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One formulation of inverse scattering theory involves the Gel’fand–Levitan equation. We present a procedure for finding exact solutions to this equation; this procedure can be applied whenever the reflection coefficient is a rational function of the wave number k, with an arbitrary number of poles. We present graphs of the potential as a function of distance, for several cases with 3, 4, 5, and 6 poles. Prior to this paper, no 4-, 5-, or 6-pole case had ever been treated successfully.

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.

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