## 3 Phenology-Based Crop Identification
### Crop Identification Model of Characteristic Ellipses
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Land Use Type** & **Indexes** & **2009** & **2010** & **2011** & **2012** & **2013** & **2014** & **2015** \\ \hline \multirow{4}{*}{Farmland} & NDVI\_max & 0.497 & 0.505 & 0.497 & 0.519 & 0.552 & 0.592 & 0.573 \\  & NDVI\_inf & 0.347 & 0.333 & 0.337 & 0.366 & 0.390 & 0.427 & 0.405 \\  & \(\Delta\)NDVI & 0.150 & 0.172 & 0.159 & 0.153 & 0.162 & 0.164 & 0.167 \\  & FGP & 39.23 & 35.00 & 32.96 & 33.47 & 31.42 & 27.36 & 32.88 \\ \hline \multirow{4}{*}{Grassland} & NDVI\_max & 0.472 & 0.467 & 0.453 & 0.487 & 0.508 & 0.547 & 0.537 \\  & NDVI\_inf & 0.333 & 0.312 & 0.310 & 0.345 & 0.361 & 0.398 & 0.387 \\  & \(\Delta\)NDVI & 0.139 & 0.153 & 0.143 & 0.143 & 0.147 & 0.149 & 0.150 \\  & FGP & 38.54 & 34.54 & 33.01 & 32.41 & 31.46 & 27.57 & 32.17 \\ \hline \multirow{4}{*}{Forest} & NDVI\_max & 0.481 & 0.476 & 0.466 & 0.496 & 0.519 & 0.554 & 0.542 \\  & NDVI\_inf & 0.339 & 0.315 & 0.320 & 0.351 & 0.368 & 0.403 & 0.387 \\  & \(\Delta\)NDVI & 0.142 & 0.161 & 0.146 & 0.145 & 0.151 & 0.151 & 0.154 \\ \cline{1-1}  & FGP & 38.74 & 34.82 & 32.49 & 32.18 & 31.35 & 27.61 & 32.65 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mean values of NDVI characteristics and phenological metrics from 2009 to 2015. Figure 6: Relationships between normalized NDVI characteristics and normalized phenological metrics selected for the identification of maize and sunflower (**(a1–c3)** correspond to classifiers in Table 5). From Figure 6, scatter points of normalized FGP vs. each normalized NDVI characteristic can be approximately covered by an ellipse. Our intention was to find out the minimum ellipse that can cover all sampling points. The standard equation of an ellipse with (0, 0) as the center and \(x\) and \(y\) axes as principal axes can be expressed as  
\[\frac{x^{2}}{a^{2}}\ +\ \frac{y^{2}}{b^{2}}\ =\ 1\ \ (a>b>0) \tag{11}\]  
where \(a\) and \(b\) are semi-major and semi-minor axes, respectively. When the ellipse center moves from (0, 0) to (\(m\), \(n\)) and the ellipse has a rotation angle of \(\theta\), the standard equation can be expressed as  
\[Ax^{2}+By^{2}+Cxy+Dx+Ey+F=0 \tag{12}\]  
where \(A\), \(B\), \(C\), \(D\), \(E\) and \(F\) are parameters, which can be calculated from \(m\), \(n\), \(a\), \(b\), and \(\theta\)[11]. The minimum ellipse was obtained by the least square method with the objective of minimizing the ellipse area on the condition that all points fell into the ellipse. Due to the limited number of sampling points, the minimum ellipse may not contain all the growth traits of maize or sunflower in the whole irrigation district. Therefore, ellipses should be amplified by increasing the semi-major and semi-minor axes to optimal values to achieve a smaller and acceptable relative error between identified crop planting areas and official statistical data. The relative area is defined as  
\[\delta=\frac{(\text{Calculated}_{\text{area}}-\text{Statistical}_{\text{area} })*100}{\text{Statistical}_{\text{area}}} \tag{13}\]  
where \(\text{Calculated}_{\text{area}}\) and \(\text{Statistical}_{\text{area}}\) are crop planting areas calculated from the classification model and from the official statistics, respectively. Furthermore, the Kappa coefficient was used to evaluate the consistency of crop spatial distribution between the identification results and the actual planting structure. The Kappa coefficient is calculated as [51]  
\[\kappa=\frac{P_{o}-P_{c}}{1-P_{c}} \tag{14}\]  
where \(P_{o}\) is the proportion of observed agreements and \(P_{c}\) is the proportion of agreements expected by chance. The years 2010, 2011 and 2013 were selected as training years, while 2009, 2012, 2014 and 2015 were the testing years. The amplification coefficients of \(a\) and \(b\) were defined as \(F_{a}\) and \(F_{b}\) and they were set to be 1.00 to 1.35 and 1.00 to 1.15, respectively, with steps of 0.01.