## 4 Principle Laws
### Linear Mixing Model  
The linear mixing model can be used to link the pixels in the coarse image and fine image by the equation below:  
\[C(x_{i},y_{i},b)=\frac{1}{m}\sum_{j=1}^{m}F(x_{ij},y_{ij},b)+\varepsilon \tag{1}\]  
For each coarse pixel, its pixel value is equal to the average of all internal fine pixels adding a bias factor \(\varepsilon\) which is the systematic difference between two sensors. It can be caused by difference in bandwidth, solar geometry, and viewing angles [11]. The basic assumption for the linear mixing model is that the variable is linearly additive. For optical satellite images, the raw images record the radiance, and is often converted to reflectance for end users. Both radiance and reflectance of individual spectral bands are linearly additive. However, some higher-level products derived from the raw data are not linearly additive, such as some vegetation indices (e.g., NDVI, EVI) and land surface temperature. Existing studies show that using linear mixing model to link these products at different scales may not introduce significant errors [72; 73], so some existing methods also use linear mixing model for fusing these high-level products, such as the NDVI-LMGM model for fusing MODIS and Landsat NDVI products [43], and the SADFAT for fusing LST products [21]. Although linear mixing model is the start point of many spatiotemporal fusion methods, it is used in two fashions. In the unmixing-based method [57; 62; 63], the internal fine pixels are aggregated into several land cover classes. Then, Equation (1) can be rewritten as:  
\[C(x_{i},y_{i},b)=\sum_{c=1}^{l}f_{c}(x_{i},y_{i})\times F(c,b) \tag{2}\]  
In this equation, fraction of land cover types \(f_{c}(x_{i},y_{i})\) can be estimated from the classification map of the given fine image at t\({}_{1}\) or any fine-scale land cover maps, and this fraction is assumed to be stable across different times. The endmember's spectral values of each land cover \(F(c,b)\) can be estimated through solving the linear equation system composed by multiple coarse pixels:  
\[\left[ \begin{array}{c}C(x_{1},y_{1},b)\\ \vdots\\ C(x_{i},y_{i},b)\\ \vdots\\ C(x_{n},y_{n},b)\end{array} \right]=\left[ \begin{array}{cccc}f_{1}(x_{1},y_{1 })&f_{2}(x_{1},y_{1})&\cdots&f_{l}(x_{1},y_{1})\\ \vdots&\vdots&&\vdots\\ f_{1}(x_{i},y_{i})&f_{2}(x_{i},y_{i})&\cdots&f_{l}(x_{i},y_{i})\\ \vdots&\vdots&&\vdots\\ f_{1}(x_{n},y_{n})&f_{2}(x_{n},y_{n})&\cdots&f_{l}(x_{n},y_{n})\end{array}  \right]\left[ \begin{array}{c}F(1,b)\\ \vdots\\ F(c,b)\\ \vdots\\ F(l,b)\end{array} \right] \tag{3}\]  
All the pixel values of coarse and fine images in Equations (1)-(3) can be replaced by the pixel value change between \(t_{1}\) and \(t_{2}\). As a result, the temporal change at fine-scale \(\Delta F(c,b)\) can be estimated by solving the linear equation system. For the unmixing-based methods, their efforts are input for how to select more appropriate coarse pixels to compose the linear equation system, how to improve the accuracy of solving the equation system, how to distribute the solved endmember values to fine pixels, how to compensate the residuals and how to add the inner-class variations to the fused results. Linear mixing model is also the basis of many weighted function-based methods. Unlike the unmixing-based methods, weighted function-based methods do not solve the linear mixing equation system. They use further assumptions to simplify the mixing equation, so that the temporal change of the internal fine pixels can be directly estimated from the corresponding coarse pixel. For example, STARFM just simply assumes that the temporal change of all internal fine pixels is equal to the corresponding coarse pixel:  
\[F_{2}(x_{ij},y_{ij},b)-F_{1}(x_{ij},y_{ij},b)=C_{2}(x_{i},y_{i},b)-C_{1}(x_{i },y_{i},b) \tag{4}\]Equation (4) is a specific case of Equation (1) if we assume that all internal fine pixels have same temporal change. This assumption is only valid when all internal fine pixels are from same land cover type, i.e., the corresponding coarse pixel is \"pure\" pixel.