A seismic data denoising method based on multi-component joint and polarization constraint

Abstract

The application is suitable for the field of geophysical exploration technology, and provides a seismic data denoising method based on multi-component joint and polarization constraint, comprising: obtaining three-component seismic data, establishing mathematical expression of the three-component seismic data, and introducing a rotation matrix to map the original data to a latent wave field representation; based on Hankel low-rank theory, constructing a Hankel matrix for a single-component spatial sequence and performing singular value decomposition; constructing a multi-component joint Hankel matrix, performing joint singular value decomposition on the multi-component joint Hankel matrix; introducing a rotation matrix, performing joint low-rank filtering on the latent wave field representation, and obtaining an optimal rotation matrix by jointly optimizing a total loss function containing polarization consistency loss and orthogonal constraint loss; and performing inverse mapping by using the optimal rotation matrix to obtain a final denoising result. The application can effectively suppress random noise and improve seismic data quality and subsequent interpretation reliability.

Description

Technical Field

[0001] This invention belongs to the field of geophysical exploration technology, and particularly relates to a method for denoising seismic data based on multi-component joint and polarization constraints. Background Technology

[0002] Multi-component seismic exploration technology simultaneously records the vibration information of particles in the subsurface medium in multiple directions (such as horizontal X, Y, and vertical Z), providing far more information than traditional single-component seismic data for oil and gas reservoir exploration, fracture prediction, and anisotropy analysis. Since seismic waves are essentially vector wave fields, different components of multi-component data possess inherent amplitude ratios, phase correspondences, and spatial structural consistency determined by the physical properties of the subsurface medium.

[0003] However, in actual acquisition, multi-component seismic data is easily affected by factors such as complex surface conditions, differences in detector coupling, and random environmental interference, resulting in a significant reduction in signal-to-noise ratio. In particular, random noise severely interferes with the effective signal, posing a huge challenge to subsequent imaging and geological interpretation.

[0004] To address this, academia and industry have proposed various seismic data denoising methods, including various transform domain filtering methods (such as Fourier transform, wavelet transform, and Curvelet transform), sparse representation, dictionary learning, and the recently emerging deep learning methods. However, existing technologies generally suffer from the following limitations when processing multi-component seismic data:

[0005] (1) Independent processing of components destroys vector characteristics: Most methods treat each component of multi-component data as an independent scalar field and process them separately. This approach is simple and easy to implement, but it ignores the inherent physical coupling relationship between components, which leads to the distortion of key vector information such as amplitude ratio and phase difference between components after denoising, destroying the vector properties of the data and affecting the accuracy of subsequent wave field separation and parameter inversion.

[0006] (2) Reliance on fixed coupling model, poor adaptability: Some methods attempt to utilize the coupling relationship between components, but often rely on a pre-set, fixed linear transformation model to characterize the relationship between different components. In the actual situation where the underground medium is highly anisotropic, complex in structure, and has drastic wave field changes, such fixed a priori assumptions are difficult to accurately describe the complexity and spatial nonstationarity of the real wave field, resulting in poor joint processing performance, especially in the low signal-to-noise ratio region.

[0007] (3) Insufficient protection of weak signals: Traditional filtering methods are based on the assumption of local smoothness or sparsity. While suppressing noise, they are prone to excessively weakening the amplitude of low-energy effective signals such as weak reflection events and converted waves, resulting in waveform distortion and loss of details.

[0008] Therefore, there is an urgent need for a new denoising method that can fully utilize the characteristics of multi-component data vectors, adaptively mine the coupling relationship between components, and effectively protect weak signals under low signal-to-noise ratio conditions. Summary of the Invention

[0009] The purpose of this invention is to provide a seismic data denoising method based on multi-component joint and polarization constraints, aiming to solve the problems mentioned in the background art.

[0010] The present invention is implemented as follows: a seismic data denoising method based on multi-component joint and polarization constraints includes the following steps:

[0011] Step 1: Acquire three-component seismic data, establish the mathematical expression of the three-component seismic data, and introduce a rotation matrix to map the original data to the latent wavefield representation;

[0012] Step 2: Based on Hankel low-rank theory, construct the Hankel matrix for the single-component spatial sequence and perform singular value decomposition to achieve single-component denoising;

[0013] Step 3: Construct a multi-component joint Hankel matrix, and perform joint singular value decomposition on the multi-component joint Hankel matrix to achieve multi-component joint low-rank filtering;

[0014] Step 4: Introduce a rotation matrix, perform joint low-rank filtering on the latent wave field representation, and construct polarization consistency loss and orthogonality constraint loss. By jointly optimizing the total loss function containing the polarization consistency loss and the orthogonality constraint loss, obtain the optimal rotation matrix. Use the optimal rotation matrix to perform inverse mapping to obtain the final denoising result.

[0015] A further technical solution, the specific steps of step 1 are as follows:

[0016] Let the three-component seismic data be:

[0017] (1)

[0018] in, For time sampling points, For spatial location, This represents the observation vector of the original three-component data at its temporal and spatial location. These are observations from three components of seismic data in three directions; within a local spatial window, the reflected wave is represented as a superposition of a finite number of plane waves:

[0019] (2)

[0020] in, This represents the scalar wave field within a local window. This represents the number of plane waves within the local window. For the first The time correlation coefficient of a plane wave For the first One spatial basis function, For the first The spatial wavenumber of a plane wave The imaginary unit is used; based on the coupling relationship between multi-component data, a rotation representation is performed through linear transformation:

[0021] (3)

[0022] in, For rotation matrix, express transpose, This represents the potential wave field.

[0023] A further technical solution, the specific steps of step 2 are as follows:

[0024] First, perform a Fourier transform on the time direction.

[0025] (4)

[0026] in, For frequency variables, Indicates the spatiotemporal signal in frequency variable Below, spatial position Complex spectral values ​​at a given frequency; at a fixed frequency variable The following spatial sequence is obtained:

[0027] (5)

[0028] in, This represents the total number of seismic traces. For fixed frequency variables Below, the spatial location is The complex spectrum value. At a fixed frequency variable. Below, the spatial sequence is simplified as ,in, for The simplified representation of ; thus constructing the Hankel matrix of the single-component spatial sequence:

[0029] (6)

[0030] in, The column number of the Hankel matrix. Let be the row number of the Hankel matrix. This represents the Hankel matrix constructed from a single-component spatial sequence; if the signal is derived from... Composed of plane waves, then:

[0031] (7)

[0032] in, This indicates finding the rank of a matrix; for a local wave field consisting of a finite number of plane waves, perform singular value decomposition on its Hankel matrix:

[0033] (8)

[0034] in, It is a left singular vector matrix. It is a singular value diagonal matrix. This is the conjugate transpose of the right singular vector matrix. The sign for conjugate transpose; retain the first one. The singular values ​​yield the reduced-rank matrix:

[0035] (9)

[0036] in, Indicates to retain the previous text A reduced-rank matrix obtained from singular values Indicates to retain the previous text The diagonal matrix after each singular value;

[0037] Finally, the reduced-rank matrix is ​​restored to a spatial sequence by averaging along the anti-diagonal:

[0038] (10)

[0039] in, Represents the first element in the recovered one-dimensional sequence. One element, express The first in One element, Indicates the first The set of matrix elements corresponding to each anti-diagonal line. The number of elements on the anti-diagonal.

[0040] A further technical solution is provided in step 2. and Selected manually, and both must meet certain conditions. .

[0041] A further technical solution, the specific steps of step 3 are as follows:

[0042] First, construct the multi-component joint matrix:

[0043] (11)

[0044] in, For a multi-component joint Hankel matrix, Here are the Hankel matrices corresponding to each of the three components;

[0045] Then, assuming that the multi-component joint matrix still has the low-rank property, we perform joint singular value decomposition on it using formula (8), and retain the previous values ​​using formula (9). A singular value is used to achieve multi-component joint denoising.

[0046] A further technical solution involves introducing a rotation matrix to enhance the low-rank structure in step 4:

[0047] (12)

[0048] In the representation of the potential wave field In the process Joint low-rank filtering:

[0049] (13)

[0050] in, To complete the joint low-rank filtering result in the latent wavefield representation, For Fourier transform operators, For inverse Fourier transform operators, Indicates rank as The joint low-rank filtering operator, For the input data in the potential wavefield representation, the number of principal singular values ​​retained is: Through inverse mapping, the final output data is:

[0051] (14)

[0052] in, The denoising result after restoring the original three-component space;

[0053] To obtain the optimal rotation matrix, let the covariance matrix of the three-component signal be:

[0054] (15)

[0055] in, Let be the covariance matrix of the three-component latent representation. Let be the mathematical expectation operator; where the eigenvalues ​​satisfy:

[0056] (16)

[0057] in, Let them represent the covariance matrix respectively. The first, second, and third eigenvalues; when the wave field has a distinct polarization direction, their relationship satisfies the following equation:

[0058] (17)

[0059] Assuming that the multi-component wavefield has stable polarization directions, the polarization uniformity loss is defined accordingly. for:

[0060] (18)

[0061] The final loss function is:

[0062] (19)

[0063] in, Represents the total loss function. This is the orthogonal constraint loss, used to constrain the stability of the rotation matrix. It is the identity matrix. It is the Frobenius norm; The weighting coefficients representing the orthogonal constraint loss; The weighting coefficients for polarization consistency loss.

[0064] This invention provides a seismic data denoising method based on multi-component joint and polarization constraints. This method can better preserve the relative amplitude relationship and wavefield polarization characteristics between different components, thus maintaining relatively consistent vector characteristics after denoising. This method has certain advantages in maintaining signal continuity. Compared with quaternion methods, this method produces smaller residuals near the reflection phase axis, indicating that the present invention introduces less signal loss during denoising. This is mainly because this method optimizes the solution of the rotation matrix to better represent the effective signal in a low-rank subspace, thereby reducing the destruction of the effective signal while suppressing noise. This method can further reduce signal loss while preserving the multi-component vector characteristics, thus obtaining higher quality denoising results. Attached Figure Description

[0065] Figure 1 A flowchart illustrating a seismic data denoising method based on multi-component joint and polarization constraints provided in an embodiment of the present invention;

[0066] Figure 2 Here is an example of three-component synthetic seismic data (where a to c are the three components of the clean synthetic data, and d to f are the three-component seismic data after random noise has been added to the three components).

[0067] Figure 3 Comparison of denoising results for component 1 (where a to c are the denoising results of the single-component low-rank method, the quaternion method, and the present method, respectively, and d to f are the differences between the corresponding denoised data and the clean data).

[0068] Figure 4 The comparison of denoising results for component 2 is shown (where a to c are the denoising results of the single-component low-rank method, the quaternion method, and the present method, respectively, and d to f are the differences between the corresponding denoised data and the clean data).

[0069] Figure 5 The comparison of denoising results for component 3 is shown (where a to c are the denoising results of the single-component low-rank method, the quaternion method, and the present method, respectively, and d to f are the differences between the corresponding denoised data and the clean data).

[0070] Figure 6 Here is an example of actual natural earthquake data with three components (where a to c correspond to components 4, 5 and 6, respectively).

[0071] Figure 7 The denoising results for component 4 are compared (where a to c are the denoising results of the single-component low-rank method, the quaternion method, and the present method, respectively, and d to f are the corresponding residual data).

[0072] Figure 8 The denoising results for component 5 are compared (where a to c are the denoising results of the single-component low-rank method, the quaternion method, and the present method, respectively, and d to f are the corresponding residual data).

[0073] Figure 9 The denoising results for component 6 are compared (where a to c are the denoising results of the single-component low-rank method, the quaternion method, and the method described in this paper, respectively, and d to f are the corresponding residual data). Detailed Implementation

[0074] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0075] The specific implementation of the present invention will be described in detail below with reference to specific embodiments.

[0076] like Figure 1 As shown, an embodiment of the present invention provides a seismic data denoising method based on multi-component joint and polarization constraints, comprising the following steps:

[0077] Step 1: Problem modeling, establishing a mathematical expression for three-component seismic data, and introducing a rotation matrix to map the original data to the potential wavefield representation;

[0078] Let the three-component seismic data be:

[0079] (1)

[0080] in, For time sampling points, For spatial location, This represents the observation vector of the original three-component data at its temporal and spatial location. These are observations from three components of seismic data in three directions. Within a local spatial window, the reflected wave can be represented as a superposition of a finite number of plane waves:

[0081] (2)

[0082] in, This represents the scalar wave field within a local window. This represents the number of plane waves within the local window. For the first The time correlation coefficient of a plane wave For the first One spatial basis function, For the first The spatial wavenumber of a plane wave The unit is the imaginary unit. Therefore, at a fixed frequency, the spatial sequence of reflected waves has a low-rank structure. Furthermore, there is a certain coupling relationship between the multi-component data, which can be represented by a linear transformation:

[0083] (3)

[0084] in, For rotation matrix, express transpose, This represents the latent wavefield, in which the spatial structure of the wavefield is more concentrated, thus exhibiting stronger low-rank properties. The rotation matrix is ​​solved through optimization. This allows for a wavefield representation that is more suitable for low-rank modeling.

[0085] Step 2: Based on Hankel low-rank theory, construct the Hankel matrix for the single-component spatial sequence and perform singular value decomposition to achieve noise reduction;

[0086] First, perform a Fourier transform on the time direction.

[0087] (4)

[0088] in, For frequency variables, Indicates the spatiotemporal signal in frequency variable Below, spatial position The complex spectrum value at a fixed frequency variable. The following spatial sequence is obtained:

[0089] (5)

[0090] in, This represents the total number of seismic traces. For fixed frequency variables Below, the spatial location is The complex spectrum value. At a fixed frequency variable. The spatial sequence can be simplified as follows: ,in, for The simplified representation of ; thus constructing the Hankel matrix of the single-component spatial sequence:

[0091] (6)

[0092] in, The column number of the Hankel matrix. Let be the row number of the Hankel matrix. and Through manual selection, both must meet the following requirements. . This represents the Hankel matrix constructed from a single-component spatial sequence. If the signal is... Composed of plane waves, then:

[0093] (7)

[0094] in, This indicates finding the rank of a matrix. A local wavefield consisting of a finite number of plane waves has a low-rank Hankel matrix. We perform singular value decomposition on this matrix:

[0095] (8)

[0096] in, It is a left singular vector matrix. It is a singular value diagonal matrix. This is the conjugate transpose of the right singular vector matrix. This is the sign for the conjugate transpose. Therefore, the first part is retained. The reduced-rank matrix can be obtained by finding a few singular values:

[0097] (9)

[0098] in, Indicates to retain the previous text A reduced-rank matrix obtained from singular values Indicates to retain the previous text The diagonal matrix after the singular values.

[0099] Finally, the reduced-rank matrix is ​​restored to a spatial sequence by averaging along the anti-diagonal:

[0100] (10)

[0101] in, Represents the first element in the recovered one-dimensional sequence. One element, express The first in One element, Indicates the first The set of matrix elements corresponding to each anti-diagonal line. Let be the number of elements on the antidiagonal. Through the above process, the low-rank matrix can be mapped back to the original spatial sequence, achieving effective suppression of random noise.

[0102] Step 3: Construct a multi-component joint Hankel matrix and perform joint singular value decomposition to achieve multi-component joint low-rank filtering;

[0103] First, construct the multi-component joint matrix:

[0104] (11)

[0105] in, For a multi-component joint Hankel matrix, Let be the Hankel matrix corresponding to each of the three components. Since the same plane wave has a related spatial structure in different components, it can be assumed that the joint matrix of the multiple components still has the low-rank property. We can use formula (8) to perform joint singular value decomposition on it, and use formula (9) to retain the first rank. Multi-component joint denoising can be achieved using a single singular value. This method leverages the structural consistency between different components to enhance the effective signal and suppress noise.

[0106] Step 4: Introduce a rotation matrix, perform joint low-rank filtering on the latent wave field representation, and construct polarization consistency loss and orthogonality constraint loss. Obtain the optimal rotation matrix by jointly optimizing the loss function, thereby obtaining the final denoising result.

[0107] To enhance the low-rank structure, a rotation matrix is ​​introduced:

[0108] (12)

[0109] In the representation of the potential wave field In the process Joint low-rank filtering:

[0110] (13)

[0111] in, To complete the joint low-rank filtering result in the latent wavefield representation, For Fourier transform operators, For inverse Fourier transform operators, Indicates rank as The joint low-rank filtering operator, For the input data in the potential wavefield representation, the number of principal singular values ​​retained is: Through inverse mapping, the final output data is:

[0112] (14)

[0113] in, The result is the denoising effect after restoring the original three-component space.

[0114] To obtain the optimal rotation matrix, let the covariance matrix of the three-component signal be:

[0115] (15)

[0116] in, Let be the covariance matrix of the three-component latent representation. Let be the mathematical expectation operator. Where the eigenvalues ​​satisfy:

[0117] (16)

[0118] in, Let them represent the covariance matrix respectively. The first, second, and third eigenvalues. When the wave field has a distinct polarization direction, their relationship satisfies the following equation:

[0119] (17)

[0120] Assuming that the multi-component wavefield has stable polarization directions, the polarization uniformity loss is defined accordingly. for:

[0121] (18)

[0122] The final loss function is:

[0123] (19)

[0124] in, Represents the total loss function. This is the orthogonal constraint loss, used to constrain the stability of the rotation matrix. It is the identity matrix. It is the Frobenius norm; The weighting coefficients representing the orthogonal constraint loss; These are the weighting coefficients for the polarization consistency loss. By jointly optimizing the above loss function, self-supervised denoising of multi-component data can be achieved.

[0125] To verify the effectiveness of this method, preliminary experiments were conducted on synthetic and actual seismic data. The experimental comparison methods included: single-component low-rank denoising, quaternion-based low-rank denoising, and multi-component polarization low-rank methods.

[0126] 1. Synthetic Data Testing: The experimental data consisted of three-component seismic gathers, with random noise artificially added to simulate a low signal-to-noise ratio acquisition environment. The three-component synthetic seismic data is shown below. Figure 2 As shown, where Figure 2 a to Figure 2 c represents the three components of the clean composite data: component 1, component 2, and component 3. Figure 2 d to Figure 2 f represents the three-component seismic data after adding random noise to the three components.

[0127] The denoising result of component 1 is as follows: Figure 3 As shown, where Figure 3 a to Figure 3 c represents the denoising results of the single-component low-rank method, the quaternion method, and the proposed method, respectively. Figure 3 d to Figure 3 f represents the difference between the denoised data and the clean data.

[0128] The denoising result of component 2 is as follows: Figure 4 As shown, where Figure 4 a to Figure 4 c represents the denoising results of the single-component low-rank method, the quaternion method, and the proposed method, respectively. Figure 4 d to Figure 4 f represents the difference between the denoised data and the clean data.

[0129] The denoising result of component 3 is as follows: Figure 5 As shown, where Figure 5 a to Figure 5 c represents the denoising results of the single-component low-rank method, the quaternion method, and the proposed method, respectively. Figure 5 d to Figure 5 f represents the difference between the denoised data and the clean data.

[0130] The signal-to-noise ratios of the denoising results from different methods are compared in Table 1 below.

[0131] Table 1. Comparison of signal-to-noise ratios of denoising results from different methods

[0132]

[0133] 2. Actual data testing

[0134] Example of three-component actual natural earthquake data: Figure 6 As shown, where Figure 6 a to Figure 6 c corresponds to components 4, 5, and 6, respectively.

[0135] The denoising result of component 4 is as follows: Figure 7 As shown, where Figure 7 a to Figure 7 c represents the denoising results of the single-component low-rank method, the quaternion method, and the proposed method, respectively. Figure 7 d to Figure 7 f represents the corresponding residual data.

[0136] The denoising result of component 5 is as follows: Figure 8 As shown, where Figure 8 a to Figure 8 c represents the denoising results of the single-component low-rank method, the quaternion method, and the proposed method, respectively. Figure 8 d to Figure 8 f represents the corresponding residual data.

[0137] The denoising result of component 6 is as follows: Figure 9 As shown, where Figure 9 a to Figure 9 c represents the denoising results of the single-component low-rank method, the quaternion method, and the proposed method, respectively. Figure 9 d to Figure 9 f represents the corresponding residual data.

[0138] Experimental results show that all three methods can effectively suppress random noise. However, the single-component low-rank method, due to its lack of utilization of the relevant information between multi-component wavefields, is prone to disrupting the vector relationships between components during denoising. In contrast, the quaternion method and our proposed method can better preserve the relative amplitude relationships and wavefield polarization characteristics between different components, thus maintaining relatively consistent vector characteristics after denoising.

[0139] Further comparison reveals that this method has certain advantages in maintaining signal continuity. Compared with the quaternion method, this method produces smaller residuals near the reflection phase axis, indicating less signal loss introduced during denoising. This is mainly because this method optimizes the solution of the rotation matrix to better represent the effective signal in the low-rank subspace, thereby reducing the destruction of the effective signal while suppressing noise.

[0140] Based on the combined image comparison and signal-to-noise ratio results, it can be seen that this method can further reduce signal loss while preserving multi-component vector features, thereby obtaining higher quality denoising results.

[0141] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A seismic data denoising method based on multi-component joint and polarization constraints, characterized in that, Includes the following steps: Step 1: Acquire three-component seismic data, establish the mathematical expression of the three-component seismic data, and introduce a rotation matrix to map the original data to the latent wavefield representation; Step 2: Based on Hankel low-rank theory, construct the Hankel matrix for the single-component spatial sequence and perform singular value decomposition to achieve single-component denoising; Step 3: Construct a multi-component joint Hankel matrix, and perform joint singular value decomposition on the multi-component joint Hankel matrix to achieve multi-component joint low-rank filtering; Step 4: Introduce a rotation matrix, perform joint low-rank filtering on the latent wavefield representation, and construct polarization consistency loss and orthogonality constraint loss. By jointly optimizing the total loss function containing the polarization consistency loss and the orthogonality constraint loss, obtain the optimal rotation matrix. Use the optimal rotation matrix to perform inverse mapping to obtain the final denoising result. In step 4, a rotation matrix is ​​introduced to enhance the low-rank structure: （12） in, Representing the potential wave field, This represents the observation vector of the original three-component data at its temporal and spatial location. It is a rotation matrix; In the representation of the potential wave field In the process Joint low-rank filtering: （13） in, To complete the joint low-rank filtering result in the latent wavefield representation, For Fourier transform operators, For inverse Fourier transform operators, Indicates rank as The joint low-rank filtering operator, For the input data in the potential wavefield representation, the number of principal singular values ​​retained is: Through inverse mapping, the final output data is: （14） in, To restore the denoising result to the original three-component space, express Transpose of; To obtain the optimal rotation matrix, let the covariance matrix of the three-component signal be: （15） in, Let be the covariance matrix of the three-component latent representation. Let be the mathematical expectation operator; where the eigenvalues ​​satisfy: （16） in, Let them represent the covariance matrix respectively. The first, second, and third eigenvalues; when the wave field has a distinct polarization direction, their relationship satisfies the following equation: （17） Assuming that the multi-component wavefield has stable polarization directions, the polarization uniformity loss is defined accordingly. for: （18） The final loss function is: （19） in, Represents the total loss function. For orthogonal constraint loss, It is the identity matrix. It is the Frobenius norm; The weighting coefficients representing the orthogonal constraint loss; These are the weighting coefficients for polarization consistency loss.

2. The seismic data denoising method based on multi-component joint and polarization constraints according to claim 1, characterized in that, The specific steps of step 1 are as follows: Let the three-component seismic data be: （1） in, For time sampling points, For spatial location, This represents the observation vector of the original three-component data at its temporal and spatial location. These are observations from three components of seismic data in three directions; within a local spatial window, the reflected wave is represented as a superposition of a finite number of plane waves: （2） in, This represents the scalar wave field within a local window. This represents the number of plane waves within the local window. For the first The time correlation coefficient of a plane wave For the first One spatial basis function, For the first The spatial wavenumber of a plane wave The imaginary unit is used; based on the coupling relationship between multi-component data, a rotation representation is performed through linear transformation: （3） in, For rotation matrix, express transpose, This represents the potential wave field.

3. The seismic data denoising method based on multi-component joint and polarization constraints according to claim 2, characterized in that, The specific steps of step 2 are as follows: First, perform a Fourier transform on the time direction. （4） in, For frequency variables, Indicates the spatiotemporal signal in frequency variable Below, spatial position Complex spectral values ​​at a given frequency; at a fixed frequency variable The following spatial sequence is obtained: （5） in, This represents the total number of seismic traces. For fixed frequency variables Below, the spatial location is The complex spectrum value; at a fixed frequency variable Below, the spatial sequence is simplified as ,in, for The simplified representation of ; thus constructing the Hankel matrix of the single-component spatial sequence: （6） in, The column number of the Hankel matrix. Let be the row number of the Hankel matrix. This represents the Hankel matrix constructed from a single-component spatial sequence; if the signal is derived from... Composed of plane waves, then: （7） in, This indicates finding the rank of a matrix; for a local wave field consisting of a finite number of plane waves, perform singular value decomposition on its Hankel matrix: （8） in, It is a left singular vector matrix. It is a singular value diagonal matrix. This is the conjugate transpose of the right singular vector matrix. The sign for conjugate transpose; retain the first one. The singular values ​​yield the reduced-rank matrix: （9） in, Indicates to retain the previous text A reduced-rank matrix obtained from singular values Indicates to retain the previous text The diagonal matrix after each singular value; Finally, the reduced-rank matrix is ​​restored to a spatial sequence by averaging along the anti-diagonal: （10） in, Represents the first element in the recovered one-dimensional sequence. One element, express The first in One element, Indicates the first The set of matrix elements corresponding to each anti-diagonal line. The number of elements on the anti-diagonal.

4. The seismic data denoising method based on multi-component joint and polarization constraints according to claim 3, characterized in that, In step 2, and By human selection, and both must meet the following conditions. .

5. The seismic data denoising method based on multi-component joint and polarization constraints according to claim 3, characterized in that, The specific steps of step 3 are as follows: First, construct the multi-component joint matrix: （11） in, For a multi-component joint Hankel matrix, Here are the Hankel matrices corresponding to each of the three components; Then, assuming that the multi-component joint matrix still has the low-rank property, we perform joint singular value decomposition on it using formula (8), and retain the previous values ​​using formula (9). A single singular value enables multi-component joint denoising.

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