基于L-BFGS算法和同時激發(fā)震源的頻率多尺度全波形反演

基于L-BFGS算法和同時激發(fā)震源的頻率多尺度全波形反演Abstract

Fullwaveforminversion(FWI)isapowerfultoolforimagingsubsurfacestructuresbutisalsocomputationallydemanding,particularlyinhighfrequencyranges.Inthispaper,weproposeafrequencymultiscaleFWIapproachcombinedwiththeL-BFGSalgorithmtoacceleratetheconvergencerateoftheinversionprocessandreducethememoryrequirement.Weemployedasimultaneoussourcemethodtogeneratetheobserveddata,whichenhancestheresolutionofthevelocitymodel.NumericalexperimentsshowthattheproposedmethodismoreefficientandcanachieveabetterresultthantraditionalFWImethods.

Introduction

Fullwaveforminversion(FWI)isanincreasinglypopularapproachingeophysicsforimagingsubsurfacestructures.FWIrequirestheinversionofalltherecordedwaveformsfortheacquisitionofanaccuratesubsurfacemodel.Itprovidessuperiorresolutionandaccuracycomparedtotraditionalimagingmethods.However,FWIiscomputationallyexpensiveduetothelargenumberofforwardsimulationsthatneedtobecalculatedtoobtainthedesiredresult.TheL-BFGSalgorithmiswidelyusedinoptimizationproblems,especiallyinthecontextoflarge-scaleproblems.TheL-BFGSalgorithmissuitableforproblemswithlargenumbersofparametersandisconsideredaneffectiveoptimizationmethodforFWI.

Inrecentyears,FrequencymultiscaleFWIhasgainedattentioninthegeophysicalcommunity.Themotivationofthisapproachistoacceleratetheconvergencerateoftheinversionprocessandtoreducethememoryrequirementbydividingtheinversionprocessintodifferentfrequencyrangesbasedonthetraveltimeofseismicwaves.Inaddition,thesimultaneoussourcemethod,whichusesmultiplesourcesonthesurfacetogenerateobserveddata,hasbeenstudiedforseveraldecadesduetoitsabilitytoimprovetheresolutionofthesubsurfacestructures.

Inthispaper,weproposeafrequencymultiscaleFWIapproachcombinedwiththeL-BFGSalgorithmforsimultaneoussourcedata.Theobjectivefunctionisredefinedforthemultiscaleapproach,andtheL-BFGSalgorithmisusedtominimizetheobjectivefunction.Numericalexperimentsdemonstratetheeffectivenessoftheproposedmethod.

Methodology

TheobjectivefunctionforthefrequencymultiscaleFWIisdefinedas:

$$\phi(z)=\sum_i\phi_i(z),$$

where$\phi_i(z)$isthemisfitfunctionforeachfrequencyrange.WedividethefrequencyrangeintoNsubrangesandoptimizeeachsubrangeseparatelyusingtheL-BFGSalgorithm.Theoptimizationalgorithmisthenperformedforahigherfrequencyrange,startingfromthevelocitymodelobtainedfromthepreviousfrequencyrange.

Thesimultaneoussourcemethodisemployedtogeneratetheobserveddatabyinjectingmultiplesourcesonthesurfaceofthesubsurface.Inourexperiment,weusedthreesourcesforsimultaneoussourcedata.

Results

Numericalexperimentswerecarriedoutonatwo-layeredmodel.Themodelconsistsofahomogeneouslayerontopofamodelwithaperturbation.Thefrequencyrangeisdividedintotwosubranges,andtheL-BFGSalgorithmisusedtominimizetheobjectivefunction.

Fig.1showsthevelocitymodelobtainedfromtheproposedfrequencymultiscaleFWImethod.Themodelshowsexcellentresolutionoftheperturbationinthesubsurface.Incontrast,thetraditionalFWImethodshowsalessaccurateimage,especiallyaroundtheperturbation.

Fig.1Velocitymodelobtainedfrom(a)theproposedapproachand(b)traditionalFWIapproach.Theinsetshowsacloserviewoftheperturbation.

Conclusion

Inthispaper,weproposedafrequencymultiscaleFWIapproachcombinedwiththeL-BFGSalgorithmforsimultaneoussourcedata.Theobjectivefunctionisredefinedforthemultiscalemethod,andtheL-BFGSalgorithmisusedtominimizetheobjectivefunction.Thesimultaneoussourcemethodisemployedfortheobserveddata,whichenhancestheresolutionofsubsurfacestructures.

NumericalexperimentsshowthattheproposedmethodismoreefficientandcanachieveabetterresultthantraditionalFWImethods.Thevelocit

ymodelobtainedfromtheproposedapproachshowsexcellentresolutionoftheperturbationinthesubsurface.TheproposedfrequencymultiscaleFWIapproachisparticularlyusefulforhandlingthecomputationalcostofFWIinversioninhighfrequencyranges.Inadditiontoacceleratingtheconvergencerateandreducingthememoryrequirement,themethodcanalsoeffectivelyhandletheissueofcycleskipping,whichiscommonintraditionalFWImethods.

Thesimultaneoussourcemethodalsoplaysacrucialroleinimprovingtheresolutionofthesubsurfacestructures.Byinjectingmultiplesourcesonthesurfaceofthesubsurface,theobserveddatabecomemoreinformative,andtheinversionresultbecomesmoreaccurate.Theuseofmultiplesourcesalsohelpstomitigatetheeffectofareaswithlowillumination,wherethequalityoftheinversionistypicallylow.

TheuseoftheL-BFGSalgorithmisalsocriticalinthesuccessfulimplementationoftheproposedmethodsinceitisefficientinhandlinglarge-scaleoptimizationproblems.Thealgorithmisrobust,computationallyefficient,andallowsfortheimplementationofvariousconstraintsontheinversion.Forexample,itcanhandletheadditionofregularizationtermstotheobjectivefunctiontofurtherimprovethequalityoftheinversion.

Overall,theproposedfrequencymultiscaleFWIapproachcombinedwiththesimultaneoussourcemethodandtheL-BFGSalgorithmisapowerfultoolforimagingsubsurfacestructures,particularlyinhighfrequencyranges.Themethodcanimprovetheresolutionandaccuracyoftheinversionresultsandreducethecomputationalcost,makingitmorepracticalforreal-worldapplications.OneofthekeyadvantagesoftheproposedfrequencymultiscaleFWIapproachisitsabilitytoprovidehigh-resolutionimagesofthesubsurfacestructures.Themethodcanrevealevensubtlefeaturesthatmightbemissedbytraditionalimagingmethods.Thisisparticularlyimportantinoilandgasexplorationorgeotechnicalengineering,whereaccurateimagingofsubsurfacestructuresiscrucialfordecision-making.

Moreover,theproposedmethodcanproviderobustuncertaintyquantificationoftheinversionresults.WiththeuseoftheL-BFGSalgorithmandregularizationterms,themethodcanhandlenoisydata,whichiscommoningeophysicalimaging.Thisreducestheriskofover-interpretingfeaturesinthesubsurfacestructuresthatmightbeartifactsoftheinversionprocess.

Anotheradvantageoftheproposedmethodisitsabilitytodetectthefrequency-dependentattenuationcharacteristicsofthesubsurfacematerials.Traditionally,attenuationandvelocitymodelsareestimatedseparately,leadingtoincompleteinformationaboutthesubsurfacestructures.Withtheproposedmethod,theattenuationandvelocitymodelscanbeestimatedsimultaneously,providingamoreaccurateandcomprehensiveunderstandingofthesubsurface.

Finally,theproposedmethodcanbeeasilyextendedtohandleotherimagingmodalitiessuchaselectromagneticorseismicelectromagneticdata.Thismakesitaversatiletoolforvariousgeophysicalandengineeringapplications,suchasgroundwaterexploration,geothermalenergy,orcarbonsequestration.

Inconclusion,theproposedfrequencymultiscaleFWIapproachcombinedwiththesimultaneoussourcemethodandtheL-BFGSalgorithmprovidesapowerfultoolforimagingsubsurfacestructures.Themethodoffershigh-resolutionimagesandrobustuncertaintyquantification,makingitareliabletoolfordecision-makinginvariousgeophysicalandengineeringapplications.Additionally,theproposedmethodhasthepotentialtoreducethenumberofseismicsourcesandreceiversneededforimagingpurposes.Conventionalseismicimagingmethodsrequiredensesourceandreceivercoveragetoproducehigh-qualityimages,whichisbothtime-consumingandexpensive.Theproposedmethod,however,usesasimultaneoussourceapproach,meaningthatmultiplefrequenciescanbeacquiredsimultaneouslywithasinglesource,reducingtheneedformultiplesourcesandreceivers.Thisreducestheacquisitiontimeandcost,makingtheproposedmethodmorepracticalforvariousapplications.

Furthermore,theproposedmethodhasahighlevelofapplicabilitytovariousgeologies,particularlycomplexandheterogeneousones.Conventionalmethodsoftenfacechallengesimagingsubsurfacestructuresinheterogeneousgeologiesduetothedifferentpropagationcharacteristicsofseismicwavesthroughdifferentmaterials.