Cauchy–Gaussian Overbound for Heavy-Tailed GNSS Measurement Errors

Zhengdao Li; Penggao Yan; Weisong Wen,; Li-Ta Hsu

doi:10.33012/navi.749

Cauchy–Gaussian Overbound for Heavy-Tailed GNSS Measurement Errors

Zhengdao Li
Zhengdao Li
Department of Aeronautical and Aviation Engineering, The Hong Kong Polytechnic University, Hong Kong, China
Penggao Yan
Penggao Yan
Department of Aeronautical and Aviation Engineering, The Hong Kong Polytechnic University, Hong Kong, China
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- For correspondence: [email protected]
Weisong Wen,
Weisong Wen,
Department of Aeronautical and Aviation Engineering, The Hong Kong Polytechnic University, Hong Kong, China
and Li-Ta Hsu
Li-Ta Hsu
Department of Aeronautical and Aviation Engineering, The Hong Kong Polytechnic University, Hong Kong, China

NAVIGATION: Journal of the Institute of Navigation
March 2026,
73
navi.749;
DOI: https://doi.org/10.33012/navi.749

Abstract

Overbounds of heavy-tailed measurement errors are essential for meeting 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 global navigation satellite system measurement errors. We develop a procedure to determine the overbounding parameters for both symmetric unimodal (SU) and non-symmetric unimodal (NSU) heavy-tailed errors and prove that the overbounding property is preserved through convolution. Experiment results on both simulated and real-world data sets reveal that our method can sharply bound heavy-tailed errors in both the core and tail regions. In the position domain, the proposed method reduces the average vertical protection level by 15% for SU heavy-tailed errors compared with the single-cumulative-density-function Gaussian overbound and by 21%–47% for NSU heavy-tailed errors compared with the navigation discrete envelope and two-step Gaussian overbounds.

Keywords

1 INTRODUCTION

The reliability and safety of positioning solutions have become critical concerns for global navigation satellite systems (GNSSs). Correspondingly, safety-of-life systems have been established, such as satellite-based augmentation systems, ground-based augmentation systems (GBASs), and receiver autonomous integrity monitoring (Brown, 1992; Federal Aviation Administration, 2010; United States. Dept. of Transportation & United States. Dept. of Defense, 2000; Walter & Enge, 1995). These systems are designed to facilitate highly robust and precise GNSS positioning solutions and to maintain integrity risks within the probability of hazardously misleading information events (P_HMI). As an alternative to integrity risk evaluation, the protection level (PL) can be derived in the position domain to measure the maximum tolerable positioning error boundary under P_HMI (Antic et al., 2023; Blanch et al., 2018; Elsayed et al., 2023).

To ensure the integrity of navigation systems, it is crucial to precisely characterize the measurement error, as the stochastic properties of the measurement error will be projected into the position domain (Lee et al., 2009). However, the limited bandwidth channels of GNSS augmentation systems hinder the transmission of complicated error profiles to users (Blanch et al., 2018). Therefore, a simpler and more conservative error boundary (namely, overbound) is commonly employed in navigation systems (Blanch et al., 2005, 2018; Gao et al., 2022; Rife et al., 2004b; Rife et al., 2006), which compromises between accurate error modeling and the complexity of broadcasting a larger number of parameters. Overbounds represent the worst error distribution in the absence of hardware failure and are designed to guarantee that the integrity risk remains below an acceptable level (Antic et al., 2023; DeCleene, 2000; Rife et al., 2004a; Xia et al., 2024). A sharp overbound (i.e., closely matching the original error distribution) helps to reduce the PL and facilitates the development of high-availability integrity-monitoring algorithms. Moreover, the overbound should also be explicitly parameterized to be conveniently broadcast to users (Blanch et al., 2015, 2018).

Gaussian overbounding techniques are commonly used to characterize GNSS error distributions. The basic single-cumulative-density-function (CDF) Gaussian overbound (DeCleene, 2000) assumes strictly symmetric unimodal (SU) errors, a limitation addressed by the paired Gaussian overbound (Rife et al., 2006) and more advanced methods such as the two-step Gaussian overbound (Blanch et al., 2018). The classic Gaussian-based methods established the framework for CDF overbounds, as detailed in Section 2. However, a persistent challenge is the conservative performance of Gaussian overbounds when applied to error distributions with heavy-tailed properties (Blanch et al., 2018; Huang et al., 2016; Yan et al., 2024), which are common in ephemeris/clock errors and multipath errors (Foss et al., 2013; Heng et al., 2011; Karaim et al., 2018; Yan et al., 2025; Zhu et al., 2018). It has been shown that typical heavy-tailed GNSS error distributions include large errors (beyond 2–3 standard deviations) that occur with a higher-than-Gaussian frequency, even though the core region may still be well represented by a Gaussian shape (Rife & Pervan, 2012). When visualized as a probability density function (PDF), a heavy-tailed distribution exhibits tails that decay more slowly than a Gaussian distribution, resulting in a sharper peak and prolonged tails. Generally, the “heavy-tailedness” of a distribution is often measured by its kurtosis (Balanda & Macgillivray, 1988; Feldmann & Whitt, 1998). To address the properties of heavy-tailed errors, Gaussian overbounds employ a large sigma (σ). While this approach may successfully bound the tails, it often creates a significant separation between the overbound and the empirical data in the core region, introducing unnecessary conservatism.

Non-Gaussian overbounding methods have been developed to address the challenges posed by heavy-tailed errors. Rife et al. (2004a) proposed the core overbounding method, which uses a Gaussian-core Gaussian-sidelobe distribution to provide less-conservative bounding on the core and tail regions separately. Blanch et al. (2008) proposed a zero-mean bimodal Gaussian mixture model (BGMM)-based method to add a heavy-tailed Gaussian component in constructing the overbounding distribution. Rife and Pervan (2012) proposed the navigation discrete envelope (NavDEN), a symmetric and discrete overbound with a tight Gaussian core and flared tails designed to model heavy-tailed distributions. However, the low grid resolution of the coarse discrete models may limit the bounding performance. Xue et al. (2017) adopted a stable distribution for bounding GBAS ranging errors and validated the feasibility using simulated error samples. Larson et al. (2019) developed the Gaussian-Pareto overbound, which applies the extreme value theory to generate a sharp tail bound. However, the Gaussian-Pareto overbound does not inherently satisfy the SU requirement defined by DeCleene (2000). Consequently, the overbound cannot be analytically proven to preserve overbounding properties through convolution, leaving its performance in the position domain unverified. Recently, Yan et al. (2024) proposed the principal Gaussian overbound (PGO) to tightly bound original errors in both the core and tail regions by strategically inflating and shifting the Gaussian components of a fitted BGMM for the error distribution. Although it demonstrates promising bounding performance for heavy-tailed errors, the PGO was developed based on a zero-mean assumption for the error distribution, which is not usually satisfied in real-world applications.

In this work, we aim to develop a systematic method to bound heavy-tailed error distributions, including both SU profiles with a zero bias and non-symmetric unimodal (NSU) profiles with a non-zero bias. The concept of core overbounding (Rife et al., 2004a) is used to separately bound the core and tail regions of the empirical error distribution. Unlike conventional non-Gaussian overbounding methods, which employ non-Gaussian distributions for tail bounding and Gaussian distributions for core bounding, we propose leveraging the intrinsically heavy-tailed properties of the Cauchy distribution to sharply bound the core region of empirical errors. The Cauchy core is then transitioned to the Gaussian distribution to tightly bound the tails of the empirical errors. The Cauchy distribution (Lorentzian in physics) was first applied to describe spectral lines due to homogeneous broadening (Born & Wolf, 2013; Cauchy, 1840). The naturally heavy-tailed property and the explicit PDF and CDF forms of the Cauchy distribution have led to the application of this distribution in various areas, including catastrophe predictions in computational finance (Mahdizadeh & Zamanzade, 2019), rainfall probability modeling in hydrology (Kassem et al., 2021), and loss functions in machine learning (Liu et al., 2022). We found that the heavy-tailed Cauchy distribution possesses significantly more prolonged tail regions and thus a sharper core region than the Gaussian distribution. This property allows the Cauchy-based overbounds to closely adhere to the core of empirical error distributions with heavy tails while diverging from the tails. In contrast, the Gaussian distribution allocates a considerably higher proportion of cumulative mass at its core, and this mass exponentially decreases as the error magnitude gradually rises, resulting in short tail regions. Hence, the Gaussian overbound can sharply bound the tails of empirical errors but exhibits large separations away from the core. Thus, following the core overbounding framework (Rife et al., 2004a), we seek to balance between bound sharpness and operational simplicity by separately generating bounds for the core and tail regions. Therefore, naturally, we seek to combine the strengths of the Cauchy distribution in the core region with those of the Gaussian distribution in the tail regions to bound heavy-tailed error distributions. This concept constitutes the main content of our work.

This paper proposes the Cauchy–Gaussian overbound for both SU and NSU error distributions, where an explicit framework is developed to determine the optimal parameters of these overbounds. Inspired by the Gaussian overbounding methods (Blanch et al., 2018; DeCleene, 2000; Rife et al., 2006), the Cauchy–Gaussian overbound is designed to function as a single-CDF bound for SU errors and as paired bounds for NSU errors. We benchmark the bounding performance of the proposed method with the single-CDF Gaussian overbound (DeCleene, 2000) for SU errors using a simulated data set and with the two-step Gaussian overbound (Blanch et al., 2018) for NSU errors using a real-world data set. Results show that the proposed Cauchy–Gaussian overbound can more tightly bound heavy-tailed errors in both the core and tail regions. In the position domain, the proposed method can reduce the vertical PL (VPL) by roughly 15% on average for SU errors. For NSU errors, the improvement is greater, with average VPL reductions of 21% and 47% compared with the NavDEN and two-step Gaussian overbounds, respectively.

The contributions of this work are three-fold:

We propose the Cauchy–Gaussian CDF overbound for both SU and NSU heavy-tailed error distributions. A procedure for determining the overbound parameters is developed.
We prove that the overbounding properties of the proposed overbound are preserved through convolution, which provides theoretical support for using the Cauchy–Gaussian CDF overbound in integrity applications.
We validate the bounding performance of the proposed method in both the range and position domains for both SU and NSU error distributions, using simulated and real data sets.

The remainder of this paper is organized as follows: Section 2 reviews three classic Gaussian overbounding methods. Section 3 develops the Cauchy–Gaussian overbound for both SU and NSU errors. Section 4 compares the bounding performance of the Cauchy–Gaussian overbound, single-CDF Gaussian overbound, and two-step Gaussian overbound in both the range and position domains through numerical simulations and real data set validations. The impacts of heavy-tailed empirical errors with non-zero bias on bounding performance are investigated in Section 4.3. Finally, Section 5 concludes this study and proposes prospective research directions.

2 REVIEW OF GAUSSIAN OVERBOUNDING METHODS

The Gaussian distribution has been widely applied in overbounding methods, owing to its simple parameterization and property of preserving overbounding nature through convolution (Larson et al., 2019; Rife et al., 2006; Yan et al., 2024). In this section, three classic Gaussian overbounding methods are reviewed, including (1) the single-CDF Gaussian overbound (DeCleene, 2000), (2) the paired Gaussian overbound (Rife et al., 2006), and (3) the two-step Gaussian overbound (Blanch et al., 2018). The paired Gaussian overbound has inspired the proposal of the Cauchy–Gaussian overbound in this paper, whereas the Gaussian single-CDF overbound and the two-step Gaussian overbound will serve as a baseline for evaluating the bounding performance in Section 4. For the sake of notation, F indicates the CDF and f indicates the PDF in the remainder of this paper.

2.1 Single-CDF Gaussian Overbound

DeCleene (2000) introduced the first successful CDF overbound on SU distributions. Specifically, the CDF overbound for SU errors is given as follows:

$\begin{matrix} F_{o b} (x) \geq F_{e} (x) & \forall F_{e} \leq \frac{1}{2} \end{matrix}$ 1

$\begin{matrix} F_{o b} (x) \leq F_{e} (x) & \forall F_{e} > \frac{1}{2} \end{matrix}$ 2

where F_ob and F_e denote the CDFs of the overbounding distribution and the empirical error distribution and both distributions are assumed to have a zero mean. When F_ob is represented by the CDF of a Gaussian distribution, the tightest single-CDF Gaussian overbound can be determined by finding the minimum scale parameter that satisfies the above inequalities. Decleene proved that if both the empirical errors and overbounds are SU distributions, the projection of the range-domain overbound to the position domain also bounds the positioning errors, which mathematically means that the overbounding properties are preserved through convolution (DeCleene, 2000). Despite laying the foundation for CDF overbounds, Decleene’s method strictly relies on SU error and overbounding distributions and is inadequate for real-world GNSS measurements, which commonly exhibit asymmetry, multi-modality, and biases (Blanch et al., 2018; Rife et al., 2006).

2.2 Paired Gaussian Overbound

The paired overbounding method expands the application scope of Decleene’s method from bounding SU errors to random error profiles, by introducing a pair of bounds (Rife et al., 2006). Specifically, the left and right (indicated by subscripts L and R, respectively) Gaussian overbounds preservable through convolution are given by the following:

$F_{L} (x) = F_{G} (x; μ_{L}, σ_{L}) \geq F_{e} (x) \forall_{x}$ 3

$F_{R} (x) = F_{G} (x; μ_{R}, σ_{R}) \leq F_{e} (x) \forall_{x}$ 4

where F_G represents the Gaussian CDF, µ and σ denote the location and scale parameters¹ (i.e., mean and standard deviation) of the Gaussian distributions, and µ_L=–µ_R. Following the definition in Equations (1) and (2), an analog single-CDF overbound F_ob based on the paired overbound can be constructed as follows:

$F_{o b} (x) = {\begin{cases} F_{L} (x) \forall F_{L} < \frac{1}{2} \\ \frac{1}{2} otherwise \\ F_{R} (x) \forall F_{R} > \frac{1}{2} \end{cases}$ 5

Notably, the formed F_ob is used only for visualization purposes and will not be used in calculating the projection of the range-domain overbound to the position domain. Overbounding via paired distributions can effectively address non-symmetric or non-unimodal empirical errors. However, finding the paired bounds using a Gaussian distribution typically results in large location parameters (µ_L and µ_R), which unavoidably increases the conservatism in bounding the measurement error and eventually increases the PL in the position domain.

2.3 Two-Step Gaussian Overbound

Based on the paired overbounding method, Blanch et al. (2018) proposed a state-of-the-art two-step Gaussian overbound to further tighten the overbounding distribution toward the empirical profile. This aim is achieved through an ad hoc SU distribution f_su(t;b_su), with a non-zero location parameter b_su. The f_su serves as an intermediate pair of left and right CDF bounds, according to the inequalities in Equations (3) and (4):

$\int_{- \infty}^{x} f_{s u_{L}} (t; b_{s u_{L}}) d t \geq F_{e} (x) \forall x$ 6

$\int_{- \infty}^{x} f_{s u_{R}} (t; b_{s u_{R}}) d t \leq F_{e} (x) \forall x$ 7

Given that both f_su and the Gaussian distribution are SU, the second step aims to find the minimum scale parameters σ_L and σ_R for the left and right regions to satisfy the following:

$F_{G} (x; b_{s u_{L}}, σ_{L}) \geq \int_{- \infty}^{x} f_{s u_{L}} (t; b_{s u_{L}}) d t \forall x \leq b_{s u_{L}}$ 8

$F_{G} (x; b_{s u_{R}}, σ_{R}) \leq \int_{- \infty}^{x} f_{s u_{R}} (t; b_{s u_{R}}) d t \forall x \geq b_{s u_{R}}$ 9

Finally, the left and right Gaussian CDF bounds are obtained as follows:

$\begin{matrix} F_{L} (x) = F_{G} (x; - b_{f}, σ_{f}) & \forall x \leq - b_{f} \end{matrix}$ 10

$\begin{matrix} F_{R} (x) = F_{G} (x; b_{f}, σ_{f}) & \forall x \geq b_{f} \end{matrix}$ 11

where b_f = max(| b_{su_L} |,| b_{su_R} |) and σ_f = max(σ_L,σ_R). Furthermore, the two-step Gaussian overbounding method maintains the overbounding properties through convolution, given that the inequalities in Equations (6)–(9) are satisfied. Compared with the conventional paired Gaussian overbound in Section 2.2, the two-step Gaussian overbounding method yields a significantly smaller location parameter b_f than the paired Gaussian overbound, while not increasing the number of parameters (i.e., location and scale) in the overbounding distributions to be broadcast. However, the involvement of the intermediate f_su can still make the error bounds fairly conservative. For heavy-tailed and biased empirical errors, the two-step Gaussian overbound would still result in either a large b_f or a large σ_f, which increases the PL in the position domain to maintain integrity.

3 CAUCHY–GAUSSIAN OVERBOUND

The intrinsically heavy-tailed properties of the Cauchy distribution can result in significantly prolonged tail regions and a sharp core region, making the Cauchy distribution a good fit for bounding the core of empirical errors. In contrast, the Gaussian distribution possesses light tails and can potentially bound errors more tightly at tail regions. Based on these properties, this section introduces a three-step procedure to construct the Cauchy–Gaussian overbound for SU and NSU error distributions. As shown by the workflow in Figure 1(a), the optimal single-CDF overbounds using zero-mean Cauchy and Gaussian distributions are first determined in Step 1 for empirical SU errors without bias. In particular, the “optimality” refers to the case in which the minimum separation between the overbound and the empirical errors is reached, given a specified overbounding framework (e.g., single-CDF Cauchy overbound or single-CDF Gaussian overbound in this context). This specific definition will be used consistently in the subsequent analysis. Processed by a tangential transition in Step 2, the two formed single-CDF overbounds are integrated in Step 3 to generate the single-CDF Cauchy–Gaussian overbound. As shown in Figure 1(b), the original NSU error distribution is optimally bounded by the paired Cauchy–Gaussian combined model (CGCM) overbound in Step 1 and the paired Gaussian overbound in Step 2. Finally, in Step 3, the right half of the synthesized paired Cauchy–Gaussian overbound is a point-wise supremum of the right-half paired CGCM overbound and right-half paired Gaussian overbound, while the left half of the proposed overbound features a point-wise infimum of the left half of the two prior overbounds. Notably, Ф^-1(·) transforms the CDFs to the equivalent standard normal quantiles; thus, Gaussian-based segments in overbounding distributions act as straight lines. We start with a short introduction to the Cauchy distribution.

FIGURE 1

Flowchart for determining the Cauchy–Gaussian overbound for (a) SU and (b) NSU error distributions

The black curve in each subfigure represents the empirical error distribution. For better visualization of the tail regions, the subfigures in (b) show the CDF values transformed into their equivalent standard normal quantiles.

3.1 Cauchy Distribution

Similar to the Gaussian distribution, the PDF of a Cauchy distribution (C(m,λ)) is a symmetric “bell curve.” The Cauchy distribution has explicit forms for both the PDF (f_C) and CDF (F_C), expressed with a location parameter m and scale parameter λ as follows:

$f_{C} (x; m, λ) = \frac{1}{λ π} . \frac{1}{1 + {(\frac{x - m}{λ})}^{2}}$ 12

$F_{C} (x; m, λ) = \frac{1}{2} + \frac{1}{π} \arctan (\frac{x - m}{λ})$ 13

The differences between the Cauchy and Gaussian distributions are explicitly reflected in their PDFs (Menon, 1962; Smelser & Baltes, 2001), where the Gaussian PDF decreases at an exponential rate with an increasing error magnitude and the Cauchy PDF decreases at a polynomial rate. Figure 2(a) shows the logarithmic-scale (log) PDF of the standard Cauchy and Gaussian distributions, with a location parameter of 0 and a scale parameter of 1 for both distributions. The inset plot shows that the standard Cauchy distribution has a sharper peak at the core region (near the symmetry line) than the standard Gaussian distribution. Moreover, the standard Cauchy distribution exhibits considerably longer and fatter tails than the standard Gaussian distribution. Correspondingly, Figure 2(b) shows the folded CDFs of the standard Cauchy and Gaussian distributions. The folded CDF combines the left half of the CDF and the right half of the complementary CDF (CCDF), i.e., 1-CDF, in one view. As can be seen, the Cauchy distribution allocates a higher proportion of accumulated mass at the tail regions than the Gaussian distribution.

FIGURE 2

Comparison between the standard Cauchy and Gaussian distributions, through (a) the PDF and (b) the folded CDF on a logarithmic scale

The extremely heavy-tailed properties of the Cauchy distribution provide an advantage in terms of bounding heavy-tailed empirical errors. Specifically, the Cauchy distribution decentralizes a higher percentage of probability mass to the tails, which makes it possible to bound heavy-tailed empirical errors without over-inflating the scale parameter. Moreover, the Cauchy distribution features a significantly sharper core region than the Gaussian. This property allows the Cauchy overbound to closely adhere to the core of an empirical error distribution.

Nevertheless, the extremely heavy-tailed properties of the Cauchy distribution become even more pronounced after multiple rounds of convolution. Therefore, the resultant positioning error distribution from the convolution of multiple Cauchy-characterized ranging error sources is expected to exhibit significantly heavy tails, which are usually associated with a large PL in the position domain. In contrast, the Gaussian distribution decreases exponentially in the PDF, guaranteeing a small probability mass in its tail region and its convolutions. To achieve tighter bounding in both the core and tail regions in the range domain and thereby reduce the PL in the position domain, it is necessary to combine the strengths of the Cauchy distribution in the core region and the Gaussian distribution in the tail regions. This is the central concept behind the proposed Cauchy–Gaussian overbound for heavy-tailed distributions, as described below.

3.2 Procedure for Bounding SU Errors

As mentioned in Section 2.1, for zero-mean SU empirical errors, the single-CDF overbound requires that we find the SU distribution (i.e., Cauchy or Gaussian distribution in this study) with the minimum scale parameter. Based on the idea of utilizing the advantages of both Cauchy and Gaussian distributions, we propose the Cauchy–Gaussian overbound with a three-step procedure as follows.

Step 1: Overbound with optimal Cauchy and Gaussian distributions

A single-CDF overbound on SU errors, subject to the constraints in Equations (1) and (2), is applied using both Gaussian (G(0,σ)) and Cauchy (C(0,λ)) distributions, respectively. We denote the optimal scale parameters of the two distributions as σ_o and λ_o, respectively. For the single-CDF Gaussian overbound, σ_o can be found as follows:

$σ_{0} = \min σ$ 14a

$\begin{matrix} s . t . & F_{G} (x; 0, σ) \geq F_{e} (x) & \forall x \leq 0 \end{matrix}$ 14b

$\begin{matrix} F_{G} (x; 0, σ) \leq F_{e} (x) & \forall x > 0 \end{matrix}$ 14c

For the Cauchy distribution, we prove in Appendix B that the heavy-tailed Cauchy distribution can well overbound a centrally aligned Gaussian distribution by $λ \geq \sqrt{\frac{2}{π}} σ$ . This result indicates that σ_o determines the upper bound of λ_o, which reduces the computational load. Specifically, the objective of finding the optimal Cauchy distribution gives the following:

$λ_{o} = \min λ$ 15a

$\begin{matrix} s . t . & F_{C} (x; 0, λ) \geq F_{e} (x) & \forall x \leq 0 \end{matrix}$ 15b

$\begin{matrix} F_{C} (x; 0, λ) \leq F_{e} (x) & \forall x > 0 \end{matrix}$ 15c

$λ \leq \sqrt{\frac{2}{π}} σ_{o}$ 15d

Inspired by the work of Blanch et al. (2008), we demonstrate an instance of heavy-tailed empirical errors by a zero-mean BGMM with the following setting:

$f_{e} (x) = 0.9 f_{G} (x; 0, 1) + 0.1 f_{G} (x; 0, 10)$ 16

The black curve in Figure 3 shows this example error profile. The optimal Cauchy and Gaussian overbounds for this empirical error distribution are shown by magenta and green curves, respectively. As can be seen, the Gaussian overbound deviates more significantly from the empirical errors in the core region while aligning more closely with the empirical errors in the tail regions, compared with the Cauchy overbound. This observation further confirms our expectation that the Cauchy overbound achieves sharper bounding at the core region and the Gaussian overbound yields tighter bounding at the tails. Therefore, it is reasonable to combine the strengths of the two models to achieve tighter bounding.

Step 2: Tangential transition to preserve overbounding properties through convolution

Using the optimal Gaussian and Cauchy overbounds, we can combine the Cauchy core and Gaussian tails. We tentatively define the combined CDF overbound as follows:

$F_{c o m b i n e d} (x) = {\begin{cases} \min (F_{C} (x; 0, λ_{o}), F_{G} (x; 0, σ_{o})) \forall x \leq 0 \\ \max (F_{C} (x; 0, λ_{o}), F_{G} (x; 0, σ_{o})) \forall x > 0 \end{cases}$ 17

The CDF of the combined distribution is plotted as a blue dashed curve in Figure 3(a). As can be seen, the Cauchy overbound intersects with the Gaussian overbound in both the left and right tail regions. The abscissae of the intersections are denoted as x_t,L and x_t,R, respectively. The zoomed-in view in Figure 3(a) also illustrates that F_combined transits from the Cauchy overbound directly to the Gaussian overbound at x_t,R in the right region. A similar transition from the Gaussian overbound to the Cauchy overbound exists in the left region, which is omitted for the present. The transitions destroy the monotonicity of the PDF for each half of the combined distribution, as shown in the zoomed-in view of Figure 3(b). A sudden jump from the Cauchy PDF to the Gaussian PDF occurs at x_t,R. This phenomenon causes the combined distribution to violate the SU property and does not preserve overbounding properties through convolutions (DeCleene, 2000). To solve this problem, a tangential transition from the Cauchy overbound to the Gaussian overbound is constructed.

FIGURE 3

Tentative combined overbounding distribution for the zero-mean BGMM error defined in Equation (16) (SU profile) in two views: (a) CDF, (b) PDF

The Gaussian and Cauchy overbounds are also plotted for reference in each subfigure.

The symmetry of the zero-mean Cauchy and Gaussian distributions ensures that the PDF of F_combined is inherently symmetric, as indicated by Figure 3(b). To further maintain the unimodality of the combined distribution, tangential line segments T_L(x) and T_R(x) are constructed as a transition from the Cauchy core to the Gaussian tails. Appendix C demonstrates that a tangential line always exists between the Cauchy and Gaussian overbounds against heavy-tailed errors. With the same SU empirical error distribution analyzed in Figure 3, the overbounding procedures using a tangential transition are exhibited in Figure 4. Taking the right half T_R(x) as an example, the two tangential points on the CDFs are denoted as (x₁, F_c(x₁)) and (x₂, F_G(x₂)) as shown in the inset plot in Figure 4(a). If we assume that 0 < x₁ < x₂, then the tangential transition should satisfy the following conditions:

FIGURE 4

Cauchy–Gaussian overbounding results for the zero-mean BGMM error defined in Equation (16) (SU profile) in three views: (a) CDF, (b) quantile-scale CDF, (c) PDF The gray-shaded rectangles denote the two tangential transition regions on the left and right. The Gaussian and Cauchy overbounds are plotted for reference in each of the subfigures.

$K = f_{C} (x_{1}) = f_{G} (x_{2}) = \frac{F_{G} (x_{2}) - F_{C} (x_{1})}{x_{2} - x_{1}}$ 18

where K is a constant that represents the slope of the tangential segments. Thus, the transition on the right half can be expressed as follows:

$\begin{matrix} T_{R} (x; x_{1}, x_{2}) = K \cdot (x - x_{1}) + F_{C} (x_{1}) & \forall x_{1} \leq x \leq x_{2} \end{matrix}$ 19

Owing to the SU property of both the Cauchy and Gaussian distributions, the tangential points on the left region are (-x₁, F_C(-x₁)) and (-x₂, F_G(-x₂)), and the left tangential transition is given as follows:

$\begin{matrix} T_{L} (x; x_{1}, x_{2}) = K \cdot (x - x_{1}) + F_{C} (- x_{1}) & \forall - x_{2} \leq x \leq - x_{1} \end{matrix}$ 20

Appendix D proves that the overbounding properties are still present within the transition region.

Figure 4 demonstrates the transition in three views, including the CDF, quantile-scale² CDF, and PDF. Notably, in Figure 4(b), the Gaussian overbound is processed by the standard normal quantile function $Ф^{- 1} (\cdot)$ and behaves as a straight line. The left transition region −x₂ ≤ x ≤ −x₁ and the right transition region x₁ ≤ x ≤ x₂ are highlighted by the gray shaded area. To illustrate the details of the transitions, a zoomed-in inset plot of the right transition is provided for each of the three subfigures. The left transition is omitted for the present owing to the similarity. In the zoomed-in CDF view in Figure 4(a), the right transition T_R (x; x₁, x₂) acts as a straight line segment, as defined in Equation (19). As shown in the inset of Figure 4(b), the shape of the tangential transition is distorted because of the transformation $Ф^{- 1} (\cdot)$ In Figure 4(c), the tangential transition in the zoomed-in PDF view acts as a horizontal line segment, which confirms the unimodality of the combined distribution.

Step 3: Synthesis of the two models and transitions

The final step synthesizes the results of the previous steps to give the final overbound of the empirical errors. The CDF of the Cauchy–Gaussian overbound for SU errors can be explicitly defined as follows:

$F_{ob} (x) = {\begin{cases} \min (F_{C} (x; 0, λ_{o}), T_{L} (x; x_{1}, x_{2}), F_{G} (x; 0, σ_{o})) & \forall x \leq 0 \\ \max (F_{C} (x; 0, λ_{o}), T_{R} (x; x_{1}, x_{2}), F_{G} (x; 0, {>σ}_{o})) & \forall x > 0 \end{cases}$ 21

Figure 4 illustrates this piecewise CDF overbound as a blue dashed line. In the right region (x>0) the finalized overbound starts with the optimal Cauchy model when the absolute error is below x₁, followed by the tangential transition from x₁ to x₂. F_ob(x) then switches to the optimal Gaussian overbound when the absolute error exceeds x₂. Because the Cauchy–Gaussian combined distribution in Equation (21) is SU, its overbounding properties can be preserved through convolutions (DeCleene, 2000).

The optimal parameters of the Cauchy–Gaussian overbound on the empirical error defined in Equation (16) are listed in Table 1. The Gaussian component of the proposed overbound represents the location and scale parameters given by the optimal Gaussian overbound, while the Cauchy component represents the counterparts yielded by the optimal Cauchy overbound. As can be seen, the Cauchy component is characterized by an optimal scale parameter, λ_o. The resulting Cauchy core is significantly sharper and more concentrated than the core of the optimal Gaussian overbound. This sharpness allows the final Cauchy–Gaussian overbound to adhere more tightly to the empirical errors in the central region, as evidenced in Figure 4.

View this table:

TABLE 1

Overbounding Parameters for the Simulated SU Errors Defined in Equation (16)

K represents the slope of the tangential transition, whereas x₁ and x₂ are the tangential points when x > 0, as discussed for Step 2 in Section 3.2.

The Cauchy–Gaussian overbound leverages both the single-CDF Cauchy and single-CDF Gaussian overbounds to capture the distributional properties of heavy-tailed empirical errors. Althoug heavy-tailed error features can also be characterized by a zero-mean BGMM (Blanch et al., 2008), the proposed overbound offers distinct advantages regarding parameter determination. Built on the core overbounding approach introduced by Rife et al., (2004a), the Cauchy–Gaussian overbound aims to balance between bound sharpness and implementation simplicity. This approach achieves an overall tight bound by piecewise synthesizing the sharp single-CDF overbounds against the error curve in the core and tail regions. Moreover, the proposed method replaces heuristic guesswork with a rigorous constrained optimization formulation. This method eliminates the subjectivity of parameter tuning, providing a strictly deterministic procedure in contrast to the manual trial-and-error efforts required for a BGMM overbound.

3.3 Procedure for Bounding NSU Errors

As discussed in Section 2.2, the overbound for the NSU profile can be handled with a pair of CDF bounds. Section 3.2 also demonstrated that the combination of a Cauchy core and Gaussian tails can more tightly bound heavy-tailed empirical profiles. Building upon these findings, this section introduces a three-step procedure to extend the Cauchy–Gaussian overbound to NSU error distributions, which are more frequently observed in real-world GNSS measurements.

In the following discussion, we use randomly generated samples from the following biased BGMM to illustrate the construction process of the proposed method:

$f_{e} (x) = 0.9 f_{G} (x; 0, 1) + 0.1 f_{G} (x; 1, 10)$ 22

where a location shift of 1 m is applied for the second Gaussian component so that the resultant distribution is NSU This empirical error distribution, with a bias of 0.02 m, is displayed by a black curve in Figure 5.

FIGURE 5

Cauchy–Gaussian overbounding results for the biased BGMM errors defined in Equation (22) (NSU profile): (a) CDF of the optimized paired CGCM overbound in Step 1, (b) CDF of the optimized paired Gaussian overbound in Step 2, (c) quantile-scale right bound (defined in Equation (30)) and (d) quantile-scale analog single-CDF (defined in Equation (31)) of the finalized Cauchy–Gaussian overbound

Step 1: Construct the optimal paired CGCM overbound

In the first step, we find the paired overbound with the Cauchy distribution. Tentatively, we assume a pair of pure Cauchy distributions to construct the two-sided overbound. According to Equations (3) and (4), the paired Cauchy overbound should satisfy the following:

$\begin{matrix} F_{C} (x; m_{L}, λ_{L}) \geq F_{e} (x) & \forall x \end{matrix}$ 23

$\begin{matrix} F_{C} (x; m_{R}, λ_{R}) \leq F_{e} (x) & \forall x \end{matrix}$ 24

However, a typical pair of Cauchy overbounds satisfying the above conditions is found to yield significantly large location parameters (|m_L|=|m_R| = 23.90) yet considerably small scale parameters (λ_L = λ_R = 0.01). Figure 5(a) visualizes the right half of the paired Cauchy overbound, with the left half being inferrable owing to symmetry. As can be seen, the paired Cauchy overbound evidently separates from the empirical errors and results in overly conservative bounding. This phenomenon is primarily caused by the extreme heavy-tailedness of the Cauchy distribution. Taking the error bounding on the right half as an example, the long and fat left tail of the Cauchy distribution hinders the Cauchy CDF from completely lying below the empirical distribution (as required in Equation (24)) with a small location shift. As a result, the paired Cauchy overbound must compromise with the constraint in Equation (24) using a significantly large location parameter, which induces excessive conservatism.

Inspired by the above analysis, we replace the left tail of the right Cauchy bound and the right tail of the left Cauchy bound with Gaussian tails to prevent over-inflation of the location parameter. Specifically, we propose the paired CGCM with its left (F_M,L) and right (F_M,R) halves constructed as follows:

F_{M, L} (x) = {\begin{matrix} F_{C} (x; - m, λ) & \forall x \leq - m \\ F_{G} (x; - m, k λ) & \forall x > - m \end{matrix}

F_{M, R} (x) = {\begin{matrix} F_{G} (x; m, k λ) & \forall x \leq m \\ F_{C} (x; m, λ) & \forall x > m \end{matrix}

Notably, we use $k = \sqrt{\frac{π}{2}}$ for the paired CGCM. This setting indicates that the centrally aligned F_C and F_G share the same value at x = −m and x = m, and their derivatives (i.e., f_C and f_G) are equal at these two locations. Therefore, each half (i.e., F_M,L, or F_M,R) is continuous and differentiable throughout the coordinate. Moreover, the setting $k = \sqrt{\frac{π}{2}}$ establishes a one-to-one relationship between the aligned Cauchy and Gaussian distributions, which avoids introducing additional parameters.

Notably, the right half (F_M,R) of the paired CGCM overbound is based on the Gaussian distribution when x ≤ m and the Cauchy distribution when x > m. Because the Cauchy distribution is heavy-tailed, the probability mass of the combined distribution is transferred rightward, causing the mean to exceed the median (i.e., right-skewed). Compared with a Gaussian tail, the Cauchy tail potentially enables the right-half CGCM to bound right-biased heavy-tailed empirical errors with a smaller location parameter. Similarly, the left-skewed F_M,L, can largely reduce the location shift when bounding left-biased heavy-tailed error distributions.

The following optimization problem is constructed to solve the optimal location (m_o) and scale (λ_o) parameters of the paired CGCM overbound:

m_{o}, λ_{i} = \underset{m, λ}{argmin} \sum_{x \in Ω} ‖ F_{M, L} (x) - F_{e} (x) ‖ + ‖ F_{M, R} (x) - F_{e} (x) ‖

27a

s . t . F_{M, L} (x) \geq F_{e} (x) \forall x \in Ω

27b

F_{M, R} (x) \leq F_{e} (x) \forall x \in Ω

27c

where ║.║denotes the Euclidean norm and Ω is the range of the empirical errors. The optimization problem minimizes the discrepancy between the empirical and overbounding distributions, thereby achieving tight bounding.

Figure 5(a) displays the optimal left and right bounds achieved using the CGCM. Each bound contains the Cauchy (magenta) and Gaussian (green) components. The empirical errors are bounded with a very small location parameter (m_o =0.04m). F_M,R, transits from Gaussian to Cauchy at x = m_o, whereas F_M,L, transits from Cauchy to Gaussian at x = −m_o. Inset 1 in Figure 5(a) highlights the smooth junction between the Gaussian and Cauchy distributions in the right (at x= m_o) and left regions (at x = −m_o). Taking the right region as an example, Figure 5(a) also validates the capability of the CGCM to generate much tighter bounds than the paired Cauchy approach.

Notably, the right tail of F_M,R and left tail of F_M,L employ the Cauchy distribution, which was proven to be significantly more heavy-tailed than the Gaussian distribution in Section 3.1. This phenomenon is also reflected in Inset 2 in Figure 5(a), where the bound of the heavy-tailed F_M,R on the empirical errors is insufficiently tight in the right tail region. The heavy-tailed characteristic of the Cauchy tail will be further enhanced after multiple ranging error sources are convoluted, eventually producing excessively conservative PLs in the position domain. To reduce this conservatism, we propose replacing these Cauchy tails with Gaussian tails from the optimal paired Gaussian overbound, as discussed in the following section.

Step 2: Construct an optimal paired Gaussian overbound

The second step aims to determine the optimal pair of Gaussian overbounds with location parameter µ_o and scale parameter σ_o. The optimal left (F_G,L(x)) and right (F_G,R(x) overbounds are those bounds that can give the minimum least-square sum of CDF differences with the empirical CDF (F_e(x)). Similar to Step 1, the optimization problem can be expressed as follows:

$μ_{o}, σ_{o} = \underset{μ, σ}{argmin} \sum_{x \in Ω} ∥ F_{G, L} (x) - F_{e} (x) ∥ + ∥ F_{G, R} (x) - F_{e} (x) ∥$ 28a

$\begin{matrix} s . t . & F_{G, L} (x) = F_{G} (x; - μ, σ) \geq F_{e} (x) & \forall x \in Ω \end{matrix}$ 28b

$\begin{matrix} F_{G, R} (x) = F_{G} (x; μ, σ) \leq F_{e} (x) & \forall x \in Ω \end{matrix}$ 28c

The CDFs of the optimal paired Gaussian overbounds are shown as cyan lines in Figure 5(b). The zoomed-in views of these overbounds confirm that the strict left and right bounds are satisfied in the tail regions according to Equations (3) and (4).

Step 3: Synthesize the optimal paired CGCM overbound and paired Gaussian overbound

The final step integrates optimal paired overbounding models from the first two steps. Specifically, the finalized Cauchy–Gaussian overbound for NSU error distributions is defined as follows:

$\begin{matrix} F_{o b, L} (x) = \min (F_{M, L} (x), F_{G, L} (x)) & \forall x \in Ω \end{matrix}$ 29

$\begin{matrix} F_{o b, R} (x) = \max (F_{M, R} (x), F_{G, R} (x)) & \forall x \in Ω \end{matrix}$ 30

Mathematically, F_ob,L(x) represents the point-wise infimum of F_M,L(x) and F_G,L(x), whereas F_ob,R(x) is the point-wise supremum of F_M,R(x) and F_G,R(x). Different from Equation (21) for bounding SU errors, the tangential transitions are unnecessary in bounding NSU error distributions. The reason for this difference is that the paired bounding approach does not require the left and right bounds to be zero-mean SU distributions to maintain the overbounding properties through convolutions (Rife et al., 2006).

Taking the right bound as an example, Figure 5(c) shows how F_ob,R(x) is constructed from the right half of the optimal paired CGCM overbound (F_M,R(x) and optimal paired Gaussian overbound (F_G,R(x)) Notably, Gaussian distributed curves F_G,R, and F_M,R(x ≤ m_o) in the quantile-scale CDF transformed by Φ^-1(·) behave as straight lines. According to Equation (30), the piecewise right bound begins with F_G,R(x) at the far end of negative error, transits to F_M,R(x) after inflection point 1, and switches back to F_G,R(x) after inflection point 2 up to the positive far end. The left bound F_ob,L(x) is omitted in Figure 5(c) but can be inferred owing to the symmetry with the right bound. Correspondingly, the following analog single-CDF overbound F_ob(x) based on the paired overbound in Equations (29) and (30) can be constructed through Equation (5):

$F_{o b} (x) = {\begin{matrix} \min (F_{M, L} (x), F_{G, L} (x)) & \forall F_{o b, L} < \frac{1}{2} \\ \frac{1}{2} & otherwise \\ \max (F_{M, R} (x), F_{G, R} (x)) & \forall F_{o b, R} > \frac{1}{2} \end{matrix}$ 31

Figure 5(d) depicts the quantile-scale CDF of F_ob and displays its relationship with the paired CGCM overbound (F_M,L, F_M,R) and paired Gaussian overbound (F_G,L, F_G,R). As can be seen, F_ob employs the paired CGCM overbound for small absolute error values. Subsequently, F_ob transits to the optimal paired Gaussian overbound in the tail region after the left and right intersection point 2. Finally, the inset plots in both Figure 5(c) and 5(d) show that the empirical error bias is bounded between −m_o and m_o in the core region.

Table 2 displays the parameters of the Cauchy–Gaussian overbound F_ob for the simulated NSU errors defined in Equation (22), which contains the optimal location and scale parameters from the paired CGCM overbound (m_o, λ_o) and the paired Gaussian overbound (μ_o, σ_o), The CGCM component features an extremely small location parameter (m_o = 0.04m),. which causes the bounding distribution to adhere closely to the empirical curves in the central region. Although the Gaussian component has a large location parameter (μ_o = 10.55m), it only contributes to the bounding in the tail regions, as reflected in Figure 5(c) and 5(d). Therefore, when bounding central-region errors, μ_o in the final Cauchy-Gaussian overbound does not introduce excessive conservatism compared with the corresponding parameter in the paired Gaussian overbound.

View this table:

TABLE 2 Cauchy–Gaussian Overbounding Parameters for the Simulated NSU Errors Defined in Equation (22)

3.4 Position-Domain Bounding

In GNSS positioning, the measurement errors are projected to the position domain. Thus, it is essential to derive the error bound for the positioning error. Let us consider N ranging error sources denoted by the vector ε, where each element ε_i has an index i ∈ {1,2,…,N}. In this paper, we assume that error sources in ε are mutually independent. The vertical positioning error (VPE) can be expressed as follows:

$VPE = \sum_{i = 1}^{N} S_{3, i} ε_{i}$ 32

where S_3,i is the element in the third row and i-th column of the projection matrix S of the least-square or weighted least-square solution. Mathematically, VPE is the linear combination of the N error sources, each of which has an individual error distribution. Thus, the PDF of the VPE, f_VPE, is a joint distribution given by multiple measurement error distributions. According to Yan et al. (2024), f_VPE can be formulated as follows:

$f_{V P E} (x) = (\prod_{i = 1}^{N} \frac{1}{| S_{3, i} |}) (\oplus_{i = 1}^{N} f_{ε_{i}} (\frac{x}{| S_{3, i} |}))$ 33

where f_{ε_i} denotes the empirical distribution related to the i-th measurement error source. The operators Π and ⊛ represent the multiplication and convolution of multiple terms, respectively. A proof of Equation (33) is provided in Appendix E. For convenience in computation, the worst-case error source ε_w (where w is a certain index among {1,2,…,N}) with the largest variance will be selected to represent each error source ε_i. In this way, the most conservative empirical distribution ${\bar{f}}_{V P E}$ is constructed as follows:

${\bar{f}}_{V P E} (x) = (\prod_{i = 1}^{N} \frac{1}{| S_{3, i} |}) (\oplus_{i = 1}^{N} f_{ε_{w}} (\frac{x}{| S_{3, i} |}))$ 34

Thus, to obtain the overbound for the most conservative VPE, only the distribution f_{ε_w,ob}, associated with the overbound against the measurement error ε_w needs to be determined. Notably, f_{ε_w,ob}, denotes the PDF of the bound for SU errors. For NSU errors, f_{ε_w,ob} represents the PDF of either the right or left half of the paired bound. The position error overbounding distribution can be expressed by the following:

${\bar{f}}_{VPE, o b} (x) = (\prod_{i = 1}^{N} \frac{1}{| S_{3, i} |}) (\oplus_{i = 1}^{N} f_{ε_{w, o b}} (\frac{x}{| S_{3, i} |}))$ 35

A discretized form of the position-domain overbound, ${\bar{x}}_{VPE, o b}$ , is constructed as follows (Yan et al., 2024):

${\bar{X}}_{V P E, o b} [k] = (\prod_{i = 1}^{N} \frac{1}{| S_{3, i} |}) (\oplus_{i = 1}^{N} X_{ε_{w, o b}})$ 36

where X_{ε_w,ab}, is the discretized f_{ε_w,ab}. For ${\bar{X}}_{VPR, o b} [k] (k \in {1, 2, …, M}), k$ , is the index, and M is the length of the discrete VPE overbounding distribution. Accordingly, the discretized range of position error values is noted as T[k],k ∈ {1,2,…,M}, with a unit interval length Δt. It is proven that this discretization method can preserve the overbounding properties through convolution. Although the discretized position error bound is calculated numerically, the convolution’s computation time can be significantly reduced by using the fast Fourier transform for practical considerations (Nussbaumer, 1982). For details, one can refer to the work by Yan et al. (2024). For a paired overbounding approach, the position error bound is typically computed by convolving both the left and right bounds. However, because the paired Cauchy–Gaussian overbound is symmetric, the left bound can be inferred from the right. Therefore, for the remainder of this discussion, we will utilize only the right bound to generate the position-domain bound ${\bar{X}}_{VPE, o b} [k]$ .

Under the requirement of integrity risk P_HMI, a particular index k_p ∈{1,2,…,M} is determined such that the following holds (Yan et al., 2024):

$\sum_{k = 1}^{k_{p} - 1} {\bar{X}}_{VPE, o b} [k] . Δ t \leq 1 - \frac{P_{H M I}}{2}$ 37

$\sum_{k = 1}^{k_{p}} {\bar{X}}_{VPE, o b} [k] . Δ t > 1 - \frac{P_{H M I}}{2}$ 38

and the VPL is obtained as follows:

$VPL = T [k_{p}]$ 39

4 NUMERICAL EXPERIMENTS

In this study, we compare the bounding performance of the proposed method with the single-CDF Gaussian overbound and the two-step Gaussian overbound. The empirical error distributions are constructed from double-differenced pseudorange measurements generated by a differential GNSS (DGNSS) model (Misra, 2006; Parkinson & Axelrad, 1987). Inspired by the work of Larson et al. (2019), this paper assumes that the DGNSS measurement errors are mutually independent, as the marginal inter-measurement correlation is expected to have a negligible impact on the computed VPL. For SU error distributions, the experiment will be conducted on a simulated DGNSS error data set because the strictly SU property is rarely satisfied in real-world measurement errors. For NSU errors, the bounding performance is validated through a real DGNSS data set collected in the Hong Kong urban environment.

4.1 Bounding Performance for SU Errors

In Section 3.2, we used an example zero-mean BGMM to illustrate the bounding performance of the proposed method on an SU heavy-tailed error distribution in the range domain. In this section, we further use this BGMM model to represent the DGNSS measurement error distribution for the entire time series of each satellite and examine the bounding performance in the position domain. Specifically, we use the open-sourced MATLAB Algorithm Availability Simulation Tool (MAAST) (Jan et al., 2001) to simulate the positions of Global Positioning System (GPS) and Galileo satellites every 100 s over 24 h. We set two imaginary locations (displayed in Appendix A) for the receiver and reference station, separated by a distance of approximately 5.58 km. For each epoch, we generate random samples using a pseudorandom number generator in MATLAB from the zero-mean BGMM in Equation (16) and add them to the true differential ranges to create the simulated DGNSS measurements. Over the 864 epochs spanning 24 h, the positioning solution based on DGNSS measurements is calculated using the least-squares method, with the VPEs shown in Figure 6. The mean VPE is 1.23 m, and the maximum reaches 12.15 m. For each epoch, we calculate the VPL of the single-CDF Gaussian and Cauchy–Gaussian overbounding methods via Equation (39), with the results plotted in Figure 6. The VPL calculation adopts the same discretization method and settings reported by Yan et al. (2024), where the unit interval length Δt is 0.01 m and P_HMI is set to be 1 10^-9. As shown in Figure 6, the VPL produced by the Cauchy–Gaussian overbound is smaller than that obtained by the single-CDF Gaussian overbound in all epochs. Table 3 summarizes the percentage of reduction in VPL achieved by the Cauchy–Gaussian overbound compared with the Gaussian overbound. Specifically, the proposed method reduces the VPL by 14.95% on average. The maximum reduction reaches 26.14%, and the minimum reduction is 6.78%. The 25th (Q1), 50th (Q2), and 75th (Q3) percentiles, which vary from 12.61% to 16.81%, are also provided for a more detailed overview of the VPL reduction.

VPL of single-CDF Gaussian and Cauchy–Gaussian overbounding methods, with PHMI = 1 × 10−9 The VPEs are also plotted for reference.

FIGURE 6

VPL of single-CDF Gaussian and Cauchy–Gaussian overbounding methods, with P_HMI = 1 × 10⁻⁹ The VPEs are also plotted for reference.

View this table:

TABLE 3 Reduction in VPL by the Cauchy–Gaussian Overbound Compared With the Single-CDF Gaussian Overbound in the Simulated SU Error Data Set Q1, Q2, and Q3 represent the 25th, 50th, and 75th percentiles, respectively.

4.2 Bounding Performance for NSU Errors

Here, DGNSS measurements collected in the Hong Kong urban environment are used to verify the feasibility of the proposed method for bounding real-world NSU errors. The receiver and the reference station are separated by approximately 4.74 km, with the locations displayed in Appendix A. This urban data set includes L1 GPS, BeiDou, and GLONASS signals obtained using u-blox ZED-F9P at a frequency of 1 Hz on June 28, 2024, producing DGNSS measurements covering 1 h. Following the settings of Yan et al. (2024), we select fault-free DGNSS measurements by filtering for observations with an elevation angle above 30° and a signal-to-noise ratio greater than 35 dB. To ensure that the data set exhibits the desired heavy-tailed properties for the bounding experiment, we select double-differenced measurement errors from satellites at low elevation angles (30°−35°), where satellite signals are known to be susceptible to multipath effects in urban scenarios (Peretic et al., 2025). After filtering, the 1-h data set generates 2887 effective epochs, where 4562 DGNSS error samples constitute the empirical distribution. The number of unique contributing satellites across the epochs is displayed in Figure 7. In the following experiments, the two-step Gaussian overbound (Blanch et al., 2018) and the non-Gaussian NavDEN overbound (Rife & Pervan, 2012) are utilized as benchmarks for comparison with the proposed Cauchy-Gaussian overbound. Details on the NavDEN model are provided in Appendix F.

FIGURE 7

Number of unique contributing satellites in each epoch from the filtered data set

The quantile-scale CDF in Figure 8(a) confirms that the error distribution is neither symmetric nor unimodal, with a bias of –0.019 m revealed by the inset plot. For comparison, a zero-mean Gaussian model (red dashed line) is fitted to the central 95% of the empirical data. Although the empirical curve (black) aligns well with this fit in the core region, it deviates significantly in the tails. This separation begins at approximately –1.5 m on the left and 2 m on the right, with the larger deviation in the left tail indicating that the left tail is fatter and longer than the right tail.

Table 4 summarizes the parameters of the NavDEN, two-step Gaussian, and Cauchy–Gaussian overbounding methods. Multiple parameters are employed to define different parts of a NavDEN overbound, and their meanings have been summarized in Appendix F. The location and scale parameters (i.e., (μ_2s, σ_2s)) of the two-step Gaussian overbound are generated by the open-source MATLAB tool described by Blanch et al. (2018). The optimal parameters of the Cauchy–Gaussian overbound defined in Equations (27) and (28) are determined by a derivative-free mesh adaptive direct search solver (Audet & Dennis, 2006). Further, the proposed overbound contains the CGCM-component parameters (i.e., (m_o, λ_o)) from the optimal paired CGCM overbound and the Gaussian-component parameters (i.e., (μ_o, σ_o)) from the optimal paired Gaussian overbound.

View this table:

TABLE 4 Overbounding Parameters for the NavDEN, Cauchy–Gaussian, and Two-Step Gaussian Methods for the Urban Data Set (NSU Profile) The tilde notations for the NavDEN overbound indicate that the parameters are divided by the fundamental spacing Δ to be unitless.

Figure 8(b) displays the left and right halves of the NavDEN overbound and presents the two-step Gaussian overbound and Cauchy–Gaussian overbound in the form of a single-CDF analog defined by Equation (5). As can be seen, each half of the NavDEN overbound is constructed using a discrete model, whereas that of the other two overbounding methods is formulated to be continuous. Clearly, the two-step Gaussian overbound is the most conservative of the three methods. The Cauchy–Gaussian overbound has the tightest bounding performance in the central region (absolute error within 2 m), and the inset plot confirms that the Cauchy-Gaussian overbound successfully bounds the empirical bias. Notably, both the two-step overbound and the Cauchy–Gaussian overbound include a Gaussian component. However, with the assistance of the Cauchy distribution in the core region, the CGCM component in the proposed overbound generates a significantly smaller location parameter (m_o =0.05 m) than the two-step Gaussian overbound (μ_2s =4.50 m). This small location parameter m_o mo enables the resultant Cauchy-Gaussian overbound to yield considerably tighter bounding in the core region, as illustrated by the analog single-CDF overbounds shown in Figure 8(b).

To analyze the bounding performance in the tail regions, we plot the log CDFs in Figure 8(c) and log CCDFs in Figure 8(d) to demonstrate the left and right tails of the overbounding distributions, respectively. Both figures demonstrate the impressively tight bounding of the NavDEN model for absolute errors of 2–11 m. However, the bounding becomes less sharp than the proposed method in the far tails (absolute error beyond 12 m) owing to the heavy-tailed decay. Additionally, while both the two-step Gaussian overbound and the Cauchy-Gaussian overbound use a Gaussian profile in the far-tail region, the proposed method produces a sharper bound, as depicted in Figure 8(c) and 8(d). This result arises because the Gaussian component of the Cauchy–Gaussian overbound has a smaller scale parameter (σ₀ =1.71 m) than the two-step Gaussian overbound (σ_2s =3.54 m).

FIGURE 8

Evaluation of overbounding performance for NSU errors in an urban data set: (a) quantile-scale CDF showing empirical DGNSS error during 1 h with an observed elevation angle of 30°−35°; overbounding results for the two-step Gaussian and Cauchy–Gaussian overbounds for the (b) CDF, (c) log CDF, (d) log CCDF, where the Cauchy–Gaussian overbound takes the analog single-CDF form defined in Equation (31)

An additional analysis was conducted to compare the position-domain bounding performance of the different overbounding methods. Figure 9 shows the time series of the VPEs, where the mean value is 0.41 m and the maximum reaches 2.89 m. For each epoch, we calculate the VPL of the NavDEN, two-step Gaussian, and Cauchy–Gaussian overbounds via Equation (39), with the results plotted in the same figure. Notably, the VPL is calculated based on the worst-case DGNSS error distribution, which is given by the error for elevation angles observed from 30°−35°. Figure 9 shows that all three methods produce VPLs that successfully bound the VPE in each epoch. The Cauchy–Gaussian overbound consistently yields the lowest VPLs because of its advantageous fit to the empirical error distribution in both the core and tail regions. The two-step Gaussian overbound produces the highest VPLs, whereas the NavDEN VPLs are at a moderate level. Table 5 quantifies the percentage of reduction in VPL achieved by the Cauchy-Gaussian overbound compared with the other two overbounds. On average, the VPL is reduced by 21.07% compared with the NavDEN overbound and by 47.66% compared with the two-step Gaussian overbound. The VPL reduction is consistent across all metrics, with a maximum percentage reduction of nearly 50% against the two-step Gaussian overbound. These results indicate that the proposed method can effectively reduce the VPL for NSU errors without compromising integrity.

VPL of the NavDEN, two-step Gaussian, and Cauchy–Gaussian overbounding methods, with PHMI = 1 × 10−9 The VPEs are also plotted for reference.

FIGURE 9

VPL of the NavDEN, two-step Gaussian, and Cauchy–Gaussian overbounding methods, with PHMI = 1 × 10⁻⁹ The VPEs are also plotted for reference.

View this table:

TABLE 5 Reduction in VPL by the Cauchy–Gaussian Overbound Compared With the Two-Step Gaussian Overbound in the Urban Data Set Q1, Q2, and Q3 represent the 25th, 50th, and 75th percentiles, respectively.

4.3 Impact of Heavy-Tailedness

This section explores how the heavy-tailedness of the NSU error distribution impacts the positioning bounding performance of the proposed Cauchy–Gaussian overbound. For comparison, we also evaluated the performance of the two-step Gaussian overbound, a method supported by an open-source toolkit for heuristically bounding errors. A biased BGMM with the following setting was used to simulate the NSU error distribution:

$f_{e} (x) = p_{1} f_{G} (x; 0.1, 1) + (1 - p_{1}) f_{G} (x; 0.1, 10)$ 40

Here, the location parameter of 0.1 m in both Gaussian components directly contributes to the bias of 0.1 m in the resultant error distribution, and p¹ represents the proportion of the first component. Notably, as p¹ increases, the resultant distribution approximates closer to the first component more centralized than the second component f_G(x;0.1,1)which is f_G(x;0.1,10). However, the second component with a larger scale parameter (10 m) always enables a fraction of data samples to deviate largely away from the center location (i.e., 0.1 m). Therefore, the increase in p₁ can enlarge the kurtosis (or heavy-tailedness) of the BGMM error distribution.

Similar to the setting in Section 4.1, we use MAAST (Jan et al., 2001) to simu late satellite positions every 100 s over 24 h. The receiver and reference locations are listed in Appendix A. The DGNSS measurements are generated by adding the randomly generated sample from the error distribution in Equation (40) to the true differential range. In the experiment, p₁ is set from 0.60 to 0.95 with an increment of 0.05. For each value of p₁, we calculate the average VPL of both overbounding methods throughout the time frame.

Figure 10(a)) depicts the average VPL given by the two-step Gaussian and Cauchy–Gaussian overbounding methods. As expected, the proposed overbound generates less-conservative bounding than the two-step Gaussian overbound in both the core and tail regions, thus yielding lower average VPLs across different settings in p₁. Moreover, both curves of average VPL calculations show an overall descending trend, with fluctuations induced by several extreme random data samples. The average VPL generated by the two-step Gaussian overbound decreases slightly from 33.80 m to 31.84 m, when p₁ increases from 0.60 to 0.95. A possible reason for this result is that the resultant distribution f_e gradually allocates a smaller proportion to the second component, which has a fatter tail than the first component. Correspondingly, the scale parameters of the Gaussian-based over bound are slightly reduced, which enables tighter bounding and thus lower VPL calculations. In contrast, the curve for the Cauchy–Gaussian overbound decreases more significantly, from 31.60 m to 24.22 m, as p₁ increases. This result can be explained by the fact that the empirical distribution becomes more heavy-tailed as p₁ increases and possesses longer tails and a sharper core region. Apart from the benefit of a smaller scale parameter given by the optimal paired Gaussian, the Cauchy–Gaussian overbound can advantageously bound the core region with a smaller location parameter by leveraging the optimal paired CGCM, thus resulting in smaller average VPLs. Figure 10(b) shows the percentage of reduction in average VPLs obtained by the Cauchy–Gaussian overbound compared with the two-step Gaussian overbound for different settings of p₁. As can be seen, the curve rises gradually from 6.53% to 23.94% as p₁ reaches 0.95. This result potentially reveals the strength of the proposed method to less conservatively bound increasingly heavy-tailed distributions, compared with the two-step Gaussian overbound.

FIGURE 10

Results for different biased and heavy-tailed BGMMs: (a) average VPLs given by the two overbounding methods, (b) percentage of reduction in average VPL yielded by the Cauchy–Gaussian overbound, compared with the two-step Gaussian overbound

5 CONCLUSIONS AND FUTURE WORK

This work employed the simply parameterized Cauchy distribution to characterize the heavy-tailed properties of empirical errors. Building on this characteristic, the proposed Cauchy–Gaussian overbound is designed to tightly bound both SU and NSU heavy-tailed error distributions. For both error types, a systematic three-step procedure was developed to determine the optimal parameters of the overbound. The proposed overbound offers a single-CDF bound for SU distributions and paired bounds for NSU error distributions. For SU errors, the Cauchy–Gaussian overbound employs a zero-mean Cauchy overbound in the core region and a zero-mean Gaussian overbound in the tail regions. In this process, the determined overbounding relationship between centrally aligned Cauchy and Gaussian distributions is applied to reduce the computational load when searching for the optimal scale parameter of the Cauchy overbound. Subsequently, a tangential transition from Cauchy to Gaussian is designed to maintain the overbounding preservation through convolution. For NSU errors, the empirical distributions are bounded with an optimal pair of half-Cauchy half-Gaussian CGCMs, which can generate significantly smaller location parameters than pure Cauchy or pure Gaussian distributions. Next, an optimal paired Gaussian overbound is determined. The paired CGCM overbound and paired Gaussian overbound are then synthesized to produce piecewise paired overbounding distributions to tightly bound the original NSU errors. The bounding performance of the Cauchy–Gaussian overbound is compared with the single-CDF Gaussian overbound for SU errors through a simulated DGNSS data set. For NSU errors, the performance is evaluated in comparison with the NavDEN o verbound and two-step Gaussian overbound through a real DGNSS data set collected in a Hong Kong urban environment. For both error types, the proposed Cauchy–Gaussian overbound provides overall tighter bounds than the baseline methods. In the position domain, the Cauchy–Gaussian method reduces the VPL by approximately 15% on average for SU errors. For NSU errors, the average VPL reduction is even more significant, reaching 21% compared with the NavDEN overbound and over 47% compared with the two-step Gaussian overbound. Furthermore, as heavy-tailedness increases in the error distribution, the Cauchy–Gaussian overbound yields a more significant reduction in average VPL.

Nevertheless, the Cauchy–Gaussian overbound has more than three parameters for both SU and NSU empirical distributions, which pose challenges for broadcasting overbound parameters in augmentation systems with limited bandwidth channels. Possible solutions involve developing a quantitative relationship between the Gaussian component and the Cauchy-based component (or the CGCM component), such that only one pair of location and scale parameters is needed for overbound computation. Additionally, the optimal location and scale parameters of the paired CGCM overbound and paired Gaussian overbound can be impacted by the objective function. Thus, it is worthwhile to analyze the bounding performance yielded by the current design compared with other cost forms, such as the discretized Jensen–Shannon divergence. To more conveniently compute tight bounds against heavy-tailed NSU errors, extended work could utilize a two-step architecture (Blanch et al., 2018) in which the Cauchy–Gaussian model is used to overbound an intermediate SU envelope generated from empirical NSU errors. Future directions may also include applying the proposed Cauchy–Gaussian overbound to fault detection algorithms in scenarios of both SU and NSU measurement errors.

CONFLICT OF INTEREST

The authors declare no conflicts of interest.

HOW TO CITE THIS ARTICLE:

Li, Z., Yan, P., Wen, W., & Hsu, L-T. (2026). Cauchy–gaussian overbound for heavy-tailed GNSS measurement errors. NAVIGATION, 73https://doi.org/10.33012/navi.749

ACKNOWLEDGMENTS

This research was supported by the project H-ZGD2 “Huawei-PolyU High-Precision Positioning with Vehicle-Mounted Multi-Sensor Fusion in Complex Scenarios (Phase II),” the project 1-BBDW “Multi-Robot Collaborative Operations in Lunar Areas for Regolith Processing,” and the Innovation and Technology Fund – Innovation and Technology Support Programme (ITF-ITSP) of the Innovation and Technology Commission, under grant No. ITS/091/23, “Improving the Smartphone Urban Positioning Accuracy Aided by AI and Probabilistic Graphical Model-Based Multi-sensory Integration for Hong Kong Smart Cities.”

A COORDINATES OF RECEIVERS AND REFERENCE STATIONS

View this table:

TABLE 6 Locations (Longitude, Latitude, Height) of Receivers and Reference Stations in the Simulation and Urban Data Set It is noted that the simulations in Sections 4.1 and 4.3 share the same pair of receiver and reference station locations.

B SUFFICIENT AND NECESSARY CONDITIONS OF THE CAUCHY OVERBOUND ON A CENTRALLY ALIGNED GAUSSIAN DISTRIBUTION

The PDF and CDF of a standard Gaussian distribution with location parameter μ and scale parameter σ are as follows:

$f_{G} (x; μ, σ) = \frac{1}{σ \sqrt{2 π}} \exp (- \frac{1}{2} {(\frac{x - μ}{σ})}^{2})$ 41

$F_{G} (x; μ, σ) = \frac{1}{2} (1 + e r f (\frac{x - μ}{σ \sqrt{2}}))$ 42

where erf (·) represents the error function. Similarly, the PDF and CDF of a Cauchy distribution with location parameter m and scale parameter λ are given as follows:

$f_{c} (x; m, λ) = \frac{1}{λ π} \frac{1}{1 + {(\frac{x - m}{λ})}^{2}}$ 43

$F_{C} (x; m, λ) = \frac{1}{2} + \frac{1}{π} \arctan (\frac{x - m}{λ})$ 44

Theorem.For a Gaussian model N(μ,σ) and a Cauchy model C(m,λ), if their medians are aligned (i.e.,μ = m = M₀, where M₀ denotes the known value of the median), then the sufficient and necessary condition of C(M₀,λ) overbounding N(M₀,σ) is $λ \geq \sqrt{\frac{2}{π}} σ$ .

It should be highlighted that the overbounding properties of SU distributions in Section 2.1 require both the error and overbounding distributions to be zero-mean (i.e.,μ = m = 0) (DeCleene, 2000). Here, an extended definition of the overbounding relationship between centrally aligned Cauchy (F_C) and Gaussian (F_G) distributions is provided:

$F_{G} (x; M_{0}, σ) \leq F_{C} (x; M_{0}, λ) \forall x \leq M_{0}$ 45

$F_{G} (x; M_{0}, σ) \geq F_{C} (x; M_{0}, λ) \forall x > M_{0}$ 46

where M₀ ∈ ℝ.

For convenience in proving this theorem, a subtraction function t(x) between the CDFs is introduced:

$t (x) = F_{G} (x; M_{0}, σ) - F_{C} (x; M_{0}, λ)$ 47

The derivative of this function gives the subtraction relationship between the PDFs:

$t^{'} (x) = f_{G} (x; M_{0}, σ) - f_{C} (x; M_{0}, λ)$ 48

If the Cauchy distribution overbounds the Gaussian distribution, their CDFs will coincide at (M₀, 1/2), which indicates the following:

$t (M_{0}) = 0$ 49

The definition of Equations (45) and (46) further gives the following:

$t (x) \leq 0 \forall x \leq M_{0}$ 50

$t (x) > 0 \forall x > M_{0}$ 51

Because the Cauchy PDF has super-exponential tails, we have the following:

$t^{'} (- \infty) < 0$ 52

$t^{'} (+ \infty) < 0$ 53

Owing to the symmetry of both Cauchy and Gaussian distributions, the following paragraphs focus on the case when x ≤ M₀ because similar procedures can be expanded to x > M₀. The proofs of both sufficiency and necessity for the theorem are provided as follows.

Proof of sufficiency. (If $λ \geq \sqrt{\frac{2}{π}} σ$ , then C(M₀,λ) overbounds N(M₀,σ).)

In the domain of x ≤ M₀, according to the properties described for Equations (49)–(51) and the heavy-tailedness of the Cauchy distribution shown in Equations (52) and (53), the conclusion that “C(M₀, λ) overbounds N(M₀, σ)” is equivalent to “∄x^∗ ∈(-∞, M₀) such that t(x^∗) > 0”. Thus, we only need to disprove the opposite statement, which specifically gives “∃x^∗ ∈ (-∞, M₀) such that t(x^∗) > 0.”

If $λ \geq \sqrt{\frac{2}{π}} σ$ , it can be proved that there is one and only one intersection between f_C(x) and f_G(x) when x < M₀ (details can be seen in the lemma). There exists $x_{0} \in (- \infty, M_{0})$ such that $t^{'} (x_{0}) = 0, t^{'} (x_{0}^{+}) > 0$ , and $t^{'} (x_{0}^{-}) < 0$ . In this case, when $x \in (- \infty, x_{0}), t^{'} (x) < 0$ , and t(x) is decreasing; when $x \in (x_{0}, M_{0}), t^{'} (x) > 0$ , and t(x) is increasing. It is inferred that x^∗ > x₀ (otherwise, the decreasing t(x) in (-∞, x₀) gives $t (x^{*}) < t (- \infty) < 0$ which is contradictory to the assumption t(x^∗). =0 Because t(x) is increasing in (x₀, M₀), and $x^{*} \in (x_{0}, M_{0})$ there will be $t (M_{0}) > t (x^{*}) = 0$ , which is contradictory to the overbounding property t(M₀) = 0.

Therefore, the opposite conclusion “∃x^∗ ∈ (-∞, M₀) such that t(x^∗) = 0.” is untenable, which proves that “∄x^∗ ∈(-∞, M₀) such that t(x^∗) > 0.” and C(M₀, λ) overbounds N(M₀, σ) 0σ. This contradiction can likewise be extended to the case when x > M₀. Thus, the sufficiency is proved.

Lemma. If $λ \geq \sqrt{\frac{2}{π}} σ$ , then f_C (x; M₀, λ)and f_G (x; M₀, λ) have one and only one intersection when x < M₀.

proof. Because $f_{G} (x) > 0 \forall x \in ℝ$ and $f_{C} (x) > 0 \forall x \in ℝ$ and they are both continuous functions, we introduce the following function:

$g (x) = \frac{f_{G} (x)}{f_{C} (x)} - 1 = \frac{λ}{σ} \sqrt{\frac{π}{2}} \exp (- \frac{1}{2} {(\frac{x - M_{0}}{σ})}^{2}) . (1 + {(\frac{x - M_{0}}{λ})}^{2}) - 1$ 54

Based on the PDF expressions for the Gaussian and Cauchy distributions in Equations (41) and (433), we know the following:

$f_{G} (x) = O (\exp (- x^{2})) a s | x | \to \infty$ 55

$f_{C} (x) = O (x^{- 2}) a s | x | \to \infty$ 56

where O(.) is the big O notation. Then, we can infer the following:

$g (- \infty) = - 1$ 57

$g (+ \infty) = - 1$ 58

Let us denote u = x - M₀; we then have u ∈(-∞, 0), and an intersection between f_C (x; M₀, λ) and f_G (x; M₀, λ) indicates that g(u) = 0 has a solution. The lemma further changes to “If $λ \geq \sqrt{\frac{2}{π}} σ$ ; then, g(u) has one and only one zero point in (-∞, 0).” The simplified gu(u) gives the following:

$g (u) = \frac{λ}{σ} \sqrt{\frac{π}{2}} \exp (- \frac{1}{2} {(\frac{u}{σ})}^{2}) . (1 + {(\frac{u}{λ})}^{2}) - 1$ 59

Its derivative with respect to u is as follows:

$g^{'} (u) = \frac{λ}{σ} \sqrt{\frac{π}{2}} . (- \frac{u}{σ^{2}} \exp (- \frac{1}{2} {(\frac{u}{σ})}^{2}) . (1 + {(\frac{u}{λ})}^{2}) + \frac{2 u}{λ^{2}} \exp (- \frac{1}{2} {(\frac{u}{σ})}^{2})) = \frac{λ}{σ} \sqrt{\frac{π}{2}} \exp (- \frac{1}{2} {(\frac{u}{σ})}^{2}) . (- \frac{u}{σ^{2}} - \frac{u^{3}}{σ^{2} λ^{2}} - \frac{2 u}{λ^{2}}) = \sqrt{\frac{π}{2}} \frac{\exp (- \frac{1}{2} {(\frac{u}{σ})}^{2})}{σ^{3} λ} . (- u^{3} - (λ^{2} - 2 σ^{2}) u)$ 60

If $λ^{2} - 2 σ^{2} \geq 0$ , which means that $λ^{\geq} \sqrt{2 σ}$ , then $g^{'} (u) > 0$ ; thus, g(u) is increasing over (-∞,0). Because g(u) is continuous, $g {(u)}_{\max} = g (0^{-}) \to g (0) = \frac{λ}{σ} \sqrt{\frac{π}{2}} - 1 \geq \sqrt{π} - 1 > 0$ . Combined with g(-∞)=-1<0, g(u) has one and only one zero point in (-∞,0).

If $λ^{2} - 2 σ^{2} < 0$ , which means that $\sqrt{\frac{2}{π}} σ \leq λ < \sqrt{2 σ}$ , when $u \in (- \sqrt{2 σ^{2} - λ^{2}}, 0)$ , then $g^{'} (u) < 0$ ; thus, g(u) is decreasing in this domain, $g {(u)}_{\min} = g (0^{-}) \to g (0) = \frac{λ}{σ} \sqrt{\frac{π}{2}} - 1 \geq 0$ and g(u) does not have a zero point on $(- \sqrt{2 σ^{2} - λ^{2}}, 0)$ . When $u \in (- \infty, - \sqrt{2 σ^{2} - λ^{2}})$ , $g^{'} (u) > 0$ ; thus, g(u) is increasing in this domain, and $g {(u)}_{\max} = g (- \sqrt{2 σ^{2} - λ^{2}}) > g (0) \geq 0$ . Given that g(x) is continuous and $g (- \infty) = - 1 < 0$ , g(u) has only one zero point in $(- \infty, - \sqrt{2 σ^{2} - λ^{2}})$ .

Combining the cases of both $λ^{2} - 2 σ^{2} \geq 0$ and $λ^{2} - 2 σ^{2} < 0$ , we can conclude that g(u) has one and only one zero point in (-∞,0). This proves that $f_{C} (x; M_{0}, λ)$ and $f_{G} (x; M_{0}; σ)$ have one and only one intersection in $(- \infty, M_{0})$ .

Proof of necessity. (If $C (M_{0}, λ)$ overbounds $N (M_{0}, σ)$ , then $λ \geq \sqrt{\frac{2}{π}} σ$ .)

Similarly, we assume that the opposite conclusion $λ < \sqrt{\frac{2}{π}} σ$ is true. It is known that the maximum of the PDF for both Cauchy and Gaussian distributions occurs when $x = M_{0}$ . Therefore, $f_{C} {(x)}_{\max} = \frac{1}{λ π} > \frac{1}{σ \sqrt{2 π}} = f_{G} {(x)}_{\max}$ .

When $x \leq M_{0}$ , there will always be a positive є such that when $x \in (M_{0} - є, M_{0})$ , $t^{'} (x) = f_{G} (x; M_{0}, σ) - f_{C} (x; M_{0}, λ) < 0$ ; thus, t(x) is decreasing in this domain. Based on the overbounding property in Equation (50), it is inferrable that $t (M_{0} - є) < 0$ . Now, with $M_{0} - є$ and decreasing t(x), we have $t (M_{0}) < t (M_{0} - є) < 0$ , which is contradictory to the premise $t (M_{0}) = 0$ given in Equation (49).

The contradiction can likewise be extended to the case in which $x > M_{0}$ ; thus, the assumption $λ < \sqrt{\frac{2}{π}} σ$ is untenable, and it can be concluded that $λ > \sqrt{\frac{2}{π}} σ$ . Thus, the necessity is proved.

C PROOF OF EXISTENCE OF A TANGENTIAL LINE BETWEEN CAUCHY AND GAUSSIAN OVERBOUNDS

In this study, a tangential link from the Cauchy overbound to the Gaussian overbound is always possible as long as the Cauchy CDF is not an overbound for the Gaussian CDF. This statement contains two parts, including (1) sufficiency: “if the Cauchy CDF is not an overbound for the Gaussian CDF, then a tangential link exists,” and (2) necessity: “if a tangential link exists, then the Cauchy CDF is not an overbound for the Gaussian CDF.” For the sake of notation, we denote the CDFs of the Gaussian and Cauchy overbounds as F_G(x) and F_C(x), respectively. Here, we focus on the positive half (i.e., when x>0), as similar procedures of proof can be inferred for the negative half (i.e., when x<0) due to the symmetry of the CDFs. In the following, sufficiency and necessity will be proven in the case of bounding SU errors.

Proof of sufficiency.We may first assume the opposite of the conclusion (i.e., a tangential link between the Cauchy and Gaussian CDFs does not exist). When x>0, both the Cauchy CDF and Gaussian CDF are strictly concave because their second-order derivatives are always negative. Because the hypothesis states that the Cauchy CDF is not an overbound for the Gaussian CDF, it is inferable that an intersection (denoted as point B in Figure (11(a)) always exists between the two CDFs when x>0, such that the following holds:

FIGURE 11

Schematics for a (a) proof of sufficiency and (b) proof of necessity

$F_{G} (x) < F_{C} (x), \forall 0 < x < x_{3}$ 61

$F_{G} (x) < F_{C} (x), \forall x > x_{3}$ 62

Because of the concavity of the CDF, for any x₁∈(0,x₃) passing through point A(x₁, F_C(x₁)), a unique tangential line l₁ for F_C(x) can always be drawn:

$l_{1} : y = F_{C}^{'} (x_{1}) (x - x_{1}) + F_{C} (x_{1})$ 63

Moreover, F_C(x₁) > F_G(x₁); thus, point A is above the Gaussian CDF. Passing through point A, a tangential line l₂ for the concave function F_G(x) can always be drawn, with a tangent point C(x₂,F_G(x₂)):

$l_{2} : y = F_{G}^{'} (x_{2}) (x - x_{2}) + F_{G} (x_{2})$ 64

Given by the premise that a tangential link between the two CDFs does not exist, we know that l₁ and l₂ cannot coincide. Equivalently, this premise indicates that the angle θ formed by l₁ (tangent to F_C(x)) and l₂ (tangent to F_G(x)) cannot be zero. Thus, we obtain the following:

$\tan (θ) = | \frac{F_{C}^{'} (x_{1}) - F_{G}^{'} (x_{2})}{1 + F_{C}^{'} (x_{1}) F_{G}^{'} (x_{2})} | \neq 0$ 65

which further gives the following:

$F_{C}^{'} (x_{1}) \neq F_{G}^{'} (x_{2}), \forall x_{1} \in (x_{3,} \infty)$ 66

Because of the strict concavity of the two CDFs, both $F_{C}^{'} (x_{1})$ and $F_{G}^{'} (x_{1})$ are decreasing functions when x>0. Therefore, we have the following:

$F_{C}^{'} (x_{1}) \in (F_{C}^{'} {(x_{1})}_{\min}, F_{C}^{'} {(x_{1})}_{\max}) = (F_{C}^{'} (x_{3}), (F_{C}^{'} (0))$ 67

$F_{G}^{'} (x_{2}) \in (F_{G}^{'} {(x_{2})}_{\min}, F_{G}^{'} {(x_{2})}_{\max}) = (\lim_{x \to + \infty} F_{G}^{'} (x), F_{G}^{'} (x_{3})) = (0, F_{G}^{'} (x_{3}))$ 68

where:

$F_{C}^{'} (0) > F_{G}^{'} (0) > F_{G}^{'} (x_{3})$ 69

We can further verify the following:

$F_{C}^{'} (x_{3}) < F_{G}^{'} (x_{3})$ 70

Let us first assume the opposite, $F_{C}^{'} (x_{3}) \geq F_{G}^{'} (x_{3})$ The intersection condition at x = x₃ gives $F_{C} (x_{3}) = F_{G} (x_{3})$ . There exists a small positive ε such that the following holds:

$\begin{array}{l} F_{C} (x_{3} + ε) = F_{C} (x_{3}) + ε * {F^{'}}_{C} (x_{3}) \\ = F_{C} (x_{3}) + ε * {F^{'}}_{C} (x_{3}) \geq F_{G} (x_{3}) + ε * {F^{'}}_{C} (x_{3}) \\ = F_{G} (x_{3} + ε) \end{array}$ 71

which contradicts with the property in Equation (62). Combining Equations (67)–(70), we obtain the following:

$\min ({F^{'}}_{G} (x_{2})) < \min ({F^{'}}_{C} (x_{1})) < \max ({F^{'}}_{G} (x_{2})) < \max ({F^{'}}_{C} (x_{1}))$ 72

Therefore, we can conclude the following:

${F^{'}}_{C} (x_{1}) = {F^{'}}_{G} (x_{2}), \exists x_{1} \in (0, x_{3}), \exists x_{2} \in (x_{3}, \infty)$ 73

which contradicts the property in Equation (66). Consequently, our initial assump tion must be false; thus, the sufficiency is proven.

Proof of necessity.We may first assume the opposite of the conclusion (i.e., the Cauchy CDF is an overbound for the Gaussian CDF). Then, we have the following:

$F_{G} (x) > F_{C} (x), \forall x > 0$ 74

When x>0, both the Cauchy CDF and Gaussian CDF are strictly concave because their second-order derivatives are always negative. As shown in Figure 11(b), passing through any point $A (x_{1}, F_{G} (x_{1})) (\forall x_{1} > 0)$ on the Gaussian CDF, there exists a unique tangential line l₁ for F_G(x), and the property of concavity gives the following:

$l_{1} \geq F_{G} (x), \forall x > 0$ 75

Combining Equations (74) and (75), we obtain the following:

$l_{1} > F_{C} (x), \forall x > 0$ 76

which means that line l₁ does not have any intersection with the Cauchy CDF F_c(x) when x>0. This result further implies that the line l₁ can never be a tangential line for F_c(x) and that the tangential link between the Cauchy CDF and Gaussian CDF does not exist, which contradicts the hypothesis (i.e., that a tangential link exists). Therefore, our initial assumption must be false, and the necessity is proven.

In conclusion, a tangential link can always exist between Cauchy and Gaussian overbounds, if and only if the Cauchy CDF is not an overbound for the Gaussian CDF. Notably, the latter condition rarely occurs unless the empirical error curve is approximately a Gaussian distribution. In that case (see Figure 11(a), for instance), points O and B will be considered coincident, and the proposed Cauchy–Gaussian overbound will become a single-CDF Gaussian overbound. Because the focus of this study is to develop an overbound for heavy-tailed error distributions, we may neglect the extreme case and conclude that a tangential link between the CDFs is always mathematically possible.

D PROOF OF OVERBOUNDING IN THE TANGENTIAL TRANSITION REGION

In the following proof, we consider only the case when x>0, because the conclusion can be similarly expanded to the domain of x≤0. Let us assume that the CDF of the original error profile is F_e(x) and that the general form of the tangential transition T(x) in CDF from Cauchy to Gaussian gives the following:

$T (x) = k_{m} x + b 0 < x_{1} \leq x \leq x_{2}$ 77

where k_m and b mathematically denote the slope and vertical intercept of the line function, respectively. Here, x₁ and x₂ are the abscissae of the two tangential points (i.e., x₁, F_C (x₁)) and x₂, F_G (x₂)) when we transition from a Cauchy to Gaussian model. According to the definition in Equation (2) and the properties of the tangential line segment, we have the following:

$T (x_{1}) = F_{C} (x_{1}) \leq F_{e} (x_{1})$ 78

$T (x_{2}) = F_{G} (x_{1}) \leq F_{e} (x_{2})$ 79

where F_C(x) and F_C(x) represent the Cauchy and Gaussian CDFs, respectively.

For an SU error, the PDF f_e(x) (i.e., $F_{e}^{'} (x)$ ) is monotonically decreasing when x > 0. For convenience in the proof, a function m(x) is introduced:

$m (x) = F_{e} (x) - T (x) 0< x_{1} \leq x \leq x_{2}$ 80

Combining with the inequalities in Equation (78) and (79), we obtain the following:

$m (x_{1}) \geq 0$ 81

$m (x_{2}) \geq 0$ 82

The first derivative of m(x) can be derived as follows:

$m^{'} (x) = F_{e}^{'} (x) - k_{m} x_{1} \leq x \leq x_{2}$ 83

If $k_{m} \leq x_{1} \leq x_{2}$ , there will be $m (x) \leq 0$ ; thus, m(x) is decreasing when $x \in [x_{1}, x_{2}]$ . The minimum $m {(x)}_{\min} = m (x_{2}) \geq 0$ , which further indicates that $F_{e} (x) \geq T (x) \forall x \in [x_{1}, x_{2}]$ .

If $x_{1} \leq x_{2} \leq k_{m}$ , there will be $m (x) \geq 0$ ; thus, m(x) is increasing when $x \in [x_{1}, x_{2}]$ . The minimum $m {(x)}_{\min} = m (x_{1}) \geq 0$ , which gives $F_{e} (x) \geq T (x) \forall x \in [x_{1}, x_{2}]$ .

If $x_{1} < k_{m} < x_{2}$ , when $x \in [x_{1}, k_{m}], m (x) \geq 0$ ; thus m(x) is increasing. When $x \in [k_{m}, x_{2}], m (x) \leq 0$ ; thus m(x) is decreasing. It is straightforward to determine the minimum $m {(x)}_{\min} = \min (m (x_{1}), m (x_{2})) \geq 0$ , which gives $F_{e} (x) \geq T (x) \forall x \in [x_{1}, x_{2}]$ .

Combining all three scenarios, we can prove that the curve F_e(x) is always located above the transition T(x) when x>0. Owing to the symmetry of CDFs, a similar conclusion can be extended that F_e(x) always lies below the transition T(x) when x>0. This conclusion indicates that the overbounding properties are guaranteed in the two transition regions (i.e., when $0 < x_{1} < x < x_{2}$ and $- x_{2} < x < - x_{1} \leq 0$ ).

E PROOF OF EQUATION (33)

According to Equation (32), the i-th component of the VPE can be expressed as follows:

${VPE}_{i} = S_{3, i} ε_{i}$ 84

Then, the CDF $F_{V P E_{i}}$ can be derived:

$F_{V P E_{i}} (x) = \Pr (V P E_{i} < x) = \Pr (ε_{i <} \frac{x}{|S_{3, i}|}) = \int_{- \infty}^{\frac{x}{|S_{3, i}|}} f_{ε_{i}} (t) d t$ 85

Let $u = | S_{3, i} | t$ then, we have $d u = | S_{3, i} | d t$ and further obtain the CDF and PDF $F_{V P E_{i}}$ as follows:

$F_{VPE i} (x) = \frac{1}{| S_{3, i} |} \int_{-\infty}^{x} f_{ε i} (\frac{u}{| S_{3, i} |}) d u$ 86

$f_{VPE i} (x) = \frac{1}{| S_{3, i} |} f_{ε i} (\frac{x}{| S_{3, i} |})$ 87

The distribution of the VPE (f_VPE) is the joint distribution of all f_VPE_i, which gives the following:

f_{VPE} (x) = \otimes_{i = 1}^{N} f_{VPE i} (x) = (\prod_{i = 1}^{N} \frac{1}{| S_{3, i} |}) \cdot \otimes_{i = 1}^{N} f ε_{i} (\frac{x}{| S_{3, i} |}))

F NAVDEN MODEL

The NavDEN model is a discrete error model designed to provide a conservative bound on heavy-tailed GNSS measurement errors. This model features a Gaussian-like core to tightly bound nominal errors and flared, non-Gaussian tails to conservatively account for the higher-than-Gaussian probability of large errors. The model is defined by a set of centrosymmetric left and right bounds on a regularly spaced grid.

The NavDEN overbound includes three regions: left tail $(K_{1})$ , core $(K_{2})$ , and right tail $(K_{3})$ . Each region contains a subset of envelope boundaries noted by integer indices as given below:

$K_{1} = [k_{\min}, - k_{t r})$ 89

$K_{2} = [- k_{t r}, k_{t r}]$ 90

$K_{3} = (k_{t r}, k_{\max}]$ 91

where $k_{\max}$ and $k_{\min}$ are the largest and smallest indices, respectively. $k_{tr}$ serves as the transition index between the core and tail regions. The normalized left bound, ${\tilde{l}}_{k}$ , for an envelope with integer index k is defined by the following piece-wise function:

${\tilde{l}}_{k} = {\begin{matrix} f l o o r (\tilde{C} \ln (1 - \frac{k + k_{t r}}{k_{\min} + k_{t r}}) - k_{t r} - k_{b i a s}) & k \in K_{1} \\ k - k_{b i a s} & k \in K_{2} \\ f l o o r ({\tilde{x}}_{\max} - k_{b i a s} - ({\tilde{x}}_{\max} - k_{t r}) e^{2 (k_{t r} - k) / \tilde{B}}) & k \in K_{3} \end{matrix}$ 92

where parameters with a tilde notation are normalized by the fundamental grid spacing, Δ. The model is defined by several parameters, including the transition index $(k_{t r})$ , asymptotes $(x_{\min}, x_{\max})$ , curvature parameters $(\tilde{B}, \tilde{C})$ , and a core off-set $(k_{b i a s})$ . The cumulative probability associated with each left bound, expressed in Gaussian quantile form $({\tilde{G}}_{l, k})$ , is given by the following:

${\tilde{G}}_{l, k} = {\begin{matrix} - k_{t r} + ψ_{1} (k + k_{t r}) & k \in K_{1} \\ k & k \in K_{2} \\ k_{t r} + ψ_{2} (k - k_{t r}) & k \in K_{3} \end{matrix})$ 93

where $ψ_{1} = \frac{{\tilde{x}}_{\min} + k_{t r}}{k_{\min} + k_{t r}}$ and $ψ_{2} = \frac{{\tilde{x}}_{\max} - k_{t r}}{k_{\max} - k_{t r}}$ . Because the model is symmetric, the right bounds $({\tilde{r}}_{k})$ and their associated Gaussian quantiles $(G_{r, k})$ are simply a reflection of the left bounds around the origin. For a detailed explanation of the NavDEN model, one may refer to the original work by Rife and Pervan (2012)

Footnotes

↵1 To maintain generality across both Gaussian and Cauchy distributions, we will use the terms “location parameter” and “scale parameter” throughout the remainder of the paper.
↵2 The term “quantile scale” indicates that the CDF values are transformed by Ф⁻¹(·) to the equivalent standard normal quantiles.

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Cauchy–Gaussian Overbound for Heavy-Tailed GNSS Measurement Errors

Zhengdao Li

Zhengdao Li

Penggao Yan

Penggao Yan

Weisong Wen,

Weisong Wen,

and Li-Ta Hsu

Li-Ta Hsu

Institute of Navigation

Mar 2026,

73

navi.749;

DOI: 10.33012/navi.749

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Cauchy–Gaussian Overbound for Heavy-Tailed GNSS Measurement Errors

Zhengdao Li

Zhengdao Li

Penggao Yan

Penggao Yan

Weisong Wen,

Weisong Wen,

and Li-Ta Hsu

Li-Ta Hsu

Institute of Navigation

Mar 2026,

73

navi.749;

DOI: 10.33012/navi.749

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Keywords

Cauchy distribution, global navigation satellite systems, heavy-tailed distributions, overbounding techniques

Cauchy–Gaussian Overbound for Heavy-Tailed GNSS Measurement Errors

Abstract

1 INTRODUCTION

2 REVIEW OF GAUSSIAN OVERBOUNDING METHODS

2.1 Single-CDF Gaussian Overbound

2.2 Paired Gaussian Overbound

2.3 Two-Step Gaussian Overbound

3 CAUCHY–GAUSSIAN OVERBOUND

3.1 Cauchy Distribution

3.2 Procedure for Bounding SU Errors

Step 1: Overbound with optimal Cauchy and Gaussian distributions

Step 2: Tangential transition to preserve overbounding properties through convolution

Step 3: Synthesis of the two models and transitions

3.3 Procedure for Bounding NSU Errors

Step 1: Construct the optimal paired CGCM overbound

Step 2: Construct an optimal paired Gaussian overbound

Step 3: Synthesize the optimal paired CGCM overbound and paired Gaussian overbound

3.4 Position-Domain Bounding

4 NUMERICAL EXPERIMENTS

4.1 Bounding Performance for SU Errors

4.2 Bounding Performance for NSU Errors

4.3 Impact of Heavy-Tailedness

5 CONCLUSIONS AND FUTURE WORK

CONFLICT OF INTEREST

HOW TO CITE THIS ARTICLE:

ACKNOWLEDGMENTS

A COORDINATES OF RECEIVERS AND REFERENCE STATIONS

B SUFFICIENT AND NECESSARY CONDITIONS OF THE CAUCHY OVERBOUND ON A CENTRALLY ALIGNED GAUSSIAN DISTRIBUTION

C PROOF OF EXISTENCE OF A TANGENTIAL LINE BETWEEN CAUCHY AND GAUSSIAN OVERBOUNDS

D PROOF OF OVERBOUNDING IN THE TANGENTIAL TRANSITION REGION

E PROOF OF EQUATION (33)

F NAVDEN MODEL

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