## 4 Results
### Comparison with the \(V_{m25}\) Obtained by BESS  
The V\({}_{\rm m25}\) calculated by BESS, BESS_P and our Two-stage linear model were compared in simulating the time series of V\({}_{\rm m25}\) inversely solved by coupling the BESS and LRC using EC data at the five flux sites (Figure 5). Unfortunately, the flux data for Daman in 2020 and Fenzidi in 2017 are incomplete; and due to the partial lack of remote sensing and meteorological data, the V\({}_{\rm m25}\) inverted by LRC for US-Ne1 and US-Ne2 sites in 2001 is also incomplete. The BESS model consistently underestimates V\({}_{\rm m25}\) throughout the growth period due to the fixed low PeakV\({}_{\rm m25}\) value, with the discrepancy becoming particularly noticeable during the middle stages of the growth period. Conversely, towards the end of the  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Site**} & \multicolumn{2}{c}{**BESS**} & \multicolumn{2}{c}{**BESS\_P**} & \multicolumn{2}{c}{**BESS\_TL**} \\ \cline{2-7}  & **RMSE** & **R\({}^{2}\)** & **RMSE** & **R\({}^{2}\)** & **RMSE** & **RMSE** & **R\({}^{2}\)** \\ \hline US-Ne1 & 18.93 & 0.23 & 25.07 & 0.23 & 7.54 & 0.85 \\ US-Ne2 & 16.58 & 0.52 & 22.67 & 0.52 & 6.82 & 0.88 \\ US-Ne3 & 14.70 & 0.67 & 19.46 & 0.67 & 6.82 & 0.87 \\ Daman & 14.55 & 0.02 & 16.99 & 0.02 & 6.44 & 0.71 \\ Fenzidi & 13.25 & 0.01 & 15.30 & 0.01 & 5.40 & 0.82 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The performance of the original BESS, BESS_P and our Two-stage linear model at simulating the V\({}_{\rm m25}\) inversely solved by coupling the BESS and LRC using EC data at the five flux sites.  
Figure 5: Comparison of V\({}_{\rm m25}\) calculated by BESS (blue lines), BESS_P (green lines) and our Two-stage linear model (red lines) in simulating the time series of V\({}_{\rm m25}\) inversely solved by coupling the BESS and LRC using EC data at the five flux sites: (**a–l**) show the verification results for US-Ne1; (**m–r**) show the verification results for US-Ne2; (**s–x**) show the verification results for US-Ne3; (**y–aa**) show the verification results for Daman; (**ab–ad**) show the verification results for Fenzidi.  
growth period when V\({}_{\rm m25}\) values are low, BESS tends to overestimate V\({}_{\rm m25}\). BESS_P, which employs the same modeling approach as BESS but adjusts the Peak\(V_{\rm m25}\) parameter, exhibiting a somewhat improved performance in estimating the trend of V\({}_{\rm m25}\) during the early growth phase. However, during the declining stage of V\({}_{\rm m25}\), BESS_P tends to significantly overestimate the values. Due to the same formula structure, the R\({}^{2}\) between the calculated V\({}_{\rm m25}\) of both methods and the verified values remains consistent, ranging from 0.01 to 0.67 (Table 2). The RMSE of the BESS model ranges from 13.25 to 18.93 \(\upmu\)mol m\({}^{-2}\)s\({}^{-1}\) and the RMSE of the BESS_P ranges from 15.30 to 25.07 \(\upmu\)mol m\({}^{-2}\)s\({}^{-1}\). Despite the adjustment in Peak\(V_{\rm m25}\), both BESS and BESS_P demonstrate poor performance in fitting the trend in V\({}_{\rm m25}\) across all sites. The observations reveal a distinct pattern of V\({}_{\rm m25}\) initially rising before declining, and our Two-stage linear model effectively captures this seasonal variation in V\({}_{\rm m25}\), with R\({}^{2}\) ranges from 0.71 to 0.88 and RMSE ranges from 5.40 to 7.54 \(\upmu\)mol m\({}^{-2}\)s\({}^{-1}\). During the early stages of crop growth, V\({}_{\rm m25}\) generally exhibits an upward trajectory, albeit with significant diurnal fluctuations. Conversely, in the descending phase, V\({}_{\rm m25}\) tends to display a more linear decline with reduced fluctuation.