The stochastic Beer–Lambert–Bouguer law for discontinuous vegetation canopies

Abstract

The 3D distribution of canopy foliage affects the radiation regime and retrievals of canopy biophysical parameters. The gap fraction is one primary indicator of a canopy structure. Historically the Beer–Lambert–Bouguer law and the linear mixture model have served as a basis for multiple technologies for retrievals of the gap (or vegetation) fraction and Leaf Area Index (LAI). The Beer–Lambert–Bouguer law is a form of the Radiative Transfer (RT) equation for homogeneous canopies, which was later adjusted for a correlation between fitoelements using concept of the clumping index. The Stochastic Radiative Transfer (SRT) approach has been developed specifically for heterogeneous canopies, however the approach lacks a proper model of the vegetation fraction. This study is focused on the implementation of the stochastic version of the Beer–Lambert–Bouguer law for heterogeneous canopies, featuring the following principles: 1) two mechanisms perform photon transport- transmission through the turbid medium of foliage crowns and direct streaming through canopy gaps, 2) the radiation field is influenced by a canopy structure (quantified by the statistical moments of a canopy structure) and a foliage density (quantified by the gap fraction as a function of LAI), 3) the notions of canopy transmittance and gap fraction are distinct. The derived stochastic Beer–Lambert–Bouguer law is consistent with the Geometrical Optical and Radiative Transfer (GORT) derivations. Analytical and numerical analysis of the stochastic Beer–Lambert–Bouguer law presented in this study provides the basis to reformulate widely used technologies for retrievals of the gap fraction and LAI from ground and satellite radiation measurements.