## 2 Materials and Methods
### Overview
#### 2.1.2 FOREST Module
\[f_{\text{swc}}=\frac{1-e^{(\beta\times swc)}}{1-e^{(\varphi)}}\] (5) \[f_{\text{swc}}=\frac{1-e^{(\beta\times\varphi_{\theta})}}{1-e^{(-\varphi_{ \theta})}}\] (6) The empirical parameter to compute soil water content modifier function, and \(\beta\) is calculated as follows: \[\beta=\left\{ \begin{array}{cl}0&\text{for}&swc<WP\\ \frac{swc-WP}{FC-WP}&\text{for}&WP<swc<FC\\ 1&\text{for}&swc>FC\end{array} \right.\] (7)
2. Temperature modifier Temperature is considered as a limiting factor for the growth of vegetation when it is not within an optimum range. The growth and dormant stages of vegetation are related to the annual cycle of air temperature. Maximum growth will happen at optimal temperatures \(T_{opt}\) and will stop when temperatures drop below or exceed certain temperature thresholds, \(T_{min}\) and \(T_{max}\), respectively. The temperature modifier is computed as follows: \[f_{T}=\bigg{(}\frac{T_{a}-T_{min}}{T_{opt}-T_{min}}\bigg{)}\bigg{(}\frac{T_{ max}-T_{a}}{T_{max}-T_{opt}}\bigg{)}^{(T_{max}-T_{opt})/(T_{opt}-T_{min})}\] (8)  
_Ta_ is the air temperature, \(T_{min}\) is the minimum temperature for the plant growth, \(T_{max}\) is the maximum temperature for the plant, and \(T_{opt}\) is the optimum temperature for the plant growth. This modifier was implemented according to the temperature modifier as proposed by [PERSON] [13]. * Age modifier The vegetation age is considered based on the factors that may affect the carbon allocation. The trees in the early stages are not growing as faster as mature trees. The age modifier is computed following [PERSON] et al. (2018). _if age \(<0.2\) age\({}_{max}\)_  
\[f_{age}=0.7+0.3\frac{age}{0.22 ag{}_{max}} \tag{9}\]  
Otherwise  
\[f_{age}=1+max\Bigg{[}0,\ \bigg{(}\frac{age-0.2 ag{}_{max}}{0.95 age{}_{max}} \bigg{)}^{3}\Bigg{]} \tag{10}\]  
where age is the effective age of the vegetation, age\({}_{\max}\) is the maximum age that the tree can achieve. 3. Vapor pressure modifier (Landsberg and waring-3 PG)  
The vapor pressure modifier (implemented according to Landsberg and Waring- 3 PG model) is calculated with the following formula:  
\[f_{VPD}=exp\big{(}-k_{g}D\big{)} \tag{11}\]  
with \(f_{VPD}\) vapor pressure modifier, kg empirical coefficient describing the relationship between stomatal and canopy conductance and \(D\) average vapor pressure deficit. The vapor pressure deficit is computed according to [PERSON] (2018):  
\[ESTAR=6.1076\times e^{\frac{(17.269\times T_{a})}{12\times 2\times 3}} \tag{12}\]  
with \(ESTAR\) is the saturation vapor pressure of the air [mbar], and \(T_{a}\) is the actual air temperature [\({}^{\circ}\)C]. 4. \(CO_{2}\) modifier  
The last modifier considered is the one relating to \(CO_{2}\) concentration in atmosphere. In fact, the presence of carbon dioxide in the atmosphere is vital to plants in order to activate the photosynthesis processes. The \(CO_{2}\) modifier \(f_{CO_{2}}\) is implemented according to the following formula ([PERSON], 1998):  
\[f_{CO_{2}}=\frac{f_{CAx}\cdot CO_{2}}{350\cdot(f_{CAx}-1)+CO_{2}} \tag{13}\]  
where \(f_{CAx}\) is a dimensionless parameter to compute the modifier, and \(CO_{2}\) is the current carbon dioxide concentration [ppm] in the atmosphere. * Carbon Allocation  
The model allows to simulate the carbon allocation to roots, stem, and foliage. This step will permit to follow the forest change in subsequent years according to the value of \(NPP\) previously calculated. The increments in roots (\(\Delta\)_Mroot_), stem (\(\Delta\)_Mstem_), and foliage (\(\Delta\)_Mleaf_) are calculated as follows ([PERSON], 1998):  
\[\Delta M_{root}=NPP\times\eta_{r} \tag{14}\]  
\[\Delta M_{stem}=NPP\times\eta_{s} \tag{15}\]  
\[\Delta M_{leaf}=NPP\times\eta_{f} \tag{16}\]  
where \(\eta_{r}\), \(\eta_{s}\), and \(\eta_{f}\) are the partitioning factors to allocate \(NPP\) to roots, stem, and leaves respectively. The allocation is based on an assumption made by [PERSON] and [PERSON] [13] that the plant will allocate more carbon to root rather than to stem under resources shortages.