Statistical Interpretation of a general framework for the analysis of raindrop size distributions and their properties

It has recently been demonstrated that all previously proposed parameterizations for the raindrop size distribution (DSD) are special cases of a general formulation, which takes the form of a scaling law. In this formulation, the DSD depends both on the raindrop diameter (D) and on the value of a so-called reference variable, in most cases taken to be the rain rate (R). The generality of this formulation stems from the fact that it is no longer necessary to impose an a priori functional form for the raindrop size distribution. Moreover, it naturally leads to the ubiquitous power law relationships between rainfall integral parameters, notably that between the radar reflectivity factor (Z) and R. Although the formulation has been successfully verified experimentally on a number of occasions, a clear physical interpretation of its scaling exponents and the general DSD function has been lacking until now. Our objective is to provide such an interpretation, to present methods to apply the general formulation in a meaningful way, and to give several examples of such applications.

Using a statistical formulation of the scaling law, we show that the scaling exponents can be expressed in terms of the variances of and the covariances between the parameters of the DSD. As a result, it is found that the values of these scaling exponents determine to what extent it is the fluctuations of the raindrop concentration (or the arrival rate) or the fluctuations of the characteristic raindrop size which control the spatial and temporal variability of the DSD. The general DSD function is a scaled form of the DSD, the intrinsic form remaining when all variability due to fluctuations of the reference variable has been filtered out. We show that if R is used as the reference variable, the general DSD function is simply an equivalent DSD at R=1 mm/h. Based on these interpretations, we have devised robust and self-consistent methods to estimate the values of the scaling exponents and to fit analytical parameterizations to the general DSD function. As such, our framework for analyzing DSDs explicitly bridges the gap between the scaling law formulation and the traditional analytical parameterizations.

Our interpretation of the scaling law formulation is an attempt to establish a direct link between the observed variability of the form of DSDs at the ground and the microphysical processes aloft that shape them. As a result, we are able to try to relate the values of the coefficients of the power laws used in radar meteorology (such as Z-R relationships) to the type of rainfall (orographic, stratiform, convective, equilibrium) and the climatic setting. We provide extensive examples of the application of our general framework for analyzing DSDs, both using data published in the literature and using data that have been collected using different types of disdrometric instruments in field experiments in the US and Europe.