1. Introduction
The radiative transfer equation (RTE), which describes the radiative transfer process, is an integro-differential equation. Many approximative methods have been developed to solve RTE (e.g., Chandrasekhar 1950; Liou 1974; Coakley and Chýlek 1975; Wiscombe 1977; Meador and Weaver 1980; Liou et al. 1988; Toon et al. 1989; Shibata and Uchiyama 1992; Li and Ramaswamy 1996; Fu et al. 1997; Zhang et al. 2010; van Oss and Spurr. 2002; Lin et al. 2013; Zhang et al. 2013; Zhang and Li 2013), since there are no exact solutions of RTE.
For solar radiation, analytical solutions of various two-stream approximations (Meador and Weaver 1980) have been widely used in climate models (Oreopoulos et al. 2012). More accurate methods of four-stream approximation and six-stream approximation have also been proposed (e.g., Liou et al. 1988; Li and Ramaswamy 1996; Zhang et al. 2010; van Oss and Spurr 2002; Lin et al. 2013; Zhang et al. 2013; Zhang and Li 2013).
Based on the invariance principle (Chandrasekhar 1950), an adding method of four-stream discrete ordinates method (DOM) (4DDA) has been proposed. The advantage of the adding method is that it can handle partial cloud in climate models (e.g., Chou and Suarez 1999; Nakajima et al. 2000; Li et al. 2005).
For solar radiation, 4DDA is much more accurate than the adding method of two-stream DOM (2DDA) in fluxes and heating rate, especially under cloudy-sky conditions, where cloud plays an important role in radiative transfer with different dynamics and microphysics (Lu et al. 2014, 2016). Zhang et al. 2015 used 4DDA in climate models to estimate aerosol direct radiative effect and forcing, showing that 2DDA underestimates the aerosol shortwave direct radiative effect and heating rate significantly.
In Zhang et al. (2015), the four-stream effect in the infrared has not been included. Compared to solar radiation, the scattering effect is much weaker for infrared radiation. Therefore, in most current climate models the method of absorption approximation is widely used in solving infrared radiative transfer (IRT) (Oreopoulos et al. 2012), in which the scattering effect is neglected. However, studies have shown that neglecting scattering can cause an overestimation of outgoing longwave radiation in climate simulations; the difference can be larger than 4 W m−2 (Fu et al. 1997; Li 2000, 2002).
The infrared scattering effect was considered in Wiscombe (1976) by a pure doubling method. First, the solutions are found for each very thin layer by applying the “thin-layer approximation,” then the doubling process is continually carried out to solve the inhomogeneous source and find the solutions for 2n layers (where n is an integer). This method in principle is a numerical approach. It lacks a corresponding adding algorithm for infrared radiation since the invariance principle in Chandrasekhar (1950) has only been applied to solar radiation. In the commonly used absorption approximation method, the downward intensity and upward intensity are decoupled as a result of no scattering effect. This makes the layer-to-layer connection simple and straightforward. When the multiple scattering is included in an infrared radiative transfer process, the downward and upward intensities are coupled. Therefore, the layer-to-layer connection should be dealt with by an adding algorithm, which follows the physics of invariance principle.
Creating an adding algorithm for infrared radiation is the main purpose of this study. First, an analytical solution of four-stream DOM for infrared radiation will be found that is consistent with the four-stream DOM for solar radiation (Liou 2002). There are two types of thermal source assumptions: Planck function exponential or linear dependence on optical depth. Both types will be considered in solutions. Second, the adding method of four-stream DOM for radiation will be built to solve infrared radiative transfer process through a multilayer atmosphere. In addition, the obtained four-stream adding algorithm can be degenerated by using in two-stream DOM.
In section 2, the single-layer solutions by four-stream DOM for infrared radiation are discussed. In section 3, the four-stream adding algorithm, 4DDA, is formulated based on invariance principle. The corresponding result of an adding algorithm to the two-stream DOM (2DDA) is shown in appendix C. In addition, the accuracy checks of δ-2DDA and δ-4DDA are performed by comparing with δ-64-stream discrete ordinates calculations in section 4. In section 5, a summary is given.
2. Single-layer solution by four-stream discrete ordinates method
a. Infrared radiation
b. Diffuse radiation
3. Adding method of infrared radiation
In solar radiative transfer, Chandrasekhar (1950) has proposed the principles of invariance. The corresponding principles of invariance in infrared radiative transfer have never been discussed before. In the following, four principles of infrared invariance are proposed. To state it clearly, a combination of two homogeneous layers with optical depth 
- The upward intensity
- The downward intensity
- The outgoing intensity
- The outgoing intensity
The schematic diagram of principles 1–4 is given in Fig. 1. These four principles of invariance are different from the four principles of invariance for solar radiation by Chandrasekhar (Zhang et al. 2013). An important characteristic of the infrared is that the thermal source emission acts everywhere. The thermal source emission is contained in the intensities of 
Schematic diagram of 
Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-15-0317.1
For the two-stream DOM of IRT, the adding method can be easily obtained in a similar way (see appendix C). For 2DDA, Planck function is only approximated exponentially in τ.
4. Comparison results and discussion
The δ-function adjustment is used to deal with the forward peak contribution in order to enhance the accuracy of radiative schemes in the following calculations (Wiscombe 1977; Liou et al. 1988; Iwabuchi 2006; Zhang et al. 2013; and others). Therefore, 2DDA and 4DDA become δ-2DDA and δ-4DDA, respectively. In this section, the accuracy of δ-2DDA and δ-4DDA will be examined by the benchmark result calculated from the discrete ordinates numerical model (Stamnes et al. 1988). A δ-64-stream scheme (δ-64S) is used in the discrete ordinates numerical model.
a. Double layer
Relative errors of the (left) δ-2DDA and (right) δ-4DDA for (top) the upward and (bottom) the downward emissivity. Τhe benchmark is δ-64S. The optical depth of the two layers 
Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-15-0317.1
The values 
As in Fig. 2, but for the single-scattering albedos 
Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-15-0317.1
To illustrate that δ-4DDA performs well in IRT, single- and multilayer results are shown in Table 1. Applying δ-4DOM to a medium with 
Emissivities for δ-4DDA vs number of sublayers with a single homogeneous layer, in which the temperature is constant; ω = 0.7771 and g = 0.7720 are considered with two optical depths (τtotal = 1.0 and 5.0).
b. Multilayer atmosphere
For a multilayer atmosphere, the radiation model uses a correlated-k distribution scheme (Li and Barker 2005) for gaseous transmission with H2O, O3, CH4, CO2, and N2O in the spectrum range. This model adopts nine infrared bands in wavenumber ranges 0–340, 340–540, 540–800, 800–980, 980–1100, 1100–1400, 1400–1900, 1900–2200, and 2200–2500 cm−1. The optical properties of ice and water clouds are calculated based on the parameterization of Yang et al. (2015) and Dobbie et al. (1999), respectively.
The U.S. standard atmospheric profile is used (McClatchey et al. 1972) and surface emissivity 
In the inhomogeneous case, we denote δ-4DDA_E for a Planck function with exponential dependence on medium optical depth and δ-4DDA_L for linear dependence, as solutions of (5) and (8). The benchmark results (the results of δ-64S) of clear-sky heating rate, downward flux, and upward flux are shown in Fig. 4 for 
(top) The profiles of (left) heating rate, (center) downward flux, and (right) upward flux for clear sky with 
Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-15-0317.1
It is interesting to find that the four-stream δ-4DDA can be less accurate than the two-stream δ-2DDA, for upward flux in the case of 
A low cloud with effective radius of re = 5.89 μm and liquid water content of LWC = 0.22 g m−3 is positioned from 1.0 to 2.0 km. In Fig. 5, the results for low clouds are presented. The maximum errors in heating rates are about 0.18, 0.1, and less than 0.1 K day−1 for δ-2DDA, δ-4DDA_E, and δ-4DDA_L, respectively. The results are not sensitive to the surface emissivity, which is different from the clear-sky case. For downward flux, the errors of δ-2DDA are generally larger in comparison with δ-4DDA_E and δ-4DDA_L. For upward flux, a large error about 1.2 W m−2 is produced by δ-2DDA, whereas errors are bounded by 0.7 W m−2 for δ-4DDA_E and δ-4DDA_L.
As in Fig. 4, but for under low-cloud conditions.
Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-15-0317.1
A middle cloud with re = 6.2 μm and LWC = 0.28 g m−3 is positioned from 4.0 to 5.0 km. In Fig. 6, the results under middle-cloud conditions are illustrated. Similar to Fig. 5, the accuracy of δ-4DDA is better than that of δ-2DDA. For 
As in Fig. 4, but for under middle-cloud conditions.
Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-15-0317.1
A high cloud with mean effective size De = 41.1 μm and ice water content IWC = 0.0048 g m−3 is positioned from 9.0 to 11.0 km. Figure 7 gives the results for an ice-cloud case. Based on Figs. 1 and 2, the errors of δ-2DDA should increase, since the optical depth of the ice cloud is much smaller than that of the water cloud. In addition, the size of ice crystals is generally much larger than that of water droplets; thus, the scattering effect is much enhanced. Compared to the two-stream method, four-stream method can handle scattering more appropriately because more scattering phase function moments are addressed in a four-stream scheme (Yi et al. 2014). It is shown in Fig. 7 that the maximum errors in heating rates for δ-2DDA is close to 0.35 K day−1. The errors in downward flux by δ-2DDA reach 1.4 W m−2, larger than the result for water clouds. The accuracy in heating rate and flux increases a lot by using the two δ-4DDA schemes.
As in Fig. 4, but for under high-cloud conditions.
Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-15-0317.1
The results of heating rate and downward and upward fluxes for low, middle, and high clouds are shown in Fig. 8. Again, δ-4DDA yields better results than δ-2DDA.
As in Fig. 4, but for the sky containing all three cloud types.
Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-15-0317.1
From the result of water and ice clouds shown in Figs. 4–8, generally δ-4DDA_L is slightly more accurate than δ-4DDA_E.
Rapid computations are required for weather and climate modeling. Table 2 illustrates the results of efficiency for δ-2DDA and δ-4DDA. The inverse matrix method used in the Fu–Liou model (Liou et al. 1988; Fu 1991; Fu et al. 1997) is also used to solve layer connection in δ-two-stream DOM scheme (denoted as δ-2DOM) and in δ-four-stream DOM scheme (denoted as δ-4DOM). From Table 2, there is no significant difference between δ-2DDA and δ-2DOM. For a pure radiative transfer calculation without considering gaseous transmission, the computational time of δ-4DDA is only about one-third of δ-4DOM. The efficiency of δ-4DDA is caused by avoiding general eigenvalue decomposition and also caused by the unrolling loop in δ-4DDA to avoid general matrix operations. For a radiative transfer calculation with gaseous transmission and cloud absorption, the computational time of δ-4DDA is about 43% of δ-4DOM. Besides, the advantage of δ-4DDA can be particularly efficient in dealing with the partial cloud in climate modeling and cloud retrieval (Nakajima et al. 2000; Gabriel et al. 2006).
Timing of IRT calculations using various approximation methods (normalized to the δ-2DOM method). Gaseous transmission and cloud absorption are included in “radiation model,” but not in “algorithm only.”
5. Summary
The main purpose of this study is to develop an adding algorithm for infrared radiation based on the invariance principle. For the Planck function with exponential and linear terms of the optical depth, the solutions for four-stream radiative transfer and their corresponding adding schemes of δ-4DDA_E and δ-4DDA_L have been created. The four-stream result can be degenerated to a two-stream result as δ-2DDA.
A wide range of accuracy checks δ-2DDA and δ-4DDA are performed in a double-layer case. It is shown that δ-4DDA is much more accurate than δ-2DDA, especially for small optical depths.
Application of δ-2DDA and δ-4DDA into a realistic atmospheric profile, the errors in heating rate by δ-2DDA are as high as 1.0 K day−1, but less than 0.1 K day−1 by δ-4DDA. For downward flux, the errors by δ-2DDA are up to 1.5 W m−2, while those by δ-4DDA are reduced to 0.8 W m−2. A dramatic improvement occurs in the upward flux, where the errors by δ-2DDA can reach 1.6 W m−2 in the middle-cloud case but are reduced to less than 0.7 W m−2 by δ-4DDA.
The computing time involved by δ-2DDA and δ-4DDA is minimal since both of them are analytical methods. For the pure algorithm, the computational time of δ-4DDA is one-third of the corresponding inverse matrix method used in Fu–Liou model.
In summary, in order to consider the cloud infrared scattering effect, the accurate solution of infrared radiative transfer and its related adding process are needed. This has been achieved by δ-2DDA and δ-4DDA. This methodology of infrared δ-4DDA matches with the methodology of solar δ-4DDA (Zhang et al. 2013); the two parts together form a complete four-stream radiative transfer scheme for climate models. The coding of δ-4DDA is available from the authors upon request.
The authors thank three anonymous reviewers for their constructive comments and Professor P. Yang for his editorial effort. The work is supported by the State Key Development Program for Basic Research of China (2015CB452805), National Natural Science Foundation of China (41305004, 91537213, and 41675003), Foundation of State Key Laboratory of Severe Weather (2016LASW-B07), and Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).
APPENDIX A
The Coefficients of Eq. (5)
The coefficients 
APPENDIX B
The Solution of Four-Stream Infrared Radiative Transfer with Linear Dependence on Optical Depth
APPENDIX C
The Adding Process for Two-Stream Infrared Radiative Transfer
The single-layer solution for the two-stream approximation based on DOM has been discussed in detail (Toon et al. 1989; Fu et al. 1997), in which the one-node Gaussian quadrature 
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