## 2 Materials and Methods
### Data Processing
#### 2.4.6 Energy Components and Variations in Water Use Efficiency (WUE)  
Net radiation \(R_{n}\) (W/m\({}^{2}\)), the difference between incident and outgoing radiation energy, was calculated as:  
\[R_{n}=(1-\alpha)R_{s}+\sigma\varepsilon_{s}\varepsilon_{a}T_{a}^{4}-\sigma \varepsilon_{s}T_{s}^{4}, \tag{7}\]  
where \(R_{s}\) is the incoming solar radiation measured by the weather station at the site, W/m\({}^{2}\); \(\alpha\) is the land surface albedo derived from Equation (4); \(\sigma=5.67\times 10^{-8}\) W/m\({}^{2}\)/K\({}^{4}\), Stefan-Boltzmann constant. \(T_{s}\), \(T_{a}\), \(\varepsilon_{s}\), and \(\varepsilon_{a}\) have the same definitions as in Equation (6).  
Latent heat flux (\(\lambda ET\), (W/m\({}^{2}\)), the \(ET\) in energy form) was estimated as the residual component of the surface energy balance:  
\[\lambda ET=R_{n}-H-G, \tag{8}\]  
where \(R_{n}\) is the net radiation from Equation (7), \(H\) [W/m\({}^{2}\)] is sensible heat flux, and \(G\) [W/m\({}^{2}\)] is the heat storage flux in soil and plants. G is estimated from net radiation \(R_{n}\) and fractional vegetation cover \(f_{c}\) using an empirical equation in [44]:  
\[G=R_{n}[0.315(1-f_{c})+0.05f_{c}], \tag{9}\]  
where \(f_{c}\) is calculated from the aforementioned directional gap fraction at the nadir-viewing direction (see Supplementary material): \(f_{c}=1-b(0)=1-e^{-0.245 LAI}\).  
The sensible heat flux \(H\) was estimated from the temperature difference between the land surface and the air using the bulk transfer equation:  
\[H=\rho C_{p}\frac{T_{s}-T_{a}}{r_{a}+r_{ex}}, \tag{10}\]  
where \(\rho\) is the air density, \(C_{p}\) is the specific heat of air at constant pressure, and \(r_{a}\) is aerodynamic resistance to heat transfer, estimated from wind speed and the average height of rice canopy following [45]; \(r_{ex}\) is an extra resistance added to \(r_{a}\) for correcting the differ ence between radiometric temperature and aerodynamic temperature in sensible heat flux calculation. \(r_{ex}\) is approximated by \(5.75/u_{*}\)[46; 47], where \(u_{*}\) is friction velocity. Note that Equation (10) uses the surface temperature as a proxy of aerodynamic temperature, even though \(r_{ex}\) is introduced to account for this, underestimates on \(r_{ex}\) may result in an overestimation of the sensible heat flux for bare soil area from an overestimated aerodynamic temperature, and subsequently result in negative evapotranspiration in Equation (10). When this happened, we assigned a zero value for evapotranspiration (reasonable for dry bare soil) and recalculated sensible heat flux \(H\) using Equation (8). The available energy (\(AE\), (W/m\({}^{2}\))) was estimated as the difference between net radiation \(R_{n}\) and the ground heat flux G, i.e., \(AE=R_{n}-G\). Evaporative fraction (\(EF\), [-]) is the latent heat flux normalized by the available energy: \(EF=\lambda ET/AE\).  
The variations in \(WUE\) after biochar application were inferred from differences between the \(GPP\) variation and the \(ET\) variation using the variation propagation method again (see Supplementary Material for further details):  
\[\frac{\Delta WUE}{WUE}=\frac{\Delta GPP}{GPP}-\frac{\Delta ET}{ET}. \tag{11}\]