On the statistical distribution of clear-sky index and solar radiation transmittances: A worldwide analysis

Accurate characterization of the statistical distributions of surface solar radiation—specifically the clear-sky index (κ), clearness index (k_t), beam transmittance (k_b), and diffuse transmittance (k_d)—is fundamental to atmospheric science and solar energy meteorology. While the presence of latent sky conditions (clear, cloudy, and overcast) justifies the use of mixture models, a consensus on the optimal component distribution remains lacking, as standard models often fail to reflect the physical bounds and asymmetric nature of radiative transfer. To bridge this gap, this study demonstrates that k-indexes are optimally modeled using a three-component truncated skew-normal (TSN) mixture. The TSN model offers critical physical interpretability: The truncation strictly enforces the physical limits of irradiance, while the skewness accommodates the asymmetric effects of heavy aerosol loading and strong forward scattering during cloud-edge enhancement. Validated against 1-min resolution data from 126 high-quality radiometric stations worldwide, the TSN mixture exhibits statistically significant superiority over traditional multi-component normal mixtures across all radiation climates. Finally, leveraging the probability densities extracted from these physically grounded models, two novel applications of the TSN mixture model are proposed, offering rigorous new frameworks for atmospheric classification and solar resource assessment.