Introduction

Investigators frequently acquire observations of raindrop sizes and seek to describe the drop size distributions (DSDs) of the underlying populations from which the samples were taken by analytical expressions, with the exponential or gamma function being most common. Although moment methods to estimate parameters for the DSD functions have become more or less traditional, Haddad et al. (1996, 1997) pointed out that such methods are biased. Hydrologists have long been aware of this bias (e.g., Wallis et al. 1974; Maidment 1993), and Smith and Kliche (2003) provided examples of the bias for the case of raindrop observations. Nevertheless, the intuitive appeal of the moment approach seems almost irresistible, and the associated mathematical manipulations lend a convincing aura. Yet the methods are indeed biased, in the statistical sense that the expected values of the “fitted” parameters differ from the parameters of the underlying raindrop populations, and so can lead to erroneous inferences about the characteristics of the DSDs being sampled.

The bias in the moment methods can be demonstrated by testing their ability to recover parameters of known DSDs from which samples are taken. This must be done by computer simulation, because DSDs in nature are inherently unknown. The simulations herein use a Monte Carlo simulation procedure similar to that described in Smith et al. (1993) and outlined below. This paper gives results for samples taken from a hypothetical exponential DSD fitted with various moment-based procedures; corresponding results for samples taken from gamma DSDs will be presented later.

Simulation of raindrop sampling

We seek to determine the sampling distributions of DSD parameters that are fitted to raindrop observations by moment methods. This is done by simulating repetitive sampling from a specified population DSD, which is exponential in form for the present paper. An exponential DSD is usually expressed in the form

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where n(D) is the number concentration of drops of diameter D per unit size interval, and n₀ and Λ are concentration and size parameters, respectively, of the DSD. On conventional semilogarithmic plots, as introduced by Marshall and Palmer (1948), n(D) becomes a straight line with intercept n₀ and slope −Λ.

For the purposes of these simulations, (1) is more conveniently expressed in terms of the total drop number concentration N_T and the mass-weighted mean diameter D_m (=4/Λ) as

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Defining a dimensionless size variable y = D/D_m, this becomes

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This can be recognized as the product of the mean number concentration N_T and the probability density function (PDF) of drop size 4 exp(−4y). For convenience herein, we also designate the mean sample size (number of drops) as N_T. This could be viewed as representing a volume-sampling instrument with a sample volume of 1 m³ (independent of the drop size). However, a sample volume of α m³ and a mean drop concentration N_t/α would lead to the same mean sample size and the same sampling statistics. Thus, with the drop sizes normalized to D_m, the results can be organized simply by the value of N_T.

The simulation proceeds from a selected value of N_T by first drawing from a Poisson distribution with a mean value N_T to determine the actual total number of drops C in a given sample. Then, C values of y drawn from the exponential PDF establish the (normalized) sizes of those drops. Normalized values for the six sample moments of M_1S through M_6S are next calculated for each sample, and then various moment-based calculations (discussed in section 4 and summarized in the appendix) are applied to estimate the DSD parameters. For purposes of the present work, we classified the drop sizes into intervals of Δy = 0.02, representing the size classification procedure that is common to drop-measuring instruments, and truncated the exponential PDF at y = 3.0. Repetition of the sampling and fitting process yields the desired distributions. We used about 1 000 000 drops (e.g., 50 000 samples with N_T = 20 and 5000 samples with N_T = 200) in the simulations; because the probability of a drop in an exponential PDF being larger than y = 3.0 is 6 × 10⁻⁶, we are lacking a total of about six larger drops from a full exponential DSD.

Characteristics of sampling distributions

Before considering the moment-based fits, we first examine some characteristics of the distributions of the sample moments M_iS = Σ_c Dⁱ [i = 3 gives the sample moment related to liquid water concentration (LWC); i = 6 gives radar reflectivity factor Z] themselves. Sampling from long-tailed DSDs, like the exponential, exhibits certain general features. Sample values of the moments are unbiased: the expected, or mean, value of M_iS corresponds to that moment of the drop population being sampled. However, the sampling distributions are skewed, as indicated by the fractional standard deviations calculated by Gertzman and Atlas (1977) and as shown in Smith et al. (1993). According to Gertzman and Atlas, the fractional standard deviation (FSD) σ_i/M_i for sample values of the DSD moment M_i as determined with a volume-sampling instrument studying an exponential DSD would be

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where V_S is the instrument-sampling volume. The general form for the moments of an exponential DSD can be written

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Thus,

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The product N_TV_S in the denominator of (6) is the mean sample size, which, for a sample volume of 1 m³, would be numerically equal to N_T. Thus, for i = 3 (LWC) and N_T = 100, (6) gives (σ₃/M₃) = 0.447. The distribution of a positive quantity (M_3S) with an FSD this large is necessarily skewed.

Inspection of (6) shows that the skewness increases with the order i of the moment M_i, and decreases with increasing sample size N_T. Figure 1 illustrates the former property for the sample moments M_3S (LWC) and M_6S (Z), for the case N_T = 100. The general tendency is for the sample moments to be lower than the corresponding population values; this behavior is the ultimate cause of the bias in the moment methods for estimating DSD parameters. As shown in section 4, the increase of the skewness with the moment order leads to greater biases when higher moments are employed. Figure 2 illustrates how the median sample moments approach the population values as the sample size increases. The skewness of the sampling distributions, and the resulting biases, decrease in a similar manner.

Sampling the numerous small drops can be a major instrumental problem for exponential DSDs, and adequately sampling the relatively rare large drops is also a concern. Fewer than one drop in 400 in an exponential DSD is larger than D = 1.5D_m, and only about one in 3000 is larger than D = 2D_m. However, the drops larger than 1.5D_m contribute more than 15% of the LWC and more than 60% of the reflectivity factor. Consequently, the relatively large but relatively rare drops tend to be important in determining the moments of physical significance. The sample values of these moments are, therefore, correlated with the size of the largest drop in each sample (e.g., Fig. 3). Table 1 demonstrates that these correlations are stronger for higher-order moments, and remain appreciable even for fairly large sample sizes. The sample moments are, in turn, correlated with each other—an artifact of the sampling variability as discussed in Smith et al. (1993).

Figure 5 of Smith et al. (1993) showed that the maximum drop size in an exponential DSD is rarely approached in samples of even hundreds of drops. The distribution of values along the abscissa in Fig. 3 here demonstrates the same thing. There is clearly no basis for assuming truncation of the underlying DSD at the maximum observed drop diameter, with samples of such sizes.

The bias in moment estimators

The essence of the moment approach for estimating parameters for DSD functions is to use the same number of moments calculated from observed raindrop size distributions as there are parameters in the function to be fitted. Analytical expressions for the selected moments of that function are solved algebraically for the needed parameters, and observed values of the sample moments are then entered into the resulting equations to estimate the parameters. The appendix summarizes the relevant mathematical expressions used here. The use of moment methods for rain DSDs evidently began with Waldvogel’s (1974) paper on the “N₀ jump” of DSDs. He used observed values of moments M_3S and M_6S (i.e., LWC and Z) along with (5), or its equivalent, to determine pairs of parameters for exponential functions like (1) that purportedly represented the observed DSDs. However, most functions that are fitted in this way do not represent well either the samples upon which they are based or the underlying populations from which the samples were taken.

The introductory section pointed out that such moment estimators are biased, and the fact that estimates of DSD parameters obtained in this way are biased was actually indicated in Smith et al. (1993). Figure 4, reproduced from that paper, compares sample estimates of D̂_m with the value of D_m for the exponentially distributed population from which the (simulated) samples were drawn. In this example, more than 80% of the values of D̂_m are underestimates, and the mean (the expected value) is about 78% of the population value. In terms of the more familiar exponential slope parameter Λ (=4/D_m), this means that Λ̂ is generally overestimated.

Estimators for exponential functions

Figure 5, based on an “ideal sample” of 100 drops from an exponential DSD, illustrates the bias in the moment-method fits. This ideal sample, unlike the randomly drawn samples used elsewhere in this paper, was constructed from the cumulative PDF of the drop size using the systematic procedure described below. It provides as close a representation of the PDF as could be achieved with a sample of 100 drops. Construction of the sample began with the “inverse cumulative PDF” N_L, showing the number of drops of diameter D or larger:

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where N_T is the total number of drops in the sample. All 100 drops have D > 0, while (with the drop sizes quantized in intervals of ΔD/D_m = 0.02) only 92 have D/D_m ≥ 0.02. Thus, we assigned D/D_m = 0.01 to eight drops in the sample. Similarly, only 85 drops have D/D_m ≥ 0.04, so D/D_m = 0.03 was assigned to the next 7 drops in the sample, and so on. The stair-step plot in Fig. 5 represents the resulting sample.

The figure includes two moment-method fits to the sample for exponential DSD functions. The two are the “Waldvogel fit,” based on moments M₃ and M₆, as employed in Waldvogel (1974), and one based on M₂ and M₃. The one using parameters based on M₃ and M₆ does not represent either the “observed” sample or the original population DSD very well; the smaller discrepancy resulting when the lower moments (M₂ and M₃) are used in the calculation is evident.

The foregoing discussion and the specific example in Fig. 5 suggest the general nature of the bias in moment estimators for parameters of exponential DSD functions: they tend to overestimate both the concentration (intercept) parameter n_o and the size (slope) parameter Λ. In terms of the parameters of (2), they tend to underestimate D_m and overestimate N_T, yielding fits having too many drops that are too small when compared with the original raindrop population. The biases are greater when higher moments are used in the procedure; Fig. 6 illustrates this behavior for n₀. Consequently, procedures that use sample values of reflectivity in the moment calculations lead to greater biases than ones employing only lower moments.

Extension of this argument would appear to suggest that using the lowest moments—M_0S, the sample size, and M_1S, related to the mean drop diameter—would yield the smallest biases of all. Such would indeed be the case in a simple mathematical exercise like that involved here. However, as noted in Testud et al. (2001), Smith (2003), and elsewhere, instrument responses to very small drops are highly variable and often suspect. Thus, trying to use moments that are lower than M_2S in the analysis would introduce another kind of uncertainty into the moment procedures. There may even be problems with using M_2S; instruments that do not adequately sense drops as small as 0.2–0.3D_m may significantly underestimate the value of M₂ (as illustrated in Table 2). If, say, D_m = 1 mm, this could be a challenging instrumentation problem. For that reason no approach using the lowest-order moments is considered here.

The skewness in the sampling distributions for the moments diminishes as the sample size increases (e.g., Fig. 2), so the bias in the moment estimators should also decrease with increasing sample size. Figure 7 shows that to be the case; with samples of hundreds or thousands of drops, the bias may become small enough to be negligible for many purposes.

Estimators for gamma functions

More recently, estimators involving three moments have been used in attempts to fit gamma distributions to DSD observations. Examples include Ulbrich (1983), Kozu and Nakamura (1991), Smith (1993), Tokay and Short (1996), and Ulbrich and Atlas (1998). While these procedures can produce closer fits to the observations, at least for part of the size spectrum, the bias that arises when moment methods are used with an assumed gamma form for the DSD is of even greater concern. A gamma DSD is often expressed in the form

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with concentration parameter n₁, distribution shape parameter μ, and size (scale) parameter λ. An alternative form using the same parameters as (2) is

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while a third form using the normalized concentration parameter N_W (Bringi and Chandrasekar 2001) is

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An exponential DSD corresponds to a gamma DSD with shape parameter μ = 0; thus, in the present simulations, gamma functions fitted to the observations should have estimates of μ that are close to zero.

The example in Fig. 8, for the same ideal sample as that in Fig. 5, suggests that not to be the case. The curve for the gamma fit based on moments M_2S, M_3S, and M_4S, as suggested in Smith (2003), has a shape parameter μ = 1.0; the actual calculated value of μ̂ was 1.013, but cumulants are not available for noninteger values of μ. Using the higher moments M_3S, M_4S, and M_6S, as done by several of the aforementioned authors, would yield μ̂ = 3.365—a more strongly biased estimate. The second gamma curve in Fig. 8, for μ = 3.0, provides a close approximation. The gamma fits match part of the sample distribution reasonably well, but in no way do they correspond to the population from which the sample was taken. As noted, the fitted value of the gamma M₂, M₃, M₄ shape parameter μ̂ is about 1.0; the expected value of μ̂ for samples of this size, drawn at random from an exponential DSD (μ = 0), is μ̂ = 1.50. For gamma fits based on moments M₃, M₄, M₆, the median value of μ̂ is about 4.5. Thus, sampling the exponential DSD with samples of this size and attempting to fit a gamma DSD function by moment methods will not reveal that the underlying DSD is exponential.

Figure 9 illustrates the bias in the moment estimators for μ, for different choices of the three moments used in the procedure. The general tendency is to overestimate μ. Here again, the lower moments yield smaller biases; use of the sample reflectivity values (M_6S) leads to the strongest bias among the examples illustrated. This bias can be quite misleading. Here, one has sampled from what is actually an exponential DSD, used moment methods in attempts to fit parameters of gamma distributions to the observations, found (biased) high values of μ̂, and probably concluded, quite erroneously, that the population DSD was gamma after all.

The hybrid approach used by Testud et al. (2001) does not employ a moment-based calculation to determine μ̂, but the estimators for their other gamma parameters (D_m and N_W) are biased. Figure 10 illustrates this bias for values of D̂_m, which as noted earlier, tend to be underestimates. Figure 11 shows corresponding results for values of N̂_W, which tend to be overestimates. (In both instances, when M_3S and M_4S are among the estimating moments, the third moment of the group has no effect on these estimates.)

As the sample size increases, the sampling variability and the skewness in the sample moments decrease, and the correlations between the various sample moments weaken (Smith et al. 1993). Consequently, the biases diminish. Figure 12 illustrates this behavior for the shape parameter μ̂. Thus, with very large samples, the moment methods may give approximations of the population parameters that become sufficiently accurate for practical purposes, even when the wrong functional form is assumed.

Related findings

The correlations of the various sample moments with the maximum drop size in a sample (section 3), and the associated correlations between moments, lead to correlation of the fitted parameters with the maximum drop size (e.g., Fig. 13). The parameters are, in turn, correlated with each other (Fig. 14), and such correlations (caused here entirely by sampling variability) could be mistaken for physical relationships. The correlations between estimated parameters are a bit less strong when lower-order moments are used in the fitting process; with N_T = 100, the n̂₀ − Λ̂ correlation is only 0.927 when sample moments M_2S and M_3S are used as compared with the 0.946 value with moments M_3S and M_6S, as illustrated in Fig. 14. The figure also illustrates the general tendency for the moment methods to overestimate both the slope and intercept parameters of an exponential function.

The n̂₀ − Λ̂ correlation actually increases as the sample size increases—to values of 0.946 (for moments M_2S, M_3S) or 0.980 (for moments M_3S, M_6S) when N_T = 1000—though the ranges of the variation of the parameters decrease. In any case, this behavior suggests that one should be wary of inferring physical relationships between such fitted parameters until the effects of the sampling variability have been taken into account.

Implications for analysis of experimental data

In trying to relate these simulations to actual raindrop observations, one should keep in mind several factors:

• The actual population DSDs in nature are not known. There is no assurance that they are exponential, though there are indications that this may be the case (Joss and Gori 1978).

• Observations with surface-sampling instruments, such as impact disdrometers, involve sample volumes that increase with the drop size, which tends to mitigate the skewness in the high-order sample moments, and, consequently, the associated biases.

• The observations include only the actual sample size (C in these simulations). The mean, or expected, sample size (N_T) is not known, though the actual sample size provides a better approximation as it increases.

• Very small drops are generally absent from many such observations, either because such small drops are not present, or because the instruments do not respond to those drops. In a full exponential DSD, 55% of the drops would be smaller than 0.2D_m, and 86% would be smaller than 0.5D_m. With typical values of D_m being 1–3 mm, this means the simulations may involve more (and in some cases much more) than twice the total numbers of drops that would be found in corresponding observations. Thus, results given above for N_T = 100 might be more applicable to observations with total drop counts of, say, 20–50.

These caveats notwithstanding, certain broad inferences as follows are applicable:

• Values of the parameters for DSD functions as estimated by moment methods will be biased.

• The bias will be stronger when higher moments are employed.

• The bias will diminish as the sample size increases.

Thus, DSD parameters that are estimated using high-order sample moments (e.g., reflectivity) with small sample sizes (say a few tens of drops) are the most highly suspect. Moreover, as suggested in the preceding section, caution should be used in attempting to impute a physical basis to relationships between DSD parameters fitted by moment methods.

Because the actual DSD parameters (even if the DSD were exponential) are unknown, it is difficult to critique in a quantitative manner any given set of results based on moment-method analyses. The most likely indication of the kind of biases discussed here appears in published frequency distributions of the gamma shape parameter μ̂, such as those reported in Kozu and Nakamura (1991) or Tokay and Short (1996). They extend to values as large as 30, which in view of Fig. 9 herein may well be a consequence of the M₃, M₄, M₆ moment approach that is used by those authors. The fact that the maximum μ̂ values reported by Kozu and Nakamura decrease as the rainfall rate increases (which should be accompanied by increases in sample size) lends further weight to this interpretation. A simple way to test this idea would be to stratify the shape parameter estimates by sample size, to look for a trend similar to that shown in Fig. 12.

Conclusions

Moment estimators for parameters of DSD functions are inherently biased. They tend to give erroneous values of the DSD parameters unless the drop samples are much larger than those commonly available. In particular, estimates of the gamma shape parameter μ tend to be far larger than the shape parameter of the underlying DSD from which the samples are taken. The bias is strongest for small sample sizes, and are also stronger when higher-order moments of the observed DSDs are used in the “fitting” process.

Moment methods may provide estimates of DSD parameters of sufficient accuracy if very large samples (hundreds, perhaps thousands) of drops are available. Failing that, some alternative approach to fitting the observed DSDs must be used. The maximum likelihood approach suggested by Haddad et al. (1996, 1997) may be satisfactory, though the maximum likelihood estimators are not without bias (Choi and Wette 1969). A variant of the L-moment approach used in hydrology (e.g., Maidment 1993) may also be worthy of consideration.