## Abstract

A novel method of retrieving the mean axis ratio (width/length) and standard deviation of orientation angles (*σ _{θ}*, which is called herein the intensity of fluttering) of ice cloud particles from polarimetric radar data is described. The method is based on measurements of differential reflectivity

*Z*

_{DR}and the copolar correlation coefficient in cloud areas with

*Z*

_{DR}> 4 dB. In three analyzed cases, the values of the retrieved axis ratio were in an interval from 0.15 to 0.4 and

*σ*found in an interval from 2° to 20°. The latter values indicate that the particles experienced light to moderate fluttering. Ambiguities in the retrievals because of uncertainties in the bulk ice density of the particles and possible presence of columnar crystals are considered. The retrieval method is applicable for centimeter-wavelength radars; the analyzed data were collected with the dual-polarization S-band Weather Surveillance Radar-1988 Doppler (WSR-88D).

_{θ}## 1. Introduction

Information on the shapes of ice cloud particles helps with the understanding of transitions among liquid, vapor, and ice in clouds. Transformations of the particles' shapes because of vapor deposition, accretion, riming, and aggregation are the key issues in cloud evolution. Ice crystals are the foundation for much precipitation worldwide and ice clouds play a significant role in the radiation balance of the earth system. Large amounts of data on particle shapes have been obtained with laboratory experiments and onboard probes (e.g., Pruppacher and Klett 1997, section 2.2.1; Korolev and Isaac 2003; Bailey and Hallett 2004, 2009; Heymsfield et al. 2011). Direct cloud measurements with onboard probes are very costly and provide data along airplane tracks (i.e., in areas with extremely small volumes compared to cloud volumes available from radar). Onboard measurements are also prone to the shedding of particles (e.g., Korolev et al. 2011). On the other hand, remote sensing of clouds with scanning weather radars does not affect particles, while at the same time it provides information from large cloud areas or volumes in real time.

Polarimetric radars have the potential of estimating particles' sizes and orientations. Empirical dependencies of observable reflectivity, differential reflectivity, and the specific differential phase upon the shapes, sizes, and density of particles have been used (e.g., Vivekanandan et al. 1991, 1994). The applicability of the empirical relations to each particular cloud frequently remains unclear. Thus, more detailed information on cloud particle characteristics is needed for detailed cloud studies.

Matrosov et al. (2001, 2005, 2012) and Reinking et al. (2002) showed that an elevation dependence of the depolarization ratio can be used to differentiate types of ice cloud particles. The authors used polarimetric radar that was capable to measure depolarization. The most prevalent radar polarization configuration at the time of this writing is a scheme with simultaneous transmission and reception (STAR) of horizontally and vertically polarized waves. Such radars cannot measure depolarization, although a proxy for that has been proposed (Matrosov 2004). Differential reflectivity *Z*_{DR} the differential phase *ϕ*_{dp}, and copolar correlation coefficient *ρ*_{hv} are measured with STAR radars. Retrievals of the shapes, axis ratios (width/length), and orientation parameters of cloud ice particles based on measured *Z*_{DR} and *ρ*_{hv} are described herein. The radar parameters, measured with STAR radars, are considered in the next section. Retrievals of axis ratios and orientation parameters from measured *Z*_{DR} (>4 dB) and *ρ*_{hv} are described in section 3, and radar data are analyzed in section 4.

## 2. *Z*_{DR} and *ρ*_{hv} of ice particles measured with STAR radar

Radar differential reflectivity and the correlation coefficient depend on the shapes of cloud particles and their spatial orientations. Orientations of cloud particles are characterized with the mean alignment angle and a distribution of orientations relative to this mean. The mean alignment angle is usually a direction of a particle's symmetry axis. For raindrops, the mean orientation angle of the droplet's minor axis relative to vertical is called the canting angle (e.g., Bringi and Chandrasekar 2001, section 3.13.2). The term “canting angle” is used herein as well for the mean alignment of the symmetry axis of ice cloud particles. Under calm air conditions, ice particles exhibit glide-pitch motions and spiral oscillations called “flutter” (e.g., Pruppacher and Klett 1997, section 10). All alternations in a particles' orientation are called flutter herein for brevity.

Positive *Z*_{DR} in observed ice clouds indicate that particles are nonspherical and predominantly oriented horizontally. The air temperature, pressure, and humidity, along with ice nuclei in clouds, determine the variety of ice particles' shapes from simple plates and columns to complex rosettes and dendrites, among other shapes as well as aggregates. A variety of crystal shapes stimulates the application of modern techniques for calculations of scattering properties of the particles (e.g., Mishchenko et al. 2000; Teschl et al. 2009) but to our knowledge there have been no studies of *Z*_{DR} and *ρ*_{hv} for cloud particles of complex shapes. Few studies consider effects of orientations of particles with complex shapes (e.g., Matrosov et al. 2001, 2005; Teschl et al. 2013). We study herein cloud areas with *Z*_{DR} > 4 dB, which implies that the particles have a platelike shape. Such particles can also have different forms from simple plates to platelike cores with different extensions (Pruppacher and Klett 1997, section 2.2.1). To account for possible difference in plate shapes, variations of bulk ice density are considered in section 4b.

The crystal shapes that can be represented by plates and columns can be approximated with oblate and prolate spheroids. The geometry of scattering particles and incident radar waves is sketched in Fig. 1. Spheroids have two principal semiaxes, *a* and *b* (*a* ≥ *b*), and the axis of rotation (OO′). The axis ratio for a spheroid is *b*/*a*. The orientation of the spheroids is characterized by two angles *θ* and *ϕ* between the axis of symmetry and the coordinate axes *X*, *Y*, and *Z*. Plane XOY is horizontal. The direction of propagation of electromagnetic waves is determined by the vector **k**, which lies along the *x* axis (i.e., horizontal sounding is considered). This case is a good approximation for antenna elevations less than 7°. A Cartesian coordinate system *X*, *Y*, *Z* is natural for a cloud particle: its terminal fall velocity is directed along the *Z* axis, and angles *θ* and *ϕ* determine the particle's orientation uniquely.

In the STAR radars, signal paths in the two radar channels with horizontally and vertically polarized waves are different so the transmitted and received waves acquire hardware phase differences *ψ _{t}* and

*ψ*, where the subscripts stand for transmit and receive. These phases are called the system differential phases upon transmit and receive. A cloud of nonspherical scatterers shifts the phase between the horizontally and vertically polarized waves by the propagation differential phase

_{r}*ϕ*

_{dp}and the differential phase upon scattering

*δ*so that the measured phase shift is

*ψ*

_{dp}=

*ψ*+

_{t}*ψ*+

_{r}*ϕ*

_{dp}+

*δ*. The phase

*δ*upon scattering by ice cloud particles at centimeter wavelengths is negligible. Thus propagation and scattering of polarized waves can be described for the STAR configuration by the following matrix equation:

where the received wave amplitudes (proportional to measured voltages) are on the left side and the transmitted wave amplitudes are denoted as *E _{h}* and

*E*. The matrix to the left from (

_{υ}*E*) describes the propagation of the waves from radar to the radar resolution volume,

_{h}E_{υ}*S*is the scattering matrix of the particles, the matrix to the left from

_{ij}*S*describes propagation of the waves on their way back to radar, and

_{ij}*C*is a constant depending on radar parameters and range to the resolution volume. This constant is omitted in the following discussion because

*Z*

_{DR}and

*ρ*

_{hv}do not depend upon it. The calibration procedure equalizes the difference in amplitudes

*E*and

_{h}*E*in (1) so we can assume that they are equal and omit them as well. We also neglect differential attenuation in nonprecipitating clouds and in the tops of precipitating systems that are considered herein. At S band, the differential phase because of propagation in such clouds also can be neglected.

_{υ}The received mean powers 〈*P _{h}*〉 and 〈

*P*〉 and signal correlation function 〈

_{υ}*R*

_{hv}〉 are

where the angle brackets stand for time averaging. Time averaging includes averaging over the orientation angles and sizes because the number of scatterers in the radar resolution volume is large. Differential reflectivity *Z*_{DR} in decibels and *ρ*_{hv} are obtained as

For a single scattering particle at a given orientation, the matrix coefficients can be represented as

where *α _{h}* and

*α*are polarizabilities along the major and minor axes of the particle [e.g., Bringi and Chandrasekar (2001), their Eq. (2.53) presented in the backscatter alignment (BSA) convention]. At centimeter wavelengths, cloud particles can be considered as Rayleigh scatterers and the polarizabilities for spheroids can be found in Bringi and Chandrasekar (2001, section A1.1). The powers and correlation function for a single particle at given

_{υ}*θ*and

*ϕ*are

where Re(*x*) stands for the real part of *x* and the asterisk denotes complex conjugate. The mean 〈*P _{h}*〉, 〈

*P*〉, and 〈

_{υ}*R*

_{hv}〉 are obtained from (5) by averaging over orientations and sizes. It is seen from (5) that the differential phase in receive

*ψ*does not affect the received powers and consequently

_{r}*Z*

_{DR}. The phase enters in

*R*

_{hv}as a multiplicative exponent, which means it does not affect the module of the function (i.e.,

*ψ*does not affect

_{r}*ρ*

_{hv}as well). In contrast, the phase upon transmit

*ψ*affects

_{t}*P*,

_{h}*P*, and

_{υ}*R*

_{hv}.

Radar data from stratiform clouds and precipitation are analyzed herein. In such clouds, electric fields are not strong enough to influence the alignment of particles so a zero mean canting angle is assumed for the particles. The fluttering of particles in the air changes orientation angles *θ* and *ϕ*. Angle *ϕ* expresses the orientation of a particle in the horizontal plane. In the absence of electric fields, this angle is random because the downfall of ice particles is accompanied with swinging/tumbling/fluttering that randomizes the *ϕ* distribution. For the random *ϕ* distribution, 〈sin*ϕ*〉 = 〈sin^{3}*ϕ*〉 = 0, 〈sin^{2}*ϕ*〉 = 1/2, and 〈sin^{4}*ϕ*〉 = 3/8, where the angle brackets stand for averaging over *ϕ.*

Orientations in *θ* and *ϕ* can be described with probability *p*(*θ*, *ϕ*) sin*θ dθ dϕ*. The terms *P _{h}*,

*P*, and

_{υ}*R*

_{hv}in (5) depend on the trigonometric functions of

*θ*and

*ϕ*so their means can be expressed via the following moments:

where 〈*x*〉_{θ,ϕ} stands for averaging of *x* over *θ* and *ϕ*.

Assuming independence of the distributions in orientations and sizes, we average (5),

where the angle brackets around polarizabilities denote averaging over sizes. It is seen that 〈*P _{h}*〉 and 〈

*P*〉 do not depend on phase in transmit

_{υ}*ψ*now, and this is a result of complete randomness in

_{t}*ϕ*distribution.

For distributions in *θ*, the Gaussian, Fisher's, and axial bell-shaped functions can be used (e.g., Bringi and Chandrasekar 2001, section 2.3.6). The Fisher's distribution is used herein as it naturally describes probabilities on a sphere (Fisher 1953). For platelike particles with a zero mean canting angle, *p*(*θ*, *ϕ*) is a function of *θ* only:

where the parameter *μ* can be represented via the width of distribution, *σ _{θ}*:

For platelike particles, the moments from (6) are

For columns oriented preferably horizontally, 〈*θ*〉 = 90° and the Fisher distribution depends on *θ* and *ϕ* [e.g., Eq. (2.73) in Bringi and Chandrasekar (2001)]. In this case, moments *J*_{1} and *J*_{2} have been obtained numerically.

Values of *Z*_{DR} and *ρ*_{hv} for platelike and columnlike particles are depicted in Fig. 2 for different *σ _{θ}*, where

*σ*= 39° corresponds to the fully random

_{θ}*θ*distribution. Figure 2 is for solid ice with a density of 0.9 g cm

^{−3}. Figure 2b is presented for

*ψ*= 82° measured by the Norman, Oklahoma (KOUN), radar. Cloud particles in the radar volume have different sizes and axis ratios. Ratios

_{t}*a*/

*b*in Fig. 2 should be considered as the mean values averaged over size parameters. For completely random distributions in

*θ*,

*Z*

_{DR}= 0 dB regardless of

*b*/

*a*; so corresponding curves in Fig. 2a coincide with the horizontal axis. It can be seen that the maximum

*Z*

_{DR}is 10 and 4 dB for plates and columns, respectively, and is attained for very thin strictly horizontal particles. It is also seen (Fig. 2a) that if the measured

*Z*

_{DR}exceeds 4 dB, the particles have platelike shapes. This was obtained for radars with alternating polarizations by Hall et al. (1984), Illingworth et al. (1987), and Hogan et al. (2002). We see that this holds for radars with the STAR configuration as well, which means that depolarization of signals in the STAR configuration does not alter the maximum value of

*Z*

_{DR}. This is a consequence of zero canting angle and the random

*ϕ*distribution. If one of the latter does not hold,

*Z*

_{DR}in the STAR and alternating configurations are different.

Two more important conclusions can be drawn from Fig. 2a for particles with *Z*_{DR} > 4 dB. 1) Values of *σ _{θ}* for such particles are less than 17.5°. The latter is obtained from a curve for plates that touches

*Z*

_{DR}= 4 dB at

*b*/

*a*= 0 (not shown in Fig. 2a). As such,

*σ*< 17.5° signifies that the particles experience small to moderate fluttering. More accurate estimation of

_{θ}*σ*is obtained in the next section. 2) The axis ratio for such particles is less than 0.4. This conclusion follows from consideration of the intersection of the horizontal line at

_{θ}*Z*

_{DR}= 4 dB and the curve for plates with

*σ*= 2° (i.e., for nearly horizontal particles). So if scatterers exhibit

_{θ}*Z*

_{DR}> 4 dB, they are platelike particles with

*b*/

*a*< 0.4 and

*σ*< 17.5°. Precise retrievals of

_{θ}*b*/

*a*and

*σ*are possible by considering also measured

_{θ}*ρ*

_{hv}, which is demonstrated in the next section.

The correlation coefficients *ρ*_{hv} for ice plates and columns are depicted in Fig. 2b for different *σ _{θ}*. At

*b*/

*a*< 0.4, the difference in

*ρ*

_{hv}for plates and columns is substantial. For very thin nearly horizontal particles,

*ρ*

_{hv}is close to unity for plates and approaches 0.97 for columns, which is due to random distribution on the horizontal plane.

## 3. The retrieval algorithm

Retrievals of the mean aspect ratios *b*/*a* and the standard deviation of orientation angles *σ*_{θ} are based on solving (3) and (7) for given *Z*_{DR} and *ρ*_{hv} that are measured with radar. The retrieval is applicable for cloud areas with *Z*_{DR} > 4 dB (i.e., for platelike particles). Parameters *b*/*a* and *σ _{θ}* cannot be expressed in the close form via

*Z*

_{DR}and

*ρ*

_{hv}using (3) and (7). This procedure is performed numerically and can be illustrated as follows. Consider the retrievals of

*b*/

*a*and

*σ*for one pair of

_{θ}*Z*

_{DR}and

*ρ*

_{hv}, say

*Z*

_{DR}= 5 dB and

*ρ*

_{hv}= 0.95, which are measured often in clouds (shown in Figs. 4 and 5). Since

*Z*

_{DR}= 5 dB, we conclude from Fig. 2a that the particles have

*σ*< 14.5° and

_{θ}*b*/

*a*< 0.4. Plots of

*Z*

_{DR}and

*ρ*

_{hv}in Fig. 3 are for

*Z*

_{DR}around 5 dB,

*ρ*

_{hv}around 0.95, and

*σ*< 10°. Values of

_{θ}*Z*

_{DR}and

*ρ*

_{hv}increase with decreasing

*σ*. The lowest curves in Figs. 3a and 3b correspond to

_{θ}*σ*= 9.2°. For this curve in Fig. 3a,

_{θ}*Z*

_{DR}= 5 dB corresponds to

*b*/

*a*= 0.195, which has

*ρ*

_{hv}of 0.945 (Fig. 3b). The latter number is lower than that required (i.e.,

*ρ*

_{hv}of 0.950). So

*σ*should be decreased further. The thick dashed line has

_{θ}*b*/

*a*= 0.20 at

*Z*

_{DR}= 5 dB and

*ρ*

_{hv}= 0.948, which is still slightly smaller than that required 0.950. The next curve in the figure is for

*σ*= 8.9°. For this curve, we obtain

_{θ}*b*/

*a*= 0.205 from Fig. 3a and

*ρ*

_{hv}= 0.950 is from Fig. 3b. So these

*b*/

*a*and

*σ*match given

_{θ}*Z*

_{DR}and

*ρ*

_{hv}and are the solution to (3) and (7). To run this procedure numerically, lookup tables of

*Z*

_{DR}(

*b*/

*a*,

*σ*) and

_{θ}*ρ*

_{hv}(

*b*/

*a*,

*σ*) with strides in

_{θ}*b*/

*a*and

*σ*of 0.005 and 0.1°, correspondingly, have been generated.

_{θ}Radar measurements are made with uncertainties Δ*Z*_{DR} and Δ*ρ*_{hv}, which depend on system calibration and parameters of observations and create uncertainties in retrieved *b*/*a* and *σ _{θ}*. A part of Δ

*Z*

_{DR}is determined by the quality of hardware and its calibration. This uncertainty is about ±0.1 dB for the Weather Surveillance Radar-1988 Doppler (WSR-88D) (Zrnic et al. 2006). The accuracy of

*ρ*

_{hv}measurements does not depend upon calibration. Radar returned signals fluctuate by nature. These fluctuations produce statistical uncertainties in the measured

*Z*

_{DR}and

*ρ*

_{hv}. The statistical uncertainties are usually characterized with the standard deviations (SD), which depend on the number of samples

*M*used in data processing, signal-to-noise ratio (SNR), intrinsic

*ρ*

_{hv}, and spectrum width

*W*in the radar resolution volume (Melnikov and Zrnic 2007). Number

*M*is known, measured

*ρ*

_{hv}is accepted as the intrinsic value, and SNR and

*W*are measured.

The 1-*σ* errors are considered herein as a measure of statistical uncertainties in measured and retrieved variables. Let *Z*_{DR(m)} and *ρ*_{hv(m)} be the values measured with radar. It is assumed that true *Z*_{DR} and *ρ*_{hv} lie in the following intervals:

where Δ*ρ*_{hv} = 0.5SD(*ρ*_{hv}) and Δ*Z*_{DR} = 0.5SD(*Z*_{DR}) + 0.1. The second addend in Δ*Z*_{DR} is from the hardware uncertainty. The retrievals of *b*/*a* and *σ _{θ}* are performed for four limiting pairs of values in (11); that is, [

*Z*

_{DR(m)}− Δ

*Z*

_{DR},

*ρ*

_{hv(m)}− Δ

*ρ*

_{hv}], [

*Z*

_{DR(m)}+ Δ

*Z*

_{DR},

*ρ*

_{hv(m)}− Δ

*ρ*

_{hv}], [

*Z*

_{DR(m)}− Δ

*Z*

_{DR},

*ρ*

_{hv(m)}+ Δ

*ρ*

_{hv}], and [

*Z*

_{DR(m)}+ Δ

*Z*

_{DR},

*ρ*

_{hv(m)}+ Δ

*ρ*

_{hv}]. These four pairs of input parameters produce four pairs of

*b*/

*a*and

*σ*from which Δ(

_{θ}*b*/

*a*) and Δ

*σ*are obtained as maximum differences of the retrieved four values of

_{θ}*b*/

*a*and

*σ*and those values retrieved from

_{θ}*Z*

_{DR(m)}and

*ρ*

_{hv(m)}. For instance, consider again

*Z*

_{DR(m)}= 5 dB and

*ρ*

_{hv(m)}= 0.95. For the area with

*Z*

_{DR}> 4 dB beyond a range of 50 km in Fig. 4b, the median SNR in the horizontal radar channel is about 10 dB and the spectrum width is 1.5 m s

^{−1}. For these parameters and

*M*= 128, SD(

*Z*

_{DR}) = 0.69 dB and SD(

*ρ*

_{hv}) = 0.026 so that Δ

*Z*

_{DR}= 0.45 dB and Δ

*ρ*

_{hv}= 0.013. Retrieved

*b*/

*a*and

*σ*for

_{θ}*Z*

_{DR(m)}= 5 dB and

*ρ*

_{hv(m)}= 0.95 are 0.205 and 8.9°, respectively. The four pairs of retrieved

*b*/

*a*and

*σ*that follow from (11) are (0.205, 10.6°), (0.160, 8.9°), (0.255, 8.6°), and (0.200, 7.1°). The maximum deviations in

_{θ}*b*/

*a*and

*σ*from the obtained 0.205 and 8.9° are 0.05 and 1.8°, respectively. The latter values are the statistical uncertainties in retrieved

_{θ}*b*/

*a*and

*σ*; given as a percent they are about 24% and 20%, respectively.

_{θ}The retrieval uncertainties in *b*/*a* and *σ _{θ}*, obtained above, are for a single radar resolution volume. To reduce the uncertainty, the standard deviations in

*Z*

_{DR}and

*ρ*

_{hv}should be decreased. This can be done with averages of radar data either spatially or temporally. The radial extent of the radar resolution volume of the WSR-88Ds is 250 m. By averaging data collected from four consecutive range gates, the mean radar parameters are obtained for a 1-km cell. Such averaging reduces the standard deviations by 2 times (4

^{1/2}) so that the axis ratios and intensity of fluttering can be retrieved with 2 times as small statistical uncertainties. For the previously considered examples, the uncertainties will be reduced to about 12% and 10% for

*b*/

*a*and

*σ*, respectively. The same reduction in the retrieval uncertainties can be achieved for a range resolution of 250 m with 4 times longer dwell time (i.e., with

_{θ}*M*= 512 samples). This requires slower antenna motion that can be afforded with research radars.

The estimation of statistical uncertainties from (11) is obtained using the 1-*σ* boundaries. About 70% of the measurements lie inside the boundaries of error, assuming that they are distributed in a Gaussian manner. The distributions of deviations in measured *Z*_{DR} and *ρ*_{hv} are close to the Gaussian one. Thus, the distributions of the uncertainties of retrieved *b*/*a* and *σ _{θ}* are close to Gaussian as well because relations between

*Z*

_{DR},

*ρ*

_{hv},

*b*/

*a*, and

*σ*are almost linear for the deviations (Fig. 3). So for the Gaussian distribution of the retrieval uncertainties, the 1-

_{θ}*σ*boundaries account for about 70% of data and we can expect some of the retrieval results to be outside the boundaries.

## 4. Retrieval results

### a. Radar data and retrievals for solid ice particles

To retrieve *b*/*a* and *σ _{θ}*, data from the dual-polarization 11-cm-wavelength KOUN WSR-88D have been used. The data (Fig. 4) were collected in nonprecipitating clouds in a true range-height indicator (RHI) mode as described by Melnikov et al. (2011). In areas close to the radar, ground clutter is so strong that the spectral leakage in the clutter filter produces residues with reflectivities less than −10 dB

*Z*. Such residues (Figs. 4d,e) at ranges within 14 km are seen as the gray background. At ranges less than 4 km, ground clutter residues are stronger (Fig. 4d).

In Fig. 4b, one can see isolated areas with *Z*_{DR} exceeding 5 dB. They are also seen in Fig. 4e, where the whole cloud exhibits very high differential reflectivities. The maximum *Z*_{DR} in both cases was about 8.5 dB. The spatial structures of reflectivity *Z* and *Z*_{DR} in Figs. 4a and 4b permit a conclusion that convection influenced the fields: structures with pockets of strong *Z* and *Z*_{DR} are evident. The cloud in Figs. 4d and 4e is more stratiform, although granularities in the fields are evident. Cases presented in Fig. 4 are called cases 1 and 2 hereafter.

The reflectivity and *Z*_{DR} (Fig. 5) in slant cross sections collected at elevation angles of 0.5°, 1.5°, and 2.5° are shown for case 3. These data were gathered in the operational radar mode using volume coverage pattern (VCP) 32. For the two lowest elevations of VCP 32, the number of samples used in data processing was *M* = 64 and for an elevation of 2.5°, *M* = 11. One can see broad areas of large *Z*_{DR} in a sector from east to south of the radar. In the operational modes, *Z*_{DR} is calculated in an interval from −7.93 to 7.93 dB with an accuracy of 0.063 dB. Values of *Z*_{DR} larger than 7.93 dB are clipped to this maximum value to fit the 1-byte format of data storing. Inspections of the *Z*_{DR} fields (Fig. 5) at an elevation of 0.5° show that 4% of data have *Z*_{DR} = 7.9 dB, which equals or exceeds the maximum value. For an elevation of 1.5°, 14% of data are clipped at *Z*_{DR} = 7.9 dB.

#### 1) Case 1: 6 January 2007

The radar data were collected with a slow antenna motion in elevation with *M* = 128. Oversampling in elevation makes the equivalent number of samples *M* = 512 (Melnikov et al. 2011). The data have been processed from an area located beyond the range of 48 km and heights higher than 2.5 km, which have *Z*_{DR} mostly larger than 4 dB, although *Z*_{DR} of lower values also are present. A distribution of measured *Z*_{DR} is presented in Fig. 6a. The median *Z*_{DR} in the area is 5.1 dB. Since the vast majority of the data have *Z*_{DR} > 4 dB, we conclude that platelike particles contribute the most to return radar signals. Figure 6b depicts median *ρ*_{hv} corresponding to given *Z*_{DR}. The median *ρ*_{hv} is about 0.95. These median *Z*_{DR} and *ρ*_{hv} were used in the previous section in the description of the retrieval algorithm.

The cloud area under consideration is not uniform; it exhibits variations in Z_{DR} and *ρ*_{hv}. These variations will be replicated in variations of retrieved parameters. To demonstrate variability in retrieved *b*/*a* and *σ _{θ}*, data from three smaller parts of the large area have been processed. In Fig. 7a, retrieved

*b*/

*a*and

*σ*are shown for three smaller areas located at distances between 51.5 and 54.5 km and two height intervals between 2.8 and 3.4 km (dots) and from 3.8 to 4.7 km (asterisks). The third area is between ranges 53.5 and 56 km and between heights of 3.8 and 4.7 km (circles) at the cloud fringe. It is seen that the retrieved

_{θ}*b*/

*a*and

*σ*in these areas are quite different with median values of 0.35, 15°; 0.25, 10°; and 0.17, 9°. Particles at the cloud fringe are more oblate than the ones that are deeper in the cloud.

_{θ}The 1-*σ* uncertainties of the retrievals are shown in Fig. 7b. Median Δ(*b*/*a*) and Δ*σ _{θ}* for the three areas are 9% and 11%, 11% and 10%, and 18% and 12%, respectively. Larger uncertainties for the third area are due to smaller signal-to-noise ratios in the cloud fringes.

#### 2) Case 2: 4 December 2008

The nonprecipitating clouds produced a weak echo with the maximum reflectivity of 5 dB*Z*. Signals with SNR larger than 1 dB in the vertical channel have been selected for the retrieval procedure. The echo exhibits high *Z*_{DR} with noticeable granularity. The distribution of *Z*_{DR} and the corresponding median *ρ*_{hv} are depicted in Figs. 6c and 6d. The retrieved *b*/*a* and *σ _{θ}* and corresponding statistical uncertainties are shown in Figs. 7c and 7d. The median retrieved

*b*/

*a*and

*σ*are 0.17 and 7°, respectively. Some data with large uncertainties are marked with the triangles and circles in Fig. 7d and corresponding

_{θ}*b*/

*a*and

*σ*are marked with the same symbols in Fig. 7c. The large uncertainties in Δ(

_{θ}*b*/

*a*), marked with circles, are due to low

*ρ*

_{hv}(<0.885). For such

*ρ*

_{hv}, the retrieved

*b*/

*a*is very small (0.05). For such axis ratios and small

*ρ*

_{hv}(<0.89), the

*ρ*

_{hv}curves lie close to each other (Fig. 2b) so the retrieval uncertainties are large.

The large uncertainties marked with triangles (Fig. 7d) correspond to very small *σ _{θ}* (about 2°; Fig. 7c) that have

*ρ*

_{hv}larger than 0.997. For such large

*ρ*

_{hv}, usual measurement uncertainties of 0.01 produce large uncertainties in

*σ*. In both cases marked with triangles and circles, the large retrieval uncertainties cannot be called outliers, and they are within the expected limits for given measured

_{θ}*Z*

_{DR},

*ρ*

_{hv}, and SNR.

The vast majority of retrieved *b*/*a* lays in an interval from 0.1 to 0.25, which corresponds to the interval *a*/*b* (i.e., length/width) from 10 to 4, which can be considered a large scatter. But a closer look at Figs. 4d and 4e reveals granular structures in reflectivity and *Z*_{DR} fields. The analysis shows that the scatter in retrieved *a*/*b* corresponds to granularities in *Z*_{DR} and *ρ*_{hv} fields so it most likely replicates natural variability of the axis ratios in the cloud.

In 3% of the measurements, *ρ*_{hv} was less than 0.84. In these cases, the retrieval uncertainty in *σ _{θ}* is very large as is seen from Fig. 2b. In such cases (3%) the retrievals have not been performed.

#### 3) Case 3: 1 February 2011

The data were collected with the KOUN radar running operational VCP 31. Distributions of measured *Z*_{DR} and *ρ*_{hv} are shown in Figs. 6e and 6f for ranges beyond 55 km from the radar and elevation of 1.5°. Despite the spike in the *Z*_{DR} distribution at 7.9 dB (Fig. 6e), which corresponds to clipped values, it is seen that the median Z_{DR} is close to 5 dB. The results of the retrievals for the two smaller areas located between azimuths 75° and 80° and distances between 60 and 65 km (the first area) and between 75 and 80 km (the second area) are shown in Fig. 7e. Height intervals for these areas are from 1.57 to 1.70 km and from 1.96 to 2.09 km, respectively. The retrieved values of *b*/*a* and *σ _{θ}* for the first area (Fig. 7e) are presented with the dots, and the results for the second area are presented with the asterisks. Each symbol corresponds to a single radar resolution volume. The median retrieved

*b*/

*a*and

*σ*for the areas are 0.21, 6° and 0.09, 6°, respectively. The 1-

_{θ}*σ*uncertainties for these data (Fig. 7f) are presented with the dots and asterisks correspondingly. These uncertainties are noticeably larger than the ones for cases 1 and 2 above, but they have been obtained with the operational data collection that used 64 radar pulses in the estimations of

*Z*

_{DR}and

*ρ*

_{hv}.

Three cases with large retrieval uncertainties are marked with circles and a triangle in Fig. 7f. Corresponding data in Fig. 7e are marked with the same symbols. Both circles in Fig. 7e correspond to the same retrieved *b*/*a* and *σ _{θ}* so there is only one circle in Fig. 7e. For these data, retrieved

*b*/

*a*= 0.015 (i.e., it is extremely small). In such cases,

*b*/

*a*is retrieved with large uncertainties and cannot be considered reliable. The data marked with the triangle have been obtained from measured

*ρ*

_{hv}of 0.998 with Δ

*ρ*

_{hv}= 0.005. Despite the small Δ

_{hv}, deviations in

*ρ*

_{hv}are large because its value is very close to 1. Such

*ρ*

_{hv}create large uncertainties in the retrieval procedure. The frequency of this for analyzed data is about 0.5% and can be tolerated.

The median statistical uncertainties in retrieved *b*/*a* and *σ _{θ}* are about 35% and 20%, correspondingly. To reduce the scatter in retrieved

*b*/

*a*and

*σ*in the radar operational modes, spatial averaging can be employed. The WSR-88D radar collects data with a radial resolution of 250 m, which beyond 18 km is smaller than the spatial resolution in the orthogonal direction (specified by the beamwidth). So if a coarser range resolution is acceptable, data from consecutive range gates can be averaged. For instance, for a range resolution of 1 km, data from four range gates can be averaged that will reduce the uncertainties by a factor of 2.

_{θ}Some overall conclusions can be drawn from the three analyzed cases. The retrieval statistical uncertainties are smaller for slower antenna motions because of the larger number of samples and smaller measurement uncertainties in *Z*_{DR} and *ρ*_{hv}. Scatter in the retrieved parameters are most likely due to two issues. 1) Radar fields of *Z*_{DR} and *ρ*_{hv} exhibit spatial nonuniformities that are transformed into the scatter of retrieved *b*/*a* and *σ _{θ}*. Thus, the retrieval scatter reflects natural variability in the values. 2) The obtained Δ(

*b*/

*a*) and Δ

*σ*are stated in the 1-

_{θ}*σ*uncertainties that encompass about 70% of scatter. The remaining 30% belong to larger uncertainties. The statistical uncertainties are large for particles exhibiting weak flutter with

*σ*≤ 2° and also for extremely thin particles with

_{θ}*b*/

*a*< 0.05. To reduce statistical uncertainties in operational data collection, a spatial averaging can be employed. If a 1-km resolution is acceptable, data from four consecutive radar range gates can be averaged.

The retrieved axis ratios span a wide interval from 0.05 to 0.4 whereas retrieved *σ _{θ}* lies in an interval from 3° to 15°, which means that the particles experience weak to moderate flutter. There is noticeable correlation between the oblateness and intensity of fluttering that can be deduced from the left panels in Fig. 7: particles with larger

*a*/

*b*(lower

*b*/

*a*) exhibit lower

*σ*. The same deduction can be drawn from Fig. 6, which summarizes data from the whole areas containing large

_{θ}*Z*

_{DR}: particles with larger

*Z*

_{DR}(larger

*a*/

*b*) exhibit larger

*ρ*

_{hv}, which corresponds to weaker flutter.

The fluttering of cloud particles depends not only on their parameters (the sizes, shapes, and densities) but also on the intensity of small-scale turbulence (e.g., Klett 1995). Photogrammetric measurements of Kajikawa (1976) indicated canting angles of 10°–25°. Zikmunda and Vali (1972) found that 40% and 90% of rimed columnar crystals fell with less than 5° and 15° canting, correspondingly, but the canting was as large as 75° for a few crystals. Matrosov et al. (2005) found the intensity of flutter from 3° to 15° for three analyzed cases. The latter numbers correlate well with those obtained in this section.

### b. Variations in ice density and mixtures of particles of different habits

In the previous subsection, the axis ratios and flutter angles were retrieved for solid ice plates. The morphology of cloud particles exhibits tremendous variability. Platelike cloud particles can be in a form of simple plates, with sectorlike branches, and can contain the extensions in a form of needles or smaller plates (e.g., Fig. 2-38 in Pruppacher and Klett 1997). The bulk density of such particles is calculated using the sizes of particles close to their maximum dimensions, which results in densities less than the density of solid ice. To obtain the sensitivity of the retrieval algorithm to ice density, we have generated dependences of *Z*_{DR} and *ρ*_{hv} (Fig. 2) for different bulk densities. The Maxwell Garnett procedure with ice as matrix and air as inclusions (e.g., Bringi and Chandrasekar 2001, section 1.6) has been utilized. Then we have applied the retrieval procedure assuming the density of solid ice. Results are in Fig. 8a for the mean *Z*_{DR} = 5 dB and *ρ*_{hv} = 0.95 obtained for cases 1–3 above. The uncertainties in measured *Z*_{DR} and *ρ*_{hv} create areas of possible *b*/*a* and *σ _{θ}*, which are shaded in Fig. 8. For solid ice particles, the retrieved

*b*/

*a*and

*σ*occupy the right shaded area in Fig. 8a. The central and left shaded patches correspond to the bulk density of 0.7 and 0.5 g cm

_{θ}^{−3}. It is seen that the uncertainty in the bulk density leads to smaller retrieved

*b*/

*a*, but does not affect the retrieved

*σ*. Thus, by applying the retrieval procedure using the density of solid ice, we obtain correct

_{θ}*σ*and the maximal possible

_{θ}*b*/

*a*. For the results represented in Fig. 8a, we conclude that the mean

*σ*= 8.9° and

_{θ}*b*/

*a*≤ 0.21.

In areas with *Z*_{DR} > 4 dB, platelike particles contribute the most to returned radar signals. But the presence of columnlike crystals cannot be excluded. It is apparent that a mixture of plates and columns has lower *Z*_{DR} and different *ρ*_{hv} compared to plates alone. The uncertainties of *b*/*a* and *σ _{θ}* retrievals are shown in Fig. 8b for measured

*Z*

_{DR}= 5 dB and

*ρ*

_{hv}= 0.95 and for different signal power contributions from columnar crystals. The shaded patches are from the statistical uncertainties in the retrieved

*b*/

*a*and

*σ*. The upper right patch in the figure corresponds to no columns with the mean values of

_{θ}*b*/

*a*= 0.21 and

*σ*= 8.9°. The presence of columns shifts the patches to the lower left corner of the domain: 30% of the signal power from columns produce

_{θ}*b*/

*a*= 0.17 and

*σ*= 7°. The latter values have been obtained by applying the retrieval procedure assuming no columns. The percentage of the signal power from columns that produces

_{θ}*b*/

*a*= 0.11 and

*σ*= 4° is 60%. We see that the uncertainty in the contribution from columns affect both

_{θ}*b*/

*a*and

*σ*. But it can be concluded that the retrievals allow obtaining the maximal

_{θ}*b*/

*a*and

*σ*. In this example it is deduced that the particles have mean

_{θ}*b*/

*a*≤ 0.21 and

*σ*≤ 8.9°.

_{θ}How can the uncertainties in the bulk ice density and mixtures of different crystal habits be reduced? Obviously, information from other sources can be used. Radar observations also have the means of reducing the uncertainties. Observations of a given cloud volume at different elevation angles are one of these. Such observations can be conducted with a dense radar network like that operated by the Atmospheric Radiation Measurement Program at the Southern Great Plains site. The site has one C-band and three X-band STAR polarimetric radars. Cloud ice particles are small enough to be considered Rayleigh scatterers at X band. Attenuation in clouds as in cases 1–3 above can be neglected. By rewriting (3) and (7) for any elevation angle, one can obtain pairs of *Z*_{DR} and *ρ*_{hv} for different known elevations, which will reduce the retrieval uncertainties. Another way of reducing the uncertainties is radar observations with a single radar in layers exhibiting spatial uniformity. The elevation dependences of *Z*_{DR} and *ρ*_{hv} could be employed in the retrievals. Case 2 above could be an example.

## 5. Conclusions

Simultaneous transmission and reception (STAR) of horizontally and vertically polarized waves is the most prevalent radar configuration nowadays. Such configuration delivers differential reflectivity *Z*_{DR}, the copolar correlation coefficient *ρ*_{hv}, and differential phase. It is shown that in cloud areas with *Z*_{DR} > 4 dB, the platelike particles contribute the most to returned radar signals and the mean axis ratios (width/length; *b*/*a*), and the standard deviation of flutter angles *σ _{θ}* can be estimated from measured

*Z*

_{DR}and

*ρ*

_{hv}.

The retrieval procedure was applied to data collected with the dual-polarization S-band KOUN WSR-88D in Norman. In three analyzed cases assuming solid ice cloud particles, the median values of *b*/*a* are from 0.15 to 0.4 and *σ _{θ}* were from 2° to 20° (Fig. 7). The median

*σ*lies in an interval from 9° to 15° (i.e., the particles experience light to moderate fluttering). The 1-

_{θ}*σ*statistical uncertainties are about 10%–30% for

*b*/

*a*and 10%–50% for

*σ*, which are acceptable errors. These uncertainties can be reduced by further spatial averaging along the radar radial or slowing the antenna rate to increase the dwell time. The former option can be utilized in operational radars, whereas the latter option can be afforded with research radars. Matrosov et al. (2005) obtained a flutter intensity of 3°–15° using depolarization radar measurements. These numbers agree well with

_{θ}*σ*obtained in section 4.

_{θ}Uncertainties in the bulk ice density do not alter retrieved *σ _{θ}* and allow the maximum possible

*b*/

*a*to be obtained. In areas with

*Z*

_{DR}> 4 dB, platelike particles return most of the radar signal. A possible presence of columnlike particles in the radar volume leads to uncertainties in retrieved

*b*/

*a*and

*σ*In such cases, the retrieval procedure allows obtaining maximal

_{θ}.*b*/

*a*and

*σ*. These parameters can be used in cloud models for the parameterization, and modeling output can also be used for reducing radar retrieval uncertainties. The retrieval uncertainties could be reduced by employing elevation dependences of

_{θ}*Z*

_{DR}and

*ρ*

_{hv}measured with dense radar networks.

## Acknowledgments

We thank our anonymous reviewers for their constructive comments that helped us to improve the manuscript. Funding for this study was provided in part by the NOAA/Office of Oceanic and Atmospheric Research under NOAA–University of Oklahoma Cooperative Agreement NA11OAR4320072, U.S. Department of Commerce, and by the National Science Foundation Grant AGS-1036237 (program officer Dr. Chungu Lu) during the course of this work.

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