NOTES AND CORRESPONDENCE On the Fractional Scattering into the Forward Peak

A hypothesis that the fractional scattering into the forward peak is related to solar zenith angle and single scattering albedo is proposed. Calculations show that this assumption can increase the accuracy of the d-Eddington approximation. For the scattering conservative case this method can improve the results in the region of thin optical depth. For the scattering nonconservative case this method can reduce the errors for reflection and absorption in the region of small solar zenith angle, where the incoming solar energy is most significant. 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 d function to represent the forward-scattering peak (Joseph et al. 1976). This method (called the d-Eddington approximation) can dramatically improve the accuracy of the Eddington approximation. The physics and the derivation of d-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: