Contents
1 Introduction 1
2 Coordinate Systems and Transformations 13
3 Gravity Field Quantities 15
3.1 Gravity field quantities in the spatial domain 15
3.2 Gravity field quantities in the spectral domain 18
3.3 Bouguer gravity field 19
4 Parameters, Data and Models 24
4.1 Parameters 24
4.2 Input data and models 25
4.2.1 Terrestrial datasets 25
4.2.2 Planetary and lunar datasets 27
5 Gravity Maps 30
5.1 Terrestrial gravity maps 30
5.2 Planetary and lunar gravity maps 34
6 Theory of Heights 40
6.1 Definitions of physical heights 40
6.2 Definitions of the geoid height and the height anomaly 42
6.3 Approximate definitions of orthometric heights 43
7 Geoid-to-quasigeoid Separation 45
7.1 Geoid-to-quasigeoid separation (accurate definition) 45
7.2 Computation in the spatial domain 47
7.2.1 Topographic component 50
7.2.2 Non-topographic component 51
7.3 Computation in the spectral domain 53
7.3.1 Topographic term (of uniform density) 54
7.3.2 Topographic term (of anomalous density) 56
7.3.3 Non-topographic term 58
7.3.4 Normal gravity term 59
7.3.5 Full spectral expression 59
7.4 Approximate definitions of the geoid-to-quasigeoid separation 60
7.5 Discussion of numerical aspects 63
7.6 Geoid-to-quasigeoid separation offshore 66
8 Comparison of Methods 69
8.1 Numerical analysis and results 69
8.1.1 Classical solution 70
8.1.2 Sj？berg’s solution 73
8.1.3 Accurate solution 74
8.2 Comparison of results 78
8.2.1 Topographic contribution differences 80
8.2.2 Non-topographic contribution differences 82
8.2.3 Complete differences 83
8.2.4 Contribution of terrain geometry 84
8.3 Sensitivity analysis 86
8.4 Discussion of results 88
9 Analysis of Gravity in the Definition of Heights 91
9.1 Differences between normal and normal-orthometric heights 91
9.2 Numerical analysis and results 92
9.2.1 Spectral analysis 106
9.2.2 Correlation analysis 108
9.3 Discussion of results 109
10 Effect of Topographic Density of the Geoid 111
10.1 Numerical analysis and results 113
10.1.1 Individual contributions to the geoid-to-quasigeoid separation 114
10.1.2 Choice of the average topographic density 121
10.2 Geoid errors due to density uncertainties 122
10.3 Discussion of results 124
11 Geoid-to-quasigeoid Separation Offshore 127
11.1 Numerical analysis and results 127
11.1.1 Methodology 127
11.1.2 Results 128
11.2 Error analysis and discussion of results 132
12 Height Systems in Planetary Geodesy 134
12.1 Physical heights for telluric planets (and moons) 136
12.2 Numerical realization and results 137
12.2.1 Topographic models 138
12.2.2 Accurate geoid and orthometric heights 141
12.2.3 Approximate geoid and orthometric heights 145
12.2.4 Comparison of accurate and approximate results 145
12.2.5 Regional study: Martian topographic features 148
12.2.6 Regional study: Lunar topographic features 148
12.3 Discussion of results 150
13 Molodensky’s Concept in Planetary Geodesy 154
13.1 Methodology 155
13.2 Results 155
13.3 Discussion of results 161
14 Concluding Summary 163
References 170
Appendix A: Topographic Potential for External Convergence Domain 187
Appendix B: Anomalous Topographic Potential for External Convergence Domain 189
Appendix C: Anomalous Topographic Potential for Internal Convergence Domain 192
Appendix D: Contribution of Uniform Topographic Density 194
Appendix E: Contribution of Anomalous Lateral Topographic Density 195
Appendix F: Contribution of Lakes and Glaciers 196
Appendix G: Sub-geoid Mass Density Contribution 197
Appendix H: Contribution of Inland Topography (offshore) 198
Appendix I: Contribution of Polar Glaciers (offshore) 200
Appendix J: Contribution of Mean Dynamic Topography (offshore) 202
Appendix K: Contribution of Sub-geoid Masses (offshore) 204
Appendix L: FFT Technique for Spherical Harmonic Analysis and Synthesis 205
Appendix M: Inverse Solutions to Boundary Value Problems 207
Appendix N: Conditionality of Inverse Solutions to Boundary Value Problems 210
Appendix O: Numerical Analysis of Conditionality of Inverse Solutions 215
Appendix P: Analytical Solution of Green Integrals 221
Appendix Q: Weak Singularity of Green Integrals in A Direct Gravity Inversion 223
Appendix R: Far-zone Contributions to A Direct Gravity Inversion 225
Appendix S: Molodensky Truncation Coefficients for Green Integrals 228
Appendix T: Least-squares Estimation Model 230
Appendix U: Iterative Method of Conjugate Gradients with Pre-conditioning 232
Appendix V: Regularization 233