Analytical Delta-Four-Stream Doubling–Adding Method for Radiative Transfer Parameterizations

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%.