## 1. Introduction

Radiative transfer is a key issue for climate modeling and remote sensing. The radiative transfer equation is an integro-differential equation. The exact solution of the radiative transfer equation in a scattering and absorbing media is impossible to be obtained in a computationally efficient manner even for the plane-parallel case; thus, approximate methods are necessary. In the last several decades, considerable attention has been paid to find simple and effective methods to solve the radiative transfer equation (Liou 1974; Liou et al. 1988; Shibata and Uchiyama 1992; Kylling et al. 1995; Li and Ramaswamy 1996; Fu et al. 1997; Li and Dobbie 1998; Thomas and Stamnes 1999; Zhang et al. 2010, 2012; and others). Among these approximative methods, the *δ*-two-stream approximation provides a simple answer to radiative transfer. Especially, the *δ*-two-stream discrete ordinates method (DOM) and *δ*-Eddington approximation are widely used in weather and climate models.

The accuracy of the different *δ*-two-stream approximations has been analyzed in both idealized medium and realistic atmospheric profiles (King and Harshvardhan 1986; Kay et al. 2001; Lu et al. 2009). It is found that both the *δ*-two-stream schemes of DOM and Eddington approximation were generally accurate under the clear-sky condition, with relative errors of less than 1.5% for heating rate and flux. However, the cloud heating might have been underestimated by as much as 10% under the cloudy-sky condition, which indicates that a four-stream or higher order approximation scheme is necessary in order to obtain the accurate solar cloud absorption in weather and climate models.

The single-layer analytical solutions of the *δ*-four-stream approximation have been obtained (Liou et al. 1988; Shibata and Uchiyama 1992; Li and Ramaswamy 1996; Li and Dobbie 1998). Among them, Liou et al. (1988) derived the solution of the four-stream DOM; Li and Ramaswamy (1996) obtained the solution of the *δ*-four-stream spherical harmonic expansion, which is the higher order extension of the Eddington approximation. However, such solutions are valid only for homogeneous single layer. Fu (1991) had extended the *δ*-four-stream DOM solution to a vertically inhomogeneous atmosphere. The DOM involves the discretization of the basic radiative transfer equation and the solution of a set of first-order differential equations. The calculation is performed by using an inverse matrix formulation to solve the connection of all the layers in radiative transfer. Generally, in current climate models, the connection of all the layers in *δ*-two-stream approximation is solved by the doubling–adding method, which is based on the invariance principle found by Chandrasekhar (1950). Doubling–adding works naturally with layered medium and yields reflection and transmission readily. In addition, the advantage of the doubling–adding method is that the calculation is performed one layer at a time, without the requirement for handling the inverse of a large matrix associated with all the model layers. Another advantage of the doubling–adding method is that it can handle partial cloud under a certain condition (e.g., Chou and Suarez 1999; Li et al. 2005). Because of these advantages, the doubling–adding method has been applied to the *δ*-four-stream approximation (Chou 1992; Li and Ramaswamy 1996; Tian et al. 2007; Ayash et al. 2008). However, the doubling–adding method used in the *δ*-four-stream approximation is that developed for the *δ*-two-stream approximation (Coakley et al. 1983; Yang 2004). In such methods, the isotropic condition is imposed on the diffuse radiation; thus, the angular characteristics of the radiance field are not represented properly. It is found that the accuracy of fluxes for the *δ*-four-stream approximation could be worse than that of the *δ*-two-stream approximation by using the same two-stream doubling–adding method. As the computer becomes faster and faster, *δ*-four-stream approximation definitely become more and more important. However, there has been no effort to derive an analytical doubling–adding method so far, which enables us to calculate the *δ*-four-stream radiative transfer through multiple layers in a vertically inhomogeneous atmosphere.

The main purpose of this work is to develop a *δ*-four-stream doubling–adding method (*δ*-4DA) based on the invariance principle. Here, we focus only on the *δ*-four-stream DOM. The extension of *δ*-4DA to the four-stream spherical harmonic expansion approximation will be addressed in a subsequent work. In section 2, the reflectance and transmittance matrices of direct/diffuse radiation are presented for the single-layer solution of *δ*-four-stream DOM. In section 3, *δ*-4DA is formulated for computation of solar radiative transfer through multiple layers. In section 4, the accuracy of *δ*-4DA is systematically investigated. Finally, a summary and conclusions are given in section 5.

## 2. Single-layer solution of four-stream approximation

*I*(

*τ*,

*μ*) is diffuse intensity;

*μ*is the cosine of the zenith angle;

*τ*is the optical depth;

*ω*is the single-scattering albedo;

*P*(

*μ*,

*μ*′) is the azimuthally averaged scattering phase function, defining the light incidence at

*μ*′, and scattered away at

*μ*;

*F*

_{0}is the solar flux at the top of the atmosphere; and

*μ*

_{0}is the cosine of the solar zenith angle. The azimuth-independent phase function

*P*(

*μ*,

*μ*′) can be written as (e.g., Liou et al. 1988)where

*P*is the Legendre function. The cosine of the scattering angle is defined by

_{l}*φ*the azimuthal angle. We can determine

*ω*from the orthogonal property of Legendre polynomials

_{l}*ω*

_{0}= 1 and

*ω*

_{1}= 3

*g*, with

*g*being the asymmetry factor.

*N*= 3, the above equation becomeswhere

*I*=

_{i}*I*(

*τ*,

*μ*), the quadrature point

_{i}*μ*

_{−i}= −

*μ*, the weight

_{i}*a*

_{−i}=

*a*,

_{i}*i*= ±1, ±2, and

*j*= ±1, ±2. The two-node Gaussian quadrature is used, by which

*μ*

_{1}= 0.211 324 8,

*μ*

_{2}= 0.788 675 2,

*a*

_{1}= 0.5, and

*a*

_{2}= 0.5 (Li 2000).

*f*

_{0}= 1/

*μ*

_{0},

*e*(

_{j}*j*= 1, 2, 3, 4) are defined asThe other terms of

*k*(

_{i}*i*= 1, 2) are defined the same as in Liou [2002, see their (6.5.48), p. 315]. For the single-layer solution, the surface albedo is zero. Therefore, the boundary condition is set to be no diffuse radiation from the top and bottom of the layer; that is,

*I*

_{−1,−2}(

*τ*= 0) = 0 and

*I*

_{1,2}(

*τ*=

*τ*

_{1}) = 0, where

*τ*

_{1}is the optical depth of the layer. Consequently, the coefficient matrix

*ψ*= exp(−

_{i}*k*

_{i}τ_{1}) (

*i*= 1, 2). Finally, the upward intensities at the top and the downward intensities at the bottom of the layer are given bywhereandEquation (7) is the single-layer solution for the four-stream approximation. For use in the next section, we define the direct reflection and transmission matrices aswhere

*i*= 1, 2). We also define the diffuse reflection and transmission matrices aswhere

*i*,

*j*= 1, 2), and the direct transmission

Matrices in (8) are called “direct” because they are related the direct incoming solar zenith angle. Matrices in (9) are called “diffuse” because they are related to a diffused angular averaged result. Actually, both the direct and diffuse reflection and transmission are used to deal with the diffuse radiation. The result of direct solar beam is represented by *T*^{dir}(*μ*_{0}), which should be added to the total transmission.

## 3. Four-stream doubling–adding method

*τ*

_{1}and

*τ*

_{2}being the optical depth of the first and second layers, respectively. Based on the definition of reflection and transmission matrices in (8), the dimensionless upward and downward internal intensities at lower boundary of the first layer (level 2) are obtained aswhere

*i*= 1, 2).

The reflection and transmission functions are denoted by *R*_{1}(*μ*, *μ*_{0}) and *T*_{1}(*μ*, *μ*_{0}) for the first layer, *R*_{2}(*μ*, *μ*_{0}) and *T*_{2}(*μ*, *μ*_{0}) for the second layer, and *R*_{1,2}(*μ*, *μ*_{0}) and *T*_{1,2}(*μ*, *μ*_{0}) for the combined two layers. The similar definition is applied to the diffuse quantities. The superscript asterisk denotes that radiation comes from below (see Fig. 1). For a homogeneous atmosphere, *R*(*μ*, *μ*_{0}) = *R**(*μ*, *μ*_{0}) and *T*(*μ*, *μ*_{0}) = *T**(*μ*, *μ*_{0}).

The above four principles of invariance can be stated as follows:

- The reflected intensity
*U*(*μ*,*μ*_{0}) at level 2 comes from two components: the reflection of the direct solar beamby the second layer and the reflection of the downward diffuse intensity *D*(*μ*,*μ*_{0}) by the second layer. - The diffusely transmitted intensity
*D*(*μ*,*μ*_{0}) at level 2 comes from two components: the direct transmission from the first layer*T*_{1}(*μ*,*μ*_{0}) and the reflection of the upward diffuse intensity*U*(*μ*,*μ*_{0}) by the first layer. - The reflected intensity
*R*_{1,2}(*μ*,*μ*_{0}) at the top of the double layer is equivalent to two components: the reflection by the first layer*R*_{1}(*μ*,*μ*_{0}) and the total transmissionof the upward diffuse intensity *U*(*μ*,*μ*_{0}) through the first layer. - The diffusely transmitted intensity
*T*_{1,2}(*μ*,*μ*_{0}) at the bottom of the double layer is equivalent to two components: the transmission of direct solar beamthrough the second layer and the total transmission of the downward diffuse intensity *D*(*μ*,*μ*_{0}) through the second layer.

*a*and

_{i}*μ*(

_{i}*i*= 1, 2) are given above], we haveFrom (12a) and (12b), we haveSubstituting (13) into (12c) and (12d), we obtain the direct reflection and transmission aswhere

*μ*

_{0}=

*μ*(

_{i}*i*= 1, 2) into (14a) and (14b), and multiplying both sides by

*μ*, we obtain the diffuse reflection and transmission asand the above two equations can be rewritten asFor a light beam incident from below (see Fig. 1),

_{i}*k*, which is obtained through a downward path calculation aswhere

*N*) to layer

*k*, which is obtained through a upward path calculation asIn addition,

*R*is the surface albedo. Thus, the dimensionless upward and downward internal intensities [

_{s}*k*+ 1 (lower boundary of the layer

*k*) are determined by the results from the downward and upward calculation paths asFinally, the upward and downward fluxes at level

*k*+ 1 (lower boundary of the layer

*k*) areand the upward and downward fluxes at the top of the atmosphere (TOA) arewhere

**is a 1 × 2 matrix, as**

*μ***= [**

*μ**μ*

_{1}

*μ*

_{2}].

*f*is separated out of the phase function. Under the

*δ*-function adjustment, the optical parameters are scaled by (Liou 2002)For

*δ*-2DA,

*l*= 0, 1 and

*δ*-4DA,

*l*= 0, 1, 2, 3, and

*δ*-function adjustment would enhance the accuracy of approximate treatments of multiple scattering.

## 4. Comparison results and discussion

### a. Double layer

First, *δ*-4DA is investigated in a double-layer case. The results of the double layer could be compared to those of the single layer, which have been well analyzed by Liou et al. (1988). Moreover, the results from the double layer help us to understand those from the multilayer shown in the next subsection.

*δ*-4DA for reflection

*r*(

*τ*

_{1,2},

*μ*

_{0}), total transmission

*t*(

*τ*

_{1,2},

*μ*

_{0}), and absorption

*a*(

*τ*

_{1,2},

*μ*

_{0}):We consider a case of the same-layer optical depth (

*τ*

_{1,2}= 2

*τ*

_{1}= 2

*τ*

_{2}), and over wide ranges of solar zenith angle, optical depth and single-scattering albedo. The asymmetry factors are set as

*g*

_{1}= 0.837 and

*g*

_{2}= 0.861 for the first and second layers, respectively. These two values correspond to the asymmetry factors of water cloud with effective radii of 6 and 15

*μ*m in the 0.25–0.69-

*μ*m spectral range (Dobbie et al. 1999). The rigorous standard model used in the following is the discrete ordinates numerical model (Stamnes et al. 1988). We use

*δ*-128-streams (

*δ*-128S) in the discrete ordinates calculations with the Henyey–Greenstein phase function.

In Fig. 2, the relative errors in reflection and transmission with respect to *δ*-128S are shown for both *δ*-2DA and *δ*-4DA under the nonabsorbing condition with single-scattering albedo *ω*_{1} = *ω*_{2} = 1. For reflection, the relative errors of *δ*-2DA generally exceed 10% for *μ*_{0} < 0.5 and *μ*_{0} > 0.9 and 20% for *μ*_{0} < 0.35 with *τ*_{1,2} < 1. Errors greater than 10% and 20% occur for the total transmission when *μ*_{0} < 0.2 and *μ*_{0} < 0.1, respectively. In general, the relative error is dramatically reduced by using *δ*-4DA. For reflection, 5%–15% of relative error occurs for *μ*_{0} < 0.25, 5%–10% of relative error occurs for 0.35 < *μ*_{0} < 0.7, and about 5% of relative error occurs for *μ*_{0} > 0.85 with *τ*_{1,2} < 1. For total transmission, an error greater than 5% occurs in the region of *μ*_{0} < 0.1. We note that these regions are associated with very small values. Thus, absolute errors are extremely small. For *δ*-4DA, the contours of error in Fig. 2 for *δ*-4DA are very similar to the result shown in Liou et al. (1988), which indicates *δ*-4DA performs well under the double-layer condition.

In Fig. 3, *ω*_{1} = *ω*_{2} = 0.9 is used as an example for an absorbing case, as well as *g*_{1} = 0.837 and *g*_{2} = 0.861. The absorbing condition leads to a larger error for *δ*-2DA in reflection compared to the nonabsorbing condition. However, the relative errors are much smaller in *δ*-4DA with relative errors mostly bounded by 5% for *τ*_{1,2} > 1. In contrast, the relative error is up to 15% in *δ*-2DA. A dramatic improvement occurs in transmission for *δ*-4DA, with errors less than 2% in most regions. In contrast, a large region is dominated by relative errors up to 10% in *δ*-2DA. For absorption by *δ*-4DA, most of the regions are bounded by error less than 2%, except for the grazing incident case of *μ*_{0} < 0.1. In contrast, the error is up to 10% in *δ*-2DA. Again, the contours of error for *δ*-4DA in Fig. 3 are very similar to the single-layer result shown in Liou et al. (1988).

To show that *δ*-4DA works in general, a multilayer result is presented in Table 1. Considering a case of *ω* = 0.9, *g* = 0.837, and *μ*_{0} = 0.5, we apply *δ*-four-stream approximation to a media as a single layer, and also apply *δ*-4DA to the same media, which is divided into 2, 5, and 10 layers. The optical depths are the same for each case, and four different optical depths are considered. In addition, exact values are computed from *δ*-128S for the single layer. It is clearly shown in Table 1 that the results for reflection, absorption, and transmission (not shown) are identical for the media containing different layers. The slight differences between the results are due to numerical roundoff errors. This indicates that the multilayer connection is properly accounted by *δ*-4DA in the four-stream radiative transfer process.

Reflection and absorption computed from *δ*-4DA vs different number of layers, which are divided from the same medium with *ω* = 0.9, *g* = 0.837, and *μ*_{0} = 0.5, four optical depths (*τ*_{1} = 0.1, 0.5, 1.0, 2.0) are considered. Exact values are computed from *δ*-128S for the single layer.

Table 2 is the same as Table 1, but the doubling–adding method, which is development for *δ*-two-stream approximation, is used in the *δ*-four-stream approximation. It shows that the results of reflection, absorption, and transmission (not shown) vary with the media containing different layers. In most cases, the more the number of sublayers is, the worse the results become. Therefore, 2DA is inappropriate for being used in *δ*-four-stream approximation.

As in Table 1, but the doubling–adding method, which is development for *δ*-two-stream approximation, is used in the *δ*-four-stream approximation.

### b. Multilayer atmosphere

The radiative transfer in the atmosphere is a complicated process. It depends not only on the single-layer direct reflection and transmission, but also on the diffuse results and the gaseous transmission, cloud/aerosol scattering and absorption, etc. It is important to evaluate errors in radiative transfer under a variety of atmospheric conditions. The Beijing Climate Center radiation model (BCC_RAD) is used in this study. This model adopts a correlated-*k* distribution method (Zhang et al. 2003) with nine solar bands in wavenumber ranges 2680–5200, 5200–12 000, 12 000–22 000, 22 000–31 000, 31 000–33 000, 33 000–35 000, 35 000–37 000, 37 000–43 000, and 43 000–49 000 cm^{−1}. All major greenhouse gases—H_{2}O, CO_{2}, O_{3}, N_{2}O, CH_{4}, and CFCs—are included. Aerosols were neglected.

In the benchmark calculations, the discrete ordinates model (Stamnes et al. 1988) with *δ*-128S is incorporated with the gaseous transmission scheme of BCC_RAD. Also, the *δ*-2DA and *δ*-4DA schemes are incorporated with the same gaseous transmission scheme. The atmosphere was vertically divided into 400 layers, each with a thickness of 0.25 km. The U.S. standard atmospheric profile (McClatchey et al. 1972) is used in the calculations of flux and heating rate. A dry land surface albedo scheme is adopted (Nakajima et al. 2000), in which the first and the second bands were set at 0.1 and 0.26, respectively, and the other bands were set at 0.3. Solar zenith angles of *μ*_{0} = 1 and *μ*_{0} = 0.5 are considered, with *μ*_{0} = 0.5 being close to the global-averaged daily-mean result.

In the left column of Fig. 4, the benchmark results of the clear-sky heating rate are shown with *μ*_{0} = 1 and 0.5. The absolute errors of *δ*-2DA and *δ*-4DA are shown in the right column. Generally, *δ*-2DA produces accurate results under the clear-sky condition, with absolute error less than 0.02 K day^{−1} at *μ*_{0} = 0.5. However, *δ*-2DA generally produces a lower heating rate, especially in the region near the surface. The result shown in Fig. 3 can partly explain why the heating rates produced by the *δ*-2DA are smaller, since the absolute error of absorption is mostly negative in sign, especially in the region of small optical depth. In Fig. 4, *δ*-4DA yields a substantially more accurate result in heating rate, which is also consistent with the result shown in Fig. 3.

The corresponding errors in upward flux at the TOA and the downward flux at the surface are shown in Tables 3 and 4 with *μ*_{0} = 1 and 0.5, respectively. The errors in flux are less than 1 W m^{−2} for *δ*-2DA and the accuracy is considerably improved by using *δ*-4DA.

Comparison of the two parameterizations and *δ*-128S for solar fluxes (W m^{−2}) at the TOA and surface for the solar zenith *μ*_{0} = 1. The numbers in parentheses give differences (W m^{−2}) between the two schemes and *δ*-128S.

As in Table 1, but for the solar zenith *μ*_{0} = 0.5.

Figures 5 and 6 show the heating rate profiles calculated by the benchmark model in the presence of the low and middle clouds, respectively. The low cloud was positioned from 1.0 to 2.0 km with a liquid-water content (LWC) of 0.22 g m^{−3} and an effective radius of 5.89 *μ*m. A middle cloud was positioned from 4.0 to 5.0 km with LWC = 0.28 g m^{−3} and effective radius of 6.2 *μ*m. These values well represent the global mean values for cloud microphysical properties (Fu et al. 1997).

In Fig. 5, when *μ*_{0} = 1, the absolute errors *δ*-2DA in heating rate become much larger than those shown in the clear-sky case, with relative errors over 6% at the cloud-top region. It is well known that the heating at the top of cloud can influence the moisture circulation inside the cloud and therefore this has significant impact on the growth of cloud. The low cloud implemented in the atmosphere was thick in optical depth and Fig. 3 shows that the *δ*-2DA scheme generally produces a lower absorption in a wide range of optical depths at a small solar zenith angle. Although it has been emphasized that the final results in heating rate are not solely determined by the absorption shown in Fig. 3 for the idealized condition, a connection between them seems to exist to some extent. When *μ*_{0} = 0.5, both the cloud heating rate and the error of *δ*-2DA are reduced, but the relative error is still about 6%. When *δ*-4DA is implemented, it is found that the relative error is considerably suppressed from that of *δ*-2DA. The relative error becomes less than 1% for the both solar zenith angles. This indicates that *δ*-4DA is accurate enough in obtaining the cloud-top solar heating.

The corresponding errors in flux are also shown in Tables 3 and 4. *δ*-2DA overestimates the upward flux at TOA by about 10 and 5 W m^{−2} for *μ*_{0} = 1 and *μ*_{0} = 0.5, respectively; these errors are much reduced to about 2 and 1 W m^{−2} by applying *δ*-4DA. The errors in the downward flux at the surface are also substantially reduced by *δ*-4DA.

In Fig. 6, the results of the middle cloud are shown. Since LWC of the middle cloud is larger than that of the low cloud, the corresponding heating rate increases compared to that of the low cloud. Similar to Fig. 5, *δ*-4DA is able to provide a much more accurate result with a relative error of less than 1%. In Tables 3 and 4, the corresponding result in flux is listed. Again, *δ*-4DA can substantially improve the accuracy of the result.

In Fig. 7, both of the low and middle clouds are considered. The result of heating rate for the middle cloud is close to that of Fig. 6, but the heating rate becomes very small for the low cloud, since the middle cloud strongly shades the solar radiation from reaching the low cloud. Also it is shown in Fig. 6 and Tables 3 and 4 that *δ*-4DA can dramatically reduce the error of *δ*-2DA.

The heating rates and fluxes in Figs. 5–7 and Tables 3–4 are the integrated results from all solar bands. Note that for each individual band, the relative errors in heating rate and flux may be larger than those of broad band result, since offsetting errors could cancel in the summation of bands. Generally, the maximum absolute error in heating rate occurs in band 1 or band 2, owing to large incoming solar energy and strong scattering. It is found the maximum relative error of *δ*-2DA is up to 10% in band 2 for the cloud case in Fig. 6 with *μ*_{0} = 1.0. However, the corresponding relative error also becomes less than 1% in *δ*-4DA.

The results of high cloud are not discussed. The optical depth of the high cloud is very thin and consequently the solar heating rate is very low, and the choice of radiative transfer method does not appear to make an obvious difference (Lu et al. 2009; Shen and Zhang 2009), as it is well known that the high cloud only plays an important role in the infrared.

For applications in three-dimensional climate models, rapid computation of the radiative flux and heating rate is required. Thus, it is important to examine not only the accuracy, but also the efficiency of radiative transfer parameterizations. To compare the efficiency, the standard discrete ordinate radiative transfer code (DISORT) in *δ*-four-stream model (*δ*-4DIS) (Stamnes et al. 1988) was also incorporated in BCC_RAD with the same gaseous transmissions scheme. Table 5 lists the comparison results for computing time in broadband heating rate and flux calculated by using the three radiative transfer approximations of *δ*-2DA, *δ*-4DA, and *δ*-4DIS. The computing time is normalized to that of *δ*-2DA. The computational time of *δ*-4DA is about 2 times more than that of *δ*-2DA. In view of the overall high accuracy and computational efficiency, it appears that *δ*-4DA is well suited for radiation calculation in weather and climate models.

Timing of heating rate and flux calculations using various radiative transfer approximations (normalized to the computing time of *δ*-2DA).

## 5. Summary and conclusions

The accuracy of the four-stream approximation compared to that of the two-stream approximation has been recognized for a long time. Several solutions of the four-stream approximation have already been found. However, no attention has been paid to derive an analytical 4DA, which enables us to calculate the radiative transfer through multilayers in a vertically inhomogeneous atmosphere. Because of the lack of 4DA, 2DA was used to handle the multilayer connection in radiative transfer with a four-stream approximation.

In this work, we have proposed an analytical method of 4DA corresponding to the solution of four-stream DOM. The derivation of 4DA follows the invariance principle by Chandrasekhar (1950). In addition, *δ*-function adjustment is used to enhance the accuracy of 4DA. Consequently the 4DA becomes *δ*-4DA.

The accuracy of *δ*-2DA and *δ*-4DA is systematically compared in a double-layer case with a wide range of solar zenith angles, optical depths, and single-scattering albedos. *δ*-4DA provides much more accurate results for a double-layer case compared to those of *δ*-2DA. When applying *δ*-4DA to an idealized medium with specified optical properties, the reflection, transmission, and absorption are the same if the medium is treated as either a single layer or dividing it into multiple layers. This indicates that *δ*-4DA is able to solve the multilayer connection properly in a radiative transfer process. It is clearly shown that the results of radiative transfer are identical between a single layer and a multilayer as long as their optical properties are the same.

The *δ*-4DA has been applied to a realistic atmospheric profile with gaseous transmission considered. It is found that the accuracy of *δ*-4DA is superior to that of *δ*-2DA in most of cases, especially under the cloudy-sky condition. The relative error of *δ*-2DA in heating rate at cloud-top layer is as high as 6%, and the error in flux is up to 12 W m^{−2}. The relative error of *δ*-4DA in heating rate at cloud-top layer could be less than 1%, and the maximum error in flux is about 2 W m^{−2}. Therefore, the errors are substantially reduced by using *δ*-4DA. The computational time of *δ*-4DA is about 2 times more than that of *δ*-2DA. The modern computer easily affords the demand on computational resources for *δ*-4DA in GCMs.

In this paper, the plane-parallel atmosphere is assumed. For a large solar zenith angle, Earth’s spherical curvature should be considered in a radiative transfer process. However, the effect of spherical curvature can be simply included by using an effective solar zenith angle (Li and Shibata 2006).

*δ*-4DA can be easily applied to a radiation model, where a fast but accurate result is required. We therefore expect *δ*-4DA will be implemented in weather (e.g., sophisticated cloud systems associated with tropical cyclone deep convection) and climate models in the near future. *δ*-4DA could be extended to include the thermal radiation. We will publish the thermal part later.

The authors thank the anonymous reviewers for their constructive comments. The work is supported by the State Key Development Program for Basic Research of China (Grants 2009CB421506 and 2009CB421500), CAS Pilot Special Project (Grant XDA05090204), Special Scientific Research Fund of Meteorological Public Welfare Profession of China (Grants GYHY201006007, GYHY201006008, GYHY201006016, GYHY201206006, and GYHY200906002), and the National Natural Science Foundation of China (Grant 40921160381 and 41075056).

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