Analytical solutions to sampling fluctuations in raindrop size distribution measurements

Fluctuations in disdrometric measurements of raindrop size distributions and derived rainfall properties are due 'both to statistical sampling errors and to real fine-scale physical variations which are not readily separable from the statistical ones' (Gertzman and Atlas, J. Geophys. Res. 82, 4955-4966, 1977). The terminology generally adopted for these two types of fluctuations is "sampling fluctuations" and "natural variability", respectively. It would be of great practical importance to be able to distinguish between both sources of variability. This is because the parameters of raindrop size distributions and the coefficients of Z-R relationships should represent the properties of the type of rainfall to which they pertain as much as possible and the properties of the raindrop sampling device from which they are derived as little as possible. It is therefore necessary to investigate to what extent rainfall fluctuations observed with different types of instruments reflect the properties of the rainfall process itself and to what extent they are merely instrumental artifacts.

We have used a statistical model of the microstructure of rainfall to derive explicit expressions for the magnitude of the sampling fluctuations in rainfall properties derived from raindrop size distribution measurements. The model is a so-called marked point process, where the points represent the drop centers and the marks their sizes. The simplest situation obviously is the case where only sampling fluctuations are present and no natural variability. As rare as this situation may be in practice, it is of more than merely academic interest. It provides a lower bound to the magnitude of the variability in a practical situation where sampling fluctuations and natural variability exist side-by-side. In the absence of natural variability, it is plausible to assume that (1) raindrops are uniformly distributed in space and (2) raindrop sizes are independent of each other and of the positions of the drops in space. This implies that the arrival process of drops at the measuring device is a so-called homogeneous Poisson process and that the numbers of drops arriving at non-overlapping time intervals have independent Poisson distributions. Within this framework, we show analytically that (and how) the sampling distribution of the estimator of any bulk rainfall variable converges to a Gaussian distribution. In addition to being useful in their own right, these results provide a theoretical confirmation and explanation of the simulation results of Smith et al. (J. Appl. Meteor. 32, 1259-1269, 1993). We also derive explicit expressions for the minimum and maximum raindrop diameters in a sample.