Modeling the scalar wave equation with Nyström methods

Transcription

1 GEOPHYSICS, VOL. 71, NO. 5 SEPTEMBER-OCTOBER 200 ; P. T151 T158, 7 FIGS / Modeling the calar wave equation with Nytröm method Jing-Bo Chen 1 ABSTRACT High-accuracy numerical cheme for modeling of the calar wave equation baed on Nytröm method are developed in thi paper. Space i dicretized by uing the peudopectral algorithm. For the time dicretization, Nytröm method are ued. A fourth-order ymplectic Nytröm method with peudopectral patial dicretization i preented. Thi cheme i compared with a commonly ued econdorder cheme and a fourth-order nonymplectic Nytröm method. For a typical time-tep ize, the econd-order cheme exhibit patial diperion error for long-time imulation, while both fourth-order cheme do not uffer from thee error. Numerical comparion how that the fourth-order ymplectic algorithm i more accurate than the fourth-order nonymplectic one. The capability of the ymplectic Nytröm method in approximately preerving the dicrete energy for long-time imulation i alo demontrated. INTRODUCTION Seimic modeling i an important foundation of exploration eimology. High-accuracy eimic modeling cheme are increaing in demand a computing capacity increae. In thi context, high-accuracy cheme are developed uing Nytröm time-tepping method and peudopectral patial dicretization. Numerical cheme for modeling of the calar wave equation involve dicretization of both pace and time. Thi dicretizing proce can be accomplihed in two way. The firt way i to dicretize both pace and time imultaneouly. The finite-difference method Claerbout, 1985 belong to thi category. The econd way i firt to dicretize pace to obtain a ytem of ordinary differential equation ODE with time a a variable and then to contruct numerical cheme by dicretizing time for the ytem of ODE. The peudopectral method Gazdag, 1981 and the finite-element method Ciarlet and Lion, 1991 fall into thi econd category. To form a unified viewpoint and facilitate our undertanding of numerical imulation cheme, we will accept that all numerical modeling cheme can be obtained by the econd approach to dicretization. Therefore, to contruct modeling cheme for the calar wave equation, we can firt dicretize pace by finite-difference method, peudopectral method, or finite-element method. Second, we numerically integrate the reulting ytem of ordinary differential equation by dicretizing time. Note that the obtained ytem of ODE i of econd order. When numerically olving thi ytem directly, we need to approximately evaluate additional tarting value in addition to initial value. The higher the order of the cheme, the more complex the evaluation of the additional tarting value. Nytröm method are numerical method deigned for econdorder differential equation. Nytröm 1925 firt conidered thi a a implification of Runge-Kutta method. Hairer et al ytematically developed Nytröm method. There are two benefit in uing Nytröm method. Firt, we do not need to evaluate the additional tarting value. Second, by introducing an intermediate variable, the calar wave equation can be cat into a Hamiltonian ytem. Uing Nytröm method, we can develop the correponding ymplectic method. The ymplectic method have remarkable capability for long-time computation. Thi i becaue the ymplectic propertie guarantee that the numerical olution evolve in the ame ytem a the olution of the original continuou differential equation Feng, 1993; Sanz-Serna and Calvo, In thi paper, I preent the peudopectral method and explore high-order time dicretization. I follow thi with a dicuion of Nytröm method. I then develop a fourth-order ymplectic Nytröm method with peudopectral patial dicretization. Finally, I illutrate the performance of thi cheme with numerical experiment that compare it with a econd-order cheme and a fourth-order nonymplectic Nytröm method. PSEUDOSPECTRAL METHODS A mentioned above, we can ue finite-difference method, peudopectral method, or finite-element method for the pace dicretization. I ue peudopectral method, for which the error reult Manucript received by the Editor July 18, 2005; revied manucript received February, 200; publihed online September 5, ChineeAcademy of Science, Intitute of Geology and Geophyic, P.O. Box 9825, Beijing , China. chenjb@mail.igca.ac.cn. 200 Society of Exploration Geophyicit. All right reerved. T151

2 T152 Chen mainly from the time dicretization. For higher accuracy, we need higher-order time dicretization. The iue of computing additional tarting value become more evident in thi ituation. Peudopectral method for modeling of the calar wave equation were preented by Gazdag The main point of the peudopectral method i to expre the wavefield under conideration in term of a complete et of orthogonal bai function whoe derivative are known exactly. In practice, Fourier peudopectral method are uually employed. Conider the calar wave equation 2 u t 2 = c2 2 u x u y u z 2, where u x,y,z,t i the wavefield and c x,y,z i the velocity. Let u = u 1,1,1,..., u Nx,N y,n z T, where T repreent tranpoe and u i,j,l are the wavefield value at dicrete location, i.e., u i,j,l u i x, j y, l z,t ; i = 1,..., N x ; j = 1,..., N y ; l = 1,..., N z. The value x, y, and z are grid increment in the x-, y-, and z-direction; and N x, N y, and N z are the number of grid line in the x-, y-, and z-direction, repectively. The emi-dicrete ytem reulting from the peudopectral method for equation 1 i 1 d 2 u dt 2 = c2 F 1 w F u, 2 where F and F 1 repreent 3D forward and invere finite Fourier tranform, repectively, and w = w 1,1,1,..., w Nx,N y,n z T with w i,j,l = k 2 xi + k 2 yj + k 2 zl, where k xi, k yj, and k zl are dicrete wavenumber in the x-, y-, and z-direction, repectively. The aterik denote array multiplication between vector. For example, uppoe that p = p 1,p 2,..., p m and q = q 1,q 2,..., q m, then p q = p 1 q 1,p 2 q 2,..., p m q m. To obtain the final modeling cheme, we further dicretize equation 2 in time. A tandard econd-order time difference i often ued and give u t 0 +2 t =2u t 0 + t u t 0 + c 2 F 1 w F u t 0 + t t 2, 3 where t i the time-tep ize and t 0 i the initial time. A uual, we take t 0 = 0. To compute u 2 t uing equation 3, we need to know u 0 and u t, which can be computed from the initial condition of equation 1: u 0 and u 0 / t. We can evaluate u 0 directly from u 0. To obtain u t, a third-order cheme baed on a Taylor erie expanion wa given by Gazdag 1981 a 3 u t = i u 0 t i i=0 t i. 4 i! In equation 4, u 0 and u 0 / t are known, 2 u 0 / t 2 i defined by equation 1, and 3 u 0 / t 3 i obtained by 3 u 2 3 = c2 t x y z 2 u t, which i obtained by ubtituting u/ t for u in equation 1. Finally, we can compute u t directly from u t. The main error in cheme equation 3 i aociated with the time differencing. Thu, to improve accuracy, we need to ue a higherorder time difference. A natural choice i the fourth-order time difference u 4 t =1u 3 t 30u 2 t +1u t u 0 c 2 F 1 w F u 2 t t 2. However, we can how that cheme 5 i unconditionally untable by uing the tandard pectral analyi. Uing the Taylor erie and the wave equation, another approach for high-order time differencing wa preented by Etgen 198. But thi approach till require the computation of additional tarting value. For example, for a fourthorder method, we need a fifth-order cheme to compute the tarting value intead of the third-order cheme 4. We are faced with a cumberome procedure for computing tarting value. Therefore, an alternative approach i needed. NYSTRÖM METHODS The baic idea of Nytröm 1925 method i a follow: Firt introduce an intermediate variable to reduce the econd-order wave equation to an equivalent firt-order ytem; then apply Runge-Kutta method to the firt-order ytem and implify by taking advantage of the pecial form of the firt-order ytem. The obtained Nytröm method do not need to evaluate tarting value and have coniderably le computational cot than Runge-Kutta method applied directly to the firt-order ytem for econd-order equation uch a equation 2. For detail, refer to Hairer et al For our purpoe, conider a econd-order ytem of ordinary differential equation written in the form d 2 y dt 2 = f y. ANytröm method for ytem read Z i = f y 0 + c i tz 0 + t 2 a ij Z j, i = 1, 2,...,, j=1 y 1 = y 0 + tz 0 + t 2 b iz i, i=1 z 1 = z 0 + t b i Z i, i=1 where z = dy/dt, y 0 = y 0, z 0 = z 0, y 1 y t, z 1 z t and where c i, a ij, b i, and b i are contant that determine the order of the method. The numerical olution y 1 i obtained through auxiliary variable Z i in cheme 7. The tarting value y 0 and z 0 = dy 0 /dt in cheme 7 are both known, and no additional tarting value are needed. Scheme 7 require computation of z 1, but thi i eay to accomplih becaue the main evaluation of Z i have been done previouly in the computation of y 1. The order condition via the algebraic equation atified by c i, a ij, b i, and b i have been obtained by uing tree theory in Hairer et al By olving thee algebraic equation, we can obtain the correponding method. Conider a table fourth-order explicit method Qin and Zhu, 1991 : 5 7

3 Modeling with Nytröm method T153 c 1 = 3+ 3 b 1 = b 1 = 3 2 3, c 2 = 3 3, b 2 = 3+ 3 a 21 = 2 3 3, a 32 =,, c 3 = 3+ 3,, b 3 = 1+ 3,, b 2 = 1 2, b 3 = 3+2 3, a 11 = a = a 13 = a 22 = a 23 = a 31 = a 33 =0. With thee coefficient, cheme 7 become Z 1 = f y tz 0, method. Therefore, Nytröm method can be applied directly. We introduce a variable v = du/dt and apply Nytröm method 8 to equation 2 to obtain V 1 = c 2 F 1 w F u tv 0, V 2 = c 2 F 1 w F u tv t 2 V 1, V 3 = c 2 F 1 w F u tv 0 + t2 V 2, u 1 = u 0 + tv 0 + t V 3, V V 2 Z 2 = f y tz t 2 Z 1, v 1 = v 0 + t V V V 3. 9 Z 3 = f y tz 0 + t2 Z 2, y 1 = y 0 + tz 0 + t Z 3, z 1 = z 0 + t Z Z 2 Z Z Z 3. Another benefit of Nytröm method i that we can develop ymplectic algorithm if equation poee a ymplectic tructure. In the pat, we only focued on the accuracy of the numerical method when olving differential equation numerically. However, with the tremendou progre in computer technology and numerical analyi theory in recent decade, computation ha become a third category of method in cientific reearch in addition to theory and experiment. Not only i the accuracy of numerical method expected, but tructure-preerving propertie alo are required. Solution to differential equation uually preerve variou tructure uch a ymplectic tructure, multiymplectic tructure, and variou geometric tructure. When olving differential equation numerically, ome numerical method alo preerve thee tructure they are uually called tructure-preerving algorithm, while other violate them. The tructure-preerving method have remarkable capability for long-time computation. The calar wave equation 1 ha a claical Hamiltonian tructure Chen, Therefore, we can develop the correponding tructure-preerving method eeappendix A. A PSEUDOSPECTRAL METHOD WITH FOURTH-ORDER NYSTRÖM TIME DIFFERENCE Now we return to the iue of higher-order time difference for equation 2. Equation 2 i a econd-order ytem of ordinary differential equation obtained from equation 1 by uing peudopectral 8 Scheme 9 i a table fourth-order explicit Nytröm method with accuracy of O t 4 Qin and Zhu, 1991; Hairer et al., Becaue we ue peudopectral patial dicretization, the patial accuracy i of exponential order O exp x Fornberg, 199. Therefore, the total accuracy of cheme 9 i O t 4 + exp x.a pointed out by Hairer et al. 1993, the Nytröm method cheme 9 i coniderably more efficient than Runge-Kutta method. In eimic modeling baed on cheme 9, we compute u 1 and v 1 from known u 0 and v 0 = du 0 /dt. Then we repeat thi proce with u 0 and v 0 replaced by u 1 and v 1, repectively. Higher-order Nytröm time difference are eaily available, and they have the ame format a cheme 9 but with more V i. NUMERICAL EXPERIMENTS In thi ection, we perform numerical experiment to tet the feaibility and performance of the preented cheme. For clarity and implicity, we only conider the 2D cae. A in Gazdag 1981, we ue the initial condition and u x,z,t =0 = exp x 2 + z z 0 2 u 0 =0. t The grid increment are x = z = 50 m. Here, z 0 i a contant that indicate the poition of the ource. In the following numerical example, we ue periodic boundary condition. In the firt example, we ue cheme 9 to imulate wave propagation in a heterogeneou medium. The velocity model Figure 1 conit of three layer with velocitie c = 1000 m/, 2000 m/, and 3000 m/; a traight interface; and a curved interface. The ource i et at x =0m,z = 200 m. The time-tep ize t i Figure 2 how the wavefield at t = 0, 0.438, 0.93, 1.8, 1.5, and The imulated reflection and tranmiion phenomena are qualitatively correct.

4 T154 Chen We now make numerical comparion between cheme 9 and the commonly ued cheme 3. In thi example, we ue the velocity c = 3000 m/ and the time tep ize t = The tarting value for equation 3 i computed through equation 4, which require evaluating the econd-order patial derivative of the initial condition. Therefore, when uing cheme like equation 3 in wave imulation, we uually require that the initial condition have a certain degree of moothne, which ometime i not atified in practice for example, dicrete initial condition. In contrat, when uing cheme 9, we Figure 1. The velocity model ued for the numerical experiment. It conit of three layer with velocitie c = 1000 m/, 2000 m/, and 3000 m/, repectively; a traight interface; and a curved interface. have no pecial requirement on the initial condition, which make cheme like 9 more practical. In Figure 3, we how the impule repone for cheme 3 and 9 after 200, 500, and 5000 time tep, repectively. For time tep equal to or le than 200, impule repone for both cheme 3 and 9 do not exhibit patial diperion error. After 500 time tep, the patial diperion become evident in the wavefront computed with cheme 3, which contrat with the clean wavefront computed with cheme 9 Figure 3c and d. After 5000 time tep t =30, the wavefront computed with cheme 3 ha blurred very badly, but the wavefront computed with cheme 9 i till harp Figure 3e and f. Thee obervation indicate that for a typical time-tep ize of 0.00, the commonly ued cheme 3 i not uitable for lengthy imulation. Now we perform imulation by cheme 3 with maller time-tep ize of 0.004, 0.003, 0.002, and To compute the wavefront at 30, the number of time tep i 7500, 10,000, 15,000, and 30,000, repectively. Figure 4 how the imulation reult. We ee that diperion error decreae a time-tep ize decreae. The reult with time-tep ize of i imilar to that obtained with cheme 9 with a time-tep ize of 0.00 Figure 3d. Thi indicate that to obtain the impule repone at 30, cheme 9 with a time-tep ize of 0.00 i more efficient than cheme 3 with a time-tep ize of We can explain thi a follow: Although for each time tep cheme 9 need to compute three pair of FFT forward and invere, which i three time a many a that of the cheme 3, the timetep ize ued in cheme 9 i ix time a large a that ued in cheme 3. Becaue the main computational cot in both cheme 9 and 3 i the computation of FFT, the overall computational cot of cheme 9 i only half that of cheme 3. To demontrate the advantage of the ymplectic method, we now make numerical comparion between the fourth-order ymplectic Nytröm cheme 9 with the following fourth-order nonymplectic Nytröm cheme: V 1 = c 2 F 1 w F u 0, V 2 = c 2 F 1 w F u tv t2 V 1, V 3 = c 2 F 1 w F u 0 + tv t2 V 2, u 1 = u 0 + tv 0 + t 2 1 V V 2, v 1 = v 0 + t 1 V V V Figure 2. Wavefield evolution with time computed with cheme 9. Ditance and time are meaured in meter and econd, repectively. Figure 5 how the impule repone computed with cheme 10 after 5000 time tep. The time-tep ize i We ee that the reult i imilar to that computed with the ymplectic cheme 9 Figure 3f. To examine more cloely the reult obtained by thee two fourth-order cheme, amplitude curve at a fixed point x = 100 m, z = 100 m over the time near time tep are hown in Figure a. We ee dicrepancy in the amplitude curve computed by the two fourth-order cheme. In Figure b, another amplitude curve i added, which i computed by the nonymplectic cheme 10 with a maller time tep ize of It can be een that the amplitude curve computed by the ymplectic cheme 9 with a time-tep ize of 0.00 agree well with the amplitude curve com-

5 Modeling with Nytröm method T155 puted by the nonymplectic cheme 10 with a maller time-tep ize of Thi indicate that the fourth-order ymplectic algorithm cheme 9 i more accurate than the fourth-order nonymplectic algorithm cheme 10 for the ame time-tep ize. To how how the amplitude curve computed by the nonymplectic cheme 10 with maller time-tep ize approach the amplitude curve computed by the ymplectic cheme 9 with a time-tep ize of 0.00, we plot the following amplitude-difference curve Figure c : AD t t = S 0.00 t NS t t, where AD t t denote the amplitude-difference curve; S 0.00 t, the amplitude curve computed by the ymplectic cheme 9 with a timetep ize of 0.00 ; and NS t t, the amplitude curve computed by the nonymplectic cheme 10 with a time-tep ize of t. Here we take t = 0.00, 0.004, 0.002, and We can ee that the magnitude of the curve of the amplitude difference diminihe with decreaing t. From the above numerical comparion, we can conclude that to achieve approximately the ame accuracy, the ymplectic algorithm cheme 9 i much more efficient than the nonymplectic algorithm cheme 10, conidering the time-tep ize ued. Thi i a great advantage. Another important advantage of a ymplectic algorithm i it ability to approximately preerve energy for lengthy imulation Feng, 1993; Sanz-Serna and Calvo, For the wave equation u tt = c 2 u xx + u zz with a periodic boundary condition, the true olution preerve the energy v M 2 + c 2 u 2 x + u 2 z dxdz, where v = u t and M i the integral domain. Now in our numerical experiment, we monitor the dicrete energy 2 G n = v n i,l 2 + c u n i+1,l i,l 2 + c u n i,l+1 z n u i,l n u i,l x 2 x z, 2 Figure 4. Impule repone at 30. The reult are computed with cheme 3 with time-tep ize of 0.004, 0.003, 0.002, and 0.001, repectively. Figure 3. Impule repone computed with cheme 3 and 9. a and b 200 time tep; c and d 500 time tep; e and f 5000 time tep. View a, c, and e are computed with cheme 3; b, d, and f are computed with cheme 9. Figure 5. Impule repone after 5000 time tep computed with cheme 10 with a tep-time ize of 0.00.

6 T15 Chen Figure 7. Dicrete energy curve G n veru n for the ymplectic cheme 9 and the nonymplectic cheme 10. a N = 5000; b N = 100,000. half of the energy i lot in the energy curve for the nonymplectic cheme 10. Thi demontrate the capability of the ymplectic algorithm for approximately preerving the dicrete energy for lengthy computation. CONCLUSIONS Figure. Amplitude and amplitude-difference curve at a fixed point x = 100 m, z = 100 m over the time near 30. a Amplitude curve computed by the ymplectic cheme 9 with t = 0.00 blue line and the nonymplectic cheme 10 with t = 0.00 red line. b Amplitude curve computed by the ymplectic cheme 9 with t = 0.00 blue line ; the nonymplectic cheme 10 with t = 0.00 red line ; and the nonymplectic cheme 10 with t = green line. c Amplitude-difference curve: t = 0.00 red line ; t = blue line ; t = yellow line ; and t = green line. where v n i,l v i x,l z,n t, u n i,l u i x,l z,n t, and n = 0,1,2,..., N. Here, N i the number of time tep. Figure 7a how the dicrete energy curve G n a a function n for both cheme 9 and 10, with N = For the ymplectic cheme 9, the energy curve fluctuate about a contant energy. For the nonymplectic cheme 10, a numerical lo in energy i oberved. The correponding reult for N = 100,000 i hown in Figure 7b. For thi very long computation, the energy curve for the ymplectic cheme 9 till fluctuate about a contant energy while nearly Nytröm time differencing with peudopectral patial dicretizaiton i preented for modeling of the calar wave equation. Two fourth-order Nytröm cheme with peudopectral patial dicretizaiton are developed; one i ymplectic and the other i nonymplectic. Numerical experiment demontrate three reult. Firt, thee fourth-order cheme can be ued for long-time imulation with a time-tep ize of 0.00, wherea the commonly ued econd-order cheme exhibit evere diperion error for thi computation time and time-tep ize. Second, the fourth-order cheme with a timetep ize of 0.00 are more efficient than the econd-order cheme with a mall time-tep ize of And third, the ymplectic algorithm i more accurate than the nonymplectic algorithm and ha a better capability of approximately preerving the dicrete energy for lengthy imulation. Preerving propagation energy i only one feature of preerving ymplectic tructure. The ymplectic algorithm alo have other feature uch a better accuracy, which can be explained by the theory of backward error analyi Hairer et al., According to thi theory, the olution of both ymplectic and nonymplectic algorithm for a Hamiltonian ytem formally atify a perturbed ytem.

7 Modeling with Nytröm method T157 However, the perturbed ytem atified by the ymplectic algorithm retain the Hamiltonian tructure; the ytem atified by the nonymplectic algorithm doe not. ACKNOWLEDGMENTS I would like to thank profeor Arcone and anonymou reviewer for valuable uggetion and correction, that led to a great improvement of thi paper. Thi work i upported by the National Natural Science Foundation of China under grant and alo partially by the Chinee Academy of Science with Key Project of Knowledge Innovation KZCX1-SW-18. du = v, dt dv dt = c2 F 1 w F u. Further, we can reformulate equation A-3 a d dt v u = 0 I 0 H I u, H v A-3 A-4 APPENDIX A HAMILTONIAN STRUCTURE In thi appendix, I preent the Hamiltonian tructure and the correponding ymplectic algorithm for the calar wave equation. Introducing an intermediate variable v = du/dt, equation 1 i equivalent to u t = v, c2 v t = 2 Equation A-1 can be reformulated a where H = 1 2 v 2 + c u x u y u z 2. t v u = u 2 x u 0 H H, v A-1 A-2 + y u 2 + z u 2 dxdy i the Hamiltonian ytem, and denote the variational derivative. Equation A-2 i an infinite-dimenional Hamiltonian ytem. The evolution of the wavefield with time i characterized by a ymplectic tructure du Ù dvdxdy, which can be viewed a an antiymmetric quadratic form. For the exact definition of ymplectic tructure, a well a the infinite-dimenional Hamiltonian ytem and variational derivative, ee Olver In olving equation A-2 numerically, we can ue finite-difference, finite-element, or peudopectral method to dicretize the pace and obtain a finite-dimenional ytem. Equation 2, which i obtained with a peudopectral method, can be cat into a finite-dimenional Hamiltonian ytem. Firt, introducing v = du/dt, we can rewrite equation A-2 a where I i an identity matrix and H = 1 2 vt v + c 2 u T Du. The matrix D i a econd-order pectral differential matrix which atifie F 1 w F u = Du. For detail about pectral differential matrix, ee Chen and Qin Equation A-4 i a tandard finite-dimenional Hamiltonian ytem. A finite-dimenional Hamiltonian ytem i jut a ytem of ordinary differential equation that ha the form of equation A-4. The true olution of equation A-4 preerve the ymplectic tructure du Ù dv. In other word, if we uppoe that the olution of equation A-4 i P = F P 0, where P = u,v T and P 0 = u t =0,v t =0 T, then the fact that F P 0 preerve the ymplectic tructure du Ù dv i equivalent to F P 0 atifying F P 0 J F T P 0 = J, where J = 0 I I 0. Here, F/ P 0 i the Jacobian of the vector-valued function F P 0. A function atifying the above equality i alo called a ymplectic mapping. Thu, it i concluded that the time evolution of the eimic wavefield i a ymplectic mapping. In eimic modeling, we need to olve equation A-4 numerically. Suppoe that F P 0 i the numerical olution of equation A-4 obtained by ome numerical algorithm. Of coure, the numerical olution F P 0 i the approximation of the true olution F P 0. Doe the numerical olution alo have other propertie? In fact, ome numerical olution are ymplectic mapping while other are not. Anumerical method i called a ymplectic algorithm if the reulting numerical olution i a ymplectic mapping. In the pat, we mainly focued on the accuracy of the numerical olution to a Hamiltonian ytem; it tructure-preerving propertie were not conidered eriouly. With the introduction of ymplectic algorithm and it great ucce in many phyical field, tructurepreerving algorithm have developed into a very active and promiing reearch area. The Hamiltonian framework developed here provide a bai for applying ymplectic algorithm in eimic modeling. Baed on thi framework, we hould develop numerical olution of equation A-4 that are alo ymplectic mapping, i.e., ymplectic algorithm. Variou ymplectic algorithm for Hamiltonian ytem have been developed. We can directly apply the ymplectic algorithm to equation A-4. Baed on a finite-difference patial dicretization, Hamiltonian ytem alo can be obtained Luo

8 T158 Chen et al., However, thi kind of patial dicretization uually caue patial diperion error. Now conider a Nytröm formulation for equation A-4: V i = c 2 F 1 w F u 0 + c i tz 0 + t 2 a ij V j, u 1 = u 0 + tv 0 + t 2 b iv i, i=1 If the coefficient in cheme A-5 atify b i = b i 1 c i, i = 1,...,, j=1 b i b j a ij = b j b i a ji, i, j = 1,...,, v 1 = v 0 + t b i V i. i=1 A-5 then cheme A-5 i a ymplectic algorithm Sanz-Serna and Calvo, It i eay to check that the coefficient ued in cheme 9 atify the above condition; therefore, cheme 9 i a ymplectic algorithm. REFERENCES Chen, J. B., 2004, Multiymplectic geometry for the eimic wave equation: Communication in Theoretical Phyic, 41, Chen, J. B., and M. Z. Qin, 2001, Multiymplectic Fourier peudopectral method for the nonlinear Schrödinger equation: Electronic Tranaction on NumericalAnalyi,, Ciarlet, P. G., and J. L. Lion, 1991, Handbook of numerical analyi: North- Holland. Claerbout, J. F., 1985, Imaging the earth interior: Blackwell Scientific Publication, Inc. Etgen, J. T., 198, High-order finite-difference revere migration with the two-way non-reflecting wave equation: SEP-48: Stanford Univerity. Feng, K., 1993, The collected work of Feng Kang: Defence Indutrial Pre. Fornberg, B., 199, A practical guide to peudopectral method: Cambridge Univ. Pre. Gazdag, J., 1981, Modeling of the acoutic wave equation with tranform method: Geophyic, 4, Hairer, E., C. Lubich, and G. Warnner, 2002, Geometric numerical integration: Springer-Verlag, Berlin. Hairer, E., S. P. Nøett, and G. Warnner, 1993, Solving ordinary differential equation I: Springer-Verlag, Berlin. Luo, M. Q., H. Liu, and Y. M. Li, 2001, Hamiltonian decription and ymplectic method of eimic wave propagation: Chinee Journal of Geophyic, 44, 0 8. Nytröm, E. J., 1925, Über die numeriche integration von differentialgleichungen: Acta Societe Scientiarum Fennicae, 50, Olver, P. J., 1993, Application of Lie group to differential equation: Springer-Verlag, New York. Qin, M. Z., and W. J. Zhu, 1991, Canonical Runge-Kutta-Nytröm method for econd order ODE : Computational Mathematic Application, 22, Sanz-Serna, J. M., and M. Calvo, 1994, Numerical Hamiltonian problem: Chapman and Hall.