One-dimensional profile inversion of a half-space bounded by a three-part impedance ground

A method which permits one to reveal the one-dimensional electromagnetic profile of a half-space over a three-part impedance ground is established. The method reduces the problem to the solution of two functional equations. By using a special representation of functions from the space L¹ (-∞, ∞), one of these equations is first reduced to a modified Riemann-Hilbert problem and then solved asymptotically. The asymptotic solution is valid when the wave used for part of the boundary is sufficiently large as compared to the wavelength of the wave used for measurements. The second functional equation is reduced under the Born approximation to a Fredholm equation of the first kind whose kernel involves the solution to the first equation. Since this latter constitutes an ill-posed problem, its regularized solution in the sense of Tikhonov is given. The accuracy of the asymptotic solution to the first equation requires the use of waves of high frequencies while the Born approximation in the second equation is accurate for lower frequencies. A criterion to fix appropriate frequencies meeting these contradictory requirements is also given. An illustrative application shows the applicability and the accuracy of the theory. The results may have applications in profiling the atmosphere over non-homogeneous terrains.