A structure-guided method for nonconvex low-rank recovery of seismic data
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-05-27
- Publication Date
- 2026-08-14
AI Technical Summary
然而，现有基于CTV的方法无法充分表征地震数据的斜率属性，而斜率属性是关键的结构特征
[0013]本发明的有益效果是：与传统的地震数据恢复方法相比，本发明利用方向梯度算子精确捕捉地震数据中由构造引导的局部倾角信息，并引入了基于RNF正则化的尺度不变非凸正则化项，以更真实地捕捉低秩先验信息，而无需引入额外的正则化权重参数。在统一的结构化低秩恢复框架下，同时利用了地震数据中固有的局部方向平滑性和相干性特征，最终得到高精度且鲁棒的地震数据恢复结果。具体来说，一是通过引入方向梯度算子，同时编码低秩性和方向一致性，从而捕捉地震数据的结构导向特征并表征其几何倾角。二是本发明与现有的非凸模型不同，通过利用方向梯度域中的RNF正则化开发了一种无参数的非凸低秩方法。三是本发明中的低秩恢复方法具有尺度不变性，确保了所提出的方法在不同的地震数据集上具有稳健的恢复性能。
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Abstract
Description
Technical Field
[0001] This invention relates to the field of seismic data recovery, and more particularly to a structure-guided method for recovering non-convex low-rank seismic data. Background Technology
[0002] The rapid development of high-performance sensing equipment and high-speed data transmission technologies has facilitated a range of applications in seismic exploration, including resource assessment, geological hazard prevention, and archaeological surveys. However, due to instrument malfunctions and data transmission anomalies, acquired seismic data is inevitably affected by missing values and noise. In practical applications, the presence of noise and missing values poses significant challenges to downstream applications of seismic data, such as seismic inversion, fault detection, and first-arrival picking. Therefore, developing effective seismic data recovery methods for joint denoising and reconstruction is of great value in the preprocessing stage.
[0003] Low-rank methods have proven to have broad application prospects in seismic data recovery due to their ability to effectively characterize the linear correlation structure in seismic events. To overcome the limitations of traditional low-rank models in representing complex seismic structures, several promising strategies for improving low-rank modeling have been proposed, mainly including models based on Hankel matrices, low-rank and smoothness fusion models, and non-convex low-rank models.
[0004] Hankel matrix-based methods have been successfully applied to seismic data denoising and reconstruction. These models are based on the fundamental assumption that the Hankel matrix constructed from clean seismic data possesses low rank. While Hankel matrix-based methods have demonstrated strong effectiveness in seismic data recovery, they typically neglect explicit modeling of local smoothness. Furthermore, because these methods are highly sensitive to parameter settings, their robustness decreases without proper parameter tuning when noise intensity and missing rates vary.
[0005] To maintain the inherent consistency of seismic data, models based on total variation (TV) are widely used. Building upon this, researchers have further proposed recovery methods that jointly utilize the low-rank and smoothness properties of seismic data. The most representative method is the separable model, which uses a weighted combination of the nuclear norm and TV terms to characterize low-rank and smoothness, respectively. However, such linear combination regularized models typically require careful tuning of the weight parameters, and the complexity of parameter selection remains a significant challenge in practical applications. Recent research has introduced a regularization technique that simultaneously considers low-rank and local smoothness, called correlated total variation (CTV), which has been proposed to effectively address the parameter selection problem. However, existing CTV-based methods cannot adequately characterize the slope attribute of seismic data, which is a key structural feature. Summary of the Invention
[0006] To address the aforementioned shortcomings of existing technologies, this invention provides a structure-guided non-convex low-rank seismic data recovery method. It proposes an directional gradient transformation operator based on local structural features of seismic data, which can accurately represent geological structures and geometric dips. Furthermore, utilizing the ratio of the nuclear norm to the Frobenius norm (RNF) as a non-convex surrogate, a novel non-convex low-rank regularizer is developed in the directional gradient domain. Within a unified structured low-rank recovery framework, this invention can simultaneously capture both global low-rank and local structure-guided smoothness without requiring additional regularization weight parameter adjustments, thereby achieving competitive seismic data recovery performance.
[0007] To achieve the aforementioned objectives, the technical solution adopted by this invention is as follows: a structure-guided method for recovering non-convex low-rank seismic data, comprising the following steps: S1: Input incomplete seismic data containing noise, initialize the seismic data to be recovered, and model variables and parameters under the alternating direction multiplier method framework; S2: The structural tensor method is used to process the seismic data to be restored, and the dip matrix corresponding to each element is calculated. S3: Apply gradient transformation to the seismic data to be recovered along the local dominant direction and the orthogonal direction of the dip matrix to obtain the seismic data in the local dominant direction gradient domain and the orthogonal direction gradient domain. S4: For seismic data in the local dominant gradient domain and orthogonal gradient domain, a low-rank constraint is applied using a non-convex regularization term with scale-invariant nuclear norm to Frobenius norm ratio, resulting in a structure-guided seismic data non-convex low-rank recovery model (DGRNF). S5: Using the alternating direction multiplier method framework, the non-convex low-rank recovery model DGRNF of seismic data is iteratively solved until convergence, and the seismic data is simultaneously denoised and reconstructed.
[0008] Furthermore, step S2 includes the following sub-steps: S21: Perform data preprocessing on the seismic data to be recovered and calculate the local gradient: , in, and For local gradients, The earthquake data to be recovered after preprocessing. and They are respectively Spatial orientation variables in the horizontal and vertical directions; S22: Constructing structural tensors based on local gradients : S23: Perform local smoothing operation on the structure tensor: in, For the reason Directional derivative and The structure tensor matrix composed of directional derivatives. For local smoothing operation, , , They represent , , ; S24: The tilt matrix of each element is calculated from the smoothed structure tensor. : .
[0009] Furthermore, the seismic data in the local dominant direction gradient domain and the orthogonal direction gradient domain in S3 are as follows: in, and Seismic data are presented in the local dominant gradient domain and the orthogonal gradient domain, respectively. For earthquake data to be recovered, and The first-order gradient operators for the horizontal and vertical directions are respectively expressed as: in, and For matrix The element index.
[0010] Furthermore, the nonconvex regularization term for the ratio of the scale-invariant nuclear norm to the Frobenius norm in S4 is: in, Let be the norm representation of the nonconvex regularization term. Let be the nuclear norm of the matrix. Let Frobenius norm be the matrix. For the first A singular value, It is a singular value. It is an L1 norm. It is an L2 norm; The structure-guided seismic data non-convex low-rank recovery model (DGRNF) is as follows: in, Norm representation of a structure-guided seismic data nonconvex low-rank recovery model.
[0011] Furthermore, for incomplete seismic data containing noise, the optimization problem using the Divergent Low-Rank Seismic Data Recovery (DGRNF) model is expressed as: in, To represent sparse components, For weight parameters, For index set The sampling operator is represented. This is incomplete seismic data that includes noise; Introducing auxiliary variables , and The optimization problem is expressed as: in, It is an auxiliary variable.
[0012] Furthermore, in S5, the alternating direction multiplier method framework is used to iteratively solve the non-convex low-rank seismic data recovery model DGRNF until convergence. The augmented Lagrangian function constructed using the alternating direction multiplier method framework is as follows: in, To augment the Lagrange function, , , For the Lagrange multipliers in the framework of the alternating direction multiplier method, This is the penalty parameter.
[0013] The beneficial effects of this invention are as follows: Compared with traditional seismic data recovery methods, this invention utilizes the directional gradient operator to accurately capture the local dip information guided by tectonics in seismic data, and introduces a scale-invariant non-convex regularization term based on RNF regularization to more realistically capture low-rank prior information without introducing additional regularization weight parameters. Within a unified structured low-rank recovery framework, it simultaneously utilizes the inherent local directional smoothness and coherence characteristics of seismic data, ultimately obtaining high-precision and robust seismic data recovery results. Specifically, firstly, by introducing the directional gradient operator, it simultaneously encodes low-rank and directional consistency, thereby capturing the structural guidance characteristics of seismic data and characterizing its geometric dip. Secondly, unlike existing non-convex models, this invention develops a parameter-free non-convex low-rank method by utilizing RNF regularization in the directional gradient domain. Thirdly, the low-rank recovery method in this invention possesses scale invariance, ensuring robust recovery performance on different seismic datasets. Attached Figure Description
[0014] Figure 1 This is a flowchart of a structure-guided method for recovering non-convex low-rank seismic data according to the present invention.
[0015] Figure 2 This is a comparison chart of the recovery results of the present invention method (DGRNF) and the comparison method in Example 1.
[0016] Figure 3 This is a comparison chart of the recovery results of the present invention method (DGRNF) and the comparison method in Example 2. Detailed Implementation
[0017] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.
[0018] like Figure 1 As shown, a structure-guided method for recovering non-convex low-rank seismic data includes the following steps: S1: Input incomplete seismic data containing noise. Initialize the seismic data to be recovered. And the model variables and parameters under the alternating direction multiplier method framework; in, Indicates the time sampling dimension. Representing spatial dimensions, incomplete data refers to structurally missing data along the spatial dimensions, and the noise it contains is random noise; this is seismic data awaiting recovery. Initialize to This embodiment uses two-dimensional synthetic seismic data, which includes 200 time sampling points, 200 spatial sampling points, a missing data ratio of 50%, and a Gaussian noise intensity of 0.1.
[0019] S2: The structural tensor method is used to process the seismic data to be restored, and the dip matrix corresponding to each element is calculated. S2 includes the following sub-steps: S21: Perform data preprocessing on the seismic data to be recovered and calculate the local gradient: , in, and For local gradients, The earthquake data to be recovered after preprocessing. and They are respectively Spatial orientation variables in the horizontal and vertical directions; S22: Constructing structural tensors based on local gradients : S23: Perform local smoothing operation on the structure tensor: in, For the reason Directional derivative and The structure tensor matrix composed of directional derivatives. For local smoothing operation, , , They represent , , ; S24: The tilt matrix of each element is calculated from the smoothed structure tensor. : .
[0020] S3: Apply gradient transformation to the seismic data to be recovered along the local dominant direction and the orthogonal direction of the dip matrix to obtain the seismic data in the local dominant direction gradient domain and the orthogonal direction gradient domain. The seismic data in the local dominant direction gradient domain and orthogonal direction gradient domain of S3 are as follows: in, and Seismic data are presented in the local dominant gradient domain and the orthogonal gradient domain, respectively. For earthquake data to be recovered, and The first-order gradient operators for the horizontal and vertical directions are respectively expressed as: in, and For matrix The element index.
[0021] Seismic data are transformed into directional gradient domain data through local dominant direction and orthogonal direction gradient transformation.
[0022] S4: For seismic data in the local dominant gradient domain and orthogonal gradient domain, a low-rank constraint is applied using a non-convex regularization term with scale-invariant nuclear norm to Frobenius norm ratio, resulting in a structure-guided seismic data non-convex low-rank recovery model (DGRNF). The nonconvex regularization term for the ratio of the scale-invariant nuclear norm to the Frobenius norm in S4 is: in, Let be the norm representation of the nonconvex regularization term. Let be the nuclear norm of the matrix. Let Frobenius norm be the matrix. For the first A singular value, It is a singular value. It is an L1 norm. It is an L2 norm; Seismic data in the local dominant gradient domain and orthogonal gradient domain and Applying RNF regularization yields the proposed structure-guided seismic data non-convex low-rank recovery model, DGRNF, as follows: in, Norm representation of a structure-guided seismic data nonconvex low-rank recovery model.
[0023] For incomplete seismic data containing noise, the optimization problem using the Divergent Low-Rank Seismic Data Recovery (DGRNF) model is expressed as: in, To represent sparse components, the proposed model can effectively handle degradation scenarios with various missing rates and noise levels. For weight parameters, For index set The sampling operator is represented. This is incomplete seismic data that includes noise; Introducing auxiliary variables , and The optimization problem is expressed as: in, It is an auxiliary variable.
[0024] S5: Using the alternating direction multiplier method framework, the non-convex low-rank recovery model DGRNF of seismic data is iteratively solved until convergence, and the seismic data is simultaneously denoised and reconstructed.
[0025] The augmented Lagrangian function is constructed using the alternating direction multiplier method framework as follows: in, To augment the Lagrange function, , , For the Lagrange multipliers in the framework of the alternating direction multiplier method, This is the penalty parameter.
[0026] Establish the following iterative formula and termination condition for the alternating direction multiplier method: in, Representing the number of iterations, using To accelerate convergence, the iteration termination condition is: .
[0027] The above problem is expressed as: The closed-form solution can be obtained by applying the singular value threshold operator: in, , Each is a matrix The left and right singular vector matrices, as well as It is singular value decomposition.
[0028] A closed-form solution can be obtained by solving the following problem: in, It is a random matrix and Auxiliary variables as well as At the same time: , in, and As an auxiliary variable; The closed-form solution can be obtained using the following formula: The closed-form solution can be obtained using the soft threshold operator: in, For soft thresholding operators, For input vectors or matrices, This is the threshold parameter.
[0029] By solving the above problem, we can obtain: in, , , Let represent the Fourier transform operator, the inverse Fourier transform operator, and the complex conjugate operator, respectively. Represents the identity matrix, auxiliary variables , .
[0030] Example 1, please refer to the experimental results. Figure 2 . Figure 2 (a) represents complete and clean seismic data. Figure 2 (b) represents seismic data with 50% missing channels and containing Gaussian noise of intensity 0.1. Figure 2In Figures (c)-(h), the results of restoration using the DMSSA, LDRR, CTV, MNN-L1, MNN-L2, and the method of this invention (DGRNF) are shown, respectively. It is evident that the restoration results of the method of this invention (DGRNF) are more effective, and the reconstructed effective data is closer to the original seismic data. The peak signal-to-noise ratios (PSNRs) of the restored data using the DMSSA, LDRR, CTV, MNN-L1, MNN-L2, and the method of this invention (DGRNF) are 31.113 dB, 31.309 dB, 30.142 dB, 29.912 dB, 29.992 dB, and 35.357 dB, respectively. The method of this invention achieves a higher PSNR, further validating the effectiveness of the proposed method.
[0031] Example 2 uses another type of synthetic seismic data, totaling 200 channels. Each channel contains 200 time sampling points, with a missing rate of 50% and includes Gaussian noise with a noise level of 0.1. The remaining procedures are the same as in Example 1. Please refer to the experimental results. Figure 3 . Figure 3 (a) represents complete earthquake data. Figure 3 (b) represents seismic data with 50% missing channels and containing Gaussian noise of intensity 0.1. Figure 3 In the table, (c)-(h) represent the results recovered using the DMSSA method, LDRR method, CTV method, MNN-L1 method, MNN-L2 method, and the method of this invention (DGRNF), respectively, with corresponding peak signal-to-noise ratios of 29.530 dB, 29.522 dB, 26.201 dB, 27.210 dB, 24.398 dB, and 33.566 dB. Figure 3 As can be seen from the peak signal-to-noise ratio results, the method of the present invention can effectively reconstruct seismic data and significantly improve the recovery accuracy.
[0032] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of the invention.
Claims
1. A structure-guided method for non-convex low-rank recovery of seismic data, characterized in that, Includes the following steps: S1: Input incomplete seismic data containing noise, initialize the seismic data to be recovered, and model variables and parameters under the alternating direction multiplier method framework; S2: The structural tensor method is used to process the seismic data to be restored, and the dip matrix corresponding to each element is calculated. S3: Apply gradient transformation to the seismic data to be recovered along the local dominant direction and the orthogonal direction of the dip matrix to obtain the seismic data in the local dominant direction gradient domain and the orthogonal direction gradient domain. S4: For seismic data in the local dominant gradient domain and orthogonal gradient domain, a low-rank constraint is applied using a non-convex regularization term with scale-invariant nuclear norm to Frobenius norm ratio, resulting in a structure-guided seismic data non-convex low-rank recovery model (DGRNF). S5: Using the alternating direction multiplier method framework, the non-convex low-rank recovery model DGRNF of seismic data is iteratively solved until convergence, and the seismic data is simultaneously denoised and reconstructed.
2. The structure-guided seismic data non-convex low-rank recovery method according to claim 1, characterized in that, S2 includes the following steps: S21: Perform data preprocessing on the seismic data to be recovered and calculate the local gradient: ， in, and For local gradients, The earthquake data to be recovered after preprocessing. and They are respectively Spatial orientation variables in the horizontal and vertical directions; S22: Constructing a structural tensor based on local gradients : S23: Perform local smoothing operation on the structure tensor: in, For the reason Directional derivative and The structure tensor matrix composed of directional derivatives. For local smoothing operation, , , They represent , , ; S24: The tilt matrix of each element is calculated from the smoothed structure tensor. : 。 3. The structure-guided seismic data non-convex low-rank recovery method according to claim 2, characterized in that, The seismic data in the local dominant direction gradient domain and orthogonal direction gradient domain of S3 are as follows: in, and Seismic data are presented in the local dominant gradient domain and the orthogonal gradient domain, respectively. For earthquake data to be recovered, and The first-order gradient operators for the horizontal and vertical directions are respectively expressed as: in, and For matrix The element index.
4. The structure-guided seismic data non-convex low-rank recovery method according to claim 3, characterized in that, The nonconvex regularization term for the ratio of the scale-invariant nuclear norm to the Frobenius norm in S4 is: in, Let be the norm representation of the nonconvex regularization term. Let be the nuclear norm of the matrix. Let Frobenius norm be the matrix. For the first A singular value, It is a singular value. It is an L1 norm. It is an L2 norm; The structure-guided seismic data non-convex low-rank recovery model (DGRNF) is as follows: in, Norm representation of a structure-guided seismic data nonconvex low-rank recovery model.
5. The structure-guided seismic data non-convex low-rank recovery method according to claim 4, characterized in that, For incomplete seismic data containing noise, the optimization problem using the Divergent Low-Rank Seismic Data Recovery (DGRNF) model is expressed as: in, To represent sparse components, For weight parameters, For index set The sampling operator is represented. This is incomplete seismic data that includes noise; Introducing auxiliary variables , and The optimization problem is expressed as: in, It is an auxiliary variable.
6. The structure-guided seismic data non-convex low-rank recovery method according to claim 5, characterized in that, In S5, the alternating direction multiplier method framework is used to iteratively solve the non-convex low-rank seismic data recovery model DGRNF until convergence. The augmented Lagrangian function constructed using the alternating direction multiplier method framework is as follows: in, To augment the Lagrange function, , , For the Lagrange multipliers in the framework of the alternating direction multiplier method, This is the penalty parameter.