# Annual Gross Primary Production from Vegetation Indices: A Theoretically Sound Approach
## 1 Introduction  
The analysis of changes in the long-term estimates of the CO\({}_{2}\) ecosystem-atmosphere fluxes is important to establish the global carbon balance, especially in forest ecosystems, the main sinks of atmospheric carbon in the biosphere. A key factor of this balance is the gross primary production (GPP), which represents the total carbon uptake through photosynthesis per unit of time and per unit of area. Although GPP is the source of carbon for all carbon fluxes in the ecosystem, there is no direct method of measurement. GPP is calculated from in situ net primary production data acquired in eddy covariance (EC) towers after correction for respiration losses [1; 2]. Ecosystem models [3] validated against EC data and combined with meteorological and remotely sensed data [4; 5] allow for the estimation of GPP across space and time and, hence, for the quantification of carbon fluxes at regional to global scales [2].  
The GPP is the rate at which vegetation converts light into chemical energy by photosynthesis. As the photosynthesis is driven by solar radiation, GPP can be estimated using the Monteith equation [6].  
\[\text{GPP}=\varepsilon f_{\text{APAPAP}}\text{PAR} \tag{1}\]where PAR is the flux density of photosynthetically-active radiation, i.e., the global incident radiation in the range 400-700 nm;\(f_{\mathrm{APAR}}\) is a structural variable that represents the fraction (between 0 and 1) of incident PAR absorbed by the canopy; and \(\varepsilon\) (g\(\cdot\)MJ\({}^{-1}\)) is the conversion efficiency (also known as light-use efficiency (LUE)). When the quantities in Equation (1) are defined on a daily basis, a subscript \(i\) ranging from 1 to 365 is added to them. The units of GPP\({}_{i}\) and PAR\({}_{i}\) are g\(\cdot\)m\({}^{-2}\cdot\)day\({}^{-1}\) and MJ\(\cdot\)day\({}^{-1}\), respectively.  
The annual GPP for the year \(Y\), in kg\(\cdot\)m\({}^{-2}\cdot\)yr\({}^{-1}\), is the sum of the corresponding daily values:  
\[\mathrm{GPP}_{\mathrm{annual}}^{Y}=\sum_{i}\mathrm{GPP}_{i}^{Y}=\sum_{i} \varepsilon_{i}^{Y}f_{\mathrm{APAR}_{i}}^{Y}\mathrm{PAR}_{i}^{Y} \tag{2}\]  
Taking into account the loss due to autotrophic respiration, the GPP\({}_{\mathrm{annual}}\) can be used to assess the carbon sequestration into dry matter during a year (annual net primary production), and hence to quantify the changes in biomass over long periods of time. However, obtaining long time series of GPP\({}_{\mathrm{annual}}\) from daily GPP values is a time-consuming process that requires a huge number of inputs that might be unavailable, especially in retrospective studies [4].  
In this work, a semi-empirical model that avoids the calculation of all the factors in Equation (1) is proposed to estimate the annual GPP. As the next section will show, our approach relies on vegetation indices (VIs) products commonly available such as the Normalized Difference Vegetation Index (NDVI) [7] and the Enhanced Vegetation Index (EVI) [8]. In this case, the two MODIS standard VI products have been used. On the one hand, MODIS NDVI is included since it is referred as the \"continuity index\" [8] to the existing 30+ year NOAA-AVHRR derived NDVI time series, which can be extended to provide a longer-term data record required in climatic studies. On the other hand, EVI has been shown to improve NDVI sensitivity in dense vegetation regions and a better performance to minimize soil and atmosphere influences [8].