Preface and acknowledgments .................................... ix
1 Introduction - modeling, migration, imaging, and
inversion .................................................... 1
1.1 Seismic data contains information ....................... 1
1.2 Models for propagation and reflection ................... 1
1.3 Going forward to go back ................................ 3
1.4 Seismic and non-seismic imaging ......................... 3
1.5 Motivation for migration ................................ 4
1.6 Time and depth migration ................................ 6
1.7 Migration velocity ...................................... 6
1.8 Full-wave and asymptotic migration ...................... 8
1.9 Seismic migration and inversion ......................... 9
1.10 True amplitude migration ................................ 9
1.11 Linear and nonlinear processes ......................... 10
2 Basic migration concepts .................................... 11
2.1 Migration as a map ..................................... 11
2.2 Ray-theoretical migration concepts ..................... 16
2.3 Downward continuation .................................. 25
2.4 Reverse time migration ................................. 33
2.5 Claerbout one-way difference-equation migration ........ 37
2.6 Frequency-wavenumber migration methods ................. 44
2.7 Other methods .......................................... 52
2.8 Time migration ......................................... 55
2.9 Summary ................................................ 60
Exercises ................................................... 60
3 Prestack migration .......................................... 62
3.1 Prestack migration concepts ............................ 62
3.2 Prestack ƒ-k migration in two dimensions ............... 63
3.3 Prestack slant-stack migration in 2.5 dimensions ....... 73
3.4 Prestack ray-theoretical migration ..................... 75
3.5 Prestack ƒ-k migration in three dimensions ............. 80
3.6 Prestack reverse-time migration ........................ 86
Exercises ................................................... 89
4 Migration limitations ....................................... 90
4.1 Perfect and imperfect migrations ....................... 90
4.2 Effects of finite bandwidth ............................ 90
4.3 Space-domain image amplitude renormalization .......... 102
4.4 Effects of spatial sampling ........................... 102
4.5 Effects of maximum dip ................................ 103
4.6 Effects of finite spatial aperture .................... 103
4.7 Extra and missing data ................................ 113
4.8 Effects of a finite time window ....................... 114
4.9 Undermigration and overmigration ...................... 118
Exercises .................................................. 121
5 Models for wave propagation and reflection ................. 122
5.1 The need for models ................................... 122
5.2 Wave equations ........................................ 123
5.3 Building reflections .................................. 128
5.4 A scattering-theory model for reflection data ......... 131
Exercises .................................................. 134
6 Green's functions .......................................... 135
6.1 The general Green's function .......................... 135
6.2 The scalar Green's function ........................... 137
6.3 The acoustic Green's function ......................... 140
6.4 The elastic Green's function .......................... 142
6.5 Local wavenumbers ..................................... 144
6.6 Multipath Green's functions ........................... 145
6.7 Green's functions in a layered medium ................. 146
Exercises .................................................. 154
7 The scattering potential ................................... 155
7.1 The scattering potential as a function of angle ....... 155
7.2 The scalar scattering potential ....................... 157
7.3 The acoustic scattering potential ..................... 158
7.4 The elastic scattering potential ...................... 159
7.5 Summary ............................................... 171
Exercises .................................................. 172
8 Reflectivity ............................................... 174
8.1 Point reflectivity .................................... 174
8.2 A scalar reflectivity function ........................ 176
8.3 An acoustic reflectivity function ..................... 179
8.4 Elastic reflectivity functions ........................ 181
8.5 A general formula relating scattering potential
and reflectivity ...................................... 186
8.6 Summary of linearized reflectivity functions .......... 187
Exercises .................................................. 189
9 Synthesizing reflection data ............................... 190
9.1 The Born model for seismic reflections ................ 190
9.2 A constant background ................................. 190
9.3 Restricted data sets .................................. 197
9.4 A depth-variable background ........................... 207
9.5 A generally variable background ....................... 212
9.6 Summary ............................................... 219
Exercises .................................................. 222
10 Frequency-wavenumber migration ............................. 223
10.1 Full constant-velocity migration ...................... 223
10.2 Partialmigration ...................................... 229
10.3 Residualf-k migration ................................. 233
10.4 Migration in a depth-variable medium .................. 236
Exercises .................................................. 248
11 Asymptotic modeling and migration .......................... 249
11.1 Migration and modeling as mapping ..................... 249
11.2 Traveltime and depth functions ........................ 250
11.3 Single-valued modeling and migration .................. 252
11.4 Relating the forward and inverse mappings ............. 255
11.5 Constant-angle migration .............................. 258
11.6 Special cases ......................................... 263
Exercises .................................................. 270
12 Residual asymptotic migration .............................. 271
12.1 Combining modeling and migration ...................... 271
12.2 Transition zones ...................................... 276
Exercises .................................................. 277
13 Asymptotic data mapping and continuation ................... 278
13.1 Combining forward and inverse migration ............... 278
13.2 Constant-velocity depth extrapolation in 2-D .......... 280
13.3 MZOandDMO ............................................. 285
Exercises .................................................. 287
14 Least-squares asymptotic migration ......................... 289
14.1 Direct versus least-squares inversion ................. 289
14.2 Multipathing .......................................... 290
14.3 The asymptotic modeling operator in three and two
dimensions ............................................ 291
14.4 Least-squares migration in 2.5-D ...................... 292
14.5 Three-dimensional common-angle least-squares
migration ............................................. 297
Exercises .................................................. 305
Appendix A Conventions and glossary of terms .................. 306
Appendix В Coordinates, vectors, and identities ............... 311
Appendix С Fourier and Radon transforms ....................... 316
Appendix D Surface and pointwise reflectivity ................. 324
Appendix E Useful filters ..................................... 327
Appendix F The phase integral and the stationary phase
approximation ...................................... 353
Appendix G The diffraction integral ........................... 366
Appendix H Wave-based, ray-based, and reflector-based
coordinates ........................................ 385
References .................................................... 397
Index ......................................................... 401
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