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  • View in gallery

    Geographical locations of Huangshan Mountain and field campaign sites.

  • View in gallery

    Averaged rain DSDs observed at three different elevations and the double-parameter GDFs with prescribed μ as 0, diagnosed μ through Diag01, Diag02, and Diag03 methods denoted as C0, D1, D2, and D3, respectively, in the legend. The triple-parameter GDF with prognostic zeroth, third, and sixth moments is denoted as 036. Panels show rain DSDs observed at the (a),(b) TOP, (c),(d) MID, and (e),(f) BOT observational sites (see the location and elevation of the sites in Table 1). Columns show observations and GDFs in the size ranges of (left) 0–3 and (right) 3–8 mm.

  • View in gallery

    Relative errors of (a) zeroth- to (g) sixth-order moments from GDFs, as well as (h) the average, using a variety of parameter solving methods. Boxes denoted as C0, C3, C6, D1, D2, and D3 are double-parameter GDFs with μ prescribed as 0, 3, and 6, and μ diagnosed through Diag01, Diag02, and Diag03 methods, respectively. The box denoted as 036 is a triple-parameter GDF with prognostic 036 moment group. Boxes indicate the range from the 25th to 75th percentiles. Whiskers indicate the range from the 10th to the 90th percentiles. Relative errors of convective and stratiform precipitation DSD samples are separately plotted in blue and green, respectively.

  • View in gallery

    As in Fig. 3, but for the triple-parameter GDFs with prognostic moment combinations of 013, 023, 034, 035, and 036 labeled along the x axis.

  • View in gallery

    The averRE values are located at two-dimensional coordinates determined by the mass concentration and number concentration of observed rain DSDs. The color contours denote averRE distributions. The contour lines denote the distribution of the frequency of samples. Panels show averRE distributions of (a),(c) convective and (b),(d) stratiform precipitation fit by the 036 GDF with (top) the solved spectral parameter and (bottom) the prescribed spectral parameter.

  • View in gallery

    As in Fig. 5, but for the averRE distributions of 034 GDFs.

  • View in gallery

    As in Fig. 5, but for the averRE distributions of 013 GDFs.

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Evaluating Errors in Gamma-Function Representations of the Raindrop Size Distribution: A Method for Determining the Optimal Parameter Set for Use in Bulk Microphysics Schemes

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  • 1 Division of Atmospheric Sciences, Desert Research Institute, Reno, Nevada
  • | 2 Interdisciplinary Program in Atmospheric Sciences, University of Nevada, Reno, Nevada
  • | 3 Department of Atmospheric Sciences, University of Wyoming, Laramie, Wyoming
  • | 4 Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters, Nanjing University of Information Science and Technology, Nanjing, China
  • | 5 Key Laboratory for Aerosol-Cloud-Precipitation, China Meteorological Administration, School of Atmospheric Physics, Nanjing University of Information Science and Technology, Nanjing, Jiangsu, China
  • | 6 Climate and Weather Disasters Collaborative Innovation Center, Nanjing University of Information Science and Technology, Nanjing, China
  • | 7 Key Laboratory of Middle Atmosphere and Global Environment Observation, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China
  • | 8 School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts
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Abstract

Significant uncertainty lies in representing the rain droplet size distribution (DSD) in bulk cloud microphysics schemes and in the derivation of parameters of the function fit to the spectrum from the varying moments of a DSD. Here we evaluate the suitability of gamma distribution functions (GDFs) for fitting rain DSDs against observed disdrometer data. Results illustrate that double-parameter GDFs with prescribed or diagnosed positive spectral shape parameters μ fit rain DSDs better than the Marshall–Palmer distribution function (with μ = 0). The relative errors of fitting the spectrum moments (especially high-order moments) decrease by an order of magnitude [from O(102) to O(101)]. Moreover, introduction of a triple-parameter GDF with mathematically solved μ decreases the relative errors to O(100). Based on further investigation of potential combinations of the three prognostic moments for triple-moment cloud microphysical schemes, it is found that the GDF with parameters determined from predictions of the zeroth, third, and fourth moments (the 034 GDF) exhibits the best fit to rain DSDs compared to other moment combinations. Therefore, we suggest that the 034 prognostic moment group should replace the widely accepted 036 group to represent rain DSDs in triple-moment cloud microphysics schemes. An evaluation of the capability of GDFs to represent rain DSDs demonstrates that 034 GDF exhibits accurate fits to all observed DSDs except for rarely occurring extremely wide spectra from heavy precipitation and extremely narrow spectra from drizzle. The knowledge gained from this assessment can also be used to improve cloud microphysics retrieval schemes and data assimilation.

Current affiliation: Department of Environment and Climate Science, Brookhaven National Laboratory, Upton, New York.

Current affiliation: School of Meteorology University of Oklahoma, Norman, Oklahoma.

Current affiliation: Department of Atmospheric Sciences, Texas A&M University, College Station, Texas.

Current affiliation: State Key Laboratory of Severe Weather and Key Laboratory of Atmospheric Chemistry, Chinese Academy of Meteorological Sciences, CMA, Beijing, China.

Current affiliation: Department of Geography, Hong Kong Baptist University, Hong Kong, China.

© 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Eric M. Wilcox, eric.wilcox@dri.edu

Abstract

Significant uncertainty lies in representing the rain droplet size distribution (DSD) in bulk cloud microphysics schemes and in the derivation of parameters of the function fit to the spectrum from the varying moments of a DSD. Here we evaluate the suitability of gamma distribution functions (GDFs) for fitting rain DSDs against observed disdrometer data. Results illustrate that double-parameter GDFs with prescribed or diagnosed positive spectral shape parameters μ fit rain DSDs better than the Marshall–Palmer distribution function (with μ = 0). The relative errors of fitting the spectrum moments (especially high-order moments) decrease by an order of magnitude [from O(102) to O(101)]. Moreover, introduction of a triple-parameter GDF with mathematically solved μ decreases the relative errors to O(100). Based on further investigation of potential combinations of the three prognostic moments for triple-moment cloud microphysical schemes, it is found that the GDF with parameters determined from predictions of the zeroth, third, and fourth moments (the 034 GDF) exhibits the best fit to rain DSDs compared to other moment combinations. Therefore, we suggest that the 034 prognostic moment group should replace the widely accepted 036 group to represent rain DSDs in triple-moment cloud microphysics schemes. An evaluation of the capability of GDFs to represent rain DSDs demonstrates that 034 GDF exhibits accurate fits to all observed DSDs except for rarely occurring extremely wide spectra from heavy precipitation and extremely narrow spectra from drizzle. The knowledge gained from this assessment can also be used to improve cloud microphysics retrieval schemes and data assimilation.

Current affiliation: Department of Environment and Climate Science, Brookhaven National Laboratory, Upton, New York.

Current affiliation: School of Meteorology University of Oklahoma, Norman, Oklahoma.

Current affiliation: Department of Atmospheric Sciences, Texas A&M University, College Station, Texas.

Current affiliation: State Key Laboratory of Severe Weather and Key Laboratory of Atmospheric Chemistry, Chinese Academy of Meteorological Sciences, CMA, Beijing, China.

Current affiliation: Department of Geography, Hong Kong Baptist University, Hong Kong, China.

© 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Eric M. Wilcox, eric.wilcox@dri.edu

1. Introduction

There are two distribution functions widely used for fitting the size distributions of atmospheric particles: the lognormal and the gamma distribution function (GDF). Previous studies have used lognormal distribution functions to represent the spectrum of small particles such as aerosols, while the GDF is a more popular approach to represent the spectra of large cloud hydrometeor categories that are represented by constant-density spheres, including rain droplets, graupel, and hail (e.g., Adirosi et al. 2015; Heintzenberg 1994). In bulk schemes, the hydrometer size distribution function N(D) is expressed by a typical GDF that is given by
N(D)=N0DμeλD,
where D is the diameter (or maximum dimension) of the hydrometer. The GDF has three varying distribution parameters: N0 is the intercept parameter, λ is the slope parameter, and μ is the spectral shape parameter. Variations of the hydrometer size distribution are reflected by changes of distribution parameters that are solved for based on the integral moments of the hydrometer distribution. The previous single-moment schemes prescribe N0 and μ, and derive λ from the predicted mass mixing ratio of hydrometeors. The double-moment schemes predict two varying distribution parameters to increase the degree of freedom of hydrometer spectrum variations. A commonly used hydrometeor size distribution function is the Marshall–Palmer distribution function (Marshall and Palmer 1948), with μ prescribed as zero and N0 and λ derived from the predicted mass mixing ratio and number concentration of cloud particles. Some observational studies have stressed that the N0, λ, and μ characterizing the rain DSDs exhibit mutual dependence (Vivekanandan et al. 2004; Zhang et al. 2001, 2003). These empirical relations between the distribution parameters were used in cloud microphysics schemes to reflect more variance of cloud particle spectra. For example, Morrison and Milbrandt (2015) used a λμ relationship obtained from in situ observations of Heymsfield (2003) to parameterize the ice crystal DSDs in the predicted particle properties (P3) cloud microphysical scheme. Additionally, Milbrandt and Yau (2005a,b) explored functional relationships between μ and mean hydrometeor size Dm as diagnostic equations by using (μ, Dm) data from one-dimensional sedimentation simulations as guidance. Also, Thompson et al. (2004, 2008) estimated these parameters based on meteorological variables such as vertical velocity and temperature.

Note that the Marshall–Palmer distribution function in the double-moment bulk scheme struggles to fit the hydrometeor DSD accurately, since this function has limited degrees of freedom to vary (Khain 2009). In response, Milbrandt and Yau (2005b) developed a closure formulation that predicts one more moment of the precipitating hydrometeor size distribution (sixth moment that is proportional to radar reflectivity) and treats μ as a variable parameter. Sensitivity experiments showed that allowing μ to vary as a diagnostic or prognostic parameter significantly improved simulations of precipitation (Milbrandt and Yau 2006a,b). For the selection of the added prognostic moment, Milbrandt and Yau (2005a,b), Loftus et al. (2014) and Paukert et al. (2019) selected the sixth moment while Chen and Tsai (2016) selected the second moment. In principle, any moment other than the zeroth and third moments can be a potential option as the additional moment to develop a triple-moment scheme.

To examine the sensitivity of rain DSD fitting to the prognostic moment selection, Wacker and Lupkes (2009) developed a pure sedimentation model and evaluated simulated hydrometeor profiles from single- and double-moment schemes against observed radar reflectivity. Additionally, Milbrandt and McTaggart-Cowan (2010) have conducted a similar investigation focusing on triple-moment schemes that are evaluated against the results from a bin-microphysical scheme. These studies, however, were limited to idealized sedimentation models and the optimal choice of prognostic moments that are suitable for simulating the entire set of cloud microphysical processes has not yet been thoroughly investigated. The evolution of the hydrometeor spectrum determined mainly by microphysical processes like evaporation, collision–breakup, self-breakup, and collision–coalescence is quite different from that by sedimentation. Therefore, further investigations should use measured hydrometeor particle size distributions (or comprehensive simulations of cloud hydrometeor spectra) determined by a complete set of cloud microphysical processes as a reference against which GDF fittings are evaluated.

In this study, the moment combination in the triple-moment scheme is termed in the “ijk” form where i, j, and k are orders of moments used to solve the parameters that determine the formula of GDF. Also, we define that the 036 GDF is a triple-parameter GDF that derives N0, λ, and μ based on zeroth (number), third (mass), and sixth (radar reflectivity) moments of hydrometeor categories. This practice will be used in the rest of this paper. Development of triple-moment schemes has generally sought to add one more prognostic moment to double-moment schemes that predict the zeroth and third moments, suggesting that a 03k prognostic moment group is desirable for a triple-moment scheme, where k denotes the additional prognostic moment other than the zeroth and third moments. Previous studies have not investigated all of the potential moment groups in the form of 03k. To address these knowledge gaps, we evaluate the ability of the GDFs of 03k form to fit observed rain droplet spectra for real meteorological cases where the evolution of the spectra is determined by diverse microphysical processes.

The objective of this article is to develop an optimal parameter-deriving method for a GDF that provides a suitable representation of rain droplet spectra as measured in the real atmosphere. We quantify the fitting abilities of GDFs by providing an error metric that compares the observed DSDs to GDFs and seeks an optimal GDF with minimum error. Evaluating the dependence of the variance of the bias in fitting the rain DSD on number and mass concentration of the raindrops is also an effective approach to examine the suitability of GDFs for different rainfall types. Discussions in the current article include physical explanations for large biases in the rain DSD, the benefits of bulk cloud microphysical schemes obtained from refined GDFs, and possible extension of the current study to future simulations and observational studies.

The remainder of this article is organized as follows. An introduction to parameter deriving methods and the observational data are presented in section 2. Section 3 examines the fitting ability of commonly used two-parameter and triple-parameter GDFs. Section 4 introduces the procedures to select the optimal prognostic moment group for a triple-parameter GDF. Section 5 discusses physical interpretations of biases in the fitting rain DSDs and introduces potential applications of the optimal GDF. Our main conclusions are summarized in section 6.

2. Data and experimental setup

a. Observed data from the Huangshan field campaign

The observational experiments were conducted at observation sites at three elevations at Huangshan Mountain located in Anhui province in eastern China (see schematic Fig. 1 and Table 1). The observed rain DSDs at Mt. Huangshan reflect microphysical characteristics of diverse cloud systems such as frontal precipitation as well as orographically forced precipitation. In addition to continuous aerosol properties and meteorological observations (Jiang et al. 2015, 2016; Zhang et al. 2015; Zhang et al. 2017), the measurements during Huangshan field campaign from 2008 to 2012 also provide a comprehensive dataset of rain droplet size distributions.

Fig. 1.
Fig. 1.

Geographical locations of Huangshan Mountain and field campaign sites.

Citation: Journal of the Atmospheric Sciences 77, 2; 10.1175/JAS-D-18-0259.1

Table 1.

Summary of geographical coordinates of three sites on Huangshan Mountain.

Table 1.

Measurements of rain droplet microphysics properties during the Huangshan 2011 field campaign conducted from June to August 2011 are described in detail by Chen (2013). Rain droplet spectrum data were collected with three particle size and velocity (Parsivel) disdrometers (Löffler-Mang and Joss 2000). Data from the disdrometers included raindrop number concentrations in 32 size bins with diameters ranging from 0.125 to 24.5 mm and 32 velocity bins from 0.05 to 20.8 m s−1. This study uses data from three observation sites with elevations spanning from 464 to 1840 m (see Table 1). Also, the data encompass 41 rainfall events and a total of 209 h of precipitation duration. The data quality control is described by Tang et al. (2014). The time resolution of this dataset is normalized to 1 min. After data filtering, we obtain 23 445 measured rain DSD samples.

Since previous studies have pointed out the distinction between convective and stratiform precipitation DSDs (Dolan et al. 2018; Niu et al. 2010), we divide all observed rain spectrum samples into either convective or stratiform precipitation based on the rainfall type classification method described by Bringi et al. (2003) and Marzano et al. (2010). The rainfall rate R at the instant time ti is denoted by R(ti) and classified as a stratiform precipitation sample if R(ti) lies in range 0.1–5 mm h−1 in the time interval from tiNs to ti + Ns; otherwise, this sample is classified as convective precipitation. Herein Ns is set to 5 min. Tang et al. (2014) show that a 5-min threshold is adequate to discriminate convective precipitation from stratiform precipitation. Table 2 shows statistics of rainfall type classification. About 60% of the observed DSDs are stratiform DSDs at the three sites. The observed DSDs exhibit comparable fractions of convective and stratiform precipitation samples, which suggests that the dataset provides a robust range of DSD shapes for evaluating the suitability of the GDF to represent a broad range of rain droplet spectra.

Table 2.

Summary of the number of samples of observed DSDs considering precipitation type classification at three sites.

Table 2.

b. Parameter deriving methods

In this section, we present an overview of the GDF and its distribution parameter yielding methods (see the appendix for more details). These methods include a parameter solving method [section 2b(1)] and parameter diagnosing methods [section 2b(2)]. Equation (1) is the general expression of the GDF used to represent the DSDs of rain droplets in cloud microphysical schemes. The pth moment of the GDF is given as
M(p)=0DpN(D)dD=N0Γ(p+μ+1)λp+μ+1.
Based on Eq. (2) where p is set as different integers, we can construct an equation group to solve parameters (i.e., N0, λ, and μ) in the GDF.

1) Parameter solving methods

Double-moment schemes usually select M(3) and M(0) as prognostic moments to derive the distribution parameters N0 and λ and hold μ as a constant (Lim and Hong 2010; Morrison et al. 2005; Morrison and Gettelman 2008; Thompson et al. 2008). For example, Morrison and Gettelman (2008) used the Marshall–Palmer distribution function (μ = 0) to represent rain droplet spectrum and derived N0 and λ from
λ=[qΓ(μ+4)NΓ(μ+1)]1/3,
N0=Nλμ+1Γ(μ+1),
where Γ is the gamma function defined as Γ(n)=0xn1exdx=(n1)!=(n1)××2×1. Note that M(0) and M(3) are proportional to number concentration N and mass mixing ratio q; that is, N = M(0)/ρa and q = [πM(3)ρw]/(6ρa), where ρa and ρw are densities of air and water, respectively.
The triple-moment schemes assume that μ is also a varying parameter suggesting that an additional equation is required to construct a closed equation group. Milbrandt and Yau (2005a,b) added an equation involving the sixth-order moment:
M(6)=G(μ)[M(3)]2M(0),
where G(μ) is expressed as
G(μ)=(6+μ)(5+μ)(4+μ)(3+μ)(2+μ)(1+μ).
We employ an iterative secant method similar to that of Loftus et al. (2014) to solve for μ in Eq. (5) and subsequently solve for N0 and λ using Eqs. (3) and (4). However, the values of μ must be limited within a certain range from 0 to 8 to guarantee computational stability of the iterative secant method. While there are some DSD samples in this study where the iterative secant method works well for solving for μ, for the other samples there are no solutions for μ in the range from 0 to 8. In these cases, μ is prescribed as 0 or 8 depending on whether the result is smaller than 0 or greater than 8 (see the appendix for details). We define the μ solved for using the iterative secant method as the solved shape parameter (SSP) and μ solved without arithmetic solutions as the prescribed shape parameter (PSP).

With known N0, λ, and μ, we can calculate the pth-order moment using Eq. (2). Milbrandt and McTaggart-Cowan (2010) divided the moments of the GDF into two classes: the prognostic moments used to derive the distribution parameters in the GDF, and others referred to as diagnostic moments. For example, if we set i, j, and k as zero, three, and six respectively, and use these moments to solve for the parameters N0, λ, and μ to yield a GDF, then, the zeroth, third, and sixth moments are the prognostic moments of this GDF. The other moments, such as the first, second, fourth, and fifth moments, are diagnostic in this case. Note that the values of the parameters N0, λ, and μ are sensitive to which moments are chosen as the prognostic moments. Therefore, it is worthwhile to examine different combinations of prognostic moments used in a triple-parameter GDF.

2) Parameter diagnosing methods

Since the iterative secant method is complicated to implement, diagnostic methods based on empirical formulas are also used to yield the value of μ. Commonly used methods for diagnosing μ are derived from the statistics of DSDs simulated by one-dimension rain droplet sedimentation models (Milbrandt and McTaggart-Cowan 2010; Milbrandt and Yau 2005a; Seifert 2008):

  1. Milbrandt and McTaggart-Cowan (2010) method (hereafter Diag01)
    μ=11.8×{1000×[M(k)M(j)]1/(kj)0.7}2+2,
    where M(k) and M(j) are kth and jth moments. In double-moment microphysics schemes, k = 0 and j = 3.
  2. Milbrandt and Yau (2005a) method (hereafter Diag02)
    μ=c1tanh[c2(Dmc3)]+c4,
    where c1, c2, c3, and c4 are constants (for rain droplets: 19.0, 0.6 mm−1, 1.8 mm, and 17.0, respectively), and Dm is the average diameter.
  3. Seifert (2008) method (hereafter Diag03)
    μ={6tanh[c1(DmDeq)]2+1,DmDeq,30tanh[c2(DmDeq)]2+1,Dm>Deq,
    where c1 and c2 are 4 and 1 mm−1, respectively. Rain droplet equivalent diameter Deq is 1.1 mm. By employing these methods to diagnose μ, we can solve for N0 and λ based on Eqs. (3) and (4).

c. Quantification of the error in the GDF fit to the observed rain DSD

We use moments of the observed rain DSDs as a reference to evaluate the ability of GDFs to fit the observations. A good representation of the rain droplet spectrum should be the result when all the moments of the GDF agree well with moments of the observed rain DSDs. We use moment relative errors (RE) to quantify the biases of GDF in fitting the observed rain DSDs. We can determine Mobs(p), the pth moment of the observed DSD; Mfit(p), the pth moment of the GDF; and RE(p), the RE of pth moment, determined
Mobs(p)=iNobs(Di)Dip,
Mfit(p)=iNfit(Di)Dip,
RE(p)=|Mobs(p)Mfit(p)|Mobs(p),
where Di is the middle diameter of ith size bin, and Nfit(Di) and Nobs(Di) are the fitted and observed raindrop number concentrations in the ith bin, respectively. We use the average relative errors of moments, averRE=(1/7)p=06RE(p), to quantify the ability of the GDF to fit the rain droplet size distributions. For the calculation of averRE, relative errors of the zeroth to sixth moments are equally weighted because all these moments are likely proportional to cloud microphysical process rates and it is challenging to quantify the importance of each moment. Low averRE or RE(p) indicate that a GDF fits well the entire observed rain DSD or the pth moment of the observed rain DSD, respectively. Note that the incomplete integrals of fitted rain DSDs represented by Eq. (11) are very close to the complete integrals used in cloud microphysical schemes, suggesting that the errors due to truncating the DSD will not produce considerable uncertainties to the examinations of rain DSD fitting.

3. Evaluations of existing gamma distribution functions

a. Characteristics of averaged rain DSDs

Figure 2 displays averages of total rain DSDs observed at three mountain sites (TOP: Figs. 2a,b; MID: Figs. 2c,d; BOT: Figs. 2d,e) and the corresponding GDFs. The curves termed as C0, D1, D2 and D3 are double-parameter GDF with μ prescribed as zero (Marshall–Palmer distribution function) and with μ diagnosed through Diag01, Diag02, and Diag03 methods, respectively. Figure 2 illustrates that the double-parameter and triple-parameter GDFs exhibit varying abilities to fit the observed spectra depending on the drop size range. For the sizes ranging from 0 to 3 mm all GDFs agree well with the observed spectra. However, the Marshall–Palmer distribution function (denoted as C0 in Fig. 2) does not represent rain droplet number concentrations in this size range well. In the size range from 3 to 7 mm, it is difficult for all of the GDFs and Marshall–Palmer distribution function to fit the observed rain DSDs that diverge from the GDFs in the largest size bins. The GDFs using μ-diagnosing methods (i.e., D1, D2, and D3) underestimate N(D) whereas the Marshall–Palmer distribution function overestimates N(D). This implies that the inverse exponential distribution (Marshall–Palmer distribution) introduces an unrealistically large number of giant particles in the tail part of rain spectrum that may lead to an overestimation of the high-order moments of rain droplets, whereas GDFs with diagnosed μ tend to underestimate these moments. Among all the examined distribution functions in Fig. 2, the 036 GDF can reproduce the averaged number concentrations of observed spectra in this 3–7-mm droplet size range, indicating that this method exhibits the best performance in fitting the rain spectra.

Fig. 2.
Fig. 2.

Averaged rain DSDs observed at three different elevations and the double-parameter GDFs with prescribed μ as 0, diagnosed μ through Diag01, Diag02, and Diag03 methods denoted as C0, D1, D2, and D3, respectively, in the legend. The triple-parameter GDF with prognostic zeroth, third, and sixth moments is denoted as 036. Panels show rain DSDs observed at the (a),(b) TOP, (c),(d) MID, and (e),(f) BOT observational sites (see the location and elevation of the sites in Table 1). Columns show observations and GDFs in the size ranges of (left) 0–3 and (right) 3–8 mm.

Citation: Journal of the Atmospheric Sciences 77, 2; 10.1175/JAS-D-18-0259.1

b. Analysis of relative errors

In this section, the relative errors of GDFs with prescribed, diagnosed, and mathematically solved μ are explored. Figure 3, suggests that the Marshall–Palmer distribution function produces the greatest RE for the zeroth to sixth moments while other GDFs do a better job of representing rain droplet spectra. The most convenient approach to reducing relative errors is to set μ as a positive constant. Table 3 shows that compared to the Marshall–Palmer distribution function, the two GDFs with μ set as 3 and 6 better reflect the shapes of observed rain DSDs and decrease the relative errors in fitting the observed spectrum. But it is difficult to determine which function provides the greater overall improvement because the GDF with μ = 6 yields fewer high-order moment relative errors compared to the GDF with μ = 3, but at the cost of greater relative errors of low-order moments.

Fig. 3.
Fig. 3.

Relative errors of (a) zeroth- to (g) sixth-order moments from GDFs, as well as (h) the average, using a variety of parameter solving methods. Boxes denoted as C0, C3, C6, D1, D2, and D3 are double-parameter GDFs with μ prescribed as 0, 3, and 6, and μ diagnosed through Diag01, Diag02, and Diag03 methods, respectively. The box denoted as 036 is a triple-parameter GDF with prognostic 036 moment group. Boxes indicate the range from the 25th to 75th percentiles. Whiskers indicate the range from the 10th to the 90th percentiles. Relative errors of convective and stratiform precipitation DSD samples are separately plotted in blue and green, respectively.

Citation: Journal of the Atmospheric Sciences 77, 2; 10.1175/JAS-D-18-0259.1

Table 3.

Average RE values (%) of the box plots in Fig. 3.

Table 3.

The GDF with diagnosed μ also provides a better fit to the rain DSDs when compared with the Marshall–Palmer distribution function. For example, both Fig. 3 and Table 3 illustrate that the mean averRE for convective precipitation samples fitted by GDFs with diagnosed μ are reduced by more than 70% compared with the averRE of the Marshall–Palmer distribution function. The improvement over the Marshall–Palmer distribution function by prescribing μ as a positive integer is comparable to the improvement from diagnosing μ, even though it is expected that a GDF with a diagnosed μ that varies with properties of the rain droplet spectrum should provide better performance. There are two possible reasons to explain this phenomenon. First, the μ-diagnosing methods are developed based on the statistics of rain DSDs produced by idealized one-dimensional rainshaft models, rather than observed DSDs (Milbrandt and McTaggart-Cowan 2010; Milbrandt and Yau 2005a). Since these rain shaft experiments only consider rain droplet sedimentation, the simulated DSDs may not reflect the characteristics of rain droplet spectra in the real atmosphere. Second, the relationships between μ and rain DSD moments (e.g., mean diameter Dm) are usually too disperse to be accurately fitted by the μ-diagnosing functions. An attempt should be made to improve μ-diagnosing methods by exploring the μDm relationship based on size resolving observations of rain DSDs (e.g., Cao et al. 2008; and Zhang et al. 2003).

The 036 triple-parameter GDF allows one to solve for the three parameters through Eqs. (3)(5). In Fig. 3, the averRE of the triple-parameter GDFs shows significant low values [down to O(100)], suggesting that the triple-parameter GDFs exhibit an apparently better fit to observed rain DSDs than the double-parameter GDFs. The accurate fit of the 036 GDF to the observed DSDs results from the mathematically derived μ using the iterative secant method in which we limit the values of error functions to be smaller than 10−5 (see the definition of error function in the appendix), even though for 36% of observed rain DSDs the numerical method cannot determine solutions of μ with values in the range from 0 to 8.

4. Development of a new moment group

The objective of this section is to examine the ability of triple-parameter GDFs with the 03k prognostic moment groups to fit observed droplet spectra and determine the optimal choice of k. The relative errors in the fit of the 03k GDF (averRE03k) is given as
averRE03k=averRE03kSSPN03kSSP+averRE03kPSPN03kPSPN03kSSP+N03kPSP,
where averRE03kSSP and N03kSSP are average relative error and sample number of rain DSD fitted by GDFs with solved μ (SSP). The averRE03kPSP and N03kPSP are the corresponding values for GDFs with prescribed μ (PSP). Note that averRE03kSSP is generally less than averRE03kPSP, since the former is produced by fitting the GDFs with mathematically solved distribution parameters, whereas the later fitting does not make the left side of Eq. (A3) equal to the right side. Additionally, the total observed sample number (N03kSSP+N03kPSP) is constant. Thus, averRE03kPSP and N03kPSP play an important role in determining averRE03k. Further investigation illustrates that N03kPSP usually increases with the value of k, suggesting that setting k as one will obtain the minimum of N03kPSP. However, setting k as one obviously increases averRE03kPSP since a GDF with only low-order prognostic moments struggles to fit the high-order moments well. Therefore, the selection of k should balance the increase of averRE03kPSP and N03kPSP to obtain the lowest values of averRE03kPSPN03kPSP and simultaneously avoid an obvious increase of averRE03kPSPN03kPSP. Consequently, we select k from a number pool of 1, 2, 4, 5, and 6, and even extend k to values larger than 6.

a. Selections of 03k moment groups

Figure 4 and Table 4 illustrate that the 013 and 023 GDFs produce small RE for the zeroth to the second moments while large RE(3) to RE(6) values are nonnegligible, especially for the stratiform precipitation DSDs. For example, the average RE(3) of stratiform DSDs fit by the 013 GDF is 7.39%, which is 4 times larger than the corresponding error (1.82%) when using the 034 GDF. Thus, the 013 and 023 moment groups with large values from RE(3) to RE(6) may fail to accurately parameterize the rain droplet mass source and sink terms that are relevant to high-order moments. For example, M(4) is approximately proportional to the sedimentation mass flux of rain droplets (see section 5b) and a large relative error in M(4) reflects a poor estimate of the fall rate of rainwater and thus the lifetime of a precipitating cloud systems.

Fig. 4.
Fig. 4.

As in Fig. 3, but for the triple-parameter GDFs with prognostic moment combinations of 013, 023, 034, 035, and 036 labeled along the x axis.

Citation: Journal of the Atmospheric Sciences 77, 2; 10.1175/JAS-D-18-0259.1

Table 4.

As in Table 3, but for Fig. 4.

Table 4.

Though the prognostic moment groups of 034, 035, and 036 show similar RE values, we propose that the 034 GDF is the best one for fitting the rain droplet spectra since it exhibits the lowest values of RE for both convective and stratiform precipitation DSDs compared to the other groups. Table 4 demonstrates that compared with the 036 GDF, the REs of 034 GDF are comprehensively lower, in some cases by up to 42%. Also, the RE values of 034 GDF are lowered by up to 29% compared to the 035 GDF. Values of 90th-percentile and 75th-percentile points of the averRE from 034 GDF are not higher than the corresponding values of other GDFs (see Fig. 5), indicating a comparable level of dispersion of the RE distribution relative to the other groups. For completeness, we investigate the fitting of 03k GDFs with k larger than 6 and find that all these fits produce larger relative errors than 036 GDF. As mentioned above, a possible reason for the high relative errors is that large k increases N03kPSP in Eq. (13), leading to a large averRE03k.

Fig. 5.
Fig. 5.

The averRE values are located at two-dimensional coordinates determined by the mass concentration and number concentration of observed rain DSDs. The color contours denote averRE distributions. The contour lines denote the distribution of the frequency of samples. Panels show averRE distributions of (a),(c) convective and (b),(d) stratiform precipitation fit by the 036 GDF with (top) the solved spectral parameter and (bottom) the prescribed spectral parameter.

Citation: Journal of the Atmospheric Sciences 77, 2; 10.1175/JAS-D-18-0259.1

b. Relative error distributions

In this section, we investigate the dependence of averRE on the number and mass concentrations of measured rain DSDs. This is done to explicitly examine the capability of GDFs to fit rain DSDs produced by precipitation featuring different intensities. We also separately plot the averRE distributions when GDFs with SSPs and with PSPs fit observed rain DSDs. The following analysis focuses on 013, 034, and 036 moment groups since the relative error distributions of 023 and 035 are similar to 013 and 036, respectively, and thus are not shown here.

The distributions of averRE of 036 GDF in the number–mass coordinate are plotted in Fig. 5. The DSD samples can be divided into two groups: stratiform DSDs predominantly characterized by lower number and mass concentrations and convective DSDs with greater number and mass concentrations. Regions where the averRE is less than 5% correspond to the cases that the DSD is accurately fit with the triple-parameter GDF. Figures 5a and 5d illustrate that the samples with averRE greater than 5% are usually convective precipitation DSDs with droplet mass concentrations greater than 5 × 10−3 kg m−3 and stratiform precipitation DSDs with droplet mass concentrations lower than 5 × 10−5 kg m−3. We speculate that these convective and stratiform samples mainly contribute to the high RE for the 036 GDF fits shown in the Fig. 3 as the RE of the rest samples is considerably low.

When compared with the 036 GDF, the 034 GDF shows advantages in fitting the frequently occurring rain DSDs with mass concentrations ranging from 10−4 to 10−3 kg m−3 (Fig. 6). As shown in Figs. 5a and 5b, the averRE values of rain DSD samples that occur most frequently (with frequencies larger than 100) usually ranges from 2.5% to 5%, whereas the corresponding values in Figs. 6a and 6b are in the range from 1% to 2.5%. Additionally, the mean averRE in each panel of Fig. 6 (ranging from 2.95% to 7.66%) is lower than that in Fig. 5 (ranging from 3.40% to 8.21%), suggesting that 034 GDF gives a comprehensively better fit than the 036 GDF, which is consistent with section 4a. The comparison between Figs. 6 and 7 demonstrates that the 034 GDF yields significantly lower aveRE compared to the 013 GDF. Figure 7d shows that all the stratiform samples fitted by 013 GDFs with prescribed μ (PSP) produce averRE greater than 10%. The 013 GDFs with solved μ (SSP) also exhibits higher averRE compared to the 034 GDF (see Fig. 7b). Based on the analysis above, we suggest that the 034 GDF exhibits the best overall performance in fitting the full range of rain DSD samples.

Fig. 6.
Fig. 6.

As in Fig. 5, but for the averRE distributions of 034 GDFs.

Citation: Journal of the Atmospheric Sciences 77, 2; 10.1175/JAS-D-18-0259.1

Fig. 7.
Fig. 7.

As in Fig. 5, but for the averRE distributions of 013 GDFs.

Citation: Journal of the Atmospheric Sciences 77, 2; 10.1175/JAS-D-18-0259.1

In this study, we have evaluated the average relative errors by applying equal weight to the relative errors of the zeroth to sixth moments. However, if microphysical processes are dominated more by the lower-order moments, as discussed further below in section 5b, then it might be appropriate to weight the lower-order moments more heavily than the higher-order moments in evaluating the error (Milbrandt and McTaggart-Cowan 2010). We note, however, as shown in Table 4, all quantities from RE(0) to RE(6) for both stratiform and convective precipitation DSD fits with the 034 moment group are lower than the corresponding relative errors of fits with the 035 and 036 moment groups. Therefore, we expect that the averRE of the 034 moment group will be lower than the averREs of 035 and 036 moment groups regardless of whether the lower-order moments are weighted higher relative to the higher-order moments in the calculation of aveRE. The 013 and 023 moment groups do show lower RE(1) to RE(3) values compared to the 034 moment group. However, even if we weight RE(0) to RE(3) of the 023 moment group to account for 85% of the aveRE and decrease the weights of RE(4) to RE(6) to 15%, the averREs of 023 and 013 moment groups are 3.82% and 4.80%, respectively, which are still higher than the averRE of 034 moment group (3.70%). Thus, we find that the lower aveRE of the 034 moment group is quite robust even to error statistics that weight more heavily the error of the lower-order moments.

5. Discussion

a. Physical interpretations of large biases in fitting rain DSDs

As shown in Figs. 57, large averRE (larger than 10%) occurs when a GDF is used to fit rain DSDs corresponding to heavy convective precipitation (with rain mass concentration greater than 5 × 10−3 kg m−3) and to weak stratiform precipitation (with rain mass concentration less than 5 × 10−5 kg m−3). Heavy precipitation usually results in wide rain DSDs with a large number of large droplets in the tail of spectrum (Konwar et al. 2014). The GDF, which exhibits a single-modal shape, struggles to fit these spectra well. As a result, the GDF tends to underestimate the number concentration of large droplets at the tails of wide spectra, which introduces large RE in fitting the higher-order moments. In addition to the extremely wide spectra, the GDF also fails to fit the truncated rain spectra (with the ends of truncated distributions lying within the 125-μm to 7-mm size range) from weak stratiform precipitation (Willis 1984). In the observational size range, the stratiform rain droplet spectra are usually truncated distributions, whereas GDFs are continuous. Consequently, the GDFs always extend beyond the truncated end of the observed rain DSDs and overestimates the high-order moments of narrow rain DSDs.

The weak stratiform precipitation DSDs (with rain mass concentrations lower than 5 × 10−5 kg m−3) are usually truncated DSDs and further investigation exhibits that these DSDs often correspond to the GDFs with prescribed μ. The iterative secant method struggles to solve for μ when the DSD is truncated. As mentioned in the appendix, μ is solved for using Eq. (A6) where the left and right sides of Eq. (A6) are represented by Eqs. (A7) and (A8), respectively. When solving for μ of a triple-parameter GDF used to fit these truncated stratiform rain DSDs, we find that the value of F*(μ*) [see Eq. (A8)] is usually smaller than F(8) [see Eq. (A7)]. Since F(μ) decreases with increasing μ, a value of μ larger than eight is required to bring Eq. (A6) to equilibrium. Unfortunately, μ must be smaller than or equal to 8 to maintain a stable calculation using the iterative scant method. Consequently, the prescribed μ is the main source of bias when fitting to stratiform DSDs. For the convective cloud rain DSDs, the truncated distributions, requiring μ greater than 8, are much less frequent than stratiform DSDs (see Figs. 57c and 7d). Furthermore, fitting convective precipitation DSDs with a GDF, even when applying a prescribed μ, does not yield greater averRE than the corresponding fits to truncated stratiform precipitation DSDs. Contrary to triple-parameter GDFs with a prescribed μ (PSP), the GDFs with solved μ (SSP) work well in fitting DSD samples shown in Figs. 57a and 7b, except for some rarely occurring heavy-convective-precipitation DSDs that are corresponding to the samples with rain mass concentrations larger than 5 × 10−3 kg m−3. Through quantitative comparisons, we conclude that the high values of relative error (averRE) are mainly contributed by the failures of the GDF to fit the frequently occurring truncated-stratiform-rain DSDs. Additionally, the challenge of fitting to the extremely wide convective precipitation DSDs is the second source of error.

b. Benefits from applying the optimal GDF in bulk cloud schemes

In this subsection we discuss how the cloud microphysical schemes benefit from accurate representation of rain spectra by a GDF with parameters derived from an optimal prognostic moment group. A comprehensive review of cloud microphysical process parameterization in bulk schemes by Straka (2009) demonstrates that almost all cloud hydrometer source and sink terms are proportional to the corresponding integral moments of the size distribution. For instance, the contribution of sedimentation to the decrease in rain droplet number concentration is parameterized as 0N(D)V(D)dD, where N(D) is the size distribution represented by the GDF and V(D) is the fall speed for an individual rain droplet with diameter D. A widely used parameterization of rain droplet fall speed is a terminal velocity–diameter power-law relation V(D) = aDb where a and b are empirical coefficients. The value of b is 0.8 (Milbrandt and Yau 2005b), indicating that 0N(D)V(D)dD is proportional to the 0.8th moment, which is approximately equal to the first moment. Hence, selection of a GDF that accurately fits the first moment is an efficient approach to improve the parameterization of sedimentation effects on the simulated rain droplet number concentration.

There are additional similar examples of linear relationships between integral moments and rain droplet source and sink terms. Sedimentation contributions to the rain droplet mass variations are proportional to the 3.8th moment (Milbrandt and Yau 2005a,b). Mass loss rates of rain droplets by evaporation are proportional to the first moment [see Eq. (6.21) in Straka 2009]. By assuming that the collection efficiencies of all falling rain droplets collecting cloud droplets are equal to one, we can parameterize the mass growth rates of rain droplets by accretion as a linear function of the 2.8th moment [see Eq. (7.67) in Straka (2009)]. Liu and Daum (2004) developed a new rain droplet collision–coalescence parameterization that judged the occurrence of coalescence growth based on the fourth or sixth moments of rain DSDs. All the source and sink terms, in addition to those mentioned above, may benefit from applying the 034 GDF since it improves upon the fit to rain DSD moments compared to the Marshall–Palmer distribution function and the 036 GDF.

c. Extension of GDF application

The conclusions of this investigation may also have applications in cloud microphysics remote sensing observations. For example, the operational lidar–radar (DARDAR) product (Delanoë and Hogan 2008, 2010) infers profiles of ice crystal effective radius and number concentration based on observed lidar extinction backscatter coefficient βext and radar reflectivity Ze. The technique uses a similar framework as presented in this study (Gryspeerdt et al. 2018; Sourdeval et al. 2018). From the standpoint of this study, the “prognostic moments” [see the special definition in section 2b(1)] in DARDAR are the second moment and sixth moment from which two parameters, N0 and k, in a modified GDF:
N(Deq)=N0Deqαexp(kDeqβ)
are derived. The other two distribution parameters α and β are prescribed as −1 and −3, respectively. Ice crystal number concentration and effective radius are therefore retrieved by calculating the zeroth moment and the ratio of third moment to the second moment. There are no other potential selections of the prognostic moments for the DARDAR product; however, RE distributions of the moments, similar to those presented in section 4b may be an appropriate method to evaluate retrieval biases in the DARDAR product. The size-resolved measurement of ice crystal size distribution (e.g., based on airborne cloud probe observations) can serve as a reference to evaluate the ability of the modified GDF [Eq. (14)] to fit the observed moments (ice crystal number concentration) and evaluate the error distributions in a βextZe space (like Figs. 57). For the ice size spectra where the fit of the modified GDF exhibits large biases, changing the prescribed parameters [α and β in Eq. (14)] may be an efficient approach to mitigate these biases and improve the retrievals.

Current data assimilation techniques are usually “single moment” and do not apply double-moment cloud microphysical schemes. For example, cloud microphysics data assimilation usually provides an initial condition of rainwater mass mixing ratio retrieved from remote sensing observations (e.g., radar reflectivity) based on which the future time evolution of rainwater mass is predicted by the cloud microphysical scheme. The rain droplet number concentration, which is also required by a double-moment cloud microphysical scheme, is not provided by the single-moment data assimilation (J. Sun, NCAR, 2019, personal communication), leading to a deficit in the information needed to determine the distribution parameters of the optimal GDF. As a result, there is no GDF that represents well the realistic rain DSD suitable for improving parameterizations of rain droplet source and sink terms. A double-moment data assimilation of rain droplets is suggested.

6. Conclusions

This study has examined the ability of double- and triple-parameter GDFs to fit measured rain DSDs observed during the Huangshan field campaign. Sufficient rain DSD samples, involving multiple precipitation types and intensities, have been used to seek an optimal GDF that is universally applicable to represent the rain DSDs in both convective and stratiform precipitation. Evaluation and potential application of the optimal GDF has been discussed. Following are the key conclusions:

Although the double-parameter GDFs with positive spectral shape parameter μ do not fit the observed spectra as accurately as the triple-parameter GDFs, the former is convenient to implement in microphysics schemes in atmospheric models. Therefore, when mass mixing ratios and number concentrations of rain droplets are substituted into a microphysical scheme module, μ can be diagnosed with empirical formulas [e.g., Eqs. (7)(9)] followed by solving for N0 and λ. It is also appropriate to set μ as a positive value like 3 or 6 to replace μ = 0 in the Marshall–Palmer distribution function. The result is a more accurate hydrometeor spectrum representation without intensive code modifications.

Examining the distribution of errors in a rain droplet number–mass coordinate (see Figs. 57) is a novel and effective method to evaluate GDF fits to rain DSDs from various precipitation events. The averRE of the moments can also be estimated based on microphysical properties of precipitation like mass, number concentration of rain droplets and rain spectrum dispersion. A rain droplet spectrum can be accurately fit with a triple-parameter GDF when the mass concentration of all rain droplets involved in this spectrum is in the range from 5 × 10−5 to 5 × 10−3 kg m−3. Due to the failure of GDFs to fit rain DSDs with extreme spectral widths, bulk cloud microphysical schemes fail to simulate the extremely heavy precipitation or weak drizzle events.

Results of sensitivity experiments demonstrate that it is appropriate to replace the 036 prognostic moment group with the 034 group to derive the distribution parameters for the triple-parameter GDF. Overall, by using the 034 group, the relative errors in the fit to observed spectra are reduced by 10%–40%, when compared with the 036 group, suggesting that the 034 GDF gives a comprehensively better fit than the 036 GDF. Note that the parameterization of rain source and sink terms in the cloud microphysical schemes are proportional to integral moments of GDF (see section 5b) and modeling errors caused by rain DSD representation will accumulate as a simulation proceeds with time. Therefore, even though the RE of the 036 group is relatively small, it is still worthwhile to further decrease the relative error by introducing the 034 group. In this study, the observed rain DSDs are used to solve GDF distribution parameters and serve as a reference to evaluate the fits. The investigation of rain droplet representation by GDFs in this study is in a “diagnostic view” since the evolution of the errors with the temporal integration in the model is neglected. For the further examination, a follow-up study will focus on developing a cloud microphysical scheme based on the 034 group and implement it into a predictive model to investigate the employment of 034 GDF in forecasting the temporal evolution of cloud microphysical properties.

In cloud microphysical retrievals, the variance of errors of retrieved moments fit with moments of a GDF (such as ice crystal number concentration retrieved by DARDAR) with prognostic moments (such as observed backscatter coefficient and radar reflectivity) is an efficient approach to judge the occurrence of questionable retrieval samples. Additionally, rain droplet simulations can be improved if data assimilation is able to introduce initial conditions of all prognostic moments, and thereby improve parameterizations of microphysical process rates. For example, double-moment cloud microphysical schemes require nudging of both mass mixing ratio and number concentrations of rain droplets to reduce the uncertainties caused by using a double-parameter GDF to represent the rain DSDs.

Note that the results of this study apply specifically to the representation of rain droplet spectra and that the observed rain droplet spectra are based on a single field campaign. Although these DSDs reflect microphysical characteristics of a wide range of precipitation events in a midlatitude location, further examination of the optimal GDF is warranted based on observations from a diversity of locations and climates. The spectrum of large particles, such as snow crystals, graupel, and rain droplets, usually exhibits a large mean size and a wide distribution. In contrast, the spectrum of small particles, such as ice crystals and cloud droplets, usually exhibits a small mean size and a relatively narrow size distribution. To fit narrow distributions, μ should be defined to be larger than eight, indicating that the iterative secant method may not be suitable for use in small particle spectra. A possible solution is to introduce a modified GDF involving four distribution parameters, two of which reflect the dispersion of the particle spectrum. Further study is required to find a suitable modified GDF for such cases and to verify the associated optimal prognostic moment groups.

Acknowledgments

This research was supported by the National Science Foundation Grant IIA-1301726, the National Natural Science Foundation of China Grant 41590873, and the National Science Foundation of China Grant 41805021. The authors thank the anonymous reviewers for their valuable comments that helped improve the presentation of the paper.

APPENDIX

Mathematical Derivation of Parameter Solving Methods

For the double-parameter GDF with a constant μ, two predictive equations of the form of Eq. (2) are needed. We substitute two integer quantities i and j into Eq. (2) and obtain
M(i)=N0Γ(i+μ+1)λi+μ+1,
M(j)=N0Γ(j+μ+1)λj+μ+1.
By assuming that μ is a constant, we construct an equation group to derive N0 and λ based on M(i) and M(j):
λ=[M(j)Γ(i+μ+1)M(i)Γ(j+μ+1)]1/(ji),
N0=M(i)λi+μ+1Γ(i+μ+1).
For the double-parameter GDF with a diagnosed μ, we use diagnostic functions such as Eqs. (7)(9) to find μ and subsequently substitute it into Eqs. (A3) and (A4) to solve N0 and λ.
For the triple-parameter GDF, we introduce one more prognostic moment M(k) given by
M(k)=N0Γ(k+μ+1)λk+μ+1.
By combining Eqs. (A1), (A2), and (A5), Milbrandt and McTaggart-Cowan (2010) obtain a predictive equation of μ:
M(k)(ji)M(j)(ik)M(i)(kj)=Γ(μ+k+1)(ji)Γ(μ+j+1)(ik)Γ(μ+i+1)(kj).
We employ an iterative secant method developed by Loftus et al. (2014) to solve μ in Eq. (A6). An assumption of the iterative secant method is that the solution for μ is in a known value range. We assume that the value range of μ is from 0 to 8, since 0 is a generally defined minimum of spectral shape parameter (Loftus et al. 2014; Morrison and Milbrandt 2015) and unstable calculations usually occur when μ is larger than 8. The right side of Eq. (A6) is defined as a function:
F(μ)=Γ(μ+k+1)(ji)Γ(μ+j+1)(ik)Γ(μ+i+1)(kj).
Therefore, for the solution μ* of Eq. (A6), there should be a relationship:
F(μ*)=M(k)(ji)M(j)(ik)M(i)(kj).
However, investigation of the measured rain droplet size distributions illustrates that for some rain spectrum samples there is no available μ value in the 0–8 range that makes the left side of Eq. (A8) to be equal to the right side, indicating that iterative secant method does not work well with these samples. Note that Eq. (A7) shows that F(μ) monotonically decreases with an increasing μ from 0 to 8. Hence, we prescribe that μ = 0, if F(μ*) > F(0) > F(8), and μ = 8, if F(μ*) < F(8) < F(0) to minimize the computational bias. Further tests show that fraction of samples with prescribed μ is about 35% and the samples with F(μ*) < F(8) < F(0) is always a large majority (larger than 95%). This fraction varies with the selection of prognostic moment combinations. Otherwise, we use iterative secant method to solve μ for the sample with F(8) < F(μ*) < F(0):
μl+1=μlG(μl)×(μl1μl)G(μl1)G(μl),
where
G(μl)=F(μl)F(μ*)=Γ(μl+k+1)(ji)Γ(μl+j+1)(ik)Γ(μl+k+1)(ji)M(k)(ji)M(j)(ik)M(i)(kj).
We use μ and μ + 0.5 values corresponding to G(μ) and G(μ + 0.5) that straddle 0 to initialize μl and μl+1. Then, Eq. (A9) is iteratively calculated until the value of error function is sufficiently small, for example ε = |(μl+1μl)/μl+1| < 10−5. Furthermore, we solve N0 and λ by substituting the solved or prescribed μ into Eqs. (A4) and (A5). Note that this is generally suitable for any moment order group containing different values of i, j, and k. Triple-moment schemes by Chen and Tsai (2016) and Milbrandt and Yau (2005a,b) are special examples with particular determined prognostic moment groups.

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