## Abstract

Recent studies have led to the statistical characterization of the flux of drops of a particular size as a doubly stochastic Poisson process (Poisson mixture). Moreover, previous papers in this series show that the fluxes at different sizes are correlated among each other both temporally and spatially over many different scales. Thus, in general, rather than being distributed evenly, significant clustering or bunching of the rain occurs. That is, regions richer in drops are interspersed with those where drops are scarcer.

This work applies these recent findings to explore the statistical characteristics of the rainfall rate itself, a triply stochastic random variable resulting from the summation over all the fluxes at different drop sizes. Among the findings, it is shown that clustering of the drops leads to increased frequencies of both smaller and larger rainfall rates. That is, because of clustering, drop rich regions boost the frequency of large rainfall rates, while the likelihood of light rainfall rates increases because of drop poor regions. These results, derived using detailed, physically based Monte Carlo simulations of clustered rain, agree with video-disdrometer observations. Moreover, it is shown that for a given mean rainfall rate, extensive averaging lengthens the tail of the probability density function (pdf) of the rainfall rate, *P*(*R*).

While the tail of the *P*(*R*) for clustered rain is sometimes reminiscent of that of the oft-used lognormal distribution, it is shown that the lognormal pdf is a poor match to the observations and simulations. It is concluded that the lognormal distribution is inconsistent with the statistical physics of natural, clustered rain.

It is also argued that for clustered rain, the relative dispersion of the rainfall rate is proportional to the relative dispersion in the total number of drops in the volumes sampled. While the constant of proportionality depends upon drop diameter, observations demonstrate that the relative dispersion in the rainfall rate is due much more to the variability in the number of drops in the sampled volumes than to variations in drop sizes. The results in this work are likely relevant to such areas of research as remote sensing and hydrology.

*Corresponding author address:* Dr. A. R. Jameson, RJH Scientific, Inc., 5625 N. 32nd St., Arlington, VA 22207.

Email: jameson@rjhsci.com