A nonperturbational inverse scattering solution for the scattering integral equation (SIE) is presented. The numerical discretization of the SIE is performed by the moment method (MM) using sinc basis functions. Previous algorithms using the alternating variable (AV) nonlinear iteration with algebraic reconstruction technique (ART) solution of the linearizations are shown to diverge for high contrast/large size acoustic scatterers. This deficiency is alleviated by the use of the Gauss-Newton (GN) nonlinear iteration with conjugate gradient (CG) solution of the linearizations. Further numerical efficiency is attained by use of the biconjugate gradient (BCG) algorithm to solve the forward scattering problems. Test problem reconstructions of circular cylinders, using the Bessel series analytic solution to generate the scattering data, demonstrate the accuracy of the method. Inhomogeneous models of human cross-sections verify the high spatial resolution and high speed of sound contrast capability of the method.