Signal-in-space range error and positioning accuracy of BDS-3

Being the first mixed-constellation global navigation system, the global BeiDou navigation system (BDS-3) designs new signals, the service performance of which has attracted extensive attention. In the present study, the Signal-in-space range error (SISRE) computation method for different types of navigation satellites was presented. The differential code bias (DCB) correction method for BDS-3 new signals was deduced. Based on these, analysis and evaluation were done by adopting the actual measured data after the official launching of BDS-3. The results showed that BDS-3 performed better than the regional navigation satellite system (BDS-2) in terms of SISRE. Specifically, the SISRE of the BDS-3 medium earth orbit (MEO) satellites reached 0.52 m, slightly inferior compared to 0.4 m from Galileo, marginally better than 0.59 m from GPS, and significantly better than 2.33 m from GLONASS. The BDS-3 inclined geostationary orbit (IGSO) satellites achieved the SISRE of 0.90 m, on par with that (0.92 m) of the QZSS IGSO satellites. However, the average SISRE of BDS-3 geostationary earth orbit (GEO) satellites was 1.15 m, which was marginally inferior to that of the QZSS GEO satellite (0.91 m). In terms of positioning accuracy, the new signals B1C and B2a are considered together with the transition signals B1I and B3I. The overall three-dimensional single-frequency standard point positioning (SPP) accuracy of BDS-3 B1C, B2a, B1I, and B3I gained an accuracy level better than 5 m. Moreover, the B1I signal exhibited the best positioning accuracy in the Asian-Pacific region, while the B1C signal set forth the best positioning accuracy in the other regions. Owing to the advantage in signal frequency, the dual-frequency SPP accuracy of B1C + B2a surpassed that of the transitional signal of B1I + B3I. Since there are more visible satellites in Asia–Pacific, the positioning accuracy of BDS-3 was moderately superior to that of GPS. The precise point positioning (PPP) accuracy of BDS-3 B1C + B2a or B1I + B3I converged to the order of centimeters, marginally inferior to that of the GPS L1 + L2. However, these three combinations had a similar convergence time of approximately 30 min.


Scientific Reports
| (2022) 12:8181 | https://doi.org/10.1038/s41598-022-12012-y www.nature.com/scientificreports/ user reception, including noise, multipath, uncorrected ionospheric delay and tropospheric delay, etc. The SISRE reflects the error of the broadcast ephemeris orbit and clock offset, which primarily showcases the performance of the space segment and the ground segment. Montenbruck et al. (2015) pointed out that the SISRE could be used as a key system indicator to analyze the overall performance of the ground segment and the space segment 10 .
The commonly used method is to select the precise orbit and clock offset as the standard to evaluate the accuracy level of the broadcast ephemeris orbit and clock offset, and then obtain the SISRE. With the development of BeiDou, many scholars conducted analysis and research on the SISRE  Table 1. Operational BeiDou System satellites in orbit. The last launched BDS-3 GEO satellite C61 is not included, because it is still in the in-orbit testing phase when the paper is completed. BDS-2   GEO  C01-C05   IGSO  C06-C10, C13, C16   MEO  C11, C12, C14   BDS-3   GEO  C59, C60   IGSO  C38-C40   MEO  C19-C30, [20][21][22][23] 27,28 . However, the above analyses were mostly conducted when the construction of BDS-3 was incomplete. Moreover, the assessment for the full constellation positioning performance of the BDS-3 new signals was limited.

System Satellite type PRN
We here present the SISRE computation method for different types of navigation satellites and deduce the differential code bias (DCB) correction method for BDS-3 new signals during the positioning data processing. Then the SISRE of BDS-3 is analyzed including GEO, IGSO, and MEO satellites. Also, the SISREs of BDS-3, BDS-2, GPS, GLONASS, Galileo, and QZSS are compared by processing measured data. Meanwhile, Standard Point Positioning (SPP) and PPP, the two positioning modes that cannot eliminate the common errors by observation difference and can best reflect the data quality of the pseudo-code and the carrier phase, are adopted to perform the in-depth analysis on the positioning performance of the new signals and the transitional signals for BDS-3.

Methods
Within this section, SISRE computation method for different types of satellites is introduced, which is convenient to compute and compare the SISREs of different navigation systems. To evaluate performance of BDS-3 SPP and PPP, the DCB correction method for B1C and B2a signals is deduced.

SISRE.
The SISRE, which involves two parts, i.e., the satellite orbit error and the satellite clock offset error, is evaluated as follows. The multi-system precise orbit and clock offset, which are calculated by the International GNSS Service (IGS) analysis centers, are taken as the standard and the broadcast ephemeris is used to derive the satellite orbit and the clock offset of the corresponding epoch, analyzing their difference to gain the SISRE. During the analysis, some issues should be processed, such as the antenna phase offset correction, the time group delay, et al 12,14,16 . In order to enhance comparison effects, the following work has been done: first of all, distinguish three different satellite types such as GEO, IGSO and MEO to reflect the SISRE differences in the two cases for different satellite types of the same system and the same satellite type of different systems; Secondly, some major navigation systems such as BDS, GPS, Galileo, GLONASS, and QZSS are simultaneously considered for mutual comparison; Finally, a long period of data is analyzed to enhance the credibility of the analysis conclusion.
The computation equation of SISRE for different types of satellites is as follows: where w R and w A,C are corresponding weight factors that are related to the satellite altitude 10 , as mentioned in Table 3; R , A , and C denote the radial, along-track, and cross-track, respectively; δt indicates the clock offset error of the satellite; and c represents the speed of light.
SPP and PPP processing for BDS-3. SPP   www.nature.com/scientificreports/ offset products are typically calculated by the B1I and B3I ionosphere-free combination. Hence, whether the broadcast ephemeris or the precise product is in use while processing B1C or B2a, the DCB correction should be noted.
Here, the B1C and B2a ionosphere-free combination for PPP is taken as an example. In this case, the B1C and B2a dual-frequency pseudo-code ionosphere-free combination 4 should be utilized as: where PC B1C_B2a indicates the B1C and B2a dual-frequency pseudo-code ionosphere-free combination observation; R indicates the geometrical distance between the receiver and the satellite; c denotes the speed of light; δt r and δt s represent the receiver clock offset and the satellite clock offset, respectively; δ trop is the tropospheric correction; f B1C and f B2a denote the frequency of B1C and B2a, respectively; τ B1C and τ B2a denote the internal signal delay of the B1C and the B2a pseudo-code, respectively.
According to (2), the satellite clock offset and the internal signal delay cannot be separated. The satellite clock offset δt st(B1C_B2a) that includes the internal signal delay is defined as: Similarly, the precise clock offset of BeiDou is often derived through the B1I and B3I ionosphere-free combination. We have Here, δt st(B1I_B3I) denotes the satellite clock offset that contains the internal signal delay of the B1I and B3I ionosphere-free combination; f B1I and f B3I denote the frequency of B1I and B3I, respectively; τ B1I and τ B3I represent the internal signal delay of the B1I and the B3I pseudo-code, respectively; δt s is the satellite clock offset as in (2) and (3).
According to (3) and (4), the relationship between δt st(B1C_B2a) and δt st(B1I_B3I) can be derived. We have can be derived from the DCB correction data released by the related organizations which is downloaded from ftp:// igs. ign. fr/ pub/ igs/ produ cts/ mgex/ dcb in the following analysis. Therefore, while using the precise clock offset of BeiDou for the PPP processing of the B1C and B2a ionosphere-free combination, it is mandatory to perform the DCB correction, as represented by "{}" in (5). In SPP processing, the similar problem need to be addressed.

Results
In this section, the actual measured data was analyzed after the official launching of the BDS-3 service. At first, the SISRE of BDS-3 was computed and compared to that of BDS-2, GPS, GLONASS, Galileo and QZSS. And then, the positioning performance of BDS-3 was analyzed in both cases of SPP and PPP. In the analysis, special attention was paid to the new signals of BDS-3.

SISRE computation.
According to the SISRE computation method specified above, the broadcast ephemeris was first used to compute the satellite position and the clock offset of 96 epochs with an interval of 900 s for each day. Later, considering the precise ephemeris and the precise clock offset at the corresponding epoch as the standard, the errors of the orbit and the clock offset were evaluated. Based on this, the SISRE was calculated. The multi-system BRDM long-filename file in the receiver independent exchange format (RINEX) provided by Multi-GNSS Experiment (MGEX) was adopted as the broadcast ephemeris. The GBM precise product was adopted as the precise orbit and the precise clock offset respectively with a sampling interval of 300 s and 30 s.
The data from August 1 to September 1, 2020, a month after the official launching of the BDS-3 service, were considered. According to the three satellite categories of GEO, IGSO, and MEO, the Root Mean Square (RMS) of the errors in all the epochs in a day was considered as the statistical accuracy of the orbit and the clock offset. The orbit accuracy (radial, R; tangential, T; normal, N) and the satellite clock offset accuracy variations were taken into account for all the in-orbit operational satellites of BDS-2 and BDS-3, as illustrated in Figs. 2, 3, and 4. To further compare the accuracy, the average RMSs of the R, T, or N orbit error and the satellite clock offset error of each day in the counting period were considered for each satellite, as highlighted in Fig. 5. In Table 4, the statistical results of orbit and clock offset accuracies are averaged for all the BDS-2 and BDS-3 satellites of each category.
The following can be seen from the above results: 1. Compared to the BDS-2 GEO satellites, the BDS-3 GEO satellites of C59 and C60 exhibited significantly better orbit accuracies in R, T, and N directions. In terms of satellite clock offset accuracy, the C59 satellite was superior to the BDS-2 satellites, while the C60 satellite demonstrated a relatively lower clock offset accuracy. Upon examination, it was concluded that C60 was launched in March 2020 while C59 was launched in December 2018, due to which the C60 satellite clock was possibly still in the process of aging and due for further improvement in its performance 29 . On average, the R-, T-, and N-direction orbit accuracy and To further compare the SISRE between BDS-3 and other systems, the analysis data above were also used. Initially, the SISRE of each epoch for every BDS-3 satellite was calculated. Then, the satellites were grouped into the three categories of GEO, IGSO, and MEO, and had the SISRE averaged in each group for all the epochs of each day. Besides, the same accuracy statistics were run for the SISRE of GPS, GLONASS, Galileo, and QZSS, to perform a better comparison. Moreover, QZSS included one GEO satellite and three IGSO satellites at the time, which were counted separately as in the case of BDS-3. As for the other systems, only MEO satellites were involved, thus demanding no categorization. Figure 6 illustrates the daily statistical result of SISRE during the analysis period for each system, while Table 5 enlists the average.
The following can be obtained from the results highlighted in Fig. 6 and Table 5:    Analysis of positioning accuracy. To enhance the presentation of the analysis result of the positioning accuracy, when selecting the analysis data, the spatial and temporal coverage was fully considered. In terms of spatial coverage, 27 multi-system observation stations distributed globally were chosen, as demonstrated in Fig. 7. Among them, 20 MGEX observation stations and seven iGMAS (international GNSS monitoring and assessment service) observation stations were included, respectively labeled as 'yellow square' and 'red triangle' . For convenient comparison, a red box is drawn in Fig. 7 that highlights the key BDS service area (55° S-55° N, 70° E-150° E). Moreover, to improve the temporal coverage of the analysis, from the days after the official service launch of the system, the observation data were chosen on the 1st date of August, September, October, November, and December in the year 2020, and on January 1, 2021, with a sampling interval of 30 s. Later, BDS-3 SPP  Figure 6. SISREs of major systems from Aug 1, 2020 to Sep 1, 2020. For comparison, BDS-3 satellites were grouped into the three categories of GEO, IGSO, and MEO. QZSS satellites are also made a similar classification.

SPP accuracy. This section mainly analyzes SPP accuracy of BDS-3 in the two cases of single-frequency
SPP and double-frequency SPP. In the former case, the single-frequency signals including B1C, B2a, B1I and B3I will be considered, and in the latter case, the double-frequency signals including B1C + B2a and B1I + B3I will be focused on.

Single-frequency SPP
The open positioning and navigation service signals of BDS-3 comprise the new signals of B1C and B2a along with the transitional signals of B1I and B3I. The former is only broadcast by the BDS-3 MEO and IGSO satellites, whereas the latter is broadcast by all the satellites of BDS-3 and BDS-2 category, as mentioned in Table 2. To assess the positioning performance of the signals, the observed pseudo-code for each single-frequency signal was utilized for SPP in each epoch. The ionospheric delay error is corrected by Klobuchar model. The satellite position and clock offset are calculated from the broadcast ephemeris. The least square estimation is used for user location estimation.
The RMSs of the east (e), north (n), up (u), and three-dimensional positioning errors were given in Fig. 8. For better comparison, the single-frequency positioning of the GPS L1 C/A code is also displayed in Fig. 8. Considering this, the positioning error RMS of each direction was averaged for all the observation stations, the results for which are enlisted in Table 6. It must be noted that single-frequency SPP has a wide range of application situations, the accuracy of which constitutes a key performance factor in the system construction.
The following can be concluded from Fig. 8 and Table 6: 1. Whether it was the Asian-Pacific region or other regions, the three-dimensional accuracy of single-frequency SPP for each signal frequency of BeiDou has always been better than 5 m on the whole. The east or north positioning result was particularly superior to that of the up direction. For all the four frequencies of BeiDou, the positioning accuracy was higher in the Asian-Pacific region than in other regions, since more satellites were visible in Asia-Pacific region because of the special BDS constellation. 2. Compared to the GPS L1 C/A code, the BDS-3 B1C, B2a, B1I, and B3I signals could achieve a comparable single-frequency SPP result. In the Asian-Pacific region, the three-dimensional positioning accuracies of B1I and B3I were comparatively better, achieving 2.30 m and 2.80 m, respectively, which were better than 2.88 m of GPS. In other regions, the three-dimensional positioning accuracy of B1C was found to be 3.55 m, which was the best among the four BeiDou signals, though marginally worse than 2.97 m of GPS. The causes for the phenomena above are as follows. B1I and B3I are transitional signals that are broadcast by both BDS-2 and BDS-3, which have an obvious advantage in the number of visible satellites in the Asian-Pacific region, thereby delivering higher positioning accuracy over there. In other regions, with only three BDS-2 MEO satellites, B1I and B3I do not possess a superior number of visible satellites. On the other hand, as a newly designed signal and being compatible with GPS L1 and Galileo E5, B1C possesses a significant advantage in the performance of capturing and tracking. However, it is the 24 BDS-3 MEO satellites that broadcast the   30 . This is also the chief reason for its positioning accuracy being slightly lower than that of GPS.

Dual-frequency SPP
Here, dual-frequency SPP processing was executed for the two dual-frequency pseudo-code combinations, namely B1C + B2a and B1I + B3I. The ionospheric delay error is corrected by using dual-frequency combination observation. The satellite position and clock offset are calculated from the broadcast ephemeris. The least square estimation is used for user location estimation. The output was compared with the L1 + L2 dual-frequency pseudo-code SPP result of GPS. The statistical approach for the accuracy was the same as that of the singlefrequency SPP. The RMSs of the east (e), north (n), up (u), and three-dimensional positioning errors were given in Fig. 9. The positioning error RMS of each direction was averaged for all the observation stations, the results for which are enlisted in Table 7.
The following can be observed from Fig. 9 and Table 7. 1. BeiDou B1C + B2a and B1I + B3I, the two dual-frequency SPP schemes, exhibited relatively high accuracy in the Asian-Pacific region. Moreover, whether in Asia-Pacific or other regions, B1C + B2a dual-frequency positioning was significantly superior to B1I + B3I. Although more satellites broadcast B1I and B3I signals in Asia-Pacific, the more the distance between the two frequencies in dual-frequency positioning, the better  PPP accuracy. The two dual-frequency pseudo-code and carrier phase combinations of B1C + B2a and B1I + B3I were adopted here. Utilizing the same observation data from the stations in the Fig. 7, the PPP experiment for BeiDou was conducted, followed by computing the positioning accuracy after convergence. The ionospheric delay error is corrected by using dual-frequency combination observation. The satellite position and clock offset are calculated from the GBM precise orbit and clock offset product. The Kalman filtering is used for user location estimation. For convenient comparison, GPS L1 + L2 dual-frequency PPP was conducted at the same time. The convergence standard was set as when the e-, n-, and u-direction deviations were all lesser than 10 cm for more than 20 epochs 31 . The PPP convergence time was also computed for each processing mode. The above results are mentioned in Fig. 10. The results below the red dash line are from the Asian-Pacific observation stations, while the ones above are from the stations in other regions. According to the comparison of the results in Fig. 9, both the new signal of B1C + B2a and the transitional signal of B1I + B3I could provide PPP results with centimeter accuracy to the observation stations globally. The Asian-Pacific region, though with a larger number of visible satellites, did not exhibit any obvious advantage. This is because the more visible satellites included some GEO satellites that had marginally lower precise orbit and clock offset accuracies, failing to improve the PPP accuracy significantly 32 . Hence, the observation stations   Table 8. The following can be inferred from the above results.
1. After the convergence of B1C + B2a, the e-, n-, and u-direction accuracies achieved 2.2 cm, 1.2 cm, and 2.4 cm, respectively, on par with the processing results of B1I + B3I; though slightly worse than that of GPS L1 + L2. The accuracy of BeiDou PPP could be further improved due to the evolution of future precise products. 2. Except for a few observation stations, the PPP convergence time for BDS-3 B1C + B2a or B1I + B3I was fundamentally equal to that of the GPS L1 + L2, being approximately 30 min.

Conclusions
In the present study, the SISRE computation method for different types of navigation satellites was present and the DCB correction method for BDS-3 new signals was deduced. In-depth analysis of the service accuracy levels have been analyzed from the perspective of BDS-3 SISRE and SPP or PPP positioning performance, leading to the following conclusions: 1. Compared to the BDS-2 satellites, the BDS-3 MEO, IGSO and GEO satellites exhibited significantly improved R, T, and N orbit accuracies. Except for a couple of newly launched satellites, the satellite clock offset accuracy was also remarkably enhanced. with an overall three-dimensional positioning accuracy level better than 5 m. Among them, the B1I signal delivered the best positioning accuracy in the Asian-Pacific region while the B1C was leading in the other regions. Owing to the advantage in signal frequency, the dual-frequency SPP of B1C + B2a expressed better positioning accuracy compared to the transitional signal of B1I + B3I. The three-dimensional positioning accuracy levels for B1C + B2a of 2.39 m and 2.63 m were achieved respectively in the Asian-Pacific region and the other regions. Since there were more visible satellites in the Asia-Pacific, the positioning accuracy of BDS-3 was marginally better than that of GPS. 4. After convergence, the PPP accuracy of BDS-3 B1C + B2a or B1I + B3I was on the level of centimeters, slightly inferior to that of GPS L1 + L2. The convergence time, however, was similar for all three, which was approximately 30 min.