The Singular Sources Method for an Inverse Transmission Problem

Abstract

We employ the singular sources method introduced in [4] to solve an inverse transmission scattering problem for the Helmholtz equation or D, respectively, where the total field u satisfies the transmission conditions on the boundary of some domain D with some constant β. The main idea of the singular sources scheme is to reconstruct the scattered field of point sources or higher multipoles (·, z) with source point z in its source point from the far field pattern of scattered plane waves. The function (z, z) is shown to become singular for z→∂D. This can be used to detect the shape D of the scattering object. Here, we will show how in addition to reconstructions of the shape D of the scattering object, the constant β can be reconstructed without solving the direct scattering problem. This extends the singular sources method from the reconstruction of geometric properties of an object to the reconstruction of physical quantities.