## 2 Method
### LEO Satellite Clock Determination in Near-Real-Time  
LEO satellite clock determination can be realized through a reduced-dynamic POD model or a kinematic POD model. The reduced-dynamic model provides higher accuracy in both the POD and the clock determination [12], while the kinematic model combined with, e.g., Kalman-filtering, achieves higher computational efficiency and better flexibility during, e.g., orbital maneuvers. The foundation for both models is the GNSS raw observation equations, with the differences of combining and not combining dynamic models. Therefore, this sub-section starts with GNSS raw observation equations, which can be formulated as:  
\[P_{i}^{s}=\rho_{i}^{s}+c(dt_{r}-dt_{s})+I_{i}^{s}+d_{i}^{s}-d_{r,i}+\varepsilon_ {i,p} \tag{1}\]  
\[L_{i}^{s}=\rho_{i}^{s}+c(dt_{r}-dt_{s})+\lambda_{i}(N_{i}^{s}+b_{r,i}-b_{i}^{s} )-I_{i}^{s}+\varepsilon_{i,L} \tag{2}\]  
where \(P_{i}^{s}\) and \(L_{i}^{s}\) represent the pseudoranged and phase observations, respectively. \(i\) indicates the frequency, and \(s\) indicates the GNSS satellite. \(c\) is the speed of light. \(dt_{r}\) and \(dt_{s}\) are the LEO satellite clock bias and the GNSS satellite clock bias, respectively. \(I_{i}^{s}\) is the slant ionospheric delay. \(d_{i}^{s}\) and \(d_{r,i}\) represent the pseudoranged hardware delays of the GNSS satellite and the LEO satellite, respectively. \(\lambda_{i}\) is the wavelength of the frequency \(i\), and \(N_{i}^{s}\) is the carrier phase ambiguity. \(b_{r,i}\) and \(b_{i}^{s}\) denote the carrier phase hardware delays of the LEO satellite and the GNSS satellite, respectively. \(\varepsilon_{i,p}\) and \(\varepsilon_{i,L}\) are the measurement noise and multipath effects of the pseudorange and carrier phase observations, respectively. Other correctable error terms such as tidal effects, phase center offsets and variations, etc., are not specifically described in Equations (1) and (2). The kinematic model determines the orbit and clocks of LEO satellites using only the GNSS observational data and related real-time GNSS products, without introducing any dynamic models. The kinematic model has the advantages of simplicity, low computational requirements, short processing time, and the ability to maintain good precision. Determining the LEO satellite clocks and orbit using the kinematic model is similar to the PPP (Precise Point Positioning) principle. However, compared to PPP, the GNSS observations received by LEO satellites are not affected by the tropospheric delays. In this study, the kinematic model is implemented using the Ionosphere-Free (IF) pseudorange and carrier phase observations. The Observed-Minus-Computed (O-C) terms can be expressed as:  
\[P_{IF}=A_{x}\Delta r_{K}+c\times df_{r}+\varepsilon_{p_{IF}} \tag{3}\]  
Figure 1: The flowchart of real-time LEO satellite clock determination. \[L_{IF}=A_{x}\Delta r_{K}+c\times df_{r}+\lambda\overline{N}_{IF}+\varepsilon_{L_{IF}} \tag{4}\]  
with  
\[d\tilde{t}_{r}=dt_{r}+\frac{d_{r,IF}}{c}+d_{rel} \tag{5}\]  
\[\Delta r_{K}=\left[\Delta r_{x},\Delta r_{y},\Delta r_{z}\right] \tag{6}\]  
\[\lambda_{IF}=\frac{c}{f_{1}+f_{2}} \tag{7}\]  
\[\bar{N}_{IF}=\frac{f_{1}^{2}}{f_{1}^{2}-f_{2}^{2}}N_{1}-\frac{f_{2}^{2}}{f_{1} ^{2}-f_{2}^{2}}N_{2}+b_{r,IF}-b_{IF}^{s}+\frac{-d_{r,IF}}{\lambda_{IF}} \tag{8}\]  
where the pseudorange 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 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.