Analysis of BDS ARAIM Integrity Support Data Parameters

3.1 Instantaneous SISRE

To evaluate ISD parameters, the SISRE must be precisely estimated. σ_URE is a direct statistic of the instantaneous SISRE, with σ_URA serving as a bound. There are several methods for calculating the instantaneous SISRE of a specified satellite. One commonly used method is based on the globally averaged instantaneous SISRE (AVE SISRE), which takes a weighted average of the orbital errors projected onto the Earth’s surface (Heng et al., 2012; Montenbruck et al., 2018). The generalized effects of orbit and clock errors on all users of the Earth’s surface can be well represented. The AVE SISRE is defined as follows:

$AVESISRE=\sqrt{{\left(\alpha R-T\right)}^2+\beta \left(A^2+C^2\right)}$ 1

Here, R,A,C are the orbital errors of the broadcast ephemeris in the radial, along-track, and cross-track directions, respectively. T is the error of the broadcast clock offset converted to length, and α and β are factors related to the satellite orbital height, which are set to 0.9823 and 0.1324 for BDS MEO satellites.

Another method is based on the instantaneous SISRE at the worst user location (WUL SISRE, also known as worst-case SISRE), which corresponds to the maximum value of all SISREs projected to every user location on the Earth’s surface with a specified elevation mask (5° is used in this work). This problem can be converted to a two-dimensional problem that consists of identifying the plane that contains the center of the Earth and the vector of the orbital error and then projecting the orbit error vector to determine the maximum SISRE value (Heng et al. 2012; Perea et al., 2017). The method for computing the WUL SISRE can be expressed as follows:

$\mathrm{WUL}\mathrm{SISRE}=\max \left(\left(\mathrm{ISISR}{E}_i\right)\right)\ast \mathrm{sign}\left(\mathrm{ISISR}{E}_i\right)$ 2

where ISISRE_i is the sum of the orbital and clock offset errors. The orbital error is the length of its vector projected to the user’s line of sight in accordance with different users’ locations on Earth.

For SISRE analysis, the AVE SISRE method obtains more correct SISRE information on accuracy; in contrast, for integrity-related purposes, the WUL SISRE bound can ensure better integrity. Thus, in this work, we use the AVE SISRE for σ_URE calculation and the WUL SISRE for σ_URA, P_sat, and P_const analysis.

In both methods, antenna phase center offsets must be corrected for orbital error calculations, as the precise orbit takes the satellite’s center of mass as a reference, whereas the broadcast ephemeris uses the antenna phase center. The antenna phase center of BDS satellites can be downloaded from the CSNO-TARC website (https://www.csno-tarc.cn/en/datacenter/satelliteparameters). Additionally, because the references for the precise clock offset and broadcast clock offset are different, these terms need to be aligned to the same reference for comparison (Montenbruck et al., 2018). Moreover, the precise ephemeris is obtained from ionosphere-free combination observations, whereas the broadcast clock offset refers to the B3I signal for BDS. Thus, the influence of the timing group delay (TGD) must be considered for evaluating the accuracy of the broadcast clock offset (Lv et al., 2019). In this work, the broadcast TGD data are used for correcting the code bias between different signals.

3.2 Fault Identification

A fault event indicates that the SISRE of a satellite is larger than the specified error threshold, but the system fails to issue warning information in a timely manner. The warning information here refers to the “unhealthy” flag in navigation messages. Hence, two conditions must be met to determine a fault for a navigation signal: first, the satellite is flagged as “healthy”; second, the satellite SISRE exceeds the error threshold. For integrity, the error threshold can be represented as follows:

$H=\kappa \cdot {\sigma }_{\mathrm{URA}}$ 3

Here, H is the SISRE threshold for fault identification, whereas κ is a multiplier for ensuring that the threshold is met with a specified confidence level.

For the B1I signal, the method for judging the “health” of a satellite is relatively simple, based on only the “SatH1” flag in the D1 navigation message; in contrast, for B1C and B2a signals, this judgment requires an integration of the status of four parameters (health status (HS), signal integrity flag (SIF), data integrity flag (DIF), and accuracy integrity flag (AIF)) in the B-CAV1/B-CAV2 navigation message (CSNO, 2017a, 2017b, 2019).

In addition, the user range accuracy (URA)/signal-in-space accuracy are broadcasted in BDS navigation messages, but these parameters are not currently designed for ARAIM. Therefore, we compute the WUL SISRE statistic for all satellites and obtain an overbound at a specified probability level using a zero-mean Gaussian distribution. The specified probability level corresponds to κ times the specified standard deviation.

4.1 SISRE Standard Deviation

AVE SISRE statistics of the B1I signal for all satellites are shown in Figure 1. It can be seen that all satellites except pseudorandom noise (PRN) 27 have a median AVE SISRE of less than 1.0 m.

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FIGURE 1

Boxplot of the B1I AVE SISRE for different satellites

The mean AVE SISRE for all satellites is 0.58 m. Satellite PRN27 has the largest error, 1.2 m, among all satellites. By assessing the orbit and clock errors of this satellite (Figure 5), we found that there was a larger deviation in the clock offset error of this satellite. That error is most likely caused by signal deformation and other error sources, which requires further study.

Although different ephemeris parameters are used (the new B-CAV1 and B-CNAV2 ephemerides have two additional parameters: the change rate of the semi-major axis and the mean motion difference)(CSNO, 2017a, 2017b), the ephemerides modulated on the B1C and B2a signals are generated from the same orbit determination results obtained for the B1I signal. The mean AVE SISRE value for the B1C and B2a signals is 0.57 m, showing an improvement of several centimeters compared with the B1I value (a figure of the B1C/B2a AVE SISRE is not shown here).

The WUL SISRE of the B1I signal is shown in Figure 2. Unlike the AVE SISRE, the WUL SISRE has positive and negative values, which can be clearly expected from a comparison of Equations (1) and (2). In general, because the error caused by orbital deviation in the AVE SISRE is primarily affected by deviations in the radial direction, the basic change characteristics of the WUL SISRE and AVE SISRE are similar. Furthermore, because more parameters are used in the B-CNAV1 and B-CNAV2 ephemerides, the WUL SISRE of the B1C and B2a signals is slightly better than that of the B1I signal, with a centimeter-level difference. The heavy black line at day 565 shows a fault event for the PRN19 satellite, which will be clarified in Section 4.4.

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FIGURE 2

B1I WUL SISRE time series for all satellites

4.2 Overbounding the SISRE

There are several methods for error bounding (Walter et al., 2018; Wang et al., 2021; Martini et al., 2020). In this work, we use the one minus Gaussian cumulative distribution function (1-CDF) method to overbound the absolute value of the WUL-SISRE for all satellites. The 1-CDF overbounding can be expressed as shown in Equation (4), where Φ is the Gaussian cumulative distribution function (CDF), Φ_o is the overbounding CDF, and Φ_a is the actual CDF:

$1-{\Phi }_{\mathrm{o}}\left(\left(x\right)\right)\ge 1-{\Phi }_a\left(\left(x\right)\right)$ 4

As the faults in different satellites can equally affect all satellites, it is reasonable to aggregate the data for all satellites and use the average fault probability for bounding analysis (Walter et al., 2019). Figure 3 shows the 1-CDF curve for WUL SISRE aggregation of all satellites. It can be seen that the WUL SISRE for all satellites (after subtracting the mean value for each satellite) can be overbounded by a zero-mean Gaussian distribution with a standard variation of 4.0 m at a probability level of 4.0 × 10^–5.

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FIGURE 3

Overbounding the WUL SISRE aggregation for all satellites

Figure 4 shows 1-CDF curves for individual satellites (colored lines). It can be seen that the curves for PRN19 (blue), PRN21 (purple), and PRN44 (green) intersect the expected bounding line. This result indicates that these three satellites do not fulfill the specified integrity requirement. An analysis of the faults for these satellites is presented in Section 4.4.

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FIGURE 4

Overbounding the WUL SISRE for individual satellites (a) B1I (b) B1C/B2a

4.3 Nominal Bias

A determination of b_nom should consider orbit and clock offset biases, TGD error, satellite antenna biases, signal distortion, and other errors (Walter et al., 2018).

For orbital and clock offset errors, the probability distribution function of the WUL SISRE (Figure 5) shows that the orbital deviation basically exhibits the characteristics of zero mean in all directions; however, the clock offset error obviously has a bias, and its magnitude varies for different satellites. The satellite with the largest SISRE (PRN27) has the largest bias, with a value of –1.16 m. This bias would be considered as the main source of b_nom in BDS applications. The composition of that bias would be complex. If we use precise TGD to evaluate the broadcast clock offset, the result would be improved. The remaining error may be introduced by the different tracking modes of the control segment and the commercial receivers used here (Chen et. al., 2021). Additionally, signal distortion would also exacerbate this bias.

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FIGURE 5

Orbit and clock offset errors and the WUL SISRE probability density function (a) PRN26 (b) PRN27

For TGD errors, because the broadcast clock offset of BDS is not calculated from an ionosphere-free combination of dual-frequency observations (it refers to the B3I signal), the TGD should be considered in BDS dual-frequency positioning. The impact of TGD on dual-frequency combination observation can be expressed as follows:

$b_{ca}=\frac{f_c^2}{f_c^2-f_a^2}{b}_c-\frac{f_a^2}{f_c^2-f_a^2}{b}_a$ 5

where b is the TGD for a single signal, f is the frequency of the carrier phase, and the subscripts c,a indicate the B1C and B2a signals. Previous work has shown that the BDS TGD has an accuracy of approximately 1.0 ns when the bias between the broadcast TGD and the IGS differential code bias (DCB) is removed, if the IGS DCB is taken as a reference for TGD evaluation (Wang et al., 2019).

For satellite antenna bias, except the fixed antenna phase center offset, the variation in the antenna phase center is on order of millimeters (Hu et al., 2022), and the effect on b_nom is small.

With regard to the impact of signal distortion, a well-designed receiver can achieve a pseudorange deviation within 0.2 m (He et al., 2020). This result is dependent on the receiver design (Hauschild & Montenbruck, 2016; Gong et al., 2022).

To summarize the above results, considering that the impacts of signal distortion and TGD have already been included in the evaluated SISRE, we simplify this analysis by setting the nominal bias of the BDS to 1.16 m. However, much work is still needed to characterize the effects of these biases and to determine their bounds. There exist more error sources such as antenna bias, code-carrier incoherence, and DCB for the receiver side (Walter et al., 2018), which require an in-depth study.

4.4 Fault Rate and Probability

While determining σ_URA to overbound the actual SISRE distribution in the prior section, we also obtain the corresponding probability level, parameter P_sat. According to the WUL SISRE bounding analysis, P_sat is set to 4.0 × 10^–5, corresponding to a bounding σ_URA of 4.0 m. The corresponding κ for a cumulative probability of 4 × 10^–5 is approximately 4.1, which can be used for assessing fault events. Three fault events were found for the B1I signal, with only one fault event for the B1C and B2a signals (listed in Table 2). Consequently, R_sat would be 7.1 × 10^–6 for the B1I signal and 2.4 × 10^–6 for the B1C and B2a signals.

For P_const, it is obvious that there are no cases in which two or more satellites are interrupted simultaneously during the studied time period. The value of 6.0 × 10^–5, which was suggested in the ICAO Standard and Recommended Practices (ICAO, 2023), may be conservative (indicating that one fault happens in 2 years, corresponding to an R_const value of 5.7 × 10^–5/h). One should note that the data set used in this analysis is limited; a longer-term data set is needed to obtain more precise results.