## 3 Method
### Linear Spectral Unmixing Using UALR  
For the MODIS images, each pixel was considered as a mixture of two endmembers, namely cropped and uncropped arable land. It was assumed that reflectance of cropped arable land and uncropped arable land is linearly mixed based on the proportions of each type. Only red band and near-infrared band reflectance was used for the spectral unmixing analysis. The red band and near-infrared band reflectance of cropped arable land for each pixel was found by solving the equations:  
\[ \begin{cases}\rho_{\text{R}}=\rho_{\text{R}_{\text{cropped}}}*(1-\text{UALR})+\rho_{\text{R}_{\text{ uncropped}}}*\text{UALR}\\ \rho_{\text{NIR}}=\rho_{\text{NIR}_{\text{cropped}}}*(1-\text{UALR})+\rho_{ \text{NIR}_{\text{uncropped}}}*\text{UALR}\end{cases}  \tag{2}\]  
where \(\rho_{\text{R}}\) and \(\rho_{\text{NIR}}\) are the reflectance of the red band and near-infrared band in a given MODIS pixel, UALR is the ratio of uncropped arable land in the pixel and \(\rho_{\text{R}_{\text{cropped}}}/\rho_{\text{NIR}_{\text{cropped}}}\) and \(\rho_{\text{R}_{\text{uncropped}}}/\rho_{\text{NIR}_{\text{uncropped}}}\) are the red band and near-infrared band reflectance of the endmembers. The average reflectance for red and near-infrared bands at each acquisition date of MODIS was calculated over the pixels for which UALR was higher than 98% to represent \(\rho_{\text{R}_{\text{uncropped}}}/\rho_{\text{NIR}_{\text{uncropped}}}\). Since Inceptisols is the dominant soil order in the North China Plain [29], it is assumed that the band reflectance of uncropped arable land is relatively constant throughout the study area. Figure 3: The decision tree for cropped and uncropped arable land mapping. Using Equation (3), \(\rho_{\text{R}\text{cropped}}\) and \(\rho_{\text{NIR}\text{cropped}}\) can be derived as follows:  
\[\begin{cases}\rho_{\text{R}\text{cropped}}=\dfrac{\rho_{\text{R}}-\rho_{\text{ R}\text{n}\text{cropped}}\text{+}\text{U}\text{A}\text{L}\text{L}\text{R}}{1-\text{U} \text{A}\text{L}\text{R}}\\ \rho_{\text{NIR}\text{cropped}}=  
land over the study area were uncropped in the winter crop growing season in 2010 and 2011, respectively. Most of the uncropped arable land is located in the north of the study region, specifically in Hebei and Shandong provinces. More uncropped arable land was observed along the west coast areas of Bohai Bay in 2011 than 2010, while less arable land was uncropped in southern Hebei in 2011 than 2010. Uncropped arable land during the winter crop growing season in southern Hebei and the neighboring areas in Shandong (red circle in Figure 4) was mainly kept for cotton and ground nut (single cropping system) cultivation during the summer. Salinization and alkalization of farmland soil are the major problems for the crops in the west coastal region of Bohai Gulf [43]. The uncropped arable land area expanded in 2011 compared to 2010 in the Yellow River delta. Only a small percentage of arable land in Henan, Anhui and Jiangsu provinces was uncropped for both 2010 and 2011. The error matrix and classification accuracy for cropped and uncropped arable land in 2010 and 2011 is shown in Tables 3 and 4. Both producer's and user's accuracy exceeded 97% in the two years, except for a user's accuracy of 92.91% for the \"cropped\" type in 2010. The overall classification accuracy was 97.81% for 2010 and 99.85% for 2011; the kappa coefficients were 0.9485 and 0.9810, which signifies that the classified cropped and uncropped arable land map was in agreement with the ground-truth data. As shown in Table 3, about 3% of uncropped pixels verified on the ground were misclassified as cropped, whereas almost 7% of cropped pixels were misclassified as uncropped pixels in 2010. For 2011, both omission and commission errors were less than 3% for uncropped and cropped. The commission error for cropped in 2010 was higher mainly due to cloudy conditions on some images. Misclassified pixels usually occurred at the intersection of different HJ-1 CCD images.