A self-consistent scattering model for cirrus. I: The solar region

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1 QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY Q. J. R. Meteorol. Soc. 133: (2007) Published online 26 October 2007 in Wiley InterScience ( A self-consistent scattering model for cirrus. I: The solar region Anthony. J. Baran a * and L.-C. Labonnote b a Met Office, Exeter, UK b Université des Sciences et Technologies de Lille, France ABSTRACT: In this paper a self-consistent scattering model for cirrus is presented. The model consists of an ensemble of ice crystals where the smallest ice crystal is represented by a single hexagonal ice column. As the overall ice crystal size increases, the ice crystals become progressively more complex by arbitrarily attaching other hexagonal elements until a chain-like ice crystal is formed, this representing the largest ice crystal in the ensemble. The ensemble consists of six ice crystal members whose aspect ratios (ratios of the major-to-minor axes of the circumscribed ellipse) are allowed to vary between unity and 1.84 for the smallest and largest ice crystal, respectively. The ensemble model s prediction of parameters fundamental to solar radiative transfer through cirrus such as ice water content and the volume extinction coefficient is tested using in situ based data obtained from the midlatitudes and Tropics. It is found that the ensemble model is able to generally predict the ice water content and extinction measurements within a factor of two. Moreover, the ensemble model s prediction of cirrus spherical albedo and polarized reflection are tested against a space-based instrument using one day of global measurements. The space-based instrument is able to sample the scattering phase function between the scattering angles of approximately 60 and 180, and a total of satellite pixels were used in the present analysis covering latitude bands between S and N. It is found that the ensemble model phase function is well able to minimize significantly differences between satellite-based measurements of spherical albedo and the ensemble model s prediction of spherical albedo. The satellite-based measurements of polarized reflection are found to be reasonably described by more simple members of the ensemble. The ensemble model presented in this paper should find wide applicability to the remote sensing of cirrus as well as more fundamental solar radiative transfer calculations through cirrus, and improved solar optical properties for climate and Numerical Weather Prediction models. Copyright 2007 Royal Meteorological Society KEY WORDS climate models; ensemble; ice crystal; phase function; polarisation; remote sensing Received 14 February 2007; Revised 31 July 2007; Accepted 29 August Introduction Cirrus is high-level cloud usually occurring at altitudes greater than about 5 km, which means that it is predominantly composed of non-spherical ice crystals of varying sizes and shapes (Baran, 2004). It covers 20 30% of the Earth s surface with 60 70% coverage in the Tropics and can occur during any season (Liou 1986; Wylie and Menzel, 1999; Stubenrauch et al., 2006). With such a temporal and spatial coverage it is unsurprising that cirrus is an important component of the earth atmosphere radiation balance (Liou and Takano, 1994; Donner et al., 1997; Kristjánsson et al., 2000; Baran, 2007; Edwards et al., 2007). Ice crystal shapes that occur in cirrus can range from simple pristine hexagonal ice columns, plates and rosettes to aggregates of these shapes which may be * Correspondence to: Anthony. J. Baran, Met Office, Corduan 2, FitzRoy Road, Exeter, Devon, EX1 3PB, UK. anthony.baran@metoffice.gov.uk The contribution of Anthony. J. Baran of Met Office, Exeter, was prepared as part of his official duties as an employee of the UK Government. It is published with the permission of the Controller of Her Majesty s Stationery Office and the Queen s Printer for Scotland. randomized. The randomization may be due to surface roughening, air aerosol inclusions and crystal distortions (Baran, 2007). The sizes of ice crystals that inhabit cirrus can range from less than 10 µm to 1000s of µm (Heymsfield and Miloshevich, 2003). With such a variation of ice crystal shape and size it is problematic to generally describe the bulk radiative and angular scattering properties of cirrus in terms of one idealised geometrical shape (Foot, 1988; Francis et al., 1999; Baran et al., 2001). This has tended to be the traditional approach where the scattering properties of cirrus have been separated from its bulk macrophysical properties by assuming just a single idealised geometrical shape. However, in order to describe cirrus solar radiative transfer fully it is necessary to seek models which not only accurately predict the scattering properties of cirrus but also its bulk macrophysical properties, thereby improving the representation of such cloud in climate and numerical weather prediction models. This is also important due to the range of space-based instrumentation that is now available to observe cirrus, which measures its radiative as well as hydrological contributions to the climate system (Stephens et al., 2002). Copyright 2007 Royal Meteorological Society

2 1900 A. J. BARAN AND L.-C. LABONNOTE In order to obtain a model of cirrus that can satisfy both macrophysical and radiative measurements it is now becoming increasingly common to use some ensemble of geometrical shapes that make up the size distribution function. McFarquhar et al. (1999) assumed an ensemble consisting of spheres, hexagonal ice columns, bulletrosettes and polycrystals. The polycrystal due to Macke et al. (1996a) is a randomization of a second generation triadic Koch fractal and is supposed to represent the more irregular shapes that are observed in cirrus. The aspect ratio of the polycrystal remains invariant with respect to size. McFarquhar et al. (1999) integrated the ice crystal shape distribution over the observed size distribution functions to obtain the bulk scattering properties to try to describe the reflectance properties of cirrus measured at four wavelengths between µm and µm obtained during the CEPEX (Central Equatorial Pacific Experiment) campaign. They found that the simulations of observed reflectances were highly sensitive to assumptions about which shapes to include in their shape distribution function, and a general description of cirrus reflectance would be difficult without prior knowledge of ice crystal shapes. In a more recent paper McFarquhar et al. (2002) changed their initial assumption regarding spheres to Chebyshev polynomials to try to better represent the smaller ice crystals in the particle size distribution function. Rolland et al. (2000) using the MODIS (Moderate-Resolution Imaging Spectroradiometer) simulator found that the assumed distribution of shapes in the particle size distribution function had the most impact on their retrieval of cirrus optical depth and ice crystal effective dimension. Moreover, Liou (2002) showed that polarization measurements are better described using an ice crystal ensemble rather than some single ice crystal shape. More recently, Baum et al. (2005) demonstrated that mixtures of shapes can better represent cirrus bulk properties such as the Ice Water Content (IWC) and median mass diameter (the size that divides the ice mass content contained in the particle size distribution function in half) than single shapes when compared against in situ measurements of IWC obtained in the midlatitudes and Tropics. The ensemble due to Baum et al. (2005) comprises droxtals (these shapes are supposed to represent the smaller ice crystals in the particle size distribution function, see Yang et al. (2003) for further details), hexagonal ice columns and plates, hollow columns, bullet-rosettes and ice aggregates (the ice aggregate consists of eight hexagonal elements arbitrarily attached, the aspect ratio of which remains invariant with respect to size, see Yang and Liou (1998)). This ensemble is supposed to represent some typical distribution of shapes that is observed in cirrus using the Cloud Particle Imager (CPI) probe due to Lawson et al. (1998). However, Westbrook et al. (2004) show that realistic ensembles of ice crystal shapes can be generated from some initial monomer crystal, which is allowed to aggregate through collisions until a distribution of aggregates is produced. It is also demonstrated by Westbrook et al. (2004) that the aspect ratio of the resulting ice aggregate tends to between 1.43 and 1.67 and is independent of initial assumptions about the shape of the initial monomer. This approach of allowing a more simple initial ice crystal to grow into a more complex spatial ice aggregate, which could better represent CPI data, was adopted by Baran (2007) in order to predict midlatitude cirrus IWC between the temperatures of 30 C and 60 C. The ensemble due to Baran (2007) consisted of a simple hexagonal ice column assuming an aspect ratio of unity to represent the smaller ice crystals in the particle size distribution function; the single element was then aggregated with respect to size by arbitrarily attaching other hexagonal ice elements, until a chain-like aggregate was produced. It was shown that the ice crystal ensemble was better able to describe the measured IWC than a single ice crystal model represented by the Yang and Liou (1998) ice aggregate model. For a self-consistent model of cirrus, not only is it necessary to obtain an ice crystal ensemble that provides a good description of the bulk properties such as the IWC but also of its angular and volume scattering properties. Since these quantities are of fundamental importance to the computation of solar radiative transfer in cirrus; their accurate prediction would enable a solution to this long-standing problem. In the paper by Mishchenko et al. (1996) it was shown that application of the correct phase function to the remote sensing of cirrus was very important when retrieving cirrus optical depth and ice crystal size, as application of an incorrect phase function could lead to significant errors in these retrieved quantities. It has been previously demonstrated that the best phase function to apply to the remote sensing of cirrus is one that is generally featureless and flat at backscattering angles (Foot, 1988; Baran et al., 1999; Francis et al., 1999; Knap et al., 1999; Doutriaux-Boucher et al., 2000; Baran et al., 2001; Labonnote et al., 2001; Baran et al., 2003; Field et al., 2003a; Jourdan et al., 2003; Ulanowski et al., 2003; Baran and Francis, 2004; Baran et al., 2005; Knap et al., 2005; Baran and Labonnote, 2006; Shcherbakov et al., 2006; Ulanowski et al., 2006). In order to theoretically generate featureless phase functions (i.e. removal of the 22 and 46 halos, the ice bow at scattering angles around 150 and the direct backscattering intensity peak at 180 ) the ice crystals can be randomized in such a way so as to remove their cylindrical symmetry that gives rise to optical features such as halos. The randomization of ice crystals in order to remove optical features may take a variety of forms such as distorting (described in section 3) or roughening the ice crystal mantles as described in Macke et al. (1996a) and Yang and Liou (1998), respectively. Adding air and/or aerosol inclusions within the ice crystal (Macke et al., 1996b; Yang and Liou, 1998; Labonnote et al., 2001; Shcherbakov et al., 2006) will also diminish or remove optical features. It is not currently known which randomization process or combination of processes dominates in the deformation of ice crystals, though CPI images such as those presented in Baran (2007) do show

3 SELF-CONSISTENT SCATTERING MODEL FOR CIRRUS. I: SOLAR REGION 1901 that ice crystals are deformed and some form of inclusions is present. In the paper by Baran and Labonnote (2006) it was shown that by using distortions of 0.4 applied to spatial models of ice crystals such as bulletrosettes or a chain-like aggregate, these were able to satisfy the POLDER (Polarization and Directionality of Earth s Reflectances) simultaneous measurements of total reflectance and polarized reflectance. The more compact ice aggregate model of Yang and Liou (1998) was not able to satisfy the measured polarized reflectance assuming a distortion parameter of Therefore, it seemed that more spatial ice crystals were better able to satisfy the POLDER measurements of polarized reflectance. In this paper a self-consistent scattering model for cirrus is presented which attempts not only to predict the bulk properties of cirrus such as IWC and the volume extinction coefficient, of fundamental importance to the computation of solar radiative transfer in cirrus, but also its general scattering and polarization properties. The ability of the model to predict the bulk cirrus properties is tested against in situ based measurements of IWC and extinction coefficient obtained in the midlatitudes and Tropics. The ability of the model to predict the angular scattering and polarization properties of cirrus is tested against one day of global POLDER data. The paper is divided into the following sections: section 2 defines the single-scattering properties used in this paper and describes the ray-tracing method applied to compute individual ice crystal scattering properties. Section 3 describes the ensemble ice crystal model and section 4 describes the basis of the midlatitude and tropical data used to test the model. Section 5 shows results of testing the model against the in situ based midlatitude and tropical data. Section 6 describes results of testing the model against POLDER measurements of total reflectance and polarized reflectance. In section 7 results are summarised and discussed. 2. The single-scattering properties and ray-tracing method Assuming incident sunlight upon an ensemble of randomly oriented non-spherical ice crystals, each possessing a plane of symmetry, the incident Stokes vector (I inc, Q inc,u inc,v inc ) is linearly related to the scattered Stokes vector (I sca,q sca,u sca,v sca )viaa4 4 scattering matrix, for each scattering angle θ, and is described by the following relation, van de Hulst (1957); I sca Q sca U sca V sca P 11 P = C sca P 21 P πr P 33 P P 43 P 44 I inc Q inc U inc V inc, (1) where in Equation (1) C sca is the scattering cross-section (scattering efficiency multiplied by the particle s geometric cross-section) of the ice crystals, and r is some distance between the scattering ice crystal and some observer. Due to the assumed symmetry properties of the ice crystals then only 6 elements of the scattering matrix given in Equation (1) are independent since P 21 = P 12 and P 43 = P 34, see van de Hulst (1957) for further information. Since incident sunlight has been assumed, which is unpolarized, then the first element of the scattering matrix, P 11, is proportional to the scattered intensity and this element is called the scattering phase function (Henyey and Greenstein, 1941). The ratio between P 12 and P 11 is called the degree of linear polarization (DLP). The phase function and DLP are very useful quantities for the remote sensing of cirrus since they depend on the shape and size of ice crystals (Baran, 2004). In this paper particular use is made of these two quantities when using the space-based instrumentation to test the ensemble model predictions of total reflectance and polarized reflectance. In order to calculate solar radiative transfer in cirrus, other single-scattering properties that are required are the scalar optical properties such as the volume extinction/scattering coefficient(s), β ext/sca, the single-scattering albedo, ω 0, (the ratio of the total scattered energy to that which has been totally attenuated), and the asymmetry parameter, g. The asymmetry parameter is a parametrization of the P 11 element over all space and takes on values between 1 and 1. If g = 1 then all incident light is scattered in the backward direction, if g = 0 then light is scattered equally in the forward and backward hemispheres, and if g = 1 then all light is scattered in the forward direction. The bulk extinction/scattering coefficient, β ext/sca, is given by the following relation; β ext/sca = Q ext/sca (q) <P(q) >n(q)dq, (2) where in Equation (2) Q ext/sca is the extinction/scattering efficiency factor (ratio between the extinction/scattering cross-section and the orientation-averaged projected area, <P (q)>) and n(q) is the particle size distribution function (PSD). Equation (2) is integrated over the vector q, which represents the various ice crystal shapes and sizes in the particle PSD. The single-scattering albedo, ω 0,isgivenby and the asymmetry parameter, g, by g = ω 0 = β sca β ext, (3) C sca (q)g(q)n(q)dq, (4) C sca n(q)dq where C sca (q) = Q sca (q) <P(q) >. The IWC is defined by IWC = ρ V(q)n(q)dq, (5)

4 1902 A. J. BARAN AND L.-C. LABONNOTE where in Equation (5) ρ is the bulk density of ice taken to be 0.92gcm 3 and V(q) is the geometric volume of the ice crystals. A useful parameter that characterizes light extinction in cloud (Wyser and Yang, 1998; Kokhanovsky, 2004) at solar wavelengths is the effective diameter, D e,defined as; D e = 3 V(q)n(q)dq. (6) 2 <P(q) >n(q)dq It is well known that Equation (6) best characterizes the solar radiative properties of cirrus over any ice crystal ensemble and PSD function (Foot, 1988; Francis et al., 1994; Fu, 1996; Yang et al., 1997; McFarquhar and Heymsfield, 1998; Wyser and Yang, 1998; Mitchell, 2002; Baran and Havemann, 2004). In this paper to calculate the terms in Equation (1) and Equations (2) (3) the Monte Carlo ray-tracing method due to Macke et al. (1996a) is used. The ray-tracing method is based on the work of Rockwitz (1989) but has been extended to any polyhedral particle shape (Macke, 1993), and the inclusion of polarization effects follows the work of Muinonen et al. (1989). Thus, Equation (1) can be fully solved using the method due to Macke et al. (1996a) for any polyhedral shape of complex refractive index, N. The method of ray-tracing may be used if the size of ice crystals is much larger than the incident wavelength, which means that the size parameter (in this paper defined as πd e /λ, whereλ is the incident wavelength) must be much greater than about 60 (Baran, 2004). Throughout this paper the geometric optics approximation is assumed since only solar wavelengths are considered and so the method of ray-tracing can be applied. In this paper the ice crystals are randomized by using the method of distortion described in Macke et al. (1996a). The crystals are distorted by allowing the normal to the mantle surface to statistically vary between 0 and 90, the tilt angle θ t, with respect to its original direction at each reflection/refraction event. The degree of distortion is therefore defined as θ t /90, which may take on values between zero and unity. However, the description of the distortion parameter originally provided by Macke et al. (1996a) has recently been revised by Shcherbakov et al. (2006). In the original code of Macke et al. (1996a) the tilt angle was described by a uniform probability density function. However, in the paper by Shcherbakov et al. (2006) they found that their measurements of the angular scattered intensities were best described by tilt angles whose random numbers were given by the Weibull statistics. The tilt angle probability density function (PDF) is described by the parameters δ and η; these two terms characterize the degree of distortion and kurtosis of the PDF, respectively. In this paper the revised ray-tracing code due to Shcherbakov et al. (2006) is utilized. There is increasing evidence that ice crystals that exist in cirrus have some form of air inclusions (Heymsfield and Miloshevich, 1995; Macke et al., 1996b; Labonnote et al., 2000; Bailey and Hallett, 2004; Schmitt et al., 2006) and the original ray-tracing code due to Macke et al. (1996a) allows for the possibility of spherical air bubble inclusions. In this paper randomization of the ice crystal ensemble is achieved through two processes. The first process involves distorting the ice crystals and the second process is represented by spherical air bubble inclusions. These two processes act simultaneously. In order to model the spherical air bubble inclusions the microphysical and optical properties assumed by Macke et al. (1996b) are used; see Table I of that paper. In Macke et al. (1996b) the mean free path length (MFPL) between two subsequent scattering events is defined. The MFPL is a variable parameter and in this paper it has been set to a particular value. The method of constraining the MFPL is described in section 3. The next section describes the ensemble ice crystal model. 3. The ensemble model As previously described in section 1, ice crystal shapes that exist in cirrus can take the form of simple pristine particles such as hexagonal ice columns, bulletrosettes and aggregates of these particles, which may form aggregate chains. The complexity of ice crystals tends to increase with increasing size, and they generally become more spatial, with the hexagonal components becoming more elongated (Heymsfield and Miloshevich, 2003). The ensemble ice crystal model, shown in Figure 1(a) (f), attempts to mimic this observed behaviour. The ensemble model consists of six members: the smallest ice crystals are represented by the more simple hexagonal ice columns (Figure 1(a)) and bulletrosettes (Figure 1(b)), whereas the largest members are represented by more complex ice crystals, formed by arbitrarily attaching other hexagonal elements, which ultimately form chain-like ice crystals (Figure 1(d) (f)). Such chains of ice crystals are observed in cirrus (Heymsfield and Miloshevich, 2003; Lawson et al., 2003; Connolly et al., 2005). Bullet-rosettes are included in the ensemble as these are commonly observed in midlatitude and arctic regions (Lawson et al., 2006; Schmitt et al., 2006). The assumed geometry of the bullet-rosette has been previously described in Macke et al. (1996a). The assumed maximum dimension of the bullet-rosette used Table I. The geometrical parameters describing the construction of the hexagonal column (HC) shown in Figure 1(a) and the maximum dimension of the bullet-rosette (BR), Figure 1(b). The geometry of the HC is described in terms of the semi-width a, length L, aspect ratio AR, and the member centre coordinates (x 0, y 0, z 0 ) expressed in relative units. Element a µm L µm AR x 0 y 0 z 0 HC BR

5 SELF-CONSISTENT SCATTERING MODEL FOR CIRRUS. I: SOLAR REGION 1903 (a) (b) (c) Smallest member Hexagonal Column Six-branched bullet rosette Three-branched (d) (e) (f) Five-branched Eight-branched Largest member Ten-branched Figure 1. The ensemble model showing (a) the smallest member of the ensemble represented by the hexagonal ice column, (b) the six-branched bullet-rosette, (c) three-branched ice crystal, (d) five-branched ice crystal, (e) eight-branched ice crystal, and (f) the largest member of the ensemble represented by the ten-branched ice crystal. This figure is available in colour online at in Figure 1(b) is given in Table I. The geometrical construction for the other five members of the ensemble is listed in Tables II V. In Tables I V the three-dimensional geometry of each hexagonal element is defined by its semi-width, a, and length, L; the aspect ratio, AR, of each hexagonal element is given by L/2a. The coordinate geometry for each member of the ensemble is taken to be the same as that Table II. Same as Table I but for the third member of the ensemble represented by the three-branched ice crystal, Figure 1(c). Element a µm L µm AR x 0 y 0 z Table III. Same as Table II but for the fourth member of the ensemble represented by the five-branched ice crystal, Figure 1(d). Element a µm L µm AR x 0 y 0 z Table IV. Same as Table III but for the fifth member of the ensemble represented by the eight-branched ice crystal, Figure 1(e). Element a µm L µm AR x 0 y 0 z described in Yang and Liou (1998) with the coordinate values for the particle centre given as x 0, y 0,andz 0 in Tables I V. The three coordinate transformations γ, θ, and φ follow the numerical values given in Yang and Liou (1998). In Table V the values of γ, θ,andφ for the last two elements are given by (29, 41, 60 ) and (19,23, 122 ), respectively. The spatial configurations and dimensions given in Tables I V are chosen so that when elements are attached together overlapping is avoided. The numerical values given in Tables I V can be appropriately scaled to give any ice crystal size. The maximum dimensions and orientation-averaged aspect ratio for each member of the ensemble shown in Figure 1 is given in Table VI. The maximum dimension, D m,in this paper is defined as literally the maximum dimension

6 1904 A. J. BARAN AND L.-C. LABONNOTE Table V. Same as Table IV but for the sixth member of the ensemble represented by the ten-branched ice crystal, Figure 1(f). Element a µm L µm AR x 0 y 0 z of the ice crystal, and the orientation-averaged aspect ratio, <AR>, of each member of the ensemble other than the first is defined as the ratio between the majorto-minor axes of the circumscribing ellipse. The ellipse is circumscribed about two of the most extreme points of the ice crystal for a number of differing orientations. As Table VI shows, the overall averaged aspect ratio of each of the elements tends from 1.0 to1.84 for the smallest and largest ice crystal, respectively. It is generally the case that geometric models previously considered for light scattering calculations such as the polycrystal and hexagonal ice aggregate have assumed that the aspect ratio remains invariant with respect to size. The final aspect ratio shown in Table VI is 1.84, which is within 10% of the upper range of asymptotic aspect ratios found in the literature. In the paper by Westbrook et al. (2004) they found that their model predicted asymptotic aspect ratios in the range for aggregated particles. Similar aspect ratios were obtained by Korolev and Isaac (2003) using CPI data and they also found values between about 1.43 and It was previously shown by Baran and Labonnote (2006) that δ = 0.40 best explained POLDER measurements of spherical albedo and polarized reflectance. Therefore, in this paper a distortion parameter of 0.4 is applied to each member of the ensemble shown in Figure 1. The MFPL is constrained by using two other phase functions, namely, the Inhomogeneous Hexagonal Monocrystal (IHM) due to Labonnote et al. (2000) and Table VI. The maximum dimension D m, and overall aspect ratio AR, for each member of the ensemble. Member D m µm <AR> the analytic phase function due to Baran et al. (2001). The IHM model consists of spherical air/aerosol inclusions within the hexagonal ice column. The analytic phase function is a linear piecewise extension of the Henyey Greenstein phase function and is entirely generated by the asymmetry parameter. The analytic phase function was originally based on laboratory measurements of the phase function from an ensemble collection of ice crystal described in Volkovitskiy et al. (1980). Both the IHM and analytic phase functions have been previously shown to be good representations of the angular total reflectance/transmittance measured from cirrus cloud (Labonnote et al., 2000; Baran et al., 2001). Therefore, these two phase functions provide a good constraint for the ensemble model shown in Figure 1, at least for the phase function. In order to constrain the phase function of the ensemble model the PSD function is taken from Fu (1996). A total number of 28 PSD functions were compiled by Fu (1996) and in this paper the PSD function giving D e = 65.0 µm is used to constrain the ensemble phase function. The size distribution function has 24 bin sizes ranging in maximum dimension between 3.0 µm and 3500 µm. The ensemble members are distributed equally throughout the PSD function. An incident wavelength of 0.87 µm was assumed with the complex refractive index for ice, N, taken from Warren (1984) and N = e 07i, where i is the imaginary index of refraction. Integration of the PSD function, using the Monte-Carlo ray-tracing method described in section 2, yielded values for P 11 (Equation 1), and g (Equation 4). The asymmetry parameter was calculated to have a value of and this value was used to generate the analytic phase function. Results of comparing the various phase functions generated assuming the IHM, analytic, the ensemble with distortions only, and the ensemble with spherical air inclusions as well as distortions are shown in Figure 2. The figure shows results of the ensemble model set with δ = 0.4, η = 0.85 for the ensemble phase function with distortions only, and δ = 0.4, η = 0.85, MFPL = µm for the ensemble phase function with spherical air bubble inclusions. The value of MFPL is important as it is necessary to choose a number which does not violate the ray-tracing and Monte-Carlo technique. In order to apply the Monte-Carlo ray-tracing technique, two conditions must be fulfilled (Mishchenko et al., 1995). Firstly, to ensure that the Snellius law and Fresnel formulas are valid then the distance between the internal spherical inclusions and crystal boundary must exceed a certain value. Secondly, the distance between the nearest neighbour internal spherical inclusions must be a few times their radius to ensure that the spherical air inclusions behave as independent scatterers. Since in this paper MFPL is set to µm then these conditions are fulfilled. The chosen values of δ, η and MFPL gave the best agreement with the IHM and analytic phase functions. As can be seen from Figure 2, the phase function representing the ensemble model with distortions and spherical air inclusions follows the IHM model very well and the analytic phase function at scattering angles

7 SELF-CONSISTENT SCATTERING MODEL FOR CIRRUS. I: SOLAR REGION 1905 greater than about 60. Note that the IHM model still shows evidence for the 22 and 46 halos. The phase function representing the ensemble model with distortions only overestimates the intensity at scattering angles between about 80 and 110, and relative to the analytic phase function between the scattering angles of about 115 and 155 there is a distinct decrease in the intensity. The action of spherical air bubble inclusions in the ensemble model is to further smooth the phase function so that it becomes essentially flat at scattering angles greater than about 120. In the rest of this paper the best value found for δ, η and MFPL is applied to each member of the ensemble shown in Figure 1(a) (f). In the next section the basis of the in situ data used to test the ensemble model predictions of bulk cirrus properties such as IWC and β ext is described. 4. The basis of the midlatitude and tropical data In this paper, to test the ensemble model prediction of IWC a parametrization of the PSD function described in Field et al. (2005) and Field et al. (2008) is used. The ensemble model prediction of β ext is tested using parametrizations described in Heymsfield et al. (2003), which link the inferred ice cloud extinction to IWC. The work of Field et al. (2005, 2008) and Heymsfield et al. (2003) is based on in situ data obtained in the midlatitudes and Tropics. In the paper by Field et al. (2005) they used many in situ measurements of ice crystal PSD functions obtained in ice stratiform cloud by the UK C-130 aircraft sampled at various locations around the British Isles between the temperatures of 0 Cand 60 C. They found that the ice crystal PSD functions can be described by a single underlying PSD from which the initial PSD can be generated from knowledge of two moments. They make use of the 2 nd moment, the IWC, and by using the in-cloud temperature, T c, power laws are obtained to link IWC to any moment. The parametrization due to Field et al. (2005) is independent of assumptions about ice crystal shape and it does not include the contribution of small ice crystals less than 100 µm due to the known uncertainties regarding the measurement of small ice particles (see Strapp et al., 2001; Field et al., 2003b). So, from given values of IWC and T c the PSD function associated with these macrophysical values can be accurately generated. In Field et al. (2008) the same analysis is performed as in Field et al. (2005) but using tropical data. The tropical data of particular interest to this paper was obtained from anvils during the CRYSTAL- FACE (Cirrus Regional Study of Tropical Anvils and Cirrus Layers Florida Area Cirrus Experiment) campaign during July 2002 using the Citation aircraft, and a total number of 9800 PSDs were sampled between the temperatures of 0 C and 60 C. Throughout the rest of this paper the PSD functions are generated from values of IWC and T c using the parametrization due to Field et al. (2005, 2008) and integrated to solve Equations (1) (6). In the paper by Heymsfield et al. (2003) they sampled a total of 13 and 6 midlatitude and tropical ice clouds in the temperature ranges 20 C to 65 C and0 C to 50 C, respectively, to investigate relationships between the visible optical depth and ice water path (product of IWC and cloud vertical geometric thickness). In this paper particular use is made of their derived relationship between the inferred cloud extinction, inf, and IWC obtained for the midlatitudes and Tropics. The term inf is not directly measured but estimated from their in situ measurements of the ice crystal total cross-sectional area of the particle population per unit volume in the limit of geometric optics. Since the limit of geometric optics has been assumed then the inferred ice cloud extinction is twice the particle geometric cross-sectional area (van de Hulst, 1957), so: Dmax inf = 2 N(D)A(D)dD, (7) D min Figure 2. The phase function (P11) plotted as a function of scattering angle for the analytic phase function (full line), IHM model (dashed-dotted line), and ensemble with distortions only (dashed line) and the ensemble with distortions plus spherical air bubble inclusions (dotted line). The ensemble model with distortions only is described by the following parameters: δ = 0.4 and η = The ensemble model with distortions and spherical air bubble inclusions is described by δ = 0.4, η = 0.85 and MFPL = 200 µm. The phase functions were computed assuming an incident wavelength of µmand N = e 07i. where N(D) is the measured PSD and A(D) is the cross-sectional area of particles of size D. They find that for the midlatitudes the best fit to their data is given by inf = 0.02(IWC) 0.89 and for the Tropics the best fit found is inf = 0.019(IWC) In the next section the parametrizations described above are used to test the ensemble model s prediction of IWC (Equation 5) and β ext (Equation 2). 5. Testing the ensemble model s prediction of IWC and extinction coefficient 5.1. The ensemble model prediction of IWC In this section the ensemble model prediction of IWC using Equation (5) is tested against the parametrization

8 1906 A. J. BARAN AND L.-C. LABONNOTE due to Field et al. (2005, 2008) for midlatitude and tropical cirrus. To test the ensemble model, two in-cloud temperatures, T c,of 30 Cand 60 C are assumed with IWC ranging from to 1.0 g m 3 for the midlatitudes and g m 3 for the Tropics. These ranges of T c and IWC were typical of the actual in situ measurements described in Field et al. (2005, 2008). For each T c and IWC a PSD function is generated; the ensemble volume is then integrated over the generated PSD function to predict IWC as described by Equation (5). The volume calculations take into account the assumed MFPL of the spherical air bubble inclusions following the description given in Macke et al. (1996b). Results of comparison for the midlatitudes between the ensemble model prediction of IWC and the true IWC (i.e. the assumed IWC to generate the PSDs) assuming a T c of 30 C and 60 C are shown in Figure 3. The figure shows that for T c = 30 C the ensemble model prediction of IWC is generally well within a factor 2. For IWC <0.01gm 3 the ensemble model underpredicts the IWC, which increases as IWC decreases, though does not exceed a factor For IWC >0.01gm 3 the ensemble model overpredicts the IWC with increasing IWC, though the predictions of IWC are generally within a factor 2. For T c = 60 C the ensemble model underpredicts the true IWC for all IWC, the underprediction ranging from a factor of 2.8 for IWC = 0.001gm 3 to a factor 1.89 when IWC = 1.0 gm 3. Results of comparison between the ensemble model prediction of IWC and the true IWC for the Tropics assuming T c values of 30 C and 60 C are shown in Figure 4. As can be seen from the figure, the behaviour of the ensemble model for these two values of T c is different to that shown in Figure 3 for the midlatitudes. For T c = 30 C the ensemble model underpredicts for IWC <0.01gm 3, which increases slightly as IWC decreases. At IWC = 0.005gm 3 the ensemble model underpredicts the IWC by a factor For values of IWC >0.01gm 3 the ensemble model increasingly overpredicts the true IWC. At most the ensemble model overpredicts the true IWC by a factor of 2.77 when IWC = 3.0 gm 3. However, for IWC values of 0.05 and 0.10 the ensemble model overpredictions are 1.44 and 1.76, respectively. For T c = 60 C the ensemble model prediction of IWC tends from an underprediction by a factor of 2.38 for IWC = 0.005gm 3 to an overprediction by afactorof1.56foriwc= 3.0 gm 3. For IWC values of 0.05 and 0.1 g m 3 the ensemble model underpredicts by factors of 1.22 and 1.01, respectively. It should also be noted here that the Field et al. (2005, 2008) measurements of the particle size distribution functions were made at the temperature of 60 C, which was at the limits of instrumental capability and the contribution of small ice crystals less than 100 µm in size were ignored The ensemble model prediction of β ext In this section the ensemble model prediction of β ext is tested against the inferred in situ β ext based on in-cloud measurements of ice crystal projected areas obtained in the midlatitudes and Tropics previously described in section 4. In order to be consistent with the estimation of inf as described in Heymsfield et al. (2003) it is assumed that the limit of geometric optics is valid, which has the consequence that the cloud extinction is just twice the ensemble model total projected area, so Equation (2) becomes: β ext = 2 <P(q) >n(q)dq. (8) The methodology used in section 5.1 to generate the PSDs is also used in this section by assuming values for IWC and T c. Results presented here assume T c = 60 C with IWC varying in the ranges to 1.0 g m 3 and to 3.0 g m 3 for the midlatitudes and Tropics, respectively. Figure 5 shows results of comparing the ensemble model predicted β ext and inf for the midlatitudes. It can be seen from the figure that the ensemble prediction of β ext underpredicts inf by varying Figure 3. The ensemble predicted IWC plotted against the true IWC for the midlatitudes assuming a cloud temperature of 30 C (diamonds) and 60 C (squares). Figure 4. Same as Figure 3 but for the Tropics.

9 SELF-CONSISTENT SCATTERING MODEL FOR CIRRUS. I: SOLAR REGION 1907 factors, the differences decreasing with larger inf,which corresponds to larger IWC. For the lowest IWC value of g m 3 (i.e. when inf 4.0e 05 m 1 ) the ensemble model underpredicts by a factor of 2.87; when IWC = 1.0 gm 3 (i.e. when inf 0.02 m 1 )the ensemble model underpredicts by a factor of only Results of comparing the ensemble model prediction of β ext and inf for the Tropics is shown in Figure 6. As in Figure 5 the ensemble model also underpredicts inf ; however, the differences are smaller than in Figure 5. For the smallest IWC value of g m 3 (i.e. when inf 1.5e 05 m 1 ) the ensemble model underpredicts by a factor of 2.10 and when IWC = 3.0 gm 3 (when inf 0.05 m 1 ) the ensemble model underpredicts by a factor of only 1.2. As shown by Figures 5 and 6 the ensemble model prediction of inf is reasonable since the problem with accurately determining inf using in situ measurements is due to the uncertainty surrounding the quantification of small ice crystals less than 100 µm in size as previously described and this uncertainty could be between a factor of 2 and 5 (Field et al., 2003b). It should also be noted here that a single ensemble model is well able to replicate results obtained from Figure 5. The ensemble predicted β ext plotted as a function of the inferred cloud extinction, inf, based on in situ measurements obtained in the midlatitudes. Figure 6. Same as Figure 5 but for the Tropics. Table VII. The range of scattering angle and latitudinal distribution of the POLDER super-pixels used in this paper. Where Latitude gives the latitudinal band, Scat range describes the angular scattering range sampled by POLDER in each of the latitude bands and Lat dist gives the percentage of super-pixels in each of the latitude bands. Latitude Scat range Lat dist % the midlatitudes and Tropics rather than needing two different ensemble models to represent each latitudinal band. Given a reasonable prediction of IWC and β ext, which are two very important quantities in the calculation of solar radiative transfer in cirrus, then the ensemble model should accurately predict solar reflection. This aspect is tested in the next section. 6. Testing the ensemble model s prediction of total reflection and polarized reflection In this paper the ensemble model s prediction of total solar reflection and polarized reflection is tested using one day of global POLDER-2 data. A description of the POLDER instrument can be found in Buriez et al. (1997). The POLDER-2 instrument was launched on 14 December 2002 onboard the satellite platform ADEOS-2 (Advanced Earth Observing Satellite) and was operational until October POLDER-2 has channels centred at 0.443, 0.490, 0.565, 0.670, 0.763, 0.765, and µm. The channels centred at 0.443, and µm are polarized. In this paper the polarized channels are utilized. The instrument can view the same scene at up to 14 viewing directions sampling scattering angles between 60 and 180. The range of scattering angles sampled by POLDER depends on the sun satellite geometry, latitude of the pixel, and the position of the pixel on the satellite track (i.e. east or west). The maximum range of scattering angle that can be sampled by POLDER in the mid to high latitudes is from 60 to 180, and about in the Tropics. The POLDER derived products are based on super-pixels composed of 3 3 single pixels, each of which corresponds to an area of km 2. Given that the POLDER products are averaged over an area of km 2 then the assumption of a plane-parallel cloud should not affect any of the results presented in this paper. The POLDER-2 data utilized to test the ensemble model were obtained on 25 June 2003, and comprise 14 overpasses. The total number of pixels over the ocean used in the analysis that follows is , which corresponds to scattering angles. The latitudinal distribution of the POLDER pixels and the scattering angle range used in this paper are shown in Table VII.

10 1908 A. J. BARAN AND L.-C. LABONNOTE The methodology used to test the ensemble model has been previously described in Labonnote et al. (2001) and the radiative-transfer model used in this section is described in Doutriaux-Boucher et al. (2000). However, a brief description of the methodology is given here. The cloud optical thickness is retrieved at the wavelength of µm from measurements of the cloud bi-directional reflectance. It should be noted here that in this paper the retrievals are only accepted if the following four conditions are met. The cloud fraction is unity in each pixel; at least seven viewing angles are satisfied with a difference between maximum and minimum scattering angles of at least 50 ; and only pixels over the ocean are considered. The retrieved optical thickness depends on look-up tables and the assumed scattering phase function (i.e. the P 11 element in Equation 1); see Doutriaux-Boucher et al. (2000) for further details. Given the retrieved optical thickness and assuming a black underlying surface, since there is a one-to-one relationship between optical thickness and cloud spherical albedo as shown in Doutriaux-Boucher et al. (2000), then the corresponding cirrus spherical albedo can be retrieved. If in the construction of the look-up tables a perfect phase function were assumed then differences between the cloud spherical albedo and directionally averaged spherical albedo should be identically equal to zero. In other words if the assumed phase function did introduce a dependence on scattering angle for the retrieved spherical albedo then, given that the spherical albedo is independent of direction, that phase function is unphysical. A description of the polarized reflection is given in Labonnote et al. (2001) but a very brief description is given here. The polarized radiance depends on the Q, U, and V parameters of the Stokes vector given in Equation (1), and the polarized radiance is normalized by the ratio of pi-to-incident solar flux density. Therefore, the polarized reflection is given by the ratio of the normalized polarized radiance-to-solar zenith angle. Given measurements of the total reflection and polarized reflection, then the ensemble model s prediction of each of the scattering matrix elements given in Equation (1) can be tested Tests against spherical albedo In this section the accuracy of the ensemble model s prediction of the P 11 element is tested using the POLDER data and methodology previously described. The phase functions are generated assuming the ensemble model previously described in section 3 for the case of distortion only (with δ = 0.4 andη = 0.85) and distortions with spherical air inclusions (with δ = 0.4, η = 0.85 and MFPL = 200 µm). The two ensemble models in this section assume the same value of D e as used in section 3, i.e. D e = 65.0 µm. Results of testing the distortion only ensemble model against the POLDER retrievals of spherical albedo are shown in Figure 7. The figure shows that this phase function does introduce some dependence on the scattering angle for the spherical albedo Spherical albedo difference june 25, 2003 Ensemble: Distortions only Scattering angle σ = Figure 7. The normalized point density of directions plotted against the scattering angle showing the differences between the mean spherical albedo and the directionally averaged spherical albedo. The data were obtained on 25 June 2003 and the mean residual spherical albedo, σ, was computed to be σ = and the standard deviation of the residual spherical albedos, σ, is Most notably for this phase function the largest deviations occur around the scattering angles of about 90, and 120 to 150. As was previously noted from Figure 2 these scattering angles correspond to the phase function overestimating and underestimating the other model phase functions, respectively. The results of comparing the POLDER retrieved spherical albedo and the ensemble model phase function with distortions and spherical air bubble inclusions is shown in Figure 8. The figure shows that this phase function minimizes the POLDER retrieved spherical albedo very well with no biases and σ = This value of σ is the smallest so far measured using POLDER. In the paper by Baran and Labonnote (2006) it is shown that other ice crystal model phase functions for the same data produce the following σ values: for the polycrystal, for the undistorted six-branched bullet-rosette, for the distorted six-branched bullet-rosette, for the distorted chain-like aggregate, for the distorted ice aggregate and for the IHM model. Figure 8 suggests that an ensemble model which includes both distortions and spherical air bubble inclusions best represents the POLDER retrieved spherical albedo. It is noted here that the IHM model also assumes spherical air and spherical aerosol inclusions. The ensemble model phase function of Baum et al. (2005) was also tested against the POLDER retrieved spherical albedo for D e values between 20 and µm, and the σ values for these were found to vary between a minimum of and a maximum of The full results for the Baum et al. (2005) ensemble model are not shown here for reasons of brevity. The ensemble model with distortions and spherical air bubble predictions of the total single-scattering properties

11 SELF-CONSISTENT SCATTERING MODEL FOR CIRRUS. I: SOLAR REGION 1909 Spherical albedo difference june 25, 2003 Ensemble: Distortions and inclusions Scattering angle σ = Figure 8. Same as Figure 7 for the ensemble model with distortions and spherical air inclusions and σ = (β ext, ω 0 and g) calculated at the wavelengths of 0.443, and µm assuming D e = 65.0 µm are shown in Table VIII. Since the ensemble model with distortions and spherical air bubble inclusions best describes the POLDER retrieved spherical albedo this model is tested in the next section against POLDER measurements of the polarized reflection Tests against polarized reflection Results of comparing the ensemble model with distortions and spherical air bubble inclusions against the POLDER measured polarized reflection are shown in Figure 9. The figure shows that at scattering angles at about 80 the ensemble model does not capture the maximum in polarized reflection, and the measured gradient between the scattering angles of about 70 to 120 is not replicated. However, at scattering angles greater than 120 the ensemble model does follow the measured polarized reflection well. The physical reason as to why the ensemble model does not reproduce the measured polarized reflection gradient could be as follows. Polarized reflection is biased towards cloud top (Platnick, 2000; Sun et al., 2006) and as indicated in the introduction the tops of cirrus are more likely to be composed of simple ice crystals. However, as shown in Figure 1(a) (c) the ensemble model is composed of simple as well as more Table VIII. The total optical properties used in the POLDER analysis for each centre channel wavelength λ, showing the volume extinction coefficient β ext, single-scattering albedo ω 0, and the asymmetry parameter g. λ µm β ext km 1 ω 0 g Polarized reflection June 25, 2003 IHM Scattering angle Single member Ensemble:distortions + inclusions Figure 9. The polarized reflection plotted as a function of scattering angle showing results for the ensemble model with distortions and spherical air inclusions (dotted line), the three-branched member of the ensemble (dashed line) and the IHM model (full line). complex ice crystals and so it would not be expected that this model would be representative of cloud top. Therefore, it might be expected that more simple ice crystal members of the ensemble model might better represent the POLDER polarized reflection measurements. Figure 9 also shows a plot representing a simpler ensemble member such as the three-element ice crystal shown in Figure 1(c). Indeed, as can be seen from Figure 9 this simpler member does better represent the POLDER measured polarized reflection (a similar result for the sixbranched bullet-rosette is obtained but not shown for reasons of brevity) than the complete ensemble. Also shown in Figure 9 for comparison purposes is the result for the IHM model. In order to further discriminate ice crystals models, polarization measurements are required at scattering angles less than 60 as demonstrated by Figure 10. In Figure 10 the P 12 element is plotted against scattering angle for (1) the IHM model and (2) the three-element ice crystal model shown in Figure 1(c). The figure shows that for scattering angles greater than 60 both models predict a featureless gradient. However, for scattering angles less than 60 the predictions of the two models are very different. The IHM exhibits very strongly the 22 and 46 halo features, whereas the three-element model due to randomization does not exhibit any optical features. Moreover, at scattering angles less than about 40 the three-element model predicts negative polarization whilst the IHM model predominantly predicts positive polarization. Therefore, as well illustrated by Figure 10, it is essential to obtain further polarization measurements at scattering angles much less than 60 in order to provide rigorous tests of ice crystal models. 7. Summary In this paper a self-consistent solar-based scattering model for cirrus has been presented. The model is based