# Annual Gross Primary Production from Vegetation Indices: A Theoretically Sound Approach
## 3 Materials and Methods
### Methodology  
Hypotheses 2 and 3 are supported by literature (see Section 1) and, therefore, they are accepted as a stating point of our methodology. Although H1 is also a sound hypothesis, its adequacy to our study area is assessed by statistical tests. They allow us to evaluate whether PAR inter-annual variations introduce negligible changes in annual GPP as compared to those due to the inter-annual variability of the VI.  
As mentioned at the end of Section 2, once H1 is confirmed, \(\text{PAR}_{i}^{\circ}\) is calculated. This is considered as a typical or representative daily PAR series for each pixel of the study area. It is obtained by averaging the daily PAR for the eight-year dataset. The representative annual PAR is computed by summing the daily values, \(\Sigma\text{PAR}_{i}^{\circ}\). For each year \(Y\) from 2005 to 2012, the average VI images (\(\overline{\text{NDVI}}^{Y}\) and \(\overline{\text{EVI}}^{Y}\)) are obtained using Equation (7) and the \(\text{GPP}_{\text{annual}}^{\text{REF}}\) images are calculated using Equation (2) with the daily GPP images at 1-km spatial resolution described in Section 3.2.2.  
Next, correlations between \(\overline{\text{NDVI}}^{Y}\) and \(\text{GPP}_{\text{annual}}^{\text{REF}}\) images, and between \(\overline{\text{EVI}}^{Y}\) and \(\text{GPP}_{\text{annual}}^{\text{REF}}\) images, are analyzed to establish two sets of coefficients \(C_{1}\) and \(C_{2}\). The calibration (obtaining \(C_{1}\) and \(C_{2}\)) and testing of the semi-empirical model are performed simultaneously by means of a cross-validation procedure [46]. This procedure allowed us to calculate and test the model parameters from the eight years data used in this study. The cross-validation method systematically removes one case in a dataset (a year data in our case), derives a forecast from the remaining cases, and tests it on the removed case. The procedure is nonparametric and can be applied to any automated model-building technique.  
To carry out the cross-validation procedure, data are divided into eight partitions corresponding to each one of the eight available annual images of GPP and weighted NDVI. The procedure is as follows. For year \(Y\) (=2005-2012), the images of years \(\
eq Y\) are used to train a linear model. The model is validated using the images of year = \(Y\). The coefficient of correlation (R), MBE (mean bias error), MAE (mean absolute error), RMSE (root mean square error), and their relative values (rMBE, rMAE, and rRMSE) are calculated. Finally, the mean and standard deviation of the regression coefficients are calculated and considered as the final regression coefficients of the model and their associated errors.  
The cross validation is carried out in two ways: (i) considering a constant value of the conversion efficiency (\(C_{1}\) and \(C_{2}\) are independent of the land-cover class); and (ii) considering that \(C_{1}\) and \(C_{2}\) depend on vegetation, according to the two thematic maps described in Section 3.2. A flowchart for GPP\({}_{\text{annual}}\) estimation using the semi-empirical model is shown in Figure 2.