To what extent can raindrop size be determined by a multiple-frequency radar?

In this paper we present an analytical treatment of the atmospheric remote sensing problem of determining the rain Drop Size Distribution (DSD) with a spaceborne multi-frequency microwave nadir-looking radar system. It is typically assumed that with two radar measurements at different frequencies one ought to be able to calculate two state variables of the DSD: a bulk quantity such as the rain rate and a distribution shape parameter. To determine if this non-linear problem can indeed be solved, we model the DSD as a Gamma distribution and use quadratic approximations to the corresponding radar-rain relations to examine the invertibility of the resulting system of equations in the case of two as well as three radar frequencies.