EVT-SIAM: A tool based on extreme-value theory for the assessment of SBAS accuracy and integrity

Following the start of WAAS extensions in Canada, in Alaska and in Mexico, as well as the start of WAAS system evolutions between 2005-2008, and considering the noticed improvements of performances and procedures, the FAA suggested widening the WAAS services up to LPV200 (aka. category I approaches), with a vertical alert limit (VAL) of 35m. This objective has also been assigned to EGNOS in Europe, and ICAO Annex 10, Volume 1 has been amended in that direction (Amendment #85). Though most of the specifications of LPV200 are identical to those for APV1 service level (where VAL = 50m) in terms of integrity, horizontal alert limit, availability and continuity, different or additional specifications have been introduced in the field of positioning accuracy. Indeed, the specification of APV1 accuracy - that the 95th centile of the horizontal error (H-NSE) be below 16 m, and the 95th centile of the vertical error (V-NSE) be below 20 m - becomes for LPV200 in the vertical domain that - V-NSE-95% < 4m - Proba(VPE-1sec > 10m) < 1E-7 in the absence of failure everywhere the operation is to be approved; - Proba(VPE-1sec > 15m) < 1E-5 in case of failure. The specification of APV1 and LPV200 integrity is Proba(VPE > VAL or HPE > HAL) < 2E-7 per 150 s where VAL (resp. HAL) the vertical (resp. horizontal) alert limits values are fixed by the International Civil Aviation Organisation for different flight phases. The failures to be taken into account are the ones that affect the used basic constellations and GNSS augmentation systems. This latter probability results from an allocation that takes into account the probability that a given failure occurs and of the probability of detection. Verifying such low probabilities with standard statistical methods require several months or years of observation data, even with a 1-second accuracy sampling. This is not realistic and compatible with industrial constraints that require system qualification time scales. For instance, such rare "events" are generally not observed within the data used to qualify a SBAS release because of their scarcity. This motivates the development of non-standard verification approaches. The statistical Extreme Value Theory (EVT) consists in an extrapolation of the error distributions tails under rough and conservative power law assumptions. The approach is not new (Fisher-Tippet, 1928), but recent developments in quantile estimation have allowed its application in numerous domains. It happens to be suited to cases regardless of the underlying measurement error distributions. It avoids in particular the questionable assumption of Gaussian error distributions, which implicitly assumes exponential fast decaying tails. EVT derives instead some properties of the distribution tails from the measured data. This allows a meaningful extrapolation into the low-probability region, even when no (or very limited amounts of) samples are available. Following two previous CNES funded studies that revealed full relevance of this approach for the verification of system integrity [4] and accuracy requirements, a tool (EVT-SIAM: Extreme Value Theory Supporting Integrity and Accuracy Measurement) has been defined, prototyped and industrialized. This allows implementing this method in the verification of Navigation system performance. Thus, the EVT-SIAM Tool is a SBAS statistical analysis tool able to characterize or verify the behaviour of the tail of an error distribution deriving from measurement data. This tool, avoiding the assumption of Gaussian error distributions, allows extrapolating meaningfully the data into the region of misleading information, even when no (or very limited amounts of) sample values in this region are available. EVT-SIAM Tool is dedicated to integrity and accuracy of SBAS requirements. With a given confidence level, a quantile can be provided by EVT-SIAM Tool for the SBAS requirements analysis mentioned above. The paper is organized in the following way. - A first part presents the main lines of the EVT approach and the Pareto-law assumption, and the conclusions of the previous studies are recalled. - The use cases (e.g. local user or global assessment, accuracy or integrity requirement assessment, orbit/clock or ionosphere correction etc areas) of the tool are described in a second part, and the main requirements that result are presented. - The tool architecture is explained, with its five modules. Two first modules monitor the input data and the configuration parameters. A third one adapts the navigation data to the constraints of assessing them against a Pareto distribution law. A fourth module validates the main assumptions (non correlation, cluster, stationary...) for the validity of the extreme value domain of attraction. The last module estimates the parameters of the error distribution law and deduces the values of the figures of merit to be compared to the requirements. - The ways to enforce representativeness and robustness of the computation results are then focused on. - The way the tool has been verified is then presented, and some results are given. Finally, recommendations are formulated for using this technique in future assessment of such demanding GNSS LPV 200 accuracy performance requirements as for new EGNOS releases for which LPV200 is required.