Direct and inverse scattering in the time domain for a dissipative wave equation. I. Scattering operators

• Published in: Journal of mathematical physics Vol. 27; no. 6; pp. 1667 - 1682
• Main Authors: Kristensson, G., Krueger, R. J.
• Format: Journal Article
• Language: English
• Published: Melville, NY American Institute of Physics 01.06.1986
• Subjects: Exact sciences and technology; Function theory, analysis; Mathematical methods in physics; Physics; ELASTICITY; MATHEMATICAL OPERATORS; INVERSE SCATTERING PROBLEM; WAVE PROPAGATION; ELECTROMAGNETIC RADIATION; VISCOSITY; DIELECTRIC PROPERTIES; DIELECTRIC MATERIALS; DISPERSION RELATIONS; Wave equation; Lossy medium; Electromagnetic wave
• ISSN: 0022-2488; 1089-7658
• DOI: 10.1063/1.527083

This is the first part of a series of papers devoted to direct and inverse scattering of transient waves in lossy inhomogeneous media. The medium is assumed to be stratified, i.e., it varies only with depth. The wave propagation is modeled in an electromagnetic case with spatially varying permittivity and conductivity. The objective in this first paper is to analyze properties of the scattering operators (impulse responses) for the medium and to introduce the reader to the inverse problem, which is the subject of the second paper in this series. In particular, imbedding equations for the propagation operators are derived and the corresponding equations for the scattering operators are reviewed. The kernel representations of the propagation operators are shown to have compact support in the time variable. This property implies that transmission and reflection data can be extended from one round trip to arbitrary time intervals. The compact support of the propagator kernels also restricts the admissible set of transmission kernels consistent with the model employed in this paper. Special cases of scattering and propagation kernels that can be expressed in closed form are presented.