Simultaneous reconstruction of approximate profiles of an inhomogeneous lossy medium through a collocation method

In this paper a one-dimensional inverse-scattering problem is considered. Two approximate inversion formulae are derived. Using these formulae, approximate permittivity and conductivity profiles of an inhomogeneous lossy medium are simultaneously generated from the partial knowledge of the reflection coefficient in the frequency domain. These formulae are deduced through the application of linear transformations to two nonlinear differential equations for the reflection coefficient in the frequency domain. Discrete versions of the inversion formulae are also developed. The transformed equations for the reflection coefficient in the frequency domain are applicable irrespective of whether the derivative of the unknown permittivity profile exists. An example to demonstrate these approximate inversion formulae is given.