## 1. Introduction

For climate models and remote sensing applications, approximate methods for solving the radiative transfer equation are required in order to obtain accurate results for reflection, transmission, and absorption for all solar zenith angles and a wide variety of optical thickness values. However, as pointed out by King and Harshvardhan (1986), to date no two-stream approximation method satisfies this criterion. Specific regions can be identified where one approximation is more accurate than another. A fundamental improvement in accuracy is available through higher-order expansions of the radiance (higher stream). In the works of Liou et al. (1988) and Li and Ramaswamy (1996) the four-stream discrete ordinate and the four-stream spherical harmonic expansion approximation have been considered.

Although improvements in accuracy are dramatic in the four-stream harmonic expansion model (Li and Ramaswamy 1996) as compared to the *δ*-Eddington approximation, the four-stream scheme can still have errors up to 10% for various optical thicknesses less than one and for many solar zenith angles. This drawback has its roots in the treatment of the phase function, since only the first few moments of phase function are preserved in the lower-order stream approximations. When considering sufficiently large optical thicknesses, photons undergo many scattering events before leaving the medium. Because of this, a highly diffuse scattering layer is established, and the single scattering directions are smoothed and appear to be more random. Therefore, the results are not very sensitive to the structure of the phase function. However, this is not the case for optically thin layers: the outgoing photons undergo only a few scattering events and the results are strongly dependent on the single scattering directions. In these circumstances, a correct description of the phase function plays an important role in the radiative transfer process.

Among all of the lower-order stream schemes (Shettle and Weinman 1970; Liou 1973; Coakley and Chýlek 1975; Meador and Weaver 1980; King and Harshvardhan 1986; Liou et al. 1988; Li and Ramaswamy 1996), only the Coakley–Chýlek (hereafter called C–C) model can keep the integrity of the phase function. Therefore, for very thin optical layers, the C–C model has more stability in the accuracy of its results in comparison with other two-stream approximation methods (King and Harshvardhan 1986). Also, only the C–C model can correctly reduce to the thin atmosphere approximation (single scattering approximation) as the optical thickness approaches zero. For this reason, the C–C model is widely used for radiative transfer involving optically thin layers and, especially, for aerosol radiative forcing studies (Charlson et al. 1992) and clear-sky radiation calculations for general circulation models (Barker and Li 1995).

The C–C scheme is not very successful for describing the radiative properties of layers with larger optical thicknesses. There can considerable disagreement between the C–C model and a precise model with increasing optical depths. Errors up to 20% or higher can occur even for optical thicknesses less than 0.5, which limits the usefulness of the C–C model. An improvement is necessary if quick, accurate results are required. The four-stream model proposed here also has the correct thin atmosphere limit and is able to substantially improve the accuracy of calculations compared to the C–C model in the optically thin domain. This model addresses radiative transfer for optical thicknesses less than unity, whereas beyond this range the problem is suitably solved by using the four-stream spherical harmonic expansion method of Li and Ramaswamy (1996).

The most important assumption in the C–C scheme is that the diffuse intensity is isotropic (independent of local zenith angle). Hence, the intensity can be taken out of the integral of the multiple scattering source term and the inherent difficulty of the radiative transfer equation is avoided; an unchanged backscattering fraction is therefore all that is retained of the phase function. In the two-stream C–C model, the radiative intensities are taken to be isotropic in the upward and downward hemispheres. These isotropic regions are further divided for a multiple stream that the diffuse intensity is isotropic within each sector. This is the basic principle for us to derive the new four-stream approximation method.

## 2. Four-stream isosector approximation scheme and solutions

*I*(

*τ, μ*) is the diffuse intensity,

*μ*= cos

*θ, θ*is the local zenith angle,

*μ*

_{0}= cos

*θ*

_{0},

*θ*

_{0}is the solar zenith angle,

*ω*is the single scattering albedo, and

*πF*

_{0}is the solar flux. Here,

*P*(

*μ, μ*′) is the azimuthally independent phase function, which can be expanded in the Legendre functionswhere

*P*

_{l}(

*μ*) is the Legendre function. The moment

*ω̃*

_{0}

*ω̃*

_{1}/3

*g*is the asymmetry factor.

In the four-stream approximation, the space is separated into four regions of *μ*_{s} < *μ* < 1, 0 < *μ* < *μ*_{s}, −*μ*_{s} < *μ* < 0, and −1 < *μ* < −*μ*_{s}, where *μ*_{s} = cos*θ*_{s} and *θ*_{s} is the angle separating a hemisphere into two portions or sectors. For convenience in derivation, set *μ*_{s} = 1/*s**s* is a real number greater than 1. The corresponding intensities in the four regions are *I*^{±1}(*τ,* ±*μ*) (0 < *μ* < *μ*_{s}) and *I*^{±2}(*τ,* ±*μ*) (*μ*_{s} < *μ* < 1). Because of the four geometrically divided regions of diffuse intensity, we call the method the four-stream isosector approximation (four-stream IA). Under this assumption, the radiative transfer equation of Eq. (1) becomes a group of equations with the different sector intensities distinguished.

*μ*<

*μ*

_{s},and for

*μ*

_{s}<

*μ*< 1,

*I*

^{±1}(

*τ,*±

*μ*) by

*I*

^{±1}(

*τ*) and

*I*

^{±2}(

*τ,*±

*μ*) by

*I*

^{±2}(

*τ*), which assumes that the intensity is independent of

*μ*in each of the regions. Integrating over

*μ*for the interval of each equation in Eq. (3), we havewhere

*s*′ =

*s*− 1,andwhere

*P*(−

*μ, μ*′) =

*P*(

*μ,*−

*μ*′) is used in the above derivation. The four-stream radiative transfer equation of Eq. (4) reduces to the two-stream C–C model II, when

*I*

^{1}=

*I*

^{2}and

*I*

^{−1}=

*I*

^{−2}are assigned. Hereafter “C–C model” refers to C–C model II.

*M*

^{±}

_{1,2}

*G*

^{±}

_{1,2}

*e*

^{−λτ}

*λ*are obtained by solvingwhere

Eigenvalues are *λ*_{1} = −*λ*_{3} = (*ζ* + (*ζ*^{2} − *ξ*)^{1/2})^{1/2}/(*s*′) and *λ*_{2} = −*λ*_{4} = (*ζ* − (*ζ*^{2} − *ξ*)^{1/2})^{1/2}/(*s*′), where *ζ* = Φ + *s*′Ξ and *ξ* = 4*s*′^{2}Δ^{+}Δ^{−}; the coefficients *G*^{±}_{1,2}

*η*

^{±}

_{1,2}

*e*

^{−τ/μ0}

*I*

^{±}

_{1,2}

*η*

^{±}

_{1,2}

*φ*

^{±}

_{1,2}

*φ,*where

*φ*=

*f*(1/

*μ*

_{0}) andThe expressions of

*φ*

^{−}

_{1,2}

*φ*

^{+}

_{1,2}

*P*

_{1,2}= (

*b*

^{+}Δ

^{−}+

*s*′

*b*

^{−}

*λ*

^{2}

_{1,2}

*W*(

*λ*

_{1,2}),

*Q*

_{1,2}=

*λ*

_{1,2}Γ

^{−}/

*W*(

*λ*

_{1,2}), and

*R*

_{1,2}=

*λ*

_{1,2}(

*s*′

*λ*

^{2}

_{1,2}

*W*(

*λ*

_{1,2}), with

*W*(

*λ*) =

*a*

^{−}

*λ*

^{2}−

*c*

^{+}Δ

^{−}and Γ

^{±}=

*a*

^{±}

*b*

^{∓}+

*s*′

*b*

^{±}

*c*

^{∓}. Here,

*P*

_{1,2},

*Q*

_{1,2}, and

*R*

_{1,2}are obtained by general method for eigenvectors (Ross 1974; Li and Ramaswamy 1996). Furthermore,

*C*

_{1},

*D*

_{1},

*C*

_{2}, and

*D*

_{2}are constants determined by the boundary conditions. For simplicity, consider boundary conditions with no downward diffuse radiance at the top of the layer and no upward diffuse radiance at the bottom of the considered layer (surface albedo is assumed to be zero). These conditions are expressed as follows; at the top

*I*

^{−}

_{1,2}

*M*

^{+}

_{1,2}

*M*

^{−}

_{1,2}

*I*

^{+}

_{1,2}

*τ*

_{0}

*M*

^{+}

_{1,2}

*τ*

_{0}

*M*

^{−}

_{1,2}

*τ*

_{0}

*τ*

_{0}is the vertical optical thickness of the considered layer. Here,

*C*

_{1},

*D*

_{1},

*C*

_{2}, and

*D*

_{2}can be obtained by solving the four linear algebraic equations of Eqs. (22) and (23). So far we have only considered the nonconservative case (

*ω*≠ 1). For the conservative case (

*ω*= 1), we can easily prove that

*a*

^{+}=

*c*

^{+}= −

*b*

^{+}by the fact of ½

^{1}

_{−1}

*P*(

*μ, μ*′)

*dμ*′ = 1 and

*P*(

*μ, μ*′) =

*P*(

*μ*′,

*μ*). Therefore, Δ

^{+}= 0 and

*λ*

_{2}= −

*λ*

_{4}= 0. For the double zero roots, the ansatz of

*M*

^{+}

_{2}

*C*

_{2}+

*D*

_{2}

*τ,*and similar ansatzes are made for the other components(Ross 1974). By substituting these ansatzesin the homogeneous part of Eq. (15) and solving the corresponding linear algebraic equation, we obtain the relationships between the coefficients in the ansatzes. This yields

The coefficients in Eq. (24) are for the solutions corresponding to the roots *λ*_{1} and *λ*_{3} (=−*λ*_{1}) and the particular solutions are the same as those for the nonconservative case with *a*^{+} = *c*^{+} = −*b*^{+}. The constants *C*_{1}, *D*_{1}, *C*_{2}, and *D*_{2} are again determined by the boundary conditions contained in Eqs. (22) and (23).

*F*

^{+}and downward

*F*

^{−}fluxes are obtained from

*F*

^{±}

*π*

*I*

^{±}

_{1}

*s*

*I*

^{±}

_{2}

*s.*

## 3. Calculation results and discussion

In order to test the accuracy of this model, a wide range of comparisons will be made for reflection, transmission, and absorption. Since both the two-stream C–C and four-stream IA models are expected to apply to the thin atmosphere, the comparisons are only taken for optical thicknesses less than two, and most of our interest will focus on the region of optical thickness less than one. The model used as our standard for comparison is the 48-stream discrete ordinate numerical model (Stamnes et al. 1988). The optical thickness is varied between 0.1 and 2 (step 0.02 in logarithmic scale), and solar zenith angle varies from 0 (0.02) to 1 (step 0.02).

The absolute error is defined as the value obtained from the approximation method minus the value obtained using the rigorous method; the relative error is the absolute error divided by the value of the rigorous method.

We choose the Henyey–Greenstein phase function, in which the moments of the phase function are simplyvarious powers of the asymmetry factor. In the following calculations, we will take a asymmetry factor of *g* = 0.75, the same as Liou et al. (1988). If the model is expected to be suitable for atmospheric aerosols and cirrus clouds alike, then the approximation must be accurate for these reduced values of the asymmetry factor. These lower values arise due to the smaller average size of aerosol particles and the nonspherical nature of ice crystals; both result in a weaker forward scattering peak. In the four-stream IA scheme, there is no so-called *δ*-*M* scaling used (Joseph et al. 1976; Wiscombe 1977), since the phase function is well represented in the solution process by applying more streams and utilizing higher-order moments of the phase function. We keep 32 moments in the following calculation. The adjustment factor *f* (fraction of forward peak) of the *δ*-*M* scaling method is proportional to *g*^{n}, where *g* is the asymmetry factor and *n* is the number of the moment kept for phase function. If *n* = 32, and *g* = 0.75, *f* is close to 10^{−4}.

The four-stream IA is based on separating each hemisphere into two portions. The cosine of the dividing angle is *μ*_{s} = 1/*s**s.* However, it is hard to find the physical principle to determine the sector range. In the following Figs. 1, 2, and 3, we will first consider the Gaussian angles of *μ*_{1} = 0.33998 (angle of about 70° and *s* = 8.6515) and *μ*_{2} = 0.86114 (angle of about 31° and *s* = 1.3615) for four-point Gaussian quadrature as the separating angle [the other two separating angles of −*μ*_{1} and −*μ*_{2} will be automatically considered; see Eq. (3)]. These two values represent the two extreme cases of large and small separating angle for a hemisphere. Later we will further test the results of the other separating angles and discuss the physics of choosing the separating angle.

First we consider the conservative case where *ω* = 1. All of the figures show comparisons evaluating both the accuracy of the four-stream IA model and the C–C model relative to rigorous calculations. The relative errors for the reflected and transmitted fluxes using the C–C model are shown in the top panels of Fig. 1, and the same radiative properties for the four-stream IA method are displayed in the middle and lower panels corresponding to the two different separating angles. Although the C–C model has been proved to reduce to the exact thin atmosphere result as the optical thickness approaches zero, it is found in Fig. 1 that even for optical thicknesses of 0.1 the error in reflection can be up to 10% for overhead solar angles using the C–C model. This is not an isolated region, either; the error gets worse as the optical depth is increased. The transmission appears more stable for the results presented in Fig. 1; however, for large solar angles the error can become quite large (15%).

The results for the four-stream IA model are shownin the middle and bottom panels of Fig. 1 for two different separating angles. For *μ*_{s} = *μ*_{1} (middle panels), the errors in reflected fluxes are substantially reduced as compared to the two-stream C–C model. Most of the regions have a less than 2% error, especially for large solar incident angles. For small solar incident angles of *μ*_{0} > 0.9, the four-stream IA can produce errors up to 15%. However, the domain of large error is much smaller and more constrained with solar angle in comparison with results from the two-stream C–C model. For *τ*_{0} < 1 the four-stream IA model also shows much more accuracy compared to the four-stream spherical harmonic expansion method of Li and Ramaswamy (1996). It is important to note that the accuracy of the four-stream IA increases with decreasing optical thickness, whereas the accuracy of the four-stream spherical harmonic expansion method decreases with decreasing optical thickness, for optical thicknesses less than one.

In the bottom panels, for *μ*_{s} = *μ*_{2}, it is found that the contour of the reflection is very similar to that of corresponding result of C–C, except with some improvement in the region of small solar zenith angle. This tells us that the small separating angle may not be a suitable choice for the four-stream IA. This will be confirmed by further comparisons.

For optically thin layers, the transmission is always more accurate than the reflection for all the two-stream approximations (King and Harshvardhan 1986). The transmission contains two parts: diffuse and direct terms [Eq. (27)]. For a thin optical depth, if the solar incident angle is not too large, the direct term dominates. This direct transmission term is not dependent on the solution of the multiple scattering processes; therefore it is very accurate in all lower-stream models. The dominance of the direct term leads to very accurate results for transmitted fluxes. If only the diffuse term is considered, then the error in transmission would be much higher in the thin optical thickness case, similar to what is observed for reflection. We find that, for large solar zenith angle and optical depths tending toward one, the error in transmission is substantial (15%) in the two-stream models (King and Harshvardhan 1986), which is due to a decreased contribution of the direct transmission term. In the optical depths region *τ*_{0} < 1, the four-stream IA method (middle panels with *μ*_{s} = *μ*_{1}) almost eliminates all errors (less than 2%) for all solar incident angles. These results imply that the diffuse radiation is more accurately characterized by the four-stream isosector technique for thin layers.

Also, similar to reflection, the results of four-stream IA with *μ*_{s} = *μ*_{2} are very similar to those of the corresponding results of C–C, only with some improvement in the small zenith angle region.

Now, in Fig. 2, we consider an absorbing medium characterized by a single scattering albedo, *ω* = 0.9. Since the top panels of C–C and the bottom panels of four-stream IA with *μ*_{s} = *μ*_{2} have very similar contours, we only address the intercomparison of the top panelsand the middle panels (four-stream IA with *μ*_{s} = *μ*_{1}). For reflection, there is a marked improvement over the two-stream C–C model. The two-stream C–C model has a thin strip of solar incident angles (around 0.5 for *μ*_{0}) for which the model is accurate. On either side of this, the error grows to values of 10%–15% for almost alloptical thicknesses shown. The four-stream IA model greatly improves upon the stability of the results. The error is reduced to 5% or lower for most optical thicknesses and solar incident angles. Although there is much improvement, the four-stream IA approximation cannot completely eliminate the region where the errors exceed15%; however, this domain is much smaller and more localized compared to the two-stream C–C model case.

For transmission, the four-stream IA has errors less than 2% in the region of *τ*_{0} < 1, which is much better than the corresponding results of the C–C model. The two-stream C–C shows errors up to 15% for large solar incident angles and moderately sized regions with errors between 5% and 10%. For *τ*_{0} between 1 and 2, the two-stream C–C model has large errors for many solar incident angles, whereas the four-stream IA method is mostly bounded by 5%.

For absorption, there is a strip of small error for incident solar angles between 0.7 and 0.9 for both models. As one looks to either side of this strip the errors typically grow. This low-error strip appears to be wider for the four-stream IA model compared to the two-stream C–C model. This property is really what characterizes these two panels. The errors up to 20% occur for largeregions of the two-stream C–C panel, whereas the four-stream IA model has errors as large as 5%.

Finally, we consider the strongly absorbing case in Fig. 3 with single scattering albedo *ω* = 0.3. In this case the absorption is dominant. At a first glance, the relative error for reflection appears unusually large. However, more importantly, the absolute errors are very small, since the reflectivity is extremely small in this case. Once again the four-stream IA (*μ*_{s} = *μ*_{1}) provides more accurate results compared to that of C–C. Therelative error in transmission shows the IA model to have wider regions of low error compared to the C–C model.

From Figs. 1–3, it is found that the four-stream IA can provide much more accurate results in comparison with two-stream C–C in thin optical depth cases. Also, it is shown that the results are dependent on the choice of the separating angle for the isosector. It is found that the larger separating angle of *μ*_{s} = *μ*_{1} generally gives much better results compared to those of *μ*_{s} = *μ*_{2}; however, further testing of dependency is needed to confirm this. In Fig. 4, three different values of *s* around *μ*_{1}(*s* = 8.6515) are considered. Only the results of single scattering albedo *ω* = 0.9 are plotted, although the results are similar for other values. It is found that the varying *s* around 8.6515 does not have much of an influence on the results. The top panel in Fig. 4 shows the case for *s* = 7 (67.8°); the middle panel is for *s* = 10 (71.6°); and the largest value of *s, s* = 12 (73.2°),is shown in the lower panel. By a comparison of the relative error for each of the radiative properties it is seen that there is not much variability between the different plots. With increasing *s,* the relative error in reflection and absorption appear to slightly improve, whereas the transmission accuracy slightly diminishes. This suggests that we cannot use too large a value of *s.* On the whole, using the first Gaussian angle as the separating angle is a suitable choice.

For the C–C model, the errors are always large in the region of large solar zenith angle. This is also true for other low-order approximation methods for radiative transfer. The physical reason seems to be that the radiation field changes more rapidly as the cosine of local zenith angle approaches zero (see Hu and Wielicki 1997). By using a large separating angle, we can improve the results by focusing more attention on the sector of large solar zenith angle. The two separated sectors are related through radiative transfer equation. The improvement in the region of large solar zenith angle also leads to the improvement in the region of small solar zenith angle.

## 4. Conclusions

Solar radiative transfer processes are governed by an integro-differential equation, and the exact solution for a scattering medium does not exist. The tricks for approximate methods are all based on transforming the integro-differential equation into a differential equation. The discrete ordinate method replaces the integral by a discrete sum of terms, whereas the spherical harmonic expansion is based on separating out the angular dependent parts of the intensity. The four-stream isosector approximation presented here is based on a more simple technique by imposing the intensity to be isotropic in a specified angular region.

For thin optical depths, very accurate radiative properties are usually difficult to obtain from lower-stream approximations. This is because the radiative processes are more reliant on an appropriate description of the phase function. Although the C–C model is regarded as the best two-stream scheme for a thin atmosphere, the error of the C–C model can be up to 10% even for *τ*_{0} = 0.1. Calculations show that errors in reflected, transmitted, and absorbed fluxes are substantially reduced by using the four-stream isosector approximation. For *τ*_{0} < 1, the four-stream IA shows better results than the two-stream C–C model. Furthermore, since atmospheric aerosols generally have optical thicknesses less than one, this model can be used to to evaluate direct aerosol radiative forcing more accurately.

All comparisons of the IA model with the C–C model were performed for two values of separating angle, equal to the first and second Gaussian angles. The larger separating angle is found to consistently provide higher accuracy. Calculations were also performed to probe the sensitivity of the results to variations in separating angle. These results show that increasing the separating angle beyond the first Gaussian angle tends to lower the relative errors in reflection and absorption, but increases error in transmission. Also the calculations show that the results are not sensitive to variations around the first Gaussian angle. Hence the first Gaussian angle is determined to provide a suitable choice for the separating angle.

Aside from small solar incident angles *μ*_{0} > 0.9, the four-stream IA model can be extended to *τ*_{0} = 2 without a large increase in error. Therefore, this model may be applied to cirrus clouds (Chýlek and Dobbie 1995). We have emphasized that the integrity of phase function is preserved in the four-stream IA; thus the integrity of the phase function of nonspherical ice crystal can be maintained in the radiative transfer process.

The four-stream IA is an analytical model, which makes it easily applicable to climate model when the aerosol, clear-sky radiation, and cirrus cloud are addressed.

## Acknowledgments

We would like to thank the anonymous reviewers for their constructive comments and J. Wong for his helpful comments on this paper. We would also like to thank Dr. S. C. Tsay for providing the discrete ordinate model.

## REFERENCES

Barker, H., and Z. Li, 1995: Improved simulation of clear-sky shortwave radiative transfer in the CCC-GCM.

*J. Climate,***8,**2213–2223.Charlson, R. J., S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, and D. J. Hofmann, 1992: Climate forcing by anthropogenic aerosols.

*Science,***255,**423–430.Chýlek, P., and S. Dobbie, 1995: Radiative properties of finite inhomogeneous cirrus clouds: Monte Carlo calculations.

*J. Atmos. Sci.,***52,**3512–3522.Coakley, J., and P. Chýlek, 1975: The two-stream approximation in radiative transfer: Including the angle of the incident radiation.

*J. Atmos. Sci.,***32,**409–418.Hu, Y., and B. A. Wielicki, 1997: A sensitivity study of angular dependence models. Preprints,

*Ninth Conf. on Atmospheric Radiation,*Long Beach, CA, Amer. Meteor. Soc., 279–282.Joseph, J. H., W. J. Wiscombe, and J. A. Weinman, 1976: The delta Eddington approximation for radiative flux transfer.

*J. Atmos. Sci.,***33,**2452–2459.King, M. D., and Harshvardhan 1986: Comparative accuracy of selected multiple scattering approximations.

*J. Atmos. Sci.,***43,**784–801.Li, J., and V. Ramaswamy, 1996: Four-stream spherical harmonic expansion approximation for solar radiative transfer.

*J. Atmos. Sci.,***53,**1174–1186.Liou, K. N., 1973: A numerical experiment on Chandrasekhar’s discrete-ordinate method for radiative transfer: Applications to cloud and hazy atmosphere.

*J. Atmos. Sci.,***30,**1303–1326.——, Q. Fu, and T. P. Ackerman, 1988: A simple formulation of the delta-four-stream approximation for radiative transfer parameterisation.

*J. Atmos. Sci.,***45,**1904–1947.Meador, W. E., and W. R. Weaver, 1980: Two-stream approximations to radiative transfer in planetary atmospheres: A unified description of existing methods and a new improvement.

*J. Atmos. Sci.,***37,**630–643.Ross, S. L., 1974:

*Differential Equation.*Xerox College Publishing, 712 pp.Shettle, E. P., and J. A. Weinman, 1970: The transfer of solar irradiance through inhomogeneous turbid atmospheres evaluated by Eddington’s approximation.

*J. Atmos. Sci.,***27,**1048–1054.Stamnes, K., S. C. Tsay, W. Wiscombe, and K. Jayaweera, 1988: A numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media.

*Appl. Opt.,***27,**2502–2509.Wiscombe, W., 1977: The delta-

*M*method: Rapid yet accurate radiative flux calculations.*J. Atmos. Sci.,***34,**1408–1422.