Direct and inverse obstacle scattering problems in a piecewise homogeneous medium

TL;DR

This paper addresses both direct and inverse acoustic obstacle scattering problems in a piecewise homogeneous medium, establishing well-posedness, deriving a priori estimates, and proving uniqueness in reconstructing obstacles and interfaces from far-field data.

Contribution

It introduces new uniqueness results for inverse scattering in layered media and employs integral equations and layer potentials for analysis.

Findings

Well-posedness of the direct scattering problem established.

Uniqueness of the inverse problem proven for obstacle and interface reconstruction.

A priori estimates of solutions are derived and utilized.

Abstract

This paper is concerned with the problem of scattering of time-harmonic acoustic waves from an impenetrable obstacle in a piecewise homogeneous medium. The well-posedness of the direct problem is established, employing the integral equation method and then used, in conjunction with the representation in a combination of layer potentials of the solution, to prove a priori estimates of solutions on some part of the interface between the layered media. The inverse problem is also considered in this paper. An uniqueness result is obtained for the first time in determining both the penetrable interface and the impenetrable obstacle with its physical property from a knowledge of the far field pattern for incident plane waves. In doing so, an important role is played by the a priori estimates of the solution for the direct problem.