## 3 Preliminaries
### Background Error Covariance Matrix  
When \(\mathbb{E}[\varepsilon]=\mathbf{0}\), indicating that the expectation of each error component is zero, \(\mathbf{B}\) can be expressed as the expectation of the outer product of error vector \(\varepsilon\):  
\[\mathbf{B}=\mathbb{E}[(\varepsilon-\mathbb{E}[\varepsilon])(\varepsilon- \mathbb{E}[\varepsilon])^{T}]=\mathbb{E}[\varepsilon\varepsilon^{T}]=\begin{bmatrix} \overline{\varepsilon_{1}\epsilon_{1}}&\overline{\varepsilon_{1}\epsilon_{2 }}&\cdots&\overline{\varepsilon_{1}\epsilon_{n}}\\ \overline{\varepsilon_{2}\epsilon_{1}}&\overline{\varepsilon_{2}\epsilon_{2 }}&\cdots&\overline{\varepsilon_{2}\epsilon_{n}}\\ \vdots&\vdots&&\vdots\\ \overline{\varepsilon_{n}\epsilon_{1}}&\overline{\varepsilon_{n}\epsilon_{2 }}&\cdots&\overline{\varepsilon_{n}\epsilon_{n}}\end{bmatrix}. \tag{7}\]  
where the overline denotes the expectation. \(\mathbf{B}\) is symmetric and positive-definite, where diagonal elements correspond to the variance of each error component:  
\[\overline{\varepsilon_{i}\epsilon_{i}}=\sigma_{i}^{2}. \tag{8}\]  
To obtain a normalized form, each element of \(\mathbf{B}\) is divided by the product of the corresponding standard deviations, yielding correlation matrix \(\mathbf{C}\):  
\[\mathbf{C}=\begin{bmatrix}1&\rho_{12}&\cdots&\rho_{1n}\\ \rho_{12}&1&\cdots&\rho_{2n}\\ \vdots&\vdots&&\vdots\\ \rho_{1n}&\rho_{2n}&\cdots&1\end{bmatrix}. \tag{9}\]  
\[\text{If }\mathbf{D}=\begin{bmatrix}\sigma_{1}&0&\cdots&0\\ 0&\sigma_{2}&\cdots&0\\ \vdots&\vdots&&\vdots\\ 0&0&\cdots&\sigma_{n}\end{bmatrix},\]  
then \(\mathbf{B}\) can be reconstructed as follows:  
\[\mathbf{B}=\mathbf{D}\mathbf{C}\mathbf{D}. \tag{10}\]  
If a single observation of an element in \(\mathbf{x}\) is introduced in Equation (6), assuming that it coincides with an analysis grid point corresponding to the \(i\)-th element of the state vector, \(\mathbf{H}\) simplifies to a row vector with all elements set to zero except for the \(i\)-th entry, which is equal to one. Under this assumption, the observation and observation error covariance are reduced to scalars, i.e., \(\mathbf{y}^{o}=y^{o}\) and \(\mathbf{R}=R_{ii}\). Consequently, the analysis increment for the \(i\)-th element follows from Equation (6):  
\[x_{i}^{a}-x_{i}^{b}=B_{ii}\frac{y^{o}-x_{i}^{b}}{B_{ii}+R_{ii}}. \tag{11}\]  
where \(B_{ii}=\sigma_{i}^{2}\), representing the variance of the background error at the \(i\)-th state variable. The relative magnitudes of \(B_{ii}\) and \(R_{ii}\) govern the assimilation outcome--if \(R_{ii}\ll\sigma_{i}^{2}\), then \(x_{i}^{a}\to y^{o}\)--indicating that the observation is dominant; conversely, if \(\sigma_{i}^{2}\ll R_{ii}\), then \(x_{i}^{a}\to x_{i}^{b}\), which indicates that the analysis remains closer to the background state. This underscores the critical role of the background error variances, which balances background and observational contributions according to their respective uncertainties.  
Since the true state \(\mathbf{x}^{t}\) is unobservable in practical scenarios, various methodologies were developed to estimate the background error covariance matrix \(\mathbf{B}\) using surrogate background error samples [19]. Among these, the NMC method [29] is widely employed by major forecasting centers due to its practical advantages. This approach estimates \(\mathbf{B}\) by computing the statistical difference between pairs of model forecasts initialized atdifferent lead times but that are valid at the same forecast time. Specifically, the covariance is approximated as follows:  
\[\textbf{B}\approx\frac{1}{2}\Big{\langle}(\textbf{x}^{48}-\textbf{x}^{24})( \textbf{x}^{48}-\textbf{x}^{24})^{T}\Big{\rangle}, \tag{12}\]  
where \(\textbf{x}^{48}\) and \(\textbf{x}^{24}\) denote 48 h and 24 h forecasts, respectively, carrying out verifications at the same forecast time.