Large wave number aperture-limited Fourier inversion and inverse scattering

Abstract

Aperture-limited Fourier operators are like pseudo-differential operators, except that the symbol of the latter is part of the unknown kernel function of the former. Such operators arise naturally in inverse scattering when one applies an integral inverse scattering operator to model data with an unknown (such as a reflection coefficient) which depends on the source/receiver location as well as on interior medium coordinates. Large wave number aperture-limited Fourier operators can be analyzed by multidimensional stationary phase. The nature of the asymptotic expansion of such operators will then depend on the properties of the kernel function to which the operator is applied. Of interest in inverse scattering are distributions with support at a point or on a surface (the singular function of a surface) and piecewise smooth functions (whose discontinuity surfaces are reflectors). The purpose of this paper is to present some features of the asymptotics of large wave number aperture-limited Fourier inversion operators and to relate these results to the application of a recently developed inverse scattering formalism as applied to Kirchhoff-approximate scattering data from a reflecting surface. It is shown how the latter problem can be reduced to the former, asymptotically. Then, the more simply derived asymptotic expansions of the former can be applied to predict the output of the latter.