The factorization method for inverse acoustic scattering in a layered medium

In this paper, we consider a problem of inverse acoustic scattering by an impenetrable obstacle embedded in a layered medium. We will show that the factorization method can be applied to recover the embedded obstacle; that is, the equation (F) over tilde g = phi(z) is solvable if and only if the sampling point z is in the interior of the unknown obstacle. Here, (F) over tilde is a self-adjoint operator related to the far field operator and phi(z) is the far field pattern of the Green function with respect to the problem of scattering by the background medium for point z. The validity of the factorization method is proven with the help of a mixed reciprocity principle and an application of the scattering operator. Due to the established mixed reciprocity principle, knowledge of the Green function for the background medium is no longer required, which makes the method attractive from the computational point of view. The paper is only concerned with sound-soft obstacles, but the analysis can be easily extended for sound-hard obstacles, or obstacles with separated sound-soft and sound-hard parts. Finally, we provide an explicit example for a radially symmetric case and present some numerical examples.