## 2 Materials and Methods
### Algorithm for Downscaling
#### 2.3.3. ATPK for Downscaling Residuals  
If the result of the first phase is perfect, there should be no bias between it and the original GPP at a 1 km resolution. However, it is unrealistic to obtain such an ideal result, and there are inevitable residuals from the first process. The residuals at a 1 km resolution are calculated as:  
\[GPP_{R\_residual}(\mathbf{x})\ =\ GPP_{R\_obs}(\mathbf{x})-(b_{0}+b_{1}H_{R} +b_{2}COS(S_{R})+b_{3}COS(A_{R})+b_{4}LAI_{R}) \tag{5}\]  
In this work, the residuals at a 1 km resolution were considered for downscaling, and the ATPK process served as the second phase when downscaling the residuals, \(GPP_{R\_residual}(\mathbf{x})\), at 1 km to those at 500 m, \(GPP_{r\_residual}(\mathbf{x})\). In the ATPK approach, \(GPP_{r2}(\mathbf{x})\) represents a linear combination of residuals from N 1 km pixel, which can be described as:  
\[GPP_{r2}(\mathbf{x})\ =\ \sum_{i\ =\ 1}^{\rm N}\lambda_{i}GPP_{R\_residual}( \mathbf{x}_{i})\ \,\ \ \ s.t.\ \ \ \ \ \sum_{i}^{\rm N}\lambda_{i}\ =\ 1 \tag{6}\]  
where \(\lambda_{i}\) represents the weight for the residual of the \(i\)th 1 km pixel, which is centered at \(\mathbf{x}_{i}\). As shown in Equation (6), the ATPK phase takes the spatial correlation among pixels into account, which is not considered in the above equations. The main issue of the ATPK phase is obtaining the weights, \(\lambda\), which are estimated by minimizing prediction error spatial variations at a 1 km resolution. The kriging system can be expressed as:  
\[\left[ \begin{array}{cccc}\mathrm{C_{RR}}(x_{1},x_{1})&\cdots&\mathrm{C_{RR}}(x _{1},x_{N})&1\\ \cdots&\cdots&\cdots&\cdots\\ \cdots&\cdots&\cdots&\cdots\\ \cdots&\cdots&\cdots&\cdots\\ \mathrm{C_{RR}}(x_{N},x_{1})&\cdots&\mathrm{C_{RR}}(x_{N},x_{N})&1\\ 1&\cdots&1&0\end{array} \right]\left[ \begin{array}{c}\lambda_{1}\\ \cdots\\ \cdots\\ \cdots\\ \lambda_{N}\\ \mu\end{array} \right]\ =\ \left[ \begin{array}{c}\mathrm{C_{rR}}(x,x_{1})\\ \cdots\\ \cdots\\ \cdots\\ \mathrm{C_{rR}}(x,x_{N})\\ 1\end{array} \right] \tag{7}\]  
where \(C_{RR}(x_{i},\)\(x_{j})\) is the residual covariance between a 1 km pixel centered at \(x_{i}\) and a 1 km pixel centered at \(x_{j}\), and \(C_{rR}(x,\)\(x_{j})\) is the residual covariance between a 500 m pixel centered at \(x\) and a 1 km pixel centered at \(x_{j}\). Variable \(\mu\) represents the range multiplier. To estimate weight \(\lambda\) according to Equation(7), these two types of residual covariance need to be calculated in advance. Assuming each pixel as a point, \(\mathbf{s}\) is intended to represent the Euclidean distance between two pixels. The 1 km to 1 km residual covariance, \(C_{RR}(\mathbf{s})\), and the 500 m to 1 km residual covariance, \(C_{rR}(\mathbf{s})\), can be estimated as:  
\[\mathrm{C_{RR}}(s)\ =\ \mathrm{C_{rr}}(s)*h_{R}(s)*h_{R}(-s) \tag{8}\]  
\[\mathrm{C_{rR}}(s)\ =\ \mathrm{C_{rr}}(s)*h_{R}(s) \tag{9}\]  
where \(C_{rr}(\mathbf{s})\) represents the 500 m to 500 m residual covariance, and \(\mathrm{h_{R}}(*)\) represents the point spread function (PSF). Variable \(-s\) indicates that the distance from a 500 m pixel (A) to another 500 m pixel (B) within a 1 km pixel is opposite to the distance from a 500 m pixel (B) to a 500 m pixel (A) (i.e., \(s\)). Supposing the value of the coarse pixel is the mean of the 500 m pixels within it, and F is the ratio between the resolutions of 1 km pixel and 500 m pixel, then, PSF can be described as:  
\[h_{R}(x)\ =\ \left\{ \begin{array}{cc}\frac{1}{F^{2}}&x\in R(x)\\ 0&otherwise\end{array} \right.