An Inverse Source Problem for the Elastic Wave in the Lower Half-Space

We consider an inverse source problem for the wave equation in a heterogeneous, isotropic, elastic media (i.e., the dynamical Lame system). We assume our data are of the form $\Lambda f := u\vert_{[0,T]\times \Gamma}$, where $u$ is the solution of the elastic wave equation with initial displacement $f$ and $\Gamma \subset \{x_n = 0\}$, and show that the singularities (wavefront set) of $f$ can be recovered in a unique and stable way. This follows the work of Stefanov and Uhlmann [Inverse Problems, 25 (2009), 075011] (partial data for an acoustic wave equation) and, more recently, of this author [Inverse Problems, 28 (2012), 055004] (full data for the same elastic system).