## 3 Methods
### Ensemble Kalman Filter  
In the 1960s, [PERSON] [48; 49] and [PERSON] [48] developed the Kalman filter algorithm for optimal estimation of system states according to a linear system state equation and combined input and output system observation data. The traditional Kalman filter algorithm is often applied to linear problems [50; 51]. It can estimate the state of a dynamic system from a series of data with measurement noise when the measurement variance is known, and is a common component in communication, navigation, guidance, and control applications. The dynamic model and observer in a Kalman filter are typically expressed as follows:  
\[x_{k+1}=M_{k}x_{k}+\omega_{k+1} \tag{9}\]  
\[z_{k}=H_{k}x_{k}+v_{k} \tag{10}\]  
where \(x_{k}\) is the dynamic model state vector; \(M_{k}\in R^{n\times n}\) is the system matrix, which changes with time; \(z_{k}\) represents observed values, and \(H_{k}\) is a time-varying observation system which transforms the state variable into the same value as the observed value. In this study, the observed values and state variable were land surface albedos. \(\omega_{k}\) is the model error and \(v_{k}\) is observed noise. The albedo has different levels of inversion accuracy over time. The predicted value of the state variable of the dynamic model at time k is defined as \(x_{k}^{f}\). The data assimilation step serves to calculate the optimal estimate \(x_{k}^{a}\) of the state variables at moment k by combining the predicted value \(x_{k}^{f}\) and the observed value \(z_{k}\) of the model state variables at time k, \(x_{k}^{a}\), the optimal estimate of \(x_{k}\), is a linear function of \(x_{k}^{f}\) and \(z_{k}\). \[x_{k}^{a}=x_{k}^{f}+K_{k}[z_{k}-H_{k}x_{k}^{f}] \tag{11}\]where \(z_{k}-H_{k}x_{k}^{f}\) is observed incrementally (innovation) and \(K_{k}\) is the Kalman gain matrix. Data assimilation is conducted to minimize the variance of \(x_{k}^{a}\) by determining \(K_{k}\). The update equations for state variables and covariance are:  
\[x_{k}^{a}=x_{k}^{f}+K_{k}[z_{k}-H_{k}x_{k}^{f}]=x_{k}^{f}+P_{k}^{f}H_{k}^{T}(H_ {k}P_{k}^{f}H_{k}^{T}+R_{k})^{-1}(z_{k}-H_{k}x_{k}^{f}) \tag{12}\]  
\[P_{k}^{a}=P_{k}^{f}-K_{k}H_{k}P_{k}^{f}=P_{k}^{f}-P_{k}^{f}H_{k}^{T}(H_{k}P_{k} ^{f}H_{k}^{T}+R_{k})^{-1}H_{k}P_{k}^{f} \tag{13}\]  
where \(x_{k}^{a}\) is the updated state variable value, \(x_{k}^{f}\) is the predicted state variable, \(P_{k}^{f}\) is the predicted covariance, and \(P_{k}^{a}\) is the posterior covariance of state variables. The ensemble Kalman filter algorithm is a complex sequential data assimilation method which produces non-linear models within a Kalman gain scheme. In the ensemble Kalman filter algorithm, x is defined as an n-dimensional model state vector and \(A=(x_{1},x_{2},\ldots x_{N})\in\Re^{n\times N}\) is composed of a collection of N model state vectors. The ensemble mean is stored in each column of \(\overline{A}\in\Re^{n\times N}\). The ensemble perturbation matrix is defined as:  
\[A^{\prime}=A-\overline{A} \tag{14}\]  
The ensemble covariance matrix is:  
\[P_{e}=\frac{A^{\prime}(A^{\prime})^{T}}{N-1} \tag{15}\]  
Given a vector of measurements \(d\in\Re^{m}\), where m is the number of measurements, the observation matrix is expressed as follows:  
\[D=(d_{1},d_{2},\ldots d_{N})\in\Re^{m\times N} \tag{16}\]  
Then, the standard analysis equation of the Kalman filter is:  
\[A^{a}=A+A^{\prime}A^{\prime T}H^{T}(HA^{\prime}A^{\prime T}H+R)^{-1}(D-HA) \tag{17}\]  
where \(H\in\Re^{n\times N}\) (m is the number of measurements) is the measurement operator relating the model state to the observations; \(R\in\Re^{m\times m}\) is the observational error covariance matrix, and \(D\in\Re^{m\times N}\) is the disturbance observation matrix set. H is a linear operator which is not suitable in cases when the operator is nonlinear. By augmenting the model state vector, \(\overline{x}^{T}=[x^{T},h^{T}(x)]\).