SUMMARY. m = {m i j R Dm ; i = 1..n y, j = 1..n x } (3) d = {d i j R D d ; i = 1..n y, j = 1..n x } (4)

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1 Nonlinar last squars invrsion of rflction cofficints using Baysian rgularization Tor Erik Rabbn and Bjørn Ursin, Norwgian Univrsity of Scinc and Tchnology SUMMARY Invrsion of sisic rflction cofficints is forulatd in Baysian frawork. In addition to th lastic paratrs two scalar quantitis quantifying th varianc lvl in th prior of th lastic paratrs and asurnt nois ar includd in th invrsion. Th axiu a postriori solution is drivd and th rsult is an adaptiv wightd last squars invrsion algorith with th ratio btwn th varianc lvls as a data drivn daping factor. Th algorith is tstd on synthtic nonlinar PP and joint PP and PS rflction data, but is valid for invrsion of any forward probl. INTRODUCTION Solving nonlinar last squars probls oftn involv ill-posd Hssian atrics, and in ordr to produc rliabl rsults so kind of rgularization is ndd. W prsnt hr a last squars invrsion algorith which has a data drivn rgularization. Th starting point is to forulat th invrsion in a Baysian frawork, hnc th na Baysian rgularization, whr w includ two scalars as invrsion paratrs. In our noral distributd prior and liklihood distributions thy ar varianc lvl factors, s Buland and Or 3. Anothr possibl solution is to xplor th postrior distribution by a Mtropolis-Hastings algorith, s.g. Tjlland and Eidsvik 5, but hr w only focus on th axiu a postriori solution. MODEL Th paratrization w ar using for th rflction cofficints is in P-wav and S-wav ipdanc and dnsity. Stovas and Ursin 3 drivd iplicit scond ordr flux noralizd xprssions for rflctions btwn two transvrsly isotropic dia. Explicit xprssions for PP and PS-rflctions siplifid for two isotropic dia, rad and 1 I α r PP = cos θ p α 4sin θ s I I tan θ p 1 4γ cos θ p + tanθ p tanθ s [4γ 1 + γ sin I θ p 4γ γ sin I θ p + γ 1 + γ sin θ p 4 r PS = tanθ p tanθ s [ 1 cosθ s cosθ s + γ cosθ p I + 1 [ 1 cosθ s cosθ s γ cosθ p I [ 1 I α 1 cos θ p + α cos 8sin I θ s θ s + 4sin θ s } tan θ p + tan θ s 1 whr γ is th background v S /v P -ratio, θ p th angl of th incoing and rflctd P-wav, and θ s th angl of th rflctd S-wav. Th paratrs ar collctd in and th asurd rflction aplituds in d, both dfind ovr a two dinsional grid = i j R D ; i = 1..n y, j = 1..n x } 3 d = d i j R D d ; i = 1..n y, j = 1..n x } 4 such that th total nubr of lnts ar n = n x n y D and n = n x n y D d. Th forward odl is th link fro to d. Our focus will b on th quadratic approxiations writtn d = f +, 5 whr f is 1 and in cas of both PP and PS rflctions. W bgin by assigning prior distributions to th odl paratrs and th nois, π σ = N;,σ Σ 6 π σ = N; µ,σ Σ. 7 Finding propr covarianc atrics is difficult, w thrfor includ th stiation of th scalars σ and σ as a part of th invrsion procdur and only spcify th structur of th covarianc. As a consqunc, w nd a prior distributions and choos th invrs gaa distribution πσ = IGσ ;α, 8 πσ = IGσ ;α, 9 whr α and ar scalar paratrs dfining th prior distributions, s.g. Buland and Or 3 for th dfinition of th distribution. This distribution is a rasonabl choic sinc it is dfind only for positiv valus and has zro probability in th origin and infinity. Fro ths four prior distributions and th forward odl w can find th liklihood and, using Bays rul, th joint postrior distribution. MAXIMUM A POSTERIORI SOLUTION Instad of trying to stiat th joint postrior π,σ,σ d w will updat ach paratr squntially in an itrativ algorith. W thrfor nd th thr postrior xprssions π d,σ,σ πd,σ π σ 1 πσ d, πd,σ πσ 11 πσ π σ πσ. 1 Each of ths could b sapld and hnc assssing both an and uncrtaintis, but a uch fastr algorith will b to only sarch for th ost likly solution, also known as th axiu a postriori MAP solution. For th postrior 1 w writ π d,σ,σ xp σ xp } [d f T Σ 1 [d f σ [ µ T Σ 1 [ µ Maxiizing th postrior is qual to iniizing th xprssion ϕ = 1 σ } 13 d f Σ σ µ Σ 1, 14

2 which is a wightd last squars probl. Th iniu is rachd whn th gradint of ϕ is zro, and by xpanding it in a Taylor sris w find th itrativ solution algorith k+1 = k H 1 k g k = k J T Σ 1 J + λ Σ 1 1 λ Σ 1 J T Σ 1 d 15 whr J = f/ T, d = d f k, = k µ, and λ = σ /σ. By skipping th first tr in th gradint and assuing Σ = Σ = I it rducs to th faous Lvnbrg-Marquardt algorith, s.g. Tarantola Th postrior 11 and 1 can, by using th dfinition of th noral and invrs gaa distribution, b writtn with s = d Σ 1 πσ d, IG σ ;α + n, + s πσ IG σ;α + n, + s and s = Σ 1. Th MAP solution of an invrs gaa distribution IGσ ;α, is σ = /α + 1, and with this rsult w find th MAP of 16 and 17. To spd up th algorith w will prfor only on updat of using 15 bfor updating th MAP of σ and σ. Th final xprssion for λ rads λk+1 = σ,k+1 σ,k+1 = + 1 s,k s,k α + 1 n 1 + α + 1 n. 18 Equations 15 and 18 constitut th basis of our invrsion algorith. IMPLEMENTATION For th covarianc atrics in 6 and 7 w choos to split th in two parts Σ = g S 19 Σ = g S. Hr, g and g ar atrics dscribing covariancs btwn paratrs and rrors within ach spatial point, and S and S ar spatial corrlation atrics. Th sybol dnots th Kronckr product. Th dinsions of g and g ar D d D d and D D rspctivly. To gnrat th corrlation atrics w us th xponntial corrlation function Si, j = xp 3 x } i x j 1 d whr d is known as th rang of th corrlation. Th lngth btwn th points x i and x j, x i x j, is dfind on a torus. This will produc dg ffcts, but thy ar liitd to a distanc proportional to th rang paratr. Th advantag of this assuption is that th atrics S and S bco circulant which in turn nabls fast calculation of th invrs. For dtails on solving circulant atrics s Ru and Hld 5. To solv 15 is th ajor part of th coputational ti. Th Hssian atrix H can asily bco too larg to vn b stord in ory. Our solution is to us an itrativ solvr and bcaus th Hssian is sytric positiv dfinit SPD w will us th conjugat gradint thod CG to solv th syst for whr = k+1 k not to b confusd with = µ. In CG th ost coputational xpnsiv is th th atrixvctor product of th typ H k p, but can b valuatd vry fast by xploiting th spars structur of th Jacobian J and that S is circulant. In addition, to iprov th convrgnc proprtis, w hav usd th diagonal of H k, D, as a prcondition atrix and solvd th prconditiond syst L 1 H k L T u = L 1 g k, = L T u, 3 whr D = LL T, th Cholsky dcoposition. For dtails on prconditioning and CG s Saad. NUMERICAL EXAMPLE W hav usd a synthtic odl to tst th invrsion algorith. Fro a chosn tru w usd th xact Zoppritz quations to gnrat synthtic asurnts d by assuing th P-wav vlocity in th uppr diu and th background v P /v S ratio to b known. Our tru was dfind on a 1 1 grid, s 3, and ranging fro. to.5. In Figurs 1 and w show th synthtic data d togthr with th bias in th linar and quadratic approxiations for PP and PS data rspctivly. It is clar that th quadratic approxiations ar suprior. For th prior an µ w hav usd half of th tru valu of, and this is also usd as initial. Th first xapl is th invrsion of four PP rflction aplituds. In Figur 3a w s th dvlopnt of λ in ach itration showing a convrgnt bhavior. In th first itration w hav liitd th axiu valu to b 5, othrwis it would hav bn svral ordrs of agnitud highr bcaus th initial guss is far fro th truth. Th valu in th final itration is approxiatly.6. Th absolut valu of th bias of th corrsponding is displayd in Figur 4, using both th linar, quadratic, and xact Zoppritz as forward odl. W s that P-wav ipdanc is wll dfind in all thr cass, whil th two othr paratrs hav bias for th two approxiat odls. Th bias is clarly corrlatd with th bias shown in Figur 1. Th scond xapl is vry siilar to th first, but this ti w also includ th thr PS rflction aplituds such that th total nubr of aplituds, D d, is qual to svn. λk for ach itration is shown in Figur 3b. Again it shows nic convrgnt bhavior and in th final itration it is.3. Copard to th prvious xapl w s that th valu is lowr and th invrsion probl is lss dapd, or, in othr words, or wight is put on th asurd data. This is also rflctd in Figur 5 whr th absolut valu of th bias is plottd, although for th linar odl thr is still a considrabl bias. CONCLUSION Fro a Baysian forulation of th invrsion probl w hav drivd a wightd last squars algorith with an adaptiv, data drivn daping factor. Th algorith was tstd on two synthtic xapls and showd a wll-posd bhavior and a low bias in th invrsion rsult. ACKNOWLEDGMENT W wish to thank BP, Hydro, Schlubrgr, Statoil, and Th Rsarch Council of Norway for thir support through th Uncrtainty in Rsrvoir Evaluation URE projct. H k = g k,

3 5 r PP θ = θ = θ = θ = Figur 1: To th lft is PP rflction cofficints fro th Zoppritz odl for 4 diffrnt incidnc angls. Th two right coluns show th bias in th linar and quadratic approxiations, rlativ to th Zoppritz odl, for th nonzro angls. For θ = th bias is zro. 5 r PS.4.3 θ = θ = θ = Figur : To th lft is PS rflction cofficints fro th Zoppritz odl for 3 nonzro incidnc angls. Th two right coluns show th bias in th linar and quadratic approxiations, rlativ to th Zoppritz odl.

4 λ Zoppritz λ Zoppritz itration a itration b 5 Figur 3: λk for a PP invrsion and b joint PP and PS invrsion. Zoppritz 1 8 Iα I Figur 4: Absolut valu of th bias in PP invrsion. Zoppritz 1 8 Iα I Figur 5: Absolut valu of th bias in joint PP and PS invrsion.

5 REFERENCES Buland, A. and H. Or, 3, Joint AVO invrsion, wavlt stiation and nois-lvl stiation using a spatially coupld hirarchical baysian odl: Gophysical Prospcting, 51, Ru, H. and L. Hld, 5, Gaussian arkov rando filds: Chapan & Hall/CRC. Saad, Y.,, Itrativ thods for spars linar systs: PWS. Stovas, A. and B. Ursin, 3, Rflction and transission rsponss of layrd transvrsly isotropic viscolastic dia: Gophysical Prospcting, 51, Tarantola, A., 1987, Invrs probl thory: Elsvir. Tjlland, H. and J. Eidsvik, 5, Dirctional Mtropolis-Hastings updat for postriors with non-linar liklihoods: Prsntd at th Gostatistics Banff 4.

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perm4 A cnt 0 for for if A i 1 A i cnt cnt 1 cnt i j. j k. k l. i k. j l. i l h 4D, 4th Rank, Antisytric nsor and th 4D Equivalnt to th Cross Product or Mor Fun with nsors!!! Richard R Shiffan Digital Graphics Assoc 8 Dunkirk Av LA, Ca 95 rrs@isidu his docunt dscribs th four dinsional