Two methods for solving a 3D acoustic inverse scattering problem

We consider the problem of finding the refractive index of a buried object by using far‐field measurements in an inhomogeneous medium. We describe two methods for solving the inverse problem. Both methods are implemented in two steps in order to better deal with the ill‐posedness of the problem. In the first method an integral equation of the first kind is derived for the far‐field operator which is solved via least‐squares and Tikhonov regularization. We then use the solution of the integral equation to derive an over‐posed boundary value problem, i.e., the Helmholtz equation in a bounded domain with Cauchy data on the boundary. The index that satisfies this over‐posed problem most closely is obtained via the Levenberg–Marquardt algorithm. The second method is an iterative method and is based on the Lippmann–Schwinger equation. It is implemented via the Newton method. The first step consists, as for the other method. Here we use a Fourier integral approach and regularization via discretization. The second step is to obtain the index by iterating the Lippmann–Schwinger equation starting with the Born approximation.