2.1. Integrity and Continuity Risk Definitions

The integrity risk, or probability of Hazardous Misleading Information (HMI), is a joint probability defined as:

(1)$${P_{HM{I^{^\ast}}}} \equiv P\left( { \vert {\varepsilon _0} \vert \gt \ell, \vert q \vert \lt T} \right)$$

where ε₀ is the error on the estimated parameter of interest (called ‘state’ of interest), ℓ  is a specified alert limit that defines hazardous situations (e.g., specified in (RTCA Special Committee 159, 2004) for aircraft approach navigation), q is the detection test statistic and T is the detection threshold.

Considering a set of n + 1 mutually exclusive, jointly exhaustive hypotheses H _i, the law of total probability can be used to express the integrity criterion as:

(2)$${P_{HMI}} = \sum\limits_{i = 0}^n {P\left( { \vert {\varepsilon _0} \vert \gt \ell, \vert q \vert \lt T \vert {H_i}} \right){P_{Hi}} \le {I_{REQ}} - {P_{NM}}} $$

where, in order to avoid introducing new notations, P _HMI (P _HMI = P _HMI* − P _NM) will be referred to in the remainder of this work (both in Parts 1 and 2) as the integrity risk, and where I _REQ  is the integrity risk requirement (also specified in RTCA Special Committee 159 (2004) for example aviation applications). P _NM  is the prior probability of two or more simultaneous faults that need not be monitored against (such that P _NM ≪ I _REQ). P _NM is further described below, P _Hi is the prior probability of H _i occurrence, H ₀ is the fault-free hypothesis, H _i  for i = 1, …, n are the fault hypotheses corresponding to faults on satellite measurement ‘i’ and n  is the number of Space Vehicles (SV) in view, i.e., the number of single-SV faults.

This work addresses single-satellite faults, and assumes that faults occur rarely and independently across SVs, so that P _NM is small with respect to I _REQ. Equation (2) expresses the integrity risk for the n fault hypotheses that must be monitored against, and conservatively accounts for the remaining fault combinations using their prior probability of occurrence P _NM (as established in Blanch et al. (Reference Blanch, Walter, Enge, Wallner, Fernandez, Dellago, Ioannides, Hernandez, Belabbas, Spletter and Rippl2013b) and Joerger et al. (Reference Joerger, Chan and Pervan2014)).

Under the fault-free hypothesis H ₀, the detection threshold T is typically set based on an allocated continuity risk requirement C _REQ (e.g., also specified in (RTCA Special Committee 159, 2004) for aviation applications) to limit the probability of false alarms (Sturza, Reference Sturza1988). T can be defined as:

(3)$$P\left( { \vert q \vert \ge T \vert {H_0}} \right){P_{H0}} \le {C_{REQ}}$$

In addition, let n and m respectively be the number of measurements and number of parameters to be estimated (i.e., the ‘states’) and let ${{\bf z}_{\ast}}$ be the n × 1 vector of stacked measurements. This work assumes that the Cumulative Distribution Function (CDF) of nominal measurement errors is bounded by a zero mean Gaussian distribution with covariance matrix ${{\bf V}_{\ast}}$ (DeCleene, Reference DeCleene2000). Vector ${{\bf z}_{\ast}}$ is pre-multiplied by ${\bf V}_{\ast}^{ - 1/2} $ to obtain the ‘normalised’ measurement equation:

(4)$${\bf z} = {\bf Hx} + {\bf v} + {\bf f}$$

where ${\bf z} = {\bf V}_{\ast}^{ - 1/2} {{\bf z}_{\ast}}$ is the normalised measurement vector, H  is the n × m normalised observation matrix, x is the m × 1 state vector, f is the n × 1 normalised fault vector and v  is the n × 1 normalised measurement noise vector composed of zero-mean, unit-variance independent and identically distributed (i.i.d.) random variables.

We use the notation:

(5)$${\bf v}\sim{\rm N}({{\bf 0}_{n \times 1}},{{\bf I}_n})$$

where ${{\bf 0}_{a \times b}}$ is an a × b matrix of zeros (in this case, it is an n × 1 vector of zeros) and ${{\bf I}_n}$ is an n × n identity matrix.

In order to avoid making assumptions on unknown fault distributions, an upper bound on the probability of HMI can be evaluated for the worst-case single-SV fault magnitude f _i under H _i:

(6)$$P\left( { \vert {\varepsilon _0} \vert \gt \ell, \vert q \vert \lt T \vert {H_i}} \right) \le \mathop {\max} \limits_{{\,f_i}} P\left( { \vert {\varepsilon _0} \vert \gt \ell, \vert q \vert \lt T \vert {\,f_i}} \right)$$

The worst-case fault magnitude f _i, which maximizes the integrity risk given H _i, is found using a straightforward line search algorithm (e.g., used in Lee (Reference Lee1995), Milner and Ochieng (Reference Milner and Ochieng2010) and Jiang and Wang (Reference Jiang and Wang2014)). For the fault-free case (i = 0), the following notation is used: f ₀ = 0. The bound on the right hand side in Equation (6) will be implemented in Section 3 of this paper, and for analytical purposes in Part 2 in Joerger et al. (Reference Joerger, Langel and Pervan2015). Also in Part 2, for practical implementations, an alternative bound will be derived, which does not require determination of f _i.

2.2. Baseline Least-Squares Estimator and Detectors Used in RAIM

This section describes the estimate error ε₀ obtained using a Least-Squares (LS) estimator, and the test statistics q for two of the most widely-implemented detection methods in RAIM: chi-squared (χ ²) RAIM and Solution Separation (SS) RAIM.

Let x be the state of interest, for example the vertical position coordinate, which is often of primary concern for aircraft approach navigation. Let α be an m × 1 vector used to extract x out of the full state vector:

(7)$${{\bi \alpha} ^{\rm T}} = \left[ {\matrix{ {{\bf 0}_{{m_A} \times 1}^T} & 1 & {{\bf 0}_{{m_B} \times 1}^T} \cr}} \right]$$

where, in the order in which states are stacked in x, m _A and m _B are the number of states respectively before and after state x. Assuming that H is full rank and that n ≥ m, the LS estimate of x is defined as:

(8)$${\hat x_0} \equiv {\bf s}_0^T {\bf z}$$

where

(9)$${\bf s}_0^T = {{\bi \alpha} ^T}{{\bf P}_0}{{\bf H}^T},\quad {\rm and}\quad {{\bf P}_0} = {\left( {{{\bf H}^T}{\bf H}} \right)^{ - 1}}$$

The LS estimate error appearing in the integrity risk Equation (2) is defined as:

(10)$${\varepsilon _0} \equiv {\hat x_0} - x = {\bf s}_0^T ({\bf v} + {\bf f})$$

(11)$${\varepsilon _0}\sim{\rm N(}{\bf s}_0^T {\bf f},\;\sigma _0^2 \equiv {{\bi \alpha} ^T}{{\bf P}_0}{\bi \alpha} )$$

In parallel, the χ ² RAIM detection test statistic is derived from the (n − m) × 1 parity vector p, which lies in the (n − m)-dimensional parity space, or left null space of H, and can be expressed as (Sturza, Reference Sturza1988; Potter and Suman, Reference Potter and Suman1977):

(12)$${\bf p} \equiv {\bf Qz} = {\bf Q}({\bf v} + {\bf f})$$

where the (n − m) × n parity matrix Q is defined as:

(13)$${\bf Q}{{\bf Q}^T} = {{\bf I}_{n - m}}\quad {\rm and}\quad {\bf QH} = {{\bf 0}_{(n - m) \times m}}$$

The χ ² RAIM detection test statistic is the square of the norm of p, and can be written as (Sturza, Reference Sturza1988; Potter and Suman, Reference Potter and Suman1977):

(14)$$q_\chi ^2 \equiv {{\bf p}^T}{\bf p}$$

$q_\chi ^2 $ follows a non-central chi-square distribution with (n − m) degrees of freedom and non-centrality parameter $\lambda _\chi ^2 $ (Potter and Suman, Reference Potter and Suman1977). The following notation is used:

(15)$$q_\chi ^2 \sim{\chi ^2}\left( {n - m,\lambda _\chi ^2} \right)$$

where

(16)$$\lambda _\chi ^2 = {{\bf f}^T}{{\bf Q}^T}{\bf Qf}$$

As an alternative to χ ² RAIM, a second integrity monitoring method called SS RAIM (Brenner, Reference Brenner1996; Blanch et al., Reference Blanch, Ene, Walter and Enge2007) is investigated. Without loss of generality, it is assumed that under the single-SV fault hypothesis H _i, the faulty measurement is the first element of z. The measurement Equation (4) can be partitioned following the equation:

(17)$$\left[ {\matrix{ {{\bf A}_i^T {\bf z}} \cr {{\bf B}_i^T {\bf z}} \cr}} \right] = \left[ {\matrix{ {{\bf A}_i^T {\bf H}} \cr {{\bf B}_i^T {\bf H}} \cr}} \right]{\bf x} + \left[ {\matrix{ {{\bf A}_i^T {\bf v}} \cr {{\bf B}_i^T {\bf v}} \cr}} \right] + \left[ {\matrix{ {{\bf A}_i^T {\bf f}} \cr {{{\bf 0}_{(n - 1) \times 1}}} \cr}} \right]$$

with

(18)$${{\bf A}_i} \equiv \left[ {\matrix{ 1 \cr {{{\bf 0}_{(n - 1) \times 1}}} \cr}} \right]\quad {\rm and}\quad {{\bf B}_i} \equiv \left[ {\matrix{ {{{\bf 0}_{(n - 1) \times 1}}} \cr {{{\bf I}_{n - 1}}} \cr}} \right]$$

Equation (17) is employed to distinguish the full-set solution ${\hat x_0}$ in Equation (8), obtained using all n measurements in z, from the subset solution ${\hat x_i}$, derived using only the (n − 1) fault-free measurements ${\bf B}_i^T {\bf z}$ under H _i. Under the assumption that n − 1 ≥ m, and assuming that ${\bf B}_i^T {\bf H}$ is full rank, ${\hat x_i}$ is defined as:

(19)$${\hat x_i} \equiv {\bf s}_i^T {\bf z},\quad {\rm for}\quad i = 1, \ldots, n$$

where

(20)$${\bf s}_i^T = {{\bi \alpha} ^T}{{\bf P}_i}{{\bf H}^T}{{\bf B}_i}{\bf B}_i^T, \quad {\rm and}\quad {{\bf P}_i} = {\left( {{{\bf H}^T}{{\bf B}_i}{\bf B}_i^T {\bf H}} \right)^{ - 1}}$$

It follows that, under H _i, the estimate error ε_i is given by:

(21)$${\varepsilon _i} \equiv {\bf s}_i^T {\rm (}{\bf v} + {\bf f}{\rm )}\sim{\rm N(}0,\;\sigma _i^2 \equiv {{\bi \alpha} ^T}{{\bf P}_i}{\bi \alpha} )$$

The solution separations are defined as (Lee, Reference Lee1986; Brenner, Reference Brenner1996):

(22)$${\Delta _i} \equiv {\hat x_0} - {\hat x_i} = {\varepsilon _0} - {\varepsilon _i},\quad {\rm for}\quad i = 1, \ldots, n$$

Δ_i can also be expressed as (Blanch et al., Reference Blanch, Ene, Walter and Enge2007; Joerger et al., Reference Joerger, Chan and Pervan2014):

(23)$${\Delta _i} = {\bf s}_{\Delta i}^T {\bf z}\quad {\rm and}\quad {\Delta _i}\sim{\rm N}\left( {{\bf s}_{\Delta i}^T {\bf f},\sigma _{{\Delta _i}}^2} \right)\quad {\rm with}\quad \sigma _{\Delta i}^2 = \sigma _i^2 - \sigma _0^2 $$

where

(24)$${{\bf s}_{\Delta i}} = {{\bf s}_0} - {{\bf s}_i}$$

In addition, the normalised solution separations can be defined as:

(25)$${q_i} \equiv {\Delta _i}/{\sigma _{\Delta i}}\sim{\rm N}\left( {{\bf s}_{\Delta i{^\ast}}^T {\bf f},1} \right)\quad {\rm for}\quad i = 1, \ldots, n$$

where

(26)$${{\bf s}_{\Delta i{^\ast}}} = {{\bf s}_{\Delta i}}/{\sigma _{\Delta i}}$$

Of particular significance in this work is the fact that for single-measurement faults, q _i can be written in terms of the parity vector p as (Joerger et al., Reference Joerger, Chan and Pervan2014):

(27)$${q_i} = {\bf s}_{\Delta i{^\ast}}^T {\bf z} = {\bf u}_i^T {\bf p}$$

where

(28)$${{\bf u}_i} = \displaystyle{{{\bf Q}{{\bf A}_i}} \over {\sqrt {{\bf A}_i^T {{\bf Q}^T}{\bf Q}{{\bf A}_i}}}} \quad {\rm for}\quad i = 1, \ldots, n$$

with

(29)$${\bf A}_i^T = \left[ {\matrix{ {{\bf 0}_{(i - 1) \times 1}^T} & 1 & {{\bf 0}_{(n - i) \times 1}^T} \cr}} \right]$$

i.e., ${\bf Q}{{\bf A}_i}$ is the i ^th column of Q. Vector ${{\bf u}_i}$ is the unit direction vector of the i ^th ‘fault mode line’, which can be represented in parity space (see Figure 1 in the next paragraphs) to visualise the noise-free impact on p of a fault on the i ^th measurement with magnitude varying from − ∞ to + ∞. Equation (27) expresses the fact that the n solution separations are projections of the parity vector on their corresponding fault mode lines.

Equations (14) and (27) show that both q _χ and q _i are derived from the parity vector p. It has been proved in Joerger et al. (Reference Joerger, Chan and Pervan2014) that for any test statistic q lying in the parity space, the least-squares estimate error ε₀ and q are statistically independent, so that the joint probability in Equation (6) can be expressed as:

(30)$$P\left( { \vert {\varepsilon _0} \vert \gt \ell, \vert q \vert \lt T \vert {\,f_i}} \right) = P\left( { \vert {\varepsilon _0} \vert \gt \ell, \vert {\,f_i}} \right)P\left( { \vert q \vert \lt T \vert {\,f_i}} \right)$$

The integrity risk bound obtained by substituting Equation (30) into Equation (6), and the result into Equation (2) is used in the remainder of this paper for both χ ² and SS RAIM. This P _HMI -bound differs from typical SS RAIM approaches (Brenner, Reference Brenner1996) because: (a) it does not use a position-domain-bound to account for the worst-case fault, and (b) it requires that the correlation between SS test statistics be evaluated in Section 3. A complete derivation of a conventional SS RAIM approach starting from Equation (1) can be found in Joerger et al. (Reference Joerger, Chan and Pervan2014). Based on this common definition for χ ² and SS RAIM, differences in P _HMI, or more precisely in probability of no-detection P(|q| <T | f _i), can be displayed in parity space.

2.3. Parity Space Representations for an Illustrative Single-State Example

In this section, both χ ² RAIM and SS RAIM are represented in parity space for an illustrative single-state example used in Potter and Suman (Reference Potter and Suman1977) and Joerger et al. (Reference Joerger, Chan and Pervan2014). Let us consider a scalar state x and a 3 × 1 measurement vector z that are expressed as:

(31)$${\bf z} = {\bf H} x + {\bf v} + {\bf f}$$

where

(32)$${\bf H} = {\left[ {\matrix{ 1 & 1 & 1 \cr}} \right]^T}\quad {\rm and}\quad {\bf v}\sim{\rm N}({{\bf 0}_{3 \times 1}},{{\bf I}_3})$$

Since m = 1 and n = 3, the (n − m) parity space is two-dimensional, which is convenient for display. The fault vector f represents three single-measurement faults, corresponding to three fault hypotheses H _i, for i = 1, …, 3, with unknown fault magnitude f _i.

(33)$${\bf f} = \left[ {\matrix{ {{\,f_1}} \cr 0 \cr 0 \cr}} \right]\quad {\rm or}\quad {\bf f} = \left[ {\matrix{ 0 \cr {{\,f_2}} \cr 0 \cr}} \right]\quad {\rm or}\quad {\bf f} = \left[ {\matrix{ 0 \cr 0 \cr {{\,f_3}} \cr}} \right]$$

Their three fault mode lines, with direction vectors ${{\bf u}_i}$ defined in Equation (28), are represented in Figure 1.

Figure 1.

Detection boundaries for χ ² (Circle) and SS (Hexagon) in parity space.

For χ ² RAIM, the probability of no detection, which is the second term of the product in Equation (30), is given by: $P(q_\chi ^2 \lt T_\chi ^2 \vert {f_i})$. In contrast, for SS RAIM, the probability of no detection is a joint probability given by:

(34)$$P\left( { \vert {q_1} \vert \lt {T_1}, \vert {q_2} \vert \lt {T_2}, \vert {q_3} \vert \lt {T_3} \vert {\,f_i}} \right)$$

It follows from Equations (14), (27) and (34) that the detection boundaries for χ ² and SS RAIM are a circle (or a hyper-sphere in higher-dimensional parity space) and a polygon (or a polytope), respectively.

Consistent with Equation (3), the SS detection thresholds can be computed using the following equation: $P\left( {\vert {q_1}\vert \ge {T_1} \cup \vert {q_2}\vert \ge {T_2} \cup \vert {q_3}\vert \ge {T_3}\vert {H_0}} \right) = {C_{REQ}}$ (where ‘$ \cup $’ captures a union of events, whereas commas in Equation (34) represent an intersection), for which a close approximation is given by:

(35)$${T_i} = {Q^{ - 1}}\left\{ {\displaystyle{{{C_{REQ,i}}} \over {2{P_{H0}}}}} \right\}\quad {\rm for}\quad i = 1, \ldots, n$$

(36)$${C_{REQ}} = \sum\limits_{i = 1}^h {{C_{REQ,i}}}, \quad {\rm e}{\rm. g}{\rm.,} \;{C_{REQ,i}} = \displaystyle{{{C_{REQ}}} \over n}$$

where the function Q ⁻¹{} is the inverse tail probability distribution of the two-tailed standard normal distribution (Q{} = 1 − Φ{}, where Φ{} is the standard normal cumulative distribution function).

Blanch et al. (Reference Blanch, Ene, Walter and Enge2007) suggests that for a practical application the difference is negligible between the probability of no detection for an optimised continuity risk requirement allocation versus the equal allocation expressed in Equation (36). In this example, because all three measurements contribute equally to estimation and detection, and because each fault occurrence is assumed equally likely (P _H1 = P _H2 = P _H3), the optimal C _REQ-allocation actually is the equal allocation in Equation (36). Following Equation (36), if the SS detection thresholds are all equal (T ₁ = T ₂ = T ₃), then the polygon is a hexagon. In Figure 1, detection happens if the parity vector p lands outside the detection boundary. Therefore, the probability of no detection is the probability of being inside the dash-dotted circle for χ ² RAIM, and inside the hexagon for SS RAIM.

Thus, χ ² RAIM and SS RAIM each generate a different detection boundary, but it is unclear which one actually provides the highest probability of detection under a given fault hypothesis. Section 3 will show that the answer to this question depends on the navigation requirements ℓ and C _REQ.

3.1. Optimal Piecewise Linear Detection Boundary

Blanch et al. (Reference Blanch, Walter and Enge2013a) recently developed an approach, based on the Neyman-Pearson (NP) theorem, to visualize the optimal no-detection region by determining the set of points in parity space that contribute to reducing the integrity risk. This NP approach does not include a process to evaluate the integrity risk P _HMI.

In contrast, in this section, a Piecewise Linear (PL) boundary is implemented to approximate the optimal detection region, which minimises the integrity risk. In this case, P _HMI can be evaluated using an algorithm similar to SS RAIM, which is described below.

With regard to computation, the two approaches are intensive: the NP method involves sampling the parity space point by point, whereas the PL approximation determines a finite number of parameters based on the P _HMI -bound defined in Section 2. The processing load for the PL approach can be reduced in future work using looser P _HMI -bounds.

In this paper, the NP and PL algorithms are exploited to make an informed decision on whether to use χ ² versus SS test statistics, depending on navigation requirements. The focus is on χ ² and SS RAIM because these methods are computationally efficient and widely implemented.

The method's main idea for establishing the optimal detection boundary is illustrated in Figure 2 for the illustrative example described in Equations (31) to (33). To design a new detector, the method first uses the n SS test statistics (n = 3 in this case), which are projections of the parity vector p on the n fault lines (represented in grey). But the method also includes new test statistics, defined as projections of p onto lines selected at regular angular intervals (displayed in black), with unit direction vectors ${{\bf u}_4}$, ${{\bf u}_5}$ and ${{\bf u}_6}$.

As illustrated in Figure 1 with the hexagonal detection region for SS RAIM, each test statistic generates two segments of the piecewise linear boundary. Let d be the total number of detection test statistics. The higher d is, the finer the resolution of the detection boundary becomes. For clarity of exposition in the figures of this section, a total of d = 6 test statistics are considered, with detection thresholds T ₁, …, T _d.

Figure 2.

Concept of Piecewise Linear Detection Region.

With this structure in place, finding the optimal detection boundary narrows down to determining the threshold values T ₁, …, T _d that minimise the integrity risk in Equation (2), while meeting the false alarm requirement in Equation (3). This problem can be formulated as a constrained minimisation problem:

(37)$$\mathop {\min} \limits_{{T_1}, \ldots, {T_d}} \left\{ {\sum\limits_{i = 0}^n {\mathop {\max} \limits_{{\,f_i}} P\left( { \vert {\varepsilon _0} \vert \gt \ell \vert {\,f_i}} \right)P\left( { \vert {q_1} \vert \lt {T_1}, \ldots, \vert {q_d} \vert \lt {T_d} \vert {\,f_i}} \right){P_{Hi}}}} \right\}$$

subject to: 1 − P(|q ₁| <T ₁, …, |q _d| <T _d | H ₀)P _H0 ≤ C _REQ

The quantity to minimise is a bound on the integrity risk expressed in Equation (37) by substituting Equation (30) into Equation (6), and the result into Equation (2).

It is worth noting that the d test statistics q _i are mutually correlated, so that the joint probability in Equation (37) cannot be written as a product of probabilities. Still, this joint probability can be numerically evaluated by stacking the test statistics q _i in a d × 1 vector q. Let ${{\bf M}_U}$ be an (n − m) × d matrix such that ${{\bf M}_U} = {\left[ {\matrix{ {{{\bf u}_1}} & \ldots & {{{\bf u}_d}} \cr}} \right]^T}$, and let ${{\bf f}_i}$ be a n × 1 single-measurement fault vector whose only non-zero element is the ith element of value f _i. The d × 1 vector of test statistics q follows a multivariate normal distribution with covariance matrix ${\bf M}_U^T {{\bf M}_U}$ and with mean vector ${\bf M}_U^T {\bf Q}{{\bf f}_i}$ that can be evaluated given f _i. In this case, the joint-probability evaluation is accomplished by integration over a hyper-box with edge lengths defined by the thresholds T _i, for i = 1, …, d, following the methods in (Drezner and Wesolowsky, Reference Drezner and Wesolowsky1989) and (Genz and Bretz, Reference Genz and Bretz2002). As mentioned earlier, this multiple integration process is computationally expensive, and in this paper, is only used for analytical purposes in two illustrative examples.

Equation (37) describes the detection region as a union of half-planes, and is consistent with the assertion proved in Blanch et al. (Reference Blanch, Walter and Enge2013a), which states that the optimal detection boundary defines a closed, convex curve. Equation (37) is also written as a ‘mini-max’ problem. The maximisation is dealt with, in this paper, using a brute force approach based on multiple line-searches over f _i, for i = 1, …, n. These line searches are repeated at each iteration of the minimisation process.

The problem formulation in Equation (37) can be modified into an unconstrained minimisation problem using the method of Lagrange multipliers (by forcing the inequality constraint to equality), or alternatively using barrier or penalty methods (Luenberger and Ye, Reference Luenberger and Ye2008). The first option is selected here, and Equation (37) becomes:

(38)$$\mathop {\min} \limits_{{T_1}, \ldots, {T_d},\theta} \left\{ \matrix{\sum\limits_{i = 0}^n {\mathop {\max} \limits_{{\,f_i}} P\left( { \vert {\varepsilon _0} \vert \gt \ell \vert {\,f_i}} \right)P\left( { \vert {q_1} \vert \lt {T_1}, \ldots, \vert {q_d} \vert \lt {T_d} \vert {\,f_i}} \right){P_{Hi}}} \hfill \cr + \theta \left[ {{C_{REQ}} - 1 + P\left( { \vert {q_1} \vert \lt {T_1}, \ldots, \vert {q_d} \vert \lt {T_d} \vert {H_0}} \right)} \right] \hfill} \right\}$$

where θ is the Lagrange multiplier, to be simultaneously determined with the optimal values of T _i.

In Equation (38), the parameters to be optimised T ₁, …, T _d are limits of integration for the objective function. This facilitates the derivation in the Appendix, using the Leibniz integral rule, of analytical expressions for the objective function's gradient vector and Hessian matrix. An iterative Newton method can then be implemented to determine the optimal parameter values (Luenberger and Ye, Reference Luenberger and Ye2008). Convergence issues are beyond the scope of this paper since the optimisation method is not currently intended for practical implementation. The computational load of this piecewise-linear approximation will be readdressed in future work.

3.2. Sensitivity to Navigation Requirements

The optimal piecewise linear detection boundary is used here to make an educated choice between χ ² RAIM and SS RAIM. The canonical example in Equations (31) to (33) is considered again, with the example parameter values: ${P_{{H_i}}} = {10^{ - 3}}$ for i = 1, …, 3, and C _REQ = 10⁻³. The analysis is carried out for a range of values of the non-dimensional parameter ℓ/σ ₀ (which is the ratio of the alert limit over the estimate error standard deviation).

First, the optimal piecewise linear boundary is represented in Figure 3 (thick black lines) for a small value of ℓ/σ ₀ (ℓ/σ ₀ = 0.3). Segments of the optimal boundary are of equal length, so that the optimal boundary approaches a circle. This can be observed again in the zoomed-in window on the upper-right hand side of Figure 3, where the optimal boundary is close to the χ ² circular boundary, and far from the corner of the SS hexagon.

This result is confirmed using the Neyman-Pearson (NP)-based approach (Blanch et al., Reference Blanch, Walter and Enge2013a), which, by sampling the parity space point by point, provides a precise depiction of the optimal no-detection region (grey area in Figure 3). In Figure 3, the optimal no-detection region closely matches the χ ² RAIM circular boundary.

Thus, the χ ² RAIM method approaches the optimal method for ℓ/σ ₀ = 0.3. However, as ℓ/σ ₀ increases, the optimal detection boundary changes. Figure 4 shows that for ℓ/σ ₀ = 15, segments of the optimal piecewise linear boundary are no longer of equal length. The optimal region starts looking like a hexagon, which corresponds to the SS detection boundary. This can be seen again in the zoomed-in window where the optimal NP boundary starts filling in the corner of the SS hexagon.

Figure 3.

Optimal Piecewise Linear (PL) detection boundary for ℓ/σ ₀ = 0.3

Figure 4.

Optimal Piecewise Linear (PL) detection boundary for ℓ/σ ₀ = 15

The next two paragraphs describe the dominating factors causing the optimal detection region to change with ℓ/σ ₀. First, the constraint in the problem statement of Equation (37) is independent of ℓ/σ ₀. Thus the optimal area is only shaped by the objective function, i.e., by the integrity risk P _HMI, which is a sum of probabilities over multiple hypotheses. The probability density function for P _HMI is represented in parity space for ℓ/σ ₀ = 0.3 and ℓ/σ ₀ = 15 in Figures 5 and 6, respectively. In this multiple hypothesis formulation, the density function is a composite of seven peaks of probability density, corresponding to the fault-free hypothesis H ₀, and to the three fault hypotheses for positive and negative values of the fault magnitude. The continuity risk requirement defines an area near the origin corresponding to relatively small fault magnitudes, which are difficult to detect.

Figure 5 shows what appears to be a single peak in probability density for small values of ℓ/σ ₀. When ℓ/σ ₀ is small, the probability of the estimate error ε₀ exceeding ℓ rises rapidly, even for small, difficult-to-detect fault magnitudes. It follows that the worst-case fault magnitudes (which maximise P _HMI) remain close to zero, and the seven peaks are all brought together near the origin. In contrast, as ℓ/σ ₀ increases in Figure 6, small-magnitude faults do not cause hazardous information so that the worst-case fault magnitudes become larger, and the six probability density peaks corresponding to fault hypotheses spread apart along the fault mode lines. Therefore, as ℓ/σ ₀ increases, the optimal detection boundary changes from a circular area (χ ²⁾ to a polygon (SS).

Figure 5.

Density function of the probability of HMI for ℓ/σ ₀ = 0.3

Figure 6.

Density function of the probability of HMI for ℓ/σ ₀ = 15

In practice, the alert limit ℓ must be substantially greater than σ ₀ or the fault-free integrity risk alone (i.e., the term of the sum in Equation (2) for which i = 0) will exceed the integrity risk requirement I _REQ. Moreover, comparing the scales of the z-axes in Figures 5 and 6 provides a clue as to the difference in orders of magnitude of P _HMI as ℓ/σ ₀ changes. Therefore for most realistic practical requirements (i.e., for large values of ℓ/σ ₀), the optimal detection boundary approaches SS. This result completes and reinforces the findings of previous papers (Joerger et al., Reference Joerger, Chan and Pervan2014; Blanch et al., Reference Blanch, Walter and Enge2013a).

3.3. Optimal detection boundary for an Example GPS Satellite Geometry

In this section, the optimal NP and PL boundaries are derived for the example GPS satellite geometry displayed in Figure 7. In this case, the state vector x in Equation (4) comprises three position coordinates and one receiver clock bias. Since the number of satellite measurements n is six, and the number of states m is four, the parity space has dimension (n − m) = 2. Example parameter values are the same as in Section 3.2: ${P_{{H_i}}} = {10^{ - 3}}$ for i = 1, …, 3, and C _REQ = 10⁻³.

Figure 8 shows the optimal detection boundaries for ℓ/σ ₀ = 10⁻⁴, and for ℓ/σ ₀ = 7. In both cases, the ‘SS boundary’ is displayed for the optimal C _REQ-allocation, which, in this example, is no longer the equal allocation. The optimal allocation and the corresponding detection thresholds are found by solving Equation (38) for the six SS RAIM test statistics. In practical applications, optimal allocation does not cause large P _HMI -reductions as compared to equal allocation (Blanch et al., Reference Blanch, Ene, Walter and Enge2007), but it does in this particular example.

In parallel, the optimal PL boundary is defined using the six SS test statistics, with parity space directions the solid grey lines in Figure 8 (along vectors ${{\bf u}_i}$, for i = 1, …, 6) as well as six additional statistics, determined by projecting the parity vector on the grey dash-dotted lines (for example, along vectors ${{\bf u}_{i + 6}} = \left( {{{\bf u}_i} + {{\bf u}_{i + 1}}} \right)/\vert\vert {{\bf u}_i} + {{\bf u}_{i + 1}}\vert\vert$ for i = 1, …, 5, and ${{\bf u}_{12}} = \left( {{{\bf u}_1} + {{\bf u}_6}} \right)/\vert\vert {{\bf u}_1} + {{\bf u}_6}\vert\vert$). These lines are placed right in-between fault lines to accentuate their impact on the detection boundary as compared to SS RAIM.

For example, in Figure 8(a), the optimal NP and PL boundaries expectedly match the circle corresponding to the χ ² circle (as explained in Section 3.2). The SS boundary does not match the circle quite as well, especially considering the corners of the SS polygon where the angular separation between fault lines is largest.

However in Figure 8(b) the optimal NP, PL, and SS boundaries are no longer circular. All three approaches provide fairly consistent boundaries, except for small differences at the corners of the polygon, where probability densities under H ₀ become extremely small. The PL method enables evaluation of the integrity risk P _HMI. Let P _HMI,χ be the integrity risk derived from χ ² RAIM. In this example, the integrity risk is reduced to 0·92 P _HMI,χ for SS RAIM versus 0·91 P _HMI,χ using the PL method. In this case, the SS RAIM integrity performance approaches that of the optimal PL method.

In Part 2 of this work (Joerger et al., Reference Joerger, Langel and Pervan2015), the Least-Squares (LS) solution separation (SS) detection test statistics are used as a starting point, because they provide a computationally-efficient, practical approximation of the optimal detection region when using a LS estimator.

Figure 7.

Azimuth-elevation sky plot for an example GPS satellite geometry.

Figure 8.

Optimal Piecewise Linear (PL) detection boundary for the example GPS satellite geometry in Figure 7, (a) for ℓ/σ ₀ = 10⁻⁴, and (b) for ℓ/σ ₀ = 7