Migration with arbitrarily wide-angle wave equations

Transcription

1 GEOPHYSICS, VOL 7, NO 3 MAY-JUNE 5; P S61 S7, 4 FIGS 1119/ Migration with arbitrarily wide-angle wave eqations Mrthy N Gddati 1 and A Homayon Heidari 1 ABSTRACT We develop a new scalar migration techniqe that is highly accrate for imaging steep dips in heterogeneos media This method is based on arbitrarily wideangle wave eqations AWWEs that are highly accrate space-domain one-way wave eqations and have a form similar to the 15 eqation The accracy of the proposed method is increased by introdcing axiliary variables, as well as adjsting the parameters of the approximation Poststack migration is carried ot by downward contination sing the AWWE, for which we have developed a stable, explicit, doble-marching scheme Up to 8 accracy is achieved by secondorder AWWE migration with only 3 times the comptational effort of the 15 eqation and reqiring almost the same storage We illstrate the performance of AWWE migration sing implse-response graphs, a single-dipping reflector, and a slice of the SEG/EAGE salt model INTRODUCTION Wave eqation based migration is enjoying significant attention W, 3; Jerzak et al, ; Sn and McMechan, 1; Gray, 1; Zho and McMechan, 1997; Zhe and Greenhalgh, 1997 In addition to academic attention, indstry is also moving from ray-based Kirchhoff methods to wave eqation based methods becase of their favorable properties with respect to applications in complex media Ritchie, 3 Among different categories of wave eqation based methods, those sing one-way wave eqations OWWE are more favored than others becase they minimize the ndesirable effects of nwanted reflections In addition to seismic imaging, OWWEs have been tilized in areas sch as nbondeddomain modeling Lindman, 1975; Gddati and Tassolas,, nderwater acostics Collins, 1989; Bamberger et al, 1988, and nondestrctive testing and evalation Chang and Chir-Cherng, OWWEs can be sed in either the space domain or the wavenmber domain Wavenmber-domain OWWEs are simple in form bt cannot handle laterally heterogeneos media Space-domain OWWEs are more desirable for heterogeneos media bt are complex in their form and comptationally expensive To alleviate the comptational expense, spacedomain OWWEs are approximated in varios ways; this paper focses on accrate approximation of space-domain OWWEs and their application to imaging Note that the exact OWWE constrcted by factorizing the fll-wave eqation is not dynamically correct for general heterogeneos media becase, ie, it does not preserve amplitde [see Zhang et al 3 for an agmented tre-amplitde OWWE] Hence, the term accracy throghot this paper refers to the traveltime kinematic accracy of the reslts Tre amplitde migration is beyond the scope of this paper and is not discssed frther Since the introdction of the 15 OWWE by Claerbot Claerbot, 197, 1985; Claerbot and Doherty, 197, many researchers have proposed varios methods to improve the accracy of OWWEs withot dramatically increasing the comptational cost eg, Lindman, 1975; Berkhot, 1979; Lee and Sh, 1985; Zhang, 1985; Zhang et al, 1988; Ristow and Rhl, 1994; Gddati and Tassolas, The basic ideas behind these methods are 1 to se higher-order rational approximation of the sqare-root operator occrring in the freqencywavenmber version of OWWE, and to write the reslting high-order differential eqation in a lower-order form amenable to nmerical implementation While the aforementioned methods are accrate for scalar imaging, they cannot be extended to imaging in more complex elastic and poroelastic media Althogh some elastic and poroelastic OWWEs have been developed in the context of ocean acostics Collins, 1993a, b; Collins et al, 1995; Lingevitch and Collins, 1998; Lingevitch et al,, they are not applicable for imaging problems becase the eqations are in terms of dilatations or displacement derivatives that are not readily available Moreover, these formlations work only for special cases, sch as Manscript received by the Editor November 1, 3; revised manscript received November 5, 4; pblished online May 3, 5 1 North Carolina State University, Department of Civil Engineering, 8 Mann Hall, Stinson Drive, Raleigh, North Carolina mngddat@eosncsed; ahheidar@ncsed c 5 Society of Exploration Geophysicists All rights reserved S61

2 S6 Gddati and Heidari transversely isotropic elasticity, with a principal material direction coinciding with the direction of one-way propagation Frthermore, the interface conditions tend to be complex in natre, reqiring great care in nmerical implementation A space-domain OWWE that can be easily tilized for imaging in complex media wold be of significant se Gddati 4 devised the arbitrarily wide-angle wave eqation AWWE, which is a highly accrate and comptationally tractable space-domain OWWE represented completely in terms of the primary field variable eg, pressre or displacements The AWWE formalism is applicable to any wave represented by second-order differential eqations in space, which incldes waves in general anisotropic elastic and poroelastic media AWWE is a system of second-order differential eqations, and it appears amenable to nmerical implementation in modeling and migration In this paper, we develop and implement a downwardcontination techniqe based on scalar AWWE First, a brief review of conventional OWWE is given, followed by the derivation of the scalar AWWE Next, the AWWE poststack migration problem is stated and solved with the help of an efficient doble-marching finite-difference scheme Finally, the accracy of AWWE migration is illstrated sing three nmerical examples, inclding poststack migration of a D slice of the SEG/EAGE salt model METHOD Classical one-way wave eqations To discss classical one-way wave eqations and to present or new approach, we start the derivation from the scalar wave eqation in a two-dimensional space x, z: 1 x,z; t x,z; t =, c t 1 where x, z; t is the field variable, and c is the wave velocity We assme that the vertical direction z isthepreferreddirection, ie, the direction of one-way propagation We also assmethat thepositivez-axis is directed downward, ie, toward increasing depth Performing Forier transforms on eqation 1 over x and t yields ûk x,z; ω z [ ] ω kx ûk x,z; ω =, c where k x is the horizontal wave nmber, ω is the freqency, and û is the Forier transform of Eqation can be factorized as { } ω z kx c { } ω z + kx ûk x,z; ω =, 3 c where the first operator represents downward-propagating waves toward increasing z, while the second is the pward propagator toward decreasing z Of particlar interest in imaging is the pward-propagating OWWE, which is sed in the downward-contination procedre for extrapolating the reflected wave back to the reflector Berkhot, 1981 By ignoring the downward propagator in eqation 3, we obtain the exact pward-propagating OWWE: û z = i ω c 1 ζ û, 4 where ζ = ck x /ω Eqation 4 forms the basis for phaseshift migration Gazdag, 1978, which is performed in the wavenmber-freqency domain However, for application to heterogeneos media, it is desirable to obtain a version of OWWE in the time-space domain The exact OWWE in the time-space domain can be written as z = 1 c t iω [ 1 1 ζ c ] e ikxx ωt dk x dω, 5 where the operator stands for convoltion with respect to x and t The comptational cost associated with the soltion of the above integral eqation is extremely high becase of the nonlocal natre of the convoltion operator, rendering the eqation impractical to solve The conventional way to overcome this difficlty is to approximate the sqare-root operator in eqation 4 sing a rational fnction and sbseqently obtain a differential form for OWWE after performing the inverse Forier transform on the approximated expression Sch an approximation is often in the form of a trncated contined-fraction approximation, which can be written in a recrsive form as r 1 = 1, r n+1 = 1 ζ, n = 1,, 3,, r n where r n is the nth order approximation of 1 ζ Thisclassical approximation has been widely sed in the context of seismic exploration Claerbot, 1985 as well as in the formlation of absorbing bondaries Engqist and Majda, 1979 The first-order approximation n = 1 is a one-dimensional wave eqation also called the 5 eqation; the second- and third-order approximations reslt in the well-known 15 and 45 eqations, respectively However, for high-order approximations n >, the reslting eqations necessitate an implicit soltion procedre that is not efficient Frthermore, there appear to be no direct stable implementations for n > 3 To implement higher-order OWWE, Bamberger et al 1988 sed an approach similar to Lindman 1975 to eliminate higher-order derivatives with the help of axiliary variables In practice, this reslts in a third-order system, implemented for modeling problems in ocean acostics by Collins 1989 A similar approach has been independently sed by Lee and Sh 1985 in the context of seismic migration Using a matrix form of the contined-fraction iteration, Gddati and Tassolas devised another approach to implement a high-order OWWE in the context of nbonded-domain modeling Their approach, which reslts in a second-order system of eqations, has sbseqently been applied to imaging problems Gddati and Heidari, 3; Heidari and Gddati, 3

3 Migration with AWWE S63 Althogh the above methods provide effective ways to se high-order OWWEs for scalar migration, they are not extendable to migration in elastic and other complex media To factorize the elastic-wave eqation into one-way propagators, it shold be rearranged in terms of a displacement component and a displacement derivative sch as dilatation or a strain component, and it cannot be sed for seismic migration Frthermore, these formlations work only for special cases, sch as transversely isotropic elasticity, with a principal material direction coinciding with the z-axis The above limitations motivated an effort to develop a general approach for obtaining accrate OWWE for elastic media in terms of displacements This led to the development of arbitrarily wide-angle wave eqations AWWE by Gddati 4, which are accrate OWWEs for general heterogeneos anisotropic elastic and poroelastic media Considering the high-accracy properties demonstrated by AWWE, we embarked on investigating their application to seismic migration This paper focses on application of AWWE for scalar migration Scalar AWWE formlation The AWWE formlation is based on a series of ideas linking one-way wave eqations to finite-element approximation of half-space stiffness relations In this section, we briefly otline the AWWE formlation in the context of the scalar-wave eqation Consider the governing eqation + C =, 7 z where C = k x ω/c We have dropped the accent over for simplicity The first step of the AWWE formlation is to link the OWWE to the half-space stiffness relationship, an idea that has been widely tilized in nbonded-domain modeling see, eg, Givoli, 199 We first observe that a half-space < z < when excited at the bondary z = wold contain only downward-propagating waves since there will be no reflections from infinity Based on this observation, the traction at the bondary F and the displacement at the bondary = z= can be linked by the exact one-way wave eqation F = z = z= i ω 1 ζ 8 c Ths, the stiffness K = F / of the half-space is given by the one-way operator [ iω/c 1 ζ ] It follows that approximating the one-way propagator is eqivalent to approximating the half-space stiffness K as K approx, and writing the approximate OWWE as z = K approx 9 To obtain the approximate stiffness, the half-space is replaced by a finite-element layer of thickness L 1 < z <L 1 and a half-space z >L 1 Within the layer, the field variable is assmed to vary linearly from at z = to 1 at z = L 1 Since the stiffness of the remaining half-space z >L 1 isk, the traction at z = can be written as { } { } { } { } F K = S +, 1 1 K 1 where S is the stiffness matrix of the finite element layer Eqation 1 is only approximate, with error reslting from discretization Gddati 4 shows that the discretization error in the half-space stiffness is eliminated by sing a simple midpoint integration rle for evalating the element stiffness matrix, ie, { K } { = S 1 } { } +, 11 K 1 with S being the stiffness of the finite-element layer obtained sing the midpoint integration rle, and given by [ ] [ ] [ ] S 11 S 1 S = 1L CL S 1 S Applying eqation 11 recrsively, ie, discretizing the entire half-space with an infinite nmber of layers with thicknesses of L j,j = 1,, 3,reslts in S K 11 1 S 1 1 S1 1 S 1 = + S 11 S 1 1 S 1 S + S where S j is the stiffness matrix corresponding to the layer [ j 1, j ] Eqation 13 is not comptationally tractable becase of the existence of an infinite nmber of nknowns Therefore, we simply trncate the half-space after discretizing with n layers and apply the Dirichlet bondary condition at the trncation bondary, ie, n = Sch a trncation reslts in a comptationally tractable, albeit approximate, stiffness relation K K approx S11 1 S1 1 S1 1 S 1 + S 11 S 1 = S 1 S n S11 n 1 n 1 14

4 S64 Gddati and Heidari Eqation 14 is obtained by discretization of a physical halfspace However, in the AWWE formlation, this expression is considered to be the approximation of the right-hand side of eqation 8 In other words, eqation 14 is a high-order approximation of the one-way propagator of eqation 9, with L j being parameters of the approximation Hence L j will no longer present a physical length in the actal propagation domain Based on eqations 14 and 9, the approximate OWWE is written as z S11 1 S1 1 S1 1 S 1 + S 11 S 1 = S 1 S n S11 n 1 15 n 1 Since is the primary field variable, we drop the sbscript The rest of the variables 1,, n 1 can be viewed as axiliary variables, which are related to throgh the eqations to n in the system Eqation 15 is eqivalent to the continedfraction approximation of the exact OWWE To show this similarity, we start from eqation 14 and eliminate all the axiliary variables from n 1 to n This reslts in the following contined-fraction form: K= ik z = S 1 11 S 1 1 S1 1 S 1 + S 11 S 1 S 1 S + S3 11 S 3 1 S3 1 S 3 + S4 11, 16 which shows that eqation 15 is a tractable form of a contined-fraction approximation of the exact OWWE Althogh eqation 15 is an approximate OWWE, Gddati 4 proves that, for any wave with its wavenmber coinciding with one of the reference wavenmbers, kz j = i, j =,,n, 17 L j eqation 15 becomes an exact propagator Therefore, if L j are set to real nmbers, eqation 15 is expected to represent all the evanescent waves as n Hence, to represent propagating waves, which is or main intent, we shold choose imaginary L j sch that it wold reslt in realvaled wavenmbers Ths, based on eqation 17, we choose imaginary lengths of L j = i/kz j,wherekj z are the reference wavenmbers, ie, the real wavenmbers for which eqation 15 is exact If both propagating and evanescent waves need to be represented, L j can be chosen as a combination of real, imaginary, and/or complex nmbers By choosing the parameters appropriately, we can adjst the accracy of the approximate OWWE for an arbitrary wide range of angles For this reason, we call the OWWE in eqation 15 the arbitrarily wideangle wave eqation AWWE Instead of L j, we can choose the reference wave nmbers kz j as the parameters of the AWWE, making its propagation characteristics more transparent Alternatively, reference phase velocities c j = ω/kz j can also be chosen as parameters Ths, by sbstitting L j = ic j /ω into eqation 1, and recalling that C = kx ω/c, we have [ ] [ ] S j = iω ik x c j c j 1 +1 ω [ ] iωc j c Sbstitting the stiffness matrix of eqation 18 into eqation 15, mltiplying by iω, and performing inverse Forier transforms over ω and k x, we obtain the time-space form of the downward-propagating AWWE: d z t + 1 c Λ 1 + Λ t cλ =, 19 x where Λ 1 = c 1 c 1 1 c c 1 c c 1 c 1 c 1 c n + 1 c n 1 1 c n 1, c n 1 c n 1 c n c 1 c 1 Λ = 1 c 1 c 1 + c c c c, cn + c n 1 c n 1 c n 1 c n 1 + c n and d T = 1 { } 1 n 1

5 Migration with AWWE S65 We se the sbscript n to represent the nmber of eqations; hence, AWWE n has n 1 axiliary variables and is defined by n reference velocities c 1,,c n Upward-propagating AWWE, which is sed for migration, is obtained by replacing z by z, ie, changing the sign of the first term in eqation 19 AWWE 1 with the reference velocity chosen as the backgrond velocity is eqivalent to the 15 eqation Accracy of AWWE The accracy of an AWWE can be assessed analytically by stdying its associated dispersion relation and comparing it with the dispersion relation of the exact OWWE The latter canbewrittenas k z ω c 1 ζ =, 3 for the pward-propagating waves In eqation 3, ζ = ck x /ω represents the dip angle as ζ = sin θ To assess the accracy of the AWWE, we start from eqation 19 and perform Forier transforms over all the variables to get k z ω [ Λ1 + 1 ζ ] 1 Λ c = 4 n 1 All of the axiliary variables in eqation 4 can be eliminated to yield the dispersion relation k z ω c F nζ,c 1,,c n =, 5 The fnctions in eqations 7 9, along with the exact sqare-root operator, are shown in Figre 1 To show the accracy in terms of the dip angle, the horizontal axis is set to θ = sin 1 ζ The figre shows that the ranges of accracy of these three AWWEs, considering a tolerance of 1, are approximately 6 for AWWE with c 1 = c = c,7 for AWWE 3 with c 1 = c = c 3 = c,and8 for AWWE with c 1 = c and c = 4c These accracy measres are confirmed nmerically by showing the implse responses of the same orders of AWWEs, whichwedolaterintheresults section Based on the above discssion, the physical meaning of c j can be explained more clearly As discssed in the previos section, each c j indicates a particlar phase velocity for which the AWWE will be exact In other words, by looking at c c j = ckj z ω = cos θ j, 3 it is clear that each c j determines a specific angle θ j where F n the approximation to the sqare-root operator matches the exact operator Therefore, for a particlar order of AWWE, by changing the vales of c j, one can achieve different levels of accracy in the neighborhood of different vales of θ j Frthermore, from eqation 3, one can immediately conclde that, in order to have better approximation for propagating waves, one mst choose c j c As an example, in AWWE with a parameter set of c, 4c, we wold expect the dispersion relation to be exact on cos 1 c/c = andoncos 1 c/4c = 755, which agrees with the corresponding crve in Figre 1 Note that c j can be less than c, which reslts in imaginary θ j, and indicates that evanescent waves are better approximated We emphasize that while AWWE is exact for points corresponding to c j, it effectively approximates the entire where F n ζ,c 1,,c n represents the fnction reslting from the elimination of axiliary variables in AWWE n with a parameter set of c 1,,c n In other words, F n ζ,c 1,,c n is the approximation to the sqare-root operator in eqation 3, or 1 ζ F n ζ,c 1,,c n 6 Hence, the accracy of any particlar order of AWWE can be assessed by comparing the reslting F n with the exact sqare-root operator As an example, we start from AWWE with parameters of c, c By writing the expression in eqation 4 for this AWWE and eliminating the axiliary variable 1, we obtain F ζ,c,c = 1 1 ζ ζ 4 8 4ζ 7 For the third-order AWWE with c i = c, we have F 3 ζ, c, c, c = 1 3/ζ + 9/16ζ 4 1/3ζ 6, 1 ζ + 3/16ζ 4 8 and finally, to show the effect of the parameters on the accracy of the AWWE, we obtain the F n for AWWE with the parameter set of c,4c, which reslts in F ζ,c,4c = 1 1 ζ ζ 4 5 ζ 9 Figre 1 Evalation of AWWE accracy by comparing the dispersion relation in terms of dip angle of AWWE F n with that of the exact OWWE 1 ζ The horizontal axis is the dip angle θ = sin 1 ζ, and the vertical axis is the exact sqare root operator for OWWE, along with its approximations F n for three cases of AWWE The parameter sets are defined in the figre legend By adjsting the parameters of AWWE, one can achieve a higher dip-angle accracy with AWWE dashdot line than AWWE 3 dotted line with its parameters set to the backgrond velocity

6 S66 Gddati and Heidari dispersion relation see, eg, AWWE 3 in Figre 1 The accracy of AWWE depends on both the nmber of axiliary variables and the specific vales of c j, indicating that high accracy can be achieved even with arbitrarily chosen c j Poststack migration with AWWE Migration with OWWE is performed as a downwardcontination procedre in which the srface trace field variable on the x t plane at z = is transformed into an image of the sorces field variable on the x z plane at t = In particlar, poststack migration with AWWE can be stated as the following initial-bondary vale problem in the timespace domain: Find the field variable x, z; t in the domain {x min < x<x max } { <z<z max } at t =, satisfying the governing eqation pward-propagating AWWE d z t 1 c Λ 1 + Λ t + cλ =, 31 x with initial conditions x,; t = x,t, 3 x,z; t max =, and the bondary conditions t x,z; t max =, 33 x min, z; t =, x max,z; t = 34 In the first initial condition, x,t is the field variable measred at the srface z = srface trace The second initial conditions are obtained assming the srface trace to be zero at t t max since the data can be padded with zeroes Dirichlet bondary conditions are applied at x = x min and x = x max Ideally, more accrate absorbing bondary conditions see, eg, Givoli, 199 representing the nbonded extent in x direction mst be designed for this prpose Devising sch bondary conditions is beyond the scope of this paper Nmerical implementation The finite-difference method is sed to solve the initialbondary vale problem of eqations Noting that AWWE propagates information downward in the positive z direction, reverse in time negative t direction, and in both positive and negative x directions, the marching procedre is sed in the positive z and negative t directions, while no marching is performed for the x direction This is consistent with the mathematical form of eqations The soltion domain is discretized into a grid sch that k i,j = x i,z j,t k, 35 where x i,z j,t k = i x, j z, k t For marching in depth, the Crank-Nicholson method is sed, and the AWWE is discretized in the z direction to obtain 1 z d j+1 t j t 1 c Λ 1 + Λ j+1/ t + cλ j+1/ x = 36 In eqation 36, sbscripts j and j + 1 are associated with vales on the lines z = z j and z = z j+1, whereas the sbscript j + 1/ represents the vales at the center of the layer z j <z<z j+1 From the Crank-Nicholson method, we have j+1 = j+1/ j 37 Sbstitting eqation 37 into eqation 36, we obtain z d j+1/ 1 t c Λ 1 + Λ j+1/ t j+1/ + cλ = x z d j t 38 From eqations 33 and 34, j+1/ satisfies the homogeneos initial and bondary conditions and j+1/ x,t max =, j+1/ x,t max =, 39 t j+1/ x min,t = j+1/ x max,t = 4 Themarchingprocedreinz can be smmarized as follows: Given the field variable j at z = z j, first obtain the axiliary variable vector j + 1/ at the center of the layer z j <z<z j+1 by solving the initial-bondary vale problem described by eqations 38 4 Next, obtain j+1 from eqation 37 In this procedre, all axiliary variables are compted at the center of the layer and are merely sed as intermediate variables that are not needed for sbseqent marching Ths, they are neither compted nor stored at the lines z =, z 1,z,,z max The soltion of the initial-bondary vale problem 38 4 is performed by the standard central-difference discretization in time The time-discrete form of eqation 38 is z d k+1 j+1/ k 1 j+1/ t 1 c Λ 1 + Λ k+1 j+1/ k j+1/ + k 1 j+1/ t = k+1 z d j k 1 j t + cλ k j+1/ x 41 Rearranging the above eqation, we obtain the eqation for reverse marching in time as [Λ 1 + Λ + α z D] k 1 j+1/ = α zd k+1 j k 1 j k+1 j+1/ Λ 1 + Λ k+1 j+1/ k k j+1/ + c t j+1/ Λ, x where D = dd T and 4 α z = c t z 43 Eqation 4 is sbseqently discretized in the x direction sing a central-difference scheme, reslting in the following

7 Migration with AWWE S67 stencil: where [Λ 1 + Λ + α z D] k 1 = α z d k+1 i,j Λ 1 + Λ i,j+1/ k 1 i,j k+1 i,j+1/ k+1 i,j+1/ k i,j+1/ + α x Λ k i+1,j+1/ k i,j+1/ + k i 1,j+1/, 44 α x = c t x 45 Note that the coefficient matrix on the left-hand side in eqation 44 is a small tridiagonal matrix Based on the implse responses and migration examples shown in the Reslts section, it appears that, for most practical prposes, the size of this matrix is less than five In light of this observation, explicit inversion is performed to obtain k 1 i,j+1/ = h 1 where and k+1 i,j k 1 i,j k+1 i,j+1/ + H k+1 i,j+1/ k i,j+1/ + H 3 k i+1,j+1/ k i,j+1/ + i 1,j+1/ k, h 1 = α z Ld, H = L Λ 1 + Λ, 46 H 3 = α x LΛ, 47 L = [Λ 1 + Λ + α z D] 1 48 Eqation 46 is explicit reverse marching in time, which is combined with marching in the z direction to obtain a doble-marching procedre Note that at any given instant of this doble-marching procedre, the only axiliary variables needed and stored are k 1 j+1/ x, k j+1/ x and k+1 j+1/ x Frthermore, the field variables stored are l j x, l k and l j+1 x,l k, and the image stored is l x,l jat t = Based on extensive nmerical stdies, we observe that the explicit doble-marching procedre is conditionally stable for c t min x, z Rigoros stability analysis may be possible bt is otside the scope of this paper One of the advantages of AWWE is its explicit implementation Generally, a conditionally stable explicit scheme is not necessarily better than an implicit scheme becase of the former s limitation on extrapolation-step size However, the AWWE s stability condition does not introdce any pper limit on the extrapolation-step size; hence, compared to an implicit scheme with the same grid size in depth, it will redce the comptational cost dramatically The reqired storage and comptational time nmber of floating-point operations for AWWE migration are and S = N x [N t + N z + 3N v + 1], 49 F = 1N x N z N t 1 + N v 14 + N v, N v >, 5 respectively, where N x,n z,andn t are the nmber of grid points in x, z, andt directions, and N v is the nmber of axiliary variables Noting that N v is sally less than 5 for practical cases, the storage needed for axiliary variables is negligible relative to that for the field variable Ths, the reqired storage is approximately of the order N x N t + N z Based on crrent implementation, the comptational cost can be written as and S S 15, 51 F = N v 14 + N v F 15, N v >, 5 where S 15 and F 15 are the comptational cost associated with the 15 OWWE The AWWE formlation and implementation can easily be extended to 3D migration To do so, one rewrites eqation 31 as d z t 1 c Λ 1 + Λ t + cλ x + =, y 53 which conseqently adds the term H 4 k i,l+1,j+1/ k i,l,j+1/ + k i,l 1,j+1/ to the right-hand side of eqation 46, where l is the index associated with the y direction, H 4 = α y LΛ,and α y = c t / y The comptational cost for the 3D method is S 3D = N y S D, F 3D = 13N y F D, 54 where the 13 factor represents the extra points in the finitedifference stencil reslting from the y derivative term in eqation 53 RESULTS The accracy of the scalar AWWE migration is first illstrated by examining the implse response of the method for varios orders of approximation Figre a shows the implse response obtained by AWWE Here, the reference phase velocities c j see eqations and 1 are assmed to be eqal to the constant backgrond velocity From Figre a, the response is accrate for a range of approximately 6 The reslt for AWWE 3, shown in Figre b, is accrate p to 7, with all the reference velocities eqal to the backgrond velocity To illstrate the effect of the phase-velocity parameters on accracy, Figre c is generated sing AWWE bt with reference velocities c 1 = c and c = 4c, wherec is the backgrond velocity As seen in the figre, the migrated image matches the semicircle for approximately 8, which is more accrate than the image obtained by AWWE 3 with constant reference velocities Frthermore, Figre c illstrates the sfficiency of

8 S68 Gddati and Heidari a few axiliary variables in this example, one for the AWWE method to be very accrate For frther illstration of its accracy, AWWE migration is sed to image a synthetic model consisting of a small reflector, with a dip angle of, in a laterally heterogeneos medim The backgrond velocity varies linearly from c = 4 m/s on the left side of the domain to c = 15 m/s on the right, and is constant over depth The exploding reflector concept Loewenthal et al, 1976 is sed to generate the zerooffset synthetic section Figre 3a shows the reslting image from migration by 15 OWWE Compared with the exact location of the reflector shown by the bold white line, the image is misplaced, and the length of the reflector is imaged incorrectly Althogh the dip angle of the reflector is within the range of accracy of the 15 OWWE, becase of bending of the rays associated with the laterally varying velocity, the angle of propagation becomes significantly more than 15,reslt- ing in a poor image The AWWE imaging method is applied to the same model, and the migration reslt is shown in Figre 3b Here, AWWE is sed with reference velocities c j = 4 m/s The accracy of AWWE migration is clearly seen from this migrated image, where the reflector is imaged correctly on the target Note that an accrate image is achieved with a comptational effort only 3 times that of a 15 one-way wave eqation Finally, as a practical benchmark test, a slice of the SEG/ EAGE salt model is migrated with the proposed method The velocity model and reflector position of this slice can be seen in Figre 4a The velocity of the model varies between 15 m/s and 45 m/s Again, an exploding reflector model is sed to obtain the zero-offset srface trace A Ricker wavelet with a dominant freqency of Hz and a plse initial time of 85 ms is sed for the exploding reflector model The density of the model is adjsted to achieve constant impedance in the forward model, ths redcing the wave-trapping effects and mltiples, since we are not dealing with the effects of mltiples in this paper A relatively fine grid of x, z, t = 3m 3m 4 ms with a second-order central-difference scheme is sed for the forward exploding reflector model to prevent dispersion However, for migration, the srface trace data are down-sampled to x, t = 15 m 65 ms, and z is kept nchanged The migration of this model sing AWWE inclding the 15 eqation reqires approximately 36 MB of storage According to eqation 54, for a 3D migration of this model assming eqal length and grid size in x and y directions, the reqired storage will be abot 31 GB Figre Implse response of the AWWE with varios parameters a AWWE with c 1 = c = c, b AWWE 3 with c 1,,3 = c, c AWWE with c 1 = c and c = 4c, wherec is the constant backgrond velocity The white semicircle is the implse response of the exact OWWE sperimposed on the images for comparison The reslts in these figres agree with observations from Figre 1 Figre 3 Migration of a small reflector in a laterally varying backgrond velocity, defined as cx, z = 4 36x for x 7 m, with a 15 OWWE AWWE 1 withc 1 = 4 m/s, and b AWWE 1 axiliary variable with c 1 = c = 4 m/s The original reflector is shown with a white line Becase of the continosly varying velocity in this model, the ray-bending effects reslt in higher propagation angles, hence the poor image with the 15 eqation In panel b, AWWE generates the correct image

9 Migration with AWWE S69 We have developed a new migration techniqe based on arbitrarily wide-angle wave eqations AWWE, which are comptationally tractable one-way wave eqations that are able to migrate steep dips The accracy of an AWWE depends on its order as well as its parameters the reference phase velocities We have shown analytically that, by adjsting these parameters, accrate coverage of p to 85 of dip angle can be achieved, even for the second-order AWWE The most attractive featre of the AWWE is that it is similar in form to the 15 eqation and is implemented sing an efficient, explicit doble-marching procedre The performance of AWWE migration is verified with the help of implse response in homogenos media as well as migration in heterogeneos media Finally, AWWE migration is applied to imaging of a D slice of the SEG/EAGE salt model The image improves significantly by sing the third-order AWWE, with a comptational time 489 times the 15 eqation, and almost eqal storage capacity Based on the generality of the derivation procedre shown in this paper, AWWE migration appears to be extendable to elastic media This possibility is crrently nder investigation ACKNOWLEDGMENTS Figre 4 Migration of a slice of the SEG/EAGE salt model a Backgrond velocity model and reflector position the velocity varies between 15 m/s and 45 m/s b Migration sing the 15 OWWE AWWE 1 with c 1 = 4 m/s c Migration sing the AWWE 3 with c j = 5, 45, 65 m/s The image is improved by sing the AWWE 3 for both steep dips and clarity nder the salt region Alternatively, one can design higher-order finite-difference schemes enabling the se of larger step sizes withot introdcing dispersion, which, in trn, wold make the 3D migration feasible on a PC However, optimization of AWWE migration for large 3D problems is otside the scope of this paper Figre 4b and 4c shows the migrated images sing the 15 OWWE AWWE 1 and AWWE 3, respectively As evident from eqation 5, AWWE 3 has a comptational time of 489 times the 15 eqation, and from eqation 51, the storage capacities reqired for the two methods are almost eqal The AWWE 3 migration has improved the image by placing the events closer to the original reflectors, captring the details of steep events and imaging the region nder the salt body more accrately The latter is associated with the ray-bending effect cased by heterogeneities arond that region, which increases the angle of propagation; hence, heterogeneities can be handled better by AWWE CONCLUSIONS This material is based pon work spported by the National Science Fondation nder Grant No 1188 Any opinions, findings, and conclsions or recommendations expressed in this material are those of the athors and do not necessarily reflect the views of the National Science Fondation We thank Pal Stoffa of the University of Texas at Astin for many fritfl discssions as well as providing s with the SEG/EAGE salt model data We also greatly appreciate constrctive comments and sggestions from anonymos reviewers and associate editor of GEOPHYSICS REFERENCES Bamberger, A, B Engqist, L Halpern, and P Joly, 1988, Higherorder paraxial wave-eqation approximations in heterogeneos media: SIAM Jornal on Applied Mathematics, 48, Berkhot, A J, 1979, Steep dip finite-difference migration: Geophysical Prospecting, 7, , 1981, Wave field extrapolation techniqes in seismic migration, a ttorial: Geophysics, 46, Chang, Y-F, and C Chir-Cherng,, Freqency-wavenmber migration of ltrasonic data: Jornal of Nondestrctive Evalation, 19, 1 1 Claerbot, J F, 197, Coarse grid calclations of waves in inhomogeneos media with application to delineation of complicated seismic strctre: Geophysics, 35, , 1985, Imaging the earth s interior: Blackwell Scientific Pblications Inc Claerbot, J F, and S M Doherty, 197, Downward contination of moveot corrected seismograms: Geophysics, 37, Collins, M D, 1989, Applications and time-domain soltion of higher-order parabolic eqations in nderwater acostics: Jornal of the Acostical Society of America, 86, , 1993a, A -way parabolic eqation for elastic media: Jornal of the Acostical Society of America, 93, , 1993b, An energy-conserving parabolic eqation for elastic media: Jornal of the Acostical Society of America, 94, Collins, M D, W A Kperman, and W L Siegmann, 1995, A parabolic eqation for poro-elastic media: Jornal of the Acostical Society of America, 98, Engqist, B, and A Majda, 1979, Radiation bondary conditions for acostic and elastic wave calclations: Commnications on Pre and Applied Mathematics, 3, Gazdag, J, 1978, Wave-eqation migration with phase-shift method: Geophysics, 43, Givoli, D, 199, Nmerical methods for problems in infinite domains: Elsevier Science Pblishing Co Gray, S H, 1, Seismic imaging: Geophysics, 66, Gddati, M N, 5, Arbitrarily wide-angle wave eqations for complex media: Compter Methods in Applied Mechanics and Engineering, pblished online March 3, 5

10 S7 Gddati and Heidari Gddati, M N, and A H Heidari, 3, Application of arbitrarily wide-angle wave eqations to sbsrface imaging and nondestrctive evalation: Proceedings of the 16th Engineering Mechanics Conference, American Society of Civil Engineers, Gddati, M N, and J L Tassolas,, Contined-fraction absorbing bondary conditions for the wave eqation: Jornal of Comptational Acostics, 8, Heidari, A H, and M N Gddati, 3, Migration with arbitrarily wide-angle wave eqations: 73rd Annal International Meeting, SEG, Expanded Abstracts, Jerzak, W, M D Collins, R B Evans, J F Lingevitch, and W L Siegmann,, Parabolic eqation techniqes for seismic waves: Pre and Applied Geophysics, 159, Lee, M W, and S Y Sh, 1985, Optimization of one-way waveeqations: Geophysics, 5, Lindman, E L, 1975, Free Space bondary conditions for the timedependent wave eqation: Jornal of Comptational Physics, 18, Lingevitch, J F, and M D Collins, 1998, Wave propagation in rangedependent poro-acostic wavegides: Jornal of the Acostical Society of America, 14, Lingevitch, J F, M D Collins, A J Fredricks, and W L Seigmann,, Forward and inverse modeling in poros media: Jornal of the Acostical Society of America, 17, 845 Loewenthal, D, L L, R Robertson, and J Sherwood, 1976, The wave eqation applied to migration: Geophysical Prospecting, 4, Ristow, D, and T Rhl, 1994, Forier finite-difference migration: Geophysics, 59, Ritchie, W, 3, Trends in depth imaging: Observations from the SEG Annal Meeting: The Leading Edge,, 6 8 Sn, R, and G A McMechan, 1, Scalar reverse-time depth migration of prestack elastic seismic data: Geophysics, 66, W, R S, 3, Wave propagation, scattering and imaging sing daldomain one-way and one-retrn propagators: Pre and Applied Geophysics, 16, Zhang, G Q, 1985, High-order approximation of one-way waveeqations: Jornal of Comptational Mathematics, 3, 9 97 Zhang, G Q, S L Zhang, Y X Wang, and C Y Li, 1988, A new algorithm for finite-difference migration of steep dips: Geophysics, 53, Zhang, Y, G Zhang, and N Bleistein, 3, Theory of tre-amplitde one-way wave eqations and tre-amplitde common-shot migration: 73rd Annal International Meeting, SEG, Expanded Abstracts, Zhe, J P, and S A Greenhalgh, 1997, Prestack mlticomponent migration: Geophysics, 6, Zho, H B, and G A McMechan, 1997, One-pass 3-D seismic extrapolation with the 45 degree wave eqation: Geophysics, 6,