## 2 Materials and Methods
### System Evaluation  
To evaluate the proposed methodology, ET, NEE, and soil moisture from the OL run and the three DA simulations are compared to a set of reference data, described in Section 2.4. Specifically, measurements from the FLUXCOM and GLEAM datasets are used to validate modeled ET and NEE, respectively. The ISMN soil moisture dataset is adopted in this work for validating model simulations of SSM and RZSM. To ease the validation procedure, the validation data were converted to the resolution of the model outputs (0.5\({}^{\circ}\)\(\times\) 0.625\({}^{\circ}\)) by simply averaging all points falling within the model grid cell. All three DA experiments were evaluated in terms of ET, NEE, SSM, and RZSM. Two metrics were used to evaluate the proposed methodology: the Normalized and Centered Root Mean Square Error (NCRMSE) and the Anomaly Correlation Coefficient (ACC). Biases were not evaluated in this study for various reasons. First off, a major assumption of the EnKF is the unbiasedness between model states and observations [57]. Secondly, satellite products, model estimates, and ground observations are different in nature (e.g., point vs. pixel estimate, soil depth for soil moisture) and should therefore be considered as indicators of a specific water/vegetation variable, rather than its perfect approximation. The NCRMSE is adopted as a measure of the random error [9; 26; 58; 59]:  
\[\text{NCRMSE}=\frac{\sqrt{\frac{\Sigma\left((\left(\hat{\kappa}-\frac{\Sigma \left(\hat{\kappa}\right)}\right)-\left(\chi_{\text{obs}}-\frac{\Sigma\left( \chi_{\text{obs}}\right)}{\Sigma}\right)\right)^{2}}{\Sigma}}}{\frac{\Sigma\left( \chi_{\text{obs}}\right)}{\Sigma}} \tag{1}\]  
where \(\hat{\kappa}\) represents the model output (DA or OL), \(\chi_{\text{obs}}\) is the related observed variable, and N represents the number of days. To determine if the DA improves or degrades the estimation of a specific variable with respect to the original (OL) model run, the Normalized Information Contribution (NIC; [24; 26]) for NCRMSE is computed as follows:  
\[\text{NIC}_{\text{NCRMSE}}=\frac{\text{NCRMSE}_{\text{DA}}-\text{NCRMSE}_{ \text{OL}}}{0-\text{NCRMSE}_{\text{OL}}} \tag{2}\]  
where \(\text{NIC}_{\text{NCRMSE}}\) is the NIC for NCRMSE, \(\text{NCRMSE}_{\text{DA}}\) is the NCRMSE of the DA output, and \(\text{NCRMSE}_{\text{OL}}\) is the NCRMSE for the OL output. A NIC value of 1 corresponds to the DA maximum possible improvement over the OL; a value of zero of NIC means that DA and OL have the same performance; and a negative NIC indicates a model degradation through DA [26]. NIC is a unitless metric. The daily ACC is calculated for the OL and DA model outputs with respect to the reference observations. Anomalies are defined here as the differences between daily values and the yearly climatological average. Each of the anomaly time series is computed relative to the mean of its respective model run. The ACC metric, regardless of potential mean biases or differences in dynamic range, captures the correspondence in phase between model estimates and the reference data [60]. ACC is defined as follows:  
\[\text{ACC}=\frac{\sum\ \left(\left(\hat{\mathbf{x}}-\frac{\sum\left(\hat{\mathbf{x}} \right)}{\mathbf{N}}\right)*\left(\mathbf{X}_{\text{obs}}-\frac{\sum\ \left(\mathbf{X}_{\text{obs}}\right)}{\mathbf{N}}\right)\right)}{\sqrt{\sum \left(\hat{\mathbf{x}}-\frac{\sum\left(\hat{\mathbf{x}}\right)}{\mathbf{N}} \right)^{2}*\sum\left(\mathbf{X}_{\text{obs}}-\frac{\sum\ \left(\mathbf{X}_{\text{obs}}\right)}{\mathbf{N}}\right)^{2}}} \tag{3}\]  
where \(\hat{\mathbf{x}}\) represents the model output (DA or OL), \(\mathbf{X}_{\text{obs}}\) is the corresponding observed variable from ground or satellite observations, and N represents the number of days.