Optimal Determination of Parameters for Gamma-Type Drop Size Distributions Based on Moments

Measured raindrop size distributions are often approximated by analytical functions. The parameters determining such functions are usually derived from measured data. This procedure can suffer from various uncertainties. The most important of which are (i) the limited detection range of measuring devices such as, for example, disdrometers, and (ii) poor statistics resulting from the rare appearance of relatively large drops. One way to derive the parameters is the moments method that has a degree of freedom in the choice of the moments. The aim of this study is to find an optimal choice of moments. To this end, numerical experiments are performed by calculating random samples from drop populations with gamma-shaped size distributions. These samples are evaluated as they were recorded by an ideal disdrometer whose single limitation is the cutoff with respect to very small and very large raindrops. From that data the parameters mentioned above are determined by the moments method. The truncation of the measurement is explicitly taken into account during the retrieval. Further, all possible combinations of three different moments used to derive the gamma parameters are impartially compared. It turns out that parameters derived on the basis of low-order moments are less affected by biases and noise than parameters derived by larger-order moments, that is, the sampling problem is more severe than the truncation problem, especially because the latter can be overcome much more efficiently within the retrieval algorithm. It is shown that by using very low-order moments and estimating the unmeasured fraction of the moments, optimal results from real measurements can be obtained.