Iterative method for multi-dimensional inverse scattering problems at fixed frequencies

Suppose that a medium with slowly changing spatial properties is enclosed in a bounded three-dimensional domain and is subjected to a scattering by plane waves of a fixed frequency. Let measurements of the scattered wave field induced by this medium be available in the region outside of this domain. We study how to extract the properties of the medium from the information contained in the measurements. We are concerned with the weak scattering case of the above inverse scattering problem (ISP). That is, the unknown spatial variations of the medium are assumed to be close to a constant. Examples can be found in studies of wave propagation in oceans, in the atmosphere and in some biological media. The iterative sequence is defined in the framework of a quasi-Newton method. Using measurements of the scattered field from a carefully chosen set of directions we are able to recover (finitely many) Fourier coefficients of the sought parameters of the model. In this method, the linearized (Born) approximation is just the first iteration, and further iterations improve the identification by an order of magnitude. Numerical experiments for scattering from a circular cylinder are presented.