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    Relationships between am and bm in (3). The points denoted by diamonds, compiled from various sources (see text), correspond to average relations obtained for ensemble of ice/snow particles. The circles show the average relations for cold high clouds (Heymsfield et al. 2004b; Schmitt and Heymsfield 2009). Two derived average relations between am and bm are superposed: from Heymsfield et al. (2010) for stratiform clouds (dotted–dashed line) and from Lin and Colle (2011) for riming degree between 0 and 0.30 (dotted line).The thin and thick solid lines are the lines of a constant ρ* equal to 0.04 and 0.10 g cm−3 for D* = 1 mm (i.e., for m* = 2 × 10−5 and 5 × 10−5 g, respectively). The thin long dashed line corresponds to ρ* = 0.04 g cm−3, but for D* = 1.75 mm (i.e., for m* = 11.2 × 10−5 g).

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    Two examples of the sensitivity of the particle density on bm. The solid lines show the particle effective density as a function of D calculated for two empirical mass-parameter sets [am, bm]: [0.000 465, 1.77] for early aggregates of planar polycrystals from Kajikawa (1982) in the upper curves (thick lines), and [0.003 59, 2.1] for synoptically generated “warm topped” clouds for temperatures higher than −25°C from Heymsfield et al. (2010) at the lower curves (thin lines). The corresponding values of ρ* for D* = 1 mm are 0.015 and 0.054 g cm−3, respectively. The results obtained with the value of bm decreased by 0.2 and increased by 0.2 are shown using the superposed dotted–dashed and long dashed lines. (left) The calculations are done with a constant am value. (right) The ρ* value is kept constant (i.e., as in the original relations).

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    Four terminal velocity–size relationships calculated for two different ρ*. The three lines—dotted–dashed, short dashed, and long dashed—show the velocity computed using the method of Heymsfield and Westbrook (2010) assuming three parameterized relations [see (10)], hence with three sets of parameters k in (9). The solid lines show the velocity obtained from velocity–mass relationship derived by Szyrmer and Zawadzki (2010). The thinner and thicker lines correspond respectively to ρ* = 0.04 and 0.10 g cm−3 with D* = 1 mm. The environmental conditions correspond to our reference conditions (i.e., −15°C and 780 hPa).

  • View in gallery

    Six polydisperse (polysized) normalized PSDs used in the retrieval. They are plotted against the scaled size (left) and (right) . The value of bm = 2 is assumed.

  • View in gallery

    Calculated relationship D23–DWR for two complete PSDs: (black) and (red), assuming three different values of bm: 1.8 (long dashed), 2.0 (solid), and 2.2 (dotted–dashed). The thin and thick lines present the results for ρ* = 0.04 and 0.10 g cm−3, respectively. All plotted lines are obtained with D* = 1 mm.

  • View in gallery

    Calculated relationship D23 for seven untruncated PSDs. The thin and thick lines present the results for ρ* = 0.04 and 0.10 g cm−3, respectively. All plotted lines are obtained for bm = 2 and with D* = 1 mm.

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    Backscattering cross sections as a function of D. Except the line illustrating the dendrites from Liu library, the remaining lines are derived for the spherical model with the assumed mass–size relationship taken from Matrosov (2007) corresponding to m* = 3.0 × 10−5 g for D* = 0.1 cm. See the text for more information.

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Snow Studies. Part III: Theoretical Derivations for the 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

As a first step toward retrieval of snow microphysics from two vertically pointing radars operating at X band and W band, a theoretical model of snow microphysics is formulated in which the number of unknown parameters is reduced to snow particle density and to two bulk quantities controlling the particle size distribution. This reduction of parameters is achieved by normalizing not only the size distribution but also the snow particle mass in the mass–size relationship as well as by using a relationship between snow density and snow terminal fall velocity. However, no single snow microphysical model could describe the observed variability in the radar measurements. The uncertainty in the developed deterministic relations that map the microphysical parameters to the observables is shown to be mainly associated with the assumed dependence of particle velocity on its mass and on the particle size distribution (PSD) representation. Hence, various mass–velocity relationships together with different generic functional forms of the PSD reported in literature are described in this paper and then used in the retrieval. The derived relations provide a reasonable range of uncertainty associated with the microphysics when used for the actual retrieval of snow properties from observations in Part IV. The uncertainty in the backscattering computations of an individual particle, performed using Mie theory assuming spherical form with nonuniform density, is not taken into account in this study.

Corresponding author address: Isztar Zawadzki, Dept. of Atmospheric and Oceanic Sciences, McGill University, 05 Sherbrooke St. West, Montreal, QC H3A 0B9, Canada. E-mail: isztar.zawadzki@mcgill.ca

Abstract

As a first step toward retrieval of snow microphysics from two vertically pointing radars operating at X band and W band, a theoretical model of snow microphysics is formulated in which the number of unknown parameters is reduced to snow particle density and to two bulk quantities controlling the particle size distribution. This reduction of parameters is achieved by normalizing not only the size distribution but also the snow particle mass in the mass–size relationship as well as by using a relationship between snow density and snow terminal fall velocity. However, no single snow microphysical model could describe the observed variability in the radar measurements. The uncertainty in the developed deterministic relations that map the microphysical parameters to the observables is shown to be mainly associated with the assumed dependence of particle velocity on its mass and on the particle size distribution (PSD) representation. Hence, various mass–velocity relationships together with different generic functional forms of the PSD reported in literature are described in this paper and then used in the retrieval. The derived relations provide a reasonable range of uncertainty associated with the microphysics when used for the actual retrieval of snow properties from observations in Part IV. The uncertainty in the backscattering computations of an individual particle, performed using Mie theory assuming spherical form with nonuniform density, is not taken into account in this study.

Corresponding author address: Isztar Zawadzki, Dept. of Atmospheric and Oceanic Sciences, McGill University, 05 Sherbrooke St. West, Montreal, QC H3A 0B9, Canada. E-mail: isztar.zawadzki@mcgill.ca

1. Introduction

In any attempt to study precipitating snow systems using remotely sensed data, we are immediately confronted with the need to represent snow by a model containing a limited number of parameters. However, because of the extreme variability of snow characteristics such models are highly unsatisfactory. An average size dependent snow density combined through a nonlinear relationship with an average particle size distribution does not give an average mass content. This is because both density and size distribution have a degree of variability as large as, or more than, the average values. This variability can be narrowed down through stratifying snow by its type, but snow type is normally unknown. Moreover, within a volume of space, different types can be present in any degree of mixture. Attempts at relating the properties of snow to external parameters—for example, temperature and precipitation depth as in the study of fall velocity by Zawadzki et al. (2010)—have a limited success in narrowing down the degree of variability of snow properties.

As an alternative to represent snow characteristics by “model” descriptors (particle size distribution, fall velocity, and density as a function of size), the present work is our first attempt at explicitly incorporating the stochastic nature of snow into interpretation of radar measurements. This is done by what we call “ensemble retrievals.” In a similar manner as in some advanced ensemble forecasting, where the uncertainties of the numerical models are used to generate possible alternative outcomes of the forecast, we will derive an ensemble of retrievals by using different combinations of snow descriptors with each combination leading to a member of the retrieval. We will avoid any additional assumptions on the snow characteristics and let the data sort out which of the combinations of snow descriptors are compatible with the observations.

The present work is divided into two parts: with the actual retrievals shown in Szyrmer and Zawadzki (2014, hereafter Part IV) and in the present paper (Part III), we will introduce the theoretical basis of the snow microphysics retrieval presented in Part IV. The observation system in this study comprises vertically pointing X-band (wavelength λ = 3.2 cm) and W-band (λ = 0.32 cm) Doppler radars. The data used in our retrievals and the retrieval methodology are described in Part IV.

The equations that represent the measurements of the two nonattenuated equivalent reflectivity factors and Doppler velocities are
e1a
e1b
for each of the two wavelengths. The value of 0.93 is used as a reference dielectric constant of water. Taking a constant value ensures the same reflectivity factor at X band and W band in the Rayleigh regime. In these equations σ(λ)(D, ρs[D]) and u(D, ρs[D]) are the particle backscattering cross section and the terminal velocity, respectively, both being functions of the snow density, ρs. The last term in (1b) describes the particle terminal velocity weighted by λ-band reflectivity, . The two measured equivalent reflectivities and Doppler velocities also depend on the unknown particle size distribution (PSD) represented in (1) by n(D). All these microphysical parameters are, in general, complex functions of size D and consequently a simplifying model of snow characteristics is required. Snow density and fall velocity are each commonly represented by a two-parameter power law (coefficient and slope)—a gross oversimplification of the representation of snow properties. Moreover, the vertical air motion w constitutes an additional unknown in (1). It immediately becomes clear that the retrieval of snow properties cannot be unambiguously achieved from (1a) and (1b) for the two wavelengths. Consequently, an effort is made in the microphysical modeling to reduce the number of unknown parameters to be retrieved.

For a given size of single particles the key quantities required to calculate the Doppler radar measurements are its mass and terminal velocity. The large variety of the published mass–size and velocity–size relationships demonstrates their natural variability and measurement uncertainties. We propose here a parameterization in which, for a given mass–area relationship taken from the literature, particle mass and velocity are controlled by only one parameter. This parameter is the mass (or equivalent density) of a snow particle having a size that is representative for the bulk quantities involved in the retrieval, and its value is not imposed but is retrieved at each range gate. Two other quantities, retrieved at each gate, define the PSD within the adopted double-moment normalization approach. They are determined by the medium-sized particles that constitute an important contributor to the quantities of interest. This choice of quantities that control the PSD diminishes the dependence of the results on the selected analytical form of the general normalized PSD.

Two critical issues with retrievals of snow properties are the significant space–time variability of snow and its microphysical complexity and the lack of direct correspondence between snow characteristics and radar observables. This introduces a large uncertainty in the established quantitative deterministic relations between observations and the unknown parameters to be retrieved, regardless of the simplifying assumption used. In the parametric representation of snow microphysics adopted in this paper, the developed relations are mainly dependent on the assumed velocity–mass relationship and assumed functional form of the particle size distribution. Therefore, the various mass–velocity relationships together with different generic functional forms of the PSD reported in literature represent here the uncertainty of the parameterized microphysics.

To fully estimate the uncertainty in the relations linking the snow microphysics to the radar observables, the uncertainty relating to the backscattering computations needs to be added. On the basis of many published studies devoted exclusively to this problem, it becomes evident that this uncertainty is in general very large owing to the dependence of the radar return on the details of the microstructure that cannot be accurately predicted on the basis of the actual knowledge.

The next three sections are devoted to describe the details of the microphysical parameterization used in the retrieval. In section 2, a new normalized representation of snow particle mass is described. Section 3 presents the methods used to derive particle velocity from particle mass. The different shapes of the generic PSD taken from the literature within a two-moment normalization framework are discussed in section 4. Based on the described representation of the snow microphysics, the relations that are used in the retrieval are given in section 5 and concluding remarks in section 6.

2. Modified mass–size relationship

The relation describing particle mass in the majority of studies is in the form of power law fitted to observational data with the particle size defined by the maximum physical dimension of the particle:
e2

Therefore, knowledge of the two power-law parameters is required. Moreover, for most particle habits, two or even three sets of coefficients in (2) are required to cover the entire range of observed sizes (Heymsfield 1972; Mitchell 1996; Matrosov 2007). In general, the set (am, bm) representing the smaller size regime is important for the microphysical studies related to the lower-order moment of the PSD that may be different from the set (am, bm) representative for the range of sizes that dominates higher-order moments such as mass content or reflectivity. However, the contribution of a single size particle to these integral quantities, such as reflectivity or total snow/ice water content Qs, depends essentially on particle density and not on the two coefficients per se. This leads to an average positive correlation between the two mass coefficients, am and bm, that has been already noticed, for example, by Heymsfield et al. (2010).

In our new mass–size relation, we introduced a reference diameter D* and its corresponding mass m* defined by
e3
From (2) and (3) we get
e4
The effective density ρ is calculated assuming spherical particles of diameter D:
e5

In general, the crystal bulk density, defined as the ratio of particle mass to its volume, is much higher than the effective density defined in (5).

With volume given by υ(D) = (π /6)(D*)3(D/D*)3 the density is expressed as
e6

Figure 1 shows various sets of am and bm found in published studies corresponding to average relations for particle populations. The filled diamonds are a compilation of average relationships for ice clouds or for precipitating snow obtained (or presumed) in different field experiments (Mitchell et al. 1990, 2010; Brown and Francis 1995; Francis et al. 1998; Heymsfield 2003a; Heymsfield et al. 2004a, 2005, 2010; Kingsmill et al. 2004 based on the data of Heymsfield et al. 2002; Evans et al. 2005; Kulie and Bennartz 2009; Matrosov and Heymsfield 2008; Brandes et al. 2007; Field et al. 2006, 2008; Szyrmer and Zawadzki 2010).1 The two relations for larger snowflakes derived from modeling studies by Ishimoto (2008) are also included in these points; one of them, describing very dense snowflakes, which is represented by the uppermost filled diamond. The two dotted–dashed and dotted lines show the parameterized relations am–bm derived by Heymsfield et al. (2010) for stratiform clouds, and the one proposed by Lin and Colle (2011) for rimed mass fraction (defined as the ratio of the rimed mass to the mass of the entire rimed snow particle) varying between 0 and 0.3 at −15°C. The following lines are representations of (3) for two values of constant reference density ρ*: the thin and thick solid lines are for ρ* = 0.04 and 0.10 g cm−3 for D* = 1 mm, i.e., for m* = 2 × 10−5 and 5 × 10−5 g, respectively. The thin long dashed line corresponds to ρ* = 0.04 g cm−3, but for D* = 1.75 mm, i.e., for m* = 11.2 × 10−5 g.

Fig. 1.
Fig. 1.

Relationships between am and bm in (3). The points denoted by diamonds, compiled from various sources (see text), correspond to average relations obtained for ensemble of ice/snow particles. The circles show the average relations for cold high clouds (Heymsfield et al. 2004b; Schmitt and Heymsfield 2009). Two derived average relations between am and bm are superposed: from Heymsfield et al. (2010) for stratiform clouds (dotted–dashed line) and from Lin and Colle (2011) for riming degree between 0 and 0.30 (dotted line).The thin and thick solid lines are the lines of a constant ρ* equal to 0.04 and 0.10 g cm−3 for D* = 1 mm (i.e., for m* = 2 × 10−5 and 5 × 10−5 g, respectively). The thin long dashed line corresponds to ρ* = 0.04 g cm−3, but for D* = 1.75 mm (i.e., for m* = 11.2 × 10−5 g).

Citation: Journal of the Atmospheric Sciences 71, 3; 10.1175/JAS-D-12-0285.1

As can be seen, the parameter sets associated with average relations, mainly corresponding to a mixture of different, often irregular, shapes are situated above the solid line of ρ* = 0.04 g cm−3 for D* = 1 mm with m* = 2 × 10−5 g. All sets [am, bm] in Fig. 1 lying on the line of constant ρ* (or equivalently of constant m*), represent the mass–size relationship in (2) giving the same mass–density for particles with size D*.

An example illustrating the sensitivity of particle density to the exponent bm, using (5) and (6), is shown in Fig. 2. In the left panel, ρ is calculated from (5) and in the right panel from (6). These curves show calculation results for two selected mass–dimensional relationships: for early aggregates of planar polycrystals from Kajikawa (1982) (thick lines), and for average relation for synoptically generated “warm topped” clouds for temperatures higher than −25°C from Heymsfield et al. (2010) (thin lines). The solid lines in all graphs show the original relation, while the dotted–dashed and long dashed lines are obtained for bm reduced and increased by 0.2 with respect to the original values, respectively. The same original values are maintained for am in the left panel and for ρ* in the right panel. The ratio of the density for bm − 0.2 to the one for bm + 0.2 at constant am is approximately 2 for larger particles and 4–5 for the smaller ones. The effect on the calculated reflectivity in the Rayleigh regime is then a difference ranging from 3 to 6–7 dB.

Fig. 2.
Fig. 2.

Two examples of the sensitivity of the particle density on bm. The solid lines show the particle effective density as a function of D calculated for two empirical mass-parameter sets [am, bm]: [0.000 465, 1.77] for early aggregates of planar polycrystals from Kajikawa (1982) in the upper curves (thick lines), and [0.003 59, 2.1] for synoptically generated “warm topped” clouds for temperatures higher than −25°C from Heymsfield et al. (2010) at the lower curves (thin lines). The corresponding values of ρ* for D* = 1 mm are 0.015 and 0.054 g cm−3, respectively. The results obtained with the value of bm decreased by 0.2 and increased by 0.2 are shown using the superposed dotted–dashed and long dashed lines. (left) The calculations are done with a constant am value. (right) The ρ* value is kept constant (i.e., as in the original relations).

Citation: Journal of the Atmospheric Sciences 71, 3; 10.1175/JAS-D-12-0285.1

It is clear that using (5) the particle density is determined by a combination of two parameters that have to be selected correctly, while with (6) the relative errors in the calculated density resulting from the bm value is very small, and ρ* is the only parameter controlling particle mass or density accurately enough for a given range of sizes. Thus, using (4), the mass–size relationship is centered on a reference diameter and in this way the mass dependence on the value of bm is significantly reduced. In other words, for an assumed D*, the same value of m* groups together the sets [am, bm] that define exactly the same mass or density for particles with D = D*. For D not too different from D*, the mass and density are similar, independently of bm.

The selected value of D* has to be chosen within the particle sizes that have an important contribution to the bulk quantities considered. In cases where an estimate of the PSD characteristic size is possible, as in our retrieval from the dual-wavelength ratio (DWR) described in Part IV, D* can be adjusted to the particle sizes that dominate the radar reflectivity. The method to estimate the size regime corresponding to D* is described in Part IV. In general, this method is based on the presumption that using the normalized mass–size relationship in (4), the dependence on bm, and PSD functional form of the relationships between bulk quantities representing radar observables is expected to be minimal.

The particle size can also be defined in terms of a liquid-equivalent diameter Dmlt (i.e., the diameter that the snow particle would have if it were completely melted having spherical shape with constant liquid water density ρw). From the mass conservation, the following relation is obtained between D and Dmlt:
e7
To ensure that the particle density does not exceeds the solid ice density ρice, the following condition has been added: Dmlt = min{[ρ(D)/ρw]1/3D, (ρice/ρw)1/3D}. In the same manner, a minimal snow particle density of 0.005 g cm−3 has been imposed.

3. Terminal velocity calculations

The terminal velocity of an individual snow particle can be taken from empirical relations for a specified form of precipitating particles (e.g., Locatelli and Hobbs 1974) and also can be derived from the environmental conditions (Zawadzki et al. 2010). However, in situations where the mass–size relationship is expected to change, the use of these relations has to be done very carefully because of the expected correlation between the particle mass and its velocity. According to observational, laboratory, and theoretical studies, density and area have a major effect on the terminal velocity of snow particles. The main work on determination of terminal velocity as a function of particle mass–density has been essentially based on hydrodynamic theory using parameterized relationships between the Reynolds number (Re) and the Best (or Davies) number X, the latter expressed in terms of mass and effective cross-sectional area (Böhm 1989, Mitchell 1996, Mitchell and Heymsfield 2005, Khvorostyanov and Curry 2005). On the other hand, velocity–mass relationships have been found from measurements, like Langleben (1954) who empirically related the snow particle velocity to its melted diameter representing the mass.

In this study the terminal velocity of a snow particle is characterized by a given mass–size relationship that is calculated using two approaches. The first approach uses the parameterized relationship X–Re as proposed by Heymsfield and Westbrook (2010) combined with three empirical relations between particle mass or density and projected area or area ratio. In the second approach, the velocity is calculated directly from the mass using velocity–mass relationship for precipitating snow derived by Szyrmer and Zawadzki (2010).

The method of Heymsfield and Westbrook (2010) is a modified version of the previous methods based on an X–Re relationship. In this version, the velocity of a particle with a given D is computed from the ratio , where the area ratio Ar is defined as the area of the particle A divided by that of a circle of diameter D: ArA/(0.25πD2). The projected area A in the most common power-law representation takes the form . Then, the area ratio is related to D as follows: . The term required for the velocity computation by the method of Heymsfield and Westbrook (2010) can be written as
e8
As for the mass, in order to reduce the sensitivity of this term to the exponent, D* is introduced, and (8) is rewritten in a modified parameterized relation between , controlling the particle velocity for given environmental conditions, and D:
e9
To determine the constants k1, k2, and k3, the three following published average relations that relate the particle mass to its area ratio are selected to be applied in the retrieval:
e10a
  • used in Heymsfield (2003a) for particle ensembles with D > 1.5 mm observed in midlatitude and tropical clouds:
e10b
e10c

All three relations [(10a)(10c)] are written in cgs units. From these relations, and with help of (3)(6), the constants k1, k2, and k3 are calculated and their values are specified for each of the three relations [(10)] in Table 1. From (9), it may be seen that for a given D*, the computed terminal velocity corresponding to D not very different from D* is determined mainly by m*, and its dependence on bm is significantly reduced.

Table 1.

The coefficients in (9) derived for relations (10a)(10c).

Table 1.

The second method used to calculate the terminal velocity as a function of particle size and mass–density is taken from Szyrmer and Zawadzki (2010) by assuming that both mass and velocity are formulated as power laws with fixed exponents: 2 for mass and 0.18 for velocity. Hence, the derived relation between the velocity prefactor and the mass prefactor is . The added superscript “sv” to the two prefactors means that this relation is valid for the side-view particle dimension. Introducing fD describing the ratio between the major side-view dimension and the maximum size D, taken from the study, the last relation becomes when applied to the snow particle described by D. The value of fD progressively decreases starting from 1 for the smallest particles and with the minimal value fixed at 0.75 as in the original study. Using relation (4) instead of (2), am is given by . The terminal velocity at the ground level is then . All these relations are in cgs units.

Figure 3 presents the velocity calculated by the two methods as a function of D. The lines dotted–dashed, short dashed, and long dashed show the velocity computed using the method of Heymsfield and Westbrook (2010) assuming three parameterized relations in (10), hence with three sets of parameters k in (9). The solid lines show the velocity obtained from velocity–mass relationship derived by Szyrmer and Zawadzki (2010). The thinner and thicker lines correspond, respectively, to ρ* = 0.04 and 0.10 g cm−3 with D* = 1 mm.

Fig. 3.
Fig. 3.

Four terminal velocity–size relationships calculated for two different ρ*. The three lines—dotted–dashed, short dashed, and long dashed—show the velocity computed using the method of Heymsfield and Westbrook (2010) assuming three parameterized relations [see (10)], hence with three sets of parameters k in (9). The solid lines show the velocity obtained from velocity–mass relationship derived by Szyrmer and Zawadzki (2010). The thinner and thicker lines correspond respectively to ρ* = 0.04 and 0.10 g cm−3 with D* = 1 mm. The environmental conditions correspond to our reference conditions (i.e., −15°C and 780 hPa).

Citation: Journal of the Atmospheric Sciences 71, 3; 10.1175/JAS-D-12-0285.1

The environmental conditions in the velocity calculations have been fixed assuming the reference level situated approximately at the midlevel of the retrieval atmospheric layer—that is, at a temperature of −15°C and a pressure of 780 hPa. When applied at another temperature and pressure, as a first approximation, the velocity is adjusted following the adjustment given in Heymsfield et al. (2007) with ρa describing the air density:
e11

4. Parameterization of the snow particle size distributions within a double-moment scaling-normalization framework

In recent years, the concept of scaling normalization has been found to be very convenient for describing the rain PSD, thereby improving the representation of PSD in microphysical schemes (e.g., Szyrmer et al. 2005). The scaling of the snow PSDs as a function of Dmlt was presented by Sekhon and Srivastava (1970) in a single-moment normalization framework. A general framework for single-moment normalization was presented in Sempere-Torres et al. (1994). Lee et al. (2004) proposed a general formulation of the double-moment normalization of raindrops that includes the two-parameter normalization proposed previously by Testud et al. (2001) as a special case. To investigate ice/snow particle population the double-moment normalization approach has been used by Delanoë et al. (2005) with Dmlt describing particle size and by Field et al. (2005, 2007), Field and Heymsfield (2003), Westbrook et al. (2004), and Szyrmer et al. (2009) by choosing D as a dimension describing snow PSDs.

In this retrieval, the double-moment PSD representation is used with both size representations Dmlt and D. In the PSD representation with Dmlt, the normalizing moments are third and fourth moments, and (the superscript m means that the moment is calculated with Dmlt), or equivalently the total ice water content: and the characteristic size representing mean mass-weighted diameter. The PSD is written as , where h is the generic function of the scaled size . The normalized concentration (characteristic number density) is given by
e12
The PSDs with the actual maximal diameter used as particle size are normalized by the second and third moments: M2 and M3. The PSD characteristic size D23 that normalizes particle size is defined as . If bm is close to 2, then D23 represents the mean mass-weighted diameter and M2 is directly related to . The PSD is written as , with . The value of is given by
e13
Our sample of the uncertainty in the PSD shape is composed of three forms of h(xmlt) describing n(Dmlt), three forms of h(x) for n(D), and a monodisperse PSD. A general analytical functional form is used to describe h(xmlt), the generalized gamma (GG), defined as
e14
with . The selection of the two shape parameters μ and is based on the results of Delanoë et al. (2005) derived from very extensive database of airborne in situ microphysical measurements. We take three options among the shape parameter sets investigated by Delanoë: 1) inverse exponential: μ = 0 and (EXP), 2) third-order standard gamma: μ = 3 and (G3), and 3) the form that is the most consistent with the data according to the authors: μ = −1 and (GG). Therefore, the three PSD functions having xmlt as spectral coordinate have a general form given in (14) with
eq18

The PSD analytical functional forms h(x), when the actual maximal diameter D is used as the particle size, are

  1. inverse exponential observed in snow (Heymsfield et al. 2008b; Szyrmer et al. 2009),
  2. lognormal showing the better fit to the observed cirrus cloud spectra in Tian et al. (2010), and
  3. a combination of exponential and gamma functions derived by Field et al. (2007) for midlatitude conditions:
    e15

All the coefficients in (15) are given in Table 2

Table 2.

The coefficients in (15) taken from Field et al. (2007) for the midlatitude conditions.

Table 2.
.

In Fig. 4, six normalized functions are shown against the scaled size: xmlt in the left panel and x in the right panel. The , , and against xmlt, and , , and against x are calculated from . As can be noticed in Fig. 4, all the distributions are relatively similar for the medium-scale-sized particles. However, the differences are more pronounced for the small size particle population, as has been observed in general in many field campaigns. As well, an important dispersion can be noticed at the PSD tail of less numerous larger particles. However, PSDF07 [average form derived by Field et al. (2007)] and [the best approximating form according to Delanoë et al. (2005)] are very similar, even if they have been derived with different spectral coordinates. Moreover, it has been highlighted that in the dataset used by Delanoë et al. (2005) the artifacts associated with shattering of ice crystal on the probe housing were not removed.

Fig. 4.
Fig. 4.

Six polydisperse (polysized) normalized PSDs used in the retrieval. They are plotted against the scaled size (left) and (right) . The value of bm = 2 is assumed.

Citation: Journal of the Atmospheric Sciences 71, 3; 10.1175/JAS-D-12-0285.1

All six PSDs described above are taken twice: either assuming they are complete (i.e., untruncated) or that they are truncated. The largest sizes of the truncated PSD are assumed to be 2.5 times larger than the PSD characteristic size D23, representing mean mass-weighted diameter. The factor 2.5 is chosen based on the observational results shown in Heymsfield (2003b) and Brandes et al. (2007). In addition, a monodisperse distribution has been used as a thirteenth PSD model.

5. Relations between the microphysical parameters and the radar observables used in the retrieval

The microphysical quantities that control snow microphysics described in the previous section are the mass-parameter m* (or ρ*) determining the density of snow particles in the size regime important for Ze (and ) as well as two bulk quantities defining the snow PSD—namely, the characteristic size D23 () and related to the normalized concentration . Their retrieval is based on the relations derived from the assumed mass–velocity relationships given in section 3 combined with different PSD representations presented in section 4 providing the range of uncertainty in the microphysical assumptions. The relations between radar observables and parameters of the microphysical model are in addition influenced by the method of calculations of particle backscattering cross section. The backscattering cross section is calculated using the Mie theory for spheres (neglecting the effects of nonsphericity). In the model that we have used, described in Fabry and Szyrmer (1999), the density is assumed to be higher in the core than in the shell representing well the aggregates, where the higher density can be expected close to the particle center, except for chain aggregates. The scattering computations assume the mixing rule of Maxwell–Garnett for ice–air mixture with air inclusions in ice in the core and ice inclusions in air in the shell. In this approach, particle size and density constitute the key factors influencing the backscattering returns. While the use of the Mie theory provides the necessary accuracy in the reflectivity calculations at X band, it constitutes an important simplification for the W-band backscattering returns for which the non-Rayleigh effects become too important. However, the accuracy of the approximated calculations from the Mie theory depends on the choice of the equivalent sphere and its effective dielectric constant. It has been shown that the values of scattering parameters of nonspherical particles lie between Mie calculation results obtained with an ice volume equivalent sphere and with a homogenous “fluffy sphere” having a volume of the circumscribing sphere (Liu 2004; Wang et al. 2005). An inhomogeneous distribution of the density inside the sphere, as the one assumed in our calculations, provides intermediate results.

As customary, instead of the measurements represented by (1), we will use in the retrievals in Part IV the variables and using (1b). Given that the measured DWR and Uz are independent of the normalized concentration, like the PSD characteristic size and mass parameter, these two microphysical quantities are derived first from DWR and .

Figure 5 shows an example of the relation between D23 and DWR calculated for two complete exponential PSDs— in black and in red—for three different values of bm: 1.8, 2.0, and 2.2. The thicker and thinner lines show the results obtained with two values of ρ*. The presented results show that for PSD defined by a physical size D, here , the relation DWR–D23 is independent on the particle density, as demonstrated in the previous studies. But it is not the same for the “melted” PSD, . For both PSDs, the value of the derived D23 from DWR is sensitive to bm, representing particle overall habit. In addition, some dependence of D23 on the PSD functional form has to be added. However, despite the uncertainty related to the assumed PSD microphysics shown in Fig. 5 (at which is attached the uncertainty of a backscattering calculation, mainly by the W-band radar), the value of measured DWR provides an important piece of information about D23.

Fig. 5.
Fig. 5.

Calculated relationship D23–DWR for two complete PSDs: (black) and (red), assuming three different values of bm: 1.8 (long dashed), 2.0 (solid), and 2.2 (dotted–dashed). The thin and thick lines present the results for ρ* = 0.04 and 0.10 g cm−3, respectively. All plotted lines are obtained with D* = 1 mm.

Citation: Journal of the Atmospheric Sciences 71, 3; 10.1175/JAS-D-12-0285.1

Besides the DWR, the mean reflectivity-weighted velocity is a function of D23 being independent on the normalized concentration. Hence, in the retrieval, the derivation of D23 is done together with ρ* from DWR and from the mean reflectivity-weighted velocity . In Part IV we show how the determination of D* based on the DWR value leads to the reduced sensitivity of the retrieved pair D23ρ* to bm, thus leaving only one parameter to describe the particle density representative for the bulk quantities of interest. In addition, the retained pair D23ρ* has to satisfy the observed DDV. Because the dynamic range of the observed DDV is relatively small, the value of DDV is used only as a constraint to be satisfied by the retrieved parameters.

Once the PSD characteristic size and effective density are determined, the ratio is computed for a given PSD form for each retrieved pair of D23 and ρ*. The values of and of the normalized concentration are then obtained from the ratio and the measured reflectivity . Figure 6 shows calculated for two different densities and for seven PSD forms: a monodisperse and six complete PSDs described in section 4 are shown in Fig. 4.

Fig. 6.
Fig. 6.

Calculated relationship D23 for seven untruncated PSDs. The thin and thick lines present the results for ρ* = 0.04 and 0.10 g cm−3, respectively. All plotted lines are obtained for bm = 2 and with D* = 1 mm.

Citation: Journal of the Atmospheric Sciences 71, 3; 10.1175/JAS-D-12-0285.1

To evaluate the reflectivity-weighted fall speeds from the measured Doppler velocity, w is required as given in (1b). Our estimation of w is based on the assumption that all water vapor above saturation, generated by w, is immediately transferred on ice and is considered to be equal to the growth of snow represented by the vertical gradient of :
e16
where describes vertical gradient of mixing ratio saturated with respect to ice and is mean mass-weighted fall speed. The ratio is estimated for a given microphysical assumptions, leading to . Introducing this last relation into (16) gives w as a function of and rsi gradients, and also measured Doppler velocity :
e17
This approach to estimate w from the vertical gradient of snow mass content is based on the methodology developed in Zawadzki et al. (2000) to diagnose the presence of supercooled clouds. The method is based on the equilibrium assumption first suggested by Kessler (1969): the rate of growth of detectable condensate equals the rate of generation of excess of water vapor with respect to saturation. Note that computing w from (17) does not preclude the coexistence of supercooled water with snow: the transfer from vapor to ice can be by direct deposition, by condensation followed by the Bergeron process, or by riming, as long as the amount of supercooled water is at a steady-state value.

6. Concluding remarks

The purpose of this paper is to introduce the derivation of the basic relations used in the retrieval described in Part IV linking the dual-wavelength Doppler radar observables and the snow microphysics parameters. These parameters are the key parameters in the presented microphysical model of snow, and they form a set of unknown quantities that are derived in the proposed algorithm. Because of a large number of microphysical parameters required to map the bulk snow microphysics to the radar observables, our goal has been to reduce the number of free parameters and to decrease the sensitivity of some of them to uncertainties in the parameterization of other parameters. This is achieved with the following steps (or assumptions):

  1. The use of conventional mass–size (or density–size) power-law relationship necessities the knowledge of two parameters. Moreover, these parameters may be different for size regimes that dominate lower-order moments and higher-order moments of the PSD. On the other hand, an assumed unique mass–size relationship may be a cause of important bias in the calculated radar observables, as has been shown in many previous studies. In our snow representation, this relation is used in a normalized form required to specify only one parameter. The normalizing diameter D* represents the PSD size regime that is an important contributor to the reflectivity and mass content. The mass–density parameter is included in the output information.
  2. Using the published relations, the snow particle velocity is expressed in terms of particle mass–density parameter, avoiding the imposition of the velocity–size relation parameters. Because of the uncertainty of the dependence of particle velocity to its mass, four different published relations have been used.
  3. The PSD is described by two bulk quantities using the two-moment normalization approach. The two normalizing quantities are controlled by the PSD segment, which is an important contributor to the radar observables and the microphysical parameters to be retrieved. In this manner the sensitivity of the integrated quantities to the assumed functional form is reduced, albeit not removed. Therefore, different PSD representations, defined for actual or melted sizes, complete or truncated, are used.

The sensitivity of the established deterministic relations providing the correspondence between observations and the free parameters has been included by the use of different methods to calculate snow particle terminal velocity from its mass and also by introducing various empirically derived functional forms of PSD. All resulting relations written for an individual pixel will be applied in the retrieval in Part IV as different possible solutions of equal probability of occurrence. Since the focus in this study lies on the methodology of the retrieval within a stochastic approach to investigate how variability in snow microphysics map to the retrieval output information, other sources of uncertainty in the developed relations are not taken into account.

The most important other source of uncertainty in the relations between snow microphysics and radar observables is related to the modeling of the W-band backscattering properties of individual snow particles, as shown in the published studies devoted to highlight and quantify these uncertainties (Hiley et al. 2011; Kneifel et al. 2011). The bias introduced by the inaccuracy of the Mie computation applied to the nonspherical ice crystals at W band is very difficult to estimate. In general, the error in the backscattering calculations using the Mie method increases with the particle sizes. But the contribution of the larger particles to the total reflectivity is limited because 1) their concentration rapidly decreases with size and 2) their backscatter cross section is significantly reduced owing to the non-Rayleigh effect. The modeled cumulative reflectivity at 94 GHz presented in Heymsfield et al. (2008a) for a given PSD shows the independence of the calculated reflectivity of 10 dBZ on the presence of particles larger than 1.5–2 mm.

The advanced, more accurate methods of backscattering calculations [e.g., discrete dipole approximation (DDA), T matrix] require the details of the ice particle geometry—mainly overall shape and orientation. Based on the actual knowledge, these details in natural ice/snow cannot be accurately associated to environmental conditions. Therefore, with no additional piece of information about representative particle shape and orientation behavior, the retrievals as the one proposed in Part IV are always subject to bias error. Within a stochastic approach, the shape and orientation of particles could be considered as two stochastic elements.

APPENDIX A

Relations between Bulk Quantities of the PSDs Formulated with Two Different Particle Size Descriptors: Dmlt and D

In the double-moment PSD representation with liquid-equivalent diameter as size parameter the power law that relates any moment to the two reference moments, here and can be written as
ea1
where Cp is the moment of order p of the generic PSD function (Lee et al. 2004, Szyrmer et al. 2005). The relations between the moments of a PSD with physical maximal diameter D describing particle size and an equivalent PSD with liquid-equivalent diameter Dmlt can be obtained from (7):
ea2
With the help of (A1) and (A2), the following relation can be obtained between representing the mass-weighted diameter for the PSD with D as size descriptor and , which is a mean mass-weighted diameter of the PSD with melted diameter as size descriptor:
ea3

The result is a function of the assumed bm. Also, D23 becomes the mass-weighted diameter if bm = 2. The dependence on the selected shape parameters of the PSD is via the term , while the dependence on the density is via m*. Since for bm = 2 the factor C4.5 is rather very close to 1, the value of D23 is very close to the physical diameter corresponding to D34(m).

The similar relations can be found between two normalized concentration parameters defined in (12) and (13) with bm = 2:
ea4

APPENDIX B

Issue of Backscattering Modeling Uncertainties

The problem of modeling of the radar scattering properties of nonspherical ice/snow particles remains an open issue. Recently, significant progress has been made in the accurate, more rigorous methods of backscattering computation. However, even if the backscattered energy can be calculated correctly, the exact solution is not well defined, because the actual knowledge is not enough to provide all details required for the computation. The natural ice/snow particles present very complex, often irregular, and highly variable shape and inhomogeneous mass distribution, even within one event.

Some studies (e.g., Matrosov et al. 2005, Hogan et al. 2012) have demonstrated in their experiments that the assumptions of horizontally oriented oblate spheroids with a uniform density distribution used within the T-matrix method give a good agreement between the computed and radar-measured quantities for the assumed aspect ratio (defined here as the ratio of minor to major size) around 0.6–0.7. As shown in these studies, the W-band backscattered intensities from larger, horizontally aligned spheroids are greater than those for spheres with a uniform density distribution. The backscattering computation used in this study, described in section 5, assuming a spherical shape but with a more realistic nonuniform density distribution, provides similar results, as shown as a function of maximum diameter D in Fig. B1. The solid line shows the backscattering values obtained from our two-layer spherical model. The mass–size relationship is taken from Matrosov (2007). For comparison, two other backscattering cross sections, calculated with the same mass–size relationship, have been added to show the sensitivity of the computation results to the assumptions about the effective medium approximations and the mass repartition inside the sphere. The dotted–dashed line illustrates the results for the sphere with a two-layer mass distribution and the same Maxwell–Garnett mixing rule, which is ice inclusions embedded within the air matrix (the solid line is derived with the same mass repartition, but the central part is modeled as air inclusions in the ice matrix). The long-dashed line is derived with the same dielectric mixing rule but with a uniform mass distribution. The dotted line shows the Rayleigh calculations. The additional line is obtained using the database developed by Liu (2008) for the dendrite category. This category has been selected by Molthan and Peterson (2011) as representative for snow in the simulations of the CloudSat radar reflectivity.

Fig. B1.
Fig. B1.

Backscattering cross sections as a function of D. Except the line illustrating the dendrites from Liu library, the remaining lines are derived for the spherical model with the assumed mass–size relationship taken from Matrosov (2007) corresponding to m* = 3.0 × 10−5 g for D* = 0.1 cm. See the text for more information.

Citation: Journal of the Atmospheric Sciences 71, 3; 10.1175/JAS-D-12-0285.1

The results of calculations with the T-matrix method assuming spheroidal shape and uniformly distributed density are sensitive, beside the particle mass and size, to the particle aspect ratio and preferential fall orientation. For example, Kneifel et al. (2011) showed that the change of the assumed aspect ratio of horizontally oriented oblate spheroids from 0.5 to 0.7 results in a reduction of the W-band reflectivity of about 5 dB calculated for the same PSD. On the other hand, for larger snowflakes, horizontal orientation is privileged, but the degree of this behavior and the associated aspect ratio are rather unclear. Different field campaigns show that this aspect ratio varies mainly between 0.6 and 1 as reviewed in Szyrmer et al. (2012). In general, observations provide rather controversial information about both the snowflake aspect ratio and the extent of the preferential horizontal orientation, which may be representative of the snow particle ensemble observed by radar in the association to the environmental conditions.

The natural variability of snow explains the fact that during certain field campaigns the spherical particle model has provided the results that fall within the range of measurement uncertainty. For example, Pokharel and Vali (2011) obtained good overall agreement between the W-band radar measurements and the reflectivity calculated using the Mie theory from in situ measured particle size distributions. Both the radar and the particle probe were carried on the same aircraft. Good correspondence of the Mie-calculated reflectivity and the radar measurements was found within the range from −18 to +16 dBZ.

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1

The relation from Brandes et al. has been recalculated taking Deq (equivalent spherical volume diameter used in their study) = 0.7D, while the relation from Brown and Francis (1995) defined for Dmean (the mean of the two orthogonal extensions) has been converted to D using the average relation Dmean = 0.8D from Hogan et al. (2012). The average relation from Szyrmer and Zawadzki (2010) obtained in terms of the side-view major size has been converted to D by assuming the ratio of 1.33 between D and the side-view size to be representative for larger particles.

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