Reconstruction of Lame moduli and density at the boundary enabling directional elastic wavefield decomposition

We consider the inverse boundary value problem for the system of equations describing elastic waves in isotropic media on a bounded domain in $\mathbb{R}^3$ via a finite-time Laplace transform. The data is the dynamical Dirichlet-to-Neumann map. More precisely, using the full symbol of the transformed Dirichlet-to-Neumann map viewed as a semiclassical pseudodifferential operator, we give an explicit reconstruction of both Lam\'{e} parameters and the density, as well as their derivatives, at the boundary. We also show how this boundary reconstruction leads to a decomposition of incoming and outgoing waves.