Analysis of the linear sampling method for imaging penetrable obstacles in the time domain

We consider the problem of locating and reconstructing the geometry of a penetrable obstacle from time-domain measurements of causal waves. More precisely, we assume that we are given the scattered field due to point sources placed on a surface enclosing the obstacle and that the scattered field is measured on the same surface. From these multistatic scattering data we wish to determine the position and shape of the target. To deal with this inverse problem, we propose and analyze the time-domain linear sampling method (TDLSM) by means of localizing the interior transmission eigenvalues in the Fourier–Laplace domain. We also prove new time-domain estimates for the forward problem and the interior transmission problem, as well as analyze several time-domain operators arising in the inversion scheme.