Preface ........................................................ xi
1 Introduction and overview of the book ...................... 1
1.1 Why passive, correlation-based imaging? .................... 1
1.1.1 Travel time estimation .............................. 2
1.1.2 Applications of travel time estimation .............. 3
1.1.3 Reflector imaging ................................... 4
1.1.4 Auxiliary array or virtual source imaging ........... 6
1.1.5 Passive synthetic aperture imaging .................. 8
1.1.6 Imaging with intensity cross correlations ........... 9
1.2 Chapter-by-chapter description of the book ................ 10
2 Green's function estimation from noise cross
correlations .............................................. 17
2.1 The scalar wave equation and its Green's function ......... 17
2.1.1 The Sommerfeld radiation condition ................. 19
2.1.2 Reciprocity ........................................ 20
2.1.3 The Helmholtz-Kirchhoff identity ................... 21
2.1.4 Application to time reversal ....................... 23
2.2 The scalar wave equation with noise sources ............... 25
2.3 Green's function estimation with a uniform distribution
of sources in a homogeneous open medium ................... 29
2.4 Green's function estimation with an extended distribution
of sources in an inhomogeneous open medium ................ 31
2.5 Green's function estimation with an extended
distribution of sources in an inhomogeneous cavity ........ 34
2.6 Green's function estimation with a limited distribution
of sources in a one-dimensional inhomogeneous medium ...... 38
2.6.1 The one-dimensional wave equation .................. 39
2.6.2 Reflection seismology .............................. 43
2.6.3 Daylight imaging ................................... 45
2.7 Conclusion ................................................ 48
Appendix: the covariance of the empirical cross
correlation ............................................... 48
3 Travel time estimation from noise cross correlations
using stationary phase .................................... 51
3.1 High-frequency wave propagation ........................... 52
3.2 High-frequency asymptotic analysis of the Green's
function in a homogeneous medium .......................... 53
3.3 High-frequency asymptotic analysis of the Green's
function in a smoothly varying medium ..................... 53
3.3.1 An introduction to geometrical optics .............. 53
3.3.2 Ray solution of the eikonal equation ............... 55
3.3.3 Fermat's principle for the travel time ............. 57
3.3.4 Properties of the travel time ...................... 58
3.4 High-frequency asymptotic analysis of the cross
correlation ............................................... 60
3.5 Conclusion ................................................ 67
4 Overview of conventional sensor array imaging ............. 68
4.1 Passive array imaging of sources ......................... 68
4.1.1 Data acquisition ................................... 68
4.1.2 Imaging function ................................... 69
4.1.3 The linear forward operator ........................ 69
4.1.4 The adjoint operator ............................... 70
4.1.5 Least squares inversion ............................ 71
4.1.6 The reverse-time imaging function .................. 73
4.1.7 Kirchhoff migration (or travel-time migration) ..... 74
4.2 Passive array imaging of sources: resolution analysis .... 74
4.2.1 Full-aperture array ................................ 75
4.2.2 Partial-aperture array ............................. 75
4.2.3 Summary of resolution analysis for passive source
imaging ............................................. 83
4.3 Active array imaging of reflectors ........................ 84
4.3.1 Data acquisition ................................... 84
4.3.2 Source and reflector array imaging: comparison ..... 85
4.3.3 Modeling ........................................... 85
4.3.4 Nonlinear inversion ................................ 86
4.3.5 Linearization of the forward problem ............... 86
4.3.6 Linearized inversion ............................... 88
4.3.7 The reverse-time imaging function .................. 89
4.3.8 Kirchhoff migration (or travel-time migration) ..... 91
4.3.9 Summary of resolution analysis for active
reflector imaging .................................. 91
4.4 A remark about time-reversal experiments .................. 92
4.5 Conclusion ................................................ 92
5 Passive array imaging of reflectors using ambient noise
illumination .............................................. 94
5.1 Imaging configurations of noise sources, sensors, and
reflectors ................................................ 94
5.2 Stationary phase analysis of the cross correlation with
reflectors ................................................ 96
5.3 Migration imaging of cross correlations ................... 99
5.3.1 Migration imaging with daylight illumination ...... 100
5.3.2 Migration imaging with backlight illumination ..... 101
5.3.3 Migration imaging with surround light
illumination ...................................... 103
5.4 Conclusion ............................................... 105
6 Resolution analysis for passive array imaging using
ambient noise illumination ............................... 106
6.1 A comparison of reflector imaging with active and
passive arrays ........................................... 107
6.2 Imaging by cross correlation of signals generated by
ambient noise sources .................................... 108
6.2.1 The wave equation with noise sources .............. 108
6.2.2 Statistical stability of the cross correlation
function .......................................... 108
6.2.3 Passive sensor imaging ............................ 109
6.2.4 Hypothesis of small decoherence time and
correlation radius for the noise sources .......... 110
6.3 Structure of the cross correlations in a homogeneous
medium ................................................... 111
6.3.1 The background Green's function ................... 111
6.3.2 The peaks of the cross correlation in the
presence of a reflector ........................... 111
6.4 Resolution analysis of correlation-based imaging ......... 115
6.4.1 The daylight imaging function ..................... 115
6.4.2 The backlight imaging function .................... 122
6.4.3 Numerical simulations ............................. 124
6.4.4 Role of illumination diversity .................... 125
6.5 Conclusion ............................................... 126
6.A Appendix: Proof of Proposition 6.2 ................... 126
6.В Appendix: Proof of Propositions 6.4-6.5 .............. 128
6.C Appendix: Proof of Proposition 6.6 ................... 132
7 Travel time estimation using ambient noise in weakly
scattering media ......................................... 136
7.1 Role of scattering in travel time estimation with
cross correlations ....................................... 136
7.2 A model for the scattering medium ........................ 138
7.3 Signal-to-noise ratio reduction and enhanced resolution
due to scattering ........................................ 140
7.4 Use of fourth-order cross correlations ................... 142
7.5 Conclusion ............................................... 145
7.A Appendix: Complete expression of the average cross
correlation .......................................... 146
7.В Appendix: Proof of Proposition 7.1 ................... 148
7.C Appendix: Proof of Proposition 7.2 ................... 149
8 Correlation-based reflector imaging using ambient
noise in weakly scattering media ......................... 152
8.1 Role of scattering in correlation-based imaging .......... 152
8.2 Passive sensor imaging in a randomly scattering medium ... 154
8.2.1 A model for the scattering medium ................. 155
8.2.2 The differential cross correlation ................ 156
8.2.3 Expansion of the clutter Green's function ......... 157
8.2.4 Expansion of the differential cross correlation ... 159
8.2.5 Statistical analysis of the differential cross
correlation ....................................... 160
8.2.6 On the trade-off between resolution enhancement
and signal-to-noise ratio reduction ............... 164
8.2.7 Numerical simulation of migration imaging with
cross correlations in the presence of
scatterers ........................................ 164
8.3 Passive sensor imaging with a reflecting interface ....... 165
8.3.1 Stationary phase analysis of the cross
correlation with a reflecting interface ........... 166
8.3.2 Numerical simulations of migration imaging with
cross correlations in the presence of an
interface ......................................... 168
8.4 Iterated cross correlations for passive imaging in
a randomly scattering medium ............................. 170
8.4.1 The coda cross correlation ........................ 170
8.4.2 Numerical simulations of migration imaging with
coda cross correlations ........................... 172
8.5 Conclusion ............................................... 172
8.A Appendix: Proof of Proposition 8.1 ................... 174
8.В Appendix: Proof of Proposition 8.2 ................... 178
8.B.1 First group .................................... 178
8.B.2 Second group ................................... 180
8.C Appendix: Statistical analysis of the cross
correlations ......................................... 182
8.C.1 The cross correlation at the difference of
travel times ................................... 182
8.C.2 The cross correlation at the sum of travel
times .......................................... 184
8.D Appendix: Proof of Proposition 8.3 ................... 185
9 Virtual source imaging in homogeneous media .............. 187
9.1 Introduction to virtual source imaging ................... 187
9.2 Ideal virtual source imaging with an infinite source
array .................................................... 190
9.3 High-frequency analysis in a homogeneous background
with a limited source array .............................. 191
9.3.1 Direct scattering problem ......................... 191
9.3.2 High-frequency analysis of the cross
correlations ...................................... 192
9.3.3 High-frequency analysis of the imaging function ... 195
9.4 Passive synthetic aperture imaging in a homogeneous
background ............................................... 197
9.4.1 High-frequency analysis of the imaging function ... 198
9.4.2 Comparison with classical synthetic aperture
imaging ........................................... 199
9.5 Conclusion ............................................... 201
9.A Appendix: Proof of Proposition 9.2 ................... 202
9.B Appendix: Proof of Proposition 9.3 ................... 203
10 Virtual source imaging in scattering media ............... 206
10.1 The auxiliary array imaging setup ........................ 206
10.2 Time-reversal interpretation of virtual source imaging ... 208
10.3 The paraxial approximation in random media ............... 209
10.3.1 The main results in the paraxial approximation .... 210
10.3.2 Validity of the paraxial approximation in
random media ...................................... 211
10.4 Analysis of virtual source imaging in the random
paraxial regime .......................................... 212
10.4.1 The cross correlation of the recorded field ....... 212
10.4.2 Migration of cross correlations ................... 216
10.5 Numerical simulations .................................... 218
10.6 Passive synthetic aperture imaging in random media ....... 219
10.7 Conclusion ............................................... 222
10.A Appendix: Proofs of Propositions 10.1-10.2 .......... 223
10.B Appendix: Proofs of Propositions 10.3-10.4 .......... 227
11 Imaging with intensity cross correlations ................ 228
11.1 The ghost imaging setup .................................. 228
11.2 The intensity correlation function ....................... 231
11.2.1 The empirical and statistical correlations ........ 231
11.2.2 Paraxial regime ................................... 233
11.2.3 Time-reversal interpretation ...................... 235
11.2.4 Averaging with respect to the random medium ....... 236
11.3 Resolution analysis ...................................... 237
11.3.1 Resolution analysis for the fully incoherent
case .............................................. 237
11.3.2 Resolution analysis for the partially coherent
case .............................................. 240
11.4 Conclusion ............................................... 242
11.A Appendix: The fields in the white-noise paraxial
regime .............................................. 243
12 A review of wave propagation in random media ............. 245
12.1 The random travel time model ............................. 245
12.1.1 Domain of validity ................................ 245
12.1.2 Statistics of the amplitude and phase
perturbations ..................................... 247
12.1.3 The moments of the Green's function ............... 250
12.2 The random paraxial model ................................ 253
12.2.1 The random paraxial regime ........................ 253
12.2.2 The random paraxial wave equation ................. 254
12.2.3 The moments of the fundamental solution ........... 255
12.3 The randomly layered model ............................... 258
12.3.1 The scaling regime ................................ 258
12.3.2 Review of wave propagation in randomly layered
media ............................................. 260
12.3.3 Statistics of the Green's function ................ 261
12.4 Conclusion ............................................... 262
12.A Appendix: Proof of Lemma 12.1 ....................... 262
12.В Appendix: Proof of Proposition 12.6 ................. 264
12.С Appendix: Proof of Proposition 12.8 ................. 267
13 Appendix: Basic facts from analysis and probability ...... 269
13.1 Fourier identities ....................................... 269
13.2 Divergence theorem ....................................... 270
13.3 Stationary phase method .................................. 270
13.4 Sampling theorem ......................................... 272
13.5 Random processes ......................................... 274
13.5.1 Random variables .................................. 274
13.5.2 Random vectors .................................... 275
13.5.3 Gaussian random vectors ........................... 276
13.5.4 Random processes .................................. 277
13.5.5 Ergodic processes ................................. 278
13.5.6 Mean square theory ................................ 279
13.5.7 Gaussian processes ................................ 281
13.5.8 Stationary Gaussian processes ..................... 282
13.5.9 Vector-and complex-valued Gaussian processes ...... 283
References .................................................... 285
Index ......................................................... 293
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