Abstract

Many practical problems of non-destructive evaluation can be solved provided we can accurately reconstruct discontinuities of the parameters of the physical medium. Seismic exploration, medical applications, crack and void detection are examples. Our concern is a mathematical formulation of the linearized inverse scattering problem, so that we can

i. obtain explicit algorithms

ii. prove that, indeed, the discontinuities are recovered.

Adopting the point of view standard in the theory of pseudodifferential operators we note that, in general, it is impossible to construct an explicit solution (say the Green's function), but a parametrix can often be obtained explicitly. The parametrix is a solution which is defined exactly as the Green’s function except that an arbitrary smooth function may have been added to the source term of the equation (see [1], for example). Mathematicians have been using these solutions to gain insight into the properties of the partial differential equations. In dealing with the problems of nondestructive evaluation, we can make use of these solutions to obtain algorithms for reconstructing the discontinuities of parameters describing the medium [2].

© 1984 Optical Society of America