3D Tensor Random Additive Footprint Noise Denoising Method Based on Total Variation Low-Rank Approximation

Abstract

The present invention discloses a denoising method based on low-rank tensor approximation, which is applied to the field of seismic data denoising. Aiming at the problem that existing denoising technologies all emphasize the suppression of random noise rather than the removal of footprints, and thus cannot completely remove mixed noise and reconstruct high-quality seismic images, the present invention proposes a new denoising model, the UTV-LRTA model. Specifically, the seismic image contaminated by footprints is decomposed into the UTV image component and the footprint component to remove footprint noise, and the low rank of the third-order seismic tensor is regularized by using the tensor nuclear norm to dilute random noise; and an algorithm based on SBI is used to effectively solve the proposed UTV-LRTA.

Description

Technical Field

[0001] The present invention belongs to the field of seismic data processing, and particularly relates to a random footprint noise attenuation technique. Background Art

[0002] The acquisition footprint is clearly described as periodic amplitude artifacts that occur on 3D seismic time or horizontal slices. These artifacts are substantially caused by amplitude variations inseparable from the measurement geometry, rather than the geology underground. These footprint artifacts significantly obscure the geological structure interpretation of seismic profiles, especially those through methods such as seismic attribute analysis or AVO (Amplitude Versus Offset). A notable fact is that random noise also inevitably accompanies the footprint and appears in real 3D field data.

[0003] For the removal of footprint noise, the key is to encounter a suitable transform domain to completely separate valuable signals and footprint noise. Recently, a large number of transform domains have been recorded in the seismic literature, which are roughly divided into three categories: frequency domain, time-frequency domain, and dictionary learning (DL). The first category treats the footprint as spatial periodic noise and uses various filtering techniques in the frequency domain to remove the footprint, including classical wavenumber filtering, adaptive wavenumber filtering, spectral notch filtering, double-pass frequency-wavenumber filtering, and wavenumber notch filtering. The second category uses time-frequency transforms to gain insights into the footprint in the time and frequency domains, thus significantly reducing the overlap between signals and footprints and facilitating filtering. Typical time-frequency transforms include wavelet transform, 3D complex wavelet transform, stationary wavelet transform (SWT), basis pursuit, and curvelet transform. However, it is a contradiction that the filtering technique can both eliminate these unwanted traces and keep the original seismic signal unaffected.

[0004] Completely different from the analytical Fourier transform and time-frequency transforms, the last category appropriately assigns the atoms of the learned dictionary (instead of the previous sparse coefficients) to represent clear image and footprint patterns. Once the atoms representing the footprint pattern are obtained, the DL-based denoising method can operate on them, perform binary pattern classification or filtering tasks on all learned atoms to reconstruct the footprint components, and then subtract the footprint from the input to obtain the denoising result. However, although the current DL-based methods have made significant progress in footprint removal, there is a key defect in this removal process: as typical matrix DL methods, they must flatten the 3-D, naturally annihilating the complex high-dimensional image structure hidden in the 3D seismic data for ease of processing, thus significantly affecting the denoising performance of the DL-based methods.

[0005] For the attenuation of random noise, the geophysics community has conducted a large amount of research work, which can be roughly divided into three research directions: spatial domain filtering, sparse representation (SR), and low rank approximation (LRA). The first direction uses the sum of the products of the filtering coefficients in the spatial mask and the relevant image pixels to generate the denoising result. Typical spatial domain filters include non-local mean filtering, median filtering, morphological filtering, and non-linear diffusion filtering. The second direction is to use the SR representation coefficients as few as possible for the noise-free seismic data in a given sparse transform; however, Gaussian random noise cannot be represented by this method. There are several sparse transforms that can be used for image transformation, such as wavelet transform, curvelet transform, wavelet transform, shearlet transform, and learned dictionary. The last direction uses various rank reduction methods such as matrix LRA for a data matrix or tensor to achieve seismic image denoising, the LRA method based on the Hankel matrix, tensor singular value decomposition (tSVD), and CANDECOMP / PARAFAC (CP) decomposition. Of course, the current random noise suppression techniques are also often used to remove footprints, such as non-linear diffusion filtering and singular value decomposition (SVD). However, it is obvious that these denoising methods cannot completely remove the mixed noise and reconstruct high-quality seismic images because the previous methods all emphasize the suppression of random noise rather than the removal of footprints. Summary of the Invention

[0006] To solve the above technical problems, the present invention proposes a three-dimensional tensor random plus footprint noise denoising method based on total variation low rank approximation, which removes random noise and footprints from three-dimensional seismic data.

[0007] The technical solution adopted by the present invention is: a three-dimensional tensor random plus footprint noise denoising method based on total variation low rank approximation, including:

[0008] S1. Denote the noise image tensor to be processed as The expression is:

[0009]

[0010] Wherein, represents the noise image tensor, represents the clear image tensor, represents the footprint, represents the random noise;

[0011] S2. Adopt the following denoising model, in and In the case of co - existence, from Estimate the clear image tensor

[0012] where is the low - rank tensor model of clear seismic data with nuclear norm minimization, represents the footprint noise UTV model;

[0013] S3. Introduce three auxiliary variables Convert in the unconstrained minimization problem into a constrained minimization problem, and then call SBI to solve the constrained minimization problem as follows:

[0014]

[0015]

[0016] S4. Further transform into the subsequent unconstrained minimization:

[0017]

[0018] where α, β, γ represent Bregman penalty parameters, is the Bregman tensor;

[0019] S5. Use the alternating minimization scheme to solve and obtain the clear image tensor

[0020] Advantages of the present invention: The present invention proposes a new UTV - LRTA three - dimensional seismic random plus footprint noise attenuation method. The method of the present invention models the denoising problem as a low - rank sparse tensor decomposition model; specifically: in the UTV - LRTA model of the present invention, the footprint noise is removed by effectively decomposing the footprint - contaminated seismic image into the UTV image component and the footprint component, while the random noise is diluted by using the nuclear norm regularization of the low - rank of the third - order seismic tensor; then, an SBI - based algorithm is used to effectively solve the proposed UTV - LRTA. The performance of the method of the present invention is verified through synthetic experiments and field experiments; the experimental results show that the algorithm of the present invention has good denoising ability for random noise plus footprint and good computational complexity. BRIEF DESCRIPTION OF THE DRAWINGS

[0021] Figure 1 is the seismic image provided by this embodiment;

[0022] Among them, (a) is the original slice of the seismic image with footprint noise, (b) is the derivative of the original slice on the x-axis, and (c) is the derivative of the original slice on the y-axis;

[0023] Figure 2 is the schematic diagram of the model of the present invention;

[0024] Figure 3 is the schematic diagram of the single-trace comparison between the method of the present invention provided in this embodiment and the prior art on the synthetic dataset;

[0025] Among them, (a) is the comparison result of the denoising result of the UTV-LRTA algorithm of the present invention and the clean synthetic data on a single trace, (b) is the comparison result of the denoising result of the FR-net algorithm and the clean synthetic data on a single trace, (c) is the comparison result of the denoising result of the SGRDL algorithm and the clean synthetic data on a single trace, and (d) is the comparison result of the denoising result of the DRR algorithm and the clean synthetic data on a single trace;

[0026] Figure 4 is the convergence curve of the SBI algorithm of the present invention on the synthetic dataset;

[0027] Figure 5 is the comparison of the denoising effects between the method of the present invention and the prior art on the synthetic dataset;

[0028] Among them, (a) is the noisy data, (b) is the footprint removal result of the UTV-LRTA method proposed by the present invention, (c) is the corresponding residual obtained by the UTV-LRTA proposed by the present invention, (d) is the real data, (e) is the footprint removal result of the FR-Net method, (f) is the corresponding residual obtained by the FR-Net method, (g) is the footprint removal result of the SGRDL method, (h) is the corresponding residual obtained by the SGRDL method, (i) is the footprint removal result of the DRR method, and (j) is the corresponding residual obtained by the DRR method;

[0029] Figure 6 is the comparison of the denoising effects between the method of the present invention and the prior art on the Time slice of the synthetic dataset;

[0030] Among them, (a) is the slice of the original 163rd time, (b) is the footprint removal result of the UTV-LRTA proposed by the present invention on the slice, (c) is the corresponding residual obtained by the UTV-LRTA method proposed by the present invention, (d) is the ground truth data, (e) is the footprint removal result of the FR-Net method on the slice, (f) is the corresponding residual obtained by the FR-Net method, (g) is the footprint removal result of the SGRDL method on the slice, (h) is the corresponding residual obtained by the SGRDL method, (i) is the footprint removal result of the DRR method on the slice, and (j) is the corresponding residual obtained by the DRR method;

[0031] Figure 7 Comparison of the denoising effect between the method of the present invention and the prior art on inline slices of the synthetic dataset

[0032] Among them, (a) is the slice of the original 30th inline, (b) is the footprint removal result of UTV-LRTA proposed by the present invention on the slice, (c) is the corresponding residual obtained by the UTV-LRTA method proposed by the present invention, (d) is the ground truth data, (e) is the footprint removal result of the FR-Net method on the slice, (f) is the corresponding residual obtained by the FR-Net method, (g) is the footprint removal result of the SGRDL method on the slice, (h) is the corresponding residual obtained by the SGRDL method, (i) is the footprint removal result of the DRR method on the slice, and (j) is the corresponding residual obtained by the DRR method;

[0033] Figure 8 Comparison of the denoising effect between the method of the present invention and the prior art on xline slices of the synthetic dataset

[0034] Among them, (a) is the slice of the original 70th xline, (b) is the footprint removal result of UTV-LRTA proposed by the present invention on the slice, (c) is the corresponding residual obtained by the UTV-LRTA method proposed by the present invention, (d) is the ground truth data, (e) is the footprint removal result of the FR-Net method on the slice, (f) is the corresponding residual obtained by the FR-Net method, (g) is the footprint removal result of the SGRDL method on the slice, (h) is the corresponding residual obtained by the SGRDL method, (i) is the footprint removal result of the DRR method on the slice, and (j) is the corresponding residual obtained by the DRR method;

[0035] Figure 9 Comparison of the denoising effect between the method of the present invention and the prior art on the Penobscot-3D dataset

[0036] Among them, (a) is the noise, (b) is the footprint removal result of UTV-LRTA proposed by the present invention, (c) is the corresponding residual obtained by the UTV-LRTA method proposed by the present invention, (d) is the footprint removal result of the FR-Net method, (e) is the corresponding residual obtained by the FR-Net method, (f) is the footprint removal result of the SGRDL method, (g) is the corresponding residual obtained by the SGRDL method, (h) is the footprint removal result of the DRR method, and (i) is the corresponding residual obtained by the DRR method;

[0037] Figure 10 Comparison of the denoising effect between the method of the present invention and the prior art on the constant time slice of the Penobscot-3D dataset

[0038] Among them, (a) is the slice of the original 163rd time, (b) is the footprint removal result of UTV-LRTA proposed by the present invention on the slice, (c) is the corresponding residual obtained by the UTV-LRTA method proposed by the present invention, (d) is the footprint removal result of the FR-Net method on the slice, (e) is the corresponding residual obtained by the FR-Net method, (f) is the footprint removal result of the SGRDL method on the slice, (g) is the corresponding residual obtained by the SGRDL method, (h) is the footprint removal result of the DRR method on the slice, and (i) is the corresponding residual obtained by the DRR method;

[0039] Figure 11 It is a comparison of the denoising effect between the method of the present invention and the prior art on the inline slice of the Penobscot-3D dataset;

[0040] Among them, (a) is the slice of the original 10th inline, (b) is the footprint removal result of UTV-LRTA proposed by the present invention on the slice, (c) is the corresponding residual obtained by the UTV-LRTA method proposed by the present invention, (d) is the footprint removal result of the FR-Net method on the slice, (e) is the corresponding residual obtained by the FR-Net method, (f) is the footprint removal result of the SGRDL method on the slice, (g) is the corresponding residual obtained by the SGRDL method, (h) is the footprint removal result of the DRR method on the slice, and (i) is the corresponding residual obtained by the DRR method;

[0041] Figure 12 It is a comparison of the denoising effect between the method of the present invention and the prior art on the xline slice of the Penobscot-3D dataset;

[0042] Among them, (a) is the slice of the original 70th xline, (b) is the footprint removal result of UTV-LRTA proposed by the present invention on the slice, (c) is the corresponding residual obtained by the UTV-LRTA method proposed by the present invention, (d) is the footprint removal result of the FR-Net method on the slice, (e) is the corresponding residual obtained by the FR-Net method, (f) is the footprint removal result of the SGRDL method on the slice, (g) is the corresponding residual obtained by the SGRDL method, (h) is the footprint removal result of the DRR method on the slice, and (i) is the corresponding residual obtained by the DRR method. Detailed implementation manner

[0043] To facilitate the understanding of the content of the present invention by those skilled in the art, the following technical terms are first explained:

[0044] Definition 1. Given tensors and is a tensor of size n1×n4×n3, defined as:

[0045]

[0046] wherein, represents all real numbers, n1, n2, n4, n3 are integers, and in this embodiment, n1×n2×n3, n2×n4×n3, n1×n4×n3 are used to represent the sizes of the tensor in three dimensions;

[0047] Generally, the fold() and unfold() operations are defined as follows:

[0048]

[0049] X (1) represents the first two-dimensional slice of X in the third dimension n3;

[0050] Define circ(·) as an n1n3×n2n3 matrix:

[0051]

[0052] Define 2(t-SVD). For its tensor SVD decomposition is

[0053]

[0054] wherein, * represents the convolution operation; and are orthogonal tensors; is a diagonal tensor, and the front slice of Θ (i.e., the slice in the n1 x n2 direction) is a diagonal matrix. t-SVD has the same characteristics as matrix SVD, that is, it adheres to and the orthogonal properties, and the diagonal structure of Θ. In the Fourier domain, each front slice of t-SVD is equivalent to a matrix SVD. This definition of t-SVD leads to the concept of the tensor nuclear norm, which is further defined below.

[0055] Definition 3. Tensor Nuclear Norm (TNN). The tensor nuclear norm of a tensor is denoted as and is defined as the average of the nuclear norms of all front slices of

[0056]

[0057] where is the block diagonal matrix of

[0058]

[0059] The meanings of the relevant parameter symbols in the present invention are shown in Table 1 as follows:

[0060] Table 1 Explanation of the Meanings of Parameter Symbols

[0061]

[0062]

[0063] The following is a detailed description of the content of the present invention in conjunction with the attached Figure 1-12 :

[0064] The processing process of the method of the present invention is as Figure 2 shown, Figure 2 in which Split Bregman Iteration Optimization represents the Split Bregman iteration optimization algorithm, Noisy Image represents the noisy image, Footprint represents the footprint noise, Clear Image represents the clean image, Random Noise represents the random noise, Denoising Image represents the denoised image, Unidirectional Total Variation represents the unidirectional total variation, and Tensor Nuclear Norm represents the tensor nuclear norm; the method of the present invention specifically includes the following steps:

[0065] 1. First, analyze the problems existing in the prior art

[0066] The footprint and the random noise effects are generally assumed to be additive effects rather than multiplicative effects, and the mixed noise degradation model can be written as

[0067]

[0068] wherein, and respectively represent the noisy image tensor and the true image tensor. Formally, the task of the present invention is to estimate the potential clear image tensor and the random noise from the given image tensor while both exist, Specifically, by minimizing the following equation:

[0069]

[0070] As can be seen from Equation (13), the task in Equation (7) is formulated as an ill-posed inverse problem. The key is to construct an appropriate regularization term to constrain the solution, thereby removing the footprint and random noise. Based on the regularization theory, the denoising model can be expressed as the combination of a data fidelity term and a regularization term:

[0071]

[0072] The first term in Equation (9) is the data fidelity term, which provides the similarity between the desired clear image tensor and the noisy image tensor. The second term is a regularization term that imposes constraints on the image, aiming to penalize the undesirable properties in the image. The last term is also a regularization term, different from the term The term represents some prior knowledge of the footprint noise τ > 0 and λ > 0 are both regularization parameters, which control the trade-off between data fidelity and the two different regularization terms.

[0073] 2. Problem Modeling

[0074] The modeling process of the present invention for the above problems includes: a low-rank tensor model and a UTV model for clean seismic data and footprints; a proposed UTV-LRTA is presented; a UTV-LRTA model capable of simultaneously removing Gaussian noise and footprints is proposed. Specifically:

[0075] 1) Low-rank tensor model of the clear image tensor: The seismic denoising problem regards obtaining the ground truth data as a low-rank matrix or tensor, and noise will increase the rank of this data matrix or tensor. Therefore, when the data is contaminated by Gaussian noise, the rank reduction method is an effective random noise attenuation method. In this work, a low-rank tensor model of clear seismic data based on the minimization of the tensor nuclear norm (TNN) can be considered, which is defined as:

[0076]

[0077] As is well known, the rank reduction method based on the minimization of the tensor nuclear norm has achieved certain noise reduction effects in seismic image denoising applications.

[0078] 2) UTV model of the footprint: The directional characteristics of the footprint can be fully exploited through the UTV model. The derivatives of the front slice of the tensor in two directions are shown in Figure 1 . The original slice is as shown in Figure 1 (a), and the derivatives of the slice on the x-axis and y-axis are respectively as shown in Figure 1 (b)-1(c). Due to the directionality of the footprint, as shown in Figure 1 (b), it can be observed that the stripes seriously damage the spatial horizontal gradient, while as shown in Figure 1(c) The spatial vertical gradient shown is completely unaffected. Therefore, the present invention proposes UTV, which utilizes the structural and directional characteristics of the footprint and has better detail preservation performance, defined as:

[0079]

[0080] where and represent the first-order derivative operators in the corresponding directions.

[0081] 3) The proposed UTV-LRTA model: Substitute and into formula (11) respectively using the low-rank tensor model and the UTV regularization method in (9), and obtain the final energy functional for matrix X:

[0082]

[0083] where λ1, λ2, and τ represent regularization parameters used to adjust the weight of each term to balance the model.

[0084] 3. Method for solving formula (12)

[0085] A. Objective function

[0086] The challenge faced when using the underdetermined equation (12) to solve is that the 1-norm term itself is non-differentiable, that is, a function that is not reachable at some points in the interval. Split Bregman iteration (SBI) is an effective tool for solving the "1-norm based regularization minimization problem (and of course the TV model). Due to its fast solution characteristics, it has recently attracted considerable attention. Therefore, SBI is naturally extended to the UTV-LRTA model proposed in formula (12) to solve the explicit third-order data tensor By introducing three auxiliary variables transform the unconstrained minimization problem in formula (12) into a constrained minimization problem, and then call SBI to solve the constrained minimization problem as follows:

[0087]

[0088] Subsequently, by strictly applying the Bregman iteration, the above constrained problem (13) can be further transformed into the subsequent unconstrained minimization:

[0089]

[0090] where α, β, and γ represent Bregman penalty parameters, and is the Bregman tensor.

[0091] B. Solution algorithm

[0092] The alternating minimization scheme is used to solve the problem in Equation (14), that is, iteratively optimize one variable while fixing other variables. Therefore, the problem in Equation (14) can be decomposed into 4 simpler minimization sub-problems, and each sub-problem can be solved independently.

[0093] 1) Update x l+1 : The sub-problem related to is

[0094]

[0095] To solve Equation (15), each positive slice of can be solved

[0096]

[0097] Note that the problem in Equation (16) is a convex function of real variable , so it is more straightforward to find the optimal solution, that is

[0098]

[0099] where performs the regular matrix transpose on the two-dimensional input slice of the tensor, and l is the iteration number. The closed-form solution of the above problem is found using the two-dimensional Fourier transform, and the solution is as follows:[[]]

[0100]

[0101] where

[0102]

[0103]

[0104] In the formula, "*" represents the complex conjugate, "o" represents the element-wise multiplication, and represent the fast Fourier transform matrix and its inverse matrix respectively.

[0105] 2) Update : The sub-problem related to is

[0106]

[0107] Obviously, the sub-problem in (21) must be solved using a tensor SVD, which is similar to the SVD of a matrix:

[0108]

[0109] Then, a closed - form solution of is obtained by using the soft - shrinkage operator as follows:

[0110]

[0111] where

[0112]

[0113] Obviously, the implementation of this shrinkage operation is very straightforward. In each iteration step, only one tSVD decomposition needs to be computed and basic tensor operations are performed.

[0114] 3) Update The sub - problem related to is

[0115]

[0116] The sub - problem in (25) can be solved by a shrinkage operator:

[0117]

[0118] 4) Update The sub - problem related to is

[0119]

[0120] Similar to the sub - problem related to the analytical solution of sub - problem (27) is:

[0121]

[0122] Table 2 Quantitatively compare the footprint removal results of the prior art and the method of the present invention on the synthetic dataset

[0123] Method Synthetic dataset DRR 13.74 dB FR-Net 20.26 dB SGRDL 21.68 dB UTV-LRTA of the present invention 37.57 dB

[0124] Table 3 Comparison of the computational time with the prior methods

[0125] Method Synthetic dataset Penobscot-3D DRR 1450.11s 410.84s FR-Net 1173.51s 358.11s SGRDL 161.79s 87.12s UTV-LRTA of the present invention 40.71s 21.26s

[0126] 5) Update the Bregman tensor and Finally, and are Bregman tensors and are updated as follows:

[0127]

[0128] The convergence condition of the solution process of UTV-LRTA proposed by the present invention is that when is satisfied, or when the number of iterations exceeds a preset threshold, the outer loop stops. In this embodiment, ∈ = 10 -5 . In addition, the SBI method can ensure the convergence of the split Bregman iteration algorithm for Equation (14), as Figure 4 shown. Figure 4 where RSE is the relative standard error, The convergence area is the convergence region, and The number of iterations is the number of iterations.

[0129] The effectiveness of the UTV-LRTA model of the present invention is determined through a series of comparative experiments on synthetic data and actual field data. First, the experimental configuration includes performance metrics, comparative algorithms used, and parameter settings, which are divided into two parts: one part uses synthetic data, and the other part uses actual field data.

[0130] A. Performance Metrics

[0131] In the synthetic data experiment, the ground truth data is available and can be compared with the footprint removal results. In addition to visual comparison, it is natural to use quantitative evaluation metrics to compare the denoising performance of FR-Net and SOTA methods. This embodiment also uses the signal-to-noise ratio (SNR) as the quantitative evaluation metric.

[0132]

[0133] where represent the clean image and the estimated image, respectively.

[0134] B. Comparative Algorithms

[0135] Since the UTV-LRTA proposed by the present invention has both classical DCAE and UTV components, we compare our proposed method with the classical total variation regularized DCAE model to verify the benefits brought by using UTV. According to the seismic literature review in the first part, we also compare our method with the recent footprint removal models and SVD-based filtering, which learn how to effectively eliminate noise from contaminated data. These SOTA methods are detailed as follows.

[0136] 1) FR-Net: Decompose the Hankel matrix vector space of the noise signal into a signal subspace and a noise subspace through SVD;

[0137] 2) Statistics-guided residual dictionary learning (SGRDL):

[0138] The SGRDL generates an algorithm framework for learning signal dictionary atoms and separates footprint features from the learned atoms. In this example, SGRDL uses a method based on statistical analysis to classify dictionary atoms as footprint-affected or footprint-unaffected.

[0139] 3) Damped rank reduction (DRR): By introducing a damping operator into the traditional truncated SVD model, DRR tends to divide the Hankel matrix vector space of the noise signal into a useful signal subspace and a noise subspace.

[0140] In the experiment, DRR and SGRDL were implemented in MATLAB, and CBDNet and FR-Net were implemented in TensorFlow3 on the Google Colab platform; remember that Colab randomly assigns GPUs to users on the platform.

[0141] C. Parameter settings

[0142] Selecting appropriate parameters is a common difficulty for many algorithms, and empirical adjustment is a common method for determining parameters. The method of this embodiment includes 6 regularization parameters of α, β, γ, τ, λ1, and λ2, and the parameters depend on the specific stripe noise level and the enhanced random noise level. Although the method of the present invention involves many parameters, the parameters have good robustness. Since this embodiment involves many stripe levels, according to the different degradation levels of the test images in the experiment, here this embodiment sets the synthetic data with α = 0.1, β = 50, γ = 50, τ = 1, λ1 = 0.1, λ2 = 0.1 and the Penobscot work area data with α = 20, β = 10, γ = 10, τ = 1, λ1 = 0.1, λ2 = 0.1.

[0143] D. Verification of synthetic data

[0144] First, apply the UTV-LRTA proposed by the present invention to a synthetic physical model data set, which contains 785×790 traces, and each trace contains 400 time samples. Random noise and footprints are deliberately added to the data without ground truth, so that the noisy seismic image becomes the input of UTV-LRTA, as Figure 5 (a), 6(a), 7(a), and 8(a) show. Note that UTV-LRTA cannot see the original seismic image at all because it is only used to calculate the signal-to-noise ratio. The present invention hopes that UTV LRTA can accurately reconstruct the noise-free original image from the corresponding noisy 3D image.

[0145] Finally, visualize the obtained denoised image. Figure 5Three-dimensional synthetic data for different methods in a mixed noise environment, with a signal-to-noise ratio of 6.02 dB. Figure 5 Shows the results of UTV-LRTA and SOTA methods for the amount of noise data, which are shown by the Madagascar software package 2. In terms of signal-to-noise ratio, the values of the FR-Net, DRR, and SGRDL methods are 21.68, 13.74, and 20.26 dB respectively, while the value of the UTV-LRTA model of the present invention is higher, at 37.57 dB, as shown in Table 2. Importantly, we found that UTV-LRTA provides findings very close to the ground truth in terms of image content, while the SOTA method results in visible deviations from the ground truth, such as Figure 5 , 6 , 7, and 8. In addition, the footprint residuals of the SOTA and FR-Net methods can be more intuitively demonstrated by single-trace comparison, as shown in Figure 3 . As observed, it is obvious that the method proposed by the present invention effectively performs footprint removal without damaging useful signals. Figure 3 In , Time represents time and Amplitude represents amplitude.

[0146] E. Verification of real field data

[0147] In this embodiment, the real Penobscot-3D dataset is used to test the proposed method, which is real turntable data publicly released by dGB Earth Sciences 3. The Penobscot-3D data is contaminated by random noise and footprints on the shallow time slices, as shown in Figure 9 (a), 10(a), 11(a), and 12(a). Each footprint pattern consists of a series of stripes of different lengths, which are arranged along the crossline direction.

[0148] Figure 9 , Figure 10 , Figure 11 , Figure 12 shows the denoising effects of all test methods. Among them, Figure 9 compares the noise reduction performance of the UTV-LRTA and SOTA methods proposed by the present invention in the entire data volume. Figure 10 , 11 , 12 are the noise reduction results of the constant time slice, xline slice, and inline slice respectively. From the noise mixed with random noise and footprints in Figure 9 , 10 , 11, 12, it can be seen that the UTV-LRTA method proposed by the present invention achieves competitive denoising results. FR-Net has good performance in removing footprints, but has poor suppression effect on random noise. DRR cannot suppress footprint noise, as shown in Figure 10(h) and (i). For SGRDL, both the randomness and the occupied space are removed; however, it is obvious that there are still quite large footprints in the noise reduction results shown in Figure 10 (f) and (g). In Figure 9 , 10 , 11, and 12, it can be observed that the visual quality of the UTV-LRTA restored images has been improved, successfully suppressing random noise and footprints and retaining the local details and structural information of the seismic images. Figure 5-12 , Time represents time, crossline represents the seismic interpretation profile of the connecting line, and inline represents the seismic interpretation profile of the main survey line.

[0149] Those of ordinary skill in the art will realize that the embodiments described herein are for helping the reader understand the principles of the present invention and should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. For those skilled in the art, various changes and modifications can be made to the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included within the scope of the claims of the present invention.

Claims

1. A three-dimensional tensor random plus footprint noise denoising method based on total variation low-rank approximation, characterized in that, Including: S1. The noise image tensor to be processed is denoted as The expression is: Among them, represents the noisy image tensor, represents the clear image tensor, represents the footprint, represents random noise; S2. Adopt the following denoising model to estimate the clear image tensor and simultaneously, and estimate the clear image tensor from where λ and τ represent regularization parameters, is a clear seismic data low-rank tensor model for norm minimization, represents footprint noise of the UTV model; S3. Introduce three auxiliary variables Convert the unconstrained minimization problem in into a constrained minimization problem, and then call SBI to solve the constrained minimization problem as follows: where λ1 and λ2 represent regularization parameters; S4. Convert into the subsequent unconstrained minimization: where α, β, γ represent Bregman penalty parameters, is a Bregman tensor; S5. Use the alternating minimization scheme to solve for the clear image tensor 2. The three-dimensional tensor random adding footprint noise denoising method based on total variation low-rank approximation according to claim 1, wherein The expression for: Among them, and represent the first-order derivative operators in the corresponding directions.

3. The 3D tensor random addition footprint noise denoising method based on total variation low-rank approximation according to claim 1, characterized in that Step S5 specifically includes: iteratively optimizing one variable while fixing other variables; decomposing into 4 sub-problems, and solving each sub-problem independently; these 4 sub-problems are respectively: updating related sub-problem; updating related sub-problem; updating related sub-problem; updating related sub-problem; updating related sub-problem; updating related sub-problem; updating related sub-problem; updating related sub-problem.

4. The 3D tensor random additive footprint noise denoising method based on total variation low-rank approximation according to claim 3, wherein Update and related sub - problems are By solving out each positive slice of will transform the sub-problem related to into a convex function of a real variable : obtain the calculation formula for solving the optimal solution: where 1 ≤ k ≤ n3, is to perform a regular matrix transpose on the two-dimensional input slice of the tensor, and l is the number of iterations; Use two-dimensional Fourier transform to find the closed-form solution of the sub-problem related to The solution is as follows: where where "*" represents complex conjugate and "o" represents component-wise multiplication, and represent the fast Fourier transform matrix and its inverse matrix, respectively.

5. The 3D tensor random plus footprint noise denoising method based on total variation low-rank approximation according to claim 3, characterized in that Update and the related sub-questions are Convert to the SVD of a matrix: Obtained using the soft shrinkage operator The closed-form solution is as follows: Among them, 6. The 3D tensor random adding footprint noise denoising method based on total variation low-rank approximation according to claim 3, characterized in that Update associated with The related sub-issues are as follows: use the shrinkage operator to solve, and the analytical solution is:

7. The 3D tensor random additive footprint noise denoising method based on total variation low-rank approximation according to claim 3, characterized in that Update associated with The sub-issues are use the shrinkage operator to solve, and the analytical solution is:

8. The 3D tensor random adding footprint noise denoising method based on total variation low-rank approximation according to claim 3, characterized in that It also includes updating the Bregman tensor according to the solution results of sub-problems and The update is as follows:

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