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Duality Methods for Waveform Inversion

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(1)Duality Methods for Waveform Inversion Guy Chavent. To cite this version: Guy Chavent. Duality Methods for Waveform Inversion. [Research Report] RR-2975, INRIA. 1996. �inria-00073723�. HAL Id: inria-00073723 https://hal.inria.fr/inria-00073723 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(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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(192) #B#    ¯ >1›  # / ¯ p:s:›ŒTlO]OUT„ˆ"[µJ]C(C!B ^ JLQRSBWO]_!Œ c TKC ¯ p % ›QlV ^ R ? Bx[ZJLVP|KMPO]Q c. ¯ pzK› Ž„QOLMPB ^ B`*TKC(XTK[ZJ]RZJ]T:V‘TlŒ ?R =B C( "QRgc “7JUa MYQO]JU¯ Rj>1_ › ™]™ HC!7T J]|y cJUŒ QRSJUTK¯ V‘>1› TŒ J][) xQ‘ˆ “KJURM? QlVKRZJ]¯ RS>1Ql› RZ J]ŽlX Bc C!T ^ JU|MYc`*QlBDRZ[J]T:Q Vc BT:ž XmBDB `*c RSQJURSŽ7T J]Rjc _ Ql9V ^  Ql¯ V=>1› B OUBD FCˆ BD? VKJLR"` ?TŒc BR ?[SJ]B C^ MPQOLQR\Q‘RZBD[S[ XYy Q:ˆ `*? B BD/VyMR[ZB? ^ B QRS:T([J]R VY? XBMPX R c J]TV=efBDR `*? RSB‹JUTKOUV¡JLVPTlBDŒ Q c ‹JU’TKB V¡^ R Œ? T B c ˆc QlQ VPc ^ |:B¶C!TlT Œ ^ BO ¯ >1› ¯ ™ >1› y Q![Z_PVKR ? BWRZJL`‹[SB`*RSJUTKV   ¯ >1› 9ˆ ? J]` ? JL[ `*O]T:[SB cŒT ˆ‡ Q cB ^ ^ O]J]JL[\VP`WBMYQ [\c [µJ]’WVYB T„^ ˆ C(R ?QlBxX=`*ˆ TK? VYBD[ZVR c MR ? `*RZBJ]T:VYV(QlRSTlM Œ¢c “7QMYOAQlT VKc RZMJ]RSVYQl[SRZ`J]ŽlQO]B)B ^C!J]c | B c ž QBRS`WJURZTKJ]VYŽ7[WJURj™ _¡¢XBRdQ c MYQl[ C!^ BWBDRSVPB Tlc RS9 B¶‹vKJL_ [ MY[SB ¯ ^ >1›µ6 R ? B ¯ p:rl› ¯ >1›   9 ¯ > =9  K› h R$? ˆQl[ c B`WTl|:VYJU’? B ^ ? Ž:B c ? _Bc Q c O]_‹JLV() w E+ ¯ JLV!c R ? B"`Ql[SBTlŒ¬Qx? O]J]VYBQ ? c J]’WB ^ Qlc `WT:MYc[SRZJL`C¯ T ^ BOœ›šQ:V ^ J]V ) p c w + ¯ J]V RTKXBNB c `QlQlRZ[ST B‹c TŒ”? Q(T„ˆdR BB ŽlB cJUy | vmBŒ`WQ:BDMY“7[SMPBBVYTŒ$`W_R ? Q:BXYXQRSRZT•BDa7VKJLC(MQQRZRSJ]JUT:TKVVm›µTP`R `WM Qlc R)J]VYR | B^ T:M Xmc J]B VP|¡QRSR T ? B!X c TK>1XY› Q‹|KˆQRSQlJUTK[)V¡Q‘TlCŒJ]R | ? BQRSŒJUT TKc V  ˆTlŒ Q ^ c ^BBQlXV B^ ŽlBDvYV:QlR\`˜[—KQ ˆc QBdc [S^ R c X T:c VPT:|:XOU_‹Q|:QQlRSRZRZB BD^ VKMŸQBRZO B^ ^ ["J]VK? ŽlBT:VYOU`WŽ:B B ^ J] V=¯ >1R ›1? BQO]`WT:QlVPO]Bd`MPJ][O]QlQ¶RZJ]ŽlT:B Vc _‹TŒXT7T  c ¯ >1C!› J]| y c R Q? RSBNJUTKQ:V‹C(T:XmXPBOUJ]c RSQM RS^ T Bc [ ™ «¬c TKc C² MP¯ Rp ˆdw ›¢cBFˆd`WBQ:V[SBWXPBc OLQ•R ? _!Qlˆ R^ J]R R ? ? BR Œ? T Bxc ˆ[\¯ `WQ Qc ^O]JLVPOUJL| VPBDIQ c JUJL’VB ^T c C(^ B Q:c X RST¯`*>1TK›1C(ˆ XJ]BDR VY?![SQlc BDRZ[SBFXmQBR`WR1vRSBDTF[ZRdR ?ŒT B c [SR `? QJLO][B ^ QRZc RSB Bž V7BMY`WQlRZRZJ]ŽKJ]T:J]RjVA_ ™ XQ QlC!BWRSB [ 9J][ BO]QlRZB RZT >1›vK_ ¯ >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*ŸmR VP6 J]RZJ]T:V w  ¯ >1›µˆ JUO]OAvB‹Q!“7MYQlVKRZJ]RSQlRZJ]ŽlBFCJ]| c QRSJUTKVJUŒ¢T:VYB¶JL[QlvPO]B¶RST!ŸmV ^ Q   ¯ >1›  ¯ >1›     ¯ >1› w ¯ % K› ²R ? _QR ` ? ¯ %Q:7VY›µ`*B `y QlV=v¯ B>1›[\ QRSJ][ZŸ¯ >1B ›"^ JLc [¶BQ:`WOU[ZTKTK[ZVYBQlvPRZTO]_‘Q ˆ^ BWJLO]QO±™|lTKVYQOšC(QR c JUa R ? J][¶J][¶ˆ ? _CJ]| c QRSJUTKVˆT c —K[A¶ [ST ^y ? c ¯ ¯ ‡ ? c c^ ^ c TlTlŒŒ ^c BDBQlXP` R ?A?BDyVO]vYBQ!MPQ R^ Œ®MP[ [ZOUyJLO1C(ˆ QlXP? `*O]BTKB¶V(MY`*[SR TKRZ? JLVY`"B¶[SJŒVP^T T B cˆc C–QlQ RZJ]JLT:V(VYC!R [µ? T[ B ? ^T„BWˆQO¢R\J]Q‹[R ? MY[SQXY[SRBQ:MY`*B)[SJ]R `WVPT:|!BC(QC(Xm[\BQl`WVYR Q[\O]J©QJLa VPRSB| [IŒT >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¥? T„Q³ˆ"V“KMR ?QlVKQlRZR$J]RRS?QlRZB¶J]Žl[SBMYvC![SXYJU| Q:c`*QlB RZJ]T:VA? ™ Q ^ > RZ? TJL[‘vmˆB"Ql` ?[‘TK`W[ZT:BDV7VNŸ T c c C!R ? B Tl^ |:T:V7VYMYQlC!OKB JLVc J]` QO]RZOU_ T‹R J]V? B)¯ [\) ,sXY+®Ql› `Wy B ˆ  ?  B Tc ŒABX J]cR‘T:ˆXYQlQl|:[‘QQRST OL[Sc T [ >    ™ > ? JL[µT c R ? Tl|:T:VYQlOUJ]Rj_QlOUO]T„ˆ"[µRSTC!JLVPJ]C!J]’WBFR ? BFR c Q•ŽlBO©ÊRZJLC!B)XmB c RSM c vYQRSJUTKVY[dJ]V ^ MY`*B ^ vK_‘R ? B QQ^YŒ^T J]c RZˆJ]T:Q Vc ^ TŒ C! T ^ RZBWT O]JLVP |‘ [Zˆ RS?BX=JL` ? ˆ JUJLR[ ? c BR “7? MPBxJ cŒ®MPB ^O]O¢RSQ:T `*T:c MBD`*[ZRST:J]V`x[ZC!R c TMY^`WRBWOÅR ™ ? BxŒ®MPO]O¢[SOUT„ˆ"VYB[\[  y X c JUT c TKVPBXmB c ŒT c C![ ^ c c^ ^ y ? c ‡ ?.   c  c Q:Q¡[S|l[STPJ]Žl`*BDJLQVRSBDB|lV ^ BT:Q RSC!T BW,9R c _=¯ >1T› Œd 9[ST:M ¯ c >1`W›BT [)`WQlQ:V=V ^ vm Bxc B`W`WT:BWC(J]ŽlQ:XYBXYMPc X ["RSB T•ˆ ^ a7J]JLRB*C(? aYXPQŸmRSO]J]VPJU`WTKJ]JURZV/RSB‘BWJLO]Ql[_ Xmy MYBŒ[ST c B cRSM Q!c QlB:[S[F™ T:M Œ> T c ?`WˆB¶BDV/Q TlŒ¢TKŸmVPVPC!BJ]RZT`BQlV/BvO Ql` V? R ^TKˆ[ZBBNJ]C(R Œ? T Qc QlR V JUa^ RC!? J]B | cJLV:QŽ:RSJUB TKc V‘[ZBŒTTcŒ

(196) CNR ? MYB"O]Q C!¯ Q:`*[SŒ [Z)Êq O]MY+®› C(™ XmB ^ C(QR c JUa ¯ >1›  ¯ >1› y ˆ ? JL` ? O]BQ ^ [RZTQx“7MYQ:V:RSJUR\QRSJUŽ:B ‹J c ` ?Y? T¨  ? c c c ¯ c^ y ? c ^ RSR J]? C!BB`*TKTl[SRZC(BJL`*`*XRSBJUTKTKVPR V BVKQl RNRy TlQ/Q·Œ “7“7 MYMYQ:JLQlV³V:VKRSRSRJUJUR\? R\QQBRSRZJUJ]J]Ž:ŽlC(B¡BQlC!|lBD[ZB(JURS| J]TC(ŒQlRZQJ]RST:JU¯ V·TK>1V¥›xT:XmTlT Œ¶B c R Q? RSBT ¯ >1[\`W› QlOU¯ B>1R ^? ›B¡c_ŠB X ž Q BDTc B‘`*RSMYJUR Ž7`*? J]BDBRj[ ?_ [SQ:ˆ9/Cˆ B„BD› ?VP™ JLBW` Ž:h?§BŒ?TKQ:QlVPXY[(B‘XPXˆO]JUc QlB T VK^ RSMY[‹RZ`*TtQ:B ^VQ BD[\`W[ZRSQlJ]OUC(B ^ QlRZc J]B T:ž VŠB`*TRSŒJUŽ7R J]? Rj_B 9 # ? @BVY $`W(B‹=R  ?  Bx TK X Bc Ql RZT 9 c y 6 T:VYB ? Ql[xRST/CNMPO]RZJLXPO]_ŠvK_  R ? B‘BD[ZRSJ]C(QlRZJ]T:V·TŒR ? B ¯ %Yw ›  ¯ >1›     ¯ >1›  JL¯ [š%YQw ›1B*aYX # c B [\[ZBD [1,R ?  B"BD“K=MYJU Ž„

(197) Q#2O]BV `*B =v B 0Rjˆd BBV( [\`WQO]JLVP |‹  R ? B 

(198) c B #2ž B `*RS

(199) JU Ž7 J]Rj_# MY V$—KVY$T„ ˆ"=V  ¯  Ql[$  ˆ B ^ J9 ^ ™ ? «B cT B•c ›šCQlMPV OL^Q XŒ®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 RZMPÊB)C!Q:J]|C(c XPQRSOUJ]JURSTKM V¡^ TKB[$XBTlc ŒAQlR RZ? T B c c B ž ¯B>1`*RS›TNc vK[ _=¯ R Q:? V/J][µQlˆC(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„+›˜ˆ ™ [SRSM ^ _R ? B `WT:VKŽlB c |:BVY`WB y ˆ ? BV   y TŒ¬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 ŒT:^ R'J]VYR ?|J][S[1TlQ"OLMP[ZRZBDJ]“7T:MPVYB[VYTl`WŒB ¯  ›"QlV ˆ ^Š? 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¬ € %œ@$ % 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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