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This paper is concerned with a direct sampling method for imaging the supportof a frequency-dependent source term embedded in a homogeneous and isotropicmedium. The source term is given by the Fourier transform of a time-dependentsource whose radiating period in the time domain is known.  The time-dependent source is supposed to be stationary in the sense that itscompact support does not vary along the time variable.  Via a multi-frequency direct sampling method, we show that the smallest stripcontaining the source support and perpendicular to the observation directioncan be recovered from far-field patterns at a fixed observation angle. Withmultiple but sparse observation directions, the shape of the convex hull of thesource support can be recovered. The frequency-domain analysis performed herecan 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 ourtheoretical findings.

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