Measurement report: Mass and Density of Individual Frozen Hydrometeors

A new precipitation sensor, the Differential Emissivity Imaging Disdrometer (DEID), is used to provide the first continuous measurements of the mass, diameter, and density of individual hydrometeors. The DEID consists of an infrared camera pointed at a heated aluminum plate. It exploits the contrasting thermal emissivity of water and metal to determine individual particle mass by assuming that energy is conserved during the transfer of heat from the plate to the particle during evaporation. Particle density is determined from a combination of particle mass and morphology. A Multi-Angle Snowflake 5 Camera (MASC) was deployed alongside the DEID to provide refined imagery of particle size and shape. Broad consistency is found between derived mass-diameter and density-diameter relationships and those obtained in prior studies. However, DEID measurements show a generally weaker dependence with size for hydrometeor density and a stronger dependence for aggregate snowflake mass.

2004; Yuter et al., 2006;Newman et al., 2009) have lent general support to this reference data set, although concerns remain about geographic and temporal specificity, measurement limitations (Yuter et al., 2006;Battaglia et al., 2010), and the ability to quantify the extent of riming (Barthazy and Schefold, 2006;Brandes et al., 2008). 25 Particle density measurements have proven more difficult to ascertain as they require, in addition to mass, an estimate of particle volume. This may be reasonably obtained for quasi-spherical particles such as lump graupel , but the task is considerably more challenging when snow particles are formed from ice crystal aggregation. A possible approach is to infer density from fallen snow using column measurements (Conger and McClung, 2009), capacitance probes (Dent et al., 1998), or a combination of a camera for snow depth and an electric scale for snow mass (Muramoto et al., 1995). 30 Muramoto et al. (1995) and Brandes et al. (2007) measured the bulk mass of snowflakes using a weighing gauge, from which the bulk volume was determined with 2D camera imagery of individual snowflakes. Tiira et al. (2016) determined a volume flux weighted snow density for a population of snowflakes using particle size distribution, fallspeed, and a weighing gauge to estimate the mass, and 2D camera imagery to determine effective diameter and volume. However, snow undergoes compaction and melting on the ground, so the relationship to individual particle density in the air is approximate (Brun et al., 1992).To 35 obtain individual snowflake particle density, Magono and Nakamura (1965) collected individual wet and dry snowflakes on a piece of dyed filter paper, from which the outline of the flake was manually measured. Individual volume was inferred from the major and minor axes of the outline of snow, and the mass from the outline of the melted snowflake. Holroyd (1971) also made measurements of the major and minor axes of powder snow and dendrites from (Magono and Nakamura, 1965). A limitation of these datasets is that they were necessarily small given the manual nature of the effort. 40 Here, we present continuous measurements of individual the masses of frozen hydrometeors using a new instrument, the Differential Emissivity Imaging Disdrometer (DEID). DEID data are combined with photographic imagery obtained using a Multi Angle Snowflake Camera (MASC) (Garrett et al., 2012) to obtain estimates of particle density.

Methods
All measurements described in this study were acquired at a meteorological measurement tower placed at the mouth of Red

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Butte Canyon (40.76857, -111.82614) in Salt Lake City at 1547 m elevation in January and February 2020. More details about the site and measurement campaign are provided by Singh et al. (2021).

Particle mass from thermal imaging
The Differential Emissivity Imaging Disdrometer (DEID) (Singh et al., 2021) consists of a thermal camera operating at a frequency between 2 and 12 Hz pointed at an aluminum plate placed atop a hotplate maintained at a self-sustained temperature 50 of 85 • C. The camera distinguishes hydrometeors as they melt and evaporate as white regions on a black background ( Figure   1) due to the contrasting infrared emissivities of water ( ≈ 0.96) and aluminum ( ≈ 0.03). A small strip of polyimide tape ( ≈ 0.95) is applied to the aluminum plate to provide a reference temperature, as well as a pixel to length dimension conversion based on the tape's known width. The effective collection area of the DEID was A ≈ 7 cm × 5 cm and the per pixel resolution of imaged particles p was 190 µm. Processing of the thermal camera imagery yields the hydrometeor area on the plate, the 55 temperature difference between the water and the plate, and the evaporation time.
Individual particle mass is determined using the DEID by employing the assumption that the heat gained by a hydrometeor is equivalent to the heat lost by the plate during the process of melting and evaporation. The heat balance equation is: where c p is the specific heat capacity of water at constant pressure, ∆T is the difference in temperature between 0 and time, t,

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M is the mass of the hydrometeor, and L eqv is the equivalent latent heat required for the conversion of the hydrometeor to gas.
For liquid precipitation L eqv = L v where L v is the latent heat of vaporization of water. For solid precipitation, L eqv = L v +L f where L f is the latent heat fusion for water. K is the thermal conductivity of the aluminum plate, H is the plate thickness, A(t) is the cross-sectional area of the water droplet at time t, T p is the temperature of the hotplate, and T w (t) is the temperature of the water at time t. Taking a single calibrated constant K d for the plate was determined experimentally by applying known masses of water from a micropipette to the plate. (See Appendix Afor the derivation of K d ). The heat balance equation was found in a laboratory setting to be highly insensitive to ambient winds, temperature, and humidity (Singh et al., 2021).
The temperature difference between the plate and water on the plate (∆T = T p − T w (t)) can be determined using the mean 70 pixel intensity of the particle in the thermal camera imagery (see Appendix B). The mass calculation then simplifies to: Three-dimensional regions integrated over particle area and evaporation time were constructed for each particle during the melting process defined by V t = A(t)dt so that from Eq: 2 M = K d V t ∆T . An example of a one-minute time interval with 3D volumetric regions representing each individual particle is shown in Figure 2 for graupel. Additional 3D representations 75 for other snowflake types are shown in Appendix B.

Hydrometeor photography
The Multi-Angle Snowflake Camera (MASC) (Garrett et al., 2012) houses three high-speed visible spectrum cameras arranged concentrically with a separation angle of 36 • . As a particle falls through the instrument collection aperture, two vertically spaced infrared detectors simultaneously trigger the cameras to take three simultaneous photographs of the particle from the 80 side and to measure the fallspeed between the two detectors. The MASC software processes the imagery to calculate the surface area, geometric cross-section, perimeter, orientation, aspect ratio, complexity, flatness, and whether the particle is a raindrop (Shkurko et al., 2016). Images taken by the Multi-Angle Snowflake Camera (MASC) are used in conjunction with the DEID to confirm precipitation type and refine density measurements. A mosaic of MASC images from February 5 and 6, 2020 is shown in Figure 3. Additional snowflake imagery catalogued from each storm is shown in Appendix C.

Particle volume and density
Immediately following the arrival of a hydrometeor on the plate, the particle cross-sectional area increases rapidly as the hydrometeor adjusts from the ambient air temperature to the temperature of the plate, melting in the process. It reaches a maximum size A max before shrinking during evaporation. Although the particle may be melted at the moment that it reaches A max , frozen particles nonetheless tend to maintain their shape such that A max is approximately representative of the frozen cross-90 sectional area in air. To obtain an estimate of particle density from particle mass, a spherical volume V S for each hydrometeor can be estimated from an effective diameter D eff derived from the maximum cross-sectional area A max V S = π 6 D 3 eff , where D eff = 4 π A max . This geometric definition is consistent with that taken by , who prescribed D eff as "the diameter of the smallest circle into which the aggregate as photographed will fit without changing its density." Unless otherwise specified, all instances of "diameter" in the text refer to D eff . An alternative diameter metric is D max as defined 95 by the circumscribed diameter from the maximum horizontal dimension of the hydrometeor as it lies on the hotplate. The measured relationship between these two metrics is described below.
For a more precise estimate of particle volume, side-viewing MASC imagery of hydrometeors can be used to determine the average aspect ratios of hydrometeors measured over a one-minute time interval. In general, by approximating hydrometeors as an ellipsoid, the volume of a snowflake is V M = π 6 D max D min D v where D max is the longest dimension as seen by the DEID, 100 D min is the shortest, and D v is bounded by the two. For example, if the hydrometeor shape is an oblate spheroid, then D v = D max , and if a prolate spheroid then D v = D min . As a snowflake falls onto the plate, the maximum dimension is expected to lie flat with respect to the plate. D eff is expected to lie between the maximum and minimum dimensions of the snowflake, so for the more general case of an ellipsoid, a reasonable assumption is that D v D eff .
The issue here is that the DEID is unlikely to provide a measure of the minimum dimension. However, it can be reasonably 105 inferred from side views of hydrometeors provided by the multiple concentric images captured by the MASC. For a given snowflake, the minimum of the aspect ratios for each hydrometeor seen by the multiple MASC cameras as captured within a one minute interval and subsequently averaged α min , can reveal a characteristic minimum dimension in the vertical direction for the time period that is otherwise invisible to the DEID. Taking D min = α min D max , the MASC-adjusted volume is then The spherical and mass-adjusted density of individual hydrometeors can then be calculated from the mass M and volume as

Results
Over the course of five storms that took place in Salt Lake City, Utah on January 14, 2020, January 17, 2020, January 26, 2020, February 2-3, 2020, and February 5-6, 2020, the DEID detected 132,459 individual hydrometeors. Of those hydrome-115 teors, 104,812 were snowflakes, and the remainder were either rain or a rain-snow mix. A filtering algorithm rejected small hydrometeors with fewer than three contiguous pixels of data in all three dimensions (See Appendix B), leaving a total dataset of 109,316 hydrometeors, of which 86,285 were snowflakes and the remainder rain. Of those snowflakes, density estimates for 43,649 were obtained using corresponding MASC imagery. The density of rain of course is known to be 1000 kg m −3 and would provide a valuable reference point for this study. Unfortunately, its measurements cannot be addressed using the 120 techniques described here as raindrops do not preserve their area after impaction on the plate. given that  found lump graupel to have densities ranging as high as 450 kg m −3 and dense graupel has also been described by others (Magono and Nakamura, 1965;Holroyd, 1971;Muramoto et al., 1995;Fabry and Szyrmer, 135 1999;Brandes et al., 2007;Tiira et al., 2016).   In the following sections, mass-diameter and density-diameter relationships are expressed with respect to D eff . By way of reference, for the 86,285 snowflakes included in this study, D max can be correlated with D eff through the power law relationship D max = aD b eff with values for a and b summarized in Table 1. In general, the two quantities are highly correlated, and the relationship is nearly linear. For the ensemble taken as a whole, the relationship is D max = 1.17D 1.11 eff with a square correlation coefficient of R 2 = 0.93. The average measured aspect ratios seen by the MASC for aggregates, densely rimed particles, and

Mass-Diameter relationships
The mass-diameter relationship for all snowflakes is shown in Figure 6. The prefactor observed in the mass-diameter relationship is 0.018, which is approximately consistent with the range of values described by  for the densely rimed snowflakes (Table 2), comprising 47% of the snowflakes observed in our study. The exponent 2.52, however, 150 generally exceeds those values obtained by  for graupel-like (2.1 to 2.4), densely rimed (2.1 to 2.3), and aggregated (1.4 to 1.9) snow.
Snowflakes were selected manually through visual classification using MASC imagery for time periods at least three minutes long where no other snow types were present, defined as aggregates (N=18,049), graupel (N=34), and densely rimed (N=15,721). Graupel occurred frequently, although there was only one period 16 minutes long that did not exhibit any other type of snow. Mass-diameter relationships M=aD b ef f for graupel, densely rimed snow, and aggregates are compared with those found by  in Figure 7 and Table 2. Values of a and b for graupel determined by the DEID (a=0.047 and b=2.73) lie within the ranges observed by  0.042 < a < 0.140 and 2.6 < b <3.0. The quantity of densely rimed and aggregate snowflakes collected by the DEID was rouhgly five orders more numerous than those described by . For densely rimed snowflakes, the prefactor a lies within the range observed by Locatelli and 160 Hobbs (1974) and the exponent b is within 10%. However, aggregate snowflakes differ by 30% in the exponent and by 43% in the prefactor. Notably, the exponent value of b = 2.75 is markedly greater than obtained previously using various theoretical and observational approaches (Dunnavan et al., 2019).  , dashed lines. Contours are for 25th, 50th, and 95th percentile bounds.

Density-Diameter relationship
Density-diameter relationships for all snowflakes in this study are shown in Figure 8 and a comparison of density-diameter 165 relationships from this work and prior studies is shown in Figure 9. Using the spherical volume approximation (Figure 8, left), the measured values for density are rarely greater than 100 kg m −3 , suggesting a possible underestimate. Using a similar assumption, Muramoto et al. (1995) observed similarly low values of density (Figure 9). The density-diameter relationships from other studies shown in Figure 9 include both dry and wet snowflakes and ice particles with densities extending into the 200-300 kg m −3 range for the lowest diameters observed.

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More refined density calculations supplemented by MASC data are shown in Figure 8, right, and include high values near the density of bulk water, as would be expected for wet snowflakes that have partially melted before reaching the hotplate.   A notable difference between the mass-diameter relationships from our study versus those described previously by  lies in fitted exponents for aggregate type snowflakes. In the earlier study these ranged from 1.4 to 1.9, while our results point to an exponent of 2.75, very similar to that observed for graupel and densely rimed snowflakes. An added implication of the larger exponent is that the masses of very small aggregates with diameters less than 1 mm are generally smaller than have previously been reported (Figure 7). While the MASC-adjusted density-diameter relationships align closely 180 with several previous studies (Figure 9) for particle sizes larger than approximately 5 mm diameter, much lower values of density tend to be observed at smaller particle sizes. Where the exponents described in previous studies (Holroyd, 1971;Fabry and Szyrmer, 1999;Heymsfield, 2003;Brandes et al., 2007;Tiira et al., 2016) are close -1, those from our study are approximately -0.5, suggesting a weaker dependence of density on particle size than is generally assumed. Indeed, even for rimed snowflakes, high resolution electron microscope imagery reveals a highly porous internal structure (Rango et al., 2003;185 Enzmann et al., 2011). On the ground, the density of snowfall at a high-elevation location in the Wasatch Mountains in Utah is typically less than 100 kg m −3 (Alcott and Steenburgh, 2010), a location where mode snowflake diameters lie between 1 mm and 2 mm (Garrett and Yuter, 2014).
Combining the weak dependence of density with size and the large exponent near 3 of the mass, a speculation may be made that frozen hydrometeors, while topologically complex, maintain a nearly Euclidean scaling with their maximum dimension 190 as they grow. The elemental crystal or frozen droplet that comprises the hydrometeor structure is not obviously affected by the collection process. So, even if the hydrometeor ultimately attains a highly non-spherical structure as it falls, the internal porosity of the structure may be expected to remain approximately fixed.

Summary
We describe measurements of the mass and density of individual frozen hydrometeors obtained using a new instrument, the 195 Differential Emissivity Imaging Disdrometer. Power-law mass-diameter relationships obtained by the DEID derived from 86,285 measured particles agree well with widely used relationships published by , which was based on a much more limited dataset. The exception is that aggregate-type flakes measured by the DEID have a significantly higher exponent close to a value of 3. To obtain hydrometeor density from the measured mass, estimates of volume are required.
Here, a simple spherical approximation for particle volume based on the particle equivalent diameter seen by a thermal camera 200 viewing the heated plate led to density estimates approximately a factor of two lower than those using a more refined calculation that incorporated concurrent MASC measurements of particle aspect ratio. For the subset of DEID measurements that included coincident MASC imagery totalling 43,649 hydrometeors, the resulting density-diameter relationships suggest substantially lower densities of particles smaller than 5 mm than has been observed in most prior studies. It may be that existing bulk microphysical parameterizations in numerical weather models tend to underestimate the masses of large frozen hydrometeors 205 while overestimating those of smaller hydrometeors. If true, any revision could have possible implications for forecasts of snow water deposition in mountain reservoirs. Future anticipated refinements to the DEID particle volume algorithm at a wider range of locations may help further refine estimates of hydrometeor density.

Appendix A: Hotplate calibration
The constant K d includes the specific heat capacity, equivalent latent heat, and the plate's thickness and thermal conductivity, 210 was calibrated experimentally for the DEID aluminum plate (Singh et al., 2021). The plate was roughened with 600 grit (P1200) sandpaper to allow for droplet spreading and more rapid evaporation. Water drops of known masses were evaporated on the plate to determine the calibrated coefficient. Combining the constants in Equation (1) yields, where M is a known mass of water. A constant is determined that includes the thermal conductivity and the thickness of the 215 hotplate using droplets of known mass M = 2 ± 0.2 × 10 −5 kg: where t 0 T p − T w (t)A(t)dt was determined from DEID measurements, c p,w = 4.28 × 10 3 J K −1 kg −1 is the specific heat of water at constant pressure, ∆T ev = 100K, and L v = 2.26 × 10 6 J kg −1 is the latent heat of vaporization. Determined through 10 trials, K/H = 4.1 × 10 3 kg s −3 K −1 .

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Including the latent and specific heat required to evaporate liquid water and ice, respectively, the derived values of K d for liquid and ice are then: = 4.1 × 10 3 2.67 × 10 6 = 1.54 × 10 −3 kg s K m 2 (A3) where c p,i = 2.10 × 10 3 J K −1 kg −1 is the specific heat of ice at constant pressure and the latent heat of fusion L f = 3.34 × 225 10 5 J kg −1 . The equation for mass becomes M = K d t 0 A(t)(T p − T w (t))dt. Experiments comparing mass calculations using t 0 A(t)T p − T w (t)dt and ∆T t 0 Adt, where ∆T = T p − T w (t) is the mean value per particle showed that the latter is a sufficient approximation and the equation for mass may then be expressed as   MATLAB Image Processing software was used to extract data from the thermal camera to determine the physical properties of individual melted hydrometeors. Experimentally relevant parameters include the hotplate temperature T p , the sampling frequency of the thermal camera f s , the physical width of each pixel in the camera imagery p, and the sampling area of the hotplate A hot . Each thermal camera image was converted to both grayscale and binary format. 3-dimensional volumetric regions, "voxels," in a product space of hydrometeor area on the DEID plate and time during the duration of the melting of 235 each particle were evaluated to yield the particle mass through Eq. A5, as illustrated in Figures B1-B5. 14 https://doi.org/10.5194/acp-2021-179 Preprint. Discussion started: 6 April 2021 c Author(s) 2021. CC BY 4.0 License. Figure B3. As for Fig. B1 but for aggregate snow Figure B4. As for Fig. B1 but for densely rimed aggregate snow Figure B5. As for Fig. B1 but for a mixture of densely rimed and aggregate snow. For the sake of processing, all partial voxel objects that bordered the edge of the 3D sampling area were removed for individual particle calculations but were included for precipitation rate calculations. A filtering threshold was used to remove small particles with fewer than three pixel data points in any dimension.
The temperature difference between the hotplate and water (∆T = T p − T w (t)) is based on the mean pixel intensity of the 240 particle during its lifetime on the hotplate. The mean pixel intensity I mean is converted to ∆T through the linear transformation ((T p − T w (t)) ≈ ∆T = T p (255 − I mean )/256 ).
The integrated cross-sectional area of the particle during its evaporation time on the hotplate is given by V t = A(t)dt. The effective diameter D eff is calculated from the point in time associated with the maximum recorded cross-sectional area of the particle A max according to D eff = 4A max /π. The evaporation time of each particle is calculated by generating a bounding 245 box, or the smallest box that contains the 3D region containing V t where the circumscribed diameter D max is the maximum in the area dimension and the evaporation time t evap is the maximum in the time dimension.
Appendix C: MASC snowflake imagery MASC Imagery of frozen hydrometeors was catalogued for each snow event described in this article with representative particles shown in Figures C1 to C4 250