Does the Hpl Bound the Hpe?

The Horizontal Position Error (HPE) must be no greater than the confidence bound, known as the Horizontal Protection Level (HPL), beyond the specified probability (integrity risk). The definition of protection levels is one of the keys to integrity in Satellite Based Augmentation Systems (SBAS) as the Wide Area Augmentation System (WAAS) and the European Geostationary Navigation Overlay Service (EGNOS), with the main application of providing guidance to aircraft during approach. In this contribution we show and explain, also by means of geometrical interpretation, that of the two defined HPLs, theoretically — with the HPE as the radial error in the two dimensional horizontal plane — one provides a lowerbound (for the probability, and hence is safe), but that the other one leads to an upperbound (hence overestimating the probability contained within the HPL-area). These effects are noticeable in the size of the integrity risk. We present, as an alternative, an exact evaluation of the circular probability, and for completeness we include the Chebyshev inequality, which provides a lowerbound for the probability regardless the actual distribution of the position error. The measures are mutually compared, and assumptions and relations are pointed out. INTRODUCTION Is the Horizontal Position Error (HPE) bounded by the Horizontal Protection Level (HPL), with the required and specified probability? That is the central question of this contribution. In operational circumstances the actual Position Error (PE) can not be assessed, only in controlled experiments an accurate ground-truth for the position or trajectory can be available. In practice a statistical measure on the position error has instead to represent (in part) navigational performance. In satellite navigation, and with a Satellite Based Augmentation System (SBAS) in particular, the use of the Protection Level (PL) as a statistical bound to the Postion Error (PE) is common place. In one dimension, setting the interval length that bounds the error in one component of the position, given the probability, is usually straightforward, once the distribution of the position error is available. In two (or more) dimensions, this may get complicated by statistical dependence between the position components (and by different possibilities for the geometric area or volume to bound the error). In aviation typically the (one-dimensional) vertical component of the user position error is of primary concern; the horizontal components are of secondary concern. This situation may be just the other way around in other applications as positioning and navigation on land and at sea. This formed the incentive to the underlying study. In general we would like to capture the two-dimensional position error in a measure or statement that relates the radial error to probability, preferably by exact means. The adjective ‘radial’ translates into a circle as the area that contains the position error in two dimensions. There is no distinction between the two components. When an exact assessment is not possible, one should underbound the probability or conversely overbound the radius, in order to be ‘on the safe side’. The performance measure shall be conservative, as in the end, it is all about ensuring user safety. There exists a large amount of literature on scalar measures to evaluate or bound multi-dimensional errors, see e.g. [1] for the two-dimensional case, and the topic is also addressed in [2]. In this contribution we review specifically two measures that have been outlined explicitly in earlier versions of the Minimum Operational Performance Standards (MOPS) for SBAS [3]. First, the probability contained in the ellipse just enclosed by the circle provides a lowerbound for the circular probability (and hence is on the safe side). Second, evaluation of the probability in only the worst case direction (and neglecting the other) leads to upperbounding the desired circular probability (and thereby does not represent guaranteed navigational performance). Then we present an exact measure of probability on the horizontal position error radius. To the authors’ knowledge this measure has not been in use yet in the context of SBAS. The basic assumption for all three measures so far is a normally distributed position error. To put this assumption into perspective also the multi-variate Chebyshev inequality is used, which provides a lowerbound for the circular probability, irrespective of the position error distribution. In the last section all four measures are systematically compared in five numerical examples.