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  • View in gallery

    (left),(middle) One hour of the four measured values and (right) their time averages. The temperature scale in the reflectivity time–height profile is obtained using Rapid Update Cycle (RUC) analysis. Note that is smoothed in time to match the lower resolution of the X-band radar and it is shown thresholded to the detectable level of . Forty minutes during the most intense period were analyzed.

  • View in gallery

    Time–height sections of the radar observables after the preprocessing described in the text. The 40-min time segment shown is between 18 h 8 min and 18 h 48 min from Fig. 1. (left) The reflectivity fields: (top) VertiX, (middle) W band in the middle, and (bottom) DWR. (right) Doppler velocity fields: (top) VertiX, (middle) W band, and (bottom) . In the two bottom images, only the region for which the retrieval is applied, i.e., with DWR > 0.4 dB, is shown.

  • View in gallery

    Dots show all our measurements in the retrieval region in the space (here, value represents , i.e., contribution of the air vertical motion neglected). Curves show the computations of these measurements for the same model descriptors as in Fig. 5 in Part III, for the same D* = 1 mm. This value of D* is taken for all observation pixels with DWR in the interval indicated by the two vertical lines. The black and red lines show the results obtained for inverse exponential PSDs with particle size defined as the liquid-equivalent diameter and actual physical diameter, respectively.

  • View in gallery

    (a) Number of model descriptors (defined by a pair of a PSD and an mu relationship; maximum 52) providing at least one solution satisfying all measurements. For all model descriptors, we take all pairs of ρ* and D23 that satisfy simultaneously the measured values of DWR, , and DDV, within measurement error. (b) The addition of all these solutions defines the number of ensemble members that simultaneously satisfy all the measurements.

  • View in gallery

    Four examples of the field of D23 obtained from four different model descriptors. The presented results are obtained for (a) untruncated with velocity calculated from (10b) in Part III, (b) untruncated with velocity calculated using the mass–velocity relation from Szyrmer and Zawadzki (2010), (c) with velocity calculated from (10c) in Part III, and (d) truncated with velocity calculated from (10a) in Part III.

  • View in gallery

    Vertical profiles of (top) the time-averaged retrieved quantities and (bottom) their standard deviations. In the bottom panels, we show the average standard deviation of all gates at this height (black) and the standard deviation from the average of the retrieved quantities at a given height (red).

  • View in gallery

    (left) The fields of ensemble averages of retrieved values and (right) the fields of their relative standard deviation (RSD; %). The retrieved variables are (from top to bottom) characteristic diameter ; liquid equivalent diameter , that is, the mean mass-weighted diameter of the distribution with size defined by Dmlt; mass content (Qs); effective density ; normalized concentration ; and normalized concentration of the distribution with size defined by Dmlt . A logarithmic scale is used for and in order to reduce the dynamic range of values.

  • View in gallery

    (top) Computed vertical air velocity w and (bottom) its time-average profile (red in the bottom panel, increased by 0.4 m s−1). Also in the bottom panel, the w ±1 SD profiles (dot–dashed red lines) and time-average reflectivity-weighted velocity profiles: (thick line) and (thin line).

  • View in gallery

    (a) A sample Doppler spectrum of the W-band radar. Similar spectra were present throughout the entire period of analysis. (b) Updraft required to generate supercooled water in the presence of snow at 800 hPa and −15°C.

  • View in gallery

    Averaged profile of the vertical gradient of the flux (from the retrieved ) and averaged profiles of the three terms in (A6). Term 1 in (A6) describes the vertical gradient of (from the measured X-band reflectivity factor). Terms 2 and 3 in (A6) are the estimated contributions to the X-band reflectivity flux from the VDF and AGG processes in the absence of nucleation, i.e., outside the layer delimited by two horizontal lines. In this layer, an important increase of (Qs) flux with falling distance cannot be explained by VDF and AGG processes only. The time averaging is over the 15-min segment containing the two major snow trails.

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Snow Studies. Part IV: Ensemble Retrieval of Snow Microphysics from Dual-Wavelength Vertically Pointing Radars

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  • 1 Department of Atmospheric and Oceanic Sciences, McGill University, Montreal, Quebec, Canada
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Abstract

Based on the theory developed in Part III, this paper introduces a new method to retrieve snow microphysics from ground-based collocated X- and W-band vertically pointing Doppler radars. To take into account the variety of microphysical relations observed in natural precipitating snow and to quantify the uncertainty in the retrieval results caused by this variety, the retrieval is formulated using the ensemble-based method. The ensemble is determined by the spread of uncertainties in the microphysical descriptions applied to map the same radar observables to the retrieved quantities.

The model descriptors use diverse assumptions concerning functional forms of particle size distribution and mass–velocity relations, all taken from previous observational studies. The mean of each ensemble is assumed to be the best estimate of the retrieval while its spread is defined by the standard deviation that characterizes its uncertainty. The main retrieved products are the characteristic size, the snow mass content, and the density parameter, as well as the vertical air motion. Four observables used in the retrieval are the difference in reflectivities and in Doppler velocities at two wavelengths, together with the equivalent reflectivity factor and Doppler velocity at X band. The solutions that are not consistent with all four observables after taking into account their estimated measurement errors are eliminated from the ensembles. The application of the retrieval algorithm to the real data yields a snow microphysical description that agrees with the snow characteristics seen in the vertical profile of the observed Doppler spectrum.

Corresponding author address: Isztar Zawadzki, Dept. of Atmospheric and Oceanic Sciences, McGill University, Montreal QC H3A 0B9, Canada. E-mail: isztar.zawadzki@mcgill.ca

Abstract

Based on the theory developed in Part III, this paper introduces a new method to retrieve snow microphysics from ground-based collocated X- and W-band vertically pointing Doppler radars. To take into account the variety of microphysical relations observed in natural precipitating snow and to quantify the uncertainty in the retrieval results caused by this variety, the retrieval is formulated using the ensemble-based method. The ensemble is determined by the spread of uncertainties in the microphysical descriptions applied to map the same radar observables to the retrieved quantities.

The model descriptors use diverse assumptions concerning functional forms of particle size distribution and mass–velocity relations, all taken from previous observational studies. The mean of each ensemble is assumed to be the best estimate of the retrieval while its spread is defined by the standard deviation that characterizes its uncertainty. The main retrieved products are the characteristic size, the snow mass content, and the density parameter, as well as the vertical air motion. Four observables used in the retrieval are the difference in reflectivities and in Doppler velocities at two wavelengths, together with the equivalent reflectivity factor and Doppler velocity at X band. The solutions that are not consistent with all four observables after taking into account their estimated measurement errors are eliminated from the ensembles. The application of the retrieval algorithm to the real data yields a snow microphysical description that agrees with the snow characteristics seen in the vertical profile of the observed Doppler spectrum.

Corresponding author address: Isztar Zawadzki, Dept. of Atmospheric and Oceanic Sciences, McGill University, Montreal QC H3A 0B9, Canada. E-mail: isztar.zawadzki@mcgill.ca

1. Introduction

Previous studies have shown that dual- (or multi-) wavelength measurements can provide estimates of the particle size distribution (PSD) characteristic size for a wide range of hydrometeors from clouds and precipitation in liquid, solid, or mixed phases. Regarding ice and snow, algorithms based on dual-wavelength radar measurements were developed for particle sizing in cirrus clouds from the ground (Matrosov 1993; Hogan et al. 2000) or from airborne platforms (Hogan and Illingworth 1999). A few studies have been devoted to the retrieval of precipitating ice/snow parameters from the ground (Matrosov 1998) or from airborne radar measurements (Liao et al. 2005, 2008; Heymsfield et al. 2005; Wang et al. 2005; Matrosov et al. 2005). Moreover, dual-wavelength radar methods that include the differential attenuation have been developed to estimate the mass contents in mixed clouds (Gosset and Sauvageot 1992; Vivekanandan et al. 1999, 2001). Also, the dual-wavelength radar techniques have been used to study melting-snow microphysics (Yokoyama and Tanaka 1984; Yokoyama et al. 1984; Liao et al. 2009). The use of two wavelengths in identifying hydrometeor phase from space has also been proposed (Liao and Meneghini 2011). Furthermore, a retrieval method has been developed for three collocated radars (Sekelsky et al. 1999; Gaussiat et al. 2003; Yoshida et al. 2006). Various combinations of wavelengths are chosen in these different studies, but at least one radar frequency needs to be outside the Rayleigh regime for ice/snow particles, and the wavelengths have to be separated to assure significant differences in the reflectivity dependence on the particle size distribution. Moreover, the feasibility of sizing snow particles using the difference in the Doppler velocity at two wavelengths has been discussed by Liao et al. (2008) and Matrosov (2011).

Even with the physically consistent simplifications presented in Szyrmer and Zawadzki (2014, hereafter Part III), uncertainties in the snow microphysics arising from the natural variability of the particle density, shape, and orientation behavior resulting in uncertainties in the particle backscattering cross section, density, and terminal velocity, as well as the natural variability in PSDs, are also important so the retrieval of snow properties must be considered a stochastic problem. The approach to the retrieval must be such as to properly derive the expected value of the retrieval and the associated uncertainty. That is, it is imperative to assess the errors in the model and their impacts on the retrievals. This is particularly important in data assimilation into numerical weather prediction (NWP). In Lee et al. (2007) we have made a first attempt at incorporating microphysical model uncertainties into generating “ensemble retrievals of precipitation.” These ensembles were based on the statistical properties of the time–space errors in the radar reflectivity–precipitation rate (ZR) relationship (the simplest model in retrieval from radar data) and assuming some scaling characteristics of precipitation fields. More recently, we have established a complete error structure of measurements of precipitation at the ground that can be used as the basis for a more realistic stochastic generation of ensemble retrievals (Berenguer and Zawadzki 2008).

In this paper we further expand on these ideas. In our attempt to properly assess the effect of model uncertainty on the retrieval, we are led to the concept of ensemble retrievals of the microphysical properties of observed snow. The ensemble is determined by the spread of the uncertainties in the microphysical models. As an example consider the density , as defined in Part III. The retrieved value is defined by the expected value of a probability distribution of , conditional to the measurements of equivalent reflectivities at two radar bands, and , and the corresponding Doppler velocities, and :
e1
where the probability density function (pdf) is obtained by sweeping all the possible realizations of the stochastic properties of snow that enter in (1). In this paper these realizations will be taken from the literature, where observed snow microphysics is described by numerous studies. Equations similar to (1) would apply to each retrieved parameter of snow properties (snow descriptors).

A similar methodology has been previously applied to estimate raindrop size distribution (Williams and Gage 2009) and recently to evaluate the uncertainty in the ZR relationship (Zawadzki and Treserras 2012) and, likewise, in Szyrmer et al. (2012) to obtain the forward model relations and the error covariance matrix within the optimal estimation technique applied to the ice cloud retrieval.

In section 2 we described the data used in the retrieval. The details of the retrieval algorithm formulation are presented in section 3. Section 4 contains results from the application of the algorithm to the real dual-wavelength radar measurements. The agreement of the retrieved fields with the measured Doppler spectrum is shown in section 5. The final section contains general remarks and a discussion of some of the retrieval limitations. The appendix presents an attempt to interpret the results using a deterministic 1D steady-state model.

2. Data and steps in data preprocessing

The two collocated Doppler radars used in this study are the vertically pointing X-band radar (VertiX, wavelength 3.2 cm) and the millimeter-wave W-band radar of 95 GHz (wavelength 3.2 mm) deployed at the McGill Radar Observatory site in Montreal, Quebec, Canada. The two radars give time–height records of reflectivity and Doppler velocity with a resolution of 45 m in range and of 1.5–2 s in time by the VertiX and of 0.2 s in time by the W-band radar. The X-band radar is calibrated with a disdrometer in rain and it has shown good stability over periods of some months. At about 5 km, the minimum detectable reflectivity of the X band is reached, for a reflectivity of −15 dBZ. At this height the calibration of the W band was adjusted assuming that Rayleigh scattering applies to the two collocated radars (after correction of W-band reflectivity for gaseous attenuation). The magnitude of the attenuation by atmospheric water vapor and oxygen is calculated using the line-by-line model of Liebe (1985). The profiles of temperature and relative humidity have been taken from soundings, with the humidity profile adjusted to the measured precipitable water vapor (PWV). The oxygen, which is relatively evenly distributed in the atmosphere, causes much less attenuation compared to the water vapor. The total two-way attenuation by atmospheric gases calculated at the echo top is around 3 dB. As an example, the calculated two-way attenuation by water vapor and oxygen at 900 hPa and temperature close to −9°C at the saturation conditions is close to 0.7 dB km−1. However, the accuracy of this value can be low. The comparison between different millimeter-wave propagation models presented in Josset et al. (2013) points to a large spread of the predicted absorption at W band by gaseous species. The presented results suggest an overestimation of absorption by water vapor at 94 GHz by the Liebe-based models. Extinction of radar signals by dry snow precipitation at X-band reflectivity lower than 15 dBZ, as in the data applied to the retrieval, is in general very small, even at 95 GHz, and can be ignored. The estimated maximum value of the extinction by dry snow of the W-band signal at the top of the retrieved layer is much less than 1 dB; however, an exact value is difficult to obtain due to the important sensitivity of the calculated extinction on the microphysical assumptions in situations of weak precipitation rates.

Figure 1 shows the case chosen for this study. The presented data correspond to the snow event that occurred in Montreal on 4 January 2011 between about 1700 and 2100 UTC. The surface air temperature was about −4°C. This case was chosen because it is sufficiently simple: Doppler velocities neither indicate the presence of convection (air velocity greater than fall velocity), nor heavily rimed particles for which the Doppler velocity would be greater than −1.5 m s−1 on the X-band radar. These conditions are needed because the equations described in Part III do not consider the appreciable content of supercooled water and consequent heavy riming. Also, liquid cloud would attenuate the W-band radar, making its information ambiguous. On the other hand, this case is not overly simple: the great variability of reflectivity in the time–height profile indicates a very structured and likely variable process of snow growth.

Fig. 1.
Fig. 1.

(left),(middle) One hour of the four measured values and (right) their time averages. The temperature scale in the reflectivity time–height profile is obtained using Rapid Update Cycle (RUC) analysis. Note that is smoothed in time to match the lower resolution of the X-band radar and it is shown thresholded to the detectable level of . Forty minutes during the most intense period were analyzed.

Citation: Journal of the Atmospheric Sciences 71, 3; 10.1175/JAS-D-12-0286.1

Since the time resolution of the two radars is not the same, an interpolation of the cloud radar measurements to the same 1.5-s time scale was performed. In the next step of the radar data preprocessing the precipitation trails were straightened by a proper time data shifting as a function of height in order to simplify the analysis of the vertical structure of the radar observables and of the retrieved fields. However, it must be pointed out that since vertically pointing radars give an indication of trails along only the plane of motion, the trail effect still has to be taken into account in the analysis of the snow microphysical evolution with height fallen. The vertically pointing radar, in general, does not observe the same particles at different heights along the radar beam.

Even though the range resolution of the two radars is the same, the beamwidth at X band is about 8 times larger than that at W band. Therefore, the sampling volumes should be matched by appropriate averaging. An example of the proposed procedure for matching the sampling volume has been presented by Sekelsky et al. (1999). Here, only time averaging is used to reduce the impact of different radar-pulse resolution volumes. Additionally, the averaging reduces the possible contribution of the air vertical motion to the measured Doppler velocity. On the other hand, it is desirable to limit the averaging in order to prevent the removal of the small-scale microphysical variability. We have chosen a degree of averaging that yields retrieval results independent of the averaging period. Finally, a moving-average algorithm was applied over a period of 1.5 min combined with smoothing over four height levels (i.e., 180 m). Once the moving average is performed, the time resolution is reduced to 1 min by averaging 40 of the 1.5-s profiles.

The algorithm has been applied to the data shown in Fig. 1 during the most intense period between 18 h 8 min and 18 h 48 min. The radar-measured fields, after the preprocessing described above for the chosen 40-min time period, are shown in Fig. 2. The data are the equivalent reflectivity factors and in the left panels and the Doppler velocities and the differential Doppler velocity in the right panels. The fields of DWR and DDV are shown only at range gates used in the retrieval. The retrieval is applied to the range gates where DWR > 0.4 dB because of the large uncertainties of the retrieval for smaller values of DWR. The values of the velocities in Fig. 2 have been adjusted to the reference height of 2 km, corresponding to −15°C and 780 hPa, given in (11) in Part III. The lowest height used in the retrieval free from ground clutter is at a range of 250 m.

Fig. 2.
Fig. 2.

Time–height sections of the radar observables after the preprocessing described in the text. The 40-min time segment shown is between 18 h 8 min and 18 h 48 min from Fig. 1. (left) The reflectivity fields: (top) VertiX, (middle) W band in the middle, and (bottom) DWR. (right) Doppler velocity fields: (top) VertiX, (middle) W band, and (bottom) . In the two bottom images, only the region for which the retrieval is applied, i.e., with DWR > 0.4 dB, is shown.

Citation: Journal of the Atmospheric Sciences 71, 3; 10.1175/JAS-D-12-0286.1

3. Retrieval procedure

After the theoretical simplifications introduced in Part III, the unknown effects associated with the backscatter modeling are neglected. The remaining uncertainty in snow characteristics that can be retrieved from radar measurements is mainly related to the assumed PSD representation and to the mass–velocity relation. Therefore, to define the spread of uncertainty introduced by model microphysical assumptions, we use the snow model descriptors: (i) the six PSDs shown in Fig. 4 in Part III in complete and truncated forms and one monodisperse PSD and (ii) the four relations that allow the computation of the fall velocity of an individual particle from (see Fig. 3 in Part III). The retrieval is done separately for each of the 13 × 4 = 52 possible combinations of the descriptors. The set of retrieved parameters contains ρ*, the PSD characteristic size (D23 and ), and the mass content Qs as the second bulk quantities describing the snow PSD. From this set of retrieved variables, the normalized concentration parameter is derived ( and ) for each of the two PSD representations. In addition, the vertical air motion is also estimated. The mean of the ensemble of the results represents the final expected value of the retrieval while the ensemble standard deviation is the retrieval uncertainty.

a. Determination of D*

In Fig. 3, we illustrate how D* is determined. The dots show all our measurements in the retrieval region in the DWR– space. The X-band reflectivity-weighted velocity is related to the Doppler velocity by . The superposed curves have been obtained for the same model descriptors as in Fig. 5 in Part III, corresponding to D* = 1 mm. The presented results show that by taking D* = 1 mm for all the observation pixels with DWR in the interval indicated by the two vertical lines, the sensitivity of the results to the mass exponent bm becomes negligible. For smaller (larger) values of D*, the intersection of the lines moves to the left (right). In this manner we determine, once and for all, five values of D* that are sufficient to make the retrieved PSD characteristic size and particle density almost insensitive to bm. This reduction of one of the sources of uncertainty shows the advantage of the normalization of the mass–size relationship using D*. The chosen values of D* corresponding to the five DWR intervals are given in Table 1. In the rest of the retrieval, we take bm = 2, thereby ensuring that the results are only very slightly affected by its variability, as shown in Fig. 1 of Part III.

Fig. 3.
Fig. 3.

Dots show all our measurements in the retrieval region in the space (here, value represents , i.e., contribution of the air vertical motion neglected). Curves show the computations of these measurements for the same model descriptors as in Fig. 5 in Part III, for the same D* = 1 mm. This value of D* is taken for all observation pixels with DWR in the interval indicated by the two vertical lines. The black and red lines show the results obtained for inverse exponential PSDs with particle size defined as the liquid-equivalent diameter and actual physical diameter, respectively.

Citation: Journal of the Atmospheric Sciences 71, 3; 10.1175/JAS-D-12-0286.1

Table 1.

The values of D* used in the calculations corresponding to the measured DWR.

Table 1.

b. Step-by-step solution

For each of the five values of D* given in Table 1 and for each possible combination of the model descriptors, lookup tables representing the relations between the measurables and microphysical parameters used in the retrieval have been generated. These lookup tables are constructed by calculating DWR, , and the difference , as a function of D23 and ρ* for a given D*. The selected values of D23 are between 0.2 and 2.4 mm with an increment of 0.022 mm, and the parameter m* has an increment of 0.25 × 10−5 g. At the same time, each pair of D23 and ρ* corresponds to one value of the mean mass-weighted diameter for a given PSD with Dmlt as the spectral coordinate.

We initially assume w = 0. For every combination of model descriptors, the following steps have been carried out independently at each pixel:

  1. Determine from the DWR value at a given pixel which of the five values of D* has to be chosen (based on Table 1).
  2. Select all pairs of D23–ρ* and that match the precalculated observables and , where and are the estimated measurement errors.
  3. From the D23ρ* pairs and the selected in step ii, reject those that do not satisfy the measured difference , where the estimated error of the Doppler velocity difference is Δ(DDV) = 3.5 cm s−1. This difference is independent of the vertical air motion.
  4. Determine for each D23ρ* pair selected in step ii and retained in step iii.
  5. Calculate from the value of and the observed .
  6. Calculate the normalized concentrations and from , D23ρ*, and .
Steps (ii)–(iv) are achieved using the precalculated lookup tables generated for a given combination of model descriptors using the value of D* determined in step (i). All five steps are repeated for all 52 combinations of model descriptors. Thus, at each pixel, ensembles of retrieved parameters are obtained. The values of these parameters must be compatible with all four available measurements for at least 1 of the 52 model descriptors. Thereafter, at each pixel satisfying the above condition, an average and a standard deviation (SD) of the obtained ensemble are computed for all parameters.

The retrieved field allows us to estimate the vertical air velocity field w using (17) in Part III. Now is calculated at each pixel from the measured and the estimated w. Then, the entire retrieval including steps (ii)–(v) and the calculations of the average parameter ensembles and the new w is repeated. The convergence of retrievals is obtained after a few iterations.

At every pixel, all the members of the generated ensembles of the retrieved parameters satisfy the four available measurements. The ensemble averages represent the expected values of the retrievals. The ensemble standard deviations are considered to be a measure of the uncertainty associated with every derived microphysical variable due to the limitations in the microphysical assumptions, as discussed in section 6, excluding the uncertainty related to the backscattering calculations.

4. Retrieval results

The time–height plots in Fig. 4a show at each pixel the numbers of model descriptors, among the 52 used, that provide at least one set of retrieved parameters. It is interesting to note that within 40 min of observations the number of model descriptors (all taken from past published studies) that are compatible with all radar measurements is quite variable in time and space. In a few points none of the included model assumptions is adequate; most likely our descriptor ensemble spread is too narrow. In about 50% of the retrieved region, only a half of the used microphysical descriptors provide at least one solution.

Fig. 4.
Fig. 4.

(a) Number of model descriptors (defined by a pair of a PSD and an mu relationship; maximum 52) providing at least one solution satisfying all measurements. For all model descriptors, we take all pairs of ρ* and D23 that satisfy simultaneously the measured values of DWR, , and DDV, within measurement error. (b) The addition of all these solutions defines the number of ensemble members that simultaneously satisfy all the measurements.

Citation: Journal of the Atmospheric Sciences 71, 3; 10.1175/JAS-D-12-0286.1

For each model descriptor combination, the derived sets of microphysical parameters are taken from the lookup tables as the ones that are consistent with the precalculated observables, taking into account the assumed observation errors. Therefore, using these high-resolution lookup tables, one model descriptor provides more than one solution for a given set of observables. In Fig. 4b the total number of ensemble members at each range gate is shown. The two fields shown in Fig. 4 are rather correlated. Hence, where more descriptors are able to provide the solutions, in general, the range of solutions is larger for each individual descriptor.

Not a single set of snow microphysics assumptions among the possible 52 allows a retrieval that is consistent with measurements at all pixels in this 40-min segment. This is illustrated in Fig. 5 showing four examples of the field of D23 obtained from four different model descriptors. Here, the ensemble at each range gate is formed by all solutions compatible with the observables assuming a unique model descriptor. Therefore, at least one member of this ensemble is required to determine D23 at each gate. The values shown in Fig. 5 represent the average of the D23 ensemble. Overall, the and combined with the velocity calculated from the relation (10b) in Part III or the mass–velocity relationship (Szyrmer and Zawadzki 2010) provide the solutions in the largest fraction of pixels. The examples are shown in Figs. 5a and 5b. Notice that a different set of microphysical assumptions is relevant for different regimes. In general, the monodisperse size distribution, , can be applied to describe the upper region; however, it completely fails in the lower part of the retrieved segment, as shown in Fig. 5c, where is combined with the mass-projected area relation from Baker and Lawson (2006) given by (10c) in Part III. On the other hand, , even when truncated, as shown in Fig. 5d where it is combined with the relation (10a) in Part III, is not able to describe the microphysics consistent with the measurements in the upper region, and yields the solutions that match the observables mainly in the middle layer. In general, the wider PSDs (i.e., inverse exponential and lognormal as one can see in Fig. 4 in Part III) succeed mainly in this middle region and fail in the higher layer. Narrow PSDs appear to be more suitable in the layer below the cloud top containing newly generated smaller particles. Depending on the relation used in the velocity calculations, the complete and complete provide a solution in less than 15% of the retrieved area in our 40-min segment.

Fig. 5.
Fig. 5.

Four examples of the field of D23 obtained from four different model descriptors. The presented results are obtained for (a) untruncated with velocity calculated from (10b) in Part III, (b) untruncated with velocity calculated using the mass–velocity relation from Szyrmer and Zawadzki (2010), (c) with velocity calculated from (10c) in Part III, and (d) truncated with velocity calculated from (10a) in Part III.

Citation: Journal of the Atmospheric Sciences 71, 3; 10.1175/JAS-D-12-0286.1

The average height-dependent evolution of retrieved quantities obtained from the ensembles formed by the solutions from the all model descriptors is shown in the top panels of Fig. 6. The average profiles of the standard deviations are presented in the bottom panels of Fig. 6. For each parameter, the black line depicts the average of the SDs at all individual range gates at a given height and represents the average uncertainty of the retrieved parameter. The red line describes the profile of internal horizontal variability of the retrieved parameter in the time segment used in the retrieval. This horizontal variability is calculated as the SD from the average of the retrieved quantities (shown in the top panels of Fig. 6) at a given height. As can be seen in the bottom panels of Fig. 6, only in the layer below about 3 km is the average SD of the derived parameters lower than the SD of their horizontal variability at a given height, except for the two normalized concentrations with the average uncertainty profile (black line) that are very close to the average variability (red line) in the entire retrieved layer. The two profiles in the bottom-right panel of Fig. 6 show a general decrease with the distance fallen, except for a small jump around 2 km, and have a similar shape as the average profiles shown in Fig. 6 (top right). The horizontal variability of with height is also similar to the vertical evolution of its average; that is, the higher variability, as expressed by larger average uncertainty, corresponds to the larger average values. Comparing the profiles of the height-dependent average uncertainty of the retrieved parameters at each pixel (the black lines in Fig. 6, bottom) with the retrieved horizontal variability of the retrieved fields (the red lines in Fig. 6, bottom), one can deduce that the retrieved pattern of the microphysical parameters is meaningful only in the lower part of the retrieved time segment.

Fig. 6.
Fig. 6.

Vertical profiles of (top) the time-averaged retrieved quantities and (bottom) their standard deviations. In the bottom panels, we show the average standard deviation of all gates at this height (black) and the standard deviation from the average of the retrieved quantities at a given height (red).

Citation: Journal of the Atmospheric Sciences 71, 3; 10.1175/JAS-D-12-0286.1

The time–height plots of the six retrieved fields representing ensemble-average values and the relative standard deviations (RSDs) of the ensemble averages are shown in Fig. 7. RSDs are expressed as a percentage and are defined as the standard deviation divided by the value of the average. In addition to the discontinuity that can be seen around 2 km (~−15°C), the averages have a nonrandom structure in space, which, in itself, is reassuring since the retrieval is done independently for each pixel. We note that the RSDs are also structured, indicating that the errors are correlated in space–time.

Fig. 7.
Fig. 7.

(left) The fields of ensemble averages of retrieved values and (right) the fields of their relative standard deviation (RSD; %). The retrieved variables are (from top to bottom) characteristic diameter ; liquid equivalent diameter , that is, the mean mass-weighted diameter of the distribution with size defined by Dmlt; mass content (Qs); effective density ; normalized concentration ; and normalized concentration of the distribution with size defined by Dmlt . A logarithmic scale is used for and in order to reduce the dynamic range of values.

Citation: Journal of the Atmospheric Sciences 71, 3; 10.1175/JAS-D-12-0286.1

The range of obtained values for the diameters and , and for the normalized concentrations and shown in Figs. 6 and 7, are in general agreement with the observed values for the given temperature interval (Field et al. 2005, 2007; Delanoë et al. 2005). The range of the derived effective density ρ*, describing the density of D* between 0.75 and 1.75 mm (see Table 1), also compares well with the empirical mass–size relationships shown in Fig. 1 in Part III. Generally, the characteristic size is increasing as the falling snow grows due to the processes of water vapor diffusion and aggregation. A general decrease of snow density with height fallen at the same time as the characteristic size increases is compatible with aggregation. An important decrease of the normalized concentration parameters, expected where the aggregation is active, is evident except around 2 km (~−15°C). Around this level, an intriguing increase in the number concentration is accompanied by a sharp increase in Qs, an important decrease in ρ*, and some reduction in the average characteristic size. A significant gradient of all the variables observed around 2 km indicates that aggregation, and eventually vapor diffusion, are not the only active processes. The discontinuity layer in the retrieved quantities can also be noticed at the same height in the input data (Figs. 1 and 2), with enhancement of the reflectivity gradient and DWR, but a decrease of the absolute value of Doppler velocity. The fields of RSDs also show a significant increase in the same layer. Only the density field displays an opposite change.

The retrieved pattern of presented in Fig. 7 does not follow a consistent pattern and the correlation with the fields of characteristic diameters is not very evident. On the other hand, the observed reflectivity pattern in Fig. 2 does not closely correspond to either Qs or the characteristic diameter patterns, but to a combination of them. This is consistent with the assumption that these two microphysical parameters have a dominant influence on the reflectivity. The fields of normalized concentrations and show very important variability of about four orders of magnitude that can be seen in Figs. 6 and 7. A high vertical rate of the reduction of ρ* of in the thin layer below the 2-km level can be partly explained by an important increase in the measured DWR in the same layer. Hence, as specified in Table 1, used in the retrieval also increases. With a snow particle density inversely proportional to , the doubling of reduces ρ* by a factor of 2.

One can see in Fig. 7 the delay of maximum values with respect to the maximum characteristic sizes. This delay implies that the largest particles arrive at the ground a few minutes before the maximal mass content and concentration parameters. This can be explained by the gravitational sorting due to the fall speed differences between the less numerous larger particles falling more rapidly and the population of smaller particles with a slower terminal velocity. The delay, which becomes evident below the level of the jump in , points to the coexistence of the larger particles with higher fall speeds with the smaller particle population containing a large fraction of . A displacement can also be noticed between the maximum DWR, containing information about the particle average size, and the maximum reflectivity factor Ze in the input data shown in Fig. 2 after the straightening of the fall streaks. This straightening of the trails in the data preprocessing is based on the reflectivity fields. It means that in the presence of gravitational sorting, the straightened trails follow the locations of the maximum reflectivity; therefore, the fall of larger or smaller particles may deviate from the vertical direction and not follow the fall streak.

As can be noted in Figs. 6 and 7, values of slightly reduce downward in the layer just above 1-km height. This decrease could be explained by the sublimation process in the ice subsaturated conditions. Indeed, the nearby soundings show an important decrease in the relative humidity in the same layer.

The RSD fields shown in the right panels of Fig. 7 are generally larger for smaller particle sizes than those for larger particles present in the lower part of the retrieved area. The large values of possible relative errors in the upper retrieved layer can be explained by DWR in this region being mainly lower than 2 dB. Another factor is the absolute measurement errors taken as constant and then relatively larger for lower DWR and Doppler velocity. The lower values of the RSD for density obtained in the upper layer can be expected since ρ* in the retrieval is mainly related to the fall speed. As shown in Fig. 3 in Part III, the spread of the velocity corresponding to the same ρ* is much smaller for smaller sizes, like the results representative of the upper layer, than for larger particles present in the lower part of the retrieved time–height cross section. In general, RSDs for are greater than those for and . In the lower layer, they are lower than 30% for and below 15% for the two diameters. The possible relative errors of the derived normalized concentrations are very high, approaching 200%. These parameters are related to the particle concentration and therefore to the small-size particle PSD regime. An important variability in this part of the snow PSD has been shown in many previously published observational studies. In any case, the observables available in our retrieval are not sensitive to these small particles.

Figure 8 shows the derived w corresponding to the retrieved variables in Figs. 6 and 7. The positive values of w describe updrafts. The top panel in Fig. 8 presents the time–height field, while the bottom panel shows the time-averaged profile of w. The time-averaged reflectivity-weighted velocity that is downward is also shown in this panel. An important maximum in the updraft can be seen in the layer around 2-km height. The mean w is about 15 cm s−1 while the maximum values are between 25 and 30 cm s−1. The estimated air velocity shown in Fig. 8 and used in the retrieval has been calculated assuming the saturation with respect to ice, that is, that the generation of vapor excess over the ice saturation value is depleted through the vapor diffusion growth calculated from the gradient as given in the relation (17) in Part III. However, in the region of higher updraft, the expected saturation could be closer to water saturation, and the assumption of zero supersaturation with respect to ice may not be valid. From the relation used, a higher vertical gradient of the vapor mixing ratio leads to a larger estimate of w. Taking the saturation with respect to water rather than ice yields maximum values of w that are larger by a few centimeters per second. Therefore, it may be expected that in the layer of maximal updraft, the calculated updraft shown in Fig. 8 could be larger and the vapor saturation could be close to water saturation. Hence, the possibility of the activation of the ice nuclei in this layer could be considered.

Fig. 8.
Fig. 8.

(top) Computed vertical air velocity w and (bottom) its time-average profile (red in the bottom panel, increased by 0.4 m s−1). Also in the bottom panel, the w ±1 SD profiles (dot–dashed red lines) and time-average reflectivity-weighted velocity profiles: (thick line) and (thin line).

Citation: Journal of the Atmospheric Sciences 71, 3; 10.1175/JAS-D-12-0286.1

5. Verification

The validation of the retrieval with independent measurements is the most difficult task in any meteorological work and in particular in remote sensing studies. Here, we use the one additional piece of information in the observations not used in the retrievals, namely Doppler spectra, to explain and verify, to a degree, certain aspects of the retrieved results. The W-band sample spectrum in Fig. 9a (representative of the entire period) shows a striking bimodality starting just above 2 km (−15°C level). At the same time, the fall velocity of snow contributing to the original mode decreases by about 15 cm s−1 (shown by the white lines), consistent with the retrieved updraft. The secondary peak shows the increase in the number concentration as a result of the activation of ice nuclei not previously activated and of the accompanying increase in water vapor. This explains the increase in the normalized concentrations and . The possibility of reaching water saturation conditions in stronger updrafts and the subsequent generation of supercooled water have been estimated as being very improbable given the values of the retrieved (Zawadzki et al. 2000). An approximate threshold value of the updraft required for generating supercooled water is shown in Fig. 9b as a function of snow content.

Fig. 9.
Fig. 9.

(a) A sample Doppler spectrum of the W-band radar. Similar spectra were present throughout the entire period of analysis. (b) Updraft required to generate supercooled water in the presence of snow at 800 hPa and −15°C.

Citation: Journal of the Atmospheric Sciences 71, 3; 10.1175/JAS-D-12-0286.1

The injection of the small crystals causes the broadening of the PSD and a local reduction of its characteristic sizes. The newly generated particles at around −15°C undergo dendritic growth and consequently fall at a slower speed than the more compact particles generated aloft. The larger particles falling from the cloud top will also be slowed down toward the velocity of the less-dense dendritic growth. The dendritic growth together with the PSD broadening leads to an important aggregation enhancement as seen in the further increase of the retrieved characteristic size. It all leads to the retrieved decrease of particle density. It may be seen in the Doppler spectrum of Fig. 9 that the second mode of the slow particles in the Doppler spectrum generated around 2 km is maintained all the way to ground level.

The activation of ice nuclei likely occurs in conditions of supersaturation with respect to ice. This contradicts the hypothesis under which the updraft is calculated. At the level of new ice nuclei activation the updraft is probably stronger than that indicated in Fig. 8, as discussed at the end of section 4.

Also to be noted is the very narrow Doppler spectrum at heights above the level where the bimodality begins. This is indicative of very narrow PSDs and is consistent with the good performance in the upper-level retrievals of the monodisperse spectrum.

6. Final remarks

An algorithm for retrieving snow microphysics from dual-wavelength vertically pointing Doppler radar measurements has been developed. The emphasis here is on the methodology of ensemble retrievals. The ensemble generated for each of the retrieved parameters at every range gate is formed by the solutions that are consistent with all four observables at this gate obtained separately for different snow model descriptors. The model descriptors are given by the combinations of the two microphysical assumptions considered as chief contributors to the uncertainty in the inferred parameters, namely the imposed generic PSD functional form and the velocity–mass relationship for an individual snow particle. The ensemble approach allows us to take into account the large variety of these relations found in natural precipitating snow and to quantify uncertainty in the retrieval results caused by this variety. The computed uncertainties expressed as SDs of the ensemble averages estimate the range of uncertainty of the model microphysics only.

The uncertainty associated with our method of determining the air vertical velocity was not discussed previously. The method is based on the equilibrium assumption: the rate of growth of detectable condensate equals the rate of generation of excess of water vapor with respect to saturation. The uncertainty due to the choice of saturation with respect to water or ice can easily be incorporated into the ensemble retrieval by doubling the number of ensembles. However, the differences are small and would not contribute significantly to the standard deviation of the retrieved parameters. The uncertainty due to the assumption of equilibrium requires more attention and is left for future work. The extreme uncertainty was evaluated by comparison of the retrieved snow parameters for and the resulting values after the converging iteration shown in Fig. 8. The comparison shows that the patterns of the time–height retrieval are not much changed. The change in the values depends on the parameter, with the characteristic size not much affected and the number concentration greatly changed by the values of . However, since the overall pattern is unchanged, the increase in at around 2 km is present even for .

The present work is a first attempt at explicitly incorporating the stochastic nature of snow into an interpretation of measurements. However, the spread of uncertainty in each of the descriptors used here is rudimentary. We have assumed that all snow descriptors are equally probable and this is clearly not correct. However, it is interesting to point out that a low probability assumption, such as a monodisperse PSD in the low levels, was found to be incompatible with observations, and ensemble members generated with this PSD were rejected in the lower levels and retained in the upper levels. On the other hand, in computing the ensemble average, we follow the assumption of an equal probability of members and we give an equal weight to all the members, and this likely biases the mean. Future advancements and refinement in this direction would require a broad communal effort in compiling all available measurements of snow for the derivation of the full probability distribution (absolute or conditioned to parameters such as temperature) of all the relevant snow parameters likely to be followed by an iteration through a set of complementary measurements. This is well beyond the scope of the present work.

The other limitation of our retrievals is the assumption of spherical particles in the computation of reflectivity. Assuming that the horizontal orientation is generally preferential for the falling nonspherical larger particles submitted to the drag force, in future work we plan to use the T-matrix method for oblate spheroids serving as possible proxies for snow particles. The particle aspect ratio and orientation behavior would be considered as additional stochastic elements and incorporated into ensemble generation in the retrieval. The use of the Mie computations is a simplification of the backscattering calculations that is applied in this first version of the retrieval where the main emphasis is on the representation of microphysical uncertainty.

A high degree of nonsphericity associated with the horizontal alignment of the particles being the main contributor to the reflectivity may affect the accuracy of the retrieval results not only via the bias introduced by the Mie calculations but also via the assumption of the same characteristic size representative for the particle fall speed and the DWR value. The particle fall speed is mainly related to the horizontal dimension, while the DWR value depends mainly on the vertical dimension. As discussed in Hogan et al. (2000, 2012), the value of the deviation from the Rayleigh approximation at the W band is mainly caused by “destructive interference from radiation scattered from the near and far sides of the particle” and, therefore, is a measure of the particle dimension in the direction of the radar radiation propagation. The notion of characteristic sizes would have to be refined.

A novel aspect of our retrieval is that the density of particles, which introduces the most important retrieval error according to the previous studies, is not assumed a priori but is retrieved using the new normalized form of the mass–size relationship. The inferred density assures the agreement with the radar measurables, and therefore, it is not expected to be representative of the sizes outside the PSD interval that dominates the information contained in the used measurements. At low reflectivity, an important uncertainty remains due to the unpredictable behavior of the PSD sizes due to small crystals that may have a nonnegligible impact on the retrieved . In general, since the radar measurables provide no information about the smallest particles, the quantities determined by the lower-order moments of the size distribution cannot be correctly derived.

The attenuation by snow that we have not taken into account may introduce some bias into the retrieval. However, comparing with other sources of uncertainty, its impact on the results is expected to be relatively very small. The assumed absence of liquid supercooled clouds seems reasonable in the situation applied in the retrieval. In the presence of the supercooled liquid cloud, the retrieval would have to be modified. We plan to add to the retrieval the detection of the supercooled cloud layer from the vertical gradient of DWR (attenuation of the W-band reflectivity) and Doppler velocity (terminal velocity increase for rimed snow particles).

The retrieval results are physically realistic, showing a great time–height variability of snow properties. No single snow model descriptor provides the solution consistent with the measurables (with their error taken into account) at all pixels in this limited sample of 40 min, underlying the question of the representativity of in situ measurements.

Some of the advantages of an ensemble retrieval procedure are 1) it allows the computation of the error covariance matrix of microphysical parameterizations used in data assimilation, 2) it renders possible the propagation of uncertainty when using retrievals in nonlinear processes, and 3) it allows the evaluation of microphysical models.

In the authors’ belief the full recognition of the stochastic nature of snow is fundamental in order to study and understand cold-weather microphysics.

APPENDIX

Analysis of the Retrieval Results Using Deterministic Steady-State Model Calculations: Contribution of VDF and AGG to the average gradient of reflectivity flux

The retrieval is based on the developed relations linking radar observables to snow microphysics independently at each pixel using a stochastic approach. On the other hand, the vertical gradients of observed and retrieved fields have to reflect the evolution of falling-snow microphysics via microphysical processes. The main processes that could be active are the aggregation (AGG), deposition/sublimation (VDF), and eventually nucleation (NUC). The nucleation process is, in general, not expected once ice particles have been initiated in the top layer of the cloud. The possibility of the riming process has been excluded in the presented case. In this appendix, we attempt to link the observed vertical variability of the radar measurements to the active microphysical processes using the approach of a steady-state one-column model. Only the region below 3 km is analyzed because of the very large errors expected at heights above 3 km. Taking some simplifying assumptions, we first attempt to determine the layer of the precipitating snow where the vertical evolution of the retrieved field cannot be explained assuming only two processes: VDF and AGG. In this layer the nucleation process is required to assure a consistency between the derived and observed fields. However, outside this nucleation zone with the two processes being active, the contribution of each of the processes can be estimated. The strategy here is to derive moment conservation equations assuming steady-state conditions for two moments of the PSD represented by liquid-equivalent diameter : describing the X-band reflectivity (Rayleigh regime) and being directly proportional to the mass content .

The total rate of vertical change of the flux of the pth moment of the snow PSD is obtained by multiplying the conservation equation of the particle-size number density by and integrating over the entire PSD. The horizontal advection is not taken into account. Assuming a steady-state condition (i.e., ), the continuity equation for any PSD moment of order p can be written as
ea1
The term on the lhs represents the vertical gradient of the pth moment flux. The rhs contains the summation of the rates of change of via different microphysical processes. Here, the nonmicrophysical processes (e.g., turbulence, radiation) are excluded. For simplicity, the term related to the variation of the air density is not included in (A1). Equation (A1) with p = 6 describes the observed gradient of the X-band reflectivity flux.
The tendency of change of in (A1) due to any microphysical process (PRC) is related to the evolution of as follows:
ea2
The rate of change of via VDF can be expressed in a general form as , where si is the ambient supersaturation with respect to ice and represents a thermodynamic function. The parameters and of the power law depend on the particle morphological details responsible for the particle capacitance and ventilation factor and are, in general, size-dependent. Taking them to be constant, considered to be representative for sizes making a dominant contribution to the moments of interest, and assuming the number concentration unchanged, a general expression of the tendency of any moment due to VDF is obtained from (A2):
ea3
The last relation is obtained by applying the integration by parts with limits set to 0 and ∞ with the assumption that the integrand vanishes at the two limits. Combining (A3) applied to p = 3 and 6, the following relation between the tendency of these two moments due to VDF can be derived:
ea4
The tendency of in (A4) can be deduced from the continuity equation [(A1)] applied to p = 3. The tendency of due to AGG in (A1) is close to 0, since the aggregation process is not accompanied by the mass increase, and the small modification of the mass flux (related to the possible modification of the terminal velocity) can be neglected. As a consequence, the continuity equation [(A1)] applied for the X-band reflectivity gives
ea5
The ratio in (A4) has been approximated by in (A5), introducing a slight underestimation (the value of being rather small). In the absence of a new ice crystal generation, (A5) becomes
ea6

Term 1 represents the gradient of the measured reflectivity flux. The expected contribution of the VDF and AGG processes to term 1 in the absence of other microphysical processes is given by terms 2 and 3. The contribution of VDF represented by term 2 can be estimated from the derived fields of and and estimated w [ can be estimated from and w, as is estimated in relation (17) in Part III]. Therefore, the unknown contribution of snow aggregation in term 3 in (A6) can be estimated as the difference between term 1 and term 2. However, (A6) is valid only if the effect of the nucleation process on the snow mass and evolution can be excluded.

In Fig. A1 the average profiles of the three terms in (A6) are shown together with the profile of the gradient of the flux. The averaging is done here over the 15-min segment corresponding to the region of the maximum precipitating snow. The flux of , term 1 in (A6), is calculated from the measured reflectivity and Doppler velocity at X band. In the layer delimited by the two horizontal lines the vertical gradient of mass ( or ) flux is very high, much larger than the one expected from the VDF alone, and in consequence, term 2 in (A6) is also very high, causing the aggregation tendency in (A6) to become negative. This nonrealistic negative AGG tendency results from the assumption that the VDF process is the only possible growth coexistent with the AGG. The addition of a process generating an increasing mass flux (NUC) is needed to explain the results. In this layer the relation (A6) fails, and the height evolution of the observables and retrieved parameters could be described instead by (A5), where the snow-mass increase is mainly from the injection of the newly initiated crystals. Close to the top of the nucleation zone, the mass flux increases much more rapidly than the reflectivity flux, showing that the nucleation effect is first on and only after on reflectivity, when the particles become larger.

Fig. A1.
Fig. A1.

Averaged profile of the vertical gradient of the flux (from the retrieved ) and averaged profiles of the three terms in (A6). Term 1 in (A6) describes the vertical gradient of (from the measured X-band reflectivity factor). Terms 2 and 3 in (A6) are the estimated contributions to the X-band reflectivity flux from the VDF and AGG processes in the absence of nucleation, i.e., outside the layer delimited by two horizontal lines. In this layer, an important increase of (Qs) flux with falling distance cannot be explained by VDF and AGG processes only. The time averaging is over the 15-min segment containing the two major snow trails.

Citation: Journal of the Atmospheric Sciences 71, 3; 10.1175/JAS-D-12-0286.1

Outside this nucleation zone, approximately evaluated as the one between the two horizontal lines in Fig. A1, the relation (A6) is expected to be valid, and therefore an average increase of flux due to VDF and AGG can be estimated. The zone of light sublimation is shown above the 1-km level in general agreement with nearby soundings. The effect of the aggregation process in this zone is very important. A high aggregation tendency is expected since the temperature interval in the conditions close to water saturation corresponds to the dendritic-growth regime where the interlocking of the ice crystal branches can enhance aggregation efficiency (e.g., Connolly et al. 2012). In addition, the presence of newly formed small crystals together with the larger snow particles initiated at the cloud top generates a very broad spread of velocity and, consequently, a high probability of collisions.

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