Suppression of multiples from complex sea-floor by a waveequation

Transcription

1 fom complex ea-floo by a waveequation appoach Dmiti Lokhtanov* *On abbatical fom Nok Hydo Reeach Cente, Been, Noway ABSTRACT We peent a new efficient wave-equation cheme fo pediction and ubtaction of wate-laye multiple and pe-le fom an abitay 2D ieula ea-floo. The cheme account fo both multiple eflection and diffaction. The method equie appoximate knowlede of the ea-floo eomety. The pedicted multiple ae plit into thee tem, whee each tem contain multiple event which equie the ame amplitude coection. Theefoe the method ha much fewe poblem with the adaptive ubtaction of pedicted multiple than the iteative SRME appoach, epecially fo data fom aea with a hallow ea-floo. In ou cheme all ouce-ide and eceive-ide multiple of all ode ae uppeed imultaneouly in one conitent tep (in one o a few time window) fom the Radon-tanfomed CMP o common hot athe. The pediction of multiple both fom eceive-ide and ouce-ide tat with data in the ame domain, theefoe no additional otin o additional Radon tanfomation ae equied. INTRODUCTION Removal of wate-laye multiple and pe-le fom had and/o ieula ea-floo i till one of the majo pocein poblem in offhoe exploation. Fo data fom uch aea, method baed on velocity dicimination ae often inefficient owin to complicated moveout of multiple event. The uface-elated multiple elimination appoach (SRME) which pedict all uface multiple fom the data themelve, hould wok at leat fo data fom aea with deep wate. Only one iteation of SRME i equied fo pediction of multiple and fewe poblem occu with diffeent amplitude coection fo intefein multiple of diffeent ode. Howeve, in pactice thi method almot neve lead to the bet eult. Pobably thi i elated to amplin iue, 3D effect, the fact that the deinatue opeato (which i ued fo adaptive ubtaction of multiple) hould be anle/offet dependent, and becaue the deinatue opeato i le compact and involve moe unknown than the eflectivity eie equied by wave-equation (WE) appoache. WE method do not taet all fee-uface multiple, but do deal with thoe that ae mot often toubleome, i.e. wate-laye multiple and pe-le. In contat to SRME, thee method equie appoximate knowlede of the ea-floo eomety. The method decibed in thi pape belon to the cla of WE method (ee efeence). The main featue of ou cheme ae the followin: 1) The adaptive ubtaction of the pedicted multiple i pefomed in the tau-p domain, theefoe we take into account the anle-dependency of the eflection coefficient fom the wate-bottom; 2) Multiple fom ouce ide and eceive ide ae uppeed imultaneouly; 3) The pediction pocedue tat fom the Radon tanfomed input CMP o common hot athe and eult in the Radon tanfomed CMP o common hot athe of the pedicted multiple; 4) The CREWES Reeach Repot Volume 15 (2003) 1

2 Lokhtanov pediction of multiple i pefomed in the ame domain a ued fo multiple uppeion. Theefoe no additional otin o additional Radon tanfomation ae equied; 5) Both multiple eflection and diffaction ae pedicted. SUPPRESSION OF WATER-LAYER MULTIPLES AND PEG-LEGS BY THE WAVE-EQUATION APPROACH The ioou deivation of ou wave-equation multiple uppeion opeato i iven in Lokhtanov (1999b). Hee we will follow a moe intuitive appoach. Denote by D the pe-tack data (in whateve domain) alon the pofile. Denote by P the opeato fo eceive-ide extapolation of the input data thouh the wate-laye. P include the popaation of the ecoded wavefield down to the ea-floo, eflection fom the ea-floo (multiplied by the eflection coefficient of the fee-uface) and popaation up to the fee-uface. A a eult of uch extapolation, the pimay wate-bottom eflection i tanfeed to the fit-ode multiple; the fit-ode multiple i tanfeed to the econd-ode multiple; each pimay eflection fom below the wate-bottom i tanfeed to the fit-ode eceive-ide pe-le; each fit-ode eceive-ide pe-le i tanfeed to the econd-ode pe-le and o on. Theefoe the opeato ( I + P ) applied to the data D emove all pue wate-laye multiple and wate-laye eceiveide pe-le: the eult F1 = ( I + P )D i fee fom all pue wate-laye multiple and eceive-ide pe-le. Suppoe now that we exchane the poition of ouce and eceive uin the ecipocity pinciple (nelectin the diffeence in ouce and eceive diectivity patten). The ouce-ide pe-le fo the oiinal eomety become the eceive-ide pe-le fo the new eomety. Similaly to the peviou tep, thee pele can be emoved fom data F 1 by the opeato ( I + P ) whee P i the ouce-ide extapolation opeato. Note that the eult F 1 contain the pimay eflection fom the wate-bottom. Conequently the opeato P applied to F 1 ceate the fit-ode pue wate-laye multiple which i aleady emoved fom F 1. Theefoe the data F without all pue wate-laye multiple and pe-le can be obtained a follow: ( I + P ) I P + D P D, F = (1) w whee D w i the pimay eflection fom the wate-bottom. Thi eflection i eaily epaated fom the et of the data in the tau-p domain. The fomula (1) can be ewitten a: F = D + D + D + D, (2) whee D = P D, D = P ( D Dw ), D = P P D. So fa we have aumed that the extapolation opeato include the eflection coefficient of the wate-bottom. In pactice, we do not know thee coefficient. Theefoe we calculate the eult 2 CREWES Reeach Repot Volume 15 (2003)

3 D, D, D (ee next ection) aumin that the eflection coefficient ae equal to one fo all anle of incidence and all eflection point alon the ea-floo. Such extapolation pedict coectly the kinematic of multiple, but not thei amplitude. Conequently, all extapolation eult in (2) hould be popely caled. The caled veion of (2) i applied tace by tace to the Radon tanfomed CMP o common hot athe. Fo each p-tace the opeato ha the fom: f ( τ ) = d( τ ) + ( τ ) d ( τ ) + ( τ ) d ( τ ) + ( τ ) d ( τ ), (3) whee d( τ ), d ( τ ), d( τ ), d ( τ ) ae p-tace fo the input data and the eult of extapolation fom the eceive-ide, ouce-ide (of muted input data) and ouce-ide afte eceive-ide epectively. The filte ( τ ), ( τ ), ( τ ) account fo anledependent eflection coefficient fom the wate-bottom and mall phae-hift due to impefect knowlede of the wate-bottom eomety. The filte ae etimated fom the citeion of minimum eney of f. Note that fo a locally 1D tuctue, the opeato (3) i tanfomed into ou inle-channel data-conitent deconvolution opeato Remul (Lokhtanov, 1999a). EXTRAPOLATION OF THE RADON-TRANSFORMED CMP GATHERS The multiple uppeion opeato (3) equie the Radon-tanfomed CMP o common hot (CS) athe of input data and of extapolation eult. The eneal cae of ieula ea-floo i conideed in the followin ection. Hee we decibe an efficient pocedue fo calculation of the equied Radon-tanfomed CMP athe aumin locally hoizontal wate-bottom and abitay 2D tuctue below it. Note that in contat to the conventional phae-hift extapolation ou pocedue doe not equie epaate FK o Radon tanfom (of common hot and common eceive athe) fo ouce-ide and eceive-ide extapolation. Aume that CMP coodinate y inceae in the hot diection. Then the input Radon tanfomed CMP athe D( p, y) can be epeented a follow (Lokhtanov, 1999b): ω D( p, y) = R( p, pd )exp{ iω pd y} dpd, 2π (4) whee R ( p, pd ) i the complex (fequency dependent) amplitude of the eflected plane wave with lowne p due to the incident plane wave with lowne p ; p = p pd 2, p = p + pd 2. With thee notation the eult of eceive-ide extapolation D ( p, y) can be obtained a: CREWES Reeach Repot Volume 15 (2003) 3

4 Lokhtanov { i ( p y + 2q h } dp = ω D ( p, y) = R( p, pd )exp ω d ) d 2π D( p, { exp{ iω[ pd ( y + 2qh] } dpd } dx, ω (5) 2π whee h i the local wate-bottom depth, while q = ( 1 c p ) i the vetical lowne; c i wate velocity. The inteal in the culy backet can be calculated by the t tationay phae appoximation. Fo each pai x, y the tationay point p d coepond to t a imple elation (fiue 1): x y = h tα whee p = cinα and p = p p d 2. The extapolation fom the ouce-ide i pefomed in a imila way. y x * R2 R1 S CMP α wate bottom FIG. 1. Illutation of the tationay phae eult: x and y ae CMP poition of the pimay and multiple event epectively. They ae elated by: x y =.5( R 1 R ) = h t α. 0 2 FIG. 2. Contant P ection befoe multiple uppeion (left), afte multiple uppeion by the new wave-equation (WE) appoach (cente) and afte Remul (iht). 4 CREWES Reeach Repot Volume 15 (2003)

5 We eneated ynthetic finite-diffeence data with all multiple fo a model with an ieula wate-bottom and a inle complex inteface below the wate-bottom. The data wee CMP-oted and Radon-tanfomed. Fiue 2 how contant P ection (the ame p-tace fo all CMP) fo the input data (left), fo data afte ou new wave-equation (WE) multiple uppeion (cente) and afte Remul (iht). The eult afte WE i almot pefect, while afte Remul we till have eidual of wate-laye pe-le fom the econd eflecto. Note the diffeence in the poitionin of the ouce-ide and the eceive-ide pe-le in the input data. The kinematic of thee pe-le i coectly pedicted by the WE appoach, but not by Remul. The diffeence in the eult of thee two appoache inceae with inceain lowne p (o inceain offet o incident anle). Fiue 3 how eal data tack befoe and afte WE multiple uppeion. Both tack wee ceated with the ame velocity and mute libaie. The eult on pe-tack level ae iven in Fiue 4. It how contant P ection (the ame p-tace fo all CMP) befoe multiple uppeion, afte WE multiple uppeion and the diffeence. Note how the weak dippin pimaie ae extacted fom below ton wate-laye pe-le fom Top and Bae Cetaceou. Othe ynthetic and eal data example can be een in Lokhtanov (2000). FIG. 3. Stack befoe multiple uppeion (left) and afte WE multiple uppeion (iht). The pink line how the expected poition of the fit-ode wate-laye pe-le (expected blue event) fom the Top Cetaceou (black line). The multiple peiod i about 140 m. CREWES Reeach Repot Volume 15 (2003) 5

6 Lokhtanov EXTRAPOLATION OF THE RADON-TRANSFORMED CS GATHERS In the cuent ection we conide the pediction of multiple fo a eneal cae of an ieula 2D ea-floo and an abitay 2D tuctue below it. Multiple fom both eceive- and ouce-ide ae pedicted fom Radon-tanfomed common hot (CS) athe. Denote by D ( p, x, ω) the Radon-tanfomed CS athe, whee p i the eceive-ide ay paamete, x i the ouce coodinate and ω i the fequency. Fo each fequency the ecoded CS wavefield can be extapolated down to the ea-floo: ω W 2 + π ( x z(, x ) = D( p, x )exp{ iω[ p ( x x ) q z( ]} dp,, whee z ( i the depth of the wate-bottom at lateal poition x ; q = ( 1 c p ) i the vetical lowne of the plane wave with hoizontal lowne p ; c i wate velocity. In the pediction pocedue we aume that the eflection coefficient ae equal to one fo all anle of incidence and all eflection point alon the ea-floo. Theefoe the eult (6) define the eflected wavefield alon the wate-bottom. Thi wavefield i contucted by upepoition of phae-hifted ecoded plane wave, theefoe the effect of multicattein alon the ea-floo ae not accounted fo. The next tep i a decompoition of eflected/catteed wavefield into plane-wave contibution. Accodin to Wenzel et al. (1990), the amplitude D ( pc, x ) of the eflected/catteed plane-wave with lowne p c i defined by the followin expeion: (6) FIG. 4. Contant P ection fo input data (left), afte WE multiple uppeion (cente) and the diffeence (iht). The coepondin vetical anle of wave popaation at the uface i about 8. Note how the weak dippin pimaie ae extacted fom below ton multiple. 6 CREWES Reeach Repot Volume 15 (2003)

7 dz pc D ( pc, x ) = W ( x, z(, x ) 1 + exp{ iω [ pc ( x x ) + qc z( ]} dx, (7) dx qc whee q = 1 c p ). Accodin to (7), each point of the bounday act a a c ( c dz pc econday plane-wave ouce with the amplitude ( ) W x, z( 1 +. Fomula (7) dx qc alo doe not account fo effect of multicattein and i deived aumin that the eflected/catteed wavefield in the wate laye conit of up-oin wave only. Fomula (6) and (7) define the hot-by-hot pocedue fo the pediction of multiple fom the eceive ide. Conide now the extapolation of input data D( p, x, ω) fom the ouce ide. Fit we ue FFT to decompoe the input data into contibution with diffeent popaation anle (wavenumbe) fom the ouce ide: R p, k, ω ) = D( p, x, ω)exp{ ikx } dx, (8) ( whee the ouce ide wavenumbe k i defined a: k the eult of decompoition down to the ea-floo: W = ω p k. Then we extapolate + 1 = z 2 + (9) π ( x, z(, p ) R( p, k)exp{ i[ k x k z( ]} dk, FIG. 5. Velocity model fo FD modellin (with PoMa. Colou velocity cale i in m/. CREWES Reeach Repot Volume 15 (2003) 7

8 Lokhtanov FIG. 6. Contant p ection (anle at the uface i about 20 deee). Multiple event in the input (left) can be identified in the eult of eceive-ide pediction (cente) o in the eult of ouce-ide pediction (iht). Note that a ton event at about 2070 m at the left ide of the input ection i a pimay eflection fom the thid bounday. FIG. 7. Input hot athe (left) and hot athe afte multiple pediction, ubtaction and invee Radon tanfom. A pimay eflection fom the thid inteface (at about 2400 m at minimum offet) i extacted fom below a ton multiple whee k ( ) 1 z = ω c k, Im( k z ) 0. Finally, we ue (7) to define eflected/catteed epone fo each k and then the invee of (8) to calculate Radon-tanfomed CS 8 CREWES Reeach Repot Volume 15 (2003)

9 athe afte ouce-ide extapolation. Note that all thee tep tatin fom (8) ae pefomed in a double loop ove p and ove ω. Alo, note that in the definition of the ea-floo eomety z ( and the in convention ued, the poitive diection of x coincide with the hootin diection fo ouce-ide extapolation and ha the oppoite diection fo eceive-ide extapolation. Fiue 6 and 7 how the pediction and ubtaction eult fom finite-diffeence ynthetic data fo a model with a tonly ieula ea-floo, Fiue 5. The multiple ae well pedicted (Fiue 6) and tonly uppeed (Fiue 7). CONCLUSIONS If tuctual vaiation in the coline diection ae not evee and the main feeuface multiple ae wate-laye multiple and pe-le, ou wave-equation (WE) appoach pefom well and i computationally efficient. Both multiple eflection and multiple diffaction ae accounted fo. All pedicted multiple of all ode ae uppeed imultaneouly in one conitent tep in one o a few time window. REFERENCES Bekhout, A. J., and Vechuu, D. J., 1997, Etimation of multiple cattein by iteative inveion, Pat 1 Theoetical conideation: Geophyic, 62, Benth, H., and Sonneland, L., 1983, Wave field extapolation technique fo petack attenuation of wate evebeation, 53d SEG Meetin, Expanded Abtact, Beyhill, J. R., and Kim, Y. C., 1986, Deep-wate pe le and multiple: Emulation and uppeion: Geophyic, 51, Hill, N. R., Lanan, R. T., Nemeth, T., Zhao, M., Bube, and K. P., 2002, Beam method fo pedictive uppeion of eimic multiple in deep wate, 72nd SEG Meetin, Expanded Abtact. Huonnet, P., 2002, Patial uface elated multiple elimination: 72nd SEG Meetin, Expanded Abtact, pape SP3.3. Jiao, J., Lee, P., and Steven, J., 2002, Enhancement to wave-equation multiple attenuation: 72nd SEG Meetin, Expanded Abtact. Levin, S., 1987, Deconvolution with patial containt, SEP 54. Lokhtanov, D., 1999a, Multiple uppeion by data-conitent deconvolution: The Leadin Ede, No.1, (fiue ae epinted in The Leadin Ede, No.5, ). Lokhtanov, D., 1999b, Pediction and ubtaction of multiple fom complex wate-bottom: 61t EAGE confeence, Extended Abtact, pape Lokhtanov, D., 2000, Suppeion of wate-laye multiple fom deconvolution to wave-equation appoach,: 70th SEG Meetin, Expanded Abtact, Moley, L., 1982, Pedictive technique fo maine multiple uppeion: SEP 29. Spitz, S., 2000, Model-baed ubtaction of multiple event in the fequency-pace domain: 70th SEG Meetin, Expanded Abtact, Vechuu, D. J., and Bekhout, A. J., 1997, Etimation of multiple cattein by iteative inveion, Pat 2: Pactical apect and example: Geophyic, 62, Welein, A., 1999, Multiple attenuation: An oveview of ecent advance and the oad ahead: The Leadin Ede, No. 1, Wenzel, F., Stenzel, K., and Zimmemann, U., 1990, Wave popaation in lateally heteoeneou layeed Media: Geoph. J. Int., 103, p Wiin, J., 1988, Attenuation of complex wate-bottom multiple by wave-equation baed pediction and Subtaction: Geophyic, 53, ACKNOWLEDGEMENTS Thank to Nok Hydo fo pemiion to publih thi pape. CREWES Reeach Repot Volume 15 (2003) 9