The influence of ground reflectance on the overcast sky luminance

Abstract

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. List of symbolsa Albedo of the Earth's surface (-) dQ a V Luminous energy emitted by an elementary surface dS into an elementary solid angle (lm s) dS Elementary surface (m 2 ) dt Elementary time interval (s) dv Elementary volume of the atmosphere (m 3 ) D 2Specific dimensionless function used in analytical formulae for sky luminance (-) E 1 , E 2 Exponential integrals (so-called E-functions) (-) E V Illuminance at the horizontally oriented ground surface (lux) gAsymmetry parameter of an aerosol poly-Altitude of a cloudy layer (m) k ext Volume extinction coefficient of an undercloud atmosphere (m À1 ) L 0VLuminance of a cloud independent of the viewing angle (cd/m 2 ) L a 0VLuminance of an Earth's surface independent of the viewing angle (cd/m 2 ) L aD 0V DirectcomponentoftheluminanceofEarth's surface observed at an altitude h (cd/m 2 ) L aS V Scattering component of the luminance of Earth's surface observed at an altitude h (cd/m 2 ) L D V Direct component of the luminance of a cloud measured at the ground (cd/m 2 ) L S V