Fluctuation Properties of Precipitation. Part II: Reconsideration of the Meaning and Measurement of Raindrop Size Distributions

Abstract

For M drop size categories, rain is frequently viewed simply as the superposition of M, statistically independent Poisson-distributed drop fluxes each described by its own mean concentration. Implicit in such a Poissonian model is the assumption of uncorrelated counts among the drops. However, it is well known that drop size distributions are the result of the processes of collision, coalescence, and breakup, which should lead to correlations. This inconsistency is resolved in this work. Using 1-min disdrometer measurements, two-point cross-correlation functions are used to show that drop counts at different sizes are correlated rather than independent. Moreover, it is argued that it is more appropriate to characterize rain statistically as a doubly stochastic Poisson process (Poisson mixture) among a collection of M correlated random variables (fluxes) each having its own probability distribution of unpredictable (random) mean values and its own coherence time, τM. It is also shown that a drop size distribution has a characteristic coherence time, τ. It is then argued that in order to preserve the purity of a size distribution of interacting drops, τ must be equivalent to the shortest τM. For sampling intervals much shorter than τ and when the observation time, T, is less than τ, the drop counts remain correlated and the drop size distribution assumes the definition of a collection of physically interacting drops. On the other hand, when T ? τ, the drop counts decorrelate and the concept of the drop size distribution changes to a formal relation among the M observed drop concentrations averaged over several different size distributions. Moreover, when T is between the longest and shortest τM, part of the observed distribution will represent the distribution of interacting drops and the other part will represent a mixture of drops from different distributions. Finally, this work suggests using multiple time series analysis techniques for estimating mean drop concentrations in order to use all the available information and to help reduce drop size distribution mixing associated with the conventional analysis based on fixed time intervals.