## 2 Methods and Data
### Quality Control for LEO Satellites' Kinematic Orbit  
KPOD does not rely on any orbit dynamics model, and the accuracy of the results is strongly influenced by the quality of the GNSS observations. However, some abrupt position offsets (termed \"jumping phenomenon\" for simplicity hereafter) can occur at certain moments. For example, by comparing the kinematic orbit of Swarm-C solved in this paper (DOY 162, 2021) with the precise orbit published by the ESA, the 3D-RMS is obtained, as shown in Figure 1. There is a significant jumping phenomenon between 14:00 and 16:00 with reaching decimeter-level. Therefore, quality control for the kinematic orbit is necessary.  
The quality control of GNSS solutions is usually carried out in the data preprocessing. For instance, [PERSON] et al. [31] performed a KPOD solution and realized a dm-level orbit accuracy by setting proper data screening options. [PERSON] and [PERSON] [13] contributed an approach for the detection and repair of cycle slips in data preprocessing, which obtained good results in Swarm orbit determination. However, none of these methods can completely eliminate the jumping phenomenon.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Model** & **Description** \\ \hline \multirow{6}{*}{Force models} & Earth gravity field & EGM2008(120 \(\times\) 120) [28] \\  & N-body & JPL DE405 \\  & Solid earth tides and pole tide & IERS 2010 Conversations [29] \\  & Ocean tides & FES2004 [30] \\  & Relativity & IERS 2010 Conversations [29] \\ \hline \multirow{6}{*}{Data sets} & Precise GPS orbits & Provided by IGS (Sampling rate: 15 min) \\  & Precise GPS clock offset products & Provided by IGS (Sampling rate: 30 s) \\  & Earth orientation & Provided by CODE \\  & GPS PCO/PCV models & isg14.4 xk \\  & Weight for phase observations & \(\cos^{2}(z)^{1}\) \\  & Sampling of GPS observations & 10 s \\  & Minimum number of observations per epoch & 6 \\  & Elevation cutoff & \(5^{\circ}\) \\ \hline \multirow{6}{*}{Estimated parameters} & Epoch state vector & Both KPOD and RDPOD will estimate \\  & Epoch clock offsets & Both KPOD and RDPOD will estimate \\ \cline{1-1}  & Phase ambiguities & Both KPOD and RDPOD will estimate with \\ \cline{1-1}  & & float-solution \\ \cline{1-1}  & Pseudo-random pulses & Only RDPOD will estimate every 15 min \\ \cline{1-1}  & RTN empirical acceleration & Only RDPOD will estimate every 6 min \\ \hline \hline \end{tabular} \({}^{1}\)\(z\) is the zenith angle of the GPS satellite.  
\end{table}
Table 1: Models, data sets and processing strategies for KPOD and RDPOD.  
In this paper, we used a simple method to control the quality of the final kinematic orbit. The RDPOD was simultaneously carried out with KPOD so that they could be compared to obtain an indicator of the orbit quality [32]. Additionally, the dynamic orbit was more accurate and smoother than the kinematic orbit, as the orbital dynamics were taken into account in the calculation. Therefore, we can make full use of the dynamic orbit and control the kinematic orbit's quality with following approaches:  
1. Taking the dynamic orbit as a reference, the orbital residuals are calculated by through subtracting the kinematic orbit and the dynamic orbit;
2. Filtering the kinematic orbit based on the 3\(\sigma\) principle, supposing the data as Gross Errors and rejecting them when the orbital residual is greater than 3\(\sigma\);
3. Using Chebyshev polynomials to replace the gaps in the kinematic orbit data. However, we only fill the gaps for less than three consecutive vacancies for reducing the effect of unnecessary fitting errors.