The Solid-Fluid Transmission Problem

Abstract: We will discuss a problem motivated from geophysics, where one is interested in understanding the propagation of seismic waves in the interior of the Earth. It is known that the interior of the Earth consists of several layers, some of which are solid and some of which are fluid. When a seismic wave meets the interface between two layers, part of its energy is reflected back (possibly with mode conversion), and, if the angle of incidence is not too large, part of it is transmitted to the other side of the interface. We are particularly interestred in understanding reflection, transmission and mode conversion of waves at the interface between a linear elastic solid and an inviscid fluid. For simplicity, we consider the case of two layers, with the fluid layer being enclosed by the solid one. The two media are described by a system of PDEs modeling the displacement in the solid and pressure-velocity in the fluid, with these quantities being coupled at the interface by transmission conditions. We study the problem microlocally: to understand the behavior of singularities of solutions of the system, we construct a parametrix for it (approximate solution up to smooth error) using geometric optics. As an application of our study, we consider the inverse problem of recovering the wave speeds in the two layers and the material density in the solid outer layer from the Neumann-to-Dirichlet map for the solid-fluid system corresponding to the exterior boundary. Based on joint work with Plamen Stefanov.

Mathematics

Audience: researchers in the topic

International Zoom Inverse Problems Seminar, UC Irvine