This article considers the issue of vector three-dimensional direct and inverse diffraction problems involving an inhomogeneous hemispherical scatterer with a piecewise constant permittivity distribution. Starting from the fundamental Maxwell equations, we recast the electromagnetic boundary value problem into an equivalent system of integro-differential equations. Novel numerical methods are introduced for addressing both the direct diffraction problem, enabling computation of electromagnetic fields across the hemisphere, as well as a two-step method for tackling the inverse diffraction problem aimed at reconstructing the permittivity. Experimental outcomes illustrating the recovery of internal structural anomalies through this two-phase technique are discussed. These findings demonstrate that the proposed two-step approach offers substantial promise for accurately determining material composition utilizing frequencies spanning between 1–10 GHz.
Smirnov et al. (2025) studied this question.