## Methods
### GPP Models: Definition, Inversion and Analysis
\begin{table}
\begin{tabular}{c c} \hline \hline
**Model** & **Equation** \\ \hline MOD1 & \(GPP=a_{1}\cdot NDVI+a_{2}\) \\ MOD2 & \(GPP=(a_{1}\cdot NDVI+a_{2})\cdot PAR\) \\ MOD3 & \(GPP=(a_{1}\cdot NDVI+a_{2})\cdot(a_{3}\cdot PRI+a_{4})\cdot PAR\) \\ MOD4 & \(GPP=a_{1}\cdot\left(\frac{a_{2}-min(max(a_{3}\cdot PVD),a_{2})}{a_{2}-a_{3}} \right)\cdot\left(\frac{max(min(a_{3}\cdot T_{\text{air}}),a_{4})-a_{4}}{a_{3} -a_{4}}\right)\cdot\left(1-e^{(-0.5\cdot(a_{6}\cdot NDVI+a_{7}))}\right)\cdot PAR\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Optically-based Gross Primary Production (\(GPP\)) parametric models inverted against observed half-hourly \(GPP\) (\(GPP_{\text{hh}}\)) and micro-meteorological variables. The least square nonlinear curve-fitting optimization method implemented in the Matlab(r) function LSQCURVEFIT. In addition, bootstrap technique [67] with 500 subsamples was applied to quantify the uncertainty of the estimated model parameters. Parameter constrain was only applied to MOD4--the MODIS algorithm--where \(a_{4}\) and \(a_{5}\), representing \(T_{\text{MINNmin}}\) and \(T_{\text{MINNmax}}\), respectively, were bounded in the ranges (\(-15\), 50) and (\(0\), 50) each with a uniform distribution. Parameters a\({}_{1\text{-}3}\) representing \(\varepsilon_{\text{max}}\), \(\mathit{VPD}_{\text{max}}\) and \(\mathit{VPD}_{\text{min}}\), respectively, as well as \(a_{6}\)--the \(\mathit{NDIVI}\)-\(\mathit{LAI}\) slope--were forced to be larger than 0. In addition, the large equifinality existing between \(\varepsilon_{\text{max}}\) and \(f_{\mathit{PAR}}\) was handled by imposing prior information of model parameters \(a_{6\text{-}7}\) describing the \(\mathit{NDIVI}\)-\(\mathit{LAI}\) relationship. To do so, a linear model was fit using the averaged data from 16 field campaigns held during the FLU\(\chi\)PEC [39] project. In each campaign, approximately \(\sim\)13 plots of 25 m \(\times\) 25 m were sampled. Grass vegetation samples and field spectroscopy data were acquired as described in [PERSON] et al. [43]; however this time only green fraction of \(\mathit{LAI}\) was used. Tree \(\mathit{LAI}\) was estimated \(\sim\)1.426 m\({}^{2}\)/m\({}^{2}\) using hemispherical cameras [68] and assumed stable over time. Tree crown \(\mathit{NDIVI}\) was the mean \(\mathit{NDIVI}\) of the \"trees\" class from all the CASI images within the 1 km pixel centered on each tower. \"Grass\" and \"trees\" variables were linearly combined with a proportion of 100% and 20%m respectively [69]. For MOD1-MOD3, the cost function used for the inversion was the sum of the squared differences between calculated and predicted \(\mathit{GPP}_{\text{hh}}\). For MOD4, the cost function accounted for the prior information in coefficients \(a_{6\text{-}7}\) and was computed as in [PERSON] et al. [70] (Equation (11)):  
\[\delta^{2}=\sum_{i=1}^{g}\left(\frac{\mathit{GPP}_{\text{hh},i}\ -\ \mathit{GPP}_{\text{ pred},i}}{\sigma_{\mathit{GPP}_{\text{hh},i}}}\right)+\sum_{j=6}^{7} \left(\frac{a_{\text{j}}\ -\ a_{j}^{\text{prior}}}{\sigma_{a_{j}}}\right), \tag{11}\]  
where \(g\) is the number of observations, \(\mathit{GPP}_{\text{hh},i}\) and \(\mathit{GPP}_{\text{pred},i}\) are each observed and predicted half hourly \(\mathit{GPP}\), and \(\sigma_{\mathit{GPP}_{\text{hh},i}}\) is the corresponding uncertainty of \(\mathit{GPP}_{\text{hh}}\) (see Section 2.3). In the second term of the equation, \(a_{j}\) are the parameters retrieved by model inversion, \(a_{j}^{\text{prior}}\) is the model parameter estimated using independent data, and \(\sigma_{a_{j}}\) the corresponding uncertainties of the priors. These are the coefficients of the abovementioned \(\mathit{NDIVI}\)-\(\mathit{LAI}\) relationship. Errors of the linear model fit were propagated to obtain the 95% confidence intervals of the coefficients, and these converted to a standard deviation assuming a normal distribution. To assess the impact of temporal mismatch between optical and EC data on \(\mathit{GPP}_{\text{hh}}\) estimation, models were fit with two different datasets.