Retrieval of Normalized Gamma Size Distribution Parameters Using Precipitation Imaging Package (PIP) Snowfall Observations during ICE-POP 2018

Ali Tokay aJoint Center for Earth Systems Technology, University of Maryland, Baltimore County, Baltimore, Maryland
bNASA Goddard Space Flight Center, Greenbelt, Maryland

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Liang Liao cMorgan State University, Baltimore, Maryland
bNASA Goddard Space Flight Center, Greenbelt, Maryland

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Robert Meneghini bNASA Goddard Space Flight Center, Greenbelt, Maryland

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Charles N. Helms dEarth System Science Interdisciplinary Center, University of Maryland, College Park, College Park, Maryland
bNASA Goddard Space Flight Center, Greenbelt, Maryland

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S. Joseph Munchak bNASA Goddard Space Flight Center, Greenbelt, Maryland

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David B. Wolff eNASA Goddard Space Flight Center Wallops Flight Facility, Wallops Island, Virginia

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Patrick N. Gatlin fNASA Marshall Space Flight Center, Huntsville, Alabama

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Abstract

Parameters of the normalized gamma particle size distribution (PSD) have been retrieved from the Precipitation Image Package (PIP) snowfall observations collected during the International Collaborative Experiment–PyeongChang Olympic and Paralympic winter games (ICE-POP 2018). Two of the gamma PSD parameters, the mass-weighted particle diameter D mass and the normalized intercept parameter NW , have median values of 1.15–1.31 mm and 2.84–3.04 log(mm−1 m−3), respectively. This range arises from the choice of the relationship between the maximum versus equivalent diameter, D mxD eq, and the relationship between the Reynolds and Best numbers, Re–X. Normalization of snow water equivalent rate (SWER) and ice water content W by NW reduces the range in NW , resulting in well-fitted power-law relationships between SWER/NW and D mass and between W/NW and D mass. The bulk descriptors of snowfall are calculated from PIP observations and from the gamma PSD with values of the shape parameter μ ranging from −2 to 10. NASA’s Global Precipitation Measurement (GPM) mission, which adopted the normalized gamma PSD, assumes μ = 2 and 3 in its two separate algorithms. The mean fractional bias (MFB) of the snowfall parameters changes with μ, where the functional dependence on μ depends on the specific snowfall parameter of interest. The MFB of the total concentration was underestimated by 0.23–0.34 when μ = 2 and by 0.29–0.40 when μ = 3, whereas the MFB of SWER had a much narrower range (from −0.03 to 0.04) for the same μ values.

This article is included in the Global Precipitation Measurement (GPM): Science and Applications Special Collection.

© 2023 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: Ali Tokay, tokay@umbc.edu

Abstract

Parameters of the normalized gamma particle size distribution (PSD) have been retrieved from the Precipitation Image Package (PIP) snowfall observations collected during the International Collaborative Experiment–PyeongChang Olympic and Paralympic winter games (ICE-POP 2018). Two of the gamma PSD parameters, the mass-weighted particle diameter D mass and the normalized intercept parameter NW , have median values of 1.15–1.31 mm and 2.84–3.04 log(mm−1 m−3), respectively. This range arises from the choice of the relationship between the maximum versus equivalent diameter, D mxD eq, and the relationship between the Reynolds and Best numbers, Re–X. Normalization of snow water equivalent rate (SWER) and ice water content W by NW reduces the range in NW , resulting in well-fitted power-law relationships between SWER/NW and D mass and between W/NW and D mass. The bulk descriptors of snowfall are calculated from PIP observations and from the gamma PSD with values of the shape parameter μ ranging from −2 to 10. NASA’s Global Precipitation Measurement (GPM) mission, which adopted the normalized gamma PSD, assumes μ = 2 and 3 in its two separate algorithms. The mean fractional bias (MFB) of the snowfall parameters changes with μ, where the functional dependence on μ depends on the specific snowfall parameter of interest. The MFB of the total concentration was underestimated by 0.23–0.34 when μ = 2 and by 0.29–0.40 when μ = 3, whereas the MFB of SWER had a much narrower range (from −0.03 to 0.04) for the same μ values.

This article is included in the Global Precipitation Measurement (GPM): Science and Applications Special Collection.

© 2023 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: Ali Tokay, tokay@umbc.edu

1. Introduction

The particle size distribution (PSD) is a critical descriptor of falling snow and provides a means to derive the total concentration NT, the snow water equivalent rate (SWER), and the equivalent radar reflectivity Ze. The accuracy of PSD measurement depends on the accuracy of particle size and concentration measurements in a given volume. The bulk descriptors of falling snow also depend on the fall velocity measurements.

The Precipitation Imaging Package (PIP) is an instrument designed to measure size, concentration, and fall velocity of precipitating particles. The instrument has seen wide use since its introduction (Newman et al. 2009) and has been improved through several hardware and software upgrades by the instrument inventor, L. Bliven of NASA Wallops Flight Facility (Pettersen et al. 2020a). A direct comparison between the PIP-calculated and gauge- or stake-field-measured snow water equivalent (SWE) provides critical information on the accuracy of the PIP measurements. Figure 2a of von Lerber et al. (2017) shows excellent agreement in SWE calculated from the PIP versus a reference Pluvio2-200 weighing bucket gauge-based SWE within a double wind fence in Finland. The total accumulation calculated from the PIP was within 5% for three different particle mass calculation methods. Pettersen et al. (2020b) demonstrated that the PIP-derived bulk liquid water equivalent accumulation was within 2.5% of that from a collocated snow stake field over several snowfall events. Tokay et al. (2022) reported less than 15% absolute bias between the PIP-calculated and stake field measured SWE for each of 30 snow events. The Pettersen et al. (2020b) and Tokay et al. (2022) studies were both conducted using the PIP dataset from Marquette, Michigan, but used different methods for estimating SWE from the snow particle measurements. This study uses the method described in Tokay et al. (2022) for the snowflake mass calculation.

The three-parameter gamma distribution has been widely used to parameterize the PSDs from aerosol and cloud droplets to liquid and frozen precipitation particles (Petty and Huang 2011 and references therein). The precipitation retrieval algorithms used for the NASA Global Precipitation Measurement (GPM) mission Core Observatory satellite active and passive sensors are no exception (Skofronick-Jackson et al. 2017). The GPM dual-frequency precipitation radar (DPR) and combined radar–radiometer algorithms have adopted a normalized gamma particle size distribution defined by mass-weighted particle diameter Dmass, normalized intercept parameter NW, and the shape parameter μ (Grecu et al. 2016; Seto et al. 2021). The shape parameter is assumed to be constant with μ = 2 for the combined radiometer–radar algorithm (Grecu et al. 2016) and μ = 3 for the DPR algorithm (Iguchi et al. 2017).

The retrieval of Dmass and NW from DPR algorithms involves six different modules that were shown by a flowchart in Fig. 2 of Seto et al. (2021). One of modules, the PSD module, uses a power-law relationship between the precipitation rate (PR) and Dmass as a constraint for the size distribution. The coefficient and exponent of the PR(Dmass) relationship rely on the coefficient and exponent of the k(Ze) relationship, where k is the specific attenuation and the relationship was calculated for Ku band (Kozu et al. 2009). The k(Ze) relationship was derived for the analysis of data from the Ku-band radar onboard the Tropical Rainfall Measuring Mission (TRMM) satellite, the predecessor of GPM, and was based on disdrometer measurements in the tropics. Under the GPM ground validation program, Chase et al. (2020) revisited the PR(Dmass) relationship using the PIP snowfall observations from Finland and found that the DPR PR(Dmass) relationship is not optimum for snowfall. Note that the PR(Dmass) relationship is based on the melted mass–weighted particle diameter Dmass.

The probability distributions of Dmass, log(NW), and μ in rain (Tokay and Bashor 2010; Tokay et al. 2016, 2017) and the two-dimensional histograms of log(NW) versus Dmass (Liao et al. 2020; Tokay et al. 2020) have been derived through the use of disdrometer measurements. Despite these studies, there are a few observational studies focusing on the PSD parameters of the falling snow. Using the PIP PSD observations from NASA Wallops Flight Facility, Wallops Island, Virginia, Liao et al. (2016) concluded that the exponential PSD model (μ = 0) is sufficiently accurate for the dual-wavelength radar retrieval of snow bulk parameters. Using the PIP PSD observations from Marquette, Michigan, Pettersen et al. (2020a) and Kulie et al. (2021) showed noticeable differences in the two-dimensional histogram of the intercept and slope parameters of exponential PSD between the deep and shallow snow events. Yu et al. (2020) presented the probability distributions of Dmass and log(NW) at different snow density intervals using the Particle Size Velocity (PARSIVEL2) disdrometer measurements collected during the International Collaborative Experiment–PyeongChang Olympic and Paralympic winter games (ICE-POP 2018) in South Korea. The distributions of Dmass and log(NW) shifted toward lower and higher values, respectively with increasing snow density. It should be noted that the snow density refers to the bulk density of falling snow within a volume, not the density of the individual snowflakes.

This study uses the PIP observations from the ICE-POP 2018 field campaign to derive the parameters of the normalized gamma PSD in snow. It has two main objectives: 1) To provide a reference for the range of Dmass and log(NW) observations in snow and their dependence on snow density ρsnow, Ku-band reflectivity Zeku, and dual-frequency ratio (DFR). 2) To evaluate the performance of the prescribed μ of the gamma PSD, particularly for μ = 2 and 3. The observation-based database makes this study original and complements the Liao et al. (2016) study, which is primarily based on theoretical calculations. The Dmass and log(NW) are retrieved from the GPM DPR algorithm and the library of the expected statistics of these parameters from in situ observations in snow is evidence of the practical significance of the study.

The paper is organized as follows. The instrumentation and the database are summarized in section 2. Section 3 is dedicated to the method, with the calculation procedure and formulations for particle mass and bulk descriptors of snowfall laid out. As mentioned, the method of this study is based on Tokay et al. (2022), and critical points are repeated here for the readers’ convenience. The results can be found in section 4, followed by a summary and conclusions in section 5.

2. Observations

a. Measurement site

ICE-POP 2018 is one of the major recent efforts to characterize the winter weather in complex terrain (Tapiador et al. 2021). A number of ground sites were located along the Pacific coast of South Korea and inland in the mountainous regions. The inland sites gathered detailed measurements within complex terrain. The PIP sites, for example, were at 175 and 789 m above the mean sea level. The precipitation at the lower elevation site was dominated by liquid and mixed phase, while the higher elevation site received abundant snowfall over a wide range of cold temperatures (0° ≥ Twb ≥ −13°C, where Twb is wet-bulb temperature). This study uses the PIP observations from the higher-elevation site, the MayHills supersite (MHS; 37.6652°N, 128.6996°E). The MHS had a wealth of in situ and remote sensing instrumentation including a W-band profiler, a Multi-Angle Snowflake Camera (MASC), a two-dimensional video disdrometer (2DVD) (Gehring et al. 2021), and a PARSIVEL2 disdrometer (Yu et al. 2020).

b. Instrumentation

This study primarily uses PIP to calculate the particle mass, PSD, and bulk descriptors of snowfall (Fig. 1a). The Vaisala WXT520 weather station provided additional measurements needed for the computation of the particle mass and the determination of snowing minutes (Twb ≤ 0°C) (Fig. 1b). A third key instrument in this study was the PARSIVEL2, a present weather sensor that is used here to verify the snowing minutes determination made by the WXT520. An important limitation of the PARSIVEL2 is that the instrument sometimes reports drizzle in the presence of light snow. This limitation, however, should have minimal impact on this study. PARSIVEL2 measures the size and fall velocity of snowflakes but the size cannot be interpreted as the equivalent diameter of the complex shape of snowflakes. The measurement uncertainty of PARSIVEL2’s fall velocity was previously reported by Tokay et al. (2013) in rain and the issue is valid for snowfall. In short, PIP is superior to PARSIVEL2 for particle size and fall velocity measurements in snow. Here, the working principles, measured quantities (e.g., size; fall velocity), and data output of PIP are briefly described, following closely the description given in Tokay et al. (2022).

Fig. 1.
Fig. 1.

Photographs of the (a) PIP and (b) Vaisala WXT520 weather station from the MHS (courtesy of K. Kim of Kyungpook National University).

Citation: Journal of Applied Meteorology and Climatology 62, 5; 10.1175/JAMC-D-21-0266.1

PIP (Pettersen et al. 2020b, 2021) is a digital video disdrometer developed as a successor to the Snowflake Video Imager (SVI; Newman et al. 2009). The instrument is composed of a high-speed video camera and a light source located 2 m apart with a focal plan located approximately 1.3 m from the camera lens. This open sample volume setup makes PIP measurements highly resistant to contamination from secondary particles generated by shattering as well as to undersampling in windy conditions. Although windy conditions can introduce motion blurring, the relative impact on the particle measurements has been found to be minor (Helms et al. 2022). Data are collected in the form of two-dimensional grayscale video at a rate of 380 frames per second with an image size of 640 by 480 pixels. Given a calibrated pixel size of 0.1 mm by 0.1 mm, the image represents a 64 mm by 48 mm field of view (FOV). Since particle that intersect the edge of this FOV are ignored, however, the actual sampling region FOV will be smaller than the full 64 mm by 48 mm FOV. The complete sampling volume will be a multiplication of this smaller particle-size-dependent FOV, a particle-size-dependent depth of field (DOF), and the number of frames over a given time period.

Although the calibrated pixel size for PIP is 0.1 mm by 0.1 mm, the instrument uses custom image compression to handle the high data rates produced by the high-speed camera. This image compression works by averaging vertically adjacent pairs of pixels prior to transmission and assigning that average to both of the pixels composing the pair after transmission. As a result, the effective pixel size is 0.1 mm by 0.2 mm when considering resolution effects. To this end, only particles with an equivalent diameter Deq of at least 0.2 mm are considered to prevent camera resolution and image compression issues from contaminating the size distributions. While Newman et al. (2009) reported the standard sizing error of SVI to be approximately 18%, the PIP setup uses a DOF that is one-half the size of the SVI DOF to reduce the uncertainty in the size measurements.

Generating particle measurements from the PIP images is handled via the National Instruments Image Acquisition (IMAQ 2004) software package. One of the major improvements from SVI to PIP is that PIP is able to measure the fall velocity of individual particles (Pettersen et al. 2020b) by using the IMAQ object tracking software to determine particle motion over the span of multiple images.

In addition to particle motion, the IMAQ software package also produces multiple measures of particle shape and size. Recent work (Helms et al. 2022) has found that PIP estimates of particle maximum dimension Dmx, which are computed by fitting an ellipse to the PIP-imaged particle, are unreliable due to the specific method the software uses to produce the ellipse fit. The PIP ellipse-fitted method that is currently (as of PIP software revision 1506) used performs the shape fitting by defining the fitted ellipse to have equal perimeter and area to the observed particle; no information about the pixel spatial distribution within the particle (i.e., the particle shape) is considered when making this fit. Although Helms et al. (2022) demonstrated that the data could be reprocessed using a more reliable shape fitting algorithm, reprocessing the data is beyond the scope of the present study. As such, the present study will estimate an appropriate Dmx from the Waddel disk diameter.

The Waddel disk diameter Deq is defined as the diameter of the disk with the same area as the particle. It is equal to 2(area/π)0.5, where the area is the total area of the shadowed pixels bounded by a box (see Table 10–1 of the IMAQ vision concepts manual; IMAQ 2004). The Dmx is expressed as a function of Deq as described in the method section. Figure 1b of von Lerber et al. (2017) shows the schematic image of a plane projection of a PIP-observed snowflake, as viewed from the side, with annotations indicating Deq and Dmx.

The PIP velocity files are generated independently of the particle files that are used to construct the PSD. They include the time stamp of Deq and the fall velocity of the particles. While most particles have positive fall velocities, it is feasible that the particle may have a negative fall velocity, indicating upward motion. To exclude upward-moving particles, only particles with a downward vertical velocity greater than 0 m s−1 are included in the analysis.

c. Database

This study uses a composite of 10 snow events over the Korean Peninsula that were recorded by the PIP during the first three months of 2018. The events are the combination of cold low (4 events), warm low (5 events), and air–sea interaction (1 event) events described in Kim et al. (2021). The PSD characteristics of air–mass interaction event was in line with the warm low events. The combined snow events resulted in a geometric snow depth of 1524 mm (60 in.) over 8230 min.

3. Method

This section describes the procedure for determining the three parameters of the normalized gamma PSD from the PIP particle size, and velocity measurements. The choice of the prescribed DmxDeq relationship and the four different approaches of particle mass calculation play important roles in determining the PSD parameters.

a. Snowflake mass

The particle mass was calculated from PIP measurements by using the hydrometeor aerodynamics outlined by Böhm (1989, hereinafter BM). A similar approach was previously used by Szyrmer and Zawadzki (2010), von Lerber et al. (2017), and Tokay et al. (2022) and the four-step process that is used in this study follows that of Tokay et al. (2022). In brief, the Dmx is prescribed as a function of the Deq:
Dmx=αDeq,
where α = 1.1, 1.2, 1.3, and 1.4 following von Lerber et al. (2017) and recent work by Helms et al. (2022). First, for each 5-min interval, the PIP fall velocity measurements are binned into 0.2-mm-wide Dmx bins whose midpoints ranged from 0.1 to 19.9 mm. As was the case in Tokay et al. (2022), the median fall velocity υ of each bin, rather than the mean fall velocity of each bin, is used here because the fall velocity measurements do not necessarily follow a normal distribution within each bin. The fall velocity distributions deviate from normal distributions mainly due to the presence of higher fall speed outliers as described in Tiira et al. (2016).
Second, the median fall velocity and the corresponding midpoint of the Dmx bin are used to calculate the Reynolds number Re for each bin using the following equation:
Re=υDmxρa/η,
where ρa and η are the density of air and the dynamic viscosity, respectively, and are computed from observations collected by the Vaisala WXT520 weather station collocated at the site.
Third, the Best number X, also known as Davies number, is calculated from Re for each size bin and is expressed as
X=(δ02C01/24{[(4Re)1/2δ0+1]21})2,
where δ0, and C0 are the boundary layer thickness and the drag coefficient and are taken as 5.83 and 0.6, respectively, following Böhm (1989). Equation (3) is a generalized theoretical relationship between Re and X considering idealized spheroidal snowflake shapes. Mitchell and Heymsfield (2005, hereinafter MH) and Khvorostyanov and Curry (2005, hereinafter KC) offered an alternative Re–X relationship for a turbulence correction to increase the drag at very large Re in the presence of large aggregates. Szyrmer and Zawadzki (2010) provided a reformulation of the MH and KC relationships in the form of an eighth-order polynomial, and it is their reformulations that will be used in this study, although the relationships will still be referred to as MH and KC, respectively. Heymsfield and Westbrook (2010, hereinafter HW) sought a better agreement between the observed and theoretical fall velocities in the presence of pristine ice crystals and aggregates. They suggested δ0 = 8.0 and C0 = 0.35 in Eq. (3) and this has been employed as the fourth alternative Re–X relationship in this study. A detailed summary of the Re–X relationship for the frozen hydrometeors is given by Rahman and Testik (2020).
Fourth, the mass of a snowflake was calculated from Re–X relationship for each size bin and is expressed as
m=πη2X8gρa(AeA)t,
where g is the gravitational acceleration, Ae is the effective particle area, and A is the area of the smallest circle or ellipse that contains Ae. The area ratio (Ae/A), which is equal to (Deq/Dmx)2, is 0.69, 0.59, 0.51, and 0.44 for α = 1.2, 1.3, 1.4, and 1.5, respectively. The exponent t in Eq. (4) is equal to 0.25 for the BM, MH, and KC methods, and is 0.5 for the HW method.
Mirroring the approach of Tokay et al. (2022), the mass of a snowflake m has also been expressed as a function of Dmx via a power-law fit. The power-law fit was applied to m(Dmx) relationships using orthogonal least squares and is expressed as
m=amDmxbm.
The coefficient and exponent in Eq. (5) are fitted using the bin-median snowflake mass, in grams, and the Dmx bin midpoint value in millimeters. The fitting is only applied if there are at least eight bins that contained 10 or more data points each.

b. PSD

The PSD is defined as the number of particles per unit volume of air in a given Deq size interval. As the true sampling volume of the PIP depends on the particle size (recall that particles that touch the edge of the image are ignored) and sampling period, the DOF and FOV for a given particle size and the sampling period T over which the PSD is being computed will factor into the PSD calculation. As such, the PIP PSD for the ith size interval is formulated as
N(Deq,i)=1TNfΔDeq,ij=1N1DOFi,jFOVi,j,
where Nf is the number of frames and ΔDeq is the width of the PSD size bin. For this study, the typical value of T is 60 s; PIP captures 380 frames per second, providing the value of Nf; ΔDeq is 0.2 mm; the FOV in square meters is approximately FOV = 10−6(48 − Deq)(64 − Deq), where Deq is expressed in millimeters; and DOF in meters is approximately DOF = (0.117/2) Deq, with Deq again expressed in millimeters.
The three-parameter gamma distribution has widely been used in atmospheric sciences community and is expressed as
N(Deq)=N0Deqμexp(ΛDeq),
where N0, Λ, and μ are the intercept, slope, and shape parameters, respectively, and Deq is the volume equivalent diameter of falling snow, which is the same as the Waddel disk diameter described above. The concept of the normalized gamma size distribution was discussed and later adopted in the scientific community following the pioneering work by Testud et al. (2001) even though it was first introduced by Willis (1984). The normalization allows the use of physical quantities of size distribution such as total concentration NT, liquid water content W, and Dmass.
This study uses the normalized gamma size distribution with respect to W and Dmass and is expressed as
N(Deq)=NWf(μ)(DeqDmass)μexp[(μ+bm+1)DeqDmass],
where Dmass is expressed as
Dmass=DminDmaxm(Dmx)DeqN(Deq)dDeqDminDmaxm(Dmx)N(Deq)dDeq,
where Dmin and Dmax are the minimum and maximum particle size, respectively. Assuming Dmin = 0 and Dmax = ∞ and the mass–diameter relationship from Eq. (5), Eq. (9) becomes
Dmass=(μ+bm+1)Λ.
The f(μ) is a dimensionless quantity and is expressed as
f(μ)=(μ+bm+1)μ+bm+1Γ(μ+bm+1).
The NW is the normalized intercept parameter (m−3 mm−1) and is expressed as
NW=Wam*Dmass(bm+1),
where am*=amαbm (g mm−bm). The W (g m−3) is given as
W=DmaxDmaxm(Dmx)N(Deq)dDeq.
If Eq. (12) adopts the standard gamma function Eq. (7) with Dmin = 0 and Dmax = ∞ first, and then uses the conversion between Λ and Dmass in Eq. (10), it becomes
W=N0am*Γ(μ+bm+1)Λ(μ+bm+1)=N0am*Γ(μ+bm+1)(μ+bm+1)(μ+bm+1)Dmass(μ+bm+1).
Equations (10) and (14) show the transformations between the standard and normalized gamma distribution parameters, respectively.

c. Shape parameter μ

The shape parameter of the gamma PSD has a wide range due to the inherent variability of the size spectrum and the boundaries are determined based on the method used to calculate the distribution. McFarquhar et al. (2015) presented five different methods to determine the parameters of gamma distribution. The shape parameter ranged from 1.62 to 2.60 for a given sample PSD in their study. Tokay et al. (2016, 2017), on the other hand, determined the shape parameter by minimizing the root-mean-square difference between the observed and gamma PSD-based rain rate. They showed that the probability distribution of μ ranges from −2 to 20 with a mode between 2 and 4.

This study uses a different method. Rather than minimizing the difference between the observed and gamma PSD based bulk descriptors of snowfall, the mean fractional bias (MFB) between the PIP- and gamma PSD-based bulk descriptors of snowfall is calculated for −2 ≤ μ ≤ 10 range with an increment of 1. The MFB is given as
MFB=1Ni=1N(Pest,iPobs,i)Pobs,i,
where P is one of the bulk parameters of the snowfall such as NT, SWE or Ze and where Pobs is calculated from PIP observations and or Pest is computed from the gamma PSD parameters. The sample size N exceeds 6000–7000 depending on the choice of bulk descriptors of snowfall. The minimum sample size thresholds to derive the m(Dmx) relationship explained above contributed to the differences in the sample size in calculating the MFB. The standard deviation of the mean fractional bias was also included in the analysis.
This method not only provides the best μ for a given range but also provides the level of accuracy for the prescribed μ values as in the GPM DPR and combined algorithms. The bulk descriptors of interest are NT, SWER, Ze at Ku band, and DFR. These quantities are a function of different moments of the PSD and therefore are sensitive to the different intervals of the size distribution. The snowfall rate (SR), the intensity of falling snow, is also needed to calculate the bulk density, which is given as
ρsnow=ρwSWERSR,
where ρw is the density of water (g cm−3).
The total concentration NT (m−3) of the particles within a volume of air is expressed as
NT=DminDmaxN(Deq)dDeq.
The SR is a measure of the intensity of falling snow and is expressed as functions of equivalent particle volume, concentration, and fall velocity and is given in units of millimeters per hour for a given minute as follows:
SR=6π104DminDmaxυ(Dmx)Deq3N(Deq)dDeq.
The SWER is the liquid equivalent rate of falling snow and is expressed as functions of mass, fall speed, and concentration of the particles in a volume of air and is in units of millimeters per hour for a given minute:
SWER=3.6ρwDminDmaxm(Dmx)υ(Dmx)N(Deq)dDeq.
The Ze is the equivalent radar reflectivity factor and is expressed as a function of backscattering cross section σbscat and concentration of the particles in a volume of air. It is calculated for Ku-band and Ka-band wavelengths (λ = 2.17 and 0.86 cm) using Mie theory (Liao et al. 2013) and is formulated as
Ze(λ)=106λ4|KW|2π5DminDmaxσbscat(Dmx,λ)N(Deq)dDeq,
where |KW|2 is the dielectric constant of water and is 0.925 and 0.880 for Ku and Ka band, respectively.
The dual-frequency ratio of Ze at Ku band to Ze at Ka band is formulated as
DFR=10log10(Zeku/Zeka).
Note that the Mie scattering–based σbscat is underestimated at Ka band, in the presence of large particles (Deq > 5 mm) (Tyynelä et al. 2011) at low bulk densities (ρsnow < 0.05 g cm−3) (Liao and Meneghini 2022). This means DFR is overestimated under these conditions. In this study, the focus is the general impact on DFR with changes in the PSD parameters rather than a quantitative assessment. These qualitative changes remain the same irrespective of whether the sphere or spheroid model is adopted.

4. Results

The findings of this study are presented by 1) histograms of Dmass versus log(NW) and 2) mean fractional bias of the shape parameter of gamma PSD. These two sets of results address two objectives of the study defined above.

a. Histograms of Dmass versus log(NW)

This section presents two-dimensional (2D) histograms of Dmass and log(NW) for four different Re–X relationships and for different DmxDeq relationships based on PIP observations during ICE-POP 2018 (Fig. 2). The combined PIP observations created a rich dataset that is adequate for the purpose of this study. The event-to-event variability of the 2D histograms of Dmass and log(NW) for a given Re–X and DmxDeq relationship (not shown) demonstrates the diversity of the snowfall characteristics though not necessarily the full diversity of global snowfall.

Fig. 2.
Fig. 2.

Two-dimensional histogram of Dmass vs log(NW) for four different DmxDeq relationships and four different Re–X relationships [(a)–(d) BM, (e)–(h) HW, (i)–(l) KC, (m)–(p) MH]. The color bar shown on the right side represents the percent occurrence for all 16 diagrams and is in a logarithmic scale.

Citation: Journal of Applied Meteorology and Climatology 62, 5; 10.1175/JAMC-D-21-0266.1

The 2D histograms of Dmass and log(NW) are generated as 40 × 40 matrices between 0 and 4 mm and between 1 and 5 log(mm−1 m−3) with a 0.1-unit increment for both parameters, respectively. The general trend with high particle concentrations [log(NW) > 2.5 log(mm−1 m−3)] at low Dmass (<0.7 mm) and low particle concentrations [log(NW) ≤ 2.5 log(mm−1 m−3)] at high Dmass (>3.2 mm) was evident in all 16 histograms (Fig. 2). The peak concentrations were bounded between 0.9 and 1.5 mm for Dmass and between 3.0 and 3.5 log(mm−1 m−3) for log(NW) in these histograms. Similar trends were previously reported for rain using 2DVD observations (Fig. 4 of Tokay et al. 2020) and for snow using PARSIVEL2 observations (Fig. 5 of Yu et al. 2020) but the range of Dmass and log(NW) values were different in those studies.

The differences between the 16 histograms were minimal. The median and maximum Dmass were lower for the HW method among the Re–X relationships and the maximum Dmass was higher for 1.4Deq among the DmxDeq relationships (Table 1). In contrast, the median and maximum log(NW) were higher for the HW method among the Re–X relationships but there were no noticeable differences in the maximum log(NW) among the DmxDeq relationships (Table 2).

Table 1.

The Dmass statistics (mm). The minimum, median, and maximum values, respectively, for four different Re–X relationships (BM, HW, KC, and MH) and for different DmxDeq relationships are given. The sample size is 7799.

Table 1.
Table 2.

The NW statistics [log(m−3 mm−1)]. The minimum, median, and maximum values, respectively, for four different Re–X relationships (BM, HW, KC, and MH) and for different DmxDeq relationships are given. The sample size ranges between 6965 and 7471.

Table 2.

The dependency of the 2D histogram of Dmass and log(NW) on ρsnow, ZeKu, and DFR are shown for BM-based Re–X relationship and Dmx = 1.2Deq (Fig. 3). For bulk density, there is minimal dependency on log(NW) but there is a clear trend of increasing ρsnow with decreasing Dmass (Fig. 3a). For ZeKu, there is an increasing trend with both Dmass and log(NW) (Fig. 3b), which is consistent with the rain study of Tokay et al. (2020). As shown in Fig. 3c, the DFR increases monotonically with Dmass and, consistent with the definition of this quantity, is essentially independent of log(NW). The slight variations of DFR with respect to log(NW) are due to the averaging procedure where each pixel value is calculated for 0.2Dmass interval.

Fig. 3.
Fig. 3.

Two-dimensional distribution of Dmass vs log(NW) as a function of (a) mean bulk ρsnow, (b) Zeku, and (c) DFR for the Dmx = 1.2Deq and the BM Re–X relationship. Note the use of a logarithmic scale for the color bar in (a).

Citation: Journal of Applied Meteorology and Climatology 62, 5; 10.1175/JAMC-D-21-0266.1

An empirical relationship between the parameters of the gamma PSD has been suggested for retrieving the PSD parameters from dual-polarization and dual-frequency radar measurements (Zhang et al. 2003). For Dmass and log(NW), it is clear that there would be a large scatter for both parameters if an empirical relationship is derived between the two parameters. Liao et al. (2020) normalized the rain rate R by NW and obtained the following power-law relationship: R/NW=(1.588×104)Dmass4.706. This relationship is virtually unchanged for the four different μ values (0, 3, 6, and 10). The additional information, R in this case, substantially reduced the scatter about Dmass.

This study normalizes both SWER and W by NW. Both SWER and W are functions of particle mass, which is calculated based on fall velocity measurements. The SWER is also a function of fall velocity as shown in Eq. (19). Unlike in rain, there is no single relationship between the fall velocity and particle size for snow and there is a wider variability in fall velocity for a given size depending on particle habit and composition. For the pair of BM Re–X relationship and Dmx = 1.2 Deq, the following expression was found between SWER/NW and Dmass using an orthogonal distance regression (Boggs et al. 1992):
SWER/NW=(4.030×104)Dmass2.598.
The spread in SWER/NW is wide, reaching 24-gate spacing in SWER/NW at Dmass = 1 mm. Despite the scatter, the power-law fit was representative of most of the observations (Fig. 4a). The distribution of SWER/NW split at Dmass > 4 mm. These two modes associate with the cold and warm low events. The merged warm low and air–sea interaction database is associated with the higher segment, while the lower segment is associated with the merged cold low database (not shown). High bulk density observations coincided with the low Dmass and high SWER/NW region, while the high Dmass and low SWER/NW region coincide with the low bulk density observations (Fig. 4b). High ZeKu and high DFR observations were found in high SWER/NW and in high Dmass regions, respectively (Figs. 4c,d).
Fig. 4.
Fig. 4.

Two-dimensional distribution of Dmass vs SWE/NW as a function of (a) percent occurrence, (b) mean bulk ρsnow, (c) ZeKu, and (d) DFR for Dmx = 1.2Deq and the BM Re–X relationship. The best-fit line between Dmass and SWER/NW following Eq. (22) is also shown. Note the use of a logarithmic scale for the color bar in (a).

Citation: Journal of Applied Meteorology and Climatology 62, 5; 10.1175/JAMC-D-21-0266.1

For the same pair of Re–X and DmxDeq relationships, the following expression was found between W/NW and Dmass using an orthogonal regression:
W/NW=(1.154×104)Dmass2.544.
The spread along the W/NW is relatively narrow, reaching 15-gate spacing in W/NW at Dmass = 1 mm, and the power-law fit represents the vast majority of the observations (Fig. 5a). Similar to the SWER normalization, there are two modes of W/NW distribution at Dmass > 4 mm (Fig. 5a) and low ρsnow observations coincide with the high Dmass and high W/NW region (Fig. 5b). The high Zeku and DFR observations are found in the region of high W/NW and high Dmass, respectively (Figs. 5c,d).
Fig. 5.
Fig. 5.

As in Fig. 4, but for Dmass vs W/NW. The best-fit line between Dmass and W/NW following Eq. (23) is also shown.

Citation: Journal of Applied Meteorology and Climatology 62, 5; 10.1175/JAMC-D-21-0266.1

The coefficients and exponents of Eqs. (22) and (23) were slightly different for the other pairs of Re–X and DmxDeq relationships but relatively more sensitive to Re–X relationships used in this study. In particular, the two-dimensional histograms of SWER/NW and Dmass as well as W/NW and Dmass in HW Re–X relationship had relatively more compact range for both variables. The spreads along the SWER/NW and W/NW were 22 and 14 gate spacing, respectively. The coefficients and exponents in Eqs. (22) and (23) are lower when HW Re–X relationships are considered.

b. Performance of shape parameter μ

This section presents the characteristics of μ of normalized gamma PSD with respect to NW and Dmass. The objective is to address two questions: 1) What is the μ within the expected range −2 ≤ μ ≤ 10 that best satisfies the four different bulk parameters of snowfall (NT, SWER, ZeKu, and DFR)? 2) What is the error if μ, for snow, is prescribed as it has been in the GPM combined radiometer radar (μ = 2) and DPR (μ = 3) algorithms? For the GPM algorithms, ZeKu and DFR are retrieved from measured reflectivity after the attenuation correction, while NT and SWER are the retrieved from PSD parameters. The retrieval of PSD parameters, Dmass and NW, was shown in a flowchart of DPR algorithm in Fig. 2 of Seto et al. (2021). The MFB described in Eq. (15) is used for answering these two questions. To be consistent with the previous section, Fig. 6 is plotted for the combination of Dmx = 1.2Deq and BM Re–X relationship. The standard deviation of the MFB is also included in the analysis. The trend of MFB with μ is the same for the other Re–X and DmxDeq relationships, which are included in the discussion.

Fig. 6.
Fig. 6.

MFB of (a) NT, (b) SWER, (c) ZeKu, and (d) DFR. The bias is the difference between when these integral parameters were calculated from PIP observed PSD and from gamma PSD parameters with prescribed shape parameter μ. The standard deviation of the MFB is shown as vertical bars, where the filled circle represents the mean.

Citation: Journal of Applied Meteorology and Climatology 62, 5; 10.1175/JAMC-D-21-0266.1

The μ = 0 had the best performance for all 16 pairs of Re–X and DmxDeq relationships when NT is the bulk parameter. The NT was underestimated when μ ≥ 0 while it was overestimated when μ < 0. When μ = 0, the MFB of NT was −0.05 and ranged from 0.18 at μ = −2 to 0.51 at μ = 10 in Fig. 6a. For all 16 pairs of Re–X and DmxDeq relationships, the range of MFB of NT was 0.23–0.33 when μ = 2 and by 0.30–0.40 when μ = 3. The combination of HW Re–X and 1.4Deq DmxDeq relationship had the lowest MFB, while the highest MFB was observed for the combination of MH Re–X and 1.1Deq DmxDeq relationship. Since the GPM algorithms derive the PSD parameters from Ze measurements, it is expected to have a large error in NT that is proportional to the zero moment of the size distribution. This study is able to quantify the error and provide a reference to the algorithm developers.

The μ = 3 had the best performance for all 16 pairs of Re–X and DmxDeq relationships when SWER is the bulk parameter. The SWER was underestimated over the entire range of μ and the absolute value of the MFB was less than 0.04 when μ ≥ 1 for all 16 pairs of Re–X and DmxDeq relationships. The MFB of SWER was −0.03 at μ = 3 and was substantially lower for μ < 0 in Fig. 6b. For μ = 2 and 3, the MFB of SWER was between −0.02 and −0.03 for all 16 pairs of Re–X and DmxDeq relationships. This suggests that the dependence of SWER on μ is small in general and can be neglected as a source of significant error in the GPM algorithms.

The μ = −1 had the best performance for all 16 pairs of Re–X and DmxDeq relationships when Zeku is the bulk parameter. Among the four bulk parameters, ZeKu had the least sensitivity to μ where the difference in MFB was ≤ 0.02 for −2 ≤ μ ≤ 10. ZeKu was underestimated for the entire μ range. The MFB of ZeKu was −0.14 at μ = −1 in Fig. 6c and this was the highest among the four bulk parameters presented here. The MFB of ZeKu was −0.19 at μ = −2 and its absolute value first decreased and then increased for μ ≥ 0, reaching to 0.27 at μ = 10 in Fig. 6c. For μ = 2 and 3, the MFB of ZeKu ranged from −0.19 to −0.21 and from −0.21 to 0.2, respectively.

The μ = −2 had the best performance for all 16 pairs of Re–X and DmxDeq relationships when DFR is the bulk parameter and the MFB at this μ value ranged from <−0.01 to −0.06. Unlike the other bulk parameters, the minimum and maximum MFB of DFR occurred for HW Re–X and 1.3Deq DmxDeq relationship and MH Re–X and 1.4Deq DmxDeq relationship, respectively. The DFR was slightly underestimated with MFB of −0.01 at μ = −2 but the underestimation increased substantially with the shape parameter reaching to MFB of −0.65 at μ = 10 in Fig. 6d. For μ = 2 and 3, the DFR was substantially underestimated with the range of MFB between −0.48 and −0.53, and between −0.52 and −0.57, respectively. This finding shows that the DFR is the least accurately retrieved at μ = 2 and 3.

5. Summary and conclusions

The PIP observations from ICE-POP 2018 field campaign were used to derive the parameters of the normalized gamma PSD in snow. The objectives of this study are to provide a reference for the range of Dmass and log(NW) observations in snow and to evaluate the performance of the prescribed μ of the gamma PSD.

This study showed that the Dmass ranged between 0.46 and 4.47 mm with median values of 1.15–1.31 mm depending on the pair of Re–X and DmxDeq relationships. It is important to note that, Dmass represents the actual snow particles diameter rather than the melted equivalent diameter. The range of log(NW) was from 1.41 to 4.39 log(m−3 mm−1), with median values of 2.84–3.03. The ρsnow increased with decreasing Dmass and was nearly insensitive to the log(NW). Higher values of ZeKu were found in the high Dmass and high log(NW) region and lower ZeKu values were found in the low Dmass and low log(NW) region. The DFR increased with Dmass and was insensitive to log(NW) as expected based on its definition.

The wide range of values in both Dmass and log(NW) demonstrate that there is no single relationship that can relate these two parameters accurately. However, using NW as a normalization of SWER and W results in power-law relationships of the form of SWE/NW=a1Dmassb1 and W/NW=a2Dmassb2, where the a1 and a2 coefficients and the b1 and b2 exponents were determined through orthogonal distance regression. The power-law relationships represent the majority of the observations where the spread in SWER/NW was higher than that in W/NW. This is caused by the fall velocity dependence of SWER where the fall velocity of snowflakes is highly variable even for a fixed size.

The optimum value of the shape parameter depends on the bulk parameter of interest. The retrieved parameters, NT and SWER yielded the best performance at μ = 0 and 2, respectively, while the measured parameters, ZeKu and DFR, had the best performance at μ = −1 and −2, respectively. All four bulk parameters were underestimated for the optimum μ except for DFR, which was 0.1%. The SWER had ≤4% underestimation for μ ≥ 0 but the MFB was as low as −0.22 at μ = −2. The ZeKu had 14% underestimation at its best μ, highest among the four bulk parameters and had the least sensitivity to the changes in μ. The NT and DFR showed the highest sensitivity across the μ spectrum. The range of MFB was 49% and 65% for those parameters, respectively. The sensitivity to the choice of DmxDeq and Re–X relationships was minimal for all four bulk descriptors of snowfall.

It is important to determine the error margins when μ is prescribed as 2 and 3 in GPM’s combined radiometer–radar and DPR algorithms, respectively. For these two μ values, the NT was underestimated by as high as 33% and 40%, respectively. In contrast, SWER was underestimated by just 3%, the least among the four bulk parameters. The ZeKu was underestimated by as high as 22% at μ = 2 and 23% at μ = 3, while the underestimation was 54% at μ = 2 and 57% at μ = 3 as being the highest in DFR.

Note that the bulk parameters presented here are directly derived from the PIP-based PSD observations. The errors in μ is therefore combination of uncertainty in PIP-based PSD measurements itself and deviations from gamma model size distribution. For the DPR algorithm, the PSD parameters, Dmass, and NW, are derived from reflectivity measurements that are functions of the higher moment of the PSD. For falling snow, there is a fairly robust relationship between the DFR and Dmass (not shown) but the scatter around the best fit line is also considerable and results in errors in the estimated Dmass.

The variability of PSD parameters in vertical is considered as a topic for future study. The PSD parameters can be derived from vertically pointing radars and compared with the PIP-based measurements. This exercise will provide guidance on the microphysical process (e.g., aggregation, accretion, sublimation) of falling snow between the cloud base and the ground.

Acknowledgments.

First and foremost, acknowledgments are given to Kwang Deun Ahn of the Korea Meteorological Administration and GyuWon Lee of Kyungpook National University for their leadership during ICE-POP 2018. Thanks are also given to Kwonil Kim of Kyungpook National University for field work and data collection during ICE-POP 2018. Specifically, the authors thank their South Korean colleagues for operating NASA’s Precipitation Imaging Package and providing the Vaisala WXT520 weather station database. Author Helms’s contribution to this study was supported by an appointment to the NASA Postdoctoral Program at NASA Goddard Space Flight Center, administered by Universities Space Research Association under contract with NASA. This study is funded through NASA’s Internal Scientist Funding Model for Precipitation Measurement Missions via Grant NNX16AE88G: Will McCarty, Program Manager (NASA Headquarters), and George Huffman, Project Scientist (NASA Goddard Space Flight Center).

Data availability statement.

The data used in this study was extracted though a secure file transformation site where the IP address, username, and password were set by the ICE-POP 2018 field campaign organizers. To access the database, please contact K. Kim (kwonil.kim.0@gmail.com).

APPENDIX

Variables Used in This Study

The variables here include environmental, snowflake mass, particle size distribution, and integral snowfall parameters. This appendix also includes variables that were mentioned but not used in this study.

Twb

Wet-bulb temperature

Deq

Equivalent diameter (a Waddel disk diameter, described in the IMAQ manual as the diameter of the disk with the same area as the particle: Deq = 2(area/π)0.5, where area is the total area of shadowed pixels bounded by a box

Dmin

Minimum particle size based on Deq measurements

Dmax

Maximum particle size based on Deq measurements

Dmx

Maximum dimension of the particle, prescribed as a function of Deq as in Eq. (1) in this study

Dmass

Mass-weighted mean diameter

Dmass

Melted mass–weighted particle diameter

N0

Intercept parameter of gamma particle size distribution

Λ

Slope parameter of the gamma particle size distribution

μ

Shape parameter of the gamma particle size distribution

NT

Total particle concentration, given as number of particles per cubic volume of air

W

Liquid water content

NW

Normalized intercept parameter of gamma size distribution with respect to W and Dmass

υ

Median fall velocity of a snowflake

m

Mass of a snowflake

am

Coefficient of the m(Dmx) power-law relationship

bm

Exponent of the m(Dmx) power-law relationship

Re

Reynolds number

η

Dynamic viscosity

ρa

Density of air at the ground level

X

Best number

δ0

Boundary layer thickness

C0

Drag coefficient

g

Gravitational constant

Ae

Effective particle area

A

The area of the smallest circle or ellipse that contains Ae

ρW

Density of water

ρsnow

Bulk density of falling snow

SR

Snow rate; this is the intensity of falling snow without melting

SWER

Melted equivalent snow rate

SWE

Melted equivalent of falling snow

R

Rain rate

PR

Precipitation rate

λ

Radar wavelength

σbscat

Backscattering cross section

|KW|2

Dielectric constant of water

ZeKu

Radar reflectivity at Ku-band wavelength

ZeKa

Radar reflectivity at Ka-band wavelength

DFR

Dual-frequency ratio

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  • Tokay, A., L. P. D’Adderio, D. B. Wolff, and W. A. Petersen, 2020: Development and evaluation of the raindrop size distribution parameters for the NASA Global Precipitation Measurement Mission ground validation program. J. Atmos. Oceanic Technol., 37, 115128, https://doi.org/10.1175/JTECH-D-18-0071.1.

    • Search Google Scholar
    • Export Citation
  • Tokay, A., A. von Lerber, C. Pettersen, M. S. Kulie, D. N. Moisseev, and D. B. Wolff, 2022: Retrieval of snow water equivalent by the Precipitation Imaging Package (PIP) over northern Great Lakes. J. Atmos. Oceanic Technol., 39, 3754, https://doi.org/10.1175/JTECH-D-20-0216.1.

    • Search Google Scholar
    • Export Citation
  • Tyynelä, J., J. Leinonen, D. Moisseev, and T. Nousiainen, 2011: Radar backscattering from snowflakes: Comparison of fractal, aggregate, and soft spheroid models. J. Atmos. Oceanic Technol., 28, 13651372, https://doi.org/10.1175/JTECH-D-11-00004.1.

    • Search Google Scholar
    • Export Citation
  • von Lerber, A., D. Moisseev, L. F. Bliven, W. Petersen, A.-M. Harri, and V. Chandrasekar, 2017: Microphysical properties of snow and their link to ZeS relations during BAECC 2014. J. Appl. Meteor. Climatol., 56, 15611582, https://doi.org/10.1175/JAMC-D-16-0379.1.

    • Search Google Scholar
    • Export Citation
  • Willis, P. T., 1984: Functional fits to some observed drop size distributions and parameterization of rain. J. Atmos. Sci., 41, 16481661, https://doi.org/10.1175/1520-0469(1984)041<1648:FFTSOD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Yu, T., V. Chandrasekar, H. Xiao, and S. S. Joshil, 2020: Characteristics of particle size distribution in the PyeongChang region of South Korea. Atmosphere, 11, 1093, https://doi.org/10.3390/atmos11101093.

    • Search Google Scholar
    • Export Citation
  • Zhang, G., J. Vivekanandan, E. A. Brandes, R. Meneghini, and T. Kozu, 2003: The shape–slope relation in observed gamma raindrop size distributions: Statistical error or useful information?. J. Atmos. Oceanic Technol., 20, 11061119, https://doi.org/10.1175/1520-0426(2003)020<1106:TSRIOG>2.0.CO;2.

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  • Fig. 1.

    Photographs of the (a) PIP and (b) Vaisala WXT520 weather station from the MHS (courtesy of K. Kim of Kyungpook National University).

  • Fig. 2.

    Two-dimensional histogram of Dmass vs log(NW) for four different DmxDeq relationships and four different Re–X relationships [(a)–(d) BM, (e)–(h) HW, (i)–(l) KC, (m)–(p) MH]. The color bar shown on the right side represents the percent occurrence for all 16 diagrams and is in a logarithmic scale.

  • Fig. 3.

    Two-dimensional distribution of Dmass vs log(NW) as a function of (a) mean bulk ρsnow, (b) Zeku, and (c) DFR for the Dmx = 1.2Deq and the BM Re–X relationship. Note the use of a logarithmic scale for the color bar in (a).

  • Fig. 4.

    Two-dimensional distribution of Dmass vs SWE/NW as a function of (a) percent occurrence, (b) mean bulk ρsnow, (c) ZeKu, and (d) DFR for Dmx = 1.2Deq and the BM Re–X relationship. The best-fit line between Dmass and SWER/NW following Eq. (22) is also shown. Note the use of a logarithmic scale for the color bar in (a).

  • Fig. 5.

    As in Fig. 4, but for Dmass vs W/NW. The best-fit line between Dmass and W/NW following Eq. (23) is also shown.

  • Fig. 6.

    MFB of (a) NT, (b) SWER, (c) ZeKu, and (d) DFR. The bias is the difference between when these integral parameters were calculated from PIP observed PSD and from gamma PSD parameters with prescribed shape parameter μ. The standard deviation of the MFB is shown as vertical bars, where the filled circle represents the mean.

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