Effect of User Antenna Group Delay Variation Error on Advanced RAIM

4.1 Considering User Antenna Errors Caused by AGDV in the Position Domain

As mentioned previously, there is no allocation for the antenna bias caused by group delay variations in the standard PL equation for ARAIM. Moreover, the manner in which the antenna-induced error should be included within the standard ARAIM user algorithm, for instance, as a deterministic bias term or as a random process, has yet to be standardized (Working Group C. ARAIM Technical Subgroup, 2016). Thus, this paper aims to evaluate how the AGDV-induced error affects ARAIM performance by investigating different methods for considering its impact and incorporating it into the PL. In this section, we first describe the modeling of position errors caused by AGDV and then directly incorporate the resulting models into the current PL equation. In subsequent sections, we present other approaches in which AGDV-associated errors are modeled in the range domain using the sigma and bias terms. Lastly, the impact of the different models on ARAIM performance are examined.

The AGDV errors for L1/L5 and E1/E5a dual-frequency measurements are estimated by using the following IF combination of single-frequency AGDV errors estimated on the different frequencies, L1/L5 (or E1/E5):

4

in which b_AGDVL1 and b_AGDVL5 are AGDV errors estimated for the L1 and L5 frequencies, respectively. This work uses the AGDV error estimates collected and analyzed in the DUFMAN project (Circiu et al., 2021) to derive the σ_AGDV bound shown in Figure 2 for characterizing position errors caused by AGDV errors. We then derive the errors that contribute to the position solution by eliminating the common bias over satellites in view that will affect the receiver clock bias estimation in order to characterize position errors caused by the antenna contribution. For this, antenna residual errors to be projected into the position domain are calculated as in Equation (5). Ie we consider GPS and Galileo constellations for ARAIM, the bias removal is performed independently for each constellation:

5

where i is the satellite index, j is the constellation index, N_j is the number of satellites in view within constellation j (j = 1, 2, i.e., GPS or Galileo), and is the antenna bias residual for the measurement from the i-th satellite within constellation j.

The AGDV residual error for each satellite at each epoch is then projected into the position domain as follows:

6

Here, e is an vector whose components are antenna error contributions projected to the states (i.e., position errors and clock biases for GPS and Galileo), G is a geometry matrix, and W is a weight matrix determined from pseudorange measurement error models such as satellite ephemeris/clock, tropospheric delay, and CNMP error models.

More specifically, we have the following:

7

where:

8

For dual-frequency-based ARAIM, variances for each measurement, , are constructed by considering the satellite ephemeris and clock error, tropospheric error, and CNMP error components (Blanch et al., 2015):

9

where i is the satellite index, σ_URA,i is the standard deviation of the clock and ephemeris error for the i-th satellite used for integrity purposes, σ_tropo,i is the standard deviation of the tropospheric delay corresponding to the signal from the i-th satellite, and σ_user,i is the standard deviation of the airborne multipath and interference error (equivalent to the user CNMP error in this paper) relevant to the signal from the i-th satellite.

The AGDV effect should be considered in addition to the existing error components in Equation (9), represented as ε_others, to construct the projection from the measurement domain to the position domain as follows:

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where v is the index for the state, e.g., the east, north, and up components. S_v is a row vector of the projection matrix for each v-th state, and Δx_v is the state estimation error, e.g., the position error in the up coordinate. and x_v are the state estimate and the true value of the state, respectively. ε is the measurement error vector, ε_others is the measurement error due to satellite ephemeris/clock error, tropospheric error, and multipath, and δb_AGDV is the antenna bias error vector.

Equation (10) implies that the two random variables (i.e., the AGDV contribution and the other contribution to the position error) can be combined into a single random variable by the root sum square of their variances, which will be applied to incorporate the AGDV effect into the PL calculation in this section.

In the remainder of the paper, for simplicity, we assume the state of interest as the vertical position coordinate. In Equation (10), the error vector, ε_others, contains pseudorange errors for different satellites due to the error sources represented in Equation (9). As shown in Equation (10), the projection by matrix S is a linear transformation. Therefore, if the antenna error, represented as δb_AGDV, is statistically independent of ε_others in the range domain, these two terms are also independent in the position domain. ARAIM provides the vertical PL (VPL), the region assured to contain the indicated vertical position with the required probability for a specific navigation mode, such that the user can assess the integrity risk at a given measurement epoch. The VPL equation for ARAIM (Blanch et al., 2015) is shown in Equation (11):

11

where Q is the tail probability of a zero-mean unit Gaussian distribution, σ₀ is the standard deviation of the all-in-view position solution (H₀), σ_k is the standard deviation of fault-tolerant position solution for H_k (k = 1, …, N), b_k is the nominal bias for H_k (k = 0, 1, …, N), P_{fault, k} is the prior probability of fault for H_k (k = 1, …, N), T_k is the threshold for the solution separation (SS) test for the k-th fault mode, and PHMI_monitored is the integrity risk allocation for monitored faults (in this example, for the vertical coordinate).

If the position error caused by the AGDV effect in the measurement domain has a Gaussian distribution for the k-th fault mode, i.e., N(b_k,AGDV, σ_k,AGDV), the standard deviation and mean can be directly incorporated in the PL equation, as shown in Equation (12). Note that, in this section, the Gaussian overbound published in the standards is applied to model the AGDV-induced position error while ensuring a margin of conservatism for integrity purposes. Because the AGDV effect can be considered as statistically independent from the other error contribution, the worst-case impact of the bias due to the AGDV error (b_k,AGDV) is considered by augmenting the nominal bias term b_k:

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To characterize the distribution of the antenna-induced error such that the PL (the term VPL) is determined from Equation (12), we first examine the antenna error contribution projected to the position error, as shown in Equation (10), by running a vertical ARAIM (V-ARAIM) service volume simulation with the ISM parameters and critical parameters listed in Table 2. All of the simulations performed in this work are based on a set of ISM parameters defined for the current baseline airborne ARAIM algorithm outlined by the Working Group C, ARAIM Technical Subgroup (2016). Particularly, the value of P_const = 10⁻⁸ for the GPS constellation-wide fault is consistent with the GPS constellation specification outlined in the fifth edition (the latest edition) of the GPS Standard Positioning Service Performance Standard. Thus, the value of 10⁻⁸ is chosen in our analysis; however, there have been discussions on other possible values because of the uncertainty regarding whether the amount of data corresponding to observed operational data during GPS history is sufficient to thoroughly verify such a low probability. Thus, future work should include further validation and updates if necessary.

For the simulation, we used the MAAST for ARAIM (MATLAB Algorithm Availability Simulation Tool, 2021) with some modifications. All of the simulations in this work were performed for a period of ten sidereal days, which corresponds to the repetition period of the Galileo constellation, with an interval of 600 s over a user grid formation of 5° by 5°. Because the position error distribution should be different for different fault types, we investigated the fraction of different fault modes separately. For the parameters and 24 GPS and 24 Galileo constellations assumed by the Working Group C, ARAIM Technical Subgroup (2016), three different types of fault modes were primarily observed in our investigation: a fault-free mode, GPS/Galileo single-satellite failure, and Galileo constellation fault, as displayed in Table 3. Table 3 also shows that the GPS and Galileo single-satellite faults are dominant for the V-ARAIM implementation, and the fault-free mode is observed as often as Galileo constellation faults.

Observing that these three fault modes are more dominant than other multiple-satellite fault modes, the error caused by AGDV is characterized by the Gaussian parameters, i.e., b_k,AGDV, σ_K,AGDV, with a focus on the three fault hypotheses in this work. Figure 5 shows a histogram of position errors caused by the AGDV residual errors in Equation (6) in the vertical position coordinate for the Galileo constellation fault (left), the probability density of the position error normalized by its mean and standard deviation (middle), and 1 – CDF of the normalized position errors (right). Note that as mentioned above, the position error resulting from the antenna contribution is our main interest in this examination.

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FIGURE 5

Histogram of vertical position errors caused by the AGDV effect obtained under the Galileo constellation fault (left); PDF of the normalized vertical position error based on the Galileo constellation fault (P_GAL = 10⁻⁵) (middle); 1–CDF of the normalized error (blue) and the overbounding curve when an inflation factor of 1.45 is applied (red) (right)

The histogram shows that the distribution has a mean of approximately 0.3 m, and the maximum position error due to the antenna error in the vertical coordinate is approximately 1.7 m for the Galileo constellation fault when the antenna-induced pseudorange error is considered.

In Figure 5 (middle), the errors are normalized by the mean and standard deviation, and the blue dotted curve indicates the probability density function (PDF) of normalized position errors on a log scale. The solid magenta curve is the PDF of a unit Gaussian distribution. Comparison of the empirical PDF and the theoretical standard Gaussian PDF shows that the position error due to the AGDV effect has a non-Gaussian tail, particularly for probabilities lower than 10⁻³. Therefore, we derive an overbounding variance for the empirical error distribution using the commonly used 1–CDF overbounding method (DeCleene, 2000). In Figure 5 (right), the empirical error distribution with the non-Gaussian tail (blue curve) is bounded by the distribution with the variance inflated by a factor of 1.45 (red curve).

Table 4 summarizes the standard deviation and mean of the position error distribution for the three fault hypotheses for the horizontal and vertical coordinates.

In addition to the position error distributions, we must consider the effect of AGDV on the fault detection threshold T in Equation (11). The ARAIM fault detection threshold, T_k, for the k-th fault mode is determined from the standard deviation of the SS, σ_SS,k, as shown in Equation (13):

13

Here, the factor K_k is obtained from the continuity requirement allocated to the k-th fault hypothesis. The SS, denoted as in Equation (14), is defined as the difference between the all-in-view solution, , and the subset solution, , which is determined by using all satellites except the faulty satellites under the hypothesis (Blanch et al., 2015):

14

Here, dS_i is the matrix used to generate the SS, and y indicates a vector whose components are pseudorange measurements for satellites in view. Because the matrix dS_i is defined as the difference S_i – S₀ and the projection by dS_i is linear, we can examine the AGDV effect on the SS domain, as the position errors due to the AGDV residual error were characterized in Equations (6) and (10). The AGVD residual errors are projected from the measurement domain to the SS domain as follows:

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Thus, we simulate position errors for different possible fault modes in different coordinates based on the same ARAIM service volume simulation and perform the same process as those for the position error results in Table 4. The corresponding results are summarized in Table 5.

The σ_SS,k term in Equation (12) is modified through the root sum square of the baseline σ_SS and the overbounding parameters, σ_SS,AGDV, shown in Table 5 for different fault modes, and the corresponding threshold T_K is then applied to the VPL equation (Equation (11)). Whereas the bias effect projected into the position domain for different fault modes is considered for integrity purposes, as presented in Table 4, the bias effect on the test is not taken into account for the σ_SS,AGDV term used to compute the SS test threshold because the monitor threshold is determined for continuity purposes, in line with the current ARAIM development framework (Blanch et al., 2015). The possible bias caused by the AGDV effect is addressed within the term b_k,AGDV in the PL calculation in Equation (12).

In this section, the antenna errors due to AGDV were considered in the position domain. However, because this approach is subject to a specific set of satellite constellation geometries and error models, including ISM parameters, it can be far from optimal for a rigorous assessment of the impact of antenna errors, particularly for safety-critical applications. Therefore, although the satellite geometries and error models assumed for the current ARAIM framework are employed for position domain modeling, in this paper, we will consider the approach for simulation purposes and compare it with other methods. Further work is needed to generalize the direct modeling of this error in the position domain.

4.2 Modeling AGDV-Induced Error as a Random Sigma Term in the Range Domain

Having considered the AGDV error effect in the user PL domain, we now describe another approach for considering the error effect, where the AGDV errors are modeled in the range domain as a random variable (i.e., a sigma term). For this purpose, the AGDV errors in Equation (4) are first collected in the range domain based on the ARAIM simulation described in the previous section. We then apply the same approach as the GNSS airborne antenna MOPS (RTCA SC-159, 2018), where AGDV-induced errors and the corresponding acceptable bounds are modeled as a function of elevation angle. Figure 6 (left) shows a two-dimensional histogram of the number of observations as a function of both the AGDV residual error and the elevation angle. The horizontal axis divides the elevation angles into bins, and the vertical axis divides observations of the residuals into bins. The color of each pixel indicates the number of measurements counted.

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FIGURE 6

The left figure shows histograms of AGDV residual errors for different satellite elevation bins. The right figure shows the empirical means (blue) and standard deviations (red) for different elevation bins and the overbounding sigma terms, σ_{AGDV,overbound} (magenta).

The AGDV varies more at low elevations, where the antenna gain pattern is not uniform; thus, Figure 6 (left) shows that the histograms at low elevations are widely spread. In contrast, the group delay variations diminish at high elevations because the patterns are more regular. Therefore, in Figure 6 (right), we obtained higher sigma values (red curve) as the elevation decreases. However, because the antenna is mounted on a metallic ground plane, according to published standards (RTCA, 2006; RTCA SC-159, 2018), small biasing effects appear at high elevations, as shown in the blue curve in Figure 6 (right).

In addition, each histogram for each elevation bin in the left figure contains different antenna group delays for different azimuth angles. Therefore, the delay does not necessarily monotonically increase or decrease with elevation angle, as shown in Figure 6. In particular, Figure 6 (left) shows that several distributions for different elevation bins have non-zero means and even two different peaks, i.e., the distribution is bimodal. The pair-bounding technique (Rife et al., 2006) was developed to create a pair of overbounding distributions for an actual error distribution that is neither symmetric nor unimodal. However, it has been found to be challenging to meet the pair-bounding condition if the actual error distribution is represented by an empirical histogram (Blanch et al., 2019). A more recent study (Blanch et al., 2019) presented an approach combining the single-CDF (DeCleene, 2000) and the pair-bounding techniques to overcome limitations on the existing pair-bounding technique. This method may be optimal, as it provides a tighter bound than the paired overbounding distribution. However, both methods provide a pair of Gaussian parameters, i.e., a bias and an overbounding sigma, representing the resulting overbounding distribution. In this section, as mentioned above, we consider the error within the random sigma term to be compliant with the current ARAIM error models (Working Group C. ARAIM Technical Subgroup, 2016). In the next section, the AGDV is considered as a bias term in the PL computation in the worst-case scenario.

Here, we take a simple and practical tail bounding approach, where the single CDF bound is applied only to the tails, not the core part, of the actual distribution because the tails lead to the most significant sigma values. In addition, to conservatively account for the non-zero mean, the absolute value of the mean is then added to the overbounding sigma such that the resulting zero-mean Gaussian distribution with the overbounding sigma can overbound the tail of the actual distribution. This simple approach for integrity purposes has often been used to characterize different GNSS errors whose empirical distributions are not symmetric or unimodal, for instance, regarding satellite ephemeris and clock errors (Walter et al., 2018) and ionospheric delay errors in the ground-based augmentation system framework (Lee et al., 2007).

Next, the sigma overbounding method is used to characterize the AGDV residual errors as follows. The mean (μ_AGDV) and standard deviation (σ_AGDV of the residuals in each bin are calculated and used to normalize the residuals; the mean is removed and the residuals are divided by their standard deviation in each bin. In this step, an additional margin for the sample means and standard deviations, i.e., μ_AGDV and σ_AGDV, is considered because the amount of data may be insufficient to ensure the stringent integrity requirement of 10⁻⁷ for ARAIM applications. The upper bound of the sample standard deviation can be derived by considering a χ²-distributed-based confidence interval with a 1 – α confidence level. In this work, a 95% confidence level was considered, and the length of the confidence level is determined as follows (Hayter, 2012):

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where n is the number of samples in each bin and σ_AGDV is the standard deviation of the sample distribution in each bin. The corresponding two-sided confidence interval for the sample mean is then constructed as follows:

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where represents the critical point of the t-distribution with n − 1 degrees of freedom for which the t-distributed random variable takes the tail probability of α/2.

In this investigation, the upper bound of the interval is taken as the worst bound of the sample mean for each bin:

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Based on the distribution of normalized AGDV residuals, the inflation factor (L) is then determined in each bin, as shown in the previous position domain analysis. Lastly, the “σ_AGDV overbound” for each bin (see Figure 6 (right)) is computed as follows:

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Figure 6 (right) shows the overbound result, i.e., σ_{AGDV,overbound}. The estimated overbounds (magenta curve with asterisks) are similar to the one-sigma values (red curve with circles) at elevation angles lower than 45°. In contrast, the estimated overbounds reach approximately 0.4 m at elevation angles higher than 45°, while the one-sigma values are below approximately 0.15 m. Table 6 shows the overbounding sigma values for each bin.

In this section, the AGDV-induced error has been considered within a random sigma value as a function of elevation angle. In the next section, the error effect is considered as the worst-case bias in the PL calculation.

4.3 Modeling the User Antenna Effect as the Worst-Case Bias Term

This section considers the AGDV error effect with respect to the VPL in the worst-case scenario by adding a new user antenna bias bound to PLs derived by the baseline user algorithm (see Equation (10)) as follows:

20

Here, VPL_user denotes the VPL computed by the baseline user algorithm. S_i indicates each measurement error projection into the vertical position domain, and b_AGDV,i represents the antenna bias error for each pseudorange measurement. In this study, two types of observables were used: AGDV errors derived from the DUFMAN work (Circiu et al., 2019) and an upper bound outlined by the antenna MOPS (RTCA SC-159, 2018). Because Circiu et al. (2021) demonstrated that the MOPS limit stated in DO-373 can overbound the observed AGDV for the L1/E1 and L5/E5a bands, this paper applies the limit in DO-373 for simulation purposes.

5.1 Impact of the New Combined Multipath and AGDV Model (Case 1 vs. Case 2)

We first assess the baseline performance based on the legacy CNMP model. Figure 7 (left) shows the 99.5% LPV-200 availability map as a function of user location when the legacy model is applied as described for Case 1. The availability indicates the percentage of time during which the user horizontal/vertical PL (H/VPL) is less than the corresponding horizontal/vertical alert limit (H/VAL). In the availability map, the coverage for users with availability higher than 99.5% corresponds to approximately 90.54% of the map. This coverage indicates the fraction of users within the region that had a time availability of 99.5% or greater, represented as purple or dark purple regions. Next, we examine the DUFMAN multipath and antenna model, as described for Case 2, by replacing the legacy multipath model (i.e., σ_MP in Equation (1)) with the modified σ_MP&AGDV. Because the receiver noise contribution was not considered in the construction of the DUFMAN multipath model (Circiu et al., 2021), the existing receiver code noise model is still included, as shown below:

21

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FIGURE 7

99.5% availability map for LPV-200 when the legacy noise and multipath model are applied (left) and a corresponding availability map for when the newly derived multipath and user antenna model (i.e., DUFMAN σ_MP&ADGV in Equation (2)) are used for the ARAIM fault detection algorithm (right)

The simulation parameters listed in Table 2, including ISM parameters, were applied.

Figure 7 (right) shows the availability map obtained when the newly derived multipath and user antenna error models are used. This figure shows that the global availability increases by approximately 5% if the new DUFMAN multipath and user antenna error models are applied for the ARAIM fault detection and PL computation. This result occurs because, as discussed earlier, the modified multipath model for GPS L1/L5 and Galileo E1/E5a has lower bounds than the legacy GPS L1 multipath model, particularly for low satellite elevation angles.

Figure 8 shows 99.5% VPL maps for the two cases, and Figures 9 and 10 present other requirements for LPV-200, i.e., σ_acc and the effective monitor threshold (EMT), which are criteria that determine the availability performance. In this study, the thresholds for σ_acc and EMT are 1.87 m and 15 m for 99.5% LPV-200, respectively (Working Group C. ARAIM Technical Subgroup, 2016). σ_acc indicates the standard deviation of the vertical position solution used for the accuracy requirement. The EMT is defined as the maximum value of detection thresholds of faults whose prior probability is equal to or higher than a given requirement (Blanch et al., 2015). As shown in Figure 8, the VPL magnitude decreases in the majority of regions in the map when the new model is applied. However, because the alert limit for LPV-200 operation is 35 m (Working Group C. ARAIM Technical Subgroup, 2016), a comparison of Figures 7 and 8 shows that effective improvements with respect to availability can be achieved in high-latitude regions while VPL reductions in mid-latitude and equatorial regions are not actually counted. Thus, despite the noticeable decrease in VPL in almost all regions, an improvement of 5% can be achieved with respect to availability.

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FIGURE 8

99.5% VPL map based on the legacy noise and multipath mode (left) and DUFMAN σ_MP&ADGV (right)

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FIGURE 9

Standard deviation of the vertical position solution (σ_acc) based on the existing noise and multipath model (left) and based on DUFMAN σ_MP&ADGV (right)

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FIGURE 10

EMT based on the legacy noise and multipath model (left) and DUFMAN σ_MP&ADGV (right)

It is evident from Figure 9 and Figure 10 that σ_acc and EMT improve in the majority of areas if the less-conservative model is applied. In particular, the coverage in which the EMT requirement, i.e., lower than 15 m, is met extends noticeably in high-latitude areas, resulting in an availability coverage improvement. In contrast, the σ_acc requirement is already met for both cases.

However, because the combined error model (σ_MP&AGDV) was considered, the improvements shown above may be due to the combined effect of multipath and AGDV contributions. Evaluating the extent of improvement that can be achieved by the new AGDV error model alone (or multipath model alone) is challenging because, unlike the new DUFMAN model in which the multipath and AGDV contributions are characterized separately, the contributions are mixed within the legacy model. Alternatively, to examine the AGDV impact in VPL, Figure 11 shows the difference between the VPL based on the combined model (VPL_σMP&AGDV) and that based on σ_MP (VPL_σMP), i.e., with _AGDV removed from σ_MP&AGDV in Equation (2). It is shown that an additional increase of more than 1.3 m in the 99.5% VPL can result from σ_AGDV in high-latitude areas, where the availability coverage predominantly expands.

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FIGURE 11

Difference between 99.5% VPL based on the combined σ_MP&AGDV (VPL_σMP&AGDV) and that based on σ_MP instead of σ_MP&AGDV (VPL_σMP)

However, this work is more focused on the AGDV error contribution, and such increases in VPL due to the AGDV error might be still optimistic because the DUFMAN model is an RMS-based model. Thus, as mentioned above, we further examine the impact of the AGDV error contribution in the worst-case scenario by taking different approaches to modeling the AGDV error in the following sections.

5.2 Impact of AGDV-Induced Error Contribution for Different Modeling Approaches (Case 3 – Case 6)

Now that we have examined the impact of the new combined model (σ_MP&AGDV) with respect to the legacy GPS CNMP model in the previous section, this section describes how the AGDV error affects the ARAIM performance when different modeling approaches are applied (see Cases 3–6 in Table 7) and evaluates the worst impact of the error.

As discussed earlier, in prior work (Circiu et al., 2021), the Gaussian term σ_MP&AGDV in Equation (2) was derived in the form of an RMS. Because this investigation is focused on the AGDV error contribution, in this case, the new overbounding sigma term for the AGDV derived in the range domain is applied for the term σ_MP&AGDV while the same RMS multipath model is used. Figure 12 shows global maps in which the increase in the 99.5% VPL (left) and 99.5% HPL (right) is represented when the overbounding σ_AGDV, instead of the RMS bound, is applied for σ_CNMP in Equation (20). The results show that the maximum increase is approximately 0.7 m in the VPL and 0.3 m in the HPL, which is not significant enough to affect the global availability coverage.

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FIGURE 12

Increase in the VPL (left) and HPL (right) magnitude for Case 3 The maximum increase in the VPL and HPL is approximately 0.7 m and 0.3 m, respectively, when the AGDV effect (σ_AGDV) is considered for the PL calculation.

Because the change in the PLs seems to be marginal for almost all regions in the map, we sought to investigate the impact of different error modeling methods on the VPL magnitude in greater detail. Thus, the VPLs computed in the availability simulation at a specific user location (110°W, 65°N) for a single day are shown in Figure 13. The VPL is calculated at this location because the 99.5% LPV-200 availability decreases/increases significantly in this region when the different error models are applied, as shown in Figure 7.

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FIGURE 13

VPLs based on different error models: the DUFMAN model (Case 2 in Table 7) (blue), overbounding in the range domain (Case 3) (green), and overbounding in the position domain (Case 4) (light blue)

In Figure 13, the results for three scenarios are shown: Case 2, Case 3, and Case 4 from Table 7. Because the DUFMAN error model considers the RMS of the multipath and AGDV errors (i.e., Case 2), it results in an optimistic VPL (solid blue line). We also investigate the extent of VPL margin that can be achieved by applying different approaches to the user antenna error modeling. For this, the VPLs based on the overbounding sigma values in the range domain (Case 3) and position domain (Case 4) are plotted in Figure 13. The green curve and light blue curve represent the VPL for Cases 3 and 4, respectively. As expected, the AGDV error bound in the range domain results in a higher VPL than the RMS model-based VPL. In addition, for the most part, the VPLs based on the position domain overbounds are slightly higher than those based on the range domain overbounds. This result primarily arises from the fact that in the position domain approach, each sigma overbound for each fault mode was determined from the worst satellite geometry for each fault mode.

In Figure 14, the black line shows the VPL when the additional position bias term was applied for the VPL computation using DUFMAN AGDV error estimates. The VPL based on the bias term is comparable to that based on the position domain overbound (light blue). Lastly, the magenta line shows the VPL when the MOPS bounds on AGDV errors were applied for the VPL calculation as the additional bias term. As expected, considering the upper bounds within the bias term results in the worst impact on the VPL. Because the antenna chosen for this investigation had the worst performance among the antennas examined in DUFMAN and the corresponding AGDV errors were at the limit of the bound defined in the MOPS, the VPL results of the actual observables are not substantially different from those of the MOPS bounds.

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FIGURE 14

Comparison of VPLs for Case 4 (light blue), Case 5 (black), and Case 6 (magenta)

The VPLs based on the six different cases are shown together in Figure 15. The results show that different improvements in the VPL can be achieved depending on the manner in which the AGDV effect is modeled and implemented at the algorithm level. As indicated in Figure 14, consideration of the MOPS bound within the bias term in the PL computation results in the worst impact on the VPL. To evaluate the impact of AGDV error in the worst-case scenario, Case 6 must be examined in greater detail. Figure 16 shows the difference between the VPL obtained from the MOPS bound (VPL_σMP&MOPS), which corresponds to the magenta line in Figure 15, and that based on the combined model (VPL_σMP&SGDV), where the AGDV error contribution is modeled as the RMS bound. An additional increase in VPL of more than 1.3 m, compared with the RMS model (i.e., VPL_σMP&AGDV), is observed when the worst impact of the AGDV error is taken into account, and the additional margin results in an availability loss of approximately 1%. Note that the DUFMAN multipath model is applied to both VPL_σMP&MOPS and VPL_σMP&AGDV.

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FIGURE 15

VPLs for the different scenarios from Cases 2–6

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FIGURE 16

Difference between 99.5% VPLs obtained by considering the MOPS bound on AGDV error within the PL computation (VPL_σMP&MOPS and the VPL based on the combined multipath and AGDV model (VPL_mp&AGDV) (left); 99.5% availability for LPV-200 when the worst PL (VPL_σMP&MOPS) is applied (right)

However, considering that the VAL for LPV-200 is 35 m (see Figure 15), the impact of AGDV error on the LPV-200 ARAIM availability performance appears to not be of major relevance if we employ the newly proposed (or less-conservative) multipath and noise models.

The impact of the different AGDV models, including the combined DUFMAN model, on V-ARAIM has been presented under various scenarios. However, considering that ARAIM should begin with horizontal service only, i.e., horizontal ARAIM (H-ARAIM), we also examine how the different models affect the HPL. For the initial implementation of H-ARAIM, the ISM parameters in Table 2 have been considered for GPS and Galileo by the community (Working Group C. ARAIM Technical Subgroup, 2016), and we applied the same parameters. Figure 17 shows HPLs for the different AGDV modeling approaches, similar to Figure 15. Only the worst-case bias scenario (Case 6) results in an HPL increase of 1–1.5 m. In contrast, the other methods result in comparable HPLs, which is in line with the VPL comparison in Figure 15. Because an HAL of 185 m is defined for required navigation performance (RNP) 0.1, which is the target performance of H-ARAIM, it is evident that an improvement in availability will not be observed. Thus, only the HPLs are compared in this section.

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FIGURE 17

HPLs for the different scenarios from Cases 2–6

In this section, we investigated the effect of AGDV on ARAIM performance in different aspects, including the protection bounds, position accuracy, and EMT, by applying a baseline user algorithm and different multipath and AGDV models. It should be noted, however, that this analysis is based on limited results (e.g., the AGDV error model for a specific antenna type), and variations in parameters such as the user range accuracy value employed in the simulations could lead to different results.