Fluctuation Properties of Precipitation. Part I: On Deviations of Single-Size Drop Counts from the Poisson Distribution

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Abstract
The traditional statistical description of the spatial and temporal distributions of cloud droplets and raindrops is the Poisson process, which tends to place the drops as uniformly as randomness allows. Yet, the “clumpy” nature of clouds and precipitation is apparent to most casual observers and well known to cloud physicists. Is such clumpiness consistent with the Poisson statistics? The authors explore the possibility of deviations from the Poisson distribution using temporal raindrop counting experiments. Disdrometer measurements during the passage of a squall line strongly indicate that a mixture of Poisson distributions (Poisson mixture) provides a better description of the frequency of drop arrivals per unit time in variable rain than does a simple Poisson model. Poisson mixture generally yields distributions different from Poissonian. While the validity of the Poisson mixture model to smaller scales requires much finer temporal resolution than available in this study, these results do show that one must carefully interpret the statistical and physical meaning of average drop concentrations when the measurements are collected through variable rain, whether observed by airborne or ground-based instruments. Statistically, the variance in the measurements is greatly increased, due to the added variability from the rain field, thus minimizing the reduction of the variance normally achieved by increasing the sample mean (N). In fact, in some cases the variance of relevant distributions scales as N2 rather than N, thereby making the relative fluctuations independent of N. Consequently, the sampling criteria proposed by Cornford are not necessarily generally applicable. Moreover, the authors conjecture that in most clouds the distribution of drop concentrations in small volumes may be more aptly described by a Poisson mixture rather than by a pure Poisson distribution. This may have significant implications with regard to the droplet growth and the evolution of rain.

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