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Hua Zhang, Feng Zhang, Qiang Fu, Zhongping Shen, and Peng Lu


The δ-two- and four-stream combination approximations, which use a source function from the two-stream approximations and evaluate intensities in the four-stream directions, are formulated for the calculation of diffuse actinic fluxes. The accuracy and efficiency of the three computational techniques—the δ-two-stream approximations, the δ-two- and four-stream combination approximations based on various two-stream approaches, and the δ-four-stream approximation—have been investigated. The diffuse actinic fluxes are examined by considering molecular, aerosol, haze, and cloud scattering over a wide range of solar zenith angles, optical depths, and surface albedos. In view of the overall accuracy and computational efficiency, the δ-two- and four-stream combination method based on the quadrature scheme appears to be well suited to radiative transfer calculations involving photodissociation processes.

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Feng Zhang, Zhongping Shen, Jiangnan Li, Xiuji Zhou, and Leiming Ma


Although single-layer solutions have been obtained for the δ-four-stream discrete ordinates method (DOM) in radiative transfer, a four-stream doubling–adding method (4DA) is lacking, which enables us to calculate the radiative transfer through a vertically inhomogeneous atmosphere with multiple layers. In this work, based on the Chandrasekhar invariance principle, an analytical method of δ-4DA is proposed.

When applying δ-4DA to an idealized medium with specified optical properties, the reflection, transmission, and absorption are the same if the medium is treated as either a single layer or dividing it into multiple layers. This indicates that δ-4DA is able to solve the multilayer connection properly in a radiative transfer process. In addition, the δ-4DA method has been systematically compared with the δ-two-stream doubling–adding method (δ-2DA) in the solar spectrum. For a realistic atmospheric profile with gaseous transmission considered, it is found that the accuracy of δ-4DA is superior to that of δ-2DA in most of cases, especially for the cloudy sky. The relative errors of δ-4DA are generally less than 1% in both the heating rate and flux, while the relative errors of δ-2DA can be as high as 6%.

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