andreas fichtner and chris=an boehm...seismology & wave physics outline part i: the...

KITSummerSchoolonFull-WaveformInversion

Lecture3

Adjointmethodsandsensi.vitykernels

AndreasFichtnerandChris=anBoehm

andtheETHSeismologyandWavePhysicsGroup

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


PARTI

Thefull-waveforminversionconcept

1.Summaryofadream


FROMTRAVELTIMESTO‘FULL’WAVEFORMS

Svelocityat150kmbeneathAustraliaFishwicketal.,2005

‘tradi=onal’travel=metomographytravel=memeasurements

Extremelysuccessful!

Canassimilateenormousquan==esofdata.

S=llTHEmostwidelyusedtomographicmethod.



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


full-waveforminversioncompleteseismicrecordings

GOALS

§  Explainbroadbandseismogramswigglebywiggle...§  ...withhardlyanyhumaninterven.on[Tarantolianblackbox]§  Be`erresolvedtomographicimages

•  thermochemicalstructureoftheEarth•  evolu=onanddynamicsoftheEarth•  improvedgroundmo=onpredic=ons•  improvedearthquakesourceinversion

•  emergencyresponse,tsunamiwarning•  tectonicinterpreta=on

•  improvedreservoircharacterisa=on•  ...


CHALLENGES

§  Seismicwavepropaga=onthroughcomplexmedia.§  Computa=onalpower.§  Nonlinearrela=onbetweenwaveformsand3DEarthstructure.§  Meaningfulmeasurementofwaveformdifferences.§  Algorithmstosearchforusefulmodels[allofthem,ideally].§  ...

2.Formula=onasanop=misa=onproblem

OPTIMISATIONPROBLEM


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






§  FindanEarthmodelmsuchthatasuitablydefinedmisfitχisminimal.

§  Thenumberofmodelparametersandthenumericalcostoftheforwardproblempreventtheapplica=onofprobabilis=cmethods.

§  Theminimisa=onproceedsitera=vely:

Comment:

Minimaldoesnotmeanthesmallestmisfitpossible!

The misfit should become about assmall as the observa=onal andforwardmodellingerrors.

!!!

GRADIENT-BASEDOPTIMISATION


The family of gradient methods:

§  method of steepst descent:

§  conjugate-gradient methods

§  Newton and Newton-like methods

§  BFGS and L-BFGS

§  …

m/hi ∂−∂= χ





hi ∝− ∂χ

∂mi







00hγ

m0 m1

χ

m







m0 m1 m2 m3 m4 …

Iteratively approach the minimum misfit by following the local descent directions.

χ

m


PARTII

Theadjointmethod

1.Problemstatement


SO,WHEREISTHEPROBLEM?


§  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


DISCRETEADJOINTMETHODSUMMARY


Lu = f LTv = −∇χ

∂χ∂mi

= vT ∂L∂mi

u

Regularwaveequa=on Adjointwaveequa=on Gradientequa=on




Adjointrecipe1.  Solveforwardproblem[regularwaveequa=on]toobtainu.2.  Evaluatemisfitχ.3.  Computeadjointsource,-∨χ.4.  Solveadjointequa=ontoobtainadjointfieldv.5.  Pluguandvintothegradientequa=on.


∂χ∂mi

= vT ∂L∂mi

u




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.


∂χ∂mi

= vT ∂L∂mi

u

3.Thecon=nuousadjointmethod


AMATTEROFNOTATION


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


Regularwaveequa.on

momentumbalance

stress-strainrela=on

ini=alcondi=ons

boundarycondi=ons

EXAMPLES


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


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



TRAVELTIMEMEASUREMENTONSPECIFICPHASES



Source-receivergeometry Seismograms



Sensi.vitykernelforPwavevelocityreceiver

source

- +




Sensi.vitykernelforSwavevelocity

- +




Sensi.vitykernelsforPwavevelocity and Swavevelocity

- +




Sensi.vitykernelforSwavevelocity

- +



MEASURINGTIME-FREQUENCYPHASEDIFFERENCES



*Mb5.1,25August2007

ver=cal-componentdisplacement,period=10s32.8milliongridpoints



§  Time-andfrequency-dependentphasedifferences

§  Basedonselec=onof=mewindowswheredataandsynthe=csaresimilar

§  Independentofabsoluteamplitudes



Sensi=vitykernels


PARTIII

AdvancedTopics

1.Localminimaandthemul=scaleapproach



THECAMEMBERTEXPERIMENT

OdileGauthier,JeanVirieux,AlbertTarantola,Geophysics1986.

Theacous.cCamembertModel§  20%velocityperturba=on§  8sourcesandreceiversaroundthemodel


THECAMEMBERTEXPERIMENT

OdileGauthier,JeanVirieux,AlbertTarantola,Geophysics1986.

Theacous.cCamembertModel§  20%velocityperturba=on§  8sourcesandreceiversaroundthemodel

InversionresultaXer5itera.ons§  Camembertnotrecovered§  Stuckinalocalminimum

THEMULTISCALEAPPROACH

§  Iden=fiescycleskippingasmainreasonfornonlinearity.§  Misfitsurfacemorecomplexthehigherthefrequency.§  Startwithlowfrequencies.§  Workyourwayuptohighfrequencies.§  Problems.ll:Lowfrequenciesmaynotalwaysbeavailable


2.Compressedwavefieldstorage



PROBLEMSTATEMENT

Sensi.vitykernelexamples


PROBLEMSTATEMENT

§  Forwardandadjointfieldsmustbeknownatthesame.me.§  Thisisnotnaturallythecase.§  Forwardwavefieldneedstobestored.§  Thisisextremelyexpensive!



PROBLEMSTATEMENT

§  Forwardandadjointfieldsmustbeknownatthesame.me.§  Thisisnotnaturallythecase.§  Forwardwavefieldneedstobestored.§  Thisisextremelyexpensive!

Canwesomehowcompressthewavefieldusuchthatthekernelintegralsares.llsufficientlyaccurate?



LOSSYCOMPRESSIONSTRATEGIES

1.  Requan=sa=on•  adjustnumberofbitstorepresentfieldvalues•  largenumberofbitsinregionswithlargeamplitudevaria=onsandviceversa




2.  p-coarsening

•  storewavefieldwithpolynomialdegreepasanewpolynomialofdegreepnew<p•  re-interpolatetoapproximatethekernelintegral




2.  p-coarsening


3.  Temporalinterpola=on

•  Storewavefieldonlyeverynth=mestep•  Splineinterpola=ontofillmissing=mestepsforkernelintegral




2.  p-coarsening


3.  Temporalinterpola=on

•  Storewavefieldonlyeverynth=mestep•  Splineinterpola=ontofillmissing=mestepsforkernelintegral

4.  Lazyforwardandadjointsimula=ons

•  Storeforwardandadjointfieldonlyinregionswheretheyoverlap


EXAMPLE[COMBININGTHESESTRATEGIES]

Boehm&Fichtner,Geophysics2016

Seismology&WavePhysicsBoehm&Fichtner,Geophysics2016



cf=compressionfactor



cf=compressionfactor


AcompressionfactorofO(1000)isozenfeasible.

3.Secondderiva=ves



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


SECONDDERIVATIVES

§  Hcannotbecomputedexplicity,andifwecould,wewouldnotbeabletostoreit!

§  ButwecancomputeH dmforanyarbitrarydm:


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)


EXAMPLE

Fréchet kernel

§  25 s Love wave

§  finite-frequency traveltime


EXAMPLE



Fréchet kernel

dm = small vs perturbation in pixel k


EXAMPLE



H dm


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

...............

...............

...............

...............

...............

H

column k

=


EXAMPLE



H dm


Twocontribu=ons:

F:First-ordersca`eringS:Second-ordersca`ering

Thanksforyoura`en=on!

andreas fichtner and chris=an boehm...seismology & wave physics outline part i: the...

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