# Modeling the Directional Clumping Index of Crop and Forest
## 3 Results and Evaluation  
### Crop Directional Clumping Index Validation  
The variation of \(\Omega(\theta,\varphi)\) with View Azimuth Angle (\(\varphi\)) under different \(\theta\) is illustrated in Figure 5. The differences between the modeled and simulated results (Figure 6) were randomly distributed with  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Column ID** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline Height of & & & & & & 0.025, & \\ trunk (m) & 0.5 & 5 & 0.5 & 0.5 & \(\begin{array}{c}0,0.25,\\ \ldots,1.75\end{array}\) & 0.5 \\ \hline Crown & & & & & & 2.07, 2.76, 3.62, \\ depth (m) & 6.5 & 1.5 & 2.28 & 2.28 & 2.28 & 4.38, 5.09, 5.75, \\  & & & & & 6.37, 6.96 \\ \hline Crown & & & & & & 0.80, 0.69, 0.60, \\ radius (m) & 0.45 & 0.75 & 0.76 & 0.76 & 0.76 & 0.55, 0.51, 0.48, \\  & & & & & 0.45, 0.43 \\ \hline H/2R ([-]) & 7.22 & 1 & 1.5 & 1.5 & 1.5 & \(\begin{array}{c}1.3,2,3,4,5,6,7,\\ 8\end{array}\) \\ \hline Radius at & & & & & & \\ breast height (m) & 0.16 & 0.15 & 0.16 & 0.16 & 0.16 \\ \hline G\({}_{\theta}\) ([-]) & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 \\ \hline Average LAI & & & & & 0.28, 0.84, & \\ ([-]) & 4.5 & 2 & & 1.40, 1.97, & 2.53, 3.09, & 2.25 & 2.25 \\  & & & & 3.65, 4.22 & & \\ \hline Area of & & & & & & \\ research & 1 & 5 & 4 & 4 & 4 \\ field (ha) & & & & & & \\ \hline Distribution & \begin{tabular}{c} [PERSON] \\ \(m_{2}=4\) \\ \end{tabular} & [PERSON] & \begin{tabular}{c} [PERSON] \\ \(m_{2}=4\) \\ \end{tabular} & \
\begin{tabular}{c} [PERSON], \\ [PERSON] \\ \(3,5,7,10,15,20\) \\ \end{tabular} & [PERSON] \(m_{2}=4\) & [PERSON] \(m_{2}=4\) \\ \hline Tree Number & 4000 & 1011 &
\begin{tabular}{c} 250,750, \\ \ldots,3750 \\ \end{tabular} & 2000 & 2000 & 2000 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Forest structure parameters used in the current paper.  
Figure 4: Upward-looking hemispherical photograph and corresponding visual (360\({}^{\circ}\)) view of Larch stand.  
an average of 0, and maximum relative discrepancy <1.5%. The MC simulated and algebraic \(\Omega(\theta,\varphi)\) are strongly consistent, which provides confidence to analyze crop \(\Omega(\theta,\varphi)\) variation while using the algebraic model.  
The different shapes of \(\Omega(\theta,\varphi)-\varphi\) curves reflect the effect of the range angle \(\theta_{c}\) of \(\Omega(\theta,\varphi)\) pattern. When \(\varphi\) increases from 30\({}^{\circ}\) to 90\({}^{\circ}\) (perpendicular to the row direction), row period width \(A\) decreases, thus \(\theta_{c}\) grows; when \(\varphi\) increases from 90\({}^{\circ}\) to 150\({}^{\circ}\), \(A\) increases, then \(\theta_{c}\) drops. When the \(\varphi\) is 0\({}^{\circ}\) or 180\({}^{\circ}\), i.e., along with the row direction, the gap fraction of row plant is obviously larger than other directions as the row gap always occupies a fraction of the field of view. The \(\Omega(\theta,\varphi)\) decreases with the growing view zenith angle, which means the crop departures more away from the random status. On the contrary, when the azimuth angle is larger, the \(\Omega(\theta,\varphi)\) starts growing with view zenith angle and reaches a stable value around one after the view zenith angle higher than the range angle. When the azimuth angle reaches 90\({}^{\circ}\), the ratio of equivalent row width to row height gets its minimum value so that the range angle is smaller than other azimuth directions. These variation trends verify the patterns of crop \(\Omega(\theta,\varphi)\) in Section 2.1.1 and they show the crop \(\Omega(\theta,\varphi)\) dispersion at different view angles from constant status.  
Figure 5: Crop clumping index at specific azimuth angles for different zenith angles (different curves). Other simulation parameters: \(a_{1}=0.6,\ a_{2}=0.4,\ H=1\), \(LAI=2\), \(r_{g}^{h}=0.18,\ \rho_{1}=T_{1}=0.05\).  
Figure 6: Relative discrepancy between simulated and modeled crop clumping index for different zenith angles at specific azimuth angles.