Exact solutions to the valley problem in inverse scattering

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Abstract
Standard approximate methods involving the Abel integral equation do not allow the ionospheric electron density to be determined in the ‘‘valley’’ between two electron density peaks. Here we present analytic solutions to the Gel’fand–Levitan equation, which occurs in the exact full-wave inverse scattering theory. These exact analytic solutions exhibit multiple peaks in the electron density as a function of height and provide a solution to the valley problem.