## 3 Data Assimilation Strategy
### Assimilation Method
#### 3.2.2 POD-Based Ensemble 4D Variational Assimilation Method  
On the basis of POD and ensemble forecasting techniques, [PERSON] et al. [21] proposed a hybrid method: the POD-based ensemble 4-D variational assimilation method (PODEn4 DVar). This method uses linear combinations of the transformed model perturbations (MPs) and observation perturbations (OPs) to express the optimal MP and its OP, respectively. Moreover, this method can assimilate multiple-time observation data and update its background error covariance by integrating the advantages of variational methods and evolving forecast ensembles. By minimizing the incremental format of the cost function in the 4 DVar algorithm, an optimal increment of initial condition can be achieved at the initial time by:  
\[J\big{(}x^{\prime}\big{)}=\frac{1}{2}\big{(}x^{\prime}\big{)}^{T}{P_{b}}^{-1} \big{(}x^{\prime}\big{)}+\frac{1}{2}\big{[}y^{\prime}\big{(}x^{\prime}\big{)} -y^{\prime}_{obs}\big{]}^{T}R^{-1}\big{[}y^{\prime}\big{(}x^{\prime}\big{)}-y^ {\prime}_{obs}\big{]}, \tag{8}\]  
where \(x^{\prime}=x-x_{b}\), \(y^{\prime}=y^{\prime}(x^{\prime})=y(x)-y(x_{b})\), \(y^{\prime}_{obs}=y_{obs}-y(x_{b})\), and \(y=H\big{[}M_{t_{0}\to t_{k}}(x)\big{]}\). Here, the \(b\) represents the background field, index \(k\) stands for the observation time, \(P_{b}\) is the background error covariance, and \(obs\) denotes observation.  
Applying the POD transformation, the optimal solution can be expressed as:  
\[x^{\prime}_{a}=P_{x}\beta\ y^{\prime}_{a}=P_{y}\beta, \tag{9}\]  
The background error covariance \(P_{b}\) can be calculated from:  
\[P_{b}=\frac{P_{x}P_{x}^{T}}{N-1}, \tag{10}\]  
where \(N\) is the size of the ensemble. By substituting Equations (9) and (10) into the Equation (8), we obtain another cost function with control variables \(\beta\) instead of \(x^{\prime}\):  
\[J\big{(}\beta\big{)}=\frac{1}{2}(N-1)\beta^{T}P_{x}^{T}\Big{(}P_{x}^{T}\Big{)} ^{-1}(P_{x})^{-1}P_{x}\beta+\frac{1}{2}\big{(}P_{y}\beta-y^{\prime}_{obs} \big{)}^{T}R^{-1}\big{(}P_{y}\beta-y^{\prime}_{obs}\big{)}, \tag{11}\]  
marking \(P_{a}^{*}=\Big{[}(N-1)I+P_{y}^{T}R^{-1}P_{y}\Big{]}^{-1}\), the incremental analysis \(x^{\prime}_{a}\) is expressed by:  
\[x^{\prime}_{a}=P_{x}P_{a}^{*}P_{y}^{T}R^{-1}y^{\prime}_{obs}, \tag{12}\]  
therefore, the final analysis \(x_{a}\) can be calculated as:  
\[x_{a}=x_{b}+x^{\prime}_{a}=x_{b}+P_{x}P_{a}^{*}P_{y}^{T}R^{-1}y^{\prime}_{obs}, \tag{13}\]  
To ameliorate the spurious long-range correlations between observation locations and model variables resulting from the ensemble-based method, a localization technique is applied to the calculations process [59]. The final analysis is rewritten as:  
\[x_{a}=x_{b}+C_{0}\bigg{(}\frac{d_{h}}{d_{h,0}}\bigg{)}\cdot C_{0}\bigg{(}\frac {d_{v}}{d_{v,0}}\bigg{)}\cdot x^{\prime}_{a}, \tag{14}\]  
where \(d_{h}\) and \(d_{v}\) are the horizontal and vertical distances between the spatial positions of state and observed variables, respectively; and \(d_{h,0}\) and \(d_{v,0}\) are the horizontal and vertical covariance localization Schur radii, respectively. The filtering function \(C_{0}\) is expressed as:  
\[C_{0}(r)=\left\{\begin{array}{ll}-\frac{1}{4}r^{5}+\frac{1}{2}r^{4}+\frac{5 }{8}r^{3}-\frac{5}{3}r^{2}+1,&0\leq r\leq 1\\ \frac{1}{12}r^{5}-\frac{1}{2}r^{4}+\frac{5}{8}r^{3}+\frac{5}{3}r^{2}-5r+4- \frac{2}{3}r^{-1},&1<r\leq 2\\ 0,&2<r\end{array}\right., \tag{15}\]  
where \(r\) is the radius of the filter.