A GPR signal denoising method based on variational mode decomposition and singular spectrum analysis

Abstract

The present application discloses a GPR signal denoising method based on variational mode decomposition and singular spectrum analysis. First, the optimal mode number K is determined based on the minimum energy loss ratio method, the original signal is decomposed by VMD and K components are obtained, and then the effective signal component is obtained by the Pearson correlation coefficient method. For the small amount of noise that still exists in the effective signal component, the singular spectrum analysis is introduced for secondary filtering to further improve the signal-to-noise ratio and reconstruct the denoised signal. Simulation experiments, numerical simulations and measured data have verified the effectiveness of the above method. The method of the present application can effectively suppress background noise and clutter interference, making the signal characteristics of the abnormal body more prominent, further confirming the effectiveness and applicability of the method for processing GPR signals, and providing a valuable reference for the data processing of GPR actual data and the secondary utilization of effective signals.

Description

Technical Field

[0001] The present invention relates to the field of geophysical technology, and in particular to a GPR signal denoising method based on variational mode decomposition and singular spectrum analysis. Background Art

[0002] Ground Penetrating Radar (GPR) is a fast and efficient geophysical detection method that studies the physical characteristics of the target body, such as its shape, size, and spatial distribution, by receiving and analyzing the target body information carried in the reflected echo. Due to its advantages such as simple operation, fast acquisition, strong adaptability, and non-destructive detection, it has been widely used in the fields of hydrological phenomenon monitoring, urban pipeline measurement, tunnel detection, and geological disaster investigation.

[0003] In the process of ground penetrating radar acquisition, affected by factors such as equipment and environment, there are different degrees of noise interference in the collected ground penetrating radar (GPR) signals, and the mixed noise signal has the characteristics of randomness and non-stationarity. In response to this, domestic and foreign scholars have adopted many denoising technologies to improve the resolution of ground penetrating radar signals, as follows:

[0004] Wavelet threshold denoising is a commonly used denoising analysis method to characterize the time-varying frequency of non-stationary signals. However, in the process of using wavelet function to denoise, the selection of wavelet basis, decomposition level and wavelet threshold is easily affected by human selection, which may lead to deviation in the results.

[0005] Empirical Mode Decomposition (EMD) is based on the time scale characteristics of the signal itself. It does not need to set basis functions or eigenvalue parameters. It has the advantage of adaptive time-frequency processing signals and is suitable for processing nonlinear and non-stationary signals. Therefore, the EMD method is better than the wavelet threshold denoising method in ground penetrating radar signal processing. However, for complex signals, the EMD method often causes mode aliasing and endpoint flying wings when performing signal decomposition. In order to solve this problem, researchers in this field further applied Ensemble Empirical Mode Decomposition (EEMD) to denoise ground penetrating radar signals, and obtained a denoising effect that is better than EMD. However, since EEMD artificially adds white noise and then averages each component, although the modal aliasing problem of EMD is solved, there is still residual white noise in the reconstructed signal, and the cyclic process increases the computational complexity of the algorithm.

[0006] Variational Mode Decomposition (VMD) is a time-frequency analysis method proposed in recent years. This method achieves noise suppression by searching for the optimal solution of the variational model to separate components with different center frequencies. However, the selection of the optimal mode number is subjective, that is, the selection of the mode number K and the extraction of effective signals still rely on the comparison of denoising effects and manual experience analysis and selection, and the denoised signal still contains some low- and medium-frequency noise, which causes the reconstructed data to have different degrees of signal oscillation.

[0007] In summary, researchers need to improve the existing ground penetrating radar signal denoising methods to solve the above problems. Summary of the invention

[0008] The purpose of this application is to provide a GPR signal denoising method based on variational mode decomposition and singular spectrum analysis to achieve efficient and accurate denoising of ground penetrating radar signals. The technical solution of this application is as follows:

[0009] A GPR signal denoising method based on variational mode decomposition and singular spectrum analysis is proposed. First, the original signal is decomposed by VMD to obtain K components. Then, the effective signal component and the noise signal component are screened by the Pearson correlation coefficient method. Finally, the singular spectrum analysis is used to further remove the residual noise in the effective signal component and reconstruct the denoised signal.

[0010] Generally speaking, the noise remaining in the effective signal component refers to the mid- and low-frequency noise, and for different GPR signals, the mid- and low-frequency noise is within 10 percent of its corresponding center frequency.

[0011] In some specific embodiments, the VMD decomposition process includes the following steps:

[0012] Step 1) using Hilbert transform to calculate the analytical signal of each modal component and obtain its corresponding single-sided spectrum;

[0013] Step 2) adjusting the estimated center frequency of each mode by adding an exponential term to each mode, and adjusting the spectrum of each modal component to its corresponding baseband;

[0014] Step 3) Construct the expression of the constrained variational model:

[0015]

[0016] Where f is the original signal, {u k} is the modal component, {w k}is{u k} is the center frequency, δ(t) is the impulse function, K is the mode number, * is the convolution operation, and ||·|| is the 2-norm;

[0017] Step 4) In solving the variational problem, the quadratic penalty factor α and the Lagrange operator λ(t) are introduced to transform it into an unconstrained equation:

[0018]

[0019] Where, <, > represents the dot product;

[0020] Step 5) Using the alternating direction multiplier iteration algorithm combined with Fourier isometric transform, the corresponding modal components and center frequencies are optimized, and the saddle point of the augmented Lagrangian expression is searched, and the solution is:

[0021]

[0022] Step 6) Repeat step 15) until the iteration condition is met and K components are output. The iteration formula is:

[0023]

[0024] In the formula, represents Fourier transform, τ is the update parameter, ε is the judgment accuracy and ε>0.

[0025] In some specific embodiments, before performing VMD decomposition, a value range of the modal number K is preset, the energy loss ratio η corresponding to each K value is calculated, and the K value corresponding to the minimum η is selected as the optimal modal number.

[0026] In some specific embodiments, the process of determining the optimal modal number K based on the energy loss ratio minimization method is as follows:

[0027] Assume that the ground penetrating radar signal collected for a finite time length L is f i (i=1,2,3...,L), after VMD decomposition, its components are u k (k=1,2,3,...,K), then the total energy of the signal is defined as:

[0028]

[0029] The sum of the energy of the signal components is:

[0030]

[0031] Due to the approximate orthogonality of VMD decomposition, that is, the better the orthogonality of each component, the less energy loss, the energy loss ratio when the modal number K takes different values ​​is compared. When η is the smallest, the corresponding K value is the most appropriate mode number.

[0032] If the modal number K is appropriate, the total energy of the decomposed components should be equal to the original signal energy. However, in the decomposition process, complete orthogonality is often not achieved. Therefore, the modal number K closest to the best, that is, the state closest to orthogonality, can be determined based on the minimum energy loss ratio. The energy loss ratio minimum method has a complete theoretical basis, has practical significance, and is simple and fast to calculate.

[0033] In some specific embodiments, the process of screening signal components using the Pearson correlation coefficient method is as follows:

[0034] Normalize the original signal and each component, and calculate the Pearson correlation coefficient of each component:

[0035]

[0036] In the formula, r i is the correlation coefficient between the i-th component and the original signal f, is the average value of the i-th component, i = 1, 2, 3, ..., K;

[0037] The standard deviation of the correlation coefficient is used as the judgment threshold to divide the effective signal and the noise signal. The calculation formula is:

[0038]

[0039] If r i >TH, the i-th component is considered to be a valid signal, otherwise it is a noise signal.

[0040] The Pearson correlation coefficient can screen out regular valid signals or messy noise signals according to the correlation between the component and the original signal. This method has low computational complexity, high computational efficiency, and good recognition effect, and can effectively separate the valid component and the noise component.

[0041] In some specific embodiments, the singular spectrum analysis process includes the following steps:

[0042] SSA denoising is performed on the effective signal component. The signal length is L, the window length is m, and n=L-m+1. Then the original time series can be lagged and arranged to obtain an m×n trajectory matrix:

[0043]

[0044] Calculate the covariance matrix S of the trajectory matrix = ff T Then perform singular value decomposition to obtain m eigenvalues ​​and sort them λ1≥λ2≥…≥λ m ≥0 and its corresponding eigenvectors U1, U2, …, U m , and there are:

[0045]

[0046] In the formula, f i represents the i-th singular spectrum value, and its eigenvalue represents the energy contribution of the corresponding singular spectrum to the original signal;

[0047] Calculate the lagged series f in {U m Projection on}:

[0048]

[0049] Where U k represents the eigenvector corresponding to the kth eigenvalue, 0<i<Lm, for The reflected time evolution weight is called the time principal component;

[0050] According to the threshold, the first t eigenvalues ​​are selected, and the corresponding {f t} to form a new subset, the m×n trajectory matrix f can be reconstructed into a time series of length L, and the reconstruction formula is:

[0051]

[0052] In some specific embodiments, the window length m is less than 0.5 times the signal length L.

[0053] The technical solution provided by this application has at least the following beneficial effects:

[0054] The GPR signal denoising method proposed in this application first uses the VMD method to remove random noise in the signal and optimizes the selection of the modal number K. Then, singular spectrum analysis (SSA) is introduced to remove a small amount of medium and low frequency noise still existing in the effective signal component, thereby improving the signal oscillation problem. The correctness and effectiveness of the method are verified by Ricker wavelet simulation and forward numerical simulation experiments. Finally, the method is applied to the ground penetrating radar signal denoising processing of actual engineering projects, which can significantly remove noise and highlight the abnormal target signal, providing a valuable reference for the data processing of GPR actual data and the secondary utilization of effective signals. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. It should be understood that the following drawings only show certain embodiments of the present application and therefore should not be regarded as limiting the scope. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0056] Figure 1It is a flow chart of the GPR signal denoising method based on variational mode decomposition and singular spectrum analysis in this application;

[0057] Figure 2 The comparison diagram of 100MHz Ricker wavelet signal denoising results in the simulation experiment of the embodiment of the present application; wherein a) is the Ricker wavelet plus noise synthesis signal, b) is the denoised signal after using the EEMD method; c) is the denoised signal after using the VMD method; d) is the denoised signal after using the method of the present application;

[0058] Figure 3 The comparison diagram of the 900MHz Ricker wavelet signal denoising results in the simulation experiment of the embodiment of the present application; wherein a) is the Ricker wavelet plus noise synthesis signal, b) is the denoised signal after using the EEMD method; c) is the denoised signal after using the VMD method; d) is the denoised signal after using the method of the present application;

[0059] Figure 4 This is a comparison chart of the denoising results of the 100th channel of the synthetic ground penetrating radar signal in the numerical simulation test of the embodiment of the present application, wherein a) is the 100th channel of the synthetic ground penetrating radar signal, b) is the denoised signal after using the EEMD method; c) is the denoised signal after using the VMD method; d) is the denoised signal after using the method of the present application;

[0060] Figure 5 The single-channel denoising effect diagram of the ground penetrating radar profile in the actual test of the embodiment of the present application, wherein a) is the 100th ground penetrating radar profile, and b) is the 200th ground penetrating radar profile. DETAILED DESCRIPTION

[0061] To facilitate the understanding of the present application, the technical solution in the present application will be described in more comprehensive and detailed manner in combination with the accompanying drawings and preferred embodiments of the specification. However, the protection scope of the present application is not limited to the following specific embodiments. Based on the embodiments in the present application, all other embodiments obtained by ordinary technicians in the field without making creative work are within the scope of protection of the present application.

[0062] Unless otherwise defined, all professional terms used below have the same meanings as those generally understood by those skilled in the art. The professional terms used herein are only for the purpose of describing specific embodiments and are not intended to limit the scope of protection of this application.

[0063] Example

[0064] See also Figure 1 , a GPR signal denoising method based on variational mode decomposition and singular spectrum analysis, the specific process is as follows:

[0065] 1. Variational Mode Decomposition

[0066] Step 1) using Hilbert transform to calculate the analytical signal of each modal component and obtain its corresponding single-sided spectrum;

[0067] Step 2) adjusting the estimated center frequency of each mode by adding an exponential term to each mode, and adjusting the spectrum of each modal component to its corresponding baseband;

[0068] Step 3) Construct the expression of the constrained variational model:

[0069]

[0070] Where f is the original signal, {u k} is the modal component, {w k}is{u k} is the center frequency, δ(t) is the impulse function, K is the mode number, * is the convolution operation, and ||·|| is the 2-norm;

[0071] Step 4) In solving the variational problem, the quadratic penalty factor α and the Lagrange operator λ(t) are introduced to transform it into an unconstrained equation:

[0072]

[0073] Where, <, > represents the dot product;

[0074] Step 5) Using the alternating direction multiplier iteration algorithm combined with Fourier isometric transform, the corresponding modal components and center frequencies are optimized, and the saddle point of the augmented Lagrangian expression is searched, and the solution is:

[0075]

[0076] Step 6) Repeat step 15) until the iteration condition is met and K components are output. The iteration formula is:

[0077]

[0078] In the formula, represents Fourier transform, τ is the update parameter, ε is the judgment accuracy and ε>0.

[0079] It should be noted that VMD decomposition requires the presetting of the modal number K. A K value that is too large will lead to excessive decomposition, causing the effective signal frequency to be reflected in multiple components. A K value that is too small will lead to incomplete decomposition, with effective signals and noise signals aliasing. Therefore, choosing the most appropriate modal number K is the key to using the VMD method.

[0080] The components obtained after VMD decomposition are orthogonal, so the sum of the energies of the components should be equal to the energy of the original signal. However, in actual operation, energy loss often occurs due to the inability to orthogonal decomposition. The less energy loss, the better the orthogonality of the decomposition. Therefore, the number of decomposition modes K can be selected by quantitatively analyzing the signal loss energy.

[0081] The energy loss ratio is introduced in the method of this application, and the K value corresponding to the minimum energy loss ratio is determined as the optimal mode number. The specific process is:

[0082] Assume that the ground penetrating radar signal collected for a finite time length L is f i (i=1,2,3...,L), after VMD decomposition, its components are u k (k=1,2,3,...,K), then the total energy of the signal is defined as:

[0083]

[0084] The sum of the energy of the signal components is:

[0085]

[0086] Comparison of energy loss ratios when the modal number K takes different values When η is the smallest, the corresponding K value is the most appropriate mode number.

[0087] The correlation coefficient method determines the effective signal and the noise signal by calculating the Pearson correlation coefficient between each component and the original signal. In order to avoid the removal of effective signals with small amplitudes, all components and the original signal must be normalized first. The normalized Pearson correlation coefficient is:

[0088]

[0089] In the formula, r i is the correlation coefficient between the i-th component and the original signal f, is the average value of the i-th component, i = 1, 2, 3, ..., K;

[0090] The standard deviation of the correlation coefficient is used as the judgment threshold to divide the effective signal and the noise signal. The calculation formula is:

[0091]

[0092] If r i >TH, the i-th component is considered to be a valid signal, otherwise it is a noise signal.

[0093] 2. Singular spectrum analysis

[0094] In the effective signal obtained by VMD decomposition, there is often a part of noise remaining, which leads to signal oscillation. In order to further remove this part of the noise signal and improve the oscillation phenomenon, the singular spectrum analysis method is introduced in the method of this application. By decomposing the trajectory matrix of the time series to be studied, the eigenvalue of the time series matrix is ​​calculated, and the smaller eigenvalue is identified as the noise subspace, while the larger eigenvalue is retained to reconstruct the signal, thereby achieving noise suppression.

[0095] Step 7) Decomposition: Perform SSA denoising on the effective signal components, select a suitable window length m, usually m < L / 2, L is the signal length, let n = L-m+1, then the original time series can be lagged and arranged to obtain an m×n trajectory matrix:

[0096]

[0097] Calculate the covariance matrix S of the trajectory matrix = ff T Then perform singular value decomposition to obtain m eigenvalues ​​and sort them λ1≥λ2≥…≥λ m ≥0 and its corresponding eigenvectors U1, U2, …, U m , and there are:

[0098]

[0099] Where U is the left singular vector, V is the right singular vector, and f i Represents the i-th singular spectrum value, and the size of its eigenvalue represents the energy contribution of the corresponding singular spectrum to the original signal. Larger eigenvalues ​​contain more effective information, while smaller eigenvalues ​​are considered to be noise;

[0100] Calculate the lagged series f in {U m Projection on}:

[0101]

[0102] Where U k represents the eigenvector corresponding to the kth eigenvalue, 0<i<Lm, for The time evolution weight reflected is called the time principal component.

[0103] Step 8) Reconstruction: Select the first t eigenvalues ​​according to the threshold, and the corresponding {f t} to form a new subset, the m×n trajectory matrix f can be reconstructed into a time series of length L, and the reconstruction formula is:

[0104]

[0105] Simulation experiment

[0106] In order to verify the denoising performance of the method of the present application for low-frequency noisy signals and high-frequency noisy signals, the denoising of the synthetic Ricker wavelet signal is compared using the integrated empirical mode decomposition method EEMD, the traditional variational mode decomposition method VMD and the method of the present application VMD-SSA. The synthetic signal consists of Ricker wavelet signals (100MHz / 900MHz) at 20ns, 50ns, and 80ns and Gaussian white noise.

[0107] See also Figure 2 and Figure 3 For the noisy Ricker wavelet signal, the three methods can suppress the noise to a certain extent and restore the signal. For signals of different frequencies, the denoising effects of the EEMD and VMD methods are unstable. For low-frequency signals, the former has smaller oscillation than the latter and is more in line with the noise-free signal; for high-frequency signals, the opposite is true, and the EEMD reconstructed signal contains more noise. The method of the present application can suppress the noise to the greatest extent for Ricker wavelet signals of different main frequencies, making the overall signal smoother and highlighting the effective signal.

[0108] Table 1 Comparison of denoising effects of three methods

[0109]

[0110] Combined with the signal-to-noise ratio analysis after denoising in Table 1, it can be seen that the method of the present application obtains the highest signal-to-noise ratio and the lowest root mean square error in both high-frequency and low-frequency cases, indicating that the method of the present application has a better noise reduction effect, and the denoised signal is closer to the original signal, and has effectiveness and stability.

[0111] Numerical simulation test

[0112] A space with a uniform background medium is established, with a relative dielectric constant of 6, a conductivity of 0.005S / m, and a relative magnetic permeability of 1. Three abnormal bodies with a buried depth of 0.2m and a radius of 0.1m are set in the dielectric layer, which are air pipelines, metal pipelines, and water pipelines. The calculation area size is set to 5m×1m, the top air layer thickness is 0.1m, the calculation grid size is 0.005m, the time step is 0.01ns, the time step is 2500 steps, and the time window length is 25ns. The transmitting antenna and the receiving antenna are both placed in the air layer, and the self-excitation and self-receiving mode is adopted. The Blackman-Harris pulse is used as the excitation source, the main frequency is 900MHz, 200 channels are simulated and collected, and the channel spacing is 0.025m. The ground penetrating radar profile is obtained by forward modeling through the FDTD method, and the synthetic ground penetrating radar signal after adding Gaussian white noise is obtained.

[0113] The signal-to-noise ratios of the denoising methods of EEMD, VMD and the method of the present application are shown in Table 2. Obviously, the method of the present application obtains the highest signal-to-noise ratio, suppresses the noise to the greatest extent, and highlights the effective echo information, which is consistent with the simulation experimental results.

[0114] Table 2 SNR results of numerical simulation B-scan scan profile after denoising

[0115]

[0116] See also Figure 4 , take the denoising result of the 100th track of the GPR profile, and zoom in on the waveform from 3ns to 6ns. It can be seen that the EEMD method and the VMD method can restore the reflected echo at this position to a certain extent, but the waveform oscillation is still serious, and the reconstructed signal still contains a lot of noise. The signal denoised by the method of this application almost coincides with the noise-free signal waveform at the reflected echo, the waveform oscillation is well improved, the overall waveform is smoother, and the denoising effect is the best.

[0117] Actual test

[0118] A GPR antenna with a center frequency of 400MHz was used to scan and detect the target area of ​​underground power bushings in Guangzhou, and the single-channel data processing results of the 100th and 200th channels of the ground penetrating radar profile were intercepted. Figure 5 The waveform in the noise-free area of ​​the first 50ns remains basically the same, and the signals of the direct wave and part of the reflected echo are retained. From about 60ns, the messy noise waveform is well suppressed, the signal curve is smoother, and the denoising effect is achieved.

[0119] In summary, the denoising effect of the method in the present application in ground penetrating radar signals is better than that of the integrated empirical mode decomposition (EEMD) method and the traditional VMD method, which further confirms the effectiveness and applicability of this method for processing GPR signals, and provides a valuable reference for the data processing of actual GPR data and the secondary utilization of effective signals.

[0120] The above description is only a partial embodiment of the present application, and does not limit the scope of patent protection of the present application. For those skilled in the art, the present application may have various modifications and changes. Within the spirit and principle of the present application, any improvement or equivalent replacement made by using the contents of the present application specification and drawings, directly or indirectly applied in other related technical fields, shall be included in the scope of patent protection of the present application.

Claims

1. A GPR signal denoising method based on variational mode decomposition and singular spectrum analysis, characterized in that: First, VMD is performed on the original signal to obtain K components. Then, the effective signal component and the noise signal component are screened by the Pearson correlation coefficient method. Finally, the singular spectrum analysis is used to further remove the residual noise in the effective signal component and reconstruct the denoised signal. The VMD decomposition process includes the following steps: Step 1) using Hilbert transform to calculate the analytical signal of each modal component and obtain its corresponding single-sided spectrum; Step 2) adjusting the estimated center frequency of each mode by adding an exponential term to each mode, and adjusting the spectrum of each modal component to its corresponding baseband; Step 3) Construct the expression of the constrained variational model: Where f is the original signal, {u k } is the modal component, {w k }is{u k } is the center frequency, δ(t) is the impulse function, K is the mode number, * is the convolution operation, and ||·|| is the 2-norm; Step 4) In solving the variational problem, the quadratic penalty factor α and the Lagrange operator λ(t) are introduced to transform it into an unconstrained equation: Where, <, > represents the dot product; Step 5) Using the alternating direction multiplier iteration algorithm combined with Fourier isometric transform, the corresponding modal components and center frequencies are optimized, and the saddle point of the augmented Lagrangian expression is searched, and the solution is: Step 6) Repeat step 15) until the iteration condition is met and K components are output. The iteration formula is: In the formula, represents Fourier transform, τ is the update parameter, ε is the judgment accuracy and ε>0; Before VMD decomposition, the value range of the modal number K is preset, the energy loss ratio η corresponding to each K value is calculated, and the K value corresponding to the minimum η is selected as the optimal modal number; The process of determining the optimal modal number K based on the minimum energy loss ratio method is: Assume that the ground penetrating radar signal collected for a finite time length L is f i (i=1,2,3...,L), after VMD decomposition, its components are u k (k=1,2,3,...,K), then the total energy of the signal is defined as: The sum of the energy of the signal components is: Due to the approximate orthogonality of VMD decomposition, that is, the better the orthogonality of each component, the less energy loss, the energy loss ratio when the modal number K takes different values ​​is compared. When η is the smallest, the corresponding K value is the most appropriate mode number.

2. The GPR signal denoising method based on variational mode decomposition and singular spectrum analysis according to claim 1, characterized in that: The process of screening signal components by Pearson correlation coefficient method is as follows: Normalize the original signal and each component, and calculate the Pearson correlation coefficient of each component: In the formula, r i is the correlation coefficient between the i-th component and the original signal f, is the average value of the i-th component, i = 1, 2, 3, ..., K; The standard deviation of the correlation coefficient is used as the judgment threshold to divide the effective signal and the noise signal. The calculation formula is: If r i >TH, the i-th component is considered to be a valid signal, otherwise it is a noise signal.

3. The GPR signal denoising method based on variational mode decomposition and singular spectrum analysis according to claim 2, characterized in that: The singular spectrum analysis process includes the following steps: SSA denoising is performed on the effective signal component. The signal length is L, the window length is m, and n=L-m+1. Then the original time series can be lagged and arranged to obtain an m×n trajectory matrix: Calculate the covariance matrix S of the trajectory matrix = ff T Then perform singular value decomposition to obtain m eigenvalues ​​and sort them λ1≥λ2≥…≥λ m ≥0 and its corresponding eigenvectors U1, U2, …, U m , and there are: In the formula, f i represents the i-th singular spectrum value, and its eigenvalue represents the energy contribution of the corresponding singular spectrum to the original signal; Calculate the lagged series f in {U m Projection on}: Where U k represents the eigenvector corresponding to the kth eigenvalue, 0<i<Lm, for The reflected time evolution weight is called the time principal component; According to the threshold, the first t eigenvalues ​​are selected, and the corresponding {f t } to form a new subset, the m×n trajectory matrix f can be reconstructed into a time series of length L, and the reconstruction formula is:

4. The GPR signal denoising method based on variational mode decomposition and singular spectrum analysis according to claim 3, characterized in that: The value of the window length m is less than 0.5 times the signal length L.

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