On the inverse problem of soil profile reconstruction: a comparison of time-domain approaches

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1 Comput Mch (8) 4:91 94 DOI 1.17/s ORIGINAL PAPER On th invrs problm of soil profil rconstruction: a comparison of tim-domain approachs Song-Won Na Loukas F. Kallivokas Rcivd: 8 May 6 / Accptd: 1 April 8 / Publishd onlin: 17 May 8 Springr-Vrlag 8 Abstract W discuss a on-dimnsional invrs matrial profil rconstruction problm that ariss in layrd mdia undrlain by a rigid bottom, whn total wavfild surficial masurmnts ar usd to guid th rconstruction. To tackl th problm, w adopt th systmatic framwork of PDEconstraind optimization and construct an augmntd misfit functional that is furthr ndowd by a rgularization schm. W rport on a comparison of spatial rgularization schms such as Tikhonov and total variation against a tmporal schm that trats th modl paramtrs as timdpndnt. W study numrically th ffcts of inact initial stimats, data nois, and rgularization paramtr choics for all thr schms, and rport invrtd profils for th modulus, and for simultanous invrsion of both th modulus and viscous damping. Our numrical primnts dmonstrat comparabl or suprior prformanc of th tim-dpndnt rgularization ovr th Tikhonov and total variation schms for both smooth and sharp targt profils, albit at incrasd computational cost. Kywords PDE-constraind optimization Invrs problm Matrial idntification Rgularization Support for this work was providd by th US National Scinc Foundation undr grant awards ATM-3515 and CMS S.-W. Na L. F. Kallivokas (B) Dpartmnt of Civil, Architctural and Environmntal Enginring, Th Univrsity of Tas at Austin, 1 Univrsity Station, C1748, Austin, TX 7871, USA -mail: loukas@mail.utas.du 1 Introduction Th rconstruction of th matrial profil of layrd soils basd on surficial masurmnts collctd as ithr th rspons to dynamic loads impartd on th soil surfac, or as sismic rcords, is cntral to gotchnical sit charactrization fforts. Th sam problm, albit at considrably diffrnt lngth scals, is also of primary importanc to sismic hazard mitigation fforts, to soil-structur-intraction problms, and to gophysics applications (.g., discovry of hydrocarbon dposits). An oftn-usd frquncy-domain tchniqu for gotchnical sit invstigations is basd on th SASW mthod (spctral analysis of surfac wavs [1]), which rlis on th analysis of surfac Rayligh wavs for dtrmining th disprsion curv that, in turn, lads to th shar wav vlocity profil. Th mthod howvr is limitd in svral ways, not th last of which is th difficulty with th rconstruction of th soil s dissipation charactristics, with th notabl cption of th works by Ri t al. [ 4]. Dspit SASW s and its variants adoption and worldwid us, it is still unclar how th mthod can b modifid so that it dos not rly on a layrd mdium assumption, and thus account for arbitrary matrial htrognity. In this articl w discuss a partial-diffrntial-quation (PDE)-constraind optimization approach for rcovring th matrial profil of layrd soils dirctly in th tim-domain, including th spatial distribution of lastic modulus as wll as of attnuation paramtrs; by and larg th lattr hav, thus far, provd to b lusiv to rconstruct. Though hrin too th tchnical dtails ar rstrictd to a horizontally layrd mdium, thus, ffctivly, giving ris to a ondimnsional problm, th mthodology, in contrast to SASW, scals to htrognous mdia in highr spatial dimnsions, and can mak us of data collctd dirctly in th timdomain.

2 9 Comput Mch (8) 4:91 94 W start with a convntional misfit functional that dscribs th diffrnc btwn computd and masurd rspons, and augmnt it via th wak imposition of th govrning PDEs. Rgularization trms ar thn addd to th augmntd functional in an attmpt to allviat solution multiplicity. Th primary intnt of this articl is to compar classical approachs such as a Tikhonov [5], and a total variation (TV) schm [6], against a tim-dpndnt schm (TD) [15] that trats, initially, th modl paramtrs as tim-dpndnt. Though computationally costly, th TD schm appars to lnd much ndd algorithmic robustnss, and typically lads to bttr profils, without bing too snsitiv to rgularization paramtr choics. A partial motivation for th TD schm stms from th fact that with both Tikhonov and TV schms th quality of th solution dpnds highly on a usr-slctabl rgularization paramtr (s,.g., Fig. 9 in [7]), and may not succssfully rconstruct profils for any valu of th paramtr, spcially in th prsnc of sharp discontinuitis or inact initial stimats. Tadi in [15] rportd that h usd a TD schm succssfully to invrs problms in population dynamics, which, subsquntly, ld to his using it also in th rconstruction of th dnsity profil of a ondimnsional rod [15]. Our own motivation for primnting with a TD schm stms from our intrst in dparting from spatial schms, whil still imposing som form of rgularization; in this sns, th only othr logical candidat was ploring rgularization in th rmaining dirction, i.., in tim. As it will b numrically shown, th TD schm is rlativly insnsitiv to th rgularization paramtr, and, in most cass, lads to bttr profils than spatial schms. Th PDE-constraind approach followd hrin was originally inspird by th optimization work in Navir Stoks control problms [8], and th similar work on acoustic wavs ld by Ghattas and his collaborators [9,1]. In [9,1], th authors primntd succssfully with matrial invrsion in acoustics using TV rgularization schms. Ovrall, w ar intrstd in rcovring both sharp and smoothly varying profils and trat th matrial proprtis as spatially picwis continuous: as it will b sn, sharp discontinuitis ar still rcovrabl. Th rcovry of sharply varying proprtis (or, quivalntly, of discontinuous PDE cofficints) has bn addrssd for lliptic problms (.g., [7]), but th litratur is thin on hyprbolic problms (but s,.g., [11 13]). Altrnativly, th problm could b posd ovr an arbitrarily prdtrmind numbr of layrs, whos thicknsss and lastic proprtis srv as unknowns. For ampl, in [14] th authors usd Lov wavs for probing, and rcovrd th (visco)lastic charactristics of th layrs; still, th layrd assumption rmains an obstacl whn gnralizations that includ latral htrognity ar of intrst. W choos a rducd-spac approach to rsolv th firstordr optimality conditions associatd with stationarity of th augmntd functional, which, in turn, rsult in stat, adjoint, ft ( ) (symbolic) Station Invrsion variabls: (), () Fig. 1 Problm configuration: soil dposits on bdrock l = and control problms (from a tmporal charactr prspctiv, thr rsult both initial- as wll as final-valu problms). W rport on our numrical primnts that dmonstrat robustnss in th rconstruction of th matrial profils, and th comptitivnss of th TD rgularization whn compard with spatial schms. W trat both singl and dual-paramtr systms (modulus and damping), and rport on th snsitivity of th rconstructd profils with rspct to inact stimats, data nois, and rgularization paramtr choics. Th forward problm W considr th rspons of a layrd (htrognous) mdium (soil dposits) ovrlying a homognous halfspac (possibly mad of rock) to surfac citation. W formally rduc th problm to a on-dimnsional on by considring, for ampl, th cas of comprssional wavs manating from th surfac of th soil du to a uniform citation applid throughout th ntir (two-dimnsional) soil surfac. Similar physical problms aris if on wr to considr only shar wavs in th sam mdium, or comprssional wavs in a rod[15]. Hr, to fi idas, w shall hncforth rstrict th discussion to comprssional wavs, which allow th rduction of th problm to on dimnsion; ultimatly, our targt application is th thr-dimnsional invrsion of highly htrognous dposits. In principl, th approach w discuss hrin, can b applid to this mor compl problm with only minor modifications to account for th highr spatial dimnsionality. Lt u(, t) dnot th (scalar) displacmnt in th dirction of th applid citation (Fig. 1). Lt l dnot th dpth of th soil dposits, and T th total obsrvation priod. Thn, th strong form of th forward problm can b statd as: Forward Problm Find u(, t), such that: u(, t) ( ) u(, t) u(, t) α() + β() =, (, t) (, l) (, T ), (1) =l

3 Comput Mch (8) 4: with u(, t) α() = f (t), () u(l, t) =, (3) u(, ) u(, ) = =, (4) whr dnots location and t dnots tim. In th abov α() dnots th soil s modulus (.g., λ + µ for comprssional wavs, with λ, µ dnoting th Lamé constants), or th squar of th wav propagation vlocity. Throughout w assum that th matrial dnsity is constant (a rasonabl assumption in gotchnical sit invstigations); in particular, in (1), w assum, without loss of gnrality, that th dnsity ρ = 1. Furthrmor, in (1),β() rprsnts viscous damping (also normalizd with rspct to th dnsity). Th attnuation charactr of soil dposits is much mor compl than simpl viscous damping could vr captur. Howvr, hr th mphasis is on th invrsion and profil rconstruction procss; w pct to tackl mor ralistic attnuation modls in th futur. For simplicity, w considr th simpl cas in which th bottom of th layrd mdium is fid (rigid) at = l (condition (3)). Th tnsion to th mor ralistic cas whr th dpth of th dposits is unknown and th computational domain nd to b truncatd will also b communicatd in th futur. W assum furthr that th systm is initially at rst (condition (4)), and that th sourc citation is at th origin (condition ()). Whras in th forward problm th citation and th matrial distributions α() and β() ar known, in th invrs problm of intrst hrin, both α() and β() ar unknown; known, howvr, is th rspons u(, t), t (, T ). 3 Invrs misfit problm For th systm dfind by th forward problm givn in (1) (4), th invrs problm can b cast as a PDE-constraind optimization problm using a (last-squars) misfit functional, as in: Minimiz: J = 1 [u(, t) u m (, t)] dt + R α (α) + R β (β), subjct to (1) (4). Hr, J dnots th misfit functional in which u m dnots th rspons masurd at th surfac and u is th computd rspons obtaind for assumd modl profils α() and β(). Th last two trms in (5) rprsnt rgularization trms for α and β, rspctivly. Thir dtaild prssions ar discussd in th nt sction. (5) 4 Rgularization schms As in all invrs problms, th minimization problm (5) associatd with th misfit trm only is inhrntly ill-posd du to, at a minimum, th incomplt data st. On way to ovrcom or allviat this difficulty is to impos additional constraints in an attmpt to rgulariz th solution, if it at all ists. Th choic of th rgularization schm is of paramount importanc. W discuss candidat schms: 4.1 Tikhonov rgularization On of th most widly adoptd schms is Tikhonov rgularization [5]. A Tikhonov-typ rgularization nforcs spatial smoothnss on th modl paramtrs. W primnt with th first-ordr Tikhonov rgularization schm, according to which th rgularization trm R Tk p is typically dfind (in on-dimnsion), as: R Tk p (p) := R ( ) p dp() d, (6) d whr R p is a rgularization paramtr that controls th amount of th pnalty th rgularization trm (6) imposs on th functional (5), and p is a modl paramtr (.g., lastic modulus). Clarly, th abov Tikhonov schm favors smooth profils sinc th pnalty trm bcoms smallr (modulo th rgularization paramtr) for smooth p distributions, whras it incrass with high (spatial) frquncy prturbations of th modl paramtrs. Thrfor, th Tikhonov schm works wll for smooth targt profils, but is not wll-suitd to sharply varying targt profils. In addition, th Tikhonov schm rquirs initial stimats which ar quit clos to th targt, sinc th schm prcluds larg prturbations from th initial gusss. Lastly, th rgularization paramtr R p should b chosn with car, sinc, as will b shown, th solution is quit snsitiv to th choic of th rgularization paramtr. 4. Total variation (TV) rgularization Th total variation rgularization trm R TV p is dfind as: (dp() ) R TV p (p) := R p + ɛ d. (7) d Th schm is similar to Tikhonov s in nforcing spatial smoothnss of th modl paramtrs, but hibits bttr prformanc in daling with discontinuitis or sharp profil changs du to th imposition of a lssr pnalty associatd with discontinuitis than th Tikhonov schm. In th abov, w hav modifid th standard TV form by adding a positiv scalar ɛ to smoothn paramtr functions that ar

4 94 Comput Mch (8) 4:91 94 not strongly diffrntiabl at discontinuitis, as is typically don in TV implmntations. 4.3 Tim-dpndnt (TD) rgularization An altrnativ choic for th rgularization trm is to us tim-drivativs of th modl paramtrs [15]. To this nd, w assum that p p(, t), thus violating th physical stting of th problm. Thn, a possibl form for th TD rgularization trm R TD p bcoms: R TD p (p) := R ( ) p p(, t) d dt. (8) Evn though th modl paramtr p is assumd to dpnd on both tim and spac, th minimization procss nforcs it to b indpndnt of tim: clarly, of all th possibl trajctoris p(, t) for tims t (, T ), th tim-indpndnt p() is th on minimizing (8). To this nd, w furthr impos that: p(, T ) p(, ) = p, =, (9) whr p is th initial stimat of th modl paramtr p. In othr words, w forc at final tim t = T, th matrial proprty distribution to b chosn as tim-indpndnt. In contrast to spatially drivn rgularizations, such as Tikhonov or TV that filtr out highr spatial frquncis of th modl paramtrs, th tim-dpndnt form (8), in principl, allows thm. In this sns, and with an y towards thrdimnsional problms, whr th spatial variability of th matrial paramtrs is gratr, and may b unduly pnalizd by a spatial rgularization schm, th TD schm shifts th mphasis from spac to tim. As statd, our motivation for primnting with (8) stms from our intrst in dparting from spatial schms, whil still imposing som form of rgularization; in this sns, th logical candidat was a tmporal schm. W radily acknowldg that in accpting (initially) tim-dpndnt moduli, w ar possibly widning th solution fasibility spac in unintndd ways. Howvr, th numrical rsults ar, thus far, promising and provid vidnc that th TD schm rgularizs th solution in both spac and tim. To dat, w know of no formal proof of th ffcts or th bnfits affordd by th TD schm, othr than thos btrayd by th numrical ampls discussd hrin. 5 PDE-constraind optimization approach 5.1 Augmntd functional To rconstruct th matrial profil w sk to minimiz (5) subjct to th govrning PDE and th boundary and initial conditions givn by (1) (4). To this nd, w first rcast th problm as an unconstraind optimization problm by dfining an augmntd functional basd on (5), whr now th govrning PDE and th boundary/initial conditions hav bn imposd (addd) via Lagrang multiplirs as sid constraints (notic, only Numann-typ conditions nd to b addd as part of th sid constraints; ssntial conditions ar plicitly nforcd). W thn sk to satisfy th first-ordr optimality conditions. W discuss first th cas of th TD rgularization; th Tikhonov and TV schms ar simplr. Th dtails follow: w dfin th augmntd functional as: A(u,λ,α,β) = 1 + R α + + [u(, t) u m (, t)] dt ( ) α d dt + R β { u λ ( α u λ(, t) ) + β u [ u(, t) α(, t) f (t) ( ) β d dt } d dt ] dt u(, ) λ(, ) d, (1) whr λ(, t) is th Lagrang multiplir (or adjoint variabl), and, whrvr appropriat, functional dpndnc has bn droppd for brvity. Notic that th originally spatially dpndnt paramtrs α() and β() hav bn modifid to α(, t) and β(, t) to account for th tmporal dpndnc, pr th TD rgularization schm. For th Tikhonov schm, th augmntd functional may b dfind in a similar way: A(u,λ,α,β) = 1 + R α + + [u(, t) u m (, t)] dt ( ) dα d + R β d { u λ λ(, t) ( α u ( ) dβ d d } ) + β u [ u(, t) α() f (t) ] dt d dt u(, ) λ(, ) d. (11) In th TV cas, th augmntd functional can b obtaind by rplacing th rgularization trms in (11) with th corrsponding trms givn by (7). Nt, th first-ordr optimality conditions ar obtaind from th variation of th augmntd functional with rspct to th stat variabl u, th adjoint variabl λ, and th modl

5 Comput Mch (8) 4: paramtrs α and β: δ λ A δ u A =. (1) δ α A δ β A 5. Th first-ordr optimality conditions W sk quadruplts (u,λ,α,β) for which th augmntd functional A bcoms stationary. Accordingly: Accordingly: δ u A = [u(, t) u m (, t)] δu(, t) dt + + { δu λ ( α δu ) + β δu } d dt λ(, t)α(, t) δu(, t) δu(, ) λ(, ) d. (18) dt 5..1 First optimality condition δ λ A = + { u δλ δλ(, t) ( α u ) + β u } d dt ] [ u(, t) α(, t) f (t) u(, ) δλ(, ) d =, (13) whr δλ dnots an arbitrary variation of λ. By taking into account th plicitly imposd homognous ssntial boundary u(l, t) =, and th initial condition u(, ) =, w rcovr th stat problm: Stat problm u ( α u ) + β u =, (, t) (, l) (, T ), (14) u(, t) α(, t) = f (t), (15) u(l, t) =, (16) u(, ) u(, ) = =. (17) Clarly th stat problm is idntical to th forward problm givn by (1) (4) Scond optimality condition Th variation of th augmntd functional with rspct to th stat variabl u yilds th scond optimality condition. 1 Whn th TD rgularization schm is usd, thn α() α(, t), and β() β(, t). dt By intgrating by parts, and taking into account th boundary and initial conditions thr rsults: { λ δ u A = δu + δu(, t) δu(l, t) λ(l, t)α(l, t) dt { δu(, T ) + λ(, T ) ( α λ ) } (βλ) d dt { [u(, t) u m (, t)] α(, t) } λ(, t) dt } + β(, T )δu(, T ) d λ(, T ) δu(, T ) d. (19) Sinc δu is arbitrary, by stting δ u A = th following adjoint problm nsus: Adjoint problm ( α λ ) (βλ) =, λ (, t) (, l) (, T ), () λ(, t) α(, t) = [u(, t) u m (, t)], (1) λ(l, t) =, () λ(, T ) = λ(, T ) =. (3) W rmark that th adjoint problm is similar to th stat problm, with, howvr, two important diffrncs: first, th right-hand-sid of (1), i.., th sourc trm, dpnds on th misfit btwn th computd and masurd valus of th stat variabl. Scondly, by virtu of (3), th adjoint problm is a final-valu problm; in addition, th damping trm s sign has changd. If th Tikhonov or TV schm wr usd, th sam stat and adjoint problm would hav bn obtaind, with, naturally, α and β bing only spatially dpndnt.

6 96 Comput Mch (8) 4: Third optimality condition W obtain th third condition as th variation of th augmntd functional with rspct to α; thr rsults δ α A = R α α λ δα d dt ( δα u ) d dt u(, t) λ(, t) δα(, t) dt [ ( ) ] α = R α δα α δα d dt [ ( λδα u ) δα λ u u(, t) λ(, t) δα(, t) dt [ α(, T ) = R α δα(, T ) α R α δα d dt [ u(l, t) λ(l, t)δα(l, t) ] u(, t) λ(, t)δα(, t) dt + δα λ u d dt λ(, t) ] d dt ] α(, ) δα(, ) d u(, t) δα(, t) dt =. (4) By taking into account that δα(, ) =, α(,t ) = and λ(l, t) = (pr (9)), (4) rducs to: δ α A = { R α α + λ u } δα d dt =. (5) Similarly, th variation of th augmntd functional with rspct to α, in th Tikhonov cas, rsults in: δ α A = [ d α T R α d + ] λ u dt δα d =, (6) whras, in th TV cas, rsults in: ( ) { dα (dα ) 3 δ α A = R α ɛ + ɛ} d d λ u + dt δα d =. (7) 5..4 Fourth optimality condition Th last condition can b obtaind similarly to th third on, by taking th variation of th augmntd functional with rspct to β; thr rsults: β δβ T δ β A = R β d dt + λδβ u d dt [ ( β = R β δβ + λ u δβ d dt [ β(, T ) = R β δβ(, T ) + β R β δβ d dt ) β δβ ] d dt ] β(, ) δβ(, ) d λ u δβ d dt =. (8) By taking into account that δβ(, ) =, β(,t ) =, (8) yilds: δ β A = { } β R β + λ u δβ d dt =. (9) If th Tikhonov rgularization wr usd, instad of (9),th last condition would hav rad: { d β T δ β A = R β d + λ u } dt δβ d =. (3) Lastly, in th TV cas, th last condition bcoms: ( ) { dβ (dβ ) 3 δ β A = R β ɛ + ɛ} d d + λ u dt δβ d =. (31) Conditions (5) and (9) constitut th control problm for th TD rgularization cas, whras (6) and (3) corrspond to th Tikhonov cas, and (7) and (31) tothtv cas. Th control quations ar usd in ordr to updat th modl paramtrs. Obviously, thy ar satisfid only for th tru profils. 6 Invrsion procss In ordr to satisfy th first-ordr optimality conditions, w adopt a rducd-spac mthod. Notic that, in principl, th stat problm (14) (17), th adjoint problm () (3), and th control conditions (5) and (9) (or(6) and (3), or

7 Comput Mch (8) 4: (7) and (31)) can b solvd as a coupld problm (by using a full-spac mthod). Howvr, th computational cost pr itration incrass, givn, upon discrtization, th rsulting matri sizs. W rmark that for th solution of th stat and th adjoint problm (ithr as a coupld systm or individually) any numrical schm may b usd (finit diffrncs, finit lmnts, tc.). In contrast to a full-spac mthod, hr w opt for a rducd-spac mthod that maps th optimization problm to th spac of th modl paramtrs (α and β), whrby liminating th stat and adjoint variabls. W start by solving th stat problm (14) (17) to obtain th stat variabl u, givn an stimat of th modl paramtrs, thrby satisfying th first condition δ λ A =. Thn, w solv th adjoint problm using th stat variabl computd in th first stp, to obtain th Lagrang multiplir λ that satisfis th scond condition δ u A =. To solv both th stat and adjoint problms, w mploy convntional finit lmnts. Thn, thr rmains to sk to updat th modl paramtrs, α and β, so that th third and fourth conditions b satisfid. W us th control quations to itrativly provid updats to th modl paramtrs. 6.1 Stat and adjoint smi-discrt forms In ordr to satisfy th first condition for an assumd st of invrsion variabls, w us a standard Galrkin approach to solv th stat problm (14) (17). Accordingly, th wak form can b obtaind by multiplying th stat Eq.(14)byan appropriat tst function υ() (with υ(l) = ) and intgrating ovr th ntir domain. Using intgration by parts, thr rsults: [ u(, t) u(, t) υ() υ() + α(, t) ] u(, t) +β(, t) υ() d = υ() f (t), (3) whr th boundary conditions hav bn takn into account. With a similar procss, whr q() is now usd as a tst function, w obtain th wak form of th adjoint problm: [ λ(, t) λ(, t) q() q() + α(, t) q() (β(, t)λ(, t)) ] d = q() [u m (, t) u(, t)]. (33) W introduc nt standard polynomial approimations for th trial functions of th stat u(, t) and th adjoint λ(, t), and thir rspctiv tst functions υ(), and q(); lt: u(, t) = λ(, t) = N u i (t)φ i (), υ() = i=1 N λ i (t)φ i (), q() = i=1 N υ i φ i (), (34) i=1 N q i φ i (), (35) i=1 whr N is th numbr of nodal points, φ ar basis functions, and u i,λ i,υ i, q i dnot nodal quantitis. Thn th smi-discrt forms of th stat and adjoint problms can b cast as: M u(t) M λ(t) whr: M ij = C ij = + K(t) u(t) + C(t) u(t) +[K(t) + Q(t)] λ(t) C(t) λ(t) φ i φ j d, K ij = β(, t)φ i φ j d, Q ij = = F(t), (36) α(, t) φ i φ j = G(t), (37) d, (38) β(, t) φ i φ j d, (39) F i = f (t)δ i1, G i =[u m (, t) u(, t)]δ i1. (4) In th abov, δ i1 dnots th Kronckr dlta, u and λ ar th vctors of th nodal stat and adjoint variabls, rspctivly, and customary notation has bn usd for th matrics. Notic that, whras th mass matri M is indpndnt of tim, th stiffnss K, and damping matrics C and Q dpnd on tim, du to th prsnc of th (assumd) tim-dpndnt moduli. 6. Tmporal discrtization To arriv at a solution first for th stat variabl and thn for th adjoint variabl, th smi-discrt forms (36) and (37) nd nt b discrtizd in tim. W not that, whras (36) is an initial valu problm for which u() = u () =, (37) is a final-valu problm for which λ(t ) = λ (T ) =. In th cas of th TD rgularization schm, th timdpndnt matrics K(t), C(t), and Q(t) nd to b appropriatly tratd. Thir tmporal dpndnc stms from th moduli, which in turn, nd also b discrtizd in both spac and tim. Accordingly, lt: α(, t) = N a j (t)ϕ j (), β(, t) = j=1 N b j (t)ϕ j (), (41) j=1 in which ϕ j ar basis functions, and a j and b j dnot nodal valus of α and β, rspctivly. At th first itration w start with th initially gussd modl paramtrs, which w nforc to b constant in tim. Upon updating th modl

8 98 Comput Mch (8) 4:91 94 paramtrs (s Sct. 6.3), thr rsult tim-dpndnt moduli α(, t) and β(, t). Using thos, on could formally procd by updating K(t), C(t), and Q(t), pr (38) (39); w rfr to this approach as Schm TD-I. Clarly, this approach is computationally costly, as it ntails th valuation of various matrics on a pr tim-stp basis. Altrnativly, on could approimat th tmporal dpndnc of th moduli by constant valus (pr invrsion itration); candidat choics includ: a j (t) a j (t), or a j (t) a j (T ), j = 1,...,N, (4) whr th formr prssion rfrs to th man valu of a j (t) ovr th priod (, T ), and th lattr prssion rfrs to its final valu (similarly, for th cofficints of β, b j ). W optd for th scond of (4) (picwis constant in spac, as wll). Thus, ffctivly, ovr an lmnt,(41) can b rwrittn as: α(, t) a (T ), β(, t) b (T ). (43) Consquntly, th lmnt matrics k, c, and q corrsponding to K, C, and Q in (38) (39), rspctivly, ar modifid to now rad: φ φ T k = a (T ) d, c = b (T ) φφ T d, (44) and q =. W rfr to this scond approach as Schm TD-II. Nt, standard tim intgration schms can b usd: hr, w optd for Nwmark s avrag-acclration schm. Through th numrical primnts, w tstd both th tim-dpndnt matrics (Schm TD-I), and th tim-indpndnt matrics (Schm TD-II) cass in solving th stat and adjoint problms; w obsrvd only minor diffrncs in th final stimatd modl paramtrs. 6.3 Modl paramtr updats By solving th stat and adjoint problms, th stat variabl u and th adjoint variabl λ satisfying th first and scond conditions, rspctivly, ar obtaind. Thn, th problm is rducd to a minimization problm with rspct to th modl paramtrs α and β. Hr, notic that th variations with rspct to th modl paramtrs of th augmntd functional, δ α A and δ β A, ar tantamount to th gradint componnts of th misfit functional α J and β J, sinc th sid constraints in th augmntd functional (1) hav alrady vanishd owing to th solution of th stat problm. Thn, what rmains to b don is to provid th mchanism for updating th modl paramtrs: this can b dirctly accomplishd via th control quations drivd for th TD, Tikhonov, or TV rgularizations. W outlin th dtails blow Tim-dpndnt (TD) rgularization Th control Eq. (5) yilds: α(, t) = 1 λ(, t) R α u(, t). (45) Th right-hand-sid of (45) can b radily computd, onc u and λ hav bn obtaind. Thn, th updat for α(, t), on a pr lmnt basis, can b computd by intgrating (45) as shown blow (suprscripts to α and a indicat nw and prvious valus btwn invrsion itrations): α (k+1) (, t) = a (k+1) (t) = a (k) (T ) t φ T R α λ(τ) + 1 t s φ T R α λ(τ) φ T u(τ) dτ φ T u(τ) dτ ds, (46) whr φ, λ, and u ar rstrictd to lmnt, and w hav nforcd conditions (9) so that: a (k+1) () = a (k) (T ), a (k+1) (T ) =. (47) Similarly, for th β updats: w us control Eq. (9) to obtain: β(, t) = 1 u(, t) λ(, t), (48) R β and thrfor, by analogy to (46): β (k+1) (, t) = b (k+1) (t) = b (k) (T ) t φ T ()λ(τ) R β + 1 t s R β φ T () u(τ) τ φ T ()λ(τ) φ T () u(τ) τ dτ dτ ds. (49) Notic that, as vidncd by th abov rlations, th us of larg valus for th rgularization paramtrs dos not distort th misfit information, sinc th rgularization trms vanish as th tim-drivativs of th invrsion variabls bcom or approach zro. Howvr, as it can b sn from (45) and (48), th acclrations of th tim-dpndnt cofficints of th modl paramtrs ar invrsly proportional to th rgularization paramtrs R α and R β. Thrfor, th us of largr rgularization paramtrs will forc th convrgnc rat to b slowr. It is, thus, bnficial to us smallr rgularization paramtr valus (as long as th rsulting optimization problm convrgs). Th ntir invrsion procss with th TD rgularization schm is summarizd in th following Algorithm 1, whr trivial stps hav bn omittd. To acclrat convrgnc it is possibl to us a lin sarch schm that optimizs th stp lngth: accordingly, in (46) and (49),

9 Comput Mch (8) 4: Algorithm 1 Invrsion algorithm using TD rgularization (Schm TD-I or TD-II) 1: Choos R α, R β : St k= 3: St initial guss of invrsion variabls, a (k) 4: St convrgnc tolranc TOL 5: St misfit = TOL + 1 6: whil (misfit > TOL) do 7: Solv th stat problm to obtain u 8: Solv th adjoint problm to obtain λ 9: Comput a using: a (k) = 1 λ u R α 1: Comput b using: b (k) = 1 λ u R β 11: Updat th modl paramtrs α using: a (k+1) (t) = a (k) a(τ) (T ) t τ t s t s dτ + a(τ) dτ ds τ 1: Updat th modl paramtrs β using: b (k+1) (t) = b (k) b(τ) (T ) t dτ + τ b(τ) τ 13: k=k+1 14: nd whil 15: Sav final stimats α(, t) and β(, t) dτ ds and b (k) th intgrals play th rol of th sarch dirction, whil th rciprocal of th rgularization paramtr plays th rol of th stp lngth. Thrfor, by combining any gradint-basd schm with a lin sarch, w could acclrat th convrgnc rat. W mploy th stpst dscnt mthod with an inact lin sarch schm; th approach is summarizd in Algorithm blow Tikhonov or TV rgularization Using th Tikhonov rgularization, th invrsion procss is similar to Algorithm ; th primary diffrnc is in th computation of th sarch dirction. Again, by solving th stat and adjoint problms, w obtain th stat variabl u and th Lagrang multiplir λ, which satisfy th first and scond optimality conditions, undr givn stimats of th modl paramtrs. Thn, th rmaining third and fourth conditions givn by (6) and (3), or by (7) and (31), rspctivly, provid, ssntially, th first-ordr variations (or componnts of th gradint) of th misfit functional with rspct to th modl paramtrs. Using ths first-ordr variations, w can adapt appropriatly any gradint-basd optimization schm. Hr w opt again for th stpst dscnt mthod combind with an inact lin sarch. In othr words, w tak th sarch dirction as th ngativ of th first-ordr variations. Th Algorithm Invrsion algorithm using TD rgularization (acclratd vrsion) 1: Choos θ, ρ, µ, R α, R β : St k= 3: St initial guss of invrsion variabls, a (k) 4: St p(t) ={a(t) b(t)} T 5: St convrgnc tolranc TOL 6: St misfit = TOL + 1 7: whil (misfit > TOL) do 8: Solv th stat problm to obtain u 9: Solv th adjoint problm to obtain λ 1: Comput a using: a (k) = 1 λ u R α 11: Comput b using: b (k) = 1 λ u R β 1: Comput th sarch dirction d k (t) = whr d α (t) = t a(τ) τ dτ + t s t s and b (k) { } dα (t) d β (t) a(τ) τ dτ ds, b(τ) b(τ) d β (t) = t τ dτ + τ dτ ds. 13: whil (J (p k + θ k d k )>J(p k ) + µθ k p k J(p k ) ) do 14: θ ρθ 15: nd whil 16: Updat th stimats p k+1 (t) = p k (T ) + θ k d k (t) 17: k=k+1 18: nd whil 19: Sav final stimats p(t ) ntir invrsion procss using ithr th Tikhonov or th TV rgularization is summarizd in Algorithm 3. 7 Numrical primnts W study th fficincy and robustnss of th algorithms dscribd in th prvious sctions via a sris of numrical primnts. For simplicity, w considr on-dimnsional problms in which svral layrs li ovr th rigid bottom. Through th numrical primnts, w tst th prformanc of th studid algorithms for both smooth and sharp targt profils. In addition, w study th ffct of th rgularization paramtr, and th algorithmic prformanc against noisy data and diffrnt initial stimats. In all cass, w us synthtic data producd in a mannr that avoids committing classical invrs crims. That is, w obtain th masurd rspons numrically by using th (act) targt profil, th prscribd sourc signal, and a vry fin msh that rsolvs mor than adquatly th physics of th problm (.g., th msh dnsity is such that thr ar approimatly 15 points corrsponding to th smallst wavlngth prsnt in th sourc signal). Th msh dnsity w us in th invrsion procss is always diffrnt than th dnsity usd to obtain th synthtic data.

10 93 Comput Mch (8) 4:91 94 Algorithm 3 Invrsion using Tikhonov or TV rgularization 1: Choos θ, ρ, µ, R α, R β : St k= 3: St initial guss of invrsion variabls, a (k) 4: St p ={ab} T 5: whil (misfit > TOL) do 6: Solv th stat problm and obtain u 7: Solv th adjoint problm and obtain λ { dα } 8: Comput th sarch dirction d k = d β whr, for Tikhonov rgularization, d α T d α, = R α λ u d dt, d β T d β, = R β d λ u dt, and for TV rgularization, ( ) [ dα ( ) ] dα 3 d α, =R α ɛ + ɛ d d and b (k) λ u dt, ( ) [ dβ ( ) ] dβ 3 T d β, =R β ɛ + ɛ d d λ u dt. 9: whil (J (p k + θ k d k )>J(p k ) + µθ k p k J(p k ) ) do 1: θ ρθ 11: nd whil 1: Updat th stimats p k+1 = p k + θ k d k 13: k=k+1 14: nd whil 7.1 On smooth profils As a first ampl problm, w considr a simpl cas, in which only th modulus (or wav vlocity) is unknown. W considr l = 1, and thus [, 1]. Th targt modulus profil is a Gaussian bll-lik distribution (Fig. ): α() = [ 1 + p ( (.5).4 )]. (5) Th sourc citation is a rapidly dcaying puls-lik signal givn blow by (51). Both th signal and its Fourir transform ar dpictd in Fig. 3a, b. For th givn targt profil and sourc citation, th masurd (synthtic) data u m (, t) ar shown in Fig. 3c. f (t) = p [ (t.1).5 ]. (51) Basd on th givn masurd data, w invrtd for th modulus profil via th TD rgularization schms (Algorithms 1,), and via th Tikhonov and TV rgularization schms (Algorithm 3). In th tim-dpndnt cas w tstd both Schms TD-I and TD-II Rgularization paramtr ffcts α().5 Targt Fig. A smooth α() targt profil W bgin by choosing an initial distribution for th modulus that is constant throughout th ntir domain and obsrvation priod, that is, w st th initial guss to α(, t) = 1.. W study th ffct th magnitud of th rgularization paramtrs has on th prformanc of th rgularization schms w considrd. For th TD rgularization cas, w trat th sam problm by two ways: first, w us th tim-dpndnt matrics in solving th stat and adjoint problms (Schm TD-I), and scond, w solv thm with th modifid matrics givn by (44) (Schm TD-II). Figur 4 a, b, d, summarizs th α profils obtaind using th TD rgularization schms for diffrnt rgularization paramtrs R α =.1 and.1 (th rsults ar rportd for th sam lvl of misfit rror tolranc, st at. 1 7 ). In addition, th diffrncs of Applid forc, f(t) Amplitud Masurd data, u m (,t) Tim, t Frquncy (Hz) Tim, t Fig. 3 Sourc signal, its frquncy contnt, and surfac data

11 Comput Mch (8) 4: α() Targt 17 rd α() Targt 169 th Diffrnc in prcntag (%) α() (d) Targt () Targt (f) 14 nd α() th Diffrnc in prcntag (%) Fig. 4 Targt, initial, and stimatd profil of α() using th TD rgularization schms th stimatd α() obtaind using Schm TD-I and Schm TD-II ar shown in Fig. 4c, f. Hr, th diffrncs D() ar prssd in prcntag as: D() = α TD II() α TD I () 1 (%), (5) α TD I () whr α TD I () and α TD II () ar th final stimatd α() using Schm TD-I and Schm TD-II, rspctivly. As it can b sn from Fig. 4, only minor diffrncs ar obsrvd btwn th two schms. For both cass, th final stimats ar almost idntical (th diffrncs ar lss than.15%), and thr is no significant diffrnc in th rquird numbr of itrations. In addition, for both cass, th solution convrgs to th targt profil, rgardlss of th valu of th rgularization paramtr, albit at highr computational cost associatd with highr paramtr choics (notic that, hr, w mployd Algorithm 1 in which th lin sarch schm is not implmntd, and, as a rsult, th rgularization paramtr rmains constant throughout th procss). Naturally, smallr paramtr valus fail to achiv th pnalty intnt of th rgularization trm: for ampl in this first problm, whras for R α =.1 satisfactory rsults wr obtaind, for R α =.1 w obsrvd divrgnc and incrasd misfit rrors. Figur 5a summarizs th convrgnc pattrns of th misfit rror for th considrd valus of th rgularization paramtrs in th TD-II cas, whras Fig. 5b, c dpicts th corrsponding convrgnc pattrns for th modulus. W rmark that th convrgnc pattrns using Schm TD-I ar narly idntical to thos obtaind using TD-II. Nt, w considr a Tikhonov rgularization schm using diffrnt valus for th rgularization paramtrs; th rsulting stimatd profils ar shown in Fig. 6. W continu th itration procss until th misfit rror convrgs. As it can b sn in Fig. 6a, whn R α =.1, th solution dos not convrg to th targt profil. Howvr, as smallr valus of th rgularization paramtr ar adoptd, th solution tnds closr to th targt profil (Fig. 6b, c). W faild to obtain a solution as clos to th targt profil (vn whn R α = ) as that obtaind using th TD rgularization. W rmark that on could improv on th Tikhonov cas rsults by, for ampl, using a sourc with a highr-frquncy contnt, or a continuation schm on th rgularization paramtr, or a numbr of othr approachs (.g., L-curv) that aim at th intllignt choic of th rgularization paramtr. W hav not implmntd such schms for ithr th Tikhonov, TV, or TD cass (in principl, all schms stand to bnfit). Nvrthlss, th rsults shown thus far suggst that th TD rgularization prforms bttr than Tikhonov, for blind, yt broad, choics of th rgularization paramtr. Figur 7a shows th misfit rror, whras Fig. 7b, d dpicts th convrgnc pattrns for th modulus, in th Tikhonov rgularization cas. Th rsults obtaind whn th TV schm

12 93 Comput Mch (8) 4: R α =.1 R α =.1 Misfit rror Numbr of itration α () Itration.5 1 α () Itration Fig. 5 Misfit rror and convrgnc pattrns of α() using th TD rgularization schm (Schm TD-II).5. Targt Targt 153 th th.5. Targt 34 th α() α() α() Fig. 6 Targt, initial and stimatd profil of α() using th Tikhonov rgularization schm was usd ar shown in Fig. 8: no major qualitativ diffrncs ar obsrvd btwn th TV and Tikhonov schms for this smooth profil cas stimats ffcts Evn though, in th prcding problm, our initial guss for th modulus distribution is a constant on throughout th domain, it is still clos to th modulus valus at th nds of th domain. To plor th algorithmic prformanc whn th initial guss, whil still constant, is not clos to th nd valus, w sk to rconstruct th profil starting with α(, t) =.7. W tst again using diffrnt rgularization paramtrs, R α = 1.,.1,.1. W st th tolranc to For th TD rgularization schm, w rport only th cas in which th modifid tim-indpndnt matrics (Schm TD-II) ar usd in solving th stat and adjoint problms, givn th comparativ prformanc rportd in th prcding sction. As it can b sn in Fig. 9, th rconstructd profil convrgd to th targt whn th TD rgularization was usd, rgardlss of th valus of th rgularization paramtr R α (for th rang considrd). Th ffct of th rgularization paramtr on th rat of convrgnc is shown in Fig. 9d. Nt, w usd th Tikhonov rgularization schm, ploring various valus for th rgularization paramtr, in an attmpt to rconstruct th profil whn th initial guss was again st at α(, t) =.7. In contrast to th cas whr th initial guss was closr to th targt profil, hr all attmpts faild to convrg as dpictd in Fig. 1. Th prformanc of th TV schm was qually problmatic, as shown in Fig Nois ffcts On of th practical difficultis arising in invrsion is associatd with th prsnc of nois in th masurd rspons (and/or on th sourc) that typically rsult in contaminatd data. It is thus of intrst to study th prformanc of th algorithms in th prsnc of noisy data. Th problm paramtrs ar th sam as thos of th first tst cas, that is, a smooth targt profil with an initial guss at α(, t) = 1. Howvr, now nois-contaminatd data ar usd. To artificially injct nois in th masurd data, w us Gaussian nois (GN) having standard dviation of 1, 5, and 1% with rspct to th maimum amplitud of th masurd data. Th nois-contaminatd data ar shown in Fig. 1. Th rconstructd profils obtaind using th tim-dpndnt rgularization schm ar shown in Fig. 13. For all cass, R α =.1 is mployd. As can b sn in th figur, th

13 Comput Mch (8) 4: Fig. 7 Convrgnc pattrn of stimatd profil of α() using th Tikhonov rgularization schm R α =.1 R α =.1 R α = Misfit rror α () Numbr of itration Misfit rror Itration 4 R α = (d) α () 1 α () Itration Itration R α =1 5 R α =.5. Targt 1 th.5. Targt 9 th.5. Targt 35 th α() α() α() Fig. 8 Targt, initial, and stimatd profil of α() using th TV rgularization schm solution convrgd satisfactorily to th targt in all cass. Howvr, as th lvl of th nois in th signal incrass, th stimations show highr lvl of fluctuations du to th ffcts of th nois. If th rgularization paramtr incrass, similar rsults to th abov can b obtaind, albit at a slowr convrgnc rat: Fig. 13d summarizs th convrgnc of th misfit rror for data contaminatd with diffrnt nois lvls. Nt, w considr again a Tikhonov schm and study th prformanc for a singl cas with 1% Gaussian nois. W adopt diffrnt valus of th rgularization paramtr. Th invrtd profils ar shown in Fig. 14. Figur 14a dpicts th rconstructd profil whn no rgularization is usd (R α = ); intrstingly, dspit th mild oscillations, th profil is crtainly comptitiv to, or vn bttr than, th on producd by th TD rgularization (s Fig. 13c). With R α = 1 5, th rconstructd profil bcoms smoothr, albit somwhat dviating from th targt (Fig. 14b) at th nighborhood of th pak. As th rgularization paramtr incrass, furthr dviation (or failur) is obsrvd (.g., Fig. 14c), sinc th Tikhonov rgularization bgins to wigh havily on th invrsion procss. Whn th TV schm is usd, as shown in Fig. 15, th solutions ar mor oscillatory than th Tikhonov cas and, in this instanc, w could not obsrv any suprior prformanc ovr th TD or th Tikhonov schms.

14 934 Comput Mch (8) 4:91 94 Fig. 9 Targt, initial and stimatd profil of α(), and misfit rror using th TD-II rgularization schm.5. Targt th.5. Targt 1466 th α() α() = α() Targt 148 th Misfit rror (d) R α =1. R α =.1 R α = Numbr of itration.5. Targt 3 nd.5. Targt 7 th.5. Targt 7 th α() α() α() Fig. 1 Targt, initial and stimatd profil of α() using th Tikhonov rgularization schm.5. Targt 6 th.5. Targt 7 th.5. Targt 7 th α() α() α() Fig. 11 Targt, initial and stimatd profil of α() using th TV rgularization schm

15 Comput Mch (8) 4: Masurd data, u m (,t) Masurd data, u m (,t) Masurd data, u m (,t) Tim, t Tim, t Tim, t Fig. 1 Masurd data contaminatd by nois Fig. 13 Targt, initial and stimatd profil of α() using th TD rgularization schm.5. Targt 5 th.5. Targt 167 nd α() α() Targt 143 nd (d) R α =1. R α =.1 R α =.1 α() Misfit rror Numbr of itration.5. Targt 5 th.5. Targt 374 th.5. Targt 153 rd α() α() α() Fig. 14 Targt, initial and stimatd profil of α() using th Tikhonov rgularization schm

16 936 Comput Mch (8) 4: Targt 8 th.5. Targt 19 nd.5. Targt 9 th α() α() α() Fig. 15 Targt, initial and stimatd profil of α() using th TV rgularization schm 7. On sharp profils In th nt sris of numrical primnts w study th prformanc of th algorithms on sharply changing profils. W start again by considring α() as th sol profil to invrt for, and adopt th following stp form for it: 1.. <.3 α() =..3 <.7. (53) 1..7 < 1. W choos α(, t) = 1. as th initial guss and rconstruct th profil using a TD, a Tikhonov, and a TV rgularization schm. W also study th prformanc of ach schm with nois-contaminatd data. For ach schm, w tstd th prformanc using, 1, 5 and 1% Gaussian nois. In Fig. 16, th masurd nois-contaminatd data ar shown. Th stimatd profils of α() using th TD rgularization schm ar shown togthr with th targt and initial profils in Figs. 17 and 18. In particular, Fig. 17a c ar th convrgd profils for diffrnt rgularization paramtrs and nois-fr data. Figur 18a c dpict th rconstructd profils obtaind using th nois-contaminatd data; hr w chos to rport rsults only for R α =.1. Th rsults ar dmd satisfactory in all cass. Nt, th stimatd profils of α() using th Tikhonov rgularization schm ar shown togthr with th targt and initial profils in Figs. 19 and. In contrast to th TD rgularization, hr w obsrv that th solutions dpnd on th choic of th rgularization paramtr. In addition, th rsults using th TV schm ar dpictd in Figs. 1 and : TV s prformanc is similar to th Tikhonov cas. For 1% nois th Tikhonov cas appars suprior to th TD cas (compar Fig. 18c against Fig. c). W sk to rconstruct th sam profil starting with α(, t) =.7 to plor th algorithmic prformanc whn th initial guss is not clos to th nd valu. W again invrt using both a TD and a Tikhonov rgularization schm and, for ach schm, w tstd th prformanc using data contaminatd with, 1, 5 and 1% Gaussian nois. Th stimatd profils of α() using th TD rgularization schm ar shown togthr with th targt and initial profils in Figs. 3 and 4. Similarly to th prvious cas, it can b sn that th profils convrg to th targt profil in all cass, but th convrgnc rat dpnds on th magnitud of th rgularization paramtr (Fig. 3). In addition, th rconstructd profils using noisy data ar dpictd in Fig. 4; hr w chos to rport rsults only for R α =.1. Th rsults ar again dmd satisfactory in all cass. In contrast, Figs. 5 Masurd data, u m (,t) Masurd data, u m (,t) Masurd data, u m (,t) Tim, t Tim, t Tim, t Fig. 16 Masurd data contaminatd by diffrnt lvls of nois

17 Comput Mch (8) 4: Targt th.5. Targt st.5. Targt 1946 th α() α() α() Fig. 17 Targt, initial and stimatd profil of α() using th TD rgularization schm; nois-fr data.5. Targt 5 th.5. Targt 3157 th.5. Targt 1645 th α() α() α() Fig. 18 Targt, initial and stimatd profil of α() using th TD rgularization schm; noisy data.5. Targt 143 th.5. Targt 3 th.5. Targt 6 th α() α() α() Fig. 19 Targt, initial and stimatd profil of α() using th Tikhonov rgularization; nois-fr data.5. Targt 143 th.5. Targt 346 th.5. Targt 66 th α() α() α() Fig. Targt, initial and stimatd profil of α() using th Tikhonov rgularization; noisy data

18 938 Comput Mch (8) 4: Targt 1 th.5. Targt 5 th.5. Targt 368 th α() α() α() Fig. 1 Targt, initial and stimatd profil of α() using th TV rgularization; nois-fr data.5. Targt 11 th.5. Targt 196 th.5. Targt 368 th α() α() α() Fig. Targt, initial and stimatd profil of α() using th TV rgularization; noisy data and 6 dpict attmpts to rconstruct th profil using th Tikhonov and TV schms, rspctivly, which faild in all cass. 7.3 Simultanous invrsion of modulus and damping W rport nt on th prformanc of th algorithm with th TD rgularization whn invrting simultanously for both th modulus α, and th attnuation mtric β. W tstd two diffrnt profils. In Cas I, w choos a sharp modulus profil (multipl layrs of diffrnt moduli), combind with a smooth linarly varying damping profil. Cas II prtains to sharply varying profils for both th modulus and damping. Th targt profils for Cas I ar: for. <.5. for.5 <.5 α() =, (54) 1. for.5 <.75.5 for.75 < 1. β() = 1..5, (55) and for Cas II: α() =, β() =.6 for. <.5, α() =., β() =.8 for.5 <.5, α() = 1., β() =.4 for.5 <.75, α() =.5, β() = 1. for.75 < 1.. (56) W choos constant α(, t) = 1. and β(, t) =.5 as initial gusss for Cas I, and α(, t) = 1. and β(, t) = 1. for Cas II. Th profils ar rconstructd using both nois-fr data, and data contaminatd with 5% Gaussian nois. Th data for ach cas ar shown in Fig. 7; th rgularization paramtrs wr st at R α =.1 and R β =.1. Th stimatd profils ar shown togthr with th targt and initial profils in Figs. 8 and 9. As it can b sn, th prformanc is quit satisfactory; notic that th dscribd procss allows, in ssnc, th rcovry of th numbr of layrs, th matrial composition of ach layr, and th thicknss (or dpth) of ach layr, without having to plicitly dclar thm as modl paramtrs. It is possibl to improv on th damping profil (β), by a judicious choic of R β rlativ to R α ; th spcifics scap th scop of this articl.

19 Comput Mch (8) 4: Targt 414 th.5. Targt 418 th.5. Targt 44 th α() α() α() R α= R α= R α=.1 Fig. 3 Targt, initial and stimatd profil of α() using th TD rgularization schm; nois-fr data.5. Targt 5 th.5. Targt 5 th.5. Targt 5 th α() α() α() Fig. 4 Targt, initial and stimatd profil of α() using th TD rgularization schm; noisy data.5. Targt 3 th.5. Targt 6 th.5. Targt 7 th α() α() α() Fig. 5 Targt, initial and stimatd profil of α() using th Tikhonov rgularization schm.5. Targt 4 th.5. Targt 5 th.5. Targt 7 th α() α() α() Fig. 6 Targt, initial and stimatd profil of α() using th TV rgularization schm

20 94 Comput Mch (8) 4:91 94 Fig. 7 Masurd data Masurd data, u m (,t) Masurd data, u m (,t) Tim, t Tim, t Masurd data, u m (,t) Masurd data, u m (,t) (d) Tim, t Tim, t Fig. 8 Targt, initial and stimatd profils of α() and β() for Cas I using th TD rgularization schm α() Targt 5186 th β() Targt 5186 th Targt 1 th 1. Targt 1 th α(). β() (d) Conclusions W discussd a PDE-constraind optimization approach for rconstructing th matrial profil in a layrd mdium basd on surficial masurmnts of its rspons to surfac citation dirctly in th tim-domain. Th primary focus was on th comparison, via numrical primnts, of spatial rgularization schms against a TD schm. To this nd, w

21 Comput Mch (8) 4: Fig. 9 Targt, initial and stimatd profils of α() and β() forcasiiusingthtd rgularization schm α() Targt 3 th β() Targt 3 th Targt 1 th 1. Targt 1 th α(). β() (d) tstd th algorithmic prformanc for both smooth and sharp profils, and discussd th ffct th rgularization paramtr, noisy data, and initial stimats hav on th invrsion procss. Whil th studis ar not haustiv, basd on th rportd numrical primnts, our obsrvations ar: robustnss to invrsion procsss similar to th on discussd hrin. Acknowldgmnts W wish to thank th rviwrs for many constructiv suggstions. Th TD schm can captur both sharp and smooth profils, whil th Tikhonov and TV rgularizations hibit difficultis whn confrontd with sharp profils. Th solutions obtaind using th TD schm appar lss snsitiv to th rgularization paramtr than th Tikhonov and TV schms. Th TD schm appars mor robust undr inact initial gusss whn compard to th Tikhonov and TV schms. In th prsnc of noisy data, th Tikhonov schm sms to outprform th TD schm only whn initial stimats ar clos to th targt profil. Lastly, w also tndd th dvlopmnt to th cas of th simultanous invrsion for both modulus and damping and rportd on th rconstruction of both smooth and sharp profils. Ovrall, th TD approach appars robust (no failur to convrg), albit at a substantial computational cost ovr ithr Tikhonov or TV rgularizations. Prliminary rsults in two dimnsions with highly htrognous domains support similar conclusions. In our opinion, th TD schm, dspit its highr computational cost, is quit promising in lnding Rfrncs 1. Stoko KH, Wright SG, Bay JA, Rosst JM (1994) Charactrization of gotchnical sits by SASW mthod. In: Woods RD (d) Gophysical charactrization of sits, pp Ri GJ, Lai CG, Spang AW Jr () In-Situ masurmnt of damping ratio using surfac wavs. J Gotch Gonviron Eng 16: Ri GJ, Lai CG, Foti S (1) In-Situ masurmnt of damping ratio using surfac wavs. Gotch Tst J 4: Lai CG, Ri GJ, Foti S, Roma V () Soil dynamics and arthquak nginring. Simultanous Mas Invrsion Surf Wav Disprsion Attnuation Curvs : Tikhonov AN (1963) Solution of incorrctly formulatd problms and th rgularization mthod. Sovit Math Doklady 4: Rudin L, Oshr S, Fatmi E (199) Nonlinar total variation basd nois rmoval algorithms. Physica D 6: Chan TF, Tai XC (3) Idntification of discontinuous cofficints in lliptic problms using total variation rgularization. SIAM J Sci Comput 5(3): Biros G, Ghattas O (1999) Paralll Nwton-Krylov Mthods for PDE-constraind optimization. In: Procdings of Suprcomputing 1999, Portland, Orgon, USA 9. Akclik V () Multiscal Nwton-Krylov mthods for invrs acoustic wav propagation. Dissrtation prsntd in partial