Implicit Splitting Finite Difference Scheme for Multi-dimensional Wave Simulation

Transcription

1 Iplicit Splittig Fiite Differece Schee for Multi-diesioal Wave Siulatio Houhu (Jaes Zhag, Yu Zhag, CGGVeritas, Housto Jaes Su, CGGVeritas, Sigapore Suar I this abstract, we propose a ew fiite-differece schee for solvig wave equatios. This schee splits the ultidiesioal sste ito differet directios ad solves each directio iplicitl. Ulie ost splittig ethods i the literature which produce uerical aisotrop i diagoal directios, this ethod gives perfect circular ipulse resposes ad allows lateral velocit variatios. I this paper, we prove that the proposed schee is ucoditioall stable. I the uerical eaples, we show soe ipulse respose tests ad copare the with the results fro soe high-order eplicit fiite-differece ethods. The ew ethod allows larger tie step ad requires less eor storage durig the reverse tie igratio. Itroductio Seisic odelig usig fiite-differece (FD ethods is a ivaluable tool for seisic iterpretatio ad plas the e role i reverse tie igratio. Geerall speaig, published fiite-differece schees ca be divided ito two classes: iplicit FD schees (Mufti, 1985 ad eplicit FD schees. For 1D wave odelig, the iplicit schees are superior for two reasos. First the ca be ade ucoditioall stable so the tie step sie ca be arbitraril chose to save coputatio tie ad, secod, the lead to tridiagoal sstes which ca be solved efficietl b forward/bacward substitutio (Trefethe, However, i higher diesios, the iplicit FD schees require solvig ulti-diesioal atrices which greatl icreases eor usage ad coputatioal cost. Therefore, i ulti-diesios, seisic wave equatio odelig algoriths rel alost eclusivel o eplicit FD ethods (Kosloff, et.al., 198; Forberg Though eplicit FD schees are eas to solve, theoretical aalsis shows that the ca ol be coditioall stable ad the archig tie step sie is liited b the stabilit coditio. Such a restrictio put a theoretical upper boud to the coputatioal speed of this tpe of FD ethods. I this abstract, we propose a ew iplicit FD schee for solvig ulti-diesioal wave equatios. This schee splits the sste ito several oe-diesioal FD schees at differet spatial directios ad solves each directio iplicitl. Ulie ost of splittig ethods i the literature which produce uerical aisotrop i diagoal directios, this ethod gives perfect circles for ipulse respose testig. The ucoditioal stabilit has bee proved ad the dispersio has bee aaled i this paper. We also tested the ew ethods with differet ipulse resposes ad copared the results with those fro the high-order eplicit FD ethods. Siilar schee was proposed b Mufti (1985. But ours ca achieve higher order accurac i space ad is sipler to ipleet i higher diesios. Iplicit Splittig Fiite Differece Schee Let us start with the three-diesioal wave equatio: 1 p p p p = + +, (1 c (,, t where c (,, is the edia velocit. Deotig p = p( l, j,, we propose the followig Iplicit Splittig Fiite Differece (ISFD schee to solve equatio (1 uericall U δ (1 3 t V δ (1 3 t W δ (1 3 t where γ ad a are chose costats, + au + av + aw, ( δ is the cetral fiite differece operator δ p( = p( + p( + p(, ad the auiliar wavefields U, V ad W are defied as U = u p + u (3 V = v p + v W = w p + w +1 The pressure wavefield p ca be coputed as u + v + w p =. (4 3 The proposed ISFD ( cosists of three decoupled iplicit fiite-differece schees alog, ad directios. Each oe of the leads to a tri-diagoal atri ad ca be efficietl solved b a forward ad bacward substitutio ethod. Therefore, uericall FD schee ( is ore cost-effective tha the covetio iplicit FD schees. Most of the eistig directioal splittig ethods for SEG/Sa Atoio 007 Aual Meetig 011

2 Iplicit Splittig Fiite Differece to guaratee stabilit. Whe a is big copared with γ, (8 allows a larger tie step tha the covetioal eplicit fiite-differece schees. Figure 1: Dispersio error of the eplicit fiite-differece schees with d, 4 th, 6 th ad 8 th order i space ad of ISFD for differet a ad γ values. Tie i d order. solvig wave equatios itroduce uerical aisotrop, which causes big ieatic errors i odelig ad igratio. But our proposed ethod does ot suffer fro this proble. Furtherore, D ad 3D ipleetatios follows the sae uerical frae ad higher order accurac i space ca be achieved b adjustig a ad γ. These ae our schee differet fro that proposed b Mufti (1985. Stabilit ad Dispersio Aalsis To aale the stabilit ad dispersio, we first trasfor the fiite differece schee ( to tie ad waveuber doai as P P + P ~ ~ ~ = c ( + + P. (5 t Where is defied as ad ~ ~ θ + = θ 1 γ 3ac t θ, (6 4si ( / = θ. (7 Siilar epressios ca be writte for Λ ad Λ. It is eas to chec that equatio (5 is secod order i tie ad is a approiatio to the origial wave equatio (1. Therefore, o splittig error is itroduced i this ethod. We ca also prove that (5 is ucoditioall stable if a 1/ 4. For arbitraril give γ ad a, t ca be chose so that c t γ ( a To aale the spatial dispersio, igorig the errors caused b tie discretiatio i (5, we have the followig equatio ~ ~ ~ ω = c + +. (9 ( The above dispersio relatio is siilar to those of the covetioal high-order space eplicit FD schees ~ ~ ~ ω = c (, +, +,, (10 ~ ~ ~ where,, ad, are the spatial dispersio ters, of stadard th ~ -order FD schees. For eaples, is, the sae as θ defied b equatio (7, ad, ~ ~ 1, 4 =,(1 + si ( /, (11 3 etc.. I Figure (1, we copare the dispersio error for the covetioal spatial FD schees fro the d order to the 8 th order ad for ISFD for various a ad γ values. The relative dispersio error is defied as ~ / 1 ~, ad / 1. Ulie the covetioal FD schees, the dispersio relatio of ISFD ca be adjusted b choosig differet a ad γ values ad it also depeds o the odelig grid sie,, tie step t ad local velocit c. B properl choosig these paraeters, ISFD ca achieve a sae dispersio as higher order eplicit fiite-differece schees ad still use larger tie step if a is selected to be larger tha γ. Nuerical Eaples The proposed algorith was first tested o a ipulse respose. Figure (a shows the sap shot at 1s coputed usig the D eplicit fiite-differece with 8 th -order i space ad d -order i tie. Figure (b shows the sap shot at the sae tie step but coputed usig ISFD. I both cases, velocit is 000/s, t=0.005sec, =0 ad =0. ISFD has less dispersio o the frot edge tha the eplicit schee. A D dispersio error is plotted i Figure (3 for 8 th order eplicit schee ad i Figure (4 for ISFD usig the sae paraeters as for producig Figure (b. As i Figure (, the dispersio give b ISFD is ver close to 8 th -order fiite-differece but soother at high waveubers. Siilar results are show i Figure (5a ad Figure (5b ecept i this case = =40. Obviousl, ISFD has less dispersio. SEG/Sa Atoio 007 Aual Meetig 01

3 Iplicit Splittig Fiite Differece a Figure 4: Dispersio error of ISFD. The D dispersio error was plotted based the followig equatios: ~ ~ ~ ~,8 +,8 + E FD = 1, E = 1. (1 ISFD + + To test how well the algorith ca hadle coplicated geolog eviroets, we desiged a salt odel with cople geoetr ad high velocit. The odel is show i Figure (6. The velocit of the right salt is 4500/s. = =0. The source is located at (,=(4000,3000. Figure : The ipulse repose of the 8 th order eplicit fiite-differece schees (a ad the ISFD ethod (b. Tie is 1 secod. Spatial grid icreet is 0. b Eighth-order eplicit fiite differece becoes stable util t is reduced to 0.005sec. Figure (7 shows the sap shot at 1sec. Figure (8 shows the correspodig result with t=0.005sec usig ISFD. The eplicit schee ad ISFD have alost idetical ieatics. But the tie step for ISFD is twice as large as that for the eplicit ethod (ad ca be bigger thereb reducig coputatio cost. Coclusios Figure 3: Dispersio error of 8 th order eplicit fiitedifferece schee. We have proposed a iplicit splittig fiite differece schee for uerical solutio of ulti-diesioal wave equatio. Ulie ost splittig ethods, this ew schee provides good approiatio i all spatial directios ad this has bee validated b the ipulse resposes. B splittig a ulti-diesioal sste ito separate directios, each of the decoupled sste leads to a tridiagoal sste which ca be solved efficietl. We have show that the dispersio error of this schee ca be iiied b properl selectig odelig paraeters. Nuerical results show that the ew ethod, b usig larger tie step, ca give results of the sae qualit as those usig eplicit high-order fiite-differece schees. SEG/Sa Atoio 007 Aual Meetig 013

4 Iplicit Splittig Fiite Differece Acowledgets We tha CGGVeritas for the perissio to publish this paper ad David Yigst for a helpful discussios. Figure 6: A cople velocit odel. a Figure 7: The sap shot at 1 secod coputed usig the eplicit fiite-differece with 8 th -order i space. t=0.005sec. b Figure 5: The ipulse repose of the 8 th order eplicit fiitedifferece schees (a ad the ISFD ethod (b. Tie is 1 secod. Spatial grid icreet is 40. Figure 8: The sap shot at 1 secod coputed usig ISFD. t=0.005sec. SEG/Sa Atoio 007 Aual Meetig 014

5 EDITED REFERENCES Note: This referece list is a cop-edited versio of the referece list subitted b the author. Referece lists for the 007 SEG Techical Progra Epaded Abstracts have bee cop edited so that refereces provided with the olie etadata for each paper will achieve a high degree of liig to cited sources that appear o the Web. REFERENCES Dablai, M. A., 1986, The applicatio of high-order differecig to the scalar wave equatio: Geophsics, 51, Etge, J. T., 1988, Evaluatig fiite-differece operators applied to wave siulatio: Staford Eploratio Project 57. Forberg, B., 1987, The pseudospectral ethod: Coparisos with fiite-differece for the elastic wave equatio: Geophsics, 5, Kidela, M., A. Kael, ad P. Sguaero, 1990, O the costructio ad efficiec of staggered uerical differetiators for the wave equatio: Geophsics, 55, Kosloff, D. D., ad E. Basal, 198, Forward odelig b a Fourier ethod: Geophsics, 47, Mufti, I. R., 1985, Seisic odelig i the iplicit odel: Geophsical Prospectig, 33, Trefethe, L. N., 1996, Fiite-differece ad spectral ethods for ordiar ad partial differetial equatios, SEG/Sa Atoio 007 Aual Meetig 015