On uniqueness and reconstruction of rough and complex obstacles from acoustic scattering data

Abstract — We deal with the inverse scattering problem by an obstacle at a fixed frequency. The obstacle is characterized by its shape, the type of boundary conditions on its surface and the eventual coefficients distributed on this surface. In this paper, we assume that the surface ∂D of the obstacle D is Lipschitz and the surface impedance, λ, is given by a complex valued, measurable and bounded function. We prove uniqueness of (∂D, λ) from the far field map under these regularity conditions. The usual proof of uniqueness for obstacles, based on the use of singular solutions, is divided into two steps. The first one consists of the use of Rellich type lemma to go from the far fields to the near fields and then use the singularities of the singular solutions, via orthogonality relations, to show uniqueness of ∂D. The second step is to use the boundary conditions to prove uniqueness of λ on ∂D via the unique continuation property. This last step requires the surface impedance to be continuous. We propose an approach using layer potentials to transform the inverse problem to the invertibility of integral equations of second kind involving the unknowns ∂D and λ. This enables us to weaken the required regularity conditions by assuming ∂D to be Lipschitz and λ to be only bounded. The procedure of the proof is reconstructive and provides a method to compute the complex valued and bounded surface impedance λ on ∂D by inverting an invertible integral equation. In addition, assuming ∂D to be C2 regular and λ to be of class C0,α, with α> 0, we give a direct formula as another method to reconstruct the surface impedance on ∂D