## 2 Method
### Theoretical Basis of EFSDAF
#### 2.2.2 Temporal Prediction (\(F^{TP}\) ) for No Land Cover Change from \(T_{B}\) to \(T_{p}\)  
* Temporal changes of each endmember at the MODIS pixels  
Figure 1: Flowchart of the proposed Enhanced Flexible Spatiotemporal DAta Fusion (EFSDAF) method. The temporal prediction of EFSDAF assumes no land cover change from \(T_{B}\) to \(T_{P}\). In other words, the endmembers and abundances of Landsat pixels will not change from \(T_{B}\) to \(T_{P}\). According to the linear mixture model [38], the values of Landsat pixels are a linear mixture of endmembers' values and abundances. Hence, the values of Landsat pixels at \(T_{B}\) and \(T_{P}\) can be expressed as:  
\[F_{B}\Big{(}x_{ij},y_{ij},b\Big{)}=\sum\limits_{m=1}^{n_{m}}A_{F}^{B}\Big{(}x_ {ij},y_{ij},m\Big{)}\times R_{F}^{B}(m,b)+\varphi \tag{1}\]  
\[F_{P}\Big{(}x_{ij},y_{ij},b\Big{)}=\sum\limits_{m=1}^{n_{m}}A_{F}^{p}\Big{(}x_ {ij},y_{ij},m\Big{)}\times R_{F}^{p}(m,b)+\varphi \tag{2}\]  
where \(n_{m}\) defines the number of endmembers and \(m\) is the \(m\)-th endmembers for one Landsat pixel, \(R_{F}^{B}(m,b)\) and \(R_{F}^{p}(m,b)\) denote the reflectance values of each endmember for all Landsat image bands at \(T_{B}\) and \(T_{P}\), respectively. \(\varphi\) is the system error. According to the above assumption, the endmembers and abundances of Landsat pixels will not change from \(T_{B}\) to \(T_{P}\), i.e., \(A_{F}^{B}=A_{F}^{P}\), and \(\varphi\) is constant. From Equations (1) and (2), we have:  
\[\Delta F\Big{(}x_{ij},y_{ij},b\Big{)}=\sum\limits_{m=1}^{n_{m}}A_{F}^{B}\Big{(} x_{ij},y_{ij},m\Big{)}\times\Delta R_{F}(m,b) \tag{3}\]  
where \(\Delta F\) denotes the temporal changes of the Landsat images. \(\Delta R_{F}\) are the reflectance changes of each endmember for all Landsat image bands. Among the variables above, if \(\Delta R_{F}\) is known, \(\Delta F\) can be calculated by a linear mixture model. Therefore, the key is to solve \(\Delta R_{F}\). Similar to the temporal changes of Landsat images, the temporal changes of MODIS images from \(T_{B}\) to \(T_{P}\) can be expressed as:  
\[\Delta C(x_{i},y_{i},b)=\sum\limits_{m=1}^{n_{m}}A_{C}^{B}(x_{i},y_{i},m) \times\Delta R_{C}(m,b). \tag{4}\]  
where \(\Delta C\) denotes the temporal changes of the MODIS images. \(\Delta R_{C}\) are the reflectance changes of each endmember for all MODIS image bands. Among the variables above, \(\Delta C\) could be calculated through \(\Delta C=C_{P}-C_{B}\). Due to the previous assumption of no land cover change from \(T_{B}\) to \(T_{P}\), the types and spatial distributions of the endmembers should be the same for Landsat and MODIS images in the same regions. Hence, \(A_{C}^{B}\) can be aggregated from \(A_{F}^{B}\) with:  
\[A_{C}^{B}(x_{i},y_{i},m)=\frac{1}{s^{2}}\sum\limits_{k=1}^{s2}A_{F}^{B}\Big{(} x_{ij},y_{ij},m\Big{)} \tag{5}\]  
where \(A_{C}^{B}(x_{i},y_{i},m)\) is calculated by averaging the endmember abundances of all Landsat pixels in one MODIS pixel. However,\(A_{C}^{B}(x_{i},y_{i},m)\) is the ratio of the number of Landsat pixels for class \(m\) to \(s^{2}\) in FSDAF. Although \(\Delta C(x_{i},y_{i},b)\) and \(A_{C}^{B}(x_{i},y_{i},m)\) from Equations (4) and (5) are known variables, Equation (4) cannot be solved because there are \(n_{m}\) unknown values of \(\Delta R_{C}(m,b)\). In this study, assuming that the land cover types are the same for a small area according to the first law of geography [39] stating that near things are more related than distant things, it is reasonable that \(\Delta R_{C}(m,b)\) are the same in a small area [20]. However, FSDAF assumes that the temporal changes of each class are the same among all MODIS pixels, which does not match the actual situation. A sliding window of MODIS pixels is introduced to perform spatial unmixing in this study, and the size of the window should be larger than the number of endmembers [40]. The equations [32] are as follows:  
\[\left\{ \begin{array}{c}\Delta C(x_{1},y_{1},b)\\ \vdots\\ \Delta C(x_{i},y_{i},b)\\ \vdots\\ \Delta C(x_{n},y_{n},b)\\ \end{array} \right\}=\left\{