Seismic data simultaneous reconstruction and denoising method based on novel weighted fast convex set projection algorithm

Abstract

The invention discloses a seismic data simultaneous reconstruction and denoising method based on a novel weighted fast convex set projection algorithm. The method aims at solving the problem that a traditional fast convex set projection algorithm cannot perform seismic data reconstruction and denoising at the same time. Firstly, curvelet transformation is adopted as a sparse base, a weighting factor is introduced in the process of a traditional fast convex set projection algorithm, and simultaneous reconstruction and denoising of seismic data based on the weighted fast convex set projection algorithm are achieved. In the process, a hard threshold operator and a novel index threshold function are adopted, and the calculation speed is increased. And then, in order to effectively suppress the residual random noise in the updated solution of the weighted fast convex set projection algorithm, an optimization strategy is adopted, the sequence of an iteration hard threshold operator and a projection operator is exchanged, and the reconstruction and denoising effects are improved, so that the seismic data simultaneous reconstruction and denoising method based on the novel weighted fast convex set projection algorithm is finally realized. According to the method, the signal-to-noise ratio after seismic data reconstruction is improved, the calculation speed is increased, weak effective wave signals are effectively recovered, and the reflected wave event is more continuous and clearer.

Description

Technical Field

[0001] The invention relates to the field of irregular missing trace seismic data reconstruction methods, and specifically discloses a seismic data simultaneous reconstruction and denoising method based on a novel weighted fast convex set projection algorithm. Background Art

[0002] Seismic data reconstruction is a method of restoring, improving or interpolating seismic data through mathematical and computational techniques, especially for incomplete or poor quality data due to various reasons such as acquisition limitations, data corruption or noise interference. Seismic data is a key source of information for underground structure, oil and gas resource exploration and seismological research. However, due to the complexity of the acquisition environment, economic cost pressure and technical limitations, it is often impossible to obtain complete and ideal seismic data. Although traditional seismic data reconstruction methods can cope with the problem of missing or incomplete data to a certain extent, when noise interference is serious, these methods usually cannot avoid large errors, which will affect subsequent data analysis. Therefore, achieving accurate reconstruction and noise suppression of noisy irregular seismic data has undoubtedly become a crucial and urgent core issue. At present, many scholars at home and abroad have proposed many types of effective seismic data reconstruction methods, such as wave equation reconstruction method (Fomel, 2003), prediction filter reconstruction method (Liu et al., 2022), reduced rank reconstruction method (Cavalcante and Porsani, 2022), sparse transform reconstruction method (Zhang et al., 2017a) and artificial intelligence reconstruction method (Chen et al., 2023), etc. Among the data reconstruction methods based on sparse transform, the most widely used is the seismic data reconstruction method based on Fourier transform (Cui et al., 2003; Gao et al., 2010). At the same time, many scholars have also developed reconstruction methods based on other sparse transforms, such as Radon transform (Xue et al, 2014), wavelet transform (Liu et al., 2024), Shearlet transform (Yang et al., 2020), Dreamlet transform (Wang et al., 2015), Seislet transform (Liu et al, 2013) and Curvelet transform (Mansour et al, 2013), etc., which have also achieved good reconstruction effects.

[0003] For the reconstruction method based on sparse transformation, when different sparse transformations are selected, the reconstruction effect has a large difference. The present invention intends to adopt the curvelet transformation method. In the process of data reconstruction, many effective reconstruction algorithms have been proposed by predecessors, and the convex set projection algorithm is one of them. This algorithm is widely used because of its simple parameter setting and significant effect. Abma (2016) and others first applied the convex set projection algorithm to seismic data reconstruction. Its outstanding features are flexible application, simple algorithm and good reconstruction effect. Gao (2010) and others proposed an exponential decay threshold model, which greatly improved the reconstruction efficiency of the POCS algorithm based on Fourier transform. In the same year, Wang (2010) and others proposed a calculation scheme for using GPU to accelerate POCS to reconstruct seismic data. Zhang Hua (2013) and others improved the exponential threshold model and proposed a new exponential threshold model to further accelerate the convergence speed. Ge (2015) and others proposed an inverse proportional threshold model. Wang Benfeng (2016) and others proposed a method that can realize interpolation and denoising based on the convex set projection algorithm. Jin Chengmei (2022) et al. proposed to introduce the convex set projection algorithm into the curvelet domain, which further accelerated the iterative convergence speed. Dong Lieqian (2023) et al. proposed a fast convex set projection algorithm based on the curvelet transform on the basis of the fast threshold iteration algorithm, which accelerated the reconstruction speed. Although a few of the above methods can achieve simultaneous reconstruction and denoising, the denoising effect is poor, which affects the quality of data reconstruction.

[0004] In order to further improve the reconstruction and noise suppression effects of seismic data, the present invention adopts the curvelet transform as the sparse basis, proposes a new exponential threshold function, and introduces a weighting factor on the basis of the fast convex set projection algorithm to realize seismic data reconstruction and denoising based on the weighted fast convex set projection algorithm. On this basis, an optimization strategy is adopted to exchange the order of iterative hard threshold operator and projection operator to further improve the denoising effect, and finally realizes the simultaneous reconstruction and denoising method based on the improved weighted fast convex set projection algorithm. Summary of the invention

[0005] The purpose of the present invention is to be able to reconstruct irregularly missing noisy seismic data in the field with high precision, remove random noise, greatly improve the reconstruction accuracy and calculation speed, protect weak effective wave signals, and make the reflection wave event axis more continuous and clear. A method for simultaneous reconstruction and denoising of seismic data based on a new weighted fast convex set projection algorithm is proposed.

[0006] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0007] A method for simultaneous reconstruction and denoising of seismic data based on a novel weighted fast convex set projection algorithm comprises the following steps:

[0008] S1. Based on the fast convex set projection algorithm, the curvelet transform is used as the sparse basis, and the weighting factor is introduced to suppress some noise during the reconstruction process, thereby realizing seismic data reconstruction and denoising based on the weighted fast convex set projection algorithm;

[0009] S2. In this process, a hard threshold operator is introduced and a new exponential threshold parameter formula is used for reconstruction and denoising;

[0010] S3. According to the fact that the weighted fast convex set projection algorithm is essentially a two-step method combining the iterative hard threshold method and the projection operator, a method for simultaneous reconstruction and denoising of seismic data based on the novel weighted fast convex set projection algorithm is realized by exchanging the order of the iterative hard threshold operator and the projection operator, thereby further improving the data reconstruction and denoising effects.

[0011] Furthermore, the relationship between the irregular missing noisy data and the complete noise-free seismic data can be expressed by the following linear equation:

[0012] d obs =Rd0+n

[0013] where d obs ∈R n represents incomplete noisy seismic data; d0∈R N , and N ≥ n, indicating the complete seismic data to be reconstructed; R∈R n×N is the sampling matrix; n is random noise. The sparsity of seismic data in the curvelet domain is used to solve the above equations and establish the target functional as follows:

[0014]

[0015] C T is the inverse curvelet transform, x is the curvelet domain coefficient vector, P(x) is the sparse constraint, and μ is the regularization factor.

[0016] Furthermore, the fast iterative shrinkage threshold algorithm is as follows:

[0017] For the solution of the above functional equation, the fast convex set projection algorithm can be used to solve it, and its iterative formula is as follows:

[0018]

[0019]

[0020] d k+1 =d obs +(IR)C T x k

[0021] d k+1represents the seismic data after the k+1th iteration, x k represents the curvelet coefficient after the kth iteration, with an initial value of zero. Indicates that the hard threshold is λ k The threshold function of , t represents the update factor.

[0022] Furthermore, the weighted fast convex set projection algorithm is as follows:

[0023] d k+1 =ad obs +(I-aR)C T x k

[0024] Where a is a weighting factor, which ranges from (0,1) and is determined by the noise level.

[0025] Furthermore, the hard threshold operator is as follows:

[0026]

[0027] in, Can be determined by different threshold parameters.

[0028] Furthermore, the new index threshold parameter formula is as follows:

[0029]

[0030] Where Max is |Cd obs The maximum value of |, that is, the maximum absolute value of the curve coefficient

[0031] Further, the iterative hard thresholding method is as follows:

[0032]

[0033] Among them, x k is the kth iterative solution in the curvelet domain. k+1 Projection on the convex observation plane {d|d obs =Rd}, and the following iterative formula is obtained:

[0034]

[0035] It can be seen that the fast convex set projection algorithm is a two-step method: first, the iterative hard threshold method is used to update the solution in the curvelet domain, and then the solution is projected onto the convex observation plane to accelerate convergence. As the number of iterations increases, the solutions of the fast convex set projection algorithm and the iterative hard threshold method are consistent when they converge.

[0036] Further, according to the analysis, the weighted fast convex set projection algorithm is essentially a two-step method that combines the iterative hard threshold method and the projection operator. The iterative hard threshold can eliminate random noise by truncating the coefficients corresponding to random noise in the curvelet domain by the threshold operator. Therefore, in order to further improve the denoising effect, it is only necessary to exchange the order of the iterative hard threshold operator and the projection operator. The new weighted fast convex set projection algorithm is expressed as follows:

[0037] d k+1 =ad obs +(I-aR)C T x k

[0038]

[0039] d k =C T x k

[0040] This formula directly uses the momentum term x k Estimate the random noise and continue to converge the solution of the equation until the accuracy requirement is met.

[0041] The traditional fast convex set projection algorithm cannot simultaneously reconstruct and denoise seismic data. Firstly, the curvelet transform is used as the sparse basis, and the weighting factor is introduced in the process of the traditional fast convex set projection algorithm to achieve simultaneous reconstruction and denoising of seismic data based on the weighted fast convex set projection algorithm. In this process, the hard threshold operator and the new exponential threshold function are used to improve the calculation speed. Then, in order to effectively suppress the residual random noise in the updated solution of the weighted fast convex set projection algorithm, an optimization strategy is adopted to exchange the order of iterative hard threshold operator and projection operator to improve the reconstruction and denoising effect, thus finally realizing a method for simultaneous reconstruction and denoising of seismic data based on the new weighted fast convex set projection algorithm.

[0042] The beneficial effects of the present invention are as follows: 1. Compared with the wave equation reconstruction method, the present invention can directly reconstruct irregular missing seismic data without changing the amplitude and phase, has good fidelity, does not require prior information on underground structures, and has less computational workload;

[0043] 2. Compared with the Fourier transform reconstruction method, the technology of the present invention can process nonlinear event axis seismic data and seismic data with large missing intervals with high precision, and has high reconstruction accuracy;

[0044] 3. Compared with the three-dimensional seismic data reconstruction and denoising based on the weighted fast convex set projection algorithm, the technology of the present invention not only improves the signal-to-noise ratio of the reconstructed signal and speeds up the calculation, but also protects the weak effective wave signal and makes the reflection wave event axis more continuous and clear;

[0045] In summary, the present invention overcomes the shortcomings of conventional reconstruction methods that cannot quickly and accurately reconstruct irregularly missing noisy seismic data, as well as the problems of low reconstruction and denoising accuracy and limited application of conventional reconstruction methods. Moreover, the inventive method can not only reconstruct irregularly missing noisy seismic data, but also reconstruct regularly missing seismic data, greatly improving the calculation speed and signal-to-noise ratio of the reconstructed signal, protecting weak effective wave signals, and making the reflection wave phase axis more continuous and clear. This has important practical value in guiding seismic data acquisition in complex areas, reconstruction of missing traces, and other aspects. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] In order to more clearly illustrate the technical solutions in the embodiments of the present application, the following is a brief introduction to the drawings required for use in the description of the embodiments. Obviously, the drawings described below are some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0047] Figure 1 is a flowchart of reconstruction and denoising of irregular noisy seismic data in an embodiment of the present invention;

[0048] Figure 2 is a theoretical seismic data diagram of the present invention;

[0049] Figure 3 It is a graph of the reconstruction and denoising results of the present invention without noise-containing seismic data;

[0050] Figure 4 The noisy simulation data and noisy sampling data of the present invention are

[0051] Figure 5 is a reconstruction and denoising error graph of different iterative algorithms of the present invention;

[0052] Figure 6 It is a comparison of the reconstruction and denoising results of single-channel data using different iterative algorithms of the present invention;

[0053] Figure 7 This is a graph showing the relationship between different weighting factors and signal-to-noise ratios of the present invention.

[0054] Figure 8 is a curve diagram showing the signal-to-noise ratio changing with the number of iterations of the present invention;

[0055] Fig. 9 This is the result of the present invention based on the curvelet transform first reconstruction and then denoising DETAILED DESCRIPTION

[0056] The technical solution of the present invention will be described clearly and completely below in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0057] Reference Figure 1 A method for simultaneous reconstruction and denoising of seismic data based on a new weighted fast convex set projection algorithm. The steps for implementing this method mainly include establishing a seismic data reconstruction model, a conventional fast convex set projection algorithm, a weighted fast convex set projection algorithm, a hard threshold operator, a new exponential threshold parameter formula, a new weighted fast convex set projection algorithm, etc.

[0058] The specific steps are as follows:

[0059] Step 1: Establishment of data reconstruction model. The relationship between irregular missing noisy data and complete noise-free seismic data can be expressed by the following linear equation:

[0060] d obs =Rd0+n

[0061] where d obs ∈R n represents incomplete noisy seismic data; d0∈R N , and N ≥ n, indicating the complete seismic data to be reconstructed; R∈R n×N is the sampling matrix; n is random noise. The sparsity of seismic data in the curvelet domain is used to solve the above equations and establish the target functional as follows:

[0062]

[0063] C T is the inverse curvelet transform, x is the curvelet domain coefficient vector, P(x) is the sparse constraint, and μ is the regularization factor.

[0064] Step 2: Conventional fast convex set projection algorithm. For the solution of the above functional equation, the fast convex set projection algorithm can be used for solution, and its iterative formula is as follows:

[0065]

[0066] d k+1 =d obs +(IR)C T x k

[0067] d k+1 represents the seismic data after the k+1th iteration, x k represents the curvelet coefficient after the kth iteration, with an initial value of zero, Tλk Indicates that the hard threshold is λ k The threshold function of , t represents the update factor.

[0068] Step 3: Weighted fast convex set projection algorithm. In order to suppress noise during data reconstruction, a weighted operator is introduced into the conventional fast convex set projection algorithm, and its expression is as follows:

[0069] d k+1 =ad obs +(I-aR)C T x k

[0070] Among them, a is a weighting operator with a value range of (0,1), which is determined by the noise level.

[0071] Step 4: Introduce the hard threshold operator. In order to better reconstruct and denoise the irregular noisy seismic data, the hard threshold operator is introduced, and its expression is as follows:

[0072]

[0073] in, Can be determined by different threshold parameters.

[0074] Step 6: Establish a new exponential threshold parameter formula. In order to further improve the speed of the weighted fast convex set projection algorithm, a new exponential threshold parameter formula is established, and its expression is as follows:

[0075]

[0076] Where Max is |Cd obs |, that is, the maximum value of the absolute value of the curvelet coefficient.

[0077] Step 7: Establish a new weighted fast convex set projection algorithm. According to the analysis, the weighted fast convex set projection algorithm is essentially a two-step method that combines the iterative hard threshold method and the projection operator. The iterative hard threshold can eliminate random noise by truncating the coefficients corresponding to random noise in the curvelet domain by the threshold operator. Therefore, in order to further improve the denoising effect, it is only necessary to exchange the order of the iterative hard threshold operator and the projection operator. The expression of the new weighted fast convex set projection algorithm is as follows:

[0078] d k+1 =ad obs +(I-aR)C T x k

[0079]

[0080] d k =CT x k

[0081] This formula directly uses the momentum term x k Estimate the random noise and continue to converge the solution of the equation until the accuracy requirement is met.

[0082] The specific operations to implement this method are:

[0083] In order to measure the effect of the method proposed in the present invention, the signal-to-noise ratio formula SNR=20log 10 ||x0||2 / ||x-x0||2 is used for comparison, and the unit is dB, where x0 represents the original model data and x represents the reconstruction result. The higher the signal-to-noise ratio, the closer the reconstruction and denoising results are to the theoretical data, and the more ideal the processing effect.

[0084] In order to compare and analyze different algorithms and their parameters, the present invention sets the following theoretical simulation data: 256 two-dimensional theoretical seismic records synthesized by 40Hz Ricker wavelet, the record has a total of 4 layers of seismic reflection waves, each layer of reflection wave energy is different, the sampling interval is 1ms, the track spacing is 4m, and each track has 1024 sampling points. Figure 2 As shown in a, Figure 2 b is the missing data formed by 50% random sampling of the theoretical data. Next, we will verify the effectiveness, denoising, and efficiency of the novel weighted convex set projection algorithm proposed in the present invention based on the simulated data.

[0085] (1) Effectiveness

[0086] In order to verify the effectiveness of the method of the present invention, the novel weighted fast convex set projection algorithm and the weighted fast convex set projection algorithm are used to reconstruct the data of 50% irregular missing seismic traces. Figure 2 In order to achieve better results, this reconstruction and denoising will use a new exponential threshold parameter and a hard threshold function model, the number of iterations is set to 25 times, the weighting factors of the two methods are set to 1, and no denoising is performed at this time. Figure 3 a and Figure 3 c are the reconstruction results of the traditional method and the method of the present invention, respectively. Figure 3 b and Figure 3 d is the corresponding residual.

[0087] Depend on Figure 3 a and Figure 3 c It can be seen that the reconstruction and denoising results of the method of the present invention are basically consistent with those of the traditional method. Figure 2Compared with the theoretical data of a, except for some edge effects, the effective wave phase axis after reconstruction is basically consistent with the theoretical data, the residuals of the two algorithms are small, and the final signal-to-noise ratio of the two algorithms is 18.6dB, which shows that the weighted fast convex set projection algorithm and the new weighted fast convex set projection algorithm have the same effect on the reconstruction of noise-free data.

[0088] (2) Noise Removal

[0089] In order to compare the advantages and disadvantages of the fast convex set projection algorithm, the weighted fast convex set projection algorithm and the method of the present invention in terms of noise resistance, Gaussian white noise with a standard deviation of 0.01 is added to the theoretical data, such as Figure 4 As shown in a, 50% random sampling is performed on it, such as Figure 4 As shown in b, the above three methods are then used to simultaneously reconstruct and denoise the noisy sampling data. Here, the maximum amplitude ratio of random noise to effective signal is 0.47, and the signal-to-noise ratio is 2.99 dB.

[0090] The fast convex set projection algorithm, the weighted fast convex set projection algorithm and the method of the present invention are used to Figure 4 The sampled data shown in b is reconstructed and denoised simultaneously, where the weighting factor is 0.35 and the number of iterations is 25. The processing results are shown in Figure 5 As shown. Among them, Figure 5 a and Figure 5 b is the reconstruction result and residual profile of the fast convex set projection algorithm. It can be seen that this method can effectively reconstruct noisy seismic data, but its recovery effect on seismic trace data with many missing traces is poor, and it cannot effectively suppress random noise; Figure 5 c and Figure 5 d is the reconstruction result and residual profile of the weighted fast convex set projection algorithm. It can be seen that this method can effectively restore the missing seismic trace information and has a certain effect on the removal of random noise. However, due to the digital space domain, part of the noise will be brought into the updated solution, resulting in unsatisfactory denoising effect. Figure 5 e and Figure 5 f is the result of simultaneous reconstruction and denoising and the residual profile of the method of the present invention. Compared with the above two methods, the method of the present invention not only achieves better noise removal results, but also has better reconstruction effects on edge areas due to the exchange of the iteration order of the hard threshold operator and the projection operator. Its reconstruction and denoising results are more consistent with the theoretical data and have the smallest error. Finally, the signal-to-noise ratio of the three methods is analyzed, and it can be seen that the signal-to-noise ratio of the fast convex set projection algorithm after simultaneous reconstruction and denoising is 4.13dB, the weighted fast convex set projection algorithm is 12.07dB, and the algorithm of the present invention is 15.25dB.

[0091] In order to compare the effects of simultaneous reconstruction and denoising of the three algorithms in more detail, the 99th and 188th seismic records before and after reconstruction were extracted for comparison, with a time window of 0 to 1 second. Figure 6 As shown. Among them, the 1st to 7th seismic traces are the original seismic data, the reconstruction results of the fast convex set projection algorithm, the weighted fast convex set projection algorithm and the method of the present invention and their residuals. The smaller the residual, the better the reconstruction and denoising effect. As can be seen from the figure, the fast convex set projection algorithm cannot effectively suppress the noise during the reconstruction process, so the corresponding residual is the largest; the residual corresponding to the weighted fast convex set projection algorithm is smaller, and the noise is not completely suppressed, indicating that its simultaneous reconstruction and denoising effect on irregular noisy seismic data is inferior to the new weighted fast convex set projection algorithm; the reconstruction and denoising results of the method of the present invention are most consistent with the original seismic data, and the error is the smallest, indicating that its effect is the best.

[0092] In order to further understand the relationship between different weighting factors and the signal-to-noise ratio after simultaneous reconstruction and denoising, the weighting factor was changed from 0.05 to 1 in increments of 0.05, and then the traditional method and the method of the present invention were used to compare the weighting factors of the image. Figure 3 b performs simultaneous reconstruction and noise removal. The processed results are as follows Figure 7 As shown, it can be seen that as the weighting factor increases, the signal-to-noise ratio of the two methods first increases and then decreases. When the weighting factor increases to 1, both methods degenerate into traditional methods and cannot effectively suppress noise. It is particularly important to note that for each weighting factor, the signal-to-noise ratio of the method of the present invention is always greater than that of the traditional method, and the two methods reach their maximum at about 0.35 and 0.25, respectively. In order to highlight the effect of the method of the present invention, the weighting factor of the present invention is selected as 0.35 when processing irregular noisy theoretical data.

[0093] (3) Efficiency

[0094] Based on the obvious advantages of the method of the present invention in denoising performance, in order to further verify the high efficiency of the method of the present invention, it is compared with the traditional weighted fast convex set projection algorithm, and different numbers of iterations are used to test its simultaneous reconstruction and denoising effects under different noise environments. We add Gaussian white noise with standard deviations of 0.01 and 0.05 to the theoretical data, and perform 50% irregular data sampling. Then, the above two methods are used to reconstruct and denoise the sampled data. The corresponding weighting factors in the two noise environments are 0.35 and 0.1, respectively. The signal-to-noise ratio under different numbers of iterations is calculated, and a curve chart of the signal-to-noise ratio changing with the number of iterations is drawn, where Figure 8 a and Figure 8b are curves showing the signal-to-noise ratio of two theoretically simulated noise data after simultaneous reconstruction and denoising as a function of the number of iterations. It can be seen that the signal-to-noise ratio of both the conventional method and the method of the present invention after simultaneous reconstruction and denoising under different noise environments increases with the increase in the number of iterations and eventually tends to a certain value. However, under the same number of iterations, Figure 8 The signal-to-noise ratio of a is generally greater than Figure 8 b, This shows that in a noisy environment, the reconstruction quality and denoising effect will be reduced. Figure 8 It can be seen that when the theoretical data contains noise with a standard deviation of 0.01, the signal-to-noise ratio of both the traditional method and the method of the present invention reaches the maximum value at 25 times, but the signal-to-noise ratio of the traditional method after reconstruction and denoising is 12dB, while that of the method of the present invention is 15.2dB. It can be seen that the method of the present invention has obvious advantages in the accuracy of simultaneous reconstruction and denoising. Overall, under different noise interference environments, when the same number of iterations is used, the reconstruction accuracy of the method of the present invention is always better than that of the traditional method; when the same signal-to-noise ratio is obtained, the number of iterations of the method of the present invention is less than that of the weighted fast convex set projection algorithm, which proves the high efficiency of the method of the present invention. Since the signal-to-noise ratio after reconstruction and denoising has reached the maximum value at 25 iterations of the method of the present invention, the number of iterations of different algorithms is set to 25 times in the processing of theoretical data.

[0095] In order to further illustrate the efficiency and denoising performance of the proposed method, we used the curvelet transform method to Figure 4 b. Perform the traditional process of reconstruction followed by denoising, such as Fig. 9 As shown, Fig. 9 a is the reconstruction result of the fast convex set projection algorithm based on curvelet transform, Fig. 9 b is Fig. 9 The residual of a, Fig. 9 c is for Fig. 9 a The result of denoising using the threshold iteration method based on curvelet transform. Fig. 9 d is Fig. 9 The residual of c. Fig. 9 aIt can be seen that when Figure 4 When only the irregular noisy data of b is reconstructed, the seismic data can be restored to a certain extent, but the noise cannot be effectively suppressed, resulting in poor continuity of the event axis and a small signal-to-noise ratio. Fig. 9 c shows that when Fig. 9 After denoising, the random noise is significantly suppressed, but there is an edge effect, the continuity of the effective signal event axis is poor, and from the error results ( Fig. 9 d), it can be found that in the process of denoising, the effective signal is also reduced, and the final signal-to-noise ratio of reconstruction followed by noise suppression is 11.36 dB, which is about 4 dB lower than that of the method of the present invention.

[0096] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein by equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for simultaneous reconstruction and denoising of seismic data based on a novel weighted fast convex set projection algorithm, characterized in that: The steps include: S1. Based on the fast convex set projection algorithm, the curvelet transform is used as the sparse basis, and the weighting factor is introduced to suppress some noise during the reconstruction process, thereby realizing seismic data reconstruction and denoising based on the weighted fast convex set projection algorithm; S2. In this process, a hard threshold operator is introduced and a new exponential threshold parameter formula is used for reconstruction and denoising; S3. According to the fact that the weighted fast convex set projection algorithm is essentially a two-step method combining the iterative hard threshold method and the projection operator, a method for simultaneous reconstruction and denoising of seismic data based on the novel weighted fast convex set projection algorithm is realized by exchanging the order of the iterative hard threshold operator and the projection operator, thereby further improving the data reconstruction and denoising effects.

2. The method for simultaneous reconstruction and denoising of seismic data based on a novel weighted fast convex set projection algorithm according to claim 1, characterized in that: The relationship between the irregular missing noisy data and the complete noise-free seismic data in the field can be expressed by the following linear equation: d obs =Rd0+n where d obs ∈R n represents incomplete noisy seismic data; d0∈R N , and N ≥ n, indicating the complete seismic data to be reconstructed; R∈R n×N is the sampling matrix; n is random noise. The sparsity of seismic data in the curvelet domain is used to solve the above equations and establish the target functional as follows: C T is the inverse curvelet transform, x is the curvelet domain coefficient vector, P(x) is the sparse constraint, and μ is the regularization factor.

3. The method for simultaneous reconstruction and denoising of seismic data based on a novel weighted fast convex set projection algorithm according to claim 1, characterized in that: The fast convex set projection algorithm is as follows: For the solution of the above functional equation, the fast convex set projection algorithm can be used to solve it, and its iterative formula is as follows: d k+1 =d obs +(I-R)C T x k d k+1 represents the seismic data after the k+1th iteration, x k represents the curvelet coefficient after the kth iteration, with an initial value of zero, T λk Indicates that the hard threshold is λ k The threshold function of , t represents the update factor.

4. The method for simultaneous reconstruction and denoising of seismic data based on a novel weighted fast convex set projection algorithm according to claim 1, characterized in that: The weighted fast convex set projection algorithm is as follows: d k+1 =ad obs +(I-aR)C T x k Where a is a weighting factor, which ranges from (0,1) and is determined by the noise level.

5. The method for simultaneous reconstruction and denoising of seismic data based on a novel weighted fast convex set projection algorithm according to claim 1, characterized in that: The hard threshold operator is as follows: in, Can be determined by different threshold parameters.

6. The method for simultaneous reconstruction and denoising of seismic data based on a novel weighted fast convex set projection algorithm according to claim 1, characterized in that: The new index threshold parameter formula is as follows: Where Max is |Cd obs |, that is, the maximum value of the absolute value of the curvelet coefficient.

7. The method for simultaneous reconstruction and denoising of seismic data based on a novel weighted fast convex set projection algorithm according to claim 1, characterized in that: The iterative hard thresholding method is as follows: k=1,2,3,···,N Among them, x k is the kth iterative solution in the curvelet domain. k+1 Projection on the convex observation plane {d|d obs =Rd}, and the following iterative formula is obtained: It can be seen that the fast convex set projection algorithm is a two-step method: first, the iterative hard threshold method is used to update the solution in the curvelet domain, and then the solution is projected onto the convex observation plane to accelerate convergence. As the number of iterations increases, the solutions of the fast convex set projection algorithm and the iterative hard threshold method are consistent when they converge.

8. The method for simultaneous reconstruction and denoising of seismic data based on a novel weighted fast convex set projection algorithm according to claim 1, characterized in that: According to the analysis, the weighted fast convex set projection algorithm is essentially a two-step method that combines the iterative hard threshold method and the projection operator. The iterative hard threshold can eliminate random noise by truncating the coefficients corresponding to random noise in the curvelet domain by the threshold operator. Therefore, in order to further improve the denoising effect, it is only necessary to exchange the order of the iterative hard threshold operator and the projection operator. The new weighted fast convex set projection algorithm is expressed as follows: d k+1 =ad obs +(I-aR)C T x k This formula directly uses the momentum term x k Estimate the random noise and continue to converge the solution of the equation until the accuracy requirement is met.

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