Uniqueness and numerical scheme for spherical shell-structured sources from the far field patterns with at most two frequencies

Abstract

It is well known that the far field patterns at finitely many frequencies are not enough to uniquely determine a general source. To establish uniqueness, we consider the acoustic scattering of spherical shell-structured sources. We show that the spherical shell-structured sources can be identified from the far field patterns with at most two frequencies. Precisely, we show that the number and the centers (i.e., locations) of the spherical shells can be uniquely determined by the far field patterns at a fixed frequency. Furthermore, the scattering strengths, the inner and outer diameters are uniquely determined by the far field patterns at two frequencies. Motivated by the uniqueness arguments, a numerical scheme is proposed for reconstructing the spherical shell-structured sources. A migration series method is designed to located the centers. The numerical simulations show that the reconstruction quality is the same as the direct sampling method. Furthermore, an iterative method is designed for computing the inner diameters and outer diameters. An important feature of the proposed iterative method is that it avoids computing any derivatives. The convergence of the iterative method is also proved. Finally, some numerical examples are presented to verify the effectiveness and robustness of the proposed numerical scheme.