1. Introduction

Optical properties of cirrus clouds are usually considered under an assumption that ice crystals constituting these clouds are randomly oriented. However the large ice crystals, especially if they are of plate-like habits, often reveal their quasi-horizontal orientation. Such orientations essentially change optical properties of cirrus clouds that should be taken into account in both light scattering and remote sensing problems. Well-known manifestations of the quasi-horizontal orientation of ice crystals in cirrus clouds are arcs and some halo phenomena, such as parhelion.

In lidar investigations, where only backward scattering direction is detected, the specular reflection by horizontally oriented plate-like ice crystals is essential that was firstly studied by Platt et al. by use of a ground-based lidar [1]. Then a scanning polarization lidar was used by Sassen and Benson [2] and Noel and Sassen [3] to investigate the quasi-horizontal orientation of ice crystals where the lidar scanned at small angles near zenith. Del Guasta et al. [4] studied the quasi-horizontal orientation of the crystals at lidar tilt 30° off zenith. As for the space-borne lidar CALIPSO, the specular reflection from horizontally oriented ice crystals and Earth's water surfaces forced to change a tilt of the lidar axis from the initial angle 0.3° to the present angle 3° off nadir [5].

Optical properties of cirrus clouds are fully determined by the (4 × 4) Mueller or scattering matrix [6]. It is important that the Mueller matrix is an additive quantity. Therefore we can calculate the Mueller matrices separately for any part of crystals and then the total Mueller matrix is found as a direct sum of the constituents.

In general, light scattering by an ice crystal particle of cirrus clouds is a difficult problem that is not solved satisfactorily yet. Though the Mueller matrices for various shapes of the crystals have been widely calculated within the geometric optics approximation (see, e.g., [7]), such an approach is not applicable to backscatter. Indeed, geometric optics approximation gives a singularity at the backward direction because of the corner-reflection effect [8]. In reality, diffraction transforms this singularity into a narrow backscattering peak depending strongly on the incident wavelength and particle orientations. Therefore only the physical optics approximation [9,10] could be effective to calculate the backscatter. We note that a few papers dealing with the problem of backscattering by horizontally oriented ice crystals [11,12] were based on some crude approximations instead of the consistent physical optics.

The purpose of this paper is to calculate the backscattering Mueller matrix for the quasi-horizontally oriented ice plates by means of the physical optics approximation [9]. Though various shapes of ice plates occur in cirrus clouds, we restrict ourselves by the hexagonal ice plate. This shape is one of the most common shapes used in numerical calculations. The numerical data are obtained for lidar tilts 0.3° and 3° off nadir corresponding to CALIPSO lidar.

2. The backscattering Mueller matrix for tilted and scanning lidars

Geometry of a space-borne tilted lidar like CALIPSO [5] is depicted in Fig. 1 . The same geometry is valid for the ground-based scanning lidars, too [1–4], replacing nadir by zenith. In Fig. 1, an arbitrary oriented hexagonal ice plate illuminated in the direction i ($\left|i\right|=1$) scatters the incident light over various scattering directions. Let us define a coordinate system where the scattered electromagnetic wave should be determined. For this purpose, the crystal particle is surrounded by the scattering direction sphere n ($\left|n\right|=1$) where the unit vector n is also described in the spherical coordinates by the zenith θ and azimuth φ angles. The pole of the spherical coordinate system is chosen in the vertical direction z ($\left|z\right|=1$) so the zenith angle $\theta =\arccos (z\cdot n)$ is counted from the pole. Every scattering direction n on the sphere is accompanied by two transverse base vectors $e_{\theta }$ ($\left|e_{\theta }\right|=1$) and $e_{\varphi }$($\left|e_{\varphi }\right|=1$). Here the vector n is directed out of the sphere, the vector $e_{\theta }$ is assumed to be directed out of the pole, and then the vector $e_{\varphi }$ is found from the equation $n=e_{\theta }\times e_{\varphi }$. In particular, in Fig. 1, these vectors are shown for the backscattering direction $n=b=-i$.

 

Fig. 1 Geometry of a tilted lidar.

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The transverse electric field $E(n,i)$ of the scattered electromagnetic wave in the wave zone is represented as the decomposition relatively to the base transverse vectors

Orientation of a hexagonal plate is determined by three values corresponding to the Euler angles Θ, Φ, and Ψ. Here Θ and Φ are the zenith and azimuth angles, respectively, for the normal N to the upper hexagonal facet in the same spherical coordinate system chosen above. The angle Ψ corresponds to rotation of the plate around the normal N. We assume that orientations of a plate are uniformly distributed over the angles Φ and Ψ in the interval of [0, 2π] and only the distribution $p(\Theta )$ over the plate tilts $\Theta =\mathrm{arc}\cos (z\cdot N)$ with the normalization of ${\displaystyle {\int }_0^{\pi }p(\Theta )\sin \Theta d\Theta }=1$ is not trivial. After averaging over orientations, such a plate becomes an axis-symmetric object relatively to the zenith direction z.

The backscattering (4 × 4) Mueller matrix for a fixed crystal orientation is the function of these three variables $M(\Theta ,\Phi ,\Psi )$. An averaging over all orientations excludes these variables, i.e. $M={(1/2\pi )}^2{\displaystyle {\int }_0^{2\pi }d\Phi {\displaystyle {\int }_0^{\pi }p(\Theta )\sin \Theta d\Theta {\displaystyle {\int }_0^{2\pi }M(\Theta ,\Phi ,\Psi )d\Psi }}}$. We shall use also the partial averaging where some variables are kept, in particular, $M(\Theta )={(1/2\pi )}^2{\displaystyle {\int }_0^{2\pi }d\Phi {\displaystyle {\int }_0^{2\pi }M(\Theta ,\Phi ,\Psi )d\Psi }}$. Then the matrices are called the totally and partially averaged matrices, respectively.

In the coordinate system determined above and for the axially-symmetric distribution of particle orientations the totally averaged backscattering Mueller matrix has the following simplest view

Equation (2) is proved as follows. For an arbitrary fixed orientation of the crystal plate, all 16 elements of the backscattering matrix can be different from zero. After averaging over crystal orientations, eight elements of Eq. (2) become zero because of the symmetry of the averaged scattering object relatively to the reference plane [6]. In addition, the reciprocity principle provides three equations: $d_{21}=d_{12}$, $d_{43}=-d_{34}$ [6] and $d_{44}\equiv 1+M_{44}/M_{11}=d_{22}+d_{33}$ [13] that are substituted in Eq. (2). Thus, in general, the backscattering Mueller matrix has five independent elements including the element $M_{11}$.

In the particular case of vertical incidence of light where $i=-z$, the additional rotation symmetry relative to the incident direction reduces the matrix of Eq. (2) to the diagonal matrix

It is worthwhile to note that Eq. (3) coincides with the well-known equation for the backscattering Mueller matrix of randomly oriented particles [6,13]. In Eq. (3), the quantity d has the physical meaning, for example, of the unpolarized part of the backscattered radiation at sounding by linearly polarized light [14].

The backscattering Mueller matrix determines all parameters of scattered light at arbitrary polarization of incident light. For example, the CALIPSO lidar emits linearly polarized light. Here a parameter of interest is the depolarization ratio $\delta =I_{\perp }/I_{\left|\right|}$ where $I_{\left|\right|}$ and $I_{\perp }$ are the signals received for the parallel and perpendicular components, respectively, relatively to the emitted light. Assume that the electric field $E_0$ of the emitted light is turned to the basic vector $e_{\theta }$ on the angle χ, i.e. $\chi =\mathrm{arc}\cos (E_0\cdot e_{\theta })$. Then the depolarization ratio is determined by three elements of the matrix and by the angle χ as follows

In the case of the diagonal matrix of Eq. (3) this value is reduced to the simple equation $\delta =d/(2-d)$ [13].

3. Microphysical model for hexagonal ice plates

A hexagonal ice plate is determined by two sizes: the circumscribed diameter D of the hexagon face and the height h. In cirrus clouds, these variables are dependent; they are connected by the empirical equation [15]

In this paper, the microphysical model means to determine a probability density function $p(\Theta ,D)$ over the plate tilts Θ and diameters D with the normalization ${\displaystyle {\int }_0^{\infty }dD{\displaystyle {\int }_0^{\pi }p(\Theta ,D)\sin \Theta d\Theta }}=1$. It is common to assume that the plates obey the gamma-distribution over the diameters [15,16] and the gaussian distribution over the tilts [17,18]. For practical calculations, we need to choose the minimum and maximum diameters and the maximum tilt. According to Eq. (5), the aspect ratio of the plates $A=h/D$ increases for small particles reaching, for example, $A\approx 0.57$ at $D=10\mu \text{m}$. The smaller particles have a uni-axial appearance and their preferential orientation in clouds becomes doubtful. In addition, our numerical code of physical optics [9] is reliable under the condition $D>>\lambda$ where λ is the incident wavelength. Therefore, we choose the minimum diameter as $D_{\min }=10\mu \text{m}$. The maximum diameter is taken as $D_{\max }=1000\mu \text{m}$ since appearance of larger plates is a rather rare event. We assume that the quantities Θ and D are statistically independent. Also, the maximum plate tilt is chosen as 20°. Thus, the model used in this paper is described by the function

This model has two parameters ${\Theta }_s$ and $D_m$ where ${\Theta }_s$ is the standard deviation of the tilts and the modal diameter $D_m$ indicates a position of the maximum in the gamma-distribution.

In nature, the typical tilts ${\Theta }_s$ depend on particle diameters D as it was discussed, for example, in [18]. Though this fact was taken into account in the previous papers [19,20] concerning light backscattering by ice crystals, in this paper we don’t include such dependence in Eq. (6) for simplicity.

4. The specular and corner-reflection components

The specific feature of the problem of light scattering by quasi-horizontally oriented plates is that the scattered field consists of two components differing qualitatively from each other as shown in Fig. 2 . The first is the specular component associated with light reflection by a plane-parallel plate truncated by the hexagonal facets. The second is the corner-reflection term since the angle of 90° between the hexagonal and rectangular facets transforms the plate into a 2D corner reflector at a specific value of the rotation angle Ψ [8,9]. It is obvious that a contribution from the specular term to the backscatter is maximal when the plate is oriented normally to the incident direction. Then this component decreases with the incident angle $\eta =\arccos (i\cdot N)$ according to the diffraction pattern on the hexagonal facet. The corner-reflection term, on the contrary, is equal to zero at the normal incidence and then it increases with the incident angle η because of the increasing projection of the plate height. Below we illustrate these regularities with numerical calculations.

 

Fig. 2 The specular and corner-reflection components in the backscatter.

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In our computer code [9], the scattered light is represented in the near zone of a crystal at a fixed orientation as a lot of plane-parallel beams leaving the crystal in various directions $n_j$ with different shapes, transverse areas and polarization properties Then, in the wave zone of the crystal, every beam is transformed by means of the Fraunhofer diffraction into a spot on the scattering direction sphere n. Since the transversal shape of any beam is a polygon, the Fraunhofer diffraction spot for the polygon is found analytically. This spot is concentrated around the scattering direction $n_j$ with the angular radius $\Delta {\rho }_j\approx \lambda /l_j$ where $l_j$ is a characteristic transverse size of the beam in the near zone of the crystal. Our computer code calculates the scattered electric field and, more generally, the Mueller matrix of every beam in any point on the scattering direction sphere. In this paper, we calculate the Mueller matrix taken only in the backward scattering direction $n=b=-i$.

For a comparison of contributions to the backscatter from the specular and corner reflection terms at different crystal tilts, consider the case where the incident direction is vertical $i=-z$ and the particle normal N is tilted at the angle Θ within the reference plane. Here the base vectors $e_{\theta }$ and $e_{\phi }$on the pole of the coordinate system can be predetermined by their extension from the zeroth meridian. This specific backscattering Mueller matrix is denoted by the separate symbol $Q(\Theta ,\Psi )$ since this matrix is important for further discussion.

The backscatter is formed by a summation of the diffraction patterns from a lot of beams scattered in various directions. Here, in general, all 16 elements of the matrix $Q(\Theta ,\Psi )$ at a fixed crystal orientation are different from zero. Then our computer code averages the matrix $Q(\Theta ,\Psi )$ over the particle rotation angle Ψ. As a result, we obtain the partially averaged matrix $Q(\Theta )$ that is of view of Eq. (2), i.e. it has five independent elements.

While the matrix $Q(\Theta )$ is obtained numerically, the partially averaged backscattering Mueller matrix $M_0(\Theta )$ for this incident direction $i=-z$ is obtained without calculations. Indeed, the matrix $M_0(\Theta )$ is the average over the azimuth particle angle Φ according to the equation

As a result, the Mueller matrix $M_0(\Theta )$ is reduced to the diagonal view of Eq. (3) where only two elements are independent. These elements are expressed through the elements of the previous matrix $Q(\Theta )$ as $M_{0°11}(\Theta )=Q_{11}(\Theta )$ and $d(\Theta )=[d_{22}(\Theta )+d_{33}(\Theta )]/2$. The first element of the both matrices has the physical meaning of the backscattering cross section $\sigma (\Theta )=M_{0°11}(\Theta )=Q_{11}(\Theta )$.

Figure 3 shows the backscattering cross section calculated separately for the specular and corner-reflection components. We remind that any averaging of the scattered radiation over crystal orientations is equivalent to some integration of the Fraunhofer fringes passing over an observation point. As a result, we see that the specular component rapidly decreases within the interval $0\le \Theta \le \lambda /D$. Then this decrease is accompanied by oscillations with the same scale $\Delta \Theta \approx \lambda /D$. It means that the diffraction fringes from the hexagon sides prove to be not completely smoothed after averaging over the rotation angle Ψ. The diffraction fringes of the corner-reflection component, on the contrary, are completely smoothed by the plate rotation as seen in Fig. 3. Thus, contributions to the backscatter from the specular and corner-reflection components depending on the plate tilt Θ are different both quantitatively and qualitatively.

 

Fig. 3 The backscattering cross section for the specular (light solid curve) and corner-reflection (dashed curve) components. The black curve is their sum.

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Magnitudes of the matrix elements are easy found analytically at $\Theta =0$, i.e. for the case of the horizontal particle orientation. Here we determine the backscattering cross section without an artificial additional factor of 4π that is often used in literature. As a result, we use the following expression

Strictly speaking, there is an exact expression for reflection of light by a plane-parallel plate that is derived from the Maxwell equations [21]. In this expression, interference among the reflected beams of arbitrary reflection orders is taken into account. In this paper, the interference phenomena can be ignored. Indeed, in the smallest plate of our model with the height $h=10\mu \text{m}$ the difference of the optical paths between the beams reflected from the upper and lower facets is equal to $2nh\approx 26\mu \text{m}$, i.e. it equals 26 wavelengths at $\lambda \approx 1\mu \text{m}$. A change of the height on, say, 10% will result in oscillations of the interference term with alternating signs. Therefore, in nature, the interference phenomena are smoothed because of the condition $h/\lambda >>1$ and the interference can be neglected. In our previous paper [9], we represented any Mueller matrix as a sum of the diffraction and interference terms. By use of this terminology, in this paper we reasonably omit the interference term and calculate the diffraction addendum by direct summation of the backscattering Mueller matrices obtained separately for all beams leaving a crystal particle.

5. Data bank of the backscattering Mueller matrices for tilted lidars

In cirrus clouds, the quasi-horizontally oriented hexagonal ice plates are distributed over their diameters D and tilts $\Theta$. If one knows such a distribution described by the probability density $p(\Theta ,D)$ then the totally averaged backscattering Mueller matrix for one particle of this statistical ensemble is found by the integral

Usually the function $p(\Theta ,D)$ is not known definitely. Therefore, in this paper, we prefer to calculate the partially averaged Mueller matrix $M(\Theta ,D)$ and presented it as a data bank. These data allows a user to calculate the integral (10) for any functions $p(\Theta ,D)$ he needs.

If a lidar is tilted at the angle $t=\arccos (i\cdot z)$, the matrix $M(\Theta ,D)$ becomes dependent on the angle t that will be denoted by the lower index $M_t(\Theta ,D)$. This matrix has also the view of Eq. (2), i.e. it has five independent elements. To find this matrix, we don't need to calculate again contributions of the specular and corner reflection components to the backscatter because this procedure was already realized calculating the previous matrix $Q(\Theta ,D)$. The desired matrix $M_t(\Theta ,D)$ is found by numerical integration of the previous matrix $Q(\Theta ,D)$ according to the following equation generalizing Eq. (7)

The data calculated are presented in our data bank open on the site ftp://ftp.iao.ru/pub/GWDT/Physical_optics/Backscattering/. It contains the partially averaged matrices $M_t(\Theta ,D)$ obtained for four parameters corresponding to the CALIPSO lidar. These parameters are: two wavelengths $\lambda =0.532\mu \text{m}$ and $\lambda =1.064\mu \text{m}$; and two lidar tilts 0.3° (initial) and 3° (present). Also the initial matrix $Q(\Theta ,D)$ for these two wavelengths is presented in the intervals $0°\le \Theta \le 20°$ and $10\mu \text{m}\le D\le 1000\mu \text{m}$.

As an example of these data, the matrix $M_{3°}(\Theta ,D)$at $\lambda =0.532\mu \text{m}$is shown in Fig. 4 . Consider Fig. 4(a) presenting the backscattering cross section ${\sigma }_{3°}(\Theta ,D)=M_{3°11}(\Theta ,D)$ for a plate with the diameter D and the fixed tilt angle Θ where the plate uniformly rotates over the angles Φ and Ψ. A main feature of this figure is a sharp ridge observed at $\Theta =3°$ that has been created by the specular component. Indeed, if a plate is tilted at the angle $\Theta =t$, the case of the normal incidence of light occurs at $\Phi =0$. Here the specular component makes the maximal contribution to the backscatter and the corner-reflection component disappears. If $\Theta \ne t$, such a contribution rapidly decreases in the interval $\Delta \Theta \approx \lambda /D$ as it is shown in Fig. 3. As a result, a width of the ridge in Fig. 4(a) decreases as $\lambda /D$ for large diameters. Note that this decrease of the specular component is accompanied by oscillations on the contrary to the corner-reflection component. Consequently, the visible oscillations near the ridge in Fig. 4(a) indicate the domain of predominance of the specular component.

 

Fig. 4 The backscattering Mueller matrix partially averaged over crystal orientations $M_t(\Theta ,D)$ as a function of crystal tilts Θ and diameters D at $\lambda =0.532\mu \text{m}$and $t=3°.$

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The similar ridge and a valley at $\Theta =3°$ appear also for the polarization elements $d_{12}$ and $d_{34}$, respectively, while the elements $d_{22}$ and $d_{33}$ prove to be monotonous functions. The magnitude of the elements $d_{22}$ and $d_{33}$ occur to be scarcely different from each other, therefore only one of them is presented in Fig. 4 for brevity. It is important that all polarization elements $d_{nm}$ prove to be small for the chosen tilt angles $t=0.3°$ and $t=3°$. Therefore they are not important for applications and we don't discuss them in details.

To illustrate an application of the matrix $M_t(\Theta ,D)$ to practical measurements, we have also calculated the totally averaged backscattering matrix of Eq. (10) where the averaging is made over both the tilts Θ and diameters D. The integral (10) is numerically calculated within the simple microphysical model determined by Eq. (6). These data are included in our data bank. As an example, Fig. 5(a) presents the averaging of Fig. 4(a) that gives the backscattering cross section as a function of the effective parameters ${\Theta }_s$ and $D_m$ of the statistical ensembles of the plates. It is obvious that the integration of the function of Fig. 4(a) results in a smooth function relative to the variable $D_m$ but it creates a jump relative to the variable ${\Theta }_s$ localized near the argument ${\Theta }_s\approx 3°$.

 

Fig. 5 The totally averaged backscattering cross section (a) and the depolarization ratio (b) versus the parameters ${\Theta }_s$ and $D_m$ of the optical model of Eq. (6) for $\lambda =0.532\mu \text{m}$ and $t=3°.$

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As a result, we can formulate the general qualitative conclusion followed from the figures like Figs. 4(a) and 5(a). The backscatter is small if the typical tilt of the plates ${\Theta }_s$ in the statistical ensemble is small in comparison with the lidar tilt t, i.e. ${\Theta }_s<t$. The backscatter increases by a sharp jump at ${\Theta }_s\approx t$ where the specular component becomes predominant. Then, at ${\Theta }_s>t$, both the specular and corner-reflection components contribute to the backscatter resulting in a smooth decrease. It is worthwhile to note that this conclusion is not true for the lidars tilted at, say, 30° [4] where ${\Theta }_s<<t$.

In interpretation of CALIPSO signals, the depolarization ratio δ is an important parameter [23–25]. We have calculated this quantity by total averaging of the other matrix elements shown in Fig. 4. Then the results were substituted in Eq. (4) at $\chi =0$, for simplicity. The depolarization ratio as a function of the parameters ${\Theta }_s$ and $D_m$is shown in Fig. 5(b). Strictly speaking, the depolarization ratio is a non additive value and the results represented are valid for the ensembles consisting of only the quasi-horizontally oriented plates. For such ensembles, as follows from Fig. 5(b), the depolarization ratio proves to be small; it is less than 1%.

The next parameter commonly used in interpretation of the CALIPSO signals is the color ratio determined by the equation $c={\sigma }_{\lambda =1.064\mu \text{m}}/{\sigma }_{\lambda =0.532\mu \text{m}}$ [19,20]. We note that the color ratio is also a non additive quantity, and its value calculated is valid for the specific ensembles consisting of only the quasi-horizontally oriented plates. It is obvious that at fixed magnitudes of the ice plate diameter D and tilt Θ, the diffraction patterns of both the specular and corner-reflection components are quickly oscillating functions with different scales for the wavelengths $\lambda =1.064\mu \text{m}$ and $\lambda =0.532\mu \text{m}$. Consequently, the color ratio is a quickly oscillation function where its magnitude can be essentially smaller and larger than c = 1. However, any averaging over either the plate diameters or tilts smoothes these oscillations.

As an example, Fig. 6 shows the color ratio at $t=3°$ where the diameters D are fixed but the cross sections are averaged over tilts Θ for the wavelengths $\lambda =1.064\mu \text{m}$ and$\lambda =0.532\mu \text{m}$. Here three domains are distinguished. First, in the strip ${\Theta }_s\approx t$ we get $c\approx 1$. This fact has a simple physical explanation. Indeed, in the strip ${\Theta }_s\approx t$ the specular component is predominant. Here the variable tilts of a particle lead factually to an integration of the diffraction pattern contributing to the backscatter. As known [21], the integral of any diffraction pattern does not depend of the wavelength that immediately leads to the result $c=1$. In the second domain ${\Theta }_s>t$, the both components are essential. A contribution of the corner-reflection component to the backscatter, unlike the specular counterpart, decreases with the wavelength because only the central part of the diffraction pattern of this component occurs to be predominant. As a result, we see a slow decrease of the color ratio in the domain ${\Theta }_s>t$ provided by the increasing contribution from the corner-reflection component. As seen in Fig. 6, the oscillations of the color ratio are saved in the third domain ${\Theta }_s<t$ where the particle tilts don't smooth the diffraction patterns received by a detector.

 

Fig. 6 Color ratio for fixed plate diameters D with an averaging over particle tilts.

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The results of the further averaging of the data over the particle size distribution of Eq. (6) are presented in Fig. 7 . We see that though the rapid oscillations of the color ratio of Fig. 6 in the domain ${\Theta }_s<t$ become smoothed, some slow oscillation still remains. So the color ratios measured by CALIPSO lidar should be essentially different for the initial 0.3° and present 3° lidar tilts if the typical ice plate tilts were less than 3°. Otherwise, if color ratios occur to be equal, this fact can be an indicator that the ice plate tilts were large, i.e.${\Theta }_s>3°$.

 

Fig. 7 Color ratio at the lidar tilts of 3° (a, b) and 0.3° (c, d) as functions of the parameters ${\Theta }_s$ and $D_m$. The right column shows selective profiles.

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6. Conclusions

The backscattering Mueller matrix for quasi-horizontally oriented hexagonal ice plates has been numerically calculated in the physical optics approximation, to our best knowledge, for the first time. It is shown that the depolarization ratio for ensembles of such ice crystals occurs to be less than 1%. Therefore a contribution of the quasi-oriented ice plates to the depolarization ratio measured by CALIPSO should be close to zero. The color ratio, on the contrary, proves to be highly sensitive to both sizes and tilts of the plates if the typical ice plate tilts are less than the lidar tilt, i.e. ${\Theta }_s<t$. If the tilts are comparable ${\Theta }_s\approx t$, the color ratio becomes close to unity. Then, if ${\Theta }_s>t$, the color ratio slowly decreases with particle tilts ${\Theta }_s$ and it does not practically depend on particle sizes $D_m$. Here the information content of the color ratio is small. Thus, the color ratio proves to be sensitive to the ratio ${\Theta }_s/t$.