1. Introduction
With near unanimity, global climate models (GCMs) employ multilayer two-stream approximations to compute solar radiative fluxes (e.g., Coakley and Chýlek 1975; Meador and Weaver 1980; Zdunkowski et al. 1980).1 The varied abilities of two streams to handle scattering by cloud droplets and ice crystals (see King and Harshvardhan 1986) has led to those that employ delta scaling of optical properties (Joseph et al. 1976; Wiscombe 1977) being the models of choice for GCMs. Paralleling over three decades of widespread usage of two streams in GCMs, numerous studies have demonstrated the shortcomings of 1D solutions of the solar radiative transfer equation in general, with particular emphases on common two-stream approximations. The focus has been on demonstrating their limitations at representing either details of particulate optical properties (e.g., King and Harshvardhan 1986; Li and Ramaswamy 1996; Räisänen 2002) or their neglect of horizontal radiative transfer for media characterized by fluctuations at scales less than a few kilometers (Marshak and Davis 2005b; Hogan and Shonk 2013). As yet, however, these two classes of errors have not been assessed on a head-to-head basis.
The purpose of this study is to provide estimates of flux and heating rate biases that can be expected from two-stream-based multilayer broadband solar transfer codes, as used in GCMs, due to their necessary neglect of both details in scattering phase functions for cloud droplets and ice crystals and horizontal transport of photons at spatial scales below the resolution of GCMs. This was achieved using cloud properties inferred from CloudSat (Stephens et al. 2002) and CALIPSO (Winker et al. 2003) data, along with ECMWF state variables. The two-stream solar code employed here was that used in the Canadian Centre for Climate Modelling and Analysis (CCCma) GCM (von Salzen et al. 2013). A 3D Monte Carlo photon transport algorithm (Barker et al. 2012), which is effectively an infinite-stream approximation, provided benchmark estimates of fluxes and heating rates. Both transfer models used the CCCma’s clear-sky optical properties, as well as common descriptions of liquid and ice cloud optical properties, derived from the Lorenz–Mie scattering theory (Wiscombe 1980) and a combination of techniques (Yang et al. 2013), respectively.
CloudSat–CALIPSO cross sections, which consist of columns taken to be infinitely wide across track and Δx = 1 km along track, were partitioned into 256-km domains to approximately represent GCM cells. The CCCma two-stream model was applied to each 1-km column and averaged over 256 km, thus affecting the independent column approximation (ICA) (Stephens et al. 1991). Each 256-km domain was then acted on by the Monte Carlo model assuming various phase functions, as well as both Δx → ∞ (i.e., ICA mode) and Δx = 1 km. While the two-stream approximation could utilize only the asymmetry parameter of the particle scattering phase functions, the Monte Carlo used the exact phase functions. It is noted here that, because horizontal variability of cloud in the across-track direction was neglected, errors due to the ICA are underestimated relative to use of fully 3D clouds, potentially by up to ±30% for some clouds under some illumination conditions (cf. Barker 1996; Pincus et al. 2005). Domain-average fluxes and heating rate profiles predicted by the models were compared for data from January 2007.
In its native configuration, the CCCma two-stream approximation executes in the Monte Carlo ICA (McICA) framework (Pincus et al. 2003), as it lacks access to explicit distributions of unresolved clouds. Errors from the McICA method arise through limitations in stochastic generators used to conjure up subgrid-scale clouds (e.g., Räisänen et al. 2004). These errors were not addressed here, so this study was restricted to assessment of limitations in two-stream approximations.
The following section provides brief overviews of the radiative transfer models and cloud optical properties used in this study. The third section shows results for single-layer clouds irradiated by monochromatic, collimated radiation. This helps set the stage for the more complicated broadband results for A-Train atmospheres, which are shown in section 4. A summary and conclusions are drawn in the final section.
2. Radiative transfer models
The primary purpose of this study was to furnish estimates of bias errors in solar fluxes computed by two-stream approximations as a result of their neglect of details in cloud scattering phase functions and cloud geometry. The two-stream approximation was exemplified here by the popular δ-Eddington approximation (Joseph et al. 1976), which is used in the CCCma GCM (von Salzen et al. 2013). Zdunkowski et al.’s (1980) practical improved flux method was also used, but differences between it and the δ-Eddington were negligible. The two-stream ICA and the Monte Carlo model both used the same clear-sky optical properties as produced by the CCCma solar radiation code’s correlated k-distribution method, which has 31 quadrature points in cumulative probability space (Li and Barker 2005). They also used the same cloud optical properties. This section’s three subsections recapitulate the δ-Eddington and Monte Carlo algorithms as well as cloud optical properties.
a. δ-Eddington approximation
Using (4), (5), and (6) in the generalized two-stream solutions (Meador and Weaver 1980) with appropriate coefficients [see King and Harshvardhan’s (1986) Table 2] leads to expressions for layer reflectance and transmittance for both isotropic irradiance and direct-beam incident at solar zenith angle 
b. 3D Monte Carlo model
The 3D Monte Carlo algorithm used here has been described elsewhere (Barker and Davies 1992; Barker et al. 1998, 2003, 2012). It employs cyclic horizontal boundary conditions and can use either simple phase functions, such as 
For domains consisting of arrays of columns, 
c. Cloud optical properties
The optical properties of ice clouds are known to be important for various applications (Bi et al. 2014; Yang et al. 2015). In this study, the general habit mixture ice cloud model was adopted (Baum et al. 2011, 2014). Spectral optical properties for ice crystals were derived from a database developed by Yang et al. (2013), which covered wide ranges of crystal habits, discrete sizes, wavelengths, and surface roughness conditions. The Amsterdam discrete dipole approximation (ADDA) (Yurkin and Hoekstra 2011) was used for small size parameters, and the improved geometric-optics method (IGOM; Yang and Liou 1996), with inclusion of the edge effect (Bi et al. 2010, 2014), was used for moderate and large size parameters (see Yang et al. 2013). Particle surface roughness was not considered for small size parameters (ADDA calculations); however, moderately and severely rough conditions were considered fully for large size parameters (IGOM simulations).
Figure 1 shows size- and spectrally integrated phase functions for the spectral interval 0.25–0.69 μm. All functions exhibit very prominent peaks for 
Scattering phase functions as functions of scattering angle integrated over wavelengths from 0.25 to 0.69 μm and size distributions for (a) liquid droplets with re = 13 and 15 μm with υe = 0.1 and (b) ice crystals with De = 40 and 80 μm. Corresponding Henyey–Greenstein functions are shown for reference (see the appendix).
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0033.1
3. Results I: Narrowband fluxes for single-layer plane-parallel clouds
Errors in the δ-Eddington approximation that arise from use of minimal information pertaining to cloud particle phase functions can be difficult to explain when looking at broadband fluxes for multilayered inhomogeneous cloud systems. Hence, narrowband results are shown in this section for single homogeneous layers that contain cloud only (see King and Harshvardhan 1986; Li and Ramaswamy 1996). Section 4 addresses the δ-Eddington’s neglect of cloud geometry at the 1-km horizontal scale.
All Monte Carlo results reported in this section used 
The most basic test of the conventional, analytic δ-Eddington approximation is to compare it to a Monte Carlo model that is equipped with both 
Relative errors for (a) albedo and (b) transmittance of single-layer homogeneous layers that consisted of cloud droplets with re = 15 μm and υe = 0.1 between the δ-Eddington two-stream approximation [δ-Edd (2-strm)] and the Monte Carlo using the δ-Eddington phase function [δ-Edd (∞-strm)] as functions of cloud optical depth and 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0033.1
As in Fig. 2, but for single-layer homogeneous layers that consisted of ice crystals with De = 80 μm (see Fig. 1).
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0033.1
Solar-zenith-angle-dependent backscattered fractions 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0033.1
The third band of the CCCma radiation model, which spans λ from 1.19 to 2.38 μm, is responsible for most of the atmospheric cloud radiative effects (see Räisänen and Barker 2004). Figure 5 shows relative differences in r, t, and absorptance a, for this band, between δ-Eddington and Monte Carlo estimates. Albedo and transmittance errors retain much of the forms seen in Figs. 2 and 3. Absorptance errors for liquid and ice resemble one another with tendencies to underestimate for most conditions, especially at middling values of 
Relative errors for (a) albedo, (b) transmittance, and (c) absorptance of single-layer homogeneous layers that consisted of cloud droplets with re = 15 μm and υe = 0.1 between the δ-Eddington two-stream approximation and the Monte Carlo using the exact Lorenz–Mie phase function [exact (∞-strm)], as shown in Fig. 1, as functions of cloud optical depth and 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0033.1
Results shown in Figs. 2, 3, and 5 reveal some of the most basic issues involving two-stream models, but they provide only part of the story pertaining to dynamical modeling, which necessarily centers on errors associated with broadband integrations over myriad cloud configurations and illumination conditions. Hence, the following section is devoted to examining broadband flux and heating rate differences between the multilayer δ-Eddington and its corresponding Monte Carlo model using numerous cloud configurations retrieved from A-Train satellite data.
4. Results II: Broadband fluxes and heating rates for A-Train clouds
Results presented in this section expand upon those shown in section 3. Cloud fields used here were represented as 2D fields of particles imbedded in scattering and absorbing gases over reflecting surfaces. Their spatial distributions and some of their properties were inferred from A-Train satellite data, which are discussed in the following subsection. The second subsection describes the setup of surface–atmosphere conditions used in the experiments. In the third subsection, values of domain-average reflected fluxes at TOA, net surface flux, and net atmospheric absorption, as well as heating rate profiles predicted by the multilayer δ-Eddington (in ICA mode) are compared to their Monte Carlo counterparts, which used both detailed phase functions and finite horizontal grid spacing.
a. A-Train satellite data
Cloud properties used here were derived from data collected between 1 January and 28 January 2007 by CloudSat’s 94-GHz cloud-profiling radar (CPR) and CALIPSO’s dual-wavelength lidar (Stephens et al. 2002; Winker et al. 2003). Estimates of cloud water content (CWC) came from CloudSat’s 2B Radar–Lidar Geometrical Profiling Product (2B-GEOPROF-lidar) data (Mace 2007; Mace et al. 2007). CloudSat columns are effectively 1.1 km along track. The radar’s circular footprints are ~1.4 km long, so integrations for ~1 km were oversampled slightly. Each layer is 0.24 km thick. Ground clutter contamination renders the lowest two or three layers unworkable. CloudSat’s ECMWF-AUX files report profiles of pressure, temperature T, and mixing ratios of water vapor and ozone for each column.
To affect A-Train samples that are meant to represent GCM grid cells, CloudSat–CALIPSO data were partitioned into 256-km domains. Near-surface precipitation was removed approximately following Barker (2008). Only domains between latitudes 70°S and 70°N with total (i.e., vertically projected) cloud fraction 
Figure 6 shows a summary of the clouds used here. Values of C are zonal averages and resemble closely those reported by others (e.g., King et al. 2013). For these domains, mean C was ~0.75 which is larger than passive imager-only estimates but close to those from A-Train data (e.g., Mace et al. 2009; Hagihara et al. 2010; Kato et al. 2010). The partition into liquid and ice differs from what gets reported for passive sensors, for if a column had both liquid and ice, both phases received a count regardless of whether they were exposed directly to space or not. Hence, liquid and ice fractions shown in Fig. 6 sum to more than C. Note, however, that when liquid and ice fractions are assumed to overlap at random, resulting total fractions are often close to C, especially between 40°S and 20°N.
Zonally averaged (a) total cloud fractions, (b) mean cloud optical depths, and (c) υ as defined in (14) for cloud water paths for liquid and ice clouds defined, respectively, by re = 13 μm and De = 40 μm (scenario B in Table 1). Values correspond to 256-km-long domains of cloud properties inferred from A-Train satellite data.
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0033.1
This way of counting clouds has an impact on estimates of cloud mean 
b. Experimental setup
Two straightforward, and somewhat extreme, cloud-surface scenarios were considered in order to supply likely bounds on estimates of errors associated with two-stream approximations. Scenario A represents oceanic clouds with relatively large particles: all ice clouds had 
Cloud characteristics for scenarios A and B. Cloud fractions and mean τ and υ corresponding to clouds averaged over all 256-km A-Train domains.
For the δ-Eddington, layer optical properties for clouds and molecular scattering and absorption were merged via extinction weighting to form effective values. For each grid cell of the Monte Carlo, a cumulative extinction function was defined as a single increment for each type of attenuator. When, in the simulation, a photon ensemble was deemed to undergo an interaction with matter, a uniform random number between 0 and 1 was generated and an attenuating species selected using the cumulative extinction function (see Barker et al. 2003).
The A-Train samples clouds over a very narrow range of time of day. To get a picture of the diurnal range and mean values of potential errors due to the use of the δ-Eddington approximation, 
To summarize, for each 256-km A-Train domain, for both scenarios A and B, domain-average radiative transfer simulations were computed for the following:
- Multilayer, analytic δ-Eddington in ICA mode,
- 1D Monte Carlo using
- 1D Monte Carlo using exact phase functions with
- 3D Monte Carlo using exact phase functions with
c. Broadband fluxes and heating rates
This section focuses on errors in broadband fluxes and heating rates incurred by the analytic δ-Eddington’s neglect of phase function details and cloud geometry. Figures 7a–c and 8a–c show 
Mean-bias (gray) and median-bias (black) errors for differences in (a) TOA upwelling flux, (b) net surface absorption, and (c) net atmospheric absorption between the δ-Eddington two-stream approximation [F1D(δ-Edd)] and the Monte Carlo [F1D(MC)] using scenario A exact phase functions (see Fig. 1 and Table 1) and Δx → ∞ for the cloudy portions of 256-km A-Train domains as functions of 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0033.1
As in Fig. 7, but for clouds defined for scenario B (see Table 1).
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0033.1
The center rows of Figs. 7 and 8 show corresponding differences between the Monte Carlo using exact phase functions with 
While there are several cases in which the solution with 
where 〈⋅〉 denotes bin mean. When differences are between a Monte Carlo simulation and the noiseless δ-Eddington, (17) represents all the Monte Carlo noise. For differences between two Monte Carlo simulations, the last term in (16) becomes ~
Figure 9 shows mean and median differences between Monte Carlo results for scenario A, with 
Mean-bias and median-bias errors for differences between the Monte Carlo using Δx → ∞ and the Monte Carlo using Δx = 1 km when both used the exact phase functions for scenario A. Black dashed lines 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0033.1
At small 
As for atmospheric absorption, when 
From Figs. 7a–f and 8a–f, it is obvious that neglecting phase function details by the δ-Eddington and cloud geometry by 1D models generally affect 
The top grouping of plots in Figs. 10 and 11 show the same data as were shown in Figs. 7 and 8 but sorted and averaged as a function of latitude and 
(top) Set of plots corresponding to those in Fig. 7 (clouds defined by scenario A in Table 1), but showing flux differences averaged over 5° wide latitude bands and 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0033.1
As in Fig. 10, but for clouds defined by scenario B in Table 1. Standard deviations were almost indistinguishable from those in Fig. 10 and are not shown.
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0033.1
The lower set of plots in Fig. 10 shows standard deviation of flux differences that correspond to the mean-bias errors shown above them. These values were determined by local variability of clouds convolved with the strengths of errors associated with neglect of phase function details and horizontal transport. For instance, areas poleward of ~50° exhibit marked reductions in standard deviations for all 
Figures 10 and 11 also show that, in the descending branches of the Hadley cells, near 30°S and 20°N, underestimation of atmospheric absorption by the δ-Eddington is greatest at intermediate values of 
(a) Profiles of mean cloud fraction and mean domain-average heating rates (for 256-km A-Train domains with clouds defined for scenario A) for the δ-Eddington two-stream approximation (δ-Edd), the Monte Carlo using exact phase functions (see Fig. 1 and Table 1) and 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0033.1
Another area with notable differences in atmospheric absorption is from 40°S to 10°N at 
As in Fig. 12, but for scenario B conditions and all domains between 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0033.1
While the heating rate bias errors just discussed are clear and explainable, they are nevertheless usually smaller than 3% of the local heating rate. The left plots in Fig. 14 show mean maximum heating rates for layers below 20 km for the δ-Eddington and the Monte Carlo models with 
(left) Mean maximum heating rates for layers at altitudes < 20 km as functions of 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0033.1
Figure 14 simply tries to put the largest bias errors into context with the estimated heating rates. The largest mean maximum heating rate differences occur between the δ-Eddington and the Monte Carlo model with 
5. Summary and conclusions
This study addressed two basic omissions in multilayer two-stream approximations of the solar radiative transfer equation when applied to cloudy atmospheres: namely, their neglect of details in scattering phase functions for cloud particles and horizontal transfer of radiation. A-Train satellite data were partitioned into a large number of 256-km segments so as to approximately represent typical GCM cells (Astin and Di Girolamo 2006). These domains were used to show that biases affected by the two mentioned omissions generally have the same sign as a function of solar zenith angle and that they are approximately additive, thus leading to overall bias errors that could be important when attempting to estimate climate sensitivities due to changes in Earth–atmosphere conditions.
Using single-layer isolated clouds, it was demonstrated that phase functions for typical cloud particle size distributions have sufficient detail that, when only their asymmetry parameters are used by two-stream models (the δ-Eddington in particular), systematic errors in reflectance, transmittance, and absorptance will follow. This general result is not new (cf. King and Harshvardhan 1986), but the provision of broadband estimates integrated over most cloud and illumination conditions is. For high-sun conditions, one can expect the δ-Eddington to underestimate surface flux for the cloudy portion of a GCM’s grid cell by ~10 W m−2 because of neglect of phase function details. Similarly, for most 
Layer heating rate errors, once averaged over all cloud conditions, are largest because of the omission of phase function details. Corresponding errors due to neglect of horizontal transfer are smaller. This is not entirely unexpected (see Cole et al. 2005). The result is that maximum heating rate errors in the lower atmosphere due to neglect of phase function details and horizontal transfer of radiation can be expected to be up to about 10%–15% of the corresponding maximum heating rates. Most errors, however, are much smaller at about ±2%.
It is recognized fully that, while the dataset used and many assumptions made in this study are not perfect, they are probably not catastrophically far from both the best available and reality. Hence, estimates reported here of flux errors associated with the multilayer δ-Eddington model, and two streams in general, as a result of neglect of details pertaining to cloud particle phase functions and horizontal transport of solar radiation are likely to be quantitatively approximate but qualitatively accurate.
The 256-km domains used in this study were supposed to represent grid cells of conventional GCMs, yet unlike GCMs, they were resolved to about the 1-km scale. Hence, this study sidestepped an additional source of flux error endemic to GCM radiation calculations: generation of unresolved clouds (Pincus et al. 2003). Whether these errors are correlated with those analyzed here remains to be seen.
To address the impacts that biases discussed here would have on GCM simulations, one would have to employ a solar transfer model, like the Monte Carlo used in this study, which is able to utilize important details of cloud particle phase functions and horizontal transport. This could be realized using a so-called superparameterized GCM (e.g., Cole et al. 2005; Randall 2013), but it would be only partially fruitful in conventional GCMs that are, as yet, unable to provide suitable descriptions of clouds at unresolved scales.
This study was supported by a grant from the U.S. Department of Energy Atmospheric Radiation Measurement Program (subcontracted to Environment Canada from The Pennsylvania State University), and a contract from the European Space Agency (subcontracted to Environment Canada from the Free University of Berlin).
APPENDIX
Biases due to the Use of the Henyey–Greenstein Phase Functions
The Henyey–Greenstein function 
Using the same input data that were used to produce Figs. 7 and 8, Fig. A1 shows differences between Monte Carlo models using 
Mean-bias and median-bias errors for differences between the Monte Carlo using 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0033.1
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For readability, the term “flux density” is replaced, hereinafter, by simply “flux,” noting that flux implies units of watts per meter squared.
