CN110838096B - Seismic image completion method based on information entropy norm - Google Patents
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Abstract
The invention discloses a seismic image completion method based on an information entropy norm, which is applied to the field of geophysical seismic data processing, aims at the problem that the low-rank matrix obtained by recovery based on a matrix rank reduction method is not really low-rank due to limitation of the existing L1 norm, replaces the nuclear norm adopted by the original SVD by an objective function based on the information entropy norm, then normalizes the rank minimization term of the objective function by the information entropy norm, and finally obtains the low-rank matrix by an iterative ADMM estimation method for approximate solution, thereby obtaining the steady recovery of a missing image.
Description
Technical Field
The invention belongs to the field of geophysical seismic data processing, and particularly relates to a seismic data completion technology.
Background
In the actual geophysical seismic exploration and acquisition, the phenomenon of seismic data channel shortage is mainly caused by the following two reasons: (1) the arrangement of the geophones is often irregular due to the influence of the surface environment (such as rivers, ponds, lakes and the like), surface obstacles (roads, houses, bridges and the like) and the like of a work area; (2) due to poor coupling effect of the detector and the earth surface, a waste channel and a waste cannon can be generated due to environmental interference or the instrument. With the technical development of seismic exploration, the quantity of acquired data is larger and larger, on one hand, densely sampled data is expected to obtain more accurate underground structures, and on the other hand, in consideration of exploration cost, sparse sampling is needed in the spatial direction, so that the obtained data is often sparse and irregular. Therefore, reconstructing complete seismic data from sparse and irregular seismic data to obtain accurate subsurface structures is an important solution to solve the above contradictions.
The method for reconstructing the seismic data has various methods, such as methods based on prediction filtering, dip interpolation and the like, and the method mainly focuses on a matrix rank reduction method. The principle of such methods is that seismic data can be represented in the frequency domain by a low rank matrix, assuming that the seismic data consists of a finite number of linear in-phase axes. Irregular seismic data can increase the rank of the frequency domain matrix, and by reducing the rank, reconstruction of the data can be achieved. By utilizing the structural characteristics of the Toeplitz matrix in the rank reduction algorithm, the direct multiplication of the Toeplitz matrix and the vector can be avoided by adopting a multi-dimensional FFT algorithm, the calculated amount is greatly reduced, and the process of the rank reduction algorithm is accelerated.
The core of the matrix rank reduction-based method is that the low rank of seismic data is described by using the L1 norm of a matrix singular value vector, but the L1 norm description has the problem of insufficient sparsity: for example, for two singular value vectors, i.e., [4,4,4,0,0] and b ═ 10,1,1,0,0], the definition according to the norm of L1 cannot be accurately distinguished, and actually, the sparsity of the singular value b vector is much higher than that of the singular value a vector, so that the L1 norm limitation causes that the low rank matrix recovered by the matrix rank reduction method is not really low rank.
Disclosure of Invention
In order to solve the technical problem, the invention provides a seismic image completion method based on an information entropy norm, and the robustness and the correctness of a low-rank matrix in the seismic image completion problem are improved by using a low-rank approximation method based on information entropy norm combination.
The technical scheme adopted by the invention is as follows: a seismic image completion method based on an information entropy norm comprises the following steps:
s1, replacing the traditional kernel norm adopted by the existing seismic image completion model truncation SVD with the kernel norm based on the information entropy norm to obtain an objective function based on the information entropy norm;
s2, regularizing a rank minimization term of the target function by adopting an information entropy norm;
and S3, carrying out iterative solution on the target function which is normalized in the step S2 and is based on the information entropy norm, thereby obtaining the robust estimation of the low-rank matrix.
Further, step S1 specifically includes the following sub-steps:
s11, the seismic image completion model expression of the seismic data matrix X is as follows:
wherein R represents the rank of the seismic data, | | · |. non-calculationFIs Frobenius norm, BΩRepresenting the actual miss matrix, XΩRepresents the observed noise-free missing matrix, and BΩ=XΩ+ E, E is additive random noise, rank (·) represents the rank of the matrix;
s12, converting the expression in the step S11 into a regular term objective function form:
wherein λ is regularized sparsity | X |*Representing a nuclear norm;
s13, introducing a robust norm to the error item of the regular item target function in the step S12 to obtain a matrix decomposition mode based on the information entropy norm:
wherein h (-) is an information entropy norm, X represents a seismic data matrix, argmin (-) represents an independent variable value when a function value is minimum, gamma represents a regular term parameter, and sigma (X) represents a singular value.
Further, step S13 is to introduce a robust norm to the error term of the regular term objective function in step S12, specifically to introduce an information entropy functionInstead of the nuclear norm:
wherein h (·) is an information entropy norm, xiRepresenting the i-th element, | x |)1Representing the L1 norm.
Further, step S3 is specifically:
s31, expressing the matrix decomposition mode based on the information entropy norm obtained in the step S13 as follows:
s.tX=M
s32, solving X, M, Y by adopting an ADMM algorithm;
s33, obtaining a robust estimate of the low rank matrix according to X, M, Y of step S32.
Further, the step S32 of solving for X includes the following substeps:
a1, knowing M and Y, when solving for X, the objective function is:
a2, solving the objective function in the step A1 by utilizing a maximum likelihood criterion to obtain an analytic solution:
further, the step S32 of solving M includes the following sub-steps:
b1, knowing X and Y, when solving for M, the objective function is:
b2, solving the objective function in the step B1 by utilizing a maximum likelihood criterion to obtain an analytic solution:
wherein w represents a weight, U, Σ, and V are SVD decompositions of the following data:
further, step S32 solves for Y, the expression is: y + μ (X-M), μ denotes a regularization term parameter.
Further, the update expression of μ is: μ ═ ρ μ.
The invention has the beneficial effects that: according to the method, a target function based on an information entropy norm replaces a nuclear norm adopted by an original SVD, then the information entropy norm regularizes a rank minimization term of the target function, and finally an iterative ADMM estimation method is used for approximately solving to obtain a low-rank matrix, so that the steady recovery of a missing image is obtained; the method adopts the information entropy function to replace the L1 norm, not only can measure the sparse value, but also can use the information entropy to count the content of the singular value, and is more in line with the essence of sparse measurement or sparsity measurement; the method combines the excellent performance of the information entropy norm, has more stable performance than the traditional rank reduction method, obviously improves the actual seismic image recovery effect, and accords with the theoretical assumption.
Drawings
Fig. 1 is a schematic diagram of low rank matrix completion according to an embodiment of the present invention;
wherein, fig. 1(a) is missing seismic data, fig. 1(b) is a low rank matrix, and fig. 1(c) is noise;
FIG. 2 is a flow chart of an embodiment of the present invention;
FIG. 3 is an exploded view of an SVD according to an embodiment of the present invention;
FIG. 4 is a schematic diagram of seismic data recovery provided by an embodiment of the present invention;
wherein, fig. 4(a) is an original seismic data slice, fig. 4(b) is a missing seismic data slice, and fig. 4(c) is a recovered seismic data slice;
FIG. 5 is a diagram illustrating an information entropy function provided by an embodiment of the present invention;
fig. 6 is a schematic diagram of a kernel norm (abbreviated as information entropy norm) based on an information entropy function according to an embodiment of the present invention;
wherein fig. 6(a) is low entropy, fig. 6(b) is high entropy, fig. 6(c) is sparse signal, and fig. 6(d) is dense signal;
FIG. 7 is a first original post-stack seismic slice provided by an embodiment of the present invention;
FIG. 8 is a first missing post-stack seismic section provided by an embodiment of the present invention;
FIG. 9 is a recovered post-stack seismic slice one provided by embodiments of the present invention;
FIG. 10 is a second original post-stack seismic section provided by an embodiment of the present invention;
FIG. 11 is a second missing post-stack seismic section provided by embodiments of the present invention;
FIG. 12 is a second recovered post-stack seismic slice provided by an embodiment of the present invention.
Detailed Description
To facilitate understanding of the present invention by those skilled in the art, the following techniques will now be explained:
matrix completion
In 2006, Donoho proposed a well-known compressive sensing Theory in "DONOHO D L.compressive sensing [ J ] Information Theory, IEEE Transactions on, 200652 (4): 1289-. The core idea of the compressed sensing theory is as follows: and accurately recovering the original high-dimensional sparse signal by using a small amount of sampling data, namely realizing the perception of the high-dimensional sparse signal. But suffer from a number of practical problems such as: text analysis, image restoration, recommendation systems and the like, data needing to be restored are often presented in a matrix. Naturally, the research object of compressed sensing extends from the research of vectors to the research of matrixes, and the low rank of the matrixes can be used for describing the sparse characteristic of singular values of the matrixes. The so-called low-rank Completion (MC) technique refers to recovering the original complete Matrix by sampling some elements in the Matrix, as shown in fig. 1, where fig. 1(a) is missing seismic data, fig. 1(b) is a low-rank Matrix, and fig. 1(c) is noise, and the standard form is as follows:
wherein M is an observation matrix, X is a low rank matrix,[m]1,2, …, m is a set of sample matrix indices, PΩ(. represents an orthogonal projection operator, MijRepresents the sampling elements, namely:
the rank function rank () of the matrix is inherently non-smooth and non-convex, so the matrix completion problem is an NP-hard problem. As the size of the matrix to be solved increases, the time complexity for solving the problem is exponentially multiplied. And the kernel function of the matrix is used for replacing the rank function of the matrix, so that the low rank of the matrix is well characterized. The definition of the matrix kernel is as follows:
σiis the ith singular value arranged from large to small. As can be seen from equation (3), the kernel function of the matrix is actually equal to the sum of the singular values of the matrix. The L1 norm of the matrix singular value vector is used for replacing the L0 norm of the matrix singular value vector, namely, the matrix kernel function is used for replacing the L0 norm of the matrix singular value vectorThe nature of matrix ranks, Recht et al in the article "Recht B.A Simpler Aproach to matrix Assembly [ J]Journal of Machine Learning Research,2011,12(Dec): 3413-:
the problem is actually a convex optimization model, also called the Lasso model of the matrix.
As shown in fig. 2, which is a flowchart of the solution of the present invention, the seismic image completion method based on the information entropy norm of the present invention includes the following steps:
s1, replacing the traditional kernel norm adopted by the existing seismic image completion model truncation SVD with the kernel norm based on the information entropy norm to obtain an objective function based on the information entropy norm;
after SVD decomposition, the clean seismic data are found to be: most of the energy of the seismic signals is concentrated on a few eigenvalues, and most of the energy is concentrated on the first 4 eigenvalues, which is called the low rank characteristic of the seismic data. Further generalizing, the energy of the effective seismic data is distributed on a part of low-rank components, and the noise is distributed on components except the low-rank components. Therefore, seismic data can be completed by using the truncated SVD method shown in fig. 3, specifically, as shown in fig. 4, fig. 4(a) is an original seismic data slice, fig. 4(b) is a missing seismic data slice, and fig. 4(c) is a recovered seismic data slice.
More specifically, the seismic image completion model for seismic data matrix X may be expressed as follows:
wherein R represents the rank of the seismic data, | | · |. non-calculationFIs the Frobenius norm,is a matrixi is 1,2, …, m, j is 1,2, …, n. The optimization problem in equation (1) is essentially a convex optimization based on the Frobenius norm. Wherein in the presence of noise, BΩAnd XΩThe following conditions are satisfied:
BΩ=XΩ+E
where E is additive random noise.
Converting equation (5) into a canonical term objective function form:
where λ is regularized sparsity. The robust norm is introduced into an error term to improve the sparse representation capability of singular value vectors, and then the formula (6) is converted into a matrix decomposition mode based on the information entropy norm:
wherein h (·) is an information entropy norm, which is specifically defined as formula (8) below. The information entropy norm is also a convex function as shown in fig. 5, so the introduction of the information entropy norm does not change the properties of the objective function (7), which is still a convex function.
S2, regularizing a rank minimization term of the target function by adopting an information entropy norm;
since the sparse eigenvalue statistics related to the kernel norm of the SVD in the prior art mainly adopts the L1 norm, and the actual statistical characteristics of the eigenvalue cannot be obtained, the distribution of the eigenvalue is counted by using the information entropy norm instead of the L1 norm, so that the method has better performance, as shown in fig. 6. The information entropy norm can be better approximated by low rank than the ordinary rank minimization method, and is stable to outliers. The information entropy norm of the matrix is a kernel norm optimization model described by formula (7), and the kernel norm optimization model is expanded to the information entropy norm:
wherein,xiindicating that the ith element is taken. 0log 0-0 and h (0) -0 since we need to iteratively solve the objective function of equation (3), it is necessary to make equation (8) continuously derivable to solve equation (7). Therefore, we need to first derive equation (8), and the derivative of the information entropy function is:
and S3, carrying out iterative solution on the target function which is normalized in the step S2 and is based on the information entropy norm, thereby obtaining the robust estimation of the low-rank matrix.
The core of the invention is to design a method based on the combination of a Huber function and a schattern p-norm to synchronously suppress random and regular noises, and simultaneously, the design of a robust inverse problem solving algorithm is also very important. To solve the problem of equation (7), the present invention chooses to use a new Alternating Direction Method of Multipliers (ADMM, alternative Direction multiplier Method). For ADMM solution convenience, equation (7) is expressed as:
s.tX=M
according to the ADMM algorithm, equation (10) can be decomposed into the following several sub-problem solutions. Lagrange augmentation is performed on the formula (10) to obtain the following formula:
1) knowing M and Y, solving for X: in this case, we have the objective function:
using the maximum likelihood criterion to solve equation (12), we can get the following analytical solution
Knowing X and Y, solving for M: in this case, the objective function (6) becomes as follows
Solving the maximum likelihood solution for equation (14) as well, the following analytical solution can be obtained:
wherein, U, Σ, and V are SVD decompositions of the following data, and the specific formula thereof is as follows:
in addition, the weight w is determined by formula (4), which is specifically defined as follows:
3) knowing M and X, solving for Y:
Y=Y+μ(X-M) (17)
the iterative equations (13), (15) and (17) belong to the convergence up to the end, which convergence is guaranteed by the ADMM framework.
As shown in fig. 7 and 10 for real seismic data without deletion, fig. 8 and 11 for seismic data after deletion, and fig. 9 and 12 for seismic data after final recovery using the method of the present invention, it can be seen from fig. 9 and 12 that the missing data is effectively recovered, and the recovered data and the missing data are very consistent, and the noise is further attenuated.
It will be appreciated by those of ordinary skill in the art that the embodiments described herein are intended to assist the reader in understanding the principles of the invention and are to be construed as being without limitation to such specifically recited embodiments and examples. Various modifications and alterations to this invention will become apparent to those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the claims of the present invention.
Claims (6)
1. A seismic image completion method based on an information entropy norm is characterized by comprising the following steps:
s1, replacing the traditional kernel norm adopted by the existing seismic image completion model truncation SVD with the kernel norm based on the information entropy norm to obtain an objective function based on the information entropy norm; step S1 specifically includes the following substeps:
s11, the seismic image completion model expression of the seismic data matrix X is as follows:
wherein R representsRank of seismic data, | · | | sweet windFIs Frobenius norm, BΩRepresenting the actual miss matrix, XΩRepresents the observed noise-free missing matrix, and BΩ=XΩ+ E, E is additive random noise, rank (·) represents the rank of the matrix;
s12, converting the expression in the step S11 into a regular term objective function form:
wherein λ is regularized sparsity | X |*Representing the traditional nuclear norm adopted by the truncation SVD of the existing seismic image completion model;
s13, introducing a robust norm to the error item of the regular item target function in the step S12 to obtain a matrix decomposition mode based on the information entropy norm:
wherein h (-) is an information entropy norm, X represents a seismic data matrix, argmin (-) represents an independent variable value when a function value is minimum, gamma represents a regular term parameter, and sigma (X) represents a singular value;
step S13, introducing a robust norm to the error term of the regular term objective function in step S12, specifically importing an information entropy function h:instead of the nuclear norm:
wherein h (·) is an information entropy norm, xiRepresenting the i-th element, | x |)1Represents the L1 norm;
s2, regularizing a rank minimization term of the target function by adopting an information entropy norm;
and S3, carrying out iterative solution on the target function which is normalized in the step S2 and is based on the information entropy norm, thereby obtaining the robust estimation of the low-rank matrix.
2. The seismic image completion method based on the information entropy norm as claimed in claim 1, wherein step S3 specifically includes:
s31, expressing the matrix decomposition mode based on the information entropy norm obtained in the step S13 as follows:
s.t X=M
where M is an observation matrix, PΩ(. h) represents an orthogonal projection operator;
s32, solving X, M, Y by adopting an ADMM algorithm;
s33, obtaining a robust estimate of the low rank matrix according to X, M, Y of step S32.
3. The seismic image completion method based on the information entropy norm as claimed in claim 2, wherein the step S32 of solving X comprises the following substeps:
a1, knowing M and Y, when solving for X, the objective function is:
a2, solving the objective function in the step A1 by utilizing a maximum likelihood criterion to obtain an analytic solution:
4. the seismic image completion method based on the information entropy norm as claimed in claim 2, wherein the step S32 of solving M comprises the following substeps:
b1, knowing X and Y, when solving for M, the objective function is:
b2, solving the objective function in the step B1 by utilizing a maximum likelihood criterion to obtain an analytic solution:
wherein w represents a weight, U, Σ, and V are SVD decompositions of the following data:
5. the seismic image completion method based on the information entropy norm as claimed in claim 2, wherein step S32 solves for Y, and the expression is: y + μ (X-M), μ denotes a regularization term parameter.
6. The seismic image completion method based on the information entropy norm as claimed in claim 5, wherein the updating expression of μ is as follows: μ ═ ρ μ.
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