An Improved RAIM Availability Assessment Method Based on the Characteristic Slope

The availability assessment is an important step for onboard application in Receiver Autonomous Integrity Monitoring (RAIM)s. It is commonly implemented using the protection level (PL)-based method. This paper analyzes the deficiencies of three kinds of PL-based methods: RAIM availability might be optimistically or conservatively assessed using the classic-PL-base method; might be conservatively assessed using the enhanced-PL-based method, and neither be optimistically nor conservatively assessed using the ideal-PL-based method with the cost of large calculation amount on-board. An improved slope-based RAIM availability assessment method is proposed, in which the characteristic slope is designed as the assessment basis, and its threshold that can exactly match the integrity risk requirement is derived. The slope-based method has the same RAIM availability assessment result as the ideal-PL-based method. Moreover, because the slope threshold can be calculated offline and searched online, the on-board calculation burden can be reduced using the slope-based method. Simulation is presented to verify the theoretical analysis of the RAIM availability assessment performances for the three PL-based and the slope-based methods.


Introduction
The integrity of the Global Navigation Satellite System (GNSS) is one of the important factors to ensure civil aviation safety.There are three categories of GNSS integrity augmentation systems: Satellite-Based Augmentation System (SBAS), Ground-Based Augmentation System (GBAS) and Aircraft-Based Augmentation System (ABAS).The first two categories are at the system level, and the latter category is at the user level [1].ABAS can be implemented with Receiver Autonomous Integrity Monitoring (RAIM), which provides a navigation solution with guaranteed integrity by consistency checking among measurements [2].
For an onboard application, RAIM is executed in two steps: the RAIM availability assessment and the satellite fault detection [1].The former (RAIM availability assessment) is used to assess in advance whether the navigation solution can meet the integrity risk requirement with the fault detection procedure.For decades, RAIM availability assessment has been achieved by calculating the protection level (PL), the upper bound of the position error corresponding to the integrity risk requirement [3].The threshold of the PL is the alert limit (AL), the upper bound of the user-allowed position error.If the PL is lower than the AL, RAIM is considered available; otherwise, it is considered unavailable.
Many studies have focused on determining how to obtain a lower PL to improve the availability of RAIM.Some of these studies were devoted to developing new navigation solution calculation methods, for example, the improved Integrity-Optimized RAIM (NIORAIM) [4] and the optimal weighted average solution (OWAS) [5] methods used for the snapshot RAIM algorithm.These methods can obviously decrease the PL with a slight Sensors 2024, 24, 3283 2 of 21 increase in nominal position error.In addition, some studies have committed to accurately modeling the stochastic measurement noise, such as the discrete error-distribution (NavDEN) model proposed by Rife and Pervan [6] and the distribution model considering both elevation angle and orbit type proposed by Fan [7].These measurement noise models are all helpful for obtaining a tight PL.
However, in the process of pursuing a lower PL, i.e., higher availability of RAIM, there is a key issue that is ignored by most researchers: whether the PL can accurately assess RAIM availability.Milner and Ochieng noted this issue [3].They qualitatively described that the classic PL, the product of the characteristic slope and the Minimum Detection Bias (MDB) proposed by Brown and Chin [8], was too optimistic for RAIM availability assessment.The slope is a geometric feature-related parameter that qualitatively describes the relationship between positioning error and pseudo-range residual [9].The reason is "PL < AL" might not mean that the integrity risk satisfies the requirement for the measurement bias less than MDB.Meanwhile, its enhancement, abbreviated as the enhanced PL in the following, which provides an additional term to protect against the variation in position error proposed by Angus [10], is too conservative.The reason is "PL ≥ AL" might not mean that the integrity risk exceeds the requirement.Here, measurement bias means the measurement error caused by the satellite fault, which is different from the measurement noise in the nominal mode.Furthermore, they proposed the ideal PL.It is the minimum PL value that guarantees the integrity risk, satisfying its requirement for arbitrary measurement bias.The ideal PL can prevent RAIM availability assessment from being optimistic or conservative.However, it cannot be solved analytically.A numerical search for the ideal PL begins with an improbably large value [3], which leads to a large amount of calculation, increasing the computational burden of a GNSS receiver or an onboard computer.
In recent years, most researchers have focused on Advanced RAIM (ARAIM), in which PL is calculated after fault detection [11].ARAIM supports multi-constellation dual-frequency GNSS integrity monitoring.Multiple hypothesis solution separation (MHSS) algorithm is used in ARAIM.How to solve the accurate PL for MHSS is a research hotspot, including the PL calculation method for each fault mode and the optimization strategy for the ultimate PL [12][13][14][15][16]. Jiang and Wang adopted the ideal PL [3] in ARAIM and verified it was more accurate than other PLs for the ARAIM availability assessment [17,18].ARAIM is still in the theoretical research stage and is not currently being applied in engineering practice.
Compared with ARAIM, RAIM has two deficiencies.The first is that RAIM is designed for a single constellation, monitoring only a single satellite fault [2].ARAIM is designed for double constellations, monitoring not only the single satellite fault but also the multiple satellite faults and the constellation fault [12].The second is classic, and the enhanced PLs are not rigorous enough for RAIM availability assessment, while the PL of ARAIM is much more rigorous.However, the on-board calculation of RAIM using the classic or the enhanced PLs is much less than that of ARAIM.For a single constellation, RAIM can still be used, but there is a problem needs to be considered, founding a RAIM availability assessment method both satisfying the rigor and maintaining the low on-board computational burden.
In this paper, a slope-based RAIM availability assessment method is proposed to solve the above problem.The characteristic slope is taken as the assessment basis.Using the ideal slope threshold, this method can achieve a consistent RAIM availability assessment with the ideal-PL-based method.The ideal slope threshold can be calculated offline and searched online because it is only related to one geometric parameter.
The remainder of this paper is organized as follows: Section 2 states some technical backgrounds.Section 3 reviews PL-based RAIM availability assessment methods, including the classic, enhanced and ideal PLs.The deficiencies of the classic and enhanced-PL-based methods can be analyzed quantitatively using the rates of optimistic or conservative assessment.Section 4 proposes the slope-based RAIM availability assessment method after the derivation of the ideal slope threshold.Section 5 gives an overview of the simulation results of the classic-PL-based, enhanced-PL-based, ideal-PL-based and slope-based methods.Section 6 concludes this work with a brief summary.The discussion of this paper takes vertical integrity as an example, using one kind of classic RAIM snapshot algorithm, i.e., the least squares residuals (LSR) algorithm.In this paper, the measurement noise is assumed to be independent white Gaussian noise (WGN).

Technical Background
Before discussing the RAIM availability assessment methods, some technical backgrounds need to be stated, including the derivation of the integrity risk requirement for the single-satellite fault mode, the definition of the vertical characteristic slope and the specific meaning of RAIM being available.

Integrity Risk Requirement for the Single-Satellite Fault Mode
The integrity risk P(HMI) [19] is the probability of undetected faults causing unacceptably large errors in the estimated position [20].HMI is short for hazardous misleading information (HMI).P(HMI) can be divided into three fault modes, the nominal mode, the single-satellite fault mode and the multiple-satellite fault mode, expressed as: In Equation ( 1), 0F, 1F and 2F respectively represent the nominal, the single-satellite fault and the multiple-satellite fault modes, where P(iF) is the prior probability of the fault mode iF and P(HMI|iF ) is the probability of HMI under the iF fault mode.P(0F) = 1 − P sat ) K and P(1F) = C 1 K P sat 1 − P sat ) K , where C 1 K means the number of combinations for choosing one element from K elements; P sat is the prior fault probability of a satellite, and K is the total visible satellite number.Taking vertical plane for example, P(HMI|iF ) is calculated with the following equation [21]: In Equation (2), VPE is the vertical position error; VAL is the vertical AL; Ts and Td are the fault detection test statistic and threshold, respectively.
To ensure the integrity of the navigation system, P(HMI) should be less than its requirement, denoted as P r (HMI).P(HMI, 0F) + P(HMI, 1F) + P(HMI, 2F) < P r (HMI) (3) Given a geometry between the user and all-in-view satellites, VAL, α and P sat , P(HMI, 0F) can be calculated following: In Equation ( 4), α is the allowable false alarm probability under the nominal fault mode, satisfying P(Ts < Td|0F ) = 1 − α; g x; 0, a 2 v is the probability density function (PDF) of the normal-distributed VPE under the nominal fault mode with mean value 0 and standard deviation value a v , where a v is explained in Appendix A.
Moreover, to ensure that P r (HMI, 1F) is nonnegative, the threshold for a v , denoted as Ta v , can be derived from the inequality P r (HMI) − P r (HMI, 2F) − P(HMI, 0F) ≥ 0.
where Φ −1 (•) represents the inverse function of the cumulative distribution function (CDF) for the standard normal distribution.a v ≥ Ta v indicates that only the integrity risk of the nominal and multiple-satellite fault modes have exceeded the total requirement, i.e., R 0 + P r (HMI, 2) ≥ P r (HMI).

Vertical Characteristic Slope
The vertical characteristic slope is defined according to these two parameters [2], s mm (7) where a 3m and s mm respectively characterizes VPE and Ts change caused by the measurement bias of the m-th visible satellite, signed as VS m , is the faulty satellite.The details of a 3m and s mm can be seen in Appendices A and B, respectively.For a specific bias value, a faulty satellite with a large slope value will present a high P(HMI|1F ), and a faulty satellite with a small slope value will present a low P(HMI|1F ).The "characteristic slope" will be abbreviated as "slope" hereafter.

Specific Meaning of RAIM Being Available
RAIM being available refers to P(HMI) < P r (HMI), i.e., P(HMI|1F ) < P r (HMI|1F ) for the arbitrary measurement bias value while RAIM being unavailable refers to P(HMI|1F ) ≥ P r (HMI|1F ) for at least one measurement bias.
A specific example is used to intuitively explain the meaning of RAIM being available.The 32-satellite GPS constellation is used in this example.The pseudorange measurement is assumed to be the dual-frequency ionosphere-free combination of L1 and L5.The standard deviation of the measurement noise for VS i , signed as σ i , is set according to the ARAIM interim report [11].For the location of 37 Figure 1 presents the base-10 logarithm of P(HMI|1F ) for PRN5, PRN6 and PRN13 with the measurement bias ξ b in the interval of (0. 30 m).These three satellites have the top three vertical slope values, as shown in Table 1.The ξ b − lgP(HMI|1F ) curves for these three satellites follow the same order as the slope values, which illustrates that a faulty satellite with a large slope will present a high P(HMI|1F ).The ξ b − lgP(HMI|1F ) curves for the other six visible satellites must be lower than that of PRN 13 because their vertical slopes are smaller.In this example, RAIM would be unavailable if PRN5 was the faulty satellite because P(HMI|1F ) is larger than P r (HMI|1F ) for ξ b values in the range from 11 m to 15 m.Therefore, the intersection between the ξ b − lgP(HMI|1F ) curve of the faulty satellite and the P r (HMI|1F ) line means that RAIM is unavailable.RAIM would be available if the faulty satellite was one of the other visible satellites except for PRN5, because its P(HMI|1F ) smaller than P r (HMI|1F ) at an arbitrary measurement bias.Therefore, the separation between the ξ b − lgP(HMI|1F ) curve of the faulty satellite and the P r (HMI|1F ) line means that RAIM is available.
Sensors 2024, 24, x FOR PEER REVIEW because its (HMI|1) smaller than  (HMI|1) at an arbitrary measuremen Therefore, the separation between the  −   (HMI|1) curve of the faulty satel the  (HMI|1) line means that RAIM is available.Because the faulty satellite is unknown in actual situations, RAIM is con available only if (HMI|1) <  (HMI|1) is true for the arbitrary measurement the worst case, i.e., the satellite with the maximum slope being faulty.

PL-Based RAIM Availability Assessment
The classic PL, enhanced PL, and ideal PL are reviewed in this section.

Classic PL and Its Enhancement
The classic PL, denoted as PL , is defined as the product of the maximum  and the minimum detectable bias  as follows:

PL = 𝑆𝑙𝑜𝑝𝑒 𝜆
where  is the noncentral parameter of the fault detection test statistic  under satellite fault mode [23].
The enhanced PL, denoted as PL , has an additional term that protects variation in the random error of the position solution on the basis of PL as follow where ( ) =  (1 −  ) and  is the standard deviation of the positio distribution.
Slope reflects the relationship between the position error and the pseud residual.PL is the projection of the pseudorange residual on the position error as in Equations ( 8) and (9)  Because the faulty satellite is unknown in actual situations, RAIM is considered available only if P(HMI|1F ) < P r (HMI|1F ) is true for the arbitrary measurement bias in the worst case, i.e., the satellite with the maximum slope being faulty.

PL-Based RAIM Availability Assessment
The classic PL, enhanced PL, and ideal PL are reviewed in this section.

Classic PL and Its Enhancement
The classic PL, denoted as PL c , is defined as the product of the maximum slope Slope max and the minimum detectable bias √ λ a as follows: where λ a is the noncentral parameter of the fault detection test statistic Ts under singlesatellite fault mode [23].
The enhanced PL, denoted as PL e , has an additional term that protects against variation in the random error of the position solution on the basis of PL c as follows [10]: where α(P MD ) = Φ −1 (1 − P MD ) and σ is the standard deviation of the position error distribution.Slope reflects the relationship between the position error and the pseudorange residual.PL is the projection of the pseudorange residual on the position error as shown in Equations ( 8) and (9).According to λ = ξ b s mm in the worst case.Similarly, the enhanced vertical PL, denoted as VPL e , is the upper quantile of the P MD of VPE distribution at s mm with the maximum slope.Therefore, P(|VPE| ≥ VPL e |1F ) = P MD at s mm in the worst case.

Separation
In this situation, there are three kinds of positional relationships among the ξ b − lgP(HMI|1F ) curve, the ξ b − lgP a (HMI|1F ) curve and the lgP r (HMI|1F ) line, as shown in Figure 2.

Separation
In this situation, there are three kinds of positional relationships amon   (HMI|1) curve, the  −   (HMI|1) curve and the   (HMI|1 shown in Figure 2. (HMI 1 ) (HMI 1 ) The ξ b − lgP(HMI|1F ) curve intersects with the lgP r (HMI|1F ) line, and is higher than the blue dashed ξ b − lgP a (HMI|1F ) curve, shown as the brown P 1 (HMI|1F ) curve in Figure 2. In this situation, RAIM is unavailable and the VPL is larger than the VAL, i.e., VPL ≥ VAL|H 1 , meaning a successful detection of "RAIM being unavailable".
The ξ b − lgP(HMI|1F ) curve is separated from the lgP r (HMI|1F ) line, and is higher than the blue dashed curve ξ b − lgP a (HMI|1F ) curve, shown as the black curve P 2 (HMI|1F ) in Figure 2. In this situation, RAIM is available but the VPL is larger than the VAL, i.e., VPL ≥ VAL|H 0 , meaning a conservative assessment.
The ξ b − lgP(HMI|1F ) curve is separated from the lgP r (HMI|1F ) line, and is lower than the blue dashed ξ b − lgP a (HMI|1F ) curve, shown as the blue curve in Figure 2. In this situation, RAIM is available, and the VPL is smaller than the VAL, i.e., VPL < VAL|H 0 , meaning a successful detection of "RAIM being available".
Therefore, for the condition that the ξ b − lgP a (HMI|1F ) curve is separated from the lgP r (HMI|1F ) line, the RAIM availability might be conservatively assessed.

Intersection
In this situation, there are also three kinds of positional relationships among the ξ b − lgP(HMI|1F ) curve, the ξ b − lgP a (HMI|1F ) curve and the lgP r (HMI|1F ) line, as shown in Figure 3.The  −   (HMI|1) curve intersects with the   (HMI|1) line, and is higher than the blue dashed  −   (HMI|1) curve, shown as the brown  (HMI|1) curve in Figure 2. In this situation, RAIM is unavailable and the VPL is larger than the VAL, i.e., VPL ≥ VAL| , meaning a successful detection of "RAIM being unavailable".
The  −   (HMI|1) curve is separated from the   (HMI|1) line, and is higher than the blue dashed curve  −   (HMI|1) curve, shown as the black curve  (HMI|1) in Figure 2. In this situation, RAIM is available but the VPL is larger than the VAL, i.e., VPL ≥ VAL| , meaning a conservative assessment.
The  −   (HMI|1) curve is separated from the   (HMI|1) line, and is lower than the blue dashed  −   (HMI|1) curve, shown as the blue curve in Figure 2. In this situation, RAIM is available, and the VPL is smaller than the VAL, i.e., VPL < VAL| , meaning a successful detection of "RAIM being available".
Therefore, for the condition that the  −   (HMI|1) curve is separated from the   (HMI|1) line, the RAIM availability might be conservatively assessed.

Intersection
In this situation, there are also three kinds of positional relationships among the  −   (HMI|1) curve, the  −   (HMI|1) curve and the   (HMI|1) line, as shown in Figure 3.
(HMI 1 ) (HMI 1 ) The  −   (HMI|1) curve intersects with the   (HMI|1) line, and is higher than the blue dashed  −   (HMI|1) curve, shown as the brown  (HMI|1) curve in Figure 3 In this situation, RAIM is unavailable and the VPL is larger than the VAL, i.e., VPL ≥ VAL| , meaning a successful detection of "RAIM being unavailable".
The  −   (HMI|1) curve intersects with the   (HMI|1) line, and is lower than the blue dashed  −   (HMI|1) curve, shown as the black  (HMI|1) curve in Figure 3.In this situation, RAIM is unavailable, but the VPL is smaller than the VAL, i.e., VPL < VAL| , meaning an optimistic assessment.
The  −   (HMI|1) curve is separated from the   (HMI|1) line, and is lower than the blue dashed  −   (HMI|1) curve, shown as the blue curve The ξ b − lgP(HMI|1F ) curve intersects with the lgP r (HMI|1F ) line, and is higher than the blue dashed ξ b − lgP a (HMI|1F ) curve, shown as the brown P 1 (HMI|1F ) curve in Figure 3 In this situation, RAIM is unavailable and the VPL is larger than the VAL, i.e., VPL ≥ VAL|H 1 , meaning a successful detection of "RAIM being unavailable".
The ξ b − lgP(HMI|1F ) curve intersects with the lgP r (HMI|1F ) line, and is lower than the blue dashed ξ b − lgP a (HMI|1F ) curve, shown as the black P 2 (HMI|1F ) curve in Figure 3.In this situation, RAIM is unavailable, but the VPL is smaller than the VAL, i.e., VPL < VAL|H 1 , meaning an optimistic assessment.
The ξ b − lgP(HMI|1F ) curve is separated from the lgP r (HMI|1F ) line, and is lower than the blue dashed ξ b − lgP a (HMI|1F ) curve, shown as the blue curve P 3 (HMI|1F ) in Figure 3.In this situation, RAIM is available, and the VPL is smaller than the VAL, i.e., VPL < VAL|H 0 , meaning the successful detection of "RAIM being available".
Therefore, for the condition that the ξ b − lgP a (HMI|1F ) curve intersects with the lgP r (HMI|1F ) line, the RAIM availability might be optimistically assessed.

Tangency
In this situation, there are two kinds of positional relationships among the ξ b − lgP(HMI|1F ) curve, the ξ b − lgP a (HMI|1F ) curve and the lgP r (HMI|1F ) line, as shown in Figure 4.
the VAL, i.e., VPL < VAL| , meaning the successful detection of "RAIM being availab Therefore, for the condition that the  −   (HMI|1) curve intersects with   (HMI|1) line, the RAIM availability might be optimistically assessed.

Tangency
In this situation, there are two kinds of positional relationships among the    (HMI|1) curve, the  −   (HMI|1) curve and the   (HMI|1) line, shown in Figure 4.The  −   (HMI|1) curve intersects with the   (HMI|1) line, and is hig than the blue dashed  −   (HMI|1) curve, shown as the brown  (HMI|1) cu in Figure 4.In this situation, RAIM is unavailable, and the VPL is larger than the VAL, VPL ≥ VAL| , meaning a successful detection of "RAIM being unavailable".
The  −   (HMI|1) curve is separated from the   (HMI|1) line, and lower than the blue dashed  −   (HMI|1) curve, shown as the black  (HMI|1 curve in Figure 4.In this situation, RAIM is available and the VPL is smaller than the VA i.e., VPL < VAL| , meaning a successful detection of "RAIM being available". Therefore, for the condition that the  −   (HMI|1) curve is tangent to   (HMI|1) line, both optimistic and conservative assessments can be prevented.
Based on the above analysis, the accuracy of PL-based RAIM availability assessm depends on the positional relationship between the  −   (HMI|1) curve and   (HMI|1) line.There is a risk of conservative assessment when the    (HMI|1) curve is separated from the   (HMI|1) line and risk of optimi assessment when the  −   (HMI|1) curve intersects with the   (HMI|1) li PL-based RAIM availability assessment is accurate only if the  −   (HMI|1) cu is tangent to the   (HMI|1) line.Because only the  (HMI|1) value at a specific i.e.,  =  , is determined for VPL or VPL , the position relationship between entire  −   (HMI|1) curve and the   (HMI|1) line is uncertain.Consequen both optimistic and conservative assessments might happen when using VPL or VP to assess whether vertical RAIM is available.Because  < VPL , the  −   (HMI|1) curve for  is much higher th that for  .Thus the possibility of intersection between the  −   (HMI|1) cu The ξ b − lgP(HMI|1F ) curve intersects with the lgP r (HMI|1F ) line, and is higher than the blue dashed ξ b − lgP a (HMI|1F ) curve, shown as the brown P 1 (HMI|1F ) curve in Figure 4.In this situation, RAIM is unavailable, and the VPL is larger than the VAL, i.e., VPL ≥ VAL|H 1 , meaning a successful detection of "RAIM being unavailable".
The ξ b − lgP(HMI|1F ) curve is separated from the lgP r (HMI|1F ) line, and is lower than the blue dashed ξ b − lgP a (HMI|1F ) curve, shown as the black P 2 (HMI|1F ) curve in Figure 4.In this situation, RAIM is available and the VPL is smaller than the VAL, i.e., VPL < VAL|H 0 , meaning a successful detection of "RAIM being available".
Therefore, for the condition that the ξ b − lgP a (HMI|1F ) curve is tangent to the lgP r (HMI|1F ) line, both optimistic and conservative assessments can be prevented.
Based on the above analysis, the accuracy of PL-based RAIM availability assessment depends on the positional relationship between the ξ b − lgP a (HMI|1F ) curve and the lgP r (HMI|1F ) line.There is a risk of conservative assessment when the ξ b − lgP a (HMI|1F ) curve is separated from the lgP r (HMI|1F ) line and risk of optimistic assessment when the ξ b − lgP a (HMI|1F ) curve intersects with the lgP r (HMI|1F ) line.PL-based RAIM availability assessment is accurate only if the ξ b − lgP a (HMI|1F ) curve is tangent to the lgP r (HMI|1F ) line.Because only the P a (HMI|1F ) value at a specific ξ b , i.e., ξ b = σ 2 m λ a s mm , is determined for VPL c or VPL e , the position relationship between the entire ξ b − lgP a (HMI|1F ) curve and the lgP r (HMI|1F ) line is uncertain.Consequently, both op- timistic and conservative assessments might happen when using VPL c or VPL e to assess whether vertical RAIM is available.
Because VPL c < VPL e , the ξ b − lgP a (HMI|1F ) curve for VPL c is much higher than that for VPL e .Thus the possibility of intersection between the ξ b − lgP a (HMI|1F ) curve and the lgP r (HMI|1F ) line for VPL c is much higher than that for VPL e , which may lead to an optimistic assessment, while the possibility of separation between the ξ b − lgP a (HMI|1F ) curve and the lgP r (HMI|1F ) line for VPL e is much higher than that for PL c , which may lead to conservative assessment.Consequently, the optimistic assessment risk of using VPL c is Sensors 2024, 24, 3283 9 of 21 higher than that of using VPL e for vertical RAIM availability; in contrast, the conservative assessment risk of using VPL e is higher than that of using VPL c .

Ideal Protection Level
According to the above analysis, the ideal positional relationship between the ξ b − lgP a (HMI|1F ) curve and the lgP r (HMI|1F ) line is tangency, which can prevent both optimistic and conservative RAIM availability assessments.The ideal VPL, denoted as VPL d , proposed by Milner and Ochieng, satisfies this condition.It matches the exact required integrity risk for the worst-case bias (WCB), the measurement bias presenting the highest integrity risk.Thus VPL d forms a ξ b − lgP a (HMI|1F ) curve tangent to the lgP r (HMI|1F ) line.If VPL d < VAL, the ξ b − lgP(HMI|1F ) curve must be separated from the lgP r (HMI|1F ) line, which means that vertical RAIM is available; otherwise the ξ b − lgP(HMI|1F ) curve must be tangent to or intersect with the lgP r (HMI|1F ) line, which means that vertical RAIM is unavailable.
VPL d is the solution of:

Slope-Based RAIM Availability Assessment
In addition to the ideal PL, there is another ideal test statistic for RAIM availability assessment: the slope.Both optimistic and conservative RAIM availability assessments can be prevented using the slope once an ideal threshold is found.The following is a deviation of this ideal threshold.

Derivation of the Ideal Threshold for the Slope
The ideal slope threshold derivation begins with searching for a condition satisfying P(HMI|1F ) < P r (HMI|1F ) for all possible measurement bias values of an arbitrary faulty satellite.To ensure P(HMI|1F ) < P r (HMI|1F ) constantly true, the maximum value of P(HMI|1F ) should be less than P r (HMI|1F ), max[P(HMI|1F )] < P r (HMI|1F ) (14) where: Substituting Equation (7) into Equation (15), , Equation ( 16) can be transformed into: Setting , P(HMI|1F ) can be taken as a function of µ s as follows: Sensors 2024, 24, 3283 10 of 21 When substituting Equation (18) into Equation ( 14), an ideal threshold must exist for Slope m with the given P r (HMI|1F ), VAL, a v , Td and K, denoted as T Slope , which satisfies the limit situation max Φ(µ s ) Slope m = T Slope = P r (HMI|1F ) (19) According to Equations ( 18) and ( 19), once P r (HMI|1F ), VAL, Td and K are given, T Slope is determined by only one parameter a v .
It can be deduced that the slope value determines the positional relationship between the µ s − lgP(HMI|1F) curve and the lgP r (HMI|1F ) line.T Slope forms a tangent µ s − lgP(HMI|1F) curve to the lgP r (HMI|1F ) line.If the slope value of a faulty satellite is larger than T Slope , its µ s − lgP(HMI|1F) curve would be intersected with the lgP r (HMI|1F ) line, meaning that P(HMI|1F ) ≥ P r (HMI|1F ) can be satisfied for at least one possible measurement bias value.If the slope value of a faulty satellite was smaller than T Slope , its µ s − lgP(HMI|1F) curve would be separated from the lgP r (HMI|1F ) line, meaning that P(HMI|1F ) < P r (HMI|1F ) can be ensured at an arbitrary measurement bias value.From the analysis of this specific example, the slope is an ideal test statistic for RAIM availability assessment, with the ideal threshold T slope calculated according to Equation (19).Therefore, for each observation epoch, the slope of a visible satellite can be considered as "large slope" if it was larger than T slope and considered as "small slope" if it was smaller than T slope .It can be deduced that the slope value determines the positional relationship between the  −   (HMI|1) curve and the   (HMI|1) line. forms a tangent  −   (HMI|1) curve to the   (HMI|1) line.If the slope value of a faulty satellite is larger than  , its  −   (HMI|1) curve would be intersected with the   (HMI|1) line, meaning that (HMI|1) ≥  (HMI|1) can be satisfied for at least one possible measurement bias value.If the slope value of a faulty satellite was smaller than  , its  −   (HMI|1) curve would be separated from the   (HMI|1) line, meaning that (HMI|1) <  (HMI|1) can be ensured at an arbitrary measurement bias value.From the analysis of this specific example, the slope is an ideal test statistic for RAIM availability assessment, with the ideal threshold  calculated according to Equation (19).Therefore, for each observation epoch, the slope of a visible satellite can be considered as "large slope" if it was larger  and considered as "small slope" if it was smaller than  .

Comparison of the Ideal Slope Threshold and the Ideal Protection Level
The functions of  and VPL are identical, forming a tangent (HMI|1) curve to the   (HMI|1) line.Referring to the equation for  , i.e., Equation ( 18), the equation for VPL , can be formulated as follows: Analyzing this equation, VPL is determined by  and  with the given  (HMI|1),  and .Therefore,  is only related to  while VPL is related to both  and  .Considering that the PDF of the noncentral  distribution is too complicated, both  and VPL should be solved numerically, which will sharply increase the computation burden of a GNSS receiver or an onboard computer.Thanks to the one-toone correspondence between  and  ,  can be calculated offline for discrete  values in the range from 1 to  and saved in a receiver.It can be on-board searched from the presaved data according to the specific  value.However, VPL needs to be calculated online after both  and  obtained.

Comparison of the Ideal Slope Threshold and the Ideal Protection Level
The functions of T slope and VPL d are identical, forming a tangent P(HMI|1F ) curve to the lgP r (HMI|1F ) line.Referring to the equation for T slope , i.e., Equation ( 18), the equation for VPL d , can be formulated as follows: Analyzing this equation, VPL d is determined by a v and Slope max with the given P r (HMI|1F ), Td and K. Therefore, T slope is only related to a v while VPL d is related to both a v and Slope max .
Considering that the PDF of the noncentral χ 2 distribution is too complicated, both T Slope and VPL d should be solved numerically, which will sharply increase the computation burden of a GNSS receiver or an onboard computer.Thanks to the one-to-one correspondence between a v and T Slope , T Slope can be calculated offline for discrete a v values in the range from 1 to Ta v and saved in a receiver.It can be on-board searched from the presaved data according to the specific a v value.However, VPL d needs to be calculated online after both a v and Slope max obtained.

Slope-Based RAIM Availability Assessment Method
Slope-based RAIM availability assessment should be implemented in the worst case to fully prevent the integrity risk.Using slope-based method, RAIM is considered available if the maximum slope is less than the ideal threshold T Slope ; otherwise, it is considered unavailable.The specific execution process for slope-based RAIM availability assessment is presented in Figure 7.
As shown in Figure 7, the inputs are the observation matrix H and the weighted matrix P. The first step is an assessment based on the total number of visible satellites K. RAIM is considered unavailable if K ≤ 4 because fault detection cannot be executed with less than 5 visible satellites.For K > 4, the second step is an assessment based on the a v value.RAIM is considered to be unavailable if a v ≥ Ta v because only the sum of P(HMI, 0F) and P(HMI, 2F) has exceed P r (HMI) when a v exceeds Ta v .For a v < Ta v , the last step is an assessment based on the maximum slope Slope max .RAIM is finally considered available if SlopeSlope max .
to fully prevent the integrity risk.Using slope-based method, RAIM is con available if the maximum slope is less than the ideal threshold ; otherwi considered unavailable.The specific execution process for slope-based RAIM ava assessment is presented in Figure 7.
Slope max calculation and T slope search RAIM unavailable RAIM available As shown in Figure 7, the inputs are the observation matrix H and the w matrix P. The first step is an assessment based on the total number of visible sate RAIM is considered unavailable if  ≤ 4 because fault detection cannot be execut less than 5 visible satellites.For  > 4, the second step is an assessment based on value.RAIM is considered to be unavailable if  ≥  because only the (HMI, 0) and (HMI, 2) has exceed  (HMI) when  exceeds  .For  the last step is an assessment based on the maximum slope  .RAIM is considered available if  . can be calculated online according to Equation ( 6) with given VAL ,  (HMI) and  (HMI|2) . should be calculated offline with discrete  small step size for different numbers of visible satellites and saved in the GNSS or the onboard computer.Assuming that the total number of visible satellites is K is in the interval of ( ,  ], where  and  are the indexes of two adjacent points [ ,  ] and [ ,  ] presaved for K visible satellites, corresponding to  should be assigned as: This value is calculated according to a linear fit for  in the interval of ( Because the  −  curve is convex as shown in Figure 5, the assigned  smaller than the real  value, which may cause a small conservative assessme Ta v can be calculated online according to Equation ( 6) with given VAL, P sat ,α, P r (HMI) and P r (HMI|2F ).T Slope should be calculated offline with discrete a v with a small step size for different numbers of visible satellites and saved in the GNSS receiver or the onboard computer.Assuming that the total number of visible satellites is K and a v is in the interval of (a v1 , a v2 ], where a v1 and a v2 are the indexes of two adjacent discrete points a v1 , T Slope 1 and a v2 , T Slope 2 presaved for K visible satellites, T Slope corresponding to a v should be assigned as: This value is calculated according to a linear fit for T Slope in the interval of (a v1 , a v2 ].Because the a v − T Slope curve is convex as shown in Figure 5, the assigned T Slope value is smaller than the real T Slope value, which may cause a small conservative assessment risk.However, if the a v step size is small enough, the assigned T Slope value would be nearly equal to the true value, thereby preventing the small conservative assessment risk.
A simulation is designed to find the desirable range a v step.1.2 × 10 6 times of a v and Slope max for different location and epochs are collected.As shown in Table 2, the times of conservative assessment increases with step size widen.Taking into account both the amount of calculation and conservative assessment rate, 0.01~0.02 is the desirable range a v step because it is the maximum step size with 0 time of conservative assessment.It should be mentioned that the calculation amount of the ideal-PL and T Slope is exactly the same.The ideal PL procedure begins with an improbably large VPL of 2000 m and halves the search step by checking if the corresponding integrity risk exceeds the requirement [3].For each step of the iteration, the integrity risk needs to be calculated for different bias values with a fixed step.This process is computationally intensive.Similarly, the T Slope procedure begins with an improbably large slope value of 15 and halves the search step.Compared with the ideal-PL-based method, the slope-based method separates the process of numerical iteration from on-board RAIM availability assessment, reducing the burden of on-board computing.

Simulation
To compare the performance of the PL-based and slope-based methods, the vertical RAIM availability assessment for a 32-satellite GPS constellation is simulated in worldwide (latitude 60 • S~60 • N and longitude 180 • W~180 • E) for a whole day (13 March 2019 0:00:00~24:00:00).The simulation area is meshed as the grid of 1 • × 1 • and the simulation time step is 300 s.The masking angle is set to 10 • .The dual-frequency ionosphere-free combination of L1 and L5 is assumed to be the pseudorange measurement.standard derivation of the measurement noise is set according to the ARAIM interim report [11].

Specific Example Analysis
Three specific examples for a single grid point and a simulation epoch are chosen to show the RAIM availability assessment using the four methods in detail.Table 3 records the VPLs, Slope max and T Slope values for these specific examples.Figures 8-11 present the real µ s − lgP(HMI|1F ) curve for Slope max , the µ s − lgP(HMI|1F ) curve for T Slope , and the µ s − lgP a (HMI|1F ) curves for VPL c , VPL e and VPL d in these examples, respectively.In Figures 8-11, the µ s − lgP(HMI|1F ) curve for T Slope and the µ s − lgP a (HMI|1F ) curve for VPL d are tangent to the lgP r (HMI|1F ) line, while the ξ b − lg a P(HMI|1F ) curve for VPL e is sep- arated from the lgP r (HMI|1F ) line.The positional relationship between the µ s − lg a P(HMI|1F ) curve for VPL c and the lgP r (HMI|1F ) line could be either intersection or separation.As shown in Figure 8, the real µ s − lgP(HMI|1F ) curve for Slope max intersects with the lgP r (HMI|1F ) line, which means that RAIM is actually unavailable in the first example.The µ s − lgP a (HMI|1F ) curve for VPL c is higher than the real µ s − lgP(HMI|1F ) curve, i.e., VPL c < VAL, which means that RAIM is assessed to be available when using VPL c .All of the µ s − lgP(HMI|1F ) curve for T Slope , the µ s − lgP a (HMI|1F ) curves for VPL e and VPL d are lower than the real µ s − lgP(HMI|1F ) curve, i.e., Slope ≥ Slope max , VPL e ≥ VAL and VPL d ≥ VAL, which means that RAIM is assessed to be unavailable when using Slope max , VPL e or VPL d .Therefore, the RAIM availability is optimistically assessed when using VPL c and successfully assessed using when using Slope max , VPL e or VPL d .As shown in Figure 8, the real  −   (HMI|1) curve for  intersects with the   (HMI|1) line, which means that RAIM is actually unavailable in the first example.The  −   (HMI|1) curve for VPL is higher than the real  −   (HMI|1) curve, i.e., VPL < VAL, which means that RAIM is assessed to be available when using VPL .All of the  −   (HMI|1) curve for  , the  −   (HMI|1) curves for VPL and VPL are lower than the real  −   (HMI|1) curve, i.e.,  ≥  , VPL ≥ VAL and VPL ≥ VAL, which means that RAIM is assessed to be unavailable when using  , VPL or VPL .Therefore, the RAIM availability is optimistically assessed when using VPL and successfully assessed using when using  , VPL or VPL .As shown in Figure 9, the real  −   (HMI|1) curve for  is separated from the   (HMI|1) line, which means that RAIM is actually available in the second example.The  −   (HMI|1) curve for  , and the  −   (HMI|1) curves for VPL and VPL are higher than the real  −   (HMI|1) curve, i.e.,  <  , VPL < VAL and VPL < VAL , which means that RAIM is assessed to be available when using  , VPL or VPL .The  −   (HMI|1) curve for VPL is lower than the real  −   (HMI|1) curve, i.e., VPL ≥ VAL, which means that RAIM is assessed to be unavailable when using VPL .Therefore, RAIM availability is conservatively assessed when using VPL and successfully assessed when using  , VPL or VPL .As shown in Figure 10, the real  −   (HMI|1) curve for  is separated from the   (HMI|1) line, which means that RAIM is actually available in the third example.Both the  −   (HMI|1) curve for  and the  −   (HMI|1) curve for VPL are higher than the real  −   (HMI|1) curve, i.e.,  <  and VPL < VAL , which means that RAIM is assessed to be available when using  or VPL .Additionally, both the  −   (HMI|1) curves for VPL and VPL are lower than the real  −   (HMI|1) curve, i.e., VPL ≥ VAL and VPL ≥ VAL, which means that RAIM is assessed to be unavailable when using VPL or VPL .Therefore, RAIM availability is conservatively assessed when using VPL or VPL , and it is successfully assessed when using  or VPL in this example.These three specific examples intuitively illustrate that both optimistic and conservative RAIM availability assessments might happen when using the classic-PLbased method; only conservative assessment might happen when using the enhanced-PL method; and both optimistic and conservative assessments can be prevented when using the ideal-PL-based or the slope-base methods.As shown in Figure 9, the real µ s − lgP(HMI|1F ) curve for Slope max is separated from the lgP r (HMI|1F ) line, which means that RAIM is actually available in the second example.The µ s − lgP(HMI|1F ) curve for T Slope , and the µ s − lgP a (HMI|1F ) curves for VPL c and VPL d are higher than the real µ s − lgP(HMI|1F ) curve, i.e., Slope < Slope max , VPL c < VAL and VPL d < VAL, which means that RAIM is assessed to be available when using Slope max , VPL c or VPL d .The µ s − lg a P(HMI|1F ) curve for VPL e is lower than the real µ s − lgP(HMI|1F ) curve, i.e., VPL e ≥ VAL, which means that RAIM is assessed to be unavailable when using VPL e .Therefore, RAIM availability is conservatively assessed when using VPL e and successfully assessed when using Slope max , VPL c or VPL d .
As shown in Figure 10, the real µ s − lgP(HMI|1F ) curve for Slope max is separated from the lgP r (HMI|1F ) line, which means that RAIM is actually available in the third example.Both the µ s − lgP(HMI|1F ) curve for T Slope and the µ s − lg a P(HMI|1F ) curve for VPL d are higher than the real µ s − lgP(HMI|1F ) curve, i.e., Slope < Slope max and VPL d < VAL, which means that RAIM is assessed to be available when using Slope max or VPL d .Additionally, both the µ s − lg a P(HMI|1F ) curves for VPL c and VPL e are lower than the real µ s − lgP(HMI|1F ) curve, i.e., VPL c ≥ VAL and VPL c ≥ VAL, which means that RAIM is assessed to be unavailable when using VPL c or VPL e .Therefore, RAIM availability is conservatively assessed when using VPL c or VPL e , and it is successfully assessed when using Slope max or VPL d in this example.
These three specific examples intuitively illustrate that both optimistic and conservative RAIM availability assessments might happen when using the classic-PL-based method; only conservative assessment might happen when using the enhanced-PL method; and both optimistic and conservative assessments can be prevented when using the ideal-PL-based or the slope-base methods.

Simulation Results Statistical Analysis
Both the rates of optimistic and conservative assessments for RAIM availability are calculated for each grid point using the classic-PL-based, enhanced-PL-based, ideal-PL-based and slope-based methods.The optimistic assessment rate is indicated by the ratio between the count of epochs in which RAIM is assessed to be available but is actually unavailable and the count of epochs in which RAIM is actually unavailable.The conservative assessment rate is indicated by the ratio between the count of epochs in which RAIM is assessed to be unavailable but is actually available and the count of epochs in which RAIM is actually available.Table 4 records the specific optimistic and conservative assessment data for all 4 kinds of RAIM availability assessment methods.As shown in Table 4, both the rates of optimistic and conservative assessments are 0 for each grid point using the slope-based and ideal-PL-based methods.These prove that both optimistic and conservative assessments can be prevented.The reason for the 0 conservative assessment rate of the slope-based method is that the a v step size is small enough (0.01) for offline calculation of T Slope , leading to the assigned T Slope value being nearly equal to its true value.For the classic-PL-based method, there are both grid points with nonzero optimistic assessment rate and grid points with nonzero conservative assessment rate.Figures 11 and 12 present the optimistic and conservative assessment rates for each grid point using the classic-PL-based method, respectively.As shown in Table 4 and Figure 11, there are 2032 grid points with nonzero optimistic assessment rate, 4.65% of the total grid points, and there are 41 grid points with a 100% optimistic assessment rate, meaning that optimistic assessment always happens when RAIM is unavailable at these grid points.As shown in Table 4 and Figure 12, there are 16,678 grid points with nonzero conservative assessment rate, 38.18% of the total grid points, and the maximum conservative assessment rate is 2.59%.Comparing Figure 12 with Figure 11, the coverage area of conservative assessment is much larger than that of optimistic assessment, but the maximum optimistic assessment rate is much higher than the maximum conservative assessment rate for a single grid point using the classic-PL-based method.
conservative assessment rate is indicated by the ratio between the count of epoch which RAIM is assessed to be unavailable but is actually available and the count of ep in which RAIM is actually available.
Table 4 records the specific optimistic and conservative assessment data for all 4 k of RAIM availability assessment methods.As shown in Table 4, both the rates of optim and conservative assessments are 0 for each grid point using the slope-based and id PL-based methods.These prove that both optimistic and conservative assessments ca prevented.The reason for the 0 conservative assessment rate of the slope-based meth that the  step size is small enough (0.01) for offline calculation of  , leading to assigned  value being nearly equal to its true value.For the classic-PL-based method, there are both grid points with nonzero optim assessment rate and grid points with nonzero conservative assessment rate.Figure and 12 present the optimistic and conservative assessment rates for each grid point u the classic-PL-based method, respectively.As shown in Table 4 and Figure 11, there 2032 grid points with nonzero optimistic assessment rate, 4.65% of the total grid po and there are 41 grid points with a 100% optimistic assessment rate, meaning optimistic assessment always happens when RAIM is unavailable at these grid point shown in Table 4 and Figure 12, there are 16,678 grid points with nonzero conserv assessment rate, 38.18% of the total grid points, and the maximum conserv assessment rate is 2.59%.Comparing Figure 12 with Figure 11, the coverage are conservative assessment is much larger than that of optimistic assessment, but maximum optimistic assessment rate is much higher than the maximum conserv assessment rate for a single grid point using the classic-PL-based method.For the enhanced-PL-based method, there are no grid points with a nonzero optimistic assessment rate.This finding illustrates that the value of P a (HMI|1F ) is small enough at being separated from the P r (HMI|1F ) line, preventing optimistic assessment.However, there are 43,038 grid points with nonzero conservative assessment rate, as shown in Table 4 and Figure 13, representing 98.53% of the total grid points.The maximum conservative assessment rate is 8.15%.Comparing Figure 13 with Figure 12, the conservative assessment using the enhanced PL is much higher than that using the classic PL, which is represented by the much larger conservative assessment coverage area and the higher conservative assessment rate for a single grid point.For the enhanced-PL-based method, there are no grid points with a nonzero optimistic assessment rate.This finding illustrates that the value of  (HMI|1) is smal enough at  =  for the enhanced PL, which leads to the  −   (HMI|1 curve always being separated from the  (HMI|1) line, preventing optimisti assessment.However, there are 43,038 grid points with nonzero conservative assessmen rate, as shown in Table 4 and Figure 13, representing 98.53% of the total grid points.The maximum conservative assessment rate is 8.15%.Comparing Figure 13 with Figure 12, the conservative assessment risk using the enhanced PL is much higher than that using the classic PL, which is represented by the much larger conservative assessment coverage area and the higher conservative assessment rate for a single grid point.According to the above simulation results, the performance of the classic PL is the worst on RAIM availability assessment for both the risks of optimistic and conservative assessments.In particular, an optimistic assessment might cause HMI, which i intolerable.The performance of the enhanced PL is better than that of classic PL because the optimistic assessment is prevented.However, the risk of conservative assessment i significant, reducing RAIM continuity.The performances of the slope and the ideal PL are optimal, preventing both optimistic assessment and conservative assessment.

Conclusions
According to the theory and simulation analysis, the RAIM availability might be optimistically or conservatively assessed using the classic-PL-based method and migh only be conservatively assessed using the enhanced-PL-based method.Using the ideal PL-based method, both optimistic and conservative assessment can be prevented However, the calculation of the ideal PL brings a heavy computational burden to the GNSS receiver or the onboard computer.The slope-based method has the same RAIM availability assessment result as the ideal-PL-based method.Because the ideal slope threshold is only related to one geometric parameter, it can be calculated offline and searched online.Thus, the on-board calculation burden can be reduced using the slope based method.This improved method can be used in RAIM for single GNSS constellation Further, a semi-physical simulation experiment will be implemented using the common on-board processor to verify the real-time performance of the slope-based method According to the above simulation results, the performance of the classic PL is the worst on RAIM availability assessment for both the risks of optimistic and conservative assessments.In particular, an optimistic assessment might cause HMI, which is intolerable.The performance of the enhanced PL is better than that of classic PL because the optimistic assessment is prevented.However, the risk of conservative assessment is significant, reducing RAIM continuity.The performances of the slope and the ideal PL are optimal, preventing both optimistic assessment and conservative assessment.

Conclusions
According to the theory and simulation analysis, the RAIM availability might be optimistically or conservatively assessed using the classic-PL-based method and might only be conservatively assessed using the enhanced-PL-based method.Using the ideal-PLbased method, both optimistic and conservative assessment can be prevented.However, the calculation of the ideal PL brings a heavy computational burden to the GNSS receiver or the onboard computer.The slope-based method has the same RAIM availability assessment result as the ideal-PL-based method.Because the ideal slope threshold is only related to one geometric parameter, it can be calculated offline and searched online.Thus, the on-board calculation burden can be reduced using the slope-based method.This improved method can be used in RAIM for single GNSS constellation.Further, a semi-physical simulation experiment will be implemented using the common on-board processor to verify the realtime performance of the slope-based method.Moreover, the RAIM availability assessment performance will be verified using massive actual measurement data.

σ 2 m 2 s
mm derived in Appendix B, the measurement bias ξ b for a faulty satellite VS m , which causes Ts to obey χ 2 (K − 4, λ a ), is ξ b = σ 2 m λ a s mm .Based on the position error derivation in Appendix A, this ξ b makes the VPE under single-satellite fault mode obey: VPE ∼ N Slope m λ a , a 2 v (10) Contrasting Equation (8) with Equation (10), the classic vertical PL, denoted as VPL c , is the expectation of VPE at ξ b = σ 2 m λ a s mm with the maximum slope.Therefore,
2) = 1.3 × 10 −8 , VAL = 50 m, P sat = 1 × 10 −5 , α = 1 × 10 −6 and K = 7, • • • , 11.As shown in Figure 5, T Slope decreases with the increase of a v .Each a v − T Slope curve exhibits a nearly constant segment at the beginning and a sharply decreasing segment at the end.The a v − T Slope curve ends when a v reaches Ta v , and the spacing between two adjacent curves obviously decreases as K increases.( ) = (HMI|1) = [1

Figure 5
presents the numerically solved  for  values with a step of 0.0  (HMI) = 1 × 10 ,  (HMI, 2) = 1.3 × 10 , VAL = 50 m ,  = 1 × 10 ,  = 1 × and  = 7, ⋯ ,11.As shown in Figure 5,  decreases with the increase of  .E  −  curve exhibits a nearly constant segment at the beginning and a sha decreasing segment at the end.The  −  curve ends when  reaches  , and spacing between two adjacent curves obviously decreases as  increases.

Figure 5 .
Figure 5. T Slope for different a v values.

Figure 6 .
Figure 6.Practical meaning of T Slope .

Figure 8 .
Figure 8. RAIM availability judgment for the first specific example.

Figure 9 .
Figure 9. RAIM availability assessment for the second specific example.

Figure 10 .
Figure 10.RAIM availability assessment for the third specific example.

Figure 8 .
Figure 8. RAIM availability judgment for the first specific example.

Figure 8 .
Figure 8. RAIM availability judgment for the first specific example.

Figure 9 .
Figure 9. RAIM availability assessment for the second specific example.

Figure 10 .
Figure 10.RAIM availability assessment for the third specific example.

Figure 9 .
Figure 9. RAIM availability assessment for the second specific example.

Figure 8 .
Figure 8. RAIM availability judgment for the first specific example.

Figure 9 .
Figure 9. RAIM availability assessment for the second specific example.

Figure 10 .
Figure 10.RAIM availability assessment for the third specific example.

Figure 10 .
Figure 10.RAIM availability assessment for the third specific example.

Figure 11 .
Figure 11.Optimistic assessment rate using the classic-PL-based method.

Figure 11 .
Figure 11.Optimistic assessment rate using the classic-PL-based method.

Figure 12 .
Figure 12.Conservative assessment rate using the classic-PL-based method.
m λ a s mm for the enhanced PL, which leads to the ξ b − lgP a (HMI|1F ) curve always Sensors 2024, 24, 3283 17 of 21

Figure 12 .
Figure 12.Conservative assessment rate using the classic-PL-based method.

Figure 13 .
Figure 13.Conservative assessment rate using the enhanced-PL-based method.

Figure 13 .
Figure 13.Conservative assessment rate using the enhanced-PL-based method.

epoch of UTC 14 March 2019 17:15:00, there are 9 visible satellites with a masking angle of 10 • . Their vertical slope values are recorded in Table 1. Table 1. Vertical slope values for different visible satellites.
• N latitude, 117 • E longitude and height 0 m and the

Table 2 .
Step and times of conservative assessment.

Table 3 .
Specific examples for RAIM availability assessment.

Table 4 .
Performance comparison for the four kinds of RAIM availability assessment methods.

Table 4 .
Performance comparison for the four kinds of RAIM availability assessment methods