## 4 Improvements in PRI Implementation
### Different Formulations of PRI  
The application of the xanthophyllic cycle and different reference bands has been discussed extensively for better understanding PRI (Table 2). [PERSON] et al. [72] used PRI calculated as (R545 - R570)/(R545 + R570), which has been associated with the rapidly conformational changes contributing to protonation in key proteins in the thylakoid membrane during reversible NPQ and has improved the correlations between NPQ and PRI, especially during early spring decoupling in conifers. A combination of a vegetation index of the xanthophyllic de-epoxidation band (531 nm) with a band associated with chlorophyllic content (701 nm from CHRIS) performed better than PRI for estimating seasonal RUE from the backscatter direction [60]. PRI515 (or PRI512) has recently been proposed to minimize the effects of canopy structure under different stresses [69; 101; 126; 117; 120] but was not significantly correlated with GPP for an olive orchard [149] and performed slightly weaker than PRI for crops under water stress [148; 150]. [PERSON] et al. [148] reported that PRI570 and PRI586 correlated with \(\Delta\)F/Fm\({}^{\prime}\) (\(R^{2}\) = 0.49 and 0.51, respectively) for cereal crops under water stress better than other PRIs based on different reference wavelengths. Additionally, PRIs based on green reference bands at 555 and 551 nm correlated mostly with RUE, and PRIs based on reference bands at 645 and 667 nm correlated better with the leaf chlorophyllic concentration than with RUE [136]. The first derivative of PRI with respect to shadow fractions viewed by the sensor (\(\Delta\)PRI\(\Delta\)\({\alpha_{s}}^{-1}\) or PRI\({}^{\prime}\)) was developed to alleviate the impacts of vegetation structure and radiometric properties on the basis of a multi-angular observation algorithm, which has been used to infer RUE across different biomes [4; 71; 151; 154]. The proposed PRI\({}^{\prime}\) technique, however, was probably not applicable to infer RUE for canopies with reduced variability in shadow fractions as noted by [PERSON] et al. [71]. [PERSON] and [PERSON] [67] showed that a combination of dark-state sampling (cPRI) and dark-to-light conversion (APRI) could experimentally isolate the two (constitutive and facultative) effects on PRI in situ without extensive destructive sampling, which could help to interpret the variation in the PRI signal from remote platforms. Differences in canopy PRI (APRI) from the minimum reference PRI near midday [110] were similarly applied and improved the correlations between RUE and PRI (\(R^{2}\) = 0.24 for PRI and 0.5 for \(\Delta\)PRI). Differences in foliar PRI (\(\Delta\)PRI, dawn PRI minus midday PRI) [30] improved the correlation with the maximum photosynthetic rate (\(R^{2}\) = 0.21 for PRI and 0.44 for \(\Delta\)PRI). Some studies also reported the use of scaled PRI (sPRI = (PRI + 1)/2) to avoid negative values and for comparable analysis [115; 129; 25; 135]. A corrected PRI (PRIc = PRI - PRI0), after subtracting an estimated PRI0 defined as PRI of perfectly dark-adapted leaves, correlated well with RUE at diurnal [86], seasonal [86; 123] and inter-annual scales [123]. This approach considered the impacts of changes in canopy structure and pigments, which greatly improved the correlations between PRI and RUE (Figure 10a) with a increment of \(R^{2}\) by 71% and 92% at the foliar level in oak and beech, respectively [86], and by 22% for a deciduous broadleaved forest [123]. The correlation between PRI and RUE, however, was not improved by correcting with PRI0 for an evergreen forest, because PRI0 varied little across the season [123]. [PERSON] et al. [107] developed an accurate deconvolution model for further improving PRI0 and PRIc to identify the influences of pigment and physiological changes on PRI variation and improved the correlation with RUE from an exponential correlation of 0.66 to a highly significant linear correlation of 0.93. [PERSON] et al. [135] similarly applied PRIc (method from [PERSON] et al.