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

TL;DR

This paper introduces a multi-frequency direct sampling method to efficiently and robustly determine the support of a stationary, frequency-dependent source in a homogeneous medium using far-field data, applicable to inverse time-dependent problems.

Contribution

It presents a novel multi-frequency direct sampling approach for imaging the support of frequency-dependent sources, including methods to recover support support and convex hull with sparse observations.

Findings

Support support can be recovered from a fixed observation angle.

Convex hull of the source support can be reconstructed with multiple sparse observations.

The method is robust against noise and computationally efficient.

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 sparse observation directions, the shape of the convex hull of the source support can be recovered. 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.