High-order kernels for Riemannian wavefield extrapolation

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1 Geophysicl Prospecting, 2008, 56, doi: /j x High-order kernels for Riemnnin wvefield extrpoltion Pul Sv 1 nd Sergey Fomel 2 1 Centre for Wve Phenomen, Geophysics Deprtment, Colordo School of Mines, Golden, CO 80401, USA, nd 2 Bureu of Economic Geology, The University of Texs t Austin, Austin, TX 78713, USA Received August 2006, revision ccepted My 2007 ABSTRACT Riemnnin wvefield extrpoltion is technique for one-wy extrpoltion of coustic wves. Riemnnin wvefield extrpoltion generlizes wvefield extrpoltion by downwrd continution by considering coordinte systems different from conventionl Crtesin ones. Coordinte systems cn conform with the extrpolted wvefield, with the velocity model or with the cquisition geometry. When coordinte systems conform with the propgted wvefield, extrpoltion cn be done ccurtely using low-order kernels. However, in complex medi or in cses where the coordinte systems do not conform with the propgting wvefields, low order kernels re not ccurte enough nd need to be replced by more ccurte, higher-order kernels. Since Riemnnin wvefield extrpoltion is bsed on fctoriztion of n coustic wve-eqution, higher-order kernels cn be constructed using methods nlogous to the one employed for fctoriztion of the coustic wve-eqution in Crtesin coordintes. Thus, we cn construct spce-domin finite-differences s well s mixed-domin techniques for extrpoltion. High-order Riemnnin wvefield extrpoltion kernels improve the ccurcy of extrpoltion, prticulrly when the Riemnnin coordinte systems does not closely mtch the generl direction of wve propgtion. INTRODUCTION Riemnnin wvefield extrpoltion Sv nd Fomel 2005) generlizes solutions to the Helmholtz eqution in Riemnnin coordinte systems. Conventionlly, the Helmholtz eqution is solved in Crtesin coordintes which represent specil cses of Riemnnin coordintes. The min requirements imposed on the Riemnnin coordinte systems re tht they mintin orthogonlity between the extrpoltion coordinte nd the other coordintes 2 in 3D, 1 in 2D). This requirement cn be relxed when using n even more generl form of Riemnnin wvefield extrpoltion in non-orthogonl coordintes Shrgge 2007). In ddition, it is desirble tht the coordinte system does not triplicte, lthough numericl methods cn stbilize extrpoltion even in such situtions Sv nd E-mil: psv@mines.edu Fomel 2005). Thus, wvefield extrpoltion in Riemnnin coordintes hs the flexibility to be used in mny pplictions where those bsic conditions re fulfilled. Crtesin coordinte systems, including tilted coordintes, re specil cses of Riemnnin coordinte systems. Two strightforwrd pplictions of wve propgtion in Riemnnin coordintes re extrpoltion in coordinte system creted by ry trcing in smooth bckground velocity Sv nd Fomel 2005) nd extrpoltion with coordinte system creted by conforml mpping of given geometry to regulr spce, for exmple migrtion from topogrphy Shrgge nd Sv 2005). Coordinte systems creted by ry trcing in bckground medium often represent well wvefield propgtion. In this context, we effectively split wve propgtion effects into two prts: one prt ccounting for the generl trend of wve propgtion, which is incorported into the coordinte system, nd the other prt ccounting for the detils of wvefield scttering C 2007 Europen Assocition of Geoscientists & Engineers 49

2 50 P. Sv nd S. Fomel due to rpid velocity vritions. If the bckground medium is close to the rel one, the wve-propgtion cn be properly described with low-order opertors. However, if the bckground medium is fr from the true one, the wvefield deprts from the generl direction of the coordinte system nd the low-order extrpoltors re not enough for ccurte description of wve propgtion. For coordinte system describing geometricl property of the medium e.g. migrtion from topogrphy), there is no gurntee tht wves propgte in the direction of extrpoltion. This sitution is similr to tht of Crtesin coordintes when wves propgte wy from the verticl direction, except tht conforml mpping gives us the flexibility to define ny coordintes, s required by cquisition. In this cse, too, low-order extrpoltors re not enough for ccurte description of wve propgtion. Therefore, there is need for higher-order Riemnnin wvefield extrpoltors in order to correctly hndle wves propgting obliquely reltive to the coordinte system. Usully, the high-order extrpoltors re implemented s mixed opertors, prt in the Fourier domin using reference medium, prt in the spce domin s correction from the reference medium. Mny methods hve been developed for high-order extrpoltion in Crtesin coordintes. In this pper, we explore some of those extrpoltors in Riemnnin coordintes, in prticulr high-order finite-differences solutions Clerbout 1985) nd methods from the pseudo-screen fmily Hung, Fehler nd Wu 1999) nd Fourier finitedifferences fmily Ristow nd Ruhl 1994; Biondi 2002). In theory, ny other high-order extrpoltor developed in Crtesin coordintes cn hve correspondent in Riemnnin coordintes. In this pper, we implement the finite-differences portion of the high-order extrpoltors with implicit methods. Such solutions re ccurte nd robust, but they fce difficulties for 3D implementtions becuse the finite-differences prt cnnot be solved by fst tridigonl solvers ny longer nd require more complex nd costlier pproches Clerbout 1998; Rickett, Clerbout nd Fomel 1998). The problem of 3D wvefield extrpoltion is ddressed in Crtesin coordintes either by splitting the one-wy wve-eqution long orthogonl directions Ristow nd Ruhl 1997) or by explicit numericl solutions Hle 1991). Similr pproches cn be employed for 3D Riemnnin extrpoltion. The explicit solution seems more pproprite, since splitting is difficult due to the mixed terms of the Riemnnin equtions. In this pper, we concentrte our ttention on higher-order kernels implemented with implicit methods. RIEMANNIAN WAVEFIELD EXTRAPOLATION Riemnnin wvefield extrpoltion Sv nd Fomel 2005) generlizes solutions to the Helmholtz eqution of the coustic wve-eqution U = 2 s 2 U, 1) to coordinte systems tht re different from simple Crtesin ones, where extrpoltion is performed strictly in the downwrd direction. In eqution 1), s is slowness, is temporl frequency, nd U is monochromtic coustic wve. The Lplcin opertor tkes different forms ccording to the coordinte system used for discretiztion. Assume tht we describe the physicl spce in Crtesin coordintes x, y nd z, nd tht we describe Riemnnin coordinte system using coordintes ξ, η nd ζ. The two coordinte systems re relted through mpping x = x ξ,η,ζ) 2) y = y ξ,η,ζ) 3) z = z ξ,η,ζ) 4) which llows us to compute derivtives of the Crtesin coordintes reltive to the Riemnnin coordintes. A specil cse of the mpping 2) 4) is defined when the Riemnnin coordinte system is constructed by ry trcing. The coordinte system is defined by trveltime τ nd shooting ngles, for exmple. Such coordinte systems hve the property tht they re semi-orthogonl, i.e. one xis is orthogonl to the other two, lthough the lter xes re not necessrily orthogonl to one-nother. Following the derivtion in Sv nd Fomel 2005), the coustic wve-eqution in Riemnnin coordintes cn be written s: 2 U c ζζ ζ + c 2 U 2 ξξ ξ + c 2 U 2 ηη η + c U 2 ζ ζ + c U ξ ξ + c U η η 2 U + c ξη ξ η = s)2 U, 5) where coefficients c ij re functions of the coordinte system nd cn be computed numericlly for ny given coordinte system mpping 2) 4). The coustic wve-eqution in Riemnnin coordintes 5) contins both first nd second order terms, in contrst with the norml Crtesin coustic wve-eqution which contins only second order terms. We cn construct n pproximte Riemnnin wvefield extrpoltion method by dropping the C 2007 Europen Assocition of Geoscientists & Engineers, Geophysicl Prospecting, 56, 49 60

3 Riemnnin wvefield extrpoltion 51 first-order terms in eqution 5). This pproximtion is justified by the fct tht, ccording to the theory of chrcteristics for second-order hyperbolic equtions Cournt nd Hilbert 1989), the first-order terms ffect only the mplitude of the propgting wves. To preserve the kinemtics, it is sufficient to retin only the second order terms of eqution 5): 2 U c ζζ ζ + c 2 U 2 ξξ ξ + c 2 U 2 ηη η + c 2 U 2 ξη ξ η = s)2 U. 6) From eqution 6) we cn derive the following dispersion reltion of the coustic wve-eqution in Riemnnin coordintes c ζζ k 2 ζ c ξξk 2 ξ c ηηk 2 η c ξηk ξ k η = s, 7) where k ζ, k ξ nd k η re wvenumbers ssocited with the Riemnnin coordintes ζ, ξ nd η. Coefficients c ξξ, c ηη nd c ζζ re known quntities defined using the coordinte system mpping 2) 4). For one-wy wvefield extrpoltion, we need to solve the qudrtic eqution 7) for the wvenumber of the extrpoltion direction k ζ, nd select the solution with the pproprite sign for the desired extrpoltion direction: k ζ = s c ζζ c ξξ c ζζ k 2 ξ c ηη c ζζ k 2 η c ξη c ζζ k ξ k η. 8) The 2D equivlent of eqution 8) tkes the form: s k ζ = c ξξ kξ 2 c ζζ c. 9) ζζ In ry coordintes, defined by ζ τ propgtion time) nd ξ γ shooting ngle), we cn re-write eqution 9) s k τ = sα α J k γ, 10) where α represents velocity nd J represents geometricl spreding. The quntities α nd J chrcterize the extrpoltion coordinte system: α describes the velocity used for construction of the ry coordinte system; J describes the spreding or focusing of the coordinte system. In generl, the velocity used for construction of the coordinte system is different from the velocity used for extrpoltion, s suggested by Sv nd Fomel 2005), nd illustrted lter in this pper. We cn further simplify the computtions by introducing the nottion = sα, 11) b = α J, 12) thus eqution 10) tking the form k τ = bk γ. 13) For Crtesin coordinte systems, α = 1 nd J = 1, eqution 13) reduces to the known dispersion reltion k z = 2 s 2 kx 2, 14) where k z nd k x re depth nd position extrpoltion wvenumbers. EXTRAPOLATION KERNELS Extrpoltion using eqution 13) implies tht the coefficients defining the problem, nd b, re not chnging sptilly. In this cse, we cn perform extrpoltion using simple phseshift opertion U τ+ τ = U τ e ikτ τ, 15) where U τ+ τ nd U τ represent the coustic wvefield t two successive extrpoltion steps, nd k τ is the extrpoltion wvenumber defined by eqution 13). For medi with lterl vribility of the coefficients nd b, due to either velocity vrition or focusing/defocusing of the coordinte system, we cnnot use in extrpoltion the wvenumber computed directly using eqution 13). As for the cse of extrpoltion in Crtesin coordintes, we need to pproximte the wvenumber k τ using expnsions reltive to nd b. Such pproximtions cn be implemented in the spce-domin, in the Fourier domin or in mixed spce- Fourier domins. Spce-domin extrpoltion The spce-domin finite-differences solution to eqution 13) is derived bsed on squre-root expnsion s suggested by Frncis Muir Clerbout 1985): ν k τ + μ ρ, 16) where the coefficients μ, ν nd ρ tke the form derived in Appendix A: ν = c 1 ) b 2, 17) μ = 1, 18) C 2007 Europen Assocition of Geoscientists & Engineers, Geophysicl Prospecting, 56, 49 60

4 52 P. Sv nd S. Fomel ρ = c 2 b. 19) In the specil cse of Crtesin coordintes, = s nd b = 1, eqution 16) tkes the fmilir form k τ s c 1 s 1 c 2 s 2, 20) where the coefficients c 1 nd c 2 tke different vlues for different orders of Muir s expnsion: c 1, c 2 ) = 0.50, 0.00) for the 15 eqution, nd c 1, c 2 ) = 0.50, 0.25) for the 45 eqution, etc. For extrpoltion in Riemnnin coordintes, the mening of 15,45 etc is not defined. We use this terminology here to indicte orders of ccurcy comprble to the ones defined in Crtesin coordintes. Mixed-domin extrpoltion Mixed-domin solutions to the one-wy wve eqution consist of decompositions of the extrpoltion wvenumber defined in eqution 13) in terms computed in the wvenumber domin for reference of the extrpoltion medium, followed by finite-differences correction pplied in the spce-domin. For eqution 13), generic mixed-domin solution hs the form: ν k τ k τ ) + μ ρ, 21) where 0 nd b 0 re reference vlues for the medium chrcterized by the prmeters nd b, nd the coefficients μ, ν nd ρ tke different forms ccording to the type of pproximtion. As for usul Crtesin coordintes, k τ 0 is pplied in the wvenumber domin, nd the other two terms re pplied in the spce domin. If we limit the spce-domin correction to the thin lens term, 0 ), we obtin the equivlent of the split-step Fourier SSF) method Stoff et l. 1990) in Riemnnin coordintes. Appendix A detils the derivtions for two types of expnsions known by the nmes of pseudo-screen Hung et l. 1999), nd Fourier finite-differences Ristow nd Ruhl 1994; Biondi 2002). Other extrpoltion pproximtions re possible, but re not described here, for simplicity. Pseudo-screen method: The coefficients for the pseudo-screen pproximtion to eqution 21) re [ ) )] b b 0 ν = 0 c 1 1 1, 22) 0 b 0 0 μ = 1, 23) ρ = 3c 2 b 0 0, 24) where 0 nd b 0 re reference vlues for the medium chrcterized by prmeters nd b. In the specil cse of Crtesin coordintes, = s nd b = 1, eqution 21) with coefficients eqution 22) 24) tkes the fmilir form k τ k τ c 1 s c 2 s 2 0 s s 0 ), 25) where the coefficients c 1 nd c 2 tke different vlues for different orders of the finite-differences term: c 1, c 2 ) = 0.50, 0.00), c 1, c 2 ) = 0.50, 0.25), etc. When c 1, c 2 ) = 0.00, 0.00) we obtin the usul split-step Fourier eqution Stoff et l. 1990). Fourier finite-differences method: The coefficients for the Fourier finite-differences solution to eqution 21) re ν = 1 2 δ2 1, 26) μ = δ 1, 27) ρ = 1 4 δ 2, 28) where, by definition, ) b 2 b0 δ 1 = 0, 29) 0 ) b 4 ) 4 b0 δ 2 = 0. 30) 0 Figure 1 Velocity mp nd Riemnnin coordinte system for the Mrmousi exmple. C 2007 Europen Assocition of Geoscientists & Engineers, Geophysicl Prospecting, 56, 49 60

5 Riemnnin wvefield extrpoltion 53 ) b) Figure 2 Coordinte system coefficients defined in equtions 11) nd 12). ) Prmeter = sα in ry coordintes. b) Prmeter b = α/j in ry coordintes. ) b) c) d) Figure 3 Migrtion impulse responses in Riemnnin coordintes. ) Extrpoltion with the 15 finite-differences eqution. c) Extrpoltion with the 60 finite-differences eqution. b) Extrpoltion with the pseudo-screen eqution. d) Extrpoltion with the Fourier finite-differences eqution. C 2007 Europen Assocition of Geoscientists & Engineers, Geophysicl Prospecting, 56, 49 60

6 54 P. Sv nd S. Fomel ) b) c) d) Figure 4 Migrtion impulse responses in Riemnnin coordintes fter mpping to Crtesin coordintes. ) Extrpoltion with the 15 finitedifferences eqution. c) Extrpoltion with the 60 finite-differences eqution. b) Extrpoltion with pseudo-screen eqution. d) Extrpoltion with the Fourier finite-differences eqution. 0 nd b 0 re reference vlues for the medium chrcterized by the prmeters nd b. In the specil cse of Crtesin coordintes, = s nd b = 1, eqution 21) with coefficients eqution 26) 28) tkes the fmilir form: k τ k τ c 2 c 1 ss s 2 ss 0 s 0 2 ) s s 0 ), 31) where the coefficients c 1 nd c 2 tke different vlues for different orders of the finite-differences term: c 1, c 2 ) = 0.50, 0.00) for 15,c 1, c 2 ) = 0.50, 0.25) for 45, etc. When c 1 = c 2 = 0.0 we obtin the usul split-step Fourier eqution Stoff et l. 1990). EXAMPLES We illustrte the high-order Riemnnin wvefield extrpoltion extrpoltors with impulse responses for two synthetic models. The first exmple is bsed on the Mrmousi model Versteeg 1994). We construct the coordinte system by ry trcing from point source t the surfce in smooth version of the rel velocity model. Figure 1 shows the velocity model with the coordinte system overlid, nd Figs 2) 2b) show the coordinte system coefficients nd b defined in equtions 11) nd 12). The gol of this test model is to illustrte the high-order extrpoltion kernels in firly complex model using simple coordinte system. In this wy, the coordinte system nd the rel direction of wve propgtion deprt from onenother, thus ccurte extrpoltion requires higher order kernels. The coordinte system is constructed from point t the loction of the wve source. This setting is similr to the cse of extrpoltion from point source in Crtesin coordintes, where high-ngle 1 propgtion requires high-order kernels. Figures 3) 3d) show impulse responses for point source computed with vrious extrpoltors in ry coordintes τ nd γ ). Pnels ) nd c) show extrpoltion with the 15 nd 60, respectively. Pnels b) nd d) show extrpoltion with the pseudo-screen eqution, nd the Fourier finite-differences 1 If the extrpoltion xis is time, the mening of higher ngle ccurcy is not well defined. We cn use this terminology to ssocite the mthemticl mening of the pproximtion for the squre-root by nlogy with the Crtesin equivlents. C 2007 Europen Assocition of Geoscientists & Engineers, Geophysicl Prospecting, 56, 49 60

7 Riemnnin wvefield extrpoltion 55 ) Figure 6 Velocity mp nd Riemnnin coordinte system for the lrge-grdient model experiment. b) Figure 5 Comprison of extrpoltion in Crtesin nd Riemnnin coordintes. ) Split-step Fourier extrpoltion in Crtesin coordintes. b) Split-step Fourier extrpoltion in Riemnnin coordintes. eqution, respectively. All plots re displyed in ry coordintes. We cn observe tht the ngulr ccurcy of the extrpoltor improves for the more ccurte extrpoltors. The finite-differences solutions pnels nd c) show the typicl behvior of such solutions for the 15 nd 60 equtions, but in the more generl setting of Riemnnin extrpoltion. The mixed-domin extrpoltors pnels b nd d) re more ccurte thn the finite-differences extrpoltors. The min differences occur t the highest propgtion ngles. As for the cse of Crtesin extrpoltion, the most ccurte kernel of those compred is the equivlent of Fourier finite-differences. Figures 4) 4d) show the corresponding plots in Figs 3) 3d) mpped in the physicl coordintes. The overly is n outline of the extrpoltion coordinte system. After re-mpping to the physicl spce, the comprison of high-ngle ccurcy for the vrious extrpoltors is more pprent, since it now hs physicl mening. Figures 5) 5b) show side-by-side comprison of equivlent extrpoltors in Riemnnin nd Crtesin coordi- ntes. The impulse response in Fig. 5) shows the limits of Crtesin extrpoltion in propgting wves correctly up to 90. The Riemnnin extrpoltor in Fig. 5b) hndles much better wves propgting t high ngles, including energy tht is propgting upwrd reltive to the physicl Crtesin coordintes. The second exmple is bsed on model with lrge lterl grdient which mkes n incident plne wve overturn. A smll Gussin nomly, not used in the construction of the coordinte system, forces the propgting wve to triplicte nd move t high ngles reltive to the extrpoltion direction. Figure 6 shows the velocity model with the coordinte system overlid. Figures 7) 7b) show the coordinte system coefficients, nd b defined in equtions 11) nd 12). The gol of this model is to illustrte Riemnnin wvefield extrpoltion in sitution which cnnot be hndled correctly by Crtesin extrpoltion, no mtter how ccurte n extrpoltor we use. In this exmple, n incident plne wve is overturning, thus becoming evnescent for the solution constructed in Crtesin coordintes. Furthermore, the Gussin nomly shown in Fig. 7) cuses wvefield tripliction, thus requiring high-order kernels for the Riemnnin extrpoltor. Figures 8) 8d) show impulse responses for n incident plne wve computed with vrious extrpoltors in ry coordintes τ nd γ ). Pnels ) nd c) show extrpoltion with the 15 nd 60 finite-differences equtions, respectively. Pnel b) nd d) show extrpoltion with the pseudo-screen eqution nd the Fourier finite-differences eqution, respectively. All plots re displyed in ry coordintes. As for the preceding exmple, we observe higher ngulr ccurcy s we increse the order of the extrpoltor. The equivlent Fourier C 2007 Europen Assocition of Geoscientists & Engineers, Geophysicl Prospecting, 56, 49 60

8 56 P. Sv nd S. Fomel ) b) Figure 7 Coordinte system coefficients defined in equtions 11) nd 12). ) Prmeter = sα in ry coordintes. b) Prmeter b = α/j in ry coordintes. ) b) c) d) Figure 8 Migrtion impulse responses in Riemnnin coordintes. ) Extrpoltion with the 15 finite-differences eqution. c) Extrpoltion with the 60 finite-differences eqution. b) Extrpoltion with the pseudo-screen eqution. d) Extrpoltion with the Fourier finite-differences eqution. C 2007 Europen Assocition of Geoscientists & Engineers, Geophysicl Prospecting, 56, 49 60

9 Riemnnin wvefield extrpoltion 57 ) b) c) d) Figure 9 Migrtion impulse responses in Riemnnin coordintes fter mpping to Crtesin coordintes. ) Extrpoltion with the 15 finite-differences eqution. c) Extrpoltion with the 60 finite-differences eqution. b) Extrpoltion with the pseudo-screen eqution. d) Extrpoltion with the Fourier finite-differences eqution. Finite-differences extrpoltor shows the highest ccurcy of ll tested extrpoltors. As in the preceding exmple, Figs 9) 9d) show the corresponding plots in Figs 8) 8d) mpped in the physicl coordintes. The overly is n outline of the extrpoltion coordinte system. Finlly, Figs 10) nd 10b) show side-by-side comprison of equivlent extrpoltors in Riemnnin nd Crtesin coordintes. The impulse response in Fig. 10) clerly shows the filure of the Crtesin extrpoltor in propgting wves correctly even up to 90. The Riemnnin extrpoltor in Fig. 10b) hndles much better overturning wves much better, including energy tht is propgting upwrd reltive to the verticl direction. DISCUSSION Accurte wve-eqution migrtion using Riemnnin wvefield extrpoltion requires choice of coordinte system tht exploits its higher extrpoltion ccurcy. An effective choice of coordinte system would be one tht minimizes the difference between the extrpoltion direction nd the direction of wve propgtion. If this condition is fulfilled, we cn chieve high-ngle ccurcy using low-order extrpoltion kernels. Otherwise, we need to extrpolte seismic wvefields with high-order kernels, like the ones described in this pper. Shot-record migrtion requires selection of coordinte systems for the source nd receiver wvefields. Optiml selection of coordinte systems in this sitution is not trivil tsk, since the source nd receiver wvefields re optimlly described by different coordinte systems which lso vry with loction. However, if we employ high-order extrpoltion kernels, different seismic experiments my shre the sme pproximtely optiml coordinte system. An esy wy to illustrte this ide is represented by imging in tilted) Crtesin coordinte systems, which re just specil cses of Riemnnin coordintes Sv nd Fomel 2006). A complete tretment of this topic remins subject for future reserch. C 2007 Europen Assocition of Geoscientists & Engineers, Geophysicl Prospecting, 56, 49 60

10 58 P. Sv nd S. Fomel ACKNOWLEDGMENT ExxonMobil provided prtil finncil support for this reserch. REFERENCES ) b) Figure 10 Comprison of extrpoltion in Crtesin nd Riemnnin coordintes. ) Split-step Fourier extrpoltion in Crtesin coordintes. b) Split-step Fourier extrpoltion in Riemnnin coordintes. CONCLUSIONS Higher-order Riemnnin wvefield extrpoltion is needed when the coordinte system does not closely conform with the generl direction of wvefield propgtion. This sitution occurs, for exmple, when the coordinte system is creted by ry trcing in medium tht is different from the one used for extrpoltion, or when the coordinte system is constructed bsed on geometricl properties of the cquisition geometry e.g. migrtion from topogrphy). Spce-domin nd mixeddomin finite-difference solutions to Riemnnin wvefield extrpoltion improve the ngulr ccurcy. 3D solutions cn be ddressed with explicit finite-differences or by using splitting nd implicit methods, similrly with the techniques used for Crtesin extrpoltion. Biondi B Stble wide-ngle Fourier finite-difference downwrd extrpoltion of 3-D wve-fields. Geophysics 67, Clerbout J Multidimensionl recursive filters vi helix. Geophysics 63, Clerbout J.F Imging the Erth s Interior. Blckwell Scientific Publishers. Cournt R. nd Hilbert D Methods of Mthemticl Physics. John Wiley & Sons. Hle D Stble explicit depth extrpoltion of seismic wvefields. Geophysics 56, Hung L.Y., Fehler M.C. nd Wu R.S Extended locl Born Fourier migrtion method. Geophysics 64, Rickett J., Clerbout J. nd Fomel S.B Implicit 3-D depth migrtion by wvefield extrpoltion with helicl boundry conditions. 68th SEG meeting, New Orlens, LA, USA, Ristow D. nd Ruhl T Fourier finite-difference migrtion. Geophysics 59, Ristow D. nd Ruhl T D implicit finite-difference migrtion by multiwy splitting. Geophysics 62, Sv P. nd Fomel S Seismic imging using Riemnnin wvefield extrpoltion. Geophysics 70, T45 T56. Sv P. nd Fomel S Imging overturning reflections by Riemnnin wvefield extrpoltion. Journl of Seismic Explortion 15, Shrgge J Non-liner Riemnnin wvefield extrpoltion. Geophysics, submitted for publiction. Shrgge J. nd Sv P Wve-eqution migrtion from topogrphy. 75th SEG meeting, Houston, TX, USA, Expnded Abstrcts, Stoff P.L., Fokkem J.T., de Lun Freire R.M. nd Kessinger W.P Split-step Fourier migrtion. Geophysics 55, Versteeg R The Mrmousi experience: Velocity model determintion on synthetic complex dt set. The Leding Edge 13, SPACE-DOMAIN FINITE-DIFFERENCES Strting from eqution 13), bsed on the Muir expnsion for the squre-root Clerbout 1985), we cn write successively: k τ = 1 [ ] 2 b A1) C 2007 Europen Assocition of Geoscientists & Engineers, Geophysicl Prospecting, 56, 49 60

11 Riemnnin wvefield extrpoltion 59 1 [ ] 2 b c 1 1 c 2 [ b ] 2 A2) If we mke the nottions ) )] ) b 2 b0 ν = 0 [c A12) 0 b 0 0 c 1 ) ) b c b ) 2. A3) 2 If we mke the nottions ) b 2 ν = c 1, A4) μ = 1, A5) ρ = c 2 b. A6) we obtin the finite-differences solution to the one-wy wve eqution in Riemnnin coordintes: ν k τ + μ ρ. A7) MIXED DOMAIN PSEUDO-SCREEN The pseudo-screen solution to eqution 13) derives from first-order expnsion of the squre-root round 0 nd b 0 which re reference vlues for the medium chrcterized by the prmeters nd b: k τ k τ 0 + k τ 0 ) + k τ 0,b 0 b b b 0 ). A8) 0,b 0 The prtil derivtives reltive to nd b, respectively, re: k τ k τ b = 0,b 0 1 = b 0 0,b b0 k γ 0 ) c 1 b0 k γ 0 1 3c 2 b0 k γ 0 ) 0 2 b0 k γ b 0 0 Therefore, the pseudo-screen eqution becomes, A9) ) b0 k 2 γ. 0 A10) k τ k τ ) [ ) )] b 0 c b0 0 b A11) 1 3c b0 2 0 μ = 1 A13) b0 ρ = 3c 2 A14) 0 we obtin the mixed-domin pseudo-screen solution to the one-wy wve eqution in Riemnnin coordintes: ν k τ k τ ) + μ ρ MIXED DOMAIN FOURIER FINITE-DIFFERENCES. A15) The pseudo-screen solution to eqution 13) derives from fourth-order expnsion of the squre-root round 0, b 0 ) nd, b): [ k τ [ k τ [ ] 2 b + 1 ) ] 4 b, A16) 8 ) b0 k 2 γ + 1 ) ] b0 k 4 γ. A17) If we subtrct equtions A16) nd A17), we obtin the following expression for the wvenumber long the extrpoltion direction k τ : ) k τ k τ ) + [ 1 2 ] 2 b b 0 k γ 0 ) + [ b 0 b 0 0 ) 4 ] k γ 0 ) 4. A18) We cn mke the nottions b b 0 δ 1 = 0, A19) ) 4 b δ 2 = 0 0 b 0 0 ) 4, A20) therefore eqution A18) cn be written s ) k τ = k τ ) ) 2 δ k γ δ k γ 2. A21) C 2007 Europen Assocition of Geoscientists & Engineers, Geophysicl Prospecting, 56, 49 60

12 60 P. Sv nd S. Fomel Using the pproximtion μ = δ 1, A25) 1 2 δ 1u δ 2u 4 2 δ2 1 u2 δ 1 1 δ 4 2u, 2 we cn write 1 2 δ2 1 k τ = k τ ) + δ 1 1 δ 4 2 If we mke the nottions ν = 1 2 δ2 1, A22). A23) A24) ρ = 1 4 δ 2, A26) we obtin the mixed-domin Fourier finite-differences solution to the one-wy wve eqution in Riemnnin coordintes: ν k τ k τ ) + μ ρ. A27) C 2007 Europen Assocition of Geoscientists & Engineers, Geophysicl Prospecting, 56, 49 60