Undifferenced Kinematic Precise Orbit Determination of Swarm and GRACE-FO Satellites from GNSS Observations

Low Earth Orbit (LEO) satellites can be used for remote sensing and gravity field recovery, while precise orbit determination (POD) is vital for LEO satellite applications. However, there are some systematic errors when using the LEO satellite orbits released by different agencies in multi-satellite-based applications, e.g., Swarm and Gravity Recovery and Climate Experiment-Follow-On (GRACE-FO), as different GNSS precise orbit and clock products are used as well as processing strategies and software. In this paper, we performed undifferenced kinematic PODs for Swarm and GRACE-FO satellites simultaneously over a total of 14 days by using consistent International Global Navigation Satellite System (GNSS) Service (IGS) precise orbit and clock products. The processing strategy based on an undifferenced ionosphere-free combination and a least squares method was applied for Swarm and GRACE-FO satellites. Furthermore, the quality control for the kinematic orbits was adopted to mitigate abrupt position offsets. Moreover, the accuracy of the kinematic orbits solution was evaluated by carrier phase residual analysis and Satellite Laser Ranging (SLR) observations, as well as comparison with official orbits. The results show that the kinematic orbits solution is better than 4 cm, according to the SLR validation. With quality control, the accuracy of the kinematic orbit solution is improved by 2.49 % for the Swarm-C satellite and 6.98 % for the GRACE-D satellite when compared with their precise orbits. By analyzing the accuracy of the undifferenced kinematic orbit solution, the reliability of the LEO orbit determination is presented in terms of processing strategies and quality control procedures.


Introduction
Low Earth Orbit (LEO) satellite is a valuable platform for Earth observation due to the low altitude, short period, and rapid connection with the ground [1]. The observation data collected by the LEO satellite have been widely used in geosciences with a wide area coverage, such as crustal deformation [2,3], ocean altimetry [4], Earth gravity field recovery [5,6], space weather [7,8] and remote sensing [9,10]. To ensure successful scientific applications undertaken by the LEO satellite, the precise orbit determination (POD) for the LEO satellite is essential. With the launch of the project Topex/Poseidon, developed jointly by the National Aeronautics and Space Administration (NASA) and the Centre National d'Etudes Spatiales (CNES) in 1992, satellite-borne GPS technology has been utilized to obtain a high-accuracy centimeter-level orbit. The on-board GNSS receivers have been applied to achieve high-accuracy orbital information in the subsequent LEO satellite missions.

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]: where P S LEO,i and L S LEO,i are the pseudorange and carrier phase observations of the LEO satellites, respectively; ρ S LEO is the geometric distance from the LEO satellite to the satellite S; c denotes the speed of light; δt LEO is the on-board receiver clock offset; δt S is the clock offset of satellite S; δρ ion,i is the ionospheric delay; δρ rel is the relativity correction; δρ pco and δρ S pco 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; λ i is the wavelength; N i is ambiguity (cycle); and ε Pi and ε 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]: 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: 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, Zhao 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. Ijssel 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.

Data Sources and Solving Strategies
In this paper, the dual-frequency GPS observation files of Swarm (Swarm-A/B/C) and GRACE-FO satellites (GRACE-C and GRACE-D) during the day of year (DOY) 152-165 (from 1 June to 14 June) in 2021 were adopted, which were provided by the ESA and the GFZ. In the processing, the GPS precise orbit products with a 15 min sampling rate and the precise GPS clock offset products with a 30 s sampling rate were provided by IGS, and the Earth Orientation Parameter (EOP) data were provided by the Center for Orbit Determination in Europe (CODE).
The RDPODs of Swarm and GRACE-FO satellites were simultaneously carried out with KPOD, while the result of RDPOD was used as a reference to control the quality of the KPOD result. We used Bernese GNSS Software to process the data, replacing an explicit modeling of non-gravitational forces by dedicated empirical orbit parametrizations [27], as the main purpose is KPOD in this paper. The force models, data sets and processing strategies of POD are summarized in Table 1. Table 1. Models, data sets and processing strategies for KPOD and RDPOD.

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, Peter et al. [31] performed a KPOD solution and realized a dm-level orbit accuracy by setting proper data screening options. Ren and Schön [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. The quality control of GNSS solutions is usually carried out in the data preprocessing. For instance, Peter et al. [31] performed a KPOD solution and realized a dm-level orbit accuracy by setting proper data screening options. Ren and Schön [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.
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 3σ principle, supposing the data as Gross Errors and rejecting them when the orbital residual is greater than 3σ; 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.

Results and Analysis
In this paper, the performance of kinematic orbits and reduced dynamic orbits solution is evaluated by carrier phase residual analysis, SLR validation, and comparison with a precise ephemeris.

Carrier Phase Residual Analysis
The residuals of the phase observations were calculated to evaluate the accuracy of the LEO satellite orbiting results. They were close to the observation noise level when the quality of the observations and the preprocessing results were satisfactory [33]. Figure 2 shows the residual RMS of the phase observations of Swarm and GRACE-FO satellites and the solar flux index F10.7 on that day. Meanwhile, all satellites' mean RMS of the residuals for all phase observations are summarized in Table 2.
According to Figure 2 and Table 2, the solar flux index F10.7 fluctuated between 70 and 80 during DOY 152 to 165 in 2021, which means that no excessively intense solar activity occurred during this time. The RMS of residuals of the phase observation for Swarm satellites KPOD solution is distributed between 4 and 5 mm, while the RMS for the RDPOD solution is mostly distributed between 6.5 and 7.5 mm (except for Swarm-C with 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 3σ principle, supposing the data as Gross Errors and rejecting them when the orbital residual is greater than 3σ; 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.

Results and Analysis
In this paper, the performance of kinematic orbits and reduced dynamic orbits solution is evaluated by carrier phase residual analysis, SLR validation, and comparison with a precise ephemeris.

Carrier Phase Residual Analysis
The residuals of the phase observations were calculated to evaluate the accuracy of the LEO satellite orbiting results. They were close to the observation noise level when the quality of the observations and the preprocessing results were satisfactory [33]. Figure 2 shows the residual RMS of the phase observations of Swarm and GRACE-FO satellites and the solar flux index F10.7 on that day. Meanwhile, all satellites' mean RMS of the residuals for all phase observations are summarized in Table 2.
According to Figure 2 and Table 2, the solar flux index F10.7 fluctuated between 70 and 80 during DOY 152 to 165 in 2021, which means that no excessively intense solar activity occurred during this time. The RMS of residuals of the phase observation for Swarm satellites KPOD solution is distributed between 4 and 5 mm, while the RMS for the RDPOD solution is mostly distributed between 6.5 and 7.5 mm (except for Swarm-C with an RMS of 15.8 mm on DOY 162). The RMS of the phase observations of GRACE-FO satellites for the KPOD solution is in the range of 5.0-6.5 mm, and the RMS for the RDPOD solution is in the range of 7.5-8.5 mm. an RMS of 15.8 mm on DOY 162). The RMS of the phase observations of GRACE-FO satellites for the KPOD solution is in the range of 5.0-6.5 mm, and the RMS for the RDPOD solution is in the range of 7.5-8.5 mm.    The phase observation residuals' RMS for the RDPOD solution of Swarm-C on DOY 162 reached 15.8 mm, which was higher than that in any other days. Analyzing the number of tracked GPS satellites after data preprocessing and the RTN accelerations estimated by RDPOD for Swarm-C on DOY 162 (shown in Figure 3), we found that lots of observations between 14:00 and 16:00 were deleted during data preprocessing, resulting in a fewer number of observations than six in many epochs. As we set the minimum number of observations when processing data, the estimation of Epoch Clock Offsets and RTN accelerations in RDPOD are affected when the number of observations is less than six, which reduces the accuracy of RDPOD. The accuracy of KPOD is also reduced for the same reason.
The phase observation residuals' RMS for the RDPOD solution of Swarm-C on DOY 162 reached 15.8 mm, which was higher than that in any other days. Analyzing the number of tracked GPS satellites after data preprocessing and the RTN accelerations estimated by RDPOD for Swarm-C on DOY 162 (shown in Figure 3), we found that lots of observations between 14:00 and 16:00 were deleted during data preprocessing, resulting in a fewer number of observations than six in many epochs. As we set the minimum number of observations when processing data, the estimation of Epoch Clock Offsets and RTN accelerations in RDPOD are affected when the number of observations is less than six, which reduces the accuracy of RDPOD. The accuracy of KPOD is also reduced for the same reason.
Therefore, the models, data sets, and solving strategies adopted in this paper are reasonable, and the results are stable. Furthermore, the number of tracked GPS satellites in many epochs between 14:00 and 16:00 is less than five, which reduces the accuracy of RDPOD. The residual RMS of the phase observation of Swarm-C for the RDPOD solution is 15.8 mm on DOY 162, 2021.

Satellite Laser Ranging Validation
Each Swarm and GRACE-FO satellite is equipped with a Laser Retro-Reflector, which consists of four corner cubes mounted in a small pyramid on the underside of the spacecraft [34]. SLR is a precise technique for ranging and orbit determination [6], which can be used for external calibration and validation of the LEO satellite orbit determination with on-board GPS receiver. Therefore, the models, data sets, and solving strategies adopted in this paper are reasonable, and the results are stable. Furthermore, the number of tracked GPS satellites in many epochs between 14:00 and 16:00 is less than five, which reduces the accuracy of RDPOD. The residual RMS of the phase observation of Swarm-C for the RDPOD solution is 15.8 mm on DOY 162, 2021.

Satellite Laser Ranging Validation
Each Swarm and GRACE-FO satellite is equipped with a Laser Retro-Reflector, which consists of four corner cubes mounted in a small pyramid on the underside of the spacecraft [34]. SLR is a precise technique for ranging and orbit determination [6], which can be used for external calibration and validation of the LEO satellite orbit determination with on-board GPS receiver.
The SLR technique for LEO satellites equipped with Retro-Reflectors is highly precise and has unambiguous distance measurements between the SLR station and the satellite. It is fully independent of the GNSS POD technique, so it can be used to validate the orbit of the LEO satellite by directly comparing the SLR range with the geometric distance between the SLR station and the LEO satellite's microwave-calculated orbit [35,36]. The path of the laser is from SLR station telescopes to the Retro-Reflector mounted on the satellite and then reflects to the detector of the station. The SLR observes the time-offlight of the laser. We will obtain a two-way range by multiplying the travel time with the speed of light [37]. The SLR range model can be described as follows [38]: where ρ is the SLR range; ρ 0 is the geometric distance between the SLR station and the LEO satellite's microwave-computed orbit; ∆ρ T is the total correction caused by solid tides, pole tides, sea tides, and atmospheric tides; ∆ρ F is the tropospheric range correction; and ∆ρ R is the relativistic correction; and ε is the observation noise. The eccentric correction of laser reflector ∆ρ M and the correction due to the station eccentricity ∆ρ O are also taken into account for the range delay.
In this paper, we collected 14 days' SLR observations of Swarm and GRACE-FO satellites stored as Normal Point (NPT) data, which were provided by the International Laser Ranging Service (ILRS). In order to ensure the quality of data processing and the reliability of the validation, the following data processing strategies (Table 3) were adopted:

1.
To reduce the effect of tropospheric refraction on SLR observations, the SLR observation is excluded with satellite altitude angles less than 25 • ; 2.
The data from the station Svetloe (1888 12350S002) and the station Irkutsk (1891 12313S007) are deducted to ensure the quality of the observations; 3.
The residuals greater than 3σ during the validation based on the 3σ principle are rejected. Table 3. SLR data processing strategies. The NPT data from selected 13 SLR stations are used, which are shown in Figure 4. Figure 5 shows the SLR residuals for each satellite's kinematic and reduced dynamic orbit from DOY 152 to 165 in 2021 at each station, while the residuals' mean value and their RMS are also summarized in Table 4.

Models/References
According to Figure 5 and Table 4, the undifferenced kinematic orbiting accuracy of Swarm-A is 3.39 cm, and the reduced dynamic orbiting accuracy is 2.25 cm. The accuracy of undifferenced kinematic orbits for Swarm-C is 3.33 cm, and the accuracy of reduced dynamic orbits is 2.52 cm. Compared with Swarm-A and Swarm-C, Swarm-B has a lower accuracy. The accuracy of Swarm-B undifferenced kinematic orbits is 3.71 cm, and the reduced dynamic orbiting accuracy is 2.69 cm. GRACE-FO satellites have higher orbiting accuracy than Swarm satellites. The undifferenced kinematic orbiting accuracy of GRACE-C is 1.99 cm, and the reduced dynamic orbiting accuracy is 1.06 cm, while the accuracy of undifferenced kinematic orbits for GRACE-D is 3.36 cm, and the accuracy of reduced dynamic orbiting is 1.20 cm.
Meanwhile, the accuracy of SLR validation for reduced dynamic orbits is higher than the accuracy for kinematic orbits as we need to interpolate the ephemeris of LEO satellite orbits recorded in integer seconds (for example, 2021-06-13T09:48:40) to match the noninteger second-time (for example, 2021-06-13T09:48:40.487596) points recorded in the SLR observations when performing the SLR validation. Since no orbital dynamics were taken into account in the kinematic orbits' calculation, some jumping phenomena may occur at certain moments, and the frequency and intensity of these phenomena affect the accuracy of the interpolation and thus reduce the accuracy of the SLR validation. For example, GRACE-D has a worse accuracy of undifferenced kinematic orbits than its reduced dynamic orbits due to more jumping phenomena, and these phenomena will become more evident in the next section.  According to Figure 5 and Table 4, the undifferenced kinematic orbiting accuracy of Swarm-A is 3.39 cm, and the reduced dynamic orbiting accuracy is 2.25 cm. The accuracy of undifferenced kinematic orbits for Swarm-C is 3.33 cm, and the accuracy of reduced dynamic orbits is 2.52 cm. Compared with Swarm-A and Swarm-C, Swarm-B has a lower accuracy. The accuracy of Swarm-B undifferenced kinematic orbits is 3.71 cm, and the reduced dynamic orbiting accuracy is 2.69 cm. GRACE-FO satellites have higher orbiting accuracy than Swarm satellites. The undifferenced kinematic orbiting accuracy of GRACE-C is 1.99 cm, and the reduced dynamic orbiting accuracy is 1.06 cm, while the accuracy of undifferenced kinematic orbits for GRACE-D is 3.36 cm, and the accuracy of reduced dynamic orbiting is 1.20 cm.
Meanwhile, the accuracy of SLR validation for reduced dynamic orbits is higher than the accuracy for kinematic orbits as we need to interpolate the ephemeris of LEO satellite orbits recorded in integer seconds (for example, 2021-06-13T09:48:40) to match the noninteger second-time (for example, 2021-06-13T09:48:40.487596) points recorded in the SLR observations when performing the SLR validation. Since no orbital dynamics were taken into account in the kinematic orbits' calculation, some jumping phenomena may occur at certain moments, and the frequency and intensity of these phenomena affect the accuracy of the interpolation and thus reduce the accuracy of the SLR validation. For example, GRACE-D has a worse accuracy of undifferenced kinematic orbits than its reduced dynamic orbits due to more jumping phenomena, and these phenomena will become more evident in the next section.  According to Figure 5 and Table 4, the undifferenced kinematic orbiting accuracy of Swarm-A is 3.39 cm, and the reduced dynamic orbiting accuracy is 2.25 cm. The accuracy of undifferenced kinematic orbits for Swarm-C is 3.33 cm, and the accuracy of reduced dynamic orbits is 2.52 cm. Compared with Swarm-A and Swarm-C, Swarm-B has a lower accuracy. The accuracy of Swarm-B undifferenced kinematic orbits is 3.71 cm, and the reduced dynamic orbiting accuracy is 2.69 cm. GRACE-FO satellites have higher orbiting accuracy than Swarm satellites. The undifferenced kinematic orbiting accuracy of GRACE-C is 1.99 cm, and the reduced dynamic orbiting accuracy is 1.06 cm, while the accuracy of undifferenced kinematic orbits for GRACE-D is 3.36 cm, and the accuracy of reduced dynamic orbiting is 1.20 cm.
Meanwhile, the accuracy of SLR validation for reduced dynamic orbits is higher than the accuracy for kinematic orbits as we need to interpolate the ephemeris of LEO satellite orbits recorded in integer seconds (for example, 2021-06-13T09:48:40) to match the noninteger second-time (for example, 2021-06-13T09:48:40.487596) points recorded in the SLR observations when performing the SLR validation. Since no orbital dynamics were taken into account in the kinematic orbits' calculation, some jumping phenomena may occur at certain moments, and the frequency and intensity of these phenomena affect the accuracy of the interpolation and thus reduce the accuracy of the SLR validation. For example, GRACE-D has a worse accuracy of undifferenced kinematic orbits than its reduced dynamic orbits due to more jumping phenomena, and these phenomena will become more evident in the next section.

Comparison with a Precise Ephemeris
The precise reduced dynamic orbit (with a sampling rate of 1 s) of Swarm satellites is provided by the ESA, which was solved by GHOST with an accuracy better than 2 cm [26]. The GFZ also provides a GRACE-FO precise orbit with a sampling rate of 1 s and accuracy better than 2 cm. In this section, the reduced dynamic orbits of Swarm satellites provided by the ESA and the precise orbits of GRACE-FO satellites provided by the GFZ are used as reference orbits, which are compared with our undifferenced kinematic orbits to assess the accuracy of the results.

Comparison with a Precise Ephemeris
The precise reduced dynamic orbit (with a sampling rate of 1 s) of Swarm satellites is provided by the ESA, which was solved by GHOST with an accuracy better than 2 cm [26]. The GFZ also provides a GRACE-FO precise orbit with a sampling rate of 1 s and accuracy better than 2 cm. In this section, the reduced dynamic orbits of Swarm satellites provided by the ESA and the precise orbits of GRACE-FO satellites provided by the GFZ are used as reference orbits, which are compared with our undifferenced kinematic orbits to assess the accuracy of the results.
To visualize the jumping phenomenon in the kinematic orbit and to verify the effectiveness of the quality control measures for the kinematic orbit, the following two groups were set up in this section for the orbit comparison concerning whether or not quality control was applied:

•
Group 1: Applying quality control for the solved kinematic orbits before comparing with reference orbits. • Group 2: Comparing with reference orbits without applying quality control. Figures 6-10 show the comparison of the undifferenced kinematic orbits with the reference orbits for Swarm and GRACE-FO satellites. We set up colorful comparisons in these figures to highlight the RMS differences between the two groups and the orbit differences in each RTN direction. The RMS of the KPOD results for each satellite in different groups is also summarized and listed in Table 5. As we can see in Figures 6-10  and Table 5, in Group 2, the RMS of the undifferenced kinematic orbits for Swarm-A/B/C are between 3.0 and 5.0 cm in RTN directions, with total 3-D RMS between 2.5 and 5.0 cm for each day. The RMS of the undifferenced kinematic orbit of GRACE-C in RTN directions ranges from 1.5 to 3.0 cm, and the total 3D-RMS are in the range of 2.0-2.8 cm for all days.
The RMS of the undifferenced kinematic orbit of GRACE-D, as the leading satellite of the GRACE-FO binary system, is slightly worse with the RMS from 3.0 to 5.5 cm. Therefore, the accuracy of the kinematic orbit calculated in this paper is better than 5 cm for Swarm satellites and better than 5.5 cm for GRACE-FO satellites. cm to 4.4 cm with about 20%. The 3D-RMSs of kinematic orbits for GRACE-D on DOY 153, 157, 164, and 165 were decreased from 5.5 cm, 4.0 cm, 3.8 cm, and 4.5 cm to 4.6 cm, 3.4 cm, 3.3 cm, and 3.4 cm, respectively with about 22 %. This shows that the quality control for kinematic orbits adopted in this paper has a significant effect.
Meanwhile, from Table 5, after quality control for the kinematic orbit, the 3D-RMS of the kinematic orbit is reduced from 3.57 cm to 3.53 cm for Swarm-A, from 4.11 cm to 4.03 cm for Swarm-B, and from 3.61 cm to 3.52 cm for Swarm-C. Therefore, the quality control method adopted in this paper improved the kinematic orbit accuracy by reducing the frequency and degree of orbital jumps, which can be seen in Figures 6-10.         Table 5. RMS with different groups comparing kinematic orbit and reference orbit.

Satellite Groups 3D-RMS (cm) R-RMS (cm) T-RMS (cm) N-RMS (cm) Improvement
Swarm   According to the comparison of Group 1 and Group 2, different levels of jumping phenomena occurred in the kinematic orbit for both Swarm and GRACE-FO satellites, some of which reached the decimeter-level, and some even reached a meter-level. However, the results of Group 1 show that the quality control method can effectively suppress the orbital jumps. The Swarm-C satellites (on DOY 162 in 2021) and GRACE-D satellites (e.g., on DOY 153, 157, 164, and 165 in 2021) with more frequent orbital jumps are taken as an example. According to Figures 8e and 10e, after we took quality control measures, the total 3-D RMS of the kinematic orbit for Swarm-C on DOY 162 was decreased from 5.5 cm to 4.4 cm with about 20%. The 3D-RMSs of kinematic orbits for GRACE-D on DOY 153, 157, 164, and 165 were decreased from 5.5 cm, 4.0 cm, 3.8 cm, and 4.5 cm to 4.6 cm, 3.4 cm, 3.3 cm, and 3.4 cm, respectively with about 22 %. This shows that the quality control for kinematic orbits adopted in this paper has a significant effect.
Meanwhile, from Table 5, after quality control for the kinematic orbit, the 3D-RMS of the kinematic orbit is reduced from 3.57 cm to 3.53 cm for Swarm-A, from 4.11 cm to 4.03 cm for Swarm-B, and from 3.61 cm to 3.52 cm for Swarm-C. Therefore, the quality control method adopted in this paper improved the kinematic orbit accuracy by reducing the frequency and degree of orbital jumps, which can be seen in Figures 6-10.

Conclusions
In this paper, the KPOD for Swarm and GRACE-FO satellites was carried out simultaneously for a total of 14 days from DOY 152 to 165 in 2021 with the same processing strategy based on an undifferenced IF linear combination and a least squares batch processing method. To suppress the jumping phenomenon of the kinematic orbit and improve the accuracy of the final solution, RDPOD was performed, along with KPOD, and the calculated reduced dynamic orbit was used as the benchmark for quality control of the kinematic orbit. The accuracy of the results was evaluated by three approaches: residual analysis of phase observations, SLR validation, and comparison with a precise ephemeris. The following conclusions were obtained: 1. The residuals of phase observations for Swarm and GRACE-FO satellites show that the models, data sets, and processing strategies are reasonable, and the orbit determination is stable.
2. According to the results of SLR validation, the accuracy of the kinematic orbit and the reduced dynamic orbit both achieved the centimeter-level. The accuracy of the kinematic orbit and the reduced dynamic orbit of Swarm satellites is better than 4 cm. The accuracy of the kinematic orbit of GRACE-FO satellites is better than 4 cm, and the accuracy of their reduced dynamic orbit is better than 2 cm.
3. After quality control for the kinematic orbit, the jumping phenomenon in the kinematic orbit was alleviated. By comparing with the precise ephemeris, the kinematic orbit accuracy was improved by 2.49% for Swarm-C and 6.98 for GRACE-D after the quality control.
Author Contributions: Conceptualization, S.J. and P.L.; methodology, P.L.; software, P.L.; validation, S.J., P.L. and Q.S.; formal analysis, P.L.; investigation, S.J. and P.L.; data curation, P.L.; writingoriginal draft preparation, S.J. and P.L.; writing-review and editing, S.J. and Q.S.; visualization, S.J.; supervision, S.J.; project administration, S.J.; funding acquisition, S.J. All authors have read and agreed to the published version of the manuscript. Data Availability Statement: The dual-frequency GPS observation data and precise dynamic orbit of Swarm satellites used in this paper can be downloaded from http://swarm-diss.eo.esa.int (accessed on 1 October 2021). The dual-frequency GPS observation data and precise orbit of GRACE-FO satellites used in this paper can also be downloaded from ftp://isdcftp.gfz-potsda.m.de/grace-fo/ (accessed on 1 October 2021).