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

Abstract

Layer stripping algorithms for inverse scattering problems are very fast but have the reputation of being numerically unstable, especially when applied to noisy data. The goal of this paper is to provide an explicitly discrete framework for layer stripping algorithms for the two-dimensional (2-D) Schrodinger equation inverse scattering problem. We determine when 2-D layer stripping algorithms are numerically stable, explain why they are stable, and specify exactly the (discrete) problem they solve when they are stable. We reformulate the 2-D Schrodinger equation as a multichannel two-component wave system by Fourier transforming the Schrodinger equation in the lateral spatial variable. Discretization results in new 2-D layer stripping algorithms which incorporate multichannel transmission effects; this leads to an important new feasibility condition on impulse reflection response data for stability of these algorithms. A 2-D discrete Schrodinger equation is defined, and analogous results are obtained. Numerical examples illustrate the new results, especially how rendering noisy data feasible stabilizes layer stripping algorithms.