Among the two-stream approximation methods for radiative transfer, the Eddington approximation (Shettle and Weinman 1970) has been the most popular. Since in that approximation it is only the first moment of the expansion of the phase function that is kept, the sharp forward-scattering peak is not well represented. This drawback has been improved by using the *δ* function to represent the forward-scattering peak (Joseph et al. 1976). This method (called the *δ*-Eddington approximation) can dramatically improve the accuracy of the Eddington approximation.

*δ*-Eddington approximation are well understood. There does not, however, seem to be sufficient physical basis upon which to determine the fractional scattering into the forward peak,

*f.*Joseph et al. (1976) require that fractional scattering into the forward peak be related to the second moment of the original phase function as follows:

*f*

*g*

^{2}

*g*is the asymmetry factor. With this requirement, the phase function of the

*δ*-Eddington approximation is always larger than the Henyey–Greenstein phase function [Eq. (7) of Joseph et al. 1976]. Since the first moments of both phase functions are the same, the large value of the phase function of the

*δ*-Eddington approximation affects the fractional scattering into the forward peak. The result of a higher fractional forward-scattering peak is that photons will pass more easily through the medium. The top panels of Fig. 1 show the relative errors of reflection and transmission for the scattering conservative case as calculated by the

*δ*-Eddington approximation. In the region of large solar zenith angle the reflection is underestimated and the transmission is consistently overestimated. This indicates that the forward scattering may be too strong, especially for the case of large solar zenith angles. With the increase of optical depth the error in reflection reduces. For a thick optical depth the path of the photon through the medium is generally longer than it is in the case of a thin optical depth. The photons must undergo more scattering events before leaving the medium. The single scattering directions are therefore smoothed and become more isotropic. Therefore, the effect of the forward scattering becomes weaker.

The top panels in Fig. 2 show the relative errors of the *δ*-Eddington approximation for reflection, transmission, and absorption in a strong absorbing case with single scattering albedo *ω* = 0.9. As in Fig. 1, less reflection occurs in the region of large solar zenith angle, but reflection in the region of small solar zenith angle becomes too large. Could this be the result of the small forward-scattering peak? However, numerical calculations indicate that the errors are also caused by overestimating forward scattering (the errors become smaller due to the reduction of the fractional scattering into the forward peak). In a strong absorbing case, when a photon penetrates into a deep region of the medium, the chance of being scattered back to the top of the medium is small. The reflection is therefore principally determined by the shallow region near the top of the medium and the result is sensitive to the structure of phase function. In the *δ*-Eddington approximation, the intensity is decoupled from the angular dependency by Legendre expansion, and only the first term of the Legendre expansion is kept. The distribution of intensity is *I* = *I*_{0} + *I*_{1} cos*θ,* where *θ* is the local zenith angle. Therefore, the distribution of intensity tends to be large in the regions near the backward direction. For an overhead sun the strong forward-scattering peak in combination with the strong backward intensity will cause some photons to be scattered back more readily. It is for this reason that the four-stream approximation (formed by keeping more terms in the Legendre expansion to make the distribution of intensity more reasonable) can dramatically improve the accuracy (Fig. 2 in Li and Ramaswamy 1996).

*f*

*g*

^{2β}

*μ*

_{0}is the cosine of solar zenith angle and

*γ*= (0.85/

*g*)

^{2}. Since the

*δ*-scaling effect would become weaker for a smaller asymmetry factor, we include a factor of

*γ*in Eq. (2) to account for this effect. Between 1 and 2,

*β*varies approximately, with the result that

*f*in Eq. (2) is generally smaller than that in Eq. (1). The corresponding results of the scattering conservative case by using Eq. (2) are shown in the lower panels in Fig. 1. We find that the accuracy is improved, especially for the region of thin optical depth with a large solar zenith angle. The region where the error exceeds 20% in reflection is almost eliminated. In addition, the error in the region of smaller solar zenith angle is also reduced. Generally, the improvement in the thick optical depth region is much less than that in the thin optical depth region. This is since the radiative transfer is not sensitive to the structure of the phase function for a thick optical depth on account of the smoothing of scattering directions by an increased number of scattering events. The results for transmission are slightly improved, mostly for the thin optical depth with a large solar zenith angle. Transmission is composed of a direct part and a diffuse part. For a very thin optical depth, with a solar zenith angle that is not too large, the direct part is dominant. This direct transmission is not dependent on the solution of the multiple scattering process. Therefore, any adjustment of the forward-scattering peak would have much less effect on the transmission than it would have on the reflection.

For the scattering nonconservative case, we consider a single scattering albedo *ω* = 0.9 in Fig. 2. The results are noticeably improved by using Eq. (2). For reflection, Eq. (2) yields substantially more accurate results for small solar zenith angles (*μ*_{0} > 0.8 or the solar zenith angle *θ*_{0} < 35°). The errors exceeding 20% are eliminated. Since the incoming solar energy is proportional to *μ*_{0}, 60% incoming solar energy is within the range of *μ*_{0} > 0.8. The improvement for small solar zenith angles is extremely important for radiative transfer processes. In contrast to the scattering conservative case, the adjustment of fractional scattering into the forward peak can influence the region of large optical depth. As mentioned previously, for a strong absorbing case the reflection is mostly determined by the photons within the shallow region near the top. For transmission, the results become slightly poorer in the region of optical depth over 10, with large solar zenith angles. However, transmission is, in this case, very small and the absolute error is also very small. Similar to reflection, the results for absorption become better in the region of small solar zenith angles.

We consider the very strong absorbing case in Fig. 3 with single scattering albedo *ω* = 0.5. In this case, since the absorption is dominant, the relative errors for reflection and transmission become very large. Equation (2) gives slightly better results for reflection. For the important case of absorption, Eq. (2) yields better results in the region of small solar zenith angle, in the same way as Fig. 2.

Next we consider different asymmetry factors of *g* = 0.6 and *g* = 0.9. For *g* = 0.6, the forward-scattering peak is weak. Therefore, the *δ*-scaling effect becomes small. Figure 4 shows the comparison results of Eqs. (1) and (2). It is found that the improvements associated with using Eq. (2) are similar to those found in Fig. 2, except smaller in magnitude.

For *g* = 0.9, the forward scattering becomes very strong. The results of a comparison of Eqs. (1) and (2) are shown in Fig. 5. Equation (2) can improve the accuracy of reflection for *μ*_{0} > 0.8. As mentioned previously, this region is very important as it represents the bulk of the incoming solar energy. Similar results are found for the case of absorption.

To summarize, since the determination of the fractional scattering into the forward peak lacks sufficient physical justification, and since this problem is hard to solve from the scattering theory or from the radiative transfer theory, we propose a simple hypothesis that the fractional forward-scattering peak is related not only to the asymmetry factor, as shown in Eq. (1), but also to solar zenith angle and single scattering albedo. The calculations show that this assumption can substantially improve results. For the scattering conservative case, the improvement occurs principally in the region of thin optical depth. For the scattering nonconservative case, this simple method gives consistently better results in the region of small solar zenith angles.

The purpose of this note is to arouse attention to the problem of fractional scattering into the forward peak. The formula proposed in Eq. (2) is not necessarily the best one. This simple method would not change the calculation procedure of the *δ*-Eddington approximation. It can therefore be easily implemented in climate models where the *δ*-Eddington approximation is used.

## REFERENCES

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.Li, J., and V. Ramaswamy, 1996: Four-stream spherical harmonic expansion approximation for solar radiative transfer.

*J. Atmos. Sci.,***53,**1174–1186.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.