Statistical treatment of the PAR variability and its application to willow coppice

Abstract

For characterization of spatial variability of the global photosynthetically active radiation (PAR) Q at different depths in a plant canopy a new statistical treatment has been developed. It is based on the use of the irradiance probability density function (IPDF) which is divided into three parts: IPDF for umbra f u Q , for penumbra f p Q and for sunflecks f s Q . In umbra, IPDF was approximated by a normal distribution, in penumbra by the Beta-distribution, and in sunflecks, by a normal distribution. Statistical treatment has been applied for the fast growing willow coppice. In the umbra area the mean value Q I of global PAR decreases exponentially with the optical path length of the direct solar radiation inside the canopy τ= L/sinh with small values of the standard deviation σ u. In the sunfleck area the mean value Q II decreases linearly with τ with small values of the standard deviation. In the penumbra area agreement with experimental data is inferior to that in umbra and sunfleck areas. Inside the canopy, the sunfleck fractional area k s, fitted by a second order polynomial, decreases with τ up to τ=3.3. The umbra fractional area k u increases nearly linearly with depth τ starting from τ=0.8. The fractional area of the penumbra k p increases rapidly with τ, reaches the maximum value ( k p=0.6–0.8) at τ=3 and then decreases nearly linearly.