vsp wavefield separation using structure tensor dip

Bollettino di Geofisica Teorica ed Applicata Vol. 61, n. 2, pp. 145-158; June 2020

DOI 10.4430/bgta0303

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VSP wavefield separation using structure tensor dip masking filter

H. HasHemi and m. GHasem FakHari

Institute of Geophysics, University of Tehran, Iran

(Received: 19 June 2019; accepted: 18 October 2019)

ABSTRACT InaVerticalSeismicProfiling(VSP)acquisition,downgoingwavefieldsandupgoingwavefields, interfere. Both wavefields are practical in various seismic applications,However,itisneededtoseparatethesetworeflectionandtransmissionfields.DifferenttechniqueshavebeenproposedforVSPwavefieldseparation.Medianfilteringaswellas2DFourier transformsarecommonlyused for separating the seismicwavefields.The former suffers from the averaging effect, generating artifacts, and amplitudemodification due to spectral energy leakage after the inverse transform. 2DFourierhastheproblemofleakageandedgeeffects.WeproposeanewapproachbasedonthestructuretensorandlocaldipestimationfollowingamaskingfilterforVSPwavefieldseparation.Thedipmaskingfilter is calculatedusing the local dip of eachpoint ofdatathatcanseparateupgoinganddowngoingwavefieldsfromtheoriginaldata.Theadvantageofthisapproachisbothinpreservingseismicamplitudesandinprovisionofasectionfreeoffakeevents;thisliterallyimpliesnoenergyleakage.Thesyntheticandrealdataexamplesaredemonstratedtoshowtheperformanceoftheproposedmethod.

Key words: VSP,waveseparation,localdip,structuretensor,dipmaskingfilter.

© 2020 – OGS

1. Introduction

VerticalSeismicProfiling(VSP)isaseismictechniquethatextractshigh-resolutioninformationfromthesubsurface.TheseismicVSPsignal isgeneratedat thesurfacerecordedbyreceiverslocatedatdifferentdepthsinaverticalwell.VSPrecordeddatahavebeenusedincalculatingthevelocityandtheattenuationinaninterval(Pevzneret al.,2011;BaharvandAhmadiandMorozov,2013;Wang,2014).

InaVSPrecording,downgoingwavefieldswithapositive-slownessandupgoingwavefieldswithanegative-slowness,interferewitheachother(SeemanandHorowicz,1983;Hardage,2000).ThesimultaneousrecordingofupgoinganddowngoingwavefieldsareregardedasanaddedvalueoftheVSPmethod.DowngoingwavesareusedtocomputeseismicP-andS-wavesvelocities,tocalculatetheseismicanelasticqualityfactorQ,tocreatetime-dependentfilters,whichareusedtorecoverhigh-frequencyseismicwaves(SudhakarandBlias,2002).UpgoingwavefieldsaretoolsforobtainingVSP-CDPimages(Blias,2005).

AsafundamentalsteptoprocessaVSPdataset,theseparationofupgoinganddowngoingwavefieldsisintroduced.Therearemanymethodstoseparateupgoinganddowngoingsignalsfromeachother.Mostlythesemethodsarebasedontwotypes:1)methodsusingaveragingfilters

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Boll. Geof. Teor. Appl., 61, 145-158 Hashemi and Ghasem Fakhari

for the separationof theupgoing anddowngoingwavesbymedianfilter; 2)wave separationtechniqueswhichtransformprimarydataintoanewdomainandseparateupgoinganddowngoingwavesinthisspaceandthereafterreturningdatatoitsprimaryspace.Themostimportantonesaref-kandτ-pfiltering.Curveletsarealsoakindofdirectional-scaletoolsusedforVSPwavefieldseparation(Heraviet al.,2012).

The median filter method (as a conventional wave separation technique) is based onflattening data for separating downgoing, then applying a classicmedian filter, shifting backandfinallysubtractingtheseparateddowngoingwavefieldfromtheoriginaldatatoobtaintheupgoingwavefields.Inthemedianfiltermethod,eachsampleisreplacedbythemedianvalueofitsneighbourhoodsamples.Therefore,duetotheaveragingeffect,thismethodsuffersfromsmoothingtheinputsignal,manipulatingprimarysignalamplitudes,anddamagingsomeoftheusefuldetails.Tocopewiththeseproblems,someimprovedversionsofmedianfiltersornon-linearmedianfiltershavebeenproposedsuchasvectormedianfiltering(KasparisandEichmann,1987;Astolaet al.,1990)whichhasbeenappliedinseismicdataprocessing(Liu,2013).Moreover,Chenet al. (2016) improved thesignalpreservingabilityofamedianfilterusingastructure-orientedspace-varyingmedianfilter.

Waveseparationtechniquessuchasf-kortheτ-p,transformdatainthenewdomainandalwaysgeneratingsomeartifactseventsinthisspaceisunavoidableandhenceamplitudemodificationduetospectralenergyleakageeffecthappens.Inexperimentalphysics,dataspectralleakageisawell-knownprobleminthedatadomaintransforms.Xuet al.(2005)proposedanantileakageFouriertransformapproachforseismicdataregularisationcase.Chenet al.(2014)proposedaniterative framework for deblending of simultaneous-source seismic data usingSeislet domainshapingregularisationandcomparedthismethodwithtwoFourierbasedtransform;f-kdomainthresholding, and f-x predictive filtering.According to their results, Fourier based transformmethodssufferfromenergyleakageandgenerateartifacts.

DowngoingwavesarecharacterisedbypositiveapparentvelocityandanegativeslopeinVSPimagedata,upgoingwavesarecharacterisedbynegativeapparentvelocityandpositiveslope.Inthispaper,itisdecidedtousethedipvarianceand,then,theoutputisusedtoseparateupgoinganddowngoingwavefieldsfromeachother.So,datawillnotbetransferredtothenewspaceandasaresult,datadomaintransferdoesnotyieldtheleakageofenergy.Inthispaper,wedecidetomakeamaskingfilterbasedonthelocaldipofeachpointofdatawhichcanseparateupgoinganddowngoingwavefieldsfromoriginaldata.Wecalledthismethod“dipmaskingfilter”andtheadvantageisthatneitheraveragingofamplitudesnorartificialeventbytransferdatahappen;thisimpliesnoenergyleakage.

An initial published work estimating dip directly from 2D seismic data is by Picou andUtzmann(1962).FinnandBackus(1986)extendeddipestimation to3Ddataasapiecewisecontinuousfunctionofspatialpositionandseismictraveltime.Marfurtet al.(1998)generalisealatersemblance-basedscanbyFinnandBackus(1986).AsdiscussedbyMarfurt(2006),thelocal reflection orientation can be described by reflection normal vector. Fomel (2002) usedplane-wavedestructionfiltertocomputereflectionslopes.vanVlietandVerbeek(1995)presentanestimatebasedonthegradientstructuretensor.ChenandMa(2014)proposedadip-separatedfiltering using an adaptive empiricalmode decomposition based on dip filter to separate theseismicdataintoanumberofdipbands.Thismethodsaysthatthedipestimationisbetterwhenitisappliedtodip-separatedprofiles.

VSP wavefield separation with structure tensor Boll. Geof. Teor. Appl., 61, 145-158

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Ourproposedlocaldipmaskingfilter,basedonthestructuretensormethod,isusedforthepurposeofcalculatingdipvariance.Thecoherentstructureofseismicimagesyieldstosupposegoodcandidatesforstructuretensorapplication(vanVlietandVerbeek,1995;Weickert,1999;FehmersandHöcker,2003),whichareacommontooltoestimatethelocalorientationofimagefeatures.Tensorstructureinseismicsisusedinestimatingtheorientationsoftheseismicstructuralcharacteristicssuchasreflectionslope,stratigraphicfeaturesorientations,horizoninterpretation,subsurfacemodelling,etc. (e.g.LiandOldenburg,2000;Bakker,2002;FehmersandHöcker,2003;Marfurt,2006;Hale,2009;WuandHale,2015a;Wu,2017a,2017b).Bakker(2002)andWuandHale(2015b)usestructuretensors(e.g.vanVlietandVerbeek,1995;Weickert,1997)toestimatereflectionorientations.WuandJanson(2017)proposeadirectionalstructuretensortoestimatetheorientationsofreflectionswithsteepandrapidlyvaryingslopes.

FarakliotiandPetrou(2005)benefitfromtheuseofstructuretensorstoanalysetheseismicdata.Inthispaper,thestructuretensoralgorithmisusedtoseparateVSPwavefieldsimagesafterfindingtheirlocaldips.

2. Methodology

2.1. Estimating local dips using structure tensorsThestructuretensoristheouterproductoftheimagegradientwithitself,whichisalsocalled

gradient-squaretensors.Structuretensorwasdevelopedprimarilyforedgedetection(FörstnerandGülch,1987),butithasbeenusedforawiderangeofproblemsinimageprocessing.

ConsideringanimageI(x.y)thestructuretensorSisdefinedas:

(1)

whereK is a smoothing kernel (e.g aGaussian kernel),* stands for convolution operator, Ix and Iy are theverticalandhorizontalcomponentsof thegradientof the image I, respectively.In2D,thestructuretensorSisfinallyofn×m×2×2wheren×misthesizeoftheinitial imageI.Effectively,tensorstructureobtainsa2×2matrixateachpixeloftheoriginalimage.BytheeigendecompositionofthematrixS,wecanobtaintheorientationinformationineachpixel.Inaccordancewiththeprinciplesofmatrixeigenvectordecomposition,theeigendecompositionofa2Dstructuretensorisasfollows:

(2)

S = λu uuT + λv vvT. (3)

where u and v are normalised eigenvectors corresponding to the eigenvalues λu, λv,respectively.

These eigenvectors provide an approximation of orientations of features in the 2D image(FehmersandHöcker,2003).Thisinformationcanbevisualisedasanellipsewithtwodiameters

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whichareequaltotheeigenvaluesanddirectedalongtheircorrespondingeigenvectorsasshowninFig.1.

AsshowninFig.1, theorthogonaleigenvectorsuandvdescribetheorientationoffeaturein each point. Individually, for each sample, the eigenvector u, corresponding to the largesteigenvalue(λu),isparalleltothedirectionsinwhichtheimagefeaturesvarymostsignificantly.AsshowninFig.1,thecomponentsoftheunitvectorûarerelatedtolocaldipsθineachsampleby:

u1 = |u| cos θ, (4)

u2 = |u| sin θ, (5)

Finally,thedominantorientation(localdip)ineachsampleiscomputedfromtheeigenvectoruassociatedwithλuas:

dip(0)=tan–1(u2/u1). (6)

2.2. Wave separation process using dip masking filterUsing dip masking filter for VSP wavefield separation based on the structure tensor, the

followingstepsareproposed:1. applicationofasmoothingfilter(e.g.withGaussiankernel)ondatatoreducerandomnoise

andnon-structuralorientations;2. calculatethepartialderivativesoftheimageandbuildstructuretensor;3. eigenvaluedecompositionofthestructuretensor;4. estimatelocaldipovereachsample;5. useamovingsimilarityfilterondipimagetoobtainthedominantslopeofeachsample;6. applyamediandipfilterondatatoeliminatenoisepointswithnon-structuraldirections;

Fig.1 -Thestructure tensoratpointOis visualised as an ellipse and its uniteigenvectorsu,vandrootedeigenvaluesλu,λvarealsodepicted.


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7. createtwodipmaskingfiltersbyseparatepositiveandnegativeobtaineddips;8. apply created dip masking filters on VSP data to separate upgoing and downgoing

wavefields;9. usingthef-kinterpolationtorecoveringintersectingpointsafterapplymaskingfilter.

Inanoise-freedata,theestimatedorientationbystructuretensorwillbeanidealapproximationfortheslopeoftheeventsreportedinsamplebysamplemanner.But,ingeneral,thepresenceofnoiseinthedatawillaffecttheestimateddirections.Inpractice,mixingthecontentofcoherentsignal and random noise in seismic data is unavoidable. Therefore, we require to computemore stabledominantorientationsby implementing a smoothingfilter to each elementof thegradient-basedstructuretensors.Thesmoothinghelpstoremovethenoiseandmuchmorestableorientationsevenifapparently, theresolutionof theinput imageisnotgood.Inthealgorithmpresented,thefirststep,relatedtosmoothingdatabeforecalculatingthedip,isdonewithseriousconsiderations.WeproposedarecursiveGaussianfilterforsmoothingtheparametersofwindowwidths both in time and spatial directions, thesemust be optimally chosen for each input. Itsmoothesafewpixelswhosevaluesdiffersignificantlyfromtheirneighborhoodwithoutaffectingtheotherpixels.Notealso thatby increasing thesizeof the smoothingwindow, the structuretensorisrobustinthepresenceofnoisydatawithlesseffectinreducingthespatialresolution[formoredetailsaboutGaussianfilterandchoosingoptimumsmoothingparameterrefertoDeriche(1992),vanVlietet al.(1998)andHale(2006,2009)].

Steps2,3,and4arerelevantincalculatinglocaldipovereachsampleusingstructuretensorsmentionedinsection2.1.Beforecreatingmaskingfiltersfromtheestimatedlocaldips,inadditiontothesmoothinginstep1,thealgorithmperformsthetwoadditionalsteps5and6todealwithnoiseandestimatedrandomorientations.

Step5issomewhatsimilartothenonlocalalgorithmfortheimagedenoising,exceptthatitdoesnotrefertotheimageandisonlyappliedtoestimateddirectionsfromstructuretensors.Aftercomputationoflocaldirectionsineachsample,amatrixwith2×1elementisconstructedateachpixeloftheoriginalimage.Next,thismatrixisdecomposedintoanumberofvertical-horizontaloverlappingsmallwindows.Forthecentralpixelofeachwindow,thecosinesimilaritybetweenthispixelandneighbourhoodsiscalculatedandonthebasisofthereportedvalue,itisdecidedtokeep/removethedirectionforthatpixel.Athresholdsimilarityvaluebytheusermustbesetinthealgorithm.TheschematicperformanceofthisstepisshowninFig.2.

Fig.2-Schematicperformance ofmovingsimilarityfilterpresentedinstep5onthedipimagetoobtainthedominantslopeofeachsample.

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Instep6,amedianfilterisappliedondatausingtheestimateddominantdipinformation,toeliminatenoisepointswhichhavenon-structuraldirections. Instep7, twodipmaskingfiltersarecreatedbyseparatepositiveandnegativecalculateddipsandthemaskingfiltersareappliedaccordinglyonVSPdata toseparateupgoinganddowngoingwavefields instep8.Finally, inthelaststep,thef-kinterpolationisusedtorecoverintersectingpointsafterapplicationofthemaskingfilter.

Ingeneral,ourproposedmethodisnotanoiseremovalmethodand,ifthereissignificantnoiseindata,itisbettertousedenoisingprocessesbefore.

3. Examples

3.1. Application on a synthetic modelTodisplaytheefficiencyoftheproposedmethod,itisfirstlytestedonasyntheticVSPdata

set.Fig.3ashowsasimplesyntheticVSPdatasetcontainingasingledowngoingandasingleupgoingwavefield.Fig.3bshowseigenvectorscomputedfromstructuretensorsineachsample

Fig.3-a)AsimplesyntheticVSPdatasetwithasingleupgoingandasingledowngoingwavefield;b)eigenvectorscomputedfromstructuretensorsineachsampleexhibitinglocalstructuralorientation;c)andd)VSPdowngoingandupgoingwavesobtainedfromapplying dipmaskingfilteronsyntheticdata.


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where structuralorientations in thesepoints are shown (for abettervisualisation,1outof20eigenvectorsinallimagesisshown).

After the creation of two dipmasking filters, namelywith positive and negative obtaineddips,andperformingthedipmaskingfilteronsyntheticVSPdata,theseparateddowngoingandupgoingwavefieldsareobtained(Figs.3cand3d,respectively).AsitisshowninFigs.3cand3d,theproposedmethodseparatesdowngoingandupgoingwavefieldsfromeachotherwithoutartifactsorsmoothing.

AsshowninFig.3b,inthisimageall theorientationsarecorrectlyestimatedbystructuraltensorandtheycontainthreedominantdirections:1)adowngoingeventwithnegativedip,2)upgoingeventwithpositivedip,and3)background.

Nowweadd theGaussianwhitenoise to theprimarysyntheticVSPdata setandcomputeeigenvectorscalculatedfromstructuretensorswithandwithouttheuseofasmoothingfilter.TheresultsforthetwonoisevaluesareshowninFigs.4and5.

Fig.4-a)NoisysyntheticVSPdatasetcorruptedbyamiddlelevelofGaussianwhitenoise;b)eigenvectorscomputedfromstructuretensorswithouttheuseofasmoothingfilter;c)eigenvectorscomputedfromstructuretensorswiththeuseofaGaussiansmoothingfilter.

AsshowninFigs.4and5,thenoiseaffectstheestimateddirectionsandcausesthedeviationoftheslopeapproximationoftheevents.But,byusingthesmoothingfilteronbothimages,theestimatedorientationsfordominantdirections(upgoinganddowngoing)arecorrectlycalculated.

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3.2. The issue of intersecting wavesStructuraltensorsareagoodestimatorforfindingthedirectionofthewavepaths,butonly

forcases inwhich theydonot intersect.Hence, thestandardstructural tensorbreaksdown inthe presence of intersecting wavefields. In order to solve the difficulty in estimating correctconflictingdips,Chen(2016)proposedadip-separatedfilteringapproach.However,inthisstudyourobjectiveistoseparateupgoinganddowngoingwavefieldsinthesimplestwaywiththelowestcost;itimpliesthatthepreciselocalslopeestimationattheintersectionpointisnottargeted.

For the case of two intersecting local orientations with different directions, the proposedapproachcansmartlyrepresentonlyoneofthedominantpathsforthecorrespondingdirectionusingf-kinterpolation.

InFig.6a,asimpleexampleoftwowaveswithdifferentlocalorientationsisshown.Figs.6band6cshowthecomputedlocaldipvectorsfromstructuretensorsineachsample.Asshowninthisfigure,attheintersectionpointonlythemostdominantdirectionisrestored.Aftercalculatingthelocaldipateachpoint,theupgoinganddowngoingmaskingfiltersarecreatedbycollectingpositiveandnegativedips,respectively.Finally,thedipmaskingfiltersareappliedontheinputdata(Fig.6a)andtheseparatedupgoinganddowngoingwavefieldsaregenerated(Figs.6dand6e,respectively).AsshowninFig.6c,onlytheupwarddirectionisobtainedattheintersectionpoint,soinfindingthemaskingfilter,thisareaisassignedonlytoupgoingwavefield.Therefore,

Fig.5-a)NoisysyntheticVSPdatasetcorruptedbyahighlevelofGaussianwhitenoise;b)eigenvectorscomputedfromstructuretensorswithouttheuseofasmoothingfilter;c)eigenvectorscomputedfromstructuretensorswiththeuseofaGaussiansmoothingfilter.


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the nearby samples are incorrectly removed in downgoing wavefields and a discontinuity isobservedindowngoingwavefieldatthisintersectingpoint(Fig.6e).

In this paper, we use the f-k interpolation technique to recover intersecting points afterapplyingamaskingfilterondata.Fig.6fshowsthedowngoingwaveafterrecoveringbythef-kinterpolationattheintersectionpoint.

3.3. Real data exampleWeconsidernowarealverticalseismicprofilefroma3Delasticsimulation.Thisdataset

hasbeenacquiredinSEAMPhaseIRPSEAbasedonthegeologyoftheGulfofMexico.Thedescribedmethodgivesusatooltoseparatedowngoingandupgoingwavefields.Fig.7showsthisVSPdatasetconsistingofdowngoingandupgoingwavefields.

Fig.6-a)Aasimpleexampleoftwowaveswithdifferentlocalorientations;b)eigenvectorscomputedfromstructuretensors in each sample; c) eigenvectors in each point; d) upgoing events obtained from the application of the dipmaskingfilteronprimarydatashowninpanela;e)downgoingeventsobtainedfromtheapplicationof thedipmaskingfilteronprimarydatashowninpanela,attheintersectionpoint,adiscontinuityisobservedindowngoingwavefield;f)downgoingwavefieldafterrecoveringbythef-kinterpolationattheintersectionpoint.

Fig. 7 - A real vertical seismic profile data setconsistingofdowngoingandupgoingwavefields.

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Fig. 8 shows eigenvectors computed from structure tensors in each sample where there is astructuralorientationinthesepoints.Thesedominantdirectionshelpustoseparatedowngoingandupgoingeventsfromeachother.

Fig.8-Eigenvectorscomputedfromstructuretensorsineachsample.Thedirectionoftheseeigenvectorscanrepresenta localstructuralorientationineachsample(ofcourse,forthepresentation, theeigenvectorofall thepointsisnotdrawnbutonlyoneouttenpoints).

AfterapplyinggenerateddipmaskingfiltersontherealVSPdatashowninFig.7,theseparateddowngoingandupgoingwavefieldsaregenerated(Figs.9aand9b,respectively).

Fig.9showstheprogressionobtainedbythedipmaskingfilterintheupgoinganddowngoing

Fig. 9 - Downgoing events obtained from applying the dipmasking filter on primaryVSP data (a) and upgoing events(b).


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separation.As it is clear in this figure, in points of discontinuity intersection in the upgoinganddowngoingeventscontinuityofeventsislost.Thesolutionpresentedhereistheuseoff-kinterpolationinintersectingpoints.

Figs.10aand10bare the separateddowngoingandupgoingwavefieldsafter interpolationattheintersectingpoints.Thesefigureshighlightthatthecontinuityofseismiceventshasbeensufficientlyrecovered.Theresultofanotherstandardwavefieldseparationmethod(flatteningandsubsequentmedianfiltering)isshowninFigs.11aand11b.

Comparingtheseparationresultsobtainedbytheapplicationofthemedianfiltermethod(Fig.11)with theproposedmethod (Fig.10), it is clear that there isnot significant interferenceof

Fig. 10 - Downgoing separated wavefields recovered in intersection points by f-k interpolation (a) and upgoingwavefieldsafterrecovered(b).

Fig.11-Downgoingseparated(a)andupgoingwavefieldsobtainedbymedianfiltermethod(b).

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downgoingeventsintheseparatedupgoingwaves(Fig.10b),butclearly,downgoingwavesarevisibleinseparatedupgoingwavefieldobtainedbythemedianfiltermethod(Fig.11b).

Comparing the separationdowngoing results, it appears thehighly coherent visible outputof standardmedian filtering algorithm. However, a further look to the computed histogramsof amplitudes (Fig.12) confirms that themedianfilteringhasconsiderablychanged thenormof amplitudevalues anddistributionconsiderably.The similarityofFigs. 12a and12bhighlyemphasisestheamplitudepreservingmanneroftheproposedmethod.

Furthermore,firstbreakpickingisafundamentalstepinVSPwavefieldseparationbymedianfiltermethod,anyerrorsofthesearrivaltimesmayhavesignificanteffectsontheresults.Anotheradvantageoftheproposedmethodisthatitdoesnotrequirethesetimes.

Fig.12-Histogramofamplitudeof:a)originalVSPdata;b)downgoingwavefieldsobtainedbydipmaskingfilter,andc)downgoingwavefieldsobtainedbymedianfiltermethod.


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4. Conclusion

Inthispaper,wehaveproposedanewapproach,structuretensorfollowedbydipmaskingfilter, to separate VSP downgoing and upgoing waves. The main idea of this method is theapplicationofslopeofeventstoseparateupwardanddownwardwaves:wefirstgenerateamaskwiththesamesizeofdatausingtheslopesderivedfromstructuretensor,then,byapplyingthemaskfilterondata,weseparatetheupwardanddownwardwaves.Thisleadstonoenergyleakage(amplitudepreservingmanner)andthetotalenergyafterseparationisequaltotheinitialinputensembleenergy,despite that,intraditionalmethods,waveseparationoftenoccurswhendataistransferredtoanewspace(f-kdomainorτ-pdomain).

Finally, itwasshownthat theapproachpresentedinthispaperperformswell inseparatingVSPdowngoingandupgoingwavesandofferstheadvantageoverconventionaltechniquesthatnoaveragingofamplitudeofdataoccurs,sonoartificialsmoothingoftheeventisdoneandhencetheleakageofenergyisnearlyzero.

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Corresponding author: Hosein Hashemi Institute of Geophysics, University of Tehran North Kargar street, Tehran, Iran Phone: + 98 21 88001115; e-mail: [email protected]

vsp wavefield separation using structure tensor dip

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