A NONLINEAR APPROACH TO INVERSE SCATTERING BY AN ACOUSTICALLY SOFT OBSTACLE (DOMAIN, PERTURBATION, ITERATIVE)

Inverse scattering associated with a fixed and bounded acoustically soft obstacle situated in a homogeneous medium is treated. The problem is to determine the shape of the obstacle from a finite number of measurements of the scattered wave. A perturbation formulation of the direct scattering problem yields a nonlinear integral equation relating the scattered field of the unknown obstacle to the scattered field produced by a comparison obstacle and the distance in the normal direction between the two obstacles. From this, the Frechet derivative of the scattered field (which links a small variation of the shape of the obstacle to a small variation of the scattered wave) is derived. This result is used to linearize the integral equation and if a finite number of measurements are made, a linear system of equations is obtained. This system is inverted and a minimization criterion is employed to obtain a unique solution. This utilizes a descent type process applied to a nonlinear functional whose absolute minimum yields a solution to the full nonlinear inverse scattering problem. An algorithm is proposed which solves the inverse problem by iteration and the stability of the process is guaranteed by regularization. The uniqueness of the solution is also examined. The method is applicable to other scattering problems in any spatial dimension