Low-rank five-dimensional seismic data reconstruction method and system based on parallel matrix decomposition

Abstract

The present invention discloses a low-rank five-dimensional seismic data reconstruction method and system based on parallel matrix decomposition. Given the known data tensor #imgabs0#, the sampling tensor #imgabs1#, the weight parameter μ of the optimization term and the regularization term, and the ranks R1, …, R of the matrices decomposed by the TT method along each direction N . Initialize the parameter Y 0 , #imgabs2# as the initial value of the algorithm; expand the low-rank reconstruction data tensor #imgabs3# of the current step, and for the matrix #imgabs4# in each direction, use randomized sampling and block coordinate descent method to calculate the decomposition matrix #imgabs6# of #imgabs5#, and obtain the next #imgabs9# through tensor reconstruction of #imgabs7# and #imgabs8# of the training samples. Repeat the above steps until the predetermined stopping criterion is met, and continuously iterate to obtain the final low-rank tensor. The use of algorithms such as sketch random sampling further speeds up the operation speed; the present invention has a good reconstruction effect, reduces the consumption of computing resources, and has broad industrial application prospects.

Description

Technical Field

[0001] The present invention belongs to the technical field of signal processing, and particularly relates to a low-rank five-dimensional seismic data reconstruction method and system based on parallel matrix decomposition. Background Art

[0002] Due to certain obstacles during the seismic data acquisition process, such as irregular surface conditions and equipment limitations, seismic data is often sampled irregularly along spatial coordinates. Missing traces in seismic records can have a negative impact on subsequent processing, imaging, and processing, imaging, and mutual masking involving multi-channel deconvolution, velocity analysis, full waveform inversion, synchronous source separation, and fault detection. Therefore, seismic data reconstruction is an important and persistent problem that has attracted extensive attention in the academic and industrial communities.

[0003] Regarding the problem of seismic data reconstruction, it is usually assumed that the complete data is low-rank, and the lack of traces and the presence of noise will lead to an increase in rank.

[0004] In recent years, the tensor decomposition theory based on tensor networks has received increasing attention and shown extraordinary prospects in processing high-order tensors, especially tensors of order three and above. Among them, the Tensor Train (TT) decomposition is the most representative, with high data compression ability and computational efficiency, and has been deeply studied in many tasks. Usually, the Singular Value Decomposition (SVD), the Alternating Least Squares (ALS) method, and the Gradient Descent (GD) algorithm are used to calculate the latent core tensors of the TT decomposition to utilize its low-rank structure. However, despite the existence of many optimization algorithms, due to the high computational cost, explicit TT decomposition is still difficult to solve for large-scale applications. The present invention implicitly utilizes the low-rank structure by minimizing the TT rank, and performs parallel low-rank matrix decomposition on different tensor unfoldings, greatly reducing the computational cost compared to traditional methods.

[0005] Prior Art

[0006] Data-driven techniques, including dictionary learning methods involving K-Singular Value Decomposition (K-SVD), and data-driven tight frame methods, can directly learn features from input data in a data-driven manner, and have strong adaptability.

[0007] Disadvantages of the Prior Art

[0008] 1. In the case of high dimensions, the computational complexity of data-driven techniques is high, and the parameter selection is too sensitive to the training results, making it unable to stably complete computational tasks.

[0009] 2. The processing capabilities of existing computers for data-driven methods are limited to three-dimensional data and are insufficient to utilize the comprehensive high-dimensional relationships inherent in five-dimensional seismic data. Summary of the Invention

[0010] The technical problem to be solved by the present invention is to provide a low-rank five-dimensional seismic data reconstruction method and system based on parallel matrix decomposition for solving the technical problem of reconstructing seismic data with missing traces in view of the above-mentioned deficiencies in the prior art.

[0011] The present invention adopts the following technical solutions:

[0012] The low-rank five-dimensional seismic data reconstruction method based on parallel matrix decomposition includes the following steps:

[0013] S1. Input the observed data tensor Sampling tensor The weight parameter μ of the optimization term and the regularization term, and the ranks R of the matrices obtained by decomposing the observed data tensor along each direction according to the TT method 1 , …, R N , initialize the parameters to obtain the initial decomposition matrices in each direction and the initial low-rank reconstruction data tensor Initialize the iteration variable k = 0 as the initial iteration step;

[0014] S2. For the k-th iteration process, due to the low-rank constraint of the real data , expand the low-rank reconstruction data tensor obtained in step S1 Use sketch random sampling and block coordinate descent method to calculate the matrix in each direction Use and the corresponding decomposition matrix Calculate the decomposition matrix of the current iteration step

[0015] S3. Use the decomposition matrix obtained in step S2 to solve the sub-optimization problem of the low-rank reconstruction data tensor . Let k = k + 1, enter the next iteration step, repeat steps S2 and S3, and use the obtained by iteration as the low-rank reconstruction result of the real data to realize the low-rank five-dimensional seismic data reconstruction.

[0016] Specifically, in step S2, due to the low-rank constraint of the real data , perform low-rank decomposition on the matrix expansion of the real data , and introduce the observed data tensor as the constraint term of the real data , and minimize to solve the approximation of the real data as follows:

[0017]

[0018] Among them, Z [n] is the TT decomposition result of the real data in the n-th direction, X n and Y n are the low-rank decomposition results of Z [n] . F represents the Frobenius norm operation, and N is the dimension of the tensor.

[0019] Furthermore, the low-rank reconstruction of the real data is solved through the observed data tensor as follows:

[0020]

[0021] Even further, the constraint of the low-rank reconstruction is as follows:

[0022]

[0023] Furthermore, in step S2, the iterative method is used to alternately optimize the low-rank decomposition matrices X n , Y n of each TT decomposition matrix, and the optimization results of X n , Y n are as follows:

[0024]

[0025]

[0026] Among them, is the TT decomposition matrix of the real data in the n-th direction.

[0027] Even further, the sketch algorithm is adopted to accelerate the calculation of the low-rank decomposition matrix X n , and the improved optimization sub-problem is as follows:

[0028]

[0029] Among them, and are the results obtained by sampling Y n and using the sketch algorithm.

[0030] Specifically, in step S2, the decomposition matrices are respectively:

[0031]

[0032] ​

[0033] Among them, is the TT decomposition result of the real data in the nth direction, is the left matrix of the low-rank decomposition result of Z [n]

[0034] Specifically, in step S3, the low-rank reconstructed data tensor of the real data is calculated iteratively as follows: as follows:

[0035]

[0036] Among them, is the all-ones tensor, α is the proportionality coefficient between the reconstructed tensor and the observed data tensor is the decomposition matrix in each direction and is the reconstructed tensor.

[0037] Furthermore, in step S3, the sub-optimization problem of the low-rank reconstructed data tensor is:

[0038]

[0039] Among them, N is the dimension of the tensor, and are the decomposition matrices, Z [n] is the TT decomposition result of the real data in the nth direction, and F represents the Frobenius norm operation.

[0040] On the second aspect, an embodiment of the present invention provides a low-rank five-dimensional seismic data reconstruction system based on parallel matrix decomposition, including:

[0041] A data module that inputs the observed data tensor the sampling tensor the weight parameter μ of the optimization term and the regularization term, and the ranks R of the matrices obtained by decomposing the observed data tensor 1 , …, R N in each direction according to the TT method, and initializes the parameters to obtain the initial decomposition matrices and the initial low-rank reconstructed data tensor Initializes the iteration variable k = 0 as the initial iteration step;

[0042] A calculation module that, for the kth iteration process, expands the low-rank reconstructed data tensor obtained by the data module according to the low-rank constraint of the real data ​​Using sketch random sampling and block coordinate descent method, calculate the matrix in each direction Use and the corresponding decomposition matrix Calculate the decomposition matrix of the current iteration step

[0043]

[0044] Reconstruction module, use the decomposition matrix obtained by the calculation module to solve the low-rank reconstruction data tensor of the sub-optimization problem Let k = k + 1, enter the next iteration step, repeat the execution, and use the obtained by iteration as the real data low-rank reconstruction result to achieve low-rank five-dimensional seismic data reconstruction.

[0045] Compared with the prior art, the present invention has at least the following beneficial effects:

[0046] The low-rank five-dimensional seismic data reconstruction method based on parallel matrix decomposition of the present invention first initializes the reconstruction tensor using the observed seismic data tensor, and utilizes the characteristic that the reconstructed seismic data has a lower rank to decompose the seismic tensor into a series of matrices using the TT method. Compared with the traditional decomposition method, it fully utilizes the high-dimensional information of the observed tensor; these series of matrices are low-rank decomposed using the sketch method, and the seismic data tensor is reconstructed using the low-rank decomposition result. Compared with directly performing matrix low-rank decomposition, using the sketch method can significantly accelerate the operation speed; iteratively solve until meeting the predetermined stopping criterion to obtain a good reconstruction result on the missing observed seismic data.

[0047] Furthermore, perform low-rank decomposition on the matrix expansion of the real data. Use the Tensor Train TT rank minimization PMF-TT method to process the data. In order to recover the missing values in the real data, it is necessary to make the rank of the real data as small as possible, and at the same time require the sampling result of the recovered data to be as close as possible to the observed data tensor.

[0048] Furthermore, through the observed data tensor solve the low-rank reconstruction of the real data By introducing the sampling operator transform the sampling process of the real data into the Hadamard product operation of tensors, which facilitates the formulation of the optimization problem and the formal representation of the calculation process.

[0049] Furthermore, in order to recover the missing values in the real data , it is necessary to make the rank as small as possible. The optimization problem representation of low-rank reconstruction, the designed optimization problem will have the constraint condition Convert it into an optimized regular term for convenient further solution.

[0050] Furthermore, optimize X n , Y n respectively to obtain the optimization results of the real data . By setting the optimization problems of the decomposition matrices X n , Y n , the optimized X n , Y n are made to be as close as possible to the corresponding components decomposed from the real data , and the reconstructed tensor results are also as close as possible to the real data.

[0051] Furthermore, the sketch algorithm can effectively reduce the running time of the program.

[0052] Furthermore, by calculating the decomposition matrices at each iteration step , the low-rank reconstruction result tensor at this iteration step can be calculated, and the low-rank real data tensor reconstruction result can be updated

[0053] Furthermore, using the and obtained from the current step iteration to solve the real data to construct an optimization sub-problem. By solving this optimization sub-problem, the optimal estimate of the real data can be obtained.

[0054] Furthermore, by designing the iteration steps for , as the number of iteration steps increases, the reconstruction result of is also closer to the real data until it meets the predetermined stopping criterion.

[0055] It can be understood that the beneficial effects of the second aspect above can be referred to the relevant descriptions in the first aspect above, and will not be elaborated here.

[0056] In summary, the present invention adopts an equilibrium tensor matrix method, has excellent reconstruction performance, and significantly reduces the calculation cost.

[0057] Next, through the drawings and embodiments, the technical solutions of the present invention will be further described in detail. Brief Description of the Drawings

[0058] Figure 1 is a schematic diagram of PMF-TT decomposition;

[0059] Figure 2 is a schematic diagram of sketch sampling;

[0060] Figure 3 It is a seismic data graph. Among them, (a) is synthetic seismic data, and (b) is 90% loss seismic data;

[0061] Figure 4 It is the reconstruction result of 90% loss seismic data. Among them, (a) is the reconstruction result of 90% loss, noiseless seismic data using the PMF method, (b) is the reconstruction result of 90% loss, noiseless seismic data using the PMF-sketch method, (c) is the reconstruction result of 90% loss, noiseless seismic data using the PMF-TT method, (d) is the reconstruction result of 90% loss, noiseless seismic data using the PMF-TT-sketch method, and (e), (f), (g), (h) are the reconstruction errors of (a), (b), (c), (d) and the real data respectively;

[0062] Figure 5 It is noisy seismic data. Among them, (a) is synthetic seismic data, and (b) is 90% loss noisy seismic data;

[0063] Figure 6 It is the reconstruction result of 90% loss noisy seismic data. Among them, (a) is the reconstruction result of 90% loss, 1dB noise-containing seismic data using the PMF method, (b) is the reconstruction result of 90% loss, 1dB noise-containing seismic data using the PMF-TT method, (c) is the reconstruction result of 90% loss, 1dB noise-containing seismic data using the PMF-TT method, (d) is the reconstruction result of 90% loss, 1dB noise-containing seismic data using the PMF-TT-sketch method, and (e), (f), (g), (h) are the reconstruction errors of (a), (b), (c), (d) and the real data respectively;

[0064] Figure 7 It is the flow chart of the present invention. Detailed implementation manners

[0065] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0066] In the description of the present invention, it should be understood that the terms "include" and "comprise" indicate the presence of the described features, wholes, steps, operations, elements and / or components, but do not exclude the presence or addition of one or more other features, wholes, steps, operations, elements, components and / or their combinations.

[0067] It should also be understood that the terms used in the specification of the present invention are merely for the purpose of describing specific embodiments and are not intended to limit the present invention. As used in the specification of the present invention and the appended claims, unless the context clearly indicates otherwise, the singular forms "a", "an" and "the" are intended to include the plural forms.

[0068] It should be further understood that the term "and / or" used in the specification of the present invention and the appended claims refers to any combination and all possible combinations of one or more of the associated listed items, and includes these combinations. For example, A and / or B can represent three situations: A exists alone, A and B exist simultaneously, and B exists alone. In addition, the character " / " in this article generally indicates that the contextually related objects have an "or" relationship.

[0069] It should be understood that although terms such as first, second, third, etc. may be used in the embodiments of the present invention to describe preset ranges, etc., these preset ranges should not be limited to these terms. These terms are only used to distinguish the preset ranges from each other. For example, without departing from the scope of the embodiments of the present invention, the first preset range may also be referred to as the second preset range, and similarly, the second preset range may also be referred to as the first preset range.

[0070] Depending on the context, the word "if" as used herein can be interpreted as "when" or "while" or "in response to determining" or "in response to detecting". Similarly, depending on the context, the phrase "if determined" or "if detected (stated condition or event)" can be interpreted as "when determined" or "in response to determining" or "when detecting (stated condition or event)" or "in response to detecting (stated condition or event)".

[0071] Schematic diagrams of various structures according to the disclosed embodiments of the present invention are shown in the drawings. These figures are not drawn to scale, where some details are enlarged for the purpose of clear expression, and some details may be omitted. The shapes of various regions and layers shown in the figures, as well as their relative sizes and positional relationships, are merely exemplary. In practice, there may be deviations due to manufacturing tolerances or technical limitations, and those skilled in the art can design regions / layers with different shapes, sizes, and relative positions according to actual needs.

[0072] The present invention provides a low-rank five-dimensional seismic data reconstruction method based on parallel matrix decomposition, and constructs a complete method for calculating the reconstructed complete signal from the observed signal; through iterative solution, the result of the previous step is subjected to tensor decomposition, and parallel matrix decomposition is performed to obtain a low-rank matrix, and the tensor is reconstructed using the low-rank matrix, and continuous iteration is performed to obtain the final low-rank tensor. In addition, the sketch random sampling method is used to further accelerate the operation speed, and the method of balanced tensor matrixization is adopted, which has excellent reconstruction performance and significantly reduces the calculation cost.

[0073] Please refer to Figure 7 , a low-rank five-dimensional seismic data reconstruction method based on parallel matrix decomposition according to the present invention, comprising the following steps:

[0074] S1. Given the known data tensor Sampling tensor The weight parameter μ of the optimization term and the regularization term, and the rank R of the matrix decomposed by the TT method along each direction 1 , …, R N ; Initialize the parameter Y 0 , as the initial value of the algorithm;

[0075] The data processed by the Tensor Train TT rank minimization PMF-TT method is high-dimensional tensor data. For 5D seismic data, if the frequency dimension is not considered, a series of remaining 4D tensors are processed. Assume that the complete true data is The observed data tensor is The dimension of the tensor is N, and it is considered that the observed data tensor is the sampling result of the true data , expressed as:

[0076]

[0077] where represents the Hadamard product, element-wise product, is the sampling operator, taking 1 at the observed position and 0 at other positions.

[0078] By solving The solution problem of low-rank reconstruction is expressed as:

[0079]

[0080] In order to recover the missing values in, it is necessary to make the rank of as small as possible. The constraint of the above formula is written as:

[0081]

[0082] Among them, is a generalization representation of the low-rank constraint.

[0083] An effective solution to the above formula is to perform low-rank decomposition on the matrix expansion of to transform the above formula into:

[0084]

[0085] Among them, is the TT decomposition in the n direction, R n is a pre-set rank of X n and Y n .

[0086] S2. Expand the low-rank reconstruction data tensor of the current step For the matrix in each direction Use randomized sampling and block coordinate descent method to calculate the decomposition matrix of

[0087] By optimizing X n , Y n respectively, solve the sub-problem to obtain the optimization result of:

[0088]

[0089]

[0090] For each X n and Y n n = 1,..., N - 1, solve iteratively through the following formula:

[0091]

[0092]

[0093] Adopt the sketch random sampling algorithm to greatly reduce the operation time. For If S n ≤ T n , at this time, solving X n is an overdetermined problem, obtained by sampling the columns of Z [n] , and vice versa. The sampling size is selected as SS n = 10R n log 10 (Rn), for Y n and Z [n]The corresponding columns of SS are randomly selected n vectors, forming and Construct the optimization subproblem for X:

[0094]

[0095] Get X n and Y n The recursive formula is:

[0096]

[0097]

[0098]

[0099] S3, the decomposition matrix obtained in step S2 and After tensor reconstruction, the next step is obtained Steps S2 and S3 are repeated until a predetermined stopping criterion is met.

[0100] Solve using the X and Y obtained from the current step iteration First write the sub-optimization problem:

[0101]

[0102] Get real data Iteration for:

[0103]

[0104] in,

[0105] In each iteration process, the termination condition of the iteration is judged, and the iteration is stopped when the termination condition is met. That is, it is regarded as an observation data tensor The low-rank reconstruction result.

[0106] In another embodiment of the present invention, a low-rank five-dimensional seismic data reconstruction system based on parallel matrix decomposition is provided. The system can be used to implement the above-mentioned low-rank five-dimensional seismic data reconstruction method based on parallel matrix decomposition. Specifically, the low-rank five-dimensional seismic data reconstruction system based on parallel matrix decomposition includes a data module, a computing module and a reconstruction module.

[0107] Among them, the data module inputs the observation data tensor Sampling Tensor The weight parameters μ of the optimization term and the regularization term, as well as the observation data tensor The rank R of the matrix obtained by decomposing in each direction according to the TT method 1 , …, R N Initialize the parameters to obtain the initial decomposition matrix in each direction and the initial low-rank reconstructed data tensor Initialize the iteration variable k = 0 as the initial iteration step;

[0108] Calculation module. For the k-th iteration process, from the real data With the low-rank constraint, the low-rank reconstructed data tensor obtained by expanding the data module Using sketch random sampling and block coordinate descent method, calculate the matrix in each direction Use and the corresponding decomposition matrix Calculate the decomposition matrix of the current iteration step

[0109]

[0110] Reconstruction module. Use the decomposition matrix obtained by the calculation module For the solution of the low-rank reconstructed data tensor of the sub-optimization problem Let k = k + 1, enter the next iteration step, repeat the execution, and use the obtained by iteration as the low-rank reconstruction result of the real data to realize the low-rank five-dimensional seismic data reconstruction.

[0111] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are some, but not all, of the embodiments of the present invention. Usually, the components described and shown in the accompanying drawings here can be arranged and designed in various different configurations. Therefore, the following detailed description of the embodiments of the present invention provided in the drawings is not intended to limit the scope of the claimed invention, but merely represents selected embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts fall within the scope of protection of the present invention.

[0112] Please refer to Figure 1 , which is the process of tensor PMF-TT decomposition; for the tensor Decompose it into a series of matrices X [n] , Similar to using the tensor Tucker rank for tensor reconstruction, it can also be extended to the TT decomposition of tensors, and use the rank of the TT decomposition matrix for tensor reconstruction. The form of the minimum TT decomposition of a tensor:

[0113]

[0114] Among them, is a series of 3D tensors with kernel sizes of 1×I 1 ×R 1 , 1×I 2 ×R 2 , …, 1×I N ×R N , and 《·》 is the multilinear product of the tensor kernels.

[0115] Denote {R 1 , R 2 , …, R N-1} as rank TT (X), which is called the TT-rank of tensor X; decompose tensor X into the following form:

[0116]

[0117] Among them, is 's i n -th slice.

[0118] The TT-rank is obtained by minimizing the TT decomposition. However, since directly calculating the TT decomposition requires a large amount of computation, it is necessary to use an efficient method to calculate the TT-rank.

[0119] The TT decomposition of X and the TT kernel have the following relationship:

[0120]

[0121] Among them, is the partial product.

[0122] The decomposition result and the TT-rank have the following relationship:

[0123]

[0124] X [n] is the lower bound of R n . When Rank(X [n] ) = R n , it is the minimum TT decomposition. Use Rank(X [n] ) to represent the TT-rank and solve the TT-rank minimization problem.

[0125] Please refer to Figure 2 , which is the sketch sampling schematic diagram. In the iterative calculation of the low-rank tensor In the process, it is necessary to calculate the low-rank decomposition of each mode matrix. Research shows that the randomized method has a significant effect on accelerating this process. The sketch sampling method we proposed accelerates the calculation by sampling the rows or columns of the given matrix Z n For When S n ≤T n it is an overdetermined problem, so sample the columns of Z n and vice versa. Better results may be achieved by adjusting the sampling distribution, but for the sake of accelerating the calculation, the present invention adopts a uniform distribution.

[0126] Please refer to Figure 3 , the size of the synthesized five-dimensional seismic data is 256×20×20×10×10, where there are 20 traces in the common midpoint dimension, 10 traces in the offset dimension, and each trace has 256 values in time. Randomly sample the data and remove 10%, 30%, 50%, 70%, and 90% of the traces respectively to form a series of incomplete data. Reconstruct the data in the range of 1 - 70 Hz, and recompose it into a five-dimensional tensor, and transform it into the time - space domain.

[0127] To quantify the reconstruction quality of each method, a reconstruction quality index Q is proposed:

[0128]

[0129] where and represent the true and reconstructed results in the time - space domain respectively.

[0130] For the noiseless case, set the stopping criterion as the maximum number of iterations K iter to be 300 or the relative error is less than 10 -4 , and the relative error can be calculated by ; Reinsert the parameter α to adjust the influence degree of the observation result on the reconstruction result. Since there is no noise, set it to 1; Finally, set the rank of the optimization matrix. For simplicity, set R = R 1 =…=R max =R N-1 , and scan the reconstruction results with the rank increasing from 1 to 12. Experiments show that in most cases, R = 4 gives the optimal result. From the experimental results, it can also be obtained that after sketch sampling, the value range of R can be expanded while maintaining a high Q value.

[0131] It is found from the experimental results that the PMF-TT method is worse than the ordinary PMF method when 70% and 90% are lost, because the fourth-order tensor is not sufficient to show the advantages of the TT method; the experimental results also show that the sketch method can obtain a higher Q value. Therefore, the sketch method also has the advantage of improving the reconstruction accuracy of noise-free signals.

[0132] In terms of running time, the running time of the PMF-TT method is less than that of the PMF method, but the Q value decreases. It can also be seen that the sketch method can improve the running efficiency.

[0133] To explore the reasons why the sketch method has a high Q value and shorter calculation time, the two methods were repeated 50 times under the frequency data block of 20 Hz. It can be found that the sketch method reduces the running time to a certain extent by reducing the number of iterations, and the total running time decreases by accelerating the running time of each iteration. In addition, it can be found that the main reason for the high running time of the PMF-TT method lies in the large number of iterations.

[0134] Please refer to Figure 4 , for a clearer explanation of the results, Figure 4 shows the reconstruction results of the 90% loss signal 100 times. It can be seen that there are slight flaws in the error map of the PMF-TT algorithm before using the sketch algorithm, and the flaws are significantly weakened after using the sketch algorithm, which confirms the previous conclusion. In addition, it also confirms the effectiveness of PMF-TT in seismic data reconstruction.

[0135] Please refer to Figure 5 , for the data with 1 dB noise added, other parameters are basically the same as those of the noise-free data. The relative error is changed to for calculation. PMF-TT gets worse reconstruction results than PMF, but the running time is greatly reduced because the arrangement process with expensive time consumption is avoided.

[0136] Please refer to Figure 6 , showing the superiority of the sketch algorithm, which not only enhances the reconstruction effect but also reduces the signal leakage.

[0137] In summary, the low-rank five-dimensional seismic data reconstruction method and system based on parallel matrix decomposition of the present invention can quickly and accurately reconstruct seismic data, reduce the influence of parameter selection on the results, improve the signal reconstruction accuracy, reduce the running time of the program, have a good reconstruction effect, reduce the consumption of computing resources, and have broad industrial application prospects.

[0138] Those skilled in the art should understand that the embodiments of the present application can be provided as a method, a system, or a computer program product. Therefore, the present application can take the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware aspects. Moreover, the present application can take the form of a computer program product implemented on one or more computer-usable storage media including but not limited to disk memory, CD-ROM, optical memory, etc. that contain computer-usable program code.

[0139] The present application is described with reference to the flowcharts and / or block diagrams of methods, apparatus systems, and computer program products according to the embodiments of the present application. It should be understood that each flow and / or block in the flowchart and / or block diagram, as well as the combination of flows and / or blocks in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to the processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing devices to generate a machine, such that the instructions executed by the processor of the computer or other programmable data processing devices generate means for implementing the functions specified in Figure 1 one or more of the flows Figure 1 or blocks.

[0140] These computer program instructions can also be stored in a computer-readable memory that can direct a computer or other programmable data processing device to work in a specific manner, such that the instructions stored in the computer-readable memory generate a manufactured article including instruction means that implement the functions specified in Figure 1 one or more of the flows Figure 1 or blocks.

[0141] These computer program instructions can also be loaded onto a computer or other programmable data processing device, such that a series of operation steps are executed on the computer or other programmable device to generate a computer-implemented process, so that the instructions executed on the computer or other programmable device provide steps for implementing the functions specified in Figure 1 one or more of the flows Figure 1 or blocks.

[0142] The above content is only to illustrate the technical idea of the present invention and cannot be used to limit the protection scope of the present invention. Any modification made on the basis of the technical solution according to the technical idea proposed by the present invention falls within the protection scope of the claims of the present invention.

Claims

1. A method for reconstructing low-rank five-dimensional seismic data based on parallel matrix decomposition, characterized in that, it includes the following steps: S1. Input the observed data tensor Sampling tensor The weight parameter μ of the optimization term and the regularization term, and the observed data tensor The ranks R of the matrices obtained by decomposing the observed data tensor along each direction according to the TT method 1 , …, R N , Initialize the parameters to obtain the initial decomposition matrices in each direction and the initial low-rank reconstructed data tensor Initialize the iteration variable k = 0 as the initial iteration step; S2. For the k-th iteration process, from the low-rank constraint of the real data unfold the low-rank reconstruction data tensor obtained in step S1 Use sketch random sampling and block coordinate descent method to calculate the matrix in each direction Use and the corresponding decomposition matrix Calculate the decomposition matrix for the current iteration step S3. Use the decomposition matrix obtained in step S2 to solve the low-rank reconstruction data tensor for the sub-optimization problem . Let k = k + 1, enter the next iteration step, repeat steps S2 and S3, and use the obtained by iteration as the low-rank reconstruction result of the real data to achieve the reconstruction of low-rank five-dimensional seismic data.

2. The method for reconstructing low-rank five-dimensional seismic data based on parallel matrix decomposition according to claim 1, characterized in that, In step S2, due to the low-rank constraint of the real data , perform low-rank decomposition on the matrix expansion of the real data , and introduce the observed data tensor as a constraint term for the real data , and minimize to solve for the approximation of the real data as follows: Among them, Z [n] is the TT decomposition result of the real data in the n-th direction, X n and Y n are the low-rank decomposition results of Z [n] , F represents the Frobenius norm operation, and N is the dimension of the tensor.

3. The method for reconstructing low-rank five-dimensional seismic data based on parallel matrix decomposition according to claim 2, characterized in that, By observing the data tensor Solve for the real data The low-rank reconstruction is expressed as follows:

4. The method for reconstructing low-rank five-dimensional seismic data based on parallel matrix decomposition according to claim 3, characterized in that, The constraint representation of low-rank reconstruction is as follows:

5. The method for reconstructing low-rank five-dimensional seismic data based on parallel matrix decomposition according to claim 2, characterized in that, In step S2, the low-rank decomposition matrices X and Y n of each TT decomposition matrix are alternately optimized using the iterative method, and the optimization results of X n and Y n are as follows: n ​ Among them, is the real data TT decomposition matrix in the nth direction.

6. The method for reconstructing low-rank five-dimensional seismic data based on parallel matrix decomposition according to claim 5, characterized in that, Accelerate the calculation of the low-rank decomposition matrix X using the sketch algorithm, and the improved optimization sub-problem is as follows: n The calculation of, the improved optimization sub-problem is as follows: Among them, and are the results obtained by sampling Y through the sketch algorithm n and ​ 7. The method for reconstructing low-rank five-dimensional seismic data based on parallel matrix decomposition according to claim 1, characterized in that, In step S2, the decomposition matrix are respectively: Among them, is the TT decomposition result of the real data in the n-th direction, is the left matrix of the low-rank decomposition result of Z [n] ​ 8. The method for reconstructing low-rank five-dimensional seismic data based on parallel matrix decomposition according to claim 1, characterized in that, In step S3, the low-rank reconstruction data tensor of the real data is calculated iteratively as follows: as follows: Among them, is a tensor of all 1s, and α is the reconstructed tensor and the proportionality coefficient with the observed data tensor . is the decomposition matrix in each direction and is the reconstructed tensor.

9. The method for reconstructing low-rank five-dimensional seismic data based on parallel matrix decomposition according to claim 8, characterized in that, In step S3, the sub-optimization problem of the low-rank reconstruction data tensor is as follows: where N is the dimension of the tensor, and are the decomposition matrices, and Z [n] is the TT decomposition result of the real data in the n-th direction, and F represents the Frobenius norm operation.

10. A system for reconstructing low-rank five-dimensional seismic data based on parallel matrix decomposition, characterized in that, it includes: Data module, input the observed data tensor Sampling tensor The weight parameter μ of the optimization term and the regularization term, and the observed data tensor The rank R of the matrix obtained by decomposing the observed data tensor along each direction by the TT method 1 , …, R N , initialize the parameters to obtain the initial decomposition matrices in each direction and the initial low-rank reconstructed data tensor Initialize the iteration variable k = 0 as the initial iteration step; Computing module, for the k-th iteration process, from the real data with low-rank constraint, the low-rank reconstructed data tensor obtained by expanding the data module Using sketch random sampling and block coordinate descent method, calculate the matrix in each direction Use and the corresponding decomposition matrix Calculate the decomposition matrix of the current iteration step A reconstruction module that decomposes the matrix obtained by the calculation module is used to solve the low-rank reconstruction data tensor of the sub-optimization problem Let k = k + 1, enter the next iteration step, and repeat the execution. The obtained by iteration is used as the real data low-rank reconstruction result to achieve the low-rank five-dimensional seismic data reconstruction.

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