Reconstruction of sources from time domain scattered waves at sparse sensors

We consider the source reconstruction from the time domain scattered waves at sparse sensors. We show that the smallest annular hull containing the unknown source can be uniquely determined by the scattered field at a fixed sensor. In particular, if the wave-form is a delta distribution, the concentric spheres passing through the point sources can be uniquely determined by the scattered field at the fixed sensor. We further show that both the locations and strengths of the point sources can be uniquely determined by the scattered field at finitely many properly chosen sensors. The uniqueness of an extended source has also be established. Motivated by the analysis, we introduce two numerical schemes, which can not only identify point sources but also reconstruct the support of an extended source. Both the theoretical results and the proposed numerical methods are then verified by the numerical examples. An extension of the results to the inverse scattering of point sources by point like scatterers is briefly sketched.