## 4 Materials and Methods
### Estimation of \(\phi_{\rm P}\) and NPQ  
In this study, \(K_{\rm D}\) and \(K_{\rm F}\) are assumed to be 0.9 and 0.1, respectively [46]. Note that NPQ should be equal to the rate coefficients of energy-dependent heat dissipation (\(K_{\rm N}\)) because NPQ = \(K_{\rm N}/(K_{\rm F}+K_{\rm D})\), and \(K_{\rm F}+K_{\rm D}\) = 1. \(K_{\rm N}\) can be estimated as [48]:  
\[K_{\rm N}={\rm a}\times\chi^{\rm c}\times\frac{1+{\rm b}}{{\rm b}+\chi^{\rm c} }\times\frac{\exp^{({\rm d}\times T_{\rm air}+{\rm e})}}{{\rm PAR}^{\rm f}}, \tag{10}\]  
where \({\rm a}\), \({\rm b}\), \({\rm c}\), \({\rm d}\), \({\rm e}\) and \({\rm f}\) are fitting parameters. Following [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], Peylin, Chevallier, Vuichard and Bastrikov [48], \({\rm b}\), \({\rm c}\), \({\rm d}\), \({\rm e}\) and \({\rm f}\) are assumed to have values of 5.74, 2.167, \(-\)0.014, \(-\)0.00437 and 0.00576, respectively. Potentially, we can use an exponential equation with three parameters to represent the relationship between the parameter \({\rm a}\) and \(\theta_{\rm SWC}\):  
\[{\rm a}={\rm g}\times\exp^{((-{\rm h})\times\theta_{\rm swC})}+{\rm j}, \tag{11}\]where g, h, and j are empirical parameters. The measurements are randomly divided into two groups, with 50% of the data used for training, and the remaining 50% for evaluating the performance of the predictions. The Matlab function 'Isqnonlin' was used to obtain the values of the parameters by minimizing a cost function on the training dataset: \(C=(M-S)^{2}\), where \(M\) is the NPQ measured with the PAM, and \(S\) is the modelled NPQ. A Trust Region algorithm was used to update the values of the parameters after each iteration step, with the iteration terminating when the improvement in the cost function was less than \(10^{-3}\). Table S1 presents the initial values, boundaries and constraints of the parameters. \(\chi\) is defined as [32; 46]:  
\[\chi=1-\frac{\Phi_{\mathrm{P}}}{\Phi_{\mathrm{Pmax}}}, \tag{12}\]  
where \(\Phi_{\mathrm{Pmax}}\) is assumed to be 0.8 and is similar among healthy plants; \(\Phi_{\mathrm{P}}\) is estimated as [32]:  
\[\Phi_{\mathrm{P}}=\frac{\mathrm{min}(A_{\mathrm{C}},A_{\mathrm{J}})}{\alpha_{ \mathrm{grm}}\times\beta_{\mathrm{PSII}}\times\mathrm{PAR}}\times\frac{4C_{ \mathrm{C}}{+}8\Gamma*}{C_{\mathrm{c}}-\Gamma*}, \tag{13}\]  
where \(A_{\mathrm{C}}\) and \(A_{\mathrm{J}}\) represent Rubisco-limited and RuBP-limited gross CO\({}_{2}\) assimilations (\(\mathrm{\mu mol~{}m^{-2}~{}s^{-1}}\)), respectively. \(\alpha_{\mathrm{grm}}\) represents the absorption efficiency of PAR by green leaves, and the value is usually fixed at 0.84. \(\beta_{\mathrm{PSII}}\) is the fraction of absorbed energy allocated to PSII, and the value is set to 0.5 [72]. \(A_{\mathrm{C}}\) and \(A_{\mathrm{J}}\) are given by [73]:  
\[A_{\mathrm{C}}=\frac{V_{\mathrm{cmax}}\times(C_{\mathrm{c}}-\Gamma*)}{C_{ \mathrm{c}}+K_{\mathrm{mC}}\times(1+O_{\mathrm{c}}/K_{\mathrm{mO}})}, \tag{14a}\] \[A_{\mathrm{J}}=\frac{J_{\mathrm{P}}}{4}\frac{(C_{\mathrm{c}}-\Gamma*)}{C_{ \mathrm{c}}{+}2\Gamma*} \tag{14b}\]  
where \(K_{\mathrm{mC}}\) is the Michaelis-Menten constants of Rubisco for CO\({}_{2}\) 270 \(\mathrm{\upmu bar}\), [73]; \(K_{\mathrm{mO}}\) is the Michaelis-Menten constants of Rubisco for O\({}_{2}\) 16,500 \(\mathrm{\upmu bar}\), [73]; \(O_{\mathrm{c}}\) is the chloroplastic O\({}_{2}\) partial pressure, assumed to equal to the oxygen partial pressure 230,000 \(\mathrm{\upmu bar}\), [74]; \(J_{\mathrm{P}}\) is the potential electron transport rate [47]:  
\[J_{\mathrm{P}}=\frac{\sigma\times\mathrm{PAR}+J_{\mathrm{max}}-\sqrt{\left( \sigma\times\mathrm{PAR}+J_{\mathrm{max}}\right)^{2}-4\times\theta\times \sigma\times\mathrm{PAR}\times J_{\mathrm{max}}}}{2\times\theta}, \tag{15}\]  
where \(\sigma\) is the product of leaf light absorptance, a fraction of absorbed photons allocated to PSII and \(\phi_{\mathrm{Pmax}}\). \(\sigma\) is set to 0.3 [72].