## 3 Methodology
### Verificaiton of Model and New Index  
Based on the assumption of the consistent relationship between the N accumulation and the canopy reflectance and ratio index across the whole growth period, the VIs (listed in Table 3) were collected from the previous literature [43; 47; 52; 53; 54; 55; 56; 57; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67; 68; 69]. Statistical analyses were performed using SPSS 13.0. Linear correlation analyses between the \(\mathrm{C_{ab}}\) and VIs were conducted individually to evaluate the feasibility of each vegetation index. The precision of the VIs was evaluated using the square of the correlation coefficient of the linear regression (\(\mathrm{R^{2}}\)) and the root mean square error (RMSE). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Indexes** & **Expressions** & **Literaries** & \(\mathbf{R^{2}}\) & **RMSE** \\ \hline NDVI & \((\rho_{\mathrm{PNT}}-\rho_{\mathrm{PNT}})/(\rho_{\mathrm{PNT}}+\rho_{\mathrm{PNT}})\) & [52] & 0.4 & 10.38 \\ VI & \(\frac{2.55\sin(\rho_{\mathrm{PNT}}-\rho_{\mathrm{PNT}})}{(\rho_{\mathrm{PNT}}+ \rho_{\mathrm{PNT}}-\gamma_{\mathrm{PNT}}+1)}\) & [53] & 0.47 & 9.75 \\ SAVI & \(1.5\times(\rho_{\mathrm{PNT}}-\rho_{\mathrm{PNT}})/(\rho_{\mathrm{PNT}}+\rho_{ \mathrm{PNT}}+0.5)\) & [54] & 0.45 & 9.95 \\ MSAVI & \(\left[2\times\rho_{\mathrm{PNT}}+1-\sqrt{2\times(\rho_{\mathrm{PNT}}+1)^{2}-8 \times(\rho_{\mathrm{PNT}}-\rho_{\mathrm{PNT}})}\right]/2\) & [55] & 0.49 & 9.64 \\ OSAVI & \(1.16\times(\rho_{\mathrm{PNT}}-\rho_{\mathrm{PNT}})/(\rho_{\mathrm{PNT}}+\rho_{ \mathrm{PNT}}+0.16)\) & [56] & 0.43 & 10.12 \\ WORKVI & \(\alpha\times(\rho_{\mathrm{PNT}}-\rho_{\mathrm{PNT}})/(\rho_{\mathrm{PNT}}+\rho_{ \mathrm{PNT}}+a)\) & [47] & 0.5 & 9.49 \\ RDVI & \((\rho_{\mathrm{PNT}}-\rho_{\mathrm{PNT}})/(\rho_{\mathrm{PNT}}+\rho_{ \mathrm{PNT}})\) & [57] & 0.45 & 9.98 \\ MSR & \((\rho_{\mathrm{PNT}}/\rho_{\mathrm{PNT}}-1)/(\rho_{\mathrm{PNT}}+1)\) & [58] & 0.52 & 9.27 \\ Viopt & \((1+0.45)\times(\rho_{\mathrm{PNT}}^{2}-1/\sqrt{1})\) & \(\rho_{\mathrm{PNT}}+0.45\) & [59] & 0.45 & 9.94 \\ GNDVI & \((\rho_{\mathrm{PNT}}-\rho_{\mathrm{PNT}})/(\rho_{\mathrm{PNT}}+\rho_{\mathrm{ PNT}})\) & [60] & 0.51 & 9.43 \\ NDVIg-b & \((\rho_{\mathrm{PNT}}-\rho_{\mathrm{PNT}})/(\rho_{\mathrm{PNT}}+\rho_{\mathrm{ PNT}})\) & [61] & 0.11 & 12.67 \\ RVI & \(\rho_{\mathrm{PNT}}/\rho_{\mathrm{PNT}}\) & [62] & 0.51 & 9.38 \\ RVI2 & \(\rho_{\mathrm{PNT}}/\rho_{\mathrm{PNT}}\) & [43] & 0.59 & 8.66 \\ NRI & \((\rho_{\mathrm{PNT}}-\rho_{\mathrm{PNT}})/(\rho_{\mathrm{PNT}}+\rho_{\mathrm{ PNT}})\) & [63] & 0.28 & 11.39 \\ NPCI & \((\rho_{\mathrm{PNT}}-\rho_{\mathrm{PNT}})/(\rho_{\mathrm{PNT}}+\rho_{\mathrm{ PNT}})\) & [64] & 0.34 & 10.96 \\ MCARI & \([(\rho_{\mathrm{PNT}}-\rho_{\mathrm{PNT}})-2\times(\rho_{\mathrm{PNT}}-\rho_{ \mathrm{PNT}})]\times\rho_{\mathrm{PNT}}/\rho_{\mathrm{PNT}}\) & [65] & 0.47 & 9.82 \\ New Index & & & 0.54 & 9.08 \\ DVI & \(\rho_{\mathrm{PNT}}-\rho_{\mathrm{PNT}}\) & [66] & 0.47 & 9.82 \\ PSRI & \((\rho_{\mathrm{PNT}}-\rho_{\mathrm{PNT}})/(\rho_{\mathrm{PNT}}+\rho_{\mathrm{ PNT}})\) & [67] & 0.02 & 13.3 \\ SIPI & \((\rho_{\mathrm{PNT}}-\rho_{\mathrm{PNT}})/(\rho_{\mathrm{PNT}}+\rho_{\mathrm{ PNT}})\) & [68] & 0.52 & 9.35 \\ GI & \(\rho_{\mathrm{PNT}}-\rho_{\mathrm{PNT}}\) & [69] & 0.41 & 10.32 \\ SRPI & \(\rho_{\mathrm{PNT}}-\rho_{\mathrm{PNT}}\) & [68] & 0.44 & 10.07 \\ MRVI & \(\frac{1}{2}\sqrt{\rho_{\mathrm{PNT}}}\times\rho_{\mathrm{PNT}}/(\rho_{\mathrm{ PNT}}-\rho_{\mathrm{PNT}})^{2}\) & 0.62 & 8.34 \\ \hline \hline \end{tabular} \(\mathrm{R^{2}}\) is the correlation coefficient of linear regression and RMSE is the root mean square error.  
\end{table} 
Table 3: The relationship between vegetation indexes (VIs) and the leaf \(\mathrm{C_{ab}}\) of the winter wheat in \(\mathrm{exp1}\) at the Yucheng Comprehensive Experimental Station. To validate the accuracy of the GPP models, the DAM and yield of the winter wheat was simulated in Hengshui City in 2017. The ground data of the DAM and yield in the corresponding pixels were used to validate the estimated results.