Impact of Ice Crystal Shapes & Sizes on Cirrus Cloud Radiative Properties, Lab Reports of Family Sociology

Download Impact of Ice Crystal Shapes & Sizes on Cirrus Cloud Radiative Properties and more Lab Reports Family Sociology in PDF only on Docsity! Sensitivity of Cirrus Bidirectional Reflectance to Vertical Inhomogeneity of Ice Crystal Habits and Size Distributions for Two MODIS Bands Ping Yang1,4, Bo-Cai Gao2, Bryan A. Baum3, Warren J. Wiscombe4, Yong X. Hu3, Shaima L. Nasiri5, Peter F. Soulen6,4, Andrew J. Heymsfield7, Greg M. McFarquhar7, and Larry M. Miloshevich7 1. Scientific Systems and Applications, Inc., Lanham, MD 2. Remote Sensing Division, Naval Research Laboratory, Washington, D.C. 3. NASA Langely Research Center, Hampton, VA 4. NASA Goddard Space Fight Center, Greenbelt, MD 5. Cooperative Institute for Meteorological Satellite Studies/University of Wisconsin, Madison, WI 6. Joint Center for Earth Systems Technology, University of Maryland Baltimore County, Baltimore, MD 7. National Center for Atmospheric Research, Boulder, CO For publication in The Journal of Geophysical Research _________________________________ Corresponding author address: Dr. Ping Yang, Code 913, NASA Goddard Space Flight Center, Greenbelt, MD 20771; Tel: 301-614-6127; Fax: 301-614-6307; email: pyang@climate.gsfc.nasa.gov. 2 Abstract A common assumption in satellite imager-based cirrus retrieval algorithms is that the radiative properties of a cirrus cloud may be represented by those associated with a specific ice crystal shape (or habit) and a single particle size distribution. However, observations of cirrus clouds have shown that the shapes and sizes of ice crystals may vary substantially with height within the clouds. In this study we investigate the sensitivity of the top-of-atmosphere bidirectional reflectances for two MODIS bands centered at 0.65 µm and 2.11 µm to cirrus models composed of either a single homogeneous layer or three distinct, but contiguous, layers. First, we define the single- and three-layer cirrus cloud models with respect to ice crystal habit and size distributions on the basis of in situ replicator data acquired during the First ISCCP Regional Experiment (FIRE-II), held in Kansas during the fall of 1991. Subsequently, fundamental light scattering and radiative transfer theory is employed to determine the single scattering and the bulk radiative properties of the cirrus cloud. For radiative transfer computations, we present a discrete form of the adding/doubling principle that introduces a direct transmission function that is computationally straightforward and efficient. For the 0.65 µm band, at which absorption by ice is negligible, there is little difference between the bidirectional reflectances calculated for the one- and three-layer cirrus models. This result suggests that the vertical inhomogeneity effect is relatively unimportant at 0.65 µm. At 2.11 µm, the bidirectional reflectances computed for both optically thin (τ = 1) and thick (τ = 10) cirrus clouds show significant differences between the results for the one- and three-layer models. The reflectances computed for the three- layer cirrus model are substantially larger than those computed for the single-layer cirrus. 5 An infrared trispectral algorithm using 8.52, 11, and 12 µm bands [Ackerman et al., 1990, 1998; Strabala et al. 1994] with a recent improvement [Baum et al. 2000a,b] form the basis of an infrared retrieval algorithm using MODIS infrared channels. The representative retrieval algorithm based on solar reflection is that developed by Nakajima and King [1990], who used a trispectral (0.75, 1.6, and 2.2 µm) method to simultaneously retrieve the optical thickness and mean effective particle size for water clouds. This approach has been applied to the retrieval of the optical thickness and mean particle size of ice crystals for cirrus [Wielicki et al., 1990]. To develop a reliable retrieval algorithm for cirrus optical and microphysical properties, it is critical to generate reliable pre-calculated look-up tables of bidirectional reflectance for cirrus clouds over a practical range of effective sizes, optical thicknesses, and viewing geometry (i.e., solar zenith angle, viewing zenith angle, and relative azimuth angle). At present, most algorithms for retrieving cirrus optical thickness and effective size assume that the ice crystals are of one specific habit, such as spheres, hexagonal plates, hexagonal columns, or fractal polycrystals [Macke et al., 1996]. In addition, a common assumption is that a single size distribution is sufficient to determine the scattering properties of the ice crystals within the cirrus layer. However, observations based on aircraft-borne two-dimensional optical cloud probe (2D-C) and balloon-borne replicator measurements [e.g., Heymsfield and Platt, 1984; Arnott et al., 1994; Mitchell et al., 1996a,b; McFarquhar and Heymsfield, 1996, 1997] demonstrate the wide range of shapes that the ice crystals in cirrus clouds may have, including bullet rosettes, solid and hollow columns, plates, and irregularly shaped aggregates. In addition, Heymsfield and 6 colleagues have showed that ice crystal habits and size distributions are vertically inhomogeneous in cirrus clouds [e.g., Heymsfield and Iaquinta, 2000]. Since satellite-based retrieval techniques essentially compare library computations of bidirectional reflectances to actual measurements in their implememtations, it is necessary to assess the effect of the vertical inhomogeneity of the ice crystal sizes and shapes within cirrus on the radiative transfer calculations for generating the reflectance libraries. Our objective is to understand the effect of the vertical inhomogeneity in the structure of cirrus clouds on their radiative properties. We employ fundamental scattering and radiative transfer theory to investigate the bidirectional reflectance of cirrus clouds for the MODIS 0.65 and 2.11 µm bands using in situ crystal habit and size distribution for a case of midlatitude cirrus on November 25, 1991, further described in Section 2. The outline of the paper is as follows. The data and models are provided in Section 2. Section 3 describes the development of the three-layer and single homogeneous cirrus models as well as the single-scattering properties associated with the two cirrus models. In section 4, we present the differences between bidirectional reflectances computed for the three-layer cirrus model and its one-layer counterpart. Also presented in this section is a sensitivity study regarding the shape effect of the small “quasi-spherically” nonspherical ice crystals on cloud reflectance. Conclusions are given in section 5. Finally, in the Appendix we present a numerically stable radiative transfer model based on the adding/doubling principle. The adding/doubling model is expressed in a discrete form for calculating reflected and transmitted intensities resulting from multiple scattering and absorption of cirrus clouds. 2. Data and Models 7 a. Data The size distributions and ice crystal habit information obtained from two cases of replicator measurements carried out in Kansas during the First International Satellite Cloud Climatology Project (ISCCP) Regional Experiment (FIRE) [Starr, 1987] phase II (hereafter, FIRE-II) are used in this study. The balloon-borne ice crystal replicators were launched at 1337 UTC on November 25 1991 and at 2045 UTC on December 5 1991. The replicator balloons had an ascent rate of approximately 4 m s-1 while passing through the cloud layers. As measured by a radiosonde connected to the balloon package, the cloud top temperature on November 25 was -57°C while on December 5, the cloud top temperature was -65°C. The replicator collects particles in a liquid plastic solution that coats a moving, 35--mm-wide transparent leader tape. The particles become imbedded in the plastic coating, and when the solvent in the solution evaporates, detailed ice crystal impressions and size spectra of crystals are recorded down to crystal sizes of approximately 10 microns. The particles generally do not break up upon impact on the replicator tape because of the slow rate of ascent of the balloon. The efficiency with which the replicator collects small crystals has been quantified theoretically and experimentally [Miloshevich and Heymsfield, 1997]. Particle size concentrations used in this study are adjusted to account for the imperfect collection efficiencies of small particles. Analysis of the ice crystal data collected during the balloon’s ascent through the cirrus provided 28 size spectra in the vertical on November 25 and 33 spectra on December 5, with each spectra representing the data collected from approximately 100 m of vertical ascent. 10 particles display less variation [Macke et al., 1996; Yang and Liou, 1998] than their counterparts for ice crystals containing smooth facets. As articulated by Mishchenko et al. [1996] on the basis of ground-based nephelometer and aircraft radiance measurement of cirrus clouds [Foot, 1988; Francis, 1995; Gayet, 1995; Posse and von Hoyningen-Huene 1995], the scattering phase functions for some ice-phase clouds can be rather featureless with no appreciable halos. Based on these studies, we account for surface roughness specifically for ice crystal aggregates in this study. In the numerical computation, the particle surface is regarded as a number of small facets whose normal direction is tilted from that in the smooth case, specified by local zenith and azimuth angles θ and ϕ , respectively. The slope of a facet along two orthogonal directions that are perpendicular to the local zenith direction, say x and y directions, can be specified by Zx = ∂Z ∂x = (µ−2 −1)1/ 2 cosϕ , (3a) Zy = ∂Z ∂y = (µ−2 −1)1/ 2 sinϕ , (3b) where µ=cosθ. The derivatives in Eqs.(3a) and (3b) are confined to the facet. The geometric configuration associated with these two equations has been illustrated by Cox and Munk [1954]. Since there is little quantitative experimental information regarding the roughness of ice crystal surfaces at present, the surface roughness is treated in a similar fashion to that of a wavy sea surface, which can be specified by the Gram-Char lie distribution [Cox and Munk, 1954]. If the tilt distribution of the roughness is azimuthally homogeneous (i.e., independent of angle ϕ ), the statistical probability density function 11 for the condition that the slopes of a facet along the two axis directions are given by the first order Gram-Charlie, or a 2-D Gaussian distribution, as follows: P(Zx, Zy) = 1 πσ2 exp[−(Zx 2 + Zy 2 )/ σ2 ], (4) where σ is a parameter determining the magnitude of roughness. Values of σ = 0-0.005, 0.005-0.05, 0.05-0.3 correspond to slight, moderate, and deep roughness in the single- scattering calculation, respectively. Further technical details concerning the treatment of surface roughness in the GOM2 light scattering computations may be found in Yang and Liou [1998]. c. Radiative Transfer Model Radiative transfer calculations for cirrus are performed using the adding/doubling method. The adding/doubling principle has been expressed mathematically in a matrix form [Twomey, 1966; Hunt and Grant, 1966] and in an integral form [Hansen and Travis, 1974]. A concise formulation in a discrete form for the adding/doubling method is provided in the Appendix. In the Appendix, some numerical concerns in the radiative transfer computation are addressed, such as the truncation of the forward-scattering peak in the phase function and a stable expansion of the phase function in terms of the re- normalized Legendre function. The discrete expression of the adding/doubling principle is straightforward and efficient in numerical realization. The present adding/doubling computational program has been validated with respect to the various cases presented by Lenoble [1985] and also in comparison with DISORT [Stamnes et al., 1988, 2000] for a number of canonical problems. 12 3. Development of Cirrus Models a. The Cirrus 3-Layer Model The vertically inhomogenous nature of cirrus clouds was observed during the FIRE-II program. Figures 1a and 1b show two different vertical profiles based on replicator images of ice crystals in cirrus clouds collected on November 25 and December 5, 1991. For these two cases, three distinct regimes of ice crystals are evident from the replicator data. In the uppermost layer, small nonspherical “quasi-spheres” are predominant. The middle layer of cirrus is composed primarily of pristine ice crystals with well-defined hexagonal shapes or bullet rosettes. The bottom layer contains larger but irregular aggregates. The edges of these irregular ice crystals seem to be rounded, perhaps due to the effect of sublimation. Roughness can also be noted from the replicator images of the irregular ice crystals. In both images, it is apparent that the particles increase in size and the shapes become more complex from the top to the base of the cirrus. The small particles in the uppermost layer have nonspherical shapes with an aspect ratio approaching 1. Sometimes the term “quasi-spherical” is used in the analysis of observed data and in theoretical studies. This term is often misleading because the optical properties of spherical and nonspherical particles are significantly different even if the nonsphericity of particle geometry is not substantial. Analysis of the replicator data has yielded detailed information on the dominant habits of ice crystals and size distributions for the two cases shown in Fig.1a and 1b. As discussed in Section 2.a, data were obtained for 28 and 33 vertical layers at approximately 100 m resolution in FIRE-II in situ observations for the November 25 and 15 b. Radiative properties of cirrus layers We employ the scattering computational model described in Section 2 to compute the extinction cross sections, single-scattering albedos, and phase functions for ice crystals. Ice crystals are assumed to be orientated randomly in the atmosphere. First, to characterize the bulk properties of size distribution, we define the mean maximum dimension for a given size distribution as follows: < D >= Dn(D)dD Dmin Dmax ∫ n(D)dD Dmin Dmax ∫ , (6) where Dmin and Dmax are the cutoffs of size distribution at small and large sizes, respectively. Studies by Foot [1988], Francis et al. [1994], Fu [1996], and Wyser and Yang [1998] have found that the details of the size distribution are not important to specifying the bulk optical properties of cirrus with respect to the effective size of ice crystals if the effective size is defined as the ratio of total volume to the total projected area. This feature has also been observed in the case of water clouds composed of liquid droplets whose scattering properties can be solved using Mie theory [Hansen and Travis, 1974; Hu and Stamnes, 1993]. Following these studies, we define the effective diameter De and effective radius re for nonspherical ice crystals with a combination of various habits as follows: De = 2re = 3 2 Vi (D) fi (D)n(D)dD i ∑ Dmin Dmax ∫ Ai(D) fi (D)n(D)dD i ∑ Dmin Dmax ∫ , (7) 16 where fi (D) is the percentage of a specific habit at size D. The summation over index i is carried out for all the ice crystal habits. We note that the preceding definition of effective radius reduces to that defined by Hansen and Travis [1974] in the case of spherical particles, i.e., re =< r 3 > / < r 2 > . The mean extinction cross section, single-scattering albedo, and phase function are given by , )()( )()()( max min max min , ∫ ∑ ∫ ∑ = D D i i D D i iie e dDDnDf dDDnDfDσ σ (8a) ˜ ω = σ s,i(D) fi(D)n(D)dD i ∑ Dmin Dmax∫ σe,i(D) fi(D)n(D)dD i ∑ Dmin Dmax∫ , (8b) ∫ ∑ ∫ ∑ Θ =Θ max min max min )()()( )()(),()( )( , , D D i iis D D i iiis dDDnDfD dDDnDfDPD P σ σ , (8c) where σs,i is the scattering cross section of habit i. Figures 3a and 3b show the bulk microphysical and optical properties for the size distributions shown in Fig.2a and 2b. The upper row shows the geometric configuration of the three layers of cirrus and the mean maximum dimension and effective size of ice crystals in these layers. The second and third rows show the extinction coefficient and single-scattering albedo. The lower two rows provide the asymmetry parameter of the phase functions and the fraction of delta transmission [Takano and Liou, 1989a] in scattered energy. Note that the delta-transmission is an artifact pertaining to the ray- tracing technique, which can be circumvented by using a more accurate physical optics 17 approach [Mishchenko and Macke, 1998]. In the present GOM2 calculation based on a simplified algorithm [Yang and Liou, 1996], we do not account for the spreading of the rays associated with delta-transmission for size parameter larger than 100. The use of either a geometric optics method or a physical optics approach in dealing with delta transmission in the single-scattering calculation for large size parameters does not make a significant difference in the radiative transfer computation. For the November 25 case (Fig.3a), the mean maximum dimension of the ice crystals is 74 µm, 112 µm, and 121 µm for the top, middle, and bottom cirrus layers. The mean effective diameters for the top, middle, and bottom cirrus layers are 65, 64, and 66 µm, respectively. For the December 5 case (Fig.3b), the mean maximum dimension of the ice crystals is 30 µm, 80 µm, and 132 µm for the top, middle, and lower cirrus layers, respectively. The corresponding mean effective diameters are 47, 92, and 89 µm for the top, middle, and bottom layers, respectively. We note that the effective diameters for the middle and bottom layers are substantially smaller for the November 25 case than for the December 5 case. The reason for this is that there are high numbers of bullet rosettes in the November 25 case, and bullet rosettes tend to have a small ratio of volume to projected area. In addition, the smaller mean effective diameter may be attributed in part to the porous structures of ice crystal aggregates which are present in a larger number concentration on November 25 than on December 5. The optical properties of ice crystals are computed for wavelengths representative of two MODIS bands centered at 0.65 and 2.11 µm. These wavelengths were chosen to represent the MODIS bands by integrating over the instrumental response functions following Baum et al. [2000a]. For the 0.65 and 2.11 µm bands, the maximum extinction 20 4. Results A. Comparison of reflectance feature for two cirrus models The radiative transfer model described in the Appendix is used to compute the bi- directional reflectances for the three-layer and single-layer cirrus models for the purpose of comparing the radiative features of each model. To understand the physics in the comparison, one needs to interpret the scattering geometry involving the sun and a satellite. For a given solar geometry specified by (θs ,ϕs ) and a viewing geometry specified by (θv ,ϕv ), the corresponding scattering angle is given by Θ = cos−1[− cosθs cosθv + sinθs sinθv cosφ], (9) where φ = ϕs −ϕv is the relative azimuth angle between sun and satellite. Note that θs and θv are the inclination angles measured from zenith. Figure 5 illustrates the contours of scattering angles versus solar- and view-zenith angles for four cases of azimuth angles. The solar zenith and viewing zenith angles range between 0° and 60°. The scattering angles for the region of view-solar geometry considered in the present study are essentially confined to side scattering and backscattering directions. The variational pattern of the scattering angle versus solar-zenith and viewing-zenith angles depends on the relative azimuth angle. A similar contour diagram of the scattering angle versus cos(θs ) and cos(θv ) has been presented by Mishchenko et al. [1996]. Figure 6 shows the relative difference between the computed bidirectional reflectances of the three-layer and one-layer cirrus models for optically thin cirrus (τ = 1) at 0.65 µm. The relative difference is defined as 21 e(θs,θv ,φ) = 100[R3(θs ,θv ,φ) − R1(θs,θv ,φ)]/ R1(θs,θv ,φ), (10) where R3 and R1 are the bidirectional reflection functions computed using the three-layer and one-layer cirrus models, respectively. The maximum difference in this case is about 5%. When the optical thickness is small, the photons originating from single scattering events dominate the total radiance. The contribution of single-scattering events to the radiance in the three-layer case is given by r(θs,θv ,φ) = 1 4 cosθs cosθv l=1 3 ∑ [ ˜ ω l∆τ lPl (θs ,θv ,φ )] , (11) where the summation is carried for all three layers of cirrus. Thus, for thin cirrus the bidirectional reflectance function is linearly proportional to the phase function. Referring to Figs. 3b and 5, the contours shown in Fig.6 can be explained as follows. For azimuth angles of 0° and 60°, the maximum difference is observed near scattering angles of 120°, which corresponds to the phase function difference at these scattering angles. For azimuth angles of 120° and 180°, the maximum difference for the three-layer and one-layer results are mainly noted near scattering angles of 155° and 180°. Figure 7 is similar to Fig.6, except that the optical thickness of the cloud is 10. The contribution of multiple scattering increases with increasing optical thickness. The differences occur at scattering angles between 90° and 120°, between 150° and 160°, and between 170°-180°. From Figs. 6 and 7, the difference between the three-layer and one- layer models is within a few percent regardless of optical thickness. Based on these results, modeling the cirrus as a single layer would seem to be sufficient at 0.65 µm. 22 Figures 8 and 9 are similar to Figs. 6 and 7, except that the calculations are performed at 2.11 µm. Absorption by ice at 2.11 µm is much higher than at 0.65 µm. The reflectance obtained using the three-layer model is always larger than that for the one- layer model at 2.11 µm. Because of ice absorption at 2.11 µm, the top layer of cirrus dominates the contribution to the cloud reflectance because photons that penetrate into the lower layers are largely absorbed. The mean size of the ice crystals in the top layer is much smaller than that associated with the general one-layer cirrus model. With the increase of optical thickness, the difference between the three-layer and one-layer models increases. For an optical thickness of 10, the differences can reach up to 50%. Because the 2.11 µm band is used for the retrieval of the mean size and optical depth of cirrus cloud, it is suggested that the vertical inhomogeneity may be important to developing more realistic cirrus retrieval algorithms. In comparing Figs. 8 and 9, it may be noted that the difference between the three-layer and one-layer models depends mainly on the scattering angle when the cloud is optically thin. However, for optically thick cirrus, the difference depends not only on scattering angle but also strongly on the viewing zenith and solar zenith angles. This is because the radiance path varies with the solar and view angles. For large solar zenith or viewing zenith angles, the ray path is large and the difference between the three-layer and one-layer cirrus models, and their associated single-scattering properties, becomes more significant. B. Sensitivity of cirrus reflectance to shapes of “quasi-spherical” particles As discussed in Section 3a regarding the replicator images of ice crystals, the small so-called “quasi-spherical” ice crystals are essentially nonspherical. We wish to clarify whether their morphologies can be treated as spheres in light scattering and 25 where Rsph and Rhex indicate the reflection functions associated with spherical and hexagonal shapes, respectively, which are assumed for the small “quasi-spherical” ice crystals. The maximum differences shown in Fig. 11, which correspond to scattering angles between 130° and 140°, are caused by the rainbow feature of ice spheres. It can also be noted that the assumption of ice spheres leads to an overestimation of reflectance near 180° (backscattering). As optical thickness increases, the contrast decreases for the rainbow feature. However, the enhanced backscattering derived using spheres as the “quasi-spherical” crystal shape is still noticeable. For optically thick cirrus, Fig. 12 shows that the assumption of sphere for the “quasi-spherical” particles leads to an underestimation of cloud reflection at 2.11 µm except for scattering angles near 180°. Figures 13 and 14 are similar to Figs. 11 and 12, except that the computations have been performed at 0.65 µm. Again, we see pronounced differences between the results associated with the hexagonal and spherical assumptions for the small ice crystals in the uppermost layer. The positive maximum near the backscattering peak noted in Figure 11, however, is not observed in the results shown in Figure 13 because the phase function value for the spheres is less than that of hexagons at 0.65 µm. Additionally at 0.65 µm, a distinct rainbow feature can be noted in the case of thin cirrus. For the optically thick cirrus, the rainbow is blurred due to multiple scattering events occurring within the clouds. Figures 11 through 14 illustrate that the influence of small particle shape in the uppermost layer of cirrus is significant at both visible and near infrared wavelengths. 26 5. Conclusions In this study, we have defined a three-layer cirrus model in terms of ice crystal habit and size distribution based on in-situ replicator data acquired during the NASA- sponsored FIRE-II field observation program. We have described a fundamental scattering model and a numerically stable radiative transfer model for the computation of the single-scattering properties of various ice crystals and the bidirectional reflection of cirrus clouds. We have found that the effect of vertically inhomogeneity within cirrus is not significant at 0.65 µm, a wavelength for which the absorption of ice is negligible. However, in comparison with the one-layer cirrus model, a vertically inhomogeneous cirrus cloud produces substantially larger reflectance at 2.11 µm, a wavelength for which absorption by ice is important. The increase in reflectance occurs because the mean size of ice crystals in the top layer in the three-layer model is smaller than in the case of the one-layer model and the total reflected radiance is dominated by the contribution from the top layer. For a given optical thickness, the reflectance increases with decreasing particle size. We also investigated the sensitivity of reflection of cirrus clouds to the particle shapes of the “quasi-spherical” ice crystals that have been often assumed to be spheres. For the two cirrus cloud cases presented in this study, the uppermost portion of the cloud tends to be predominately composed of very small ice crystals. Numerical results have demonstrated that the bidirectional reflection function of cirrus is very sensitive to the shape of these particles at both visible and near-infrared wavelengths. 27 Acknowledgements The authors thank Steve Platnick, Michael Mishchenko, David Mitchell, Klaus Wyser, and Erik Olson for their comments and suggestions. This research has been supported by a grant of NASA’s MODIS project and partially by the Office of Naval Research. This study was also supported by the Atmospheric Radiation Measurement (ARM) program sponsored by the U.S. Department of Energy (DOE) under Contract DE- AI02-00ER62901 and by NASA/EOS grant (contract No. S-97894-F). Appendix : A Discrete Expression of Adding/Doubling Principle The adding/doubling method is one of the most robust approaches to solve the radiative transfer equation for multiple scattering events. The standard mathematical expression of this method involves various tedious angular integrals although it can be written symbolically in a very simple form. In this section we present a discrete form of the method by introducing a direct transmitting function. As an improved mathematical expression from a practical viewpoint, the discrete adding/doubling equations are straightforward and more efficient in numerical implementation. In addition, the discrete form of the adding/doubling method is more suitable for addressing some numerical concerns, such as the numerical singularity of adding/doubling calculation and the performances of various quadrature schemes. To economize computational cost and memory requirements, we apply a Fourier expansion over the azimuth angle for radiance and bidirectional reflection and transmission functions: 30 ˜ t kj (m) = ˜ t (m)(µ j , µk ), ∆kj (m) = ∆(m)(µ j , µk ) . (A4c) Since the continuous region [0,1] is discretized by using a set of points, it is required that an integration of a function f(µ) with respect to its argument defined in [0,1] be replaced by a discrete summation in the form of f (µ)dµ → f (µi ) i=1 n ∑ 0 1 ∫ Wi , (A5) where Wi are the weights in the summation. For an integral involving the Dirac delta function, the definition of the delta function and Eq.(A5) lead to the following relationship: f (µi ) = f (µ)δ (µ − µi)dµ → f (µ j ) j =1 n ∑ 0 1 ∫ Wjδ(µ j − µi ) . (A6) Evidently, to guarantee the equality in Eq.(A6) in the discrete procedure, the Dirac delta function should be replaced by Kronecker symbol in the form of δ(µ j − µi ) → δ ji / Wj = 1/ Wj for j = i 0 for j ≠ i    . (A7) Thus, the direct transmission function in discrete form is given by ∆ij (m)(τ ) = 1 (1 +δm0 )Wj µ j exp(−τ / µ j )δ ij . (A8) For the discrete quantities defined with respect to the set of discrete points [µ1,µ2,...µn ] , we introduce a mathematical operator ⊗ defined by 31 Aij (m) ⊗ Bjk (m) = (1 + δm0 ) Aij (m) j =1 n ∑ Bjk(m)µ jWj , (A9a) Cj (m) ⊗ Bjk (m) = (1+ δm0) Cj (m) j=1 n ∑ Bjk(m)µ jWj . (A9b) The operator ⊗ is similar to an ordinary matrix multiplication except that a weight is included in the former. Thus, for one homogeneous layer, the reflected, diffusely transmitted, and total transmitted radiances are related to the incident radiation via the following relationships: Ik r(m) = I j i(m) ⊗ rjk (m) , Ik t(m) = I j i(m) ⊗ tjk (m) , ˜ I k t(m) = I j i(m) ⊗ ˜ t jk (m) . (A10) One of the interesting features of using the operator ⊗ is the variation of subscripts in the expressions in Eq.(A10): the incident beam denoted by subscript j is re-directed to the direction denoted by subscript k after interacting with the scattering layer. Similarly, for two layers indicated by superscripts a and b, we have the following relationships: Ik r(m) = I j i(m) ⊗ [rjk a(m) + ˜ t jl a(m) ⊗ Uln (m) ⊗ ˜ t nk a*(m)] , (A11a) and ˜ I k t(m) = I j i(m) ⊗[˜ t jl a(m) ⊗ ˜ t lk b(m) + ˜ t jl a(m) ⊗ Dln (m) ⊗ ˜ t nk b(m)] , (A11b) where the asterisk indicates that the transmission function corresponds to the case of illumination coming from below. The quantities D and U in Eqs.(A11a) and (A11b) are given by Dij (m) = Dij (m),(n) n=1 N ∑ , Uij(m) = rijb(m) + Dil(m) ⊗ rljb(m) , (A12a) 32 Dij (m),(n+1) = Dil (m),(n) ⊗ Dlj (m),(1) , Dij (m),(1) = ril b(m) ⊗ rlj a*(m) . (A12b) According to the sensitivity study by Hansen and Travis [1974], we use N=12, 5, and 3 in the summation involved in the first expression in Eq.(A12a) for m<10, 10<m<100, and m>100, respectively. The remaining terms are approximated by a geometric series. The physics of the adding/doubling principle can be viewed clearly in terms of the variations of the subscripts from left to right in the right-hand sides of Eqs.(A11a) and (A11b). The reflection and transmission functions for the combined layer are given by Rjk (m) = rjk a(m) + ˜ t jl a(m) ⊗ Uln (m) ⊗ ˜ t nk a*(m) , (A13a) and ˜ T jk (m) = ˜ t jl a(m) ⊗ ˜ t lk b(m) + ˜ t jl a(m) ⊗ Dln (m) ⊗ ˜ t nk b(m) . (A13b) The transmission function given in Eq. (A13b) contains the contribution due to direct transmission, which is implicitly in the form of a delta function, and it may potentially cause inaccuracy in numerical computation. Thus, it is necessary to separate the diffusive and direct components in Eq.(A13b). It is noted that ˜ t jl a(m) ⊗ ˜ t lk b(m) = [ tjl a(m) + ∆jl (m)(τ a )]⊗ [tlk b(m) + ∆lk (m)(τb )] = t jl a(m) ⊗ tlk b(m) + exp(−τ a / µ j )t jk b(m) + t jk a(m) exp(−τ b / µk ) + ∆jk (m)(τ a +τ b ). (A14) Thus, the diffusive transmission function for the combined layer is given by Tjk (m) = tjl a(m) ⊗ tlk b(m) +t jk a(m) exp(−τ b / µk ) + exp(−τa / µ j )tjk b(m) + ˜ t jl a(m) ⊗ Dln (m) ⊗ ˜ t nk b(m) . (A15) 35 Eq.(A19). Thus phase function information at the exact forward and backward directions is accounted for. The function ˜ P l m in Eqs.(A18a) and (A18b) are the “re-normalized” ( or the “normalized” called in DISORT [Stamnes et al. 2000]) associated Legendre polynomials first introduced by Dave and Armstrong [1970], defined as ˜ P l m(µ) = (l − m)! (l + m)! Pl m (µ) , (A20a) where pl m is the ordinary Legendre function. The normalized associated Legendre functions can be calculated on the basis of the following recurrence relationship: ˜ P l+1 m (µ) = 2l +1 ( l + m +1)(l − m +1) µ ˜ P l m (µ) − (l + m)(l − m) (l + m +1)(l − m + 1) ˜ P l−1 m (µ) , (A21a) with the two initial values for the preceding recurrence given by ˜ P m m(µ) = (−1)m (2m −1)!! (2m)!! (1 − µ2)m / 2 , ˜ P m+1 m (µ) = µ 2m +1 ˜ P m m (µ) . (A21b) Note that alternatives for initializing the preceding recurrence can be found in the paper by Dave and Armstrong [1970] and a technical report for DISORT [Stamnes et al. 2000]. It should be pointed out that in many references the phase function expansion based on Legendre polynomials is given in the form of P(m)(µi ,µ j ) = (2 − δm0 )˜ ω l l=m M ∑ (l − m)!(l + m)! Plm (µ i )Plm (µ j ) , (A22a) where the associated Legendre polynomial or Legendre functions can be calculated on the basis of the following recursive relationship 36 Pl+1 m (µ) = (2l +1)µPl m (µ) − (l + m)Pl−1 m (µ) l − m + 1 . (A22b) The factor (l − m)!/(l + m)! in Eq.(A22a) rapidly reduces to zero while the values of Legendre functions are very large for a large m with l ≥ m , as noted by Dave and Armstrong [1970]. For this reason, the preceding approach given by Eqs.(A22a) and (A22b) for the expansion of the phase function in radiative transfer simulations is not numerically stable, in particular, when the asymmetry of phase function is substantial and higher order Legendre functions are required in the phase function expansion. Since the predominant sizes of ice crystals in cirrus clouds are much larger than visible and near infrared wavelengths, there is a strong forward peak in the corresponding phase function. To include this forward peak in numerical computations, thousands of terms may be required in the Fourier expansions involved in Eqs.(A1a)-(A1c) for a general solar-view geometry, and also in the phase function expansion given by Eqs. (A18a) and (A18b). It should be pointed that the number of radiance streams used in the adding/doubling calculation needs to increase with the increase of the terms used in the phase function expansion so that the orthogonality of the selected finite set of Legendre functions in a discrete form can be guaranteed. Thus, the strong forward peak makes the numerical computation impractical. For this reason, the strong forward peaks of phase functions are truncated in practice. The schemes used for the truncation are diverse, which have been evaluated in an extensive discussion by Wiscombe, [1977], who has further developed the δ − M method in order to avoid the shortcomings pertaining to various ad hoc empirical approaches. For truncating the forward peak of the phase function involved in this study, we employ the method developed by Hu et al. [2000], 37 which is an extension of the δ − M method. After the phase function is truncated, the single-scattering properties need to be adjusted on the basis of the similarity principle [Wiscombe, 1977; van de Hulst, 1980; Takano and Liou, 1989b]. For example, optical depth and single scattering albedo are adjusted as follows: τ’= (1− f ˜ ω )τ , ˜ ω ’= (1 − f ) ˜ ω /(1 − f ˜ ω ) , (A23) where f is the fraction of energy associated with the truncated forward peak. 40 Heymsfield, A. J. and C. M. R. Platt, A parameterization of the particle size spectrum of ice clouds in terms of the ambient temperature and the ice water content, J. Atmos. Sci., 41, 846-855, 1984. Heymsfield, A. J. and J. Iaquinta, Cirrus crystal terminal velocities, J. Atmos. Sci., 57, 916-938, 2000. Hildebrand, E. B., Introduction to Numerical Analysis, Dover Publications, New York, 1974. Hu, Y. X., and K. Stamnes, An accurate parameterization of the radiative properties of water clouds suitable for use in climate models. , J. Climate, 6, 728-742, 1993. Hu, Y. X., B Wielicki, B. Lin and G. Gibson, S. C. Tsay, K. Stamnes, and T. Wong, Delta-fit: A fast and accurate treatment of particle scattering phase functions with weighted singular-value decomposition least-square fitting, , J. Quant. Spectrosc. Radiat. Transfer, 65, 681-690, 2000. Hunt, G. E., and I. P. Grant, Discrete space theory of radiative transfer and its application to problems in planetary atmospheres, J. Atmos. Sci., 26, 963-972, 1966. Inoue, T. On the temperature and effective emissivity determination of semi- transparent cirrus clouds by bispectral measurements in the 10 µm region, J. Meteorol. Soc. Japan, 63, 88-99, 1985. King, M. D., Number of terms required in the Fourier expansion of the reflection function for optical thick atmospheres, J. Quant. Spectrosc. Radiat. Transfer, 30, 143-161, 1983. 41 King, M. D., Y. J. Kaufman, W. P. Menzel, and D. Tanre, Remote sensing of cloud, aerosol, and water vapor properties from the Moderate Resolution Imaging Spectrometer (MODIS), IEEE Trans. Geosci. Remote Sens., 30, 2-27, 1992. King, M. D., S.-C. Tsay, S. E. Platnick, M. Wang, and K. N. Liou, Cloud Retrieval algorithms for MODIS: Optical thickness, effective particle radius, and thermodynamic phase, MODIS Algorithm Theoretical Basis Document, 79pp., 1997, [available at http://eospso.gsfc.nasa.gov/atbd/modistables.html]. Lenoble, J. Ed., Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures. Deepak, Hampton, VA. 1985. Liou, K. N., Influence of cirrus clouds on weather and climate processes: A global perspective, Mon. Wea. Rev., 114, 1167-1199, 1986. Liou, K. N., S. C. Ou, Y. Takano, F. P. J. Valero, and T. P. Ackerman, Remote sounding of the tropical cirrus cloud temperature and optical depth using 6.5 and 10.6 µm radiometers during STEP, J. Appl. Meteor., 29, 716-726, 1990. Mace, G. G., T. Ackerman, E. E. Clothiaux, and B. A. Albrecht, A study of composite cirrus morphology using data from a 94-GHz radar and correlations with temperature and large-scale vertical motion, J. Geophys. Res., 102, 13,581- 13,593, 1997. Macke, A., J. Muller, and E. Rascke, Single-scattering properties of atmospheric crystals, J. Atmos. Sci., 53, 2813-2825, 1996. McFarquhar, G. M., and A. J. Heymsfield, Microphysical characteristics of three cirrus anvils sampled during the central equatorial pacific experiment (CEPEX), J. Atmos. Sci., 52, 4143-4158, 1996. 42 McFarquhar, G. M., and A. J. Heymsfield, Parameterization of tropical cirrus ice crystal size distributions and implications for radiative transfer: Results from CEPEX, J. Atmos. Sci., 54, 2187-2200, 1997. Miloshevich, L. M., and A. J. Heymsfield, A balloon-borne continuous cloud particle replicator for measuring vertical profiles of cloud microphysical properties: Instrument design, performance, and collection efficiency analysis. J. Atmos. Ocean. Tech., 14, 753-768, 1997. Minnis, P., K. N. Liou, and Y. Takano, Inference of cirrus cloud properties using satellite-observed visible and infrared radiances, I. Parameterization of radiance fields, J. Atmos. Sci., 50, 1279-1304, 1993a. Minnis, P., K. N. Liou, and Y. Takano, Inference of cirrus cloud properties using satellite-observed visible and infrared radiances, II, Verification of theoretical cirrus radiative properties, J. Atmos. Sci., 50, 1305-1322, 1993b. Mitchell, D. L., S. K. Chai, Y. Liu, A. J. Heymsfield, Y. Dong, Modeling cirrus clouds: I. Treatment of bimodal size spectra and case study analysis, J. Atmos. Sci., 53, 2952-2966, 1996a. Mitchell, D. L., A. Macke, Y. Liu, Modeling cirrus clouds: II. Treatment of radiative properties of radiative properties, J. Atmos. Sci., 53, 2967-2988, 1996b. Mishchenko, M. I., W. B. Rossow, A. Macke A. A. Lacis, Sensitivity of cirrus cloud albedo, bidirectional reflectance and optical thickness retrieval accuracy to ice particle shape, J. Geophys. Res., 102, 16,973-16,985, 1996. 45 Takano, Y., K. N. Liou, and P. Minnis, The effects of small ice crystals on cirrus infrared radiative properties, J. Atmos. Sci., 49, 1487-1493, 1992. Twomey, S., Matrix methods for multiple scattering problems, J. Atmos. Sci., 23, 289- 296, 1966. van de Hulst, H. C., Light Scattering by Small Particles, Wiley, New York, 470pp, 1957. van de Hulst, H. C., Multiple Light Scattering. Academic Press, 739pp, 1980. Wielicki, B. A., J. T. Suttles, A. J. Heymsfield, R. M. Welch, J. D. Spinhire, M. L. C. Wu, and D. O. Starr, The 27-28 October 1986 FIRE IFO cirrus case study: Comparison of radiative transfer theory with observations by satellite and aircraft, Mon. Wea. Rev., 118, 2356-2376, 1990. Wiscombe, W. J., The delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase functions, J. Atmos. Sci., 34, 1408-1422, 1977. Wyser, K. and P. Yang, Average ice crystal size and bulk short-wave single-scattering properties of cirrus clouds, Atmos. Res., 49, 315-335, 1998. Yang, P. and K. N. Liou, Light scattering by hexagonal ice crystals: Comparison of finite-difference time domain and geometric optics model, J. Opt. Soc. Am. A, 12, 162-176, 1995. Yang, P. and K. N. Liou, Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals, Appl. Opt., 35, 6568-6584, 1996. Yang, P., and K. N. Liou, Light scattering by hexagonal ice crystals: Solution by Ray-by- Ray integration algorithm, J. Opt. Soc. Am. A., 14, 2278-2289,1997. Yang, P., and K. N. Liou, Single-scattering properties of complex ice crystals in terrestrial atmosphere, Contrib. Atmos Phys. 71, 223-248, 1998. 46 Figure Captions Figure 1a. Replicator images of ice crystals from a cirrus cloud observed on November 25 1991 during the FIRE-II field experiment (after Heymsfield and Iaquinta, 2000). Note the three-layer structure with small quasi-spherical crystals in the top layer, and columns and bullet rosettes in the second layer. The third layer is composed mostly of large aggregated crystals. Figure 1b. Same as Figure 1a, except that the observation was made on December 5, 1991 and the top layer is dominated by pristine columns. Figure 2a. Size distribution modeling the cirrus observed on November 25, 1991 that is shown in Figure 1a. Panels A, B, and C show the size and habit distributions for the top, middle, and bottom layer, respectively. Panel D shows the mean size distribution averaged over height. Figure 2b. Same as Figure 2a except for the case of December 5, 1991 Figure 3a. The mean size and single-scattering properties for the three-layer (November 25, FIRE-II) cirrus model. The vertical lines indicate the results computed using the one-layer model mean size distribution (i.e., the cloud is assumed to be vertically homogeneous). Figure 3b. Same as Fig.3b except for the December 5 case that is shown in Fig.2b. Figure 4. Phase function corresponding to the single-scattering properties shown in Figure 3. Figure 5. The scattering angle versus solar zenith and view zenith angles for four azimuthal angles. Note that the scattering angles are essentially for side-scattering and backscattering directions. 47 Figure 6. The percent relative difference of bidirectional reflectance computed using the three- and one-layer models at MODIS 0.65 µm band for thin cirrus (τ=1). The maximum difference for this case is about 5% and depends mainly on scattering angle. Figure 7. Same as Figure 6 except for thick cirrus (τ=10). Figure 8. Same as Figure 6 except for MODIS 2.11 µm band. Note the relative difference is much higher (up to 12%) than at 0.65 µm wavelength due to absorption by ice. Figure 9. Same as Figure 7 except for MODIS 2.11 µm band. The differences for large optical thickness reach up to 50% and depend also on viewing and solar zenith angles. Figure 10. Comparison of the top and middle layer phase functions computed by assuming that the small “quasi-spherical” ice crystals are either spheres or non- spherical hexagons with an aspect ratio of unity. Note the presence of the ice sphere rainbow feature between 130° and 140°. Figure 11. The percent relative difference of the bidirectional reflectances computed assuming spherical and hexagonal shapes for the small “quasi-spherical” ice crystals. The difference contours shown are for thin cirrus (τ=1) at MODIS 2.11 µm band. Note the large differences at the ice rainbow and backscattering angles. Figure 12. Same as Figure 11 except for thick cirrus (τ=10). Note the smoothing of the rainbow maximum. Figure 13. Same as Figure 11 except for MODIS 0.65 µm band. Note the absence of the positive backscattering angle maximum. Figure 14. Same as Figure 13 except for thick cirrus (τ=10). 0 0.1 0.2 0.3 0.4 0 100 200 300 400 500 600 0 0.1 0.2 0.3 0 100 200 300 400 500 600 0 0.1 0.2 0.3 0 100 200 300 400 500 600 0 0.2 0.4 0.6 0 100 200 300 400 500 600 Z=9.62-10.41 km (Dz=0.79 km) T=-56.9 to -50.6oC Z=8.89-9.62 km (Dz=0.73 km ) T=-45.3 to -50.6oC Z=7.71-8.89 km (Dz=1.18 km) T=-37.1 to 45.3oC Averaged (Dz=2.7 km) A B C D N um be r C on ce nt ra tio n (# /L /µ m ) Maximum Dimension (µm) -40-3-20-10 120340 -4 0 -3 0 -2 0 -1 0 0 10 20 30 40 -5-4-3-2 -1012 345 -4 -3 -2 -1 0 1 2 3 4 D<100 µm D>100 µm -403-21020 34 -1 50 -1 00 -5 0 0 50 10 0 15 0 -5-4-3-2 -1012 345 -4 -3 -2 -1 0 1 2 3 4 % + 70% -5-4-3-2 -1012 345 -4 -3 -2 -1 0 1 2 3 4 20% + 80% Fig.2a ( ) ( ) D<100 µm D>100 µm -40-3-20-10 120340 -4 0 -3 0 -2 0 -1 0 0 10 20 30 40 -403-21020 34 -1 50 -1 00 -5 0 0 50 10 0 15 0 -5-4-3-2 -1012 345 -4 -3 -2 -1 0 1 2 3 4 2 % + 1% +35% +32% -5-4-3-2 -1012 345 -4 -3 -2 -1 0 1 2 3 4 -403-21020 34 -1 50 -1 00 -5 0 0 50 10 0 15 0 5% + 46% +39% Fig.2b ( ) ( ) ( ) 0 2 4 6 8 0 100 200 300 400 500 600 0 0.1 0.2 0.3 0 100 200 300 400 500 600 0 1 2 3 0 100 200 300 400 500 600 0 0.2 0.4 0.6 0 100 200 300 400 500 600 Z=11.39-12.63 km (Dz=1.24 km) T=-55.2 to -65.4oC Z=10.27-11.39 km (Dz=1.12 km) T=-45.0 to -55.2oC Z=9.1-10.27 km (Dz=1.17 km) T=-38.5 to -45.0oC Averaged (Dz=3.53 km) D<50 µm -40-3-20-10 120340 -4 0 -3 0 -2 0 -1 0 0 10 20 30 40 -403-21020 34 -1 50 -1 00 -5 0 0 50 10 0 15 0 90% + 3% +7% D>50 µm -403-21020 34 -1 50 -1 00 -5 0 0 50 10 0 15 0 75% + 25% -403-21020 34 -1 50 -1 00 -5 0 0 50 10 0 15 0 30% + 70% -40-3-20-10 120340 -4 0 -3 0 -2 0 -1 0 0 10 20 30 40 D<50 µm -403-21020 34 -1 50 -1 00 -5 0 0 50 10 0 15 0 D>50 µm -40-3-20-10 120340 -4 0 -3 0 -2 0 -1 0 0 10 20 30 40 D<50 µm -403-21020 34 -1 50 -1 00 -5 0 0 50 10 0 15 0 D>50 µm A B C D N um be r C on ce nt ra tio n (# /L /µ m ) Maximum Dimension (µm) ( ) ( ) 9 10 11 12 13 0.99 1 H ei gh t ( km ) be (1/km) 9 10 11 12 13 H ei gh t ( km ) 9 10 11 12 13 0 40 80 120 160 H ei gh t ( km ) <D> ( µm ) 9 10 11 12 13 0 40 80 120 H ei gh t ( km ) De ( µm ) 9 10 11 12 13 0.2 0.3 0.4 0.5 H ei gh t ( km ) be (1/km) fd fd w̃ Top Layer Middle Layer Bottom Layer 0.65 µm 0.65 µm 0.65 µm 0.65 µm 2.11 µm 2.11 µm 2.11 µm 2.11 µm 9 10 11 12 13 0.2 0.3 0.4 0.5 H ei gh t ( km ) 9 10 11 12 13 0.8 0.85 0.9 0.95 H ei gh t ( km ) 9 10 11 12 13 0.75 0.8 0.85 H ei gh t ( km ) g 9 10 11 12 13 0.8 0.85 0.9 H ei gh t ( km ) g 9 10 11 12 13 0.04 0.08 0.12 0.16 H ei gh t ( km ) 9 10 11 12 13 0.02 0.06 0.1 H ei gh t ( km ) w̃ Fig.3 View Zenith Angle ( ° ) View Zenith Angle ( ° ) 60 F 50 40 30 20 60 50 40 30 20 10 0 0 A=0.65 um, Thin Cirrus (tT=1) $=60° p=0° 10 20 30 40 50 $=120° 10 20 30 40 50 Solar Zenith Angle ( ° ) 60 60 60 F 50 40 30 0 10 20 30 40 50 $=180° 10 20 30 40 50 Solar Zenith Angle ( ° ) Fig.6 60 60 4.0 3.5 3.0 25 2.0 15 1.0 0.5 0.0 0.5 -1.0 View Zenith Angle ( ° ) View Zenith Angle ( ° ) 60 50 40 30 20 10 0 60 50 40 30 20 10 0 0 A=0.65 um, Thick Cirrus (T=10) $=60° g=0° 10 20 30 40 50 $=120° 60 0 10 20 30 40 50 Solar Zenith Angle ( ° ) 60 60 50 40 30 20 10 0 60 50 40 30 20 10 0 0 10 20 30 40 50 60 $=180° 0 10 20 30 40 50 60 Solar Zenith Angle ( ° ) Fig.7 3.0 25 2.0 15 1.0 05 0.0 0.5 -1.0 15 -2.0 View Zenith Angle ( ° ) View Zenith Angle ( ° ) A=2.11 pum, Thin Cirrus (tT=1) $=60° g=0° 60 50 40 30 20 10 0 QO 10 20 30 40 50 60 $=120° 60 50 40 30 20 10 0 0 10 20 30 40 50 60 Solar Zenith Angle ( ° ) 60 12.0 50 40 30 20 9.0 0 10 20 30 40 50 60°° 60 ~=180° 70 50 6.0 40 30 5.0 20 4.0 10 0 3.0 0 10 20 30 40 50 60 Solar Zenith Angle ( ° ) Fig.8 View Zenith Angle ( ° ) View Zenith Angle ( ° ) A=2.11 pum, Thin g=0° 60 50 40 30 20 10 0 QO 10 20 30 40 50 60 $=120° 60 50 40 30 20 10 0 0 10 20 30 40 50 Solar Zenith Angle ( ° ) 60 Cirrus (T=1) =60° 60 50 40 30 20 20 30 40 50 $=180° 60 50 40 30 20 -25.0 10 0 -30.0 0 10 20 30 40 50 Solar Zenith Angle ( ° ) 60 Fig.11 View Zenith Angle ( ° ) View Zenith Angle ( ° ) A=2.11 pum, Thick Cirrus (T=10) g=0° 60 50 40 30 20 QO 10 20 30 40 50 60 $=120° 60 50 40 30 0 L \ 0 10 20 30 40 50 Solar Zenith Angle ( ° ) 60 $=60° 60 5.0 50 40 0.0 30 5.0 20 20 30 40 50 $=180° 60 50 40 30 -30.0 -35.0 0 10 20 30 40 50 Solar Zenith Angle ( ° ) 60 Fig.12 View Zenith Angle ( ° ) View Zenith Angle ( ° ) A=2.11 pum, Thick Cirrus (T=10) g=0° 60 50 40 30 20 QO 10 20 30 40 50 60 $=120° 60 50 40 30 0 L \ 0 10 20 30 40 50 Solar Zenith Angle ( ° ) 60 $=60° 60 5.0 50 40 0.0 30 5.0 20 20 30 40 50 $=180° 60 50 40 30 -30.0 -35.0 0 10 20 30 40 50 Solar Zenith Angle ( ° ) 60 Fig.12