A Gamma Parameterization for Precipitating Particle Size Distributions Containing Snowflake Aggregates Drawn from Five Field Experiments

George Duffy aJet Propulsion Laboratory, California Institute of Technology, Pasadena, California

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Derek J. Posselt aJet Propulsion Laboratory, California Institute of Technology, Pasadena, California

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Abstract

In this study, we investigate the tendencies of gamma parameters for particle size distributions (PSDs) containing snowflake aggregates in orographic, convective, and stratiform clouds, above snowstorms and above rainstorms, in temperatures ranging from 0° to −45°C. We find a strong relationship between μ and Λ but no dependence on temperature. Higher μ are observed during the experiments sampling winter snowstorms, and lower μ are observed during experiments sampling frozen clouds above convective and orographic storms. We find that a gamma function with a μ of −1.25 provides the best average representation of PSD shape and the most accurate representation of PSD moments related to mass and reflectivity. We also provide a lookup table of maximum particle size boundaries that can be used to parameterize incomplete gamma functions with negative μ values.

Significance Statement

In many weather models and satellite retrieval algorithms, frozen clouds and precipitation are governed by the same assumptions even though they develop through different growth processes. This paper provides recommendations for snowflake aggregate size distributions that reflect natural conditions, and these recommended assumptions are demonstrated to improve estimates of mass and radar reflectivity. We studied a variety of storms, such as thunderstorms, snow storms, and winter rainstorms, and we found that our model for snowflake aggregates was nearly identical in all observed conditions.

© 2022 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: George Duffy, gduffy01@syr.edu

Abstract

In this study, we investigate the tendencies of gamma parameters for particle size distributions (PSDs) containing snowflake aggregates in orographic, convective, and stratiform clouds, above snowstorms and above rainstorms, in temperatures ranging from 0° to −45°C. We find a strong relationship between μ and Λ but no dependence on temperature. Higher μ are observed during the experiments sampling winter snowstorms, and lower μ are observed during experiments sampling frozen clouds above convective and orographic storms. We find that a gamma function with a μ of −1.25 provides the best average representation of PSD shape and the most accurate representation of PSD moments related to mass and reflectivity. We also provide a lookup table of maximum particle size boundaries that can be used to parameterize incomplete gamma functions with negative μ values.

Significance Statement

In many weather models and satellite retrieval algorithms, frozen clouds and precipitation are governed by the same assumptions even though they develop through different growth processes. This paper provides recommendations for snowflake aggregate size distributions that reflect natural conditions, and these recommended assumptions are demonstrated to improve estimates of mass and radar reflectivity. We studied a variety of storms, such as thunderstorms, snow storms, and winter rainstorms, and we found that our model for snowflake aggregates was nearly identical in all observed conditions.

© 2022 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: George Duffy, gduffy01@syr.edu

1. Introduction

More than one-half of all precipitation on the planet either falls as, or is generated by, falling snow. Therefore, it is important to research and realistically simulate snow clouds. Particle size distribution (PSD) functions are fundamental components of weather models and satellite retrieval algorithms. The current most common recommendation for PSDs containing snowflake aggregates would likely be the exponential model,
n(D)=N0eΛD,
with N0 and Λ representing the intercept and slope parameter, respectively.
Equation (1) represents the first observed model to fit measured snowflakes at the ground (Gunn and Marshall 1958) and it also represents a theoretical prediction of snow-aggregate size distributions based off of fractal statistic theory (Field and Heymsfield 2003). The exponential model is used in radar retrievals of the properties of falling snow (Wood and L’Ecuyer 2021; Leinonen et al. 2018; Liao et al. 2005; Matrosov 1998) and in observations of snowing clouds (Sekhon and Srivastava 1970; Woods et al. 2008; Heymsfield et al. 2008). The exponential distribution can also be considered as an aggregate “tail” for a more complete model of frozen hydrometeors that is better described by a gamma distribution,
n(D)=N0DμeΛD.

PSD models for falling snow have traditionally been generated from datasets that combine distributions of cloud-ice crystals and monomer snowflakes along with PSDs containing aggregate snowflakes (Borque et al. 2019; Field et al. 2007; Gunn and Marshall 1958; Brandes et al. 2007; Wood and L’Ecuyer 2021). In PSDs dominated by pristine crystals or precipitating monomers, gamma fits are often positive to fit the Gaussian or lognormal shaped PSD curves (Delanoë et al. 2005; Martinez and Bennartz 2015). In PSDs containing snowflake aggregates, however, gamma functions are most often exponential or superexponential (μ < 0) to reflect the monotonically decreasing portions of a PSD that dominate mass and reflectivity. Therefore, results drawn from all-inclusive frozen hydrometeor datasets may be detrimental for inferring properties of frozen precipitation clouds where aggregates are expected to be present—for example, in retrieval algorithms for the Global Precipitation Measurement (GPM) dual-frequency precipitation radar, which has a minimum detectable reflectivity of 12 dBZ. If one wants to make a realistic model for PSDs containing snowflake aggregates, one would want to draw from results that focus on PSDs containing snowflake aggregates.

The goal of this study is to determine the best gamma parameterization to represent precipitating PSDs containing snowflake aggregates. We use a dataset consisting of measurements from five field experiments representing stratiform, convective, and orographic clouds, in different seasons and covering different surface types, with temperatures ranging from 0° to −45°C. We restrict the size range of our PSDs to focus on snowing aggregates as best as we can without sacrificing diversity of frozen precipitation environments. The method used to determine these models is provided in section 2. The first section of results in section 3 determines a new gamma parameterization for PSDs containing aggregates and investigates the sensitivities to μ and variety of PSD shapes provided in the dataset. The second set of results in section 3 evaluates this new parameterization against the exponential model. The study is summarized in section 4.

2. Method

PSDs measurements first require collections of particle sizes. In this study, PSDs are generated from images captured by probes attached to an airplane flying through precipitating clouds. Ice crystal images with D > 1 mm come from the high-volume precipitation spectrometer (HVPS). The HVPS has a sampling frequency of at least 0.4 m3 s−1 for typical aircraft speeds above 100 m s−1. Ice crystal images with D < 1 mm are imaged by the cloud-imaging probe or the 2D cloud-imaging probe, depending on the experiment. In this study, D is defined as the maximum measurable dimension of a two-dimensional ice crystal image, or as the minimum diameter of a circle that completely encompasses a reconstructed image of a partially imaged ice particle (Heymsfield and Parrish 1978). Ice crystal images are reduced to a D and sorted into PSDs by the System for Optical Array Probe Data Analysis (SODA) processing code at the National Center for Atmospheric Research. These experiments used antishattering tips, and the processing code accounted for interarrival time errors that could indicate particle shattering. PSDs also start at D > 100 μm to further remove the influence of shattered ice crystals. These PSDs are generated by sorting all measured D into size bins and dividing the number of sampled particles per bin by a sample volume representing the area of the HVPS collection window, the true airspeed, and adjustments for out-of-frame particle images. Size bins increase on a quasi-logarithmic scale, such that the largest-size bins are larger than the smallest-size bins by more than an order of magnitude. PSDs are averaged over a 15-s collection time period to reduce sampling uncertainties. We also eliminate PSD bins that contained less than 10 particles per collection sample in an attempt to remove unmeasurable low concentrations of large particles. To focus on PSDs that contain snowflake aggregates, we restrict our data to PSDs that have a maximum particle diameter Dmax > 4 mm. This threshold mainly eliminated spurious high-μ measurements from PSDs containing large monomers and low concentrations of aggregates during GCPEX and IMPACTS. We are ultimately left with 4428 measured PSDs across the five experiments to use in this study.

Since PSDs span several orders of magnitude, and since PSD boundaries, collection time, sample volume, and fitting technique can all influence the shape of a PSD, it is impossible to define a single “best fit” to a measured particle size distribution. For this study, we generate a gamma PSD that best reduces the sum of errors between some set of measured and fit moments of a size distribution
Dn=DminDmaxn(D)DndD,
with error being represented by a χ2:
χn2=(DnmeasDnfitDnmeasDnfit)2.
In this study we fit the PSD to the 1.5th, 3rd, and 4.5th moments of the size distribution to capture the range of moments that will likely be responsible for mass and reflectivity. Fitting to lower and higher moments becomes problematic, since sensitivity tests determined that particles beyond the resolution of imaging probes become increasingly influential for these moments. Best-fit gamma distributions for each measured PSD are generated by cycling through a large volume of coarsely resolved candidate gamma parameters. In this study, μ has a resolution of 0.25, Λ has a resolution of 0.1 mm−1, and N0 is drawn from 300 lognormally distributed bins between 10−10 and 1025.

PSD measurements come from five field experiments. The GPM Cold Season Precipitation Experiment (GCPEX; Skofronick-Jackson et al. 2015) sampled winter snowstorms from extratropical cyclones and lake-effect snow storms over Barre, Ontario, Canada. The Midlatitude Continental Convective Clouds Experiment (MC3E; Jensen et al. 2015) sampled summer convective thunderstorms over Oklahoma. The Integrated Precipitation and Hydrology Experiment (IPHEX; Barros et al. 2016) sampled summer convective and orographic thunderstorms over the Great Smoky Mountains. The Olympic Mountains Experiment (OLYMPEX; Houze et al. 2017) sampled frontal and orographic rain and snow storms over the Olympic Mountains. The Investigation of Microphysics and Precipitation for Atlantic Coast-Threatening Snowstorms (IMPACTS) experiment has provided one year of field data in northeastern U.S. snowstorms as of the time of writing (McMurdie 2020). In situ observations come from probes attached to the University of North Dakota Citation aircraft.

3. Results

a. Observed μ in measured PSDs containing snowflake aggregates

First, we look at the tendency of μ from all experiments in Fig. 1. Since different experiments provided different numbers of measurements, we randomly draw a smaller subset from the four more robust experiments such that each experiment is providing an equal amount of measurements (N = 269) for this figure. All remaining figures in this study use the total measured dataset. The histogram of the best-fit μ reveals that the majority (83%) of PSDs in this dataset had negative μ values. The 25th, 50th, and 75th percentile of observed μ values were −1.75, −1.25, and −0.5, respectively.

Fig. 1.
Fig. 1.

Cumulative distribution function representing μ values from IPHEX, GCPEX, MC3E, IMPACTS, and OLYMPEX. The black dashed line represents the second quartile (median); red dashed lines represent the first and third quartiles. This CDF represents an equal amount of data randomly drawn from each experiment.

Citation: Journal of Applied Meteorology and Climatology 61, 8; 10.1175/JAMC-D-21-0131.1

Measured PSDs from IPHEX, MC3E, and OLYMPEX provide similar distributions to each other, with median μ values of −1.5 (Fig. 2a). The smallest range of μ values comes from IPHEX, followed by MC3E and then OLYMPEX. PSDs from GCPEX and IMPACTS have higher μ values (comparing μ values can be confusing because their range crosses 0. In this study, “higher” can be synonymous with “closer to 0” or “more positive,” so −1 would be higher than −2, and 1 would be higher than −1) and a wider range of μ values. GCPEX has the highest median μ value (μ = −0.5) with IMPACTS in a close second (μ = −0.75), but the range of μ values is larger during IMPACTS.

Fig. 2.
Fig. 2.

Distributions of μ stratified (a) by experiment and by ranges of (b) Λ, (c) temperature, and (d) LWC.

Citation: Journal of Applied Meteorology and Climatology 61, 8; 10.1175/JAMC-D-21-0131.1

Figures 2b–d do not reveal any more noteworthy or unexpected sensitivities to μ. There is some influence of LWC on μ, but it is minor (correlation coefficient squared r2 = 0.1). Median values of μ drop from −1 for LWC ≈ 0 to −1.75 for LWC ≥ 0.06. There is also little correlation between temperature T and μ. While there is a wider variety of μ available at T > −20°C, these high μ measurements seem to come exclusively from GCPEX and IMPACTS. In MC3E, OLYMPEX, and IPHEX, μ generally stayed below 0 at all temperatures. The Λ and μ are inversely proportional, as expected. The Λ approaches 0 as μ becomes close to −3, and the median Λ is set to −1.25 for μ = 0.

Figure 1 implies that a gamma function with a μ set to 1.25 would be the best average gamma function to represent frozen precipitation in models and retrieval algorithms. To demonstrate how well this function actually captures the shape of measured PSDs, Fig. 3 provides some selected PSDs matched to integer-rounded μ values ranging between −3 and 2. A μ = −1.25 gamma function, an exponential function, and a best-fit μ are plotted on top of each of these measured PSDs. In all cases, the two PSDs provide similar approximations of the measured PSD for D > 1 mm, albeit with slight changes of curvature. As expected, when μ is ≤ −1, the μ = −1.25 function provides a closer fit to the complete PSD than the exponential function, and vice versa for μ ≥ 0.

Fig. 3.
Fig. 3.

Examples of measured PSDs corresponding to six sequential μ values. Exponential, μ = −1.25, and best-fit gamma functions are overlaid as dashed lines.

Citation: Journal of Applied Meteorology and Climatology 61, 8; 10.1175/JAMC-D-21-0131.1

In some cases (e.g., Fig. 3b), the gamma fit to a function can provide a direct interpretation of PSD shape as some combination of a power law and an exponential distribution. More often, however, a PSD is complicated in some way that cannot be captured by a single gamma function. This is particularly common for PSDs with positive μ, which should be defined by some moderate positive curvature to follow the curvature of a gamma function. When such curvature is observed however, as in Fig. 3e, the curvature is too sharp to be described by its corresponding gamma distribution. More often, as in Fig. 3f, no positive curvature is observed. In these cases (and arguably in the case of Fig. 3d), the PSD follows a shallow slope up until D is around 1 mm, after which the PSD drops precipitously. In some cases, multimodel distributions with high concentrations of small particles are fit to gamma distributions with very low μ (Fig. 3a). The takeaway of these figures is that, even though the μ = −1.25 assumption provides a PSD that can reduce the χ2 error more than any other two-parameter gamma distribution, it should not be interpreted to represent the typical overall shape of measured size distributions. There is considerable room for improvement if one could find functions to conditionally fit these more precise PSD shapes.

b. Evaluation of the μ = −1.25 gamma fit for snowflake aggregate distributions

We compare the performance of the μ = −1.25 assumption with the exponential fit for all measured PSDs in Fig. 4. The two models are evaluated for their ability to recreate the ⟨D2⟩, ⟨D3⟩, and ⟨D4⟩ and the second-moment weighted mean diameter D2, with
Dn=Dn+1/Dn.
The ⟨D2⟩ is associated with the mass of ice aggregates, and ⟨D4⟩ is thus associated with the Rayleigh reflectivity (Smith 1984). The ⟨D3⟩ is not associated with any traditional property for frozen nonspherical particles, but it may be closer to the reflectivity for particularly dry particles at non-Raleigh wavelengths than ⟨D4⟩. The D2 is close to a characteristic average particle size, which is often utilized for its direct relation to Λ and relatively easy retrieval from multiple frequency reflectivity (Borque et al. 2019; Seto et al. 2013; Williams et al. 2014). The evaluation is presented as a cumulative distribution function of percentage errors between the fit and measured integrated PSD properties, and the distributions are capped at the 95th percentile for visibility purposes.
Fig. 4.
Fig. 4.

Evaluation of exponential and μ = −1.25 fit functions in recreating (a) ⟨D2⟩, (b) ⟨D3⟩, (c) ⟨D4⟩, and (d) D2.

Citation: Journal of Applied Meteorology and Climatology 61, 8; 10.1175/JAMC-D-21-0131.1

The μ = −1.25 fit function leads to improvements for estimated ⟨D2⟩ and ⟨D3⟩ but has little impact on ⟨D4⟩ or D2. The median percentage errors from the sensitivity tests of Fig. 4 corresponding to the exponential distribution are 11.3%, 21.1%, 5.5%, and 1.6% for ⟨D2⟩, ⟨D3⟩, ⟨D4⟩ and D2, respectively. The corresponding errors from the μ = −1.25 estimated properties were 7%, 10.8%, 6.3%, and 1.4% in the same order. Thus, the superexponential fit provides no significant improvement to Rayleigh reflectivity, implying that all of the information required for this property lies in the tail of the distribution that is similar among all gamma fit functions (Fig. 3). For non-Rayleigh reflectivity, which may be closer to ⟨D3⟩, however, the superexponential region of PSDs may have an impact on simulated reflectivity. This distinction could be especially important for PSD assumptions in orbiting W-band radars such as CloudSat.

For many modeling and retrieval purposes, complete PSDs are generated from single measurements, and in these cases a two parameter PSD will be underconstrained. Oftentimes, N0 will be fixed to a constant value to constrain one of the gamma parameters. We used an N0 of is 5.9 × 107 m−4 for our study. We generate fixed N0 parameterizations from the average of the log-transformed parameters in the database corresponding to μ = 0 and μ = −1.25 values. The fixed N0 corresponding to μ = 0 is 5.9 × 107 m−4, and the fixed N0 corresponding to μ = −1.25 is 4.1 × 103 m−4. One can also draw a relationship between Λ and N0 (Figs. 5b,d) such that a prescribed N0 can be more specifically tailored to a given size distribution. We draw equations in the form of N0 = 10aΛ+b from Figs. 5b and 5d. When μ is equal to 0, a and b are equal to 5.1 × 10−3 and 6.7, respectively. When μ is equal to −1.25, a and b are equal to 4.5 × 10−3 and 3.1, respectively. Relationships between N0 and T have also been suggested, but we find no correlation between the two in our dataset (Fig. 5a,c).

Fig. 5.
Fig. 5.

Measured sensitivity of N0 to (left) T and (right) Λ for gamma PSDs with μ set to (a),(b) 0 and (c),(d) −1.25.

Citation: Journal of Applied Meteorology and Climatology 61, 8; 10.1175/JAMC-D-21-0131.1

The results from the one-moment fit functions (Fig. 6) are similar to those from the two-moment functions in that the μ = −1.25 function provides superior estimates of ⟨D2⟩ and ⟨D3⟩ relative to the exponential distribution but the differences between the two are minor for ⟨D4⟩ and D2. There is some slight improvement from the Λ-dependent parameterization in recreating ⟨D4⟩, while the improvement is largely inconsequential for the second and third moments. As expected, the one-moment fit is less accurate than the two-moment fit.

Fig. 6.
Fig. 6.

As in Fig. 4, but evaluating single-moment fits; N0 is either prescribed to a constant value or as a function of Λ.

Citation: Journal of Applied Meteorology and Climatology 61, 8; 10.1175/JAMC-D-21-0131.1

One practical issue with a μ = −1.25 gamma function, and any gamma function with a negative μ for that matter, is that it leads to a divergent integral. In this study we have used the same Dmax from the measured PSD in closing our fit PSDs, but in retrievals the Dmax of a target PSD will be unknown. As we demonstrate in Fig. 7, however, there is a smooth function of the mean Dmax in Λ–μ coordinates, leading to a visible function of Λ and Dmax when we fix μ to −1.25. Unfortunately, we could not fit a satisfactory curve to the Λ and Dmax in Fig. 7b, so a lookup table of mean Dmax corresponding to different combinations of μ and Λ is provided as a “csv” file in the online supplemental material.

Fig. 7.
Fig. 7.

(left) Demonstration of Dmax, the maximum boundary for a PSD integration, as a function of μ and Λ. (right) Boxplots of Dmax associated with various Λ (note the units of Λ).

Citation: Journal of Applied Meteorology and Climatology 61, 8; 10.1175/JAMC-D-21-0131.1

4. Conclusions

In this study, we draw gamma PSDs from more than 4000 measured particle size distributions from five field experiments to determine the variability and best representation of typical PSDs in frozen precipitation. The experiments provided broad differences in storm type—such as the orographic winter storms over the Olympic mountains in OLYMPEX and the continental summer convective storms over Oklahoma in MC3E—as well as finer differences in cloud microphysics—such as temperatures ranging from 0° to −45°C. A μ of −1.25 was the most common incomplete gamma fit to measured PSDs, and 83% of all PSDs had μ < 0. The negative μ parameterizations contrast with most other studies on gamma parameterizations of frozen PSDs, and this is most likely a result of our deliberate exclusion of PSDs with Dmax < 4 mm. The μ = −1.25 model provides more accurate estimates of ⟨D2⟩ and ⟨D3⟩ but has little impact on ⟨D4⟩ and D2 when single- and double-parameter fit functions are used. There is little appreciable difference between single-parameter fit functions using constant N0 or an N0(Λ). An N0(T) was considered, but we found that N0 and T had no correlation in our exclusive dataset of PSDs containing aggregates.

The μ was higher during the winter snowstorm experiments IMPACTS and GCPEX. We did not notice any substantial trends or deviations with respect to temperature or LWC. We found the expected strong relationship between μ and Λ, and we found that the Dmax of a PSD could be inferred for any size distribution based on the combination of these two variables. Thus, there should be no issue integrating an incomplete gamma PSD with a negative μ even if the function itself is not integrable as a complete gamma PSD.

One of the most surprising results of this study was how often the superexponential curvature of low-μ PSDs extended beyond the range of particle sizes that could be associated with small ice crystals, shattered particles, liquid water, or snowflake monomers. In these cases, commonly seen when μ was less than −2, it would appear that aggregating particles are growing along a power-law curve instead of an exponential distribution. Since the exponential distribution of snowflake aggregates is supposed to be a natural result of the aggregation process, these PSDs may represent special cloud processes that are leading to different statistical aggregation curves. We hope to investigate the specific causes of different PSDs shapes in future studies.

Acknowledgments.

The work described in this paper was performed at NASA’s Jet Propulsion Laboratory. This research was supported by NASA’s Advanced Information Systems Technology (AIST) program (NNH18ZDA001N). Publication support was provided by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.

Data availability statement.

All in situ measurements from GCPEX, MC3E, IPHEX, and OLYMPEX are available online (https://ghrc.nsstc.nasa.gov/pub/fieldCampaigns/gpmValidation), as are the IMPACTS data (https://ghrc.nsstc.nasa.gov/uso/ds_details/collections/impactsC.html).

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  • Wood, N. B., and T. S. L’Ecuyer, 2021: What millimeter-wavelength radar reflectivity reveals about snowfall: An information-centric analysis. Atmos. Meas. Tech., 14, 869–888, https://doi.org/10.5194/amt-14-869-2021.

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Supplementary Materials

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  • Fig. 1.

    Cumulative distribution function representing μ values from IPHEX, GCPEX, MC3E, IMPACTS, and OLYMPEX. The black dashed line represents the second quartile (median); red dashed lines represent the first and third quartiles. This CDF represents an equal amount of data randomly drawn from each experiment.

  • Fig. 2.

    Distributions of μ stratified (a) by experiment and by ranges of (b) Λ, (c) temperature, and (d) LWC.

  • Fig. 3.

    Examples of measured PSDs corresponding to six sequential μ values. Exponential, μ = −1.25, and best-fit gamma functions are overlaid as dashed lines.

  • Fig. 4.

    Evaluation of exponential and μ = −1.25 fit functions in recreating (a) ⟨D2⟩, (b) ⟨D3⟩, (c) ⟨D4⟩, and (d) D2.

  • Fig. 5.

    Measured sensitivity of N0 to (left) T and (right) Λ for gamma PSDs with μ set to (a),(b) 0 and (c),(d) −1.25.

  • Fig. 6.

    As in Fig. 4, but evaluating single-moment fits; N0 is either prescribed to a constant value or as a function of Λ.

  • Fig. 7.

    (left) Demonstration of Dmax, the maximum boundary for a PSD integration, as a function of μ and Λ. (right) Boxplots of Dmax associated with various Λ (note the units of Λ).

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