## 2 Materials and Methods
### The Ensemble Kalman Filter  
The Ensemble Kalman Filter (EnKF) is an optimal recursive data assimilation method introduced by [PERSON] [39]. There are several different versions of the EnKF. In this work, we used the EnKF of [PERSON] et al. [40]. Therefore, the observations also needed to be treated as random variables and it is commonly assumed that observation errors have a Gaussian distribution. The EnKF performs model forecasting where the model responses (state variables) are propagated forward in time based on the model dynamics and a filter update in which the ensemble of the model state is adjusted through incorporating available observations [30; 41; 42]. The EnKF can be considered as an approximate version of the Kalman Filter, in which the state uncertainty is represented by the ensemble spread [43]. If we assume that observations are related to the true data \(x_{t}\),  
\[y=Hx_{t}+\varepsilon, \tag{6}\]  
\[x_{t}=Ax_{t-1}+\
u, \tag{7}\]  
where \(y\) is the observation vector, \(\varepsilon\) and \(\
u\) are Gaussian random error vectors with a mean of zero, \(H\) is the observation operator that relates to \(y\), \(A\) represents a linear state-transition model that links \(x_{t}\) and \(x_{t-1}\), and in the crop model data assimilation system it represents the crop model. The forecast of \(x_{t}\) at \(=k\) is Gaussian, mean, \(\pi_{t=k}^{f}\), and error covariance, \(P_{t=k}^{f}\), can be calculated as Equations (8)-(9), and \(y_{t=k}\) is observation vector at \(t=k\), as follows:  
\[x_{t=k}^{a}=x_{t=k}^{f}+K\big{(}y_{t=k}-Hx_{t=k}^{f}\big{)}, \tag{8}\]  
\[P_{t=k}^{a}=(I-KH)P_{t=k}^{f}, \tag{9}\]  
where \(f\) and \(a\) are indices of the prior and posterior estimates, respectively, \(I\) is the identity matrix, and \(K\) is the Kalman gain matrix, defined as Equation (10), where \(R_{t=k}\) is the error covariance of the observation ensemble. \[K=P_{t=k}^{f}H^{T}\Big{(}HP_{t=k}^{f}H^{T}+R_{t=k}\Big{)}^{-1}. \tag{10}\]  
The EnKF forecast and analysis error covariance come directly from an ensemble of model simulations, as follows:  
\[P^{f}H^{T}=(N_{e}-1)^{-1}\sum_{n=1}^{N_{e}}(x_{n}^{f}-\overline{x}^{f})(Hx_{n }^{f}-H\overline{x}^{f})^{T}, \tag{11}\]  
where \(N_{e}\) is the number of ensemble members, \(n\) is a running index for ensemble member, and \(\overline{x}^{f}\) represents the ensemble mean, calculated as Equation (12). The observation data should be perturbed and the variance used in the perturbation is based on the uncertainty of the observation data. \[\overline{x}^{f}=N_{e}^{-1}\sum\
olimits_{n=1}^{N_{e}}x_{n}^{f}. \tag{12}\]  
In this study, the soil moisture retrieved by the water cloud model was the only state variable that was directly assimilated into the WOFOST model. So, \(H\) can be taken as an identity matrix and the optimal estimation of state variable \(x_{t}\) at the time \(k\) (Equation (8)). The Kalman gain matrix (Equation (10)) and the EnKF forecast and analysis error covariance (Equation (11)) can be rewritten as Equations (13)-(15). \[x_{t=k}^{a}=x_{t=k}^{f}+K(y_{t=k}-x_{t=k}^{f}), \tag{13}\]  
\[K=P_{t=k}^{f}\Big{(}P_{t=k}^{f}+R_{t=k}\Big{)}^{-1}, \tag{14}\]  
\[P^{f}=(N_{e}-1)^{-1}\sum_{n=1}^{N_{e}}(x_{n}^{f}-\overline{x}^{f})(x_{n}^{f}- \overline{x}^{f})^{T}. \tag{15}\]  
The uncertainties of the crop model parameters and the remotely sensed observations are an important part of the EnKF assimilation system. There are two common ways to account for model uncertainty in the EnKF assimilation system. One is adding a Gaussian perturbation to the model output state variables. Another is to perturb model input parameters to generate ensemble members. In this study, we used the first method and an uncertainty level of 10% was introduced to WOFOST model output soil moisture and perturbed using a Gaussian distribution [25]. Additionally, the uncertainty of remotely sensed soil moisture was set to 7%.