andreas fichtner and chris=an boehm...seismology & wave physics outline part i: the...
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KITSummerSchoolonFull-WaveformInversion
Lecture3
Adjointmethodsandsensi.vitykernels
AndreasFichtnerandChris=anBoehm
andtheETHSeismologyandWavePhysicsGroup
Seismology&WavePhysics
Seismology&WavePhysics
OUTLINEPARTI:Thefull-waveforminversionconcept
• Summaryofadream• Formula=onasanop=misa=onproblem• Gradient-baseddescentmethodsPARTII:Theadjointmethod
• Problemstatement• Discreteadjointmethod• Con=nuousadjointmethod• Sensi=vitykernelsØ Break.Timeforques=onsandshortdiscussion.PARTIII:AdvancedTopics
• Localminimaandthemul=scaleapproach• Compressedwavefieldstorage• Secondderiva=ves
Seismology&WavePhysics
PARTI
Thefull-waveforminversionconcept
1.Summaryofadream
Seismology&WavePhysics
FROMTRAVELTIMESTO‘FULL’WAVEFORMS
Svelocityat150kmbeneathAustraliaFishwicketal.,2005
‘tradi=onal’travel=metomographytravel=memeasurements
Extremelysuccessful!
Canassimilateenormousquan==esofdata.
S=llTHEmostwidelyusedtomographicmethod.
Seismology&WavePhysics
FROMTRAVELTIMESTO‘FULL’WAVEFORMS
GOALS
§ Explainbroadbandseismogramswigglebywiggle...§ ...withhardlyanyhumaninterven.on[Tarantolianblackbox]§ Be`erresolvedtomographicimages
• thermochemicalstructureoftheEarth• evolu=onanddynamicsoftheEarth• improvedgroundmo=onpredic=ons• improvedearthquakesourceinversion
• emergencyresponse,tsunamiwarning• tectonicinterpreta=on
• improvedreservoircharacterisa=on• ...
‘tradi=onal’travel=metomographytravel=memeasurements
full-waveforminversioncompleteseismicrecordings
FROMTRAVELTIMESTO‘FULL’WAVEFORMS
full-waveforminversioncompleteseismicrecordings
GOALS
§ Explainbroadbandseismogramswigglebywiggle...§ ...withhardlyanyhumaninterven.on[Tarantolianblackbox]§ Be`erresolvedtomographicimages
• thermochemicalstructureoftheEarth• evolu=onanddynamicsoftheEarth• improvedgroundmo=onpredic=ons• improvedearthquakesourceinversion
• emergencyresponse,tsunamiwarning• tectonicinterpreta=on
• improvedreservoircharacterisa=on• ...
Seismology&WavePhysics
CHALLENGES
§ Seismicwavepropaga=onthroughcomplexmedia.§ Computa=onalpower.§ Nonlinearrela=onbetweenwaveformsand3DEarthstructure.§ Meaningfulmeasurementofwaveformdifferences.§ Algorithmstosearchforusefulmodels[allofthem,ideally].§ ...
2.Formula=onasanop=misa=onproblem
OPTIMISATIONPROBLEM
Seismology&WavePhysics
1. Start from initial Earth model
2. Update according to with
step length descent direction
iiii hmm 1 γ+=+ )()( 1 ii mm χχ <+
§ FindanEarthmodelmsuchthatasuitablydefinedmisfitχisminimal.
§ Thenumberofmodelparametersandthenumericalcostoftheforwardproblempreventtheapplica=onofprobabilis=cmethods.
§ Theminimisa=onproceedsitera=vely:
OPTIMISATIONPROBLEM
Seismology&WavePhysics
1. Start from initial Earth model
2. Update according to with
step length descent direction
iiii hmm 1 γ+=+ )()( 1 ii mm χχ <+
§ FindanEarthmodelmsuchthatasuitablydefinedmisfitχisminimal.
§ Thenumberofmodelparametersandthenumericalcostoftheforwardproblempreventtheapplica=onofprobabilis=cmethods.
§ Theminimisa=onproceedsitera=vely:
Comment:
Minimaldoesnotmeanthesmallestmisfitpossible!
The misfit should become about assmall as the observa=onal andforwardmodellingerrors.
!!!
GRADIENT-BASEDOPTIMISATION
Seismology&WavePhysics
The family of gradient methods:
§ method of steepst descent:
§ conjugate-gradient methods
§ Newton and Newton-like methods
§ BFGS and L-BFGS
§ …
m/hi ∂−∂= χ
1. Start from initial Earth model
2. Update according to with
step length descent direction
iiii hmm 1 γ+=+ )()( 1 ii mm χχ <+
hi ∝− ∂χ
∂mi
GRADIENT-BASEDOPTIMISATION
Seismology&WavePhysics
1. Start from initial Earth model
2. Update according to with
step length descent direction
iiii hmm 1 γ+=+ )()( 1 ii mm χχ <+
00hγ
m0 m1
χ
m
GRADIENT-BASEDOPTIMISATION
Seismology&WavePhysics
1. Start from initial Earth model
2. Update according to with
step length descent direction
iiii hmm 1 γ+=+ )()( 1 ii mm χχ <+
m0 m1 m2 m3 m4 …
Iteratively approach the minimum misfit by following the local descent directions.
χ
m
Seismology&WavePhysics
PARTII
Theadjointmethod
1.Problemstatement
Seismology&WavePhysics
SO,WHEREISTHEPROBLEM?
Seismology&WavePhysics
§ Thefullgradient–withallitscomponents-isneededineachitera=on.
§ Themoststraigheorwardapproach:approximatethegradientbyfinite-differences:
§ Examplewith500,000modelparameters:
500,001forwardsimula=ons
× 0.5hpersimula=on
× 126computecores
× 50sources(earthquakes)
× 50conjugategradientitera=ons
78e9cpuhours≈8,900,000cpuyears
mmmm
mm kk
k δχδχχ ,...)(...,,...)(...,)( −+≈
∂∂
2.Thediscreteadjointmethod
Seismology&WavePhysics
DISCRETEADJOINTMETHODSUMMARY
Seismology&WavePhysics
Lu = f LTv = −∇χ
∂χ∂mi
= vT ∂L∂mi
u
Regularwaveequa=on Adjointwaveequa=on Gradientequa=on
DISCRETEADJOINTMETHODSUMMARY
Seismology&WavePhysics
Regularwaveequa=on Adjointwaveequa=on Gradientequa=on
Adjointrecipe1. Solveforwardproblem[regularwaveequa=on]toobtainu.2. Evaluatemisfitχ.3. Computeadjointsource,-∨χ.4. Solveadjointequa=ontoobtainadjointfieldv.5. Pluguandvintothegradientequa=on.
Lu = f LTv = −∇χ
∂χ∂mi
= vT ∂L∂mi
u
DISCRETEADJOINTMETHODSUMMARY
Seismology&WavePhysics
Regularwaveequa=on Adjointwaveequa=on Gradientequa=on
Adjointrecipe1. Solveforwardproblem[regularwaveequa=on]toobtainu.2. Evaluatemisfitχ.3. Computeadjointsource,-∨χ.4. Solveadjointequa=ontoobtainadjointfieldv.5. Pluguandvintothegradientequa=on.
Comments1. Noneedtoexplicitlycomputethederiva=veofthewavefieldu[byconstruc=on].2. Gradientisen=relydeterminedbythedefini=onofthemisfit[adjointsourceistheonlythingthatexplicitlydependsonthemisfit].3. Computa=onofgradientrequiresstorageofforwardwavefieldu.
Lu = f LTv = −∇χ
∂χ∂mi
= vT ∂L∂mi
u
3.Thecon=nuousadjointmethod
Seismology&WavePhysics
AMATTEROFNOTATION
Seismology&WavePhysics
Lu = (−ω2M+K)u L(u)=ρ!!u−∇⋅(C:∇u)= f
Discretecase[frequencydomain] Con=nuouscase[=medomain]
∇χ = vT∇Lu ∇χ
' = vT∇L(u)dt∫
§ Thesameformalderiva=onfromthediscretecasecanbeusedinthecon=nuouscase.• MatrixLbecomesoperatorL.• ScalarproductaTbbecomesintegral∫a(x)b(x)dx.
§ Insomewhatlooseterms,∨χiscalledasensi.vityorFréchetkernelandsymbolisedbyK.§ Theonlyques=on:WhatisLTinthecon=nuouscase?...SeeRusselHeweQ’slecture!
EXAMPLES
Seismology&WavePhysics
Regularwaveequa.on
momentumbalance
stress-strainrela=on
ini=alcondi=ons
boundarycondi=ons
EXAMPLES
Seismology&WavePhysics
Regularwaveequa.on Adjointwaveequa.on
momentumbalance adjointmomentumbalance
stress-strainrela=on adjointstress-strainrela=on
ini=alcondi=ons terminalcondi=ons
boundarycondi=ons boundarycondi=ons
v = u†nota=on
EXAMPLES
Seismology&WavePhysics
Regularwaveequa.on Adjointwaveequa.on
momentumbalance adjointmomentumbalance
stress-strainrela=on adjointstress-strainrela=on
ini=alcondi=ons terminalcondi=ons
boundarycondi=ons boundarycondi=ons
Comments§ Adjointequa=onisawaveequa=on[samecodecanbeusedforitssolu=on].
§ Solvingterminalcondi=onscanbedonebyrunningcodeinreversed=me.
4.Sensi=vitykernels
Seismology&WavePhysics
Seismology&WavePhysics
TRAVELTIMEMEASUREMENTONSPECIFICPHASES
Seismology&WavePhysics
TRAVELTIMEMEASUREMENTONSPECIFICPHASES
Source-receivergeometry Seismograms
Seismology&WavePhysics
Source-receivergeometry Seismograms
Sensi.vitykernelforPwavevelocityreceiver
source
- +
TRAVELTIMEMEASUREMENTONSPECIFICPHASES
Seismology&WavePhysics
Source-receivergeometry Seismograms
Sensi.vitykernelforSwavevelocity
- +
TRAVELTIMEMEASUREMENTONSPECIFICPHASES
Seismology&WavePhysics
Source-receivergeometry Seismograms
Sensi.vitykernelsforPwavevelocity and Swavevelocity
- +
TRAVELTIMEMEASUREMENTONSPECIFICPHASES
Seismology&WavePhysics
Source-receivergeometry Seismograms
Sensi.vitykernelforSwavevelocity
- +
TRAVELTIMEMEASUREMENTONSPECIFICPHASES
Seismology&WavePhysics
MEASURINGTIME-FREQUENCYPHASEDIFFERENCES
Seismology&WavePhysics
MEASURINGTIME-FREQUENCYPHASEDIFFERENCES
*Mb5.1,25August2007
ver=cal-componentdisplacement,period=10s32.8milliongridpoints
Seismology&WavePhysics
MEASURINGTIME-FREQUENCYPHASEDIFFERENCES
§ Time-andfrequency-dependentphasedifferences
§ Basedonselec=onof=mewindowswheredataandsynthe=csaresimilar
§ Independentofabsoluteamplitudes
Seismology&WavePhysics
MEASURINGTIME-FREQUENCYPHASEDIFFERENCES
Sensi=vitykernels
Seismology&WavePhysics
PARTIII
AdvancedTopics
1.Localminimaandthemul=scaleapproach
Seismology&WavePhysics
Seismology&WavePhysics
THECAMEMBERTEXPERIMENT
OdileGauthier,JeanVirieux,AlbertTarantola,Geophysics1986.
Theacous.cCamembertModel§ 20%velocityperturba=on§ 8sourcesandreceiversaroundthemodel
Seismology&WavePhysics
THECAMEMBERTEXPERIMENT
OdileGauthier,JeanVirieux,AlbertTarantola,Geophysics1986.
Theacous.cCamembertModel§ 20%velocityperturba=on§ 8sourcesandreceiversaroundthemodel
InversionresultaXer5itera.ons§ Camembertnotrecovered§ Stuckinalocalminimum
THEMULTISCALEAPPROACH
§ Iden=fiescycleskippingasmainreasonfornonlinearity.§ Misfitsurfacemorecomplexthehigherthefrequency.§ Startwithlowfrequencies.§ Workyourwayuptohighfrequencies.§ Problems.ll:Lowfrequenciesmaynotalwaysbeavailable
Seismology&WavePhysics
2.Compressedwavefieldstorage
Seismology&WavePhysics
Seismology&WavePhysics
PROBLEMSTATEMENT
Sensi.vitykernelexamples
Seismology&WavePhysics
PROBLEMSTATEMENT
§ Forwardandadjointfieldsmustbeknownatthesame.me.§ Thisisnotnaturallythecase.§ Forwardwavefieldneedstobestored.§ Thisisextremelyexpensive!
Sensi.vitykernelexamples
Seismology&WavePhysics
PROBLEMSTATEMENT
§ Forwardandadjointfieldsmustbeknownatthesame.me.§ Thisisnotnaturallythecase.§ Forwardwavefieldneedstobestored.§ Thisisextremelyexpensive!
Canwesomehowcompressthewavefieldusuchthatthekernelintegralsares.llsufficientlyaccurate?
Sensi.vitykernelexamples
Seismology&WavePhysics
LOSSYCOMPRESSIONSTRATEGIES
1. Requan=sa=on• adjustnumberofbitstorepresentfieldvalues• largenumberofbitsinregionswithlargeamplitudevaria=onsandviceversa
Seismology&WavePhysics
LOSSYCOMPRESSIONSTRATEGIES
1. Requan=sa=on• adjustnumberofbitstorepresentfieldvalues• largenumberofbitsinregionswithlargeamplitudevaria=onsandviceversa
2. p-coarsening
• storewavefieldwithpolynomialdegreepasanewpolynomialofdegreepnew<p• re-interpolatetoapproximatethekernelintegral
Seismology&WavePhysics
LOSSYCOMPRESSIONSTRATEGIES
1. Requan=sa=on• adjustnumberofbitstorepresentfieldvalues• largenumberofbitsinregionswithlargeamplitudevaria=onsandviceversa
2. p-coarsening
• storewavefieldwithpolynomialdegreepasanewpolynomialofdegreepnew<p• re-interpolatetoapproximatethekernelintegral
3. Temporalinterpola=on
• Storewavefieldonlyeverynth=mestep• Splineinterpola=ontofillmissing=mestepsforkernelintegral
Seismology&WavePhysics
LOSSYCOMPRESSIONSTRATEGIES
1. Requan=sa=on• adjustnumberofbitstorepresentfieldvalues• largenumberofbitsinregionswithlargeamplitudevaria=onsandviceversa
2. p-coarsening
• storewavefieldwithpolynomialdegreepasanewpolynomialofdegreepnew<p• re-interpolatetoapproximatethekernelintegral
3. Temporalinterpola=on
• Storewavefieldonlyeverynth=mestep• Splineinterpola=ontofillmissing=mestepsforkernelintegral
4. Lazyforwardandadjointsimula=ons
• Storeforwardandadjointfieldonlyinregionswheretheyoverlap
Seismology&WavePhysics
EXAMPLE[COMBININGTHESESTRATEGIES]
Boehm&Fichtner,Geophysics2016
Seismology&WavePhysicsBoehm&Fichtner,Geophysics2016
EXAMPLE[COMBININGTHESESTRATEGIES]
Seismology&WavePhysicsBoehm&Fichtner,Geophysics2016
cf=compressionfactor
EXAMPLE[COMBININGTHESESTRATEGIES]
Seismology&WavePhysicsBoehm&Fichtner,Geophysics2016
cf=compressionfactor
EXAMPLE[COMBININGTHESESTRATEGIES]
AcompressionfactorofO(1000)isozenfeasible.
3.Secondderiva=ves
Seismology&WavePhysics
Seismology&WavePhysics
WHYAREWEINTERESTEDINSECONDDERIVATIVES?
§ Quadra=capproxima=onofthemisfitfunc=onalneartheop=malmodel[approximatelyvanishingfirstderiva=ve].
§ TheHessianH[second-deriva=vematrix]:
• Localgeometryofthemisfitsurface• resolu=onandtrade-offs• H=inverseposteriorcovariance
Ø Hcontainsinforma.ononuncertain.es!
mHmmmm Toptopt δδχδχ +≈+ )()(
misfit functional optimal Earth model Hessian at mopt
Seismology&WavePhysics
SECONDDERIVATIVES
§ Hcannotbecomputedexplicity,andifwecould,wewouldnotbeabletostoreit!
§ ButwecancomputeH dmforanyarbitrarydm:
Seismology&WavePhysics
SECONDDERIVATIVES
§ Hcannotbecomputedexplicity,andifwecould,wewouldnotbeabletostoreit!
§ ButwecancomputeH dmforanyarbitrarydm:
§ Secondderiva=ve=firstderiva=ve(firstderiva=ve)
§ Finite-differenceapproxima=onofsecondderiva=ve=differenceoffirstderiva=ves:
§ H dmcantriviallybeapproximatedbysubtrac=ngtwosensi=vitykernels.
§ Alsopossiblewithoutapproxima=on[beyondscopeofthislecture,details:Fichtner&Trampert,GJI2011].
H(m)dm∝K(m+dm)−K(m)
Seismology&WavePhysics
EXAMPLE
Fréchet kernel
§ 25 s Love wave
§ finite-frequency traveltime
Seismology&WavePhysics
EXAMPLE
§ 25 s Love wave
§ finite-frequency traveltime
Fréchet kernel
dm = small vs perturbation in pixel k
Seismology&WavePhysics
EXAMPLE
§ 25 s Love wave
§ finite-frequency traveltime
H dm
dm = small vs perturbation in pixel k
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
=
...............
...............
...............
...............
...............
H
column k
=
Seismology&WavePhysics
EXAMPLE
§ 25 s Love wave
§ finite-frequency traveltime
H dm
dm = small vs perturbation in pixel k
Twocontribu=ons:
F:First-ordersca`eringS:Second-ordersca`ering
Thanksforyoura`en=on!