# Modeling the Directional Clumping Index of Crop and Forest
## 2 Modeling and Evaluation
### Analytical Method for Crop and Forest
#### 2.1.2 Forest Model  
The derivation of forest \(\Omega(\theta)\) uses the average gap fraction (3), which is referred to [PERSON]'s model [PERSON] (1966). We modelled the tree crown as vertical ellipsoid for three reasons. First, the vertical ellipsoid is axially symmetrical, which accords with our assumption that the variation of clumping index along with azimuth angle is not considered. Second, the crowns of different species can be discriminated though the ratio between crown depth and crown width, which is related to the variation of the  
Figure 1: Illustration of effective path length variation with view zenith angles.  
Figure 2: Crop \(\Omega(\theta)\) along with zenith angle for different leaf area index (LAI). Other simulation parameters: \(a_{1}=0.2,\,\,a_{2}=0.3,\,\,H=1,\,\,r_{g}^{h}=0.22,\,\,\rho=0.0999,\,\,T=0.1303\).  
clumping index along with zenith angle. Third, the ellipsoid can be easily parameterized using crown depth and crown width in the model.  
For randomly distributed forest, the gap fraction at canopy level can be expressed as  
\[p_{\theta}=e^{-n\pi R^{2}\sqrt{1+\left(\frac{h}{2R}\right)^{2}\tan^{2}\theta} \left(1-p_{1}(\theta)\right)} \tag{6}\]  
where \(p_{i}(\theta)\) is the intra-crown directional gap fraction of the tree, \(h/2R\) is a crown shape factor that discriminates different trees types or species, and \(n\pi R^{2}\) is the crown closure where the overlapped regions of crown projections have been multiply counted, i.e., \(C_{CR}=n\pi R^{2}\). Combining (3) and (4),  
\[\Omega(\theta)=\frac{C_{CR}\sin\theta\sqrt{\tan^{-2}\theta+\left(\frac{h}{2R} \right)^{2}}\left(1-p_{1}(\theta)\right)}{G\ LAI_{\mathrm{a}}} \tag{7}\]  
However, (6) and (7) do not consider the trunk, which can affect optical measurements for some high trees with tall or wide trunks [43]. The average gap fraction at canopy level, incorporating trunk effects, can be expressed as  
\[p_{\theta}=e^{-n\left(\pi R^{2}\sqrt{1+\left(\frac{h}{2R}\right)^{2}\tan^{2} \theta}+d_{t}\ h_{t}\ \tan\theta\right)\ \left(1-p_{1}(\theta)\right)} \tag{8}\]  
and the directional \(\Omega(\theta)\) can be expressed as  
\[\Omega(\theta)=\frac{C_{CR}\sin\theta\bigg{(}\sqrt{\tan^{-2}\theta+\left( \frac{h}{2R}\right)^{2}}+\frac{d_{t}\ h_{t}}{\pi R^{2}}\bigg{)}(1-p_{1}(\theta ))}{G\ LAI_{\mathrm{a}}} \tag{9}\]  
Alternatively, if we count the overlapped regions of crown projections only once in the canopy closure calculation, then \(C_{CA}=1-e^{-n\ \pi R^{2}}\), since it is equivalent to the canopy gap fraction at the nadir direction. Then, the canopy average gap fraction and \(\Omega(\theta)\) can be expressed as, respectively,  
\[p_{\theta}=(1-C_{CA})^{\sqrt{1+\left(\frac{h}{2R}\right)^{2}\tan^{2}\theta} \left(1-p_{1}(\theta)\right)} \tag{10}\]  
and  
\[\Omega(\theta)=-\frac{\ln(1-C_{CA})\ \sin\theta\sqrt{\tan^{-2}\theta+\left( \frac{h}{2R}\right)^{2}}\left(1-p_{1}(\theta)\right)}{G\ LAI_{\mathrm{a}}} \tag{11}\]  
if considering the tree trunk,  
\[p_{\theta}=(1-C_{CA})^{\left(\sqrt{1+\left(\frac{h}{2R}\right)^{2}\tan^{2} \theta}+\frac{d_{t}h_{t}}{\pi R^{2}}\ \tan\theta\right)\ \left(1-p_{1}(\theta)\right)} \tag{12}\]  
and  
\[\Omega(\theta)=-\frac{\ln(1-C_{CA})\sin\theta\bigg{(}\sqrt{\tan^{-2}\theta+ \left(\frac{h}{2R}\right)^{2}}+\frac{d_{t}h_{t}}{\pi R^{2}}\bigg{)}\left(1-p_ {1}(\theta)\right)}{G\ LAI_{\mathrm{a}}} \tag{13}\]  
Equations (7) and (9) are \(\Omega(\theta)\) expressions for multiple counted canopy closure, and Equations (11) and (13) are \(\Omega(\theta)\) expressions for once counted canopy closure.