## 1. Introduction

Snowfall intensity is an extremely difficult quantity to measure by any means. However, at high latitudes and in many high terrain locations, snow is the dominant form of precipitation and therefore must be measured as accurately as possible for forecasting and water resource applications. Similar to rainfall, single-wavelength radar estimates of snow rate have traditionally been accomplished through power-law relationships of the particle size distribution and radar reflectivity (e.g., Sekhon and Srivastava 1970). However, snowfall is more difficult to quantify than rain because of the large variability in density and size that must be accounted for (Matrosov 1998). Radar reflectivity–snow rate relationships have been developed by measuring snow spectra and assuming either a constant bulk density for ice (Smith 1984) or by using previously published mass–size relationships (Loffler-Mang and Blahak 2001). The Loffler-Mang and Blahak (2001) study utilized a particle size and velocity (PARSIVAL) optical disdrometer to measure snow size distributions (SSDs).

The two-dimensional video disdrometer (2DVD) was first used for snow measurements, mainly the fall speed and shape, by Hanesch (1999), who also developed the first snow “matching” algorithm. Using the fall speed data, Hanesch (1999) used the theoretical method in Böhm (1989) to derive the density versus size power-law relation. Subsequently, Schönhuber et al. (2000) used 2DVD data to compare the liquid equivalent snow rates (SR) with a collocated tipping-bucket gauge with modest success for one event. They introduced the concept of “apparent” volume. The scan lines (SL) recorded by the front- and side-view cameras divide the particle into several slices. In each of these slices, the particle is assumed to form an elliptical cylinder. By summing the volume of all slices, we can obtain the volume of the particle. Because the shape of snowflake is irregular and the disdrometer is not able to detect the internal structure of particle, this volume is obviously overestimated and is termed as the apparent volume. The apparent diameter is the equivolume spherical diameter *D*_{app} (Schönhuber et al. 2008).

More recently, Brandes et al. (2007) analyzed 2DVD data from a number of snow events near Boulder, Colorado, with a collocated heated tipping-bucket gauge. Their main focus was on statistical analysis of the gamma snow size distributions (mainly the median volume diameter *D*_{0}) as well as fall speed from a variety of snow types and environmental conditions. They were able to obtain a bulk density versus *D*_{0} relation by comparing the liquid equivalent snow amounts from a collocated heated tipping-bucket gauge with the apparent volume of snow from the 2DVD over periods of 5 min. They defined the median volume diameter *D*_{0} from the snow size distribution in the usual manner. The average relation they obtained was *ρ* = 0.178*D*_{0}^{−0.922} (units for *D*_{0} are mm, whereas for *ρ* they are in g cm^{−3}). In their Table 2, they compared the density versus size relation from five previous articles with their relation (assuming their *D*_{0} was a proxy for *D*_{app} or size). Some of the variability in the coefficients/exponents was attributed to different measures of size of a snow particle and different snow types. Nevertheless, except for two outliers (Magono and Nakamura 1965; Muramoto et al. 1995), the other three relations fell within the scatter about the Brandes et al. (2007) average relation. Here, we use the average density–*D*_{app} relation from Brandes et al. (2007) as an initial guess for our methodology.

There are other optical, line-scan camera disdrometers that have been used to measure snow size and fall speed, such as the hydrometeor velocity and shape detector (HVSD; Barthazy et al. 2004), which is similar in design to the 2D video, except that two orthogonal views are not provided. Recently, Tollman et al. (2008) used the precursor instrument to the HVSD [called the hydrometeor shape detector (HSD)] to estimate the bulk density variations for several snow events near Helsinki, Finland. The radar used was a collocated precipitation occurrence and sensor system (POSS), and the constant bulk density was estimated by matching the radar-measured *Z _{e}* with that calculated from the HSD, similar to Tokay et al. (2007).

In this paper, we develop a method to estimate the effective snow density as a function of *D*_{app}, as well as *Z _{e}*–SR relations. The measurements we use are from the 2D video disdrometer and the

*Z*from the King City C-band radar. Our procedure is to minimize the difference between (i) the calculated equivalent reflectivity from the snow size distribution measured by the 2D video disdrometer and (ii) the measured radar reflectivity from King City radar in a least squares sense. Moreover, we develop the

_{e}*Z*–SR relations for seven selected snow days from the Canadian CloudSat Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) Validation Program (C3VP; Hudak et al. 2007; Petersen et al. 2007). The double-fence international reference (DFIR) gauge is used to validate our estimated liquid equivalent snow amounts (from the derived

_{e}*Z*–SR relations) for the various events.

_{e}## 2. Data sources

During the winter of 2006/07, a number of in situ and remote sensing precipitation measuring devices were operated at the Center of Atmospheric Research Experiments (CARE) site located near Egbert, Ontario, about 30 km to the northwest of the King City C-band operational dual-polarized radar. The CARE site is a well-instrumented facility including the DFIR gauge for accurate liquid equivalent snow amounts, which were manually derived twice per day. Although the experiment was originally designed to measure winter precipitation for the C3VP, the National Aeronautics and Space Administration (NASA) Global Precipitation Measurement (GPM) ground validation program joined the efforts (Petersen et al. 2007) bringing the 2DVD and a multifrequency radar to the CARE site.

### a. King City radar

The King City radar, operated by Environment Canada, is a C-band (5.5 GHz) dual-polarized radar located north of Toronto, Ontario, Canada. A description of the radar system is given by Hudak et al. (2006). Table 1 lists the main system characteristics. Notable is the narrow 3-dB beamwidth of 0.62° for a C-band system. The radar is on top of a 30-m tower and the radar horizon in the direction of the CARE site is −0.3°. Briefly, the radar transmits in the “simultaneous” mode and receives the H and V components of backscattered power in two separate receivers. Although the radar is a dual-polarized Doppler system, here we only focus on measurement of the equivalent radar reflectivity factor *Z _{e}* at H polarization. During C3VP, the King City radar periodically inserted special scans [polarization plan position indicator (POLPPI) and polarization range–height indicator (POLRHI) to augment the C3VP data collection into the normal operational scan sequence. Table 2 lists the scanning modes. Data from the POLPPI mode were generally used herein that had a time sampling of 10–30 min for the events described in this study. POLRHI (not listed in Table 2) scans were also done periodically over the CARE site and these data were also used as available to improve the time sampling.

The reflectivity data were averaged (in units of mm^{6} m^{−3}) over a small polar area centered around the 2DVD location. For the lowest POLPPI sweep (elevation angle of 0.2°), the polar area selected was 3 beams and 8 gates per beam (range interval 32–34 km; gate spacing of 125 m), whereas for the RHI the range interval was the same except only gates in the height interval 0.2–0.5 km were used for averaging the *Z _{e}* data. The radar averaging “cell” is based on prior intercomparisons of reflectivity between the King City radar and the 2DVD (Thurai et al. 2007). Vertical profiles of the copolar correlation coefficient

*ρ*

_{hv}over the CARE site showed that the values were typically >0.99 down to 200 m above ground level (AGL). Though clutter contamination was nearly negligible over the CARE site area, additional thresholds were imposed; that is, only gates with

*ρ*

_{hv}> 0.9 and standard deviation of

*ϕ*

_{dp}(differential propagation phase) over a moving 8-gate (2 km) window <12°.

The absolute *Z _{e}* calibration of the King City radar was excellent (within ±1 dB or better; e.g., Thurai et al. 2007) as determined by comparison with both the Precipitation Occurrence Sensor System (POSS; Sheppard and Joe 2008) and the 2DVD on 30 November 2006 during a long-duration stratiform event. In summary, the King City radar data were processed to extract the time series of

*Z*over the 2DVD site for the seven snow days during the C3VP campaign. Except as noted, no attempt is made here to analyze the spatial structure of the various snow events from radar or a meteorological perspective.

_{e}### b. 2D video disdrometer

The 2D video disdrometer was originally developed by Joanneum Research to measure rain drop shapes for comparison with dual-polarized radar measurement of differential reflectivity (a full history, detailed measurement principles and performance specifications are given in Schönhuber et al. 2008; see also Kruger and Krajewski 2002). A schematic of the main principles of the instrument is given in Fig. 1. The 2D video disdrometer has two orthogonally placed line-scan cameras and two “bright” lamps, which give two views of the particle as it falls through the virtual measuring area.

In the case of rain drops, it is relatively straightforward to match the drop image from, for example, camera A as it falls through the upper light plane with the same drop image from camera B as it falls through the lower light plane, thus giving the important vertical fall speed measurement. The matching of snow particles is more complicated because of their complex shapes in the two orthogonal views. The only real criterion is that the height *H* of the particle images (as measured in the direction perpendicular to the light planes) from the two cameras must ideally be the same. The matching software originally provided by Joanneum Research is proprietary, and the exact algorithm used cannot be described here. However, their procedure uses the criteria that *H* from the two images be close enough within a certain tolerance and selects the first match as opposed to selecting an optimal match based on all the possible pairs within an a priori time window. Appendix A illustrates via simulations how total mismatching of pairs of spherical particles would affect the fall speed estimation and the *H* values (for spheres this translates to errors in the equivolume spherical diameter *D*). The main points from appendix A are as follows:

- The mismatched fall speed mostly overestimates the “true” values (see Fig. A2).
- For the simulations involving identical spheres, the histogram of
*D*values are spread out significantly from the true fixed*D*= 3 mm with positive skewness (long tail; see Fig. A3). - For the simulations involving the more realistic gamma SSD, the mismatched
*N*(*D*) also has a long tail for*D*> 8 mm, which alters the calculated*Z*and liquid equivalent snow rate from the true values (see Fig. A4)._{e}

^{−1}herein to have more snowflakes for a possible match. For each possible match, weights are attached to the Hanesch (1999) set of criteria (see Table 3), and the optimal match is selected based on the maximum score defined here as the summation of the matching factors (

*f*;

_{i}*i*= 1, 2, and 3) multiplied by the weights (i.e., the score is equal to 0.6

*f*

_{1}+ 0.2

*f*

_{2}+ 0.2

*f*

_{3}). Note that we give the highest weight (0.6) to the ratio of difference of two heights to maximum scan lines, because it is the only real criterion directly applicable to the data provided in the two views of the 2DVD. The mode of the ratio of width to height is 0.8 consistent with that obtained by Hanesch (1999, her Fig. 2.10). See Barthazy et al. (2004) for a more detailed discussion of a different matching algorithm they used for a similar instrument, the HVSD which gives parallel views (i.e., not the orthogonal views of the 2DVD).

To evaluate the original matching algorithm with the new rematched algorithm, the case of 22 January 2007 (a synoptic-scale snow event) was used. Figure 2 shows the fall speed versus *D*_{app} [defined later in (1)] for (i) the original matching algorithm and (ii) the new rematching algorithm. Note that, for plotting purposes, a threshold of 6 m s^{−1} was applied in (i); otherwise, too many values exceeding 6 m s^{−1} make it difficult to compare with (ii). It is clear that the rematching algorithm has improved the data quality by greatly reducing the unreasonably high fall speeds (for snow).

As a further evaluation of the algorithms, we compare in Fig. 3 the snow size distribution *N*(size), which is averaged over the whole event on 22 January 2007, between the snow video imager (SVI; L. Blivens 2008, personal communication), the original matching algorithm, and the new rematching algorithm. Because the SVI-based *N*(*D*) does not have the matching problem and is less sensitive to the undercatchment problem, we have used it as a reference to compare with our rematching-based *N*(*D*). Note that size now refers to the maximum dimension of the image (for SVI) and for the 2DVD it refers to max(*W*_{r1}, *W*_{r2}, *H*_{1}, and *H*_{2}) where *W*_{r1,2} (rectangle width) and *H*_{1,2} are defined in Fig. 4. Note that the long tail in the size distribution when the original matching algorithm is used has now been reduced (in agreement with the simulations in appendix A), and the slope of the rematched *N*(size) is nearly the same as that from the SVI. However, the concentration is more or less uniformly reduced compared with SVI, but the 2DVD detects particles as large as 22 mm, compared with 16 mm for the SVI.

## 3. Methodology

The inputs to our algorithm are (i) the time series of radar-measured *Z _{e}* at the location of the 2DVD and (ii) the

*N*(

*D*

_{app}) from the 2DVD after rematching. Here, we use the notation

*D*

_{app}for apparent diameter of the snow particle, because it is based on the apparent volume based on the two orthogonal views. Note that the true volume of the snow particle cannot be determined from such imaging instruments. For each (orthogonal) view, as illustrated in Fig. 4, we have the following: the height

*H*(which should be very close in value), the width

*W*, and the area

*A*of the “shadowed” pixels. The width is defined as the maximum distance from the set of horizontal scan lines (not shown) in each image (i.e., from the first shadowed pixel on the left to the last shadowed pixel on the right in each line).

*W*

_{1,2}are calculated assuming the shadowed area equals the area of an equivalent ellipse with widths of

*W*

_{1,2}, respectively. The term

*D*

_{app}is the equivolume spherical diameter, where the volume is given in (1a).

To calculate the backscattering properties of the particles measured by the 2DVD, we consider snow to be a mixture of ice and air. Using the Maxwell-Garnet (1904) mixture formula, the effective permittivity of the snow particle can be expressed in terms of the fractional volume concentration *c* of the ice part and assuming that ice is the inclusion and air is the matrix material. It is straightforward to show that the dielectric factor *K*_{snow} = *cK*_{ice}; furthermore, *c* (from physical grounds) equals the ratio of snow density to ice density.

*K*

_{ice}and

*K*are the dielectric factors for ice and water and

_{w}*ρ*

_{ice}and

*ρ*

_{snow}are the densities of ice and snow, respectively. Note that

*K*

_{ice},

*K*, and

_{w}*ρ*

_{ice}are known; the former two are functions of frequency and temperature, whereas

*ρ*

_{ice}= 0.917 g cm

^{−3}at 0°C.

*ρ*

_{snow}is the power-law form

*ρ*

_{snow}=

*α*(

*D*)

^{β}. Because we only have

*D*

_{app}, the

*ρ*

_{snow}–

*D*power law is also referred to as an apparent relation, as opposed to the true relation. Henceforth this distinction will not be explicitly stated. Because

*β*≈ −1, the

*Z*is the fourth moment of

_{e}*N*(

*D*). SR (mm h

^{−1}) from the 2DVD is calculated from where

*A*is the virtual measurement area, Δ

*t*is the integration time, and the summation is for all particles within Δ

*t*. Note that

*D*and

*V*are apparent values, as previously defined. It is clear that, if

*β*≈ −1, then SR is the second moment of

*N*(

*D*).

In previous work using this methodology (Tokay et al. 2007; Bringi et al. 2008), we have estimated *α* and *β* as follows: we calculate the equivalent radar reflectivity factor at *H* polarization from the 2DVD (*Z*_{ssd} at each 1-min interval) using the T-matrix scattering method at C band with the following assumptions:

- oblate spheroid shape, with fixed axis ratio = 0.8;
- Gaussian canting angle distribution, with (0;
*σ*= 40°) to simulate tumbling; and - permittivity of snow using Maxwell-Garnet model, with
in units of g cm ^{−3}.

*Z*

_{ssd}that is coincident with the radar samples (

*Z*

_{radar}). The

*α*and

*β*values are adjusted to minimize the difference between

*Z*

_{ssd}and

*Z*

_{radar}in a least squares sense. The initial guesses are 0.178 for

*α*and −0.922 for

*β*(Brandes et al. 2007). Once the optimal

*α*and

*β*values are found, we calculate the SR as in (3) and the final

*Z*

_{ssd}at 1-min intervals by rerunning the T-matrix code. This procedure allows for a sufficient number of points to fit a power law of the form

*Z*

_{ssd}=

*a*(SR)

*using a weighted total least squares (WTLS) technique (Draper and Smith 1981). Validation was provided by the DFIR gauge at the CARE site, which is used to manually determine the liquid equivalent snow accumulation at 1300 and 2100 UTC. The intercomparison between the DFIR and the accumulation from the optimal*

^{b}*Z*

_{ssd}–SR relation for seven snow events is provided in Bringi et al. (2008). Although the comparisons in Tokay et al. (2007) and Bringi et al. (2008) were good in terms of fractional standard deviation (20%), the normalized bias (−11%) was somewhat higher than expected. This may have been due to the errors in the original matching algorithm, strong horizontal winds, or some combination of the two. In appendix B, the role of horizontal winds and/or particle loss resulting from imperfect matching algorithm, both of which reduce

*N*(

*D*), are discussed within the context of the effect on the optimal

*α*and

*β*values and the SR. The main point of appendix B is as follows:

- Reducing
*N*(*D*) by a constant factor of 50% produced higher optimal*α*= 0.21 and*β*= −0.80 values. This resulting SR was underestimated by 34%. See Figs. B1 and B2.

*α*and

*β*values has a self-adjusting feature that reduces the magnitude of the error in SR from what it would have been if the initial

*N*(

*D*) was used. Because we have already rematched the data, we now extend the methodology to include a concentration adjustment parameter (

*γ*> 1) that is jointly estimated along with

*α*and

*β*(i.e., a three-parameter simultaneous estimation for minimizing the difference between

*Z*

_{ssd}and

*Z*

_{radar}).

## 4. Measurements and results

### a. The event of 20 January 2007

To illustrate the methodology, we chose the 20 January 2007 event, which gave the maximum snow accumulation during the C3VP period (12 mm of liquid equivalent as measured by DFIR at the CARE site). Figure 5 shows the time series of 1-min-averaged *N*(*D*_{app}) for the entire event. This particular event was a lake effect snowband with surface temperatures decreasing nearly linearly from −2.5°C at 0000 UTC to −14°C at 1830 UTC and then rising to −10°C at 0000 UTC 21 January. The wind direction shifted at 0600 UTC abruptly from the northwest (300°) to the north (350°). Figure 6 shows the radar-measured *Z*(*Z*_{radar}) and the *Z*_{ssd} based on optimal *α*, *β*, and *γ*: 0.115, −1.188, and 1.92, respectively. The *Z*_{radar} time series is highly correlated with *Z*_{ssd} without any need for a time shift.

Figure 7 shows the scatterplot of *Z*_{ssd} versus SR [from Eq. (3), using optimal values of *α* and *β* and scaled by *γ*]. The power-law fit using the weighted total least squares is *Z _{e}* = 204 SR

^{1.58}for this event. As a validation, we compute the liquid equivalent snow accumulation using the SR from the power law and compare with the DFIR data, as shown in Fig. 8. As can be noted, the agreement is excellent for this event. Figure 9 shows the 24-h snow accumulation map by using the

*Z*

_{radar}from the King City radar to estimate SR using the derived

*Z*

_{ssd}–SR relation. The low elevation angle (0.2°) POLPPI sweeps were spaced ∼10 min apart during this event. The snow accumulation band is oriented northwest–southeast, with maximum accumulation occurring in a small region near the CARE site. The maximum accumulation at the CARE site is close to 12.5 ± 1.25 mm, which is in excellent agreement with the DFIR reading shown in Fig. 8.

### b. Summary for seven events

The seven snow days from C3VP were chosen based on DFIR accumulations and ranged from 0.4 to 12 mm. Table 4 shows the optimal values of *α*, *β*, and *γ* and the *Z*–SR power-law coefficient and exponent *a* and *b*, respectively, for each day. The 7 December 2006 event was split into two periods, whereas the 14 February 2007 event was split into four periods. These periods were chosen to allow for higher correlation between *Z*_{radar} and *Z*_{ssd} as opposed to using the entire duration. However, the *Z*_{ssd} and SR from these split periods were all combined together to arrive at the final *Z*–SR power law.

The validation of the *Z*–SR relation is done by comparing the liquid equivalent accumulation with the DFIR manual records. Figure 10 shows the scatterplot of the accumulations from DFIR (made twice per day) with the various daily *Z*–SR relations. The fractional standard error is 26%, and the normalized bias is 2.1% (excluding the outlier from 17 January 2007). The normalized bias is very low and justifies our use of the new additional *γ* parameter in our methodology. The fractional standard deviation is also reasonable, considering the various other sources of error such as point versus volume measurement (2DVD versus radar), height of the radar resolution above ground, and the “representativeness” error of the 2DVD itself.

The results in Fig. 10 show overall excellent agreement between the DFIR and 2DVD. The high correlation is somewhat surprising, given that our methodology only permits an effective density–size relation to be derived and the variations in *α* would likely not be well correlated with snow types (e.g., aggregates versus rimed ice crystals). During the long events (e.g., 20 January 2007), the snow types would likely change depending on temperature, humidity, degree of riming, etc. Although there are a few outliers for *α* and *β* in Table 4, the rest of the values appear within the range quoted in the literature (Brandes et al. 2007). This is encouraging, considering that the *D*_{app} defined here is different from other measures of size, which are not based on having two orthogonal views.

Fujiyoshi et al. (1990) have compiled *a* and *b* values from 11 separate studies and have presented them as a scatterplot. The range of *a* is [100–3000]; for *b*, it is [1–2.3]. One caveat is that the range of *a* may be larger, because some of the studies do not clearly distinguish between *Z _{e}* and

*Z*(reflectivity factor). The range in our study is much narrower for

*a*and

*b*and closer to the values obtained by Ohtake and Henmi (1970), Marshall and Gunn (1952), and Boucher and Wieler (1985), as summarized by Fujiyoshi et al. (1990). We note that Environment Canada uses a climatological

*Z*–SR relation [

*Z*= 1780SR

^{2.21}based on Eq. (13) of Sekhon and Srivastava (1970)] for their operational weather radar network, where

*Z*is the reflectivity factor, which is 6.5 dB larger than the equivalent radar reflectivity factor

*Z*(Smith 1984). As a future validation, we expect to use our

_{e}*Z*–SR relations for the seven days and compare the daily accumulations with other climatological gauges operated by Environment Canada within ∼100 km of the King City radar.

_{e}## 5. Summary and conclusions

A methodology has been described to derive a *Z _{e}*–SR power-law relation using a 2D video disdrometer (2DVD) and a well-calibrated weather radar. The 2DVD is used to calculate the

*N*(

*D*

_{app}). A new rematching algorithm was developed to improve the quality of the fall speed estimates from the original matching algorithm used in prior work (Tokay et al. 2007; Bringi et al. 2008). The snow density in our study is parameterized as

*γ*has been introduced to account for possible imperfect matching or undercount of particles, most likely because of strong horizontal wind. The set of three parameters (

*α*,

*β*, and

*γ*) are estimated by minimizing the difference between the radar-measured equivalent reflectivity factor

*Z*and the 2DVD calculated equivalent reflectivity in a least squares sense. Once the optimal

_{e}*α*,

*β*, and

*γ*values are determined, the

*Z*–SR power law is derived.

_{e}Seven snow days have been analyzed; for validation, the derived liquid equivalent accumulations were compared with a manually recorded DFIR gauge amounts. The normalized fractional standard error was 26%, and the normalized bias was 2.1%. This is the first time that such accuracy has been achieved using an imaging disdrometer and a scanning weather radar. The *Z _{e}*–SR power-law coefficient and exponent appear to be in the expected range as in the conventional

*Z*–

_{e}*R*relations. This is not surprising, because we are relating radar reflectivity to liquid equivalent snow rate. However, the

*Z*for snow is proportional to the fourth moment of the size distribution, whereas SR is proportional to the second moment (for rain, it is 6 and 3.67, respectively). Hence, the errors resulting from size distribution variability are less for

_{e}*Z*–SR relations than for

_{e}*Z*–

_{e}*R*relations.

Clearly, more validation is needed, and this is the subject of future work, where the King City radar PPI scans will be used for the seven days to generate snow accumulation maps, which can be compared with other snow gauges in the vicinity of the radar. It may also be possible to do hydrometeor classification (e.g., between pristine ice crystal, dry snow, and wet snow; Ryzhkov et al. 2005) using polarimetric radar measurements prior to application of a *Z _{e}*–SR relation. There is also the question if the

*Z*–SR relation developed using data at one site can be generally applied to the whole area scanned by the radar. But these questions plague conventional

*Z*–

*R*relations as well. It would be preferable to have a collocated vertically pointing radar with the 2DVD to achieve better time sampling, which should improve the results of the methodology. For example, the microrain radar (MRR; Peters et al. 2005) is a possible relatively inexpensive choice.

The 2D video deployment at the CARE site for C3VP was made possible by the NASA Ground Validation Program under the management of Matt Schwaller (NX06AG89G). GJH and VNB acknowledge support under the NASA PMM Science Program via NNX07AD47G. WAP acknowledges support under both NASA Ground Validation Science and PMM Science funds (NNX07AK39G). The authors are grateful to Peter Rodriguez and Steven Brady of Environment Canada for their 2DVD installation and subsequent maintenance. We are grateful to all the participants involved in the C3VP experiment who made the field program such a success.

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# APPENDIX A

## Matching Problem

The basic design of the 2DVD (Fig. 1) includes two line-scan cameras that are aligned perpendicular from each other and separated (in the vertical plane) by a specified distance (Cd). The precise distance between the two cameras is calibrated [for the Colorado State University (CSU) low-profile 2DVD, the separation distance is nominally 6 mm]. Because two light planes are separated in the vertical direction, it is necessary to match the images from the two cameras (see Barthazy et al. 2004 for details for the HVSD instrument). For particles with regular shapes and constant density, such as rain drops, we can apply the characteristic properties, such as a size–velocity relation, the symmetric shape, and the axis ratio to the matching algorithm. Unfortunately, the shapes of snowflakes are complex, and their fall speeds are only weakly related to their size. Therefore, the mismatching problem is more significant for snow than for rain. The only physical criterion for matching snow images is the height of the particle (inferred from the number of affected scan lines) from the two orthogonal images. If the snowflake falls vertically, the heights from the two cameras should be nearly identical (the change of orientation resulting from turbulence within the 6-mm separation or digitization error may cause a slight difference in height). In this appendix, we study how mismatching affects the SSD measurement and the effective size–density estimation via simulations assuming total mismatch.

*W*falling with different fall speeds (

*V*

_{1}and

*V*

_{2}) and separated in the vertical by a distance

*D*. Particle 1 is sampled by the light plane of camera A at

*t*

_{10}= 0 and the light plane of camera B at

*t*

_{11}= Cd/

*V*

_{1}. Particle 2 is sampled by camera A at

*t*

_{20}=

*D*/

*V*

_{2}and the light plane of camera B at

*t*

_{21}= (Cd +

*D*)/

*V*

_{2}. Because of mismatching, the two simulated fall speeds of particles 1 and 2 are It is assumed that the fall speeds of snowflakes are within a reasonable range (∼0–6 m s

^{−1}). In the matching algorithm, an acceptable velocity range (

*V*

_{min}–

*V*

_{max}) is therefore set. From this velocity range, we can decide the matching time widow and the possible maximum distance between two particles

*D*

_{max}, which is given by The number of shadowed or affected scan lines of the two particles are equal to their heights (

*H*=

*W*for spherical particles) divided by the falling velocity (

*V*

_{1}or

*V*

_{2}) and then divided by the scan period of the camera T. The vertical resolution of the image (or pixel height) is equal to the fall speed multiplied by the scan period of camera T. The value of T for the CSU 2DVD is 18

*μ*s (equivalent to a scan frequency of 55.555 kHz). So, the measured heights

*W*of 3 mm) whose fall speeds are distributed uniformly from 0.2 (

*V*

_{min}) to 6 m s

^{−1}(

*V*

_{max}). The distances between each pair were distributed uniformly from 0 to

*D*

_{max}. Figure A2 shows the scatterplot of (

*V*

_{1}

*−*

^{m}*V*

_{1})/

*V*

_{1}versus (

*V*

_{2}

*−*

^{m}*V*

_{2})/

*V*

_{2}. Because the data are restricted to the second and fourth quadrants, it means that the errors of the two simulated speeds are in opposite directions (i.e.,

*V*

_{1}

*is overestimated when*

^{m}*V*

_{2}

*is underestimated). In this case, the amount of overestimation is much more than the underestimation. The example shows that, on average, the fall speeds will be overestimated because of mismatch. The overestimated fall speeds will cause an overestimation in heights and volumes; subsequently, the*

^{m}*D*

_{app}will be overestimated in this situation. Figure A3 shows the histogram of simulated

*D*

_{app}. The long tail shows the overestimation of

*D*

_{app}resulting from mismatch.

To better understand the impact of image mismatching for a distribution of sizes, we further simulated 5584 pairs of spherical particles whose diameters ranged from 0.125 to 25.125 mm with a bin width of 0.125 mm. These particles were assumed to follow a Gamma distribution with *N _{w}* = 21 325,

*D*

_{0}= 3.6 mm, and

*μ*= 1 (Bringi and Chandrasekar 2001). For each diameter interval, we know the number of particle pairs whose value of

*D*is at the midpoint of the bin interval. For each pair the simulation described earlier [(A1)–(A3)] was performed assuming the minimum (

*V*

_{min}) and maximum (

*V*

_{max}) in (A2) to be 0.2 and 6 m s

^{−1}, respectively. The actual fall speeds [

*V*

_{1}and

*V*

_{2}in Eq. (A2)] were uniformly distributed in the interval 0.5–4 m s

^{−1}(the fall speed range from Hanesch 1999). We apply the simulation for all the pairs within each diameter interval in the gamma size distribution, which results in a new size distribution resulting from simulated total mismatch. We further assumed the Brandes et al. (2007) density–size relation and applied the T-matrix method to get the true reflectivity (

*Z*around 36 dB

_{e}*Z*) for the input gamma parameter values. Using the methodology described in section 3, we adjust

*α*and

*β*to match the

*Z*from the simulated distribution to the true. These new

*α*and

*β*values and the mismatched volume

*V*for each diameter interval were then used to compute SR according to (3).

The simulations were repeated 100 times. Figure A4 shows the comparison of true or input *N*(*D*_{app}) with simulated *N*(*D*_{app}) from one of the 100 simulations. As expected, the simulated *N*(*D*_{app}) has a long tail resulting from total mismatch. To reduce the effect of spurious large size particles, the *Z* matching procedure would lower the density of the large particles. This means that the exponent in the density–size relationship will be smaller than the true one. However, these large spurious particles are still part of the distribution and will contribute to the *Z* value; hence, the coefficient will also be smaller than the true one. Figure A5 shows the coefficients and exponents of the effective density–size relations for all 100 simulations. The value of *α* was 0.1742 ± 0.0044, and the value of *β* was −0.9635 ± 0.0151. The SR was overestimated by 11.7 ± 0.0566%. Note that this amount is for the situation of a total mismatch between the two particles in each diameter interval of the distribution and therefore can be assumed as the upper bound of SR error resulting from the mismatch.

# APPENDIX B

## Other Factors Affecting the Accuracy of the Snow Size Distribution Measurement by 2DVD

It is well known that snow measurements under windy conditions result in varying amounts of inaccuracies. In the case of the low-profile 2DVD, the wind-induced errors that exist in the tall 1-m unit [simulated by Neŝpor et al. (2000) for raindrops] are believed to be much smaller. However, as in all disdrometers that use a horizontal virtual sensing area located below the orifice, the shadowing effect caused by partial filling of the virtual sensing area resulting from horizontal movement of the particles in windy conditions reduces the concentration. This shadowing effect can clearly be observed in real time, because the 2DVD calculates the (*x*, *y*) position of each particle falling through its virtual sensing area. Although empirical corrections for rain rate have been explored by Godfrey (2002) for the tall 2DVD in rainfall, the snow underestimation can be more severe because of the smaller fall speeds. Brandes et al. (2007) used a double-fence wind shield for their 1-m-tall 2DVD unit. In addition, they only analyzed events with ambient wind speeds <4 m s^{−1}. In the case of our deployment of the low-profile 2DVD unit at the CARE site, we built a partial “makeshift” double-fence wind shield that extended around a semicircle (from southwest to northeast).

In addition, because the matching algorithm used herein is not “perfect,” some snowflakes simply cannot be matched and are rejected by the matching algorithm. Subsequently, the measured concentration will be lower than the actual one. To understand the impact of the wind and/or the imperfect matching algorithm on the resulting *Z*–SR relation, we analyzed a 3-min segment of 2DVD data and examined the information from each particle. The data segment was from 20 January 2007 (0251–0254 UTC), and these data were used as the true SSD with the Brandes et al. (2007) density–size relation assumed as the true density. From this, the T-matrix scattering method gives true *Z _{e}* and true SR. Now, a random percentage

*r*(ranging from 0% to 50% with an interval of 2%;

*r*is referred to as the particle loss factor) of snowflakes was then removed, and the resulting SSD was computed. After applying the

*Z*matching procedure, coefficient

_{e}*α*and exponent

*β*of the effective density–size relation were retrieved. Figure B1 shows the results of the simulations. Because the concentration is smaller than the true concentration, coefficient

*α*and exponent

*β*artificially increase to match the true

*Z*. Note that, in the T-matrix scattering calculations, the permittivity of the particle is not allowed to exceed 0.917 g cm

_{e}^{−3}; that is,

*γ*= 1/(1 −

*r*0.01), where

*r*is in percent.

Figure B2 shows the corresponding errors of SR versus *r* (using the *α* and *β* values from Fig. B1). It shows that the errors of SR are less than the particle loss factor (e.g., 50% particle loss only results in a 34% SR underestimate). It is clear that the two-parameter *Z _{e}* matching procedure can only partially compensate the error due to particle loss because

*Z*is approximately related to the fourth moment of the SSD whereas SR is related to the second moment.

_{e}From Eq. (2) one could define *α*′ = (*α*^{2}*γ*)½ thus reducing the *Z _{e}* matching procedure to a two-parameter minimization problem with

*α*′ and

*β*. Following the

*Z*matching procedure, which estimates

_{e}*α*′, we cannot compute SR from Eq. (3), because it now involves the parameter

*α*″ =

*αγ*, and

*α*″ cannot be obtained from

*α*′. Hence, it is not possible to derive a

*Z*–SR relation. It also follows that one must use a three-parameter (

_{e}*α*,

*β*, and

*γ*) minimization, as described in section 3.

Characteristics of the King radar.

Scan strategy of the King radar. CONVOL refers to an operational scan sequence that is collected every 10 min. POLPPI refers to a special scan (1° tilt PPI) that was implemented during C3VP.

The matching criteria used by Hanesch (1999) along with the matching factors and weights using in the rematching algorithm.

Best-fit *α*, *β*, and *γ* for the density–size relation, best-fit values of *a* and *b* for the *Z*–SR relation, and the DFIR accumulations for all events used in this study. Note that several of the events were split into multiple time periods, as described in the text.