MarieLPruchaandBiondoLBiondiStanfordUniversityWilliamWSymesRiceUniversitySUMMARY ShotandoffsetdomaincommonimagegathersencounterproblemsincomplexmediaTheycanplaceeventsthatcomefromdifferentpo ID: 94440
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Angle-domaincommonimagegathersbywave-equationmigration MarieL.Prucha*andBiondoL.Biondi,StanfordUniversity,WilliamW.Symes,RiceUniversitySUMMARY Shot-andoffset-domaincommonimagegathersencounterproblemsincomplexmedia.Theycanplaceeventsthatcomefromdifferentpointsinthesubsurfaceatonesubsurfacelocationbasedonidenti-calarrivaltimesandhorizontalslownesses.Angle-domaincommonimagegathersuniquelyde®neraycouplesforeachpointinthesubsur-face,thereforeeacheventinthedatawillbeassociatedwithonlyonesubsurfacelocation.Itispossibletogenerateangle-domaincommonimagegatherswithwave-equationmigrationmethodsandtheseangle-domaincommonimagegathersmaybeusedforvelocityanalysisandamplitude-versus-angleanalysis.INTRODUCTIONCurrentdepthimagingtechnologyworksverywellinareasthathaveslowvelocityvariationsbutmayfailinmorecomplexareas(Claer-bout,1985)foravarietyofreasons,suchasmultiplere¯ections,badvelocities,andspatialaliasing.Onepotentialcauseofimagingfailureisre¯ectorlocationambiguityduetomultipathingofre¯ectedenergy:itispossiblethatasingleeventrecordedinthedataatonesurfacelocationcouldcomefromre¯ectorsattwoormoresubsurfaceloca-tions.Besidescontributingtoimagingartifacts,re¯ectorambiguitycontributesnon-¯ateventstocommonimagegathers,thusrenderingvelocityanalysisambiguous(NolanandSymes,1996).Severalauthorshavesuggestedangledomainimagingasasolutionforthere¯ectorambiguity(Xuetal.,1998;Brandsberg-Dahletal.,1999).Angledomainsectionscollecttheenergyinadatasetwhichhasscatteredoveraspeci®cre¯ection(ªopeningº)angle .Wewillarguebelowthataneventinananglesectionuniquelydeterminesaraycouple,whichinturnuniquelylocatesthere¯ector.Thusimagingartifactsandvelocityupdateambiguityduetomultipathingareeliminatedinthisdomain.Multipathingisbetterhandledbywave-equationmigrationmethodsthanKirchhoffones,thereforetheformerareanaturalchoiceforpro-ducingangle-domaincommonimagegathers(CIGs).WepresentasimplemethodforextractingCIGsfrom3-DprestackdatadownwardcontinuedusingtheDoubleSquareRootequation(DSR).Themethodisbasedonaslant-stackdecompositionofthedownwardcontinuedwave®eldateachdepthlevel.OurmethodisthusdifferentfromthemethodproposedbyOttoliniandClaerbout(1984),thatappliestheDSRtodownwardcontinueprestackdataslant-stackedatthesurface.Inlayeredmediathetwomethodsshouldproduceequivalentresults,butinpresenceoflateralvelocityvariationsplane-wavedownwardcon-tinuationisnotstrictlyvalidandtrueangle-domainCIGscanonlybeproducedbywave®elddecompositionatdepth.MigrationmethodsbasedonDSRoperatorshavebeenappliedto2-Dprestackmigrationforlongtime(Claerbout,1985).However,thedirectapplicationofDSRmigrationmethodsto3-Dprestackdatahavebeenpreventedbythetremendouscomputationalcost.Onlyrecentlycomputationallyef®cientmethodstocontinue3-Dprestackdatahavebeenpresented(BiondiandPalacharla,1996;Mosheretal.,1997).Inparticular,common-azimuthmigrationisanattractivealternativetoKirchhoffmigrationforsub-saltimagingbecauseofitsrobustnesswithrespecttothecomplexmultipathingthatisinducedbysaltbodies.Thisabstractwillexplainhowsomewidelyusedcommonimagegatherscancontainre¯ectorambiguityduetomultipathingandwhyangledomaincommonimagegatherswillnot.Thenwewilldemonstratetheconstructionofangle-domaincommonimagegathersfromwaveequationdownwardcontinueddataforuseinvelocityanalysisandamplitude-versus-re¯ectionangleanalysis.KINEMATICSOFMULTIARRIVALSINTHESHOTANDOFFSETDOMAINSThekinematicsofshotdomaincommonimagegathersandoffsetdo-mainimagegathersarewellunderstoodinconstantvelocityand z media.Dif®cultiesarisewhenwebeginconsideringcomplexsub-surfaceswithrapidlateralvelocityvariations.Evenin2-Ditiseasytoconstructamodelforwhichanindividualcommonshotgatherorcommonoffsetgathercancontaintwoeventsfromtwopointsinthesubsurfacethatarriveatthesametimeandareindistinguishable.Letusinvestigateparticularcasesoftheseoccurrences.Anindividualcommonshotgatherisparameterizedbythereceiverlo-cationr.Iftworaypathsbetweenthesamesourceandreceiverexistsuchthattheyhavethesamereceiverhorizontalslownessprandthetwo-waytraveltimealongeachisthesame,itisimpossibletodistin-guishbetweenthetwore¯ectorlocations(NolanandSymes,1996).Figure1showsaverysimplecaseofthis. Figure1:Individualshotgather:thecircularlensisalowvelocityanomalysothetraveltimesandprareidentical.Anindividualcommonoffsetgatherisparameterizedbymidpointm.Supposethatthehorizontalmidpointslownesspm ps pristhesamefortworaypathswiththesametraveltimes.Onceagain,thetworaypathsrepresentthesameevent,andthelocationofthere¯ectorcausingthiseventiscompletelyambiguous.Figure2showsacasewhereps1 pr1andps2 pr2sothatpm1 pm2 0.ByshootingafanforraysfrombothxlocationsinFigure2,itispossibletoobtaintraveltimecurvesforacommonoffset.ThesecurvesareshowninFigure2.Thetraveltimecurveforthediffractorinsidetheanomalyisnicelyhyperbolic.Thecurveforthedeeperdiffractorisdiscontinuousduetotheshadowzonesthatoccurabovetheedgesoftheanomalybutthesymmetricaltriplicationscanbeseen.Atthisoffset,thetraveltimesforbothcurvesareidenticalforthemidpointdirectlyabovethediffractors.Fromthegeometryofthemodel,itisclearthatthemidpointslownessesarethesame,thereforetheeventsinthedatawillbeindistinguishable.KINEMATICSOFMULTIARRIVALSINTHEANGLEDO-MAINWecon®neourdiscussiontothe2Dcase;the3Dcaseissimilar,pro-videdthatcompletesurfacecoverageisavailable.Specularre¯ec-tionconnectsare¯ectorelement,consistingofasubsurfaceposition Re¯ectionanglegathers Figure2:Individualoffsetgather:thecircularlensisalowvelocityanomalysothetraveltimesareidenticalandthemidpointslownessesareequal. Figure3:TraveltimecurvesforthemodelinFigure2.Thethickcurveisforthedeeperre¯ectorandisdiscontinuousduetoshadowzones. Figure4:Simplere¯ection:thevector isnormaltothere¯ector, isone-halfthere¯ectionangle,and isrelatedtothedipangle.xcontainingthemidpointlocationmandthedepth zanddipvector representingthere¯ectornormalatthatpositionwithaneventele-ments px r pr tconsistingofsourceandreceiverpositionssandr,sourceandreceiverhorizontalslownessespsandpr,andtwowaytimet.Theconnectionviaincidentandre¯ectedraysisdepictedinFigure4,whichalsoshowstheopeningangle .Notethatgivenx, ,and ,theincidentandre¯ectedraysandtheeventelements ps r pr tarecompletelydetermined:thereforethelatterarefunctionsofx, ,and .Theangletransformofadataset d s r t isa x d x d s r t Theprincipleofstationaryphaseshowsthataneventinasingleanglepanel,i.e.apositionyandanangledomaindipvector ,arisewhenincidentandre¯ectedraysmeetatyandarebisectedby ;theseraysdetermineonceagainaneventelements ps r pr t,andthiseventmusthavebeenpresentinthedatafortheeventinquestiontobepresentintheangledomain.Ofcoursetheeventelements ps r pr tcompletelydeterminestheraysinthesubsurfacecarryingtheenergyoftheevent.WeassumetheTraveltimeInjectivityCondition(tenKroodeetal.,):apairofraysandatotal(two-way)timedeterminesatmostonere¯ectorelement.Inthatcase,theeventintheangledomainiscompatiblewithatmostonere¯ectorelement.Notethecontrastwiththeconstantoffsetdomainasdescribedintheprecedingsectionwhereaneventelementinthedatacouldcorrespondkinematicallytomorethanonere¯ectorelement.Thevelocity®eldusedtogeneratetheraysusedintheformationoftheangletransformdoesnotnecessarilyneedtobethesameasthevelocity®eldwhichgaverisetothemoveoutinthedata-whichisfortunate,aswedon'tknowthelatterattheoutsetofthemigration/velocityanalysisprocess,andhaveonlyanapproximationofitattheend!Whenthetwovelocity®eldsaredifferent,theangletransformeventswillnotnecessarilymatchthere¯ectorsintheEarth:thetwowilldifferbyaresidualmigration.Whenthetwovelocity®eldsarethesame,theimageisperfect,i.e. x y .ANGLE-DOMAINCIGBYWAVE-EQUATIONMIGRATIONIntheprevioussectionswediscussedtheadvantagesofangle-domainCIGsoveroffset-domainCIGswhenacomplexvelocityfunctionin-ducesmultipathingandeventtriplication.Inthissectionweshowhowtoextractangle-domainCIGsfromdownward-continuedprestackdata.Recorded3-Dseismicdatacanbeorganizedasafunctionofmidpointcoordinates(m)andoffsetcoordinates(h).Prestackdataareef®cientlydownwardcontinuedusingtheDSRequationinthefrequency( )do-main.Furthermore,sinceweeitheruse2-Ddownwardcontinuationor3-Dcommon-azimuthdownwardcontinuation,theoffsetspaceisrestrictedtothein-lineoffsethx,andthusweexpresstherecordedwave®eldasP m hx z 0 ,wherezisdepthandz=0indicatesdatarecordedatthesurface.Theprestackwave®eldatdepthisobtainedbydownwardcontinuingtherecordeddatausingtheDSR,andisimagedbyextractingthevaluesatzerotimeP m hx z 0 DSR P m hx z (1)P m hx z Imaging P t 0 m hx z (2)Thedownward-continuationprocessfocusesthewave®eldtowardszerooffset(leftpanelinFigure5)andifthecontinuationvelocityiscorrect,amigratedimagecanbeobtainedbyextractingthevalueofthewave®eldatzerooffset.However,thezero-offsetwave®eldhas Re¯ectionanglegathers Figure5:Left:Offsetpanelafterdownwardcontinuation.Right:Angle-domainCIGlimiteddiagnosticinformationforvelocityupdating,andnoinforma-tionontheamplitudeofthere¯ectionsversusre¯ectionangle(AVA).Wethereforeperformaslantstackalongtheoffsetaxisbeforeimagingandobtainanimageasafunctionoftheoffsetrayparameter phx,asP m hx z 0 DSR P m hx z (3)P m hx z Slantstack P m phx z (4)P m phx z Imaging P 0 m phx z (5)(6)Angle-domainCIGsaresubsetsofP 0 m phx z at®xedmid-pointlocation.TherightpanelinFigure5showstheangle-domainCIGgathercorrespondingtothedownward-continuedoffsetgathershownintheleftpanel.Noticethatbecauseindownward-continuedoffsetgatherstheenergyisconcentratedaroundzerooffset,theslantstackdecompositiondoesnotsufferfromtheusualartifactscausedbytheboundaryconditions.Strictlyspeaking,theCIGgathersobtainedbytheproposedprocedurearefunctionoftheoffsetrayparametersphxandnotoftheapertureangle .However,phxislinkedto bythefollowingsimpletrigono-metricrelationshipphx 2sin cos V z m (7)where isthegeologicaldipalongthein-linedirectionandV z m isthevelocityfunction.Angle-domainCIGandvelocityAngle-domainCIGscanbeusedtoupdatethevelocityfunctionaftermigrationsimilarlytothewaythatoffset-domainCIGsarecurrentlyused(Brandsberg-Dahletal.,1999)orforwave-equationMigrationVe-locityAnalysis(BiondiandSava,1999).Asforoffset-domainCIGs,ifthevelocityfunctioniscorrectthere¯ectionsarealignedalongtheangleaxis.Ifthevelocityfunctionistoolowthere¯ectionswillsmileupward;ifthevelocityfunctionistoohighthere¯ectionswillfrowndownward.Thisbehaviorisdemonstratedbytheanalysisofthegath-ersinFigure6.Thegatherswereextractedfroma3-Dprestackwave-®eldfocusedusingcommon-azimuthdownwardcontinuation.Theleftgatherwasobtainedusingthecorrectvelocity.Therightgatherwasobtainedusingalowconstantvelocity.Figure7showstheinlinemi-gratedsectionthatpassesthroughthegathershowninFigure6.NoticethattheCIGgathersshowonlythe®rstkilometeroftheimage. Figure6:Left:Angle-domainCIGwithcorrectvelocity.Right:Angle-domainCIGwithtoolowconstantvelocity. Figure7:In-linesectionof3-Dmigratedcube Re¯ectionanglegathers Figure8:Right:Angle-domainCIGobtainbywave-equationmigra-tion.Left:Offset-domainCIGgatherobtainedbyKirchhoffmigrationAngle-domainCIGandAVAAngle-domainCIGcanalsobeusedtoanalyzethere¯ectivityasafunctionofthere¯ectionangletoestimaterockand¯uidpropertiesinthesubsurface.ThispotentialuseisillustratedbythegathersshowninFigure8.Theleftpanelshowsanangle-domainCIGgatherwhiletherightpanelshowsthecorrespondingoffset-domainCIGgatherobtainedbyanamplitudepreservingKirchhoffmigration(Eckeretal.,1996).Theamplitudebehaviorasafunctionofoffsetrayparameter(leftpanel)isinqualitativeagreementwiththetheamplitudebehaviorasafunctionofoffset(rightpanel).CONCLUSIONSAngle-domainCIGgathershaveattractivepropertieswhenacomplexvelocitymodelcausesmultipathingofre¯ectedenergy.Theyarefreeoftheimagingartifactscausedbyre¯ector-positionambiguitythatdegradetheimageobtainedfromeithershotgathersorcommon-offsetgathers.Wepresentedasimplemethodforextractingangle-domainCIGsfromtheprestackwave®elddownward-continuedusingthewaveequation.Ourmethodproduceshigh-qualityCIGgathersthatcanbereadilyusedforeithervelocityanalysisorAVAanalysis.REFERENCESBiondi,B.,andPalacharla,G.,1996,3-Dprestackmigrationofcommon-azimuthdata:Geophysics,61,1822±1832.Biondi,B.,andSava,P.,1999,Wave-equationmigrationvelocityanal-ysis:69thAnnualInternat.Mtg.,Soc.Expl.Geophys.,ExpandedAbstracts,submitted.Brandsberg-Dahl,S.,deHoop,M.V.,andUrsin,B.,1999,Sensitivitytransforminthecommonscattering-angle/azimuthdomain:EAGE61stAnnualConference,ExpandedAbstracts,submitted.Claerbout,J.F.,1985,ImagingtheEarth'sInterior:BlackwellScien-ti®cPublications.Ecker,C.,Dvorkin,J.,andNur,A.,1996,Sedimentswithgashydrates:Internalstructurefromseismicavo:67thAnnualInternat.Mtg.,Soc.Expl.Geophys.,ExpandedAbstracts,1767±1770.Mosher,C.C.,Foster,D.J.,andHassanzadeh,S.,1997,Commonangleimagingwithoffsetplanewaves:67thAnnualInternat.Mtg.,Soc.Expl.Geophys.,ExpandedAbstracts,1379±1382.Nolan,C.J.,andSymes,W.W.,1996,Imagingandconherencyincomplexstructure:Proc.66thAnnualInternationalMeeting,359±363.Ottolini,R.,andClaerbout,J.F.,1984,Themigrationofcommon-midpointslantstacks:Geophysics,49,no.03,237±249.tenKroode,A.P.E.,Smit,D.-J.,andVerdel,A.R.,Linearizedinversescatteringinthepresenceofcaustics:WaveMotion.Xu,S.,Chauris,H.,Lambare,G.,andNoble,M.,1998,Commonangleimagegather-astrategyforimagingcomplexmedia:DepthImagingofReservoirAttributes,X012.