## 2 Materials and Methods
### Methods
#### 2.3.4 Structural Equation Modeling (SEM)  
SEM is a linear statistical model that encompasses a set of multivariate statistical techniques, including factor analysis, multiple regression analysis, path analysis, and simultaneous equation modeling. SEM uses hypothesis testing procedures in statistics to analyze internal structures by observing external indicators of the system. This method not only quantifies the effect of a single variable by removing the influence of other variables, but also effectively quantifies the combined effect of multiple influencing factors on the target variable by considering the interactions between independent variables [48]. The main approach to constructing an SEM is as follows: first, based on prior experience, hypothesize the potential causal relationships between all variables to build a conceptual model. Then, the conceptual model is converted into a mathematical model, which is continuously adjusted and fitted. Once the fitting results are satisfactory, it is considered the final constructed SEM [49]. The criteria for an acceptable SEM are RMSEA < 0.08, CFI > 0.80, and GFI > 0.90. The structural equation model mainly consists of two parts: the measurement model and the structural model.  
The measurement model, also known as the factor model, is used to analyze the relationships between observed variables (variables that can be directly observed) and latent variables (variables that cannot be directly observed and need to be inferred from other directly observed variables). The model mainly reflects the fit, reliability, and validity by measuring the relationship between latent variables and the integrated observed variables. The principle expression is as follows:  
\[X=\Lambda_{x}\xi+\delta \tag{6}\]  
\[Y=\Lambda_{y}\eta+\varepsilon \tag{7}\]  
In the equation, \(X\) represents the observed variables of \(\xi\), and \(\delta\) is the residual term of \(X\). \(\Lambda_{x}\) is the factor loading matrix (q \(\times\) n) of \(X\) on \(\xi\); \(Y\) represents the observed indicators of \(\eta\), and \(\varepsilon\) is the residual term of \(Y\). \(\Lambda_{y}\) is the factor loading matrix (p \(\times\) m) of \(Y\) on \(\eta\). \(q\) is the number of observed variables in \(X\), and \(p\) is the number of observed variables in \(Y\).  
The structural model, also known as the causal model, is used to measure the relationships between latent variables and is an important component of the SEM.  
\[\eta=B\eta+\Gamma\xi+\zeta \tag{8}\]  
In the equation, \(\eta\) represents the endogenous latent variables, \(\xi\) represents the exogenous latent variables, and \(\zeta\) represents the error term, which is the part of \(\eta\) and \(\xi\) that is not explained. B is the (m \(\times\) m) coefficient matrix of the relationships between multiple \(\eta\) variables, and \(\Gamma\) is the (m \(\times\) n) coefficient matrix of the relationships between multiple \(\xi\) variables. m is the number of \(\eta\) variables, and n is the number of \(\xi\), variables.  
Temperature (Temp) and precipitation (Prec) are the fundamental hydrothermal factors for vegetation growth, directly affecting respiration and transpiration. Soil moisture (SM) and soil temperature (ST) influence vegetation growth by affecting root activity and water absorption. The vapor pressure deficit (VPD) indirectly affects vegetation growth by influencing the hydrothermal conditions. Photosynthetically active radiation (PAR) is an important energy source for vegetation growth and development. Therefore, this study selected six hydrothermal and radiative factors--Temp, Prec, SM, ST, VPD, and PAR--that are closely related to vegetation carbon-water coupling (NPP, ET, and WUE) to construct the SEM. The schematic diagram of the constructed SEM is shown below (Figure 2):