• Coakley, J. A., Jr., and P. Chylek, 1975: The two-stream approximation in radiative transfer: Including the angle of the incident radiation. J. Atmos. Sci.,32, 409–418.

  • Diermendjian, D., 1969: Electromagnetic Scattering on Spherical Polydispersions. Elsevier, 290 pp.

  • 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.

  • Hansen, J. E., 1971: Multiple scattering of polarized light in planetary atmosphere. Part I: The doubling method. J. Atmos. Sci.,36, 508–518.

  • Liou, K. N., 1974: Analytic two-stream and four-stream solutions for radiative transfer. J. Atmos. Sci.,31, 1473–1475.

  • 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.

  • Schaller, E., 1979: A Delta two stream approximation in radiative flux calculations. Contrib. Atmos. Phys.,52, 17–26.

  • 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, 1048–1055.

  • Stamnes, K., S. C. Tsay, W. J. Wiscombe, and K. Jayaweera, 1988: Numerically stable algorithm for discrecte ordinate method radiative transfer in multiple scattering and emitting layered media. Appl. Opt.,27, 2502–2509.

  • Zdunkowski, W. G., R. M. Welch, and G. Korb, 1980: An investi-calculation of solar fluxes and heating rates in clouds. Contrib. Atmos. Phys.,53, 147–166.

  • Zhu, X., and A. Arking, 1994: Comparison of daily averaged reflection, transmission, and absorption for selected radiative flux transfer approximations. J. Atmos. Sci.,51, 3580–3592.

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Modified Delta-Eddington Approximation for Solar Reflection, Transmission, and Absorption Calculations

Jinhuan QiuInstitute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China

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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 ϖ, (ground) surface reflectance A, and cosine of solar zenith angle μ0. 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 ⩽ μ0 ⩽ 1.0, and 0.5 ⩽ ϖ ⩽ 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.

Corresponding author address: Dr. Jinhuan Qiu, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China.

Email: jhqiu@mimi.cnc.ac.cn

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 ϖ, (ground) surface reflectance A, and cosine of solar zenith angle μ0. 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 ⩽ μ0 ⩽ 1.0, and 0.5 ⩽ ϖ ⩽ 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.

Corresponding author address: Dr. Jinhuan Qiu, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China.

Email: jhqiu@mimi.cnc.ac.cn

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