Inverse scattering inverse source theory

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Abstract
The inverse scattering inverse source problem associated with the inhomogeneous Helmholtz wave equation, the (special case) Sturm–Liouville (acoustic wave) equation, and the time-independent Schrödinger equation is treated . To this end, the concepts of a reference wave velocity and an associated free reference space Green’s function spectrum are introduced. A modified Kirchhoff surface integral, containing only the gradient of the real part of this free reference space Green’s function spectrum and the fields on a measurement surface is formulated, yielding an integral equation for the unknown fields and sources in the interior of the closed piecewise smooth surface on which the (remotely sensed) fields are known. A well-posed, analytic closed form solution of this integral equation for the unknown fields and their Laplacians is obtained with the aid of a (modified) spatial Fourier transform in which the reference velocity is continually varied in such a fashion that the Ewald sphere shell sweeps to fill the entire transform space. The unknown potential or medium properties and the unknown sources are then determined algebraically for the inverse scattering and inverse source problems, respectively. The effects of finite sampling density and incomplete observation domain are discussed briefly.

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