Mathematical models for passive imaging I: general background

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Yves Colin De Verdière

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Mathematical models for passive imaging I: general background. Yves Colin de Verdiere ? September 25, 2006 Abstract Passive imaging is a new technics which has been proved to be very efficient, for example in seismology: the correlation of the noisy fields be- tween different points is strongly related to the Green function of the wave propagation. The aim of this paper is to provide a mathematical context for this approach and to show, in particular, how the methods of semi- classical analysis can be be used in order to find the asymptotic behaviour of the correlations. Introduction Passive imaging is a way to solve inverse problems: it has been succesfull in seismology and acoustics [2, 3, 11, 15, 16, 20, 21, 23]. The method is as follows: let us assume that we have a medium X (a smooth manifold) and a smooth, deterministic (no randomness in it) linear wave equation in X. We hope to recover (part of) the geometry of X from the wave propagation. We assume that there is somewhere in X a source of noise f(x, t) which is a stationary random field. This source generates, by the wave propagation, a field u(x, t) = (u?(x, t))?=1,··· ,N which people do record on long time intervalls.

dispersion relation

relation between

pseudo-differential equations

high frequency limit

wave equations

schrodinger equation

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MathematicalmodelsforpassiveimagingI:generalbackground.YvesColindeVerdie`re∗September25,2006AbstractPassiveimagingisanewtechnicswhichhasbeenprovedtobeveryeﬃcient,forexampleinseismology:thecorrelationofthenoisyﬁeldsbe-tweendiﬀerentpointsisstronglyrelatedtotheGreenfunctionofthewavepropagation.Theaimofthispaperistoprovideamathematicalcontextforthisapproachandtoshow,inparticular,howthemethodsofsemi-classicalanalysiscanbebeusedinordertoﬁndtheasymptoticbehaviourofthecorrelations.IntroductionPassiveimagingisawaytosolveinverseproblems:ithasbeensuccesfullinseismologyandacoustics[2,3,11,15,16,20,21,23].Themethodisasfollows:letusassumethatwehaveamediumX(asmoothmanifold)andasmooth,deterministic(norandomnessinit)linearwaveequationinX.Wehopetorecover(partof)thegeometryofXfromthewavepropagation.WeassumethatthereissomewhereinXasourceofnoisef(x,t)whichisastationaryrandomﬁeld.Thissourcegenerates,bythewavepropagation,aﬁeldu(x,t)=(uα(x,t))α=1,∙∙∙,Nwhichpeopledorecordonlongtimeintervalls.WewanttogetsomeinformationonthepropagationofwavesfromBtoAinXfromthecorrelationmatrix1TZ1?CA,B(τ)=limu(A,t)⊗u(B,t−τ)dtT∞→T0(equivalentlyTZ1βααCA,B(τ)=Tli→m∞u(A,t)uβ(B,t−τ)dt)T0∗InstitutFourier,Unite´mixtederechercheCNRS-UJF5582,BP74,38402-SaintMartind’He`resCedex(France);http://www-fourier.ujf-grenoble.fr/˜ycolver/1Foreverymatrix(aij),wewrite(aij)?:=(aji).1

whichcanbecomputednumericallyfromtheﬁeldsrecordedatAandB.ItturnsoutthatCA,B(τ)iscloselyrelatedtothedeterministicGreen’sfunctionG(A,B,τ)ofthewaveequationinX.Itmeansthatonecanhopetorecover,usingFourieranalysis,thepropagationspeedsofwavesbetweenAandBasafunctionofthefrequency,or,inotherwords,theso-calleddispersionrelation.Ifthewavedynamicsistimereversalsymmetric,thecorrelationadmitsalsoasymmetrybychangeofτinto−τ;thisobservationhasbeenusedforclocksynchronizationintheocean,see[17].ThegoalofthispaperistogivepreciseformulaeforCA,B(τ)inthehighfrequencylimitassumingarapidedecayofcorrelationsofthesourcef.Moreprecisely,wehave2smallparameters,oneofthementeringintothecorrelationdistanceofthesourcenoise,theotheroneinthehighfrequencypropagation.Thefactthatbothareofthesameorderofmagnitudeiscrucialforthemethod.Letusalsomentiononthetechnicalsidethat,ratherthanusingmodede-compositions,weprefertoworkdirectlywiththedynamics;inotherwords,weneedreallyatimedependentratherthanastationaryapproach.Modedecompo-sitionsareoftenusefull,buttheyareofnomuchhelpforgeneraloperatorswithnoparticularsymmetry.Forclarity,wewillﬁrstdiscussthenon-physicalcaseofaﬁrstorderwaveequationliketheSchro¨dingerequation,thenthecaseofamoreusualwaveequa-tions(acoustics,elasticity).Themainresultexpresses,forτ>0,CA,B(τ)astheSchwartzkernelofΩ(τ)◦ΠwhereΠisasuitablepseudo-diﬀerentialoperator(aΨDO),whoseprincipalsymbolcanbeexplicitelycomputed,andΩ(τ)isthe(semi-)groupofthe(damped)wavepropagation.Itimpliesthatwecanrecoverthedispersionrelation,i.e.theclassicaldynamics,fromtheknowledgeofalltwo-pointscorrelations.Inordertomakethepaperreadablebyalargesetofpeople,wehavetriedtomakeitself-containedbyincludingsectionsonpseudo-diﬀerentialoperatorsandonrandomﬁelds.InSection1,westartwithaquitegeneralsettinganddiscussageneralformulaforthecorrelation(Equation(4)).Section2isdevotedtoexactformulaeincaseofanhomogeneouswhitenoise.InSection3,wediscusstheimportantpropertyoftimereversalsymmetrywhichplaysaprominentpartintheapplicationsandisalsousefullasanumerical.tsetInSection4,weintroducealargefamilyofanisotropicrandomﬁeldsandshowtherelationbetweentheirpowerspectraandtheWignermeasures.Section5containsthemainresultexpressingthecorrelationinthecaseofaSchro¨dingerwaveequation.Section6doesthesameincaseofawaveequation.TheshortSection7isaproblemsection.Section8isaaboutaquiteindependentissuerelativetocorrelationsofscat-teredwaves.2

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