# A scattering phase function for ice cloud: Tests of applicability using aircraft and satellite multi-angle multi-wavelength radiance measurements of cirrus

Anthony Baran, P. Francis, Laurent C.-Labonnote, Marie Doutriaux-Boucher

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## To cite this version:

Anthony Baran, P. Francis, Laurent C.-Labonnote, Marie Doutriaux-Boucher. A scattering phase function for ice cloud: Tests of applicability using aircraft and satellite multi-angle multi-wavelength radiance measurements of cirrus. Quarterly Journal of the Royal Meteorological Society, 2001, 127 (577), pp.2395-2416. 10.1002/qj.49712757711. hal-00811757

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Q. J. R. Meteorol. Soc. (2001), **127**, pp. 2395-2416

# A scattering phase function for ice cloud: Tests of applicability using aircraft and satellite multi-angle multi-wavelength radiance measurements of cirrus

By A. J. BARAN¹*, P. N. FRANCIS¹, L.-C. LABONNOTE² and M. DOUTRIAUX-BOUCHER²

¹Met Office, UK

²Université des Sciences et Technologies de Lille, France

(Received 21 December 2000; revised 23 May 2001)

## SUMMARY

A parametrization of the Henyey-Greenstein scattering phase function is presented for application to aircraft and satellite remote sensing of cirrus-cloud bulk and microphysical properties at non-absorbing and absorbing wavelengths. The phase function is based initially around a non-absorbing laboratory-measured phase function between scattering angles of 30° and 180°, although at scattering angles greater than 95° the phase function is made constant with scattering angle. This 'analytic' phase function has an asymmetry parameter value of 0.80 at non-absorbing wavelengths. This analytic phase function is tested using aircraft-based transmission radiance measurements at the wavelengths of 0.55 µm, 0.87 µm, 1.6 µm and 3.7 µm between scattering angles of 10° and 120°, and satellite retrievals of cirrus-cloud spherical albedo at the wavelength of 0.67 µm between scattering angles of 60° and 180°. The analytic phase function reproduces both aircraft and satellite measurements well, and is shown to be a distinct improvement over single-crystal model phase functions representing both small and large crystals in terms of angular scattering pattern and retrieved optical thickness.

**KEYWORDS:** Cirrus phase function Remote sensing Scattering

## 1. INTRODUCTION

Cirrus clouds cover on average 20% of the earth's surface (Liou 1986; Hartmann et al. 1992). Due to their spatial and temporal coverage it is now well established that cirrus clouds play an important role in the earth-atmosphere radiation balance (Liou 1986; Ramanathan 1987; Mitchell et al. 1989; Liou and Takano 1994; Donner et al. 1997; Kristjánsson et al. 2000). However, the overall radiation balance of cirrus is still very uncertain due to an incomplete knowledge of their microphysical and radiative properties. This lack of understanding is due to cirrus being generally composed of non-spherical ice crystals (McFarquhar and Heymsfield 1998) which creates difficulties characterizing these clouds in terms of an effective crystal dimension (McFarquhar and Heymsfield 1998) or a consistent set of single-scattering properties (Francis et al. 1999). To emphasize the point, recent calculations of cirrus short-wave forcing show that if two different crystal habits are assumed in the model calculations then the difference in short-wave forcing between the habits can be significant (Kristjánsson et al. 2000). Correct calculation of cirrus radiative properties is, therefore, very important in terms of climate modelling.

The single-scattering properties that are required to calculate the radiative impact of cirrus are the single-scattering albedo (the ratio of scattered energy to the total amount of attenuated energy) which at non-absorbing wavelengths is unity, the scattering phase function, usually parametrized in terms of the asymmetry parameter, and the cloud extinction coefficient. In this paper we concern ourselves only with the phase function and asymmetry parameter at non-absorbing and absorbing wavelengths. The asymmetry parameter, *g*, is an important quantity since its value determines how much radiation is scattered in the forward and backward hemispheres. In climate models the influence of cirrus is strongly determined by the choice of *g* (Stephens et al. 1990; Kristjánsson et al. 2000). The most appropriate value of *g* to represent scattering from cirrus within a

* Corresponding author: Met Office, Meteorological Research Flight, Building Y46, DERA, Farnborough, Hampshire GU14 0LX, UK. e-mail: anthony.baran@metoffice.com

© Crown copyright, 2001.
climate model is currently an open question. Theoretical calculations of *g* for different single-crystal shapes tend to predict *g* values which are usually between 0.74 and 0.85 at non-absorbing wavelengths (Takano and Liou 1989, 1995; Macke *et al.* 1996; Yang and Liou 1998; Liou *et al.* 2000). If, however, the ice crystal has within it air inclusions then the asymmetry parameter can be as low as 0.55 (Macke and Mishchenko 1999) for maximum air density. Interpretation of airborne and satellite-based radiometric measurements using visible and infrared wavelengths obtained over midlatitude cirrus have suggested *g* values in the range 0.50 to 0.85 (Chepfer *et al.* 1999; Sauvage *et al.* 1999) and 0.70 to 0.80 (Stephens *et al.* 1990; Wielicki *et al.* 1990; Francis *et al.* 1994). Recently, Mitchell *et al.* (1998) argued against cirrus generally having a *g* value less than 0.77. Instead they find that *g* values as represented by hexagonal columns (*g* = 0.80–0.84) provide the best fit to their visible/near-infrared radiance data obtained over tropical anvil cirrus. Measurements of the asymmetry parameter are now possible using an instrument called the Cloud Integrating Nephelometer (CIN) which is described by Gerber *et al.* (2000), and which has been used to measure the asymmetry parameter of Arctic ice clouds at visible wavelengths. The measurements suggest that for an ice cloud consisting of complex bullet rosettes, *g* is about 0.74, but the average value of *g* obtained over all the ice clouds came out to be 0.82. Clearly, uncertainty as to which *g* value best represents cirrus remains. In this paper a representative value for the asymmetry parameter of cirrus is further explored.

In terms of remote sensing, application of the correct scattering phase function at non-absorbing and absorbing wavelengths is important when retrieving cirrus microphysical and bulk properties. Application of an incorrect scattering phase function can lead to significant errors in the retrieval of optical thickness and/or microphysical properties of the cloud (Mishchenko *et al.* 1996; Baran *et al.* 1999; Francis *et al.* 1999). Generally, observations of cirrus suggest a relatively smooth and featureless phase function (Foot 1988; Francis 1995; Posse and von Hoyningen-Huene 1995; Gayet *et al.* 1998; Crépel *et al.* 1997; Francis *et al.* 1999). A smooth and featureless phase function can be generated by representing a complex ice crystal by a randomized second-generation triadic Koch fractal, such a structure commonly being called the ‘polycrystal’, due to Macke *et al.* (1996). The asymmetry parameter of the polycrystal is 0.74 at non-absorbing wavelengths and remains constant with crystal maximum dimension (literally the maximum extent of the crystal) since its aspect ratio is invariant with size. Alternatively, an aggregate ice crystal can be represented as a collection of randomized hexagonal elements with rough surfaces (Yang and Liou 1998). This structure predicts an asymmetry parameter of 0.77 and also produces a featureless phase function. It is shown in Yang and Liou (1997) that hexagonal columns with size parameters (ratio of particle size to incident wavelength) less than 50 can also produce featureless phase functions. Moreover, Labonnote *et al.* (2000) have incorporated spherical air bubbles within hexagonal columns which also results in a featureless phase function having an asymmetry parameter of 0.76.

The polycrystal phase function is used routinely for retrieval of cirrus visible optical thickness as part of the International Satellite Cloud Climatology Project (ISCCP) (Rossow *et al.* 1996). The introduction of the polycrystal phase function into the ISSCP retrieval scheme has improved retrievals of cirrus optical thickness relative to previous retrievals which assumed a spherical geometry (Rossow and Schiffer 1999). Some studies have shown that, when only one or two particular scattering angles are sampled, the polycrystal phase function is consistent with reflectance measurements made at near-infrared wavelengths using aircraft and satellite-based radiometers (Francis *et al.* 1998; Baran *et al.* 1999). However, if more scattering angles are sampled the polycrystal
phase function does not generally fit transmission or reflectance measurements (Francis *et al.* 1999; Doutriaux-Boucher *et al.* 2000). Recent studies made by Francis *et al.* (1999, hereinafter referred to as F99) and Doutriaux-Boucher *et al.* (2000, hereinafter referred to as D-B2000) have noted errors in the polycrystal phase function when comparison is made between multi-angle radiative-transfer model calculations, which are based on the polycrystal, and multi-angle transmission/reflectance measurements. In the paper by F99, the angular distribution of scattered radiance was measured at the wavelength of 0.87 µm by flying an aircraft-mounted radiometer beneath cirrus at midlatitudes and sampling scattering angles between 20° and 125°. The results contained in F99 demonstrate that the polycrystal does not generally describe the multi-angle transmission measurements well, in particular the polycrystal fails to fit the aircraft measurements at scattering angles between about 50° and 125°. Moreover, they also found that the polycrystal underestimated optical thickness by approximately a factor of two when compared with independent retrievals of optical thickness at the wavelength of 11.0 µm. Reflectance measurements obtained from the Polarization and Directionality of the Earth's Reflectances (POLDER) instrument between scattering angles 60° and 180° could not be matched by the cirrus model if a polycrystal phase function were assumed in the retrieval of cirrus-cloud spherical albedo (D-B2000).

There have also been several laboratory measurements of phase functions. Perhaps the most widely used is that of Volkovitskiy *et al.* (1980, hereinafter referred to as VPP). This phase function was measured, at scattering angles greater than 10°, by 0.63 µm light scattered from a collection of non-spherical ice crystals having maximum dimensions of around 20–25 µm. The collection of non-spherical particles produces a relatively featureless phase function which has significantly lower side scattering than the polycrystal between the scattering angles of 50° and 120° (Francis *et al.* 1999). Since the VPP phase function does not include measured intensities less than 10° the asymmetry parameter can only be estimated. The strong forward peak at the scattering angle of 0° had to be estimated by Foot (1988) using Mie theory applied to the same measured size distribution function. Using this method Foot (1988) estimated the VPP asymmetry parameter to be 0.85. Other laboratory measurements of the angular scattering pattern from a variety of particle shapes with differing compositions also show relatively smooth and featureless phase functions (e.g. Gustafson and Kolokolova 1999).

Although laboratory measurements of phase functions have shown good agreement when applied to the real atmosphere, we clearly still need to be able to model the ice-crystal single-scattering properties, since laboratory measurements will typically only cover a small number of discrete wavelengths, and represent a relatively small number of crystal size distributions, whereas our requirement is to know the scattering properties of any size distribution at any required wavelength. Thus, it should be stressed that phase functions such as the VPP, valid for non-absorbing wavelengths in the solar spectrum, are of limited value unless models can be developed which reproduce these properties, and which can also be used for calculations at other wavelengths and for other size parameters. It seems clear that scattering from cirrus cannot be adequately represented by using single-crystal shapes such as the hexagonal column, hexagonal plate, polycrystal or bullet rosette (Baran *et al.* 1999; F99; D-B2000). An alternative to a single-crystal model is to use some mixture of crystal shapes such as hexagonal columns, hexagonal plates, bullet rosettes, polycrystals or aggregates. For each shape the single-scattering properties are calculated then integrated over some specified size distribution function to obtain the total scattering contribution (McFarquhar *et al.* 1999; Rolland *et al.* 1999). Another method, adopted by this paper, is to model the collection of habits by some simple set of analytic expressions which capture the essentials of the
angular scattering pattern, which is a relatively smooth and featureless phase function, enabling its rapid computation at non-absorbing and absorbing wavelengths. In the sections that follow, a simple analytic phase function, based on the Henyey-Greenstein model but with some modifications, is described and tested using aircraft and satellite data chiefly from F99 and D-B2000, respectively.

## 2. THE CIRRUS PHASE-FUNCTION PARAMETRIZATION

In section 1 it was argued that an appropriate scattering phase function for cirrus is one which is generally smooth, featureless, and represents angular scattering from a collection of non-spherical particles. A phase function which is straightforward to calculate, smooth and featureless at all scattering angles is the Henyey-Greenstein function, which for its generation depends only on the value of g. The Henyey-Greenstein phase function is given by the following equation:

$$P(\theta) = \frac{1 - g^2}{(1 + g^2 - 2g \cos \theta)^{3/2}} \quad (1)$$

where P(θ) is the scattering phase function and g is the asymmetry parameter.

Due to its simplicity, the Henyey-Greenstein phase function has been used previously in the remote sensing of cirrus-cloud properties by Smith *et al.* (1993), and Duda (1994) applied the double Henyey-Greenstein phase function (particles generally have forward and backward peaks in their phase functions, the double Henyey-Greenstein phase function is a simple representation of such phase functions) to study the radiative properties of water clouds. Recently in a paper by Miller *et al.* (2000) it is shown that absolute errors in the double Henyey-Greenstein phase function can reach about 40% when it is compared with a more representative phase function for an ice aggregate. If a simple parametrization of the cirrus-cloud phase function is to be sought then some modification of the Henyey-Greenstein phase function is required. In this paper we base such a modification on the VPP phase function between the scattering angles of 30° and 180°, since this phase function represents angular scattering from a collection of non-spherical ice particles and has shown good agreement with aircraft measurements in the past (e.g. Foot 1988; F99). However, at scattering angles greater than 95° the phase function is made constant with scattering angle. The reasons for doing this are based on the analysis of D-B2000, where the use of a similarly 'flat' phase function was shown to fit POLDER data well at these scattering angles. At scattering angles less than 30° it is shown in F99 that the VPP phase function underestimates aircraft-based radiometric measurements of the transmitted radiance at 0.87 µm. The aircraft measurements are used in this paper to modify the Henyey-Greenstein phase function at scattering angles between 20° and 30°.

The resulting set of parametrized equations that describe the modified Henyey-Greenstein phase function (hereinafter referred to as the 'analytic' phase function) are given by Eqs. (2)–(5). The phase function, P(θ), is calculated at scattering angles between 0° and 180°, and is normalized to 4π.

The equations are as follows. For g < 0.70:

$$P(\theta) = \begin{cases} \frac{1 - g^2}{(1 + g^2 - 2g \cos \theta)^{3/2}} \alpha \cos \theta & : \theta \le 54.8^\circ \\ \frac{1 - g^2}{(1 + g^2 - 1.5g \cos \theta \sin \theta)^{3/2}} & : \theta > 54.8^\circ \end{cases} \quad (2)$$
<table><caption>TABLE 1. VALUES OF THE COEFFICIENTS $\beta$, A, B, C, AND $\sigma$ FOR VARIOUS VALUES OF THE ASYMMETRY PARAMETER, g</caption><thead><tr><th>g</th><th>β</th><th>A</th><th>B</th><th>C</th><th>σ</th></tr></thead><tbody><tr><td>g &lt; 0.3</td><td>1.25</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>0.3 ≤ g &lt; 0.45</td><td>1.50</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>0.45 ≤ g &lt; 0.6</td><td>1.23</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>0.6 ≤ g &lt; 0.7</td><td>1.095</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>0.7 ≤ g &lt; 0.8</td><td>-</td><td>0.1481 × 10<sup>3</sup></td><td>0.2025 × 10<sup>3</sup></td><td>0.4949 × 10<sup>2</sup></td><td>0.68</td></tr><tr><td>0.8 ≤ g &lt; 0.9</td><td>-</td><td>0.2771 × 10<sup>3</sup></td><td>0.5102 × 10<sup>3</sup></td><td>0.2329 × 10<sup>3</sup></td><td>0.68</td></tr><tr><td>g ≥ 0.9</td><td>-</td><td>0.4219 × 10<sup>3</sup></td><td>0.8271 × 10<sup>3</sup></td><td>0.4063 × 10<sup>3</sup></td><td>0.71</td></tr></tbody></table>

and for $g \ge 0.70$:

$$P(\theta) = \begin{cases} \frac{1 - g^2}{(1 + g^2 - 2g \cos \theta)^{3/2}} \alpha \cos^{128} \theta & : \theta \le 3^\circ \\ \frac{1 - g^2}{\{1 + g^2 - 2g \cos(1.3\theta)\}^{1.2}} \cos \theta & : 3^\circ < \theta \le 30^\circ \\ \frac{1 - g^2}{\{1 + g^2 - 2g \cos(\Delta\theta)\}^\alpha} \cos \theta & : 30^\circ < \theta \le 54.8^\circ \\ \frac{1 - g^2}{(1 + g^2 - 1.5g \cos \theta \sin \theta)^{3/2}} & : 54.8^\circ < \theta \le 95^\circ \\ P(\theta = 95^\circ) & : \theta > 95^\circ \end{cases} \quad (3)$$

where $\Delta$ is given by:

$$\Delta = \left( \frac{1 - g}{4.6} \right) + g \quad (4)$$

and $\alpha$ is given by the following equations for the values of g shown,

$$\alpha = \begin{cases} \frac{\beta}{\sqrt{1-g}} & : g < 0.3 \\ \frac{1}{\sqrt{\beta g}} & : 0.3 \le g < 0.7 \\ \frac{\text{Norm}}{\sqrt{g}} & : g \ge 0.7. \end{cases} \quad (5)$$

In Eq. (5), Norm is a polynomial fit to the asymmetry parameter to ensure that $P(\theta)$ is correctly normalized to $4\pi$, and Norm = $A - Bg + Cg^2$. The values for each of the coefficients $\beta$, A, B, C and $\sigma$ for various ranges of g are given in Table 1. Equations (2)–(5) describe a phase function which is determined only by the asymmetry parameter. In section 1 it was argued that the phase function of cirrus clouds should be essentially featureless, with little variation in the phase function at backscattering angles. Such a phase function is produced by complex particles, which have aspect ratios close to unity, and as such their asymmetry parameters tend to remain constant with respect to crystal size at non-absorbing wavelengths. Given these essential properties, it is physically meaningful to model the phase function of cirrus clouds in terms of the asymmetry parameter, since cirrus clouds tend to be dominated by complex particles.
<figure><img src="image_3.png" /><figcaption>Figure 1. A comparison between the parametrized analytic phase function ($P(\theta)$) (full line) and the experimentally determined Volkovitskiy *et al.* (1980) phase function (dot-dashed line), as a function of scattering angle.</figcaption></figure>

Figure 1 shows a comparison between the analytic and the VPP phase functions at scattering angles between 0° and 180°. In the case of the analytic phase function a value of $g = 0.80$ was input into Eqs. (3)–(5). As shown by Fig. 1 the analytic phase function does follow the VPP for scattering angles between about 30° and 95°, and at scattering angles greater than 95° the analytic phase function is constant. At scattering angles between about 3° and 30° the VPP has generally less forward scattering than the analytic phase function.

A test of the set of Eqs. (2)–(5) is shown in Fig. 2 for two values of $g$ which are $g = 0.37$ and $g = 0.91$, these values of $g$ represent the extremes of small and large ice crystals at infrared wavelengths. These two values of $g$ were input into Eqs. (2)–(5) to obtain the scattering phase function, and were obtained from exact T-matrix calculations (Mishchenko 1991). The T-matrix calculations assumed a circular cylinder with the diameter-to-length ratio set to unity, and equal-surface-area sphere radii of 2.5 and 25 µm were used in the calculation of $g$. The incident wavelength was assumed to be 11 µm using a complex refractive index, $N$, where $N = 1.0 + 0.5j$ and $j$ is the imaginary part of the complex refractive index. This value of $N$ was chosen to minimize the diffraction oscillations that appear in the circular-cylinder phase function at scattering angles between about 10° and 30°. As shown in Fig. 2 the analytic phase function for both sets of $g$ values is well behaved and generally follows the functional form of the T-matrix calculations, though for $g = 0.37$ there is some divergence between the exact and approximate calculations at scattering angles greater than about 140°. For $g = 0.91$ the parametrization describes the exact calculation well between scattering angles of 50° and 180°. At scattering angles less than 50° divergence between the parametrization and exact calculation is due to diffraction oscillations present in the phase function of the circular cylinder which are not completely damped out for this value of $N$. In the
<figure><img src="image_4.png" /><figcaption>Figure 2. Comparison of exact T-matrix calculations of the phase function for a circular cylinder with the analytic phase function for two values of the asymmetry parameter assuming a complex refractive index $N = 1.0 + 0.5j$ at the wavelength of 11 µm. The dashed and dot-dashed lines are the T-matrix results for $g = 0.37$ and $g = 0.91$, respectively, and results for the analytic phase function are represented by the full line.</figcaption></figure>

next two sections the analytic phase function is tested against multi-wavelength multi-angular aircraft and satellite observations of cirrus.

### 3. A TEST AGAINST AIRCRAFT OBSERVATIONS

On 9 November 1995 the Met Office C-130 aircraft sampled an element of detached frontal cirrus off the north-eastern coast of England. This cirrus was reported to be homogeneous and stable. The details of this case are fully reported in F99, but a brief summary of the microphysical data is given here. The cloud microphysical measurements were made with Particle Measuring Systems (PMS) two-dimensional optical array probes. The visible-extinction optical thickness was estimated from these measurements as being in the range 0.8 to 1.5, with the *in situ* crystal effective diameter (see below) estimated as being between 47 and 71 µm. The measurement uncertainties and the impact of these uncertainties on the microphysical and bulk *in situ* information are also described fully in F99. The shape information revealed that aggregates were the most common type of ice crystal inhabiting the cloud. The main difference between this paper and F99 is that in F99 the VPP phase function is applicable only at non-absorbing wavelengths, whereas in this paper the analytic phase function is applied and tested not only at non-absorbing wavelengths but also at absorbing wavelengths. Also, the single-scattering properties calculated in this paper for the different particle shapes were computed using the improved geometric-optics method of Yang and Liou (1996). The improved geometric-optics method improves upon classical geometric optics by incorporating wave theory.

The aircraft flew a level run beneath the cirrus, with a corresponding scattering angle of 74.3°, and also in a banked orbit such that the transmitted solar radiance was measured between the scattering angles of approximately 20° and 125°. For this paper we
<table><caption>TABLE 2. SINGLE-ICE-CRYSTAL MODELS</caption><thead><tr><th>Single-ice-crystal model</th><th><i>D</i><sub>e</sub> (μm)</th><th><i>g</i></th></tr></thead><tbody><tr><td>Small hexagonal column</td><td>9.3</td><td>0.67</td></tr><tr><td>Small bullet rosette</td><td>4.0</td><td>0.73</td></tr><tr><td>Large bullet rosette</td><td>79</td><td>0.83</td></tr><tr><td>Small aggregate</td><td>5.3</td><td>0.76</td></tr><tr><td>Large aggregate</td><td>134.0</td><td>0.77</td></tr></tbody></table>

Single-ice-crystal models used in the aircraft retrievals at the wavelength of 0.87 µm with their values of effective diameter, $D_e$, and asymmetry parameter, $g$.

make use of the radiometric measurements obtained during the flight at the wavelengths of 0.87 µm, 1.6 µm, 3.7 µm and 11.0 µm. The orbit radiance measurements provide a good test of various crystal models. The single-crystal models tested are summarized in Table 2 with their associated $\lambda = 0.87$ µm $g$ values and effective diameters. In this paper the effective diameter, $D_e$, is defined as (e.g. Yang *et al.* 1997):

$$D_e = \frac{\text{IWC}}{\rho_{\text{ice}} A_c} \qquad (6)$$

where IWC is the ice water content, $\rho_{\text{ice}}$ is the density of bulk ice, and $A_c$ is the total cross-sectional area of ice crystals per unit volume. The midlatitude size distribution functions used in the integration of the single-scattering properties are taken from Fu *et al.* (1998). The single-scattering properties integrated over the midlatitude size distribution functions are for $D_e$ ranging from 5.3 to 134.0 µm. At the wavelengths of $\lambda = 0.87$, 1.6 and 3.7 µm the single-scattering properties, extinction coefficient, single-scattering albedo, and $g$ are based on the aggregate particle (Yang and Liou 1998), and were calculated using improved geometric optics. The $g$ parameters determined at the wavelengths of 1.6 µm and 3.7 µm are used to generate the analytic phase function by inputting them into Eqs. (2)–(5). At the wavelength of 0.87 µm the $g$ value used to generate the analytic phase function is 0.8. This value of $g$ at non-absorbing wavelengths is used since the analytic phase function agrees well with the VPP phase function at side-scattering angles as shown in Fig. 1.

For the level aircraft run below the cirrus in the 9 November 1995 case, the mean retrieved extinction optical thickness at $\lambda = 11.0$ µm was found to be $1.40 \pm 0.57$, the standard deviation representing the variability during the run. This 11.0 µm retrieval is based on the absorption approximation, the validity of which is discussed in F99. The retrieved absorption optical thickness, $\tau_{\text{abs}}$, at 11.0 µm can be related to the extinction optical thickness, $\tau_{\text{ext}}$, via the relation $\tau_{\text{abs}} = (1 - \omega_0)\tau_{\text{ext}}$. For the crystal effective diameters encountered during this flight, $\omega_0 \sim 1/2$, so that $\tau_{\text{ext}} = 2\tau_{\text{abs}}$ is a very good approximation. As mentioned above, from *in situ* data collected during a profile through the cirrus layer the visible-extinction optical thickness was estimated to lie between 0.8 and 1.5. These results are shown in Fig. 3, where the 11.0 µm retrieval is shown as the hatched area and the *in situ* range of extinction optical thickness is indicated by the bars on the extreme right-hand side of the diagram.

The mean retrieved extinction optical thickness at 0.87 µm assuming the aggregate model, shown as the dotted line in Fig. 3, was found to be $0.60 \pm 0.37$, more than a factor of two less than the infrared retrieval and generally smaller than the *in situ* range of values. However, the extinction optical thickness retrieval at 0.87 µm using the analytic phase function (shown as the full line in Fig. 3) was found to give a mean value of $1.39 \pm 0.89$, in much better agreement with the infrared retrieval and
<figure><img src="image_5.png" /><figcaption>Figure 3. Retrieved optical thickness for the level aircraft run below cirrus obtained on 9 November 1995 off the north-east coast of England. The retrieved optical thickness was obtained at the wavelength 0.87 µm assuming the analytic phase function shown as the full line, the aggregate is represented by the dotted line, and mixed habit by the dashed line. The hatched area shows the variation in retrieved optical thickness at 11.0 µm due to uncertainties in cloud emitting temperature.</figcaption></figure>

the *in situ* measurements. Also shown in Fig. 3 is the retrieval assuming a mixture of habits (shown as a dashed line), the habit mixture consists of the following particle shapes. For maximum dimensions less than 70 µm the mixture consists of 50% bullet rosettes + 25% hexagonal plates + 25% hollow columns, and for maximum dimensions greater than 70 µm the mixture consists of 30% aggregates + 30% bullet rosettes + 20% hollow columns + 20% hexagonal plates (Baum *et al.* 2000). The retrievals based on a mixture of shapes and the analytic phase function, which could be regarded as a representation of an ensemble of shapes, agree very well with each other and with the *in situ* measurements whereas the retrieval based on a single shape significantly underestimates the *in situ* measurements.

Orbital radiance measurements obtained on 9 November 1995 below the cirrus are shown in Fig. 4. Each set of results are offset by a factor of 10 to aid clarity. The retrieved extinction optical thicknesses for each of the shapes was obtained via a chi-squared minimization procedure: when minimization between model and observation has been achieved for all scattering angles simultaneously then the best-fit optical thickness is obtained. For each crystal model the best-fit optical thickness is shown on the extreme right-hand side of Fig. 4. The mean retrieved extinction optical thickness at 11.0 µm for the orbital run was found to be 0.75 ± 0.08.
<figure><img src="image_6.png" /><figcaption>Figure 4. Aircraft-measured multi-angle radiance measurements obtained from the aircraft orbit beneath the cirrus on 9 November 1995 off the north-east coast of England. The radiance measurements plotted as a function of scattering angle are shown as filled circles and the optimal best-fit phase functions for each of the crystal types are shown as full lines. For each crystal type the optimally derived optical thickness ($\tau$) obtained from the orbit is also shown. The crystal models are the small column, small bullet rosette (b/r), large b/r, small aggregate (aggr), large aggr, Volkovitskiy *et al.* (1980), and analytic.</figcaption></figure>
It is clear from Fig. 4 that single-crystal model phase functions do not correctly describe the angular transmitted radiances or retrieve an extinction optical thickness that is consistent with an independently derived extinction optical thickness at 11 µm. However, it is interesting to note that the single-crystal models that correspond to small crystal effective diameters do describe the angular transmitted radiances better than single-crystal models associated with larger effective diameters. For instance, the fit to the angular radiance data obtained using the small hexagonal column is similar to the VPP, and the small aggregate describes the angular data well. If the small bullet rosette is compared with the large bullet rosette, the large bullet rosette retains its primary halo feature at around scattering angles of 20° to almost 40°, with a secondary halo appearing at scattering angles around 42°. The small bullet rosette does not have such distinct halo features on its phase function so it appears the generally better fit to the angular radiance data. The small-crystal models have in common phase functions which are relatively featureless which is why these models describe this particular dataset well. If the only data available were the angular transmitted radiance measurements then one could conclude that the small-crystal models are an adequate representation of this particular cloud. However, the retrievals have a further constraint, namely the extinction optical thickness which was independently retrieved at 11 µm. The retrieved extinction optical thickness obtained using the small-single-crystal models tends to under predict the extinction optical thickness by a factor of between 1.4 to 1.8, and these small-crystal models have *g* values ranging from 0.67 to 0.76. The large bullet rosette and VPP *g* values are 0.83 and 0.85 respectively, and these models tend to over predict the extinction optical thickness by a factor of about 1.2. The analytic phase function also shown in Fig. 4 not only describes, better than other models, the angular transmitted radiance but also retrieves an optical thickness of 0.78 which is consistent with the best-fit infrared optical thickness. The *g* value associated with the analytic phase function is 0.80. Although these results do not represent an accurate or precise measure of *g*, since scattering angles less than 20° are not sampled, they do indicate that an asymmetry parameter of 0.80 does retrieve an extinction optical thickness which is consistent with an independently derived value. Figure 4 demonstrates the essential need for obtaining further angular transmission measurements at scattering angles much less than 20° in order to test model phase functions and their associated asymmetry parameters.

To further test the angular scattering of the analytic phase function, use is made of other airborne radiometer measurements of cirrus that were obtained on the 23 April 1992 and 8 July 1993, referred to as A189 and A263, respectively. These other cases are described in Francis (1995). The radiometer measurements of the angular transmitted radiance were obtained at 0.55 µm by flying the aircraft in an orbit within and below the cirrus, sampling the phase function between scattering angles of approximately 10° and 80°. For these comparisons, the model radiances are scaled to the radiometer measurements at a scattering angle of 60°, where they were assigned an arbitrary value of unity (the precise reasons for this are described in Francis (1995)).

Figure 5 shows the A189 case obtained on 23 April 1992, the scattering angles sampled during this flight ranging from about 30° to 80°. Also shown on the figure are model results for a small hexagonal column, large aggregate and small aggregate, the results for the analytic phase function being shown as the full line. Generally, between the scattering angles of 30° and 80° the single-crystal models diverge from the measurements. However, the small-effective-diameter single-crystal models are again in better agreement with the angular measurements. The analytic phase function generally produces the best fit to the angular measurements with some slight divergence from the measurements at scattering angles around 80°. Figure 6 shows the cirrus case A263 that
<figure><img src="image_7.png" /><figcaption>Figure 5. Aircraft-measured multi-angle radiance measurements obtained from the aircraft orbit beneath the cirrus on 23 April 1992 (case A189) at the wavelength of 0.55 µm. The radiance measurements are plotted as a function of scattering angle (shown as filled circles) and the optimal best-fit phase functions for each of the crystal types are shown in the upper right-hand side of the figure.</figcaption></figure>

occurred on the 8 July 1992. This time scattering angles were sampled between about 10° and 60°. The same crystal models as used in Fig. 5 are also employed in Fig. 6. Figure 6 shows that at scattering angles between about 15° and 60° the analytic phase function (shown as the full line) is again a generally better fit to the observations than the other crystal models, though at scattering angles around 10° it does underpredict transmission. Interestingly, the large aggregate at scattering angles around 22° has a pronounced halo feature, which does not agree with the observations at all. This is not surprising since the aggregate is based on a collection of eight hexagonal elements, and although the 22° halo feature is diminished it is not removed. In Yang and Liou (1998) it is shown that in order to remove the hexagonal column halo feature at 22°, the surfaces of each hexagonal element comprising the aggregate must be very rough.

Figures 4, 5 and 6 show results for the analytic phase function at the wavelengths of 0.87 and 0.55 µm. For the 9 November 1995 case other angular transmitted radiance measurements were also obtained at the wavelengths of 1.6 and 3.7 µm. In Fig. 4 at non-absorbing wavelengths the retrievals were constrained by the angular measurements and extinction optical thickness. At the absorbing wavelengths of 1.6 and 3.7 µm the retrievals, based on the phase function and single-scattering properties, can be further constrained not only by the angular measurements but also by the crystal effective diameter. Tests of the analytic phase function in terms of the angular transmission and retrieval of effective diameter are shown in Fig. 7. The single-scattering properties used in Fig. 7 are summarized in Table 3 and the corresponding phase functions are shown in Fig. 8. Firstly, the analytic phase function between the scattering angles of 20° and 120° at all three wavelengths is well behaved and reasonably describes the
<figure><img src="image_8.png" /><figcaption>Figure 6. Same as Fig. 5 but for case A263 which occurred on 8 July 1992.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 7. Aircraft multi-angle radiance measurements at the wavelengths of 0.87 µm, 1.6 µm, and 3.7 µm, obtained from the aircraft orbit beneath the cirrus on 9 November 1995 off the north-east coast of England. The radiance measurements are plotted as a function of scattering angle (shown as filled circles) and the optimal best-fit analytic phase function at each of the three wavelengths is shown as the full line, with the best-fit extinction optical thickness ($\tau_{0.87}$) and effective diameter ($D_e$).</figcaption></figure>
<table><caption>TABLE 3. THE BEST-FIT SINGLE-SCATTERING ALBEDO, $\omega_0$, AND ASYMMETRY PARAMETER, $g$</caption><thead><tr><th>Wavelength (μm)</th><th>ω<sub>0</sub></th><th>g</th></tr></thead><tbody><tr><td>0.87</td><td>0.9999</td><td>0.80</td></tr><tr><td>1.6</td><td>0.921</td><td>0.81</td></tr><tr><td>3.7</td><td>0.742</td><td>0.84</td></tr></tbody></table>

Representing an effective diameter of 48.9 µm, to the aircraft data at the wavelengths of 0.87, 1.6 and 3.7 µm.

<figure><img src="image_10.png" /><figcaption>Figure 8. The three analytic phase functions at the wavelengths of 0.87 µm, 1.6 µm, and 3.7 µm used in the retrieval described in Fig. 7.</figcaption></figure>

angular measurements at absorbing wavelengths. At the wavelength of 3.7 µm, between scattering angles of 60° and 100°, there is some difference between the analytic phase function and measured values, and at 1.6 µm the parametrization slightly underestimates the transmitted radiances between scattering angles of 20° and 40°. Also shown in Fig. 7 are the best-fit 0.87 µm extinction optical thickness and retrieved $D_e$, both of which are consistent with the *in situ* measurements. With this analytic phase function and single-scattering albedo (the single-scattering albedo being based on the aggregate), consistent retrievals were obtained between non-absorbing and absorbing wavelengths in terms of the angular scattering pattern and $D_e$. In the next section the analytic phase function is tested against multi-angular satellite observations at the wavelength of 0.67 µm.

## 4. A TEST AGAINST SATELLITE OBSERVATIONS

In section 3 the analytic phase function was rigorously tested against aircraft observations for three different cirrus cases. In this section, it is tested against satellite retrievals of cirrus-cloud spherical albedo which cover a large number of observations.
<figure><img src="image_11.png" /><figcaption>Figure 9. Normalized density of selected pixel directions as a function of scattering angle showing differences between the retrieved spherical albedo and the mean spherical albedo assuming the analytic phase function.</figcaption></figure>

The satellite observations utilized in this section are from the POLDER instrument which was on board the Japanese Advanced Earth Observing System satellite from August 1996 until June 1997. A description of the POLDER instrument and products derived from POLDER can be found in Buriez *et al.* (1997) and a description of the radiative-transfer model and calculations used in this section can be found in D-B2000. However, a brief summary of the basis of the POLDER products is given here. The POLDER products are based on ‘super pixels’, the ‘super pixels’ are composed of 9×9 single pixels, each of which correspond to an area of approximately 60×60 km². The POLDER instrument samples each of the pixels at up to 14 different viewing directions, with each direction corresponding to a particular scattering angle. The range of scattering angles sampled depends upon the latitude of the pixel, its position on the satellite track (i.e. east or west), and the solar zenith angle. The maximum range
<figure><img src="image_12.png" /><figcaption>Figure 10. Same as Fig. 9 but for the polycrystal phase function.</figcaption></figure>

of scattering angles sampled at midlatitudes and high latitudes is from 60° to 180°, and from 120° to 160° in the tropics. The cloud optical thickness is retrieved at the wavelength of 0.67 µm from measurements of the cloud bidirectional reflection (based on look-up tables calculated using a particular scattering phase function and assuming a plane-parallel cloud, see D-B2000 for details). Given the retrieved optical thickness, and assuming a black underlying surface, the corresponding cirrus-cloud spherical albedo is retrieved, since there is a one-to-one relationship between optical thickness and cloud spherical albedo. If in the construction of the look-up tables a perfect scattering phase function and cloud model were used, and given that the cirrus-cloud spherical albedo is the same at all scattering angles, then the difference between the cloud spherical albedo at a particular scattering angle and the mean cloud spherical albedo should be identically equal to zero. The assumption of a plane-parallel cloud should not affect the results presented in this paper since the measured POLDER reflectances are averaged over 9×9 pixels. The difference between the cloud spherical albedo at a particular scattering angle and the mean cloud spherical albedo provides a rigorous test of the scattering phase function, since if the difference deviates strongly from zero then the phase function used in the construction of the look-up tables cannot be correct. The range in scattering angles provided by the POLDER instrument enables a strong test of the analytic phase
<figure><img src="image_13.png" /><figcaption>Figure 11. Same as Fig. 10 but for the large bullet-rosette phase function.</figcaption></figure>

function, and further extends the largest scattering angle sampled from 120°, this being the greatest scattering angle extent in section 3, out to the maximum scattering angle of 180°. In this section we utilize POLDER data principally from midlatitude cirrus which occurred on 10 November 1996, giving the maximum range in scattering angle from 60° to 180°. The total number of observations made during that day is 9000. Cirrus cloud is identified from polarization measurements described in Parol *et al.* (1999), and reflectance data at 0.67 µm is only used if the cloud fraction is one and at least seven viewing angles are satisfied with a difference between maximum and minimum scattering angles of at least 50°. In this paper only cirrus cases which occur over the sea are considered. The phase-function models tested in this section are the analytic, polycrystal, large bullet rosette, and small hexagonal column. Optical properties of the large bullet rosette and small hexagonal column are summarized in Table 3 and the polycrystal was previously described in section 1.

If the above criteria are satisfied then the phase functions are tested as previously described. Figures 9–12 show the differences between the cirrus-cloud spherical albedo at a particular scattering angle and the mean cirrus-cloud spherical albedo (averaged over all scattering angles) as a function of scattering angle for each of the phase-function
<figure><img src="image_14.png" /><figcaption>Figure 12. Same as Fig. 11 but for the small hexagonal column.</figcaption></figure>

models. Figure 9 shows that the analytic phase function is a good fit to the POLDER data, being particularly good between the scattering angles 100° and 180°. However, at scattering angles less than 100° there is some departure between model and observation, though the maximum difference is at most only 5% at the scattering angle of 60°, with most differences between model and observation lying within ±2%. The total standard deviation of the residual spherical albedos at scattering angles between 60° and 180° for the analytic phase function is 0.0153, this compares with 0.0313 for the polycrystal model shown in Fig. 10.

Figure 10 shows a systematic bias in the cloud spherical-albedo difference for the polycrystal with scattering angle, with differences between model and observation reaching almost 15% at scattering angles around 80° and 160°. Although the polycrystal is generally featureless, it does not appear to have desirable scattering properties between the scattering angles of 60° to 100°, and 140° to 180°. Results for the large bullet rosette are shown in Fig. 11. At scattering angles between 140° and 160° there are significant differences between model and observation, which are due to the ‘ice bow’ feature appearing on the phase function of large bullet rosettes, and at scattering angles around 70° differences between model and observation are about 10%. The total standard deviation of the residual spherical albedos at scattering angles between 60° and
180° for the large bullet rosette is 0.0201, which is a better minimization relative to the polycrystal. The 'ice bow' feature shown in Fig. 11 is typical for large crystals such as bullet rosettes, hexagonal columns and hexagonal plates, however, small crystals would not exhibit such a feature and so should better fit the POLDER data. Figure 12 shows such an example of the effect of small crystals on the spherical-albedo differences. In that figure the phase function is based on small hexagonal columns. As expected for such a small crystal, differences between model and observation are minimized at scattering angles around the ice-bow feature. However, at scattering angles between 70° and 140° there are still differences between model and observation. The resulting total standard deviation of the residual spherical albedos at scattering angles between 60° and 180° for small hexagonal columns is 0.0168. The total standard deviation for small hexagonal columns is an improvement over the polycrystal and large bullet rosette, but the analytic phase function better minimizes differences between model and observation.

## 5. DISCUSSION AND CONCLUSIONS

In this paper a parametrization of the Henyey-Greenstein phase function, called the analytic phase function, has been presented. The phase function, initially modelled on an experimentally derived phase function based on a collection of non-absorbing non-spherical ice particles, requires as input the asymmetry parameter to calculate the angular scattering pattern. The phase function can be applied at non-absorbing and absorbing wavelengths and should find wide applicability in the remote sensing of cirrus cloud. At non-absorbing wavelengths the asymmetry parameter of the analytic phase function is 0.80.

The analytic phase function was tested using multi-angular observations of cirrus cloud from both aircraft and satellite radiometric measurements. The aircraft observations were used to test the phase function over the angular scattering range 10° to 125° at the wavelengths of 0.55 µm, 0.87 µm, 1.6 µm and 3.7 µm, and this phase function was found to be a better fit than other model phase functions in terms of both the angular scattering pattern and the mean retrieved extinction optical thickness at the wavelength of 0.87 µm. At the absorbing wavelengths of 1.6 µm and 3.7 µm the analytic phase function matched the angular transmitted radiances well (the single-scattering albedo being based on the aggregate particle) between scattering angles of 20° and 125° and retrieved an effective diameter that was consistent with *in situ* measurements. For a level run below the cirrus the analytic phase function agreed very well with a phase function based on a mixture of habits in terms of the retrieved extinction optical thickness, and both agreed reasonably well with *in situ* measurements of extinction optical thickness. However, phase functions based on single-crystal models did not agree well with either the angular measurements of transmitted radiance or with independent measurements of the extinction optical thickness. The analytic phase function was also tested over the angular scattering range 60° to 180° by taking differences between POLDER retrievals of spherical albedo at a particular scattering angle and the mean spherical albedo. The POLDER test of the analytic phase function shows that the maximum departure from a perfect model was only ±0.05, and the total standard deviation of the residual spherical albedos between the scattering angles of 60° and 180° was found to be 0.0153. The total standard deviation for other phase-function models such as the polycrystal was found to be 0.0313, 0.0201 for the large bullet rosette and 0.0168 for the small hexagonal column. The analytic phase function best minimized the POLDER observations over the range of scattering angles observed, which indicates that the analytic phase function does appear
to capture the true angular reflection characteristics of cirrus, at least between the scattering angles 60° and 180°. Presently, the analytic phase function has been predominantly applied to midlatitude and high-latitude cirrus cases. Future work will consist of testing the analytic phase function at tropical latitudes. Although the present paper considers the radiative properties only, a future paper will relate the analytic phase function to cirrus microphysical observations in terms of surface-area and ice-mass measurements.

The aircraft observations presented in this paper begin at a scattering angle of about 10°. It should be of high priority to obtain further measurements of transmitted radiance at scattering angles much less than 10° since in this region we can become more confident in testing whether the asymmetry parameter value of 0.80 is truly representative of cirrus.

## ACKNOWLEDGEMENTS

Anthony Baran is supported by the UK Department of the Environment, Transport and the Regions, under contract PECD 7/12/37. Thanks are also due to the POLDER science team at the University of Lille, the aircraft technicians and aircraft scientists at the Meteorological Research Flight, Farnborough, Qiang Fu for providing the midlatitude size distribution functions, and Ping Yang for the crystal-model phase functions.

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</table>