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Paul L. Smith and Donna V. Kliche

Abstract

The moment estimators frequently used to estimate parameters for drop size distribution (DSD) functions being “fitted” to observed raindrop size distributions are biased. Consequently, the fitted functions often do not represent well either the raindrop samples or the underlying populations from which the samples were taken. Monte Carlo simulations of the process of sampling from a known exponential DSD, followed by the application of a variety of moment estimators, demonstrate this bias. Skewness in the sampling distributions of the DSD moments is the root cause of this bias, and this skewness increases with the order of the moment. As a result, the bias is stronger when higher-order moments are used in the procedures. Correlations of the sample moments with the size of the largest drop in a sample (D max) lead to correlations of the estimated parameters with D max, and, in turn, to spurious correlations between the parameters. These things can lead to erroneous inferences about characteristics of the raindrop populations that are being sampled. The bias, and the correlations, diminish as the sample size increases, so that with large samples the moment estimators may become sufficiently accurate for many purposes.

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Paul L. Smith and Donna V. Kliche
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Donna V. Kliche, Paul L. Smith, and Roger W. Johnson

Abstract

The traditional approach with experimental raindrop size data has been to use the method of moments in the fitting procedure to estimate the parameters for the raindrop size distribution function. However, the moment method is known to be biased and can have substantial errors. Therefore, the L-moment method, which is widely used by hydrologists, was investigated as an alternative. The L-moment method was applied, along with the moment and maximum likelihood methods, to samples taken from simulated gamma raindrop populations. A comparison of the bias and the errors involved in the L-moments, moments, and maximum likelihood procedures shows that, with samples covering the full range of drop sizes, L-moments and maximum likelihood outperform the method of moments. For small sample sizes the moment method gives a large bias and large error while the L-moment method gives results close to the true population values, outperforming even maximum likelihood results. Because the goal of this work is to understand the properties of the various fitting procedures, the investigation was expanded to include the effects of the absence of small drops in the samples (typical disdrometer minimum size thresholds are 0.3–0.5 mm). The results show that missing small drops (due to the instrumental constraint) can result in a large bias in the case of the L-moment and maximum likelihood fitting methods; this bias does not decrease much with increasing sample size. Because the very small drops have a negligible contribution to moments of order 2 or higher, the bias in the moment methods seems to be about the same as in the case of full samples. However, when moments of order less than 2 are needed (as in the case of modelers using moments 0 and 3), the moment method gives much larger bias. Therefore a modification of these methods is needed to handle the truncated-data situation.

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Paul L. Smith, Donna V. Kliche, and Roger W. Johnson

Abstract

This paper complements an earlier one that demonstrated the bias in the method-of-moments (MM) estimators frequently used to estimate parameters for drop size distribution (DSD) functions being “fitted” to observed raindrop size distributions. Here the authors consider both the bias and the errors in MM estimators applied to samples from known gamma DSDs (of which the exponential DSD treated in the earlier paper is a special case). The samples were generated using a similar Monte Carlo simulation procedure. The skewness in the sampling distributions of the DSD moments that causes this bias is less pronounced for narrower population DSDs, and therefore the bias problems (and also the errors) diminish as the gamma shape parameter increases. However, the bias still increases with the order of the moments used in the MM procedures; thus it is stronger when higher-order moments (such as the radar reflectivity) are used. The simulation results also show that the errors of the estimates of the DSD parameters are usually larger when higher-order moments are employed. As a consequence, only MM estimators using the lowest-order sample moments that are thought to be well determined should be used. The biases and the errors of most of the MM parameter estimates diminish as the sample size increases; with large samples the moment estimators may become sufficiently accurate for some purposes. Nevertheless, even with some fairly large samples, MM estimators involving high-order moments can yield parameter values that are physically implausible or are incompatible with the input observations. Correlations of the sample moments with the size of the largest drop in a sample (D max) are weaker than for the case of sampling from an exponential DSD, as are the correlations of the MM-estimated parameters with D max first noted in that case. However, correlations between the estimated parameters remain because functions of the same observations are correlated. These correlations generally strengthen as the sample size increases.

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Roger W. Johnson, Donna V. Kliche, and Paul L. Smith

Abstract

When fitting a raindrop size distribution using a gamma model from data collected by a disdrometer, some consideration needs to be given to the small drops that fail to be recorded (typical disdrometer minimum size thresholds being in the 0.3–0.5-mm range). To this end, a gamma estimation procedure using maximum likelihood estimation has recently been published. The current work adds another procedure that accounts for the left-truncation problem in the data; in particular, an L-moments procedure is developed. These two estimation procedures, along with a traditional method-of-moments procedure that also accounts for data truncation, are then compared via simulation of volume samples from known gamma drop size distributions. For the range of gamma distributions considered, the maximum likelihood and L-moments procedures—which perform comparably—are found to outperform the procedure of method-of-moments. As these three procedures do not yield simple estimates in closed form, salient details of the R statistical code used in the simulations are included.

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Paul L. Smith, Roger W. Johnson, and Donna V. Kliche

Abstract

Use of the standard deviation σ m of the drop mass distribution as one of the three parameters of raindrop size distribution (DSD) functions was introduced for application to disdrometer data supporting the Global Precipitation Measurement dual-frequency radar system. The other two parameters are a normalized drop number concentration N w and the mass-weighted mean diameter D m. This paper presents an evaluation of that formulation of the DSD functions, in two parts. First is a mathematical analysis showing that the procedure for estimating σ m, along with the other DSD parameters, from disdrometer data is in essence another moment method. As such, it is subject to the biases and errors inherent in all moment methods. When the form of the DSD function is specified, it is inferior (like all moment methods) to the maximum likelihood technique for fitting parameters to sampled data. The second part is a series of sampling simulations illustrating the biases and errors involved in applying the formulation to the specific case of gamma DSDs. It leads to underestimates of σ m and consequently to overestimates of the gamma shape parameter—with large root-mean-square errors. Comparison with maximum likelihood estimates shows the degree of improvement that could be obtained in the estimates of the shape parameter. The propensity to underestimate σ m will be pervasive, and users of this DSD formulation should be cognizant of the biases and errors that can occur.

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