WaveInversionTechnology, ReportNo. 4, pages87-99

Aspectsof the unified approachtheory to seismicimaging: theory, implementation,and application

T. Hertweck, M. RiedeandA. Goertz1

keywords: seismicimaging, true-amplitude, migration,demigration

ABSTRACT

This contribution presentstheoretical and practical aspectsof the challenge, how tohandle2D and3D datain a true-amplitude, efficient,andtime-savingwayfor a later-ally andvertically inhomogeneousisotropicearthmodel.Theunderlyingbasicimag-ing processes— Kirchhoff-typetrue-amplitudemigration anddemigration — are ex-plainedand an implementationof the unifiedapproach theory is shown. Practicalaspectssuch as aliasing, tapering, the migration and demigration apertures or thecreationof Green'sFunctionTables(GFT)areconsideredand,finally, somesyntheticdatasetexamplesareshown.

INTRODUCTION

In the past two decades3D reflectionseismicshasbecomea powerful tool in theworld of seismicexploration. This developmentis drivenby theaccuracy of imagesof the subsurfaceobtainedby 3D seismicscomparedwith the imagesof 2D seismiclines,asonly 3D seismicscanhandleout-of-planereflectionscorrectly. Many of thebasicconceptsthat have beendevelopedfor 2D seismicimagingare still valid for3D imaging. However, 3D imagingpresentsseveral new problemsthat are causedby higher dimensionality– prestackseismicdata is definedin a 5D space(time,two componentsof the sourceposition, two componentsof the receiver position),comparedwith the 3D space(time, onesourcecoordinate,onereceiver coordinate)of 2D prestackdata. An obvious problemis the amountof datathat is recordedinthefield andthathasto beprocessed.Thus,an importantaspectis theright strategyfor reducingthe time andresourceswhich arenecessaryto obtainan imageof thesubsurfacefrom theoriginal largeamountof datawithoutcompromisingtheaccuracyof theresults.

During the last years,processingmethodswhich do not only accountfor the1email: Thomas.Hertweck@gpi.uni-karlsruhe.de

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kinematic(relatedto traveltimes)but for thedynamic(relatedto amplitudes)aspectsaswell have becomemoreandmore important. Suchmethodsareable to provideimagesthat can be usedto determinegeologicaland quantitative physicalproper-ties of the subsurfacewhich are useful for reservoir characterizationby meansofamplitude-variation-with-offset (AVO) and amplitude-versus-angle(AVA) analyses.An AVO/AVA analysisaimsto point out amplitudeanomalieswhich may be causedby hydrocarbonreservoirs. Thequantitativephysicallywell-definedamplitudevaluesof imagesarereferredto as' trueamplitudes'.A trueamplitudeis definedasnothingmore than the amplitudeof a recordedreflection compensatedby its geometricalspreadingfactor. Moreover, theconstructionof a true-amplitudereflectionimpliesnotonly a scalingof the consideredreflectionamplitudewith the geometricalspreadingfactorbut alsothereconstructionof theanalyticsourcepulse.

To achievetheabovementionedobjectives,theunifiedapproachtheoryby (Hubralet al., 1996)(basicconcepts)and(Tygel et al., 1996)(theory) is presentedwhich isbasedon ray theory and simple geometricalconcepts. This approachis composedof firstly a weightedKirchhoff-type diffractionstackintegral to transformseismicre-flection datafrom the time domaininto the depthdomain,andsecondlya weightedKirchhoff-type isochrone-stackintegral to transformthemigratedseismicimagefromthedepthdomainbackto thetime domain.By cascadingor chainingbothprocesses,we areableto solvevariouskindsof seismicreflectionimagingproblems,e.g.migra-tion to zerooffset (MZO) or remigration(RM). Due to the fact thatall methodspre-sentedherearetarget-oriented,efficient,andhighly parallelizable,they canbeusedtoperform4D seismicmonitoringby meansof modellingby demigration.Thatmeanstime-dependentvariationsof subsurfacepropertiescan be investigated,e.g. the ex-ploitationof anoil field andtheeffecton recordedseismograms.

THEORY

The taskof depthmigration is the transformationof seismicdataacquiredwith anarbitraryshot-receiver configurationat a certainmeasurementsurfacefrom the timeto the depth domain. If this transformationis performedin such a way that thedynamicaswell asthekinematicpartsaretreatedcorrectly, it is calledtrue-amplitudemigration. A true-amplitudetrace is a seismictrace where the amplitudeis freeof geometricalspreadingeffects, i.e., where the amplitude is correctedby thegeometricalspreadingfactor. Therefore,the amplitudeof a true-amplitudemigratedreflectorimageis a measureof the angle-dependentreflectivity andcanbe usedtodeterminegeologicaland(quantitative) physicalpropertiesof themediaby meansofan AVO/AVA analysis. It is importantto recognizethat a true-amplitudemigrationis not able to provide the true reflectioncoefficient, althoughthe nameimplies it,becausesomefactors– includingfor examplesourceandreceivereffects(e.g.strength

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and coupling, geophonesensitivity), transmission,attenuation,thin-layer effects,scattering,andanisotropy – mayhave falsifiedtheamplitudes.The lattermentionedeffects,however, areusually small. Furthermore,we aremainly interestedin rela-tivechangesof themediumparametersataninterfacenotatglobalquantitativevalues.

Demigrationis the (asymptotic)mathematicalinverseto migration and undoeswhat the migrationprocesshasdoneto the original seismicsection,i.e., the aim ofdemigrationis thereconstructionof a seismictime section(primaryreflectioneventsonly) from a depthmigratedsection. This transformationcan also be performedin considerationof geometricalspreadingeffects which will then also be calledtrue-amplitude.

The seismicrecordis expectedto consistof analytic tracesthat containthe se-lectedanalyticelementaryreflectionevents  ¢¡�£¤f¥i¦i§ where £¤ is aconfigurationparame-tercharacterizingtheshot-receivergeometry(Schleicheretal., 1993).Analytic tracesof reflectioneventsarenecessaryto correctlyaccountfor thephaseshift of primaryre-flections;they areconstructedfrom therealtracesby addingtheirHilbert transformasanimaginarypart.Oneprimaryreflectionevent  ¢¡�£¤f¥�¦i§ canbeexpressedin zero-orderrayapproximationas >¡ £¤f¥�¦i§j¨  ª©«¡ £¤�§j¬ ¡ ¦!­A®n¯�§j¨O°²±n³ ´µ¬ ¡ ¦!­A®n¯�§T¥ (1)

where¬ ¡ ¦i§ representstheanalyticpoint-sourcewaveletwhichhasto bereproducible

for all sourcepoints ¶ . Thefunction®n¯

providesthetraveltimealongtheray ¶¸· ¯¹°where· ¯ is anactualreflectionpointand

°is thereceiver locatedat thesurface.The

parameter°²±

is the plane-wave reflectioncoefficient at the reflectionpoint, ³ is thetotal lossin amplitudedueto e.g.transmissionacrossall interfacesalongtheray, and´

is thenormalizedgeometricalspreadingfactor.

True-amplitude migration

The true-amplitudemigrationprocedureis performedby a weightedKirchhoff-typediffraction stack. The kinematicpart of the transformationcaneasilybe performedin the following way: assumea denserectangulargrid of points · in the depthdo-main in which we want to constructthedepth-migratedreflectorimage(target zone)from thegiventime section.Moreover, let all depthpoints · on thegrid be treatedlike diffractionpoints(which primarily led to thename'diffraction-stackmigration')in thegiven(or alreadyestimated)macro-velocitymodel.Theirdiffractiontraveltimesurfaces– whichwouldpertainto actualdiffractionresponsesif thepoints · wereac-tualdiffractionpointsin thegivenmodel– canbecalculatedusingthemacro-velocitymodel. Now, a Kirchhoff-type depthmigrationinvolvesin principlenothingelsebut

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performinga summation(also called stack)of all the seismictraceamplitudesen-counteredalongeachdiffraction-timesurface(Huygenssurface)in the time section,andplacingthesummationvalueinto thecorrespondingpoint · . Thedynamicpartwhich we refer to astrue-amplitudecanbeperformedby weightingall seismictraceamplitudesalongthediffraction-timesurfaceduringthestackingprocessby avaryingtrue-amplitudeweightfactor. Thesummationcanmathematicallybeexpressedbyº ¡»· §¸¨ ­ �¼«½¿¾�¾À Á ¤«Â Á ¤nÃ�¬UÄ4Å ¡ £¤'¥ · §«Æ  ¢¡ £¤�¥�¦i§Æ ¦ ÇÇÇÇÇ ÈÊÉ�Ë»Ì�ÍÏÎÐ�Ñ ÒÔÓ � (2)

Thestackingsurfaceis theHuygenssurfacegivenby¦j¨Õ®ÖÄ ¡ £¤�¥ · §¸¨Q® ¡×¶Ø¡ £¤�§�¥ · §wÙV® ¡×· ¥�° ¡ £¤?§i§�¥ (3)

with the vector £¤ varying over the apertureÚ . Note, that to eachsubsurfacepoint· , onegenerallyassignsthevalueReÛ º ¡»· §�Ü to displaythedepthmigratedimage.However, to permit the extraction of complex reflection coefficients in caseofovercritical reflections,one has to considerthe full complex value

º ¡×· § instead.Thechoiceof Ý�ÞÝ È is neededin orderto correctlyrecover thesourcepulse.Theregionof integration Ú should ideally be the total ¡ ¤«Â�¥�¤nÃ�§ -plane. This is, of course,im-possibledueto thelimitationof theapertureof thedataacquisition(Tygeletal.,1996).

Thetrue-amplitudeweightfunction,is givenby¬UÄ4Å ¡�£¤'¥ · §j¨ ß�àaá ÃÒ¼ �n��� à � Ò ´ Å Ò ´ Ò ¯â¥ (4)

where ß�à ¨ ß�à ¡ £¤f¥ · § is the Beylkin determinant(Jaramilloet al., 1998). It isimportantto notethattheweightfunctionon its own doesnot removethegeometricalspreadingeffect. Only thetrue-amplitudeweightfunctionandthesummationprocessalong the diffraction surface accountfor the whole geometricalspreadingeffect.Surprisingly, theevaluationof thestackingintegral by meansof thestationaryphasemethodshows thatthestackautomaticallycorrectsfor theFresnelfactor, i.e. reflectorcurvatureeffects on the total geometricalspreadingfactor are automaticallytakeninto account. The chosenweight function (4) simply correctsfor the remaininggeometricalspreadingeffects causedby ray segmentsin the reflectoroverburden.This is thereasonwhy theweightfunctionis independentof thereflector's curvature.

True-amplitudeweight functions for specialshot-receiver configurationsand acomparisoncanbefoundin (Hanitzsch,1997).

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True-amplitude demigration

Thetrue-amplitudedemigrationprocedureis performedby aweightedKirchhoff-typeisochronestack. The kinematicpart of the transformationcaneasilybe performedin a completelyanalogousway to true-amplitudemigrationasdescribedin the lastsection.Ratherthandefininga grid of points · in thedepthdomain,we now definea densegrid of points ã in the time domain, i.e., in the ¡ £¤f¥�¦i§ -volume in whichwe desireto constructthe time sectionfrom the depth-migratedsection. Eachgridpoint ã togetherwith the macro-velocity model definesan isochronein the depthdomain.Theisochronedeterminesall subsurfacepointsfor which a reflectionfrom apossibletruereflectorwouldberecordedat ã aftertravelingalongaprimaryreflectedray from ¶ to

°. We now have to perform a stackalong eachisochroneon the

depth-migratedsectionamplitudes.Then,we placethe resultingstackvalueinto thecorrespondingpoint ã . As in Kirchhoff migration,the isochrone-stackdemigrationwill only resultin significantvaluesif thepoint ã is in thenearvicinity of anactualreflection-traveltimesurface. Elsewhereit will yield negligible results.Thedynamicpartcanonceagainbeperformedby amultiplicationof theamplitudeswith avaryingweightfunction.

Thesummationcanmathematicallybeexpressedbyº ¡»ã §j¨ �¼ä½å¾�¾æ Á�çèÁ�é ¬ëêiÅ ¡ £ì ¥ ã § Æzí ¡ £ì ¥�î�§Æ î ÇÇÇÇÇ ï�Éfðòñ�Í Îó Ñ ôªÓ � (5)

Thestackingsurfaceis theisochroneîZ¨Oõ�ê ¡ £ì ¥ ã § givenby®nÄ ¡ £¤'¥ · §j¨c® ¡×¶Ø¡ £¤�§%¥ · §1Ù/® ¡»· ¥�° ¡ £¤�§�§l¨c¦ (6)

for all points · with £ì in ö . As we canseehere,the isochroneand the Huygenssurfacearedefinedby the sametraveltime function

®nÄ ¡�£¤f¥ · § . The function í ¡ £ì ¥�î�§representsthemigratedreflectorimageto bedemigratedandcanbeexpressedasí ¡ £ì ¥�î�§j¨ í ©«¡ £ì §l¬ ¡ò÷ Ä ¡ îJ­Võn¯ø§�§�¥ (7)

where¬ ¡ ¦i§ representsthe analytic point-sourcewavelet, ÷ Ä ¡ £ì § is the vertical

stretchfactor, andpoints · ¯ù¨ ·ú¡ £ì ¥�îû¨üõ�¯ ¡ £ì §i§ definethe actualreflector.¬Uê�Å

is the true-amplitudeweight function to be specified. The valueReÛ º ¡ýã §�Ü is thedemigrationresultwhich is generallyassignedto the point ã . However, if we wantto chaintrue-amplitudemigrationanddemigration,we have to usethecomplex valueº ¡ýã § (Tygeletal., 1996).

Thetrue-amplitudeweightfunctionfor demigrationrelatesto theweightfunctionfor migrationbecausethedemigrationshouldundowhatthemigrationhasdoneto the

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originalseismicsection(providedthesamemacro-velocitymodel,thesamemeasure-mentconfiguration,andthesameray codesareused;only primaryeventsareconsid-eredhere).Theweightfunctionfor demigrationcanbecalculatedin ananalogouswayto thepreviouslydescribedmigrationweightfunction,andreads¬ëê�Å ¡ £ì ¥ ã §j¨ �¼ ´ Å Ò ´ Ò ¯ �n��� Ã�þ Ò ¥

(8)

whereÅ Ò and Ò ¯ denoteonceagainthepoint-sourcegeometricalspreadingfactors

for the ray segments¶j· and · ° , respectively;þ Ò is the anglebetweenthe half-

angledirection– definedby thetwo raysegments– andtheverticalat · (Jaramilloetal., 1998).

IMPLEMENT ATION

As we have shown in theprevioussection,theknowledgeabouttheHuygenssurfaceand the isochrone(stackingsurfaces)are fundamentalfor Kirchhoff-type migrationand demigration. Thesesurfacescan be built by the sum of traveltimesalong theray branchesfrom the source ¶ to the depthpoint · and from · to the receiver°

. Thus,we needa priori informationaboutthe traveltimes(for the calculationofthe Huygenssurfaceor isochrone)and dynamicparameters(for the calculationofthe weight function) to performa true-amplitudemigrationor demigration. This iswherethe macro-velocity modelcomesinto operation. The necessaryinformation,i.e., the Green's functionof the medium,is usuallyobtainedby asymptoticdynamicray tracingfor a specifiedray-codein themacro-velocityearthmodel,andstoringtheparametersin a databasewhich we call Green's FunctionTable(GFT).This databaseis thelink betweentheinput andoutputspace,seeFigure1. In otherwords,we needinformationaboutthe wavefield from every shotandreceiver positionat the surfaceto every diffraction point · in the subsurface. Obviously, this is a large amountofinformation,andtheGFTcanexceedtheinputdataby severalordersof magnitude.Inorderto avoid computationandstorageof redundantinformation,methodsto optimizethecreationof theGFTareexplainedby (Riedeetal., 2000)and(Hertweck,2000).

Our imaging tool is written in C++ in an object-orientedway. According tothe previously mentionedstacking integrals, we have to calculate the analyticalsignaland its derivative (3D) or half-derivative (2D). This is donein the frequencyor wavenumberdomain, respectively. Taperingof the input data is usedto avoidboundaryeffects at the borderof the apertureduring the summationprocess. Wemultiply theinput data(alongtheshotcoordinates)by thefunction ÿ ¡ ç §T¨ � ­ �n��� çover a small rangeat the border, usuallysometraces.No taperingis neededwithinthe shotgathersalongthe offset dimension.The actualnumberof traceswhich areusedfor taperingmaybespecifiedby theuser. Thecompletetaperfunction in 3D is

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0

0

ksi1

Seismogramt

Green’s Function Table(GFT)

ksi1

Traveltime

z

z

Depth-migrated image

migration

demigration

x

x

Figure1: Schemeshowing themigrationanddemigrationprocedurefor 2D data.Theinput datais transformedto theoutputdomainwith thehelpof theGreen's FunctionTable(GFT) which containsall necessaryinformationto constructthe stackingsur-facesaswell astheweightfunctions.

plottedin Figure2.

The weight functionsfor true-amplitudemigrationor demigrationarecalculatedduringruntime.Thenecessaryinformationis obtainedfrom theGFT.

To avoid operatoraliasing,the operatoritself is taperedor, in other words, thepossiblereflectordip which canbeimagedis limited. Thelimit canbedefinedby theuseraccordingto hisneedsandtheinputandoutputgrid spacings.

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tapervalue

1

0

ksi1

ksi2

Figure2: Taperfunctionin 3D. In this plot, thenumberof traceswhich aretaperedisenlargedcomparedto theoverallnumberof tracesto show theeffectmoreclearly.

APPLICATION

We have appliedour imaging tool to several syntheticdatasetsto show how true-amplitudemigrationanddemigrationwork. Thedevelopmentandtheapplicationonsyntheticandrealdatasetsaregoingon. Thecurrentversionof theprogramis abletohandle2D and3D true-amplitudepoststackmigrationanddemigration,and2D true-amplitudeprestackmigration. The3D prestackcaseis not consideredat themomentdueto limitations in availablecomputingfacilities. We cannotreally expectthat thetheorypresentedhereandthe respective weight functionsandalgorithmswork wellfor all earthmodels.However, thefollowing resultsshow thatour imagingtool is ableto handlevariousinputdatasetsin aflexible andaccurateway.

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Figure3: Isochrone.This curve is obtainedby migratinga singleevent,herefor zerooffset (ZO) configurationand(for simplicity) constantvelocity. In a homogeneousmedium,theZO isochroneis asemicircle(2D) or ahemisphere(3D).

Figure4: Huygenscurve. This curve is obtainedby demigratinga singlediffractionpoint,herefor zerooffset(ZO) configurationand(for simplicity) constantvelocity. Ina homogeneousmedium,theZO Huygenscurve is a hyperbola(2D) or a hyperboloid(3D).

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01

23

45

Depth [km

]

0 1 2 3 4 5 6 7 8 9 10CMP [km]

Fault Model

2.5 2.7 2.9 3.1 3.3 3.5 3.7 3.9P-velocity [km/s]

Figure5: A syntheticmodelusedfor testingmigrationanddemigrationin 2D. ThecolorsdenoteP-wavevelocities.For simplicity, thedensityof all layersis constant.

Figure6: 2D poststacktrue-amplitudedepthmigration. The waveletsarecorrectlymappedfrom the time to the depth domain, their amplitudescorrespondto theimpedancecontrastateachinterface.

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Figure7: 2D artificial migratedimageof theupperthreelayers(modelFig. 5). Thissectionis createdby attachingto eachreflectorelementacorrectlystretchedwavelet.

Figure8: 2D demigratedseismogramof theartificial migratedsectionshown in Fig.7. Due to the limited demigrationaperture,the seismogramcanonly be recoveredroughlyabove thetargetzone.

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CONCLUSIONAs a generalizationof classicalkinematicseismicreflectionmappingprocedures,wehavepresentedacompletewave-equationbased,highfrequency, amplitude-preservingtheoryfor seismicreflectionimaging. Thebasictoolsaretwo weightedstackingin-tegrals,namelythe diffraction andisochronestacks. Both stackingintegralscanbeappliedin sequenceusingdifferentmacro-models,measurementconfigurationsor raycodes,thusallowing us to solve variouskindsof imagingproblems.All transforma-tionscanbecarriedout in a true-amplitudesense,thatmeans,thegeometricalspread-ing factorsare correctly transformedfrom the input to the output domain. This isa featureof the Kirchhoff-type stackingstructureof the integralsandthe adequatelydeterminedstackingsurfaces.The theorydoesnot dependon any assumptionaboutthe mediumandthe reflectorcurvaturebesidesthe obvious conditionthat all quan-tities vary smoothlyfor the zero-orderray theoryandthe asymptoticevaluationstobe valid. The implementationis straightforward andshows that we areableto han-dle true-amplitudemigrationanddemigrationin a flexible way. Furtherresultsarepresentedby Goertzetal. (this issue).

REFERENCES

Hanitzsch,C., 1997,Comparisonof weightsin prestackamplitude-preservingKirch-hoff depthmigration:Geophysics,62, no.6, 1812–1816.

Hertweck,T., 2000,Practicalaspectsof theunifiedapproachto seismicimaging:Mas-ter's thesis,Universityof Karlsruhe,Germany.

Hubral, P., Schleicher, J., andTygel, M., 1996,A unified approachto 3-D seismicreflectionimaging,PartI: Basicconcepts:Geophysics,61, no.3, 742–758.

Jaramillo,H., Schleicher, J.,andTygel,M., 1998,Discussionanderratato: A unifiedapproachto 3-D seismicreflectionimaging,PartII: Theory:Geophysics,63, no.2,670–673.

Riede,M., Goertz,A., andHertweck,T., 2000,CascadedSeimsicMigrationandDem-igration:70thAnn. Internat.Mtg., Calgary/Canada,Soc.Expl.Geophys.,ExpandedAbstracts,Session:MIG P1.4.

Schleicher, J.,Tygel,M., andHubral,P., 1993,3-D true-amplitudefinite-offsetmigra-tion: Geophysics,58, no.8, 1112–1126.

Tygel, M., Schleicher, J., andHubral, P., 1996,A unified approachto 3-D seismicreflectionimaging,PartII: Theory:Geophysics,61, no.3, 759–775.

PUBLICATIONS

Detailedresultswerepublishedby (Hertweck,2000)and(Riedeetal., 2000).