To validate the effectiveness of the proposed method in improving the multipath error of GNSS-RTK, both simulated data and real GNSS-RTK data were used to analyze the feasibility and effectiveness of the new method. The simulated data contained a variety of component signals and noise mixtures, which were used to analyze the denoising effect of the proposed method and validate its effectiveness in extracting the multipath error of GNSS-RTK. The real data were obtained by continuously observing data from a GNSS receiver for two consecutive days and were used to analyze the effectiveness of the proposed method in extracting and correcting multipath error in practical applications. In this experiment, the signal-to-noise ratio (SNR), root-mean-square error (RMSE), and Pearson’s correlation coefficient (R) were used to provide a more intuitive description of the effectiveness of the proposed method in reducing multipath error.
3.1. Data Simulation
The simulation data in the simulation experiment had a sampling interval of 1 s and consisted of a total of 5000 samples. The simulation data were composed of multiple harmonic signals, and different levels of Gaussian white noise were added during the simulation process to simulate different signal-to-noise ratios. Their expression was as follows:
where
is the synthesized simulation data;
,
, and
are different frequency signals; and
is Gaussian white noise. The simulation results are shown in
Figure 5.
The following experimental methods were set up separately for comparison tests to better demonstrate the benefits of the proposed algorithm:
- (1)
Use CEEMDAN algorithm;
- (2)
Use WPTD algorithm;
- (3)
Use ICEEMDAN-AWPTD algorithm.
In this experiment, the parameters of CEEMDAN in method 1 were the same as those of ICEEMDAN in
Section 2.1. In method 2, the number of wavelet packet decomposition layers was set to five, the wavelet basis function was set to Sym6, the threshold rule was set to
, and the threshold function was set to the hard threshold function.
First, three methods were employed to denoise the noisy signal, and the processed results are shown in
Figure 6. In order to highlight the differences in the denoising results obtained with different algorithms, the residual results obtained by subtracting the denoised results of the three methods from the original signal are shown in
Figure 7.
Table 2 presents the SNR, RMSE, and R values of the CEEMDAN, WPTD, and ICEEMDAN-AWPTD algorithms after denoising. As shown in
Table 2, compared to the CEEMDAN algorithm, ICEEMDAN-AWPTD improved the RMSE by 22.8%, R by 2.8%, and SNR by 16.5%. Compared to the WPTD algorithm, ICEEMDAN-AWPTD improved the RMSE by 18.3%, R by 2.1%, and SNR by 12.4%. These results indicate that the proposed combined algorithm outperforms the single CEEMDAN and WPTD algorithms in denoising performance and has a good denoising effect. Therefore, it can be applied to extract multipath errors in GNSS observation data.
3.2. Analysis of Actual GNSS Measurement Data
The experiment used two low-cost u-blox M8T receivers. One was used as the base station, and the other was used as the monitoring station. The sampling epoch was set to 1 s, and the satellite elevation cut-off angle was set to 15°. The measured data were first processed using RTKLIB, and the coordinate residuals were obtained by taking the difference between the calculated coordinates and the known coordinates. It is generally believed that the coordinate residuals obtained using the carrier-phase double difference under an ultra-short baseline only consist of random noise and multipath error [
16].
To compare the effects of multipath errors on GNSS-RTK observations in this experiment, the monitoring station was first placed in an open area for data collection with both stations fixed in place and a baseline length of approximately 7 m. The GNSS-RTK deployment is shown in
Figure 8, and the observation results are shown in
Figure 9.
In
Figure 9, it can be observed that the horizontal and vertical accuracies of static GNSS-RTK observations in an open environment were approximately 5 mm and 10 mm, respectively, indicating high precision.
The satellite antenna is shown in
Figure 10 below with a choke added, and the correlation between the residual coordinate data of the two adjacent days was relatively low.
The next experiment placed the monitoring station in an environment with obstacles, with the monitoring station approximately 2 m from the west wall and 3 m from the north wall, while the reference station remained in the same position. The baseline length remained at 7 m, and the deployment is shown in
Figure 11.
The experiment was conducted during the same time period on two consecutive days: 27–28 October 2022 (DOY 270–271). The observation results are shown in
Figure 12.
The results shown in
Figure 11 indicate that when conducting GNSS-RTK observations in complex environments, the fluctuation of coordinate residuals increases and there is a certain correlation between the trends of coordinate residuals on adjacent days. In actual bridge monitoring, observation conditions are even more complex, and the low-frequency dynamic deformation and displacement of structures require high precision. However, due to the existence of multipath error, true low-frequency structural deformation data are obscured, making it difficult to meet the requirements of bridge health monitoring [
17].
In this experiment, the GNSS-RTK monitoring station remained nearly stationary on two consecutive days: DOY 270 and DOY 271. As previously noted, multipath error exhibits a periodic repetition characteristic under static observation conditions. Therefore, the coordinate residuals between adjacent days have a strong correlation, and the time series of the first day relative to the second day is subject to a delay.
Table 3 summarizes the maximum correlation coefficient of the coordinate residuals for DOY 270–DOY 271, along with the index value of the epoch where the maximum correlation coefficient occurred.
In
Table 3, the correlation coefficients of the ENU coordinate residual series for adjacent days (DOY 270 and DOY 271) are all around 0.6, indicating a significant correlation between the coordinate series for the two days. Moreover, the DOY 270 coordinate series was delayed by about 4 min relative to the DOY 271 coordinate series, which was consistent with the theoretical time [
18]. Therefore, the multipath errors in the DOY 270 coordinate residual can be extracted, and the extracted multipath error sequence can be used to correct the coordinate series observed on DOY 271, based on the delay amount at the epoch of maximum correlation. In short, the multipath errors in the reference day coordinate residual are first extracted, and then the subsequent day’s coordinate is corrected using the sidereal filtering method.
As a way of demonstrating the advantages of the proposed algorithm, we selected CEEMDAN, WPTD, and ICEEMDAN-AWPTD for comparison and analysis in the extraction of multipath errors from the DOY 270 coordinate residual sequence. In the comparative experiment, the parameter settings for CEEMDAN and WPTD were consistent with the simulation experiment in
Section 3.1. The multipath error extracted using the three methods is shown in
Figure 13.
Figure 11 indicates that the ICEEMDAN-AWPTD algorithm extracts multipath error with significantly less noise compared to the CEEMDAN and WPTD methods, and the extracted multipath error sequence does not jump, retaining more characteristic signal information. To provide a more intuitive analysis of the effectiveness of the three algorithms in extracting multipath error,
Table 4 presents the coordinate residuals of the original DOY 270 data in the ENU direction and the RMSE values of the multipath error extracted using the three methods. Compared to the results, the RMSE values of the multipath error extracted using all three methods were smaller than those of the original observations, and the ICEEMDAN-AWPTD algorithm outperformed the CEEMDAN and WPTD methods.
This experiment also calculated the maximum correlation coefficient R values of DOY 270 and DOY 271 after the extraction of multipath error using CEEMDAN, WPTD, and ICEEMDAN-AWPTD, and the results are shown in
Table 5. A larger R value indicates a higher correlation between the adjacent two-day coordinate sequences and more characteristic information extracted from the multipath error. The results in
Table 5 show that the correlation coefficient of the multipath error model extracted using ICEEMDAN-AWPTD was slightly higher than those of the CEEMDAN and WPTD algorithms.
After conducting two comparative experiments, it can be demonstrated that the ICEEMDAN-AWPTD algorithm is effective in extracting multipath error from reference days while preserving the original information components in the signal as much as possible. Therefore, it is suitable for multipath error suppression.
The effectiveness of the ICEEMDAN-AWPTD algorithm for multipath error extraction from the coordinate residual sequence of the reference day (DOY 270) was demonstrated in the preceding comparison test, and subsequent experiments will validate the correction effect of the multipath error extracted by using this algorithm on the coordinate sequence of the subsequent second day (DOY 271). Similarly, we compared the performances of three methods (CEEMDAN, WPTD, and ICEEMDAN-AWPTD) in correcting the coordinate residual sequence of DOY 271 for multipath error, as shown in
Figure 14.
The results in
Figure 12 demonstrate that among the three methods compared, the ICEEMDAN-AWPTD algorithm was more effective in suppressing the amplitude of multipath error on the second day, and the remaining residual amount was almost random noise error.
To quantitatively analyze the correction accuracy of the three algorithms for multipath error,
Table 6 presents the RMSE values of the coordinate residual sequences after processing for DOY 271. It is evident from
Table 6 that the ICEEMDAN-AWPTD algorithm outperformed the CEEMDAN and WPTD algorithms for the multipath error reduction, with improvements of 49.2%, 65.1%, and 56.6% in the RMSE values of the original E, N, and U directions, respectively. The improvement was most noticeable in the N direction, which was due to the largest correlation coefficient of the adjacent two-day coordinate series in the N direction, indicating that it was more influenced by the multipath effect.
The results shown in
Figure 12 and
Table 6 indicate that the ICEEMDAN-AWPTD algorithm can extract multipath error with high accuracy and effectively mitigate the influence of multipath error in short-baseline GNSS-RTK observations.