## 1 Introduction
Many of these approaches use flux tower data to train or calibrate their models with a limited number of sites or PFTs and have not been tested globally. Some of the noted problems using VIs are (1) a decrease in sensitivity at high values, particularly in managed croplands (e.g., [25; 32]) and (2) a positive offset that occurs when photosynthetic activity should be very low or zero (e.g., [20; 26; 33; 34]) (i.e., these parameters have positive values when photosynthesis is not occurring). [PERSON] et al. [35] found that LUE models in general tended to overestimate GPP at low values and underestimate at high values and that most LUE models do not account for the fact that LUE is higher on cloudy days with more diffuse radiation (e.g., and referencestherein [36]). However, some LUE model formulations have included the radiation effect on LUE [37; 38]. [PERSON] et al. [39] and [PERSON] et al. [40] found that the use of potential PAR (PAR\({}_{\text{pot}}\), i.e., the PAR at top of canopy that would be present in a cloudless atmosphere) in a simple model, i.e.,  
\[\text{GPP}\propto\text{VI}\times\text{PAR}_{\text{pot}}, \tag{3}\]  
produced superior results as compared with PAR\({}_{\text{in}}\) in cropland vegetation. The improvement obtained by the use of PAR\({}_{\text{pot}}\) in place of PAR\({}_{\text{in}}\) with constant LUE implicitly accounts for the diffuse radiance effect on LUE by producing an effectively higher LUE at lower values of PAR\({}_{\text{in}}\). [PERSON] et al. [40] notes that this model does not account for variations in GPP produced by short time scale (minutes to hours) stresses. They also suggested testing of this model at other geographic locations and for other vegetation types. Here, we expand on this approach by training and evaluating models based on Equations (2) and (3) with available global GPP estimates from flux tower observations. We then test the satellite-based trained model using independent global GPP from flux towers (i.e., not used in the training). This is a simple and computationally-efficient method to estimate global GPP with a limited number of remotely-sensed parameters with a focus of daily to monthly time scales. We specifically examine whether a modified function of NDVI or a linear combination of band reflectances can accurately represent FAPAR\({}_{\text{chl}}\) to mitigate the above-mentioned shortcomings of VIs. We further explore whether satellite measurements of chlorophyll solar-induced fluorescence (SIF) may be used to improve estimation of global GPP. SIF originates from the photosynthetic machinery inside leaves and has been measured with several satellite instruments [41; 42; 43; 44; 45; 46; 47; 48]. The spectrum of chlorophyll fluorescence is spread over a range of wavelengths with two broad peaks, centered near 685 and 740 nm, referred to as the red and far-red emissions, respectively. Radiative transfer and physiological modeling suggests that canopy-level far-red SIF is strongly related to APAR [49]. Most satellite SIF studies have been conducted using far-red SIF, and henceforth, in this work, SIF will refer to far-red SIF. Several studies have shown a highly linear relationship between far-red canopy-level SIF and GPP on monthly time scales [32; 46; 50; 51; 52; 53; 54] that can be explained by modeling results [55]. Far-red SIF also closely follows seasonal [33] and daily to diurnal [54; 56] variations in GPP. SIF measurements also alleviate the above-mentioned short-comings of VIs [32; 33; 57]. However, unlike VIs, SIF is a very small and difficult to measure signal. As a result, satellite SIF measurements are relatively noisy, prone to systematic errors [44; 50; 58] and currently limited in terms of both spatial and temporal resolution. Here, we use attributes of both satellite-based reflectances and SIF in a computationally-efficient way to estimate global GPP accurately at high temporal and spatial resolution.