Bollettino di Geofisica Teorica ed Applicata Vol. 56, n. 3, pp ; September 2015 DOI /bgta0156

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1 Bollettino di Geofisic Teoric ed Applict Vol. 56, n. 3, pp ; Septemer 2015 DOI /gt0156 The importnce of the Vp/Vs rtio in determining the error propgtion, the stility nd the resolution of liner AVA inversion: theoreticl demonstrtion M. AleArdi Erth Sciences Deprtment, University of Pis, Itly (Received: Novemer 13, 2014; ccepted: Jnury 25, 2015) ABSTRACT The liner Amplitude-Versus-Angle (AVA) inversion hs ecome stndrd tool in deep-sediments hydrocron explortion since its introduction in the oil nd gs industry. However, in the lst decdes, with the increse of offshore construction ctivity, pplictions of this method hve een lso extended to predict overpressured zones nd/or to evlute the geotechnicl properties of shllow se ottom lyers. Among the input prmeters requested y liner AVA inversion there is the ckground Vp/Vs rtio cross the reflecting interfce nd Vp/Vs rtio of two is frequently ssumed. This vlue is usully very close to the true rtio in cse of deep, compcted sediments ut it cn e gross underestimtion of the true vlue in cse of shllow or overpressured sediments. Despite tht, the importnce of the ckground Vp/Vs rtio in AVA inversion is frequently underrted nd thus I consider two frequently used pproximtions of the Zoeppritz equtions to study their impct on the outcomes of liner AVA inversion: the three-term Aki nd Richrds eqution nd the two-term Ursench nd Stewrt formul. These equtions re then nlysed, vrying the Vp/ Vs vlue, using tools frequently pplied in sensitivity nlysis. It turns out tht the ckground Vp/Vs rtio controls the error propgtion from dt to model spce nd determines the cross-tlk etween the inverted prmeters. Moreover, n incresing Vp/Vs rtio cuses decrese of stility of the AVA inversion nd worsens the estimte of the Vs contrst t the reflecting interfce. Key words: AVA inversion, Vp/Vs rtio. 1. Introduction Amplitude-Versus-Angle (AVA) methods exploit the vrition in seismic reflection mplitudes with incresing incidence ngle to infer the contrsts in seismic velocities nd densities t the reflecting interfces (Cstgn et l., 1998). For this chrcteristic, AVA techniques hve een extensively used worldwide for lithology nd fluid prediction in hydrocron explortion (e.g., Ostrnder, 1984; Rutherford nd Willims, 1989; Mzzotti, 1990, 1991; Grion et l., 1998; Mzzotti nd Zmoni, 2003). Most AVA methods re sed on the Zoeppritz equtions (Zoeppritz, 1919) which descrie the vrition in seismic mplitude with incresing ngle of incidence for plne wve incident 2015 OGS 357

2 Boll. Geof. Teor. Appl., 56, Alerdi on n interfce seprting two semi-infinite hlf spces. The system of equtions formulted y Zoeppritz is lgericlly complex nd mny different pproximted formuls hve een derived to simplify nd linerise the inversion. These simplified equtions, vlid under certin ssumptions, re those frequently used in AVA inversion nd interprettion (Wng, 1999; Ursench nd Stewrt, 2008). Performing liner AVA inversion, n verge Vp/Vs rtio cross the interfce equl to two is usully ssumed (Cstgn et l., 1998). This rtio is good pproximtion for clssicl deep hydrocron explortion, ut generlly it is n underestimtion of the true verge rtio in cse of seed sediments. Therefore, this pproximtion my constitute source of errors when AVA inversion is used for investigting shllow lyers which re usully chrcterized y high Vp/ Vs rtios. In prticulr, due to the increse of offshore construction ctivity in mrine res, relile chrcteriztion of shllow sediments is of gret interest (Theilen nd Pecher, 1990; Ayres nd Theilen, 1999). For this, to identify sfe zone where instlling underwter structures, the outcomes derived from AVA method re frequently used for shllow hzrd ssessment nd well site nlysis (Riedel nd Theilen, 2001). In these explortion phses the elstic properties derived from AVA inversion (P- nd S-wve velocities nd ulk density) re often converted into geotechnicl properties (e.g., sher strength nd elstic moduli) needed for engineering purposes. Therefore, in this work I wnt to ssess the impct of the ssumed Vp/Vs rtio on the expected resolution nd uncertinties ssocited with ech inverted prmeter. To this end, I nlyze the three-term Aki nd Richrds (Aki nd Richrds, 1980) eqution nd the two-term Ursench nd Stewrt eqution (Ursench nd Stewrt, 2008), mking use of the sensitivity nlysis tools pplied to the inversion kernel. Firstly, I study how the Vp/Vs vlue influences the condition numer, the mgnitude of the eigenvlues nd the orienttion of ssocited eigenvectors in model spce, then, studying the model resolution nd covrince mtrices, I nlyze how the Vp/Vs rtio determines oth the expected resolution of ech inverted prmeter nd the error propgtion from dt spce to model spce. 2. Inverse prolems, sensitivity nlysis nd SVD decomposition A seismic inverse prolem ims to estimte model prmeters (m) from collected dt (d) minimizing the misfit etween predicted nd oserved dt (Trntol, 2005). If we ssume tht the fundmentl physics is dequtely understood, function G, my e specified relting m nd d: The simplest inverse prolems re those tht cn e represented y n explicit liner eqution d = Gm, where G tkes mtrix form. Mny importnt seismic inverse prolems re liner, such s the AVA inversion performed y pplying pproximtions of the Zoeppritz equtions. One commonly used mesure of misfit etween oserved dt nd modelled dt in solving n inverse prolem is the L 2 norm of the residuls. A model tht minimizes this L 2 norm is clled lest-squres solution. The lest-squres solution for liner inverse prolems cn e derived using the following eqution (lso clled the norml equtions solution): (1) (2) 358

3 The importnce of the Vp/Vs rtio in liner AVA inversion Boll. Geof. Teor. Appl., 56, where the superscript T indictes the trnspose. In compct form, the solution of liner inverse prolem cn e written s follow: (3) where G -g is clled the generlized inverse. For common overdetermined lest-squres prolem, this mtrix is equl to:. (4) However, to solve n inversion prolem, one must not only find solution tht est fits the oserved dt ut should lso investigte the reltion etween the estimted model nd the true model or, in other words, nlyze which properties of the true model re resolved in the estimted model. This issue cn e pproched with the sensitivity nlysis method. For liner inverse prolems this nlysis essentilly consists in computing the model covrince nd model resolution mtrices.the model resolution mtrix (R) descries how well the predicted model mtches the true one. It cn e demonstrted (Aster et l., 2005) tht the resolution mtrix for liner inverse prolem cn e computed s follows:. (5) If R is equl to n identity mtrix ech model prmeter is perfectly resolved nd uniquely determined. When R is not equl to the identity mtrix some prt of the prolem is not perfectly resolved nd the finl solution is lso influenced y the off-digonl terms [see Aster et l. (2005) for complete discussion].to understnd how n error in the dt propgtes s n error in the estimted model, it is useful to define the model covrince mtrix (C m ). If the dt re ssumed to e uncorrelted nd ll hve equl vrince, the covrince mtrix (unit covrince mtrix) is given y:. (6) The unit covrince mtrix is mesure of how uncorrelted noise with unit vrince in the dt is mpped into uncertinties in the estimted model prmeters. The digonl terms indicte the vrince ssocited with ech model prmeter, wheres the off-digonl terms indicte covrinces. The model resolution nd model covrince mtrices re functions of only the dt kernel (the G mtrix in Eq. 1) nd the -priori informtion dded to the prolem. Another useful tool in pproching inverse prolems is the Singulr Vlue Decomposition (SVD). According to this method the mtrix G is roken down into the product of three mtrices: (7) where S is digonl mtrix of singulr vlues, V is the mtrix of eigenvectors in model spce nd U contins the eigenvectors in dt spce. The SVD decomposition is essentil in sensitivity nlysis ecuse it permits to get etter understnding of the physicl mening of the G mtrix. Moreover, the SVD method is lso powerful tool for solving ill-conditioned lestsqures prolems. In these prolems, the process of computing n inverse solution is extremely unstle nd smll chnge in the mesurements cn led to lrge chnge in the estimted model. In these cses the G mtrix is chrcterized y high condition numer, which is the rtio etween the highest nd the smllest singulr vlues of the G mtrix. Therefore, in order to stilize the inversion, the Truncted SVD method (T-SVD) cn e pplied. This method is imed t eliminting the smllest singulr vlues of the G mtrix nd t reducing the condition numer. We py price for this stility in tht the regulrized solution hs decresed resolution. Very detiled informtion out geophysicl inverse prolems cn e found in Aster et l. (2005) nd Trntol (2005). 359

4 Boll. Geof. Teor. Appl., 56, Alerdi 3. The Aki nd Richrds nd Ursench nd Stewrt pproximtions Strting from the Zoeppritz equtions, Aki nd Richrd (1980) provided pproximtion for P-P wve reflection coefficients tht is vlid for smll physicl contrsts nd smll incidence ngles (generlly less thn degrees). This eqution cn e written s where R pp is the P-wve reflection coefficient, θ is the verge of P-wve incidence nd P-wve trnsmission ngles cross the interfce, nd α, β, nd ρ, indicte the P-wve velocity, S-wve velocity nd density, respectively. In Eq. 8, Δx is the difference of property x cross the reflecting interfce (x 2 -x 1 ) nd indictes the verge property cross the interfce (x 2 -x 1 )/2, wheres γ is the reciprocl of the ckground Vp/Vs rtio: (8) (9) where the suscripts 1 nd 2 refer to the overlying nd underlying medi, respectively. The Aki nd Richrds eqution is inverted to retrieve the reltive contrsts t the reflecting interfce tht cn e conveniently written s In this form R p, R s nd R d indicte the P-wve, S-wve nd density reflectivity, respectively. To reduce the physicl miguity inherent to the AVA method (Drufuc nd Mzzotti, 1995) nd to stilize the inversion process, the numer of unknowns cn e reduced. To this end twoterm pproximtions of the Zoeppritz equtions re frequently used. In prticulr in this work I consider the Ursench nd Stewrt eqution (Ursench nd Stewrt, 2008): (10) (11) where the density term is incorported into the P nd S-impednce reltive contrsts t the reflecting interfce expressed y R I nd R J, respectively: where Ip nd Is represent the P nd S-impednce, respectively. These liner pproximtions of the Zoeppritz equtions enle the description of the reltionship etween the oserved AVA response (R pp ) nd the model prmeters (m) in liner, compct, mtrix form:. (13) In this form the G mtrix contins the three- or the two-term eqution, wheres the vector m contins the inverted prmeters (elstic or impednce contrsts t the reflecting interfce). The singulr vlue decomposition of the G mtrix (G=USV T ; see Eq. 7) splits the reflectivity R pp (θ) into three orthogonl components in oth dt spce nd model spce. (12) 360

5 The importnce of the Vp/Vs rtio in liner AVA inversion Boll. Geof. Teor. Appl., 56, The energy of ech component is given y the corresponding eigenvlue. If the orders of mgnitude of the eigenvlues re significntly different from ech other, then high signlto-noise rtio is needed to estimte the signl in the low-energy directions. It is interesting to consider the physicl mening of the decomposition. The eigenvectors V re sis in the model spce. The eigenvlues S represent the reflected energy due to medium perturtions long the eigenvectors in model spce. The mplitude versus ngle effects of the reflections re descried y the eigenvectors in dt spce (U), which re three orthogonl functions (De Nicolo et l., 1993). 4. Condition numer, eigenvlues nd eigenvectors in model spce I now compre the condition numer for the three- nd the two-term inversions y vrying the ckground Vp/Vs rtio (in ll the following considertions when referring to Vp/Vs>> 2 Vp/Vs rtio equl to 8 is ssumed). I remind tht high condition numers indicte n illconditioned prolem. Therefore, I cn determine how the Vp/Vs rtio influences the stility of the inverse prolem. The threshold of stility of liner AVA inversion cn e pproximtely estlished t condition numer etween 200 nd 500 (round -40 to -50 db). If we fix this threshold t 300 (dshed line in Fig. 1), we cn see tht in cse of Vp/Vs=2 (or Vs/Vp=0.5, common rtio used in deep sediment explortion), the inverse prolem ecomes stle s we pss from the three-term pproximtion (red curve in Fig. 1) to the two-term pproximtion (lue curve in Fig. 1). Conversely, when the Vp/Vs rtio is very high (or Vs/Vp pproches 0), s it occurs for shllow or seed sediments, the inverse prolem is ill-conditioned even if two-term pproximtion is considered. Therefore, in the cse of liner AVA inversion with very high Vp/Vs rtios, regulriztion is needed to stilize the inversion. A common method used for this purpose is the truncted singulr vlue decomposition (T-SVD) tht consists in the suppression of the smllest singulr vlues of the G mtrix. Let us consider the sensitivity nlysis, reminding tht the threshold of stility rnges etween 200 nd 500. I strt with the three-term Aki nd Richrds eqution. Fig. 2 shows the singulr vlues of the G mtrix, where we cn see tht, independently from the Vp/Vs rtio, the first singulr vlue contins lmost ll of the signl energy; the second one is negligile for smll incidence ngles nd, lthough it increses t higher ngles, is lwys db elow the first singulr vlue. The third singulr vlue is very smll for ll of the ngle rnge nd, in prcticl cses, will e covered y noise nd should e eliminted to stilize the inversion. These results evidence tht for oth Vp/Vs=2 nd Vp/Vs>>2, only one liner comintion of prmeters (the comintion tht corresponds to the first eigenvector) cn e relily estimted t low ngles. The estimtion of two independent comintions (the first nd second eigenvector) requires wider ngles nd is chrcterized y poorer signl-to-noise rtio in the direction of the second singulr vlue. The estimtion of three independent comintions of prmeters is clerly n ill-conditioned prolem. Concerning the effect of the ckground Vp/Vs on the stility of the inversion, we cn compre the results for Vp/Vs= 2 (Fig. 2) with those for Vp/Vs>>2 (Fig. 2): the vlues ssocited with the second nd third singulr vlues decrese s the Vp/Vs rtio increses nd this fct explins why the stility of the inversion decreses for incresing Vp/Vs rtios. 361

6 Boll. Geof. Teor. Appl., 56, Alerdi Fig. 1 - Condition numer for the three-term Aki nd Richrds eqution (red line) nd the two-term Ursench nd Stewrt eqution (lue line) for vrying ckground VS/VP rtios. The dotted line represents the ssumed threshold of stility for the liner AVA inversion. Fig. 2 - Singulr vlues of the G mtrix for the three-prmeter inversion. Pnels nd correspond to VP/VS =2 nd VP/VS >>2, respectively. Fig. 3 - Eigenvectors in model spce versus the mximum incidence ngle for three-term inversion. Pnels nd correspond to VP/VS =2 nd VP/VS >>2, respectively. For ech cse, the first, second nd third eigenvector re represented from top to ottom. Now I move on to descrie the orienttion of the eigenvectors in model spce for the threeterm inversion. Firstly, I nlyze the Aki nd Richrds eqution ssuming Vp/Vs rtio of two (Fig. 3). For low ngles, the Rp nd Rd components re equl nd Rs is zero. Therefore, the vector points in the direction of P-impednce perturtions. This result is ovious: it is known tht the norml incidence reflection coefficient depends on the coustic impednce contrst only. The Rs component ecomes significnt for higher ngles. The second eigenvector points, pproximtely, in the direction of S-impednce perturtions, wheres the third eigenvector is difficult to interpret ecuse it depends y comintion of different perturtions nd does not hve ny prticulr physicl mening. 362

7 The importnce of the Vp/Vs rtio in liner AVA inversion Boll. Geof. Teor. Appl., 56, Fig. 4 - Singulr vlues of the G mtrix for two-prmeter inversion. Pnels nd represent the V P /V S =2 nd V P /V S >>2 cses, respectively Fig. 5 - Eigenvectors in model spce versus the mximum incidence ngle for two-term inversion. Prts nd correspond to V P /V S =2 nd V P /V S >>2, respectively. For ech cse, the first nd second eigenvector re represented from top to ottom. In the Vp/Vs>>2 cse (Fig. 3), oth the first nd second eigenvectors, ssocited with the first nd second singulr vlues, point towrd the P-impednce. Conversely, only the third eigenvector, ssocited with the smllest singulr vlue, points entirely in the R s direction. This fct indictes tht this component spns the null-spce of the G mtrix nd thus the S-wve velocity plys very minor role in determining the AVA response. Moreover, y compring the first nd second eigenvectors for Vp/Vs=2 nd Vp/Vs>>2, we cn see tht n incresed Vp/Vs rtio increses the cross-tlk etween the P-velocity term R p nd the density term R d : smller distnce is oserved etween the R p nd R d components s the Vp/Vs rtio increses. This indictes tht n independent estimtion of these two prmeters is more prolemtic in the cse of high Vp/Vs vlues. These oservtions llow us to drw some importnt conclusions. First, the difficulty of chieving relile R s estimtion with incresing Vp/Vs vlues; second, the cross-tlk etween R p nd R d lso increses s the Vp/Vs rtio increses. Let us now consider the sensitivity nlysis for the two-term Ursench nd Stewrt eqution. Bsed on the singulr vlues of the G mtrix (Fig. 4), it is cler tht the stility of the prolem is gin influenced y the Vp/Vs rtio, confirming the oservtion mde on the condition numer (Fig. 1): the two-term liner AVA inversion ecomes stle if the Vp/Vs vlue is sufficiently low. For wht concerns the eigenvectors in model spce for the two-term pproximtion we cn see tht in the cse of Vp/Vs=2 (Fig. 5), the first eigenvector points towrd the P-impednce for smll ngles, wheres the R J component is not null only if lrge incidence ngles (greter thn 20 degrees) re considered. Conversely, if we increse the Vp/Vs rtio (Fig. 5), the first eigenvector points towrd the P-impednce for the entire ngulr rnge. In this cse, the R J prmeter spns the null spce of the G mtrix, indicting tht, to try estimting the R J term, sufficiently low Vp/Vs rtio is needed. Note tht the two-term inversion is stle for sufficiently low Vp/Vs vlues only (see Fig. 1), nd in these cses, the use of the second eigenvector llows the inversion to 363

8 Boll. Geof. Teor. Appl., 56, Alerdi extrct the R J prmeter. In the Vp/Vs=2 cse, this eigenvector cn e used in the inversion nd the R J informtion cn e recovered with good degree of ccurcy. Insted, in cses of Vp/Vs>>2, to stilize the inversion the trunction of the second singulr vlue (nd the ssocited eigenvector) is needed nd this renders the estimtion of R J impossile. 5. Model resolution nd unit covrince mtrices The model covrince nd resolution mtrices descrie how the error in the dt spce propgtes in the model spce nd how well the estimted prmeters mtch the true ones, respectively. I strt y nlyzing the unit covrince mtrix (computed y ssuming n identity dt covrince mtrix) for the lest-squres inversion, for which the model resolution mtrix is equl to n identity mtrix [see Aster et l. (2005) for rigorous mthemticl demonstrtion]. Fig. 6 shows the unit covrince mtrices computed for Vp/Vs=2 (Figs. 6 nd 6c) nd Vp/Vs>>2 (Figs. 6 nd 6d) nd for oth the three- nd two-term pproximtions. Note tht the order of mgnitude of the errors decreses pssing from the three- to two-term inversion (for ny ckground Vp/Vs vlue), nd pssing from Vp/Vs>>2 to Vp/Vs=2, for oth prmetriztions. Also note tht the Vp/Vs rtio determines the mount nd the distriution of error propgtion from the dt to the model spce. In fct, for high Vp/Vs vlues (Figs. 6 nd 6d), the prmeters most contminted y noise re those ssocited with the S-wve velocity (R s nd R J ). Insted, if the Vp/Vs is equl to two (Figs. 6 nd 6c), the error is more homogeneously distriuted lthough, even in this cse, the error most strongly ffects R s nd R J. Now I eliminte the smllest singulr vlue of the G mtrix (pplying the T-SVD method) nd recompute the unit covrince nd the model resolution mtrices. Let us first consider the model resolution mtrices (Fig. 7). For the three-term inversion nd in the cse of Vp/Vs=2, the three prmeters cn e recovered with lmost the sme resolution, even if the lowest resolution is lwys relted to R s (Fig. 7). Conversely, it is cler tht for the Vp/Vs>>2 cse (Fig. 7) we otin null resolution for the R s prmeter nd good resolution for oth R p nd R d (note tht the resolution is expressed y the digonl terms). If we reduce the dimension of the model spce considering the two-term eqution, we cn see tht for oth cses (Figs. 7c nd 7d), the R I prmeter is chrcterized y the highest resolution. Also in this cse, the resolution of the Vs-relted prmeter R J decreses with the incresing Vp/Vs rtio. Now I descrie the unit covrince mtrix, which is otined fter pplying the T-SVD method (Fig. 8) to eliminte the smllest singulr vlue of the G mtrix nd to stilize the inversion. I strt with the three-term inversion. For high Vp/Vs rtios (Fig. 8), the error is mpped onto the R p nd R d prmeters ecuse the third eigenvlue, pointing towrd the R s prmeters, hs een eliminted y the trunction. We lso oserve strong negtive covrince (expressed y the off-digonl terms nd indicting correltion) etween R p nd R d, which confirms the strong cross-tlk etween these two unknowns nd the difficulties of chieving n independent estimtion. As expected, oth the correltion etween R p nd R d nd the error mgnitude decrese if we consider Vp/Vs rtio equl to two (Fig. 8). In this cse, the error is more homogeneously distriuted mong the three prmeters. Also y oserving the unit -covrince mtrix for the two-term inversion, we see tht the error mgnitude decreses from the Vp/Vs>>2 (Fig. 8d) cse to the Vp/Vs=2 cse (Fig. 8c). Moreover, the trunction of the 364

9 The importnce of the Vp/Vs rtio in liner AVA inversion Boll. Geof. Teor. Appl., 56, c d Fig. 6 - Unity covrince mtrices in the cse of lest-squres inversion. Pnels nd c represent the V P /V S =2 cse nd the ssocited three- nd two-term inversions, wheres the V P /V S >>2 cse nd the ssocited three- nd two-term inversions re shown in pnels nd d. c d Fig. 7 - Model resolution mtrices fter pplying the T-SVD method. Pnels nd c represent the V P /V S =2 cse nd the ssocited three- nd two-term inversions, wheres the V P /V S >>2 cse nd the ssocited three- nd two-term inversions re shown in pnels nd d. Fig. 8 - Unit covrince mtrices fter pplying the T-SVD method. Pnels nd c represent the V P /V S =2 cse nd the ssocited three- nd two-term inversions, wheres the V P /V S >>2 cse nd the ssocited three- nd two-term inversions re shown in pnels nd d. second singulr vlues (in the cse of Vp/Vs>>2) results in the error eing mpped entirely onto the R I prmeters, wheres in the cse of Vp/Vs=2, the error lso ffects the R J vlues. Finlly, y compring Figs. 6 nd 8, we cn see tht in ny cse the T-SVD method reduces the order of mgnitude of the error ssocited with ech prmeter estimtion. 6. Conclusions The sensitivity nlysis highlights the strong influence of the ckground Vp/Vs rtio on oth the stility of the liner AVA inversion nd on the physicl mening expressed y the G mtrix. Specificlly, I hve nlyzed how the Vp/Vs vlue influences the condition numer, the orienttion of eigenvectors in model spce, the resolution for ech inverted prmeter nd the error propgtion from dt to model spce. From the nlysis of the condition numer, I note tht if Vp/Vs is equl to 2 the inverse prolem ecomes stle s I pss from the three-term (contrsts in P-wve velocity, S-wve velocity nd density) to the two-term (contrsts in P-wve nd S-wve impednces) pproximtion. Conversely, when the Vp/Vs rtio is very high (s occurs for overpressured or shllow seed sediments), the inverse prolem is ill-conditioned even if two-term pproximtion is considered. Therefore, in the cse of liner AVA inversion with very high Vp/Vs rtios, the ppliction of regulriztion method (i.e., the T-SVD method) is needed to stilize the inversion process. Moreover, the orienttion of the eigenvectors in 365

10 Boll. Geof. Teor. Appl., 56, Alerdi model spce shows tht for high Vp/Vs rtios the eigenvectors ssocited with the Vs-relted prmeter (R s nd R J ) spn the null-spce of the inversion kernel. This fct, comined with the oservtion of the resolution mtrices, highlights tht the determintion of the Vs contrst (or the S-impednce contrst) for shllow sediments or t se ottom ecomes hopelessly non-unique prolem in the cse of high Vp/Vs vlues. Finlly, I oserve tht when incresing the Vp/Vs vlues the error propgtion from dt to model spce ecomes more nd more severe. The sme hppens to the cross-tlk etween R p nd R d, mking their independent estimtion impossile. Therefore, it emerges tht liner AVA inversion is not suitle to investigte underconsolidted or overpressured sediments tht re usully chrcterized y very high Vp/Vs rtios. In those cses, it is likely tht non liner nd wide-ngle inversion pproches re needed. RefeReNceS Aki K. nd Richrds P.G.; 1980: Quntittive seismology: theory nd methods. WH Freemn, Sn Frncisco, CA, USA, vol. 1 nd 2, 557 nd 373 pp. Aster R.C., Borchers B. nd Thurer C.H.; 2005: Prmeter estimtion nd inverse prolems. Elsevier Acdemic Press, London, Englnd, 296 pp. Ayres A. nd Theilen F.; 1999: Reltionship etween P- nd S-wve velocities nd geologicl properties of ner-surfce sediments of the continentl slope of the Brents Se. Geophys. Prospect., 47, Cstgn J.P., Swn H.W. nd Foster D.J.; 1998: Frmework for AVO grdient nd intercept interprettion. Geophys., 63, De Nicolo A., Drufuc G. nd Rocc F.; 1993: Eigenvlues nd eigenvectors of linerized elstic inversion. Geophys., 58, Drufuc G. nd Mzzotti A.; 1995: Amiguities in AVO inversion of reflections from gs-snd. Geophys., 60, Grion S., Mzzotti A. nd Spgnolini U.; 1998: Joint estimtion of AVO nd kinemtic prmeters. Geophys. Prospect., 46, Mzzotti A.; 1990: Prestck mplitude nlysis methodology nd ppliction to seismic right spots in the Po Vlley, Itly. Geophys., 55, Mzzotti A.; 1991: Amplitude, phse nd frequency versus offset pplictions. Geophys. Prospect., 39, Mzzotti A. nd Zmoni E.; 2003: Petrophysicl inversion of AVA dt. Geophys. Prospect., 51, Ostrnder W.; 1984: Plne-wve reflection coefficients for gs snds t non-norml ngles of incidence. Geophys., 49, Riedel M. nd Theilen F.; 2001: AVO investigtions of shllow mrine sediments. Geophys. Prospect., 49, Rutherford S.R. nd Willims R.H.; 1989: Amplitude-versus-offset vritions in gs snds. Geophys., 54, Trntol A.; 2005: Inverse prolem theory nd methods for model prmeter estimtion. Soc. Ind. Appl. Mth., Phildelphi, PA, USA, 342 pp., doi: / Theilen F. nd Pecher I.A.; 1990: Assessment of sher strength of the se ottom from sher wve velocity mesurements on ox cores nd in-situ. In: Hovem J.M., Richrdson M.D. nd Stoll R.D. (eds), Sher Wves in Mrine Sediments, Kluwer Acdemic Pulishers, Dordrecht, the Netherlnds, pp Ursench C.P. nd Stewrt R.R.; 2008: Two-term AVO inversion: equivlences nd new methods. Geophys., 73, Wng Y.; 1999: Approximtions to the Zoeppritz equtions nd their use in AVO nlysis. Geophys., 64, Zoeppritz K.; 1919: Erdeenwellen VII. Nchrichten von der Gesellschft der Wissenschften zu Göttingen, Mthemtisch-Physiklische Klsse, corresponding uthor: Mtti Alerdi Diprtimento di Scienze dell Terr, Università Vi Snt Mri 53, Pis, Itly Phone: ; fx: ; emil: mtti.lerdi@for.unipi.it 366