Acoustic imaging in a shallow ocean with a thin ice cap

Transcription

1 Inverse Problems 16 (2000) Printed in the UK PII: S (00) Acoustic imaging in a shallow ocean with a thin ice cap Robert P Gilbert and Yongzhi Xu Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA Received 4 January 2000, in final form 17 April 2000 Abstract. This paper considers the determination, from scattered sound, of a distributed inhomogeneity in a shallow, two-dimensional, ocean with a thin ice cap. Assuming that we know the acoustic properties of the ice cap, we determine the unknown inhomogeneity by sending in incident waves from point sources in prescribed locations, and detect the total acoustic field over a line. In this paper we consider the case the wavenumber, k, is small. Under these circumstances, we obtain a representation for the solution to the direct problem, and prove the uniqueness and existence of the direct scattering problem. The inverse problem is formulated as a regularized minimization problem. Numerical examples illustrating the procedure are presented. (Some figures in this article are in colour only in the electronic version; see 1. Introduction In our previous researches we have discussed a number of models of underwater acoustics in which the effect of various models for the basement were considered. These seabed models might be characterized, physically, into three types: reflecting, elastic and poroelastic. The reflecting seabed case has been investigated extensively by Gilbert and Xu in a sequence of papers [16,22 24,32,33,37,38], they have considered both direct and inverse problems, see also [28, 29]. Physically speaking the reflecting case corresponds to a seabed consisting of a rigid rock formation. Seabeds consisting of a tightly packed sediment might be modelled as being elastic; see, e.g., [9, 10, 13 15, 17]. Buchanan and Gilbert [5 8] have constructed the Green function, using modal solutions, for an acoustic system with a homogeneous water-column over a poroelastic basement [3, 4, 25 27, 30, 31, 35, 36]. Subsequently, Gilbert and Lin [15] extended this approach to the case of a non-homogeneous ocean by using the transmutation methodology [11, 12, 20, 21]. In this paper we extend our methodology to investigate an inverse scattering problem in a shallow ocean having a thin ice cap. 2. A thin-plate, acoustic, approximation for the ice cap In order to formulate a reasonable inverse problem it is necessary to idealize the direct problem. As we want to determine, by acoustics, an unknown inhomogeneity, we idealize the ice cap as a thin elastic plate. We further assume that the ocean is two dimensional and has a uniform depth; we describe this by saying it occupies the region R 2 b := R1 [0,b]. The basement is /00/ $ IOP Publishing Ltd 1799

2 1800 R P Gilbert and Y Xu designated by Ɣ b := R 1 {b}, and the surface by Ɣ 0 := R 1 {0}. If we describe the acoustic displacement vector u, in the ocean, in terms of a potential φ, u = φ, (2.1) the acoustic pressure is given by [1] P = λφ, (2.2) is the two-dimensional Laplacian and λ(z) is the stratified bulk modulus. P then satisfies λ(z)φ + ρ 0 (z)ω 2 φ = 0. (2.3) At the ocean bottom (z = b) we have the acoustic hard condition P = 0, which implies z φ = 0. We model the ice cap as a Kirkhoff plate (rod) (see [34]), and we match the rod z displacement in the vertical direction (at z = 0) with the water column displacement by setting D d4 w dx ρ shω 2 w = P(0) = ω 2 ρ 4 0 (0)φ(0). (2.4) In this equation D is the rod stiffness and ρ s is the ice density. (See also the model of Bedanin and Belinskii [2], for a fluid in a finite region with an elastic boundary.) For our purposes it is necessary to construct the acoustic Green function for the watercolumn ice cap system. To this end, it is convenient to work with the pressure instead of the displacement potential. For the case ρ 0 and λ are constant we are led to consider P + ω2 ρ 0 λ P = 1 2π δ(x x 0)δ(z z 0 ). (2.5) By Fourier transforming this equation we obtain ( z 2 P ˆ ω 2 ) ρ 0 + λ k2 P ˆ = 1 2π δ(z z 0), (2.6) k denotes the modulus of the Fourier transform. For the source at z 0 (0,b)the solution to this equation has the form ( )[ P(z,k) ˆ = cos [b z > ] k 20 k2 A sin ) )] (z < k 20 k2 + B cos (z < k 20 k2, (2.7) k0 2 := ω2 ρ 0 /λ and z > := max[z, z 0 ], and z < := min[z, z 0 ]. In order to determine the coefficients A and B, we use the jump in the derivative of P(z,k)at ˆ z 0, namely P ˆ (z 0 + ) P ˆ (z 0 ) = 1 2π, (2.8) which leads to ( ) ( ) A cos b k 20 k2 B sin b k 20 k2 = 1 1. (2.9) 2π k 20 k2 Another condition is provided by the water-column ice cap boundary conditions, which we derive using the rod displacement, which we obtain from the transformed rod equation as P(0) ˆ ŵ = (Dk 4 ρ s hω 2 ). (2.10) We obtain another condition from P = λφ = ρ 0 ω 2 φ for z z 0. Hence, at z = 0, z P(0) ˆ = ρ 0 ω 2 z ˆφ(0) = ρ 0 ω 2 ŵ,or ( ) P ˆ (0) = A k0 2 k2 sin [b z 0 ] k 20 k2 = ρ 0 ω 2 ŵ, (2.11)

3 Acoustic imaging in shallow ocean with thin ice cap 1801 or A k0 2 k2 = ρ 0ω 2 B Dk 4 ρ s hω. (2.12) 2 Using the above equations to solve for A and B to obtain ( ) cos [b z > ] k 20 k2 P(z,k) ˆ = 2π k0 2 k2 ) ) sin (z < k 20 k2 k0 2 k2 (Dk 4 ρ s hω 2 )/(ρ 0 ω 2 ) cos (z < k 20 k2 cos ( ) b k 20 k2 + k0 2 k2 (Dk 4 ρ s hω 2 )/(ρ 0 ω 2 ) sin ( ). b k 20 k2 (2.13) The acoustic Green function P(x,z) may then be found by performing the inverse Fourier transform, using the Mittag Leffler expansion of P(z,k) ˆ in the complex k-variable. Using standard arguments as in [15], we find a representation for P(x,z,z 0 ) ( ) P(x,z,z 0 ) = i cosh [z > b] ξ j k0 2 L(ξ j,z < ) e i ξ j x 4 j=0 ξ M(ξ j,b) ξ j k0 2 and L(k, z) := sinh ( ) M(k,z) = cosh z ξ k0 2 + ξ k0 2 (Dξ 2 ρ s hω 2 ( ) ) sinh z ξ k 2 ρ 0 ω 2 0 ( ) ( ) z ξ k0 2 ξ k0 2(Dξ 2 ρ s hω 2 )/(ρ 0 ω 2 ) cosh z ξ k0 2. Compare this result with that of Gilbert and Lin [18]. We can also obtain the Green function, using the method of separation of variables, for a water-column with a Kirkhoff-rod type ice cap. Recall that the acoustic pressure satisfies p + k0 2 p = 1 2π δ(x)δ(z z 0), with k0 2 = ω2 ρ 0, (2.14) λ 0 and the equation describing the rod displacement is d 4 w dx ρ shω 2 4 D w + 1 p(0) = 0. (2.15) D We would like to make the problem (2.15) homogeneous of second order now. This is achieved by introducing an additional quantity g and splitting the rod equation into d 2 dx g ρ shω 2 2 D w + 1 p(0) = 0, D d 2 (2.16) dx w g = 0. 2 We use the separation of variables method, seeking solutions in the form p(x, z) = f(z)e ikx, w(x) = αe ikx, g(x) = βe ikx, (2.17)

4 1802 R P Gilbert and Y Xu which leads to the eigenvalue system f + (k0 2 k2 )f = 0, f (b) = 0, (2.18) k 2 β 2 + mα nf (0) = 0, f (0) = ρ 0 ω 2 α, (2.19) k 2 α + β = 0, (2.20) m := ρ shω 2 D, and n := ρ shω 2 D. Theorem 2.1. There exist eigenvalues k i and corresponding eigentriplets {f i,α i,β i } such that under the inner product {f i,α i,β i }, {f j,α j,β j } := (f i,f j ) L 2 [0,b] + β i α j β j α i (2.21) the triplets corresponding to different eigenvalues are orthogonal. Proof. The usual partial integration of (2.18) gives (f i,f j ) L 2 [0,b] + (f i,k0 2 f j ) L 2 [0,b] + f i f j b 0 = k2 i (f i,f j ) L 2 [0,b]. Substituting in (2.19) yields (f i,f j ) L 2 [0,b] + (f i,k0 2 f j ) L 2 [0,b] α i f j (0) = ki 2 (f i,f j ) L 2 [0,b]. Interchanging i and j then produces α j f i (0) α i f j (0) = (ki 2 kj 2 )(f i,f j ) L 2 [0,b]. Now from (2.19) ki 2 β iα j kj 2 β j α i = n(f i (0)α j f j (0)α i ). Using (2.20) ki 2 α iβ j kj 2 α j β i = 0. These last two equations combine to give n(f i (0)α j f j (0)α i ) = (ki 2 kj 2 )(β iα j β j α i ) which in turn leads to (ki 2 kj 2 )(n(f i,f j ) L 2 [0,b] + β i α j ) = 0. As {f i,α i,β i }, {f i,α i,β i } = (f i,f i ) L 2 [0,b] (2.22) this suggests that the Green function expansion for the ocean seabed rod system is given by G(r,z,z 0 ) = i f j (z)f j (z 0 ) 4 j=0 f j 2 e ik j x. (2.23) L 2 [0,b] We have established an appropriate framework allowing us to solve acoustic problems in a finite-depth ocean with a thin ice cap. For the constant index case the Green functions have been calculated via Fourier transforms and separation of variables. These results are to be incorporated into numerical schemes for the direct and inverse scattering problem in the next sections.

5 Acoustic imaging in shallow ocean with thin ice cap Figure 1. Acoustic field in a shallow ocean with a thin ice cap Figure 2. Acoustic field in a shallow ocean with a rigid bottom. 3. The direct scattering problem Now let us consider the case there exists a given inhomogeneity in the water-column R 2 b. Moreover, we assume that the inhomogeneity is contained in a bounded domain with a C 2 boundary having the outward normal vector, ν. The propagating solution { p1 (x, z), if (x, z) R 2 b p(x, z) = \ p 2 (x, z), if (x, z) (3.1) satisfies p 1 + k 2 p 1 = 1 2π δ(x x s)δ(z z s ) in R 2 b \, (3.2) p 2 + k 2 n 2 (x, z)p 2 = 0 in, (3.3) p 1 (x, 0) = 0 on Ɣ 0. (3.4)

6 1804 R P Gilbert and Y Xu Here k 2 n 2 (x, z) = ω 2 /c 2 (x, z) ω is the frequency and c = λ is the celerity. As usual λ ρ is the bulk modulus and ρ the density. d 4 w dx ρ shω 2 4 D w + 1 D p 1(x, h) = 0 on Ɣ b, (3.5) ρ s ω 2 w = p 1 z (x, h) on Ɣ b, (3.6) ρ 1 p 1 = ρ 2 p 2 on, (3.7) p 1 ν = p 2 ν on, (3.8) and p 1 satisfies the outgoing radiation condition. Here (x s,z s ) R 2 b \ is the location of acoustic source, k0 2 the wavenumber in the water and n(x, z) the refraction index of the object, ρ 1,ρ 2 and ρ s are densities of the water, the object and the ice cap, respectively. We rewrite the equations (3.2) and (3.3) in the form p + k 2 p = k 2 m(x, z)p 1 2π δ(x x s)δ(z z s ), (x, z) R 2 b, (3.9) { 0, if (x, z) R 2 m(x, z) = b \ 1 n 2 (x, z), if (x, z). Let G(ξ, ζ ; x,z)be the Green function for an ice capped water-column the acoustic source is located at (ξ, ζ ). Multiplying both sides of (3.9) by G(ξ, ζ ; x,z) and integrating over, we obtain G(ξ, ζ ; x,z)[ p(ξ, ζ ) + (k) 2 p(ξ, ζ )]dξ dζ [ = G(ξ, ζ ; x,z) mp(ξ, ζ ) 1 ] 2π δ(x x s)δ(z z s ) dξ dζ. (3.10) If (x, z) R 2 b \, then G(ξ, ζ ; x,z)[ p(ξ, ζ ) + (k) 2 p(ξ, ζ )] p(ξ, ζ )[ G(ξ, ζ ; x,z) + (k) 2 G(ξ, ζ ; x,z)]dξ dζ = [G(ξ, ζ ; x,z) p(ξ, ζ ) p(ξ, ζ ) G(ξ, ζ ; x,z)]dξ dζ [ = G(ξ, ζ ; x,z) p n (ξ, ζ ) p (ξ, ζ ) G ] (ξ, ζ ; x,z) ds (3.11) n p n (ξ, ζ ) and p (ξ, ζ ) are the limits of p (ξ, ζ ) and p(ξ, ζ ) as (ξ, ζ ) approaches n from the interior of. Similarly, we use p + n (ξ, ζ ) and p +(ξ, ζ ) to denote the limits of p n (ξ, ζ ) and p(ξ, ζ ) as (ξ, ζ ) approaches from the exterior of. For(x, z) /, G(ξ, ζ ; x,z)and G n (ξ, ζ ; x,z) are continuous across. Since p satisfies p +(k)2 p = 0 for (x, z) R 2 b \, we have by the Green formula [ p(x, z) = p + (ξ, ζ ) G n (ξ, ζ ; x,z) p ] + (ξ, ζ )G(ξ, ζ ; x,z) dξ dζ. (3.12) n From (3.10) (3.12) and by (3.7), (3.8), we obtain the representation [ G p(x, z) = p + n p ] + n G ds

7 Acoustic imaging in shallow ocean with thin ice cap 1805 [ G = p n p ] n G ds + (p + p ) G n ds = (p G G p) dξ dζ + (p + p ) G n ds = (p[ G + (k) 2 G] G[ p + (k) 2 p]) dξ dζ + (p + p ) G n ds = G[ p + (k) 2 p]dξ dζ + (p + p ) G n ds = k 2 G(ξ, ζ ; x, z)mp(ξ, ζ ) dξ dζ + G(x s,z s ; x,z) + (p + p ) G n ds, (x, z) R2 b \. (3.13) Similarly, p(x, z) = k 2 G(ξ, ζ ; x, z)mp(ξ, ζ ) dξ dζ +G(x s z s ; x,z) + (p + p ) G ds, n (x, z). (3.14) Combining (3.13) and (3.14), we obtain the representation of the wave field in R 2 b. Now we deduce the integral equations that determine the displacement, u(x, z). Let φ(ξ,ζ) = p + (ξ, ζ ) p (ξ, ζ ). (3.15) Equations (3.13) and (3.14) become p(x, z) + k 2 G(ξ, ζ ; x, z)mp(ξ, ζ ) dξ dζ φ(ξ,ζ) G (ξ, ζ ; x,z)ds n = G(x s,z s ; x,z), (x,z). (3.16) Using (3.7), from (3.13) and (3.14), we have for (x, z), φ(x,z) + 2(ρ 1 ρ 2 ) φ(ξ,ζ) G (ξ, ζ ; x,z)ds ρ 1 + ρ 2 n 2(ρ 1 ρ 2 ) k 2 G(ξ, ζ ; x, z)mp(ξ, ζ ) dξ dζ ρ 1 + ρ 2 = 2(ρ 1 ρ 2 ) G(x s,z s ; x,z), ρ 1 + ρ 2 (x,z). (3.17) Here we have used the facts that G(x s,z s ; x,z) k 2 G(ξ, ζ ; x, z)mp(ξ, ζ ) dξ dζ is continuous across, and v(x,y) = satisfies the jump conditions v + (x, z) = 1 2 φ(x,z) + v (x, z) = 1 2 φ(x,z) + φ(ξ,ζ) G (ξ, ζ ; x,z)ds n φ(ξ,ζ) G (ξ, ζ ; x,z)dξ dζ, n φ(ξ,ζ) G (ξ, ζ ; x,z)dξ dζ, n (x,z). v + (x, z) v (x, z) = φ(x,z), The above analysis establishes the following theorem.

8 1806 R P Gilbert and Y Xu Theorem 3.1. If (p, φ) satisfies the direct scattering problem (3.1) (3.8), then (p, φ) satisfies the integral equations (3.16) and (3.17). Conversely, if (p, φ) C() C( ) is a solution of the integral equations (3.16) and (3.17), then (p, φ) is a solution of the direct scattering problem. Theorem 3.2. If k 2 m := max{k2 m} and ρ 1 ρ 2 are small enough, then the system of integral equations (3.16) and (3.17) has a unique solution. Proof. Rewrite (3.16) in the form p + T p = f, (3.18) T p(x, z) = k 2 G(ξ, ζ ; x, z)mp(ξ, ζ ) dξ dζ and f(x,z)= φ(ξ,ζ) G n (ξ, ζ ; x,z)ds + G(x s,z s ; x,z). If k m is small enough, then the operator I + T has a bounded inverse (I + T ) 1, I denotes the identity operator in L 2 (). Substituting p = (I + T ) 1 f into (3.17), we obtain φ(x,z) + 2(ρ 1 ρ 2 ) Sφ(x,z) 2(ρ 1 ρ 2 ) T (I + T ) 1 f(x,z) ρ 1 + ρ 2 ρ 1 + ρ 2 = 2(ρ 1 ρ 2 ) G(x s,z s ; x,z), ρ 1 + ρ 2 (3.19) Sφ(x,z) = φ(ξ,ζ) G (ξ, ζ ; x,z)ds. n The operators S and T (I + T ) 1 are bounded in L 2 ( ) and L 2 (), respectively. Consequently, the composite operators are also bounded. If ρ 1 ρ 2 is small enough, then (3.19) has a unique solution φ L 2 ( ). Substituting this φ into (3.16), we may determine p uniquely. For given, m, ρ 1, ρ 2 and source location (x s,z s ), we can determine p(x, z) on and φ = p + p on by integral equations (3.16) and (3.17). Hence, we can compute the wave field from (3.13) and (3.14). When ρ 1 = ρ 2 then (3.17) becomes φ(x,z) = 0, (x,z). (3.20) The system of integral equations reduces to a single integral equation p(x, z) + k 2 G(ξ, ζ ; x, z)mp(ξ, ζ ) dξ dζ = G(x s,z s ; x,z). (3.21) In what follows we develop an iterative algorithm for the integral equation (3.21). Under the our assumption the pressure field can be determined in R 2 b if we solve p(x, z) p(x,z,x s,z s ) such that p(x, z) = G(x s,z s ; x,z) k 2 G(ξ, ζ ; x,z)mp(ξ,ζ; x s,z s ) dξ dζ. (3.22) We use the following algorithm. Let p(ξ, ζ ; x s,z s ) = G(ξ, ζ ; x,z), (ξ,ζ) (3.23)

9 Acoustic imaging in shallow ocean with thin ice cap Figure 3. Acoustic field in a homogeneous shallow ocean with a thin ice cap Figure 4. Acoustic field in a inhomogeneous, shallow ocean with a thin ice cap. and for n = 1, 2, 3,..., and (x, z), let p n+1 (x, z) = G(x s,z s ; x,z) k 2 G(ξ, ζ ; x,z)mp n (ξ, ζ ; x s,z s ) dξ dζ. (3.24) This algorithm converges if km 2 is small. The implementation of the above algorithm for a two-dimensional water-column has been carried out using Matlab. Figure 3 is the propagating field in a shallow ocean with an ice cap without an inhomogeneity in the water-column, while figure 4 is the propagating field in a shallow ocean with an ice cap having an inhomogeneity located in the region [, 90] [, ]. 4. The inverse scattering problem: numerical experiments The inverse problem considered is the following: let Ɣ d be a subset of Ɣ 1 :={(x, z 1 ) x R 2 b,z 1 = constant}, and Ɣ s be a subset of Ɣ 2 :={(x, z 2 ) x R 2 b,z 2 = constant}. Given u(x, z; x s,z s ) for (x, z; x s,z s ) Ɣ d Ɣ s, determine the inhomogeneity m(x, z).

10 1808 R P Gilbert and Y Xu Here we assume that Ɣ 1 and Ɣ 2 are strictly above the inhomogeneity ; i.e., max (x,z) {z} < min{z 1,z 2 }. In the following we reformulate the inverse problem as an overdetermined linear system, and use a nonlinear optimization scheme to solve the regularized, nonlinear, least squares problem. We consider only the case ρ 1 = ρ 2 in this paper. Moreover, we assume that m C(R 2 b ). From (3.13) we can represent the acoustic field detected on Ɣ d with sources on Ɣ s as p(x, z; x s,z s ) = F (mp)(x, z; x s,z s ) + G(x s,z s ; x,z), (x,z) Ɣ d, (x s,z s ) Ɣ s (4.1) F (mp)(x, z; x s,z s ) := G(ξ, ζ ; x,z)mp(ξ,ζ; x s,z s ) dξ dζ p(ξ, ζ ; x s,z s ), (ξ, ζ ) satisfies (3.27) with φ = 0; i.e., u + T (mu) = G, (4.2) T (mp)(x, z) = G(ξ, ζ ; x, z)mp(ξ, ζ ) dξ dζ and G = G(x s,z s ; x,z). Using the measured data p = p (x, z; x s,z s ), (x, z; x s,z s ) Ɣ d Ɣ s, we reformulate the problem of determining the inhomogeneity as the minimization problem, namely, find m(x, z), (x, z) which minimizes J ɛ (m) for some suitable chosen ɛ, J ɛ (m) := u F (m(i + T ) 1 G) G 2 L 2 (Ɣ d ) + ɛ m 2 L 2 (). (4.3) Since k0 2 is small, we have the approximation (I + T ) 1 I T + T 2 T 3, which allows us to solve the inverse problem by minimizing J ɛ (ˆk 2 ) := u F (ˆk 2 (I T + T 2 T 3 )G) G 2 L 2 (Ɣ d ) + ɛ ˆk 2 2 L 2 (). (4.4) The difference between the minimization of (4.3) and the minimization of (4.4) is that in (4.4) it is not required to solve an integral equation. We present some illustrations of this method for solving the inverse problem by the following numerical examples. Example 1. The inhomogeneity is (z 77.)π (x 53.33)π x sin sin m(x, z) = z , otherwise. The original inhomogeneity is shown in figure 5 (top). The reconstruction by the regularized Born approximation method is shown in figure 5 (bottom). The regularization parameter ɛ = Note that the measured field is generated by an iteration algorithm described in section 2, and the inversion is performed by the regularized Born method. This prevents the so-called inverse crime. To test the robustness of the scheme, we add 5% normally distributed noise to the signal and perform the inversion.

11 Acoustic imaging in shallow ocean with thin ice cap Figure 5. Example 1 of inverse problems: top, original; bottom, reconstruction Figure 6. Example 2 of inverse problems: top, original; bottom, reconstruction.

12 1810 R P Gilbert and Y Xu Example 2. The inhomogeneity is given by 0.2((5 16 (z 82.5) 16 )( m(x, z) = (x ) 16 )){ } 1 (x, z) [53.3, 66.6] [77.5, 87.5] 0 otherwise. The signal is measured on Ɣ d ={(x d,z d ) : x d = 140:0.5 :26,z d = }. Note that Ɣ d is on a line located in the water. The original inhomogeneity is shown in figure 6 (top), and the reimaging is shown as figure 6 (bottom). Here we use ɛ = for the regularization parameter. Acknowledgments This research was supported in part by the National Science Foundation through grants BES and INT YX s research was supported in part by NSF grant INT and by grants from CECA of the University of Tennessee at Chattanooga. References [1] Achenbach J D 1976 Wave Propagation in Elastic Solids (Amsterdam: North-Holland) [2] Badanin A V and Belinskii B P 1993 Oscillations of a liquid in a bounded cavity with a plate on the boundary Comput. Math. Math. Phys [3] Biot M A 1956 Theory of propagation of elastic waves in a fluid-saturated porous solid I. Lower frequency range J. Acoust. Soc. Am Biot M A 1956 Theory of propagation of elastic waves in a fluid-saturated porous solid II. Higher frequency range J. Acoust. Soc. Am [4] Biot M A 1962 Mechanics of deformation and acoustic propagation in porous media J. Appl. Phys [5] Buchanan J and Gilbert R 1997 Transmission loss in the far field over a one-layer seabed assuming a Biot sediment model ZAMM [6] Buchanan J and Gilbert R 1996 Transmission loss in the far field over a seabed with rigid substrata assuming the Biot sediment model J. Comput. Acoust [7] Buchanan J and Gilbert R 1998 Transmission loss over a two-layer seabed Int J. Solids Struct [8] Buchanan J, Gilbert R, Xu Y and Xu S 1996 Green function representation for acoustic pressure over a poroelastic seabed Appl. Anal [9] Collins M D 1990 Higher-order and elastic parabolic equations for wave propagation in the ocean Computational Acoustics vol 3, ed D Lee, A Cakmak and R Vichnevetsky (Amsterdam: Elsevier) [10] Collins M D 1991 Higher-order Padé approximations for accurate and stable elastic parabolic equations with application to interface wave propagation J. Acoust. Soc. Am [11] Gelfand I M and Levitan B M 19 On the determination of a differential equation from its spectral function Trans. Am. Math. Soc. I [12] Gilbert R P 1969 A method of ascent Bull. Am. Math. Soc [13] Gilbert R P, Lin Z Y and Buchanan J L 1995 Acoustic waves in a shallow inhomogeneous ocean Mathematical Modelling of Flow through Porous Media (Singapore: World Scientific) [14] Gilbert R P and Lin Z Y 1996 Acoustic waves in a shallow inhomogeneous ocean with a layer of sediment Acustica [15] Gilbert R P and Lin Z Y 1997 Scattering in a shallow ocean with an elastic seabed Comput. Acoust [16] Gilbert R P and Xu Y 1989 Starting fields and far fields in ocean acoustics Wave Motion [17] Gilbert R P and Lin Z Y 1998 The fundamental singularity in a shallow ocean with an elastic seabed Appl. Anal [18] Gilbert R P and Lin Z 1997 Acoustic waves in shallow inhomogeneous oceans with a poro-elastic seabed ZAMM [19] Gilbert R P, Scotti T, Wirgin A and Xu Y S 1998 The unidentified object problem in a shallow ocean J. Acoust. Soc. Am [20] Gilbert R P and Wood D 1986 A transmutational approach to underwater sound propagation Wave Motion

13 Acoustic imaging in shallow ocean with thin ice cap 1811 [21] Gilbert R P 1986 Transmutations occurring in ocean acoustics Proc. Conf. on Partial Differential Equations and Applied Mathematics (Oakland University, 1986) (London: Pitman) [22] Gilbert R P and Xu Y 1989 Dense sets and the projection theorem for acoustic harmonic waves in homogeneous finite depth oceans Math. Methods Appl. Sci [23] Gilbert R P and Xu Y 1990 The propagation problem and far-field patterns in a stratified finite-depth ocean Math. Methods Appl. Sci [24] Gilbert R P and Xu Y 1992 Generalized Herglotz functions and inverse scattering problems in finite depth oceans Inverse Problems (Philadelphia, PA: SIAM) [25] Hackl K 1997 Asymptotic methods in ocean acoustics Generalized Analytic Functions Theory and Applications to Mechanics ed H Florian, K Hackl, F J Schnitzer and W Tutschke (Dordrecht: Kluwer) [26] Hamilton E L 1980 Geoacoustic modelling of the sea floor J. Acoust. Soc. Am [27] Holland C W and Brunson B A 1988 The Biot Stoll model: An experimental assessment J. Acoust. Soc. Am [28] Scotti T and Wirgin A 1995 Shape reconstruction using diffracted waves and canonical solutions Inverse Problems [29] Scotti T and Wirgin A 1995 Location and shape reconstruction of a soft body by means of canonical solutions and measured scattered sound fields C.R. Acad. Sci., Paris B [30] Stoll R D 1974 Acoustic waves in saturated sediments Physics of Sound in Marine Sediments ed L Hampton (New York: Plenum) [31] Stoll R D and Bryan G M 19 Wave attenuation in saturated sediments J. Acoust. Soc. Am [32] Xu Y 1990 Scattering of acoustic wave by obstacle in stratified medium IMA Preprint 734 [33] Xu Y 1999 Inverse acoustic scattering problems in ocean environments J. Comput. Acoust [34] Timoshenko S P and Woinowsky-Krieger S 1959 Theory of Plates and Shells (New York: McGraw-Hill) [35] Yamamoto T 1983 Acoustic propagation in the ocean with a poro-elastic bottom J. Acoust. Soc. Am [36] Yamamoto T 1983 Propagator matrix for continuously layered porous seabeds Bull. Seism. Soc. Am [37] Xu Y 1990 Direct and inverse scattering in shallow oceans PhD Thesis University of Delaware [38] Xu Y 1991 An injective far-field pattern operator and inverse scattering problem in a finite depth ocean Proc. Edinburgh Math. Soc