Using snowflake surface-area-to-volume ratio to model and interpret snowfall triple-frequency radar signatures

The snowflake microstructure determines the microwave scattering properties of individual snowflakes and has a strong impact on snowfall radar signatures. In this study, individual snowflakes are represented by collections of randomly distributed ice spheres where the size and number of the constituent ice spheres are specified by the snowflake mass and surface-area-to-volume ratio (SAV) and the bounding volume of each ice sphere collection is given by the snowflake maximum dimension. Radar backscatter cross sections for the ice sphere collections are calculated at X-, Ku-, Ka-, and W-band frequencies and then used to model triplefrequency radar signatures for exponential snowflake size distributions (SSDs). Additionally, snowflake complexity values obtained from high-resolution multi-view snowflake images are used as an indicator of snowflake SAV to derive snowfall triple-frequency radar signatures. The modeled snowfall triple-frequency radar signatures cover a wide range of triple-frequency signatures that were previously determined from radar reflectivity measurements and illustrate characteristic differences related to snow type, quantified through snowflake SAV, and snowflake size. The results show high sensitivity to snowflake SAV and SSD maximum size but are generally less affected by uncertainties in the parameterization of snowflake mass, indicating the importance of snowflake SAV for the interpretation of snowfall triplefrequency radar signatures.

taneously at three microwave frequency bands has shown that the range of observed snowfall triple-frequency radar signatures is much larger than the total range of modeled snowfall radar signatures when representing snowflakes by soft spheroids; especially triple-frequency radar signatures of snowfall characterized by large aggregate snowflakes fall outside the modeled range (Leinonen et al., 2012;Kulie et al., 2014;Kneifel et al., 2015). Using detailed 3D shape models instead of soft spheroids leads to a wider range of modeled snowfall triple-frequency radar signatures and indicates better agreement between observed and 5 modeled snowfall radar signatures.
Due to the large variety of (visually distinct) snow types defined by characteristic geometric shapes resembling the snowflake microstructure, such as planar dendrites or aggregates of plates (Magono and Lee, 1966;Kikuchi et al., 2013;Fontaine et al., 2014), and the high natural variability of snowflake microstructural properties like size and aspect ratio (e.g., Brandes et al., 2007;Gergely and Garrett, 2016), modeling microwave backscatter in snowfall based on detailed snowflake 3D shape models 10 requires significant computational resources and time, e.g., when determining backscatter cross sections for a large number of snowflake models with the widely used discrete dipole approximation (Draine and Flatau, 1994). Therefore, it would be desirable to identify 'effective' microstructural parameters that quantify snowflake shape independent of snow type and still explain important features of observed and modeled snowfall radar signatures, thus further constraining snowflake shape for snowfall remote sensing. 15 In materials science, four basic characteristics play a central role for an objective and quantitative description of 3D microstructures: volume fraction or equivalently (mass) density, surface area per volume, integrated mean curvature per volume, and integrated Gaussian curvature per volume (Ohser and Mücklich, 2000). Physical and chemical properties strongly depend on these characteristics and can often already be analyzed faithfully when the 3D microstructure is quantified through all or some of these four characteristics. Ice volume fraction or snow density and the ratio of ice surface area to volume are 20 crucial for modeling light scattering and radiative transfer at optical wavelengths in falling and deposited snow, for example (Grenfell and Warren, 1999;Grenfell et al., 2005;Kokhanovsky and Zege, 2004;Picard et al., 2009;Gergely et al., 2010).
Besides snowflake density, however, none of these four basic characteristics have been investigated to evaluate the impact of snowflake shape on snowfall microwave scattering signatures.
In this study, snowflake density and surface-area-to-volume ratio (SAV) are used to model snowflake backscatter cross 25 sections at X-, Ku-, Ka-, and W-band frequencies and then derive snowfall triple-frequency radar signatures for realistic snowflake size distributions. The impact of snowflake SAV on snowfall triple-frequency radar signatures is analyzed based on high-resolution snowflake imaging data collected with the Multi-Angle Snowflake Camera (MASC; Garrett et al., 2012), a pre-established density-diameter relationship for deriving snowflake mass from snowflake maximum dimension according to Heymsfield et al. (2004), and the snowflake SAV range given by Honeyager et al. (2014). 30 First, MASC measurements are presented in Sect. 2. The applied method for modeling snowflake backscatter cross sections and snowfall triple-frequency radar signatures is described in Sect. 3. Individual snowflakes are represented by collections of ice spheres where the size and number of the constituent ice spheres are specified by the snowflake mass and SAV and the bounding volume of each ice sphere collection is defined by the snowflake maximum dimension. Backscatter cross sections of these collections of ice spheres are calculated with the generalized multiparticle Mie solution (Xu, 1995;Xu and Å. S. Gustafson, is described by a low snowflake complexity of χ ≈ 1 and large aggregate snowflakes are characterized by higher complexity values (see examples in Fig. 1). Lower χ values are then expected to indicate stronger snowflake riming in general (see also Garrett and Yuter, 2014, who used a definition of snowflake complexity which additionally included brightness variations within each MASC image to classify snowflakes according to their degree of riming).
One MASC was installed at Alta Ski Resort at 2590 m above sea level (a.s.l) in Collins Gulch within the Wasatch Mountain 5 Range. A second MASC was located at Barrow at the North Slope of Alaska Atmospheric Radiation Measurement (ARM) site at 10 m a.s.l., approximately 500 km north of the Arctic Circle on the coast of the Arctic Ocean. Figure 2 shows the distributions of snowflake diameter D, complexity χ, and orientation angle θ derived from all qualifying MASC observations with realistic complexity values of χ ≥ 1 that were collected at Alta from December 2013 to April 2014 10 and at Barrow in April and May 2014, resulting in a MASC data set of 4.4 · 10 5 sampled snowflakes. Snowflake size distributions N (D) are expressed as frequency size distributions and reflect the number of snowflakes sampled at Alta (4.3 · 10 5 ) and

Measurement results
at Barrow (10 4 ). For snowflake complexity and orientation, the presented relative distributions are normalized with respect to the maximum values N max (χ) and N max (θ) of the respective frequency distributions N (χ) and N (θ).
The distributions of snowflake diameters and complexities in Fig. 2   Similar to previous studies that have used exponential snowflake size distributions to describe snowfall (e.g., Matrosov, 2007;Kneifel et al., 2011), snowflake (frequency) size distributions N (D) [mm −1 ] in this study are expressed through 5 where Λ is the exponential slope parameter specifying the width of the distribution and N 0 [mm −1 ] denotes the scaling factor determined by the snowflake sample size. Commonly, N (D) and N 0 are additionally normalized with respect to atmospheric volume to account for the atmospheric snow water content, giving N (D) and N 0 in units of mm −1 m −3 . As the normalization of N (D) has no impact on the analyzed dual-wavelength ratios of modeled Z e in Sect. 4.2, the scaling factor N 0 is ignored in the analysis and exponential distributions are specified only through the exponential slope parameter Λ.
10 Exponential snowflake size distributions N (D) were fitted to MASC data restricted to D > 1 mm and collected for 47 snowstorms at Alta and 7 snowstorms at Barrow. These snowstorms lasted between 4 h and 24 h, and 10 2 to 10 4 snowflakes were recorded during each snowstorm. Small sample sizes of 10 2 snowflakes correspond to snowstorms at Barrow marked by very low snowfall liquid equivalent of less than about 1 mm and by strong crosswinds that affected overall sampling efficiency.
Large sample sizes of up to 10 4 snowflakes correspond to high-intensity snowfall at Alta. For each snowstorm, the sampled 5 snowflakes were divided into 20 size bins according to their diameter D. An exponential snowflake size distribution N (D) was then determined by the non-linear least squares method for fitting Eq.
(1) to the binned snowflake size distribution.
For uniform visualization in Fig. 2, N (D) curves illustrating the total range of exponential size distributions fitted to the MASC data from Alta and from Barrow were rescaled to the total number of snowflakes sampled at the respective location.
At Alta, N (D) are characterized by exponential slope parameters of Λ min = 0.5 mm −1 ≤ Λ ≤ Λ max = 3.1 mm −1 with mean 10 Λ = 1.2 mm −1 . At Barrow, the range of N (D) is given by 0.6 ≤ Λ ≤ 3.6 mm −1 with mean Λ = 1.5 mm −1 . The derived exponential slope parameters yield snowflake size distributions N (D) that are in line with previously presented snowflake size distributions using different measurement methods, e.g., by Brandes et al. (2007) and Tiira et al. (2016), with their reported median volume diameters D 0 of the derived snowflake size distributions converted to Λ = 3.67/D 0 for N (D) given by Eq. (1).
For each analyzed snowstorm, the sampled snowflakes were also divided into 20 bins according to their complexity χ, 15 and an exponential snowflake complexity distribution N (χ) = N 1 exp(−Λχ) was fitted to the binned distribution by the nonlinear least squares method. At Alta, the range of N (χ) is characterized by exponential slope parameters of Λ min = 1.5 ≤ Λ ≤ Λ max = 5.5 with mean Λ = 3.3. At Barrow, a range of 1.2 ≤ Λ ≤ 5.0 is found with mean Λ = 2.2 (see Fig. 2).
To illustrate the correlation between snowflake diameter D and complexity χ, Fig. 3 shows a logarithmic 2D histogram of the frequency distributions for D and χ at Alta and Barrow (Fig. S1 in the Supplement shows the corresponding non-20 logarithmic 2D histogram). Calculated mean complexity values χ per size bin ∆D are shown separately for both MASC data sets collected at Alta (χ/∆D given for D ≤ 15 mm) and at Barrow (χ/∆D given for D ≤ 10 mm) to indicate typical snowflake complexities at the two locations. Despite the skewed distribution of χ within the size bins, the choice of whether typical snowflake complexities are quantified through the mean or through the median complexity per size bin has only a minor influence on the derived results in this study and does not affect the drawn conclusions. 25 As already seen in Fig. 2, small values of D and χ dominate the distributions in Fig. 3. Additionally, mean complexity χ generally increases with increasing snowflake diameter. Notably, snowflake complexities of χ = 1.0 are not observed for snowflake diameters of D 3 mm. These results are consistent with previous observations suggesting that larger snowflakes are generally aggregates characterized by a high complexity of the snowflake microstructure (Garrett and Yuter, 2014).
Based on the mean snowflake complexity values χ per size bin ∆D shown in Fig. 3, a snowflake complexity-diameter 30 relationship is then defined by a modified power law as with parameters a and b. Power laws have been applied to parameterize a variety of snowflake properties y with respect to snowflake size, illustrated by the density-diameter relationship in Eq. (3), for example. In Eq.
(2), the constant of χ 0 = 1 is  large snowflakes and thus follow the observed increase in χ with increasing snowflake diameter, but also reflect the observed convergence of χ → 1 for small snowflakes. Furthermore, the mean complexity values χ per size bin ∆D and the two derived χ(D) curves generally indicate lower snowflake complexities (at a given snowflake diameter) for the MASC data recorded at Alta.

Modeling method 10
In this study, snowflakes are specified by their diameter, mass, and surface-area-to-volume ratio (SAV). Snowflake diameters were derived from a large set of MASC observations in Sect. 2. In Sect. 3, the quantification of snowflake mass and SAV is described, and the approach for modeling snowflake backscatter cross sections and snowfall triple-frequency radar signatures is presented.

Snowflake mass
No coincident measurements of snowflake mass are available for the analyzed MASC data in Sect. 2. Therefore, snowflake mass is derived from measured snowflake diameter D following a previously determined density-diameter relationship that uses a similar definition of snowflake diameter , abbreviated as 'H04' throughout the text). H04 determined effective ice-cloud particle densities by combining observations by airborne 2D optical array probes with coincident 5 measurements of cloud ice water content. According to their results, snowflake density ρ f [g cm −3 ] and mass m f [mg] are calculated from snowflake maximum dimension D [mm] for a spherical snowflake bounding volume V f of diameter D: and 10 Here, derived ρ f (D) values are limited to the density of pure ice ρ ice = 0.917 g cm −3 , leading to densities of ρ f (D) = ρ ice for snowflakes with D ≤ 0.1 mm.
With Eqs. (3) and (4), snowflake mass m f can alternatively be expressed through the radius r eq of a single mass-equivalent ice sphere given by

Snowflake surface-area-to-volume ratio
The normalized snowflake surface-area-to-volume ratio ξ is defined as the ratio of snowflake surface-area-to-volume ratio SAV f to the surface-area-to-volume ratio SAV s of a mass-equivalent ice sphere: This definition of ξ for snowflake 3D microstructures is conceptually similar to the definition of complexity χ applied to 25 snowflake 2D projection images in Sect. 2. For a given ice volume or mass, an ice sphere has the smallest surface area of any 3D microstructure and a surface-area-to-volume ratio of SAV f = SAV s = 3/r with ice sphere radius r, leading to a normalized SAV of ξ = 1. Increasing values of ξ > 1 then imply a larger deviation of the snowflake shape from an ice sphere, and thus an increasing complexity of the snowflake 3D microstructure.  Snowflake SAV is quantified from the total range of ξ values determined by Honeyager et al. (2014). They used a Voronoi cell-based approach to define an effective SAV by Eq. (6) for their database of snowflake 3D shape models and found values of 1 ≤ ξ ≤ 5.
The impact of snowflake SAV on snowfall radar signatures is analyzed based on synthetically generated expressions ξ(D). surface-area-to-volume ratio affects modeled snowfall triple-frequency radar signatures in Sect. 4.2 focuses on these constant ξ(D). Nonetheless, non-constant ξ(D) given by Eq. (7) with exponents q > 0 are included in the analysis to outline the total range of modeled snowfall triple-frequency radar signatures and to establish a relationship between normalized snowflake surface-area-to-volume ratio ξ and snowflake complexity χ that reflects the similarity of these two characteristics and can be applied to estimate ξ(D) relationships for the recorded MASC data at Alta and at Barrow.
The method for relating ξ to χ uses the two complexity-diameter relationships χ(D) fitted to the MASC data in Fig. 3.
To estimate ξ(D) relationships at Alta and at Barrow, it is assumed that the snowflake complexity range of 1 ≤ χ(D) ≤ χ(D max ) = χ max at each location corresponds to the full snowflake SAV range of 1 ≤ ξ ≤ 5 with After inserting Eq.
(2) for χ(D), Eq. (8) leads to a modified power law for ξ(D) given by Eq. (7), with power-law exponent of Only the scaling factor a in Eq.
High values of q ≫ 1 in Eq. (7) lead to ξ(D) relationships marked by a steep increase from ξ = 1 to ξ = 5 for large snowflake diameters (see Fig. 4), corresponding to a sudden change in snowflake shape from ice spheres to complex 3D shapes found for 10 aggregates, for example. This is an unrealistic description of snowflake shape because such an abrupt transition is not seen in snowflake observations. Figure 3 showed power-law exponents of b 1, leading to q 1 according to Eq. (8). Nonetheless, ξ(D) with q ≫ 1 are also included for completeness. Section 4.2 indicates that these ξ(D) relationships contribute only a small fraction to the total range of modeled snowfall triple-frequency radar signatures and do not affect the drawn conclusions.

15
Microwave backscatter by a snowflake is modeled at X-, Ku-, Ka-, and W-band frequencies of 10, 14, 35, and 94 GHz, respectively. Here, the (radar) backscatter cross section σ b is calculated for mass-and SAV-equivalent collections of non-overlapping ice spheres with the generalized multiparticle Mie (GMM) solution (Xu, 1995;Xu and Å. S. Gustafson, 2001). Calculated σ b values correspond to the differential scattering cross sections at backscatter multiplied by 4π (see Bohren and Huffman (1983) for a discussion on commonly applied conventions for expressing backscatter by a particle). The modeling approach is outlined 20 in Fig. 5 and described in this section.
A snowflake defined by the diameter D, the mass m f (D), and the normalized surface-area-to-volume ratio ξ is represented by a collection of ice spheres where the radius r cl and the number N cl of the constituent ice spheres are specified by m f , or equivalently by r eq through Eq. (5), and by ξ: The snowflake diameter D specifies the (spherical) bounding volume V f of the collection of ice spheres according to Eq. (4).
Limitations of this representation and implications for the modeling results presented in Sect. 4 are discussed in Appendix A.
Sets of 500 realizations were chosen for averaging because mean values of σ b (D; ξ) stabilize to within relative differences 10 of less than 0.1 once 10 1 to 10 2 collections of randomly distributed ice spheres are included (see Fig. S2 in the Supplement for details). These uncertainties in σ b (D; ξ) are small compared to the impact of ξ on modeled σ b (D; ξ), characterized by relative differences of up to a factor of 10 2 in Sect. 4.1. The presented methodology then quantifies the impact of normalized SAV on the calculated backscatter cross sections without including effects due to the spatial distribution or clustering of the N cl ice spheres inside the bounding volume V f .

15
To analyze the impact of snowflake surface-area-to-volume ratio on modeled backscatter cross sections for a given snowflake For comparison, the analysis also includes mass-equivalent soft (mixed ice-air) oblate spheroids and snowflakes modeled according to the self-similar Rayleigh-Gans approximation (SSRGA; Hogan and Westbrook, 2014;Hogan et al., 2017).
Backscatter cross sections of randomly oriented soft spheroids with major axis length D are calculated with the T-matrix 10 method (Waterman, 1971), using the implementation of Mishchenko and Travis (1998) within the PyTMatrix software package of Leinonen (2014). Aspect ratios of α = 1, 0.6, and 0.2 are considered, representing soft spheres with α = 1, spheroids that are characterized by typical average values of α = 0.6 found in snowflake observations (e.g., Korolev and Isaac, 2003;Gergely and Garrett, 2016) and used for the interpretation of snow-and ice-cloud radar measurements (Matrosov et al., 2005;Hogan et al., 2012), and spheroids described by extreme values of observed snowflake aspect ratios of α = 0.2. Effective re- tions of the snowflake 3D microstructure with the same snowflake diameter D, for 50 random orientations of each snowflake 3D shape model, and for then illuminating each of the reoriented 3D shape models along its three orthogonal directions. Here, the SSRGA is applied to snowflake masses derived by Eqs. (3) and (4) and for complex refractive indices n ice,λ of pure ice, using the parameterizations listed by Hogan et al. (2017) for synthetic aggregate snowflakes that were generated according to Westbrook et al. (2004), abbreviated as 'W04' throughout the text, and according to Nowell et al. (2013), abbreviated as 'N13'.

Snowfall triple-frequency radar signatures
In this study, snowfall triple-frequency radar signatures are defined by the two dual-wavelength ratios of modeled snowfall radar reflectivity factors at (i) Ka and W band and at (ii) either X and Ka band or Ku and Ka band, where X, Ku, Ka, and W band refer to frequencies of 10, 14, 35, and 94 GHz, respectively. The selected frequencies are within ±1 GHz of X-, Ku-, Ka-, and W-band frequencies commonly used for the analysis of snowfall triple-frequency radar signatures (e.g., Leinonen et al.,30 2012; Kulie et al., 2014;Kneifel et al., 2015Kneifel et al., , 2016Yin et al., 2017).
To derive snowfall triple-frequency radar signatures at X, Ka, and W band and at Ku, Ka, and W band, snowflake (radar) backscatter cross sections σ b modeled according to Sect. 3.3 are first integrated for exponential snowflake size distributions N (D) expressed through Eq. (1), yielding the corresponding snowfall (equivalent) radar reflectivity factors Z e (e.g., where n w,λ denotes to the complex refractive index of liquid water at wavelengths of λ = 30.0, 21.4, 8.6 and 3.2 mm for the analyzed frequencies of 10, 14, 35, and 94 GHz, respectively. Here, n w,λ is determined for pure water at a temperature of 0 • C following Meissner and Wentz (2004).

5
Snowfall triple-frequency radar signatures are then given by dual-wavelength ratios (DWRs, Kneifel et al., 2011) of where λ 1 /λ 2 indicate the pairs of analyzed radar frequency bands of X/Ka, Ku/Ka, and Ka/W. diameters of D ≤ 23.6 mm, for all considered snowflake models, and for frequencies of 10 and 14 GHz is included in Fig. S3 in the Supplement. 25 For soft spheres, Figs. 6 and S3 show strong resonances in calculated σ b typical for applying Mie scattering theory to large particles (Mie, 1908;Bohren and Huffman, 1983). The higher the frequency, and thus the larger the effective size of a spherical particle with diameter D relative to the wavelength, the more oscillations are observed within the total diameter range. Oscillations in σ b are heavily dampened for spheroids due to orientation averaging of σ b and for SSRGA results due to averaging over an ensemble of many different realizations of non-spherical snowflake shape models. Collections of randomly   distributed ice spheres inside the (spherical) snowflake bounding volume also lead to a much weaker oscillation pattern in σ b than soft spheres of diameter D because the refractive index n ice,λ of pure ice generally differs significantly from the effective refractive indices of soft spheres determined with the Maxwell-Garnett mixing rule (real and imaginary parts of soft-sphere effective refractive indices are smaller and thus closer to one and zero, respectively) and because the ice spheres are characterized by a radius of r cl ≪ D/2 and therefore by a much smaller effective size relative to the wavelength (see Sect. 3.3).
In Fig. 6, calculated backscatter cross sections σ b (D; ξ) for collections of 1 ≤ N cl = ξ 3 ≤ 125 randomly distributed ice spheres inside the snowflake bounding volume cover a maximum range of over 2 orders of magnitude for r eq ≈ 0.85 mm or D ≈ 6.3 mm at 35 GHz and for r eq ≈ 0.44 mm or D ≈ 2.4 mm at 94 GHz. Outside the Mie resonance regions, σ b (D; ξ) 5 decrease with increasing normalized surface-area-to-volume ratio ξ. This trend is consistent with results of Honeyager et al.
(2014) who found smaller backscatter cross sections for greater snowflake surface complexity when modeling microwave backscatter for their snowflake 3D shape models with the discrete dipole approximation.
A comparison of the σ b curves in Figs. 6 and S3 shows that differences in σ b associated with the choice of snowflake model generally increase with increasing snowflake diameter and microwave frequency. In Fig. 6, σ b curves can be distinguished Notably, snowfall triple-frequency radar signatures modeled according to the SSRGA for N13 and W04 snowflake parameterizations and snowflake size distributions truncated at D max = 10.0 mm in Sect. 4.2 show similar characteristic differences with respect to triple-frequency radar signatures modeled for collections of randomly distributed ice spheres and for soft spheres and spheroids as the differences found for snowflake size distributions spanning the total analyzed range of diameters 30 up to D max = 23.6 mm. Therefore, application of the two SSRGA snowflake parameterizations beyond the size range they were originally derived for by Hogan et al. (2017) is not expected to significantly affect the corresponding analysis results and conclusions in this study.  7). Colored rectangles are adapted from Kneifel et al. (2015) and roughly outline regions associated with the presence of large aggregate snowflakes (cyan) and rimed snowflakes (graupel; magenta) that were inferred by relating snowfall triplefrequency radar reflectivity measurements at X, Ka, and W band to coincident in situ snowflake observations. Corresponding triple-frequency radar signatures for snowflake size distributions limited to D ≤ 10.0 mm and to D ≤ 5.0 mm are shown in Figs. S7 and S8, respectively.

Snowfall triple-frequency radar signatures
An overview of the snowfall radar reflectivity factors Z e derived from the modeled snowflake backscatter cross sections in Sect. 4.1 is included in Fig. S4 in the Supplement but not discussed in this study. Snowfall triple-frequency radar signatures are shown in Fig. 7. For all considered snowflake models, using DWR Ku/Ka to quantify triple-frequency radar signatures in combination with DWR Ka/W leads to compressed triple-frequency curves by ∆DWR 3 dB compared to using DWR X/Ka.
Triple-frequency curves for soft spheres and spheroids with aspect ratios of α = 1, 0.6, and 0.2 in Fig. 7 are characterized by strictly increasing DWRs with decreasing exponential slope parameters Λ of the snowflake size distribution. For a given value of Λ, DWRs determined for the three aspect ratios are generally within 3 dB from each other.
Modeled triple-frequency radar signatures for the N13 and W04 snowflake parameterizations according to the SSRGA roughly follow the shape of the curves determined for soft spheres and spheroids for high values of Λ, but show a maximum in 5 DWR Ka/W near Λ ≈ 0.5 mm −1 . A further decrease of 0.5 ≥ Λ ≥ 0.3 mm −1 then leads to a decrease in DWR Ka/W by less than 1 dB, resulting in triple-frequency curves roughly shaped like a comma sign. Based on synthetic aggregate snowflakes generated according to W04, Stein et al. (2015) related the maximum in DWR Ka/W to the fractal geometry of the modeled aggregate snowflakes.
For collections of randomly distributed ice spheres inside the (spherical) snowflake bounding volume, triple-frequency Modeling snowfall triple-frequency radar signatures for collections of randomly distributed ice spheres inside the snowflake bounding volume also leads to a much wider range of triple-frequency radar signatures in Fig. 7 than the region between the triple-frequency curves derived for soft spheres and spheroids or for the N13 and W04 snowflake parameterizations according to the SSRGA. Modeled triple-frequency curves for 1 ≤ ξ ≤ 5 cover a range of up to about 10 dB in DWR X/Ka, 8 dB 30 in DWR Ku/Ka, and 7 dB in DWR Ka/W (see also Fig. S5). In contrast, soft spheres and spheroids or the N13 and W04 snowflake parameterizations according to the SSRGA show DWR ranges of generally about 3 dB and less.
The total range of triple-frequency radar signatures modeled for collections of randomly distributed ice spheres in Fig. 7 covers a large part of all observed triple-frequency signatures in snowfall radar reflectivity measurements by Kulie et al. (2014), Kneifel et al. (2015), andYin et al. (2017). This modeled range also includes many of the triple-frequency radar signatures 35 that Stein et al. (2015) observed in their radar reflectivity measurements at 3, 35, and 94 GHz and modeled based on synthetic aggregate snowflakes generated according to W04. In the present study, the overlap between modeled W04 triple-frequency curve and the total range of triple-frequency radar signatures modeled for collections of randomly distributed ice spheres increases for small Λ, and thus for broad snowflake size distributions characterized by larger snowflakes, when higher normalized surface-area-to-volume ratios of ξ > 5 are also included in the modeling approach (see Fig. 8 for the effect of including ξ = 6).

5
Modeled triple-frequency radar signatures in Fig. 7 for intermediate and high values of ξ combined with small exponential slope parameters Λ of the snowflake size distribution correspond to triple-frequency radar signatures that were related to the presence of large aggregate snowflakes by Kneifel et al. (2015). The region of triple-frequency radar signatures that they related to snowfall characterized by rimed snowflakes, denoted as graupel in Fig. 7, contains triple-frequency curves modeled for low normalized surface-area-to-volume ratios of ξ ≈ 1 in this study. High values of ξ indicate high complexity of the snowflake 10 microstructure (Sect. 3.2), as expected for aggregate snowflakes. Furthermore, broad snowflake size distributions characterized by small Λ in Eq.
(1) contain a higher amount of large snowflakes, consistent with the observation of large aggregates for triple-frequency radar signatures that correspond to small Λ. Extensive snowflake riming, on the other hand, is associated with a coarsening or rounding of the snowflake microstructure due to the accretion of supercooled water droplets. This reduction in the complexity of the snowflake microstructure for strongly rimed snowflakes is reflected in the applied modeling approach 15 by low normalized surface-area-to-volume ratios, leading to relatively flat triple-frequency curves with consistently low dualwavelength ratios DWR X/Ka and DWR Ku/Ka for ξ ≈ 1.
In contrast, snowfall triple-frequency radar signatures that were modeled by Leinonen and Szyrmer (2015) based on detailed 3D shape models of rimed snowflakes extend to higher values of DWR X/Ka and DWR Ku/Ka and roughly span the region between the W04 and N13 triple-frequency curves shown in Fig. 7 for small exponential slope parameters Λ, depending on the 20 amount of riming assigned to the snowflake 3D shape models. Nonetheless, Fig. S6 indicates that truncated size distributions of the rimed snowflake 3D shape models analyzed by Leinonen and Szyrmer (2015), i.e., snowflake size distributions excluding large snowflakes, again lead to flat triple-frequency curves characterized by consistently low DWR X/Ka and DWR Ku/Ka, in line with the snowfall triple-frequency radar signatures related to snowflake riming by Kneifel et al. (2015) and modeled for low normalized snowflake surface-area-to-volume ratios of ξ ≈ 1 in this study (see also the discussion below of how truncated 25 snowflake size distributions and different parameterizations of snowflake mass affect modeled snowfall triple-frequency radar signatures).
Triple-frequency curves determined for soft spheres and spheroids and for the N13 and W04 snowflake parameterizations according to the SSRGA cover a much smaller region of the indicated range of observed snowfall triple-frequency radar signatures in Fig. 7 than the triple-frequency radar signatures modeled for collections of randomly distributed ice spheres 30 inside the snowflake bounding volume and do not explain the distinct regions related to the presence of large aggregates and rimed snowflakes that were observed by Kneifel et al. (2015). Notably, even if various combinations of snowflake gamma size distributions, mass-diameter relationships, aspect ratios, and distributions of preferentially horizontally oriented snowflakes are used to model snowfall triple-frequency radar signatures for soft spheroids, the range of modeled triple-frequency radar signatures does not show significantly better agreement with the observed range of snowfall triple-frequency signatures in radar reflectivity measurements (Leinonen et al., 2012;Kneifel et al., 2015).
Comparing radar reflectivity measurements and in situ snowflake observations, Kneifel et al. (2015) also found that a clear distinction between different snow types was not feasible for combinations of low DWR Ka/W and low DWR X/Ka. Here, this ambiguity can be explained by the similarity of all triple-frequency curves in Fig. 7 for high exponential slope parameters Λ, In general, truncation at smaller D max leads to an 'un-hooking' or flattening of the derived triple-frequency curves. For D ≤ 10.0 mm, modeled snowfall triple-frequency radar signatures in Fig. 9 follow the corresponding triple-frequency curves derived for D ≤ 23.6 mm down to snowflake size distributions characterized by exponential slope parameters of Λ ≈ 1.0 mm −1 before splitting off (toward higher values of DWR Ka/W for N cl = ξ 3 = 27, 125 and toward lower DWR X/Ka and DWR Ku/Ka for N cl = ξ = 1). Triple-frequency curves derived for D ≤ 5.0 mm already start to deviate visibly from the two correspond-10 ing curves determined for D ≤ 23.6 mm and for D ≤ 10.0 mm at higher values of Λ ≈ 2.0 mm −1 . Additionally, truncating snowflake size distributions at D max = 5.0 mm leads to a smaller total range of modeled DWR X/Ka and DWR Ku/Ka, consistent with DWR modeling results by Kneifel et al. (2011) for truncated snowflake size distributions of various snowflake 3D shape models. For low normalized surface-area-to-volume ratios, indicated by N cl = ξ = 1 in Fig. 9, truncation at D max = 5.0 mm also leads to a smaller range of modeled DWR Ka/W. The comparison of the triple-frequency curves in Fig. 9 shows 15 the strong impact of the maximum diameter D max of the snowflake size distribution on modeled snowfall triple-frequency radar signatures.
For snowflake size distributions limited to diameters of D ≤ D max = 10.0 mm, modeled snowfall triple-frequency radar signatures based on the MASC measurements of snowflake complexity χ at Alta and at Barrow are included in Fig. S9.
Compared to Fig. 8, truncation at D max = 10.0 mm leads to an increase in modeled DWRs of up to about 3 dB. These 5 differences are caused by the strong influence of D max on the value of χ(D max ) = χ max calculated with Eq.
(2), i.e., χ max of the two χ(D) relationships illustrated in Fig. 3 decreases for snowflake size distributions truncated at smaller D max , which translates into higher normalized snowflake surface-area-to-volume ratios ξ(χ) for D ≤ D max = 10.0 mm following Eq. (8).
A reliable determination of D max is therefore also important for modeling snowfall triple-frequency radar signatures based on snowflake complexity measurements.

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Combining the hook shape of triple-frequency curves derived for high normalized surface-area-to-volume ratios in Figs. 7 and 8 with the flattening of triple-frequency curves due to the truncation of snowflake size distributions at smaller maximum diameters as illustrated in Fig. 9, modeled triple-frequency radar signatures for snowfall characterized by high snowflake surface-area-to-volume ratios and small snowflake diameters can resemble snowfall triple-frequency radar signatures modeled for soft spheroids. This explains why some non-spheroidal snowflake shape models may lead to similarly high val-15 ues of modeled DWR Ka/W > 10 dB as soft spheroids, e.g., for the aggregates of needle-shaped ice crystals analyzed by Leinonen et al. (2012). According to Fig. 9, values of DWR Ka/W > 10 dB are expected for snowfall characterized by normalized snowflake surface-area-to-volume ratios of ξ ≈ 5 and exponential snowflake size distributions limited to snowflake diameters of D ≤ D max = 5.0 mm with exponential slope parameters of Λ 1.0 mm −1 . Higher values of ξ > 5 already lead to DWR Ka/W > 10 dB for less restrictive snowflake size distributions with respect to D max and Λ. terized by ξ = 1 additionally yields consistently higher (lower) DWR Ka/W for all Λ and thus an increase (decrease) in the modeled DWR Ka/W range (see Fig. S11 for truncation at D max = 10.0 mm; and extreme differences ∆DWR for ξ = 1 are illustrated in Fig. S12 by comparing triple-frequency radar signatures determined for the H04 snowflake density-diameter relationship with triple-frequency radar signatures determined for a snowflake mass-diameter relationship which was derived by Locatelli and Hobbs (1974) specifically to describe lump graupel). These results demonstrate that modeled DWRs are less 5 sensitive to uncertainties associated with the parameterization of snowflake mass than modeled Z e at a single wavelength but can still be affected significantly by these uncertainties, especially at low normalized surface-area-to-volume ratios.
Nonetheless, even high differences of ∆DWR > 1 dB associated with changes in ρ f (D) and m f (D) of ±50 % are generally much smaller than the differences ∆DWR of up to about 10 dB in DWR X/Ka, 8 dB in DWR Ku/Ka, and 7 dB in DWR Ka/W associated with the range of normalized surface-area-to-volume ratios of 1 ≤ ξ ≤ 5 (compare Fig. 10 with Fig. S5). The presented analysis then highlights the importance of snowflake surface-area-to-volume ratio for a detailed interpretation of observed and modeled snowfall triple-frequency radar signatures. 5

Conclusions
In this study, snowflake (radar) backscatter cross sections were modeled at X-, Ku-, Ka-, and W-band radar frequencies of 10, 14, 35, and 94 GHz based on representing individual snowflakes by collections of randomly distributed ice spheres. The size and number of the constituent ice spheres are defined by the snowflake mass derived from the snowflake maximum dimension or diameter D and by the snowflake surface-area-to-volume ratio (SAV); the bounding volume of each collection of ice spheres 10 is given by a sphere of diameter D. SAV was quantified through the normalized ratio ξ of snowflake SAV to the SAV of a single mass-equivalent ice sphere for a range of 1 ≤ ξ ≤ 5.
Snowfall triple-frequency radar signatures were then determined from dual-wavelength ratios (DWRs) of the snowfall equivalent radar reflectivity factors Z e that were calculated using the modeled snowflake backscatter cross sections. Based on nearsurface snowflake observations collected by high-resolution multi-view imaging at Alta (UT, USA) and at Barrow (AK, USA),

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Z e and DWRs were calculated for exponential snowflake size distributions with snowflake diameters of D ≤ D max = 23.6 mm and exponential slope parameters of 0.3 ≤ Λ ≤ 5.0 mm −1 .
The analysis focused on the impact of snowflake SAV on modeled snowfall triple-frequency radar signatures. Additionally, snowflake complexity values obtained from the snowflake images and averaged over one winter season were used as an indicator of snowflake SAV to derive snowfall triple-frequency radar signatures at Alta and at Barrow. Finally, the effect of truncating 20 snowflake size distributions at D max = 10.0 mm and at D max = 5.0 mm on modeled triple-frequency radar signatures was investigated, and the impact of the parameterization of snowflake mass on modeled DWRs was evaluated by uniformly increasing and decreasing all snowflake densities, and thus all snowflake masses, by up to 50 %. Important findings are summarized by the following bullet points: -Average snowflake complexity increases with increasing snowflake size.

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-Modeled snowflake backscatter cross sections generally decrease with increasing snowflake surface-area-to-volume ratio (SAV).
-Modeled snowfall triple-frequency radar signatures cover a wide range of snowfall triple-frequency signatures previously determined from radar reflectivity measurements.
-Snowflake SAV and truncated snowflake size distributions offer a physical interpretation of snowfall triple-frequency 30 radar signatures that is consistent with previously observed differences in snowfall triple-frequency radar signatures related to the presence of large aggregate snowflakes and rimed snowflakes and that may explain why some snowfall triple-frequency radar signatures apparently point to a spheroidal snowflake shape.
-While modeled Z e show high sensitivity to the parameterization of snowflake mass, with typical differences of ∆dBZ e 6 dB for the analyzed snowflake density range, derived DWRs are less sensitive, with corresponding differences of ∆DWR 1 dB except for low SAV.

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-The analyzed impact of the parameterization of snowflake mass on modeled snowfall triple-frequency radar signatures is generally much smaller than the analyzed impact of snowflake SAV.
Overall, the results indicate a strong influence of snowflake SAV on modeled snowfall radar signatures that may be exploited in the interpretation of snowfall triple-frequency radar measurements, e.g., to distinguish snow types characterized by different snowflake SAV. For a detailed analysis of snowfall triple-frequency radar signatures based on snowflake SAV, however, 10 a more comprehensive quantification of snowflake SAV will be needed. This should include characteristic differences and similarities in snowflake SAV among various snow types and reveal potential relationships between snowflake SAV and other microstructural parameters important for the interpretation of snowfall radar signatures like snowflake size and mass.
Accordingly, current and future databases of microwave scattering properties determined for detailed snowflake 3D shape models would benefit from incorporating snowflake surface area as additional microstructural parameter (besides snowflake 15 size and mass). Common features and differences in modeled scattering properties could then be related not only to visually distinct snow types (and snowflake size and mass) but also to snowflake surface-area-to-volume ratio, providing a quantitative description of the snowflake microstructure across all snow types and thereby helping to further constrain snowflake shape for snowfall remote sensing.
Based on a more comprehensive quantification of snowflake surface-area-to-volume ratio that reflects characteristic differ-20 ences among snow types, the outlined approach for relating normalized snowflake surface-area-to-volume ratio ξ to snowflake complexity χ obtained from snowflake images could be applied to derive ξ(D) relationships for a variety of snowfall conditions. Snowfall triple-frequency radar signatures could then be modeled from these ξ(D) relationships and compared to triple-frequency radar reflectivity measurements. Such comparisons would show whether ξ(D) relationships derived from snowflake imgaging data can adequately describe snowflake surface-area-to-volume ratio for the interpretation of snowfall 25 triple-frequency radar signatures and may therefore lead to a parameterization of snowflake shape by ξ(D) relationships similar to the parameterization of snowflake mass by snowflake density-diameter or mass-diameter relationships commonly used in snowfall remote sensing.

Appendix A: Representation of snowflakes by collections of randomly distributed ice spheres
In this study, snowflakes defined by the maximum dimension or diameter D, the mass m f (D), and the normalized surfacearea-to-volume ratio ξ are represented by collections of randomly distributed ice spheres where the radius r cl and the number N cl of the constituent ice spheres are specified by Eq. (9) and the diameter of the (spherical) bounding volume V f of each ice sphere collection is given by D. The Appendix discusses limitations of this representation and implications for the modeled 5 radar signatures.
To generate collections of non-overlapping ice spheres inside V f according to Eq. (9), ξ 3 = N cl has to be an integer and the snowflake mass m f (D) has to be sufficiently low. Backscatter cross sections σ b (D; ξ) were calculated for collections of

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To determine radar reflectivity factors Z e with Eq. (10), σ b (D; ξ) for collections of multiple ice spheres were calculated only for snowflake diameters of D > 0.55 mm, corresponding to (single) mass-equivalent ice sphere radii of r eq > 0.16 mm.
For smaller snowflakes, Eqs. (3) and (4) lead to high snowflake masses that could not be reached consistently by randomly placing non-overlapping ice spheres given by Eq. (9) inside the snowflake bounding volume V f . Here, σ b was calculated only for a single mass-equivalent ice sphere specified by ξ = 1, and the value of σ b (D; ξ = 1) was then assigned to all ice 20 sphere collections, leading to σ b (D; 1 ≤ ξ ≤ 5) = σ b (D; ξ = 1) for D ≤ 0.55 mm or r eq ≤ 0.16 mm. This simplification has no significant impact on modeled snowfall triple-frequency radar signatures in Sect. 4.2 because radar reflectivity factors determined with Eq. (10) are only affected weakly by the backscatter cross sections of small snowflakes. Even when snowflake diameters of D ≤ 0.55 mm are ignored completely, modeled Z e decrease and DWRs increase by less than about 0.3 dB for snowflake size distributions with exponential slope parameters of Λ ≤ 2.0 mm −1 . Slightly higher changes in modeled 25 Z e and DWRs are noted for snowflake size distributions characterized by higher values of Λ, with a maximum decrease of 1.7 dB in modeled Z e at 94 GHz and a maximum increase of 0.8 dB in DWR Ka/W found for an extreme slope parameter of Λ = 5.0 mm −1 . These differences are generally much smaller than the impact of normalized SAV on modeled Z e and DWRs discussed in Sect. 4.2.
At 10 and 14 GHz, all σ b (D; 1 < ξ ≤ 5) for 0.55 < D ≤ 1.4 mm or 0.16 < r eq ≤ 0.3 mm were additionally replaced by 30 σ b (D; ξ = 1) to obtain smooth spline interpolants of σ b (D; ξ) across the entire range of D (see Sect. 4.1). The effect of these modifications on modeled snowfall triple-frequency radar signatures in Sect. 4.2 is again small, with associated differences in modeled Z e and in DWR X/Ka and DWR Ku/Ka of less than about 0.1 dB for Λ ≤ 2.0 mm −1 and slightly increasing differences for higher Λ up to a maximum of 0.7 dB at Λ = 5.0 mm −1 .
As N cl non-overlapping ice spheres were placed randomly inside the spherical bounding volume V f specified by the snowflake diameter D (see Sect. 3.3), the maximum dimension or diameter D cl of each generated ice sphere collection is always smaller than D (D cl = 2 r cl ; r cl quantifies the size of each individual ice sphere within the collection, D cl indicates the size of the entire collection). Figure S13 illustrates the relation between snowflake diameter D and the mean diameter D cl of 500 generated collections of randomly distributed ice spheres inside V f . High values of N cl = ξ 3 , i.e., collections of many small 5 ice spheres, lead to small relative differences between D and D cl of less than 5 %. For collections of N cl = 4 (and thus fewer but larger) ice spheres, D cl is up to about 25 % smaller than the snowflake diameter. Nonetheless, the calculated backscatter cross sections σ b for the ice sphere collections show only a weak correlation with the diameter D cl (see examples in Fig. S14), and Fig. S15 illustrates the weak influence of the differences between D and D cl on the modeled snowfall triple-frequency radar signatures. Here, dual-wavelength ratios DWR X/Ka, DWR Ku/Ka, and DWR Ka/W generally change by less than 1 dB 10 when the mean diameter D cl of the generated ice sphere collections is used instead of the snowflake diameter D to determine the corresponding snowfall radar reflectivity factors Z e with Eq. (10). These differences are again small compared to differences in DWRs associated with the range of normalized surface-area-to-volume ratios of 1 ≤ ξ ≤ 5.
Competing interests. T. J. Garrett is a member of the editorial board of the journal and has a financial interest in Particle Flux Analytics which sells the MASC.