Lovejoy and Schertzer (1990a) presented a statistical analysis of blotting paper observations of the (twodimensional) spatial distribution of raindrop stains. They found empirical evidence for the fractal scaling behavior of raindrops in space, with potentially far-reaching implications for rainfall microphysics and radar meteorology. In particular, the fractal correlation dimensions determined from their blotting paper observations led them to conclude that "drops are (hierarchically) clustered" and that "inhomogeneity in rain is likely to extend down to millimeter scales". Confirming previously reported Monte Carlo simulations, we demonstrate analytically that the claims based on this analysis need to be reconsidered, as fractal correlation dimensions similar to the ones reported (i.e. smaller than the value of two expected for uniformly distributed raindrops) can result from instrumental artifacts (edge effects) in otherwise homogeneous Poissonian rainfall. Hence, the results of the blotting paper experiment are not statistically significant enough to reject the Poisson homogeneity hypothesis in favor of a fractal description of the discrete nature of rainfall. Our analysis is based on an analytical expression for the expected overlap area between a circle and a square, when the circle center is randomly (uniformly) distributed inside the square. The derived expression (πr2-8r 3/3+r4/2, where r denotes the ratio between the circle radius and the side of the square) can be used as a reference curve against which to test the statistical significance of fractal correlation dimensions determined from spatial point patterns, such as those of raindrops and rainfall cells.@en

A stochastic model of the microstructure of rainfall is used to derive explicit expressions for the magnitude of the sampling fluctuations in rainfall properties estimated from raindrop size measurements in stationary rainfall. The model is a marked point process, in which the points represent the drop centers, assumed to be uniformly distributed in space. This assumption, which is supported both by theoretical and by empirical evidence, implies that during periods of stationary rainfall the number of drops in a sample volume follows a Poisson distribution. The marks represent the drop sizes, assumed to be distributed independent of their positions according to some general drop size distribution. Within this framework, it is shown analytically how the sampling distribution of the estimator of any bulk rainfall variable (such as liquid water content, rain rate, or radar reflectivity) in stationary rainfall converges from a strongly skewed distribution to a (symmetrical) Gaussian distribution with increasing sample size. The relevant parameter controlling this evolution is the average number of drops in the sample ns. For a given sample size, the skewness of the sampling distribution is found to be more pronounced for higher order moments of the drop size distribution. For instance, the sampling distribution of the normalized mean diameter becomes nearly Gaussian for ns > 10, while the sampling distribution of the normalized rain rate remains skewed for ns ∼ 500. Additionally, it is shown analytically that, as a result of the mentioned skewness, the median Q50 as an estimator of a bulk rainfall variable always underestimates its population value Qp in stationary rainfall. The ratio of the former to the latter is found to be Q50 / Qp = 1 - b / ns + O (ns- 2), where b is a constant depending on the drop size distribution. For bulk rainfall variables this constant is positive and therefore the median always underestimates the population value. This provides a theoretical confirmation and explanation of previously published simulation results. Finally, relationships between the expected number of raindrops in the sample ns and the rain rate are established for different parametric forms of the raindrop size distribution. These relationships are first compared to experimental results and then used to provide examples of sampling distributions of bulk rainfall variables (in this case rain rate) for different values of the average rain rate and different integration times of the disdrometric device involved (in this case a Joss-Waldvogel disdrometer). The practical relevance of these results is (1) that they provide exact solutions to the sampling problem during (relatively rare) periods of stationary rainfall (e.g., drizzle), and (2) that they provide a lower bound to the magnitude of the sampling problem in the general situation where sampling fluctuations and natural variability co-exist.@en

Normalization of drop size distributions (DSDs) is reexamined here. First, an extension of the scaling normalization that uses one moment of the DSD as a scaling parameter to a more general scaling normalization that uses two moments as scaling parameters of the normalization is presented. In addition, the proposed formulation includes all two-parameter normalizations recently introduced in the literature. Thus, a unified vision of the question of DSD normalization and a good model representation of DSDs are given. Data analysis of some convective and stratiform DSDs shows that, from the point of view of the compact representation of DSDs, the double-moment normalization is preferred. However, in terms of physical interpretation, the scaling exponent of the single-moment normalization clearly indicates two different rain regimes, whereas in the double-moment normalization the two populations are not readily separated. It is also shown that DSD analytical models (exponential, gamma, and generalized gamma DSD) have the same scaling properties, indicating that the scaling formalism of DSDs is a very general way of describing DSDs.

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