Comparative Accuracy of Selected Multiple Scattering Approximations

Michael D. King Laboratory for Atmospheres, Goddard Space Flight Center, NASA, Greenbelt, MD 20771

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Harshvardhan Laboratory for Atmospheres, Goddard Space Flight Center, NASA, Greenbelt, MD 20771

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Abstract

Computational results have been obtained for the plane albedo, total transmission and fractional absorption of plane-parallel atmospheres composed of cloud droplets. These computations, which were obtained using the doubling method, are compared with comparable results obtained using selected radiative transfer approximations. Both the relative and absolute accuracies of asymptotic theory for thick layers and delta-Eddington, Meador–Weaver and Coakley–Chýlek approximations are compared as a function of optical thickness, solar zenith angle and single scattering albedo. Asymptotic theory is found to be accurate to within 5% for all optical thickness greater than about 6. On the other hand, the Coakley–Chýlek approximation is accurate to within 5% for thin atmospheres having optical thickness less than about 0.2 for most values of the solar zenith angle. Though the accuracies of delta-Eddington and Meador-Weaver approximations are less easily summarized it can generally be concluded that the delta-Eddington approximation is the most accurate for conservative scattering when the solar zenith angle is small, while the Meador–Weaver approximation is the most accurate for nonconservative scattering (ω0 ≤ 0.9). Selected results from the Eddington approximation are presented to illustrate the effect of delta function scaling in the delta-Eddington approximation. In addition, selected results from the single scattering approximation and asymptotic theory are presented in order to help explain the strengths and limitations of the various approximations.

Abstract

Computational results have been obtained for the plane albedo, total transmission and fractional absorption of plane-parallel atmospheres composed of cloud droplets. These computations, which were obtained using the doubling method, are compared with comparable results obtained using selected radiative transfer approximations. Both the relative and absolute accuracies of asymptotic theory for thick layers and delta-Eddington, Meador–Weaver and Coakley–Chýlek approximations are compared as a function of optical thickness, solar zenith angle and single scattering albedo. Asymptotic theory is found to be accurate to within 5% for all optical thickness greater than about 6. On the other hand, the Coakley–Chýlek approximation is accurate to within 5% for thin atmospheres having optical thickness less than about 0.2 for most values of the solar zenith angle. Though the accuracies of delta-Eddington and Meador-Weaver approximations are less easily summarized it can generally be concluded that the delta-Eddington approximation is the most accurate for conservative scattering when the solar zenith angle is small, while the Meador–Weaver approximation is the most accurate for nonconservative scattering (ω0 ≤ 0.9). Selected results from the Eddington approximation are presented to illustrate the effect of delta function scaling in the delta-Eddington approximation. In addition, selected results from the single scattering approximation and asymptotic theory are presented in order to help explain the strengths and limitations of the various approximations.

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