Table Of ContentLecture Notes ni
Earth Sciences
Edited yb Somdev Bhattacharji, Gerald .M ,namdeirF
Horst .J Neugebauer dna Adolf Seilacher
52
E )~snaS .R Rummel ).sdE(
yroehT of
etilletaS Geodesy dna
ytivarG dleiF noitanimreteD
galreV-regnirpS
Berlin Heidelberg NewYork London oykoTsiraP Hong Kong
Editors
Prof. Dr. Fernando Sansb
Polytecnico di Milano
Istituto di Topografia, Fotogrammetria e Geofisica
Piazza Leonardo da Vinci 32, I-Milano 20133, Italy
Prof. Dr. Reiner Rummel
Delft University of Technology, Faculty of Geodesy
Thijsseweg ,11 NL-2629 JA Delft, The Netherlands
ISBN 3-540-51528-3 Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-38?-51528-3 Springer-Verlag New York Berlin Heidelberg
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© Springer-Verlag Berlin Heidelberg 1989
Printed ni Germany
Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr.
2132/3140-543210- Printed on acid-free paper
DROWEROF
This book is the collection of the Lecture Notes of na International remmuS loohcS
of Theoretical ysedoeG held in Assisi (Italy) from yaM 32 to June 3 -1988. ehT
loohcS saw sponsored yb the International Association of ysedoeG dna organized yb .R
lemmuR dna yb the writer.
I still rebmemer nehw I first conceived the idea of organizing such a School dna I
realized that the goal saw really very ambitious. Talking for the first time to emos
of the persons ohw afterwards emaceb lecturers of the School, I got very different
reactions: emos very positive, from Reiner lemmuR for instance, ohw saw enthusiastic
dna agreed to supply his substantial support to organize the school, emos very
cautious or even sceptical.
I ma proud to yas that at the dne of the courses all the lecturers claimed they erew
very happy about the work they did.
Therefore, I must first of all give ym sincere thanks to all the teachers for the
fine job they did which, I believe, will remain for long in the form of these
Lecture Notes, sa basic material for scientists studying satellite .ysedoeg
sknahT era also eud to the students ohw really participated, concentrating their
interest dna intelligence no the subject, cooperating with the teachers in a very
pleasant .erehpsomta
Special thanks are eud to Jozsef madA ohw prepared the Introduction to the school
that immediately follows this .draweroF
eW must also recognize the foundamental role of the secretary of the school, Miss.
ardnaS Marescalchi, ohw helped os hcum in finding the right organization dna
equilibrium neewteb scientific labour dna social events, solving all the practical
problems, emos of them very hard like na unexpected transport strike, right no the
last yad of the school.
In this work of organization, ew have been greatly supported yb all the staff of
our host organization, aL" Cittadella", dna primarily yb .rM Marco Marchini. Their
active presence dna their cordiality have facilitated establishing the right doom
gnoma people during.the School.
eW musta lso recognize that ynam organizations have substantially contributed to the
realization of the loohcS yb supporting it in various forms, from fellowships to
grants. oT ,GGUI IAG, ,ASE NSP (Italian ecapS Plan), TESIC Co., Salmoiraghi Co.,
Wild Co., Zeiss Co., og our gratitude dna our thanks.
iV
Moreover, sincere thanks are eud to the Municipality of Assisi, to the "Azdiie nda
enoizomorP Turistica" of Assisi, Todi, dna Orvieto, ohw provided no different
occasions, support in the organization of excursions dna specially the beautiful
concert of Medioeval music dna .sgnos
Last but not least I want to dda personal thanks to ym friend Reiner ,lemmuR ohw did
os hcum for the scientific margorp of the School; it saw a pleasure for em to work
with mih dna I hope it will last for long.
odnanreF bsnaS
STNETNOC
Introduction
J. Adam ....................................................................... 1
SERUTCEL
Introduction to Classical scinahceM
.H Moritz
Introduction .................................................................. 9
Lecture i. NOTWEN ............................................................. 11
1.1 Motion of a Mass Point .................................................. 11
1.2 Central Forces .......................................................... 41
1.3 Planetary Motion ........................................................ 51
1.4 Free Motion ............................................................. 51
1.5 ehT Many-Body Problem ................................................... 61
1.6 Motion of a Rigid Body .................................................. 71
1.? Inertial Navigation and Surveying ....................................... 91
1.8 Error Propagation ....................................................... 20
1.9 Unstable Convergence in Integration ..................................... 20
Lecture 2.LAGRANGE ...................... .. .................................... 23
2.1 Statics; Principle of Virtual Displacements ............................. 23
2.2 Mass Point in Equilibrium on a Surface .................................. 25
2.3 D'Alembert's Principle .................................................. 27
2.4 Lagrange's Equations of First Kind ...................................... 28
2.5 Lagrange's Equations of Second Kind ..................................... 29
2.6 Geometrical Interpretations ............................................. 13
2.7 The Principle of Least Action ........................................... 34
Lecture 3. NOTLIMAH ........................................................... 63
3.i The Legendre Transformation ............................................. 63
3.2 Hamilton's Canonical Equations .......................................... 38
3.3 Canonical Transformations ............................................... 14
3.4 Symplectic Geometry ..................................................... 42
3.5 Cyclical Variables ...................................................... 46
3.6 Perturbation Theory ..................................................... 48
3.7 Keplerian Motion ........................................................ 49
Lecture 4. ERACNIOP .... ...................................................... 55
4.1Liouville's Theorem, Measure-Preserving Transformations and
Stochastic Processes .................................................... 55
4.2 The Kolmogorov-Arnold-Moser Theorem ..................................... 95
4.3 The Example of H~non and Heiles ......................................... 60
4.4 Chaos and Order ......................................................... 16
Suggested Additional Reading .................................................. 66
Lectures in Celestial scinahceM
J. Kovalevsky
1 General Introduction ........................................................ 69
2 The owT Body Problem ........................................................ 70
2.1 Elliptical Solution of the owT Body Problem ............................. 07
2.2 Kepler's Equation ....................................................... 27
2.3 Expansions in Mean Anomaly .............................................. 74
2.4 Orbital Elements ........................................................ 67
IIIV
3 Equations of Perturbed Motion .............................................. 87
3.1 ehT Disturbing Function ................................................ 87
3.2 Osculating Elements .................................................... 97
3.3 Lagrange Planetary Equations ........................................... 08
3.4 Gauss Equations ........................................................ 08
3.5 Canonical Osculating Elements .......................................... 18
4 General Perturbation Techniques ............................................ 28
4.1 Development of the Disturbing Function ................................. 28
4.2 Artificial Satellite Disturbed by the nuS .............................. 38
4.3 Method of Solution ..................................................... 58
4.4 Solution in Canonical Variables ........................................ 78
4.5 Form of the Solution ................................................... 98
5 Motion of na Artificial Satellite .......................................... 98
5.1 Earth's Gravitational Potential ........................................ 98
5.2 Development of the Disturbing Function .................................. 29
5.3 First Order Solution ................................................... 39
5.4 Second Order Solution .................................................. 69
5.5 Effects of Other Zonal Harmonics ....................................... 69
5.6 Treatment of Tesseral Harmonics ........................................ 79
5.7 Other Perturbations .................................................... 89
6 Resonances ................................................................. 99
6.1 General Presentation ................................................... 99
6.2 A Simplified Problem Resonance ......................................... 101
6.3 Applications to Artificial Satellites .................................. 301
6.4 ehT Critical Inclination ............................................... 401
7 Numerical Methods .......................................................... 401
7.1 Semi-Numerical General Analytical Theories ............................. 501
7.2 Secular Semi-Numerical Theories ........................................ 801
7.3 Numerical Integration .............................................. :... 801
7.4 Single Step Methods .................................................... 901
7,5 Multistep Methods ...................................................... 011
7.6 Discussion of Numerical Integration Methods ............................ 311
General Bibliography ......................................................... 411
ruoF Lectures no Special dna lareneG Relativity
.W.E Grafarend
Lecture I. Flat spacetime, pseudo-Euclidean space, the Lorentz transformation
yrammuS ......................................... .......................... 511
i Flat spacetime, pseudo-Euclidean space, the Lorentz transformation ...... 511
2 Examples ................................................................ 221
3 Exercises ............................................................... 721
3.1Lorentz transformation in spinor notation ........................... 721
3.2 ehT Sagnac effect ................................................... 821
Lecture II-IV. Curved spacetime, space, pseudo-Riemann the affine
transformation ................................................ 921
yrammuS ................................................................... 921
1 Curved Spacetime, space, pseudo-Riemann the affine transformation ....... 921
2 Examples ................................................................ 731
3 Exercises ............................................................... 841
3.1 ehT Kerr metric ..................................................... 841
3.2 Schwarzschild metric in isotropic coordinates ....................... 941
Literature ................................................................... 051
ecnerefeR etanidrooC :smetsyS nA etadpU
I.I. Mueller
1 Introduction ............................................................... 351
2 Conventional Inertial Systems (CIS) of Reference ........................... 651
XI
2.1 Basic Considerations ................................................... 651
2.2 Inertial Systems in Practice ........................................... 951
2.21 Extragalactic Radio Source System .................................. 951
2.22 Stellar System .................................................... 161
2.23 Dynamical Systems ................................................. 361
2.3 Conclusions ............................................................ 661
3 Conventional Terrestrial (CTS) Systems of Reference ........................ 761
3.1 Brief History of the Past edaceD ....................................... 961
3.2 ehT weN STC ............................................................ 071
3.3 Reference emarF Ties ................................................... 271
3.31 Ties Between the SIC semarF ....................................... 271
3.32 Ties Between the STC semarF ....................................... 471
4 Modeling the Deformable Earth .............................................. 571
4.1 Precession )P( ......................................................... 771
4.2 Nutation )N( ........................................................... 771
4.3 Earth Rotation )S( ..................................................... 081
4.4 Deformations (L') ...................................................... 181
4.41 Tidal Deformations ................................................ 281
4.42 Plate Tectonic ssaM Transfer ...................................... 281
4.43 Other Deformations ................................................ 381
4.44 Current (1988) Practice ........................................... 381
4.5 Recent Developments .................................................... 481
4.51 Expected segnahC in the Adopted Series of Nutation ................ 481
4.52 Expected egnahC in the Constant of Precession ..................... 581
4.53 Intermediate Reference emarF Issues ............................... 581
5 ehT International Earth Rotation Service ................................... 681
5.1 ehT SETOC-TIREM Programs .......... . ..................................... 681
5.2 ehT International Earth Rotation Service ............................... 781
References ................................................................... 981
Appendix 1. Principal snoitadnemmoceR of the dna TIREM SETOC
gnikroW Group ..................... L .............................. 491
Appendix 2, Resolution of International Astronomical Union (1985) ............ 691
Appendix 3. Resolution 1 of the International Union of ysedoeG dna Geophysics,
XIX General Assembly, Vancouver, 12 August 1987 .................. 691
Gravity Field yrevoceR from Satellite Tracking ataD
.C Reigber
1 Introduction ............................................................... 791
2 Principles of Gravity Parameter Determination .............................. 991
2.1 Linear Observation Equations ........................................... 302
3 Gravity Induced Linear Orbit Perturbations ................................. 602
3.1 Secular Perturbations .................................................. 012
3.2 Periodic Perturbations ................................................. 112
4 Adjustment Procedures ....................................................... 712
4.1 Single Arc Solution .................................................... 712
4.2 Solution from denibmoC Normal Equations ................................ 812
4.3 Constraint Equations ................................................... 912
4.4 Light Constraint Solutions ............................................. 022
4.5 Parameters Considered for Adjustment ................................... 122
5 Tracking Data ............................................................... 222
5.1 Existing Data .......................................................... 222
5.2 DatSae lection ..... : ................................................... 322
6 Processing Steps ........................................................... 422
7 Special Topics ............................................................. 822
8 Global Gravity Field Models ................................................ 032
8.1 Recent Gravity Field Models ............................................ 032
8.2 weN Gravity Model Developments ......................................... 232
References ................................................................... 432
slatnemadnuF of Orbit noitanimreteD
B.D. Tapley
Introduction ................................................................. 235
The Orbit Determination Problem .............................................. 235
Linearization of the Orbit Determination Process ............................. 238
The Least Squares Solution ................................................... 239
The Minimum Norm Solution .................................................... 240
Weighted Least Squares Solution .............................................. 241
The Minimum Variance Estimate ................................................ 242
Propagation of the Estimate .................................................. 244
Minimum Variance Estimate With A Priori Information .......................... 244
The Sequential Estimation Algorithm .......................................... 246
The Extended Sequential Estimation Algorithm ................................. 247
State Noise Compensation Algorithm ........................................... 247
Batch and Sequential Estimation Compared ..................................... 248
Error Sources ................................................................ 249
Solution Methods for the Orbit Determination Problem ...... • ................... 250
Cholesky Decomposition ....................................................... 251
Least Squares Solution via Orthogonal Transformation ......................... 251
Appendix A. The Primary Forces on a Near-Earth Satellite ..................... 255
Gravitational Perturbations .............................................. 255
Gravitational Potential for the Earth .................................... 255
Solid Earth Tides ........................................................ 256
N-Body ................................................................... 256
Ocean Tides .............................................................. 257
General Relativity ....................................................... 258
Nongravitational Perturbations ........................................... 258
Atmospheric Drag ......................................................... 258
Solar Radiation Pressure ................................................. 259
Earth Radiation Pressure ................................................. 259
References ................................................................... 260
noitanibmoC of Satellite, Altimetric dna Terrestrial Gravity ataD
R.H. ppaR
1.0 Introduction ............................................................. 162
2.0 Representation of the Gravitational Potential ............................ 262
2.1 Spherical Harmonics ...................................................... 262
2.1.1 Spherical Potential Coefficients dna Gravity Anomalies ............. 362
2.1,2 Spherical Harmonics dna Orthogonality Relationships ................ 662
2.2 Ellipsoidal Harmonics .................................................... 762
2.2.1Ellipsoidal Harmonics dna Gravity Anomalies ........................ 862
3,0 Data Definition .......................................................... 962
3.1 Satellite Data ....................................................... 962
3.2 Terrestrial Gravity Data ............................................. 962
4.0 Data Combination ......................................................... 172
4.1 General Principles ................................................... 172
4.2 Least Squares Principles ............................................. 172
4.3 Optimal Estimation ................................................... 472
5.0 Observation Equation Formation ........................................... 572
5.1 Combination Procedure - Method A ..................................... 572
5.2 Combination Procedure - Method B ..................................... 672
5.3 tnemmoC .............................................................. 872
6.0 ehT Development of High Degree Potential Coefficient Models .............. 972
7.0 ehT Role of Satellite Altimeter Data ..................................... 972
8.0 Comparisons of Satellite dna Terrestrial Gravity Anomaly Fields .......... 182
382
9.0 Conclusion ...............................................................
References ................................................................... 382
lX
remmuS loohcS Lectures no Satellite Altimetry
C.A. rengaW
Lecture 1. Purposes dna Motivation, ehT Altimetric Equation, Radial
Perturbations ..................................................... 582
Lecture 2. Frequency Classification dna Observability of Radial
Variations ........................................................ 203
Lecture 3. Determination of Permanent aeS Topography morF Altimetry 1:
lavomeR of Orbit Error ............................................. 213
Lecture 4. Determination of TSP from Altimetry 2: Simulation of a
Subtraction Method ................................................ 513
Lecture 5. Determination of TSP from Altimetry 3: Simulation of a
Simultaneous Solution for the Geoid ............................... 323
A Footnote no weN Results from the Subtraction Method ........................ 923
References ................................................................... 233
Techniques Advanced for High-Resolution Mapping of the
Gravitational Field
.O obmoloC
I Basic Techniques for Gathering Data no a Global Basis ...................... 533
yhW esu a gradiometer? .................................................. 533
latnemadnuF problems .................................................... 633
Accuracy required ....................................................... 733
Problems limiting the accuracy .......................................... 733
ehT null-point principle ................................................ 833
Every-day examples ...................................................... 833
ehT precision balance used in laboratories .............................. 833
emoS basic relationships ................................................ 143
hcihW yaw is pu in freee fall? .......................................... 143
Prospecting in the Asteroids .................................. .......... 243
Dealing with orbit error dna attitude / Rotation to estimate
gravitation ............................................................. 343
a) ehT orbit error ...................................................... 343
)b Dealing with attitude dna rotation errors ............................ 346
2 Global Data Analysis ........................................................ 943
3 Mission Error Analysis for a 01 2 E.U., Full-Tensor Instrument ............. 253
3.1 Time series representation of the second gradients for a circular,
polar, repeating orbit ................................................. 353
3.2 ehT general element of the normal matrix H for the full-tensor
gradiometer ............................................................ 653
3.3 Rescaling for different altitude accuracies dna mission lengths ........ 753
3.4 Calculating global SMR errors of area naem anomalies ................... 753
4 Implications for the Study of the Earth of the Results of a Global
Error Analysis of a Full-Tensor Gradiometer Mission ........................ 953
References ................................................................... 863
SRANIMES
ehT Integrated hcaorppA to Satellite ysedoeG
.B Betti, .F 6snaS
1 Introduction ............................................................... 373
2 The Typical Form of Satellite Observation Equations ........................ 374
3 The "Spherical Field-Circular Motion" approximation ........................ 380
4 The Solution of Hill's Equation in the Circular Motion Approximation ....... 683
5 The Covariance Function and the Integrated Scheme .......................... 593
6 Sub-Optimal Solutions ...................................................... 104
Xll
Appendix ..................................................................... 405
References ................................................................... 416
Determination of a Local Geodetic Network'by Multi-Arc Processing
of Satellite Laser segnaR
.A Milani, .E Melchioni
1 Introduction and summary ............................................. ...... 714
2 Choice of the arc length ................................................... 419
3 Symmetries and rank deficiency ............................................. 224
4 ehT multi-arc algorithm .................................................... 034
5 Experimentalresults ....................................................... 534
References ................................................................... 444
yradnuoB Value smelborP dna Invariants of the Gravitational
Tensor in Satellite Gradiometry
.P Holota
yrammuS ...................................................................... 744
1 Introduction ............................................................... 447
2 Differential Accelerometry ................................................. 448
3 Invariants of the Gravitational Tensor ..................................... 054
4 Reduction and Linearization ................................................ 254
5 Separation of the Field and Orbit Perturbations ............................ 454
References .................................................................. 754
A Possible Application of the Space VLBI Observations for
Establishment of a weN Connection of Reference Frames
J. madA
yrammuS ...................................................................... 459
1 Introduction ............................................................... 064
2 ehT role of space VLBI in reference frames tie ............................. 264
3 Rank deficiencies within a space VLBI network .............................. 564
4 Conclusions ................................................................ 374
References ................................................................... 474
Optimization of the Reordering Algorithm for Least Squares Problems
Relevant to Space Geodesy
.M Crespi, .G Forlani, L. Mussio
yrammuS ...................................................................... 774
1 ehT Problem ................................................................ 774
2 ehT Method ................................................................. 478
3 ehT Program ................................................................ 084
4 ehT Test ................................................................... 184
5 ehT Example ................................................................ 184
6 Remarks .......................................................... ~ ......... 484
References ....................................................... ............ 194
