To get a blocky interval velocity estimate that is more geolog ically reasonable a regularized Dix inversion needs to be done using an 1like norm In this paper we compare di64256erent 2D regularizations using both the norm and the hybrid L 2 norm to ID: 62211
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Almomin2Blockyvelocityinversion DIXINVERSIONASANL1-OPTIMIZATIONPROBLEMTheDixequationcanbemadelinearbyrelatingthesquareofintervalvelocityvtothesquareofRMSvelocityV,v2 = V2 �( �1)V2 �1;(1)where isthetwo-waytraveltime.Bydeningu =v2 andd = V2 ,wecansetuptheDixinversionprobleminanL1senseasfollows:jjWd(Cu�d)jjhybrid0;(2)whereWdisaweightfunctionproportionaltothepickstrengthinthevelocityscandividedby ,Cisthecausalintegrationoperator,anduanddarevectorscontainingallthevaluesofu andd ,respectively.Thedivisionby reducesthestrengthofthelatereventstobalancethedatattingstrengthalongthetimeaxis.Thehybridnormabovedenesthecostfunctionasfollows:C(r)=p r2+R2�R;(3)whereristheresidualandRisathresholdwhichdenesasmoothtransitionbetweentheL1andL2norms(Claerbout,2009).Fittinggoal(2)isnotenoughtofullyconstraintheinversion,becauseithasalargenullspace(LiandMaysami,2009).Moreover,pickingerrorscanleadtoincorrectRMSvelocitiesandunreasonableintervalvelocities.Therefore,asecondttinggoal(i.e.aregularizationterm)isrequiredtoconstrainthisinversion.Theregularizationtermcanbewrittenasfollows:jjAujjhybrid0;(4)whereAistypicallyarougheningoperator,andisascalartobalancethetwottinggoals.Noticethatthenorminttinggoal(2)hasadi erente ectthanthenorminttinggoal(4).Usingthehybridnormindatattingmakestheinversionlesssensitivetooutliers.Ontheotherhand,usingthehybridnorminmodelstylinga ectsthegeneralshapeoftheestimatedmodel,whichisthegoalofthispaper.LiandMaysami(2009)successfullyproducedblockinessin1Dwhenusingtherstderivativeasaregularizationoperator.Inthefollowingsections,wewilltrydi erentregularizationoperatorstoachievethesamegoalsin2D.REGULARIZATIONBYTHELAPLACIANOPERATORFirst,wewillusetheGulfofMexicodataprovidedbyWesternGeco.Thisisafour-seconddataset,atasamplingintervalof4ms.Theo setaxishas24tracesstarting SEP{140 Almomin4Blockyvelocityinversion lineartrendinvelocitywillalsoresultinazerosecondderivative.Inthenextsectionweattempttomorecloselyapproachtherstderivativebyusingthehelixderivative. Figure3:TheWGdataset.(a)TheintervalvelocityestimatedbyusingtheLaplacianoperatorasaregularizationintheL2norm.(b)ThereconstructedRMSvelocity.[ER] REGULARIZATIONBYTHEHELIXDERIVATIVEOPERATORNowweconsiderthehelixderivative(Claerbout,1997)asaregularizationoperator.Figure5showstheresultsofusingtheL2norm,andFigure6showstheresultsofusingthehybridnorm.Inthiscase,weseeadramaticdi erencebetweenthetworesults.Inthehybridnormcase,wecanseethebeginningsofblockiness,butonlyinonedirection(towardtheright).ThereasonforthisasymmetryisthatusinganL1-likenormissimilartoapplyingtheregularizationonlyonce.Ontheotherhand,wedonotseethise ectintheL2normresults,becausetheregularizationinthatnormissimilartoapplyingtheforwardandtheadjointofanoperator,whichisasymmetricprocedure.REGULARIZATIONBYTHEFIRSTDERIVATIVEOPERATORINTWODIRECTIONSThepreviousregularizationsshowthatonlyarstderivativecancreateblockiness.However,usingtherstderivativemeansthatwemustpickadirectioneachtime SEP{140 Almomin5Blockyvelocityinversion Figure4:TheWGdataset.(a)TheintervalvelocityestimatedbyusingtheLaplacianoperatorasaregularizationinthehybridnorm.(b)ThereconstructedRMSvelocity.[ER] Figure5:TheWGdataset.(a)TheintervalvelocityestimatedbyusingthehelixderivativeoperatorasaregularizationintheL2norm.(b)ThereconstructedRMSvelocity.[ER] SEP{140 Almomin6Blockyvelocityinversion Figure6:TheWGdataset.(a)Theintervalvelocityestimatedbyusingthehelixderivativeoperatorasaregularizationinthehybridnorm.(b)ThereconstructedRMSvelocity.[ER] weapplythederivative.Asarsttest,wepicktwodirections:theverticalandhorizontalasfollows:jjDzujjhybrid0;(5)jjDxujjhybrid0;(6)whereDzandDxaretherstderivativeoperatorsalongthez-andx-axis,respectively.Thederivativeofeachdirectionisappliedinaseparateregularizationequation(i.e.wehavetworegularizationequationsinthiscase)inordertomaintainsymmetry.Combiningthesetwoltersinoneregulariztionwillcauseanasymmetryinblockiness,similartothepreviousresultfromthehelixderivativeregularization.Figure7showstheresultsofusingtheL2normwithtworstderivativeapplica-tions,andFigure8showstheresultsofusingthehybridnorm.Blockinessisclearlypresentinthehybridnormresults.However,thereseemstobeapreferenceforthesharpboundariestobeeitherhorizontalorvertical,whichisduetothedirectionsofthederivativeswechose.REGULARIZATIONBYTHEFIRSTDERIVATIVEOPERATORINFOURDIRECTIONSToreducethebiasinblockinessdirections,weincreasedthedirectionsoftherstderivativetofour:theprevioustwodirections,plustwodirectionsat45degreestotheverticalandhorizontalaxes.Figure9showstheresultsofusingtheL2norm, SEP{140 Almomin7Blockyvelocityinversion Figure7:TheWGdataset.(a)TheintervalvelocityestimatedbyusingtherstderivativeoperatorintwodirectionsasaregularizationintheL2norm.(b)ThereconstructedRMSvelocity.[ER] Figure8:TheWGdataset.(a)Theintervalvelocityestimatedbyusingtherstderivativeoperatorintwodirectionsasaregularizationinthehybridnorm.(b)ThereconstructedRMSvelocity.[ER] SEP{140 Almomin8Blockyvelocityinversion andFigure10showstheresultsofusingthehybridnorm.BycomparingFigure10toFigure8,wecanclearlyseealargeimprovementinthemodel.Thedetailsandblockinessarestillpreserved.However,theinversionnowhasmore\freedom"topickthedirectionofblockinessoutoffourdirectionsinsteadoftwodirections. Figure9:TheWGdataset.(a)TheintervalvelocityresultedbyusingtherstderivativeoperatorinfourdirectionsasaregularizationintheL2norm.(b)ThereconstructedRMSvelocity.[ER] ELFDATASETInthissection,wewillestimateanintervalvelocitymodeloftheNorthSeadataprovidedbyELF.Thereisaknownsaltbodyinthemiddleofthisdata,whichmakesitapropertestcaseforourblockinessgoals.Also,thisdatasethasbetterspatialsamplingthanthepreviousdataset,andcanthusbetterillustratethedi erencesbetweenthedi erentinversionresults.Thisisalsofour-seconddataset,atasamplingof5.9ms.Theo setaxishas143tracesstartingat0mwithanincrementof25m.Thereare537CMPgatherswithaspacingof25m.Figure11(b)showstheresultsofautopickingthedataset,whichwasconstrainedbythebackgroundRMSvelocity,andFigure11(a)showstheresultsofdirectDixconversion,asdenedabove.Figure12showsthestrengthofthepicksinthevelocityscans.Forthisdataset,wewillonlyrepeatthelasttworegularizations,whicharetherstderivativeintwoandfourdirections,sincetheyshowedthebestresultswiththemostsymmetricblockiness.Figure13showstheresultsofregularizingintwodirectionsintheL2normandFigure14showstheresultsofthesameregularizationinthehybridnorm.Figures15and16showtheresultsofusingfour-directionregularization. SEP{140 Almomin9Blockyvelocityinversion Figure10:TheWGdataset.(a)Theintervalvelocityresultedbyusingtherstderivativeoperatorinfourdirectionsasaregularizationinthehybridnorm.(b)ThereconstructedRMSvelocity.[ER] Figure11:TheELFdataset.(a)TheinputRMSvelocitywhichisautomaticallypickedfromtheCMPgathers.(b)TheintervalvelocitybydirectDixconversion.[ER] SEP{140 Almomin10Blockyvelocityinversion Figure12:ThestrengthofthepicksinthevelocityscanofELFdataset,whichisusedastheweightbeforedividingbytime.[ER] Sincethedatasetislargerwithsmallersampling,theimprovementofusingmoredirectionsisevident.Forcingtheinversiontopickbetweentwodirectionshasanapparente ectofreducingtheresolution.Oneobviousexampleisthechalklayer,whichlooksveryhorizontalinFigure14butmoredetailedinFigure16.Inallcases,L2alwaysgivessmoothresults,whichsmearthemodelandlowertheresolutionoftheinversion. Figure13:TheELFdataset.(a)TheintervalvelocityestimatedbyusingtherstderivativeoperatorintwodirectionsasaregularizationintheL2norm.(b)ThereconstructedRMSvelocity.[ER] SEP{140 Almomin11Blockyvelocityinversion Figure14:TheELFdataset.(a)Theintervalvelocityestimatedbyusingtherstderivativeoperatorintwodirectionsasaregularizationinthehybridnorm.(b)ThereconstructedRMSvelocity.[ER] Figure15:TheELFdataset.(a)TheintervalvelocityestimatedbyusingtherstderivativeoperatorinfourdirectionsasaregularizationintheL2norm.(b)ThereconstructedRMSvelocity.[ER] SEP{140 Almomin12Blockyvelocityinversion Figure16:TheELFdataset.(a)Theintervalvelocityestimatedbyusingtherstderivativeoperatorinfourdirectionsasaregularizationinthehybridnorm.(b)ThereconstructedRMSvelocity.[ER] CONCLUSIONSANDDISCUSSIONSWesuccessfullyachievedblockyvelocitymodelsbyusingthehybridnorm.Wealsoshowedthatthechoiceoftheregularizationoperatorhasagreatimpactonhowblockytheresultsare.Nonetheless,thehybridnormalwaysshowedmoredetailandresolutionthantheL2norm,evenwhenblockinesswasnotachieved.AnexampleofthiscanbeseenbycomparingFigures4and9.AlthoughtherstFigureusesaLaplacianoperatorforregularizationandthelaterFigureusestherstderivativeinfourdirections(whichweshowedhasthebestresults),thehybridnormwasstillsuperiortotheL2norminpreservingmoredetailsandshowinghigherresolution.Anotherpointtokeepinmindisthatthehybridnormhasa exibilethreshold.Inallpreviouscases,wesetthatthresholdto0.20,meaningthat80percentofresiduals(bothdataresidualsandmodelresiduals)aregoingtobeintheL1regionandtherestintheL2region.However,thisisaparameterthatcanbeadjustedbasedonthedesireddegreeofblockiness.FUTUREWORKAswehaveseen,increasingthenumberofdirectionalderivativeswillincreasetheresolutionand exibilityoftheinversionresults.However,increasingthenumberofdrectionswillalsoslowtheinversion,becauseeachdirectionhasamodelresidualthesizeofthemodel.Insteadofusingmanydirections,itispossibletousesteeringlters SEP{140 Almomin13Blockyvelocityinversion topre-denethelocaldirectionofmaximumvarianceandthenusethatinformationtoalignthedirectionsoftheregularizationtobeparallelandperpendiculartoit.Thisway,wemightonlyneedtwodirectionsintheregularization.ACKNOWLEDGMENTSWethankJonClaerbout,RobertClapp,AntoineGuitton,LouisVaillant,ElitaLiandYangZhangfortheirhelpfulsuggestionsanddiscussionsthroughoutthisproject.WealsothankWesternGecoandTotalFinaElfforprovidingthedata.REFERENCES Claerbout,J.F.,1997,Multidimensionalrecursiveltersviaahelix:SEP-Report,95,1{13. ||{,2009,Blockymodelsviathel1/l2hybridnorm:SEP-Report,139,1{10. Clapp,R.G.,2001,Geologicallyconstrainedmigrationvelocityanalysis:PhDthesis,StanfordUniversity. Dix,C.H.,1952,Seismicprospectingforoil. Harlan,W.S.,1999,Constraineddixinversion. Koren,Z.andI.Ravve,2006,Constraineddixinversion:Geophysics,71. Li,Y.E.andM.Maysami,2009,Dixinversionconstrainedbyl1-normoptimization:SEP-Report,139,23{36. Maysami,M.andN.Moussa,2009,Generalized-normconjugatedirectionsolver:SEP-Report,139,11{22. SEP{140