## 2 Description of USTARFM
### Unmixing Data  
Mixed pixel decomposition based on the linear spectral mixture model is a popular method for estimating land cover fraction [15]. The surface reflectance of mixed pixels can be calculated as the sum of mean reflectance values of the different land-cover classes within the pixel, weighted by the corresponding abundance [33]. The abundance of classes and surface reflectance within coarse pixels (usually in window size) as input parameters can be solved with the ordinary least squares technique, producing the mean surface reflectance value (\(\overline{r}\)) for each land cover component as shown in Equation (1). Then mean reflectance values are assigned to each fine-resolution pixel of the classification map according to the class labels.  
The spectral unmixing model is sensitive to the co-linearity caused by high correlations between endmembers, leading to inversion of ill-posed matrices [34]. This problem could be decreased by increasing the number of equations through enlarging the search window to adopt more classes, _i.e._, increasing \(n\) in Equation (1) [22]. Furthermore, due to the existence of spatial heterogeneity, the solution of Equation (1) should also be conducted in a rectangular window, which can preserve spatial heterogeneity to the extent [31]. The oversize window scale containing too many pixels to make the model weakens spatial variation and leads to degradation in the accuracy of mixed pixel decomposition [1; 31]. Hence, an appropriate window size is crucial to the decision of the coarse pixels decomposition. A set of different window sizes were tested to determine the optimal window size, and the optimal window size was measured with three metrics: the correlation coefficient (\(\gamma\)), Root Mean Square Error (RMSE), and the Erreur Relative Globale Adimensionalle de Synthese (ERGAS) between the synthetic image and the reference fine-resolution image [1; 35; 31]:  
\[\left[\begin{array}{c}R(1,t)\\ \ldots\\ R(i,t)\\ \ldots\\ R(n,t)\end{array}\right]=\left[\begin{array}{cccc}f_{c}(1,1)&\ldots&f_{c}(1,c )&\ldots&f_{c}(1,k)\\ \ldots&\cdot&\cdot&\cdot&\ldots\\ f_{c}(i,1)&\ldots&f_{c}\left(i,c\right)&\ldots&f_{c}\left(i,k\right)\\ \ldots&\cdot&\cdot&\cdot&\ldots\\ f_{c}(n,1)&\ldots&f_{c}\left(n,c\right)&\ldots&f_{c}\left(n,k\right)\end{array} \right]\left[\begin{array}{c}\overline{r}(1,t)\\ \ldots\\ \overline{r}\left(c,t\right)\\ \ldots\\ \overline{r}\left(k,t\right)\end{array}\right]+\left[\begin{array}{c} \xi(1,t)\\ \ldots\\ \xi\left(c,t\right)\\ \ldots\\ \xi\left(k,t\right)\end{array}\right] \tag{1}\] \[\text{constraints}:\sum\limits_{c=0}^{k}f_{c}(i,c)=1;\,0<\overline {r}(c,t)<1\]  
where \(R\left(i,t\right)\) is the reflectance of coarse-pixel \(i\) at time \(t,f_{c}(i,c)\) is the abundance of class \(c\) in the \(i_{th}\) coarse pixel, \(\overline{r}\left(c,t\right)\) is the mean reflectance of class \(c\) at time \(t,\xi\left(i,t\right)\) is the residual error term, and \(k\) is the number of classes, \(n\) is the total coarse pixels number within the predefined window.  
Apparently, the class number and the window size in Equation (1) are key parameters for the unmixing-based method. Therefore, by varying the class number (\(k\)) and the sliding window size (\(W\)), a series of unmixing data can be obtained and assessed by the metrics of quantative assessment indices. The optimal combination (\(k,W\)) of class number and window size was finally determined to unmix the MODIS scale resolution images [1].