## 2 Materials And Methods
### Calibration: Hierarchical and Multi-Objective Approach  
In this study, we calibrated the surface fluxes (\(Rr\), \(H\), \(LE\), and \(NEE\)) to solve a multi-criteria optimization problem [10; 12; 13]. INLAND parameters were automatically calibrated using the genetically adaptive multi-objective calibration method (AMALGAM) [11], which allows finding a set of parameters that defines the \"Pareto-optimal\" curve corresponding to a vector of multiple objective functions. The AMALGAM combines several genetic algorithms that adapt to the problem of multiple optimization to simultaneously optimize more than one objective function.  
Mathematically, the problem involves minimizing \(m\) (\(m\geq 1\)) objective functions with \(d\) parameters  
\[F=\{f_{1}(\theta),f_{2}(\theta),...,f_{m}(\theta)\} \tag{36}\]  
where \(\theta=\{x_{1},x_{2},...,x_{d}\}\) denotes the vector with the set of parameters (decision vector) and where \(F\) is the objective space. The search space of values of \(x\) can be quite large, but it must be delimited in \(\theta\in\Re^{d}\). Thus, rather than finding a single optimal solution, we find a Pareto-optimal set of solutions. The objective functions used in the calibration included the root mean square error (\(RMSE\)) and the amplitude of the daily average cycle error (\(ACD\)) (Table 1). However, AMALGAM allows the use of only one metric if calibration involves more than one variable (output of model). On the other hand, if we use more than one metric, we must choose only one variable.  
The different hierarquical framework of the processes represented in INLAND are used to define the hierarchical levels of calibration and model output: _level 1_--radiation fluxes (for parameters used in reflected solar radiation--\(Rr\)), and _level 2_--turbulent fluxes (for parameters used in \(LE\), \(NEE\), and \(H\)). The metrics used to calibrated \(Rr\) were \(RMSE\) and \(ACD\). On the other hand, the metric used for \(LE\), \(H\), and \(NEE\) was \(RMSE\) for a better representation of these fluxes in average terms. The parameters calibrated in this work are presented in Table 2 following the studies developed by [9; 13; 25]  
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Criteria** & **Name** & **Formulation** & **Optimal Value** \\ \hline \hline \(RMSE\) & Root-mean-squared error & \(\sqrt{\frac{1}{n}\sum_{t=1}^{n}(S_{t}-O_{t})^{2}}\) & 0 \\ \hline \(ACD\) & Amplitude of the daily average cycle error & \(\left|\frac{\Delta S_{h}-\Delta O_{t}}{\Delta O_{h}}\right|\times 100\) & 0 \\ \hline \(MAE\) & Mean absolute error & \(\frac{1}{n}\sum_{t=1}^{n}|S_{t}-O_{t}|\) & 0 \\ \hline \(NSE\) & Nash–Sutcliffe efficiency & \(1-\frac{\sum_{t=1}^{n}(S_{t}-O_{t})^{2}}{\sum_{t=1}^{n}(S_{t}-O)^{2}}\) & 1 \\ \hline \(PBIAS\) & Percentual Bias & \(\frac{\sum_{t=1}^{n}(S_{t}-O_{t})}{\sum_{t=1}^{n}(O_{t})}\times 100\) & 0 \\ \hline \(R2\) & Coefficient of determination & \(\frac{(\sum_{t=1}^{n}(O_{t}-\overline{O})(S_{t}-\overline{S}))^{2}}{\sum_{t=1} ^{n}(O_{t}-\overline{O})^{2}\sum_{t=1}^{n}(S_{t}-\overline{S})^{2}}\) & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Metrics used to evaluate the performance of the calibration and its mathematical formulations. \(n\) is the length of the series, \(S\) and \(O\) correspond to the simulated and observed data, respectively, \(\Delta\) denotes the average daily cycle amplitude, and \(\overline{S}\) is the mean of \(S\).