A Modified Beer–Lambert–Bouguer Law for Nonrandom Distributions and Its Application in Gap Probability Calculations for Heterogeneous Canopies

Abstract

AbstractThe three‐dimensional (3D) structure of vegetation canopy strongly influence the absorption and reflection of radiation at the land surface. The gap probability is one primary indicator of the canopy structure. For homogeneous canopies, where the plant elements are distributed completely randomly and statistically independent, the gap probability can be modeled with the Beer–Lambert–Bouguer law. However, this method is not suitable for nonrandom distributions. A physically based and computationally efficient algorithm is developed to address this limitation, by introducing the pair correlation function to describe the nonrandom distribution pattern. Pair correlation function is based on the distribution of distances of pairs of points, and can therefore describe random, clumped, or regular distribution patterns, and detect mixed patterns at different scales. In this paper, we take tree crown as the basic structure element, and apply the new method to calculate the gap probability. The simulations agree well with results obtained by the ray casting method for various distribution patterns. The results show that horizontal distribution of trees is one of the key factors impacts the gap probability. For the case when different patterns coexist on various scales, we can ignore the distribution pattern at larger scale, and focus on the pattern at smaller scale. When zenith angel is large, the azimuthal dependency of the gap probability should be noted.