## 2 Materials and Methods
### Data Analysis  
The Pearson correlation (\(p<0.001\)) was used to verify the correlation between the VIs and winter wheat maturity during the 5 consecutive periods and to verify the overall correlation for each field. The development speed of crop yield formation in different stages is generally different, and usually, a growth curve model can be used to achieve better fitting and prediction effects. In order to enhance the stability of the monitoring model, the numerical samples of the wheat grain-filling process indicators in five periods were randomly divided into model development and model validation according to a ratio of 3:1. On the basis of the correlation analysis, the logistic model was selected to adjust the VI value to the grain-filling process according to the winter wheat growth development (Equation (1)), and linear and polynomial relationships were also established for each significantly correlated vegetation index for comparison.  
\[W=W_{f}/[1+exp^{-k(g-M)}] \tag{1}\]  
where \(W\) represents the response surface (wheat grain-filling process indicators), \(W_{f}\) is the maximum value (upper asymptote), \(k\) is the relative growth rate in \(M\), \(g\) is the variable (VIs), and \(M\) is the VIs at which the growth rate is maximized.  
Using the samples of the model development and model validation, the model was evaluated by plotting the 1:1 relationship graph between the predicted and measured values of the indicators of the wheat grain-filling process. The root-mean-square error (RMSE) and R\({}^{2}\) were used to measure the accuracy and precision of each monitoring model. Monitoring models with a low RMSE and high R\({}^{2}\) indicate that a vegetation index has high accuracy and precision in estimating the wheat grain-filling process. The R\({}^{2}\) and RMSE were calculated using Equations (2) and (3):  
\[R^{2}=1-\Sigma_{i=1}^{n}{(y_{i}-\hat{y_{i}})}^{2}/\Sigma_{i=1}^{n}{(y_{i}- \overline{y})}^{2} \tag{2}\]  
\[RMSE=\sqrt{\Sigma_{i=1}^{n}{(y_{j}-\hat{y_{i}})}^{2}/n} \tag{3}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**VI** & **Formula** & **References** \\ \hline NDVI & (NIR-Red)/(NIR+Red) & [33] \\ GNDVI & (NIR-Green)/(NIR+Green) & [43] \\ PSRI & (Red-Blue)/NIR & [44] \\ NDRE & (NIR-Rededge)/(NIR+Rededge) & [45] \\ NDVI-rededge & (Rededge-Red)/(Rededge+Red) & [46] \\ NLI & (NIR2-Red)/(NIR2+Red) & [47] \\ \hline \hline \end{tabular}
\end{table}
Table 2: Vegetation indices used to monitor the grain-filling process of winter wheat.  
where \(y_{i}\) and \(\hat{y}_{i}\) represent measured values and predicted values of indicators of the wheat grain-filling process, respectively, \(\overline{y}\) is the average value of indicators of the wheat grain-filling process, and \(n\) is the number of samples.  
Vegetation indices with the highest performance (RMSE and R\({}^{2}\)) in the overall analysis were selected to represent the variability in indicators of the wheat grain-filling process. The best performing model equations were used to estimate indicators of the wheat grain-filling process from VIs to create wheat maturity maps. For that, VI values for each pixel in the downloaded imageries were inputted into the model equation and the wheat grain-filling process was estimated for each pixel at each sample date.