McMaster University. Advanced Optimization Laboratory. Authors: Dinghui Yang, Ming Lu, Rushan Wu and Jiming Peng

Transcription

1 McMaser versy Advaced Opmao Laboraory Tle: A Opmal Nearly-Aalyc Dscree Mehod for D Acousc ad Elasc Wave Equaos Auhors: Dghu Yag Mg Lu Rusha Wu ad Jmg Peg AdvOl-Repor No. 004/9 July 004 Hamlo Oaro Caada

2 A Opmal Nearly-Aalyc Dscree Mehod for D Acousc ad Elasc Wave Equaos Dghu Yag ad Mg Lu Deparme of Mahemacal Sceces Tsghua versy Beg P. R. Cha (dhyag@mah.sghua.edu.c) Rusha Wu Isue of Tecocs versy of Calfora Saa Cru CA SA (rwu@es.ucsc.edu) Jmg Peg 3 3 Deparme of Compug ad Sofware McMaser versy Hamlo Caada (peg@mcmser.ca) Bulle of he Sesmologcal Socey of Amerca ( press)

3 Absrac I hs arcle we prese he so-called opmal early-aalyc dscree mehod (ONADM) whch s a mproved verso of he NADM proposed recely BSSA (Yag e al. 003a). We compare umercally he error of he ONADM wh hose of he NADM ad oher FD mehods for D ad D cases ad gve he wave-feld modelg -D soropc meda. We also dscuss he valdy of he -mes absorbg boudary codo (ABC) whe he absorbg boudary codos are corporaed ONADM. We show ha compared wh he orgal NADM he ONADM for he -D case ca reduce sgfcaly he sorage space ad compuaoal cos. The me accuracy of he opmal mehod s also creased from -order of he orgal NADM o 4-order whle he space accuracy remas he same as ha of he orgal oe. Promsg umercal resuls sugges ha he ONADM s suable for large-scale umercal modelg as ca suppress effecvely umercal dspersos caused by dscreg he wave equaos whe oo-coarse grds are used. Iroduco Aalyss of sesmc daa o deerme earh srucure ad sesmc source parameers requres accurae ad effce mehods for compug syhec sesmograms. Whe s hard o solve he wave equaos we usually refer o wo ds of appromae mehods. Oe appromae mehod s he perurbao mehod ad he oher s he dscreao mehod whch we frs dscree he wave equaos ad he solve he resulg fe-dfferece equaos. Tme-doma mehods dscree he spaal ad emporal dervaves acousc ad elasc wave equaos. Several wdely used mehods such as secod-order ceer scheme (Alford e al. 974; Kelly e al. 976; Igel e al. 995) ad varous hgh-order compac schemes or so-called La-Wedroff scheme (Dabla 986; Blach ad Robersso 997; Wag e al. 00) are based o he fe-dfferece (FD) mehod. There are also oher mehods ha use he pseudo-specral (PS) mehod o compue spaal dervaves (e.g. Kosloff ad Baysal 98). FD mehods have bee prove o be successful ad provded useful ools for eplorg sesmology. Due o hs reaso umerous fe-dfferece schemes have bee wdely employed o solve acousc ad elasc wave equaos (Alford e al. 974; Kelly e al. 976; Dabla 986; Vreu 986; Taeuch ad Geller 000; ohers). These FD mehods are also appled o asoropc ad vscoelasc problems (Robersso e al. 994; Igel e al. 995; Blach ad Robersso 997; Zhag e al. 999; Taeuch ad Geller 000). However should be meoed ha some FD mehods le he coveoal fe-dfferece mehods wh - ad 4-orders ofe suffer from serously umercal dsperso whe oo few samples per wavelegh are used or whe he models have large velocy coras or arfacs caused by source a grd pos (Fe ad Larer 995; Yag e al. 00). Numercal dspersos affec o oly he performace of he lower-order FD mehods bu also he performace of he so-called hgh-order FD mehods. For eample he 0-order compac FD schemes (e.g. Wag e al. 00) whch

4 usually use more grds ha low-order schemes also suffer from umercal dspersos. The demad of more grds hgh-order FD mehods preves he algorhms from effce parallel mplemeao ad arfcal boudary reame. O he oher had alhough he umercal dspersos or udesrable rpples ca also be suppressed usg he so-called flu-correced raspor (FCT) echque he FCT mehod s ypcally uable o fully recover he los resoluo by he umercal dsperso whe he spaal samplg becomes oo coarse (Yag e al. 00). The pseudo-specral mehod (Kosloff ad Baysal 98) s aracve as he space operaors are eac up o he Nyqus frequecy bu requres he Fourer rasform of wave-feld o be made whch s compuaoally epesve for 3-D asoropc models ad has dffculy hadg sharp boudares (Mua e al. 000). Moreover ag he Fourer rasform meas ha each po eracs wh every oher po. To some ee hs s cosse wh he physcal pheomea as eraco dyamc elascy s of a local aure. Recely Yag developed a perurbao mehod called early aalyc dscree mehod (NADM) for acousc ad elasc equaos (Yag e al. 003a). The NADM was ally suggesed by Koddoh e al.(994) for solvg parabolc ad hyperbolc equaos. The mehod based o he so-called hough aalyss crero (Koddoh e al. 994) uses a rucaed Taylor epaso wh respec o me o aalycally appromae he wave dsplaceme ad s frs-order paral dervaves a grd-pos. The uses some erpolao relaos based o he fuco of he rucaed Taylor epaso wh respec o spaal cremes o deerme he hgh-order space dervaves volved hese rucaed Taylor formulae. O he bass of such a srucure he mehod eables effecvely o suppress he umercal dsperso caused by dscreg he wave equao by usg he local erpolao compesao for he rucaed Taylor seres whle he coveoal FD schemes suffer from umercal dsperso ear large velocy coras or whe oo few samples per wavelegh are used (Yag e al. 003a). Furher by usg he frs-order spaal dervaves ad he wave dsplaceme smulaeously we ca deerme hgh-order spaal dervaves based o he erpolao relaos ha ca be eplcly hadled. Ths s dffere from oher FD mehods ha use a dscree epresso o appromae he orgal wave equao. I he followg secos we shall furher dscuss he dfferece bewee he NADM ad La-Wedroff/compac/opmal FD mehods. The ma purpose of hs paper s o dscuss he effce mplemeao of he NADM. For hs we prese a modfed verso of NADM called opmal NADM (or ONADM bref). Our heorecal aalyses shows ha he ONADM ca mprove sgfcaly over he orgal oe umerous perspecves cludg umercal errors sorage spaces ad compuaoal cos. The accuracy of he ONAMD me also creases from -order of he orgal NADM o 4-order. Promsg umercal resuls verfy our heorecal coclusos. Meawhle we also dscuss he valdy of he -mes absorbg boudary codo developed he ced refereces (Hgdo 99; Yag e al. 003b). Basc Nearly Aalyc Dscree Mehod To llusrae he dervao of he NADM we brefly revew ad summare he ey 3

5 deas. For he -D case he wave equao ca be wre as σ ρ u + f = =3 () where subscrp aes he values of ad 3 ρ = ρ( ) s he medum desy u deoes he dsplaceme compoe he h dreco f s he force-source compoe he h dreco adσ s he sress esor. We use he same oao as ha he orgal NADM (Yag e al. 003a).e. = ( u ) T u y u P T T P P = P = P ad W =. Noe ha W s he me dervave of boh he dsplaceme compoe ad s frs-order spaal dervaves whch s called smply he velocy W. sg he above oao wh values a he me ad he rucaed Taylor seres epaso Yag e al. (003a) obaed he followg formulae + = + W ( ) + P ( ) P ( ) P () ( ) P + W = W + P + ( ) where deoes he me creme. P Obvously he more he seres erms ad (3) he more accuracy he compued values of 3 ( ) P + ( ) (3) 6 ( = 0 ) are ep formulae () + ad + W are. foruaely s mpossble o use he fe Taylor seres epaso. As a alerave we use he rucaed Taylor seres epasos () ad (3) ha have lower compuaoal accuracy regardg + ad + W because of he loss of he sesmc formao cluded he hgher-order erms he Taylor seres. To capure hese los sesmc formao ad furher crease he compuaoal accuracy accordg o he aalyss hough (Koddoh e al. 994) we ca appromaely corporae he los formao by roducg he erpolao fuco ad usg he coeco relaos. I order o acheve hs arge ad avod he mplc schemes ad cosg sorage derved from drec 4

6 ceral-dfferecg of he hgh-order me dervaves of ha cluded mplcly P (=) (equaos () ad (3)) we cover hese hgh-order me dervaves o he spaal dervaves + l W l + l ( + l 5 ) l for equaos () ad (3) ad ( + l 3) for equao (3). The rasform used here s smlar o he hgh-order FD mehods (Dabla 986) or he so-called La-Wedroff correco (LWC) mehods (Blach ad Robersso 997). However he way ha he NADM appromaes he hgh-order spaal dervaves s compleely dffere from he laer oes. I s also dffere from he opmal FD scheme based o a predcor-correcor mehod (Geller ad Taeuch 998; Taeuch ad Geller 000). The hgh-order FD mehods (Dabla 986) (or he so-called LWC) uses oly he wave dsplaceme o deerme he hgh-order spaal dervaves whle he NADM uses boh he wave dsplaceme ad s grades o deerme he hgh-order spaal dervaves. Compared wh hgh-order FD schemes he NADM eeds fewer grds o acheve he same accuracy. For eample he NADM usg 3 grd pos a dreco has 4-order space accuracy he same as ha of he hgh-order compac FD scheme (Dabla 986; Wag e al. 00) wh 5 grds. To be more specfc le us cosder he NADM based o he Taylor seres epaso o varables X ad Z. We use he followg erpolao fuco G( X Z) = M r= 0 ( X + Z ) r! r. (4) To deerme he hgh-order spaal dervaves we use formulae (4) a he grd po ( ) ad s egh eghborg odes: (- ) (+ ) ( -) ( +) (- -) (- +) (+ -) ad (+ +). Le us ae he grd po (- ) as a llusrag eample for he erpolao relaos we have [ ] G( 0) = X G( 0) = ( ) ad (5) where G Z ( 0) = ( ) s he spaal creme he -as dreco. From hese relaos we ca fd he aalyc epressos of + l ( + l 5 ) l erms of he dsplaceme ad s frs-order spaal paral dervaves a he mesh po ( ) ad s eghborg 5

7 grds (Yag e al. 003a). Followg he above-meoed scheme whe compug + he NADM uses o oly he values of he dsplaceme a he mesh po ( ) ad s eghborg grd pos bu also he values of he paral dervaves of wh respec o me ad space (see equaos () ad (3)). Ths allows he algorhm o capure more sesmc formao boh he fuco ad s paral dervaves. Therefore he NADM ca effecvely suppress he loss of formao cluded he hgher-order erms of he Taylor epaso furher resulg grea umercal accuracy ad less umercal dspersos. The roduco of he local coeco relaos (5) grealy mproves he couy ad dervably of he appromae fuco (because s a appromae varable durg daa processg) ad cosequely sables he NADM. Opmal Nearly-Aalyc Dscree Mehod Now le us ae a closer loo a formula () he NADM. To compue he velocy W a he grd-po ( ) formula (3) ha s ecessary for compug + we have o calculae + l W ( + l l 3) because of he volveme of he hgher-order paral dervaves formula (3). I our earler wor (Yag e al. 003a) he velocy W s calculaed by usg he followg bacward dfferece mehod: + l W + l + l l l l = + l 3 (6) Ths leads o several dsadvaages: () addoal compuaoal cos for compug W ; () lower me accuracy because of he use of low-order dfferece scheme (6) whch s oly -order me whle he space accuracy s 4-order for he NADM (Yag e al. 003a); (3) hgher sorage requreme. For eample we roduce a ew vecor wh 9 compoes ad hus we eed 8 arrays o sore + ad. Sce + W = / we also eed 8 arrays o sore W ad W. Sorg ( + l 3) requres arrays ad hus we eed 39 arrays o sore all he formao for compug + l W ( + l l + l l 3). Therefore he umber of oal arrays volved NADM s 57. I order o mprove he orgal NADM regardg he above-meoed pos we ry o reduce he addoal compuaoal cos ad sorage for compug he velocy 6

8 W. Acually by usg he values a me ad he Taylor seres epaso we ca oba he appromae soluo of a me as follows:. 4 ) ( 6 ) ( ) ( 4 3 P P P W + + = (7) Addg equao (7) ad equao () ogeher we oba he followg formula: P ) ( + = +. ) ( 4 P + (8) Because boh P ad P ca be epressed va ad s paral dervaves l l + (Yag e al. 003a) he opmal formula (8) does o volve he velocy W. Ths dcaes ha f we use formula (8) o compue + we do o eed o compue boh W ad P hus he sorage for + W W ad + l l ( + l 3) ca also be waved. The roduco of he opmal algorhm (8) reduces sgfcaly he compuaoal cos ad sorages whou sacrfcg he accuracy as he ONADM volves oly + ad. As a cosequece he umber of oal arrays requred for compuao he ONADM reduces o 7 whle he orgal NADM requres 57 arrays as saed earler. Ths saves he sorage space roughly 53 perce compared wh he orgal oe for he -D homogeeous case. Meawhle he me accuracy creases from -order of he orgal NADM o 4-order because here s o eed o compue he dervave l l W + he opmal formula (8). Moreover he ONADM eoys he same space accuracy as he NADM does as show he error aalyss he ced referece (Yag e al. 003a) ad formula (8). From formula (8) we ca fd ha he opmal mehod s que smlar o he so-called La-Wedroff or compac schemes or hgh-order opmal FD schemes where he orgal wave equao s used o cover he hgh-order error erms Taylor epasos o spaal dervaves ha ca be hadled eplcly hereby creasg he accuracy of he mehod sgfcaly (Dabla 986; Blach ad Robersso 997; Taeuch ad Geller 000). However he ONADM ad he NADM are dffere from 7

9 he above-meoed mehods he ways of appromag he hgher-order spaal dervaves. Boh he ONADM ad he NADM use he dsplaceme ad s grades o deerme he hgh-order spaal dervaves whle he above hgh-order FD mehods use oly he wave dsplaceme o deerme he hgh-order spaal dervaves ad hus hard o capure he sesmc formao characered by he grade of wave dsplaceme. I he ONADM we also employ he rucaed Taylor seres fuco (4) of spaal cremes ad he coeco relao (5) o deerme he hgh-order spaal dervaves a he grd-po ( ) he same echque as we used he orgal NADM. Therefore he ONADM eoy several desrable properes as he orgal NADM. I ca suppress effecvely he loss of sesmc formao cluded he hgher-order erms of he fe Taylor epaso leadg o grea umercal accuracy ad less umercal dspersos as verfed by our umercal epermes. Moreover as show our earler aalyss he ONADM s much more effce compuao ha he NADM as he compuaoal cos ad sorage requreme he ONADM s much less ha ha he NADM whle he me accuracy s mproved from -order o 4-order. Absorbg Boudary Codos Due o he lmao compuaoal source he compuaoal doma s arfcally resrced he umercal modelg of acousc ad elasc waves propagag ubouded meda. Ths resuls uphyscal reflecos derved from he arfcal boudares. As a remedy for hs ssue absorbg boudary codos (ABCs) are developed ad wdely used he umercal smulaos of wave moos ubouded meda o reduce he spurous umercal reflecos. I wha follows we dscuss he ssue of how o elmae spurous reflecos from arfcal boudares he ONADM for smulag sesmc propagao a ubouded doma. For hs we frs eed o choose a suable ABC o rea wh he boudary reflecos. Amog he mehods developed he perfecly mached layer (PML) echque orgally used he elecromagec wave equao s recely appled popularly acousc ad elasc wave smulaos (Chew ad Lu 996; Lu ad Tao 997). The PML echque alhough successful may applcaos as a maeral ABC requres some specal reames such as he sreched coordae rasformao ad/or splg he velocy feld ad roducg some ew varables for he elasc wave case whle he decoupled ABC does o requre o roduce ew varables ad o do specal reames. Therefore we choose he followg decoupled ABC a he boudary =0 (Hgdo 99) ha s mplemeed usg he sable based-ceer scheme preseed he ced referece (Yag e al. 003b) where = v absorbg mes. v ( ) u cosθ = 0 = 0 (9) s he wave velocy θ ( 0 θ < π / ) s he cde agle deoes 8

10 Numercal Errors -D case To es he accuracy of umercal resuls we frs choose he followg -D al problem: u u = (0a) α πf u( 0 ) = cos( ) α ad (0b) u(0 ) πf = πf s( ) (0c) α where f deoes he frequecy ad α s he wave velocy. Obvously he aalycal soluo for he al problem (0) s u ( ) = cos πf ( ) α. For comparso we use he so-called LWC (fourh-order compac scheme (Dabla 986)) for equao (0). I our calculao he parameers are chose he followg way. The grd umber N=00 he frequecy f=5 H ad he wave velocyα = 4000 m/sec. The relave error (E r ) s he rao of he RMS of he resdual ( u u )) ad he RMS of he ( eac soluo u ). Is eplc defo s as follows: ( N E r (%) = [ u u( )] 00. N = [ ( )] u = () 9

11 .0E-.0E- Relave Error (%). 0E-3.0E-4.0E-5 le le le 3.0E Tme Fgure. The relave errors of he La-Wedroff correco (le ) he NADM (le ) ad he ONADM (le 3) measured by E r (formula ()) are show a sem-log scale for he -D al problem (0). The spaal ad emporal cremes are 0 m ad 0-4 s respecvely..0e+.0e+0.0e-.0e-.0e-3 le le le 3.0E Tme (sec) Fgure. The relave errors of he La-Wedroff correco (le ) he NADM (le ) ad he ONADM (le 3) measured by E r (formula ()) are show a sem-log scale for he -D al problem (0). The spaal ad emporal cremes are 30 m ad 0-3 s respecvely. Fgures ad show he compuaoal resuls of he relave error E r a dffere mes for cases of dffere spaal ad me cremes where hree les of E r for he fourh-order LWC (le ) he NADM (le ) ad he ONADM (le 3) are show a sem-log scale. From Fgures ad we ca coclude ha he error roduced by he ONADM measured by E r s less ha hose of he NADM ad he fourh-order LWC. I Fgure he mamum E r of he ONADM s 0.063% whle he mamum relave errors 0

12 are 0.069% for he NADM ad % for he LWC respecvely. Comparg Fgure wh Fgure we ca see ha he relave errors crease wh he crease of he spaal ad me cremes. The mamum relave errors show Fgure are.5336% % ad 6.78% for he ONADM NADM ad he so-called LWC respecvely. -D case To furher compare he accuracy of umercal resuls we choose he followg -D al problem ha s smlar o ha of he -D case: u u u + = (a) α πf πf u( 0 ) = cos( cosθ sθ 0 α α u(0 ) πf πf = πf s( cosθ 0 α α 0 whereα s he velocy of he plae wave ad θ 0 ) ad sθ 0 (b) ) (c) s a cde agle a me = 0. We meo ha he aalycal soluo for he al problem ca be foud he ced referece (Yag e al. 003a). I our umercal epermes all he parameers are chose as follows: he grd umber N=00 he frequecy f=5 H α = 4000 m/sec ad θ = / 4 error -D case s defed by: 0 π. The relave E r (%) = = = N N [ u( )] N N = = [ u u( )] 00. (3).0E+.0E+0.0E-.0E-.0E-3.0E-4.0E-5.0E-6.0E-7.0E-8 le le le Tme Fgure 3. The relave errors of he secod-order FD (le ) he NADM (le ) ad he

13 ONADM (le 3) measured by E r are show a sem-log scale for he -D al problem (). The spaal ad emporal cremes are 0 m ad 0-4 s respecvely..0e+.0e+.0e+0.0e-.0e-.0e-3 le le le 3.0E Tme Fgure 4. The relave errors of he secod-order FD (le ) he NADM (le ) ad he ONADM (le 3) measured by E r are show a sem-log scale for he -D al problem (). The spaal ad emporal cremes are 30 m ad 50-4 s respecvely..0e+3.0e+.0e+.0e+0.0e-.0e- le le le 3.0E-3.0E Tme Fgure 5. The relave errors of he secod-order FD (le ) he NADM (le ) ad he ONADM (le 3) measured by E r are show a sem-log scale for he -D al problem (). The spaal ad emporal cremes are 40 m ad 0-3 s respecvely.

14 .0E+3.0E+.0E+.0E+0.0E-.0E-.0E-3 le le le 3.0E-4.0E Tme Fgure 6. The relave errors of he secod-order FD (le ) he NADM (le ) ad he ONADM (le 3) measured by E r are show a sem-log scale for he -D al problem (). The spaal ad emporal cremes are 50 m ad 0-4 s respecvely. Table. Comparso of mamum E r (%) for dffere cases ad dffere mehods. Case : Mehods Secod-order FD NADM ONADM h = 0m = 0 Case : h = 30m = s s Case 3: Case 4: h = 40m = 0 = 0 3 h = 50m 4 s s Fgures ad 6 are he curves of he error E r versus me correspodg o dffere spaal ad me sep ses where hree les of E r for he secod-order ceer scheme (le ) he NADM (le ) ad he ONADM (le 3) are show a sem-log scale. I hese Fgures he mamum relave errors for dffere cases are lsed Table. From hese error curves ad Table ( = = h ) we fd ha E r creases correspodg o he crease he me ad/or spaal cremes for all he hree mehods. Fgures 3 4 ad 5 show ha he ONADM has he hghes accuracy amog hem. I should be oed ha Fgure 6 preses wo decal curves for he case wh he same small me creme ( =0 4 s) ad he large spaal creme (h=50 m) 3

15 compued by he ONADM ad he NADM respecvely whle he relave error E r geeraed by he ONADM s smaller ha he error E r of he NADM for he case wh he fe spaal grds ad he same me creme as case 4. I shows ha for he specal case (case 4) he errors of boh he ONADM ad he NADM deped maly o he spaal creme. Ths cocdes wh he fac ha he ONADM ad he NADM have he same spaal accuracy. Furher from Fgures 3 4 ad 5 he cocluso ha he accuracy of he ONADM me s hgher ha ha of he NADM ca be verfed as he relave errors of he ONADM are smaller ha hose of he NADM for cases ad 3. Our umercal resuls are cosse wh wha we derved our heorecal aalyses. Wave-Feld Modelg I hs seco we compare he soropc wave-feld smulaos compued by he NADM ad he ONADM. I hs case equao () ca be specfed o u ρ u ρ u ρ y u u u = ( λ + µ ) + µ + ( λ + µ ) + f u y u y = µ ( + ) + f u u u = ( λ + µ ) + µ + ( λ + µ ) + f whereλ ad µ are Lamé cosas. 3 (4a) (4b) (4c) I he frs model (model ) we choose λ =.75 (GPa) µ =6.5 (GPa) ad he desy ρ =.g / cm 3. The compuaoal doma s m m. We choose he spaal creme h=50 m he me creme = 3 0 sec ad he umber of grd pos The source s a eplosve source ha s a coordae ) = (4.95 m 4.95 m) ad has a Rcer wavele wh a pea frequecy ( s s 3 of =5 H. The me varao of he source fuco s s f 0 ( πf 0 ) ep(-4 π f /6 ) 0 (Zahrad e al. 993). Fgures 7 ad 8 are he wave-feld sapshos a =.4 sec geeraed by he orgal NADM ad he ONADM respecvely. From he hree-compoe sapshos we ca see ha hese wo wave-feld resuls are decal ad he P- ad SV-waves preseed he horoal compoe u (Fgs. 7a ad 8a) ad vercal compoe u (Fgs. 7c ad 8c) are very clear. The wave-feld sapshos geeraed by boh he NADM ad he ONADM show ha he ONADM has less umercal dspersos eve f he space creme chose s 50 m whou ay addoal reames. The compuao o geerae hree-compoe 4

16 resuls Fgures 7 ad 8 were performed o a Peum 4 PC wh 8MB RAM. The ONADM oo abou 3.4 m o geerae Fgure 8 whle he orgal NADM oo abou 5 m o geerae Fgure 7. Ths cocdes wh our heorecal cocluso ha he ONADM ca save he compuaoal cos abou 3% compared wh he NADM. 0 Dsace (Km) 9.95 (a) (b) (c) 9.95 Deph (m) Deph (m) Deph (m) 0 Fgure 7. Sapshos of sesmc wave felds for hree-compoes a me.4 sec soropc meda geeraed by NADM for (a) u compoe (b) u compoe (c) y u compoe. For furher comparso we prese he waveforms compued usg he NADM he ONADM ad he secod-order FD. I hs model we use he spaal sep se = = 0 m he me creme =0 3 s ad he grd pos The 5

17 compuaoal doma s chose as m m. The source s locaed a ) = ( m 3 m) ad he recever s a ( ) = (3 m 3 m). The res ( s s model parameers are same as hose model. 0 Dsace (Km) 9.95 (a) (b) 9.95 Deph (m) Deph (m) 0 (c) 9.95 Deph (m) 0 Fgure 8. Sapshos of sesmc wave felds for hree-compoes a me.4 sec soropc meda geeraed by he ONADM for (a) u compoe (b) u compoe y (c) u compoe. Fgure 9 preses he waveforms of hree compoes he soropc medum geeraed by he NADM he ONADM ad he secod-order FD. Fgures 9a ad 9b show ha he waveforms geeraed by he ONADM are decal wh hose compued by he NADM ecepg for margal dfferece behd wave peas. However he compuaoal mes requred by he ONADM ad he NADM are dffere. I oo he 6

18 ONADM abou 5 mues CP me o geerae Fgure 9b whle geerag Fgure 9a wh he same codos coss he NADM abou 37 mues. Ths also cofrms our heorecal coclusos. Meawhle comparg Fgures 9a ad 9b wh Fgure 9c we ca observe ha boh he ONADM ad he NADM have farly less umercal dspersos whle he secod-order FD suffers from serously umercal dspersos. (a) NADM u u y u s (b) ONADM u u y u s (c) Secod-order FD u u y u s 7

19 Fgure 9. A comparso of he hree-compoe waveforms a homogeeous soropc medum. The syhec sesmograms (a) (b) ad (c) are geeraed by he NADM he ONADM ad he secod-order FD respecvely. 0 Dsace (m).98 (a).98 Deph (m) 0 (b).98 Deph (m) 0 (c).98 Deph (m) 0 Fgure 0. Sapshos of sesmc wave felds for hree compoes a me 0.6 sec for he u ad u compoes ad a me 0.95 sec for he u compoe soropc meda geeraed by he ONADM wh he 4-mes absorbg boudary codo. The goal of he fal eample s o vesgae he valdy of he -mes decoupled absorbg codo (9) whle corporag he -mes absorbg boudary codo he ONADM. I hs model we choose he absorbg boudary codo wh orders of =4 equao (9). The compuaoal doma s chose as 0.98 m 0.98 m. The source s locaed a (0.66 m.4 m). The compuaoal parameers are chose by h=0 m = y sec he umber of grd pos 8

20 50 50 ad oher parameers are he same as hose used model. Fgure 0 coas hree-compoe sapshos geeraed by he ONADM wh he 4-mes absorbg boudary codo. The wavefeld sapshos (Fgs. 0a ad 0c) ad he sapsho show Fgure 0b are ae a propagao me 0.6 sec ad 0.95 sec respecvely. From he wavefeld sapshos we ca observe ha he reflecos of P- ad S-waves from he arfcal boudares are effecvely absorbed ad he compuao s sable. Ths llusraes ha corporag he 4-mes absorbg boudary codo he ONADM s effecve. Coclusos We opmally modfy he NADM developed by Yag e al. (003a) for solvg acousc ad elasc wave equaos. The opmal mehod eoys 4-order accuracy boh space ad me whle he NADM has oly -order accuracy me ad 4-order accuracy space. Ths cocluso s also verfed by our umercal epermes of compug he relave error E r va formula () for -D case ad formula (3) for -D case. Boh he umercal errors produced by he ONADM ad NADM are smaller ha ha based o he fourh-order La-Wedroff correco (Dabla 986) ad he coveoal secod-order FD mehod. Compared wh he orgal NADM he ONADM does o volve o he so-called velocy W. Therefore ca subsaally save sorage reduce he compuaoal cos ad mprove grealy he umercal accuracy. Our heorecal coclusos are verfed by promsg umercal resuls whch demosrae ha he opmal mehod ca save compuaoal coss-by abou 3% ad sorage space-by abou 53% as compared wh he orgal NADM. Wave-feld modelg also llusraes ha he opmal mehod ca effecvely suppress umercal dspersos whe oo-coarse compuao grds are used. Ths dcaes ha he opmal mehod eables wave propagao o be smulaed large-scale models hrough usg he coarse compuao grds. Besdes he umercal vesgao of boudary codos shows ha he decouplg 4-mes absorbg boudary codo developed by Hgdo (99) ad dscreed by Yag e al. (003b) wors well. Acowledgmes We ha Joha O.A. Roberso for valuable commes ha grealy corbued o mprovg he mauscrp. We also ha Dr. Fred F. Poll (Assocae Edor) for helpful commes. Ths wor was suppored by he Naoal Naural Sceces Foudao of Cha (Gra No ) ad he Foudao of Tsghua versy (Gra No. JC00038). Par of hs wor was doe whe he frs auhor vsed Deparme of Compug ad Sofware McMaser versy. Ths vs was parally suppored by he MITACS proec ``New eror po mehods ad sofware for cove coc lear opmao wh applcaos o solvg VLSI crcu layou problems. The research of he las auhor was parally suppored by he gra #RPG of he Naoal Sceces ad Egeerg Research Coucl of Caada (NSERC) a PREA award from Oaro ad he above-meoed MITACS proec. 9

21 Refereces Alford R. M. K. R. Kelly ad D. M. Boore (974). Accuracy of fe-dfferece modelg of he acousc wave equao Geophyscs Blach J. O. ad J. O. A. Robersso (997). A modfed La-Wedroff correco for wave propagao meda descrbed by Zeer elemes Geophys. J. I Chew W. C. ad Q. H. Lu (996). Perfecly mached layers for elasodyamcs: A ew absorbg boudary codo J. Compuaoal Acous Dabla M. A. (986). The applcao of hgh-order dfferecg o he scalar wave equao Geophyscs Fe T. ad K. Larer (995). Elmao of umercal dsperso fe-dfferece modelg ad mgrao by flu-correced raspor Geophyscs Geller R. J. ad N. Taeuch (998). Opmally accurae secod-order me-doma fe dfferece scheme for he elasc equao of moo: oe-dmesoal case Geophys. J. I Hgdo R. L. (99). Absorbg boudary codos for elasc waves Geophyscs Igel H. P. Mora ad B. Rolle (995). Asoropc wave propagao hrough fe-dfferece grds Geophyscs Kelly K. R. Ward S. Treel ad R. Alford (976). Syhec sesmograms: a fe-dfferece approach Geophyscs 4-7. Koddoh Y. Y. Hosaa ad K. Ish (994). Kerel opmum early aalycal dscreao algorhm appled o parabolc ad hyperbolc equaos Compuers Mah. Appl Kosloff D. ad E. Baysal (98). Forward modelg by a Fourer mehod Geophyscs Lu Q. H. ad J. Tao (997). The perfecly mached layer for acousc waves absorpve meda J. Acous. Soc. Am Mua H. R. J. Geller ad N. Taeuch (000). Comparso of accuracy ad effcecy of me-doma schemes for calculag syhec sesmograms Phys. Earh Plae. I Robersso J. O. A. J. O. Blach ad W. W. Symes (994). Vscoelasc fe-dfferece modelg Geophyscs Taeuch N. ad R. J. Geller (000). Opmally accurae secod order me-doma fe dfferece scheme for compug syhec sesmograms -D ad 3-D meda Phys. Earh Plae. I Vreu J. (986). P-SV wave propagao heerogeeous meda: Velocy-sress fe-dfferece mehod Geophyscs Wag S. Q. D. H. Yag ad K. D. Yag (00). Compac fe dfferece scheme for elasc equaos J. Tsghua v. (Sc. & Tech.) ( Chese) Yag D. H. E. Lu Z. J. Zhag ad J. Teg (00). Fe-dfferece modellg wo-dmesoal asoropc meda usg a flu-correced raspor echque Geophys. J. I Yag D. H. J. W. Teg Z. J. Zhag ad E. Lu (003a). A early-aalyc dscree mehod for acousc ad elasc wave equaos asoropc meda Bull. Sesm. 0

22 Soc. Am Yag D. H. S. Q. Wag Z. J. Zhag ad J. W. Teg (003b). -mes absorbg boudary codos for compac fe dfferece modelg of acousc ad elasc wave propagao he -D TI Medum Bull. Sesm. Soc. Am Zahrad J. P. Moco ad T. Hro (993). Tesg four elasc fe-dfferece schemes for behavor a dscoues Bull. Sesm. Soc. Am Zhag Z. J. G. J. Wag ad J. M. Harrs (999). Mul-compoe wave-feld smulao vscous eesvely dlaacy asoropc meda Phys. Earh Plae. Ier