Improving GNSS Positioning in Dense Urban Areas by Diffraction Feature Matching

Hai Di; Ng Hoi-Fung; Zhang Guohao,; Hsu Li-Ta

doi:10.33012/navi.759

Abstract

The performance of global navigation satellite systems (GNSSs) in urban canyons is degraded by signal scattering from buildings. Three-dimensional mapping-aided (3DMA) GNSS methods have been developed using feature matching to mitigate non-line-of-sight (NLOS) reception errors, but are ineffective when NLOS features are limited. Moreover, diffraction can attenuate signal strength, introducing faulty matches. Diffraction attenuation depends on the signal propagation geometry, with potential to serve as a new location-dependent feature for positioning. In this study, the carrier-to-noise density ratio attenuation of diffracted signals is modeled by the uniform geometrical theory of diffraction, as a new feature for positioning. The performance of this approach is validated by assessing location-dependent capability within a feature-matching framework. The proposed method is validated in dense urban canyons, reducing the two-dimensional root-mean-square error from the 10-m level of shadow matching to less than 5 m by using only three or four satellites with diffracted signals, enabling meter-level accuracy applications for pedestrian positioning and future integration with existing 3DMA GNSS methods.

Keywords

1 INTRODUCTION

Since their establishment, global navigation satellite systems (GNSSs) have been widely used to provide position, velocity, and time (PVT) information on a global scale. Currently, with the rapid expansion of autonomous vehicles, unmanned aerial vehicles, and portable devices in civil life, available and reliable GNSSs have become indispensable. Differential GNSS methods allow low-cost GNSS receivers to achieve meter-level positioning accuracy in outdoor environments (Monteiro et al., 2005), which satisfies the needs of most civil applications. However, such methods are unable to eliminate errors introduced by the local environment surrounding antennas (Hsu et al., 2023), such as signal blockage, reflection, and diffraction from environmental objects like buildings, degrading the quality of transmission signals and the corresponding measurements (Schön et al., 2022). More specifically, the reception of both direct and reflected signals can cause multipath effects, while receiving only the reflected signal will lead to non-line-of-sight (NLOS) reception (Groves, 2013). Both cases can introduce a significant bias in GNSS pseudorange measurements and consequently lead to a degradation in positioning accuracy (Zhu et al., 2020). Moreover, diffraction effects may occur when the transmission path of a satellite signal is close to a building’s edge (Bradbury, 2007). Although the pseudorange measurement delay due to diffraction effects is only several decimeters, the signal strength can be significantly attenuated by diffraction, which degrades the measurement quality, hindering various positioning methods that rely on signal strength (Zhang & Hsu, 2021). Given the density and complexity of buildings in urban scenarios, multipath, NLOS reception, and diffraction occur frequently and have become widely concerning issues for GNSSs.

Various methods have been proposed to mitigate the degradation that arises from signal reflections. Because the polarizations of direct and reflected signals are usually different, a dual-polarization antenna can be employed to detect and mitigate NLOS reception and multipath interference (Jiang & Groves, 2014; Sun et al., 2021). Additionally, various consistency-verification-based positioning methods can be deployed to detect and alleviate multipath effects (Bhamidipati & Gao, 2020). However, this approach is less effective when multiple outliers are present and may lead to poor satellite geometry in dense urban areas (Groves & Jiang, 2013). Because NLOS reception can introduce positioning errors greater than 10 m, many positioning algorithms have been proposed to de-weight or exclude NLOS receptions (Groves & Jiang, 2013; Suzuki & Kubo, 2014). NLOS receptions can be detected and excluded by either using a building model database combined with a priori position estimates (Peyraud et al., 2013) or applying an omnidirectional infrared camera for obstacle sensing (Meguro et al., 2009), thereby enhancing urban positioning accuracy. Some studies have detected and excluded multipath-contaminated satellites based on corresponding signal quality (Smolyakov et al., 2020). However, the complete exclusion of NLOS and multipath measurements not only reduces the number of available satellites but also hampers the satellite distribution geometry during positioning. Thus, the effectiveness of NLOS exclusion in dense urban areas is limited.

Recently, the three-dimensional mapping-aided (3DMA) GNSS method has become prevalent for improving positioning accuracy in dense urban environments. This methodology employs three-dimensional (3D) building models to simulate signal propagation characteristics across different candidate locations, considering NLOS receptions as a location-dependent feature for feature-matching positioning rather than excluding them (Bradbury et al., 2007; Groves, 2011). 3DMA GNSS techniques can be broadly categorized into two representative groups: satellite-visibility-based methods and ranging-based methods. Satellite-visibility-based techniques, such as shadow matching, predict line-of-sight (LOS)/NLOS satellite visibility at candidate positions using 3D building models and match these predictions with visibility inferred from the measured signal-to-noise-density ratio (C/N₀) (Wang et al., 2015). Ranging-based approaches, including likelihood-based ranging (Groves et al., 2020) and ray-tracing (Hsu, 2017), exploit pseudorange measurements as location-dependent features, modeling NLOS delays through 3D building geometries and comparing these delays with measurements. Recent advancements integrate auxiliary sensors such as lidar to dynamically reconstruct environmental geometry without pre-existing 3D city models (Pugliese et al., 2023), while collaborative frameworks (Zhang et al., 2020) and multi-receiver configurations (Strandjord et al., 2020) further enhance positioning accuracy. State-of-the-art 3DMA GNSS algorithms achieve sub-10-m average positioning errors in urban areas (Ng et al., 2021). However, the performance of 3DMA GNSS methods cannot be guaranteed for all environments. In an environment with very few signal obstructions, the information from satellite visibility for positioning is limited, introducing a large ambiguity during positioning. Furthermore, GNSS signal diffraction is prone to occur in urban areas and affects all types of measurements, including LOS, multipath, and NLOS receptions (Icking et al., 2020; Schaper et al., 2022). Although C/N₀ serves as a critical location-dependent feature in many 3DMA GNSS frameworks, its reliability is systematically compromised by diffraction effects near building edges. Specifically, diffracted signals exhibit C/N₀ levels intermediate between LOS and NLOS thresholds, corrupting visibility classification and pseudorange error models (Nicolás et al., 2012; Wang et al., 2015). Consequently, the degraded positioning accuracy of current 3DMA GNSS performance is insufficient for applications requiring meter-level accuracy, such as pedestrian positioning. Thus, a key challenge of current 3DMA GNSS frameworks lies in the distinct attenuation pattern of diffracted signals, which deviates from the C/N₀ characteristics of LOS and NLOS receptions, necessitating specific research to improve urban positioning.

The GNSS signal attenuation caused by diffraction is related to the geometrical relationship between the GNSS receiver, satellites, and obstacles (Nicolás et al., 2012; Zhang & Hsu, 2021). Specifically, the obstruction level of the first Fresnel zone (FFZ) for GNSS signals can be employed to distinguish satellites that are potentially influenced by diffraction effects (Zimmermann et al., 2018). Several studies have modeled the C/N₀ of diffracted signals by applying the uniform geometrical theory of diffraction (UTD) with building models, which can simulate C/N₀ values consistent with real GNSS measurements (Zhang & Hsu, 2021). Thus, searching for location-dependent features from diffraction measurements has the potential to enhance 3DMA GNSS methods. In recent work, the UTD and GNSS reflectometry principles were employed with 3D building models to estimate the signal strength of both reflection and diffraction at different locations and were compared with the observed signal-to-noise ratio to determine user position (Suzuki, 2012). However, this study did not distinguish diffraction from other phenomena to specify its role in aiding positioning. Moreover, the feasibility and effectiveness of using diffraction for positioning still requires investigation. State-of-the-art shadow-matching techniques (Wang et al., 2013) mitigate diffraction effects by de-weighting measurements within a certain range of C/N₀ that are likely to be diffracted; thus, these techniques cannot benefit from diffraction effects. Therefore, there is a need to investigate the potential and performance of employing diffraction measurements as a location-dependent feature for feature-matching.

In this study, a new location-dependent feature related to diffraction effects is explored, and the feasibility of applying the proposed feature for positioning is investigated. First, the ray-tracing technique and UTD are applied to model signal propagation and behaviors under diffraction effects for different hypothetical location candidates. Candidates with better consistency between simulated diffraction attenuations and measurements are assigned higher weighting scores, which are applied in a weighted average among all candidates for positioning. The effectiveness of the proposed diffraction feature for positioning is analyzed by its location dependency. The proposed diffraction-based feature-matching method is verified by three experiments in urban areas and compared with the conventional least-squares positioning method, the receiver’s commercial algorithm solution, and the prevalent 3DMA GNSS shadow-matching method. The contributions of this paper are threefold: (1) the effectiveness of using diffraction effects as a feature for positioning is comprehensively evaluated through an analysis of its location dependency in urban areas, (2) a novel diffraction-based feature-matching method is developed for urban areas, and (3) the performance of the proposed diffraction-based feature-matching method is validated in three experiments, achieving accuracy with a two-dimensional (2D) root-mean-square error (RMSE) of less than 5 m by using only diffracted satellite signals.

The remainder of this paper is structured as follows. Section 2 introduces the GNSS diffraction model based on ray-tracing and the UTD. Section 3 introduces the framework of the proposed diffraction-based feature-matching method. In Section 4, the location dependency of diffraction effects is analyzed, and the effectiveness of applying diffraction features in positioning is evaluated in terms of building edge orientations and time series. Section 5 validates the proposed method through different experiments in urban areas. Finally, conclusions are drawn in Section 6.

2 GNSS DIFFRACTION MODELING BY UTD

The strength of a diffracted GNSS signal is closely related to the signal propagation geometry between the satellite, receiver, and obstructions (Bradbury, 2007). By assessing the geometrical parameters related to diffraction via ray-tracing, it is feasible to evaluate whether a signal undergoes diffraction as well as its diffraction attenuation.

2.1 Ray-Tracing Technique

Ray-tracing is a popular method for modeling signal propagation paths based on geometric information (de Sousa & Thomä, 2018). GNSS signal scattering induced by buildings can be considered as a local effect that depends on the scattering characteristics in the vicinity of a stationary point (e.g., a reflection point on the building surface), given that the wavelength of the GNSS signal is much smaller than the size of environmental objects such as buildings (McNamara et al., 1990). A diffracted signal can be approximated as a ray consisting of two line segments connected at a turning point (namely, the diffraction point) on the building edge according to Fermat’s principle (Keller, 1962). At this point, the signal has the shortest propagation path from the satellite to the receiver. The characteristics of a diffracted signal can be determined by the geometrical parameters associated with its diffraction point (Zhang & Hsu, 2021). The geometrical parameters for diffraction modeling are illustrated in Figure 1. Based on the shortest propagation path, the diffraction point can be determined as follows:

$Illustration of ray-tracing on a diffracted GNSS signal and definitions of geometrical parameters for the diffraction model based on the UTD$

FIGURE 1

Illustration of ray-tracing on a diffracted GNSS signal and definitions of geometrical parameters for the diffraction model based on the UTD

$x_{m, diff}^{i} = \underset{x \in within building edge}{\arg min} ‖ x^{i} - x ‖ + ‖ x_{m} - x ‖$ 1

where i and m are indices denoting the satellite and the candidate site, respectively. xⁱ and x_m denote the position of the satellite and candidate, respectively. There may exist multiple feasible diffraction points on different edges. In this study, the signal with the shortest delay compared with the LOS path is regarded as the dominant diffraction signal, as it is processed first by the receiver (Zhang & Hsu, 2021) and is likely to have the minimum attenuation (Hsu et al., 2016). Thus, the diffraction point $x_{m, diff}^{i}$ and its propagation geometry information are obtained by ray-tracing to all surrounding building edges, as follows:

${x_{m, diff}^{i}, η_{edge}} = RayTracing (x^{i}, x_{m}, X_{b})$ 2

where X_b is the 3D building model containing corner coordinates of all surrounding buildings within a certain area and $η_{edge}$ denotes the unit vector of the building edge at which diffraction occurs. Parameters obtained from ray-tracing are applied with the UTD to estimate diffraction attenuation.

2.2 Diffraction C/N₀ Modeling by UTD

The UTD provides an effective method for asymptotically representing a diffracted field of electromagnetic signals, including GNSS signals. The principle of the UTD is based on modern geometrical optics (GO) derived from classical GO and Maxwell’s equations. By extending the geometrical theory of diffraction with several diffraction field coefficients related to propagation geometrical parameters, the UTD ensures the continuity of the resultant diffraction field, even at shadow boundaries (Kouyoumjian & Pathak, 1974; Zhang & Hsu, 2021).

The UTD decomposes the electromagnetic field at any propagation path location into parallel (‖) and vertical (⊥) components relative to the diffraction edge. These components are defined using orthonormal unit vectors associated with the incidence and diffraction planes (the plane contains a diffraction edge and incident or diffraction ray, as shown in Figure 1). Specifically, $u_{‖}^{i}$ and $u_{⊥}^{i}$ are unit vectors orthogonal to the incident ray, parallel and perpendicular to the incidence plane, respectively. Similarly, $u_{‖}^{d}$ and $u_{⊥}^{d}$ correspond to the diffraction plane. The attenuation due to diffraction is governed by the vertical (D_⊥) and parallel (D_‖) diffraction coefficients, derived from the UTD as formulated in Equation (3):

$D_{⊥, ‖} = D_{1} + D_{2} \pm (D_{3} + D_{4})$ 3

Here, D₁ and D₂ are related to the diffracted field when the o-face and n-face (the two planar surfaces of the building edge, as shown in Figure 1) are shadowed, whereas D₃ and D₄ correspond to reflections from the n-face and o-face, respectively. As reflection and diffraction are unlikely to occur simultaneously at the same building, based on the geometry (Zhang & Hsu, 2021), the diffraction coefficients can be simplified as follows:

$D_{⊥, ‖} = D_{1} + D_{2}$ 4

Based on the UTD (McNamara et al., 1990), D₁ and D₂ are calculated from the diffraction-related geometrical parameters illustrated in Figure 1:

$D_{1} = \frac{- e^{- \frac{j π}{4}}}{2 n \sqrt{2 π k} \sin β} \cot [\frac{π + (ϕ - ϕ^{'})}{2 n}] T [k L a^{+} (ϕ - ϕ^{'})]$ 5

$D_{2} = \frac{- e^{- \frac{j π}{4}}}{2 n \sqrt{2 π k} \sin β} \cot [\frac{π - (ϕ - ϕ^{'})}{2 n}] T [k L a^{-} (ϕ - ϕ^{'})]$ 6

$T (x) = 2 j \sqrt{x} e^{j x} \int_{\sqrt{x}}^{\infty} e^{- j τ^{2}} d τ$ 7

where j represents the imaginary unit and n, a parameter of the building’s wedge exterior angle, is set to n = 1.5 for a 90^∘ building edge (Kouyoumjian & Pathak, 1974). k is the wavenumber, and β, the diffraction angle, defines the angle between the diffraction edge and the diffraction ray. Angular parameters ϕ^′ and ϕ measured from the o-face specify the location of the incidence and diffraction planes, respectively. The transition function T(⋅) is governed by the wavenumber k, the distance parameter L, and the geometry-related function $a^{\pm} (ϕ - ϕ^{'})$ . For GNSS applications, satellites are far away from the buildings and receiver, allowing the incident wave to be approximated as a plane wave (Kouyoumjian & Pathak, 1974). Under this assumption, the distance parameter L is calculated as Equation (8), where s_d denotes the distance between the diffraction point $x_{m,diff}^{i}$ and the receiver x_m:

$L = S_{d} {sin}^{2} β$ 8

The geometry-related function $a^{\pm} (ϕ - ϕ^{'})$ is defined as in Equation (9):

$a^{\pm} (ϕ - ϕ^{'}) = 2 \cos^{2} {\frac{[2 n π N^{\pm} - (ϕ - ϕ^{'})]}{2}}$ 9

where N^± is the integer that most nearly satisfies $2 n π N^{\pm} - (ϕ - ϕ^{'}) = \pm π$ (Kouyoumjian & Pathak, 1974). Because GNSS signals are designed with right-hand circular polarization, the co-polarization diffraction coefficient $D_{R R}$ is obtained by combining two orthogonal linear polarization diffraction coefficients, as shown in Equation (10) (Zhang & Hsu, 2021):

$D_{R R} = \frac{- u_{‖}^{d} u_{‖}^{i} D_{‖} - u_{⊥}^{d} u_{⊥}^{i} D_{⊥}}{2}$ 10

In urban propagation scenarios, both direct signals (LOS) and diffracted fields may coexist when the FFZ is partially obstructed. Under such conditions, the field of total received signals is determined by the superposition of the diffracted and LOS fields. Thus, two types of diffraction effects can occur, as illustrated in Figure 2, where the corresponding coefficients are obtained as follows:

$D_{total} = {\begin{matrix} \frac{D_{R R}}{\sqrt{s_{d}}} e^{- j k s_{d}}, & LOS obstructed \\ e^{- j k (s_{d} - ε_{d})} + \frac{D_{R R}}{\sqrt{s_{d}}} e^{- j k s_{d}}, & LOS unobstructed \end{matrix}$ 11

$Two types of diffraction effects depending on the LOS obstruction$

FIGURE 2

Two types of diffraction effects depending on the LOS obstruction

Here, ε_d is the signal propagation delay resulting from diffraction. For a diffracted GNSS signal, the relationship among the diffraction attenuation coefficient, the C/N₀ under diffraction ( $C / N_{0, diff}$ ), and the C/N₀ without any obstruction ( $C / N_{0, open}$ ) can be described as in Equation (12):

$C / N_{0, diff} = C / N_{0, open} + 10 \log {‖ D_{total} ‖}^{2}$ 12

3 DIFFRACTION-BASED FEATURE-MATCHING POSITIONING

3.1 System Architecture

As diffraction effects are dependent on the propagation geometry, i.e., the receiver position with respect to the building, these effects can be used as a feature for positioning through the standard feature-matching framework shown in Figure 3. An offline database can be generated beforehand for practicality, containing the diffraction coefficients corresponding to different satellite positions in the sky-plot for different candidate positions in a certain area (Liu et al., 2018). This study employs a database that contains different candidate positions from 2-m-resolution grid points and different satellite positions, with 1^∘ resolution for the elevation angle and azimuth angle. The resolution of candidate positions and satellite geometries is determined through a tradeoff of computational efficiency, storage constraints, and accuracy requirements for urban navigation applications. Based on each candidate position, satellite position, and the surrounding building model, ray-tracing is applied to obtain the diffraction propagation geometry and parameters, which are further used to derive the diffraction coefficient from the UTD. In real applications, the GNSS receiver measurements are collected to extract diffraction features, i.e., the C/N₀ attenuations of different satellites under diffraction effects. Additionally, the expected C/N₀ attenuation of the received satellite signals under diffraction can be retrieved, corresponding to different candidate positions from the database. Here, the candidates within 24 m around the user’s initial position from the National Marine Electronics Association (NMEA) solution is considered to include the user’s actual position. For each candidate position, the expected C/N₀ attenuations from the database are compared with the actual C/N₀ attenuations from measurements, providing a matching score. The final position can be determined by taking the weighted average of the candidate positioning based on the matching score.

$System architecture of the proposed diffraction-based feature-matching method$

FIGURE 3

System architecture of the proposed diffraction-based feature-matching method

3.2 Diffraction-Based Feature Matching

In the 3DMA GNSS algorithm, location-dependent features are critical in determining the user position. 3DMA GNSS shadow-matching (Wang et al., 2015) and range-based (Hsu, 2017) methods use satellite visibility and NLOS delay, respectively, as location-dependent features to match with features from real measurements for positioning. In this study, the C/N₀ attenuation caused by diffraction in each epoch is used as the location-dependent feature for the matching process.

For actual receiver measurements, the C/N₀ diffraction attenuation of the i-th satellite signal with diffraction can be estimated as follows:

${\tilde{D}}^{i} = C / N_{0}^{i} - C / N_{0, open}^{i}$ 13

$C / N_{0, open}^{i} = f_{regression model} (θ^{i})$ 14

Here, $C / N_{0, open}$ is correlated with the satellite elevation angle; this trend can be modeled by a polynomial regression fit based on long-term collections of C/N₀ from different positions of satellites in open-sky areas (Strode & Groves, 2016; Suzuki, 2012). Then, the expected $C / N_{0, open}$ of a satellite can be estimated by its elevation angle θⁱ and the regression model $f_{regression model}$ . For the m-th selected candidate position from the database, the expected C/N₀ diffraction attenuation can be obtained based on the satellite elevation angle θⁱ and azimuth angle ψⁱ :

${\hat{D}}_{m}^{i} = 10 \log {‖ D_{total, m}^{i} ‖}^{2}$ 15

$D_{total, m}^{i} = database (m, θ^{i}, ψ^{i})$ 16

For each candidate position, the corresponding expected attenuation of the diffracted satellite signal is compared with the attenuation from the actual measurements, as follows:

$Δ d_{m}^{i} = | {\hat{D}}_{m}^{i} - {\tilde{D}}^{i} |$ 17

The difference between these two values is rescaled by Equation (18), where $Δ D_{m}^{i}$ represents the normalized difference in diffraction attenuation between measurements and simulations, namely, the normalized ΔC/N₀:

$Δ D_{m}^{i} = \frac{Δ d_{m}^{i} - Δ d_{min}^{i}}{Δ d_{max}^{i} - Δ d_{min}^{i}}$ 18

In Equation (18), $Δ d_{min}^{i}$ and $Δ d_{max}^{i}$ are the minimum and maximum values of the expectation/measurement inconsistencies among all candidate positions for a specific satellite. Note that a satellite with no valid diffraction points in the database, i.e., a satellite for which no diffraction occurs according to ray-tracing and the UTD, will be assigned $Δ d_{max}^{i}$ .

3.3 Hypothetical Feature-Matching Positioning with Diffraction Features

The user position can be determined by searching for candidates with the expected diffraction attenuations most consistent with those from the actual measurements. The normalized ΔC/N₀ calculated from Equation (18) can be converted into a matching score, by considering the averaged differences among different satellites, $Δ {\overset{―}{D}}_{m}$ , following a normal distribution, as follows:

$Λ_{m} = e^{- \frac{{(Δ {\overset{―}{D}}_{m})}^{2}}{σ^{2}}}$ 19

$Δ {\overset{―}{D}}_{m} = \frac{1}{I} \sum_{i}^{I} Δ D_{m}^{i}$ 20

where I denotes the total number of satellites being classified under diffraction effects and σ is the standard deviation, heuristically set to 0.1 in this study. A higher matching score indicates a better consistency between the database and measurements, where the corresponding candidate position is more likely to be the user’s true position. Finally, the user’s position can be estimated by applying a weighted average of the candidate positions x_m based on their corresponding matching score:

${\hat{x}}_{u s e r} = \frac{\sum_{m} (Λ_{m} x_{m})}{\sum_{m} (Λ_{m})}$ 21

4 LOCATION DEPENDENCY AND SENSITIVITY OF THE DIFFRACTION FEATURE

4.1 Experimental Setup

A long-term experiment was performed in Hong Kong to analyze the location dependency of diffraction effects and the effectiveness of the proposed diffraction-based feature-matching process. Only one building is on one side of the antenna to ensure that the GNSS signal propagation path adjacent to the building edge is subjected only to diffraction effects, as shown in Figure 4 (a). Schematics of diffracted signal propagation paths from satellites G27 and B10 related to the building edges are displayed in Figure 4 (b). A low-cost GNSS receiver, u-blox EVK-M8T, with a patch antenna, u-blox ANN-MB-01, is used to collect Global Positioning System (GPS) and BeiDou measurements at 1 Hz for approximately 3500 epochs in Receiver Independent Exchange (RINEX) format. The 3D building models were obtained based on 3D spatial data from the Hong Kong Lands Department (Hong Kong Lands Department, 2024). The user’s true position was derived from real-time kinematic (RTK) positioning results using a high-precision Novatel PWRPAK7 receiver equipped with a Novatel GPS-703-GGG antenna. In this study, the satellites with diffracted signals were pre-classified according to satellite trajectories and signal strength performance, to evaluate the full potential of the proposed diffraction-based feature-matching algorithm.

FIGURE 4

(a) Environment of the standalone building experiment, (b) illustration of signal propagation paths for satellites G27 and B10

4.2 Performance Validation of Diffraction Modeling

During validation, the signal propagation paths for satellites G27 and B10 were adjacent to building edges, as shown in Figure 5 (a), and were thus classified as being under diffraction effects according to their FFZ blockage and signal performance. Ray-tracing and the UTD from Section 2 were used to model the diffraction-attenuated C/N₀ at the user’s true location. Figures 5 (b) and (c) illustrate the modeling results for the G27 and B10 satellites at the true user position with the real-time satellite position. As the open-sky C/N₀ regression model could have a certain level of error for different constellations, an overall bias of 6 dB-Hz was manually adjusted for B10 based on a comparison with the data before diffraction attenuation. The trend of the modeled C/N₀ attenuation shows good consistency with the real measurements, verifying the accuracy of the diffraction modeling.

(a) Positions of satellites G27 and B10 on the sky-plot with the building obstruction (colored in gray) at the user’s true position, (b) C/N0 of G27 from the UTD simulation and measurement, (c) C/N0 of B10 from the UTD simulation and measurement

FIGURE 5

(a) Positions of satellites G27 and B10 on the sky-plot with the building obstruction (colored in gray) at the user’s true position, (b) C/N₀ of G27 from the UTD simulation and measurement, (c) C/N₀ of B10 from the UTD simulation and measurement

4.3 Location Dependency of Diffraction Features

According to the principles of the UTD, field attenuation coefficients are highly dependent on signal propagation geometry, which varies for different positions for the same satellite. Thus, the resulting C/N₀ attenuation also varies for different positions. In this subsection, ray-tracing and the UTD are applied to simulate the diffracted C/N₀ for different locations around the truth to evaluate the location dependency of this feature. Two sets of eight locations, located 2 or 10 m from the user’s true location in the direction perpendicular or parallel to the building edge, were sampled for the analysis on satellite G27 and B10, as shown in Figures 6 (a) and 7 (a), respectively.

$(a) Eight sampled positions located 2 m or 10 m from the true position in the direction parallel or perpendicular to building edge 1, labeled as 1-8; simulation result of C/N0 from satellite G27 based on the diffraction model for (b) positions 1-4 and (c) positions 5-8$

FIGURE 6

(a) Eight sampled positions located 2 m or 10 m from the true position in the direction parallel or perpendicular to building edge 1, labeled as 1-8; simulation result of C/N₀ from satellite G27 based on the diffraction model for (b) positions 1-4 and (c) positions 5-8

$(a) Eight sampled positions located 2 m or 10 m from the true position in the direction parallel or perpendicular to building edge 1, labeled 1-8; simulation result of C/N0 from satellite B10 based on the diffraction model for (b) positions A-D and (c) positions E-H$

FIGURE 7

(a) Eight sampled positions located 2 m or 10 m from the true position in the direction parallel or perpendicular to building edge 1, labeled 1-8; simulation result of C/N₀ from satellite B10 based on the diffraction model for (b) positions A-D and (c) positions E-H

For satellite G27, the diffraction modeling results for different sampled locations are shown in Figures 6(b) and (c). Because locations 1-4 are distributed perpendicular to building edge 1 at different distances, the C/N₀ attenuation starts earlier at locations 1 and 2, which are closer to the edge compared with the true location, and vice versa for locations 3 and 4. When the satellite crosses the building edge at epoch 1621 for the true location, the diffraction attenuation differences exceed 5 dB-Hz for locations 2 and 3 compared with the center, despite being located only 2 m from the center. According to the UTD, these locations have different diffraction geometries, resulting in variations in the Fresnel zone obstructions and thus significant differences in their C/N₀ attenuations. These differences are distinguishable when compared with the thermal noise in the C/N₀ attenuations (Kaplan & Hegarty, 2017). However, for locations 5-8, which are parallel to the building edge, the C/N₀ attenuations are not distinguishable, as shown in Figure 6 (c), even though locations 5 and 8 are separated by 20 m. Note that location 8 shows a different pattern at the end of the time series, as the diffraction edge changes.

For satellite B10, the C/N₀ attenuations at different sampled locations are shown in Figure 7. Similar to G27, the diffraction attenuation shows a strong location dependency at the meter level among locations A−D in the direction perpendicular to the diffraction edge, whereas the attenuation results are not distinguishable among locations E-H. In summary, the diffraction attenuation exhibits location dependency in the direction perpendicular to the diffraction edge, which is distinguishable with a resolution finer than 2 m, demonstrating its potential for use as a feature to determine user position.

4.4 Performance of Diffraction Feature Matching

For the single-building experiment, the simulation results of C/N_0,diff at different locations for satellites G27 and B10 are shown on a color scale in Figures 8 (a) and (b), respectively. Here, locations are sampled with 2-m resolution within a 24-m range around the receiver’s true location, where black indicates that no valid diffraction points are identified. The distribution of the scale of C/N₀ has a clear geometry similar to the building edges where diffraction occurs, confirming that the diffraction attenuation is location-dependent in terms of the diffraction edge orientation.

$Simulated C/N0 at different positions sampled with 2-m resolution for (a) satellite G27 and (b) satellite B10 The color bar denotes the value of C/N0, with black denoting no valid diffraction propagation from ray-tracing. The star marker denotes the receiver’s true location.$

FIGURE 8

Simulated C/N₀ at different positions sampled with 2-m resolution for (a) satellite G27 and (b) satellite B10

The color bar denotes the value of C/N₀, with black denoting no valid diffraction propagation from ray-tracing. The star marker denotes the receiver’s true location.

The proposed feature obtained from Equation (18), the normalized ΔC/N₀ between the measurements and simulations of diffraction attenuation, denoted as $Δ D_{m}^{i}$ , is shown at different locations for G27 and B10 in Figures 9 (a) and (b), respectively. For both satellites, the normalized ΔC/N₀ is close to zero at the true location (star marker), which demonstrates a good consistency between model simulations and actual measurements. More specifically, the C/N₀ difference around the true location is less than 3 dB-Hz. Consistent with the location dependency analysis, there is a band of area in which the normalized ΔC/N₀ is similar to the true value. Using this feature from one satellite will introduce ambiguity in determining the receiver’s location. Because this ambiguity is associated with diffraction building edges, the integration of features from multiple satellites related to differently oriented edges can mitigate the ambiguity. As Figure 9 (c) shows, the integrated features from G27 and B10 have a unimodal distribution, correctly indicating the receiver’s true location. Thus, the normalized C/N₀ attenuation difference is an effective diffraction feature for positioning.

Normalized C/N0 difference between simulation and measurement at epoch 2000 for (a) G27 and (b) B10; (c) averaged difference of G27 and B10 The receiver’s true location is marked by a star at the center of sampling positions. The color bar indicates the normalized C/N0 difference.

FIGURE 9

Normalized C/N₀ difference between simulation and measurement at epoch 2000 for (a) G27 and (b) B10; (c) averaged difference of G27 and B10

The receiver’s true location is marked by a star at the center of sampling positions. The color bar indicates the normalized C/N₀ difference.

4.5 Effectiveness of Diffraction Features for Feature Matching

In the proposed feature-matching method, candidates for which the difference in diffraction attenuation features between measurements and simulations is less than 5 dB-Hz can be regarded as high-score candidates that determine the user’s position. A small cluster of high-score candidates indicates that the diffraction effect varies significantly at different locations (location dependency), which can provide a better positioning accuracy. In contrast, a large cluster indicates that the diffraction effect is similar at different locations, introducing ambiguity during positioning. As shown by the location dependency analysis in Section 4.3, the effectiveness of diffraction features in positioning varies in terms of the diffraction edge orientation. Here, the effectiveness of diffraction features is evaluated by the cluster width of high-score candidates adjacent to the true position in the direction perpendicular or parallel to the diffraction building edge for satellite G27 and B10, as shown in Figure 10. Candidate positions remain clustered and centered at the user’s true location, with a resolution of 2 m within a 24-m range.

$Cluster widths of high-score candidates during the matching process in the direction perpendicular and parallel to the diffraction building edge: (a) an example illustration on a map, (b) variation for G27 during the experiment, and (c) variation for B10 during the experiment$

FIGURE 10

Cluster widths of high-score candidates during the matching process in the direction perpendicular and parallel to the diffraction building edge: (a) an example illustration on a map, (b) variation for G27 during the experiment, and (c) variation for B10 during the experiment

The area with green background in Figures 10 (b) and (c) indicates the period during which the FFZ of the GNSS signal at the true location is obstructed by the building, i.e., the actual received signal is subject to diffraction. As illustrated in Figures 10 (b) and (c), the cluster width is much lower in the perpendicular direction than in the parallel direction for both satellites. When the FFZ begins to be obstructed, the diffraction attenuation varies among different candidates, as their corresponding building edge elevation angles are different. Thus, the diffraction attenuation can be used as a feature to indicate the user’s position in the direction perpendicular to the building edge, with a smaller size of high-score candidates. In particular, when more than half of the FFZ has been obstructed for G27 during epoch 1700-2000 or B10 during epoch 2200-3000, the high-score candidate cluster width is less than 5 m in the perpendicular direction. In other words, the signal exhibits a strong diffraction attenuation, which is sufficiently sensitive to determine the user’s position at a resolution of less than 5 m.

By applying the proposed feature-matching framework with satellite G27 and B10, the 2D positioning error during the diffraction stage is determined, as shown in Figure 11. After epoch 1300, signals from G27 and B10 experience diffraction attenuation, providing location-dependent features correlated with building edges that enhance positioning accuracy. For epochs 1700-2000, the strong sensitivity of the feature from G27 significantly improves the positioning performance, substantially reducing the 2D errors to less than 3 m. After epoch 2000, the sensitivity of the diffraction feature from G27 degrades as the FFZ becomes fully obstructed, resulting in low signal strength and higher noise. However, the diffraction features from B10 can still indicate the user’s position in the direction perpendicular to the related building edge with diffraction. As a result, during the period of diffraction, the proposed feature-matching method can achieve a 2D RMSE of 5.9 m, with a minimum 2D error below 3 m.

$Positioning error of the proposed diffraction-based feature-matching method and NMEA solution during the experiment$

FIGURE 11

Positioning error of the proposed diffraction-based feature-matching method and NMEA solution during the experiment

5 EXPERIMENTAL VALIDATION IN URBAN ENVIRONMENTS

In addition to the standalone building scenario, three dense urban environments were selected to validate the performance of the proposed diffraction-based feature-matching methods: an urban square and dense residential blocks A and B, as illustrated in Figures 12 (a), (b), and (c), respectively. All three experiments include significant signal obstructions, reflections, and diffractions. For these three experiments, the receiver’s NMEA solution was used as the initial position for candidate sampling. The receiver configuration and method for obtaining the user’s ground truth position were the same as those described in Section 4.1. Five minutes of static data were collected for positioning. The positioning performance was validated by comparing the following methods:

(1) Least squares: Conventional least-squares positioning method without fault detection and exclusion, serving as a baseline to illustrate the positioning challenges encountered in select dense urban environments.
(2) Shadow matching: State-of-the-art 3DMA GNSS algorithm (Wang et al., 2015).
(3) NMEA: The receiver’s commercial positioning solution from NMEA data.
(4) Diffraction-based feature matching: The proposed feature-matching method.

5.1 Positioning Performance in an Urban Square

For the urban square experiment, the average building elevation angle is 36.64^∘, where satellite trajectories of G18, G24, B20, and B29 intersect building edges with diffraction effects, as illustrated in Figure 12 (d). The C/N₀ attenuation of these satellites was used as a feature to apply the proposed diffraction-based feature-matching positioning method. Note that the open-sky C/N₀ regression model was adjusted with an overall bias of 7 dB-Hz for all satellites in this experiment. The least-squares positioning achieves a 2D RMSE of 14.9 m, with a maximum error exceeding 40 m in this dense urban experiment. The positioning solutions and 2D errors of the proposed method are shown in Figures 13 (a) and (d), compared with the shadow-matching and NMEA solutions. The shadow-matching performance is degraded by diffracted signal misclassification, yielding a 2D RMSE of 6.8 m and a maximum error of 12.3 m. The NMEA solution demonstrates stability and optimal performance in the urban square, achieving a 2D RMSE of 2.4 m (maximum error: 5.3 m). The NMEA solution achieves high accuracy owing to the wide-open urban square environment, which maintains consistent LOS satellite availability. The proposed diffraction-based feature-matching method achieves a 2D RMSE of 4.6 m (maximum 2D error: 7.2 m) by using only four satellites with diffracted signals for positioning, while maintaining stable performance, demonstrating better performance compared with the conventional least-squares method and 3DMA GNSS shadow matching. The positioning performance of the proposed method compared with conventional methods is summarized in Table 1.

$Three experiments in urban areas, including (a) an urban square, (b) dense residential block A, and (c) dense residential block B, with corresponding sky-plots (d-f) of the satellites with diffracted signals and building obstructions$

FIGURE 12

Three experiments in urban areas, including (a) an urban square, (b) dense residential block A, and (c) dense residential block B, with corresponding sky-plots (d-f) of the satellites with diffracted signals and building obstructions

$Positioning performance in the urban square experiment achieved via shadow matching, the NMEA solution, and the proposed feature-matching method: (a) solutions on a map, (b) diffraction feature-matching performance for satellites G18, G24, B20, and B29 in the urban square, (c) diffraction feature-matching performance achieved by integrating different satellites, and (d) epoch-wise 2D positioning errors$

FIGURE 13

Positioning performance in the urban square experiment achieved via shadow matching, the NMEA solution, and the proposed feature-matching method: (a) solutions on a map, (b) diffraction feature-matching performance for satellites G18, G24, B20, and B29 in the urban square, (c) diffraction feature-matching performance achieved by integrating different satellites, and (d) epoch-wise 2D positioning errors

View this table:

TABLE 1 Positioning Performance of Diffraction-Based Feature Matching Compared With Conventional Least-Squares, Shadow-Matching, and NMEA Solutions in Terms of 2D Horizontal RMSE, Maximum Error, and Standard Deviation (STD) for Three Experiments

The proposed feature-matching method was verified through an analysis of the diffraction feature-matching performance. Figure 13 (b) illustrates the consistency of C/N₀ attenuation between the measurements and database at different candidate positions for individual satellites and their integration at the first epoch. The color on the map indicates the matching level of each candidate position, where red represents a good consistency and is more likely to be the user’s position. All heatmaps show excellent consistency in the red results at the user’s true position. According to Figure 12 (d), the matching level (as the heatmap) reflects the same pattern as the building boundaries close to the satellite. The results can distinguish the user position at a resolution of 2-6 m in the direction perpendicular to the building edge, as the C/N₀ attenuations differ significantly for the candidates in this direction. Note that B29 has two rows of matching candidates related to two parallel edges from the buildings on two sides. Although the user position is ambiguous at the candidate positions along the building edge orientation, this ambiguity can be resolved by integrating the matching results from different satellites diffracted by building edges with different orientations. As shown in Figure 13 (c), upon integration of the matching results from G18, G24, B20, and B29 with different azimuth directions, the overall matching peak area is centered around the true position, providing a unique indication to determine the user’s position.

5.2 Positioning Performance in Dense Residential Block A

For dense residential block A, shown in Figures 12 (b) and (e), the antenna is surrounded by buildings with an average elevation angle of 47.25^∘. Here, signals from satellites G07, G08, and B07 are expected to experience diffraction and were selected for diffraction-based feature-matching positioning. In this experiment, no bias is adjusted when the C/N₀ attenuation is estimated with the open-sky C/N₀ regression model. Conventional least-squares positioning yields a 2D RMSE of 18.5 m, with a maximum error of 37.9 m, highlighting the complexity of the environment. The positioning solutions from shadow matching, NMEA data, and the proposed method are shown in Figures 14 (a) and (d). Both the shadow matching and NMEA solutions yield unsatisfactory results, with a 2D RMSE of 9.6 m and 15.0 m, respectively. However, the diffraction-based feature-matching method significantly reduces the 2D RMSE to less than 3 m by using only three satellites with diffracted signals. The feature-matching performances of individual satellites and their integration at the first epoch are shown in Figures 14 (b) and (c), respectively. For each satellite, the high-score candidates are distributed with a pattern similar to the building exterior edges. Although the true position is always correctly indicated with a high score, there is still a large region of ambiguity for positioning. Note that the diffraction of B07 has a better location dependency to achieve a narrower ambiguous area. By integrating the high-score candidates from G07, G08, and B07 with different building edge patterns, a distinct red area can be obtained near the true position, as illustrated in Figure 14 (c). Thus, an accurate positioning solution can be obtained by applying a weighted average on all candidates when considering the integration of multiple satellites.

$Positioning performance in dense residential block A: (a) solutions from different methods, (b) matching performance for individual satellites, (c) matching performance achieved by integrating multiple diffracted satellite signals, and (d) epoch-wise 2D positioning errors$

FIGURE 14

Positioning performance in dense residential block A: (a) solutions from different methods, (b) matching performance for individual satellites, (c) matching performance achieved by integrating multiple diffracted satellite signals, and (d) epoch-wise 2D positioning errors

5.3 Positioning Performance in Dense Residential Block B

For dense residential block B, shown in Figures 12 (c) and (f), the average building elevation angle is 49.60^∘, with a very limited sky-view. In this experiment, no bias is adjusted when the C/N₀ attenuation is estimated with the open-sky C/N₀ regression model. Under severe signal obstruction and multipath conditions, conventional least-squares positioning achieves a 2D RMSE of 20.0 m, with a maximum error of 31.7 m. Additionally, both the shadow-matching and NMEA solutions experience significant performance degradation, exhibiting 2D RMSEs of 11.4 m and 13.6 m, respectively, as shown in Figures 15 (a) and (d). The proposed method makes use of the diffraction attenuation of satellites G18, B06, and B19, whose signal propagation paths are adjacent to the building edges, as features for positioning, achieving an accurate position solution with a 2D RMSE of 2.8 m. The candidates during the matching process at the first epoch are demonstrated in Figures 15 (b) and (c). The diffraction features from each satellite can indicate the receiver’s position, with the highest matching score occurring near the true position. Note that the matching process for B19 exhibits two bands of high-score candidates and is thus unable to determine the user’s position. By integrating the matching results from G18 and B06 with different patterns, the ambiguous area is significantly reduced, resulting in a unimodal cluster of high-score candidates centered at the true position. Thus, an accurate position can be obtained by averaging those candidate positions. Overall, for these three experiments in dense urban areas, the proposed method achieves a positioning accuracy with a 2D RMSE of less than 5 m, outperforming the conventional least-squares method and shadow-matching techniques in all three experiments and demonstrating the superior accuracy of the proposed method over the NMEA solutions in both dense residential test environments.

$Positioning performance in dense residential block B: (a) solutions from different methods, (b) matching performance for individual satellites, (c) matching performance by integrating multiple diffracted satellite signals, and (d) epoch-wise 2D positioning errors$

FIGURE 15

Positioning performance in dense residential block B: (a) solutions from different methods, (b) matching performance for individual satellites, (c) matching performance by integrating multiple diffracted satellite signals, and (d) epoch-wise 2D positioning errors

6 CONCLUSION AND FUTURE WORK

This paper has presented a comprehensive investigation of the effectiveness and performance of using diffraction features of GNSS signals for positioning. By using ray-tracing and the UTD, the expected GNSS signal attenuation at different positions in an urban area can be simulated; this attenuation is uniquely related to the signal propagation geometry associated with the antenna position. Thus, a feature-matching algorithm was proposed based on diffraction attenuation as a new location-dependent feature. Through a real experiment, the location dependency of diffraction features was analyzed, enabling the receiver position to be distinguished in the direction perpendicular to the building edge introducing diffraction. During the matching process, the inconsistency between diffraction features increases to over 5 dB-Hz for positions located 2 m from the ground truth, which is sufficiently sensitive to determine the receiver’s position at the meter level. Although the diffraction features are similar along the building edge direction, the positioning ambiguity can be effectively reduced by integrating multiple satellites diffracted by different building edges. The performance of the proposed diffraction-based feature-matching positioning method was validated by three experiments in dense urban areas. The proposed method was able to achieve a 2D RMSE of less than 5 m using only 3-4 diffraction measurements, outperformed the conventional least-squares method and shadow-matching solutions in all experiments, and performed better than the commercial NMEA solutions in two dense residential blocks. Therefore, the proposed method provides a new approach to employ diffraction as additional information for accurate positioning in urban areas.

However, several limitations must be addressed before this method can be applied in practice. First, the diffraction attenuation in this study was estimated via an elevation-angle-based regression model from open-sky data, which may have a certain level of bias and requires adjustment. A better modeling method for estimating the unobstructed C/N₀ requires further investigation. Second, whether a satellite signal undergoes diffraction is assumed to be known in this study to evaluate the full potential of this effect on positioning. Thus, the development of an intelligent diffracted measurement classification method is needed for the proposed algorithm. In addition, owing to the limited number of satellites with diffracted signals, integration with existing 3DMA GNSS techniques is required to ensure sufficient availability and continuous localization. Future research will integrate the proposed location-dependent features with existing 3DMA GNSS techniques (e.g., shadow matching) to employ all satellite observations for better positioning performance. Furthermore, the method’s applicability to dynamic scenarios requires systematic experimental investigation.

HOW TO CITE THIS ARTICLE

Hai, D., Ng, H-F., Zhang, G., & Hsu, L-T. (2026). Improving GNSS positioning in dense urban areas by diffraction feature matching. NAVIGATION, 73. https://doi.org/10.33012/navi.759

ACKNOWLEDGMENTS

The work described in this paper was substantially supported by a grant from the National Natural Science Foundation of China (Grant No. 62303391) and partly supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. PolyU15228023).

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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Improving GNSS Positioning in Dense Urban Areas by Diffraction Feature Matching

Abstract

1 INTRODUCTION