# Annual Gross Primary Production from Vegetation Indices: A Theoretically Sound Approach
## 2 A Theoretically Sound Approach  
The incident irradiance at the surface depends on the extraterrestrial irradiance and on the atmospheric conditions. The brightness variation of the Sun due to the sunspot cycle (with a period of ca. 11 years) provokes changes in the extraterrestrial global irradiance smaller than 0.1% [9]. Hence, the extraterrestrial irradiance is practically constant. However, the surface global irradiance at a given location might be highly variable since it is affected by random changes due to atmospheric conditions, which introduce an inter-annual variation. Regarding the long-term trends in the global irradiance at the surface, a general decrease (\"global dimming\") has been reported during the last half century both in the northern and southern hemisphere, which might be related to a decrease of atmospheric transmittance due to changes in cloud characteristics and in atmospheric aerosols [10; 11]. Nevertheless, in the last three decades, these changes have been less significant or even ceased [12]. According to the above considerations, the inter-annual variations of the annual PAR should be negligible. Although PAR\({}_{i}^{Y}\) in Equation (2) might show random inter-annual variations associated to atmospheric state, it can be approximated (at each location) by a typical or representative value PAR\({}_{i}^{\circ}\) that shows the main seasonal periodicity (associated exclusively to geometric factors). This assumption means that the effect of the changes of solar irradiance on annual GPP is less important than that due to vegetation changes (as discussed in Section 4). Radiation explains most of the variation in canopy photosynthesis over short periods, but environmental factors such as water availability become more important over long periods [12; 13; 14; 15]. Since alterations in global solar irradiance only have a small impact on GPP\({}_{\mathrm{annual}}\), the observed inter-annual variability in GPP\({}_{\mathrm{annual}}\) has to be related to structural changes that affect the\(f_{\mathrm{APAR}}\), such as changes in leaf area index (LAI) or leaf area duration. The \(f_{\text{APAR}}\) is related to the vegetation structure and pigment content. It is frequently obtained from remote sensing data using vegetation indices (VIs) [16; 17; 18; 19; 20]. Some authors use the VIs as proxies of both \(\varepsilon\) and \(f_{\text{APAR}}\) and rewrite Equation (1) as GPP = VI \(\times\) VI \(\times\) PAR [21; 22]. In the latter case, both broad-band VIs and narrow-band indices can be considered. The Photochemical Reflectance Index (PRI) is a narrow-band VI [23] that is potentially connected with \(\varepsilon\)[24]. The GPP has also been directly related to PAR through the chlorophyll content, which is in turn closely related to green \(f_{\text{APAR}}\)[25; 26]. In any case, a linear relation with a vegetation index, such as the NDVI and the EVI, is a realistic approximation for the evaluation of \(f_{\text{APAR}}\). This has been sufficiently shown by several authors, as mentioned above, and does not require further analysis. As commented by some authors [27], whereas \(f_{\text{APAR},i}\) is related to the vegetation structure and pigment pools, the conversion efficiency is related to the physiology. The daily conversion efficiency \(\varepsilon_{i}\) can vary spatially between biomes, ecosystems, and plant species, and temporally during the growing season due to environmental and physiological limitations [28; 29; 30; 31; 32]. The daily conversion efficiency is generally taken as the product of a biome-specific maximum value (g\(\cdot\)M)\({}^{-1}\)) and several dimensionless factors accounting for the efficiency reduction due to different types of stress, such as the thermal and the water stress. Our study area is characterized by different Mediterranean ecosystems where the water stress introduces most of the inter-annual variability in \(\varepsilon_{i}\)[4; 13].