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 Delta u + kappa(2)u = 0 in R-m(D) over bar or D, respectively, where the total field u satisfies the transmission conditions u(+) = u(-) and partial derivative u(+)/partial derivative nu = beta partial derivative u/partial derivative nu on the boundary of some domain D with some constant beta. The main idea of the singular sources scheme is to reconstruct the scattered field of point sources or higher multipoles Phi(s) (.; z) with source point z in its source point from the far field pattern u(infinity) (., d), d is an element of S of scattered plane waves. The function Phi(s)(z, z) is shown to become singular for z --> partial derivative 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 b 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.