Characterizing and Modeling the BDS-3 Time Group Delay Error for ARAIM

2 TGD PARAMETERS IN THE DUAL-FREQUENCY OBSERVATION EQUATION

As noted earlier, the B3I signal is used to compute the satellite clock in BDS-3, and different TGD parameters provide the group delay differential between different signals and the B3I signal. For single-frequency users (except those using only the B3I signal), the group delay differential correction must be applied. In this case, the observation equation between satellite i and receiver j on a specified signal s1 is given by:

$P_{s1}^{i,j}=\rho +d{t}_j-(d{t}_i-G_{s1})+T+I_n+{\epsilon }_{s1}$1

where $P_{s1}^{i,j}$ is the raw measurement of the pseudorange at s1, ρ is the geometric range between the satellite and the receiver, dt_j is the receiver clock offset, and dt_i is the broadcast satellite clock offset. T and I_n are the tropospheric and ionospheric delays, respectively. G_s1 represents the group delay differential parameter of signal s1, and ε_s1 is the observation noise.

For dual-frequency users combining any two BDS-3 frequencies, the group delay differential correction is more complicated:

$P_{(s1,s2)}^{i,j}=\rho +{dt}_j–({dt}_i–G_{(s1,s2)})+T+{\epsilon }_{(s1,s2)}$2

Here, the ionospheric delay has been eliminated, and ε(_s1,s2) is the noise of the combined observation of signals s1 and s2. $G_{(s1,s2)}=\frac{f_{s1}^2}{f_{s1}^2–f_{s2}^2}{G}_{s1}–\frac{f_{s2}^2}{f_{s1}^2–f_{s2}^2}{G}_{s2}$ represents the group delay differential corrections for the two signals, where f_s1 and f_s2 are the frequencies of signals s1 and s2, respectively. For example, the ionosphere-free pseudorange observation of the linear combination of signals B1C and B2a is affected by the following correction:

$G_{(B1C,B2a)}=\frac{f_{B1C}^2}{f_{B1C}^2–f_{B2a}^2}{\mathrm{TGD}}_{B1Cp}–\frac{f_{B2a}^2}{f_{B1C}^2–f_{B2a}^2}{\mathrm{TGD}}_{B2ap}$3

where TGD_B1Cp and TGD_B2ap are the group delay differentials of the B1C and B2a signals, which are broadcast by the satellite (CSNO, 2017a, 2017b). All other signal combinations have the same form as Equation (3), with the exception of combinations containing B3I, in which case the correction is:

$G_{(s1,B3I)}=\frac{f_{s1}^2}{f_{s1}^2-f_{B3I}^2}{G}_{s1}$4

Based on the above analysis, the TGD parameters in BDS-3 are used to correct measurements for different signals. However, any errors in the TGD parameters can appear as a clock difference in the measurement equation, as shown in Equation (2). As noted in the Introduction, B3I is not permitted for use in aviation navigation, so ionosphere-free signal combinations must be comprised of the remaining four signals (i.e., B1I, B1C, B2a and B2bI). For combinations that include B1I, both TGD₁ and the TGD parameter corresponding to the second signal must be used; for combinations without B1I, TGD₁ does not need to be used, as shown in Equation (3). Considering only the error in TGD₁ (as is done for the existing BDS-3 model in ARAIM) is therefore not sufficient and could result in a loss of integrity. Instead, the TGD error should be more fully considered and analyzed in different ionosphere-free combinations.

As noted earlier, both the TGD and DCB parameters describe the same delay. The relationship between the two parameters can be given as follows:

$\begin{array}{ccc}TG{D}_1& =& DC{B}_{C2I-C6I}\\ TG{D}_{B1Cp}& =& DC{B}_{C1P-C6I}\\ TG{D}_{B2ap}& =& DC{B}_{C1P-C6I}-DC{B}_{C1P-C5P}\\ TG{D}_{B2bI}& =& DC{B}_{C2I-C6I}-DC{B}_{C2I-C7D}\end{array}$5

where TGD₁ is the group delay differential of B1I and TGD_B2bI is the group delay differential of B2bI. The four parameters shown in Equation (5) are broadcast by the satellites. C2I, C6I, C1P, C5P and C7D are the RINEX observation codes of B1I, B3I, B1C, B2a, and B2bI, respectively.

Because TGD and DCB are generated with different strategies, there is a systematic offset between the broadcast TGD parameter and the DCB product. One common practice for removing this offset is realigning the bias across the entire constellation and time entire period, which is typically done by subtracting the average difference between TGD and DCB before comparison (Wang et al., 2019; Zhang et al., 2020). Based on Equation (3), using frequency combinations with smaller frequency inflation factors avoids enlarging the TGD error and measurement noise. We therefore do not consider the combinations of B2a/B2bI and B1I/B1C due to their large frequency inflation factors (i.e., $\frac{f_{2b}^2}{f_{2b}^2–f_{2a}^2}=19.92$, $\frac{f_{1C}^2}{f_{1C}^2–f_{1I}^2}=55.25$. The frequency inflation factors for the other four combinations (i.e., B1I/B2a, B1C/B2a, B1I/B2b, and B1C/B2b) are much smaller, at $\frac{f_{1I}^2}{f_{1I}^2–f_{2a}^2}=2.31$, $\frac{f_{1c}^2}{f_{1c}^2–f_{2a}^2}=2.26$, $\frac{f_{1I}^2}{f_{1I}^2–f_{2b}^2}=2.48$ and $\frac{f_{1c}^2}{f_{1c}^2–f_{2b}^2}=2.42$, respectively. For these four combinations, the TGD error can be evaluated as follows:

$\begin{array}{c}{E}_{1I\_2a}=\frac{f_{1I}^2(TG{D}_1-DC{B}_{C2I-C6I})-f_{2a}^2(TG{D}_{B2ap}-DC{B}_{C1P-C6I}+DC{B}_{C1P-C5P})}{f_{1I}^2-f_{2a}^2}\\ E_{1C\_2a}=\frac{f_{1C}^2(TG{D}_{B1Cp}-DC{B}_{C1P-C6I})-f_{2a}^2(TG{D}_{B2ap}-DC{B}_{C1P-C6I}+DC{B}_{C1P-C5P})}{f_{1C}^2-f_{2a}^2}\\ E_{1I\_2b}=\frac{f_{1I}^2(TG{D}_1-DC{B}_{C2I-C6I})-f_{2b}^2(TG{D}_{B2bI}-DC{B}_{C2I-C6I}+DC{B}_{C2I-C7D})}{f_{1I}^2-f_{2b}^2}\\ E_{1C\_2b}=\frac{f_{1C}^2(TG{D}_{B1Cp}-DC{B}_{C1P-C6I})-f_{2b}^2(TG{D}_{B2bI}-DC{B}_{C2I-C6I}+DC{B}_{C2I-C7D})}{f_{1C}^2-f_{2b}^2}\end{array}$6

where E_{1I_2a}, E_{1C_2a}, E_{1I_2b}, and E_{1C_2b} represent the TGD errors for the different signal combinations.

3 DATA SETS

We analyzed the TGD parameters of four BDS-3 signals stored in the broadcast message (i.e., B1I, B1C, B2a, and B2bI) using data from a one-year period spanning January 1, 2022 to December 31, 2022. The B1I signal adopts the D1 legacy broadcast message (CSNO, 2019), and the daily broadcast message is provided by MGEX of the International GNSS Service (IGS) (MGEX of IGS, 2022a). In contrast, the B1C, B2a, and B2bI signals use civil navigation (CNAV) message format (i.e., CNAV-1, CNAV-2 and CNAV-3) (CSNO, 2017a; 2017b; 2020). For B1C and B2a, the Test and Assessment Research Center (TARC) of the China Satellite Navigation Office (CSNO) provides the broadcast message and extends the RINEX3 format for CNAV1 and CNAV2 (TARC of CSNO, 2023a). Starting in January 2022, CNAV-3 data for B2bI is generated by the German Aerospace Center (DLR) and stored in the BRD400DLR product file (DLR, 2022). Because the broadcast message can contain some erroneous records, such as incorrect Pseudorandom Noise (PRN) codes and inconsistent reference clock times (Heng et al., 2010), several criteria were used to exclude invalid broadcast messages:

1. The effective value of PRN should be between 1 and 63.

2. The reference time of clock and the reference time of ephemeris should be integers, as ephemeris parameters are updated at the start of every hour (CSNO, 2019).

3. The healthy flag should be 0.

During the period covered by our data set, the DCB product was determined by two analysis centers, the Institute of Geodesy and Geophysics at the Chinese Academy of Sciences (CAS) and the DLR, and made available through the IGS site (MGEX of IGS, 2022b). The CAS product is generated daily with a latency of two to three days, whereas the DLR product is only updated every three months, meaning that the DCB product cannot be used for ARAIM in real time. Moreover, considerably more stations contribute to the DCB product of CAS than to DLR (Montenbruck et al., 2022). This paper accordingly uses the DCB product of CAS to assess the quality of TGD₁, TGD_B1Cp and TGD_B2ap, with the caveat that the DCB products of CAS are missing for five days in the one-year observation period (January 20, August 8 and the last three days of the year). However, because CAS lacks the ground stations to observe B2bI, this paper uses the DCB product of DLR to evaluate TGD_B2bI.

4 THE ABNORMAL CHANGES OF TGD PARAMETERS

Before building the model of TGD error, we identified and excluded abnormal changes of TGD parameters that occurred during the analysis period. These anomalies are divided into two types, the first of which is an abrupt change at one epoch. Figure 1 shows two examples of this first type of anomaly.

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FIGURE 1

Two examples of the first type of TGD anomaly, where an abrupt change is observed at a single epoch (a) PRN33 on May 9, 2022 (b) PRN45 on April 4, 2022

Because the BDS-3 broadcast message is updated every hour (Zhang et al., 2023), the TGD parameter is also updated hourly, as indicated in Figure 1. Figure 1(a) shows a case where the TGD_B2ap broadcast by satellite PRN33 was suddenly set to 0 nanoseconds (ns) at 14 h on May 9, while the corresponding value from the DCB product is 39.1120 ns. The same anomaly also occurs in Galileo (Martini et al., 2020). Figure 1(b) shows a case where the TGD_B1Cp broadcast by satellite PRN45 increases by almost 40 ns at 21 h, while the corresponding DCB value remains stable at −18.9590 ns on April 4. Similar abrupt changes also occurred in other satellites during the analysis period. Because the TGD parameter is generated by the ground center and then broadcast by the satellite, this first type of anomaly likely arises due to the instability of onboard equipment. It should be noted this type of anomaly is only observed in TGD_B1Cp and TGD_B2ap.

The second type of anomaly is an abnormality in the daily update process of the TGD parameter. Two cases of this anomaly, in satellites PRN45 and PRN46, are shown in Figure 2.

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FIGURE 2

Two examples of the second type of TGD anomaly, where an abnormal update process results in an inaccurate measurement (a) The update process of PRN45 from September 19 to 21, 2022 (b) The update process of PRN46 from September 14 to 16, 2022

To illustrate the complete update process, Figure 2(a) shows TGD_B1Cp, DCB_C1p–C6I, and the corresponding error over the 72-hour period from September 19 to 21, 2022. On September 19, the broadcast message is missing from 5 h to 8 h and flagged as unhealthy from 9 h to 10 h, resulting in the first gap in the blue TGD_B1Cp line. The second gap, between 33 h and 34 h, is due to the unhealthy broadcast message at 9 h and 10 h on September 20. Although TGD_B1Cp is updated twice over this period, these two updates are inaccurate, resulting in the error of ±6 meters on September 20 as shown in the right-hand panel of Figure 2(a).

As a second example, Figure 2(b) shows the same measurements for satellite PRN46 over the 72-hour period from September 14 to 16. Here, the broadcast message is missing from 6 h to 9 h and flagged as unhealthy at 10 h on September 14, resulting in the first gap in the blue TGD_B1Cp line. The second gap, between 32 h and 35 h, arises because the broadcast message is flagged as unhealthy at 8 h, missing at 9 h, and unhealthy at 10 h and 11 h on September 15. In this case, the update is later than the change of DCB value, resulting in the error of nearly ±6 meters on September 15.

As shown in Figure 2, these changes in TGD_B1Cp are reflected in DCB_C1P–C6I, which indicates that the onboard changes can be attributed to the reconfiguration of the PRN45 and PRN46 transmitters. The figure also shows that both PRN45 and PRN46 encounter the missing and unhealthy broadcast messages before updating TGD to its normal value. Although the ground control center was aware of the onboard changes on these two satellites and attempted to update the TGD parameter, it failed. This second type of anomaly therefore arises because BDS-3 cannot update TGD parameters accurately and in a timely manner.

5 BUILDING THE TGD ERROR MODEL FOR BDS-3

We have established that accounting only for the TGD₁ error does not sufficiently represent the actual TGD error encountered by dual-frequency users. Here, we outline a method to accurately model the actual TGD error of BDS-3 for aviation users. This method can be used to ensure signal integrity in ARAIM. We evaluate and bound the TGD error for all reasonable combinations of the four BDS-3 signals permitted for dual-frequency ARAIM measurements (i.e., B1I, B1C, B2a and B2bI) as well as their corresponding bias components. However, before detailing the full model, we first consider the characteristics of the TGD parameter.

Figure 3 shows the TGD parameters broadcast by all MEO and IGSO BDS-3 satellites during the one-year analysis period, including the group delay differentials of the B1I, B1C, B2a and B2bI signals.

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FIGURE 3

TGD parameters broadcast by all MEO and IGSO satellites over a one-year analysis period from January 1 to December 31, 2022 (a) TGD₁ (b) TGD_B1Cp (c) TGD_B2ap (d) TGD_B2bI

In Figure 3, the colors (from blue to red) represent the different satellites in order of ascending PRN, and any gaps in the lines indicate missing broadcast messages. Most of the gaps occur in Figure 3(d) because DLR just started providing CNAV-3 data for B2b at the beginning of 2022. In addition, the valid TGD_B1Cp parameter for satellite PRN26 is unavailable from February 25 to September 5, 2022. For all four TGD parameters shown, the ground control center conducted two updates, one in February and one in September, for almost all satellites. These two updates are most easily visible in the TGD₁ and TGD_B1Cp parameters, as the magnitude of the TGD_B2ap and TGD_B2bI updates is smaller. The September update has a significant effect on the error of the signal combinations given in Equation (6); these will be analyzed and shown in Section 6.

The broadcast message represents the ground control station’s prediction for the satellite. As one of broadcast parameters, the TGD reflects the station’s ability to predict the onboard situation. However, updates to reflect onboard changes are not always immediate, which can affect the accuracy of the TGD correction. An example of this issue is shown in Figure 4 for satellite PRN24.

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FIGURE 4

The TGD and corresponding DCB for PRN24 (a) Inter-frequency bias (b) Error in TGD_B1Cp

In Figure 4(a), gaps in the yellow and blue lines indicate missing DCB products and missing valid broadcast messages, respectively. The absolute mean error of TGD_B1Cp, as shown in Figure 4(b), decreases by about 0.2 m immediately after an appropriate update at 3 h on day 56 (i.e., February 25, 2022). However, the updates before day 56 do not decrease the mean error for PRN24, indicating that BDS-3 could still improve its ability to predict TGD in a timely manner.

The mean and standard deviation for the four TGD parameters for each satellite are illustrated in Figure 5, with each parameter shown in a different color and line style.

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FIGURE 5

The mean and standard deviation of TGD₁, TGD_B1Cp, TGD_B2ap and TGD_B2bI calculated across all satellites

On the horizontal axis of Figure 5, the CR satellites are manufactured by the China Aerospace Science and Technology Corporation (CASC) and equipped with a Rubidium atomic clock (TARC of CSNO, 2023b); the SH satellites are manufactured by the Shanghai Engineering Center for Microsatellites and equipped with a Hydrogen clock; and the CH satellites are manufactured by CASC and equipped with a Hydrogen clock. The mean values are also similar for TGD_B2ap and TGD_B2bI because of the close frequency of B2a and B2b.

However, compared with the other two types of satellite, the SH satellites have the smallest standard deviation in TGD parameters: 0.42 m, 0.14 m, 0.11 m, and 0.15 m for TGD₁, TGD_B1Cp, TGD_B2ap, and TGD_B2bI, respectively. This low standard deviation indicates that the TGD parameters broadcast by SH satellites are more stable across satellites. In contrast, the standard deviations of TGD₁ and TGD_B1Cp for the CH satellites PRN45 and PRN46 are more than 20 ns, which is much larger than for any other satellite. To identify the cause of this high stand deviation, Figure 6 shows the TGD₁ and TGD_B1Cp parameters for PRN45 and PRN46 over the one-year observation period.

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FIGURE 6

The TGD₁ and TGD_B1Cp parameters broadcast by PRN45 and PRN46 over the one-year observation period

Based on Figure 6, the high standard deviation for these two satellites is caused by TGD₁ and TGD_B1Cp being updated by almost 50 ns in late September. For PRN45, the two small blue and green segments in the enlarged image show the updates on September 19 that were described in Figure 2(a).

In the following paragraphs, we outline our model of TGD error based on the TGD parameters discussed above.

As shown in Equation (2), the broadcast message causes two other two errors in addition to the orbit error: the satellite clock offset in dt_i and the TGD error in G_(s1,s2). The correlation between the satellite clock offset and the TGD error in BDS-3 must be considered in the pseudorange error model used for integrity evaluation in ARAIM.

We therefore begin with a method to generate the clock correction and TGD parameters. Clock correction parameters can be calculated for each BDS-3 satellite using the Ka-band Inter-Satellite Link and L-band Two-way Satellite Time Frequency Transfer method (Pan et al., 2018; Liu et al., 2019). This process does not involve any inter-frequency bias. In contrast, the broadcast TGD parameter is initially determined by the manufacturer before launch and then updated by the ground control center to reflect actual onboard changes (Liu et al., 2014). By differencing the pseudorange measurement equations at different frequencies, the inter-frequency bias can be calculated using the least squares method (Wang et al., 2016). The satellite clock offset calculated by the clock correction parameters is eliminated in the process of obtaining the TGD parameter, so there should be no correlation between these two errors. We tested this by assessing the correlation between the satellite clock offset and TGD parameter error for the data collected in May. The results of this analysis are shown in Table 1. For BDS-3, the satellite clock offset is assessed by comparing the precise and broadcast clock offsets after using the DCB product to align the reference between them.

The correlation coefficients shown in Table 1 were calculated using the Pearson formula (Ly et al., 2018). The coefficients for all satellites are small (|r| < 0.2), confirming that the clock and TGD errors can be treated as independent contributions to the pseudorange error. In ARAIM, the pseudorange error for aviation users is bounded by a Gaussian distribution obtained from the root-sum-square of different error components, as defined by the following equation (Blanch et al., 2015):

$C_{\mathrm{int},i}=UR{A}_i^2+{\sigma }_{T,i}^2+{\sigma }_{u,i}^2,$7

where C_int,i is the ith value of the diagonal in the pseudorange error covariance matrix. URA_i is the URA of satellite i. σ_T,i and σ_u,i are the standard deviations of the tropospheric error and user noise, respectively, of satellite i.

To rigorously characterize each error component for the BDS-3 model of ARAIM, the TGD error contribution is separated from the URA according to the following equation:

${\sigma }_{URA\&G,i}^2=UR{A}_i^2+{\sigma }_{G,i}^2,$8

where σ_G,i represents the standard deviation of the TGD error in the measurement. Because the DCB product is used when evaluating the URA, the existing $UR{A}_i^2$ term in Equation (7) can be replaced as follows:

$C_{\mathrm{int},i}={\sigma }_{URA\&G,i}^2+{\sigma }_{T,i}^2+{\sigma }_{u,i}^2\cdot$9

6 CHARACTERIZING THE TGD ERROR

Due to the limitations of the frequency inflation factor, as discussed earlier, we evaluated four reasonable signal combinations. Figure 7 shows the error for the B1C/B2a signal combination for each satellite over the analysis period, as calculated from Equation (6). The errors for the other three combinations are shown later.

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FIGURE 7

Error of the combination of the B1C and B2a signals

In Figure 7, each satellite is shown in a different color, and gaps in the lines indicate a missing valid broadcast message and/or missing DCB product. Most of the errors are between −1 m and 0 m. However, two obvious variations, indicated by red dashed rectangles, occurred in February and September for all satellites. Overall, the error appears to decrease after the variation in February but increase after the variation in September. These variations significantly affect the bounding. To investigate the cause of these two variations, Figure 8 shows the differences between the two TGD parameters and the DCB for each satellite, hereafter referred to as the TGB difference.

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FIGURE 8

The difference between broadcast TGD parameters and the DCB for each satellite (a) TGD_B1Cp (b) TGD_B2ap

The two obvious variations highlighted in Figure 7 occurred at the same time as the variations highlighted by black dashed rectangles in Figure 8(a). In addition, the frequency inflation factor for the TGD_B1Cp error $(i.e.,\frac{f_{1C}^2}{f_{1C}^2-f_{2a}^2}=2.26)$ is larger than for TGD_B2ap $(i.e.,\frac{f_{2a}^2}{f_{1C}^2-f_{2a}^2}=1.26)$. These two observations indicate that the error in the B1C/B2a combination is mainly affected by TGD_B1Cp. Before examining these two obvious variations in the TGD_B1Cp difference, we first consider the abnormally large value in Figure 8(a) (indicated by the red arrow) and the vertical line (indicated by the red dashed circle), shown in detail in Figure 9.

In Figure 9(a), there are two gaps in the blue line representing TGD_B1Cp: first, where the broadcast message is missing from 28 h to 31 h and then flagged as unhealthy at 32 h, and second, where the broadcast message is missing from 56 h to 58 h and flagged as unhealthy at 59 h and 60 h. Because the update lags by twelve hours, the resulting error is close to 1.5 m, which causes the large error indicated by the red arrow in Figure 8(a). There are also two gaps in Figure 9(b). The first gap, between 32 h and 37 h, occurs because the broadcast message is missing from 8 h to 11 h and flagged as unhealthy at 12 h on July 24. The second gap occurs because the broadcast message is missing from 60 h to 62 h and flagged as unhealthy at 63 h and 64 h. Due to these gaps and the corresponding onboard changes, the TGD_B1Cp difference from 48 h to 49 h appears as a vertical line on the yearly scale, indicated by the red box in Figure 8(a).

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FIGURE 9

Two examples of TGD_B1Cp error (a) The large TGD difference for PRN32 from July 19 to 22, 2022 (b) The vertical line for PRN42 from July 23 to 26, 2022

The causes of the two obvious variations of the TGD_B1Cp difference, highlighted by the black boxes in Figure 8(a), are analyzed in detail in Figure 10.

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FIGURE 10

Detailed analysis of two variations in the TGD_B1Cp difference (a) The TGD and DCB for all satellites (b) The TGD_B1Cp difference for PRN24

As shown in Figure 10(a), the February and the September updates to TGD_B1Cp (highlighted in the black and red solid rectangles, respectively) coincide with the two variations shown in Figure 8(a). In addition, DCB_C1P–C6I also changes in September, as indicated by the red dashed rectangle. The two TGD_B1Cp updates are examined in more detail for PRN24 in Figure 10(b), which shows that the first update to TGD_B1Cp reduces the error while the second update attempts to reflect the actual onboard change but lacks sufficient accuracy. These updates explain why the TGD_B1Cp difference shown in Figure 8(a) decreases in February and then increases in September. Multiplying by the frequency inflation factor results in the two obvious variations of the error for the B1C/B2a signal combination shown in Figure 7.

To further examine the effect of these two updates on each satellite, Figure 11 shows the absolute mean TGD_B1Cp difference for each satellite before and after these two updates.

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FIGURE 11

The absolute mean TGD_B1Cp difference before and after the two TGD_B1Cp updates (a) Comparison of the absolute mean TGD_B1Cp–DCB_C2I–_C6I before and after the first TGD_B1Cp update in February 2022 (b) Comparison of the absolute value of TGD_B1Cp–DCB_C2I–_C6I before and after the second TGD_B1Cp update in September 2022

Data from PRN26 is missing from Figure 11 because this satellite was missing a valid TGD_B1Cp parameter for up to six months. In Figure 11(a), the TGD_B1Cp difference is lower after the February update for all satellites except PRN19, 20, 36 and 46, indicating that this update generally improved the performance of BDS-3. In contrast, the TGD_B1Cp difference shown in Figure 11(b) is higher after the September update for more than two-thirds of the satellites, with only four satellites (PRN19, 27, 28 and 44) showing a decrease. The inaccurate second update therefore degrades the performance of BDS-3.

The errors for the other three valid signal combinations (i.e., B1C/B2b, B1I/B2a, and B1I/B2b) are shown in Figure 12. Colors indicate the different satellites following the same legend as Figure 7.

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FIGURE 12

The error for the other three signal combinations (a) B1C and B2b (b) B1I and B2a (c) B1I and B2b

In Figure 12, two obvious variations occur in February and September for all three signal combinations, consistent with the variations observed for the B1C/ B2a signal combination in Figure 7. Figure 13 further illustrates the corresponding difference between TGD and DCB for each satellite.

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FIGURE 13

The difference between the broadcast TGD parameters and the DCB for each satellite (a) TGD₁ (b) TGD_B2bI

In Figure 13(a), variations occur in February and September, as before. However, compared with the TGD_B2ap difference in Figure 8(b), Figure 13(b) only shows one larger variation in September. The difference after September is larger for the B1C/B2b combination, shown in Figure 12(a), than for the B1C/B2a combination, shown in Figure 7. The reasons for this variation in the TGD₁ difference are similar to the reasons for the variation in the TGD_B1Cp difference, so detailed descriptions will not be provided here.

7 BOUNDING THE TGD ERROR

We used the folded cumulative distribution function (FCDF) overbounding method to characterize the standard deviation of the TGD error in each signal combination. In FCDF, the empirical CDF is compared with different theoretical Gaussian CDFs with various standard deviations (DeCleene, 2000; Khanafseh et al., 2012), and the Gaussian CDF that overbounds all empirical values is then chosen. The bounding results are illustrated in Figure 14.

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FIGURE 14

The FCDF overbounding results for the four different signal combinations (a) Combination of B1C and B2a (b) Combination of B1C and B2b (c) Combination of B1I and B2a (d) Combination of B1I and B2b

In Figure 14, red lines represent the expected CDF value corresponding to a Gaussian distribution with a mean of zero and different variances. For all four signal combinations, the blue line representing the actual error is below the red line, indicating that the TGD error for each signal combination is correctly bounded by the Gaussian assumption. Of the different signal combinations, the B1I/B2a combination has the smallest standard deviation (1.95 m), whereas the B1C/B2b combination has the largest (3.19 m). However, these standard deviations are due, in part, to the inaccurate updates to TGD_B1Cp and TGD₁ in September, which were discussed in Section 6 and cause the bulge indicated by the dashed green circles in Figure 14. If the data after this inaccurate September update is excluded, the standard deviations of these four signal combinations, in the order presented in Figure 14, are reduced to 0.78 m, 0.88 m, 0.85 m, and 0.94 m, respectively. Given that this inaccurate update only occurred once in a one-year period, we recommend using the B1C/B2a signal combination (with a standard deviation of 0.78 m) as the dual-frequency measurement in ARAIM.

In addition to the standard deviation, we also considered the bias component from the TGD error to b_nom. The monthly mean value of the TGD error for each tested signal combination is shown in Figure 15, where separate colors indicate different signal combinations.

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FIGURE 15

The monthly mean TGD error for each signal combination

As shown in Figure 15, the monthly mean TGD error for all signal combinations increases after September following the inaccurate update of TGD_B1Cp and TGD₁. The errors for the different signal combinations otherwise vary little between January and August. The largest biases for the B1C/B2b and B1I/B2b signal combinations are close to 2 m, whereas the largest biases for the B1C/B2a and B1I/ B2a signal combinations are closer to 1.5 m. However, because these large biases are transient and only appear after the inaccurate update, we do not recommend using these values for the contribution of TGD error to b_nom. Instead, based on the average TGD errors from January to August, we set the bias component from the TGD error as 0.71 m, 0.88 m, 0.77 m, and 0.91 m, respectively, for the four signal combinations as listed in the legend of Figure 15. Based on these results, the TGD error for the B1C/B2a signal combination can be bounded by a Gaussian distribution with a standard deviation of 0.78 m and a mean of 0.71 m. We suggest that this combination be used in ARAIM.

8 CONCLUSION

Of the five civil signals in BDS-3 (i.e., B1I, B3I, B1C, B2a and B2bI), B3I is not within the bandwidths permitted by ARNS and therefore cannot be used in aviation navigation. As a result, estimation approaches that only consider the error in TGD₁ do not capture the actual TGD error encountered by users of the dual-frequency measurements generated by the remaining four signals. This underestimation of the TGD error in the existing BDS-3 model used by ARAIM can result in an underestimated SIS error. More importantly, the corresponding URA and b_nom values cannot bound the actual measurement error, resulting in a loss of integrity. To avoid this risk, this paper developed a Gaussian model to bound the TGD error for BDS-3 ionosphere-free measurements in ARAIM.

We first analyzed the TGD parameters of all BDS-3 MEO and IGSO satellites from January 1, 2022, to December 31, 2022. As part of this analysis, two types of abnormal changes in TGD parameters were identified and described in detail. Due to the limitations imposed by their large frequency inflation factors, two signal combinations (B2a/B2bI and B1I/B1C) were not included in the analysis, so we then characterized and bounded the TGD errors for four reasonable signal combinations (i.e., B1C/B2a, B1C/B2b, B1I/B2a, and B1I/B2b).

After demonstrating that the satellite clock offset and the TGD error were independent of each other, we modeled the TGD error as a Gaussian distribution separately from URA. The smallest standard deviation (0.78 m) was obtained from the B1C/B2a signal combination, and this combination is therefore recommended for ARAIM. The corresponding bias component contributing to b_nom is 0.71 m. Notably, if we included data from after the inaccurate update in September, the standard deviation and the bias component for the B1C/B2a signal combination would increase to 2.34 m and 1.5 m, respectively. This highlights the critical importance of properly modeling the TGD error of BDS-3 in ARAIM.