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

Transcription

1 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

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3 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 Su carre ou a par of Reearch ace - BRGM 7 PDR7ARN4 Duceller A., Aoch H. Checke b: Name: Phlppe Joue Dae: Sgnaure: Approe b: Name: Hormo Moare Dae: Sgnaure: BRGM' qual managemen em cerfe ISO 9: b AFAQ IM 3 ANG Aprl 5

4 Kewor: Semc wae propagaon, Aborbng conon, Perfecl Mache Laer, Conoluonal Perfecl Mache Laer, Non-reflecng conon, Fne-fference. In bblograph, h repor houl be ce a follow: Duceller A., Aoch H. (7) Implemenng a Conoluonal Perfecl Mache Laer n a fne-fference coe for he mulaon of emc wae propagaon n a 3D elac meum, BRGM echncal repor, BRGM/RP-559-FR. BRGM, 5. No par of h ocumen ma be reprouce whou he pror permon of BRGM.

5 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon Snop One of he mo popular meho o mulae numercall he emc wae propagaon n an elac mea he fne-fference meho. In he cone of numercal moellng n unboune mea, he wae nee o be aborbe a he arfcal bounare of he compuaonal oman an herefore, necear o efne non-reflecng conon a hee bounare o mmc an unboune meum. The Perfecl Mache Laer (PML) conon ha he remarkable proper of hang a ero reflecon coeffcen for all angle of ncence an all frequence before creaon an ha become wel ue (e.g. Collno an Togka, ). Howeer, h reflecon coeffcen no ero anmore afer he creaon an become een er large a grang ncence. Therefore, an mproe eron of he PML conon ha been eelope: he Conoluonal Perfecl Mache Laer (CPML) conon (e.g. Komach an Marn, 7). The am of h repor o ecrbe how are mplemene hee wo non-reflecng conon n a 3D fne-fference compuaonal coe for an elac meum. We wll fr eplan he heorecal formulaon of he PML an CPML conon. Then, we wll epoe he rucure of our compuaonal coe. Fnall, we wll preen he reul of ome numercal e performe o erf he effcenc of he meho. BRGM/RP-559-FR Progre repor 3

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7 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon Conen. Inroucon...7. The CPML formulaon CLASSICAL PML FORMULATION CPML FORMULATION Implemenaon n he 3D coe GRID CELL BOUNDARY CONDITIONS SOURCE FONCTION STRUCTURE OF THE CODE Numercal e PML CPML Concluon Reference...35 BRGM/RP-559-FR Progre repor 5

8 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon L of lluraon Fgure Elemenar gr cell of he hree-menonal aggere paal fnefference meho... 9 Fgure - Source me funcon mpoe a pon ource... 3 Fgure 3 Dampng profle ()... 4 Fgure 4 Tme eoluon of he, an componen of he hree-menonal eloc ecor a he fr aon of he numercal oluon wh clacal PML bounare for he hn lce an he large lce Fgure 5 Tme eoluon of he, an componen of he hree-menonal eloc ecor a he econ aon of he numercal oluon wh clacal PML bounare for he hn lce an he large lce Fgure 6 Tme eoluon of he, an componen of he hree-menonal eloc ecor a he hr aon of he numercal oluon wh clacal PML bounare for he hn lce an he large lce Fgure 7 Snapho of he, an componen of he hree-menonal eloc ecor of he numercal oluon wh clacal PML bounare for he hn lce, a me ep 375, 65, 875, 5, 375 an 65, a 39 m... 8 Fgure 8 Tme eoluon of he, an componen of he hree-menonal eloc ecor a he fr aon of he numercal oluon wh CPML bounare for he hn lce an he large lce Fgure 9 Tme eoluon of he, an componen of he hree-menonal eloc ecor a he econ aon of he numercal oluon wh CPML bounare for he hn lce an he large lce Fgure Tme eoluon of he, an componen of he hree-menonal eloc ecor a he hr aon of he numercal oluon wh CPML bounare for he hn lce an he large lce Fgure Snapho of he, an componen of he hree-menonal eloc ecor of he numercal oluon wh CPML bounare for he hn lce, a me ep 375, 65, 875, 5, 375 an 65, a 39 m BRGM/RP-559-FR Progre repor

9 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon. Inroucon One of he mo popular meho o mulae numercall he emc wae propagaon n an elac mea he fne-fference meho. In he cone of numercal moellng n unboune mea, he wae nee o be aborbe a he arfcal bounare of he compuaonal oman an herefore, necear o efne non-reflecng conon a hee bounare o mmc an unboune meum. The Perfecl Mache Laer (PML) conon ha he remarkable proper of hang a ero reflecon coeffcen for all angle of ncence an all frequence before creaon an ha become wel ue (e.g. Collno an Togka, ). Howeer, h reflecon coeffcen no ero anmore afer he creaon an become een er large a grang ncence. Therefore, an mproe eron of he PML conon ha been eelope: he Conoluonal Perfecl Mache Laer (CPML) conon (e.g. Komach an Marn, 7). The am of h repor o ecrbe how are mplemene hee wo non-reflecng conon n a 3D fne-fference compuaonal coe for an elac meum. We wll fr eplan he heorecal formulaon of he PML an CPML conon. Then, we wll epoe he rucure of our compuaonal coe. Fnall, we wll preen he reul of ome numercal e performe o erf he effcenc of he meho. BRGM/RP-559-FR Progre repor 7

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11 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon. The CPML formulaon One of he mo popular meho o mulae numercall he emc wae propagaon n an elac mea he fne-fference meho (Grae, 996, Leaner, 988). In he cone of numercal moellng n unboune mea, he wae nee o be aborbe a he arfcal bounare of he compuaonal oman an herefore, necear o efne non-reflecng conon a hee bounare o mmc an unboune meum. The Perfecl Mache Laer (PML) conon ha he remarkable proper of hang a ero reflecon coeffcen for all angle of ncence an all frequence before creaon an ha become wel ue (e.g. Collno an Togka, ). Howeer, h reflecon coeffcen no ero anmore afer he creaon an become een er large a grang ncence. Therefore, an mproe eron of he PML conon ha been eelope: he Conoluonal Perfecl Mache Laer (CPML) conon (Komach an Marn, 7). We wll fr eplan he clacal PML formulaon, an hen we wll nrouce he CPML formulaon... CLASSICAL PML FORMULATION The fferenal form of he 3D elac wae equaon can be wren a: ρ ρ ρ for he eloce (, an ) an: µ BRGM/RP-559-FR Progre repor 9

12 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon BRGM/RP-559-FR Progre repor µ µ µ µ µ for he re (,,,, an ), where ρ en an an µ are elac coeffcen. In he parcular cae of a homogenou meum, h equaon ha plane wae oluon of he form ( ) ( ) k A r r r. ep. Le u coner a regular oman locae n. In orer o ao reflecon a, we nee o efne an aborbng laer locae n >. Th laer wll be an aborbng laer f he amplue of he wae ecreae eponenall n he recon for >. To efne h laer, we wll efne a new wae equaon, uch ha oluon a plane wae n he regular oman an an eponenall-ecang wae n he PML. We nrouce hen a new comple coornae ~. Thu, he PML can be ewe a an analcal connuaon of he real coornae n he cople pace (Collno an Togka, ). The coornae ~ efne b: ( ) ( ) ~ Equaon Equaon ge u upon fferenang: ~ ϖ Equaon where

13 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon BRGM/RP-559-FR Progre repor We wll now change he orgnal wae equaon wren n erm of arable, an no a new wae equaon wren n erm of arable ~, an. We fr rewre he elac wae equaon n he frequenc oman: ρ ρ ρ for he eloce an: µ µ µ µ µ µ for he re. We hen replace he wae equaon wren n erm of wh a generale wae equaon wren n erm of ~. Ine he regular oman, boh equaon are encal becaue.

14 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon BRGM/RP-559-FR Progre repor ~ ρ ~ ρ ~ ρ for he eloce an: ~ ~ µ µ ~ µ ~ ~ µ ~ µ µ for he re. In he PML, he new wae equaon ha eponenall-ecang plane wae oluon of he form: ( ) ( ) ( ) ( ) ( ) k k A k k k A ep. ep ~ ep r r r r n he recon.

15 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon BRGM/RP-559-FR Progre repor 3 Ung Equaon : ~, we rewre he wae equaon n erm of raher han ~ (n he regular oman, ): ρ ρ ρ for he eloce an: µ µ µ µ µ µ for he re. The eloc an re fel are hen pl no wo par, an, uch ha:

16 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon 4 BRGM/RP-559-FR Progre repor ρ ρ ρ ϖρ ρ ϖρ for he eloce an: µ µ µ

17 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon BRGM/RP-559-FR Progre repor 5 µ µ ϖ µ µ ϖ µ for he re. Conerng back o he me oman, we fnall ge: ρ ρ ρ ρ ρ ρ for he eloce an:

18 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon 6 BRGM/RP-559-FR Progre repor µ µ µ µ µ µ µ µ for he re. Th new wae equaon perm o hae an eponenall-ecang plane wae oluon n he PML laer ( > ). Therefore he bounar a appear o be a non-

19 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon reflecng bounar. We hen ue he ame echnque for he fe oher bounare of he regular oman. In each ege of he oman, we appl h echnque n wo recon. In each corner, we appl h echnque n he hree recon. Th formulaon perm o hae numercal oluon wh reuce arfcal reflecon bu effcenc become poor a grang ncence. The CPML formulaon wa eelope o are h ue... CPML FORMULATION The man ea of he CPML formulaon o chooe a more general efnon of b nroucng new arable α en κ : κ α Equaon 3 A h epreon epen on frequenc, when we go back n he me oman, we wll ge a me conoluon on each mofe paal erae. Denong () he nere Fourer ranform of ( ) *, we can wre: ~ Equaon 4 * beng a conoluon operaor. We hae from Equaon 3: ( ) δ ) H ep κ κ ( ( ) ( ( κ α ) ) Equaon 5 where δ() an H() are he Drac an Heae rbuon. If we enoe: ζ κ ( ) H ( ) ep( ( κ α ) ) Equaon 6 we can ee from Equaon 4, 5 an 6 ha: ~ ζ κ ( ) * BRGM/RP-559-FR Progre repor 7

20 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon 8 BRGM/RP-559-FR Progre repor Le u enoe n ϕ he conoluon erm compue a me ep n, wh beng he me ep: ( ) ( ) τ τ ζ ζ ϕ τ n n n * Equaon 7 Equaon 7 ge u b creaon: ( ) ( ) ( ) ( ) ( ) n m m n n m m m n n m n n n m m m n m Z ϕ τ τ ζ ϕ τ τ ζ ϕ τ Equaon 8 wh: ( ) ( ) ( ) m m m Z τ τ ζ Equaon 9 From Equaon 6 an 9, we oban: ( ) ( ) ( ) ( ) ( ) ( ) m a m Z m m α κ τ τ α κ κ ep ep, wh ( ) ( ) b a α κ κ an ( ) ( ) b α κ ep We coner hen φ a a memor arable whoe me eoluon goerne a each ep b: n n n a b ϕ ϕ From a praccal pon of ew, o mplemen he CPML echnque n a compuaonal coe, we ju nee o replace he paal erae b: ϕ κ ~ an upae φ a each me ep.

21 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon 3. Implemenaon n he 3D coe 3.. GRID CELL We ue a clacal econ-orer aggere gr n pace an me (Maaraga, 976, an Vreu, 986): Fgure Elemenar gr cell of he hree-menonal aggere paal fne-fference meho of Maaraga (976) an Vreu (986) ue clacall o cree he equaon of elaonamc. 3.. BOUNDARY CONDITIONS We pu non-reflecng bounare on he e of he compuaonal oman (Komach an Marn, 7) o a o mulae wae propagaon n nfne elac meum. In h cae, he bounar conon are: BRGM/RP-559-FR Progre repor 9

22 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon (,.,. ) ( mn ma,.,.) (., jmn,. ) (., jma,. ) (.,., kmn ) (.,., k ) ma an (,.,. ) ( mn ma,.,.) (., jmn,. ) (., jma,. ) (.,., kmn ) (.,., k ) ma an (,.,. ) ( mn ma,.,.) (., jmn,. ) (., jma,. ) (.,., kmn ) (.,., k ) ma mn, ma, j mn, j ma, k mn an k ma beng he lm of he regular oman an beng he number of cell n he (C)PML laer SOURCE FONCTION The ource gen b a eloc ecor n he plane k k ource, he ame a Komach an Marn (7) for alaon. A he pon ource, we ju a: ( ) ρ ( α ) a ( ) ep a ( ) n f * Equaon o ( ource,j ource,k ource ) an: ( α ) a ( ) ep a ( ) ( ) ρ co f * Equaon o ( ource,j ource,k ource ), beng he me, α, f, a an beng ource parameer gen n he coe an ρ beng he en STRUCTURE OF THE CODE The rucure of he compuaonal coe he followng: Inalaon BRGM/RP-559-FR Progre repor

23 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon Calculaon of he alue of (), α (), κ (), (j), α (j), κ (j), (k), α (k) an κ (k) Begnnng of he me loop Tme (n/) (*) for XMIN-,,XMAX { for j YMIN-,,YMAX { for k ZMIN-,,ZMAX { f (,j,k, n he CPML) { Calculaon of he j n he CPML wh n orer FD } ele { Calculaon of he j n he regular oman wh n orer FD } } } } Tme n for XMIN-,,XMAX { for j YMIN-,,YMAX { for k ZMIN-,,ZMAX { f (,j,k n he CPML) { Calculaon of he n he CPML wh n orer FD } ele { Calculaon of he n he regular oman wh n orer FD } } BRGM/RP-559-FR Progre repor

24 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon } } Ang a eloc ecor a he ource Impong he bounar conon Sang he alue n fle for he emogram an he napho Incremen of n an reurn o (*) BRGM/RP-559-FR Progre repor

25 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon 4. Numercal e 4.. PML To e our compuaonal coe, we coner he ame moel a Komach an Marn (7). We coner wo moel: he e of he fr moel *64*64 meer, repreenng a oman much longer han we (.e., a hn lce). In h moel, he wae wll propagae a grang ncence along he ege, whch ma caue purou reflecon. To hghlgh hee reflecon, we wll coner a econ, larger moel (6*64*64 meer), whch bounare are far enough from he ource o ao an purou reflecon, an we wll compare he reul of he wo moel. The hn moel cree ung a gr comprng pon * 64 pon * 64 pon. The large moel ecree ung a gr comprng 6 * 64 * 64 pon. The e of a gr cell m. The compreonal wae pee Vp 33 m. -, he hear wae pee V Vp/ m. - an he en ρ 8 kg.m -3. The me ep.6 an we perform he mulaon for 5 me ep,.e. four econ. The pon ource a eloc ecor locae n ource 79 m, ource 47 m an ource 39 m. The me funcon gen b Equaon an. The parameer for he ource are: α 35, f 7, a π*f,. / f an f 7 H Source funcon 5 ource (m/) 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 () Fgure - Source me funcon mpoe a pon ource The aon where we compue emogram are locae a he followng pon: BRGM/RP-559-FR Progre repor 3

26 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon Saon number X coornae (m) Y coornae (m) Z coornae (m) Saon locae a he mle of he moel where here no purou reflecon. Saon an 3 are locae along he ege a grang an er grang ncence, where purou reflecon coul or he emogram. For he ampng coeffcen n he PML, we ue he followng profle: ( ) L N where he ance beween he pon where compue an he begnnng of he PML laer, an L he hckne of he PML: L *, N an: ( N ) log( R ) p c wh Rc.. L Dampng profle () 6 4 () (m) Fgure 3 Dampng profle (). In he regular oman ( < ), he ampng profle null an he wae equaon no mofe. In he aborbng bounar ( > ), he ampng profle ncreae wh an he wae equaon mofe n orer o hae eponenall-ecang plane wae oluon. Aborbng laer of hckne gr pon are mplemene on he e of he moel. 4 BRGM/RP-559-FR Progre repor

27 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon The reul are hown hereafer n Fgure 4 o 7. Ob - V - FD Ob - V - FD,, Thn lce Large lce,,,8 Thn lce Large lce,6,4 V -, V, -, -, -,4 -,3 -,6,,4,63,84,5,6,47,68,89,,3,5,7,93 3,4 3,35 3,56 3,77 3,98 me,,4,63,84,5,6,47,68,89,,3,5,7,93 3,4 3,35 3,56 3,77 3,98 me Ob - V - FD,,8,6,4 V, -, -,4 -,6 Thn lce Large lce,,44,67,89,,33,56,78,,45,67,89 3, 3,34 3,56 3,78 me Fgure 4 Tme eoluon of he (lef), (rgh) an (boom) componen of he hreemenonal eloc ecor a he fr aon of he numercal oluon wh clacal PML bounare for he hn lce (blue) an he large lce (pnk). A he fr aon, relael far from he begnnng of he PML laer an wh non-grang ncence, he agreemen almo perfec for an. For, here numercal peron an he eloc no eacl null. BRGM/RP-559-FR Progre repor 5

28 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon Ob - V - FD Ob - V - FD,, Thn lce Large lce,,8,6 Thn lce Large lce,4 V V, -, -, -, -,4 -,3 -,6,,4,63,84,5,6,47,68,89,,3,5,7,93 3,4 3,35 3,56 3,77 3,98 me,,43,64,85,6,8,49,7,9,3,34,55,77,98 3,9 3,4 3,6 3,83 me Ob - V - FD,, V -, -, -,3 -,4 Thn lce Large lce,,44,66,88,,3,53,75,97,9,4,63,85 3,7 3,9 3,5 3,73 3,95 me Fgure 5 Tme eoluon of he (lef), (rgh) an (boom) componen of he hreemenonal eloc ecor a he econ aon of he numercal oluon wh clacal PML bounare for he hn lce (blue) an he large lce (pnk). A he econ aon, purou ocllaon, n parcular for he S-wae of he componen, ar o appear for an. For, here numercal peron an he eloc no eacl null. 6 BRGM/RP-559-FR Progre repor

29 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon Ob 3 - V - FD Ob 3 - V - FD,5 Thn lce,5 Large lce, Large lce,4 Large lce,3,5, V V, -,5 -, -, -, -,5 -,3,,43,64,85,6,8,49,7,9,3,34,55,77,98 3,9 3,4 3,6 3,83 me,,43,64,85,6,8,49,7,9,3,34,55,77,98 3,9 3,4 3,6 3,83 me Ob 3 - V - FD,3,, V -, -, -,3 -,4 Thn lce Large lce,,44,66,88,,3,53,75,97,9,4,63,85 3,7 3,9 3,5 3,73 3,95 me Fgure 6 Tme eoluon of he (lef), (rgh) an (boom) componen of he hreemenonal eloc ecor a he hr aon of he numercal oluon wh clacal PML bounare for he hn lce (blue) an he large lce (pnk). A he hr aon, he ocllaon become large, he P-wae no correcl calculae an he hape of he S-wae compleel ore for an. For, here numercal peron an he eloc no eacl null. BRGM/RP-559-FR Progre repor 7

30 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon Fgure 7 Snapho of he (lef), (mle) an (rgh) componen of he hreemenonal eloc ecor of he numercal oluon wh clacal PML bounare for he hn lce, a me ep 375, 65, 875, 5, 375 an 65 (op o boom), a 39 m. The hree o ncae he poon of he aon a whch he emogram repreene n Fgure 4 o 6 are recore. Spurou wae appear a grang ncence along he ege of he moel an en purou energ back no he man oman. We fn ha ome large reflecon wae reman epecall a aon an 3. In fac, he napho how ome rong reflecon a he bounar cloe o he ource n he begnnng. We alo obere he ocllaon ocke n he PML laer, whch propagae parallel o he bounar. Th he problem of he PML preoul pone ou (e.g. Komach an Marn, 7). 8 BRGM/RP-559-FR Progre repor

31 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon 4.. CPML To e he CPML, we ue he ame approach a for he PML. We chooe o make α ar n a lnear fahon beween mamal alue α ma a he begnnng of he CPML an ero a op. We ake α ma π*f. We chooe κ conan, equal o. The reul are hown hereafer n Fgure 8 o. Ob - V - FD Ob - V - FD,, Thn lce Large lce,,,8 Thn lce Large lce,6,4 V -, V, -, -, -,4 -,3 -,6,,4,63,84,5,6,47,68,89,,3,5,7,93 3,4 3,35 3,56 3,77 3,98 me,,4,63,84,5,6,47,68,89,,3,5,7,93 3,4 3,35 3,56 3,77 3,98 me Ob - V - FD,,8,6,4 V, -, -,4 -,6 Thn lce Large lce,,44,67,89,,33,56,78,,45,67,89 3, 3,34 3,56 3,78 me Fgure 8 Tme eoluon of he (lef), (rgh) an (boom) componen of he hreemenonal eloc ecor a he fr aon of he numercal oluon wh CPML bounare for he hn lce (blue) an he large lce (pnk). A he fr aon, relael far from he begnnng of he CPML laer an wh non-grang ncence, he agreemen almo perfec for an. For, here numercal peron an he eloc no eacl null. BRGM/RP-559-FR Progre repor 9

32 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon Ob - V - FD Ob - V - FD,, Thn lce Large lce,,8,6 Thn lce Large lce,4 V V, -, -, -, -,4 -,3 -,6,,4,63,84,5,6,47,68,89,,3,5,7,93 3,4 3,35 3,56 3,77 3,98 me,,43,64,85,6,8,49,7,9,3,34,55,77,98 3,9 3,4 3,6 3,83 me Ob - V - FD,, V -, -, -,3 -,4 Thn lce Large lce,,44,66,88,,3,53,75,97,9,4,63,85 3,7 3,9 3,5 3,73 3,95 me Fgure 9 Tme eoluon of he (lef), (rgh) an (boom) componen of he hreemenonal eloc ecor a he econ aon of he numercal oluon wh CPML bounare for he hn lce (blue) an he large lce (pnk). A he econ aon, wh grang ncence an raher cloe o he begnnng of he CPML, he agreemen reman ecellen for an. For, here numercal peron an he eloc no eacl null. 3 BRGM/RP-559-FR Progre repor

33 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon Ob 3 - V - FD Ob 3 - V - FD,5, Thn lce Large lce,5,4 Thn lce Large lce,3,5, V V, -,5 -, -, -, -,5 -,3,,43,64,85,6,8,49,7,9,3,34,55,77,98 3,9 3,4 3,6 3,83 me,,43,64,85,6,8,49,7,9,3,34,55,77,98 3,9 3,4 3,6 3,83 me Ob 3 - V - FD,3,, V -, -, -,3 -,4 Thn lce Large lce,,44,66,88,,3,53,75,97,9,4,63,85 3,7 3,9 3,5 3,73 3,95 me Fgure Tme eoluon of he (lef), (rgh) an (boom) componen of he hreemenonal eloc ecor a he hr aon of he numercal oluon wh CPML bounare for he hn lce (blue) an he large lce (pnk). A he hr aon, n he ffcul cae of er grang ncence, of a long ance of propagaon an of a aon locae cloe o he begnnng of he CPML laer, he agreemen reman er afacor for an. For, here numercal peron an he eloc no eacl null. BRGM/RP-559-FR Progre repor 3

34 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon Fgure Snapho of he (lef), (mle) an (rgh) componen of he hreemenonal eloc ecor of he numercal oluon wh CPML bounare for he hn lce, a me ep 375, 65, 875, 5, 375 an 65 (op o boom) a 39 m. The hree o ncae he poon of he aon a whch he emogram repreene n Fgure 8 o are recore. No purou wae of gnfcan amplue ble, een a grang ncence. In he CPML cae, we are able o ge he encal reul a Komach an Marn (7) well mproe n erm of elmnaon of purou reflecon wh repec o he preou one (PML). We o no hae an more he arfcal wae reflecon een n he napho. We confrm here ha he CPML aborbng conon work much beer. 3 BRGM/RP-559-FR Progre repor

35 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon 5. Concluon The behaour of he Perfecl mache Laer a grang ncence well mproe wh he new Conoluonnal PML approach. I can be ueful when ung wae propagaon n hn lce or when he ource locae near he ege of he moel. The co of CPML n memor orage mlar o he clacal PML. BRGM/RP-559-FR Progre repor 33

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37 Implemenng a CPML n a 3D fne-fference coe for he mulaon of emc wae propagaon 6. Reference Collno F., Togka C. () - Applcaon of he PML aborbng laer moel o he lnear elaonamc problem n anoropc heerogeneou mea. Geophc, 66(), Grae R.W. (996) - Smulang Semc Wae propagaon n 3D Elac Mea Ung Saggere-Gr Fne Dfference. Bull. Sem. Soc. Am., 86, 9-6. Komach D., Marn R. (7) - An unpl conoluonal Perfecl Mache Laer mproe a grang ncence for he emc wae equaon. Geophc, n pre, -. Leaner A.R. (988) Fourh-orer fne-fference P-SV emogram. Geophc, 53, Maaraga R. (976) - Dnamc of an epanng crcular faul. Bull. Sem. Soc. Am., 65, Vreu J. (986) - P-SV wae propagaon n heerogeneou mea: eloc-re fne-fference meho. Geophc, 5, BRGM/RP-559-FR Progre repor 35

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