## 3 Results
### Overall Performance of SIF Emission in Predicting Hourly and Daily WUEs  
#### 3.2.1 Performance in Predicting Hourly WUEs  
The coefficients of determination (\(R^{2}\)) show that SIF\({}_{687}\) is a weak predictor of the selected WUEs. It shows almost no correlation with all the four WUEs, and its best \(R^{2}\) is only 0.17 when it is used to estimate IWUE (Table 1). In comparison to SIF\({}_{687}\), SIF\({}_{720}\) obtains an obvious improvement in predicting IWUE and uWUE and it accounts for 48% and 36% variances, respectively, by using the GPR models (Table 1). However, SIF\({}_{720}\) has weak correlations with WUE and WUE\({}_{i}\), as they can only explain at most 8% and 14% of the variance in them (Table 1). Among the three single SIF bands, SIF\({}_{761}\) shows the greatest strength in predicting hourly IWUE and uWUE and it determines 52% and 34% of the variability in them, but SIF\({}_{761}\) again shows poor correlations with WUE and WUE\({}_{i}\) even when the GPR models are used (Table 1).  
Our results confirm that the SIF band combinations provide better predictive power than the single SIF bands when they are used individually. Both of the two combinations have the best performance in estimating IWUE. The LR results show that the combination of SIF\({}_{687}\) and SIF\({}_{761}\) account for 49% variability in hourly IWUE (Table 1). The more SIF bands are used in the combination, the greater the likelihood of yielding a stronger predictive power. The combination of SIF\({}_{687}\), SIF\({}_{720}\), and SIF\({}_{761}\) produces further improvement, which governs about 50% variability in IWUE (Table 1). Both of the two \(R^{2}\) values are better than that provided by SIF\({}_{761}\), which is the best single SIF band. Also, the GPR models further enhance the predictive power of the band combinations. When the GPR models are used, the two combinations can determine 62% and 65% of the IWUE variances, respectively (Table 1).  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**SIF bands**} & \multicolumn{2}{c}{**WUE**} & \multicolumn{2}{c}{**WUE\({}_{i}\)**} & \multicolumn{2}{c}{**IWUE**} & \multicolumn{2}{c}{**uWUE**} \\ \cline{2-9}  & **LR** & **GPR** & **LR** & **GPR** & **LR** & **GPR** & **LR** & **GPR** \\ \hline SIF\({}_{687}\) & 0.09 & 0.14 & 0.07 & 0.12 & 0.13 & 0.17 & 0.02 & 0.05 \\ SIF\({}_{720}\) & 0.00 & 0.08 & 0.11 & 0.14 & 0.42 & 0.48 & 0.23 & 0.36 \\ SIF\({}_{761}\) & 0.00 & 0.08 & 0.14 & 0.17 & 0.44 & 0.52 & 0.25 & 0.34 \\ SIF\({}_{687}\), SIF\({}_{761}\) & 0.24 & 0.44 & 0.15 & 0.20 & 0.49 & 0.62 & 0.39 & 0.55 \\ SIF\({}_{687}\), SIF\({}_{720}\), SIF\({}_{761}\) & 0.23 & 0.46 & 0.15 & 0.28 & 0.59 & 0.65 & 0.36 & 0.61 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The correlation coefficient (\(R^{2}\)) between the SIF bands and water use efficiency (WUE), intrinsic water use efficiency (WUE\({}_{i}\)) inherent water use efficiency (IWUE), and underlying water use efficiency (uWUE), by using linear regression analysis (LR) and Gaussian processes regression (GPR), respectively. All datasets have an hourly time step and are from between 7 a.m. and 5 p.m. local time.  
Figure 2: The time series of hourly (**a**) SIF\({}_{687}\) (mW/m\({}^{2}\)/sr/nm), (**b**) SIF\({}_{720}\) (mW/m\({}^{2}\)/sr/nm) and (**c**) SIF\({}_{761}\) (mW/m\({}^{2}\)/sr/nm) at the Harvard Forest site in 2014. The red lines represent 4-day daytime averages. The values in the top \(x\)-axes represent months.  
In fact, the band combinations also produce considerable improvements in \(R^{2}\) with the other three WUEs over using the single SIF bands (Table 1). For example, the combination of \(\text{SIF}_{687}\), \(\text{SIF}_{720}\), and \(\text{SIF}_{761}\) achieves \(R^{2}\) of 0.46, 0.28, and 0.61, with WUE, \(\text{WUE}_{\text{i}}\), and uWUE, respectively, whereas single \(\text{SIF}_{761}\) yields \(R^{2}\) of 0.08, 0.17 and 0.34 for these three WUEs (Table 1).