to integrity monitoring of GNSS navigation signals ... solution, the separability of the integrated GPS/Galileo ... Monitoring the integrity of a navigation system is.
GNSS Receiver Autonomous Integrity Monitoring: A Separability Analysis Steve Hewitson School of Surveying and Spatial Information Systems The University of New South Wales Australia
BIOGRAPHY Steve Hewitson is currently enrolled as a full-time Ph.D. student with the School of Surveying and Spatial Information Systems, the University of New South Wales (UNSW), Australia. His current research interests are concentrated around applications of GNSS and Inertial Navigation Systems. ABSTRACT Receiver Autonomous Integrity Monitoring (RAIM) refers to integrity monitoring of GNSS navigation signals performed by a receiver independent of external reference systems, except for the navigation signals themselves. Quality measures need to be used to evaluate the GNSS RAIM performance at different locations and under various navigation modes, such as GPS only and GPS/Galileo integration, etc. The quality measures should include both the reliability and separability. The reliability measure is used to evaluate the capability of GNSS receivers to detect the outliers, while the separability measure is used to assess the capability of GNSS receivers to correctly identify the outlier from the measurements processed. In this paper, the mathematical models for RAIM and its measures of reliability and separability are described. Detailed simulations have been carried to evaluate the reliability and separability performance of GPS only and integrated GPS/Galileo navigation solutions. Simulation results show that, in comparison with the GPS-only solution, the separability of the integrated GPS/Galileo solution is improved by over 150%. INTRODUCTION During the past decade, the Global Navigation Satellite Systems (GNSS) have been playing a more and more important role in surveying, geodesy and other positionsensitive disciplines, such as transportation, personal location and telecommunications. Whilst American GPS and Russian GLONASS are currently undergoing their modernisation, the European Commission (EC) and European Space Agency (ESA) have established plans for a new satellite navigation system called GALILEO.
However, for all these satellite-based navigation systems, the integrity of navigation solutions is still one of major concerns. Integrity, as defined in the International Civil Aviation Organisation’s GNSS Standards and Recommended Practices (SARPS): is a measure of the trust which can be placed in the correctness of the information supplied by the total system. Integrity includes the ability of a system to provide timely and valid warnings to users (ESA, 2002). Monitoring the integrity of a navigation system is essential to ensure that the navigation solution is within tolerable constraints. Ideal integrity monitoring involves the detection, isolation and the removal of faulty measurement sources from the navigation solution. Receiver Autonomous Integrity Monitoring (RAIM) refers to integrity monitoring of GNSS navigation signals performed by a receiver independent of external reference systems, except for the navigation signals themselves. RAIM performance is measured in terms of the maximum allowable alarm rate and the minimum detection probability; both are dependent on the failure rate of measurement sources, range accuracies and measurement geometry. Optimal RAIM algorithms should exhibit high detection rates and low false alarm rates. Significant efforts have been made to develop and analyse RAIM methods and algorithms over the past decade. Among others, Gao (1993) has investigated a GPS integrity test procedure with reliability assurance to offer real-time precision and reliability checks on navigation solutions. Walter and Enge (1995) presented a versatile weighted form of RAIM where measurement sources are weighted based on a priori information or broadcast weighting information. A probabilistic approach to the determination of geometrical criteria for the evaluation of GPS RAIM availability has been discussed in Sang and Kubik (1997). More recently, Romay et al. (2001) investigates the availability of RAIM computed for GPS, Galileo and combined GPS/Galileo constellations through simulations. Current RAIM methods are usually based on the statistical testing procedures. The reliability of detecting fault measurements or outliers is highly dependent on the redundancy and the geometry in the navigation solution.
RAIM is only available when there is a minimum of five satellites visible with sufficiently strong measurement geometry as redundancy is essential to permit the measurement consistency checks. However, with only five satellites available, one outlier can be detected but it is impossible to identify which measurement in the solution is the outlier. It is also noted that more redundant measurements and stronger geometry will increase the capability of GNSS RAIM procedures in both detecting and identifying the outliers. Therefore, some quality measures need to be used to evaluate the GNSS RAIM performance at different locations and under various navigation modes, such as GPS only and GPS/Galileo integration, etc. The quality measures should include both the reliability and separability. The reliability measure is used to evaluate the capability of GNSS receivers to detect the outliers, while the separability measure is used to assess the capability of GNSS receivers to correctly identify the outlier from the measurements processed. In this paper, the mathematical models for RAIM and its measures of reliability and separability are described. The detailed simulations have been carried to evaluate the reliability and separability performance of GPS only and integrated GPS/Galileo navigation solutions. RAIM ALGORITHMS Mathematical Models GNSS positioning requires a set x of m parameters to be estimated from a set y of n measurements, where the function pertaining to this relationship is:
F ( x) = l
(2)
where A is the n x m design matrix, x is the vector of m parameters, l is the vector of n measurements and v is the vector of n residuals. Complementing the above functional model is the stochastic variance covariance (VCV) matrix of measurements Ql. This matrix, which is assumed to be known, describes the ‘noise’ characteristics related to the measurements. Define P =Ql-1
(3)
as the n x n weight matrix of measurements. For the estimation of the unknown parameters and measurement n residuals the following least squares formulae are used:
xˆ = ( AT PA) -1 AT Pl
(5)
where xˆ is the vector of m least squares estimates of x, is the vector of n least squares estimates of v.
vˆ
Outlier Detection and Identification When dealing with copious amounts of data, which has been obtained and processed automatically, the need to implement statistical methods of quality control is absolutely essential. Positioning and navigation methods reliant upon systems such as GNSSs, require very little interaction with the user who, as a result, can only possess limited knowledge of the quality of the measurements. Through proper data analysis and the implementation of quality control measures the user can put much more faith in the quality of the measurements. Measurements containing gross errors, outliers, cannot be described or accounted for by the stochastic model and will therefore undesirably affect parameter estimations and related variances. As a result there is an extremely high risk that an outlier will go undetected unless statistical testing methods are employed. One widely accepted procedure is called Detection, Identification and Adaptation (DIA), which is discussed below. The variance factor test is used to indicate whether or not the adjustment model is satisfactory and hence detects the presence of any gross anomalies. The ‘a posterior’ variance factor, or unit variance:
sˆ 02 =
vT Pv n-m
(6)
(1)
The linearised adjustment model of the above function can be expressed as:
Ax = l + v
vˆ = Axˆ - l
(4)
can be tested against the two-tailed test limits derived from the Chi-squared distribution:
c (21-a ) / 2, n - m n-m
£ sˆ 02 £
c a2 / 2, n - m n-m
(7)
where n – m is the degrees of freedom in the solution and _ is the significance level of the test. If the variance factor exceeds the test limit (critical value), the adjustment model is considered as invalid. The expected value of the variance factor is equal to one and is often referred to as the unit variance statistic. Possible reasons for the failure of the variance factor test could be input or programming errors, the presence of outliers in measurements, model errors and poor estimation of the a priori covariance matrix. In this study, we assume that only the outliers may cause the failure of the variance factor test. For the failure occurring purely as a result of a poor estimation of the ‘a priori’ VCV matrix it maybe corrected by multiplying the estimated VCV matrix by the variance factor to obtain the corrected VCV matrix.
The ‘a posteriori’ VCV matrices Q xˆ and Qvˆ for the estimated parameters and estimated residuals respectively are derived as:
Q xˆ = ( AT PA) -1
(3)
Qvˆ = Ql - AQ xˆ AT
(4)
Q xˆ is used to determine the standard deviations of all position-fixes. A VCV matrix for the corrected measurements can also be calculated as:
QL = AQ xˆ AT
(5)
Assuming that an outlier —Si exists, the adjustment model may be extended to:
Axˆ + ei —Si = l + vˆ
(6)
where ei is a unit vector in which the ith component has a value equal to one and dictates the measurement to be tested: T
ei = [0 0 ... 1 0]
A least squares estimation of the magnitude of the outlier —Si can be determined by:
—Sˆi = -(eiT PQvˆ Pei ) -1 eiT Pvˆ
(7)
with a variance of:
s 2 ˆ = (eiT PQvˆ Pei ) -1 —S i
(8)
Statistical testing for the identification of an outlier —Si relies on the null hypothesis, which shows that the measurements are outlier free, and the alternative hypothesis indicating the existence of outlier of magnitude —S : Null Hypothesis:
H o : E (—Sˆ ) = 0 ~ N (0,1)
(9)
Alternative Hypothesis:
H a : E (—Sˆ ) = —S ≠ 0 ~ N (d i ,1) with:
(10)
d i = —Sˆi eiT PQvˆ Pei = —Sˆi Q—Sˆ i
(11)
where _i is the non-centrality parameter of the alternative hypothesis H a and is dependant on the chosen a and b probabilities. Provided that the adjustment model is correct, the w-test can be used to identify an outlier. The test statistic is (Baarda, 1968; Cross et al., 1994; Teunissen, 1998)
wi =
—Sˆi Q—Sˆ
(12) i
The critical value for w i to be tested against is N1-a / 2 (0,1) . For the situation where:
| wi |> N1-a / 2 (0,1)
(13)
the ith measurement is a ‘supposed’ outlier. The test is carried out with respect to each measurement and the largest value exceeding the critical value is deemed an outlier and is removed from the model. The w test is performed again to see if any more outliers exist. If another outlier is found it should be removed from the model and the measurement that was first regarded as an outlier should be included and the model retested and so on until no more outliers are detected. Hypothesis testing will result in one out of five possible outcomes occurring. Only two of these outcomes are correct. Four of these outcomes are rejecting the null hypotheses when it is true (Type I error), rejecting the null hypotheses when it is indeed false, accepting the null hypotheses when it is in fact true and accepting the null hypotheses when it is actually false (Type II error). The probabilities of committing Type I and Type II errors are, respectively a and b. a is the probability of a false alarm and b is the probability that an outlier is undetected. a and b are inversely proportional. The power of the test is the probability of detecting an outlier, g =1-b . The fifth possible outcome is referred to as a Type III error and occurs when an outlier is detected but the wrong measurement is removed (Hawkins, 1980). The Adaptation phase refers to the treatment or disposal of the outlier so that the null hypothesis may be legitimately accepted. The measurement regarded as an outlier may be eliminated from the adjustment computation or the resulting bias may be included as a parameter to be estimated and accounted for. If the latter approach is chosen then a new null hypothesis must be selected that accounts for the new error parameter. RELIABILITY AND SEPARABILITY Reliability describes the probability of performing a certain function without failure under given conditions for
a specified period of time. More specifically, in terms of GNSS RAIM, reliability comprises the ability of the system to detect outliers, referred to as internal reliability, and the measures of the influence of outliers below the detectable threshold on parameter estimation, referred to as external reliability (Baarda, 1968). Separability refers to the ability to separate or distinguish any two measurements from one another. This ability is of the upmost importance as poorly separated measurements encumber the reliability of a navigation solution by manifesting a high risk of incorrectly flagging a ‘good’ measurement as an outlier (Type III error). Internal Reliability Internal reliability is dictated by the lower bound for detectable outliers and is expressed by the Minimal Detectable Bias (MDB). The MDB is the magnitude of the smallest error that can be detected for a specific level of confidence and is determined, for correlated measurements (Baarda, 1968; Cross et al., 1994) by:
— 0 Si =
d0 eiT PQvˆ Pei
(14)
In the calculation of the MDB the power of the test is usually held fixed at a reference value of say 80% (Baarda, 1968; Cross et al., 1994). External reliability A measure for external reliability is derived by evaluating the effect of the MDB on the estimated parameters. Such evaluation is achieved using (Baarda, 1968; Cross et al., 1994):
— 0 xˆ = Q x AT Pei — 0 S i
(15)
External reliability is solely concerned with the effect of undetected outliers upon the final solution. So one can only quantify the probability of detecting them if they do occur, and the effect they will have if they are not detected. Separability For the case where a blunder is large enough to cause many w-test failures, resulting in many alternatives, it is essential to insure that any two alternatives are separable. A measure of the separability of H ai and H aj is given by (1-r_), where r_ is the probability of committing a Type III error and depends on the correlation of the test statistics wi and w j. The degree of correlation of the two test statistics is determined through derivation of the correlation coefficient (Förstner, 1983; Tiberius, 1998):
r ij =
s ij s i2s 2j
=
eiT PQvˆ Pe j eiT PQvˆ Pei ⋅ eTj PQvˆ Pe j
(16)
where |_ij| ≤ 1. A correlation coefficient value of one and zero respectively indicates that the two test statistics are fully correlated and uncorrelated, respectively. The greater the correlation between two test statics, the more difficult it is to separate the outliers in the corresponding measurements. Therefore, there is a stronger probability for the ‘good’ measurement being flagged as an outlier when the w-test statistics are highly correlated. The degree of correlation of the w-test statistics is dependant on the strength of the geometry. A strong geometry will deliver weakly correlated w statistics. For situations where only five satellites are available, all measurements are fully correlated to one another. The separability (1-r_) between the alternatives Hai and Haj can be calculated for a given significance level a0 and non-centrality parameter _o and is then used in the derivation of the separability multiplying factor K_i j to extend the internal reliability to the localizability of outliers. The separability multiplying factor K _ij is determined as (e.g., Wang & Chen, 1994, Moore et al., 2002):
K r ij =
d 0, r ij d0
= K (a 0 , g 0¢ , r0¢ , r ij )
(17)
where _o,_ij is the critical value of the non-centrality parameter _o satisfying the conditions of the power of the combined test __0 of wi and w j and the separability of (1r_0) for the rejection of the null hypothesis. In other words, the non-centrality parameter _o must be large enough to permit the rejection of the null hypothesis for the assumed power and separability of the combined test. _o,_ij will always be greater than or equal to _o due to the correlation between the two test statistics. Localizability is expressed as:
— 0 S ij = K r ij — 0 S i
(18)
To obtain the lower bound value for an MDB which, can just be separated by the combined test, it is sufficient to consider only the maximum correlation coefficient _ijmax ("j ≠ i ) . In that case equation (18) becomes:
— 0 S ij max = K r ij max — 0 S i
(19)
GNSS RAIM PERFORMANCE STUDIES In order to analyse GNSS separability, a series of intensive simulation tests were conducted. UNSW GNSS (GPS and Pseudolite) measurement simulation and analysis software (Lee et al, 2002), originally designed for GPS and Pseudolite simulation, was modified by the
author to enable the following simulations involving the Galileo system. The analyses are actually based on the GPS and Galileo satellite and receiver coordinates. The GPS satellite coordinates were calculated by ephemeris (converted from the almanac files). The constellation that was implemented for Galileo was compiled from the following papers: Lucas and Ludwig (1999), Tytgat and Owen (1999), Ryan and Lachapelle (2000), Verhagen (2002) as it is still in the defining phase. The pseudoconstellation consists of 30 medium Earth orbit satellites in three perfect circular orbital planes with an inclination of 54 degrees and an altitude of 23,000km. The simulations also assume that the satellites are equally spaced with 10 satellites on each orbital plane.
Satellite Availability for Portland 20
16 s V S 14 f 12 o # 10 8 6 4 0:00 2:00 4:00 6:00 8:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 24:00 UT (h)
RAIM Performance in Sydney and Portland The simulations for locations in Sydney and Portland were carried out for 24 hours at a sample rate of 1 Hz commencing at 0:00 h on the 16th January, 2003. 24 hour simulations were chosen to show the time based variations of the simulated measures; maximum and minimum correlation coefficients and availability. Figures 1 and 2 display the satellite availability of the existing GPS constellation and the combined GPS/Galileo constellation at Sydney and Portland. From these results it is clear that the combined system offers far greater availability as the combined constellation consists of more than double the number of satellites in the GPS constellation.
Figure 2: Satellite Availability at Portland over 24hrs for GPS and combined GPS/Galileo. The results from the 24 hour Sydney and Portland simulations (see Figures 4 and 6) also indicate that there is a strong connection between satellite availability and the correlation of the w test statistics. The GPS/Galileo constellation offers a much better separation of outliers in measurements, as the system measurement redundancy and measurement geometry are far greater and stronger than for the existing GPS system. These results show that, in comparison with the GPS-only solution, the correlation coefficients of the integrated GPS/Galileo solution are improved by over 150%. GPS Internal Reliability for Sydney
Satellite Availability for Sydney
100
20 18
GPS GPS/Galileo
18
GPS GPS/Galileo
s 16 V S 14
s e r t e M
4 0:00 2:00 4:00 6:00 8:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 24:00 UT (h)
Figure 1: Satellite Availability at Sydney over 24hrs for GPS and combined GPS/Galileo. As expected, the internal and external reliability values for the GPS-only and combined GPS/Galileo solutions shown in Figure 3 are consistent with the satellite availability trends shown in Figure 1. The more satellites available, the more reliable the solution becomes. Huge improvements are apparent for the integrated GPS/Galileo system when compared with the GPS-only system. The integrated results are much more stable.
40 20
GPS/Galileo Internal Reliability for Sydney
# 10
6
60
0
f 12 o
8
Maximum Minimum Average
80
100 s e r t e M
80 60
Maximum Minimum Average
40 20 0 0:00 2:00 4:00 6:00 8:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 24:00 UT (h)
Figure 3: Internal and external reliability results at Sydney over 24hrs for GPS (top) and combined GPS/Galileo (bottom).
GPS Correlation Coefficients for Sydney 1 0.8 0.6 Maximum Minimum
0.4 0.2 0 GPS/Galileo Correlation Coefficients for Sydney 1
Maximum Minimum
0.8 0.6
‘Gaps’ in the GPS correlation coefficient data can be found for instances when the number of satellites is less than five (eg. at approximately 2:30 h for the GPS system, Sydney, Figure 4 and 8:20 h for the GPS system, Portland, Figure 6). Both sole GPS based scenarios display results showing periods, of about 20 minutes, where satellite availability is lower than five and hence RAIM and separability are unavailable. Again, the combined GPS/Galileo system is unaffected, in terms of RAIM and separability, as the combined satellite availability does not fall below ten.
0.4
Worldwide RAIM Separability Analysis
0.2 0 0:00 2:00 4:00 6:00 8:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 24:00 UT (h)
Figure 4: Correlation coefficients at Sydney over 24hrs for GPS (top) and combined GPS/Galileo (bottom). The localizability results (see Figure 5) are considerably improved for the combined system and appear identical to the internal reliabilities (see Figure 3) regarding the combined case. GPS Localizability for Sydney 100 s e r t e M
Maximum Minimum Average
80 60 40 20 0 GPS/Galileo Localizability for Sydney
The global results were obtained from snapshot simulations for 0:00 h on the 16th January, 2003 at 1 degree intervals of latitude and longitude and an altitude of 50m. Snapshot results permit analysis based on spatial variations as time is held constant. Simulated measures include maximum correlation coefficients and availability. The results from the global snapshot scenario are presented as orthographic global colour maps. ) g e D ( e d u t i t a L
Availability GPS 90 10 s V S 8 f o
60 30 0 -30
6 #
-60
100 s e r t e M
Maximum Minimum Average
80 60 40 20
0 0:00 2:00 4:00 6:00 8:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 24:00 UT (h)
Figure 5: Localizability results at Sydney over 24hrs for GPS (top) and combined GPS/Galileo (bottom) GPS Correlation Coefficients for Portland 1
) g e D ( e d u t i t a L
-90
-45 0 45 90 Longitude (Deg) Availability GPS/Galileo
135
180
4
90 22
s 20 V S 18 f 16 o
60 30 0 -30
14
#
-60 12 -90 -180 -135
0.8 0.6
Maximum Minimum
0.4
-90
-45 0 45 Longitude (Deg)
90
135
180
Figure 7: Availability for GPS (top) and combined GPS/Galileo (bottom)
0.2 0 GPS/Galileo Correlation Coefficients for Portland 1 0.8
-90 -180 -135
Maximum Minimum
0.6 0.4 0.2 0 0:00 2:00 4:00 6:00 8:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 24:00 UT (h)
Figure 6: Correlation coefficients at Portland over 24hrs for GPS (top) and combined GPS/Galileo (bottom).
Figure 8 is a presentation of the spatial variations of maximum correlation coefficients between two satellites for both constellations, GPS and GPS/Galileo. The results for the GPS/Galileo system show that the w-test statistics are far less correlated than those of the GPS system are. For the combined case, the maximum correlation coefficient is below 0.5 for the majority of the earth’s surface. The maximum correlation coefficient results for the GPS system cover the majority of the earth with values of 0.7 or greater. The more correlated the w-test
statistics are, the more poorly separated they are. Poor separation means that it may be impossible to distinguish an outlier in the measurements from another or others. If an outlier cannot be distinguished with a reasonable level of certainty, there is a large risk that, for a case where an outlier has been detected, the wrong measurement could be identified as the outlier. If this occurred then a good measurement would be removed and the outlier would remain and bias the navigation solution. It is also undesirable to remove more measurements than is 1absolutely necessary as this would weaken the measurement geometry and dilute the precision of the solution. Correlation coefficients between all possible outliers are equal to one for situations were only five satellites, or less, are available. If the minimum correlation coefficient is large, all the statistics for outlier detection at that point must be highly correlated and thus inseparable. This situation occurs when only five satellites are available (compare Figures 4 and 6 for the GPS only solutions). For the combined system, the minimum correlations are equal or close to zero worldwide for these instants. ) g e D ( e d u t i t a L
) g e D ( e d u t i t a L
CONCLUSIONS A RAIM algorithm with reliability and separability assurance measures has been presented and the abilities of the current and existing GPS and the anticipated combined GPS/Galileo system to distinguish and separate measurements have been investigated and compared. Through simulations it has been shown that the advent of Galileo will markedly lower correlations between the statistics for the outlier detection through the provision of greater satellite availability. This, in turn, will improve RAIM performance, as it will enable another parameter, namely separability, to be monitored. Separability monitoring will provide users with a level of assurance that a good measurement will not be flagged as an outlier, and more importantly, the outlier can be easily excluded, ensuring the robustness of the GNSS navigation solutions. Based on the results presented in this paper, separability of the integrated GPS/Galileo solution is expected to improve by over 150% in comparison with the GPS-only solution. ACKNOWLEDGEMENTS
Max Correlation Coefficient GPS 90
1
60
0.9
30 0.8
0 -30
0.7
-60
0.6
-90 -180 -135
-90
-45 0 45 90 135 Longitude (Deg) Max Correlation Coefficient GPS/Galileo
REFERENCES
180
90
0.8
60
Baarda W. (1968) A testing procedure for sse in geodetic networks, Netherlands Geodetic Commission, New Series, Vol. 2, No. 4.
0.7
30 0.6
0
0.5
-30
Cross P.A., Hawksbee D.J., & Nicolai R. (1994) Quality measures for differential GPS positioning, The Hydrographic Journal, Hydrographic Society, 72, 17-22.
0.4
-60 -90 -180 -135
The author would like to thank Dr. Jinling Wang, Prof. Bill Kearsley and Prof. Chris Rizos at UNSW for their encouragement and support, my fellow students HungKyu Lee and Michael Moore for their assistance, and Ms. Sandra Verhagen from Delft University of Technology for her help in presenting the results over the world map.
-90
-45 0 45 Longitude (Deg)
90
135
180
0.3
Figure 8: Maximum correlation coefficient (_) for GPS (top) and combined GPS/Galileo (bottom) As these results are snapshots in time it should be noted that the areas affected by insufficient availability to permit RAIM and separability functions (ie Satellites less than five), move and vary in size and number. The results from Figures 4 and 6 also illustrate the benefits of the advent of Galileo, where marked improvements in satellite availability and subsequently, greater separability of outliers in measurements will occur.
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