Transformation of light backscattering phase matrices of crystal clouds depending on the zenith sensing angle

Yury Balin; Bruno Kaul; Grigorii Kokhanenko; David Winker

doi:10.1364/OE.21.013408

1. Introduction

High-level (cirrus) crystal clouds influence considerably the radiation balance of the atmosphere. Radiation reflection by and transmission through these clouds depends essentially on particle shapes and orientations. For example, a layer of oriented ice plates strongly reflects solar radiation, which is sometimes manifested as a well-known optical subsun phenomenon that can be observed when aircrafts fly above the clouds. To estimate the radiation reflection, it is necessary to consider the particle orientation [1]. Even a non-dense cloud of highly oriented particles contributes significantly to radiation reflection. This was repeatedly observed for laser sensing of crystal clouds [2–6]. The effect of particle orientation is important for optical sensing of the atmosphere from space. The orientation of the lidar optical axis should be chosen so that specular reflections did not interfere with estimations of geometrical and optical cloud thicknesses [7].

This is a very simple problem if all crystals are oriented horizontally. Then it is sufficient to deviate the sensing angle from the nadir direction by the value slightly exceeding λ/d, where d is the particle diameter (of the order of 50 μm). In this case, deviation angle 0.5–1° will be sufficient. However, in addition to the orientation effect of the aerodynamic momentum, particles are subject to the destructive effect of turbulence and flutter [8, 9]. When the sensing angle deviates from the nadir, most of the particles cease to give specular reflections in the direction of scattering by 180°. However, some particles that previously gave no specular reflections give them. It is clear that sufficient deviation angle in this case should be increased. In [4], the change of the depolarization of scattered radiation attendant to changes in the zenith sensing angle by seven degrees was experimentally determined. Based on the experimental results, a relationship between oriented and non-oriented particles was estimated. Del Guasta et al. [10] investigated the depolarization ratio for zenith sensing angles up to 30°. However, the experimental results refer to particular cloud fields and do not provide a clear understanding of reasons for this or that depolarization ratio at various orientations of cloud particles. Therefore, a theoretical investigation of the behavior of the BackScattering Phase Matrices (BSPM) is expedient for sensing of clouds comprising partially oriented particles.

Among publications devoted to this subject [11, 12], should be mentioned, where results of calculations of the radiation intensity and of the BSPM elements were presented for radiation reflected from a polydisperse ensemble of ice plates for sensing angle of 3° to the nadir direction. A disadvantage of these works is the assumption that the distribution function over the orientation angles is the same for particles having different sizes, which of course, is not the case.

In the present work, we use another approach to the problem. On an example of monodisperse ensembles of ice plates, we show how simply the BSPM can be calculated for arbitrary zenith sensing angle and arbitrary distribution over the orientation angles given that the BSPM value of a particular particle has been calculated. We also present a possible method of calculating the BSPM of a polydisperse ensemble at arbitrary zenith sensing angle taking into account the effect of particle sizes on the distribution function over the orientation angles.

2. Main special features of the BSPM

For the BSPM, we take a $4 \times 4$ matrix M relating the Stokes vectors of radiation scattered in the direction toward the source S with the Stokes vector S₀ of radiation incident on an ensemble of particles contained in an elementary volume ΔV:

S = \frac{1}{r^{2}} M S_{0} Δ V .

Generally, all 16 BSPM elements are nonzero. This was indicated for experimentally measured matrices [13, 14]. At the same time, in [13] it was shown that within the error limits, the experimental BSPM M_exp is reduced to a simpler form by the following transformation:

M_{\mod} = R (- φ_{0}) M_{\exp} R (- φ_{0}) = M_{11} (\begin{matrix} 1 & m_{12} & 0 & m_{14} \\ m_{21} & m_{22} & 0 & 0 \\ 0 & 0 & m_{33} & m_{34} \\ m_{41} & 0 & m_{43} & m_{44} \end{matrix}),

where

m_{i j} = M_{i j} / M_{11}

,

R (φ) = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos (2 φ) & \sin (2 φ) & 0 \\ 0 & - \sin (2 φ) & \cos (2 φ) & 0 \\ 0 & 0 & 0 & 1 \end{matrix})

. Transformation (2) represents a virtual rotation of the coordinate system

e_{x} \times e_{y} = e_{z}

affixed to the lidar in which the Stokes vector of scattered radiation is defined. The wave vector of radiation coincides with the е_z direction. In practice, the procedure of finding the angle φ₀ involves variation of the rotation angle φ to minimize the sum of the squared elements which in Eq. (2) are represented by zeros for the ideal case that would take place without experiment errors. This condition is satisfied when the angle φ₀ has been found at which the reference plane (е_xе_z) coincides with the symmetry plane of cloud particles. Or in other words, it coincides with the preferred orientation direction in the plane perpendicular to the wave vector. The possibility of reduction of the experimental BSPM to the form given by Eq. (2) is caused by the symmetry properties that are a consequence of the fact that any plane comprising the z axis can be chosen for the reference backscattering plane. For any arbitrary BSPM, conditions

m_{12} = m_{21}, m_{13} = - m_{31}, m_{14} = m_{41}, m_{23} = - m_{32}, m_{24} = m_{42}, m_{34} = - m_{43},

m_{11} - m_{22} + m_{33} - m_{44} = 0

are always satisfied. The last equality is a consequence of the reciprocity theorem [15]. A detailed analysis of the symmetry properties can be found in [16, 17]. For further presentation, we need the following consequences for a cloud of particles. According to [18, 19], if the cloud possesses rotational symmetry, that is, is transformed into itself when rotated through an arbitrary angle φ about the wave vector (the z axis), the BSPM can be written in the following form:

M = M_{11} (\begin{matrix} 1 & 0 & 0 & m_{14} \\ 0 & 1 - d & 0 & 0 \\ 0 & 0 & d - 1 & 0 \\ m_{41} & 0 & 0 & 2 d - 1 \end{matrix}),

where d is the parameter which determines the depolarized part of scattered radiation when the cloud is illuminated by linearly polarized light [20]. Such matrix type should be expected for 3D and 2D randomly orientated cloud particles. In the latter case, particles can be oriented in the horizontal plane. In this case, it is implied that the BSPM is determined for sensing in the zenith or nadir direction. Here it can be seen that the condition

m_{33} = - m_{22}

is satisfied.

If there is no rotational symmetry and there is the direction of preferred azimuth orientation, then the BSPM assumes the form of Eq. (2) given that the reference plane (e_x, e_z) coincides with this direction. In this case, unlike Eq. (5), $m_{33} \neq - m_{22}$ and, as indicated in [13], the quantity

χ = \frac{m_{22} + m_{33}}{1 + m_{44}}

characterizes the degree of orientation in the azimuth direction. If the reference plane and the preferred orientation direction do not coincide, all BSPM elements can differ from zero. The elements

m_{14} = m_{41}

are nonzero when asymmetric particles are encountered in the cloud. However, if equal numbers of such particles and their specular reflections are presented in the cloud,

m_{14} = 0

. It is pertinent to note that experiments [13] have shown that zero values of the elements

m_{14}

and

m_{34}

are most probable. And one more BSPM property important for further presentation is that its angular elements are invariant under transformation of Eq. (2). That is, elements M₁₁, M₁₄, M₄₁, and M₄₄ do not change when the angle φ is changed.

3. Modeling of the BSPM

The above-indicated properties allow us to model the BSPM for arbitrary ensembles of particles. Here we restrict our consideration to the case of hexagonal particles.

Let a single hexagonal particle of the given size be located in unit volume. The angles that specify its position in the reference frame are shown in Fig. 1. Let us rotate the particle through the angle φ so that its symmetry axis lie in the xz reference plane where φ = 0 and calculate its BSPM M(0, θ, γ). Then this matrix should be averaged with equal probabilities over every possible angle γ taking into account the sixth-order symmetry:

M (0, θ) = \frac{3}{π} \int_{0}^{π / 3} M (0, θ, γ) γ d γ .

As a result of averaging, the BSPM M(0,θ) assumes the form of Eq. (2) with zero elements

m_{14}

and

m_{41}

. To determine the BSPM of the particle for arbitrary angle φ, the rotational transformation

M (φ, θ) = R (- φ) M (0, θ) R (- φ)

should be used that has already been used in Eq. (2). The matrix of a cloud with n identical particles in unit volume and probability density of angular distribution

f (φ, θ)

can be calculated by integration over the total solid angle:

Fig. 1 Hexagonal crystal orientation angles: θ, φ are polar and azimuth angles and γ is the angle of rotation about the hexagonal axis. Radiation is propagated along the z axis.

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M (n) = n \int_{0}^{2 π} \int_{0}^{\frac{π}{2}} R (- φ) M (0, θ) R (- φ) f (φ, θ) \sin θ d θ d φ .

From Eq. (7)-(8) it follows that to calculate the BSPM of a cloud of identical arbitrary oriented particles, the matrix $M (0, θ)$ should be calculated with sufficiently small step in the angle θ and then stored in a tabular form. In this case, for each θ value, averaging should be performed as in Eq. (7). Below we take advantage of one result of such calculations. If analogous calculations are available for particles of other sizes, BSPM models can be constructed for polydisperse clouds of hexagonal particles by summation of matrices taking into account the contribution of each particle type to the total concentration. Undoubtedly, this method is vary laborious and requires the creation of a vast data bank of crystal particle matrices.

4. Orientation of particles

The main orientation process involves aerodynamic forces of free fall particles that orient their major diameters in the horizontal direction. In the given article, we neglect the less pronounced azimuth orientation of particles under the action of wind velocity pulsations or electric fields described in [21, 22]. The angular distribution of orientations is the result of competing processes of particle orientation during free fall and disorientation due to random turbulence cells. The influence of turbulence on particle disorientation can be the subject of discussion. Thus, in [23] the influence of turbulence on the falling particle orientation was denied. Then Klett [8] analyzed in detail the influence of turbulence having different energy dissipation rates on the particle distribution over the orientation angles. References to this work can be found in many articles. However, we believe that the influence of turbulence in that article was somewhat underestimated. According to [8], particles with sizes 30–40 μm are strongly oriented under conditions of weak and moderate turbulence, which causes doubt. Observations of particle orientation using television technique demonstrated that orientation of ice columns becomes noticeable starting from particle sizes 50–60 μm [24]. Probably, the fall velocity of smaller particles (u < 6 cm/s) is insufficient for the creation of noticeable orientation moments of forces. In the present article, we take advantage of the distribution over the polar angle θ suggested in [9]:

f (θ)d θ = \frac{{2e}^{k \cos (2 θ)}}{π I_{0} (k)} d θ, θ \in [0, π / 2] ​,

where θ is the deflection angle of the normal to the hexagonal plate base from the zenith direction or from the horizontal direction in the case of columns, I₀(k) is the zeroth order modified Bessel function of the first kind. The distribution parameter here is

k = Λ_{p, c} A^{2} d^{2 b} / 4 \sqrt{ν ε},

where Λ_p,c is the form factor for plates (p) or columns (c), d is the maximum particle diameter, ν is the kinematic viscosity of air, ε is the energy dissipation rate, and A_p,c and b_p,c are empirical constants for the fall velocity u specified by the formula

u = A d^{b}

. Numerical values of these parameters can be found in [25–27] for plates and columns of various shapes and sizes. Information on the influence of turbulence on the orientation of particles of various sizes can be found in Table 1 based on calculations by Eq. (9), (10). Table 2 presents the degree of particle orientation (their distribution is given by Eq. (9)) versus values of the parameter k.

Table 1. Dependence of the distribution parameter k on the maximum particle diameter d for five values of the energy dissipation rate ε. The upper values here are for plates, and the lower values are for columns.

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Table 2. Standard deviation σ of plates from horizontal position, in degrees, for the indicated values of the distribution parameter k.

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5. Model BSPM of monodisperse cloud of ice plates versus the orientation parameter k and zenith sensing angle

To construct a concrete model, we used the data presented in [28] in which results of calculations of the matrix M(0, θ) for hexagonal plates with circumscribed circle diameter of 400 μm and thickness of 30.64 μm are presented. Figure 2 illustrates some of this data. In this case, small and frequent interference peaks of the original figure have been smoothed. We believe that this is justified, because the position of these peaks depends on the change of the crystal sizes by the value of the order of the wavelength.

Fig. 2 Element М₁₁ of the BSPM for hexagonal ice plates (height L = 30.64 μm, base diameter а = 200 μm, λ= 0.55 μm, and refractive index n = 1.311; the left ordinate) and normalized elements М₁₂ and М₄₄ (the right ordinate).

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For the known matrix M(0, θ), elements of the BSPM M(n) can be calculated from Eq. (7) for different values of the parameter k in the distribution given by Eq. (8), and the matrix obtained corresponds to sensing in the zenith (or nadir) direction. In this case, there is no preferred direction for the azimuth angle φ. The distribution over the orientation angles is written in the following form:

f (φ, θ) \sin θ d θ d φ = \frac{1}{2 π B (k)} \frac{2 \exp [k \cos 2 θ]}{π I_{0} (k)} \sin θ d θ d φ,

where

B (k) = \frac{1}{2 π} \int_{0}^{2 π} \int_{0}^{\frac{π}{2}} \frac{2 \exp [k \cos (2 θ)]}{π I_{0} (k)} \sin (θ) d θ d φ

is obtained from the normalization condition. Results of calculations are shown in Fig. 3, where the increase of the reflected radiation intensity (element М₁₁) and the behavior of the diagonal BSPM elements normalized by М₁₁ with increase in the orientation parameter k are displayed. It can be seen that they are close to asymptotic values already at k = 10.

Fig. 3 Dependence of the reflected signal intensity (element М₁₁) and normalized diagonal BSPM elements with increase in k – the parameter of the distribution function over the polar orientation angles – for sensing in the zenith or nadir direction.

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We now analyze how the BSPM changes when the sensing angle deviates from the zenith direction by the angle β. In this case, the rotational symmetry is broken. The preferred plane z0β arises that comprises the zenith and sensing directions. For clarity, we now assume that all plates are oriented horizontally. Then projections of horizontally oriented plates onto the plane perpendicular to the sensing direction differ for directions lying in the z0β plane and transverse to it. For example, the projection of a round plate is an ellipse. All major axes of ellipses are oriented in the same azimuth direction transverse to the z0β plane. As a result, the rotational symmetry about the direction z inherent in the particle ensemble is not retained for rotation about the direction β.

Formally, the BSPM of the particle ensemble can be written in the spherical system of coordinates as Eq. (8) with θ substituted by θ′ and φ substituted by φ′, where θ′ is counted from the direction β, and φ′ is counted from the points of intersection of planes θ′ = const with the z0β plane:

M^{'} (n) = n \int_{0}^{2 π} \int_{0}^{\frac{π}{2}} R (- φ^{'}) M (0, θ^{'}) R (- φ^{'}) f (φ^{'}, θ^{'}) \sin θ^{'} d θ^{'} d φ^{'} .

For the strict zenith orientation of the plate normals, calculations by Eq. (12) would be equivalent to calculations by Eq. (8), but for the distribution over the angles φ, θ of the following form:

f (φ, θ) = f (π, β) .

This implies that the normals of all plates are deflected from the zenith direction by the angle β in the azimuth direction φ = π. Generally, the distribution function written in the coordinates φ′, θ′ is transformed into a function of coordinates φ, θ. For this purpose, we now write down the following transformation:

(\begin{matrix} \cos β & 0 & \sin β \\ 0 & 1 & 0 \\ - \sin β & 0 & \cos β \end{matrix}) (\begin{matrix} \sin θ \cos φ \\ \sin θ \sin φ) \\ \cos θ \end{matrix}) = (\begin{matrix} \cos β \sin θ \cos φ + \sin β \cos θ \\ \sin θ \sin φ \\ - \sin β \sin θ \cos φ + \cos β \cos θ \end{matrix}) .

On the left side of Eq. (14), we have the Euler matrix of the coordinate transformation written for the particular case of rotation of the z axis about the y axis through the angle β and the vector-column of coordinates x, y, z of an arbitrary unit vector. On the right side, there is the vector-column of coordinates x′, y′, z′ of the same vector in the rotated system of coordinates. We further find the direction of this vector in the spherical system of coordinates with the z′ axis coinciding with the direction β:

θ^{'} (φ, θ) = a \cos (- \sin β \sin θ \cos φ + \cos β \cos θ),

φ^{'} (φ, θ) = a \cos (\frac{\cos β \sin θ \cos φ + \sin β \cos θ}{\sqrt{{(\cos β \sin θ \cos φ + \sin β \cos θ)}^{2} + (\sin θ {(\sin φ)}^{2}}}) .

The dependences

θ (φ^{'}, θ^{'})

and

φ (φ^{'}, θ^{'})

are determined from Eq. (15), (16) by replacement of the sign of β. These functions show the weight of the normals of plates with orientation

φ^{'}, θ^{'}

in the distribution given by Eq. (11). They should be substituted into Eq. (11) and then into Eq. (8). Results of calculations for three values of the distribution parameter k and several zenith sensing angles β are presented in Table 3. For the limiting case of almost complete orientation (k = 500), the dependences of the same parameters on the zenith sensing angle are shown in Fig. 4.

Table 3. Elements of the normalized BSPM and parameter χ (see Eq. (6)) as functions of the zenith sensing angle β (in degrees) for the indicated values of the parameter k of the distribution function over orientation angles.

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Fig. 4 Dependence of the normalized BSPM elements and parameter χ (see Eq. (6)) on the zenith sensing angle β for a strong degree of particle orientation. Closed symbols – k = 10, open ones – k = 500.

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6. Discussion

For sensing in the zenith (or nadir) direction, the signal intensity increases with orientation parameter k by more than two orders of magnitude (for k = 500) in comparison with the intensity of reflection from the ensemble of randomly oriented particles shown in Fig. 3. The normalized diagonal BSPM elements change even faster. From Fig. 3 it can be seen that with increasing k, they monotonically approach their asymptotic values $| m_{i, i} | = 1$ and almost reach them already at k = 10. From Table 2 it follows that k = 10 implies not too strict orientation: the standard deviation is about 9°. It is obvious that the polarization is influenced to a greater degree by particles that at the present moment are close to the horizontal position and bring the main contribution to the intensity.

For small values of the orientation parameter (k < 3), the zenith sensing angles up to 6° have practically no effect on the polarization characteristics of scattered radiation. In particular, for sensing by polarized radiation, the BSPM is described by the matrix Eq. (5) with m₁₄ = m₄₁ = 0. The situation changes when it is impossible to neglect the nonzero value of the element m₁₂. Then Eq. (5) loses its meaning, since $m_{33} \neq - m_{22}$ . However, in this case the relationship between the parameter d and the depolarization ratio $δ = I_{⊥} / I_{I I}$ , presented in [20], holds true:

d_{l} = \frac{2 δ}{1 + δ},

if d_l is taken to mean the depolarization for sensing by linearly polarized radiation. Indeed, for sensing by radiation with unit power and the Stokes vector

S = {(\begin{matrix} 1 & 1 & 0 & 0 \end{matrix})}^{T}

, we can write

δ = \frac{1 - m_{22}}{1 + 2 m_{12} + m_{22}},

and for the second normalized Stokes parameter q and depolarization d_l, we can write

q = \frac{m_{12} + m_{22}}{1 + m_{12}}, d_{l} = 1 - q .

It can be easily seen that Eq. (17) follows from Eq. (18), (19).

For sensing with circularly polarized radiation, attention should be paid to the sign of the fourth Stokes parameter or hence the element m₄₄. In [29,30] attention has already been paid that the element m₄₄ becomes positive in the case of chaotic particle distribution for large values of the depolarization parameter d > 0.5, and that the direction of rotation of the polarization plane changes. In our calculations, this is also manifested for random particle orientation (see the first line of Table 3). This effect can also be manifested for large zenith sensing angles and strong orientation of plates in the horizontal direction. For example, this can be seen in the last line of Table 3. In Fig. 4 it is also seen that for $45 ° \leq β \leq 80 °$ , where m₄₄ is positive.

For strong particle orientation (k = 500), scanning of the sensing direction is in fact equivalent to measurement of the scattering matrix M(θ) for a fixed particle position. The situation illustrated by Fig. 4 is close to that expected for distribution Eq. (13). The behavior of the elements m_ij (β) in this case reproduces almost completely the behavior of the matrix M(0, θ)/M₁₁ (see Eq. (7) and Fig. 3).

Summary

In this article, on the example of the monodisperse ensemble of ice plates partially oriented in the horizontal direction, we have shown how the backscattering phase matrix can be calculated for arbitrary sensing angle. The initial condition in this case is the BSPM M(0, θ) calculated with sufficiently small step in the polar angle θ for zero azimuth angle. If there is a databank of such matrices for particles of various shapes and sizes, the BSPM transformation with changes of the zenith sensing angle can be calculated for composite polydisperse ensembles of particles. The distribution over the polar orientation angles used in this article is not commonly accepted, but it is not a matter of principle. The method presented above is applicable to any arbitrary distribution. The assumption on the random azimuth orientation is not obligatory as well. In the article, it is shown that the proximity of the diagonal matrix elements to their asymptotic values $| m_{i i} | = 1$ does not obligatory imply a high degree of orientation within the limits 1–3°. As can be seen from Fig. 3 and Table 2, the condition $| m_{i i} | ≅ 1$ is fulfilled for flutter of the order of 10°. In the case of a monodisperse ensemble of plates, the deviation of the sensing angle by 6° from the zenith (or nadir) direction does not practically influence the backscattered radiation polarization.

Acknowledgments

This work was supported in part by the Russian Foundation for Basic Research (grant No. 13-05-00096-a), the Ministry of Education and Science (State Contracts Nos. 11.519.11.6033 and 14.518.11.7063), and CRDF grant RUGI-7053-TO-11.

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d, μm ε, m²/s³	5	10	20	30	50	100	200	400	500	750
	0.20	0.71	1.61	2.97	6.38	18.0	51.0	144	202	372
1.10 ⁻⁴	0.04	0.14	0.50	1.06	2.71	19.9	34.7	124	187	395
	0.09	0.25	0.72	1.33	2.85	8.07	22.8	64.5	90.2	166
5. 10 ⁻⁴	0.02	0.06	0.22	0.47	2.21	4.34	15.5	55.6	83.8	176
	0.03	0.08	0.23	0.42	0.90	2.55	7.22	20.4	28.5	52.4
5 10 ⁻³	0.01	0.02	0.07	0.15	0.38	1.37	4.91	17.6	26.5	55.9
	0.01	0.03	0.07	0.13	0.28	0.81	2.28	6.45	9.02	16.6
1.10 ⁻²	-	0.01	0.02	0.05	0.12	0.43	1.55	5.56	8.38	17.7
5. 10 ⁻¹	-	0.01	0.02	0.04	0.09	0.25	0.72	2.04	2.85	5.24
	-	-	-	0.01	0.04	0.14	0.49	1.77	2.65	5.59

k	β	m₂₂	m₃₃	m₄₄	m₁₂	χ
0	0	0.492	−0.492	0.018	0	0
3	0	0.873	−0.873	−0.746	0	0
	3	0.873	−0.873	−0.749	0	0
	6	0.876	−0.875	−0.752	0.007	0.008
	12	0.878	−0.867	−0.745	−0.049	0.039
	24	0.868	−0.799	−0.667	−0.182	0.205
6	0	0.960	−0.960	−0.920	0	0
	3	0.960	−0.960	−0.921	−0.003	0.002
	6	0.960	−0.959	−0.919	−0.013	0.016
	12	0.958	−0.946	−0.904	−0.088	0.119
	24	0.939	−0.850	−0.790	−0.260	0.424
10	0	0.983	−0.983	−0.965	0	0
	3	0.983	−0.983	−0.965	−0.004	0.004
	6	0.982	−0.981	−0.963	−0.029	0.035
	12	0.980	−0.967	−0.947	−0.109	0.234
	24	0.962	−0.857	−0.819	−0.308	0.583
	45	0.885	−0.284	−0.169	−0.655	0.724
	60	0.761	0.097	0.336	−0.681	0.642
	70	0.602	−0.009	0.388	−0.553	0.427
	80	0.422	−0.388	0.189	−0.270	0.028
	90	0.449	0.449	0.102	−0.136	0

d, μm ε, m²/s³	5	10	20	30	50	100	200	400	500	750
	0.20	0.71	1.61	2.97	6.38	18.0	51.0	144	202	372
1.10 ⁻⁴	0.04	0.14	0.50	1.06	2.71	19.9	34.7	124	187	395
	0.09	0.25	0.72	1.33	2.85	8.07	22.8	64.5	90.2	166
5. 10 ⁻⁴	0.02	0.06	0.22	0.47	2.21	4.34	15.5	55.6	83.8	176
	0.03	0.08	0.23	0.42	0.90	2.55	7.22	20.4	28.5	52.4
5 10 ⁻³	0.01	0.02	0.07	0.15	0.38	1.37	4.91	17.6	26.5	55.9
	0.01	0.03	0.07	0.13	0.28	0.81	2.28	6.45	9.02	16.6
1.10 ⁻²	-	0.01	0.02	0.05	0.12	0.43	1.55	5.56	8.38	17.7
5. 10 ⁻¹	-	0.01	0.02	0.04	0.09	0.25	0.72	2.04	2.85	5.24
	-	-	-	0.01	0.04	0.14	0.49	1.77	2.65	5.59

k	β	m₂₂	m₃₃	m₄₄	m₁₂	χ
0	0	0.492	−0.492	0.018	0	0
3	0	0.873	−0.873	−0.746	0	0
	3	0.873	−0.873	−0.749	0	0
	6	0.876	−0.875	−0.752	0.007	0.008
	12	0.878	−0.867	−0.745	−0.049	0.039
	24	0.868	−0.799	−0.667	−0.182	0.205
6	0	0.960	−0.960	−0.920	0	0
	3	0.960	−0.960	−0.921	−0.003	0.002
	6	0.960	−0.959	−0.919	−0.013	0.016
	12	0.958	−0.946	−0.904	−0.088	0.119
	24	0.939	−0.850	−0.790	−0.260	0.424
10	0	0.983	−0.983	−0.965	0	0
	3	0.983	−0.983	−0.965	−0.004	0.004
	6	0.982	−0.981	−0.963	−0.029	0.035
	12	0.980	−0.967	−0.947	−0.109	0.234
	24	0.962	−0.857	−0.819	−0.308	0.583
	45	0.885	−0.284	−0.169	−0.655	0.724
	60	0.761	0.097	0.336	−0.681	0.642
	70	0.602	−0.009	0.388	−0.553	0.427
	80	0.422	−0.388	0.189	−0.270	0.028
	90	0.449	0.449	0.102	−0.136	0

Transformation of light backscattering phase matrices of crystal clouds depending on the zenith sensing angle

Abstract

1. Introduction

2. Main special features of the BSPM

3. Modeling of the BSPM

4. Orientation of particles

5. Model BSPM of monodisperse cloud of ice plates versus the orientation parameter k and zenith sensing angle

6. Discussion

Summary

Acknowledgments

References and links

Cited By

Figures (4)

Tables (3)

Equations (19)

Optics Express