheless, we have identified a systematic error in estimated ZTD parameters between GPS and GLONASS. For all stations in the network, the GLONASS ZTDs resulted in approx. 1–3 mm lower values than ZTDs estimated by GPS, Fig. 129.8. All GNSS ZTDs then provide approx. 1 mm lower values than GPS ZTDs, which roughly correspond to the smaller GLONASS impact on GNSS product due to a smaller data volume. In post-processing results, already Bruyninx (2006) found a 0–2 mm (in most cases 0.5–1 mm) underestimation of ZTDs for all the stations provided by GLONASS+GPS compared to GPS. This stands in agreement with our results clearly approved by the effect in the stand-alone GLONASS ZTD solution. The impact of GLONASS on ZTD estimation will increase with completing the system and with increased number of observations, as well as when it is supported by the precise ultra-rapid orbit products from IGS. The higher benefit is expected mainly in gradient estimation or slant tropospheric path delay estimation.
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The most important result of the development is that the estimation of GPS orbits and Earth rotation parameters was not degraded by using GNSS solution in our generalized procedure. This allows us to consider an inclusion of GLONASS data directly into the existing GOP system, which routinely contributes to the IGS ultra-rapid orbit combination. The limitation of the current GLONASS product still lies in the poor global hourly/real-time observation coverage, in the estimation of solar radiation pressure parameters and in the lack of integer ambiguity resolution strategy. Acknowledgements The author thanks the International GNSS Service as well as all contributing organizations for providing global GNSS data. This work was supported by the Czech Science Foundation (project no. 205/08/0969) and by the Ministry of Education, Youth and Sport of the Czech Republic (CEDR, project no. LC506).
References
Fig. 129.8 Inter-comparisons of ZTDs for all stations (x-axis) estimated using data from GPS, GLONASS and GNSS (GPS+GLONASS) data
Conclusion
The GOP ultra-rapid global solution for GPS satellite orbit determination has been extended to the GLONASS system. The procedure is ready for the stand-alone GPS or GLONASS as well as for GNSS combined solution. When incorporating the incomplete GLONASS system, the overall robustness of the procedure has been further improved. Given the current results, the GLONASS orbit determination still benefits from the combined solution due to coordinate, troposphere and ERP estimates using mainly GPS data. The resulted GLONASS orbits were approx. 20% less accurate in case of the stand-alone solution. The first GOP routine GLONASS orbit product achieved overall accuracy of 10 cm and 15–30 cm for fitted and 12 hpredicted satellite position, respectively, with an exception of R06 and R10, which provide worse predictions.
Bevis M, Businger S, Herrin TA, Rocken C, Anthes RA, Ware RH (1992) GPS-meteorology: remote sensing of atmospheric water vapor using the global positioning system. J Geophys Res 97(15):15787–15801 Bruyninx C (2006) Comparing GPS-only with GPS+GLONASS positioning in a regional permanent GNSS network. GPS Solut 11(2):97–106. doi:10.1007/s10291-006-0041-9 Dach R, Hugentobler U, Fridez P, Meindl M (2007) Bernese GPS software version 5.0, Astronomical Institute, University of Berne Dousa J, Mervart L (2001) On hourly orbit determination. Phys Chem Earth (A) 26(6–8):555–560 Dousa J (2001) The impact of ultra-rapid orbits on precipitable water vapor estimation using ground GPS network. Phys Chem Earth (A) 26(6–8):393–398 Dousa J (2002) On the specific aspects of precise tropospheric path delay estimation in GPS analysis. In: Adam J, Schwarz KP (eds) Vistas for geodesy in the new millenium, IAG symposium, vol 125. Springer, Berlin, pp 285–290 Dousa J (2004) Precise orbits for ground-based GPS meteorology: processing strategy and quality assessment of the orbits determined at geodetic observatory pecny´. J Meteorol Soc Jpn 82(1B):371–380 Dow JM, Neilan RE, Gendt G (2005) The international GPS service (IGS): celebrating the 10th anniversary and looking to the next decade. Adv Space Res 36(3):320–326. doi:10.1016/j.asr.2005.05.125 Elegered G (2001) An overview of COST Action 716: exploitation of ground-based GPS for climate and numerical weather prediction applications. Phys Chem Earth (A) 26 (6–8):399–404 RIA Novosti (2008) http://en.rian.ru/russia/20080321/ 101957980.html
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Vedel H et al. (2003) Targeting optimal use of GPS humidity measurements in meteorology. In: Proceedings of the international workshop on GPS meteorology, January 14–17, 2003, Tsukuba, Japan Weber G, Dettmering D, Gebhard H (2006) Networked transport of RTCM via internet protocol (NTRIP). In: A window on the future of geodesy, international association of
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geodesy symposia, vol. 128. Springer, Berlin, pp 60–64, doi: 10.1007/b139065 IERS (2006) IERS annual report (2006) In: Wolfgang R. Dick, Bernd Richter (eds) International earth rotation and reference systems service, Central Bureau. Verlag des Bundesamts f€ur Kartographie und Geod€asie, Frankfurt am Main, p 187 (online at www.iers.org)
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AGrav: An International Database for Absolute Gravity Measurements
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H. Wziontek, H. Wilmes, and S. Bonvalot
Abstract
The steadily growing number of absolute gravimeters and absolute gravity measurements all over the world emphasizes the demand of an overview about existing locations, observations, instruments and institutions involved. As a contribution to the International Gravity Field Service (IGFS), a relational database was designed and implemented in a joint development of BKG and BGI and is in operational status now. Two objectives are aimed at: With freely available metadata and contact details, the database should give an overview about existing stations and observations, serve as a platform for multidisciplinary cooperation and allow the coordination of forthcoming measurements. Among contributing groups or within international projects, an exchange of gravity values and processing details is possible. The database will function as a data inventory, assuring long term availability of the data. Prospectively, the database will be the foundation for a future international gravity reference system and will serve as a pool for geophysical interpretation of absolute gravity observations on a global scale.
130.1 Information Levels To allow sharing the data while respecting the data property of the participating groups, three different levels of providing information content to the database are distinguished: 1. The lowest level is to inform simply about the AG station locations. This involves only the station
H. Wziontek (*) H. Wilmes Federal Agency for Cartography and Geodesy (BKG), RichardStrauss-Allee 11, 60598 Frankfurt/Main, Germany e-mail: [email protected] S. Bonvalot Bureau Gravime´trique International, Observatoire MidiPyre´ne´es, 14 av. Edouard Belin, 31450 Toulouse, France
coordinates. Information about the station owner can be added, but no contact details can be provided unless an observation is stored in the database. This is for groups who established absolute gravity sites and want to enable cooperation by providing access to other teams. 2. The next level of information is to include observation epoch, instrument details and owner – but still without the observation results. Contact to the data owner is possible for everyone. This is important for groups who need to exploit their measurements before sharing the data with others. 3. At the highest level the gravity results are included. The data owner still controls the data, since it is possible to edit the submitted data. To preserve data property, the database provides the gravity values
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_130, # Springer-Verlag Berlin Heidelberg 2012
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with reduced precision via its public access. The complete value is shared only among the other contributors of gravity data. Although level 3 is favoured as standard, level 2 is proposed as a compromise in case of difficult data property. However, if it is impossible to provide observation information at all, level 1 is still considered as valuable information for the gravity community. The data retrieval distinguishes two different views to the data accordingly and foresees the selective handling of level 3 in particular. Meta data or Who measured Where and When: This comprises information about locations, instruments, institutions and periods of station occupation, which are freely available. Gravity values (if present) are provided with a reduced precision. No processing details are available. Links between different data views, allowing a fast navigation, lead to contact details for further information. The intention is to leave the decision of a public release of measurement results in the responsibility of the data providing institution. Complete datasets: All existing information is available, but the access is restricted to contributing groups and enables to share processing results. The user interface allows editing of the data (usually restricted to the own data) and access to supplemental information such as station descriptions.
130.2 Data Exchange Format Prior to a database setup, an exchange format must be defined which requires a compromise between completeness of the dataset, traceability of the processing scheme and additional efforts for the user. A proposal for an absolute gravity format was already made by Barriot et al. (2004) and comprises main aspects. Within the development of the European Combined Geodetic Network (ECGN) (Ihde et al. 2005) standards for absolute gravimeter data have been suggested which are based upon the so-called absolute gravity “project files”. These files are routinely generated by the absolute gravity processing software “g” of micro-g-LaCoste, which is a standard processing software for FG5 and A10 gravimeters and therefore is available for most potential contributors without additional effort. The simple
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structured text file, which can be easily adopted, contains most of the information requested by BGI and additional processing details. The project file includes information about – Location: site code, site name, coordinates and height – Instrument: type and serial number, rubidium frequency, laser model and frequencies – The observation epoch – Details of the data acquisition and observation and processing details like – Gravity value and error estimates – Gravity gradient and reference height – Applied reduction models. Table 130.1 describes in detail the identifiers of the proposed exchange format. The line-based structure of the text file is simple. Each identifier is followed by the values, delimited by a colon. In cases where the values exceed a single line, e.g. tidal parameters, the data can be expanded over the following lines. However, such data cannot be extracted automatically while inserting new data into the database. Since data and meta-data are closely related, a strict separation is not possible. The gravity value and related parameters are considered as primary data, while information like station location and measurement epoch are addressed as meta-data. Information about the gravimeter or supplemental data used for gravity corrections have an intermediate position, but are treated as meta-data as well. Due to independent operation of individual institutions and the lack of a central coordination, meta-data about the station location are heterogeneous and difficult to harmonise. Therefore this information must be specified supplemental to the project file to ensure consistency. Two further definitions are necessary to characterize location and observation exactly. A site (station) is defined as a location with one or more gravimetric observation points/monuments in close spatial relationship with equal environmental (atmospheric, hydrologic) influences. Usually the distance between such points should not exceed a few 10 m. Most of the sites have only one observation point. A site is unambiguously identified by its country- and site-code. An observation epoch is defined as a period of an instrumental setup and continuous data acquisition. Short interruptions due to technical demands are accepted within one epoch. More than one processing
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Table 130.1 Description of project file used for data exchange Identifier in project file General settings Project name Institution Operator Comments Station data Name Site code Position Instrument data Meter type Meter S/N Factory height Rubidium frequency Laser Laser locks Laser modulation frequency Acquisition settings Acquisition version Set interval Drop interval Number of sets Number of drops Transfer to reference height Setup height Transfer height Gravity gradient Gravity corrections Nominal air pressure Barometric admittance factor Polar motion coordinates Earth tide Ocean load Processing results Processing version Date/time Gravity Set scatter Measurement precision Total uncertainty Number of sets collected Number of sets processed Number of drops/set Total drops accepted Total drops rejected Total fringes acquired Fringe start Processed fringes
Description Label of observation/processing Institution performing observation Name(s) of operator(s)
Used Database table
x x x
tbProcs tbInstitutions tbObsEpoch tbObsEpochs
Name of station Station code Latitude, longitude (GRS 80), height (physical)
x x x
tbStations tbStations tbPoints
Gravimeter type (acronym) Gravimeter serial number Reference height of gravity value referred to the instrument base Frequency of time reference Laser type and serial number List of available laser locks (WEO) Laser modulation frequency
x x x x x
tbMeters tbMeters tbProcs tbProcs tbObsEpochs
x
tbProcs
Software version used for measurement Waiting time between acquisition of sets Waiting time between individual drops Total number of sets Total number of drops
x
tbProcs
Vertical distance between instrument base and monument Vertical distance used to transfer gravity value to reference height Linear gravity gradient to at least four significant digits
x x x
tbProcs tbProcs, tbObsEpochs tbProcs, tbObsEpochs
Height dependent reference air pressure, from standard-atmosphere x Air pressure admittance factor used for atmospheric correction x Polar motion coordinates (provided by IERS) x potential, wave groups, delta-factors filename, wave groups, amplitudes, phases Software version used for processing Date and time of measurement referred to mean observation epoch Gravity value Scatter between set means Error estimate obtained from set scatter Uncertainty, including systematic effects Number of sets acquired Number of sets used during processing Number of drops per set Total number of reliable drops Total number of faulty drops First interference maximum used Number of zero crossings used
tbProcs tbProcs tbProcs
x x x x x x x x
tbObsEpochs tbProcs, tbObsEpochs tbProcs tbProcs, tbObsEpochs tbProcs, tbObsEpochs tbObsEpochs tbProcs tbObsEpochs
x x
tbProcs tbProcs
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result per observation epoch is possible and a final gravity value may be assigned outside automated processing schemes under consideration of external corrections. The observation period will be characterized by a mean point in time. Further, three main types of stations will be distinguished: field stations, stations with laboratory conditions and reference stations. Usually, absolute gravity stations are indoors, have a concrete foundation and are insulated against most environmental influences. Such conditions are denoted as laboratory conditions. A reference station is a regional gravity comparison site or a station where the national gravity standard is realized. Extended applications result from the concurrent existence of additional geodetic or geophysical sensors at the site, like permanent GNSS, SLR, VLBI, superconducting gravimeter, tide gauge, hydrological sensors etc. These observations make the stations valuable for investigations like the uplift or subsidence induced by geodynamic, climate or human processes. Initiatives like GGOS, GMES or GGP propose the definition of data interfaces to include the gravimetric observations. Criteria should be developed and fixed in the Working Group on Absolute Gravimetry and in the Global Geodynamics Project (GGP). Based on the information provided by the project files together with the above definitions, the following data model was set up.
130.3 Data Model The various relationships between different groups of data suggest a relational database design. The main objects are four tables which hold information about stations, instruments, the involved institutions and finally the observations. This approach avoids redundancies when data appear more than once in different context. The tables then are related to each other by sets of matching keys. A further advantage of the relational concept is the option of a stepwise development and flexible future extension. The four main tables and their features are: 1. Stations (tbStations): Name and site code of the station are stored. According to the definition above, at least one pier is assigned to each station, using the child table tbPoints. Coordinates are
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specified separately for each pier. Different network identifiers can be assigned to each station, using a n:m relation. Further, a class should be assigned to characterize the measurement conditions. Currently three classes are distinguished: reference stations and stations with laboratory and field conditions. 2. Gravimeters and their components (tbMeters): This table contains the list of instruments. Only type, serial number and a log are available here. Because an absolute gravimeter consists of several interchangeable parts (Laser, Dropping chamber etc.), a more detailed description of the meter components and its history can be stored optionally with the child table tbComponents. 3. Institutions (tbInstitutions): Used to store name and contact information of institutions which carry out measurements and/or own a gravimeter. This table will also be used to organize the database user access. 4. Observations (tbObsEpochs): Here, reference time and final result (gravity value) for an observation epoch are saved. Different processing results can be stored for each epoch, using the child table tbProcs. Perspectively, the child tables tbSetData and tbDropData can be used to store mean values of sets and the results for every drop experiment. Supplemental information, such as station descriptions, measurement protocols or log-files, which cannot be stored efficiently inside a database are provided as individual data-files which are stored in a file system, are linked to the respective records and can be downloaded directly.
130.4 Web-Based User Interface The database is set up on a MySQL server and is accessed with an apache/php based web front-end. It is installed on two mirrored systems, one located at the Federal Agency for Cartography and Geodesy (BKG) in Frankfurt/Main (Germany)1, and the other located at the Bureau Gravime´trique International (BGI) in Toulouse (France)2.
1 2
http://agrav.bkg.bund.de http://bgi.dtp.obs-mip.fr
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Fig. 130.1 Frontpage of the web-interface of the AGrav database
Both servers provide identical content. At the frontpage, a Java-script based map application3 is incorporated to show the geographic distribution of stations and allow for a location-based selection of the particular information (see Fig. 130.1). Starting from the four main sections, institutions, stations, instruments and observations, the web interface offers two main styles, a list view and a form to show or edit single datasets (Fig. 130.2). The list view offers an overview based on selected information shown as table, where sorting by different columns is possible. A search function is implemented to limit the displayed rows to match single criteria which will be applied to one of the columns. Each row has a hyperlink which leads to the detailed information for the specific record. In detailed view, all fields of a particular record are shown. Information in related tables appears in list view again. Both of these styles exist in slightly modified versions for meta-data and the complete data content. The meta-data version has no editing capability and makes use of links for fast navigation between different datasets. It is accessible without authentication. Gravity values are shown with reduced accuracy to preserve data ownership. The view with full access
3
Currently using the Google Maps API.
Fig. 130.2 Web interface for station meta-data. (a) List view of all stations, (b) details for a specific station
to the database is password protected and offers editing capabilities. Editing is possible for individual datasets and is restricted to those records which were created by the actual user. Data of other users are set to be read-only. Editing comprises changes of textual information, of related values and the creation of new records. The deletion of records is restricted to avoid accidental loss of data due to the cascaded links of entries in related child-tables, which is necessary to preserve consistency. Auxiliary information, which is too complex to be stored inside the database itself can be added as files and retrieved later via hyperlink. In this way it is possible to add station descriptions, photos, instrument logs or other documentation.
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130.5 Upload of New Data The development work of this database was initiated by the International Gravity Field Service (IGFS) and carried out by BKG and in cooperation with BGI. It was coordinated within the IAG Intercommission Working Group on Absolute Gravimetry. The international community of absolute gravimeter owners and the owners of absolute gravity measurements are kindly asked to support this initiative and to contribute to the database. To enable the user to insert new or update existing data, a web form was set up which enables the upload of project files directly by the user. In this way, the data transfer remains under control of the user, and the user may decide to include the gravity values or not. The upload is organised in a three steps process. First, the station must be specified. This can be an existing station, chosen from a list in the database or a new station. In the next step, the observation point at this station must be selected or a new entry will be created. Then, one or more project files for this point can be specified which will be uploaded to the database server and parsed by a script after pressing the upload button. Next, the user has the choice to limit the upload to meta-data only. Already uploaded datasets can be updated or modified. Then, some basic checks will be done, ensuring consistency of the station names and codes in the database. Within a country, a site code can appear only once, and it is excluded to create a new station in the vicinity of an existing one. Further, the gravity value will be checked against normal gravity to avoid gross errors. The results of the upload script are shown on a final page, informing the user about details or possibly problems. After successful upload, links to the newly created data are provided for checks or further editing. There might be cases where this upload procedure is inconvenient, especially if a large number of stations and observations are involved. In these cases it might be more efficient, to send the project-files to the database operator4 who will upload the data directly on the server. For this purpose a supplemental list with basic station information such as country and site code and pier is required, since this information is not contained sufficiently in the project file. If absolute gravity measurements are already compiled
4
[email protected]
H. Wziontek et al.
in tabulated form or project files not available, it is still possible to import these data, provided that all basic information about station and instrument are present. Conclusions
A database for absolute gravity measurements was set up and put into operation in joint cooperation between BKG and BGI. The database is capable to store information about stations, instruments, observations and involved institutions. By this, it allows the exchange of meta data and the provision of contact details of the responsible institutions on the one hand and the storage and long term availability of gravity data and processing details on the other hand. The database can be accessed by a web based interface which provides publicly available meta-data as well as complete datasets for the community of users contributing to the archive. A simple exchange format was selected which includes all relevant information and is existent for the majority of users avoiding additional effort. In this way the upload of data to the database is possible, using a web based upload form. Acknowledgements We gratefully acknowledge data contributed by: BEV, Bundesamt f€ur Eich-und Vermessungswesen, Austria (D. Ruess); EOST, Ecole et Observatoire des Sciences de la Terre, Strasbourg, France (J. Hinderer, M. Amalvict); FTSC, University of Luxembourg (O. Francis); Geosciences Universite´ Montpellier, France (R. Bayer, N. Lemoigne); IfE, Universit€at Hannover, Institut f€ur Erdmessung, Germany (L. Timmen, O. Gitlein); IGGA WUT Politechnika Warszawska, Poland (A. Pachuta, T. Olszak); IRD, Institut de Recherche pour le De´velopement, France (S. Bonvalot); NRCan, Natural Resources Canada (J. Liard); ON-COGE, Observatorio Nacional – Coordenacao de Geofisica, Brasilia (M. de Sousa); ROB, Royal Observatory of Belgium (M. Van Camp); swisstopo, Bundesamt f€ur Landestopografie, Switzerland (U. Marti); VUGTK, Geodetic Observatory of Pecny, Czech Republic (V. Palinkas, J. Kostelecky).
References Barriot J-P, Sarrailh M, Liard J, Boedecker G (2004) AGMAF03: a new archiving format to store absolute gravity measurements at the BGI, gravity, geoid and space missions 2004, IAG International Symposium, Porto, Portugal, Poster Presentation, Aug 30–Sep 3, 2004 Ihde J, Baker T, Bruyninx C, Francis O, Amalvict M, Luthardt J, Liebsch G, Kenyeres A, Makinen J, Shipman S, Simek J, Wilmes H (2005) The implementation of the ECGN stations – status of the 1st call for participation, EUREF Publication, No. 14. Mitteilungen des Bundesamtes f€ur Kartographie und Geod€asie, Frankfurt/Main 35:49–58
Index
A Abarca-del-Rio, R., 549 Abreu, M.A.., 915, 965 Abt, T., 247 Acun˜a, G., 899 Aguirre, M., 223 Ahola, J., 59 Albarici, F., 731 Alcantarilla, I., 971 Almeida, F.G.V., 883 Altamimi, Z., 19, 95 Amos, M., 341 Andersen, O.B., 363 Andritsanos, V.D., 409, 479 Angermann, D., 87, 993 Antico, P., 603 Aquino, M., 617 ´ ngel, A., 751 Arago´n-A Arias, E.F., 263, 1021 Azevedo, J.B., 867 Azpilicueta, F., 751, 767, 971 B Bacaicoa, L., 899 Bacˇic´, Zˇ., 681 Barbosa, A.C.B., 883 Barbosa, K., 965 Barbosa, S.M., 565 Barrios, M., 899 Bar-Sever, Y.E., 1011 Basˇic´, T., 559, 681 Bastos, L., 255 Becek, K., 723 Becker, M., 19, 263 Behrend, D., 997 Bergeot, N., 95 Beutler, G., 139, 147, 161, 985 Bingley, R.M., 663 Biskupek, L., 519 Blitzkow, D., 883, 891 Bock, O., 793, 1011 B€ohm, J., 1005 B€ohm, S., 1005 Bonvalot, S., 1035 Borghi, A., 625 Bos, M., 255
Bosser, P., 793 Bouin, M.-N., 11, 95 Bouman, J., 993 Brockmann, E., 687 Brunini, C., 3, 751, 759, 767 Bruyninx, C., 19, 27, 95 Brzezin´ski, A., 497 Buga, A., 105 Buist, P.J., 939 C Calvao, J., 131 Campos, I.O., 891 Cannizzaro, L., 625 Capitaine, N., 51 Caro, J., 971 Cesanelli, A., 603 Cesare, S., 223 Cezo´n, A., 971 Charles, Ph., 305 Christophe, B., 215 Chuerubim, M.L., 617 Cioce, V., 899 Claessens, S.J., 433 Codallo, H., 899 Combrinck, L., 19 Combrink, A., 19 Cordua, K.S., 363 Corey, B.E., 997 Costa, S.M.A., 3, 851, 857, 867, 915, 965 Coulot, D., 51 Craymer, M., 19, 703 Crespi, M., 759 Creutzfeldt, B., 305 Cˇunderlı´k, R., 331 Czarnecka, K., 955 D Dach, R., 161 Dawson, J., 19 Dayoub, N., 321 de Freitas, S.R.C., 907 de Jager, C., 505 de Matos, A.C.O.C., 883, 891 Del Cogliano, D., 907
S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1, # Springer-Verlag Berlin Heidelberg 2011
1043
1044 Denys, P., 695 Deurloo, R., 255 Dietrich, R., 19, 455, 579 Dill, R., 497 Ding, X.L., 711 Ditmar, P., 171 Dobslaw, H., 305, 497 dos Santos, D.P., 907 Dousa, J., 1027 Drewes, H., 3, 67, 87, 655, 875 Duhau, S., 505 Duquenne, H., 463 E Eberlein, L., 579 Edwards, K.R., 471 Edwards, S.J., 321 Eicker, A., 153 Eisner, S., 297 F Fagard, H., 43 Faggion, P.L., 907 Fagiolini, E., 147 Fairhead, J.D., 891 Falk, R., 273 Fasˇkova´, Z., 331 Featherstone, W.E., 525 Ferhat, G., 671 Fernandes, R., 19 Ferraccioli, F., 455 Ferreira, V.G., 907 Flechtner, F., 147 F€oldva´ry, L., 589 Fonseca Junior, E.S., 731 Forsberg, R., 363, 441, 611 Fortes, L.P.S., 915, 965 Foulon, B., 215 Fourie, P., 305 Francis, O., 263 Fund, F., 571, 641 G Galo, M., 867 Galva´n, L., 349 Gao, Y., 711 Garayt, B., 43 Gende, M., 751 Gerstl, M., 993 Ghidella, M.E., 579 Giorgi, G., 939 Gobinddass, M.L., 43 Gomes, E., 965 Govind, R., 19 Grejner-Brzezinska, D., 471, 633, 955 Grigoriadis, V.N., 479 Gruber, T., 147, 993 Guarracino, L., 603
Index Guimara˜es, G.N., 883 Gunter, B.C., 171 G€untner, A., 305 H Haagmans, R., 223 Habrich, H., 27 H€akli, P., 59, 77, 105 Hamayun, 387 Hansen, D.N., 663 Hauschild, A., 933 Heck, B., 671, 737 Heidbach, O., 655 Heiker, A., 535 Henkel, P., 923 Herna´ndez-Pajares, M., 751 Herring, T., 19 Hobiger, T., 779 Hofmann, F., 519 Holota, P., 195 Hoyer, M., 899 Huang, O., 239, 247 Hugentobler, U., 113, 993 Huinca, S.C.M., 737 I Ichikawa, R., 779 Ilk, K.H., 179, 187 Ineichen, D., 687 Infante, C., 349 J J€aggi, A., 139, 147, 161 Jahr, T., 297 Jansson, T.R.N., 441 Jekeli, C., 239, 247 Jiang, Z., 263 Jin, S., 711, 823 Jokela, J., 59, 105 Jordan, T., 455 Juan, J.M., 751 Ju´nior, N.J.M., 915, 965 K Kallio, U., 35, 59 Keller, W., 525 Kenyeres, A., 19, 27 Kern, M., 223 Kessler-Schulz, K.U., 263 King, R., 19 Kingdon, R., 425 Klees, R., 171, 379, 399 Kn€opfler, A., 671, 737 Koivula, H., 77, 105 Kondo, T., 779 Kosek, W., 511, 543 Koyama, Y., 779
Index Krause, P., 297 Kreemer, C., 19 Kroner, C., 297, 305 Krueger, C.P., 737 Kuhn, M., 525 Kurtenbach, E., 153 Kutterer, H., 535 L Laurı´a, E., 349 Lavalle´e, D., 19 le Bliguet, G., 641 Lebedev, S.A., 831 Legrand, J., 9, 95 Leo´n, J., 899 Leone, B., 223 Lim, S., 773 Lima, M.A.A., 915 Lobianco, M.C.B., 891 Longuevergne, L., 291 Lux, N., 273 Luz, R.T., 907 M Machado, W.C., 731 Mackern, V., 3 Madsen, K.S., 565 Marenssi, S.A., 579 Marjanovic´, M., 681 Markiel, J.N., 955 Marque, J.P., 215 Marques, H.A., 617 Marti, U., 687 Martı´nez, W., 3 Masson, F., 671 Massotti, L., 223 Mayer, M., 671, 737 Mayer-G€urr, T., 153 Mayrhofer, R., 231 Mazzoni, A., 759 Mentes, G., 717 Mervart, L., 161 Meyer, U., 139, 161 Mikula, K., 331 Miranda, S., 611 Moafipoor, S., 955 Mocquet, A., 571 Monachesi, L., 603 Monico, J.F.G., 617, 731, 965 Moore, P., 321 Morel, L., 571, 641 Mtamakaya, J., 703 M€uller, J., 519, 535, 595 M€uller, Ja., 273 N Nastula, J., 489 Naujoks, M., 297 Nesvadba, O., 195
1045 Nicolas, J., 641 Niedzielski, T., 543 Niell, A.E., 997 Nielsen, J., 441 Nievinski, F., 807 Nilsson, T., 1005 O Odijk, D., 743 Oja, T., 313 Olesen, A.V., 363 Oliveira, L.C., 867 Ostolaza, J., 971 P Pacino, M.C., 611, 891 Pail, R., 231 Palinkas, V., 263 Panet, I., 463 Pany, A., 1005 Pelon, J., 793 Penna, N.T., 321 Pereira, A., 611 Pereira, R.A.D., 907 Perosanz, F., 641 Peterseim, N., 595 Petrachenko, W.T., 997 Pflug, H., 305 Plank, L., 1005 Polezel, W.G.C., 731 Pollet, A., 51 Poutanen, M., 35, 59 Prange, L., 161 Prutkin, I., 379 Putrimas, R., 105 R Radicella, S.M., 767 Ramos, A.M., 915 Ramos, R., 349 Reguzzoni, M., 205 Reiterer, A., 717 Repanic´, M., 559 Retscher, G., 711, 747, 947 Richter, B., 993 Rigo, A., 641 Rizos, C., 773 Rodrigues, J., 131 Rodriguez-Solano, C.J., 113 Rosa, G.P.S., 617 Rothleitner, C., 263 Royero, G., 899 Ro´zsa, S., 787, 815 Rummel, R., 985 S Salstein, D.A., 489 Sampietro, D., 205
1046 Sa´nchez, L., 3, 19, 843, 875 Santamarı´a-Go´mez, A., 11, 19, 95 Santos, M.C., 425, 703, 711, 807, 867, 965 Santos, M.F., 867 Sanz, J., 751 Scanlon, B., 291 Schaer, S., 687 Schaffrin, B., 633 Scheinert, M., 449, 455, 579 Schlatter, A., 687 Schmidt, T., 147 Sch€on, S., 799 Schrama, E.J.O., 371 Schuh, H., 997, 1005 Schulz, H.R., 263 Schwabe, J., 455 Schwarz, G., 147 Seem€uller, W., 3, 843, 875 Seitz, M., 87, 843, 875 Sella, G., 19 Shabanloui, A., 179, 187 Sharp, J., 291 Shen, Z., 19 Shimada, S., 649 Shin, J., 773 Sideris, M.G., 123, 417 Silva, A.L., 3, 851, 857, 915, 965 Silvestrin, P., 223 Smith, D., 471 Sneeuw, N., 371 S€ohne, W., 27 Spicakova, H., 1005 Stangl, G., 27 Steffen, H., 595 Steigenberger, P., 113, 993 Stevenson, M., 695 Stolk, W., 171 Strykowski, G., 363 T Teferle, F.N., 663 Teke, K., 1005 Tenzer, R., 341, 379, 387, 399, 695 Te´treault, P., 965 Teunissen, P.J.G., 743, 939 Thom, C., 793 Thomas, M., 305, 497 Tierra Criollo, A.R., 907 Tisserand, L., 263 Tocho, C., 417, 603 Toth, C.K., 955 Touboul, P., 215 Trautmann, T., 147
Index Tscherning, C.C., 441 Tuchband, T., 787 Tziavos, I.N., 123, 409, 479
U Ulrich, P., 671 Ultmann, Z., 281 Urquhart, L., 807 V Vajda, P., 387 Valty, P., 463 van Bree, R.J.P., 933 Vanı´cˇek, P., 425 Vatrt, V., 341 Vaz, J.A., 851, 857 Va´zquez, G.E., 633 Vennebusch, M., 799 Vergos, G.S., 123, 417, 479 Verhagen, S., 711, 743, 933, 939 Viarre, J., 549 Villiger, A., 687 Visser, P.N.A.M., 371 Vitti, A., 625 V€olgyesi, L., 281 V€olksen, C., 27
W Wadhams, P., 131 Wei, M., 355 Weigelt, M., 371 Weise, A., 297 Werth, S., 305 Wickert, J., 147 Wildermann, E., 899 Williams, S.D.P., 663 Willis, P., 43, 793, 1011 Wilmes, H., 273, 1035 Wilson, C.R., 291 Wittwer, T., 171, 399 W€oppelmann, G., 11, 19, 95 Wresnik, J., 997 Wu, H., 291 Wziontek, H., 273, 1035 Z Zakrajsek, A.F., 579 Zenner, L., 147 Zhang, K., 773
