An SVM is a supervised learning tool that generates input–output mapping functions from a set of labeled training data (Wang, 2005). In this study, the SVM classifier outputted LOS and NLOS signals and defined LOS signals as negative and NLOS signals as positive. We implemented an SVM classifier using Scikit-learn (Pedregosa et al., 2011), an open-source machine learning library for the Python programming language. The SVM classifier implemented by Scikit-learn requires tuning of the parameters called “hyper-parameters.” We chose a radial basis function kernel (RBF), which is generally used for nonlinear classification, while the other parameters were selected using a grid search (Min and Lee, 2005).

Xu et al. (2020) previously used four features—the signal-to-noise ratio (SNR), elevation angle (EA), normalized pseudorange residual (NPR), and pseudorange rate consistency (PRC). As for the SNR and EA, it is well known that both are closely related to the NLOS signal (Tokura and Kubo et al., 2017). SNR can be obtained by RINEX, and EA can be calculated by the satellite position estimated by the ephemeris and the approximate user position.

The pseudorange residual is valid for use with machine learning (Hsu, 2017). It is calculated by the least-squares method and usually becomes large for NLOS signals. However, the positioning result from the least-squares method has large errors under multipath environments and the pseudorange residual does not perfectly indicate the difference between LOS and NLOS signals. For this reason, the pseudorange residual was normalized at each epoch. The NPR can be expressed as N P R s = P r s − P r m i n P r m a x − P r m i n , where P r m a x and P r m i n were calculated by the maximum and minimum pseudorange residuals at each epoch, respectively, and P r s is the pseudorange residual of satellite s .

PRC is defined as the difference in the changing rate of pseudorange from the pseudorange and Doppler shift (Hsu, L. (2017). The changing rate of pseudorange from the pseudorange can be expressed as Δ P = P ( t i ) − P ( t i − 1 ) , where P ( t i ) is the pseudorange in the i th epoch. The changing rate of the pseudorange can also be calculated by the Doppler shift as Δ ρ = ( − λ s f s ) Δ t , where λ s is the wavelength of satellite s , f s is the Doppler shift of satellite s in unit of Hz, and Δ t is the time difference of t i and t i − 1 . PRC can be expressed as P R C = Δ P − Δ ρ

The pseudorange can be estimated by the receiver-code tracking loop, and the Doppler shift can be estimated by the frequency tracking loop in the GNSS receiver. For an LOS signal, the PRC is 0, but for an NLOS signal, the PRC is not always 0 because the receiver-code tracking loop is more affected by multipaths than by the frequency tracking loop.

Previous studies have shown that there are more methods other than just these four to characterize NLOS signals. Tokura and Kubo (2017) proposed an SNR-based satellite selection method that uses the magnitude of variability in the SNR (hereafter, the “SNR fluctuation magnitude” or SFM). Because NLOS signals exhibit large fluctuations in their SNR time series, the SFM is an indicator of the magnitude of such fluctuations. Tokura and Kubo (2017) calculated the SFM using the moving standard deviation of the difference between the observed and estimated SNRs, which sets the elevation-dependent threshold of the SNR (Suzuki et al., 2004; Kubo et al., 2005; Shirai & Kubo, 2012). The disadvantage of this approach is that it is difficult to set this threshold. Instead, we calculated the SFM in this study by moving the standard deviation of the SNR, expressed as S F M ( i ) = 1 N ∑ n = i − N i ( S N R ( t i − n ) − S N R m e a n ) 2 , where N is the window size and S N R ( t i − n ) is the SNR in the i − n th epoch and S N R m e a n is the mean of the SNR. Figure 1 shows a comparison of the LOS and NLOS SFM time series in a multipath environment. The window size was set to 120 epochs. The LOS of the mean residual was 1.56 m, but that of the NLOS was 55.90 m. These results indicate that SFM can be a good indicator of LOS versus NLOS signals.

We generated a classifier with five features—SNR, EA, NPR, PRC, and SFM. For the SFM, the window size for calculating the moving standard deviation influenced the accuracy of the SVM classifier. In this study, we set various window sizes (30, 60, 90, 120, 180, and 240 epochs) and considered the best window size for the accuracy of the SVM classifier. The SVM classifier was then used to classify all signals that could be used for positioning at each epoch.

Three datasets were used in this study; Figure 2 shows the data collection locations. The area around Tokyo Station is surrounded by skyscrapers over 100 m in height, and the DGNSS positioning error easily reached 100 m. Locations A and B are marked in Figure 2. For all datasets, a U-blox F9P and a standard patch antenna (ANN-MB-00–00) installed on the roof of a parked car (U-blox, Switzerland) were used. In the reference station, a U-blox F9P was used and the GNSS antenna was a Trimble Zephyr 2 Geodetic (Trimble, Inc., United States). All datasets were recorded at 1 Hz.

The upper part of Figure 3 shows detailed images around location A and the antenna. As can be seen, it was likely that NLOS signals would be received from the higher buildings at the azimuth of 290°. There were several high-rise buildings of different heights and several trees on both sides of the antenna. The lower part of Figure 3 shows detailed images around location B and the antenna. As in location A, it was also likely at this location that NLOS signals would be received from the higher buildings at the azimuth of 180°, and there were several high-rise buildings of different heights on both sides of the antenna.

Datasets (1) and (2) were collected at location A but were 50 cm apart when comparing the precise position. Dataset (3) was collected at location B. All datasets were accompanied by reference station data obtained at the Tokyo University of Marine Science and Technology Etchujima Campus located ∼3 km from Tokyo Station. Table 1 shows the dates and times when the datasets were collected. As for the reference position, we adopted an RTK-GNSS Fix solution outputted by U-blox F9P, and we confirmed that this position was correct to an accuracy of approximately 5 cm in post-processing by positioning software developed by our laboratory.

Dataset (1) and reference station data were used for training the SVM classifier, and datasets (2) and (3) were used for testing the SVM classifier. For datasets (1) and (2), the locations of data collection were almost the same, but the collection times differed. The locations and times of data collection were different for datasets (1) and (3). Therefore, these datasets were suitable for evaluating the SVM classifier.

In the testing and evaluation of our SVM classification, we estimated accurate pseudorange residuals and determined the labels for classification. Kubo et al. (2020) proposed an estimation of the precise receiver clock error and pseudorange residuals. Pseudorange measurements include errors from the receiver clock, satellite clock, ionosphere, troposphere, and multipath + noise. In pseudorange positioning, the multipath + noise error is the residual of the pseudorange P  = ρ + c ( d t rcv − d T sat ) + ion + tropo + mp + ε , where P is the pseudorange (m), ρ is the geometrical range (m), c is the speed of light (m/s),   d t r c v is the receiver clock error (m),   d T s a t is the satellite clock error (m), ion is the ionospheric error (m), tropo is the tropospheric error (m), mp is the multipath error (m), and   ε is the noise error (m).

In this study, the predicted geometrical range was set as the distance between the reference position of the antenna and the satellite position estimated by ephemeris. The satellite clock, ionospheric, and tropospheric errors are considered to be in part eliminated by the DGNSS correction data, which were calculated at the base station. Additionally, the clock bias of GPS and other satellite systems was eliminated by the DGNSS correction data. Therefore, a receiver clock was needed to estimate the multipath + noise. If the receiver clock was estimated within a few meters, then the multipath + noise can be estimated with the same accuracy. Usually, the receiver clock error was estimated by pseudorange positioning and estimation accuracy if the receiver clock errors were within several meters under open sky conditions. However, in multipath environments, the estimation accuracy of receiver clock errors deviated over tens of meters because of multipath errors. In this study, we tracked a strong signal from the Japanese QZSS even near high-rise buildings in Tokyo as at least one QZSS remained at a very high EA > 80°. In practice, the accuracy of the pseudorange of the highest satellite is within 1.0 m. Using the receiver clock error estimated with the pseudorange of the highest EA and reference positions of the antenna, the multipath + noise could be expected within 1.0 m. The NLOS signal multipath errors usually exceed 10 m. Here, if the multipath errors exceeded 10 m, they represented an NLOS signal.

Multipath errors caused by NLOS signals generally have a greater impact on the pseudorange than the carrier phase, and the accuracy of carrier-phase positioning, such as real-time kinematic, is affected by the pseudorange positioning. In this study, we adopted DGNSS positioning as it is suitable for evaluating positioning errors because it is less affected by tropospheric, ionospheric, and satellite clock errors. Table 2 shows the common parameters of our analysis.

Figure 4 shows the workflow of this study. We extracted five features (SNR, EA, NPR, PRC, and SFM) and determined labels for classification during training. To validate the SFM, we generated two SVM classifiers—SVM classifier (1), which used four features (SNR, EA, NPR, and PRC), and SVM classifier (2), which used five features (SNR, EA, NPR, PRC, and SFM). During the testing phase, signals classified as NLOS were excluded by DGNSS positioning. There were five outputs.

・Output (1) was the DGNSS position from satellites that could be observed and used to assess the effect of pseudorange residual check.

For the SFM, we set various window sizes (30, 60, 90, 120, 180, and 240 epochs) and considered the best window size from the results of the SVM classifier accuracy across all datasets.

The positioning software developed by our laboratory was used. As for the pseudorange residual check, the position and pseudorange residuals were first calculated by the least-squares method using all satellites. If the maximum pseudorange residual exceeded the threshold, the satellite with the maximum pseudorange residual was excluded. This process continued until the maximum pseudorange residual fell below the threshold or the number of satellites was insufficient (Jiang et al., 2011).

The parameters of the SVM classifier, the regularization parameter, and the kernel coefficient were set equal to 10, and the scales were selected via grid search (Min & Lee, 2005). There are four types of statistics used to evaluate binary classifications—true positive (TP), true negative (TN), false positive (FP), and false negative (FN) (Sokolova & Lapalme, 2009). In this study, accuracy was chosen as the metric for evaluating the SVM classifier. Accuracy can be expressed as Accuracy = TP + TN TP + TN + FP + FN

Table 3 shows the accuracy of the SVM classifier. Using the SFM, the accuracy of the classifier increased by >15% with the testing. Regarding the best SFM window size, there was no significant difference in the accuracy when the window size was 60 epochs or more. Therefore, we selected 120 epochs, which maximized the accuracy in dataset (2).

We evaluated the horizontal DGNSS error using six statistics: mean error, maximum error, standard deviation (SD), and the percentages of positioning errors within 3 m, 10 m, and 30 m.

To find the limitations of the proposed method, we conducted a kinematic test. The GNSS antenna was installed in a car that traveled along the route shown in Figure 7. We collected data and analyzed the DGNSS positioning. The route included skyscrapers around Tokyo Station in Tokyo, Japan, and the DGNSS positioning error easily reached several tens of meters or more in this area. The total data recording period was 1,811 s, and the data were recorded at 1 Hz. GPS-703-GGG-HV (Novatel, Inc., Canada), which is a survey-grade GNSS antenna, was used for this experiment. The reference position was obtained using POS LVX (Applanix, Inc, Canada). The other settings were the same as in Table 2.

Table 6 shows the accuracy of the SVM classifier in the kinematic test and dataset (1). Compared to the experiment under the static conditions, there was no contribution of SFM to the accuracy. This is because the positioning environment changes from time to time under the kinematic conditions, and the signal frequently changes from LOS to NLOS or NLOS to LOS. Hence, we used SVM classifier (1) in the kinematic test.

Table 7 shows a comparison of the horizontal positioning errors in the kinematic test, and Figure 8 shows the DGNSS positioning error with the DGNSS + residual check, DGNSS + SVM classifier (1), and DGNSS + SVM classifier (1) + residual check. Comparing the DGNSS and DGNSS + residual check, the pseudorange residual check was as effective as in the static test. As for DGNSS + SVM classifier (1), SVM classifier (1) was effective in positioning. However, it did not exceed the values of the DGNSS + residual check. For the combination of the DGNSS + SVM classifier (2) + residual check, all statistics were better as compared with the other methods.