A Wasserstein distance-based technique for the evaluation of GNSS error characterization

Abstract

The characteristics of residual errors in GNSS positioning are crucial for fault detection and integrity monitoring. Despite the wide use of the zero-mean Gaussian assumption in the navigation community, studies highlight non-Gaussian traits and heavy-tailed patterns in residual errors. The problem will be even more challenging for users in difficult environments where residual errors consist of a combination of multiple modes with high complexity and cannot be fitted with known distributions or empirical models. To address these issues, our work introduces a novel approach leveraging the Wasserstein distance for assessing the performance of error characterization and fault modeling. However, relying solely on the Wasserstein distance value for direct similarity assessment is hindered by its dependency on dimensionality. We propose a second-order Gaussian Wasserstein distance-based precision metric to offer a quantitative evaluation of GNSS error models in terms of both goodness-of-fit and underlying assumptions. We also establish a robust scoring criterion to distinguish between various GNSS error models, ensuring comprehensive evaluation. The proposed method is validated through a known high-dimensional Gaussian model, achieving a score of 99.95 over 100 with a sample size of 10,000. To demonstrate the capability in dealing with complexity, two multivariate complex GNSS models incorporating copula functions to capture intricate inter-dimensional correlations are established and assessed by our approach. Experimental results show that the method can effectively deliver the evaluation of goodness-of-fault models using the establishment of a universal criteria with different dimensions. It provides a quantitative measure on the goodness of fittings and enhances the modeling to reflect the reality, therefore solving the problems raised above. In addition, with this technique, the close-to-reality fault models can be chosen to generate simulated faulty datasets, thus benefiting algorithm testing and improvement. This is also beneficial to more accurate integrity risk assessment to avoid overbounding- or underbounding-resulted false or missed alert.