# Assimilation of Earth Observation Data Over Cropland and Grassland Sites into a Simple GPP Model
## 3 Methods
### GPP Model
The leaf lifespan (f(i) of LAI produced on day i is computed as a function of _GDD_(i) using the following linear equation:  
\[J\;(\mathrm{i})=c+d\;GDD(\mathrm{i}) \tag{5}\]  
where the parameters \(c\) and \(d\) control the magnitude of the lifespan as a function of the _GDD_ at emergence. A positive slope (_d_ > 0) increase the lifespan of the leaves emitted later in the season. It is noted that senescence is modulated only as a function of accumulated temperature while other possible stress factors (i.e., water deficiency) are neglected. In order to select a parametric function to model the temporal evolution of partitioning into leaves we inspected the partitioning look-up tables of the crop growth model WOFOST [34]. Such look-up tables are crop specific, derived from experimental data, and express partitioning into leaves as a function of crop development stage, in turn function of _GDD_. A preliminary analysis of such data showed that partition into leaves could be well fitted by a three parameters logistic decay function of _GDD_ (data not shown). Although the functional form was selected to mimic partitioning in crops, it appears generic enough to be used for grassland as well. \[Pl(GDD)=Pl_{max}\frac{1}{1+e^{k(GDD-GDD_{0})}} \tag{6}\]where \(Pl_{max}\) is the curve's asymptotic maximum value, \(k\) is the steepness of the curve, and \(GDD_{0}\) is the x-value of the logistic midpoint. For the intended application it was convenient to express \(k\) and \(GDD_{0}\) as a function of the \(GDD\) values \(GDD_{min}\) and \(GDD_{max}\) at which the function takes the actual values of \((1-co)*Pl_{max}\) and co * \(Pl_{max}\), where \(co\) is cutoff value used to numerically deal with the asymptotic function. The cutoff value \(co\) is set to 0.05 to yield function values close to its asymptotic maximum (\(Pl_{max}\)) and minimum (0). \(GDD_{max}\) is the \(GDD\) at which there is (nearly) no more partitioning to leaves. \(GDD_{min}\) is the \(GDD\) at which partitioning to leaves is at its maximum, here assumed to be at emergence, where \(GDD\) is equal 0. By fixing this latter we reduce the parameters of the logistic curve to two. Using the following intermediary variables:  
\[C_{min}=ln\left(\frac{1}{1-co}-1\right),\ C_{max}=ln\left(\frac{1}{co}-1\right) \tag{7}\]  
We can rewrite \(k\) and GDD\({}_{0}\) of Equation (6) as:  
\[k=\frac{C_{max}-C_{min}}{GDD_{max}},\ GDD_{0}=\frac{1}{k}C_{max} \tag{8}\]  
The stress factor \(\varepsilon_{ws}\) is computed to account for possible water limitations. The formulation is similar to that of the CASA model [35] where photosynthesis is limited by water availability up to a maximum of 50%:  
\[\varepsilon_{ws}=\min\ (1,\ 0.5+0.5\ \text{AET}/\text{PET}) \tag{9}\]  
where AET and PET are the actual and potential evapotranspiration. We assume that all the precipitation can be used for evapotranspiration (i.e., AET equals precipitation). Differently from the CASA model and similar applications (e.g., [36; 37]) that compute AET and PET over a period of one to two months (extending backwards in time) we extend back in thermal time, i.e., \(GDD\). In this way, by fixing a \(GDD\) period over which precipitation and potential evapotranspiration are considered, we extend back in time over a shorter (longer) period if the temperature was warmer (colder). In addition, considering that more recent weather conditions have a greater effect on the physiological status of the plant we average PET and precipitation with exponentially declining weights when going back into the past (in the \(GDD\) dimension). For instance, for the precipitation we compute the following exponentially weighted precipitation (EWP, used in Equation (9) in place of AET):  
Figure 1:0  
where \(x\) is the thermal time extending back from \(GDD\) at time i, \(CAP\) is the maximum value of daily precipitation (set to 200 mm, above this threshold water is assumed to exceed storage capacity), and \(hl\) is the exponential decay half-life that controls the steepness of the weights decay.