32nd Conference on Broadcast Meteorology/31st Conference on Radar Meteorology/Fifth Conference on Coastal Atmospheric and Oceanic Prediction and Processes
Wednesday, 6 August 2003
The Bias in Moment Estimators for Parameters of Drop-Size Distribution Functions
Paul L. Smith, South Dakota School of Mines and Technology, Institute of Atmospheric Sciences, Rapid City, SD; and D. V. Kliche
Poster PDF (502.1 kB)
Investigators frequently acquire observations of raindrop sizes and seek to describe the drop-size distributions (DSDs) by analytical expressions, the exponential or gamma function being most common. Moment methods to estimate parameters for the DSD functions have become more or less traditional since first introduced in Waldvogel's 1974 paper on the "N0 jump" of DSDs. The intuitive appeal of such methods is almost irresistible, and the associated mathematical manipulations lend a convincing aura. Unfortunately, the methods are biased - in the statistical sense that the expected values of the "fitted" parameters differ from the parameters of the underlying raindrop population. Moreover, plots of the "fitted" DSDs do not look like ones of either the drop samples on which they are based or the underlying population. In other words, with these moment methods, one should expect to get the wrong answer.

This is not a trivial problem, and the procedure is rife with opportunities for self-delusion. For example, one can take samples of about 50 drops from what is actually an exponential DSD; hypothesize that the DSD is gamma; use the third (LWC), fourth (bracketing R) and sixth (Z) moments to "fit" gamma parameters; find a mean value of about 6 for the gamma shape parameter; and thus convince oneself, quite erroneously, that the DSD is gamma after all. This and other findings have been obtained by Monte Carlo simulations of sampling from known DSDs, which allows:

(1) The "fitted" distribution functions to be compared with the original distribution from which the samples were drawn (something that can never be done in practice) as well as with the sample distributions;

(2) The process to be repeated to assess the effects of sampling variability; and

(3) Aspects (such as the bias) not accessible through error propagation analysis to be investigated.

Some general conclusions from the work thus far:

(a) The bias is greater when higher-order moments are employed (implication: use of Z in the procedure is especially unwise).

(b) The size of the largest drop in the sample is an excellent predictor of the calculated gamma shape parameter.

(c) Samples in the hundreds of drops are required to reduce the bias to any manageable size.

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