## 1. Introduction

In the past three decades, great strides have been made in the simulation of the single-scattering properties (i.e., the scattering phase matrix, extinction cross section, and single-scattering albedo) of nonspherical ice particles within ice clouds [see review by Yang and Liou (2006)]. The motivation for studying ice clouds is their importance for understanding the earth’s hydrological cycle. Additionally, ice clouds have a significant impact on the radiation budget in the earth–atmosphere system, and hence on climate (Liou 1986; Stephens et al. 1990; Lynch et al. 2002, and references cited therein). At present, the representation of these clouds in climate models is still quite crude in comparison with the case of water clouds that are assumed to consist of spherical liquid water droplets. Significant differences are found when simulations of ice clouds by climate models are compared to satellite observations (Zhang et al. 2005). To improve the parameterization of the bulk radiative properties of ice clouds, which is a necessary component of the radiative transfer schemes used in climate models, it is important to learn more about the optical properties of highly complex, nonspherical ice particles. The optical properties are determined fundamentally by the ice habits, particle size distributions, and refractive indices. Furthermore, modeling the optical properties of complex ice particles is also an interesting but challenging research topic in the discipline of electromagnetic scattering and computational physics.

In synoptic cirrus, ice particles tend to range from tens to hundreds of microns in size. In regions of intense convection, however, ice particles can become much larger. Thus, the principles of geometric optics, particularly the ray-tracing technique based on Snell’s law, the Fresnel formulas, and the Fraunhofer diffraction approximation, can be used to derive the scattering properties of these particles at visible and near-infrared wavelengths because of the ensuing large size parameters. At infrared wavelengths, the strong absorption of ice complicates the implementation of the ray-tracing technique because of the inhomogeneous properties of the localized waves within the scattering particles (Yang et al. 2001; Chang et al. 2005), although this effect is usually neglected in many studies reported in the literature. After the pioneering studies of Jacobowitz (1971), Wendling et al. (1979), and Cai and Liou (1982), the ray-tracing technique has been employed extensively to investigate the interaction between electromagnetic radiation and nonspherical ice particles with size parameters in the geometric optics regime. However, the scattering of light by nonspherical ice particles has also been investigated with other approaches (Mishchenko and Sassen 1998; Yang and Liou 1996a; Sun et al. 1999; Baran et al. 2001; Mitchell et al. 2006). The early ray-tracing studies focused on the case of pristine hexagonal ice plates and columns that were solid ice particles with smooth faces (e.g., Takano and Jayaweera 1985; Takano and Liou 1989; Rockwitz 1989; Hess and Wiegner 1994). Such pristine particles are found in the Antarctic atmosphere (Tape 1994), but elsewhere more complex particles are common. Later, the single-scattering characteristics associated with complex habits, such as bullet rosettes, hollow columns, polycrystals, aggregates, and clusters (or aggregates) of bullet rosettes, were investigated using the ray-tracing technique (e.g., Macke 1993; Macke et al. 1996b; Takano and Liou 1995; Iaquinta et al. 1995; Yang and Liou 1998; Zhang et al. 2004; Um and McFarquhar 2007). The ray-tracing technique in combination with the Monte Carlo method has been applied to the computation of the scattering properties of inhomogeneous ice particles, that is, ice particles with the internal inclusion of air bubbles (Macke et al. 1996a; C.-Labonnote et al. 2000, 2001). Furthermore, improved algorithms for the geometric optics method have been developed for light-scattering calculations involving nonspherical ice particles (Muinonen 1989; Yang and Liou 1996b). The results from these efforts over the past two decades have been applied to the parameterization of the radiative properties of ice clouds (Fu et al. 1998; McFarquhar et al. 2002; Key et al. 2002), the remote sensing of ice clouds (Minnis et al. 1993a, b; King et al. 2004; Baum et al. 2005a, b), and the interpretation of the spectral signature of cirrus clouds (Wendisch et al. 2007).

In this study we investigate the scattering and polarization characteristics of hollow bullet rosette ice particles. Bullet rosettes are frequently observed in midlatitude ice clouds (Heymsfield et al. 2002), and are also common in snowfall on the Antarctic Plateau (Walden et al. 2003). The bullet elements of these particles usually have hollow ends, as evident from in situ measurements recently reported by Schmitt and Heymsfield (2007). Based on the modification of a ray-tracing computational model developed by Iaquinta et al. (1995), Schmitt et al. (2006) investigated the scattering properties of hollow bullet rosettes, but did not report the complete phase matrix. As polarimetric quantities (e.g., the backscattering depolarization ratio) contain important information about cloud optical and microphysical properties (Yang et al. 2003; Hu et al. 2007) that can be exploited in remote sensing studies, it is necessary to understand the complete polarization configuration associated with the scattering of light by bullet rosettes. The hollowing of the bullets also has consequences for the absorption of near-infrared solar radiation by ice clouds; the hollowing decreases the mass-to-surface-area ratio of the particles, a quantity that is proportional to the mean absorption pathlength for the radiation transmitted through the bullet rosette.

As a follow-up of the study by Schmitt et al. (2006), we report the complete phase matrix of hollow bullet rosettes. Unlike Schmitt et al. (2006), who assume that the tip of a bullet element has three faces, we define a bullet tip as a pyramid that has six triangular faces, a geometry that is more in line with observations. This paper is organized as follows. In section 2, we define the geometries for bullet rosettes that have various numbers of bullets. We also illustrate the dependence of the projected area of a bullet rosette, expressed as a function of the particle’s maximum dimension, aspect ratio, and number of bullets. In section 3, we briefly describe the scattering computational model used in this study. In section 4, we discuss the computed results for the phase matrix, backscattering depolarization ratio, and asymmetry factor of hollow bullet rosettes. In section 4, we also illustrate the optical effects that result from the inclusion of hollow structures in the individual bullets, and their influence on the prediction of ice cloud microphysical and optical properties (particle effective size and optical thickness, in particular). The conclusions of this study are given in section 5.

## 2. Geometry of hollow bullet rosettes

A recent analysis of balloonborne replicator data acquired during three ascents through midlatitude cirrus clouds shows that 50%–80% of bullets are not solid (Schmitt and Heymsfield 2007). High-resolution photographs of the particle habits have been obtained by collecting ice particles naturally falling to the surface on the Antarctic Plateau. Walden et al. (2003) analyzed 84 photographs taken on 84 days during the winter of 1992. Figure 1 is typical of the particles falling during snowfall from nimbostratus clouds. In all the bullet clusters photographed during the winter, the bullets were hollow (Table 4 of Walden et al. 2003). Hollow bullet rosettes are more likely to form at rapid particle growth rates and levels of supersaturation similar to the conditions that lead to the development of hollow columns. Since they formed at the low temperatures in the stable atmosphere of the Antarctic Plateau, hollow bullet rosettes could be expected in bullet clusters elsewhere, as Schmitt and Heymsfield (2007) found. The interior walls of the hollow portion of the rosettes are often corrugated as seen in Fig. 1, but in this paper we consider only flat-walled hollow structures. For the particles shown in Fig. 1, the corresponding size parameters at visible and near-infrared wavelengths are clearly within the geometric optics regime.

*L*,

*t*,

*H*, and 2

*a*, respectively. The angle between the axis of symmetry and the pyramidal faces is indicated by

*α*following the notation used by Greenler (1980). Note that only two parameters among

*t*,

*a*, and

*α*are independent. Specifically, we have the following relationship:Historically, the value of

*α*was suggested to be either 25° or 28° in the atmospheric optics literature (e.g., Besson 1923). However, Goldie et al. (1976) argued that

*α*= 28° was required to explain an unusual halo display observed over England. Furthermore,

*α*= 28° leads to a value of 31.5° for the angle between the bullet symmetry axis and the edges of the pyramidal faces, which is consistent with that observed for some bullet ice particles (Tape and Moilanen 2006). For this reason, we assume

*α*to be 28°, and note that this value was also used by Um and McFarquhar (2007).

*OX*

_{b}

*Y*

_{b}

*Z*

_{b}in Fig. 2a. A particle coordinate system,

*OX*

_{p}

*Y*

_{p}

*Z*

_{p}, is employed to define the geometry of the rosette. The position of a bullet branch in

*OX*

_{p}

*Y*

_{p}

*Z*

_{p}is specified by three Euler angles

*φ*

_{b},

*θ*

_{b}, and

*γ*

_{b}shown in Fig. 2b. The rotational transformation between the two coordinate systems is given as follows:where (

*x*

_{b},

*y*

_{b},

*z*

_{b}) indicate the coordinates in

*OX*

_{b}

*Y*

_{b}

*Z*

_{b}, whereas (

*x*

_{p},

*y*

_{p},

*z*

_{p}) are the corresponding coordinates in

*OX*

_{p}

*Y*

_{p}

*Z*

_{p}.

Figure 3 shows representations of a single hollow bullet and various hollow bullet rosettes consisting of between 2 and 12 hollow bullet branches. Different values have been reported in the literature for the mean number of bullet branches, 〈*n*_{b}〉, for bullet rosettes observed in cirrus clouds. For example, an analysis of 869 ice particle images by Heymsfield et al. (2002) that were acquired over the U.S. Department of Energy–Atmospheric Radiation Measurement Program (ARM) site in Oklahoma on 9 March 2000 shows a value of 〈*n*_{b}〉 = 5.8 ± 0.6, whereas the number of bullet branches per rosette ranges from 2 to 12. From an analysis of the ice particle replicator data (Arnott et al. 1994) acquired near Coffeyville, Kansas, during the First International Satellite Cloud Climatology Project (ISCCP) Regional Experiment (FIRE-I) held in 1991, Mitchell and Arnott (1994) estimated that the average number of branches per rosette is approximately 11. The Antarctic rosettes had 1–9 branches (except for one outlier with 16), averaging 4 per particle (Walden et al. 2003, Table 2; V. P. Walden 2004, personal communication). In many reported studies, with the exception of Iaquinta et al. (1995), 4- or 6-branched bullet rosettes are usually assumed (Takano and Liou 1995; Yang and Liou 1998). Um and McFarquhar (2007) also used 4-branched and 6-branched bullet rosettes to define aggregates of bullet rosettes. While Iaquinta et al. (1995) investigated the effect of branch number on the single-scattering properties of solid bullet rosettes, a similar sensitivity study should be carried out in the case of hollow bullet rosettes.

The top panel of Fig. 4 compares the aspect ratio expressions given by Eqs. (3) and (4). It is evident that the bullet branches specified by Eq. (3) are thinner than their counterparts based on Eq. (4). The differences are especially pronounced for moderate particle sizes (i.e., *L* ∼ 50 *μ*m). For large (*L* > 400 *μ*m) bullet rosettes, the two relations converge. The bottom panel of Fig. 4 shows two relationships between *L* and *D* that are calculated using Eqs. (3) and (4). The *L–D* relationships are similar because the small aspect ratio of the bullets means that *D* is insensitive to *a*. In particular, the *L–D* relationships derived from Eqs. (3) and (4) are essentially the same when *L* is larger than approximately 1200 *μ*m. In this limit, Eq. (5) approximates to *D* ≈ 2(*t* + *L*).

Figure 5 shows the projected areas calculated from Eqs. (3) and (4) as a function of the maximum dimension for bullet rosettes that are assumed to have 6 or 11 branches. For comparison, the measurement data are shown from Arnott et al. (1994). Note that each measurement data point shown in Fig. 5 represents an average of a number of ice particle images (see Arnott et al. 1994 for details). In the case of 11-branched bullet rosettes, the projected areas computed from either Eq. (3) or Eq. (4) are substantially overestimated in comparison with the measurements when *D* is larger than 400 *μ*m. The overestimation is more pronounced in the case of the results based on Eq. (4). However, if 6 branches are assumed for bullet rosettes, the projected areas simulated on the basis of the aspect ratio relationship defined in Eq. (4) agree better with the observations than those based on Eq. (3). Furthermore, the limited datasets in Fig. 5 suggest that large bullet rosette ice particles have fewer branches than small particles.

## 3. Scattering computational model

Observations reported by Arnott et al. (1994) and Mitchell and Arnott (1994) show that bullet rosettes are generally larger than 200 *μ*m. Thus, the optical properties of these particles cannot be derived from rigorous numerical approaches such as the discrete dipole approximation (DDA; Draine and Flatau 1994) and the finite-difference time domain (FDTD) method (Yee 1966; Taflove 1995; Yang and Liou 1996a; Sun et al. 1999). The large size parameters prevent the application of these methods because of their tremendous demand on computer CPU time and memory. Note that comprehensive reviews of various rigorous and approximate methods for light-scattering computations can be found in Mishchenko et al. (2000) and Kahnert (2003). When size parameters are larger than approximately 80, the conventional geometric optics method, also called the ray-tracing method, can be applied to the computation of the single-scattering properties of dielectric particles (Macke et al. 1995; Mishchenko and Macke 1999; Wielaard et al. 1997).

For our calculations, we use an improved geometric optics method (IGOM) developed by Yang and Liou (1996b) to compute the single-scattering properties of hollow bullet rosettes. Unlike the conventional ray-tracing technique, IGOM does not separate the extinction of the incident radiation into equal contributions from diffraction and reflected–refracted Fresnelian rays. In IGOM, the ray-tracing calculation is carried out to compute the near field on the particle surface, which is subsequently mapped to the far field using a rigorous electromagnetic relationship that relates the near field to the far field. The concept of mapping the near field to the corresponding far field was first reported by Muinonen (1989). A previous study compared the IGOM results with the FDTD solutions for light scattering by small ice particles (Yang and Liou 1996b). For computational efficiency, we employ the simplified IGOM algorithm (Yang and Liou 1996b) that remaps the phase matrix computed from the conventional ray-tracing technique by taking into account the ray spreading effect. The technical details of the simplified IGOM are tedious and are reported by Yang and Liou (1996b). Most recently, Yang et al. (2007) showed that the simplified IGOM offers a reasonable approximation of the phase matrix of randomly oriented spheroids with size parameters larger than 30–40, as compared with that computed from the T-matrix method (Mishchenko and Travis 1994). As the size parameters involved in this study are on the order of 1000, the simplified IGOM is applicable. For these large size parameters, the solutions from the simplified IGOM are essentially the same as those from the conventional ray-tracing technique; the scattering peaks corresponding to halos are especially pronounced.

*A*whose position vector is denoted by

**r**

_{A}(not shown in the diagram). A unit vector

**n̂**denotes the outward normal direction for a particle face. The position vector of the apex of the pyramidal tip of the hollow structure is denoted by

**r**

_{0}. To determine the next impinging point of the refracted ray, it is necessary to determine whether the ray intercepts with the pyramidal hollow end. A necessary condition for the interception of the ray by a face of the pyramidal hollow section at the end of the bullet is as follows:If the condition in Eq. (6) is satisfied, an impinging point denoted as point

*B*or by a position vector

**r**

_{B}(not shown in the diagram) on this pyramidal face can be determined as follows:After point

*B*is determined, a new unit vector

**ê**′ can be defined as follows:where

**r**

_{A′}denotes the position vector of the center (point

*A*′ in the diagram) of the hollow end. Among all the faces that satisfy

**ê**′ ·

**n̂**< 0 there is one face that has the shortest distance from point

*A*′. Let the intercepting point on this face be

*B*′, as shown in Fig. 6. If points

*B*and

*B*′ are on the same pyramidal face such as in the case shown in Fig. 6a, then the ray impinges on the hollow end at point

*B*. Unlike the case illustrated in Fig. 6a, point

*B*′ and

*B*in Fig. 6b are not coincident, and thus the ray does not intercept the hollow end. In this case, we can find the impinging point by considering those faces other than the hollow faces with

**ê**·

**n̂**> 0 and search for the one that has the shortest distance from point

*A*as determined by Eqs. (7a) and (7b). Note that Eqs. (6)– (8) do not depend on a specific coordinate system. Thus, this algorithm can be easily implemented in either a laboratory coordinate system or the particle coordinate system. Furthermore, we use the approach reported in Zhang et al. (2004) to determine whether an outgoing ray is blocked by a bullet branch. If the ray is not blocked, the ray will be considered as a scattered ray; otherwise, the reflection–refraction event is considered with respect to that bullet branch.

## 4. Results and discussion

Figure 7 shows the comparison of the phase matrices of 6-branched solid and hollow bullet rosettes for an incident wavelength of 0.532 *μ*m. This wavelength could be considered as representative of a lidar measurement such as from the Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) on the Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO). CALIOP takes measurements at 0.532 and 1.064 *μ*m (Winker et al. 2007).

*β*in Fig. 7 is defined as follows:Thus,

*β*= 0 indicates a solid bullet rosette whereas

*β*= 1 represents the maximum hollow structure depth. It is evident from Fig. 7 that the 22° halo peak observed in the phase function is substantially reduced for hollow bullet rosettes because the hollow structures alter the paths of a substantial number of rays associated with the 22° minimum deviation angle responsible for this scattering peak.

The most significant effect of the hollow structures in the bullet branches is the reduction of the phase function values at scattering angles near the backscattering direction. Specifically, the backscattered intensity associated with hollow bullet rosettes is approximately 10 times smaller than its counterpart in the solid case. This may have an implication for active remote sensing of ice clouds.

Another effect of including hollow structures at the ends of the bullets is to increase the magnitude of the phase function at scattering angles between ∼1° and 20°. The behavior of the other phase matrix elements is generally similar for solid and hollow bullet rosettes. However, at scattering angles larger than approximately 50°, the *P*_{22}/*P*_{11} and *P*_{44}/*P*_{11} values are quite different for hollow and solid bullet rosettes. For −*P*_{12}/*P*_{11}, often referred to as the degree of linear polarization, the differences between the solutions for the hollow and solid bullet rosettes are observed primarily in a region of approximately 40°–120° as well as in the 22° halo region.

*f*) is equivalent to the delta transmission discussed by Takano and Liou (1989) and Mishchenko and Macke (1998). The total scattering phase function that includes this effect can be expressed as follows:where

_{δ}*δ*is the Dirac delta function, and

*P*

_{11}(

*θ*) is the continuous component of the phase function shown in Fig. 7. The values of

*f*

_{δ}are 0.1222 and 0.003 37 for the solid and hollow bullet rosettes, respectively. The significant reduction of

*f*

_{δ}for the hollow bullet rosettes occurs because the hollow structures alter the ray paths for many rays that would otherwise transmit directly through two parallel faces if the bullets were solid.

The hollow structures in the bullets increase the scattering phase function values within the forward scattering angle region from 1° to 20°, a conclusion similar to that of Schmitt et al. (2006), but decrease values at side- and backscattering angles of 60°–180°. As a result, the presence of the hollow structures tends to increase the asymmetry factor. The values of the asymmetry factor associated with hollow and solid bullet rosettes whose phase functions are shown in Fig. 7 are 0.865 and 0.829, respectively.

Figures 8a,b show the asymmetry factor values for various aspect ratios and hollow depths for bullet rosettes with *L* = 50 and 300 *μ*m. For a given *β* and *L*, the asymmetry factor decreases with increasing aspect ratio in a range of 2*a*/*L* = 0.1 − 1. This is consistent with a phenomenon noticed by Grenfell et al. (2005) and Fu (2007), who found a minimum in the asymmetry factor when 2*a*/*L* ≈ 1 (i.e., compact ice particles) for the case of hexagonal pristine ice particles. The results in Figs. 8a,b for hollow bullet rosettes demonstrate that the asymmetry factor increases with an increase of the hollow structure depth, except in the case of 2*a*/*L* = 0.1 and *β* > 0.8. Schmitt et al. (2006) also showed that the asymmetry factor is quite sensitive to the presence of the hollow structures on the bullets as well as the aspect ratios of bullet branches. Specifically, Fig. 4 in Schmitt et al. (2006) shows *g* increasing with hollowness for 2*a*/*L* = 0.5 − 1, but not for longer and thinner bullets.

Another outstanding effect of the hollow bullets is to increase the single-scattering albedo. Figures 8c,d show the single-scattering albedos of hollow rosettes with various aspect ratios as a function of *β* at a near-infrared wavelength of 2.11 *μ*m for particle sizes of 50 and 300 *μ*m, respectively. The single-scattering albedo generally increases with *β*; that is, the absorption of a rosette decreases with the increase of the hollow structure depth. This occurs because the hollow structures at the ends of bullet branches reduce the mass-to-surface-area ratio of the particle, which in turn reduce the mean absorption pathlength for the rays transmitted through the particle.

A commonly used approach to retrieve cloud microphysical and optical properties (specifically, optical thickness and effective particle size) from reflected sunlight during daytime is a bispectral method (Nakajima and King 1990; Platnick et al. 2003) that is based on radiometric measurements at a nonabsorbing band (e.g., 0.66- or 0.86-*μ*m band) and a slightly absorbing band (e.g., 2.11- or 1.64-*μ*m band). The absorbing band provides the information content for retrieving the effective particle size. For this reason, we also investigate the optical properties of hollow bullet rosettes at a near-infrared wavelength. Figure 9 shows the phase matrix computed for an incident wavelength of 2.11 *μ*m. The particle geometry for Fig. 9 is the same as that for Fig. 7. The effect of the hollow bullet structures on the phase matrix for the near-infrared wavelength is similar, but not as pronounced, as that for a visible wavelength.

In Figs. 7, 8 and 9, the angles between the bullet branches of a 6-branched bullet rosette ice particle are assumed to be 90°. Observations show that the frequency distribution of angles between two bullet branches peaks primarily at 70°, although 90° has also been observed (Kobayashi et al. 1976; Takano and Liou 1995; Iaquinta et al. 1995, and references cited therein). Figure 10 shows the phase functions of 4-branched hollow bullet rosettes with *β* = 0.5. The angle between any two branches is specified as 70° (or its supplement angle, 110°) or 90°. Evidently, a choice of 70° or 90° has little effect on the phase matrix.

Figure 11 shows the phase matrices of 1-, 6-, and 12-branched hollow bullet rosettes at a wavelength of 0.532 *μ*m. The hollow structure depth is *β* = 1. The differences are small in the optical properties of these three particles. Iaquinta et al. (1995) previously noted that the scattering properties of bullet rosettes depend only weakly on the number of bullet branches. At scattering angles larger than 90°, the phase function values tend to increase slightly with an increase in the number of bullets. The effect of the number of bullet branches on the other phase matrix elements is also small except for *P*_{22}/*P*_{11} at scattering angles larger than 150°. A scattering peak was observed at approximately 55° in the phase function of 6-branched bullet rosettes as computed by Schmitt et al. (2006; see the lower panel in their Fig. 3). However, this peak is not observed in the present study. This may be a result of the different bullet geometries assumed in these two studies and warrants further investigation.

Figure 12 shows the phase functions computed from the present model in comparison with those computed from the ray-tracing models reported by Iaquinta et al. (1995) and Schmitt et al. (2006). The model reported in Iaquinta et al. (1995) was designed for solid bullet rosettes and Schmitt et al. (2006) extended this model to hollow rosettes. In the case of solid bullet rosettes (i.e., *β* = 0), it is evident from Fig. 12 that our results generally agree well with the solution from the model developed by Iaquinta et al. (1995), except that our phase function values are lower at a scattering angle of 180° (i.e., backscattering). In the case of hollow bullet rosettes, there are significant differences between the present results and those from Schmitt et al. (2006), particularly for *β* = 0.5 and 1. Additionally, the phase functions computed from the present method have lower values for scattering angles larger than approximately 40°, and a scattering peak at ∼55° is not observed.

Figure 13 shows the phase functions for solid (*β* = 0) and hollow (*β* = 0.1, 0.5, and 1) bullet rosettes. The backscattered intensity decreases with the increase of the hollow structure depth. In the case of *β* = 1, the scattering maximum at 154° is not observed. The locations of the scattering peaks at scattering angles less than 22°, which are associated with rays that interact with the pyramidal tips of the bullet branches, are sensitive to the hollow structure depth. The scattering angle where the scattering peak is observed decreases with an increase of the hollow structure depth.

*D*and

*A*are the maximum dimension and projected area of an ice particle, respectively. Here

*n*(

*D*) specifies the number density of ice particles with the maximum dimension between

*D*and

*D*+

*dD*;

*D*

_{min}and

*D*

_{max}are the lower and upper boundary of the PSD; and

*Q*

_{e}, ϖ,

*g*, and

*P*are the extinction efficiency, single-scattering albedo, asymmetry factor, and scattering phase matrix, respectively. The symbol 〈〉 denotes that the relevant scattering property is averaged for a population of ice particles. The effective particle size in this study is proportional to the volume-to-projected-area ratio as follows (Foot 1988; King et al. 2004):where

*V*is the volume of an ice particle. In this study the volume of a hollow rosette is defined to exclude that associated with the hollow structures. Note that for a monodispersion of solid spheres,

*D*is just the diameter of the spheres. The PSDs used in this study are obtained from several major field campaigns of in situ ice cloud observations over the last few decades. The same datasets were also used in the development of ice cloud optical models for the Moderate Resolution Imaging Spectroradiometer (MODIS) operational (collection 5) cloud retrievals. Detailed discussions of the PSD data are provided in Baum et al. (2005a) and the references cited therein.

_{e}Figure 14 shows the bulk scattering asymmetry factor for a size distribution of hollow bullet rosettes as a function of *D _{e}* at a wavelength of 0.532

*μ*m. In the calculation, 6 or 11 branches are assumed for bullet rosettes, and the aspect ratio relations reported by Mitchell and Arnott (1994) and Um and McFarquhar (2007) are used. Similar to the case of individual hollow bullet rosette particles, the inclusion of hollow structures substantially increases the asymmetry factor of a population of hollow bullet rosettes, particularly for rosettes with 11 branches. We note that the phase functions shown earlier in Fig. 11 are not sensitive to the number of bullet branches, except in the forward direction where the magnitude of the diffraction contribution is proportional to the particle projected area. In Fig. 11, the phase function is expressed as a function of particle maximum dimension. However, the results shown in Fig. 14 are expressed as functions of the effective particle size. In the calculation involved for Fig. 11, the aspect ratio is specified by Eq. (3) or Eq. (4) for a given maximum dimension. The weighting of the particle in the computation of the effective particle size is different for 11- and 6-branched bullet rosettes. Thus, the asymmetry factor depends both on the number of bullet branches, which is illustrated in Fig. 14, and the effective particle size. The asymmetry factor values calculated on the basis of the aspect ratio reported by Mitchell and Arnott (1994) monotonically increases with

*D*in both the hollow and solid bullet rosette cases. However, if the aspect ratio relationship of Um and McFarquhar (2007) is used, hollow bullet rosettes do not show this monotonic variation for small

_{e}*D*(10–50

_{e}*μ*m). The results shown in Fig. 14 suggest that the asymmetry factor values based on these two aspect ratio relationships are quite different, particularly for solid bullet rosettes with

*D*< 120

_{e}*μ*m.

*δ*) as a function of effective particle size for solid (

*β*= 0) and hollow (

*β*= 1) bullet rosettes at a wavelength of 0.532

*μ*m. The backscattering depolarization ratio is defined as follows:Inspection of Fig. 15 shows that

*δ*of hollow bullet rosettes is substantially smaller than that of solid bullet rosettes. Moreover, the variation of

*δ*as a function of the effective particle size for solid and hollow bullet rosettes is significantly different.

Figure 16 shows the bulk scattering phase functions of solid and hollow bullet rosettes at wavelengths of 0.86 and 2.11 *μ*m. As mentioned previously, the spectral bands centered at these two wavelengths can be used for inferring cloud effective particle size (*D*_{e}) and optical thickness (*τ*). The results in Fig. 16 indicate that the phase function values at scattering angles near 180° are much lower for hollow bullet rosettes than for solid bullet rosettes. For small effective particle sizes (e.g., *D _{e}* = 35

*μ*m), the 22° halo peak is not pronounced for hollow bullet rosettes, but the halo becomes better defined as particle size increases (e.g.,

*D*= 135

_{e}*μ*m). These results suggest that the phase function associated with a population of hollow bullet rosettes is more sensitive to the effective particle size than its solid counterpart at visible and near-infrared wavelengths.

To investigate the effects of hollow bullet rosettes on the retrieval of *D*_{e} and *τ*, two sets of lookup tables (LUT) based on the bulk scattering properties of hollow and solid bullet rosettes are computed from the Discrete Ordinates Radiative Transfer (DISORT) model developed by Stamnes et al. (1988). A Lambertian surface with an albedo of 0.03 is assumed for calculations over ocean. Bullet rosettes are assumed to have 11 branches with aspect ratios as specified by Eq. (3). Figure 17 shows the comparison of the two sets of LUTs for two sun-satellite configurations. The dashed maroon lines in Fig. 17 indicate the LUT for solid (*β* = 0.0) bullet rosettes and the solid blue lines indicate the LUT for hollow (*β* = 1.0) bullet rosettes. The top panel of Fig. 17 shows a case with a solar zenith angle of *θ*_{0} = 30° and a viewing zenith angle of *θ* = 0° (note that the view azimuth angle is not required for specifying the observed radiance in this case). For the bottom panel of Fig. 17, the solar zenith angle, the satellite-viewing zenith angle, and the relative azimuth angle between the sun and satellite are 26°, 40°, and 42°, respectively.

It is evident from the LUTs shown in Fig. 17 that retrievals based on hollow bullet rosettes will result in larger values of optical thickness and smaller values of effective particle size in comparison with the retrievals based on solid bullet rosettes. To illustrate this, consider a pair of radiances for the two bands centered at 0.86 and 2.11 *μ*m, which are denoted by the red dots in Fig. 17. For the red dot in the top panel, the inferred optical thickness and effective size is 7.8 and 34 *μ*m, respectively, if hollow bullet rosettes are assumed. However, the inferred optical thickness and effective size is 5.2 and 60 *μ*m, respectively, if solid bullet rosettes are assumed. For the red dot in the bottom panel, the retrieval values based on hollow and solid bullet rosettes are (*τ* = 7.8, *D*_{e} = 34 *μ*m) and (*τ* = 6.2, *D*_{e} = 52 *μ*m), respectively. From the example shown in Fig. 17, it is obvious that the optical effects resulting from the hollow structures in bullet rosettes can have an important implication for the remote sensing of ice cloud properties from measurements at visible and near-infrared wavelengths.

## 5. Summary

We compute the complete phase matrix of randomly oriented hollow bullet rosette ice particles using the improved geometric optics method previously developed by Yang and Liou (1996b). A hollow bullet rosette ice particle is assumed to have between 2 and 12 branches with a hollow structure at the end of each bullet; each hollow structure extends into the bullet and has a pyramidal tip. We also discuss the bulk optical properties of a population of hollow bullet rosette ice particles in comparison with their solid counterparts.

In comparison with solid bullet rosettes, the primary optical effect of including hollow structures in each bullet is to decrease the amount of backscattering in the phase function. The magnitude of the decrease of backscattering is sensitive to the depths of hollow ends. Conversely, the magnitude of the forward scattering in the phase function is higher for hollow bullet rosettes than for their solid counterparts. The asymmetry factor is larger for hollow bullet rosettes than for solid bullet rosettes. Simulations are performed at two wavelengths, 0.86 and 2.11 *μ*m, to investigate the differences between hollow and solid bullet rosettes and investigate the impact of the habit on the inference of effective particle size and optical thickness. With a bispectral technique based on these two wavelengths, we find that use of hollow bullet rosettes results in larger value of optical thickness and a smaller effective particle size in comparison with the case of solid bullet rosettes.

For active lidar measurements, the depolarization ratio is of great importance. The depolarization ratio for hollow bullet rosettes is much lower than its counterpart in the solid case. Moreover, the behavior of the depolarization ratio as a function of the effective particle size is significantly different for solid and hollow bullet rosettes.

In future work, we will extend this study to further examine the implications of the hollow bullet rosette habit to ice cloud properties inferred from infrared hyperspectral measurements as well as from other sensors.

Ping Yang’s research is supported by the National Science Foundation Physical Meteorology Program (ATM-0239605), and a research grant (NNL06AA23G) from the National Aeronautics and Space Administration (NASA). George Kattawar’s research is supported by the Office of Naval Research under Contracts N00014-02-1-0478 and N00014-06-1-0069.

## REFERENCES

Arnott, W. P., , Y. Dong, , J. Hallett, , and M. R. Poellot, 1994: Role of small ice crystals in radiative properties of cirrus: A case study, FIRE II, November 22, 1991.

,*J. Geophys. Res.***99****,**1371–1381.Baran, A. J., , P. Yang, , and S. Havemann, 2001: Calculation of the single-scattering properties of randomly oriented hexagonal ice columns: A comparison of the T-matrix and the finite-difference time-domain methods.

,*Appl. Opt.***40****,**4376–4386.Baum, B. A., , A. J. Heymsfield, , P. Yang, , and S. M. Bedka, 2005a: Bulk scattering properties for the remote sensing of ice clouds. I: Microphysical data and models.

,*J. Appl. Meteor.***44****,**1885–1895.Baum, B. A., , P. Yang, , A. J. Heymsfield, , S. Platnick, , M. D. King, , Y. X. Hu, , and S. M. Bedka, 2005b: Bulk scattering properties for the remote sensing of ice clouds. II: Narrowband models.

,*J. Appl. Meteor.***44****,**1896–1911.Besson, L., 1923: Concerning haloes of abnormal radii.

,*Mon. Wea. Rev.***51****,**254–255.Cai, Q., , and K. N. Liou, 1982: Polarized light scattering by hexagonal ice crystals: Theory.

,*Appl. Opt.***21****,**3569–3580.Chang, P. C., , J. G. Walker, , and K. I. Hopcraft, 2005: Ray tracing in absorbing media.

,*J. Quant. Spectrosc. Radiat. Transfer***96****,**327–341.C.-Labonnote, L., , G. Brogniez, , M. Doutriaux-Boucher, , J-C. Buriez, , J-F. Gayet, , and H. Chepfer, 2000: Modeling of light scattering in cirrus clouds with inhomogenous hexagonal monocrystals: Comparison with in-situ and ADEOS-POLDER measurements.

,*Geophys. Res. Lett.***27****,**113–116.C.-Labonnote, L., , G. Brogniez, , J-C. Buriez, , M. Doutriaux-Boucher, , J-F. Gayet, , and A. Macke, 2001: Polarized light scattering by inhomogeneous hexagonal monocrystals: Validation with ADEOS-POLDER measurements.

,*J. Geophys. Res.***106****,**12139–12153.Draine, B. T., , and P. J. Flatau, 1994: Discrete-dipole approximation for scattering calculations.

,*J. Opt. Soc. Amer. A***11****,**1491–1499.Foot, J. S., 1988: Some observations of the optical properties of clouds. II: Cirrus.

,*Quart. J. Roy. Meteor. Soc.***114****,**145–164.Fu, Q., 2007: Generalized parameterization of asymmetry factor of cirrus clouds for climate models.

,*J. Atmos. Sci.***64****,**4140–4150.Fu, Q., , P. Yang, , and W. B. Sun, 1998: An accurate parameterization of the infrared radiative properties of cirrus clouds for climate models.

,*J. Climate***11****,**2223–2237.Goldie, E. C. W., , G. T. Meaden, , and R. White, 1976: The concentric halo display of 14 April 1974.

,*Weather***31****,**304–311.Greenler, R., 1980:

*Rainbows, Halos, and Glories*. Cambridge University Press, 195 pp.Grenfell, T. C., , S. P. Neshyba, , and S. G. Warren, 2005: Representation of a nonspherical ice particle by a collection of independent spheres for scattering and absorption of radiation: 3. Hollow columns and plates.

,*J. Geophys. Res.***110****.**D17203, doi:10.1029/2005JD005811.Hess, M., , and M. Wiegner, 1994: COP: A data library of optical properties of hexagonal ice crystals.

,*Appl. Opt.***33****,**7740–7746.Hu, Y., and Coauthors, 2007: The depolarization–attenuated backscatter relation: CALIPSO lidar measurements vs. theory.

,*Opt. Express***15****,**5327–5332.Heymsfield, A. J., , S. Lewis, , A. Bansemer, , J. Iaquinta, , L. M. Miloshevich, , M. Kajikawa, , C. Twohy, , and M. R. Poellot, 2002: A general approach for deriving the properties of cirrus and stratiform ice cloud particles.

,*J. Atmos. Sci.***59****,**3–29.Iaquinta, J., , H. Isaka, , and P. Personne, 1995: Scattering phase function of bullet rosette ice crystals.

,*J. Atmos. Sci.***52****,**1401–1413.Jacobowitz, H., 1971: A method for computing the transfer of solar radiation through clouds of hexagonal ice crystals.

,*J. Quant. Spectrosc. Radiat. Transfer***11****,**691–695.Kahnert, F. M., 2003: Numerical methods in electromagnetic scattering theory.

,*J. Quant. Spectrosc. Radiat. Transfer***79–80****,**775–824.Key, J. R., , P. Yang, , B. A. Baum, , and S. L. Nasiri, 2002: Parameterization of shortwave ice cloud optical properties for various particle habits.

,*J. Geophys. Res.***107****.**4181, doi:10.1029/2001JD000742.King, M. D., , S. Platnick, , P. Yang, , G. T. Arnold, , M. A. Gray, , J. C. Riedi, , S. A. Ackerman, , and K. N. Liou, 2004: Remote sensing of liquid water and ice cloud optical thickness, and effective radius in the Arctic: Application of airborne multispectral MAS data.

,*J. Atmos. Oceanic Technol.***21****,**857–875.Kobayashi, T., , Y. Furukawa, , K. T. Takahashi, , and H. Uyeda, 1976: Cubic structure models at the junctions in polycrystalline snow crystals.

,*J. Cryst. Growth***35****,**262–267.Liou, K. N., 1986: Influence of cirrus clouds on weather and climate process: A global perspective.

,*Mon. Wea. Rev.***114****,**1167–1199.Lynch, D. K., , K. Sassen, , D. O’C. Starr, , and G. Stephens, 2002:

*Cirrus*. Oxford University Press, 480 pp.Macke, A., 1993: Scattering of light by polyhedral ice crystals.

,*Appl. Opt.***32****,**2780–2788.Macke, A., , M. I. Mishchenko, , K. Muinonen, , and B. E. Carlson, 1995: Scattering of light by large nonspherical particles: Ray-tracing approximation versus T-matrix method.

,*Opt. Lett.***20****,**1934–1936.Macke, A., , M. I. Mishchenko, , and B. Cairns, 1996a: The influence of inclusions on light scattering by large ice particles.

,*J. Geophys. Res.***101****,**23311–23316.Macke, A., , J. Mueller, , and E. Raschke, 1996b: Single scattering properties of atmospheric ice crystal.

,*J. Atmos. Sci.***53****,**2813–2825.McFarquhar, G. M., , P. Yang, , A. Macke, , and A. J. Baran, 2002: A new parameterization of single-scattering solar radiative properties for tropical anvils using observed ice crystal size and shape distributions.

,*J. Atmos. Sci.***59****,**2458–2478.Minnis, P., , P. W. Heck, , and D. F. Young, 1993a: Inference of cirrus cloud properties using satellite-observed visible and infrared radiances. Part II: Verification of theoretical cirrus radiative properties.

,*J. Atmos. Sci.***50****,**1305–1322.Minnis, P., , Y. Takano, , and K-N. Liou, 1993b: Inference of cirrus cloud properties using satellite-observed visible and infrared radiances. Part I: Parameterization of radiance fields.

,*J. Atmos. Sci.***50****,**1279–1304.Mishchenko, M. I., , and L. D. Travis, 1994: Light scattering by polydispersions of randomly oriented spheroids with sizes comparable to wavelengths of observations.

,*Appl. Opt.***33****,**7206–7225.Mishchenko, M. I., , and A. Macke, 1998: Incorporation of physical optics effects and computations of the Legendre expansion for ray-tracing phase functions involving delta-function transmission.

,*J. Geophys. Res.***103****,**1799–1805.Mishchenko, M. I., , and K. Sassen, 1998: Depolarization of lidar returns by small ice crystals: An application to contrails.

,*Geophys. Res. Lett.***25****,**309–312. doi:10.1029/97GL03764.Mishchenko, M. I., , and A. Macke, 1999: How big should hexagonal ice crystals be to produce halos?

,*Appl. Opt.***38****,**1626–1629.Mishchenko, M. I., , W. J. Wiscombe, , J. W. Hovenier, , and L. D. Travis, 2000: Overview of scattering by nonspherical particles.

*Light Scattering by Nonspherical Particles: Theory, Measurements, and Geophysical Applications,*M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Eds., Academic Press, 29–60.Mitchell, D. L., , and W. P. Arnott, 1994: A model predicting the evolution of ice particle size spectra and radiative properties of cirrus clouds. Part II: Dependence of absorption and extinction on ice crystal morphology.

,*J. Atmos. Sci.***51****,**817–832.Mitchell, D. L., , A. J. Baran, , W. P. Arnott, , and C. Schmitt, 2006: Testing and comparing the modified anomalous diffraction approximation.

,*J. Atmos. Sci.***63****,**2948–2962.Muinonen, K., 1989: Scattering of light by crystals: A modified Kirchhoff approximation.

,*Appl. Opt.***28****,**3044–3050.Nakajima, T., , and M. D. King, 1990: Determination of the optical thickness and effective particle radius of clouds from reflected solar radiation measurements. Part I: Theory.

,*J. Atmos. Sci.***47****,**1878–1893.Platnick, S., , M. D. King, , S. A. Ackerman, , W. P. Menzel, , B. A. Baum, , J. C. Riédi, , and R. A. Frey, 2003: The MODIS cloud products: Algorithms and examples from Terra.

,*IEEE Trans. Geosci. Remote Sens.***41****,**459–473.Rockwitz, K-D., 1989: Scattering properties of horizontally oriented ice crystal columns in cirrus clouds.

,*Appl. Opt.***28****,**4103–4110.Schmitt, C. G., , and A. J. Heymsfield, 2007: On the occurrence of hollow bullet rosette and column shaped ice crystals in midlatitude cirrus.

,*J. Atmos. Sci.***64****,**4515–4520.Schmitt, C. G., , J. Iaquinta, , and A. J. Heymsfield, 2006: The asymmetry parameter of cirrus clouds composed of hollow bullet rosette–shaped ice crystals from ray-tracing calculations.

,*J. Appl. Meteor. Climatol.***45****,**973–981.Stamnes, K., , S-C. Tsay, , W. Wiscombe, , and K. Jayaweera, 1988: Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media.

,*Appl. Opt.***27****,**2502–2509.Stephens, G. L., , S. C. Tsay, , P. W. Stackhouse, , and P. J. Flatau, 1990: The relevance of the microphysical and radiative properties of cirrus clouds to climate and climate feedback.

,*J. Atmos. Sci.***47****,**1742–1753.Sun, W., , Q. Fu, , and Z. Chen, 1999: Finite-difference time-domain solution of light scattering by dielectric particles with perfectly matched layer absorbing boundary conditions.

,*Appl. Opt.***38****,**3141–3151.Takano, Y., , and K. Jayaweera, 1985: Scattering phase matrix for hexagonal ice crystals computed from ray optics.

,*Appl. Opt.***24****,**3254–3263.Takano, Y., , and K. N. Liou, 1989: Solar radiative transfer in cirrus clouds. Part I: Single-scattering and optical properties of hexagonal ice crystals.

,*J. Atmos. Sci.***46****,**3–19.Takano, Y., , and K. N. Liou, 1995: Radiative transfer in cirrus clouds. Part III: Light scattering by irregular ice crystals.

,*J. Atmos. Sci.***52****,**818–837.Taflove, A., 1995:

*Computational Electrodynamics: The Finite-Difference Time-Domain Method*. Artech House, 599 pp.Tape, W., 1994:

*Atmospheric Halos*. Antarctic Research Series, Vol. 64, Amer. Geophys. Union, 143 pp.Tape, W., , and J. Moilanen, 2006:

*Atmospheric Halos and the Search for Angle X*. American Geophysical Union, 238 pp.Um, J., , and G. M. McFarquhar, 2007: Single-scattering properties of aggregates of bullet rosettes in cirrus.

,*J. Appl. Meteor. Climatol.***46****,**757–775.Walden, V. P., , S. G. Warren, , and E. Tuttle, 2003: Atmospheric ice crystals over the Antarctic Plateau in winter.

,*J. Appl. Meteor.***42****,**1391–1405.Wendisch, M., , P. Yang, , and P. Pilewskie, 2007: Effects of ice crystal habit on the thermal infrared radiative properties and forcing of cirrus clouds.

,*J. Geophys. Res.***112****.**D08201, doi:10.1029/2006JD007899.Wendling, P., , R. Wendling, , and H. K. Weickmann, 1979: Scattering of solar radiation by hexagonal ice crystals.

,*Appl. Opt.***18****,**2663–2671.Wielaard, D. J., , M. I. Mishchenko, , A. Macke, , and B. E. Carlson, 1997: Improved T-matrix computations for large, nonabsorbing and weakly absorbing nonspherical particles and comparison with geometrical-optics approximation.

,*Appl. Opt.***36****,**4305–4313.Winker, D. M., , W. H. Hunt, , and M. J. McGill, 2007: Initial performance assessment of CALIOP.

,*Geophys. Res. Lett.***34****.**L19803, doi:10.1029/2007GL030135.Yang, P., , and K. N. Liou, 1996a: Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space.

,*J. Opt. Soc. Amer. A.***13****,**2072–2085.Yang, P., , and K. N. Liou, 1996b: Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals.

,*Appl. Opt.***35****,**6568–6584.Yang, P., , and K. N. Liou, 1998: Single-scattering properties of complex ice crystals in terrestrial atmosphere.

,*Contrib. Atmos. Phys.***71****,**223–248.Yang, P., , and K. N. Liou, 2006: Light scattering and absorption by nonspherical ice crystals.

*Light Scattering Reviews: Single and Multiple Light Scattering,*A. Kokhanovsky, Ed., Springer-Praxis, 31–71.Yang, P., , B-C. Gao, , B. A. Baum, , W. Wiscombe, , M. I. Mishchenko, , D. M. Winker, , and S. L. Nasiri, 2001: Asymptotic solutions of optical properties of large particles with strong absorption.

,*Appl. Opt.***40****,**1532–1547.Yang, P., , H. L. Wei, , G. W. Kattawar, , Y. X. Hu, , D. M. Winker, , C. A. Hostetler, , and B. A. Baum, 2003: Sensitivity of the backscattering Mueller matrix to particle shape and thermodynamic phase.

,*Appl. Opt.***42****,**4389–4395.Yang, P., and Coauthors, 2007: Modeling of the scattering and radiative properties of nonspherical dust particles.

,*J. Aerosol Sci.***38****,**995–1014.Yee, S. K., 1966: Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media.

,*IEEE Trans. Antennas Propag.***14****,**302–307.Zhang, M. H., and Coauthors, 2005: Comparing clouds and their seasonal variations in 10 atmospheric general circulation models with satellite measurements.

,*J. Geophys. Res.***110****.**D15S02, doi:10.1029/2004JD005021.Zhang, Z., , P. Yang, , G. W. Kattawar, , S-C. Tsay, , B. A. Baum, , Y. Hu, , A. J. Heymsfield, , and J. Reichardt, 2004: Geometrical-optics solution to light scattering by droxtal ice crystals.

,*Appl. Opt.***43****,**2490–2499.