## 1. Introduction

Nowadays, civil structures, such as bridges, have been growing ever faster to support, e.g., economical activities and attracting tourists. Long-span bridges, such as cable-stayed and suspension bridges, have been constructed to reduce their overall cost [

1]. Lack of precise information on the health of their structure may lead to incorrect decision-making in repairing, retrofitting, and strengthening the structure. For example, an inadequate assessment of the structure’s health may lead to retrofit and replacement, while they are still healthy. Additionally, bridges are prone to environmental effects leading to their corrosion. Aktan et al. [

2] defined Structural Health Monitoring (SHM) as the measurement of the operating and loading environment as well as the critical responses of a structure to track and evaluate the symptoms and operational incidents, anomalies, and deterioration of damage indicators, serviceability, and safety or reliability of structures. Considering their slender geometry, bridges can be affected by wind, and loads imposed by vehicles, pedestrians, etc. In addition, changes in environmental variables such as temperature might lead to changes in their deflections and corrosions, which result in changing their normal dynamic characteristics [

3,

4]. Therefore, it is necessary to perform health monitoring to guarantee the safety and service life of bridges, for which various techniques and methods have been proposed (see examples in [

5,

6]).

Generally speaking, the bridge’s movements are characterized by its static, semi-static, and dynamic components. Therefore, it is necessary to extract these components precisely to be able to model their behavior. The static component does not change over time, in other words, it shows the trend of movement, and semi-static is referred to as the long-period component. The short-period component consists of two parts: the dynamic displacements and noise, where the first changes rapidly in time and is caused by a physical process, and the latter is randomly distributed.

Among the monitoring systems, the application of the Global Navigation Satellite System (GNSS), which is weather-independent and easy to use, is ever-increasing to monitor civil structures. GNSS can be used to measure static and dynamic movement components in real-time, whereas the conventional monitoring system using accelerometers or strain gauges cannot measure static and semi-static displacements [

1]. The movement estimation using geotechnical sensors such as accelerometers has their own difficulty such as their contamination with errors, e.g., drift, and the fact that they can only measure relative displacements of a structure as shown by [

5,

6,

7]. Furthermore, the rapid advancements in GNSS devices and processing algorithms can mitigate positioning errors. In addition, integrating GNSS with supplementary sensors, such as acceleration sensors, strain gauges, and laser displacement sensors, can improve the accuracy of position estimation (see examples in [

8,

9]).

From various strategies of GNSS data processing, the application of the Real Time Kinematic (RTK) GNSS with a high sampling rate (>1 HZ) has already been addressed in previous studies [

1,

10,

11,

12,

13]. However, the RTK system contains errors and noise of various statistical distribution (i.e., colored noise and white noise), which needs to be filtered before the precise displacement/deformation monitoring [

14]. This is achieved in previous works by implementing time series analysis techniques that enable the extraction of semi-static and dynamic components.

For example, Moschas and Stiros [

11] used the Moving Averaging (MA) filter to extract the semi-static movements of the Global Positioning System (GPS) measurements. Yu et al. [

15] proposed a double filtration method to detect the dynamic movements of structures. Other processing techniques such as Autoregressive Integrated Moving Average (ARIMA) and Autoregressive Moving Average (ARMA) and Generalized Autoregressive Conditional Heteroskedasticity (GARCH) are investigated in [

10,

16,

17,

18,

19,

20,

21]. Le and Nishio [

21] found that the ARIMA model parameters can be used for evaluating the temporal changes of the movement components. Kaloop et al. [

10] evaluated the behavior of a railway high-speed bridge and showed that the ARMA and GARCH models can be used to detect the behavior change of a bridge. Xin et al. [

22] provided a new integrated method that combines the Kalman Filter (KF), ARIMA, and GARCH, where the first KF is applied on the raw position estimates to reduce the noise. Then, the ARIMA model is used to analyze and predict the structure deformation, and in the last step, the nonlinear recursive GARCH model is introduced to improve the accuracy of the prediction. Their evaluation revealed that the mean absolute error of the prediction using the KF-ARIMA-GARCH model was 10.2% less than the other time series analysis techniques. In addition to the linear models (e.g., ARMA, and ARIMA) [

3,

16,

23,

24], nonlinear models (e.g., Neural Network (NN) method, Extreme Learning Machine (ELM), and Ant Colony Optimization algorithm (ACO)) [

18,

25,

26,

27,

28,

29,

30] have also been applied to predict the structural behaviors. These studies demonstrated that the monitoring data of huge structures (e.g., dam, bridge, and tower) contain nonlinear characteristics due to the uncertain environment [

29]. As a result, the linear models represent some limitations, for example, they are only able to predict stationary (or simple non-stationary) changes. Therefore, their prediction accuracy is very limited for revealing the dynamic components.

Most of the previous studies have applied their processing approach in the time domain (see e.g., [

13,

14,

31,

32]), while only a few provided their processing steps in the frequency domain (e.g., [

31,

33]). The latter is performed mostly by implementing techniques such as Fast Fourier Transformation (FFT) and Wavelet Analysis, where FFT was used to estimate the natural frequency of structures and to eliminate the positioning noise, and the wavelet technique was applied to detect changes in the frequency behavior of structures [

33,

34,

35,

36,

37]. However, the efficient application of FFT requires an understanding of its assumptions and limitations. For example, FFT decomposes a signal into sine and cosine functions of different frequencies. When these signals cannot be divided into the predefined cycles, amplitude-scaling errors are expected (see more details in [

30]). For the RTK-GNSS derived position time series, where the signals consist of the “harmonic” and “noise” part, a more advanced methodology must be considered.

In this paper, we introduce an efficient combination of time series analysis and prediction techniques to model movements of a bridge, which is monitored by the RTK-GNSS technique. The goal is to extract the full behavior of the bridge (its static, semi-static, and dynamic components), which is achieved by applying different filtering methods. The Least Square Harmonic Estimation (LS-HE) [

38] and the Neural Network (NN) [

18] methods are applied to estimate the dominant frequency and to obtain the predictive movement model, respectively.

This paper is organized as follows: In

Section 2, we describe the RTK GNSS-based monitoring system and characteristics of the stations used. The proposed algorithm in order to extract the bridge displacement components, as well as the displacement model are presented in

Section 3. The evaluation results and numerical experiments are discussed in

Section 4, which are essential for the assessment of the full behavior of the bridge, and finally, this study is concluded in

Section 5.

## 4. Results and Discussions

I In this section, the results of the bridge’s movement components are described, the performance of the proposed dynamic model is discussed, and the behavior of the stations along the bridge is also compared with a permanent station.

#### 4.1. Data collection and Preparation

In this study, GNSS measurements of the Tabiat bridge (latitude: 35°45′16.15″ and longitude: 51°25′13.64″) and M022 (a permanent SAMT station, latitude: 35°45′56.97″ and longitude: 51°11′45.55″) are collected during 28–29 November 2018 with a 30 s sampling rate in order to examine the capability of the proposed algorithm. The X-direction refers to the east of the bridge (Taleghani Park), the Y-direction is perpendicular to the X-direction and is stretched along the Modares highway, and the Z-direction represents the vertical direction movement, as shown in

Figure 1b. The Bernese software in a double-difference mode was utilized to process the GPS raw data of these three stations. The processing strategy is summarized in

Table 3. It is worth mentioning that we used the result of the kinematic station coordinates derived from Bernese. These time-series are expressed in the local geodetic reference frame (i.e., North, East, and Up local directions).

Figure 3 shows the coordinates time-series of three stations (1 fixed and 2 monitoring stations) in the Universal Transverse Mercator (UTM) grid system.

#### 4.2. Data Pre-Processing

The KF method is used to de-noise the random errors of coordinates time-series derived from

Section 4.1. The vector of coordinates time-series has an average accuracy of 0.019, 0.016 and 0.039 m during 24 h in the North, East, and Up component, respectively, which are estimated by the variance-covariance matrix of the Bernese software. After implementing the KF, the accuracy of unknown parameters reduces to 0.0013, 0.0012 and 0.0019 m in the N, E, and U directions, respectively.

#### 4.3. Bridge Movement Evaluation

To assess the bridge movements, the semi-static movement is considered as the first component that can be extracted by applying the MA filter to the de-noised time-series. The accuracies of semi-static components in the North, East, and Up are 0.013, 0.012, and 0.019 cm, which are determined using the least squares error propagation.

Figure 4 represents the semi-static component of stations 1, 2, and M022. The results indicate that the correlation between the semi-static component and de-noised time-series is 99% at each station for the N and E components and 98% for the U component.

To assess the semi-static movement, the maximum and Standard Deviation (STD) of this component for the bridge monitoring stations are presented in

Table 4. Based on

Figure 4 and

Table 4, the maximum movement of the stations is detected in the Up direction (0.2553 and 0.1934 m at Stations 1 and 2, respectively). Moreover, the STD value of Stations 1 and 2 in the Up direction (0.057 and 0.058 m, respectively), which is higher than in the East and North directions. The movement variations along the Up direction can be related to the load factor. In the semi-static component of the permanent station (M022), the maximum value of movement (0.1948 m) and STD (0.065 m) are found in the Up direction.

The statistical assessments provide similar results for Stations 1 and 2, which indicates that both stations have a safe semi-static behavior due to changing in the loads.

Figure 5 illustrates the correlation between Stations 1 and 2, in the North, East, and Up directions. The correlation between the North and East was found to be −18.5% at Station 1 and 4.2% at Station 2. However, in

Figure 5, we found that the correlation between the movements of Stations 1 and 2, in three directions is high for the same directions (e.g., the correlation between the North of Station 1 and North of Station 2 is 73.1%), which indicates that the structure movements are controlled.

As a second component, the static movement of stations is extracted using the mean of 5 min of the semi-static movement, which is also shown in

Figure 4. The static behavior of Stations 1 and 2, in the East versus the North, is presented in

Figure 6a, which reveals horizontal displacements. Although the static responses are found to be similar and correlated, the magnitude of this component in Station 1 is exhibited higher than those of Station 2. As

Figure 6a indicates, the static response of the points is approximately similar at the end of the monitoring time, which can be associated with the negligible effect of traffic load on the bridge.

Similarly, according to

Figure 6b, the static behavior of the two stations in the Up was found to be almost the same. The maximum static deviation in Station 1 (0.231 m) and Station 2 (0.167 m) occurred at time 28.59 (unit in day), which indicates the correlated behavior of two stations. Since Stations 1 and 2 are located on the left and right sides of the bridge, the situation in the location can counteract the effect of the correlated behavior. Therefore, the rigidity of the bridge structural under the traffic load effect can be concluded.

After removing the semi-static components using the MA filter from the de-noised time-series, the short-period component can be extracted which includes the dynamic component and the residual noise.

Following the description in

Section 3.2.3, a low-pass filter is applied to estimate the dynamic behavior from the short-period component. In order to select the optimal filter, the evaluation criteria are evaluated, as shown in

Figure 7. In this figure, it is obvious that the Median filter is the optimal method to estimate the dynamic component due to the high NRMSE and low RMSE. The dynamic behavior of the monitored points which are derived from the Median filter is presented in

Figure 8.

The correlation between the dynamic components in Stations 1 and 2 are found to be 1.22%, 2.31%, and 10.28% in the North, East, and Up directions, respectively, which indicate that this component is most correlated with the Up compared to the other directions. The maximum values of dynamic movement in the Up direction are found to be 0.0464 and 0.0389 m for Stations 1 and 2, respectively. From these results, it can be concluded that the monitored points are characterized by small dynamic components, which means that the bridge is safe under the traffic loads.

#### 4.4. Frequency Domain Evaluation

In this section, the LS-HE method is used to evaluate the time-series response of the monitored points in the frequency domain. The short-period component is employed and LS-HE is applied to extract the dominant frequencies. The dominant frequency magnitudes for the stations along the bridge and the permanent station during the monitoring period are summarized in

Table 5.

From

Figure 9, it can be seen that the investigation of the short-period movement component in the time domain cannot provide any specific information. Therefore, the evaluation of this component in the frequency domain can be interesting. A comparison between the dominant frequencies derived from the three stations is shown in

Figure 9, in which the log-log plot is used for the power spectrum versus frequency. Periodogram diagrams in

Figure 9 and dominant frequencies values in

Table 5 illustrate the similar pattern in frequencies between the station along the bridge and permanent station, which can be considered as a proof that the bridge does not show an irregular high-frequency dynamic behavior.

According to the results, we found that the mean deflections of the bridge (the mean of semi-static and short-period components) are 4.2, 3.2, and 5.2 cm in the North, East, and Up directions, respectively, and the norm of three component displacements is 7.4 cm. Therefore, the full behavior of the Tabiat bridge using GPS measurements is within the safety limits of the bridge design ([−22–22] cm).

#### 4.5. Evaluation of the Bridge Movement Prediction Model

As mentioned in

Section 3.3, the NN method is used to determine and predict the bridge movements. In the present study, a repetitive process is applied to determine the number of hidden layers and delay, and finally, the parameters with the highest level of fitting and least error was selected as the most optimal model. As a result, a neural network with five hidden layers and five delays is considered to stimulate the semi-static and dynamic components. In this study, to perform the NN method, the data set is divided into three parts randomly: Training, validation, and test data. In addition, 80%, 10%, and 10% are regarded as the training, validation, and test data set, respectively. For the semi-static component, the RMSEs between the model and the actual values in station 1 are found to be 3.78 × 10

^{−7}, 2.945 × 10

^{−7}, and 3.108 × 10

^{−7} m in the North, East, and Up directions, while these values for Station 2 are found to be 7.623 × 10

^{−7}, 2.651 × 10

^{−7}, and 6.425 × 10

^{−6} m. For the dynamic component, the RMSEs calculated between the model and the actual dynamic component, in Station 1, are found to be 2.478 × 10

^{−6} m in the North direction, 1.983 × 10

^{−6} m in the East, and 4.2 × 10

^{−6} m in the Up direction. For Station 2, the dynamic model is fitted with 2.335 × 10

^{−6}, 2.19 × 10

^{−6}, and 3.477 × 10

^{−6} m errors in the North, East, and Up direction, respectively. The abilities of the NN model in the prediction of semi-static and dynamic components are estimated with the RMSE values, which are summarized in

Table 6. We also simulate the semi-static and dynamic components of the M022 station using this NN. The results of fitting in the prediction mode demonstrate the RMSEs of 3.894 × 10

^{−5}, 6.98 × 10

^{−5}, and 2.727 × 10

^{−4} m in the North, East, and Up directions, respectively.