Two-Stream Approximation to the Radiative Transfer Equation: A New Improvement and Comparative Accuracy with Existing Methods

• Published in: Advances in atmospheric sciences Vol. 41; no. 2; pp. 278 - 292
• Main Authors: Temgoua, F. Momo, Nguimdo, L. Akana, Njomo, D.
• Format: Journal Article
• Language: English
• Published: Heidelberg Science Press 01.02.2024; Springer Nature B.V; Department of Electrical and Electronic Engineering,Faculty of Engineering and Technology,University of Buea,PO Box 63 Buea,Cameroon; Environmental Energy Technologies Laboratory (EETL),Faculty of Sciences,University of Yaoundé,PO Box 812 Yaoundé,Cameroon%Environmental Energy Technologies Laboratory (EETL),Faculty of Sciences,University of Yaoundé,PO Box 812 Yaoundé,Cameroon
• Subjects: Accuracy; Approximation; Atmosphere; Atmospheric Sciences; Delta function; Divergence; Earth and Environmental Science; Earth Sciences; Geophysics/Geodesy; Mathematical models; Meteorology; Methods; Optical thickness; Original Paper; Polynomials; Quadratures; Radiative transfer; Reflectance; Rivers; Solar radiation; Zenith; 勒让德多项式; 透过率; moments of specific intensity; conversion function; reflectance; 二流方法; transmittance; two-stream method; Legendre polynomial; Radiative Transfer Equation; 反射率; 转换函数; 辐射传输方程; 比强度矩; 光学厚度; optical thickness
• ISSN: 0256-1530; 1861-9533
• DOI: 10.1007/s00376-023-2257-9

Mathematical modeling of the interaction between solar radiation and the Earth’s atmosphere is formalized by the radiative transfer equation (RTE), whose resolution calls for two-stream approximations among other methods. This paper proposes a new two-stream approximation of the RTE with the development of the phase function and the intensity into a third-order series of Legendre polynomials. This new approach, which adds one more term in the expression of the intensity and the phase function, allows in the conditions of a plane parallel atmosphere a new mathematical formulation of γ parameters. It is then compared to the Eddington, Hemispheric Constant, Quadrature, Combined Delta Function and Modified Eddington, and second-order approximation methods with reference to the Discrete Ordinate (Disort) method ( δ –128 streams), considered as the most precise. This work also determines the conversion function of the proposed New Method using the fundamental definition of two-stream approximation (F-TSA) developed in a previous work. Notably, New Method has generally better precision compared to the second-order approximation and Hemispheric Constant methods. Compared to the Quadrature and Eddington methods, New Method shows very good precision for wide domains of the zenith angle μ 0 , but tends to deviate from the Disort method with the zenith angle, especially for high values of optical thickness. In spite of this divergence in reflectance for high values of optical thickness, very strong correlation with the Disort method ( R ≈ 1) was obtained for most cases of optical thickness in this study. An analysis of the Legendre polynomial series for simple functions shows that the high precision is due to the fact that the approximated functions ameliorate the accuracy when the order of approximation increases, although it has been proven that there is a limit order depending on the function from which the precision is lost. This observation indicates that increasing the order of approximation of the phase function of the RTE leads to a better precision in flux calculations. However, this approach may be limited to a certain order that has not been studied in this paper.