Simulating-Seismic-Wave-Propagation--in-3D-Elastic

BulletinoftheSeismologicalSocietyofAmerica,Vol.86,No.4,pp.1091-1106,August1996SimulatingSeismicWavePropagationin3DElasticMediaUsingStaggered-GridFiniteDifferencesbyRobertW.GravesAbstractThisarticleprovidesanoverviewoftheapplicationofthestaggered-gridfinite-differencetechniquetomodelwavepropagationproblemsin3Delasticmedia.Inadditiontopresentinggeneralized,discreterepresentationsofthediffer-entialequationsofmotionusingthestaggered-gridapproach,wealsoprovidede-tailedformulationsthatdescribetheincorporationofmoment-tensorsources,theimplementationofastableandaccuraterepresentationofaplanarfree-surfacebound-aryfor3Dmodels,andthederivationandimplementationofanapproximatetech-niquetomodelspatiallyvariableanelasticattenuationwithintime-domainfinite-differencecomputations.Thecomparisonofresultsobtainedusingthestaggered-gridtechniquewiththoseobtainedusingafrequency-wavenumberalgorithmshowsex-cellentagreementbetweenthetwomethodsforavarietyofmodels.Inaddition,thisarticlealsointroducesamemoryoptimizationprocedurethatallowslarge-scale3Dfinite-differenceproblemstobecomputedonaconventional,single-processordesk-topworkstation.Withthistechnique,modelstorageisaccommodatedusingbothexternal(hard-disk)andinternal(core)memory.Toreducesystemoverhead,acas-cadedtimeupdateprocedureisutilizedtomaximizethenumberofcomputationsperformedbetweenI/Ooperations.Thisformulationgreatlyexpandstheapplicabilityofthe3Dfinite-differencetechniquebyprovidinganefficientandpracticalalgorithmforimplementationoncommonlyavailableworkstationplatforms.IntroductionModemcomputationalefficiencyhasadvancedtoastatewherewecanbegintocalculatewave-fieldsimulationsforrealistic3Dmodelsatfrequenciesofinteresttobothseismologistsandengineers.Themostgeneralofthesenu-mericalmethodsaregrid-basedtechniquesthattrackthewavefieldonadense3Dgridofpoints,e.g.,thefinite-difference(FD),finite-element(FE),andpseudospectral(PS)methods.Variousalgorithmshavebeendevelopedtoimple-mentthesetechniques,andwhiletherewillalwaysbedebateastowhichoneisthebesttechnique,eachmethodhasitsmeritsandpitfalls.Ourapproachusesastaggered-gridfinite-differenceal-gorithmtomodelthefirst-orderelastodynamicequationsofmotionexpressedintermsofvelocityandstress.Inseismicapplications,thevelocity-stressformulationwasfirstusedbyMadariaga(1976)tomodelfault-rupturedynamics.Vi-rieux(1984,1986)andLevander(1988)havesinceextendedthetechniquetomodelseismicwavepropagationin2Dme-dia,andtheformulationfor3DmediaisoutlinedbyRandall(1989)andYomogidaandEtgen(1993).Theadvantagesofthestaggered-gridformulationare(1)sourceinsertionisstraightforwardandcanbeexpressedintermsofvelocity(viabodyforces)orstress;(2)astableandaccuraterepre-sentationforaplanarfree-surfaceboundaryiseasilyimple-mented;(3)sincethefinite-differenceoperatorsarelocal,theentiremodeldoesnothavetoresideincorememoryallatonce;(4)itiseasilyextendedtohigh-orderspatialdiffer-enceoperators;(5)themethodcanbeinterfacedwithothermodelingtechniquesbyexpressingtheinputwavefieldalongaboundaryofthefinite-differencegrid;and(6)thealgorithmiseasilyimplementedonscalar,vector,orparallelcomputers.Inthefollowingsections,weoutlinethenumericalap-proachbeginningwiththeequationsofmotionandthende-scribetheirdiscreteformulationusingthestaggered-gridapproach.Wedonotprovideadetailedanalysisofthedevelopmentofabsorbingboundaryconditions,stabilityofthenumericalsystem,orissuesregardingnumericalgriddis-persion,asthesetopicsarealladequatelycoveredinthebefore-mentionedarticles.Wedo,however,discussindetailsomenewideasrelatedtoincorporatingearthquake(double-couple)sources,free-surfaceboundaryimplementation,andmodelingspatiallyvariableanelasticity(usingQ).Inaddi-tion,wedescribeamemoryoptimizationtechnique,whichallowsthecomputationoflarge-scale3Dfinite-differenceproblemsusingonlyasingle-processordesktopworkstation.10911092R.W.GravesThekeytothismemoryoptimizationtechniqueisacascadedtimeupdatealgorithmthatmaximizesthenumberofcom-putationsperformedbetweensuccessiveI/Ooperations.Alongwiththepresentationanddiscussionofthesenewideas,wealsoshowaseriesofcomparisonsbetweenresultsgeneratedbythestaggered-gridfinite-differencemethodwiththosegeneratedusingafrequency-wavenumberinte-grationtechnique.Thesecomparisonsdemonstratethehighdegreeofaccuracyandutilityofthestaggered-gridtech-nique.EquationsofMotionThefollowingsetsofequationsdescribewavepropa-gationwithinthree-dimensional(3D),linear,isotropicelas-ticmedia.Equationsofmomentumconservation:POttUx=Ox'Cxxq-Oy'Cxy-[-OzZ'xzq-fx,POttUy=3x'Cxy+OyTyy+OzTyz-[-fy,tgOnUz=Ox'Cxz+Oy'Cyz+Oz'Czz+fzO)Stress-strainrelations:r~x=(2+2/~)OxUz+2(ayUy+OzUz),ryy=(,l+2/OOyUy+2GUx+OzUz),rzz=(2+2/l)Ozuz+2(OxUx+OyUy),rxy=/4OyU~+O~Uy),rxz=~(Ozu~+O~Uz),Tyz=]l(Ozlgy-'~Oyl~z).(2)Intheseequations,(u~,Uy,uz)arethedisplacementcom-ponents;(r~,ryy,Zzz,%y,%z,ryz)arethestresscomponents;(fx,fy,fz)arethebody-forcecomponents;pisthedensity;2and/tareLam6coefficients;andthesymbolsOx,Oy,0z,andOttareshorthandrepresentationsofthedifferentialoperatorsO[Ox,O/3y,O/Oz,and32/0t2.Theseequationscanbeformulatedintoasetoffirst