On recovery of an inhomogeneous cavity in inverse acoustic scattering

Fenglong Qu,Jiaqing Yang

doi:10.3934/ipi.2018012

Abstract

Consider the time-harmonic acoustic scattering of an incident point source inside an inhomogeneous cavity. By constructing an equivalent integral equation, the well-posedness of the direct problem is proved in $L^p$ with using the classical Fredholm theory. Motivated by the previous work [ 10 ], a novel uniqueness result is then established for the inverse problem of recovering the refractive index of piecewise constant function from the wave fields measured on a closed surface inside the cavity.

Highlights

In this paper, we study an inverse scattering problem of determining an inhomogeneous cavity from many measurements inside the cavity.Precisely, let D denote the inhomogeneous cavity, which is described by a bounded connected domain in R3 with the refractive index n(x) ∈ L∞(D)
Consider an incident field ui which is induced by a point source located at y ∈ D0, i.e., (1)
Different from the previous works on the inverse cavity problem (ICP), we will focus in the current paper on the uniqueness issue on the refractive index n(x) and the cavity D in Problem (2) by interior measurements generated by incident point sources (1)

Summary

Introduction

We study an inverse scattering problem of determining an inhomogeneous cavity from many measurements inside the cavity. Different from the previous works on the ICP, we will focus in the current paper on the uniqueness issue on the refractive index n(x) and the cavity D in Problem (2) by interior measurements generated by incident point sources (1) To this end, we first transfer Problem (2) into an equivalent integral equation by introducing a Dirichlet-Green function. We can provide a generalized symmetry property of the solution to Problem (2) with incident fields located at two different points Based on these analysis, we can propose a novel technique to deal with the ICP, especially for the unique recovery of the index of refraction in the inverse cavity scattering problem. We are devoted to study the solvability of Problem (2) with the data induced by a general point source wave

Inverse Problems and Imaging

Lq into