# Annual Gross Primary Production from Vegetation Indices: A Theoretically Sound Approach
## 2 A Theoretically Sound Approach
On a daily scale, the different types of stress must be accounted for at quasi-real time, which is rather difficult and frequently requires actual meteorological data. Nevertheless, it has been shown that after relatively long stress periods, the plant photosynthetic apparatus is damaged and ultimately leads to structural canopy alteration (i.e., a decrease in green LAI and therefore in \(f_{\text{APAR}}\)). This alteration can be noticed, with a time delay of one or two months [33; 34], in broad-band spectral indices such as the NDVI and narrow-band indices such as the PRI. In particular, numerous studies carried out in different environments have shown that NDVI is sensitive to vegetation water stress [35; 36]. It can be safely assumed that the effects of the different down-regulating factors at annual scale affect the \(f_{\text{APAR}}\), since stresses acting on the canopy finally produce either less chlorophyll and/or less foliage and reduce the VI value. Hence, at annual scale, the conversion efficiency \(\varepsilon\) can be considered to be independent of time, and approximated by the ecosystem maximum efficiency. As \(f_{\text{APAR},i}\) and \(\varepsilon_{i}\) can be confounded to varying degrees depending on the underlying dynamic biological processes and on their exact definition [27], with this approach we are avoiding a further insight in these concepts and in their exact operational definition. In fact, this approach takes into account the coordinate response or functional convergence [27] of both variables in the presence of nutrient or water constraints that limit plant growth as well as plant structure and pigment content. Table 1 summarizes the hypotheses concerning the estimation of \(\text{GPP}^{Y}_{\text{annual}}\) (H1 is analyzed in next section). Hence, the variations from one year \(Y\) to another are attributed to those of \(f_{\text{APAR}}\), i.e., to VI. From Equation (2) and the hypotheses in Table 1, an approximated \(\text{GPP}^{Y}_{\text{annual}}\) can thus be calculated as  
\[\text{GPP}^{Y}_{\text{annual}}=\varepsilon_{\text{max}}{\sum_{i}}\Big{[} \left(a+b\,\text{VI}^{Y}_{i}\right)\text{PAR}^{{}^{\circ}}_{i}\Big{]} \tag{3}\]  
\[\text{GPP}^{Y}_{\text{annual}}=C_{1}+C_{2}\overline{\text{VI}}^{Y} \tag{4}\]  
\[C_{1}=\varepsilon_{\text{max}}a\underset{i}{\sum}\text{PAR}^{{}^{\circ}}_{i} \tag{5}\]  
where  
\[C_{2}=\varepsilon_{\text{max}}b\underset{i}{\sum}\text{PAR}^{{}^{\circ}}_{i} \tag{6}\]  
That is, \(C_{1}\) and \(C_{2}\) are constant for a specific vegetation type, and \[\
abla\Pi^{Y}=\frac{\sum\limits_{i}\Pi^{Y}_{i}\mathrm{PAR}^{\circ}_{i}}{\sum \limits_{i}\mathrm{PAR}^{\circ}_{i}} \tag{7}\]  
This average VI is physically sound as it is an average over the energy received per surface area. Equations (3)-(7) require the series \(\Pi^{Y}_{i}\) and \(\mathrm{PAR}^{\circ}_{i}\) (which is the same for all years), and the parameters, \(a,b\) and \(\epsilon_{\mathrm{max}}\). In this work, the annual GPP is obtained at 1-km spatial resolution from the MODIS VI standard products (NDVI and EVI) using the semi-empirical model in Equation (4). After a brief analysis of H1, the coefficients \(C_{1}\) and \(C_{2}\) are obtained and discussed as a function of the vegetation type. For the sake of clarity, \(\mathrm{GPP}^{\mathrm{REF}}_{\mathrm{annual}}\) refers to annual GPP values obtained using Equation (2) since they are considered our \"reference values\", whereas \(\mathrm{GPP}_{\mathrm{annual}}\) refers to estimated values using the semi-empirical model. Data from 2005 to 2012 are employed in the study. The performance of the model is assessed in its ability to reproduce annual GPP inter-annual variations in close agreement with those shown by the reference data.