Photosynthetically-active radiation: Sky radiance distributions under clear and overcast conditions

The photosynthetically active radiation (PAR), defined as the wavelength band of 0.400 μm to 0.700 μm, represents most of the visible solar radiation. Although the proportion of global irradiance that originates from diffuse sky radiation is higher for PAR than for all solar shortwave radiation, it is often assumed that the PAR diffuse sky radiation is distributed identically to that of all shortwave solar radiation. This assumption has not been tested. PAR sky radiance measurements were made in a rural area over a wide range of solar zenith angles. The distribution of PAR sky radiance was modeled using physically-based, non-linear equations. For clear skies, the normalized sky radiance distribution (N) was best modeled using the scattering angle (Ψ) and the zenith position in the sky (Θ) as N(Θ,ψ) = 0.0361[6.3 + (1 + cos²ψ)/(1 - cosψ)][1 - e(-0.31secΘ)]. The angle Ψ is defined by cosψ = cosΘcosΘ* + sinΘsinΘ*cosΦ, where solar zenith angle is Θ* and the difference in azimuth between the sun and the position in the sky is Φ. Modeling of the overcast sky depended on the visibility of the solar disk. The translucent middle/high cloud overcast conditions (cloud base greater than 300 m above ground level) were best modeled as: N(Θ*,ψ) = 0.149 + 0.084Θ* + 1.305e(-2.5ψ) while the translucent low cloud overcast conditions (cloud base less than 300 m above ground level) were best modeled as: N(Θ*,ψ) = 0.080 + 0.058Θ* + 0.652e(-2.1ψ). The obscured overcast sky condition (solar disk obscured) was best modeled as: N(Θ) = 0.441[1 + 4.6cosΘ]/[1 + 4.6]. The unit of N for all equations is π Sr^-1, so that integration of each function over the sky hemisphere yields 1.0. These equations can be applied directly to the sky diffuse irradiance on the horizontal, I(diff), to provide radiance distributions for the sky. Estimates of actual sky radiance distribution can be estimated from N(a)(Θ,ψ) = I(diff)N(Θ,Φ).