A Novel Integral Equation for Scattering by Locally Rough Surfaces and Application to the Inverse Problem

Abstract

This paper is concerned with the direct and inverse acoustic or electromagnetic scattering problems by a locally perturbed, perfectly reflecting, infinite plane (which is called a locally rough surface in this paper). We propose a novel integral equation formulation for the direct scattering problem which is defined on a bounded curve (consisting of a bounded part of the infinite plane containing the local perturbation and the lower part of a circle) with two corners. This novel integral equation can be solved efficiently by using the Nyström method with a graded mesh introduced previously by Kress and is capable of dealing with large wave number cases. For the inverse problem, we propose a Newton iteration method to reconstruct the local perturbation of the plane from multiple-frequency far-field data, based on the novel integral equation formulation. Numerical examples are carried out to demonstrate that our reconstruction method is stable and accurate even for the case of multiple-scale profiles.