Implementing a Convolutional Perfectly Matched Layer in a finite-difference code for the simulation of seismic wave propagation in a 2D elastic medium

Size: px

Start display at page:

Download "Implementing a Convolutional Perfectly Matched Layer in a finite-difference code for the simulation of seismic wave propagation in a 2D elastic medium"

Roy Johnson
5 years ago
Views:

1 Implemenng a Conoluonal Perfecl Mached Laer n a fne-dfference code for he smulaon of sesmc wae propagaon n a D elasc medum Progress Repor BRGM/RP-559-FR December 7

3 Implemenng a Conoluonal Perfecl Mached Laer n a fne-dfference code for he smulaon of sesmc wae propagaon n a D elasc medum Progress Repor BRGM/RP-559-FR December 7 Sud carred ou as par of Research aces - BRGM 7 PDR7ARN4 Duceller A., Aoch H. Checked b: Name: Phlppe Jousse Dae: Sgnaure: Approed b: Name: Hormoz Modaress Dae: Sgnaure: BRGM's qual managemen ssem s cerfed ISO 9: b AFAQ IM 3 ANG Aprl 5

4 Kewords: Sesmc wae propagaon, Absorbng condon, Perfecl Mached Laer, Conoluonal Perfecl Mached Laer, Non-reflecng condons, Fne-dfference. In bblograph, hs repor should be ced as follows: Duceller A., Aoch H. (7) Implemenng a Conoluonal Perfecl Mached Laer n a fne-dfference code for he smulaon of sesmc wae propagaon n a D elasc medum, BRGM echncal repor, BRGM/RP-559-FR. BRGM, 5. No par of hs documen ma be reproduced whou he pror permsson of BRGM.

5 Implemenng a CPML n a D fne-dfference code for he smulaon of sesmc wae propagaon Snopss One of he mos popular mehods o smulae numercall he sesmc wae propagaon n an elasc meda s he fne-dfference mehod. In he cone of numercal modellng n unbounded meda, he wae needs o be absorbed a he arfcal boundares of he compuaonal doman and herefore, s necessar o defne non-reflecng condons a hese boundares o mmc an unbounded medum. The Perfecl Mached Laer (PML) condon has he remarkable proper of hang a zero reflecon coeffcen for all angles of ncdence and all frequences before dscrezaon and has become wdel used (e.g. Collno and Tsogka, ). Howeer, hs reflecon coeffcen s no zero anmore afer he dscrezaon and becomes een er large a grazng ncdence. Therefore, an mproed erson of he PML condon has been deeloped: he Conoluonal Perfecl Mached Laer (CPML) condon (e.g. Komasch and Marn, 7). The am of hs projec s o mplemen hese wo non-reflecng condons n a D fne-dfference compuaonal code for an elasc medum. We wll frs eplan he heorecal formulaon of he PML and CPML condons. Then, we wll epose he srucure of our compuaonal code. Fnall, we wll presen he resuls of some numercal ess performed o erf he effcenc of he mehod. BRGM/RP-559-FR Progress repor 3

7 Implemenng a CPML n D fne-dfference code for he smulaon of sesmc wae propagaon Conens. Inroducon...7. The CPML formulaon CLASSICAL PML FORMULATION CPML FORMULATION Implemenaon n he D code GRID CELL BOUNDARY CONDITIONS SOURCE FONCTION STRUCTURE OF THE CODE Numercal ess PML CPML Concluson References...3 BRGM/RP-559-FR Progress repor 5

8 Implemenng a CPML n a D fne-dfference code for he smulaon of sesmc wae propagaon Ls of llusraons Fgure - Elemenar grd cell of he wo dmensonnal saggered spaal fnedfference mehod... 7 Fgure - Source me funcon mposed a pon source... Fgure 3 Tme eoluon of he and componens of he wo-dmensonal eloc ecor a he frs, second and hrd saons of he numercal soluon wh classcal PML boundares for he hn slce and he large slce Fgure 4 Snapshos of he and componens of he wo-dmensonal eloc ecor of he numercal soluon wh classcal PML boundares for he hn slce, a me sep 3, 5, 7, 9, and Fgure 5 Tme eoluon of he and componens of he wo-dmensonal eloc ecor a he frs, second and hrd saons of he numercal soluons wh CPML boundares for he hn slce and he large slce Fgure 6 Snapshos of he and componens of he wo-dmensonnal eloc ecor of he numercal soluon wh CPML boundares for he hn slce, a me sep 3, 5, 7, 9, and BRGM/RP-559-FR Progress repor

9 Implemenng a CPML n D fne-dfference code for he smulaon of sesmc wae propagaon. Inroducon One of he mos popular mehods o smulae numercall he sesmc wae propagaon n an elasc meda s he fne-dfference mehod. In he cone of numercal modellng n unbounded meda, he wae needs o be absorbed a he arfcal boundares of he compuaonal doman and herefore, s necessar o defne non-reflecng condons a hese boundares o mmc an unbounded medum. The Perfecl Mached Laer (PML) condon has he remarkable proper of hang a zero reflecon coeffcen for all angles of ncdence and all frequences before dscrezaon and has become wdel used (e.g. Collno and Tsogka, ). Howeer, hs reflecon coeffcen s no zero anmore afer he dscrezaon and becomes een er large a grazng ncdence. Therefore, an mproed erson of he PML condon has been deeloped: he Conoluonal Perfecl Mached Laer (CPML) condon (e.g. Komasch and Marn, 7). The am of hs projec s o mplemen hese wo non-reflecng condons n a D fne-dfference compuaonal code for an elasc medum. We wll frs eplan he heorecal formulaon of he PML and CPML condons. Then, we wll epose he srucure of our compuaonal code. Fnall, we wll presen he resuls of some numercal ess performed o erf he effcenc of he mehod. BRGM/RP-559-FR Progress repor 7

11 Implemenng a CPML n D fne-dfference code for he smulaon of sesmc wae propagaon. The CPML formulaon One of he mos popular mehods o smulae numercall he sesmc wae propagaon n an elasc meda s he fne-dfference mehod. In he cone of numercal modellng n unbounded meda, he wae needs o be absorbed a he arfcal boundares of he compuaonal doman and herefore, s necessar o defne non-reflecng condons a hese boundares o mmc an unbounded medum. The Perfecl Mached Laer (PML) condon has he remarkable proper of hang a zero reflecon coeffcen for all angles of ncdence and all frequences before dscrezaon and has become wdel used (e.g. Collno and Tsogka, ). Howeer, hs reflecon coeffcen s no zero anmore afer he dscrezaon and becomes een er large a grazng ncdence. Therefore, an mproed erson of he PML condon has been deeloped: he Conoluonal Perfecl Mached Laer (CPML) condon (Komasch and Marn, 7). We wll frs eplan he classcal PML formulaon, and hen we wll nroduce he CPML formulaon... CLASSICAL PML FORMULATION The dfferenal form of he D elasc wae equaon can be wren as: ρ ρ for he eloces ( and ) and: λ µ λ µ BRGM/RP-559-FR Progress repor 9

12 Implemenng a CPML n a D fne-dfference code for he smulaon of sesmc wae propagaon µ for he sress (, and ), where ρ s dens and λ and µ are elasc coeffcens. In he parcular case of a homogenous medum, hs equaon has plane wae r r r Aep k. ω. soluons of he form ( ( ) Le us consder a regular doman locaed n. In order o aod reflecons a, we need o defne an absorbng laer locaed n >. Ths laer wll be an absorbng laer f he amplude of he wae decreases eponenall n he drecon for >. To defne hs laer, we wll defne a new wae equaon, such ha s soluon s a plane wae n he regular doman and an eponenall-decang plane wae n he PML. We nroduce a new comple coordnae ~. Thus, he PML can be ewed as an analcal connuaon of he real coordnaes n he comple space (Collno and Tsogka, ). The coordnae ~ s defned b: ~ ω ( ) d ( s) ds Equaon wh d () beng a dumpng profle n he PML regon, such ha d nsde he regular doman and d > n he PML. Equaon ges us upon dfferenang: ω ~ ϖ d s Equaon wheres ω d ω d ω We wll now change he orgnal wae equaon wren n erms of arable and no a new wae equaon wren n erms of arables ~ and. We frs rewre he elasc wae equaon n he frequenc doman: ωρ ωρ BRGM/RP-559-FR Progress repor

13 Implemenng a CPML n D fne-dfference code for he smulaon of sesmc wae propagaon BRGM/RP-559-FR Progress repor for he eloces and: µ λ ω µ λ ω µ ω for he sress. We hen replace he wae equaon wren n erms of wh a generalzed wae equaon wren n erms of ~. Insde he regular doman, boh equaons are dencal because d. ~ ωρ ~ ωρ for he eloces and: ~ ~ µ λ ω µ λ ω ~ ~ µ ω for he sress. In he PML, he new wae equaon has eponenall-decang plane wae soluons of he form: ( ) ( ) ( ) ( ) ( ) ds s d k k A k k A ep. ep ~ ep ω ω ω r r r r

14 Implemenng a CPML n a D fne-dfference code for he smulaon of sesmc wae propagaon BRGM/RP-559-FR Progress repor n he drecon. Usng Equaon : s ~, we rewre he wae equaon n he whole doman n erms of raher han ~ (n he regular doman, s ): s ωρ s ωρ for he eloces and: s s µ λ ω s µ λ ω s µ ω for he sress. The eloc and sress felds are hen spl no wo pars, and, such ha: s ωρ ωρ s ωρ ϖρ

15 Implemenng a CPML n D fne-dfference code for he smulaon of sesmc wae propagaon BRGM/RP-559-FR Progress repor 3 for he eloces and: s s µ λ ω λ ω s λ ω µ λ ω s µ ω µ ϖ for he sress. Conerng back o he me doman, we fnall ge: d ρ ρ d ρ ρ for he eloces and: d λ µ

16 Implemenng a CPML n a D fne-dfference code for he smulaon of sesmc wae propagaon λ d λ λ µ d µ µ for he sress. Ths new wae equaon perms o hae an eponenall-decang plane wae soluon n he PML laer ( > ). Therefore he boundar a appears o be a nonreflecng boundar. We hen use he same echnque for he hree oher boundares of he regular doman. Ths formulaon perms o hae numercal soluons wh reduced arfcal reflecons bu s effcenc becomes poor a grazng ncdences. The CPML formulaon was deeloped o address hs ssue... CPML FORMULATION The man dea of he CPML formulaon s o choose a more general defnon of s b nroducng new arables α and κ : s d κ α ω Equaon 3 As hs epresson depends on frequenc, when we go back n he me doman, we wll ge a me conoluon on each modfed spaal derae. Denong s () he nerse Fourer ransform of s, we can wre: 4 BRGM/RP-559-FR Progress repor

17 Implemenng a CPML n D fne-dfference code for he smulaon of sesmc wae propagaon BRGM/RP-559-FR Progress repor 5 ( ) s * ~ Equaon 4 * beng a conoluon operaor. We hae from Equaon 3: ( ) ( ) ( ) ( ) d H d s α κ κ κ δ ep ) ( Equaon 5 where δ() and H() are he Drac and Heasde funcons. If we denoe: ( ) ( ) ( ) ( ) d H d α κ κ ζ ep Equaon 6 we can see from Equaons 4, 5 and 6 ha: ( ) * ~ ζ κ Le us denoe n ϕ he conoluon erm compued a me n, wh beng he me sep: ( ) ( ) n n n d * τ τ ς ζ ϕ τ Equaon 7 Equaon 7 ges us b dscrezaon: ( ) ( ) ( ) ( ) ( ) n m m n n n m m m m n n n m m m n n m Z d d ϕ τ τ ς ϕ τ τ ς ϕ τ Equaon 8 wh: ( ) ( ) ( ) m m d m Z τ τ ς Equaon 9 From Equaons 6 and 9, we oban:

18 Implemenng a CPML n a D fne-dfference code for he smulaon of sesmc wae propagaon d ( m ) Z ( m) ep ( ( d ) ) m κ α τdτ a κ ( b ) m wh a d ( ) ( b ) and b ep ( ( d κ α ) ) d κ α κ We consder hen φ as a memor arable whose me eoluon s goerned a each sep b: ϕ n b ϕ n a n From a praccal pon of ew, o mplemen he CPML echnque n a compuaonal code, we jus need o replace he spaal derae b: ~ κ ϕ and updae φ a each me sep. 6 BRGM/RP-559-FR Progress repor

19 Implemenng a CPML n D fne-dfference code for he smulaon of sesmc wae propagaon 3. Implemenaon n he D code 3.. GRID CELL We use a classcal second-order saggered grd n space and me (Madaraga, 976, and Vreu, 986): Fgure - Elemenar grd cell of he wo dmensonnal saggered spaal fne-dfference mehod of Madaraga (976) and Vreu (986) used classcall o dscreze he equaons of elasodnamcs. 3.. BOUNDARY CONDITIONS We pu non-reflecng boundares on he four sdes of he compuaonal doman (Komasch and Marn, 7) so as o smulae wae propagaon n nfne elasc medum. In hs case, he boundar condons are: (,. ) ( mn ma,.) (., jmn ) (., j ) ma and (,. ) ( mn ma,.) (., jmn ) (., j ) ma BRGM/RP-559-FR Progress repor 7

20 Implemenng a CPML n a D fne-dfference code for he smulaon of sesmc wae propagaon mn, ma, j mn and j ma beng he lms of he regular doman and beng he number of cells n he (C)PML laer SOURCE FONCTION The source s gen b a eloc ecor, he same as Komasch and Marn (7) for aldaon. A he pon source, we jus add: ( ) d ρ ( α ) a ( ) ep a ( ) sn f * Equaon o ( source,j source ) and: ( α ) a ( ) ep a ( ) ( ) d ρ cos f * Equaon o ( source ½, j source ½), beng he me, α, f, a and beng source parameers gen n he code and ρ beng he dens STRUCTURE OF THE CODE The srucure of he compuaonal code s he followng: Inalzaons Calculaon of he alues of d (), α (), κ (), d (j), α (j) and κ (j) defned n Equaon 3 Begnnng of he me loop Tme (n/) (*) 8 BRGM/RP-559-FR Progress repor

21 Implemenng a CPML n D fne-dfference code for he smulaon of sesmc wae propagaon for j { for { f (,j n he CPML) { Calculaon of and n he CPML } else { Calculaon of and n he regular doman } } } for j { for { f (,j n he CPML) { Calculaon of n he CPML } else { Calculaon of n he regular doman } } } Tme (n) for j { for { f (,j n he CPML) { Calculaon of n he CPML } else { Calculaon of n he regular doman } } } for j { BRGM/RP-559-FR Progress repor 9

22 Implemenng a CPML n a D fne-dfference code for he smulaon of sesmc wae propagaon for { f (,j n he CPML) { Calculaon of n he CPML } else { Calculaon of n he regular doman } } } Addng a eloc ecor a he source Imposng he boundar condons Sang he alues n fles for he sesmograms and he snapshos Incremen of n and reurn o (*) BRGM/RP-559-FR Progress repor

23 Implemenng a CPML n D fne-dfference code for he smulaon of sesmc wae propagaon 4. Numercal ess 4.. PML To es our compuaonal code, we use he same approach as Komasch and Marn (7). We consder wo models: he sze of he frs model s 8*64 meers, represenng a doman much longer han wde (.e., a hn slce). In hs model, he wae wll propagae a grazng ncdence along he edge, whch ma cause spurous reflecons. To hghlgh hese reflecons, we wll consder a second, larger model (34*64 meers), whch boundares are far enough from he source o aod an spurous reflecons, and we wll compare he resuls of he wo models. Ths hn model s dscrezed usng a grd comprsng 8 pons * 64 pons. The large model s dscrezed usng a grd comprsng 34 * 64 pons. The sde of a grd cell s m. The compressonal wae speed s Vp 33 m.s -, he shear wae speed s Vs Vp/ m.s - and he dens s ρ 8 kg.m -3. The me sep s. s and we perform he smulaon for me seps. The pon source s a eloc ecor locaed n source 6 m and source 48 m. The me funcon s gen b Equaons and. The parameers for he source are: α 35, f 7, a π*f d,. / f d and f d 7 Hz Source funcon 5 source (m/s) 5-5,8,36,54,7,9,8,6,44,6,8,98,6,34,5,7,88 3,6 3,4 3,4 3,6 3,78 3, me (s) Fgure - Source me funcon mposed a pon source The saons were we compue sesmograms are locaed a he followng pons: BRGM/RP-559-FR Progress repor

24 Implemenng a CPML n a D fne-dfference code for he smulaon of sesmc wae propagaon Saon number X coordnae Y coordnae Saon s locaed a he mddle of he model where here s no spurous reflecon. Saon and 3 are locaed along he edge a grazng and er grazng ncdence, where spurous reflecons could dsor he sesmograms. For he dumpng coeffcen n he PML, we use he followng profle: d ( ) d L N where s he dsance beween he pon where d s compued and he begnnng of he PML laer, and L s he hckness of he PML: L *, N and: d ( N ) log( R ) p c wh Rc.. L Absorbng laers of hckness grd pons are mplemened on he four sdes of he model. The resuls are shown hereafer n Fgures 3 and 4. Saon - V - PML Saon - V - PML,5,8,5,6,4 (m/s) -,5 - Thn slce Large slce (m/s), Thn slce Large slce -,5 -, - -,4 -,5 -,6,,4,6,8,3,4,44,65,85,6,7,47,68,88 3,9 3,3 3,5 3,7 3,9 me (s),,4,6,8,3,4,44,65,85,6,7,47,68,88 3,9 3,3 3,5 3,7 3,9 me (s) BRGM/RP-559-FR Progress repor

25 Implemenng a CPML n D fne-dfference code for he smulaon of sesmc wae propagaon Saon - V - PML Saon - V - PML,5,8,5,6,4 (m/s) -,5 - Thn slce Large slce (m/s), c Thn slce Large slce -,5 -, - -,4 -,5 -,6,,4,6,8,3,4,44,65,85,6,7,47,68,88 3,9 3,3 3,5 3,7 3,9 me (s),,4,6,8,3,4,44,65,85,6,7,47,68,88 3,9 3,3 3,5 3,7 3,9 me (s) Saon 3 - V -PML Saon 3 - V - PML,8,5,6,4 (m/s) -,5 Thn slce Large slce (m/s), Thn slce Large slce -, - -,4 -,5 -,6,,4,6,8,3,4,44,65,85,6,7,47,68,88 3,9 3,3 3,5 3,7 3,9 me (s),,4,6,8,3,4,44,65,85,6,7,47,68,88 3,9 3,3 3,5 3,7 3,9 me (s) Fgure 3 Tme eoluon of he (lef) and (rgh) componens of he wo-dmensonal eloc ecor a he frs (op), second (mddle) and hrd (boom) saons of he numercal soluon wh classcal PML boundares for he hn slce (blue) and he large slce (pnk). A he frs saon, relael far from he begnnng of he PML laer and wh non-grazng ncdence, he agreemen s almos perfec. A he second saon, spurous oscllaons, n parcular for he S-wae of he componen, sar o appear. A he hrd receer, he oscllaons become large, he P-wae s no correcl calculaed and he shape of he S-wae s compleel dsored. BRGM/RP-559-FR Progress repor 3

26 Implemenng a CPML n a D fne-dfference code for he smulaon of sesmc wae propagaon Fgure 4 Snapshos of he and componens of he wo-dmensonal eloc ecor of he numercal soluon wh classcal PML boundares for he hn slce, a me sep 3, 5, 7 (lef, op o boom), 9, and 3 (rgh, op o boom). The hree dos ndcae he posons of he saons a whch he sesmograms represened n Fgure are recorded. Spurous waes appear a grazng ncdence along he edge of he model and send spurous energ back no he man doman. We fnd ha some large reflecon waes reman especall a saons and 3. In fac, he snapshos show some srong reflecon a he boundar closes o he source n he begnnng. We also obsere he oscllaon socked n he PML laer, whch propagaes parallel o he boundar. Ths s he problem of he PML preousl poned ou (e.g. Komasch and Marn, 7). 4 BRGM/RP-559-FR Progress repor

27 Implemenng a CPML n D fne-dfference code for he smulaon of sesmc wae propagaon 4.. CPML To es he CPML, we use he same approach as for he PML. We choose o make α ar n a lnear fashon beween mamal alue α ma a he begnnng of he CPML and zero a s op. We ake α ma π*f d. We choose κ consan, equal o. The resuls are hereafer n Fgures 5 and 6. Saon - V - CPML Saon - V - CPML,5,8,5,6,4 (m/s) -,5 - Thn slce Large slce (m/s), Thn slce Large slce -,5 -, - -,4 -,5 -,6,,4,6,8,3,4,44,65,85,6,7,47,68,88 3,9 3,3 3,5 3,7 3,9 me (s),,4,6,8,3,4,44,65,85,6,7,47,68,88 3,9 3,3 3,5 3,7 3,9 me (s) Saon - V - PML Saon - V - PML,5,8,5,6,4 (m/s) -,5 - Thn slce Large slce (m/s), c Thn slce Large slce -,5 -, - -,4 -,5 -,6,,4,6,8,3,4,44,65,85,6,7,47,68,88 3,9 3,3 3,5 3,7 3,9 me (s),,4,6,8,3,4,44,65,85,6,7,47,68,88 3,9 3,3 3,5 3,7 3,9 me (s) Saon 3 - V - CPML Saon 3 - V - CPML,8,5,6,4 (m/s) -,5 Thn slce Large slce (m/s), Thn slce Large slce -, - -,4 -,5 -,6,,4,6,8,3,4,44,65,85,6,7,47,68,88 3,9 3,3 3,5 3,7 3,9 me (s),,4,6,8,3,4,44,65,85,6,7,47,68,88 3,9 3,3 3,5 3,7 3,9 me (s) BRGM/RP-559-FR Progress repor 5

28 Implemenng a CPML n a D fne-dfference code for he smulaon of sesmc wae propagaon Fgure 5 Tme eoluon of he (lef) and (rgh) componens of he wo-dmensonal eloc ecor a he frs (op), second (mddle) and hrd (boom) saons of he numercal soluons wh CPML boundares for he hn slce (blue) and he large slce (pnk). A he frs saon, relael far from he begnnng of he CPML laer and wh non-grazng ncdence, he agreemen s almos prefec. A he second saon, wh grazng ncdence and close o he begnnng of he CPML laer, he agreemen remans ecellen. A he hrd receer, n he dffcul case of er grazng ncdence, of a long dsance of propagaon and of a saon locaed close o he begnnng of he CPML laer, he agreemen remans er sasfacor. 6 BRGM/RP-559-FR Progress repor

29 Implemenng a CPML n D fne-dfference code for he smulaon of sesmc wae propagaon Fgure 6 Snapshos of he and componens of he wo-dmensonnal eloc ecor of he numercal soluon wh CPML boundares for he hn slce, a me sep 3, 5, 7 (lef, op o boom), 9, and 3 (rgh, op o boom). The hree dos ndcae he posons of he saons a whch he sesmograms represened n Fgure 4 are recorded. No spurous wae of sgnfcan amplude s sble, een a grazng ncdence. In he CPML case, we are able o ge he dencal resuls as Komasch and Marn (7) well mproed wh respec o he preous ones (PML). We do no hae an more he arfcal wae reflecon een n he snapshos. We confrm here ha he CPML absorbng condon works much beer. BRGM/RP-559-FR Progress repor 7

31 Implemenng a CPML n D fne-dfference code for he smulaon of sesmc wae propagaon 5. Concluson The behaour of he Perfecl mached Laer a grazng ncdence s well mproed wh he new Conoluonnal PML approach. I can be useful when sudng wae propagaon n hn slces or when he source s locaed near he edge of he model. The cos of CPML n memor sorage s smlar o he classcal PML. BRGM/RP-559-FR Progress repor 9

33 Implemenng a CPML n D fne-dfference code for he smulaon of sesmc wae propagaon 6. References Collno F., Tsogka C. () - Applcaon of he PML absorbng laer model o he lnear elasodnamc problem n ansoropc heerogeneous meda. Geophscs, 66(), Graes R.W. (996) - Smulang Sesmc Wae propagaon n 3D Elasc Meda Usng Saggered-Grd Fne Dfferences. Bull. Sesm. Soc. Am., 86, 9-6. Komasch D., Marn R. (7) - An unspl conoluonal Perfecl Mached Laer mproed a grazng ncdence for he sesmc wae equaon. Geophscs, 7, SM55- SM67. Madaraga R. (976) - Dnamcs of an epandng crcular faul. Bull. Sesm. Soc. Am., 65, Vreu J. (986) - P-SV wae propagaon n heerogeneous meda: eloc-sress fne-dfference mehod. Geophscs, 5, BRGM/RP-559-FR Progress repor 3

36 Scenfc and Techncal Cenre Deelopmen Plannng and Naural Rsks Dson 3, aenue Claude-Gullemn - BP Orléans Cede France Tel.: 33 ()

Implementing a Convolutional Perfectly Matched Layer in a finite-difference code for the simulation of seismic wave propagation in a 3D elastic medium

Implementing a Convolutional Perfectly Matched Layer in a finite-difference code for the simulation of seismic wave propagation in a 3D elastic medium Implemenng a Conoluonal Perfecl Mache Laer n a fne-fference coe for he mulaon of emc wae propagaon n a 3D elac meum Progre Repor BRGM/RP-559-FR December, 7 Implemenng a Conoluonal Perfecl Mache Laer n