Evaluation of Path Length Correction for Forest Canopies Over Sloping Terrains: Theoretical Derivations and Computer Simulations

Topography distorts the angular distribution of the canopy gap fraction (GF). Path length (PL) correction is a simple and effective method to harmonize this distortion and improve canopy reflectance modeling and in situ leaf area index (LAI) measurements for vegetation, including both continuous (e.g., grass and crop) and discrete (e.g., forests) canopies, over sloping terrains. The rigorously theoretical derivation of PL correction for continuous canopies has been implemented. However, for discrete canopies, the PL shows a serious heterogeneity, making it nearly impossible to calculate. In this regard, there is still a need to develop theoretical derivation to evaluate and improve PL correction for forests over sloping terrains. In this study, first, PL correction is proven to be equivalent to the correction of the canopy GF over sloping terrains, and our strategy concerns the canopy GF as a proxy of PL. Second, PL correction is first proven to be completely valid for forests with the Poisson trees distribution; yet, it may produce uncertainty in certain directions for forests with tree distribution deviating from the Poisson model, especially for forests with regular tree distribution. Finally, an improved model based on Nilson and Peterson’s GF model for correcting PL for forests is given in this study. The results show that the error produced by the PL correction for some forests can be effectively decreased by the improved model. The variation of directional tree distribution parameter $c_{B}(\theta)$ with slope is the main cause of error produced by PL correction for forests. The study is of importance for better understanding and more accurate application of PL theory in topographic corrections and in situ LAI measurements for forest canopies over sloping terrains.