Research ArticleRegular Papers
Open Access

Improving the Prediction of GNSS Satellite Visibility in Urban Canyons Based on a Graph Transformer

Shaolong Zheng, Kungan Zeng, Zhenni Li, Qianming Wang, Kan Xie, Ming Liu, and Shengli Xie
NAVIGATION: Journal of the Institute of Navigation December 2024, 71 (4) navi.676; DOI: https://doi.org/10.33012/navi.676

Abstract

Signals from global navigation satellite systems (GNSSs) in urban areas suffer from serious multipath errors caused by building blockages and reflections. The use of deep neural networks offers great potential for predicting and eliminating complex multipath/non-line-of-sight (NLOS) errors. However, existing methods for predicting the original signals face two remaining challenges. The first challenge is an inability to effectively exploit irregular GNSS measurement data caused by an inconsistent number of visible satellites in different epochs. The second challenge is degradation in the generalization performance of the multipath/NLOS prediction model when using data collected from different locations and periods. To address these challenges, this paper proposes a novel graph transformer neural network (GTNN) for predicting satellite visibility that effectively learns environment representations from irregular GNSS measurements to both alleviate multipath interference and improve the generalization performance of the multipath prediction model. To learn from irregular GNSS measurements, a sky satellite graph is constructed as input to a graph neural network by using satellites captured in the same epoch, which can represent the spatial relationships between satellites and enable the model to learn satellite-related features sufficiently well. To improve the generalization ability of our multipath prediction model, a multihead attention mechanism is introduced to aggregate satellite node information by computing the correlation between satellites to extract the environment representation around the receiver. Based on the constructed sky satellite graph and the multihead attention mechanism, our novel GTNN for predicting satellite visibility can not only handle irregular GNSS measurements but can also learn an environment representation via graph attention. Comparative experiments were conducted on real-world GNSS measurement data in urban areas, demonstrating that the proposed method can achieve an accuracy exceeding 96% for satellite visibility prediction and obtain better generalization performance than existing multipath prediction methods. Moreover, the attention weights among satellites were visualized to demonstrate the environment representation learned by the GTNN from the sky satellite graph.

Keywords

1 INTRODUCTION

Global navigation satellite systems (GNSSs) are widely used as the primary navigation source in various intelligent transportation systems and location-based services, such as unmanned aerial vehicles (da Silva et al., 2020; Mohamed et al., 2021), vehicle navigation (Liu et al., 2022; Zhao et al., 2023), and robot technology (Shetty et al., 2023; Shuai & Yu, 2021). However, GNSS positioning performance relies on the quality of the received GNSS signals. When a user receives non-line-of-sight (NLOS) signals, the positioning results estimated by the receiver may have significant errors, caused by reflection or diffraction of the signal by building surfaces producing unpredictable multipath interference (Zabalegui et al., 2020). This interference is more significant in urban environments with towering buildings and has become the primary issue for GNSS urban positioning (Zhang et al., 2021). Unlike interference from atmospheric delays or satellite orbit bias, multipath errors in urban canyons are difficult to model via physical and/or statistical approaches because they are closely related to the particular environment surrounding the receiver. The key to improving positioning performance in such scenarios lies in accurately predicting the original satellite signals that have been affected by multipath interference, such that reliable line-of-sight (LOS) satellites can be appropriately considered in the GNSS position estimation.

Numerous attempts to mitigate multipath interference in urban areas have been reported in the literature. Machine learning is an effective method for detecting multipath signals. For example, in the methods of Qin et al. (2022) and Kim et al. (2022), some satellite features are input into a machine-learning model and each satellite signal is partitioned as LOS, multipath, or NLOS. However, the impact of the multipath effect is closely related to the environment surrounding the receiver (Haigh et al., 2019). When the environment is complicated by varying structures of buildings or trees, this method of inputting single satellite features for modeling fails because of the difficulty in capturing such changes. Recently, deep neural network (DNN)-based methods have brought new opportunities for GNSS multipath signal detection because of their substantial advantages in extracting hidden features from complex data. Some studies have utilized multipath information contained in satellite signal correlation images by constructing a deep convolutional neural network (Blais et al., 2022; Gonzalez et al., 2022). Furthermore, Zhang et al. (2021) and Li et al. (2023) have demonstrated that feeding the measurements of satellites captured in the same epoch into a DNN can effectively model environmental changes. However, the fixed input model of a DNN is not amenable to a variable number of measurement sets. Therefore, existing multipath/NLOS prediction models still face the following two challenges.

  • Existing methods fail to effectively exploit irregular GNSS data sets because of the inconsistent number of visible satellites in different epochs. The number of satellites in GNSS measurement sets collected in different epochs will be inconsistent because of the changing environment around the GNSS receiver. However, most existing neural network frameworks are designed to learn from input data with a fixed shape and a preset order, resulting in low utilization of the training data and excessive learning of sequential patterns. Therefore, there is a pressing need for a method that can effectively exploit irregular GNSS measurement sets.

  • The generalization performance of existing multipath and NLOS detection models is limited. Existing GNSS data sets are typically collected from a few specific locations and periods worldwide. However, when a multipath and NLOS detection model trained on such data sets is applied to new GNSS satellite measurement data collected from different locations and periods, the detection accuracy of the model may sharply decline because of environmental variations between the training locations and testing locations. Therefore, there is a pressing need to develop an effective approach for learning environment representations and improving the generalization performance of the model.

To address these two challenges, this paper proposes a graph transformer neural network (GTNN) for improving the prediction of GNSS satellite visibility. Here, “satellite visibility” refers to determining whether a satellite signal is LOS or NLOS. Specifically, to learn from irregular GNSS measurements, we first construct a sky satellite graph as the input to a graph neural network (GNN) to enhance the model’s ability to learn satellite-related features sufficiently well, where satellites in the airspace of the same epoch are represented as nodes and edge connections are established between satellites that belong to the same constellation or are adjacent. Then, to improve the generalization performance of our multipath prediction model, we employ a multihead attention mechanism as an aggregation method for adjacent satellite nodes in the propagation process of GNNs to learn the environmental characterization. The proposed method for predicting satellite visibility using a GTNN is shown in Figure 1. Our approach can not only learn irregular GNSS measurements from the graph structure to extract the global environment representation around the receiver sufficiently well, but can also improve the generalization performance of multipath/NLOS recognition models and the accuracy of satellite visibility prediction. To the best of our knowledge, this work is the first to apply a GNN to multipath/NLOS signal detection. The contributions of this study can be summarized as follows:

  • To learn hidden information from irregular GNSS measurement sets, we construct a sky satellite graph as the input to a GNN, where satellites are represented as nodes of the graph and connections between pairs of satellites are determined from neighbor and constellation attributes. The sky satellite graph represents the spatial relationships between satellites and enables the model to learn satellite-related features sufficiently well.

  • To improve the generalization performance of the GNSS multipath detection model, we introduce a multihead attention mechanism derived from a transformer architecture to aggregate the features of adjacent satellites by computing the relationships between satellites. This approach enhances the information interaction between satellites and the learning of environment representations, which improves the expressive ability of the model across a variety of urban environments.

  • Based on the constructed sky satellite graph rule and the proposed attention aggregation method, we develop a GTNN to improve the prediction of GNSS satellite visibility in urban canyons. This GTNN can not only learn irregular GNSS data sets sufficiently well but can also extract environmental representations via graph attention. Our experimental results show that, compared with state-of-the-art (SOTA) multipath detection methods, the proposed method has a stronger generalization capability. In addition, we discuss the representations learned from the sky satellite graph by the GTNN.

FIGURE 1
FIGURE 1

The proposed method uses a GTNN to predict satellite visibility. The measurements from the GNSS receiver are first constructed into a sky satellite graph, which is then fed into the GTNN to extract the environment representations and improve the prediction of satellite visibility.

The remainder of this paper is organized as follows. Section 2 reviews previous work related to multipath detection. Section 3 presents the background of GNNs. Section 4 describes our proposed method, including details of the construction rules for the sky satellite graph and the method for using the GTNN to predict satellite visibility. Section 5 gives details of our experimental investigations based on measurement sets collected in urban canyons. Finally, our conclusions are summarized, together with our plans for future work, in Section 6.

2 RELATED WORK

In recent years, to improve the service quality of GNSSs, a variety of studies have been conducted to mitigate multipath or NLOS errors. Broadly speaking, existing multipath detection methods can be partitioned into three categories: software-based methods, machine-learning-based methods, and DNN-based methods.

Software-Based Methods

Numerous software-based approaches for multipath mitigation have been developed (Kim et al., 2021; Xu & Rife, 2019; Zhu et al., 2020). For instance, Kim et al. (2021) assessed the quality of satellite signals based on the carrier-to-noise ratio (C/N0). If the C/N0 exceeds a certain threshold, the signal is deemed LOS; otherwise, the signal is classified as NLOS. Multipath error is closely related to the position of satellites. Satellite tracks exhibit periodicity over time, which leads to similar periodic variation in the corresponding multipath errors. According to this characteristic, sidereal filtering (SF) is employed to mitigate multipath effects (D’Antonio et al., 2021; C. Liu et al., 2021). Considering the spatial repeatability of multipath, Dong et al. (2016) proposed to establish a multipath hemispherical map (MHM) by using satellite altitude and azimuth as independent variables. This method is easy to implement and appears to effectively mitigate multipath in a stable environment (Lu et al., 2021). Recently, the three-dimensional (3D) matching algorithm (3DMA) has been shown to have a positive effect on mitigating multipath interference, with the development of a 3D architectural model for urban environments (Anat Schaper & Schön, 2022; Ng, 2022; Ng et al., 2021). This algorithm is utilized to support GNSS real-time kinematic positioning, attaining sub-10-m positioning accuracy in an urban setting (Ng et al., 2021). Although software-based methods are widely used for mitigating the multipath of fixed antennas, they usually become unstable when applied to changing environments. In addition, the 3DMA, SF, and MHM approaches all have the problem of high computational complexity, presenting challenges in utilizing these models for real-time multipath mitigation.

Machine-Learning-Based Methods

With the rapid development of computer science, it is possible to learn representations from existing data using computational models. Machine-learning algorithms are based on big-data-driven approaches for learning representations, and their multipath detection performance is better than that of traditional methods. Some traditional machine-learning models, such as the support vector machine (SVM), random forest (RF), and decision tree (DT), have achieved preliminary results in the field of multipath/NLOS signal recognition (Kim et al., 2022; Qin et al., 2022; Wang et al., 2021). Qin et al. (2022) proposed using SVM to classify GNSS signals based on features such as the C/N0, satellite elevation, and pseudorange residual of BeiDou Navigation Satellite System (BDS) signals into LOS, multipath, and NLOS categories and found that the multi-feature classification method achieved the best performance. Kim et al. (2022) used five features collected by dual-polarized antennas, including the C/N0, temporal difference of C/N0, and elevation angle of the satellite, and input the features into gradient-boosted DT, RF, DT, and k-means models to classify multipath signals. Their approach achieved high accuracy in the test set, but exhibited a significant decline in performance for data in different locations. Although using well-trained machine-learning models can effectively reduce computational complexity, their performance rapidly deteriorates in constantly changing environments, with difficulties in effectively extracting the features of complex multipath interference signals.

DNN-Based Methods

With the rise of artificial intelligence, DNNs have brought new opportunities for GNSS multipath signal detection arising from their substantial advantages in extracting hidden features from complex data (Blais et al., 2022; Geragersian et al., 2022; Gonzalez et al., 2022; Quan et al., 2018). Quan et al. (2018) applied the sequential C/N0 features of GNSSs to a convolutional neural network for multipath mitigation. A deep convolution neural network has been constructed to learn multipath information contained in satellite signal correlation images (Blais et al., 2022; Gonzalez et al., 2022). Geragersian et al. (2022) proposed an LOS/NLOS detection method based on the gated recurrent unit using five features, including the pseudorange, elevation, and Doppler shift, as input to a neural network. Then, the authors inputted the satellite signal classified as LOS into the positioning estimation to improve accuracy. Unfortunately, the visibility of a satellite is closely related to the environment, but the above methods do not consider the impact of the environment on multipath errors. Recent studies have focused on learning environmental information to improve the generalization ability of models. Such studies have used long short-term memory networks and transformer architectures to learn environmental representations from measurements of captured satellites in the same epoch and have combined fully connected neural networks to predict satellite visibility (Li et al., 2023; Zhang et al., 2021). However, most DNN architectures are designed for a fixed shape of inputs supplied in a pre-determined order. When the above-described DNN-based methods are applied to satellite measurements with varying input number, they usually need to pad zero to maintain the same length of data, which will weaken the learning capability of the model. In addition, improvements are still needed in the generalizability of the above-described DNN models.

To address the challenges in the above-mentioned multipath detection methods, this study uses a GNN with a multihead attention mechanism to learn environmental representations for improving satellite visibility prediction. GNNs have the advantage of being able to flexibly accept data structures with variable inputs and diverse attributes. Moreover, because GNNs are permutation-invariant, the order among input nodes does not affect the prediction results. With this property, GNNs are well-suited for modeling GNSS measurements, which may have varying numbers of satellites in different epochs. In addition, by introducing a multihead attention mechanism in the GNN, our method can adequately extract environmental representations, which is key for strengthening the generalization ability of the model.

3 BACKGROUND OF GNNS

GNNs have recently emerged as a powerful class of deep learning architectures for analyzing complex irregular structured (graph-structured) data sets. GNNs usually have connected graphs G = (V, E) as the input, where V represents the nodes and E represents the edges. An adjacency matrix A describes the relationships between nodes based on their connections. The adjacency matrix A is generated as follows: if there is a connection between two nodes, then the corresponding value in the adjacency matrix is set to 1; otherwise, the corresponding value is set to 0. In the process of forward propagation, the GNN aggregates the information of neighboring nodes through an adjacency matrix. The original structure of the graph convolutional network (GCN) is shown in Figure 2, and its corresponding update formula is as follows:

H(l+1)=σ(D˜−12A˜D˜−12H(l)W(l))1

where σ is an activation function and H(l) represents the node features of the l-th layer. à = A + I, A is the adjacency matrix, and à is the adjacency matrix with self-connection. To normalize Ã, a degree matrix D˜ is generated by Ã, where D˜(i,i)=∑j=1nA˜(i,j). W(l) refers to the trainable weight in the l-th GCN layer.

FIGURE 2
FIGURE 2

Structure of the GCN (Kipf and Welling, 2016)

The general form of Equation (1) can also be expressed as follows:

hvk+1=σ(aggregate{huk}Wk+bk),u∈N(v)2

where hvk is the feature vector of the central node v at the k-th layer and huk is the feature vector of the adjacent node u at the k-th layer. Wk denotes the projection matrix, and bk denotes the bias vector. N(x) is the set of adjacent nodes of x. The term aggregate indicates the aggregation function; existing research generally selects the sum, maximum, or mean for aggregation. However, there are also some novel methods of node feature aggregation. In this study, we use the multihead attention mechanism in transformer architecture for aggregation, which is discussed in Section 4.2. After learning useful features through multi-level updating, the GNN can perform regression prediction or classification on the node, edge, and graph:

Y=fout({h1,h2,…,hn})3

where hi is the feature vector of the node and the output mapping fout depends on the specific task.

FIGURE 3
FIGURE 3

A simple graph and representation of the adjacency matrix (a) Nodes and edges in the graph (b) Adjacency matrix

4 PROPOSED METHOD

In this section, we describe our approach of using a GTNN to predict satellite visibility. First, we formulate the rules for constructing a graph on satellite measurements that are important inputs for GNNs. Then, we describe the architecture and training process of our GTNN, which employs a multihead attention mechanism to strengthen the ability to learn environmental representations. Figure 6 shows the overall architecture of the proposed GTNN.

4.1 Construction of a Sky Satellite Graph

In this subsection, we introduce the construction of a sky satellite graph G = (V, A) with node feature vector Vi and adjacency matrix Aij. First, the selection of the satellite-related feature vector is discussed, and then, the rules for connecting edges are defined.

Satellite Feature Selection

Various features can be derived from raw GNSS receiver data for analysis. Following Zhang et al. (2021) and Li et al. (2023), we utilized four features associated with multipath detection as the information representation of the satellite nodes. These features include the elevation angle, azimuth angle, C/N0, and pseudorange residual.

  • Elevation Angle. The elevation angle of a GNSS satellite is a critical determinant of signal quality. The signal strength and quality of low-elevation satellites are often poor because of interference from buildings, trees, or other obstacles. Thus, high-elevation satellites are typically prioritized in positioning calculations by GNSS receivers to minimize signal degradation. Moreover, the elevation angle can impact satellite visibility in different areas; satellites with low elevation angles may not be visible in urban canyons. Figure 4 shows that the majority of LOS satellites are concentrated at higher elevations, highlighting the importance of satellite elevation angles for successful GNSS applications.

  • Azimuth Angle. The azimuth angle is a crucial environmental parameter that, together with the elevation angle, describes the satellite orientation. As illustrated in Figure 4(left), when the azimuth angle is between 135° and 200°, this corresponds to a relatively low elevation angle for satellite 30, which is an LOS satellite. Moreover, satellites 8 and 9 in this range are also LOS satellites, suggesting that the direction is highly likely to have an unobstructed view. The same trend can also be observed in the range of 225° to 315° in Figure 4(right). By including the azimuth angle as a node feature, our model can learn about the degree of environmental obstruction in a particular direction, given that it takes the sky satellite graph of a specific epoch as input.

  • Carrier-to-Noise Ratio (C/N0). The feature C/N0 directly reflects the quality of the GNSS satellite. A number distribution of LOS and NLOS satellites based on C/N0 in the collected data sets is shown in Figure 5. It is evident that most LOS satellites have relatively high C/N0 values, whereas NLOS satellites are distributed in positions with relatively low C/N0 values. This trend can be attributed to the attenuation of satellite signals when they are blocked and reflected.

  • Pseudorange Residual. Pseudorange residuals are a critical metric for assessing the quality of GNSS satellite signals. The pseudorange residuals reflect multipath errors induced by environmental factors. The iterative least-squares estimation method, commonly used to determine GNSS receiver positions, is expressed as follows:

    Δx=(HTH)−1HTΔρ4

    where Δx includes the bias from the initial guess and the receiver clock. H is usually referred to as a “geometry matrix.” Δρ is the vector of the difference between pseudorange measurements and the geometric distance from the initial guess to the satellites. All available measurements are iteratively optimized to enhance accuracy. The convergence is determined by the pseudorange residual as follows:

    ε=Δρ−H⋅Δx5

    Generally, a larger absolute value of the pseudorange residual indicates poorer satellite observation quality and a higher likelihood of the signal being NLOS.

FIGURE 4
FIGURE 4

Sky plots captured at two different epochs with the satellite visibility labeled from ground truth

B denotes BDS satellites, G denotes GPS satellites, and A denotes GAL satellites.

FIGURE 5
FIGURE 5

Distribution of C/N0 and proportion of NLOS/LOS signals

Rules for Connecting Edges

In a graph, nodes gather neighborhood information by exchanging messages through their edges. To facilitate the use of collected GNSS satellite measurements as input for GNNs, we establish rules for connecting satellite nodes in the same epoch based on their neighborhood and constellation attributes. This enables the network to learn features from the satellite data.

  • Neighborhood Attributes. In a GNSS measurement epoch, close satellites tend to share similar properties and can provide each other with valuable information. For instance, in Figure 4(left), satellites 15 and 29 are in proximity, and if satellite 15 is an LOS signal, the probability of satellite 29 being an LOS signal is high. To enable our model to learn such relationships, we establish edge connections based on neighborhood attributes of the satellites. Assuming that the satellites in the same epoch are distributed on a hemisphere centered at the receiver, elevation and azimuth angles are used to represent their positions on the sphere. To calculate the distance between two satellites on the sphere, their position from the elevation and azimuth angles are first converted to 3D Cartesian coordinates:

    {z=|r⋅sin(El)|y=|r⋅cos(El)|⋅sin(Az)x=|r⋅cos(El)|⋅cos(Az)6

    where (x, y, z) denotes the 3D coordinates in Euclidean space, r denotes the radius of the mapped sphere, and El and Az denote the elevation and azimuth angles, respectively. Next, 3D Euclidean coordinates are used to calculate the spherical distance between the two satellites. Let Pi(xi, yi, zi) and Pj(xj, yj, zj) represent the positions of two satellites in space. Using the cosine formula, we can obtain the following:

    cosα=xixj+yiyj+zizjxi2+yi2+zi2·xj2+yj2+zj27

    where α represents the angle between the P1 and P2 vectors. Because P1 and P2 are on the sphere, we have the following relationship:

    {S=α⋅rr=xi2+yi2+zi2=xj2+yj2+zj28

    where S denotes the spherical distance between two points. Combining Equations (7) and (8), we finally obtain the following:

    S=r⋅arccos(xixj+yiyj+zizjr2)9

    In this study, the unit sphere is taken as the standard, i.e., r = 1. When the S value between two satellites is less than a threshold γ, we assign an edge connection to the two satellites.

  • Constellation Attributes. Satellite receivers have the ability to receive signals from various constellations, such as the Global Positioning System (GPS), BDS, and Galileo satellite navigation system (GAL). However, these constellations are distributed in orbits of different altitude. For instance, the GPS is spread over six track planes with an average altitude of 20,200 km, whereas the BDS is spread over three track planes with an altitude between 20,000 km and 36,000 km. Consequently, for multi-constellation GNSS measurement sets collected at the same epoch, data from the same constellation have similar properties. To enable the model to learn information about the same constellation, we establish connections between satellites within the same constellation.

Finally, the sky satellite graph rule that we have constructed can be expressed as follows:

Aij={1,ifviandvj∈CorSij<γ0,otherwise10

where C refers to the constellation attribute and v denotes the satellite nodes on the sky satellite graph.

4.2 GTNN for Improving Satellite Visibility Prediction

To determine satellite visibility from the constructed sky satellite graph, we develop a GTNN based on GNNs and the multihead attention aggregation method. The architecture of the GTNN is presented in Figure 6. The model’s output is the visibility of satellites captured at the same epoch. We begin by explaining the satellite vector embedding layer. Then, we describe how the adjacency satellite-related features are aggregated via the multihead attention mechanism. Finally, the construction of the cross-layer residual concatenating branch is demonstrated.

FIGURE 6
FIGURE 6

Architecture of the proposed GTNN for improving satellite visibility prediction The satellite node vector is represented by four features: elevation angle, azimuth angle, C/N0, and pseudorange residual. Edge connnections between satellites in the sky satellite graph are established when satellites either have a close spherical distance as determined by a threshold or belong to the same constellation. The graph is provided as input for the model to predict the visibility (LOS or NLOS) of satellites in the sky. The entire architecture is composed of an embedding layer for projecting, five GAblocks with multihead attention aggregation, and two cross-layer concatenating branches to alleviate over-smoothing.

Embedding

The input data of the model consist of the sky satellite graph G = (V, A) constructed in Section 4.1. An embedding layer will project the node features into a high-dimensional space to facilitate better learning of the model at the initial stage. The calculation formula of an embedding layer is similar to that of a linear projecting layer, as follows:

ℋ=Embedding(Vi)=ReLu(ViWe)11

where ℋ∈ℝn×64 denotes the output of the embedding layer and Vi∈ℝn×4 denotes the original node features. We∈ℝ4×64 is a linear projecting matrix, and RelU represents a nonlinear activation function, which ensures that the output of the model is positive (Varshney & Singh, 2021).

Graph Attention (GA) Block

To facilitate effective interactions between central node features and adjacent node features, a GAblock with the multihead attention mechanism in the transformer architecture (Vaswani et al., 2017) is constructed for node aggregation. Specifically, given a node feature set Hl={h1l,h2l,…,hnl}, the multihead attention mechanism between satellite nodes can be calculated according to the edge connections. The m-th head attention is calculated as follows:

qm,il=hilWm,ql+bm,qlkm,jl=hjlWm,kl+bm,klαm,ijl=softmax((qm,il)Tkm,ild)12

where d represents the number of neurons in the hidden layer of each attention head. First, the central feature hil and the adjacent feature hjl are transformed into a query vector qm,il∈ℝd and a key vector km,il∈ℝd, respectively, using different trainable parameters Wm,ql,Wm,kl,bm,ql,bm,kl. Here, an adjacency matrix containing edge information must be taken as an additional input. The attention weight matrix αm,ijl is obtained by applying the softmax function to the result of query vector transpose and key vector multiplication.

After the multihead attention mechanism operation of the graph is complete, the information learned by the multiple heads is fused together for message passing, so as to obtain the update result of the next layer:

vm,jl=hjlWm,vl+bm,vlhil+1=hilWcl+‖m=1M(∑j∈N(i)αm,ijl·vm,jl)13

where || denotes the operation of tensor concatenating and M is the total number of heads. Wcl is used to balance the central satellite features and the neighborhood satellite features. The obtained attention weight is multiplied by the learned value vector vm,jl, and the operation of multihead information concatenation is then carried out to obtain updated information for the central node after the aggregation of neighboring nodes. The traditional simple aggregation approach, such as the sum, mean, or maximum approach, is replaced with the multihead attention mechanism, which enables the model to learn the neighborhood environment representation of the central satellite by attending to the state of adjacent satellites. This design is beneficial for the task of predicting satellite visibility.

Cross-Layer Residual Connection

To enhance the learning ability and generalization performance of the model, five GAblocks are stacked to build the GTNN. As shown in Figure 7, the central satellite node can only learn the environment representation of the first neighborhood through a single-layer GAblock (Hamilton et al., 2017). When multi-layer GAblocks are used to build the model, after multiple steps of message passing and aggregation between nodes, each satellite node can learn global information, i.e., the environmental representation of the current epoch. However, node representations become indistinguishable (known as over-smoothing) and prediction performance is severely degraded when we stack many layers (Oono & Suzuki, 2019). Therefore, a cross-layer residual concatenating branch is introduced as follows:

Hl+1=GAblock(Concat(Hl,Hl−2))14

where l ∈ 3, 4. Before the data are inputted to the later GAblocks, we concatenate the information of the previous layer with the current input, so that the later layer of the model will not lose the original characteristics of the input features and the network can be stronger in generalization.

FIGURE 7
FIGURE 7

Red represents the central satellite node, the dotted circle represents the k-th neighborhood, and different colors correspond to different neighborhood satellite nodes. Nodes with a direct edge connection with the central node are called first neighborhood nodes. The first layer of the GTNN can only learn the representation of the first neighborhood, but when the number of layers is sufficient, the central node can aggregate to the features of the global nodes.

Loss Function

The aim of this work is to predict the visibility of satellites, i.e., to classify LOS and NLOS satellites. Therefore, to effectively train our model, we adopt the binary cross entropy as the loss function:

ℒloss=−1N∑i=1N(yilog(pi)+(1−yi)log(1−pi))15

where y denotes a binary label of 0 or 1, 0 denotes the LOS signal, 1 denotes the NLOS signal, and p denotes the prediction probability of the NLOS signal. This loss function is often used in binary classification. The closer the prediction result is to 1, the smaller the loss function value is.

5 EXPERIMENTAL STUDIES

In this section, real-world measurements are used to validate the performance of our network. To demonstrate the generalizability of the proposed method, we compare the accuracy of our proposed method with that of various SOTA methods, such as transformer (Li et al., 2023), multilayer perceptron (MLP) (Min et al., 2022), and other machine-learning-based methods. We discuss the environment representations learned by our model in Section 5.4 by analyzing the attention weights between satellites. We first present details regarding the collection of real-world measurements and plot the data distribution of the collected data sets. Then, the experimental parameters utilized in implementing the model are introduced. Next, we present the performance of the GTNN using GNSS measurement data collected in urban canyons in comparison with the performance of SOTA methods. Finally, we analyze the attention weights between adjacent satellites and discuss the impact of the number of GAblocks and the spherical distance threshold on classification accuracy.

5.1 GNSS Data Set Construction

GNSS measurement data sets collected in several urban environments in Guangzhou were used to train the proposed model. The process of constructing data is illustrated in Figure 8. GNSS measurements were collected by using two U-blox F9 high-precision GNSS receivers, two GNSS antennas, a fisheye camera, and a laptop for storing the original data of the satellite. A differential GNSS satellite signal acquisition method was used to obtain accurate measurement data. First, we collected medium-Earth-orbit satellite signals from BDS, GPS, and GAL constellations at a frequency of 1 Hz in an urban canyon environment. Satellite measurements captured in the same epoch from the receiver were obtained, including the four features used in this paper: elevation angle, azimuth angle, C/N0, and pseudorange residual. Then, the sky image taken by the fisheye camera was binarized into a sky segmentation map to mark LOS/NLOS labels using the elevation angle and azimuth angle. Finally, we used the GNSS measurements and labels to construct a sky satellite graph.

FIGURE 8
FIGURE 8

The process of collecting GNSS measurements and constructing training and testing data

We collected GNSS data from ten locations within urban canyons and under overpasses. The GNSS measurements were taken from five urban canyon locations Ci (i = 1, 2, 3, 4, 5) and five overpass locations Oi (i = 1, 2, 3, 4, 5), each for a duration of half an hour. To illustrate the data collection environments, corresponding sky images for these ten locations are presented in Figure 9. The images reveal that most of the sky is obscured by tall buildings or overpasses, indicating a challenging environment for signal reception. In such obstructed scenarios, the receiver predominantly captures NLO signals.

FIGURE 9
FIGURE 9

Sky images for 10 collection locations

Ci (i = 1, 2, 3, 4, 5) represent locations in urban canyons, and Oi (i = 1, 2, 3, 4, 5) represent locations under an overpass.

After data collection, we use Equations (6)–(10) to construct a sky satellite graph as the input of our GTNN. Table 1 lists the numbers of LOS satellites, NLOS satellites, and sky satellite graphs. It can be seen that the numbers of NLOS and LOS satellites in each data set are relatively balanced, which can prevent the output from leaning toward a certain category.

View this table:
TABLE 1

Number of NLOS/LOS Samples and Corresponding Graphs for the Collected GNSS Data

By feeding a sky satellite graph into the GTNN, we can obtain a series of visibility probabilities between 0 and 1, which represent the confidence level of NLOS for each satellite at the current time. In practice, the weight of a satellite in positioning estimation can be determined based on its NLOS confidence, which is an effective way to improve positioning accuracy.

5.2 Experimental Setup

In this section, we first introduce the data set settings of the collected GNSS measurements and the implementation details for training and testing.

Data Set Settings

We divide the collected GNSS measurements into three scenarios: “urban canyon scenario,” “overpass scenario,” and “hybrid scenario.” The details of the training and testing sets for the three scenarios are as follows:

  • Urban canyon scenario. The first 60% of the data collected at C1, C2, C3, and C4 were used as the training set. The next 20% of data (from 60% to 80%) from these locations were designated as Ctest1, and the remaining 20% of data were designated as Ctest2. The data collected at C5 were designated as Ctest3. Ctest1 and Ctest2 were used to validate the performance of the NLOS detection model over time. Additionally, Ctest3 was used to validate the model’s performance in a different location.

  • Overpass scenario. Similar to the urban canyon scenario, the first 60% of the data collected at O1, O2, O3, and O4 were used as the training set. The next 20% of data (from 60% to 80%) from these locations were designated as Otest1, and the remaining 20% of data were designated as Otest2. The data collected at O5 were designated as Otest3.

  • Hybrid scenario. We combined the data from two scenarios to create a training set and tested the model separately in each scenario. Specifically, the urban canyon data collected at C1, C2, C3, C4 and the overpass data collected at O1, O2, O3, O4 constituted the training set. The data collected at C5 were used as COtest1, and the data collected at O5 were used as COtest2.

Implementation Details

Our experiments were conducted on an NVIDIA GeForce RTX 3090 with a batch size of 256 for both training and testing. Adaptive moment estimation was employed to optimize the gradient update, and the initial learning rate was set to 0.01. The total training process consisted of 200 epochs, and the learning rate was decayed to its original value of 0.2 every 50 epochs. We adopted one embedding layer and five GAblocks to construct our GTNN. The dropout rate was set to 0.3.

5.3 Satellite Visibility Prediction Results

To demonstrate the effectiveness of our GTNN, we fed the training sets into the proposed model and the SOTA methods for training and compared their satellite visibility prediction accuracy on the testing sets. A total of five SOTA models were considered: the transformer-based model (TBM) (Li et al., 2023), stacked fully connected neural network (MLP) model (Min et al., 2022), DT model (Wang et al., 2021), RF model (Kim et al., 2022), and SVM (Qin et al., 2022). Based on the data set settings in Section 5.2, we conducted three sets of experiments in different scenarios to evaluate the generalization capacity of the proposed method.

Results in the Urban Canyon Scenario

The test results of the GTNN and SOTA methods in the urban canyon scenario are shown in Figure 10. The proposed GTNN achieves 96.29% prediction accuracy on Ctest1, which is 6%–25% higher than other machine-learning-based methods. On the data set Ctest2 for different time periods, the accuracy of our method decreased slightly, but still reached 86.15%. On Ctest3, the prediction accuracy of all models decreased to different extents because of the large differences in the environment; however, the GTNN still achieved the best performance among all of the methods, indicating that the proposed method has a more robust generalization performance. Overall, the GTNN and TBM have outstanding performance in the three test sets because of the addition of the attention mechanism and the consideration of environmental changes, whereas the other methods achieved accuracies of only 65%–80%. Furthermore, compared with the TBM, the GTNN has the highest accuracy in the three test sets because it sufficiently learns environmental information from the sky satellite graph. Therefore, our model’s performance is superior to that of the SOTA methods, benefiting from the graph learning mode and attention-based aggregation, which enhance its generalization power on different test sets.

FIGURE 10
FIGURE 10

Performance comparison between the proposed method and existing multipath detection methods for the urban canyon scenario

Results in the Overpass Scenario

Experimental results for the overpass scenario are presented in Figure 11, which shows that our proposed GTNN method achieved the best performance across all three test sets. In Otest1, our method outperformed the other methods by 4.35%–11.36%. The accuracies in Otest2 show that the performance decreased over time, likely because of satellite movement. On out-of-domain data (Otest3), the accuracy of the other methods fell below 80%, whereas our method maintained an accuracy of 81.85%, demonstrating its robustness and strong generalization capability.

FIGURE 11
FIGURE 11

Performance comparison between the proposed method and existing multipath detection methods for the overpass scenario

Results in the Hybrid Scenario

To evaluate the performance of our model, we combined data from the two scenarios for training and tested the model separately in COtest1 and COtest2. The experimental results for these two testing sets are presented in Figure 12. The proposed GTNN demonstrated superior performance in both scenarios. Additionally, the accuracies in the overpass scenario were higher than those in the urban canyon, likely because of the more complex surrounding environment in urban canyons. The accuracy of our method in COtest1 was 7.32% higher than in Ctest3, and its accuracy in COtest2 was 1.92% higher than in Otest3. These results indicate that a rich training data set can enhance the generalization performance of the GTNN.

FIGURE 12
FIGURE 12

Performance comparison between the proposed method and existing multipath detection methods for the hybrid scenario

5.4 Attention Weight Analysis

To further illustrate the representations learned from the sky satellite graph by our GTNN, the attention weights of some satellites are visualized. The connections between the central satellite 29 and adjacent satellites are shown in Figure 13(a), and the corresponding middle-layer weight values are shown in Figure 13(c). Similarly, the connections between the central satellite 12 and adjacent satellites are shown in Figure 13(b), and the corresponding middle-layer weight values are summarized in Figure 13(d). From the weight values, it can be seen that the weight changes between different layers are relatively large, because the previous layers and the deep layers focus on different content in the process of learning. Because the weight of the last layer directly affects the classification results, its weights are taken for analysis. For convenience, we use abbreviations for the satellites, such as S29 for satellite 29. The analysis of Figure 13 is as follows:

  • Among the weights between S29 and its neighborhood satellites, the weights for S30, S4, and S23 are relatively large. Although S12 and S15 are also close to S29, the weights for these two satellites are the smallest because their NLOS/LOS labels are different. Among the weights between S12 and its neighborhood satellites, the weights for S19, S32, and S15, which have the same label and a close distance, are large. Although S2 is close to S12, the weight for S2 is small because these two satellites have different NLOS/LOS labels. The above phenomena indicate that our model learned the NLOS/LOS information hidden in satellite-related features and the distance information between satellites to determine the contribution of neighboring nodes and to promote the process of learning from each satellite. In addition, from the distribution of LOS and NLOS satellites on both sides of S12, it can be seen that the left neighborhood tends to be open, whereas the right neighborhood tends to be occluded. Although S5 is an LOS signal, it is distributed in the right neighborhood of S12; thus, its weight with S12 is greater than that of other LOS satellites. This finding indicates that the GTNN can explore the environmental representation of the neighborhood by ensuring that the central node utilizes the distributions of NLOS and LOS satellites in different directions.

FIGURE 13
FIGURE 13

Connection and attention weights of a sky satellite graph in GNSS measurements (a) Neighborhood connectivity of satellite 29. (b) Neighborhood connectivity of satellite 12. (c) Attention weight of satellite 29 and adjacent satellite. (d) Attention weight of satellite 12 and adjacent satellite.

5.5 Ablation Study

Number of GAblocks

The number of GAblocks is an important parameter in the model. If the depth of the GNN is too small, the representation ability of the model will be affected; however, if the number of layers is too large, over-smoothing problems can easily arise. To determine the optimal number of GAblocks, we varied the number of GAblocks from 2 to 7. When the number was 2 or 3, we did not use the cross-layer residual connection. The test performance for different network depths is shown in Figure 14. When the network depth is 5, the accuracy is relatively high on the three test sets. As the depth increases further, the performance of the model tends to decline. Therefore, five GAblocks are the best choice for our model.

FIGURE 14
FIGURE 14

Accuracy of the proposed GTNN with different numbers of GAblocks in the three test sets

Spherical Distance Threshold γ

To construct the sky satellite graph, we must determine the spherical distance threshold γ between two satellites. If γ is too small, satellite information in the neighborhood cannot be obtained; in contrast, if γ is too large, interference from distant satellites will be introduced. To obtain the best threshold, numerical values between 0.5 and 1.1 were selected to construct the sky satellite graph training model. The results for different thresholds γ on three test sets are shown in Figure 15. When γ = 0.8, the accuracy rate for the three test sets is better, and the constructed graph allows the model to achieve optimal performance. Therefore, in this paper, the value of γ is taken as 0.8.

FIGURE 15
FIGURE 15

Accuracy of the proposed GTNN with a range of γ values in the three test sets

5.6 Discussion

In the experiment section, we validated the proposed GTNN in comparison with SOTA multipath detection methods using real-world measurements and visualized the attention weights between satellites. Based on the obtained results, the effectiveness of the proposed method can be discussed from the following two aspects:

  • Improvement of Generalization Ability in Multipath Detection.

    Section 5.3 presents a performance comparison with SOTA multipath detection methods to validate the generalization ability of the proposed GTNN. As shown in Figures 10 and 11, the proposed GTNN can achieve better performance than the SOTA methods in the three test sets. In the urban canyon scenario, the accuracy of our method is 5%–14% higher in the data collected at different times and 3%–6% higher in the data collected in different locations, compared with the other methods. These results demonstrate that our method has superior generalization ability owing to its advantage in modeling complex irregular GNSS measurements. The accuracy of all methods decreased to varying extents in Ctest2 (or Otest2) and Ctest3 (or Otest2). The degradation of prediction performance in Ctest2 (or Otest2) is most likely due to changes in the environment around the receiver over time, whereas the degradation of prediction performance in Ctest3 (or Otest2) may be attributed to the difference in building style around the receiver. Furthermore, the proposed GTNN, which combines a GNN and a transformer architecture, can achieve better accuracy than the TBM, indicating that the GTNN can learn more features from irregular GNSS measurements by constructing a sky satellite graph.

  • Effective Environment Representations Learned by the GTNN. The attention weights between satellites in a graph have been analyzed in Section 5.4. From the results shown in Figure 13, we can see that the attention weights are greater between satellites with the same label and close distance. Moreover, LOS satellites tend to obtain information from satellites with large altitude angles and satellites in open directions because of the greater weight between them. These results show that the proposed GTNN can learn environment representations hidden within GNSS measurements captured in the same epoch to improve the visibility prediction of center satellites, which may be attributed to the role of the attention mechanism as an aggregation method to promote the learning of relationships between satellites.

6 CONCLUSION AND FUTURE WORKS

In this paper, we presented a method for predicting satellite visibility in urban canyons based on a GNN with a multihead attention mechanism, i.e, a GTNN. In contrast to existing NLOS detection methods, the proposed method focuses on learning environmental representations by utilizing a graph structure and the multihead attention aggregation method, thus promoting interactions between satellite vectors. To learn hidden information in irregular GNSS measurement sets, we proposed a construction rule for the sky satellite graph, where the satellites are represented as nodes of the graph and the connections between pairs of satellites are determined from both neighbor attributes and constellation attributes. To improve the generalization performance of the model, we introduced a satellite node aggregation method based on the multihead attention mechanism, which focuses on the similarity between nodes via multihead attention, so as to determine relationship weights between nodes.

We validated our proposed method on real-world GNSS measurements by comparing our method with SOTA multipath detection methods in three different scenarios. The results show that the proposed GTNN has good generalization performance for data collected in different periods and locations, compared with existing methods for satellite visibility prediction. By visualizing the attention weights of some satellites, we found that the GTNN can learn environment representations from existing GNSS measurements, such as the openness of the direction and satellite distance information. Moreover, we determined that the model achieves optimal performance when the number of GAblocks is 5 and the spherical distance threshold γ is 0.8.

To the best of the authors’ knowledge, this study is the first to apply a GNN approach to the field of multipath detection. However, our experimental results show that the generalization ability of our method in different locations still has room for improvement. In the future, we plan to introduce satellite-related features in the time dimension to further improve the generalization performance of our model by constructing a spatiotemporal satellite graph.

HOW TO CITE THIS ARTICLE

Zheng, S., Zeng, K., Li, Z., Wang, Q., Xie, K., Liu, M., & Xie, S. (2024). Improving the prediction of GNSS satellite visibility in urban canyons based on a graph transformer. NAVIGATION, 71(4). https://doi.org/10.33012/navi.676

CONFLICT OF INTEREST

The authors declare that they have no conflicts of interest.

ACKNOWLEDGMENTS

This research was supported in part by the National Natural Science Foundation of China under grants 62273106, 62203122, and 62320106008 and in part by the GuangDong Basic and Applied Basic Research Foundation under grants 2023A1515011480 and 2023A1515011159.

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 the Prediction of GNSS Satellite Visibility in Urban Canyons Based on a Graph Transformer
Shaolong Zheng, Kungan Zeng, Zhenni Li, Qianming Wang, Kan Xie, Ming Liu,, Shengli Xie
NAVIGATION: Journal of the Institute of Navigation Dec 2024, 71 (4) navi.676; DOI: 10.33012/navi.676
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