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

TL;DR

This paper introduces a new integral equation approach for direct and inverse scattering problems involving locally rough surfaces with Neumann boundary conditions, utilizing RCIP for efficient computation and demonstrating stable surface reconstruction from far-field data.

Contribution

It extends previous Dirichlet boundary condition methods to Neumann conditions, proposing a novel integral equation and a Newton-based inverse reconstruction algorithm.

Findings

The integral equation formulation is effective for large wave numbers.

The inverse reconstruction is stable and accurate with multiple frequency data.

The method successfully reconstructs multi-scale surface profiles.

Abstract

This paper is concerned with direct and inverse scattering by a locally perturbed infinite plane (called a locally rough surface in this paper) on which a Neumann boundary condition is imposed. A novel integral equation formulation is proposed 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 and some closed smooth artificial curve. It is a nontrivial extension of our previous work on direct and inverse scattering by a locally rough surface from the Dirichlet boundary condition to the Neumann boundary condition [{\em SIAM J. Appl. Math.}, 73 (2013), pp. 1811-1829]. In this paper, we make us of the recursively compressed inverse preconditioning (RCIP) method developed by Helsing to solve the integral equation which is efficient and capable of dealing with large wave numbers. For the inverse problem, it is proved that the locally rough surface is uniquely determined from a knowledge of the far-field pattern corresponding to incident plane waves. Further, based on the novel integral equation formulation, a Newton iteration method is developed to reconstruct the locally rough surface from a knowledge of multiple frequency far-field data. Numerical examples are also provided to illustrate that the reconstruction algorithm is stable and accurate even for the case of multiple-scale profiles.