Computing K Factors for all Integrity Needs
Julie Antic, Odile Maliet, Sébastien Trilles, Thales Alenia Space
Date/Time: Thursday, Sep. 22, 10:40 a.m.
Integrity risk (IR) is the probability that an hazardously misleading information lasts for more than time to alert, at least once over a time interval T (typically, T equals 150 seconds for aviation precision approach). At user level, integrity is ensured thanks to Protection Levels (PL) computation described in appendix J of MOPS standard: the PL is the product of the estimated variance of errors (provided by SBAS) and a scaling factor, referred as K factor or K value. The K factor can also be seen as a quantile of the normalized residual errors (also called SFI for SaFety Index) distribution.
K factor plays a crucial role on SBAS service performance: decreasing K factor improves SBAS availability and continuity, but could lead to a unacceptable IR. MOPS standard introduced K factor values for several aviation approaches: for instance, 5.33 for the CATI vertical protection level (IR=10-7/150s). However, MOPS standard (Appendix §J5) recognized that “the values of KH,NPA, KH,PA, and KV were originally chosen to be consistent with certain assumptions on the distribution of position error and on error correlation time. It was then realized that these assumptions may not hold under all conditions, but that the choice of values is somewhat arbitrary.”. The arbitrary of MOPS K factors made them difficult to generalize to IR or time interval length differing from the aviation standards.
Today, there is a growing interest for generalization of integrity concept to additional safety of life applications. They include more stringent aviation approaches, and, beyond aviation, maritime, rail, automotive, and unmanned aerial vehicles users. These new applications involve new integrity risk targets, such as 10-5/150s for aviation special authorized CATI, 10-5/3 hours for maritime, 10-9/h for rail… The problem is to derive K factors that are compliant with each specific integrity risk, and coherent with a common integrity risk at service level.
This paper describes a rigorous method to compute K factor for any integrity risk (like 10-5, 10-7, …) and time interval T (like 150 seconds, 1 hour or 3 hours), that is coherent with a common assumption at service level: stationarity and Gaussianity of SFI.
The proposed method is divided in two steps. The first step is the over-bounding of SFI power spectral density using a Gauss-Markov process with correlation time ?. We demonstrate that such over-bounding guarantees that a K factor computed for the Gauss-Markov process bounds integrity risk even if SFI is not a Gauss-Markov process. In practice, this over-bounding step could be performed once for all at system level and documented in standards. The second step is the computation of K factor for a Gauss-Markov process with correlation time ?. To our knowledge, no explicit formulation is available for Gauss-Markov process but we use theoretical results on the maximum of its the time-continuous counterpart, the Ornstein-Uhlenbeck process. We provide easy-to-implement formulations for integrity risk related to errors in one dimension (like vertical or horizontal cross-track errors) and two dimensions (like horizontal errors). These formulations have been validated with respect to alternative formulations, when available, and simulations. They are then applied to several new integrity needs like aviation special authorized CATI, maritime and any integrity risk per hour between 10-4 and 10-9.
The new method proposed for K factor computation has several strengths. It relies on rigorous theoretical grounding with solid probability justification, which makes it applicable to any integrity requirement. No restrictive assumption is needed on the time correlation pattern of the errors. Finally, the method is easy-to-implement and applicable to any non-integrity risk and time interval T. This method is currently under implementation in the frame of ESA H2020 H044.01 EGNOS Next studies.
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