Quantifying GNSS Multipath Statistics by Signal Properties in Urban, Suburban, and Rural Environments

Maximilian von Arnim; Damien Vivet,; Yoko Watanabe

doi:10.33012/navi.790

Abstract

Positioning solutions based on global navigation satellite systems remain vulnerable to local multipath and non-line-of-sight conditions, a challenge for autonomous systems that must operate reliably across diverse environments. To support context-aware navigation, we quantify pseudorange multipath effects across four environmental settings: canyon, urban, trees, and open sky. Given a known receiver position, the range-free double difference (RFDD) can be used to isolate multipath errors. We propose a refinement accounting for clock biases in a commercial off-the-shelf receiver, which otherwise distort geometric range estimates—a key input to the RFDD. Applying our method to data collected in Toulouse, France, we statistically analyze how multipath errors can be modeled as a function of carrier-to-noise ratio (C/N₀) within each labeled environmental context. Histograms and the first four moments—mean, variance, skewness, and kurtosis—offer complementary perspectives, showing that the multipath component of the RFDD may be described by a chi-square-like distribution.

Keywords

1 INTRODUCTION

Global navigation satellite systems (GNSSs) form the backbone of numerous navigation applications, including commercial aviation, uncrewed aerial vehicles (UAVs), automotive systems, and portable devices. Under ideal line-of-sight (LOS) conditions, state-of-the-art GNSS receivers can achieve horizontal positioning accuracies better than 1 cm, both in postprocessing and real time using real-time kinematics (RTK) techniques (Stern & Kos, 2018). This level of accuracy has already been commercially leveraged, for instance, in precision agriculture (Lange & Peake, 2021), and could theoretically satisfy the positioning requirements of autonomous driving (Kealy & Moore, 2017).

However, centimeter-level kinematic positioning remains difficult outside specific environments favorable to GNSSs such as open fields. In more obstructed areas, particularly in urban or tree-covered settings, GNSS-only positioning accuracy can degrade to meter-level errors (Stern & Kos, 2018). GNSS signal disturbances can arise from various sources, including atmospheric delays, multipath reflections, non-line-of-sight (NLOS) reception, unintentional interference (Humphreys, 2017), and deliberate spoofing or jamming (Morales-Ferre et al., 2020; Psiaki & Humphreys, 2021). This study focuses on multipath and NLOS effects-caused by reflection, diffraction, and absorption of signals by nearby surfaces-as they are difficult to model owing to their complex and localized nature, yet found in virtually all environments. These features set multipath and NLOS effects apart from atmospheric errors, which are spatially smooth over several kilometers and can be addressed by differential GNSS approaches including RTK methods. In contrast, spoofing and jamming remain severe and unpredictable, yet rare, occurrences that require countermeasures adapted to evolving threats (Renaudin et al., 2025).

These degradations are especially pronounced in devices constrained by size, cost, or power, such as UAVs and smartphones (Stern & Kos, 2018). Real-time connectivity to RTK base stations is likewise affected by these constraints, as is the use of precise carrier-phase measurements required for RTK or Galileo’s new High Accuracy Service positioning (Suzuki, 2023). The latter’s wide-area correction services may be unavailable in harsh environments such as urban canyons (Cucchi et al., 2024). Multi-frequency receivers promise enhanced stand-alone performance through higher chipping rates and ionosphere-free combinations (Ng et al., 2020), but low-cost devices have only recently entered the market and, especially on smartphones, perform markedly worse at L5 than at L1 frequencies (van Diggelen, 2021).

As a result, multipath poses a persistent challenge despite being recognized as a critical error source since the early days of GNSS development (Goblick, 1966). While advanced mitigation techniques within correlators or tracking loops exist, their integration in low-cost commercial off-the-shelf (COTS) receivers is often challenging owing to commercial and technical constraints. Simultaneously, such receivers rarely expose the intermediate steps, which prevents users from adding these techniques themselves. It is also difficult to separate effects caused by the environment from those stemming from the receiver design. The desire to nevertheless work with COTS parts in demanding applications has inspired the exploration of context-adaptive navigation strategies (Groves et al., 2013).

Although GNSS sensor fusion is common, even the addition of state-of-the-art visual-inertial navigation systems cannot yet solve the urban canyon problem entirely (Bai & Hsu, 2025). Terrestrial radio positioning can be useful but is equally subject to environmental effects (Li et al., 2025). Jiang et al. (2025) added adaptive integrity monitoring but observed significant latency of the adaptation owing to its reliance on implicit contextual information within the estimator innovation. Critically, multi-sensor fusion and integrity depend on well-characterized error models, which motivates this empirical study of environment-specific GNSS multipath effects.

In this paper, we aim to characterize multipath and NLOS effects on code-phase measurements from Global Positioning System (GPS) and Galileo constellations across distinct environmental contexts. By analyzing signal distortions across varied settings, we seek to support the development of environment-aware pseudorange error models that can ultimately enhance navigation robustness in real-world deployments.

Understanding the statistical characteristics of multipath errors is essential for their reliable detection and mitigation. Many downstream applications such as integrity monitoring, sensor fault detection, and multi-sensor fusion rely on accurate measurement error models. While prior work has examined signal propagation in various environments and proposed context-agnostic multipath models based on GNSS observables, these approaches often overlook the impact of environmental context on pseudorange measurements. To our knowledge, this is the first study that systematically analyzes the statistical effects of multipath on GNSS pseudoranges across four distinct environmental conditions.

Our contributions are as follows:

We extend the method described by Matera et al. (2018, 2019), which isolates the multipath component from other error sources via range-free double differences (RFDDs), to better accommodate non-synchronized receivers.
The modified method is applied to a low-cost COTS receiver and a geodetic receiver using a GNSS/vision data set (Feriol, 2022) that includes cameraaided environmental context classifiers.
We quantify the first four statistical moments—mean, variance, skewness, and kurtosis—of multipath-affected pseudoranges for GPS L1 coarse/acquisition (C/A) and Galileo E1 OS signals, segmented by environmental context.
We analyze the relationship between GNSS observables and multipath severity and propose context-specific pseudorange error models to support improved positioning algorithms.

2 METHODOLOGY

Improving GNSS robustness against multipath errors in the pseudorange observable requires models of their effects. One method for isolating the multipath component was introduced by Matera et al. (2018), which relies on the known receiver position to compute the RFDD. This technique effectively separates multipath from other error sources under ideal synchronization. However, in practice, clock biases between the receiver under test and the reference system introduce a constant offset in the satellite position computation, thereby biasing the resulting range differences.

For completeness, we also briefly compare this approach to state-of-the-art multipath modeling and measurement techniques.

2.1 Background

Approaches to modeling GNSS multipath effects can be broadly categorized into two classes: physics-based (or model-driven) methods and data-driven (or probabilistic) techniques. Physics-based models rely on geometric ray-tracing or signal propagation theory to simulate multipath effects in specific environments. In contrast, data-driven methods typically use statistical learning or empirical modeling to capture multipath behavior from observations, often without explicitly modeling the physical signal environment. Throughout this work, we use the term "multipath" to encompass not only classical reflected signals but also related effects such as NLOS reception, diffraction, and scattering by the surrounding environment.

Physics-based models are grounded in the physical interaction of radio-frequency signals with solid surfaces. A detailed environment model permits simulations using ray-tracing (Bradbury et al., 2007; Li et al., 2024) or electromagnetic differential equations (Hannah, 2001). A key prerequisite is precise knowledge of the geometry of the environment (Bradbury et al., 2007) and the electromagnetic characteristics of the materials found therein. Coefficients for common materials can be readily found in the literature (Braasch, 2017).

Data-driven approaches avoid this dependence on high-fidelity environmental models by treating multipath as a stochastic process. Especially diffuse reflections at rough surfaces are difficult to model deterministically (Hannah, 2001). The traditional method for quantifying the code multipath component utilizes the code-minus-carrier (CMC), as the carrier-phase measurements are far less affected by multipath. The CMC is not trivial, however, for cheap single-frequency receivers and dynamic scenarios owing to frequent cycle slips or simply missing carrier-phase data (Massarweh et al., 2020).

These challenges motivated Matera et al. (2018) to propose a multipath isolation method without the need for carrier-phase measurements, yet comparable to the CMC in performance (Matera et al., 2024). In this method, a ground truth position, obtained from a precise reference receiver, and a base station help remove the other terms affecting the pseudorange. Peretic et al. (2025) demonstrated that a nearby International GNSS Service station can replace a user’s own base station, expanding the range of locations to study. One remaining challenge, which we address in this work, is correcting for clock biases between unsynchronized receivers and the position reference system.

Based on either physics- or data-driven modeling, several studies have proposed environment-specific multipath models. Such efforts include models for urban (Lima Filho & Moraes, 2021; Steingass & Lehner, 2008; Turin et al., 1972; Viandier et al., 2008), urban canyon (Peretic et al., 2025), suburban (Steingass & Lehner, 2008; Viandier et al., 2008), airport (Braasch, 1996), forest (Schubert et al., 2012; Zhou et al., 2025), and general rural (Lima Filho & Moraes, 2021; Schubert et al., 2012) environments. Many more examples exist, but few cover more than two distinct environments, making a comparison of results difficult. Urban environments have received much attention, whereas vegetation effects have been somewhat neglected. A common observation is a strong dependence on environment class.

Li et al. (2024) remarked that models focus on different aspects: some describe the signal before entering the correlator or discriminator (Lima Filho & Moraes, 2021; Schubert et al., 2012; Steingass & Lehner, 2008; Turin et al., 1972), others provide models of the pseudorange and other observables after this stage (Braasch, 1996; Li et al., 2024; Matera et al., 2019; Viandier et al., 2008), and a few focus on the position, velocity, and time (PVT) estimate (Stern & Kos, 2018).

It is important to note that multipath is often studied for specific waveforms and correlators, as their interactions are well understood and error envelopes for common modulation/correlator combinations exist. For example, recent waveforms such as the binary offset carrier (BOC) used by the Galileo constellation are expected to outperform the simpler binary phase shift keying of GPS L1 C/A. Correlators with narrower bandwidth and advanced techniques offer stronger multipath rejection, while antennas can mitigate ground reflections (Braasch, 2017).

In contrast, double-difference residuals reflect a combination of receiver-internal random noise and environment-induced multipath. Although we previously identified the receiver-internal noise as normally distributed under benign environmental conditions (von Arnim et al., 2025b), typically with a much smaller magnitude than multipath errors, this characteristic limits how well the multipath component can be characterized. Consequently, transferring a multipath model between receivers, e.g., from a u-blox to a NovAtel unit, proves difficult because of differing receiver architectures, as shown in prior work (von Arnim et al., 2025b). Thus, generalizing one experimental setup to all possible applications has its limits. Still, COTS receivers share similar technologies, and we believe that the data collected by Feriol (2022) are representative of many common use cases.

Matera et al. (2018) used the RFDD to directly obtain models of observables. This approach circumvents the challenge of combining signal and correlator models, which is notoriously difficult if the inner workings of COTS "black box" receivers are unknown. Peretic et al. (2025) and Zhou et al. (2025) successfully applied this approach in recent work focused on urban canyon and vegetated scenarios, respectively.

On the environmental side, certain contexts, such as signal loss and reacquisition while driving under bridges, present challenges beyond mere multipath and are omitted from our study. These cases may warrant the introduction of more fine-grained context classes, as proposed by Mayolle et al. (2025), beyond the four classes discussed in this contribution. Peretic et al. (2025) further remarked on the influence of individual satellites or obstacles on the statistics of smaller data sets, motivating the study of larger and diverse data sets.

Beyond the well-known impact of building materials on multipath, vegetation may have varied effects depending on species and season, posing a challenge for the transfer of models between climatic regions (Zhou et al., 2025). Additionally, recent work on computer vision permits the identification of trees as opposed to solid obstacles in contrast to the simpler sky/non-sky segmentation applied in our work. A third class, vegetation, beyond binary LOS and NLOS labels has been proposed to model multipath effects on signals that pass through an obstacle along the LOS path (Marais et al., 2025; Zhou et al., 2025). Consequently, a sky-facing camera to obtain this context information is poised to become an increasingly valuable tool in research settings, although further work may be necessary for commercial deployment.

The results obtained here are directly applicable to the navigation solution of the studied system and have been exploited by the authors in a context-adaptive estimator (von Arnim et al., 2025a).

2.2 Measuring the Multipath Component

For COTS receivers, the user typically has access only to the observables, that is, the code phase or pseudorange, the carrier phase, the Doppler shift, and the signal strength or C/N₀, in addition to the receiver’s internal PVT estimate. Matera et al. (2018) proposed a multipath estimator that uses the test receiver’s precisely known position and a fixed base station receiver to obtain a residual, the RFDD. Many errors are correlated between receivers or satellites and thus cancel in the RFDD, such that only the uncorrelated multipath component, random noise, and minor modeling errors remain. The overall RFDD process is illustrated in Figure 1.

FIGURE 1

Flowchart of the RFDD algorithm

Prior work is shown in black; novel contributions are shown in red. The original RFDD approach compensates geometric, atmospheric, clock-related, and hardware-related signal delays; our work seeks to further correct errors of the satellite position calculation caused by receiver timestamp biases. LS: least squares

2.2.1 Theoretical Model

First, we summarize the prior RFDD method (Matera et al., 2018) and discuss some of the details of our implementation. To analyze the contribution of multipath to GNSS measurements, we adopt the standard pseudorange observation model (Odijk, 2017):

$P_{r}^{i} = R_{r}^{i} + b_{r} c - b^{i} c + h_{r} + h^{i} + A_{r}^{i} + M_{r}^{i} + η_{r}^{i}$ 1

where the parameters (in meters) are

${\begin{cases} P : & pseudorange \\ R : & true geometric range \\ b c : & clock bias (b in s) \times speed of light (c = 2.99792458 \times 10^{8} m / s) \\ h : & hardware bias \\ A : & atmospheric (ionospheric + tropospheric) delay \\ M : & multipath delay \\ η : & random noise, ephemeris, and other unmodeled errors \end{cases}$

A superscript i is used for the individual satellite, with a subscript r for each receiver, where r is D for the device under test, B for the ground base station, or R for the reference system. x_r values depend on the time of reception t_r, xⁱ depend on the time of transmission tⁱ, and $x_{r}^{i}$ depend on both (Hauschild, 2017). Estimated values are denoted by a hat, $\hat{x}$ . All positions $\vec{X}$ are given in a common Earth-centered Earth-fixed (ECEF) frame.

First, the range-free pseudorange $Δ P R_{r}^{i}$ is obtained by subtracting the geometric range estimate from the pseudorange:

$Δ P R_{r}^{i} (t_{r}) = P_{r}^{i} (t_{r}) - R_{r}^{i} (t_{r}, {\hat{t}}_{i})$ 2

where we obtain the estimated emission time ${\hat{t}}_{i} (t_{r})$ from the geometric algorithm of Subirana et al. (2013). Applying this approach for both the mobile receiver D and the base station B yields the single-difference $ϵ_{D, B}^{i}$ for each satellite i at the time of reception t_D:

$\begin{matrix} ϵ_{D, B}^{i} (t_{D}) & = Δ P R_{D}^{i} (t_{D}) - Δ P R_{B}^{i} (t_{D}) \\ = (b_{D} c + h_{D} + M_{D}^{i} + η_{D}^{i}) - (b_{B} c + h_{B} + M_{B}^{i} + η_{B}^{i}) \end{matrix}$ 3

which eliminates the satellite-dependent delay terms and the atmospheric delay from Equation (1) by assuming $A_{D}^{i} \approx A_{B}^{i}$ . At our mid-latitude site (43.5° N), worst-case differential errors may reach up to 11 ppm, 27 ppm, and 0.05 ppm for ionospheric, tropospheric, and ephemeris errors, respectively (Odijk & Wanninger, 2017). Consequently, their contribution to $ϵ_{D, B}^{i}$ is limited to less than 25 cm on our baseline of up to 6 km, which is small compared with pseudorange multipath. Further, we assume $M_{B}^{i} \approx 0$ for our rooftop base station antenna. Then, the RFDD $Δ ϵ_{r, b}^{i, j}$ is derived by taking a difference of the single difference with a chosen pivot satellite j:

$Δ ϵ_{D, B}^{i, j} (t_{D}) = ϵ_{D, B}^{i} (t_{D}) - ϵ_{D, B}^{j} (t_{D})$ 4

which eliminates the receiver-dependent delay terms from Equation (3), leaving only random noise and the mobile receiver’s multipath delay:

$\begin{matrix} Δ ϵ_{D, B}^{i, j} (t_{D}) & = (M_{D}^{i} + η_{D}^{i} - η_{B}^{i}) - (M_{D}^{j} + η_{D}^{j} - η_{B}^{j}) \\ = (M_{D}^{i} (t_{D}) - M_{D}^{j} (t_{D})) + ((η_{D}^{i} - η_{B}^{i}) - (η_{D}^{j} - η_{B}^{j})) \\ = Δ M_{D}^{i, j} (t_{D}) + η_{D, B}^{i, j} \end{matrix}$ 5

2.2.2 Choosing the Pivot Satellite

The choice of pivot satellite plays a key role in this method, giving the relative multipath $Δ M_{D}^{i, j}$ between it and the other satellites. Ideally, the pivot satellite is unaffected by multipath; in practice, the satellite with minimal multipath effects should be selected, such that the following holds:

$M_{D}^{i} (t_{D}) ≫ M_{D}^{j} (t_{D})$ 6

Matera et al. (2018) proposed to combine all data above a certain C/N_0,min threshold as a joint pivot satellite, presuming that such averaged multipath is more Gaussian than that of a single satellite. However, that approach complicates the receiver clock offset correction in the following step, as it does not address the temporal offset between mobile and base station receivers. In our data set (Feriol, 2022), there is generally at least one high-elevation satellite with minimal multipath impact, permitting us to select a single pivot satellite.

The preferred pivot satellite j is found by counting, for each satellite, the number of measurements with a C/N₀ above a threshold C/N_0,min. Short instances with $C / N_{0, D}^{j} \leq C / N_{0, min}$ where the pivot satellite is strongly affected by multipath, for example, under bridges, are replaced by interpolated values ${\bar{ϵ}}_{D, B}^{j}$ , assuming small short-term temporal variability of the non-multipath receiver biases. The pivot satellite is removed from our analysis as its RFDD is, by definition, 0 m for almost all samples. Nevertheless, its pseudorange noise is incorporated into $η_{D, B}^{i, j}, i \neq j$ .

To obtain Galileo statistics while using a GPS pivot satellite, Matera et al. (2019) included the GPS-to-Galileo clock offset. The expansion of Galileo since 2018 permits us to select one pivot satellite per constellation, allowing us to better study constellation-specific effects.

As the sampling of the receiver under test and that of the reference system are not synchronized, linear interpolation of ${\vec{X}}_{R} (t_{R})$ obtains the true position ${\vec{X}}_{D}$ at the time of reception t_D such that $t_{R, k} \leq t_{D} < t_{R, k + 1}$ .

We have summarized the standard approach from the literature for estimating the multipath component of pseudorange measurements, including a discussion of other contributing error sources. To mitigate receiver-related biases, we introduced a pivot satellite strategy that allows their relative elimination. However, an indirect contribution from the receiver bias still remains, which requires further consideration in the subsequent analysis.

2.2.3 Dealing with Receiver Clock Bias

To lay the foundation for the following discussion, let us summarize how the RFDD method allows the study of any receiver by making use of a base station (B) and a pivot satellite j to calculate double-differenced pseudoranges. Additionally, we require the previously discussed geometric range estimate $R_{r}^{i} (t_{r}, {\hat{t}}_{i})$ . The receiver under test (D) by itself provides noisy estimates of its position ${\vec{X}}_{D}$ , leading to errors in the range estimate. To circumvent this issue, the position estimate is instead taken from the reference system (R), which is the ground truth.

Unfortunately, the receiver’s clock bias introduces an error in the calculated true range due to satellite movement. This contribution proposes a novel approach that corrects this error without requiring precisely synchronized measurements. This approach bears some similarity to range-rate corrections of differential GNSS services (Odijk & Wanninger, 2017).

Each GNSS receiver maintains its own internal clock, which assigns a common timestamp T_r,k to all observables from the tracked satellites at each measurement epoch k. This timestamp is derived from the received time of week (TOW), but differs from the true reception time t_r,k by the receiver’s clock bias b_r,k. This timestamp can be expressed as follows:

$T_{r, k} = t_{r, k} + b_{r, k}$ 7

The bias is typically on the order of milliseconds, unless the receiver is equipped with a steered clock. The latter is the case for the reference receiver and its associated base station. Therefore, in this study, we assume that their clocks are nearly bias-free and, for a differential GNSS system, T_R = T_B. Time transfer based on phase measurements regularly achieves better than 10-ns accuracy (Defraigne, 2017), which results in millimeter-level errors on the geometric range estimate (Subirana et al., 2013). In contrast, the unsteered clock of a low-cost receiver D may contribute a bias of up to 60 m (Subirana et al., 2013).

We do not know the true time of reception t_r; instead, the timestamp T_r must be used to process the data. The true time would be part of the PVT estimate, but this estimate is itself affected by multipath and should therefore be avoided. The clock bias incurs a small but non-negligible error in the estimated geometric range ${\hat{R}}_{r}^{i} = R_{r}^{i} (t_{r}, {\hat{t}}^{i})$ if the timestamp T_r rather than the true time of reecption t_r is used for Equation (2), as the geometric range estimated with the true time of reception can be approximated to first order as follows:

$\begin{matrix} R_{r}^{i} (t_{r}, {\hat{t}}^{i} (t_{r})) & = R_{r}^{i} (T_{r} - b_{r}, {\hat{t}}^{i} (T_{r} - b_{r})) \\ \approx R_{r}^{i} (T_{r}, {\hat{t}}^{i} (T_{r})) - b_{r} {\dot{R}}_{r}^{i} (T_{r}) \end{matrix}$ 8

Therefore, we must correct the resulting error term $b_{r} {\dot{R}}_{r}^{i} (T_{r})$ . The range rate ${\dot{R}}_{r}^{i}$ with respect to the time of reception in Equation (8) can be estimated by differentiating the calculated ranges $R_{r}^{i} (T_{r}, {\hat{t}}^{i} (T_{r}))$ for the two epochs k and k + 1:

${\hat{\dot{R}}}_{r}^{i} (T_{r, k}) \approx \frac{R_{r}^{i} (T_{r, k + 1}, {\hat{t}}^{i} (T_{r, k + 1})) - R_{r}^{i} (T_{r, k}, {\hat{t}}^{i} (T_{r, k}))}{T_{r, k + 1} - T_{r, k}}$ 9

Equation (9) would be an exact solution if the unit vector from the receiver to the satellite as well as the receiver’s and satellite’s velocity vectors were constant. Given the slow movement of GNSS satellites across the sky and the low vehicle dynamics of the receiver, this assumption is reasonable for millisecond-order time frames. A receiver under multi-g acceleration could face millimeter- to decimeter-level RFDD errors given similar clock offsets and sampling rates. An expanded discussion of the range rate is provided in Appendix A. We can insert Equations (8) and (9) into Equation (2) to give the corrected range-free pseudorange, supposing a clock bias estimate ${\hat{b}}_{r}$ :

$\begin{matrix} Δ P R_{r}^{i} (t_{r}) & \approx P_{r}^{i} (T_{r}) - R_{r}^{i} (T_{r}, {\hat{t}}^{i} (T_{r})) + {\hat{\dot{R}}}_{r}^{i} (T_{r}) {\hat{b}}_{r} \\ = Δ P R_{r}^{i} (T_{r}) + {\hat{\dot{R}}}_{r}^{i} (T_{r}) {\hat{b}}_{r} \end{matrix}$ 10

where $P_{r}^{i} (T_{r})$ is the measured pseudorange assigned a timestamp T_r, which corresponds to the true position ${\vec{X}}_{r} (t_{r})$ and the true range $R_{r}^{i} (t_{r}, t^{i})$ . Given the steered base station clock, we further approximate $Δ P R_{B}^{i} (t_{B}) \approx Δ P R_{B}^{i} (T_{B})$ , from which we obtain $Δ P R_{B}^{i} (T_{D})$ by linear interpolation. Then, the single difference of range-free pseudoranges in Equation (3) is as follows:

$ϵ_{D, B}^{i} (t_{D}) \approx Δ P R_{D}^{i} (T_{D}) + {\hat{\dot{R}}}_{D}^{i} (T_{D}) {\hat{b}}_{D} - Δ P R_{B}^{i} (T_{D}) = ϵ_{D, B}^{i} (T_{D}) + {\hat{\dot{R}}}_{D}^{i} (T_{D}) {\hat{b}}_{D}$ 11

Subsequently, Equation (11) is inserted into Equation (4) to obtain the RFDD and its estimate $Δ \hat{ϵ}$ :

$Δ ϵ_{D, B}^{i, j} (t_{D}) \approx ϵ_{D, B}^{i} (T_{D}) - ϵ_{D, B}^{j} (T_{D}) + [{\hat{\dot{R}}}_{D}^{i} (T_{D}) - {\hat{\dot{R}}}_{D}^{j} (T_{D})] {\hat{b}}_{D} = Δ ϵ_{D, B}^{i, j} (T_{D}) + Δ {\hat{\dot{R}}}_{D}^{i, j} (T_{D}) {\hat{b}}_{D}$ 12

From this expression, we can estimate a clock bias ${\hat{b}}_{D}$ . As the RFDD derived in Equation (4) eliminates the direct impact of the clock and hardware biases, including their temporal variability, we will assume that the offset b_r between the receivers remains constant for the duration of the experiment. During each experiment, we observe a residual clock bias δb_D of up to ±0.5 ms on the low-cost receiver D, which, combined with an observed $Δ {\dot{R}}_{D}^{i, j} < 1 \frac{km}{s}$ , limits the residual range error $Δ {\dot{R}}_{D}^{i, j} δ b_{D}$ to 0.5 m. As the range rate is zero at zenith and the pivot satellite is typically close to zenith, this residual error is largest at low satellite elevations, which remains small compared with a low-cost receiver’s multipath and random noise components under those conditions.

Consider a set S of GNSS data samplings with a high C/N₀ value, which is expected to be free of extreme multipath:

$S = {i \neq j, k ∣ (C / N_{0, k}^{i} > C / N_{0, min}) \land (C / N_{0, k}^{j} > C / N_{0, min})}$ 13

The RFDD components, multipath $Δ M_{D}^{i, j}$ , and random noise $η_{D, B}^{i, j}$ in Equation (5) can be assumed to follow a zero-mean Gaussian distribution over this set (Matera et al., 2018). Then, with the samples in S, we model Equation (12) as a linear system with additive white Gaussian noise:

$Δ ϵ_{D, B}^{S, j} = - Δ {\hat{\dot{R}}}_{R}^{S} \cdot b_{D} + Δ M_{D}^{S, j} + η_{D, B}^{S, j}$ 14

Finally, after solving Equation (14) for the receiver clock bias b_D via least squares, its estimate ${\hat{b}}_{D}$ is passed to Equation (12) to correct the RFDD for all measurements.

Thus far, we have shown how a known receiver position can be used to estimate the multipath term by RFDDs. An existing method can lead to significant RFDD biases stemming from biased local timestamps and large satellite range rates. Our proposed modification estimates the clock bias between the receiver under test and the position reference system and corrects the estimated geometric range from each satellite to the receiver.

3 EXPERIMENTAL STUDY

Experimental data from Feriol (2022) were processed using our proposed method. Here, we briefly present the data set and how the environmental context labels were created.

3.1 Data Set Description

A public data set from Feriol (2022) was recorded on three paths in downtown Toulouse, France, and the nearby countryside in February and March 2022. From this data set, we studied two subsets acquired on March 24, 2022 (dubbed Morning and Afternoon here). These subsets are shown in Figure 2. Each experiment contains approximately 2–3 h worth of data, as summarized in Table 1.

FIGURE 2

The two studied trajectories in and near Toulouse, with ground truth environmental context labels shown. (a) Morning data set, (b) Afternoon data set.

The base station receiver is marked by a white pin. The two experiments are similar, but were performed at different times of day.

View this table:

TABLE 1 Number of Measurements in the Two Analyzed Data Sets from Each of the Sensors Mobile equipment is mounted on the vehicle, whereas the base station remains stationary.

The reference system consists of two NovAtel PwrPak7-E1 receivers (geodetic GNSS receiver plus Micro-Electromechanical System (MEMS) inertial measurement unit (IMU)); one serves as the stationary base station (B), and the other is installed on the vehicle (R). The two sensors combined provide the vehicle position ground truth. As the COTS receiver under test (D), the u-blox M8T is chosen, although the pseudoranges recorded by the mobile NovAtel PwrPak7-E1 are also analyzed. The u-blox M8T can receive GNSS signals at and near the L1 frequency. Measurements from the GPS L1 C/A and Galileo E1 OS signals were recorded. The NovAtel PwrPak7-E1 recorded GPS and Galileo signals at the L1, L2, and L5 frequencies; however, we examined only the L1 C/A and E1 OS signals. For brevity, we refer to these two signals as "GPS" or "L1" and "Galileo" or "E1," respectively. The receivers were sampled at a rate of 5 Hz, measuring pseudoranges, Doppler shift, carrier phase, and C/N₀. Further data such as elevation, broadcast ephemeris, and the number of satellites visible (NSV) was also recorded. The two receivers on the vehicle share a NovAtel VEXXIS GNSS-804 antenna. The base station uses a Septentrio PolaNt Choke Ring B3/E6 antenna. Additionally, a sky-facing fisheye camera mounted on top of the vehicle recorded images at a sampling rate of 2 Hz. Both the GNSS antenna and the camera were placed along the vehicle center line, with a horizontal and vertical separation of 39 cm and 20 cm, respectively (Feriol, 2023).

The data set contains a position ground truth based on postprocessing software that makes use of the carrier phase recorded by the mobile NovAtel PwrPak7-E1 and the base station, plus an IMU contained in the mobile device. This ground truth system is specified to have centimeter-level accuracy along the entire trajectory. A maximum distance of 6 km between the base and mobile station was reached. For more information on the test campaign, see the report by Feriol (2023). The GNSS observables measured by the reference system are not part of the public data set. These observables were retrieved from the experiment records and can be provided upon request.

3.2 Environmental Contexts and Multipath-Related Parameters

The data were labeled by a context detector (Feriol et al., 2022) as one of five environmental context categories: (A) Mixed, including static, (B) Canyon (urban), (C) Urban, (D) Trees, and (E) Open Sky. The labels rely on a sky-facing fisheye camera, semantic image segmentation, and the classification scheme depicted in Figure 3.

FIGURE 3

Deterministic environmental context detection heuristic (Feriol et al., 2022)

Approximately 25% of the measurements are labeled as B-E; the remainder remain unlabeled in the Mixed category (A). As the context labels depend on the images, they are not available at the GNSS sampling rate. Associations between the pseudoranges and a context were determined by choosing the closest context in time, so that each context label at 2 Hz may be assigned to up to four GNSS records at 5 Hz. The data are separated by constellation and signal and further grouped with respect to four observable parameters. Table 2 details the broad high, medium, and low groupings chosen to evaluate statistical properties with sufficiently large sample sizes. Specifically, the elevation range was divided into three equally spaced bins. The first C/N₀ threshold of 35 dB Hz represents a typical lower bound for low-multipath LOS signals (Matera et al., 2018), whereas the second threshold of 20 dB Hz was selected after it had been observed that the RFDDs of weaker signals appeared more challenging to characterize. Lastly, a minimum of four satellites is required for single point positioning, whereas the upper NSV threshold permits the study of specific GPS-only high-NSV samples, as Galileo remained at NSV < 8. Thus, the data were grouped according to the following parameters:

(a) LOS–NLOS classification (per labeled pseudorange)

(b) Received signal strength C/N₀ (per pseudorange)

(d) NSV (per timestamp)

View this table:

TABLE 2

Binning Strategies for Each Numeric Parameter

Binary LOS/NLOS labels directly provide distinct data bins. Each parameter was treated independently; for example, a given pseudorange can be simultaneously considered as NLOS, medium-C/N₀, high-elevation, and low-NSV.

The LOS-NLOS classification uses a fisheye camera and the NLOS detection method of Feriol (2023). Although camera-based classification methods are sensitive to changing scenes and lighting conditions (Gakne & Petovello, 2015), the sky segmentation’s high Intersection-over-Union (IoU) of 90.8% (Feriol, 2023) inspires confidence in the LOS–NLOS labels. Given a typical distance to obstacles on the order of 10 m to 100 m, the GNSS-camera lever arm has little impact.

Concerning sensor synchronization, the label is only available at the camera’s sampling rate. As the local geometry changes too quickly to reliably interpolate this label, the inverse approach was chosen rather than the environmental context: For each image, only the nearest GNSS record was labeled and used for analysis, resulting in a downsampling from the original 5 Hz to 2 Hz for this parameter. For very weak signals whose navigation message is not contained in the data set, no LOS–NLOS classification was assigned.

3.3 Data Set Discussion

Figure 4 compares the C/N₀ of each pseudorange recorded by the two mobile receivers to the C/N₀ of the matching observation recorded by the stationary B receiver, where both the u-blox and NovAtel receiver are treated as D. The vertical cutoff shows that the NovAtel receiver stops tracking the GPS signal at C/N₀ ≈ 25 dB Hz, whereas the u-blox receiver only does so at C/N₀ ≈ 10 dB Hz. Similar results were observed for Galileo tracking. The base station’s C/N₀ plotted on the y-axis shows the same cutoff, this time horizontally, as R. Our prior work (von Arnim et al., 2025a) suggested that this cutoff is related to the tracking loop bandwidth of the receivers.

Received C/N0 of the fixed base station B over received C/N0 of the two mobile receivers treated as D (including the reference receiver R), colored by the elevation of the associated satellite (Morning data set) A simultaneous observation by D and B would lie on the axis diagonal if both receivers measure the same C/N0 at the same TOW.

FIGURE 4

Received C/N₀ of the fixed base station B over received C/N₀ of the two mobile receivers treated as D (including the reference receiver R), colored by the elevation of the associated satellite (Morning data set)

A simultaneous observation by D and B would lie on the axis diagonal if both receivers measure the same C/N₀ at the same TOW.

Nonetheless, we obtained matching observations from the base station for nearly every observation made by the u-blox receiver, which is essential for the proposed RFDD method. Even if a signal experiences deep fading in challenging environments, exemplified by high-elevation, low-C/N₀ measurements in the top left corners, we can trust the base station to provide an undisturbed observation at the nominal high C/N₀. Simultaneously, the base station easily tracks low-elevation satellites in the lower portions of Figure 4. The base station consistently records slightly higher maximum C/N₀ values than the mobile receivers at high elevations and slightly lower values at low elevations. The "line of equal C/N₀" in Figure 4 separates measurements by the relative C/N₀ between base and mobile receivers. Points to the right of this line are more strongly received at the mobile receiver than at the base station. Comparing the right edge of the point clouds to this line, some low-elevation samples extend to the right of this line whereas high-elevation samples stay to its left. This trend is likely explained by slightly different antenna radiation patterns and is not expected to impact the subsequent multipath study. The u-blox further shows lower C/N₀ values for Galileo E1 than for GPS L1 in otherwise equal conditions, visible as an offset of the point cloud to the left.

Despite identical sampling rates, the high C/N₀ cutoff combined with fading reduces the quantity of recorded pseudoranges for the mobile reference receiver compared with the u-blox receiver under test, as illustrated by Table 1. In Figure 5, this trend is most evident in the Urban and Canyon contexts, where the u-blox tracks on average at least two satellites at C/N₀ < 35 dB Hz, while the mobile NovAtel only does so sporadically. Consequently, the reference receiver frequently has fewer than four concurrently received satellites of either constellation in challenging contexts (Canyon). Yet, both receivers record equal quantities of high-C/N₀ pseudoranges, confirming that these differences stem from the receiver, not the environment. This result must be kept in mind when comparing medium- and low-C/N₀ results between the two receivers.

FIGURE 5

Average count of GPS L1 pseudoranges obtained for each sample per detected context in the Afternoon data set for Toulouse, binned by the associated C/N₀

Data for the u-blox receiver are shown as solid bars; data for the reference NovAtel receiver are shown as hatched bars.

The proposed algorithm necessitates the selection of a less disturbed pivot satellite to remove receiver clock bias. In line with the discussion on the data set, we chose the same selection criterion as Matera et al. (2018): C/N_0,min = 35 dB Hz.

Finally, the u-blox receiver recorded pseudoranges including Galileo pseudorandom noise (PRN) 14, 20, and 34. The NovAtel receivers did not track these three satellites but did track PRN 36, which is missing from the u-blox data. PRNs 14 and 34 broadcasted a non-zero signal health status, i.e., "unhealthy," whereas PRN 20 did not broadcast any ephemeris. Thus, these satellites were excluded from our analysis.

4 RESULTS

In this section, first, the RFDD correction for the receiver clock bias is validated. Then, the RFDDs calculated by Equation (12) are assessed via selected histograms after being binned according to Table 2. Subsequently, each bin’s first four sample moments – mean, variance, skew, and kurtosis – quantify the observed multipath distributions.

4.1 Receiver Clock Bias Correction

We observed multimodal behavior in our initially obtained RFDDs that could not be explained by multipath alone, prompting the development of a correction method. Figure 6 demonstrates the effect of correcting the RFDD using an estimated receiver clock bias. Calculating the RFDD with and without the proposed method, we generated example RFDD histograms from all GPS L1 samples of the u-blox (D) receiver in the Morning data set. Their x-axis was scaled from the 0.01th quantile to the 0.99th quantile to highlight its main modes while also including the distribution tails. The histogram bin width was chosen as 0.4 m to balance resolution and interpretability.

Histograms of the RFDD ∆ϵ^ for all u-blox GPS L1 samples in the Morning data set with and without the proposed correction for the receiver clock bias. (a) Original data without the proposed correction, (b) Original data with the proposed correction. The distribution without the correction is highly multimodal owing to a satellite-specific RFDD error, increasingly so for higher clock bias (c). The proposed correction provides a single mode near zero regardless of clock bias (d), as expected.

FIGURE 6

Histograms of the RFDD $∆ \hat{ϵ}$ for all u-blox GPS L1 samples in the Morning data set with and without the proposed correction for the receiver clock bias. (a) Original data without the proposed correction, (b) Original data with the proposed correction.

The distribution without the correction is highly multimodal owing to a satellite-specific RFDD error, increasingly so for higher clock bias (c). The proposed correction provides a single mode near zero regardless of clock bias (d), as expected.

As a baseline, the uncorrected RFDD histogram (Figure 6(a)) exhibits a clear multimodal distribution. The modes are scattered at offsets of several meters around zero, reflecting the magnitude of the needed correction $Δ {\dot{R}}_{D}^{i, j} b_{D}$ . In contrast, Figure 6(b) shows the histogram obtained after the proposed correction. This histogram presents a unimodal distribution; its single mode is located near zero RFDD, consistent with the expectations for a large and environmentally diverse data set (Matera et al., 2018). The least-squares solution to the receiver clock bias estimation problem, Equation (14), yields an estimated receiver clock bias $\hat{b} = 7.68 ms$ on the recorded data.

To further verify the clock bias estimation, we added an artificial bias of 10 ms to the timestamps. Subsequently, the estimated bias increased to $\hat{b} = 17.68 ms$ , verifying the accuracy of the bias estimation. Figures 6(c) and 6(d) show the histograms obtained after an artificial bias has been introduced. The uncorrected RFDDs (Figure 6(c)) present even more widely scattered modes, whereas the corrected results (Figure 6(d)) remain nearly unchanged compared with Figure 6(b), demonstrating the method’s effectiveness.

4.2 Pseudorange Residual Histograms

To better understand the multipath distribution described by the obtained RFDD $∆ \hat{ϵ}$ , we first study the context-dependent histograms of LOS- and NLOS-classified signals. These labels were obtained from a camera-based classification scheme. We then discuss the statistics of signals at a certain C/N₀ or elevation.

Once more, the x-axis of the histograms was scaled from the 0.01th quantile to the 0.99th quantile for a given receiver-signal combination, i.e., for all samples in each row of Figure 7 plus those without context labels. The maximum and minimum values are displayed in each plot, documenting the presence of rare but potentially extreme outliers. Striking a balance between granularity and statistical significance, histogram bins were chosen with a width of 0.1 m for the NovAtel receiver (R) and 0.4 m for the u-blox receiver (D) to account for large differences in RFDD magnitude between the two receivers.

Histograms of the RFDD ∆ϵ^ for each context label and LOS/NLOS category By changing one parameter at a time, we can compare the effect of different receivers (cf. Figures 7(a)–7(d) and Figures 7(e)–7(h)), different satellite constellations and signals (cf. Figures 7(a)–7(d) and Figures 7(i)–7(l)), and varying time of day (cf., Figures 7(i)–7(l) and Figures 7(m)–7(p)). Additionally, the number of measurements N for each context and their proportion of the entire row’s data set, the percentage of measurements labeled as LOS and NLOS in each context, and the maximum and minimum value of ∆ϵ^ are presented for each context.

FIGURE 7

Histograms of the RFDD $∆ \hat{ϵ}$ for each context label and LOS/NLOS category By changing one parameter at a time, we can compare the effect of different receivers (cf. Figures 7(a)–7(d) and Figures 7(e)–7(h)), different satellite constellations and signals (cf. Figures 7(a)–7(d) and Figures 7(i)–7(l)), and varying time of day (cf., Figures 7(i)–7(l) and Figures 7(m)–7(p)). Additionally, the number of measurements N for each context and their proportion of the entire row’s data set, the percentage of measurements labeled as LOS and NLOS in each context, and the maximum and minimum value of $∆ \hat{ϵ}$ are presented for each context.

Figure 7 shows an initial comparison of the RFDD statistics by context, including different signals, receivers, and times of day. Subsequently, Figure 8 presents the statistics in relation to the observed C/N₀ and elevation. Specifically, RFDDs of GPS L1 sorted by LOS and NLOS labels are shown in Figures 7(a) to 7(d) and Figures 7(e) to 7(h) for the two receivers, u-blox (D) and NovAtel (R), respectively. For the u-blox, Galileo E1 RFDDs are featured in Figures 7(i) to 7(1) and Figures 7(m) to 7(p) from the Morning and Afternoon data sets, respectively. Finally, Figures 8(a) to 8(d) and Figures 8(e) to 8(h) present the RFDD for different C/N₀ values and elevations, respectively, to offer a new perspective on the u-blox’s Morning GPS L1 results.

Histograms of the RFDD ∆ϵ^ for each context label and C/N0 or elevation bin We can compare the effect of the received C/N0 (Figures 8(a)–8(d)) and of the satellite elevation (Figures 8(e)–8(h)); also cf. Figures 7(a)–7(d) for the same data but binned by LOS and NLOS labels. Additionally, the number of measurements N for each context and their proportion of the entire row’s data set, the percentage of measurements labeled by Table 2 as high, medium, or low C/N0 and elevation in each context, and the maximum and minimum value of ∆ϵ^ are presented for each context.

FIGURE 8

Histograms of the RFDD $∆ \hat{ϵ}$ for each context label and C/N₀ or elevation bin We can compare the effect of the received C/N₀ (Figures 8(a)–8(d)) and of the satellite elevation (Figures 8(e)–8(h)); also cf. Figures 7(a)–7(d) for the same data but binned by LOS and NLOS labels. Additionally, the number of measurements N for each context and their proportion of the entire row’s data set, the percentage of measurements labeled by Table 2 as high, medium, or low C/N₀ and elevation in each context, and the maximum and minimum value of $∆ \hat{ϵ}$ are presented for each context.

The distribution of the Mixed context multipath appears to be a mix of all other contexts, similar to a distribution without considering context at all; thus, we focus on the four labeled environments Canyon, Open Sky, Trees, and Urban. This trend is observed in all cases and highlights the value of context-aware studies. Both Mixed and Urban contexts contain several extreme outliers of more than 100 m, both positive and negative. Such outliers are less frequent—albeit not entirely absent—in the other contexts. Therefore, outlier detection and exclusion remain important components of PVT estimators.

4.2.1 LOS- and NLOS-Labeled Pseudoranges

With regard to the LOS and NLOS labels, the RFDD histograms for both categories resemble each other in the Trees context. A certain semblance remains under Open Sky but with visibly increased noise in the NLOS case. The other two environments, Urban and Canyon, show clear distinctions for LOS and NLOS signals. This result suggests a connection to the opacity to radiation at GNSS frequencies for typical materials found in each environment. Our camera-based labeling only indicates whether a satellite is visible in a sky segment of the image to determine the LOS status.

To illustrate the impact of the receiver model, we can compare the u-blox (D) shown in Figures 7(a) to 7(d) to the NovAtel (R) in Figures 7(e) to 7(h). The 0.01th quantile to the 0.99th quantile extends over a much reduced range of values for the NovAtel receiver, narrowing the x-axis scale with respect to the u-blox plots. We further note that the NovAtel’s maximum and minimum RFDDs are consistently much smaller than those of the u-blox, indicating smaller outliers. The qualitative effect of each environment remains similar to that observed for the u-blox (D). We nonetheless note some minor differences: The NovAtel’s histograms appear erratic in the Canyon context, likely caused by its smaller sample size and linked to the low NSV illustrated by Figure 5. Meanwhile, in the Urban context, in contrast to the u-blox results, the long right-sided tail of the NLOS signals is greatly reduced, highlighting the impact of receiver design choices on multipath-induced errors in urban scenarios.

Returning to the u-blox in Figures 7(i) to 7(1), we compare GPS and Galileo pseudoranges. In the Open Sky and Urban contexts, a surprising amount of sub-zero RFDD appears for the Galileo-LOS grouping, as shown in Figures 7(j) and 7(1). Differing E1 and L1 performance may be explained by the receiver implementation for each signal, or it may be inherent to each modulation and its respective interaction with complex environments, although the hypothesis of improved multipath resistance by BOC remains unconfirmed. Indeed, both Figures 7(i) to 7(1) and Figures 7(a) to 7(d) show heavy resemblance in the multipath-heavy Urban and Canyon environments. The x-axis scales of the two signals are similar, showing that 98% of measurements lie in the same interval. We conclude that the environmental context is the primary factor responsible for the histogram shape, rather than the signal modulation or receiver model, even if quantitative differences remain.

The NLOS-labeled RFDDs in Open Sky, shown in Figures 7(b), 7(f), 7(j), and 7(n), appear nearly Gaussian. This result is likely due to diffraction rather than reflection being the primary NLOS mechanism in this environment. In radio propagation, the first Fresnel zone plays a key role for both effects. This zone forms an ellipsoid with a radius of approximately $r_{F} = \sqrt{λ d_{F}}$ around the LOS between the receiver and satellite, with wavelength λ and distance to the receiver d_F along the LOS (Braasch, 2017). The elevation-dependent vertical extent $h_{F}^{\pm} = d \tan (e l) \pm a$ determines the impact of an obstacle at distance d, where the intersection of a plane with the ellipsoid creates an ellipse with semi-major axis a (Zimmermann et al., 2018). We will neglect the antenna height of our ground vehicle to simplify this discussion; the antenna height does have a quantitative impact but does not change the conclusions. Typical tree and building heights in the experiment area are less than 20 m (Colin-Koeniguer & Trouve, 2014; Morin et al., 2023). Feriol (2023) showed that measurements in Open Sky are detected as NLOS at low elevations only, generally less than 20°. Given this upper limit of el = 20°, an obstacle, such as that illustrated in Figure 9, starts to infringe on the first Fresnel zone at a distance of 66 m (a), leads to a camera-based NLOS classification at 55 m (b), and fully obstructs the first Fresnel zone at a distance of 46 m distance (c). Taller structures or lower elevations would increase those values and the spacing between them.

FIGURE 9

First Fresnel zone and obstacles starting to infringe on it (a), fully blocking it (c), and blocking the LOS with partial blockage of the first Fresnel zone (b)

The orange zone marks distances at which obstacles cause a camera-based NLOS classification but the first Fresnel zone is only partially blocked, while the purple zone marks distances at which the signal would be classified as LOS despite partial blockage. At low elevations, common in the Open Sky context, this creates a large range of potential obstacle locations at which signals are visually classified as NLOS but see a minimal added path length.

In this case, the line between LOS and NLOS signals is blurred. Zhang and Hsu (2021) observed that sensitive low-cost receivers can even receive diffracted signals if the first Fresnel zone is fully blocked, creating a large range of locations where a satellite signal is still received but already considered NLOS. Conversely, signal reflection requires a second obstacle providing a well-angled reflecting surface in clear LOS conditions. Such precise geometries are rare in Open Sky contexts, although they may still be responsible for the few observed large outliers, such as the maximum RFDD value of 125 m shown in Figure 7(b). Consequently, we propose that diffraction is the primary mechanism of NLOS signals in Open Sky conditions. Diffraction contributes relatively little pseudorange error, especially if the first Fresnel zone is not completely obstructed (Zhang & Hsu, 2021), but severely impacts C/N₀ with an associated increase in random noise.

The qualitative effect of the environment and the difference between LOS and NLOS signals remains similar for Figures 7(i) to 7(1) (Morning) and Figures 7(m) to 7(p) (Afternoon). Differences are visible for LOS signals in Canyon and Urban environments. Not only does the Afternoon data set contain a much greater share of LOS data, but it also shows a more pronounced peak. A likely explanation of the qualitative difference is the change in satellite elevation over time. One Galileo satellite had moved to a very high elevation by the time of the Afternoon experiment. LOS signals at high elevation are subject to less multipath than at low elevation. However, the dependence on time of day is generally less relevant than context and the GNSS observables, justifying our focus on the Morning data set.

4.2.2 Satellite Elevation

Another perspective is provided in Figure 8, where u-blox GPS L1 signals have been grouped by C/N₀ (Figures 8(a) to 8(d)) or elevation (Figures 8(e) to 8(h)) into the high/medium/low categories defined in Table 2. These figures can be directly compared with the previous LOS/NLOS classification of Figures 7(a) to 7(d).

Elevation is frequently considered in the form of elevation masks. Figures 8(a) to 8(d) show that a high elevation of ≥ 60° can be clearly associated with little bias and few outliers relative to each environment. In contrast, a low elevation of <30° shows the strongest multipath effects in all environments. The RFDD variance visibly increases over that of higher-elevation satellites, even in the Open Sky and Trees contexts. However, the environment also strongly affects the results for intermediate elevations between these two limits. The distribution is clearly skewed and one-sidedly long-tailed in Canyon and Urban contexts for both lowand medium-elevation signals. In the Canyon context, the diminished proportion of low-elevation observations further illustrates a natural masking effect that is less pronounced in Trees and Urban environments. A context-aware algorithm could implement a context-dependent elevation threshold, but our results show that only very high thresholds can reliably exclude strong multipath in Urban and Canyon environments. Consequently, there is a risk of insufficient NSV and poor geometries for GNSS-only PVT estimates.

These results lead to the conclusion that elevation-based multipath mitigation will struggle to reject strongly non-Gaussian measurements without discarding too many normally distributed observations. Additionally, locations with overhanging features such as bridges were not studied, in which case we would expect strong effects even on very high-elevation satellites (Mayolle et al., 2025). Such features become more important in so-called "vertical cities" with multi-level infrastructure, such as downtown Hong Kong or Tokyo, limiting the transferability of purely elevation-based models between data sets. Lastly, such locations often have strongly reflecting surfaces, such as coated glass, compared with the more common brick surfaces found in Toulouse, permitting reception even after multiple reflections and thus reducing the canyon’s masking effect.

4.2.3 Measured C/N₀

The challenge of elevation-based models motivated the C/N₀ grouping shown in Figures 8(a) to 8(d). We immediately see distinct shapes for medium- and high- C/N₀ signals in each context. High- C/N₀ (≥ 35 dB Hz) signals appear near-Gaussian and unbiased in all environments, whereas the RFDDs at medium C/N₀ are more spread out and exhibit a heavy tail toward positive values. This result confirms that signals under this threshold experience strong multipath effects. Only in Open Sky environments do these medium- C/N₀ signals yield a near-Gaussian distribution, albeit with greater variance than stronger signals. This increase is an expected result of the random noise internal to the receiver being a function of C/N₀ (Hauschild, 2017). Noting the qualitatively similar effects on the two C/N₀ groups, although still specific to each context in quantity, we consider it feasible to derive a statistical multipath model as a function of only C/N₀ and a context-specific tuning parameter.

A few very weak signals (C/N₀ < 20 dB Hz) were recorded by the u-blox receiver; these pseudoranges appear to be extremely poor, with some far outliers. The sole exception for which the signals remain relatively acceptable is the Canyon context. The performance of weak signals in Canyon is comparable to that of intermediate-strength Urban signals. Given their negligible occurrence in the Open Sky context, these weak signals are clearly caused by strong environmental effects. Owing to the diversity of real environments, each low- C/N₀ RFDD may be the product of specific circumstances unique to our data set. While this behavior poses a challenge to our C/N₀-based model proposed for higher C/N₀ values, an overbounding approach may still be possible.

Both C/N₀ and the NLOS labels have thus revealed important information on the magnitude and shape of multipath effects, warranting a direct comparison of Figures 7(a) to 7(d) and Figures 8(a) to 8(d).

The similarity of the histograms for the medium-C/N₀ and NLOS categories in the Canyon context (Figures 7(a) and 8(a)) implies similar effects on signals in both categories. The same trend is observed in Urban environments (Figures 7(d) and 8(d)). The chosen threshold of 35 dB Hz is a typical lower limit for undisturbed LOS signals. Lower C/N₀ values require either strong LOS attenuation, heavy destructive multipath interference, or pure NLOS. We conclude that the camera labels primarily identify true NLOS signals in these contexts. By extension, the more spread-out NLOS histograms in the Urban context imply that open urban spaces admit reflections at greater distances from the receiver compared with the more constrained Canyon context. This trend makes NLOS identification especially valuable in Urban environments. The Canyon environment can still profit from such labels, e.g., by excluding those satellites, but risks insufficient NSV owing to a lack of LOS signals. Thus, a GNSS user must make a trade-off between accuracy and availability here.

In contrast, the Trees environment yields similar histograms for both LOS and NLOS categories in Figure 7(c), yet clear differences between high- and medium-C/N₀ signals in Figure 8(c). As foliage is optically dense yet transparent at GNSS frequencies, diffuse scattering causes multipath errors and attenuates the signal but does not lead to true NLOS signals. Both the LOS and NLOS histograms present increased variance in the Trees context over Open Sky but remain relatively free of bias or skew, in stark contrast to the building-dominated Urban and Canyon environments. Our binary LOS/NLOS labels are therefore of limited utility in the Trees context, whereas C/N₀-based models remain informative. These findings support the introduction of a new camera-based label, Vegetation, to better model signals in the Trees environment, as proposed by Zhou et al. (2025) for signals that pass through obstacles along the LOS path. The inclusion of this label is left to future work.

Finally, considering the NSV, we did not observe any clear relationship between the number of concurrently received satellites and the distribution of residuals for a given context.

The qualitative approach for Figures 7 and 8 shows that C/N₀ is a better signal quality indicator in the Trees context than the LOS/NLOS classification. Both labels are a good indicator in other environments. The gold standard for identifying signals with little to no multipath in Urban and Canyon environments remains the camera-based LOS label. We can model sufficiently strong signals as quasi-Gaussian with rare outliers. In comparison, elevation is more challenging to integrate in statistical multipath models. While high elevation signals are less perturbed, most measurements are found at ≤ 60°, where they can already be subject to strong multipath leading to non-Gaussian distributions in all except Open Sky contexts. Our findings can be summarized as follows:

Environmental context determines histogram shape and scale for a given grouping (see Table 3).
Receiver model, signal modulation, and time of day have less impact on qualitative pseudorange error distributions.
High-C/N₀ ≥ 35 dB Hz- or LOS-labeled pseudoranges produce approximately Gaussian histograms, albeit with different variance in each environmental context (broadly $σ^{Open Sky} < σ^{Trees} \leq σ^{Urban} \leq σ^{Canyon}$ ).
Low/medium-C/N₀ < 35 dB Hz-labeled pseudoranges produce unbiased Gaussian RFDD distributions only in the Open Sky context, whereas bias and non-Gaussianity visibly increase from Canyon over Trees to Urban contexts.
NLOS and low/medium-C/N₀ histograms are similar except in the Trees context, indicating poor performance of visual LOS detection under vegetation but good LOS labeling in Urban and Canyon contexts.

View this table:

TABLE 3 Qualitative Interpretation of RFDD Histograms and the Approximate RFDD Distribution, Summarized

4.3 Statistical Sample Moments

To quantitatively assess the statistical properties of the multipath, the first four sample moments—mean, variance, skew, and excess kurtosis—of the RFDD were calculated. The results for the first two moments are shown in Figure 10, with the other two shown in Figure 11.

After the RFDD ∆ϵ^ has been grouped by signal properties (color), as defined in Table 2, and context (marker shape), each group’s standard deviation σ is plotted over the absolute value of its mean μ. Further, Equation (15) is fitted to the plotted points and shown as a dashed line with fit results m, c and fit residual R. Each row contains results of the u-blox receiver observing one GNSS signal (GPS L1, Galileo E1) in the Morning data set. See Figure 11 for the third- and fourth-order moments.

FIGURE 10

After the RFDD $∆ \hat{ϵ}$ has been grouped by signal properties (color), as defined in Table 2, and context (marker shape), each group’s standard deviation σ is plotted over the absolute value of its mean μ. Further, Equation (15) is fitted to the plotted points and shown as a dashed line with fit results m, c and fit residual R. Each row contains results of the u-blox receiver observing one GNSS signal (GPS L1, Galileo E1) in the Morning data set. See Figure 11 for the third- and fourth-order moments.

After the RFDD ∆ϵ^ has been grouped by signal properties (color), as defined in Table 2, and context (marker shape), each group’s excess kurtosis γ2 is plotted over the absolute value of its skewness γ1. Each row contains results for the u-blox receiver observing one GNSS signal (GPS L1, Galileo E1) in the Morning data set. Pearson’s inequality defines a lower bound for γ2, with the forbidden area shown in gray. See Figure 10 for the first- and second-order moments.

FIGURE 11

After the RFDD $∆ \hat{ϵ}$ has been grouped by signal properties (color), as defined in Table 2, and context (marker shape), each group’s excess kurtosis γ₂ is plotted over the absolute value of its skewness γ₁. Each row contains results for the u-blox receiver observing one GNSS signal (GPS L1, Galileo E1) in the Morning data set. Pearson’s inequality defines a lower bound for γ₂, with the forbidden area shown in gray. See Figure 10 for the first- and second-order moments.

The samples were again grouped by the "latent variables": environmental context label, time of day, receiver, and signal, as well as the observable parameters according to Table 2. For each group, statistical moments were obtained. As we found little dependence of the qualitative results on the time of day and receiver, we show only the u-blox receiver’s Morning data. Each row of diagrams corresponds to one signal, while each column groups the data according to the observable parameter. The parameter groups given in Table 2 are denoted by the color of each point, while the shape corresponds to the environmental context. We represent Mixed by ◯, Canyon by □ , Open Sky by Δ, Trees by ◇, and Urban by ★.

Figure 10 presents the sample standard deviation σ over the sample mean μ of all samples of the Morning data set. The higher-order moments for the same data set are shown in Figure 11, which displays the sample excess kurtosis γ₂ over the sample skewness γ₂. Both axes are shown on a logarithmic scale to account for large differences in magnitude between the sample moments of the different groups.

We will show that the RFDD can be described as the sum of normally distributed random noise η ∼ N and chi-square — or similar — distributed multipath ΔM ∼ χ². The distribution parametrization is determined by the environmental context class.

4.3.1 Mean and Variance

From Figure 10, we see that a larger mean error corresponds to a larger standard deviation. This trend applies to all contexts, parameter groupings, signals, and receivers; it is visible in all diagrams. Thus, the underlying process driven by these latent variables acts jointly on both the mean and variance of any sample.

The three columns of Figure 10 display a certain similarity for each group of Table 2. We can see that LOS, high-C/N₀, and high-elevation samples are found in the lower left corner, with medium-C/N₀ and medium-elevation data in the center and NLOS, low-C/N₀, and low-elevation data in the upper right. This result confirms the intuitive correlation among C/N₀, elevation, and LOS/NLOS effects; thus, an estimator may choose any of the three or a combination thereof to evaluate the expected signal error.

Interestingly, each environmental context is clearly distinguishable in Figure 10; for example, the medium-C/N₀ points in Figures 10(b) and 10(e) are spread over multiple orders of magnitude for different contexts. Generally, the mean and variance within a grouping are the highest in the Urban context, followed by the Mixed, Canyon, and Trees contexts. The mean and variance are consistently the lowest in the Open Sky context, to the point that such medium-C/N₀ measurements approach the accuracy of high-C/N₀ data in other contexts.

Widely considered a challenge for GNSS, the Canyon context regularly has a lower RFDD mean and variance than Urban environments within a grouping. Under NLOS conditions, Figures 10(a) and 10(d) show that Canyon environments have RFDD values similar to those of the Open Sky and Trees environments. This effect is likely caused by the proximity of reflecting surfaces in very narrow streets in Toulouse and the complete blockage of low-elevation satellites, which lead to fewer reflections on far obstacles than in more open Urban settings. In contrast, Figures 10(a) and 10(b) remind us that the Canyon context still exhibits an increased variance in LOS or high-C/N₀ conditions, albeit with near-zero mean. Thus, obstacles covering large parts of the sky contribute multipath to nearly all signals in the Canyon context, both destructive and constructive, whereas other environments retain some multipath-free signals.

Finally, the Open Sky and Trees contexts display similar statistics for NLOS-classified satellites in Figure 10(d). In contrast, the results move further apart in the medium-C/N₀ group (Figures 10(d) and 10(e)). Even when the NLOS label separates the two environments clearly, as in Figure 10(a), their separation only grows when grouped by C/N₀ in Figure 10(b). This result supports our hypothesis that, in these environments where the geometry or materials of obstacles can allow GNSS signals to pass, NLOS labels obtained by vision-based systems reliably capture true—i.e., reflected— NLOS signals but may include many useful quasi-LOS signals. In the Trees context, pseudorange exclusion based on such labels would be overly aggressive, leading to suboptimal positioning performance, while C/N₀-based error models permit more fine-grained evaluation. Nonetheless, the clear separation of LOS and NLOS points in Figures 10(a) and 10(d) highlights their potential value, especially for lower-cost receivers such as the u-blox.

We will conclude with a remark on possible multipath noise models. On a log-log scale, most points in Figure 10 appear to indicate an approximately linear relationship between the two measures; thus, we could apply the following model:

$σ = c μ^{m} \Leftrightarrow \log σ = m \log μ + \log c, where m, c const.$ 15

After an unweighted fit, we add Equation (15), the obtained parameters, and the norm of the residuals R to Figure 10. The constants m and c vary significantly on our data: m = 0.25–0.61 (Figures 10(d) and 10(e)) and c = 3.88 – 5.72 (Figures 10(c) and 10(d)), depending on the data and its grouping strategy. Values slightly beyond these ranges were noted for the NovAtel receiver, up to m = 0.74 (R = 0.61). We observed some outliers with a large impact on the fit in the LOS groups in Figure 10(d) (Trees, Canyon). The variance of these outliers is relatively high given their very low mean RFDD, possibly because each receiver has a minimum variance threshold regardless of the bias or because those environments can create unbiased noise on LOS signals. Especially good fits were observed for the C/N₀ grouping of GPS measurements (u-blox: R = 0.46, NovAtel: R = 0.20). Conversely, the elevation grouping in Figure 10(f) shows a large spread (R = 0.88) around the model, underlining our finding that the C/N₀ is more informative regarding the pseudorange error distribution. For the C/N₀ groupings, the GPS data (Figure 10(b)) exhibit a smaller slope than the Galileo data (Figure 10(e)), which implies an effect of the signal structure: the more modern Galileo E1 modulation produces slightly less biased pseudoranges in the presence of multipath, whereas elevated variance cannot be avoided.

Such linkage of mean and variance is found in many non-normal distributions, notably the chi-square (m = 1/2, log c = log(2)/2), Rayleigh (m = 1), Gamma (m ∈ [1/2, 1]), and related distributions. While our m ≈ 0.5 could suggest a χ² distribution of the RFDD, our c ≈ 4 is much larger, indicating a greater variance than χ² for a given mean. We thereby show that the two components of the RFDD, multipath and random noise, each contribute a separate distribution: ΔM ∼ χ² or similar and $η \sim N (0, σ_{η}^{2})$ . Their parametrization, whether the degrees of freedom of χ² or the variance of N, remains a function of the latent variables. Because the random noise η is known to primarily depend on the receiver architecture, signal design, and received C/N₀ (Hauschild, 2017), that leaves each environment to contribute uniquely to the multipath term ΔM.

4.3.2 Skewness and Kurtosis

The mean and variance are key properties for traditional non-robust estimators, which can capture only the first two statistical moments. It is nonetheless worthwhile to examine the normalized third and fourth moments, skew γ₁ and (excess) kurtosis γ₂, shown in Figure 11. Both measures are zero for normal distributions, which is preferable for most estimators. Kurtosis provides a measure of "tailedness," while skewness measures the (a)symmetry of a distribution. Kurtosis is commonly understood to also measure outlier occurrence, although it does not distinguish between rare, but far, and frequent, but near, outliers. Skew and kurtosis are related. As non-zero skew indicates an asymmetric distribution, such a distribution has at least one heavy tail. With kurtosis being sensitive to the presence of any heavy tail, Pearson’s inequality $γ_{2} \geq γ_{1}^{2} - 2$ creates a lower bound on kurtosis as a function of skewness, a result of the definitions of the two moments (Pearson, 1916). Figure 11 shows this forbidden area in gray.

Figures 11(a) to 11(c) and Figures 10(a) to 10(c) show properties of the same measurements; similarly, Figures 11(d) to 11(f) correspond to Figures 10(d) to 10(f). The points in Figure 11 are more dispersed than the mean and standard deviation in Figure 10. In comparison, higher-order moments require greater sample sizes to be representative and are more susceptible to experimental outliers (DeCarlo, 1997). As some of our groups, particularly the low-C/N₀ group, are quite small, with fewer than 1000 samples, we focus on overarching trends rather than individual points. Despite the variability, Figure 11 supports the conclusion that the multipath term is approximately χ²-distributed, with the distribution parametrization driven by the environment class.

First, most points in Figure 11 lie close to Pearson’s inequality, the lower bound on kurtosis given by the skewness. This trend indicates that their kurtosis, i.e., their heavy-tailedness, is already well explained by their skewness, which may be interpreted as a single heavy tail with mostly one-sided outliers. Second, similar to the mean and variance, correlation is observed between the two plotted measures in all graphs of Figure 11. Beyond the bounds set by Pearson’s inequality, certain non-normal distributions follow γ₂ = f(γ₁). An example is the χ² distribution with $2 γ_{2} = 3 γ_{1}^{2}$ and an inversely proportional relation $γ_{1}^{2} = 8 / μ$ to the mean. Indeed, we find an almost inverted picture between Figures 10(b) and 11(b): low-C/N₀ measurements with high mean and variance correspond to points with low skew and kurtosis, whereas high-C/N₀ data deemed accurate in Figure 10(b) yield large higher-order moments. Figures 10(e) and 11(e) show a similar phenomenon for medium and low C/N₀. The negative correlation of log μ and log γ₁ reinforces our hypothesis that the multipath term ΔM of the RFDD may be χ²-distributed or similar. Follow-up work may consider three-dimensional simulations to connect these observations with physical signal propagation models.

Unfortunately, this relationship of mean and skewness/kurtosis does not hold consistently across the other grouping methods. On the contrary, Figures 10(d) and 11(d) exhibit great similarity, where most LOS points — excluding the unlabeled Mixed class — are clustered in each diagram’s lower left corner. This positive correlation for the Galileo samples suggests that the multipath distribution changes between different signal structures; in fact, one could approximate its LOS samples as unbiased Gaussian. Better multipath rejection under LOS conditions may account for the observed difference, a design objective of the Galileo E1 OS signal. Yet, its NLOS signals as well as medium- and low-C/N₀ groups remain consistent with our χ² approximation.

We conclude with some context-specific observations. NLOS and medium-C/N₀ measurements in Open Sky, shown in Figures 11(a) and 11(b), generally lie above or to the right of other labeled contexts in their groupings. Similarly, LOS and high-C/N₀ data, as well as the high- and medium-elevation groups in Figure 11(c), exhibit a kurtosis much larger than the Pearson inequality. Tightly spread data with some outliers will give a much greater sample kurtosis than another data set that contains the exact same samples within a noisier baseline. The above groups were shown to feature benign multipath environments in terms of the mean error and variance, yet the higher-order moments prove that outlier testing and rejection remain crucial—especially for GPS L1—and likely effective.

Conversely, the Canyon context typically produces low skew and kurtosis within each grouping. This result again highlights our finding that confined terrain geometries limit the presence of multipath to the extent predicted by the variance and mean error, whereas multipath remains a major risk in the more open Urban context, for which LOS and high-C/N₀ points tend to the upper right. Finally, concerning the Trees environment specifically, a comparison of Figures 11(a) and 11(b) confirms similar statistics for samples labeled as LOS and high C/N₀ in all contexts except Trees, again underlining the difficulty of proper LOS labeling in the context of vegetation.

5 CONCLUSION

Our new approach to multipath error isolation augments the existing RFDD method with a receiver clock correction. This approach avoids multimodality caused by clock bias-induced errors in the geometric range estimate, thereby enabling more accurate statistical analysis for any receiver. By applying the proposed method to a multi-constellation GNSS data set (GPS L1 C/A, Galileo E1 OS) using both geodetic and low-cost receiver observations with labeled environmental contexts in Toulouse, France, we quantified the statistical properties of RFDD residuals in each environmental context.

The carrier-to-noise density ratio (C/N₀) emerged as a strong indicator of signal quality, especially when combined with environmental context. Histograms and sample moments demonstrate that the Gaussian distribution approximates pseudorange errors well given high C/N₀ ≥ 35 dB Hz, although mild skew and rare outliers persist. However, this approximation breaks down at lower C/N₀ values outside the Open Sky context, giving rise to non-Gaussianity. LOS labels from a sky-facing camera reliably identify signals with near-Gaussian errors. Conversely, NLOS labels capture both strongly biased or non-Gaussian errors and, depending on the environment, varying proportions of quasi-LOS signals with unbiased Gaussian errors. In Open Sky and Trees contexts, sky segmentation-based NLOS labels easily misclassify diffracted and scattered signals, whose added path length is much smaller than that caused by NLOS reflection. Vision-based LOS/NLOS detection is confirmed as a valuable tool in reflection-dominated Urban and Canyon contexts.

Environmental context degrades the signal quality in the following order: Open Sky, Trees, Canyon, and Urban. Counterintuitively, Canyon environments show lower variance and bias than general Urban settings. Despite their narrower, more challenging streets, the close proximity of obstacles reduces delay spread. Additionally, canyons fully block low-elevation satellites, which contribute complex, distant reflections in open urban areas. Indeed, signal blockage and limited LOS availability pose the primary challenges in constrained canyons, necessitating multi-sensor fusion and adaptive exploitation of biased NLOS signals. In contrast, open urban areas provide ample high-C/N₀ LOS signals, which facilitate the rejection of non-Gaussian outliers by statistical tests.

Unlike prior work focused on variance and distribution tail modeling, our higher-order moment analysis reveals heteroscedasticity: a linear log-log relationship between mean and variance, as well as between skewness and kurtosis, whose slope indicates a scaled χ² or Gamma distribution. We surmise that the distance and number of reflections in each environment drive the statistical differences. The semantic classifier appears to encode geometric information traditionally obtained from costly three-dimensional ray-tracing. Knowing the distribution family paves a way toward non-Gaussian error modeling to adapt multiple statistical moments simultaneously as a function of environmental context, C/N₀, and LOS/NLOS labels. These models improve sensor fusion consistency, integrity, and accuracy, helping system designers choose a lightweight, context-appropriate sensor suite. For example, properly modeled GNSS measurements can bound visual-inertial odometry drift over extended urban trajectories, leveraging complementary sensor strengths to meet positioning requirements.

While this study enables models of multipath-induced pseudorange errors in Toulouse’s diverse urban and rural environments, critical gaps remain for its generalization. Vertical cities (e.g., Seoul, Tokyo) present frequent overhanging features—excluded from our study—and narrower canyons, both of which may cause stronger high-elevation, multi-bounce, and multi-satellite multipath. When traditional multipath treatment struggles, good environmental models become crucial. Yet, the high likelihood of blockage or multipath affecting the pivot satellite simultaneously poses challenges for the RFDD method. Added variability comes from vehicular motion dynamics and seasonal or regional foliage differences, implying potential time-variant errors. We would like to enrich our analysis with more diverse environment classes, including mixtures. Higher C/N₀ resolution can provide more precise thresholds, below which biased or non-Gaussian errors dominate. At the brink of a widespread roll-out of autonomous vehicles, our context-adaptive approach paves a path toward safe positioning across even the most challenging urban and forested environments.

HOW TO CITE THIS ARTICLE:

von Arnim, M., Vivet, D., & Watanabe, Y. (2026). Quantifying GNSS multipath statistics by signal properties in urban, suburban, and rural environments. NAVIGATION, 73. https://doi.org/10.33012/navi.790

CONFLICT OF INTEREST

The authors declare no conflicts of interest.

A | JUSTIFICATION OF THE RANGE RATE APPROXIMATION

Equation (8) is derived as follows:

$\begin{array}{l} R_{r}^{i} (t_{r}) = R_{r}^{i} (t_{r}, {\hat{t}}^{i} (t_{r})) & = R_{r}^{i} (T_{r} - b_{r}, {\hat{t}}^{i} (T_{r} - b_{r})) \\ = {R_{r}^{i} (t_{r}) |}_{t_{r} = T_{r}} + {(t_{r} - T_{r}) \frac{d}{d t_{r}} R_{r}^{i} (t_{r}) |}_{t_{r} = T_{r}} + {\frac{1}{2} {(t_{r} - T_{r})}^{2} \frac{d^{2}}{d t_{r}^{2}} R_{r}^{i} (t_{r}) |}_{t_{r} = T_{r}} + \dots \\ = R_{r}^{i} (T_{r}) + (t_{r} - T_{r}) {\dot{R}}_{r}^{i} (T_{r}) + \frac{1}{2} {(t_{r} - T_{r})}^{2} {\ddot{R}}_{r}^{i} (T_{r}) + \dots \\ = R_{r}^{i} (T_{r}) - b_{r} {\dot{R}}_{r}^{i} (T_{r}) + \frac{1}{2} b_{r}^{2} {\ddot{R}}_{r}^{i} (T_{r}) - \dots \\ \approx R_{r}^{i} (T_{r}, {\hat{t}}^{i} (T_{r})) - b_{r} {\dot{R}}_{r}^{i} (T_{r}) \end{array}$ 16

In the first line, we recall the relationship among timestamp, clock bias, and receiver time. Subsequently, we expand the range function as a Taylor series around the recorded timestamp, followed by a simplified notation and the substitution of the clock bias. By truncating the series, we obtain a first-order approximation of the range.

We use the definition of the geometric range:

$R = ‖ {\vec{R}}_{r}^{i} (t_{r}, t^{i} (t_{r})) ‖ = ‖ {\vec{X}}^{i} (t^{i} (t_{r})) - {\vec{X}}_{r} (t_{r}) ‖$ 17

and a signal travel time model:

$t_{r} - t^{i} = \frac{1}{c} (R_{r}^{i} + h_{r} + h^{i} + A_{r}^{i})$ 18

From this, we derive the exact geometric range rate ${\dot{R}}_{r}^{i} (t_{r})$ :

$\begin{matrix} {\dot{R}}_{r}^{i} (t_{r}) = \frac{d R_{r}^{i} (t_{r}, t^{i} (t_{r}))}{d t_{r}} & = \frac{d}{d t_{r}} ‖ {\vec{R}}_{r}^{i} (t_{r}, t^{i} (t_{r})) ‖ \\ = \frac{{\vec{R}}_{r}^{i} (t_{r}, t^{i} (t_{r}))}{‖ {\vec{R}}_{r}^{i} (t_{r}, t^{i} (t_{r})) ‖} \frac{d {\vec{R}}_{r}^{i} (t_{r}, t^{i} (t_{r}))}{d t_{r}} \\ = u_{r}^{i} \cdot \frac{d {\vec{R}}_{r}^{i} (t_{r}, t^{i} (t_{r}))}{d t_{r}} \\ = u_{r}^{i} \cdot \frac{d}{d t_{r}} ({\vec{X}}^{i} (t^{i} (t_{r})) - {\vec{X}}_{r} (t_{r})) \\ = u_{r}^{i} \cdot (\frac{d {\vec{X}}^{i} (t^{i})}{d t^{i}} \frac{d t^{i} (t_{r})}{d t_{r}} - \frac{d {\vec{X}}_{r} (t_{r})}{d t_{r}}) \\ = u_{r}^{i} \cdot ({\vec{V}}^{i} (t^{i}) \frac{d}{d t_{r}} (t_{r} - \frac{1}{c} (R_{r}^{i} + A_{r}^{i} + h^{i} + h_{r})) - {\vec{V}}_{r} (t_{r})) \\ = u_{r}^{i} \cdot ({\vec{V}}^{i} (t^{i}) (1 - \frac{1}{c} ({\dot{R}}_{r}^{i} (t_{r}) + {\dot{A}}_{r}^{i} + {\dot{h}}^{i} + {\dot{h}}_{r})) - {\vec{V}}_{r} (t_{r})) \\ = u_{r}^{i} \cdot ({\vec{V}}^{i} (t^{i}) - {\vec{V}}_{r} (t_{r}) - \frac{{\vec{V}}^{i} (t^{i})}{c} ({\dot{R}}_{r}^{i} (t_{r}) + {\dot{A}}_{r}^{i} + {\dot{h}}^{i} + {\dot{h}}_{r})) \end{matrix}$ 19

where $u_{r}^{i}$ is the unit vector from the receiver to the satellite and $\vec{V}$ denotes the velocity. We make use of the total derivative. Similar to the purely geometric range algorithm (Subirana et al., 2013), the derivative remains recursive. We can identify two separate components: the relative velocity of the satellite and receiver, projected onto the unit vector, and a much smaller second-order correction term that accounts for time derivatives of the propagation delay. Subirana et al. (2013) entirely ignored the propagation-related terms $\dot{A}, \dot{h}$ in the transmission time model ${\hat{t}}^{i} = t_{r} - {\hat{R}}_{r}^{i} (t_{r}) / c$ to obtain the explicit solution:

${\dot{R}}_{r}^{i} (t_{r}) \approx \frac{u_{r}^{i} \cdot ({\vec{V}}^{i} ({\hat{t}}^{i}) - {\vec{V}}_{r} (t_{r}))}{1 + \frac{1}{c} u_{r}^{i} \cdot {\vec{V}}^{i} ({\hat{t}}^{i})}$ 20

Taking into account that $u_{r}^{i} \cdot {\vec{V}}^{i} ≪ c$ , the following approximation is analogous to the range rate models commonly used in Doppler shift calculations (Hauschild, 2017):

${\dot{R}}_{r}^{i} (t_{r}) \approx u_{r}^{i} \cdot ({\vec{V}}^{i} ({\hat{t}}^{i}) - {\vec{V}}_{r} (t_{r}))$ 21

Evidently, integrating the derivative over a time interval (T_r,k, T_r,k+1) will yield the range increment $R_{r}^{i} (T_{r, k + 1}) - R_{r}^{i} (T_{r, k})$ . Dividing the range increment by the time interval, Equation (9) represents the mean range rate, which is equal to the instantaneous rate if all components of Equation (19) remain constant over this interval. The correction factor ${\dot{R}}_{r}^{i} (T_{r}) b_{r}$ itself approximates the integral of $\dot{R}$ over an interval (T_r - b_r, T_r). This approximation is exact for any $\dot{R}$ —constant or not— if T_r,k+1 = T_r,k - b_r. In all other cases, the higher-order derivatives of Equation (16) cause errors. Akin to Equation (16), we make use of the Taylor expansion of $\dot{R}$ to solve the integral and define ΔT = T_r,k+1 – T_r,k. The exact solution permits us to calculate the error:

$\begin{array}{l} {\hat{\dot{R}}}_{r}^{i} b - \int_{T_{r, k} - b}^{T_{r, k}} \dot{R} d t & = \frac{b}{Δ T} \int_{T_{r, k}}^{T_{r, k} + Δ T} \dot{R} d t + \int_{T_{r, k}}^{T_{r, k} - b} \dot{R} d t \\ = \frac{b}{Δ T} (\dot{R} (T_{r, k}) Δ T + \frac{1}{2} \ddot{R} (T_{r, k}) Δ T^{2} + \frac{1}{6} \overset{⃛}{R} (T_{r, k}) Δ T^{3} + \dots) \\ + (- \dot{R} (T_{r, k}) b + \frac{1}{2} \ddot{R} (T_{r, k}) b^{2} - \frac{1}{6} \overset{⃛}{R} (T_{r, k}) b^{3} + \dots) \\ = b (\dot{R} (T_{r, k}) + \frac{1}{2} \ddot{R} (T_{r, k}) Δ T + \frac{1}{6} \overset{⃛}{R} (T_{r, k}) Δ T^{2} + \dots) \\ + b (- \dot{R} (T_{r, k}) + \frac{1}{2} \ddot{R} (T_{r, k}) b - \frac{1}{6} \overset{⃛}{R} (T_{r, k}) b^{2} + \dots) \\ = b (\frac{1}{2} \ddot{R} (T_{r, k}) (Δ T + b) + \frac{1}{6} \overset{⃛}{R} (T_{r, k}) (Δ T^{2} - b^{2}) + \dots) \end{array}$ 22

With the velocity terms canceling, the error is dominated by the acceleration. By assuming (ΔT + b) = 10 ms. b = 10 ms, and an acceleration of 10 g, we obtain a negligible range error of approximately 5 mm. Given a sampling rate of our ground truth position of 125 Hz and a clock bias in the single-digit millisecond range, this bound is realistic. This bound is also justified by our observation that the error histograms after correction shown in Figures 6(b) and 6(d) are not perturbed by the approximation error. Caution is warranted, however, in case of low sampling rates, large clock bias, and high receiver dynamics. If both (ΔT + b) and b increase to 100 ms, a constant acceleration of 10 g causes a potentially significant range error of approximately 50 cm. In the case of a low sampling rate, we can estimate the range rate by using Equation (21) instead of approximating it by the mean range rate in Equation (9). Although it gives a more precise range rate estimation, this alternative approach requires instantaneous satellite velocity information. If, instead, we were to estimate $\hat{\dot{R}} = \dot{R} (T_{r, k})$ using Equation (21), the dependency on the sampling rate ΔT would disappear in Equation (22), which can be advantageous in the case of a low sampling rate and small bias.

ACKNOWLEDGMENTS

This work was supported by the Defense Innovation Agency (AID) of the French Ministry of Defense (research project CONCORDE no. 201965 0090004707501).

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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View Abstract

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Quantifying GNSS Multipath Statistics by Signal Properties in Urban, Suburban, and Rural Environments

Abstract

1 INTRODUCTION

2 METHODOLOGY

2.1 Background

2.2 Measuring the Multipath Component

2.2.1 Theoretical Model

2.2.2 Choosing the Pivot Satellite

2.2.3 Dealing with Receiver Clock Bias

3 EXPERIMENTAL STUDY

3.1 Data Set Description

3.2 Environmental Contexts and Multipath-Related Parameters

3.3 Data Set Discussion

4 RESULTS

4.1 Receiver Clock Bias Correction

4.2 Pseudorange Residual Histograms

4.2.1 LOS- and NLOS-Labeled Pseudoranges

4.2.2 Satellite Elevation

4.2.3 Measured C/N0

4.3 Statistical Sample Moments

4.3.1 Mean and Variance

4.3.2 Skewness and Kurtosis

5 CONCLUSION

HOW TO CITE THIS ARTICLE:

CONFLICT OF INTEREST

A | JUSTIFICATION OF THE RANGE RATE APPROXIMATION

ACKNOWLEDGMENTS

REFERENCES

4.2.3 Measured C/N₀