## 3 Results and Analysis
### 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]. Figure 3: The number of observations and RTN accelerations in RDPOD for Swarm-C on DOY 162, 2021. (**a**) The number of tracked GPS satellites after data preprocessing for Swarm-C; (**b**) Estimated RTN accelerations of Swarm-C. 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-of-flight 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]:  
\[\rho=\rho_{0}+\Delta\rho_{T}+\Delta\rho_{F}+\Delta\rho_{R}+\Delta\rho_{M}+ \Delta\rho_{O}+\varepsilon \tag{7}\]  
where \(\rho\) is the SLR range; \(\rho_{0}\) is the geometric distance between the SLR station and the LEO satellite's microwave- computed orbit; \(\Delta\rho_{T}\) is the total correction caused by solid tides, pole tides, sea tides, and atmospheric tides; \(\Delta\rho_{F}\) is the tropospheric range correction; and \(\Delta\rho_{R}\) is the relativistic correction; and \(\varepsilon\) is the observation noise. The eccentric correction of laser reflector \(\Delta\rho_{M}\) and the correction due to the station eccentricity \(\Delta\rho_{O}\) 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\({}^{\circ}\);
2. The data from the station Svetloe (1888 123505002) and the station Irkutsk (1891 123135007) are deducted to ensure the quality of the observations;
3. The residuals greater than 3\(\sigma\) during the validation based on the 3\(\sigma\) principle are rejected. 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.  
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.