9404283 Symes This project will focus on inverse problems arising in geophysics, in particular, the determination or estimation of the index of refraction, or wave velocity, of a medium from measurements made at or near the surface of the body. The field equations for such problems are hyperbolic, or are closely related to hyperbolic systems. Several key mathematical questions will be investigated, each with substantial practical implications: 1) global estimates for a linearized inverse problem for the wave equation in the presence of caustics; 2) the relation between nonsmooth propagation of singularities and linearization; 3) design of objective functions to give quasiconvex optimization formulations of inverse problems; and 4) accommodation of attenuation and dispersion; for example, inverse problems in viscoelasticity. The remote sensing techniques of pure and applied geophysics use measurements of an acoustic or electromagnetic field near the earth's surface to infer the internal structure of the earth. These fields are modeled by the solutions of some appropriate field equations in a domain representing a portion of the earth. The coefficients of these equations represent distributed physical parameters such as conductivity, stiffness, density, etc. Thus remote sensing techniques pose inverse problems: to estimate coefficients of field equations in the interior of a domain from measurements of fields near the boundary. This research concerns models of those experimental designs and parameter ranges which cause the probe fields to consist of propagating waves.
