Adaptive Fast Independent Component Analysis Methods for Mitigating Multipath Effects in GNSS Deformation Monitoring

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Introduction
GNSS has been one of the leading high-precision positioning technologies and is widely used in dynamic deformation monitoring of engineering constructions [1,2], such as dams, bridges, high-rise buildings, and railway roadbeds [3][4][5][6][7]. In short-baseline GNSS measurements for deformation monitoring, differential GNSS techniques can largely eliminate common-mode errors between the reference and the rover GNSS stations resulting from ionospheric and tropospheric refraction and delays, satellite and receiver clock biases, and orbital errors. However, carrier-phase multipath effects cannot be removed with this approach and still have significant effects on GNSS position estimates [8]. Due to the reflection or diffraction caused by nearby obstacles, GNSS multipath phenomenon occurs when signals travel from a satellite to a receiver via several paths. Multipath is mainly decided by the satellite constellation, receiver environment, and reflector positions. In deformation monitoring, the operation of the satellites directly affects the multipath since the receiver and reflector positions are relatively stable. Thus, multipath features can be extracted and used to reduce positioning errors by analyzing the operation of the satellites. For example, as the satellite operating cycles are usually stable, the corresponding multipath also has a similar periodic variation. By using the previous observation measurements to model different satellites separately, Sidereal Filtering (SF) was employed to mitigate the multipath [9][10][11][12]. But it is necessary to preset the session duration of the static baseline solution in classic deformation monitoring, which reduces the timeliness of the method. Dong et al. [13] constructed a single-difference observation equation based on a single receiver with multiple antennas and proposed to establish a multipath hemispherical map (MHM) with satellite altitude and azimuth as independent variables. However, when these methods are used to mitigate the multi-GNSS multipath, the multipath model will have a very large matrix, which makes it impossible to achieve lightweight calculation. Moreover, an increase in the number of satellite and signal frequency types [14] by using multi-GNSS observation could make the multipath more complicated [15].
In order to mitigate the multi-GNSS multipath result from different types of GNSS satellites, a novel filtering method called AF-ICA is proposed to mitigate multipath in this paper, which can improve the timeliness of dynamic deformation monitoring by combining signal processing technologies and the principle of SF. In the AF-ICA method, the multi-GNSS single epoch solution is input as a signal with multipath to be processed, and the desired output signal is used as the multipath model. First, mean smoothing filter (MSF) method is used to denoise by calculating the mean value of a window. Then, the Fast Independent Component Analysis (FastICA) is adopted to separate the multipath signal generated by different types of satellites from the source signals by combining prior engineering information. Next, to improve the timeliness of the method, first-order Fourier fitting formula is used to get the signal period. We can recover the complete multipath signal by the linear fitting formula according to the period of the separated signal. Finally, we can obtain the final deformation monitoring results by removing the multipath signal. Contributions of this paper can be summarized as follows: Considering the complicated multipath signal generated by different types of satellites in the GNSS deformation monitoring, we propose to use FastICA to separate the multipath signal from the source signals. In order to improve the timeliness, we design the AF-ICA method: obtain the prior period range of the separated signal (i.e., multipath signal) according to the actual engineering experience, and then use the prior period information as the basis to judge whether FastICA can separate the signal, so as to determine the smallest MSF window. Our method mitigates the multipath in the coordinate domain and can automatically get the session duration of mean filtering by extracting multiple multipath signal periods corresponding to the BDS orbit through the ICA algorithm without estimating the orbit period in advance like classical SF-OD method.
Our proposed method can automatically get the session duration of mean filtering by extracting multiple multipath signal periods corresponding to the BDS orbit through the ICA algorithm without estimating the orbit period in advance to realize automatic deformation monitoring and improve engineering efficiency. Through the automated monitoring solution, the AF-ICA method in this paper can be applied to the natural disaster monitoring in the Internet of Things and provides real-time data monitoring information for disaster early warning.

Methodology
Considering the complicated multipath signal generated by different types of satellites and the timeliness of traditional SF methods, the AF-ICA method is presented to address the problems in this paper. In our case, the multi-GNSS single epoch solution is input as a signal with multipath to be processed, and the desired output signal is used as the multipath model. First, the multipath signal is weak in the source signals. Thus, the MSF method needs to be first used for denoising to get the multipath signal with enough strength. Then, FastICA is adopted to separate the multipath signal generated by different types of satellites from the source signals by combining prior engineering information. The environmental information includes two aspects: one is that the observation data contains GPS satellites whose orbital period is close to a sidereal day, another is that three multipath signals need to be separated because BDS satellite has three types of orbits. Next, to improve the timeliness of the system, first-order Fourier fitting formula is used to get the signal period. We can recover the complete multipath signal by the linear fitting formula according to the period of the separated signal. Finally, we can obtain the final deformation monitoring results by removing the multipath signal. As shown in Figure 1, AF-ICA method contains two parts: first is the MSF using the single epoch GNSS static positioning solution to wipe off the influence of Gaussian white noise, and second is the FastICA method for extracting multiple multipath signal periods. After removing the high frequency signal with MSF method, the remaining low frequency signal is the multipath signal to be separated.
2.1. MSF for Denoising. Classic GNSS deformation monitoring algorithm uses the static baseline solution method to obtain the positioning accuracy of millimeter level. The static baseline solution needs to set the session duration in advance, and then use the static postprocessing method to solve the baseline. In this paper, the method of single epoch solution combined with MSF is used to obtain millimeter positioning accuracy. The main purpose of MSF is to provide clean multipath signals for subsequent ICA methods. There is no need to preset the session duration, and the duration is automatically controlled by the AF-ICA method. Meanwhile, the final result of MSF algorithm is equivalent to the static postprocessing baseline solution. MSF algorithm using the single epoch solution is described as follows.
Normal equation of fixed solution in the static postprocessing baseline solution can be written as

Journal of Sensors
Least-square method can be used to calculate the fixed solution X in (1) where X can be calculated by n epoch iterations. The n-th iteration could be written as where X ðnÞ represents the result of the n-th iteration; X n = ðH T n P n H n Þ −1 ⋅ ðH T n P n Z n Þ represents the least-squares result of the n-th epoch; and μ = ðH T 1 P 1 H 1 +⋯+H T n P n H n Þ −1 represents the inverse of the coefficient matrix in normal equation.
Iterative formula can be obtained by further decomposing (3): where α, β is regarded as the smoothing coefficient and can be expressed as where I is a unit matrix.

AF-ICA for Mitigating Multipath Effects.
In the static baseline solution, the duration of the session needs to be set in advance. While the AF-ICA is proposed in this paper can automatically analyze the session duration by using the single epoch solution. Before introducing the AF-ICA method, let us first introduce the FastICA method. FastICA is a signal processing method for transforming an observed multidimensional random vector into components that are statistically as independent from each other as possible [16]. The standard ICA model could be written as where x = ½x 1 , x 2 ,⋯,x m T is regarded as the observed mixtures, s = ½s 1 , s 2 ,⋯,s n T represents the unknown sources, and A = ½a 1 , a 2 ,⋯,a m T is the unknown mixing matrix of size m × n. With the assumption that m ≥ n and that sources s are mutually independent, the ICA could optimally estimate a demixing matrix, say B, to separate the original signals s based on some rules of optimization (such as the least squares). Then the best approximation vector of s can be derived from where y = ½y 1 , y 2 ,⋯,y n T , which is the best approximation vector of s. Generally, the process of the ICA algorithm can be divided into three steps: first, center x, i.e., subtract its mean vector m = Efxg so as to make x a zero-mean variable; second, whiten the observed mixtures to get the whitening signals z = Vx, where V is the whitening matrix, and Efzz T g = I (I is a unit matrix); third, get a rotation matrix W, such that y = Wz, by the specific independence optimization rule. In the ICA algorithm, the sources s consist of one Gaussian source at most, but the separated components y are uncertain in amplitude and order.
FastICA is based on a fixed-point iteration for finding a maximum of the non-Gaussianity of y. To begin with, we first show the one-unit version of FastICA [17]. Denote the derivative of the nonquadratic functions are where 1 ≤ a 1 ≤ 2 is a suitable constant, often taken as a 1 = 1.
Object of GNSS deformation monitoring can be expressed by the baseline solution results in three components E, N, and U of the topocentric coordinate system. When using FastICA for multipath signal separation (here we think that the source signal is a multipath signal), we regard the three coordinate components of ENU as mixed signals x, m = 3. According to engineering experience and GNSS satellite orbital period [18], we can obtain three kinds of prior information:

Results and Discussion
In this section, a comprehensive experimental study has been conducted to inspect the performance of our proposed method. The content consists of three parts: the experimental setup, comparative experiments using the MSF, the SF-OD, and the AF-ICA methods, and the discussion.

Experimental Setup.
To test our proposed methods, data from an experiment with a railway roadbed deformation monitoring project were used. This experiment was 4 Journal of Sensors conducted on the Guangshan railway. It monitored the impact on the completed roadbed during construction and used GNSS receivers to monitor the deformation. In the monitoring area, we selected a firm position that is not easy to deform at the top of the mountain as the reference point, marked as B001. Four monitoring points noted as M001, M002, M005, and M006 were arranged along the railway in the monitoring area. The distribution of each observation station is shown in Figure 2.
The location was chosen because it was clearly subject to multipath effects from the smooth reflective surfaces of the railway roadbed. ComNav GNSS L1/L2 dual-frequency K506 receivers with Harxon HX-CSX601A survey antennas were used in the base and monitoring stations. The baselines between the base station and monitoring station were about 100-500 m. Due to this short baseline, satellite clock errors, receiver clock errors, satellite orbit error, ionospheric delay, tropospheric delay, and other common errors are removed by double differencing. The test began on March 3, 2020, and lasted for 8 days. The detailed data processing strategy is first analyze the positioning accuracy after MSF denoising, then use the classic SF-OD method to analyze the positioning accuracy after removing the multipath, and finally, use the AF-ICA method to analyze the positioning accuracy after the multipath is removed.

Experiment with MSF.
Position accuracy obtained from the relative positioning of GNSS single epoch (i.e., interval = 1 s) is cm level, which cannot meet the mm level accuracy requirements of deformation monitoring, and MSF processing is required. In order to analyze the relationship between positioning accuracy and session duration in MSF, we designed 16 kinds of session duration: 1 s, 15 s, 30 s, 60 s, 3 min, 5 min, 15 min, 30 min, 45 min, 1 hour, 2 hours, 3 hours, 4 hours, 6 hours, 12 hours, and 24 hours and then used MSF algorithm for processing. The average positioning accuracy of the four monitoring stations' baseline results after MSF was counted in the ENU geographical coordinate system.
In this test, the 3-sigma principle (99.7% confidence probability) is used to count the positioning accuracy. The statistical results are shown in Table 1, and the relationship between positioning accuracy and session duration is shown in Figure 3.
As can be seen from Table 1 and Figure 3: the longer the session duration, the higher the positioning accuracy. After more than 12 hours, the positioning accuracy is better than 1 mm, and then increasing the duration has little improvement on the positioning accuracy. When the session duration is within 0-1 hour, the effect of increasing duration on the positioning accuracy is very obvious, and it enters a turning point near 1 hour. After the     Journal of Sensors duration exceeds 1 hour, the horizontal positioning accuracy is better than 3 mm and the altitude positioning accuracy is better than 4 mm. In summary, we suggest that in GNSS deformation monitoring using MSF method, it is more appropriate to select 1-12 hours for the duration, which can not only obtain mm level positioning accuracy but also ensure a fast deformation monitoring result.

Experiment with SF-OD.
We use the SF-OD method to eliminate multipath signals and analyze the positioning accuracy after removing multipath. The data processing strategy of SF-OD is first collect data for one sidereal day, then get the difference between second sidereal day with the first sidereal day in observation domain, and finally, obtain the deformation of the second sidereal day relative to the first sidereal day. The SF-OD method uses a 2 hours interval for processing. The first day as a reference and calculate the single difference residuals. The positioning solution is carried out after removing the single difference residuals in the same period of the next day. Due to the adoption of the sidereal day period, we only process the GPS dual-frequency data. Figure 4 shows the ENU positioning error of M001 station after removing the multipath signal, Table 2 lists the positioning accuracy statistics using 2 hours of MSF and SF-OD methods.
Compared with MSF, the positioning accuracy of SF-OD increases by -9%, -10%, and 2% in the three directions of  7 Journal of Sensors ENU. The corresponding peak-to-peak value increases are -23%, -17%, and -16%. It can be seen that although SF-OD can effectively deal with the influence of sidereal day multipath signals, the positioning accuracy and peak-to-peak value are worse than those of the MSF method. The possible reason is that SF-OD only processes GPS data, while MSF processes GPS+BDS data which contains more satellites.
3.4. Experiment with AF-ICA. According to the data results of the MSF test, the session duration range of MSF is controlled within 1-12 hours. Firstly, four monitoring stations are tested through a priori information in AF-ICA, and it is concluded that the minimum session duration that can separate multipath signals is 2 hours. Subsequent AF-ICA analysis is processed based on the MSF results of 2 hours. Figure 5 shows the period and graph of separating the multipath signal of M001 station by FastICA method. The input signal is the E, N, and U coordinate components after MSF, and the output signal is the separated three independent multipath signals. Through the first-order Fourier fitting formula, the periods of the three multipath signals are 8 hours, 12 hours, and 24 hours, respectively. It can be seen that compared with the sidereal day period in GPS data, the GPS+BDS data used in this test adds two multipath signals with smaller cycles, which may be mainly due to the use of multiple orbit types of satellites in GNSS relative positioning.
After the multipath signal is separated by the FastICA method, the complete multipath signal is generated by the linear fitting formula. Figure 6 plots the positioning error of M001 station before and after multipath signal removal  Journal of Sensors by FastICA method. In Figure 6, the results after MSF are represented by asterisk points, fitting multipath represents the complete multipath signal, and finally represents the positioning error after multipath signal removal. Table 3 lists the accuracy statistics after removing multipath signals by AF-ICA compared with MSF. Figure 7 plots the positioning error of M001 station before and after multipath signal removal by AF-ICA. Compared with MSF, the positioning accuracy of the AF-ICA method is improved by 44%, 14%, and 31% in the three directions of ENU. The corresponding peak-to-peak value increases were 36%, 17%, and 29%. It can be seen that the AF-ICA method can effectively deal with the influence of multipath signals during multi-GNSS fusion positioning, and the positioning accuracy is significantly improved compared with the simple MSF method.
3.5. Discussion. In the above experiments, we use two different methods to mitigate the multipath in deformation monitoring. To compare the MSF method and sidereal day filtering method, we use the SF-OD method to process GPS dual-frequency data. Although SF-OD effectively processes the influence of sidereal day multipath on the positioning accuracy, peak-to-peak values are worse than those of the MSF method. On the other hand, through the analysis of 8-day static test results by the AF-ICA method, according to a priori information of multipath signal, we determined that the minimum session duration meeting the multipath signal separation is 2 h, and successfully separated the multipath signals of four stations. Compared with the simple MSF method, the positioning accuracy of the AF-ICA method is improved by 44%, 14%, and 31% in the three directions of ENU. The corresponding peak-to-peak value increases were  9 Journal of Sensors 36%, 17%, and 29%. The AF-ICA method can effectively solve the problem of multiperiod multipath signals caused by multiple satellite orbits of GNSS. The AF-ICA can automatically identify the session duration, to realize automatic deformation monitoring.

Conclusion
In this study, we proposed a novel method called the AF-ICA method to address the complicated multipath in the deformation monitoring of the GNSS that contains multiple types of satellites and improve the timeliness. The experimental results show that our method can achieve mm level positioning accuracy and meet the requirements of GNSS deformation monitoring. At the same time, by analyzing the value range of session duration, it is found that the session duration of 1-12 hours is more appropriate, which can not only obtain ideal positioning accuracy but also ensure fast deformation monitoring. Compared with the simple MSF method and the SF-OD method, the proposed AF-ICA method is able to improve both the positioning accuracy and peak-to-peak values. In a word, our method is more appropriate for mitigating multipath than the traditional sidereal filtering method, especially in multi-GNSS fusion positioning.

Data Availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author for academic purposes.

Conflicts of Interest
The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.