1. Introduction

To properly represent the radiative properties of ice clouds in cloud and climate models, accurate optical properties of individual ice crystals are needed. For example, Vogelmann and Ackerman (1995) estimate that scattering asymmetry parameters must be known to within about 2%–5%, or about 0.02–0.04 in absolute terms, to constrain computed shortwave fluxes to within about 5%. For ice crystal scattering properties, in particular the asymmetry parameter, the shape and geometry of ice crystals and their level of surface roughness are especially important. Several databases and parameterizations are available that attempt to provide ice optical properties for a large part of the virtually limitless variation of ice crystal geometries in natural clouds leading to simulated asymmetry parameters of ice crystals that vary from about 0.7 to over 0.9 depending on crystal geometry and surface roughness (e.g., Fu 1996; Mitchell 1996; Fu 2007; Baran and Labonnote 2007; Yang et al. 2013; van Diedenhoven et al. 2014a). For example, the database of Yang et al. (2013) provides optical properties for a range of individual ice crystal habits with preselected geometries and surface roughness levels. Others (e.g., Baran and Labonnote 2007; Baum et al. 2005, 2014; Liu et al. 2014b) aim to construct mixtures of two or more predetermined crystal habits that provide the best match to a range of available data on average. Another approach is taken by Fu (2007) and van Diedenhoven et al. (2014a), who use the fact that the scattering properties, in particular the asymmetry parameter, of individual ice crystals mainly depend on the aspect ratio of their hexagonal components and on their level of microscale crystal surface roughness or distortion¹ (Iaquinta et al. 1995; Macke et al. 1996; Um and McFarquhar 2007, 2009; Yang and Fu 2009; Baran et al. 2009). Thus, the approaches of Fu (2007) and van Diedenhoven et al. (2014a) use hexagonal columns or plates as radiative proxies for more complex polycrystals consisting of hexagonal components. A commonality of all these databases and parameterizations is that, to represent the scattering properties of natural ensembles of ice crystals with varying ranges of geometries and crystal distortion levels occurring in a given cloud volume, they use one or a limited set of ice crystal geometries and distortion levels. For example, to represent the mean scattering properties of natural ice columns within a certain size bin, a single hexagonal column is used with a single roughness level and an aspect ratio that is the approximate mean value derived from observations at those sizes. Thus, such approaches use ensemble-average values of ice crystal aspect ratios and distortion levels that aim to represent the scattering properties of natural ensembles of ice crystals. However, the use of such ensemble-average aspect ratios and distortion levels to estimate the ensemble-average scattering properties has not been appropriately evaluated to date.

Until recently, very little information about the natural variation of roughness and aspect ratios of hexagonal ice prisms and components of complex ice crystals has been available (e.g., Auer and Veal 1970; Mitchell and Arnott 1994; Yang and Liou 1998). Recent developments in laboratory and in situ observations provide techniques to determine detailed information about the geometry and roughness of individual ice crystals and their components (e.g., Ulanowski et al. 2014; Castellano et al. 2014; Um et al. 2015). Furthermore, recent electron-microscopic imaging studies on ice crystals (Neshyba et al. 2013; Magee et al. 2014) show potential to provide physical ice surface roughness metrics and to determine how crystal roughness varies with temperature and humidity. Such measurements help constrain optical models for climate models or radiative closure studies such as conducted by Kindel et al. (2010). Usually, measured or estimated ice properties, such as mass and area, are reported after being averaged over size bins and/or time and temperature ranges (e.g., Lawson et al. 2010; Jackson et al. 2015). As demonstrated below, however, different strategies for averaging measured ice crystal geometry and roughness may yield inconsistent results. In addition, recent developments in ice cloud modeling include single-particle ice-growth treatments that allow ice crystal aspect ratios to evolve continuously as a result of vapor growth and riming (Sulia and Harrington 2011; Jensen and Harrington 2015). Implementations of such models allow ensembles of ice crystals of several classes with continuously varying aspect ratios to evolve over time in each model grid box. Thus, the common practice of using precalculated lookup tables of ice optical properties based on predetermined size-dependent ice crystal geometries is not suitable for such models; their optical properties need to be computed “on the fly” instead.

Appropriate scattering properties for ensembles of ice crystals are calculated by averaging the scattering properties of all members of such ensembles weighted by their scattering cross sections (Baum et al. 2005). However, since calculating the scattering properties of every ensemble member can be impractical and can lead to a large computational burden, as an alternative scattering properties can be estimated using average properties of the ensemble members (Fu 2007). Furthermore, to link geometries of ice crystals measured in situ or in the laboratory or those represented in cloud models to ensemble-averaged values used in the radiation and remote sensing applications, information is needed about the optimal approach to average aspect ratios and distortion parameters of ensemble members to obtain radiatively equivalent aspect ratios and distortion parameters. In this paper, we will explore different strategies for computing ensemble-average distortion parameters and aspect ratios for mixtures of plate- and columnlike particles to estimate their ensemble-average asymmetry parameter.

The recent advances in in situ measurement discussed above can provide valuable datasets to evaluate retrievals of characteristic ice crystal shapes and roughnesses at cloud tops using airborne- or satellite-based multidirectional polarization measurements (van Diedenhoven et al. 2012, 2013, 2014b; Baum et al. 2014; Cole et al. 2014). For a given sensor pixel, such remote sensing techniques allow inference of a single ice crystal model that is consistent with the average properties of the ensemble of ice crystals in the tops of clouds within the field of view. To evaluate such retrievals with collocated in situ measurements, an appropriate strategy is needed to consistently average the in situ–measured crystal geometries and roughnesses. Furthermore, it needs to be determined whether the inferred ensemble-averaged crystal geometry and roughness are consistent with the ensemble-averaged scattering properties. Here, we focus on the approach by van Diedenhoven et al. (2012) to infer crystal-component aspect ratios, distortion parameters, and asymmetry parameters from satellite or aircraft measurements. The accuracy of the retrieved asymmetry parameters were evaluated using simulated measurements assuming bullet rosettes, complex crystals, and mixtures of complex crystals. Furthermore, the accuracy of the retrieved aspect ratios of arms of bullet rosettes was evaluated using simulated measurements. However, the ability to retrieve radiatively appropriate, ensemble-average aspect ratios, and distortion parameters has not been evaluated yet. This will be another subject of this paper.

This study focuses on the calculation and retrieval of effective asymmetry parameters for particles that are large enough for geometric optics to apply, which is generally assumed to be true for size parameters larger than about 100 (Bi et al. 2014). Small ice crystals are generally found to be compact and thus variation in crystal geometry is of less importance (Baum et al. 2011). Effective extinction coefficients and single scattering albedos are not considered here, since, for size parameters larger than about 100, extinction coefficients are close to 2 and independent of particle geometry (e.g., Bi et al. 2014), while single scattering albedos only minimally depend on particle geometry and are not affected by crystal distortion (van Diedenhoven et al. 2014a). Finally, although an evaluation of effective phase functions from ensemble-average aspect ratio and distortion parameter is beyond the scope of this paper, of primary concern for such effective phase functions is the accuracy of their asymmetry parameters.

The definition of ensemble-average aspect ratio and distortion parameter is discussed in section 2. Section 3 describes the evaluation of using ensemble-average aspect ratios and distortion parameters for estimating ensemble-average asymmetry parameters. The ability of the remote sensing technique using multidirectional polarization measurements to retrieved ensemble-average aspect ratio, distortion parameters, and asymmetry parameters is evaluated in section 4. We conclude the paper in section 5.

2. The definition of ensemble-average aspect ratio and distortion parameter

For a single hexagonal ice particle, aspect ratio α is commonly defined as

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where L is the hexagonal prism length and D is the prism width. This definition yields aspect ratios below unity for plates and greater than unity for columns. However, this definition is arbitrary in that aspect ratio is sometimes defined as the inverse of Eq. (1) (e.g., Fu 2007; Yang and Fu 2009). A similar definition as Eq. (1) can be used to define aspect ratios of hexagonal components of complex crystals, such as the bullets of a bullet rosette (e.g., Iaquinta et al. 1995; Um et al. 2015). Alternatively, an aspect ratio that has unity as an upper bound for both columns and plates can, for example, be defined as

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(We use “log” to denote the logarithm base 10 throughout.) With such a definition, it needs to be noted separately whether crystals are column or plates.

Using the parameterization of van Diedenhoven et al. (2014a), Fig. 1 shows that the asymmetry parameter of a hexagonal prism at 865 nm increases as aspect ratio increasingly deviates from approximately unity. This is because of the increase of parallel surface areas leading to greater probability of light passing through the particle with a minimal change of direction (Yang and Fu 2009). Furthermore, Fig. 1 shows that the asymmetry parameter systematically decreases with increasing distortion parameter, since crystal distortion increases the probability of light changing direction as it passes through the particle (Macke et al. 1996). Here, crystal distortion parameter δ is defined by the parameterization included in the geometrics optics code developed by Macke et al. (1996). This ray-tracing code takes crystal distortion into account in a statistical manner by perturbing, for each interaction with a ray, the normal of the crystal surface from its nominal orientation by an angle varied randomly with uniform distribution between 0° and δ × 90°. This distortion parameterization can be considered as a proxy for the randomization of the angles between parts of crystal facets caused by crystal surface roughening or other distortions of the solid hexagonal structure of ice crystals (Yang et al. 2008; Liu et al. 2014a). Comparisons between various definitions and parameterizations of crystal surface roughness are provided by Neshyba et al. (2013) and Geogdzhayev and van Diedenhoven (2015, manuscript submitted to J. Quant. Spectrosc. Radiat. Transfer). The increase of asymmetry parameter with aspect ratio deviating from unity is somewhat stronger for plates than for columns when plotted on a logarithmic scale as in Fig. 1 owing to the relatively larger parallel surfaces of plates (Macke et al. 1996; Yang and Fu 2009). Interestingly, the minimum asymmetry parameter occurs not quite at unity aspect ratio, but at for pristine crystals, decreasing to when the distortion parameter is 0.7. Previous results by, for example, Yang et al. (2008) and Yang and Fu (2009), also show a minimum asymmetry parameter at aspect ratios associated with thick plates rather than at unity, although this was not noted in those studies. Analysis provided by Yang and Fu (2009) suggests that this offset is mostly related to rays that undergo two sequential refractions without undergoing an internal reflection [ in the terminology used by Yang and Fu (2009)]. Also, we note that a plate with an aspect ratio of 0.86 is the most compact hexagonal prism—that is, with the lowest ratio of volume to projected area. Further discussion and analysis of this offset is beyond the scope of this study.

Asymmetry parameters of hexagonal ice crystals at a wavelength of 865 nm as a function of aspect ratio α. The solid, dashed, and dashed–dotted lines are for distortion parameters of 0, 0.4, and 0.7, respectively.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0150.1

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Asymmetry parameters of hexagonal ice crystals at a wavelength of 865 nm as a function of aspect ratio α. The solid, dashed, and dashed–dotted lines are for distortion parameters of 0, 0.4, and 0.7, respectively.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0150.1

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Asymmetry parameters of hexagonal ice crystals at a wavelength of 865 nm as a function of aspect ratio α. The solid, dashed, and dashed–dotted lines are for distortion parameters of 0, 0.4, and 0.7, respectively.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0150.1

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Natural clouds consist of ensembles of ice crystals with a range of aspect ratios and crystal distortion levels. To estimate the asymmetry parameter of an ensemble of ice crystals with a distribution of aspect ratios, Fu (2007) defined an ensemble-average aspect ratio as

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where N and denote the size distribution and the projected area, respectively. The aspect ratio is weighted by the projected area of the crystals, assuming random orientation, since ensemble-average scattering properties are weighted with the scattering cross section, which is proportional to projected area in the geometric optics approximation at nonabsorbing wavelengths (Macke et al. 1996; Baum et al. 2005). Note that Eq. (3) assumes aspect ratio varies with crystal size. However, aspect ratio can vary independently of crystal size. A more general, discretized formula for the total area and number weighting of any quantity x is given by

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Here, we will denote the estimates of the ensemble asymmetry parameters that are computed using the ensemble-average aspect ratio by effective asymmetry parameters . Note that the correct ensemble-average asymmetry parameter for any distribution of particles can be computed using Eq. (4) and (cf. Baum et al. 2005).

As mentioned above, the definition of aspect ratio α given by Eq. (1) is arbitrary, but the result of averaging described by Eqs. (3) and (4) (with ) depends on the definition of α. For example, consider two hexagonal columns with equal area and with aspect ratios [defined by Eq. (1)] of 1.25 and 10. Following Eq. (4), the ensemble-average aspect ratio would be simply . However, when the aspect ratio is defined as the inverse of Eq. (1), the ensemble-average aspect ratio would be 0.45, which is equivalent to 2.22 in the definition of Eq. (1). This dependence on aspect ratio definition can be avoided by taking the geometric mean of the aspect ratio instead of the arithmetic mean by taking x = logα; that is,

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In section 3a, we will investigate which of the three averaging strategies (arithmetic means of or or geometric means) is optimal for computing average aspect ratios for crystal ensembles to estimate their ensemble-average scattering asymmetry parameters.

Owing to conditions in cloud volumes that vary over time or to migration of ice crystals within clouds, crystals with plate- and columnlike components can coexist in the same cloud volume. For the same reasons, platelike and columnlike components can be internally mixed within single particles (Bailey and Hallett 2009). The arbitrariness of the definition of α has even greater implications when considering such mixtures of crystals with plate- and columnlike components. For example, when two prisms with equal area and aspect ratios of 0.1 and 10 are considered, Eq. (4) using yields an average aspect ratio of 5.05. This average is obviously largely determined by the aspect ratio value larger than unity. Alternatively, using geometric means (i.e., x = logα) yields an average aspect ratio of unity for our example, although the asymmetry parameters at aspect ratios of 0.1 and 10 are both substantially larger than the value at an aspect ratio of unity (Fig. 1). The optimal strategy for computing average aspect ratio for mixtures of plates and columns will be discussed in section 3b.

Equation (4) can also be used to define an ensemble-average distortion parameter by using . Here crystal distortion parameter δ is defined as by Macke et al. (1996), but the same ensemble-average distortion parameter can be defined using other distortion definitions. Generally, crystal ensembles that assume only a single distortion parameter are considered for remote sensing and simulation purposes (e.g., Yi et al. 2013; van Diedenhoven et al. 2014b; Cole et al. 2014), although crystal ensembles in natural ice clouds likely have a variety of distortion parameters (van Diedenhoven 2014). The ensemble-average distortion parameters for estimating the asymmetry parameter for mixtures of hexagonal prisms with varying distributions of distortion parameters will be evaluated in section 3c. Finally, ensemble-average aspect ratios and distortion parameters for estimating the asymmetry parameter of ensembles with aspect ratios and distortion parameters both varying are evaluated in section 3d.

3. Ensemble-average aspect ratios and distortion parameters for estimating asymmetry parameters

a. Ensembles of plates or columns

First, we will evaluate the use of ensemble-average aspect ratios for ensembles of either hexagonal plates or columns. Let us return to the simple example posed in section 2 assuming an ensemble of only two hexagonal columns with equal area assuming random orientation and with aspect ratios [defined by Eq. (1)] of 1.25 and 10. The crystals are assumed large enough for geometric optics approximations to apply. We use the parameterization of van Diedenhoven et al. (2014a) to calculate the asymmetry parameters for such crystals at a wavelength of 865 nm and with a distortion parameter of zero, yielding and for and , respectively. Since the particles have equal area, and scattering properties for crystal ensembles are computed through area-weighted averages (Baum et al. 2005), the asymmetry parameter of this two-particle ensemble is simply the arithmetic mean value (i.e., 0.831). Using arithmetic means with and and geometric means (x = logα), Table 1 shows the resulting values of and the corresponding values of the effective asymmetry parameter . Furthermore, the difference between the ensemble-average asymmetry parameter and is also given in Table 1. From this example it is clear that, for columns, using yields unfavorable results of , while the absolute errors in are smaller and comparable for this example when using and x = logα. Note that, for columns, using is effectively the same as using the aspect ratio defined by Eq. (2). Next, the exercise is repeated with hexagonal plates with aspect ratios of 0.1 and 0.8. For hexagonal plates, Table 1 shows that using either or gives unfavorable results for , while the absolute errors in are small when using x = logα. Similar results are obtained at other visible and shortwave infrared wavelengths. From this simple demonstration it is already obvious that using arithmetically averaged aspect ratios that are greater than unity yields inappropriate ensemble-average aspect ratios and consequently erroneous effective asymmetry parameters. Therefore, a definition of aspect ratio as given by Eq. (2) is more appropriate than that given by Eq. (1).

Table 1.

Two-particle examples of computations of effective asymmetry parameters from ensemble-average aspect ratios obtained by Eq. (4) using different methods. The aspect ratios α and asymmetry parameters g of the two ensemble particles are given as well as the average asymmetry parameter and the difference between and .

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Next, we investigate the accuracy of for ensembles with lognormally distributed aspect ratios. To the best of our knowledge, there is little to no information on shape of distributions of aspect ratios of ice crystals (or the components of complex crystals) in natural clouds. However, our conclusions are independent of the shape of the aspect ratio distribution used. Using the aspect ratio definition given in Eq. (2), the mode of the lognormal distribution is varied between 0.05 and 1 and the geometric standard deviation is varied from 0.005 to 0.5. Values of are removed from the distributions and the distributions are renormalized. Here, a distortion parameter of 0.4 is assumed, but similar results are obtained for other distortion parameter values. Furthermore, similar results are generally obtained for visible and shortwave infrared wavelengths other than the 865 nm assumed here, although smaller errors are obtained at low single-scattering albedos since crystal absorption reduces the contribution of the refraction plus refraction asymmetry parameter to the total asymmetry parameter (Macke et al. 1996). The single-scattering albedo itself only varies minimally with aspect ratio and is not affected by crystal distortion (van Diedenhoven et al. 2014a). For simplicity, all crystals are assumed to have equal projected areas. Assuming projected areas that vary with aspect ratio would only change the weighting of both aspect ratio and asymmetry parameters per Eq. (4), but would not influence our conclusions. The ensemble asymmetry parameter for such distributions of particles can be computed using Eq. (4) and (Baum et al. 2005). For columns and plates, Fig. 2 shows the difference between the ensemble-average average asymmetry parameter and when using Eq. (4) and . Since the parameterization of van Diedenhoven et al. (2014a) assumes the definition of aspect ratio given by Eq. (1), the computed ensemble-average aspect ratios for columns are inverted before calculating . Consistent with our simple test shown in Table 1, Fig. 2 shows that using leads to relatively accurate values of for columns with maximum absolute values of 0.0065, but not for plates, for which absolute values vary to up to 0.019. Figure 3 shows that using geometric means (x = logα) leads to much better results for plates with maximum absolute values of 0.0067, although maximum absolute values for columns increase slightly to 0.0083.

Differences between the ensemble-average asymmetry parameter and calculated using arithmetic averages of aspect ratios for ensembles with lognormally distributed aspect ratios having varying modal values and geometric standard deviations . Ensembles of (left) plates and (right) columns. A distortion parameter of 0.4 is assumed.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0150.1

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Differences between the ensemble-average asymmetry parameter and calculated using arithmetic averages of aspect ratios for ensembles with lognormally distributed aspect ratios having varying modal values and geometric standard deviations . Ensembles of (left) plates and (right) columns. A distortion parameter of 0.4 is assumed.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0150.1

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Differences between the ensemble-average asymmetry parameter and calculated using arithmetic averages of aspect ratios for ensembles with lognormally distributed aspect ratios having varying modal values and geometric standard deviations . Ensembles of (left) plates and (right) columns. A distortion parameter of 0.4 is assumed.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0150.1

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As in Fig. 2, but using geometric averages of aspect ratios .

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As in Fig. 2, but using geometric averages of aspect ratios .

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As in Fig. 2, but using geometric averages of aspect ratios .

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The results presented here are based on ensembles of hexagonal prisms. However, it has been shown elsewhere that complex habits consisting of hexagonal components have scattering phase functions that closely resemble the scattering phase functions of the individual components (Iaquinta et al. 1995; Um and McFarquhar 2007, 2009; Fu 2007; Baran et al. 2009). Thus, our conclusions are likely qualitatively applicable to ensembles of more complex structures such as aggregates of columns, aggregates of plates, and bullet rosettes constructed of components with a range of aspect ratios.

b. Mixtures of plates and columns

Next, we investigate the optimal strategy for computing ensemble-average aspect ratios for mixtures containing both plates and columns. For this we construct plate ensembles with lognormally distributed aspect ratios α as described in section 3a. The mode of the lognormal distribution is varied between 0.05 and 1 in 10 steps and the geometric standard deviation is varied from 0.005 to 0.5 in 10 steps. Values of are removed from the distributions and the distributions are renormalized. Values of remain in the distributions and are considered columns. For column ensembles, the same distributions are constructed and then inverted to yield modal values of α greater than 1. Subsequently, for each mixture, the plate and column ensembles are added together using a weighting factor varied between 0.1 and 1 in 10 steps. In total, 10⁵ combinations are included. Again, all crystals are assumed to have equal projected areas and a distortion value of 0.4.

As discussed in section 2, using the definition of aspect ratio given by Eq. (1) would lead to mean aspect ratios that are primarily determined by the column aspect ratio values greater than unity. Hence, we restrict ourselves to using arithmetic or geometric means of for computing ensemble-average aspect ratios with Eq. (4) ( or , respectively). Subsequently, when the fraction of columns contributing to a given mixture’s total area is lower or greater than 50%, the computed ensemble-average aspect ratio is converted into an aspect ratio with the definition of Eq. (1) assuming plates or columns (i.e., or ), respectively, before the effective asymmetry parameter is calculated using the parameterization of van Diedenhoven et al. (2014a).

In Fig. 4, we show box-and-whisker plots for the differences between the ensemble-average asymmetry parameters and values of when using either arithmetic or geometric averages of . Mean values of using arithmetic or geometric averages are −0.0067 and −0.0012, respectively, absolute maximum values are 0.053 and 0.032, respectively, and root-mean-squared values are 0.0094 and 0.0061, respectively. Thus, while errors resulting from the use of a mean aspect ratio calculated using arithmetic averaging of are generally small, better results are obtained calculating geometric averages. Figure 4 shows that largest errors when using arithmetic averages are obtained for distributions with ensemble-average aspect ratios between about 0.2 and 0.5, mostly for distributions with a strong contribution of plates, as could be expected from Fig. 2. When using geometric averaging of , the largest outliers are obtained for ensembles with relatively small ensemble-average aspect ratios, as could be expected from the results shown in Fig. 3. Further perusing the results, we find that the largest errors occur for cases that have approximately equal contributions of plates and columns.

Box-and-whisker plots of differences between the ensemble-average asymmetry parameter and calculated using arithmetic (green) and geometric (red) averages of aspect ratios for mixtures of columns and plates with varying distributions of aspect ratios. Results are shown for several intervals of ensemble-average aspect ratio . Whiskers and boxes show the total and interhalf ranges, respectively. Horizontal lines and stars show median and mean values, respectively. A distortion parameter of 0.4 is assumed.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0150.1

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Box-and-whisker plots of differences between the ensemble-average asymmetry parameter and calculated using arithmetic (green) and geometric (red) averages of aspect ratios for mixtures of columns and plates with varying distributions of aspect ratios. Results are shown for several intervals of ensemble-average aspect ratio . Whiskers and boxes show the total and interhalf ranges, respectively. Horizontal lines and stars show median and mean values, respectively. A distortion parameter of 0.4 is assumed.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0150.1

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Box-and-whisker plots of differences between the ensemble-average asymmetry parameter and calculated using arithmetic (green) and geometric (red) averages of aspect ratios for mixtures of columns and plates with varying distributions of aspect ratios. Results are shown for several intervals of ensemble-average aspect ratio . Whiskers and boxes show the total and interhalf ranges, respectively. Horizontal lines and stars show median and mean values, respectively. A distortion parameter of 0.4 is assumed.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0150.1

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c. Ensembles with varying distortion parameters

Similarly to the approach used for ensemble-average aspect ratios in section 3a, next we evaluate effective asymmetry parameters calculated from the ensemble-average distortion parameters obtained using Eq. (4) with [see Macke et al. (1996) for the definition of δ]. Asymmetry parameters are calculated with the parameterization of van Diedenhoven et al. (2014a) for 1000 mixtures of plates with aspect ratio of 0.5 and Gaussian-distributed distortion parameters between 0 and 0.7. Again, to the best of our knowledge, there is no reliable information on the shape of distributions of ice crystal distortion in natural clouds. The mode values and standard deviations of the Gaussian distributions are varied between between 0 and 0.7 and 0 and 0.4, respectively. Distortion parameter values outside the considered range () are removed from the distributions and the distributions are renormalized. Figure 5 shows the difference between the ensemble-average asymmetry parameter and calculated using . The ensemble-average asymmetry parameter is generally overestimated using the ensemble-average distortion parameter. For the considered distributions, the largest values of remain below 0.01 and are obtained for distributions with large modal values and large standard deviations. Similar results are obtained for crystals with other aspect ratios.

Differences between the ensemble-average asymmetry parameter and calculated using average distortion parameters for ensembles of plates with Gaussian-distributed distortion parameters δ with varying modal values and standard deviations . An aspect ratio of 0.5 is assumed.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0150.1

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Differences between the ensemble-average asymmetry parameter and calculated using average distortion parameters for ensembles of plates with Gaussian-distributed distortion parameters δ with varying modal values and standard deviations . An aspect ratio of 0.5 is assumed.

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Differences between the ensemble-average asymmetry parameter and calculated using average distortion parameters for ensembles of plates with Gaussian-distributed distortion parameters δ with varying modal values and standard deviations . An aspect ratio of 0.5 is assumed.

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d. Ensembles with varying aspect ratios and distortion parameters

Finally, we test the use of ensemble-average aspect ratios and distortion parameters for estimating the asymmetry parameters of ensembles of hexagonal ice crystals with distributions of in which aspect ratio and distortion vary both. For this, the same distributions as used in section 3b are used, but additionally Gaussian-distributed distortion parameters are used with modal values between 0 and 0.7, varied in eight steps, and standard deviations between 0.01 and 0.4, varied in five steps. In total, combinations are included.

The box-and-whisker plot in Fig. 6 shows that absolute maximum errors in asymmetry parameter and their interquartile ranges generally increase somewhat as compared to the results with a fixed distortion parameter shown in Fig. 4. Furthermore, owing to the general overestimation of asymmetry parameter when using ensemble-average distortion parameters, as shown in Fig. 5, mean errors are generally positively offset compared to the values shown in Fig. 4. This offset somewhat compensates the general underestimation of asymmetry parameters obtained when using arithmetic averages of that are seen in Fig. 4, although this compensation depends on the mode and width of the crystal distortion distribution. Mean values of using arithmetic and geometric averages are −0.0025 and 0.0055, respectively, while absolute maximum values are 0.059 and 0.041, respectively, and root-mean-squared values are 0.0079 and 0.0088, respectively. Thus, when both aspect ratios and distortion parameters vary, compensating errors lead to smaller mean errors in estimated asymmetry parameters when using arithmetic averages of compared to geometric averages, although maximum errors are larger when using arithmetic averages.

As in Fig. 4, but for ensembles with both aspect ratios and distortion parameters varying.

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As in Fig. 4, but for ensembles with both aspect ratios and distortion parameters varying.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0150.1

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As in Fig. 4, but for ensembles with both aspect ratios and distortion parameters varying.

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4. Remote sensing of ensemble-average aspect ratios, distortion parameters, and asymmetry parameters

A method to infer crystal-component aspect ratios, distortion parameters, and asymmetry parameters from satellite- or aircraft-based multidirectional polarization measurements was introduced by van Diedenhoven et al. (2012). This method uses single columns and plates with a virtually continuous range of aspect ratios and distortion parameters as radiative proxies of complex crystals consisting of hexagonal components. In short, the aspect ratio and distortion parameter of a proxy hexagonal ice prism are inferred by determining a best fit to multidirectional polarization measurements within a lookup table of corresponding simulated measurements calculated by assuming individual hexagonal particles with varying aspect ratios (α = 0.02–50, 51 in total) and distortion parameters (δ = 0–0.7, 15 in total). The best fit is determined by the combination of ice crystal aspect ratio and distortion parameter that leads to the lowest relative root-mean-squared difference between polarized reflectance measurements and simulated values over scattering angles between 100° and 165°. The aspect ratio, distortion value, and asymmetry parameter of the proxy ice crystal that yield the best fit are then considered the retrieved values.

In van Diedenhoven et al. (2012), this approach was evaluated using simulated measurements based on a large collection of solid and hollow and pristine and roughened ice crystal habits (Baum et al. 2011; Yang et al. 2013), and asymmetry parameters were found to be retrieved within about 0.04 (5%) for individual simulated measurements. Moreover, particles with plate- and columnlike components were found to be generally correctly identified. The aspect ratio of the components of bullet rosettes were found to be retrieved within about 20% in general, while the distortion parameter was shown to be retrieved within 0.05 on average in absolute terms. Furthermore, van Diedenhoven et al. (2012) demonstrated that the asymmetry parameters of smooth and roughened mixtures of complex crystals defined by Baum et al. (2011) are also generally retrieved within 0.04 (5%). Thus, in the case of ensembles of ice crystals, the retrieved aspect ratios and distortion parameters are those that lead to a retrieved asymmetry parameter that is generally consistent with the arithmetic-mean asymmetry parameter of the ice crystals in the targeted cloud. As could be concluded from the previous section, the retrieved aspect ratios and distortion parameters are therefore expected to be close to the ensemble averages of the aspect ratios () and distortion parameters of the ice crystals in the targeted cloud. To evaluate the technique’s ability to retrieve ensemble-average aspect ratios and distortion parameters for mixtures of ice crystals, here retrievals are performed on simulated measurements of clouds consisting on ensembles of ice crystals with varying aspect ratios and distortion parameters.

For this, 1000 mixtures of columns and plates are constructed, each consisting of two separate ensembles of columns and plates, respectively. The plate ensembles have lognormally distributed aspect ratios α with mode values randomly selected between 0.05 and 1 and geometric standard deviations randomly picked between 0.01 and 0.4. Values of are removed from the distributions and the distributions are renormalized. For columns, similar random distributions are constructed and then inverted to yield modal values of α greater than 1. Subsequently, for each mixture, the plate and column ensembles are added together using a weighting factor randomly selected between 0 and 1. Furthermore, Gaussian-distributed distortion parameters are used with mode values and standard deviations that are randomly varied between 0 and 0.7 and between 0 and 0.4, respectively. The distributions are set to zero outside the considered range of distortion parameters () and renormalized. Simulated measurements at a wavelength of 865 nm are calculated using the database of optical properties of hexagonal ice crystals described by van Diedenhoven et al. (2012). In this database, the aspect ratio of columns in this database 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 roughness parameter is varied between 0 and 0.7 in steps of 0.05. The optical properties in this database are integrated over the 1000 ensemble distributions. Again, crystals are assumed to be large enough for the geometric optics approximations to apply and the projected areas of all crystals are assumed to be equal. Simulated measurements are calculated assuming uniform cloud layers with cloud optical thickness of 5. Details about the forward simulations, the radiative transfer code and an analysis how retrieval results generally vary with observation geometry, cloud optical thickness, and instrument characteristics are provided by van Diedenhoven et al. (2012).

Figure 7 shows the box-and-whisker plot for the differences between the ensemble-average and retrieved asymmetry parameters for all simulated cases. As expected from the evaluation presented by van Diedenhoven et al. (2012), asymmetry parameters are generally retrieved to within 0.04. However, outliers occur when plates with low aspect ratios are a prominent component of the ensemble, which is consistent with conclusions by van Diedenhoven et al. (2012). Such thin plates have polarization phase functions with relatively weak features, even for pristine particles, leading to a large uncertainty in distortion parameter and subsequently a large uncertainty in asymmetry parameter. Nonetheless, within each asymmetry parameter bin, the absolute mean and median errors in retrieved asymmetry parameter are below 0.01 and absolute values of the upper and lower quartiles are below 0.02.

Box-and-whisker plots of errors in asymmetry parameters retrieved from simulated multidirectional polarization measurements as compared to the corresponding ensemble-average asymmetry parameters.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0150.1

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Box-and-whisker plots of errors in asymmetry parameters retrieved from simulated multidirectional polarization measurements as compared to the corresponding ensemble-average asymmetry parameters.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0150.1

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Box-and-whisker plots of errors in asymmetry parameters retrieved from simulated multidirectional polarization measurements as compared to the corresponding ensemble-average asymmetry parameters.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0150.1

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The box-and-whisker plot for the differences between the ensemble-average and retrieved aspect ratios is shown in Fig. 8. Here, geometric averaging is used to calculate the ensemble-average aspect ratios. Furthermore, retrieved aspect ratios are converted into the definition of Eq. (2) before the comparison to be consistent with the definition of the ensemble-average aspect ratios. Aspect ratios are mostly overestimated in the retrieval. The average absolute errors between retrieved and ensemble-average aspect ratio are all below 0.1 and the absolute values of the upper and lower quartiles are below 0.2. Further perusing the results, we find that the largest outliers occur for mixtures with approximately equal contributions of columns and plates. We note, however, that errors in asymmetry and distortion parameters for such equal mixtures of columns and plates are not particularly large. Figure 9 demonstrates the technique’s ability to determine whether a given crystal ensemble is dominated by columnar or platelike crystals. Generally the approach tends to bias toward columnlike retrievals, although for over 70% of the cases the dominant geometry is correctly retrieved. Furthermore, considering only cases that are dominated by over 3/4 plates or columns, the dominant geometry is correctly determined in about 90% of the cases.

As in Fig. 7, but for errors in retrieved ensemble-average aspect ratios.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0150.1

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As in Fig. 7, but for errors in retrieved ensemble-average aspect ratios.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0150.1

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As in Fig. 7, but for errors in retrieved ensemble-average aspect ratios.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0150.1

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Fraction of retrievals indicating columnlike aspect ratios () as a function of fraction of columns contributing to the simulated measurements. The dotted line is the 1:1 relationship.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0150.1

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Fraction of retrievals indicating columnlike aspect ratios () as a function of fraction of columns contributing to the simulated measurements. The dotted line is the 1:1 relationship.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0150.1

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Fraction of retrievals indicating columnlike aspect ratios () as a function of fraction of columns contributing to the simulated measurements. The dotted line is the 1:1 relationship.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0150.1

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The box-and-whisker plot in Fig. 10 shows that the mean and median differences between the ensemble-average and retrieved distortion parameters are generally within 0.1 and the maximum absolute values of the upper and lower quartiles are below 0.2. As discussed above, large absolute errors up to about 0.4 can occur for cases with a strong contribution of thin plates.

As in Fig. 7, but for errors in retrieved ensemble-average distortion parameters.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0150.1

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As in Fig. 7, but for errors in retrieved ensemble-average distortion parameters.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0150.1

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As in Fig. 7, but for errors in retrieved ensemble-average distortion parameters.

Citation: Journal of the Atmospheric Sciences 73, 2; 10.1175/JAS-D-15-0150.1

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In summary, the retrieval approach of van Diedenhoven et al. (2012) is shown to generally yield aspect ratios, distortion parameters, and asymmetry parameters that are largely consistent with the ensemble-average values. However, for mixtures with approximately equal contributions of columns and plates relatively large errors in retrieved aspect ratio can occur, while relatively large errors in distortion parameters and asymmetry parameters can occur for ensembles with a large contribution of thin plates (). These results also indicate that generally the linear polarization for an ensemble of ice crystals is closely matched by that of a single crystal with an aspect ratio and distortion value close to that of the ensemble mean, at least at scattering angles between 100° and 165°.

5. Conclusions

The shape of ice crystals and their level of crystal distortion substantially affect their scattering properties, in particular the asymmetry parameter. Simulated asymmetry parameters of ice crystals generally range from about 0.7 to over 0.9 depending on crystal geometry and distortion parameter, while it is estimated that scattering asymmetry parameters must be known to within about 2%–5%, or about 0.02–0.04 in absolute terms, to constrain computed shortwave fluxes to within about 5% (Vogelmann and Ackerman 1995). It is generally assumed that the scattering properties of natural ensembles of ice crystals with a range of geometries and crystal distortion levels occurring in a given cloud volume can be sufficiently represented using a limited number of ice crystal geometries and distortion levels. This paper discusses the accuracy of using ensemble-average values of aspect ratio and the distortion parameter of hexagonal ice prisms for the computation of ensemble-average scattering asymmetry parameters. Here, ensembles of either plates or columns, mixtures of plates and columns, ensembles of plates with distributions of distortion values, and ensembles with aspect ratios and distortion parameters both varying are considered.

It is shown that using crystal aspect ratios greater than unity generally lead to ensemble-average values of aspect ratio that yield effective asymmetry parameters that are inconsistent with the ensemble-average asymmetry parameters. When a definition of aspect ratio is used that limits the aspect ratio to unity for both plates and columns, the ensemble-average aspect ratio yields effective asymmetry parameters that are generally consistent with the ensemble-average asymmetry parameters. Furthermore, differences between ensemble-average asymmetry parameters and the effective values calculated from ensemble-average aspect ratios are generally considerably smaller when is averaged geometrically rather than arithmetically, especially for ensembles of plates. In the case of mixtures of plates and columns, we recommend geometrically averaging the plate and column aspect ratios in the ensemble and calculating the effective asymmetry parameter using a column or plate geometry when the contribution by columns to a given mixture’s total projected area is greater or less than 50%, respectively. Using this strategy, our tests on mixtures of plates and columns yield mean differences between ensemble-average asymmetry parameters and those calculated from ensemble-average aspect ratios of 0.0012, respectively, absolute maximum differences of 0.032, and root-mean-squared differences of 0.0061. The largest errors occur for some cases that have relatively small ensemble-average aspect ratios and approximately equal contributions of plates and columns.

Effective asymmetry parameters based on arithmetic averages of distortion parameters generally overestimate ensemble-average asymmetry parameters. For our tests, the largest overestimates were below 0.01 and were obtained for distortion distributions with large modal values and large standard deviations.

For cases where both aspect ratios and distortion parameters are varied, mean differences between ensemble-average asymmetry parameters and effective values calculated from ensemble-average aspect ratios and distortion parameters are 0.0055, while absolute maximum differences are 0.041, and root-mean-squared differences are 0.0088, when the aspect ratios are averaged geometrically. In contrast to other tests where only aspect ratios are varied, smaller mean errors of −0.0025 are obtained when aspect ratios are averaged arithmetically owing to compensation of errors from the use of ensemble-average aspect ratios on the one hand and ensemble-average distortion parameters on the other. However, absolute maximum errors in asymmetry parameter are larger, at 0.059, when using arithmetic averages.

This paper focuses on effective asymmetry parameters for particles that are large enough for geometric optics to apply. Smaller ice crystals are generally found to be compact and thus variation in crystal geometry is of less importance. The results presented are obtained for a wavelength of 865 nm, but generally similar results are obtained for other visible and shortwave infrared wavelengths. However, at strongly absorbing wavelengths errors are reduced as the contribution of the refraction plus refraction asymmetry parameter to the total asymmetry parameter decreases with decreasing single-scattering albedo. Furthermore, single-scattering albedo only varies minimally with aspect ratio and is not affected by crystal distortion (van Diedenhoven et al. 2014a).

The ability of the approach of van Diedenhoven et al. (2012) to retrieve ensemble-average aspect ratios, distortion parameters, and asymmetry parameters from simulated multidirectional polarization measurements was also evaluated. The approach was shown to generally yield retrieved aspect ratios and distortion parameters that are largely consistent with the ensemble-average values. The absolute mean error in retrieved asymmetry parameter is below 0.01. The average absolute errors in retrieved ensemble-average aspect ratio are below 0.1, and mean errors in the ensemble-average distortion parameters are also within 0.1. However, for mixtures with approximately equal contributions of columns and plates relatively large errors in retrieved ensemble-average aspect ratio can occur. Furthermore, relatively large errors in distortion parameters and asymmetry parameters can occur for ensembles with a large contribution of thin plates (). Generally the approach tends to be somewhat biased toward retrieving columnlike crystals, although for about 75% of the test cases the dominating geometry was correctly determined. Furthermore, only considering cases that are dominated for over 3/4 by either plates or columns yielded the correct dominating geometry in about 90% of the cases.

As discussed by Fu (2007) and van Diedenhoven et al. (2014a), the scattering properties of complex particles consisting of hexagonal components can be represented by single hexagonal columns or plates that serve as radiative proxies. Thus, although the study in this paper is based on ensembles of hexagonal prisms, the conclusions are likely qualitatively applicable to ensembles of more complex structures such as aggregates of columns, aggregates of plates, and bullet rosettes with a range of aspect ratios of their components. Moreover, the conclusions from this paper suggest that the asymmetry parameters of ensembles of complex particles can be sufficiently estimated from the asymmetry parameters of single hexagonal columns or plates with the corresponding ensemble-average aspect ratio and distortion parameter. As discussed by Fu (2007), a mean aspect ratio can also be defined for complex shapes such as bullet rosettes and aggregates by considering their individual components. While the calculations in this paper were made for external mixtures of plates and columns, the conclusions should also largely apply to internal mixtures of plates and columns, such as asymmetric bullet rosettes, aggregates of varying plates, and/or columns or plate-capped columns. However, this hypothesis should be evaluated in future work.