## 2 Methods and Data  
### Satellite-Based GNSS Observation Model  
Both Swarm and GRACE-FO satellites are equipped with on-board dual-frequency GNSS receivers [20]. The pseudorange and carrier phase observations for LEO satellites relative to satellite \(S\) (one of GNSS satellites) at a frequency of \(i\) can be expressed as [21]:  
\[P_{LEO,i}^{S}=\rho_{LEO}^{S}+c\cdot\delta t_{LEO}-c\cdot\delta t^{S}+\delta\rho_ {ion,i}+\delta\rho_{rel}+\delta\rho_{po}+\delta\rho_{po}^{S}+M_{Pi}+\varepsilon _{Pi} \tag{1}\]  
\[L_{LEO,i}^{S}=\rho_{LEO}^{S}+c\cdot\delta t_{LEO}-c\cdot\delta t^{S}+\delta\rho _{ion,i}+\delta\rho_{rel}+\delta\rho_{po}+\delta\rho_{po}^{S}-\lambda_{i}\cdot N _{i}+M_{Li}+\varepsilon_{Li} \tag{2}\]  
where \(P_{LEO,i}^{S}\) and \(L_{LEO,i}^{S}\) are the pseudorange and carrier phase observations of the LEO satellites, respectively; \(\rho_{LEO}^{S}\) is the geometric distance from the LEO satellite to the satellite \(S\); \(c\) denotes the speed of light; \(\delta t_{LEO}\) is the on-board receiver clock offset; \(\delta t^{S}\) is the clock offset of satellite \(S\); \(\delta\rho_{ion,i}\) is the ionospheric delay; \(\delta\rho_{rel}\) is the relativity correction; \(\delta\rho_{po}\) and \(\delta\rho_{po}^{S}\) are the antenna Phase Center Offset (PCO) of the LEO satellite and satellite \(S\), respectively; \(M_{Pi}\) and \(M_{Li}\) are the pseudo-range and carrier phase multi-path correction, respectively; \(\lambda_{i}\) is the wavelength; \(N_{i}\) is ambiguity (cycle); and \(\varepsilon_{Pi}\) and \(\varepsilon_{Li}\) are the noises of the pseudorange and carrier phase observation, respectively.  
The ionospheric delay is one of main errors for GNSS positioning and orbit determination [22; 23]. To eliminate the effect of ionospheric delay, a typical ionosphere-free (IF) combination for dual-frequency pseudorange and carrier phase observation can be derived [24]:  
\[P_{3}=\frac{f_{1}^{2}}{f_{1}^{2}-f_{2}^{2}}\cdot P_{LEO,1}^{S}-\frac{f_{2}^{2 }}{f_{1}^{2}-f_{2}^{2}}\cdot P_{LEO,2}^{S} \tag{3}\]  
\[L_{3}=\frac{f_{1}^{2}}{f_{1}^{2}-f_{2}^{2}}\cdot L_{LEO,1}^{S}-\frac{f_{2}^{2 }}{f_{1}^{2}-f_{2}^{2}}\cdot L_{LEO,2}^{S} \tag{4}\]  
where \(f_{i}\) represents the frequency \(i\); \(P_{3}\) and \(L_{3}\) are the pseudorange and carrier phase IF observation. If we combine the equations from (1) to (4), we can obtain the IF observation equation as:  
\[P_{3}=\rho_{LEO}^{S}+c\cdot\delta t_{LEO}-c\cdot\delta t^{S}+\delta\rho_{rel} +\delta\rho_{po}+\delta\rho_{po}^{S}+\frac{f_{1}^{2}\cdot M_{Pi}-f_{1}^{2} \cdot M_{P2}}{f_{1}^{2}-f_{2}^{2}}+\varepsilon_{P} \tag{5}\]  
\[L_{3}=\rho_{LEO}^{S}+c\cdot\delta t_{LEO}-c\cdot\delta t^{S}+\delta\rho_{rel} +\delta\rho_{po}+\delta\rho_{po}^{S}+\frac{f_{1}^{2}\cdot M_{Li}-f_{1}^{2} \cdot M_{L2}}{f_{1}^{2}-f_{2}^{2}}-\lambda_{3}\cdot N_{3}+\varepsilon_{L} \tag{6}\]  
where \(N_{3}\) is ambiguity of the IF observation equation. The least squares batch processing method is commonly used in LEO satellites' POD, which can achieve high accuracy and stability of the solution. For example, [PERSON] et al. [25] used the least squares batch processing method to achieve centimeter-level POD of GPS and CHAMP satellites based on PANDA (Position And Navigation Data Analysis) software developed independently by Wuhan University. [PERSON] et al. [26] applied GNSS High-precision Orbit determination Software Tools (GHOST) to estimate the kinematic orbits of Swarm satellites with undifferenced observations and standard Bayesian weighted least squares estimation and realized a centimeter-level accuracy.  
In this paper, a least squares batch processing method based on an undifferenced IF combination and quality control are adopted when simultaneously performing KPOD and RDPOD for Swarm and GRACE-FO satellites.