Time domain inverse source problem and fluid-saturated porous media scattering problem

摘要

This dissertation applies Corones and Krueger's invariant imbedding and wave splitting techniques to two time domain direct and inverse scattering problems. In the first problem, invariant imbedding and wave splitting are extended to the case of a transient electric source J(t) inside a dispersive or inhomogeneous dielectric slab. Representations of composite transmission operators are obtained. These operators are used to establish a delay Volterra type integral equation, which is used to infer the transient source J(t) from the transmitted field. One analytical frequency-domain example and two numerical time-domain examples are presented. Also, Green's operators that map the source J(t) to the field at an arbitrary observation point are defined and used to determine the internal E field. For the Green's operator kernels, we obtain linear integrodifferential equations with various initial, boundary and jump conditions. In the second problem, representations of reflection and transmission matrix operators are found, and integrodifferential equations for the operator kernels are derived from the Biot system of compressional wave equations for a finite slab of dispersive, dissipative, fluid-saturated porous medium. Some properties of these operator kernels, such as reciprocity relations and the multiple modes of propagation of discontinuities, are discussed. A numerical scheme for solving the inverse problem is described, and specific numerical computations for a half-space direct and inverse scattering problem are presented.