Chapter 4 Seismic Attenuation Problem - maths-in-industry.org

1 Chapter4 SeismicAttenuationProblemProblemPresente dBy:KennethJ. Hedlin(Husky Energy);GaryMargrave (UniversityofCalgary)Mentors:NilimaNigam (McGillUniversity);TobiasSchaefer(Univer sityofNorthCarolinaatChapelHill)StudentP articipants:Mohammadal-Khaleel(McGill);L inpingDongandCarlosMontana(Uni-versityof Calgary);WanChen,CatherineDupuis,GillesH ennenfend,FelixHermannandPeymanPoorMogha ddam(UniversityofBritishColumbia);Heejeo ngLeeandJinwooLee(SeoulNationalUniversit y);JooheeLee(UniversityofNorthCarolinaat ChapelHill);NamyongLee(MinnesotaStateUni versity);YanWu techniqueinwhichthereflectionsofa sourceseismicwave arerecordedasitpassesthroughtheearth,is a profileofthematerialpropertiesoftheearth below thesurface,andis Energyconcernsseismicattenuation:theloss ofenergyasa seis-micwave ,attenuationeffectshave usefulintwo ways:asa meansofcorrectingseismicdatatoenhanceres olutionofstandardimagingtechniques,andas a directhydrocarbonindicator.

2 Theoreti-cally, a subsurfacereservoirfullofhydrocarbonswil ltendtobeacousticallysofterthana porousrockfilledonlywithwater;Kumaret al show thatattenuationis highestina physicalprocessescanleadtotheattenuation ofa ,weignoreattenuationeffectssuchasspheric aldivergenceorscattering,andconcentrateo nintrinsicat-tenuationeffectsexclusively . Thelatterarecausedbyfriction,particularl yinporousrocksbetweenfluidandsolidpartic les,see[2,7]. meansofcomputingseismicattenuationfromre lativelyshortwindowsofseismicimagingdata , beginbya , ,weconsidertheuseoffrequency-shifttechni questo identifyanomalousattenuation;two ,differentwavelet-baseddenoisingtechniqu esareusedtoidentifytheattenuationanomaly . ,wepresentthemathematicalideasbehindanex tensionofa ,includingtheuseofa materialtoattenuateseismicwavesis measuredbya dimensionlessquantityQ, calledtheattenuationfactor, byQ:=energyofseismicwaveenergydissipated percycleofwave=2 E4 EwhereEistheenergyofthewave, (dirt)through100(rock)to10,000(steel).

3 Inwhatfollows,weassumethatthisattenuatio nfactoris independentoffrequency! is directlylinkedtothedifferentlayersthatco mposetheEarth,sothatwheneverchangesinthe compositionoflayersoccur, whywewouldlike tobeabletodetectchangesinattenuation,asi t wouldenableustoidentifys :To , werestrictourattentionto1-Dmodels, , assumethatthereceiveris positionedat thesurfaceoftheearth(inotherwords,wedono tconsiderverticalseismicprofiles).Fora mediumwithlinearstress-strainrelation,it is knownthatwave amplitudeAis proportionaltopE. Hence,1Q= A A( )fromwhichwecanobtaintheamplitudefluctua tionsdueto ,giveninitialamplitudeA0, let bethewave lengthgivenintermsoffrequency!andphaseve locitycby = 2 c=!, then A= (dA=dz). Hence,equation( )becomes,dAdz= !2cQA( ) (!; z) =A0(!) exp !z2cQ :( )Now, fromobservationofexponentiallydecayingva luesofA(!; z), ,from( ),wehaveln AA0 = !

4 Z2cQ = ! t2Q ( )Hereweassumethatthephasevelocitycdoesno tdependonfrequency, ie, ,byrecordingtheln(A=A0)versus!graph,andt henestimatetheaverageslope,wecanrecovert hevalueofQ. Thisideais knownaslog ,whichincludestheeffectofreflectionsofth esignalfromvariouslayers, (t)denotetheseismictraceobtainedfromlaye rk, receivedattimetkatthereceiver(assumingth esourcesignals0wasemittedat timet= 0). Supposethecoefficientofreflectionat thekthlayerisrk, andthethesourcesignalisso. Ingeneral,rkis unknown,asisso. Indeed,thesourcesignalis usuallygeneratedbya denotation;characterizingthissignalis frequency!, wemaywritej^sk(!)j=rkj^so(!)je p!t=Q( )wherepis a constant,andrkis similarinformationabouta seismictracereflectedfromlayerj, thenthelogspectralratiomethodestimatesth eattenuationQas:log j^skjj^sjj = logjrkj logjrjj+p!Q(tj tk) ConvolutionalModelofAttenuationLets(t)de noteanunattenuatedseismictracereceivedat timetata receiver.

5 Ifw(t)wasthesourcewaveformandr(t)is thereflectivityasa functionofdepth(equivalentlytime),thenwe maywrites(t) =w r:=Z1 1w( )r(t )d :( )Two key assumptionsaremaderegardingthesourcesign alandthereflectivity: thewhitereflectivityassumption. Thewhitereflectivityassumptionsimplymean sthereflectivityrsatisfiesZ1 1~r(s)r(t s)ds= (t)where (t)is theDiracmeasure. theminimumphaseassumption: thismeansthatthesourcesignalw(t)is causal,invertible,andpossessesminimumpha seinthesensethatif wewritethesignalinthefrequency domain^w(!) =A0(!)ei (!)wecanfindthephase (!)byusinga ,andintheabsenceofattenuation,weareablet orecoverthesourcesignalwfroma giventracesusingtheWienerprocessonequati on( ):s ~s= (w r) ( ~w ~r) = (w ~w) :Takinga Fouriertransformoftheabove expression,wegettheamplitudeA(!)ofthesou rcesignalw;theminimumphaseassumptionnow , theWienerprocessdoesnotapplyinquitesucha straight-forwardfashiontothecasewherethe signalis describedbytheactionofa pseudodifferentialoperator:theattenuated tracesais nowsa(t) :=Z1 1w ( ; t )r( )d ( )wherew (u;v) :=Z1 1 (u; )ei v^w( )d ;( ) (u; ) = exp u2Q exp iu2QZ1 1e ede :( ) canalsodescribe,inthesettingofthisconvol utionalmodel,thewindowedlogspectralratio techniquewhichis 1and 2betwo expectthat^sa( 1) ^weff1 ^reff1whereweff1is theeffective signaloverwindow 1, andreff1is theeffective alsoexpectthatj^weff2j=j^weff1jexp( !)

6 2Q)where is thewindow ,thelogspectralratioislog j^sa( 1)jj^sa( 2)j =! 2Q+ log j^reff1jj^reff2j : DiscreteModelInpractice,seismictracedata is sampledatdiscretetimeintervals,fora discreteversionoftheconvolutionalmodelab ove:supposeweknowtheinitialsourcesignal, ; t2; : : : tn. Fromthis,wecanconstructa matrixW , anda vectorofreflectivitiesr= (r1; r2; : : : rn)T, whereriis thereflectivityofthelayerat depthcti. Then,thediscreteversionofequation( )isW r= (w1jw2j: : :jwn)r=s:=(s1; s2; : : : sn)T:TheentrieswijofmatrixW have thefollowingproperties: Ifti> tj,wji= 0(causalityassumption) Ifti< tj,wji=w (ti; tj ti), wherew wasdefinedbyequations( )and( ).Therefore,W islowertriangular, seismicproblemis:givenW ,r, :givens, findW ,r. Inourspecificcase,wehave tofindW ,specificallytheamountofattenuationbetwe entheamplitudespectraofthecolumnsofW.

7 Asis easytosee,theinverseproblemis SimpleModelAsa firstapproachtosolvingtheattenuationretr ieval Problem ,webeganbycreatingsomesimulatedda taandsolvingtheforwardproblem,wheretheQ- profileis toseehow thepresenceoftheanomalychangesa 2s,atintervalsoftime0 drawnfroma 20Hz,is generatedonce, (a): Thenormalattenuationcase,wheretheattenua tionis a constantQ= 100foralldepths; Theanomalousattenuationcase,whereQchange sat depthct= 1toQ= 40, andthenchangesbacktoQ= 100at depthct= 1:1(weassumethespeedofpropagationhasbeen normalizedtoc= 1).We calculatetheresultantseismictracess(t)as it propagatesthroughthetwo (b)weshowthecomputedseismictraceswithand withouttheanomaly. Ascanbeseen,thetwo tracesappearnearlyidentical;onlywhenwesu btractthemcanweclearlyspottheonsetofthea nomaly(attimet= 1). (a)Profileof normal and anomalous attenua-tion.(b)Left right:normaltrace,anomaloustrace,tracess uperimposed, :Theseismictracecorrespondingto theattenuationanomalyis nearlyidenticalto ( )ofseismicattenuationthatweareusing,it is clearthattheamplitudesofhigherfrequency componentsattenuatemoreoverthesamedeptht handolowerfrequency thatthereis ared-shiftinthesignalasit , theamplitudespectrumofthesourcesignalinb lue, is to lookat theamplitudespectrumoftheseismictraceove r many overlappingwindowsintime,andlookat thechangesinthemeanfrequency givenseismicdatas(t), wecantake a windowedFouriertransform( )toseethelocalspectralpropertyofthedata: ^sg(t; !)

8 =Z1 1s( )g(t )e i! d ;( )wheregis a Gaussianfunctionusedasa window. We canthencomputetheaveragewithrespecttothe frequency toobtainthecentroidfrequency,fc(t), bytheformula:fc(t) =R!j^sg(t; !)jd!Rj^sg(t; !)jd! Pk!kj^sg(t; !k)jPkj^sg(t; !k)j:( )Ina similarfashion, ,wecomputedthesecondmoment, (t) =R!2j^sg(t; !)jd!Rj^sg(t; !)jd! Pk!2kj^sg(t; !k)jPkj^sg(t; !k)j:( ) :Shiftinmeanfrequency:amplitudespectrumi nredis suddendecayofoverallfrequency amplitudeswhensharpanomaliesoccur,wemayo bserve lowervaluesoffc(t)attheabnormality. We , ,welookat testedtheattributesfc; decreasingfasteraftert= 1thanthatofthenormalsignal,actingasa thevariationinthesecondmomentfsforthesei smictracesinthenormal(blue)andanomalous( red) alsochecktherobustnessofthisattributeton oise;forverylowlevelsofnoise,thesecondmo mentis stilla goodpredictoroftheonsetoftheinstability.

9 Asthenoiseincreases,theamplitudespectrab ecometoopollutedinthehighfrequency referencedatato ,if weusetheseattributes,thereshouldbeintrin sicchangesin thevaluesoftheseattributeswhichallowsust o experimentsweinitiallysuggestedthatthegr aphofthecentroidfrequencyfcasafunctionof depthbecomesconcave neartheonsetofananomaly. Unfortunatelywefoundthatwhilethisconcavi tynecessarilyhappensnearabnormality, it alsohappenseveninnormalregions, testtheabilityofthecentroidalfrequency techniquestofindattenuationanomaliesinre aldata,weusedtwo Thesewereseismicprofilestakenat two geographical : normaltrace aregreaterthanthe anomaloustrace .AA SSiimmppllee IIddeeaaWWee ccoommppuuttee tthheesseeccoonndd mmoommeenntt :Syntheticsignalsandthesecondmomentoffre quency. Addingnoiserendersthedetectionoftheanoma lylessrobust. DETECTIONUSINGMOMENTSOFFREQUENCY61locati ons:Pike PeakandBlackfoot,respectively.

10 Datasets:ineach,many source-receiverpairswerelocatedalonga horizontalline,anddatawascollectedovera ,thesurfaceoftheearthis ontop;they axisindicatesdepth(equivalentlytimeoftra vel ofthesignal).(a)PikesPeak:anomalyatdepth 350,locationx=400.(b)Blackfootreservoir: anomalyatdeptht=600,locationx= :Actualseismictraces:PikesPeakandBlackfo ot.(a)PikesPeak:actualtrace.(b)Centroida lfrequenciesfc.(c) :ThePike Peaksdataset:theactualsection, (b) (b),theredindicatethatthecentroidalfrequ encyfcovera givenwindowis high,andblueindicatesa lowfc. (b),oneseesa regionofsuddendecayoffcat depthapproximately350, (b),onecanclearlyseea regioninyellow (lowerfcatdepth600, location45 50, identifyingthepresenceofanattenuationano maly. Thisregioncorrespondswellwitha knowngasreservoir. (a) (b)Centroidalfrequenciesfc.(c) :TheBlackfootdataset:theactualsection, 600, (c) (c), , (c) (b);it wouldbeinterestingtocross-checkthiswitha ny , (c)picksouta widebandofanomalousattenuationatdepth300 ,ana narrower,morelocalizedbandat ,westartedwiththeassumptionthattheFourie rtransformoftheseismictracesisproportion altotheFouriertransformofthereflectivity r, wheretheproportionalityfactoris positiveandcontainsinformationonthesourc esignalandtheattenuationfactor.)