A novel integral equation for scattering by locally rough surfaces and application to the inverse problem

TL;DR

This paper introduces a new integral equation approach for direct scattering problems involving locally rough surfaces and applies it to develop a stable, accurate Newton iteration method for reconstructing surface perturbations from far-field data.

Contribution

It presents a novel integral equation formulation on a bounded curve for scattering problems and a Newton-based inverse method for reconstructing local surface perturbations.

Findings

Efficient solution of the integral equation using Nystrom method with graded mesh.

Capability to handle large wavenumber cases.

Stable and accurate reconstruction of multi-scale surface profiles.

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 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.