In the radar remote sensing of precipitation, a two-parameter raindrop size distribution (DSD) model is needed to facilitate an observation-based retrieval because both the network of dual-polarization Weather Surveillance Radar-1988 Doppler (WSR-88D) instruments and the Global Precipitation Measurement dual-frequency radar each provide two independent measurements. A two-parameter DSD model is also needed for two-moment microphysics parameterizations in numerical weather prediction (Zhang et al. 2006). Since the *μ*–Λ relation was introduced to reduce the three parameters of the gamma distribution model to the two parameters of the constrained-gamma (C-G) model for rain DSDs (Zhang et al. 2001, 2003), there have been many papers, including this one (Williams et al. 2014), publishing new constraints and debating the validity of the C-G model in remote sensing applications. It is great to see these papers because they further our understanding of the C-G DSD model, but there is some confusion regarding how the C-G models are similar/different and how to accurately derive and use a valid constraint. The confusion needs to be clarified, which is the motivation for this comment.

My first point is that the gamma DSD model with a *σ*_{m}–*D*_{m} relation (Williams et al. 2014) is essentially equivalent to the C-G model with a *μ*–Λ relation that has already been introduced (Zhang et al. 2001), which can be shown in both the equations and the plots of the parameters (*μ*–Λ vs *σ*_{m}–*D*_{m}). People should think of the gamma DSD model constrained by either a *μ*–Λ relation or a *σ*_{m}–*D*_{m} relation as just another two-parameter DSD model, like the exponential distribution model, the gamma distribution model with a fixed *μ*, or the gamma distribution model with a *μ*–*D*_{m} relation. The C-G model is equivalent to the normalized DSD model with two parameters (Testud et al. 2001), as shown in Figs. 9 and 10 of Cao and Zhang (2009). It is also similar to the gamma DSD model that is represented by three uncorrelated/independent parameters through parameter transformation in which the three DSD parameters are transformed into a set of three new parameters but one of new parameters is fixed/predetermined (Haddad et al. 1996, 1997, 2006).

For a better understating of the model similarity, let us first provide some fundamentals about the gamma DSD model. For a gamma DSD of *N*(*D*) = *N*_{0}*D*^{μ} exp(−ΛD), the *n*th moment of the DSD (ignoring truncation effects for simplicity) is

A general mean characteristic size can be defined as a ratio between the (*n*+1)th and the *n*th moments as follows:

The standard deviation (or spectrum width) of the distribution *p*_{n}(*D*) = *D*^{n}*N*(*D*)/*M*_{n} is

If the *σ*_{n} and *D*_{n} are related by

then a relation between *μ* and Λ can be derived by substituting Eqs. (1) and (2) into Eq. (3). After simplification, we obtain

This means that the width–size (*σ*_{n}–*D*_{n}) relation in Eq. (3) is equivalent to the shape–slope (*μ*–Λ) relation in Eq. (4) for a gamma distribution. Both of them serve as a constraint of a gamma DSD and reduce the three parameters of the gamma distribution to two parameters.

In the case of mass-weighted mean diameter *D*_{m} for a rain DSD, we have *n* = 3 in Eqs. (1)–(4), yielding *D*_{m} ≡ *D*_{3} = (*μ* + 4)/Λ and *σ*_{m} = (*μ* + 4)^{1/2}/Λ. Therefore, the *σ*_{m}–*D*_{m} relation of *σ*_{m} = [Eqs. (22) and (23) in Williams et al. 2014] becomes

that is,

So, the *σ*_{m}–*D*_{m} relations presented in Williams et al. (2014) are essentially a *μ*–Λ relation but just in a different functional form when compared with the previously introduced *μ*–Λ relations in quadratic form (Zhang et al. 2001).

The equivalence between a *μ*–Λ relation and the *σ*_{m}–*D*_{m} relation was established previously, and it was updated and shown in Figs. 6 and 7 of Cao et al. (2008). The updated *μ*–Λ relation derived using the method of sorting and averaging with two parameters (SATP) in Cao et al. (2008),

was not used in Williams et al. (2014). The results of Cao et al. (2008) and Williams et al. (2014), represented by Eqs. (6) and (5), respectively, are now compared in Figs. 1a and 1b. The original figures of Fig. 7a in Cao et al. (2008) and Fig. 5a in Williams et al. (2014) are copied for comparison in Figs. 1c and 1d of this comment, respectively. The red solid line is the solid line of Fig. 1d, the blue line is the line of Fig. 1c, except for ignoring truncation effects, which is generally valid if *μ* is not too small (Vivekanandan et al. 2004).

It can be seen from the two domains (*μ*–Λ in Fig. 1a and *σ*_{m}–*D*_{m} in Fig. 1b) that the results from the two approaches follow the same trend well, and especially that the two sets of results from Eqs. (6) and (5b) agree very well. To obtain Eq. (5b), the upper bound *σ*_{m}–*D*_{m} relation of Eq. (23) in Williams et al. (2014) is treated as a mean relation as an alternate example. The main difference between the mean relations (solid lines) is that Eq. (5a) gives a smaller mass spectrum width *σ*_{m} and a larger *μ* than does Eq. (6). This is likely due to more aggressive thresholds that were used to filter the DSD data in Williams et al. (2014), which may exclude the DSDs with a long tail of a few large drops, hence yielding narrower DSDs with smaller *σ*_{m} and larger *μ* (more on this will be given in my second point).

My second point is about the data/thresholds used in Williams et al. (2014) to filter out data and about the results of Figs. 4, 5, and 7 of that paper, which are derived from such data/thresholds. Williams et al. (2014) use DSD data collected by two-dimensional video disdrometers (2DVD) in Huntsville, Alabama. Three stages of thresholds were used to filter out data with large errors: The first stage “consisted of retaining spectra with 1) at least 50 raindrops in at least 3 different diameter bins, 2) reflectivity factor greater than 10 dB*Z*, and 3) rain rate greater than 0.1 mm h^{−1}. The rain estimates were not divided by rain regime. A total of 29 705 min of raindrop spectra passed these criteria.” In the second stage, “[a]pproximately 84% (24 872 min) of the original Huntsville observations had *X*_{max} > 1.5 and were used for further analysis.” In the last stage, “[u]sing observations with *D*_{m} > 1.0 mm (a total of 18 969 observations), power-law curves are estimated with the form ” [all three quotations are from Williams et al. (2014), p. 1287].

These thresholds may have excluded more than one-half of the original data. This exclusion is not justified for disdrometer measurements because the occurrence of large drops is small because of the small sample volume (a few cubic meters) for 1-min 2DVD data. Just because 2DVDs only sense a small number of large drops (thus measuring them with large sampling errors) does not mean that these data can be filtered out, because these large drops contribute greatly to radar measurements. The DSD with a few large drops in large size bins may have large sampling error, which has been quantified with side-by-side 2DVD measurements (Fig. 2 in Cao et al. 2008), but these measurements are real and are very important in comparing with radar measurements because they are the ones that contribute most to radar reflectivity and differential reflectivity, and they should not be thrown out. Figure 2 shows 3D plots of rain DSDs for a squall line observed by NCAR’s 2DVD in Oklahoma on 13 May 2005. Plotted are DSD (Fig. 2a), mass distribution (Fig. 2b), radar reflectivity distribution (Fig. 2c), and differential reflectivity distribution (Fig. 2d). Whereas DSDs are dominated by small drops and mass/water content mainly contributed from median size drops, radar reflectivity and differential reflectivity are contributed to mainly by a few, but large, drops. Hence, DSD data with a few large drops should not be filtered out.

The thresholds used in Williams et al. (2014) may have excluded the DSDs with a long tail of a few large drops, hence yielding narrower DSDs with smaller *σ*_{m} and larger *μ.* This can be seen in their Fig. 5a (repeated as Fig. 1d of this comment): although Williams et al. (2014) use a bigger dataset, the ranges of *σ*_{m} and *D*_{m} are much smaller than that in Fig. 7 of Cao et al. (2008) (repeated as Fig. 1c of this comment). Not having a significant difference in estimated mean and standard deviation of *σ*_{m} with/without the filtering does not mean that the derived *σ*_{m}–*D*_{m} relation is representative for all rain DSDs. This is because each DSD was equally weighted in Williams et al. (2014) in estimating *σ*_{m} statistics. These statistics are dominated by light rain DSDs because of their high occurrence and more data points. This situation results in missing a few heavy rain DSDs (with large drops) and would not change the *σ*_{m} mean and standard deviation by much. The DSDs with large drops can exist with low occurrence, however, and they are physically important. If, for example, a DSD of rain rate 100 mm h^{−1} were to receive a weight equal to that of a DSD of rain rate <1 mm h^{−1}, each contributes one sample in calculating the mean of *σ*_{m}. It is obvious that in terms of physical significance the heavy rain (100 mm h^{−1}) DSD and the light rain (<1 mm h^{−1}) DSD are not the same. Therefore, the low-occurrence (low probability) heavy-rain events are physically important and cannot be ignored. That is why we estimated *σ*_{m} and *D*_{m} values directly from DSDs (Zhang et al. 2003), and the updated *μ*–Λ relation derived with the SATP in Cao et al. (2008) agrees better with the *σ*_{m}–*D*_{m} relation derived from DSD-estimated *σ*_{m} and *D*_{m} values than with those from single-DSD-estimated *μ* and Λ values, because the data points for light-rain events were substantially reduced in SATP and the contribution from heavy rain becomes important in estimating the statistics and relations. SATP is similar to the water-content (or rain rate) weighted average that makes the few heavy-rain data points important, and direct estimation of *σ*_{m} and *D*_{m} values from DSDs can avoid error propagation in the DSD-fitting procedure.

Furthermore, missing very small and very large drops is an intrinsic issue for disdrometer measurements, which is another error source that yields large *μ*. It is also noted that Eq. (5a) [which is derived from Eq. (22) of Williams et al. (2014)] gives no negative *μ* when *D*_{m} < 4 mm, meaning that all DSDs have a convex shape. This is not representative of observed natural rain DSDs: we see many DSDs with a concave shape in our 2DVD data, especially those representing strong convection in which large drops are present. Rain DSDs shown in Fig. 2 are replotted in Fig. 3 in 2D plots. It is clear that some of the DSDs are in a concave shape in the semilogarithmic plots, meaning negative *μ*s if they are represented by the gamma distributions. The *μ*–Λ relations [Eq. (6) and Eq. (5b), which is derived from the upper bound of the *σ*_{m}–*D*_{m} relation] have the capability to represent concave DSDs by using a C-G model with negative *μ*s, as shown in Fig. 1a.

Because Williams et al. (2014) and Zhang et al. (2003) use different datasets from different locations, use different thresholds to filter data, and use different ways to derive the constrained relations, it is not surprising to see that Eq. (5a) gives smaller biases—this result is due to the fact that the mean relation/model used in the estimation was actually derived from the same data from Huntsville, and the thresholds and fitting procedures in Williams et al. (2014) were used to evaluate and compare the performance of the two constrained relations. However, the *μ*–Λ relations in Zhang et al. (2003) and Cao et al. (2008) were derived from Florida and Oklahoma datasets, respectively, and using different fitting procedures and without excluding data with a few large drops. In the ideal case, for a fair comparison, one would use an independent dataset that spans the wide range of conditions to test/evaluate the two methods rather than using one dataset and approach to test the other.

My third point concerns the usage of the *μ*–Λ relation in probabilistic retrieval. Although it is convenient to use a variable that is Gaussian distributed, I cannot agree that a *μ*–Λ relation cannot be used as a mean relation in probabilistic estimation as stated in Williams et al.: “Even without concerns over mathematical artifacts, single-value *μ*–Λ relationships as in Eq. (2) cannot be used in probabilistic rainfall retrieval algorithms because they only provide the expected (or initial) value of a DSD constraint. Probabilistic algorithms need the expected value *plus* [italics in original] a range of acceptable values to converge to a final solution (Haddad et al. 2006; Munchak and Kummerow 2011)” (Williams et al. 2014, p. 1283).

It is true that a *μ–*Λ relation is a mean (expected) relation, which is compared with the mean *σ*_{m}–*D*_{m} relations in Fig. 1. Although the constrained *μ*–Λ or *σ*_{m}–*D*_{m} relations are mean relations, they are needed for a two-parameter DSD model in either the deterministic or the statistical case. For example, a Gaussian probability density function (PDF) of needs a mean and standard deviation to be defined (Papoulis 1991). Moreover, the mean relations can be relaxed or adjusted to be a statistical C-G DSD model, by introducing an adjustment term as in Cao et al. (2008) or by introducing the Gaussian distribution for the coefficient in the *σ*_{m}–*D*_{m} relation in Williams et al. (2014). Furthermore, a distribution can be introduced for the DSD parameters and used in probabilistic retrieval as in Cao et al. (2010).

In Eq. (16) of Cao et al. (2008), an adjustment term of *C*Δ*Z*_{DR} is added to the *μ*–Λ relation to account for the DSD effects of big drops at leading edges of a squall line. There is no reason that the adjustment term cannot be a random variable with a prior distribution (or range) if the C-G model is used as a statistical model. Using a random adjustment term in the *μ*** –**Λ relation (Cao et al. 2008) is equivalent to that of randomizing the coefficient

*a*(or ) in the

*σ*

_{m}–

*D*

_{m}relation, that is, Eq. (21) of Williams et al. (2014). There is no essential difference for the abovementioned two statistical C-G DSD models except that the random term (

*C*Δ

*Z*

_{DR}) may have a different PDF from that of the random coefficient ().

With respect to probabilistic algorithms, as a matter of fact, the C-G model has been used in Bayesian retrieval for rain DSD parameters from dual-polarization radar measurements in Cao et al. (2010), where the two DSD parameters of *N*_{0} and Λ were transformed to two parameters of = log_{10}(*N*_{0}) and Λ′ = Λ^{1/4}, to reduce their dynamic ranges and mitigate nonlinear/non-Gaussian effects. The joint PDF of () was directly obtained from 2DVD measurements, as shown in Fig. 4 of Cao et al. (2010). This joint prior PDF was then used in Bayesian retrieval to find the statistics of the DSD parameters, as well as the statistics of rain-physics parameters such as rain rate and mass-weighted mean diameter.

It was shown that the probabilistic retrieval allows the errors of the radar measurements to be specified and linked to retrieval results. The retrieved rain rates agree with gauge measurements better than do those with fixed empirical radar rain estimators, shown in Fig. 13 and Table 2 in Cao et al. (2010). In summary, the C-G model with a *μ*–Λ relation can be used as a statistical DSD model and has been used in probabilistic retrieval—it is just a matter of how to represent the parameter/relation distributions correctly.

My fourth point is about the coefficient in rain-rate Eqs. (12), (28), and (29), which is wrong by a factor of 10. This might be due to a unit conversion error. Nevertheless, from the units provided in the paper [*N*(*D*) in number of raindrops per meter cubed per millimeter, *υ*(*D*) in meters per second, and *D* in millimeters], the units in the equation are

where the units are given in square brackets. To obtain a unit of millimeters per hour from a unit of meters per second for the rain rate, a factor of 1000 × 3600 = 36 × 10^{5} is needed from the second line to the third line in Eq. (7) to yield the coefficient of 6*π* × 10^{−4}, instead of the 6*π* × 10^{−3} that is given in Williams et al.

Given the errors in derived *μ*–Λ relations, it is a matter of how to derive a valid relation using high-quality data and the right procedures. The performance of a C-G model depends on the 1) accuracy of the instrumentation (disdrometer), 2) measurement environment, 3) data quality control, 4) fitting procedures, and so forth. The bottom line is that using a derived C-G model with the right approach tends to yield smaller bias and error in the estimation of rain-physics parameters than does the gamma model with a fixed *μ*. What is presented in Williams et al. (2014) is not an exception and is consistent with what is proposed in the C-G model—the slight differences are due to the different datasets and different quality-control thresholds and fitting procedures. The different methods or constraints used in the C-G DSD model are just different ways to achieve the same ends because one can transform from one set of parameters into the other. The choice of DSD parameters to use depends on preference and simplicity for a specific application, just as one can choose to use Cartesian or polar coordinates to solve a geometry problem. The C-G DSD model is just a model that is useful in representing rain microphysics and facilitates DSD retrievals from two remote measurements. A model is not “truth,” and there is no absolute truth with a DSD model; we can only approximate and approach the truth (which is not a two-parameter model) with a model and with further studies/understanding.

## Acknowledgments

This work was supported by NSF Grant AGS-1046171. The author appreciates the editor’s suggestions for improving the presentation of his views. The author is also thankful for the data collection and processing by Drs. Brandes and Cao and for helpful discussions with Drs. Richard J. Doviak and Dusan S. Zrnić.

## REFERENCES

*Probability, Random Variables and Stochastic Processes.*3rd ed. McGraw-Hill, 666 pp.

*Proc. 31st Conf. on Radar Meteorology*, Seattle, WA, Amer. Meteor. Soc., 2A.4a. [Available online at https://ams.confex.com/ams/pdfpapers/63993.pdf.]

## Footnotes

The original article that was the subject of this comment/reply can be found at http://journals.ametsoc.org/doi/abs/10.1175/JAMC-D-13-076.1.