Robust mixed norm constraint high-dimensional seismic data reconstruction method and device

Abstract

The invention relates to the technical field of oil and gas exploration and development, and particularly discloses a robust mixed norm constraint high-dimensional seismic data reconstruction method and device, and the method comprises the steps: carrying out the Fourier transformation of five-dimensional data along a time axis, and obtaining the Fourier transformation data; unfolding 4D data of each frequency slice of the Fourier transform data along a mode-(m, n) to obtain a near-square matrix; a non-convex Frobenius / Nuclear mixed norm and L1 norm combined regularization constraint is applied to the near-square matrix, and a target functional of observation seismic data and reconstruction data is constructed; and alternately solving the target functional by adopting an ADMM algorithm, and carrying out inverse Fourier transform on a solving result along a frequency axis. According to the method, a high-dimensional tensor completion method under the robust Frobenius / Nuclear mixed norm constraint is introduced, accurate representation of nonlinear high-dimensional seismic data is achieved, the seismic data reconstruction and denoising precision is improved, and therefore regularized reconstruction and abnormal noise suppression of the seismic data with the low signal-to-noise ratio are achieved.

Description

Technical Field

[0001] This invention relates to the field of oil and gas exploration and development technology, specifically to a robust hybrid norm-constrained high-dimensional seismic data reconstruction method and apparatus. Background Technology

[0002] With the continuous improvement of oil and gas exploration in my country, seismic exploration targeting concealed and lithologic oil and gas reservoirs, as well as deep and ultra-deep complex oil and gas reservoirs, has gradually become routine, placing higher demands on the quality of seismic data. However, the acquisition of seismic data in the field is limited by various factors, such as complex surface conditions, rugged terrain, and industrial and agricultural facilities, leading to irregular spatial distribution or missing data. Furthermore, the surrounding environment also affects the acquisition process, introducing various types of interference noise. This noise reduces the signal-to-noise ratio of the acquired data and can even cause excessively large noise amplitudes to overwhelm the effective reflection signals. This negatively impacts subsequent processing steps such as seismic wavelet estimation, surface-related multiple suppression, first arrival picking, migration, and imaging. Therefore, in the seismic data preprocessing stage, noise suppression and interpolation reconstruction of missing data are necessary to improve the quality of seismic data. This is extremely important for the current trend towards seismic wave inversion imaging.

[0003] In recent years, numerous publications have proposed various methods to improve the signal-to-noise ratio and reconstruction accuracy of seismic data, mainly categorized into filtering, sparse transform, low-rank matrix / tensor decomposition, and deep learning methods. Filtering-based methods primarily utilize the predictability of signals in the time-space, frequency-space, and frequency-wavenumber domains to attenuate random noise or perform interpolation reconstruction of seismic data. Sparse transform methods, in seismic data processing, leverage the sparsity of data mapped to the transform domain via basis functions to suppress noise or perform reconstruction. Besides considering the sparsity of seismic data, it is assumed that noise-free, complete data possesses low-rank characteristics. The random missing data from seismic traces and the presence of random noise significantly increase the rank of the data matrix or tensor, thus allowing the use of low-rank matrix or tensor decomposition methods for reconstructing 2D or higher-dimensional seismic data and suppressing random noise. Ulrych et al. (1999) proposed a domain-specific feature image method, applying singular value decomposition (SVD) to 2D data after dynamic correction (NMO). Kreimer and Sacchi (2012) used the High-Order Singular Value Decomposition (HOSVD) method to reconstruct pre-stack 5D seismic data after dynamic correction (NMO). Gao et al. (2017) proposed a fast and efficient low-rank tensor completion algorithm—Parallel Square Matrix Decomposition (PSMF)—which expands the fourth-order tensor in the time-space domain along the mode-, making the expanded matrix closer to a square matrix, thus enabling more accurate recovery of the data tensor. Zhang et al. (2023) proposed a non-convex but smooth substitution function (logDet function) and applied it to solve the 5D seismic data reconstruction problem.

[0004] However, when the underground structure is complex or there is lateral speed variation, the in-phase axis of the acquired CMP data may not satisfy the linearity assumption after dynamic correction, but it still satisfies the low-rank property.

[0005] Based on this technical background, this invention studies a robust method and apparatus for reconstructing high-dimensional seismic data with hybrid norm constraints. Summary of the Invention

[0006] To address the shortcomings of existing technologies, this invention provides a robust method and apparatus for reconstructing high-dimensional seismic data under hybrid norm constraints. This method introduces a high-dimensional tensor completion method under robust Frobenius / nuclear hybrid norm constraints, which not only achieves accurate representation of nonlinear high-dimensional seismic data but also improves the accuracy of seismic data reconstruction and denoising, thereby enabling regularized reconstruction and noise suppression of low signal-to-noise ratio seismic data.

[0007] To achieve the above objectives, a first aspect of the present invention provides a robust method for reconstructing high-dimensional seismic data with hybrid norm constraints, comprising:

[0008] Fourier transform data is obtained by performing a Fourier transform on the five-dimensional data along the time axis;

[0009] The 4D data of each frequency slice of the Fourier transform data is expanded along the pattern-(m,n) to obtain a near-square matrix.

[0010] A non-convex Frobenius / nuclear mixed norm and L1 norm joint regularization constraint is applied to the near-square matrix, and an objective functional is constructed from the observed seismic data and the reconstructed data.

[0011] The objective functional is solved alternately using the ADMM algorithm, and the solution is then subjected to an inverse Fourier transform along the frequency axis.

[0012] A second aspect of the present invention provides a robust hybrid norm-constrained high-dimensional seismic data reconstruction apparatus, comprising:

[0013] The Fourier transform module is used to perform a Fourier transform on five-dimensional data along the time axis to obtain Fourier transform data;

[0014] The pattern expansion module is used to expand the 4D data of each frequency slice of the Fourier transform data along the pattern-(m,n) to obtain a near square matrix.

[0015] The regularization constraint module is used to apply non-convex Frobenius / nuclear mixed norm and L1 norm joint regularization constraints to the near-square matrix, and to construct the target functional of the observed seismic data and the reconstructed data.

[0016] The solution module is used to alternately solve the objective functional using the ADMM algorithm and perform an inverse Fourier transform on the solution results along the frequency axis.

[0017] A third aspect of the present invention provides an electronic device, the electronic device comprising:

[0018] Memory, which stores executable instructions;

[0019] A processor that executes the executable instructions in the memory to implement the robust hybrid norm-constrained high-dimensional seismic data reconstruction method described in the first aspect.

[0020] A fourth aspect of the present invention provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the robust hybrid norm-constrained high-dimensional seismic data reconstruction method described in the first aspect.

[0021] The beneficial effects of this invention include:

[0022] (1) The robust hybrid norm-constrained high-dimensional seismic data reconstruction method proposed in this invention introduces a high-dimensional tensor completion method under robust Frobenius / nuclear hybrid norm constraints. This not only realizes the accurate representation of nonlinear high-dimensional seismic data and improves the accuracy of seismic data reconstruction and denoising, but also realizes the regularized reconstruction and noise suppression of low signal-to-noise ratio seismic data (especially suitable for anomalous noise suppression).

[0023] (2) The robust hybrid norm-constrained high-dimensional seismic data reconstruction method proposed in this invention is a high-dimensional seismic data reconstruction and denoising method based on the joint constraints of Frobenius / nuclear hybrid norm and L1 norm, with low-rank tensor completion as the core. By applying a non-convex Frobenius / nuclear hybrid norm to the mode-(m,n) expansion matrix of the frequency spatial domain 4D data tensor, the established model can more accurately approximate the rank function of the matrix. This model is then applied to the noise suppression and regularization reconstruction process of high-dimensional seismic data to improve the quality of subsequent seismic wave migration images and inversion imaging, thereby improving the accuracy of reservoir description and drilling success rate.

[0024] (3) The robust hybrid norm-constrained high-dimensional seismic data reconstruction method proposed in this invention breaks the constraints of linear or quasi-linear assumptions of seismic data in complex exploration areas by introducing the Frobenius / nuclear hybrid norm and L1 norm-constrained high-dimensional seismic signal accurate characterization method, thereby improving the accuracy of seismic data processing (denoising, reconstruction, etc.), laying a solid foundation for high-precision seismic imaging, and accelerating the complex exploration and development process of Sinopec.

[0025] Other features and advantages of the present invention will be described in detail in the following detailed description section. Attached Figure Description

[0026] The above and other objects, features and advantages of the present invention will become more apparent from the more detailed description of exemplary embodiments of the invention in conjunction with the accompanying drawings.

[0027] Figure 1 This is a flowchart illustrating the robust hybrid norm-constrained high-dimensional seismic data reconstruction method proposed in this invention.

[0028] Figure 2 This is a flowchart illustrating a specific implementation of the robust hybrid norm-constrained high-dimensional seismic data reconstruction method proposed in this invention.

[0029] Figure 3 This diagram illustrates the bilinear factorization of the three modes-(m,n) expansion matrix and the expansion matrix of the fourth-order tensor X in a specific implementation of the robust hybrid norm-constrained high-dimensional seismic data reconstruction method proposed in this invention.

[0030] Figure 4 This is a schematic diagram of the data reconstruction result in the CMP domain in a specific implementation of the robust hybrid norm-constrained high-dimensional seismic data reconstruction method proposed in this invention. Detailed Implementation

[0031] Preferred embodiments of the invention will now be described in more detail. While preferred embodiments of the invention are described below, it should be understood that the invention can be implemented in various forms and should not be limited to the embodiments set forth herein.

[0032] This invention provides a robust method for reconstructing high-dimensional seismic data with hybrid norm constraints, such as... Figure 1 As shown, it includes:

[0033] Fourier transform data is obtained by performing a Fourier transform on the five-dimensional data along the time axis;

[0034] The 4D data of each frequency slice of the Fourier transform data is expanded along the pattern-(m,n) to obtain a near-square matrix.

[0035] Non-convex Frobenius / nuclear mixed norm joint regularization constraints are applied to the near-square matrix, and objective functionals of observed seismic data and reconstructed data are constructed.

[0036] The ADMM algorithm is used to solve the objective functional alternately, and the solution is then subjected to an inverse Fourier transform along the frequency axis.

[0037] This invention introduces a high-dimensional tensor completion method under robust Frobenius / nuclear hybrid norm constraints, which not only achieves accurate representation of nonlinear high-dimensional seismic data and improves the accuracy of seismic data reconstruction and denoising, but also realizes regularized reconstruction and noise suppression of low signal-to-noise ratio seismic data (especially suitable for anomaly noise suppression).

[0038] According to the present invention, the five-dimensional data is a 5D CMP gather;

[0039] 5DCMP gathers are represented as d(t,m) x ,m y ,h x ,h y ), where m x ,m y The x and y coordinates of the common center point are respectively, h x ,h y t represents the offset distance on the x and y axes, and t represents time.

[0040] The expression for Fourier transform data is:

[0041] X = D ω (ω,mx ,m y ,h x ,h y );

[0042] X is the abbreviation for Fourier transform data.

[0043] According to the present invention, the expression for the near-square matrix is:

[0044]

[0045] Among them, X [n] For a near-square matrix, U n and V n It is a factor matrix;

[0046] X is defined when n=1. [1] The tensor pattern is an (1,2) matrix expansion;

[0047] When n = 2, X is defined. [2] The tensor pattern is an expansion of the (1,3) matrix;

[0048] X is defined when n=3 [3] For tensor pattern - (2,3) matrix expansion.

[0049] According to the present invention, the expression for the non-convex Frobenius / nuclear mixture norm is:

[0050]

[0051] Where X is a matrix of rank R.

[0052] The formula used for the objective functional is:

[0053]

[0054] in, Let ||X|| be the Frobenius norm of the matrix. * =∑ i σ i (X) is the nuclear norm of matrix X, and the variable... ε and ε represent noiseless complete data, anomalous noise, and random noise, respectively; the symbol ||·||1 represents the tensor element's Norm, For sampling operators;

[0055] The sampling operator in the formula used by the objective functional It applies to variables On, that is, when When the value is 0, the formula has already adjusted the variables. Since constraints have been imposed on ε, there is no need to further constrain the variables. Apply additional constraints.

[0056] In this invention, a high-dimensional seismic data reconstruction and denoising method based on Frobenius / nuclear mixing and L1 norm joint constraints, with low-rank tensor completion as its core, applies a non-convex Frobenius / nuclear mixing to the mode-(m,n) expansion matrix of the frequency spatial domain 4D data tensor. This allows the established model to more accurately approximate the rank function of the matrix. This model is then applied to the noise suppression and regularization reconstruction process of high-dimensional seismic data to improve the quality of subsequent seismic wave migration images and inversion imaging, thereby improving the accuracy of reservoir description and drilling success rate.

[0057] According to the present invention, the objective functional is solved alternately using the ADMM algorithm, and the inverse Fourier transform of the solution results along the frequency axis is performed as follows:

[0058] Introducing auxiliary variables and Lagrange multiplier variable D n and F n and Where 1≤n≤3, the objective functional constraint problem is transformed into an unconstrained optimization problem, resulting in the augmented Lagrangian function;

[0059] The ADMM algorithm is used to alternately solve for variable U. n V n ε, D n F n and on variables Perform an inverse Fourier transform along the frequency axis.

[0060] Preferably, the expression for the augmented Lagrange function is:

[0061]

[0062] variable U n The solution equation is:

[0063]

[0064] variable U n The least squares solution is:

[0065]

[0066] variable V n The solution equation is:

[0067]

[0068] variable V n The least squares solution is:

[0069]

[0070] variable The solution equation is:

[0071]

[0072] variable The solution is:

[0073]

[0074] in, The soft threshold operator is defined as follows:

[0075]

[0076] The equation for solving variable S is:

[0077]

[0078] The least squares solution for variable S is:

[0079]

[0080] The equation for solving the variable ε is:

[0081]

[0082] The least squares solution for variable ε is:

[0083]

[0084] variable The update formula is:

[0085]

[0086] According to the present invention, variables The solution equation is:

[0087]

[0088] tensor The least squares solution is:

[0089]

[0090] in, for The symbol mat2ten(X,j) n) represents moving matrix X along pattern -j n Folding, with variables pattern-j n The matrix expansion method is exactly the opposite;

[0091] Variable D n and F n And the update formula for variable μ is:

[0092]

[0093] μ = max(ρμ, μ) max );

[0094] Wherein, the parameter ρ takes values ​​in the range of (1.0, 1.8], and the initial value of μ is set to 0.005-0.02. max Set to 10 7 -10 9 .

[0095] This invention introduces a high-dimensional seismic signal accurate characterization method under the combined constraints of Frobenius / nuclear hybrid and L1 norm, breaking the constraints of linear or quasi-linear assumptions on seismic data in complex exploration areas, improving the accuracy of seismic data processing (denoising, reconstruction, etc.), laying a solid foundation for high-precision seismic imaging, and accelerating the complex exploration and development process of Sinopec.

[0096] The present invention will be described in more detail below through embodiments.

[0097] Example 1:

[0098] like Figure 2 As shown, this embodiment proposes a robust method for reconstructing high-dimensional seismic data with hybrid norm constraints.

[0099] First, a Fourier transform is performed on the five-dimensional data along the time axis. Second, the 4D data of each frequency slice is expanded along mode-(m,n) rather than mode-n, so that the shape of the expanded matrix is ​​as close to a square matrix as possible. Then, a non-convex Frobenius / nuclear mixed norm and L1 norm joint regularization constraint is applied to the mode-(m,n) expanded matrix, so that the estimated matrix is ​​closer to the original true solution, and the target functional of the observed seismic data and the reconstructed data is constructed. Finally, the ADMM algorithm is used to solve the target functional alternately, and the result is subjected to an inverse Fourier transform along the frequency axis, so as to realize the regularized reconstruction and noise suppression of high-dimensional seismic data.

[0100] The specific implementation process of this method is as follows:

[0101] 1. Perform a Fourier transform on the five-dimensional data along the time axis:

[0102] Actual 3D seismic acquisition yields a 5D data volume. For shot gathers, the five dimensions correspond to the x and y coordinates of the shot point, respectively. x ,s y The x and y coordinates of the receiver point g x ,g y And time t; for a common center point (CMP) gather, the five dimensions correspond to the x and y coordinates m of the common center point, respectively. x ,m y Offset from the horizontal and vertical coordinates h x ,h y And time t. Currently, most earthquake data noise suppression or reconstruction methods process 2D or 3D data. These methods are difficult to effectively utilize the spatial structure information of high-dimensional data, and the computational load increases rapidly with the increase of dimension, making it impossible to efficiently process large-scale earthquake data. However, directly modeling 4D data in the frequency spatial domain can make full use of the spatial structure information of high-dimensional data, achieve the best prediction of signals, and obtain better denoising or reconstruction results.

[0103] Therefore, the 5DCMP gather d(t,m) x ,m y ,h x ,h y Performing a Fourier transform (FFT) along the time dimension yields D(f,m). x ,m y ,h x ,h y If the frequency ω is in the 4D frequency space domain, then the frequency ω can be written as D. ω (ω,m x ,m y ,h x ,h y ), abbreviated as X;

[0104] 2. Expand the 4D data of each frequency slice along pattern -(m,n):

[0105] For noise-free complete data X containing hyperbolic in-phase axes or other types of nonlinear in-phase axes (data in the frequency spatial domain is inseparable in some dimensions), its expansion along mode -(m,n) yields:

[0106]

[0107] Define X [1] For the tensor pattern - (1,2) matrix expansion, X [2] For the tensor pattern - (1,3) matrix expansion, X [3]The tensor pattern is an expansion of the (2,3) matrix; the factor matrices are also distinguished by their subscript notation; in addition, for the convenience of subsequent descriptions, vectors j1 = (1,2), j2 = (1,3) and j3 = (2,3) are defined. Figure 3 'a' is a schematic diagram of the three modes-(m,n) expansion matrices of the 4th-order tensor X. Figure 3 b is a schematic diagram of bilinear factorization of the pattern-(1,2) expansion matrix of the 4th order tensor. The approximate square matrix after expansion along the pattern-(m,n) is also low-rank; b is a schematic diagram of bilinear factorization of the pattern-(1,2) expansion matrix of the tensor X; since it is only possible to expand the tensor into an approximate square matrix when the data dimension N≥4, it is possible to model the 5D seismic data (data reconstruction is performed on the 4D frequency slice data).

[0108] 3. Apply non-convex Frobenius / nuclear mixed norm and L1 norm joint regularization constraints to the pattern-(m,n) expansion matrix:

[0109] For any matrix of rank R Its Frobenius / nuclear mixture norm is defined as follows:

[0110]

[0111] The problem of high-dimensional seismic data reconstruction and denoising under the joint constraints of the Frobenius / nuclear mixed norm and L1 norm can be written as:

[0112]

[0113] in For the factor matrix, Let ||X|| denote the Frobenius norm of the matrix. * =∑ i σ i (X) is the nuclear norm of matrix X, where the tensor ε and ε represent noise-free complete data, anomalous noise, and random noise, respectively;

[0114] 4. Solve formula (3) using the ADMM algorithm, and perform an inverse Fourier transform on the result along the frequency axis:

[0115] To solve the constraint optimization problem of the above formula (3), auxiliary variables are introduced. and Lagrange multiplier variable D n and F n and 1≤n≤3, transforming the above equation-constrained optimization problem into an unconstrained optimization problem, we obtain the augmented Lagrangian function corresponding to problem (3):

[0116]

[0117] For the unconstrained optimization problem described above, the ADMM algorithm can be used to alternately solve for the variable U. n V n , ε, D n F n The following section will detail the solution process for each variable in step 4:

[0118] 1) Regarding the factor matrix U in step 4 n The subproblems are:

[0119]

[0120] Then matrix U n The least squares solution is:

[0121]

[0122] 2) In step 4, regarding the factor matrix V n The subproblems are:

[0123]

[0124] Then matrix V n The least squares solution is:

[0125]

[0126] 3) Step 4 regarding the matrix The subproblems are:

[0127]

[0128] Then matrix The solution is:

[0129]

[0130] The symbol T α (·) is the soft threshold operator, defined as follows:

[0131] T α (x) = sign(x)max{|x|-α,0};

[0132] 4) The subproblem of step 4 concerning the tensor matrix S is:

[0133]

[0134] The solution to the tensor matrix S is:

[0135]

[0136] 5) The solution equation for matrix ε in step 4 is:

[0137]

[0138] The least squares solution to matrix ε is:

[0139]

[0140] 6) Regarding step 4 The update formula is:

[0141]

[0142] 7) Regarding tensors The subproblems are:

[0143]

[0144] Then tensor The least squares solution is:

[0145]

[0146] Among them, tensor for The symbol mat2ten(X,j) n ) indicates that matrix X is moved along pattern -j n Folding occurs, which corresponds to the mode -j of tensor X. n The matrix expansion method is exactly the opposite;

[0147] 5) Matrix D n and F n The update formula for variable μ is as follows:

[0148]

[0149] μ = max(ρμ, μ) max );

[0150] The parameter ρ is selected from the interval (1.0, 1.8], and the initial value of μ is generally set to 0.01. max Set to 10 8 Table 1 details the update process of each variable in the high-dimensional seismic data reconstruction and denoising problem under the joint constraints of Frobenius / nuclear mixed norm and L1 norm.

[0151] In this embodiment, the problem of high-dimensional seismic data reconstruction and denoising under the joint constraints of Frobenius / nuclear mixed norm and L1 norm is solved.

[0152] Input: Observed data Y, sampling operator P, estimated mode -j n The rank of the matrix expansion is -(R1,R2,R3), and the hyperparameter is λ.

[0153] Output: Reconstructed data X;

[0154] 1. Randomly initialize matrix U n and V n Set parameters k = 1, μ = 0.01, ρ = 1.2, μ max =10 7 ;

[0155] 2. If the condition k≤100 is met, repeat step 2; otherwise, skip to step 3. The following is a detailed process for the kth iteration:

[0156] a) Update the matrix using the aforementioned formulas

[0157] b) Update the factor matrix U using the aforementioned formulas. n :

[0158] c) Update the factor matrix V using the aforementioned formulas. n ;

[0159] d) Update tensor X using the aforementioned formulas;

[0160] e) Update the tensor using the formula

[0161] f) Update the tensor ε using the formula;

[0162] g) Update matrix D using the aforementioned formulas. n F n and G;

[0163] μ = max(ρμ, μ) max );

[0164] k = k + 1;

[0165] 3. Output the reconstructed data X.

[0166] In this embodiment, high-dimensional seismic data containing anomalies, random noise, and missing data is used to illustrate the effectiveness of the invention. First, the method for generating anomaly noise on a single channel is described: for each seismic channel, m sampling points are randomly determined (m is a random number); then, the values ​​of the m sampling points are generated using a uniform distribution function within the interval (-1, 1); the reflection coefficient containing the anomalies is obtained by multiplying the values ​​of the m sampling points by 6 times the maximum amplitude value; convolving this with the Ricker wavelet yields the anomaly noise; numerical experiments are conducted on the first synthetic data (composed of four linear phase axes); when synthesizing data containing anomaly noise, -1dB Gaussian white noise is added to the data, and then anomaly noise is added to 20% of the seismic channels, and 30% of the seismic channels are randomly missing; when reconstructing the synthetic data containing anomaly noise, the number of iterations N of the method of the present invention is... iter Set it to 100, and set both the regularization parameters β and λ to 1; Figure 4 (Data reconstruction with anomalies, random noise, and missing data: (a) shows the original data slices in the CMP domain (CMPx=8, CMPy=8 and Hy=2,5,8,10); (b) shows the input data contaminated with -1dB random noise, with 20% of the seismic traces contaminated by anomalous noise and 30% of the seismic traces randomly missing; (c) shows the result after reconstruction using the FN-TC method, with an SNR of 23.4dB) This shows the data reconstruction results (waveform display) of the present invention in the CMP domain. The SNR after data reconstruction is 23.4dB, indicating that the method of the present invention can effectively achieve regularized reconstruction and noise suppression of high-dimensional seismic data.

[0167] Example 2:

[0168] This embodiment provides a robust method for reconstructing high-dimensional seismic data with hybrid norm constraints, such as... Figure 1 As shown, it includes:

[0169] Fourier transform data is obtained by performing a Fourier transform on the five-dimensional data along the time axis;

[0170] The 4D data of each frequency slice of the Fourier transform data is expanded along the pattern-(m,n) to obtain a near-square matrix.

[0171] A non-convex Frobenius / nuclear mixed norm and L1 norm joint regularization constraint is applied to the near square matrix, and an objective functional is constructed from the observed seismic data and the reconstructed data.

[0172] The ADMM algorithm is used to solve the objective functional alternately, and the solution is then subjected to an inverse Fourier transform along the frequency axis.

[0173] In this embodiment, the five-dimensional data is a 5DCMP gather;

[0174] 5D CMP gathers are represented as d(t,m) x ,m y ,h x ,h y ), where m x ,m y The x and y coordinates of the common center point are respectively, h x ,h y t represents the offset distance on the x and y axes, and t represents time.

[0175] The expression for Fourier transform data is:

[0176] X = D ω (ω,m x ,m y ,h x ,h y );

[0177] X is the abbreviation for Fourier transform data;

[0178] In this embodiment, the expression for the non-convex Frobenius / nuclear mixture norm is:

[0179]

[0180] Where X is a matrix of rank R.

[0181] The formula used for the objective functional is:

[0182]

[0183] in, Let ||X|| be the Frobenius norm of the matrix. * =∑ i σ i (X) is the nuclear norm of matrix X, and the variable... ε and ε represent noiseless complete data, anomalous noise, and random noise, respectively; the symbol ||·||1 represents the tensor element's Norm, For sampling operators;

[0184] The sampling operator in the formula used by the objective functional It applies to variables On, that is, when When the value is 0, the formula has already adjusted the variables. Since constraints have been imposed on ε, there is no need to further constrain the variables. Impose additional constraints;

[0185] In this embodiment, the ADMM algorithm is used to alternately solve the objective functional, and the inverse Fourier transform of the solution along the frequency axis is performed, including:

[0186] Introducing auxiliary variables and Lagrange multiplier variable D n and F n and Where 1≤n≤3, the objective functional constraint problem is transformed into an unconstrained optimization problem, resulting in the augmented Lagrangian function;

[0187] The ADMM algorithm is used to alternately solve for variable U. n V n , ε, D n F n and on variables Perform an inverse Fourier transform along the frequency axis;

[0188] In this embodiment, the expression for the augmented Lagrange function is:

[0189]

[0190] variable U n The solution equation is:

[0191]

[0192] variable U n The least squares solution is:

[0193]

[0194] variable V n The solution equation is:

[0195]

[0196] variable V n The least squares solution is:

[0197]

[0198] variable The solution equation is:

[0199]

[0200] variable The solution is:

[0201]

[0202] in, The soft threshold operator is defined as follows:

[0203]

[0204] The equation for solving variable S is:

[0205]

[0206] The least squares solution for variable S is:

[0207]

[0208] The equation for solving the variable ε is:

[0209]

[0210] The least squares solution for variable ε is:

[0211]

[0212] variable The update formula is:

[0213]

[0214] In this embodiment, variables The solution equation is:

[0215]

[0216] tensor The least squares solution is:

[0217]

[0218] in, for The symbol mat2ten(X,j) n ) represents moving matrix X along pattern -j n Folding, with variables pattern-j n The matrix expansion method is exactly the opposite;

[0219] Variable D n and F n And the update formula for variable μ is:

[0220]

[0221] μ = max(ρμ, μ) max );

[0222] Wherein, the parameter ρ takes values ​​in the range of (1.0, 1.8], and the initial value of μ is set to 0.005-0.02.max Set to 10 7 -10 9 .

[0223] Example 3:

[0224] This embodiment provides a robust hybrid norm-constrained high-dimensional seismic data reconstruction apparatus, comprising:

[0225] The Fourier transform module is used to perform a Fourier transform on five-dimensional data along the time axis to obtain Fourier transform data;

[0226] The pattern expansion module is used to expand the 4D data of each frequency slice of Fourier transform data along the pattern-(m,n) to obtain a near-square matrix.

[0227] The regularization constraint module is used to apply non-convex Frobenius / nuclear mixed norm and L1 norm joint regularization constraints to the near-square matrix and to construct the target functional of the observed seismic data and the reconstructed data.

[0228] The solver module is used to alternately solve the objective functional using the ADMM algorithm and perform an inverse Fourier transform on the solution along the frequency axis.

[0229] In this embodiment, the five-dimensional data is a 5DCMP gather;

[0230] 5DCMP gathers are represented as d(t,m) x ,m y ,h x ,h y ), where m x ,m y The x and y coordinates of the common center point are respectively, h x ,h y t represents the offset distance on the x and y axes, and t represents time.

[0231] The expression for Fourier transform data is:

[0232] X = D ω (ω,m x ,m y ,h x ,h y );

[0233] X is the abbreviation for Fourier transform data;

[0234] In this embodiment, the expression for the near-square matrix is:

[0235]

[0236] Among them, X [n] For a near-square matrix, Un and V n It is a factor matrix;

[0237] X is defined when n=1. [1] The tensor pattern is an (1,2) matrix expansion;

[0238] When n = 2, X is defined. [2] The tensor pattern is an expansion of the (1,3) matrix;

[0239] X is defined when n=3 [3] For tensor pattern - (2,3) matrix expansion;

[0240] In this embodiment, the expression for the non-convex Frobenius / nuclear mixture norm is:

[0241]

[0242] Where X is a matrix of rank R.

[0243] The formula used for the objective functional is:

[0244]

[0245] in, Let ||X|| be the Frobenius norm of the matrix. * =∑ i σ i (X) is the nuclear norm of matrix X, and the variable... ε and ε represent noiseless complete data, anomalous noise, and random noise, respectively; the symbol ||·||1 represents the tensor element's Norm, For sampling operators;

[0246] The sampling operator in the formula used by the objective functional It applies to variables On, that is, when When the value is 0, the formula has already adjusted the variables. Since constraints have been imposed on ε, there is no need to further constrain the variables. Impose additional constraints;

[0247] In this embodiment, the ADMM algorithm is used to alternately solve the objective functional, and the inverse Fourier transform of the solution along the frequency axis is performed, including:

[0248] Introducing auxiliary variables and Lagrange multiplier variable D n and F n and Where 1≤n≤3, the objective functional constraint problem is transformed into an unconstrained optimization problem, resulting in the augmented Lagrangian function;

[0249] The ADMM algorithm is used to alternately solve for variable U. n V n , ε, D n F n and on variables Perform an inverse Fourier transform along the frequency axis;

[0250] In this embodiment, the expression for the augmented Lagrange function is:

[0251]

[0252] variable U n The solution equation is:

[0253]

[0254] variable U n The least squares solution is:

[0255]

[0256] variable V n The solution equation is:

[0257]

[0258] variable V n The least squares solution is:

[0259]

[0260] variable The solution equation is:

[0261]

[0262] variable The solution is:

[0263]

[0264] in, The soft threshold operator is defined as follows:

[0265]

[0266] The equation for solving variable S is:

[0267]

[0268] The least squares solution for variable S is:

[0269]

[0270] The equation for solving the variable ε is:

[0271]

[0272] The least squares solution for variable ε is:

[0273]

[0274] variable The update formula is:

[0275]

[0276] In this embodiment, variables The solution equation is:

[0277]

[0278] tensor The least squares solution is:

[0279]

[0280] in, for The symbol mat2ten(X,j) n ) represents moving matrix X along pattern -j n Folding, with variables pattern-j n The matrix expansion method is exactly the opposite;

[0281] Variable D n and F n And the update formula for variable μ is:

[0282]

[0283] μ = max(ρμ, μ) max );

[0284] Wherein, the parameter ρ takes values ​​in the range of (1.0, 1.8], and the initial value of μ is set to 0.005-0.02. max Set to 10 7 -10 9 .

[0285] Example 4:

[0286] This invention provides an electronic device including a memory and a processor, comprising:

[0287] Memory, which stores executable instructions;

[0288] The processor runs executable instructions in memory to implement robust hybrid norm-constrained high-dimensional seismic data reconstruction methods.

[0289] This memory is used to store non-transitory computer-readable instructions. Specifically, the memory may include one or more computer program products, which may include various forms of computer-readable storage media, such as volatile memory and / or non-volatile memory. The volatile memory may, for example, include random access memory (RAM) and / or cache memory. The non-volatile memory may, for example, include read-only memory (ROM), hard disk, flash memory, etc.

[0290] The processor may be a central processing unit (CPU) or other form of processing unit with data processing capabilities and / or instruction execution capabilities, and may control other components in the electronic device to perform desired functions. In one embodiment of the invention, the processor is used to execute computer-readable instructions stored in the memory.

[0291] Those skilled in the art should understand that, in order to solve the technical problem of how to achieve a good user experience, this embodiment may also include well-known structures such as communication buses and interfaces, and these well-known structures should also be included within the protection scope of this invention.

[0292] For a detailed description of this embodiment, please refer to the corresponding descriptions in the foregoing embodiments, which will not be repeated here.

[0293] Example 5:

[0294] This invention provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements a robust hybrid norm-constrained high-dimensional seismic data reconstruction method.

[0295] A computer-readable storage medium according to embodiments of the present invention stores non-transitory computer-readable instructions. When these non-transitory computer-readable instructions are executed by a processor, all or part of the steps of the methods described in the foregoing embodiments of the present invention are performed.

[0296] The aforementioned computer-readable storage media include, but are not limited to: optical storage media (e.g., CD-ROM and DVD), magneto-optical storage media (e.g., MO), magnetic storage media (e.g., magnetic tape or portable hard drive), media with built-in rewritable non-volatile memory (e.g., memory card), and media with built-in ROM (e.g., ROM cartridge).

[0297] The robust hybrid norm-constrained high-dimensional seismic data reconstruction method proposed in the embodiments of the present invention introduces a high-dimensional tensor completion method under robust Frobenius / nuclear hybrid norm constraints. This not only achieves accurate characterization of nonlinear high-dimensional seismic data and improves the accuracy of seismic data reconstruction and denoising, but also realizes regularized reconstruction and noise suppression of low signal-to-noise ratio seismic data.

[0298] The various embodiments of the present invention have been described above. These descriptions are exemplary and not exhaustive, nor are they limited to the disclosed embodiments. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and spirit of the described embodiments.

Claims

1. A robust method for reconstructing high-dimensional seismic data with hybrid norm constraints, characterized in that, include: Fourier transform data is obtained by performing a Fourier transform on the five-dimensional data along the time axis; The 4D data of each frequency slice of the Fourier transform data is expanded along the pattern-(m,n) to obtain a near-square matrix. A non-convex Frobenius / nuclear mixed norm and L1 norm joint regularization constraint is applied to the near-square matrix, and an objective functional is constructed from the observed seismic data and the reconstructed data. The objective functional is solved alternately using the ADMM algorithm, and the solution is then subjected to an inverse Fourier transform along the frequency axis.

2. The method according to claim 1, characterized in that, The five-dimensional data is a 5D CMP gather; The 5DCMP gather is represented as d(t,m) x ,m y ,h x ,h y ), where m x ,m y The x and y coordinates of the common center point are respectively, h x ,h y t represents the offset distance on the x and y axes, and t represents time. The expression for the Fourier transform data is: X=D ω (ω,m x ,m y ,h x ,h y ); X is the abbreviation for Fourier transform data.

3. The method according to claim 2, characterized in that, The expression for the near-square matrix is: Among them, X [n] For a near-square matrix, U n and V n It is a factor matrix; X is defined when n=1. [1] The tensor pattern is an (1,2) matrix expansion; When n = 2, X is defined. [2] The tensor pattern is an expansion of the (1,3) matrix; X is defined when n=3 [3] For tensor pattern - (2,3) matrix expansion.

4. The method according to claim 3, characterized in that, The expression for the non-convex Frobenius / nuclear mixture norm is: Where X is a matrix of rank R. The formula used for the objective functional is: in, Let ||X|| be the Frobenius norm of the matrix. * =∑ i σ i (X) is the nuclear norm of matrix X, and the variable x, ε and ε represent noiseless complete data, anomalous noise, and random noise, respectively; the symbol ||·||1 represents the tensor element's Norm, For sampling operators; The sampling operator in the formula used by the target functional It applies to variables On, that is, when When the value is 0, the formula already imposes constraints on variables x and ε, so there is no need to impose further constraints on the variables. Apply additional constraints.

5. The method according to claim 4, characterized in that, The objective functional is solved alternately using the ADMM algorithm, and the inverse Fourier transform of the solution along the frequency axis is performed, including: Introducing auxiliary variables and Lagrange multiplier variable D n and F n and Where 1≤n≤3, the objective functional constraint problem is transformed into an unconstrained optimization problem, and the augmented Lagrangian function is obtained; The ADMM algorithm is used to alternately solve for variable U. n V n , x, ε, D n F n And perform an inverse Fourier transform on the variable χ along the frequency axis.

6. The method according to claim 5, characterized in that, The expression for the augmented Lagrange function is: The variable U n The solution equation is: The variable U n The least squares solution is: The variable V n The solution equation is: The variable V n The least squares solution is: The variable The solution equation is: The variable The solution is: in, The soft threshold operator is defined as follows: The equation for solving the variable S is: The least squares solution for the variable S is: The equation for solving the variable ε is: The least squares solution for the variable ε is: The variable The update formula is:

7. The method according to claim 6, characterized in that, The equation for solving the variable x is: The least squares solution to the tensor x is: in, for The symbol mat2ten(X,j) n ) represents moving matrix X along pattern -j n Folding is performed, with the pattern -j of variable x. n The matrix expansion method is exactly the opposite; The variable D n and F n And the update formula for variable μ is: μ=max(ρμ,μ max ); Wherein, the parameter ρ takes values ​​in the range of (1.0, 1.8], and the initial value of μ is set to 0.005-0.

02. max Set to 10 7 -10 9 .

8. A robust hybrid norm-constrained high-dimensional seismic data reconstruction device, characterized in that, include: The Fourier transform module is used to perform a Fourier transform on five-dimensional data along the time axis to obtain Fourier transform data; The pattern expansion module is used to expand the 4D data of each frequency slice of the Fourier transform data along the pattern-(m,n) to obtain a near square matrix. The regularization constraint module is used to apply non-convex Frobenius / nuclear mixed norm and L1 norm joint regularization constraints to the near-square matrix, and to construct the target functional of the observed seismic data and the reconstructed data. The solution module is used to alternately solve the objective functional using the ADMM algorithm and perform an inverse Fourier transform on the solution results along the frequency axis.

9. An electronic device, characterized in that, The electronic device includes: Memory, which stores executable instructions; A processor that executes the executable instructions in the memory to implement the robust hybrid norm-constrained high-dimensional seismic data reconstruction method according to any one of claims 1-7.

10. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a computer program that, when executed by a processor, implements the robust hybrid norm-constrained high-dimensional seismic data reconstruction method according to any one of claims 1-7.

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