Supplement of Unshielded precipitation gauge collection efficiency with wind speed and hydrometeor fall velocity

Abstract. Collection efficiency transfer functions that compensate
for wind-induced collection loss are presented and evaluated for unshielded
precipitation gauges. Three novel transfer functions with wind speed and
precipitation fall velocity dependence are developed, including a function
from computational fluid dynamics modelling (CFD), an experimental fall
velocity threshold function (HE1), and an experimental linear fall velocity
dependence function (HE2). These functions are evaluated alongside universal
(KUniversal) and climate-specific (KCARE) transfer functions with
wind speed and temperature dependence. Transfer function performance is
assessed using 30 min precipitation event accumulations reported by
unshielded and shielded Geonor T-200B3 precipitation gauges over two winter
seasons. The latter gauge was installed in a Double Fence Automated
Reference (DFAR) configuration. Estimates of fall velocity were provided by
the Precipitation Occurrence Sensor System (POSS). The CFD function reduced
the RMSE (0.08 mm) relative to KUniversal (0.20 mm), KCARE (0.13 mm), and the unadjusted measurements (0.24 mm), with a bias error of
0.011 mm. The HE1 function provided a RMSE of 0.09 mm and bias error of
0.006 mm, capturing the collection efficiency trends for rain and snow well.
The HE2 function better captured the overall collection efficiency,
including mixed precipitation, resulting in a RMSE of 0.07 mm and bias error
of 0.006 mm. These functions are assessed across solid and liquid
hydrometeor types and for temperatures between −22 and 19 ∘C. The results demonstrate that transfer functions incorporating
hydrometeor fall velocity can dramatically reduce the uncertainty of
adjusted precipitation measurements relative to functions based on
temperature.



S1.2 Collection efficiency based on hydrometeor fall velocity and wind speed
For each wind speed, monodispersed hydrometeors were injected from a horizontal plane upstream and above the gauge orifice ( Fig. S1). Hydrometeor types were characterized by their fall velocity, diameter, density, and mass (Table S2). For dry snow 35 and wet snow, the hydrometeor sizes are related to the fall velocity by a general power law (Rasmussen et al., 1999). The fall velocity for dry snow ( dry U ) in cm s -1 is a function of the size ( dry D ) in cm, with a similar relationship for wet snow (Eqs. S1a and b). For dry snow and wet snow, the hydrometeor density was chosen such that the size and fall velocity followed the power law relationship of Rasmussen et al. (1999), with the drag coefficient for spherical hydrometeors given by Henderson (1976) based on the relative hydrometeor to air velocity. This drag formulation closely matches that of Haider and Levenspiel (1989), and has been used in previous studies . 45 Table S2. Characteristics for rain, dry snowflake, wet snowflake, and ice pellet hydrometeors. For dry snow, hydrometeor fall velocities between 0.5 m s -1 to 1.0 m s -1 were included, representing sizes up to 7 mm and densities below that of ice (Table S2). Fall velocities for wet snow were selected between 1.0 m s -1 to 2.5 m s -1 for sizes between 0.2 to 21.8 mm. Spherical ice pellets with a density of 916.7 kg/m 3 were also included in the analysis for fall velocities 50 of 5.0 m s -1 and 10.0 m s -1 . Spherical hydrometeors with a density of 998.2 kg/m 3 , representing rain, were included for fall velocities from 0.25 m s -1 to 10 m s -1 for comparison.
The hydrometeor trajectory was derived from the drag force, gravitational force, and buoyancy forces acting on the hydrometeor as it moves through the flow using Lagrangian analysis. A dilute two-phase flow was assumed, where the influence of the hydrometeors on the fluid flow was negligible and hydrometeor interactions including potential hydrometeor 55 coalescence were ignored. Combining these forces gave the net hydrometeor acceleration ( p a ) as a function of the drag coefficient ( D C ), hydrometeor cross sectional area ( p A ), density of air ( a  ), relative hydrometeor to air velocity ( p a  u u ), hydrometeor density ( p  ), hydrometeor volume ( p V ), and acceleration due to gravity ( g ) acting in the negative ẑ direction as shown in Eq. (S2).
The hydrometeors were injected into the flow uniformly at equilibrium with an initial velocity ( p1 u ) equal to the free-stream wind speed ( w U ) along the x direction and hydrometeor fall velocity ( f U ) in the negative ẑ direction (down).
In the free-stream region under steady-state conditions, the hydrometeor fall velocity and terminal velocity will be equivalent.
Hydrometeor interactions with the inside of the gauge orifice were assumed to be ideal reflections. 65 The collection efficiency ( CE ) for a given free-stream wind speed ( w U ) and hydrometeor fall velocity ( f U ) corresponds to the ratio of the number of hydrometeors collected ( C N ) over the horizontal gauge orifice area ( C A ) to the number of hydrometeors injected from the horizontal injection plane above ( I N ) over the horizontal injection plane area ( I A ), as shown in Eq. (S4).

S1.3 Integral collection efficiency with wind speed
The collection efficiency presented in Sect. S1.2 was for monodispersed hydrometeors with identical size, mass, density and fall velocity. In this section, we define the integral collection efficiency as that derived over the entire hydrometeor size distribution and associated characteristics (e.g. fall velocity, density and volume).

S1.3.1 Rainfall 75
The total precipitation intensity ( Total P ) is a function of the hydrometeor size distribution ( R N ), density ( p  ), volume ( p V ), fall velocity ( f U ), and density of water ( w  ).
The hydrometeor size distribution (number of hydrometeors per unit size per unit volume) for raindrops ( R N ) can be expressed as a gamma distribution defined by the 0R N (m -3 cm -1- ) parameter, exponential factor ( R  ) in cm -1 , unitless exponent (  ), 80 and hydrometeor diameter ( D ) in cm as given by Ulbrich (1983).
Assuming the product of the exponential factor and the maximum hydrometeor diameter is large, the exponential factor can be expressed in terms of the exponent and median volume diameter ( 0 R D ) in cm.
The median volume diameter is determined based on the rainfall intensity ( R ) in mm/hr and the empirical constants  and  for the specific rain type.
The value of the exponent (  ) will be positive or negative depending on the rain type (orographic, thunderstorm, stratiform, or showers), corresponding to a concave up or down distribution when plotted on a   log R N versus D plot. Orographic rain 90 with an exponent less than zero corresponds to a concave up distribution with small droplets and low fall speed (Ramana et al., 1959) (Table S3). Thunderstorm rain with a concave down distribution corresponds to large drops and high fall speed (Blanchard, 1953).
The hydrometeor fall velocity for rainfall is given by Beard (1976). At standard air temperature and pressure, the rainfall hydrometeor fall velocity ( f U ) in m s -1 is a function of the equivalent hydrometeor diameter ( D ), acceleration due to gravity 95 ( g ), raindrop hydrometeor density (density of water   Blanchard (1953) The integral collection efficiency is the ratio of the precipitation intensity that is captured by the gauge to that which is falling 100 in the free-stream airflow for a given hydrometeor size distribution, following the approach of Nešpor and Sevruk (1999). The collection efficiency is implicitly dependent on the equivalent hydrometeor diameter through the hydrometeor fall velocity.

S1.3.2 Snowfall
The total precipitation intensity ( S,Total P ) for snowfall is a function of the hydrometeor size distribution for snowfall ( S N ), 105 , fall velocity ( f U ) and density of water ( w  ), integrated over the range of equivalent diameters ( Taking the equivalent snowfall diameter as that for a spherical water droplet with the density of water gives the total precipitation intensity integral as a function of the snowfall size distribution. Using this approach, both the size distribution 110 and fall velocities of snowflakes are defined as a function of the equivalent spherical diameter of water droplets. The size distribution for snowflakes ( S N ) can be expressed by the Gunn and Marshall size distribution (Gunn and Marshall, 1957) as a function of the size distribution parameter ( 0 S N ) in m -1 mm -1 , exponential factor ( S  ) in cm -1 , and equivalent spherical water drop diameters ( D ) in cm above 0.1 cm. 115 The size distribution parameter and exponential factor vary with the precipitation intensity ( R ) in mm h -1 .
The median volume diameter ( 0S D ) in cm varies with the exponential factor as shown by Atlas (1953). 120 The fall velocity f U in cm s -1 for various snowfall types, based on the equivalent spherical diameter of water droplets ( D ) in cm, is given by a general power law.
The fall velocity power law coefficients for dendrites, rimed dendrites, and a mixture of dendrites and aggregates of plates is 125 summarized in Table S4 based on the work of Langleben (1954).  (1954) The integral collection efficiency at a given wind speed is the ratio of the precipitation intensity that is captured by the gauge 130 to that which is falling in the free-stream airflow for a given crystal habit and size distribution. As with that for rainfall, the collection efficiency is implicitly dependent on the equivalent hydrometeor diameter through the hydrometeor fall velocity.

S2.2.1 Wind speed dependence of integral collection efficiency 155
For specified hydrometeor types and precipitation intensities, the integral collection efficiency (Eqs. S9 and S17) was derived for wind speeds from 0 to 10 m s -1 using the CFD transfer function (Eq. 6) based solely on wind speed and hydrometeor fall velocity. The collection efficiency, derived using the CFD transfer function, decreases nonlinearly with wind speed, decreasing more rapidly at lower wind speeds and more gradually at higher wind speeds above approximately 5 m s -1 (Fig. S3). A wide range of integral collection efficiency results with wind speed is apparent, depending on the hydrometeor type and precipitation 160 intensity. Figure S3. Integral Geonor unshielded gauge collection efficiency with wind speed for thunderstorm rain at 20 mm h -1 , 10 mm h -1 , and 1 mm h -1 precipitation intensities (R); orographic rain at 10 mm h -1 , 1 mm h -1 , and 0.1 mm h -1 precipitation intensities (R); and aggregates of 165 plates, rimed dendrites, and dendrites at 2.5 mm h -1 , 1.5 mm h -1 , and 0.5 mm h -1 precipitation intensities (R).
Lower integral collection efficiencies are observed for snowfall relative to orographic and thunderstorm rain. Across snowfall types, a wide range of integral collection efficiency values are apparent. Dendrites show the greatest nonlinearity with wind speed and the lowest integral collection efficiency, with the latter decreasing to 0.01 at 6.7 m s -1 wind speed and 0.5 mm h -1 170 precipitation intensity. The integral collection efficiency continues to decrease with increasing wind speed, remaining small, but non-zero, up to 10 m s -1 wind speed. For 1.5 mm h -1 and 2.5 mm h -1 intensities, integral collection efficiencies are higher, with the magnitude of the increase varying with the wind speed. It is important to note the more gradual decrease in integral collection efficiency for wind speeds above ~6 m s -1 compared with the results in Fig. 2 for a given hydrometeor fall velocity.
The integral collection efficiency includes the contribution from a range of hydrometeor sizes, including larger hydrometeors 175 with higher fall velocities that are still able to be captured by the gauge, providing small but non-zero integral collection efficiencies.
For comparison, integral collection efficiency results were computed for dry snow and wet snow using the same hydrometeor size distribution parameters as Colli et al. (2016b) for sizes between 0.25 mm to 20 mm. The results from the different models show good agreement (Fig. S4). For dry snow, the integral collection efficiency decreases more gradually with wind speed in 180 the present study than the Colli et al. (2016a) results, with collection efficiency values up to 0.18 higher at 3 m s -1 wind speed using the present model. For wet snow, the results of Colli et al. (2016a) show a nonlinear decrease in collection efficiency above 5 m s -1 wind speed that is not apparent in the results from the present study. As a result, the present study predicts up to 0.16 higher collection efficiency at 8 m s -1 wind speed.
185 Figure S4. Integral Geonor unshielded gauge collection efficiency with wind speed for present study and Colli et al. (2016b) k- SST model for dry snow and wet snow with Colli et al. (2016b) size distribution.

S2.2.2 Precipitation intensity dependence of integral collection efficiency
Using the CFD transfer function, integral collection efficiencies were derived for 0.5 mm h -1 , 1.5 mm h -1 , and 2.5 mm h -1 precipitation intensities for three different snowfall types, between 0.1 mm h -1 and 10 mm h -1 for orographic rain, and between 1 mm h -1 and 10 mm h -1 for thunderstorm rain. The results are shown in Fig. S5 for selected wind speeds (1 m s -1 , 3 m s -1 , and 6 m s -1 ). 195 Figure S5. Integral Geonor unshielded gauge collection efficiency with precipitation intensity for rainfall and snowfall types at 1 m s -1 , 3 m s -1 , and 6 m s -1 wind speeds.

200
Integral collection efficiencies increase with precipitation intensity and decrease with wind speed. For thunderstorm rain at 3 m s -1 wind speed, the integral collection efficiency increases from 0.97 to 0.99 when the precipitation intensity increases from 1 mm h -1 to 10 mm h -1 . For orographic rain, a sharp decrease in the integral collection efficiency is apparent with decreasing precipitation intensity below 1 mm h -1 . 205 For all snowfall types, the integral collection efficiency is shifted to lower values relative to rain. The integral collection efficiency increases with precipitation intensity from 0.42 at 0.5 mm h -1 to 0.70 at 2.5 mm h -1 for dendrites and aggregates of plates at 3 m s -1 wind speed. Increasing the wind speed to 6 m s -1 further decreases the integral collection efficiency from 0.15 at 0.5 mm h -1 to 0.49 at 2.5 mm h -1 .
For dendrites and aggregates of plates, rimed dendrites, and dendrites, integral collection efficiencies are within 0.09 to 0.10 210 of one another for 0.5 mm h -1 precipitation intensities at 1 m s -1 wind speed. This range increases to 0.17 for 0.5 mm h -1 precipitation intensity and 0.34 for 2.5 mm h -1 precipitation intensity at 6 m s -1 wind speed. This provides an estimate of the overall variability in integral collection efficiency due to crystal habit if the precipitation intensity, wind speed, and the occurrence of snowfall are all known, but not the specific snowfall type. In cases where only the precipitation intensity and wind speed is known, and the hydrometeor phase (rainfall or snowfall) and type is uncertain, the range of possible integral 215 collection efficiencies grows dramatically. For example, at 6 m s -1 wind speed, the integral collection efficiencies for rain exceed 0.75, while that for dendrites is below 0.16; hence, the catch efficiency in this case can vary by ~ 0.74 depending on the hydrometeor phase and type.

S3.1 Fall velocity impacts on collection efficiency
The time-averaged numerical model describes the three-dimensional airflow around the unshielded Geonor gauge, including the updraft above the leading edge of the gauge orifice and downdraft at the back of the gauge orifice shown in previous studies (Thériault et al., 2012;Colli et al., 2016b;. The updraft velocity increases sharply with height above the leading edge of the gauge orifice, which appears to play an important role in the horizontal spreading and capture of 225 hydrometeors, particularly for lower fall velocity hydrometeors (Fig. S2). These velocities scale with the wind speed, as shown previously by Colli et al. (2016b), and the relative magnitudes of the wind speed and hydrometeor fall velocity influence the collection efficiency. The hydrometeor fall velocity influences both the free-stream approach angle of hydrometeors before they encounter the local airflow around the gauge and the degree of coupling between the hydrometeor trajectories and the local airflow. Hydrometeors with fall velocities above 2 m s -1 fall more vertically, and their paths show less deviation with the 230 updraft and local airflow around the gauge orifice (Fig. S2). Hydrometeors with lower fall velocities have a smaller approach angle and are more closely coupled to the local airflow around the gauge orifice.
The numerical results for this study are based on a 5 % inlet turbulence value that acts as a bulk turbulence in the atmosphere (Panofsky and Dutton, 1984) but may underestimate experimental results (Armitt and Counihan, 1968). A slip boundary condition was modelled at the surface following the approach of previous studies Baghapour 235 et al., 2017). Further study with a no-slip wall condition under different turbulence conditions could lead to further insights into the influence of turbulence intensity on precipitation gauge collection efficiency.

S3.2.1 Wind speed dependence of integral collection efficiency 240
The integral collection efficiency decreases nonlinearly with wind speed depending on the hydrometeor type and fall velocity.
Large differences in the integral collection efficiency dependence with wind speed are apparent across different hydrometeor types and intensities. Previous studies have shown similar differences across liquid (Nešpor and Sevruk, 1999;Jarraud, 2008) and solid hydrometeor types (Colli et al., 2016a;Colli et al., 2020;Thériault et al., 2012). The fall velocities of snowflakes are generally smaller than those of raindrops; accordingly, the collection efficiency for snowfall at a given wind speed is lower 245 than that for rainfall (Fig. S3). Similarly, dendrites have lower fall velocities than rimed dendrites and columns and plates, and lower collection efficiency.
The integral collection efficiency results decrease continuously with increasing wind speed as the magnitude of the updraft at the leading edge of the gauge increases, free-stream hydrometeor trajectories decrease, and hydrometeors trajectories become more closely coupled with the local airflow around the gauge. For dendrites, the nonlinearity in the integral collection 250 efficiency is more pronounced, as collection efficiencies decrease to small but finite values at higher wind speeds. This is due to the smaller number of hydrometeors with sufficient fall velocity to be captured by the gauge at higher wind speeds.
The differences in collection efficiency for different precipitation characteristics (type, habit, precipitation intensity) illustrate the large variability that can be expected when the characteristics or fall velocity are not considered. This variability presents a particular challenge for mixed precipitation conditions, in which the precipitation type may not be well defined and can 255 change rapidly over time. The proposed expression for the collection efficiency as a function of the wind speed and hydrometeor fall velocity (Eq. S18) provides a means of estimating the collection efficiency over different hydrometeor types and intensities, even if the precipitation type is not well defined.
The integral collection efficiency results using the CFD transfer function developed in the present study show good overall agreement with the results of Colli et al. (2016a) for wet snow and dry snow, as shown in Fig. S4. Integral collection efficiency 260 values in the present study are slightly higher than those of Colli et al. (2016a), attributed in part to the differences in the gauge geometry and hydrometeor drag model as discussed by . The gauge geometry in the present study includes a refined orifice wall thickness and full-length orifice extending down into the gauge housing (Fig. S1). The peak velocities above the gauge in the present study are similar to those observed by , who also used a refined orifice wall thickness and observed reduced peak velocities compared to the results of Colli et al. (2016a). Increases in 265 the velocity magnitude over the gauge would be expected to decrease the collection efficiency in a manner similar to that for increased wind speed; hence, the higher peak velocities above the gauge in the results of Colli et al. (2016a) Fig. S4. Nešpor and Sevruk (1999) used a similar empirical approach for rain by developing an expression for the partial wind-induced error based on free-stream velocity and drop diameter applicable to Mk2, Hellman and ASTA gauges. The integral collection efficiency results of Colli et al. (2016b) were derived directly from numerical CE results for dry snow and wet snow at discrete sizes and wind speeds. 275

S3.2.2 Precipitation intensity dependence of integral collection efficiency
Knowledge of the precipitation type, intensity, and wind speed can provide a means for adjusting gauge catchment totals. For rainfall, the precipitation intensity has been shown to be an important parameter for the estimation of integral collection efficiency (Nešpor and Sevruk, 1999;Jarraud, 2008). A gradual increase in integral collection efficiency with precipitation intensity is observed for intensity values above 1 mm h -1 in Fig. S5. Below this intensity, the integral collection efficiency 280 decreases more rapidly, with the rate of decrease depending on the rainfall type and wind speed. This is in general agreement with the results of Nešpor and Sevruk as presented in Jarraud (2008), who showed a sharper increase in the conversion factor (inverse of integral collection efficiency) below 1 mm h -1 .
Integral collection efficiencies for snowfall also increase with precipitation intensity, as higher intensities correspond with larger numbers of hydrometeors with higher fall velocities and increased collection efficiencies, as shown by Colli et al. (2020). 285 Integral collection efficiencies for snowfall can be much lower than for rain, depending on the wind speed. Differences are apparent across different snowfall crystal habits (e.g. dendrites vs. dendrites and aggregates of plates), with the magnitude of differences increasing with wind speed. This illustrates the difficulty of adjusting snowfall measurements if the crystal habit is not known. The range of possible integral collection efficiency values is even larger under conditions when solid, liquid, and mixed precipitation can all be present. An additional challenge is presented by the measurement of low precipitation 290 intensities for snowfall, where accumulations can be small relative to gauge uncertainties due to environmental factors (e.g. wind, temperature).

S3.2.3 Hydrometeor fall velocity dependence of integral collection efficiency
Integral collection efficiency differences across precipitation types are much smaller when stratified by wind speed and hydrometeor fall velocity (Fig. 5) than when stratified by wind speed and precipitation intensity (Fig. S5) or by wind speed 295 alone (Fig. S3). This results from the ability of the hydrometeor fall velocity to capture differences in the integral collection efficiency across different hydrometeor types and precipitation intensities. The small differences in collection efficiency across different hydrometeor types with the same fall velocity are attributed to the varying contribution from higher fall velocity hydrometeors, with collection efficiencies approaching 1, in the mass-weighted distribution of hydrometeor fall velocities.