## 2 Materials and Methods
### Flux and Meteorological Observations
#### 2.1.2 Transpiration Data Acquisition  
This study used three sets of daily \(T\) estimates from uWUE, Perez-Priego, and TEA methods [52].  
[PERSON] et al. and [PERSON] et al. assumed that an ecosystem has an actual underlying water use efficiency (uWUE\({}_{\mathrm{a}}\)), which is maximized or reaches its potential underlying water use efficiency (uWUE\({}_{\mathrm{p}}\)) when \(T/\mathrm{ET}\) approaches 1 [22; 23]. Therefore, \(T/\mathrm{ET}\) can be calculated based on the ratio of uWUE\({}_{\mathrm{a}}\) to uWUE\({}_{\mathrm{p}}\), using the optimal assumption for both.  
\[\mathrm{uWUE}_{\mathrm{p}}=\frac{\mathrm{GPP}\sqrt{D_{\mathrm{a}}}}{T} \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Site ID** & **Latitude** & **Longitude** & **Climate** & **Plant Functional** & **Measurement Height** & **Depths1 (cm)** & **Depth2** & **Temporal** & **Cover** \\  & **(\({}^{\circ}\)N)** & **(\({}^{\circ}\)E)** & **Type** & **Type** & **(Z, m)** & **Depth3 (cm)** & **Depth2** & **Temporal** \\ \hline BR-Sa3 [41] & \(-3.0180\) & \(-54.9714\) & Am & EBF & 64 & 10 & 20 & 2000–2004 \\ US-AR1 [42] & \(36.4267\) & \(-99.4200\) & Das & GRA & 2.84 & 10 & 30 & 2009–2012 \\ US-AR2 [43] & \(36.6383\) & \(-99.5975\) & Das & GRA & 2.95 & 10 & 30 & 2009–2012 \\ US-AR4 [45] & \(35.5497\) & \(-98.0402\) & Cra & GRA & 3.45 & 10 & 30 & 2005–2006 \\ US-ARM [45] & \(36.6058\) & \(-97.4888\) & Cfa & CRO & 4.28 & 10 (5 before 13 & 20 (25 before 13 & 2002–2013 \\ US-IB2 [46] & \(41.8406\) & \(-88.2410\) & Dfa & GRA & 3.76 & 2.5 & 10 & 2004–2011 \\ US-SRM [47] & \(31.8214\) & \(-110.8661\) & Bk & WSA & 7.82 & 2.5 & 12.5 & 2004–2014 \\ US-Ton [48] & \(38.4309\) & \(-120.9660\) & Cas & WSA & 23 & surface & 20 & 2001–2014 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of the 8 FLUXNET sites.  
\[\mathrm{uWUE_{a}}=\frac{\mathrm{GPP}\sqrt{D_{a}}}{\mathrm{ET}} \tag{3}\]  
GPP is gross primary productivity, and \(D_{a}\) is vapor pressure deficit (kPa).  
[PERSON] et al. used optimality theory for a more complex partitioning of ET using a big-leaf canopy model (big-leaf scheme) in which parameters were optimized using half-hourly data from a 5-day window so that the parameters for each 5-day window maximally satisfied the fit between the model and the observed GPP and also minimized water loss per carbon gain [19]. \(T\) was then calculated from the model using \(G_{\mathrm{s}}\), and \(E\) was calculated as the residual (ET-\(T\)).  
\[g_{\mathrm{surf}}=g_{0}+g_{1}\frac{\mathrm{GPP}}{D_{\mathrm{L}}^{\mathrm{m}}} \tag{4}\]  
where \(g_{0}\) (corresponding to soil conductance), \(g_{1}\) (corresponding to vegetation conductance), and m are optimization parameters, \(D_{\mathrm{L}}\) is the inferred leaf-level VPD, and \(g_{\mathrm{surf}}\) is assumed to represent the ecosystem conductance to water vapor flux.  
The method used independent soil moisture data associated with EC to classify the data with \(g_{0}\), \(g_{1}\), and m optimized in each classification to account for changes due to moisture limitation. The partitions were then calculated with the following equations:  
\[\frac{T}{\mathrm{ET}}=\frac{g_{1}}{g_{\mathrm{surf}}} \tag{5}\]  
\[\frac{E}{\mathrm{ET}}=\frac{g_{0}}{g_{\mathrm{surf}}} \tag{6}\]  
[PERSON] circumvented the assumption that \(T/\mathrm{ET}\) is close to unity at certain times by directly estimating ecosystem conductance. Meanwhile, the TEA of [PERSON] et al. [20] utilized a nonparametric model with a few assumptions, the Random Forest (RF). However, TEA must assume that \(T/\mathrm{ET}\) is close to 1, achieved by excluding observations when the surface may be wet. In a validation study using model outputs as a synthetic EC dataset with known \(E\) and \(T\), TEA could predict \(T/\mathrm{ET}\) patterns in space and time.