Inverse Scattering Solutions by a Sinc Basis, Multiple Source, Moment Method -- Part I: Theory

Abstract

A new method for solving the inverse scattering problem for the scalar, inhomogeneous, exact, Helmholtz wave equation is presented. No perturbation approximations are used and the method is applicable even for many cases where weak to moderate attenuation and moderate to strong refraction of incident fields occur. The ill-posed nature of the inverse scattering problem for a single monochromatic source is known. However, the use of multiple sources, the collection of redundant (i.e., overdetermined) data, and the constraining of the fields and complex refractive index to be spatially band limited constitutes a new problem. The cases we have tested by computer simulation indicate that the new problem is well posed, has a unique solution, and is stable with noisy data. The method is an application of the well-known method of moments with sinc basis and delta testing functions to discretize the problem. The inverse scattering solution may be obtained by solving the resulting set of simultaneous, quadratic, multivariate equations. Several algorithms for solving these equations are given.