Today, large model discrepancies exist in estimated cloud radiative effects (CREs) and irradiances across 1-D radiative transfer schemes aimed for climate models. The primary purpose of this study is to understand physical causes of such model discrepancies, especially in CREs under partly cloudy sky. To achieve this goal, the unique Cloud-Aerosol-Radiation (CAR) ensemble modeling system was employed, offline driven by the ERA-Interim global data for July 2004 with no feedback considered. For evaluating each individual contribution from the existing scheme diversity of cloud horizontal inhomogeneity, cloud optical properties, cloud vertical overlap, and gas absorptions, several sets of numerical experiments were conducted. It is the first time to explicitly demonstrate that after removing most of the disagreement in cloud fields, model spreads of CREs among the CAR's seven major radiation schemes, as well as those of radiative fluxes, dramatically diminish. Taking global mean CREs for example, their current model ranges can decrease to <4 W m−2 from about 10 W m−2 for shortwave and also to <4 W m−2 from 5–8 W m−2 for longwave. Dominant roles of subgrid-scale cloud structures (including vertical overlap and horizontal variability) were proven in general, explaining about 40–75% of the total model spreads. We have also found that model spreads of CREs are very sensitive to cloud cover fractions. Such nonlinear sensitivity can be largely reduced after removing the model difference in the treatments of cloud vertical overlap.
 Credible climate simulations require accurate radiation transfer calculations, which have been reinforced for more than 20 years, especially under cloudy and aerosol-laden conditions [e.g., Ellingson and Fouquart, 1991; Barker et al., 2003; Collins et al., 2006; Oreopoulos et al., 2012a; Randles et al., 2013]. Clouds exert significant radiative effects on climate. The different numerical representations of cloud-related processes in global climate models (GCMs) account for much of the uncertainty in estimates of climate sensitivity [e.g., Intergovernmental Panel on Climate Change, 2007; Randall et al., 2007]. Today, in most GCMs, the horizontal grid sizes typically exceed 50–100 km, in which various clouds exist. The Independent Column Approximation (ICA) has long been recognized as a very good approximation to the full 3-D solutions [Cahalan et al., 1994; Barker et al., 1999, 2003]. However, the full ICA calculation of cloudy sky flux needs double integral over both wavelength and cloud state, which makes this method far too time-consuming to be practical in GCMs. In fact, 1-D radiation transfer schemes with various simplifications and parameterizations are commonly used, such as the conventional maximum-random overlap [Geleyn and Hollingsworth, 1979; Tian and Curry, 1989; Stubenrauch et al., 1997]. Therefore, cloud subgrid-scale structures are generally unresolved in current climate modeling, although their large impacts on the simulated radiation budgets and cloud radiative effects (CREs) have been demonstrated by a lot of studies [Morcrette and Fouquart, 1986; Cahalan et al., 1994; Liang and Wang, 1997; Stubenrauch et al., 1997; Barker et al., 1999; Pincus et al., 1999; Morcrette and Jakob, 2000; Barker and Räisänen, 2005; Cole et al., 2005; Wu and Liang, 2005; Shonk et al., 2010; Shonk and Hogan, 2010]. Our inability to capture in 1-D radiation transfer schemes what we understand about the overarching 3-D processes has further reinforced the need of accurate radiation transfer calculation in climate studies.
 Under partly cloudy sky, a quite large inconsistency in models appears due to the commonly used, unnatural coupling between assumptions about cloud vertical overlap and methods for computing radiative transfer. Even different implementations of the same cloud overlap assumptions (e.g., the widely used maximum-random overlap) may generate substantial intermodel discrepancies and biases [Barker et al., 2003]. Hence, almost all techniques or schemes show large dependence on the host models, for example, whether the details of cloud vertical structure matter much for radiation [Oreopoulos et al., 2012b]. Therefore, given the complexity introduced by partial clouds, all existing intercomparisons of radiation codes in climate models except Barker et al.  either omitted clouds, e.g., Radiative Transfer Model Intercomparison Project [Collins et al., 2006; Iacono et al., 2008], or just considered very simple clouds, i.e., plane-parallel, homogeneous overcast ones, e.g., the most recent Continual Intercomparison of Radiation codes [Oreopoulos et al., 2012a]. Barker et al.  is the only intercomparison effort to assess the performances of 1-D solar radiation algorithms with regard to the impact of the partial clouds. They presented the large model spreads for each of the four genres that were partitioned according to the different assumptions about unresolved clouds. However, they did not further investigate the detailed physical reasons for the model discrepancies. The role of unresolved cloud structures in current model spreads of estimated irradiances and CREs is far from being understood, preventing us from obtaining an accurate Earth's radiation budget.
 Two advanced methods are now available for explicitly representing cloud subgrid-scale structures in 1-D radiation codes, i.e., the mosaic treatment of subgrid scale cloud-radiation interaction (MOSAIC) [Liang and Wang, 1997] and the Monte Carlo Independent Column Approximation (McICA) [Barker et al., 2002; Pincus et al., 2003; Räisänen et al., 2004]. Recently, both approaches have been successfully incorporated into the Cloud-Aerosol-Radiation (CAR) ensemble modeling system, along with seven radiative transfer schemes commonly used in the major climate prediction centers worldwide [Liang and Zhang, 2013; Zhang et al., 2013]. Decoupling the assumption of cloud subgrid-scale structures from the calculation of radiative transfer not only allows for more flexible, complex, and realistic cloud descriptions but also simplifies the radiative transfer algorithms. Based on the modular design of the CAR system, all built-in cloud parameterizations including MOSAIC and McICA methods are fully compatible with any of the built-in radiation codes. So the CAR system has the unique ability to largely simplify the intercomparisons of 1-D radiation codes, especially under partly cloudy sky.
 Our major objective in this study is to understand why large model discrepancies in estimated CREs and irradiances exist among current 1-D radiation transfer codes aimed for climate studies. To this end, the unified CAR system was employed. The detailed physical causes of the current model spreads among the CAR's seven major radiation transfer codes, and meanwhile whether the current model spreads can be significantly reduced, were studied. It is the first time that the individual contributions from the current scheme diversity of cloud horizontal inhomogeneity, cloud optical properties, cloud vertical overlap, and gas absorptions are quantitatively investigated. This kind of ability is far beyond prior model intercomparisons, such as Barker et al. . Although seven radiation transfer codes seem to be not enough for one intercomparison effort, some reasonable and general conclusions still can be obtained due to the large intermodel diversity among them [Randles et al., 2013; Zhang et al., 2013]. In this study, only purely diagnostic calculations were conducted with no feedback considered.
 In sections 2 and 3, the CAR system and detailed experiment will be described. Results and analyses are shown in section 4. Section 5 presents the discussion and conclusions.
2 CAR System
 A general description of the model and a basic evaluation of the skill of the CAR system were presented in Liang and Zhang . Below is a brief summary of the system for the purposes of this study.
 In the CAR system, all alternate parameterizations for cloud physical properties, aerosol physical properties, and radiation transfer processes are handled, respectively, in cloud, aerosol, and radiation drivers, with three couplers for the cloud optics, aerosol optics, and aerosol impacts on cloud droplet nucleation, respectively. In the radiation driver, there are seven major radiation transfer schemes commonly used in the key operational centers and research institutions worldwide, including NASA GSFC (U.S. National Aeronautics and Space Administration, Goddard Space Flight Center), NOAA GFDL (U.S. National Oceanic and Atmospheric Administration, Geophysical Fluid Dynamics Laboratory), NCAR (U.S. National Center for Atmospheric Research), NCEP (U.S. National Centers for Environmental Prediction), CCCma (Canadian Centre for Climate Modeling and Analysis), CAWCR (Centre for Australia Weather and Climate Research), and ECMWF (European Centre for Medium-Range Weather Forecasts) (Table 1). In the cloud driver, different modules have been designed to deal with parameterizations of cloud cover fraction, water path, particle effective size/radius, and geometry including vertical overlap and horizontal subgrid-scale variability. In the aerosol driver, the CAR has the ability to corporate either observed or modeled aerosol loadings or optical properties with different vertical profile schemes if needed.
Table 1. The Original Radiation Transfer Schemes Used in This Study
cop, cloud optical property members, i.e., combination of schemes for cloud optical properties plus those for cloud particle effective size.
cop1 ~ cop7, the different cops used by each rad (Table 2).
Conventional maximum-random overlap: for cam and cawcr, adjacent cloudy layers share maximum overlap while discrete clouds are randomly overlapped; for gsfc and flg, mix overlap among high/middle/low cloud blocks.
Homogeneous clouds mean that the grid-mean cloud water fields are used without subgrid variability considered.
δ-Ed, delta Eddington.
Inhomogeneous clouds mean the horizontal subgrid variability is included.
An inhomogeneity factor = 0.7 is used by flg to account for the cloud inhomogeneous effects [Gu et al., 2003].
NASA, USA; Chou and Suarez  and Chou et al. 
δ-Ed, SW: 19 spectral and pseudospectral intervals, 0.2–5.0 µm, gas absorptions of H2O, O3, and CO2; LW: 1 band, 0.0–2000 cm−1, absorptivity-emissivity approach, gas absorptions of H2O, O3, CO2, N2O, CH4, and CFCs
UCLA, USA; Fu and Liou , Liou et al. , and Gu et al. 
δ-Ed, ESF, SW: 9 bands, 0.2–5.0 µm, 27 g-points, gas absorptions of H2O, O3, O2, CO2, CH4, and N2O; LW: 8 bands, 0.5–4000 cm−1, 31 g-points, gas absorptions of H2O, O3, CO2, N2O, CH4, and CFCs
 Based on the modular design of the CAR system, all built-in cloud and aerosol parameterizations can be selected and fully exchanged with all major radiation transfer schemes. Especially, both the MOSAIC [Liang and Wang, 1997] and McICA [Barker et al., 2002; Pincus et al., 2003; Räisänen et al., 2004] have been consistently applied to each CAR major radiation transfer scheme. The MOSAIC method distinguishes the geometric associations of cloud genera by the different formation mechanisms. It is one kind of ICA, in which generally 15 ~ 20 different cloudy states (may include one clear subcolumn) are enough for an acceptable computation accuracy. The McICA eliminates the integral over cloud state in the ICA by using a different subcolumn for each g point or k point that is usually produced by some stochastic cloud generators [Räisänen et al., 2004]. Though the McICA method generates random, uncorrelated errors in the modeled irradiances, the long-time averaged estimates are unbiased with respect to the ICA [Pincus et al., 2003]. These two approaches define cloud subgrid-scale structures outside radiative transfer calculations. Thus, in the CAR, most of the disagreement in cloud fields can be removed before they are inserted into different radiation transfer schemes.
 In addition, one external module has been designed to manage all external forcings introduced by natural or anthropogenic climate agents, such as variations in solar insolation, variations in the Earth's orbit, changes in radiative gas concentrations (CO2, CH4, N2O, CFCs, CCl4, O2, CH3Cl), changes in aerosol loading, and changes in surface albedo and surface emissivity caused by land use and land changes. All these can be predicted via coupling with other modules or prescribed.
 With all the above features, the CAR system has the ability to significantly simplify the objective quantification of the contributions from different factors related to cloud aerosol/radiation. When coupled with a climate model, it will be a very useful tool to further study the cloud-aerosol-radiation interactions and feedbacks. Details of the individual schemes and their respective references are available online at http://car.umd.edu.
3 Experiment Design and Observation Data
 Our major suites of experiments are summarized in Table 3. Six major sets of global experiments were conducted for July 2004, offline driven by the 6-hourly ERA-Interim (ERI) observational analysis data with a horizontal resolution of 1.5° × 1.5° [Uppala et al. 2008]. They are referred to as dcop_dovp1, dcop_dovp0, scop_dovp0, scop_sovp0 (i.e., McICA_MRO), scop_sovp1, and gas_exp, respectively. Here, “d” means different treatments used among different radiation transfer schemes, while “s” the same treatments consistently adopted; “cop” denotes cloud optical properties, and “ovp” refers to cloud vertical overlap; “0” indicates grid-mean cloud water fields used, while “1” cloud horizontal variability considered. These sets of experiments were designed according to the level of difficulty in realizing an intercomparison. The dcop_dovp1 employed the original codes without any changes; the dcop_dovp0/scop_dovp0 can be easily performed by different scientific groups; while to conduct scop_sovp0, scop_sovp1, and gas_exp remains difficult without a system like the CAR. As we know, today, the conventional maximum-random overlap is commonly used by 1-D radiation transfer codes [Barker et al., 2003, Table 1], which makes those codes hard to generalize. In fact, it took us more than 4 years to build the CAR system.
 To show the similar results among different methods used for cloud subgrid-scale structures, along with McICA_MRO (i.e., scop_sovp0) in which the McICA method with the maximum-random overlap generator was adopted, another two sets of experiments were performed: (1) McICA_Fgen, the McICA method with generalized overlap generator, was used. The decorrelation length for overlapping fractional cloud Lα is set to 2 km, gotten from some prior work [Räisänen et al., 2004; Morcrette et al., 2008; Oreopoulos et al., 2012b]; (2) MOSAIC, the MOSAIC method with 15 subcolumns, was employed [Liang and Wang, 1997]. No cloud inhomogeneity effects were considered in McICA_MRO, McICA_Fgen, and MOSAIC. In addition, to include the cloud inhomogeneous effects, one extra set of experiments, i.e., McICA_Fgen1, was also carried out. Here the rank correlation-decorrelation length Lγ = 1 km [Räisänen et al., 2004; Morcrette et al., 2008; Oreopoulos et al., 2012b], and the γ distribution was adopted with the normalized standard deviation of cloud condensates defined by the layer cloud cover fractions (J. Cole, personal communication, 2012).
 Each set of experiments has 98 global runs, combining 7 radiative transfer schemes (Table 1) and 14 cloud cover members (Table 4). Clearly, from Table 3, while all experiments share the same profiles of mean cloud condensate, the treatments of cloud optical properties and subgrid-scale structures are different from set to set. In this study, to minimize the model discrepancy caused by different built-in band structures, the solar insolation at the top of the atmosphere and the Planck fluxes evaluated at 233.15 K were used as the weights for the SW (shortwave) and LW (longwave) spectral integrals of the calculated cloud optical properties (including cloud optical depth, single scattering albedo, and asymmetry factor) among the different radiation schemes, respectively. In scop_sovp0 (McICA_MRO), scop_sovp1, gas_exp, McICA_Fgen, McICA_Fgen1, and MOSAIC, the same cloud subgrid-scale structures were applied to each radiation algorithm. In this study, with no feedback considered, schemes of cloud particle effective radius (or size) only influence the calculations of cloud optical properties and hence are included into cops as shown in Table 2.
Table 2. The cops Used by the CAR Original Radiation Transfer Codesa
rel: scheme for cloud liquid droplet effective radius; dei: scheme for cloud ice particle effective size; lwl: scheme for cloud liquid water LW optical properties; lwi: scheme for cloud ice water LW optical properties; swl: scheme for cloud liquid water SW optical properties; swi: scheme for cloud ice water SW optical properties.
rel schemes are unavailable with all radiation transfer codes except cam and gfdl. Martin et al.  is used by gfdl; however, this scheme now cannot be used offline due to the missing cloud droplet number concentrations in the ERA-Interim (ERI) data. Here, Szczodrak et al.  is used.
dei schemes are unavailable with cccma, rrtmg, and cawcr. Here McFarquhar  is used.
 Moreover, as shown by Räisänen et al. , in a single application of the McICA, the random errors with fewer subcolumns might be expected to be larger than those with more subcolumns, while mean values of sufficient multiple (e.g., >5000) realizations were essentially independent of the number of subcolumns. Hence, to verify the similarity for monthly zonal means of modeled irradiances between two McICA implementations with different numbers of subcolumns, another four sets of experiments with 15 subcolumns for scop_sovp0 (McICA_MRO), scop_sovp1, gas_exp, and McICA_Fgen were conducted, too.
 In addition, based on different treatments of cloud subgrid-scale structures, the inter-cop discrepancies of radiative components have also been investigated among the seven cops originally used by each radiation transfer code (Table 2). Two extra sets of experiments were conducted, referred to as dovp1 and sovp0. The former used original treatments of cloud subgrid-scale structure (Table 1), while the latter adopted those in McICA_MRO (Table 3). Each set has 49 global runs, combining seven radiative transfer schemes and seven cops. Here only cld1 (Table 4) was taken for example. Note that cam has only one LW spectral band when solving the LW radiative transfer equation, which largely reduces the random sampling in the McICA/MOSAIC methods and makes these methods ineffective. Therefore, cam and cop3 were excluded from the intercomparison of LW radiation below.
Table 3. Experiment Design
Methods for Cloud Vertical Overlap
Whether Including Cloud Horizontal Subgrid-Scale Variability
The different original schemes used for each rad (Table 2; dcop)
The different original treatments for cloud vertical overlap used by each rad (Table 1, dovp). cccma, gfdl, and rrtmg adopt the different maximum-random cloud generators (the number of subcolumns = 100) for the used McICA method [Räisänen et al., 2004], while others use the conventional treatments, i.e., maximum/random overlap among high/middle/low cloud blocks for gsfc and flg, and among adjacent/nonadjacent layers for cam and cawcr
In the original rads, only cccma, flg, and gfdl consider horizontal inhomogeneous clouds. Here, “1” indicates the fact that some rads include the cloud subgrid-scale inhomogeneity, while the others do not
The same as dcop_dovp1
The same as dcop_dovp1
Only homogeneous clouds are considered across all rads, i.e., grid-mean cloud water fields used. “0” indicates the same cloud homogeneity
The same cop, i.e., cop1 (Table 2) used across all major rads (scop)
The same as dcop_dovp1
The same as dcop_dovp0
The same as scop_dovp0
For the adopted McICA method, the same implementation of the maximum-random overlap generator provided by Räisänen et al.  is consistently used by each rad. The number of subcolumns = 100
The same as dcop_dovp0
The same as scop_dovp0
The same as scop_sovp0
The β distribution is adopted with shape factors both set to 5 [Räisänen et al. 2004]. Here, “1” means the same cloud horizontal inhomogeneity
The same as scop_dovp0
The same as scop_sovp0
The same as scop_sovp1
Including only H2O and O3 for SW and only H2O, O3, and CO2 for LW calculations
The same as scop_dovp0
For the adopted McICA method, the same implementation of the generalized overlap generator is applied with the decorrelation length for overlapping fractional cloud Lγ = 2 km [Räisänen et al., 2004]. The number of subcolumns = 100
The same as dcop_dovp0
The same as scop_dovp0
The same as McICA_Fgen except that the rank correlation-decorrelation length Lγ = 1 km is added for cloud vertical inhomogeneity
The γ distribution [Räisänen et al. 2004] is adopted with normalized cloud condensate standard deviation defined by the layer cloud cover fractions (J. Cole, personal communication, 2012)
The same as scop_dovp0
The MOSAIC method is used by different rads [Liang and Wang, 1997]. Here the number of subcolumns = 15
The same as dcop_dovp0
Table 4. The Cloud Cover Members (Total 14) Used in This Studya
Cloud Cover Members
Schemes for Stratiform
Schemes for Deep Cumulus
Here, Slingo  is used for cirrus and boundary layer clouds.
 To avoid the model discrepancy caused by the different surface boundary conditions, surface emissivity was prescribed as 1.0 globally, and one broadband surface albedo from ERI data was used by all SW radiative transfer calculations. In addition, the cloud water path was directly diagnosed from the ERI cloud water mass mixing ratio. Thus, all experiments mentioned above used the same 3-D cloud condensate fields. Although only one month (i.e., July 2004) is evaluated in this study, globally more than 105 grid columns are occupied by different clouds, indicating sufficient cloud variety.
 In this study, for radiation fields, three global radiation data sets were used to depict the observation uncertainty range (i.e., interobservation discrepancy): the Clouds and the Earth's Radiant Energy System (CERES) mission [Wielicki et al., 1996] data, International Satellite Cloud Climatology Project Flux data (ISCCP-FD) [cf. Zhang et al., 2004], and Global Energy and Water Cycle Experiment Surface Radiation Budget (SRB) data [Cox et al., 2004; Stackhouse et al., 2011]. The SRB data are based on released version 3.0 for both LW and SW. The CERES data are from the Edition 2D/2A products of Terra/Aqua SRBAVG (Surface Radiation Budget Averages) and CERES EBAF (Energy Balanced and Filled). For cloud fields, the ISCCP, CERES SRBAVG, and CERES ISCCP-like data were used.
 To show the large scheme differences, comparisons of global mean total cloud cover fraction among different cloud cover fraction members (clds) are presented in Figure 1 for July 2004, as well as the vertical sum of visible and 10.5 µm grid-mean cloud optical depth among different cloud optical property members (cops). Note that the total cloud cover fractions were obtained from the assumption of maximum-random overlap among adjacent/nonadjacent layers, and cop3 was not used for LW comparisons due to the cam's coarse LW spectral resolution. Clearly, different cloud cover fraction members generate quite different total cloud cover fractions with the spread about 10%. Compared with the observation of 59.88–64.69%, only cld2/cld4/cld6/cld13 has smaller total cloud cover fractions. However, considering that our major purpose is to understand the model disagreement in estimated CREs, we have included all the clds here. As implied by total cloud cover fractions <65%, partial clouds prevail globally for grid domains of 1.5° × 1.5° adopted by this study. In addition, the obvious differences of total grid-mean cloud optical depth among different cops are also shown with ranges about 7.76–9.01 for SW and 7.19–8.13 for LW. Currently, we have found that it is difficult to directly compare the modeled cloud optical depth with those satellite retrievals. As well known, no direct observations are available for cloud water fields and cloud optical depth, and hence, different inverse algorithms are used to retrieve them, assuming large uncertainties. For example, the retrieved cloud visible optical depth from Moderate Resolution Imaging Spectroradiometer (MODIS) is about 29 for July 2004 [King et al., 2003], while that from ISCCP is 4.47. This large difference is maybe due to the MODIS values being means of pixel-level retrievals while the ISCCP ones being radiatively weighted means [Rossow et al., 2002]. Moreover, some necessary simulators used to facilitate comparisons have not yet been coupled into the CAR. Here, Figure 1b is just to indicate the large variation among different cops. If no obvious differences existed among different cops, some conclusions in this study should be meaningless. In this study, by using these modeled cloud optical depth mentioned above, quite good results for estimated CREs and irradiances can be obtained.
4 Results and Analyses
4.1 General Picture Based on Zonal Means
 Figure 2 shows comparisons of zonal mean TOA/SFC SW/LW CREs among the CAR's seven major radiative transfer schemes for dcop_dovp1 (original), dcop_dovp0, scop_dovp0, scop_sovp0, and scop_sovp1, respectively, where TOA means top of atmosphere and SFC means ground surface. The results for July 2004 are shown. Different cloud cover members (cld1–cld14) generate similar CREs. Here we took cld1 for example. The CRE is defined as , where F is the net (downward minus upward) flux, clr designates clear skies, all denotes a mixture of clear and cloudy skies.
 As shown in Figure 2, for dcop_dovp1 (original), when different radiation algorithms employ different approaches for cloud optical properties and subgrid structures, although the same 3-D distributions for cloud cover fraction and water path are used, large intermodel diversities exist for both SW and LW CREs. For SW, the spreads of 25–30 W m −2 are found in both the tropical and middle latitude areas, while for LW, the large diversities are most remarkable for TOA CREs over tropics with ranges of about 30 W m−2. For SFC LW CREs, since their values of 10–40 W m −2 are quite small, the intermodel differences of about 5–10 W m −2 cannot be ignored.
 In dcop_dovp1, the weakest cloud radiative effects for SW CREs are produced by cccma and gfdl, while those for TOA LW CREs are from flg. By removing the cloud inhomogeneity effects, in dcop_dovp0, these three are just located in the middle of the model ranges with stronger CREs. The cloud inhomogeneity effects weaken cloud radiative effects for both SW and LW. We have also found that flg has the similar SW CREs as rrtmg, cam, and cawcr in dcop_dovp1, indicating the similarity between the mixing overlap (i.e., max-random overlap) among high/middle/low cloud blocks for inhomogeneous clouds (i.e., an inhomogeneity factor = 0.7 was employed) and that among adjacent/nonadjacent layers for homogeneous clouds. In brief, the cloud horizontal variability has a large impact on modeled CREs and may affect those model spreads.
 The scop_dovp0 is the same as dcop_dovp0 except that the same cop1 is applied to each radiation code. In scop_dovp0, the seven radiation codes form two groups. Regardless of whether the conventional or the McICA method is used, those using the same assumption of maximum/random overlap among adjacent and nonadjacent layers, i.e., cccma, cam, gfdl, rrtmg, and cawcr, converge together, while flg has the similar performance as gsfc because these two adopt the same mixing overlap among high/middle/low cloud blocks. In each group, the influence of cloud optical properties on model spreads is large. Especially in the tropics, the differences of LW and SW CREs among cccma, cam, gfdl, rrtmg, and cawcr are reduced from more than 20 W m−2 in dcop_dovp0 to about 6–8 W m −2 in scop_dovp0 by using the same cop1. In a word, the effects of using the same treatment of cloud optical properties are substantially limited by the different assumptions of cloud vertical overlap, although they are not small as shown in each group.
 In scop_sovp0 and scop_sovp1, by eliminating most of the disagreement in cloud fields, the model discrepancies in dcop_dovp1 dramatically diminish. Similar SW and LW CREs are shown globally across all seven radiation codes. At most latitudes, model spreads are no more than 5 W m−2. Clearly, the large existing intermodel discrepancies in CREs primarily come from different cloud treatments that are originally adopted by different radiation codes.
4.2 Intermodel Discrepancy Based on Different Cloud Cover Members (clds)
4.2.1 Individual Effects on SW Radiation Calculations
 Figure 3 shows the comparisons of global mean TOA SW CREs and SWUPT (upward SW fluxes at TOA) among 14 different cloud cover members (cld1–cld14) for different sets of experiments defined above. The results for July 2004 are shown here. Green shading indicates the observation ranges. For all clds, the quite large intermodel discrepancies in dcop_dovp1, dcop_dovp0, and scop_dovp0 are comparable to the full ranges of about 15 W m−2 caused by different clds, as clearly shown in scop_sovp0 and scop_sovp1. This emphasizes the important roles of cloud cover fractions in SW radiation transfer calculations, as well as the total contributions from the other cloud factors.
 In dcop_dovp1, the model differences between each other also significantly vary with different cld schemes. For example, for cld9 and cld12, cawcr has almost the same TOA SW CREs and SWUPTs as gsfc; however, when cld1 or cld7 or cld8 is used, large differences of about 4–6 W m −2 between these two appear. In this study, different 3-D distributions of cloud cover fractions are generated by different clds (Figure 1), and in dcop_dovp1, different cops and/or different treatments of cloud subgrid-scale structures were employed by different radiation transfer codes (Table 1). Hence, the combination of the nonlinear roles of these different cloud factors in radiation transfer calculations is the major reason for such variation with different cld schemes.
 More strikingly, in dcop_dovp1, dcop_dovp0, and scop_dovp0, due to the large intermodel diversity, all radiation codes can achieve quite good calculation accuracies by using the different combination of cloud schemes, and the performance of each cloud scheme just depends on the host model. For instance, when cld1 is used in dcop_dovp1, the performance of cccma is poor, which can be improved either by applying cld3 or cld10 or cld14 as shown in dcop_dovp1, or by ignoring the cloud horizontal inhomogeneity in dcop_dovp0. These methods do not work for most other radiation schemes, e.g., gsfc. In fact, as shown in dcop_dovp1, cld2/cld4/cld5/cld9/cld11/cld13 is the best for gsfc. In addition, whether cloud horizontal variability is included or not further complicates such situations. Taking cccma for example, cld1/cld7/cld9/cld11/cld12/cld13 generates accurate results in dcop_dovp0, while cld3/cld10 is the best in dcop_dovp1. Clearly, large model spreads obscure the true roles of clouds in radiation calculations. This largely prevents us from reaching the accurate understanding of Earth's radiation budget. Hence, it is necessary for us to investigate the detailed physical causes of current model spreads.
 Figure 4 is to quantitatively compare the intermodel discrepancy across the CAR's seven major radiation transfer codes of the global mean TOA SW CRE, SWUPT, SFC SW CRE, and SWDNS (SW downward fluxes at surface) among different sets of experiments. The medians and min-max ranges are among 14 intermodel discrepancies generated by applying the same cloud cover member (cld1–cld14) to each intercomparison of the CAR's seven major radiation transfer codes. “obs” denotes the interobservation range among ISCCP, SRB, CERES, and CERES_EBAF. The results for July 2004 are shown. Note that to simplify the following descriptions, the median is used to show the general intermodel discrepancy for each set of experiments.
 Clearly, as cloud fields become similar, the intermodel discrepancies, as well as the min-max ranges among 14 cloud cover members (cld1–cld14), are significantly reduced, even to smaller than the corresponding interobservation ranges. Different treatments of cloud inhomogeneity effect account for 19.5–44.2% of intermodel diversities, as indicated by the reduction of the medians from 10–14 W m −2 in dcop_dovp1 to 7–8 W m −2 in dcop_dovp0. As shown in Figure 3, in dcop_dovp1, based on the same 3-D distributions for cloud condensate and cloud cover fraction, using the mixing overlap among adjacent/nonadjacent layers for inhomogeneous clouds generally produces the weakest SW CREs and smallest SWUPT, e.g., cccma and gfdl. Both can be enhanced by removing the cloud inhomogeneity effects in dcop_dovp0, which then decreases the model spreads. Moreover, using the similar cloud optical properties seems to have no positive effects on the reduction of the SW model discrepancies, although it reduces the min-max ranges of ~7 W m−2 in dcop_dovp0 to <5 W m−2 in scop_dovp0. As shown in Figure 3, in scop_dovp0, two groups are formed based on the different assumptions of cloud vertical overlap: one, cccma, cam, gfdl, rrtmg, and cawcr; and the other, flg and gsfc. In each group, different cops largely contribute to the current intermodel spreads. Taking cccma, cam, gfdl, rrtmg, and cawcr for example, using the same cop (i.e., cop1) decreases the model spreads among them from 4.33–7.06 W m−2 and 5.09–9.05 W m−2 in dcop_dovp0 to 1.87–3.28 W m−2 and 1.50–3.73 W m−2 in scop_dovp0 for TOA SW CREs and SWUPT, respectively. However, such roles are greatly limited by the variety in the existing assumptions of cloud vertical overlap originally adopted by different radiation codes. Furthermore, when the same treatment of cloud vertical overlap is consistently applied in scop_sovp0, both the medians and the min-max ranges are greatly reduced. All the medians (i.e., intermodel discrepancies) further decrease to <3–4 W m −2 for CREs and SWUPT and <7 W m−2 for SWDNS, and all the min-max ranges are < 2–3 W m −2. Those model spreads are even smaller than their corresponding interobservation ranges. Hence, the dominant role of cloud vertical overlap is demonstrated in general as the key contributor not only to the current model diversities (44.4–50.6%) but also to the sensitivity of model ranges to different clds (i.e., min-max ranges) except SWDNS. For SWDNS, in scop_sovp0 and scop_sovp1, all radiation codes converge except gsfc (not shown). The specific performance of gsfc maintains the quite large model spread and may partly compensate the contribution from the different treatments of cloud vertical overlap. In gas_exp, by removing the discrepancies caused by the different gas absorptions of CH4, CO2, N2O, and O2, the medians are generally smaller than those in scop_sovp1, especially for SWDNS. This indicates that the uncertainties in gas SW absorptions still remain large, playing substantial roles in model spreads of SWDNS (~17.5%). In summary, for SW, about 50–75% of the current model disagreement results from the different treatments of cloud subgrid structures. The percentages of the model diversities contributed by different treatments for different factors are summarized in Table 5.
Table 5. The Percentage of the Model Diversities Contributed by Different Treatments for Different Factors (%)a
UPT: upward fluxes at TOA; DNS: downward fluxes at SFC; - : no effective roles can be gotten at this time from Figures 4 or 6, which is mainly due to the limitations from the roles of key factors. For example, the variety in the existing assumptions of cloud vertical overlap largely obscures the roles of different cloud optical properties.
Trace gaseous effects (i.e., CO2, O2, CH4, and N2O for SW; CH4, N2O, CFCs, etc. for LW)
4.2.2 Individual Effects on LW Radiation Calculations
 Figure 5 is the same as Figure 3 except for LW, where LWUPT is the upward LW fluxes at TOA. For LW, different cloud cover members usually generate quite small LW spreads, illustrating their small influences on LW calculations, while the different cloud optical property schemes show significant impacts on estimated LW CREs and irradiances. Therefore, due to the large existing intermodel diversities, generally different cops can be used to improve the different model's performances. For example, in dcop_dovp1/dcop_dovp0, the cop4 is used in flg (Table 1), which generates both poor TOA LW CREs and poor LWUPT. As shown in scop_dovp0, flg performs better by using cop1 (Table 2). However, cop1 worsens the performance of gfdl.
 Figure 6 is the same as Figure 4 except for TOA LW CRE, LWUPT, SFC LW CRE, and LWDNS (downward LW fluxes at surface). For LWUPT, using the same treatment of cloud optical properties (i.e., cop1) largely decreases the min-max ranges among 14 clds from about 3 W m−2 in dcop_dovp1/dcop_dovp0 to < 1 W m−2 in the other sets of experiments, such as scop_sovp0. Cloud optical properties play the key roles in the sensitivity of model spreads to clds for LWUPT. However, the large intermodel discrepancies denoted by the medians are kept even in gas_exp with values of 9 W m−2. This is mainly due to the specific performance of gfdl as shown in Figure 5. All other radiation codes produce the similar LWUPTs in scop_dovp0 except gfdl. Now we do not know why and are still investigating the reasons. For the other LW radiative components, as the cloud fields become similar, the medians (i.e., intermodel discrepancies), as well as the min-max ranges among 14 cloud cover members (cld1–cld14), decrease, even to smaller than the interobservation ranges.
 Table 5 also lists the percentages of the model diversities (i.e., the medians) contributed by different LW factors. Clearly, the different cloud subgrid structures usually account for 40–67% of the intermodel discrepancies for LW, playing the dominant roles. For TOA LW CREs, the substantial impacts (16.4%) of cloud optical properties are limited and suppressed by the variety in the existing assumptions of cloud vertical overlap. As shown in Figure 5, in scop_dovp0, two groups are formed, although they are not as distinct as those for SW. The model spread among cccma, gfdl, rrtmg, and cawcr that adopt the same assumption of max-random overlap among adjacent/nonadjacent layers of homogeneous clouds decreases from 3.94–7.97 W m−2 in dcop_dovp0 to 2.64–3.28 W m−2 in scop_dovp0. For LWDNS, different gas absorptions of CH4, N2O, and CFCs explain 17.9% of its intermodel diversity. Different cloud inhomogeneity effects are found to be critical to model ranges of SFC LW CREs and LWDNS (34–39%). As indicated by our results (not shown), based on the same 3-D distributions for cloud condensate and cloud cover fraction, the radiation schemes using the mixing overlap among adjacent/nonadjacent layers for inhomogeneous clouds generally produce the weakest SFC LW CREs and smallest LWDNS in dcop_dovp1, such as cccma and gfdl. Both can be enhanced in dcop_dovp0 by removing the cloud inhomogeneity effects.
4.2.3 Comparisons of Different Methods Used
 As we know, the results may vary with the different random cloud generators. To clarify this point, Figure 7 compares the intermodel discrepancies across the CAR's seven major radiation transfer codes among original (i.e., dcop_dovp1), McICA_MRO (i.e., scop_sovp0), McICA_Fgen, and MOSAIC. McICA_MRO, McICA_Fgen, and MOSAIC are defined in Table 3, along with the McICA method based on 100 subcolumns; in McICA_MRO, the maximum-random overlap generator is used, while in McICA_Fgen, besides the McICA method, the generalized overlap is adopted with the decorrelation length for overlapping fractional cloud Lγ = 2 km; in MOSAIC, the MOSAIC method with 15 subcolumns is used. The difference between scop_sovp0 and scop_sovp1 is quite small, as shown by Figures 4 and 6. So to simplify the comparisons, the cloud inhomogeneity effects are excluded. Clearly, despite the explicit methods, for all SW and LW radiative components, removing most of the disagreement in cloud fields significantly reduces both the general intermodel discrepancies (i.e., the medians) and the sensitivity of such discrepancies to different cloud cover members (i.e., the min-max ranges). LWUPT is the only exception, whose intermodel discrepancy remains large because of the specific performance of gfdl as shown in Figure 5. In brief, the small differences among McICA_MRO, McICA_Fgen, and MOSAIC further validate our conclusions in this study.
 Moreover, to make our conclusions more robust, we also investigated the similarity between zonal monthly means from McICA with 100 subcolumns and those from McICA with 15 subcolumns. Here, for each radiation transfer code listed in Table 1 and for each radiative component shown in Figure 7, a total of 8470 (= 5 sets of experiments (i.e., scop_sovp0, scop_sovp1, gas_exp, McICA_Fgen, and McICA_Fgen1) × 14 cloud cover members × 121 latitudes) pairs were used for this comparison. Despite the radiative components and the host radiation codes, these two have high correlations (>0.999) and extremely small root-mean-square differences (~0.0). Qualitatively, there are no significant differences in monthly zonal/global mean fields when changing the number of subcolumns from 100 to 15 in McICA. The McICA noise of zonal/global means for sufficiently long runs (at least a month) is quite small, the same conclusions as Oreopoulos et al. [2012b].
4.2.4 More Explanation
 As shown in Figures 3 and 5, in scop_sovp1, compared with the observations, the consistently weaker LW CREs and larger LWUPT, as well as the weaker SW CREs and smaller SWUPT, are found for most clds. Although this is partly due to the prescribed surface emissivity and simplified surface albedo, the small model ranges also suggest at least three other consistent solutions to this problem: (1) increasing the cloud condensate field, (2) increasing total cloud cover fractions, and (3) increasing cloud optical depth for both SW and LW. Usually, these solutions are quite ambiguous in dcop_dovp1 due to the large model spreads. In this section, an example is shown to illustrate the improvement of model performances by increasing the total cloud cover fractions. The generalized overlap generator produces larger total cloud cover fractions than the maximum-random overlap generator [Räisänen et al., 2004]. Thus, the results from the McICA method with the generalized overlap generator (McICA_Fgen1; Table 3) are shown in Figure 8. Compared with the results in scop_sovp1, i.e., the McICA method with the maximum-random overlap generator for inhomogeneous clouds, McICA_Fgen1 produces larger cloud radiative effects for both LW and SW, along with larger SWUPT and smaller LWUPT. The performances of all radiation transfer codes are significantly improved for most clds. Hence, after removing most of the disagreement in cloud fields, it becomes clearer on how to further improve the estimated CREs and irradiances.
 The remaining model discrepancies are mainly from different treatments of gas absorptions of H2O and O3 for SW and H2O, O3, and CO2 for LW; different spectral resolutions; different number of streams for scattering approximation; and different implementations of radiative transfer algorithms, such as whether scattering effects are included in LW calculations or not. For example, LW scattering effects have been explicitly considered in flg and cccma, but not in gfdl and rrtmg. Due to the smaller intermodel spreads in scop_sovp0/scop_sovp1 than the observation ranges, different treatments of LW scattering effects seem not to be a key factor in current model ranges.
 Moreover, when the same cloud optical property schemes are consistently applied in one intercomparison, there still exist some differences in cloud optical properties due to the different spectral bands adopted by each radiation code, although the energy-weighted methods are used in spectral integrals. Fortunately, these differences seem to have small impacts on the existing model ranges because the model spreads are quite small in scop_sovp0/scop_sovp1. Furthermore, the sampling errors of the McICA method for each radiation code may vary with different numbers of bands/subbands adopted. However, for a long-time integration (at least 1 month), the results from McICA assume no bias to those from ICA [Barker et al., 2002; Pincus et al., 2003; Räisänen et al., 2004]. As a result, the remaining model diversities of monthly global means in scop_sovp1/McICA_Fgen1 are usually smaller than those observation ranges, showing the small impacts of different sampling errors on model ranges.
4.3 Inter-cld Discrepancy
 Figure 9 shows the comparisons of the inter-cld discrepancy across 14 CAR cloud cover members (cld1–cld14) of the global mean (a) TOA SW CRE, (b) TOA LW CRE, (c) SFC SW CRE, and (d) SFC LW CRE among the different sets of experiments. The inter-cld discrepancy denotes the full spreads generated by different clds. The medians and min-max ranges are among the seven inter-cld discrepancies generated by applying the same radiation transfer code to each intercomparison of cld1–cld14. The results are for July 2004. In dcop_dovp1, the large min-max ranges of about 5 W m−2 for SW CREs and about 2–3 W m −2 for LW CREs show the obvious dependence of inter-cld discrepancy on different original radiation transfer codes. This emphasizes the complicated nonlinear nature in the coupling of cloud/radiation processes. Fortunately, as shown in scop_dovp0, such min-max ranges are greatly reduced to < 1–2 W m −2 by using the similar cloud optical properties for SW CREs and TOA LW CREs, and by removing the different cloud inhomogeneity effects for SFC LW CREs. The remaining nonlinear effects between cloud and radiation significantly decrease.
 In addition, including the cloud inhomogeneity effects in scop_sovp1 significantly decreases the inter-cld discrepancies (i.e., the medians), which is mostly due to the smaller cloud radiative effects in scop_sovp1 than in scop_sovp0.
4.4 Inter-cop Discrepancy
 Figure 10 compares the inter-cop discrepancy across seven CAR cloud optical property members (cop1–cop7) of the global mean TOA/SFC SW/LW CREs among different sets of experiments. The inter-cop discrepancy denotes the full ranges generated by different cloud optical property members (cops). The medians and min-max ranges are among the seven inter-cop discrepancies generated by applying the same radiation transfer code to each intercomparison of cop1–cop7. In “dovp1,” the original treatments of cloud subgrid-scale structures are used, while in “sovp0,” those in McICA_MRO are employed. Here, only cld1 was used, and the results are for July 2004.
 Clearly, for all SW CREs, using the same cloud subgrid-scale structures has very small impacts on the inter-cop discrepancies, as shown by almost the same medians in sovp0 as in dovp1. However, it significantly reduces the min-max ranges from about 2–3 W m −2 in dovp1 to < 1 W m−2 in sovp0, indicating the major roles of different cloud subgrid-scale structures in the sensitivity of inter-cop discrepancy on different radiation codes for SW. It is reasonable because different assumptions of cloud vertical overlap may produce a quite different clear-sky portion among different radiation transfer codes, which has strong nonlinear influences on different cops and hence generates such sensitivity. While for LW CREs, different cloud subgrid structures have quite small effects on both the inter-cop discrepancies and their sensitivities to different radiation codes. In addition, it is found that cloud optical property schemes have large impacts on TOA LW CREs as indicated by the large inter-cop discrepancies of ~15 W m−2, and meanwhile, very small effects on SFC LW CREs are shown by the medians < 1.5 W m−2. For TOA LW CREs, the variation of high cloud optical properties means much, while for SFC ones, without changes in the position of cloud base, the strong dependence of SFC downwelling LW fluxes on both near-surface water vapor and near-surface air temperature makes cloud presence rather irrelevant in many locations, especially in the tropics.
5 Discussion and Conclusions
 By using the CAR system, the current model spreads of CREs among seven radiation codes were significantly reduced, even to smaller than the observation ranges. Taking global July mean (2004) CREs for example, their model spreads can be decreased from about 10 W m−2 for SW and 5–8 W m −2 for LW both to 2–4 W m −2 by removing most of the disagreement in cloud fields. Furthermore, it is the first time that the detailed physical causes of current model spreads were investigated. Several sets of numerical experiments were performed to elucidate each individual contribution from the existing scheme diversity of cloud optical properties, cloud horizontal variability, cloud vertical overlap, and gas absorptions, respectively.
 The discrepancies in the treatments of cloud fields that are originally adopted by different radiation codes are the major reason for the current model spreads. The dominant roles of cloud subgrid-scale structures in general were demonstrated, accounting for 40–75% of the total model ranges. The key factors are usually different for different radiative components, especially for LW. Generally speaking, using different assumptions of cloud vertical overlap is critical to model spreads of SW components and TOA LW CREs (>35%), inferring the importance of both the position of cloud top level and the total cloud cover fractions, while different cloud inhomogeneity effects is key to those of SFC LW CREs and LWDNS (34–39%), indicating the roles of cloud optical properties. The differences in some gas absorptions also substantially contribute to the model diversities of SWDNS (~17.5%) and LWDNS (~17.9%). For SWDNS, this is most likely due to the long direct optical path to surface as opposed to the TOA where the diversities due to gaseous absorptions are smaller; while for LWDNS, gaseous absorptions and emissions below clouds play important roles. In addition, the variety in the existing assumptions of cloud vertical overlap largely limits the roles of different cloud optical properties.
 The complicated nonlinear nature existing in the coupling of cloud/radiation processes is highlighted in this study. The current intermodel diversities among different radiation codes vary strongly with different cloud cover members (clds), and meanwhile, the dependences both of the existing inter-cld discrepancies (i.e., discrepancies among different cloud cover members) and of the inter-cop discrepancies (i.e., discrepancies among different cloud optical property members) are also shown on different radiation transfer codes. The major reasons have been investigated in this study. In general, using the different assumptions of cloud vertical overlap accounts for most of the sensitivity of the intermodel diversity to different clds. Different clds produce different vertical profiles of cloud cover fractions, which determine whether the different assumptions of cloud vertical overlap matters much for radiation. Then, the quite different ranges of clear-sky portions among different radiation codes generate the large nonlinear sensitivity to different clds. Moreover, for SW CREs/TOA LW CREs, the nonlinear sensitivities of inter-cld discrepancies to host models mainly result from the different cloud optical properties used, and for SFC LW CREs, applying different cloud inhomogeneity effects is the major reason. Both two factors are closely related to cloud optical properties, indicating the strong nonlinear influences of cloud cover fractions and cloud optical properties on the radiative transfer calculations. Using the different cloud subgrid-scale structures explains well the sensitivities of inter-cop discrepancies to host radiation codes for SW CREs. In a word, by removing the contributions from the different treatments of some key cloud factors mentioned above, the remaining nonlinear effects between cloud and radiation are largely reduced.
 In this study, we have demonstrated that our conclusions are commonly valid despite the explicit methods (i.e., McICA with maximum-random cloud generator, McICA with generalized cloud generator, and MOSAIC method). For monthly zonal/global means, the number of subcolumns adopted by the McICA method has ignorable impacts on estimated CREs and irradiances. The McICA noise of zonal/global means for sufficiently long runs (at least a month) is quite small with small impacts on model spreads.
 Given the large intermodel diversity, generally some specific methods can be used to improve the performance of each radiation code. Those methods are model dependent, i.e., those working well for some radiation codes may not work for others at all. This largely obscures the roles of clouds in radiation. As shown in this study, we now have the ability to significantly diminish the large model discrepancies by using a system like the CAR. Hence, without distractions from the large intermodel diversities, to achieve the accurate radiation transfer calculations, more work independent on host radiation transfer codes can be done to consistently improve cloud water fields, cloud optical property schemes, or/and treatments of cloud subgrid-scale structures. For example, as shown by this study, after removing most of the disagreement in cloud fields, combined with cop1 (i.e., the cloud optical property member 1), McICA with the generalized overlap generator produces better modeled CREs and irradiances for most cloud cover members than McICA with maximum-random overlap generator. The search for an optimal model would enhance our physical understanding of cloud and radiation processes and their interaction and feedback in climate models.
 This work was supported by the U.S. DOE Office of Biological and Environmental Research (BER) DE-SC0001683, the National Basic Research Program of China (2010CB951901), NSFC40830103, and the “Strategic Priority Research Program-Climate Change: Carbon Budget and Relevant Issues” of the Chinese Academy of Sciences (XDA05110101). The ERA-Interim data were obtained from the ECMWF. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under contract DE-AC02-05CH11231. During the initial period (2008–2009), Feng Zhang was a visiting scholar from the International Center for Climate and Environment Sciences, Institute of Atmospheric Physics, Chinese Academy of Sciences. This research also used the Evergreen computing cluster at the Pacific Northwest National Laboratory. Evergreen is supported by the Office of Science of the U.S. Department of Energy under contract DE-AC05-76RL01830. We thank Shenjian Su for his great help on the experiment conduction. We appreciate the Editor-in-Chief of JGR-Atmosphere and our three invaluable reviewers.