## 3 Mathematical Models for the RT POD of LEO Satellites
### Kalman Filtering Model  
In RT POD, Kalman filtering is adopted to calculate all the parameters to be solved considering the model dynamics. The calculated parameters are  
\[X=\left[\left(o_{0}^{leo}\right)^{T},\delta t_{leo,i}^{G},ISB^{G/C},\left(B_{ leo,IF}^{s}\right)^{T}\right]^{T} \tag{11}\]  
The Kalman filter is first initialized. The initial state estimate \(\hat{X}_{0}\) is set as \(X_{0}^{ref}\) calculated from the kinematic solution based on code observations. The initial state covariance \(\hat{P}_{0}\) is set as \(\hat{P}_{0}^{ref}\) in Table 4[20]. The fourth-order Runge-Kutta scheme with Richardson extrapolation is adopted to calculate the position \(\left[x_{0}^{leo},y_{0}^{leo},z_{0}^{leo}\right]^{T}\) and velocity \(\left[x_{0}^{leo},y_{0}^{leo},z_{0}^{leo}\right]^{T}\) in the predicted state. The empirical acceleration prediction is  
\[\overline{a}_{i}=e^{-\left[i_{i}-t_{i-1}\right]/\tau}\cdot\hat{a}_{i-1} \tag{12}\]  
where \(\tau\) is the time corresponding to the empirical accelerations. \(\hat{a}_{i-1}\) is the prediction of the empirical acceleration at epoch time \(t_{i-1}\) and \(\overline{a}_{i}\) is the empirical acceleration at epoch time \(t_{i}\). In a continuous observation arc, the prediction of the coefficients \(C_{R}\) and \(C_{D}\), the prediction of the spaceborne GPS receiver clock offset \(\delta t\), and the prediction of the inter-system biases \(ISB^{G/C}\) and the ambiguity parameter \(\left(B_{leo,IF}^{s}\right)^{T}\) are all constants during the filter propagation process. It is thus concluded that  
\[\left(\overline{C}_{R},\overline{C}_{D},\delta t_{leo,i}^{G},ISB^{G/C},\left( B_{leo,IF}^{s}\right)^{T}\right)_{i}=\left(\overline{C}_{R},\overline{C}_{D}, \delta t_{leo,i}^{G},ISB^{G/C},\left(B_{leo,IF}^{s}\right)^{T}\right)_{i-1} \tag{13}\]  
Firstly, the transition matrix \(\Phi_{i,i-1}=\alpha_{X_{i}^{ref}}/\alpha_{X_{i-1}^{ref}}\) is calculated. The predicted state \(\overline{X}\) is then computed as  
\[\overline{X}_{i}=X_{i}^{ref}+\Phi_{i,i-1}(\hat{X}_{i-1}-X_{i-1}^{ref}) \tag{14}\]  
where \(\hat{X}_{i-1}\) denotes the state estimate at the previous epoch \(i-1\) and \(X_{i}^{ref}\) is the reference state matrix obtained from the observation information at epoch \(i\). The covariance matrix \(\overline{P}_{i}\) is then computed as  
\[\overline{P}_{i}=\Phi_{i,i-1 where \(R_{i}\) is the measurement noise covariance matrix. The non-negativity of the covariance matrix is guaranteed by Equation (18). The design matrix is \(H\). \(L_{i}\) denotes the observed-minus-calculated code and phase measurements.  
To solve the divergence problem caused by traditional linear Kalman filtering, extended Kalman filtering is adopted for the RT POD of the LEO satellite in this study. Here, the reference state vector \(X_{i}^{ref}\) is computed through the orbital integration of satellite dynamics as  
\[\widehat{X}=X_{i}^{ref}+x_{i} \tag{19}\]  
where \(x_{i}={{{(H_{i}^{T}{P_{i}H_{i})}^{-1}}}}{H_{i}^{T}{P_{i}L_{i}}}\) and \(P_{i}=R_{i}^{-1}\) is the weight of the observations. In this study, through high-precision pseudo-range observations, navigation satellite orbits, and clock offsets obtained from the broadcast ephemeris corrected by PPP-B2b, the high-precision 3D coordinate increment of LEO satellites is obtained, and the high-precision orbits are transferred to the next epoch.