One-dimensional inverse scattering problem

Abstract

The inverse problem of electromagnetic scattering of media with one-dimensional permittivity distribution is considered. Two approaches are applied in the study. First is based on the solution of non-linear integral equation for the scattered field; second-involves in analysis the Lagrange formalism applied to initial differential equations (Maxwell's equations). Based on the developed theory, solution algorithms have been worked out for diagnostics of subsurface permittivity inhomogeneities into multilayer periodic dielectric structures. Keywords: inverse scattering problem, permittivity profile, multilayered periodic structures INTRODUCTIONInverse problems of scattering are widely used in various methods of sounding and tomography of media parameters in electromagnetism, acoustics and quantum mechanics. For one-dimensional (1D) distributions of media parameters, they can be reduced to the known Gelfand-Levitan-Marchenko equation (1). However, such generalizations are inapplicable to layered or absorbing media. This simplest problem for the ultra low-frequency electromagnetic geomagnetic sounding of earth crust permittivity profiles has been formulated firstly by Tikhonov [2] in frameworks of his general theory of ill-posed problems. The frequency dependence of the effective depth of the received signal formation (skin-depth) of measured fields was in use in this method, applied further in the magnetotelluric exploration [3].In [4][5][6], in frameworks of electromagnetic perturbation theory, the inverse scattering problem in various statements has been reduced to the non-linear integral equation of the 1 st kind that should been solved iteratively, beginning with the Born approximation. Based on this equation, the one-dimensional problem of low-frequency earth crust profiling has been solved with the use of Tikhonov's method of generalized discrepancy [7]. Results of the numerical study [7] for low-frequency conductivity sounding have demonstrated serious limitations of such approach for large perturbations, when the Born approximation (first guess of iterative method) is inapplicable. To overcome these restrictions of perturbation theory, the new method of dual regularization based on the Lagrange approach in the optimization theory [8] has been applied in this problem to solve initial Maxwell equations [9]. Results show its ability to retrieve very strong variations of conductivity profiles.The approach, based on the solution of the non-linear integral equation has been also applied to diagnostics of permittivity inhomogeneities in multilayer periodical structures that are basic elements of the modern X-ray optics [10]. Since their invention in 1976 [11], they have been widely used as reflectors, polarisers and filters in the "soft" X-ray range, where crystals are irrelevant. Their parameters were optimized for different purposes, but some deviations from a desired perfect meander structure appear at the synthesis. For diagnostics of these structures, measurements of X-ray scattering are in use. This method ha...

Inverse scattering problems in subsurface diagnostics of inhomogeneous media

“…However, our results of our study of more simple one-dimensional problems [7][8][9] demonstrate serious limitations of such iterative approach for large perturbations, when the Born approximation (used as the first guess of iterative method) gives considerable deviations from true solutions. To overcome these restrictions of perturbation theory, the new method of dual regularization based on the Lagrange approach in the optimization theory has been proposed and applied to solve such problems [4][5][6][7][8][9].…”

“…In the dual regularization algorithm for 3D problems proposed here by analogy with those for onedimensional problems [7][8][9][10], it is necessary to satisfy the condition …”

“…To overcome these restrictions of perturbation theory, the new method of dual regularization based on the Lagrange approach in the optimization theory has been proposed and applied to solve such problems [4][5][6][7][8][9]. It can be based on initial Maxwell equations, and results of this application to one-dimensional problems of low-frequency sounding of Earth crust conductivity profile and to retrieve profiles of diffuse perturbations of permittivity in multilayer structures of Xray optics, show its ability to retrieve very strong variations of sounded parameters.…”

“…Among them, inverse scattering problems can be also applied in various methods of electromagnetic tomography 3D distributions of media parameters [2][3][4][5] and in profile retrieval of one-dimensional inhomogeneities [6][7][8][9][10]. In frameworks of electromagnetic perturbation theory, inverse scattering problems in has been reduced to the non-linear integral equation that can be solved iteratively at each step as linear Fredholm integral equations of the 1 st kind, beginning with the Born approximation using, for example, Tikhonov's method of generalized discrepancy.…”

Near-field inverse scattering problems in noninvasive diagnostics of subsurface dielectric structures

“…Unfortunately, there are no universal methods to solve the considered here nonlinear ill-posed problems. Nevertheless, some algorithm based on especial schemes of measurements, suitable approximations, and new achievements in mathematical theory have been worked out and used in this new area [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18].…”