Applied geodesy: global positioning system : networks, particle accelerators, mathematical geodesy

Lecture Notes in Earth Sciences Edited by Somdev Bhattacharji, Gerald M. Friedman, Horst J. Neugebauer and Adolf Seilacher

12 Stuart Turner (Ed.)

Applied Geodesy Global Positioning System - Networks - Particle Accelerators - Mathematical Geodesy

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo

Editor Stuart Turner LEP Division, CERN CH- 1211 Geneva 23, Switzerland

Originally published as Internal Report under the Title: Proceedings of the CERN Accelerator School of Applied Geodesy for Particle Accelerators, Editor: S. Turner, Geneva 1987 ISBN 3-540-18219-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-38?-18219-5 Springer-Verlag New York Berlin Heidelberg

This work Is subject to copyright. All rights are reserved, whether the whole or part of the material Is ooncerned, specKflcally the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or m other ways, and storage In data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, Jn ~ts version of June 24, 1985, and a copyright fee must always be paid. Violat~ons fall under the prosecution act of the German Copyright Law. © Sprlnger-Verlag Berlin Heidelberg 198"7 Printed in Germany Pnntmg and binding Druckhaus Beltz, Hemsbach/Bergstr 213213140-543210

PREFACE

The CERNAccelerator School (CAS) was founded in 1983 with the aim to preserve and disseminate the knowledge accumulated at CERN (European Organization for Nuclear Research) and elsewhere on part i c l e accelerators and storage rings.

This is being achieved by means of a biennial programme of

basic and advanced courses on general accelerator physics supplemented by specialized and topical courses as well as Workshops. The chapters included in this present volume are taken from one of the specialized courses, Applied Geodesyfor Particle Accelerators, held at CERNin April 1986. When construction of the f i r s t large accelerators started in the 1950's, i t was necessary to use geodetic techniques to ensure precise positioning of the machines' components. Since that time the means employedhave constantly evolved in line with technological progress in general, while a number of specific developments - manyof them achieved at CERN - have enriched the range of available instruments.

These techniques and precision instruments are used for most of the world's accelerators

but can also be applied in other areas of industrial geodesy: surveying of c i v i l engineering works and structures, aeronautics, nautical engineering, astronomical radio-interferometers, metrology of large dimensions, studies of deformation, etc. The ever increasing dimensions of new accelerators dictates the use of the best geodetic methods in the search for the greatest precision, such as distance measurementsto 10-7, riqorous evaluation of the local geoid and millimetric exploitation of the Navstar satellites.

At the same time, the

powerful computer methods now available for solving d i f f i c u l t problems are also applicable at the instrument level where data collection can be automatically checked. Above a l l , measuring methods and calculations and their results can be integrated into data bases where the collection of technical parameters can be e f f i c i e n t l y managed. In order to conserve the logical presentation of the different lectures presented at the CAS school, the chapters presented here have been grouped under four main topics.

The f i r s t and the

fourth deal with spatial and theoretical geodesy, while the second and third are concerned with the work of applied geodesy, especially that carried out at CERN. Readers involved in these subjects will find in the following chapters, i f not the complete answer to their problems, at l e a s t the beginning of solutions to them. J. Gervaise

P.J. Bryant, Head of CAS

M. Mayoud

S. Turner, Editor

Applied GeodesyGroup CERN

LIST OF AUTHORS

BAKER, L.S. BEUTLER, G. BOUCHER. C. BORRE, K. BURKI, B.

8612 Fox Run, Potomak, USA University of Berne, Switzerland Inst. G~ographiqueNational, Salnt-Mand~, France Aalborg University, Aalborg Ost, Denmark Institute for Geodesy & Photogram~etry, ETH Zurich, Switzerland

CAMPBELL, J. CASPARY,W.F.

Geodetlc Inst. University of Bonn, FRG Univ. der BundeswehrMunchen, Neubiberg, FRG

COOSE~%~NS,W. DUFOUR, H.M.

CERN, Geneva, Swltzerland Inst. G~ographiqueNational, Saint-Mand~, France

FISCHER, J.C. HAYOTTE,M.

CERN, Geneva, Switzerland CERN, Geneva, Switzerland CERN, Geneva, Switzerland National Geodetic Survey, NOAA,Rockville, USA Astronomical Institute, University of Berne, Switzerland CERN, Geneva, Switzerland

GERVAISE, J., GOAD, C.C. GURTNER,G., HUBLIN, M. ILIFFE, J. LASSEUR, C. MAYOUD,M. MORITZ, H. OLSFORS, J. QUESNEL, J.P. TROUCHE,G. UNGUENDOLI,M. WELSCH, W.M. WILSON, E.J.N.

University College London, United Kingdom CERN, Geneva, Swltzerland CERN, Geneva, Switzerland Technical University, Graz, Austrla CERN, Geneva, Switzerland CERN, Geneva, Switzerland CERN, Geneva, Switzerland University of Bologna, Italy. Univ. der BundeswehrM~nchen, Neubiberq, FRG CERN, Geneva, Switzerland

CONTENTS

I.

GLOBALPOSITIONING SYSTEMAND V.L.B.I.

L.S. BAKER GPS Its Development and Deployment . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

C. BOUCHER GPS Receiver Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

C.C. GOAD Precise Positioning with the GPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

G. BEUTLER GPS Orbit Determination Using the Double Difference Phase Observable

. . . . . . . . . . . .

31

W.M. WELSCH Accuracy Problems when Combining Terrestrial and S a t e l l i t e Observations

..........

47

J o CAMPBELL Very Long Base Interferometry II.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

SURFACEGEODETIC NETWORKSAND UNDERGROUNDGEODESY

J. GERVAISE, J. OLSFORS The LEP T r i l a t e r a t i o n Network

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111

W. GURTNER, B. BURKI Deviation of the Vertical G. BEUTLER Co~Darison between Terrameter and GPS Results - and How to Get There

. . . . . . . . . . . .

125

Geodetic Networks f o r Crustal Movements Studies . . . . . . . . . . . . . . . . . . . . . . .

135

P. BALDI, M. UNGUENDOLI

W.F. CASPARY Gyroscope Technology, Status and Trends

. . . . . . . . . . . . . . . . . . . . . . . . . .

163

J.C. FISCHER, M. HAYOTTE, M. MAYOUD, G. TROUCHE Underground Geodesy . . . . . . . . . . . . . . . . . . . . . . . . . .

.. . . . . . . . . . .

181

III.

APPLIEDGEODESYFOR PARTICLEACCELEPJ~TORS

J. GERVAISE, E.J.N. WILSON High Precision GeodesyApplied to CERNAccelerators . . . . . . . . . . . . . . . . . . . . .

209

J. ILIFFE Three-Dimensional Adjustments in a Local Reference System . . . . . . . . . . . . . . . . . .

247

M. HUBLIN Computer Aided Geodesy (I) LEP Installation Pro<:edure . . . . . . . . . . . . . . . . . . . .

269

W. COOSEMANS Computer Aided Geodesy ( I I ) Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . .

279

M. MAYOUD Applied Metrology for LEP (1) Computing and Analysis Methods . . . . . . . . . . . . . . . .

293

J.P. QUESNEL Applied Metrology for LEP (II) Data Logging and Managementof Geodetic Measurements with a Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

309

C. LASSEUR Metrolo~Lv for Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.

317

MATHEMATICALGEODESY

H. MORITZ New Trends in Mathematical ~odesy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

345

H.M. DUFOUR Inverse Regional Reference Systems: A Possible Synthesis between Three- and Two-Dimensional Geodesy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

359

K. BORRE On Continuous Theory for Geodetic Networks . . . . . . . . . . . . . . . . . . . . . . . . . APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

375 387

I. Global Positioning System and V.L.B.I.

GPS ITS DEVELOPMENT AND DEPLOYMENT Leonard S. Baker Retired, Director, National Geodetic Survey

ABSTRACT The Navstar Global Positioning System (GPS) is an all-weather, space-based navigation system under development by the US Department of Defense.

The m i l i t a r y needs and requirements

are well documented, but the combined u t i l i z a t i o n , savings, and deployment by the c i v i l i a n users throughout the world w i l l surpass the benefits to be reaped by DoD.

1.

DEVELOPMENT Scientists have always reached for goals that at the time seemed unobtainable. When

those goals were near, new goals were envisioned to stretch our imagination, deepen our thoughts, and yes, day dream a l i t t l e

- "WHAT IF?".

This simplistic view is so very true of the Physical Sciences.

When the f i r s t break-

through was made concerning matter, i t only opened one door along the way, and you immediately planned the attack on the next door.

The i n s t a l l a t i o n s , at each of the laboratories

around the world, have monuments to this path of discoveries.

Here at CERN there are the

Proton Synchrotron, the Intersecting Storage Rings, the Super Proton Synchrotron;

and

now, the Large Electron Positron Collider, LEP, is under construction. In the United States, there is renewed interest in the Super Colliding Accelerator, that may have a diameter two or three times the size of LEP.

As a geodesist, who has

worked on an accelerator ring, I have gained a greater appreciation of the extremely high surveying accuracies required.

Each new plan for the next generation of accelerators

makes me stop and wonder, " w i l l

they be able to meet the surveying accuracy require-

ments?".

GPS w i l l play a major role in meeting those stringent demands.

Humans have never been content to accept the norm for very long, and in 1957, when the Russians launched SPUTNIK, the gestation period of GPS began. has grown quite rapidly in i t s short l i f e . limited preview of what GPS w i l l be.

GPS has been born and

However,we have been treated to only a very

The real GPS w i l l not occur until the f u l l constel-

lation of GPS s a t e l l i t e s are in o r b i t , in 1988, or 1989. The world has changed dramatically since SPUTNIK. Everyday, each of us enjoys the benefits of the "Space Age" technology, in our work, in our play, and in our normal living.

The world is moving towards being a better place to l i v e in, not only for us, but

for all

humans on this planet we call earth, and GPS w i l l

various ways.

contribute in so many and

GPS is an all weather, space based, navigation system, under development by the United States Department of Defence (DoD), to satisfy the requirements of the m i l i t a r y forces.

Requirements include, accurately determined positions, velocity, and time in a

common reference system, anywhere on or near the Earth, on a continuous basis.

The Air

Force Systems Command'sSpace Division acts as the executive agent for the DoD in managing the GPS programme. All branches of the United States m i l i t a r y are represented, as well as the Department of Transportation, and NATO. The Navstar GPS navigation and time-transfer system operates on two L-band frequencies, L1 (1575.4 MHz) and L2 (1227.6 MHz). The system consists of three major segments; a space segment, s a t e l l i t e s that transmit radio signals;

a control segment, ground-based

equipment to monitor the s a t e l l i t e s and update their signals;

and a user segment, equip-

ment which passively receives and converts s a t e l l i t e signals into positioning and navigation information. The GPS system has already completed Phase I, concept validation.

Phase I I , engi-

neering development, is nearing completion, and a constellation of six development satell i t e s are being maintained to support testing of both the m i l i t a r y and c i v i l i a n applications.

The contract for the development of the operational s a t e l l i t e s has been awarded

and, barring any additional delays such as the shuttle disaster, operational s a t e l l i t e s are scheduled to be placed in orbit starting this year, with a planned completion of the constellation in 1988. The f u l l constellation w i l l contain 18 s a t e l l i t e s , in six orbital planes, 60 degrees apart, and each plane w i l l have three s a t e l l i t e s .

The s a t e l l i t e s w i l l operate in circular

20,200 km orbits, at an inclination of 55 degrees, with 12-hour periods.

Spacing is

planned to ensure that at least four s a t e l l i t e s w i l l be in view to the user at any time on a worldwide basis. This is a DoD programmethat could serve as a model for other programmes, anywhere. It

is well planned, funded, tested, and is now being put into place.

The over-riding

questions that have yet to be answered are - what level of accuracy, in real time, w i l l users other than DoD, be allotted, and - w i l l the signals be distorted? 2.

GEODETICRECEIVERS A number of GPS geodetic receivers are already being used to gather data, while

others are undergoing tests in preparation for their employment. these include:

At the present time

MACROMETER V-IO00, the f i r s t GPS geodetic receiver developed for precise surveying by Macrometrics Inc.

The Aero Service Division of Western Geophysical Companyof America,

Houston, Texas, acquired the Macrometics Company, in March 1984.

TEXAS INSTRUMENT TI 4100, developed by Texas Instruments, with funding from U.S. Govern-

-

ment Agencies. This was the f i r s t dual frequency geodetic receiver to be produced u t i l i z ing both the P- and the S-code. - GPS LAND SURVEYOR, MODEL 2002, developed by ISTAC Inc., Pasadena, California.

I t is an

outgrowth of the SERIES technology, o r i g i n a l l y conceived and developed by MacDoran and his co-workers at the Jet Propulsion Laboratory, Pasadena, California. - TRIMBLE 4000S, a second generation GPS instrument developed by Trimble Navigation, Ltd., Sunnyvale, California.

The f i r s t GPS instrument developed by Trimble was the 4000A, a

navigation instrument, developed primarily for the shipping industry. WM 101, a j o i n t venture between the Wild Co., Heerbrugg, Switzerland, and Magnavox,

-

Torrance, California.

The instrument is presently undergoing testing, prior to being put

into service. - TR5S, developed by Sercel, France. These units have already collected data, but no additional information is available. There w i l l future.

undoubtedly be numerous new geodetic receivers available in the near

Manyof these receivers w i l l be second generation receivers.

They w i l l shrink in

size and decrease in price as production increases to meet the demand of the multitude of users. 3.

DEPLOYMENT No one can dispute the needs, requirements, and planned use DoD has for GPS. The

accomplishments w i l l be too numerous to tabulate.

However, the combined u t i l i z a t i o n ,

savings, and deployment by the c i v i l i a n users, throughout the world, w i l l

surpass the

benefits to be reaped by DoD, perhaps not in the near term, but within a very short span after the system is f u l l y operational. Even in the develoment phase, GPS is being u t i l i z e d to monitor seismic actions, both horizontal and v e r t i c a l .

I t eliminates the time delay factor, and because the s a t e l l i t e s

are in space, ground truths can be located in regions completely outside the areas under study. One of the major problems the scientist working at sea has always been confronted with, is accurate location while underway. Geophysical p r o f i l i n g is now, with the aid of computers, three-dimensional.

When the profiles show a promising structure, the scien-

t i s t s can be confident that the coordinates are accurate, thereby ensuring that the d r i l ling crew w i l l be able to pierce the structure.

U t i l i z i n g the GPS translocation mode,

ships can now be assured of one metre accuracies, within a range of about 300 kilometres from the GPS receiver over a known location.

The most excitement GPS has generated, has been in the geodesy community. Using methods that w i l l

be discussed l a t e r by other speakers, geodesists

achieve extremely high accuracies between two points.

have been able to

These are greater than the normal

accuracies obtainable with t r a d i t i o n a l surveying equipment and approach, or sometimes exceed, the accuracies of special surveys at i n s t a l l a t i o n s such as CERN. When the e l e c t r o n i c distance measuring equipment became a v a i l a b l e to the surveying and geodetic communities, i t

broke the bond the steel tape had had on them.

Formerly,

distances could only be measured i f a clear l i n e of sight was a v a i l a b l e and the angles had to be determined in order to permit the computations to be made.

A l l t h a t has changed.

No l i n e of sight is required, no angles are needed, only a window to the sky, a piece of space-age technology,

an i n n o v a t i v e

computer program,

and the

radio

signals

from

a

m i l i t a r y programme s a t e l l i t e . 3.1

Mappin9 With the advent of s a t e l l i t e s , mapping of the earth's surface and resources, has

become a new science, a science that was only dreamed of 30 years ago. Remotesensing has been responsible for many new discoveries and has allowed new maps of the world to be produced, at least in regions where aerial photography is allowed.

But the science of remote

sensing has not reached the level of accuracy, or scale, required for both developed, and the developing nations.

W i t h GPS, employed in the translocation mode, and the other

antenna mounted in an airplane, positions can be computed for the airplane, either in f l i g h t , or post-processed, to an accuracy of less than one metre within a range of 300 kilometres.

With very l i t t l e

additional equipment, other than an aerial camera, the pic-

tures taken can be used to produce maps to a scale of 1/40,000, with l i t t l e control.

or no ground

Mapping on this scale is more than adequate for all developing nations who need

to plan and establish the basic road systems, access the natural v i s i b l e resources, manage the water resources, and i n s t i t u t e major land reforms. Most of the developing nations do not enjoy an adequate network of coordinated cont r o l , the existing system being established by methods now labelled as inadequate. GPS instruments

The

allow control to be established faster and less expensively than with

other means. T h e y can also be used to up-grade the existing f a c i l i t i e s and to t i e

in

separated areas to provide a single coordinated framework. 3.2

Air navigation I f the Precise Positioning Service (PPS) would be available to the c i v i l i a n users, an

accuracy of 15 metres in x, y, flight.

and z would be available for positioning a i r c r a f t

in

T h a t accuracy represents a distance that is less than the wing span of almost

every a i r c r a f t in existance. The present method for f l i g h t plans for commercial a i r c r a f t , is to assign a f l i g h t a l t i t u d e in a designated corridor, with check points along the way.

The a i r c r a f t is

tracked by ground control systems that relay this information to a i r t r a f f i c controllers, assigned to monitor that particular f l i g h t .

Noneof the i n - f l i g h t and the ground tracking

instrumentation can provide the p i l o t with real time locations as accurately as the PPS GPS system. Barring the problems all systems experience, a p i l o t , flying an airplane equiped with GPS, could take off in zero v i s i b i l i t y , navigate to the destination, and land in near zero v i s i b i l i t y without any outside assistance.

Whenthe coordinates, x, y, and z

are established by GPS at the end of every runway, and PPS GPS is available, each of them becomes an emergency landing site. The accuracy of the Standard Positioning Service (SPS), to be available to the worldwide community of c i v i l users, w i l l be 100 metres, in x, y, and z with a 95% confidence level.

This is s t i l l

in f l i g h t ,

and i t

as much accuracy as a p i l o t , travelling at 600 knots/hour, can use

is only the z that is so useful to them in landing with reduced

visibility. One can speculate on the adverse side of GPS in aviation. Will inexperienced p i l o t s , not proficient in flying by instrument, substitute the advantages of GPS for experience and attempt to f l y in adverse weather conditions or after dark?

Even cautious pilots

become lost, perhaps due to weather fronts moving into the f l i g h t path, and f a i l to reach their destination even during daylight hours.

GPS can add a measure of safety.

We are so prone to think only of the flying conditions of our respective countries where we have excellent f a c i l i t i e s .

Why should less fortunate people, living in distant

lands without adequate air t r a f f i c control or improved airports, be subject to more hazardous f l i g h t s when flying is often their only link to the outside?

With GPS they w i l l

be able to take a quantum leap forward in air navigation and safety, even with the SPS, 100 metre accuracy. 3.3

Sea navigation Shipping has used Transit s a t e l l i t e generated positions for many years.

Transit has

a 90 minute interval between fixes, therefore is not an adequate system for navigating in close quarters. seas".

WhenGPS is available world wide, i t may spell the end of "freedom of the

No ship has the right to endanger the coastline of another nation, by polluting

i t s shores with an oil s p i l l .

GPS w i l l allow the nations of the world to assign dangerous

cargo vessels to certain shipping lanes that w i l l not cross, except at designated points. Very few ships collide when going in the same direction; in passing or crossing situations.

the majority of collisions occur

GPS can not alleviate the crossing situation, but can

eliminate the chance of a passing c o l l i s i o n , i f shipping lanes are established, and GPS is a requirement. 3.4

Multipurpose cadastre There is a very c r i t i c a l requirement for a land information system in the United

States.

The system is needed to improve land-conveyance and to provide for equitable

taxation, information for resource management and environmental planning.

We need a

MULTIPURPOSE CADASTRE. We have needed i t for more years than we care to admit. The concept of the multipurpose cadastre, is a framework that supports continuous, r e a d i l y available and comprehensive land-related information at the individual parcel level.

The f i r s t component is a reference framework, consisting of a geodetic network.

The geodetic control must be an integrated network of points on the ground, over the entire area of the parcels that are related to each other.

For a true cadastre, the con-

tinental United States must have a single, densely spaced, related geodetic network.

Most

of the populated European countries have a densely spaced control network, that supports t h e i r cadastre.

These nations have u t i l i z e d , and benefited from the use of t h e i r multi-

purpose cadastre system for many, many years. GPS presents to the United States, the f i r s t chance to have a t r u l y continental integrated network of control, at an acceptable price, and within a reasonable time frame, a requirement for our cadastre.

The continental network w i l l

be a part of the world net-

work, to respond to the intercontinental needs. 3.5

Commercial application One United States automobile manufacturer is

models.

planning to offer GPS in i t s

1989

A GPS receiver w i l l be connected to a computer and digitized map system and w i l l

display on a small screen a continuous, detailed, map of the location of the automobile. This w i l l be a very useful tool for t r a v e l l e r s to a new location, and to those individuals who have d i f f i c u l t y finding t h e i r way. The system w i l l allow a delivery firm to prepare a route map for the driver, increasing the efficiency of the driver and vehicle, and eliminating the,

"I

got l o s t " ,

excuse.

(How many more distractions can the human driver

master?) 3.6

Rescue GPS w i l l become a very important element for a i r and sea rescue.

When the search

a i r c r a f t or vessel knows i t s exact location within a few metres, there is no doubt that the suspected area has been covered.

I f the a i r c r a f t , vessel, or person knows the exact

location, and can relay this information, finding them becomes almost as easy as following the vehicle on the screen in the car. 3.7

Boundarylocations There are some countries that continue to have boundary disputes with t h e i r neigh-

bours.

Surprisingly enough, many of the disputes are not over the physical location, they

are over the coordinates of the commonturning points of the boundaries, such as the mountain peak.

GPS may eliminate many of these problems by placing each country on an identi-

cal coordinate system, thereby establishing commoncoordinates for boundary points.

3.8

Da~ dreamin9 I f we are allowed to day dream, there are some excellent humanitarian projects where

GPS could play a major role.

What i f we were able to build a GPS unit including a d i g i -

tized map of Geneva on a small micro disk and small enough to f i t

into a back-pack.

This

could acquire the s a t e l l i t e signals, compute the position, and feed this information to the computer which would search the digitized map and then, verbally inform the person of his exact location. I do not believe this is a "what i f " ,

but rather a "when" idea.

When w i l l

i t be

available to all those people who need i t now? Such a device would give them the freedom to move about with ease, with confidence, and with the assurance that they could go i t alone and not be dependent upon others for guidance. 4.

SUMMARY The original GPS scientists may never hear the many thanks that surely w i l l be voiced

on behalf of t h e i r invention.

They may never receive the recognition r i g h t f u l l y due to

them, but they can take a great deal of pride in knowing that t h e i r efforts have benef i t e d and w i l l continue to benefit mankind, a l l over the earth, in so many ways, for so many years to come. THE USE OF GPS IS ONLY LIMITED TO OUR IMAGINATION AND INGENUITY.

REFERENCES R.W. King, E.G. Masters, C. Rizos, A. Stoltz, J. Collins;

Surveying with GPS; Monograph

No. 9, School of Surveying, The University of New South Wales, Kensington NSW, Australia 2033. Need for a Multipurpose Cadastre;

Panel on a Multipurpose Cadastre, Committee on Geodesy,

Assembly of Mathematical and Physical

Sciences, National Research Council;

National

Academy Press, Washington, D.C. 1980. Positioning with GPS -

1985;

Proceedings, First

International Symposium on Precise

Positioning with the Global Positioning System, Volume I, 15-19, 1985.

Rockville, Maryland, April

GPS RECEIVERTECHNOLOGY C. Boucher Institut G~ographiqueNational, Saint~and~, France ABSTRACT The NAVSTAR/GPS system is already widely used for geodetic positioning, although the f u l l

capability is not yet available.

Various types of

receivers, using the C/A and/or P codes or codeless, are available as prototypes or commercially. This paper reviews these receivers. 1. THE GPS SYSTEMCONCEPT The NAVSTAR/GIobal Positioning System (GPS) is a satellite-based system developed by the US DOD for worldwide navigation of any system moving in the Earth's environment (ground, maritime or aerospace vehicle). The capability of this system for geodesyhas been illustrated manytimes.

Although

i t will only be f u l l y operational in the late eighties, a provisional constellation of satellites, available for a few hours per day, enables the geodesists not only to prove the efficiency of this system for precise positioning but also to use i t for operational purposes, replacing efficiently many of the conventional procedures. We shall review here the various receivers available for geodetic use, either prototypes, commercially available or under development. Similar surveys have already been madeby J.D. Bossler and C.W. ChallstromI), J. Hannah2), P. Hartl et al. s) and B. Remondi4). 2. THE NAVSTARSPACESEGMENT(Fig. 1) The GPS system uses an L-Band down-link from a cluster of several satellites in the c ~ n of a receiver which can perform several tj@es of measurementsas described below.

--•

P-COI?E (F)

EXCLUSIVE OR MIXER SU~tIATIOR

Fig. 1 GPS satellite signals

view

12

Each s a t e l l i t e transmits a pair of carriers at L1 : 1575.42 MHz and L2 = 1227.6 MHz. These are modulated by three levels of codes; namelyC/A (also called S in the final stage of the system), P (called Y in the final stage) and message. Each code is a flow of bits, 0 or 1, represented by a ± 180 degree phase shift. the message code 50 b/s.

The P code has a bit rate of 10.23 Mb/s, the C/A code of 1.023 I~)/s and The message and the P Code exist on both L1 and L2.

C/A code exists only

on L1 in quadrature with the P Code. For more details see Ref. 3. 3.

MEASUREMENTTECHNIQUES Three types of techniques can be conceived to perform measurementsof geodetic interest:

3.1 Codecorrelation (Fig. 2) This technique was adopted for normal use. The receiver produces a replica of the code (C/A or P) from a local oscillator.

The code stream can be shifted in time and i t is mixed with a beat sig-

nal from the received GPS signal and a pure carrier locally generated.

The code is then shifted in

order to obtain a maximum correlation between the received and the generated codes.

The shift

expresses directly the sum of the offset between s a t e l l i t e and receiver clocks, and the group propagation delay from the s a t e l l i t e to the receiver.

This quantity, expressed in length through the

velocity of light, is namedpseudo-range. The continuous alignment of codes cleans the received carrier on which phase or Doppler measurement now gives useful information. OUTPUT

] (~

FREQUENC MYULTIPLIER MIXER

Fig. 2 GPS correlation channel 3.2 Signal squarin9 (Fig. 3) The principal is to multiply the received signal by i t s e l f . is done d i r e c t l y .

In the case of the macrometer, this

As the code is in phase opposition, this technique removes i t and produces a pure

carrier with double frequency on which phase measurements can be performed.

13

7. .y

j"

"'-,

v

/" ".,. ".,

OUTPUT

]

FREQUENCY MULTIPLIER

I OSCILLATOR

Fig. 3 GPS squaring-type channel In the SERIES technique developed at the Jet Propulsion Laboratory, one signal is shifted in time by half the period of the code before being mixed with the other.

I t can be shown that epochs

of code transition can be obtained precisely (see Ref. 5). 3.3

Interferometry This last technique is identical to the one used in VLBI: each received signal is down conver-

ted and recorded.

Later, signals are correlated amongtwo or more simultaneous trackings.

To m6' knowledge, this has been used only for test purposes at the Haystack Observatory using a Macrometer prototype. 4.

ERRORBUDGETAND MEASUREMENTMODELLING A typical error for the pseudorangemeasur~nent is at the meter level while, for the phase, i t

is assumedto be millimetric (5 x 10-s cycle).

At present, the limiting factor of the GPS technique

(as far as the 1 ppm application is concerned) is not the measurement in i t s e l f .

The error due to

the orbits, to the insufficiently accurate propagation models and to the internal clocks are larger. If the 0.1 ppm application is required all these error factors must be removed at the sametime, then the problem of the measurementuncertainties becomesmore crucial. 5.

REVIEWOF EXISTING RECEIVERS Several receivers are already available, either as prototypes or comnercially.

Table 1 lists

only instruments which can be used for sub-metric geodetic positioning and which had been announced at the time of writing.

14

Table 1 Presently available receivers and their characteristics Type

Manufacturer

Model

No. Chan Frequency Mode

No.SAT

Codeless receivers I~AC

Macrometrics

MACROMETER1000

6

L1

SQ and CT

6

MAC2

Western Geophys.CyAmer. Aero Service Div.

MACROMETERII

12

L1/L2

SQ amd CT

6

SERIES

Jet Propulsion Lab.

SERIES

3

L1/L2

SD

n

SERX

Jet Propulsion Lab.

SERIES-X

3

LI/L2

SD

n

IST2002

ISTAC, Inc.

Model 2002

1(3)

L1(/L2)

SD

35

GLS1991

GEO/HYDRO

GPS Land Surveyor Model 1991

3

L1/L2

SD

n

C/A and P code STI5010

Stanford Telecom. Inc.

STI 5010

2

LI/L2

CC and LS

i

TI4100

Texas Instruments

TI 4100

i

L1/L2

CC and MX

4

1

L1

CC and FS

4

L1

CC

4 9

C/A code 400(IA

Trimble

Model 4000 A

4000S

Trimble

Model 4000 S

WMI01

Wild-Magnavox

WM101

4

L1(/L2)

CC and FS

ll~5S

SERCEL

TR5S

5

L1

CC and CT

5

LGSS

Litton Aero Products

Litton GPS Survey Set LGSS

L1

CC

8

SEL

Standard Electrik Lorenz "low cost"

L1

CC and FS

4

5.1

1

Codeless receivers The measurement technique is basically the signal squaring technique mentioned in Section 3.2.

Several receivers are available on the market and as knowledgeof the codes is not needed, they are usually double-frequency devices.

On the other hand, they need to be synchronized before every

experiment and the orbits need to be known from an external source. 5.2 C/A and P code receivers ll~e principle of these receivers is based on the code correlation.

Both codes (C/A and P) must

be known. 5.3 C/A code receivers The technique is the same but only the C/A code (non-restrictive) needs to be known. As only one frequency is available the ionospheric d e l ~ must be removedby the use of models.

15

6. CONCLUSION Even though the GPS system is not yet f u l l y operational, i t is widely used by the geodetic community.

Many different receivers are actually available on the market.

measurement techniques are very different. cient for the 1 ppm applications.

Their capabilities and

At present, the precision of these instruments is suffi-

In the future, i f significant improvementsare realized in propo-

gation medels and in orbit determinations, this precision may be improved.

REFERENCES i)

J.D. Bossler and C.W. Challstrom, GPS instrumentation and federal policy, Proceedingsof the 1st Int. S~p. on Precise Positioning with GPS, Rockville, April 15-19, 1985, NOAN(1985).

2)

J. Hannah, The Global Positioning System - The positioning tool for the future, New Zealand Surveyor (1985), pp. 268-281.

3) P. Hartl, W. Scholler and K.H. Thiel, Review paper: GPS technology and methodology for geodetic applications, Proceedings of the Joint Meeting of Study groups 5B and 5C, Munich, July 1-3, 1985, Univ. der BundeswehrMunich, Scheiltenreihe, Munich (1985)o 4) B.W. Remondi, GPS Geodetic Receivers - A status update report, American Congress of Surveying and Mapping. 5) L.A. Buennagel et a l . , SERIES project:

Final report on research and development phase, 1979 to

1983. JPL Publ. 84-16, NASA/UPLPasadena (March 1984).

PRECISE POSITIONING WITH THE GPS

C. C. Goad National Geodetic Survey C h a r t i n g and G e o d e t i c S e r v i c e s N a t i o n a l Ocean S e r v i c e , R o c k v i l l e , Maryland

NOAA

20852

U.S.A.

ABSTRACT A r e v i e w o f c u r r e n t measurement r e d u c t i o n t e c h n i q u e s f o r p r e c i s e positioning

using carrier

Positioning

System s a t e l l i t e s

each a r e d i s c u s s e d . slips,

is presented.

station

slips

are given.

of

o v e r s h o r t t i m e g a p s when d a t a from a It is then argued that

w i l l be u s e f u l f o r c y c l e s l i p

detection

Optimal estimation

this

algorithm

in precise positioning

of a moving p l a t f o r m a s w e l l as i n a s t a t i c mode.

The q u a l i t i e s

The main c a u s e of d a t a p r o b l e m s , c y c l e

i s a l s o g i v e n a l o n g w i t h an e x a m p l e , and t h e n a p o s s i b l e

algorithm to correct single

p h a s e o b s e r v a b l e s from t h e G l o b a l

relative

positioning

i n t h e p r e s e n c e of unmodeled e r r o r s

is discussed.

i.

INTRODUCTION

The use of the Global Positionlng System for precise surveys is now a reality. Several government, university, and commercial agencies are using the various carrier phase measuring receivers to obtain data which are processed interferometrically to provide base llne vector estimates which far exceed minimum flrst-order standards.

The p r o c e s s i n g of t h e s e d a t a can be s e p a r a t e d i n t o two g e n e r i c c l a s s e s

depending

on

the handling of cycle slips which can be caused by problems inside the tracking hardware and by environmental causes.

I call these two categories batch and interactive techniques.

The batch techniques do not require decisions to be made by an analyst during the actual data,reduction.

They provide fast and accurate results but do not take advantage

integer nature of the ambiguities (more about ambiguities later). do not yield the most precise estimates of base line vectors.

of the

These batch techniques

Interactive techniques

do

try to take advantage of the integer nature of the ambiguities, but because of the human intervention these solutlons require far more time and thus more expense. may change, however, as processing schemes and algorithms are improved. automation is desired from a cost per base line point of view.

This situation Obviously,

more

Also, such improvements in

the static relative positioning processing might impact moving-platform applications as well.

18

The basic phase observable will be presented in Section I,

Section 2 will discuss the

problem of cycle slips in phase data and how to fix them while Section 3 will address cycle slip filter

p ro blem u s i n g d a t a o n l y from a s J n g ] e s t a t i o n .

single station

In S e c t i o n 4 one t e c h n i q u e

d a t a w i l l be p r e s e n t e d w i t h p o s s i b l e a p p l i c a t i o n

d i s c u s s e d i n S e c t i o n 5,

to

t o moving p l a t f o r m s

In S e c t i o n 6, t h e problem of o p t i m a l e s t i m a t i o n

o f unmodeled s y s t e m a t i c e r r o r s

the

in the

presence

is discussed.

2. THE PHASE OBSERVABLE

The b a s i c c a r r i e r transmitted

phase observable is the phase difference

at the satellite

and a l o c a l r e c e i v e r ' s

oscillator.

be t w e e n t h e c a r r i e r

signal

Mathematically

this

is

expressed as

¢~(t R) = ¢i{t t) - Cj(t R) where s u p e r s c r i p t s receiver;

refer

to a particular

and t h e a r g u m e n t i s e i t h e r

E q u a t i o n (1) i s now r e w r i t t e n

(I)

satellite;

subscripts

the transmit time, t t,

i n a more u s a b l e form.

The r i g h t

t h e t i m e s a r e e x p r e s s e d i n t e r m s of t h e r e c e i v e d t i m e , t R. to receiver

i s g i v e n by ~ ( t t ) .

Therefore the travel

where c i s t h e s p e e d of l i g h t . f,

gq.

(1) l s now r e w r i t t e n

refer

to a particular

or t h e r e c e i v e d t i m e ,

s i d e i s w r i t t e n so t h a t

The d i s t a n c e

from t h e s a t e l l i t e

t i m e i n a vacuum i s g i v e n by

Assuming t h a t t h e o s c i l l a t o r s

ground

t R.

4

(tt/c)

run a t a c o n s t a n t f r e q u e n c y ,

as:

¢~(tR) = ¢iCtR - p~(tt)/c) - Cj(tR) (2)

i t) - ~j(t R) + Nj. i = @l(tR) - f/c pj(t

Notice that an integer N~ ] has been inserted in Eq. (2) to acknowledge that @~(tl) the phase observable at the first

measurement t i m e ,

i s d e t e r m i n e d modulo one c y c l e .

I f lock i s

maintained thereafter, j and s a t e l l i t e as t h e e f f e c t s

1.

t h e same N~ w i l l a p p e a r i n e v e r y p h a s e o b s e r v a b l e b e t w e e n r e c e i v e r 3 Some s m a l l t e r m s have been i g n o r e d i n t h e d e r i v a t i o n of Eq. (2) such

of r e l a t i v i t y ,

i o n o p s h e r i c and t r o p o s p h e r i c

Some o f t h e s e w i l l be a d d r e s s e d l a t e r . or e x p l i c i t l y

Since all

a f u n c t i o n of t h e r e c e i v e d t i m e t a g ,

with the understanding that hereafter

refraction

and t i m e t a g e r r o r s .

t i m e a r g u m e n t s In Eq, (2) a r e i m p l i c i t l y tR, t h e R d e s i g n a t i o n

the time tag r e f e r s

to the r e c e i p t

will time,

he d r o p p e d Thus,

(2) i s r e w r i t t e n

~(t k) = ¢i(t k) - f/c p~- ~j(t k) + Ni. 3 where k r e f e r s

to the k - t h sample time.

(3)

Eq.

19

1.1 Differenced

data t_~p~

Geodesists recovery. function

of the satellite

difference

term,

differencing thus

are mainly interested

This information

differencing clock)

and receiver

~-(tk)

these

simplify

-

~(tk),

reduction

For e x a m p l e ,

phase difference

these

phase observables for base line vector i P i i n Eq. (3) w h i c h i s a

o n l y from t h e t e r m - f / c locations.

and

phase observables

the

the

The o t h e r

initial

o v e r a common s a t e l l i t e

these

9 and t h e l o c a l

non-geodetic

first

to eliminate

1 and 2, a t

the

{3) a r e t h e c l o c k i g i. Selectively

ambiguity,

Goad a n d R e m o n d i 1)

two s t a t i o n s ,

between satellite

t e r m s i n Eq.

integer

i s one way t o e l i m i n a t e

algorithm.

phase observables

term.

in using

is extracted

(almost)

the satellite instant,

oscillators.

terms

and

showed r e s u l t s

Using

from

phase (or

t k, m e a s u r e t h e Eq.

(3),

this

is

expressed as 9

~92, l ( t k ) = ¢1 ( t k ) Equation 1 and 2 a t

- ~2(tk)

(4) i s now r e f e r r e d

the k-th

the satellite

+ f/c

to as the single

epoch to satellite

phase value

between ground receivers The c l o c k d i f f e r e n c e s

9

$9(tk)

9.

initial

difference

difference

~l(tk)

be t h e same s h o u l d a n o t h e r

difference

processing.

The t e r m i n v o l v i n g and s a t e l l i t e

positions.

contributors

possesses

values

rather

a structure

easily 2).

thus these results

using

and no s p e c i a l

(for

differences

from

values

differences.

one epoch

Furthermore, satellite

to

the

the receiver clock

participate

t o two d i f f e r e n t

4 = ¢2,1(tk)

example,

satellites.

9 - ~2,1(tk)

basis.

treatment will

be c o r r e l a t e d .

o f two s i n g l e

4 and 9 a t e p o c h k a r e c h o s e n .

4,9 ¢2,1(tk)

clock)

in

the

The l e a s t - s q u a r e s n o r n ~ l

no p h a s e o b s e r ~ ] a b l e s a p p e a r more t h a n once,

the phase observables

in general,

double differences

from two r e c e i v e r s satellites

are considered,

by taking

(or

and d i s t a n c e

one t o s o l v e f o r t h e epoch r e c e i v e r c l o c k d i f f e r e n c e

For a given base line,

measurements will,

are obtained

basis.

on an e p o c h - b y - e p o c h

which allows

the measurements are uncorrelated

t h a n one b a s e l i n e

in phase

stations

does not contain

9 9 p ] - O2 c a n be c a l c u l a t e d b a s e d on a k n o w l e d g e o f t h e s t a t i o n 9-9 N2-N 1 i s t h e same f o r a l l epochs. Thus t h e s e two terms a r e "common"

and n e e d n o t be c o n s i d e r e d

matrix

thus

the

between

observable

h a v e a known r e l a t i o n s h i p

on an e p o c h - b y - e p o c h

will

observable this

differences,

n e x t and t h u s m u s t be e s t i m a t e d - ¢2(tk)

(4)

differences

ambiguity

do n o t n e c e s s a r i l y

9

+ N1 - N2'

As a n t i c i p a t e d ,

b u t now c o n t a i n s

1 and 2,

9

(pl-P2)

Then

are required. appear

more than

Some i n v e s t i g a t o r s

Goad a n d Remondl 1J," B o c k e t difference For

observables

an example,

But when more once, have

al.3))."

and given These

a t a common e p o c h

receivers

1 and

2 and

20

9

Now a l l four

satellite

ranges

observables,

the double

differences this

clock

comblnation

the

are

Iess

whether

differences,

This

differences has the

the

than a step

as

Losses

characteristic,

editor

especially

results

are

not

required.

between

single-,

double-,

In

Eq.

only base base

line

lines

indeed are

(5)

nature

the

very

double

to

orbit

values

the recovery

can be determined,

of base

line

more receivers

collecting

of the data

One d a t a

reduction

this data

at

the

s c h e m e when a l l

scheme whose least-squares

how m a n y r e c e i v e r s

undifferenced

phases. data,

integer

It

are is

in time.

to compute in data rather

outliers)

are

when the

easily desirable

most

precise

were given.

a very

nature

desirable

quality--the

removed in the differencing

that

will

of processing the

estimates

deteriorate

ionospheric

data

process, from

many

of the biases are

as the base line lengths

and tropospheric

these

biases

of solution

should

phase

collect

same t i m e , are

weight

tracking

data

vector

refraction

data

obvious the

point

keep

their

strengthens

become

more

l o c a t i o n s w i l l become

of view,

but

with

3 or

together.

will

remain

nature

uncorrelated

(diagonal)

is one which analyzes

a phase-o~ly

integer

receivers

from several

simultaneously that

to

This greatly

c l o s u r e i s an i m p o r t a n t c h a r a c t e r i s t i c

processed

matrix

be forced

parameters.

from an economical

b u t how d o e s o n e e x p l o i t

w l l l now b e a n s w e r e d .

are

As c a r r i e r

collecting quite

by

solutions,

then

data

differences

is an extremely

or

The r e s u l t s

such as

desirable

This

to possess

30 km a r e

to simultaneously

is

(or

by Eq. given

observables.

triple-difference

spikes

clock

process

given

model

difference

double

initially

terms

say,

the

in the

results

seen

from the set

components.

the opportunity Not only

uncorrelated

contaminate

as

difference

the number of ranges

scheme.

data

remain.

effects

To o p t i m a l l y

b y Goad a n d R e m o n d i 1) w h e r e a c t u a l c o m p a r i s o n s

clock

than,

This

and be excluded

is

all

effects

less

integers.

more routine.

no matter

bias are

errors

If the i n t e g e r s

available,

Since

and non-cancelling

and satellite

integer

difference

and integer

close

the

This was discussed

(no e p o c h - t o - e p o c h

double

These

detection at

to handle

of the double

a spike

and triple-differencing

whose lengths

increased

or outlier

side contains

single-difference

correlated.

two successive

cause

the

a more complex model

nature

processing.

when l o o k i n g

of the bias.

are

and increasing

of lock will

in double-difference

removed by an automatic

integer

the bias

The r i g h t

Unlike

simpler

by differencing

of removing

the model to eight.

sets

to go with

or to choose

are generated

effect

removed.

complicated

Eq. (5) and t h e n be f o r c e d t o model t h e c o r r e l a t e d Triple

are

ambiguitltes.

but the data

one must decide

single

differences

of four

differences

to model or estimate),

Information

(4) for

and receiver

and a linear

(5)

algorithm

of the biases?

will

the

consider

This question

21

1 . 2 The b a s e s t a t i o n

- base satellite

The b a s e s t a t i o n

- base satellite

that

a t t h e k - t h epoch a s u f f i c i e n t

a double difference. 4 and 9.

Again l e t

Then from Eq. ( 3 ) ,

¢~(tk)

Again generate

concep_t

c o n c e p t can b e s t

us assume there

= *4(tk)

- f/c

P~ - * l ( t k )

¢ ~ ( t k) = ¢ 9 ( t k) - f / c

P~ - ¢ l ( t k )

+ N~

¢~(tk)

P~ - 0 2 ( t k )

+ N~,

~ = .~ + , l , j

= ¢9(tk)

- f/c

be emphasized

that

Arbitrarily,

station

Notice

that

hfter~ubstitutlng

That

Is,

the

1 and s a t e l l i t e

no t i m e

argument

for the N's,

4 will

be c h o s e n a s

the

base

station

and base

~

9

is

included

in the K variable.

This is intentional.

one g e t s

(7)

_

K varlable

consists

of the

linear

c o m b i n a t i o n of t h e I n i t i a l

I t r e m a i n s t h e same t h r o u g h o u t an o b s e r v a t i o n

This Is the essence of the algorithm. the fourth

(8)

O~ + N ~ ( t k ) .

Then l e t u s d e f i n e

9

ambiguities.

f o u r p h a s e m e a s u r e m e n t s c o u l d be u s e d t o

Now t h e above r e l a t i o n s a r e r e w r i t t e n s u b s t i t u t i n g

i(t k)

f/c

respectively.

these

One gets

- - f/c

=-

1 and 2, and two s a t e l l i t e s ,

- 0 2 ( t k) + N~

observable.

in each e.ua~ion.

¢~(tk)

Suppose

+ N~

~

should

by an example.

the measurements are given as follows:

one d o u b l e - d i f f e r e n c e

satellite,

a r e two s t a t i o n s ,

¢ ~ ( t k) = 0 4 ( t k ) - f / c

it

be p r e s e n t e d

number of p h a s e m e a s u r e m e n t s a r e c o l l e c t e d to c o n s t i t u t e

measurement actually

session

Now t h e f o u r m e a s u r e m e n t s a r e

changed.

Integer

if lock is maintained. rewritten

with

only

22

4 @l(tk) = - f/c Pl4 + ~ ( t k ) 4 4 ~2(tk) = - f/c P2

$~(tk) = -

+

N~(tk)

p~ + N~(tk)

f/c

(B)

{~(tk) = - f/c P& + K 9 2 + ~ ( t k ) + ~ ( t k ) - ~1(tk) 9 The key idea is that ~2(tk)

is the only phase observable which has neither u base

station subscript nor a base satellite superscript. with the K formulation. if double

differences

When this occurs,

the N is replaced

The K values are indeed the same integer biases one would were generated.

But this

obtain

formulation allows the data to remain

uncorrelated and thus simplify the overall process, no mutter how many stations collect data simultaneously.

Some base station and base satellite must be chosen.

This selection

is

not at all crucial to the data reduction.

Now the technique can be stated formally. satellite.

(I) Choose the base station

(2) When a phase measurement is encountered,

if it involves

and the base

either

the base

satellite or the base station, then the mathematical formulation free of the K's is used. Otherwise the K formulation given in the last equation of (8) is used. of N must be estimated. all epochs,

Epoch-to-epoch values

The station posJtlons and K variables are common contributors

Thls formulation also allows for a very particular

structure

to

of the set of

least-squares normal equatlons 2) .

1.3

Ionospherlc

At radio inversely in the

refraction

frequencies

proportionally T h u s Eq.

at

two f r e q u e n c i e s ,

154 t i m e s

ionospheric

of the

square

ionopshere

of the

(3) m u s t be a u g m e n t e d

the fundamental

MHz w h i c h i s e x a c t l y the

to the

phase,

transmit

for

the effect

L 1 a n d L 2. P-code

120 t i m e s

chipping

the P-code

on g r o u p v e l o c i t y

frequency. to account

The L I f r e q u e n c y rate

A corresponding for is

o f 1 0 . 2 3 MHz.

rate,

Equation

(3)

this

effect.

is

to

retard

it

advance is created The GPS s a t e l l i t e s

1 5 7 5 . 4 2 MHz w h i c h i s The L 2 f r e q u e n c y

is

i s now a u g m e n t e d

to

exactly 1227.6 account

effect,

#;(tk) = ~i(tk) - f/c p~-~j(tk)

+ Ni~ +3

(9)

A/f.

Equation (9) must be written twice, once each for the L I and L 2 frequencies. time we recognize that both the station and satellite L 2 phase values should

At this

be equal

to

the corresponding L I phase values scaled by the frequency ratio f2/fl since they are both based on the same fundamental rate of 10.23 MHz.

23

i

"

Cj(tk)L1 = @l(tk) - Cj(tk) - fl/C P~ + N~{L1) + A/f I t

(io)

Cj(tk)L2 = f2/fl [¢i(t k) - Cj(tk) - fl/C P~] + N~(L 2) + A/f 2

We desire to eliminate the ionospheric effect by combining the two observations above i n a l i n e a r c o m b i n a t i o n as f o l l o w s :

~(tk)

i i . = 51 ~j(tk)Ll + 5 2 S](tk)L2

Choosing 2

=

~2

2

fl/(f 1

=-

_

f~) (ii)

flf2/(f~ - f~)

yields the following "corrected" observation:

¢ij(tk) = ¢i(t k) - f l / c

pij- Cj(tk)+ alN~.(L 1) + C~2 NijtL2).

(12)

Equation (12) now looks exactly as the original phase equation given in (3) except the integer bias term is replaced wlth the linear combination of the L 1 and L 2 integer biases. Thus when ionospherically corrected data are processed, no programming changes linearly combine the dual frequency data into one measurement are required.

except to

Once combined,

the data reduction continues without change.

2. DATA PROBLEMS

The most prevalent problem associated with GPS carrier phase observation is the occurrence of cpcle slips.

This occurs when the satellite slgnal is lost and reacqulsltlon Is required.

Upon reacqulrlng the carrier,

the integer ambiguity almost always takes on a different value

which appears in the data as an abrupt jump in the carrier phase history. be ~ccommodated in the analysis.

It can be manually fixed.

Additional

This jump must differences

can

isolate the Jump as is done in triple differencing, or, additional unknowns can be introduced whenever a jump occurs.

As is usually the case, the technique which yields the most precise

base line determination requires the most labor, and the most automatic procedure does not make use of the integer nature of the ambiguity.

One possible exception will be the analysis

of data over short base lines where the integer values of the biases are easily obtained. For these, cycle slips can be accommodated automatically by simply cycle count.

ignoring

By using only the fractional part of the measurement residuals

the integer (rounded

to

zero), phase measurement processing with biases set to integers should proceed easily after a good estimate has been made available from some other technique such as triple differencing.

24

The c y c l e s l i p s

can o c c u r on e i t h e r

a t t h e same t i m e s o r d i f f e r e n t

times.

t h e L 1 c h a n n e l or t h e L2 c h a n n e l .

They can o c c u r

The s o u r c e s can be i n t h e h a r d w a r e or due t o s i g n a l

blockages such as trees, bulldlngs, mountains, etc.

Large cycle s l i p s can be i d e n t i f i e d when looking a t single s t a t i o n data, but to date these Jumps usually cannot be repaired to better than a cycle or two except when the data are combined interferometrically.

The optimum solution

is to fix the cycle slips on a

statlon-by-statlon basis (unless the slip is due to an abnormally

large gap in the data

history), and a possible solution to this problem will be discussed next.

S. SINGLE SITE CYCLE FIXING

A possible availability

technique for repairing

cycle slips

of d u a l f r e q u e n c y d a t a .

terms (receiver for the factor

phase, f2/fl.

terms are related

satellite

We s t a r t

at a single

w i t h Eq.

station

is

b a s e d on t h e

(10) and n o t e t h a t t h e c l o c k - b a s e d

p h a s e , and r a n g e t e r m ) a r e t h e same f o r L 1 and L 2 e x c e p t

The L1 and L2 i n t e g e r a m b i g u i t i e s a r e n o t r e l a t e d , and t he ionospheric

by f l / f 2 .

Thus when t h e l i n e a r

i ~k = * j ( t k ) L 1 _ f l / f 2

combination

i Cj(tk)L2

(13)

is formed, all the clock terms drop out and then

~k = N~(LI) - fl/f2 N~(L2) + (l-f12/f2 2)"

Since the integers

N1 s h o u l d r e m a i n t h e same, e p o c h - t o - e p o c h c h a n g e s in t h i s

a r e o n l y due t o t h e i o n o s p h e r e .

The term i n v o l v i n g A r e p r e s e n t s

t h e L1 p h a s e measurement ( w i t h o p p o s i t e s i g n ) . phase history

(14)

while collecting

F i g u r e 1 was g e n e r a t e d from a dual frequency

d a t a i n A l a s k a i n 1984.

s e c o n d s , and t h e d a t a were c o l l e c t e d

The e p o c h s w e r e s e p a r a t e d

a t a p p r o x i m a t e l y 9 : 0 0 a.m. l o c a l t i m e .

was c h o s e n f o r I t s o b v i o u s c y c l e s l i p s

a t e p o c h s 6, 7, and 8.

data is given in Table 1 along with the differenced removed b a s e d on t h e p s e u d o - r a n g e d e t e r m i n a t i o n

quantity

65% of t h e c o n t r i b u t i o n to

data.

A listing

by 120

This data set

of t h e r a w p h a s e

The D o p p l e r c o n t r i b u t i o n has been

of a b s o l u t e

position.

What i s n o t i c e d

immediately is the difference

i n dynamic r a n g e between t h e D o p p l e r c o r r e c t e d L1 and L2 phase

v a l u e s and d i f f e r e n c e d

The a v e r a g e c h a n g e from epoch t o epoch of the d i f f e r e n c e d data

data.

is 0.I CyCles while it is 17 cycles for L 1 and 13 cycles for L 2 (disregarding the cycle slips). 12O s.

Also remember that these epoch-to-epoch differences are over a time interval of What is proposed here is to track the difference signal and use it to detect and

fix short term cycle slips. is -1.28.

The L 1 coefficient in Eq. (13) is 1.0 and the L 2 coefficient

Thus an isolated cycle slip would cause the 6k to change by either l.O or 1.28.

If the same sllp occurred on both channels, the uncertainty that one could predict.

the 6 k would change by 0.28, much greater than

Also, if data closer in time were collected,

the

25

epoch-to-epoch changes would be reduced proportionally to the decreased interval, cycle sllp changes would remain at the same large values.

Another advantage

the difference quantity is being able to follow the contribution

but the

to tracking

of the ionosphere

(see

Bender and Larden 4) .

8.80E+05

Z.00E+05,

--J

m 1.50E+05

~ I.OOE+O~ 5.00E+04

O.00E+00

x

I

0

Fig.

4.

I

zo

~o

I

,

I

I

30 ~ 50 EPOCH N U M B E R

I

eo

70

Raw Phase Difference [L !- (FI/F2)L2]

1

n~TZRISS~Zg.NIqUZS A Kalman filter Is easily implemented to follow the signals and the difference quantity.

I have chosen to represent the state transitions of the signals as a thlrd-order (quadratic polynomial).

This model is linear in the polynomial

coefficients

process

with state

transitions computed as follows:

~/2nt21 x'(tnq

!,.o.,q (tn+lq (tn+l~

0

X' (tn)~

(zs)

OP

X(tn+ 1) = O(At) X ' ( t n) where the prime denotes previously updated values.

(16) The c o v a r i a n c e of the p r e d i c t i o n

is

26

Table

1

LI/L 2 phase differences a nd Doppler-corrected phase values.

Epoch

Difference (cycle)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2O 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4o 41 42 43 44 45 46 47 48 49 5O

L1 Phase (cycle)

.14 .99 .11 -17.00 .11 -34.49 .10 -51,47 .10 -67.58 39129.58 39045.94 261022.78 197508,41 79421.22 79303.44 -.51 -134.33 -.51 -149.09 -.52 -163.76 -.57 -178.04 -.56 -192.07 -.56 -206.28 -.59 -220.72 -.58 -235.21 -.60 -249.48 -.63 -264.19 -.65 -279.17 -.67 -294.03 -.68 -309.23 -.74 -324.26 -.75 -339.50 -.83 -355.06 -.85 -371.08 -.86 -387.03 -.92 -403.58 -.96 -420.36 -1,03 -437.15 -1.04 -454,38 -1.17 -472.02 -1,15 -489,78 -1.18 -507,66 -1.24 -526.18 -1.25 -543.86 -1.29 -561,57 -1.38 -579.04 -1.45 -596.62 -1.54 -614.30 -1.62 -631.96 -1.72 -649.58 -1.79 -667.15 -1.90 -684,36 -1.95 -701.65 -2.01 -718.30 -2.16 -735.47 -2.27 -752.41 -2.34 -769.38 -2,42 -786.50 -2.55 -803,56

L2 Phase (cycle) .66 -13.33 -26.96 -40,19 -52.74 -65.17 -49491.71 91,77 -104.27 -115.77 -127.19 -138.28 -149.22 -160.30 -171.53 -182.82 -193.93 -205.36 -217.02 -228.58 -240.42 -252,09 -363.96 -276.02 -288.49 -300.91 -313.76 -326.80 -339.83 -353,25 -366.93 -380.74 -394,65 -409.94 -422.81 -436.58 -450.12 -463.76 -477.47 -491.17 -504.82 -518.46 -531.78 -545.21 -558.15 -571.40 -584.52 -597.68 -610.97 -624.16

27

calculated as

[Pn+1] = ~(~t)

[Pn' ]

cT(At) +

[Q]

(17)

where the matrix [Q] represents the uncertainty in the predictions due to errors or omissions in the transition matrix.

The [Q] matrix can take on a rather dynamic behavior whereby the

magnitude of the elements can be chosen to be small when smooth behavior and large when the quadratic model cannot adequately

model

Is encountered,

the state transitions.

The

measurements (L1, 52 phase; phase differences) are then used to update the predicted quantities by the standard Kalman formulae.

Finally the eovariance matrices are updated.

Again, the vectors X represent the values of measured carrier phase dlfferencebetween the satellite transmitted phase fronts and the internal receiver oscillator values or the differenced quantities ~(LI) - (fl/f2) ¢(L2).

The latter values

are the least changing

quantltles (see Figs. 2, 3, and 4) and are used for deciding when lock was not maintained. if a loss-of-lock occurs,

the differenced quantity and the L 2 prediction are used to obtain

integer corrections to the L I and L 2 phase measurements.

And the process continues.

0 -1

-Z

m-3

-5 -e -7 i

I

i

W

I

I

I

10

20

30

40

50

80

70

EPOCH N U M B E R Fig.

2

Filtered Phase Differenoe [L I- (FI/F2) L2]

28

-ZOO

-,°° I -600 -60¢ -100(] -1200 I

I

0

0

I

I

I

I,,

I

I

ZO

30

40

50

60

70

EPOCH NUMBER Fig.

3

Filtered Phase Data (LI) *

-200

-400

-a00

-800

-t000

I

I

|

!

0

10

I~O

80

I ,

40

I

i

I

50

60

70

EPOCH NUMBER

Fig.

4

Filtered Phase Data (L2)*

* Doppler contribut~_~ has been r e ~ e d .

,,

29

Experience with several data sets has revealed that the correcfions are good to 1 cycle or better.

This alone justified preprocessing the data prior to interferometrlc analysis.

However, should a data sampling interval much smaller than 120 seconds be chosen, then the predlctor/filter performance would improve.

This does not imply that the increased sampling

must be used in the interferometric processing step.

The single site, p r e p r o c e s s e d

could be sampled to reduce the computational workload in s u b s e q u e n t

data

data r e d u c t i o n s .

A

typical sampling rate might be from 3 to 30 seconds.

5. APPLICATION TO DYNAMIC PROBLEMS

Returning satellite receiver

t o Eq.

range

(14),

value

was i n m o t i o n

it

is again

emphasized

has been removed in the (say

in an airplane),

that

the

differencing the

term containing process.

term would not

appear

gradually changing signal as given in Fig. 2 would be expected,

the

receiver

Obviously, and

However,

thus

to

if the

the same

the a b i l i t y

predict the L 2 phase values will be degraded by the arbitrary motion of the platform.

to But

still, thls Is one very important quantity that can be used to measure whether or not lock has been maintained.

6.

UNMODELED ERRORS

No matter which technique one chooses, it is clear that unmodeled errors are a l w a y s present and will contaminate the data reductions to some ( h o p e f u l l y small)

level.

combining data from more than two stations, it is also clear that c a n c e l l a t i o n such as e n v i r o n m e n t a l

unmodeled

errors,

stations.

Expressed mathematically we say that the measurement errors are correlated based

on station separation.

effects,

When

of these

will be more complete for the closer

Given same gain matrix, K, the linear estimate can be written

~Ky where x is the vector of estimated parameter values, y is the vector of phase measurements. The key idea here is to realize that the vector y is composed of more than just parameters x and noise e.

Let us express y as

y =Ax

+ B z + e

where the design matrices A and B map the effects of the parameters to be estimated, x, and the unadjusted parameters (but which have error), z respectively. where the W matrix is the inverse of the "noise" covariance matrix.

Usually K = (ATWA)-IATw But in this case, the

properly modeled covarlance matrix consists of both noise and contribution of the unadjusted errors.

The proper weight matrix should be modeled as

30

W -I = B coY(z) B T + coy(e)

w h e r e C o v ( e ) = E ( e e T ) ', Coy(z) = E(zzT),

a n d t h e mean v a l u e s

o f z a n d e a r e a s s u m e d t o be z e r o .

This is almost identical to the approach used in collocation 5}

By properly modeling

correlation between phase measurements,

line components

the estimation

of base

integer ambiguities could accommodate systematic error contribution.

the

and the

Put a different way,

modeling correlations based on station separation may keep errors present on long base lines from contaminating integer bias recovery on short base lines when simultaneous with many stations' data are attempted.

More rigorous

error

statistics

solutions

should

also be

available from the data reduction.

Most often residual errors due to environmental sources ionosphere are difficult,

such as the troposphere

if not impossible, to model in the matrix B.

or

Thus, some rule is

usually used and is expressed mathematically as a function of distance b e t w e e n

stations.

In this regard, any rule that is used should not destroy the posltive-deflnlte structure of the weight matrix, W 6).

This is identical to the standard collocation implementation.

REFERENCES

1)

C. C. Goad, B. W. RemondJ, Global Positioning System.

2)

Initial relative positioning results using the Bull Geoid,, 58, 193-210,

(1984).

C. C. Goad, Precise relative positioning using GPS carrier phase measurements in a nondifference mode, in Goad, C. C.(ed) Proceedings of the First International Symposium on Precise Positioning with the Global Positioning System, Volume I, National Geodetic Survey, Rockville, Maryland, pp 347-356,

(1985).

3)

Y. Bock, C. C. Counselman,

S. A. Gourevitch,

R. W. King, A. R. Paradis,'

Geodetic Accuracy of the Macrometer TM Model V-IO00.

58, 2 1 1 - 2 2 1 ,

4)

Bull. Good.,

(1984).

P. L. Bender, D. R. Larden, GPS Carrier Phase Ambiguity Resolution Over Long Base lines.

Proceedings of the First International Symposium on

Precise Positioning with Global Positioning System, Rockville, Maryland,

April 15-19, (1985). 5)

H. Moritz, Physical Geodesy, W. H. Freeman and Company, San Francisco, 364 p p . ,

6)

(1967).

S. K. Jordan, Self-consistent statistical models for the gravity anomaly, vertical deflections,

and undulation of the geold, J. Geophys. Res., 77

(20), 3660-3670, (1972).

GPS ORBIT DETERMINATION USING THE DOUBLE DIFFERENCE PHASE OBSERVABLE

G. Beutler Astronomical Institute,

After

University of Berne, Switzerland

discussing

vehicles

orbital

precision

a review of simple but

orbital biases elimination-)

requirements

correct

procedures

orbit

for

GPS

space

determination

is given.

The

(and

1984 Alaska

GPS experiment and the 1985 High Precision Baseline Test (HPBL-Test) are

presented

techniques

as

typical

are mandatory.

examples

The results

where

orbit

determination

indicate that we may expect

accuracies of 1 part in I0" in the near future for GPS-derived regional

and

continental

networks

-- provided

orbit

determination

techniques are combined with the network processing m o d e

i. INTRODUCTION In this article we summarize theory and applications of GPS orbit determination facilities developed at the Astronomical

Institute of the University of Berne,

Switzerland

Most

of the material presented here may be found in Refs. 1-3. Although only partially deployed, ploited

for high precision

interferometric

the Global Positioning System (GPS) is already being ex-

geodetic

surveys

The Macrometer R V-1000,

receiver using only the L1 carrier

demonstrated its capability in differential positioning Ref. 2, in

a first

generation

phase of the GPS signal~ has

clearly

In 1984 the Alaska GPS Experiment

1985 the March HPBL-test (High Precision Baseline-Test,

see e.g. Ref. 3, clearly

demonstrated the potential of measuring networks with GPS on a regional or even continental scale with an accuracy of a few centimeters - provided the orbits are known or are being estimated with an accuracy of 1-2 meters.

These

orbital

precision

requirements

suggest

that

an

extended

network equipped with high-precision receivers will be required. available to civilian users nor will one be available several

regional

civilian

tracking

networks

worldwide

in the immediate future

are presently

tracking

Such a network is not now

being

developed

Plans for

and/or

imple-

32

mented, and these may provide acceptable orbits over certain regions. However, the geodesist using GPS for large-scale,

high-precision

surveys may not be able to assume that the GPS

orbits are known to sufficiently high accuracy. biases along with the parameters

He may have to estimate so-called orbital

in which he is actually

interested,

i.e., the relative

ooordinates of the receivers.

Methods to estimate such orbital biases have been developed

and widely used in the

processing of Transit Doppler data. With the exception of same of the so-called short arc procedures, these algoFit~ns do not describe the orbits by physical parameters. For example, one technique is to paFallely shift and rotate the orbit. ~]hat usually results is non-physical in the sense that the resulting orbit is not a particular solution of the equations of motion of the satellite.

Here we attempt to demonstrate that there is no need

for such non-physical methods in

the determination of GPS satellite orbits. As a matter of fact, it is quite simple to model the orbital biases for these satellites to any precision required in a purely physical way. Because of the orbital characteristics of GPS satellites (almost circular orbits with s~aimajor axes of about 26,500 km), we may assuae the earth's gravity field to be known. As a rigorous modelling of the gravitational serious problem,

attraction due to the sun and the moon

is not a

the only significant external force on the satellites which is not adequ-

ately known a priori is radiation pressure.

The subsequent

discussion

is divided into three sections.

ments for geodetic applications

Orbital precision require-

are discussed in Section 2. Then the necessary theory is

developed in Section 3. In Section 4 we present applications.

2. PRECISION REQUIREMENTS

The accuracy of orbits

needed

to obtain

baseline

estimates

depends mainly on the length of the baseline (see Ref. 4, Eq.

db b

where b

._ d_!r p

of

a certain

accuracy

(84)):

(I)

is the length of the baseline

P

is the range (receiver to satellite)

dr

is the orbit error

db

is the induced baseline error.

For GPS satellites we have approximately

p = 25000 km .

(2)

33

Accepting, for example,

db = 1 ca

(3)

as a maximtF, for the baseline error introduced by the orbit, we obtain the values in Table 1 for the maximunorbit error, dr, allowed.

Table 1

Maximum permissible orbit error, dr, of 1 ca for an accuracy in a baseline of length b.

b (kin)

dr (m)

0.I

2500.0

1.0

250.0

i0.0

25.0

I00.0

2.5

I000.0

•

25

Rather than asstFaing a specific value for a baseline error independent of baseline length, we may wish to talk about a relative error expressed as parts per million

(ppm) of the

baseline length. Table 2 gives the values of dr for a number of relative baseline errors.

Table 2

Maxim%~ permissible orbit error, dr, for a certain relative accuracy in the baseline

db/b (ppm)

dr (m)

5

125.0

1

25.0

0.5

12.5

0. I

2.5

Stmmarizing Tables 1 and 2: a)

The orbital accuracy required for surveying depends highly on the length of the base lines to be measured.

(b)

for

"local surveys"

(diameter

of surveyed region

smaller than

i00 km), an orbital

34

accuracy of 5 m to i0 m will be sufficient inmost cases.

(c)

for regional or even continental surveys, orbital accuracies of 1 m to 4 m will be required.

3. PRINCIPLES OF ORBIT DETERMINATION

The orbit of every satellite is a particular solution of a system of second-order differential equations:

(4)

?(~) = ?(t;~,~ ( ' ) , p,,p~, . . . . pn) where r = r(t) is the position of the satellite in a nonrotating

(with respect to inertial

space), geocentric coordinate system.

~(i), i=I,2, are the first and second time derivatives of ?(t) Pi'

i=i'2 ..... n

("dynamical"

are parameters

defining

the

forces

To define an orbit uniquely (one particular solution of tion has to be supplied.

acting

on

the

satellite

parameters, e.g. parameters describing radiation pressure).

Eq.

(4)), additional informa-

Normally the problem is formulated as an initial value problem:

?(t O) = ~o(kl,k2 . . . . . k6) (5)

?(1)(to) = ?O(1)(kl,k2 . . . . . k6) where k i,

i=1,2 ..... 6 are six parameters uniquely specifying the vectors on the right-hand sides of ~q.

(5).

Possible choices for these parameters are:

Components of vectors r O, rO (I) Osculating orbital elements at time t O.

If we know the right-hand sides of

Eq.

(5) (parameters k i, i=I,2 ..... 6) and the dynamical

parameters Pi' i=1,2 ..... n, the orbit of a satellite is uniquely defined.

I~e therefore may

state: (a)

Orbit

determination

in its usual,

more

problem of determining the six parameters values on the right-hand sides of Eqs.

(b)

restricted,

sense

is

defined

as the

k i, i=1,2 ..... 6 defining the initial

(5).

Orbit determination in its most general sense is the problem of determining the six parameters k i, i=1,2 ..... 6 defining the initial values or boundary values and

35

the dynanical parameters Pi' i=1,2 ..... n.

For the application we have in mind here (modelling the orbits of GPS satellites) most of the dynamical parameters earth's gravity field, however,

in Eq.

(4) may be assumed to be known

gravitational force of the sun and the moon).

(coefficients

of the

For utmost accuracy,

we will have to estimate some of those parameters defining radiation pressure.

To solve an orbit

determination

problem,

at some point one needs observations.

observation may be defined as a value of a function of satellite positions, tions, and nuisance parameters, clocks. More specifically,

An

ground posi-

like clock offsets or clock drifts of satellite and receiver

the following GPS observables have been instrumented in presently

available receivers:

- pseudoranges using the C/A-code - pseudoranges using the P-code - Doppler measurements - phase measurements of carriers.

The carrier phase measurements

are potentially the most powerful measurements

can be made with the greatest precision.

as they

We therefore assume that we will be dealing with

measurements of this kind for orbit determination problems.

Every orbit determination

is actually an orbit improvament process using observations

such as those given above. These observations are nonlinear functions of the satellite position ?(t i) (and possibly of the velocity ?(1)(ti)) at observation time t i. ?(t i) in turn is a nonlinear function of the parameters of the orbit. mination problem is therefore done in two steps: as a linear function of ~(ti),

The linearization of the orbit deter-

(a) the observation has to be approximated

which is straight forward;

and (b) ?(t i) has to be repres-

ented by a linear function of the unknown parameters k i (and possibly some of the pi ). This is done in the following way. Since Eq. mine our orbit iteratively,

(4) is nonlinear, we must linearize it and deter-

where in each iteration step we assume we have a known approxim-

ate orbit ~a(t) at our disposal.

?a (2) =

f(t;{a,?a(1),pl,P2 .....pn )

~a(t0 ) = ~a0(kal,ka2 ..... ka6)

(6)

~a(1)(t0 ) = ?a0(1)(kal,ka2 ..... ka6)

where the kai, i=l,l

. . . . .

6 are approximate values of the unknown parameters.

The true, initially unknown, parameters k i, i=I,2 ..... 6:

orbit is now assumed to be a linear(ized)

function of the

36 B

?(t) = ~a(t) + Z zi(t)(ki-kai) l=i

(7)

where

Ti -zi(t) = aa{a I k=ka

i=i,2 ..... 6

If we deal with the more general problem, parameters appear on the Fight-hand side of

additional terms involving the dynamical

Eq. (7):

(8)

~(t) = ~a(t) + i =Z, z~(t)(k.-k " i a l.) + £ : Z 1 z i ( t ) ( P i - P a i )

wher~

~(t) a~a I aPi =--

,

i=i,2

.....

n

(9)

P=Pa

The functions zi(t), ~ ( t )

are solutions of an initial (or boundary) value problem,

which follow from the primary problem (6) by taking the (total) derivatives of all the equations in (6) with respect to k i, i=2,3 ..... 6 and Pi' i=1'2 ..... n. The resulting set of differential equations are usually called the systems of variational equations.

It is ele-

mentary to show (see Ref. I) that each of the zi or zi* is a particular solution of a linear second order differential equation system. Let us summarize: In every iteration step of the orbit improvement process, we have to solve one system of nonlinear differential equations of type

(6), and,

for each orbit

parameter we have to estimate (for each of the zi' zi *) one linear differential equation system. In our GPS software system we solve the nonlinear systems (Eqs. technique of numerical integration,

(4), (5) with the

the partial derivations ~i' i=I,2 ..... 6 are computed

approximately: (a) the k i, i=i,2 ..... 6 are chosen to be the osculating elements pertaining to the initial epoch to; (b) in formula (9) the approximate orbit ~a is replaced by the Kepler orbit defined by these osculating elements.

It is then possible to compute the par-

tials analytically using the well known formulae of the t~o-body problem. Let us conclude this section by briefly

reviewing our modelling of the force field

acting on GPS-satellites. (In Ref. 2 the influences of the constituents of the force field on the GPS orbits are discussed in some detail): The earth's gravity field is expressed by spherical harn~nics, where we may select the coefficients of either the GEM-10 or the GRIM-3LI model. development after degree and order 8).

(Usually we cut off the

37

Point mass attractions from sun and moon are taken into account. The radiation pressure model we use at present is very simple from the point of view of application:

0nly two parameters define the model, only t~o parameters m a y b e

from observation data.

In Fi B . 1 we define three unit vectors:

estimated

e, is the antenna

axis, e 2 is the solar panels axis and e o is the unit vector pointing to the sun.

Fig. 1

Model of GPS Space Vehicle

Leqend

e i : Antenna axis e~ : Axis of solar panels e o : Unit vector to sun

We model the acceleration a excerted by radiation pressure as

(I0)

The first term is called direct solar radiation pressure (it would be the only one if the space vehicles were spherically symmetric or if the surfaces were totally absorbing the radiation), the second is usually referred to as "y-bias". Several effects may contribute to this latter term. Misalignement of the e~-axis tributes to this term. Fliegel et al. vehicle as another candidate.

(e2 not perpendicular to e o) certainly con-

(see Ref. 5) discuss thermal radiation of the space

38

Parameters

Po and p~ may be estimated

from observation

data.

This means

that

the

partial derivatives of the orbit with respect to these parameters have to be computed too (see

Eqs. (9))~ At present we do that by ntm~erical integration.

Very often it is useful to introduce a priori knowledge for some parameters, adjustment.

If we have e.g. precise ephemerides at our disposal as

into the

a priori orbit source,

it would be a pity not to use the fact, that these orbits are good to - let us say - I0 meters along track and 2 meters in the other directions.

In

our

parameter estimation program

we take a priori information for parameters into account by adding an artificial observation equation for the (improvement of) parameter under consideration.

To this observation equa-

tion a weight

a priori

proportional

to the

inverse of the

(estimated)

variance of the

parameter is assigned.

4. APPLICATIONS

The results for two campaigns are stmlnarized here:

(a) 1984 Alaska GPS Campaign (see Ref. 6). (b) March 1985 High Precision Baseline Test (HPBL-Test)

(Ref. 3).

In the first case we are looking at a typical regional network (~ 2500 km × I000 km) where orbit improvement had to be done as a byproduct of the analysis, lying on the estimation of site coordinate. the estimation of analysis.

the main interest

It became clear that network-processing mode an__dd

"orbital biases" were the two keys to the success of this particular

The situation was quite different for the second analysis.

was the main topic, the quality of receiver coordinates of the orbits estimated from GPS observations

from

Here orbit estimation

"only" served to prove the quality

four fiducial sites whose coordinates

were known from VLBI observations.

4-I Th e 1984 Alaska GPS C~mpaiqn

During the simmer of 1984, the U.S. National Geodetic Survey (NGS) and the Jet Propulsion Laboratory operated Mobile VLBI systems in Alaska and Canada as part of a series of measurements

by

the

National

Aeronautics

and

Space

Administration's

Crustal

Dynamics

Pro~ect. Given the great expense and complex logistics of moving mobile VLBI systems to and around such a distant

location,

NGS decided to measure the same baselines

by GPS in the

hopes that as GPS technology and techniques mature and a history of GPS/VLBI emerge,

comparisons

these measurements may be performed solely by the less expensive GPS systems.

sites occupied by both the mobile VLBI and GPS systems are shown in

Fig,

The

2, the approxim-

ate distance between the sites may be found in Table 3.

During this GPS/VLBI original

GPS

observing

interccnkDarison, plan

was

to

five TI-4100 GPS receivers were available.

occupy

Nome,

Fairbanks,

Sourdough,

Yakataga,

The and

39

Whitehorse for 2 consecutive days. The second scheduled session kept the same receivers at Nome and Fairbanks and moved the other receivers to Sandpoint,

Kodiak and Yellowknife for 2

additional observing

days. The principal observing

session on each day consisted

satellite scenarios.

The first began in early afternoon local time,

of two

included SV's 6, 8, 9

and Ii and lasted almost 3 hours. This was immediately followed by the second scenario which included SV's 4, 9, ii, and 13 and lasted about 1 hour. scheduled for each day approximately 12 hours after

A second observing

session was

the primary one, but it turned out that

these sessions did not produce useful results.

The first observing occupation scheme went as planned and produced data that contained only a few cycle slips.

The second occupation

scheme had to be severely modified

as it

lasted for the next 2 weeks due to reciever failures in Alaska. Coincidentally these data are heavily burdened with cycle slips.

Fig. 2

The 1984-Alaska-GPS-Campaign

Alaska

i

i I I I °6 I

°2

Canada

O8

1

F-~°

2

\

3

\

s

6

Yakataga Falrbanks Kodiak Nome Sandpolnt Sourdough Yellowknlfe

Whitehorse 500

km !

\

40

Table 3

Approximate distances between stations in kilometers

~ation

Fair

Kodiak

849

Nome

Sand

Sour

White

Yaka

Yellow

848

1285

276

789

603

1631

557

671

1044

632

2135

1006

1004

1591

1276

2460

1179

1598

1188

2681

591

329

1575

414

1105

1025

Kodiak Nome Sand Sour White

1510

Yaka

Fair:

Fairbanks

White

:

Sand:

Sandpoint

Yaka

:

Sour:

Sourdough

Yellow :

After the pre-processing

step (see Ref.

Whitehorse

Yakataga Yellowknife

6) roughly 20'000 double difference observa-

tions (8000 from scheme I, 12000 from scheme 2) could be processed by the parameter estimation program.

In the program run combining the two observation

entire Alaska network with eight sites as main result, estimated of which 180 were orbit parameters.

schemes,

thus giving the

a total of 330 parameters had to be

Since we had precise ephemerides at our dis-

posal, very small variances could be associated with the artificial observations constraining the orbit parameters (Io errors of 15 m along track and 3 m in the other directions were used, see Ref. 6 for more information).

Consistency and integrity of both, the GPS and VLBI solutions could be demonstrated by a 7-parameter Helmert transformation between the two final networks.

Table 4 gives residuals

and transformation parameters of this transformation.

4 . 2 T h e March 1985 High-Precision BaselineTest

(HPBL-Test)

This campaign took place from March 29 to April 5 in the United States of/g~erica.

In March/April 1985 space vehicles 4, 6, 8, 9, Ii, 12 and 13 could be observed. daily observation period started approximately at 3 a.m. and lasted till II a.m.

The

Nine TI-

4100 receivers, three AFGL dual frequency instruments and two SERIES-X receivers were distributed over nine sites (see Fig. 3 and Table 5), among them Haystack, Fort Davis, Richmond and Mojave which we used as fiduclal points (coordinates known from VLBl-surveys). Hat Creek and Owens Valley VLBI results are available too.)

(For

41

Table 4

Residuals of a Helmert Transformation between the GPS Solution for the 1984 Alaska GPS Campaign and the VLBI-Solution transformed into ~GS-72 Observations Schemes 1 and 2

VLBI - GPS (m)

Station x

y

z

YAKATAGA

0.146

-0.027

-0.147

FAIRBANKS

0.023

0.098

-0. I06

KODIAK

-0.301

0.039

-0.038

N(3ME

-0.085

0.083

0.115

SANDPOINT

0. 132

-0. 153

0.253

SOURDOUGH

0.157

0.030

-0.173

-0.112

-0.106

0.237

0.041

0.036

-0.140

YELLO[.~
=

0.16

Rotation around x-axis

of Transformation

=

0.18

m +-

0.03

arc sec

Rotation around y-axis

=

-0.01

+-

0.03

arc sec

;°;;7;;;;°-°°'--~" ; _;:;; ;: ;:;; ;;2°° Fig. 3

Receivers and sites in the 1985 HBPL test

~'-- - - -

r

k

/

<

\

~

o .°r~oo-x

/)

42

T~le5

Distances in ~

BIGP

FTDA

BIGP

Mar~

l~SHP~-Test

HAYS

HAT(:

M~

598

2678

2417

2100

1866

2081

1778

1508

3930

484

71

245

3557

3742

2035

AU~

bet~en~itesofthe

3136

FTDA HAYS

MDJA

NS~

RICH

1933

1577

1314

2586

2363

4033

3959

3904

669

2045

416

729

3731

4066

313

3597

3800

3498

2046

HA~ MAMM MOJA

1442

NSWC

Station names: AUSTAustinARL

HATCHat Creek, CA

BIGP Owens Valley Big Pine

M~MMMammoth Lake, Casa 1956, Ca

FTDA Fort Davis

MOJAMo)ave,

HAYS Haystack O b s . , Westford

NST~C Dahlgren, Va

Barstow, Ca

RICH Richmond, Perrine F

Table 6

Observation ~chedule for TI-4100 (TI) and AFUL (MM) receivers

Day

AUST

BIGP

Mar 29

TI

TI

Mar 30

TI

TI

TI,MM

TI

TI,MM

TI

TI,MM TI,MM

Mar 31 Agr 01

TI

FTDA

TI

TI

&or 03

TI

TI

&or 04

TI

TI

TI

&or 05

TI

TI

TI,MM

see

HATC

MAMM

MOJA

NSW~

RICH

TI

TI

TI

TI

TI

TI,MM

TI

TI

TI,MM

TI

TI,MM

TI

TI

TI

TI,MM

TI

TI

TI

TI

M~f TI,MM

~

&or 02

Station names:

HAYS

~

MM MM

TI

TI

TI

TI

MM

TI

TI

TI

TI

TI,~t4

TI,NM

TI

TI

TI

TI

TIAVM

TI,M~f

TI

TI

TI

TI

TI,~

Table 5

Because we did not have a ccmplete set of SERIES-X data at our disposal, only Macrometer and TI-observations

were used for the investigation. The observation

schedule for the

43

two instrument types are given in Table 6. The TI-receivers (which leads to impressive ntm~bers of observations be used per

day),

the Macr~meter

recorded

recorded data each 30 seconds,

(5000 to 12000 double differences could

data at a much

lower

rate

(one epoch per

6

minutes).

The present analysis was severly handicapped by a n ~ b e r of circumstances,

the most im-

portant being that

(a)

No surface weather data had been available to us when we processed the campaign.

(b)

The GESAR software implemented in the TI-4100 receivers was not working properly.

(The

consequence was that one clock offset had to be estimated for each TI-receiver for each session.)

In order to study repeatability of the estimated parameters the campaign was artificially divided into two parts:

Part A: March 29 - A p r i l 1 Part B: April

2 -April 5

Either all observations of part A o_rrall observations of part B were processed. Tests were made with 1 day arcs, 2 day arcs, 4 day arcs. available as a priori information.

"Only" broadcast ephemerides were

Each arc was characterized by six osculating elements and

by t~o radiation pressure paraneters

(see Eq.

(i0)). For the latter parameters the follow-

ing a priori values were used for all satellites Po = 0-94"10-7 m/s2 P2 = 0

m/s 2 .

(Ii)

The perigee was constrained by

o(~) : 0:'i .

(12)

No constraints have been imposed o n the other elements.

Results:

(a)

Radiation Pressure Results

It soon became evident that for short arcs

(I day) it is not necessary (and

does not make sense) to improve the a priori values for Po' P2 given in (II). 2-day arcs only Po could be estimated with a satisfactory reliability. arcs the y-bias ( P2 ) gave a significant contribution.

For

Only for 4-day

44

The estimates for Po were quite consistent:

The results obtained in parts A and B of

the campaign differ from each other by only 1 z - 2 z, a similar statement holds for the different arc-lengths used. The situation is somewhat less satisfactory for p~, the y-bias parameter (see Ref. 3).

(b)

Osculatinq Elements Two f a c t s ace worth being mentioned:

(I)

During

the

quality:

campaign,

the

the positions

only for

broadcast

derived

ephemerides

were

of

a

surprisingly

frc~ our a posteriori orbit estimates

I0 - 25 m frcm the braodcast predictions.

high

differed

(The exception was satellite

4 where differences of up to i00 m occured.)

(2)

Fig.

4

illustrates the "state of the art" of our orbit modelling capabilities:

There a'two-day arc (April 2, 3, solution B21)

is compared with a four day arc

(April 2-5), solution B4~2). In solution B21 for each arc seven parameters were estimated

(six osculating

elements and po),

in solution B42 8 parameters

(six

elements and Po' P2 )- As one may see the differences between the two solutions are of the order of a few meters, which indicates that the orbits should be good to a few parts in i0-' (see Section 2).

Fig. 4

SOLUTION SUN"

B21

8

- SOLUTION

DATE(START)" 1985

4

2

B42 3.0

DIFF. RADIAL(+),ALOHG TRACK(X),OUT OF PLAHE(*)

7' 6 ¸

)<x x

p

P

x

1 4

I

3"

I

a"

IxXx

×

x,

X

x

>~I

X

X ~ l]~X ,X~x

×

x

+

I.÷+ ++*

+

++++

,

":+÷I+

+ I÷

+

*

~

!

i

200

400

i 600

I 8e0

l

1000

+~

4

I I

I 0

x

I

xX

ii,x, x +++

x

Ix

I

120~

'3

I

140e

I

16ee

18eo

'~

aeee

TIME IN MIM FROMSTART p: observation period

45

Table 7 D i f f e r e n c e s in t h e e s t i m a t e d s t a t i o n c o o r d i n a t e s w i t h respect to s o l u t i o n B21 (only s t a t i o n s o t h e r than f i d u c i a l stations)

Station

Difference

d(latitude)

d(longitude)

cm

c~n

d (height) frn

AI0 A21 A42 BI0 B42

-

B21 B21 B21 B21 B21

-2.4 -4.9 -1.6 -2.4 0-I

0.3 -8.0 3.9 0.3 2.3

3.3 -0.6

Owens Valley

AI0 A21 A42 BI0 B42

-

B21 B21 B21 B21 B21

1.2 -1.2 0.8 I.i 0.0

-1.7 -5.1 0.6 -1.4 1.3

3.6 4.6 4.4 3.6 -0.4

Mammoth Lake

AI0 A21 A42 BI0 B42

-

B21 B21 B21 B21 B21

-1.3 -1.7 -1.2 -1.3 0.0

-0.4 -4.8 2.2 -0.4 1.6

-2.3 -0.8 -0.3 -2.3 -0.5

Austin

AI0 A21 A42 BI0 B42

-

B21 B21 B21 B21 B21

2.1 2.3 4.6 -2.1 0.2

5.6 5.3 -0.6 5.6 -2.0

5.8 8.2 12.0 5.8 2.5

Dahlgren (NSWC)

AI0 A21 A42 BI0 B42

-

B21 B21 B21 B21 B21

4.0 2.4 4.9 4.0 -3.1

0.4 0.5 -1.5 -0.4 7.0

0.5 5.4 15.6 0.5 8.7

Hat Creek

Where:

Aik:

- Solution u s i n g all o b s e r v a t i o n s

3.3 6.1

7.7

f r o m M a r c h 29 to A p r i l

i.

- i = arc length in days. - k = number o f r a d i a t i o n p r e s s u r e p a r a m e t e r s

Bik:

- S o l u t i o n u s i n g all o b s e r v a t i o n s

estimated.

f r o m A p r i l 2 t o A p r i l 5.

- i = arc length in days. - k = number of r a d i a t i o n p r e s s u r e p a r a m e t e r s

estimated.

46

(c)

Coordinates

An independent check of the quality of the estimated orbits the coordinates

(as obtained

in the various

solutions)

is possible by comparing

for the stations Hat Creek,

Owens Valley, Mammoth Lake, Austin and Dahlgren, because these reciever locations were estimated without constraints in all program runs. If our orbits would be good to - let us say - i ppm, we would expect differences of up to 4 meters in the different solutions,

since we have baselines of up to 4000 km. The

results in Table 7 show differences that are almost two orders of magnitude smaller! Table 7 also shows, that the differences between the three different orbit types (I-, 2-, 4-day arcs, different radiation pressure models) cause only coordinate changes on the sub-decimeter level, which indicates adequate modelling. We also see that the repeatability "A-solutions minus B-solutions" of magnitude.

is of the sane order

(Remember that solutions A and B are indpendent in the sense that they

have no observations in ccmmon and that the orbits are completely independent from each other.) It is interesting (but not surprising),

that the largest differences occur in height -

hopefully surface weather data and water vapour radiometer data will improve the situation somewhat. e

e

REFERENCES I)

G.

Beutler, D.A. Davidson, R. Langley,

oretical System

and

R. Santerre,

using

Carrier Phase Difference Observations, Dept.

Technical Report No. i09, University of New Brunswick,

2)

G.

Beutler,

W. Gurtner, I. Bauersima,

Positioning with GPS, Vol. i, 99-111,

3)

G.

Beutler,

P. Vanicek,

Practical Aspects of Geodetic Positioning with

D.E. Wells, Some

the of

The-

Global

Positioning

Surveying

Engineering

Canada (1984).

R. Langley, Proc. First Int. Symp.

on

Precise

GPS

Orbits

(1985).

W. Gurtner, M. Rothacher, T. Schildknecht, Determination

of

using Double Difference Carrier Phase Observations from Regional Networks, Fourth Geodetic Symp. on Satellite Positioning,

4)

I.

Bauersima,

Navstar/Global

Satellitenbeobachtungsstation

5)

H.F.

Positioning

Zimmerwald,

Int,

Austin, Texas (1986).

System

(GPS)

If,

Mitteilungen

der

No. I0, University of Berne (1983).

Fliegel, W.A. Feess, W.C. Layton, N.W. Rhodus, Proc- First Int. Symp. on

Precise

Positioning with GPS, Vol. I, 113-119 (1985).

6)

W.

Gurtner,

GPS papers presented by the Astronomical Institute of the

Berne in the Year 1985, Mitteilungen der Satellitenbeobachtungsstation I0, University of Berne (1983).

University Zimmerwald,

of No.

ACCURACY PROBLEMS WHEN COMBINING TERRESTRIAL AND SATELLITE OBSERVATIONS W.M. Welsch I n s t i t u t fur Geod~sie, Universit~t der Bundeswehr MUnchen, D-8014 Neubiberg

ABSTRACT The combination of t e r r e s t r i a l and s a t e l l i t e aided network observations requires hybrid models of two kinds:

a hybrid functional

model and a stochastic model being able to process the heterogene~ ous observations a p p r o p r i a t e l y .

An appropriate combination of

stochastic information means to adjoin proper weights to the v a r i ous types of observations on one hand and on the other hand to separate the relevant internal accuracy from that accuracy which is affected by a number of network external influences.

The f i r s t

task can be overcome by the technique of estimating variance components of the individual observation groups, the second one can be handled by applying S-transformations to covariance matrices.

The

paper deals with problems of the hybrid stochastic model. Examples are given.

i.

INTRODUCTION The combination of c l a s s i c a l l y performed t e r r e s t r i a l

and s a t e l l i t e aided geodetic

network observations rank among the many tasks being attended with the i n t r o d u c t i o n of the new observation techniques.

Doing so, several aspects are important.

functional connection of the heterogeneous observations.

One of them is the

This is the problem of knotting

together d i f f e r e n t reference systems whose i n t e r r e l a t i o n s are not c l e a r l y defined. rule the task is mastered by s i m i l a r i t y transformations.

As a

I t has often been treated, as an

extensive bibliography proves. The stochastic hybrid model, however, has been hardly ever d e a l t w i t h . the f o l l o w i n g representation intends to make a c o n t r i b u t i o n , f i r s t racies of i n d i v i d u a l l y adjusted t e r r e s t r i a l and s a t e l l i t e networks. of two kinds:

Therefore,

by analyzing the accuThese accuracies are

accuracies which are influenced by numerous network external e f f e c t s , and

internal accuracies which are relevant f o r r e l a t i v e accuracy statements w i t h i n the network. The combination of those internal and external accuracies in a hybrid stochastic model has to be treated with care.

This is one problem.

The other problem is harmonizing the

stochastic model by adjoining proper weights to the i n d i v i d u a l and heterogeneous groups of t e r r e s t r i a l and s a t e l l i t e aided observations. Therefore, when comparing or combining t e r r e s t r i a l vations the f i r s t

and s a t e l l i t e aided network obser-

task is to get a deeper i n s i g h t i n t o the accuracies of the i n d i v i d u a l

and heterogeneous observations.

48

2.

THE ACCURACY OF SATELLITE AIDED NETWORK OBSERVATIONS F i r s t i n f o r m a t i o n about the accuracies of s a t e l l i t e

aided network observations can be

taken from the various data processing software packages.

The covariance matrices pro-

vided are, however, affected with a l l kinds of network external e f f e c t s such as i n s t r u mental and propagation medium influences, ephemeris e r r o r s etc.

In many cases, e s p e c i a l l y

in cases of small networks and short observation periods w i t h m u l t i - s t a t i o n instrumentat i o n s , those e f f e c t s exercise a systematic influence by s h i f t i n g , network

r o t a t i n g and scaling the

both the coordinates and the elements of the covariance m a t r i x

CX .

In case

only the mutual or r e l a t i v e p o s i t i o n s of the network points are o f i n t e r e s t , t h i s " e x t e r n a l " covariance m a t r i x

CX does not provide an appropiate measure of the r e l a t i v e or i n -

t e r n a l accuracies of the p o i n t p o s i t i o n s . have to be s p l i t o f f .

To achieve such a measure the external e f f e c t s

According to the " I n t e r n a l Error Theory of Networks" (Ref. I ; Ap-

pendix A) t h i s can be done by t r a n s l a t i o n s and r o t a t i o n s and a change of scale common to a l l network points.

That means the covariance m a t r i x

CX o f the network points provided

by those data processing software packages and representing e x t e r n a l l y affected accuracies has to undergo a s i m i l a r i t y transformation (S-transformation) in order to reveal the " r e a l " i n t e r n a l accuracy

CXi

of the network p o i n t determination.

The diagonal block s t r u c t u r e o f the covariance m a t r i x o f the GPS measurements of the INN VALLEY Network (Appendix C) shows the f o l l o w i n g c h a r a c t e r i s t i c s : sCx :

(subscript

sx = _+ 1.5 cm,

s"

Sy = _+ 1,3 cm,

for satellite).

sh = _+ 2.0 cm

(I)

These values contain external e f f e c t s .

After their

re-

moval by S-transformation the i n t e r n a l accuracies are

S

3.

C± : x

s

x

= + 1.3 cm, -

s

y

= +_ 1.2 cm,

sh = + 1.6 cm

(2)

THE ACCURACY OF TERRESTRIAL NETWORK OBSERVATION RESULTS If,

f o r instance, horizontal d i r e c t i o n s , s p a t i a l distances and zenith angles are i n -

troduced i n t o a f r e e three~imensional network adjustment, freedom of the geodetic datum e x i s t s only with respect to 3 t r a n s l a t i o n s and the r o t a t i o n around the h-axis.

The re-

maining 3 datum d i s p o s i t i o n s are established by the scale o f distances and the o r i e n t a t i o n o f the z e n i t h angles (plumbline or normal o f the e l l i p s o i d ) .

Due to these d i s p o s i t i o n s o f

3 datum c o n s t r a i n t s which are network external e f f e c t s the network is somehow r e s t r a i n e d from being a b s o l u t e l y f r e e . adjusted coordinates, too.

The r e s t r a i n t s are r e f l e c t e d in the covariance m a t r i x of the Thus the covariance m a t r i x provided by a " f r e e " network ad-

justment of t h i s kind provides external accuracies which are2in the case of the t e r r e s t r i al INN VALLEY Network)

49

C

:

T x

s

x

( s u b s c r i p t T"

=

± 0.4 cm,

s

= ± 0.4 cm,

y

for terrestrial).

sh = ± 3.6 cm

(3)

A f t e r S - t r a n s f o r m a t i o n the corresponding r e s u l t s of the

i n t e r n a l accuracies are

Ci

:

T x

S

x

:

± 0.4 cm,

s

y

= ± 0.4 cm,

sh = ± 2.1 cm

(4)

As expected~the S-transformation influences especially the accuracy in heights by rotations around the x- and y-axes since strong datum effects are f i l t e r e d o f f .

4.

COMPARISONOF SATELLITE AND TERRESTRIAL OBSERVATION RESULTS The f i r s t

possibility

to achieve a s s e r t i o n s on accuracies of s a t e l l i t e

observations in comparison to t e r r e s t r i a l

aided network

network observations is the simple comparison of

s p e c i f i c r e s u l t s obtained from the i n d i v i d u a l adjustments of the t e r r e s t r i a l

and s a t e l l i t e

observations, respectively. The coordinates of those adjustments can be compared with each o t h e r a f t e r a 7-parameter s i m i l a r i t y

t r a n s f o r m a t i o n which s p l i t s o f f network e x t e r n a l e f f e c t s caused f o r i n -

stance by a d i f f e r e n t

T_sdX

=

T_sdX T

=

Tx -

geodetic datum.

mRsX

(5)

Idx dy dh 1 coordinate differences to be investigated a f t e r transformation

Tx

centre of g r a v i t y related t e r r e s t r i a l coordinates

Sx

centre of g r a v i t y related GPS coordinates

m

scale factor

R

rotation matrix .

Applied to the example of the INN VALLEY Network the f o l l o w i n g r e s u l t s have been obtained (Ref. 2): The c o o r d i n a t e d i f f e r e n c e s in -3.2 cm to +2.8 cm, in height t i o n is

dh

dx

range from -2.4 cm t o +3.0 cm, in

from -12.8 cm to +12.6 cm.

Sp = ± 2.6 cm , w h i l e the r . m . s , e r r o r in h e i g h t i s

excellent result,

sh

can be questioned, i t

sh = ± 8.0 cm .

Sp

i s an The un-

i s much higher than the r . m . s , value

sh = ± 2.0 cm of the GPS d e r i v e d heights and also higher than heights.

from

i t corresponds q u i t e n i c e l y to what can be d e r i v e d from Eq. 1.

c e r t a i n t y in height restrial

dy

The r . m . s , e r r o r in p o s i -

sh = ± 3.6 cm of the t e r -

The conclusion may be dr~wn t h a t the GPS heights cannot be burdened

with the discrepancy.

I t i s most l i k e l y

t h a t the r e f r a c t i o n i n f l u e n c e d t e r r e s t r i a l

heights are much weaker and not so c o n s i s t e n t ; t h e r e f o r e , they cannot compete with the satellite

aided height o b s e r v a t i o n s .

50

Most suitable for comparison purposes are also those elements which are by themselves independent of network external influences and geodetic datum invariant, i . e . spatial distances. The differences T_sdS = Ts - sS

(6)

of the t e r r e s t r i a l and s a t e l l i t e network distances ar~with only one exception~positive, ranging from +0.2 cm to +9.8 cm; related to the length of the distances a scale factor of ms = +2.47 ppm can be derived.

The differences of the scaled distances range from

-2.6 cm to +6.6 cm having a relative accuracy of ± 2.0 ppm. The above-mentionedstatements are the usual ones when t e r r e s t r i a l and s a t e l l i t e observations are compared. However, the rigorous combination of both the observation types yields accuracy statements which go much further. The f i r s t step is some considerations on the functional and stochastic hybrid network models.

5.

THE COMBINATIONOF TERRESTRIAL AND SATELLITE AIDED NETWORKOBSERVATIONS

THE FUNCTIONAL MODEL The observation equations of a combined or hybrid adjustment model have to be formulated with respect to a commonreference system. For the followin~ models the " s a t e l l i t e " coordinates, i.e. the network point coordinates determined by s a t e l l i t e aided observations, and their covariance matrix were transformed onto the t e r r e s t r i a l coordinate system refering to the national geodetic datum. 5.1

Gauss-Markovmodel A r e l a t i v e l y simple model regards the s a t e l l i t e aided observations as direct coordi-

nate observations which are added to the observation equations of the t e r r e s t r i a l measurements. Therefore, the combined system of observation equations is

T1 + T V

=

AX

,

TPI

=

TCI 1 ,

s£ + s v

( ~

=

Ix

,

SPx

=

C -I

s x

t e r r e s t r i a l observations, s l

CI

=

TCI

0

0

sCx

(7)

s a t e l l i t e coordinates, mPz , TCL and sPx , sCx

weight and covariance matrices respectively of the observations).

51

The v e c t o r

z

o f unknowns contains, among others, the p o i n t c o o r d i n a t e s

x , y , h

common

to both types of networks. In case the s a t e l l i t e

c o o r d i n a t e s are used as approximate c o o r d i n a t e s ,

sZ = 0 .

Then the normal equations are (AT TPZA + sPx)X

where the s a t e l l i t e

AT TPZ r z

=

,

(8)

c o o r d i n a t e s are o n l y represented by t h e i r weight m a t r i x

sPx .

The

c o f a c t o r m a t r i x of the adjusted c o o r d i n a t e s Qx

: (AT TPZA + SPx )+ = (TPx + sPx )+ :

demonstrates t h a t the weights o f the t e r r e s t r i a l added.

The mutual i n t e r a c t i o n

partners.

will

T+s~

(9)

and s a t e l l i t e

c o o r d i n a t e s are d i r e c t l y

depend on the size and on the "nature" of the two

With p o i n t c o o r d i n a t e s observed by s a t e l l i t e

aided t e c h n i q u e s ,

Qx

is r e g u l a r

in general. 5.2

Gauss-Helmert model Both the sets of t e r r e s t r i a l

and s a t e l l i t e

c o o r d i n a t e s are considered o b s e r v a t i o n s

w i t h i n a system o f c o n d i t i o n equations w i t h unknowns. parameters

p

(Eq. A . I ) .

The unknowns are the t r a n s f o r m a t i o n

The System o f equations is

Tx + T v - (S x + S v) +

G.p

=

0

(10a)

or Bv + G'p + w

=

0

(10b)

wi t h

=

i

0

0

-1

0

0

0

I

0

0

-1

0

0

0

0

1

0

0

-1

0

uT :

ITV x TVy TVh sVx sVy sVh

wT :

ITX - Sx

TY - Sy

T h - Sh

0 ...

"''I ...I

The weight m a t r i x o f the o b s e r v a t i o n s r e s u l t s from the i n d i v i d u a l l y and s a t e l l i t e

networks in

adjusted t e r r e s t r i a l

52

AT TPIA P

0

=

TPx

0

0

sPx

(11)

=

x

sPx

0

The solution is according to the well-known algorithms. inversion of

A t t e n t i o n has to be paid to the

Px which is possibly positive semi-definit.

I f one transforms model Eq. 10 owing to Bv

(12)

=

i n t o the Gauss-Markov model w+~+G.p

0

(13)

,

the weight matrix Px results from

Qx

=

TQx + sQx

(14)

'

i f the Q-matrices are the inverses of the corresponding P-matrices.

6.

THE COMBINATIONOF TERRESTRIALAND SATELLITEAIDED NETWORKOBSERVATIONS THE STOCHASTICMODEL There are two major problems involved in the proper combination of the stochastic in-

formation of terrestrial and s a t e l l i t e aided network observation. 6.1

The combination of heterogeneousobservations The stochastic information contained in geodetic observations can often not be u t i -

lized in its entirety in "usual" adjustments especially in cases of heterogeneousobservation material since with the unit variance s~ only one single quantity of the stochastic model is estimated on a global scale. I f , however, some knowledgeabout the stochastic structure of the observations ispre~iven, the estimation of the variance components s~ of individual groups of observations is made possible. This a posteriori estimation technique (Appendix B) is suitable to adjoin proper weights to the various observation groups. Due to the linear structure of the stochastic model in Eq. 7

C1

=

2 ~0Q~

=

2 2 ~TTQI + ~ S S %

(is) 2 GO

=

a2 T

+ OS S~x

53

the unit variance o~ as well as the variance components ~$ and o~ can be estimated (Eq. B.16). Even a refinement is possible i f one splits

TQI into its components

2 TQ 1 2 2 42 T = Odir % i r + ~dis % i s + zen % e n (dir

directions, dis

(16)

distances, zen zenith angles)

and estimates the respective variance components. For the combinationof the terrestrial observations with the internally highly accurate GPS measurementsof the INN VALLEYNetwork there were 6 separate groups of observations established for the VCE: horizontal directions, zenith angles, 3 groups of spatial distances and the GPS observations. By this means the interaction of the satellite measurements and the individual groups of terrestrial observations can be studied thoroughly (Ref. 3). Applying VCE to the Gauss-Markov model (Eq. 13) the linear structure (Eq. 14) to be considered is

CI

2= ~0QI

2= ~0%

2 2 = (TT T % + ~S S % (17)

The estimation procedure is according to Eq. B.13. 6.2

Combinationof internal and external stochastic information There are several possibilities of combining the stochastic information in terms of

internal and external accuracies.

terrestrial accuracies internal

I

external

internal

I

external

satellite accuracies Combination of stochastic information

54

The combination of o n l y i n t e r n a l accuracies provides, a f t e r proper weighting by VCE, i n t e r n a l accuracies w i t h i n the combined network.

An analogous conclusion can be, drawn i f

e x c l u s i v e l y external accuracies are combined.

I t can be questioned, of course, whether

such a combination is reasonable in t h i s case

-

are apples and oranges assembled?

same question arises i f i n t e r n a l accuracies are combined with external ones.

The

However,

a f t e r S-transformation in a l l cases the same i n t e r n a l accuracy of the combined network can be revealed.

This is i l l u s t r a t e d

in the f o l l o w i n g (see Appendix A):

= HT%H + Hs%H

:

(19)

H(T% + ~ % ) .

Equation 19 shows t h a t the i n t e r n a l covariance m a t r i x of the combined network can be achieved e i t h e r by the sum of the i n t e r n a l covariance matrices of the t e r r e s t r i a l

and sat-

ellite

Due to

networks or by the S-transformed sum o f the external covariance matrices.

the idempotence of

H the same is v a l i d i f an external covariance m a t r i x is combined with

an i n t e r n a l one. Applied to Eq. 9

T+s< :

HT+S%H :

(H(TP + sPx)")+ :

(~T~ H + HsP.")+

'

(20)

and applied to Eq. 14

:

H~.

:

"(T% + s%)"

:

"T% H + "5%"

(21)

the same statement holds (Eq. A.11).

7.

THE COMBINATION OF TERRESTRIAL AND SATELLITE AIDED NETWORK OBSERVATIONS RESULTS The INN VALLEY Network serves again as an example f o r numerical experiments (Appen-

d i x C). 7.1

The accuracy o f coordinate observations As the f i r s t

measure the accuracy o f the s a t e l l i t e

aided coordinate observations as

obtained from a hybrid network adjustment is i n v e s t i g a t e d . Introducing the external GPS covariance m a t r i x

sCx

(Tab. I , row 2) leads to coordi-

nate accuracies of the combined network which cover the t e r r e s t r i a l

accuracies (Tab. 1,

55

row 1) t o t a l l y .

After applying VCE the a p r i o r i weights of the GPS coordinate observa-

tions turn out to be by a factor 4.5 less accurate than before (Tab. 1, row 3 vs. 2).

The

accuracy situation of adjusted coordinates of the combined network (Tab, I , row 4) which represents at the same time the accuracy of the adjusted coordinate observations is governed by the external accuracy of the GPS coordinates (Eq.1).

For further processing, see

next paragraph. Table 1 Accuracies of coordinate observ.ations

accuracies of coordinates status

Sx

Sy

Sh

[cm]

[cm]

[cm]

t e r r e s t r i a l network 0.4

1 ,,,

0.4

3.6

,,,

combined t e r r e s t r i a l and GPS (sCx) network 2

i

1.5

1.3

2.0

3

2

6.7

5.8

8.9

4

3

1,7

1.5

3.4

combined t e r r e s t r i a l and GPS (sC~) network 5

I

1.3 !

Status:

1.2

1.6

....

6

2

5.2

4.8

6.4

7

3

0.4

0,4

3.1

1 = a p r i o r i input standard deviations of coordinate observations; 2 = a p r i o r i output standard deviations after VCE; 3 = standard deviations of the adjusted coordinate observations after VCE; they represent also the standard deviations Of the adjusted netpoint coordinates. All stanflard deviations are r.m.s, values.

Using the internal GPS covariance matrix (Eq. 2; Tab. 1, row 5) one obtains after VCE the internal observational accuracies of the GPS measurements (Tab. 1, row 6) which lead to standard deviations of adjusted coordinate observations and coordinates (Tab. 1, row 7) being the same as the t e r r e s t r i a l ones. An exception is the standard deviation

sb de-

monstrating the good influence of GPS measurements to an increased (external) accuracy of heights in this combined network.

56

7.2

Accuracy of adjusted coordinates The results of the combined t e r r e s t r i a l and GPS-network are: sCx (Eq. 1) and TCx (Eq. 3) lead to

s+TCx

Ci

S x Ci S+T x

:

sX = ± 1.7 cm,

(Eq. 2) and :

sy = ± 1.5 cm,

O± (Eq. 4) lead to

Tx

S x = + 0 . 4 cm, -

s+TCx after VCE (Tab. 1, row 4):

s

y

(22)

sh = ± 3.4 cm . ± after VCE: S+TOx

= ± 0 . 4 cm,

s h = ± 1.6 c m

(23)

t

S-transformation of Eq. 22 leads to Eq. 23 as does S-transformation of any other combination of external and internal accuracies, for instance of the results as given in Tab. 1, row 7.

Note that the internal

accuracy of heights has been improved by the in-

troduction of GPS-measurements. A more detailed investigation of the behaviour of the accuracies of the original and adjusted t e r r e s t r i a l observations and the coordinate observations is given by Ref. 3.

8.

CHANGESIN POSITIONS AND HEIGHTS DUE TO THE NETWORKCOMBINATION The coordinates based on s a t e l l i t e aided observations were transformed onto the ad-

justed coordinates of the t e r r e s t r i a l network prior to the network combination.

The t e r -

r e s t r i a l network coordinates were serving as approximate coordinates in all cases of hybrid adjustments.

However, in contrast to the t e r r e s t r i a l adjustment with a datum defini-

tion based on the national geodetic datum, the s a t e l l i t e coordinates and t h e i r covariance matrices determine the geodetic reference system within the combined adjustment models. I f one considers the mean values of coordinate unknowns resulting from the various combinations of stochastic information, one can realize that systematic changes in positions and heights result from the application of the external covariance matrices.

This

effect is due to the external influences which govern the external covariance matrices; in contrast, the use of the internal covariance matrices leaves on an average the hybrid network coordinates in the position of the t e r r e s t r i a l network. This effect is the clearest with the height coordinates.

Therefore, the behaviour of

the changes in heights w i l l be discussed in the following. The observed GPS heights d i f f e r from the t e r r e s t r i a l heights between -12.9 cm . . . +12.3 cm (mean ± 0 cm) before the combination.

After combination with

sCx (Eq. 1) and

VCE,height changes occur from -7.4 cm . . . +3.6 cm (mean -2.5 cm). Particularly interesti ing are the results of the combination with the internal GPS covariance matrix sCx (Eq. 2):

they reach from -11.0 cm . . . +6.5 cm (mean ± 0 cm).

51/

Considering the essential results of the combination of t e r r e s t r i a l with GPS heights, the d i s t i n c t influence of the GPS heights to the height determination of the network points becomes clear.

Especially the extreme values -11.0 cm and +6.5 cm are highly sig-

n i f i c a n t improvements of two, t e r r e s t r i a l l y only weakly determined heights.

9.

CONCLUSIONS An appropriate combination of the stochastic information obtained from t e r r e s t r i a l

and s a t e l l i t e aided network observation requires -

a hybrid functional and stochastic model of adjustment

-

variance component estimation techniques

-

S-transformation of covariance matrices.

Some results are -

various combinations of external and internal covariance matrices can be applied;

-

a f t e r variance component estimation an identical internal covariance mat r i x of the adjusted coordinates of the hybrid network can be obtained from d i f f e r e n t combinations by S-transformations;

-

the covariance matrices of s a t e l l i t e aided coordinate observations as obtained from various standard software packages are strongly affected by external influences;

-

f o r comparison of accuracies only internal covariance matrices (or invariant functions) are suitable;

-

GPS observations can contribute to an improvement of the internal accuracy of even very precise local t e r r e s t r i a l networks.

58

Appendix

A

S-TRANSFORMATION The S-transformation parameters

p

are 3 t r a n s l a t i o n s , 3 r o t a t i o n s and 1 scale f a c t o r

f o r a threeK~imensional network pT

:

I t x t y t z cx ~y ~z ml

the c o e f f i c i e n t m a t r i x 1

GT

=

G is f o r

0

0

0

i

0

0

0

i

n

0

...

I

0

I

0

...

0

0

I

... ".

0

0

-Zl

Yl

0

0

-X 1

Z2

-Yl

XI

0

Xl

Y]

Zl

-Z2 Y2 0

-Y2 X2 X2

Cx

points

0

Z1

The transformation o f a H =

1

(A.I)

'

Y2

0

0

0

i

0

0

0

1

(A.2)

-Zn Yn

-X2

o..

zn

0

...

-Yn Xn

0

Z2

...

Xn

Zn

0

Yn

-x n

covariance m a t r i x is c a r r i e d out with

( I - G(GTG)- I GT)

(A.3)

according to Ref. 1 to

Ci

HC H

=

x

The matrices r{G} (r{.}

(A.4)

x

G and

=

H have the f o l l o w i n g important p r o p e r t i e s :

d

(A.5)

r a n k ) , i f the number of transformation parameters applied is r{G~G}

so t h a t

:

(GTG)

:

d

(A.6)

,

is r e g u l a r and

G(OTG)-~ GT = ~T

i s symmetric and idempotent

d .

(GTG)- I

exists. (A.7)

59

GG = (tr{.}

G ,

trace)

r{G}

=

tr{G}

=

(A.8)

.

The t r a n s f o r m a t i o n m a t r i x H

d

HT ,

HH =

H ,

H

r{H}

i s symmetric and idempotent =

tr{H}

and the Moore-Penrose g e n e r a l i z e d i n v e r s e H+

=

tr{l}

H+

- tr{G}

:

3n-d ,

(A.9)

i s the m a t r i x i t s e l f

H

(A.IO)

From the idempotence (Eq. A.9) i t a p p l i e d to a c o v e r i a n c e m a t r i x

f o l l o w s t h a t a S - t r a n s f o r m a t i o n can o n l y once be

CX ; a second S - t r a n s f o r m a t i o n o f the same m a t r i x would not

change i t . The rank d e f e c t o f the r e s u l t i n g m a t r i x

C~

a t the most in case o f a t h r e e - d i m e n s i o n a l n e t w o r k . verse o f

Cix :

Cx

is

d , if

r { H } = 3n-d ~ r{C x }

On t h i s c o n d i t i o n ,

with

Px

;

d = 7

as the i n -

and w i t h Eq. A.IO a l s o

HCxH =

holds (Ref. 4 ) .

(HPxH) +

(A.I1)

60

Appendix

B

VARIANCE COMPONENTESTIMATION In the f o l l o w i n g only a short review i s given.

For d e t a i l e d studies see f o r instance

Ref. 5 - 7. The stochastic part of the Gauss-~arkov model 1

+ v

=

Ax

,

Cz

(B.I)

1

vector of observations

v

vector of residuals

(n×l)

A

design matrix

x

vector of unknowns (uxl)

(nxl)

(nxu)

Consists of the covariance matrix

Cl

of the observations which can be divided in two

2

C1

=

(B.2)

~0~ 1

2

theoretical unit variance

~0

Ql = p~l

cofactor matrix of the observations

Qz is considered prc~given,

~

is a f t e r the adjustment (a p o s t e r i o r i ) unbiasedly e s t i -

mated by the e m p i r i c a l u n i t v a r i a n c e

2

so , i.e.

vTPI v

2

(B.3)

S0

redundancy of the observations r (tr{ •}

= tr{PzQw }

= n-

(B,4)

u

trace operator),

Q~

:

% - A ( A T P ~ A ) + AT

=

%-A~A

T

(B.5)

cofactor matrix of the residuals (n×n) , v

and

:

(A0_ATP~ - I ) Z

=

- Q~PIZ

(B.6)

61

v~pzu = CplQWpzQWpzl = Sotr{PzQ 2 w piQvv}

(B.7)

Frequently observations of different types have to be adjusted in one hybrid model, i . e . horizontal directions, spatial distances, zenith angles drop in the adjustment model of a t e r r e s t r i a l network, or even t e r r e s t r i a l and s a t e l l i t e aided network observations have to be combined. In such a case the original Gauss-Markov model is s p l i t into several parts corresponding to the

11

c different groups of observations

AI

VI =

+

1c

• x

(B8)

Ac

Vc

Also the stochastic model f a l l s into groups 2

CI

=

2 =

~0~i

+ "'" +

OcO.c

2p-i

2 -i

2

~Q~

=

~iP1

+

"'" +

While the model was homogeneous, only the estimation of the unit variance terest.

(B.9)

c c

o~ was of in-

In the inhomogeneouscase, however, the wish may exist to estimate the variance

components ~

, . . o~ as w e l l .

This can be achieved by dissecting Eqs. B.5 and B.7 according to vv

Q~I ' ' '

Qlc

(B.IO)

°

with

Q•q=

i = 1...c

Ai%AT

i,j = i...c , i # j

and C

vTPIV

>7 VTPc v ± i=1 i

=

(B.11)

with C

v~Piv i

=

Etr{ j=l c

~rv

2

PjQjiPiQi."j } sj

2

El t i j s j j=

(B.12)

@2

A f t e r composing to one system of equations

vTP1vz

tii =

2 SI

... tic

(B.13)

•

tcl

VcPcv C

2 "c

tcc 2

the variance components sj

can be solved for by inversion. The solution has to be found 2 i t e r a t i v e l y until all components converge s~ ~ sj . The final estimation result is unbiased. Eq. B.13 is the general formula which has to be applied i f the linear structure of the

% matrix

is "overlapping", i.e. ~I

+ ~2

(B.14)

or "sequential"

00

=

01

+

In t h i s case the equation system B.13 can be s i m p l i f i e d .

v~Piv i

(B.15)

02

The estimation formula is

= S~ tr{PiQ~.V~}

for each variance component s~ independentof the others.

(B.16)

The estimation process is

iterative, too. One very important fact of the estimation of variance components is its independence of the geodetic datum or other network external effects.

63

Appendix

C

THE INN VALLEY NETWORK T e r r e s t r i a l observations The INN VALLEY Network of the I n s t i t u t e of Geodesy of the Bundeswehr U n i v e r s i t y Munich covers a space of 25 km x 15 km x 1.3 km. pillars.

The t e r r e s t r i a l

All points but one are monumented by concrete

coordinates were determined by 48 s p a t i a l distances (three clas-

ses of accuracies), 44 h o r i z o n t a l d i r e c t i o n s , 44 z e n i t h distances and 8 x 2 components o f the d e f l e c t i o n s o f the v e r t i c a l .

4 (1566m)

Error ellipse

mH = 36ram

mp~ 5,gmm

Fig. I

The INN VALLEY Network:

0

terrestrial

~' I cm

5kin

observations

64

GPS observations In November 1984 coordinate differences of the INN VALLEY Network points were determined by Macrometer V 1000 measurements (Ref. 2).

2

1

8

3

/+

6 Fig. 2

The INN VALLEY Network: ( Q

Macrometer measurements

simultaneous observations)

65

REFERENCES

1)

P. Meissl, Zusammenfassungund Ausbau der Inneren Fehlertheorie eines Punkthaufens, in: K. Rinner, K. K i l l i a n , P. Meissl, Beitr~ge zur Theorie der geod~tischen Netze im Raum, Deutsche Geod~tische Kommission, Reihe A, Heft 61, MUnchen 1969

2)

H. Heister, A. Sch~dlbauer, W. Welsch, Macrometer Measurements 1984 in the INN VALLEY Network, in: First International Symposiumon Precise Positioning with the Global Positioning System, proceedings, 567-578, Rockville 1985

3)

W. Welsch, W. Oswald, The hybrid adjustment of t e r r e s t r i a l and s a t e l l i t e aided network observations - investigation of accuracies, XVlII. International FIG Congress, invited paper 503.1, Toronto 1986

4)

A. Albert, Regression and the Moore-Penrose Pseudoinverse, and London 1972

5)

E. Grafarend, A. Kleusberg, B. Schaffrin, An introduction to the variance-covariancecomponent estimation of Helmert-Type, Zeitschrift fur Vermessungswesen 105 (1980)4, 161-180

6)

H. Pelzer, Grundlagender mathematischen S t a t i s t i k und der Ausgleichungsrechnung, in: H. Pelzer (ed.), Geod~tische Netze in Landes- und Ingenieurvermessung I I , KonradWittwer-Verlag, Stuttgart 1980, 3-120

7)

W. Welsch, Grundlagen, Gebrauchsformeln und Anwendungsbeispiele der Sch~tzung von Varianz- und Kovarianzkomponenten, Vermessung, Photogrammetrie und Kulturtechnik 82 (1984)9, 296-301

Academic Press, New York

67

VERY LONG BASELINE INTERFEROMETRY

J. Campbell Geodetic Institute U n i v e r s i t y of Bonn, Fed. Rep. of G e r m a n y

ABSTRACT This c o n t r i b u t i o n reviews the technique of very Long Baseline Interferometry as a tool for high p r e c i s i o n measurements of relative point positions and spatial baseline orientation.

The geodetic and geophysical

a p p l i c a t i o n s of

these m e a s u r e m e n t s are d i s c u s s e d in r e l a t i o n to the objectives of global and regional programs of Earth dynamics research.

Following a d e s c r i p t i o n of the m e t h o d and the

instrumentation,

the systematic errors limiting the accu-

racy of VLBI baseline v e c t o r d e t e r m i n a t i o n s are d i s c u s s e d and the d i f f e r e n t approaches of error e l i m i n a t i o n are indicated.

Some of the most interesting results recently ob-

tained by the different groups involved in geodetic VLBI are shown.

I.

INTRODUCTION Radio i n t e r f e r o m e t r y o r i g i n a t e d as an a s t r o n o m i c a l o b s e r v a t i o n m e t h o d

with the aim of increasing as much as p o s s i b l e the low angular resolving power of single telescopes.

In its simplest form a radio i n t e r f e r o m e t e r con-

sists of two antennas separated by a given distance D, the baseline,

and

c o n n e c t e d via the receivers and a phase stable e l e c t r i c a l connection to a phase meter.

The response of such a system to the o b s e r v e d radio source

carries information on its angular size and structure, interest to astronomers.

w h i c h are of a prime

For the study of extremely c o m p a c t sources,

the r e s o l u t i o n of the connected e l e m e n t radio i n t e r f e r o m e t e r s

however,

(CERI) proved

i n s u f f i c i e n t and some means to increase the b a s e l i n e length, w h i c h is proportional to the resolving power, made in 1967, when independent

had to be found. A major b r e a k t h r o u g h was

stations both in Canada and in the U.S.A.

were able to produce interference fringes by using very precise oscillators and tape recorders, et al. 19671);

thus eliminating

BARE et al. 19672).

the need for a cable connection

First experiments

aimed at achieving geodetic accuracy took place line b e t w e e n the Haystack Observatory,

in 1969 on the 845-km base-

Massachusetts

of the National Radio A s t r o n o m y O b s e r v a t o r y

(BROTEN

that were e x p l i c i t l y

and the 43-m antenna

in Greenbank,

West V i r g i n i a

( H I N T E R E G G E R e t al. 19723)). In order to reach the high group delay resolution of ± 1 ns attained in the above experiments, had been developed

a bandwidth

(ROGERS 1970~)). With this technique

synthesis

technique

it became possible

to

68

use tape recording equipment with a limited bandwidth and sample the broadband receiver w i n d o w at a set of widely spaced frequencies.

Both the intro-

duction of independent high stability frequency standards and the b a n d w i d t h synthesis technique have c o n t r i b u t e d to t r a n s f o r m radio a geodetic m e a s u r i n g system of u n p r e c e d e n t e d as Very Long Baseline I n t e r f e r o m e t r y

into

now commonly known

(VLBI).

The following d e s c r i p t i o n of the VLBI technique, achievements,

interferometry

accuracy,

its potential and its

is primarily o r i e n t e d towards the geodetic a p p l i c a t i o n s with

emphasis on the achievement of high group d e l a y accuracy, w h i l e o m i t t i n g the a s t r o n o m i c a l

2.

source imaging techniques.

F U N D A M E N T A L S OF THL V L B I - T E C H N I Q U E

The most striking difference b e t w e e n Very Long Baseline and other space techniques

Interferometry

is, that the interferometric observables are ob-

tained a posteriori by the alignment in a processor of the two identical signal streams received at different times at the two telescopes.

Thus the

telescopes plus the processor could be seen to embody one instrument and the baseline separating the telescope sites may be

(and often is) named an in-

strumental c a l i b r a t i o n constant.

Fig.

I

Functional d i a g r a m of a VLBI - system

tape

transport ~L

tape ~tfanspor t

I PROC~SSORI ~ I cross-correlation j

~

R(T,t)

(on computer tape)

89

The main elements coming

from a given

at a p r e s e l e c t e d frequency) Before

and r e c o r d e d

mation derived

(usually

the s i g n a l

in a d i g i t i z e d

in the p l a y b a c k

the a c t u a l

During

standards.

processor,

aligned

signal

of the o b s e r v e d

correlation

to b a s e b a n d

streams

Later,

compensated

the p h a s e of w h i c h

w h e n the t a p e s are

for a c e r t a i n

correla-

interval.

Be-

is d o n e at a

by AT = I/2B s w h e r e B s is the s y n t h e -

channels.

the d a t a s t r e a m f r o m one s t a t i o n

frequency,

tape.

time infor-

a coherent

the c o r r e l a t i o n

in such a w a y that the c h a n g i n g

completely

stations

(video

on h i g h d a t a - r a t e

with precise

this p e r m i t s

time lag is s t i l l u n k n o w n ,

n u m b e r of d i f f e r e n t d e l a y s s e p e r a t e d sized bandwidth

form)

1: T h e r a d i o s i g n a l s

at two or m o r e

converted

streams are provided

f r o m the local f r e q u e n c y

together

continuously

simultaneously

in the G H z - r e g i o n ,

tion of the a p p r o x i m a t e l y cause

a r e s h o w n in ~ig.

s o u r c e are o b s e r v e d

frequency

registration

brought

of a V L B I - s y s t e m

geometric

for. T h i s g i v e s r i s e tO a r a t h e r slowly varies

is d e l a y e d

quasi-

d e l a y T9 is a l m o s t low residual

fringe

o n the s c a l e of a f e w t u r n s p e r

minute. The o u t p u t of the c o r r e l a t o r correlation meter

function

R(T,t)

which

is u s u a l l y

described

translates

the r e s p o n s e

s y s t e m to a r a d i o source, see e. g.

by the c o m p l e x

(MORAN 19765)).

RECEIVED SIGNALS

÷~ Station

I : x1(t)=/XI

Station

2 : xa(t-T)

(~) e j ( ~ -

=/X2

CROSSCORRELATION FUNCTION ~T

a) computed:

b) model:

1 / xl(t) 2T -T

B s

(T

amplitude

K = constant Bs= m a x i m u m

(t

t-roT] de

-

T i) dt

[~t

- ~ B s (T - Ti) ]

~

~r

T i) J

term

proportional

The c r o s s c o r r e l a t i o n ~Bs(T-Ti)

x2

phase

T i = discrete function

2

term

to SNR, T = i n t e g r a t i o n

spanned bandwidth,

T = true delay,

m a i n peak.

-

v

- ~o)

R(t,Y i) =

K sin ~ B s (T - T i) ej •

~ ° ) t d~

(~) e j [ ( ~

~f

= fringe

correlation

amplitude

= ~, h e n c e AT = I/B s, w h i c h

cross-

of an i n t e r f e r o -

time

frequency, lags

has its f i r s t

is the h a l f w i d t h

zero at of t h e

70

Fig. Response

2

of a Mk II - V L B I

i

i

I

- System

i

~

.o

~,,P~.",,,,"-.,'

"r.

:;'*.v

-v'',,,'~- 2.50..- ~ .......

":

."

2.75.......

:

300

.'/-,,.'V'....s.,,,'~.~.'...::%~./...o~'./~°.,,'--

,,., - ~ - , , . ' , . , , . - - , - - . - , . ~

In Fig. single

the d e l a y

spacing

The analysis geodetic vables

I

I

1

1

i

15

2

2.5

I/2

of the

information.

delay

~I

= 0.25

pattern

as f o u r

travelling

time

group

formed

(phase

of p h a s e

unambiguously

function,

which

versus

yields

a full b a s e l i n e

solution

in g e o d e t i c

VLBI.

term

dependent

wavefront

in two ways,

Here

and

- obser-

signals

delay

in the b a n d

the

(group

delay)

constituent

is d e f i n e d

of

as the

a n d c a n be e s t i -

of the c r o s s c o r r e l a t i o n

bandpass above).

between

i. e. as the

monochromatic

the maximum

square

(see a m p l i t u d e

of a s t r o n o m i c a l

~(t) = fringe phase A(t) fringe amplitude f(t) fringe rate T(t) time delay

The ~roup

frequency

f o r an i d e a l

function

role

delay).

by a

is shown.

function:

by the w i d e - b a n d of a g i v e n

by f i n d i n g

sinx/x

portant

a wealth

T of a p a r t i c u l a r

of a w a v e

stream

source

partially

~

c a n be e x p r e s s e d

difference

as p r o d u c e d

a point

yields

- albeit

sites

derivative

pattern

Us.

at two

signal

mated

fringe

observing

General Purpose Computer Software Package

a n d as t h e p h a s e the

fringe

UT (mini

f r o m the c r o s s c o r r e l a t i o n

I

The actual antennas

• 2 MHz

As m a n y

c a n be d e r i v e d

R (T, t)

of a t y p i c a l

interferometer

is

z ~

--,,--.-- 3.75 ..............

|

2 the a s p e c t

..........

.........

3.25 -.. ~ ......

0.5

2 MHz-chann~l

~(e,~)

2.25 " " ' ~ .......

-./.,~%-,,...V~,:vV~,~.'k.~,,v .2

source

i

•" , ~ . ' ~ . ' - ~ - " , - " " " - " ~ . " ~ ' ~ - - . ' - ~ " - ~ , , " , " " • - .1 ,.,'-,- - - . . , ' ~

to a p o i n t

assumes

the

The delay

and therefore

plays

f o r m of a

observable

the most

im-

71

The fringe phase and the fringe rate

(fringe frequency)

are ob-

tained from the sine and cosine parts of the c r o s s c o r r e l a t i o n function.

Due to the close

r e l a t i o n s h i p of ~ and f, these obser-

vables are d e t e r m i n e d simultaneously,

either from an ordinary

sine wave a d j u s t m e n t or using the fourier t r a n s f o r m into the frequency domain, where the m a x i m u m of methods,

w h i c h allow to establish the function

certain interval of time rupted source scan) methods

S(f) is estimated.

These

#(t) over a

(usually the d u r a t i o n of an uninter-

are often referred to as "phase tracking"

(THOMAS 19726)).

In local i n t e r f e r o m e t r y

(CERI) this

method is the common way of c a l i b r a t i n g the short baselines and m e a s u r i n g source positions.

Due to large phase f l u c t u a t i o n s

on longer baselines the phase o b s e r v a b l e can be used in VLBI only under special conditions.

The fringe rate is unambiguous, z-component of the baseline.

but it is insensitive to the

So, compared to the d e l a y obser-

vable it plays a less important role in geodetic b a s e l i n e determinations.

The fringe amplitude is of essential importance for source structure mapping.

Figure 3 shows a typical example of the output of the Mk III fringe analysis

software package developed by A.R. W H I T N E Y Fig.

3

M u l t i c h a n n e l d e l a y resolution function

: "COARSE"

~1976)7).

(Mk III fringe analysis)

: "FINE" DELAY t . Bs= 40 MHz

DELAY

÷

~

"COARSE" P4XTE

o ! i ::" o

*

""1 ""'"

~. x'. 1 •°" ltl'~

i

*,,,

."

t

,H

~* • • • •,,

9

*,1 •

.. 9

,

l 1 i.~

UPppppp

9

:

o~ x ~* x xx x xx xxx xxx o ] xxxxxxxxx-xxxxx-xxx-xxx--xxxx--x---x-x-x---X 0

I

xx x

xxxxx

~ ..........

~

xx x

xx

xx;

xxxxx

xxxx

x

x

x

XLXLXLX

X UX

~-X-X---X-X-X-X--XX--X-XXX-XXXLXXX~X-XXXXXXXXl

72

The accuracy w i t h w h i c h the time delay y is e s t i m a t e d from the crossc o r r e l a t i o n function depends chiefly on the halfwidth of the main peak, which is given by AT

(see above).

There are different possible methods of delay estimation 19767)I,

all of which yield a precision ~

of roughly 1 %

of course on the SNR 6signal to noise ratio)

(WHITNEY et al.

of A~, depending

and the available

integration

time per observation. The s i g n a l - t o - n o i s e

ratio after averaging a total of 2 B0T c o r r e l a t e d

signal samples can be expressed by SNR

= S

•

/At

2k

" A~

•

/2

B0T

,

TS

where k is Boltzmann's constant.

This e x p r e s s i o n shows that, apart from the fundamental r e l a t i o n s h i p AT = I/B s, the delay a c c u r a c y is p r o p o r t i o n a l to the flux density S point source, square

of a

the geometric mean of the antenna apertures AI and A2, the

root of the r e c o r d e d b a n d w i d t h B0 and integration time ~ and inverse-

ly p r o p o r t i o n a l to the geometric mean of the system n o i s e temperatures at both stations.

If the ratio I/D becomes very small,

i. e. at high observing

frequencies~nd on long baselines, many of the compact sources are resolved. This results in a marked decrease of the correlated flux density with the effect that sources w h i c h show strong fringes on b a s e l i n e s of a few hundred k m become very weak on i n t e r c o n t i n e n t a l baselines al. i9718)J. particular

Therefore

(see e. g. K E L L E R M A N N et

in the latter case a high system sensitivity

is of

importance.

Efforts to improve the sensitivity of a V e r y - L o n g - B a s e l i n e

interfero-

meter have b e e n c o n c e n t r a t e d on the sampling rate 2B0. With the newly d e v e l o p e d Mk III system it is possible to record a data stream of 112 Megabit per second on 28 tracks of 2 MHz b a n d w i d t h each

(CLARK 197991). This has

p r o d u c e d an increase of the sensitivity by a factor of 5.3 over the commonly used M k II system.

The available c o h e r e n t integration time T depends o n the stability of the f r e ~ u e n c ~ s t a n d a r d s ionosphere).

and o n the state of the a t m o s p h e r e

(troposhere and

Due to the latter the ultimate stability that is achievable is

limited to about 10 -15 over time scales of 102 - 104 seconds.

Hydrogen maser

frequency standards guarantee a stability of 10 -14 over the same periods of time w h i c h is acceptable for geodetic applications. sion for the SNR the instrumental phase error ~# = (SNR) -I

U s i n g the above expres-

73

and the group

delay

=

uT can be computed. VLBI

system

error

(2z S N R B s)

To i l l u s t r a t e

consisting

acquisition

terminals

X - band

of two

these

: 8 channels

f l u x of r a d i o

(uncooled

spanned

This

extremely

high

= 0.025

degree

as w i l l

be d i s c u s s e d

The baseline

by systematic

fundamental and

~

for a

III d a t a

later

each;

T = 300 sec

and

50

% eff.)

= ± 3°

T

= 360 MHz: ns

(~0.7 cm)

instrumental

certain

example

paramps)

bandwidth T

with Mk

I Jansky

of 2 0 m d i a m e t e r

S N R = 18.2 With

a typical

equipped

2 MHz

sources:

e A = 0.057 ° (two a n t e n n a s 0 R = 160 ° K

expressions,

20m antennas

is given:

(8.4 GHz)

Correlated

-1

precision

is,

instrumental

of c o u r s e ,

curtailed

and environmental

error

to a

sources

in t h i s s e c t i o n .

observation

source vectors

equation

relating

may be written

the time delay

with

the

as

c where • S(t)

= b cos6cosh x

with

the b a s e l i n e

and

the G r e e n w i c h

the r a d i o

h = GST

The baseline rotation

axis.

vector

At this in a

stage

the c o o r d i n a t e s

In o r d e r

there

~,

to m a k e

a high

degree

effects

components

angle

are

of t h e

(see Fig.

are

~, of the

with

referred

source

source

to the

points

the

three

listed below. while

Earth

position

baseline

parameters components

to be d e t e r b x, by,

b z and

sources.

consistent terms

instantaneous

to t h e a p p a r e n t

4).

3 + 2 - n fundamental

fit:

theory

of a c c u r a c y

bx, by, b z

position

hour

~ of n o b s e r v e d

has to be s u p p l e m e n t e d instrumental

source

The unit vector

least-squares

components

+ b sin6 z

-

at t h e t i m e of o b s e r v a t i o n

mined

+ b cos6sinh y

with observation,

allowing

for a n u m b e r

S o m e of t h e s e

others

have

effects

the

above

of p h y s i c a l

model and

c a n be p r e d i c t e d

to be p a r a m e t e r i z e d

or s u p p l e -

to

74

m e n t e d with additional measurements.

In Fig. 5 an example of a geodetic VLBI

software system is shown. Fig. 4 Geometric VLBI Model

Model refinements: a) Effects of p r e c e s s i o n and nutation Precession and n u t a t i o n are motions of the Earth's axis with respect to the celestial system r e p r e s e n t e d by the positions of the o b s e r v e d compact radio sources. by external forces, of the solar system,

the g r a v i t a t i o n a l

These motions are caused a t t r a c t i o n of the members

acting upon the non-spherical~

inhomogeneous

and v i s c o - e l a s t i c body of the Earth. The main c o e f f i c i e n t s of the standard model have been d e t e r m i n e d e m p i r i c a l l y by long astronomical observing series and therefore are not error-free. In turn, any m i s a l i g n m e n t of the Earth's axis due to these errors will be d e t e c t e d by the highly sensitive VLBI observations.

Significant corrections

to the p r e c e s s i o n c o n s t a n t and

to some of the nutation coefficients have been d e r i v e d from the analysis of only a few years of VLBI data

(HERRING et al.

198510~.

75

Fig. Flow diagram

5

of a g e o d e t i c

VLBI

software

system

Corre Lot ion ] F

(delays

~

,

=o priori's" / statLon coordlnates/ source poslLions / precession, nutotlon

.

delay rotes %")

aberrotlon

I

instrumental

]

calibration dotaJ pole c o o r d l n a t e s ] ~ UTl-vQrlation J period. earth tides] JUTJ-var.

troposphere idr~ I [ground meo- i wet[ surements, radiometer)

....

J

ionosphere (S-X differences)

I J

ocean Loodlng I 4pole tlde I

Faraday-, Doppler- J

measurements

J

retativlstic deftectlon ~ Light (sun,earth,planets)

movements of the J reference point: -otmosph. Loading -changes of groundwater-Level -pillar torsion -loadlng of wind, snow, ice ;

source structure I UTC [time ~LheoretlcoL observationsI Of observ.]

I

I

-baseline components / -clock parameters / [polgnomlaL] /

Least squares solutlon [single-, muLtl-baseLlne sol.)

-otmosph. parometers/e -source positions / -pole coordlnates / -DUTI /

/ b)

Relativistic - special

(diurnal - general

~

during

Due

observation

additional

aberration

relative

the

delays

finite

which

motion speed

of the V L B I

of l i g h t

can be precisely

causes modelled

etc.). The e f f e c t waves

delay

orders

m a y be m a n y error).

Einstein's

geometry)

to the

of e l e c t r o m a g n e t i c

verify

~ive ]

/

(space-time

relativity.

measurement

~tX

-nutatlon amplitudes -~-llght-deflection factor (~I) -parameters of a plate tect. model

effects

significant

tlstliat tests I

global solution -Love numbers h , l -precesslon constant

relativity.

antennas

]

VLBI

theory

of g r a v i t y

is c o n s i d e r a b l e of m a g n i t u d e observations

on the p r o p a g a t i o n (near the

larger have

to an a c c u r a c y

than

been

of a b o u t

sun the the

used

to

0.1%.

76

c) I n s t r u m e n t a l

effects

- local o s c i l l a t o r duced

differential offsets

- change

drift

rate

in the z-component

delay

d e l a y changes bration

reference

and cable delays.

The effect

of the t e l e s c o p e

problem,

on V L B I - o b s e r v a t i o n s

because

of the t e l e s c o p e s

which

is a highly

the radio

frequency

observing

frequencies.

ably with

increasing

to the ionosphere

band,

Moreover,

its i n f l u e n c e

therefore

at 2.3 GHz

the dual

is applied. presents

radio

adds up to an e x t r a - z e n i t a l

changes

servations. meter

consists

method

how-

using

atmosphere, in VLBI on

p a t h of 2.0 - 2.5 m. stable

appears

of m e a s u r i n g 22 GHz

of the baseline

of accuracy

the r e g u l a r l y

(NI

m)

published

and can be although during

much

the ob-

to be the radiothe m i c r o - w a v e

thermal

in the line-of-sight.

observations

have p r o v e n

the p r o b l e m

These

are effects

but not its length.

these v a r i a t i o n s

bulletins

to the o b s e r v a b l e s

Therefore

method

path due

effect,

Its influence

and has to be m o n i t o r e d

and UT1-variations.

the components

tions"

consider-

effects

polar m o t i o n

degree

which

T h e neutral

The wet component,

from w a t e r v a p o r near

e) G e o p h y s i c a l -

rapidly

frequency

in

two different

diminishes

observations.

of the dry air is rather

The most p r o m i s i n g

technique,

emission

satellite

models.

The

for radiation

the same problems

and radar

by suitable

great-

2 dm. The remaining

as in D o p p l e r

The c o n t r i b u t i o n

the

differ

At 8 GHz the e x t r a - z e n i t a l

to about

the troposphere,

stations

themselves.

medium

can be dealt w i t h by using

reduces

frequency

signals

supplied

is c o n s i d e r e d

separated

conditions

dispersive

frequency.

is very unstable;

at w i d e l y

aimed at the same source

ly as well as the m e t e o r o l o g i c a l

essentially

of the

by models

measurements.

of the atmosphere

ionosphere,

displacements

can be c h e c k e d

effects

serious

look-angles

smaller,

a any

b y a p h a s e and d e l a y cali-

structure,

These effects

geodetic

d) E n v i r o n m e n t a l

described

While

by the clock offset parameter,

have to be m o n i t o r e d

points.

local

a second

clock

errors

of the baseline.

circuitry

is a b s o r b e d

Unmodelled

systematic

is introand a

system.

- deformations

ever,

a clock model

epoch difference

(two parameters).

in the electronic

constant

the m o s t

Usually

for the unknown

and rates may lead to large

especially

with

instabilities.

that accounts

can be o b t a i n e d

of the B I H to apply

(or to the model). to be more a c c u r a t e

has b e e n

that change To a certain

inverted,

In recent

from

"correcyears

the

than the corrections.

the changes

showing

in

77

the b a s e l i n e

components

Earth

rotation

Earth

tides.

and diurnal

The

tides

tides

effects

the b a s e l i n e

used

as p r e c i s e

of the

components

significant

solid

Earth

of a f e w d e c i m e t e r s

are p r e d i c t a b l e

derive

being

polar

motion

and

data.

as w e l l

as the

length.

to a f e w cm, b u t

estimates

For

stations

close

to the

t i o n due

to the o c e a n

tides

has

The

it is a l s o

of L o v e ' s

vations.

are m a i n l y

semidiurnal

in a m p l i t u d e

numbers

shores

recently

that

solid

possible

from VLBI

an a d d i t i o n a l been

change Earth

included

to

obsercorrec-

in the

model. crustal

motions.

that

the E a r t h ' s

that

are

sally

accepted. motions

placements

rates

to e a c h

Predictions

derived

source

positions

are carried

source

catalogues

The

accuracy

are

located

out

the

level

about

(~0.01

which

geodetic

thirty

in t h e

astronomy,

compact

dis-

of the p r e s e n t

the

simultaneously astrometric

consistent optical

of g e o d e t i c

radio

hemisphere,

to i m p r o v e

compact

sources,

cam-

radio-

counterprograms.

most

of w h i c h

is n o w at a l e v e l

of

from VLBI-experiments. source

coverage

over

to the e x i s t i n g

sources

structure, system.

c a n be d o n e

error and

budget

the

optical

the

of the

of a b o u t

These

and crustal

1000 k m

high

efforts

of s u c h

motion.

of

of

struc-

data,

in the

components

extremely

increasing

of t h e

VLBI

errors

km show errors

for the d e t e r m i n a t i o n rotation

geodetic

of t h e s e

at

in p a r t i c u l a r

monitoring

individual

6000

to s h o w s t r u c t u r e effect,

a l i m i t o n the a c c u r a c y

on b a s e l i n e s

3.5 c m in the

f o r m the b a s i s

in the E a r t h ' s

poses

by u s i n g

of a b o u t

tend This

Permanent

the c o n t r i b u t i o n

baselines

of the V L B I - t e c h n i q u e tions

special

support

as d e t e r m i n e d

10 c m in the c o m p o n e n t s .

ppm)

evidence

small

of a c o n t i n u e d

determined

to e s t a b l i s h

of a f e w m i l l i a r c s e c o n d s .

I cm in l e n g t h

and

plates

univer-

system.

to r e d u c e

Intercontinental length

These

interests

so,

a firm connection

the r a d i o - r e f e r e n c e

The total

Even

the

in the n o r t h e r n

of the o b s e r v e d

ture,

from geophysical p e r year.

usually

in o r d e r

a n d for

is in p r o g r e s s

any changes

help

now been

at the d e t e c t i o n

for f u n d a m e n t a l

of some

sky a n d to p r o v i d e reference

are

components.

identification

Most

has

stipulates

separate

effects

2 - 3 milliarcseconds

-

aimed

which

of

of m o t i o n .

paigns

Work

other

of the m a i n

experiments

the b a s e l i n e

part

one

tectonics

by a m o s a i c

relative

constitute

of V L B I

Precise with

of p l a t e

is f o r m e d

of a few c e n t i m e t e r s

f) R a d i o - a s t r o m e t r i c -

theory

crust

in m o t i o n

yield

series

The

will

future.

sums (see

up to Fig.

6).

2 - 3 c m in

accuracies

invested

subtle

in the

effects

use

as v a r i a -

78

Fig. Error Budget

6

of Geodetic

VLBI A c c u r a c y

3.5cm

sYs,,M NO,SE

I

,o.oSP.EI~

,~OPOSP.ERE SO~CE O~y WE. ~OSI,|ON

CLOCK INSTABILITY

r

UTI ANTENNA

I

J I

LOCATION

POLAR MOTION

RSS

INSTR. DELAY CALIBR.

SCALED • BY IJJ~ELINE LENGTH

~" IOOO km

1.0 c m

~

2.

SCIENTIFIC

INTEREST;

A detailed

description

presented

as early as

quake d i s p l a c e m e n t 197011)).

fields

Geodetic

sources.

Earth's

goals

of the o n g o i n g

vectors

crust.

the needs

2. the m o n i t o r i n g enable

a better

geodetic

understanding

system and the structure

3. the estimation

of the Earth's

in VLBI

is b a s e d

compact

and some of

in time b e t w e e n

distant

system

points

on the

observations

in order

are:

to satisfy

systems;

in the Earth's

of the k i n e m a t i c s of the Earth's

on the

radio

very accurately

to use VLBI

reference

and changes

fundamentally

extragalactic

to m e a s u r e

objectives

elasticity

or

In the follow-

are summarised

and n a v i g a t i o n a l

of polar m o t i o n

KNIGHT

could be realized

verification.

it is possible

global

on "Earth-

are shown.

of highly

the p r i m a r y

of VLBI has been Canada

(SHAPIRO,

all of the goals

and projects

interest

of a u n i f i e d

of global

in London,

of the Earth"

campaigns

frame

applications

held

successful

and their changes

Therefore

I. the r e a l i z a t i o n

Moon

virtually

With the V L B I - t e c h n i q u e

the b a s e l i n e

RSS

A N D RESULTS

of initial

reference

I

INSTR. DELAY CALIBR.

and the rotation

and g e o p h y s i c a l

use of an inertial

ANIENNA LOCATION

of the g e o p h y s i c a l

the scientific

results

N SOURCE

WET POSITION

PROGRAMS

years

at least reach the stage ing p a r a g r a p h

DRY

1969 at a conference

In subsequent

the recent

r-1

lltI~OSPHERE

SYSTEM J | ONONOISE SPHERE CLOCK INSTABILIn'

rotation

and dynamics

rate to

of the Earth-

interior;

parameters

from d i r e c t l y

measured

tidal deformations; 4. the d e t e r m i n a t i o n standing

of global

of plate m o t i o n plate

tectonics;

and plate

stability

to improve

the under-

79

5. the i n v e s t i g a t i o n of regional crustal movements

in order to deduce the

b u i l d i n g up of strain and p r o v i d e input to e a r t h q u a k e p r e d i c t i o n programs. Other important activities

are:

6. the d e t e r m i n a t i o n of improved values constants,

for the p r e c e s s i o n and n u t a t i o n

and

7. the precise v e r i f i c a t i o n of the effects of general relativity. The accuracy requirements n e c e s s a r y to attain these goals are ± 5 to ± 10 cm in each baseline component for items I and ± I to ± 3 cm in baseline

and 2

(± 0.1 ms for UTI)

length r e p e a t a b i l i t y for items 3

By now these accuracies have been d e m o n s t r a t e d

through 5.

in hundreds of VLBI campaigns

on baselines connecting nearly all m a j o r continents of the globe. C o r r e s p o n d i n g to the international nature of the a n t i c i p a t e d goals several programs of multilateral

c o o p e r a t i o n have been initiated,

w h i c h the following most important projects

I. NASA Crustal Dynamics Project

among

should be mentioned:

(CDP).

This project is part of a US Federal p r o g r a m involving several government agencies for the application of space technology to crustal dynamics and earthquake research.

Major c o o p e r a t i v e a r r a n g e m e n t s have been made with

European and other countries extending the project to a global research program

(NASA 198312)).

The VLBI part of the CDP comprises regular experiments

(10 - 20 a year)

of one to three days duration b e t w e e n the m a j o r geodetic VLBI facilities in the US, Europe and in and around the n o r t h e r n Pacific.

In addition, so-

called bursts of o b s e r v a t i o n s using the mobile VLBI units are carried out each year in the t e c t o n i c a l l y active

zones of C a l i f o r n i a and Alaska.

These

campaigns are aimed at the c r e a t i o n of a detailed p i c t u r e of the local crustal m o t i o n pattern to assist the i n v e s t i g a t i o n of e a r t h q u a k e mechanisms. The global station d i s t r i b u t i o n

is shown in Fig.

7. Some of the stations

are still in the process of b e i n g fully equipped for Mk III VLBI. The CDP in its present form is planned to extend through

1988. F o l l o w - u p

programs will be set up in order to insure the r e p e t i t i o n of the measurements at regular intervals of one or two years. 2. Project IRIS

(~nternational R a d i o ~ n t e r f e r o m e t r i c Surveying).

The goal of project IRIS is to c o n d u c t (US National Geodetic Survey)

joint activities b e t w e e n the NGS-

P O L A R I S - N e t w o r k and other international

stations that are dedicated to full-time geodetic work, station of Wettzell,

such as the VLBI-

for the regular m o n i t o r i n g of Polar M o t i o n and UTI.

80

xl

rjo~

o~

<

u 5r.x ~

P O ¢ff't

MO

o +.~

~.~-~

~,

H

O

r~ 0

I ID 0~ ~

H

°p,,~

g

~

or,4

o M

°/

& -,-t

81

The IRIS activities consist of a series of VLBI o b s e r v i n g sessions of 24 hours duration at five-day intervals.

The IRIS o b s e r v a t i o n s normally

involve three stations in the U n i t e d States Massachusetts,

(the W e s t f o r d telescope

the George R. Agassiz Station

Richmond O b s e r v a t o r y zell Observatory

in Florida)

(Ft. Davis)

in

in Texas and the

and the 20 m r a d i o t e l e s c o p e of the Wett-

in the Federal Republic of Germany.

One session per

m o n t h also includes the Onsala Space O b s e r v a t o r y in Sweden

(see Fig.

8).

Fig. 8 IRIS earth rotation network

The National Geodetic Survey

(NGS)/Rockville,

Md. regularly analyses all

of the IRIS data to obtain polar motion and UTI time series, which are published m o n t h l y

in the IRIS Bulletins A. Thus,

since Jan.

5, 1984, the

date that the W e t t z e l l O b s e r v a t o r y b e g a n regular operations,

the IRIS

system has b e e n routinely p r o v i d i n g the x and y components of p o l a r m o t i o n w i t h an a c c u r a c y of I to 2 marcsecs, 0.1 msecs

(CARTER, ROBERTSON,

MACKAY,

and UTI w i t h an a c c u r a c y of 0.04 to ]98513)).

Figures 9 and i0 show the r e m a r k a b l y smooth trace of the instantaneous pole of rotation from Sept. campaign)

26, 1980

until March 15, 1986.

(the time of the M E R I T p r e l i m i n a r y

In particular,

of accuracy can be seen from the moment

a significant

improvement

that the full IRIS network w i t h

the station of Wettzell has become operational

in J a n u a r y

1984

(Fig. I0).

In addition to the regular 5-day o b s e r v a t i o n s a special campaign of daily o b s e r v a t i o n s on the baseline W e s t f o r d - Wettzell

is b e i n g p e r f o r m e d

82

Figs• Merit

9 and

I0

and P O L A R I S / I R I S Sept.

1980

pole

- April

POLARIS/IRIS

Pole

•

•

1984

Position i

i

•

position

ee.

bQseL;n~. 4- sln~te

• e ÷

4-

+

-,.-+

÷÷

*

+

.+

+

.+

"4-

'++ '*

,4~ 2G. 5 Q p t . 8 0

+

4+

UO L <

~

4"

+

÷

4-

e

,11.

÷

,~.

+

+

•

4-

~ . hfov, 8 3 5~ar~

w~eez~It

÷ mO ~O ~-

÷

e

e ÷

~,

.I-

.

÷~

4

4+

4.

÷

"*4"

•

ee +

.I-

4

4.

4"

.4

+ •

+

• ÷

s;o

4;0

+

"~

~

300 Mssc

of

42 ,0 0

0

I vO

Arc

l •

*

o

° "O •

o

? O

hl 0

Eo

v

1983 - March 1986

Nov•

._~ <X Io

"n B

~:

XO

.2 ..

o Cq

Q

..O

• . ®

"...

......

..'" o

o •

.

.

.

•

'

200

Y--Axls ( m s e c arc)

'

100

0

-- 1100

83

w i ~ h the aim of looking

at the short p e r i o d U T 1 - v a r i a t i o n s .

lysis on 90 days of daily VLBI o b s e r v a t i o n s

already

of the p r e d i c t e d

13.6 day and 9.1 day periods

the polar m o m e n t

of inertia

and

0.2 msec)

These

and phases

results

ly derived which

represent

tion of the BIH data

The N A S A - C D P length

results

changes

of the campaign.

cession

factors

long series

period

rate of (Fig.

models)

indepently

tion of the m u c h more

an a p p r e c i a b l e

behaviour

frequently

on a routine

At p r e s e n t

could

m a y still c a u s e

problems across

c o v e r a long e n o u g h of a m e a n i n g f u l

can be drawn

and the E u r a s i a n evolu-

- Wettzell

(Ft. Davis,

theories

rela-

the same annual

the further

Westford

- GRAS

(Fig.

pre-

only two b a s e l i n e s

baseline

from

as an i n d e p e n d e n t

seem to indicate

confirm

from

are free

the a s s o c i a t e d

- Westford)

Westford

baseline

system b e c a m e

series

the N o r t h - A m e r i c a n

observed

which

basis,

Shanghai,

(Fig.

active

since the b e g i n n i n g

Kwajalein,

of

a contraction

The

Thus

should

Texas)

also

of an elastic

accuracy

campaigns.

1984.

levels

are made

to include

South A f r i c a

Japan

and Kauai,

On this baseline,

on a n o t h e r is the

Pacific

largest

with the p r e d i c t e d HEKI,

the VLBI

amounts

and IRIS n e t w o r k s

technique

Hawaii

where

have been

large m o t i o n s from the first

baseline,

rates d e r i v e d

Kashima

-

The agree-

from g e o p h y s i c a l

198616~.

has lived up to its e x p e c t a -

of data have p r o v e n

can be e f f e c t i v e l y

n e w stations,

so far observed.

TAKAHASHI,

has b e e n

and Atibaia,

of 5 cm per year has been d e r i v e d

(KONDO,

in recent years

By now large

efforts

of Kashima,

of -8 cm in one year

is very good

of the CDP

Hartebeesthoek,

length v a r i a t i o n

m e n t of these o b s e r v a t i o n s evidence

great

7). The stations

are expected, two epochs.

section

China,

Brazil

cipated

effects

series

198615) .

in the error budget

first e s t i m a t i o n s

baseline

change,

length

shown

speed,

resolu-

is i n d e p e n d e n t

(including

and Onsala

observed

the N o r t h - A t l a n t i c

in p a r t i c u l a r

tions.

positions

values.

of the plates.

While observed

as has been

any firm c o n c l u s i o n s

The continental

vector

baseline

and a t m o s p h e r i c

- Haystack

Before

the Mk III VLBI

and can be t r e a t e d

I - 2 cm per year b e t w e e n

12).

be awaited.

However,

to be able to make

Both

plate

when

SCHUH

have been p r o v i d i n g

of a b a s e l i n e

are to be interpreted.

(4 years)

drift

campaigns

temporal

CAMPBELL,

of (0.9

of t h e o r e t i c a l -

rotational

and the poor

198514),

parameters

such as source

(Onsala

tive motion.

et al.

confirmation

in the Earth's

in the noise

the o b s e r v e d

rotation

and n u t a t i o n

the A t l a n t i c

oscillations

hidden

in its orientation,

other

shows

the t h e o r e t i c a l

As the length

result

when

with

empirical

since the late seventies

in the E a r t h

6),

are in good a g r e e m e n t

(ROBERTSON

errors

deformation

Both the amplitudes

and the IRIS VLBI

fully operational.

by tidal

11).

tidal

now had been

caused

ana-

the d e t e c t i o n

(Fig.

a first purely

short period

until

of the Earth

A frequency

permitted

conclusively

realized

that the anti-

in routine

observing

84

Fig. 11 Short period UTI variations (obtained form 90 days of daily VLBI observations)

q

t 0~

E,.-;

In

o.

L

I

Apt,

1984

O~2~k

I . . . . . . .

'~'s

1

MOy

POWERSPECTRUM ITERATION: t. z,~ j 13,6d

J,

1s

~

Jun

I%

| : O.C]

POWERSPECTRUMITERATION:6 9.I d

o

/,00 ANGULARFREQUENCY

"~

o,1-

~.. 0,0-

!

~oo AN6ULARFREOUENCY

!

1 Ju!

85

FiN. Transcontinental

and

12

intercontinental

baseline

~

evolution

L

'° ~

4

I el /

i

-3

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t

:

3

"'

tt

t -1~

SLOPE = 1.36 ± 0,~.i CM/YR

~ /

-15

- 1985

II,

-.

~"

1981

1 SOpt;

Jo~

t

lgBO

t981

1 N~y

| Sept

I

Jsn

I , HSy

t SeDt.

I, , JSf~

tOO2

, ,,

• Jsf~

Sept

1983 T

/ W e s t f o r d - Onsoto

o

I May

-I'~

I 1

~904 I

T

+ 2 . 0 _+ 0 . 3 c m / y r

o

i

o

{

Jul 19B 1

,.,Ion 19B2

Westford -

Jul

Jon 1983

GRAS

Jul

-1.3

Jon 1984

+ 0.1 c m / y r

JuI

Jon 19B5

86

REFERENCES i)

BROTEN, N. W. et al.:

Long Baseline Interferometry:

A New Technique.

Science, Vol. 156, p. 1592 - 1593, 1967. 2)

BARE, C. et al.:

Interferometer Experiments with Independent Local Oscillators.

3)

HINTEREGGER et al.:

Science, Vol. 157, p. 189 - 191, 1967.

Precision Geodesy via Radio Interferometry.

Science, Vol. 178, p. 396 - 398, 1972. 4)

ROGERS, A. E. E.:

5)

MORAN, J. M.:

Very Long Baseline Interferometry with Large Effective Bandwidth for Phase Delay Measurements. Radio Science, Vol. 5, p. 1239 - 1248, 1970.

Very Long Baseline Interferometric Observations and Data Reduction.

In:

Methods of Experimental Physics,

Vol. 12, Part C, Radio Observations, Academic Press, New York, 1976. 6)

THOMAS, J. B.:

An Analysis of Long Baseline Radio Interferometry.

In:

The Deep Space Network Progress Report, Technical Report, 32 - 1526, Vol. VII, VIII, XVI, Jet Propulsion Laboratory, Pasadena, Calif., 1972. 7)

WHITNEY, A. R. et al.:

A Very-Long-Baseline Interferometer for Geodetic

Applications.

Radio Science, Vol. ii, p. 421 - 432,

1976. 8)

KELLERMANN et al.:

High-Resolution Observations of Compact Radio Sources at 6 and 18 cm.

Astrophys. Journ., Vol. 169,

p. 1-24, 1971. 9)

CLARK, T. A.:

Mark III System Overview.

Proc. Radio Interferometry

Techniques for Geodesy.

Cambridge, Mass., June 19-21,

1979, NASA Conf. Publ. 2115. i0)

HERRING, T. A., GWINN, C. R., SHAPIRO, I. I.: ferometry:

Geodesy by Radio Inter-

Studies of the forced Nutations of the

Earth. Journ. of Geophys. Research, 1985. Ii)

SHAPIRO, I. I., KNIGHT, C. A.:

Geophysical applications of long base-

line radio interferometry.

In:

Earthquake Displace-

ment Fields and the Rotation of the Earth, edited by L. Mansinha, D. E. Smylie, and A. E. Beck, p. 284, Springer, New York, 1970. 12)

NASA:

The NASA Geodynamics Program:

An Overview, NASA

Geodynamics Program Office, Technical Paper 2147, Jan. 1983. 13)

CARTER, W. E., ROBERTSON, D. S., MACKAY, J. R.: ferometric Surveying:

Geodetic Radio Inter-

Applications and Results.

Journ. of Geophys. Research, Vol. 90, p. 4577 - 4587, 1985. 14)

ROBERTSON, D. S., CARTER, W. E., CAMPBELL, J., SCHUH, H.:

Daily Earth

Rotation determinations from IRIS very long baseline interferometry.

Nature, Vol. 316, p. 424 - 427, 1985.

87

15)

CAMPBELL, J., SCHUH, H.:

Short-period Variations of Earth Rotation

determined by VLBI.

Proc. Tenth. Int. Symp. on Earth

Tides, Sept. 1985, Madrid, 16)

KONDO, T., HEKI, K., TAKAHASHI, Y.:

(in press) 1986.

Pacific Plate Motion detected by

the VLBI experiments conducted in 1984-1985.

Proc.

Symp. on Application of Space Techniques to Astronomy and Geophysics, p. 98 - 107, University of Tokio Press, 1986. 17)

CLARK, T. A. et al.:

Precision Geodesy using the Mark-III Very-Long-

Baseline Interferometer System.

IEEE Transactions on

Geoscience and Remote Sensing, Vol. GE-23, No. 4, p. 438 - 449, 1985.

II. Surface Geodetic Networks and Underground Geodesy

THE LEP TRILATERATION NETWORK

J. Gervaise and J. Olsfors CERN, Geneva, Switzerland

ABSTRACT The

installation

require Large

and

a very high

Electron

reliable

the

Positron

control

alignment

accuracy. ring

network

For (LEP)

has

been

of

a

the of

particle

new

accelerator

CERN project,

27 km

created.

the

circumference, Measurements

a

have

been made with the two-colour Tetrameter, giving an accuracy of 10 -7 ,

and

precision

levelling.

Computations

took

into

account a very fine assessment of the geoid, through collocation on

a

large

zenithal ~ 1.5

mass

model

and

Resulting

mm

planimetry

for

above a local ellipsoid. network

astro-geodetic

camera.

were

satellite

local

were

o ( 5

for

mm

with good

a to

altimetry

In December 1984, seven points of this

observed

using

receivers.

and

measurements

coordinates

The

GPS

(Global

measurements

Positioning

were

System)

carried

out

by

GEO-HYDRO Inc. with three Macrometer V-lO00 Surveyor instruments. Six independent baselines of this GPS network were measured twice during three consecutive days.

Although the observation constel-

lation was

rather poor, a common evaluation of all observations

(with

Bernese

the

roughly

1 - 2 mm

consistency

of

GPS

per the

Software

coordinate, GPS

Package) showing

solution.

The

gave

rms-errors

an excellent comparison

of

internal with

the

terrestrial solution was established through a Helmert transformation.

i.

The overall rms error of the transformation is 4 n~a.

INTRODUCTIOM

LEP

(Large

Electron-Positron

Collider)

is

intended

to

develop

more

exactly

the

extensive new synthesis of electro-weak interaction born out of the discoveries of the W and

Zo

bosons.

In

such

a machine,

accelerated simultaneously.

Whilst

electrons

and

their

antiparticles,

turning in opposite directions,

positrons,

are

they collide head-on

and are annihilated and transformed into energy which, in turn, results in new particles and

antiparticles.

France, machines implies

LEP

is

a

27 tun-long underground

ring

located

in the Pays

and partially in Switzerland, with eight experimental areas. will that

act

as

the LEP

networks (SPS).

injectors geodetic

to

LEP,

network

which

should

represents be

adapted

an

important

conveniently

de

Gex,

The existing CER~ economy. to

This

the previous

92

Although

geodesy,

topography

and

cartography

may

be

but

a

small

part

of

the

preliminary studies for a large civil engineering project, they are an indispensable part, and the work of the geodesist must precede that of the other engineers. more

true for the construction of a 27 km particle collider,

highest precision.

In search for greater accuracy,

This is all the

a project

requiring

the

the geodesist must look beyond the

causes and effects that are already appreciated to other influences, albeit small ones, at the very limits of measurement.

The

complexity of

certain

influences

accelerators.

The

the geodesy has grown with each new accelerator, which

become

influence

ever-increasing complexity.

of

more the

significant

earth's

with

curvature

the

is

a

notably due to

increasing

first

For LEP there is another influence to consider,

deviation of the vertical,

size

exaraple of

of this

that of the

which produces effects that were not foreseen at the start of

the pro~ect.

Due

to the curvature

of

the earth,

the normal

height

of

the centre

(Po) of

Proton Synchrotron (PS) is 0.8 mm above the horizontal plane of the synchrotron.

the

Because

the Linac is aligned in the horizontal plane of the in~ection point, the origin of the levelling had to be 0.3 ram above the height of this injection point.

For the Intersecting

Storage Rings (ISR) transfer tunnels the corrections reached 9.5 mm for TTI and 15.8 nm for TT2.

The

corrections (SPS)

ISR being

around

transfer

virtually circular,

the ring

tunnels

were

as is also

itself.

The corrections

made

like

in

manner.

the PS,

there were no height

for the Super Proton Synchrotron However,

due

to

the

existence

straight sections of 256 m, the machine can no longer be considered circular. accelerator

in

the

plane,

the height

corrections

in

these

straight

of

To keep the

sections

reached

3.5 ram. The geometry of LEP has been calculated in the CERN tridimensional system (x, y, z) but, as z is not measurable,

all z-coordinates have been converted to normal heights

(h) above the reference ellipsoid.

The surrounding topography gives rise to distortions of the equlpotentlal surfaces. The

extent

of

these

distortions

has

been

measurements of the deviation of the vertical.

calculated

theoretical altitudes (h) of all the points of LEP. by the non-homogeneity of the surroundins

by

theory

and

confirmed

by

This has allowed the definition of the The deviation of the vertical, caused

topography,

necessitates a correction for the

determination of points at the bottom of the pits in relation to the geodetic points on the surface.

For a 140 m pit at the foot of the Jura, this correction is 6 mm.

The deviation of the vertical also affects the gyroscope measurements which provide the orientation for the excavation of the LEP tunnel.

The

precision

necessitates, standard.

required

as was

the

This is why,

during

case

for

the

various

the SPS,

steps

a surface

of

the

geodetic

construction network

of

LEP

of very high

as soon as the project was approved, the decision was taken to

equip the Applied Geodesy Group of CERIg with an LDH 2 Tetrameter.

This two-wavelength

electromagnetic distance measurement device arrived at CERN in Autumn 1982.

93

2.

PRINCIPLE OF MEASUREMENT VISIBLE SPECTRUM

The

use

of

two-colour

NITH

INSTRUMENTS

electromagnetic

USING

distance

TWO

DIFFERENT

measuring

NAVELEMGTHS

instruments

IN

will

THE

modify

considerably the approach to the problems of very high accuracy geodesy.

Rather than consider the time dt taken by the light to cover a length element dl, we use

the proportional

quantity

vacuum durin~ a time dt.

c.dt,

called

the optical path,

the

length covered

in the

The ratio of the speed of an electromagnetic wave in a vacuum to

that in matter is known as the absolute index of refraction and given by n = c l v . index is actually frequency-dependent.

This

The dependence of n on the wavelength of light is

a well-known effect called dispersion.

The optical path becomes

: n . v . dt = n . dl.

Thus, the optical path L, going from A to B, is defined by :

L(AB)=

The proposed

principle

two-colour

electromagnetic

by Prilepin in 1957 and Bender & Owens

measurements variable

of

n dl

less dependent

for long paths

one-frequency

EDM

on the refractive

(I)

distance

in 1965,

measuring

instruments

was

in order to make the distance

index of air.

This

index

is essentially

through the atmosphere and impossible to measure accurately with

instruments,

optical path with different

Along

the

frequencies

same

allow

path,

simultaneous

the desired correction

measurements

of

the

for the refractive

index to be largely determined.

~" i", "i \

~ /'~ ~ .,'~ i' i"', '"! i ~

4EoI

~'~

," ,"": i"', -"~ i~ "~ "t

A/~

i~i'~/\

~'1J'2/(~'2 -- )'l)

E°2(

: x

Fig. 1

Definition of group velocity u

94

The two-colour instruments measure two different optical paths due to the dispersion of

the

refractive

index

in

relation

to

the wavelength.

length L of 15 km between light source and reflector,

We

assume

a geometrical

path

i.e. 30 km outgoing and incoming.

The one-way optical length will be L + S0 where S is the additional contribution due to the

atmosphere.

velocity

u.

sufficient v 2,

The

To to

total

Light

being

study

the

consider

modulated,

addition

two

disturbance

of

S

is inversely proportional

waves

of

kI

and

k2

from

the

waves

arising

L +

slightly

different

travelling

combination

of

at

to

the group

frequency,

velocities

these waves

it vI

having

is and

equal

amplitude and zero epoch an~les may be regarded as a travelling wave of frequency fm with modulated amplitude E

The

irradiance

o

(x,t) (Fig. i).

is

proportionnal

to

E2 o

2 E2 . The disturbance E (x,t) consists o amplitude modulated by a cosine function. advances is known as the Kroup velocity u.

(x,t) of The

frequency

at which

Specifically,

: n

g

about (e)

a

value

carrier

the modulation

of

wave,

envelope

in a vacuum, v = u = c.

the refractive index increases with

(dv/dk); u is smaller than v.

also a group index of refraction n.

oscillates

rate

optical media in re~ions of normal dispersion,

frequency inasmuch as u = v

and high

In a non-dispersive media, in which the phase

velocity v is independent of wavelength, u = v. For

a

Clearly, one should define

= c/u whlch must be carefully distinguished from

So, S can be defined by :

S =/~

(ng-

For red light alone, S is about 400 cm.

I) dl.

(2)

If the red line of a helium-neon laser (632.8

nm) and the blue line of a helium-cadmium laser (441.6 ran) are chosen, the extra optical paths

SB and SR

difference

&S =

for the blue and red respectively will differ by about 10%, 8B

-

SR

of about

40 cm.

atmosphere is a number A without dimension

By

convention,

giving a

the dispersive power of

the

:

A = (nB - n R) / g g

(nR- i),

(3)

g

its inverse I/A is called the constringency of the atmosphere.

One can thus write the difference of the optical path :

AS =

A

being

independent

of

the

R L 0 A (ng-

atmospheric

(4)

i) dl.

density

along

the

path

and

only

weakly

dependent on the atmospheric composition, we can replace it with a good approximation by its average value ( A ) and take it outside the integral sign :

&S = ( A

)

0 (n

- 1) dl = ( A

) SR,

(5)

95

then

SR = S / ( A )

and

(6)

L = L R - S R = L R - (L B - L R) / ( A ),

which and

shows

that

L R over

the true distance

the

same path

thus

is obtained

removing

the

by measuring

effect

of

(7)

only

the

the optical

atmosphere

from

lengths L B the

optical

measurement.

We

call

standard

refractivity

atmospheric

the

value

N

=

(n - i)

conditions,

temperature

( A

the

106

being

and

Ng °

15°C

and

the

group

refractivity

in

atmospheric

pressure

being

Tetrameter

- 632.8

run in

760 mm of mercury.

We

can

calculate

) for

the red, 441.6 nm in the blue. visible wavelengths

k

is

calculated

used

in

the

The Barrell & Sears formula is particularly convenient for

:

N

where

wavelengths

go

106 = 287.604 + 3 (1.6288/k 2) + 5 (0.0136/k 4),

the wavelength

of

the radiation

in nm.

This

allows

(8)

the constringency

to be

: I/A = 21.12.

If we use Edlen's equation with the Cauchy form for visible and near infra-red light,

N

106 = 272.600 + 4.608/k 2 + 0.066/k 4

go

(9)

In this case, IIA = 21.16.

In with

P

the Tetrameter, the partial

the

following

pressure

simplified

of water

formula

DS =

P s / T [ 1 + Ps ( 5 7 . 9

Dw =

Pw/T

a 0.03% CO 2 atmosphere,

[1

+

Pw

(1

+

3.7

the partial

. 10 8 = (C 1 D s + C 2 D w)

. 10-8-9,325

7 1 0 , 7 9 2 / T 2 + 7.75141

and P

assumes

pressure w s P is atmospheric pressure in millibars; T is temperature in degrees Kelvin.

~ g O = (n-l)

vapour,

• 10 8

of

dry

air.

(10)

. 10-4/T + 0.25844)/TZ 1

.

10 - 4

Pw )

(-

2,37321

. 1041T3) 1

C1 =

(He-Cd)

80.874255

. 10 - 6

(He-~e)

84.735399

. 10 - 6

C2 =

(He-Cd)

69.094806

. 10 - 6

(He-Ne)

73.700935

. 10 -6

X

10 -3

+

2.23366/T

-

96

The value of P

can be determined from the relative humidity RH and the saturated water w vapour pressure; i.e. :

P

In this case, I/A

w

= RH . P saturated.

= 20.86.

The measurement

of gS gives the atmospheric correction to the optical path length.

To measure gS, it is necessary

to modulate

simultaneously at frequencies of 3 GHz. to be

obtained

(ii)

and also

the

the polarisation of

the red and blue light

The signal-to-noise ratio allows a great accuracy

reduction

of

systematic

errors,

gS is determined with a

precision equal to a small fraction of the modulation wavelength under good atmospheric conditions.

3.

LDM 2 TERRAMETER (TerratechnoloKY Corporation)

The

first

prototype,

presented

in

June

1969

at

Boulder,

"International Symposium on Electromagnetic Distance Measurements"~ limited index

series.

of

air

Unlike along

conventional

EDM

measurement

path

the

pressure and water vapour,

instruments by

sample

which

Colorado,

at

the

is now produced in a

approximate

measurements

of

the

refractive

temperature~

air

the Tetrameter makes a direct and precise measurement of the

refrative index by using simultaneous distance measurements

at two optical wavelengths,

one in the red part of the spectrum and the other in the blue.

The instrument calculates

correction terms from the optical path length difference between the two wavelengths and computes the corrected base line distance. effects of temperature,

This automatically eliminates the first-order

air pressure and water vapour.

The basic principle of operation is the same as that of the Fizeau velocity of light experiment.

In that experiment, light was returned from a distant retro-reflector to the

photodetector only if there existed a proper temporal and spatial relationship between the outgoing and incoming light.

Light returned to the detector only if the transit time of

the light was exactly equal to the integral number of modulation periods.

The principle

of the Fizeau cog-wheel was adapted first by Michelson and then by Bergstrand in 1950.

A

KD*P (potassium dideuterium phosphate) crystal cell is used in the Tetrameter to modulate the polarization of light by a linear electro-optical effect, known as Pockel's effect, inasmuch as the induced birefringence is proportional to the applied electrical field and, therefore, lack

a

the applied voltage.

centre

of

symmetry.

The

Pockel's effect exists only in certain crystals which operating

principle

for

such

a device

is

that

birefringence is varied electrically by means of a controlled applied electric field. retardance can be altered as desired, incident linear wave.

the The

thereby changing the state of polarization of the

In this way, the system functions as a polarization modulator.

A

Pockel cell is simply an appropriate non-symmetric, oriented, single crystal immersed in a controllable electric field.

97

IF

,

HE-NELASER

RETROREFLECTOR

TELESCOPE I OPTICAL MODULATOR

]I

..........

OPTICALPATH

I

~[ FREQI!ENCY] ~

STANDARD

I

~

FREQUENCY COUNTER

SIGNAL PROCESSING t 1 DISPLAY ~ t

MICROCOMPU'[ER

tP : inlerfcrencefihcr

Fig. 2

Scheme of the LDM2 Tetrameter

In the Tetrameter (Fig. 2), red and blue light from the He-Ne and He-Cd lasers enters a Rochon prism at the proper angle and polarization to make the outgoing beams co-linear. The

light passes

through

a microwave

polarized light at 3 GHz.

modulator

that

modulates

the path being measured and is returned by the retro-reflector. the same optics used for transmission, where

the ellipticity of

The light, transmitted by a Cassegrainian telescope,

The beam is received by

and passes through the modulator a second time,

the ellipticity of the polarization

phase of the modulator excitation.

the

traverses

is increased or decreased depending upon the

The beams emerging from the prism are separated by

colour and directed to the photodetectors.

The analogue outputs of the photodetectors are processed and used to control one YCO (voltage controlled oscillator) for each colour. and

5

MHz,

depending

on

input

frequency

The VCO output can be varied between 4

giving

a

2995.5

MHz

signal

which

after

amplification to 4 Watts drives the optical modulator.

Because

there

is

only

one

modulator

for

the

two

colours,

it

must

be

switched

alternatively between the red and blue frequencies.

In order to adjust the modulating frequency so that a minimum of light is received at the photodetectors, each VCO is switched between two frequencies, one below and the other above the centre frequency (Fig. 3).

As can be seen on the figure, the minima are much better defined than the maxima.

The

frequency difference between the minimas is :

fc - fc'

= v/2D,

(12)

98

where v is the speed of light in the air, and D is the one-way distance.

The amount of

VCO frequency shift &f (determined by the "Approximate distance" set by the operator on the control panel) is about one quarter of fc - fc'

C

=

Fig. 3

The internal calculate parameters,

the

fm

Received light intensity vs. modulator frequency

frequencies are measured and these values used in the microcomputer to

true

distance.

rough end-point

Since

values

of

A

is

a

temperature,

weak

function

pressure

entered usin~ thumbwbeels on the instrument panel (Fi~. 4).

Fig. 4

Back view of the Tetrameter

and

of

the

relative

meteorological humidity

are

99

With

these data

the microcomputer determines

the true distance, which

displayed on a liquid crystal readout on the back panel of the instrument. all input data,

is digitally

A hard copy of

internal frequencies and true distance may be obtained every ten seconds

by connecting a terminal or a printer to the Terrameter's computer interface. frequency

standard

is

used

as

the

reference

for

counters

frequencies that are used for the distance calculations. of

magnitude

more

accurate

than

ever

required

and

that

measure

A rubidium

the

internal

This standard is several orders no

subsequent

recalibration

is

necessary.

The repeatability is 2 . I0 -I0 after I0 minutes of wa~m-up, with a long term -II stability of 4 . i0

The 5).

An

laser beams

from the instrument are directed to a distant retro-reflector (Fig.

aligning telescope

then adjust

is used to acquire

the fine pointing

of

the reflected beam,

the instrument by viewing

and the operator can

the frosted glass aperture

mounted on the rear,

Fig. 5

Retrorefleetor

The Terrameter's range is from 350 m to 15 km. is 5 m W and that of the He-Cd laser is I0 ~W. a modulation wavelength of I0 cm. 2 . 10 -4 radians,

The outgoing power of the He-Ne laser

The polarisation is modulated at 3 GHz, i.e

The beam dispersion

in the atmosphere

is limited to

100

The

frequencies

are

servo-controlled

opposite phase to the outgoin~ signal.

in

and return path is an integer plus a half. that N is equal for the two colours. of the two frequencies.

order

that

instrument

is

automatic,

meteorological values.

apart

from

signal

arrives

in

The red and blue frequencies are adjusted such

Due to the dispersion,

the distance

return

The ratio of the two indices is equal to the ratio it is possible

indices, all the other parameters being known except N. an approximate value of

the

Thus the number N of wavelengths on the outward

to better the

to calculate n B and n R

It is thus necessary to determine

than half

introduction

of

a wavelength the

(0.05 m).

approximate

distance

The and

However, due to the elaborate optical and electronic systems, it

needs maintenance and internal adjustments which require the presence of an electronics engineer throughout the observation period. use

in

all

field geodetic

deformations landslips.

of

the

measurements.

earth's

crust,

the

Also the bulkiness and weight inhibit easy It has

been used principally

stability

of

large

to measure

construction

sites

the and

It can, with difficulty, be adapted to the measurement of particular geodetic

networks such as those of CERN.

4.

MEASUREMENT OF THE LEP GEODETIC NETWORK WITH THE TERRAMETER

The device.

first

critical

operation

was

to

determine

the

instrumental

constant

of

the

The Applied Geodesy Group has at its disposal a 500-m outdoors baseline which has

a pillar with forced centring at every 50 m.

With the new model of the Distinvar,

the

absolute accuracy over a distance of 50 m is 0.03 mm, which means, on the 500 m of the base,

a precision of O.I mm (relative precision of 2 . 10-7).

It has been necessary to

increase the length of this base by building two more pillars, one located 478 m to the north and the second 549 m to the south.

Figure 6 shows all the measurements carried out

with the Tetrameter.

Values measured with the Distinvar between pillars

been

adjustment,

included

in

the

the

results

of which

have

given

instrument-reflector set of 0.0275 m.

Fig. 6

2 and 12 have

a constant

Determination of the constant of the Tetrameter

for

the

101

The constant of the instrument-reflector set having been determined with the maximum accuracy, it was possible to measure a geodetic flgure, the nucleus of the LEP network, in order to carry out the acceptance tests of the Tetrameter.

Figure

7

shows

the Iocatlon of

the geodetlc points which

surround the LEP

ring.

Measurements were made from all points except the plllar of point 232 located on the water tower o£ Collex-Bossy.

The wooded, hilly and rather dense urbanization of the area meant

that not all pillars could be positioned at ground level.

•

233

/

/

i t

I

,#

/ / /

230

23,/

/

/

-,g, ....

/

/

'.'t ,,."

/

'., i i

..

t

/

./":

t/1/ 22- '' "/<'~" ............ Fig. 7

measured more

than

a

the

pillar

locatin$

100 times. a quarter

ATt i\t itt

i i

t. "/.Z

point

\ it \ it Vt

/

/

~. / / N .z / W",.~//

has

/ ~ "., ,.,

";L..

ti, ,

each

/

~•

/\

At

\

!i 1 /

-..x.::::~j---i.. .../ ,,,"--7.-~. %..,

3

Tetrameter

I

is

an hour

of the day and over several

days.

per

234

LEP test geodetlc network

equiped

cylinder

One m e a s u r e m e n t of

....

with is

with

a

forced

an expanding displayed

distance.

every

centring

locking ten

Measurements

reference

system.

seconds, were

taken

socket.

Each distance

The was

which means a little at

different

times

102

Table 1 LEP test geodetic network, June 1983; Comparison of reciprocal observations with the Tetrameter (without addition of the instrument constant)

(-

Observed distances (m)

226 - 230

-) (-

7482.33805 7482.33803

0.02

0.03

226 - 231

-) (-

7314.84645 7314,84691

0.46

0.6

226 - 232

-)

8808.13369

226 - 233

-) (-

13663.40556 13663.40524

0.32

0.2

226 -

-) (-

7755.90733 7755.90860

1.27

1.6

230 - 231

-) (-

3543.57321 3543.57230

0.91

2.6

230 -

-)

8110,89990

230 - 233

-) (-

8869.18119 8869.18148

0.29

0.3

230 - 234

-) (-

11001.47017 11001.46911

1.06

1.0

231 - 232

-)

4593.22481

231 - 233

-) (-

6541.98778 6541.98669

1.09

1.7

231 - 234

-) (-

8080.72270

0.49

0.6

233 - 232

-)

6458.90541

233 - 234

-) (-

11294.64935 11294.64892

0.43

0.4

234 - 232

-)

4861.91680

-)

Pillars

234

232

Distance difference (mm)

Precision x 10 -7

8080,72221

Note : Pillar 232, water tower of Collex-Bossy, non-stationned

Table value

of

i

shows

these

the

comparison

differences,

of

all

expressed

as

reciprocal a

relative

distance value

measurements. is

0.9 . 10 -7 .

The

r.m.s.

Therefore,

the relative precision of independent measurements carried out under different conditions is as stated by the manufacturer.

The network has been computed by a tridimensional

least squares adjustment programme

constrainin~ at each point the measured altitude above the XY plane of the CERN cartesian system.

This transformation makes use of a local reference ellipsoid with the parameters

recommended by IUGG (Canberra 1979) and includes the computation of ~eoidal heights.

103

The

residuals between measured and adjusted distances ere

r.m.s, values of these residuals is 1.3 mm.

less than 2.5 ram end the

The axes of the confidence ellipses obtained

from the covariance matrix are less than 1.5 mm (Table 2).

Table2 Results of the LEP ~eodetic measurements

Fixed point and orientation point

Pillars

X (m)

226

1288.2656

723.7339

Y (m)

2500.8843

Z (m)

DX(mm)

233

-1150.7310

14165.7641

2729.9655

DY(mm)

-13.0

EQX(mm)

EQY(mm)

71.5

1.5

Fixed points in Z

X (m)

Pillars

Y (m)

Z (m)

DY(mm)

EQX(mm)

230

-4176.6449

5828.7078

2745.4066

DX(mm) -29.0

13.3

1.7

EQY(mm) 1.8

231

-1130.9828

7626.9147

2527.4834

-15.4

48.7

1.8

1.5

232

3132.0103

9336,7623

2491.8598

32.0

40.3

1.8

1.6

234

6815,9375

6164.0805

2456.1840

37.5

0.5

1.6

2.0

Distances

Pillars

Observed distances

Residues(ram)

231 - 226

7314,87420

-0.60

7314.87360

231 -

230

3543.60020

-0.82

3543.59938

231 -

232

4593.25230

-1.01

4593.25129

231 -

233

6542.01470

-1.22

6542.01348

231 -

234

8080,75000

-0.67

8080.74933

230 -

232

8110.92740

1.83

8110.92923

226 -

230

7482.36550

0.24

7482,36574

226 -

232

8808.16120

1,72

8808.16292

226 -

234

7755,93550

-0.57

7755.93493

234 - 230

11001.49710

2.51

11001.49961

234 -

232

4861.94430

-0.01

4861,94429

233 -

226

13663.43290

-1.44

13663.43146

233 -

230

8869.20880

2.05

8869,21085

233

232

6458.93290

0.90

6458,93380

234

11294.67660

-0.69

11294.67591

233

-

-

The actual surface geodetic network for LEP 1983.

Compensated distances

It

is

measurements

not

possible

between all

to

publish

the pillars

of

here

the

(Fi~.

8) was measured in October

comparison

the network,

but

of

the

reciprocal r.m.s,

1.3 mm and the axes of the confidence ellipses are less than 1.5 mm. are the same as those obtained in the measurement of the test network.

distance

difference

is

These results

104

Fi~. 8

This

network

including showin~

226.

the

was The

quality

adapted

to

displacement of

the

LEP geodetic network (1983)

that

of

vectors

two networks

the

SPS,

range and

which

from 0.25

the

shares mm

five

geodetic

to 6.10 mm

stability of

points

(8 m pillar),

the points,

as

the SPS

surface network has not been measured since 1976.

In summer 1985, a new measurement of the LEP geodetic network (Fig. 9) has been made, in order to check the stability of points and to determine new pillars at the vicinity of access pits to the tunnel.

105

4233 •

|

*230

\ 4231

\358 232; t

A 357 °

........

Ae:'.......

\

%'

%

352~ i

\

V--" ~.\ %

358 A

4234

•

: Points of the principal network

A

: Points situated near the pits

4

: Points measured by G.P.S.

Fig. 9

The main

results of

this

LEP geodetic network (1985)

trilateration campaign are shown in Tables 3 and 4.

comparison with 1983 network is given in Table 5.

The

106

Table 3 LEP test Keodetic network. 1985 Free figure adjustment

Fixed point :

226

Orientation point :

233

Number of observations Number o f unknowns :

: 31 18

All points constrained at their measured altitude H

[6]

[1]

[2]

[3]

[4]

[5]

Point

X (m)

Y (m)

Z (m)

SX (nun)

215

1200.67301

2077.57816

2484.5225

1.23

.32

223

1821.90857

4555.77730

2476.2915

1.10

1.53

226

1288.26220

723.730g0

2500.8724

-

230

-4176.63933

5828.70653

2745.4962

.88

.92

231

-1130.98254

7626.91154

2527.4999

1.01

.73

232

3132.00429

9336.76067

2491.8226

.80

1.19

233

-1150.72829

14165.76389

2729.9498

.15

.84

234

6815.93367

6164.07295

2456.1702

1.02

1.50

236

-3886.99035

1148.46141

2602.9765

.98

1.14

353

-2796.83180

4648.59131

2527.0570

1.17

1.23

Columns

[2] and [3] : adjusted coordinates

Columns

[5] and [6] : standard-deviation

(io) on adjusted coordinates

sY

(ms)

107

Table 4 LEP test geodetic network,

1985

A d j u s t e d Distances of LEP n e t w o r k

[1]

[2]

Point 1

Point 2

[3] Observed

[4] (m)

Sisma (mm)

[5] Adjusted

[ 6 ]

(m)

Residual

223

230

6138.0287

1.43

6138.0287

223

231

4260.7574

.84

4260.7574

223

353

4619.9517

.93

4619.9517

-

.05

226

215

1356.8156

.38

1356.8156

-

.02

226

230

7482.3613

1.43

7482.3636

-2.34

226

231

7314.8718

1.25

7314.8723

-

226

234

7755.9323

2.94

7755.9314

226

236

5193.6560

1.37

5193.6558

.25

226

353

5665.0880

1.05

5665.0867

1.20

230

215

6561.7170

1.69

6561.7173

-

.28

230

232

8110.9225

.58

8110.9226

-

.13

230

233

8869.2101

1.46

8869.2111

.93

230

353

1828.7323

1.22

1828.7308

1.50 .48

-

.01 .02

.51 .86

231

230

3543.5988

.79

3543.5983

231

215

6019.4543

1.79

6019.4545

.18

231

226

7314.8740

2.73

7314.8723

1.68

231

232

4593.2468

.65

4593~2465

.32

231

233

6542.0165

1.75

6542.0154

i.I0

231

234

8080.7475

2.62

8080.7464

I.II

231

236

7040.7109

1.85

7040.7096

1.30

233

226

13663.4334

1.43

13663.4330

.36

233

232

6458.9298

.98

6458.9299

- .07 - .65

233

234

11294,6758

1.96

11294.6764

233

236

13302.3839

i.II

13302.3839

.07

234

215

6944.8709

4.13

6944.8711

- .25

234

223

5246.6471

1.30

5246.6472

- .08

234

232

4861.9497

.43

4861.9496

.02

236

215

5173.2320

1.40

5173.2318

236

353

3666.7583

1.61

3666.7586

236

234

11820.7631

3.84

11820.7654

-2.25

236

232

10785.4941

.72

10785.4942

-

Mean value of residuals

: 0.07 mm

S t a n d a r d - d e v i a t i o n of residuals

: 0.9

Column

[4] : standard-deviation

(I ~) of m e a s u r e m e n t s

mm

Column

[6] : difference between o b s e r v e d and adjusted values

.16 -

.28

.16

(mm)

108

Table 5 Comparison

between

Scale Factor

Transformed

(Helmert Transform)

Rotation

Ech = 1.000000

Transformed

1985 and 1983 networks

Translation

(gon)

Alpha = - .00001

coordinates

of active

(stable)

TX(m)

= - .000

[1]

[2]

[3]

[4]

[5]

X (m)

Y (m)

dX (mm)

dY Gram)

215

1200.6729

2077.5789

-1.05

.35

223

1821.9083

4555.7780

.91

.74

226

1288.2622

723.7316

.25

-1.80

231

-1130.9831

7626.9120

.45

- .56

233

-1150.7293

14165.7644

- .73

.99

234

6815.9333

6164.0741

-1.42

- .70

236

-3886.9903

1148.4617

1.06

1.19

353

-2796.8321

4648.5917

.53

- .22

of passive

= .001

points

Point

coordinates

TY(m)

(unstable)

points*

[i]

[2]

[3]

[4]

[5]

Point

X (m)

Y (m)

dX (nun)

dY (ram)

230

-4176,6397

5828.7068

2.40

.30

232

3132.0036

9336.7615

-6.00

I.i0

Columns

[2] and [3] : transformed

Columns

[4] and [5] : dX and dY with respect

1985 coordinates to 1983 coordinates

* The unstable character of some points results from b o t h external considerations and iterative runs of the Helmert transform program. The point 232 is at the top of a 36 m high water-tower. The point 230 is located on a steep side of the Jura mountain and its displacement vector, oriented along the line of greatest slope may correspond to a slippage of the ground.

109

5.

CONCLUSION

The results shown in Table 5 are self-explanato~y trilateration

network.

These

results

will

be

and show the quality of the LEP

compared

with

those

obtained

from

the

satellites of the Global Positionning System but, to be able to make this comparison it is necessary

to

mountains

around

take

into

account

the site

so

the

Newtonian

allowing

attraction

the calculation

of

of the

the

nearby

and

true heights

distant

above

the

reference ellipsoid.

BIBLIOGRAPHY

P.L. Bender & J.C. Owens, "Correction of Optical Distance Measurements for the Fluctuative Atmospheric

Index of Refraction",

Journal of Geophysical Research, Vol.

70, No I0,

(p.

2461-2462), The American Geophysical Union, Washington D.C., 1965.

J. Cruset,

"Cours d'Optique Appliqu~e et de Photographie", Vol. 6, (p. 501-508),

I.G.N.,

Paris, 1948.

K.D. Froome & L. Essen, "The Velocity o f Light and Radio Waves", Academic Press, London, 1969.

J. Gervaise,

"Applied Geodesy for CERN Accelerators", Chartered Land Surveyor / Chartered

Minerals Surveyor, Vol. 4, No 4, RICS Journal Limited (p. 10-36), London, 1983.

J. Gervaise, "Instruments ~lectroniques de Mesure de Distances & deux Longueurs d'Ondes", Proceedings XVII Con~r&s de la FIG, No 503.2, Sofia, 1983.

J.

Gervaise,

"Premiers

R4sultats

de Mesures G4od~siques

avec

le Terram&tre

- Appareil

~lectronique de Mesure de Distances & deux Longueurs d'Ondes", XVIII Assembl4e g4n~rale de l'Union G~od~sique et G~ophysique Internationale, Hamburg, 15-27.9.1983.

J. Gervaise, "First Results of the Geodetic Measurements carried out with the Tetrameter, Two-wavelength

Electronic

Distance

Measurement

Instrument",

Geod~tisches

Seminar

~ber

Electrooptisch Pr~zisionstreckenmessung, M~nchen, 23.9.1983.

J.

Gervaise,

Tetrameter,

"Results a

of

the

two-wavelength

Geodetic

Measurements

Electro-magnetic

Distance

carried

out

Measurement

at

CERN

with

Instrument",

Commission 6, Technical Papers p. 23-32, Washington D.C., March I0-II, 1984.

the FIG,

110

J. Gervaise, de

Mesure

"R~sultats de Mesures g~od4siques de

Distances

~

deux

longueurs

avec le Terram~tre, d'Ondes",

Kulturtechnik, Mensuration, Photogramm~trie, G~nie Rucal, Vol 6.84,

J. Gervaise,

"Applied Geodesy for CERN Accelerators",

Appareil

Ver~nessung,

~lectronique

Photogrammetrie,

189-194,

Z~rich,

1984.

Seminar on "High Precision Geodetic

Measurement", Facolta di Ingeneria di Bologna, October 16-17, 1985.

M.T. Prilepin, "Light Modulatin~ Method for Determinin8 the Average Index of Refraction of Air alon E a line", Translation, 114, (p. 127-130),

E.N.

Hernandez

Institute of Geodesy,

Aeronomy and Cartography,

URSS, No

1957.

& G.R.

Huggett,

"Two Color Tetrameter

- Its Application

and Accuracy",

Technical papers, The American Congress on 8uweying and Mapping, Washington D.C., 1981.

DEVIATION OF THE VERTICAL

Herner Gurtner Astronomical Institute,

University of Berne, Switzerland

Beat Burki Institute for Geodesy and Photogrammetry,

ETH Zurich, Switzerland

ABSTRACT The

presentation

gives

a

short

overwiew

of

the

most

important

formulae, definitions used in the computation of vertical deflections and geoidal undulations,

a summary of the known computation methods.

As an example the results of the respective ccsioutations for the CERN LEP control network are presented.

1. INTRODUCTION

The ultimate goal of the control survey for a particle accelerator

is to ensure that

the particle beam will follow a path as close to the theoretical figure as possible.

One critical point is the realisation of a true geometrical plane of the size necessary for the Large Electron Positron Collider (LEP, diameter ca. 8.5 km) using classical geodetic methods as precise levelling:

Since levelling is related to the equipotential

surfaces of

the earth's gravity field, any undulations and divergences of these surfaces will show up in a distortion of the plane deduced from the levelled heights,

if they are not known or not

taken into account.

In the presented paper we first summarize the most important definitions and formulae of the physical geodesy used or referred to later on. Then we will give an overview of the methods of observation and computation of the vertical deflections and the geoidal heights. As an example we will show the main results of the respective computations performed for the control network of the LEP.

2. THE GRAVITY FIELD OF THE EARTH

The notations used here are mostlytaken

from Ref. I.

The potential W of the gravity is the sum of the potential V of the gravitational force and the potential ~ of the centrifugal force:

Potential of the gravitational force:

112

(i)

Potential of the centrifugal force:

(2)

Potential of the gravity:

W

=

W(x,y,z)

=

V

(3)

+

with:

k : gravitational constant p : density of the volume element dv 1 : distance between dv and the point P(x,y,z) : angular velocity of the earth

x~-++y2 : distance from the axis of rotation.

The

gradient g of the gravity field i$ called gravity,

its direction the directio,

of

the plumb line ("Vertical").

dW

:

grad H . dx

The equipotential

dW

i.e.

=

:

surfaces

g - dx

=

0

is perpendicular

g • dx

H(x,y,z)

(dxe

with:

dx

:

(4)

(dx,dy, dz).

= const are called level surfaces,

so:

level surface)

to the level surface.

The g e o i d i s a u n i q u e l e v e l s u r f a c e ~ c o i n c i d i n g w i t h t h e s u r f a c e o f t h e o c e a n s . The d i r e c t i o n latitude

¢ and t h e

of the ph~b

line

in a Point P is usually

l o n g i t u d e X. ( S i n c e t h e pltaub l i n e

is

expressed by t~o angles,

slightly

curved,

the

¢ and X c h a n g e

w i t h t h e h e i g h t o f P.) The h e i g h t o f P above t h e g e o i d i s c a l l e d o r t h o m ~ r i c The d i f f e r e n c e

h e i g h t h.

between the values of the gravity potential

t h e g e o i d i s c a l l e d g e o p o t e ~ c i a l number. I t c a n b e o0mputed a s :

a t P and a t i t s f o o t p r i n t

in

I13

H

C

=

W0 - W

=

Jn g " dh

(5)

(W0 at the geoid)

V

or

W

C

(6)

Y Wo

~,k,h or 4~,k,C are called natural coordinates. A first approximation of the earth by a mathematical surface is a sphere. A much better approximation is an ellipsoid of re~)lution. We can look for a "best" fitting ellipsoid either locally (e.g. "Bessel-Ellipsoid" in S~itzerland) or globally (e.g. NWLS-E, WGS-?2).

opogrop ~

~

Geoid

( Fig. 1 Approximation of the Geoid If the ellipsoid is an equipotential surface of a normal gravity field it is called level ellipsoid: U

=

U(x,y,z)

:

Potential of normal gravity

: Normal gravity. U is completely determined by: - the ellipsoidal dimensions a,b - the total mass M - the angular velocity Ellipsoidal Coordinates: The reference ellipsoid be given by its semi-ma~or axis a and the excentricity e, with: e ~ = (a2-b=)/a ~

114

z

P

!i

a

xA Fig. 2 P(x,y,z)

with:

=

Ellipsoidal coordinates

P(B,L,H)

(B,L) : Direction of the normal n H

: Ellipsoidal height

x

=

(N + H) • cosB • cosL

y z

= =

(N + H) • cosB • sinL (N-(l-e2) + H) • sinB N

=

(7)

1 a.(l- e2.sin2B) -~

Undulations, deflections, anomalies

P W

~P~, ~

Q

Fig. 3

0

=

Wp

Seoid: W

~

h:

Orthometric height

H:

Ellipsoidal height

= W0

Ref. Ellipsoid:

U

= W0

115

Some definitions: geoidal height (undulation)

N

gP.

gravity anomaly vector

o

Ag = gp. - ~'O.

T

=

~ n

=

:

gravity anomaly

8o, e

deflection of the vertical

9 - 9o

curvature of the plumb line

W-U

disturbing potential

¢-B } (A L) -cosB :

components of the deflections in NS-, EW-direction

An important relation between the disturbing potential T and the geoidal height is the so called Bruns Theorem:

N

=

T

~

(8)

Diverqenosof the Equipotential Surfaces

@=WA=W h

f

B Ah

:

hB - hA ¢ 0

A W = W0

A

B0

(Geoid)

AW = WB - I~A = 0

Fig. 4 Important consequences of the non-pacallelity of the equipotential surfaces are the dependancy of levelled height differencies on the chosen path of the levelling; closure of a levelled loop:

dh

~

0

and the curvature of the p l L ~ lines:

(¢B' XB) ~ (¢Bo' XBo)

(~B' nB) ~ (~Bo' ~Bo)

the mis-

116

3.

SOME EXAMPLES

3.1

Detailed structure of the qeoid The first example shows a small part of the geoid in the Bernese 0berland.

Undulations

on the decimeter level over a few kilometers can be found.

(Equidistance:

5 cm, altitudes in m above sea level) 2350

20 KM

Fig. 5

3,2

Geoidal undulations

Deflections in a profile The figure shows a north-south profile of the deflections (~-component)

at the earth's

surface and in the geoid. Variations of i0 seconds in the surface values can be found within a few kilometers only. TeilGotthardmeridian: Oberfl~chen-LA und LA im Geoid

--2~

e

Fig.

Puekte m i t beobachteten L A

6: North-South profile "Gotthard"

(Deflections ~)

117

3.3

Gravity anomalies The largest gravity anomalies

630000

650000

i

i

in Switzerland

670000

i

i

6~0000

~

i

are caused by the Ivrea mass

710000

t

t

730000

t

i

anomaly.

750000

i

i

i

.....

t

170000

170000

t6000O

160000

150000

150000

140000

140000

]30000

130000

120000

120000

tZOOOO

110000

I00000

100000

90000

90000

80000

e0000 70000

70000 i

i

1

650000

630000

Fig.

7:

1

!

1

670000

T

aaooao

1

i

710000

i

i

~

"730000

1

!

750000

Gravity anomaly caused by the Ivrea Body

Way do we need to know the geoid and the deviations of the vertical? Most of all because of the difference between the "physical"

-

and the "geometrical world". A few examples:

Levelling is related to eguipotential surfaces Zenith and horizontal angles measured with a theodolite are related to the local directions of the pltmab line

-

Astrogeodetic point positioning gives the direction of the physical plumb line

-

Network

adjustments

are done in the geometrical

space:

The natural coordinate

systems have to be bound together! -

Only distances are not influenced by the disturbing potential

-

Every site has its own local (natural) coordinate system

-

GPS-results ace given in a pure three dimensional o0ordinate system.

GPS "sees

neither geoidal undulations nor deviations of the vertical". -

Particles in an accelerator will not follow the equipotential have to orbit in a true plane.

surface but they

118

4.

DETERMINATION OF (~,Q), ~%q, N

4.1

By Observation

4. i. ~ Overview

Quantity

Instrument

Reference

@ , k

Theodolite Zenith camera

B,L

~A ' nA

g

Gravimeter

~

Z~g

z~H Ah

Doppler, GPS Levels

R ~M'

Result

R ~M

Z~

4. I. 2 Example As an example we give a short description of the transportable zenith cauera used by the Institute for Geodesy and Photogrammetry of the ETH Zurich. Principles of operation: By taking a photograph

of the fixstars by means of a special

cauera in a vertical setup the astronomical parameters ~ and A of the observation point can be determined.

Two essential quantities must be known to fulfill the requirements

of

the

method: -

The epoch (local sidereal time) oorresponding to the exposures (start and end of each

single exposure as fed back from the shutter) -

The readings of the electronic levels mounted in perpendicular position on the camera

tube to determine the spacial orientation

of the camera and

its optical axis with

respect to the local pltmlb line. The camera as it can be seen in Fig. 8 and 9 is equipped with a set of electronic levels,

a high precision

internal

clock,

an electronic

shutter

and

a dedicated

micro-

processor system for the monitoring of the camera and the data access and storage. After development,

the films (size 6.5 x 9 cm) are measured on a high precision compa-

rator to obtain the plate coordinates of the star images.

Applying an appropriate transfor-

mation model between the theoretical and the measured plate coordinates of the known stars one can determine a set of astronomical parameters ~ and I referring to the camera position. The components of the vertical deflection are easily obtained by the equations x=~-B, ~=(l-L)cosB.

119

Fig. 8 Transportable

zenith

camera

of the Institute of Geodesy and Photogrammetry,

ETH

Zu-

rich. Beside the optical tube a pair of electronic levels of "Talyvel 2" type is mounted in perpendicular orientation.

Fig. 9 C<m%plete camera

system

in

"ready-to-use" position. The electronics on the right hand side

contains

the

power

supply, the clock with timesignal

receiver,

the

level

electronics, the hardware to monitor

the

shutter

and

a

microprocessor system for the data handling and storage.

120

4.2

By ccakoutation

4.2.1 Overvie W

Ag

Result

Method

Input Vening-Meinesz _ 1

dS. ~I' nl

o Stokes Ag

N

= ~ ;

Ag-S(*)'do

NI

o Astroge~detic Levelling ~A' qA Z

%A, nA,Ag

~A'nA (Ag,N) Mass Model

= - 1%-ds A

AN I

Reciprocal Zenith Distances Astrogravimetric Levelling: Combination of Vening-Meinesz and Astrogeod. Levelling Indirect Methods using Collocation

~I 'nl 'Lu'II (Ag I )

4.2.2 Exanlole: Indirect Computation Using Collocatiqn This method has been described in more detail mass model of (a part of) the earth's

e,g, in Ref. 2 or Ref. 3: Provided a

crust be given,

it is possible to compute the

differences between observed derivatives of the disturbing potential vertical,

(deviations of the

gravity anomalies etc. ) and the corresponding values defined by the mass model.

These differences show a much smoother behaviour than the original observations. Therefore they are more suited for interpolation.

Using collocation (appropriate covariance functions

between all the quantities are necessary) we can not only interpolate between the given types of observation, but compute e.g. equipotential surfaces 6N I of the reduced disturbing potential. Adding the geoidal heights given by the mass model (N~) leads us to the final geoidal heights N I.

121

I

Reference Sites

/

I

Observations

Mas~ model

R 4

~Ro nR'o (ag~)

~M'

'

'4'

I 6~R, 6nR, (6~gR)

I New Sites Collocation

'I I

6~ I, ~n I, (sAgl). 8N I

Mass model

i 4 ,4, 4

I ~I, nI

(AgI), N I

The astrogeodetic geoid in Switzerland has been cce~puted using (Ref. 2): a set of more than 200 cc,ioonents of observed deviations of the vertical a mass model of the earth's crust in and around Switzerland, containing information about the topography, the depth of the Mohorovicic discontinuity, and the most prominent mass disturbancy, the "Ivrea Body".

5.

APPLICATIONS ON THE CERN-LEP CONTROL NETWORK In order to increase the relative accuracy of the interpolated deviations and geoidal

height% the Geodetic Institute of the ETH ZUrich observed additional deviations of the vertical in the LEP area using the above described transportable zenith camera. The field measurements within the LEP-area took place in August 1983. All observations were carried out in only two nights. This particular work within the LEP project was the first field experiment with the camera, which was built at the Institut fur Erdmessung, Technical University Hannover. Meanwhile the system has been used in the southern part of Switzerland and Northern Italy where, so far, a set of about I00 vertical deflection points have been measured (c.f.Ref. 6).

122

These additional vertical deflections in the LEP area have been introduced as additional reference values into the computation program. The results of the computations can be found in detail in Ref. 4 and 5. They can be summarized as follows:

-

The vertical deflections range from 0 to 15 arc seconds at surface level and from 0 to 9 arc seconds at zero level.

-

The resulting separation between the

local reference ellipsoid and the geoid

reaches 200 mm at I0.5 km from the origin. -

The altimetric corrections along the LEP machine - to obtain a true plane in space - vary from -40 mm to +i00 ram.

Figures I0, ii, and 12 show the locations of the astrogeodetic stations, deflection intensities,

the total

and the equipotential heights in the LEP area. (The local reference

ellipsoid coincides with the geoid at PO" )

~km

Fig. I0: Astrogeodetic ~tations

123

Am

Fig, Ii: Total deflection intensity (arc seconds)

Fig. 12: Equipotential keights (ram)

124

REFERENCES i)

W.A.Heiskanen, H- Fbritz, Physical Geodesy, W.H. Freeman and Co., San Francisco and London, 1967.

2)

W. Gurtner, Da$ Geoid in der Schweiz, Astronomisch-geodatische Arbeiten in der Schweiz, Band 32, 1978.

3)

W. Gurtner,

A. Elmiger, Computation of Geoidal Heights and Vertical Deflections in

Switzerland 7 XVlII General Assembly of the IUGG, Hamburg, 1983. 4)

B.A. Bell, A Simulation of the Gravity Field around LEP, Private communication, 1985.

5)

J. Gervaise, M. Mayoud, G. Beutler, W. Gurtner, Test of GPS on the CERN LEP Control Network.

Proceedings of the Joint Meeting of FIG Study Groups 5B and 5C, Inertial,

Doppler and GPS Measurements for National and Engineering Surveys. Munchen, 1985.

6)

B. Burki, H.G. Kahle, [~. Torge, Plans for the Application of the Hannover-type Zenith CameFa System in the Central Alpine Region.

Proceedings of the 2nd

Symposium on the Geoid in Europe and Mediterranean Area, Rome, 1982.

International

COMPARISON BETWEEN TERRAMETER AND GPS RESULTS -- AND HOW TO GET THERE

G. Beutler Astronomical Institute, University of Berne, Switzerland.

ABSTRACT After a short historical introduction accuracy limiting influences on GPS-derived engineering type networks are discussed in section i, sections 2 and 3 are devoted to the 1984 Macrometer- and the 1985 Sercel- GPS Campaigns conducted on the CERN - LEP control network.

i. MEASURING ENGINEERING TYPE NETWORKS

In the early days of GPS (1982-1984)

measurement c~mpaigns usually were organized with

t~o receivers operating simultaneously. The processing software was baseline- and sessionoriented, which means that one three-dimensional vector (from the first to the second receiver)

was estimated for each session.

The scientific objectives may be sLmmarized as

follows:

Repeatability demonstrations of baseline measurements

-

- Study of figure misclosures - Accuracy demonstrations over very short baselines (invar-wire measurements)

Accuracy demonstrations of the order of 3 to 5 ppm were

Ottawa-Macrometer

campaign ')

considered

a success.

If a GPS campaign is organized today, it is almost certain that m~re than two are

operated

development.

simultaneously.

The

may serve as a typical example of a "pioneer-age" campaign.

Also,

the

processing

software

went

through

a

receivers remarkable

The First International S y m p o s i ~ on Precise Positioning using GPS in Rockville

1985 and the Fourth International Geodetic Symposium on Satellite Positioning in Austin 1986 clearly showed the trend from baseline- and session- to network- and campaign- oriented software systems ~-* )

.

The scientific objectives may be summarized as follows: - Repeatability demonstrations on entire networks (no figure misclosures) -

Subdivision of a large network:

If a network has to be observed in several sessions,

how many points of the network should be in common from session to session? - Accuracy demonstrations on terrestrial high precision networks. - Studies of accuracy-limiting effects such as troposphere,

ionosphere,

orbits and re-

ceiver clocks.

The ultimate accuracy-goal for GPS-established engineering type networks certainly is 0. I ppm -- but it is doubtful whether this will be achievable at all. What certainly is possible on a routine basis with good processing software are accuracies of the order of 1 ppm in networks of the order of I0 km x I0 kin.

126

Let us now have a look at the accuracy-limiting

influences

on GPS derived

small scale

networks.

i.I Orbits

The influence of orbit errors on baseline and network estimates discussed in the contribution

is one of the topics

"GPS Orbit Determination" also included in these proceedings.

It is therefore sufficient to stm~aarize here our experiences with different orbit and processing types in small scale applications.

Two orbit types are -- ~ r e

or less -- readily available: Broadcast ephemerides trans-

mitted by the satellites and precise ephemerides produced by the U.$. Department of Defence in an a posteriori analysis.

It has been our experience during the past few

years that broadcast ephemerides usually are of a surprisingly good quality. one may expect orbit accuracies of the order of 20 meters or better, being satellites with oscillator problems

Generally

the exceptions

(such as space vehicle 4) where errors of

the order of I00 meters may be present in broadcast ephemerides.

The few specimens of

precise ephemerides we had at our disposal do not allow a general )udgement of their orbit quality,

but there are indications that the accuracies are of the order of I0

meters for all satellites.

- Our parameter program has the capability of estimating orbits s).

According to our ex-

perience one may not expect significantly better network coordinates

if this program

option is used i_~f, at least, orbits of the broadcast quality are available.

It must be

pointed out that this statement does not hold for networks larger than ~ lOkmx lOkm. -

For the 1984 CERN Macrometer

campaign we had both orbit types

at our disposal.

No

significant difference in the quality of the GPS solution could be detected when replacing one orbit type by the other. This again demonstrates the high standard of the broadcast orbits. - This

comfortable

situation

could

change

considerably

ephemerides were artificially deteriorated. Then it ~ u l d

if,

in

future,

broadcast

be mandatory to estimate at

least some of the orbital elements in order to produce high quality results.

1.2 Ionosphere

As probably first pointed out by Campbell et al 8} the main systematic effect of neglecting a "well behaving" ionosphere (no dramatic changes in space and time) when processing GPS carrier phase data networks. tenths to

(single frequency)

is a scale-factor

in the GPS-derived

baselines

or

Baselines measured with single frequency GPS receivers tend to be short by a few -- may be -- three ppm depending on the electron content in the ionosphere.

should be mentioned that an agitated ionosphere (e.g. travelling may have an unpredictable influence on the

coordinates

It

ionospheric disturbances)

derived from single frequency GPS

observations. We had a closer look at this scale-phenomenon when we processed the 1984 CERN Macrometer campaign.

This part of the 1984 CERN GPS analysis may be summarized as follows:

127

- The study was motivated by an unexplained scale factor of 1.8 ppm between the GPS and the terrestrial solutions when the ionosphere was completely neglected. -

It seemed appropriate

(night time observations

in moderate

latitude)

to model the

ionosphere for the Cern campaign with the single layer model, a s s ~ i n g that all electrons are concentrated in a thin layer at a height H above ground.

Moreover

it was

asstm~ed that the electron density E was a constant (in time and within the layer). -

Four program runs were made using elect~on densities of E = 0~i016, i0~I016, 20.1016, 30.1016 electrons per m 2, and it could be demonstrated that each change of AE=IO~IO 16 introduced a change of 0.5 ppm in the scale of the GPS solution.

- It should be stated clearly that we had no hope of estimating the electron content from the Macrometer

observations

errors of double differences unlikely however,

(only marginal

differences

in the

estimated

rms

fo~ different electron contents occu[ed). It seemed most

that an electron content of more than E=IO~IO 16 should have been

present.

More conclusive statements concerning the ionosphere could be made when analyzing the material of the 1984 Alaska GPS Campaign ,) (a) the size of the net was

The situation was quite different in this case:

~ lO00km x 2000km,

(b) the observations were made during the afternoon, (c) both carrier frequencies L1 and L2 were measured by the TI-4100 receivers.

It turned out that the L2 derived solution had a scale of 1.9 p ~

and the L1 1.4 ppm

with respect to the solution using the so called ionosphere-free linear combination of L1 and L2. (For the sake of completeness I should mention that the scale of this combined solution was identical with the scale of VLBI). From our point of view these results show that an ionosphere induced scale factor has to be expected if baselines or networks are estimated from single frequency phase data and if the ionosphere is neglected.

1 . 3 Troposphere

When processing GPS observations, usually the models developed for the TRANET - Doppler system are used; the best known ones may be the Hopfield- and the $aastanoinen n~dels.

In

order to give an impression of the order of magnitude of the tropospheric refraction correction we reproduce here the Saastamminen formula'):

Ar = 2.28 * ( P + (1255/(273+T)

+ 0.05 ) * e )

where Ar is the tropospheric refraction correction in zenith direction in mm and P,T,e are atmospheric pressure (mbar), temperature (°C) and water vapOur pressure (mbar). The water vapour pressure in turn may be expressed as a function of relative h ~ i d i t y H (expressed in z) and temperature:

128

e = H/IO0 * exp(-37.2465+.21366 * (T+273)-.000256988 * (T+273)2).

Using the two formulae it is quite easy to compute the bias A(AF)

introduced into the

GPS observable (in the zenith direction) by an error in temperature of l°C, in pressure of 1 mbar and in humidity of 1 z,

A(&F) =

3 mm

,

at

00 C,

2 mm

,

I000 mbar and

0.6 mm

50z htmnidity:

for errors of

1° C

I mbar

,

These biases are even more pronounced if the surface temperature is rising. for example

1

If we have

T=30°C, an error of l°C would introduce a bias of 14 mm while an error of iz in

humidity would introduce 14 r~u into the observable in the zenith direction! that these meteorological biases coordinates,

in the observable induce biases

It may be shown

in the estimated height

where the above figures will be anplified by a factor of about 3. (For a mere

detailed treatment see Ref- 9).

Effects of this order of magnitude are of no importance in the "TRANET-Doppler world", but they are of vital importance in GPS where we are hunting the millimeter! Keeping in mind the accuracy limits of standard meteo equipment and the severe problems involved with this kind of measu[ement (e.g. surface effects),

it is not really surprising that in GPS process-

ing of small scale networks it is usually preferable not to use the weather data recorded at the receiver atmosphere".

sites directly,

bu___ttto use them to

compute

some

kind of

local

"standard

This has been our experience and that of others too *°) . During the last few

months we have tested atmosphere models of the kind used for classical EDM - measurements (e.g. model by Essen and Froome). Although first applications are looking promising **) , we are not yet in a position to draw any conclusions. For the two CERN campaigns to be discussed in the subsequent pages we used the surface weather

data to

generate

a

local

atmosphere,

where

temperature,

pressure

and

humidity

gradients were assumed to be known as a function of the heights of the sites. Consequently the meteorological measurements uniquely served to define temperature, pressure and humidity at a reference height H O.

2. THE 1984 CERN M~3ROMETER CAMPAIGN

A detailed report concerning this campaign was presented at the 1985 FIG meeting Munich *~),

in

Therefore I only stmmarize the main characteristics and results.

Points P226, P230, P231, P232, P233, P234 and P236 of the CERN-LEP control network (see Fig. i) were observed with three Macrometer V-IO00 S~/veyor instruments from II December to 13 December 1984. The daily observation periods started at ~ Oh and ended at ~ 3h30m (UT).

129

Fig. 1 CERN - LEP Control Network

Each observation period was divided into two observation sessions:

The first lasted for

two hours, the second for 80 minutes. Therefore two independent baselines could be measured per session,

four per day and twelve during the entire campaign.

Roger Jones from GEO/HYDRO

Inc. organized the campaign in order to measuFe twice those six baselines starting from the central point P231.

(This

implies

that P231 was permanently occupied by one of the

re-

Since Navstar 4 was moved by the U.S. DoD during the week preceeding the campaign,

the

ceivers).

Macrometer instruments could not find this satellite. three satellites could be observed

simultaneously,

The consequence was that usually only a fourth satellite being visible only

during 10-15 minutes at the end of the first and during the first 10-15 minutes of the second, optimum,

session.

Considering

five

to six

simultaneously

observed

satellites

to be

an

it is clear that the Macromete~ Campaign was severely handicapped by the "absence"

of NavstaF 4,

130

The observations were processed with the Bernese GPS Software Package ~).

The main char-

acteristics of the processing step may be s[mmarized as follows:

-

Double differences were used as basic observables.

- Precise ephemerides could be used to produce the final solution. to

No attempt was made

improve these orbits.

- The central point P231 was held fixed.

Its ~GS-72 coordinates were transformed from

the Swiss Dat~m~ "CH - 1903" using TRANSIT-Doppler derived parameters. - Initially 54 parameters (6*3 coordinates,

36 ambiguities) had to be estimated.

- Of these 36 ambiguities 33 could be resolved (related to integer numbers) by our automatic ambiguity resolution algorithm. - After ambiguity resolution the adjustment was repeated with only 21 unknowns coordinates,

- Four different p r o g r ~ solutions.

(6*3

3 ambiguities) to give the final solution. runs (with different electron contents)

gave four different

In this report we only present the solution for E = i0 * 1016 electrons per

m 2, which seems to be a hypothesis that makes sense.

In Table 1 we reproduce what we believe is "the best" GPS solution. The I 0 the coordinates

errors for

(actually the coordinate differences with respect to point P231)

given in this Table. They are typically 1 mm for latitude,

are not

2 mm for longitude and 3 to 4 ram

for height.

Table 1 Results of 1984 CERNMacrometer Campaign Datu~n: CH 1903

Station

Latitude

Longitude

Height

P226

46

13

36. 24140

6

O~

54. 44865

498.821

P230

46

17

32.26122

6

O0

36. 07847

747. 743

46

17

25. 41472

6

03

21. 02502

428. 774

P231

*)

P232

46

16

54.05284

6

06

50.77772

494. 033

F233

46

20

21. 28416

6

06

ii.26072

740. 512

P234

46

14

22. 10559

6

07

50. 34466

457. 019

P236

46

15

21. 30264

5

58

45. 18231

603. 758

*) Fixed point

This brings us to the last and most interesting part of the analysis, between the GPS and the Tetrameter results.

the ccmparison

In order to make a meaningful comparison,

the

terrestrial coordinates -- given in a local Cartesian system -- had to be transformed into the ~GS - 72 coordinate system. The parameters of this transformation are: (a) astronomical latitude and longitude of the z - axis,

13t

(b) astronomical azimuth of the x - axis and (c) WGS - coordinates (Doppler derived) of the origin of the local systam.

For a more explicit definition of these parameters see Ref. 12.

The transformation between the GPS- and the Tetrameter- derived K~3S - 72 coordinates was then

performed

parameters,

through

three

a

seven - parameter

rotation parameters,

Helmert

one

scale

transformation

factor).

The

(three

latter

translation

four parameters

(transformation GPS --> Tetrameter) were estimated as:

~ns = 0.25"

±

0.08"

(rotation about ns - axis)

~ew = 0.52"

±

0.08"

(rotation about ew - axis)

~z

±

0.05"

(rotation about z

s

The three orientation

= 1.3mm/km ± O ~ 3 m m / k m

rotation

and

the

= 0.72"

(scale factor).

angles are all smaller than

orientation

deduced

from

- axis)

the

I", which means terrestrial

that the GPS based

observations

match

quite

nicely. The scale factor of 1.3 ppm still lacks an explanation.

Table 2 gives the residuals in the coordinates that remain after the Helmert transformation between the terrestrial and the GPS - solutions.

The figures in Table 2 demonstrate the

high consistency and integrity of both the Tetrameter and the GPS solution.

Table 2

Residuals of Helmert Transformation (in m m)

Station

Latitude

Longitude

Height

P226

3

-3

6

P230

1

2

4

P231

3

3

-6

P232

-3

-8

-i

P233

-3

2

3

P234

1

2

-i

P236

-2

2

-4

3. THE 1985 CERN SERCEL CAMPAIGN

Eight points of the CERN - LEP control network -- actually the seven points already surveyed by Macrometer in 1984 plus point "PM 15" (see Fig. I) -- were equiped with three Sercel TR5S receivers *~) from 14-18 October, 1985. The campaign was organized in a way very similar to that of the CERN Macrometer: The daily observation period (from 3h30m to 7h LIT) was divided into two sessions,

each lasting for approximately 90 minutes. P231, the central

132

point, was always occupied and always

by the same receiver unit. As opposed to the 1984

campaign, the Sercel measurements lasted for five nights. Since the receivers were not moved between the two daily sessions, three points could be surveyed per night (namely P231, P230, P236 on the 14th, P231, P226, P234 on the 15th, P231, P232, P233 on the 16th, P231, P236, PM 15 on the 17th and finally P231, P226, PM 15 on the 18th of October). Although the three Sercel receivers had no problems tracking all GPS satellites available at the time of the campaign,

the configuration was not much better than during the

1984 campaign, due to low maximum elevations of two of the satellites. Moreover it became obvious -- unfortunately only during the processing phase four months after the campaign -- that one of the receivers had had serious oscillator problems.

As a

matter of fact these problems turned out to be so serious that its phase observations could not be used to produce high quality coordinates.

Being familiar with E. Murphy's law I was

not surprised to see that the faulty receiver was the one always used at the central point P231! Therefore it is clear that the Sercel campaign could not produce a network comparable to the Macrometer net.

What

remained

is stmmarized

in the following Table,

where we only compare baseline

lengths (established with Tetrameter, Macrometer and Sercel) from that part of the net which could be covered by the two properly working Sercel receiver units.

The conclusions are quickly drawn: (a)

If the Sercel receivers are working properly they promise to produce high quality results (the formal errors being of the same order of magnitude as those for Macrometer).

(b)

In order

to

demonstrate the

Sercel

receivers'

capability to produce GPS

derived networks of a consistency and integrity cc~parable to the Macrometer V-1000 receivers

it is certainly necessary for Sercel to repeat the CERN

campaign (or a canpaign in a net with a comparable ground truth.

Table 3 Ccmparison of Baseline lengths (meters) (T = Tetrameter,

Baseline

S

M = Macrometer, S = Sercel)

S - T

S - M

M - T

P226-P234

7755.932

-.002

.017

P226-P236

5193.633

-.022

-.012

-.010

P232-P234

6458.931

-.004

.003

-.007

P234-P236

11820.738

--032

.000

-.032

P226-PM15

2204.744

-.002

P234-PM15

5757.046

-.005

P236-PM15

6248.884

-.021

-.019

133

REFERENCES I)

G.

Beutler,

D.A.

Davidson,

R.

Langley,

R.

Santerre,

P. Vanicek,

D.E.

Bells,

Some

theoretical and practical aspects of geodetic positioning with the Global Positioning System using carrier

phase

difference

observations,

Dept.

of

Technical Report No. 109, University of New Brunswick, Canada, 2)

g. Gurtner, G. Beutlec, I. Bauecsin~, T. Schildknecht, Positioning

uith GPS,

Vol.

I, Rockville,

Surveying

Engineering

(1984).

Proc. First Int. S ~ .

1985 (National

on Precise

Geodetic Survey,

Rockville,

1985), pp. 363-372. 3)

C.C. Goad, Ibid., pp. 347-356.

4)

Y. Bock, R . I . Abbot, C.C. Counselman, R.W. King, A.R. Paradis, Processing o f GPS Data i n the Network Mode, Fourth I n t e r n a t i o n a l Austin, Texas (to be printed 1986).

5)

G. Beutler, W. Guctner, Positioning

uith GPS,

I. Bauersima, Vol.

Geodetic Symposium on S a t e l l i t e

R. Langley,

I, Rockville,

Proc. First

Positioning,

Int. Symp. on Precise

1985 (National Geodetic Survey,

Rockville,

1985), pp. 99-111. 6)

J.

Campbell,

H.

Geodynamics,

Cloppenburg,

Vol. I, Sopron,

F.-J. Lohmar,

Int. Symp.

on

Space

1984 (Hungarian Academy of Sciences,

Techniques

Sopron,

for

1984), pp.

196-206. 7)

W. Gurtner,

GPS papers presented by the Astronomical

Institute of the University of

Berne in the Year 1985, Mitteilungen dec Satellitenbeobachtungsstetion

Zim~erwald No.

18, 53-70, University of Berne (1985).

8)

I. Bauersima, Navstar/Global Positioning System (GPS) II, Mitteilungen dec Satellitenbeobachtungsstation Zinmerwald, No. i0, University of Berne (1983).

9)

G. Beutler, B. Gurtner,

Influence of Atmospheric Refraction on the Evaluation

Carrier Phase Observations,

Berichte der Satellitenbeobachtungsstation

of GPS

Zimmerwald,

No.

16, University of Berne (1986). i0) Y. Bock, private communication. 11) M. Rothacher,

G. Beutler,

Suiss GPS Campaign,

A. Gelger,

B. Gurtner,

H.G. Kahle, D. Schneider,

Fourth Int. Geodetic Symp. on Satellite Positioning,

The 1985

Austin, Texas,

(to be printed 1986). 12) J. Gervaise, G. B e u t l e r , •. tial,

Gurtner, M. Mayoud, Proc. o f FIG Groups 5b and 5c on I n e r -

Doppler and GPS Measurements for National and Engineering Surveys,

Munich,

1985,

( U n i v e r s i t ~ t dec Bundeswehr, M~nchen, 1985), pp. 337-358. 13) C. Boucher, G. Nard, Proc. F i r s t

Int.

Syrup. on Precise P o s i t i o n i n g w i t h GPS, R o c k v i l l e ,

1985 ( N a t i o n a l Geodetic Survey, R o c k v i l l e ,

1985), pp. 135-146.

GEODETIC NETWORKS FOR CRUSTAL MOVEMENTS STUDIES P. Baldi and M. Unguendoli University of Bologna, Italy. ABSTRACT Starting from a short review of the geophysical phenomena representing the major causes of crustal movements, we analyze some problems related to the geodetic networks used for the detection of the movement them selves. Some examples of real networks set up in Italy are also shown and discussed.

I.

INTRODUCTION

The application of Geodesy to the obser~lation of crustal deformation can be divided into two main categories: the continental scale monitoring of the relative movement of tectonic plates, and the regional or local scale observation of deformation connected with seismicity, volcanic activity and subsidence. On an even smaller scale one might study the phenomena of landslides and land settlement etc. Since the reIative movement of continental blocks is estimated to be within the order of a Few centimetres a year, the relative positions o£ the points need to be defined with an accuracy of 1 0 - 8 1 0 - 9 This is now made possible by the use of Satellite Laser Ranging and ~=rc~,~e~~r~" (Ve:'f Long Base T............... ~ ÷ ~ ~ -~,-,,.,-,'.~,~ * - , ' .j v ~j~ Extragalactic Source Radio Int=~ --~^ On the other hand, the use of artificial satellites, particularly those using the GPS (Global Positioning System), will in the next Few years furnish us with an exceptionally versatile instrument for small and medium scale geodetic applications; the use of a constellation of 18 satellites permitting one to determine the 3~lative positions of points on the Earth's surface with an accuracy in the order of 1 crn " However, classical geodetic techniques, i.e. the repeated measurements of distances, angular height differences, direction and intensity of the gravity Field, remains, at least for the time being, the most effective way of observing and checking defor~lations associated with seismic activity and/or other local or regional scale geophysical phenomena. Here crustal movement is generally subdivided into horizontal and vertical deformations. The techniques corsnonly employed consist of triangulation and trilateration of reasonably complicated networks For the Former, whilst vertical deformations are studied by means of geometric or trigonometric levelling integrated with gravimetric, with clinometric, extensimetric and mareographic measurements. In some cases, such as mountainous areas, plano-altimetric networks can also be used, integrating the various types of measurement, and including astronomic observations in order to obtain thr~e-dimensional networks. There is no doubt about the importance of studying surface deformations that may be connected with geological and geophysical phenomena taking place beneath the Earth's crust; one need only think, For example, about the possibility of monitoring in detail perhaps the most Fascinating, though still not well-defined, phase of the seismic cycle taking place before an earthquake and characterised by the presence of the phenomena known as precursors. This phase, observed prior to several earthquakes, can be com_nected with changes in the physical properties of rock subjected to tectonic forces, according to models amply dicussed by various authors. Without doubt, anomalous deformations of the terrain seem to be the most easily recorded signals of such activity among the various other effects observable, (namely the velocity variations o£ seismic waves, changes in rock resistivity,etc. ). Monitoring of the deformations in volcanic areas using geodetic techniques in association with seismic and geochemical measurements etc., is particularly important and useful when it facilitates the making of "deterndnistic" previsions. Let us finally remember the possibility of monitoring and thus modelling subsidence phenomena, which, if not forecast or controlled, may result in damage to or destruction o£ surface Facilities, the irrigation system and ground water regime, etc.

136

2.

SOME EXAMPLES OF GROUND DEFORMATIONS

2.1

Seismic ar~as

The pattern of crustal deformation in a seismic area can provide essential inforT~ation on the state o£ stress, and thus furnish further details concerning the danger level of the area. Generally speaking, a stationary deformation may be correlated with the seismic risk of the zone, the level o£ which depends upon the speed of deformation, and the time elapsed since the preceding earthquake; in those cases where the entire deformation characteristics over the whole area are known, the crust~l movements measured can be used to estimate the period in which such events might recur 4 ) Clearly an affirmation of this nature would be based upon the hypothesis that the deformation cycle o£ large seismic events is not complicated by complex phenomena, and is repetitive. In the schematisation o£ Fig. I this is shown in the presence of a pos_tvseismic transient and permanent deformations modify the simple scheme proposed by Reid b ). At the present moment the data available for various seismic areas in the world indicate a post-seismic transient in the overall coseismic deformation, notwithstanding the fact that a part of the period concerned shows constant deformations. Many cases seem to show that a large number o£ seismic deforTnations take place in a short time (several years ), and are then followed by a slow and gradual transition to pre-seismic deformation levels.

;¢

O n,,

O k~ tU El

• PERIVIANENT DEFORMATION TIME

Fig.

1 Diagrams of seismic cycle

Many authors have developed models to explain time-dependant crustal deformation observed in regions of ongoing tectonic activity. They found evidence for the existance o£ viscoelastic relaxtion in the asthenosphere. In particular, 'visco-elastic stress relaxation was found to be the cause o£ the long-terTn inter-seismic/aseismic deformation occuring between major earthquakes in several regions of Japan. In this case the deformation takes the form o9 subsidence in the source region, with a compensating uplift on the flanks of the subsiding region. The width o£ the subsidence profile seems to be roughly proportional to the elastic plate thickness, and its time development was found to depend on the characteristic relaxation time ( T = 2 ~ / ~ ), where ~ is the asthenospheric viscosity and ~ is the asthenospheric shear modulus).

137

InFormation concerning structural deformation on a geological scale allows one to point out permanent deformations. It is, however, necessary to bear in mind the Fact that some of the deformation over the centuries may not be strictly connected with the storing of elastic energy, and that anomalous crustal movements, defined as precursor phenomena, may be adding their presence to the deformation cycle; a presence which in many instances shows different characteristics, overall behaviour and duration From that oF the characteristic seismotectonic changes in the area. As an example of the possible complexity of deformation see Fig. 2 in which are laid out the altimetric variations in the region of Honshu (japan). Three different Fields of deformation Following three different processes can be identified: the dominant process produces a constant subsidence related to the westerly subsidence o£ the Pacific Plate along the Japanese trench. Local effects are added to this movement, namely uplift o£ the Quaternary structure, and sinking in the locality of the Fault associated with the 1896 earthquake (M=7.3). In Fig. 2 the heights over time variations are repeated in relation to two different zones A and B. Whilst zone B, which is Far From the Fault, shows a linear process, in zone A the deformations agree with those Forecast by the theory of stress / relaxation in a viscoelastic asthenosphere following a Faulting 6 . Important Features of the earthquake mechanism include the size and shape of the rupture surface, its orientation, the Faulting motion on this surface, and the time history of the process. Seismic data, in general, provide information about seismic source mechanisms; geodetic surveys could plxxluce important advances in our knowledge of earthquake source models. Theoretical expressions for surface and subsurface deformations accompanying Faulting, obtained on the bas~.oF the elastic theory of dislocation 7,8) have been given by Press 9) , M~_nsi.~a ~nd Smylie ) and others. For example, the horizontal displacement components are derived For a Finite rectangular strike-slip Fault in a uniform elastic halE-space with the geometry shown in Fig. 3. In particular the horizontal displacement component U along a median line perpendicular to the Fault strike is shown in Fig. 4.

)

A CA .J:

42. E 40° 1900 1975

/

~cm)

-2o0

1 lOOK,./

138°

Fig. 2

144 °

8.

. - ~ ' % ~

.'" ~ . ~ i

B

......i

C"

_3ooi' .....

1900

1920

1940 TIME

1960

Level changes, 1900-1975, in Northern Honshu, Japan, and region of surface Faulting in the 1896 M=7.3 Riku-u earthquake (Thatker, 1981 )4)

138

,~

,~

r

,---v--

Z.

z.

z.

z.

z..

~Y

7/

,

¢

/

// FAULT PLANE

/

?

V

/

Z

Fig. 3

Horizontal components of displacement, for a finite rectangular strike slip fault

U

X d= O k m

O= 9 0 °

d=5km

D=15km =~)=~n=

d =5kin J D = 15 km

5/

0-

75 °

d=5km

d-5km

D = t5km

D= 25 km

=-

| /

Fig. 4

¥(km)

d-t5km D= 25 k m 8 -75"

U component o£ displacement for different paraneters o£ a strike-slip fault . The displacement is given as a fraction o£ the slip o£ the fault.

139

In general, the assymetry o£ the deformations at the two sides of the fault is strictly correlated to the slip and inclination o£ the fault plane, whereas the displacement fields near to and distant from the fault are indicative repectivelyof the upper and lower iL~dts o£ the fault. Obviously, if we consider a more realistic model, in which the uniform elastic half space is substitued with a variable shear modulus, we obtain different results. In fact asstmdng that rigidity (z) is a continous and smooth function of depth: ~(z)

: C (I - Te

-Z/Zoj

where the value of -,.T1i~2)derivedFrom seismic data (T=0.9) as well as From laboratory measurements (T=O,5) ; Mahrer and Nur 13) obtain theoretical displacements almost identical For different values of T in the case of surface Faults; in contrast, in the case of a buried Fault, a notable dependance is present (Fig. 5).

1.-

T=O.

o}

DEPTH (knl) FAULT

b

-g0.5-[

UJ

5

0,5

-I

-°'2 E I-.

DISTANCE FROM FAULT qkin}

"~T=O,9

c

z

uJ LU

(..)

5n.. _

{ DISTANC2 FROM3FAULT {kin) Fig. 5

Normalized shear modulus as a function of depth (a);surface displacement as a function of distance from the £ault:(b) surface fault case;(c) buried case. (Mahrer and Nut, 1979) 13)

140

It is evident that the quantitative information obtained by geodetic measurement should be interpreted in the light of one's total knowledge regarding the phenomenon being studied, and o£ the various factors that might strikingly influence the surface effects associated with a state o£ stress in the crust, or more simply a slip along a fault plane. One thinks, for example, of the possible correlations bebmeen morphology and disturbances in surface deformations using a simple elastic model. Starting in fr~n a model with topography o£ small slope, McTigue and Stein 14 ) analyse those deformations associated with a variation in the uniform tectonic force (Fig. 6 ), establishing that topography gives size to local departures from regional strain of the order o£ the characteristic slope (-~). Considering instead structural discontinuities caused by surface structures (alluvial deposits), one sees that these can introduce notable disturbances (dependant upon rigidity contrasts) into the general field of surface deformations, even if, in general, they are limited to areas of similar dimensions to those of the disturbed tract Fig. 6. To these sources o£ disturbance one should add the local, changing effects produced by the weath~, and related to rainfall, changes in atmospheric pressure and temperature, etc., which can generate a background noise detectable using modern geodetic instrumentation.

-LII

=e~1

I H

-"k~OPO

GRAp H Y ,,,-- x

U ~II_,V2j

HORIZONTAL DISPLACEMENT

-0,5 - 1.0

_2

V

0.25 0.0

_X_ L

.2

~1

VERTICAL

~ (l+'l'Xl+2~')

DISPLACEMENT

-025 .2

X

,2

L Fig. 6

~xample o£ dimensionless perturbation displacement field, due to a Far-field tectonic stress (compression); the surface relief is assumed to be characterized by a horizontal length scale, L, and a vertical length scale H. (from McTingue and Stein, 1984) 14)

141

2.2

Volcanic areas

A classic case where the integration of a multitude of different techniques is indispensable, is undoubtedly the control of deformations related to volcanic activity, due to their speed of change, and the real possibility of making previsions. Subsurface movements of magma generally cause rapid movements of the surface, with uplifting and subsidence that alternate and vary the magma level and pressures involved° We have laid out the case of Campi Flegrei (in the region of Naples, Italy) as an example where periods o£ rapid uplift, accompanied by intense seismic activity, alternate with periods of calm. The main features of the expansion can be sur~narised as £ollc~gs: the expanding area of quasi-circular shape, and radius o£ about 7 kin, shows maximum uplift near the town of Pozzuoli (Fig. 7). The uplift practically vanishes at the edge of the caldera. The speed of uplift (as high as 4-5 nvn/day) and its change over time constrained the operators £oll(~#ing the phenomenon both to set up mareographic and clinometric networks in the area, which permit continual monitoring of the phencmenon, and to carry out the periodic checks by geometric levelling using a high number of working teams. Horizontal measurements provided additional information on the strain field. The displacement vector referred to a point located in the area of maximum uplift, appearing to be roughly perpendicular to the contour lines of ground uplift, with an average horizontal strain value of about 10 -4. To this, microgravimetric measurements were added because of the magnitude and speed of the deformations; these being extremely useful for monitoring, as well as for supplying an important quantitative element in the definition of the source model. Cm

I

80 60 .1=

40

~°:'.:

•

"a ~ J

j.

"

.

20 0

1970

1975

1980

b

2~

cm

~

~

Km

Fig . 7 (a) Vertical ground deformation at Pozzuo!i ( I t a l y ) during 1970-83 period ; (b) area d i s t r i b u t i o n of v e r t i c a l ground deformation, and horizontal displacement for the period 1980-83 (from Berrino et al,, 1985)15)

t42

2.3

Subsidence

It is a well known fact that extensive withdrawals of fluid from subsurface resources is the major cause of land subsidence. This is due to a subsurface compaction which represents the response of a compressible, porous, medium to changes in the fluid flow field operating within it. The artificial withdrawal of fluid modifies the natural balanced condition by causing a decrease in pore pressure and a consequent increase in intergranular stress. Under the influence of this increase in the effective stress, the Formations compact and the land surface subsides. Subsidence may in general be connected with vertical and horizontal surface displacements, cracks and fissures in soft ground, changes in the hydrological regime, contamination o£ fresh water aquifer, changes in local tectonic regimes due to altered stress and strain conditions with a consequential risk o£ increased seismicity, etc. The qualitative and quantitative determination of the parameters governing the subsidence process involve measurements and analyses of geological, geotechnical and geophysical quantities, as well as surface and near surface measurements o£ vertical and horizontal movements, such as levelling and trilateration surveys, tilt-meter and extensometer surveys. If a complete set of the above-mentioned information is available, the complex natural system will be a mean o£ those mathematical and numerical prediction models formulated from the equation of subsurface flow combined with the elastic equilibrium equations. Major subsidence in oil and gas fields, or due to ground water pumping, has been extensively covered in the literature 16 ~ For example, a surface lowering of about 8m from 1937 to 1962 has been observed in the harbour area of Los Angeles and Long Beach. A more recent case o£ particular interest is the subsidence of the Po Plain (Italy) and Venice 17 ), where the lowering, though modest, is nevertheless a matter of concern because it compounds the effect of the exceptionally high tides in the city.

3.

GEODETIC FZASUREMENTS

Having started with this short review of the geophysical phenomena representing the major causes of crustal movement, we will now move on to consider the geodetic method used For the detection of such ~ovemlent, basing our lecture more on practical than on theoretical considerations. We will thus describe classical networks, not~,ithstanding our awareness that soon much will change with the extensive use of spatial techniques (GPS), since at the present moment we do not have sufficient experience with such techniques to be able to supply exhaustive answers to all the problems related to their application. As we have mentioned, we will also subdivide our exposition into three different subjects: planimetric, alt£metric and plano-altimetric networks.

3.1

Planimetric networks

Horizontal crustal deformation mey be measured by means of a survey involving mngles and distances; electromagnetic distance-measuring inst~oments are able to determine distances with an accuracy of I ppm or less, according to the extent to which one is able to estimate the average propagation speed o£ light in the atmosphere, using either meteorological measurements ID,I 9 ! or by simultaneously measuring optical path length at two or three different wavelengths 20,21 ) With a view to developing adequate geodetic networks, various criteria, such as accuracy, reliability and expense, need to be balanced in order to optimise the network. Such optimisation is usually classified in Four different orders, starting with a parametric adjustment based on the observation equation: Ax-d=v which, by means of a least-squares solution and following the free adjustment scheme, gives:

143

X = ( ATpA)-1ATpd or following the free adjustement scheme x : (ATPA)+AT~d

and with the co-variance matrix: K

= (~ATpA)+

(~2 =

xx

~x

2 where A is the configuration matrix, P the weight matrix, a is the variance o£ the unit weight and ( ) + denotes the Moore-Penrose inverse. In the case of planimetric networks, for example, disregarding the "Zero-order design" relating to the optimal reference system, the optimization process is carried out by means of 'First, Second and Third-order design", which relate respectively to the configuration of the network and observation plan, to the search for an optimal distribution of observation weighting, and to the improvement of the network by including additional points and/or observations 22 ) One of the criteria used to evaluate the reliability o£ a network, is based on the analysis o£ the diagonal elements r of the matrix R, where: R = A~xA

T

P-I

These elements may vary From O to I ; good reliability is obtained For high values of z, as well as with a h ~ e n e o u s distribution of such values throughout the entire network. As regards "First-order design", the theoretical approach is severely limited by topography and the available information relating to the geophysical phenomenon in question. As an example o£ theoretical trilateration networks we show you those in Fig. 8 which correspond to differing needs: the first is used when it is impossible to localise the direction of expected movements, and the others to concentrate research on the deformation along a traverse of the strike fault. 10

11

A

°

'

tO

Fig. 8

'D

11

'

,

-,~

~'

"E

10

Diffe~nt sch~es o£ trilateration ne~orks.

11

B

t44

On the basis o£ point distribution, measurements should be planned seeking the greatest possible reliability and precision, without forgetting the question o£ cost. The most practical approach is the interactive simulation. Once the network has been drawn up and the theoretical aspect o£ the measurements established, one can take steps to improve the accuracy o£ single measurements, with a view to obtaining a better overall network design. This may be done by seeking a method for weighting the measurements themselves which respond more satisfactorily to the needs for which the measurements are taken (second-order design ). In effect, the analytic approach is founded on the solution o£ the following equation, as it relates to P: T (i% PA)

= Qxx

in which the criterion matrix O must be defined in advance. In general, a geodetic ne~ork should tend towards homogeneity and isotropy. In this case, 0 is normally constructed using the Tailor-Karman structure solution 23 ), the simplest case o~Xwhich may be reduced to the equation: 0

xx

where I is the unit matrix, involving the coordinates is assumed to be zero.

=I drawback

that the

co-variance

between

the

A different approach, which appears to yield results more in line with real needs, consists in the use o£ the singular value decomposition o£ the matrix 0 of the network:

0= = vAv T where the diagonal matrix j contains the eigenvalues o£ Qxx ' and V is the matrix o£ the corresponding orthonormalised eigenvectors. A possible derlvation o£ the criterion matrix might consist in contracting the eigenvalue spectrum, without changing the eigenvectors, with a consequent diminution in the semi-axes o£ the error ellipses. A different approach, o£ particular interest in the study o£ deformations, mlght be to obtain a variation in the orientations o£ the error ellipses semi-axes by rotating the eigenvectors matrix v 24) . In any case, these methods are severely limited by the shape of the network and also, especially in distance measurements, by instrument limitations. For these reasons practical results are generally negligible, but second-order design nevertheless supplies us with much useful information for the planning and execution of networks. We have extensively used second-order design and reliability criteria in an undognatic way to reduce the measurements in a rational manner involving only a slight loss in accuracy, with a consequent reduction in costs and execution time. In trilateration networks, for example, starting with the maximum number o£ measurements possible, we eliminate, on the basis of second-order design, the least weighted sides one after another, even i£ they are characterised by high reliability. This method, which may be easily programmed for automatic processing, gives good results and can be stopped when the points error becomes greater than a pre-fixed value and/or the minimum reliability is less than an acceptable value (Fig. 9 ). Second-order design can also be useful for solving some of the problems related to the t'hird order, such as adding measurements o£ different kinds. As an example o£ this, see trilateration network E o£ Fig. 8 in which the error ellipses are very flat, and may therefore be improved by the addition o£ angular measurements, in this example the angles around points 4,5 and 6 improve the isotrephy of the net. They were found by applying an empirical trial and error procedure, taking into account only those angles with maximum weighting• %~nen a network has different types of observation, (angles, distances, Laplace azimuth... ), correct weighting o£ the observations presents a problem. This is a very difficult matter and at present we think that the best method is an empirical approach based •

,

145

on experience, also if we look with interest to the method proposed by Prof. Kubik 25), for an a posteriori estimate of the various measurements' weighting. This method has been tested in my institute with good results 26).

I

r 0.5

(cm)

0.4

2.0

0.3

rml n

.., . . . - ' / i i

0.2

1,5

0.1 I

I

I

I t

t

I

55 Fiz. 9

30

SIDES

E F f e c t of the r e d u c t i o n of m e a s u r e d sides on r e l i a b i l i t y and p r e c i s i o n (net B, Fig. 8)

We shall describe at length the measurements themselves because they are so strictly related to the instruments used. For instance, atmosphere related problems can be overcome using a two-wavelength instrument such as the tetrameter, while they represent the most limiting factor i£ a one wavelength model is used over long distances. Although much research has been devoted to this problem 27 ) we think that any atmospheric model used to correct measurements is likely to be Far £ ~ perfect. We have obtained good results using the method proposed by Robertson, based on the relation between two lengths measured from the same point in two directions with a similar topographical profile in the same period of time 28,29). For short or medium sides one tends towards limitation of the instrument's base error by using a graduated slide for the reflectors 30,31 ) As regards the adjustment and analysis o£ data we have tended towards procedural rationalisation using a series o£ well organised progrmrmes: I ) Handling and organisation of the data. 2) Adjustment by indirect observation. For the free net we use the bordering technique or -~ --| the expression N = N (NN) or the singular value decomposition N = USV T , N+=V I' SU. For the inversion of the regular matrices we use the modified Cholesky method. 3 ) The detection o£ the outlayers can be performed using data snooping and considering weighting matrix to be a diagonal using the variable

the

V, 1 W

=

av i I if lW. I > F~ ~.~ ~ (normally 3.29) the observation is eliminated and the adjustment is 1 i--~,i, , r e p e a t e d . I t i s p o s s z b l e t o have an i n t e r a c t i v e v e r s i o n o£ t h i s p r o c e d u r e . One can a l s o use an alternative approach based on the minimisation o£ the sum o£ the absolute value of residuals jz), ~which in some situations seems to be more efficient 33 )

146

4 ) The analysis of the network is carried out by considering the absolute and relative error ellipses, distance error, and that o£ the bearings between pairs o£ points. The ellipses can be plotted automatically. 5) Other programs for similarity transformation, for testing the significance of supposed movements, for studying the local strain field etc. are needed. From amongst the various networks installed in Italy to study crustal movement, let us show you as an example the one we set up in the Cassino Area (south of Rome ) in 1984. Starting form the geophysical need to connect three mountainous blocks separated by two supposed Fault zones, the work sequence has been as follows: I

to search, using suitable cartography, for those intervisible points which might be to form a very compact network (within a circle),

used

to mount a field campaign in order to verify the supposed possible connections by means of powerful lamps; to verify the possible routes and transportion needs (special cars, mules, helicopters.. ); to obtain permission from the owners o£ the land; to study the geological and lithological features o£ those places selected in order to choose the foundations o£ the bench-marks (reinforced concrete pillars ); and so on. After that we planned various possible networks on the basis o£ optimisation criteria, taking into account all the information collected during on-the-spot investigations. In this manner we chose the network in Fig. 10 consisting o£ three points on every three blocks. 4) Another field campaign was necessary to construct the pillars, built upon rock foundations, and with a self-centring device on the top. On a geologically stable block we installed two pillars in order to have a base line for the calibration of the instruments before, during and after every measurement campaign. 5 ) During the first measurements campaign we experienced difficulty in measuring some o£ the long sides, and thus decided to add another three points in order to eliminate such long connections. The final network is that shown in Fig. 11 in which the error ellipses are laid out. As you can see the network is characterised by very good homogeneity and isotrophy, which is most important since the direction of the expected movements is not known.

3.2

Alt£metric networks

A series of high-precision levelling c&mpaigns is the most fruitful method o£ investigating vertical crustal movements; using this technique it is possible to examine in detail and with great accuracy displacements extending over even the largest areas. Vertical control networks should be planned bearing in mind a considerable number of parameters, such as the area affected by deformation, the characteristics o£ the displacement and its supposed magnitude and velocity, the type and location o£ the bench marks and their density, the possible level routes and the degree of accuracy expected, also taking into account the high cost of such measurements. It is not possible here to give a detailed examination o£ all the problems associated with precise levelling, such as instrumentation, operation methods, refraction, magnetism, rod calibration, bench-mark stability and so on. We will only emphasise the fact that interest in this technique has increased to such an extent that entire conferences are dedicated to it, many interesting suggestions and ideas are given in their proceedings 34,35,36) . As far as accuracy is concerned, we have no problems in the detection o£ coseismic deformations, large scale subsidence or bradeisism. However, we

147

N

o,.,,

E~,'I Fig.

10

8 k,,,.

o '!' ~' c,..

Scheme of the trilateration network o£ Cassino and comparison between the error ellipses obtained in the First c~npain and the expected ones (dashed lines)

N

5 I

Icm, Fig. 11 Scheme of a modified network in Cassino

10 km. I

148

encounter some difficulties in other cases, such as pre-seismic or post-seismic deformations, or with a slow subsidence or uplift where the small degree o£ movement concerns a large area. Amongst the various souses o£ error we shall consider the following two in particular: refraction, and movement occuring during the measurement phase. Regarding refraction, the greatest limiting factor for lines in hilly areas, there are essentially two ways in which it may be taken into account: the first is based on the well-known Kukkumaki 's formula R = -10-6 A (5~----) (~h (~t L = length of sight in metres 6 h = difference in level in metres ~t = difference in temperature in degrees at two hights, normally 0,5 m and 2.5 m. A

1190 c c z 2- z 1

1

I~7T

, C+1 C+1 ~zI - z2 ) -z 0 (z I - z 2)I

where Z is the height o£ the instruments, ~ and Z 2 6L~e the sight heights, and C is a 0 coefficient the value of which is normally assumed to be equal to - I/3. The second way is based upon the creation of atria)sphericmodels which provide a means o£ adjusting the measurements using one temperature and general data regarding insolation, wind, etc. Different techniques can be applied to check on any possible movellents taking place during the measuring campaign, or to obtain a periodic indication of whether or not it is necessary to repeat the measurements over the entire network. One o£ these is based on the continous recording of tilt components by tiltmeter; another method we have used extensively is based on the use of double levels at night 37). This technique makes it possible to ascertain with accuracy, and within a few hours, the difference in height between points at almost the same height, but located at a considerable distance (3-6 km) from each another. By way o£ example we have set out the closing error o£ a control triangle used in various campaigns. ~

I

97___7

1978

1980

-0,9

1,4

It must be remembered that the length of the precise levelling lines necessary to join the three points is over 20 km, with routes characterised by average differences in height o£ 50 mlbn. Having an idea o£ the movements occuring during the measurement period, it is possible for one to take them into account during the data processing phase 38 ). Neverthless, we think that the best method is to limit the entire network's measurement period to as short a time as possible. This can be achieved by contracting small areas of the network to a variety o£ companies working simultaneously. The need to employ several firms is typical o£ levelling networks, and this fact should be taken into account when preparing the specifications for the contracting finns concerning instrumentation, rod calibration, observation procedures, refraction, tolerances for a single section, for a line, for a loop etc. 39) In addition we must emphasise that when all the blocks, already analysed separately, are put together, other errors arise; it by no means being a simple matter to complete a serious data analysis or the adjustment o£ a whole network consisting of hundreds or thousands of measured sections. As an example of this kind o£ work we present in Table I a flow diagram o£ the program prepared in my institute by Prof. Barbarella.

149

Table I Flow diagram for network adjustment

"I

I

Field Data

1

Handling of baba (Code number, coordinates ) r . . . . . . . . . . .

ICorrection Or . . . . . .

L o£

t" . . . . . . . . .

[Search !eri'ors

,,

I,

4

Automatic set up o£ condition equations ; Adjustment ;

using

griph

I I

theory '

,.

Itio[

"- ----'I

data

L- . . . . . . . . . . . . .

j

,~

Anal;sis o£ closing errors

[

Are they acceptable?

I

L

yes IStorage of. lllf.or-!

Imation of" the bench-marks o£

[.4~--

Ithe liiles

i

U- ......

-~uM_ap of. the net]

Search of independent loops[

', - I

I . . . . . .

~

"i .....

I

! I'elneasuremeut

--'

•. . . . . .

_J

1

I

,.

;.*.....,

heights using graphs

Alternatively

Computatioll of pseudoobservations :diFFerences in height a.d weight of. the lines

I

Adjustment of. quasi-

]

[

iF-Rejectio,,

or

~--~, remeasuremeut I !o£ some l i n e s t L .... l .....

-1 1

I I

J

] observatious by parametric me thod

t

m- 1

I t e r a t i v e data S | l O O p i n g Or

It°bust identi{ £ication

Adjusted h e i g h t s and m . s . e , o£ nodal p o i n t s

Computation o£ h e i g h t s and m.s.e, o£ all the benclrmarks I . . . . . . . . r- . . . . . . . . . . . .

"I

IPrevious heights ~ Lo£ bench-mar~s .]

i

T

Computation o£ the differences I and automatic design Of [

--[ [contour

lines o£ equal movements[

One can also apply other statistical tests For the study o£ error behaviour in order to obtain information about trends due to systemlatic errors. The tests are baser] on the random variable

150

m~

iJ.

ij

R.. mJ

in which is the discrepancy between the direct and reverse measurements of the relative height of two consecutive bench-marks, and R is the distance between them. The method is based on tests concerning each line, and cumulative tests regarding the entire network, preferably using non-parametric tests because of their generality. Many tests can be used, such as, for example: randcmness tests, nonnality tests, the Wilcoxon test, the Person P. Test, the Kurskal Walis Test, the Bertlett Test 40). At the end of this short dscription o£ altimetric networks we must briefly make one or two conments concerning gravity. Gravity remeasurements may to some extent be used as a substitute for a levelling survey. The accuracy obtainable is less satisfactory, but costs are lower and the method is quicker. Simultaneous monitoring of elevation and gravity change D~ovides us with further e±ements for the interpretation of the phenomenon in question 41) For example, generally speaking the gravity variation corresponding to a vertical displacement caused by an acct~nulating stress field in a seismic area, can be expressed as a sum o£ the free air and Bauguer effects, plus the gravity contribution made by the subsequent density change in relation to the volumetric strain 42) With the use of m_~ern ~crogravimeters, (e.g.: la Coste-Romberg rood.D), standard errors smaller than 5.1(7~ m/sec may be obtained 43 [. Improvements in accuracy depend mainly on the el Jr/nation of effects caused by earth tides, tidal loading, atmospheric mass movements, and so on.

Fig,12

Scheme of the altimetric network in Bologna (Italy)

151

As an example of an altimetric net we shall show you the one set up in the Bologna area, (our home city), where over the last ten years we have had a differential subsidence velocity of at least about 11 on/year, causing many problems regarding the stability o£ buildings, and water management. The network planned in my institute by Prof. Pieri and Dr. Russo was conceived in order to give diF£erent levels o£ information. In Fact it consists o£ a large network, (460 km 2 ), in which the bench-marks are spaced from between I to 0.5 km along the lines, and the distance between the nodal points ranges from 7 to 3 kin, the density increasing towards the town; (areas I ,2 and 3 o£ Fig. 12). Another network, (Fig. 13), covers the old town, with the bench-marks spaced at intervals o£ 0.25 km. along the lines, and with a distance between nodal points o£ about 0.5 kin. Finally there are some local networks set up £or the purpose of studying the movement of buildings or monuments, often in conjunction ~
Fig. I 3 Scheme o£ the altimetric network in the town o£ Bologna

152

~r~:h

i t88/5

Now.1979~

)

/

5

_..4---

MI

10 15 20

~,c.19a31i//'/j'/" 0 I

Fig. 14

3.3

100

'

2'''f0 0

m

25 o111

Differential subsidence along a street of the old town (Bologna)

Plano-altimetric networks

In order to give an idea o£ some of the problems related to plano-altimetric networks we shall present an example o£ a network in which we have applied many different techniques in order to obtain the greatest possible information about movements in the seismic area of Ancona, and to create a test network for the study o£ these various techniques. As you can see in Fig. 15, it consists o£ a planimetric network of 10 stations, signalized by means o£ reinforced concrete pillars using 30 optical connections which provide a ver~y good redundancy (1.76), £urther~nore using length measurements only performed by the AGA geodimeter reed. 8. Seven o£ these stations are also connected by a precise levelling network, consisting o£ over 80 km o£ levelling lines with more than 100 bench-marks, and connected with the national first-order levelling network and with a tide gauge located at Ancona. A gravimetric network was set up consisting o£ nine base stations situated on levelling bench-marks. Another two stations were set up on a bell tower for the time monitoring o£ the gravity gradient. The measurements are taken using a La Coste-Romberg rood. D gravimeter. In addition to these base networks we have also set up two altimetric control triangles using double levels (Zeiss Ni2). Astrogeodetic measurements have also been taken at Five planimetric stations For the direct measurement o£ vertical deflection. Consideration o£ all the measurements together in order to make a tridimensional adjustment in a general or local Framework presents many problems. However, this question will be treated at length in a later lecture in these proceedings. Instead we should now like to draw to your attention two problems that need to be solved when dealing with trigonometric levelling. The problem is to improve trigonometric levelling so that this technique may be more widely used. In some cases trigonometric levelling may be combined with, or may even partially substitute, spirit levelling, with a considerable reduction in costs, and with an increased rapidity of data acquisition. To accomplish this we have to reduce the effects o£ refraction, and take into account the influence o£ vertical deflection without astrDgeodetic measurements.

153

N

~P~ .¢

......... 7 "-........ /

0

1

2

km

\-,.,...

Measured side '%

Local levelling lines

\

I.G.M. levelling lines +.÷.÷.4.

÷.÷.÷.÷,

~.÷. ÷.÷. +

,~ Control points double levels • Gravimetric station

(~) Astrogeodet ic station Fig. 15

Plano-altimetric network in the Ancona area

To carry out reciprocal and simultaneous zenithal angles we always make use oF First-order theodolites, (Wild T3, DKM3 ), equipped with a small but powerful lamp For collimation at day or night. The lamp is mounted directly on the telescope and allows it to rotate around its horizontal axis. To link points that are at approximately the same height, (less than 1.5 rn dif£erence), we use, as we have said, double levels with special rods designed for collimation at night. Atmospheric data, ( temperature, pressure, vapour pressure, wind speed), were collected during the measurement period by means o£ small meteorological stations.

154

As is well known, the accuracy of trigonometric levelling is severely limited by the effects of refraction. Essentially there are t~o possible ways of reducing these effects. The first is based on two wavelength angular measurements; and the second approach uses atmospheric models in order to find the variation of refractivity., Since the first method does not seem to be feasible for the near' future 46,47; we have devoted our attention to the extensive use of the second. The theory is well known. S t a ~ t i n g ~ t h the fomnula

hA+hB hA - hB : D (I+ - - ~

) tan ½ (ZAB.- ZBA + dZAB - dZBA)

following Prof. Moritz 48)we have I0-6 ZAB:--~--

sin Z

(9~ a~ ( S - x )

A~

dx

in which the refractivity gradient may be expressed: dT - - : -0,2696 N --~ dN P (0,0342 + ~z ) + 11.25 dz 0

e dT

11.25 T

+

dz

de dz

where the temperature gradient is the most important Factor. As regards the evaluation of the temperature gradient, one can T,~sl<e use of the heat flux estLmate, a pam&meter that can be deduced from the radiation budget estimate. This approach involves the consideration of a large n~mber of additional ,meteorological and ground parameters, such as heat flux into and out of the ground, evaporation, wind spreed, cloudiness, sun height, terrain structure, vegetation and so on. We have applied this technique extensively following the suggestions of many authors ~II ~nov.~ for their interest in refraction probl~uns, such as Angus-Leppan and Brunner 49-52), with good results. Table II Example of result of computation of refraction angles for recirpocal and simultaneous trigonometric leveling measurements ( from Achilli, Baldi and Unguendoli, 1985 ) 54 )

topographic

50

~Nr

.~"~

profile

%~£

6.6 km

.

.

.

Ilch:tclt.tl EqLi~plllclIl

.

.

.

h. I111

Theraclelile

822.9

|3.0

30

--10.64

--8.61 -- 1.19

819.7

*,

82h "~

16.0

27

--10.64

--8.78

817.9

.

81~.0

20.0

25

--19.07

--22.0.I

4.6.3

816.7

.

810.9

2.0

20

--19.26

--2.3.84

10.3-1

82h2

81-1.2

21.0

I!

--24.51

--28.4.1

6.16

819.4

Do.ble levels

Tc:mp~t~tur~'

a

b

I C|

dffl~..,c.cc IoIII

- - 3.27

SO

C,.ri'ctl'd M ~ , I ,d d~.. Mu~n . ! ih~.

"rim,e Ihl

dlll~',c.~Jcs

" I

(cml

(¢ml

lahlcs (~v.I

816l(~

819.0

sahl~s

155

AS an example of how the method may be applied, Table II sets out the results of five series o£ reciprocal and simultaneous measurements o£ zenithal angles relating to a link 6 ~n in length belonging to a triangle, taken in a variety o£ atmospheric conditions. The corrected values listed in Table II show that the deviations from the mean are much smaller than in the previous ease. Furthermore, correcting the other two sides of the triangle in the same way, it should be pointed out that the mean value of the corrected determinations shows an appreciable L~mprovement in the misclosure error. a

geo~d ¢; = o c a l a n o m a l y of vertical deflection

6~ = v e r t i c a l

Fig. 16

et "/~

deflection

o'

,s

%. "o"

Scheme of the local anomalies of vertical ~5lection (from Achilli, Baldi and Unguendoli, 1985 )

Strictly speaking, if we want to refer the trigonometric height to the same surface, (the ellipsoid ), we have to know the vertical deflection, obtainable by astronomic measurement, which are a little troublesome. In control networks where the area involved is always small enough, the problem of the reference surface may be solved in a local way, taking into account only the local ananalies of vertical deflection, and obtaining a smoothed reference surface that we call a "smoothed geoid" (Fig. 16 ). These anomalies are mainly due to the distortion of the gravitational field caused by the distribution of the surrounding masses. For the computation of these anomalies we make the "mass model" with a program prepared by Achilli and Baldi 53 ) and consisting of an evaluation o£ the disturbing gravitational attraction exerted by the surrounding topography, and by density anomalies. The area is subdivided into a regular quadratic grid, with the side lengths constant or diminishing towards the point considered. The area is therefore represented by a set of parallelepipeds with a density determined from geological information, the gravitational components of which can be calculated by means o£ the Formula: T;=

n ~ i=I

t . (P); xl

T~ =

n ~ i=I

t . (P) yl

in which:

tX 0G /Xl x2fjzi2 -I

x

l(x_a)2+(y_b)2+(z_c)2,i,,3/~ dxdydz .

156

The components of the local anomalies in vertical deflection are thus: T £ = arctan

T ;

T

~ = arctan

g

To verify the correction methods for refraction and local vertical deflection, we carried out, on a quadrilateral o£ the Ancona network, a series o£ trigonometric height measurements in different atmospheric conditions, paying particular attention during the measurement stage, and collecting atmospheric data. The sides range From 5 to 11.5 km., with a maximum difference in height O~ 400 m. The mean results o£ three series of measurements are listed in the table, and as you can see the closing error of the independent triangles improves markedly with the application o£ the corrections described above.

Table III Differences in height obtained by means of trigonometric levelling and correction to take account of refraction and i ~ anomalies in the deflection of the vertical (from ichilli, Baldi and Unguendoli, 1985)

side

~h(m) , measurea

correction (.,)

I-2

386.89

-0.0[

l-]

54.51

O.

1-4

Lt5.79

O.OL

2-3

-332.4t

0.06

2-4

-27[.18

0.09

3-4

6L.L9

0.03

Z

11,6

3

MISCLOSUEE ERRORS

triangle

measured (m)

corrected (m)

t-2-3

-0.09

-0.04

l-]-~,

0.03

-O.Oi

1-2-4

-0.08

-0.0[

As we have said, seven plan[metric stations of the Ancona network belong to a precise levelling network, and their heights were obtained both by trigonometric and spirit levelling, so that one could make an interesting comparison between the two means of measuregent. This kind of comparison is only possible if the two types o£ measurement refer to the same surface, which, given the small size of the area involved, can be identified with the smoothed geoid. We have used the mass model, which permits calculation of the local anomalies o£ the geoid (Fig. 17 ). By means of such geoidal undulation the heights obtained by spirit levelling were corrected, while the trigonometric measurements were corrected for refraction and for the local anomalies o£ vertical deflection. The results set out in the Table IV show a high level of congruence between the two height measurements, with a maximum difference o£ 2 cm.

157

13'26'

13"38'

4

43"38'

43"30' I;J ;JU*

l~p z u

Fig, 17 Local anomalies of geoid in the Ancona area (nun) Table IV Comparison between height measurements obtained by means o£ trigonometric levelling and spirit levelling (from Achilli, Baldi and Unguendoli, 1985) 54) N

4

3

TR[GONOMETRI.C

1.EV£LL[NG

$P|R[T

L£VELL[NC

c o t' z'e c I::e d a . d

L 2 3 4

|95.03 238.23 $82.77 121.57

5 6

2SI.L~ 3LL.GL

7

250.40

L?5 83 238 L9 ~82 74 L2L ~5 251 12 311 65 250,~L

[95.83 238.18 582.75 12L.~5 25t.13 3tL.67 250.37

158

4.

CONCLUSIONS

As a conclusion to this short lecture on deformation measurement, we should like to consider the £oll~ing: Firstly, concerning methodology, the philosophy behind the m ~ e in which we take our measurements is itself conceptually different from the classical determination o£ points location. In fact we measure large quantities to obtain small ones, and those probl~ms related to precision, measurement conditions, instr%~nentation, and data analysis are more difficult and more delicate. Secondly, considering the type o£ measurements to carry out, we think that the best way is to move towards an integrated application of temporally spaced measurements, (plano-altimetric networks, gravimetric networks, astrogeodetic measurements... ) and continous recording, (tiltmeters, strain meters, hydrostatic levelling), in order to acquire a better knowledge of the phenomenon studied. Finally, we think that in this Field firm cooperation between different disciplines is absolutely essential. Geodesists, Geophysicists ~ d Geologists have to cooperate strictly in the various steps o£ the work; i.e. in the planning and execution o£ measurements, and in the analysis and interpretation o£ data.

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