Massimiliano Ignaccolo

The Droplike Nature of Rain and Its Invariant Statistical Properties

Abstract This study looks for statistically invariant properties of the sequences of inter-drop time intervals and drop diameters. The authors provide evidence that these invariant properties have the following characteristics: 1) large inter-drop time intervals (≳10 s) separate drops of small diameter (≲0.6 mm); 2) the rainfall phenomenon has two phases: a quiescent phase, whose contribution to the total cumulated flux is virtually null, and an active, nonquiescent, phase that is responsible for the bulk of the precipitated volume; 3) the probability density function of inter-drop time intervals has a power-law-scaling regime in the range of ∼1 min and ∼3 h); and 4) once the moving average and moving standard deviation are removed from the sequence of drop diameters, an invariant shape emerges for the probability density function of drop diameters during active phases.

Detrended fluctuation analysis of scaling crossover effects

Detrended fluctuation analysis (DFA) is one of the most frequently used fractal time series algorithms. DFA has also become the tool of choice for analysis of the short-time fluctuations despite the fact that its validity in this domain has never been demonstrated. We adopt an Ornstein-Uhlenbeck Langevin equation to generate a time series which exhibits short-time power-law scaling and incorporates the fundamental property of physiological control systems ?negative feedback. To determine the scaling exponent, we derive the analytical expressions for the standard deviation of the solution X(t) of this equation using both the ensemble of statistically independent trajectories and the ensemble obtained by partitioning a single trajectory. The latter approach is used in DFA and many other physiological applications. Surprisingly, the formulas for the standard deviations are different for these two ensembles. We demonstrate that the partitioning amounts to building up deterministic trends that satisfy the trend?+?signal decomposition assumption which is characteristic of DFA. Consequently, the dependence of the rms of DFA residuals F(?) on the length ? of data window is the same for both ensembles. The growth of F(?) is significantly different from that of the standard deviation of X(t). While the DFA estimate of the short-time scaling exponent is correct, the polynomial detrending delays the approach of F(?) to the asymptotic value by as much as an order of magnitude. This delay may underlie the gradual change of the DFA scaling index typically observed in time series that exhibit crossover between the short- and long-time scaling.

Phase Space Parameterization of Rain: The Inadequacy of Gamma Distribution

We show that the Gamma distribution is not an adequate fit for the probability density function of drop diameters using the Kolmogorov-Smirnov goodness of fit test. We propose a different parametrization of drop size distributions, which not depending by any particular functional form, is based on the adoption of standardized central moments. The first three standardized central moments are sufficient to characterize the distribution of drop diamters at the ground. These parameters together with the drop count form a 4-tuple which fully describe the variability of the drop size distributions. The Cartesian product of this 4-tuple of parameters is the rainfall phase space. Using disdrometer data from 10 different locations we identify invariant, not depending on location, properties of the rainfall phenomenon.

The discrete charm of rain

Weather reports and many scientific models treat rainfall as a continuous process, but to truly understand rain and its effects, one must consider its fundamentally discrete nature.