## 2 Mathematical Principles
### Ensemble Kalman Filter  
The Ensemble Kalman Filter (EnKF) is a DA method designed for state estimation in nonlinear systems. This method was proposed by [PERSON] et al. [18] to address the limitations of the traditional Kalman Filter (KF) when applied to nonlinear systems. The EnKF uses an ensemble of samples to represent the probability distribution of the system's state and adjusts the samples to more closely align with experimental observations by calculating their covariance matrix and the Kalman gain [19]. The subject of study is represented by a set of state variables, denoted collectively as the vector \(X_{f}\). Experimental observations are represented as the vector \(y_{\text{exp}}\). The algorithm assumes a specific relationship exists among state variables, experimental observations, and predictive models. The state-space model describing the relationship between the experimental observations and prediction model is:  
\[X_{f}=F(X_{0},v) \tag{6}\]  
\[y_{\text{exp}}=HX_{f}+W \tag{7}\]  
In the formula:  
\(F\): prediction model function. \(X^{0}\): initial state vector of the samples. \(v\): prediction model error (neglected in this study). \(H\): observation matrix mapping the state vector to observation space. \(W\): experimental observation error. 1. Forecasting process During the prediction process, the state parameter vector within each ensemble member will be iteratively computed using the RNG k-\(\epsilon\) model starting from the initial state until the numerical simulation of turbulence converges. The updating of the state parameters follows the following equation:  
\[X_{f}^{i}=F\left(X_{0r}^{i},v^{i}\right) \tag{8}\]  
Here, \(F\) represents the governing equations of the RNG k-\(\epsilon\) turbulence model, and the state parameters of each ensemble member are expressed in the following form:  
\[X_{f}^{i}= \begin{pmatrix}Q^{i}\\ A^{i}\end{pmatrix} = \begin{pmatrix}Q^{i}\\ C_{u}^{i}\\ C_{1e}^{i}\\ C_{2e}^{i}\\ Pr_{t}^{i}\end{pmatrix}  \tag{9}\]  
Here, \(Q\) denotes the gas volumetric flow rate predicted by the RNG k-\(\epsilon\) model, and \(A\) is the vector of turbulence model constants, expressed as: \(A=(C_{u}\ C_{1e}\ C_{2e}\ Pr)^{T}\). The superscript \(i\) denotes the index of the ensemble member. Mean value of the ensemble members:  
\[\overline{x}_{f}=\frac{1}{l}\sum_{i=1}^{l}x_{f}^{i} \tag{10}\]  
where \(l\) represents the total number of set members, and \(\overline{x}_{f}\) denotes the average value of the set members. 2. Analysis process The analysis phase is a critical step in the ENKF algorithm. In this phase, the Kalman gain is determined, and the ensemble members are updated by synthesizing the uncertainty of observational data with the statistical information of the ensemble members. The specific process is as follows:  
Prediction error analysis:  
\[P=\frac{1}{l-1}\sum_{i=1}^{l}\left(x_{f}^{i}-\overline{x}_{f}\right)\left(x_{f }^{i}-\overline{x}_{f}\right)^{T} \tag{11}\]  
\[R=\frac{1}{l-1}WW^{T} \tag{12}\]  
Here,  
\[W=\left( \begin{array}{cccc}w^{1}&w^{2}&\cdots&w^{n}\end{array} \right) \tag{13}\]  
Kalman gain calculation:  
\[K=PH^{T}\left(HPH^{T}+R\right)^{-1} \tag{14}\]  
Updating collection members:  
\[X_{\text{a}}^{i}=X_{f}^{i}+K\left(y_{\text{exp}}+w^{i}-HX_{f}^{i}\right) \tag{15}\]Mean value of the corresponding new set member:  
\[\overline{x}_{a}=\frac{1}{l}\sum_{i=1}^{l}x_{a}^{i} \tag{16}\]  
For highly nonlinear models, a relatively accurate prediction is often not achieved by performing the analysis process only once. Consequently, this study employs a multi-iteration EnKF algorithm for both prediction and analysis. By iterating multiple times, the uncertainty of observational data and the statistical properties of the ensemble members can be more comprehensively incorporated, thereby enhancing the precision of predictions. At each iteration, the convergence of the algorithm is assessed based on the standard deviation of the ensemble members and the maximum number of iterations, which is set to 10,000.