Direct sampling method to inverse wave-number-dependent source problems: determination of the support of a stationary source

Abstract

This paper is concerned with a direct sampling method for imaging the support of a frequency-dependent source term embedded in a homogeneous and isotropic medium. The source term is given by the Fourier transform of a time-dependent source whose radiating period in the time domain is known. The time-dependent source is supposed to be stationary in the sense that its compact support does not vary along the time variable. Via a multi-frequency direct sampling method, we show that the smallest strip containing the source support and perpendicular to the observation direction can be recovered from far-field patterns at a fixed observation angle. With multiple but discrete observation directions, the shape of the so-called Θ-convex hull of the source support can be recovered. We even extend the proposed algorithm to the case of near-field data in three dimensions. The frequency-domain analysis performed here can be used to handle inverse time-dependent source problems. Our algorithm has low computational overhead and is robust against noise. Numerical experiments in both two and three dimensions have proved our theoretical findings.