Exact G-functions for various leaf normal distribution functions applied to analytical two-stream radiative transport in turbid vegetation media

Abstract

In the radiative transfer theory for turbid vegetation media one defines leaf normal distribution (LND) functions to describe the orientation of the model leaves’ normal vectors. For simulation purposes, standard LNDs are frequently assumed, e.g. based on measurements, which have to fulfil a normalisation condition. Unfortunately, the literature reveals various discrepancies with respect to the proper definition of normalised LNDs. So-called G-functions are derived from the LNDs, and thus they are an important part of the radiative transfer equation for turbid vegetation media. Up to now, analytical G-functions have been derived for the most simple LND cases, only. We will revisit the normalisation problem and present an extended set of analytical expressions of G-functions for the most commonly used LNDs, which are then compared to approximations in literature. Moreover, these considerations allow us to extend the two-stream theory in a stringent analytical manner to various LND scenarios ranging from purely horizontal to purely vertical model leaves. Finally, we applied these very fast radiative transfer schemes to simulate canopy reflectances and transmittances as a function of the ratio of direct and diffuse light impinging on the vegetation medium at its upper boundary.