1. Introduction

Cirrus clouds influence noticeably the radiation balance in the atmosphere. Therefore, much attention has been given to a study of their morphological, microphysical, and optical properties [1]. The problem of optical cirrus cloud anisotropy caused by spatial orientation of cloud particles under the action of aerodynamic forces was also discussed [2]. As a result of anisotropy, the light extinction and scattering coefficients depend not only on polar but also on azimuth scattering angles and polarization state and direction of radiation incidence on a cloud layer. In the case of lidar sensing, anomalously high backscattering coefficients were observed in [3]. The problem of particle orientation is one of the most debatable [4–6]. The main point of discussion is a destructive effect of the air turbulence on the particle orientation. For a long time, the only source of data on the particle orientation in crystal clouds had been the observations of halos of all types, for example, solar columns. However, such observations could be carried out only in the daytime under favorable conditions. In addition, halos can be formed by particles of definite shapes and sizes that make, probably, an insignificant part of the number of cloud particles. Therefore, observations of halos cannot give detailed information on the orientation of an ensemble of particles.

In [7], results of lidar investigations of the total light backscattering phase matrix (BSPM) were presented, and in terms of this matrix, the parameters characterizing the orientation of cloud particle ensemble were defined. From the results of statistical processing of 450 experimental BSPM it follows that in the presence of a preferred orientation of cloud particles, lidar depolarization measurements can yield ambiguous results. In particular, the depolarization can depend on the sensing direction. This fact was experimentally confirmed in [8], where an increase in the zenith sensing angle from 0 to 7° resulted, on average, in the corresponding increase of depolarization from 0.1 to 0.3.

Based on the results obtained in [7], we give some recommendations to avoid the ambiguity of estimations of horizontal particle orientation which can take place for the case of linear polarization. It is demonstrated that in the experiment whose complexity is comparable with that of the widely used technique of determination of the lidar depolarization from scattering of linearly polarized radiation, the element a ₄₄ of the normalized (BSPM) can be measured. Moreover, results of measurements are independent of the presence or absence of azimuthal particle orientation. For sensing in the zenith or nadir, this element characterizes the degree of horizontal orientation of long particle diameters under the action of aerodynamic forces arising during free fall of particles. This offers the possibilities for detecting layers of ice particles with high reflectivity, for example, the layers responsible for the optical phenomenon of the lower Sun observed from a board of airborne vehicles above crystal clouds due to light reflection from horizontally oriented crystal faces. Exactly the possibility of detecting such situations is the main advantage of application of circular polarization. In this case, the distribution of scattered radiation differs significantly from the distribution observed for random orientation of particles.

It should be emphasized that the present article considers effects of spatial anisotropy of an ensemble of ice particles, in other words, the effects caused by the presence of a preferred orientation of nonspherical particles. If particles are randomly oriented, measurements with linear and circular polarization yield mutually unambiguous results. We demonstrate this below.

We have realized the suggested sensing technique in a lidar operating within the framework of the international QPQ program on validation of data of the spaceborne CALIOP lidar placed onboard the CALIPSO satellite [9].

2. Backscattering phase matrix and orientation parameters.

For the BSPM, we take a 4×4 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:

Because the optical density of crystal clouds is often small, we cannot neglect molecular scattering. Therefore, M should be considered as a matrix of the two-component medium:

where A denotes the BSPM of the aerosol component, and ∑ denotes the BSM of the molecular component.

It is assumed that the volume ΔV contains sufficiently large number of independently scattering ice particles, so that the microphysical volume parameters characterize the cloud as a whole. The BSPM of the ensemble of independently scattering particles is equal to the sum of BSPMs of individual particles. By virtue of the symmetry of the diagonal BSPM elements, the relation [10],

holds true, and for the nondiagonal BSPM elements,

As demonstrated in [7], the experimental BSPM can be reduced with acceptable accuracy to the following form:

The primed elements indicate that the matrix A′ is obtained from the experimental matrix A using the transformation

where R(Φ) is the operator of rotation of the coordinate system around the wave vector of incident (and scattered) radiation through the angle Φ:

Here we note that the angular BSPM elements are invariant under rotation and remain unchanged after transformation in Eq. (6). In particular, A ₄₄ = A′₄₄.

We call the matrices A′, obtained using transformation in Eq. (6), the reduced matrices. It should immediately be noted that the reduction operation makes sense for the BSPMs with nonzero nondiagonal elements. This is the case when a preferred azimuthal direction is present in the cloud. For example, long axes of columns are partly oriented in one preferable direction. The physical meaning of operation in Eq. (6) consists in virtual cloud rotation around the z axis of the lidar coordinate system in which the Stokes parameters are determined. The coordinates are specified by three unit vectors e _x × e _y = e _z. The optical lidar axis coincides with the z axis. The x0z plane is taken for the plane of reference.

A mathematical description of the reduction operation can be found in [7, 11]. It consists in a search for an argument Φ₀ of the operator R at which the matrix is described by Eq. (5). The operation is ambiguous. If instead of Φ₀ we substitute Φ₀±π/2, the matrix will be described by Eq. (5) again, but the signs of the elements A ₁₂, A ₂₁ and A ₃₄,A ₄₃ will change. In [7], the Φ₀ value with negative elements A ₁₂ and A ₂₁ was taken. Then the angle Φ₀ counted from the x axis specifies the azimuthal direction perpendicular to which the long particle diameters are oriented.

If the ensemble of cloud particles possesses a rotational symmetry about the wave vector of incident radiation, that is, there exists no preferred azimuthal direction, the BSPM assumes the following form [10]:

where the condition A ₃₃ = −A ₂₂ is satisfied in addition to relation in Eq. (3) and, as experimentally demonstrated in [7], the probability of obtaining nonzero values of elements A ₁₄ and A ₄₁ is sufficiently low.

Matrix Eq. (8) can be reduced to the following form:

To this end, designating A ₂₂ = A ₁₁ −d from Eq. (3) and using the equality A ₃₃ = −A ₂₂, we write down A ₃₃ = d − A ₁₁ and A ₄₄ = 2d − A ₁₁ and then normalize by A ₁₁. The meaning of the designation is obvious. The element a ₂₂ of the normalized matrix is numerically equal to the degree of polarization of radiation by a medium upon exposure to radiation linearly polarized in the reference plane or perpendicular to it. In this case, d is the commonly used definition of this quantity.

According to [10], zero values of elements A ₁₄ and A ₄₁ mean that the particles forming the ensemble possess a mirror symmetry or that the ensemble comprises equal numbers of particles being mirror reflections of one another. As already mentioned above and shown in Fig. 1, zero values of elements A ₁₄ and A ₄₁ are most probable. Then a diagonal matrix is obtained the form of which was suggested in [12]. It was used, for example, in [13,14].

Matrices (8) and (8′) characterize the medium with either three-dimensional random orientation of particles or with their two-dimensional orientation when particles are preferably oriented in the plane perpendicular to the wave vector of incident radiation and randomly oriented in the azimuthal direction. By way of example, zenith or nadir sensing of clouds comprising particles whose long diameters are preferably oriented in the horizontal direction and randomly oriented in the azimuthal direction can be considered. In most studies devoted to lidar polarization measurements, exactly this BSPM form is assumed explicitly or implicitly. However, in [7], it was demonstrated that a preferred azimuthal orientation is observed in 30% for crystal clouds. The reason for the occurrence of azimuthal orientation, by our opinion, is pulsations of the wind velocity [15,16]. The destructive influence of turbulence on particle orientation does not lead to a completely chaotic particle ensemble. Here it is pertinent to note that when the optical lidar axis deviates from the zenith, matrix (8) is transformed into matrix (5). In this case, elements A ₁₂ = A ₂₁ have the minus signs if the x0z reference plane coincides with the lidar tilt plane. This occurs because of breaking of the rotational symmetry. Projections of particles onto the optical lidar axis are less than those on the direction perpendicular to the lidar tilt plane. For example, a round plate is represented by an ellipse extended in this direction. The effect arises of apparent orientation of the long particle diameters perpendicular to the reference plane. Negative values of the above-indicated matrix elements are confirmed by the BSPM calculated in [17], for hexagonal plates and columns. Exactly this effect was experimentally observed in [8], when the optical lidar axis deviated from the zenith. As an example of ambiguity of depolarization measurements, we can refer to Fig. 6 in [18], where results of simultaneous measurements of depolarization for linearly and circularly polarized radiation were presented.

Below we consider the BSPM elements normalized by the element A ₁₁:

Owing to invariance of the angular BSPM elements under rotation operation in Eq. (6), the following important relationships:

are satisfied, from which it follows that a ₁₄ and a ₄₄ are independent of the azimuth of the lidar reference plane and hence of the presence or absence of azimuthal orientation of cloud particles. The importance of this BSPM property is caused by the fact that the element a ₄₄ in the case of zenith or nadir sensing acts as a parameter describing the horizontal orientation of long particle diameters [7].

The azimuthal orientation is characterized by the parameter χ defined by the following combination of elements of the reduced BSPM:

The parameter χ changes from 0 (for random orientation of particles) to 1 (for strict orientation of particles in a given direction). The modal value of the experimental χ distributions is 0.1. The presence of azimuthal orientation is additionally indicated by the nonzero value of the element a′₁₂.

Figure 1 shows distributions of relative recurrence frequencies for values of elements a′₄₄, a′₁₂, and a′₁₄ of the reduced normalized BSPM obtained in [7], by statistical processing of 450 measurements of the total scattering phase matrix of crystal clouds. From the figure it can be seen that a preferred horizontal orientation of particles (a′₄₄ < −0.4) is observed almost in 20% of all cases. The average value of a′₁₂ is 〈a′₁₂〉 = −0.22, and the distribution mode is −0.15. The probability that the a′₁₂ value lies in the interval [−0.4, −0.6] is approximately equal to 0.15. Values −0.6 > a′₁₂ > −1 are observed in several fractions of percent.

 

Fig. 1. Relative recurrence frequencies f for values of the indicated elements of the reduced normalized backscattering phase matrices for crystal clouds obtained by processing of 450 signal realizations.

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The element a′₁₂ of the reduced BSPM corresponds to such mutual orientation of the lidar and cloud when the x0z lidar reference plane is parallel to the direction perpendicular to which the long particle diameters are oriented. If the lidar is rotated around the z optical axis through the angle φ, the element a ₁₂ of the nonreduced BSPM, that is, of the experimental BSPM, will be

and for random φ, it can change from a′₁₂ to −a′₁₂. As demonstrated below, this leads to ambiguity of the depolarization measured in lidar experiments with linearly polarized laser radiation.

3. Depolarization of backscattered radiation

The physical meaning of radiation depolarization is determined by the formula

where P is the polarization degree and q, u, and v are the Stokes parameters normalized by the radiation intensity. Let us consider two methods of depolarization measurements in lidar research. The first and the most widespread method assumes that laser radiation is linearly polarized, and two components of the scattered radiation intensity are measured, one of which (I _∥) is polarized in the plane of laser radiation polarization, and another component (I _⊥) is cross-polarized. The second method assumes that laser radiation is circularly polarized, and a λ/4 phase plate is placed in the receiver in front of the analyzer of linear polarization. The fast axis of the plate is rotated about the x axis counterclockwise through an angle of 45°.

The Stokes vectors of laser radiation of unit intensity are written as column-vectors S ^L ₀ = (1 1 0 0)^T for radiation linearly polarized in the x0z reference plane, S ^c ₀ = (1 0 0 −1) for circularly polarized radiation.

We describe the action of the receiving polarization devices by row-vectors representing the first rows of the Müller matrices L ^∥ and L _⊥ of linear polarizers that transmit radiation in the reference plane and in the perpendicular direction:

and

The vectors G here are the first lines of the matrices G ^∥ = L ^∥ P and G ^⊥ = L ^⊥ P, where P is the Müller matrix of the λ/4 plate placed at an angle of 45°, as described above.

To determine the polarization, we take advantage of the normalized BSPM a assuming that A ₁₁ = 1 and that laser radiation has unit intensity. The intensities of the parallel and cross-polarized components in the first method are described by the formulas

where a ₁₁ ≡ 1, a ₁₂ = a ₂₁, and k is the proportionality factor unimportant for the examined case. The quantity

is the second Stokes parameter normalized by the radiation intensity according to the physical definition.

Let us assume that the parameters u = v = 0. Then from Eqs. (13), (16), and (17) we obtain the depolarization value

The definition of depolarization on the right side of Eq. (18) is commonly used in studies devoted to measurements of the Stokes vectors and scattering matrices and coincides with expression given in [13], for the case of random orientation of particles.

However, by virtue of Eq. (12), it appears ambiguous when ∣a′₁₂∣ ≡ 0. The ambiguity is caused by the fact that for sensing of an ensemble of particles oriented in an azimuthal direction, the parameters u and v can differ from zero and hence must be substituted into Eq. (13); however, they are not determined in the experiment. For vivid presentation, we now estimate the uncertainty interval for the modal value a′₁₂ =−0.15 mentioned in Section 2. Taking a ₂₂ = 0.6 as the most probable value, we obtain that according to Eq. (12), the depolarization can change within the limits of 0.35–0.47 depending on the lidar orientation relative to the cloud. However, much more uncertain situations can be realized. Probably, the reader engaged in lidar polarization research of clouds will recall cases in which the intensity of the cross-polarized component was equal or even exceeded that of the parallel component. According to Eq. (18), the equality I _⊥ = I _∥ implies the complete depolarization, and for I _⊥ > I _∥, definition (18) loses its meaning at all. The similar situation is possible when the lidar reference plane is tilted at an angle close to 45° to the direction of preferable azimuthal orientation of particles. Scattered radiation can have high enough degree of polarization, but small value of the second and large value of the third Stokes parameters. One of such BSPM presented in [7], will describe the similar situation after transformation in Eq. (6) with Φ = 45°.

For sensing of a layer with two-dimensional random orientation of particles, the equality u = v = 0 is satisfied, but ∣a′₁₂∣ depends on the tilt angle of the optical lidar axis. Therefore, the ambiguity of lidar depolarization is retained. Exactly for this reason we suggest to use the circular polarization which excludes the risk of obtaining ambiguous estimates of horizontal particle orientation and gives values of the a′₄₄ BSPM element independent of the azimuthal orientation of particles.

Let us consider the second method. By analogy with formulas (16), we obtain

The advantages of this method are obvious. The matrix elements a ₄₄ and a ₁₄ = a ₄₁ are invariant under transformation in Eq. (6) and hence are independent of the azimuthal lidar orientation. In addition, the probability that the value of the element a ₁₄ is nonzero is sufficiently small. In Fig. 1, the histogram of distribution of this element is well approximated by a Gaussian distribution with zero average and standard deviation σ = 0.05. This means that the element a ₄₄, which, as already indicated above, characterizes the degree of particle orientation relative to the plane perpendicular to the optical lidar axis, can be determined experimentally with a small probability of error. For the random orientation of particles, a ₄₄ values so obtained are unambiguously related to the element d of matrix (8′).

4. A lidar for measurements with circularly polarized radiation

Figure 2 shows the external view and the optical scheme of a lidar. An LS-2137 laser (Nd: YAG, 532 nm) with energy per pulse of 300 mJ and pulse repetition frequency of 10 Hz was used. The polarization plane of linearly polarized radiation of the laser forms the x0z reference plane. To change the polarization from linear to circular, a λ/4 quartz plate is inserted into the beam path. The fast axis of the plate is at an angle of 45° to the reference plane. The angle is counted counterclockwise as viewed counter to the laser beam.

The receiver antenna is the Cassegrainian telescopic system (D = 0.2 m and f = 2 m). The field stop forms a field-of-view angle of 1 mrad. The Wollason prism (WP), forming two beams with mutually orthogonal polarization states, is inserted into the scattered beam path. The plane in which the ordinary and extraordinary beams lie is perpendicular to the x0z plane. The x axes of the transmitter and receiver coordinate systems coincide, and their y and z axes have opposite directions. In front of the Wollaston prism, a λ/4 quartz plate is placed. Its fast axis is at an angle of 45° to the reference plane. The angle is counted from the x axis counterclockwise as viewed counter to incident radiation. Two FÉU-84 photomultipliers were used as receivers. The error in calibration of the relative sensitivity of photodetectors did not exceed 3%. This lidar configuration allows the column-vector S ^c ₀ considered above and row-vectors G _∥ and G _⊥ determined by Eq. (15) to be obtained.

 

Fig. 2. Optical scheme of the LOSA-S lidar.

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5. Calculation of the element a₄₄ normalized by the element a₁₁ for the BSPM of the aerosol component from lidar signals

The element a ₁₁ defines the backscattering coefficient of natural light. Let us write down the equation of laser sensing in the following form:

where P(h) is lidar return signal power and s(h) is the Stokes vector of scattered radiation normalized by the intensity; c is the velocity of light; W ₀ is the laser pulse energy; D is the receiving antenna area; h = ct/2 is the distance to the scattering volume at the moment of time t/2; M(h) is the BSPM of the ensemble of particles occupying this volume; T ² = exp[−2∫^h ₀(h′,φ,θ)dh′]; and ε(h′,φ,θ) is the extinction coefficient.

We act on Eq. (22) first by one and then by another operator in Eq. (15) and proceed from the radiation power to the detector responses by multiplication of both sides of Eq. (22) into the ampere-watt sensitivities κ _∥ and κ _⊥ of the detectors in the corresponding channels. As a result, we obtain a pair of equations for the detector currents

We now obtain from Eq. (13) the following equation:

where α = κ_⊥ / κ_∥.

The matrix of two-component medium in Eq. (2) is written in the following form:

where a(h) = A(h)/A ₁₁ (h) is the normalized BSPM of the aerosol component, a ₁ is the row-vector, representing the first row of the matrix a (the quantity A ₁₁(h)a ₁ S ^c ₀ is equal to the aerosol backscattering coefficient β_a for circularly polarized radiation); σ is the BSPM of molecular scattering normalized by the element ∑₁₁ (σ₁₁ = 1, σ₂₂ = 0.97, σ₃₃ = σ₄₄ = −0.97, σ_ij = 0);

and R(h) is the backscattering ratio. This quantity is determined by the well-known methods of reconstructing the optical parameters from lidar signals in the two-component molecular-aerosol atmosphere. For example, lidar signal calibration against a model profile of the molecular scattering described in [19], can be used with the reference point chosen at altitudes at which the influence of the atmospheric aerosol is minimal.

After substitution of Eq. (25) into Eq. (24) and matrix and algebraic transformations, we obtain the formula

where K(h) = (C(h) − 1)/ α(C(h) + 1)

It is easy to check that when the conditions of equality of the quantum detector efficiencies (α = 1), dense cloud γ(h) → 0, and a ₁₄ = 0 are satisfied, a ₄₄ = C(h). That is, the required parameter is simply equal to the value determined directly from lidar return signals. The terms containing γ(h) are corrections for the molecular scattering contribution. For R(h) → 1, the error in determining R(h) can lead to inadmissibly large errors in determining the element a ₄₄. The acceptable accuracy can be obtained for R(h) > 3.

Formula (26) requires a priori determination of the element a ₁₄. It is natural to take its average value ∣a ₁₄∣ = 0 obtained in [7], for this distribution. The probability that ∣a ₁₄∣ > 0.1 does not exceed 0.05. The risk of an error is much lower than in lidar experiments with linearly polarized laser radiation when by default one takes a ₁₂ = 0. If the risk is excessive, the complete experiment should be carried out to determine elements a ₁₄ and a ₄₄. To this end, additional measurements analogous to the above-described but with circular laser radiation polarization of opposite sign S ^c ₀ = (1 0 0 1) should be carried out. As a result, the second equation similar to Eq. (24) will be obtained. From the system of two equations, elements a ₁₄ and a ₄₄ can correctly be determined.

By way of example, Fig. 3 shows results of lidar sensing of crystal clouds during a 20-min measurement session. Lidar signals were averaged over 1-s periods for a 10-Hz pulse repetition frequency of the transmitted signal. An altitude-time map of signal F _∥ (in units of 12-bit ADC code) is shown at the top of the figure. A 2D distribution of the a ₄₄ matrix element is shown at the bottom of the figure. It can be seen that the matrix element changes from a ₄₄ = −0.95 at altitudes beyond the cloud layer (between 6–7 and 8–9 km), which is typical of molecular scattering, to a ₄₄ = 0 within dense cloud layers. Significant time variations can be traced at fixed altitudes and significant altitude variations can be traced at fixed times. In this case, the fine structure of the a ₄₄ element not always coincides with that of the backscattering coefficient, which demonstrates the temporal and spatial variability of the orientation of crystal particles.

 

Fig. 3. Structure of a layer of crystal clouds recorded on April 10, 2008 with a circular polarization lidar. The signal from the photodetector oriented in the reference plane is shown at the top, and the a ₄₄ element is shown at the bottom of the figure. Scales in artificial colors for the corresponding signals are shown to the right of the figure.

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6. Conclusion

In this article, we have demonstrated that small additional modification of the lidar optical scheme by using two λ/4 phase planes increases significantly the reliability of depolarization measurements in the case of sensing of crystal clouds. In addition, this lidar modernization has opened up fresh opportunities for regular estimation of particle orientation from measurements of the element a ₄₄ of the normalized BSPM and thereby elucidation of a debatable question on the role of particle orientation in modeling the radiative transfer in the atmosphere. Nowadays, a lidar whose characteristics have allowed us to start measurements coordinated with the schedule of the CALIPSO program is operated at the IAO SB RAS.