Wavefront healing operators for improving reflection coherence

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1 Wavefon healing opeaos fo impoving eflecion coheence David C. Henley Wavefon healing ABSTRACT Seismic eflecion image coninuiy is ofen advesely affeced by inadequae acquisiion o pocessing pocedues by he inefeence of vaious kinds of noise o by he iegula naue of he ecoding suface. In siuaions whee convenional mehods fail a simple pocessing echnique called wavefon healing can incease eflecion coninuiy enough o significanly impove he inepeaion. The opeaion can be applied o eihe ace gahes o sacked secions. Two diffeen opeaos based on he Huygens wavele consucion model of seismic wavefon popagaion have been implemened in PoMAX a CREWES and ae demonsaed hee on boh eal and model daa. INTRODUCTION Someimes in spie of bes effos saisfacoy images of seismic eflecos canno be obained by convenional means. One of he moe common causes of his siuaion is he pesence of iegulaiies in he eah jus beneah he suface. Even if a seismic wavefon euns o he viciniy of he suface as a elaively coheen efleced even i may be disuped by avesing he las few mees o he suface when he neasuface maeial is iegula o ess on a ugged bedock suface. This causes saic delays in small poions of a euning wavefon as well as scaeing and mode convesions fom disconinuiies in he eah maeials. As a esul he wavefon image exhibis sho-wavelengh saic jie and ampliude flucuaions due o inefeence by vaious noises. Figue 1 poays his disupion schemaically. Diffacions on seismic secions ae familia o geophysiciss and indicae he need o migae he daa o emove he smoohing effecs of he physical wavefons fom he eflecion images. Less familia is he idea of delibeaely inoducing diffacion ino image pocessing. Huygens wavele consucion is a hisoical geomeical echnique based on diffacion fo popagaing eflecion images on ime secions. I can also be used o inoduce he smoohing effecs of diffacion ino eflecion images. A diffacion esuls physically wheneve he local cuvaue of an ineface exceeds he cuvaue of an inciden wavefon. Consequenly Huygens wavele consucion using waveles wih cuvaue smalle han ha of eflecion image iegulaiies can be used o popagae eflecion images and smooh hem simulaneously. Physically he pocess simulaes buying he eah s suface beneah a laye of unifom maeial and ecoding a wavefon afe i popagaes o he new suface via waveles oiginaing a poins coinciden wih eceive locaions on he oiginal suface he wavefon being smoohed o healed by is ansi. The acual ecoding suface can be moved afe he daa ae ecoded bu he suface can be effecively moved fuhe away duing pocessing using he Huygens wavele consucion in a simple pocessing algoihm. The echnique descibed is simila o CREWES Reseach Repo Volume 1 (000)

2 Henley wave equaion dauming (Beyhill 1979) (Yang 1999) o o modelling by demigaion (Jaamillo e al 1997) (Jaamillo e al 1999) bu is simple han eihe. To popagae a eflecion image wih Huygens waveles a velociy is chosen (which deemines he wavele adius cuvaue and popagaion ime) and cicula waveles ae consuced using closely spaced poins on he eflecion image as oigins of diffeen waveles of he same adius (see Figue 1). The envelope (summaion) of he esuling wavele ampliudes in he diecion of popagaion foms he popagaed image of he eflecion. The eflecion image inceases in smoohness and coheence wih each popagaion sep depending diecly upon he chosen popagaion velociy. Figue shows he ineface in Figue 1 afe wo seps of Huygens wavele popagaion. The inceased smoohness wih espec o he disoing ineface is significan. This heuisic model hen is he basis fo a pai of pocessing opeaions called wavefon healing as deailed and shown below. ALGORITHM DETAILS Wavefield images in digial seismic daa ae no by naue coninuous bu ae disceised using gids defined by aces and ime samples. The wavefon healing algoihm is consained by his fundamenal sampling; is sep size and wavele adius ied in a basic way o he sample incemen in boh he X and T dimensions. Each of he algoihms descibed hee has been oganised as a gaheing opeaion in which a seleced oupu poin is he sum of veco ampliudes fom all poins a a consan (ealie) delay ime on a Huygens wavele suface cened on he oupu poin. In he hee-poin algoihm each ace akes ampliudes only fom iself and is wo neaes neighbou aces) o compue oupu while in he five-poin algoihm ampliudes ae also used fom he nex-neaes neighbou aces. The Huygens consucion fo he hee-poin algoihm and he elaionships beween vaious disances and delay imes is shown in Figue 3 while he discee fomula is expessed in equaion 1: A( n T + d) = W{ A( n T ) + cos( α ) + cos( α ) l [( / d ) A( n 1 T + d) + ( / d ) A( n 1 T )] dl [( / d ) A( n + 1 T + d) + ( / d ) A( n + 1 T )] d vl v } ; (1) cos( α ) = l vl / cos( α ) = v / vl hl dl = = [ x( n) x( n 1) ]/ v = [ x( n + 1) x( n) ] = d vl hl d = d v v h = h / v = d W = 1/ [ 1+ cos( α ) + cos( α )] l CREWES Reseach Repo Volume 1 (000)

3 Wavefon healing whee A(nT) is he wavefield ampliude on ace n a ime T; d is he sample ineval; x(n) is he souce-eceive offse of ace n; hl h ae hoizonal delay imes beween poins idenified in Figue 3; vl v dl d ae veical delay imes beween poins idenified in Figue 3; is he Huygens adius; v is he popagaion velociy; and W nomalises he veco summaion wih espec o uniy. The deails fo he five-poin wavefon healing algoihm ae illusaed in Figue 4 and he discee fomula is given in equaion : A( n T + d) = W { A( n T) + cos( α ) l1 + cos( α ) l + cos( α ) 1 + cos( α ) [( / d ) A( n 1 T + d) + ( / d ) A( n 1 T) ] dl1 [( / d ) A( n T + d) + ( / d ) A( n T + d) ] dl [( / d ) A( n + 1 T + d) + ( / d ) A( n + 1 T )] d1 [( / d ) A( n + T + d) + ( / d ) A( n + T + d) ] d vl1 v1 vl v } ; () cos( α ) = ( l1 vl1 + d) / cos( α ) = ( 1 v1 + d) / cos( α ) = l vl / cos( α ) = v / vl1 = hl1 d v1 = h1 d vl hl1 hl dl1 = = = [ x( n) x( n 1) ]/ v = [ x( n + 1) x( n) ] [ x( n) x( n ) ]/ v = [ x( n + ) x( n) ] = d vl1 hl v h1 v d1 = = d v1 h / v / v dl = d = d vl W = 1/ [ 1+ cos( α ) + cos( α ) + cos( α ) + cos( α )] l1 d = d v l 1 whee A(nT) is he wavefield ampliude on ace n a ime T; d is he sample ineval; x(n) is he souce-eceive offse of ace n; hli hi i=1 ae hoizonal delay imes beween poins idenified in Figue 4; vli vi dli di i=1 ae veical delay imes beween poins idenified in Figue 4; is he Huygens adius; v is he popagaion velociy; and W nomalises he veco summaion wih espec o uniy. The choice of popagaion velociy as well as he X co-odinaes of he aces deemines he adius of he wavele and hence he inesecions of he wavefon wih he aces. Eihe algoihm chooses he minimum adius ha will foce he wavele o inesec boh neaes neighbou aces (and nex-neaes aces in he case of he five-poin algoihm) and hus include hei ampliudes. The oupu of he CREWES Reseach Repo Volume 1 (000)

4 Henley healing opeao consiss of a veco sum of all ace ampliudes ineseced by he wavele assigned o he poin a he oigin of he wavele. The veco sum equies ha ace ampliudes ae weighed by he cosine of hei incidence angle wih espec o he wavele oigin. This paicula implemenaion of wavefon healing is appopiae fo land seismic daa in which paicle velociy o acceleaion vecos ae he usual seismic measuemens bu no fo pessue scala measuemens which occu in maine daa. Alhough boh wavefon healing algoihms ae designed fo unifomly gidded daa hey can accommodae iegula offse spacing on he ace gahes o which hey ae applied. Time delays cosine weighs and inepolaion weighs all depend upon specific souce-eceive offses and ae compued individually fo each ace. Also hough inended pimaily fo upwad coninuaion in which evens ae popagaed o geae ime as hey heal boh modules have he opion o un in evese downwad coninuaion mode. The fome is quie simila o a local demigaion while he lae has no paicula physical analogy and hus no much acual jusificaion fo use excep ha i impoves eflecion coninuiy. Like many gid-based echniques wavefon healing is subjec o sabiliy consains elaed o he sampling ineval in boh dimensions of he inpu daa. Specifically he velociy chosen fo he wavele should be lage enough ha he poduc of velociy and ime sample incemen exceeds he nominal disance incemen beween aces. When he velociy is lowe han his sabiliy cieion aliasing and dispesion will occu and become appaen afe moe han one o wo seps of healing. Because wavefon cuvaue is invesely popoional o velociy vey lage velociy values cause wavefon healing o appoach laeal ace-mixing in is effecs. Anohe effec associaed wih wavefon healing is he loss of highe fequencies fom eflecion evens being healed he loss being geaes fo low velociies. Since hese echniques ae inended only fo applicaion o daa wih vey poo coheen signal-o-noise aio some loss of enegy a highe fequencies may well be accepable in ode o enhance oveall even coheence. SOFTWARE The wavefon healing opeaos descibed above in equaions (1) and () and Figues 3 and 4 have been implemened as PoMAX opeaions called Wavefon healing and Wavefon healing II especively whee he lae is he five-poin algoihm. Boh modules ae included in he 000 CREWES sofwae elease and ae fully documened in he help files. The paamees of which hee ae only hee ae he same fo eihe module. Specifically he fis paamee is he popagaion velociy o be used fo he Huygens wavele consucion he second is he numbe of popagaion seps o apply and he hid is simply he choice of upwad (defaul) o downwad coninuaion. The lae em may cause some confusion alhough used hee in he convenional sense as upwad coninuaion acually popagaes he eflecions downwad (geae in ime) on he secion o gahe. CREWES Reseach Repo Volume 1 (000)

5 Wavefon healing EXAMPLES To demonsae he acion of wavefon healing on seismic aces hee synheic examples ae pesened followed by wo field daa examples. Figue 5 shows he hee synheic models buil o illusae he wavefon healing opeaions. Figue 5a is an isolaed bandlimied spike useful fo illusaing he wavefon healing opeao impulse esponse ; Figue 5b is a simulaed plane efleco wih a gap; and Figue 5c is a simulaed plane efleco wih vaious saic displacemens. The lae wo models exhibi some of he kinds of even disubances ha wavefon healing is inended o addess. The hee-poin wavefon healing algoihm is illusaed fis in Figue 6 whee he opeao has been applied fo one sep and in Figue 7 whee wo seps of healing have been applied. A convenien way o hink of he impulse esponses in Figues 6a and 7a is as Huygens wavefons popagaing downwad o geae ime fom a diffacing poin. The speading and smoohing acion of he opeao is eadily eviden no only in he impulse esponses bu in he model eflecion esponses in Figues 6b 6c 7b and 7c. Noe as well ha each sep of healing shifs he eflecion even o geae ime. This simulaes buying he old suface beneah a laye of unifom maeial and placing new deecos on op of he new laye. Figues 8 and 9 feaue he impulse and model esponses fo he five-poin wavefon healing algoihm fo one and wo seps especively. I is obvious ha his algoihm has oughly wice he smoohing acion pe sep of he hee-poin algoihm fo he same velociy. Wha may be less appaen howeve is he fac ha he five-poin algoihm also exhibis geae aenuaion of high fequencies pe sep. Alhough no shown hee unning eihe algoihm in he evese diecion (downwad coninuaion insead of upwad coninuaion) eveses he cuvaue of he impulse esponses and shifs eflecions o smalle ime. The acion of he algoihms is exacly he same in eihe diecion. To illusae he acion of wavefon healing on field daa a sho gahe was seleced fom he Blackfoo -D 3-C suvey of 1995 and is shown in Figue 10. This gahe is elaively good qualiy daa and would no odinaily waan he use of wavefon healing alhough adial ace fileing would be consideed. Thee ae howeve enough disubances of vaious kinds exhibied on he eflecions o illusae wavefon healing effecively. These include inefeing paens of coheen noise andom noise small amouns of saic jie a low level of andom noise a ace ovewhelmed by 60 Hz hydo-line pickup and a couple of aces consising of high fequency andom noise. Figues 11a and b show he effec of applying one sep of hee-poin and fivepoin wavefon healing especively using a velociy of 3000 m/s. The diec aival velociy was chosen hough a wide ange of ohe velociies can be used. As can be seen in he figues wavefon healing wih eihe algoihm damaically educes ace-o-ace flucuaions along eflecions wih he effec being geaes fo he fivepoin algoihm as expeced. CREWES Reseach Repo Volume 1 (000)

6 Henley Since wavefon healing can also be applied o sacked secions a sack of he veical componen daa fom he Blackfoo suvey is shown in Figue 1 while Figues 13a and b show he esuls of applying one sep of wavefon healing fo he 3-poin and 5-poin algoihms especively. As can be seen wavefon healing acs as a vey effecive cleaning agen fo esidual noise paiculaly noise which vaies apidly laeally. DISCUSSION I is ineesing o noe ha he acion of wavefon healing is elaively genle alhough as demonsaed on he synheic models one o wo seps of healing can have a significan effec on he appeaance of a gahe o secion of eal seismic daa. A vey good example of his is shown in an accompanying chape (Henley 000) whee wo vey maginal sacked secions have hei laeal coninuiy damaically impoved by only one o wo passes of he hee-poin wavefon healing algoihm. Fuhemoe expeience has shown ha seismic daa need no coespond o he model on which wavefon healing is based (emeging wavefons disuped by a ugged ineface) in ode o benefi fom he echnique. In fac anohe siuaion ha benefis fom wavefon healing is inefeence by coheen noise if he noise wavefons slope significanly. Wavefon healing in his case popagaes only he hoizonal veically avelling wavefons and aenuaes he es. One side effec of wavefon healing ha should be eieaed is he educion in esoluion in boh X and T dimensions. Evey individual se of daa will be affeced diffeenly by wavefon healing so each new se should be esed wih an opeao ying diffeen velociies and numbes of seps. In geneal howeve he wavefon healing opeao inceases laeal coheence a he expense of esoluion in boh dimensions. Highe popagaion velociies lead o moe laeal smeaing and less bandwidh loss while lowe velociies lead o less laeal smeaing bu geae high fequency loss and someimes causes aliasing if he sabiliy cieion has been violaed. Finally all effecs of wavefon healing ae cumulaive each healing sep compounds he effecs of pevious seps whehe inceased coheence o los esoluion. CONCLUSIONS When convenional seismic acquisiion and/o pocessing mehods fail o povide seismic daa of sufficien qualiy o fom usable images an unconvenional echnique like wavefon healing can someimes be used o foce a ade-off beween even esoluion and coheence deceasing he fome in favou of he lae. In paicula fo vey noisy daa gaheed wih vey high esoluion (bu someimes low fold) wavefon healing can povide a way o make use of disappoining suvey esuls. ACKNOWLEDGEMENTS The auho wishes o acknowledge CREWES saff and sponsos fo suppo and discussion of he maeial pesened. CREWES Reseach Repo Volume 1 (000)

7 REFERENCES Wavefon healing Beyhill J.R Wave equaion dauming: Geophysics 44 no Henley D.C. 000 Hash imaging echniques fo shallow high esoluion seismic daa CREWESeseach epo 1. Jaamillo H.H. and Bleisein N Demigaion and migaion in isoopic inhomogeneous media: Annual Meeing Absacs Sociey of Exploaion Geophysiciss Jaamillo H.H. and Bleisein N The link of Kichhoff migaion and demigaion o Kichhoff and Bon modelling: Geophysics 64 no Yang K Wave equaion dauming fom iegula suface using finie-diffeence scheme: Annual Meeing Absacs Sociey of Exploaion Geophysiciss CREWES Reseach Repo Volume 1 (000)

8 Henley FIGURES Suface Receives Disubed wavefon Huygens waveles Rugged ineface Emeging wavefons Figue 1. Coheen wavefons fom deep eflecions being disuped by passage hough a ugged ineface. Huygens wavele consucion illusaes how he disubed wavefon can be consuced by consideing he ineface o be composed of poin souces. Paially healed wavefon Suface Receives Disuped wavefon Rugged Ineface Emeging wavefons Figue. Disuped wavefon afe one fuhe sep of popagaion. Huygens wavele consucion used o popagae disuped wavefon fom Figue 1 o he new posiion. Noe he inceased smoohness of he popagaed wavefon. CREWES Reseach Repo Volume 1 (000)

9 Wavefon healing Huygens avelime cuve hl h T + d d vl α l α v dl d n - 1 n n + 1 T Figue 3. The vaious imes and angles used in he hee-poin wavefon healing algoihm expessed in equaion 1. Huygens avelime cuve hl1 h1 hl h T + d d vl v dl α l α l1 α 1 α d T + d d vl1 v1 n - dl1 n - 1 d1 n n + 1 n + T Figue 4. The vaious imes and angles used in he five-poin wavefon healing algoihm expessed in equaion. CREWES Reseach Repo Volume 1 (000)

10 Henley S p ik e Figue 5a. Bandlimied spike o es wavefon healing algoihms G a p Figue 5b. Even wih gap o es wavefon healing algoihms Saics Figue 5c. Even wih simulaed saics o es wavefon healing algoihms CREWES Reseach Repo Volume 1 (000)

11 Wavefon healing S p ik e Figue 6a. Spike esponse of one pass of he hee-poin wavefon healing opeao G a p Figue 6b. Response of even wih gap o hee-poin wavefon healing opeao Saics Figue 6c. Response of even wih simulaed saics o hee-poin wavefon healing opeao CREWES Reseach Repo Volume 1 (000)

12 Henley Spike Figue 7a. Spike esponse o wo passes of hee-poin wavefon healing opeao G a p Figue 7b. Response of even wih gap o wo passes of hee-poin wavefon healing opeao S a ic s Figue 7c. Response of even wih simulaed saics o wo passes of wavefon healing opeao CREWES Reseach Repo Volume 1 (000)

13 Wavefon healing Spike Figue 8a. Spike esponse o one pass of five-poin wavefon healing opeao G a p Figue 8b. Response of even wih gap o one pass of five-poin wavefon healing opeao S a ic s Figue 8c. Response of even wih simulaed saics o one pass of five-poin wavefon healing opeao CREWES Reseach Repo Volume 1 (000)

14 Henley Spike Figue 9a. Spike esponse o wo passes of five-poin wavefon healing opeao G a p Figue 9b. Response of even wih gap o wo passes of five-poin wavefon healing opeao S a ic s Figue 9c. Response of even wih simulaed saics o wo passes of five-poin wavefon healing opeao CREWES Reseach Repo Volume 1 (000)

15 Wavefon healing mees seconds.0 Figue 10. Raw sho gahe fom he Blackfoo -D 3-C suvey used o es wavefon healing mees seconds.0 Figue 11a. Sho gahe fom Figue 10 afe one pass of hee-poin wavefon healing CREWES Reseach Repo Volume 1 (000)

16 Henley mees seconds.0 Figue 11b. Sho gahe fom Figue 10 afe one pass of five-poin wavefon healing mees seconds.0 Figue 11c. Sho gahe fom Figue 10 afe wo passes of hee-poin wavefon healing CREWES Reseach Repo Volume 1 (000)

17 Wavefon healing mees seconds.0 Figue 11d. Sho gahe fom Figue 10 afe wo passes of five-poin wavefon healing 0 0 mees seconds.0 Figue 1. Bue sack of Blackfoo -D 3-C suvey daa CREWES Reseach Repo Volume 1 (000)

18 Henley 0 0 mees seconds.0 Figue 13a. Sack of Figue 1 afe one pass of hee-poin wavefon healing 0 0 mees seconds.0 Figue 13b. Sack of Figue 1 afe one pass of five-poin wavefon healing CREWES Reseach Repo Volume 1 (000)

, on the power of the transmitter P t fed to it, and on the distance R between the antenna and the observation point as. r r t

, on the power of the transmitter P t fed to it, and on the distance R between the antenna and the observation point as. r r t Lecue 6: Fiis Tansmission Equaion and Rada Range Equaion (Fiis equaion. Maximum ange of a wieless link. Rada coss secion. Rada equaion. Maximum ange of a ada. 1. Fiis ansmission equaion Fiis ansmission