Solution to reverse refraction problem

The reverse refraction problem (determination of radial profile of refractive index in planetary atmospheres, such as Earth, from radio probe measurements) is formulated as a bistatic radar problem for a spherically symmetric medium. The modified refractive index n(r)r (a-radius at which the refraction angle as function of relative distance is measured) is assumed to reach extreme values at the upper boundary r sub 1 or at observation level. Before the corresponding Fredholm equation of the first kind can be solved, it must be well-conditioned in the Tikhonov sense. This is done here by two quasi-optimum integral transformation variants with respect to the measurement function and subsequent simplified regularization. The first method is two successive Fourier cosine transformations followed by an Abel transformation, with the possibility of discrete Fourier transformations and numerical Abel transformation. The second method is twofold discrete Fourier transformation. Both yield solutions readily evaluated by simple algorithms. Regularization is effected by approximating functions satisfying the two fundamental conditions for convergence required of the measurement function.