  Assume a time-harmonic elastic wave is incident onto a penetrable anisotropic
body embedded into a homogeneous isotropic background medium. We propose an
equivalent variational formulation in a truncated bounded domain and show the
uniqueness and existence of weak solutions by applying the Fredholm alternative
and using properties of the Dirichlet-to-Neumann map in both two and three
dimensions. The Fr\'echet derivative of the near-field solution operator with
respect to the boundary of the scatterer is derived. As an application, we
design a descent algorithm for recovering the interface from the near-field
data of one or several incident directions and frequencies. Numerical examples
in 2D are presented to show the validity and accuracy of our methods.
