## 2 Materials and Methods
### Methodology
#### 2.2.2 Accuracy Evaluation  
\(\bullet\) On-the-ground GPP observations-based evaluation \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Data Type** & **Temporal Resolution** & **Spatial Resolution** & **Timespan** & **References** \\ \hline FPAR\({}_{\text{MCD15A}}\) & 8 days & 500 m & 2000– & [9] \\ FPAR \({}_{\text{FPAR}}\) & FPAR\({}_{\text{BNU}}\) & 8 days & \(1/12^{\circ}\) & 1981–2015 & [30] \\ GIMMIS NDVI3g & 15 days & 500 m & 2000– & [12; 39] \\ Land use and cover data & - & 1 km & 2005 & [35] \\ Annual mean air temperature (TAVG) & 8 days & 1 km & 1980–2018 & [42] \\ Annual total precipitation (PRCP) & 8 days & 1 km & 1980–2018 & [42] \\ Annual total shortwave radiation & 8 days & 1 km & 1980–2018 & [42; 44] \\ (SWRad) & 8 days & 1 km & 1980–2018 & [42; 44] \\ Nitrogen deposition & Yearly & 0.1\({}^{\circ}\) & 1980–2010 & [49] \\ CO\({}_{2}\) concentration & Monthly & - & 1990–2016 & [45] \\ Daily gross ecosystem exchange (GEE) & Daily & - & 2004–2010 & [41] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Data information. In terms of seasonal changes, the GPP was applied to evaluate the FPAR estimation in this study. On the other hand, FPAR was assumed to be linearly correlative with GPP in the light use efficiency model [51]. Therefore, a linear regression was applied to evaluate the FPARANN as the predictive variable for the on-the-ground GPP observations, as the dependent variable. The coefficient of determination (\(R^{2}\)) from regression was used to measure the performance of the FPARANN. The formula for \(R^{2}\) is:  
\[R^{2}=1-\frac{\Sigma\Big{(}y-\hat{y}\Big{)}^{2}}{\Sigma(y-\overline{y})^{2}} \tag{4}\]  
where, \(\hat{y}\) and \(\mathbf{y}\) are the values of FPARANN and GPP for each site, respectively. \(\overline{y}\) is the mean GPP. * Consistency with FPARMCD15A2  
Consistency was evaluated for the FPARANN estimations to determine whether they reproduce the spatiotemporal features of FPARMCD15A2. Specifically, the following matrices were calculated on the pixel scale by comparing the FPARANN with the FPARMCD15A2 as a benchmark for the same growing season date (day 225 of the year, as an example) across different years (with 2005, 2010, and 2015, as an example). The matrices were root mean square error (_RMSE_), Kling-Gupta efficiency (_KGE_), and structural similarity index (_SSIM_). The _RMSE_ is generally used to measure the error of estimations compared with observations. A smaller _RMSE_ represents a higher accuracy. It is formulated as follows:  
\[RMSE=\sqrt{\frac{\sum_{t=1}^{n}\Big{(}\hat{y_{t}}-y_{t}\Big{)}^{2}}{n}} \tag{5}\]  
where, \(n\) is the number of samples, \(\hat{y_{t}}\) and \(y_{t}\) are the values of the \(t\)-th sample for FPARANN and FPARMCD15A2, respectively. The _KGE_ is a measure of agreement between estimations and observations and its value ranges from -Inf to 1; the closer its value is to 1, the stronger the agreement:  
\[KGE=1-\sqrt{\left(r-1\right)^{2}+\left(\frac{\sigma_{est}}{\sigma_{obs}}-1 \right)^{2}+\left(\frac{\mu_{est}}{\mu_{obs}}-1\right)^{2}} \tag{6}\]  
where, the \(\sigma_{est}\) and \(\sigma_{obs}\) are the standard deviations of estimations (FPARANN and FPARBNU) and observations (FPARMCD15A2), respectively, and \(\mathbf{r}\) is the linear correlation between them. The \(\mu_{est}\) and \(\mu_{obs}\) are the means of estimation and observations, respectively [52]. The _SSIM_, a measure to quantify image quality changes caused by data processing between a reference image and a processed image, ranges from \(-1\) to 1; a higher value means a higher similarity [53]. \[SSIM(x,y)=\frac{\big{(}2\mu_{x}\mu_{y}+C_{1}\big{)}\big{(}2\sigma_{xy}+C_{2} \big{)}}{\left(\mu_{x}^{2}+\mu_{y}^{2}+C_{1}\right)\left(\sigma_{x}^{2}+ \sigma_{y}^{2}+C_{2}\right)} \tag{7}\]  
where, \(\mu_{x}\) and \(\mu_{y}\) are the local means, \(\sigma_{x}\) and \(\sigma_{y}\) are the standard deviations for images \(x\), \(y\), and \(\sigma_{xy}\) is the cross-covariance for them.