## 4 Improvements in PRI Implementation
### Combining with Other Parameters to Evaluate Carbon Fixation  
Combining PRI with other vegetation indices (VIs) was a promising approach in the review by [PERSON] et al. [20] to improve the detection of carbon uptake. The conventional and widely used RUE model expresses GPP as the product of APAR and RUE. PRI is a proxy of RUE, so some VIs (e.g., normalized difference spectral indices (NDSI), NDVI, the MERIS terrestrial chlorophyll index (MTCI) and EVI) correlate well with the fraction of APAR (\(\mathrm{\it{GAP}}\)) [10; 14; 15; 16; 17; 20; 24; 129; 136]. GPP can thus be efficiently assessed with the combination of MODIS VIs and PRI, and the correlations were  
Figure 10: Comparison of the relationships between RUE/GPP vs. PRI and RUE/GPP vs. improved/combined PRI. (**a**) PRIc was used for \(Q\). _robur_ and _F. sylvatica_[86], a mixed canopy of \(Q\). _robur_, _F. sylvatica_ and _P. sylvestris_[107] and a deciduous broadleaved forest [123]; \(\mathrm{PRI_{R1}}\) and \(\mathrm{PRI_{R2}}\) were used for _T. aestivum_[109] and rPRI for _T. aestivum_ (mixed non-linear model, \(\mathrm{rPRI=PRI-(0.15\times(1-\exp(-0.5\times\mathrm{LAI}))-0.2)}\)) [124]; (**b**) Tracking RUE combining PRI with SIF (red) or SIF (far-red) yield in \(Z\). _maps_ (RUE = a + b \(\times\) PRI + c \(\times\) SIF + d \(\times\) PRI \(\times\) SIF) [81] and with VPD in a dipterocarp forest (RUE = 0.153 \(\times\) PRI \(-\) 0.00067 \(\times\) VPD + 0.029) [77]; (**c**) Tracking GPP combining PRI with SIF (red) or SIF (far-red) in \(Z\). _maps_ (GPP = a + b \(\times\) PRI + c \(\times\) SIF + d \(\times\) PRI \(\times\) SIF) [81], with NDSI (GPP = (a0 \(\times\) sPRI + a1) \(\times\) (a2 \(\times\) NDSI + a3) \(\times\) PARi, sPRI = (1 + PRI)/2) and SIF (far-red, GPP = (a0 \(\times\) sPRI + a1) \(\times\) (a2 \(\times\) SIF + a3), sPRI = (1 + PRI)/2) in \(O\). _sativa_[129] and with VI (MTCI, NDVI and EVI) and In(PAR) (GPP = (a0 \(\times\) PRI + b0) \(\times\) (a \(\times\) VI + b) \(\times\) In(PAR)) [136] in a subalpine grassland.  
22%-77% higher than with PRI (Figure 10c) [136]. Estimation of seasonal GPP based on MTCI-related fAPAR and PRI-related RUE (PRI551, MODIS band 4 as the reference band) modeled a subalpine grassland best (\(R^{2}\) = 0.90) [136].  
SIF, another photoprotective mechanism successfully sensed from space, is also an indicator of vegetation function and is used for carbon modeling [157; 158]. The combination of SIF with PRI is strongly correlated with RUE and GPP for crops (Figure 10b,c). Combining PRI and SIF in a linear regression model improved GPP estimation for a cornfield (\(R^{2}\) = 0.8 for SIF at red band and 0.78 for SIF at far-red band, Figure 10c) [81]. Gross ecosystem productivity (GEP) for a rice crop was best estimated based on APAR as a function of steady-state SIF computed at 760 nm (SIF(far-red)) and on RUE as a function of sPRI (Figure 10c) [129], which was the first study that modeled seasonal courses of GEP based on measurements of remotely sensing fluorescence. [PERSON] et al. [77] demonstrated that the accuracy of estimating RUE (Figure 10b) and the estimation error were improved for an evergreen tropical rainforest by applying a regression model using PRI and VPD, probably due to the influence of the water conditions on unseasonal variation in RUE.