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RADIO SCIENCE, VOL. 31, NO. 1, PAGES 51–65, 1996

Wavelet-based methods for the nonlinear inverse scattering problem using the extended Born approximation

Eric L. Miller

Center for Electromagnetics Research, Department of Electrical and Computer Engineering Northeastern University, Boston, Massachusetts


Alan S. Willsky

Laboratory for Information and Decision Systems, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts


Abstract

In this paper, we present an approach to the nonlinear inverse scattering problem using the extended Born approximation (EBA) on the basis of methods from the fields of multiscale and statistical signal processing. By posing the problem directly in the wavelet transform domain, regularization is provided through the use of a multiscale prior statistical model. Using the maximum a posteriori (MAP) framework, we introduce the relative Cramér-Rao bound (RCRB) as a tool for analyzing the level of detail in a reconstruction supported by a data set as a function of the physics, the source-receiver geometry, and the nature of our prior information. The MAP estimate is determined using a novel implementation of the Levenberg-Marquardt algorithm in which the RCRB is used to achieve a substantial reduction in the effective dimensionality of the inversion problem with minimal degradation in performance. Additional reduction in complexity is achieved by taking advantage of the sparse structure of the matrices defining the EBA in scale space. An inverse electrical conductivity problem arising in geophysical prospecting applications provides the vehicle for demonstrating the analysis and algorithmic techniques developed in this paper.

Received 30 March 1995; accepted 11 September 1995.

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Citation: Miller, E. L., and A. S. Willsky (1996), Wavelet-based methods for the nonlinear inverse scattering problem using the extended Born approximation, Radio Sci., 31(1), 51–65.