Numerical performance of layer stripping algorithms for two-dimensional inverse scattering problems

Abstract

Numerical results of implementing a two-dimensional layer stripping algorithm to solve the two-dimensional Schrodinger equation inverse potential problem are presented and discussed. This is the first exact (all multiple scattering and diffraction effects are included) numerical solution of a multi-dimensional Schrtidinger equation inverse potential problem, excluding optimization-based approaches. The results are as follows: (1) the layer stripping algorithm successfully reconstructed the potential from scattering data measured on a plane (as it would be in many applications); (2) the algorithm avoids multiple scattering errors present in Born approximation reconstructions; and (3) the algorithm is insensitive to small amounts of noise in the scattering data. Simplifications of layer stripping and invariant imbedding algorithms under the Born approximation are also discussed.

On the feasibility of impulse reflection response data for the two-dimensional inverse scattering problem

“…However, this approach is quite different from ours since our decomposition into waves is quite different; our results are derived in the discrete as well as in the continuous domain, and we obtain a feasibility result not evident using other approaches. Layer stripping algorithms for inverse scattering problems are very fast but have the reputation of being numerically unstable, especially when applied to noisy data, although the 2-D layer-stripping algorithm proposed in [4] was numerically implemented successfully in [5] and shown to be robust in the presence of small noise levels. The goal of this paper is to provide an explicitly discrete framework for layer-stripping algorithms for the 2-D Schrodinger equation inverse scattering problem.…”

An Iterative Method for the Inversion of the Two-Dimensional Wave Equation with a Potential

“…5, where x ∈ [0, 3], z ∈ [0, 1], and t ∈ [0, 2.25], the numbers of grid points are N x = 40, N z = 40, and N t = 45, and x = 0.075, = 0.025, and n = 10. It was reported in [22] that for such a potential which drops off rapidly to zero in the deep part, the linearized reconstruction, based on the Born approximation, always has a "tail" which goes to zero very slowly. This is due to the multiple reflection in the response which is considered as the primary reflection in the linearized inversion.…”

“…For the original inverse problem (1.1), we can also obtain the following conditions for p by singularity propagation analysis, which were used to construct the layerstripping algorithm in [22],…”

“…This problem has several applications. We refer readers to the discussion in [22] for further references. What motivates us to study this problem is its application to petroleum prospecting in reconstructing an acoustic medium with density and wave speed from surface measurements of the displacement response to a line source.…”

“…We refer readers to the survey papers [1,10]. In [21], Yagle and Levy suggested a layer-stripping method for the inverse problem (1.1) and in [22] Yagle and Raadhakrishnan presented the results of some numerical experiments. Their main idea is to regularize the ill-posed problem by cutting the lateral wave numbers.…”