# Modeling the Directional Clumping Index of Crop and Forest
## 1 Introduction
[32] applied angularly dependent leaf clumping index in determining LAI, which led to a 30% higher estimation when compared to the indirect measurements using LAI-2000, but matched precisely with the direct measurements using litter traps. [PERSON] et al. [33] tested the effects of ignoring foliage clumping characteristics in global GPP mapping, and found that neglecting clumping index will increase the uncertainty in GPP estimation, particularly in tropical regions. Using accurate LAI, but ignoring clumping index, causes an overestimation of global GPP by 12%; applying the effective LAI without considering the clumping effect would underestimate global GPP by 9%. Comparing the effect of clumping index on LAI/GPP estimation and the directional variation magnitude of \(\Omega\)[2; 21], it is necessary to apprehend that if angular variation is not provided as well as the average, large errors can occur in the inverted leaf angle distribution and LAI for canopies with direction dependent clumping [34]. Many studies have discussed the directional variation of \(\Omega\) using the gap size distribution measurements from in situ instruments, such as TRAC or digital hemispherical photography. The efficiency in data measurement is the bottleneck in wide application of the gap size distribution model. [PERSON] [22] developed a simulation model that had extended the understanding of directional clumping effect, but it was difficult to be applied for practical calculations, as many input parameters cannot be measured. [PERSON] [21] modeled the directional clumping index using an empirical equation to extend its availability for more species, although the accuracy of coefficients is restricted in the regression hull. This paper aims to propose analytical models to estimate directional clumping index of crop and forest at canopy level. According to the Beer's law [35], the clumping index can be estimated as a function of gap fraction of vegetation canopy. Crop canopy illustrates an apparent clumping index angular dependence due to its gap fraction variation in discrete periodic row structures [22; 36; 37]. [PERSON] et al. [37] proposed an analytical model for the crop clumping index using the directional average gap fraction, where each hedgerow was regarded as the integration of four parts with different average LAI. This assumption led to an accurate expression of crop clumping index, although the integration component is incorrect in concept. [PERSON] et al. [38] rectified the method and applied the directional clumping index in retrieving crop albedo. The average gap fraction of forest was estimated according to the interaction between light regime and the canopy, such as the widely used Geometric-Optical (GO) model that was proposed by [PERSON] et al. [39], [PERSON] [7], and [PERSON] [40]. In the GO model, forest gap fraction is the complement of the projected area fraction on the ground. [PERSON] [7] proposed the gap probability was determined by the number of trunks occurring in the shadowed region of a single tree. It is assumed each tree crown was ellipsoid, which fits most practical cases. The measurements and simulations [21] suggest that the largest gaps between crowns are a major cause of the canopy clumping effect. This paper proposed analytical models for crop and forest directional clumping index based on gap fraction estimation. To extend the applicability of the model, we emphasized the angular variation of \(\Omega\) due to the first-level structure, i.e., the row structure of crop, and the abundance of crown shapes coupled with complex tree spatial distribution of forest. Section 1 introduced the importance of clumping index and the necessity of directional clumping index according to the previous studies. Section 2 introduces the derived analytical model of angular clumping index variation for crop and forest. Section 3 demonstrates validation of the model, while using in situ results to validate the reliability of the proposed methods, and we evaluated the variation range of \(\Omega(\theta)\).