1 Introduction

Bridges have become a key pivot in transportation infrastructure. Nowadays, numerous bridges are in service with the increasing sophistication of modern technology, like Hong Kong-Zhuhai-Macao Bridge, Akashi Kaikyo Bridge, and Golden Gate Bridge. However, some extreme environments may affect the safety of bridges [1, 2]. Therefore, bridge health monitoring has attracted widespread attention in recent years [3]. To ensure the proper function of bridges, it is necessary to assess the bridge structures in time.

The dynamic responses of a bridge are the results of the interaction between the external excitation and the structure, and contain various types of information about the structural state. With the development of sensor technology, a growing number of sensor monitoring systems are installed to collect structural dynamic responses. Among them, the global navigation satellite system real-time kinematic (GNSS-RTK) and accelerometers are commonly employed [4,5,6]. They can acquire three-dimensional displacement and acceleration of targets in real-time under almost any weather. One of the purposes of measuring response data is to identify the structural modal parameter (i.e., frequency, damping ratio, and mode shape), where the frequency is easier to obtain and is the most common modal parameter for bridge assessment. For example, Yu et al. [7] adopted a multi-positioning mode GNSS system to get the displacement and frequency of a mid-span bridge. Moschas et al. [8] determined the dynamic deflections and dominant frequencies of a pedestrian bridge through multiple sensors based on GNSS and accelerometers. Xiong et al. [9] extracted the first five-order modal frequencies of a suspension bridge from the vibration monitoring data of the GNSS and accelerometer. Although GNSS and accelerometer have advantages in bridge response measurement, there are still some challenges. First, the response data (especially measured by high sampling rate sensors) is affected by noise pollution, which influences the identification of the modal parameters as well. Second, internal and external factors such as sensor failures, extreme loads, poor weather, and so on make it difficult or even impossible to collect complete response data and thus may fail to capture structural modal information.

For the first problem, nowadays the time–frequency domain analysis methods can decompose the signal responses into several discrete intrinsic mode functions (IMF) to filter the noisy components. For instance, empirical mode decomposition (EMD) [10,11,12], empirical wavelet transform (EWT) [13,14,15], empirical adaptive wavelet decomposition [16], ensemble EMD (EEMD) [17,18,19,20], complete EEMD with adaptive noise (CEEMDAN) [21,22,23], and variational mode decomposition (VMD) [24,25,26,27]. He et al. [28] extracted the modal frequencies of the bridge via EMD and random decrement technique. Li et al. [29] investigated the time-varying characteristics of bridge frequencies under vehicle-bridge interactions based on the improved EWT and ridge detection method. Xiao et al. [30] applied CEEMDAN to capture the bridge frequency characteristics from the acceleration response. Nevertheless, the spectral segmentation of EWT is usually susceptible to noise. One major problem with EMD is mode mixing. EEMD alleviates mode mixing but does not eliminate the added auxiliary noise, which makes it unable to reach the desired decomposition. As for CEEMDAN, although it avoids mode mixing, the emerging problem is that it decomposes redundant IMF components and leads to a lag in the information components.

Fortunately, the VMD proposed by Dragomiretskiy et al. [31] is an innovative method that decomposes the signal response into inherent IMF components. It has no mode mixing and has better robustness than EMD and other algorithms. Yang et al. [32] presented VMD with a band-pass filter to deal with the contact point response of vehicles and bridges. The study demonstrated the ability of the approach to extract structural frequencies. Mazzeo et al. [33] estimated the frequency and damping ratio of a cable-stayed bridge by VMD. The only issue with VMD is the difficulty in determining the preset number of decomposition parameter K. If a reliable and convenient criterion can be set up to decide the optimal number of K, the adaptiveness of the VMD will be enhanced. Although some studies proposed enhanced VMD algorithms, they are usually found in bearing fault diagnosis. Consequently, this study develops an improved VMD algorithm for large bridge structures.

For the second problem, machine learning algorithms are utilized for prediction to recover/increase the completeness of the response data, such as random forest [34], Bayesian [35,36,37], long short-term memory (LSTM) [38,39,40,41], and other neural networks [42,43,44,45,46,47], etc. Among numerous machine learning techniques [48, 49], neural networks are particularly popular in response prediction. Oh et al. [50] discussed the data recovery performance of neural networks in the numerical study and frame structure analysis. Betti et al. [51] evaluated the natural frequencies of a three-story steel frame at each damage level using artificial neural networks and genetic algorithms. However, some studies ignore whether the information that helps to assess the structural characteristics, such as modal parameters, can be extracted from the predicted response data. In other words, if the structural characteristic is not available from the predicted response data, the prediction results lose some practicality. Therefore, this paper applies the genetic algorithm-back propagation (GA-BP) neural network to the prediction of the bridge response data. Meanwhile, the predicted responses are used for modal frequency estimation to characterize the usefulness of prediction results.

Inspired by the above research, a combination approach of the optimized VMD (OVMD) and the GA-BP neural network is proposed and applied to the predictive evaluation of dynamic response and modal frequencies of a cable-stayed bridge. The OVMD algorithm aims to reduce the noise in the data and address the shortcomings of the VMD algorithm. In the GA-BP operation, to improve prediction accuracy, the selected IMF components are predicted individually to form the predicted responses. The frequency of the structure is finally estimated from the predicted restructured responses. Section 2 describes the methodology. Section 3 is a damaged arch bridge test to primarily examine the performance of OVMD. Section 4 processes the data responses of the field bridge experiment adopting the combined OVMD and GA-BP approach and performs a comparative analysis to verify the superiority of the proposed method. Section 5 summarizes the main conclusions.

2 Methodology

2.1 VMD

VMD divides the signal into a specified number of mode components, then determines the center frequency and bandwidth of different mode component signals [52].

  1. 1.

    The mode component signal has the following form:

    $$ u_{k} (t) = A_{k} (t)\cos \left[ {\varphi_{k} (t)} \right]\quad(k = 1,\;2,\;3,\;...,\;K) $$
    (1)

    where \(u_{k} (t)\) is each mode component; \(A_{k} (t)\) is the instantaneous amplitude; \(\varphi_{k} (t)\) is the instantaneous phase.

  2. 2.

    The unilateral spectrum of each mode component is calculated through the Hilbert transform and is tuned to the estimated center frequency by multiplying the exponential term. The square of the signal gradient norm after frequency mixing is calculated. The bandwidth of the mode component after the frequency shift is estimated. Thus a constrained variational problem is constructed:

    $$ \left\{ {\begin{array}{*{20}c} {\mathop {\min }\limits_{{\left\{ {u_{k} } \right\},\;\left\{ {\omega_{k} } \right\}}} \left\{ {\sum\limits_{k = 1}^{K} {\left\| {\partial_{t} \left[ {\left( {\delta (t) + \frac{{\text{j}}}{{{\uppi }t}}} \right) * u_{k} (t)} \right]{\text{e}}^{{ - {\text{j}}\omega_{k} t}} } \right\|_{2}^{2} } } \right\}} \\ {s.\;t.\sum\limits_{k = 1}^{K} {u_{k} (t) = } f(t)} \\ \end{array} } \right. $$
    (2)

    where \(\left\{ {u_{k} } \right\}\) and \(\left\{ {\omega_{k} } \right\}\) are the set of K mode component signals and the corresponding center frequencies; \(*\) represents the convolution symbol; \(\delta_{t}\) is the Dirac function; \(\partial_{t}\) denotes the partial derivative of t; \(f(t)\) is the input signal response.

  3. 3.

    The Lagrange multiplier \(\lambda\) and the quadratic penalty factor \(\alpha\) are introduced to replace the constrained variational problem with an unconstrained variational problem:

    $$ L(\left\{ {u_{k} } \right\},\left\{ {\omega_{k} } \right\},\lambda ) = \alpha \sum\limits_{k = 1}^{K} {\left\| {\partial_{t} \left[ {\left( {\delta (t) + \frac{{\text{j}}}{{{\uppi }t}}} \right) * u_{k} (t)} \right]{\text{e}}^{{ - {\text{j}}\omega_{k} t}} } \right\|_{2}^{2} } + \left\| {f(t) - \sum\limits_{k = 1}^{K} {u_{k} (t)} } \right\|_{2}^{2} + \left\langle {\lambda (t),f(t) - \sum\limits_{k = 1}^{K} {u_{k} (t)} } \right\rangle $$
    (3)
  4. 4.

    The alternating direction method of multipliers is adopted to update \(u_{k}\), \(\omega_{k}\), and \(\lambda\). Their expressions are as follows:

    $$ \hat{u}_{k}^{n + 1} (\omega ) = \frac{{\hat{f}(\omega ) - \sum\nolimits_{i \ne k} {\hat{u}_{i} (\omega ) + \frac{{\hat{\lambda }(\omega )}}{2}} }}{{1 + 2\alpha (\omega - \omega_{k} )^{2} }} $$
    (4)
    $$ \omega_{k}^{n + 1} = \frac{{\int_{0}^{\infty } {\omega \left| {\hat{u}_{k}^{n + 1} (\omega )} \right|^{2} {\text{d}}\omega } }}{{\int_{0}^{\infty } {\left| {\hat{u}_{k}^{n + 1} (\omega )} \right|^{2} {\text{d}}\omega } }}, $$
    (5)
    $$ \hat{\lambda }^{{n + 1}} (\omega ) = \hat{\lambda }^{n} (\omega ) + \tau \left[ {\hat{f}(\omega ) - \sum\limits_{{k = 1}}^{K} {\hat{u}_{k}^{{n + 1}} (\omega )} } \right] $$
    (6)

    where n is the number of iterations; τ is the noise tolerance; ‘\(\wedge\)’ represents the Fourier transform.

  5. 5.

    The iteration termination condition is:

    $$ \sum\limits_{k = 1}^{K} {\frac{{\left\| {\hat{u}_{k}^{n + 1} - \hat{u}_{k}^{n} } \right\|_{2}^{2} }}{{\left\| {\hat{u}_{k}^{n} } \right\|_{2}^{2} }} < \varepsilon } $$
    (7)

    where ε is the convergence accuracy and ε = 10–7. Once the accuracy condition is satisfied, the K mode components, namely IMF components, are output.

2.2 OVMD

It is difficult for the VMD algorithm to set the decomposition parameter K reasonably depending on subjective experience. Improper K impacts the signal decomposition effect. In addition, the VMD only focuses on signal decomposition and does not involve signal reconstruction. Based on this, the OVMD algorithm for optimal decomposition and efficient reconstruction of signal responses is proposed. The specific steps are below:

  1. i.

    Apply VMD on the input signal response and determine the IMF component with the highest correlation to the signal.

  2. ii

    Compute the RRMSE to update K iteratively. When RRMSE is a local minimum value, the corresponding K is optimal. That is, the signal response is optimally decomposed into K IMF components. Otherwise, it is needed to continue updating K iteratively. The expression of RRMSE is:

    $$ {\text{RRMSE}} = \sqrt {\frac{{\sum\nolimits_{t = 1}^{N} {\left[ {f(t) - C_{\max } {\text{IMF}}_{i} (t)} \right]^{2} } }}{{\sum\nolimits_{t = 1}^{N} {\left[ {f(t) - \overline{f} } \right]^{2} } }}} $$
    (8)

    where \(f(t)\) is the input signal response; \(C_{\max } {\text{IMF}}_{i} (t)\) denotes the IMF that has the highest correlation with \(f(t)\); \(\overline{f}\) is the mean value of \(f(t)\).

  3. iii

    As for a noisy signal, the noise is dispersed to the IMF components during the decomposition. Therefore, the correlation coefficient is presented to eliminate the noise components and choose valid IMF components to retain the signal information adequately. The correlation coefficient denotes the degree of correlation of each IMF component with the input signal response. Its expression is:

    $$ C = \frac{{\sum\nolimits_{t = 1}^{N} {({\text{IMF}}_{i} (t) - \overline{{{\text{IMF}}_{i} }} )} (f(t) - \overline{f} )}}{{\sqrt {\sum\nolimits_{t = 1}^{N} {{\text{(IMF}}_{i} (t) - \overline{{{\text{IMF}}_{i} }} )^{2} \sum\nolimits_{t = 1}^{N} {(f(t) - \overline{f} )^{2} } } } }} $$
    (9)

    where \({\text{IMF}}_{i} (t)\) is each IMF component; \(\overline{{{\text{IMF}}_{i} }}\) is the average value of \({\text{IMF}}_{i} (t)\).

The range of the correlation coefficient is 0 to 1. The closer the value is to 1, the stronger the correlation between the IMF and the input signal. When 0 ≤   C≤ 0.3, the IMF components are very weakly correlated with the input signal response and are considered noise-dominant components to remove. The rest of the components are retained. Figure 1 depicts the OVMD procedure.

Fig. 1
figure 1

Flowchart of the OVMD algorithm

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2.3 GA-BP neural network model

BP neural network [53] is a multilayer feed-forward neural network trained by an error backpropagation algorithm. The procedure of the BP neural network is divided into two main parts, i.e., forward propagation of the sample signal and backward feedback of the error. As depicted in Fig. 2, a typical BP structure includes the input layer, the hidden layer, and the output layer. Their number of nodes is expressed as n, L, and m, respectively. The input and predicted output values are expressed as Xi (i = 1, 2, …, n) and Yk (k = 1,2, …, m). Wij and Wjk mean the weights.

Fig. 2
figure 2

BP neural network structure

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The training procedure of the BP neural network is given below.

The BP network structure is established. The values of n, L, and m are determined; the thresholds of the hidden layer and output layer, i.e., a and b, are initialized. The learning rate and the activation function are decided. The expression of the activation function is:

$$ f(x) = \frac{2}{{1 + {\text{e}}^{ - 2x} }} - 1 $$
(10)

The output H of the hidden layer is calculated:

$$ H_{j} = f\left( {\sum\limits_{i = 1}^{n} {W_{ij} X_{i} } - a_{j} } \right)\;\;\;\;(j{ = }1,\;2,\;...\;,\;L) $$
(11)

where L is the number of hidden layer nodes.

The predicted output of the output layer Y is calculated according to H, b, and Wjk:

$$ Y_{k} = \sum\limits_{j = 1}^{L} {H_{j} W_{jk} } - b_{k} \;\;\;\;(k{ = }1,\;2,\;...\;,\;m) $$
(12)

The expected output O subtracts Y to obtain the prediction error e:

$$ e_{k} = O_{k} - Y_{k} $$
(13)

The weights are updated based on the prediction error:

$$ W_{ij} = W_{ij} + \eta H_{j} (1 - H_{j} )X_{i} \sum\limits_{k = 1}^{m} {W_{jk} e_{k} } $$
(14)
$$ W_{jk} = W_{jk} + \eta H_{j} e_{k} $$
(15)

where η is the learning rate.

The thresholds are updated:

$$ a_{j} = a_{j} + \eta H_{j} (1 - H_{j} )\sum\limits_{k = 1}^{m} {W_{jk} e_{k} } $$
(16)
$$ b_{k} = b_{k} + e_{k} $$
(17)

If the convergence condition is satisfied, the iteration ends. Otherwise, return to Eq. (11) and repeat the above steps.

The training process of the BP neural network is prone to trap the local optimum problem. The GA can optimize the structure of the BP model to improve its performance. In the GA-BP operation, firstly, the GA is employed to encode the initial values of weights and thresholds in the BP neural network. Subsequently, the optimal weights and thresholds are acquired by selection, crossover, and mutation. Finally, the optimal weights and thresholds from the GA are fed into the BP neural network for iterative operations to get the best prediction values. The GA-BP neural network model is shown in Fig. 3.

Fig. 3
figure 3

GA-BP neural network model

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2.4 Framework of the proposed approach

The application of the proposed OVMD combined with the GA-BP approach in Fig. 4 includes the following steps:

Fig. 4
figure 4

Application flow of the proposed approach

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  1. (a)

    The measured displacement and acceleration are taken as the input signal responses.

  2. (b)

    The signals are optimally decomposed based on RRMSE in the OVMD algorithm to yield a series of IMF components, and the correlation coefficient in OVMD is employed to eliminate the noisy components and retain the valid IMF components. The performance of OVMD is tested via comparative analysis.

  3. (c)

    Each retained IMF component is predicted by GA-BP, and the prediction results are superimposed to form the predicted reconstructed signals.

  4. (d)

    The structural frequencies and their characteristics are evaluated from the predicted reconstructed signals by the fast Fourier transform (FFT).

3 Damaged arch bridge test

The signal's proper decomposition and reconstruction is the key to the predictive evaluation. Here, a test analysis of a damaged bridge is made before the formal experiment on the target bridge. Meanwhile, the following factors are mainly considered: (1) check the behavior of GNSS-RTK under only natural excitation (no vehicle loads) at the same sampling rate as the target bridge; (2) verify the advantages of OVMD in the decomposition and reconstruction for the actual measured signal; (3) evaluate the noise reduction effect of OVMD in strong noisy signal response.

The Rainbow Bridge, situated in Tianjin, China, was built in 1998. The overall appearance is shown in Fig. 5. The structure is a through concrete-filled steel tube arch bridge. The length of the main bridge is 504 m; the width of the longitudinal beam is 29 m, as shown in Fig. 6. On September 23, 2022, it was found that the arch foot of the structure was abnormal and part of the tie bar steel strands was broken. The bridge began repairs in September 2022 with the replacement of all load-bearing rods. The bridge reopened to traffic on June 2, 2023, after nearly 9 months of maintenance. The test was conducted on March 10, 2023, and lasted 12 h. The GNSS-RTK receiver at measuring point P in Fig. 6 as a mobile station is placed in the middle of the main span downstream of the river. The RTK sampling rate is the same as the target bridge, i.e., 50 Hz. Partial vertical displacement response is intercepted for analysis, as seen in Fig. 7.

Fig. 5
figure 5

General view of the bridge

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Fig. 6
figure 6

Structure diagram (unit: m) and measurement point arrangement

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Fig. 7
figure 7

Displacement signal response

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During the test, the damaged bridge was under repair and closed to traffic. Compared with the dead weight of the structure, workers and construction vehicles on the bridge deck can be negligible. Because of the lack of vibration generated by vehicle loads, the signal in Fig. 7 is a weak dynamic displacement excited only by natural environments such as wind load and ground pulsation. Naturally, the influence of noise on the weak dynamic displacement is also more pronounced.

To attenuate the noise effect, a comparative noise reduction analysis of the displacement signal response is made employing OVMD and CEEMDAN. With the procedure of OVMD, the K is updated iteratively using RRMSE. Table 1 gives the results. When K is 7, the RRMSE shows a local minimum, which means that decomposing the signal into 7 IMF components is optimal. Simultaneously, we decompose the signal via CEEMDAN to yield 14 IMF components. The decomposition results of the two methods are depicted in Figs. 8 and 9. Obviously, CEEMDAN generates redundant IMF components; this is not conducive to the selection of valid components. In contrast, OVMD decreases the redundancy of IMF components.

Table 1 RRMSE values when K is iterated
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Fig. 8
figure 8

OVMD decomposition

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Fig. 9
figure 9

CEEMDAN decomposition

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Then, this paper adopts the correlation coefficient in OVMD to select the valid IMF components for reconstruction. Table 2 lists the correlation coefficients of each IMF component with the signal response in the two methods. The IMF components with correlation coefficients higher than 0.3, i.e., IMF1–IMF2 in OVMD, IMF7–IMF9 and IMF11–IMF13 in CEEMDAN, are taken as valid information and reconstructed. Furthermore, the wavelet soft threshold and EEMD-wavelet soft threshold are adopted to denoise the signal response. To estimate the performance of four methods, signal-to-noise ratio (SNR), root mean square error (RMSE), and correlation coefficient are treated as criteria. The expressions of SNR and RMSE are:

$$ {\text{SNR = 10log}}_{10} \left\{ {\frac{{\sum\nolimits_{t = 1}^{N} {x^{2} (t)} }}{{\sum\nolimits_{t = 1}^{N} {\left[ {x(t) - x^{\prime}(t)} \right]^{2} } }}} \right\} $$
(18)
$$ {\text{RMSE = }}\sqrt {\frac{1}{N}\sum\limits_{t = 1}^{N} {\left[ {x(t) - x^{\prime}(t)} \right]^{2} } } $$
(19)

where \(x(t)\) and \(x^{\prime}(t)\) denote the real signal and denoised signal. The larger the SNR, the stronger the noise reduction ability; the lower the RMSE, the smaller the signal error before and after noise reduction; the higher the correlation coefficient, the better the correlation between two sequences. As shown in Table 3, the denoised signal response after OVMD has the largest SNR (9.3209 dB), the smallest RMSE (0.0032 m), and the highest correlation (0.9374). This indicates that using the correlation coefficient presented in OVMD effectively reconstructs the signal with fewer IMF components. Also, the proposed OVMD algorithm has the best noise reduction performance.

Table 2 Correlation coefficients of each IMF component with the displacement response
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Table 3 Performance evaluation results of different methods
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The power spectral density (PSD) function of the signal reconstructed by OVMD is plotted in Fig. 10. It can be found that the first two-order frequencies (0.695 and 1 Hz) of the damaged bridge are different from those (0.6169 and 1.0960 Hz) before the damage [54]. Probably mainly structural damage and lack of moving vehicles cause variations in the frequency characteristics of the bridge.

Fig. 10
figure 10

PSD function detected based on OVMD

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In conclusion, the damaged bridge test reveals that the GNSS-RTK with 50 Hz can obtain the structural dynamic signal response even without significant vibration excitation and extracts structural frequency based on OVMD. OVMD method facilitates the optimal decomposition and reconstruction of the signal and shows superior noise reduction capability.

4 Practical application of a long-span cable-stayed bridge

This section applied the proposed method to the predictive evaluation of the responses and frequencies of a long-span cable-stayed bridge named Haihe Bridge using GNSS-RTK and acceleration data.

4.1 Bridge overview and experimental scheme

The Haihe Bridge, located in Tianjin, China, consists of two bridges with the same structure but built in different years, as shown in Fig. 11a. The old bridge was built in 2002 and the new bridge was officially opened to traffic in 2011. Two bridges in the Haihe Bridge are single-tower double cable-stayed bridges and are reinforced concrete composite structures. Haihe Bridge has eight lanes in both directions, the main girder is 3 m high. The total lengths of the old and new bridges are 2650 and 2033 m; the spans of the two main bridges are (310 + 3 × 48 + 46) m and (310 + 2 × 50 + 2 × 40) m; the tower heights are 167.3 and 164.8 m, respectively, as seen in Fig. 11b. Haihe Bridge has a heavy traffic flow every day. Given the old bridge was completed earlier, thus it is taken as the object of this study.

Fig. 11
figure 11

Haihe Bridge: a panoramic view of Haihe Bridge; b structure schematic (unit: m) and measuring position

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The GNSS-RTK mobile station and triaxial accelerometer with high sampling rates of 50 and 100 Hz, as displayed in Fig. 12a–b. They are positioned in the middle of the old main bridge (point M in Fig. 11b). The triaxial accelerometer is installed on the iron guardrail of the bridge sidewalk. Another GNSS receiver as a reference station (Fig. 12c) is placed on the smooth ground about 100 m away from the bridge. The erection heights of the mobile station and the reference station are 1.50 and 1.66 m, respectively. The experiment lasts from 9:00 a.m. to 6:00 p.m. on March 11, 2023, local time. In the monitoring period, the wind speed is 0.04–9.04 m/s, the temperature is 7.30–12.24 ℃, and the humidity is 25.71–43.27%RH.

Fig. 12
figure 12

Equipment arrangement: a triaxial accelerometer; b mobile station; and c reference station

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4.2 OVMD processing

This study selects part of vertical GNSS displacement and acceleration dynamic signal responses for analysis. In Fig. 13, under the vehicle-dominated excitation, the displacement is − 0.0797 to 0.0563 m, and the acceleration is − 0.0334 to 0.0339 m/s2. It is observed that two signals are subject to noise; this impacts the precision of the response prediction and frequency evaluation. The excellent performance of the OVMD is demonstrated in Sect. 3. Consequently, the OVMD algorithm is adopted to process two signal responses to determine the valid IMF components.

Fig. 13
figure 13

Original GNSS displacement and acceleration signals

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Following the OVMD operation rules, Table 4 illustrates the RRMSE values corresponding to different K when the two signals are decomposed. As can be seen, K is 11 and 9 are the optimal decomposition parameters because the corresponding RRMSE has local minimum values. That is, the GNSS displacement and acceleration signals are decomposed into 11 and 9 IMF components, respectively.

Table 4 Corresponding RRMSE values when iterating K
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The correlation coefficients between two signals and their corresponding IMF components are computed via Eq. (9), as summarized in Table 5. The IMF components with correlation coefficients greater than 0.3 are effective components, so IMF1–IMF2 decomposed from the GNSS displacement signal and IMF1–IMF3 decomposed from the acceleration signal are retained.

Table 5 Correlation coefficients of two signals with their corresponding IMF components
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4.3 GA-BP prediction and analysis of dynamic signal responses

Subsequently, instead of restructuring the valid IMF components, each retained IMF component is used as the input to the GA-BP neural network. That is, the IMF components retained by OVMD are predicted using GA-BP, and then the predicted IMF components are restructured. 70% of the sample data is considered for model training, and the remaining 30% is used for test validation. The parameters in the BP neural network are set as follows: there are five hidden layers and one output layer; the number of training epochs is 2000. A learning rate of 0.001 is most appropriate. In the GA method, if the crossover probability is too large, the individuals with high fitness in the population will be destroyed due to rapid updating. But too small a value can bring the search to a standstill. Thereby, the crossover probability after attempts here is 0.7. The rest of the parameters in the GA approach are as follows: the number of evolution is 150; the population size is 60; the mutation probability is 0.05. Figures 14 and 15 display the predicted IMF components and signals reconstructed by the predicted IMF components. It can be observed that the predicted results are in good agreement with the actual ones. The noise in the predicted reconstructed signals is attenuated compared with the original signals in Fig. 13. The amplitudes of the two signals decrease, within −0.0712 to 0.0491 m and − 0.0329 to 0.0330 m/s2, respectively.

Fig. 14
figure 14

Actual and predicted IMF components: a actual and predicted IMF components of GNSS displacement; and b actual and predicted IMF components of acceleration

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Fig. 15
figure 15

Actual and predicted restructured signals

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Besides, other three methods are adopted for comparison with the proposed method. For the three cases, the signals are also first processed via OVMD. Then in method 1, the BP neural network is used to predict the retained IMF components and then reconstruct the predicted IMF components. In methods 2 and 3, BP and GA-BP neural networks are applied to predict the reconstructed signals. Namely, recombination followed by prediction. The parameters of the BP model are the same in each case. The mean error (ME), mean absolute error (MAE), mean square error (MSE), normalized root mean square error (NRMSE), and coefficient of determination (R2) are utilized as precision indicators. Their expressions are expressed as:

$$ {\text{ME = }}\left| {\frac{1}{N}\sum\limits_{t = 1}^{N} {\left[ {y(t) - \hat{y}(t)} \right]} } \right|, $$
(20)
$$ {\text{MAE = }}\frac{1}{N}\sum\limits_{t = 1}^{N} {\left| {y(t) - \hat{y}(t)} \right|} , $$
(21)
$$ {\text{MSE = }}\frac{1}{N}\sum\limits_{t = 1}^{N} {\left[ {y(t) - \hat{y}(t)} \right]^{2} } $$
(22)
$$ {\text{NRMSE = }}\frac{{\sqrt {\frac{1}{N}\sum\limits_{t = 1}^{N} {\left[ {y(t) - \hat{y}(t)} \right]^{2} } } }}{\max (y(t)) - \min (y(t))} $$
(23)
$$ R^{2} { = 1} - \frac{{\sum\limits_{t = 1}^{N} {\left[ {y(t) - \hat{y}(t)} \right]^{2} } }}{{\sum\limits_{t = 1}^{N} {\left[ {y(t) - \overline{y}} \right]^{2} } }} $$
(24)

where \(y(t)\) is the actual signal; \(\hat{y}(t)\) is the predicted signal; \(\overline{y}\) is the mean value of \(y(t)\). The smaller the ME, MAE, MSE and NRMSE are, and the larger the R2 is, the better the precision is.

The prediction precision of the four methods is illustrated in Fig. 16. In general, the ME values of the predicted acceleration signal are smaller than that of the predicted GNSS displacement signal, while other indicators are the opposite. This is understandable, because the expression of ME in Eq. (20) is different from other indicators in that it is averaged before taking the absolute values. The acceleration with more sample data reduces the ME values. Notice that method 2 has the poorest indicators in the GNSS displacement prediction. In the acceleration signal prediction, method 1 appears to have the largest MAE (7.7573 × 10–4), MSE (0.9995 × 10–6), and NRMSE (1.5003 × 10–2); method 2 has the largest ME (1.4146 × 10–5) and the smallest R2 (0.9249). In short, the prediction precision of the BP neural network (i.e., methods 1 and 2) is inferior to that of the GA-BP neural network (i.e., methods 3 and proposed). Most notably, the performance of the proposed method is the best in all indicators, especially the R2 (0.9898) is the highest. The above indicates the proposed method has the best prediction precision.

Fig. 16
figure 16

Precision indicators of different methods: a ME; b MAE; c MSE; dNRMSE; and e R.2

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4.4 Predictive identification and evaluation of modal frequencies

In Sect. 4.3, the signal responses (Fig. 15) reconstructed by the predicted IMF components are precisely obtained with the proposed approach. This section will detect structural modal frequencies from predicted reconstructed signal responses. The FFT is executed on the responses to get the PSD functions, as depicted in Fig. 17.

Fig. 17
figure 17

PSD functions: a PSD function of the predicted reconstructed GNSS displacement; and b PSD function of the predicted reconstructed acceleration

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Two frequency peaks, i.e., 0.3708 and 1 Hz, are observed from the PSD function of the predicted recombined GNSS displacement signal. However, there are five frequency peaks (i.e., 0.3692, 0.6462, 0.9538, 1.1385, and 1.8308 Hz) in the PSD function of the predicted recombined acceleration signal. This implies that the predicted reconstructed GNSS displacement signal contains less structural modal information. In theory, the predicted recombined GNSS data with a sampling rate of 50 Hz has the potential to measure more modal frequencies of interest synchronously. But this may be because the vertical dynamic positioning accuracy of GNSS is not as good as that of accelerometers and makes it fail to do so in a short time. In other words, the predicted restructured acceleration contains more order modal frequencies.

In the same way, this paper summarizes the structural frequency variation characteristics identified from the predicted reconstructed signal responses, as displayed in Fig. 18. Two order frequencies statistically derived from the predicted restructured GNSS displacements vary between 0.3663–0.3758 and 0.9938–1.0038 Hz. The five order frequencies counted from the predicted restructured accelerations change between 0.3692–0.3846, 0.6462–0.6615, 0.9385–0.9692, 1.1385–1.1846, and 1.8308–1.8769 Hz. On the whole, each order frequencies vary insignificantly. In contrast to the first- and third-order frequencies in Fig. 18b, the frequencies in Fig. 18a fluctuate less, but the frequencies of the corresponding orders in the two figures have an acceptable agreement. Table 6 lists the average frequencies calculated from Fig. 18 and frequencies measured in the literature [55]. The first-order average frequency determined from the predicted restructured GNSS displacements is slightly lower than that obtained from the predicted recombined accelerations. The opposite is true for the third-order frequency. However, the difference between them is minor. The first-order average frequencies and the second-order frequency from the predicted restructured accelerations are essentially the same as the frequencies in the literature [55]. Table 6 reveals the frequencies identified from the predicted recombined signal responses have satisfactory accuracy and also reflect the usefulness of the prediction. In addition, the structural higher-order frequencies are captured in this study than in the literature [55].

Fig. 18
figure 18

Structural modal frequencies: a frequencies captured from the predicted reconstructed GNSS displacements; and b frequencies captured from the predicted reconstructed accelerations

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Table 6 Statistical results of frequency evaluation
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Thus far, the precise prediction of the structural dynamic responses and the evaluation of the modal frequencies have been successfully obtained by applying the proposed approach.

5 Conclusions

In this study, an approach of OVMD combined with a GA-BP neural network is proposed for the predictive evaluation of the structural dynamic responses and frequencies based on GNSS displacement and acceleration at high sampling rates. The effectiveness of the approach was illustrated by field bridge experiments. The following conclusions are summarized:

  1. (1)

    The capability of the OVMD to address the shortcomings of VMD is proven through the damaged arch bridge testing. Compared with CEEMDAN, OVMD effectively reduces the number of IMF components by optimal decomposition and realizes reasonable reconstruction of signals. Among the four methods (i.e., wavelet soft threshold, CEEMDAN, EEMD-wavelet soft threshold, and OVMD), the OVMD has the best noise reduction effect and applicability. Furthermore, the modal frequencies of the damaged bridge can be extracted from the 50 Hz GNSS-RTK displacement response handled by OVMD.

  2. (2)

    The IMF components from GNSS displacements and accelerations processed by OVMD are used as inputs to GA-BP and realize the accurate prediction of dynamic responses of a long-span cable-stayed bridge. The comparative analysis indicates that the developed method enhances the prediction accuracy and its prediction indicators are better than the other three methods. This confirms the reliability and superiority of the proposed approach.

  3. (3)

    The frequencies and their variation characteristics of the long-span cable-stayed bridge are determined successfully from the predicted restructured GNSS displacements and accelerations. The two-order and five-order average frequencies calculated from two predicted recombined responses are 0.3704, 0.9996 Hz and 0.3723, 0.6457, 0.9518, 1.1508, 1.8395 Hz, respectively. The frequency results are well consistent with the measurement in the previous study. Furthermore, this study detects higher-order frequencies from the predicted restructured acceleration response. The identified results reveal that the frequency estimation adopting the proposed method has satisfactory accuracy and ensures the usefulness of the predicted information.

The results obtained in this study are beneficial in providing data support for bridge health assessment. In future work, the analysis of the approach in other data sources (such as temperature and strain) or other large structures (such as super-high-rise buildings and offshore platforms) will be explored in depth to more comprehensively assess the potential integration of the proposed method.