Analytical solutions to sampling effects in drop size distribution measurements during stationary rainfall: Estimation of bulk rainfall variables

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 n_s. 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 n_s > 10, while the sampling distribution of the normalized rain rate remains skewed for n_s ∼ 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 Q_p in stationary rainfall. The ratio of the former to the latter is found to be Q₅₀ / Q_p = 1 - b / n_s + O (n_s^{- 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 n_s 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.