1. Introduction
Ice clouds remain one of the key uncertainty sources in the study of the atmospheric radiation budget and atmospheric remote sensing (Liou 1986; Lynch et al. 2002; Wendisch et al. 2007; Minnis et al. 1993a,b; Baum et al. 2000, 2005; Baran 2009, and references cited therein). These clouds also pose a challenge to atmospheric radiative transfer and remote sensing studies. As the optical properties of ice crystals are fundamental to quantifying the radiative properties of ice clouds, the atmospheric research community has made a significant effort to investigate the single-scattering properties of ice particles (Takano and Liou 1989a; Macke et al. 1996; Yang and Liou 1996; Borovoi and Grishin 2003; Baran et al. 2001; Baran 2004; Um and McFarquhar 2007; Zhang et al. 2009). For practical applications to atmospheric radiative transfer and remote sensing involving ice clouds, the single-scattering properties (i.e., the phase matrix, single-scattering albedo, asymmetry factor, and extinction cross section) of individual ice crystals need to be integrated over particle size and shape distributions to derive the bulk optical properties (Ebert and Curry 1992; Fu 1996; Key et al. 2002; McFarquhar et al. 2002; Baum et al. 2000, 2005). For example, Field et al. (2005, 2007) developed a moment estimation parameterization of the particle size distribution for ice clouds, which had been used by Baran (2009) and Baran et al. (2011) to compute the bulk scattering properties for ice clouds.
In situ measurements (Heymsfield and Platt 1984; Heymsfield et al. 2002) of the ice crystal size distributions are essentially provided in terms of discrete bins. For each bin, the number concentration of ice crystals is assumed to be a constant. The conventional method of integrating the single-scattering properties of ice crystals over the size distribution is to use the microphysical properties (volume and area) and the single-scattering properties at bin centers, although some bins are quite large (e.g., the bin width may be as large as several hundred microns). In the literature, a coarse bin resolution of 5 bins (Takano and Liou 1989b; King et al. 2004) and relatively fine bin resolutions of 27 bins (Nasiri et al. 2002), 38 bins (Fu 1996), and 45 bins (Baum et al. 2005) have been used to discretize the size distribution. Nasiri et al. (2002) compared the bulk optical properties of ice clouds derived from using 5 and 27 bins to simulate the ice cloud microphysical and optical properties. Their results indicate that the ice cloud effective particle size and optical properties calculated from a given size distribution tends to be sensitive to the number of size bins. The intent of our study is to investigate the impact of resolving the bin variation of the single-scattering properties of ice crystals on the corresponding bulk scattering properties by utilizing the scattering kernel technique, which has been used to calculate the bulk optical properties of dustlike particles in the studies of King et al. (1978), Dubovik and King (2000), and Dubovik et al. (2006). This paper is organized as follows: Section 2 presents the formulas for computing the bulk optical properties of an ice crystal ensemble by the kernel technique, section 3 contains the results and discussion, and section 4 is a summary of the study.
2. Bulk optical properties of ice clouds and the kernel technique
The ensemble (or population) averaged single-scattering properties of individual ice crystals within ice clouds are usually referred to as the bulk optical properties of the clouds. In this study, the number concentration of ice crystals is indicated in terms of n(L), where L is the maximum dimension of an ice crystal. The quantity n(L) is specified in terms of discrete bins and Fig. 1a shows one of the size distributions used in deriving ice cloud optical properties used for the Moderate Resolution Imaging Spectroradiometer (MODIS) collection 4 cloud product (King et al. 2004). Figure 1b shows one of the size distributions used by Fu (1996) that compiled 28 size distributions from the literature (Heymsfield and Platt 1984; Mitchell and Arnott 1994). Figures 1c and 1d are the same as Figs. 1a and 1b, respectively, except that a logarithmic abscissa scale is used. From Fig. 1, it is evident that some of the size bins, particularly those for large particles, are quite coarse. Thus, using ice crystal microphysical (volume and projected area) and optical properties at the bin centers as surrogates for their bin-averaged counterparts may lead to substantial errors in deriving the bulk optical properties. To avoid the potential errors, the kernel technique can be employed. Dubovik and King (2000) and Dubovik et al. (2006) have formulated this technique for application to the calculation of the bulk optical properties of nonspherical dustlike particles. Following Dubovik and King (2000) and Dubovik et al. (2006), we briefly outline the kernel technique for application to the bulk optical properties of ice clouds.
Examples of size distributions discretized with (a) 5 bins (King et al. 2004) and (b) 38 bins (Fu 1996). (c),(d) As in (a),(b), respectively, but logarithmic scales are applied to the abscissas of (c),(d).
Citation: Journal of Atmospheric and Oceanic Technology 29, 1; 10.1175/JTECH-D-11-00034.1
Similarly, the kernels for the extinction cross section, volume, and projected area can be defined and are denoted as 
Because the conventional approach is, in principle, to approximate the kernels, it is a simplified kernel technique. However, in this paper, we refer to the kernel method as a special term for the approach based on Eqs. (5) and (6). To avoid ambiguity, the kernels calculated from the conventional method based on Eqs. (3) and (4) are referred to as “coarse kernels” and their counterparts based on Eqs. (5) and (6) as “fine kernels.”
3. Results and discussion
Figure 2 shows the extinction and absorption efficiencies computed from the ADDA (red line), the IGOM (green line), and their adjusted IGOM counterparts (blue line). Ice has negligible absorption at the 0.66-μm wavelength, and thus the absorption efficiency is not shown. Figure 2 indicates that the edge effect is important, and the incorporation of this effect leaves no discontinuity between the ADDA and IGOM results.
Single-scattering properties [(top) extinction efficiency and (bottom) absorption efficiency] of randomly oriented hexagonal ice particles at specific wavelengths of (left to right) 0.66, 3.78 and 12.0 μm. The complex refractive indices are 1.3078 + i1066 × 10−8, 1.384306 + i7.055 × 10−3, and 1.2762 + i4.133 × 10−1, respectively.
Citation: Journal of Atmospheric and Oceanic Technology 29, 1; 10.1175/JTECH-D-11-00034.1
In the present study, a single-scattering property database is developed with a fine resolution with respect to the particle size (4390 size bins, specifically). The particle sizes, in terms of the maximum dimension L, range from 1 to 3500 μm with a step of 0.1 μm when L ≤ 100 μm and a step of 1 μm when L > 100 μm. The bin resolutions are also used in the numerical evaluation of the integrals in Eqs. (5) and (6). Based on 4390 size bins, “exact” bulk-scattering properties can be obtained and are employed as a reference to assess the accuracy of the results computed from the conventional approach and the kernel technique with different numbers of size bins.
Ice particle size distributions based on in situ observations and corresponding fitted ones. The effective particle sizes De associated with the fitted size distributions are (a) 20.1, (b) 116.1, and (c) 185.2 μm, respectively.
Citation: Journal of Atmospheric and Oceanic Technology 29, 1; 10.1175/JTECH-D-11-00034.1
The effective particle sizes associated with the three generated size distributions are 20.1, 116.1, and 185.2 μm, respectively. The values of the three effective particle sizes are not the same as those reported by Fu (1996) because, although the ice crystal shapes are also assumed to be hexagonal, the aspect ratios defined in Fu (1996) and in the present study are different.
To test the errors for both methods in the cases of different numbers of size bins, we discretize the size distributions into a number of size bins. To be more specific, three continuous size distributions as shown in Fig. 3 are discretized into 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, and 100 size bins in equal logarithmic steps from 1 to 3500 μm. We recalculate the bulk properties using the kernel method and the conventional method for each number of size bins. By comparing the results with the benchmark based on 4390 size bins, we investigate the relative errors of both methods and examine at which number of size bin the conventional method and the kernel method give similar results.
Figure 4 shows the comparisons of scattering cross section kernels as a function of size parameter calculated by using the kernel technique and the conventional method for the cases of 5 and 30 size bins, respectively, at 0.66-, 3.78-, and 12.0-μm wavelengths. The scattering cross section kernels (smooth lines) based on 4390 size bins calculated from the conventional method are also included. The step lines show the fine kernels calculated from the kernel technique and the circles show the coarse kernels computed from the conventional method. The colors represent different wavelengths. The kernels computed from the conventional method and the kernel technique are different values within a wide size bin. The solution based on the kernel method is an accurate bin-averaged value, whereas its counterpart from the conventional method is an inaccurate approximation. However, the differences of the two solutions decrease with the decrease of the size-bin width, as is illustrated by the lower panel of Fig. 4.
The scattering cross-section kernel as a function of size parameter at three wavelengths for (top) 5 and (bottom) 30 bins. The smoothed curved lines show the kernel of 4390 size bins, the step lines show the fine kernel calculated from the kernel technique, and the circles show the coarse kernel computed from the conventional method.
Citation: Journal of Atmospheric and Oceanic Technology 29, 1; 10.1175/JTECH-D-11-00034.1
Figure 5 shows the scattering phase function kernel at the scattering angles 1° and 60° as a function of size parameter at three wavelengths. From Fig. 5, the differences between the kernels from the conventional and the present methods, especially for coarse size bins, can be seen. Therefore, the advantage of the present kernel method over the conventional method is that more accurate bulk-scattering properties can be obtained even if a coarse bin resolution is used.
The scattering phase function kernel at the scattering angles of (top) 1° and (bottom) 60° as a function of size parameter at three wavelengths for (left) 5 and (right) 30 bins. The smoothed curved lines show the kernel of 4390 size bins, the step lines show the fine kernel calculated from the kernel technique, and the circles show the coarse kernel computed from the conventional method.
Citation: Journal of Atmospheric and Oceanic Technology 29, 1; 10.1175/JTECH-D-11-00034.1
Figure 6 shows the comparison between the relative errors of the effective particle sizes calculated from the kernel technique and the conventional approach as functions of the number of discrete size bins for the three size distributions shown in Fig. 3. The relative error of the effective particle size is defined as 
Comparison of relative errors of the effective particle size calculated from the kernel technique and the conventional approach for De = (top) 20, (middle) 116, and (bottom) 185 μm.
Citation: Journal of Atmospheric and Oceanic Technology 29, 1; 10.1175/JTECH-D-11-00034.1
Figures 7–9 show the comparison of the relative errors of the extinction efficiency, asymmetry factor, and albedo, calculated from both the kernel technique and the conventional approach as functions of the number of discrete size bins for a set of three particle size distributions at the three wavelengths of 0.66, 3.78, and 12.0 μm. The absorption of ice particles is negligible at the 0.66-μm wavelength and the single-scattering albedo is 1.0, which is not shown in Fig. 9. The relative error is defined as 
Comparison of relative errors of the extinction efficiency calculated from the kernel technique and the conventional approach for (top to bottom) three particle size distributions at (left to right) the 0.66-, 3.78-, and 12-μm wavelengths.
Citation: Journal of Atmospheric and Oceanic Technology 29, 1; 10.1175/JTECH-D-11-00034.1
As in Fig. 7, but for the asymmetry factor.
Citation: Journal of Atmospheric and Oceanic Technology 29, 1; 10.1175/JTECH-D-11-00034.1
As in Fig. 7, but for the single-scattering albedo.
Citation: Journal of Atmospheric and Oceanic Technology 29, 1; 10.1175/JTECH-D-11-00034.1
As an important quantity in radiative transfer simulation, the asymmetry factor (g), the first moment of the scattering phase function, indicates the relative strength of forward scattering. The radiative fluxes, especially for shortwave fluxes, at the surface and the top of the atmosphere are highly sensitive to the uncertainties in the asymmetry factor. Vogelmann and Ackerman (1995) investigated the sensitivity of net shortwave fluxes to the uncertainties in cirrus cloud scattering properties and found that uncertainties on the order of 1% in a typical cirrus cloud asymmetry factor (g ~ 0.8) could lead to uncertainties on the order of 2.5% in net surface fluxes. Fu (2007) showed that the uncertainties in the global cloud albedo are approximately ±0.06 in conjunction with the uncertainties of ±0.05 in an asymmetry factor of 0.8. Figure 8 is similar to Fig. 7 but compares the relative errors associated with the bulk asymmetry factor. The differences in the asymmetry factor derived from the two methods are obvious for the size bins less than 30 and indicate that the kernel technique can also improve the accuracy of the bulk asymmetry factor when using fewer size bins. The difference decreases with an increase in the number of size bins used. The relative differences between both methods can be under 0.2%, which is much less than the requirements determined in previous studies (Vogelmann and Ackerman 1995; Fu 2007), when the number of size bins is larger than 40. Both methods get similar results when the number of size bins is larger than 50.
From Fig. 9, it can be seen that the relative errors of the single-scattering albedo calculated from the kernel technique are less than those from the conventional approach, and the differences of the single-scattering albedo between the two methods are obvious for the size bins less than 30. The kernel technique can improve the accuracy of the single-scattering albedo when using fewer size bins. The relative errors from both methods are under 0.1% for the number of size bins larger than 30, but both methods get similar results when the number of size bins is larger than 40.
Figures 10–12 show the comparison of the bulk phase functions calculated from both methods for various numbers of size bins at the 0.66-, 3.78-, and 12.0-μm wavelengths. It can be seen that the biggest difference of the bulk phase functions from both methods occurs with 10 size bins, and the difference tends to be smaller with an increase in the number of size bins. When the number of size bins is 60, the phase functions obtained from both methods are approximately the same.
Comparison of the bulk phase function calculated from the kernel technique and the conventional approach at the wavelength of 0.66 μm for 10, 20, 30, and 60 bins. The effective particle size is 20 μm.
Citation: Journal of Atmospheric and Oceanic Technology 29, 1; 10.1175/JTECH-D-11-00034.1
As in Fig. 10, but for the 3.78-μm wavelength.
Citation: Journal of Atmospheric and Oceanic Technology 29, 1; 10.1175/JTECH-D-11-00034.1
As in Fig. 10, but for the 12.0-μm wavelength.
Citation: Journal of Atmospheric and Oceanic Technology 29, 1; 10.1175/JTECH-D-11-00034.1
Figures 7–12 show the deviations of the scattering properties from both methods for three given size distributions with the so-called true effective particle sizes of 20, 116, and 185 μm. However, since the effective particle sizes calculated from two methods are different and also deviate from the true values, the compared scattering properties actually correspond to different effective particle sizes. Figure 13 shows the single-scattering albedo versus the effective particle size calculated from both methods for different numbers of size bins at the wavelengths 3.78 μm (left column) and 12 μm (right column). The red circles indicate the results calculated from the kernel technique and the blue squares indicate the results based on the conventional approach. When the effective particle sizes calculated from two methods are similar, the corresponding single-scattering albedo values are in agreement. However, the ranges of the effective particle sizes computed from both methods are quite different. Nasiri et al. (2002) showed that using different bin resolutions for a given size distribution in the simulation of the bulk scattering properties of ice clouds can modify the reflectance simulation. However, since the effective particle size and single-scattering properties vary consistently, as clearly illustrated by Fig. 13, the net effects may have little impact on the relation between the simulated reflectance and effective particle size (i.e., the dependence of the simulated reflectance on the effective particle size) (King et al. 2004).
The single-scattering albedo vs (top to bottom) the effective size calculated from both methods for different number of size bins at wavelengths of (left) 3.78 and (right) 12 μm.
Citation: Journal of Atmospheric and Oceanic Technology 29, 1; 10.1175/JTECH-D-11-00034.1
4. Summary
To study the differences of the bulk single-scattering properties of ice crystals derived from the kernel technique and the conventional approach, three wavelengths at 0.66, 3.78, and 12.0 μm, corresponding to cases with slight, moderate, and strong absorption, respectively, are considered. The shape of an ice crystal is assumed to be a hexagonal column. The optical properties of individual ice crystals are calculated from the ADDA code and an improved geometric optics method (IGOM). The edge effects are incorporated in the IGOM solutions for the extinction and absorption efficiencies.
“Exact” bulk-scattering properties are first obtained based on 4390 size bins. Both the kernel technique and the conventional approach are utilized to calculate the bulk optical properties of ice clouds based on various size bins from 5 to 100 and then compared with the exact results. It is demonstrated that the kernel technique can uncover more delicate kernel information and improve the accuracy of the bulk microphysical and optical properties of ice crystals when the number of size bins is less than approximately 30. However, the differences are much smaller when using more than 40 size bins and both methods give similar results, which implies that more than 40 size bins should be used to calculate the bulk optical properties of ice clouds by the conventional method.
It is not efficient to obtain bulk scattering properties in various applications based on the single-scattering database of fine-bin resolution. A more proper approach is to compute the scattering kernels from the scattering database of fine bin resolution. Consequently, bulk scattering properties can be efficiently obtained from the database of scattering kernels without losing accuracy.
X. D. Liu acknowledges support from the National Basic Research Program of China (2011CB403406), CAS Strategic Priority Research Program (XDA05110101), and NSFC (40825008, 41075067). P. Yang acknowledges support from the endowment funds associated with the David Bullock Harris Chair in Geosciences, College of Geosciences, Texas A&M University.
REFERENCES
Auer, A. H., Jr., , and Veal D. L., 1970: The dimension of ice crystals in natural clouds. J. Atmos. Sci., 27, 919–926.
Baran, A. J., 2004: On the scattering and absorption properties of cirrus cloud. J. Quant. Spectrosc. Radiat. Transfer, 89, 17–36.
Baran, A. J., 2009: A review of the light scattering properties of cirrus. J. Quant. Spectrosc. Radiat. Transfer, 110, 1239–1260.
Baran, A. J., , Yang P., , and Havemann S., 2001: Calculation of the single-scattering properties of randomly oriented hexagonal ice columns: A comparison of the T-matrix and the finite-difference time-domain methods. Appl. Opt., 40, 4376–4386.
Baran, A. J., , Connolly P. J., , Heymsfield A. J., , and Bansemer A., 2011: Using in situ estimates of ice water content, volume extinction coefficient, and the total solar optical depth obtained during the tropical ACTIVE campaign to test an ensemble model of cirrus ice crystals. Quart. J. Roy. Meteor. Soc., 137, 199–218.
Baum, B. A., , Kratz D. P., , Yang P., , Ou S., , Hu Y., , Soulen P. F., , and Tsay T. C., 2000: Remote sensing of cloud properties using MODIS airborne simulator imagery during SUCCESS. I. Data and models. J. Geophys. Res., 105, 11 767–11 780.
Baum, B. A., , Yang P., , Heymsfield A. J., , Platnick S., , King M. D., , Hu Y., , and Bedka S. T., 2005: Bulk scattering properties for the remote sensing of ice clouds. Part II: Narrowband models. J. Appl. Meteor., 44, 1896–1911.
Bi, L., , Yang P., , and Kattawar G. W., 2010: Edge-effect contribution to the extinction of light by dielectric disks and cylindrical particles. Appl. Opt., 49, 4641–4646.
Bi, L., , Yang P., , Kattawar G. W., , Hu Y. X., , and Baum B. A., 2011: Scattering and absorption of light by ice particles: Solution by a new physical-geometric optics hybrid method. J. Quant. Spectrosc. Radiat. Transfer, 112, 1492–1508.
Borovoi, A. G., , and Grishin I. A., 2003: Scattering matrices for large ice crystal particles. J. Opt. Soc. Amer., 20A, 2071–2080.
Dubovik, O., , and King M. D., 2000: A flexible inversion algorithm for retrieval of aerosol optical properties from sun and sky radiance measurements. J. Geophys. Res., 105, 20 673–20 696.
Dubovik, O., and Coauthors, 2006: The application of spheroid models to account for aerosol particle nonsphericity in remote sensing of desert dust. J. Geophys. Res., 111, D11208, doi:10.1029/2005JD006619.
Ebert, E. E., , and Curry J. A., 1992: A parameterization of ice cloud optical properties for climate models. J. Geophys. Res., 97, 3831–3836.
Field, P. R., , Hogan R. J., , Brown P. R. A., , Illingworth A. J., , Choularton T. W., , and Cotton R. J., 2005: Parametrization of ice-particle size distribution functions for mid-latitude stratiform cloud. Quart. J. Roy. Meteor. Soc., 131, 1997–2017.
Field, P. R., , Heymsfield A. J., , and Bansemer A., 2007: Snow size distribution parameterization for midlatitude and tropical ice cloud. J. Atmos. Sci., 64, 4346–4365.
Foot, J. S., 1988: Some observations of the optical properties of clouds: II. Cirrus. Quart. J. Roy. Meteor. Soc., 114, 145–164.
Fu, Q., 1996: An accurate parameterization of the solar radiative properties of cirrus clouds for climate models. J. Climate, 9, 2058–2082.
Fu, Q., 2007: A new parameterization of an asymmetry factor of cirrus clouds for climate models. J. Atmos. Sci., 64, 4140–4150.
Heymsfield, A. J., , and Platt C. M. R., 1984: 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, 846–856.
Heymsfield, A. J., , Bansemer A., , Field P. R., , Durden S. L., , Stith J. L., , Dye J. E., , Hall W., , and Grainger C. A., 2002: Observations and parameterizations of particle size distributions in deep tropical cirrus and stratiform precipitating clouds: Results from in situ observations in TRMM field campaigns. J. Atmos. Sci., 59, 3457–3491.
Key, J. R., , Yang P., , Baum B. A., , and Nasiri S. L., 2002: Parameterization of shortwave ice cloud optical properties for various particle habits. J. Geophys. Res., 107, 4181, doi:10.1029/2001JD000742.
King, M. D., , Byrne D. M., , Herman B. M., , and Reagan J. A., 1978: Aerosol size distributions obtained by inversion of spectral optical depth measurements. J. Atmos. Sci., 35, 2153–2167.
King, M. D., , Platnick S., , Yang P., , Arnold G. T., , Gray M. A., , Riedi J. C., , Ackerman S. A., , and Liou K.-N., 2004: Remote sensing of liquid water and ice cloud optical thickness and effective radius in the Arctic: Application of airborne multispectral MAS data. J. Atmos. Oceanic Technol., 21, 857–875.
Liou, K.-N., 1986: Influence of cirrus clouds on weather and climate processes: A global perspective. Mon. Wea. Rev., 114, 1167–1199.
Lynch, D. K., , Sassen K., , Starr D. O., , and Stephens G., Eds., 2002: Cirrus. Oxford University Press, 480 pp.
Macke, A., , Mueller J., , and Raschke E., 1996: Single scattering properties of atmospheric ice crystal. J. Atmos. Sci., 53, 2813–2825.
McFarquhar, G. M., , Yang P., , Macke A., , and Anthony J. B., 2002: A new parameterization of single scattering solar radiative properties for tropical anvils using observed ice crystal size and shape distributions. J. Atmos. Sci., 59, 2458–2478.
Minnis, P., , Takano Y., , and Liou K. N., 1993a: Inference of cirrus cloud properties using satellite-observed visible and infrared radiances, Part I: Parameterization of radiance fields. J. Atmos. Sci., 50, 1279–1304.
Minnis, P., , Heck P. W., , and Young D. F., 1993b: Inference of cirrus cloud properties using satellite-observed visible and infrared radiances. Part II: Verification of theoretical cirrus radiative properties. J. Atmos. Sci., 50, 1305–1322.
Mitchell, D. L., , and Arnott W. P., 1994: A model predicting the evolution of ice particle size spectra and radiative properties of cirrus clouds. Part II: Dependence of absorption and extinction on ice crystal morphology. J. Atmos. Sci., 51, 817–832.
Nasiri, S. L., , Baum B. A., , Heymsfield A. J., , Yang P., , Poellot M., , Kratz D. P., , and Hu Y., 2002: The development of midlatitude cirrus models for MODIS using FIRE-I, FIRE-II, and ARM in situ data. J. Appl. Meteor., 41, 197–217.
Nussenzveig, H. M., , and Wiscombe W. J., 1980: Efficiency factors in Mie scattering. Phys. Rev. Lett., 45, 1490–1494.
Purcell, E. M., , and Pennypacker C. R., 1973: Scattering and absorption of light by nonspherical dielectric grains. Astrophys. J., 186, 705–714.
Takano, Y., , and Liou K.-N., 1989a: Solar radiative transfer in cirrus clouds. Part I: Single-scattering and optical properties of hexagonal ice crystals. J. Atmos. Sci., 46, 3–19.
Takano, Y., , and Liou K.-N., 1989b: Solar radiative transfer in cirrus clouds. Part II: Theory and computation of multiple scattering in an anisotropic medium. J. Atmos. Sci., 46, 20–36.
Um, J., , and McFarquhar G. M., 2007: Single-scattering properties of aggregates of bullet rosettes in cirrus. J. Appl. Meteor. Climatol., 46, 757–775.
Vogelmann, A. M., , and Ackerman T. P., 1995: Relating cirrus cloud properties to observed fluxes: A critical assessment. J. Atmos. Sci., 52, 4285–4301.
Warren, S. G., , and Brandt R. E., 2008: Optical constants of ice from the ultraviolet to the microwave: A revised compilation. J. Geophys. Res., 113, D14220, doi:10.1029/2007JD009744.
Wendisch, M., , Yang P., , and Pilewskie P., 2007: Effects of ice crystal habit on the thermal infrared radiative properties and forcing of cirrus. J. Geophys. Res., 112, D08201, doi:10.1029/2006JD007899.
Yang, P., , and Liou K.-N., 1996: Geometric-optics–integral-equation method for light scattering by nonspherical ice crystals. Appl. Opt., 35, 6568–6584.
Yurkin, M. A., , Maltsev V. P., , and Hoekstra A. G., 2007: The discrete dipole approximation for simulation of light scattering by particles much larger than the wavelength. J. Quant. Spectrosc. Radiat. Transfer, 106, 546–557.
Zhang, Z., , Yang P., , Kattawar G. W., , Tsay S.-C., , Baum B. A., , Hu Y., , Heymsfield A. J., , and Reichardt J., 2004: Geometric optics solution to light scattering by droxtal ice crystals. Appl. Opt., 43, 2490–2499.
Zhang, Z., , Yang P., , Kattawar G. W., , Riedi J., , Labonnote L. C., , Baum B. A., , Platnick S., , and Huang H. L., 2009: Influence of ice particle model on retrieving cloud optical thickness from satellite measurements: Model comparison and implication for climate study. Atmos. Chem. Phys., 9, 7115–7129.
