## 2 Method
### Theoretical Basis of EFSDAF
#### 2.2.4 Residual Distribution by Using a New Residual Index (\(RI\))  
From Equation (11), the predicted Landsat-like image is calculated by adding a reasonable residual based on the temporal prediction. Therefore, the residual distribution is the key step for the final prediction. In FSDAF, a homogeneity index (\(HI\)) derived from the classification map of the Landsat image at \(T_{B}\) was introduced to guide the residual distribution. Because the \(HI\) only depends on the classification map at \(T_{B}\), it will be unreasonable when the land cover type has changed from \(T_{B}\) to \(T_{P}\) or the classification map is wrong [33]. Hence, a new residual index (\(RI\)) is proposed to reasonably guide the residual distribution in this study, and more residual will be assigned to the area of land cover change. According to Section 2.2.3, the spatial variation \(\Delta F_{BP}\Big{(}x_{ij},y_{ij},b\Big{)}\) can be expressed as:  
\[\Delta F_{BP}\Big{(}x_{ij},y_{ij},b\Big{)}=\Big{|}F_{P}^{SP}\Big{(}x_{ij},y_{ ij},b\Big{)}-F_{B}^{SP}\Big{(}x_{ij},y_{ij},b\Big{)}\Big{|} \tag{15}\]  
where \(\Delta F_{BP}\Big{(}x_{ij},y_{ij},b\Big{)}\) defines the changes of interpolation results of MODIS images from \(T_{B}\) to \(T_{P}\), and the larger \(\Delta F_{BP}\Big{(}x_{ij},y_{ij},b\Big{)}\), the larger the changes of land cover type. In EFSDAF, we assume the changes of land cover type occur in a small area. If \(\Delta F_{BP}\big{(}x_{ij},y_{ij},b\big{)}\) in one Landsat pixel is greater than the average of the \(\Delta F_{BP}\), and we think the Landsat pixel \(\big{(}x_{ij},y_{ij}\big{)}\) is the area of land cover change. The \(RI\) can be calculated as follows:  
\[RI\big{(}x_{ij},y_{ij},b\big{)}=\frac{\sum\limits_{k=1}^{n}I_{k}}{n_{w}} \tag{16}\]  
where \(I_{k}\) is equal to 1 when the \(k\)-th Landsat pixels is determined to the pixel of land cover change within a sliding window; otherwise, \(I_{k}\) is equal to 0. \(n_{w}\) defines the window size. The \(RI\) is the ratio of the numbers of \(I_{k}=1\) in the window to \(n_{w}\). The range of \(RI\) is 0-1, and a larger \(RI\) indicates more residual to the area of land cover change.  
Then we use the same weighting function to distribute the final residual as FSDAF. The weighting function is defined as follow:  
\[CW\big{(}x_{ij},y_{ij},b\big{)}=E_{hc}\big{(}x_{ij},y_{ij},b\big{)}\times RI \big{(}x_{ij},y_{ij},b\big{)}+E_{ho}\big{(}x_{ij},y_{ij},b\big{)}\times\Big{(} 1-RI\big{(}x_{ij},y_{ij},b\big{)}\Big{)} \tag{17}\]  
where \(E_{hc}\big{(}x_{ij},y_{ij},b\big{)}=R(x_{i},y_{i},b)\), and \(E_{hc}\big{(}x_{ij},y_{ij},b\big{)}\) is the residual for the area of land cover change. \(E_{ho}\big{(}x_{ij},y_{ij},b\big{)}=F_{P}^{SP}\big{(}x_{ij},y_{ij},b\big{)}-F ^{TP}\big{(}x_{ij},y_{ij},b\big{)}\), and \(E_{ho}\big{(}x_{ij},y_{ij},b\big{)}\) is the residual for area of no land cover change.  
The final residual \(\big{(}x_{ij},y_{ij},b\big{)}\) can be calculated as follows:  
\[r\Big{(}x_{ij},y_{ij},b\big{)}=s^{2}\times R(x_{i},y_{i},b)\times W\big{(}x_{ ij},y_{ij},b\big{)} \tag{18}\]  
where \(W\big{(}x_{ij},y_{ij},b\big{)}\) is the normalized \(CW\big{(}x_{ij},y_{ij},b\big{)}\).  
According to Equations (10) and (11), the final changes of the Landsat images between \(T_{B}\) and \(T_{P}\) can be obtained from Equation (19):  
\[\Delta F_{f}\big{(}x_{ij},y_{ij},b\big{)}=r\big{(}x_{ij},y_{ij},b\big{)}+ \Delta F\big{(}x_{ij},y_{ij},b\big{)} \tag{19}\]  
where \(\Delta F_{f}\big{(}x_{ij},y_{ij},b\big{)}\) and \(\Delta F\big{(}x_{ij},y_{ij},b\big{)}\) are the final changes of Landsat images and the changes of temporal prediction from \(T_{B}\) to \(T_{P}\), respectively.