Cross-equalizationprocessing for time-lapseseismic reservoir monitoring data: a case ...sep · 2015. 5. 27. · Cross-equalizationprocessing for time-lapseseismic reservoir monitoring

Cross-equalizationprocessingfor time-lapseseismicreservoirmonitoring data: a casestudy fr om the Gulf of Mexico

JamesE. [email protected]

Stanford Exploration Project, Mitchell Building, Department of Geophysics,Stanford University, Stanford, CA 94305-2215

David E. [email protected]

4th Wave Imaging Corp., 850 Glenneyre Street, Laguna Beach, CA 92651formerly of Chevron Petroleum Technology Co., La Habra, CA

ManuscriptonJuly 6, 20001

ABSTRACT

Non-repeatablenoise,causedby differencesin vintagesof seismicacquisitionandpro-cessing,canoftenmake comparisonandinterpretationof time-lapse3D seismicdatasetsfor reservoir monitoringmisleadingor futile. In thisGulf of Mexico casestudythemajorcausesof non-repeatablenoisein thedatasetsaredueto differencesin survey acquisitiongeometryandbinning,temporalandspatialamplitudegain,waveletbandwidthandphase,differentialstatictimeshifts,anddynamicmispositioningof imagedreflectionevents.Wemitigatetheseacquisitionandprocessingdifferencesby developingandapplyingacross-equalizationdataprocessingflow for time-lapseseismicdata.Thecross-equalizationflowconsistsof regridding the two datasetsto a commongrid, applyinga spaceand time-variantamplitudeenvelopebalance,applyinga first passof matchedfilter correctionsforglobal amplitude,bandwidth,phaseandstaticshift corrections,followed by a dynamicwarpto alignmispositionedevents,and,finally, asecondpassof constrainedspace-variantmatchedfilter operators.Dif ferencesectionsobtainedby subtractingthetwo datasetsaf-ter eachstepof the cross-equalizationprocessingflow show a progressive reductionofnon-repeatablenoiseanda simultaneousimprovementin time-lapsereservoir signal.

INTRODUCTION

Time-lapseseismicreservoir monitoringaims to usemultiple 3D seismicsurveys acquiredat differentcalendartimesto directly imagefluid movements,pressure/temperaturefrontsor

1This paperwasfirst presentedat the 68th Annual InternationalMeetingof the Societyof ExplorationGeophysicistsin New Orleans,1998.

Cross-equalization case study 2 Rickett & Lumley

othereffectsof productionin thesubsurface.Early casestudies(Pullin et al., 1987;GreavesandFulp, 1987;Johnstonet al., 1992;Lumley, 1995;Eastwood et al., 1996)describedtheseismicmonitoringof fieldsundergoingthermallyenhancedrecoveryprocessessuchassteamor fire-flood.Laboratoryexperiments(WangandNur, 1989)haveshown temperaturechangesin reservoir conditionsmayinduceextremelylargevelocitychangesof up to 40%.

Fields undergoing enhancedrecovery representonly a small percentageof producingfields. However, thesuccessof thesestudies,andthepromiseof directimagesof fluid move-ments,hasled researchersto begin to monitornon-thermalreservoir productionmechanisms,which typically have a subtlerseismicsignature(Lumley et al., 1994;Sønnelandet al., 1997;Johnstonetal., 1998;Eastwoodet al., 1998;MacCleodet al., 1999).

Non-repeatability of seismicreflectionexperiments

If significantreservoir changesoccurdueto productionin the time interval betweenrepeatseismicsurveys, thenreal seismicimagedifferencesmay be seenwithin the reservoir zone.However, afterstandardprocessing,imagedifferencesareoftenvisible throughouttheentirecube,including thenon-reservoir zonewheretheearthhasnot changed.In this case,whereimagedifferencesare seenboth inside and outsidethe reservoir, it is doubtful that imagedifferencescanbemeaningfullyinterpreted.

Thereasonthattime-lapsesurvey imagesmayexhibit undesirabledifferencenoiseis dueto an inability to repeatthe exact sameseismicsurvey eachtime, or to processthe seismicdatain exactly thesamerepeatablemanner. Even in thebest-casescenarios,surveys shotatdifferent timescannever be completelyrepeatable,and thereis alwayssomelevel of non-repeatablenoise.

Somecommonreasonsfor non-repeatablenoisedue to survey-to-survey acquisitionin-cludedifferentacquisitiongeometries(grid orientation,bin sizes,offset-azimuthsetc.), dif-ferent sourcewaveformsor shootingdirections,different receiver hardwareor deploymentmethods,positioningerrorsfor sourceandreceiverlocations,includingcable-feather, differentacquisitioncrewsandequipment,andchangingnear-surfaceconditions(weather, watertable,tides,ambientnoiseetc.). Publishedcasestudiesof repeatabilityundercontrolledconditions(Beasley et al., 1997;Rennieet al., 1997;Porter-HirscheandHirsche,1998)demonstratethepotentialimpactof acquisitiondifferencesonseismicrepeatability.

Commonreasonsfor non-repeatablesurvey-to-survey dataprocessingnoiseincludediffer-entprocessingcontractorsandalgorithmimplementations,inaccuraciesin processingnaviga-tion dataanddefininggeometries,differencesin processingflows,datadependentprocessing,differencesin processingparameters,(e.g. velocity models,staticstime picks, etc.), or ad-vancesin seismicprocessingmethods.

Previously, RossandAltan (1997)warnedof possibledangerswith statisticaldata-dependentoperators,suchasdeconvolution,demultipleandscaling;however, Porter-HirscheandHirsche(1998)observedthatwhile statisticalsignal-enhancingprocesses,suchasspikingdeconvolu-tion, mayhurt repeatability, statisticalnoise-attenuationalgorithms,suchas f � x deconvolu-


tion, mayactuallyincreaseseismicrepeatability.

Cross-equalization

In principle, different generationsof 3-D seismicshould always be reprocessedfrom theprestackdatatapesin a consistentmanner. Matchingmay thenevenbedonein theprestackdomain(HarrisandHenry,1998). In caseswherelow signal-to-noiselevels in prestackdatameanmatchingis not feasible,comparisonscanbe madefor quality control (QC) purposes.However, in many cases,accessingraw, prestackdatacanbe prohibitively expensive, espe-cially if dealingwith legacy datasets,wheresomeof theoriginal datatapesmaybemissing,corruptedor otherwiseirretrievable. In othercases,beforereprocessingprestackdata,it isoften worth doing a preliminarystudyon easilyavailablepoststackdata,to estimatetime-lapsesignal-to-noiseandidentify potentialproblemareasof low-repeatability. Both of thesescenariosrequirecarefulcross-equalizationof poststackdata.

Cross-equalization(Rossetal., 1996)is generictermgivento statisticalprocessesthatre-movesystematicdifferencesbetweensurveys thatarepresumablyduenon-repeatableseismicacquisitionor processing.Sincereservoir monitoringis a relatively new technologyandin-dividual casestudiestendto vary significantly, theindustryhasnot developeda standard4-Dcross-equalizationprocessingflow. However, cross-equalizationof poststackseismicdatasetstypically includesthefollowing genericelements:

1. Survey resamplingto a commongrid, includingspatialandtemporalre-registrationtocompensatefor differentgeometriesandacquisitionparameters.

2. Globalbandwidthandphaseequalizationto compensatefor differentsourcewavelets,for example.

3. Amplitudebalancingto scalethedatato thesameamplitude(or energy) level.

Furthermore,it is oftennecessarytoapplyadditionalprocesses:for example,to balancespatialfrequency or dip-content,or performresidualmigrationto co-locatereflectors,imagedwithdifferentmigrationvelocities. Publishedcross-equalizationcasestudiesincludeBurkhartetal. (1997),RossandAltan (1997),Rickett andLumley (1998),Eastwoodet al. (1998,1999),andHarrisandHenry(1998).

Fromabusinesspointof view, atime-lapseprojectis only successfulif thevalueof thead-ditional informationit providesoutweighstheprojectcosts.Underthatcriteria,Lumley et al.(1997)classifiedthefactorsthatdeterminethesuccessof a time-lapsestudyinto two groups:reservoir parametersandseismicparameters.Thereservoir parametersconsistmainly of therockphysicspropertiesthateffecttheseismicsignatureof production.Theseismicparametersconsistof factorssuchseismicrepeatability. Inevitably thereis a trade-off betweenthe twosetsof parameters;however, if eithertherockphysicsor seismicqualityareunfavorable,thenthechancesof successareheavily reduced.


In thispaper, weillustratetheproblemsthatneedto beaddressedby thecross-equalizationprocessfor a field in the Gulf of Mexico. We then review elementsof a typical cross-equalizationprocessingflow, andintroduce‘warping’ asa residualmigrationoperatorto co-locatereflectorsimagedat differentpositions.Finally, we describethecross-equalizationoftheGulf of Mexico example.

GULF OF MEXICO EXAMPLE

Figure1 shows panelsfrom two very differentmarinedatasetsshotover the samereservoirneara saltdiapir in theGulf of Mexico. We would like to imageproductionrelatedchangesby studyingthedifferencesbetweenthetwo surveys.

Thetargetreservoirsarelocatedin thefaultblockdippingto theleft ataboutCDPX1� 2500m,

andabout2.8 s traveltime depth. ClassicGulf of Mexico rock physicsmeansthe reservoirparametersarefavorablefor a successfultime-lapseseismicreservoir monitoringstudy, andwould scorewell in a technicalrisk assessment(Lumley et al., 1997).However, questionsofseismicrepeatabilitycastdoubtover theusefulnessof thetwo vintage3-D datasetsfor asucha reservoir monitoringstudy.

Figure 1(a-b) shows a very early 3-D datasetshot in 1979. The survey shown in Fig-ure 1(c-d) wasshot in 1991. Improvementsin 3-D acquisitionandprocessingtechnologiesover the12yearsbetweensurveyshaveresultedin thelargedifferencesbetweenthepoststackcubes.Accessto theprestackdata-tapesfor the1979survey wasprohibitively expensive,andsonon-productionrelateddifferencesbetweenthesesurveys have to beattenuatedby cross-equalizationin thepoststackdomain.

Surveygeometryand binning

Thefirst stepin thecross-equalizationprocessingflow is to identify physicalsourcesof non-productiondifferences.Althoughsimilar featuresarevisible in thetwo surveysshown in Fig-ure1,directcomparisonsbetweenthemarenotpossible,sincethetwo surveysareondifferentgrids. The two surveys wereneithershotnor processedwith seismicreservoir monitoringinmind,andconsequentlythey have significantlydifferentgeometries.

The left sideof Figure2 shows the arealcoverageof the two Gulf of Mexico datasets,andtheir 34

�differencein azimuth.The1979survey coversa largerarea,but thesignificant

overlapbetweensurveys includestheproducingintervals.

Nominalbin parametersalsodiffer significantly, asillustratedin theright sideof Figure2,whichshowsaclose-upof thebin-centersfor partof thesurvey. The1979survey wasshotandprocessedwith bin-spacingsof 25 m (inline), and75 m (crossline);whereasthe1991surveywasprocessedwith 12.5m bin spacingsin boththeinline andcrosslinedirections.

The 1991survey hasa muchhigherspatialbandwidth,anda larger dip range,thanthe1979survey. Part of the differencein spatialbandwidthis a consequenceof the difference


Figure1: Two 3-D poststackvolumesover the samereservoir in the Gulf of Mexico. Thedatasetshown in panels(a-b)wasshotin 1979,thedatasetshown in panels(c-d) wasshotin1991. The time-slices(traveltime depth,2.76s) cut throughthe high amplitudefault-blockthatcontainsthereservoir intervals. before2 [NR]


Figure2: Poststackgeometries.Left panel:thespatialcoverageof the1979survey (dashed-line), and the 1991survey (solid-line). Right panel: close-upof the differentbin-spacingsbetweenthe1991(denser)and1979(lessdense)surveys. geometries[CR]

in spatialsampling. However, differencesin field arrayfiltering andsubsequentdifferencesin processingwill also effect the spatial resolutionand dip contentof the migratedimagecubes.For example,the1991cubewasmigratedwith a sophisticatedturning ray migrationalgorithmin orderto imagetheover-hangingsalt-flank,whereassteepeventswerenot imagedin the 1979survey. Unfortunately, differencesin spatialbandwidthfrom thesesourcesareverydifficult to completelyremove from thepoststackdata.

Time-varying gain

Anotherdifferencebetweentheseismicprofilesin Figure1 is thatthetwosurveyshaveslightlydifferentamplitudeversustraveltimedepthprofiles.Figure3 illustratesthispointmoreclearlyby showing theroot meansquared(RMS) amplitudeasa functionof traveltimedepthfor thetwo surveys.

We remainunsureof the exact gain processingparameters;however, both datasetsap-pearto containconsistentamplitudecharacterthatwouldnothavesurvivedif ashort-windowautomaticgaincontrol(AGC) hadbeenapplied.Althoughit maybeimpossibleto fully com-pensatefor differencesin time-varying gain, part of the cross-equalizationprocessingflowshouldattemptto statisticallycorrectfor systematicallydifferenttime-varyinggainfunctions.

Waveletdiffer ences:bandwidth, phaseand statics

In partdueto their differentacquisitionparameters,the two surveys have two very differenteffectivewavelets.Figure4 shows amplitudespectrafrom thetwo datasets.The1991survey


Figure3: RMS energy asa functionof depth. No verticalsmoothingwasapplied,ratherspatialaveragingovertheentireseismicvolumes.Thesolidline correspondsto the 1979 surveyand the dashedline to the 1991sur-vey. gain [NR]


clearlycontainsinformationovera greatertemporalbandwidththanthe1979survey.

A secondwaveletproblemcomesfrom thephasedifferences.Typical seismicprocessingattemptsto zero-phaseseismicdatabeforestacking;unfortunately, however, thisprocessdoesnot work perfectly, and inevitably thereareresidualphasedifferencesin waveletsbetweensurveys.

Figure 4: Amplitude spectrabeforematched-filtering. The solid linecorrespondsto the 1979 survey andthe dashedline to the 1991 survey.fspecbefore[NR]

The effective wavelet of a survey may alsovary both spatiallyandtemporallyover theextent of a survey. For example,in the upperpart of a section,NMO stretchmay impactthe seismicwavelet significantly. As a direct consequence,the choiceof mute parametersmayalter theeffective waveletof thepoststackdata.In thelower partof thesection,seismicattenuationmaylower theeffectivefrequency of thewavelet.Sincetheindividualwaveletsofthetwo surveysvarybothspatiallyandtemporallyin anunknown manner, theresidualwaveletdifferencewill alsovaryspatiallyandtemporally.

Differ ential statics

Staticdifferencesbetweensurveysmayarisefrom a numberof differentcauses:for example,datumingproblems,tides,or staticsolutions.Figure5 illustratesdifferentialstaticspresentintheGulf of Mexico example,andshows thesignificantspatialvariability over thearealextentof thetwo surveys. Thestaticshiftsshown in Figure5 stronglycorrelatewith thesubsurfacegeology, especiallynearthe flanksof the salt. Thesekinds of staticshifts arelikely to be aresultof migration/imagingdifferencesratherthantheresultof near-surfacephenomena.

Migration imaging mispositioning

Thefinal importantdifferencebetweensurveys is causedby incorrectpositioningby themi-grationprocess.Ideally, regardlessof prestackgeometry, a singlemigratedreflectionshouldappearat the samespatiallocationandtraveltimedepthin the two surveys. However, if thetwo migrationswereperformedwith differentvelocity functions,thena single imagepointcanbe locatedat different locationsin the two surveys. This point is clearly illustratedin


Figure5: Areal distribution of differ-entialstaticsshiftsderivedfrom Gulfof Mexico example. The areawithhighfrequency noiseis thesaltdiapir.statics [NR]

Figure6: thesamefault is pickedin boththe1979survey (a) andthe1991survey (b). Therearesignificantlateralandtemporalshiftsbetweenfault interpretations.

Figure 6: Illustration of spatialmispositioningof eventsbetweensurveys due to differentvelocity functionsand/ormigration algorithms. Target interval and interpretedfault in (a)1979survey and(b) 1991survey. coloc [NR]

MATCHED FILTERING

Matchedfiltering, or shapingfiltering (Claerbout,1976; Robinsonand Treitel, 1980), cansimultaneouslyestimateacorrectionfor static,phaseandspectraldifferencesbetweensurveys.We designa convolutionalshapingfilter, A, to match“operatordesignwindows”, d1 andd2,


by minimizing thenormof theresidualvector,

r � Ad1� d2 (1)

with respectto the operator, A. In this paper, we parameterizeA in the time domain,andminimize the residualin the usual leastsquares(L2) sensewhich amountsto solving thesystemof normalequations:

D1T D1a � D1

T d2. (2)

Following RobinsonandTreitel’s (1980)notation,A is the matrix representingconvolutionwith filter, a. Similarly, D1 andD2 arematricesrepresentingconvolution with vectorsd1 andd2 respectively.

Oncewe have calculatedthefilter coefficients,we canapply the operator, A, to a largerareaof survey 1, possiblyincludingtheareaof interest.

Fromequation(2), it is clearthatfilter coefficientsof a aregivenby thecross-correlationofd1 with d2, filteredby theinverseof theautocorrelationof d1. Spectralcomponentsof a maybecomeunstableif d1 containsspectralzeros.However, in practicethis is not importantsincewe areinterestedin the product,Ad1, which is constrainedby equation(1), andso remainswell behaved.

The degreeof spectralmatching,essentiallythe numberof degreesof freedom,is con-trolled by the length of the time domainoperator. By working with a short operatorof asimilar lengthto thetwo waveletsbeingmatched,theoperatorcanprovidethe“right amount”of spectralshaping:a closeenoughspectralandphasematchto compensatefor differencesin waveletsanddifferential staticsbetweenthe two surveys, while avoiding an over-matchthatcanzerooutdifferencesin thedatasetscausedby petrophysicalchangesduringreservoirproduction.

Equation(1) illustratesthat matchedfiltering is not a symmetricprocess:the choiceofwhich survey to matchto theotherwill affect theoutcomeof thecross-equalizationprocess.Typically thesurvey with higherbandwidthis matchedto thesurvey with thelowerbandwidth;trying to increasethebandwidthof asurvey will tendto increasenoiselevels,andmake time-lapseinterpretationmoredifficult.

AMPLITUDE BALANCING

WhetheranAGCwindow or a morecarefulgeometricspreadingcorrectionhasbeenapplied,two generationsof seismicsurvey will, in general,have differenttime-varyinggainfunctionsappliedto them.If notcompensatedfor correctly, thismayleadto asystematicleakageof non-reservoir eventsinto thedifferencesection.Althoughanamplitudecorrectionmayneedto betimeandspace-varying,it shouldbeconstrainedto varyveryslowly, soit is not influencedbychangesin thereservoir zone,or effect therelativeamplitudenatureof thedata.

Thesimplestapproachto amplitudebalancingis to scalethedatabasedontheRMSenergyin the two surveys. However, this assumesthat theenergy presentin thenoisefieldsarethesamein bothdatasets,or of muchsmallermagnitudethanthesignalenergy.


As anillustrationwe canconsidertwo normalizeddatasets,d1 andd2, to consistof somesharedsignal,s, anduncorrelated“noise” components,n1 andn2, which includethereservoirdifferenceanomalyweseek:

d1� 1

�s � n1

� (s � n1) , and (3)

d2� 1

�s � n2

� (s � n2) . (4)

In orderto rescalethesignalsto thesamelevel,weneedto applyascalefactor, � to d1, where

� ��s � n1

��s � n2

� , (5)

or againassumingthenoisefieldsareweaklycorrelatedwith thegeologicalsignal

��s2 � n1

2

s2 � n22�

1 � 1s21

1 � 1s22

(6)

wheres1 ands2 arethe signal-to-noiselevels in the two datasets.If the signal-to-noiselev-els in eitherdatasetvariesspatiallyor temporally, � may alsovary spatiallyandtemporally.However, in practice,estimatingsucha function is difficult, andsowe assumes1 ands2 areconstantthroughoutthe two datasets.For high (s1 � 1 and s2 � 1), or similar (s1 � s2),signal-to-noiselevels � reducesto unity, andtheequalenergy conditionis valid.

For theearlyamplitudebalancingstepsin thefield examplein this paper, we appliedtheequalenergy condition(� � 1), sincethis is areasonableassumptionfor many cases,anddoesnotrequireindependentestimatesof thesignal-to-noiseratio. However, for thefinal differenceimage,wescannedthroughseveralvaluesof � , andchosethevaluewhich containedtheleastcoherentenergy in thedifferencesection.For this example,thedifferencesectioncontainedtheleastcoherencewith a valueof � � 1, indicatingtheequalenergy conditionwasvalid.

RESIDUAL MIGRA TION AND WARPING

RossandAltan (1997)notedthatdifferentNMO andmigrationvelocity functionsmaycauseboththetraveltimedepthandspatialpositioningof imagedreflectorstodifferbetweensurveys,causingartifactsto appearin differencesections.Moreover, thedegreeof this mispositioningwill vary throughoutthe3-D seismicvolume. Althoughstatictime correctionsmayprovidea partial solution, static correctionswill not co-locatereflectorsimagedat different lateralpositions,nor will they accountfor dynamic(asopposedto static)time-shiftsthat vary asafunctionof traveltimedepth.

Evensmall shifts (lessthana sampleinterval in magnitude)cancauseenoughmisalign-mentthat falseeventsappearin thedifferencesections.Without accessto theprestackdata,therearea limited numberof wayssuchdifferencescanbecorrected.


Residualpoststackmigration(Rothmanet al., 1985),or velocity continuation(Claerbout,1987;Fomel,1997),provideoperatorsthatmapbetweendifferentmigrationvelocities.How-ever, withoutdetailedknowledgeof thevelocityfieldsusedfor NMO and/ormigration,andincaseswherethemigrationalgorithmsdiffer significantlybetweensurveys, it will bedifficultto determinethecorrectresidualmigrationoperatorto applyapriori. Insteadany operatorwillhave to beestimatedfrom thedata(Eastwoodet al., 1999).

As an alternative to standardresidualmigration, we usea warping operator(Wolberg,1990)to correctfor kinematicdifferencesbetweensurveys. We estimatethechangeof coor-dinatesthatmapsonesurvey ontoanother. Applying thischangeof coordinates,warpsoneofthedatasets,andthesetof vectorsthatdescribesthechangeof coordinatesis calledthe‘warpfunction’.

Figure7 illustratestheprocedurefor calculatingthewarpfunction.Weextractsmallcubesof dataatnodepointsthroughouttheseismicvolume,andcalculatelocal3-D cross-correlationfunctionsbetweensurveys. Pickingthemaximaof thesefunctionsproducesa sparsecubeof3-D shift vectorsthatdescribesthechangeof coordinatesnecessaryto maponesurvey ontotheother. Beforeapplyingthewarp,wemedian-filter, thensmooth,theninterpolatethewarp-functionto fill thevolume. To apply thewarp,we simply resampletheoneof thedatasetsinthenew, warped,coordinatesystem.Becauseof thesymmetryimplied by cross-correlations,the warp-functionthat mapssurvey A to survey B is just the negative of that which mapssurvey B to survey A.

GrubbandTura (1997)useda similar algorithmto estimateuncertaintyin AVO migra-tion/inversionresults. They migratedthe samedatasetmany times with slightly differentvelocity fields, andthenco-locatedreflectorswith a warpingalgorithm. This allowed themto separatethe kinematicandamplitudeeffectsof the differentmigrationvelocities. Morerecently, GerhardtandFilpo (1999)describedanapplicationof warpingto fine-tuneadaptivemultiplesubtraction.

Warping asresidualmigration

Warpingprovidesa linear mappingbetweendifferentmigrationvelocitiesthat is kinemati-cally equivalent to velocity continuationfor plane-wave events. Fomel (1997)showed thisdirectly from the zero-offset velocity continuationequation,but it is apparentintuitively ifyouconsidertheeffectmap-migration(Claerbout,1993)hasonaplanardippingevent. In thiscontext, warpingbearsthe samerelationshipto residualmigrationas‘map-migration’bearsto conventionalzero-offset migration. Map-migrationand warping are both point-to-pointoperators;whereasconventionalzero-offsetmigrationandresidualmigrationarebasedon aconvolutionalmodel.Warping,therefore,canbethoughtof as‘residualmap-migration’.

The relationshipbetweenwarp-functionandvelocity changecanbe derived from kine-maticmap-migrationequations.The following threeequationsdescribemigrationof a zero-offsetplanareventat (xzo, tzo) dippingwith slowness,pzo, with half velocity :

t � tzo 1 � 2 p2zo, (7)


x lag

time lag

(c)

x lag

time lag

(b)

cmp_xtraveltim

e depth

(d)

traveltime depth

cmp_x

traveltime depth

(a)

Figure 7: Schematiccartoonof two-dimensionalwarping process: (a) Extract coincidentpatchesfrom seismicdatacubes.(b) Calculatemulti-dimensionalcross-correlationfunctions.(c) Pick vectorassociatedwith maximaof cross-correlationfunction. (d) Repeatprocesstobuild warpfunctionthatdescribesrelativeshift betweencoordinatesystems. warpflow [NR]


x � xzo� 2tzo pzo, and (8)

p � pzo

1 � 2 pzo.pzo

. (9)

Dif ferentiatingwith respectto , andeliminatingthezero-offset variablesleadsto theequa-tions thatdescriberesidualmap-migrationalongFomel’s velocity rays,providing a link be-tweenthewarp-functionandtheresidualvelocitycorrection:

x � � 2 t p

(10)

t � tp2 . (11)

Applying analgorithmbasedon map-migrationmayseemquestionablewhenwe areconsid-eringanamplitude-sensitive issuesuchasreservoir monitoring.Indeed,in somesituations,itmaybebetterto try andestimatea full residualmigrationoperatorfrom thedatathanapplyapurelykinematicwarpcorrection.However, thedangeris thatanincorrectresidualmigrationoperatormaydefocuscorrectlyimagedevents,smearamplitudes,andchangethecharacterofreflectors.In contrast,warpingwill preservereflectoramplitudes.Thereis nowayof knowingwhetheramplitudechangesfollowing residualmigrationarecorrector incorrect,andso, inthefaceof uncertainty, preservingtheoriginal amplitudesmaybethebestcourseof action:ifthereis doubtabouttheresidualvelocity function,warpingwill actuallybepreferableto fullconvolutionalresidualmigration.

Separatingkinematicsand dynamics

In somefields,notablythoseundersteam-flood,reservoir changeshave beenshown to havelargekinematiceffectsontheseismicresponseof thereservoir (Lumley, 1995).If the‘patches’(3-D designwindows) usedto calculatethe cross-correlationsare small enough,the shift-functionsmaythemselvescontainhigh-frequency informationthatprovidesinformationaboutfluid changes(Pepperet al., 1997;Eastwoodet al., 1998).Warping,therefore,mayprovide away to separatethedynamicandthekinematiceffectsof production.

Conversely, if thecross-correlationpatchesarelargeenough,thentheshift-functionswillbe smooth,andsmall localizedchangesdueto fluid productionwill not influencethe warp.This is the casein thedataexamplepresentedhere,sincedespikingandsmoothingkept theshift-functionsconservatively smooth. Despitethis, warpingled to a significantdecreaseintheoverall amplitudeof thedifferencesection,andmadedirectcomparisonsbetweensurveyssimplerandmoreinsightful.

PROCESSINGFLOW

We demonstratethe applicationof thesepoststackcross-equalizationoperatorson the Gulfof Mexico datasetintroducedabove. The processingflow is robust, and addressesnon-stationaritythroughtheuseof spatially-variableoperators.


After eachstep,asa quality control measure,we displaydifferenceimages. Ideally wewould like differenceimagesto containlow amplituderandomnoiseuncorrelatedwith geol-ogy in areasnotaffectedby production,while producingintervalsshouldcontainmorecoher-entsignals.

Althoughquality control from differenceimagesis somewhatqualitative, it is difficult todraw statisticallysignificantinferencesfrom simple numericalassessmentsof repeatabilitysuchascorrelationcoefficient, or differenceenergy. Harris andHenry (1998)suggestusingspectralcoherencefunctions(White,1973)to providemorethancompleteassessmentof dataquality includingbandwidthandfrequency dependentsignal-to-noiseestimates.

Spatial realignment

To comparethe two surveys, the first stepis to realignthemboth onto a commongrid. Inthis case,we applieda spatialanti-aliasingfilter to time-slicesof the 1991 dataset,beforeregriddingto 1979grid. Theanti-aliasingfilter wasa low-passradialHanningwindow withan elliptical shapein the (kx ,ky) domaindue to the anisotropicsamplingof the 1979grid.Sincethe 1991survey wasspatially-sampledso muchmore finely than the 1979survey, alinearinterpolationalgorithmwasusedfor theregridding,withoutcausingartifacts.Theanti-aliasingfilter hadthesecondaryeffect of partiallyequalizingthedip contentbetweenthetwodatasets,sincethesteepdips in the1991survey would bespatiallyaliasedif sampledon the1979grid.

The differencebetweenthe two surveys after spatial realignmentis displayedin Fig-ure 8(a). Large amplitudedifferencesarevisible throughoutthe section,which areclearlynot productionrelated.Figure8(a) illustrateswhy cross-equalizationis neededfor this kindof study.

Global (stationary) corrections

As a first-ordercorrectionfor thewaveletdifferencesdescribedabove, we designeda singleleast-squarestime-domainmatched-filteroperatorto matchtheeffectivewaveletof thehigherresolution1991survey to thatof the1979survey.

For the single,240 ms long, operator, we choosea designwindow from 0.5 s to 2.0 sdepth(significantlyabove the reservoir zones)that coveredthe spatialextent of the survey,but excludedthesaltregion sinceit containsno signal.Closeinspectionof thematchedfilterrevealeda phase-rotationof approximately90

�, anda residualstaticcorrectionof aboutone

time sample(6 ms). Figure 9 shows the amplitudespectraof the two datasetscalculatedover theentirevolumeafterglobalmatchedfiltering. Dif ferencesbetweenspectraaresmall,indicatingthedesignwindow wasappropriate.Beforeandafter theglobalmatchedfilter, weappliedasingleamplitudescalefactorto equalizeRMS energy betweensurveys.

Figure8(b) shows a sectionfrom thedifferencecubeafterglobalmatchedfiltering. Theoverall amplitudeof the sectionis decreasedsignificantly comparedwith Figure 8(a). As


Figure8: Dif ferencesbetween1991survey and1979survey during cross-equalization:(a)initial differencesectionafter realignmentto commongrid, (b) after matchedfiltering withsingle(global)filter, (c) differencesectionafterwarping,and(d) final differencesectionafternon-stationarymatchedfiltering andtime-variableenergy balancing.Cross-sectionshown isthesameasFigure1(a). difference[NR]


Figure 9: Amplitude spectraaftermatchedfiltering with a singleglobalfilter. The solid line correspondstothe 1979survey and the dashedlineto the1991survey. fspecafter[NR]

expectedthedifferenceamplitudeis especiallyreducedin theflat layersin theupper2 secondsof the survey, which coincideswith the operatordesignwindow. Large residualdifferencesstill remainin theeventsdippingup againstthesalt,andthroughoutthe lower portionof thesection.

Warping

Thenext stepof thecross-equalizationflow wasto addressinconsistentimagingof reflectorswith the warping algorithm describedabove. We calculatedcross-correlationfunctions inoverlappingwindows measuring900 ms � 1500 m � 225 m. In the inline direction, weobservedreliableshiftsof up to 100m. However, thehorizontalcomponentsof thewarparelesswell constrainedthanthetime shifts,andin thecrosslinedirection,wheretherewaslessdip andthegrid spacingwasmuchgreater(75m vs25m), thesmallshiftsweestimatedwerenot consistentfrom window to window. Therefore,we restrictedthecrosslinecomponentsofthewarpvectorsto zero.

We warpedthe 1979 survey to matchthe 1991 survey, sincereflectorpositionsin the1991survey aremoreaccurate.Figure10 shows a two-dimensionalslice throughthe three-dimensionalwarp-functionderivedto mapthe1979survey onto the1991survey. As we ex-pect,verylittle warpingis neededin theuppertwo secondsof thesurvey. However, significantshiftsarerequiredto co-locatethedippingfault blockcontainingtheproducinginterval.

Figure8(c)showsthedifferencesectionafterthewarping.Warpinghasreducedtheenergythroughoutthedifferencesection,especiallyin areaswith largedip bothcloseto thesaltandlower in thesection.

Local (non-stationary) corrections

A single,global filter will sufficiently matchtwo datasetsif bandwidth,phaseandstaticdif-ferencesarestationaryover the entiresurvey volumes. In this context, stationarityis takento meanthat thestatisticaldifferencesbetweensurveys arespatiallyinvariant. Similarly, the


Figure 10: Cross-sectionthrough3-D warp function applied to the1979dataset,matchingit to the1991dataset. Vector lengths have beenmultiplied by a factor of two forclarity. This is correspondsto thesame cross-sectionas displayed inFigure8. warpfn [NR]

singleamplitudescalefactorassumesthesameratio of amplitudesbetweensurveys over theentiresurvey area,andfor theentire6 secondsof recordeddata.

As a secondmatched-filterstep,to addressnon-stationarydifferencesbetweensurveys,we designeda setof 140ms long spatially-variableresidualmatchedfilters. Separatefilterswerethendesignedfor eachtrace,by consideringa designwindow from 0.5 s to 2.0s depth(above the reservoir zones)and threetraceswide in the inline direction. Figure 11 showsexampleoperators.Sincefirst-orderstaticandphasedifferenceswereremovedby theglobaloperator, the residualoperatorsareessentiallyimpulsefunctions,with small residualphaseandstaticshifts. Thefilters becomenoisy in thesaltat aboutCDPX1

� 4000m, sincethereareno reflectorspresent.Justbeforethesaltyou canseeresidualnon-stationarystaticshiftsassociatedwith imagingdifferencestherewastoohighfrequency to beremovedby thesmoothwarpingoperator.

Figure11: Matchedfilters from the sameinline sectionasshown in Figure1(a). The saltbeginsat approximatelyCDPX1

� 4000m, coincidingwith thenoisyfilters. filters [NR]


After non-stationarymatchedfiltering, weappliedannon-stationaryenergy balancingcor-rection. The spatialextent of the energy balancingwindows was the sameas that of thematchedfilter designwindows,andtheir temporalextentwas1.2s.

Figure8(d)showsasectionfrom thedifferencecubeafternon-stationarymatchedfilteringandenergy balancing.Figure8(d) appearsto bea significantimprovementover Figure8(c).Specifically, differenceenergy is attenuatedin the layersdipping up againstthe salt dome,andin eventsbetween2.0 s and3.0 s. Onedifferenceevent,however, remainsanomalouslystrongbothup-dipanddown-dip in thereservoir interval. It is unclearwhetherthis is genuineproductiondifference(pressurechange,for example),or anartifactdueto non-repeatability.Unfortunately, it is difficult to resolve this ambiguitywith the poststackseismicdataalone.The cross-equalizationprocessis finished: an interpreterwith full knowledgeof the field’sproductionhistoryneedsto inspecttheremainingdifferenceanomalies.

The order of cross-equalizationelementscan be important. We comparedresultswiththewarpingoperatorappliedbeforeandafterthenon-stationarymatchedfiltering corrections,andfoundthemto besimilar. However, theflow describedhere(with warpingbeforematchedfiltering) did giveslightly betterresults.

DISCUSSION

Figure12 shows the two datasetsaftercross-equalization.The time-slicescorrespondto thedepthof the producingintervals,wherewe would expectthereto be productionrelateddif-ferences.Figure13 shows the relative differenceenergy in differentareasof thecube,afterdifferentstagesin the cross-equalizationprocessingflow. Zero decibelscorrespondsto thesameenergy in thedifferenceimageastheaverageof thetwo inputsections.If thereis nocor-relationbetweenthetwo surveys, theenergy in thedifferencesectionincreasesto about3 dB.Negativevaluescorrespondto reducedenergy in thedifferenceimage.With carefullyplannedandprocessedtime-lapsesurveys, it is possibleto obtaindifferenceenergy levels lower than� 10dB’s. However, whenvintagesurveysareinvolved,anoiselevelbetween� 5 and � 10dB(Eastwoodet al., 1998)is morecommon,asin theshallow sectionof thisexample.

While thetwosurveysinitially appearto becompletelynon-repeatable,aftercross-equalizationthe largestdifferencesarealmostentirely restrictedto the reservoir interval andbelow, sug-gestinga successfulcross-equalization.However, it is importantto realizethe limitationsofthepoststackcross-equalizationprocess:someacquisitionandprocessingdifferenceswill beimpossibleto remove at thepoststackstage,anddifferencesdeepin thesectionmaystill bedueto non-productioncauses.

To fully assesswhetherthetwo datasetscontainusefultime-lapseinformation,acompleteinterpretationof theproducingintervals is neededin conjunctionwith a studyof productionhistories.


Figure12: After cross-equalization:panelsfrom the1979survey (a-b),panelsfrom the1991survey (c-d). Time-slicescorrespondto a traveltimedepthof 2.76s. after [NR]


Deep, reservoir

Deep, non-reservoir

Shallow & dipping

Shallow & flat

-6-2

42

0-4

Aftercolocation

Afterglobal

equalization

Afterwarping

Afterlocal

equalization

diffe

renc

e en

ergy

(dB

)

Figure13: Relativedifferenceenergy in areasof thesurvey afterdifferentstagesof thecross-equalizationprocess. rms2 [NR]

CONCLUSIONS

Wehavepresentedaprocessingflow suitablefor cross-equalizingtime-lapsemigratedseismicdatacubesfor reservoir monitoring,andtestedtheflow on two seismicsurveys from theGulfof Mexico. After cross-equalization,amplitudesmaybecomparedmoredirectly making4-Dinterpretationeasier. Theflow consistedof four basicelements,eachof which hasa physicalbasis:spatialrealignment,matched-filtering,amplitudebalancing,anda new warpingoper-ator. Thewarpingoperatoractsasa data-dependentresidualmigrationalgorithm,correctingfor spatialmispositioningby differentmigrationvelocities.

ACKNOWLEDGEMENTS

The authorswould like to thankHarry Martin, Greg Surveyer, andDamienLynch for theirhelpwith this project,andChevron for providing thedata.We would alsolike to thankKenCraft, JohnEastwoodandChristopherRossfor their constructive feedbackduringthereviewprocess.


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