## 1. Introduction

The radiative transfer equation is an integro-differential equation. The exact solution of the radiative transfer equation in a scattering and absorbing media generally does not exist; thus, the approximate methods are necessary. In the last several decades considerable attention has been paid to find the simple and effective methods for solving 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; Lu et al. 2009; Zhang et al. 2013a,b; and others). Among these methods, the *δ*-two-stream approximations provide a simple answer to radiative transfer, especially the *δ*-Eddington approximation and *δ*-two-stream discrete ordinates method (DOM), are widely used in climate models.

The applications in climate modeling and remote sensing require accurate results in reflection, transmission, and absorption for a wide range of solar zenith angles and optical thicknesses. However, as shown in King and Harshvardhan (1986), even for the case of single layer, no *δ*-two-stream method satisfies this criterion. Recently, the accuracy of the *δ*-two-stream approximations has been compared under the real atmospheric profile condition with inclusion of gaseous transmission and cloud (Kay et al. 2001; Lu et al. 2009; Zhang et al. 2013b). It is found that both the *δ*-two-stream DOM and *δ*-Eddington approximation were generally accurate under the clear-sky condition, with a relative error of less than 1.5% in heating rate and flux. However, under the cloudy-sky condition, the cloud heating might have been underestimated by as much as 10%, which indicates that a four-stream or higher-order approximation is needed in order to obtain an accurate cloud solar absorption in climate models.

The single-layer analytical solutions of *δ*-four-stream approximations have been found (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 *δ*-four-stream DOM. Li and Ramaswamy (1996) obtained the solution of the *δ*-four-stream spherical harmonic expansion method (SHM), which is the higher-order extension of *δ*-Eddington approximation. However, all these solutions are only applicable to homogeneous single layer. In current climate models, the layer connection in two-stream radiative transfer is solved generally by the doubling–adding method, which is based on the invariance principle found by Chandrasekhar (1950). The doubling–adding method allows accurate solution of the radiative transfer equation for anisotropic scattering and complicated boundaries, and works naturally with layered media and yields reflection and transmission readily. In addition, the partial cloud can be handled by doubling–adding method 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 *δ*-Eddington approximation (referred as *δ*-2SDA) and *δ*-two-stream DOM (referred as *δ*-2DDA). Also, 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), but the doubling–adding method used in *δ*-four-stream approximation is that developed for *δ*-Eddington approximation (Coakley et al. 1983). We refer this scheme as *δ*-2/4SDA. In *δ*-2/4SDA, the isotropic condition is imposed on the diffuse radiation; thus, the angular characteristics of radiance field are not represented properly. It is found that the accuracy of *δ*-2/4SDA can be even worse than *δ*-2SDA, especially for flux.

Based on the invariance principle, recently Zhang et al. (2013b) have derived a four-stream doubling–adding method for *δ*-four-stream DOM (referred as *δ*-4DDA). It is clearly shown that *δ*-4DDA can dramatically reduce the errors in flux and heating rate compared to *δ*-2DDA, especially in cloudy-sky cases. Since the *δ*-four-stream DOM and *δ*-four-stream SHM are two popular methods with similar accuracy (Li and Ramaswamy 1996), a further effort is needed to generate an analytical doubling–adding algorithm for *δ*-four-stream SHM (referred as *δ*-4SDA), which enables us to calculate the four-stream radiative transfer through a vertically inhomogeneous atmosphere with multilayers. In addition, it is found that *δ*-4SDA has a higher computational efficiency compared to *δ*-4DDA. Therefore, *δ*-4SDA is easier to implement in climate models. In the following section 2, the reflectance and transmittance matrix of the direct/diffuse radiation are presented for a single-layer solution of four-stream SHM. In section 3, the invariance principle is applied to SHM and four-stream spherical harmonic expansion doubling–adding method (4SDA) is proposed. In section 4, the accuracy of *δ*-4SDA is systematically investigated. Finally, a summary and conclusion is given in section 5.

## 2. Single-layer solution of four-stream spherical harmonic expansion approximation

### a. Direct incident radiation

*P*(

*μ*,

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

*μ*′, which is scattered in the direction

*μ*;

*F*

_{0}the incoming solar flux; and

*μ*

_{0}is the cosine of the solar zenith angle. The quantity

*P*(

*μ*,

*μ*′) can be written aswhere

*P*is the Legendre function. The

_{l}*ω*can be determined from the orthogonal property of Legendre polynomials

_{l}*ω*

_{0}= 1 and

*ω*

_{1}= 3

*g*, with

*g*being the asymmetry factor.

*μ*≤ 1, we obtain the general relationWe consider a solution with a truncation at

*N*= 3, which corresponds to a four steam case. Equation (4) yieldswhere

*a*= [(2

_{l}*l*+ 1) −

*ωω*] and

_{l}*l*= 0, 1, 2, 3). Written in matrix form, (5) becomesThe 4 × 4 matrix accounts for the scattering–absorption in radiative transfer.

*f*

_{0}= 1/

*μ*

_{0},

*C*

_{1}

*D*

_{1}

*C*

_{2}

*D*

_{2}]

^{T}with the superscript T indicating a matrix transpose, and the constants in matrix

*j*= 1, 2, 3, and 4 are defined as follows:The coefficients

*Q*

_{1,2},

*R*

_{1,2},

*k*

_{1,2}, and

*η*

_{0,1,2,3}are defined in the appendix.

*P*

_{2m+1}(

*μ*) for

*m*= 0, 1, 2 … are orthogonal in both hemispheres with −1 ≤

*μ*≤ 0 and 0 ≤

*μ*≤ 1, a physical quantity that is limited within a hemispheric range can be decomposed by

*P*

_{1}(

*μ*) and

*P*

_{3}(

*μ*) in a four-stream case. We therefore definewhere

*F*

^{+}and

*F*

^{−}are the upward and downward fluxes, respectively.

*τ*= 0), there is no downward diffuse intensity, and thus

*F*

^{−}(

*τ*= 0) =

^{−}(

*τ*= 0) = 0; at the bottom of the layer, there is no upward diffuse intensity, and thus

*F*

^{+}(

*τ*=

*τ*

_{0}) =

^{+}(

*τ*=

*τ*

_{0}) = 0. Hence,where

*i*= 1, 2). Finally, the upward fluxes at the top of the layer and downward fluxes at its bottom are given bywhere

*I*(

*τ*,

*μ*), the corresponding reflection and transmission areThe reflection and transmission can be decomposed based on

*P*

_{1}(

*μ*) and

*P*

_{3}(

*μ*):We category the result into a 2 × 1 matrix based on the modes of

*P*

_{1}(

*μ*) and

*P*

_{3}(

*μ*) as

In four-stream SHM, the parameters of *i* = 1, 2) are dependent on eight basic simple parameters of *a _{i}* and

*b*(

_{i}*i*= 0, 1, 2, 3). However, the same parameters

*c*

_{i}_{,j}and

*b*(

_{i}*i*,

*j*= −2, −1, 1, 2) (Liou et al. 1988; Liou 2002; Zhang et al. 2013b), each of which contains a Gaussian quadrature summation with four terms. Therefore, it is expected that four-stream SHM is superior to four-stream DOM in computational efficiency.

### b. Diffuse incident radiation

*A*(

*μ*)/

*π*is the incoming diffused intensity. Based on the orthogonal basis of

*P*

_{2m+1}(

*μ*) (

*m*= 0, 1, 2, …) for −1 ≤

*μ*≤ 0, there can be an orthogonal expansion as

*A*(

*μ*)/

*π*=

*α*

_{1}

*P*

_{1}(

*μ*)/

*π*+

*α*

_{3}

*P*

_{3}(

*μ*)/

*π*+ … , where

*α*

_{1},

*α*

_{3}, … are constant. At the truncation of

*N*= 3, (16a) can be written as four equations, which are the same as (5) but with

*b*= 0 (

_{i}*i*= 0, 1, 2, 3). Also similar to (9), there are two types of boundary conditions corresponding to the first two terms in the orthogonal expansion of

*A*(

*μ*):

*a*:

*F*

^{−}(

*τ*= 0) = 1 and

^{−}(

*τ*= 0) = 0] and

*b*:

*F*

^{−}(

*τ*= 0) = 0 and

^{−}(

*τ*= 0) = 1]. Without loss of generality, the boundary conditions are normalized. For the both types of

*a*and

*b*, the lower boundary conditions are

*F*

^{+}(

*τ*=

*τ*

_{0}) = 0 and

^{+}(

*τ*=

*τ*

_{0}) = 0.

*a*boundary condition,where

_{3}= [1 0 0 0]

^{T}. Therefore, the upward fluxes at the top and downward fluxes at the bottom of the layer are given bywhere the subscript

*a*denotes the diffuse radiation under the type

*a*boundary condition in order to distinguish it from that in (12). Similarly, under the type

*b*boundary condition,where

_{4}= [0 1 0 0]

^{T}.

*I*

_{a}_{,b}is the intensity corresponding to the

*a*and

*b*types of boundary condition, and the direct beam transmission of

*P*

_{1}(

*μ*) and

*P*

_{3}(

*μ*),Similar to (15), the 2 × 2 matrices for diffuse reflection and transmission are created for using in the next sectionMatrices in (15) are called “direct” because they are related to the direct incoming solar zenith angle

*μ*

_{0}. Matrices in (23) are called “diffuse” because they are related to diffuse incident radiation. Actually, both the direct/diffuse reflection and transmission are used to deal with the diffuse radiance field. The result of direct solar beam is represented by

## 3. Doubling–adding method for four-stream spherical harmonic expansion method

### a. Two layers

#### 1) Direct incident radiation

*τ*

_{1}and

*τ*

_{2}are the optical depths of the first and second layers. At the interface between the two layers, the dimensionless up and down intensities areWe can decompose the dimensionless intensities based on the orthogonal base of

*P*

_{1}(

*μ*) and

*P*

_{3}(

*μ*):and categories the result into 2 × 1 matrices,

*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 the radiation comes from below. For a homogeneous atmosphere,

*R*(

*μ*,

*μ*

_{0}) =

*R**(

*μ*,

*μ*

_{0}) and

*T*(

*μ*,

*μ*

_{0}) =

*T**(

*μ*,

*μ*

_{0}). The above equation is slightly different from Chandrasekhar (1950) and Liou (1992), since the direct beam transmissions of

The physical meaning of the above four principles of invariance has been stated clearly in Liou (1992) and Zhang et al. (2013b).

*P*

_{1}(

*μ*) and

*P*

_{3}(

*μ*) in the range of 0 ≤

*μ*≤ 1 can help us to achieve this. We multiple 2

*P*

_{1}(

*μ*)

*d*μ in both sides of (27) and do integration from 0 to 1. Also we multiple 2

*P*

_{3}(

*μ*)

*d*μ in both sides of (27) and do integration from 0 to 1. Then by using (15), (23), and (26), we obtainThe detailed proof of (28) is shown in the appendix. From (28a) and (28b), we haveSubstituting (29) into (28c) and (28d), we obtain the direct reflection and transmissionwhere

#### 2) Diffuse incident radiation

*a*and

*b*types of boundary condition arewhere the direct beam transmission of

*P*

_{1}(

*μ*) and

*P*

_{3}(

*μ*), we obtainWe also category the result into matrices as

*P*

_{1}(

*μ*)

*d*μ in both sides of (35) and do integration from 0 to 1; also, we multiple 2

*P*

_{3}(

*μ*)

*d*μ both sides of (35) and do integration from 0 to 1. Then by using (23) and (34), we obtainFrom (36a) and (36b), we obtainSubstituting (37) into (36c) and (36d), we obtain the diffuse reflection and transmissionSimilarly for the diffuse beam incident from below, we obtainEquations (30) and (31) [(38) and (39)] constitute a closed set of iterative equations to compute the direct (diffuse) reflectance and transmittance for a combined layer.

### b. Multiple layers

*k*, which is obtained through a downward path calculation aswhere

*N*) to layer

*k*, which is obtained through a upward path calculation asIn (41),

*R*is the surface albedo. The dimensionless upward and downward internal intensities of

_{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 areand the upward and downward fluxes at the top of the atmosphere (TOA) arewhere

*u*

_{k}_{+1},

*d*

_{k}_{+1}, and

*r*

_{1,N}are the first element of matrices

_{k+1}(

*μ*

_{0}),

_{k+1}(

*μ*

_{0}), and

_{1,N}(

*μ*

_{0}), respectively.

## 4. Comparison results and discussion

In the following, the accuracy of 4SDA will be systematically investigated. The results will be compared to another doubling–adding scheme of 4DDA. In addition, the results of comparison to 2DDA and 2SDA are included as well. As in Zhang et al. (2013b), the δ-function adjustment of scattered energy residing in the forward peak is considered in the following calculations. Therefore, the four kind of doubling–adding schemes become *δ*-2DDA, *δ*-2SDA, *δ*-4DDA, and *δ*-4SDA.

### a. Double layer

First, the accuracy of *δ*-2DDA, *δ*-2SDA, *δ*-4DDA, and *δ*-4SDA is investigated in a double-layer case. The results of the double layer could be compared to those of the single layer, which has been well analyzed by Liou et al. (1988) and Li and Ramaswamy (1996). Moreover, the results from the double layer help us to understand the multilayer results shown in the next subsection. For a double layer (labeled 1 and 2 for the first and second layers), based on (43) and (44) we have reflection *a*(*τ*_{1,2}, *μ*_{0}) = 1 − *r*(*τ*_{1,2}, *μ*_{0}) − *t*(*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 angles and optical depths. The asymmetry factors are *g*_{1} = 0.837 and *g*_{2} = 0.861 for the first and second layers, respectively, following Zhang et al. (2013b). The values of single-scattering albedo are set the same (*ω*_{1} = *ω*_{2} = 0.9). The rigorous standard model used in the following is the discrete ordinates numerical model (Stamnes et al. 1988). The *δ*-128 stream scheme (*δ*-128S) are used in the discrete ordinates calculations with the Henyey–Greenstein phase function.

In Fig. 2, the relative errors in reflection and absorption are shown for the four methods. For reflection, *δ*-2SDA yields better results for a thick optical depth, but slightly poorer results for a thin optical depth, in comparison with *δ*-2DDA. The errors of both are up to 20% and higher in the region of thin optical depth and small *μ*_{0}. The errors are dramatically reduced in δ-4DDA and *δ*-4SDA. The relative errors of both are bounded by 5% for *τ*_{1,2} > 1. Although there are significant improvements, δ-4DDA and δ-4SDA cannot completely eliminate the region where the errors exceed 15%.

For absorption, the relative errors are generally less than 15% for both *δ*-2DDA and *δ*-2SDA, except in the region of a very large solar zenith angle. Generally *δ*-2SDA is slightly more accurate than *δ*-2DDA. Again, a dramatic improvement happens for both *δ*-4DDA and *δ*-4SDA, with errors less than 5% in most regions. Generally *δ*-4DDA is slightly more accurate than *δ*-4SDA.

To show that *δ*-4SDA 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 SHM to a media as a single layer, and also apply *δ*-4SDA to the same media, which is divided into 2, 5, and 10 layers. Two different optical depths (*τ*_{total} = 0.1 and *τ*_{total} = 2.0) are considered. In addition, the exact values are computed from δ-128S for the single layer. It is clearly shown in Table 1 that the results of 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*-4SDA in radiative transfer process.

Reflection and absorption (from *δ*-4SDA and *δ*-2/4SDA) vs number of sublayers, which are divided from the medium with *ω* = 0.9, *g* = 0.837, and *μ*_{0} = 0.5; two optical depths (*τ*_{total} = 0.1, 2.0) are considered. Exact values are obtained from *δ*-128S for the single layer.

In Li and Ramaswamy (1996), the two-stream doubling–adding method was used to connect the multilayer medium with δ-four-stream spherical harmonic expansion solution (as *δ*-2/4SDA). The corresponding results of *δ*-2/4SDA are also listed in Table 1. It is shown that the results of reflection, absorption, and transmission (not shown) vary with the media containing different layers. In general, the greater the number of sublayers, the worse is the result. Therefore, δ-2/4SDA is inappropriate to calculate the radiative transfer through a multilayer atmosphere.

### b. Multilayer atmosphere

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 radiation under a variety of atmospheric conditions by using a radiation algorithm. The Fu–Liou radiation model (Fu and Liou 1992) is used in this study. This model adopts the correlated-*k* distribution method for gaseous transmission, including five major greenhouse gases H_{2}O, CO_{2}, O_{3}, N_{2}O, and CH_{4}.

In benchmark calculations, the discrete ordinates model (Stamnes et al. 1988) with a *δ*-128-stream scheme is incorporated with the gaseous transmission scheme of Fu–Liou radiation model by replacing the existing radiative transfer algorithm in the model. Also the *δ*-2DDA, *δ*-2SDA, *δ*-4DDA, and *δ*-4SDA schemes are incorporated with the same gaseous transmission scheme. The atmosphere was vertically divided into 280 layers, each of thickness 0.25 km. The midlatitude winter of atmospheric profile (McClatchey et al. 1972) is used with surface albedo of 0.2 for each band. The mixing ratios of CO_{2}, N_{2}O, and CH_{4} are assumed to be uniform throughout the atmosphere with concentrations of 330, 0.28, and 1.6 ppmv, respectively. For ice clouds, the optical properties are parameterized in terms of ice water content (IWC) and mean effective size (*D _{e}*) (Fu and Liou 1993). For water cloud, the optical properties are parameterized in terms of liquid water content (LWC) and effective radius (

*r*) (Fu 1991). Five calculations are performed: 1) clear sky; 2) a low cloud (LWC = 0.22 g m

_{e}^{−3},

*r*= 5.89

_{e}*μ*m) positioned from 1.0 to 2.0 km; 3) a middle cloud (LWC = 0.28 g m

^{−3},

*r*= 6.2

_{e}*μ*m) positioned from 4.0 to 5.0 km; 4) a high cloud (IWC = 0.0048 g m

^{−3},

*D*= 41.1

_{e}*μ*m) positioned from 10.0 to 12.0 km; and 5) low, middle, and high clouds together. The visible optical depths for low, middle, and high clouds are ~60, ~72, and ~0.8, respectively. Three solar zenith angles of

*μ*

_{0}= 1, 0.5, and 0.25 are generally considered.

In the first row of Fig. 3, the benchmark results of clear-sky heating rate are shown for three solar zenith angles. The absolute errors of *δ*-2DDA, *δ*-2SDA, *δ*-4DDA, and *δ*-4SDA against the benchmark results are shown in the second row. Generally, both *δ*-2DDA and *δ*-2SDA are relatively accurate under the clear-sky condition. However, *δ*-2DDA and *δ*-2SDA can produce relatively large error in heating rate in the region near the surface, especially for *δ*-2DDA. The relative error of *δ*-2DDA is about 2.0% near the surface. The *δ*-2SDA is generally better than *δ*-2DDA in clear-sky heating rate. It is shown in Fig. 3 that both *δ*-4DDA and *δ*-4SDA yield substantially more accurate results in heating rate.

The corresponding errors in upward flux at TOA and the downward flux at the surface are shown in Tables 2–4 for *μ*_{0} = 1, 0.5, and 0.25, respectively. For a small solar zenith angle (e.g., *μ*_{0} = 1.0), *δ*-2SDA generally produces larger relative errors compared to *δ*-2DDA. The results become opposite for a large solar zenith angle (e.g., *μ*_{0} = 0.25). Similar to the result of heating rate, the accuracy in flux is considerably improved by *δ*-4DDA and *δ*-4SDA.

Comparison of upward flux at TOA and downward flux at the surface (W m^{−2}) for the four doubling–adding schemes. 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 2, but for the solar zenith *μ*_{0} = 0.5.

As in Table 2, but for the solar zenith *μ*_{0} = 0.25.

For the low and middle clouds, Fig. 4 shows that the absolute errors in heating rate for *δ*-2DDA and __δ__-2SDA become much larger, compared to the result of clear sky. When *μ*_{0} = 1, the absolute errors of δ-2DDA are up to about 1.5 and 2.3 K day^{−1} (relative errors about 6%) for the low and middle clouds. The error of *δ*-2SDA is smaller than *δ*-2DDA. When *μ*_{0} = 0.25, the result becomes opposite as the error of *δ*-2SDA becomes larger than *δ*-2DDA. By using *δ*-4DDA and δ-4SDA, the relative errors are much suppressed. In general the relative error becomes less than 1% for the three solar zenith angles. This indicates both *δ*-4DDA and *δ*-4SDA are accurate enough to obtain the cloud-top solar heating.

The corresponding errors in flux are also shown in Tables 2–4. Here *δ*-2DDA overestimates the upward flux at TOA by ~11, ~6, and ~3 W m^{−2} for *μ*_{0} = 1, 0.5, and 0.25, respectively. Generally the error in flux for *δ*-2SDA is smaller than that of *δ*-2DDA. The only exception is in the case of *μ*_{0} = 0.25, where the errors of downward flux at the surface by *δ*-2SDA is larger than *δ*-2DDA. When *δ*-4DDA and *δ*-4SDA are implemented, the errors are substantially reduced. Generally the relative errors are less than 0.8%.

The results of high cloud are shown in Fig. 5. The *δ*-4DDA is better than *δ*-4SDA in heating rate, especially for the large solar zenith angle of *μ*_{0} = 0.25. The optical depth of the high cloud is very thin and, consequently, the solar heating rate is very low. Therefore, the choice of radiative transfer method does not appear to make a big difference. The corresponding results in flux are also shown in Tables 2–4; both *δ*-4DDA and *δ*-4SDA are very accurate with relative error less than 0.8%. Different from the result of heating rate, *δ*-4SDA is not inferior to *δ*-4DDA in flux.

In Fig. 5, the results of heating rate for all the low, middle, and high clouds are presented as well. The corresponding results in flux are shown in Table 2–4. Again, both *δ*-4SDA and *δ*-4DDA are superior to *δ*-2DDA and *δ*-2SDA in accuracy.

In Fig. 2, it is shown that the relative errors of *δ*-2DDA and *δ*-2SDA are considerably large in the region of small optical depth. Generally the aerosol optical depth in the atmosphere is very small. The global mean dust optical depth is less than 0.1 (Peng et al. 2012). To investigate the error in aerosol radiative forcing, we consider a dust aerosol profile with a column optical thickness of *τ _{a}* = 0.1, which is uniformly distributed from the surface to 2 km. The asymmetry factors and single-scattering albedo are set as

*g*= 0.679 and

_{a}*ω*= 0.988, which corresponds to optical property of dust aerosol with effective radii of 0.5

_{a}*μ*m at a wavelength of 0.55

*μ*m (Fu et al. 2009).

The results in flux are listed in Table 5. As expected, both *δ*-2DDA and *δ*-2SDA produce relatively large errors in the upward flux at TOA and downward flux at the surface. For example, the error of upward flux at TOA by *δ*-2SDA is up to 5.4 W m^{−2} at *μ*_{0} = 1. The corresponding error in the clear sky without aerosol is 3.9 W m^{−2} (see Table 2). Thus, the aerosol radiative forcing is overestimated by 1.5 W m^{−2}. When *μ*_{0} = 0.25, the errors of upward flux at TOA are −0.59 W m^{−2} and −3.58 W m^{−2}, respectively, for the clear sky without and with aerosol (Table 4). Thus the aerosol radiative forcing is underestimated by 2.99 W m^{−2}. Since *δ*-2SDA is widely used in current climate models, the model estimated aerosol forcing is not reliable. The results of *δ*-2DDA are slightly better than *δ*-2SDA for a small solar zenith angle but become worse for a large solar zenith angle. It is shown that the errors in flux are dramatically reduced by applying *δ*-4DDA or *δ*-4SDA.

Comparison of upward flux at TOA and downward flux at the surface (W m^{−2}) for the four doubling–adding schemes. The aerosol optical depth *τ _{a}* = 0.1. The value in the

*μ*

_{0}column is the cosine of the solar zenith angle. The values in parentheses are the differences between the four schemes and

*δ*-128S.

The computational efficiency is a key issue for climate modeling. Table 6 lists the comparison results of computing time for *δ*-2SDA and *δ*-4SDA. The computational efficiency of *δ*-2SDA is not shown since it is close to that of *δ*-2SDA. In Fu–Liou model, an inverse matrix formulation is used to solve the connection of all the layers in *δ*-four-stream DOM radiative transfer (denoted as *δ*-4SMODEL). As proposed by Zhang et al. (2013b), *δ*-4DDA has the very similar efficiency as *δ*-4SMODEL. For the pure radiative transfer calculations without considering the gaseous transmission, it is found that the computational time of *δ*-4SDA is about 70% of *δ*-4SMODEL. As discussed above, the single-layer solution of *δ*-four-stream SHM is much simpler than that of *δ*-four-stream DOM. For the whole model process including the calculations of gaseous absorption and cloud optical properties, the computational time of *δ*-4SDA is about 80% of δ-4SMODEL. In view of the overall high accuracy and computational efficiency, it appears that *δ*-4SDA is well suited for radiation calculation in weather and climate models.

The timing for the radiative transfer calculation (normalized to the timing ratio of Fu–Liou model) by the algorithm only and by the radiation model.

## 5. Summary and conclusions

Even though several single-layer solutions of δ-four-stream approximation have been found in the last several decades, less attention has been paid to the derivation of an analytical result for a corresponding doubling–adding calculation, which would enable us to calculate the radiative transfer through a vertically inhomogeneous atmosphere with multilayers. The recent work of *δ*-4DDA (Zhang et al. 2013b) was the first effort in this aspect. Among the various δ-two-stream approximations, the *δ*-Eddington approximation and δ-two-stream DOM are widely used in radiation algorithms. The higher-order extensions of the two methods are the *δ*-four-stream SHM and *δ*-four-stream DOM. Therefore, it is natural to seek a doubling–adding scheme for δ-four-stream SHM. In this work, first the simplified single-layer solution of four-stream SHM is shown; second, the invariance principle is applied to four-stream SHM and an analytical method of *δ*-4SDA is obtained.

In a double-layer case, the accuracy of *δ*-4SDA is systematically compared with a wide range of solar zenith angles and optical depths. Similar to the comparison for a single layer, *δ*-4SDA provides much more accurate result compared to *δ*-2DDA and *δ*-2SDA. The idealized medium test shows that the *δ*-4SDA scheme can exactly solve the multilayer connection in a radiative transfer process.

The *δ*-4SDA, along with *δ*-2DDA, *δ*-2SDA, and *δ*-4DDA, has been applied to a realistic atmospheric profile with inclusion of gaseous transmission. It is found the accuracy of *δ*-4SDA is comparable to *δ*-4DDA; both of them are superior to *δ*-2DDA and *δ*-2SDA in most cases, especially under the cloudy-sky condition. For *δ*-2DDA and *δ*-2SDA, the relative error in heating rate at cloud-top layer can be as large as 6%, and the errors in flux are up to 12 W m^{−2} and 9 W m^{−2}, respectively. These errors are substantially reduced by using *δ*-4SDA or *δ*-4DDA. The errors of both are less than 1.5 W m^{−2}.

It is found that the aerosol radiative forcing can be significantly misestimated, owing to the large errors from *δ*-2DDA and *δ*-2SDA under the thin optical depth condition. For a typical case of dust, the error of aerosol radiative forcing at TOA can be over 3 W m^{−2}. The only way to eliminate the large error in aerosol radiative forcing is to use *δ*-4DDA or *δ*-4SDA. The errors of both are less than 0.72 W m^{−2}.

Both *δ*-4DDA and *δ*-4SDA are analytical method so that the computing time involved is minimal. It is shown that *δ*-4SDA is more efficient than *δ*-4DDA. Therefore, *δ*-4SDA could be more easily implemented in climate models, where both the accuracy and efficiency are required.

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 2011CB403405), the key program of CAMS (Grant 2012Z001), the National Natural Science Foundation of China (Grants 41221064, 41075056, 41205116, and 41075015), and the Program of China Meteorological Administration (Grant GYHY200906002).

# APPENDIX

## The Coefficients of (2) and the Proof of (28)

This appendix contains two parts: 1) the coefficients of (7) and 2) the proof of (28).

*R*(

*μ*,

*μ*′) and

*P*

_{1}(

*μ*) and

*P*

_{3}(

*μ*) in the hemisphere, The following subparts of (25)indicate a hemispherical harmonic expansion of

*D*(

*μ*′,

*μ*

_{0})

*μ*′ in the range of 0 ≤

*μ*≤ 1,where the terms beyond truncation are neglected.

*R*(

*μ*,

*μ*

_{0}) ~

*I*(

*μ*)/

*μ*

_{0}, where

*I*(

*μ*) is the reflected direct intensity. In (A1),

*μ*

_{0}is replaced by the local

*μ*′ for all direction of 0 ≤

*μ*′ ≤ 1. From (21),

*R*

^{a}^{,b}~

*I*

_{a}_{,b}, where

*I*

_{a}_{,b}is the reflected diffused intensity. It is shown in (26a) of Zhang et al. (2013a) that the angular averaged

*R*(

*μ*,

*μ′*)

*μ*′ (as integral to

*μ*′) corresponds to the diffused reflection. In four-stream SHM,

*R*(

*μ*,

*μ′*)

*μ*′ should be further decomposed to

*P*

_{1}(

*μ*′) and

*P*

_{3}(

*μ*′) according to the

*a*and

*b*types of boundary condition as shown in (19)–(21). Therefore, following (22),

*P*

_{1}(

*μ*) and

*P*

_{3}(

*μ*), by (A4a) and (A4c) we haveand by (A4b) and (A4d) we haveFrom (A5) and (A6), based on the orthogonal relation of

*P*

_{1}(

*μ*) and

*P*

_{3}(

*μ*), we obtainIn (A5)–(A7), the terms beyond truncation are neglected. Finally, it is proven thatEquations (28b) and (28d) can also be proven in a similar way as was (28a).

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