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

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 Nystrom method with a graded mesh introduced previously by Kress and is capable of dealing with large wavenumber 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.