Duality Methods for Waveform Inversion
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Duality Methods for Waveform Inversion Guy Chavent. N ˚ 2975 September 11, 1996. THEME 4. ISSN 0249-6399. apport de recherche.
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(191) ) # : = 5
(192) #B# ¯ >1 # / ¯ p:s:TlO]OUT"[µJ]C(C!B ^ JLQRSBWO]_! c TKC ¯ p % QlV ^ R ? Bx[ZJLVP|KMPO]Q c. ¯ pzK QOLMPB ^ B`*TKC(XTK[ZJ]RZJ]T:VTl ?R =B C( "QRgc 7JUa MYQO]JU¯ Rj>1_ ] HC!7T J]|y cJU QRSJUTK¯ V>1 T J][) xQ KJURM? QlVKRZJ]¯ RS>1Ql RZ J]lX Bc C!T ^ JU|MYc`*QlBDRZ[J]T:Q Vc BT: XmBDB `*c RSQJURS7T J]Rjc _ Ql9V ^ Ql¯ V=>1 B OUBD FC BD? VKJLR"` ?Tc BR ?[SJ]B C^ MPQOLQR\QRZBD[S[ XYy Q: `*? B BD/VyMR[ZB? ^ B QRS:T([J]R VY? XBMPX R c J]TV=efBDR `*? RSBJUTKOUV¡JLVPTlBD Q c JUTKB V¡^ R ? T B c c QlQ VPc ^ |:B¶C!TlT ^ BO ¯ >1 ¯ >1 y Q![Z_PVKR ? BWRZJL`[SB`*RSJUTKV ¯ >1 9 ? J]` ? JL[ `*O]T:[SB cT Q cB ^ ^ O]J]JL[\VP`WBMYQ [\c [µJ]WVYB T^ C(R ?QlBxX=`* TK? VYBD[ZVR c MR ? `*RZBJ]T:VYV(QlRSTlM ¢c 7QMYOAQlT VKc RZMJ]RSVYQl[SRZ`J]lQO]B)B ^C!J]c | B c QBRS`WJURZTKJ]VY7[WJURj _¡¢XBRdQ c MYQl[ C!^ BWBDRSVPB Tlc RS9 B¶vKJL_ [ MY[SB ¯ ^ >1µ6 R ? B ¯ p:rl ¯ >1 9 ¯ > =9 K h R$? Ql[ c B`WTl|:VYJU? B ^ ? :B c ? _Bc Q c O]_JLV() w E+ ¯ JLV!c R ? B"`Ql[SBTl¬Qx? O]J]VYBQ ? c J]WB ^ Qlc `WT:MYc[SRZJL`C¯ T ^ BOQ:V ^ J]V ) p c w + ¯ J]V RTKXBNB c `QlQlRZ[ST Bc T? Q(TdR BB lB cJUy | vmB`WQ:BDMY7[SMPBBVYT$`W_R ? Q:BXYXQRSRZTBDa7VKJLC(MQQRZRSJ]JUT:TKVVmµTP`R `WM Qlc R)J]VYR | B^ T:M Xmc J]B VP|¡QRSR T ? B!X c TK>1XY Q|KQRSQlJUTK[)V¡QTlCJ]R | ? BQRSJUT TKc V Tl Q ^ c ^BBQlXV B^ lBDvYV:QlR\`[KQ c QBdc [S^ R c X T:c VPT:|:XOU_Q|:QQlRSRZRZB BD^ VKMQBRZO B^ ^ ["J]VK? lBT:VYOU`W:B B ^ J] V=¯ >1R 1? BQO]`WT:QlVPO]Bd`MPJ][O]QlQ¶RZJ]lT:B Vc _TXT7T c ¯ >1C! J]| y c R Q? RSBNJUTKQ:VC(T:XmXPBOUJ]c RSQM RS^ T Bc [ «¬c TKc C² MP¯ Rp dw ¢cBFd`WBQ:V[SBWXPBc OLQR ? _!Ql R^ J]R R ? ? BR ? T Bxc [\¯ `WQ Qc ^O]JLVPOUJL| VPBDIQ c JUJLVB ^T c C(^ B Q:c X RST¯`*>1TK1C( XJ]BDR VY?![SQlc BDRZ[SBFXmQBR`WR1vRSBDTF[ZRdR ?T B c [SR `? QJLO][B ^ QRZc RSB B V7BMY`WQlRZRZJ]KJ]T:J]RjVA_ XQ QlC!BWRSB [ 9J][ BO]QlRZB RZT >1vK_ ¯ >1 ¯ >1 ¯ p{K . :. 1. . . . . . . . . . . . . . . . . . .
(193). . . . . . . . . . . . . . ì1ìÏ4FHG ç*÷ø. . .
(194). . . .
(195) wE. # . Q:V ^ R ? B `WT c\c BD[SXmT:V ^ J]VP|(C!JU| c QlRZJ]T:V¡T:XmB c QRZT c [Q c B c BO]QlRZB ^ vK_ 6 ¯ >1 ¯ pqK ¯ >1 H[\7`WT Qly OUJLQlVP`| `*T c ^ JL[\VPMY|N` ?RST R ?^ QlB*mR VP6 J]RZJ]T:V w ¯ >1µ JUO]OAvBQ!7MYQlVKRZJ]RSQlRZJ]lBFCJ]| c QRSJUTKVJU¢T:VYB¶JL[QlvPO]B¶RST!mV ^ Q ¯ >1 ¯ >1 ¯ >1 w ¯ % K ²R ? _QR ` ? ¯ %Q:7VYµ`*B `y QlV=v¯ B>1[\ QRSJ][Z¯ >1B "^ JLc [¶BQ:`WOU[ZTKTK[ZVYBQlvPRZTO]_Q ^ BWJLO]QO±|lTKVYQOC(QR c JUa R ? J][¶J][¶ ? _CJ]| c QRSJUTKVT c K[A¶ [ST ^y ? c ¯ ¯ ? c c^ ^ c TlTl ^c BDBQlXP` R ?A?BDyVO]vYBQ!MPQ R^ ®MP[ [ZOUyJLO1C( QlXP? `*O]BTKB¶V(MY`*[SR TKRZ? JLVY`"B¶[SJVP^T T B cc CQlQ RZJ]JLT:V(VYC!R [µ? T[ B ? ^TBWQO¢R\J]Q[R ? MY[SQXY[SRBQ:MY`*B)[SJ]R `WVPT:|!BC(QC(Xm[\BQl`WVYR Q[\O]J©QJLa VPRSB| [IT >1c X R c ? T:B)XmT|lcB>1RZT:J]C!T:J]V[ BWQR V7OYc MYJLRS`WC!TQlOYB[\K[SJLXM`WcQ QBDcO]QO]B _^ c JLT:T7VPMPT:| RR y RS[ T¥? TQ³"VKMR ?QlVKQlRZR$J]RRS?QlRZB¶J]l[SBMYvC![SXYJU| Q:c`*QlB RZJ]T:VA? Q ^ > RZ? TJL[vmB"Ql` ?[TK`W[ZT:BDV7VN T c c C!R ? B Tl^ |:T:V7VYMYQlC!OKB JLVc J]` QO]RZOU_ TR J]V? B)¯ [\) ,sXY+®Ql `Wy B ? B Tc ABX J]cRT:XYQlQl|:[QQRST OL[Sc T [ > > ? JL[µT c R ? Tl|:T:VYQlOUJ]Rj_QlOUO]T"[µRSTC!JLVPJ]C!J]WBFR ? BFR c QlBO©ÊRZJLC!B)XmB c RSM c vYQRSJUTKVY[dJ]V ^ MY`*B ^ vK_R ? B QQ^Y^T J]c RZJ]T:Q Vc ^ T C! T ^ RZBWT O]JLVP | [Z RS?BX=JL` ? JUJLR[ ? c BR 7? MPBxJ c®MPB ^O]O¢RSQ:T `*T:c MBD`*[ZRST:J]V`x[ZC!R c TMY^`WRBWOÅR ? Bx®MPO]O¢[SOUT"VYB[\[ y X c JUT c TKVPBXmB c T c C![ ^ c c^ ^ y ? c ?. c c Q:Q¡[S|l[STPJ]l`*BDJLQVRSBDB|lV ^ BT:Q RSC!T BW,9R c _=¯ >1T d 9[ST:M ¯ c >1`WBT [)`WQlQ:V=V ^ vm Bxc B`W`WT:BWC(J]lQ:XYBXYMPc X ["RSB T ^ a7J]JLRB*C(? aYXPQmRSO]J]VPJU`WTKJ]JURZV/RSBBWJLO]Ql[_ Xmy MYB[ST c B cRSM Q!c QlB:[S[F T:M > T c ?`WB¶BDV/Q Tl¢TKmVPVPC!BJ]RZT`BQlV/BvO Ql` V? R ^TK[ZBBNJ]C(R ? T Qc QlR V JUa^ RC!? J]B | cJLV:Q:RSJUB TKc V[ZBTTc
(196) CNR ? MYB"O]Q C!¯ Q:`*[S [Z)Êq O]MY+® C( XmB ^ C(QR c JUa ¯ >1 ¯ >1 y ? JL` ? O]BQ ^ [RZTQx7MYQ:V:RSJUR\QRSJU:B J c ` ?Y? T¨ ? c c c ¯ c^ y ? c ^ RSR J]? C!BB`*TKTl[SRZC(BJL`*`*XRSBJUTKTKVPR V BVKQl RNRy TlQ/Q· 77 MYMYQ:JLQlV³V:VKRSRSRJUJUR\? R\QQBRSRZJUJ]J]:lC(B¡BQlC!|lBD[ZB(JURS| J]TC(QlRZQJ]RST:JU¯ V·TK>1V¥xT:XmTlT ¶B c R Q? RSBT ¯ >1[\`W QlOU¯ B>1R ^? B¡c_B X Q BDTc B`*RSMYJUR 7`*? J]BDBRj[ ?_ [SQ:9/C BBD ?VP JLBW` :h?§B?TKQ:QlVPXY[(BXPXO]JUc QlB T VK^ RSMY[RZ`*TtQ:B ^VQ BD[\`W[ZRSQlJ]OUC(B ^ QlRZc J]B T: VB`*TRSJU7R J]? Rj_B 9 # ? @BVY $`W(B=R ? Bx TK X Bc Ql RZT 9 c y 6 T:VYB ? Ql[xRST/CNMPO]RZJLXPO]_vK_ R ? BBD[ZRSJ]C(QlRZJ]T:V·TR ? B ¯ %Yw ¯ >1 ¯ >1 JL¯ [%YQw 1B*aYX # c B [\[ZBD [1,R ? B"BDK=MYJU
(197) Q#2O]BV `*B =v B 0Rjd BBV( [\`WQO]JLVP | R ? B
(198) c B #2 B `*RS
(199) JU 7 J]Rj_# MY V$KVY$T "=V ¯ Ql[$ B ^ J9 ^ ? «B cT Bc CQlMPV OL^Q X®mQ:T:`*[SRSR)T cC MPOURS J]XYQlOUJ]_7C!JLVPB ^|¡QlR ?R Bc BDMY[Z[\RSMYT Qlc J]O¢VY| |Fc RQ ? ^ B JUBDR V:c RZMPÊB)C!Q:J]|C(c XPQRSOUJ]JURSTKM V¡^ TKB[$XBTlc AQlR RZ? T B c c B ¯B>1`*RSTNc vK[ _=¯ R Q:? V/J][µQlC(Q:XP[O]R J]?RSM B¶^ X B!c BD`*[ZT BDcSVKc RSBQ`WRSRZJUJ]TKT:VV M[ZB ^ JLB BV `W)Qlq V+ VP) E T+ [SRSM ^ _R ? B `WT:VKlB c |:BVY`WB y ? BV y T¬R ? B c BW|:MYO]Q c J]WB ^ [ZT:O]MYRZJ]T:V > 9 B `W`WT:T VYc\c [SBDJ ^[SXmB T:c V T:^ R'J]VYR ?|J][S[1TlQ"OLMP[ZRZBDJ]7T:MPVYB[VYTl`WB ¯ "QlV ^? B¯ / V y
(200) Q:V y ^ Q:V v:^_ ^ BDVPTlR RS? BµBvK_C(> Ql R c J©a > ¯ > =9 6 9 . . . . . x ½ ! ½¬ %@$ % ½
(201) # = 0 ( ¯ w ¯ p: ¯ % # , 5 ¯ >1 # 0 ¯ >1 w ¯ % p: # > . . . . . . . . . . . . J. . . . . . . . . . . . . . . . . . . . &5. . . 5. . . . L. E). . . . &5. 5. . . .
(202).
(203). . . . E).
(204) C. . . 0. . . . 0.
(205).
(206). . . B. . L. F5. . .
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