A Breviary of Seismic Tomography: Imaging the Interior of the Earth and Sun

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 9
Preface......Page 13
1 Introduction......Page 17
1.1 Early efforts at seismic tomography......Page 18
1.3 Global tomography......Page 19
1.4 Some major discoveries......Page 20
1.5 Helioseismology......Page 23
1.6 Finite-frequency tomography......Page 24
2.1 The stress tensor......Page 27
2.2 Forces in continuous media......Page 28
2.3 Newton\'s law and the elastodynamic equations......Page 29
2.4 The acoustic wave equation......Page 32
2.5 The ray approximation......Page 36
2.6 Ray solutions in layered and spherical systems......Page 38
2.7 Geometrical spreading......Page 41
2.8 Rays in an isotropic, elastic Earth......Page 42
2.9 Fermat\'s Principle......Page 45
2.10 Huygens, Fresnel and Green......Page 46
2.11 Flow: solar p-waves or ocean acoustic waves......Page 50
2.12 Appendix A: Some elements of Fourier analysis......Page 52
3.1 The shooting method......Page 56
3.2 Ray bending......Page 58
3.3 Other raytracing algorithms for 3D media......Page 62
3.4 Ray-centred coordinates......Page 65
3.5 Dynamic ray tracing......Page 66
3.6 Ray tracing on the sphere......Page 69
3.7 Computational aspects......Page 70
4 Wave scattering......Page 74
4.1 The acoustic Green\'s function......Page 76
4.2 An acoustic point scatterer......Page 78
4.3 Green\'s functions for elastic waves......Page 79
4.4 Green\'s functions in the ray approximation......Page 84
4.5 The Born approximation......Page 86
4.6 Scattering of a plane wave......Page 89
4.7 The scattering matrix......Page 93
4.8 Appendix B: The impulse response......Page 95
5.1 Geometrical spreading......Page 98
5.2 The quality factor Q......Page 100
5.3 The correspondence principle......Page 102
5.4 Attenuating body waves......Page 104
5.5 Scattering......Page 107
6 Travel times: observations......Page 109
6.1 Phase picks......Page 112
6.2 Matched filters......Page 116
6.3 Wavelet estimation......Page 119
6.4 Differential times......Page 123
6.5 Signal and noise......Page 125
6.6 Time–distance analysis in helioseismology......Page 126
7.1 The ray theoretical interpretation......Page 132
7.2 Cross-correlation of seismic arrivals......Page 137
7.3 Forward scattering......Page 141
7.4 Finite frequency sensitivity: a simple example......Page 142
7.5 Finite frequency kernels: general......Page 145
7.6 Alternative arrival time measurements......Page 151
7.7 Alternative methods for kernel computation......Page 154
7.8 Computational aspects......Page 155
8 Body wave amplitudes: observation and interpretation......Page 161
8.1 Amplitude observations......Page 162
8.2 t* observations......Page 165
8.3 Amplitude healing......Page 167
8.4 Boundary topography......Page 168
8.5 Finite-frequency Q tomography......Page 170
9 Normal modes......Page 174
9.1 The discrete spectrum......Page 175
9.2 Rayleigh\'s Principle......Page 182
9.3 Mode splitting......Page 184
9.4 Observations of mode splits......Page 191
10 Surface wave interpretation: ray theory......Page 194
10.1 The theory of surface waves......Page 196
10.2 Love and Rayleigh waves......Page 198
10.3 Measuring fundamental mode dispersion......Page 203
10.4 Measuring higher mode dispersion......Page 206
10.5 Waveform fitting......Page 208
10.6 Partitioned waveform inversion (PWI)......Page 210
10.7 Appendix C: Asymptotic theory......Page 213
11 Surface waves: finite-frequency theory......Page 224
11.1 Phase and amplitude perturbations......Page 225
11.2 Practical considerations......Page 230
11.3 Phase velocity maps: an incompatibility......Page 233
12 Model parametrization......Page 235
12.1 Global parametrization......Page 236
12.2 Local parametrization......Page 238
12.3 Numerical considerations......Page 244
12.4 Spectral analysis and model correlations......Page 245
13.1 Ellipticity corrections......Page 249
13.2 Topographic and bathymetric time corrections......Page 254
13.3 Crustal time corrections......Page 256
13.4 Surface wave corrections......Page 259
13.5 Source corrections......Page 261
13.6 Amplitude corrections for body waves......Page 264
13.7 Dispersion corrections......Page 266
13.8 Instrument response......Page 267
13.9 Clock corrections......Page 269
14 Linear inversion......Page 271
14.1 Maximum likelihood estimation and least squares......Page 272
14.2 Alternatives to least squares......Page 276
14.3 Singular value decomposition......Page 277
14.4 Tikhonov regularization......Page 281
14.5 Bayesian inference......Page 282
14.6 Information theory......Page 286
14.7 Numerical considerations......Page 288
14.8 Appendix D: Some concepts of probability theory and statistics......Page 291
15.1 Resolution matrix......Page 293
15.2 Backus-Gilbert theory......Page 297
15.3 Sensitivity tests......Page 301
16 Anisotropy......Page 305
16.1 The elasticity tensor......Page 306
16.2 Waves in homogeneous anisotropic media......Page 311
16.3 S-wave splitting......Page 312
16.4 Surface wave anisotropy......Page 317
17.1 Beyond Born......Page 322
17.2 Adjoint methods......Page 323
17.3 Global coverage of seismic sensors......Page 326
17.4 Helioseismology and astroseismology......Page 328
References......Page 329
Author index......Page 350
General index......Page 355