JOURNAL OF THE CANAOIAIN SOCIETY OF EXPLORATION tEOPH”SlClSTS “OL 12 NO ? ,DFC 1986, P 14~55

INVERSE VELOCITY STACKING FOR MULTIPLE ELIMINATION

DAN HAMPSON’

Inverts Velocity Stxcking i% Ihe rums :ipplied 10 :I procedure devised hy Thorson (IYX4). and dcscr-ihcd ill Thorson and Clacrhouc c 19X5). for modelling hypcr- holic cwxls ori seismic wxd\. In Ihis algorithm ihe recorded seismic pn)filc is modelled ilsil linearc~,n-,hina~ tiun of simple hypcrholic CVCII~S ~fconswni ;m~plitudc. A bet of \rcighling coefficient\ is dcrivcd such thet the rcsulling model ;~ppn)sim;itc? the input protilc ill the lenst-squawssense. ICxh hypc~holiccvencischal-;,c~cl-- izcd by two parametrx\: the Lcro-offset lime of the CVCIII. ‘T,,. :tnd the st~u.rw\\.

p = 1:” III

where V is lhc KMS velocity associxcd with that cvc~~t. As discussed by Thorson (IYX4). rhc \ct of weighting

cocfticicnlscnn hc inlet-prcrcd dil.cctlyiogivc lhccncrg) iwoci:mxl with the vill~iou\ vcloci~ic~ iit diffcfel-cnr /crw offset times. In this scnsc‘. the cocfficient~ form WI ‘idc;diLcd’ consli!r~l docity \blck. 11scf111 for vclocil) an:dy\i\. The derived ~modcl mxy he furthel. LISCCI fur noise reducG0n :ind intcrpol:ltion of missing d:11:1.

An ;Ipplic;ifion that WIS n<,t discussed by Thorwil i\ that of long-pet-id mrdGple elimination. ‘I‘his xpplic:~. lion depc~lds Ott the fxct Ih;ll prim;lry ;!nd long-period multiple events 111 :I given /cro-offv2t time will gcner;~lly hi~vc diffcl.cnt RMS vclocitic\ and will snap ,<I diffcwuc rcgi0n\ of the tl0mGr~ \p;mned hy lhe v3 of weighting cocfticicnts. Invet-\ctr-anclol-min~:~ftel-rcl-oin~sclcctc~l set\ ~fcocfficicl~t> will then iwlillc plCrn;\r\ ;III~ Imulti- pie encrgy. :I% is c~~mm~~~lly d~~nc with F-K lillcl-ing (liyu. I’IXO). This prowdu~-c W:IS u\cd with wmc ~IIC- ccss by Ilamp\on and I~11nham , IYXi ,. hut lhC cw,Iprll;!- liowl effwl ~wde lhc ;1pp~~0;1ch impl~x3ic;ll liw volume p,-occs\l”g.

4 I;~t-gc p;~rt ufthe coniplltational cffut-1 cxr> it1 fxct hc 11.aced IO the hypcrholic f~l-m ofthc move-out equation. Modifyin~thisequation 1,,inc~,~-pol-atc;1p;ll-;lhulic wthct rhan II hypcrholic fu~-rn I-CSIIIIS in ;I signifiwnr simplifica- 11on d the liww by\lcm. which mu\1 hc wlvd. xrxl ii cot-re\p0niling reduction it7 c0mp~bt:1linn;1l ellilrI. 111 lhe rcmaindet- of this p:\pcr-. I de!-ivc lhc m<dificd linc;ll sybtcm. .itlrtify ilb use on NhlO-co~~~~cc~cJ di11i1. i~ml dcmonsttxle it5;Ipplic;llion lothe pl-ohlem~,flon~-pul-i~,d multiple climillatiorl.

The hxic modelling equ:ltioll pl-opovxl I>) Thorwn i\

dih.t) = ~jdpd~LJ~p.~);iI-;~\ t~~-p~~h-I i tGh.l) (21

whcl-c d 1h.t) = mc;~su~-cd xi\mogu-;lm xt offset h and Iw-wiy ll!llC I

U (p.~) hypc[holic tr;lllrf(>rm cocfficicr~t ;~t glow ners p :mtl /cl-id-offwl lime 7

n lh.0 = mcasul-cmcnt nDisc xl offwt h ;~nd tw~w;\) t1111c t

1vrrit;l\ S,,ftuiiw I .A,,.. ,>I’ I,K ;,\c ,111 c S,\\‘.. C’;,lga, >, .,,h c,na

14

lNVEKSE “I.I.OCI’I’Y SrAc:KINc; 4s

Therearetw~,waysofintelpretingthiseqllation. One way is to ,view it as a modelling system. Our seismic datasrcdCh~.t). whichcould beashot profileoracommon- depth-point gather. In equation (21 we we attempting to model thcsc data as a linear combination of functions ot the form

6 IT \,:(i’] (3)

The real numbers U(P.T) arc the weights associated witheachofthcfunctions.Thesefilnctionsinf~tct repre- sent hypcrbolae. as can be seen by imagining a situation in which all the ueights are zero except for that associ- ated with p = p,, and T = T,) :

U(P.TI =I A 6 (p-p,?) 8 C-~-T,,) 141

Inserting this into equation (2). and ignoring the noise component. momcntarily.we get

d(h.t) = A 6 [T,> \:t’~ p<,‘hr] IS)

This represents a hyperbola with amplitude A and a trajectory defined hy

1’ = T,,’ + p,,‘h’ (6)

In gcner:d, the double integral in equation (2) i\ a sum ovcrallpossible hyperbolae.each with itsownwcighting fxtor.

The prcscnce ofthe noise term. nth.0. indicates that WC do not expect this modelling to he perfect; that is we do not expect the linear combination to yield prcciscly the input data. One t-ciwn for this is that the integrals will be performed over :L limited range of slowncsses. p. and times. T. The noise term rcprcsents the misfit between our ‘bat model and the input data. As we shall see below. thil; ‘best- model is derived by minimizing the cncr-gy in this noise term.

The second way to interpt-et equation (2) is to view it as a translixm. analogous, for example, to the 7-p transform. In fxcl. the 7-p transform can be written as (Thorson, 19X4):

d(h.t) = IIdpd~IJ(p,~)6[~-(t-ph)l + n(h.0 (7)

Just as the 7-p transform can be calculated by sum- mingalon~Iincnrtr~jcctories,thecalculationofU(p,~)in equation (:I) involves summing along hyperbolic trajec- tories. From this point of view, equation (2) is closely analogous to the T-P transform. The presence of the noise term now implies that the transform space gcncr- ated by the basis functions (3) is not a complete space. That is, if we transform R data set from the (h.t) domain

to the hypet-bolic (p.7) domain and hack. WC do not retrieve the original data. The noise n(h.t) i\ the compo- nent of dCh.t) that is ‘lost’ in the transformation or. equivalently. the component of d(h,t) that is orthogonal to the space gcncratcd by the basis functions (3).

Regardless of which intcrpt-ctation WC use. the proh- lem lo he solved is: given the mcasurcd seismogram d(h.t). find the weights U(p.7) that satisfy equation (2) whiletninimiringthetnisfit n(h.t).ThorsonandClae~hout (19851 showed that this problem can be solved as an ~)verdeterminedlincarsy,tcm. Assumingth;ltd(h.t)and U(P.T) are defined at discrctc values of h,t.p. and T. equalion (2) can hc roduccd to the matrix equation:

d=Lu+n (8)

where d is a vector containing all values in the mcusurcd seismogram, u a vector containing all the weights. n the vector of noise ulues and L. the matrix that incorpo- rates the discrete form ofthc delta-function in (2). The standard least-squares solution to (8) is:

u = (L.‘L)- ’ 1.~‘.d (9)

While the form of equation (9) suggest> that this wlw lion is straightforward, the dimensions of the matricw involved make an explicit solution impractical. L is a matrix of dimension:

(N-, x NH) by (N.,- x NP)

where N.r = number of time samples in the input seismogram

NH = number of offsets in the input sis- mogram

N ,I = numherofslownessesused in Ulp.~)

In a typical 96.tract seismic shot of 3.second do&on. 1. would hc of the order of lSO,OOO hy lStJ.000. .Thorson and Clacrhout (1985) suggested an itcrativc solution, which is still prohibitively time-consuming.

In order to reduce the computational load. I propose lo replace the hyperbolic equation (2) by the parabolic form:

d(h,t) = IIdpdTU(p,T)6[7-(t-phZ)] + n(h,t) (IO)

where p is no longer the slowness defined by equation (I). hut simply a coefficient whose significance will hecomeapparentlater. Justascquation(?)modelledthe input seismogramasalinearcomhinationofhyperholae, equation (IO) models the seismogram as a linear comhi- nationofparaholae. Thcappropriatencssofthisfunction will be discussed below. The important feature to note here is that we can take the Fourier Transform of both sides of equation (IO) and show that each Fourier com-

.M> I, ,I \\ll’iOY

poncnI cim hc cidc~~li~lcd illdependc~111y h! v>lvitig 21 fX”l! re<lilccd \y\,en,. I .c,:

illlil

dch.li \IJltl.\ilc~“~ 1121 M

dctr.i\, 1 llip.ui c l~w”l j Illh.i\i II31 I’

Nolc thill. in Ihi\ equation. the unknou 11 coctficicnti 1Jlp.w) 11lr;t given frcqilcncy w depend only 03 Ihc da1;1 c~,~np~~~nc~\l\~ll(h.~~ I (\>rth;d l’~vxp~ncy. ‘I’hix ~dcc<lc~pting’ ~11th~ il-c~j~~c-ni) components OCCLII\ hec:mc cctil:llion ( IO/ i5 linc:ll. in both I :stld 7. t~qu;itit~n (2). on the t~thc~ hitnd. i\ 1101 linci~r~ itncl 170 \llch dcurupling uoutd 0ccIfi in Ihc fvcctltcncy ilom;i,iIi.

<I I.11 n , I-l)

whca~c (I i\ lhc cornplc\ \cclu~~ CI~ Al ihc comtx~i~c‘l~l\ d(ll.\i / 11w il \,wil~iC Li. I1 IhC mllplc\ Vi’ClUl~ 01’ Ill~iW c~II11,~ImIII\ 8, tl~c‘,“rnc~ \\. II lhC \CC,,II 01~ c~ln,,~~l-

ncn1\ IIl,~.U ,. illl<l I. 1hc Illilll~i\ ddincd I>\:

1t.‘t.1 II I.‘<1 (Ihi

tihhlllg ihccounl of lhc tilcl 1h:LI wc i~w nc1iv dr,;~lifig with c~mplcy quanlilic5. ‘t~t~c~tirrli!n\i~~n~~l’lhc~~~n~plc\ ~niatrix. t.‘t._ which ~n,u\t hc invcl.lcll. i\ I\,. I>! I\!.. nhcrc N,. is Ihc nltnit,~~.otiti\cl~~l~ v;ltrle\ 01 t’o\~cr \vIbich LLL‘ pcl-lllm~ Ihc irllcyrmlic~n in / 10)~ ;A\ dixil3wd hcl~~i~. lhii is 115r1:1lly t>f the 01n1c.1, or 211. VI that invcrling t .‘I, i\ cq~~~v;dcnl I0 in\~cl-tin!: ;A0 lh) JO rc.i~l IlliillG\. it I;l\h I hill i\ ~~~I~III~IIIC~~ very ~~llii‘icnlly in :in xw~!; prn,cc\\iw Fc>I~ :I giwn \h,,l pr,,tit<:. sy\tar, ( Ihi ,,,,,‘.I the \C’I I,,> :,,,<I ~1t~lvccl N,, lime\. v lhcw N,, i\ 111~ II~IIII~CI~ oI’iti\cI.t.t~. I’lrcqllcnc~ \.illllC\ 01 inlel.e\r.

vcr-y lilllc cncl-gy itI ncighlx~unng vctocIIIc\. tn I;~cl. a\ ‘l’ho~~~on p~~iclicd C)LL~. lhc conventionill (‘L’S ix prcciwly lhc t .‘<I ~;~cIcI~- in cquxiion (4). ~l’hr, c,cwKicicnl\ t’ip.;i :~re c:~lc~~l;~lcd h) nrultipl)inL: lt?l\ I;IC~OI. lh\ (t.‘l.i ! wllick ctTcclivcl\ dtx01I~~1I~c~ lllc I;llcldty \illciil~cd

cvcnt\ ;tnd \h:~rpcn\ thc‘ir im:igie. tn Ihi\ f,~~.rn 1hcl-c i\ :I hepawlion 01 cvcnt\ al similar timc5 xcurding ttr lhcil ~~t~~cilic~,i~ll~l~~t~;~l~i~tio~~~~t’~~l~lllipl~I’~~~~l~l priiliilrycllc1~,g\ ~lmdd lx pwrihle.

.r,h) ‘I- / \ T‘ t,‘:\‘~’ \ T’ , t,‘:V<- 117)

‘1 I, ~I~ \ ( +~ \,?/~“r-’ \ , i ,,2/\/c2’,“\ (,s)

t)cliGng the rcsidu:i/ vclcxil) VIM thy:

,;\fI-‘ _ I;,!’ ,jL’<.’ ,I’)1

i~ncl c\t~;In~tinrcctll;~li~lil ( I Xl in ;~Ta~Icwwric~ ill lt1\11~Tl. tic gc,

‘l.ih! T ,,‘;2fv,~‘ I 1201

It’lldVr-f)~ ~~~ I. wearc.i~~s~ilic~l in dropping rhc highct~. ordct~ 1cnus. .11111\. (0 lhc cxlenl lh;ll cqu:llion (201 is

INVEKSI: \‘,:I.OCI’I’Y YICKIZ~; 47

valid, WE can cxpcct the NMO-correcied events on the 2.This~cp~cscnts1lnNMO.corrcctedshot profile. Expe- input sci:imogl-am 10 map 10 discrctc point\ on the plot ricncc has shown that. when pt-occssin$ split-qread of UCp.v dcfincd by equakm C IO). As even,\ deviaw shot profiles. it is best to procc~ each Gdc of the spread fromthadeal p;lraholicfortn.~ccnnexpectasmca~ing. scparatcly. Accordingly. thesynthetic in Figure? repre- with a I-c:culkmt dcgr-adation in the ability to distinguish scnts one side of a split-spt-cad profile. Both primary multiple from primary cncrgy. and multiple enel-gy can be xcn in a strongly intcl-fcring

pttU”.

MOIEI. RESLII.‘I’S

The application of equation I IO) will now he demon- stratcd WI the model data set shown on the left of I;igurc

Figure 2 dcmonstrakx the modelling intcrprctation of equation (IO) - the input profile is I-cprcscntcd as a weighted wrn of simple peraholx ofconstant amplilude. One might at first wonder how a combination of con-

Constant Velocity Stack W-L PI 11500 “ELcmTY (ftkx, 6500 8.7 x 10-5 mecm, 1.54x 10

Fig. 2. In the modelling interpretation, each input NMO~corrected profile is represented as a linear combination of weighted parabolic event.5

h~mt unplilutc curves ~wltl possibly rept-cscn! t-c.;11 seismic data with ils variation\ in amplitude and w~avcfwm. The qucslion is cnlircly cqrliulcnt 10 thow a series of unifwm sine waves can modct 2 compte~ time xl-its in the ~ouricr ‘II-ansl’orm. The ~II\M’CI i\ lhill KC use i, VC,-? lar~c numbc~- OS clo\~Iy rpaccd paraholac which consli-uclively interfcl-c 10 prod~~ce the t-c\ulling imx~e. Ceure 3 demonslrale~ the case of four pxaholic curves Sor each of three z~~-o-ofl’x~ time ~mplc\. In pl-aclicc. 20 10 30 curvc‘~ are used over wmc specified ~-:m~c of rnovc-o~~Is. With an X00-m\ dal:~ window al 1- ms s:rmplc rate. this would mean that 40(1 Y 10 X000 pnraholac xc king used 10 model the profile of I$wrc 2.

Offset -

t’igurc 4 showy rhc ICSIIII ofwlvin$ the Ic;~st-~q~~xcs system ( 16) fwthis model. t’.ach puahola i\ piwim~c~cr- i/cd by its ~cro-~~fl’~c~ lime :lnd ils .IIIOVC-out‘. or lime diffcrcnce hctwcen ta-off\er and ~c~-~~-ollet lime:

1~ 1 P2 T2 P3

T3 P4

In this cusc. the variahlc. p \\a\ disclKli/cd at 20 ~alucs cot-responding Iomwc-ou(\ I-;m~in~ffl~uu IO rms to + 200 ms. ‘l‘hc right hand side ofl;iguw 4 plots the co- cfficicnts U(p. 7). III fxzl. cqultion C 16) uik wlvcd at a wxics 0fdiscrcIc li-cqucncic\ 1‘1~~17 IOtt 10 1001 II and U(P.T) wax c:~lculated hy using equarion C I I).

Fig. 3. For each ~ero-011set time. T. a range of paraboIae is used. etkctively spanning the profile with a Iarge number 01 paraboIae.

INPUT PROFILE

x(W

Model Calculation W-L P)

200 145 89 34 -10 a

Fig. 4. Plot of parabolic coefficicnis Uip.i) caIcuIated for input model by using equation 116).

p: Moveout at ‘al offset in ms

ISVEKSL VEL.O~‘ITY ST;\(‘K,r;<; 4’)

The transform U(p.~)cnn now be interpreted in much the ramc way as the constant-v&city stack of Figure I, except that ‘move-out replaccs velocity on lhc hwiron- tal axis. Since the input model is NMO-corrected, we can cxpcc’l pr-inxwy entxgy lo lnap to events at around O-ms mow-WI. while undercorrccrcd mulliples should map to higher move-outs. This has in fact taken pl;~~e. and primary energy can he discerned at 7X0 ms, 1000 ms, and I I IO ms. Strong multiples appcwat YX ms. YXO m\, 1(1X0 ms, and below 1200 ms. The input model was calculated and NM0 corrected by using exact hypcl-- holic move-outs. w the shxpnesr ofthc imaged cncl-gy on the Utp.7) plot is evidence (11 lhe validity of lhe parabolic ;approximation.

Having calcuk~tcd the cocfficicnts UCp.7). the for- ward model can now lx gencratcd by using equation (IO) with nth,tI set to LCTO. This result is shown in Figure 5. The left panel of the figure shows the ot-iginal synthetic data set, the cents-e pxnel shows the model result, anJ the t-ight p;mcl shows the residwd erw calculated hy subtracting the model I’-om the input. As expected. the model is a close approximation to rhc input. In fact, it is cay to show that the residual crt-o~ could he reduced to LWC by using :I sufficiently Iargc range of moYc-out wlucs p.

For the purpose of multiple elimim~tion. it is now ncccssary to product a model containing the multiple encl-gy only. This is done by ‘filtering’ the coefficient\ of Figure&that i\. setting a \~llxec ofthc cocSficicnl\ to zero. The procedure is directly comparable to filterins in the F-K~ domain, and one would expect that I-&\ ot thumb appropriate to that domain should he applicable here. In particular. sharpcutoffs have been ohset-ved to

f.enerate artifacts analogous to the Gihb‘s phenomena well known in F-K t’iltcring. Although it has not been tcs~cd. it should hc possible 10 reduce [his pl-ohlcm by tapering between the pass and reject zones in the trans- fwm domain. Fw these enamplcs a shwp cutoff wax xlually used without any apparent ill effects. Setting to zero all coefficients with move-outs Icss than 30 ms produced the model result in Figure 6. Comparing it with Figwc 5. we can xc thal the pt-imary cvcnts at 780 ms. IWtl ms and I I Ill ms are largely misjing and the multiple cvcnts arc isolated. Subtracting the multiple model from the input data produces the final multiplc- attenuated rcstllt in the right panel of l+re 6.

The model result in Figul-c 6 confirms that the algo- rithm is XI-Y powclful at ~~rtcnuarin~llonppcl-iod multiple\. A very satisf:xtory feature is that. where multiples have been attcnuatcd. the attenuation is equally good at all offscrs. This is in contrast to the ux of F-K filtering form~~lripleelimination. where it iscommonlyohscrved that the srmd-offset 01 ‘near trace\ iire handled lew than satisfactorily. Thcrc is ofcourse nu mystel-y in the fact that these two algorithms petform differently on this data set. ‘~l’hc inverse velocity stacking algorithm i\ based on a parabolic model and is ideally suited to modelling parabolic orapproximatcly pat-nholic cvrnts. The F-K algorithm is b:~scd on a plane-wave model and is ideally suited to modelling Iinewevents. The trajccto- I-& of long-period multiple\ aftel- NMO-cwrcction XI the primary velocity at-e mwe closely approximated hy pxaholnc than by straight lines.

Another important fcaturc ofthc algorithm is evident in Figute 6: while the primarics should he NMO-corrected. it is not necessary that the corl-wtion hc exactly right.

INPUT MODEL OUTPUT ,p = -mm3 to 200m*,

RESIDUAL (INPUT- MODEL 0ulPul-J

Fig. 5. Input synthetic. model caIculated by applying iorward equation (101 ~~5th coefficients displayed in previous figure. and residuaI noise resulting from subtracting model from input data.

50 I). ,I,\111’SON

‘l-he cvcnt al IOOI~ 11,s i\ ohviouily und~~c~,l-t-~~~~(l, ‘fhc pcrli~rmance or the dgoriltml i\ in IW way dcgmdcd lh) Ihis cmwas tang :I\ the clegrcc o111tlilerc~)l.l-ccti,,n is not such a\ 10 GIIISC the pr-imary (0 appwr in Lhc mut[iple /one: i.~.. ~reaie~~th;ln ill-ms inlovc-0LII. II ix now pcr\si- htc (~1 pcrt’o~-m II residual vcloci( anidyis on I~hc oulphl! <>f~igurc 6 and refine the vclocit! CSIIIII:~~~ wflhout Ihc strong inlcrfcl-ins muttiptc~.

A Sinal poinl \houtd hc ndc ahoul lhc ;dgorilhnl. which is nc,l p:~r~icul:~rly ohviow in I;igwc 6. Since the L~matt-ixdcllncd hycquation( t Slc~~l,t;lin~l”zci~cinl;)l-~ nmation cmwxning1hc 11111nhc1~am1 di\ll-ihulim ~I~‘~I~‘~‘YCI\. no xtifkc!\ will he genera~cd cithw h) the houndarics ofthc model or- hy uneven imcc >pxinp. Thi\ i\ anolt]cI~ feature in contuse b\ith 1:-K algorilhm\. which a\sumc periodic h0uncla1.y i~clndilions :md Ilniti)ml \p;lcing. It makes the pxahulic ;~lgor-ilhm lpavticululy clllr:ictivc for u\c 011 cl.(,oke~l~\pl~~;,,l or 1-I) d:11;1 \c,\.

RIL\l, I):\,‘:\ til:sr~l.l~s

‘l’hc invc‘rsc vcl<>cily SI:IC~ :dgorilhnl will he :lpplicd to two real data sets. :I shot pl-olitc t’rom the first data XI i\ shcnvn on lhc tct’t of t’ifllrc 7. ‘I’lli\ pr~rfitc i\ N MO-correcled :md lhc /WIG ol’inlercst CCIIIICY :II.OIIII~ lhe rctlector at ahout I I00 imh. .A sIl-c,ng multiple with ahout IOIl ,I,\ rc\idu;d II,,IVC-o,,, appear\ ill ;I /cl-c~-olt\cl lime ot’ 1020 m\ xnd cut\ thrc~ugh bhc tprimmy even\ at an c,lT5~21 II~ ;IppI-o*im;licli; Iu cl-third\ 01 Lhc spt~cid length. .Thc \hot pl-ol.ilc II;IY had dcconvol~lli,>n ilnd 1lwl--3ul-face \t;liic cowcclionx appticd. and i5 f111ly p10- ce~wd with the cxc<~plion c,Ilhi‘ fin:ll filler.

INPUT

‘I’hc lrigh! side of t;igurc 7 \how\ the IOI~ ~mmlcl g!cnualcci hy \l~lv~~r~eqri;llil)n (10) f’or~ Ihi\ tl;~la wt. .Ihc diwl-ete WI of pal-ahotac uwd was iclcnlic:ll 10 thal in the pwccding n~~del rc\utt. A timi~ed \~indow of input d;lt;l. fNlm 650 ,114 10 1500 111,. \\il\ ;In;lty/ed t<r KdllCC ccm~ptc~- ruwlinlc. t’igrare 7 shows the 1.cwl1 ol’m~dct- ling hclth primal-y and multiple cm!-g!. A\ cxpl;~incd ahow. each dc d’lhc ymxd i\ nrdellcd icdcpcndcnlly which lm~li~cc\ 2 dighI diswnlinuily itt lhc ccnln~ of Ihc ~n~odcl protitc. ‘1’0 ev:~lu:~le how well thih model tit? 1hc inpul dilti~. 17iglla~c X \tlon\ lhc rc5utI ~~l‘~i~hllacting one Ir~om !he t,lher. Ohviolr\ly. ;~ny wg!mc‘nt~ ~~Iwllec- IOP 01~ coh\,rcnl energy that the eye perce~\c\ as signal on lhc input prolilc have lhcer~ mdcllcd. iI\ cvidcnccd hv lhc (<)I;11 lath of cohcrcnl Ggnal within lhc ;maty+ n’indou of the r-csidual lp~~~l’ile 01 I;iylw X. M’h:l1 nou hti~nclh out is lhc I~~u-frcq~~cncy gl-c~uncl roll. which has IXXII ctl’cctivcly ignored th! Ihc ;Ilgorilhcn heca~~x dims di\\imil;ll-il); 10 rhc rricxlclliug p;\m:ltxllac.

l:or ihi\ pal.ticuI:ll. dxt:~ xI. Ihe ml~t~iplc cu-ol’~ ha\ hecn SCI a! HO 111s nic~vc-c~ul. and lhc mllttiptc xflzn~~i~~cd vc\utt i\ \ho\vn i/l Fig~l-u ‘I. ‘I’he ni:~.j,rt.illlllliple event :II IO?0 I,,\ hii\ lhccn Iwgely :~ftcnui~tcd. ill have \oinc \haltc,wcl- ~m~~l~iplc~ ;I, ;m~~~nd 900 in\. ‘I‘IIc pl~inl;ir) c\cnI iit I llll~~~~sh;~\ lhccn l~~~cwr~ed;~nd ill t’zlcl cnh;!nced h); xuppl-coxing lhc inlerfcrin~ rmb~lliple enel-gi) ;,n the 1;u o(tsels. II appc.i~bihi~l Irc\idllill mrllliplccncrg) I~C~V.C.CII I IO0 and 1200 113s ha3 hecn lell. p~w\r~~n;~hl) thix.:~lIx ilx. lmovco~~t i\ lcs\ th;ln X0 In>\.

MULTIPLE MODEL ,p = mm* to 200ms,

FINAL OUTPUT (INPUT-MULTIPLE MODEL,

Ins

180

s*n

696

,ss

ISS

IBA

611

5BI

baa

Fig. 6. lnpul synlhetic, muiti~le model calculaled by applynq forward equatiur il Oj wttk coeff~r:~enls lb parabolai: with move-outs from 30 ms 10 200 ms. and rnul,iple a,,enuateil ,CS”!I

INVl-KSI,~ Vb, OC’, I Y S,~~\rh,Nc; il

common~dcpth~point gnthcl-> and stacking twx~ \~ith identical ,,rsinlil:11-\orlrcc-rcceivcl-ofl\cls. The res~111 is an’aven~fc‘conlmon-dcplh-poinl gather-when-ethe stxk- ing hasenhanced the signal-lo-noix Iratio while p,-exl-\;- ing offset-dcpcndcnl chiwactcristic~. The thwe p;mcl\ of Figure IO I-especrively \hovv lhe common offset st:xk\ over the SUTIL d:rta tracts before mulliplc ctimination. atic~~ applying the invcrsc docity stack atgot~irhm 10

INPUT PROFILE

the \hot profiles as &XI-ihcd above. ;md after applymg the I:-K muttiplc elimination p~rnxlurc dRyu (IYXW IO thcshot p~otilcs.Thctatterproccdul-econsistsofNMO- cwrecting the data at a velocity intemediare between that of lhc pl-imat-y and multiple vclocilics. imd uing an F-K fitlcr ro renwv~ down-dipping cncrg);. lloth procr- dures have heen very effective in \uppre\binf the strong IllLllliptc cvcnts hC1MCCIl 1000 md 1200 ,115. In I;lCl. 1hc

TOTAL MODEL (I) = -10ms lo 20cms,

Fig. 7. In~sut real-data Shot profile ww ma, mode, calculated within data window from 650 ms to ,500 ms

INPUT PROFtLE RESIDUAL

(INPur TOTIlL MODEL,

<I Il. ,,,\\II’%iN

two mstltts are surpl-isingl!, 5Inlitiw ~11 the t’zr ol’l’wt~ Mlh~I.Cpl.itlliil.?illldllillltiplCCIICI.~\ ~nte~~~~~cJc\il~ac~i\~~l\. ‘lhc rn;(ior~ diffwcncc\ 0cc111~ 011 I the iwwoffw~ mui,\

INPUT PROFILE

M hew the I:-K ;dgorilhm has lcfl significant nruttiplc cnc~gy. bhilc 111~ InvcInc vclocilk 5lach ;,lg~,riihm ha5 tx~‘un etYccli\‘c at xl1 clff\cl\

Common Offset Stacks

FINAL RESULT

INPUT

AFTER INVERSE VELOCITY STACK

MULTIPLE ATTENUATION

~. .‘~:‘.!,.,.,.~?.J.;~~~ ,,“I. !?!T

:y;,,; ,,,, .,, ~, .,,/*:T,;,,:::I:;

“Bn..-e ,I,.t..~.*Pl.>t*,,,

::::1,:,1::::::::.::: , ‘c::x:::<::JAx,

AFTER F-K MULTIPLE ATTENUATION

.~, ~.~.~...----&..& I,,,, ..,.

,, ,4,:/ i,il::L:i:!,

“““:;q!gJg

*Wi,..P,;& ~, __._~ ‘.~ ),,) ~

,:~,;;w,:f:!;, ~*~~!;:,:.>,,..W -hmrrmnm

,:;;t;T;,::::;::!i:i .“;&~;~;;;:-~;~‘;

f_m ,,,..,.,, ,, “*y!l!:: ~..,.,,pe.......,,m..,

Il.,r**rm.,.*,.,.,,.... ‘,” -‘~-~‘--.~-~~~;~~~~~~

,.,..,.**:, ,,,,)),. * .,,, “‘.,“‘),,: I)...“l,l ,,I,,, I . . . . . . ,IIIutI, .,,.., ,;I:!!.~!..!:.~‘;:~.~“““” 2 .J.,umcw fi,,rur

I.,,*... ,,,*,,,, ‘WW.I... .,1..,+: ),)),.,,,.,,,~ >,,, ;I,i;~ ,J;l~**launu,,i,~,‘,,,~,,,w,,-

*Wi,..P,;& ~, __._~ ‘.~ ),,) ~

,:~,;;w,:f:!;, ~*~~!;:,:.>,,..W -hmrrmnm

,:;;t;T;,::::;::!i:i .“;&~;~;;;:-~;~‘;

f_m ,,,..,.,, ,, “*y!l!:: ~..,.,,pe.......,,m..,

Il.,r**rm.,.*,.,.,,.... ‘,” -‘~-~‘--.~-~~~;~~~~~~

,.,..,.**:, ,,,,)),. * .,,, “‘.,“‘),,: I)...“l,l ,,I,,, I . . . . . . ,IIIutI, .,,.., ,;I:!!.~!..!:.~‘;:~.~“““” 2 .J.,umcw fi,,rur

I.,,*... ,,,*,,,, ‘WW.I... .l,..lllll “,l,;l,i;!iLsguL ,J;l~;;;;ii;j;,~,~,‘,,,,,,,,W,,~

e ‘;!:::::!:,:~.,:,,..,,,ru*,,~~~~~,~~~;~, , ,I~‘~..,~,.~~,‘,.! ..,.... ,),,) p.,,; )),,,, j,,>,...~l’,‘.‘~‘.‘,) “‘II! “n,,,,,I,,IIIL~~ >l.,,. ,I,,. .,,,.,,b> - -.

,,,.,,,,,),,.).,)(l,I, ,Y>1b,>~~ #,I, S...c*rrttr*rnrl,.‘““~,“,~~,,, <, . bl~~w~~,,~b~, ‘)V “““’ .” ,* .,.,.), ;,,b,,> .+TTyx$x~:~i~~ n,~,,!~.,,!~,:;!:.:~;,!,~,:l,.,!”””.””” :.‘:I:,, :-....,,,,J!l~~~~,!~‘~:1:~,”

Fig. 10. Common dkcr viiwi :wr:x ,, I’,%(, i.(;,l’ ,,a,ii rx? ,,A e Pi I,,,,, Jlf: ;ii,rn,la,,<m at,?, ,,,“e‘SP “Plocily 5111Ck rllllllipk allenUa,i”l,. and aflel F-K multiple aNenuamil

The second re;&data resmelt i\ shown in Figures I I to 14. The first two. Figures I I and 12. show comn~on offset stacks hcforc and ;ifter~ multiple eliminaiiow The zone ofintercst. from approximately 700 to 1000 my. is strongly contaminated with undcrcorr-cctcd multiple energy as in the prrviou\ example. Again the inwnc velocity stack :dgorithm M’:IS :lpplicd 10 :I limited time window: from hO0 to 1200 ms in this cast. The ran~c 01 pat-abolic mwc-011t~ win ~51 fi-om IO ms 10 + 200 m5. with the cut-off move-out sepnraling mulliple (~-or pri- mary energy scl at + 30 ms. Cornpal-ing FiglIt-cs I I and IL!. WC xc a very significant change in the reilcc~n~ quality as a rewlt of multiple attenuation. It should he noted that each common ot&ct stack was sc:dcd with :I short wilndow ,ZC;c‘. This accoutres for the inwcarc in amplitude ohscrved on pCmary cvcm~. particularly between 700 and YOO ms.

Figures Ii and I4 compar-c the final stacked section hcl’orc and ilflcr multiple elimination. The pl-occaing tlow w::n identical in each USC except for the invcrsc velocity stack step. which was applied to the shot PI-“- filcsjust hcforc final stack. The NMOand stilticscowcc- tions MCI-C idcntica. as UCI’C lhc twcc sculling\ hcforc stack. The final stacked traces were xalcd hy deriving a single mulliplicr fat- each trace IO equlize the RMS ampliludle over the zone of inrere\t. III fenu;~l. MC WC a very signific:ml improvement in continuity 0\ic1 n~an)

xrcxs of the zone of interest. which i\ consista with Lhc changes olxcrvcd on the common ol’fset stacks of Fig~11-c5 I I and 12. Expel-iencc has \hou.n Ihal Ihis level of inrpr”vement on the ~l:~chcd section i\ not achicvcd in every cazc ~ stacking is itself a powerful multiple attcnuxlion ~mcchani~n~. H~nrcvc~~. it i\clu~-that whcl-c multiple\ d~minxle Ihe ~-ecol-ded seiw~ogl-am\ 10 the cxtcnt visihlc in IGgurc I I. an :dgwithm such as inverse velocity stacking can ?ubstanti;~lly impr-o\,c the imaging of weak pl.imar\; reflectors.

Both the ~nodcl and twill dala cxamplcs have shown the pcl-form;mcc ofthc xlgwithm on \ho-wdered profiles. Thcorctically Ihi\ approach should he applicable only to plane-l:~ycux. rcrwdip d:~l:~. I’l-aclical experience has shown that. in fact. the algorithm is quite rohus! in the pi-c\cncc ol’~~~odc~~a~c \~rw~:t~~rwl devi;llion\ from ihc ideal. :!s c:~n he wen in the previolls cxamples. One reason SOIL this is that. lhy modelling the multiple and subtracting it fl-om Ihc input I%rofitc. SII-IIC~~~I-~~ anonx- lie\ \uch :IS I;~r!lls will lend 10 lx pw~rly modt~llcd :md hence left in the data as noise. In wrnc (it-cumslanccs. howc\;c~. ~I~n~cl~~~~iil ;Inomillic,; like difl‘filction patterns may he modelled ah milltiplc\ and suppressed hy the ;~lgwithm. In ~lvzxca~cs. I hi% ;Ilgol-ithm should hc applied in the C‘DI’ domain. whc.~.c alI pt-imar-!; c’nwgy should id~~~lly;~ppc;~l-a~~cl-~~-dipcv~nlsaftcl-n~~~~m~~lmo~c-or~t.

)iBO

Zone of Interest

‘,BB

54 ,i~ ,,\Al,‘>‘i;U

?nR~- 7 TV--- ;:~r:;;,, :yb,:I.,,

!,!i y,!yr;~

~BLI lJ~];,p +‘,~I ,**:,:l./,L>,/~.

BnB ,,,’ ,, .,i ‘P,~‘,~ v ,yp,’ (I ,,,,b ,,,q r~o~g;,;>l&&!&

“~~ ~y:~,---~-s~: :~:~~~I~ ~:;:,/;,,

:!!r!‘n ~~ !, ,;,,i’,,,ic

”

Fig. 12. Sccmd rcal~~c!ata :;e: com:r:m d/se. a:, L ““3 -k aim #,11,:‘~:,;: \-ol’v lb<’ !S!RW lW,!d., f 1) Jo ~,,,.r,i,iit,,l~,

7RR ~__ z iih8

Zone of Interest 1vu

‘+f

HYI Zone of Interest

SilM

+ nun -

iBI:

ina

,188 +$+;i(!;e : ;:,,,;,l;i;,ir~i*,.i),;,~‘,,,;,~ii,l; ,,,)I,,) ~,,!,.,“,;;‘:;,,::;,;,:;;i” / ‘))I”‘,“’ ’

~llil~l~‘~ill~!!~~~liI:~;~~ ,,,,,,,, * ,,,, , ,,...,,,.,,, ..,. .,., ,, ,.,ab.db ~, ,,,I ,‘~:,,!,,‘:L,I)‘,,,.‘I.Y,,” ‘)I )

,i~o~:;.~l)l,~/~i~l,l,,,~“~~~~~“l~ ‘la+*, I”“,‘:,“:‘;ilr..;;!l;t::;,l!i!~~~,~,!,,~~,!~~ I,), *, /be: ‘8,) bb’,,‘“‘,q) II “:,;;,;‘,!,,,,) (, )) .!!,n,,,r,:,,,!!

” lW.I,,,,,, ‘I, ,,‘.i,i,.,,‘, #II .,!,fW,,

b’Lsill ,mm,,,,, ,,),,),, ‘,“ill’[~?,.!,, ,,

4

Zone of Interest

Fig. 14. Second real~data set stack after inverse velocity stack multiple alenuation

This paper has shown how the Inverw Velocity Stack algorithm u1‘Thot-son can be modified to incorporate :I parabolic z~pproximation. and thus provide :I useful fool for multiply.: climinatiw. Some ofthc features that ~makc It particularly attractive we:

I) It achieves multiple attenuation at all offsets equally. 2) It rcquir-cr no knuwlcdg ofthc mllltiplc~~cnci-ati~~~

mechanism. 3) It requires no detailed knowledge of the multiple

and primat-y velocities. 4) It will attenuate 11 w#ide range of mulliplcs with

variable movco~~~s. 5) It will xcommodate vat-iahlc or nonunifwm acqu-

s11wn geometry. while minimiring ‘edge cl’l’ccts‘xssoci- aled with lilnitetldala;~pertul-e. This m;~kesthealgorirhm attractive lix application to 3-D data sets. Limitntiotn associated with the met hod arc:

II Computation lime is significantly gvstct- than for standard techniq\lc\.

2) Multiples must have xdficirnr IIIOVC-,wt di\cGln- nalion to hu attenuated. This issue wx not xldt-essed in

this paper. hut cxpcrience has xhown Ihat. Mhile vcrq fine discrimination may bc achieved on ~modcl data. I-eal data. with their variable amptitudcs and w~vcfol-ms and their additive ~wix. demand at least 30-111s mow-o~~~ St-orn nwr tr-xc to Sal- tr~c to he el’l’ccrivc.

31 As with 1:-K filtering. \h:lrp cut-offs in the trans. Swm domain may pl-oducc arlifactr a\ a result ol’Gibh’r pt1c110111c11011.