Overbounding Revisited: Discrete Error-Distribution Modeling for Safety-Critical GPS Navigation

For a growing number of aviation applications of the Global Positioning System (GPS) and of other Global Navigation Satellite Systems (GNSS), it is essential to establish a rigorous bound on measurement error. Most existing bounding methods rely on representing the actual measurement error distribution with a conservative, continuous model (e.g., a Gaussian "overbound"). We propose a conservative, discrete model as a practical alternative. A key limitation of continuous error models is validation, particularly in the distribution tails where comparatively little statistical data is available. With a discrete model, it is easy (1) to define a minimally conservative core region, where data are plentiful, and (2) to introduce a highly conservative tail region, where data are sparse. The trade-off is increased computational complexity, as no closed-form expression exists for convolution of non-Gaussian error distributions. We propose a particular form of a discrete error distribution, which we call the NavDEN model. Through application to a heavy-tail GPS data set, we demonstrate that the NavDEN model compares favorably to Gaussian models, both in providing more margin for tail uncertainty and, at the same time, in providing generally tighter protection levels (PLs) when multiple distributions are convolved.