The approximate analytical expressions for overcast sky luminance depending on both the physical state of the atmosphere and the reflectance of an underlying ground surface are derived and verified against numerical calculations. The physical parameters (such as ground reflectance, scattering efficiency of the atmosphere and/or atmospheric optical depth) make the model easily-scalable and feasible for describing sky luminance under various meteorological conditions. Despite the significant simplifications applied in the theoretical derivations, the approximate formulae can provide satisfactorily accurate data (contrary to empirical models in current use). Because the analytical equations enable extremely fast calculations, the presented approach is suitable for rapid modelling of daylight availability.
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