The PIP video disdrometer measures hydrometeor size, fall velocity, an estimate of particle shape and PSD. In this study PIP data are used for characterizing the microphysical properties of the snowfall, which include estimates of the mass-dimensional m(D) relations. The PIP instrument works in the same way as its predecessor, the Snow Video Imager (SVI) , but using a camera with a higher frame rate of 380 frames per second. The 2-D grayscale images of falling particle are obtained, when it falls between the camera and the lamp (distance between the two is 2 m) and from these multiple images the particle fall velocity is derived. The camera focal plane is at 1.3 m and the field of view is 64 mm × 48 mm with a resolution of 0.01 mm2. Sampling volume of PIP depends on particle size and fall velocity . For each particle, the PIP processing software automatically records the disk-equivalent diameter DDeq, which is the diameter of a disk with the same area as the particle shadow.

Particles smaller than 14 pixels (approximately DDeq < 0.2 mm) or particles only partly observed or out of focus (blurred) are rejected by the software . Because of the blurring effect, the sizing standard error is estimated to be 18 % . Also, other shape-descriptive particle parameters are retrieved with the image processing software (National Instruments IMAQ) such as particle orientation, total area, and bounding box width and height. Particle fall velocities are recorded as a function of DDeq and values are considered reliable if there are more than two observations of the identified particle and values are higher or equal to 0.5 ms-1 (PIP software release 1308). In later software versions the fall velocity threshold is removed. The PIP dataset includes PSD in m-3mm-1 for every minute. The PSD is also determined as a function of DDeq and subdivided into 105 bins (from 0.125 to 25.875 mm) with the last bin containing particles larger than 25.875 mm. The observed maximum diameter Dmax for each particle is determined by fitting an ellipse inside the bounding box with considering the particle orientation angle as explained in , and the mean ratio between Dmax and DDeq is approximately 1.38. For simplicity, D will be used hereinafter to replace DDeq.

In this study 5 min time series of the observed PSD, the fitted v(D) and the retrieved m(D) relations are utilized  as a function of the diameter D. Typically during the 5 min period 103 particles are observed. The PSD is averaged from 1 min observations, after spurious particle records are filtered out. The v(D) relation is derived by a linear regression fit in the log space for the observed particles during every 5 min . The m(D) relation is retrieved by utilizing the general hydrodynamic theory , where a snow particle mass is computed from the observed dimension, fall velocity and area ratio of a snow particle. The PIP observes falling particles from the side, whereas the particle dimensions projected to the flow are needed for the hydrodynamic calculations. In , errors associated with the observation geometry, and also with the measured PSD were addressed by devising a simple correction procedure; the value of the correction was chosen for each snow event by comparing the estimated liquid water equivalent (LWE) accumulation to precipitation gauge measurements. Similar to the v(D) relation, the power-law m(D) = amDbm fit is determined by a linear regression fit in the log space for the computed particle masses every 5 min. The uncertainty in the retrieved factors of m(D) relation are discussed in detail in .

The weighing precipitation gauge, OTT Pluvio2 200, records every minute the bucket weight expressed in mm. The gauge is located on a platform at 2 m height surrounded by a double wind fence similar to Double Fence Intercomparison Reference (DFIR) fence . In addition, the gauge has a Tretyakov wind shield. The Hyytiälä measurement site is surrounded by boreal forest, and therefore the wind effects are usually moderate. The PIP measurement volume is open and typically affected less by the wind than instruments with enclosed sampling volumes . Therefore, in these wind conditions, the expected wind-induced errors are expected to be small.

The AMF2 two-channel MWR, located 20 m away from PIP, is a sensitive microwave receiver that provides time-series measurements of column-integrated amounts of water vapor and liquid water. Two channels, respectively 23.8 and 31.4 GHz, allow simultaneously to obtain water vapor and liquid water along line-of-sight (LOS) path. The LWP is estimated on a weighted difference of the optical thicknesses of the two channels. In  the LWP was used as a proxy of riming, and in this study we use the LWP as the driven observable for the k-means clustering of the dataset.

The Ikaalinen dual-polarization Doppler weather radar (IKA), used for cross-calibration analysis, belongs to the Finnish weather radar network . It operates at C band and is located circa 64 km west of Hyytiälä. The antenna has a half-power beam width of 1∘. The radar performs volume scans, repeated every 5 min, and range height indicator (RHI) scans over the Hyytiälä site every 15 min.

The IKA data are quality-controlled and calibrated using a number of techniques. The engineering calibration, where different radar components are characterized, is performed during the radar installation and after major system modifications . In addition to the engineering calibration, the radar receiver and antenna pointing are monitored using sun observations . The differential reflectivity calibration is monitored using a combination of vertically pointing scans and sun observations. During the summer months, the IKA radar absolute calibration was checked using the polarimetric self-consistency principle .

Given the continuous monitoring of the radar stability and regular calibration, we use the IKA observations as the calibration standard for the ARM radars. This approach allows us to cross-calibrate the ARM radars even in the presence of radome attenuation caused by, for example, large snow accumulation.

The ARM cloud radar systems operating at X, Ka and W band are integral part of the BAECC snowfall IOP. The antennas of the XSACR and KAZR are mounted on top of two containers located 17 m away from each other. The MWACR is mounted on the same container as KAZR. All the ARM radars make zenith-pointing observations. Looking at the radar technical properties in Table , the range gate spacing and the temporal sampling are comparable, but there is a difference in the beam width between XSACR and the other two systems. To reduce the beam mismatch and to facilitate the intercomparison with the ground-based sensors, all the radar data are averaged to 5 min. To derive consistent X-, Ka- and W-band Ze–S relations, the measured radar reflectivity factors were calibrated and corrected for attenuation. The absolute calibration of ARM cloud radars has been performed at the beginning and during the BAECC IOP using engineering calibration and external standard target procedure.

We have also performed a cross-calibration in order to reduce biases between different radar systems. The cross-calibration method is based on the assumption that in the low reflectivity region at the cloud top the small crystals basically scatter in the Rayleigh regime . We have compared the radar measurements in regions close to cloud top (height higher than 5 km) in non-precipitating ice clouds. The selected radar reflectivity profiles have reflectivity values of less than 0 dB. Furthermore, only cases where no lower clouds or precipitation were detected were used for calibration. The cross calibration was performed for all the cases before and after the snowfall events. Only events where the cross-calibration values did not change are used in this study. As mentioned in Sect. , the IKA radar observations are considered to be the reference for this analysis. The main reason for this selection is that the IKA radar is very stable and its performance is well monitored. Additionally, given its operating frequency, it does not suffer from attenuation during winter storms. Figure a shows the profile of 15 February 2014 at 17:13 UTC in which we performed the calibration between 4 and 6 km and in Fig. b the histograms of the three different calibration errors. The calibration error, measured as the standard deviation (SD) of the histograms in Fig. b, shows that the best result is for the error between Ka and W band and the worst is for C and X band, this being related mostly to the beam width. Looking at Table , it can be seen that the larger the beam width, greater the measured dispersion and vice versa.

One of the reasons for differences in reflectivity measurements can also be attributed to the radome attenuation. For example, the flat shape of the KAZR radome increases the possibility of snow accumulation during a storm. Consequently, when the temperature rises above the melting point of ice, the melting snow could produce heavy attenuation that should be monitored. On the other hand, the conical shape of the MWACR radar limits the amount of accumulated snow, but because of the higher operating frequency it is more sensitive to the freezing rain/drizzle. To monitor the radome attenuation sky-noise analysis has been performed for the millimeter-wavelength radars, KAZR and MWACR. The sudden changes in the sky-noise temperature could have resulted from the increased surface temperature, which may indicate snow melting, and thus increased radome attenuation. The data in these cases are discarded. The stability analysis made with the sky noise is shown in Fig.  as a histogram of sky-noise power measured during 10 snowfall days of BAECC IOP. The SD is around 0.25 and 0.14 dBm, respectively, for KAZR and MWACR radar; according to these values, cases during the 10 snowfall events are excluded from the cross-calibration. This is shown in the Ka-band histogram in Fig. , where a secondary Gaussian-like peak is visible centered around -68.06 dBm.

During the BAECC IOP, radiosondes were launched four times a day. Using these observations as the input to the millimeter-wave propagation model , the two-way gaseous path attenuation was computed for the dataset. This computation has been performed for all the dataset. For example, for 15 February 2014 at 17:24 UTC, the Ka-band two-way gas attenuation is 0.4334 dB. For the same time sample, the W-band two-way gas attenuation is 1.0206 dB. As expected, the attenuation for the W band is about twice as large as for Ka band. By taking into account the gaseous attenuation, the radar calibration offsets during the snowfall experiment were estimated as 2.9 dB for the XSACR, 3.9 dB for the KAZR and 4 dB for the MWACR. The results shown for the case study of 15 February 2014 have also been checked for the other snow events inside the dataset, confirming the consistency of the calibration analysis.

The equivalent reflectivity factor Ze, measured by the radar systems at different wavelengths, and the liquid-water-equivalent snowfall rate S, evaluated from PIP, are the two related variables. The S (in mm h-1) is derived from mass flux as S=3.6ρw∫m(D)v(D)N(D)dD, where m is the mass (in g), v is the velocity (in ms-1), N is the particle size distribution (PSD, in mm-3m-1) and ρw is the liquid water density (in gcm-3). In Eq. () all quantities are expressed in terms of the disk-equivalent diameter D=DDeq and derived from PIP measurements .

The radar data used in this study were collected in the vertical pointing mode. To match radar and in situ measurements, the radar data at the lowest meaningful altitude were used. Given the different radar specifications (see Table ), the Fraunhofer far-field distance for the radars is different. This distance defines the near-field of the radars and is related to the radar antenna size. The beam width difference is related to the antenna diameter, which is respectively 1.82, 1.82 and 0.9 m for XSACR, KAZR and MWACR, so that the Fraunhofer distance (2D2/λ) is approximately 214 m for XSACR, 773 m for KAZR and 514 m for MWACR. Taking into account the near-field influence, all radar data are selected at 400 m .

Another important aspect is related to the different time acquisitions for the various instruments. In Table  we note that the temporal sampling of the radars is 2 s, whereas for the PIP instrument it is 1 min. To ensure similar sampling, we have decided to average data over 5 min. This results in PIP sampling volume of roughly 1 m3 for ice particles falling with a fall velocity of 1 ms-1. As mentioned in Sect. , the averaging is also useful to tackle the differences in radar beam widths.

The Ze–S is expressed in a power-law form, Ze=aSb, where Ze is in mm6m-3 and S is in mmh-1 . In order to estimate the regression coefficients, we can choose nonlinear least squares in the variable linear space or linear least squares in the log–log variable space. We have adopted the latter approach by applying a linear regression as in . The applied log–log model is given by log⁡10Ze=blog⁡10S+log⁡10a, where Ze can be either the time-averaged range-resolved co-polar radar measurement (disregarding the near-field effects) or the numerically simulated backscattering radar response.

Single-scattering computations for spheroids are performed using Python implementation  of the TMM code . The spheroidal particle model has been widely used for describing raindrops but also for approximating more complex particles such as snowflakes . In this study the spheroid model is initiated by using retrieved snowflake masses and maximum dimensions. This leaves the spheroid aspect ratio as a free parameter that adjusts volume, density and therefore the refractive index.

The aspect ratio is defined as rs=bs/as, where as and bs are the horizontal and vertical dimensions of the spheroid (rs=1 spherical particle, rs≥1 prolate particle and rs≤1 oblate particle) . The snowflakes, due to aerodynamic forcing, typically fall with the major axis preferentially oriented horizontally . We have modeled the spheroids preferentially horizontally oriented with 10∘ SD of the canting angle distribution following  and . It should be noted that while snowflakes in the nature may have wider orientation angle distributions, the goal of the particle models used for scattering computations is to provide a link between radar observation and cloud/precipitation properties such as snowfall intensity or ice water content. This goal does not necessary imply that all of the particle model properties coincide with properties of naturally occurring snowflakes. Our studies show that use of wider canting angle distributions results in worse agreement between measured and computed radar reflectivity values .

To test whether the spheroidal model can produce consistent multi-frequency radar observations, the TMM computations are performed using different aspect ratios. If the computations with the same aspect ratio value can explain measured Ze–S relations at all the frequencies, then the spheroidal model can be considered adequate. If different aspect ratios are needed, then the model has failed. As stated above, the aspect ratio defines particle density as ρ=mπ/6DVeq3, in which the mass m is defined as in  and DVeq is the volume equivalent diameter defined from Dmax, the maximum diameter obtained by PIP , as DVeq = rs1/3Dmax. The presence of the aspect ratio inside the density reflects its influence on the complex refractive index of snow mS that is defined through the Maxwell Garnett EMA.

The Γ-size distribution (in mm-1m-3) is assumed to model the PSD: N(DVeq)=Nw,Veqf(μVeq)DVeqD0,VeqμVeqexp⁡(-ΛVeqDVeq), where Nw,Veq is the intercept parameter (in mm-1m-3), f(μVeq) and μVeq parameters are dimensionless, ΛVeq is the slope of the distribution in 1 mm-1 and D0,Veq is the median volume diameter in mm. This Γ-size distribution can be expressed starting from the moments of the snowflake distributions measured by PIP, as in , taking into account the variable changing from Dmax to DVeq as follows: Nw,Veq=Nw,maxdDmaxdDVeq=Nw,max1rs1/3, with D0,Veq=D0,maxrs1/3,ΛVeq=Λmax⁡dDmaxdDVeq=Λmax⁡1rs1/3,μVeq=μmax⁡.

In the computations we have used the Γ-modeled size distribution, with the maximum dimension of 2.5 D0,veq.

The DDA model is used to characterize the single-scattering properties of snowflakes when described with complex and more realistic shape models. Because of computational reasons, here DDA is not used to compute the scattering properties of the observed snowflakes, but rather the pre-calculated lookup tables (LUTs) are utilized for realistically shaped particles. have published an extensive LUT of backscattering properties for realistically modeled unrimed and rimed snow particles. The shape model is obtained by accurately simulating the microphysical processes that lead to snowflake growth. In particular, the snowflake formation is simulated by aggregation of pristine dendrites and subsequent or simultaneous riming of those aggregates using multiple values of equivalent LWP which in turn determine the degree of riming. The simulation of the riming process provides the scattering database to span through a large range of particle masses and sizes allowing to use those microphysical features to constrain the ice particle scattering properties. have shown that during BAECC experiment snow particles were moderately to heavily rimed; therefore the selection of the database that includes rimed particles appears to be justified. The scattering properties of the simulated particles are in fact picked from the LUT by finding the entries that most closely match the retrieved particle size and mass in .

According to the PSD bin sizes of the PIP, the LUT is filtered to find entries which falls within each bin category. Then, using the retrieved m(D) relation determined in  the LUT entries are sorted with respect to the difference between their mass, and the expected particle mass is computed using the retrieved m(D) relation. An arbitrary number of 10 entries that most closely match the retrieved m(D) relation are selected and their scattering properties are averaged in order to define the representative backscattering cross section of that particular size range. Larger number of particles can be picked from the LUT in order to represent a larger variability of particle mass, but the effects of including heavier and lighter particles tend to cancel out in the averaging and do not produce notable differences in the final integrated reflectivity value.

It is worth noting that the particles of are partially horizontally aligned where orientation of their shortest principal axis is, being normally distributed, with the SD of 40∘.

The dataset was divided into three riming classes – lightly, moderately and heavily rimed (LR, MR and HR) snow – following the same logic used presenting the m(D) relations in . Case studies were divided into classes using LWP values for the direct correspondence to the degrees of riming. Since the ice particle mass growth rate due to riming is proportional to LWP along the particle fall trajectory, the LWP can be seen as a proxy for riming e.g.. Given the growth rate, the riming degree of snowfall may also be influenced by the average particle size, such as D0. To take all of this into account, the presented four snowfall events were classified into three subgroups using a k-means cluster analysis trained by LWP and D0. The results are presented in Fig.  where the three clusters are identified in the LWP-D0 space. It is worth noting that riming is strongly related to LWP but almost not dependent on the estimated size D0 of snow particles. In summary, we have analyzed four events for Ka and W bands and three cases (excepted 20 March 2014) for X band. For the Ka and W band radar observations we have 282 data samples, which correspond to 1410 measurement minutes, of which 50.35 % are LR, 37.23 % MR and 12.42 % HR. For X band we have 174 data samples, 870 measurement minutes, divided into 49.42 % of LR, 35.06 % of MR and 15.52 % of HR. In Figs. – the derived Ze–S relations for all radar frequencies and riming classes are presented.

The LR snowfall samples are plotted in Fig. , showing the retrieved liquid-water-equivalent snowfall rate S from PIP (see in Eq. ) with respect to the measured equivalent reflectivity factor Ze from the ARM radars. A representation with Ze in dBZ and S in base-10 logarithm has been chosen to adhere to the log–log model in Eq. (). The parameters of the three Ze–S relations are given in the Table . The accuracy of Ze–S relations has been evaluated using the root-mean-square error (RMSE) in dB (where the error is defined as the difference between observed Ze and estimated from PIP, using the regression coefficients). The normalized RMSE (NRMSE), i.e., RMSE values normalized by the observed reflectivity range, is also presented in the table. The NRMSE in percentages of the regressions shown in Fig.  is about 13 and 10 % for X and Ka/W band, respectively. Both prefactors and exponents of the Ze–S relations tend to decrease with the radar frequency increase similar to what is presented in  and . The regression coefficients are very close to those of  and and are rather different from those of  and . The literature values of the Ze–S relations are summarized in Table .

Similar to the light riming class, the MR snowfall Ze–S observations are shown in Fig. . The prefactor, a, and exponent, b, are slightly different with respect to the LR snowfall class; they decrease with the frequency but they have lower values, especially for X band. The trend of the a coefficient can also be considered in line with  and , while the b coefficient is close to  and  only for W band.

The last result is for HR snowfall, presented in Fig. . The values of the a and b coefficients are lower than those of the previous two riming regimes and than those of  and , having a worse NRMSE accuracy of about 17 % (X band), 14 % (Ka band) and 16 % (W band).

Using data of the studied snowfall cases, the frequency behavior of the a and b power-law coefficients in Table  may be useful to suggest a general trend of the Ze–S relation, even though only three frequency at X, Ka and W band are available. Figure  shows the spectral variation of the a and b coefficients, splitting the results between lightly rimed snowfall (black triangles), moderately rimed (black circles) and heavily rimed snowfall classes (black squares). The spline interpolation has been introduced for the three riming classes to outline a possible trend for these two coefficients. Considering all the limitations of the presented analysis it is still worth noting that (i) the monotonic decrease in the a coefficient with the frequency has been noted for all the three classes in Fig. a (the slope is higher for the LR with respect to the MR and HR), and (ii) the different spectral trend of the b coefficient in Fig. b decreases with the frequency, but it could be also used to separate the three regimes.

While analyzing the presented Ze–S relation trends, we should understand that these relations depend on PSD parameters, such as Nw, m(D) and corresponding single-scattering ice particle properties. The difference between Ze–S obtained for different radar frequency bands arises from the changes in the snowflake scattering properties. In the Rayleigh regime, the dependence of radar cross section (RCS), on D, is given by m(D)2. For higher frequencies the exponent of RCS(D) relation will become smaller, and therefore the exponent of Ze–S relation should decrease as well. However, the relations derived for different snowfall riming regimes are influenced not only by changes in m(D) but also by changes in PSD. Furthermore, here not only changes in average values of, for example, Nw are important but also PSD parameter variations during the recorded events . Therefore, some of the changes in the a and b coefficients between the riming classes are probably caused by the PSD values and variations.

Time series of multi-frequency radar measurements can provide a further insight into the analysis of snowfall regime and the capability to simulate its behavior. Figure  shows the equivalent reflectivity factor Ze as a function of time for the snow case study of the predominantly LR 12 February 2014 case (100 % for X band, 91.67 % for Ka/W band). The black triangles and circles correspond to ARM-radar mean Ze, whereas the bars are related to the variation between their minimum and maximum values within the same averaging time interval of 5 min. A total of 8.33 % of the measurements (black circles in Fig. b and c) at the beginning of the event correspond to the MR snow data (0 % for X band, 8.33 % for Ka/W band) and they are disregarded since the variation index (defined as the ratio between minimum–maximum variability interval and its mean value) is considered to be too high. The different colored lines refer to Ze, simulated using TMM from PIP data with a variable aspect ratio rs between 0.2 and 1 with a step of 0.2. The smaller value rs = 0.2 (red line) indicate very oblate particles, whereas rs = 1.0 (blue line) correspond to spherical snowflakes. By comparing ARM measurements and TMM simulations, the optimal aspect ratio value seems to decrease when increasing the frequency: X-band data are better represented by TMM-derived spherical particles (rs=1), whereas Ka- and W-band results are in agreement with an aspect ratio of 0.6. After 07:00 UTC within the heavy precipitation period, no data are available for X-band radar in this case study, but the optimal aspect ratio tend to change to a value around of rs=0.4 for the millimeter-wave radars (Ka and W band). This frequency dependence of the aspect ratio indicates that the soft-spheroid model is not consistent across the frequencies, for this snow event. This finding is in line with and , who showed that for low-density aggregates the soft-spheroid model may not be adequate.

Figure  again shows Ze as a function of time for the mixed LR–MR 15–16 February 2014 snowfall case (61.02 % for LR and 38.98 % for MR). We can distinguish two main intervals: before the heavy precipitation around 22:10 UTC, as for the previous case, the optimal aspect ratio decreases when increasing the radar frequency, whereas during the heavy precipitation interval (from 22:50 UTC on) the optimal aspect ratio seems to be around 0.6 independent of the frequency that corresponds to the LR time period. Figure  shows the time behavior for 21–22 February 2014, a snow case with all three regimes present (X band: 13.33 % for LR, 56.67 % for MR and 30 % for HR; Ka/W band: 10.14 % for LR, 65.94 % for MR and 23.91 % for HR). Until 22:00 UTC, in the presence of MR snow, the optimal aspect ratios seem to be around 1, 0.8 and 0.8 at X, Ka and W band, respectively, whereas during the heavy precipitation period (23:00–00:00 UTC), in the presence of LR snow, it is constant around rs=0.6 irrespective of the frequency. These considerations are also valid for 20 March 2014 in Fig. , in which the optimal aspect ratio is about rs=0.6 for the millimeter-wave radars (Ka and W band) and in fact it was a predominantly LR case (72.41 % for LR, 24.14 % for MR and 3.45 % for HR). For this case X-band data are not available and thus they are not shown in the figure.

As a general comment on Figs. –, we note that measured data fall within the computed range of uncertainty. The incremental difference in terms of Ze due to an increase of 0.2 in the particle aspect ratio is about 1.7 dBZ at X band, 2.5 dBZ at Ka band and 6 dBZ at W band. The difference between the value for rs=0.2 and rs=1 is on average 5.5 dBZ for X band, 7 dBZ for Ka band and 12 dBZ for W band. By increasing the frequency from X to W band, the radar reflectivity seems to be, in general, more sensitive to the non-spherical shape of the snowflakes, with rs decreasing from 1 to 0.6.

To investigate how the soft-spheroid model performs in terms of reproducing the observed Ze–S relations, the TMM computed reflectivity factors were used to derive multi-frequency Ze–S relations. The relations were computed using different values of the soft-spheroid model aspect ratio. The derived relations are summarized in Table  and shown in Figs. , , and . Similar to the analysis of the Ze time series, presented above, we may conclude that to reproduce the observed Ze–S relations at different frequencies, different spheroid aspect ratios may need to be used. This effect is clearest for the LR class (see Fig. ), where for X band the best fitting aspect ratio is 1 and for W band it is closer to 0.6. For heavier rimed particles this difference becomes less pronounced.

It should be noted that the observed differences between observed Ze and ones computed using TMM are not as large as was previously expected. To investigate why this is the case, single-scattering properties computed using DDA were compared to the TMM results for the three cases shown in Fig. . The computations are performed for the W band. TMM simulations are given by red, green and blue lines referring to different aspect ratios (rs=0.2, 0.6, 1, respectively), whereas DDA results are given by the black line. The dotted line shows the product between the snowflake PSD and the RCS computed using TMM with aspect ratio of 0.6 (green dotted line) and the DDA (black dotted line). This figure shows that TMM computations using lower aspect ratios agree better with DDA. Furthermore, it indicates that there is a compensating effect, where TMM overestimates RCS for smaller snowflakes and underestimates it for larger particles. This may explain smaller differences between the DDA and TMM calculations.

This compensation effect depends on the integration limits used to compute the Ze. In this study we have integrated from 0 to 2.5 D0. To check whether this integration limit is valid, in Fig.  measured and fitted PSDs are shown. As can be seen, the assumed upper integration limit appears to be valid.