Reconstruction of multiscale elastic sources from multifrequency sparse far field patterns

We consider the inverse time-harmonic elastic source problems with multifrequency sparse far field patterns. The unknown multiscale object is a combination of point sources and an extended source with compact support. Even if the extended source is unknown, we prove that the locations and the polarization strengths of the point sources can be uniquely determined by the multifrequency far field patterns at sparse observations. The least number of observation directions is given in terms of the number of the point sources. Having identified the point sources, under certain conditions, we further show that the multifrequency far field patterns at a fixed observation direction uniquely determine the narrowest strip containing the source support. With the increase of the number of observation directions, a convex hull of the source support can be obtained. Based on the constructive uniqueness proof, we introduce a novel direct sampling method both for locating the point sources and for reconstructing the support of the extended source. A formula for computing the polarization strengths is also given. Numerical examples in two dimensions are presented to verify the accuracy and robustness of the proposed methods for multiscale elastic sources.