Principal Gaussian Overbound for Heavy-Tailed Error Bounding

A sharp yet conservative overbound for heavy-tailed error distributions is essential in integrity monitoring applications due to availability and continuity constraints. This article proposes the principal Gaussian overbound (PGO) for heavy-tailed error distributions by leveraging the characteristics of the Gaussian mixture model. The overbounding property of the PGO is proved to be preserved through convolution, which makes it possible to derive pseudorange-level requirements from the position domain integrity requirements. Experimental results on two datasets show that the PGO provides the most competitive bounding performance for heavy-tailed differential global navigation satellite system (GNSS) pseudorange errors when compared to the two-step Gaussian overbound and Gaussian-Pareto overbound, yielding a sharp bound in both the core and tail parts of the error distribution. The proposed method reduces the mean vertical protection level (VPL) by more than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$78\%$</tex-math></inline-formula> compared to the two-step Gaussian overbounding method on the urban dataset. In addition, the mean computation time of VPL is only <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$0.08 \,\,\mathrm{s}$</tex-math></inline-formula> with 15 measurements by employing fast Fourier transforms, suggesting the substantial potential of the PGO in GNSS applications with strict integrity and real-time requirements. Furthermore, the feasibility of the PGO in fault detection is discussed.