Contents I. Introduction
Forests generally have a high degree of natural variability and typically exhibit complex levels of organization, such as crowns, that cause their foliage to be grouped or clumped [1]. Clumping index (CI) is defined as the ratio of the effective leaf area index (LAI) to the true LAI and quantifies the degree of deviation of the leaf spatial distribution from the random case (i.e., the Poisson model) [2], [3]. CI is of great significance in terrestrial carbon and water cycle studies, as it determines radiation absorption and distribution within plant canopies. CI is closely linked to canopy gap fraction (GF) from which the ratio of sunlit LAI to shaded LAI can be calculated [2]. For a canopy with a random distribution of leaves, the relationship between canopy GF and LAI can be described with the well-known Beer–Lambert theory [4] \begin{equation*} P_{\text {Poisson}} \left ({\theta }\right)=e^{-LG\left ({\theta }\right)/\cos \theta }\tag{1}\end{equation*} where is the probability of the transmission of a beam of light at the zenith angle through the canopy, i.e., the canopy GF; is the extinction coefficient, which is 0.5 for canopies with a spherical distribution of leaf angles; and is the LAI, which is most commonly defined as one half the total (all sided) leaf area per unit ground surface area [5], [6]. As leaves often have nonrandom distributions in reality, i.e., they often aggregate in crowns in forests, CI was first introduced in the modified Beer–Lambert theory for calculating canopy GF by Nilson [4] \begin{equation*} P\left ({\theta }\right)=e^{-L\Omega G\left ({\theta }\right)/\cos \theta }\tag{2}\end{equation*} where is canopy GF. expresses the zenith angle dependence of the CI [1]. Combining (1) and (2), we can obtain [7] \begin{equation*} \Omega \left ({\theta }\right)=\frac {\log \left [{P\left ({\theta }\right)}\right]}{\log \left [{P_{\text {Poisson}} \left ({\theta }\right)}\right]}.\tag{3}\end{equation*}
