# Modeling the Directional Clumping Index of Crop and Forest
## 1 Introduction  
Canopy clumping affects the gap fraction and redistributes radiation interception within the canopy [1; 2; 3; 4]. The assumption of random distributions causes underestimation of leaf area index (LAI) as compared to considering clumping, since the clumping phenomenon of natural forest increases the gap fraction [5; 6; 7]. Therefore, the effective LAI that was indirectly obtained from gap fraction measurements needs to be interpreted together with the clumping condition to approach the actual LAI, i.e., the one-sided leaf area per unit horizontal ground surface area [8; 9; 10; 11]. Clumping index (\(\Omega\)) is a commonly used vegetation dispersion parameter that quantifies the level of foliage distribution non-randomness in real case [12], with \(\Omega=1\) for completely randomly distributed foliage, \(\Omega>1\) for regular spatially distributed foliage, and \(0<\Omega<1\) for clumped foliage [13; 14]. The difficulties in assigning \(\Omega\) are due to its multiple scale and directional variation. First, the clumping effect exist at different scales, such as at shoot level, among canopies level, and at theecosystem level [7; 15; 16; 17; 18; 19; 20]. Second, canopy level \(\Omega\) varies with zenith angle so that the directional change of clumping index has attracted more and more attention [16; 19; 21; 22; 23]. Due to this complexity, the study and application of \(\Omega\) faces accuracy-efficiency-robustness trade-off [19]. In field campaigns, \(\Omega\) is derived from gap size statistics or gap fraction, as measured by many ground based optical instruments, such as the Tracing Radiation and Architecture of Canopies (TRAC) system [18] or fish-eye photographs [24]. In situ measured \(\Omega\) can accurately depict the site-specific \(\Omega\) variation, but lack efficiency. Nevertheless, it is risky to extrapolate the values or relationships beyond the site. In operational retrieval of MODIS LAI product, the algorithm assigns a single clumping factor to each biome type [25]. This is the most simplified method but it guarantees the robustness. [PERSON] et al. [4] developed a satellite level regression model for hemispherical average clumping index from an angular index that is called the normalized difference between hotspot and dark spot (NDHD), which made it possible to retrieve global time series average clumping index [9; 26; 27]. The NDHD method balances the need for computational efficiency while including the pixel specific information. However, the directional variation of clumping index has not been included. The angular variation of directional clumping index (\(\Omega(\theta)\)) contains abundant vegetation structure information [28]. Discontinuous crop and forest are two representative examples. The directional variation pattern of the clumping effect reflects various vegetation canopy morphologies [29; 30]. Furthermore, the directionality of clumping index is necessary to be taken into account in vegetation monitoring for two reasons. One reason is the observed clumping varies along with the view direction, which results in the uncertainty in using single clumping index value. Another reason is the accuracy of \(\Omega\) estimation has an apparent influence on LAI and Gross Primary Production (GPP). \(\Omega\) estimation error directly and linearly affects LAI retrievals. Normally, LAI measurements from optical instruments and sensors are referred to as effective LAI (\(LAI_{c}\)), the product of \(\Omega\) and actual LAI (\(LAI\)) (half the total leaf area) (\(LAI_{c}=\Omega\)_-\(LAI\))_. [PERSON] et al. [31] tested LAI retrieval sensitivity while using digital hemispherical photography over 10 crop types. The clumping index induced difference in LAI estimation at 30-m scale was as high as 31% (relative root-mean-square error or relative RMSE). [PERSON] [14] suggested that the accurate estimation of clumping index for each zenith angle will increase LAI estimation accuracy. [PERSON] et al.