A simple regularization method for solving acoustical inverse scattering problems

The problem of determining the shape of an object from far-field data is considered. We present a method, originally formulated in Ref. 1 and furtherly modified in Ref. 3, for the solution of this ill-posed nonlinear inverse problem whose main features are: • the method is exact, that is no low- or high-frequency approximation is considered; • it is not necessary to know the number of scatterers and whether or not the scatterers are penetrable by the waves; • if the medium is not penetrable, it is not necessary to know whether the obstacle is sound-hard or sound-soft; • in the case of an inhomogeneous scatterer, the method provides the shape of the inhomogeneity. The method is particularly simple since it requires only the solution of a linear Fredholm integral equation of the first kind whose integral kernel is the far-field pattern. The numerical instability due to ill-conditioning can be reduced by using regularization algorithms such as Tikhonov method where the regularization parameter is chosen by using Morozov's discrepancy principle generalized to the case where the noise affects the kernel of the integral operator.