# Improving a Process-Based Model to Simulate Forest Carbon Allocation under Varied Stand Density
## 2 Materials and Methods
### Simulations
#### 2.3.2 Model Evaluation  
The mean absolute error (\(MAE\)) and normalized root mean square error (\(NRMSE\)) were used to evaluate the differences between the predicted and observed values.  
\[MAE=100\frac{\sum_{i=1}^{n}\lvert P_{i}-O_{i}\rvert}{n\overline{O}}\]  
\[NRMSE=(O_{max}-O_{min})^{-1}\left[\frac{\sum_{i=1}^{n}(P_{i}-O_{i})^{2}}{n} \right]^{0.5}\]  
where \(P_{i}\) is the \(i\)th predicted value; \(O_{i}\) is \(i\)th the observed value; \(\overline{O}\) is the mean observed value; \(O_{max}\) and \(O_{min}\) are the maximum and minimum observed values, respectively; and \(n\) is the number of observations. The lower the value of the two indicators, the smaller the difference between the model predictions and observations, indicating a more accurate simulation.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Description** & **Note** \\ \hline Absorb = 0.15 & Atmospheric absorption factor & [37] \\ Cloud = 0.45 & Time fraction of cloudy days & This study \\ PAR factor = 0.65 & Solar radiation fraction & This study \\ BLond = 0.12 & Canopy boundary layer conductance (mL m\({}^{-2}\) s\({}^{-1}\)) & This study \\ MaxCond = 0.26 & Max. canopy conductance (mL m\({}^{-2}\) s\({}^{-1}\)) & This study \\ StomCond = 0.012 & Stomata conductance (mL m\({}^{-2}\) s\({}^{-1}\)) & This study \\ ExtCoef = 0.46 & Radiation extinction coefficient & [38] \\ TaMin = 7 & Min. temperature for growth (\({}^{\circ}\)C) & This study \\ TaMax = 40 & Max. temperature for growth (\({}^{\circ}\)C) & This study \\ Topt = 25 & Optimum temperature for growth (\({}^{\circ}\)C) & This study \\ N factor = 0.2 & N factor for tree growth & [38] \\ Na = 3 & Effects of age on GPP & [38] \\ Sla = 13 & Specific leaf area (m\({}^{2}\) kg\({}^{-1}\)) & [39] \\ GamaF = 1 & Leaf turnover per year & This study \\ GamaR = 0.21 & Fine root turnover per year & This study \\ Lnr = 26 & Lignin–N ratio & [38] \\ Ls = 0.215, 0.215, 0.2351, 0.255, 0.255 & Lignin in leaf, fine root, coarse root, branch, and wood & [38] \\ A1, A2, A3 = 15, 45, 125 & Soil water depth of layer 1, 2, and 3 (cm) & [38] \\ AWL1, 2, 3 = 0.5, 0.3, 0.2 & Relative root density for layer 1, 2, and 3 & [38] \\ KF = 0.5 & Fraction of H\({}_{2}\)O flow to stream & [40] \\ KD = 0.5 & Fraction of H\({}_{2}\)O flow to deep storage & [40] \\ KX = 0.3 & Fraction of deep storage water to stream & [40] \\ AWater = 250 & Max. soil water (mm) & [40] \\ MiuNorm = 0 & Normal mortality & This study \\ CSP = 0.26 & Wood C density (t C m\({}^{-3}\)) & This study \\ CD = 20 & Crown to stem diameter ratio & This study \\ AlphaC = 0.08 & Canopy quantum efficiency & This study \\ MaxHeight = 30 & Max. height (m) & This study \\ AgeMax = 100 & Max. stand age (year) & This study \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters used within the model simulations.  
The Willmott index of agreement (_d_) was used as an indicator of modeling efficiency, with values ranging from 0 to 1.0, with 1.0 indicating perfect agreement; it is expressed as follows [41]:  
\[d=1-\left[\frac{\sum_{i=1}^{n}\left|P_{i}-\mathcal{O}_{i}\right|^{2}}{\sum_{i=1} ^{n}\left(\left|P_{i}-\overline{\mathcal{O}}\right|+\left|O_{i}-\overline{ \mathcal{O}}\right|\right)^{2}}\right]\]