A green function reconstruction method combining s-transform and dictionary learning

Abstract

The application discloses a Green function reconstruction method combining S transform and dictionary learning. The method is as follows: S transform denoising is performed on a background noise cross-correlation function containing noise to obtain time-frequency spectrum coefficients and extract real coefficient matrix and imaginary coefficient matrix of the time-frequency spectrum coefficients; dictionary learning is performed on the real coefficient matrix and the imaginary coefficient matrix respectively to obtain a real coefficient matrix dictionary and an imaginary coefficient matrix dictionary; real coefficients after dictionary learning and imaginary coefficients after dictionary learning are obtained; reconstructed spectrum coefficients are obtained, and S inverse transform is performed on the reconstructed spectrum coefficients to obtain reconstructed Green functions. The application can eliminate the interference of noise sources with different directions in the seismic background noise cross-correlation function, and good results can also be achieved for background noise cross-correlation data with low signal-to-noise ratio; the application has low complexity, high timeliness and high reconstruction precision.

Description

Technical Field

[0001] This invention relates to a function reconstruction technique in the field of signal processing, and more particularly to a Green's function reconstruction method that combines S-transform and dictionary learning. Background Technology

[0002] Earthquake background noise is mainly generated by ocean tidal fluctuations, friction and collision between the ocean and seabed, atmospheric disturbances, and friction between the atmosphere and the Earth's surface. With the development of theoretical research, the Green's function can be approximated by cross-correlation calculations of long-term earthquake background noise. In seismology, the Green's function can be understood as the displacement field generated by a unit concentrated impulse force, which is valuable for many applications, such as noise surface wave imaging, monitoring changes in subsurface media, monitoring subsurface attenuation structures, and earthquake location. In actual observational data, due to the uneven distribution of noise sources, the amplitudes of the left and right branches of the earthquake background noise cross-correlation function exhibit asymmetry; that is, the amplitude factor of the earthquake background noise cross-correlation function differs from that of the theoretical Green's function. Therefore, reconstructing the Green's function is necessary.

[0003] Currently, there are several main methods for obtaining the Green's function. One method involves superimposing background noise cross-correlation functions over a long period, but this requires very long time series data, thus demanding high-quality background noise data acquisition. Another method is to approximate the Green's function through linear superposition or phase-weighted superposition of background noise cross-correlation waveforms, but this method is affected by seasonal variations and cannot guarantee the accuracy of the reconstructed Green's function. Since the higher-order cross-correlation of the seismic background noise cross-correlation tail is also symmetrical, further cross-correlation calculations on the tail of the background noise cross-correlation can yield a higher-quality Green's function, but the large computational cost is the main limitation to its application. To address the shortcomings of these methods, a method has been developed to obtain the Green's function by denoising the seismic background noise cross-correlation function, such as S-transform denoising and sliding window weighted singular value decomposition filtering. However, this method is less effective for processing seismic background noise cross-correlation functions with low signal-to-noise ratios. Summary of the Invention

[0004] To overcome the shortcomings of the above methods, this invention provides a Green function reconstruction method that combines S-transform and dictionary learning. This invention denoises the background noise cross-correlation function using S-transform, performs dictionary learning on the real and imaginary coefficients of the denoised S-spectrum, and finally obtains the reconstructed Green function through inverse S-transform, thus realizing the reconstruction of the Green function in the time-frequency domain.

[0005] The technical solution adopted in this invention is:

[0006] This method includes the following steps:

[0007] S1. Perform S-transform denoising on the cross-correlation function of the noisy background noise to obtain the filtered time-spectrum coefficients and extract the real and imaginary coefficient matrices of the filtered time-spectrum coefficients.

[0008] S2. Perform dictionary learning on the real part coefficient matrix and the imaginary part coefficient matrix respectively to obtain the dictionary of the real part coefficient matrix and the dictionary of the imaginary part coefficient matrix;

[0009] S3. Using the dictionary of the real coefficient matrix and the dictionary of the imaginary coefficient matrix, as well as the sparse matrix of the real coefficient matrix and the sparse matrix of the imaginary coefficient matrix, we obtain the real coefficients and the imaginary coefficients after dictionary learning.

[0010] S4. Combine the real coefficients and imaginary coefficients learned from the dictionary to obtain the reconstructed spectral coefficients, and then perform an inverse S-transform on the reconstructed spectral coefficients to obtain the reconstructed Green's function.

[0011] The dictionary learning is specifically determined based on the following model:

[0012]

[0013]

[0014] In the formula, RFS represents the real part coefficient matrix; RD represents the dictionary of the real part coefficient matrix; RX represents the sparse matrix of the real part coefficients; IFS represents the imaginary part coefficient matrix; ID represents the dictionary of the imaginary part coefficient matrix; IX represents the sparse matrix of the imaginary part coefficients; rx i ix represents the column vector of the i-th column of the sparse matrix RX with real coefficients; i The column vector representing the i-th column of the sparse matrix IX with imaginary coefficients; ||·|| F represents the Frobenius norm; ||·||0 represents the l0 norm; ε represents the error threshold; st represents the constraint condition; min() represents the minimum value of the objective function.

[0015] Step S1 specifically involves:

[0016] S3.1. Time window t is applied to the background noise cross-correlation function, setting the group velocity window to 2 km / s to 5 km / s, and setting the signal amplitude outside the time window to 0. This determines the time window t of the background noise cross-correlation function. min ~t max Specifically, it is determined according to the following formula:

[0017] t min =dist / 5

[0018] t max =dist / 2

[0019] In the formula, dist represents the station spacing;

[0020] S3.2. Perform an S-transform on the background noise cross-correlation function truncated in step S3.1, specifically determined according to the following formula:

[0021]

[0022] In the formula, S(τ, f) represents the time-frequency coefficients; τ represents the center point of the window function, controlling the position of the window function on the time axis; f represents the frequency; x(t) represents the background noise cross-correlation function after the time window is truncated; i represents the imaginary unit; t represents time; exp() represents the exponential function;

[0023] S3.3. Set the threshold coefficient λ = 0.08, and determine the element with the largest absolute value in the spectral coefficients S(τ, f) |S max |, the spectral coefficients S(τ,f) whose absolute values ​​are less than the threshold λ|S max Setting the elements of | to 0 yields the filtered time-frequency coefficients FS;

[0024] S3.4. Determine the real and imaginary coefficient matrices of the filtered time-frequency coefficients FS, specifically according to the following formula:

[0025] RFS = Re(FS) = RD * RX

[0026] IFS = Im(FS) = ID * IX

[0027] In the formula, RFS represents the real part coefficient matrix; IFS represents the imaginary part coefficient matrix; Re() represents the real part of the complex number; Im() represents the imaginary part of the complex number; RD represents the dictionary of the real part coefficient matrix; ID represents the dictionary of the imaginary part coefficient matrix; RX represents the sparse matrix of the real part coefficients; and IX represents the sparse matrix of the imaginary part coefficients.

[0028] Step S2 specifically involves:

[0029] First, the dictionary learning algorithm is iteratively updated. Then, based on the iteratively updated dictionary learning algorithm, the updated dictionary of real coefficient matrices and the updated dictionary of imaginary coefficient matrices are obtained. The dictionary learning algorithm is specifically iteratively calculated according to the following formula:

[0030]

[0031] In the formula, RT represents the training sample matrix with real part coefficients; ||·|| F The Frobenius norm is represented by rd; N represents the number of samples in the training set. j This represents the column vector of the j-th column of the dictionary of real part coefficients; rd represents the row vector of the j-th row of the sparse matrix with real coefficients; n This represents the column vector of the nth column of the dictionary of real part coefficients. Represents the row vector of the nth row of the sparse matrix with real coefficients; RE n Represents the real residual; IT represents the training sample matrix of the imaginary coefficients; id j This represents the column vector of the j-th column of the dictionary of imaginary coefficients. Let IE represent the row vector of the j-th row of the sparse matrix containing the imaginary coefficients. n Represents the imaginary residual, id n This represents the column vector of the nth column of the dictionary of imaginary coefficients. Represents the row vector of the nth row of the sparse matrix containing the imaginary coefficients;

[0032] The real residual RE n The column vector rd of the nth column of the dictionary of real part coefficients n The row vector of the nth row of the sparse matrix with real coefficients. Imaginary residual (IE) n The column vector id of the nth column of the dictionary of imaginary coefficients n The row vector of the nth row of the sparse matrix with imaginary coefficients Specifically, the update and iteration are determined based on the following formula:

[0033]

[0034]

[0035] The updated real residual and the column vector rd of the nth column of the dictionary of real coefficients after iterative update. n The row vector of the nth row of the sparse matrix with real coefficients. The column vector id of the nth column of the dictionary of imaginary residuals and imaginary coefficients. n The row vector of the nth row of the sparse matrix with imaginary coefficients Specifically, it is determined according to the following formula:

[0036]

[0037]

[0038] In the formula, RE′ n Let RU denote the real part residual after sparsification; R∑ denote the left singular matrix after real part residual decomposition; RV denote the right singular matrix after real part residual decomposition; IE′ nIU represents the imaginary residual after sparse decomposition; I∑ represents the unitary matrix after imaginary residual decomposition; IV represents the right singular matrix after imaginary residual decomposition; RU(:,1) represents the column vector of the first column of matrix RU; RV T (1,:) represents the row vector of the first row after the transpose of matrix RV; IU(:,1) represents the column vector of the first column of matrix IU; IV T (1,:) represents the row vector of the first row after the IV transpose.

[0039] Step S3 specifically involves:

[0040] S5.1. Combine the dictionary with the updated real coefficient matrix The real part coefficients are reconstructed to determine the real part coefficients after dictionary learning using DRFS; specifically, they are determined according to the following formula:

[0041]

[0042] S5.2. Combine the dictionary with the updated imaginary coefficient matrix. The imaginary part coefficients are reconstructed to determine the DIFS imaginary part coefficients after dictionary learning; specifically, they are determined according to the following formula:

[0043] .

[0044] Step S4 specifically involves:

[0045] The reconstructed spectral coefficients DFS(τ, f) are obtained by combining the real part coefficients DRFS and the imaginary part coefficients DIFS after dictionary learning, and are specifically determined according to the following formula:

[0046] DFS(τ, f) = DRFS + j * DIFS

[0047] Then, an inverse S-transform is performed on the reconstructed spectral coefficients to obtain the reconstructed Green's function, which is determined according to the following formula:

[0048]

[0049] In the formula, Y(t) represents the reconstructed Green's function.

[0050] The beneficial effects of this invention are:

[0051] This invention is based on experiments using the cross-correlation function of earthquake background noise from actual observations. It has achieved significant results even for earthquake background noise cross-correlation functions with low signal-to-noise ratios, restoring the symmetry of the theoretical Green's function amplitude, reducing model complexity, and improving timeliness, thus laying the foundation and providing a guarantee for subsequent applications.

[0052] This invention combines dictionary learning with time-frequency analysis to reconstruct the Green's function in the time-frequency domain. Existing techniques cannot reconstruct a good Green's function when the background noise cross-correlation function is low. This invention also performs well for earthquake background noise cross-correlation functions with low signal-to-noise ratios, and the model has low complexity. Attached Figure Description

[0053] Figure 1 This is a flowchart of the present invention;

[0054] Figure 2 This is the original time-domain signal diagram of waveform 1;

[0055] Figure 3 The image shows the denoising result of the S-transform of waveform 1.

[0056] Figure 4 The result of the S-transform and dictionary learning reconstruction of waveform 1 is shown in the figure.

[0057] Figure 5 The spectrum of waveform 1 during S-transform denoising;

[0058] Figure 6 The spectrum of waveform 1 during S-transform and dictionary learning reconstruction;

[0059] Figure 7 This is the original time-domain signal diagram of waveform 2;

[0060] Figure 8 The image shows the denoising result of the S-transform of waveform 2.

[0061] Figure 9 The result of the S-transform and dictionary learning reconstruction of waveform 2 is shown in the figure.

[0062] Figure 10 The spectrum of waveform 2 when reconstructed by the S-transform;

[0063] Figure 11 The spectrum of waveform 2 during S-transformation and dictionary learning reconstruction. Detailed Implementation

[0064] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0065] like Figure 1 As shown, the method includes the following steps:

[0066] S1. Perform S-transform denoising on the cross-correlation function of the noisy background noise to obtain the time-frequency coefficients and extract the real and imaginary coefficient matrices of the time-frequency coefficients;

[0067] S2. Perform dictionary learning on the real part coefficient matrix and the imaginary part coefficient matrix respectively to obtain the dictionary of the real part coefficient matrix and the dictionary of the imaginary part coefficient matrix;

[0068] S3. Reconstruct the sparse matrices of the real coefficients and the imaginary coefficients using the dictionary of the real coefficient matrix and the dictionary of the imaginary coefficient matrix, respectively, to obtain the real coefficients and the imaginary coefficients after dictionary learning.

[0069] S4. Combine the real coefficients and imaginary coefficients learned from the dictionary to obtain the reconstructed spectral coefficients, and then perform an inverse S-transform on the reconstructed spectral coefficients to obtain the reconstructed Green's function.

[0070] Specifically, dictionary learning is determined based on the following model:

[0071]

[0072]

[0073] In the formula, RFS represents the real part coefficient matrix; RD represents the dictionary of the real part coefficient matrix; RX represents the sparse matrix of the real part coefficients; IFS represents the imaginary part coefficient matrix; ID represents the dictionary of the imaginary part coefficient matrix; IX represents the sparse matrix of the imaginary part coefficients; rx i ix represents the column vector of the i-th column of the sparse matrix RX with real coefficients; i The column vector representing the i-th column of the sparse matrix IX with imaginary coefficients; ||·|| F represents the Frobenius norm; ||·||0 represents the l0 norm; ε represents the error threshold; st represents the constraint condition; min() represents the minimum value of the objective function.

[0074] Specifically, step S1 is as follows:

[0075] S3.1. Perform time windowing on the cross-correlation function of the background noise, such as... Figure 2 and Figure 7 As shown, the group velocity window is set to 2km / s to 5km / s, and the signal amplitude outside the time window is set to 0. The time window t of the background noise cross-correlation function is then determined. min ~t max Specifically, it is determined according to the following formula:

[0076] t min =dist / 5

[0077] t max =dist / 2

[0078] In the formula, dist represents the station spacing;

[0079] S3.2. Perform an S-transform on the background noise cross-correlation function truncated in step S3.1, specifically determined according to the following formula:

[0080]

[0081] In the formula, S(τ, f) represents the time-frequency coefficient; τ represents the center point of the window function, controlling the position of the window function on the time axis; f represents the frequency; x(t) represents the background noise cross-correlation function after the time window is truncated; i represents the imaginary unit; t represents time; π represents pi; exp() represents the exponential function.

[0082] S3.3. Set the threshold coefficient λ = 0.08, and determine the element with the largest absolute value in the spectral coefficients S(τ, f) |S max |, the spectral coefficients S(τ,f) whose absolute values ​​are less than the threshold λ|S max Setting the elements of | to 0 yields the filtered time-frequency coefficients FS;

[0083] S3.4. Determine the real and imaginary coefficient matrices of the filtered time-frequency coefficients FS, specifically according to the following formula:

[0084] RFS = Re(FS) = RD * RX

[0085] IFS = Im(FS) = ID * IX

[0086] In the formula, RFS represents the real part coefficient matrix; IFS represents the imaginary part coefficient matrix; Re() represents the real part of the complex number; Im() represents the imaginary part of the complex number; RD represents the dictionary of the real part coefficient matrix; ID represents the dictionary of the imaginary part coefficient matrix; RX represents the sparse matrix of the real part coefficients; and IX represents the sparse matrix of the imaginary part coefficients.

[0087] Specifically, step S2 is as follows:

[0088] First, the dictionary atoms and sparse coefficients in the dictionary learning algorithm are iteratively updated. Then, based on the iteratively updated dictionary learning algorithm, the updated dictionary of real coefficient matrix and the updated dictionary of imaginary coefficient matrix are obtained. The dictionary learning algorithm specifically performs iterative calculations according to the following formula:

[0089]

[0090] In the formula, RT represents the training sample matrix with real part coefficients; ||·|| F The Frobenius norm is represented by rd; N represents the number of samples in the training set. j This represents the column vector of the j-th column of the dictionary of real part coefficients; rd represents the row vector of the j-th row of the sparse matrix with real coefficients; n This represents the column vector of the nth column of the dictionary of real part coefficients. Represents the row vector of the nth row of the sparse matrix with real coefficients; REn Represents the real residual; IT represents the training sample matrix of the imaginary coefficients; id j This represents the column vector of the j-th column of the dictionary of imaginary coefficients. Let IE represent the row vector of the j-th row of the sparse matrix containing the imaginary coefficients. n Represents the imaginary residual, id n This represents the column vector of the nth column of the dictionary of imaginary coefficients. Represents the row vector of the nth row of the sparse matrix containing the imaginary coefficients;

[0091] Real residual RE n The column vector rd of the nth column of the dictionary of real part coefficients n The row vector of the nth row of the sparse matrix with real coefficients. Imaginary residual (IE) n The column vector id of the nth column of the dictionary of imaginary coefficients n The row vector of the nth row of the sparse matrix with imaginary coefficients Specifically, the update and iteration are determined based on the following formula:

[0092]

[0093]

[0094] In the column vector rd of the nth column of the dictionary of real part coefficients n During updates, to ensure sparsity, the real residuals (REs) need to be updated. n The row vector corresponding to the nth row of the sparse matrix with real coefficients. Discarding positions with values ​​of 0, we obtain the sparsed real residual RE′. n And the real residual RE′ after sparsification n Perform singular value decomposition.

[0095] The updated real residual and the column vector rd of the nth column of the dictionary of real coefficients after iterative update. n The row vector of the nth row of the sparse matrix with real coefficients. The column vector id of the nth column of the dictionary of imaginary residuals and imaginary coefficients. n The row vector of the nth row of the sparse matrix with imaginary coefficients Specifically, it is determined according to the following formula:

[0096]

[0097]

[0098] In the formula, RE′ nLet RU denote the real part residual after sparsification; R∑ denote the left singular matrix after real part residual decomposition; RV denote the right singular matrix after real part residual decomposition; IE′ n IU represents the imaginary residual after sparse decomposition; I∑ represents the unitary matrix after imaginary residual decomposition; IV represents the right singular matrix after imaginary residual decomposition; RU(:,1) represents the column vector of the first column of matrix RU; RV T (1,:) represents the row vector of the first row after the transpose of matrix RV; IU(:,1) represents the column vector of the first column of matrix IU; IV T (1,:) represents the row vector of the first row after the IV transpose.

[0099] Specifically, step S3 is as follows:

[0100] S5.1. Combine the dictionary with the updated real coefficient matrix The real part coefficients are reconstructed to determine the real part coefficients after dictionary learning using DRFS; specifically, they are determined according to the following formula:

[0101]

[0102] S5.2. Combine the dictionary with the updated imaginary coefficient matrix. The imaginary part coefficients are reconstructed to determine the DIFS imaginary part coefficients after dictionary learning; specifically, they are determined according to the following formula:

[0103]

[0104] Specifically, step S4 is as follows:

[0105] The reconstructed spectral coefficients DFS(τ, f) are obtained by combining the real part coefficients DRFS and the imaginary part coefficients DIFS after dictionary learning, and are specifically determined according to the following formula:

[0106] DFS(τ, f) = DRFS + j * DIFS

[0107] Then, an inverse S-transform is performed on the reconstructed spectral coefficients to obtain the reconstructed Green's function, which is determined according to the following formula:

[0108]

[0109] In the formula, Y(t) represents the reconstructed Green's function.

[0110] The dictionary learning process is as follows: First, the data is processed using a sliding window, with a sliding window size of 8*8. The data within each sliding window is then organized into a 64*1 column vector, and the sliding step size is set to 3. Next, the dictionary is initialized. In this invention, the number of dictionary atoms k = 256 is set, and 256 columns of data are randomly selected from the data as the atoms of the dictionary. Then, the standard deviation of the noise within the noise window of the original data is calculated, and the standard deviation is set as the threshold ε for dictionary learning, where ε = std(noise). Due to the large number of data samples, 70% of the samples are set as the training set, and the number of iterations is set to 10.

[0111] Then, the sparse matrix is ​​solved using the Orthogonal Matching Pursuit (OMP) algorithm: first, the real and imaginary coefficient matrices are solved using OMP; then, the real and imaginary residuals are solved, and singular value decomposition is performed on the real and imaginary residuals respectively. The dictionaries of the real and imaginary coefficient matrices are updated column-wise. The original data is then approximated using the obtained dictionaries and their corresponding sparse coding matrices. The real and imaginary coefficients obtained after dictionary learning are restored to time-spectrum coefficients. Finally, the time-spectrum coefficients after dictionary learning are subjected to an inverse S-transform to obtain the reconstructed Green's function. The reconstruction result of the embodiment is as follows: Figures 4-6 and Figures 9-11 As shown.

[0112] In this context, both the real coefficient matrix RFS and the imaginary coefficient matrix IFS contain both noise and useful signals. Therefore, during dictionary learning, an overcomplete dictionary matrix d containing k atoms is used. k Determine the dictionary RD of the real coefficient matrix and the dictionary ID of the imaginary coefficient matrix, where RD, ID ∈ R. m×k , where m represents the dimension of the dictionary and k represents the number of atoms in the dictionary.

[0113] Specifically, if the filtered time-frequency coefficients FS are directly subjected to an inverse S-transform to obtain the reconstructed time-domain signal S(t), the specific formula is determined according to the following formula:

[0114]

[0115] In the formula, S(τ, f) represents the reconstructed spectral coefficients; S(t) represents the reconstructed time-domain signal; the results of the embodiment are as follows. Figure 3 and Figure 8 As shown.

[0116] In summary, through the transformation and denoising results, reconstruction results, reconstruction time spectrum, and spectrum of waveforms 1 and 2 in the embodiments, it can be seen that this method achieves the reconstruction of the Green's function in the time-frequency domain, and achieves good results when the background noise cross-correlation function is low.

Claims

1. A method for reconstructing the cross-correlation function and Green's function of seismic background noise by combining S-transform and dictionary learning, characterized in that: The method includes the following steps: S1. Perform S-transform denoising on the cross-correlation function of the noisy seismic background noise from actual observations to obtain the filtered time-spectrum coefficients and extract the real and imaginary coefficient matrices of the filtered time-spectrum coefficients. S2. Perform dictionary learning on the real part coefficient matrix and the imaginary part coefficient matrix respectively to obtain the dictionary of the real part coefficient matrix and the dictionary of the imaginary part coefficient matrix; S3. Using the dictionary of the real coefficient matrix and the dictionary of the imaginary coefficient matrix, as well as the sparse matrix of the real coefficient matrix and the sparse matrix of the imaginary coefficient matrix, we obtain the real coefficients and the imaginary coefficients after dictionary learning. S4. Combine the real coefficients and imaginary coefficients learned from the dictionary to obtain the reconstructed spectral coefficients, and then perform an inverse S-transform on the reconstructed spectral coefficients to obtain the reconstructed Green's function.

2. The method for reconstructing the seismic background noise cross-correlation function and Green's function by combining S-transform and dictionary learning according to claim 1, characterized in that: The dictionary learning is specifically determined based on the following model: ； ； In the formula, Represents the real part coefficient matrix; A dictionary representing the real part coefficient matrix; A sparse matrix representing the real coefficients; Represents the imaginary part coefficient matrix; A dictionary representing the imaginary part coefficient matrix; A sparse matrix representing the imaginary part coefficients; Sparse matrix representing the real coefficients The column vector of the i-th column; Sparse matrix representing the imaginary coefficients The column vector of the i-th column; Denotes the Frobenius norm; Norm; Error threshold; st represents the constraint condition; () represents the minimum value of the objective function.

3. The method for reconstructing the seismic background noise cross-correlation function and Green's function by combining S-transform and dictionary learning according to claim 1, characterized in that: Step S1 specifically involves: S3.

1. Time window truncation is applied to the cross-correlation function of the seismic background noise, and the group velocity window is set to... Furthermore, the signal amplitude outside the time window is set to 0 to determine the time window for the cross-correlation function of the seismic background noise. Specifically, it is determined according to the following formula: ； ； In the formula, dist represents the station spacing; S3.

2. Perform an S-transform on the seismic background noise cross-correlation function extracted in step S3.1, specifically determined according to the following formula: ； In the formula, Represents the time-frequency coefficients; This indicates the center point of the window function, controlling its position on the time axis; Indicates frequency; This represents the cross-correlation function of seismic background noise after time window truncation; Represents the imaginary unit; Indicates time; Represents an exponential function; S3.

3. Set the threshold coefficient λ=0.08 and determine the spectral coefficients. The element with the largest absolute value , spectral coefficient The absolute value is less than the threshold Setting the elements to 0 yields the filtered time-spectrum coefficients. ; S3.

4. Determine the time-frequency coefficients after filtering. The real and imaginary coefficient matrices are determined using the following formulas: ； ； In the formula, Represents the real part coefficient matrix; Represents the imaginary part coefficient matrix; Represents the real part of a complex number; Represents the imaginary part of a complex number; A dictionary representing the real part coefficient matrix; A dictionary representing the imaginary part coefficient matrix; A sparse matrix representing the real coefficients; This represents a sparse matrix representing the imaginary coefficients.

4. The method for reconstructing the seismic background noise cross-correlation function and Green's function by combining S-transform and dictionary learning according to claim 3, characterized in that: Step S2 specifically involves: First, the dictionary learning algorithm is iteratively updated. Then, based on the iteratively updated dictionary learning algorithm, the updated dictionary of real coefficient matrices and the updated dictionary of imaginary coefficient matrices are obtained. The dictionary learning algorithm is specifically iteratively calculated according to the following formula: ； In the formula, The training sample matrix representing the real part coefficients; Denotes the Frobenius norm; This indicates the number of samples included in the training sample. This represents the column vector of the j-th column of the dictionary of real part coefficients; Represents the row vector of the j-th row of the sparse matrix containing the real coefficients; This represents the column vector of the nth column of the dictionary of real part coefficients. Represents the row vector of the nth row of the sparse matrix containing the real coefficients; Represents the real residual; The training sample matrix representing the imaginary part coefficients; This represents the column vector of the j-th column of the dictionary of imaginary coefficients. Let j represent the row vector of the j-th row of the sparse matrix containing the imaginary coefficients. Indicates the imaginary residual. This represents the column vector of the nth column of the dictionary of imaginary coefficients. Represents the row vector of the nth row of the sparse matrix containing the imaginary coefficients; The real residual The column vector of the nth column of the dictionary of real part coefficients The row vector of the nth row of the sparse matrix with real coefficients. Imaginary residuals The column vector of the nth column of the dictionary of imaginary coefficients The row vector of the nth row of the sparse matrix with imaginary coefficients Specifically, the update and iteration are determined based on the following formula: ； ； The updated real residuals and the column vector of the nth column of the dictionary of real coefficients after iterative updates. The row vector of the nth row of the sparse matrix with real coefficients. The column vector of the nth column of the dictionary of imaginary residuals and imaginary coefficients. The row vector of the nth row of the sparse matrix with imaginary coefficients Specifically, it is determined according to the following formula: ； ； In the formula, This represents the real residual after sparsification. Represents the left singular matrix after decomposition of the real part residual; R Represents the unitary matrix after decomposition of the real part residual; Represents the right singular matrix after decomposition of the real part residual; This represents the imaginary residual after sparsification. Represents the left singular matrix after decomposition of the imaginary residual; I Represents the unitary matrix after decomposition of the imaginary residual; Represents the right singular matrix after decomposition of the imaginary residual; Representation matrix The column vector of the first column; Representation matrix The row vector of the first row after transposition; Representation matrix The column vector of the first column; express The row vector of the first row after transposition.

5. The method for reconstructing the seismic background noise cross-correlation function and Green's function by combining S-transform and dictionary learning according to claim 4, characterized in that: Step S3 specifically involves: S5.

1. Combine the dictionary with the updated real coefficient matrix The real coefficients are reconstructed to determine the real coefficients after dictionary learning. ; Specifically, it is determined according to the following formula: ； S5.

2. Combine the dictionary with the updated imaginary coefficient matrix. The imaginary coefficients are reconstructed to determine the imaginary coefficients after dictionary learning. Specifically, it is determined according to the following formula: 。 6. The Green's function reconstruction method combining S-transform and dictionary learning according to claim 5, characterized in that: Step S4 specifically involves: Real coefficients learned from the dictionary The imaginary part coefficients learned from the dictionary Obtain the reconstructed spectral coefficients Specifically, it is determined according to the following formula: ； Then, an inverse S-transform is performed on the reconstructed spectral coefficients to obtain the reconstructed Green's function, which is determined according to the following formula: ； In the formula, This represents the reconstructed Green's function.

Images