## 3 Processing Methods
### Filter-Based Real-Time LEO Satellite Clock Determination  
In this study, the LEO real-time clock is determined using a kinematic model processed using a Kalman filter to achieve both high precision and high computational efficiency. The orbits can be estimated together with the clocks or introduced based on a short-term prediction [30]. This study employs the former case. The Observed-Minus-Computed (O-C) terms of the Ionosphere-Free (IF) pseudo-range and carrier phase GNSS observations collected onboard LEO satellites can be expressed as follows:  
\[P_{IF}=A_{x}\Delta r_{K}+c\times d\overline{t}_{r}+\varepsilon_{P_{IF}} \tag{1}\]  
\[L_{IF}=A_{x}\Delta r_{K}+c\times d\overline{t}_{r}+\lambda\overline{N}_{IF}+ \varepsilon_{L_{IF}} \tag{2}\]  
with  
\[d\overline{t}_{r}=dt_{r}+\frac{d_{r,IF}}{c}+d_{rel} \tag{3}\]  
\[\Delta r_{K}=\left[\Delta r_{x},\Delta r_{y},\Delta r_{z}\right] \tag{4}\]  
\[\lambda_{IF}=\frac{c}{f_{1}+f_{2}} \tag{5}\]  
\[\overline{N}_{IF}=\frac{f_{1}^{2}}{f_{1}^{2}-f_{2}^{2}}N_{1}-\frac{f_{2}^{2}} {f_{1}^{2}-f_{2}^{2}}N_{2}+b_{r,IF}-b_{IF}^{s}-\frac{d_{r,IF}}{\lambda_{IF}} \tag{6}\]  
where the pseudo-range and phase observations O-C terms for the IF combinations are represented by \(P_{IF}\) and \(L_{IF}\), respectively. \(f_{1}\) and \(f_{2}\) correspond to the first and the second frequencies used, respectively. \(A_{x}\) is the design matrix containing partial derivatives of the observations with respect to the orbital coordinates. \(\Delta r_{K}\) is the kinematic orbital correction vector based on the initial orbits, with \(\Delta r_{x}\), \(\Delta r_{y}\), and \(\Delta r_{z}\) representing the increments of the LEO satellite position in the \(X\), \(Y\), and \(Z\) directions of the Earth-Centered Earth-Fixed (ECEF) system, respectively. No dynamic model is applied to the orbit. \(d\overline{t}_{r}\) represents the estimable receiver clocks with the inclusion of the IF receiver code bias \(d_{r,IF}\) and the relativistic effects \(d_{rel}\). Recall that \(d\overline{t}_{r}\) denotes the LEO satellite clocks. Note that \(d\overline{t}_{r}\) is estimated as independent epoch-by-epoch parameters without applying constraining clock models. It is assumed here that the IF code hardware delay of the GNSS satellite is already \begin{table}
\begin{tabular}{c c c} \hline \hline
**ACs** & **Continuity** & **Latency** \\ \hline WHU & 80.69\% & \(\leq\)5 s \\ \hline GMV & 67.53\% & \(\leq\)30 s \\ \hline CNE & 78.07\% & \(\leq\)20 s \\ \hline BKG & 87.60\% & \(\leq\)20 s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The continuity and latency of the received real-time GPS satellite clocks from different analysis centers. included in the GNSS satellite clock products. Depending on the systems and types of signals used, Differential Code Bias (DCB) and Inter-System Biases (ISBs) may apply [5; 25]. Compared to the stricter data preprocessing approach in the BLS-based reduced-dynamic model, in the Kalman filter-based kinematic model, cycle slip detection is performed using only the GF combination, with the threshold set to 0.05 m. Once a cycle slip is detected, the ambiguity parameters are reset. The above-mentioned LEO satellite clocks are determined in two ways in this study; the first method involves continuous filtering calculations, as seen in the left panel of Figure 2, while the second method periodically starts the filtering process for a certain arc length and connects different rounds of real-time clock products based on near-real-time determination and short-term prediction [18], as seen in the right panel of Figure 2.  
The difference between the two methods lies in the observation data input, as shown in the contents within the red box in Figure 2. The left panel of Figure 2 describes the traditional Kalman filtering process, where the observation data are input epoch by epoch, and the clocks are estimated epoch by epoch, referred to as \"epoch-estimate\" in this study.