# Modeling the Directional Clumping Index of Crop and Forest
## 2 Modeling and Evaluation
### Analytical Method for Crop and Forest  
#### 2.1.1 Crop Model  
We regard the crop as clumps of vegetation elements at the canopy level [37]. The vegetation elements are randomly distributed within hedgerows [41]. Following the Beer's law, the average gap fraction of the vegetation canopy is a function of view zenith angle and its structural parameters, including clumping index, LAI, and leaf angle distribution [42],  
\[\overline{p(\theta,\varphi)}=\exp(-\Omega(\theta)\ \frac{G_{\theta}}{\mu}\ LAI_{a}) \tag{1}\]  
where \(\overline{p(\theta,\varphi)}\) is the gap fraction; \(\Omega(\theta)\) refers to the directional clumping index; \(G_{\theta}\) is a parameter to describe canopy structure; \(\mu\) is the cosine of zenith angle \(\theta\); and, \(LAI_{a}\) denotes the spatially-averaged leaf area index.  
The average gap fraction of hedgerow crop canopy can be estimated using [PERSON] et al.'s algorithm [41].  
\[\overline{p(\theta,\varphi)}=\frac{1}{A}\left[\begin{array}{c}\left(A_{1}-H \ \tan\theta-2\ \frac{\sin\theta}{\cos\theta}\ u\right)\ e^{-\frac{G_{\theta}\ u}{\cos \theta}}\\ +\left(A_{2}-H\ \tan\theta+2\ \frac{\sin\theta}{\cos\theta}\ u\right)\end{array}\right] \tag{2}\]  
where \(A\) is a row unit consisting of canopy width (\(A_{1}=a_{1}/\sin\ \varphi\)) and row spacing (\(A_{2}=a_{2}/\sin\ \varphi\)) at a relative azimuth angle of \(\varphi\) with the row direction; \(a_{1}\) and \(a_{2}\) represents the canopy width and row spacing that vertical to the row direction. \(H\) is the height of the crop. Then, \(\Omega(\theta)\) can be calculated from the average gap fraction, \(\overline{p(\theta,\varphi)}\), i.e.,  
\[\Omega(\theta)=-\frac{\mu\ \ln\left(\overline{p(\theta,\varphi)}\right)}{G_{ \theta}\ LAI_{a}} \tag{3}\]  
where the equivalent LAI (\(LAI_{a}\)), includes the influence of foliage and non-foliage elements.  
[PERSON] et al. [16; 21] proposed that the canopy clumping index is a function of path length through the canopy. More specifically, it is a function of the effective path length in proportion to the encountered vegetation elements, as it is related to the energy attenuation in the canopy. For crop canopy, this effective path length does not count the row gap. Therefore, the maximum effective path length shows ups and downs while penetrating more rows and row gaps at increasing zenith angles, which is illustrated as the line A-E in Figure 1.  
Accordingly, the variation of \(\Omega(\theta)\) with zenith angle can be pinpointed while using the local feature angles, \(\theta_{ci}\), corresponding to the peaks of the \(\Omega(\theta)\) curve with zenith angle (Figure 2). The feature angles are determined by the row structure,  
\[\theta_{ci}=\tan^{-1}\left(\frac{iA}{H}\right) \tag{4}\] \[\theta_{di}=\tan^{-1}\left(\frac{iA+A_{1}}{H}\right)\]where \(i=1,\,2,\,3,\,\ldots\,\,i\) denotes the number of rows that photons penetrated through in the crop canopy; and \(A=A_{1}+A_{2}=a_{1}/\sin\varphi+a_{2}/\sin\varphi\). The subscript c and d corresponds to the local peak and valley value.  
\(\Omega(\theta)\) increases along with zenith angle before reaching the first feature angle (Figure 2). In practice, if the fluctuation (normally \(<0.05\)) can be tolerated for \(\Omega(\theta)\), we can regard \(\Omega(\theta)\) as approximately stable after the first turning point \(\theta_{c}\). Therefore, \(\theta_{c}\) can be used to predict the main \(\Omega(\theta)\) trend, which is regarded as the range angle of \(\Omega(\theta)\),  
\[\theta_{c}=\tan^{-1}\biggl{(}\frac{A}{H}\biggr{)} \tag{5}\]