Full-Seismic-Waveform-Inversion-The-Adjoint-Method

ShortCourseFullSeismicWaveformInversionTheAdjointMethodAndreasFichtnerOUTLOOK1.Introductionseismictomography–purposesanddifficulties2.Waveformtomographysetupasoptimisationproblem3.TheAdjointMethodmotivation&generalconceptsensitivitykernelsobjectivefunctionals:cross-correlationtimeshifts,rmsamplitudedifferences,…4.Solutionoftheoptimisationproblempreparatorysteps&initialmodelconjugate-gradientalgorithmpreconditioning5.Resultsfromourwork1.Introduction•EarthquakesoractivesourcesexcitewavesthattravelthroughtheEarth.•Recordedasseismograms.•WaveformsdependontheEarth‘sstructure.•Seismictomography:InferEarthstructurefromseismograms.t=200st=600sINTRODUCTION:seismictomography•HighPerformanceComputing:SimulationofelasticwavepropagationthroughhighlyheterogeneousEarthmodels•Exploitmoreinformationfromseismograms→MoreaccurateEarthmodels.wavefieldsnapshots,horizontalslices@100kmdepthINTRODUCTION:objectivesPrincipalobjectivesofseismictomography:•inferthepresentdynamicsandevolutionoftheEarth•produceaccurateEarthmodelsforreliabletsunamiwarnings…•…andthemonitoringoftheComprehensiveNuclearTest-BanTreaty•searchfornaturalresources•assessmentofconstructiongroundproperties•monitoringofsubsurfaceprocesses(CO2storage,magmachambers)•…Principalobjectivesofseismictomography:•non-destructivetestinginmaterialsciences•medicalimaging•…INTRODUCTION:difficultiesofseismictomographyWhatmakesseismictomographydifficult?•receivers:unevenlydistributed•sources:fewandpoorlydistributed•sourcemechanisms:unknown•wavefield:highlycomplex•measurement:verychallenging•inversion:non-linearandill-posed•receivers:welldistributed•sources:welldistributed•sourcemechanisms:known•wavefield:rathersimple•measurement:rathersimple•inversion:ratherwell-posed,linearseismictomographymedicaltomographyWorldseismicmap2.Waveformtomography(general)WAVEFORMTOMOGRAPHY:SetupasoptimisationproblemSeismictomographyisusuallysetasanon-linearoptimisationproblem.Solutionoftheforwardproblem-Elasticwaveequation-NoanalyticalsolutionsinrealisticallyheterogeneousEarthmodel-Numericalmethods(FD,FEM,SEM,…)1Comparisonofdataandsynthetics-Physicallymeaningfulmeasuresofmisfitthatareapplicabletoimperfectdata-Physicalintuitionrequired2Gradientofmisfitfunctionalw/rmodel-Sensitivitydensities(Fréchetkernels)viatheadjointmethod-Choiceofthemodelparameterisation3Iterativemisfitminimisation-steepestdescent,conjugategradients,…-regularisation-pre-conditioning4WAVEFORMTOMOGRAPHY:Characterandsizeoftheproblem≈500000freeparametersModelData≈3000waveforms500000freeparameters+gradientmethodsformisfitminimisation500000partialderivativesofthemisfitwithrespecttothemodelparametersWAVEFORMTOMOGRAPHY:CharacterandsizeoftheproblemDifferentwaysofcomputingpartialderivatives:•Finitedifferencing:500,001forwardsimulationsx0.5hpersimulationx126processorsx50earthquakesx4simulationsperconjugategradientiterationx10conjugategradientiterations6.3e10cpuhours≈720,000cpuyearsInaccuratebecausewedividesmallnumbersbysmallnumbers.•Automaticdifferentiation:knowsnothingaboutphysics→inefficient•Adjointmethod:iΔparameterameters)misfit(par)iΔparameterametersmisfit(pariparameterameters)misfit(par−+≈∂∂WAVEFORMTOMOGRAPHY:Howtocomputehalfamillionderivatives3.Theadjointmethod3.1.GeneralconceptTheadjointmethodisamathematicaltrickthatallowsustocomputetheexactpartialderivativeswith2insteadof500,001simulations.Verygeneralandwidelyused(meteorology,groundwatermodelling,optimalcontrolproblems,…)Therelevantequationscanbederivedindifferentways:1)Lagrangemultipliermethod(e.g.Liu&Tromp,GJI2008)2)Perturbationtheory(e.g.Tarantola,PAGEOPH1988)3)Operatorformulation(Fichtneretal.,PEPI2006)-verycompact-applicabletoanyPDE,includingtheelasticwaveequationADJOINTMETHODADJOINTMETHOD:Applicationtotheelasticwaveequation∫∞∞−τ∇τ−⋅∇−∂=dt),(:)t,(t),()ρ()ρ,,(2txuxCxuxCuL&0==0tt|t),(xu0=∂=0ttt|t),(xu∫∞−∈=ττ∇τ−⋅tΓ|d),(:)t,(0xxuxCn&Elasticwaveequation:Subsidiaryconditions:f=),(CuLρ,Adjointwaveequation:∫∞∞−τ∇−τ⋅∇−∂=dt),(:t),(t),()ρ(ρ),(2ttxuxCxuxCuLttt&,ttρ),(f=CuLt,←determinedbythemisfitmeasureAdjointsubsidiaryconditions:0xut==1tt|t),(0xut=∂=1ttt|t),(∫∞−∈=ττ∇−τ⋅tΓ|d),(:t),(0xuxCnxt&3.2.DerivativeswithrespecttoselectedstructuralparametersADJOINTMETHOD:DerivativesinanisotropicmediumjkiljlikklijijklδμδδμδδλδC++=Elastictensorforisotropicmedia:λ,μ:LaméparametersPartialderivatives:ThetermsunderthevolumeintegralarethesensitivitykernelsorFréchetkernels…ADJOINTMETHOD:DerivativesinanisotropicmediumFréchetkernelsforanisotropicmedium:ADJOINTMETHOD:DerivativesinanisotropicmediumPartialderivativeswithrespecttodifferentsetsofparameters:ADJOINTMETHOD:DerivativesinamediumwithradialsymmetryTheelastictensorinamediumwithradialsymmetryaxis:[→Polarisationanisotropy]ADJOINTMETHOD:DerivativesinamediumwithradialsymmetryFréchetkernelsforamediumwithradialanisotropy:⎟⎟⎠⎞⎜⎜⎝⎛∂+∂=ijjiijdxudxu21εstraintensor3.3.Derivativeswithrespecttotheright-handside(thesource)ADJOINTMETHOD:Derivativeswithrespecttotheright-handside•HowdoesthemisfitchangeasIchangethesource?•InsteadofL(u,m)=f,weconsiderL‘(u,m,f)=L(u,m)-f=0.•Repeat