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Lapped transformsand hiddenMarkov modelsforseismi data ltering
LaurentDUVAL
Te hnology,ComputerS ien eandAppliedMathemati sDepartment, InstitutFran aisduPetrole,
92500Rueil-MalmaisonCedex,Fran e laurent.duvalifp.fr
CarolineCHAUX
InstitutG.Monge,UniversitedeMarne-la-Vallee 77454Marne-la-Vallee,Fran e
aroline. hauxuniv-mlv.fr
Re eived(DayMonthYear) Revised(DayMonthYear) Communi atedby(xxxxxxxxxx)
Seismi explorationprovidesinformationaboutthegroundsubstru tures.Seismi im-agesaregenerally orruptedbyseveralnoisesour es.Hen e,eÆ ientdenoising pro e-duresarerequiredtoimprovethedete tionofessentialgeologi alinformation.Wavelet basesprovidesparserepresentationforawide lassofsignalsandimages.Thisproperty makesthemgood andidatesforeÆ ient lteringtools,allowingtheseparationofsignal andnoise oeÆ ients.Re entworkshaveimprovedtheirperforman ebymodellingthe intra-andinter-s ale oeÆ ientdependen iesusinghiddenMarkovmodels,sin eimage featurestendto lusterandpersistinthewaveletdomain.Thisworkfo usesontheuse oflappedtransformsasso iatedwithhiddenMarkovmodelling.Lappedtransformsare traditionallyviewedasblo k-transforms, omposedofMpass-band lters.Seismi data presentos illatorypatternsandlappedtransformsos illatorybaseshavedemonstrated goodperforman esforseismi data ompression.Adyadi likerepresentationoflapped transform oeÆ ientispossible,allowingawavelet-likemodellingof oeÆ ients depen-den ies.Weshowthatthe proposed lteringalgorithmoftenoutperformsthewavelet performan ebothobje tively(intermsofSNR)andsubje tively:lappedtransform bet-ter preservethe os illatoryfeatures presentinseismi data at low to moderate noise levels.
Keywords:seismi data ltering;lappedtransforms;hiddenMarkovmodels. AMSSubje tClassi ation:22E46,53C35,57S20
1. Introdu tion
Seismi explorationaimsatprovidinginformationaboutthegroundsubstru tures. This information is addressed indire tly by disturban es, arti ially reated by seismi energy sour es. The disturban es propagate through the ground, where geophysi al strata re e t the spreading wave front. Portions of the re e ted (or refra ted)wavesarethen olle tedbysensors,oftensituated neartheground sur-fa e. The one-dimensional signal a quired by a single sensor is alled a seismi tra e.Inthesimplest onvolutiveearthmodel,atra eisatime-basedsignal om-posedofthegenerateddisturban e onvolvedwiththere e tion oeÆ ientsatthe stratainterfa es.Seismi pro essing isthetask ofinferring substru tureslo ation andpropertiesfrom the olle tedsignals,withthehelpof geologi almodels. Seis-mi signalsgenerallyde rease in energyas thewavefront propagatesdeeperand ares atteredbysubsurfa eheterogeneities.Thesignalsarealso orruptedby sev-eralnoisesour esthatredu ethepossibilitytodete tessentialinformationsu has strataorfaults.Seismi data lteringisthusaprominenttaskinseismi pro essing, espe iallyas explorationaimsatimagingdeepertargets, ingeologi allydisturbed zones.
Althoughthetermwavelet(fromtheFren hondelette,orlittlewave 1
)was orig-inallyusedinseismi fortheshortsupportdira shapeddisturban e,waveletshave re-emergedonlyre entlyin geophysi saseÆ ient ompression
2
andnoise ltering tools
3 .
1.1. Related work
Duetothelargevolumesofseismi data,thedis retewavelettransform(DWT)has generallybeenpreferredtoits ontinuousintegral ounterpart.Someauthorshave nevertheless remarked that, although seismi tra es usually appear as naturally madeofphysi alwavelets,seismi imagesaresometimesmoreeÆ ientlyrepresented by other short lo al bases.It as been shown in the ontext of ompression with lappedtransforms
4
(LT)seenas lterbanksorwiththeLo alCosineTransform 5 (LCT).Theseshortlo albasesarebelievedtobemoreeÆ ientat apturingseismi os illatorypatterns,whi hbearsomesimilaritieswith texturesin naturalimages. A omparisonofvariouslo al osinetransformsforimage ompressionisgivenin
6 . Goingba kto aboutmorethan 15yearsofdevelopments,thedis retewavelet transformprovides sparsebases for naturalsignaland images. As a onsequen e, numerousDWT-basedalgorithmshavebeenproposedinthepastyearsforeÆ ient signalandimagestatisti al analysis.Forinstan e,wavelet-domain thresholdingor shrinkageisknownto provideasymptoti allyoptimalperforman e
7
observedthatwaveletde ompositionsexhibittwoheuristi propertiesoftentermed as " lustering" and "persisten e": feature-related wavelet oeÆ ients (near edges or singularities)tend to luster lo ally in asubband and to persist a rosss ales, throughthe lassi alwaveletparent- hildquad-treestru ture.Re ently,algorithms adopted tree-adaptedsubband-dependentshrinkage
8;9
. Also,sophisti atedmodels ofthejointstatisti smaybeusefulfor apturingkey-featuresinreal-worldimages. Re entapproa hesrelyonMarkovrandom elds.WerefertoL. Rabiner
10 andA. Willsky
11
foranri hoverviewoftheiruseinsignalandimage pro essing. Re ently, M. Crouse et al.
12;13
have proposed anew framework based on the hidden Markov tree. Based on
12
, H. Choi et al. 14
have proposed eÆ ient image denoising
15
as well as robust SAR segmentation 16
. HMM-based algorithms seem totakemoreadvantageofthe" lustering"and"persisten e"propertiesofwavelet oeÆ ientsaround imagefeatures. Theyyield animprovedmodelling ofthe oef- ients' statisti al dependen ies and their non Gaussian behavior.They have re- entlybeen notablyimprovedbytheiruse in the ontextof texturesegmentation withdual-tree omplexwavelets
17
andsteerablepyramids 18
.
1.2. Main ontribution
We propose in the present work to extend the use of hidden Markov models to a lapped transform domain for seismi data ltering. This work has been partly presentedin
19
. LT are usually viewed as blo k-transforms, omposed of M pass-band lters. The superiority of lapped transforms over wavelets may ome from additionaldesign exibilityandshortlo albases.
Lappedtransforms were generallynotoften used in ompression on denoising algorithms,duetothesuperiorityoftheinter-s ale oeÆ ientdependen yobtained fromthewaveletdyadi de omposition.Though,T.Tranetal.
20
havedemonstrated thatwell-designedLTareabletoimproveonDWTfornaturalimage ompression, in the Embedded Zero-tree framework
21
. In the ontext of denoising, the LT o-eÆ ientsare rearrangedinto an o tave-likerepresentation. Theresulting "s ales" bearthesame lusteringandpersisten epropertiesasinthewaveletrepresentation. Moreover,LTdesignmayenfor ebothorthogonalityandlinear-phase(in ontrast to non-Haar 1D wavelets), as well as attra tiveadditional degrees of freedom in design.
The superiority of lapped transformsover wavelets may ome from these de-sign degrees of freedom and sharper frequen y attenuation properties of the M lters(potentiallyredu ingaliasinge e ts a rossthesubbands). Oneother inter-esting featureisbasedonZ. Xionget al.
22
transforms,relyingon 12;15
. Spe ial areis taken in the designof the LTused, to assesstheanisotropi shapeofsomeseismi surveys.
Inthefollowing,we rstaddressthephilosophybehindthere entlydeveloped wavelet-domainhiddenMarkovmodelsinSe tion2.Then,wefo usonthe ombi-nationoflappedtransformsandhiddenMarkovmodelsinSe tion3,wherelapped transformsdesignandpropertiesarereviewed,aswellasthedyadi representation ofblo ktransforms.Se tion 4brie y des ribesproperties ofseismi datawith an emphasisontheiros illatorynature.Obje tiveandsubje tivedenoisingresultsare dis ussedin Se tion5.Finally, wedrawsome on lusions ontheproposed lapped transformbasedhiddenMarkovmodeldenoisingalgorithm,aswellaspossible im-provements.
2. Wavelet-Domain HiddenMarkovModels
Undertheadditivenoiseassumption,animagex anditsnoisyobservationx n
are usuallywrittenas
x n
(i;j)=x(i;j)+n(i;j); (2.1) where n is a random noise. The joint probability density fun tion of the family of imagesthat x belongs to is oftenunattainable. Basedon wavelet approximate de orrelation, simpler models have been proposed for oeÆ ient modelling. The simplestindependentGaussianmodelsgenerallyobtainimprovementsfromresidual inter- oeÆ ientsdependen ies.
M.Crouseetal. 12
havere entlyproposedanewframeworkforstatisti alsignal pro essing,basedonwavelet-domainhiddenMarkovmodels(WD-HMM).
Let w j;k
denote a wavelet oeÆ ient at level j, 1 j J, with j = 1 orre-spondingtothe oarsestwavelets ale.Themarginalpdf fortheasso iatedrandom variableW ismodelledasaGaussianmixtureofN
S
omponents.Intheframework ofhiddenMarkovmodels
23
,adis retehiddenstateS j;k
isasso iatedtoea hw j;k withaprobabilitymass fun tion P(S
j;k
=s)givenforea h states, 1sN S
. Whilethevaluesw
j;k
areobserved,thevalueofthestateS isgenerallyunknown. Depending on the a tual state s the oeÆ ient, the onditional pdf of W given S =sis given byf
WjS
(wjS =s),modelled as aGaussian distribution,des ribed inEq.2.2: g s (w; s j;k ; s j;k )= 1 p 2 s j;k exp 0 w s j;k s j;k ! 2 1 A (2.2) where x and 2 x
arethemeanandthevarian eofg.Thepdf ofW isgivenby:
p W (w j;k )= NS X s=1 P(S j;k =s)f WjS (wjS =s): (2.3)
Based onheuristi s developed forimage ompression 21
belongingtoeitheralargeLorsmallS statedependingonwhetherthe oeÆ ient is lo ated near a dis ontinuity or not.The asso iated probabilities p
L = p
1 j
(the supers ript 1referringto therootnode) andp
S
=1 p 1 j
possesslarge andsmall varian esrespe tively. Sin ethe oeÆ ientsw areobtainedbypass-bandor high-pass lters,theyareassumedtohavezeromean.Themodel anbefurtherredu ed by onsideringthatthevarian esare onstanta rossea hs alej,foragivenstate s.Asa onsequen e,Eq. 2.3resultsin thefollowingmarginaldistribution:
p W (w j;k )= X s2fL;Sg p s g s (0; s j;: ): (2.4)
TheHMT(hiddenMarkovtree)modelisoftendes ribedasaquad-tree stru -tured probabilisti graph that aptures the statisti al properties of the wavelet transformofimages.TheHMTmaterializesthe ross-s alelinkbetweenthehidden states.It drawsinspirationfrom zero-treeor hierar hi altrees image ompression systems
21;24
.An illustrationofanHMTis depi tedin Figure1.
Fig. 1. Diagram of a hidden Markov tree in a quad-tree. White dots represent hidden states witharrows asdependen ies,bla kdotsthe transformed oeÆ ients.Thebla kdot onthe top representsaparent oeÆ ientwithitsfour hildren.
Thepersisten epropertyismodelledbyamarkoviandependen ybetween par-ent and hildren hidden states at onse utive s ales. A state S
j
asso iated with the hild oeÆ ient w
j;k
at s ale j depends only on the state S (j)
of its parent oeÆ ientw
(j)
ats alej 1.Thetransitionprobabilitiesbetweenthetwostates s
1
(parent)and s 2
( hild) anbedes ribedbythetransition matrix j ,givenby: s 1 !s 2 j;(j) =p S j jS (j) (S j =s 2 jS (j) =s 1 );s 1 ;s 2 2fL;Sg: (2.5) TheWD-HMT is ompletelyde ned bythesetofmodelparameters:
Expe tationMaximizationalgorithmsfor ttingaHMTusingtheMinimumLength Des ription riterion.Wereferto
12;15;25
fordetailsontheimplementationofhidden Markovtrees.
3. Lapped transformsand HMT models 3.1. Generalities onLapped Transforms
Fig.2. Blo k diagram of a M- hannel maximally de imated lter bank (H k
(z): analysis and F
k
(z):synthesis).
The LappedOrthogonal Transform 26
(LOT) hasbeen developed to over ome annoyingblo kingartifa tsarisingfromnonoverlappingblo ktransformssu has theDis rete Cosine Transform. Moregenerally, lapped transformsare de ned as linear phase paraunitary lter banks (FB). Figure 2 shows a typi al M- hannel maximally de imated lter bank. The kth analysis and synthesis subband lters are denoted by H
k
(z) and F k
(z), respe tively. The lter bank may be eÆ iently representedbyitspolyphase formbyE(z)(type-Ianalysispolyphasematrix)and R(z)(type-IIsynthesispolyphasematrix),de nedby:
[H 0 (z)H 1 (z):::H M 1 (z)℄ T =E(z M ) 1z 1 :::z 1 M T : (3.7) and [F 0 (z)F 1 (z):::F M 1 (z)℄= z 1 M z 2 M :::1 R(z M ): (3.8) TheanalysisandthesynthesisM-band FBpolyphasematri esR(z)andE(z) (representedinFig.3)provideperfe tre onstru tionwithzerodelayifandonlyif:
R(z)E(z)=I M
; (3.9)
whereI M
istheidentitymatrix 27
.LTmaybeparameterizedthrougheÆ ient lat-ti estru turesfor ost-drivenoptimization.Wereferto
26;27;20
trans-Fig. 3. Blo k diagram of the M- hannel maximally de imated lter bank from Fig. 2 with polyphaseimplementation.
anevennumberM of hannels;
FIR lterswithlinearphaseandlengthLmultiple ofM (L=KM);
a large lass of LT, alled Generalized Lapped BiorthogonalTransforms (GLBT) mayberewritteninthefollowingform:letU
i andV i beinvertiblematri es, i and W de nedfori21;:::;nas:
i = U i 0 0 V i ; (3.10) W = I I I I ; (3.11) (z)= I I I z 1 I ; (3.12) K 0 = 0 W;andK i (z)= 1 2 i W(z)W: (3.13) Thentheanalysispolyphasematrix anbefa toredas:
E(z)= 0 Y K 1 K i (z); (3.14)
with anappropriate hoi e ofinvertiblematri es U i
and V i
. Theinversesynthesis polyphase matrix followsby element-wise inversionof matri es in Formula 3.10{ 3.13.
3.2. Lapped transform optimization
orthogonal matrix of size M=2M=2 in a produ t of M(M 2)=8 elementary rotations
28
.Inaddition,everyinvertiblematrixU anbefa toredintotheprodu t U l U U r ,whereU l andU r
aretwoorthogonalmatri esandU
isadiagonalmatrix with non-negative elements
i
. Su h a de omposition is summarized in Figure 4 fora44invertiblematrix.
Fig.4.Givensde ompositionofan44invertiblematrixU.
Sparsetransformsaregenerallydesiredforsignaldenoising.Itisalsodesirable to design transforms with redu ed aliasing in the transform domain. Transforms an be obtained using un onstrained non-linear optimization of a weighted sum of popular ost riteria for ompression: generalized oding gain G
, stop-band attenuation for the analysis and synthesis lter bank A
a sb and A s sb , DC leakage A d
and attenuationatmirrorfrequen iesA mf ,detailed inEquations3.15{3.19. G =10log " M 1 Y i=0 x i x kF i k 2 2 # 1=M ; (3.15) A a sb = M 1 X i=0 Z !2 i H i e j! 2 d!; (3.16) A s sb = M 1 X i=0 Z !2 i F i e j! 2 d!; (3.17) A d = M 1 X i=2;4;::: L 1 X j=0 H i ( j) ; (3.18) A mf = M 1 X i=0 H i e j! i 2 : (3.19) We referto 20
fordetails on LT optimization. It was shown in 4
that a lapped transformoptimization allowedsuperiorseismi data ompressionresults,as om-paredto wavelet oding, as proposed by P. Donohoet al.
2
joint optimization of the oding gain and the other ost riteria. Di erent are estimated for di erent types of seismi data, in both the horizontal and verti al dire tion.
3.3. Dyadi remapping of Lapped Transforms
Sin ehiddenMarkovtreemodelsarebasedonaquad-treestru ture,theiruseina lappedtransformframeworkrequiresasimilararrangementfortheLT oeÆ ients. A LT proje tssignalsonto M equally spa ed frequen ybands, in ontrast to the o tave-bandwaveletrepresentation.Fortunately,su haarrangementispossibleif the number of hannels M is a power of 2 (typi ally 8 or 16). The transformed oeÆ ientsbearano tave-likegrouping,with J =log
2
M de ompositionlevels 22
. Inonedimension,foronegroupofM transformed oeÆ ients,theDC omponent ( orresponding to the average of the signal oeÆ ients) is assigned to the lower s ale subband. Then, from low to high frequen ies, the kth subband is formed respe tively from the next groupof 2
k
=2 oeÆ ients. The J +1 groupsare then asso iatedwithrespe tto theblo kpositionin thesignal.Figure 5illustrates the dyadi rearrangementfor two onse utive blo ks of M =8 oeÆ ients. The two blo ksof8=2
3
oeÆ ients(dotsontopofFigure5)arerearrangedintoJ+1=4 groupsandyieldathree-levelde omposition(dots onbottomof Figure5).Intwo
Fig.5.Dyadi rearrangementof1DLT oeÆ ients:(Top)blo k-transformwithuniformfrequen y partition,(Bottom)o tave-likerepresentation.
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Fig.6.Dyadi equivalen e between 2Dwavelet andLT oeÆ ients: (Left) two-level o tave-like representation,(Right)four- hannelblo k-transformwithuniformfrequen ypartition.
4. Generalities on Seismi Data
Seismi explorationaimsatprovidinginformationaboutthegroundsubstru tures. This information is indire tly addressed by disturban es, arti ially reated by seismi energy sour es. The disturban es propagate through the ground, where geophysi alstratare e tthespreadingwavefront.Portionsofthere e ted(or re-fra ted)wavesarethen olle tedbysensors(geophones,representedbysquaresin Fig.7),oftensituatednearthegroundsurfa e.Theone-dimensionalsignala quired byasinglesensoris alledaseismi tra e.Inthesimplest onvolutiveearthmodel, atra eisatime-basedsignalmadeofthegenerateddisturban e onvolvedwiththe re e tion oeÆ ientsatthestratainterfa es.There e tionanda quisitionof seis-mi signalsisrepresentedonFigure7.Disturban esprovokedbytheseismi sour e (depi ted by triangles on Fig. 7) propagate along the ray-paths (represented by dashedlines).Ea hlo ationonthere e torisilluminatedbyseveralpropagations between ouplesofsour eandre eptor.
Seismi pro essing is the task of inferring substru ture lo ation and proper-ties from the olle tedsignals,with thehelp of geologi almodels.Seismi signals generallyde reasein energyas the wavefrontpropagatesdeeperand is s attered by subsurfa eheterogeneities.Thesignalsare also orrupted severalnoisesour es thatredu e thepossibilityto dete tessentialinformationsu has strataorfaults. Seismi data lteringisthus aprominenttask inseismi pro essing,espe iallyas explorationaimsatimagingdeepertargets,ingeologi allydisturbedzones.We re-ferto thebook by
O.Yilmaz 29
Ground surface
mirror point
raypath
source
geophone
Legend:
Reflector
Fig.7.Seismi a quisitionforonehorizontallayerandthreedi erentshotpoints.
Fig.8.Exampleofsta kedseismi datain lassi alimageform(left)andwithawiggleplot(right).
Figure8thusprovidesase tionofthegroundsubstru ture.Theverti aldire tion isa fun tion of time, sin eit depends onthe time ofarrivalof the wavefrontto ea hsensor. Itisnot orre tedwiththevelo ityin ea hstrata,andthusdoesnot providedire tlyinformationonthedepthofthesubstru tures.Figure9depi tsthe rstverti altra eobtainedfromFigure8.Theos illatorybehaviorofseismi data learlyappearsfrom Figure9.It justi estheuse oftransforms apableof aptur-ing these os illations. The rossings appearing on the seismi image are zones of interest, whi h shall not be blurredby denoising pro edures. The de omposition oeÆ ientsmagnitudeforthe30-taporthogonalCoi etanda32-taplapped trans-form,for one verti al signal, are depi ted in Figure 10, from the low-passto the high-passsubband (left to right). Figure 11 displaysthe same oeÆ ients, sorted by de reasingmagnitude. We remark that the smallest oeÆ ients (on the right hand side)with the Lapped Transformare substantiallysmallerthan that of the wavelet (dotted blue). This behavior illustrates how a lapped transform yields a sparser de omposition,with generallyless large and more small oeÆ ients. The sparsityofthetransformisillustratedin 2Din12{14.
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Fig.9.Firstseismi tra e(verti aldire tion)fromFig.8.
5. Experimentalresults
5.1. Comments on the dyadi remapping
Figures12{14illustratethee e tsofde ompositionontheseismi data.The trans-formed oeÆ ients
k
are res aledby a geometri alfa tor following sign( k
)j k
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Fig.10.Waveletandlappedtransform oeÆ ientsobtainedfromthede ompositionofthesignal fromFig.9.
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Fig.11.Waveletandlappedtransform oeÆ ientsfromFig.10,sortedinde reasingorder.
tothenoise.Figure13representstheblo k-wisede ompositionofseismi data(see thediagramontherightofFig.6). Thetransformed oeÆ ientsarerearrangedin Figure14.ComparedtoFigure12,thisrepresentationexhibitslesshighmagnitude oeÆ ients(brightdots),yieldingasparserde omposition.Theanisotropi ontent oftheseismi image suggeststhat separatemodels an beusedforthehorizontal, diagonalandverti alsubbandsofthewavelettree,in ontrasttowhatisobserved innaturalimages,forinstan einM.Doetal.
18
,whereitissuggestedthatwavelet oeÆ ientsatthesames aleandlo ationbut di erentorientationsshouldbetied uptogethertohavethesamehiddenstate.
Fig.12.Dyadi representation ofseismi data fromFig. 8obtained froma three-level wavelet de omposition(Coi et30-tap lters).
5.2. Choi e of the lapped transform
Fig.13. Blo krepresentation ofseismi data fromFig.8obtainedfroma eight- hannellapped transform.
Fig.14.Dyadi representationofseismi data fromFig.8obtainedfromaeight- hannellapped transform(Fig.13afterdyadi remapping).
5.3. Results
The denoising results are addressed in both the obje tive and subje tive sense. Obje tiveresults aredes ribed in termsofsignal-to-noiseratio(SNR): lets
k , s
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Index
Basis vectors
Fig.15.Basisve torsforaneight- hannel32-taporthogonallappedtransform.
s d k
bethesamplesoftheoriginal,thenoisyandthedenoiseddatarespe tively.
SNR=20log 10 X k s 2 k (s k s d k ) 2 ! : (5.20)
Table1.Obje tivedenoising results omparisonat var-iousinitialsignal-to-noiselevelsindB.
Noisydata Wavelet(Coif30) (a) LT(832) (b) 21.9 30.1 29.9 24.4 31.7 32.1 26.0 32.7 33.2 29.1 34.8 35.3 34.0 38.3 39.0 40.0 42.4 43.2 43.0 44.6 45.4 Note: Tablenotes
a
Two- hannel30-tapCoi et lterbank b
Eight- hannel32-taporthogonallappedtransform
Subje tive results also are of spe i importan e for seismi data quality as-sessment.Itisparti ularlyimportantthatdenoisingdoesnotblurtheground sub-stru ture. Therefore,it isuseful to arefullyobservethedenoised data,as wellas thenoiseremovedbythe lteringpro edure,asillustratedinFigures 16{19.Ea h gure representsthedenoised data s
d k
(left panel)and the removednoise(or dif-feren e se tion),i.e. s
n k
s d k
,ontherightpanel. Themajorrequirementsarethat featuresremain learin thedenoisedimageandthatthedi eren ese tionexhibits as fewstru turednoiseas possible.
Figures16{17displaythedenoisedimageswithaninitialmoderate40dBnoise. Clearer stru ture preservation is apparent at the top of the seismi se tion after lappedtransformdenoising:theutmosttopalignmentsontheseismi image have apparentlymergedafterwaveletdenoising.Thisfeatureismorepronoun ed(more oherent on neighboring tra es) on the wavelet denoised di eren e se tion. We on ludethatat moderateSNRs, lappedtransformsgenerallypreserveseismi in-formationbetterthatwavelets,whileobje tivemeasuresdonotdi erbymorethan 1dB.
Seismi featurespreservationis leareratlowerSNRs, asillustratedinFigures 18{19.Oversmoothingisobservedafterwaveletdenoisingontheleftof19,espe ially at the bottom of the image. Crestand valley alignmentsin thewigglesalign less evidentlythanin theLT ase.Di eren ese tionsfrom Figures 18{19(righthand side) learlyshowthatalotmoreofstru turedinformationisremovedwithwavelet, as ompared to lappedtransform denoising.Similarobservationswere derivedon texturepreservationinnaturalimages
30
.Texturesandseismi seemtosharesimilar os illatory ontent, givingan advantage onlapped transform de omposition over waveletbases fordenoising.
6. Con lusions and dis ussion
We propose to extend the use of hidden Markov models to a lapped transform domain for seismi data ltering. Lapped transforms are onverted to a dyadi likerepresentation, to a ount forinter-s ale oeÆ ient dependen ies.Due to the os illatorynature ofseismi data,os illatoryproje tionbasesyieldsparer de om-position of the data. Moreover, lapped transform enjoy improved design degrees of freedom. They allow to design data adapted transforms. Sharper attenuation between the lter frequen y bands also redu es aliasing e e ts in the frequen y domain. We show that lapped transform based denoising generally outperforms wavelet denoising using an obje tive SNR measure.More important, we demon-strate that lapped transforms better preserve seismi information (subje tively), sin e they ause less blurring than wavelet and the removed noise ontains less oherentgeologi stru tures.
Fig.16. Seismi image (left)and di eren e se tion(right) withlapped transformbased HMT denoisingat40.0dB.
to the low-passsubband. Future works will fo us on a better ontrol of the low-passapproximationimage,possiblybyahierar hi allappedtransformwithshorter support, to redu e edge artifa ts on the smaller approximation. Improvement is alsopossiblewiththeuseofmoreinvolveddire tionaltransformsorshift-invariant implementation,sin ethelappedtransformsusedinthisworkaremaximally de -imated.
A knowledgment
The rstauthorwouldliketothankH.Elloumiforprogrammingsomeofthe lter bankroutinesusedin thiswork.
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Fig.17.Seismi image(left)anddi eren ese tion(right)withwaveletbasedHMTdenoisingat 40.0dB.
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