In seismology, people use waves generated by earthquakes, artificial explosions, or even “noises”, to detect the Earth’s interior structure. The waves traveling in rocks, which are the main components of Earth’s crust and mantle, are elastic waves. A key problem in seismology is whether the interior structures of interest could be uniquely determined by the measurements people can collect. Since in most situations, people
can only record the vibrations at the surface, such problems usually are formulated as inverse boundary value problems for the elastic wave equation. We study several inverse boundary value problems for the elastic wave equation and give some uniqueness and stability results for them. In Chapter 2, we consider the inverse problem of determining the Lamé parameters and the density of a three-dimensional elastic body from the time-harmonic Dirichletto-Neumann map. We prove uniqueness and Lipschitz stability of this inverse problem when the Lamé parameters and the density are assumed to be piecewise constant on a given domain partition. In Chapter 3 and 4, we study the recovery of the density and stiffness tensors of a three-dimensional domain from the dynamical Dirichlet-to-Neumann map. We give explicit reconstruction schemes for the determination of stiffness tensors and the density at the boundary under certain assumptions. In Chapter 4, we prove uniqueness of piecewise analytic parameters in the interior for which we have boundary determination. In Chapter 5, we give a semiclassical description of surface waves or modes in an elastic medium that is stratified near its boundary at some scale comparable to the wave length. The analysis is based on the work of Colin de Verdière  on acoustic surface waves. In Chapter 6, we consider a Riemannian manifold in dimension n ≥ 3 with strictly convex boundary. We prove the local invertibility, up to potential fields, of the geodesic X-ray transform on tensor fields of order four near a boundary point. Under the condition that the manifold can be foliated with a continuous family of strictly convex surfaces, the local invertibility implies a global result.