A Combined Noise Reduction Method for Floodgate Vibration Signals Based on Adaptive Singular Value Decomposition and Improved Complete Ensemble Empirical Mode Decomposition with Adaptive Noise

Abstract

To address the issue of the vibration characteristic signals of floodgates being affected by background white noise and low-frequency water flow noise, a noise reduction method combining the improved adaptive singular value decomposition algorithm (ASVD) and the improved complete ensemble EMD with adaptive noise (ICEEMDAN) is proposed. Firstly, a Hankel matrix is constructed based on the collected discrete time signals. After performing SVD on the Hankel matrix, the ASVD algorithm is used to automatically select the effective singular values to filter out most of the background white noise and retain the useful frequency components with similar energy in the signal. Then, ICEEMDAN combined with the Spearman correlation coefficient method is used to further filter out residual white noise and low-frequency water flows. The noise reduction performance of this combined method is verified through simulation experiments. Filtered by the ASVD-ICEEMDAN method, the signal-to-noise ratio of the simulation signal (50% noise level) is increased from 4.417 to 16.237, and the root mean square error is reduced from 2.286 to 0.586. Based on the practically measured vibration signals of a floodgate at a large hydropower station, the result shows that the ASVD-ICEEMDAN method exhibits good noise reduction performance and feature information extraction abilities for floodgate vibration signals, and can provide support for operational mode analysis and damage identification of practical structures under complex interference conditions.

1. Introduction

Recently, the safety diagnosis of hydraulic structures using vibration signals under flood discharge excitation has become one of the hot spots in the research of hydraulic engineering [1,2,3]. Due to the long-term effect of huge fluctuating loads of water flow, floodgates are prone to problems such as excessive amplitudes and even fatigue damage. Vibration monitoring of floodgates plays a significant role in the safe operation of structures. However, the collected vibration response signal is usually accompanied by the interference of background white noise and low-frequency water flow noise [4,5], which makes the structural vibration feature information easily submerged in noise and greatly affects the extraction of the vibration information of floodgates. Therefore, effective signal processing methods need to be adopted to filter out the noise in the collected signal and retain the characteristic vibration information of floodgates.

The vibration signal of floodgates is a kind of typical nonlinear and non-stationary signal [6], which commonly can be processed by methods including wavelet threshold [7,8], singular value decomposition (SVD) [9,10,11], empirical mode decomposition (EMD) [12,13,14], its improved algorithms [15,16], etc. Among them, EMD and its improved algorithms automatically decompose a signal into multiple intrinsic mode functions (IMFs) as well as a residual (RES) based on its own characteristics. Due to the advantages of strong adaptability and no need for predetermined basis functions [17,18], EMD and its improved algorithms are widely used to process nonlinear and non-stationary signals. Zhang et al. [19] used the EMD method to filter high-frequency and low-frequency noise from blasting vibration signals, achieving principal component extraction of blasting vibration signals. However, the traditional EMD algorithm is prone to the phenomenon of mode mixing [20], which makes the IMFs lose their physical significance. In response to this issue, Huang et al. [21] proposed the ensemble empirical mode decomposition (EEMD) algorithm, which introduces white noise with a mean of zero to mask the noise of the signal itself through a multiple-noise-adding process, thereby obtaining more accurate upper and lower envelopes and suppressing the mode mixing phenomenon to a certain extent. Li et al. [22] adopted the EEMD algorithm to filter out the vibration noise of rotating machinery and successfully extracted the vibration components that reflect the true characteristics of the system. In addition, due to the problems of residual noise, as well as the low computational efficiency, in EEMD, Torres et al. [23] proposed the complete ensemble EMD with adaptive noise (CEEMDAN) method, which adds white noise to each EMD decomposition process, making the decomposition process complete and greatly reducing signal reconstruction errors as well as improving the decomposition efficiency. Peng et al. [24] used CEEMDAN to establish a noise reduction model, retaining key information while successfully filtering out high-frequency noise in the collected vibration signals. Colombia et al. [25] highlighted the problem of the weak residual noise and false modes in CEEMDAN; thus, they proposed an improved complete ensemble EMD with adaptive noise (ICEEMDAN) algorithm that adds IMF components of white noise during each EMD process and introduces the concept of the average of local means to further improve the CEEMDAN algorithm. Yuan et al. [26] utilized the ICEEMDAN algorithm to effectively remove random noise and wave noise of the original signal, obtaining high-precision seabed geomagnetic field data. For vibration signals of hydraulic structures, EMD and its improved methods are effective at filtering out low-frequency water flow noise, but are unsatisfactory in filtering white noise. Therefore, some scholars [27,28,29] proposed combined noise reduction methods. Among them, Li et al. [29] combined CEEMDAN and wavelet threshold methods to filter out noise from the guide wall of a hydropower station. However, the wavelet thresholding method has drawbacks, such as the difficulty in selecting threshold functions and its inability to automatically decompose signals. Zhang et al. [27] combined the CEEMDAN and SVD methods to successfully filter out white noise in the dynamic displacement signals of the Laxiwa arch dam and to extract key vibration information. However, ordinary SVD [30,31,32,33] denoising methods rely on certain experience to choose effective singular values (ESVs) rather than automatic selection.

Aiming at the noise characteristics of the collected floodgate vibration signals, a combined adaptive singular value decomposition (ASVD) and ICEEMDAN noise reduction method is proposed. Firstly, the discrete time signal is converted into a Hankel matrix. After performing SVD on the Hankel matrix, the improved hyperparameter ACC method is used to select ESVs. Via signal reconstruction, a signal with removed background white noise can be obtained. Secondly, the ICEEMDAN method is used to decompose the signal into several IMF components, whose center frequencies are arranged from high to low. Effective IMF components are selected through the correlation coefficient method for reconstruction to remove low-frequency water flow noise and residual white noise.

The main structure of this paper is as follows: Section 2 introduces the basic principles of ASVD and ICEEMDAN as well as the process of the combined ASVD–ICEEMDAN noise reduction method. Section 3 validates the noise reduction performance of the ASVD–ICEEMDAN method through simulation experiments. In Section 4, this method is applied to a real-life floodgate structure. Section 5 contains the conclusions of this paper.

2. Basic Principles

2.1. ASVD Noise Reduction

SVD has a strong filtering ability for background white noise, and its principles can be described as follows: convert a one-dimensional signal $\mathit{X}=(x_1,x_2,\cdots ,x_L)$ into a Hankel matrix H:

$$\mathit{H}=\left[\begin{array}{cccc}{x}_1& x_2& \cdots & x_c\\ x_2& x_3& \cdots & x_{c+1}\\ \cdots & \cdots & \cdots & \cdots \\ x_r& x_{r+1}& \cdots & x_L\end{array}\right]$$

(1)

The above equation satisfies $1<c<L$ and $L=r+c-1$. Based on engineering experience [27,33], in order to achieve good noise reduction effects, the matrix H is usually constructed as a square matrix or a matrix close to a square matrix. Therefore, the variable r needs to satisfy Equation (2).

$$r=\{\begin{array}{ll}L/2,& \mathrm{When}L\mathrm{is}\mathrm{odd};\\ (L+1)/2,& \mathrm{When}L\mathrm{is}\mathrm{even}.\end{array}$$

(2)

Then, the matrix H is decomposed via SVD:

$$\mathit{H}=\mathit{U}\mathit{D}{\mathit{V}}^{\mathit{T}}$$

(3)

where $\mathit{U}\in {\mathit{R}}^{r\times r}$, $\mathit{V}\in {\mathit{R}}^{r\times c}$, and $\mathit{U}$ and $\mathit{V}$ are both unit orthogonal matrices. $\mathit{D}\in {\mathit{R}}^{r\times c}$ can be stated as Equation (4).

$$\mathit{D}=\{\begin{array}{ll}\left[diag({\sigma }_1,{\sigma }_2,\cdots ,{\sigma }_a),0\right]& ,r<c\\ diag({\sigma }_1,{\sigma }_2,\cdots ,{\sigma }_a)& ,r=c\\ {\left[diag({\sigma }_1,{\sigma }_2,\cdots ,{\sigma }_a),0\right]}^T& ,r>c\end{array}$$

(4)

where a is the minimum value of m and n; ${\sigma }_1,{\sigma }_2,\cdots ,{\sigma }_a$ are a series of singular values arranged from large to small, that is, ${\sigma }_1\ge {\sigma }_2\ge \cdots \ge {\sigma }_a\ge 0$; and 0 represents a zero matrix. Then, by setting the singular values representing noise to 0 and reconstructing a new Hankel matrix, the denoised signal can be obtained by the diagonal averaging method.

In the process of SVD noise reduction, it is a crucial step to select the ESVs. Li et al. [33] proposed a hyperparameter ACC method after noting the link between the singular values and the frequency components of the signal, which can be summarized into the following three points:

(1)

Each frequency component of the signal relates to a singular value group consisting of two adjacent singular values that have small differences between them.

(2)

Due to significant energy differences between different frequency components of the signal, a jump phenomenon is observed between adjacent groups of ESVs.

(3)

The energy of white noise is uniformly distributed in the broadband frequency domain, which makes the jump phenomenon of singular value inconspicuous.

To sum up, by observing the jump phenomenon between singular values groups, the singular values group can be judged to represent a useful signal or noise. Due to the first few singular values relating to the effective signal components and the latter relating to noise components, at the boundary between the signal and noise, the jumping phenomenon changes from conspicuous to inconspicuous. In the study of Li et al. [33], singular values are processed by differencing and normalization (SVDN), so that the change in singular values is amplified and the singular value boundary can be found more easily. In addition, to enhance the adaptability of the algorithm, the hyperparameter ACC (0.1 suggested) is proposed to automatically select the minimum sequence number that satisfies “acc1 > ACC and acc2 < ACC”.

However, the frequency components of the vibration signals of floodgates are usually complex (as shown in Section 4), such as the presence of many frequency components with similar energy levels, which is not consistent with the second law. In order to explain this problem and propose an improvement, a simulation signal with white noise is constructed as follows:

$$\{\begin{array}{l}{S}_1(t)=4\sin (5\mathsf{\pi }t)+8\cos (6\mathsf{\pi }t)+10\sin (7\mathsf{\pi }t)+4\sin (8\mathsf{\pi }t)+4\cos (9\mathsf{\pi }t)\\ Nois{e}_1(t)=std(S_1)\times randn\left[size(S_1)\right]\\ X_1(t)=S_1(t)+Nois{e}_1(t)\end{array}$$

(5)

The noisy signal $X_1(t)$ consists of a signal with five frequency components and white noise. In addition, the first (2.5 Hz), fourth (4 Hz) and fifth (4.5 Hz) frequency components possess the same energy. After performing SVD on signal $X_1(t)$, the top 20 singular values are shown in Figure 1. Two adjacent singular values of a singular group are connected by the blue line and the singular groups are connected by red straight lines or dotted lines in Figure 1. It is obvious that the singular groups from one to five represent the five most useful frequency components and other singular groups represent the noise components. After the calculation of SVDN values shown in Figure 2, the hyperparameter ACC is adopted to select ESVs. According to the ACC threshold (0.1), the number of ESVs is 4, which is different to the presumed condition of 10 (twice the number of effective frequencies).

By observing Figure 1, we can see that the jump phenomenon can be observed clearly between singular groups 1 and 2, 2 and 3, and 5 and 6, but it is not obvious between groups 3 and 4 and 4 and 5. Therefore, when the sequence numbers are six and eight, the SVDN values are lower than the ACC threshold, which results in the wrong selection. However, the jump phenomenon is still obvious between groups 5 and 6. Thus, an improved hyperparameter ACC method is developed according to the upper analysis: select the largest sequence number that satisfies “acc1 > ACC and acc2 < ACC”.

The flowchart of this new algorithm is shown in Figure 3, and the process is described as follows:

Step 1. Compute the signal length L of the one-dimensional signal $\mathit{X}$, then construct a Hankel matrix H according to Equations (1) and (2);

Step 2. Decompose Hankel matrix H via SVD and record singular values from large to small as $\sigma \left(1\right),\sigma \left(2\right),\cdots ,\sigma \left(r\right)$.

Step 3. Process the singular values by differencing and normalization.

Step 4. Set the initial value of the iteration variable N = 2 and the hyperparameter ACC = 0.1.

Step 5. Assign singular values to acc1 and acc2, that is, acc1 = $\sigma \left(N\right)$ and acc2 = $\sigma (N+2)$.

Step 6. Determine whether the condition “acc1 > ACC and acc2 < ACC” is met. When the condition is met, record and update the variable N; otherwise, proceed to the next step.

Step 7. Decide whether the condition “N + 3 ≥ r” is met. When the condition is met, perform the next step; otherwise, set N = N + 2 and jump to step 5.

Step 8. Set the singular values whose sequence numbers are greater than N to 0, reconstruct the Hankel matrix, and use the diagonal averaging method to obtain the denoised signal.

2.2. ICEEMDAN Method

Aiming at the problem of weak residual noise and false modes in CEEMDAN, ICEEMDAN improves CEEMDAN by changing the way white noise is added and introducing the concept of local means. Let $W(i)(\mathrm{i}=1,2,\cdots ,n)$ represent white noise with zero mean unit variance; let $<\cdot >$ represent the operator that averages the different signals; and let $E_k(\cdot )$ represent the kth IMF component after performing EMD on the bracketed signal. ${\beta }_0,\cdots ,{\beta }_{k-1}$ are the parameters controlling the magnitude of the added noise. Let $M(\cdot )$ represent the local means for the bracketed signals, that is, $M(\mathit{S})=\mathit{S}-E_1(\mathit{S})$. The flowchart of the ICEEMDAN algorithm is shown in Figure 4, and the decomposition steps are presented as follows:

Step 1. Add n groups of noise ${\beta }_0{E}_1(W(i))$ to the original signal X to obtain the noise-added signal $x(i)=x+{\beta }_0{E}_1\left(W(i)\right)$, and perform EMD on the noise-added signal to obtain n local means $M\left(x\left(i\right)\right)$. Then, average the n local means to obtain the first-stage residual value $r_1$:

$$r_1=<M\left(x\left(i\right)\right)>$$

(6)

Step 2. Compute the IMF value $I{C}_1$ for the first stage:

$$I{C}_1=x-r_1$$

(7)

Step 3. Calculate the local means of $r_1+{\beta }_1{E}_2\left(w(i)\right)$ and average them to get the second stage residual value:

$$r_2=<M\left(r_1+{\beta }_1{E}_2\left(W(i)\right)\right)>$$

(8)

Step 4. Calculate the IMF value $I{C}_2$ for stage 2:

$$I{C}_2=r_1-r_2$$

(9)

Step 5. For $k=3,\cdots ,K$, compute the residual for the kth stage:

$$r_k=<M\left(r_{k-1}+{\beta }_{k-1}{E}_k\left(W(i)\right)\right)>$$

(10)

Step 6. Calculate the kth IMF value:

$$I{C}_k=r_{k-1}-r_k$$

(11)

Step 7. Repeat Step 5 to 6 until the signal is less than three extremes, then the last signal that cannot be decomposed is the RES, and ICEEMDAN is complete.

2.3. Combined ASVD–ICEEMDAN Noise Reduction

In the measured dynamic floodgate vibration signal, the floodgates’ characteristic information is usually overwhelmed by the background white noise as well as the low-frequency water flow noise. Aiming at these two noise characteristics, the ASVD–ICEEMDAN method is proposed for combined noise reduction. The flow chart is shown in Figure 5, and the noise reduction steps are as follows:

Step 1. The original signal X₀ is constructed as a Hankel matrix and then decomposed via SVD to obtain the singular values.

Step 2. The improved hyperparameter ACC method is used to automatically search ESVs. Then, the new signal X₁ can be reconstructed via the ASVD algorithm.

Step 3. Decompose X₁ via ICEEMDAN to yield several IMF components whose center frequencies are arranged from high to low.

Step 4. Select the effective IMF components based on the Spearman correlation coefficient [34,35,36] as in Equation (12)

$$r_s=1-\frac{6{\displaystyle \sum _{i=1}^n{d}_i^2}}{L(L^2-1)}$$

(12)

where $d_i$ is the level difference between elements $X(i)$ and $Y(i)$, L is the length of the signal. The value of $r_s$ is between −1 and 1. When the absolute value of $r_s$ approaches 1, it indicates a strong correlation between the two signals; when the absolute value of $r_s$ approaches 0, it indicates a weak correlation between signals. In addition, if the absolute values of $r_s$ of the signal before decomposition and the IMF component are greater than 0.4 [36], the IMF component can be regarded as an effective IMF component.

Step 5. Reconstruct the effective IMF components to obtain the denoised signal X.

3. Numerical Simulation

To clarify the denoising performance of the ASVD–ICEEMDAN method, a pure simulation signal $S_2(t)$ was constructed in MATLAB R2019a. Furthermore, it is assumed that the collected signal $X_2(t)$ is mixed with low-frequency noise (LN) and white noise (WN), and the function expression is shown in Equation (13):

$$\{\begin{array}{l}{S}_2(t)=6e^{-\frac{t}{4}}\sin (3\pi t)+8e^{-\frac{t}{3}}\sin (8\pi t)+9e^{-\frac{t}{3}}\sin (20\pi t)\\ Nois{e}_2(t)=4e^{-\frac{t}{4}}\sin (\pi t)+P\times std(S_2)\times randn\left[size(S_2)\right]\\ \\ X_2(t)=S_2(t)+Nois{e}_2(t)\end{array}$$

(13)

where P is the noise level and $std(S_2)$ is the standard deviation of $S_2$; $randn\left[size(S_2)\right]$ represents a white noise with a zero mean, a standard deviation of 1 and a normal distribution. The sampling frequency is 50 Hz, the total sampling time is 10 s, and the noise level P is 50%. The time waveform and power spectrum curves of pure signals and noisy signals are shown in Figure 6.

After constructing a Hankel matrix for the simulation signal, the SVD operation of the Hankel matrix is performed to calculate its SVDN values. The top 20 values are shown in

Table 1. Top 20 singular values and SVDN values.

Sequence Number	1	2	3	4	5	6	7	8	9	10
Singular value	332.520	327.151	306.594	299.108	260.133	245.444	185.043	156.517	58.924	58.764
SVDN value	0.055	0.211	0.077	0.399	0.151	0.619	0.292	1.000	0.002	0.029
Sequence number	11	12	13	14	15	16	17	18	19	20
Singular value	55.940	55.672	55.038	54.870	52.886	52.711	51.616	51.524	51.410	51.344
SVDN value	0.003	0.006	0.002	0.020	0.002	0.011	0.001	0.001	0.001	0.008

, and the line graph is plotted in Figure 7.

According to Figure 7, the number of ESVs is computed as 8. Next, the signal is reconstructed, and the noise reduction result is shown in Figure 8.

From Figure 8, it can be observed that the signal denoised by ASVD is smoother, and most of the background white noise in the signal has been filtered out after ASVD filtering. However, low-frequency strong noise at 0.5 Hz still exists in the filtered signal. Next, the CEEMDAN and ICEEMDAN methods are used to decompose the pre-filtered signal, respectively. To ensure that the decomposition results are comparable, the same initial parameters were used in these two methods: the white noise amplitude is set to 0.2, the average number of signals is 100, and the maximum iterations number is 1000. The decomposition results are shown in Figure 9.

In Figure 9, the signal is decomposed into eight IMFs and a RES by the CEEMDAN method, while only seven IMFs and a RES are generated via ICEEMDAN, which proves that ICEEMDAN can reduce the number of invalid components and inhibit the generation of mode mixing phenomena to a certain extent compared to CEEMDAN.

In addition, to calculate the residual noise of the two decomposition methods, all IMFs and the RESs are summed and reconstructed, and the signal deviation, RRSE, between the original signal and the reconstructed signal is calculated via Equation (14):

$$RRSE=\frac{\Vert {\mathit{S}}_1-{\mathit{S}}_2\Vert }{\Vert {\mathit{S}}_2\Vert }$$

(14)

where ${\mathit{S}}_1$ is a vector that sums all IMFs and RESs decomposed by ICEEMDAN or CEEMDAN and ${\mathit{S}}_2$ is the original signal vector. The RRSE value of the CEEMDAN method is 1.90 × 10⁻¹⁶, while that of the ICEEMDAN method is only 1.39 × 10⁻¹⁶, indicating that the signal decomposed by the ICEEMDAN method has lower residual noise.

To select the effective IMFs to reduce noise, the absolute values of $r_s$ of IMF1-IMF7 and RES were calculated and are shown in

Table 2. The absolute values of $r_s$ of IMF1-IMF7 and RES.

IMF Component	$\mathbf{Absolute}\mathbf{Values}\mathbf{of}{\mathit{r}}_{\mathit{s}}$	IMF Component	$\mathbf{Absolute}\mathbf{Values}\mathbf{of}{\mathit{r}}_{\mathit{s}}$
IMF1	0.536	IMF5	0.397
IMF2	0.578	IMF6	0.303
IMF3	0.498	IMF7	0.094
IMF4	0.458	RES	0.076

. The absolute values of $r_s$ of IMF1-IMF4 are all greater than 0.4, while the remaining components are less than 0.4. Therefore, IMF1-IMF4 were selected for reconstruction, and the signal denoised by ASVD–ICEEMDAN is shown in Figure 10.

The noise is almost completely filtered out, and the waveforms of the denoised signal and the pure signal are rather similar, as shown in Figure 10. To verify the denoising performance of the combined ASVD–ICEEMDAN method compared with other methods, the signal-to-noise ratio (SNR) and the root mean square error (RMSE) were calculated via Equations (15) and (16):

$$SNR=10\log \left[\frac{{\displaystyle \sum _{t=1}^T{X}_0^2(t)}}{{\displaystyle \sum _{t=1}^T{(X(t)-X_0(t))}^2}}\right]$$

(15)

$$RMSE=\sqrt{\frac{1}{T}{\displaystyle \sum _{t=1}^T{\left[X_0(t)-X(t)\right]}^2}}$$

(16)

where T is the sampling number. The calculation results are shown in

Table 3. Comparison of evaluation indexes of noise reduction performance of different methods.

Noise Reduction Method	Index
	SNR/dB	RMSE
Original signal	4.417	2.286
Moving average	5.414	2.038
IIR digital filters	5.544	2.008
ASVD	9.387	1.290
ICEEMDAN	5.711	1.969
ASVD-CEEMDAN	16.104	0.595
ASVD–ICEEMDAN	16.237	0.586

. The signal filtered by the ASVD–ICEEMDAN noise reduction method presents the highest SNR and the lowest RMSE, proving the superiority of the combined noise reduction method. The simulation data and MATLAB R2019a script are available at: https://github.com/RabbitInTheNorth/Noise-reduction-simulation.git (accessed on 9 December 2023).

4. Engineering Example

The studied floodgate is located in the upper reaches of the Hanjiang River, and is a part of a large-scale hydropower station. The hub project is divided into two parts by a longitudinal pier in the middle of the river. The left side is the powerhouse dam section, and the right side is the flood discharge dam section, where there are six holes. From left to right, the six holes are floodgates 1#~4#, a vertical ship lift (also used as a floodgate), and a surface hole in the right auxiliary dam. The actual project is shown in Figure 11a.

During flood discharge operations in the past, the staff of the project found that there was an obvious vibration phenomenon at the top of the floodgate pier, especially in hole 2#. To study the influence of flood discharge on the pier structure, a vibration test was carried out under various working conditions. The sampling frequency of this test was 50 Hz and the sampling time was 81.92 s. The dynamic displacement sensor layout diagram is shown in Figure 11, and the parameters of the sensors are shown in

Table 4. Performance table of DP seismic low frequency vibration sensors.

Sensor Number	Model	Range of Frequency Response	Sensitivity
1~10	DPS-0.5-15-H	0.5~200 Hz	5 mv/μm

.

The collected vibration signal was processed by the ASVD–ICEEMDAN method introduced in Section 3. Due to length limitations, only the process of measuring point 10 under working condition 1 is described. The time waveform and power spectrum curves of the vibration signal are shown in Figure 12. In the power spectrum figure, there are three obvious frequency peaks (plotted in orange and red wireframes) and many “small burrs” (plotted in green wireframes). These “small burrs” represent white noise components, which would be filtered by the ASVD method. In addition, the main energy of peak 1 is within 1 Hz [37,38], which represents low-frequency noise and would be filtered by the ICEEMDAN method.

After constructing the Hankel matrix of the signal, the ASVD algorithm was performed, determining the number of ESVs, which is 68, as shown in Figure 13. Then, the white-noise-filtered signal was obtained after SVD reconstruction. In order to filter the low-frequency water flow noise, the signal processed by ASVD was decomposed via ICEEMDAN, and the decomposition result is shown in Figure 14. After removing the components whose main frequencies are less than 1 Hz and IMFs whose absolute values of $r_s$ are less than 0.4, the signal was reconstructed. The denoised signals at measuring point 10 under conditions 1 and 2 are shown in Figure 15 and Figure 16, respectively.

In Figure 15 and Figure 16, the white noise and the low-frequency water flow noise are considerably filtered out. From the power spectrum curves after windowing, it can clearly be seen that the two-order natural frequencies of the structure under condition 1 are 2.35 Hz and 3.76 Hz, while they are 2.47 Hz and 3.82 Hz under condition 2. The noise reduction results were compared with the results of the eigensystem realization algorithm (ERA) modal identification method [37] in

Table 5. Results comparison of ASVD–ICEEMDAN and ERA.

Working Condition	Order	Suggested Method/Hz	ERA/Hz	Relative Error */%
Condition 1	1	2.35	2.39	1.67
	2	3.76	3.77	0.27
Condition 2	1	2.47	2.43	1.65
	2	3.82	3.82	0

Note: * $\mathrm{Relative}\mathrm{error}=\frac{\left|\mathrm{Suggested}\mathrm{method}\mathrm{value}-\mathrm{ERA}\mathrm{method}\mathrm{value}\right|}{\mathrm{ERA}\mathrm{method}\mathrm{value}}\times 100\%$.

. The maximum error of the two frequency identification results is only 1.67%, indicating that the method can filter out the noise of the floodgate and restore the structural feature information successfully.

5. Conclusions

The ASVD–ICEEMDAN combined noise reduction method is proposed to filter background white noise and low-frequency water flow noise in the vibration signals of a floodgate structure. The following main conclusions are drawn through simulation experiments and an actual engineering case.

(1)

An ASVD method is proposed to select effective singular values automatically based on the relationship between singular values and signal components. It avoids the uncertainty of manual selection and improves the efficiency of noise reduction.

(2)

The ICEEMDAN algorithm can automatically decompose the signal into several IMF components whose center frequencies are arranged from high to low. Compared to the CEEMDAN algorithm, ICEEMDAN algorithm performs better in terms of suppressing mode mixing and reducing residual noise.

(3)

The ASVD–ICEEMDAN method successfully filters out white noise and low-frequency noise in simulated signals, which increases the SNR of the signal (50% noise level) from 4.417 to 16.237 and reduces the RMSE from 2.286 to 0.586. In the engineering case study, the ASVD–ICEEMDAN method effectively filters out the noise and accurately extracts the structural characteristic vibration information, proving it can provide support in operational modal analyses and damage identification in actual structures.

(4)

Vibration signals of discharge structures are mainly affected by white noise and low-frequency noise; thus, this method has the potential to be extended to other hydraulic structures under discharge excitation, such as arch dams, gravity dams, and guide walls.