Cauchy-Gaussian Overbound for Heavy-tailed GNSS Measurement Errors

2026-01-12

AI Summary

This study addresses the challenge of constructing tight upper bounds for heavy-tailed GNSS measurement errors in navigation integrity monitoring, which are difficult to model accurately with conventional approaches. To this end, the authors propose a novel piecewise hybrid bounding method that integrates a Cauchy distribution to characterize the central region and a Gaussian distribution to control the tails. This approach provides, for the first time, a unified and tight envelope for both symmetric and asymmetric unimodal heavy-tailed error distributions while preserving the upper-bound property under convolution, thereby ensuring theoretical robustness. Experimental results demonstrate that, in the position domain, the proposed method reduces the vertical protection level by an average of 15% under symmetric unimodal errors and achieves reductions of 21%–47% compared to existing methods under asymmetric unimodal errors.

Technology Category

• Intelligent Robots: State Estimation
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Overbounds of heavy-tailed measurement errors are essential to meet stringent navigation requirements in integrity monitoring applications. This paper proposes to leverage the bounding sharpness of the Cauchy distribution in the core and the Gaussian distribution in the tails to tightly bound heavy-tailed GNSS measurement errors. We develop a procedure to determine the overbounding parameters for both symmetric unimodal (s.u.) and not symmetric unimodal (n.s.u.) heavy-tailed errors and prove that the overbounding property is preserved through convolution. The experiment results on both simulated and real-world datasets reveal that our method can sharply bound heavy-tailed errors at both core and tail regions. In the position domain, the proposed method reduces the average vertical protection level by 15% for s.u. heavy-tailed errors compared to the single-CDF Gaussian overbound, and by 21% to 47% for n.s.u. heavy-tailed errors compared to the Navigation Discrete ENvelope and two-step Gaussian overbounds.

Problem

• heavy-tailed errors
• GNSS measurement errors
• integrity monitoring
• overbound
• Cauchy-Gaussian

Innovation

• Cauchy-Gaussian overbound
• heavy-tailed errors
• integrity monitoring
• GNSS
• protection level