Abstract - Much work has been done on inverse scattering problems for one-dimensional acoustic media. Unfortunately, the linear method of the Gel’fand - Levitan integral equations is developed only for inversion from reflection data in case of negligible dissipative effects. On the contrary, measurement data, in geophysical prospecting particularly, are often influenced by these effects. The main result of this paper is a necessary and sufficient condition of solvability of the inverse dissipative scattering problem with data given on a finite time interval. We also consider a bilinear procedure for recovering the dissipation coefficient from the reflection data. The approach is based on a step-by-step split method for numerical solution of a system of the Gel’fand - Levitan integral equations under nonlocal boundary conditions. Besides, we develop a method of inversion of difference schemes for solving the inverse dissipative scattering problems.

On local solvability of inverse dissipative scattering problems

The dynamic inverse seismics problem is considered in a generalized setting. We investigate whether the wave propagation problem in a vertically nonhomogeneous medium is well-posed. We show that the regular part of the solution is an L2 function and the inverse problem, i.e., the determination of the reflection coefficient, is thus reducible to minimizing the error functional. The gradient of the functional is obtained in explicit form from the conjugate problem, and approximate formulas for its evaluation are derived. A regularization algorithm for the solution of the inverse problem is considered; simulation results using various excitation sources are reported.

Solving the Inverse Generalized Problem of Vertical Seismic Profiling

A three-dimensional inverse scattering problem for the acoustic wave equation is studied. The task is to determine the density and acoustic impedance of a medium. A necessary and sufficient condition for the unique solvability of this problem is established in the form of an energy conservation law. The interpretation of the solution to the inverse problem and the construction of medium images are discussed.

Solution of an inverse scattering problem for the acoustic wave equation in three-dimensional media

The solvability of two-dimensional inverse scattering problems for the Klein-Gordon equation and the Dirac system in a time-local formulation is analyzed in the framework of the Galerkin method. A necessary and sufficient condition for the unique solvability of these problems is obtained in the form of an energy conservation law. It is shown that the inverse problems are solvable only in the class of potentials for which the stationary Navier-Stokes equation is solvable.

On t-local solvability of inverse scattering problems in two-dimensional layered media

Two-dimensional (2D) inverse scattering problems for the acoustic wave equation consisting of obtaining the density and acoustic impedance of the medium are considered. A necessary and sufficient condition for the unique solvability of these problems in the form of the law of energy conservation has been established. It is proved that this condition is that for each pulse oscillation source located on the boundary of a half-plane, the energy flow of the scattered waves is less than the energy flux of waves propagating from the boundary of this half-plane. This shows that for inverse dynamic scattering problems in acoustics and geophysics when the law of energy conservation holds it is possible to determine the elastic density parameters of the medium. The obtained results significantly increase the class of mathematical models currently used in solving multidimensional inverse scattering problems. Some specific aspects of interpreting inverse problems solutions are considered.

Imaging of Layered Media in Inverse Scattering Problems for an Acoustic Wave Equation

The problem of finding the variable characteristics of an inhomogeneous layer from the scattered field when the plane wave incident on the layer is unknown, is considered Equations are obtained whereby both the wave shape and the parameters of the medium can be found. The equations are solved by an iterative regularization method.

A. V. Bayev

Papers

A method of solving the inverse scattering problem for the wave equation