RECONSTRUCTION PROCEDURES FOR TWO INVERSE SCATTERING PROBLEMS WITHOUT THE PHASE INFORMATION

This is a continuation of two recent publications of the authors [J. Inverse Ill-Posed Probl, 23 (2015), pp. 415-426; J. Inverse Ill-Posed Probl, 23 (2015), pp. 187-193] about reconstruction procedures for three-dimensional (3-D) phaseless inverse scattering problems. The main novelty of this paper is that, unlike [J. Inverse Ill-Posed Probl, 23 (2015), pp. 187-193], the Born approximation for the case of the wave-like equation is not considered. It is shown here that the phaseless inverse scattering problem for the 3-D wave-like equation in the frequency domain leads to the well known inverse kinematic problem. The uniqueness theorem follows. Still, since the inverse kinematic problem is very hard to solve, a linearization is applied. More precisely, geodesic lines are replaced with the straight lines. As a result, an approximate explicit reconstruction formula is obtained via the inverse Radon transform. The second reconstruction method is via solving a problem of the integral geometry using integral equations of the Abel type.