## 2 Materials and Methods
### Satellite-Based ET Modeling Framework
#### 2.3.2 TSEB-PM  
In the current study, the TSEB-PM model was run according to the iterative procedure described above, except that the calculation of \(\mathrm{T_{c1}}\) (Equation (11)), \(\mathrm{\lambda E_{c}}\) (Equation (12)), and subsequent iterations used a formulation for canopy transpiration based on the Penman-Monteith (PM) equation:  
\[\mathrm{T_{cl}}=\mathrm{T_{a}}+\frac{\mathrm{R_{nc}}\mathrm{r_{a}}\gamma^{*}} {\rho\mathrm{C_{p}}(\mathrm{\Delta}+\gamma^{*})}-\frac{\mathrm{e_{s}}- \mathrm{e_{a}}}{\mathrm{\Delta}+\gamma^{*}}, \tag{11}\]  
\[\mathrm{\lambda E_{c\_PM}}=\frac{\mathrm{\Delta R_{nc}}}{\mathrm{\Delta}+ \gamma^{*}}+\frac{\rho\mathrm{C_{p}}(\mathrm{e_{s}}-\mathrm{e_{a}})}{\mathrm{ r_{a}}(\mathrm{\Delta}+\gamma^{*})}, \tag{12}\]  
In Equations (11) and (12), \(\mathrm{r_{a}}\) is the aerodynamic resistance between the canopy and the air above canopy, \(\mathrm{e_{s}}\) and \(\mathrm{e_{a}}\) are the saturation and actual vapor pressures of the air, respectively, \(\gamma^{*}=\gamma(1+\mathrm{r_{c}}/\mathrm{r_{a}})\), \(\mathrm{r_{c}}\) is the bulk canopy resistance, and all other terms are as defined previously. Following [38], \(\mathrm{r_{c}}\) is set at 50 s m\({}^{-1}\) (see [67] for explanation). Analogous to TSEB-PT and the throttling back of \(\mathrm{\alpha}\) when \(\mathrm{LE_{s}}<0\), \(\mathrm{r_{c}}\) is incrementally increased by 10 s m\({}^{-1}\) under similar conditions until \(\mathrm{LE_{s}}\geq 0\). The substitution of PT for PM also required changes to the original \(\mathrm{T_{c}}\), \(\mathrm{T_{s}}\), and \(\mathrm{T_{ac}}\) calculations outlined in Appendix A of [27]. A detailed explanation of these changes is found in Appendix B of [38]. However, we include the core equations below for completeness. \(\mathrm{T_{c}}\) is calculated according to Equation [13] as the summation of linear (\(\mathrm{T_{c,LIN}}\), \(\mathrm{T_{s,LIN}}\); Equations (14) and (15) and small correction (\(\mathrm{\Delta T_{c}}\); Equation (16)) components.  
\[\mathrm{T_{c}}=\mathrm{T_{c,LIN}}+\mathrm{\Delta T_{c}}, \tag{13}\]  
\[\mathrm{T_{c,LIN}}=\frac{\frac{\mathrm{T_{a}}}{\mathrm{r_{a}}}+\frac{\mathrm{ T_{r}}}{\mathrm{r_{s}}(1-\mathrm{\ell_{c}})}+\left[\frac{\mathrm{r_{c}} \gamma^{*}\mathrm{R_{nc}}}{\rho\mathrm{C_{p}}(\mathrm{\Delta}+\gamma^{*})}- \frac{\mathrm{r_{s}}}{\mathrm{r_{a}}}\frac{(\mathrm{e_{s}}-\mathrm{e_{a}})}{( \mathrm{\Delta}+\gamma^{*})}\right]\left[\frac{1}{\mathrm\[\Delta\mathrm{T_{c}}=\frac{\mathrm{T_{r}^{4}}-\mathrm{f_{c}T_{c,LIN}^{4}}-(1- \mathrm{f_{c}})T_{s,LIN}^{4}}{4\mathrm{f_{c}T_{c,LIN}^{4}}+4(1-\mathrm{f_{c}})T_{ s,LIN}^{3}\Big{(}1+\frac{\mathrm{f_{a}}}{\mathrm{f_{a}}}\Big{)}}, \tag{16}\]  
All terms included in Equations (11)-(16) are previously defined above in this manuscript. Once \(\mathrm{T_{c}}\) is acquired, \(\mathrm{T_{s}}\) is calculated according to Equation (1) and \(\mathrm{T_{ac}}\) is calculated in terms of only temperatures and resistances that are previously defined and calculated (Equation (17)).  
\[\mathrm{T_{ac}}=\frac{\frac{\mathrm{T_{a}}}{\mathrm{f_{a}}}+\frac{\mathrm{T_{ s}}}{\mathrm{f_{a}}}-\frac{\mathrm{T_{c}}}{\mathrm{f_{x}}}}{\frac{1}{\mathrm{f_{a}}}+ \frac{1}{\mathrm{f_{a}}}+\frac{1}{\mathrm{f_{x}}}}, \tag{17}\]