Dual regularization in one-dimensional inverse scattering problem

The considered inverse problem of electromagnetic scattering is widely applied in the subsurface profiling of media permittivity. In previous works, mainly the non-linear integral equation for the scattered field has been in use. It has been solved in the Born approximation or, sometimes, iteratively - beyond this approximation. However, the solution of this ill-posed problem at each step of iterations faced difficulties. To overcome these difficulties, we propose to use the new approach based on the Lagrange formalism applied to initial differential equations (Maxwell's equations). That gives a possibility to obtain the solution of one-dimensional inverse problems of scattering beyond the range of applicability of the perturbation theory. Based on the developed theory, the solution algorithm has been worked out and applied to the simplest one-dimensional problem of low frequency geomagnetic profiling of conductivity of the earth crust.