Bridge Measuring Point Displacement Denoising Method Based on Microwave Radar Echo Energy and VMD

Abstract

The present invention discloses a method for denoising the displacement of bridge measurement points based on microwave radar echo energy and VMD. When using microwave radar to synchronously monitor the displacement time history of multiple measurement points on a bridge, snapshot data is collected and FFT transformation is performed to obtain the echo energy information of each measurement point; the measurement point with the maximum echo energy is selected as the reference measurement point, and the VMD algorithm is used to decompose the displacement time history information of the reference measurement point to obtain several intrinsic mode functions. The correlation between each order of intrinsic mode function and the original signal is defined, and a threshold is set as the judgment condition to determine the decomposition mode order. Based on the frequency domain information of each level of intrinsic mode function after decomposition of the reference measurement point, the VMD decomposition is performed on the displacement time history data of the remaining measurement points to obtain the intrinsic mode functions after decomposition of the remaining signals; the effective displacement signals of each measurement point on the bridge are restored by using the intrinsic mode functions after decomposition of each measurement point, improving the signal-to-noise ratio of the displacement signals of each measurement point on the bridge monitored by a unified microwave radar.

Description

Technical Field

[0001] The invention belongs to the technical field of bridge monitoring, and particularly relates to a method for denoising displacements of multiple measuring points of a bridge based on microwave radar echo energy and VMD. Background Art

[0002] Structural displacement is an important index for evaluating the performance of a bridge, but it is difficult to measure accurately. In recent years, the theory of bridge structural health monitoring using various advanced sensor technologies has developed rapidly, and they are highly expected in ensuring structural safety and realizing the whole life cycle management of the structure. At present, various structural health monitoring systems have been installed on many long-span bridges at home and abroad, and they play an active role in early warning of emergencies such as earthquakes, typhoons, ship collisions, etc. At the same time, the sensors used in the existing health monitoring systems have problems such as a wide variety of types and high costs. Among the existing displacement monitoring technologies, the wire or rod type displacement meter has high accuracy, but it is difficult to apply in actual bridge tests due to the lack of fixed base points. GPS displacement measurement has its advantages, but its accuracy needs to be improved; the communicating pipe or pressure transmitter is easily affected by the vibration of the vehicle flow, resulting in its deformation signal being easily severely clipped. Laser ranging has the advantage of non-contact measurement, but it is easily limited by the short laser penetration distance. As a new type of non-contact non-destructive testing technology, microwave radar has played an important role in the displacement monitoring of many bridges. According to its working principle and data characteristics, microwave radar is mainly divided into microwave interferometric radar and inverse synthetic aperture radar. Among them, the microwave interferometric radar can collect the displacement of the structural target along the radar line of sight at a relatively high frequency, and is suitable for short-term dynamic and static displacement monitoring of the structure; the inverse synthetic aperture radar is divided into spaceborne and ground-based types, which obtain a two-dimensional image by repeatedly observing the ground structural target, and use the phase difference to obtain the displacement of the structural target along the radar line of sight. However, it should be noted that its acquisition frequency is relatively low, and it is suitable for long-term monitoring of structural deformation.

[0003] In the research on microwave interferometric radar, the Italian IBIS-S radar, as a relatively mature commercial product, has been applied in many bridge projects at home and abroad. However, it should be noted that although microwave radar has been applied in bridge deformation monitoring, it has not been widely accepted by the engineering community, and the above research mainly focuses on the application of radar core technology in bridge displacement monitoring. The main reason is that in actual engineering applications, there are problems such as inconsistent signal-to-noise ratios of long-distance and short-distance targets monitored by microwave interferometric radar, and due to the limitations of the navigable height of the bridge and the beam angle of the equipment antenna, the microwave interferometric radar cannot synchronously perform a full-bridge cross-section test at one time, and there is a lack of corresponding data processing methods. Summary of the Invention

[0004] To solve the above technical problems, the invention provides a method for denoising displacements of multiple measuring points of a bridge based on microwave radar echo energy and VMD.

[0005] The specific solution is as follows:

[0006] A method for denoising the displacement of bridge measurement points based on microwave radar echo energy and VMD, the specific steps are as follows:

[0007] S1. Use a microwave radar to transmit microwaves and receive the microwave echo signals of each measurement point of the bridge, and synchronously obtain the original displacement time history signal matrix X(t) with inconsistent signal-to-noise ratios at each measurement point of the bridge based on the phase interference method;

[0008] S2. According to the original displacement time history signal matrix X(t) of each measurement point monitored by the microwave radar, extract a column of snapshot signals in the signal matrix for FFT transformation to obtain the microwave radar echo energy values of each measurement point of the bridge, and select the measurement point with the maximum microwave radar echo energy as the reference point;

[0009] S3. Based on VMD, perform modal decomposition on the original reference signal to obtain multiple intrinsic mode functions IMF;

[0010] S4. Define the spectral correlation coefficient ρ i between each decomposed intrinsic mode function IMF i and the original reference signal, set the threshold σ = 0.01, and determine that when the correlation coefficient ρ i < σ, the modal decomposition stops, and determine the decomposition layer k of the original reference signal;

[0011] S5. Extract the frequency domain information of each order of intrinsic mode function signals of the reference measurement point after decomposition, and based on its frequency domain information as a reference, decompose the displacement time history signals of the remaining measurement points of the bridge structure based on the VMD algorithm to obtain the corresponding intrinsic mode functions of each order of the remaining signals;

[0012] S6. Superimpose the intrinsic mode functions of each measurement point of the bridge after VMD decomposition to obtain the effective displacement time history signals after denoising at each measurement point

[0013] Step S1 specifically includes: To achieve accurate measurement of the displacement of bridge measurement points, a corner reflector needs to be installed at the position of the bridge measurement point before measurement. Ignoring the amplitude of the microwave radar transmission signal, the transmission signal of the microwave radar can be expressed as:

[0014]

[0015] In the formula, f0 represents the starting frequency; t represents the time change within the scanning period, also known as the snapshot; φ0 represents the initial phase; B is the microwave radar bandwidth, T is the microwave radar modulation period, j is the imaginary part. Assuming that the echo signal of the measurement point is received by the microwave radar receiving antenna after a time τ, the microwave radar echo signal after ignoring the amplitude change can be expressed as:

[0016]

[0017] Where τ = 2R / c, R represents the distance from the target to the radar, and c represents the speed of light. After amplification, down-conversion, and low-pass filtering, the obtained baseband beat signal can be expressed as:

[0018]

[0019] Where (·) * represents the complex conjugate. Based on the phase interference method, the displacement of the target in the radar line-of-sight direction can be expressed as:

[0020]

[0021] Where λ represents the wavelength, represents the phase difference between adjacent moments of the actually measured target signal. Among them, is the phase difference caused by target deformation, and represent the phase difference and phase noise caused by atmospheric environmental interference respectively. Due to the different distances of each measurement point relative to the microwave radar, the two are different.

[0022] The microwave radar can achieve range resolution for each measurement point. The original displacement time-history signal matrix X(t) with inconsistent signal-to-noise ratios for each measurement point can be expressed as: X(t) m×n ={x LOS,1 x LOS,2 ... x LOS,m} T ; where x LOS,i ={x i,1 x i,2 ... x i,n} represents the displacement time-history signal of the i-th range cell, m represents the number of samples of a periodic signal of the microwave radar, and n represents the number of periods measured by the microwave radar.

[0023] Step S4 specifically includes: First, define the decomposition layer number based on the spectral correlation coefficient between each order of the intrinsic mode function and the original signal, which can be defined as:

[0024]

[0025] Where U j represents the Fourier transform of the j-th order intrinsic mode function u j after decomposition, j = 1, L, k; X i represents the Fourier transform of the original reference displacement signal.

[0026] The specific steps of the method are:

[0027] First, set k = 2 and perform iterative update with k = k + 1. According to formula (5), the spectral correlation coefficients (ρ1, ρ2,..., ρ k+1 ) of each decomposed independent component and the original signal can be obtained. Define ρ min = min{ρ1, ρ2,..., ρ k+1}. If ρ min is significantly small, it can be determined that the decomposed u k+1 is significantly a fictitious component, which means that the decomposition layer number k needs to be reduced. Therefore, set the threshold to σ = 0.01. If ρ min < σ, the iteration stops and the decomposition layer number k is determined.

[0028] Step S5 specifically includes: performing VMD decomposition on the displacement signal of the reference measuring point according to the determined decomposition layer number k to obtain k-order intrinsic mode functions, performing Fourier transform on each of the above intrinsic mode functions to obtain the frequency-domain information of each intrinsic mode function, that is, the central frequency. Taking the central frequency of each order of the intrinsic mode function of the reference measuring point as a reference, perform VMD decomposition on the displacement signals of the remaining measuring points, and determine the decomposition layer number according to the one-to-one correspondence between the central frequencies of each order of the intrinsic mode functions after decomposition and the central frequencies of each order of the intrinsic mode functions of the reference measuring point.

[0029] The bridge multi-measurement point displacement denoising method based on microwave radar echo energy and VMD proposed by the present invention is essentially different from the existing signal denoising methods. The method of the present invention first considers the parameters of the microwave radar equipment used, extracts the maximum value point of the microwave radar echo energy as the reference measuring point for analysis, and then respectively uses the frequency-domain correlation coefficient between the intrinsic mode function and the original reference signal and the central frequency of the reference measuring point to determine the VMD decomposition layer number of each measuring point, rather than the single specification of the decomposition layer number in the traditional signal denoising method. The above method can be applied to the denoising and unification of the bridge multi-measurement point displacement signals based on microwave radar.

[0030] The advantages and technical effects of the present invention are as follows:

[0031] (1) The present invention considers the problem of non-uniform signal-to-noise ratio caused by inconsistent distances of each measuring point when measuring the target displacement by microwave radar, selects the measuring point with the maximum microwave radar echo energy as the reference point for VMD decomposition, and can effectively remove the influence of atmospheric interference and thermal noise;

[0032] (2) When the present invention performs VMD decomposition on the remaining measuring points with the central frequency of the reference measuring point signal as a reference, the problem of determining the decomposition layer number is avoided, so that the signal-to-noise ratios of the displacement signals of each measuring point of the bridge are unified. Description of the Drawings

[0033] Figure 1 Implementation flowchart.

[0034] Figure 2 Original displacement signal diagram of multiple measurement points of the bridge measured by microwave radar in the embodiment.

[0035] Figure 3 Microwave radar echo energy diagram of each measurement point of the bridge in the embodiment.

[0036] Figure 4 Correlation coefficient diagram of each order of intrinsic mode function and the original reference signal after VMD decomposition of the reference measurement point in the embodiment.

[0037] Figure 5 VMD decomposition diagram of the reference measurement point signal in the embodiment.

[0038] Figure 6 Displacement signal diagram of each measurement point of the bridge after denoising based on VMD in the embodiment. Detailed implementation manners

[0039] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0040] The present invention provides a method for denoising the displacement of bridge measurement points based on microwave radar echo energy and VMD. First, the present invention uses a microwave radar to synchronously monitor the displacement time history of multiple measurement points of a bridge, and performs FFT transformation on the collected snapshot data to obtain the echo energy information of each measurement point. Then, the measurement point with the maximum echo energy is selected as the reference measurement point, and the VMD algorithm is used to decompose the displacement time history information of the reference measurement point to obtain k intrinsic mode functions. Among them, the correlation between each order of intrinsic mode function and the original signal is defined, and a threshold value σ = 0.01 is set as the judgment condition to determine the modal order of decomposition. Secondly, based on the frequency domain information of each order of intrinsic mode function after decomposition of the reference measurement point, the displacement time history data of the remaining measurement points are subjected to VMD decomposition to obtain the intrinsic mode functions after decomposition of the remaining signals. Finally, the effective displacement signals of each measurement point of the bridge are restored by using the intrinsic mode functions after decomposition of each measurement point, and thus the signal-to-noise ratio of the displacement signals of each measurement point of the bridge monitored by a unified microwave radar can be improved.

[0041] As Figure 1 shown, the method for denoising the displacement of bridge measurement points based on microwave radar echo energy and VMD is specifically as follows:

[0042] S1. Use a microwave radar to transmit microwaves and receive the microwave echo signals of each measurement point of the bridge, and synchronously obtain the original displacement time history signal matrix X(t) with inconsistent signal-to-noise ratios of each measurement point of the bridge based on the phase interference method;

[0043] When obtaining the displacement time - history signals \(X(t)\) of multiple measurement points of the bridge structure, first, select reasonable measurement positions, and use microwave radar to synchronously monitor the displacement time - history signals of multiple measurement points of the bridge structure. The displacement time - history signals \(X(t)\) of each measurement point obtained are as Figure 2 shown.

[0044] S2. According to the original displacement time - history signal matrix \(X(t)\) of each measurement point monitored by the microwave radar, extract a column of snapshot signals in this signal matrix for FFT transformation to obtain the microwave radar echo energy values of each measurement point of the bridge. Select the measurement point with the maximum microwave radar echo energy as the reference point; the microwave radar echo energy diagrams of each measurement point are as Figure 3 shown. Select the measurement point with the maximum microwave radar echo energy as the reference measurement point for the next step of processing.

[0045] S3. Based on VMD, perform modal decomposition on the original reference signal to obtain multiple intrinsic mode functions IMF; perform VMD decomposition on the original displacement signal of the reference measurement point, and calculate the frequency - correlation coefficients of each order of intrinsic mode functions and the original reference signal. Specifically, as Figure 3 shown. Determine the decomposition layer number according to the threshold, and then the time - frequency domain information of each order of intrinsic mode functions after VMD decomposition of the reference measurement point can be obtained. Specifically, as Figure 4 shown.

[0046] S4. Define the spectral - correlation coefficient \(\rho\) i between each decomposed intrinsic mode function IMF i and the original reference signal. Set the threshold \(\sigma = 0.01\). When the correlation coefficient \(\rho\) i < \(\sigma\), stop the modal decomposition and determine the decomposition layer number \(k\) of the original reference signal;

[0047] Taking the central frequency of the 3 - order intrinsic mode function after decomposition of the reference measurement point as a reference, perform VMD decomposition on the original displacement signals of the remaining measurement points. Determine its decomposition layer number by making the central frequency domain of the decomposed intrinsic mode function consistent with the central frequency of the 3 - order intrinsic mode function after decomposition of the reference measurement point, and obtain the intrinsic mode functions of each order after VMD decomposition of the remaining measurement points.

[0048] S5. Extract the frequency - domain information of each order of intrinsic mode function signals of the decomposed reference measurement point, and based on this frequency - domain information as a reference, perform decomposition on the displacement time - history signals of the remaining measurement points of the bridge structure using the VMD algorithm to obtain the corresponding intrinsic mode functions of each order of the remaining signals; as Figure 5 shown

[0049] S6. Superimpose the intrinsic mode functions of each measurement point of the bridge after VMD decomposition to obtain the effective displacement time - history signals with noise reduction for each measurement point After decomposing the VMD of each measurement point of the bridge, the effective displacement time history signal is obtained by superimposing the inherent mode functions of each order, specifically as Figure 6 shown.

[0050] Step S1 specifically includes: To achieve accurate measurement of the displacement of the bridge measurement points, corner reflectors need to be installed at the bridge measurement point positions before measurement. Ignoring the amplitude of the microwave radar emission signal, the emission signal of the microwave radar can be expressed as:

[0051]

[0052] where f0 represents the starting frequency; t represents the time change within the scanning period, also known as the snapshot; φ0 represents the initial phase;

[0053] B is the microwave radar bandwidth, T is the microwave radar modulation period, j is the imaginary part. Assuming that the echo signal of the measurement point is received by the microwave radar receiving antenna after time τ, the microwave radar echo signal after ignoring the amplitude change can be expressed as:

[0054]

[0055] where τ = 2R / c, R represents the distance from the target to the radar, and c represents the speed of light. After amplification, down-conversion, and low-pass filtering processing, the obtained baseband beat signal can be expressed as:

[0056]

[0057] where (·) * represents the complex conjugate. Based on the phase interference method, the displacement of the target in the radar line-of-sight direction can be expressed as:

[0058]

[0059] where λ represents the wavelength, represents the phase difference between adjacent moments of the actual measured target signal. Among them, is the phase difference caused by target deformation, and respectively represent the phase difference and phase noise caused by atmospheric environment interference. Due to the different distances of each measurement point relative to the microwave radar, the two are different.

[0060] The microwave radar can achieve range resolution for each measurement point. The original displacement time history signal matrix X(t) with inconsistent signal-to-noise ratios for each measurement point can be expressed as: X(t) m×n ={x LOS,1 x LOS,2 ... x LOS,m} T . Among them, x LOS,i ={x i,1 xi,2 ... x i,n} represents the displacement time history signal of the i-th distance unit, m represents the number of sampling points of a cycle signal of the microwave radar, and n represents the number of cycles measured by the microwave radar.

[0061] Step S4 specifically includes: First, define the decomposition layer number based on the spectral correlation coefficient between each order of intrinsic mode function and the original signal, which can be defined as:

[0062]

[0063] In the formula, U j represents the Fourier transform of the j-th order intrinsic mode function u j after decomposition, j = 1, L, k; X i represents the Fourier transform of the original reference displacement signal.

[0064] The specific steps of the method are as follows:

[0065] First, set k = 2 and perform k = k + 1 iterative update. According to formula (5), the spectral correlation coefficients (ρ1, ρ2,..., ρ k+1 ) of each independent component after decomposition and the original signal can be obtained. Define ρ min = min{ρ1, ρ2,..., ρ k+1}}, if ρ min is significantly small, it can be judged that the decomposed u k+1 is significantly a fictitious component, which means that the decomposition layer number k needs to be reduced. Therefore, set the threshold to σ = 0.01. If ρ min < σ, the iteration stops and the decomposition layer number k is determined.

[0066] Step S5 specifically includes: Perform VMD decomposition on the displacement signal of the reference measuring point according to the determined decomposition layer number k to obtain k-order intrinsic mode functions, perform Fourier transform on each order of the above intrinsic mode functions to obtain the frequency domain information of each order of intrinsic mode functions, that is, the center frequency. Taking the center frequency of each order of intrinsic mode functions of the reference measuring point as a reference, perform VMD decomposition on the displacement signals of the remaining measuring points, and determine the decomposition layer number according to the one-to-one correspondence between the center frequencies of each order of intrinsic mode functions after decomposition and the center frequencies of each order of intrinsic mode functions of the reference measuring point.

[0067] The method for denoising multi - measurement - point displacements of bridges based on microwave radar echo energy and VMD proposed by the present invention is essentially different from existing signal denoising methods. The method of the present invention first considers the parameters of the microwave radar equipment used, extracts the maximum value point of the microwave radar echo energy as a reference measurement point for analysis, and then determines the VMD decomposition layers of each measurement point by using the frequency - domain correlation coefficient between the intrinsic mode function and the original reference signal and the central frequency of the reference measurement point respectively, rather than the single specification of the decomposition layer number in traditional signal denoising methods. The above - mentioned method can be applied to the denoising and unification of multi - measurement - point displacement signals of bridges based on microwave radar.

[0068] The present invention considers the problem of non - uniform signal - to - noise ratio caused by inconsistent distances of each measurement point when the microwave radar measures the target displacement, selects the measurement point with the maximum microwave radar echo energy as the reference point for VMD decomposition, and can effectively remove the influence of atmospheric interference and thermal noise;

[0069] Taking the central frequency of the reference measurement - point signal as a reference, when performing VMD decomposition on the remaining measurement points, the present invention avoids the problem of determining the decomposition layer number, thereby unifying the signal - to - noise ratio of the displacement signals of each measurement point of the bridge.

[0070] The technical means disclosed in the solution of the present invention are not limited to the technical means disclosed in the above - mentioned embodiments, but also include technical solutions composed of any combination of the above - mentioned technical features. It should be noted that for those of ordinary skill in the art of this technology, without departing from the principle of the present invention, several improvements and refinements can be made, and these improvements and refinements are also regarded as the protection scope of the present invention.

Claims

1. A method for denoising the displacement of bridge measurement points based on microwave radar echo energy and VMD, characterized in that, The specific steps are as follows: S1. Use a microwave radar to transmit microwaves and receive the microwave echo signals of each measuring point on the bridge. Based on the phase interference method, synchronously obtain the original displacement time history signal matrix X(t) with inconsistent signal-to-noise ratios at each measuring point on the bridge; S2. According to the original displacement time history signal matrix X(t) of each measuring point monitored by the microwave radar, extract a column of snapshot signals in the signal matrix for FFT transformation to obtain the microwave radar echo energy values of each measuring point on the bridge. Select the measuring point with the maximum microwave radar echo energy as the reference point; S3. Based on VMD, perform modal decomposition on the original reference signal to obtain multiple intrinsic mode functions IMF; S4. Define each Intrinsic Mode Function (IMF) after decomposition i The spectral correlation coefficient ρ with the original reference signal i , set the threshold σ = 0.01, and determine the correlation coefficient ρ i When ρ < σ, the mode decomposition stops, and determine the decomposition layer number k of the original reference signal; S5. Extract the frequency domain information of each order of the intrinsic mode function signals of the decomposed reference measuring point, and based on its frequency domain information, perform decomposition on the displacement time history signals of the remaining measuring points of the bridge structure using the VMD algorithm to obtain the corresponding intrinsic mode functions of each order of the remaining signals; S6. Superimpose the intrinsic mode functions of each measuring point of the bridge after VMD decomposition to obtain the effective displacement time history signal after noise reduction at each measuring point 2. The method for denoising the displacement of the bridge measurement point based on the microwave radar echo energy and VMD according to claim 1, wherein, Step S1 specifically includes: To achieve accurate measurement of the displacement of the bridge measuring points, corner reflectors need to be installed at the bridge measuring point positions before measurement. Ignoring the amplitude of the microwave radar transmission signal, the transmission signal of the microwave radar can be expressed as: In the formula, f0 represents the starting frequency; t represents the time change within the scanning period, also known as the snapshot; φ0 represents the initial phase; B is the microwave radar bandwidth, T is the microwave radar modulation period, and j is the imaginary part. Assuming that the echo signal of the measuring point is received by the microwave radar receiving antenna after a time τ, the microwave radar echo signal after ignoring the amplitude change can be expressed as: In the formula, τ = 2R / c, R represents the distance from the target to the radar, c represents the speed of light. After amplification, down-conversion, and low-pass filtering, the obtained baseband beat signal can be expressed as: where (·) * denotes the complex conjugate. Based on the phase interference method, the displacement of the target in the radar line-of-sight direction can be expressed as: where λ represents the wavelength, represents the phase difference between adjacent moments of the actually measured target signal; among which, is the phase difference caused by target deformation, and respectively represent the phase difference and phase noise caused by atmospheric environmental interference. Since the distances of each measuring point relative to the microwave radar are different, the two are different; The microwave radar can achieve resolution of each measuring point in the range direction. The original displacement time history signal matrix X(t) with inconsistent signal-to-noise ratios at each corresponding measuring point can be expressed as: X(t) m×n ={x LOS,1 x LOS,2 ...x LOS,m} T ; where x LOS,i ={x i,1 x i,2 ...x i,n} represents the displacement time history signal of the i-th range cell, m represents the number of sampling points of a periodic signal of the microwave radar, and n represents the number of periods measured by the microwave radar.

3. The method for denoising the displacement of bridge measurement points based on microwave radar echo energy and VMD according to claim 1, characterized in that Step S4 specifically includes: First, define the decomposition layer number based on the spectral correlation coefficient between each order of the intrinsic mode function and the original signal, which can be defined as: where U j denotes the Fourier transform of the j-th order intrinsic mode function u j after decomposition, j = 1, ..., k; X i denotes the Fourier transform of the original reference displacement signal; The specific steps of the method are as follows: First, set k = 2 and perform k = k + 1 iterative update; According to formula (5), the spectral correlation coefficients (ρ1, ρ2,..., ρ k+1 ) between each independent component after decomposition and the original signal can be obtained; define ρ min = min{ρ1, ρ2,..., ρ k+1}, set the threshold to σ = 0.01; if ρ min < σ, the iteration stops and the decomposition layer number k is determined.

4. The method for denoising bridge measurement point displacement based on microwave radar echo energy and VMD according to claim 1, characterized in that, Step S5 specifically includes: Perform VMD decomposition on the displacement signal of the reference measuring point according to the determined decomposition layer number k to obtain k-order intrinsic mode functions. Perform Fourier transform on the above-mentioned intrinsic mode functions of each order to obtain the frequency domain information of each order of the intrinsic mode functions, that is, the center frequency. Based on the center frequencies of the intrinsic mode functions of each order of the reference measuring point, perform VMD decomposition on the displacement signals of the remaining measuring points, and determine the decomposition layer number so that the center frequencies of the intrinsic mode functions of each order after decomposition correspond one by one to the center frequencies of the intrinsic mode functions of each order of the reference measuring point.

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