1. Introduction
The BeiDou navigation satellite system (BDS) independently developed by China follows three construction phases, including the demonstration system (BDS-1) consisting of three Geostationary Earth Orbit (GEO) satellites, the regional system (BDS-2) composed of five GEO satellites, seven Inclined Geosynchronous Satellite Orbit (IGSO) satellites and three Medium Earth Orbit (MEO) satellites, and the global system (BDS-3) containing three GEO, three IGSO and 24 MEO satellites [
1,
2,
3,
4]. With the centralized launch of BDS-3 satellites, all BDS-3 MEO satellites were launched on or before December 16, 2019, and the core constellation deployment of the BDS-3 global system has been completed. The performance of global positioning, navigation and timing (PNT) services will be further improved.
With the gradual deployment of the BDS-3 constellation, the BDS constellation will include the satellite systems of BDS-2 and BDS-3. So, the observations of BDS-2 and BDS-3 need to be fully used to take full advantage of the BDS potential. Therefore, it is meaningful for global BDS users to assess the accuracy of BDS-2 and BDS-3 broadcast ephemeris and compare their differences. About the analysis and assessment of BDS broadcast ephemeris, Montenbruck and Steigenberger assessed the GPS, BDS-2, GLONASS, Galileo and QZSS in terms of broadcast orbit errors, clock offsets errors and signal-in-space range error (SISRE), indicating that the SISRE values of BDS-2 are 1.5 m with a monthly scatter of 0.1 m, which is better than GLONASS and the (nonoperational) Galileo IOV system [
5]. The initial assessment of BDS-3 broadcast ephemeris in terms of BDS-3 orbit, clock offsets, SISRE and orbit-only SISRE is presented by Lv and Geng, concluding that the 3D orbit errors are better than 0.6 m, and the root mean square (RMS) of SISRE and orbit-only SISRE are 0.5 m and 0.1 m, respectively [
6]. Zhang and Kubo used the precise product provided by the GNSS Research Center of Wuhan University (WHU) to evaluate the BDS-2 and BDS-3, which shows that the BDS-3 SISRE is superior to that of BDS-2, and the corresponding values are 0.71 m and 0.97 m, respectively [
7].
For the assessment of BDS signal-in-space accuracy, especially for BDS-3, most researchers only use the precise orbit and clock offsets provided by WHU to assess the accuracy of BDS-3 broadcast ephemeris. The RMS of orbit errors and the accuracy of clock offsets and SISRE (including the orbit-only SISRE) used to evaluate the accuracy of BDS broadcast ephemeris will inevitably be affected by the accuracy and the datum of precise product. The conclusions drawn by using the precise satellite product obtained from one analysis center (AC) lack slight reliability. Hence, the precise products provided by the Helmholtz Centre Potsdam German Research Center for Geosciences (GFZ), Wuhan University (WHU) and the International GNSS Monitoring and Assessment System (iGMAS) ACs of Shanghai Astronomical Observatory (SHA) are used to assess the BDS signal-in-space accuracy to draw more scientific and reliable conclusions.
In fact, the broadcast ephemeris of BDS-2 and BDS-3 are calculated separately. Although the corrections of unifying the datum of BDS-2 and BDS-3 satellite clock offsets are considered in the BDS information processing, an obvious systematic bias between BDS-2 and BDS-3 is found by comparing the broadcast ephemeris and precise products in terms of satellite orbit and clock offsets. The previous research indicated that there is a pseudorange bias (hardware delay bias) between BDS-2 and BDS-3 in some receivers [
8]. The pseudorange bias of the receiver will still produce the inconsistency of the datum of clock offsets for BDS-2 and BDS-3 during the processing of broadcast clock offsets. Most references ignore the inconsistency of the datum of satellite clock offsets in evaluating the accuracy of BDS broadcast ephemeris and positioning [
7,
9]. Only Wang and Li found a datum deviation of 3.8 ns between BDS-2 and BDS-3 by comparing the differential code bias (DCB) and time group delay (TGD) of BDS-3 satellites [
10]. Zhang and Wang further attributed this inconsistency to satellite-dependent TGD bias of the BDS broadcast ephemeris, and corrected this inconsistency by calculating TGD bias [
11,
12]. Although the corrections of TGD bias can weaken the impact of the inconsistency of the datum of satellite clock offsets on the assessment of BDS-2 and BDS-3 signal-in-space accuracy, the inter-system bias (ISB) between BDS-2 and BDS-3 is affected by both the datum of satellite clock offsets and receiver pseudorange bias [
13,
14,
15,
16]. The receiver pseudorange bias and residuals of these corrections affected by the number of tracking stations and the accuracy of satellite orbits and ionospheric products will continue to affect the assessment of BDS-2 and BDS-3 signal-in-space accuracy and single point positioning (SPP).
With this background, we give the evaluation methods of broadcast clock offsets and SISRE and the new BDS-2 and BDS-3 combined SPP model in
Section 2. The datasets needed for the assessment of BDS broadcast ephemeris and the SPP processing strategy are presented in
Section 3. In
Section 4, we first analyze the accuracy of broadcast orbit and clock offsets, SISRE and orbit-only SISRE, and prove that the datum of broadcast clock offsets for BDS-2 and BDS-3 are inconsistent by comparing average broadcast-minus-precise (BMP) clock values of BDS-2 and BDS-3 during the assessment of broadcast clock offsets. Then, we investigate the impact of the inconsistent datum of broadcast clock offsets on BDS-2 and BDS-3 combined SPP. Finally, some conclusions and recommendations are drawn.
3. Datasets and Processing Strategies
Due to the different solution strategies adopted by different ACs in terms of the precise attitude model, the solar radiation pressure model and the strategy of selecting the reference station during the estimation of precise clock offsets [
28,
29,
30,
31,
32], the release of BDS-3 satellite precise products is also different.
Figure 2 shows the missing of precise products obtained from GFZ, WHU and SHA.
The 10 IGS multi-GNSS experiment (MGEX) tracking stations [
24] and 10 iGMAS tracking stations are selected to assess the BDS SPP performance and research the impact of the inconsistency datum of broadcast clock offsets and the receiver pseudorange bias on BDS-2 and BDS-3 combined SPP.
Figure 3 shows the distribution of the selected stations. In order to descript the BDS service performances in these tracking stations, the corresponding mean position dilution of precision (PDOP) is marked in the geographical distribution. Incidentally, the global PDOP grid interval is 2.5° × 5°, and the cutoff elevation is set to 7.5°, which is consistent with SPP. All selected tracking stations can receive BDS-2 and BDS-3 observations, and the time interval is 30 s. As with previous studies, the receiver type and the environmental factors (temperature and humidity, etc.) play important roles in the hardware delay [
8,
33,
34,
35].
Table 2 shows the properties of the selected stations.
These selected IGS MGEX and iGMAS tracking stations are used to perform SPP. The least-square method is applied in SPP, the antenna phase center offset (PCO) and phase center variations (PCVs) are corrected by using IGS ANTEX files [
36,
37]. About the tropospheric delay, the modified Hopfield model based on the Global Pressure and Temperature 3 (GPT3) model is used to correct the dry and wet tropospheric delay, and the mapping functions of both dry and wet parts are obtained by Vienna mapping functions 3 (VMF3) according to the elevation angle of each satellite [
38]. The ISB processing strategies of not estimating ISB and estimating ISB are performed.
Table 3 summarizes the processing strategies for BDS SPP.
5. Conclusions
This contribution focused on the analysis and evaluation of BDS-2 and BDS-3 broadcast ephemeris in terms of the accuracy of the orbit and clock offsets, SISRE, orbit-only SISRE and the SPP performances. We prove that the datum of broadcast clock offsets for BDS-2 and BDS-3 are inconsistent and propose the SISRE processing strategies of calculating SISRE of BDS-2 and BDS-3 separately and the new BDS-2 and BDS-3 combined SPP model.
From the viewpoint of the assessment of broadcast clock offsets, selecting one and the same average BMP clock values of satellites of the same constellation can eliminate the datum of the satellite clock offsets, and the average BMP clock values of BDS-2 and BDS-3 are not equal, which can indirectly prove that the datum of broadcast clock offsets for BDS-2 and BDS-3 are inconsistent.
To eliminate the datum of the satellite clock offsets and avoid the inconsistency between BDS-2 and BDS-3, the average BMP clock values of BDS-2 and BDS-3 satellites are calculated separately, which can obtain the objective and scientific evaluation results. The mean STD of three ACs for BDS-2 and BDS-3 broadcast clock offsets are 1.22 ns and 0.58 ns, and the improvement rate of BDS-3 mean STD is 52.49% compared to BDS-2. Meanwhile, the mean RMS for BDS-2 and BDS-3 are 2.90 ns and 1.73 ns, respectively, and that of BDS-3 can be improved by 40.34% compared to BDS-2.
About the broadcast orbits, the mean RMS and STD of all BDS-2 satellites are (1.46, 3.04, 1.21) m and (0.34, 1.00, 0.99) m, and that of all BDS-3 satellites are (0.20, 0.32, 0.28) m and (0.07, 0.23, 0.23) m for the average of three ACs in radial, along-track and cross-track directions, respectively. The broadcast orbit accuracy of BDS-3 is roughly improved by one order magnitude compared to BDS-2. The RMS and STD of BDS-3 broadcast orbit errors are improved by (86.47%, 89.47%, 76.86%) and (79.41%, 77.00%, 76.78%) compared to BDS-2 in three directions, respectively.
Due to the inconsistency of the datum of clock offsets for BDS-2 and BDS-3, BDS-2 and BDS-3 cannot solve SISRE together. We propose the SISRE processing strategies of calculating SISRE of BDS-2 and BDS-3 separately. Without additional bias correction, the proposed SISRE processing strategies can more scientifically and accurately reflect the accuracy of the current broadcast ephemeris of BDS-2 and BDS-3. For BDS-2 signal-in-space accuracy, the mean RMS and STD of three ACs are (1.78 and 0.40) m for SISRE, and (1.72 and 0.34) m for orbit-only SISRE. As for BDS-3, the mean RMS and STD of three ACs are (0.50 and 0.14) m for SISRE, and (0.17 and 0.04) m for orbit-only SISRE, respectively, and the mean RMS and STD of BDS-3 are improved by (71.91% and 65.00%) for SISRE and (90.12% and 88.24%) for orbit-only SISRE compared to BDS-2, respectively. The BDS-3 signal-in-space accuracy has reached 0.50 m, and its accuracy will be further improved with the development of BDS.
In BDS-2 and BDS-3 combined SPP, ISB is affected by the inconsistency of the datum of BDS-2 and BDS-3 broadcast clock offsets and receiver pseudorange bias. Under this background, we analyze the ISB characteristics and SPP performance of different types of receivers separately. For JAVAD TRE_3 receivers, the ISB is relatively minor, and setting the parameter to estimate ISB will reduce the stability of the SPP model and degrade the positioning accuracy of the SPP. However, for the TRIMBLE ALLOY, SEPT POLARX5, CETC-54-GMR-4016, CETC-54-GMR-4011 and UB4B0-13478 receivers, estimating ISB can improve the positioning accuracy by 20.15%, 19.81% and 12.76% in three directions, respectively.
Finally, we recommend that BDS-2 and BDS-3 should be treated separately when assessing the signal-in-space accuracy of BDS-2 and BDS-3. Furthermore, the performances of BDS-2 and BDS-3 combined SPP can be improved by setting parameters to estimate ISB for receivers with relatively larger ISB values.