Statistical tools for drop size distributions: Moments and generalized gamma

摘要

Several problems associated with drop size distributions are treated. For rainfall rate R or radar reflectivity Z high powers of the drop diameters must be taken into account. This paper suggests methods to deal with the relevant moments and to approximate the distributions by a generalized gamma distribution P(x) = gamma[ u, (x/c)(r)]/ Gamma (u). Retaining the power r as a parameter is the difference to a gamma distribution. It leads to a large flexibility, enabling one to fit the distribution to the moments of interest. The three parameters r, u, and c can be fitted exactly to three moments of the order alpha, 2 alpha, and 3 alpha, respectively, where alpha defines the power of the variable. The expressions for the moments and the transformation to another power a are simple for generalized gamma distributions. A quickly converging algorithm to calculate the parameters r, u, and c is shown. The solutions can be obtained in a spreadsheet or from the internet. The proposed technique is particularly powerful for describing sampling distributions with R or Z from a known drop size distribution. Whether the sample size is a fixed number or whether it varies according to a Poisson distribution makes a difference, which is treated here. However, the processes associated with precipitation and drop size distributions seem to be more complex, leading to larger variations than expected from simple models of random sampling. [References: 18]