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, ie, 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

Direct and inverse scattering in the time domain for a dissipative wave equation. Part 1: Scattering operators

The technique for splitting waves into up and down components is applied to solutions of the generalized wave equation in three dimensions. In particular, the splitting associated with the principal part of the equation is applied to the fundamental solution (Green function). The equations for the up and down wave components are obtained and the singular parts removed. The use of these reduced wave components together with their equations, boundary data and initial data, in the inverse problem (where the reflected field is produced by a point-impulse source, exterior to the scattering medium) is discussed.

https://robots.iopscience.iop.org/article/10.1088/0266-5611/9/6/013/pdf

Wave splitting of the telegraph equation in R3 and its application to inverse scattering

The direct scattering problem for an inhomogeneous lossy medium is examined for the one‐dimensional case in which the phase velocity profile is discontinuous at the boundaries of the medium. Scattering operators (or impulse responses) and propagation operators are defined and equations that govern their behavior are developed. Knowledge of the scattering kernels for one round trip in the medium implies that the scattering kernels can be determined on any time interval. Numerical examples are presented. It is also shown that this scattering problem is reducible to one in which there are no phase velocity mismatches. This reduction provides considerable numerical advantage in the solution of the direct scattering problem. The inverse problem is examined in a companion paper.

Direct and inverse scattering in the time domain for a dissipative wave equation. III. Scattering operators in the presence of a phase velocity mismatch

The focus of this thesis is on the use of wave splitting in electromagnetic direct and inverse scattering problems. Wave splitting offers a decomposition of wave fields into appropriate input and output wave constituents. Several different wave splittings are studied including one-dimensional, multi-dimensional energy-flux, and multi-dimensional locally exact wave splittings. The Bremmer series is naturally connected to wave splitting as a method to decompose a complex scattering problem into a sequence of single scattering problems. The one-dimensional Bremmer series is reviewed and time-domain convergence is shown for the acoustic locally exact wave splitting. The emphasis of the inverse scattering problems is on the identification of the spatial structure of complex medium models in multi-dimensions from time-domain data. The parameter identification is determined in an iterative fashion with a conjugate-gradient algorithm where the least-squares error of the output field is minimized. The gradient is determined from the solution of an additional adjoint problem. The energy-flux split fields are shown to give a good representation of the boundary fields in the inverse scattering problem. Several multi-parameter identifications are performed in two spatial dimensions. A detailed analysis is included about electromagnetic modeling. The non-unique-ness of the instantaneous response and the long-time behavior is specially emphasized. Finally, time-reversal mirrors and time-reversal cavities are discussed. (Less)

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Wave Splitting in Direct and Inverse Scattering Problems

Wave splitting and invariant imbedding is used to obtain a one-parameter family of scattering problems and the associated reflection operator for the wave equation in three dimensions. The velocity is assumed to be at least twice differentiable. Existence and smoothness properties of the reflection operator are proven and a smoothed version of the imbedding equation for the kernel of the reflection operator is developed. The practical use of these imbedding equations in a layer-stripping approach (that can be implemented numerically) to the direct and inverse problem is shown.

https://iopscience.iop.org/article/10.1088/0266-5611/6/6/014/pdf

Invariant imbedding for the wave equation in three dimensions and the applications to the direct and inverse problems

The dispersion of transient electromagnetic waves in a homogeneous medium can be characterized by expressing either the complex permittivity as a function of frequency or the susceptibility kernel as a function of time. In this paper, a time domain technique is used to derive a nonlinear integrodifferential equation which relates the susceptibility kernel for a one‐dimensional homogeneous slab to the reflection operator for the medium. Thus, the susceptibility kernel (which is a function of time) can be determined from reflection data. A numerical implementation of this technique is shown. The more general case of a medium consisting of a stack of homogeneous dispersive layers is also addressed.

R. J. Krueger

Papers

An electromagnetic inverse problem for dispersive media