Regularization of Marchenko’s integral equation by total least squares

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Abstract
Succinctly, the inverse scattering problem is to infer the shape, size, and constitutive properties of an object from scattering measurements which result from either seismic, acoustic, or electromagnetic probes. Under ideal conditions, theoretical solutions exist. However, when the scattering measurements are noisy, as is the case in practical scattering experiments, direct application of the classical inverse scattering solutions results in numerically unstable algorithms. In this paper, we discuss an optimization technique called total least squares, which provides a regularization to the one-dimensional inverse scattering problem. Specifically, we show how to use multiple data sets in a Marchenko-type inversion scheme and how the theory of total least squares introduces an adaptive spectral balancing parameter which explicitly depends on the scattering data. This is in strong contradistinction to ordinary least squares techniques which utilize nonexplicit and nonadaptive spectral balancing parameters, generally derived by ad hoc considerations.

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