A sampling method for inverse scattering in the time domain

We consider a near-field inverse scattering problem for the wave equation: find the shape of a Dirichlet scattering object from time domain measurements of scattered waves. For this time-domain inverse problem, we propose a linear sampling method, a well-known technique for corresponding frequency domain inverse scattering problems. The problem setting and the algorithm incorporate two basic features. First, the data for the method consist of measurements of causal waves, that is, of waves that vanish before some moment in time. Second, the inversion algorithm directly works on the time-domain data without using a Fourier transformation. The first point is related to the applications we have in mind, which include for instance ground-penetrating radar imaging. The second feature allows us to naturally incorporate multiple (in fact, a continuum of) frequencies in the inversion algorithm. Consequently, it offers the potential of improving the quality of the reconstruction compared to frequency domain methods working with a single frequency. We demonstrate this potential by several numerical examples.