Higher order numerical schemes for paraxial approximations of thewave equationE. B�ecache � F. Collino � P. Joly �AbstractIn a previous work, Collino and Joly [9] have constructed new 3D paraxial approximationsthat are compatible with the use of splitting methods without loss of accuracy. We presenthere new higher order numerical schemes to solve these equations in an heterogeneous medium.The dispersion e�ects observed by using classical second-order schemes can be considerablyattenuated, even with a very few number of discretization points per wavelength. We also showhow to adapt the method developed by Berenger for Maxwell equations in order to get absorbinglayers on the lateral sides of the domain.1 IntroductionParaxial approximations constitute a good approximation of the wave equations when the wavespropagate around a privileged direction. In many applications, one of the variable appears naturallyas the privileged direction (z in geophysics, [6], r in ocean acoustics, [19]) and becomes an evolutionvariable. In the following, as we are mainly interested in geophysical applications, the evolutionvariable is the depth z. The classical way to discretize these equations consists in using a Crank-Nicolson scheme in the depth direction and second order �nite di�erences for the derivatives in thelateral variables. In the 3D case, each extrapolation step requires the inversion of quite di�cultand expensive linear systems (see Joly and Kern [13] and Kern [14]). This can be avoided byusing the new higher order paraxial approximations constructed by Collino and Joly [9], whichare compatible with splitting methods. The novelty of their approach in comparison with classicalalternate direction methods ([5, 11, 17, 10]) is to introduce other directions for the splitting thanthe usual cross-line and in-line directions which allow to get forty-�ve degree and sixty-degreeapproximations. It reduces the problem to a series of 2D extrapolations in each direction of splitting.They have presented these new equations for constant velocities, but there is no di�culty to extendthem to the case of heterogeneous velocities, following the criteria given by Bamberger and al. [2].The observations of results obtained with the classical second-order numerical schemes have pointedout some dispersion e�ects of these numerical schemes, especially for large angles of propagation(with respect to the depth direction) and we propose here some new higher order numerical schemesthat attenuate these e�ects.�INRIA, Domaine de Voluceau-Rocquencourt, BP 105, F-78153 Le Chesnay C�edex1

The aim of this paper is to describe a systematic way to get accurate discretizations, both indepth and in the lateral variables to solve the 3D paraxial equations with splitting methods, inheterogeneous media.The paper is organized as follows. In the �rst section we make some brief recalls on the classicalparaxial approximations and set the PDEs associated to the new paraxial approximations. We alsoshow how the method developed by Berenger [4] for Maxwell equations can be adapted for theparaxial equations, to achieve a quasi-perfect absorption of the waves on the lateral boundaries.In section 3, we explain how to get higher order discretizations in the depth variable with theprocedure presented by Kern (see [14]) for the full 2D paraxial equations based on a conservativeRunge-Kutta method. Section 4 presents the discretization in the lateral variable with variational�nite di�erences techniques. With these techniques, the variable coe�cients are easily taken intoaccount. The classical way to improve the accuracy is to use higher order discretizations of thederivatives in the lateral variables. However, this increases quickly the band-width of the linearsystem to be solved. We present a family of \modi�ed" schemes of order 2n that cost the sameprice than the classical (2n� 2) order scheme. Numerical experiments show how the dispersion isattenuated with higher order schemes, especially with the modi�ed schemes.2 The continuous problem2.1 The classical paraxial equationsParaxial equations are approximations of the one way up-going wave equationdv̂dz + i!c 1� c2 jkj2!2 !12 v̂ = 0; (1)where v̂ denotes the Fourier transform of v with respect to t (time) and x1; x2. They are obtainedby replacing the square root in (1) with rational fractions so as to transform this non local integro-di�erential equation into a local Partial Di�erential Equation, this approximation being valid as longas cjkj=! remains small enough, i.e. as long as the wave propagates close enough to the z-direction.A general class of well-posed high order approximations has been proposed by Bamberger and al[1] and leads to the following system of (L+1) coupled PDEs , written in the frequency domain(since paraxial equations are usually handled in the frequency domain)8>>>><>>>>: @v@z + i!c v � i!c LX̀=1 b`'` = 0!2c2 '` + a`�'` = ��v ` = 1; : : : ; L:Here L speci�es the degree of the approximation and '` are auxiliary unknown functions introducedso as not to deal with a high-order PDE.Classically, we handle the transport term exactly, with the Claerbout change of unknown func-tions u = vei!z=c, and after elimination of the auxiliary unknown functions, end up with the2

following system (written in operator form)8>>>><>>>>: @u@z � i!c LX̀=1A`u = 0A` = �b`(a`�+ !2c2 )�1�: (2)Since the evolution operator is written as a sum of simpler operators, it is a natural idea touse a splitting method to solve (2). Section 2.4 describes the algorithm in more details. Let usjust mention here that, after discretization in depth, we have to solve, at each depth step, a linearsystem, with a large, sparse, complex valued, non-hermitian matrix. Kern [16] has proposed to usemodern iterative methods to solve it. We investigate here an alternative way to avoid the di�cultyaltogether, by introducing a new class of paraxial equations suitable for use with splitting in thelateral variables, and requiring only the solution of 1D PDEs.2.2 New paraxial approximationsUsing splitting in the horizontal variables is not a new idea. In order to avoid the inversion ofthe large matrix when using the full paraxial approximation, several authors [5, 11, 17, 10] haveadvocated approximating the square root with(1� j�j2) 12 � 1� 12 �211� 14�21 � 12 �221� 14�22 ( error : O(�21�22));which is consistent with the forty-�ve degree equation (i.e., with an error � O(j�j6)) only in the�1 = 0 and �2 = 0 directions.The basic idea to get better accurate approximations involving only one variable per fraction isto introduce more than two directions of splitting. The paraxial equations constructed in Collinoand Joly [9] are derived from an approximation of the square root with rational fractions of thefollowing form (1� j�j2)1=2 = (1� (�21 + �22))1=2 � 1�R(�1; �2);with R(�) = NDXj=1 LX̀=1 bj̀(�:nj)21� aj̀(�:nj)2 ; (3)where ND corresponds to the number of directions, L the number of fractions per direction, andnj the unit vector associated to the jth direction (nj = (cos�j ; sin�j)). It has been shown in [9]that the conditions on the coe�cients aj̀ > 0 ; bj̀ � 0 ensures the well-posedness of these paraxialequations. Also, in order to keep the approximation error in the square root from blowing-up incertain directions inside the unit disk �21 + �22 � 1, it is natural to impose 0 < aj̀ � 1.Collino and Joly [9] have constructed several families of approximations of the above type so asto achieve comparable accuracy to the classical forty-�ve (error � O(j�j6)) or sixty degree (error �O(j�j8)) approximations. In particular, they obtained a family of forty-�ve degree approximationsdepending on one parameter using four directions of splitting (ND = 4) uniformly distributed -namely x1; x2; x1 + x2; x1 � x2- and one fraction per direction (L = 1). For instance, the choice3

aj = 1=3; bj = 1=4; i = 1; : : : ; 4 gives them the \maxi-isotropic" forty-�ve degree approximation.More examples are given in the above paper.The paraxial equation corresponding to (3) can now be written in a form analogous to (2):8>>>><>>>>: @u@z � i!c NDXj=1 LX̀=1Aj̀u = 0Aj̀u = �bj̀(!2c2 + aj̀D2j )�1D2ju; (4)where Dj = nj � ~r is the derivative in the jth direction. Each of the operators Aj̀ in (4) only involvesa one-dimensional di�erential operator. Thus this new family equation lends itself to a splittingmethod in the horizontal variables, but as mentioned before, the splitting has the advantage ofbeing consistent with the forty-�ve degree equation (for a proper choice of coe�cients).In practice, a mesh being given in the (x1; x2) plane, there will be two main choices for ND:ND = 4 if the mesh is built from squares, and then the directions are given by the two coordinateaxes and the two main diagonals, or ND = 3 if the mesh is built from equilateral triangles, and the3 directions are 60� apart. Using more than 4 directions permits to get higher accuracy but wouldimply additional di�culties for the discretization which are beyond the scope of this paper.In the following, for the sake of simplicity, we only consider the forty-�ve degree approximationfamily with ND = 4 directions and L = 1 fraction per direction and we denote the correspondingcoe�cients and operator by aj; bj and Aj .2.3 Extension to heterogeneous media and design of absorbing boundary con-ditionsThe extension to heterogeneous media as well as the treatment of the absorbing boundaries aresimply obtained by using similar modi�cations on the operators.Paraxial equations in heterogeneous media have been proposed and analyzed by Bamberger andal. [2]. Their approach was to de�ne several criteria (both mathematical and geophysical), and toselect among a general class of possible candidates the one that satis�ed those criteria.Their result gives a recipe which allows one to extend any paraxial equation to heterogeneousmedia. Thus equations (4) keep the same form, with a new de�nition for the operators Aj8>>>><>>>>: @u@z � i!c NDXj=1Aju = 0Aju = �bj(!2c2 + aj�cj)�1�cj ; (5)and we have de�ned �cj = 1cDj(cDj):A problem of practical importance is the treatment of the lateral boundaries. It must bedesigned in such a way that the waves are absorbed when they reach the boundaries. Recently,Collino [8] proposed to adapt a novel technique, introduced for electromagnetism by B�erenger[4]. This technique consists in designing an absorbing layer model called perfectly matched layer(PML). It possesses the astonishing property to generate no re exion at the interface between the4

free media and the arti�cial lossy medium, and the re ected waves are only due to the discretizationof the model. This property allows one to use a very high damping parameter inside the layer, andconsequently a short layer length, while still achieving a quasi-perfect absorption of the waves.Practically, the PML is very easy to implement : we replace the velocity c in the operators Ajand �cj de�ned previously by cd(x) with d(x) = i!i! + c�(x) and �(x) is a positive function withsupport in the damped area. We close the system with a Dirichlet boundary condition at the endof the layer. The e�ciency of the method has been assessed by numerical experiments : a quasi-perfect absorption has been obtained with only 4 extra-nodes at the boundary and an appropriatechoice for � (see �gure 2).2.4 SplittingWe brie y recall some classical points about splitting methods (see [18]) which are speci�callydesigned for the solution of ODEs in the form (5). The exact solution of (5) satis�esu(z +�z) = exp(Z z+�zz i!c NDXj=1Ajdz) u(z) :The splitting methods consist in approximating the exponential of the sum of operators with theproduct of exponentialsuap(z +�z) � exp Z z+�zz i!c ANDdz!� � � � � exp Z z+�zz i!c A1dz! u(z); (6)which is second order accurate with respect to the discretization in the z variable, in the generalcase of an heterogeneous medium (in such a case the operators Aj do not commute). Approximation(6) then leads naturally to de�ne ND intermediate unknowns wj ; j = 1; : : : ; ND satisfying:8><>: dwjdz � i!c Ajwj = 0wj(0) = wj�1(�z) � wj�1 : (7)Finally we have uap(z +�z) = wND :Problem (7), to be solved at each step, is still an evolution problem in depth, but with a singleoperator. In the next section, the discretization in the evolution variable is performed in order toget a high order semi-discretized scheme.3 Semi discretization in depthWe assume in the following the velocity to be independent on z between z and z + �z (forinstance, c(x1; x2; z) = cn(x1; x2) for z 2 [zn; zn+1]). The discretization in the depth variable usedhere has been initially proposed by Joly and Kern [13]. It is based on the fact that the exactsolution of (7) satis�es wj = ei!c Aj�zwj�1, and on the relationship between Runge-Kutta methodsand Pad�e approximations to the exponential (see for instance [12]). In order to get conservative5

schemes of order 2K, the exponential is replaced with a Pad�e approximant on the following formexp(ix) � KYk=1 1 + rkx1 + �rkx;where rk are appropriate complex numbers and �rk is the complex conjugate of rk. The integrationfrom 0 to �z is then done formally as followswj = KYk=1(I + �rk�zAj)�1(I + rk�zAj)wj�1:This procedure leads us to de�ne K intermediate unknowns wj�1k associated to each fraction,solutions of (I + �rk�zAj)wj�1k+1 = (I + rk�zAj)wj�1k :We then set wj = wj�1K .The classical Crank-Nicolson second-order scheme is obtained for K = 1 and r1 = i=2. Kern[15] has shown that this process can also be used to get non conservative schemes, which generalizethe �-scheme to higher order schemes.Finally, one elementary step requires the solution of systems of the form�I + dj(!)�cj�U = F :There remains the task of discretizing the 1D operator with respect to the lateral variable. Wedevote next section to specifying how this approximation is achieved.4 Discretization in the lateral variablesIn view of section 2.4, we now take for simplicity one particular direction, which we denoteby x. The domain is an interval in R and for clarity we make the presentation with Dirichletconditions (but there is no di�culty to do the same with the PMLs). Thus we have to solve8>>>>>>><>>>>>>>:@w@z � i!c ' = 0 in !2c '+ @@x �c @@x (a'+ bw)� = 0 in w � ' � 0 on @: (8)We base the discretization of (8) on a variational formulation, which we recall below. This pro-vides us with a systematic treatment of heterogeneous media, in a way that insures good numericalproperties (i.e., convergence and stability). We set(u; v) = Z u�v dx ; m(u; v) = Z 1cu�v dx ; k(u; v) = Z c@u@x @�v@x dx6

With these de�nitions, problem (8) admits the variational formulationFind (w;') such that8><>: ddz (w;�) � i!m('; �) = 0 8�!2m('; ) � k(a'+ bw; ) = 0 8 :The discretization is built upon a variational �nite di�erences method. It consists in approx-imating the operators with operators of �nite di�erence type. Let us introduce some notations.The domain is discretized by a regular grid (xi = i�x) and we de�ne the shifted grid by the nodes(xi+1=2 = (i + 1=2)�x = (i + 1=2)h). The approximation is then decomposed on the functions�i where �i is the characteristic function on [xi�1=2; xi+1=2]. Since the functions �i are constantsper element, one can de�ne the (diagonal) mass matrix as M?ij = m(�i; �j), but one has to give ameaning to the derivative terms in the sti�ness matrix. This can be done by approximating thederivative with a �nite di�erences operator. Let D" be the usual second-order �nite di�erencesapproximation of d=dx, i.e., D"�(x) � �(x + ") � �(x � "), we de�ne an approximation of d=dx,denoted by @[2n]h , as a linear combination of the form @[2n]h = 1h nXp=1 �pD(2p�1)h=2 which is of order2n provided that the coe�cients satisfy the Van der Monde like systemnXp=1 �p(2p� 1)2k�1 = �k1 for 1 � k � n:This leads to the sti�ness matrixK [2n]ij � (c@[2n]h �i; @[2n]h �j) � (k[2n]�i; �j) (here k[2n] is the notationfor the operator (@[2n]h )?(c@[2n]h )). The 2n-order classical scheme uses these two matrices and thusleads to the solution of the linear system8>><>>: dwhdz � i!M?'h = 0�!2M? � aK [2n]�'h = bK [2n]wh;We propose here another family of schemes, the modi�ed schemes, that can be seen as anextension of Claerbout's scheme [6] to higher orders. It consists in using a linear combinationM [2n]� = �M?+(1��)U [2n], of two di�erent mass matrices M? and U [2n], where U [2n] comes froma 2n-order �nite di�erences approximation of the operator identity (see below). The linear systemassociated to the modi�ed schemes can then be written as8>><>>: dwhdz � i!M?'h = 0�!2M [2n]� � aK [2n]�'h = bK [2n]wh; (9)It is important to keep the diagonal mass matrix M? in the �rst equation as we will have to invertit when we eliminate 'h. On the other hand, one will show that a proper choice of � allows us togain two orders of accuracy when compared to the classical discretization. The classical schemescorrespond to the particular choice � = 1 in (9).7

Let us explain in more details how we derive U [2n]. We use the same kind of procedure thanfor the sti�ness matrix, and set I" the �nite di�erence operator approximating the identity, i.e.,I"�(x) = 12(�(x + ") + �(x � ")). We de�ne an approximation of I, denoted by �[2n]h , as a linearcombination of the form �[2n]h = 1h nXp=1�pI(2p�1)h=2 which is of order 2n provided that the coe�cientssatisfy nXp=1�p(2p� 1)2(k�1) = �k1 for 1 � k � n:There is a very simple relation between the coe�cients �p for the approximations of @[2n]h and �pfor the approximations of �[2n]h �p = (2p� 1)�p 1 � p � n: (10)This gives the approximate mass matrix U [2n]ij = (1c �[2n]h �i; �[2n]h �j), and the resulting combinationM [2n]� = �M? + (1 � �)U [2n] is still a 2n-order approximation (again we denote by m[2n]� theassociated operator equal to �c I + (1 � �)(�[2n]h )?(1c �[2n]h )). To determine the order of the resultingscheme (9), we consider a smooth solution (w;') of (8), and perform a Taylor expansion, to get�!2m[2n]� � ak[2n]�'� bk[2n]w = !2c '+ c @2@x2 (a'+ bw)+h2n @2n@x2n "!2c (1� �)C [2n]1 '+ cC [2n]2 @2@x2 (a'+ bw)#+O(h2n+2);where C [2n]1 and C [2n]2 are constants computed thanks to Taylor expansions with respect to h (weomit the details). The �rst term in the RHS is exactly the equation satis�ed by (w;') and thusvanishes. The order of the scheme is therefore at least equal to 2n. Moreover, one again recognizesthe same equation between the brackets for an appropriate choice of � which is � = �[2n] = 1�C [2n]2C [2n]1and hence we obtain a 2n+2-order scheme with that particular choice for �. Finally, relation (10)provides an explicit expression for this value, �[2n] = 2n2n+ 1 : From a practical point of view,relation (10) is very useful, since it is su�cient to compute a 2n-order approximation of d=dx toget at the same time the 2n-order approximation of the identity.With n = 1, the modi�ed scheme corresponds to the classical Claerbout's scheme [6], with therelation = (1��)=4. The corresponding value �[2] = 2=3 leads then to the well known = 1=12choice and gives a fourth-order scheme with a matrix of bandwidth equal to 3 instead of 7 for theclassical fourth-order scheme. For n = 2 and � = 4=5, we get a sixth-order one with a bandwidthequal to 7 instead of 11 for the classical sixth-order scheme. The drawback is that the analysisthrough a priory energy estimates, that can be made for the classical schemes to show their well-posedness, is not valid anymore for these modi�ed schemes in heterogeneous media (although itstill applies to homogeneous media)(see Collino [7]).Before closing this section, let us write the total discretization. System (9) can be rewritten8

after elimination of the auxiliary function asdwhdz = i!M?(!2M [2n]� � aK [2n])�1bK [2n]wh: (11)The discretization in depth is introduced as in section 3, and we �nally end up with a set of linearsystems of the form �Skwk+1 = Skwk ;with Sk = (!2a M [2n]� �K [2n])(M?)�1 + rk!�zba K [2n] and �Sk its complex conjugate.5 Numerical resultsWe will illustrate the method with two examples. For both examples, we consider the migrationof a point source, the source being located on the surface. The computational domain is 1250 min each of the horizontal directions, and 625 m in the vertical direction. The grid sizes are h =�z = 12:5 m. The �rst simulation is done in a 2D heterogeneous medium (see �gure 1), theBELOW 1000

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(b) 5 PML ; 4th-order scheme in zFigure 4: Modi�ed 6th-order schemes in xThe second simulation is done in a 3D homogeneous medium with velocity equal to 1000m/s.We use the forty �ve degree paraxial equation with 4 directions and 1 fraction per direction,characterized by the coe�cients a1 = a4 = 0:27; b1 = b4 = 0:3 in the directions x1 and x2 anda2 = a3 = 0:41; b2 = b3 = 0:2 in the diagonal directions. The central frequency of the source isaround 20 Hz, and the cuto� frequency around 50 Hz. Both tests are relatively severe since thenumber of points per wavelength at the cuto� frequency is around 1.5.As announced in section 2.3, �gures 2 show that the re exions on the lateral boundary due toDirichlet boundary conditions have almost completely disappeared using PMLs of size equal to 5h(notice that this experiment corresponds to the heterogeneous case).As explained in section 4, the main drawback of modi�ed schemes compared to the classicalones is to achieve the same accuracy for a lower price (lower bandwidth linear systems). Moreover,the numerical experiments show their superiority from a dispersion point of view as can be seen on�gures 5,6 and this is con�rmed by a dispersion analysis (see B�ecache et al [3], for further numericalexperiments and for the dispersion analysis).The simulations in the heterogeneous 2D medium, and for �x = �z, show a still good improve-ment when we use a modi�ed 4th-order discretization in x instead of a 2nd-order one (compareFigures 2(b) and 3(a)) and even better when we also use a 4th-order discretization in z (see Figure3(b)). Nevertheless, the improvement is not clear when we increase the discretization order in xuntil the order 6 (see Figure 4). 10

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In the second example (3D homogeneous), again the gain is really important by using a 4th-orderdiscretization in the lateral variables instead of a 2nd-order one. But this time the improvementusing a 4th-order discretization in z is not obvious (compare �gures 5(b) and 7(a) for the 4th-order modi�ed scheme and �gures 6(b) and 7(a) for the 6th-order modi�ed scheme) although thedispersion is much more attenuated with the modi�ed 6th-order schemes (see �gure 6(b) for the 2nd-order discretization in z). Also one should notice in the homogeneous case the quite good isotropyobtained with these new paraxial equations despite of the introduction of particular directions usedfor the splitting.These two examples lead to di�erent conclusions concerning the optimal choice for the scheme.In order to better understand the behavior of the di�erent schemes, we analyze in [3] their dispersionrelations. This helps to choose the scheme with respect to the frequency, the number of points perwavelength... Of course, since we want the result in the time domain, we have to handle witha quite large number of frequencies and the optimal choice is not the same for all of them. Apossible way for further improvements could be to adapt the choice of the numerical scheme to thefrequency.6 CONCLUSIONWe have presented a way to solve e�ciently and accurately new 3D paraxial equations thatlend themselves to splitting with respect to the lateral variables in a consistent manner. Thisgives rise to the solution of a sequence of 1D linear systems for each splitting direction. This isdone in a rather general context that handles heterogeneous media as well as the treatment of thelateral boundaries. The numerical dispersion occuring with the classical second-order schemes canbe considerably attenuated with the use of the modi�ed higher order schemes, even with coarsediscretization grids.One should also compare the e�ciency of these equations with the full paraxial equation. Thisis work in progress and will be the subject of a forthcoming paper.7 ACKNOWLEDGMENTSThis research was carried out as part of the Prestack Structural Interpretation consortiumproject (PSI). The authors hereby acknowledge the support provided by the sponsors of this project.

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References[1] A. Bamberger, B. Engquist, L. Halpern, and P. Joly, Higher order paraxial approxi-mations for the wave equation, SIAM J. on Appl. Math., 48 (1988), pp. 129{154.[2] , Paraxial approximations in heterogeneous media, SIAM J: Appl. Math., (1988), pp. 98{128.[3] E. B�ecache, F. Collino, and P. Joly, Higher order numerical schemes for the paraxialwave equation. Yearly report, PSI Consortium, 1995.[4] J. P. B�erenger, A Perfectly Matched Layer for the Absorption of Electromagnetic Waves,J. of Comp. Phys., 114 (1994), pp. 185{200.[5] D. L. Brown, Applications of operator separation in re ection seismology, Geophysics, 3(1983), pp. 288{294.[6] J. F. Claerbout, Imaging the earth's interior, Blackwell Scienti�c Publication co., 1983.[7] F. Collino, Numerical analysis of mathematical models for wave propagation. Report, PSIConsortium, IFP, Rueil-Malmaison, France, 1993.[8] , Perfectly matched absorbing layers for paraxial equations. Yearly report, PSI Consor-tium, 1995.[9] F. Collino and P. Joly, Splitting of operators, alternate directions and paraxial approxima-tions for the 3-D wave equation, SIAM J. on Scienti�c and Stat. Comp., (September, 1995).[10] P. Froideveaux, First result of a 3-d prestack migration program. Annual Report, PSIConsortium, IFP, Rueil-Malmaison, France, 1990.[11] R. W. Graves and R. W. Clayton, Modeling acoustic waves with paraxial extrapolators,Geophysics, 55 (1990), pp. 306{319.[12] E. Hairer and G. Wanner, Solving ordinary di�erential equations : 2 : sti� and di�erential-algebraic problems, Springer Verlag, 1991.[13] P. Joly and M. Kern, Numerical methods for 3-d migration. Annual Report, PSI Consor-tium, IFP, Rueil-Malmaison, France, 1990.[14] M. Kern, Numerical methods for the 3-d paraxial wave equation in heterogeneous media,.Annual Report, PSI Consortium, IFP, Rueil-Malmaison, France, 1991.[15] , Numerical methods for the 3-d paraxial wave equation in heterogeneous media - part iii,.Annual Report, PSI Consortium, IFP, Rueil-Malmaison, France, 1992.[16] , A nonsplit 3d migration algorithm, in Mathematical Methods in Geophysical ImagingIII, S. Hassanzadeh, ed., SPIE, 1995. 13

[17] Z. Li, Compensating �nite-di�ences errors in 3-d migration and modelling, Geophysics, 56(1991), pp. 1650{1660.[18] G. I. Marchuk, Splitting and alternating direction methods, in Handbook of Numerical Anal-ysis, P. G. Ciarlet and J. L. Lions, eds., vol. I (Finite Di�erence Methods), Elsevier SciencePublishers B.V. (North{Holland), Amsterdam, 1994.[19] F. Tappert, The parabolic approximation method, in Wave propagation and underwateracoustics, Lecture Notes in Physics, Vol. 70, J.B. Keller and J. Papadakis, eds., 1977.

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