1. Introduction

Cirrus clouds composed mostly of ice crystal particles have an essential impact on the Earth’s radiative budget and, hence, on the climate [1,2]. At present, this impact has not been completely understood because both the radiative properties (i.e., scattering and extinction coefficients and phase functions) and microphysical properties (i.e., sizes, shapes, and orientation of particles) of ice clouds are not known with necessary accuracy. Therefore, many ground-based, airborne, and spaceborne measurements of ice cloud parameters have been performed for the last three decades by use of various tools ranging from the visible to microwaves. For example, we refer to the papers concerning some field campaigns [3,4] and interpretations of data obtained by satellite radiometers [5,6].

Among the facilities used for studying ice crystal clouds, lidars provide not only unique information on the vertical profiles of particle number density, but also they prove to be the most sensitive tools for retrieving the size, shape, and orientation of ice crystals. Ground-based lidars have been used for such measurements for a long time [7,8]. The spaceborne lidars LITE (The lidar in-space Technology Experiment) [9] and CALIPSO (The Cloud-Aerosol Lidar Infrared Pathfinder Observations) [10,11] were launched for these purposes, too.

While most scattering media encountered in physics are composed of nonspherical particles with random orientation, ice crystals in the atmosphere reveal a tendency toward horizontal orientation. Here an area of particle projection on the horizontal approaches its maximum value because of the aerodynamic forces appearing for a falling particle. Of course, the perfect horizontal orientation for a particle ensemble is impossible, and such crystals have certain small oscillations of their orientation relative to the horizon that are called flutter. The horizontal orientation of ice crystals essentially distorts the optical properties of ice clouds as compared with randomly oriented crystals. However, both the optical properties of the preferably oriented ice crystals and characteristics of their flutter occurring in the atmosphere have not been well studied.

It was known earlier that depolarization of lidar signals could be used for discrimination between ice and water clouds. Indeed, light backscatter by randomly oriented ice crystals for linearly polarized incident light is essentially depolarized, unlike in the case of water drops. However, horizontally oriented ice crystals do not depolarize lidar signals too, if a lidar is pointed in the vertical. To avoid this uncertainty, Platt et al.[12,13] scanned the lidar up to a few degrees from the vertical. They obtained a strong decrease in signal intensity along with a sharp increase of depolarization when the lidar deviated from the vertical. These phenomena allowed them to distinguish the case of the horizontally oriented crystals and to estimate their flutter. At present, the same technique using scanning polarization lidars is used to study the flutter if horizontal orientation of ice crystals occurs [14,15].

In this paper, a new technique for studying horizontally oriented ice crystals is discussed. Our technique differs from the previous one [13-15] in two ways. In the previous technique, the monostatic scheme was used, where a source and a detector of light were united and only backscatter by particles was detected. The first difference of our technique is the bistatic scheme, where a light source and a detector are separated in space. In this case, not only backscatter but bidirectional scattering, i.e., phase functions for various incident directions, can be measured. In addition, in the monostatic scheme the temporal evolution of a lidar signal should be processed to find the distance to a scattering volume. In the bistatic scheme, the scattering volume can be identified by the angle between the horizon and the point seen by an observer. Therefore, any continuous radiation sources can be used, and our further discussion is applicable when a sounding light beam is generated by both continuous and pulse lasers as well as by any thermal light source. In particular, in the experiments presented in this paper we use a floodlight beam produced by a standard beam projector with a xenon lamp as a light source.

The second difference of our technique from the previous works [13-15] is that our detector measures not the light power but the ray intensity I(r, n) or radiance, i.e., the distribution of photons over propagation directions n at the detector localization r. The ray intensity can be measured by use of any optical system forming an image. In particular, we use a standard CCD camera to digitize the ray intensity I(r, n). Note that the same experimental scheme, bistatic + imaging, is already used for lidar sounding of aerosols [16].

The physical idea proposed in this paper is to separate the specular component of scattered light that appears at the horizontal ice crystal orientation. An advantage of the specular component is that its angular structure is connected with both ice crystal sizes and their flutter by rather simple equations. Consequently, the angular structure of the specular component proves to be an effective tool for retrieving these microphysical parameters of ice clouds.

2. The specular and diffuse components of scattered light

 

Fig. 1. Experimental schemes with (a) vertical and (b) tilted illumination of a scattering layer.

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Light scattering by ice crystal clouds in the atmosphere can be described by the conventional concepts of optics of scattering media, where every particle scatters incident light according to its differential scattering cross section s(n, n ₀). Here, n and n ₀ are the scattering and incident directions, respectively. If ice crystals are randomly oriented, both the differential scattering cross section s(n, n ₀) and the ray intensity I(r, n) at any observation point r are rather smooth functions of their variables. However, if the ice crystals reveal their preferably horizontal orientation, the ray intensity has been separated into two qualitatively different parts:

where the diffuse part I_d is a smooth function of its variables as before, and the specular part I_s has a sharp peak in the direction corresponding to the reflection of light by horizontally oriented small mirrors.

The specular component can be found easily with simple geometrical optics drawings. In particular, consider the experimental scheme shown in Fig. 1. Here, an ice crystal cloud is assumed to be a thin layer situated at height h, and both a light source and an observer are localized on the ground at some distance from each other. The incident light is a narrow diverging beam with the angular radius α (α≪1). In this case, an observer can see a spot of radius R≈hα localized at height h that corresponds to the diffuse component of the scattered light of Eq. (1). On the contrary, the specular component will be seen as a bright dot like a star that is localized at double height 2h. This dot has a physical meaning as the virtual image of the point source of light denoted as point V in Fig. 1. This image is formed by a horizontal mirror localized at height h.

Our experiments in the atmosphere have realized the scheme of Fig. 1 We used a beam projector with a xenon lamp that produced an inhomogeneous diverging beam. Its bright core had the angular radius of about 1° and a weak lateral part spread up to 10°. Figure 2 presents a movie where the beam was scanned near the zenith. In the movie, we see that the diffuse spot follows the beam axis, while the specular spot is motionless because its position is independent of both the axis tilt β and the angular structure of the beam (see Fig. 1).

 

Fig. 2. (2.65 MB) Movie of the floodlight beam scanning near the zenith. The scan direction is perpendicular to the line connecting the beam projector and an observer. The right picture is an enlargement of the left one. [Media 1]

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The bright dot of the specular component always has to be watched on the background of the spot of the diffuse component. It is illustrated in Fig. 3. Sometimes the specular and diffuse spots look like spatially separated pictures. This phenomenon is easy understandable, for example, in the scanning presented in Fig. 2. Here, brightness of the specular spot decreases with the scan angles according to the decreasing intensity of the inhomogeneous diverging beam in the proper direction. There are the scan angles where the bright core of the diffuse spot does not already overlap the specular spot. However, the intensity of the lateral part of the beam is still enough to produce brightness of the specular spot such that it is comparable with that of the diffuse core spot.

In the atmosphere, situations where ice crystals are oriented horizontally do not occur often and depend on both the meteorological state of the atmosphere and its turbulence. In the boundary layer of the atmosphere, the horizontal orientation of ice crystals in snowfalls reveals itself as light pillars seen over some ground light sources (e.g., [17]). Ice crystals in cirrus clouds usually create thin layers at high altitudes that correspond to Figs. 2 and 3. Wide-angle beams produced by thermal light sources have some advantages for detection of such high layers in comparison with the narrow beams emitted by lasers. Indeed, as seen in Fig. 1, the specular component is detected on the ground only within the circle of the radius of R≈2hα. In the case of a narrow beam, this circle may not overlap an observer if the beam is not pointed strictly to the zenith or if the layer with the oriented crystals is not strictly horizontal because of large-scale, nonhorizontal motion of the air. Note that this separation of the diffuse and specular components by use of a beam projector was first watched by the co-authors (V.G. and A.M.) and published earlier in [18].

 

Fig. 3. Specular and diffuse spots seen by an observer in the experimental scheme of Fig. 1(a).

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3. Angular structure of the specular component

3.1 Horizontally oriented crystals

In this section, a simple analytical equation for the angular width of the specular component is derived. It is convenient to begin with the following simple physical model. Let us consider an ensemble of small mirrors randomly positioned within a horizontal plane at height h. The term mirror means a planar boundary of two media that reflects partly incident light. Sizes of mirrors D are taken greater than wavelength λ, i.e., D≫λ, so reflection of incident light by these mirrors is well-described by geometric optics. In particular, in the scheme shown in Fig. 1, the scattered (i.e., reflected) light is described by the simple Eq. (2)

where δ is the Dirac delta function, n ₀ is the direction from the virtual point source V shown in Fig. 1 to an observation point r, and an explicit equation for the function c(r) is of no interest for further discussion The function c(r) is nonzero if a ray between the virtual point source and the point r intersects a mirror; otherwise it is zero.

Equation (2) is true within the geometric optics approximation that is valid only in the near zone of the mirrors, i.e., at distance z≪D ²/λ. The typical size of cirrus ice crystals is, e.g., 30µ, so that puts estimates of their near zone at about 1 mm for visible wavelengths. Nevertheless, it is important to emphasize that consideration of scattered light in the near zone is very fruitful. It provides deep insight in all physical regularities that appear. If the near-zone field is known, the conventional scattered light of the wave zone (i.e. z≫D ²/λ) is directly obtained from this field by well-known equations like the Fraunhofer diffraction equation.

Under the natural restriction of D/h≪1, we can neglect a curvature of the wavefronts of both incident and reflected waves in the vicinity of a mirror. As a result, Eq. (2) corresponds to a lot of plane-parallel beams of transversal sizes of D. Every beam propagates in the directions n ₀, where n ₀ means a direction from virtual point source V to the center of a mirror. At distance z≈D ²/λ, these plane-parallel beams are spread in the transversal direction due to the Fresnel diffraction. In wave-zone z≫D ²/λ, every beam is transformed into a spherical wave that diverges from the mirror center and is concentrated within a narrow cone of scattering directions n around the initial direction n ₀. Intensity of the wave scattered in the direction n corresponds to the differential scattering cross section s(n,n ₀)=S(n-n ₀). The function S(n-n ₀) is easy determined for a given shape of mirror by the Fraunhofer diffraction equation [19]. The explicit equation for the function S(n-n ₀) is not needed for further discussion. We only refer to the angular radius of the diffraction cone (Δθ)_d that is equal to

Let us apply the aforesaid model to real ice crystal particles occurring in the atmosphere. Mostly the specular component of the scattered light is created by thin crystal plates oriented horizontally. In this case, the lower horizontal facets are just the mirrors considered above. In addition, the light transmitted inside any crystal is reflected and refracted by other facets of the crystal. As a result, this light leaves a crystal as a set of various plane-parallel beams like the above-considered beam reflected by the lower facet. If propagation directions of those beams are different from the direction n ₀, such beams create the diffuse part of scattered light [see Eq. (1)]. However some of the beams can obtain the same propagation direction n ₀. For example, they are the beams created by several re-reflections between the lower and upper facets of the crystal plate. These beams should be included in the specular component as well. Appearance of these additional beams results only in correction of the quantity D that should be ultimately treated as an effective transversal size of the beams contributing to the specular component.

In addition to the horizontally oriented ice plates, there is also a rare case of the appearance of the specular component. This case corresponds to hexagonal columns with their long axes oriented horizontally, but whose two rectangular facets are also fixed in the horizontal plane. This type of orientation is referred to as Parry (see, e.g., [20]). It is obvious that the aforesaid discussion is applicable for the Parry-oriented columns, too.

Let us consider a transformation of the scattered light by any optical imaging system, including the eye of an observer. If geometric optics were true at any distance from the particles and were described by Eq. (1), the imaging system would gather the specular component to a dot in the plane corresponding to the image of the virtual point source V shown in Fig. 1. This imaging plane can be replaced approximately by the focal plane under the weak restriction f/h≪1, where f is the focal length. Any point ρ of the focal plane gathers all photons on a receiving lens coming at those propagation directions n that obey the condition ρ=m f, where m is the projection of direction n on the receiving lens. Thus, the distribution of intensity of scattered light in the focal plane would be described by just Eq. (1) if geometric optics approximation were valid, where n ₀ is the direction from the virtual point source V to the lens center. In practice, the beams reflected by particles are smoothed around scattering directions n ₀ because of the Fraunhofer diffraction. Therefore, the Dirac delta function of Eq. (2) should be replaced by differential scattering cross section S (n - n ₀) that yields

Usually the radius hλ/D of the Fraunhofer diffraction footprint produced by a crystal on the ground is larger than the radius of a receiving lens. In this case, a contribution from one particle is reduced to a dot in the focal plane, but intensity of this dot is proportional to the function S(n-n ₀). Consequently, contributions from all particles positioned in the circle of the radius hλ/D in an ice cloud result just in the diffraction pattern of Eq. (4). If the incident beam shown in Fig. 1 is narrow, i.e., α<λ/D, an imaging system detects only the shortened Fraunhofer pattern that is cut off by the angular aperture α of the beam.

3.2 Flutter

In nature, ice crystals of a statistical ensemble are not perfectly oriented in the horizontal plane. Their orientations oscillate relative to the horizon; that is called flutter. In the case of large flutter, the specular and diffuse components are mixed, and these concepts become senseless. However, flutter is often small where a deviation of the normal to the facet responsible for the specular component is of a few degrees relative to the vertical [13-15]. Here, the bright dot of the specular component remains distinguishable on the background of the diffuse spot. In this case, a separation of the specular component provides us a tool for measuring both the sizes and flutter of the ice crystals that is discussed below.

A change of a crystal orientation leads to deviation of the propagation directions n ₀ for the near-zone plane-parallel beams creating the specular component. Therefore, an impact of flutter on the scattered light is strictly determined within the geometric optics approximation. Within geometric optics, it is obvious that flutter smoothes the virtual point source V shown in Fig. 1. It transforms the perfect point source at height 2h into a spot around the initial point. This spot is seen by an observer on the ground as a spot of certain angular radius (Δθ)_f. We denote the zenith angle between the vertical and the normal to the facet responsible for the specular component as γ and assume that flutter is determined by the probability density function P (γ) with the normalization ∫P(γ)d(cosγ)dφ=1, where φ is azimuth angle. In the simplest case of vertical direction of incident light, it is obvious that the tilt of a crystal at angle γ leads to the zenith scattering angle θ=2γ. It can be proven that the differential scattering cross section of such a fluttering crystal is proportional to P(θ/2). In particular, if the flutter angle γ runs the interval [0, F], the specular component is distributed relative to the zenith scattering angle θ in the interval [0, 2F], i.e.,

For the case of small values of the angles α and β shown in Fig. 1, when incident directions of light scarcely deviate from the vertical, Eq. (5) can be considered as an approximation for all crystal particles illuminated.

4. Retrieval of ice crystal sizes and flutter parameters from the specular spot

We have shown that the Dirac delta function of the specular component of Eq. (2) is spread into a specular spot relative to both variables n and n ₀. Here, diffraction smoothes the delta function over the variable n while flutter disperses the variable n ₀. Because these smoothings are more or less independent of each other, the resulting angular radius of the specular spot Δθ becomes the following sum:

Equation (6) provides us a promising and simple method to retrieve the effective horizontal size D of the crystals and their maximum flutter angle F by measuring the angular radius Δθ of the specular component at two wavelengths. The main advantage of this method is that Eq. (6) does not depend on the height h of ice clouds. In addition, the tilt β, the angular aperture α, and the structure of an incident beam are not of principal importance for this equation. In nature, cirrus clouds often occur as thin layers. The finite thickness of the layers Δh leads to a vertical smoothing of the virtual point source V of Fig. 1 and, consequently, to an additional smoothing of the specular spot in the focal plane of a receiving lens (Δθ)h. Under condition Δh/h≪1, the quantity (Δθ)_h can be ignored in Eq. (6).

Figure 4 presents an example of the retrieval procedure according to Eq. (6). Here, the upper part of the figure shows the specular spot of Fig. 3 obtained by a standard CCD camera. This spot is digitized along its diameter taken between the two horizontal lines drawn in Fig. 4. The results of the digitization are presented in the lower part of the figure for the red light (λ₁≈0.7µ) and blue light (λ₂≈0.4µ) that is provided with the standard code of the CCD camera. The background noise at these two wavelengths was assumed as two constants that were obtained by digitization of the picture of Fig. 4 at some distance from the specular spot. This noise is not shown in Fig. 4. The numerical data presented in Fig. 4 are already the difference between the measured quantities and the noise. In this method, we do not care about the central part of the specular spot, which is strongly dependent on the spectral sensitivity of the detector. We are interested only in the border of the spot. The border means a point in Fig. 4 where the signal becomes zero. Of course, a determination of the border is a rather subjective procedure. Nevertheless, such a procedure is defensible for estimations. In the case of Fig. 4 we can assume that (Δθ)₂≈36 mrad and (Δθ)₁≈54 mrad. As a result, the effective horizontal size of the crystals and their maximum flutter angle are estimated as D≈17µ and F≈7 mrad≈0.4°.

In general, an ensemble of horizontally oriented ice crystals is described by the probability density function P (D, γ) over both particle sizes D and tilts γ where dependence of flutter on particle sizes could be taken into account. It is possible to write down the specular component I_s(r, n) as an integral transformation of function P (D, γ). Conversely, the inverse transformation determines the inverse scattering problem, i.e., the retrieval of either the function P (D, γ) or its certain parameters from the experimentally measured function I_s(r, n). Such a task is out of the scope of this paper.

 

Fig. 4. Specular spot and its angular structure obtained by digitization of the spot along its diameter.

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5. Conclusions

A new method to retrieve microphysical parameters of ice crystals when they have preferably horizontal orientation is proposed and realized. The method consists of separation of the specular component in the bistatic sounding scheme. This technique is realized with cheap equipment: the light sources both can be continuous, and pulse lasers as well as floodlight beams and detectors are standard CCD cameras.