Estimation of Gamma Raindrop Size Distribution Parameters: Statistical Fluctuations and Estimation Errors

Abstract

The gamma distribution is often used to characterize raindrop size distributions (DSDs). However, the estimation of measured raindrop distributions suffers from the shortcomings of statistical sampling errors, which become increasingly significant when the collecting surface of the measuring instrument and the integration time are small. Different estimators of the three parameters (N0*, ?, and Dm) that characterize a normalized gamma distribution have been computed from simulated DSD. A database has been established, containing 22 950 simulated DSDs, corresponding to a wide set of various rainfall situations. Moment, least squares, and maximum likelihood estimators have been evaluated. Error measurement considerations are discussed, in particular the difficulty encountered in measuring small drops (diameter <0.5 mm) with a disdrometer. Modified estimation approaches are proposed to compensate for the lack of small drops accounted for by real measurements. For each of the different methods, systematic error analysis is performed, and the estimation error is quantified in terms of its bias and standard deviation. The sensitivity of the various methods to instrumental characteristics is also evaluated. A case study is run to highlight correlation effects in the estimated DSD parameters, resulting from the use of various retrieval techniques. Finally, a criterion is derived that enables the hypothesis of gamma-distributed DSD to be tested. When applied to real data recorded by an optical disdrometer, this criterion shows that approximately 91% of DSDs are of the gamma type. Real gamma DSDs are then used to compare adapted maximum likelihood estimators with the more commonly used methods.