Bridge Tower Warning Method Based on Improved Multi-Rate Fusion Under Strong Wind Action

Abstract

The displacement of bridge towers is relatively large under strong wind action. Changes in tower displacement can reflect the usage status of the bridge towers. Therefore, it is necessary to conduct performance warning research on tower displacement under strong wind action. In this paper, the triple standard deviation method, multiple linear regression method, and interpolation method are used to preprocess monitoring data with skipped points and missing anomalies. An improved multi-rate data fusion method, validated using simulated datasets, was applied to correct monitoring data at bridge tower tops. The fused data were used to feed predictive models and generate structural performance alerts. Spectral analysis confirmed that the fused displacement measurements achieve high precision by effectively merging the low-frequency GPS signal with the high-frequency accelerometer signal. Structural integrity monitoring of wind-loaded bridge towers used modeling residuals as alert triggers. The efficacy of this proactive monitoring strategy has been quantitatively validated through statistical evaluation of alarm accuracy rates.

1. Introduction

Nowadays, more and more long-span bridges are being built. During long-term service of long-span bridges, structural damage and performance degradation are prone to occur under strong wind action [1]. The failure of large structures can lead to significant casualties and economic damage [2,3,4]; however, precise bridge damage warnings can effectively prevent collapses. Structural health monitoring systems (SHMSs) strategically deployed on long-span bridges continuously generate extensive operational datasets. Analyzing these datasets to establish performance alerts for critical structural elements like bridge towers is vital for safeguarding operational safety and ensuring long-term structural integrity [5,6,7]. As wind-induced forces represent a predominant environmental excitation for large-scale civil infrastructure [8,9], investigating the variation patterns of pseudo-static displacements in bridge towers under severe wind conditions becomes essential for both structural assessment and preventive maintenance strategies.

Analyzing bridge structural responses under strong winds can effectively identify early degradation [10,11]. For example, Wen et al. [12] assessed the fatigue life of cables on the Jintang cable-stayed bridge under wind–rain conditions. Recent research has further advanced methodologies for wind-related bridge monitoring and fatigue evaluation. Ye et al. [13,14] developed a fatigue damage evaluation framework for Jiupao Bridge using structural health monitoring data, demonstrating through principal S-N curve analysis that medium–long hangers exhibit a predicted service life of 366.32 years under wind-induced stresses. Kim et al. [15,16] also found that abrupt deck width transitions at bridge entrances can amplify wind loads on vehicles by three to four times, highlighting a significant aerodynamic interaction relevant to structural monitoring. Huang et al. [17] introduced a strain-based diagnostic approach for girder performance evaluation, utilizing long-term operational data collected under variable environmental conditions including wind, thermal fluctuations, and traffic loads. Huang et al. [18] implemented a multivariate statistical approach incorporating temperature, humidity, and wind velocity parameters, employing partial least squares regression for environmental factor dimensionality reduction to establish a frequency-domain warning system for structural integrity. Salvatori et al. [19] developed a computational framework to estimate transient wind loads during downburst events, showing that it can predict significant variations in bridge deck angle-of-attack caused by medium-scale downburst systems.

The data fusion technology of machine learning and Bayesian methods is an important approach in the field of structural health monitoring. The framework shows considerable potential as a practical and reliable AI-powered digital tool for advancing next-generation structural health monitoring systems. Vibration-based structural damage detection methods can solve various engineering problems, thereby extending the service life of structures [20]. Gomez-Cabrera et al. [21] identified the machine learning algorithms currently implemented in the construction of structural health monitoring systems. By combining ANN, CNN, and SVM with preprocessing techniques, 68 articles from 2011 to 2022 were analyzed. The application of these techniques in structural condition monitoring has enhanced the reliability and performance of the systems. Zhang et al. [22] proposed an online detection method for structural health monitoring data anomalies based on a Bayesian dynamic linear model. To overcome the initialization problem of the expectation maximization algorithm, a subspace identification method was introduced. The structural health monitoring data of two long-span bridges verified the proposed method.

The reconstruction of monitoring data, structural health monitoring, and damage identification based on artificial intelligence are important research directions in structural health monitoring. Su et al. [23] proposed an LSTM-based method for reconstructing missing vibration data that segments frequency bands to fuse multi-frequency information. Comparative analyses demonstrate the efficacy and practicality of this Bayesian-optimized, split-band approach. Mao et al. [24] introduce a novel ResNet50-based approach for anomaly detection in bridge vibration data, prioritizing time efficiency and transfer learning. Their method extracts features using a pre-trained ResNet50 network and then classifies anomalies via k-means clustering on these feature vectors. Wang et al. [25] present a lightweight damage identification framework featuring an optimized extreme learning machine (ELM). This approach utilizes a dramatically reduced parameter set derived from structural vibration frequency features, demonstrating strong performance.

This paper proposes a novel methodology for warning of structural performance degradation in bridge towers during strong wind events, addressing shortcomings in current methods. The principal contributions are structured as follows: First, an overview of the bridge structure and its monitoring system is provided, followed by comprehensive data preprocessing techniques including the three-sigma criterion, multiple linear regression analysis, and interpolation algorithms to enhance data quality. Subsequently, an enhanced multi-rate data fusion framework is developed with theoretical foundations, successfully implemented on field monitoring data, accompanied by spectral analysis of the fused displacement signals. Third, correlation studies between environmental factors and structural responses are conducted to quantify relationship coefficients and establish critical warning indicators. The proposed methodology is then validated through a case study demonstrating its effectiveness in early warning applications for cable-stayed bridges. Compared to conventional methods, this investigation concludes with detailed findings showing significantly improved warning accuracy. These improvements stem from key advancements: refined modeling of environmental effects, capabilities for correcting acceleration data during fusion, and enhanced correlation analysis to minimize false negatives in structural health monitoring.

2. Bridge Monitoring System and Preprocessing of Monitoring Data

2.1. Description of the Bridge Monitoring System

This research explores early warning indicators for structural performance using monitoring data from a long-span cable-stayed bridge. The bridge’s structural health monitoring (SHM) system instrumentation is shown in Figure 1. This five-span, continuous, symmetrical cable-stayed bridge features twin pylons and crosses China’s Yangtze River. Its spans are arranged as 50 m + 215 m + 510 m + 215 m + 50 m. Both the southern and northern towers rise to 181.53 m in height. The bridge deck accommodates a dual-carriageway four-lane highway system designed for 100 km/h traffic speeds. The monitoring system includes wind speed and direction instruments, air temperature and humidity sensors, structural temperature sensors, road temperature sensors, visibility meters, strain sensors, displacement sensors, and acceleration sensors. The monitoring system of the bridge can monitor wind speed, wind direction, air humidity, air temperature, structural temperature, road temperature, visibility, strain, displacement, and acceleration in real time.

This research analyzes structural monitoring data from the elevated sections of bridge piers.

Table 1. Detailed locations and information of the sensors.

Monitoring Subject	Position	Serial Number	Sampling Frequency (Hz)	Unit
Wind speed	North tower	FS01	1	$\mathrm{m}/\mathrm{s}$
	South tower	FS02	1	$\mathrm{m}/\mathrm{s}$
Wind direction	North tower	FD01	1	$\mathrm{Degree}$
	South tower	FD02	1	$\mathrm{Degree}$
Displacement	North tower	GPS01	1	$\mathrm{mm}$
	South tower	GPS02	1	$\mathrm{mm}$
Acceleration	North tower	ACC01	20	${\mathrm{m}/\mathrm{s}}^2$
	South tower	ACC02	20	${\mathrm{m}/\mathrm{s}}^2$

systematically details comprehensive specifications, including installation coordinates, device IDs, measurement intervals, and parameter units. This study specifically investigates wind measurement records from the two-year period 2014–2015. The implemented wind velocity detectors feature a temporal resolution of 1 sample per second, with complementary wind orientation instruments deployed at identical spatial coordinates. The sampling rates of the GPS and accelerometer sensors arranged at the same position on the bridge tower are 1 Hz and 20 Hz, respectively. The correlation modeling is carried out using the modified displacement data and wind speed to give performance warnings to bridge towers under strong wind action.

The GPS sensor adopts the GR10 sensor produced in Switzerland. The antenna of the GPS monitoring station at the top of the Sota Tower is installed at the top of the antenna column to ensure that the GPS antenna can see the sky at a level of more than 10 degrees horizontally. A metal base needs to be pre-embedded on the top platform of the cable tower. The base should be stable and level, and the screw holes required for installing the antenna column should be reserved. The product selects the 81000U type three-directional ultrasonic anemometer produced by R.M. Young Company of the United States. Moreover, the installation position of the anemometer should not be obstructed. The anemometer should be installed on the extended arm protruding from the outer edge of the box girder, and the height between the anemometer and the outermost edge of the box girder should be greater than 5.0 m. To ensure that the vibration of the bridge under strong wind conditions does not adversely affect the measurement results of the anemometer, the extension arm or support rod of the three-directional ultrasonic anemometer in this monitoring system all have sufficient rigidity.

2.2. Repair Methods for Skipped Abnormal Monitoring Data

2.2.1. Repair Method for Skipped Abnormal Monitoring Data Based on Eliminating the Maximum Value

Skipped-point anomaly monitoring data are a common monitoring data error, which refers to the presence of particularly large sampling anomaly points in the data. When skipped points significantly exceed normal monitoring values, the normal data appear flattened to a baseline. The maximum-elimination-based skip-abnormal repair method is suitable for monitoring data that are predominantly normal but contain sparsely scattered large abnormal sampling points. Figure 2 shows the results of the acceleration monitoring data with skipped abnormal points, where Figure 2a is the original acceleration time-history curve with skipped abnormal points, and Figure 2b–d are acceleration time-history curves with the largest abnormal point eliminated 2, 4, and 6 times, respectively. It can be seen that the acceleration monitoring data with skipped abnormal points eliminated 6 times are repaired.

2.2.2. Repair Method for Skipped Abnormal Monitoring Data Based on Triple Standard Deviation

Triple standard deviation is simple, intuitive, and computationally efficient. It is a classic and commonly used method for detecting significant outliers in numerical data. It is simpler and faster than more complex density-based methods (such as LOF) or distance-based methods (such as KNN anomaly detection). Generally, triple standard deviation is used by default, that is, $3\sigma$ [26]. The method based on triple standard deviation is also suitable for monitoring data with normal overall data and some abnormal, large sampling points. For a piece of monitoring data, $\sigma$ is the standard deviation of the data, $\mu$ is the mean of the data, and $a$ is a multiple of the standard deviation. The data within the range of $(\mu -a\sigma ,\mu +a\sigma )$ are regarded as normal data, and the data outside the range of $(\mu -a\sigma ,\mu +a\sigma )$ are regarded as abnormal data. A more reasonable $a$ value can also be set according to the actual data situation or some kind of monitoring data processing experience. Figure 3 presents air temperature monitoring data rectified using the triple standard deviation method. Figure 3a displays the original time-history curve containing outlier points (skips), while Figure 3b shows the corrected time-history curve following repair. These results demonstrate the efficacy of the triple standard deviation method in rectifying air temperature data affected by skip-type anomalies.

2.3. Repair Methods for Missing Abnormal Monitoring Data

2.3.1. Repair Method for Missing Abnormal Monitoring Data Based on Multiple Linear Regression

The missing abnormal monitoring data repair method based on multiple linear regression is suitable for the repair of missing data due to its clear principle, high computational efficiency, and strong engineering applicability. The application condition of the multiple linear regression method is that there are more than 10 sampling points or across the time scale of signal features [27]. The repair method of missing abnormal monitoring data based on multiple linear regression is suitable for the missing monitoring data recorded by multiple similar sensors. The regression equations of multiple similar sensors are constructed by using the multiple linear regression method. When the data from a certain sensor are missing, the constructed regression equation can be used to estimate the missing data. Assuming that the monitoring system has similar sensors {${\mathit{x}}_1$, ${\mathit{x}}_2$, $\cdots$ ${\mathit{x}}_i$}, and the sensor with missing data is ${\mathit{y}}_i$, the regression equation is as follows:

$${\mathit{y}}_i=b_0+b_1{\mathit{x}}_1+b_2{\mathit{x}}_2+\cdots +b_i{\mathit{x}}_i+u_i$$

(1)

where ${\mathit{y}}_i$ is the variable that needs to be estimated; ${\mathit{x}}_1$, ${\mathit{x}}_2$, $\cdots$ ${\mathit{x}}_i$ are the monitoring data of similar sensors; $b_0$, $b_1$, $\cdots$ $b_i$ are the coefficients of the multiple linear regression equation, and $u_i$ is a random variable.

The abnormal temperature monitoring data repaired using the multiple linear regression method are illustrated in Figure 4. Figure 4a depicts the temperature time history curve for the period from 1 to 10 January 2014. ${\mathit{y}}_i$ represents the monitoring data of structural temperature sensor WD0101 in section NO7 (name of layout location section), ${\mathit{x}}_1$ represents the monitoring data of structural temperature sensor WD0201 in section NA6, and ${\mathit{x}}_2$ represents the monitoring data of structural temperature sensor WD0301 in section North Tower. The multiple linear regression equation for this example is $y_i=0.1282+1.0026x_1+0.0003x_2$. Figure 4b presents the reconstructed temperature time-history curve between 11 and 20 January 2014, generated through the established multiple regression model. The calculated WD0101 curve demonstrates strong alignment with the actual monitoring data, particularly evident in their comparable temporal patterns and fluctuation characteristics. This consistency between the regression-estimated values and empirical measurements validates the effectiveness of multiple linear regression methodology for reconstructing incomplete monitoring datasets. The model’s capability to accurately reconstruct missing structural health monitoring data, leveraging multivariate correlation analysis, is validated by its successful replication of the temperature profile over the 10-day period.

2.3.2. Repair Method for Missing Abnormal Monitoring Data Based on Interpolation

The repair method for missing abnormal monitoring data based on interpolation is applicable to monitoring data with less missing data. The interpolation method is not suitable for monitoring data with a large amount of missing data. The interpolation method is applied when the amount of missing data is less than 10 sampling points or does not span the time scale of signal features [27]. This is because repairing large amounts of missing data often leads to peak loss, compromising correction accuracy. Figure 5 shows the missing abnormal displacement monitoring data repaired using interpolation. Due to the sensor failure causing a relatively short loss of mid-span displacement, the interpolation method can be used for repair. Figure 5a,b show the interpolation method repair results of the displacements in the longitudinal and vertical directions on 1 June 2014. It can be seen that the interpolation method can repair the monitoring data without similar sensors, but with less missing data.

2.4. Statistics of Wind Speed Monitoring Data After Preprocessing

Wind load represents a significant environmental force acting on bridge towers. Utilizing wind speed and direction sensors enables real-time monitoring of wind conditions. This study provides a statistical analysis of wind speed patterns recorded over the years 2014 and 2015. The trend in average wind speed (AWS) fluctuations is presented in Figure 6a, while Figure 6b comparatively displays the evolving patterns of both 10 min average wind speed (10 min AWS) and peak instantaneous wind speed (IWS) measurements.

Table 2. Statistical features of wind speed on a monthly basis for the years 2014 and 2015.

Month	Statistical Analysis of Wind Speeds in 2014 (m/s)			Statistical Analysis of Wind Speeds in 2015 (m/s)
	AWS	10 Min AWS	IWS	AWS	10 Min AWS	IWS
1	3.16	9.42	15.66	3.19	9.46	15.73
2	4.19	10.26	16.01	4.25	10.35	16.21
3	3.52	9.11	17.39	3.71	9.21	17.45
4	3.37	11.16	17.34	3.26	11.16	17.21
5	2.48	7.50	12.34	2.50	7.50	12.46
6	2.97	8.08	12.88	2.79	8.12	12.70
7	2.60	15.35	23.07	2.55	15.45	20.12
8	3.05	11.97	15.75	3.15	11.77	15.15
9	3.45	8.55	15.94	3.46	8.56	15.66
10	3.19	9.72	16.09	3.21	9.89	16.21
11	3.20	8.23	15.44	3.26	8.11	15.61
12	3.17	9.95	17.15	3.22	9.81	17.31

shows the detailed monitoring data values corresponding to Figure 6. As depicted in Figure 6a and Table 2, the annual average wind speeds for both 2014 and 2015 maintained a consistent range of 2–5 m/s. The recorded extremes showed remarkable similarity between years, with 2014 reaching peak and minimum values of 4.19 m/s and 2.48 m/s, respectively, while 2015 exhibited comparable extremes of 4.25 m/s and 2.50 m/s. A more pronounced fluctuation pattern emerges when examining the 10 min average wind speeds in Figure 6b and Table 2. The 2014 data revealed significant variations between 7.50 m/s and 15.35 m/s, with 2015 demonstrating nearly identical parameters (7.50 m/s to 15.45 m/s). Notably, instantaneous wind speeds reached their annual peaks during July in both years, recording 23.07 m/s (2014) and 20.12 m/s (2015), while May witnessed the lowest instantaneous speeds at 12.34 m/s and 12.46 m/s, respectively. The consistent elevation of instantaneous wind speeds above their 10 min average counterparts suggests the substantial impact of wind gust phenomena on short-duration wind patterns.

Considering that a relatively high wind field occurred in July, a statistical analysis of the wind speed in July 2014 was then conducted. Figure 7a,b respectively show the conventional wind speeds in July and on 23 July 2014. The monitoring system data presented in Figure 7a show significantly elevated standard wind speed readings on multiple dates in July 2014, specifically on the 2nd, 7th, 12th, 17th, 24th, and 31st of the month. As depicted in Figure 7b, the conventional wind speed reached its maximum from 10:00 to 17:00, with measured velocities surpassing 15.75 m/s.

3. Improved Multi-Rate Fusion Method

3.1. Improved Multi-Rate Fusion Method Theory

Correlation modeling between environmental factors and structural responses underpins performance warning systems. However, displacement responses are susceptible to noise contamination, degrading the accuracy of the correlation data. Traditional multi-rate fusion techniques prove inadequate for reconstructing acceleration measurements [28,29,30]. This study proposes a modified multi-rate fusion approach incorporating dual Kalman gain coefficients, specifically designed to enhance the adaptability of the method for bridge structural health monitoring applications. A novel multi-sensor synchronization framework was developed to enhance measurement accuracy. It employs hybrid signal processing to cross-calibrate collocated displacement and acceleration recordings, reconciling temporal–spatial data and optimizing signal fidelity in both time and frequency domains. The measurement equations governing acceleration and displacement can be written as follows:

$$\left[\begin{array}{l}\dot{x}\\ \ddot{x}\\ \stackrel{⃛}{x}\end{array}\right]=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right]\left[\begin{array}{l}x\\ \dot{x}\\ \ddot{x}\end{array}\right]+\left[\begin{array}{l}0\\ 0\\ 1\end{array}\right]z_c$$

(2)

$$\left[\begin{array}{l}{x}_d\\ {\ddot{x}}_a\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{l}x\\ \dot{x}\\ \ddot{x}\end{array}\right]+\left[\begin{array}{l}{z}_d\\ z_a\end{array}\right]$$

(3)

where $x$, $\dot{x}$, and $\ddot{x}$ are the fused acceleration, velocity, and displacement, respectively; $z_d$, $z_a$, and $z_c$ are the associated measurement noise of displacement and acceleration; and $x_d$ and ${\ddot{x}}_a$ are the measured displacement and acceleration, respectively.

Time update of the improved multi-rate fusion method:

$$\widehat{x}(k+\left.1\right|k)=F_d\widehat{x}(\left.k\right|k)$$

(4)

$$P(k+\left.1\right|k)=F_{d}P(\left.k\right|k)F_d^T+Q_d$$

(5)

where $\widehat{x}(k+\left.1\right|k)$ and $\widehat{x}(\left.k\right|k)$ are the a posteriori estimates of the state vector before and after the time update, respectively; $P(k+\left.1\right|k)$ and $P(\left.k\right|k)$ are the a posteriori estimates of the system variance before and after the time update, respectively; and $F_d$ and $Q_d$ are the measured system matrix and the covariance matrix, respectively.

Measurement updates for the improved multi-rate fusion method:

$$\widehat{x}(k+\left.1\right|k+1)=\widehat{x}(k+\left.1\right|k)+K_d(k+1)\left[z_d(k+1)-H_d\widehat{x}(k+\left.1\right|k)\right]$$

(6)

$$P^{-1}(k+\left.1\right|k+1)=P^{-1}(k+\left.1\right|k)+H_d^T{R}_d^{-1}{H}_d+H_a^T{R}_a^{-1}{H}_a$$

(7)

where $\widehat{x}(k+\left.1\right|k+1)$ and $\widehat{x}(k+\left.1\right|k)$ are the a posteriori estimates of the state vector before and after the measurement update, respectively; $P(k+\left.1\right|k+1)$ and $P(k+\left.1\right|k)$ are the a posteriori estimates of the system variance before and after the measurement update, respectively; $F_d$ and $Q_d$ are the measured system matrix and the covariance matrix, respectively; $z_d$ is the measured displacement value; $H_d$ and $H_a$ are the measurement matrices for displacement and acceleration, respectively; and $R_d$ and $R_a$ are the variances of displacement and acceleration measurement noise, respectively.

The Kalman gain matrices of Equations (6) and (7) are

$$K_d(k)=P(\left.k\right|k)H_d^T{R}_d^{-1}(k)$$

(8)

$$K_a(k)=P(\left.k\right|k)H_a^T{R}_a^{-1}(k)$$

(9)

To address the discrepancy in sampling rates between acceleration and displacement sensors in bridge monitoring systems, this study introduces a multirate Kalman filtering-based fusion approach for synchronizing heterogeneous measurements from acceleration sensors (with higher sampling rates) and displacement sensors (with lower sampling rates). The displacement sensor sampling rate is $T_a$, the acceleration sensor sampling rate is $T_d$,$T_d$/$T_a$=$M$, and $M$ is a positive integer. In non-$k{T}_d$, where $R_d$ approaches infinity and $K_d$ approaches zero, only time and measurement updates are required for acceleration. At time $k{T}_d$, the displacement and acceleration measurements are updated simultaneously.

$$\widehat{x}(k+\left.1\right|k+1)=\widehat{x}(k+\left.1\right|k)+K_a(k+1)\left[z_a(k+1)-H_a\widehat{x}(k+\left.1\right|k)\right]$$

(10)

$$P^{-1}(k+\left.1\right|k+1)=P^{-1}(k+\left.1\right|k)+H_a^T{R}_a^{-1}{H}_a$$

(11)

To enhance the precision of state estimation, this study introduces a hybrid smoothing methodology that synergistically integrates fixed-point smoothing and fixed-interval smoothing techniques to refine the fused outputs. The augmented state vector within the unified smoothing framework can be mathematically formulated as follows:

$$\widehat{x}(\left.k\right|M)=\widehat{x}(\left.k\right|k)+A(k)\left[\widehat{x}(k+\left.1\right|M)-\widehat{x}(k+\left.1\right|k)\right]$$

(12)

$$A(k)=P(k+\left.1\right|k)F^T{P}^{-1}(k+\left.1\right|k)\hspace{1em}\hspace{1em}k=M-1,M-2,\cdots 0$$

(13)

where $X(\left.k\right|M)$ is the estimator for the smooth estimate of Kalman at the $k+1$ step, and $A(k)$ is the smoothing gain matrix.

3.2. Comparison of the Simulation Data Corrected by the Traditional Multi-Rate Fusion Method and the İmproved Multi-Rate Fusion Method

Compared with the mechanical engineering department and the Bayesian method, this paper selects the multi-rate Kalman method for fusion because the input terms of the Kalman method are displacement and acceleration data, which are consistent with the monitoring data. Compared with the traditional multi-rate fusion method, the introduction of two Kalman gain coefficients and other measures has improved the traditional multi-rate fusion method. The advantage of the improved multi-rate fusion method is that it corrects both displacement and acceleration simultaneously and ultimately improves the modeling accuracy of displacement and wind speed.

This study uses an improved multi-rate fusion method to calibrate simulation data, validated specifically with a linearly modulated sinusoidal chirp signal. In the simulated multi-rate sensor system, acceleration is sampled at 1 kHz while displacement is sampled at 100 Hz, resulting in a decimation factor of 10 between the two signals. To realistically replicate real-world bridge structural monitoring conditions, the synthetic dataset incorporates additive white Gaussian noise (AWGN) with a relative intensity corresponding to 10% of the signal’s nominal amplitude, consistent with typical noise characteristics observed in civil engineering instrumentation systems. The simulated signal, the velocity equation, and the acceleration equation are represented as follows:

$$x(t)=\sin [(at+b)t]+ct$$

(14)

$$\dot{x}(t)=(2at+b)\cos [(at+b)t]+c$$

(15)

$$\ddot{x}(t)=2a\cos [(at+b)t]-{(2at+b)}^2\sin [(at+b)t]$$

(16)

where $a$ = $2\pi (f_2-f_1)$/$T$; $f_1$ and $f_2$ are the start and end frequencies, respectively; $T$ is the sampling time; and $c$ is the linear term coefficient.

The results presented in Figure 8 illustrate a fundamental constraint in conventional multi-rate fusion methodologies applied to signal correction. The algorithm, while yielding displacement and velocity outputs (Figure 8a–c), is incapable of providing a complete reconstruction of the system state. More critically, as evidenced by the substantial error magnitudes shown in Figure 8b–d, the conventional approach demonstrates unsatisfactory correction accuracy, particularly in dynamic response characteristics and parameter consistency maintenance. Figure 9 demonstrates the performance of the enhanced multi-rate fusion algorithm in calibrating simulated signals. As illustrated in Figure 9a,c,e, the refined methodology successfully generates integrated outputs for displacement, velocity, and acceleration measurements. A comparative analysis between the error profiles in Figure 8b–d and their counterparts in Figure 9b,d,f reveals a marked reduction in calibration inaccuracies when employing the upgraded fusion approach. This improvement highlights the enhanced precision achieved through the optimized multi-rate data integration technique.

Table 3. RMSE of improved multi-rate Kalman fusion and smoothing results.

Methods	Measured Value	Multi-Rate Fusion	Smoothed
Traditional multi-rate fusion	Displacement	0.128	0.116
	Velocity	0.229	0.213
Improved multi-rate fusion	Displacement	0.032	0.027
	Velocity	0.201	0.190
	Acceleration	1.247	1.124

compares the RMSE of conventional and enhanced multi-rate fusion methods for simulated signal correction, shown in Figure 8 and Figure 9. The evaluation incorporates both direct error measurements from Equations (10) and (11) and smoothed error results derived from Equations (12) and (13). Notably, while the conventional approach only provides displacement and velocity outputs with respective RMSE values of 0.116 and 0.213, the upgraded methodology expands measurement capabilities to include acceleration estimation alongside improved displacement (0.027 RMSE) and velocity (0.190 RMSE) precision. This comparative analysis demonstrates the enhanced multi-rate fusion method’s superior performance in data correction accuracy across multiple measurement dimensions.

3.3. Improved Multi-Rate Fusion Method Corrects the Displacement and Acceleration Monitoring Data

This study analyzes four sets of monitoring data gathered during intense wind conditions. An 8 h dataset was recorded from 02:00 to 10:00 on 4 July 2015. The subsequent recording lasted 7 h, from 01:00 to 08:00 on 5 July 2015. The third monitoring session occurred between 10:00 and 15:00 on 18 July 2014 (5 h), with the final dataset captured from 08:00 to 13:00 on 22 July 2014 (5 h). Collectively, these four monitoring periods encompass 25 h of displacement measurements. Employing displacement sensors with a 1 Hz sampling frequency, the combined dataset contains 90,000 individual data points. The integrated processing outcomes of the comprehensive displacement monitoring are shown in Figure 10. As evidenced in Figure 10, the fusion methodology effectively adjusts lateral displacement measurements for both the bridge’s northern and southern towers.

While traditional multi-rate fusion outputs displacement and velocity, its enhanced version also estimates acceleration, enabling simultaneous calibration of acceleration monitoring data. Figure 11 demonstrate that the upgraded methodology effectively adjusts lateral acceleration measurements for both the north and south tower apexes. These refined acceleration values exhibit sufficient precision for subsequent engineering applications, particularly in structural modal analysis and potential damage detection for bridge structures. The improved system’s capacity for triaxial motion parameter estimation (displacement, velocity, and acceleration) significantly expands its utility in structural health monitoring applications.

The frequency-domain outcomes of the enhanced multi-rate fusion approach are presented in Figure 12, with Figure a-b illustrating the spectral analyses for the north and south towers, respectively. Both subfigures demonstrate that the fused displacement effectively integrates low-frequency GPS data and high-frequency accelerometer measurements. The fused displacement signal effectively combines the low-frequency stability of GPS measurements with the high-frequency resolution of accelerometer data, leveraging the complementary strengths of both sensors. The frequency-domain verification confirms that the improved multi-rate fusion methodology successfully calibrates and enhances the reliability of bridge monitoring data through this hybrid signal integration.

4. Correlation Modeling of Lateral Wind Speed and Displacement of Bridge Towers

4.1. Correlation Analysis of Lateral Wind Speed and Displacement of Bridge Towers

Figure 13 displays the integrated correlation analysis of lateral wind speed and displacement, based on 10 min averaged monitoring data. The analysis encompasses 25 h of continuous monitoring (yielding 150 data points after 10 min interval averaging) for both the north and south towers. In the graphical representation, dashed lines indicate wind speed measurements while solid lines denote displacement values, with Figure 13 illustrating the analytical outcomes for the northern and southern towers, respectively. The effectiveness of the enhanced multi-rate data fusion method is clear, with strong positive correlations between lateral wind forces and structural displacements evident in both tower configurations after correction.

Table 4. Correlation coefficients of lateral wind speed and displacement of bridge towers before and after fusion.

Position	Correlation Coefficient
	Before Fusion	After Fusion	Difference Value	Percentage Increase
North tower	0.753	0.818	0.065	8.63%
South tower	0.781	0.842	0.061	7.81%

displays the Pearson correlation coefficients for lateral wind speeds and bridge tower displacements, both prior to and following data fusion. As a statistical measure ranging from −1 to 1, these coefficients quantitatively reflect the strength of association between wind loads and structural responses. The analysis reveals that the north tower’s correlation values increased from 0.753 to 0.818 post-fusion, demonstrating a 0.065 absolute difference and 8.63% relative enhancement in parameter association. Similarly, the south tower exhibited a 7.81% improvement in correlation strength following data integration. These statistically significant enhancements confirm that the multi-rate sensor fusion methodology enables more precise structural modeling by effectively combining heterogeneous measurement data. The elevated correlations, particularly in lateral wind–displacement relationships, suggest improved system identification accuracy through the implemented fusion approach.

Table 5. Statistical significance p-values and mean absolute errors (MAEs) of lateral wind speed and displacement of bridge towers before and after fusion.

Position	Statistical Significance p-Value		Mean Absolute Error (MAE)
	Before Fusion	After Fusion	Before Fusion	After Fusion
North tower	$6.421\times {10}^{-33}$	$2.738\times {10}^{-37}$	6.06	5.16
South tower	$3.145\times {10}^{-37}$	$1.876\times {10}^{-41}$	6.17	5.23

shows that the statistical significance p-values and mean absolute errors (MAEs) of the fused data become smaller in the north tower and the south tower, indicating stronger statistical significance.

Table 6. Coefficients of determination $R^2$ of lateral wind speed and displacement of bridge towers before and after fusion.

Position	$\mathbf{Coefficient} \mathbf{of} \mathbf{Determination} {\mathit{R}}^2$
	Before Fusion	After Fusion	Difference Value	Percentage Increase
North tower	0.567	0.669	0.102	17.99%
South tower	0.610	0.709	0.099	16.23%

shows that the coefficient of determination after fusion decreases by 17.99% in the north tower and 16.23% in the south tower. The data modeling accuracy after the same surface fusion is higher.

4.2. Regression Model of Lateral Wind Speed and Displacement of Bridge Towers

Linear regression was performed following correlational analysis of lateral wind velocity and displacement data corrected using multi-rate fusion. Figure 14 illustrates the regression model’s comparative results between lateral wind speed and bridge tower displacement. In Figure 14a,b, which, respectively, display the north and south tower analyses with 95% confidence intervals, the horizontal axis represents lateral wind speed while the vertical axis indicates the corresponding lateral displacement of bridge towers. The results demonstrate a significant positive correlation between the lateral wind speed and the fusion-corrected lateral displacement of both bridge towers.

Table 7. The fitting equations and the confidence interval widths of the lateral bridge wind speed and displacement of bridge towers.

Position	Fitting Equations	Width of Confidence Interval
		Before Fusion	After Fusion	Difference Value	Percentage Decrease
North tower	D = 2.243 V − 3.859	4.676	4.233	0.443	9.47%
South tower	D = 2.340 V − 4.086	5.281	4.853	0.428	8.10%

presents the regression models and confidence interval (CI) widths for lateral wind speed and displacement of the bridge towers. The fitted equations for the north and south towers are D = 2.243 V − 3.859 and D = 2.340 V − 4.086, respectively. The residuals of these regression models serve as performance warning indicators for the bridge. Prior to data fusion, the CI widths for the north and south towers were 4.676 and 5.281, respectively. After fusion, these decreased to 4.233 and 4.853, reflecting reductions of 0.443 and 0.428. This corresponds to 9.47% and 8.10% reductions in CI width for the north and south towers, respectively. The narrower confidence intervals indicate enhanced modeling precision post-fusion.

5. Bridge Tower Damage Performance Warning

The displacement response of the bridge tower increases with the occurrence of cracks or stiffness degradation [31,32]. Wang et al. [33,34] set the modeling residuals as warning indices and applied them to the monitoring of mid-span and cable warnings. The bridge tower, functioning as a compression-bearing component, necessitates an investigation into its mechanical characteristics, including compressive resistance to ensure operational reliability [35,36]. In this study, structural failure was modeled by progressively applying lateral displacements to both the north and south bridge towers. The critical displacement thresholds associated with structural damage are mathematically defined as follows:

$$S_{de}=S-\Delta$$

(17)

where $S$ is the actual displacement of the bridge, $\Delta$ is the degradation of the displacement, and $S_{de}$ is the simulated value of the displacement after damage. The concluding flowchart of the performance warning method is shown in Figure 15.

Table 8. Bridge tower performance warning cases.

North Bridge Tower Degradation Simulation (mm)				South Bridge Tower Degradation Simulation (mm)
Case	Before Fusion	Case	After Fusion	Case	Before Fusion	Case	After Fusion
1	$\epsilon$ = 0	7	$\epsilon$ = 0	13	$\epsilon$ = 0	19	$\epsilon$ = 0
2	$\epsilon$ = 5	8	$\epsilon$ = 5	14	$\epsilon$ = 5	20	$\epsilon$ = 5
3	$\epsilon$ = 10	9	$\epsilon$ = 10	15	$\epsilon$ = 10	21	$\epsilon$ = 10
4	$\epsilon$ = 15	10	$\epsilon$ = 15	16	$\epsilon$ = 15	22	$\epsilon$ = 15
5	$\epsilon$ = 20	11	$\epsilon$ = 20	17	$\epsilon$ = 20	23	$\epsilon$ = 20
6	$\epsilon$ = 25	12	$\epsilon$ =25	18	$\epsilon$ = 25	24	$\epsilon$ = 25

presents a bridge tower performance warning simulation using Equation (17), encompassing 24 damage scenarios. Case 1 corresponds to the baseline (undamaged) condition of the north tower before any damage is introduced. Structural degradation severity increases progressively from Case 2 to Case 6. Case 7 corresponds to the north tower’s baseline state following fusion, followed by incrementally increasing structural deterioration from Case 8 to Case 12. Similarly, Case 13 denotes the initial intact condition of the south tower before fusion, with successive degradation stages spanning Case 14 to Case 18. Case 19 establishes the post-fusion reference state for the south tower, after which structural integrity progressively diminishes from Case 20 through Case 24.

This study defines three significance levels (0.05, 0.01, and 0.003), corresponding to Threshold 1, Threshold 2, and Threshold 3, respectively. Threshold 1, with the narrowest interval, is the most sensitive for triggering warnings, while Threshold 3, characterized by the widest interval, requires the strongest statistical evidence to activate a warning.

Table 9. Warning rates of north tower performance degradation.

Case	North Tower Warning Rate Before Fusion (mm)			Case	North Tower Warning Rate After Fusion (mm)
	$\mathit{\alpha }$ = 0.05	$\mathit{\alpha }$ = 0.01	$\mathit{\alpha }$ = 0.003		$\mathit{\alpha }$ = 0.05	$\mathit{\alpha }$ = 0.01	$\mathit{\alpha }$ = 0.003
1	0	0	0	7	0	0	0
2	5.71	0	0	8	11.43	8.57	5.71
3	31.43	20.00	17.14	9	31.43	31.43	22.86
4	48.57	40.00	40.00	10	68.57	62.86	62.86
5	74.29	71.43	60.00	11	88.57	88.57	85.71
6	91.43	85.71	82.86	12	100	100	100

shows the damage warning rate of the north tower. From Case 1 to Case 6 in Table 9, it can be seen that no warning occurred before fusion. It can be seen from Table 9 that the warning rate reaches 100% after fusion in Case 12, indicating that damages larger than 25 mm can be detected.

Table 10. Warning rates of south tower performance degradation.

Case	South Tower Warning Rate Before Fusion (mm)			Case	South Tower Warning Rate After Fusion (mm)
	$\mathit{\alpha }$ = 0.05	$\mathit{\alpha }$ = 0.01	$\mathit{\alpha }$ = 0.003		$\mathit{\alpha }$ = 0.05	$\mathit{\alpha }$ = 0.01	$\mathit{\alpha }$ = 0.003
13	0	0	0	19	0	0	0
14	8.57	2.857	0	20	8.57	5.71	5.71
15	34.29	25.71	20.00	21	31.43	31.43	22.86
16	51.43	42.86	40.00	22	65.71	60.00	57.14
17	74.29	71.43	68.57	23	88.57	88.57	82.86
18	88.57	82.86	82.86	24	100	100	100

shows the warning rate of the south tower. After fusion, the warning rate reaches 100% in Case 24, indicating that the south tower can also detect damage larger than 25 mm.

Figure 16 displays the fused performance warning results for the north tower at a significance level of 0.01. Figure 16a (Case 7) depicts the structure’s baseline condition during normal operation. Figure 16b–f (Cases 8–12) show a progressive increase in alarm frequency. Particularly in Figure 16f, all monitored parameters have surpassed the predefined safety threshold, achieving full alarm activation (100% warning rate). This comprehensive alert response confirms the north tower’s monitoring system possesses sufficient sensitivity to identify structural damages exceeding 25 mm in magnitude. Figure 17 shows the south tower performance warning results after fusion, with a significance level of 0.01. It can be seen that the warning rate reaches 100% in Figure 17f, indicating that the south tower can also detect damage greater than 25 mm.

6. Conclusions

This study investigates the monitored data from a cable-stayed bridge exposed to severe wind conditions. The precision in modeling environmental effects and structural response influences the effectiveness of performance alerts. Improved measurement accuracy of wind speed and displacement parameters enhances modeling reliability. This paper proposes a precise modeling method for environmental effects and structural response using an enhanced multi-rate fusion technique, first applied to bridge tower performance warning using measured data. Validation procedures involve both synthetic test cases featuring linear-superimposed sinusoidal sweep signals and empirical datasets comprising bridge displacement, acceleration, and wind speed measurements. The principal findings of this research are summarized as follows:

A repair method for skipped points and missing abnormal monitoring data is proposed, along with a statistical analysis of bridge wind speed monitoring data. During the period of 2014 to 2015, the average wind speed consistently maintained fluctuations within the 2–5 m/s range, with both the 10 min mean wind speed and peak instantaneous wind speed exhibiting identical temporal profiles in their time-series patterns.

The improved multi-rate data fusion approach presented in this study, which incorporates two Kalman gain coefficients, enables simultaneous calibration of both displacement and acceleration measurements for bridge tower monitoring. Frequency-domain analysis demonstrates that the synthesized displacement data exhibit alignment with GPS measurements in the low-frequency range and converge with accelerometer recordings in the high-frequency spectrum. The fused displacement simultaneously possesses the low-frequency advantages of GPS and the high-frequency advantages of an accelerometer.

The precise simulation of lateral wind speed and displacement dynamics for the bridge towers was successfully achieved. Following data fusion, correlation evaluations between wind speed and displacement revealed respective enhancements of 8.63% and 7.81% in correlation coefficients for the north and south towers, respectively. Subsequent regression modeling demonstrated improved predictive reliability, evidenced by 9.47% and 8.10% reductions in confidence interval ranges for the north and south towers, respectively. These numerical gains confirm the improved accuracy of the combined wind speed–displacement model.

A study was conducted on the structural performance alerts of a cable-stayed bridge tower under strong wind conditions, with 24 test cases validating the effectiveness of a data fusion methodology. Both the warning rate table and the warning control chart show that the pre-fusion model generated no warnings. However, the enhanced fusion model demonstrated exceptional performance, achieving 100% warning accuracy for the north tower in Case 12 and the south tower in Case 24. These results confirm the model’s enhanced capability in reliably identifying structural damage exceeding 25 mm displacement thresholds.