Main Menu SUMMARY INVERSE ACOUSTIC SCATTERING THEORY. Renormalization of the Lippmann-Schwinger equation

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1 of the Lippmann-Schwinger equation Anne-Cecile Lesage, Jie Yao, Roya Eftekhar, Fazle Hussain and Donald J. Kouri University of Houston, TX, Texas Tech University at Lubbock, TX SUMMARY We report the extension of the inverse acoustic scattering approach presented in (Kouri and Vijay, 003) from the use of both reflection and transmission data (R k /T k ) to the sole use of the reflection data (R k ). The approach consists in combining two ideas: the renormalization of the Lippmann-Schwinger equation to obtain a Volterra equation framework (Kouri and Vijay, 003) and the formal series expansion using reflection coefficients (Moses, 1956). The benefit of formulating acoustic scattering in terms of a Volterra kernel is substantial. Indeed the corresponding Born-Neumann series solution is absolutely convergent independent of the strength of the coupling characterizing the interaction. We derive new inverse acoustic scattering series for reflection data which we evaluate for test cases both analytically and numerically (Dirac-δ interaction and the square well or barrier). Our results compare well to results obtained by (Weglein et al., 001) for the square barrier and to previous results obtained in (Kouri and Vijay, 003) using both transmission and reflection data. INTRODUCTION Inverse acoustic scattering methods pioneered by Weglein and co-workers (Weglein et al., 1997, 001) have a significant advantage in comparison to other data inversion methods like Full Waveform Inversion (Tarantola, 1984; Pratt, 1999). Indeed they are the only methods for which the actual and the estimated reference are not assumed to be equal. The inversion for medium properties uses only reflection data and reference medium properties. The methods are non-iterative in the sense that the reference medium is never updated. The approach uses the Born-Neumann power series solution of the acoustic Lippmann-Schwinger equation and a related expansion of the interaction in orders of data (Jost and Kohn, 195; Moses, 1956; Razavy, 1975). Nevertheless the approach is limited by the finite radius of convergence of the Born-Neumann series of the acoustic Lippmann Schwinger equation. Moreover, it depends on the square of the frequency. Despite this limitation, Weglein and coworkers have made significant progress by separating the Born- Neumann series terms into task-related subseries. In (Weglein et al., 1997), they derived an inverse scattering method to separate series for internal multiple events from the one for primaries (reflector imaging subseries). In (Weglein et al., 001), they illustrated the method for 1D acoustic scattering for a square well assuming multiples have been removed. In (Kouri and Vijay, 003), Kouri and co-workers proposed to apply the renormalization transformation of the Lippmann- Schwinger equation into a Volterra equation form to tackle the convergence limitation of the inverse acoustic scattering series. The Volterra form allows for the derivation of a Born- Neumann expansion that converges absolutely independent of the strength of the scattering interaction. It results from the property that the Fredholm determinant of the corresponding integral equation can be shown to be equal to one. Nevertheless, the inverse series was derived for the case of R k /T k data which is not useful for oil prospecting. This paper provides the extension of the approach to the case where one only has reflection data. We compare the results of this extension to those of R k /T k data (Kouri and Vijay, 003) and those of Weglein and co-workers (Weglein et al., 001). INVERSE ACOUSTIC SCATTERING THEORY Renormalization of the Lippmann-Schwinger equation Here we illustrate our method for a 1-D acoustic medium but the approach is completely general and extends to three dimensions. For a normal incident wave upon a 1-D acoustic medium, the pressure P(z, ω) is governed in the frequencydomain formulation by the Helmholtz equation: ( z + ω ω c )P = 0 (z) c (1 c 0 (z) c 0 )P (1) (z) ( z + k )P = k V P () with c 0 (z) the reference acoustic P-wave velocity, c the velocity to invert for, k = ω and V = (1 c c 0 ) the spatial part of c 0 the perturbation interaction. The Lippmann-Schwinger integral equation for the pressure reads as follows: P k + (z) = eikz i dz e ik z z k V (z )P k k + (z) (3) with the usual causal free Green function (including the k factor coming from the interaction), given as: G + 0k = ik eik z z. (4) The renormalization transformation of the Lippmann-Schwinger equation to a Volterra equation results from eliminating the z z argument in the free Green s function in equation (3). This is done by dividing the integration over z into segments from to z and from z to : P k + (z) = eikz ik dz e ik(z z ) V (z )P k + (z ) 013 SEG DOI SEG Houston 013 Annual Meeting Page 4645

2 ik dz e ik(z z ) V (z )P k + (z ) (5) z One then adds and substracts ik z dz e ik(z z) V (z )P k + (z ) to obtain a Volterra equation. P k + (z) = eikz ik dz e ik(z z) V (z )P k + (z) ik dz [e ik(z z ) e ik(z z ) ]V (z )P k + (z ) P k + (z) = eikz + R k e ikz + k dz sink(z z )V (z )P k + (z ) (6) The new Green operator G 0k is given by: G 0k (z,z ) = ksin[k(z z )] η(z z ) (7) with η(z) being the heaviside function. Because of the triangular nature of G 0k, the Born-Neumann series converges absolutely and uniformly on any compact set of z, provided V decays faster than 1 for large z, and has only z integrable singularities. Inverse series for R k /T k data We recall the Volterra inverse series in orders of T the auxiliary transition operator (cf. (Kouri and Vijay, 003)): T = (V G 0k ) n V (8) We express V as a power series in orders of T (cf Weglein) V = j=1 ε j Ṽ j. Then we collect coefficients of each power of ε j ε 1 : T = Ṽ 1, (9) ε : 0 = Ṽ +Ṽ 1 G 0k Ṽ 1, (10) ε 3 : 0 = Ṽ 3 +Ṽ G 0k Ṽ 1 +Ṽ 1 G 0k Ṽ +Ṽ 1 G 0k Ṽ 1 G 0k Ṽ 1, etc., with Ṽ 1 ( k,k) = ir k kπt k. Inverse series for R k data (11) To develop an approach to inversion using solely the reflection data R k, we consider the following expression (Moses, 1956): dz e ikz V (z )P k + (z ) (1) with P k + (z) = eikz + R k e ikz + k dz sin[k(z z )]V (z )P k + (z ) (13) We search for a solution of the form: P + k (z) = U 1k(z) + R k U k (z) (14) which gives U 1k (z) = e ikz + k dz sin[k(z z )]V (z )U 1k (z ) U k (z) U1k (z ) (15) We now can write dz e ikz V (z )[U 1k (z ) + R k U1k (z )] (16) We write the Born-Neumann series U 1k (z) = ( G 0k V ) n k > (17) We substitue in equation (16): dz e ikz V (z )[ ( G 0k V ) n k > + R k ( G 0k V ) n k >] (18) Next, assuming that R k is the sole measured data, we express the perturbation as V (z) = j=1 V j(z) where V 1 is first order in R k,..., V j is jth order in R k. dz e ikz V j (z)[ ( G 0k V j (z)) n k > n=1 n=1 +R k ( G 0k V j (z)) n k >] (19) n=1 with k > e ikz, k > e ikz. This leads to the following expressions which determine V 1, V,.... The first order is given as: dz < k V 1 (z ) k > V 1 (z) = W 1 (k) = R(k) πik dkw 1 (k)e ikz The second order is given as: V (z) = dkw (k)e ikz 1 [ W (k) = dz < k V 1 (z ) G π 0k V 1 k > ] + dz V 1 (z )R k (0) (1) The third order is given as: V 3 (z) = dkw 3 (k)e ikz W 3 (k) = 1 [ dz < k V (z ) G π 0k V 1 k > + dz < k V 1 (z ) G 0k V k > + dz < k V 1 (z )( G 0k V 1 ) k > + dz < k V (z )R k k > + dz < k V 1 (z )R k G 0k V 1 k > ] () 013 SEG DOI SEG Houston 013 Annual Meeting Page 4646

3 We point out that the V 1 expression, equation (0), is the same as arises in (Weglein et al. 001). The higher order terms differ in two ways. First they contain the triangular Green function G 0k. Secondly, there are extra terms that also contain the data R k. ANALYSIS OF NUMERICAL AND ANALYTICAL RE- SULTS FOR THE INVERSE SERIES We test the new Volterra inverse acoustic scattering series in R k (equations (19, 0, 1, )) on two cases: the Dirac δ-function interaction and the case of sound scattering by either a square well or barrier. Application to the Dirac δ-function interaction In the case of the application to scattering by the Dirac δ- function interaction located at depth z = z 0, the reflection and transmission coefficients can be computed analytically. They are given as follows: R k = T k = ikγ + ikγ eikz 0 + ikγ. (3) Using the Volterra inverse series in R k and T k (equations (8, 9,10,11)), we recall the results obtained in (Kouri and Vijay, 003): Ṽ 1 (z) = γδ(z z 0 ), Ṽ (z) = 0. (4) Thus, this Volterra series converges to the exact answer in a single term, with all higher terms being zero. Using the Volterra inverse series in R k, we can demonstrate for the first three orders computing analytically the integrals of equations (0, 1, ), that: V 1 (z) = 4e 4 γ (z 0 z) η(z0 z) (5) V (z) = γδ(z 0 z) 16e 8 γ (z 0 z) η(z0 z) (6) V 3 (z) = 11 γδ(z 0 z) + 63e 1 γ (z 0 z) η(z0 z) e 4 γ (z z 0) η(z z0 ) + 3γ 8 δ (z z 0 ) (7) The new Volterra inverse series has not yet fully converged after three terms. The analytical and numerical evaluation of higher order terms will be the goal of further investigations. However, we anticipate that one will obtain higher derivatives of the Dirac-δ function which, as is the case in the approach of (Weglein et al. 001) for the square barrier, are associated with a Taylor expansion of Heaviside functions. Application to the square well or barrier In the case of the application to scattering by a finite width square well or barrier (V (z) = V 0 η(z)η(a z)), the reflection and transmission coefficients can again be computed analytically. They are given as follows: R k = V 0 ( V 0 ) + i 1 V 0 cot(ak 1 V 0 ) (8) T k = 1 V 0 ie ika V 0 sin(ka 1 V 0 ) R k (9) V 0 is the perturbation interaction amplitude and a is its width. The case V 0 < 0 corresponds to a square well (decrease of the acoustic velocity so that c < c 0 ). The case 0 < V 0 < 1 corresponds to a barrier. Using the Volterra inverse series in R k and T k (equations (8, 9,10,11)), we recall the results obtained in (Kouri and Vijay, 003): Ṽ 1 (z) = V 0 1 V0 η(z z min )η(z max z) (30) with z min = a (1 1 V 0 ) and z max = a (1 + 1 V 0 ). Using the new Volterra inverse series in R k, we can compute the first order V 1 : V 1 (z) = dkw 1 (k)e ikz (31) W 1 R (k) = k (3) πik iv 0 sin(ak 1 V 0 ) = πk[( V 0 )sin(ak 1 V 0 ) + i 1 V 0 cos(ak 1 V 0 )] We can evaluate it numerically by a trapezoidal rule quadrature from the following equation: V 0 V 1 (z) = π(v 0 ) dk sin(ak 1 V 0 ) 1 V 0 cos(ak 1 V 0 )cos(kz) + ( V 0 )sin(ak 1 V 0 )sin(kz) k[1 V 0 (V 0 ) cos (ak 1 V 0 )] We observe that V 1 can be derived analytically as a infinite series of decreasing height barriers: V 1 (z) = C n η(z)η(z n ) (34) n=1 with z n = na 1 V 0. The C n expression can be obtained from equation (33) by expanding the denominator with the Taylor 1 series 1 x = 0 xn. In practice, we use m = 4 as truncation order which gives a good convergence. For different values of V 0 (cf Figures 1, ), we compare the plots of the exact barrier, the first order term Ṽ 1 (z) obtained through the Volterra inverse series with R k, T k data and the first order term V 1 (z) obtained through the new Volterra inverse series in R k both numerically and analytically (equations (33, 34)). As for the Volterra inverse series for R k /T k (Kouri and Vijay, 003), the first-order result has the correct analytical form of a square well or barrier but it has incorrect width and height (or depth). For a barrier (0 < V 0 < 1), the first order result using only R k has higher barrier than the true one (cf Figures 1). While for a well (V 0 < 0), the first order result has higher barrier than the true one (cf Figure ). The height is slightly closer to the exact value than for the R k /T k data results. The R k /T k series spreads the error in the width equally onto both sides of (33) 013 SEG DOI SEG Houston 013 Annual Meeting Page 4647

4 the barrier while the R k series has all the error on the right (this is the same result as Weglein and co-workers (Weglein et al., 001)). Using equation (1), we evaluate the second order V (z) for the R k Volterra inverse series. We compute the first term V 1 (z) and the second term V (z) analytically with: V 1 (z) = dkw 1 (k)e ikz W 1 (k) = 1 dz dz e ik(z +z ) V 1 (z ) G0k (z,z )V 1 (z ) π V 1 (z) 3 ( C j η(z z j)η(z j+1 z) j=0 + C j z 1 4 [δ(z j+1 z) + δ(z j z)] + C jz 1 4 Figure 1: Comparison of V 1 (z) obtained through the new Volterra inverse series in R j 1 k, Ṽ 1 (z) obtained through the Volterra inverse series with R k and the exact barrier. Square C k [δ(z k+ z) δ(z k+1 z) + δ(z k z)]) barrier test case: V 0 = 0.5, a = 1.0. k=0 V (z) = dkw (k)e ikz 1 W (k) = dz V 1 (z )R π k. V (z) = 3 j=0 (35) D( 1 1 V 0 1 V0 + 1 ) j+1 [δ(z z j ) δ(z j+1 z)] (36) with D = z 1 3 j=0 C j. Figure 3 shows the plot of V 1 (z) (Heaviside part). We observe that the heaviside part of V 1 (z) partially corrects the error on the height barrier. V 1 (z) and V (z) both contain Diracδ functions which correspond to Taylor expansion of Heaviside functions. As mentionned in (Weglein et al 001), those terms will correct for the barrier/well width error. Figure : Square barrier test case: V 0 =.0, a = 0.5. CONCLUSION We have reported the extension of the inverse acoustic scattering approach presented in (Kouri and Vijay, 003) to the sole use of the reflection data R k. The first two terms are encouraging but higher order terms are needed to clearly demonstrate convergence. Future works will include the computation both analytically and/or numerically of higher order terms and extending the Volterra approach to D and 3D. ACKNOWLEDGMENTS We thank Total and PGS for their support and the authorization to present this work. The author D.J.K. thanks A.B. Weglein for introducing him to inverse scattering based on the Born- Neumann expansion. Figure 3: Comparison of V 1 (z) +V (z) obtained through the new Volterra inverse series in R k and the exact barrier. Square barrier test case: V 0 =.0, a = SEG DOI SEG Houston 013 Annual Meeting Page 4648

5 EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 013 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES Jost, R., and W. Kohn, 195, Construction of a potential from a phase shift: Physical Review, 87, no. 6, , Kouri, D. J., and A. Vijay, 003, Inverse scattering theory: renormalization of the Lippmann-Schwinger equation for acoustic scattering in one dimension: Physical Review, 67, no. 4, , Moses, H. E., 1956, Calculation of the scattering potential from reflection coefficients: Physical Review, 10, no., , Pratt, R. G., 1999, Seismic waveform inversion in the frequency domain, Part 1: Theory and verification in a physical scale model: Geophysics, 64, , Razavy, M., 1975, Wave velocity in an inhomogeneous medium: The Journal of the Acoustical Society of America, 58, no. 5, , Tarantole, A., 1984, Inversion of seismic reflection data in the acoustic approximation: Geophysics, 49, , Weglein, A. B., F. A. Gasparotto, P. M. Carvalho, and R. H. Stolt, 1997, An inverse-scattering series method of attenuating multiples in seismic reflection data: Geophysics, 6, , Weglein, A. B., 001, An inverse-scattering sub-series for predicting the spatial location of reflectors without the precise reference medium and wave velocity: Presented at the 71 st Annual International Meeting, SEG. 013 SEG DOI SEG Houston 013 Annual Meeting Page 4649