| description abstract | The fractional factor f of δ-function scaling in the δ-Eddington approximation modifies the fractional scattering into the forward peak. As shown in this paper, reasonably choosing the factor f can yield a great improvement of transmission, reflection, and absorption calculations in the condition of the optical depth τ ? 1. Based on this fact, a modified δ-Eddington approximation is empirically and mathematically developed using a parameterization model of the factor f that mainly depends on asymmetry factor g0, total optical depth τ, single scattering albedo ?, (ground) surface reflectance A, and cosine of solar zenith angle ?0. There are 69?120 sets of comparative numerical tests, covering seven aerosol and two cloud size distributions, as well as three Henyey?Greenstein phase functions. Among the exiting two-stream approximations, δ-Eddington generally has better transmission, reflection, and absorption accuracy as τ ? 1. In an average sense, in the condition of A ? 0.6, τ ? 1, 0.1 ? ?0 ? 1.0, and 0.5 ? ? ? 1, the modified δ-Eddington approximation can reduce transmission, reflection, and absorption errors by a factor of about 2, compared with the results of the δ-Eddington. For the conservative atmosphere, much greater improvement of transmission and reflection accuracy is obtained. | |
