Tu Nonlinear partial Functional Derivative and Nonlinear LS Seismic Inversion

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1 Tu-7-1 Nolear partal Fuctoal Dervatve ad Nolear LS Sesmc Iverso R.S. Wu* (Uversty o Calora), L. Ye (Uversty o Calora, Sata Cruz) & Y. Zheg (Massachussets Ist. o Techology) SUMMARY Taylor expaso o the ull olear partal dervatve (NLPD) operator s drectly related to the ull scatterg seres (Bor seres) whch has a serous covergece problem or strog scatterg. The reormalzato procedure appled to the Taylor-Fréchet seres leads to the De wol approxmato o NLPD, whch chages the Fredholm type seres to a olterra type seres so that reormalzed Fréchet seres has a guarateed covergece. Numercal smulatos demostrate the deret covergece behavors o the two types o seres or NLPD the case o strog perturbatos. Prelmary study o the LS verso theory usg the olear kerel leads to a verso scheme o smultaeous updatg both the model parameters ad propagators. 75 th EAGE Coerece & Exhbto corporatg SPE EUROPEC 213 Lodo, UK, 1-13 Jue 213

2 Itroducto The Fréchet dervatve s wdely used geophyscal verse problems. We kow that the lear Fréchet dervatve correspods to the Bor modelg (Taratola, 1984; Pratt et al., 1998), For the real Earth, the wave equato s strogly olear wth respect to the medum parameter chages. Thereore, t s terestg to see how the hgher order terms o the olear uctoal dervatves,.e. the hgher order Fréchet dervatves luece the verso procedure ad ts covergece. Wu ad Zheg (212) troduced the hgher order Fréchet dervatves or the acoustc wave equato, ad termed the ull uctoal dervatve as the olear Fréchet dervatve. Because o the possble couso wth the covetoal otato, whch the Fréchet dervatve s detcal wth lear Fréchet dervatve, here we ame the sum o all terms as olear uctoal dervatve or olear partal dervatve (NLPD). I ths paper we wll dscuss the reormalzato o the Taylor-Fréchet seres to a De Wol-Fréchet seres ad the compare the covergece propertes o these deret Fréchet seres wth umercal demostrato. Fally we pot out pot out the potetal o applcato o olear partal dervatve to least-square waveorm verso. Hgher order Fréchet dervatves ad olear partal dervatve operator We wrte the orward problem to a operator orm, d A( m ) (1) where d s the data vector (pressure eld the case o acoustc wave equato), m s the model vector, ad A s the orward modelg operator. Assume a tal model m, we wat to quaty the sestvty o the data chage d (also called data resdual ) to the model perturbato m, ddd ( ) ( ) ( ) Fm m A m m A m (2) We kow that F( m) s a olear deretal operator, ad ca be expaded to Taylor expaso at the curret model (Kwo ad Yazc, 21; Wu ad Zheg, 212): F( m) A' m m A m m A mm (3) 2!! ( ) where A ', A '', ad A are the rst, secod, ad the th order Fréchet dervatves. It has bee show that the wave equato case, the hgher order Fréchet dervatves ca be realzed by cosecutve applcatos o the scatterg operator ad a zero-order propagator to the source, ad the Taylor-Fréchet dervatve seres s closely related to the Bor scatterg seres (Wu ad Zheg, 212): ( ) A ( m ) m! GS p! GSGS 1... GS1p (4) where p s the cdet eld, g s the backgroud Gree's ucto, G s the backgroud Gree s operator ad S s a local scatterg operator (scatterg patter). I we dee a olear partal dervatve operator A ( m, m ) based o the olear deretal operator Fm (, m) through A ( m, m) m F( m, m), the we have 1 1 A ( m, m) A' m 1 A m m A m m. (5) 2!! Note that A s m( x) -depedet because o the olear mutual teractos (multple scatterg) betwee perturbatos. Compare wth the tradtoal lear perturbato model F ( m) A' m m, (6) we call equato (5) wth the operator (or kerel) A ( m m ) as olear perturbato model. Reormalzato o the Taylor-Fréchet dervatves seres ad the De wol approxmato I we splt the scatterg operator to orward scatterg ad backscatterg parts b SS S (7), 75 th EAGE Coerece & Exhbto corporatg SPE EUROPEC 213 Lodo, UK, 1-13 Jue 213

3 ad substtute t to the Fréchet seres, we ca have all combatos o hgher order orward ad backward dervatves. The De Wol approxmato scatterg seres correspods to eglectg multple backscatterg (reverberatos),.e. droppg all the terms cotag two or more backscatterg operators but keepg all the orward scatterg terms utouched (De Wol, 1971, 1985; Wu, 1994, 23; Wu et al., 212). To demostrate the prcple o the olear partal dervatve, we treat a smple problem o trasmsso tomography smooth meda. I ths case, there s o relecto ad we have oly orward scatterg due to velocty perturbatos. The hgher order Fréchet dervatves oly volve orescatterg operator S, ad ts applcato to the model perturbato yelds, 1 ( ) A ( m) m GS mgs m... G 1 S mg 1 S m( x) GS m... G 1 S m 1 g! (8) 1 G( G) S m S m( x) ( GS m) g Followg the reormalzato procedure the De wol approxmato (De Wol, 1985;, Delamotte, 24; Wu, 23; Wu et al., 212), we sum up all the hgher order terms the Taylor seres rstly or the multple orescatterg operators o the let-had sde o m( x ) (recever path) ad the or that o the rght-had sde o m( x ) (source path) the above equato, resultg l l, 75 th EAGE Coerece & Exhbto corporatg SPE EUROPEC 213 Lodo, UK, 1-13 Jue G G S mg g G S m g (9) l l Whe ad the step legth becomes tely small, we reach the reormalzed G ad g. Uder ths approxmato, the Taylor seres o NLPD s reormalzed to NLPD, ( ) A ( ) m m m x GS m x g (1) We see that whe NLPD applyg to a perturbato ucto uder the De Wol approxmato, all the olear teractos due to orescatterg are corporated to g ad G. Covergece property o the reormalzed Fréchet seres (NLPD) uder the De Wol approxmato Compared wth the Taylor expaso (Bor seres) the De Wol-Fréchet seres has the stablty ad ececy advatages. Frst look at the stablty (seres covergece) problem. The orgal Taylor seres (Bor seres), derved by applyg the Bor-Neuma teratve procedure to the Lppma- Schwger equato, s a Fredholm type, ad has the well-kow problem o lmted rego o covergece (slow covergece ad dvergece). The teratve procedure based o the seres usg gradet or Newto method wll have o guaratee o covergg to a correct soluto. I cotrast, we wrte out explctly the th term o g (1) (same or G ) the te seres (9) l l g G S m g g g g... g g g( xx, s ),... g GS m g G( xr, x) Smx ( ) GS mx ( 1) GS mx ( 1) g( x1, xs) (11) dzdydxg ( xx ; ) Smx ( )... dz2dy2dx2g( x3; x2) S m( x2) dz1dy1dx1g ( 2; 1) x x S mx ( 1) g( x1, xs ), x x x 1... x2 x1 xs where x s take as the orward marchg drecto. We see that each term o the seres (11) s a olterra type tegral (Trcom, 1985; Schetze, 198). Thereore, seres (11) s a olterra seres NLPD whch coverges absolutely ad uormly (bd). Thereore, the seres or A, m mhas a guarateed covergece. Physcally, the reormalzato procedure ca be uderstood as a way o

4 rearragg the order o mutual cacelatos (destructve tererece) betwee deret terms. I Bor seres, each term volves multple whole volume tegral ad the mutual cacellatos oly act betwee each terms the al stage. So orward scatterg or the whole volume may become very strog or sgular or each term. I each dvdual term does ot blow up the process, the the al summato may get a approxmate soluto due to mutual cacellatos. However, the errors o the dvdual terms become too bg, the al summato may blow up or gve a wrog result. I comparso, the mutual cacellatos or the reormalzed seres (11) are realzed step-by-step durg the orward marchg process so that the orward-scatterg accumulato s ot allowed to develop to a ull-blow catastrophe. Now we show some examples to compare the covergece o the Taylor-Fréchet seres ad the reormalzed Fréchet seres. The source s located o the top ad recevers are dstrbuted alog the bottom as show o the upper-rght pael Fgure 1. Source has a Rcker wavelet, cetered at 2 Hz 2 2 ( 1m ). The model s a ast Gaussa aomaly dvx,z = v exp / r 2 a, embedded a costat backgroud v 2 k m/ s. The Gaussa ball has parameters a 3, ad the perturbato s gve as 5%, 8% & 1%, respectvely. We see that or weak scatterg (5%) the Bor seres coverges very ast, ths case oly 1 terms; For medum-stregth perturbatos, t coverges slowly (17 terms); However, or strog scatterg ( a 3, 1% ), the seres dverges! Ths demostrates the lmted rego o covergece or the Bor seres ad the related terato process. I comparso, we plot the correspodg results usg the De wol approxmato Fgure 2. We see that t has a guarateed covergece ad the results are early the same as the FD smulato results. The other advatage o NLPD the orm o (1) s the ececy. Although t s the orm o a te seres, t ca be mplemeted ecetly by the th-slab propagator or GSP (geeralzed scree propagator) (Wu, 1994, 23), whch s a oe sweep algorthm or orward scatterg problem. The equvalece o the multple orward scatterg seres ad the th-slab propagator has bee proved (Wu et al., 212). Fgure 1 Taylor-Fréchet (Bor) seres test or a Gaussa ball wth a=3, 5%, 8% & 1%. Blue wggles are rom FD calculatos; red: Bor seres summg up to certa orders. 75 th EAGE Coerece & Exhbto corporatg SPE EUROPEC 213 Lodo, UK, 1-13 Jue 213

5 Fgure 2 Comparso o results rom Bor approxmato (lear Fréchet dervatve) (let), FD (md) ad De Wol (rght) or 5%, 8% & 1%. Cocluso ad dscusso Taylor expaso o the ull olear partal dervatve (NLPD) operator s drectly related to the ull scatterg seres (Bor seres) whch has a serous covergece problem or strog scatterg. The reormalzato procedure appled to the Taylor-Fréchet seres leads to the De wol approxmato o NLPD, whch has a guarateed covergece. Prelmary study o the LS verso theory usg the NLPD kerel leads to a verso scheme o smultaeous updatg both the model ad propagator whch may reduce the tal model depedece o sesmc verso. Ackowledgemet The research s supported by the WTOPI (Wavelet Trasorm o Propagato ad Imagg or sesmc explorato) Research Cosortum at the Uversty o Calora, Sata Cruz. Reereces Delamotte,., 24. A ht o reormalzato, Am. J. o Phys., 72, No 2, De Wol, D.A., Reormalzato o EM elds applcato to large-agle scatterg rom radomly cotuous meda ad sparse partcle dstrbutos. IEEE Tras. Ateas ad PropagatosAP-33, Kwo, K. ad Yazc, B., 21. Bor expaso ad Frechet dervatves olear duse optcal tomography, Computers ad Mathematcs wth Applcatos, do: 1.116/j.camwa Schetze, M., 198, The olterra ad Weer Theores o Nolear Systems, Wley. Taratola, A., Iverso o sesmc-relecto data the acoustc approxmato, Geophyscs, 49(8), Trcom, F.G., 1985, Itegral Equatos, Dover Publcatos, Ic.. Wu, R.S., 23. Wave propagato, scatterg ad magg usg dual-doma oe-way ad oe-retur propagators, Pure ad Appl. Geophys., 16(3/4), Wu, R.S., Xe, X.B. ad J, S., 212. Oe-Retur Propagators ad The Applcatos Modelg ad Imagg, Chapter 2 Imagg, Modelg ad Assmlato Sesmology, Hgher Educato Press Lmted Compay, Bejg, Wu, R.S. ad Y. Zheg, 212. Nolear Fréchet dervatve ad ts De Wol approxmato, Expaded Abstracts o Socety o Explorato Gephyscsts, SI th EAGE Coerece & Exhbto corporatg SPE EUROPEC 213 Lodo, UK, 1-13 Jue 213