Multi-constellation ARAIM exploiting satellite motion

3.1 Tropospheric delay

We model the tropospheric delay for SV i at time k as follows:

3

where ⁱv_ZTD,k is the zenith tropospheric delay

ⁱθ_k is the elevation angle for SV i at time k

ⁱc_T,k is the tropospheric zenith-to-slant mapping coefficient:

Equation (3) is the instantaneous error contribution assumed in snapshot ARAIM^1-3: ⁱv_ZTD,k is zero-mean normally distributed with variance . We use the notation where σ_ZTD = 0.12 m.¹ In this paper, ⁱv_ZTD,k is a random time sequence (white noise) independent across SVs i, for i = 1,…,n. Unlike in previous work,^22,23 we do not model the tropospheric error correlation over time or across satellites. This is a conservative assumption that will be refined in future work using tropospheric error data.

3.2 Satellite clock and orbit ephemeris error

The satellite clock and orbit ephemeris error equation for satellite i at time k is as follows:

4

where ⁱb_E is an unknown, constant bias

ⁱg_E is an unknown, constant gradient (ramp)

t₁ is the first time epoch of the batch interval

t_k represents any time k during the batch interval, such that 1 ≤ k ≤ q, where q is the last epoch of the batch interval

ⁱε_RES,k residual errors not captured by the “bias plus ramp” model

The first term in Equation (4) captures the instantaneous uncertainty in ⁱε_E,k. It is therefore set according to ARAIM assumptions as , nominally, σ_URA = 1 m.^2,3 It can easily be shown that a time-invariant bias ⁱb_E affecting a sequence of raw code measurements input to a Hatch filter causes the same error ⁱb_E in the resulting output CSC .

However, the ARAIM error model does not specify whether nominal SV clock and orbit errors can be assumed constant over any particular time interval. To account for temporal error variation, a second term in Equation (4) is added, which is a ramp over time with an unknown but constant gradient ⁱg_E, accounting for linear variations from the initial value over (t_k − t₁). Joerger et al.¹¹ give evidence that periodic variations of orbit errors are on the order of the MEO GNSS orbital period, which supports the use of a simple linear model over 10 minutes. It also cites Parkinson and Spilker,²⁴ Warren and Raquet,²⁵ and Pervan and Gratton²⁶ to establish a distribution on the rate of change of orbit/clock errors: . In addition, Section 3.5.2 of the GPS Standard Positioning Service Performance Standard (SPS PS)²⁷ describes the signal-in-space (SIS) user range rate error (URRE), which supports the model of linear orbit and clock variations. The GPS SPS PS²⁷ does not specify an integrity performance standard yet.

To validate Equation (4), a preliminary experimental data analysis is carried out. Similar to prior work,^2,25,28 precise GPS satellite orbit and clock estimates from the National Geospatial-Intelligence Agency (NGA) are considered truth reference data and are compared to broadcast GPS ephemerides archived by the Crustal Dynamics Data Information System (CDDIS). The focus in this work is on the validation of error model characteristics over time.

In this analysis, a set of data from January 4, 2015, to September 19, 2015, is processed, for Block IIF satellites including PRNs 1, 3, 6, 9, 25, 26, 27, and 30 (the total number of orbit data points is 572 520). The bias-plus-ramp model in Equation (4) is fit to truth-minus-broadcast data. Because NGA orbit data are only provided every 5 minutes, the fit interval is selected to be 30 minutes long. This is larger than the example 10- and 20-minute batch periods assumed in the next sections of this paper. It will provide conservative results because residual fitting errors are larger over 30 minutes than they would be over 20 minutes.

Based on this data, empirical parameter distributions for ⁱb_E and ⁱg_E are established. These are then bounded in the cumulative distribution function (CDF) sense^20,21 using Gaussian distributions. The standard deviations of the overbounding Gaussian functions are given in Table 1 for the clock error contribution and for the three-dimensional orbit errors. Orbit errors are expressed in a local-level, satellite-fixed reference frame, in terms of the in-track, cross-track, and radial components.²⁴ Because GPS satellites are at altitudes of about 20 000 km, user receivers near Earth’s surface are affected by ranging errors that are mostly due to the radial orbit and clock components. Fortunately, orbit radial and clock errors are significantly smaller than orbit in-track and cross-track components, as shown in Table 1.

To get a conservative estimate of the error parameter variance for ⁱb_E while taking into account the worst-case geometry between an SV and a user receiver near Earth’s surface, we use the following equation:

5

where 0.24 is a multiplier accounting for the worst-case projection of non-radial orbit error components; , , and , respectively, are the variances of the clock, orbit radial, in-track, and cross-track error components contributing to ⁱb_E. The same can be done for the variance of ⁱg_E.

Equation (5) is a formulation similar to the ones found in Warren and Raquet,²⁵ Department of Defense GPS NAVSTAR,²⁷ and Heng²⁹ but modified to evaluate variances instead of errors. The worst-case error projections are often referred to signal-in-space range error (SISRE) for ⁱb_E and SIS range rate error (SISRRE) for ⁱg_E. Our 9-month data analysis provides values of σ_BE = 0.61m and σ_GE = 1.8 · 10⁻⁴m/s, which are both smaller than the assumptions mentioned above. It is therefore conservative, in the upcoming availability analysis, to use the values assumed in prior work^1-3,11: σ_BE = σ_URA = 1m and (in future work, we will process more data to check whether it is safe to assume lower σ_BE and σ_BE values).

In addition, the distribution of residual errors ⁱε_RES,k is plotted for the orbit radial and in-track components in Figures 3 and 4, respectively. ⁱε_RES,k is obtained by removing the best fit bias-plus-ramp model from the truth-minus-broadcast data. The figures show that ⁱε_RES,k is not negligible as compared to other error sources affecting carrier measurements. Since we do not know whether these residual errors are due to errors in the NGA truth data, we will account for them in the availability analysis, assuming

6

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FIGURE 3

Folded CDF (empirical and bounding Gaussian) for the orbit radial error component of the best-fit bias-plus-ramp model

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FIGURE 4

Folded CDF (empirical and bounding Gaussian) for the orbit in-track error component of the best-fit bias-plus-ramp model

where the standard deviation of 0.056 m was obtained using the same formula as in Equation (5) but applied to ⁱε_RES,k, using the bounding standard deviations in the rightmost column of Table 1. Future work will include a detailed analysis of the time characteristics of ⁱε_RES,k, which, for now, is modeled as white noise for samples taken several minutes apart (we will use a 5-minute sample time interval in Section 6 later in this paper).

3.3 Multipath and antenna group delay error

Time-correlated raw code and carrier measurement errors due to multipath reflections are modeled as first-order Gauss-Markov processes (GMP) with time constant T_MP. Using this model, Chan et al.³⁰ provide an expression for the steady-state CSC multipath error and dual-frequency antenna group delay variance in terms of T_MP, the Hatch filter (HF) smoothing time constant T_HF, the HF sampling interval T_S, and the raw code and carrier error variances and , respectively. is given by

7

where α = T_HF/T_S and.

To date, ARAIM analyses have assumed T_HF = 100 seconds and an associated elevation-dependent model for . For GPS, the following equation is given in Working Group C¹:

8

where

is expressed in meters for SV i at time k

ⁱθ_k is the elevation angle in degrees

c_IF is the ionosphere-free measurement combination multiplier:

f_L1 and f_L5, respectively, are L1 and L5 frequencies.

Equation (8) accounts for the combination of multipath and antenna group delay.^31,32 Dual-frequency antennas assumed in this work have much lower group delay than single-frequency antennas.³³

A similar model is provided in Working Group C¹ for Galileo. Equation (8) was established using experimental data collected and analyzed in Murphy et al.¹⁸ using T_HF = 100 seconds and T_S = 0.5 second. We assume, based on experimental data analysis in Joerger et al.¹¹ and Parkinson and Spilker,²⁴ that is typically 100 times smaller than . Equation (7) still contains two unknowns: and T_MP.

Fortunately, a second set of experimental data was analyzed in Murphy et al.,¹⁹ which provides a different value of the CSC standard deviation using T_HF = 30 seconds. This second relationship provides the means to solve for both unknowns in Equation (7). It follows that

9

and

10

This model assumes a fixed T_MP even though T_MP values actually vary.³⁴ In future work, we will evaluate the sensitivity of our proposed batch ARAIM integrity risk bound to T_MP values (preliminary analyses suggest that the bound is not very sensititive to T_MP because ⁱε_T,k is the dominating term in the carrier phase error budget). In parallel, we will use the method in Langel³⁵ to derive an upper bound on the positioning error based on unknown, bounded values of T_MP.

3.4 Receiver noise

The CSC receiver noise standard deviation assumed in ARAIM¹ is given by

11

where is expressed in meters for SV i at time k

ⁱθ_k is the elevation angle in degrees

Receiver noise is time uncorrelated. We follow the same approach as for multipath errors in Equations (7) to (9), which is simplified because T_MP no longer needs to be determined and β = 0. Raw code and carrier standard deviations are thus expressed as follows:

12

Throughout this section, we made the assumption that the error models are robust and accurate over the entire 10- to 20-minute–long batch interval. References are cited in Joerger et al.¹¹ to support this assumption, but further experimental validation will have to be carried out in future work. Also, undetected and unrepaired cycle slips that impact both snapshot and batch ARAIM are not accounted for in this paper, the emphasis being on comparing snapshot to batch ARAIM. The only sources of error left unaddressed in Equation (2) are ⁱb_ρ,k and ⁱb_φ,k, which are dealt with in the next section.

4.1 Batch measurement equation

For each SV i, for i = 1,…,n, raw code and carrier measurements from times 1 to q are, respectively, stacked in q × 1 vectors ⁱρ and ⁱφ. These vectors are then arranged in a batch measurement vector:

13

This vector can be expressed in terms of state variables and measurement error vectors as follows:

14

where, for an example dual-constellation GPS/Galileo implementation, u is a vector of positions and GPS and Galileo receiver clock offsets at all times

η is an n × 1 vector of cycle ambiguities

s_ERR is a 2n × 1 vector of constant error states

0_{a × b} is an a × b matrix of zeros

v_T,E,RNM,φ and v_T,E,RNM,ρ respectively, are raw code and carrier errors due to residual troposphere delay, residual clock and orbit ephemeris errors, receiver noise, and multipath.

Vector u is constructed using the following equations:

15

where τ_GPS,k and τ_GAL,k are the receiver clock offsets for GPS and Galileo, respectively, assuming that the time offset between the two constellations is unknown.

The vector of constant cycle ambiguities is defined as follows:

16

The vector of error states is given by

where b_E and g_E are the n × 1 vectors of constant clock and orbit ephemeris biases and gradients for all n satellites constructed following the exact same pattern as η in Equation (16).

Error states s_ERR are included in the state vector, not because their estimated values are of particular interest but because state augmentation is a practical way to incorporate measurement error dynamics. Prior knowledge on these error states is captured in a diagonal state information matrix (inverse of covariance matrix) with diagonal elements:

where1_{a × b} is an a × b matrix of ones.

This a priori knowledge of state estimate errors can directly be incorporated in the estimator by adding up information matrices as described in Section 2.1.2 of Joerger et al.,¹¹ or it can be included by measurement vector augmentation, ie, assuming pseudo-measurements on s_ERR as in Joerger et al.¹¹ and Joerger.³⁶

For clarity of presentation, the batch state coefficient matrices G, H_N, and H_ERR are constructed for example GPS/Galileo geometries where satellites are visible throughout the batch duration. Procedures to handle satellites coming in and out of sight are straightforward in batch implementations and are described in Joerger.³⁶

Position and receiver clock state coefficients for SV i at time k are arranged in the following vectors:

if SV i is a GPS satellite,

if SV i is a Galileo satellite, where line-of-sight vectors were defined in Equation (2). We then build geometry matrices over time, one satellite at a time:

17

and we stack these matrices for all satellites:

Next, the cycle ambiguity and error state coefficient matrices take the following forms:

. The measurement error covariance matrix V of vector requires additional steps to account for code/carrier measurement error correlation; both are described in Appendix B.

The 2qn × 1 vector represents nominal carrier and code biases for all SVs at all times for vector [φ^T ρ^T]^T. The elements of b_φ are significantly smaller than those of b_ρ because we assume that signal deformation does not affect the carriers. b_φ is included to account for non-Gaussian errors following the paired overbounding method.²¹ b_ρ and b_φ are unknown, but their elements are bounded by known values.

Next, we consider the batch measurement Equation (14), with an additive 2qn × 1 batch fault vector f, which will have to be detected. The resulting batch observation equation can be rewritten in a standard form as follows:

18

where z is the batch measurement vector

H is the batch observation matrix

x is the batch state vector

v is the batch measurement error vector:

v~N(0,V)b is the batch bias-bounded vector accounting for non-Gaussian components of errors:

4.2 Batch estimator design

The batch estimate for the state of interest (eg, for the vertical position coordinate, which is of primary interest in aircraft approach navigation) at the current time q is defined as follows:

19

where s₀ is the 2qn × 1 vector of batch estimator coefficients. Subscript “0” designates the all-in-view or “full-set” solution (the same notations, with additional details, are used in Joerger et al.³⁷) The estimator is a weighted least-squares estimator modified using Equations (35) to (37) in Working Group C³ to reduce integrity risk. The full-set estimate error is noted , where x is the true value of the state of interest. ε₀ is such that

20

where b₀ is the impact on state estimation of nominal bias-bounded errors. The ARAIM error model in Working Group C^1,2 assumes that the elements of b for CSC measurements can be bounded by a maximum value noted b_nom: b_nom = 0.75 m in MS2.² In this paper, we conservatively assume that nominal ranging biases can be different for all satellites at all times, that they mostly affect code due to signal deformation, and that they also account for small biases in the common clock and orbit ephemeris paired bounding process. It follows that b₀ can be bounded by

21

where || denotes the element-wise absolute value operator. Matrix A is defined as follows:

where I_a is the a × a identity matrix.

4.3 Batch detector design

A multiple-hypothesis solution separation (MHSS) batch RAIM method^1-3,12,37 is adopted for detection of f. Let h be the number of fault hypotheses that need to be monitored against (refer to Working Group C³ and Joerger et al.³⁷ for details on how to determine h). A set of mutually exclusive, exhaustive hypotheses H_i, for i = 0,…,h, is considered. Under H_i, a number n_i of measurements are simultaneously impacted by the fault. The fault-free subset solution, which excludes these n_i measurements, is written as follows: , where s_i is the 2qn × 1 vector of the subset solution’s batch estimator coefficients with zeros for elements corresponding to the n_i faulted measurements.³⁷ Under H_i, the estimation error ε_i of is such that

22

where

23

The batch MHSS test statistics are then defined as follows:

24

Δ_i is normally distributed with variance .³ In snapshot ARAIM, hypotheses H_i, for i = 1,…,h, are defined by satellites being faulted or not. Batch ARAIM additionally opens the possibility for a sub-subset of data being either faulted or fault free over time. It is then worth asking whether a larger number of measurements being faulted over time necessarily cause a higher integrity risk. On the one hand, more faulted measurements may increase the estimation error, but, on the other hand, it may also provide more opportunities for detection.

4.4 Computationally efficient batch implementation

In this section, two additional parameters are introduced. Let T_B be the batch period, ie, the finite time interval over which measurements are processed: T_B determines the amount of change in SV geometry (eg, we will use T_B = 10 minutes and T_B = 20 minutes in the next sections of the paper). Also, let T_S be the sampling interval, ie, the time between raw samples within the batch (eg, T_S = 0.5 second).

Batch Equations (13) to (25) assume that multi-GNSS measurements are processed at regular T_S intervals over T_B. The dimensions of the resulting raw batch matrices and vectors are therefore very large, and matrix inversions, needed, for example, to calculate s₀ in Equation (19), are extremely time-consuming. Alternatives to batches were considered, including Kalman filters (KF). But KF innovation-based RAIM requires computation of the P_HMI maximizing failure mode slope,^38,39 and KF-based MHSS requires banks of KFs, which is cumbersome.¹²

In response, we derive a computationally efficient, reduced batch implementation, illustrated in Figure 5, based on raw carrier phase and CSC measurements taken at few, infrequent sample times during T_B. This algorithm exploits the same two fundamental estimation principles as the raw batch: (a) Code noise is averaged out using CSC obtained from Hatch filters (HF); (b) redundant satellite motion is captured using raw carrier phase measurements at few, temporally separated sample times. This implementation is a straightforward augmentation of snapshot ARAIM that incorporates a few samples over time instead of a set of data at one instant in time.

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FIGURE 5

Computationally efficient batch ARAIM implementation

Let T_RBS be the reduced batch sampling interval within the batch period T_B. Because in this reduced batch process, noise averaging is performed in separate HFs, T_RBS can be selected much larger than the raw measurement sampling period T_S (T_RBS >> T_S).

For example, in Section 5, we use T_B = 10 minutes and T_RBS = 5 minutes. The resulting number of sample times drops from q = 1200 for the raw batch (for T_B = 10 minutes, T_S = 0.5 second) down to q = 3 for the reduced batch.

The reduced batch ARAIM MHSS method is the same as for raw batch in Equations (13) to (25), except for two differences. First, Equation (14) becomes

25

where q is the number of sample times, much smaller for the reduced batch than for the raw batch

is a qn × 1 vector of CSC measurements

is a qn × 1 vector of residual ephemeris and tropospheric error and of CSC receiver noise and multipath error

Then, off-diagonal elements of the covariance matrix of vector account for the correlation of raw carrier measurements with CSC. Complete analytical expressions of the correlation between CSC and raw carrier measurements are given in Appendix C. It is worth noting that the 5-minute T_RBS value used in the next sections of this paper is very large as compared to the multipath correlation time constant (T_MP = 80 seconds) and to the HF smoothing time constant (T_HF = 100 seconds). It follows, as explained in Appendix C, that the expression of can be simply and accurately approximated using blocks of diagonal matrices.

In summary, the reduced batch ARAIM MHSS method presents the following characteristics.

• a. It is much more computation and memory efficient than the raw batch.

• b. Since it is also based on CSC, it only requires a minor augmentation of the existing snapshot ARAIM algorithms.

The reduced batch implementation in Figure 5 is used in Section 6. From now on, it is referred to as “reduced-batch ARAIM” or simply “batch ARAIM,” as opposed to the “raw batch” in Equation (14), which will be used shortly to analyze raw measurement error models.

4.5 Integrity and continuity risk evaluation

The integrity risk, or probability of hazardous misleading information P_HMI, is upper bounded as follows³⁷:

26

where ℓ is the alert limit (AL) that defines hazardous situations: in MS2,² the vertical AL is ℓ =35m

P_Hi is the prior probability of H_i occurrence

H₀ is the fault-free hypothesis

H_i for i = 1,…,h are the fault hypotheses corresponding to faults on subset measurement i (including single-satellite and multi-satellite faults)

Under fault-free hypothesis H₀, the detection threshold τ_i is set based on an allocated continuity risk requirement C_REQ (specified in Working Group C^2,3) to limit the probability of false alarms. τ_i can be defined as follows:

27

where the function Q⁻¹{} is the inverse tail probability distribution of the standard normal distribution.

The following section analyzes the probability distributions of ε₀, ε_i, and Δ_i for raw batch versus snapshot ARAIM and for varying values of T_B.

5.1 “Frozen geometry” analysis: Raw versus CSC measurement error model comparison

The comparison between snapshot ARAIM and raw batch ARAIM in Figure 2 illustrates that the main difference between the two approaches is the use of SV motion. Assuming a constant, “frozen geometry” should therefore provide identical performance for both implementations. This observation can be used to confirm that the raw measurement error models in Equations (1) to (12) are consistent with ARAIM CSC models in the ARAIM MS1-2.^1,2

To do so, the transient response of the Hatch filter (HF) must also be accounted for, because CSC error models in^1,2MS2[2], Murphy et al.¹⁹ and De Cleene²⁰ are specified assuming that the HF has reached steady state. The steady-state CSC multipath error standard deviation is expressed in Equation (7) in terms of α = T_HF/T_S. In parallel, in appendix A of Joerger,³⁶ the variance of the raw batch position estimate assuming frozen geometry is expressed in terms of the number of samples n_S (n_S=T_B/T_S). These two expressions are used to find that equivalent performance is expected when T_B = 2T_HF, ie, T_B = 200 seconds since T_HF = 100 seconds.¹⁹

In the following Figures 6–8, the fault-free means and standard deviations of the random variables in Equation (26) are evaluated over 24 hours at an example Chicago location (25.5^°N, −80.1^°E), assuming dual-frequency measurements from GPS and Galileo. Nominal simulation parameters are further described in the next section where they become relevant. The means and standard deviations for raw batch ARAIM are σ₀, b₀, σ_i, b_i, and σ_Δi, which are defined in Equations (20) to (25). The same quantities for snapshot ARAIM are, respectively, noted , and . The legend is the same in all three figures. Figure 6 shows the results for T_B = 200 seconds, for frozen geometries. Snapshot versus raw batch results match closely, which confirms that the raw measurement error models derived earlier are consistent with ARAIM assumptions in MS2.²

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FIGURE 6

Comparing snapshot and raw batch ARAIM to show that raw measurement error models are consistent with ARAIM assumptions on CSC in Working Group C^1,2 (frozen geometry, T_B = 200 seconds)

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FIGURE 7

Impact of satellite motion: all parameters identical to Figure 6, but geometry is no longer frozen (T_B = 200 seconds)

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FIGURE 8

Impact of T_B: all parameters identical to Figure 7, except T_B = 600 seconds

5.2 Impact of satellite motion

The fault-free estimation error means and standard deviations are plotted in Figure 7 assuming the same simulation parameters as in Figure 6, but the geometry is no longer frozen. Figure 7 shows the substantial impact of satellite motion over T_B = 200 seconds on estimation performance. It confirms that bias observability using SV motion is a performance-driving principle. This effect is further accentuated in Figure 8 (especially for b₀ and b_i), where the batch interval is increased from T_B = 200 seconds to T_B = 600 seconds.

Figure 9 also shows the significant P_HMI reduction obtained using raw batch ARAIM as compared to snapshot ARAIM. In Figure 9, the fraction of time where the P_HMI curves are below the dashed line is the availability. In this case, availability is 99.7% for raw batch ARAIM and is much lower for snapshot ARAIM (65%). In addition, Figure 9 shows that raw batch ARAIM and reduced batch ARAIM have overlapping P_HMI curves, even though reduced batch ARAIM is much more computation and memory efficient.

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FIGURE 9

Integrity risk bound using snapshot vs batch ARAIM: all parameters identical to Figure 8 (T_B = 600 seconds, T_RBS = 300 seconds)

The integrity risk P_HMI is plotted in Figure 9 for raw batch ARAIM (T_B = 600 seconds, T_S = 0.5 seconds) and for reduced batch ARAIM (T_B = 600 seconds, T_RBS = 300 seconds), over 24 hours, at the example Chicago location. The dashed horizontal line in Figure 9 is the integrity risk requirement I_REQ, specified in Working Group C² to be I_REQ = 0.98 · 10⁻⁷ for the vertical position coordinate. Figure 9 also assumes that the vertical alert limit in Equation (24) is ℓ = 10m, that prior probabilities of satellite and constellation faults, noted P_Hi in Equation (24), respectively, are P_sat = 10⁻⁵ and P^const = 10⁻⁸, and that the continuity risk requirement in Equation (27) is C_REQ = 3.9 · 10⁻⁶.

6.1 Batch vs snapshot ARAIM availability using depleted constellations

Figures 10 and 11 display availability maps for a 10^° × 10^° latitude-longitude grid of locations, for “depleted” 24-1 GPS and 24-1 Galileo constellations, for satellite geometries simulated at regular 10-minute intervals over a 24-hour period. Availability is computed at each location as the fraction of time where the P_HMI bound is lower than I_REQ.

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FIGURE 10

Availability map for snapshot ARAIM using depleted constellations, P_const =10⁻⁴, ℓ = 35 m (coverage of 99.5% availability is 87.2%)

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FIGURE 11

Availability map for reduced batch ARAIM (T_B = 600 seconds, T_RBS = 300 seconds) using depleted constellations, P_const = 10⁻⁴, ℓ = 35 m (coverage of 99.5% availability is 93.3%)

In the figures, availability is color coded: White corresponds to a value of 100%, and black represents 80%. Constant availability contours are also displayed. The gray areas in Figure 10 indicate that snapshot ARAIM is clearly outperformed by batch ARAIM in Figure 11. Even more substantial differences will be observed in the next subsection.

The worldwide availability metric given in the figure captions is the weighted coverage of 99.5% availability: Coverage is defined as the percentage of grid point locations exceeding 99.5% availability. The coverage computation is weighted at each location by the cosine of the location’s latitude because grid point locations near the equator represent larger areas than near the poles. Figures 10 and 11 show that the coverage of 99.5% availability increases from 87.2% for snapshot ARAIM to 93.3% for batch ARAIM, assuming T_B = 600 seconds for ℓ = 35 m. For context, based on the assumptions described in Section 4 of MS3,³ coverage of 99% availability using Satellite Based Augmentation Systems (SBAS) is 11% as of 2015 and may reach 45% in 2026 using dual-frequency SBAS.

6.2 Batch vs snapshot ARAIM availability for a 10-m alert limit

Figures 12 and 13 evaluate the potential of batch ARAIM to meet requirements that are more stringent than LPV200, including a tight alert limit: ℓ = 10 m. One key assumption to achieve high availability is that the prior probability of constellation-wide faults must be reduced to P_const = 10⁻⁸. This assumption is also made for GPS in en route operation using horizontal ARAIM (see appendix B in Working Group C²) and may be justifiable using the guidelines in Walter and Blanch.⁴¹ These guidelines exploit additional information from the Air Navigation Service Provider (ANSP) ground segment. In Figures 12 and 13, availability maps assume nominal 24-satellite GPS and Galileo constellations. The same color code as in Figures 10 and 11 is employed. Again, batch ARAIM with T_B = 1200 seconds in Figure 13 provides a dramatic improvement as compared to snapshot ARAIM in Figure 12.

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FIGURE 12

Availability map for snapshot ARAIM using nominal constellations, P_const = 10⁻⁸, ℓ = 10 m (coverage of 99.5% availability is 0%)

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FIGURE 13

Availability map for reduced batch ARAIM (T_B = 1200 seconds, T_RBS = 300 seconds) using nominal constellations, P_const = 10⁻⁸, ℓ = 10 m (coverage of 99.5% availability is 75.5%)

Table 2 lists worldwide coverage of 99.5% availability and of 95% availability (given in parentheses), for the above configurations, for T_B = 600 seconds and assuming T_B = 1200 seconds, and for depleted, nominal, and optimistic constellations. The table quantifies the global performance improvement brought by batch ARAIM as compared to snapshot ARAIM (snapshot ARAIM results match the coverage reported in tables 2 and 3 in Working Group C² for ℓ = 35 m). This improvement comes at the cost of slightly higher computation and memory loads.

Further improvement can be obtained by lengthening the batch duration. For example, for T_B = 1800 seconds, ℓ = 10 m, P_const = 10⁻⁸, and assuming a nominal constellation, the coverage of 99.5% availability reaches 95% as compared to 75.5% using T_B = 1200 seconds and to 0% for snapshot ARAIM. But the batch duration is limited by mission duration and by the period of validity of the error models. Another source of improvement is the potential future addition of other GNSS constellations (eg, GLONASS and BeiDou), which will increase the accumulated impact of redundant satellite motion. In this case, new measurement error and fault models would have to be derived and validated because current assumptions on GPS and Galileo performance may not hold for other constellations (eg, the definition of GLONASS satellite faults is not rigorously consistent with that of GPS⁴²).

It must also be noted that, for aircraft navigation standards that are more stringent than LPV-200 and that include ℓ = 10 m, additional requirements are typically involved, eg, on the communication link between ANSP ground segment and aircraft. Such considerations are beyond the scope of this paper.