Direct sampling methods for inverse medium scattering problems of elastic waves

This paper concerns the inverse elastic scattering problem of determining an unknown penetrable obstacle from far-field data. Using the Helmholtz decomposition, the coupled boundary value problem is reformulated as a coupled scalar Helmholtz system. We prove the uniqueness of the associated coupled Helmholtz boundary value problem and of the corresponding boundary integral equation system, which is then discretized by a Nystr\"om method for efficient numerical computation. Leveraging the relation between the compressional and shear far-field patterns of the Navier system and those of the coupled Helmholtz system, we employ three indicators to reconstruct the location and shape of the obstacle. Furthermore, we analyze the decay properties of these indicators and establish corresponding stability estimates. Numerical experiments are presented to illustrate the effectiveness and robustness of the proposed method, even for limited-aperture data.