## Introduction

Marshall and Palmer (1948) approximated the size distributions of raindrops with exponential functions. Conventional semilogarithmic plots of observed drop size distributions often exhibit deviations from the straight lines that represent such functions on those scales. Many of the observations show fewer drops of small diameter, less than a millimeter or so, than an exponential function would indicate (e.g., Cataneo and Stout 1968). That has stimulated interest in other functional forms, among which the gamma distribution is a favorite (e.g., Ulbrich 1983; Wong and Chidambaram 1985), to describe the size distributions. At the cost of greater mathematical complexity, such functions can represent a variety of departures from the simple exponential form.

If the primary interest is in bulk quantities such as liquid water concentration (LWC), rainfall rate, or reflectivity factor *Z,* however, it usually turns out that the added complexity only yields “improvements” that are within the experimental uncertainties of the raindrop size observations. As a consequence, the gamma distribution in particular offers little practical advantage over the simpler exponential approximation.

## Mathematical background

*D*represents drop diameter, and

*n*and Λ are concentration and size parameters, respectively. The last form is expressed in terms of the mass-weighted mean diameter

_{o}*D*

_{m}, determined as the ratio of the fourth to third moments of the distribution. On the common semilogarithmic plots of experimental data, this function would be represented as

*n*

*D*

*n*

_{o}

*D*

*D*

_{m}

^{1}

*n*

_{1},

*μ*, and

*λ*are concentration, distribution shape, and size parameters, respectively. On a semilogarithmic plot, this function would be

*n*(

*D*) for small drop sizes only in the region

*D*<

*D*

_{m}; the mode of ln[

*n*(

*D*)] (for positive values of

*μ*) is at [

*μ*/(

*μ*+ 4)]

*D*

_{m}. The small-drop region is poorly observed by many drop-sensing instruments and, as discussed by Wong et al. (1988) and shown in section 4, is of little consequence where values of the significant moments of

*n*(

*D*) are concerned.

Figure 1 illustrates the various terms in (2) and (4) on a common semilogarithmic plot, for the particular case *μ* = 2. The curve represents the middle term on the right-hand side of (4), which produces a positive slope on *n*(*D*) for *D* < [*μ*(*μ* + 4)]*D*_{m} (or *D* < *D*_{m}/3 in this example) with a rightward translation from the straight-line term for larger sizes. More important for current purposes are the various moments of the size distribution functions; Table 1 summarizes those quantities for values of *μ* > 0. The second column of the table indicates relevant physical attributes associated with some of these moments.

Of importance to the subsequent discussion are various weighted mean diameters, each defined by the ratio of the (*m* + 1)th to the *m*th moment. For example, the mass-weighted mean diameter is the ratio of the fourth to third moments, *D*_{m} in each case. The area-weighted mean diameter *D*_{a}, known in the radiation community as the effective diameter, is the ratio of the third to second moments. For the exponential distribution, *D*_{a} = 3*D*_{m}/4; for the gamma distribution, *D*_{a} = (*μ* + 3)*D*_{m}/(*μ* + 4).

## Fitting parameters by moment methods

Procedures attempting to fit these functions to observed drop size distributions by moment methods are widely used [but see Haddad et al. (1996, 1997) for a contrary view]. Waldvogel (1974) provides the earliest example; he used the third (LWC) and sixth (*Z*) moments to determine the two parameters *n*_{o} and Λ of the Marshall–Palmer distribution. Weber (1976; or see Smith et al. 1976) used the third and fourth moments [the latter designated as *R** by Joss and Gori (1978)] to determine those two parameters. Kozu and Nakamura (1991), Tokay and Short (1996), and Tokay et al. (2001) chose the third, fourth, and sixth moments to determine the three parameters of the gamma distribution; Smith (1993) used the zeroth, first, and second moments; and Ulbrich and Atlas (1998) used the second, fourth, and sixth moments. Sempere Torres et al. (1994) presented a more general procedure, in effect using the first six moments to fit a generalized function. Testud et al. (2001) present another generalized approach and then use the third and fourth moments to determine two parameters for the gamma distribution, along with a curve-fitting procedure to determine the third.

Table 1 offers a wide range of possible choices for the two moments needed to fit the two-parameter Marshall–Palmer distribution, or the three needed for the gamma distribution, and the list of moments there is not exhaustive. However, the practical options are more restricted. As shown in Gertzman and Atlas (1977) or Smith et al. (1993), the sampling statistics for high-order moments like *Z* are unfavorable at best. Thus, use of high-order moments in the fitting process is inadvisable, the work of Waldvogel (1974) and others notwithstanding. In a similar way, the low-order moments for observed drop size data involve substantial uncertainties because of deficiencies in the observations of the smallest raindrops. A hint of this appears in the classic Marshall and Palmer (1948) paper, where the semilog plot of Laws and Parsons's data decreases monotonically as the drop size increases while that of their own data suggests a possible mode at drop sizes around 1 mm.

Some instruments, for example, the Illinois State Water Survey drop camera, yield observations indicating modal character (Cataneo and Stout 1968); others, for example, some optical disdrometers (Stow and Jones 1981; Willis 1984), yield data with monotonic behavior. In some cases, the observations of small drops are known to be deficient, for example, because of wind effects (Rinehart 1983; Nešpor et al. 2000). The results may even vary depending upon the rainfall rate (Tokay et al. 2001) or the degree of small-drop evaporation in the subcloud layer. At best, the issue is unresolved, and it is therefore safest to avoid using low-order moments in fitting the distributions. That consideration negates the practical value of the elegant treatment in Smith (1993).

*M*

_{i}represents the

*i*th moment of an observed drop size distribution, the foregoing discussion suggests that

*M*

_{2}and

*M*

_{3}, or

*M*

_{3}and

*M*

_{4}, would be most appropriate for the Marshall–Palmer distribution. The moments

*M*

_{2},

*M*

_{3}, and

*M*

_{4}would be appropriate for the gamma distribution. For the exponential distribution, the parameters would be, using

*M*

_{2}and

*M*

_{3},

*M*

_{3}and

*M*

_{4},

*μ*often appears to be somewhat unstable in experimental fits (e.g., Kozu and Nakamura 1991): it involves a ratio of differences between products of moments that, according to Table 1, will not be greatly different. (Similar expressions for

*μ*result with other choices of the three moments.) For example, the ratio of the positive to negative terms in the numerator of the expression for

*μ*would be (3

*μ*+ 12)/(4

*μ*+ 12) for a gamma distribution. This ratio would vary from unity when

*μ*= 0 (i.e., in sampling from an exponential distribution) to 5/6 when

*μ*= 6. Similarly, the ratio of the two terms in the denominator, (

*μ*+ 3)/(

*μ*+ 4), would vary from 3/4 when

*μ*= 0 to 9/10 when

*μ*= 6. Thus, it is clear that sampling limitations and other observational uncertainties will have a substantial effect on values of

*μ*determined in this way. (The range of the corresponding ratio for terms in the numerator of the

*μ*expression based on

*M*

_{3},

*M*

_{4}, and

*M*

_{6}would be 1.0–1.36, but this slightly greater range would be offset by the greater sampling uncertainty involved in measuring

*M*

_{6}.) Testud et al. (2001) wisely chose a separate curve-fitting procedure to determine their values of

*μ.*

## Comparison of bulk quantities

A main point of this analysis is the insensitivity of important bulk quantities to the choice between the exponential and the gamma distribution functions. That can be illustrated by comparing those quantities for each function; positive-integer values for *μ,* spanning the range likely to be encountered, are used here. For this purpose, we compare the exponential fit in (6), using *M*_{3} and *M*_{4}, to the gamma fit in (7). The comparisons are expressed in terms of ratios of the bulk quantities calculated from the two fitted functions.

*M*

_{o}

*e*

*n*

_{o}

*D*

_{m}

*M*

^{4}

_{3}

*M*

^{3}

_{4}

*μ*= 0, for which the exponential and gamma distributions are identical, to 5.38 when

*μ*= 6, with an asymptote of 10.67 as

*μ*→ ∞. This represents a considerable range of variation in the calculated total drop concentration (though the variation is less than might be expected from inspection of Fig. 1); that quantity, however, is governed by the small drops, which in any case are poorly observed. The total drop concentration, moreover, is a quantity of relatively minor physical importance.

*μ*= 0 to 1.20 when

*μ*= 6, with an asymptote of 1.33. Such variation is well within the uncertainty of likely drop samples (Gertzman and Atlas 1977), and so we may conclude that, in terms of this bulk quantity, it makes no important difference whether the observations are fit with an exponential or a gamma function.

Because *M*_{3} and *M*_{4} were used in the fitting process for each function, they will agree identically whether the exponential or gamma representation is used. Thus, the LWC will be identical for either function. In a similar way, the ratio *M*_{4}/*M*_{3}, the mass-weighted mean diameter, will agree identically. The ratio *M*_{3}/*M*_{2}, the area-weighted mean or “effective” diameter, will fall in the same ratio as the *M*_{2} values. The rainfall rate *R* is related to a moment somewhere between the third and fourth, and so the values of *R* will also agree almost exactly.

*μ*= 0 to 1.42 when

*μ*= 6 (i.e., a difference of only 1.5 dB), with an asymptote of 1.88 (i.e., a difference of less than 3 dB). Testud et al. (2001) noted a similar insensitivity to the value of

*μ*in the context of

*Z*–

*R*relationships. Again, the result is well within the range of plausible sampling uncertainty, and so the distinction between the exponential and gamma functions is of little practical consequence.

## Conclusions

This analysis shows that if observations of raindrop size distributions are fit by using “central” moments of the observed distributions to determine parameters of the fitted functions, then it makes little practical difference whether exponential or gamma distribution functions are employed. Differences among the bulk quantities of importance, for example, effective drop size, LWC, rainfall rate, or radar reflectivity factor, between the two fitted functions fall within the observational uncertainties. The principle of Occam's razor is applicable here: The exponential (Marshall–Palmer) distribution function is simpler to use, and so the extra effort involved with the gamma distribution is not often justified by the results that can be obtained. However, the physical processes of concern (Willis 1984) or the need to treat polarimetric variables (e.g., Illingworth and Blackman 2002) may influence this conclusion in some cases.

## Acknowledgments

This material is based upon work supported by the National Science Foundation under Grants ATM-9509810 and ATM-9907812.

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Moments of exponential and gamma size distribution functions. Total surface area and total drop volume are expressed per unit volume

^{1}

Here both functions are written as drop size distributions appropriate to the physical sciences, as opposed to the normalized probability density functions that statisticians might prefer.