## 1 Introduction  
Global terrestrial gross primary productivity (GPP), the amount of carbon dioxide (CO\({}_{2}\)) that is assimilated by plants through photosynthesis, plays a critical role in the global carbon cycle as it can moderate the amount of CO\({}_{2}\) from anthropogenic sources that remains in the atmosphere. Despite the collection of ground-based data at hundreds of individual sites, as well as vast amounts of remotely-sensed data from satellites and the development of sophisticated process-based models, there still remains significant uncertainty in global GPP estimates including their global annual mean and inter-annual variability (IAV) [1]. Estimation of the variability in the land sink for global carbon budget assessments relies upon dynamic global vegetation models that are driven by observed environmental changes [2]. [PERSON] et al. [3] note that the understanding of the land sink is currently limited in part by a dearth of spatially-explicit observations of vegetation changes and recommend systematic benchmarking of the models, including productivity, for improvement. GPP is not measured directly either at the ground or from satellite. Using the eddy covariance technique, tower-based measurements of net ecosystem exchange (NEE) are partitioned into GPP and ecosystem respiration using assumptions regarding the latter (e.g., [1; 4; 5] and references therein). Despite the assumptions used in this approach, tower-based estimates are considered a standard for GPP estimates. Approaches to scale up these measurements and extrapolate globally involve first deriving relationships between tower-based GPP estimates and parameters from satellite-based instruments and meteorological analyses and then applying these parameterizations globally (e.g., [6; 7; 8; 9; 10]). [PERSON] et al. [9] demonstrated that such approaches that use only remotely-sensed data perform almost as well as the best model that also incorporates meteorological inputs. Other data-driven approaches to estimate GPP globally use satellite data in combination with models and various assumptions (see for comprehensive reviews [1; 11]). Many of these are based on the light-use efficiency (LUE) concept [12; 13], i.e.,  
\[\mathrm{GPP}=\mathrm{LUE}\times\mathrm{FAPAR}_{\mathrm{chl}}\times\mathrm{ PAR}_{\mathrm{in}}, \tag{1}\]  
where PAR\({}_{\mathrm{in}}\) is photosynthetically-active radiation incident at the top of canopy and FAPAR\({}_{\mathrm{chl}}\) is the fraction of PAR\({}_{\mathrm{in}}\) absorbed by chlorophyll. Several studies have used the LUE model with space-based estimates of FAPAR\({}_{\mathrm{chl}}\) or a proxy such as from satellite-derived vegetation indices (VI), PAR\({}_{\mathrm{in}}\) estimates from a data assimilation system or other measurements, and parameterizations of LUE that use meteorological data as inputs to estimate GPP globally or on a smaller scale (e.g., [14; 15; 16; 17; 18; 19; 20; 21]). VIs are computed with a limited number of broad-band channels on satellite imagers. VIs that have been used or considered for estimating GPP include the dual wavelength Normalized Difference Vegetation Index (NDVI) [22] and the Near-Infrared Reflectance of terrestrial Vegetation (NIR\({}_{V}\)) [23] that is defined as the product of NDVI and reflectance from the near-infrared channel used in the NDVI. The potential benefit of using VIs for GPP estimation is that they can be simply derived from satellite reflectances. In particular, an NDVI record is available for over three decades [24]. VIs have been used to compute GPP with an approximation to Equation (1), i.e.,  
\[\mathrm{GPP}=S\times\mathrm{VI}\times\mathrm{PAR}_{\mathrm{in}} \tag{2}\]  
(e.g., [25]) or other similar forms (e.g., [26]) where VI is used as a proxy for FAPAR\({}_{\mathrm{chl}}\). LUE, here represented as \(S\), may be treated a constant [27; 28] or as a more complex function of VI itself [29; 30] or other parameters where \(S\) may vary with plant functional type (PFT) (e.g., [31]).