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SECTION I

SATELLITE-based retrievals of cloud optical depth and effective size are sensitive to a priori assumptions of particle shape [1] and orientation [2]. In most radiative transfer models, ice crystals are assumed to be randomly oriented. However, in environments with small vertical velocities, ice particles with certain shapes may become preferentially oriented due to aerodynamic effects [3]. If a substantial fraction of the ice crystals in an ice cloud is oriented, it might cause errors in cloud properties retrievals for some specific measurements unless properly accounted for. Therefore, quantifying the occurrence of horizontally oriented ice crystals in ice and mixed-phase clouds is necessary. We would particularly like to know the fraction of ice and mixed-phase clouds containing horizontally oriented ice crystals and the average percentage of horizontally oriented ice crystals among the cloud particles.

Ice crystal orientation is generally hard to detect, but satellites and ground lidars [4] can detect horizontally oriented plates (HOPs) from the specular reflection by their prism facets. Therefore, this letter focuses on the distribution of HOPs within cloud particles.

Previous studies have provided estimates of the fraction of clouds containing HOPs using different methods and data sets, but the uncertainties remain quite large. From observations made by the Polarization and Directionality of the Earth Reflectances (POLDER) satellite, Chepfer *et al.* [5] estimated that HOPs exist in more than 40% of ice clouds, and Breon and Dubrulle [6] suggested that HOPs exist in more than 50% of the ice cloud pixels. Using Cloud–Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) data, Noel and Chepfer [7] suggested that HOPs appear in approximately 6% of optically thin (i.e., $\tau < 3$) ice cloud layers. Ground-based Doppler lidar measurements indicate that the fraction of clouds showing specular reflection is 26%–40% [8].

Estimates of the percentage of HOPs among cloud particles in ice and mixed-phase clouds are also subject to large uncertainties. Breon and Dubrulle [6] suggested, based on POLDER data, the typical fraction of HOPs in ice clouds to lie between 0.1% and 1%. Noel and Chepfer [9] estimated, also from POLDER data, the percentage of HOPs to be less than 10% in most cases. From CALIPSO observations, Noel and Chepfer [7] later estimated the fraction of HOPs in optically thin cloud layers to be 1%–5% and the overall HOP percentage in clouds to lie between 0.1% and 0.5%.

As a follow-up to these previous studies, statistical methods are employed to analyze CALIPSO data in order to provide an estimate of the HOP fraction in optically thick ice and mixed-phase clouds.

SECTION II

Clouds containing HOPs feature strong backscatter and low depolarization ratios [10] when the lidar beam is perpendicular to the main facets of the ice crystals due to specular reflection. At the 532-nm wavelength, optically thick clouds $(\tau > 3)$ are opaque to the CALIPSO lidar, and the backscattering intensity is independent of optical depth. If all ice crystals are randomly oriented, no statistically significant change should occur in the joint probability density function (PDF) of the layer-integrated attenuated backscatter (defined as $\gamma^{\prime} = \int_{\rm top}^{\rm base}\beta_{\Vert}^{\prime} (r) + \beta^{\prime}_{\perp}(r)dr$, where $\beta_{\Vert}^{\prime}(r)$ and $\beta^{\prime}_{\perp}(r)$ are the parallel and perpendicular components of the attenuated backscatter, respectively) or in the layer-integrated depolarization ratio (defined as $\delta = \int_{\rm top}^{\rm base}\beta^{\prime}_{\perp}(r)dr/ \int_{\rm top}^{\rm base}\beta^{\prime}_{\Vert}(r)dr$), when the lidar off-nadir angle increases from 0.3° to 3°.

However, the existence of HOPs enhances the lidar backscatter and reduces the depolarization ratio significantly when the off-nadir angle is 0.3°, but the effect is negligible when the off-nadir angle is 3°. This occurs because specular reflection associated with HOPs gives rise to strong attenuated backscatter and small depolarization ratios for a lidar off-nadir angle close to zero [11].

There are substantial uncertainties in the scattering properties of HOPs largely due to the uncertainty in surface roughness. In this letter, we calculate the equivalent percentage of fine ice plates to match the observed specular reflection. Fine ice plates are defined as horizontally oriented, pristine planar hexagons without surface roughness and with small deviation angles. Furthermore, we assume a cloud to be a mixture of fine HOPs and randomly oriented particles. The $\delta{-}\gamma^{\prime}$ relationship for optically thick ice and mixed-phase clouds can be described as follows [12]: TeX Source $$\gamma_{{\rm offnadir} = 0.3}^{\prime} = {\delta_{{\rm offnadir} = 3}\gamma_{{\rm offnadir} = 3}^{\prime} \over 1 + \delta_{{\rm offnadir} = 3}}{1 + \delta_{{\rm offnadir} = 0.3} \over \delta_{{\rm offnadir} = 0.3}}\quad\hbox{(1)}$$ where $\gamma_{{\rm offnadir} = 0.3}^{\prime}$ and $\delta_{{\rm offnadir} = 0.3}$ are the layer-integrated attenuated backscatter and the layer-integrated depolarization ratio, respectively, at the 0.3 ° off-nadir angle, and $\gamma_{{\rm offnadir} = 3}^{\prime}$ and $\delta_{{\rm offnadir} = 3}$ are the counterparts for the 3° off-nadir angle. The percentage of fine HOPs in ice clouds specified in the form of TeX Source $$W_{h} = C\left(\gamma_{{\rm offnadir} = 0.3}^{\prime} - \gamma_{{\rm offnadir} = 3}^{\prime}\right) \times 100\%\eqno{\hbox{(2)}}$$ where $W_{h}$ is the relative area fraction of fine HOPs in percentage, and the value of $C$ is decided by the particle size distribution and derivation angles. To estimate the value of $C$, we first calculate the scattering properties of HOPs with the physical–geometric optics hybrid method [13]. Afterward, we simulate the variation of $\gamma^{\prime}$ versus $\delta$ using the Monte Carlo method [12] and calculate the $C$ value with simple linear regression. In this letter, the size distribution of fine HOPs used to match the specular reflection of HOPs is $n(D > 0.1) = N_{0}D^{1.026}\exp(-8.7D)$, where $D$ (in millimeters) is the maximum dimension of HOPs, and $N_{0}$ is a constant used to normalize the probability distribution function. Considering that only ice plates with maximum dimensions greater than 0.1 mm are likely to be quasi-horizontally oriented [14], we set $n(D < 0.1) = 0$. The tilt angles of the HOPs are assumed to follow the normal distribution with a standard deviation of 0.8°, and the characteristic tilt angle of HOPs is less than 1° in most ice clouds [6]. The simulation results indicate a $C$ value of 0.1 (sr). By varying the size distribution and tilt angle distribution, we can estimate an uncertainty interval for $C$ of 0.05–0.2 (sr). In this letter, the value of $C$ for fine HOPs is set to be 0.1 (sr) to match the specular reflection from HOPs. For this value, the backscatter of HOPs at the 0.3° off-nadir angle is approximately 370 times as strong as for randomly oriented ice crystals, very close to the value provided by Sassen and Benson [4].

The global average $\delta{-}\gamma^{\prime}$ PDF is used to evaluate the distribution of HOPs. The CALIPSO level-2 cloud-layer products (5-km resolution) used in this analysis are as follows: the layer-integrated attenuated backscatter $\gamma^{\prime}$, the layer-integrated volume depolarization ratio $\delta$, the opacity flag, the midlayer temperature, the ice/water phase, and the lidar off-nadir angle. To reduce the bias brought about by the diurnal circle [15], both daytime and nighttime observations are included.

Fig. 1(a) shows the global $\delta{-}\gamma^{\prime}$ PDF of single-layer opaque clouds observed between July 2006 and October 2007 at a lidar off-nadir angle of 0.3° based on from more than $10^{7}$ global observations. The geographic pattern of the crystal orientation does not cycle noticeably over the year, and the calculations use almost all the data available at the 0.3° off-nadir angle, similar to Noel and Chepfer [7]. Pixels in the branch with a positive slope represent water clouds, and pixels in the branch with a negative slope represent ice and mixed-phase clouds [10], [16]. Fig. 1(b)–(h) shows the $\delta{-}\gamma^{\prime}$ PDF for cloud layers at various temperatures.

Fig. 2 shows the $\delta{-} \gamma^{\prime}$ PDF of single-layer opaque clouds observed between July 2009 and October 2010 at a lidar off-nadir angle of 3°. The long tail of the ice and mixed-phase cloud branch above the water cloud branch, which represents clouds containing HOPs, does not exist in Fig. 2.

The differences between the $\delta{-}\gamma^{\prime}$ PDF at the 0.3° and 3° off-nadir angles are shown in Fig. 3. It is evident that the $\delta{-}\gamma^{\prime}$ PDF for different off-nadir angles is because of specular reflection by the HOPs. Fig. 3(b)–(h) indicates that cloud layers containing HOPs are present over a wide temperature range.

The distribution of HOPs is sensitive to the cloud-layer temperature. From a comparison between the $\delta{-}\gamma^{\prime}$ PDF in Fig. 1(b)–(g) and Fig. 2(b)–(g), HOPs apparently occur frequently in ice and mixed-phase cloud layers warmer than $-45\ ^{\circ}\hbox{C}$. However, for ice clouds below $-45\ ^{\circ} \hbox{C}$, the difference in the $\delta{-}\gamma^{\prime}$ PDF is small (the difference accounts for $\sim$15% of the observations) and verifies that HOPs appear less frequently in extremely cold ice clouds.

SECTION III

To estimate the distribution of HOPs in ice and mixed-phase clouds, water cloud signals are removed from the $\delta{-}\gamma^{\prime}$ PDF based on the CALIPSO ice/water phase discrimination product. The present approach of removing water cloud signals works quite well for both the Cloud–Aerosol Lidar with Orthogonal Polarization 0.3° and 3° off-nadir angles [17].

Using the $\delta{-}\gamma^{\prime}$ PDF for ice and mixed-phase clouds, we calculated the equivalent percentage of fine HOPs with (1) and (2). The normalized probability distribution function of fine HOPs is shown in Fig. 4(a), and the global mean equivalent percentage of fine HOPs among particles in ice and mixed-phase clouds was estimated to be approximately 0.16%. If the uncertainty of $C$ in (2) is considered, the uncertainty interval for the equivalent percentage of fine HOPs is 0.08%–0.3%. The result is consistent with some previous studies [6], [7].

Fig. 4(b) shows the zonal mean equivalent percentage of fine HOPs as a function of latitude. HOP fraction is a minimum in the tropics and Antarctic and reaches a maximum in the Arctic. The difference in the HOP percentage with latitude may result from differences in the thermodynamic environment of clouds.

The percentage of HOPs occurring in clouds critically depends on the temperature [see Fig. 4(c)]. As illustrated in the figure, fine HOPs exist mainly in warm clouds, particularly in the mixed-phase clouds above $-20\ ^{\circ}\hbox{C}$, and the concentration sharply decreases as the temperature decreases.

The equivalent percentage of fine HOPs within cloud layers can be expressed as the summation of exponential distributions [for example, Fig. 4(a)] and is quantitatively specified as TeX Source $$P_{\rm HOP} = A_{1}\exp(-k_{1}W_{h}) + A_{2}\exp(-k_{2}W_{h})\eqno{\hbox{(3)}}$$ where $P_{\rm HOP}$ is the probability that the equivalent percentage of HOPs within a cloud layer is $W_{h}$. Factors $A_{1}$, $A_{2}$, $k_{1}$, and $k_{2}$ are constants.

The fraction of ice clouds containing HOPs can be estimated with a statistical approach. Assume $x$ of clouds contain HOPs, and $(1 - x)$ of clouds do not contain HOPs, then TeX Source $$\eqalignno{\hbox{PDF}_{{\rm offnadir} = 3}\! = &\, x\! \bullet\! \hbox{PDF}_{{\rm HOP}, {\rm offnadir} = 3}\! +\! (1\! -\! x)\! \bullet\! \hbox{PDF}_{\rm ROIC}\cr&&\hbox{(4)}\cr \hbox{PDF}_{{\rm offnadir} = 0.3}\! = &\, x\! \bullet\! \hbox{PDF}_{{\rm HOP}, {\rm offnadir} = 0.3}\! +\! (1\! -\! x)\! \bullet\! \hbox{PDF}_{\rm ROIC}\cr&&\hbox{(5)}}$$ where $\hbox{PDF}_{{\rm offnadir} = 0.3}$ is the $\delta{-}\gamma^{\prime}$ PDF of ice and mixed-phase clouds at the 0.3° off-nadir angle, $\hbox{PDF}_{{\rm offnadir} = 3}$ is the $\delta{-}\gamma^{\prime}$ PDF of ice and mixed-phase clouds at the 3° off-nadir angle, $\hbox{PDF}_{{\rm HOP}, {\rm offnadir} = 0.3}$ is the $\delta{-}\gamma^{\prime}$ PDF for clouds containing HOPs at the 3° off-nadir angle, $\hbox{PDF}_{{\rm HOP}, {\rm offnadir} = 3}$ is the $\delta{-}\gamma^{\prime}$ PDF for clouds containing HOPs 3° off-nadir angle, and $\hbox{PDF}_{\rm ROIC}$ is for clouds that do not contain HOPs.

Therefore, the difference between the two $\delta{-}\gamma^{\prime}$ PDFs is TeX Source $$\eqalignno{\Delta \hbox{PDF} =&\, \hbox{PDF}_{{\rm offnadir} = 0.3} - \hbox{PDF}_{{\rm offnadir} = 3}\cr =&\, x \bullet (\hbox{PDF}_{{\rm HOP}, {\rm offnadir} = 0.3} - \hbox{PDF}_{{\rm HOP}, {\rm offnadir} = 3}).\qquad &\hbox{(6)}}$$ Assume $\hbox{PDF}_{{\rm HOP}, {\rm offnadir} = 3}$ is equal to $\hbox{PDF}_{{\rm offnadir} = 3}$, then the term $\Delta \hbox{PDF}$ can be calculated with any given $x$ by (1)–(3). Using the least squares method and for each temperature interval, parameters $x$, $A_{1}$, $A_{2}$, $k_{1}$, and $k_{2}$ are chosen to minimize the differences between the $\Delta \hbox{PDF}$ expressed by (6) and the $\Delta \hbox{PDF}$ observed by CALIPSO. The estimated results and uncertainty intervals for $x$ are shown in Fig. 4(d). The results suggest that HOPs exist in about 88% of optically thick ice and mixed-phase clouds warmer than $-30\ ^{\circ}\hbox{C}$ and in approximately 84% of optically thick ice and mixed-phase clouds between $-30\ ^{\circ}\hbox{C}$ and $-45\ ^{\circ}\hbox{C}$. In ice clouds colder than $-45\ ^{\circ} \hbox{C}$, approximately 29% contain HOPs.

SECTION IV

Using the difference of the $\delta{-}\gamma^{\prime}$ PDFs at 0.3° and 3° off-nadir angles, we estimate that HOPs exist in approximately 60% of optically thick ice and mixed-phase cloud layers, a value consistent with previous estimations from POLDER [5], [6]. The fraction of clouds containing HOPs is larger than found by Noel and Chepfer [7], because, in their estimation, only clouds with very strong specular reflection were considered to contain HOPs. On the other hand, the equivalent percentage of fine HOPs needed to match observed specular reflection of horizontally oriented crystals is 0.08%–0.3%, consistent with the estimation of HOPs in optically thin ice clouds using CALIPSO [7].

Although HOPs can be frequently found in cold ice clouds, the equivalent percentage of HOPs in these clouds is much smaller than in warm ice and mixed-phase clouds. In mixed-phase clouds, high supersaturation enables the ice crystals to quickly grow with relatively smooth crystal facets, which generate strong specular reflection, despite the fact that crystals may be partially covered with trans-prismatic strands [18]. However, in subsaturated ice clouds, evaporation roughens the ice crystal surfaces [8], and the crystal facets are almost entirely covered with trans-prismatic strands [18], which results in very weak specular reflection. As a result, the attenuated backscatter of HOPs in cold ice clouds is much weaker than that in warm mixed-phase clouds, which implies that the total weight of HOPs may be much greater than the equivalent percentage of fine HOPs needed to match the observed specular reflection. Therefore, although the equivalent percentage of fine HOPs needed to match the observed specular reflection of horizontally oriented crystals is small, considering HOPs in cloud property retrievals in future radiative transfer models may be important.

This work was supported by the National Aeronautics and Space Administration under Grant NNX10AM27G and Grant NNX11AK37G.

C. Zhou, P. Yang, and A. E. Dessler are with the Department of Atmospheric Sciences, Texas A&M University, College Station, TX 77843 USA (e-mail: pyang@tamu.edu).

F. Liang is with the Department of Statistics, Texas A&M University, College Station, TX 77843 USA.

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

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