In this section, we first assess the ambiguity resolution performance for different SRP models in the experiment. Then we evaluate the precision of real-time predicted orbits based on orbit overlap differences with post-processed observation arcs. Finally, we present the accuracy of microwave-based predicted orbits by satellite laser ranging validation.
3.1. Ambiguity Resolution Performance
Ambiguity resolution can significantly improve the GNSS orbit determination. To assess the impact of different SRP models, we first analyze the ambiguity fixing rate of the baselines of various lengths for GPS, GLONASS, Galileo, and BDS, respectively. A higher fixing rate indicates a tighter distribution of double-differenced ambiguities around integers and a better orbit solution.
Table 2 gives the average number of GPS independent double-differenced ambiguities and the fixing rates for different baseline lengths. It shows that the relation between GPS ambiguity fixing rate and the baseline length was not obvious, which indicates that GPS orbit precision was high enough for ambiguity resolution at long baselines. Overall, about 95% of the independent ambiguities were fixed, which was consistent with previous studies [
46]. The fixing rate is almost the same for ECOM 1 and ECOM 2, which indicates that the two SRP models can achieve similar and high GPS orbit precision.
Table 3 gives the number of GLONASS independent double-differenced ambiguities and the fixing rate for different baseline lengths. Considering the relatively lower precision of real-time solution and the short wavelength (about 5.3 cm) of GLONASS ionospheric-free ambiguities, the fixing rate for baselines shorter than 1500 km was analyzed. It shows that the GLONASS fixing rate decreased gradually with the baseline length, which indicates that the geometric errors biased the ambiguity parameters, especially for the long baselines. The longer the baseline is, the better ECOM 2 performs compared to ECOM 1. Overall, the fixing rate is 86.5% and 89.3% for ECOM 1 and ECOM 2, respectively, which implies that ECOM 2 performs better than ECOM 1 for GLONASS ultra-rapid orbits.
Table 4 gives the average number of Galileo independent double-differenced ambiguities and the fixing rate for different baseline lengths. The relation between Galileo fixing rate and the baseline length is obvious. The longer the baseline is, the lower the fixing rate. It indicates that the Galileo orbit accuracy was not high enough, and the geometric errors largely biased the ambiguity parameter. The longer the baseline, the better the ECOM 2 performs compared to ECOM 1. Overall, the fixing rate is 80.0% and 83.1% for ECOM 1 and ECOM 2, respectively, which implies that ECOM 2 performs better than ECOM 1 for Galileo ultra-rapid orbits.
Table 5 gives the average number of BDS independent double-differenced ambiguities and the fixing rate for different baseline lengths. It shows that the BDS fixing rate decreased steeply with the baseline length. It indicates that BDS orbit accuracy was not high enough, and geometric errors hindered the ambiguity resolution, especially for longer baselines. Considering the relatively small number of independent ambiguities, the difference in fixing rate between ECOM 1 and ECOM 2 is not obvious. Overall, the fixing rate is 76.3% and 77.9% for ECOM 1 and ECOM 2, respectively, which is slightly lower than that of Galileo.
The ambiguity resolution comparison shows that for GLONASS and Galileo satellites, ECOM 2 achieves higher fixing rates than ECOM 1; and for GPS and BDS IGSO/MEO satellites, ECOM 1 and ECOM 2 achieved similar fixing rates. The different ambiguity resolution performance of ECOM 1 and ECOM 2 is also reflected in the following orbit overlap precision comparison and satellite laser ranging validation.
3.2. Orbit Overlap Precision
We first assess the precision of ultra-rapid orbits using the difference of satellite positions in the overlap arcs between two orbit solutions. The current IGS GLONASS final orbits are the combination of mainly float solution orbits from the analysis centers and cannot be used to evaluate the fixed solution orbits accurately. The IGS analysis center CODE enables fixing of all GLONASS ambiguities for baselines shorter than 200 km, but for longer baselines (below 2000 km), only the ambiguities between satellites with the same frequency are fixed [
56]. In contrast, the GLONASS ambiguity resolution method in this research can fix the ambiguities with no restrictions on frequency and receiver type for long baselines [
53]. For Galileo and BDS, the orbit products of different IGS MGEX analysis centers show some inconsistencies, indicating that it is still difficult to achieve the same accuracy level as GPS and GLONASS. Moreover, for the new GNSS, the lack of enough stations with even global distribution has a negative impact on the IGS MGEX orbit products, especially for BDS. Well-distributed BDS stations in the Asia-Pacific region are still lacking, which can degrade the IGS MGEX orbit products, such as CODE and GFZ products.
To evaluate the orbit precision more objectively, Griffiths and Ray [
57] show that the discontinuities in overlap arcs is a better metric. In addition, the overlap arc difference can effectively avoid the biases in inter-AC comparison caused by different strategies and models, such as satellite attitude, antenna phase center, and ambiguity resolution strategies. Thus, the single factor of the solar radiation pressure model on orbits can be analyzed more properly. Therefore, we used this indicator of overlap comparison to evaluate the orbits. Because the overlap difference is calculated by two adjacent orbits, the precision of each orbit is small by about sqrt(2). The predicted orbit arc of the previous daily solution was compared with that of the observed arc of the next daily solution. The predicted orbit arcs of 0 h (00:00 point), 1, 3, 6, and 24 h were analyzed, respectively. The seven-parameter Helmert transformation was used in orbit comparison to remove possible systematic differences between orbits.
The mean RMS of overlap differences in the along-track, cross-track, and radial direction for GPS satellites are shown in
Figure 2. ECOM 2 generally obtained higher precision than ECOM 1 for GPS predicted orbits. The GPS predicted orbit precision degraded gradually with the predicted arc length. For the predicted orbit arc of 1 h, ECOM 1 obtained the overlap precision of 3.3 cm, 2.1 cm and 2.6 cm in the along-track, cross-track and radial direction, respectively; ECOM 2 obtained the overlap precision of 3.0, 2.0, and 2.2 cm in the along-track, cross-track, and radial direction, respectively, which was a slight improvement of 9.1%, 4.8%, and 15.4% compared with ECOM 1, respectively.
The mean RMS of overlap differences in the along-track, cross-track, and radial direction for GLONASS satellites are shown in
Figure 3. ECOM 2 model generally obtained higher 3D precision than the ECOM 1 model for GLONASS predicted orbits. The GLONASS predicted orbit precision degraded gradually with the predicted arc length. For the predicted orbit arc of 1 h, ECOM 1 model obtained the overlap precision of 7.0, 5.0, and 2.7 cm in the along-track, cross-track, and radial direction, respectively; ECOM 2 model obtained the overlap precision of 6.8, 4.3, and 2.5 cm in the along-track, cross-track, and radial direction, respectively, which was a slight improvement of 2.9%, 14.0%, and 7.4% compared with ECOM 1, respectively. The 1-h predicted GLONASS orbits using ECOM 2 obtained the along-track and cross-track precision worse by a factor of about two compared to GPS, and the radial precision close to GPS, which is the main component of the signal-in-space range error.
The mean RMS of overlap differences in the along-track, cross-track, and radial direction for Galileo satellites are shown in
Figure 4. The ECOM 2 model generally obtained higher precision than the ECOM 1 model for Galileo predicted orbits. The Galileo predicted orbit precision degraded rapidly with the predicted arc length, especially for the along-track direction, which implies certain mismodeling in observation or force models. For the predicted orbit arc of 1 h, the ECOM 1 model obtained the overlap precision of 11.7, 8.5, and 7.2 cm in the along-track, cross-track, and radial direction, respectively; ECOM 2 model obtained the overlap precision of 8.8, 6.2, and 5.9 cm in the along-track, cross-track, and radial direction, respectively, which was a significant improvement of 24.8%, 27.1%, and 18.1% compared with ECOM 1, respectively. The 1-h predicted Galileo orbits using ECOM 2 were worse by a factor of about three compared to GPS.
The mean RMS of overlap differences for BDS-2 IGSO satellites is shown in
Figure 5. The ECOM 1 model generally obtained higher precision than ECOM 2 for BDS-2 IGSO predicted orbits. The BDS-2 IGSO predicted orbit precision degraded rapidly with the predicted arc length, especially for the along-track direction, which also implies certain mismodeling in the observation or force models. For the predicted orbit arc of 1 h, the ECOM 1 model obtained the overlap precision of 28.4 cm and 18.4 cm in the 3D and radial components, respectively; ECOM 2 model obtained the overlap precision of 32.9 cm and 24.7 cm in the 3D and radial components, respectively, which was a significant degradation of 15.8% and 34.2% compared to ECOM 1, respectively. The 1-h predicted BDS-2 IGSO orbits using ECOM 1 were worse by a factor of about seven compared with GPS.
The mean RMS of overlap differences for BDS-2 MEO satellites are shown in
Figure 6. The ECOM 1 model generally obtained higher precision than the ECOM 2 model for BDS-2 MEO predicted orbits. The BDS-2 MEO predicted orbit precision degraded rapidly with the predicted arc length, especially for the along-track direction, which also implies certain mismodeling in observation or force models. For the predicted orbit arc of 1 h, the ECOM 1 model obtained the overlap precision of 15.6 cm and 5.7 cm in the 3D and radial components, respectively; ECOM 2 model obtained the overlap precision of 16.1 cm and 6.6 cm in the 3D and radial components, respectively, which was a degradation of 3.2% and 15.8% compared with ECOM 1, respectively. The 1-h predicted BDS-2 MEO orbits using ECOM 1 were similar to Galileo, and worse by a factor of about three compared with GPS in the radial direction.
For the BDS GEO satellites in the experiment, the ECOM 2 model generally obtained better overlap precision than the ECOM 1 model in the 3D and radial components. For the predicted orbit arc of 24 h, the ECOM 1 and ECOM 2 models obtained the radial overlap precision of 282.4 cm and 220.6 cm, respectively, and ECOM 2 showed an improvement of 21.9% compared with ECOM 1. For BDS GEO satellites, the SRP model should accurately describe the SRP force while reducing the correlation between parameters, which needs further SLR validation.
The multi-GNSS ultra-rapid orbits in the experiment obtained comparable precision with the current predicted orbit products [
5,
6,
7,
8,
9]. The orbit overlap comparison shows that for GPS, GLONASS, Galileo, and BDS GEO satellites, the ECOM 2 model generally achieved better orbital prediction than the ECOM-1 model, while for BDS IGSO and MEO satellites, the ECOM 1 model generally achieved better orbital prediction than the ECOM 2 model.
3.3. Satellite Laser Ranging Validation
Satellite laser ranging (SLR) observables are usually used as external validation to evaluate GNSS satellite orbit quality. The SLR residuals, i.e., the differences between the SLR observables and the range calculated from microwave-based satellite positions, including the observed and predicted orbit arc, mainly show the GNSS orbit accuracy in the radial direction. In the experiment, the GLONASS, Galileo, and BDS GEO C01, IGSO C08, and C10, MEO C11 satellites equipped with laser retroreflector arrays were observed by the SLR stations from the International Laser Ranging Service (ILRS) [
58]. The SLR station coordinates were fixed to the a priori reference frame, and the station displacements were corrected consistently with the microwave-based solutions. The tropospheric delays, relativistic effects, and the Laser Retro-Reflector Arrays offsets with respect to the satellites’ center of mass were corrected in the SLR observables. Outliers exceeding 0.6 m were excluded for GLONASS, Galileo, and BDS MEO satellites, and those exceeding 3.0 m and 10.0 m were excluded for BDS IGSO and GEO satellites, respectively. This outlier check excluded approximately 2.5%, 3.7%, 9.3%, and 1.2% of the data points for GLONASS, Galileo, BDS MEO, and BDS IGSO satellites, respectively, and no data points were excluded for BDS GEO satellites. After the removal of outliers, the number of normal points for observed orbit arcs of 24 h (predicted 0 h) and various predicted orbit arcs is shown in
Table 6.
The RMS of the SLR residuals for the observed and predicted orbits are given in
Figure 7. The predicted orbit precision of GLONASS, Galileo, and BDS MEO satellites generally degraded gradually with the predicted arc length. The RMS of the SLR residuals was slightly larger than the RMS of the orbit overlap differences in the radial direction in the previous section. Considering SLR as the external validation from a different technique, the SLR residuals generally agreed with the orbit overlap precision.
For the GLONASS predicted orbit arc of 1 h, the SLR residuals RMS was 6.7 cm for the ECOM 1 model and 6.4 cm for the ECOM 2 model, which corresponds to an improvement of 5.2% of ECOM 2 compared to ECOM 1. For the GLONASS predicted orbit arc of 24 h, the SLR residuals RMS was 9.8 cm for ECOM 1 and 8.9 cm for ECOM 2, which corresponds to an improvement of 9.2%. For the Galileo predicted orbit arc of 1 h, the SLR residuals RMS was 11.1 cm for the ECOM 1 model and 7.0 cm for ECOM 2, which corresponds to an improvement of 36.9% of ECOM 2 compared to ECOM 1. For the Galileo predicted orbit arc of 24 h, the SLR residuals RMS was 15.9 cm for ECOM 1 and 11.5 cm for ECOM 2, which corresponds to an improvement of 27.7%. For the BDS MEO predicted orbit arc of 24 h, the SLR residuals RMS was 12.8 cm for ECOM 1 and 13.2 cm for ECOM 2, which corresponds to a degradation of 3.1% of ECOM 2 compared to ECOM 1.
For the BDS IGSO satellites, the ECOM 1 model obtained the SLR residuals RMS of 6.8 cm and 74.2 cm for the observed and predicted orbit arc of 24 h, respectively, whereas ECOM 2 model obtained the SLR residuals RMS of 7.7 cm and 99.7 cm, which was a degradation of 13.2% and 34.4% compared to ECOM 1. For the BDS GEO satellites, the respective values for the ECOM 1 model are 27.5 cm and 508.5 cm, and for the ECOM 2 model, they are 13.2 cm and 166.4 cm, which was an improvement of 52.0% and 67.3% compared to ECOM 1.
The satellite laser ranging comparison shows that for GLONASS, Galileo, and BDS GEO satellites, the ECOM 2 model generally achieved better predicted orbits than the ECOM 1 model while for BDS IGSO and MEO satellites the ECOM 1 model generally achieved better predicted orbits than the ECOM 2 model. The satellite laser ranging results generally agreed well with the orbit overlap precision comparison, which confirms the findings with the external validation.