A derivation of the diffuse fraction's dependence on the clearness index

When averaged over many events having the same k_t, the average diffuse fraction carries an important quasi-universal functional dependence on the clearness index k_t. This paper presents a theoretical derivation of k(k_t). The atmosphere is modeled as having two homogeneous, nonselectively absorbing layers: an upper layer (roughly, the ozonosphere) with zero scattering and beam transmittance τ_u, and a lower layer with isotropic, nonselective scattering characterized by an albedo ω_l. After an analysis which traces generations of scattered photons, this model predicts a function which, upon adjusting ω_l and τ_u, fits the measured data closely.