Snow Studies. Part III: Theoretical Derivations for the Ensemble Retrieval of Snow Microphysics from Dual-Wavelength Vertically Pointing Radars

As a first step toward retrieval of snow microphysics from two vertically pointing radars operating at X band and W band, a theoretical model of snow microphysics is formulated in which the number of unknown parameters is reduced to snow particle density and to two bulk quantities controlling the particle size distribution. This reduction of parameters is achieved by normalizing not only the size distribution but also the snow particle mass in the mass–size relationship as well as by using a relationship between snow density and snow terminal fall velocity. However, no single snow microphysical model could describe the observed variability in the radar measurements. The uncertainty in the developed deterministic relations that map the microphysical parameters to the observables is shown to be mainly associated with the assumed dependence of particle velocity on its mass and on the particle size distribution (PSD) representation. Hence, various mass–velocity relationships together with different generic functional forms of the PSD reported in literature are described in this paper and then used in the retrieval. The derived relations provide a reasonable range of uncertainty associated with the microphysics when used for the actual retrieval of snow properties from observations in Part IV. The uncertainty in the backscattering computations of an individual particle, performed using Mie theory assuming spherical form with nonuniform density, is not taken into account in this study.