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.

GNSS 定位中的残余误差特征对于故障检测和完整性监测至关重要。尽管零均值高斯假设在导航界得到广泛使用，但研究强调了残差误差中的非高斯特征和重尾模式。对于处于困难环境中的用户来说，这个问题将更具挑战性，因为残差误差由多种模式的组合组成，具有很高的复杂性，并且无法用已知的分布或经验模型进行拟合。为了解决这些问题，我们的工作引入了一种利用 Wasserstein 距离来评估错误表征和故障建模性能的新方法。然而，仅仅依靠 Wasserstein 距离值进行直接相似性评估会因其对维度的依赖性而受到阻碍。我们提出了一种基于二阶高斯 Wasserstein 距离的精度度量，以根据拟合优度和基本假设对 GNSS 误差模型进行定量评估。我们还建立了稳健的评分标准来区分各种 GNSS 误差模型，确保综合评估。所提出的方法通过已知的高维高斯模型进行了验证，在样本量为 10,000 的情况下，获得了 100 分以上的 99.95 分。为了证明处理复杂性的能力，我们的方法建立并评估了两个多元复杂 GNSS 模型，其中结合了 copula 函数来捕获复杂的维度间相关性。实验结果表明，该方法通过建立不同维度的通用标准，可以有效地实现过错良性模型的评估。它提供了拟合优度的定量测量，并增强了建模以反映现实，从而解决了上述问题。此外，通过该技术，可以选择接近真实的故障模型来生成模拟的故障数据集，从而有利于算法的测试和改进。这也有利于更准确的完整性风险评估，以避免超出或低于范围导致错误或错过警报。