## 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 (*Z*–*R*) 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).

*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

A similar methodology has been previously applied to estimate raindrop size distribution (Williams and Gage 2009) and recently to evaluate the uncertainty in the *Z*–*R* 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 dB*Z*. 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 dB*Z*, 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.

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

## 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 *ρ**, the PSD characteristic size (*D*_{23} and *Q*_{s} as the second bulk quantities describing the snow PSD. From this set of retrieved variables, the normalized concentration parameter is derived (

### 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–*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 *b*_{m} 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 *b*_{m}. 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 *b*_{m} = 2, thereby ensuring that the results are only very slightly affected by its variability, as shown in Fig. 1 of Part III.

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

### 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, *D*_{23} and *ρ** for a given *D**. The selected values of *D*_{23} 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 *D*_{23} and *ρ** corresponds to one value of the mean mass-weighted diameter *D*_{mlt} 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:

- Determine from the DWR value at a given pixel which of the five values of
*D** has to be chosen (based on Table 1). - Select all pairs of
*D*_{23}*–ρ** andthat match the precalculated observables and , where and are the estimated measurement errors. - From the
*D*_{23}–*ρ** pairs and theselected 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. - Determine
for each *D*_{23}–*ρ** pair selected in step ii and retained in step iii. - Calculate
from the value of and the observed . - Calculate the normalized concentrations
and from , *D*_{23}–*ρ**, and.

*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 *w* using (17) in Part III. Now *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.

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 *D*_{23} 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 *D*_{23} at each gate. The values shown in Fig. 5 represent the average of the *D*_{23} ensemble. Overall, the

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

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.

The range of obtained values for the diameters *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 *Q*_{s}, 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 *Q*_{s} 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 *ρ** of *ρ** by a factor of 2.

One can see in Fig. 7 the delay of *Z*_{e} 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

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

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 *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.

## 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

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

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

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

*p*th moment of the snow PSD is obtained by multiplying the conservation equation of the particle-size number density

*p*can be written asThe term on the lhs represents the vertical gradient of the

*p*th moment flux. The rhs contains the summation of the rates of change of

*p*= 6 describes the observed gradient of the X-band reflectivity flux.

*s*

_{i}is the ambient supersaturation with respect to ice and

*p*= 3 and 6, the following relation between the tendency of these two moments due to VDF can be derived:

*p*= 3. The tendency of

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 *w* [*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

In Fig. A1 the average profiles of the three terms in (A6) are shown together with the profile of the gradient of the

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

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