1. Introduction
A substantial uncertainty in properly representing ice clouds and consequently their climate feedbacks is characterizing their radiative properties (Stephens et al. 1990; Stackhouse and Stephens 1991; Schlimme et al. 2005; Fu 2007). The fundamental radiative properties of atmospheric ice crystals for atmospheric models are the ice extinction cross section, single-scattering albedo, and the first moment of the scattering phase function, commonly referred to as the asymmetry parameter (Coakley and Chylek 1975; Fu 1996; Yang et al. 2000; Fu 2007). An increasing number of parameterizations for these optical properties of ice clouds are available. Such parameterizations generally relate the optical properties in selected wavelength bands in terms of predicted or imposed bulk characteristics of the ice, such as effective size, shape, and ice water content (e.g., Fu and Liou 1993; Fu 1996; Wyser and Yang 1998; Kristjánsson et al. 1999; Yang et al. 2000; McFarquhar et al. 2002; Key et al. 2002; Edwards et al. 2007; Xie et al. 2012), sometimes in combination with atmospheric conditions, such as temperature and ice supersaturation (Kristjánsson et al. 2000; Baran 2012). Although ice crystals are found with a virtually countless variation of shapes (Baran 2009), the extinction cross section and single-scattering albedo of randomly oriented individual ice crystals mainly depend on their projected areas and volumes, while the asymmetry parameter is shown to be determined mainly by the aspect ratios (defined throughout as the ratio of hexagonal prism height to width) of the ice crystal hexagonal components and their microscale surface roughness or crystal distortion (Iaquinta et al. 1995; Macke et al. 1996; Fu 2007; Yang and Fu 2009; Baran 2009; van Diedenhoven et al. 2012a).
Parallel to advances in understanding ice optical properties are efforts to improve ice microphysics modeling schemes. Some of these schemes are becoming increasingly complex and calculate or parameterize ice fall speeds and capacitances (an electrostatic analog used to compute ice deposition and sublimation rates) in terms of ice mass, projected area, and aspect ratio (Böhm 1989, 1992; Heymsfield and Iaquinta 2000; Wood et al. 2001; Westbrook 2008; Westbrook et al. 2008; Sulia and Harrington 2011; van Diedenhoven et al. 2012b). Since mass m can be related to bulk ice volume V via V = m/ρi, where ρi is the density of bulk ice, the ice characteristics determining fall speeds and capacitances are largely the same as those determining ice optical properties. Only crystal distortion does not affect fall speeds and capacitances substantially while being important for ice crystal optical properties. Self-consistency within a model dictates that the same ice volume, area, and aspect ratio used in an ice microphysics scheme should also be used in a model’s radiative transfer scheme (Mitchell et al. 2008; Baran 2012). However, there is at present a lack of optical property parameterizations flexible enough to accommodate all possible combinations of ice volume, area, aspect ratio, and crystal distortion. Furthermore, optical properties parameterizations are usually given for a predetermined set of wavelength bands, which are not necessarily the same as those used in a model’s radiative transfer calculations. Finally, most current optical property schemes require many and large tables to be read by the radiative transfer algorithm.
In this paper, we present a simple yet flexible parameterization that provides the extinction cross section, single-scattering albedo, and asymmetry parameter of ice crystals for any combination of volume, projected area, aspect ratio (of crystal components), and crystal distortion and at any wavelength in the shortwave. Similar to previous parameterizations (e.g., Fu 1996, 2007), our scheme makes use of geometric optics approximations and the observation that optical properties of complex, aggregated ice crystals can be approximated by those of single hexagonal crystals with varying size, aspect ratio, and crystal distortion.
After describing the reference calculations, the theory, and the parameterization in section 2, we assess its accuracy in section 3. We provide an example of an application in section 4 and present conclusions in section 5.
2. Theory and parameterizations
Here, we first discuss the dependencies of extinction cross section, absorption cross section, and asymmetry parameter on ice crystal volume, projected area, aspect ratio, and crystal distortion before parameterizing those dependencies.
a. Reference calculations
The reference calculations are based on single hexagonal plates and columns. The optical properties for hexagonal plates and columns with random orientation are calculated using the geometric optics (GO) code developed by Macke et al. (1996). In this standard geometric optics implementation, the extinction cross section is twice the projected area of crystals in random orientation. This Monte Carlo ray-tracing code takes crystal distortion of ice crystals into account in a statistical manner by perturbing the normal of the crystal surface from its nominal orientation by an angle that, for each interaction with a ray, is varied randomly with uniform distribution between 0° and δ × 90°, where δ is referred to as the distortion parameter. Thus, δ does not represent a single realization of a distorted crystal but rather the stochastic large-scale distortion of a collection of ice crystals (Macke et al. 1996). For a large collection of ice crystals, microscale surface roughness and large-scale particle distortion both lead to a similar randomization of the angles between crystal facets, which in turn leads to the suppression of features in the scattering phase matrix (Yang et al. 2008; Neshyba et al. 2013). Increasing the number of impurities within ice crystals also has a similar effect (Hess et al. 1998). Thus, the distortion parameter used here can be considered as a proxy for the randomization of the angles between crystal facets caused by any of these effects.
Calculations are made for bands with the average wavelengths and complex refractive indices listed in Table 1. Various sets of calculations including different dimensions of the hexagonal particles are used to construct and evaluate the parameterization, as detailed in the next subsections. The relations between volume V, orientation-averaged projected area Ap, maximum half-width of the hexagonal planes ahex, and aspect ratio α of hexagonal plates and columns are given in appendix A.
Average wavelengths λa and real (mr) and imaginary (mi) parts of the refractive index used in the reference calculations. Values of mr, mi (Warren 1984; Warren and Brandt 2008), and λa are weighted by solar irradiance at the top of the atmosphere within each wavelength band. Band edges are given in Table 1 of Kato et al. (1999), with bands 9–15 there corresponding to band 9 here.
The reference calculations and thus the parameterization presented here are based on conventional geometric optics calculations. It is generally accepted that GO approximations are applicable to scattering size parameters above about 100 (e.g., Macke et al. 1995; Yang et al. 2000, 2004, 2013). (Here we define the particle scattering size parameter χscat as 2πrsph/λ, where rsph is the radius of a sphere with the equivalent projected area; i.e., 
b. Extinction cross section and single-scattering albedo
Single-scattering albedo for compact hexagonal ice crystals as a function of (a) effective distance and (b) absorption size parameter. Colors indicate different wavelength bands. The gray–red line represents the exponential fit described in the text. The blue dashed line is the result using ADT.
Citation: Journal of the Atmospheric Sciences 71, 5; 10.1175/JAS-D-13-0205.1
As in Fig. 1b, but for (a) hexagonal plates and (b) columns with various aspect ratios, as indicated by the colors. All bands listed in Table 1 with weak to moderate absorption (mi < 0.02) are included. Gray–red dashed lines indicate fits.
Citation: Journal of the Atmospheric Sciences 71, 5; 10.1175/JAS-D-13-0205.1
Differences between the parameterized single-scattering albedo for particles with an aspect ratio of unity and the calculated single-scattering albedos for several (a) plates and (b) columns with various aspect ratios. Gray–red dashed lines indicate fitted lognormal functions as described in the text.
Citation: Journal of the Atmospheric Sciences 71, 5; 10.1175/JAS-D-13-0205.1
Coefficients ci,j used by Eq. (9) to compute coefficients li in Eq. (8). See also Fig. 4, box 3.
Summary of the parameterization of single-scattering albedo (box 4), with the absorption size parameter in box 1, parameterization of single-scattering albedo for particles with aspect ratio of unity (ωα=1) in box 2, and the parameterization of the relative difference between ωα=1 and scattering albedo for particles with other aspect ratios in box 3. The geometric optics approximation for the extinction cross section is given in box 5.
Citation: Journal of the Atmospheric Sciences 71, 5; 10.1175/JAS-D-13-0205.1
Parameterized single-scattering albedo as a function of particle aspect ratio and absorption size parameter.
Citation: Journal of the Atmospheric Sciences 71, 5; 10.1175/JAS-D-13-0205.1
As given by Eq. (2), the extinction cross section is always twice the projected area of crystals in random orientation in the standard geometric optics implementation used here (see Fig. 4, box 5). This approximation generally holds for scattering size parameters above about 50 (e.g., Yang et al. 2000, 2004, 2013). For particles with smaller scattering size parameters (1–50), so-called edge effects and wave interference have to be taken into account (Mitchell 2000; Yang et al. 2013), which lead to a general increase of the extinction coefficient up to about 3 with decreasing size parameter and to oscillations in the extinction coefficient in size parameter space of about 20% or smaller. For yet smaller size parameters (χscat < 1), particles are comparable to or smaller than the wavelength and transition to the Rayleigh regime, associated with a decrease of extinction coefficient with decreasing size. Since the parameterizations presented in this paper are based on calculations that are based on the standard geometric optics approximations, these effects are neglected and Qe is assumed to be equal to 2 for all sizes. Thus, substantial errors in the extinction coefficient owing to the geometric optics approximation can be expected for small ice crystals. However, the contribution of these small crystals to the total extinction cross section is expected to be generally minor under many conditions, owing to their small projected area. To our best knowledge, a proper quantification of the errors in extinction cross sections and ice cloud radiation simulations resulting from the approximation Qe = 2 is not yet available. As discussed in section 2a, such an investigation is beyond the scope of the current paper and left for future work.
c. Asymmetry parameter
Diffraction asymmetry parameter of hexagonal ice crystals with an aspect ratio of unity as a function of scattering size parameter. The gray dashed line represents the exponential fit described in the text.
Citation: Journal of the Atmospheric Sciences 71, 5; 10.1175/JAS-D-13-0205.1
Summary of the parameterization of asymmetry parameter gtot (box 9), with parameterization of the diffraction asymmetry parameter in boxes 1 and 2, parameterization of ray-tracing asymmetry parameter at 862 nm in boxes 3–5, and correction factors for real refractive index and absorption in boxes 6–8.
Citation: Journal of the Atmospheric Sciences 71, 5; 10.1175/JAS-D-13-0205.1
The refraction-plus-reflection asymmetry parameter depends on particle volume, projected area, aspect ratio, distortion parameter, and complex refractive index. However, at any given nonabsorbing wavelength, gRT of single hexagonal ice crystals mainly depends on aspect ratio and crystal distortion (cf. Fu 2007; Yang et al. 2008; Yang and Fu 2009; van Diedenhoven et al. 2012a). For example, Fig. 8 shows the total asymmetry parameter at a nonabsorbing wavelength of λ = 862 nm as a function of δ for several aspect ratios. Note that these calculations are made for large particles (χscat ≫ 100), and thus gdif = 1 and the dependencies of the total asymmetry parameter on aspect ratio and distortion parameter seen in Fig. 8 are solely attributable to variations in gRT [see Eq. (13)]. Here, the wavelength band at 862 nm is used as a reference because it corresponds to similar bands commonly used in aircraft and satellite retrievals of ice cloud properties (e.g., Chepfer et al. 2001; King et al. 2004; van Diedenhoven et al. 2012a,b). As described by van Diedenhoven et al. (2012a), sizes are varied so that the projected areas of the particles, assuming random orientation, correspond to the projected areas of spheres with radii of 14, 20, 28, 40, 56, 80, 113, 160, 226, and 320 μm. The aspect ratio of columns is varied between 1 and 50 with 26 geometrically increasing steps. The aspect ratios of plates are the inverse of those for columns, for a total of 51 aspect ratios. The distortion parameter is varied between 0 and 0.8 in steps of 0.05. Since GO calculations are independent of size for large particles, results are simply averaged over all sizes in order to decrease numerical noise. In the following, we aim to first parameterize the total asymmetry parameter at λ = 862 nm from which we derive gRT at λ = 862 nm using Eq. (13). Subsequently, we derive factors to scale the resulting gRT to other wavelengths—that is, for other complex refractive indices.
Ice crystal total asymmetry parameters at 862 nm as a function of distortion parameter for various aspect ratios. Dashed lines represent the fits described in the text.
Citation: Journal of the Atmospheric Sciences 71, 5; 10.1175/JAS-D-13-0205.1
Figure 8 shows that asymmetry parameters increase as aspect ratio departs from unity owing to the increase of parallel surface areas, leading to greater probability of light passing through the particle with low orders of refraction plus reflection and a minimal change of direction (Yang and Fu 2009). Increase of crystal distortion increases the chance of light refracted away from forward directions, leading to decreased asymmetry parameters (Yang et al. 2008).
Figure 9 shows the parameterized gtot at 862 nm and the differences from the reference calculations (cf. Fig. 1 in van Diedenhoven et al. 2012a). The asymmetry parameter at 862 nm is seen to be approximated with an accuracy better than 0.004. The refraction plus reflection asymmetry parameter at 862 nm can be derived from this parameterization using gRT = 2gtot − 1, since gdif = 1 and ω = 1 [see Eq. (13)]. Next, we will demonstrate how this parameterized asymmetry parameter for λ = 862 nm can be scaled to other wavelengths—that is, for other complex refractive indices.
(top) Parameterized total asymmetry parameters at 862 nm and (bottom) the differences from the reference calculations.
Citation: Journal of the Atmospheric Sciences 71, 5; 10.1175/JAS-D-13-0205.1
The refraction-plus-reflection asymmetry parameter of nonabsorbing ice crystals as a function of real refractive index, relative to its value at 862 nm, or mr = 1.3038. Calculations for several plate aspect ratios are indicated by color. Dashed lines represent fits as described in the text with indicated fit parameters ϵ.
Citation: Journal of the Atmospheric Sciences 71, 5; 10.1175/JAS-D-13-0205.1
(a) Relative differences between calculated total asymmetry parameters and those estimated by using Eq. (12) and the refraction-plus-reflection asymmetry parameter for nonabsorbing particles (ω = 1). Calculations for different wavelength bands are indicated by colors. Filled circles represent compact particles (α = 1) and open diamonds indicate thin plates (α = 0.1). Only two aspect ratios are shown to reduce clutter. (b) Relative differences in the refraction-plus-reflection asymmetry parameter for particles with ω = 1 and values for particles with lower single-scattering albedos. Red lines indicate fits for α = 1 (dashed) and α = 0.1 (dashed–dotted) as described in the text. (c) As in (a), but using the derived factors as described in the text to correct the refraction-plus-reflection asymmetry parameter for absorption effects.
Citation: Journal of the Atmospheric Sciences 71, 5; 10.1175/JAS-D-13-0205.1
3. Results
a. Optical properties of individual particles
The parameterization is tested on GO calculations for hexagonal crystals with aspect ratios varying between 0.05, 0.1, 0.5, 1, 2, 5, and 20; effective distances of 10, 30, 50, 70, 90, 110, 150, and 190 μm; and distortion parameters of 0, 0.2, 0.4, and 0.6 and at wavelength bands listed in Table 1. For all wavelength bands with substantial absorption, absolute differences between calculated and parameterized single-scattering albedos are generally below 0.015, as seen in Fig. 12. Only for bands with mi > 0.02 (at 2.6 and 3.3 μm) are absolute differences up to 0.05 obtained, especially for small single-scattering albedos, which is expected considering the different dependence of ω on χabs for these bands seen in Fig. 1. Though not evident in Fig. 12, the smallest errors are obtained for relatively compact particles with 0.5 ≤ α ≤ 2.
Absolute difference between parameterized and calculated single-scattering albedos for several wavelength bands as indicated by colors. Circles, diamonds, and squares indicate hexagonal columns, plates, and compact hexagons with aspect ratio of unity, respectively.
Citation: Journal of the Atmospheric Sciences 71, 5; 10.1175/JAS-D-13-0205.1
Figure 13 shows the absolute differences between calculated and parameterized asymmetry parameters for all wavelength bands listed in Table 1. For nonabsorbing bands, the accuracy of the parameterization is better than 0.01. For moderately absorbing wavelengths, absolute differences between calculated and parameterized asymmetry parameters do not exceed 0.015, but the parameterization is slightly biased toward underestimating the asymmetry parameter, especially for larger, more absorbing particles. For strongly absorbing wavelengths (mi > 0.02), errors in the parameterized asymmetry parameter up to 0.05 are seen, which is attributable to the deviations of single-scattering albedo from the parameterization for these bands, and the fact that the correction of gRT for its dependence on the single-scattering albedo for these bands are not well represented by the fits (Fig. 11). The smallest errors are again obtained for relatively compact particles with 0.5 ≤ α ≤ 2.
As in Fig. 12, but for asymmetry parameter.
Citation: Journal of the Atmospheric Sciences 71, 5; 10.1175/JAS-D-13-0205.1
b. Application to flux calculations
Figure 14 shows errors in reflectance, transmittance, and absorptance resulting from the parameterization. Relative errors in reflectance generally increase with decreasing reflectance and the same is generally true for the error in transmittance. The error in absorptance peaks around absorptance values of 0.1, which occur mostly at optical thicknesses of 20–60. Errors in reflectances are mostly caused by errors in the parameterization of asymmetry parameter, while errors in transmittance and absorptance are mostly affected by parameterization errors in the single-scattering albedo, as might be expected. Overall median errors and interquartile ranges in reflectance, transmittance, and absorptance are listed in Table 4 for several aspect ratio ranges. Lowest errors and interquartile ranges are generally obtained for relatively compact crystals (0.5 ≤ α ≤ 2), while more extreme aspect ratios yield somewhat larger errors. The overall relative root-mean-squared errors in reflectance, transmittance, and absorptance are 1.4%, 1.1%, and 3.4%, respectively. Crystal distortion is found to have no significant effect on the resulting errors (not shown). Similar errors are also obtained for other solar zenith angles (not shown; cf. Fu 1996).
Relative parameterization errors in transmittance, reflectance, and absorptance of uniform cloud layers. The box-and-whisker plots show median values, 25th and 75th percentiles, and full ranges of the errors in several transmittance, reflectance, and absorptance bins. Cloud optical thicknesses range from 0.01 to 100 and wide ranges of different size distributions, aspect ratios, and distortion parameters are included, as described in the text. A solar zenith angle of 60° is assumed. Further details provided in text.
Citation: Journal of the Atmospheric Sciences 71, 5; 10.1175/JAS-D-13-0205.1
Medians and interquartile ranges (within parentheses) of relative errors in fluxes and absorptance owing to the proposed parameterization for several aspect ratio ranges. All values are percentages.
To put our results in perspective with those of other parameterizations, we note that the averaged absolute values of the relative errors in reflectance, transmittance, and absorptance of 1.1%, 0.7%, and 2.5%, respectively, resulting from our parameterization, are comparable to the corresponding values obtained by the parameterization of Fu (1996), which are 1.2%, 0.3%, and 2.9%, respectively, for a solar zenith angle of 60° (their Fig. 6). The absolute maximum errors in reflectance, transmittance, and absorptance of 6%, 4%, and 11%, respectively, are also comparable or somewhat larger than the corresponding maximum errors reported by Fu (1996), which are about 6%, 2%, and 8%. However, the parameterization scheme presented here is evaluated over a much wider range of particle size distributions, cloud optical thicknesses, aspect ratios, and distortion values than the scheme by Fu (1996). Moreover, our parameterization scheme is much more flexible than previous schemes.
Finally, we note that the errors given here do not include any possible errors owing to the application of the standard geometric optics approximations—the most important being the approximation Qe = 2. As stated above, quantification of such errors is currently lacking and left for future work.
4. Practical application in atmospheric models
It can now be considered conventional wisdom that shortwave optical properties of complex aggregates of hexagonal ice crystals at a given wavelength are mainly determined by their effective distance, distortion, and the aspect ratios of their components (e.g., Fu 1996, 2007; Um and McFarquhar 2007, 2009; van Diedenhoven et al. 2012a). Although the parameterization proposed here is based on single hexagonal plates and columns, it can provide optical properties of particles with any combination of volume, projected area, aspect ratio, and distortion parameter. In the case of complex aggregates, such as bullet rosettes and aggregates of columns or plates, the aspect ratio to be used in the parameterization is the mean aspect ratio of the components of the complex particle, as described by Fu (2007). For example, in the case of bullet rosettes, which are commonly defined as a structure of a number of bullets (usually six) that all have the same aspect ratio (Iaquinta et al. 1995; Yang and Liou 1998; Garrett et al. 2001), the aspect ratio to be used in the parameterization is the aspect ratio of the individual bullets, while the volume and projected area are quantities determined for the whole rosette. These assumptions are essentially the same as those made by the parameterization by Fu (1996, 2007).
The parameterization presented here can be used to compute the ice crystal shortwave optical properties that are consistent with the assumed (or predicted) particle mass, area, and aspect ratios. Crystal distortion is the only parameter important for ice shortwave optical properties that is not considered to affect ice microphysics. Although the parameterization is designed to be combined with microphysical schemes that use assumptions about particle mass, area, and aspect ratio, in the case of microphysics parameterizations that do not include assumptions about area and/or aspect ratio, reasonable assumptions for those quantities can be made that are consistent with the assumed mass–dimension relations and observations (e.g., Mitchell et al. 1996; Baker and Lawson 2006; Auer and Veal 1970; Um and McFarquhar 2007, 2009).
The parameterization described in this paper yields the shortwave optical properties of individual ice crystals that need to be integrated over size distributions. Such an integration can be efficiently performed by binning the size distribution (Baum et al. 2005b). Since the parameterization can yield optical properties for any particle size, an unlimited number of bins can be used for this numerical integration, although Liu et al. (2012) showed that using about 40 bins is generally sufficient.
To illustrate such an application of the parameterization, we calculate fluxes for a uniform cloud layer consisting of ice crystals with varying aspect ratios, distortion parameters, and projected-area and mass relationships. First, we assume mass–dimension and projected area–dimension relationships consistent with aggregates of side planes as given by Mitchell et al. (1996); that is, 
Illustration of variations of (a),(b) upwelling- and (c),(d) downwelling fluxes and (e),(f) cloud absorptance (left) as a function of aspect ratio and distortion and (right) as a function of aspect ratio and effective diameter. The cloud layer has an optical thickness of 4 and is embedded in a tropical atmosphere between 10 and 11 km. The solar zenith angle is 60°.
Citation: Journal of the Atmospheric Sciences 71, 5; 10.1175/JAS-D-13-0205.1
Second, to illustrate the additional dependence of radiative fluxes and cloud absorptance on projected-area and mass relationships, we artificially scale the projected area of the ice crystals to obtain bulk effective diameters between 25 and 200. [Essentially the same results are obtained by scaling masses as apparent from Eqs. (5), (21), and (22).] Figures 15b and 15d show the dependence of radiative fluxes on effective diameter and aspect ratio for smooth crystals (δ = 0). The dependence of the fluxes on effective diameter is seen to be weaker than the dependence on aspect ratio. Cloud absorptance, however, is strongly dependent on effective diameter but only weakly dependent on aspect ratio (Fig. 15f), as might be expected. Note that the dependence of absorptance on effective diameter and aspect ratio resembles the variation of single-scattering albedo seen in Fig. 5. Similar dependencies are obtained for crystals with greater distortion (δ > 0).
The example seen in Fig. 15 illustrates that ice crystal volume, projected area, aspect ratio, and distortion parameter that are inconsistent with a model’s microphysics can yield substantial errors in radiative fluxes and absorptance (cf. Baran 2012). Our parameterization provides for a consistent treatment that avoids such errors. Our approach uses the geometric optics approximations, which are less applicable for longwave radiation calculations. However, longwave radiation is mainly determined by absorption, which in turn is determined by particle volume and projected area, largely independent of particle shape (Fu et al. 1999). For longwave flux calculations, optical properties that are largely consistent with the shortwave optical properties obtained with our parameterization may be calculated by applying Lorentz–Mie theory to collections of spheres that conserve ice crystal volume and projected area (Grenfell and Warren 1999; Fu et al. 1999; Neshyba et al. 2003), although such an approach may lead to errors in cloud emissivity of order 10% mainly attributable to the overestimation of tunneling effects (Fu et al. 1999; Mitchell et al. 2006). The modified anomalous diffraction approximation (MADA; Mitchell et al. 2006) partly accounts for tunneling effects using shape-dependent tunneling efficiency factors.
5. Conclusions
The parameterization scheme for optical properties of individual ice crystals presented in this paper is based on the same principles as the well-known scheme developed by Fu (1996, 2007), namely, that optical properties of complex, aggregated ice crystals can be approximated by those of single hexagonal crystals with varying size, aspect ratio, and distortion parameter. For example, in the case of a bullet rosette, the aspect ratio used in such a parameterization is the aspect ratio of the individual bullets, while the volume and projected area are the quantities determined for the whole rosette. Previous schemes by Fu (1996, 2007) and others are restricted to preselected wavelength bands, particle geometries, and distortion levels. Our scheme further extends these concepts to provide single-scattering albedos and asymmetry parameters for individual ice crystals with any combination of volume, projected area, aspect ratio (of particle components), and distortion parameter and at any wavelength in the shortwave.
Similar to the scheme developed by Fu (1996, 2007), the parameterization of single-scattering albedo and asymmetry parameter is based on geometric optics calculations for single hexagonal crystals with varying size, aspect ratio, and crystal distortion. In the standard geometric optics implementation used here, the extinction cross section is simply twice the projected area of crystals in random orientation. The ice crystal single-scattering albedo is parameterized in terms of particle aspect ratio and an absorption size parameter that is proportional to the ratio of the imaginary part of the refractive index to the wavelength multiplied by the ratio of particle volume to projected area. The parameterization of single-scattering albedo uses 26 coefficients. The asymmetry parameter of hexagonal ice crystals is parameterized in terms of particle aspect ratio, crystal distortion, projected area, single-scattering albedo, and wavelength, using 62 coefficients in total.
The scheme is tested for a large variety of hexagonal crystals in several wavelength bands from 0.2 to 4 μm, revealing absolute errors in both single-scattering albedo and asymmetry parameter that are generally below 0.015. Furthermore, using a large variety of particle aspect ratios, size distributions, and cloud optical thicknesses, the relative root-mean-square errors in cloud reflectance, transmittance, and absorptance are shown to be 1.4%, 1.1%, and 3.4%, respectively. Note, however, that these errors given do not include any possible errors owing to the application of the standard geometric optics approximations, as a quantification of such errors is currently lacking and left for future work.
As we demonstrate, this parameterization scheme is flexible enough to obtain ice crystal optical properties that are consistent with any assumptions about ice crystal mass (equivalent to bulk volume), projected area, and aspect ratio, which might be used in modern ice microphysics schemes to compute fall speeds and depositional growth rates of ice particles. The scheme can also be used to estimate optical properties of ice crystals of which the mass, projected area, and component aspect ratio are derived from measurements of in situ probes or remote sensing (e.g., Mitchell et al. 1996; Korolev and Isaac 2003; Um and McFarquhar 2007; Lawson et al. 2010; Mauno et al. 2011; Lindqvist et al. 2012; van Diedenhoven et al. 2012a, 2013). Appropriate distortion parameter values can be derived from satellite and aircraft remote sensing measurements (van Diedenhoven et al. 2012a,b, 2013).
A Python computer code is provided as supplemental material.
Acknowledgments
This material is based upon work supported by the NASA ROSES program under Grant NNX11AG81G. We are grateful for the contributions from three anonymous reviewers. We thank William Martin for his help in coding the Python program. We would like to thank Dr. Petri Räisänen for reporting a typographical error in Eq. (8) in the early online release of this paper.
APPENDIX A
Geometrical Relations of Hexagonal Prisms
Maximum dimension D is equal to 2ahex for plates and 2αahex for columns, where ahex is the side length of the hexagon, which is the same as the maximum half-width of the hexagonal planes, and α is aspect ratio, determined throughout as the ratio of prism height to width.
Appendix B
Two-Stream Radiative Transfer
For wavelength bands that include high spectrally variant absorption lines, the correlated-k distribution method (Lacis and Oinas 1991) is used.
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