ApplicationofWeightedEarly-ArrivalWaveformInversiontoShallowLandDataHa - PDF document

1 ApplicationofWeightedEarly-ArrivalWavefo
ApplicationofWeightedEarly-ArrivalWaveformInversiontoShallowLandDataHanYu1,DongliangZhang1andXinWang11KingAbdullahUniversityofScienceandTechnology,DivisionofPhysicalScienceandEngineering,Thuwal23955-6900.SaudiArabia ABSTRACTRecentstudieshaveshownthatinvertingtracesweightedbytheenergyoftheearly-arrivalscanimprovetheaccu-racyofestimatingshallowvelocities.Thisisexplainedbyshowingthattheassociatedmistgradientfunctiontendstobesensitivetothekinematicsofwavepropagationandinsensitivetothedynamics.Asyntheticexampleveriesthetheoreticalpredictionsandshowthattheeectsofnoiseandunpredictedamplitudevariationsintheinversionarere-ducedusingthisweightedearlyarrivalwaveforminversion(WEWI).Wealsoapplythismethodtoa2Dlanddatasetforestimatingthenear-surfacevelocitydistribution.Thereversetimemigrationimagessuggestthat,comparedtothetomograminverteddirectlyfromtheearlyarrivalwave-forms,theWEWItomogramprovidesamoreconvincingvelocitymodelandmorefocusedreectionsinthedeeperpartoftheimage. 1.INTRODUCTIONThenear-surfacevelocitydistributioniscrucialforimagingthedeeperpartsoftheEarth.Complexvelocityvariationsatthenearsurfaceareoftenassociatedwithundulatingtopographyorirregulargeologyinthenear-surfaceweatheredlayers(Amor-iumetal.,1987;Taneretal.,1998).Ifthenear-surfacevelocitydistributionisnotaccuratelyestimated,thecoherencyofthedeepermigratedreectionscanbestronglydegraded(White,1989;Marsden,1993).Topartlyremedythisproblem,thenear-surfacevelocitymodelwithsmoothvariationscanbees-timatedbytraveltimetomography(ZhuandMcMechan,1989;PrattandGoulty,1991;AkiandRichards,2002)thatinvertstherst-arrivaltraveltimes.However,ingeologicallycomplexareas,amorehighlyresolvedvelocitymodelisneededforimagingdeeperreectors.Inthisregard,waveforminversion(Tarantola,1984;Mora,1987;Zhouetal.,1995)wasdevel-opedtoinvertformoreaccuratetomogramsbynite-frequencyseismicwavepropagation.Toreducethecomputationaltimeandlocalminimaprob-lems(SirgueandPratt,2004),early-arrivalwaveforminversion(EWI)wasproposedbyShengetal.(2006)inthespace-timedomainandlaterappliedtomarinedata(Boonyasiriwatetal.,2010).Inthiswork,wecarryouttheinversiononlanddatabyfollowingtheconventionalEWImethodbutusingarecentlydevelopedobjectivemistfunction(Shen,2010),whosegra-dientismorerobustandfocusesmoreonmatchingthephaseratherthantheamplitudeinthedata.However,theassociatedgradientdoesnothaveanimportantenergynormalizationtermwhichisimportantforoptimalimaging.Inthiswork,thegradi-entassociatedwiththisweightedearlyarrivalwaveforminver-sion(WEWI)isproperlynormalizedandshowntosignicantlyimprovetheaccuracyofthenaltomogram.Insteadofreplac-ingtheamplitudespectrumofacalculatedtracewiththatofthecorrespondingobservedtrace(SunandSchuster,1993),weimplementWEWIinthetimedomainbynormalizingboththeobservedandcalculatedearlyarrivalsusingtheL2normofthetrace,wherethisapproachavoidsthephasewrappingprobleminthefrequencydomain(ShinandMin,2006).OursyntheticresultsdemonstratethatcomparedtoEWI,WEWIcanmi

2 ti-gatetheeectsofnoiseandunpredictedamp
ti-gatetheeectsofnoiseandunpredictedamplitudevariationsinthedataandrobustlyinvertforhighlyresolvednear-surfacetomogram.Moreover,alanddatatestillustratesthatWEWIproducesamoreaccurateshallowsubsurfacetomogramwheretheenergyisfocusedinthedeeperpart.Thispaperisorganizedintofoursections.Therstpartistheintroduction,andthesecondpartanalyzesthemistfunc-tionassociatedwithitsgradientinourapproach.Insection3,numericalresultsareshownforinvertingdataassociatedwiththeMarmousimodelandaeldexperimentinSaudiArabia.Thelastsectionpresentstheconclusions.2.THEORYInmanyeld,particularlylanddatasets,therearestrongelas-ticarrivalssuchassurfacewavesthatcannotbemodeledbythe61 62Yuetal.acousticwaveequation.Inaddition,theamplitudesofsometracesaredistortedduetounexplainedenvironmentalsourcesandnotexplainedbygeometricspreading.Someelasticeectsinthedatacanbereducedbyapplyinganearlyarrivalwindowtomutethelaterarrivals.Therefore,theconventionalwave-forminversionmistfunctionismodiedbyShen(2010)andexpressedinthetimedomainasE=1 2Xs;rjj
p(xr;tjxs)jj22;=1 2Xs;rjjpcalc(xr;tjxs) jjpcalc(xr;tjxs)jj2pobs(xr;tjxs) jjpobs(xr;tjxs)jj2jj22;(1)wherep(xr;tjxs)denotesthepressureeldtracerecordedatthereceiverpositionxr,withlisteningtimetandasourceatxs;jjpjjdenotestheL2normoftheN1vectorp,namelyp pTp,whereNisthenumberoftimesamplesinthetrace.Here,pobsrepresentstherecordedtracewithwindowedearlyarrivalsandpcalcrepresentsthesyntheticearlyarrivals.Thesyntheticdataarecalculatedbysolvingtheconstant-densityacousticwaveequation,1 c2(x)@2p(x;tjxs) @t2r2p(x;tjxs)=s(x;tjxs);(2)wherec(x)representsthevelocitymodelatpositionx.Theso-lutiontoequation2iscalculatedbyasecondorderintimeandeighthorderinspacestaggered-gridmethod(Levander,1988).Equation1normalizestheobservedandthesyntheticearlyarrivalssothattheirenergycanbecomparedatthesamescale,andthewaveforminversioninthiscaseismoresensitivetophasedierencesinthemistfunction.Tounveilthisfact,theFr´echetderivative[grad(x)=@E @c(x)]ofthefunctionalEwithrespecttoc(x)iscalculatedbygrad(x)=
pT@(pcalc=jjpcalcjj2) @c=1 jjpcalcjj2(
p
pTpcalc jjpcalcjj22pcalc)T@pcalc @c:(3)Here,ifthescalar1=jjpcalcjj2isignored,therstterminequa-tion3exactlycorrespondstothegradientofconventionalwave-formmistfunction.Inthispaper,weinverttheelddatausingthisgradientbutignore1=jjpcalcjj2becauseitisincludedincal-culatingthesteplengthwhenupdatingthevelocitymodel.Pre-vioussynthetictests(Shen,2010)missedthesuperscriptterm”2”inthedenominatorjjpcalcjj22ofequation3,thusmakingthephasematchlessaccurate.Thismistfunctionbecomesmoresignicantiftheterm
pTpcalc jjpcalcjj22isnotsmall,whichcanbecausedbycomplicatedgeologicalconditions,becauseitattachesmoreimportancetoaccuratelypredictingthephasesratherthantheamplitudes.Note,thatthegradientterm
p
pTpcalc jjpcalcjj22pcalcisorthogonaltopcalcsince(
p(
p)Tpcalc jjpcalcjj22pcalc)Tpcalc=
pTpcalc
pTpcalc=0;(4)whichindicatesthat
p
pTpcalc jjpcalcjj22pcalceliminatesthephasein-formationofpcalcinpobs

3 .Thereforethecommonphasesinpcalcandpobsw
.Thereforethecommonphasesinpcalcandpobswillnotbeselectedtomatchagaininthenextiteration.Figure1showsfactthefactthisnewvirtualsourceisperpendiculartotherecordeddata.Itthusweakenstheef-fectsofpcalcandstrengthensthephasedierencebetweenpobsandpcalcinthebackpropagatedwaveelds,thereforemakingWEWIrobustandsignicant. Figure1:ConstructionofdataresidualsasbackpropagatingsourcesforEWIandWEWI.Thevelocitymodelisestimatedbyaniterativeconjugategradientmethodwhereck+1(x)=ck(x)+kdk(x);(5)andtheconjugatedirectionsaredenedbydk=Pkgk+kdk1;(6)foriterationsk=1;2;:::;kmax,g=[grad(x)],andPisthecon-ventionalgeometrical-spreadingpreconditioner(Causseetal.,1999).Thescalarkisthesteplengthwhichcanbedeter-minedbyaquadraticline-searchmethod(NocedalandWright,1999),anddk(x)isthecomponentofthedirectionvectordk(x)indexedbyx.Fortherstiteration,wesetd0=g0.Thepa-rameterkiscalculatedbythePolak-Ribi´ereformula(NocedalandWright,1999)k=gTk(PkgkPk1gk1) gTk1Pk1gk1:(7)Tocomputethegradientdirectionateachiterationreducestocomputingthereversetimemigrationoperation.Additionalforwardmodelingsarerequiredforthelinesearch.Theinitialvelocitymodelc0(x)isthetraveltimetomograminvertedbypickedrstarrivals(Nemethetal.,1997),andequation5isiterativelyapplieduntiltheobjectivefunctionalEsatisesastoppingcriterion. WeightedEWItoShallowLandData633.NUMERICALTESTSOFWEWI3.1SyntheticDataExampleTheMarmousimodelisusedtotesttherobustnessandqual-ityofWEWIbeforeitisappliedtoalanddatasetinthenextsection.First,asyntheticdatasetisgeneratedbasedontheMarmousimodel(Figure2(a))witha576(horizontal)by184(vertical)griddedmeshwitha6.0mgridinterval.Thereare60shotswitha54mshotspacing,andforeachshot,thenumberofreceiversis190witha6mreceiverspacing.Therecord-inglengthis1.5swithasamplingrate0.5ms.Whitenoiseisaddedtoeachtraceofeveryshotgather.Thenonzeromean-valuednoiseconsistsoftwoparts:randomreceivernoiseisaddedtoeveryCSGandarandomstaticamplitudeshiftisalsoappliedtoeachtrace.Figures3(a)and3(b)showonecommonshotgatherbeforeandafteraddingthenoise,andtheyaredis-playedwiththesameamplitudescale.Figure3(c)showsthenoisemaskaddedtoFigure3(a)thatsimulatesdeadtraces.Fi-nally,WEWIandconventionalEWIareusedtoinvertthenoisydataupto0.5saftertherstarrivalusingthesameinitialve-locitymodel(seeFigure2(b)).Theresultingtomogramsafterthe30thiterationarepresentedinFigures4(a)and4(b),whichprovesthatWEWIislesssensitivetothenoise;thematchedphaseinformationinthepredicteddataisexcludedincalcu-latingthegradientusingreversetimemigration(RTM).TheirmistgradientFigures5(a)and5(b)fortherstiterationfur-thervalidatetheadvantageoftheWEWImethod,followedbytwoobviouslydierentconvergenceratesofthedataresidualshowninFigure5(c).ThecomparisonofthetwotomogramsapparentlyindicatesWEWIcaninvertforamoreaccurateve-locitymodelwithunpredictednoiseinthedata,andimplytherobustnessoftheWEWIcomparedtoEWI. Distance X (m)Depth Z (m)(a) Marmousi Velocity Model 600 1800 3000 240 480 720 960 2.5 3.5 4.5 km/s

4 Distance X (m)Depth Z (m)(b) Initial Vel
Distance X (m)Depth Z (m)(b) Initial Velocity Model 600 1800 3000 240 480 720 960 Figure2:TheMarmousimodelswith(a)thetruevelocitydis-tributionand(b)theitssmoothedversionastheinitialvelocitymodelforwaveforminversion. (a) CSG #10 before Adding the NoiseReceiver IndexTime (s) 20 60 100 140 180 0.5 1.0 1.5 (b) CSG #10 after Adding the NoiseReceiver IndexTime (s) 20 60 100 140 180 0.5 1.0 1.5 (c) The Noise Mask for CSG #10Receiver IndexTime (s) 20 60 100 140 180 0.5 1.0 1.5 Figure3:TheCSG#10generatedbytheMarmousimodel(a)before,(b)afteraddingthenoise,and(c)itsnoisemask.3.2LandDataExample3.2.1AcquisitionandProcessingA2DseismicsurveyiswascarriedoutnearKAUSTwiththeacquisitiongeometryillustratedinFigure6.The2Dacqui-sitionlineconsistsof1279shotsand240verticalcomponentgeophonespershot,withauniformspacingof30mforbothshotsandreceivers.Foreachchannel,therecordlengthis2switha4mssamplingrate.Foracommonshotgather(CSG),theshotpositionisinthemiddleofthe240receivers,sothelargestosetis3600m.WEWIisappliedtotherst180CSGs,andthehorizontaldistanceforourinversionisrestrictedbetweenshot#1and#180(redcrossesinFigure6).Thetopog-raphyofthese180shotsisshowninFigure7,whichisalmostatifthehorizontaldistanceXandthedepthZareofthesamescale.Inthiscase,wechooseZ=25masthesurfacefortheinversionandignoretheelevationvariationsofthegeophonesandshotpoints.Priortodataprocessing,itisusefultoestimatethenear-surfacevelocitydistributionfromthepickedrstarrivals.Fig-ure8showsCSG#17withpickedrstarrivalsmarkedbyredcrosses,andthedirectwaveandtherefractionsarerespectivelymarkedbythewhiteandgreendashedlines.Theslopesofthe 64Yuetal. Distance X (m)Depth Z (m)(a) WEWI Tomogram 600 1800 3000 240 480 720 960 2.5 3.5 4.5 km/s Distance X (m)Depth Z (m)(b) Conventional EWI Tomogram 600 1800 3000 240 480 720 960 Figure4:Theinvertedtomogramsusing(a)WEWIand(b)conventionalEWI.twolinessuggestthatthevelocitycorrespondingtotherstlayerisapproximatelyv1=750m=0:2s=3750m=s,whereasthevelocityforthesecondlayerisaboutv2=2800m=0:5s=5600m=s.WerstapplytheF-KltertoremoveapparentsurfacewavesinalltheCSGs.Asanexample,therawCSG#11isshowninFigure9(a),withmostofthesurfacewaveseliminatedinFig-ure9(b)byF-Kltering.Theelddataarethentransformedfrom3Dto2Dformatbyapplyingthelterp i=!toallthetracesinthefrequencydomainandscaledbyp tfor3Dgeo-metricalspreadinginthetimedomain.Inthemeantime,thedataarealsolteredwithapassbandfrom10to20Hz.TheresultsafterlteringofCSG#11areshowninFigure9(c).ThespectraofatraceinCSG#11beforeandaftertheprocessingstepsarepresentedinFigure10.3.2.2ApplyingWEWItoTheLandDataSetWEWIisimplementedusingastartingmodelcalculatedbyrst-arrivaltraveltimetomgoraphywitha757(horizontal)by97(vertical)griddedmeshanda7.5mgridinterval.First,aray-basedtraveltimetomographymethodisusedtoinverttherstarrivaltraveltimesforasmoothvelocitymodel.Thetravel-timetomogramisshowninFigure11,wherethehigh-velocitylayerisabout80mdeep,asillustratedbytherefractiondata

5 inFigure8.Second,WEWIisusedtoinverttheda
inFigure8.Second,WEWIisusedtoinvertthedatarestrictedbydynamicallyincreasingtimewindows.Thetimewindowstretchesfromthebeginningoftherecordto0.24saftertherstarrivalsfortherst20iterationsanditssizequadraticallyincreasesupto0.5sinthenext10iterations.Allthetracesboundedbytheirtimewindowsintheobservedandcalculateddataarenormalizedaccordingtoequation1duringeachitera-tion.The1=jjpcalcjj2terminequation3isdroppedandtherestofthetermsareexactlyfollowedwhencalculatingthegradient. Distance X (m)Depth Z (m)(a) WEWI Gradient for the First Iteration 600 1800 3000 240 480 720 960 Distance X (m)Depth Z (m)(b) EWI Gradient for the First Iteration 600 1800 3000 240 480 720 960 1 10 20 30 0.2 0.4 0.6 0.8 1 1.1 Iteration NumberNormalized L2 Data Residual(c) Data Residual Convergence Rate Comparison for Synthetic Data WEWI Convetional EWI Figure5:(a)WEWIand(b)conventionalEWIgradientsfortherstinteration,and(c)thecomparisonoftheirconvergencerate. 5 10 15 20 25 30 35 40 45 0 1.6 X [km]Y [km]Acquisition Geometry with Equal Scales in Distance Red Crosses: the first 180 CSGs for inversion Figure6:Theacquisitiongeometrywithequalscalesinthehorizontalandtheverticaldirections.TheinvertedWEWItomogramatthe30thiterationisshowninFigure12(a).Atthe20thiteration,theobserveddataandcal- WeightedEWItoShallowLandData65 0 750 1500 2250 3000 3750 4500 5250 0 500 Distance X (m) Depth Z (m)The Topography of Shot 1 to 180 Figure7:Topographyoftherst180shots. Receiver IndexTime (s)CSG # 17 with Picked First Arrivals 40 80 120 160 200 240 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Figure8:Acommonshotgatherwithitspickedrstarrivals.Redcrosses:pickedrstarrivals;greendashedline:theesti-mateddirectwave;whitedashline:theestimatedrefraction.culateddatarecorded0.24saftertherstarrivalsmostlyagreewithoneanotherinFigures13(a)and13(b).Notethatnear-osettraceswithin150mfromthesourceandfar-osettracesmorethan3300mfromthesourcesarenotinvertedduetothepoorqualityoftherecordeddata.Afterthedynamicwindowsareapplied,manylaterarrivaleventsalsomatchaccordingtoFigures14(a)and14(b)atthe30thiterationwithoutharmingthematchedpreviousarrivals.Figure15depictstheresidualvs.iterationnumberplotwhenapplyingWEWIfor30itera-tions.Notethattheresidualgraduallyincludesmoredataforcomparisonwhilethewindowsizegrowsveryslowly.Toverifytheinvertedvelocitymodel,wemigratethehighfrequencyportion(45Hz)oftheobserveddatausingthetraveltimeandtheWEWItomogramsasmigrationvelocitymodels.Thereversetimemigration(RTM)imagesabove720mindepthareexhibitedinFigures16(a)and16(b)andtheirassociatedcommonimagegathersarealsoshowninFigures17(a)and17(b).TheRTMimagebasedontheWEWItomo-gramshowsamorecontinuousstructureintheshallowpartandmorefocusedenergyinthedeeppartcomparedtotheRTMim-agecomputedwiththetraveltimetomogram.TheCIGsbasedontheWEWItomogramalsoareatterforboththenear-surfaceandthedeeperreectors.TheEWIistomogramshowninFigure12(b).Someshallowreectorscanstillbe

6 detectedaccordingtothistomogrambuttheyar
detectedaccordingtothistomogrambuttheyaremuchstrongercomparedtothetomograminFigure12(a).Figure12(b)couldbelessrealisticbecausetheesti-matednear-surfacevelocityatZ=80misaround5600m=sandnothigherthan6000m=saccordingtothepickedrstar-rivalsinFigure8.ThecalculatedCSG#11forthedegradedtomogrampresentedinFigure13(c)doesnotmatchwellwith Receiver IndexTime (s)(a) Raw CSG #11 50 100 150 200 0.4 0.8 1.2 1.6 2 Receiver IndexTime (s)(b) F-K Filtered CSG #11 50 100 150 200 0.4 0.8 1.2 1.6 2 Receiver IndexTime (s)(c) CSG #11 after Bandpass Filtering and 3D to 2D Transformation 50 100 150 200 0.4 0.8 1.2 1.6 2 Figure9:Araw(a)CSG#11,(b)withitssurfacewavesre-movedbythediplter,and(c)theCSG#11afterbandpasslteringand3Dto2Dtransformation.Figure13(a)evenfortheearlyarrivalsattheintermediate20thiteration.Whenthetimewindowincreasesto0.5sinthe30thiteration,thecalculatedCSG#11withconventionalEWI(14(c))showsgreaterdierencefromtheobservedCSG#11thanitscounterpartwithWEWI(Figure14(b)).TheRTMim-age(Figure16(c))usingthistomogramisalsoinferiorcom-paredtoFigure16(b),andsoaretheCIGs(Figures17(c)and18(c))associatedwithit.Moreover,thenormalizedEWIdataresidualonlydecreasesto0.85after30iterationsasshowninFigure15.4.CONCLUSIONSAmodiedmistfunctionisappliedanditsassociatedFr´echetderivativeisexactlyfollowedtocalculatethegradientforup-datingthevelocitymodel.Thismodiedfunctionusingtrace-by-tracenormalizationattachesmoreimportancetoaccurately 66Yuetal. 10 20 30 40 50 60 70 0.2 0.4 0.6 0.8 1 Frequency (Hz)Normalized Amplitude(a) Spectrum of a Trace of Raw CSG #11 10 20 30 40 50 60 70 0.2 0.4 0.6 0.8 1 Frequency (Hz)Normalized Amplitude(b) Spectrum of the Trace in (a) after Filtering Figure10:ThespectrumofatraceinCSG#11(a)beforeand(b)afterprocessing. Distance X (m)Depth Z (m)Traveltime Tomogram 750 2250 3750 5250 180 360 540 720 2.5 3.5 4.5 5.5 km/s Figure11:ThetraveltimetomogramforthepickedrstarrivalsfromCSGs#1#180. Distance X (m)Depth Z (m)(a) WEWI Tomogram 750 2250 3750 5250 180 360 540 720 2.5 3.5 4.5 5.5 km/s Distance X (m)Depth Z (m)(b) Conventional EWI Tomogram 750 2250 3750 5250 180 360 540 720 Figure12:(a)TheWEWI,and(b)theconventionalEWIto-mogramsinvertedfromtheearlyarrivalsofCSGs#1#180.predictingthephasethantheamplitudeoftherecordeddata.Thegradientassociatedwiththisobjectivefunctionmitigates Receiver IndexTime (s)(a) Observed CSG #11 20 40 60 80 100 120 0.2 0.4 0.6 0.8 1.0 1.2 Receiver IndexTime (s)(b) Calculated CSG #11 with WEWI 20 40 60 80 100 120 0.2 0.4 0.6 0.8 1.0 1.2 Receiver IndexTime (s)(c) Calculated CSG #11 with Conventional EWI 20 40 60 80 100 120 0.2 0.4 0.6 0.8 1.0 1.2 Figure13:(a)TheprocessedCSG#11,(b)thecalculatedCSG#11withtheWEWIgradient,and(c)thecalculatedCSG#11computedwithconventionalEWI.theeectsofnoiseandunpredictedamplitudevariations,anditstrengthensthefrequencydierencebetweentheobservedandcalculateddatainthebackpropagatedwaveelds.Therobustnessandthephasematchingpro

7 pertyofWEWIarethenvalidatedbyinvertingfo
pertyofWEWIarethenvalidatedbyinvertingfortheMarmousimodelusingapol-luteddataset.ThismethodisalsotestedonarealcasebyinvertingCSGsofalanddatasetafterregularprocessing.Adynamictimewindowisalsousedtoinverttheearlyarrivals.SothatbothrefractionsandsomeearlyarrivingreectionsareincludedintheWEWIapproach.OurelddataresultssuggestthatWEWIcangenerateamoreaccurateandhighlyresolvedvelocitymodelcomparedtotheconventionalEWItomogram.Althoughthedatawithpeakfrequencyaround15Hzarerstusedintheinversion,thedrawbacksofWEWIforlanddatalargelycomefromthelackoflowerfrequencydatafrom1-5Hz.Partofthisproblemcancomefromtheltersthatre-movelowfrequencyinformationpollutedbysurfacewavesorothernoise,andthismightberemediedbybetterprocessing WeightedEWItoShallowLandData67 (a) Observed CSG #11 Muted by A Dynamic Time WindowReceiver IndexTime (s) 20 40 60 80 100 120 0.2 0.4 0.6 0.8 1.0 1.2 (b) Calculated CSG #11 Muted by A Dynamic Time WindowReceiver IndexTime (s) 20 40 60 80 100 120 0.2 0.4 0.6 0.8 1.0 1.2 (c) Calculated CSG #11 with Conventional EWIReceiver IndexTime (s) 20 40 60 80 100 120 0.2 0.4 0.6 0.8 1.0 1.2 Figure14:(a)TheprocessedCSG#11,(b)thecalculatedCSG#11withWEWI,and(c)thecalculatedCSG#11withEWImutedbyadynamictimewindow. 1 10 20 30 0.7 0.8 0.9 1 Iteration NunberNormalized L2 Data ResidualData Residual Convergence Rate Comparison for Land Data WEWI Conventional EWI Figure15:TheconvergencerateofWEWIfor30iterations.techniques.Also,WEWImaystillencounterthecycleskip-pingproblemalthoughitbetterutilizesthephaseinformationinthedata.Cycleskippingproblemscanbepartlyovercomebyonlyinvertingthenearosettraces,andthenwithlaterit-erationsinvertthelargerosettraces.Forreectionseventsfromdeepreservoirgeologyinoilindustry,itisalsonecessary Distance X (m)Depth Z (m)(a) RTM Image Using Traveltime Tomogram 1125 2250 3375 4500 5625 180 360 540 720 Distance X (m)Depth Z (m)(b) RTM Image using WEWI Tomogram 1125 2250 3375 4500 5625 180 360 540 720 Distance X (m)Depth Z (m)(c) RTM Image Using Conventional EWI Tomogram 1125 2250 3375 4500 5625 180 360 540 720 Figure16:RTMimagesbasedon(a)thetraveltimetomogram,(b)theWEWItomogram,and(c)theconventionalEWItomo-gram.tocontinuewideningthetimewindoworinvertingtheeventsfromdeeperreectors.5.ACKNOWLEDGMENTSWewouldliketothankthe2013sponsorsoftheCSIMConsor-tium(http://csim.kaust.edu.sa/web/)fortheirnancialsupport.ThecomputationresourceShaheen(http://shaheen.hpc.kaust.edu.sa/)forinversionprovidedbythehighperformancecomputing(HPC)centerofKingAbdullahUniversityofScienceandTechnology(KAUST)isgreatlyappreciated.WealsothankProf.Schus-terandanonymousCSIMmembersfortheirprofessionalcom-mentsinthedevelopmentofthiswork.REFERENCESAki,K.andP.G.Richards,2002,Quantitativeseismology:2ndedition.UniversityScienceBooks.Amorium,W.N.D.,P.Hubral,andM.Tygel,1987,Computingeldstaticswiththehelpofseismictomography:Geophys-icalProspecting,35,907–919.Boonyasiriwat,C.,G.T.Schuster,P.Valasek,andW.Cao,2010

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