Fundamental Definition of Two-Stream Approximation for Radiative Transfer in Scattering Atmosphere

The two-stream approximation (TSA) is a primary method for tackling radiative transfer in a scattering atmosphere, which has wide applications in radiation balance evaluation, atmospheric simulation, and remote sensing data assimilation. This article defines fundamental TSA (F-TSA) by introducing the TSA conversion function, which reveals the causal characteristics of various existing specific TSA (S-TSA) models. The unified diffuse irradiance equations and their solutions are obtained based on F-TSA. The properties of the conversion function and the coefficients of irradiance equations are analyzed. It is proven that the weighted average of different conversion functions is also a conversion function. The specified conversion functions and scattering phase function treatments of six existing and four newly proposed S-TSA models are described. These S-TSA models and their typical combinations are evaluated and compared for incident lights from different directions and atmospheric layers with different optical depths. The main results are: 1) the accuracy of an S-TSA model depends on which one of transmissivity and reflectivity is concerned, and is affected by multiple factors such as the specified conversion function, the phase function processing, the single scattering albedo, the optical depth, and the direction of incident light; 2) the proper combination of different S-TSA models can improve the TSA accuracy significantly, and both the delta function and phase function B models combined with the modified Eddington model are recommended; and 3) applying proper combined S-TSA models instead of the delta transformed S-TSA models to calculate radiation flux may be beneficial.