Forward and inverse scattering for discrete one-dimensional lossy and discretized two-dimensional lossless media.

Abstract

Forward and inverse scattering for discrete media is developed in two parts. First, the one-dimensional (1-D) scattering problem for lossy (absorbing) media is formulated. A complete digital signal processing (DSP) theory is presented for forward and inverse scattering in discrete (piece wise-constant) lossy media. This generalizes previous work for discrete lossless and continuous lossy media. New feasibility and sufficiency conditions for impulse reflection data are introduced. In addition, new derivations for the discrete Gel'fand-Levitan-Marchenko and Krein equations for a discrete lossy media are presented. The latter is solvable using the asymmetric Levinson algorithm to completely reconstruct media in a recursive fashion. Application to electromagnetic wave propagation in 1-D and 2-D layered media is discussed along with new reconstruction techniques. Second, a discrete formulation of the two-dimensional (2-D) inverse scattering problem is introduced. The inverse problem for the discretized 2-D Schrodinger equation is formulated and a complete DSP theory is presented. This leads to an important new feasibility condition for scattering data used in 2-D layer stripping algorithms. A new variable mesh finite difference approximation applicable to the variable wave speed problem is developed. Both 1-D and 2-D DSP formulations, consisting of fast algorithms and systems of equations, are discrete counterparts of integral equations and allow exact solutions, without discretization approximations, and include all multiple reflections, transmission scattering losses, and absorption effects.