Inverse Problems in One Dimension

Abstract

The scattering problem in one dimension on the entire real line — ∞ < x < ∞ is known to have some fundamental differences from scattering in three dimensions, and its radial one-dimensional counterpart on the half-line 0 ≤ r < ∞. It has given rise to numerous studies because of its applications in many electrical engineering problem, among which we may mention: 1. Nonuniform transmission line in which either the inductance or the capacitance is constant, and in which we would like to reconstruct the line from the knowledge of the reflection coefficient r(ω) supposed to be known for all frequencies. 2. Under certain conditions, the reconstruction of a waveguide of variable cross section from some part of the scattering matrix, usually the reflection coefficient. 3. The problem in which a variable stratified dielectric is to be constructed so that the reflection coefficient of an electromagnetic wave will vary in a prescribed way with frequency. 4. Determination of the ionization density of a stratified ionosphere from the time delay of a pulse radio wave which has been transmitted from the earth and reflected back to the earth by the ionosphere.