Foreword ...................................................... vii
List of Figures ............................................... xix
List of Tables ................................................ xxi
Acknowledgments ............................................. xxiii
1 Science of seismology ...................................... 1
Preliminary remarks ........................................ 1
1.1 Purpose and methodology: Historical sketch ................. 2
1.2 Classification ............................................ 10
Closing remarks ........................................... 11
1.3 Exercises ................................................. 14
2 Seismology and continuum mechanics ........................ 19
Preliminary rcnnarks ........................................... 19
2.1 On axiomatic formulation .................................. 21
2.2 Kinematic descriptions .................................... 23
2.2.1 Spacetime .......................................... 23
2.2.2 Motion ............................................. 24
2.2.3 Coordinates ........................................ 27
2.3 Field equations ........................................... 28
2.3.1 Balance equations .................................. 28
2.3.2 Continuity equation ................................ 29
2.3.3 Cauchy equation of motion .......................... 31
Closing remarks ........................................... 33
2.4 Exercises ................................................. 34
3 Hookeaii solid: Material symmetry ......................... 41
Preliminary remarks ....................................... 41
3.1 Hookoan solids ............................................ 42
3.2 Material symmetry ......................................... 45
3.2.1 On symmetries ...................................... 45
3.2.2 On tensor rotations ................................ 48
3.2.3 Finite and infinitesimal elasticities .............. 49
3.2.3.1 Deformation gradient ........................... 49
3.2.3.2 Elasticity tensor .............................. 52
3.2.3.3 Prestressed linearly elastic materials ......... 54
3.2.3.4 Material symmetry: Finite elasticity ........... 57
3.2.3.5 Material symmetry: Relation between finite
and infinitesimal elasticities ................. 58
3.2.4 Symmetry classes ................................... 61
3.2.4.1 Material-symmetry conditions ................... 61
3.2.4.2 Hooke's law in and ............................. 64
3.2.4.3 Index symmetries ............................... 65
3.2.4.4 Kelvin notation ................................ 67
3.2.4.5 Monoclinic tensor .............................. 69
3.2.4.6 Orthotropic tensor ............................. 75
3.2.4.7 Tetragonal tensor .............................. 76
3.2.4.8 Transversely isotropic tensor .................. 78
3.2.4.9 Trigonal tensor ................................ 79
3.2.4.10 Cubic tensor ................................... 79
3.2.4.11 Isotropic tensor ............................... 79
3.2.4.12 Relations among elasticity parameters .......... 80
3.2.4.13 Diclinic solids ................................ 85
3.2.4.14 Hexagonal solids ............................... 87
Closing remarks ........................................... 88
3.3 Exercises ................................................. 89
4 Hookean solid: Effective symmetry and equivalent medium ... 95
Preliminary remarks ....................................... 95
4.1 Effective symmetries ...................................... 96
4.1.1 On accuracy ........................................ 96
4.1.2 Fixed orientation of coordinate system ............ 103
4.1.2.1 Monoclinic tensor ............................. 104
4.1.2.2 Orthotropic tensor ............................ 105
4.1.2.3 Tetragonal tensor ............................. 106
4.1.2.4 Transversely isotropic tensor ................. 107
4.1.2.5 Trigonal tensor ............................... 107
4.1.2.6 Cubic tensor .................................. 108
4.1.2.7 Isotropic tensor .............................. 109
4.1.3 Optimal orientation of coordinate system .......... 110
4.2 Equivalent media ......................................... 113
4.2.1 Introduction ...................................... 113
4.2.2 Equivalence parameters for isotropic layers ....... 116
4.2.2.1 Formula ....................................... 116
4.2.2.2 Justification ................................. 118
4.2.2.3 Interpretation ................................ 129
4.2.3 Equivalence parameters for TI layers .............. 130
Closing remarks .......................................... 132
4.3 Exercises ................................................ 134
5 Body waves ............................................... 159
Preliminary remarks ...................................... 159
5.1 Wave equations ........................................... 160
5.1.1 Assumptions and formulation ....................... 160
5.1.2 Particular case: Isotropy and homogeneity ......... 161
5.1.3 Particular case: Inhomogeneous string ............. 167
5.1.4 Particular case: String with friction ............. 171
5.2 Solutions of wave equation ............................... 171
5.2.1 Introduction ...................................... 171
5.2.2 Product solution .................................. 172
5.2.3 d'Alembert solution ............................... 173
5.2.3.1 d'Alembert's approach ......................... 173
5.2.3.2 Euler's approach .............................. 174
5.2.3.3 Spherical-symmetry approach ................... 177
5.2.4 Fourier-transform solution ........................ 178
5.2.5 Green's-function solution ......................... 182
5.3 On approximations ........................................ 185
Closing remarks .......................................... 188
5.4 Exercises ................................................ 189
6 Surface, guided and interface waves ...................... 197
Preliminary remarks ...................................... 197
6.1 Introduction ............................................. 198
6.2 Surface waves: Homogeneous elastic halfspace ............. 200
6.3 Guided waves: Homogeneous layer above halfspace .......... 209
6.3.1 Elastic layer above rigid halfspace ............... 209
6.3.2 Elastic layer above elastic halfspace ............. 212
6.4 Existence of surface and guided waves .................... 216
6.4.1 Introduction ...................................... 216
6.4.2 Elasticity parameters and mass densities .......... 216
6.4.3 On Love waves in homogeneous halfspace ............ 217
6.4.4 On P waves in homogeneous halfspace ............... 217
6.5 Interface waves: Homogenous halfspaces ................... 219
6.5.1 Introduction ...................................... 219
6.5.2 Elastic and liquid halfspaces ..................... 220
6.5.3 Liquid halfspaces ................................. 231
6.6 Existence of interface waves ............................. 235
6.6.1 Introduction ...................................... 235
6.6.2 Elasticity parameters and mass densities .......... 236
6.6.3 On SH waves as interface waves .................... 236
Closing remarks .......................................... 238
6.7 Exercises ................................................ 239
7 Variational principles in seismology ..................... 241
Preliminary remarks ...................................... 241
7.1 Historical comments ...................................... 242
7.2 Fermat's principle ....................................... 243
7.2.1 Isotropic layered medium .......................... 243
7.2.2 Isotropic ('ontinuously inhomogeneous medium ...... 246
7.2.3 Global optimization and causality ................. 249
7.2.4 Stationarity versus minimization .................. 251
7.2.5 Mathematical justification ........................ 252
7.2.5.1 Fermat's principle ............................ 252
7.2.5.2 Head waves .................................... 254
7.2.6 Physical interpretation ........................... 257
7.2.6.1 Macroscopic interpretation .................... 257
7.2.6.2 Microscopic interpretation .................... 259
7.2.6.3 Phase consideration ........................... 259
7.2.7 On teleology of Fermat's principle ................ 260
7.3 Hamilton's principle ..................................... 264
7.3.1 Action ............................................ 264
7.3.2 Wave equation ..................................... 265
7.3.3 Mathematical justification ........................ 266
7.3.4 Physical interpretation ........................... 267
7.4 Conserved quantities ..................................... 268
7.4.1 Introduction ...................................... 268
7.4.2 Ray parameter ..................................... 268
7.4.2.1 Isotropy ...................................... 268
7.4.2.2 Anisotropy .................................... 270
7.4.3 Hamiltonian and Lagrangian ........................ 272
7.4.3.1 Ray theory .................................... 272
7.4.3.2 Classical mechanics ........................... 274
Closing remarks .......................................... 275
7.5 Exercises ................................................ 276
8 Gravitational and thermal effects in seismology .......... 283
Preliminary remarks ...................................... 283
8.1 Gravitation .............................................. 284
8.1.1 Body forces ....................................... 284
8.1.2 Wave speeds ....................................... 287
8.2 On weak gravitational waves .............................. 291
8.3 Temperature .............................................. 298
8.3.1 Propagation and diffusion ......................... 298
8.3.2 Isothermal and adiabatic formulations ............. 299
8.3.2.1 Lame parameters ............................... 299
8.3.2.2 Bulk moduli ................................... 301
Closing remarks .......................................... 301
8.4 Exercises ................................................ 303
9 Seismology as science .................................... 307
Preliminary remarks ...................................... 307
9.1 Hypotheticodeductiv(! formulation ........................ 308
9.1.1 Hypotheses ........................................ 308
9.1.2 Deductive argumentation ........................... 310
9.2 Theory versus data ....................................... 313
9.2.1 Introduction ...................................... 313
9.2.2 Theory-ladenness of data .......................... 313
9.2.3 Underdetermination of theory by data .............. 314
9.3 Bayesian inference ....................................... 315
9.4 Predictions versus explanations .......................... 318
9.4.1 Introduction ...................................... 318
9.4.2 Covering-law model ................................ 319
9.4.3 Inference to best explanation ..................... 321
9.5 Realistic approach versus instrumental approach .......... 321
9.6 Coherence theory of justification ........................ 323
Closing remarks .......................................... 324
9.7 Exercises ................................................ 326
Appendix A. On covariant and contravariant transformations ... 331
Preliminary remarks ...................................... 331
A.l Contravariant transformations ............................ 332
A.2 Covariant transformations ................................ 333
A.3 Mixed transformations .................................... 334
A.4 Transformations in Cartesian coordinates ................. 334
Closing remarks .......................................... 335
Appendix В On covariant derivatives .......................... 337
Preliminary remarks ...................................... 337
B.1 Metric tensor ............................................ 338
B.2 Christoffel symbol ....................................... 341
B.3 Covariant derivative ..................................... 342
Closing remarks .......................................... 344
Appendix С List of symbols ................................... 347
C.l Mathematical relations and operations .................... 347
C.2 Physical quantities ...................................... 348
C.2.1 Greek letters ..................................... 348
C.2.2 Roman letters ..................................... 348
Bibliography .................................................. 351
Index ......................................................... 363
About the Author .............................................. 380
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