## 3 Methods  
### Model Introduction  
#### 3.1.1 ESTARFM  
ESTARFM was proposed to improve the STARFM algorithm for the accurate prediction of surface reflectance in heterogeneous landscapes using the observed reflectance trend between two points in time and spectral unmixing theory [38]. According to the linear mixture model, the changes in coarse-resolution reflectance from \(t\)0 to _t_k can be expressed as:  
\[R\big{(}t_{k}\big{)}-R\big{(}t_{0}\big{)}=\sum_{i=1}^{M}\frac{f_{i}}{a}\big{(} r\big{(}t_{k}\big{)}-r\big{(}t_{0}\big{)}\big{)} \tag{1}\]  
where \(R\big{(}t_{k}\big{)}\) and \(R\big{(}t_{0}\big{)}\) are the coarse-resolution reflectances from time \(t_{0}\) to \(t_{k}\), \(r\big{(}t_{k}\big{)}\) and \(r\big{(}t_{0}\big{)}\) are the fine-resolution reflectances from time \(t_{0}\) to \(t_{k}\), \(M\) is the total number of endmembers, \(f_{i}\) is the fraction of the \(i\)-th land type and \(a\) is the sensor calibration coefficient. ESTARFM assumes that thechange in the reflectance of each land type is linear with time:  
\[r\left(t_{k}\right)=h_{k}\times\Delta t+r\left(t_{0}\right) \tag{2}\]  
where \(\Delta t=t_{k}-t_{0}\) and \(h_{k}\) is the rate of change of the \(k\)-th endmember, which was assumed to be constant from time \(t_{0}\) to \(t_{k}\). Then, Equations (1) and (2) can be rewritten as:  
\[\Delta t=\big{(}r\left(t_{k}\right)-r\big{(}t_{0}\big{)}\big{)} \tag{3}\]  
\[R\left(t_{k}\right)-R\left(t_{0}\right)=\Delta t\sum_{i=1}^{M}\frac{f_{i}h_{i }}{a} \tag{4}\]  
Substituting Equation (3) into Equation (4), the ratio,\(v_{k}\), of the change in reflectance for the \(k\)-th endmember to the change in reflectance for a coarse pixel can be described as:  
\[\frac{r\left(t_{k}\right)-r\left(t_{0}\right)}{R\left(t_{k}\right)-R\left(t_ {0}\right)}=\frac{h_{k}}{\sum_{i=1}^{M}\frac{f_{i}h_{i}}{a}}=v_{k} \tag{5}\]Equation (5) can be rewritten as:  
\[r\big{(}x,y,t_{{}_{k}}\big{)}=r\big{(}x,y,t_{{}_{0}}\big{)}+v_{{}_{k}}\times \big{(}R\big{(}x,y,t_{{}_{k}}\big{)}-R\big{(}x,y,t_{{}_{0}}\big{)}\big{)} \tag{6}\]  
where (_x_,_y_) is the position of the target pixels.  
By introducing additional information from the neighbouringpixels to reduce the influence of land cover changes, surface heterogeneity and solar geometrybi-directional reflectance distribution function (BRDF) changes, a weighted ESTARFM can be determined as Equation (7):  
\[r(x,y,t_{{}_{k}})=r(x,y,t_{{}_{k}})+\sum\limits_{{}_{i=1}}^{w}\sum\limits_{{}_{ j=1}}^{w}\sum\limits_{{}_{k=1}}^{w}W_{{}_{ijk}}\times v_{{}_{ijk}}\big{(}R \big{(}x,y,t_{{}_{k}}\big{)}-R\big{(}x,y,t_{{}_{k}}\big{)}\big{)} \tag{7}\]  
where \(w\) is the size of the search window; \(W_{{}_{ijk}}\) is the weight determined by the spectral difference \(S_{{}_{ijk}}\) and the temporal difference \(T_{{}_{ijk}}\) between the fine- and low-resolution data, and by the location distance \(D_{{}_{ijk}}\) between the target pixel and the candidate pixel; \(k\) is the number of pixels (_x_,_y_) in window \(w\). These parameters are calculated as follows:  
\[W_{{}_{i}}=\big{(}1\big{/}D_{{}_{i}}\big{)}\bigg{/}\sum\limits_{{}_{i=1}}^{n} \big{(}1\big{/}D_{{}_{i}}\big{)} \tag{8}\]  
\[D_{{}_{i}}=\big{(}1+S_{{}_{i}}\big{)}\times d_{{}_{i}} \tag{9}\]  
\[d_{{}_{i}}=1+\sqrt{\big{(}x-x_{{}_{i}}\big{)}^{2}+\big{(}y-y_{{}_{i}}\big{)}^ {2}}\ /\big{(}w\,/\,2\big{)} \tag{10}\]  
\[S_{{}_{i}}=\frac{E\big{[}\big{(}r_{{}_{i}}-E\big{(}r_{{}_{i}}\big{)}\big{)} \times\big{(}R_{{}_{i}}-E\big{(}R_{{}_{i}}\big{)}\big{)}\big{]}}{\sqrt{D\big{(} r_{{}_{i}}\big{)}\times\sqrt{D\big{(}R_{{}_{i}}\big{)}}}} \tag{11}\]