Evaluation and Refinement of the Beer-Lambert Extinction Law in the Discontinuous Vegetation Canopy

The classical Beer-Lambert (BL) law of exponential decay in direct transmission is widely used for modeling the photon propagation in optical media and has been employed for retrieving vegetation structure parameters (e.g. leaf area index, LAI). However, BL law assumes that these absorbing obstacles are distributed in the space independently, which is the main reason of model-observation-inconsistency and arises many studies of so-called sub- and super-exponential extinction for spatially correlated media. Discontinuous vegetation canopy is the typical case of the extinction field with spatial correlations. Because of many practical difficulties, the uncertainty of the BL law used in vegetation canopy still lacks quantitive assessment. In this paper, we carry out this task by utilizing a ray-tracing-based 3-Dimensional radiative transfer model (3D RT) to simulate the photo propagation process in real vegetation canopy scenes. We confirm that the classical BL law is only suitable for both horizontally and vertically homogenous canopy (e.g. dense grasses) and shows increasing discrepancy with the decrease of the fraction of vegetation cover (FVC). When canopy clumping occurs (FVC<1), absorbing obstacles (i.e. leaves) become to be spatially correlated and lead to a slower-than-exponential (sub-exponential) extinction with propagation distance, which will result in an underestimation of LAI when classic BL law is employed. To solve this problem, we propose a new 1st order scattering extinction model by modifying the classic BL law by introducing a pair-correlation-function. This attempt is based on the stochastic radiative transfer theory and shows good performance when compared with the reference from computer 3D simulation.