Higher order variational �nite di�erenceschemes for solving 3D paraxial waveequations using splitting techniquesEliane B�ecache a Francis Collino a Patrick Joly aaINRIA, Domaine de Voluceau-Rocquencourt, BP 105, F-78153 Le Chesnay C�edexAbstractWe construct numerical schemes for solving 3D paraxial equations, using splittingtechniques. The solution can be reduced to a series of 2D paraxial equations ineach direction of splitting. The discretization along the depth is done using higherorder conservative schemes, and the one along the lateral variable is done usinghigher order �nite di�erence variational schemes. Numerical experiments show thesuperiority of higher order schemes, much less dispersive, even for a small numberof discretization points per wavelength.Key words: seismic migration, paraxial equations, splitting, higher-order �nitedi�erences, �nite elements, numerical dispersion.1 IntroductionParaxial approximations of the wave equation are commonly used in manyapplications when the waves propagate in directions close to a privileged di-rection. We focus here on the frequency-domain solution of paraxial approxi-mations (see e.g. [11] for time domain methods). For waves propagating neara particular direction, the Helmholtz equation can be seen as the factorizationof two non-local one-way wave equations, involving a square root operator,for which this direction plays the role of the evolution variable. The squareroot can be approximated in several ways leading to local parabolic partialdi�erential equations. The accuracy depends on the order of magnitude of theerror between the exact square root and its approximation: the better theapproximation the wider the permitted propagation angle. That is how sev-eral families of approximation, with the denomination of 15, 45 or 60 degreeapproximations, have been designed, e.g. [10]. One of the main applicationsof paraxial approximations is the solution of range-dependent ocean acousticPreprint submitted to Elsevier Preprint 16 August 1999

propagation problems, and the range r becomes the evolution variable. Tap-pert's parabolic equation [20] was the �rst paraxial approximation introducedto the underwater acoustics community and numerous contributions have beenmade since then, see [17]. The other application, the one we have in mind inthis paper, is migration in geophysics, the evolution variable being the depthz. In this area Claerbout [7] was the �rst to introduce 15� and 45� equationsfor the extrapolation of 2D seismic data.Concerning the discretization of these equations, two kinds of variables appearnaturally: the depth variable z and the lateral variables x1 for 2D problems,(x1; x2) for 3D problems. There are essentially two di�erent approaches forsolving these equations. The �rst one is based on the use of discrete extrap-olation operators, e.g. [14]. The second approach consists in approximatingthe equation with Finite Di�erences or Finite Elements. These equations arecommonly used for solving 2D problems, e.g. [6]. Their extension to 3D prob-lems requires the solution at each extrapolation step of a 2D problem in thetransverse plane, that gives rise to a large linear system, which can be solvedusing modern iterative methods [15, 16].To avoid solving a transverse 2D problem, two of the authors have constructednew families of paraxial approximations that are compatible with splittingmethods, [10] . The novelty of their approach compared to classical alternatedirection methods, [5, 13], is to introduce other directions for the splittingthan the usual cross-line and in-line directions. This allows one to get 45� and60� accurate approximations and the undesirable anisotropic e�ects, observedwith Brown's approximation [5], disappear. The problem is then reduced to aseries of 2D extrapolations in each direction. Recently and independently ofthis work, D. Ristow and T. R�uhl [19] used the same idea of operator splittingin alternate directions.The aim of this paper is to describe a systematic way of getting accurate dis-cretizations, both in the depth and in the lateral variables, for solving the 3Dparaxial equations, introduced in [10], with splitting methods. It is well knownthat lower order numerical schemes can give rise to numerical dispersion, see[8] for the classical discretization of 2D paraxial equations. The dispersion be-comes even more important in 3D problems, that is why we propose some newhigher-order numerical schemes that attenuate these e�ects. We construct ourschemes in general heterogeneous media, study their accuracy via a dispersionanalysis and compare them with numerical experiments.The paper is organized as follows. Section 2 is devoted to the continuous equa-tions: in x2.1, we recall some properties of the classical paraxial approximationsand in x2.2 we brie y describe the paraxial approximations, introduced in [10],which are compatible with 4 directions of splitting. Section 3 is the main partof the paper and concerns the discretization of a model 2D equation. Higher-2

order discretizations in the depth variable are presented in x3.2. Section 3.3presents the discretization in the lateral variable with variational �nite dif-ferences techniques. In x4, we compare the dispersion of several particularschemes. Section 5 presents numerical experiments.2 Multiway splitting for 3D paraxial equations2.1 The classical paraxial equationsThe solution of the wave equation in the whole homogeneous space1c2 @2v@t2 � @2v@x21 � @2v@x22 � @2v@z2 = 0; (1)with appropriate boundary and initial conditions, can be split into two waves,an up-going wave and a down-going wave. The up-going wave, that we areinterested in, satis�es the one-way wave equationdvdz + i!c 1� c2 jkj2!2 ! 12 v = 0; z � 0; (jkj2 = k21 + k22); (2)where v is the Fourier transform of v with respect to both t (time) and (x1; x2)v(k1; k2; z; !) = Z Z Z v(x1; x2; z; t)ei(k1x1+k2x2�!t)dx1dx2dt: (3)Paraxial equations are approximations of (2) obtained by replacing the squareroot, (1� j�j2)1=2 (where � = (�1; �2) and �1 = ck1=!; �2 = ck2=!), by ra-tional fractions, that we denote (1� j�j2)1=2ap , so that the non local integro-di�erential equation (2) is changed into a system of PDE's. This approx-imation is designed in order to be valid as long as cjkj=! remains smallenough, i.e. for propagation directions close to the z-direction. Let e(�) theerror e(�) = (1� j�j2)1=2 � (1� j�j2)1=2ap : The 15� paraxial equation, based onthe Taylor expansion p1�X = 1�X=2 +O(X2), corresponds to an errore(�) = O(j�j4) and the 45� one, based on the �rst Pad�e expansion p1�X =(4 � 3X)=(4 � X) + O(X3), to an error e(�) = O(j�j6). Figure 1 shows theaccuracy of these two classical approximations. It represents the variations ofthe error e(�1; �2), with respect to �1 and �2 and can be understood as follows:the larger the white region (e(�) � 10�3) the better the approximation. It is3

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(b) 45� approximationFig. 1. Errors for the classical approximations, e(�1; �2) with respect to �1 and �2.now easy to rewrite the equation in the physical space corresponding to the15� approximation (still denoting v the frequency domain solution)@v@z + i!c v + ic2!�v = 0; (4)where � = @21 + @22 is the 2D Laplacian operator (@j � @=@xj). One handlesthe transport term i!v=c exactly, with the Claerbout change of unknownu = vei!z=c, and get @u@z + ic2!�u = 0: (5)The operator � is the sum of two di�erential operators @21 and @22 : this is aparticularly interesting property on the numerical point of view, since it allowsus to solve (5) with a splitting technique, as explained in the next paragraph.Splitting methods. The splitting methods [18], are speci�cally designed forthe solution of ODE's involving a sum of Ns operators:8>>><>>>: @u@z (x; z)� i NsXj=1Aj(z)u(x; z) = 0; z � 0; x = (x1; x2) 2 IR2 ;u(z = 0) = u0 in IR2: (6)The exact solution of (6) satis�es u(z0 +�z) = eR z0+�zz0 iPNsj=1 Aj(z)dz u(z0), forall z0 � 0. With splitting methods, we use the approximationuap(z0 +�z) � eR z0+�zz0 iANs(z)dz � � � � � eR z0+�zz0 iA1(z)dzu(z0); (7)4

which is exact when the operators Aj commute and second-order accuratewith respect to �z when they do not. Knowing u(z0), the computation ofuap(z0 + �z) from (7) leads naturally to set w0 = u(z0) and to de�ne Nsintermediate unknowns wj; j = 1; : : : ; Ns satisfying8>>>>><>>>>>: @wj@s (x; s)� iAj(z0 + s)wj(x; s) = 0; 0 � s � �z; x 2 IR2;wj(s = 0) = wj�1 in IR2;wj = wj(�z) in IR2; (8)wj being nothing but the result of the product of the j �rst exponentialsapplied to u(z0). In particular we have uap(z0 + �z) = wNs. Problem (8) isstill an evolution problem in z, but with a single operator.It is straightforward to apply this technique to the 15� equation (5) withAj = �c=2!@j, Ns = 2 and this leads to solve 1D problems, alternatively inthe x1 and in the x2 direction. The 45� approximation can be written as@v@z + i!c v � i!2c j�j21� j�j2 =4 v = 0() @u@z � i!c jkj2 =2!2=c2 � jkj2 =4 u = 0; (9)which corresponds to the following ODE@u@z � iAu = 0; with A = � !2c(!2c2 + 14�)�1�; (10)where the meaning of (!2=c2+1=4�)�1 can be precized thanks to the limitingabsorption principle [12]. There is no more obvious decomposition of operatorA into the sum of simple 1D operators, which would remain consistent withthe 45� approximation order. The classical way to approximate (10) posed ina bounded domain, subject for instance to Dirichlet boundary conditions, isto introduce a discrete version of the Laplacian �h on a uniform grid N �N@u@z � iAhu = 0; with Ah = � !2c(!2c2 + 14�h)�1�h: (11)Using for instance an implicit and second-order accurate Crank-Nicolson schemefor the z-discretization, leads to solve:(I + d(!)�h) uj+1 = �I + d(!)�h�uj; where d(!) = c24!2 + ic�z4! : (12)This a linear system, with a large, sparse, complex valued, non-hermitianmatrix, that can be solved using modern iterative methods [16]. To avoid5

the inversion of the large matrix, when using the full paraxial approximation,Brown [5] has suggested approximating the square root with(1� j�j2) 12 = 1� 12 �211� 14�21 � 12 �221� 14�22 +O(�21�22); (13)and this has been used by several authors, e.g. [13]. But this is consistent withthe 45� equation (e(�) = O(j�j6)) only in the �1 = 0 and �2 = 0 directions.In the other directions, the approximation is of the same order as the 15�approximation, as one can see in Fig 2-(a). One will note the anisotropy of theresult and the loss of accuracy due to the loss of one order in the approximationfor directions which are not parallel to the axes.2.2 The four way splitting equations� Approximations in a homogeneous unbounded medium.-1.0 -0.5 0.0 0.5 1.0

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(b) Maxi-isotropic (b1 = 0:25)Fig. 2. Error e(�). (a): 2 directions (b): 4 directions.The basic idea developed in [10], to get more accurate approximations involv-ing only one lateral space variable per fraction, is to introduce more than twodirections of splitting. The simplest family of paraxial equations is derivedfrom an approximation of the square root with rational fractions, based on 4directions of splitting, x1; x1 + x2; x1 � x2; x2(1� j�j2)1=2 � 1� R(�); with R(�) = 4Xj=1 bj(�:nj)21� aj(�:nj)2 ; (14)where nj is the unit vector associated to the jth direction, nj = (cos�j; sin�j),with �j = (j � 1)�=2 for j = 1; 4. The conditions aj > 0, bj � 0 must besatis�ed to ensure the well-posedness. Moreover, to prevent the approximationerror in the square root from blowing-up in certain directions inside the unit6

disk �21 + �22 � 1, one has to impose 0 < aj � 1. Looking for an approximationconsistent with the 45� approximation, i.e. such that e(�) = O(j�j6), gives riseto a system satis�ed by the coe�cients aj and bj. This leads to a family of 45�approximations depending on one parameter b1 that has to be chosen in theinterval [1=12; 5=12] in order to ensure the conditions 0 < aj � 1 and bj � 0:b4 = b1 ; b2 = b3 = 1� 2b12 ; a1 = a4 = 112b1 ; a2 = a3 = 112b2 : (15)In [10], several criteria, based on the error e(�), are proposed for the choiceof b1. We present in Fig. 2-(b) the error for the so-called maxi-isotropic 45�approximation, corresponding to b1 = 1=4. The quality of the approximationis comparable to the classical 45� approximation not only in the �1 and �2directions, as Brown's approximation, but also in the other directions. Theparaxial equation corresponding to (14) can be written as@u@z � i!c 4Xj=1Aju = 0; with Aju = �bj(!2c2 + ajD2j )�1D2ju; (16)Dj = nj � ~r corresponding to the derivative in the jth direction. Each Ajinvolves a 1D di�erential operator. Thus, this family of equations lends itselfto a splitting method in the horizontal variables, see x2.1: it leads to determineNs = 4 intermediate unknowns wj, related to each operator Aj.Remark 2.1 In [10], it was shown how to design other families of paraxialapproximations of a given order of accuracy, introducing either more than 4 di-rections for the splitting, or more than one fraction per direction. However, forthe numerical exploitation of these equations, the most appropriate familiesare those using only 4 directions, when regular grids are used for the com-putations. Actually, this is not a strong restriction, because 4 directions aresu�cient to get accurate approximations (e.g., 60� approximations), providedthat one increases the number of fractions per direction.� Extension to heterogeneous media. Paraxial equations in heterogeneousmedia have been proposed and analyzed in [2]. Their approach was to de�neseveral criteria, both of mathematical and physical nature, and to select amonga general class of possible candidates the one that satis�ed those criteria.Their result gives a recipe which allows one to extend any paraxial equationto heterogeneous media. Thus equations (16) keep the same form, with a newde�nition for the operators Aj (c now depends on (x1; x2; z)):Aju = �bj(!2 + aj�cj)�1�cju with �cj = cDj(cDj): (17)� Design of absorbing boundary conditions: the PML approach. For7

the numerical computation, the problem has to be set in a cylindrical do-main D � fz � 0g. When D is unbounded, it is necessary to introduce arti�-cial lateral boundaries, designed such that the waves are absorbed when theyreach the boundaries. Recently, Collino [9] has adapted a new technique, in-troduced for Maxwell's equations [4], which consists in designing an absorbinglayer model called perfectly matched layer (PML). It possesses the astonishingproperty to generate no re exion at the interface between the free media andthe arti�cial lossy medium (this can be shown through a plane wave analysis).Furthermore, the wave is exponentially damped in the absorbing layer. Withthe 2D paraxial equations, the PML is easy to implement: we change the oper-ator �cj (see (17)) to e�cj = cd(x)Dj (cd(x)Dj) with d(x) = i!i!+c�(x) and �(x),the damping, a positive function with support in the \damped area". In 3D,the damped area is around the domain of interest i in the (x; y)-plane, andthe di�culty is to de�ne �(xji ) on each line. We explain in [3] how we proceed.3 Higher-order schemes for a 2D paraxial equation3.1 The 2D model paraxial equation with Dirichlet boundary conditionsWe now write down the series of 2D paraxial equations to be solved afterapplying the splitting technique to (16) with Dirichlet boundary conditions.Reduction to a series of 2D paraxial equations. The knowledge of u(z0+�z) from u(z0), after splitting, consists in solving ND = 4 problems like (8). Ina heterogeneous medium, it can be rewritten thanks to the auxiliary unknowns'j(x; s) = Aj(x; z0 + s)wj(x; s); 1 � j � ND as8>>>>>>>>><>>>>>>>>>:@wj@s (x; s)� i!c 'j(x; s) = 0; x = (x1; x2) 2 D; 0 � s � �z;!2c 'j +Dj (cDj(aj'j + bjwj)) = 0; (x; s) 2 D � [0;�z];wj(s = 0) = wj�1; in Dwj = wj(�z): (18)

Let Gj be a grid composed on lines parallel to direction j (Fig. 3), (18) canbe decomposed as several problems, posed on each line independently. On oneparticular line , the problem can be written as (18), except that x variesonly on : it is nothing but a 2D paraxial equation.The 2D model paraxial equation. Without loss of generality, we focus on8

j = 1�x1 = �xL1 lines j = 2�x2 = p2�xL2 lines j = 3�x3 = �xL3 lines j = 4�x4 = p2�xL4 linesFig. 3. Computational grid for the 45� approximation with 4 directions of splitting.a particular direction, x, =]� L; L[ , and we want to solve:8>>>>>>>>>><>>>>>>>>>>:@w@z � i!c ' = 0; in � [0; Z] (a);!2c '+ @@x c @@x (a'+ bw)! = 0; in � [0; Z] (b);w(z = 0) = w0; in (c);w � ' � 0 on @ � [0; Z]; (d); (19)

with a > 0 and b � 0 and w is related to the 2D wave �eld v via the change ofunknown w = ei!c zv . To give a variational formulation of (19), we introducethe following notations�(u; v)0; = Z u�v dx; kuk0; = (u; u)1=20;; ; j�j1; = 0@Z �����@�@x �����2 dx1A1=2 ;� H = L2() = 8<:u; Z juj2 dx <19=; ;� V = H10 () = (u 2 H; dudx 2 H; u = 0 on @) ;� m(u; v) = Z 1cu�v dx; 8(u; v) 2 H;� k(u; v) = Z c@u@x @�v@x dx; 8(u; v) 2 V;(20)

m(:; :) and k(:; :) are the mass and sti�ness bilinear forms. Let w0 be in V . Thevariational formulation of (19) consists in �nding (w; ') : [0; Z] �! V � Vsuch that 8>>>>><>>>>>: ddz (w; �)� i!m('; �) = 0; 8� 2 V!2m('; �)� k(a'+ bw; �) = 0; 8� 2 Vw(z = 0) = w0: ; (21)9

We recall the classical L2-stability result (e.g., [15]): any solution (w; ') to(21) satis�es the energy conservation kw(z)k0; = kw0k0; ; 8z:3.2 Semi-discretization in depthWe assume that c(x1; x2; z) = cm(x1; x2) for z 2 [zm; zm+1]. In each interval[zm; zm+1], (19) is rewritten as follows8><>: dwdz = iCmw; zm � z � zm+1;w(zm) = wm; (22)with Cm = �!=cm(!2 + a�cm)�1b�cm and �cm = cm@=@x (cm@=@x). We usethe discretization in depth, proposed in [15], based on the expression of theexact solution of (22), w(zm+1) = eiCm�zw(zm) and on the Pad�e approximantof the exponential:KYk=1 1 + rkx1 + �rkx = NK(x)NK(x) ; with rk such that eix = NK(x)NK(x) +O(jxj2K); (23)where NK(x) = QKk=1(1 + rkx). The integration from zm to zm+1 is thenformally done as follows: wm+1 = QKk=1(I+�rk�zCm)�1(I+rk�zCm)wm: Thisprocedure leads to de�ne K + 1 intermediate unknownswm0 = wm; (I + �rk�zCm)wmk = (I + rk�zCm)wmk�1; 1 � k � K: (24)We then set wm+1 = wmK. The Crank-Nicolson second-order scheme is obtainedfor K = 1 and r1 = i=2.3.3 Discretization in the lateral variable with higher-order variational �nitedi�erence schemesThe discretization in x of (19) is based on the variational formulation (21),which provides us with a systematic treatment of heterogeneous media andinsures stability thanks to energy estimates. The usual way is to use standardGalerkin �nite elements Pk, e.g. [1], but this would yield several drawbacks,especially in view of the use of the splitting method to get the 3D solution, see[3]. We prefer to use a variational �nite di�erence approach. In this section,the proofs are systematically omitted but they can be found in [3].10

3.3.1 Presentation of the discretizationThe domain is discretized with a regular grid, xi = ih, and we de�ne theshifted grid by the nodes xi+1=2 = (i + 1=2)h. We look for an approximatesolution of (21), (wh; 'h) : [0; Z] �! (Hh)2 whereHh = nvh 2 L2(); vh=[xi�1=2;xi+1=2] 2 P 0; vh(x0) = vh(xN+1) = 0o ; (25)is a �nite dimensional space which is only included in L2() and not in V (P 0is the space of constant functions). Therefore, k(:; :) needs to be approximatedby a bilinear form kh(:; :) de�ned on Hh (see below), and we have to solve8><>: ddz (wh; �h)� i!m('h; �h) = 0; 8�h 2 Hh;!2m('h; �h)� kh(a'h + bwh; �h) = 0; 8�h 2 Hh: (26)Let (�i)i=1;:::;N be the basis of Hh de�ned such that �i(xj) = �ij=ph for1 � i; j � N and (�i; �j)0 = �ij. The approximate solution is decomposed aswh(x; z) = PNi=1Wi(z)�i(x), 'h(x; z) = PNi=1 �i(z)�i(x) and (26) is equivalentto �nd the vector functions (Wh;�h) satisfying:dWhdz � i!Mh�h = 0; �!2Mh � aKh��h = bKhWh; (27)where Mh is a diagonal de�nite positive matrix, called the mass matrix,(Mh)ij = m(�i; �j), Kh is the sti�ness matrix, (Kh)ij = kh(�i; �j), and khwill be explicited below.3.3.2 De�nition of the approximate sti�ness bilinear form khTo derive an approximate sti�ness bilinear form, the derivative @=@x is ap-proximated with a �nite di�erence operator de�ned on Hh. Let D" denotethe usual 2nd-order �nite di�erence D"�(x) � �(x + ") � �(x � "); andD�" its adjoint, de�ned as D�" = �D". For � 2 Hh, and " = h=2, we de�neDh=2�(x) = �(xi)� �(xi�1), for x 2 [xi�1; xi], so thatDh=2� 2 H1=2h = nvh 2 L2() such that vh=[xi�1;xi] 2 P 0; 81 � i � N + 1o :More generally, if we consider the �nite di�erence operator D(2p�1)h=2, we haveD(2p�1)h=2� 2 H1=2h , where � is extended by zero outside . We introduce some11

real numbers �p and set @h = 1h nXp=1 �pD(2p�1)h=2 : (28)In particular we have @h�(xj+1=2) = Pnp=1 �p (�(xj+p)� �(xj�p+1)) =h; and if� 2 Hh, then @h� is in H1=2h . We have the following lemma (see [3]).Lemma 3.1 The approximation (28) is of order 2n- i.e. for any regular �,@h�(x) = d�dx(x) +O(h2n), 8x- provided that the coe�cients �p satisfynXp=1 �p(2p� 1)2k�1 = �k1; for 1 � k � n: (29)In that case it is denoted by @[2n]h and for any regular function � we have8>>><>>>: @[2n]h �(x) = d�dx(x) + h2nR[2n]S �(2n+1)(x) +O(h2n+2) ;R[2n]S = Pnp=1 �p(2p� 1)2n+1(2n+ 1)! 22n : (30)We now de�ne the approximate sti�ness bilinear form and sti�ness matrix8><>: k[2n]h (�; �) = (c@[2n]h �; @[2n]h �); 8(�; �) 2 H2h;(Kh)[2n]ij = k[2n]h (�i; �j): (31)3.3.3 The classical schemesa - Description. With the above de�nitions, we de�ne the 2n-order clas-sical scheme as followsdWhdz � i!Mh�h = 0; (a) ; �!2Mh � aK [2n]h ��h = bK [2n]h Wh; (b): (32)After elimination of the auxiliary unknown, we get the evolution system�!2Mh � aK [2n]h � (Mh)�1 dWhdz = i!bK [2n]h Wh: (33)This requires to invert Mh which is easy since Mh is diagonal. This propertyis important to keep in mind when constructing the new modi�ed schemes.12

b - Order of the classical schemes in a homogeneous medium. Equa-tion (32) can be rewritten as8>>><>>>: dwhdz (xj; z)� i!c 'h(xj; z) = 0; 8j; (a);!2c 'h(xj; z)�Xi (K [2n]h )ji (a'h(xi; z) + bwh(xi; z)) = 0; (b): (34)To analyze the \quality" of this approximation, we choose as a criterion ofquality the truncation error which quanti�es at which order the exact solution(w; ') of (19) satis�es the scheme. The �rst equation is obviously satis�ed bythe exact solution. The error comes thus from the second equationEclassh = !2c '(xj; z)�Xi (K [2n]h )ji(a'(xi; z) + bw(xi; z)): (35)Lemma 3.2 The scheme (32) is of order 2n. The �rst term of the truncationerror can be explicited (with R[2n]S de�ned in (30))Eclassh = 2cR[2n]S h2n@2n+2(a'+ bw)@x2n+2 (xj; z) +O(h2n+2): (36)3.3.4 The modi�ed schemesa - Description. The idea of the modi�ed schemes is to approximate Mhin the �rst term of (33) with a matrix M [2n]h;� which provides a better accuracywhile preserving the bandwidth of the system (to keep the same computationalcost). This corresponds to modify the mass matrix only in (32)-(b):8><>: dWhdz � i!Mh�h = 0; (a);�!2M [2n]h;� � aK [2n]h ��h = bK [2n]h Wh; (b0): (37)b - Construction ofM [2n]h;� . The mass matrix is de�ned as (Mh)ij = (1=c�i; �j)0:As done for approximating @=@x, we introduce an approximation I"�(x) =(�(x+ ") + �(x� "))=2 (I�" = I") of the identity operator I. Again for � 2 Hhand " = (2p� 1)h=2, we get I"� 2 H1=2h . We setIh = nXp=1�pI(2p�1)h=2 ; where �p are real numbers: (38)Lemma 3.3 The approximation (38) is of order 2n -i.e., Ih�(x) = �(x) +O(h2n) 8x, for any regular function �- provided that the coe�cients �p satisfy13

the Vandermonde systemnXp=1�p(2p� 1)2(k�1) = �k1; for 1 � k � n: (39)In this case, it is denoted I [2n]h and for any regular function � we have8>><>>:I [2n]h �(x) = �(x) + h2nR[2n]M �(2n)(x) +O(h2n+2);R[2n]M = Pnp=1 �p(2p� 1)2n(2n)! 22n : (40)We can now construct an approximate symmetric and positive mass matrix(M[2n]h )ij = (1cI [2n]h �i; I [2n]h �j)0: (41)Remark 3.1 For the same value of n, the approximations I [2n]h �i and @[2n]h �iuse the values of the function at the same points. Therefore the matricesM[2n]hand K [2n]h have the same bandwidth.c - Order of the modi�ed schemes in a homogeneous medium. Theidea is now to introduce a convex combination of both mass matricesM [2n]h;� = �Mh + (1� �)M[2n]h ; 0 � � � 1 ; (42)and to look for a particular value of � so as to gain accuracy when compared tothe classical discretization (corresponding to � = 1). From remark 3.1, we seethat !2M [2n]h;� � aK [2n]h keeps the same bandwidth as in the classical scheme.In a homogeneous medium the modi�ed scheme can be rewritten as8>><>>: dwhdz (xj; z)� i!c 'h(xj; z) = 0; (a);!2Xi (M [2n]h;� )ji'h(xi; z)�Xi (K [2n]h )ji (a'h + bwh) (xi; z) = 0; (b0); (43)and the truncation error still comes from the second equationEmodh = !2Xi (M [2n]h;� )ji'(xi; z)�Xi (K [2n]h )ji (a'+ bw) (xi; z) ; (44)where (w; ') is the exact solution of (19), assumed regular enough.14

Proposition 3.1 The modi�ed scheme (37), with the matrix M [2n]h;� de�ned in(42) is of order 2n+ 2 in a homogeneous medium with the choice� = �[2n] = 2n2n+ 1 : (45)The proof of proposition 3.1 is a consequence of the following lemma, (see [3]),Lemma 3.4 The coe�cients �p and �p satisfy the relation�p = (2p� 1)�p; 1 � p � n: (46)Remark 3.2 From a practical point of view, relation (46) is very useful,since it is su�cient to compute a 2n-order approximation of @=@x to get at thesame time the 2n-order approximation of the identity. Moreover, coe�cients�p satisfy a Vandermonde system and are known explicitely, for all 1 � p � n,�p = Qm6=p(2m� 1)2Qm6=p ((2m� 1)2 � (2p� 1)2) : (47)Remark 3.3 - Interpretation of the modi�ed schemes as an exten-sion of Claerbout's scheme. The classical Claerbout's scheme [7] is usuallyseen as a modi�cation of the sti�ness matrix in (27). The 2nd-order approxi-mation K [2]h is replaced by (I � h2K [2]h )�1K [2]h which is still of 2nd-order andbecomes of 4th-order with the value = 1=12. In the modi�ed schemes, themodi�cation plays on the mass matrix, but it can also be interpreted as amodi�cation on the sti�ness matrix by rewriting the modi�ed mass matrixas M [2n]h;� = �I + (1� �)(M[2n]h �Mh)M�1h �Mh ; and multiplying the secondequation by �I + (1� �)(M[2n]h �Mh)M�1h ��1 in order to reobtainMh for themass term. The sti�ness matrix is then modi�ed and becomesA[2n]h;� = �I + (1� �)(M[2n]h �Mh)M�1h ��1K [2n]h : (48)The modi�ed scheme can then be rewritten as follows8><>: dWhdz � i!Mh�h = 0; (a)!2Mh�h � A[2n]h;� (a�h + bWh) = 0: (b0) (49)For n = 1, the 4th-order modi�ed scheme is equivalent to Claerbout's schemewith the relation = (1� �)=4. The higher order modi�ed schemes can thusbe interpreted as an extension to higher orders of Claerbout's scheme.15

3.3.5 Stability analysisIt is convenient to analyze the well-posedness of the schemes written in theform (49) (instead of (37)) which is very close to the continuous equationsProposition 3.2 . � If the matrix A[2n]h;� de�ned in (48) satis�es the conditionA[2n]h;�Vh:Vh 2 IR; 8Vh; (50)(: denotes the vector scalar product), then the approximate problem (49) hasa unique solution (Wh;�h) that satis�es the energy conservation: kWh(z)k =kWh(0)k ; 8z.� The classical schemes (� = 1) satisfy condition (50), for any medium.� The modi�ed schemes satisfy (50) in the case of a homogeneous medium.For the proof we refer to [3]. We only want to underline here that for provingthe stability of the modi�ed schemes, we end up with a quantity on the form(c@hcIh 1cIhth; @hth)0 which has to be real. The key point in the homogeneouscase is that the constant operator c commutes with the other operators, sothat this quantity is equal to c(@hI2hth; @hth)0 and since operators Ih and @halso commute, there is no di�culty to conclude. In the heterogeneous case,the velocity does not commute any more with the other operators. However,numerical experiments show that the modi�ed schemes are stable, even inheterogeneous media.Notation - In the following, we only consider the 2n+2 order modi�ed schemesobtained with the particular choice of � = �[2n]. A scheme obtained using aclassical (resp. modi�ed) 2n-order discretization in x and a 2K-order dis-cretization in z will be called (2nxclass � 2Kz)-scheme (resp. 2nxmod � 2Kz).4 Dispersion analysis� Dispersion relations of the numerical schemes. The dispersion anal-ysis consists in analyzing the propagation of plane waves in a homogeneousmedium, v(x; z) = exp�i(kxx+kzz) with k = (kx; kz). For the harmonic waveequation, the dispersion relation is k2x + k2z = !2=c2, which is equivalent to(p2e)wave = p2a=(1� p2a), if we set pa = ckx=! = tan �a and pe = kx=kz = tan �e.The dispersion relation for the 45� paraxial equation, (19), is (ckz=!)cont =1 � bp2a=(1 � ap2a) = pa=pconte . We make a similar plane wave analysis for theschemes by looking for solutions on the form vmj = exp�i(kxj�x + kzm�z),but this time pnume , not only depends on pa but also on !: this is the numerical16

dispersion. The weaker this dependence, the better the scheme. To evaluatethe quality of the scheme, we de�ne the dispersion error E = j�conte � �nume j.This quantity depends on (i) the ratio rzx = �z=�x and the number G =2�c=(!�x) of discretization points per wavelength in x, (ii) on the propagatingangle pa < 1 and (iii) on the coe�cients a and b of the paraxial approximation.� Comparison between several schemes. We restrict the analysis toa = 1=4, b = 1=2, pa = tan(32�) and rzx = 1 (otherwise mentioned), so thatE becomes only a function of H = 1=G. We have compared the (2nx� 2Kz)classical and modi�ed schemes for K = 1; 2 and n = 1; 2; 3 and we give someconclusions of these comparisons (see [3] for more details).(a) Classical and modi�ed schemes. For a given order of accuracy, theless expensive modi�ed schemes have the unexpected property to be alwaysless dispersive than the classical ones. We illustrate this in Fig. 4 (a), for thefourth order schemes in x and 2nd order in z.(b) Re�nement and higher-order discretization in z. To get betteraccuracy, it is better to increase the order of the scheme than to re�ne themesh. We see in Fig. 4 (b) that even if we divide the step size by 4, the2nd-order z-discretization is more dispersive than the 4th-order one.(c) Comparison between 4xmod � 4z and 6xmod � 2z. Although the z andx directions do not play the same role in the equations, it is better to takethe same order for both discretizations, at least for su�ciently �ne meshes(more than 2.5 points per wavelength) . We illustrate this property with thecomparison between the schemes 4xmod � 4z and 6xmod � 2z in Fig. 4 (c).0

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

E

H

4x_{class}-2z 4x_{mod}-2z

(a) Modi�ed and Clas-sical schemes (4th-orderin x and 2nd-order in z). 0

0.05

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

E

H

6x_{mod}-2z , Dz= Dx/4 6x_{mod}-4z, , Dz= Dx

(b) 6xmod � 2z withrzx = 1=4 and 6xmod�4zwith rzx = 1. 0

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4x_{mod}-4z 6x_{mod}-2z

(c) 4xmod � 4z and6xmod � 2z.Fig. 4. Behaviour of the dispersion error with respect to the inverse of number ofpoints per wavelength.17

5 Numerical experimentsWe illustrate the method by several experiments in the time domain, for whichwe specify the surface data v0(x; t), solve the problem for each frequency,recover the transient result through a Fourier transform and indicate the cuto�frequency, Fc. The computational domain is 1250 m long in each horizontaldirection, and 625 m in the vertical one. The grid sizes are h = �z = 12:5 m.We handle 120 equidistributed frequencies. We represent the solution at timeTobs, instant when the wave reaches 80% of the total depth, i.e. 500m. In allthe experiments, the use of higher order schemes permits us to get satisfactoryresults even for a small number of discretization points per wavelength.� Filtered point source in a 2D heterogeneous medium. The initialcondition at z = 0 is a �ltered point source (Fx;t is the (x,t) Fourier trans-form) v0(x; t) = d2dt2 (g!S(t)) �(x � xS) (x;t)� F�1x;t (1jckxj<j!j) � d2dt2 (g!S(t)) (t)�F�1t (2 sin(!jx�xS j=c)jx�xS j ) ; where g!S(t) = exp(�!2St2=4). The �ltering process elim-inates the evanescent modes present in the wave equation. The simulation isdone in a smoothly varying velocity medium, see Fig. 5. The central frequencyof the source is FS = 2�c=!S = 28 Hz and the cuto� frequency FC = 76 Hz.The mesh contains about 3 points per wavelength for FS. Figures 6 representthe solution at time Tobs = 0:34s. In the �rst experiment, the point sourceis located in the center of the domain. Figures 6 (a), (b) and (c) show thatthe modi�ed schemes give very good results, in heterogeneous media, anda very good improvement concerning the dispersion, compared to the 2nd-order scheme. The second experiment is devoted to testing the PML absorbingboundaries, x2.2. We shift the location of the source close to the left bound-ary of the domain and present the results obtained with scheme 4xmod � 4z.The strong re exion obtained with Dirichlet BC in Fig. 6 (d) has completelydisappeared with an absorbing layer of depth 5h, Fig. 6 (e). This illustratesthe capacity of the PML in heterogeneous media. The extra cost due to thePML is negligible as the layer represents 5% of the total length.BELOW 1000

1000 - 1143

1143 - 1286

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velocity model

Fig. 5. 2D smooth heterogeneous medium: level lines of the velocity.� Filtered point source in a 3D homogeneous medium. This simulation18

(a) 2xclass � 2z. (b) 4xmod � 4z. (c) 6xmod � 4z.(d) Dirichlet BC.4xmod � 4z. (e) PMLs of 5 �x.4xmod � 4z.Fig. 6. 2D smooth heterogeneous medium. Snapshots at Tobs = 0:34s.

(a) 2xclass � 2z. (b) 4xmod � 2z. (c) 6xmod � 4z.Fig. 7. 3D homogeneous medium. Snapshots at zobs = 375m and Tobs = 0:51s.is done in a 3D homogeneous mediumwith c = 1000m=s. We consider a �lteredpoint source, the di�erence with the 2D case comes from the inverse Fouriertransform F�1x1;x2(1jckj<j!j)(x1; x2) = j!j = j2�cxjJ1 (j!xj=c) ; where J1 denotesthe Bessel function. The central frequency of the source is FS = 20 Hz andthe cuto� frequency FC = 50 Hz. The number of points per wavelength isabout 4 for FS along the x1 direction, but only 3 along the diagonal. Figure7 shows sections of the solution at a �xed depth, zobs = 375m observed attime Tobs = 0:51s. One should notice the quite good isotropy despite of theintroduction of particular directions used for the splitting.� Filtered point source in a 3D heterogeneous medium. The last sim-ulation is done in a smooth varying velocity medium, see a slice in Fig. 8. The19

number of points per wavelength for the central frequency is about 5 in the xdirection and 3.5 along the diagonal. In Fig. 9 we represent again the sectionsof the solution at time Tobs = 0:254s for two �xed depth (zobs = 20 in (a),(b) and (c) and zobs = 30 in (d) and (e)). The improvement on the dispersionusing a higher order scheme is still very good in this heterogeneous medium(compare Fig. (a) and (c)). The PML technique is also used here. Althoughthe extra cost is a bit higher and the absorption with only 6 layers is not asaccurate as in 2D, the results compared to the Dirichlet BC are still quitegood (compare Fig. (a) and (b) at zobs = 20 and Fig. (d) and (e) at zobs = 30).BELOW 1500

1500 - 1714

1714 - 1929

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ABOVE 3000

Slice of the 3D velocity model

Fig. 8. (x; z) slice of a 3D smooth heterogeneous medium: level lines of the velocity.BELOW -0.002

-0.002 - -0.001

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ABOVE 0.002

(a) zobs = 20, 4xmod � 2z (Claer-bout) with PMLs of 6 �x. (b) zobs = 20,4xmod � 2z withDirichlet BC. (c) zobs = 20,6xmod � 4z withPMLs of 6 �x.(d) zobs = 30, 4xmod� 2zwith PMLs of 6 �x. (e) zobs = 30, 4xmod � 2zwith Dirichlet BC.Fig. 9. 3D heterogeneous medium. Sections of snapshots at Tobs = 0:254s.20

References[1] G. Akrivis, V. Dougalis, N. Kampanis, \Error estimates for �nite elementmethods for a wide-angle parabolic equation", Appl. Num. Math. 16 (1-2),(1994).[2] A. Bamberger, B. Engquist, L. Halpern, P. Joly, "Paraxial approximationsin heterogeneous media", SIAM J. Appl. Math. 48, 98{128, (1988).[3] E. B�ecache, F. Collino, P. Joly, \Higher-order numerical schemes and op-erator splitting for solving 3D paraxial wave equations in heterogeneousmedia", INRIA's report 3497, (1998).[4] J.P. B�erenger, \A Perfectly Matched Layer for the Absorption of Electro-magnetic Waves", J. Comp. Phys. 114, 185-200, (1994).[5] D.L. Brown, "Applications of operator separation in re ection seismology",Geophysics 3, 288-294, (1983).[6] H. Brysk, "Numerical analysis of the 45� �nite di�erence equation formigration", Geophysics 48 (5), 532{542, (1983).[7] J. Claerbout, Imaging the earth's interior, Oxford (1983).[8] F. Collino, "Numerical analysis of mathematical models for wave propa-gation", PSI Consortium Report, (1993).[9] F. Collino, "Perfectly Matched Absorbing Layers for the Paraxial Equa-tions", J. Comp. Phys. 131 (1), 164-180, (1997).[10] F. Collino, P. Joly, "Splitting of operators, alternate directions and parax-ial approximations for the 3-D wave equation", SIAM J. Sci. Comput. 16(5), 1019-1048, (1995).[11] M.D. Collins , "The time-domain solution of the wide-angle parabolicequation including the e�ects of sediment dispersion", J. Acoust. Soc. Am.84 (6), 2114-2125 (1988).[12] D.M. Eidus, "The principle of limiting absorption", Amer. Math. Soc.Transl. 47, 157-191, (1965)[13] R. Graves, R. Clayton, "Modeling Acoustic Waves with Paraxial Extrap-olators", Geophysics 55, 306-319, (1990).[14] D. Hale, "Stable explicit depth extrapolation of seismic wave�eld", Geo-physics 56 (11), 1770-1777, (1991).[15] P. Joly, M. Kern, "Numerical methods for 3-D migration",Annual Report,PSI Consortium, (1990).[16] M. Kern, \A nonsplit 3D migration algorithm", in S. Hassanzadeh ed.,Proc. Math. Meth. in Geoph. Imaging III, SPIE, (1995).[17] D. Lee, A.D. Pierce, \ Parabolic Equation Development in RecentDecade", J. Comp. Ac. 3 (2), 95-173, (1995).[18] G. Marchuk, "Splitting and Alternating Direction Methods", Handbookof Numerical Analysis, I, Elsevier, North{Holland, (1994).[19] D. Ristow, T.D. Ruhl, "3-D implicit �nite-di�erence migration by multi-way splitting", Geophysics 62 (2), 554-567, (1997).[20] F.D. Tappert, "The parabolic approximation method",Wave propagationand underwater acoustics, Lecture Notes in Physics, Vol. 70, (1977).21