Feasibility Study of GBAS/INS and RRAIM for Airport Surface Movement Under Low-Visibility Conditions

3.1 Integrity Allocation

Integrity must be assured throughout the surface operation, both before and after the transition to RRAIM coasting. The allocation of integrity risk must cover all considered fault modes in both cases. The baseline for integrity risk allocation, used for GBAS CAT I precision approaches, is shown in Figure 1(a) (Pullen, 2012). Modifications of this allocation are presented in Figure 1(b) and (c), which address the GBAS/INS+RRAIM cases for t < t₀ and t ≥ t₀, respectively. An additional branch is included to account for INS failure, which is not addressed in this paper and is left for future work, leveraging DO-384 and considering the short operation times of surface movement. With the integrity risk allocated to INS failure, the remaining integrity risk assigned to GBAS and RRAIM decreases. As a result, a more conservative protection level (PL) is calculated.

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FIGURE 1

(a) Integrity risk allocation for a GBAS CAT 1 precision approach (Pullen, 2012) and (b, c) integrity risk allocation for surface movement (b) during pre-coasting (t < t₀) and (c) during coasting (t ≥ t₀)

PA: precision approach, SV: space vehicle

The allocation tree is applied for each sub-phase of the surface movement. The tree begins with an allocation to the specific sub-phase, which is then divided into H0, H1, and H2 conditions, representing fault-free, single-reference-receiver failure, and all other conditions, respectively, with allocations of 12.5%, 12.5%, and 37.5%, respectively. The branch for all other conditions encompasses major ground subsystem failures, which are not addressed in this paper as such failures are assumed to be managed by the manufacturer, as well as undetected satellite failures.

To verify whether the solution supports surface movement prior to coasting, t < t₀, the requirements defined in Figure 1(b) must be met. These requirements include nominal conditions (H0), receiver failure for a single local area augmentation system ground facility (LGF) (H1), and for all other conditions (H2). The H2 condition is further divided into H2a and H2b, representing single-satellite failures and all other failures, respectively. For the coasting case t ≥ t₀, the conditions are subdivided depending on the failure conditions of both the GBAS/INS and RRAIM. In Figure 1(c), the nominal condition (H0) accounts for cases in which no failure occurs during either the pre-coasting phase (t < t₀) in GBAS/INS mode or the coasting phase (t ≥ t₀) in RRAIM mode. The H1 hypothesis indicates the condition in which the initial position of RRAIM is computed under a single-LGF-receiver failure and no failure has occurred during coasting. All other conditions (H2) include scenarios in which no failure has occurred during pre-coasting (t < t₀), but a single-satellite failure occurred during the coasting period (t ≥ t₀) (H2a), as well as situations in which a single satellite is faulty during pre-coasting (H2c). All remaining conditions are denoted as H2b’.

3.2 Pre-Coasting (GBAS/INS)

The state vector estimate consisting of position and receiver clock (δτ) estimates based on smoothed and GBAS-corrected pseudoranges can be expressed as follows:

2

where the hat symbol indicates the estimated value and H and R are the observation and measurement covariance matrices, respectively. The vector z represents the linearized pseudorange residual after GBAS corrections have been applied, and the projection matrix is denoted by S. β represents the differential nominal bias, which can be caused by nominal signal deformation errors as well as satellite and airborne antenna group delays (Macabiau et al., 2015). Because the airborne antenna group delay is included in the multipath budget and the satellite antenna group delay cancels out when the GBAS correction is applied to the airborne measurement, signal deformation is the main contributor to the nominal bias in the pseudorange residual. The covariance matrix of the estimated solution is denoted by Q, which is defined as (H^TR⁻¹H)⁻¹.

To consider the impact of GBAS/INS tightly coupled integration, the following INS mechanization error equation and measurement equation are used in our analysis (Rhee et al., 2004). In addition, to consider time-correlated errors in the pseudorange residual resulting from GBAS smoothing and correction, the noise of the pseudorange residual is assumed to consist of Gaussian white noise (ε_z) and colored noise (ξ), which can be modeled as a first-order Gauss–Markov process (Khanafseh et al., 2010). The colored noise term is then augmented to the state vector (Crassidis & Junkins, 2004). Thus, the modified system error equation and measurement equation can be expressed as follows:

3

4

Here, δx, δv, δq, δb_a, δb_g, ż, and indicate the position error, velocity error, quaternion error, accelerometer bias error in the body frame, gyro bias error in the body frame, Doppler measurement, and clock drift, respectively. The superscript e denotes the Earth-centered Earth-fixed (ECEF) frame. ω_a, ω_g, ω_ba, ω_bg, ω_δτ, , and ω_ξ are the process noises of the accelerometer, gyro, accelerometer bias, gyro bias, clock, clock drift errors, and colored noise, respectively. The terms , , , and Ω indicate the gradient of the gravitational force, the bias-compensated accelerometer measurement, a transformation matrix from the body to ECEF frame, and the skew symmetric matrix of the Earth’s rotation vector, respectively. In addition, τ_mp indicates the time constant of the colored noise in the smoothed pseudorange residual error after the GBAS correction has been applied. Because GBAS reference receivers are equipped with a multipath-limiting antenna to reduce multipath errors, temporal correlation mainly arises from multipath in the smoothed pseudorange of the aircraft. Although determining the exact time constant of multipath in the smoothed code is challenging, it is possible to estimate a range of potential time constants in advance (Pervan et al., 2017). Khanafseh et al. (2010) and Langel et al. (2021) have addressed the impact of mismodeling and proposed a covariance bounding method. In particular, to consider the impact of the uncertainty of time constants for colored noise, Langel et al. (2021) proposed a method to ensure that the Kalman filter estimate error covariance bounds the true estimate error covariance matrix. The authors suggested setting the time constant in the Kalman filter system error equation and the initial variance of the colored noise using the maximum and minimum time constants and the maximum variance of the steady-state colored noise. Our study adopted this method to provide practical results. To obtain the maximum and minimum values of the time constants of smoothed multipath of the aircraft, a Monte Carlo simulation was conducted. According to Pervan et al. (2017), the time constants of raw multipath for Boeing aircraft are 7, 14, and 170 s at the 5^th, 50^th, and 95^th percentiles, respectively. Assuming a log-normal distribution for the time constants, we generated a total of 105 time histories of multipath using the first-order Gauss–Markov process. We then applied a carrier smoothing filter to obtain the range of time constants for smoothed multipath. Maximum and minimum time constants of 54 and 2894 s, respectively, were obtained when a carrier smoothing filter with a 100-s time constant was applied. The maximum steady-state variance of multipath is set to the square sum of the receiver models of the GBAS ground station and aircraft (RTCA, 2004; Circiu et al., 2021) because these models provide variances that bound the receiver error after the carrier smoothing filter has been applied. A detailed analysis is provided in Appendix A. The values obtained in Appendix A, which are used in the surface movement simulation, are also shown in Table 2 in Section 4.1.

The benefit of INS integration is a continuous solution, even during GNSS outages; moreover, the solution is smoothed by the high-rate and high-precision INS observations. However, the addition of INS cannot enhance the positioning accuracy in terms of the bias error. Therefore, we assume that the bias error in the GBAS/INS state estimates is equal to that of the GBAS solution, expressed as Sβ. To simplify the simulation procedure, only the covariance matrix of the states, denoted as Q′, is updated via the discretized Equations (3) and (4). The standard deviations of the GBAS/INS position estimates from Q′ are smaller than those from the GBAS-only covariance matrix Q because the use of INS further smooths out the position estimates. The bias error in the position domain, Sβ, and the covariance matrix associated with the position and receiver clock errors, denoted as , are used to compute the NSE and horizontal PL (HPL) to assess the feasibility of the GBAS/INS pre-coasting solution (t < t₀).

Recall that cases H0, H1, H2a, and H2b must be addressed. For the nominal condition (H0) and the single-LGF-receiver failure condition (H1), the HPL is computed as follows, based on the GBAS PL computation defined in RTCA DO-253D (RTCA, 2017) with the inclusion of an additional nominal bias term, β_hor :

5

The subscripts i and j represent the indices of the satellite and ground reference receiver, respectively. The term s corresponds to the projection coefficient, with . σ_i and σ_i,H1 denote the standard deviations of the pseudorange residual of satellite i under the nominal and single-LGF-failure conditions, respectively. B_hor,j represents the position domain B-value in the horizontal direction to account for the impact of a possible LGF receiver failure at the aircraft. This term can be computed as . The B-value, B_value,i,j, is broadcast from the ground subsystem through type 1 and 11 GBAS messages. For simulation purposes, we considered the impact of the B-value and the nominal bias comprehensively by using β_hor for different levels of nominal bias. The constants K_{f fmd} and K_md represent the multiplier for a fault-free missed detection and a receiver-fault missed detection probability, respectively. These terms are computed based on the approach provided in RTCA DO-245 (RTCA, 2004), assuming that there are four ground receivers and using the continuity and integrity risk requirements for surface movement. For example, K_{f fmd} = 6.54 and K_md = 4.06 are used for a rapid exit taxiway. The term β_hor signifies the worst-case impact of the nominal bias in the horizontal domain, computed as follows (van Graas & Soloviev, 2004):

6

where β_max indicates the maximum magnitude of the nominal bias. The computed HPL for each sub-phase of surface movement from Equation (5) can be compared with the horizontal alert limit (HAL) derived in Table 1 to assess the feasibility of the GBAS/INS approach.

Next, we must examine the compliance to the integrity risk requirement for the remaining condition, referred to as H2a, which considers the case in which a single-satellite failure occurred during the pre-coasting period. The integrity risk, or equivalently, the probability of hazardously misleading information (HMI), for H2a can be expressed as follows:

7

where pe, TSC, q, and Th are the horizontal position error (HPE), the TSC determined by the recommended layout of the airport, the test statistic, and the threshold of the test statistic, respectively. The probability that the position error exceeds the TSC can be computed via Equation (8) by using the cumulative distribution function of the standard normal distribution Q:

8

Here, σ_TSE indicates the standard deviation of the TSE, which can be calculated by using the standard deviations of the NSE (GBAS/INS solution), FTE, and PDE, which are defined in the standards and the reference literature (RTCA, 1999; Amielh et al., 2019). The terms E_hor and β_hor are the bias error in pe caused by the fault and nominal bias, respectively. P(fault) in Equation (7) indicates the probability of fault during pre-coasting or at t₀ for coasting, depending on the operation mode. It is important to note that for GAST D and GAST F ground monitors, only the single-failure case is considered.

To compute the probability of the fault being present at the current time, we must consider the impact of the GBAS ground monitors in the past. Because of the time correlation in the test statistics, it is assumed that a fault present at time t must have occurred after the last independent test, as the probability of two independent missed detections may be neglected. Here, we define this interval as the effective decorrelation time (τ_effective). Now, P(fault) can be computed as follows:

9

where r_fault and N_sat are, respectively, the fault rate expressed per hour and the number of satellites, which is set to 18 according to the capacity of the VDB data-link (ICAO, 2009). The term τ_effective is set to 100 s, according to the temporal correlation analysis of the GBAS ground monitors, as shown in Appendix B.

Finally, the limit of the probability that the test statistics fail to detect the fault, P_MD,limit, for surface movement can be computed for different fault biases under a given nominal bias error as follows:

10

The range domain fault bias, E_R, can be obtained by dividing the horizontal position domain fault bias, E_hor, by the projection coefficient s_hor. The term P_MD,limit computed from Equation (10) is a derived requirement expressed as a bound of the PMD under the H2a condition for surface movement. GBAS provides monitoring performance in terms of the differential range error for GAST D (ICAO, 2018) and GAST F (SESAR, 2015). In GAST D and F, this PMD requirement is provided according to the range domain bias error, which is derived based on the vertical domain because the effects in the vertical domain result in a more stringent requirement than in the lateral domain, as stated in the SARPs (ICAO, 2009). As we assess the monitoring performance related to the HPE, we map the range domain performance to the horizontal domain using the horizontal projection factors defined by SESAR (2021). If P_MD,limit bounds this guaranteed GBAS monitoring performance (expressed as a requirement in the standards but used here as a minimum performance guarantee), we can take advantage of the integrity provided by the GBAS to meet the surface movement requirement. In our simulation, the maximum allowable magnitude of the nominal bias will be determined, which still allows this condition to be met. Then, Equation (10) can be used to account for the undetected fault in the GBAS/INS solution at t₀ included in hypothesis H2a for coasting.

The remaining condition, H2b, includes a major subsystem failure, which is assumed to be managed by the manufacturer and is not addressed in this paper.

3.3 Coasting (GBAS/INS+RRAIM)

RRAIM was first proposed to support the instrument approach operation known as lateral precision with vertical guidance at 200 feet (LPV-200) (Lee & McLaughlin, 2008). Lee and McLaughlin (2008) proposed to use RRAIM with other integrity-assured systems such as satellite-based augmentation systems (SBASs) that meet the LPV-200 requirement, where RRAIM is used to fill the gap during any possible SBAS integrity outages. In our analysis, a GBAS/INS solution is used with RRAIM at the airport. This study investigates the conditions in terms of the maximum allowable nominal bias and the coasting period duration under which GBAS/INS+RRAIM meets the requirement for each surface movement sub-phase. A complete technical description of RRAIM can be found in the literature (Lee & McLaughlin, 2008; Gratton et al., 2010); this paper describes only the necessary information. As shown in Equation (1), the change in position, , based on the TD carrier phase measurement can be expressed as follows:

11

where Δ indicates the time difference operator and i denotes the index of a satellite. The terms ϕ, r, x, l, and f represent the carrier phase measurement, satellite position, aircraft position, line-of-sight (LOS) vector, and fault bias, respectively. The position estimated by the GBAS/INS is used to compensate in Equation (11). This leads to a propagation of the nominal bias contained in the pseudorange residual to the TD position solution. The covariance matrix and estimation bias can be expressed as follows (Lee & McLaughlin, 2008):

12

13

where R_Δϕ is the measurement covariance of the TD carrier phase and S_Δϕ indicates the projection coefficient based on the TD observation matrix . Finally, the covariance matrix and bias at time t in Equation (1) can be expressed by Equations (14) and (15):

14

15

The RRAIM HPL can be computed as follows, based on a solution separation (SS) integrity monitoring approach under a single-satellite failure assumption (Lee & McLaughlin, 2008) for a nominal condition (H0), a single-LGF failure (H1), and the single-satellite failure condition (H2c):

16

For the single-LGF condition (H1), the subscript j denotes the index of the GBAS reference receiver being excluded. Under this condition, the H1 covariance matrix of the GBAS/INS solution is used to compute the covariance matrix of RRAIM, Q_RRAIM,H1, whilst the impact of the B-value in the RRAIM solution, which can be computed as |MS| B_value,j, is also included. For the single-satellite failure condition (H2c), the subscript i denotes the satellite that is excluded from the full-set solution to compute the subset solution. Consequently, Q_RRAIM,i indicates the covariance matrix of the subset solution of RRAIM with satellite i being excluded. The worst-case impact of a nominal bias on the full-set solution (β_hor,RRAIM) and the subset solutions (β_hor,i,RRAIM) can be computed in the same manner as Equation (6) with different projection matrices M and M_i, respectively. M_i is the projection matrix computed with satellite i excluded. The term Th_hor,i is the threshold of each SS test statistic when satellite i is excluded. The constants k_FF and k_FC are multipliers of fault-free coasting and faulty coasting, which can be computed according to the reference literature using the integrity and continuity risks derived for surface movement requirements (Amielh et al., 2019).

The hypothesis H2b’ includes the case in which a fault is not detected by the GBAS ground monitors and is thus present in the GBAS/INS solution at t₀ and is undetected by the RRAIM but impacts the TD solution. Because these monitors are independent, this case is considered highly improbable and may be neglected. To verify the undetected satellite fault in the GBAS/INS solution at t₀ (H2a), P_md,limit is assessed by the approach given in Section 3.2.

To consider the impact of multiple simultaneous failures, advanced receiver autonomous integrity monitoring (ARAIM) has been proposed (Blanch et al., 2015). In this paper, we extend RRAIM to consider multiple failures not only for the Global Positioning System (GPS) but also for Galileo by adopting the ARAIM approach. Further details on the derivation of the HPL in ARAIM have been reported by Blanch et al. (2015), and the following equation defined for east or north directions ( q ) is used in our analysis to compute the RRAIM HPL:

17

The term PHMI denotes the probability of an HMI event, which is set to the PHMI allocated to a single-satellite failure in Figure 1(c) for comparison with the results from Equation (16) when a single-satellite failure is considered. We assume that PHMI is halved for the allocation to the east and north components. The term N_{fault modes} is the number of possible fault combinations for n satellite faults and a constellation fault per a single constellation. The probability P_fault,k is the prior probability of each fault mode k. The term P_{not monitored} denotes the sum of the prior probabilities of all fault modes that are not monitored. The PLs in the east and north directions are determined iteratively by Equation (17), and the HPL is computed from the root-sum square of the east and north PLs.

Figure 2 shows the flowchart of the navigation at the aircraft during surface movement based on a GBAS/INS initial position and the RRAIM approach described in this section.

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FIGURE 2

Overall methodology of accuracy and integrity assessment

4.1 Measurement Error Models

The standard deviations of the GBAS-corrected pseudorange residual and the TD carrier phase measurements can be expressed as follows:

18

19

20

The terms σ_gnd, σ_tropo, σ_iono, and σ_air indicate the standard deviations of the ground receiver noise, tropospheric residual, ionospheric residual, and airborne receiver noise, respectively. The subscript i represents the index of a satellite. Note that GAST F uses an ionosphere-free (IF) measurement combination, where the ionospheric delay is eliminated, whereas the receiver noise is inflated by a factor of approximately 2.6 compared with the SF measurement. σ_Δtropo, σ_Δclock, and σ_air,ϕ are the standard deviations of the spatial decorrelation of the tropospheric delay, the temporal decorrelation of the receiver clock bias, and the carrier phase noise at the airborne receiver, respectively. A scaling factor of 1/20 is applied to the airborne model so that the standard deviation of the carrier phase noise is conservatively set to approximately 2 cm at an elevation angle of 5°, a factor of 10 greater than the theoretical value achieved by aeronautical-class receivers (Xu, 2007). The variance of the noise in the TD carrier phase is conservatively expressed as twice the variance of the carrier phase noise, assuming no temporal correlation. It is important to note that regardless of the type of GBAS corrections applied, an airborne user can access the DF carrier phase.

Table 2 provides details of the error model described above. The standard deviations of the ground and airborne receiver noise after application of the carrier smoothing filter with a 100-s time constant are denoted as σ_gnd and σ_air, respectively. To estimate the standard deviation of the receiver noise after application of the smoothing filter with a 30-s time constant for GAST D, a factor of is applied (Murphy et al., 2021). γ, el, and v represent the squared ratio of L1 over L5 frequencies, the elevation angle, and the aircraft speed, respectively. The maximum speed of the aircraft for each sub-phase (RTCA, 1999) is used as v to accurately model the spatial decorrelation of the tropospheric delay. In the simulation, the differential nominal bias varies from 0.1 to 0.5 m with a 0.1-m interval. The largest differential nominal bias is conservatively assumed to be 0.5 m, which is smaller than the recommended value of 0.75 m for absolute nominal bias in ARAIM. It should be noted that the bias error cancels out when the GBAS correction is applied, except for the nominal signal deformation component, which is expected to be less than 0.3 m (Wong, 2014). The parameters τ_mp,max, τ_mp,min, and are used to ensure that the Kalman filter estimate error covariance bounds the true estimate error covariance (Langel et al., 2021). A detailed explanation of these parameters is given in Appendix A.

4.2 Flight Trajectory and IMU Specifications

To develop the INS mechanization error equation and run the extended Kalman filter (EKF), navigation-grade inertial measurement unit (IMU) sensors are assumed in the simulation. The default sensor performance level is based on the Honeywell HG5700: the 1-sigma bias, the random walk, and the scale factor of the gyro sensor are set to 0.035°/h, 0.006°/√h, and 40 ppm, respectively. In addition, the 1-sigma bias and the scale factor of the accelerometer error are set to 0.035 mg and 120 ppm, respectively. For the process noise of the receiver clock bias and drift, we used Allan variance values for a temperature-compensated crystal oscillator given by Brown and Hwang (1996), and we assumed that the clock dynamic model reflects the actual clock behavior. In our analysis on the accuracy, IMU sensors are considered as fault-free. For the integrity analysis, IMU failure (H3) is considered in the integrity allocation tree; however, its impact is not assessed here, and we leave it as future work.

To simulate the convergence of the covariance matrix of the GBAS/INS, we used the synthesized aircraft waypoints and time information from Wichita to the Chicago airport provided by MATLAB in the sensor fusion and tracking toolbox (MATLAB, 2021). An aircraft trajectory and attitude information at a desired sampling rate are obtained by a generic MATLAB function called “geoTrajectory.” In the simulation, 10 Hz and 1 Hz are assumed for the IMU and GNSS outputs, respectively. Figure 4 shows the trajectory of the aircraft in east–north representation and a time history of the height and attitude of the aircraft. The red dotted circle in Figure 4(a) indicates the coverage of the GBAS service.

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FIGURE 4

(a) Airborne trajectory represented by east-north axes, (b) time history of aircraft height, and (c) time history of attitude information

To test the satellite geometries observed at the GBAS research prototype installed at the Barcelona airport within the Single European Sky ATM Research (SESAR) project, the generated aircraft trajectory in Figure 4 is translated to this location. Satellite geometries were recorded from January 6, 2019 to January 8, 2019 with 10-min intervals. The different initialization times of the GBAS/INS EKF were tested to account for the impact of satellite geometry on the convergence of the covariance matrix.

4.3 Simulation Configuration for Integrity Analysis

The parameters used to compute the HPL are summarized in Table 3. To include multiple-failure modes, the source code from the MATLAB Algorithm Availability Simulation Tool developed by Stanford University was adopted and modified (Stanford University, 2022). The Galileo satellite fault rate, r_sat, is set to two possible values in the absence of published values: r_sat = 3 × 10⁻⁵ (Wallner et al., 2021) when a mean fault duration (MFD) of 1 h is assumed, and r_sat = 10⁻⁵ where the GPS value (GPS SPS, 2020) is assumed for Galileo.

5.1 Pre-Coasting (GBAS/INS) Assessment

A performance assessment of the GBAS/INS during pre-coasting is needed to determine the sub-phase in which RRAIM should be initiated. We expect that the severity of the multipath error will grow as the aircraft approaches the stand lead-in line, but we assess the GBAS/INS performance for all sub-phases and determine the level of allowable differential nominal bias to meet the requirement. Figure 5 shows the HPE against the HPL based on the steady-state covariance matrix from the GBAS/INS tightly coupled filter. Also shown are the NSE requirement and the HAL derived for the taxiway phase. As shown in Figure 5, even with a nominal bias of 0.5 m (cyan), the HPL for both GAST D and F can meet the requirement in the taxiway. In contrast, the HPE is only compliant to the NSE requirement when the magnitude of the nominal bias is less than 0.4 m (light green) for GAST D and 0.5 m (cyan) for GAST F. GAST F allows slightly larger nominal bias values than GAST D because the first-order ionospheric delay residual is completely eliminated in the IF measurement. In addition, despite the performance loss due to the inflated receiver noise in the IF measurement, the improvement in satellite geometry in GPS/Galileo GAST F yields a more significant gain. Table 5 summarizes the maximum allowable nominal bias error for compliance in all sub-phases.

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FIGURE 5

HPE and HPL of GBAS/INS during pre-coasting computed for the taxiway using (a) GPS-only GAST D and (b) GPS/Galileo GAST F corrections

To account for the integrity allocated to the H2a hypothesis, the P_md,limit value of the GBAS/INS during pre-coasting is compared with the monitoring performance of the GBAS, which is expressed as the PMD according to the differential fault bias in the range domain, E_R. The limit of the PMD derived for the GBAS/INS during coasting can be expressed as the differential range error, E_R, by using the worst-case projection coefficients in the horizontal direction. These coefficients are set to 2 and 0.7 for GAST D and F (SESAR, 2021), respectively. We also validated these values by simulating all possible satellite geometries observed globally for a duration of 10 days, using the optimized 24 GPS and 24 Galileo constellations (Guilbert, 2016). As mentioned previously, for the satellite fault rate of Galileo, two possible values are considered: 3×10⁻⁵/h from the state probability of the Galileo fault, as stated in the draft SARPs (Wallner et al., 2021) under the assumption of a 1-h MFD, and 1×10⁻⁵/h, the GPS fault rate. These two cases are denoted by A and B in Figure 6 and Table 6. Figure 6 shows the limit of the PMD, which accounts for the H2a hypothesis, computed for the taxiway with various magnitudes of the differential nominal bias, as well as the GBAS monitoring performance. The results demonstrate that a larger nominal bias is allowed when GAST F is used because of the relaxed GBAS monitoring performance resulting from improved satellite geometry. As mentioned above, the projection factor is greatly improved from 2 to 0.7 when a dual constellation is considered (SESAR, 2021). In addition, case B, which has a smaller Galileo fault rate, allows larger differential nominal biases whilst still bounding the monitoring performance required for the GBAS.

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FIGURE 6

PMD limit of GBAS/INS during pre-coasting computed for the taxiway with respect to the GBAS monitoring performance for (a) GAST D, (b) GAST F under Case A (r_sat = 3×10⁻⁵/h), and (c) GAST F under Case B (r_sat = 1×10⁻⁵/h)

Table 6 summarizes the maximum allowable nominal bias for all sub-phases. It is feasible to use GBAS/INS on the airport surface in the rapid exit taxiway and the taxiway phases, with a sufficient margin in the case of GAST F. For the apron taxiway and taxi lane, the performance is met by GPS/Galileo GAST F with zero nominal bias. From these results, RRAIM based on the GAST D correction should be initiated before the apron taxiway. In the case of GAST F, RRAIM can be initiated at any sub-phase before reaching the stand lead-in line sub-phase.

5.2 Coasting (GBAS/INS+RRAIM) Assessment

In this section, the performance during RRAIM coasting is assessed. Although the results in Section 5.1 introduce some restrictions, we evaluate the performance of RRAIM for all sub-phases. To analyze the degradation of RRAIM performance, coasting times ranging from 10 to 600 s are tested in the simulation. In addition, coasting times longer than the exposure time of each sub-phase of the surface movement operation specified in the standards are also considered, to account for cases in which coasting is started prior to the sub-phase and for possible standby. Figure 7(a) presents the maximum allowable differential nominal bias to ensure full compliance to the 95% NSE and integrity requirements. The white boxes labeled as “NaN” indicate that the requirement is not met even with a differential nominal bias of zero. For a comparative analysis, results for GPS-only GAST D, GPS-only GAST F, and GPS/Galileo GAST F processing modes are shown together. The results suggest that GPS/Galileo GAST F allows the largest differential nominal bias whilst achieving compliance, whereas GPS-only GAST D exhibits slightly better performance than GPS-only GAST F. This observation suggests that the improvement in satellite geometry from GPS/Galileo outweighs the performance loss caused by the inflated noise in the IF measurement. When comparing the results between the SF and DF modes (GPS-only GAST D vs. GPS-only GAST F), the performance loss due to the inflated noise is greater than the loss caused by ionospheric residual error, primarily because the distance between the aircraft and the GBAS reference point at the airport is small.

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FIGURE 7

Maximum allowable nominal bias (m) for complete compliance to (a) 95% NSE, (b) integrity risk (sf), and (c) integrity (mf) requirements for each sub-phase and GBAS modes sf: single-failure consideration; mf: multiple-failure consideration

Figure 7(b) shows the maximum allowable differential nominal bias to ensure that the RRAIM HPL is inferior to the HAL in each sub-phase and for various coasting times under the single-failure assumption. Because only a single-satellite failure is considered and because the prior probabilities of GPS and Galileo satellite fault are almost the same, the magnitude of acceptable nominal bias mainly depends on the satellite geometry and measurement residual errors. Therefore, GPS/Galileo GAST F allows the largest differential nominal bias, preceded by GPS-only GAST D and GPS-only GAST F.

Figure 7(c) summarizes the integrity performance results for a multiple-failure consideration. Compared with the results in Figure 7(b), the allowable differential nominal bias has decreased throughout the parameter space. This result occurs because the integrity risk is allocated to the multiple fault modes in which the subset solution performance is greatly degraded compared with the full-set solution, whilst the HPL must still bound the error with respect to the corresponding PHMI allocation. Under multiple-failure consideration, GPS/Galileo GAST F does not always allow the largest differential nominal bias. For relatively short coasting times, GPS-only GAST-D allows a larger nominal bias than GPS/Galileo GAST F mode in the stand lead-in line, whereas for larger coasting times, GPS/Galileo GAST F exhibits superior performance. For longer coasting times, the spatial and temporal error decorrelation becomes significant in the SF mode (GAST D); however, in the DF mode (GAST F), the impact is less significant because of the elimination of the ionospheric delay. In contrast, for the apron taxiway with a coasting time of 60 s and for the taxi lane with a coasting time of 20–60 s, GPS-only GAST F allows a slightly larger nominal bias than GPS/Galileo GAST F, even for worse satellite geometry. We attribute this result to the higher prior probability of Galileo constellation fault compared with that of GPS.

Table 7 shows the allowable differential nominal bias for P_md,limit defined in Equation (9) to meet the integrity risk allocated to the H2b’ hypothesis during coasting. The allowable bias is slightly decreased compared with the results during pre-coasting shown in Table 6, resulting from the increased standard deviation of the position estimates due to the spatial and temporal decorrelation errors from RRAIM coasting and the reduced PHMI allocation to the H2 hypothesis during coasting.