Table Of ContentPASSIVE IMAGING WITH AMBIENT NOISE
Wavesgeneratedbyopportunisticorambientnoisesourcesandrecordedbypassivesensor
arrays can be used to image the medium through which they travel. Spectacular results
have been obtained in seismic interferometry, which open up new perspectives in acous-
tics, electromagnetics, and optics. The authors present, for the first time in book form,
a self-contained and unified account of correlation-based and ambient noise imaging. In
ordertofacilitateunderstandingofthecorematerial,theyalsoaddressanumberofrelated
topicsinconventionalsensorarrayimaging,wavepropagationinrandommedia,andhigh-
frequency asymptotics for wave propagation. Taking a multidisciplinary approach, the
bookusesmathematicaltoolsfromprobability,partialdifferentialequations,andasymp-
totic analysis, combined with the physics of wave propagation and modeling of imaging
modalities.Suitableforappliedmathematiciansandgeophysicists,itisalsoaccessibleto
graduatestudentsinappliedmathematics,physics,andengineering.
JOSSELIN GARNIER is a professor in the Mathematics Department at the Université
ParisDiderot,France.Hisbackgroundisinappliedprobabilityandhehasmanyyearsof
research experience in the field of wave propagation and imaging in random media. He
received the Blaise Pascal prize from the French Academy of Sciences in 2007 and the
FelixKleinprizefromtheEuropeanMathematicalSocietyin2008.
GEORGE PAPANICOLAOU istheRobertGrimmettProfessorinMathematicsatStan-
ford University, USA. He specializes in applied and computational mathematics, partial
differentialequations,andstochasticprocesses.HereceivedtheJohnvonNeumannprize
from the Society for Industrial and Applied Mathematics in 2006 and the William Ben-
ter prize in Applied Mathematics in 2010. He was elected to the National Academy of
Sciencesin2000andhebecameafellowoftheAmericanMathematicalSocietyin2012.
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PASSIVE IMAGING WITH
AMBIENT NOISE
JOSSELIN GARNIER
UniversitéParisDiderot,France
GEORGE PAPANICOLAOU
StanfordUniversity,USA
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(cid:2)c JosselinGarnierandGeorgePapanicolaou2016
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Firstpublished2016
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Names:Garnier,Josselin,author. | Papanicolaou,George,author.
Title:Passiveimagingwithambientnoise/JosselinGarnier(Université
ParisDiderot,France),GeorgePapanicolaou(StanfordUniversity,USA).
Description:Cambridge,UnitedKingdom;NewYork,NY:CambridgeUniversity
Press,2016. | (cid:2)c 2016 | Includesbibliographicalreferencesandindex.
Identifiers:LCCN2015051482 | ISBN9781107135635(hardback;alk.paper) |
ISBN110713563X(hardback;alk.paper)
Subjects:LCSH:Imageprocessing–Mathematics. | Noise. | Green’s
functions. | Waveequation.
Classification:LCCTA1637.G372016 | DDC621.36/78–dc23
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Contents
Preface pagexi
1 Introductionandoverviewofthebook 1
1.1 Whypassive,correlation-basedimaging? 1
1.1.1 Traveltimeestimation 2
1.1.2 Applicationsoftraveltimeestimation 3
1.1.3 Reflectorimaging 4
1.1.4 Auxiliaryarrayorvirtualsourceimaging 6
1.1.5 Passivesyntheticapertureimaging 8
1.1.6 Imagingwithintensitycrosscorrelations 9
1.2 Chapter-by-chapterdescriptionofthebook 10
2 Green’sfunctionestimationfromnoisecrosscorrelations 17
2.1 ThescalarwaveequationanditsGreen’sfunction 17
2.1.1 TheSommerfeldradiationcondition 19
2.1.2 Reciprocity 20
2.1.3 TheHelmholtz–Kirchhoffidentity 21
2.1.4 Applicationtotimereversal 23
2.2 Thescalarwaveequationwithnoisesources 25
2.3 Green’sfunctionestimationwithauniformdistributionofsourcesin
ahomogeneousopenmedium 29
2.4 Green’sfunctionestimationwithanextendeddistributionofsources
inaninhomogeneousopenmedium 31
2.5 Green’sfunctionestimationwithanextendeddistributionofsources
inaninhomogeneouscavity 34
2.6 Green’sfunctionestimationwithalimiteddistributionofsourcesin
aone-dimensionalinhomogeneousmedium 38
2.6.1 Theone-dimensionalwaveequation 39
v
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vi Contents
2.6.2 Reflectionseismology 43
2.6.3 Daylightimaging 45
2.7 Conclusion 48
2.A Appendix:thecovarianceoftheempiricalcrosscorrelation 48
3 Travel time estimation from noise cross correlations using
stationaryphase 51
3.1 High-frequencywavepropagation 52
3.2 High-frequencyasymptoticanalysisoftheGreen’sfunctionina
homogeneousmedium 53
3.3 High-frequencyasymptoticanalysisoftheGreen’sfunctionina
smoothlyvaryingmedium 53
3.3.1 Anintroductiontogeometricaloptics 53
3.3.2 Raysolutionoftheeikonalequation 55
3.3.3 Fermat’sprincipleforthetraveltime 57
3.3.4 Propertiesofthetraveltime 58
3.4 High-frequencyasymptoticanalysisofthecrosscorrelation 60
3.5 Conclusion 67
4 Overviewofconventionalsensorarrayimaging 68
4.1 Passivearrayimagingofsources 68
4.1.1 Dataacquisition 68
4.1.2 Imagingfunction 69
4.1.3 Thelinearforwardoperator 69
4.1.4 Theadjointoperator 70
4.1.5 Leastsquaresinversion 71
4.1.6 Thereverse-timeimagingfunction 73
4.1.7 Kirchhoffmigration(ortravel-timemigration) 74
4.2 Passivearrayimagingofsources:resolutionanalysis 74
4.2.1 Full-aperturearray 75
4.2.2 Partial-aperturearray 75
4.2.3 Summaryofresolutionanalysisforpassivesourceimaging 83
4.3 Activearrayimagingofreflectors 84
4.3.1 Dataacquisition 84
4.3.2 Sourceandreflectorarrayimaging:comparison 85
4.3.3 Modeling 85
4.3.4 Nonlinearinversion 86
4.3.5 Linearizationoftheforwardproblem 86
4.3.6 Linearizedinversion 88
4.3.7 Thereverse-timeimagingfunction 89
4.3.8 Kirchhoffmigration(ortravel-timemigration) 91
4.3.9 Summaryofresolutionanalysisforactivereflectorimaging 91
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Contents vii
4.4 Aremarkabouttime-reversalexperiments 92
4.5 Conclusion 92
5 Passivearrayimagingofreflectorsusingambientnoiseillumination 94
5.1 Imagingconfigurationsofnoisesources,sensors,andreflectors 94
5.2 Stationaryphaseanalysisofthecrosscorrelationwithreflectors 96
5.3 Migrationimagingofcrosscorrelations 99
5.3.1 Migrationimagingwithdaylightillumination 100
5.3.2 Migrationimagingwithbacklightillumination 101
5.3.3 Migrationimagingwithsurroundlightillumination 103
5.4 Conclusion 105
6 Resolution analysis for passive array imaging using ambient noise
illumination 106
6.1 Acomparisonofreflectorimagingwithactiveandpassivearrays 107
6.2 Imagingbycrosscorrelationofsignalsgeneratedbyambientnoise
sources 108
6.2.1 Thewaveequationwithnoisesources 108
6.2.2 Statisticalstabilityofthecrosscorrelationfunction 108
6.2.3 Passivesensorimaging 109
6.2.4 Hypothesis of small decoherence time and correlation
radiusforthenoisesources 110
6.3 Structureofthecrosscorrelationsinahomogeneousmedium 111
6.3.1 ThebackgroundGreen’sfunction 111
6.3.2 Thepeaksofthecrosscorrelationinthepresenceofareflector 111
6.4 Resolutionanalysisofcorrelation-basedimaging 115
6.4.1 Thedaylightimagingfunction 115
6.4.2 Thebacklightimagingfunction 122
6.4.3 Numericalsimulations 124
6.4.4 Roleofilluminationdiversity 125
6.5 Conclusion 126
6.A Appendix:ProofofProposition6.2 126
6.B Appendix:ProofofPropositions6.4–6.5 128
6.C Appendix:ProofofProposition6.6 132
7 Travel time estimation using ambient noise in weakly scattering
media 136
7.1 Roleofscatteringintraveltimeestimationwithcrosscorrelations 136
7.2 Amodelforthescatteringmedium 138
7.3 Signal-to-noise ratio reduction and enhanced resolution due to
scattering 140
7.4 Useoffourth-ordercrosscorrelations 142
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viii Contents
7.5 Conclusion 145
7.A Appendix:Completeexpressionoftheaveragecrosscorrelation 146
7.B Appendix:ProofofProposition7.1 148
7.C Appendix:ProofofProposition7.2 149
8 Correlation-basedreflectorimagingusingambientnoiseinweakly
scatteringmedia 152
8.1 Roleofscatteringincorrelation-basedimaging 152
8.2 Passivesensorimaginginarandomlyscatteringmedium 154
8.2.1 Amodelforthescatteringmedium 155
8.2.2 Thedifferentialcrosscorrelation 156
8.2.3 ExpansionoftheclutterGreen’sfunction 157
8.2.4 Expansionofthedifferentialcrosscorrelation 159
8.2.5 Statisticalanalysisofthedifferentialcrosscorrelation 160
8.2.6 On the trade-off between resolution enhancement and
signal-to-noiseratioreduction 164
8.2.7 Numerical simulationofmigrationimaging withcross
correlationsinthepresenceofscatterers 164
8.3 Passivesensorimagingwithareflectinginterface 165
8.3.1 Stationaryphaseanalysisofthecrosscorrelationwitha
reflectinginterface 166
8.3.2 Numericalsimulationsofmigrationimagingwithcross
correlationsinthepresenceofaninterface 168
8.4 Iterated cross correlations for passive imaging in a randomly
scatteringmedium 170
8.4.1 Thecodacrosscorrelation 170
8.4.2 Numericalsimulationsofmigrationimagingwithcoda
crosscorrelations 172
8.5 Conclusion 172
8.A Appendix:ProofofProposition8.1 174
8.B Appendix:ProofofProposition8.2 178
8.B.1 Firstgroup 178
8.B.2 Secondgroup 180
8.C Appendix:Statisticalanalysisofthecrosscorrelations 182
8.C.1 Thecrosscorrelationatthedifferenceoftraveltimes 182
8.C.2 Thecrosscorrelationatthesumoftraveltimes 184
8.D Appendix:ProofofProposition8.3 185
9 Virtualsourceimaginginhomogeneousmedia 187
9.1 Introductiontovirtualsourceimaging 187
9.2 Idealvirtualsourceimagingwithaninfinitesourcearray 190
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Contents ix
9.3 High-frequency analysis in a homogeneous background with a
limitedsourcearray 191
9.3.1 Directscatteringproblem 191
9.3.2 High-frequencyanalysisofthecrosscorrelations 192
9.3.3 High-frequencyanalysisoftheimagingfunction 195
9.4 Passivesyntheticapertureimaginginahomogeneousbackground 197
9.4.1 High-frequencyanalysisoftheimagingfunction 198
9.4.2 Comparisonwithclassicalsyntheticapertureimaging 199
9.5 Conclusion 201
9.A Appendix:ProofofProposition9.2 202
9.B Appendix:ProofofProposition9.3 203
10 Virtualsourceimaginginscatteringmedia 206
10.1 Theauxiliaryarrayimagingsetup 206
10.2 Time-reversalinterpretationofvirtualsourceimaging 208
10.3 Theparaxialapproximationinrandommedia 209
10.3.1 Themainresultsintheparaxialapproximation 210
10.3.2 Validityoftheparaxialapproximationinrandommedia 211
10.4 Analysisofvirtualsourceimagingintherandomparaxialregime 212
10.4.1 Thecrosscorrelationoftherecordedfield 212
10.4.2 Migrationofcrosscorrelations 216
10.5 Numericalsimulations 218
10.6 Passivesyntheticapertureimaginginrandommedia 219
10.7 Conclusion 222
10.A Appendix:ProofsofPropositions10.1–10.2 223
10.B Appendix:ProofsofPropositions10.3–10.4 227
11 Imagingwithintensitycrosscorrelations 228
11.1 Theghostimagingsetup 228
11.2 Theintensitycorrelationfunction 231
11.2.1 Theempiricalandstatisticalcorrelations 231
11.2.2 Paraxialregime 233
11.2.3 Time-reversalinterpretation 235
11.2.4 Averagingwithrespecttotherandommedium 236
11.3 Resolutionanalysis 237
11.3.1 Resolutionanalysisforthefullyincoherentcase 237
11.3.2 Resolutionanalysisforthepartiallycoherentcase 240
11.4 Conclusion 242
11.A Appendix:Thefieldsinthewhite-noiseparaxialregime 243
12 Areviewofwavepropagationinrandommedia 245
12.1 Therandomtraveltimemodel 245
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