In this talk, it is shown how an equation may be derived for the mean backscatter cross section of an ensemble of aggregate snowflakes at centimeter and millimeter wavelengths. It uses the Rayleigh-Gans approximation, which has previously been found to be applicable at these wavelengths due to the low density of snow aggregates. Although the internal structure of an individual snowflake is random and unpredictable, we find from simulations of the aggregation process that their structure is self-similar and can be described by a power law. This enables an analytic expression to be derived for the backscatter cross section of an ensemble of particles as a function of their maximum dimension in the direction of propagation of the radiation, the volume of ice they contain, a variable describing their mean shape and two variables describing the shape of the power spectrum. The slope of the power spectrum is found to be the Kolmogorov value of -5/3, a curious result given that simulated aggregation process involved no turbulence. The values of the last three variables describing the internal structure are found to be almost invariant to the shape of the monomer crystals from which the aggregates are formed. We refer to the new model as the Self-Similar Rayleigh-Gans (SSRG) approximation.
For particles larger than the wavelength, the SSRG equation predicts increasingly higher backscatter cross sections with size than the commonly used "soft sphere" and "soft spheroid" approximations (where particles are treated as a homogeneous ice-air mixture). This is because SSRG includes scattering by structures within the particle at the scale of half the wavelength. To illustrate the importance of this effect at 94 GHz, we consider distributions of snowflakes reported in the literature. From those measured by Heymsfield et al. (2008), the SSRG equation predicts reflectivity factor 5 dB higher than the soft spheroid model. For the distributions reported by Lawson et al. (1998), this difference rises to 17 dB. The impact on CloudSat-Calipso retrievals of ice water content will be reported in the talk.