  We address the inverse problem of recovering the stiffness tensor and density
of mass from the Dirichlet-to-Neumann map. We study the invariance of the
Euclidean and Riemannian elastic wave equation under coordinate
transformations. Furthermore, we present gauge freedoms between the parameters
that leave the elastic wave equations invariant. We use these results to
present gauge freedoms in the Dirichlet-to-Neumann map associated to the
Riemannian elastic wave equation.
