Three-dimensional inverse scattering: Plasma and variable velocity wave equations

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Abstract
Exact equations governing three-dimensional time-domain inverse scattering are derived for the plasma wave equation and the variable velocity classical wave equation. This derivation was announced for the plasma wave equation in a short note by the authors. That work was motivated by Newton’s three-dimensional generalization of Marchenko’s equation. This paper gives the details of the new derivation and extends it to the classical wave equation. For the time domain derivation in this paper, the scattering region is assumed to have compact support and smoothly joins the surrounding three-dimensional infinite medium. The derivation contains the following ingredients: (1) a representation of the solution at a point in terms of its values on a large sphere, (2) the far-field form of the Green’s function, (3) causality, and (4) information carried in the wave front of the solution. The derivation of the classical wave inverse scattering equation requires that the velocity in the scattering region be less than that of the surrounding medium. This condition is natural, for example, in the scalar wave model of electromagnetic scattering from dielectric nonconducting bodies in free space. Finally, an experiment to verify the inverse scattering equations is proposed.

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