Unified Formulation of Single- and Multimoment Normalizations of the Raindrop Size Distribution Based on the Gamma Probability Density Function

This study offers a unified formulation of single- and multimoment normalizations of the raindrop size distribution (DSD), which have been proposed in the framework of scaling analyses in the literature. The key point is to consider a well-defined “general distribution” g ( x ) as the probability density function (pdf) of the raindrop diameter scaled by a characteristic diameter D c . The two-parameter gamma pdf is used to model the g ( x ) function. This theory is illustrated with a 3-yr DSD time series collected in the Cévennes region, France. It is shown that three DSD moments ( M 2 , M 3 , and M 4 ) make it possible to satisfactorily model the DSDs, both for individual spectra and for time series of spectra. The formulation is then extended to the one- and two-moment normalization by introducing single and dual power-law models. As compared with previous scaling formulations, this approach explicitly accounts for the prefactors of the power-law models to yield a unique and dimensionless g ( x ), whatever the scaling moment(s) considered. A parameter estimation procedure, based on the analysis of power-law regressions and the self-consistency relationships, is proposed for those normalizations. The implementation of this method with different scaling DSD moments (rain rate and/or radar reflectivity) yields g ( x ) functions similar to the one obtained with the three-moment normalization. For a particular rain event, highly consistent g ( x ) functions can be obtained during homogeneous rain phases, whatever the scaling moments used. However, the g ( x ) functions may present contrasting shapes from one phase to another. This supports the idea that the g ( x ) function is process dependent and not “unique” as hypothesized in the scaling theory.